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Phil. Trans. R. Soc. A (2008) 366, 2145–2159
doi:10.1098/rsta.2008.0004
Published online 18 March 2008
Foam: a multiphase system with many facets
B Y S ASCHA H ILGENFELDT 1,2, * , S HEHLA A RIF 1,2
AND
J IH -C HIANG T SAI 1,2
1
Department of Engineering Sciences and Applied Mathematics, and
2
Department of Mechanical Engineering, Northwestern University,
2145 Sheridan Road, Evanston, IL 60208, USA
Liquid foams are an extreme case of multiphase flow systems: capable of flow despite a very
high dispersed phase volume fraction, yet exhibiting many characteristics of not only
viscoelastic materials but also elastic solids. The non-trivial, well-defined geometry of
foam bubbles is at the heart of a plethora of dynamical processes on widely varying length
and time scales. We highlight recent developments in foam drainage (liquid dynamics) and
foam rheology (flow of the entire gas–liquid system), emphasizing that many poorly
understood features of other materials have precisely defined and quantifiable analogues in
aqueous foams, where the only ingredients are well-known material parameters of
Newtonian fluids and bubble geometry, together with subtle but important information on
the surface mobility of the foam. Not only does this make foams an ideal model system for
the theorist, but also an exciting object for experimental studies, in which dynamical
processes span length scales from nanometres (thin films) to metres (foam continuum
flows) and time scales from microseconds (film rupture) to minutes (foam rheology).
Keywords: liquid foam; drainage; coarsening; foam rheology
1. Introduction
When the gas volume fraction of a gas/liquid multiphase flow (i.e. a bubbly
liquid) is increased, interactions between bubbles become a more and more
prominent feature in describing the dynamics and physical properties of the
system. Hydrodynamic forces between bubbles can become important even at
relatively low gas fraction f (Wijngaarden 1972; Mazzitelli et al. 2003), and
the speed of sound in a bubbly liquid is famously lowered dramatically even for
f/10K3 (Caflisch et al. 1985; Commander & Prosperetti 1989). When f reaches
a critical value, bubbles start to interact directly by contact, i.e. they start to
touch. For monodisperse bubbles, this happens around the density of random
close packing of spheres, frcpz0.64 (Torquato et al. 2000). For fT frcp , we can
start to think of the bubbly liquid as a foam, where bubbles touch continually
and, crucially, are therefore deformed from their individual spherical equilibrium
shapes. The foam systems discussed in this work will thus be liquid foams, where
the continuous phase between bubbles is a liquid, as opposed to solid or solidified
* Author and address for correspondence: Department of Mechanical Science and Engineering,
University of Illinois at Urbana-Champaign, 1206 W Green St, Urbana, IL 61801, USA
([email protected]).
One contribution of 11 to a Theme Issue ‘New perspectives on dispersed multiphase flows’.
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surface rheology
M
foam
gas flow through liquid:
foam rheology
liquid flow through
gas: drainage
continuum description and
discrete description
Figure 1. Foam multiphase flow is viewed from different perspectives in the fields of drainage and
foam rheology, but is in both cases rigorously described by only a few parameters.
foams such as polyurethane or styrofoam. Industrial and everyday applications
for foams abound: the foam head on beer or soda and soapy dishwater come to
mind immediately, but there is also a huge market for liquid foams as oil recovery
agents or fractionation media for refining metallic ores in a process called foam
flotation (Gibson & Ashby 1997; Evans et al. 1998). Studies of multiphase flow in
foams can focus on either the liquid moving by bubble interfaces (foam drainage)
or the bubbles moving through the liquid in some externally defined geometry
(foam rheology). In this contribution, we highlight recent key results in both fields,
pointing out how foam multiphase dynamics, despite being typically confined to
creeping flow, gives rise to non-trivial behaviour that can be understood in terms
of very few, well-known parameters (figure 1).
(a ) Geometry parameters
The deformations from spherical bubble shape usually lead to the formation of
bubble–bubble contact areas that appear as flat, or slightly curved, facets shared
by the contacting bubbles that, strictly speaking, do not touch but are still
separated by a thin film of liquid. If the gas–liquid interfaces are clean, this is a
very unstable situation. As figure 2b shows, the curvature of the bubbles in the
non-flat areas necessitates a Young–Laplace pressure difference with respect to
the fluid, while the fluid pressure in the thin film is equal to that of the bubble.
Consequently, a pressure gradient in the fluid is set up that draws liquid from the
thin film into the interstitial space between bubbles. The film rapidly thins and
eventually breaks as thickness perturbations bridge the film width. This process
of film drainage takes only a few milliseconds for millimetre-sized air bubbles in
water (Narsimhan & Ruckenstein 1996).
This is why foam needs to be stabilized by means of a surfactant. The
surfactant molecules cover all liquid–gas interfaces; in particular, they reside on
both sides of the thin film. The two surfactant layers repel each other by one of a
number of possible processes (Bergeron 1999), most prominently Coulomb
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Foam
(a)
thin films
Plateau borders
(b)
Figure 2. (a) Geometry of a polyhedral bubble in a dry foam, from a surface evolver simulation
(Brakke 1992; courtesy of A. Kraynik). Depicted are the PBs containing most of the liquid and the
thin films. (b) In the thin films, surfactant molecules exert a disjoining pressure P that balances the
Young–Laplace pressure difference pLYwg/a between the bubbles and the liquid adjacent to a
curved bubble portion.
(a)
(b)
(c)
Figure 3. Geometry of PB. (a) Experimental image (angled view) of a quasi-two-dimensional foam.
The bubbles are confined between glass plates at the top and bottom. Vertical PBs span the gap
between the plates, whereas horizontal PBs lie along either plate. (b) Geometry of vertical
(internal) PBs. (c) Geometry of horizontal (external) PBs (rotated, the plate is indicated with dotdashed lines) (see Koehler et al. (2004) for details).
repulsion or steric interaction. The repulsive force can be interpreted as an
effective pressure inside the film, the disjoining pressure, counteracting the
suction of the Young–Laplace pressure. Typically, equilibrium is reached for
aqueous films of thickness 20–100 nm (Mysels et al. 1959; Bergeron 1999). With
surfactant-stabilized films, bubble coalescence is very effectively inhibited and
the gas volume fraction can be pushed beyond 99%. In these cases of dry foams,
we prefer to give the liquid volume fraction as eh1Kf.
Figure 2a shows the typical geometry of a dry foam bubble: it is a polyhedron
whose faces are made up of stabilized thin films and whose edges are liquid
channels known as Plateau borders (PBs). A Plateau border has a characteristic
scalloped-triangle cross section (figures 2a and 3) with a radius of curvature a
related to a typical edge length L via the liquid fraction by e zde ða=LÞ2 , where
dez0.171, as shown by Koehler et al. (2000). An important special case is a
quasi-two-dimensional foam, e.g. a single layer of bubbles confined between rigid
plates (figure 3a). Seen through the plates, such bubbles have a polygonal cross
section. The vertical PBs spanning the gap between plates (figure 3b) are called
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internal PBs; the horizontal PBs (those spread out on one of the plates; figure 3c)
are called external PBs and relate to the liquid fraction via a different dez0.43
(see Koehler et al. 2004). These foam geometries provide the framework for
dynamical studies in the foam system: note that the knowledge of bubble size
(and thus L) and liquid volume fraction e is enough to quantify in detail the nontrivial geometrical features displayed in figures 2 and 3.
(b ) Material parameters
The presence of most surfactants does not measurably compromise either the
character of water as a Newtonian fluid or even the value of its viscosity, so that
liquid flow is always Newtonian fluid dynamics. As bubbles in experiments tend
to be millimetre-sized and the cross-sectional dimensions of PBs are consequently
10–200 mm for typical ew10K3K10K2, the resulting Reynolds numbers are small
and liquid flow in foams is almost always viscous (Stokes), entirely characterized by
the viscosity of water, mz10K3 Pa s. In foam rheology, bubbles move as a whole and
will generally deform. These deformations result in restoring forces due to the
Young–Laplace pressure, so that the surface tension g is the parameter that governs
elastic properties of the foam dynamical system. We can infer here that foams
generally behave as viscoelastic materials, yet unlike other viscoelastic systems
(such as clays, polymer solutions, polymer melts, slurries and many others; see
Larson (1999)) there are no hard-to-determine parameters depending on intricate
properties of macromolecules or other complicated constituents of the system—m
and g, together with bubble size, will usually describe the system entirely.
One caveat needs to be made with respect to the above statement: the boundary
conditions of the fluid flow crucially depend on the surfactant properties, more
specifically the surfactant mobility. This will be discussed in more detail in §2.
2. Liquid dynamics: foam drainage
As the film equilibrium thickness is very small, liquid drainage in dry foams occurs
predominantly through the PB between bubbles, which form a network of channels
as in a porous medium; however, PB cross sections are variable. In foam drainage,
we seek a description of the temporal and spatial variation of the liquid fraction e.
(a ) Nonlinear flow behaviour in Stokes flow
The driving forces for drainage flow are gravity and capillarity. Note that the
capillary forces (for a given bubble size L) are dependent on the Young–Laplace
pressure and thus, via the radius of curvature a, on e. These driving forces can
formally be combined into an effective pressure gradient
G ZKVp C rg:
ð2:1Þ
As is appropriate for Stokes flow, this driving is opposed by viscous friction in the
fluid of viscosity m (flowing at velocity v), which in porous media is typically
expressed as mv/k, with a permeability k, which is a function of the cross section
of liquid-carrying channels. In foams, however, this permeability is not a
constant, but does itself depend on e, as the cross section of the PBs is liquid
fraction dependent. Making use of the continuity equation in the form
ve=vtC V$ðevÞZ 0, we can therefore write
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Foam
(a)
2149
(b)
Figure 4. Liquid flow through PBs during drainage is dependent on surface rheology. (a) Immobile
surfactants lead to rigid PB walls and a Poiseuille flow profile. Viscous dissipation occurs along the
entire structure. (b) Mobile surfactants induce a flow close to a plug flow. Viscous dissipation is
dominated by the node regions (shaded), where PBs meet.
1=2
ve
gde
m C rg$VðkðeÞeÞK
ð2:2Þ
V$ðkðeÞVe1=2 Þ Z 0:
vt
L
Clearly, the resulting PDE for e will be nonlinear in both the advection and the
diffusion terms, posing a non-trivial problem. It was shown that this foam
drainage equation sustains solitary wave solutions (Koehler et al. 1999) as well as
subdiffusive spreading of pulses (Koehler et al. 2000, 2001).
The exact form of the permeability e(k) depends on the nature of flow
resistance in the PB. Interestingly, the most important factor here is the mobility
of the interface.
(b ) Interfacial rheology
A clean gas–liquid interface will readily move in-plane if the liquid is driven to
flow. In the case of a Plateau border, the bulk liquid inside the PB can eliminate its
resistance by flowing with uniform velocity (plug flow, figure 4b). This is, of course,
in contradiction to the assumption of Stokes flow: without viscous dissipation, the
flow would have to accelerate until inertial terms become important. Such an
acceleration is never observed and two different mechanisms can prevent it.
Surfactants at a gas–liquid interface will, in general, affect its mobility. In
particular, large molecule surfactants such as proteins tend to interlock at the
interface forming a ‘skin’ that is not easily moved in-plane. In such a case,
exemplified by the use of bovine serum albumin (BSA) in Koehler et al. (2002),
the PB walls behave as if they were rigid, providing no-slip boundary conditions.
The resulting flow is a Poiseuille flow, the resistance of which can be accurately
calculated from the shape of the PB cross section (figure 4a). The permeability
becomes k(e) fe and leads to nonlinear exponents 2 and 3/2 for the advection
and diffusion terms of the foam drainage equation, respectively,
1=2
ve
ve2 gde K1 L v2 e3=2
m C K1 rgL2
Z 0:
ð2:3Þ
K
vt
3
vz
vz 2
Here, K1z0.0063 is a permeability constant following from the geometry
outlined above. Because the viscous resistance in this case of immobile interfaces
is dominated by the PB (channels), this has been called the channel-dominated
foam drainage equation. It was first derived by Gol’dfarb et al. (1988) and was
further analysed by Verbist et al. (1996).
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(a)
bubble
bubble
bubble
bubble
bubble
bubble
(b)
Figure 5. Flow rheology in a Hele-Shaw cell is dependent on surface rheology. (a) Mobile surfactants,
discussed in the classic work of Bretherton (1961), lead to viscous dissipation dominated by the
junction regions between bubbles. (b) Viscous dissipation with immobile surfactants is dominated by
the flat film regions between bubbles and plates (cf. Denkov et al. 2005).
Other classes of surfactants (in particular small, soap-like molecules) do not
provide great resistance to in-plane motion of the interface. Indeed, in some
cases, the rheology of a clean interface may be almost recovered. Here, the flow is
stopped from accelerating because all PBs have a finite length (of order L) and
meet in junctions (called nodes). Plateau’s rules (Plateau 1873) demand that
each node is the meeting point of precisely four PBs and, furthermore, that the
angles under which they meet are maximally symmetric (i.e. tetrahedral angles
in three-dimensional foams and 1208 angles in a quasi-two-dimensional foam,
with the fourth PB spanning the gap between the plates; figure 3). This ensures
that the liquid flowing from one PB to one or several connected PBs must change
direction and has to split into several flows while merging with others. As a
result, even if the flow in the PBs themselves is essentially a plug flow, viscous
dissipation is present in the nodes. The difference is that the dissipation occurs
over a much smaller volume than with the channel-dominated case above
(namely, the volume of the nodes only). Consequently, the effective permeability
for a foam with mobile interfaces changes qualitatively to k(e) fe1/2, which
results in the node-dominated foam drainage equation,
1=2
ve
ve3=2 K1=2 de Lg v2 e
Z 0;
ð2:4Þ
C K1=2 rgL2
K
vt
2
vz
vz 2
where K1/2z0.0023 is another permeability constant (see Koehler et al. 2000).
Note that the nonlinearity is now restricted to the advection (gravity) term—the
liquid in a perfectly mobile foam shows pure diffusion dynamics perpendicular to
the direction of gravity.
To quantify the mobility of the interfaces, it is convenient to introduce a
dimensionless parameter that is the ratio of viscous bulk resistance in the PB
to interfacial resistance, the latter described by an effective surface viscosity ms.
The interfacial mobility parameter MZma/ms is /1 for cases where the
channel-dominated equations hold, while mobile-interface surfactants such as
sodium dodecyl sulphate (SDS) show MO1. These mobilities have been tested
experimentally by Koehler et al. (2002), directly measuring the liquid flow profiles
and verifying that BSA leads to Poiseuille flow, while SDS approximates a plug
flow. Experimentally, mobilities do not rise above Mz2, so that the velocity
profile still retains a sizable drop towards the edges of the PB (figure 5a). This is
m
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(a)
(b)
Figure 6. Illustration of T1 transitions in the (a) (quasi-) two-dimensional and (b) three-dimensional
cases. In both the cases, the total number of bubbles remains the same, but the number of neighbours
changes, with two bubbles gaining one neighbour and two bubbles losing one.
also reflected in the fact that the permeability k(e)fex for SDS is best fit with an
exponent xz0.6, larger than that for an ideally mobile interface. The same is true
for the mobile surfactant in the commercial detergent Dawn, which is used in many
experiments, including those discussed below. The work of Durand et al. (1999)
showed that, by adding a co-surfactant to SDS, the effective permeability exponent
could be varied between 0.6 and nearly 1, thus changing M and the drainage
behaviour continually from almost node dominated to entirely channel dominated.
(c ) T1 dynamics
The same importance of interfacial rheology has been demonstrated in the
dynamics of relative bubble motion ( Weaire & Hutzler 2000): when two
neighbouring bubbles in a foam are subject to sufficiently different forces to move
one bubble with respect to the other, these bubbles will at some point cease to be
neighbours and become neighbours to other bubbles in the foam. Such a neighbourswitching (T1) process is outlined in figure 6. Its dynamics is that of an activated
process: usually, the nearly symmetric configuration between the initial and end
states (where a number of PBs greater than four would have to meet in a node) is
energetically much less favourable than either the initial or end configurations.
Once it is surpassed, though, the T1 process proceeds spontaneously towards the
end configuration. This spontaneous dynamics was investigated by Durand &
Stone (2006) and found to be strongly dependent on interfacial rheology.
3. Foam dynamics: a first-principle study in rheology
The insights gained in experiments on foam drainage (§2) have been discussed
extensively in the literature pertaining to drainage processes in foam flotation or
physical chemistry (Prud’homme & Khan 1996) and have been extended to
include shear-thinning and viscoelastic fluids (Safouane et al. 2006). However,
the principal outcomes (nonlinear rheology and dependence on interfacial
mobility) are just as valid when the bubbles are moving with respect to the fluid.
In this section, we will concentrate on pressure-driven quasi-two-dimensional
foams, i.e. exactly the case discussed by Bretherton (1961) in his famous study of
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viscous resistance of a bubble driven in a tube or between parallel plates. For
Bretherton’s case of ideally mobile interfaces, it is the flow around the front and
back regions of the bubble (analogous to the nodes in node-dominated drainage)
that provides the viscous resistance, not the flow in the uniform film between the
bubble and the tube wall (figure 5a). One important result is that the viscous
resistance of the moving bubble scales with Ca2/3, where CaZmv/g is the
capillary number. For realistic surfactants, one expects the mobility of the
interface to be at least somewhat impaired, so that dissipation in the thin film
regions between bubbles and plate (figure 5b) becomes important. Indeed,
Denkov et al. (2005) showed theoretically and experimentally that, in the
opposite limit of rigid interfaces, viscous resistance scales as Ca1/2.
Thus, in both the mobile and immobile limits, the quasi-two-dimensional foam
is a rheological system in which the viscous resistance (balancing the driving
pressure) depends on the power of the velocity (or capillary number). As this
power is less than 1, it is a shear-thinning power-law fluid. Other fluids, such as
certain polymer solutions (Larson 1999), have been characterized as power-law
fluids, but often with only empirical evidence to back up this assumption of a
constitutive relation. With quasi-two-dimensional foams, we have a system that
can be shown, from first principles, to be a power-law fluid.
(a ) Hele-Shaw flows
Of the experiments used to study the rheology of foam, most have used sheardriven flows, e.g. in rheometers. Denkov et al. (2005) have measured the shear
stress of wet foam (ez10%) with small bubble size (Lw30 mm) in a parallel-plate
rheometer. By contrast, we use a pressure-driven flow set-up, namely a HeleShaw cell (Hele-Shaw 1898), and reproduce the quasi-two-dimensional set-up
theoretically introduced by Bretherton (1961), whose experimental results were
for tubes, not parallel plates.
In our set-up (figure 7), two large parallel rectangular glass plates enclose a
rectangular channel of approximately LcZ1.0 m length and wZ10 cm width.
The gap between the plates is of bZ1 mm thickness and can be filled through an
inclined feeder channel with a foam of uniform, controlled bubble size (size
monodispersity better than 5%) and controlled wetness (liquid fraction eZ0.01).
Details of the set-up will be reported elsewhere. The aqueous foam is made with
Dawn dishwashing detergent, the same surfactant that was found to result in
nearly completely mobile bubbles in foam drainage experiments (Koehler et al.
1999, 2000). The experiment combines a number of features of important
previous work: the pressure driving of Bretherton’s original studies of bubble
flow resistance, the ability to measure viscous resistance (but via an applied
pressure rather than applied shear as in Denkov et al. (2005)) and the general
aspect of a Saffman–Taylor fingering experiment, which has been tried for
aqueous foams in circular-plate set-ups (Park & Durian 1994; Lindner et al.
2000), but not for a controlled rectangular channel. Once filled into the gap, the
foam is driven from one end of the channel by injection of air at a fixed pressure
(applied by a hydrostatic pressure head). For very low pressures, this leads to
uniform displace-ment of the entire foam. For pressures larger than a critical
value, fingering of air in the foam is observed (see §3b).
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Foam
high-speed imaging
V
100 cm
CZ
CX
10 cm
soap
water
solution
P
top plate
bubble
bubble
bottom plate
bubble
~ 3 mm
Figure 7. Experimental set-up (Hele-Shaw cell) for rheology studies. Monodisperse bubbles made
from Dawn dishwashing detergent are fed through the inclined feeder channel into the horizontal
main channel, where the liquid fraction is a uniform ez0.01. The plate separation bZ1 mm is
below the capillary length to ensure uniformity between top and bottom plates. Bubbles are of
typical diameter Dz3 mm, larger than b, resulting in a quasi-two-dimensional foam (illustrated in
cross section). Foam flow is induced by feeding air into one end of the channel, either at defined
volume flow or at controlled hydrostatic pressure overhead. Bubble deformation and motion is
observed with a high-speed camera (PHANTOM v. 5.1).
We take care to fill the gap between the plates with only one layer of bubbles,
i.e. we generate a quasi-two-dimensional foam with bubble diameters
Dz3 mmOb. The bubble geometry in side view is sketched in figure 7. Can
we verify the power-law rheology for the pressure-driven foam? Can we
determine the coefficients of viscous friction, and are they compatible with the
existing results on shear flows and foam drainage?
The material parameters of the foam are as follows: the viscosity is very close to
that of pure water (mz0.001 Pa s) and the surface tension was determined, by
means of the pendant-drop method1, to be gz0.025 N mK1. The bubble size D
translates into Plateau border lengths of Lz2 mm. Using the geometry of bubbles
and both exterior and interior PBs (see figure 3 and Koehler et al. 2004), we derive
that the radius of curvature of the PBs is az140 mm. The foam is driven by
pressure heads of between PZ10 and 100 cmH2O, i.e. PZ1000–10 000 Pa,
resulting in bubble velocities relative to the plate walls of between vZ0.1 and
5 cm sK1 for these measurements of viscous resistance. The capillary number Ca
therefore ranges between 4!10K5 and 2!10K3, well within the applicable range of
small Ca theories such as those in Bretherton (1961) and Denkov et al. (2005).
1
We thank Prof. Ken Shull for the opportunity to obtain these accurate surface tension
measurements.
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100
90
80
70
60
50
(cmH2O)
40
30
20
10
9
0.1
0.2
0.3
0.4
0.5 0.6 0.7 0.8 0.9 1.0
(cm s–1)
2.0
3.0
Figure 8. Experimental results of bubble velocity versus pressure overhead, P, in a pressuredriven Hele-Shaw experiment. The exponent b in PZAv b is 0.63G0.03, compatible with the
exponent 2/3 for ideally mobile surfactants, and possibly indicating slightly limited mobility.
The prefactor A is discussed in the main text (lines: solid (pluses), P2 0 Lfoam Z80 cm; dashed
(inverted triangles), P3 0 Lfoam Z70 cm; dot-dashed (circles), P4 0 Lfoam Z60 cm; dotted (triangles),
P5 0 Lfoam Z50 cm).
(b ) Measuring rheology
A total of 16 runs with different driving pressure P were analysed, each at
eight different positions along the Hele-Shaw cell. It was found that data closer
than approximately 15 cm to the inlet or 20 cm from the outlet of the Hele-Shaw
cell were influenced by end effects, but the centre portion of the foam yielded
consistent results. The functional dependence of P on v was determined via a
least-squares best fit. The resulting exponent b in PZR(Lfoam)v b is bz0.63 (see
figure 8). This is just slightly below 2/3, indicating—in agreement with the foam
drainage results—that the Dawn foam has almost entirely mobile interfaces. The
resistance prefactor R itself depends on the length Lfoam of the foam pushed
by the pressure. It fitted well with the linear relation RZaLfoamCC, where
the constant offset C is due to the fact that the air not only pushes bubbles
in the main, horizontal Hele-Shaw cell (where Lfoam is variable), but also down
the attached feeder channel (figure 7), which presents a significant additional
resistance that does not change. The prefactor is determined as az0.20G0.05,
with the driving pressure given in cmH2O, Lfoam in cm and v in cm sK1. This
should now be compared with the Bretherton (1961) theory.
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(c ) Evaluating rheology
Bretherton (1961) derived an expression for the viscous resistance of a bubble
in a circular tube as well as for a bubble between parallel plates. The latter
results in a dynamic pressure drop per bubble of
dP Z 4:70pc Ca2=3 ;
ð3:1Þ
where pcZgk is the capillary pressure with the mean curvature k and CaZmv/g
is a capillary number (cf. Denkov et al. 2005). In Bretherton’s work, kZ1/R b for
a cylindrical bubble between plates, where R b is the bubble radius. In our case,
where the foam is dry, the radius of curvature determining the capillary pressure
is much smaller, namely kZ1/a. Taking this into account, we conclude that each
of our bubbles carries a pressure drop of dPz840Ca2/3 Pa. Given a length Lfoam
of foam and a bubble diameter Dz3 mm, the total pressure drop is PZ(Lfoam/
D)dP and the proportionality factor in R becomes az280 kPa mK1. Expressed in
our experimental units, we obtain az0.15, very close to the experimental value.
Note that the estimate presented here contains uncertainties about the role of
overall bubble deformation (as compared with the Bretherton case) as well as the
role of finite bubble size in the direction perpendicular to the motion. It is
noteworthy, though, that our pressure-driven quasi-two-dimensional experiments
do not seem to need a significant correction factor, in contrast to the shear stress
experiments by Denkov et al. (2005), where an additional factor of approximately
4 was needed to reconcile experiments with Bretherton’s results. This is perhaps
to be expected as the current experiment (other than being devised for
tightly packed foam bubbles) is much more closely related to Bretherton’s.
The remaining discrepancy in a can be at least partially explained by the
finite mobility of the interfaces, as a smaller b!2/3 leads to a compensating
larger prefactor.
4. Foam dynamics: discrete processes and solid behaviour
Beyond a characterization as a continuum, viscoelastic medium, foam offers other
important insights: viscoelasticity is usually a consequence of a mesostructure
being present in the medium, i.e. a non-trivial organizational level of discrete
objects between the atomic or molecular and the macroscopic length scales
(cf. Witten 1999). For polymer solutions or melts, the mesoscale is that of
macromolecules; for clays, it is microscopic platelets; and for granular media,
grains. For foam, this is obviously the scale of individual bubbles, which is
conveniently accessible in experiments (on length scales of mm).
(a ) Foam dynamics by T1 processes
This allows for unprecedented access to the solid characteristics of the
viscoelastic medium. The geometrical arrangement of the mesostructure is important in setting the shear modulus, yield stress and other quantities associated
with solids. The order of magnitude of the yield stress sY in the foam is simply
given by the generic pressure scale g/L, while it was shown in Kraynik & Hansen
(1986) that the O(1) prefactor c in sYZcg/L is determined by the relative
orientation between the principal axes of the foam and the direction of shear.
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(a)
(b)
(c)
Figure 9. (a–c) An example of discrete foam flow dynamics in quasi-two-dimensional foams. The
invading air advances by T1 transitions (neighbour swapping), equivalent to edge dislocations in a
solid accompanying its plastic deformation. The speed of air advance is slow enough, so no films are
broken. Arrows indicate the locations of some T1 transitions (others can be found in the images).
Interframe distance is approximately 40 ms.
This yielding process is, microscopically, associated with neighbour flips of
individual bubbles, i.e. T1 processes (see §2c). In fact, foam rafts have long been
used to quantitatively study the formation and motion of dislocations in solids
(always associated with neighbour changes of atoms), starting with Bragg & Nye
(1947) and continuing until recently (Gouldstone et al. 2001). We have also
observed dislocation formation and motion in the fingering experiments
mentioned above. Figure 9 illustrates how T1 processes advance the air/foam
boundary by one bubble length at a time.
Together with the power-law shear-thinning behaviour characterizing any
bubble motion involving liquid viscous dissipation, the yield stress property
characterizes a quasi-two-dimensional foam as a Herschel–Bulkley fluid in
in-plane flow (as opposed to the pure power-law behaviour for mere motion with
respect to the confining plates), leading to the constitutive relation
s Z sY C RCab ;
ð4:1Þ
determined from first principles through surface tension, liquid viscosity and
bubble geometry only.
(b ) Foam dynamics by film rupture
While the fingering process described above proceeds without breakage of films,
a foam can also yield by breaking the thin films between bubbles. One way of
transitioning between the T1 yielding and the film breakage is by applying forces
at different rates (Arif et al. 2008). Another is by applying driving forces strong
enough to overcome the cohesive elastic forces due to surface tension as follows.
We again make use of Bretherton’s formula for viscous resistance, taking into
account now that the radius of curvature (and implicitly the thickness of the
liquid film along the plate) varies, within one bubble, owing to the shape of the
(horizontal) PB lying along the plates. Figure 10a illustrates how these PBs
experience differential viscous resistance force, largest at the thinnest part of the
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Foam
(a)
(b)
film elongates
and breaks
bent
PBs
rupture
air
Figure 10. Illustration of the velocity limit for gliding bubbles. (a) The PBs shown are horizontal
(along one plate) and the driving air pushes from right to left (thick arrow). The PBs are subject to
varying viscous forces along their length (dotted arrows), larger in the thinner regions and smaller
in the wider node regions. The resulting force imbalance is compensated by surface tension (dashed
arrow) and results in bending of the horizontal PBs. This counterintuitive bending away from the
driving air is seen in the experimental snapshots of (b) in the bubbles directly in front of
the advancing air. When the PB and the attached film between the plates (thick dotted lines) are
bent and stretched too much, the film fails and bubbles rupture (see (b)). Interframe distance in (b)
is approximately 1.1 ms.
PB and smallest at the ends of the PBs, where they are connected to others. For
the same driving force, therefore, the ends tend to move faster than the centres of
the PBs. In figure 10b, we show an experimental confirmation of that fact,
showing a ‘counter-intuitive’ curving of the PBs opposite to the direction of
driving. Such a curvature, however, results in surface tension providing a
restoring force trying to straighten the PB. As long as surface tension is strong
enough to overcome the viscous force differential, all parts of the bubble still
move at the same speed, and bubble integrity is preserved. But if the viscous
force differential is greater than the maximum surface tension force (e.g. that
occurring for a semicircular PB), the PB and the attached film will be elongated
due to the velocity differential between the centre and end parts, resulting in the
rupture of the film.
The mean curvature varies between approximately 1/a at the centre of the PB
to approximately 1/(2a) at the end (cf. Koehler et al. 2000). With (3.1), we
expect the viscous stress difference to be 4.70(g/2a)Ca2/3 for a single bubble.
However, the maximum Laplace–Young pressure able to balance this force is
reached when the horizontal PB has formed a semicircle (of radius L/2),
for which pLYw2g/L. Equating both stresses yields a purely geometric criterion
for a maximum capillary number without rupture
4a
C ac Z
4:70L
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3=2
:
ð4:2Þ
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With the experimental values (Lz2 mm for the hexagonal bubbles of DZ3 mm),
we obtain a critical velocity of vcz0.36 m sK1. Film breakage (propagation
without T1 transitions) was observed for bubbles with local speed above
vz0.4 m sK1, in good agreement with the estimate (4.2). Figure 10b shows that
the curved PBs are indeed attached to films for which breakage is imminent.
5. Conclusions
We have emphasized here the many unique qualities of a foam both as a model
system and as a soft material in its own right: it is a non-Newtonian, viscoelastic
material, the rheology of which is described by a power-law fluid model (when
driven uniformly between parallel plates) and a Herschel–Bulkley model
(when deformed in the plane between the plates). The models can be derived
from first principles, where yield stress and the power of shear thinning are
directly related to simple material parameters (surface tension and viscosity) and
information about the mobility of the gas–liquid interfaces. This mobility is a
crucial feature of a liquid foam: although set on a very small scale by the physical
chemistry of surfactant molecules, it determines macroscopic behaviour such as
liquid drainage (fundamentally altering the power laws of the drainage
dynamics) as well as microscopic behaviour on the single-bubble scale (such as
the dynamics of T1 processes). We have shown that Bretherton’s formulae give
very good estimates for pressure-driven quasi-two-dimensional foam rheology.
The accessibility of the single-bubble scale, including the positions and shapes of
all bubbles, allow for observation and interpretation of discrete processes, from
T1s to film rupture, as an analogue to what happens to atoms in a solid. In this
fashion, foams not only model complex fluids, but also allow an unprecedented,
detailed look into the flowing and yielding behaviour of solids, bridging the gap
between multiphase flow phenomena and the solid mechanics of yielding.
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