Viscous Friction in Foams and Concentrated Emulsions under Steady Shear
N.D. Denkov,1S. Tcholakova,1K. Golemanov,1K.P. Ananthapadmanabhan,2and A. Lips2
1Laboratory of Chemical Physics & Engineering, Faculty of Chemistry, Sofia University, Bulgaria
2Unilever Global Research Center, Trumbull, Connecticut 06611, USA
(Received 30 April 2007; revised manuscript received 6 January 2008; published 3 April 2008)
We present a model for the viscous friction in foams and concentrated emulsions, subject to steady
shear flow. First, we calculate the energy dissipated due to viscous friction inside the films between two
neighboring bubbles or drops, which slide along each other in the flow. Next, from this energy we
calculate the macroscopic viscous stress of the sheared foam or emulsion. The model predictions agree
well with experimental results obtained with foams and emulsions.
DOI: 10.1103/PhysRevLett.100.138301 PACS numbers: 83.80.Iz, 47.57.Bc, 82.70.Kj, 82.70.Rr
The rheological properties of foams and concentrated
emulsions, with a volume fraction of the dispersed phase
? > ?CP(where ?CPis the volume fraction of closely
packed spheres), are usually described [1–7] by the
Herschel-Bulkley equation, ? ? ?0? ?V? _ ?? ? ?0? k _ ?n.
Here _ ? is shear rate, ? is total stress, ?V? _ ?? is its rate-
dependent part, ?0is yield stress,n is power-law index, and
k is consistency. Theoretical and experimental studies
showed that ?0scales with the capillary pressure of the
bubbles or drops and depends on ? [1–5,9,10]. The rate-
dependent term ?V? _ ??, related to the viscous dissipation of
energy, is not well understood. Various values of n and k
were measured [1–6] or estimated [2,11–13] without a
clear understanding of why and how they depend on spe-
In this Letter we describe a simple theoretical model of
the viscous friction in steadily sheared foams and concen-
trated emulsions. In such systems, drops and bubbles are
compressed against each other due to their high volume
fraction, so that planar foam or emulsion films are formed.
For brevity, in the model formulation we discuss explicitly
only bubbles in foams. However, the model is equally
applicable to emulsions (unless the drop viscosity is so
high that the drop deformation time is comparable to 1= _ ?)
and could be upgraded to other types of dispersions of soft
particles, such as suspensions of vesicles and gel particles.
The current model differs from the other rheological
models proposed in literature in several aspects: First, we
consider steadily sheared foam, in which foam films are
perpetually forming and disappearing between the bubbles,
which slide along each other dragged by the flow. The
bubble dynamics in such foams is qualitatively different
from the case of small oscillatory deformations of the foam
around its equilibrium configuration. Therefore, the mod-
els of the oscillatory deformation [2,11,12] are inappli-
cable to describe the steady foam flow. Second, we are
interested in shear rates characterizing the foam or emul-
sion transportation (ca. 0.1 to 200 s?1). Hence, we assume
that bubble size and volume fraction are known, and we do
not consider Ostwald ripening and water drainage, which
are important at longer time scales [5,14–17]. Third, we
assume that the mainviscous friction occurs in planar films
formed between sliding bubbles. Hence, the current model
is applicable only to foams with volume fraction ? > ?CP
and could not be compared to models and experimental
results obtained at ? < ?CP[3,4,18,19].
We developed an extended version of our model, in
which we account also for the friction in the Plateau bor-
ders (PBs) surrounding the films and for the surface dissi-
pation at bubble surface. Numerical estimates by the ex-
tended model showed that the friction in PBs is detectable,
but usually smaller than the friction in foam films—below
30% for all systems considered in the current manuscript,
except for the data at the highest capillary numbers, Ca ?
10?3, where the contribution of PBs increases to ?50%.
The extended model predicts that surface dissipation is
important for systems with high surface modulus, ES, in
good agreement with the experimental results in , where
foams stabilized by different surfactants were studied and
much higher viscous stress was measured when fatty acid
salts with ES> 100 mN=m were used (compared with
<5 mN=m for typical synthetic surfactants). Because of
limited space, we present the basic model and comparison
with experimental data in this Letter, while the extended
model is presented in subsequent study .
For simplicity, we assume that the static foam contains
monodisperse bubbles arranged in an fcc lattice, with given
? and bubble volume, VB? ?4=3??R3
tween the centers of two neighboring bubbles is lS?
1:812R0=?1=3, where the numerical factor is calculated
for bubbles with the shape of rhombic dodecahedron (for
other shapes this factor is similar within a few percent).
The film radius RFScan be estimated from expressions
derived by Princen [21,22], RFS? ?4SF=pS?1=2R0, where
p ? 12 is the number of planar films per bubble, SF??? is
the area of the bubble surface occupied by films, and S???
is the total surface area of the deformed bubble. From
Princen’s approach one can determine also the capillary
pressure of the bubbles, PC??? ? 2?S?=R0S0?? ?
f????, where ? is interfacial tension and S0? 4?R2
0. The distance be-
PRL 100, 138301 (2008)
PHYSICAL REVIEW LETTERS
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© 2008 The American Physical Society
the area of the nondeformed bubbles. Explicit expressions
for the functions SF???, S???, and f??? are given in .
Thus we determine the geometrical parameters of the
deformed bubbles in the static foam and the bubble capil-
lary pressure PC??? which are needed in the following
We consider now the viscous friction between two bub-
bles located in two neighboring planes of the assumed
fcc structure of the foam. These planes are assumed to
slide with constant velocity u along each other in the
sheared foam; see Fig. 1. The relative position of the
bubbles can be described by the distance between their
geometrical centers, l?t? ? ?l2
angle ??t?, where sin??t? ? lm=l?t? . Here lm?
=2 is the minimal distance between the bubble cen-
ters, x0? ?l2
the moment of film formation.
The sliding bubbles form a transient foam film with
radius RF?t?. For convenience we introduce an effective
radius of the deformed bubbles, REFF? ?R2
defined as the radius of a spherical surface with just one
film (instead of the 12 filmsin the fcc structure) that has the
same ratio RFS=lS, as the deformed bubbles in the real
foam. Thus we replace the real polyhedral bubbles by
‘‘imaginary’’ bubbles having just one film, and the instan-
taneous film radius becomes RF?t? ? ?R2
To describe the liquid flow in the film between the
bubbles, we use the lubrication equation ?dP=dr? ?
??@2Vr?r;z?=@z2?, written in a radial rz-coordinate system
(located in the center of the film with z axis perpendicular
to the film plane); ? is liquid viscosity, and P?r? is pressure
in the film [24,25]. Assuming tangentially immobile bub-
ble surfaces (due to high viscosity of the emulsion drops or
to surface Marangoni stress created by surfactants), the
boundary conditions for the fluid velocity at the film sur-
faces are Vr?r;z ? ?h=2? ? ?usin??t?cos’=2, where
h?t? is the instantaneous film thickness, usin??t?cos’ is
the projection of the relative bubble velocity in the plane of
the film, and ’ is polar angle in the film plane. By using
standard procedures, one can show that the film thinning is
governed by the known Reynolds equation :
m? ?ut ? x0?2?1=2, or by the
m?1=2, and l0? l?t ? 0? is the distance in
?dh=dt? ? ?2?PC? ??h??h3=3?R2
Here ??h? is the disjoining pressure accounting for the
surface forces between the two film surfaces (electrostatic,
van der Waals, etc. ). For simplicity, we neglect below
the surface forces ??h? ? PC, although they can be in-
corporated straightforwardly in the calculations .
Using the fact that h ? l, one can integrate Eq. (1) to
obtain an explicit expression for the film thickness h?t?:
where h0? h?t ? 0? is the film thickness in the moment of
From the lubrication equation, one derives the following
expression for the fluid velocity in the film, which is used
below to calculate the dissipated energy:
Vr?r;z? ? ?2PCr?z2? h2=4?=?R2
V’?r;z? ? usin??t?zsin’=h:
For the calculation scheme we should specify the initial
moment of collision between the bubbles (viz., the moment
of planar film formation) and the respective quantities,
such as h0, RF0, l0, etc. Following , we assume that
the planar foam film is formed when the dynamic pressure
in the gap between the colliding bubbles becomes compa-
rable to bubble capillary pressure P?r ? 0? ? PC. As
shown in , this condition leads to the following ex-
pression for the initial film thickness, h0? F=2??, where
F is the force pushing the bubbles against each other. At
low Reynolds number, this pushing force could be esti-
mated by Taylor’s formula, F ? 3??R2
is the radius of curvature of the approaching bubble sur-
faces and uz?t? ? ucos??t? is the velocity component
along the line connecting bubble centers. Thus we obtain
h0? ??3=4?cos?0~ u?1=2RN, where ?0? ??t ? 0? is the
angle in the moment of film formation and ~ u ? ??u=??
is the dimensionless relative velocity of the two neighbor-
ing bubble planes in the foam. We assume that RNis
approximately equal to the radius of curvature of the
nodes in the static foam, RN? 2?=PC, where PC??? is
known . The force balance shows  that the initial
film radius is RF0? ?h0RN?1=2, and the distance between
bubble centers in the moment of film formation is l0?
equations for the foam film formation and thinning be-
tween two neighboring bubbles in sheared foam.
Nuz=2h, where RN
F0?1=2. Thus we obtained a complete set of
FIG. 1 (color online).
motion of neighboring planes of bubbles in sheared foam, and of
the process of film formation between two bubbles, sliding along
each other (upper line: side view; bottom line: projection onto
the plane of bubbles).
Schematic presentation of the relative
PRL 100, 138301 (2008)
4 APRIL 2008
The rate of energy dissipation inside the foam film is
where the subscript ‘‘DF’’ denotes dissipation inside the
films. To determine the energy dissipated inside one foam
film,~ EDF? EDF=?R2
contact time of the two bubbles to derive [see also
0?~ u1=2?, we integrate Eq. (4) over the
~ EDF? ?
? ??sin??~ t??2??F?~ t??2
where the dimensionless quantities ?N? RN=R0, ?F?
RF=R0,~ t ? tu=R0, and ? ? h=hCare introduced, where
hC? ~ u1=2R0is the scaling factor for the film thickness.
The scaling for h is chosen to comply with the dependence
1=h2?? ? ?=2? / ?=?u ? 1=~ u, predicted by Eq. (2), in
which the term with h0is negligible and the second term in
the brackets is identically zero at ? ? ?=2. This particular
scaling leads to numerical values of~ EDF, which depend
very weakly on the relative bubble velocity,~ EDF/ ~ u?0:035.
Next, we determine the time-averaged energy dissipa-
tion rate per unit foam volume h_Ei, which is equal to the
macroscopic viscous stress ?VFmultiplied by the shear
rate _ ?. The shear rate and the conventional capillary num-
ber Ca ? ?? _ ?R0=?? are proportional to the relative veloc-
ity of the bubble planes, _ ? ? u=m and Ca ? ~ uR0=m ?
0:676~ u?1=3, where m??? ? ?2=3?1=2lS? 1:479R0=?1=3
is the distance between the planes. To find h_Ei, we consider
the motion of the bubble plane as a sequence of equivalent
steps with length lS. During one such step, the six contacts
of the ‘‘central’’ bubble with its neighbors undergo partial
cycles of type ‘‘film formation-thinning-disappearance,’’
like those expressed by Eq. (5). Geometrical consideration
shows that these partial cycles could be summed up to 4
equivalent full friction cycles. Thus, the viscous stress of
the foam is
?VF_ ? ? h_Ei
) ~ ?VF? ?VF=??=R0?
? 0:39~ u1=2?~ EDF
? 0:474Ca1=2?5=6~ EDF;
where the multiplier 1=2 accounts for the sharing of
the dissipated energy inside one film by two neigh-
boring bubbles, and ~ ?VF is the respective dimension-
less stress. The subscript ‘‘VF’’ denotes viscous friction
inside foam films. The respective dimensionless effective
viscosity of the foam is ~ ?EF? ?EF=? ? ~ ?VF=Ca ?
0:474?5=6Ca?1=2~ EDF. Therefore, the model predicts that
the viscous stress is approximately / Ca1=2, whereas
?EF/ Ca?1=2, just as observed experimentally in .
This scaling is ultimately related to the dependence h /
Ca1=2, predicted by Eq. (2) as explained above.
To account more precisely for the effects of ? and Ca,
we calculated numerically~ EDFin the ranges 10?6? Ca ?
10?2and 0:80 ? ? ? 0:99. The numerical results were
fitted by empirical function,
??0:5, which was combined with Eq. (6) to obtain the final
model expression for the contribution of the friction in the
foam films into the foam viscous stress
~ EDF? 1:7Ca?0:035=?1 ?
~ ?VF? 0:806Ca0:465?5=6=?1 ? ??0:5:
Equation (7) should be used in its range of validity only.
Extrapolation to ? ! 1 is not justified, because the films
become very thin at high volume fractions (due to high
capillary pressure) and the surface forces, which were
neglected here, become important . Therefore, the
upper limit of using these equations is set by the compari-
son of the thickness of the dynamic films, h ? 0:2hC?
Ca1=2R0=4, with the range of surface forces (typically
between 1 and 10 nm). The lower limit of ? is set mainly
by the model assumption that the bubbles form planar films
while sliding along each other . The comparison of the
Capillary number, Ca
Foam - ionic surfactant SLES
Foam - commercial body wash Axe
Emulsion - nonionic surfactant ROX
Emulsion - nonionic surfactant ROX
log(τ τ-τ τ0)
Φ Φ = 0.83
Φ Φ = 0.96
; (b) foams and emulsions stabilized with synthetic surfactants. The solid curve in (b) is drawn according to Eq. (7), whereas the
dashed curves are drawn with account for the dissipation in plateau borders (all curves—without adjustable parameter).
Comparison of model predictions with experimental results (symbols): (a) oil-in-water emulsions with ? ? 0:83 and 0.96
PRL 100, 138301 (2008)
4 APRIL 2008
model predictions with experimental data (see Fig. 2 be- Download full-text
low) shows that Eq. (7) is applicable at least in the range
0:80 < ? < 0:98.
Let us compare model predictions with the experimental
results of Princen and Kiss , who measured the viscous
stress, ?V?Ca?, for a series of concentrated emulsions with
different ?. The comparison showed a reasonably good
agreement with all experimental data without using any
adjustable parameter—no difference larger than 25% was
found (for most of the data <10%). In Fig. 2(a) we show as
illustration our theoretical curves and the experimental
data for two of the samples studied in . Next, we
compare the model predictions with experimental results
for sheared foams and emulsions, obtained by us with the
procedure from Ref. . This comparison also showed
reasonably good agreement without adjustable parameter
for both foams and emulsions, stabilized by various syn-
thetic surfactants; see Fig. 2(b).
Note that the procedure used for comparing the model
with experiment (assuming ?V? ? ? ?0) implies that the
elastic contribution to total stress in sheared foam is a weak
function of _ ?. This assumption is strongly supported by the
fact that our model describes equally well both foams and
emulsions, for which the relative contribution of the elastic
term is rather different (higher for emulsions due to the
smaller drop size, as compared to bubbles in foams). If the
elastic term was contributing significantly to the rate-
dependent part of ?, one should expect different depen-
dences ?V? _ ?? for foams and emulsions, which is not ob-
served with the systems studied.
In conclusion, we present a theoretical model for the
viscous friction inside foams and concentrated emulsions,
subject to steady shear. The model predicts that the macro-
scopic viscous stress is approximately proportional to
Ca1=2, when the contributions of the disjoining pressure
??h? between the surfaces of the sliding bubbles or drops
and of the surface dissipation are negligible. The model
predictions are compared with experimental data and a
very good agreement is found without adjustable parame-
ters for both emulsions and foams.
As shown in a related study , the effect of ? leads to
stronger dependence of ?Von Ca (1=2 < n < 1), whereas
the surface dissipation and possible shear thinning of the
continuous phase lead to n < 1=2. These extensions of the
current model explain why different values of n are also
Let us note that our model could be upgraded to other
types of dispersions of soft particles, such as suspen-
sions of vesicles and gel particles, for which similar scal-
ing of the viscous stress with Ca1=2was reported [26,27];
see, e.g.,  for a possible approach. Therefore, the
model makes a link between the rheological properties of
all these systems (emulsions or foams, gel particles,
vesicles), which are usually studied by independent re-
This study is supported by Unilever GRC,Trumbull, CT.
We are grateful to Professor I.B. Ivanov and Dr. V.
Subramanian for useful discussions.
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