Rheology of emulsions

1. ELSEVIER
Current Opinion in Colloid& Interface Science 4(1999) 231-238 www.elsevier.nl/locate/cocis
New fundamental concepts in emulsion rheology
T.G. Mason
CorporateResearch Science Laboratory,Exxon Research and Engineering Co., Route 22East,Annandale, NJ 08801, USA
Abstract
The field of emulsion rheology is developing rapidly due to investigations involving monodisperse emulsions having narrow
droplet size distributions. The droplet uniformity facilitates meaningful comparisons between experiments, theories, and
simulations. 0 1999 Elsevier Science Ltd. All rights reserved.
Keywords: Monodisperse emulsions; Droplet size distribution;Coalescence
1. Introduction
Emulsions consist of droplets of one liquid dis-
persed in another immiscible liquid. By contrast to
microemulsion phases, emulsions are not thermo-
dynamic states. Instead, emulsions are metastable dis-
persions; external shear energy is used to rupture
large droplets into smaller ones during emulsification.
Surfactants that provide a stabilizinginterfacial repul-
sion are typically introduced to inhibit droplet coales-
cence [l]. If the liquids are highly immiscible,
molecules of the dispersed phase cannot be ex-
changed between droplets, so coarsening of the
droplet size distribution due to Ostwald ripening is
negligible. When coalescence and ripening are sup-
pressed, the emulsion can remain stable for years
even when osmotically compressed to form a biliquid
foam.
Emulsions exhibit highly varied rheological behav-
ior that is useful and fascinating [2', 3-51. An emul-
sion's macroscopic constitutive relationships between
the stress and strain depend strongly on its composi-
tion, microscopic droplet structure, and interfacial
interactions. By controlling the droplet volume frac-
tion, +, an emulsion can be changed from a simple
viscous liquid at low + to an elastic solid having a
substantial shear modulus at high +, as shown
schematically in Fig. 1. This elasticity results from the
work done against interfacial tension, (T, to create
additional droplet surface area when the shear fur-
ther deforms the already compressed droplets. The
elasticity of foams [6'], the gas-in-liquid counterpart
to concentrated emulsions, results from the same
mechanism, although Ostwald ripening of gas bubbles
usually causes the foam to age and its elasticity to
become weaker over time. The rheological properties
of such products as lotions, sauces, and creams are
typically adjusted by varying the composition or the
emulsification process to alter the droplet size dis-
tribution and hence packing. Additives such as po-
lymers can also modify emulsion rheology by raising
1359-0294/99/$ - see front matter 0 1999Elsevier Science Ltd. All rights reserved.
PII: S 1 35 9 - 0 2 9 4 ( 9 9 ) 0 0 0 3 5 - 7

2. 232 T.G. Mason / CurrentOpinionin Colloid& Interface Science 4 (I999)231-238
Figure 1
I Dilute Concentrated
Uncaged Caged Packed Compressed
0
I 0 0
I
0
I I
' @
Viscous Elastic
1& =0.58 &CP 0.64
Current Opinion in Colloid & InterfaceScience1
Schematic diagram of droplet positional structure and interfacial
morphology for disordered monodisperse emulsions of repulsive
droplets as a function of the volume fraction, 4,of the dispersed
phase. In the dilute regime at low q5, the droplets are spherical in
the absence of shear. As 4 is raised near the hard sphere glass
transition volume fraction, q5g = 0.58, the droplets become tran-
siently caged by their neighbors. As q5 is further increased into the
concentrated regime, the droplets randomly close pack at q5RCP =
0.64, and become compressed with deformed interfaces for larger
4. As 4 + 1, the droplets become nearly polyhedral in shape and
form a biliquid foam. Dilute emulsions behave as viscous liquids,
whereas concentrated emulsions exhibit solid-like elasticity.
the viscosity of the continuous phase or by causing
adhesion between droplets without coalescence [71.
Emulsions comprised of viscoelastic polymeric liquids,
or blends, exhibit a rich rheological complexityarising
from the interplay of bulk and interfacial elastic con-
tributions [PI.
For years, measurements of emulsion rheology
[9-131 were not quantitatively understood because the
droplet size distributions had not been controlled and
no two emulsions had either the same distribution of
Laplace pressures, IIL= 2u/a, where a is the droplet
radius, or the same critical volume fractions, +,, at
which droplet packing would occur. Recently, mea-
surements using monodisperse emulsions have es-
tablished a conceptual foundation for quantitatively
understanding emulsion rheology, especially at high +
[2', 14", 15", 161. In contrast to a recent opinion
[17], these studies show that polydispersity is impor-
tant in emulsion rheology. The monodispersity has
facilitated comparisons between rheological experi-
ments, theories, and simulations, and sparked a com-
parison with uniform hard sphere (HS) suspensions
for +<+, and foams as ++ 1.
2. Monodisperse emulsions
Traditional methods of emulsification, such as
stirring and shaking typically lead to droplet size dis-
tributions that are uncontrolled and have a large
polydispersity, defined as Pa= Sa/ii, where is the
average droplet radius and Sa is the S.D. However,
many methods for making monodisperse emulsions
with Pa = 0. 1 now exist. These include depletion
flocculation fractionation [MI, controlled shear rup-
turing [19', 201, controlled coalescence [21"], mem-
brane emulsification [22'], phase-separating binary
mixtures under shear [23], and classic Bragg extrusion
of the dispersed phase through a pipette into a flowing
continuous phase [24].An example of a monodisperse
silicone oil-in-water emulsion stabilized by sodium
dodecylsulfate (SDS) with ii = 0.5,Pa= 0.1, and +=
0.6 is shown in Fig. 2. The emulsion can be diluted to
lower +, or an osmotic pressure, II, can be applied
through centrifugation or dialysis to raise +. If II is
applied rapidly, the disordered positional structure of
the droplets at low + can be quenched in. Light
scattering experiments on index-matched bulk emul-
sions at high + have demonstrated this disordered
glassy structure [2'1.
3. Droplet interactions
Interactions between the deformable interfaces of
droplets play an important role in emulsion rheology.
For incompressible dispersed phases, the most basic
interaction is that of excluded volume. The second
basic repulsive interaction results from work done
against u to create additional droplet surface area
when two droplets deform as they are forced together.
Finally, the surfactant typically provides a short-range
repulsion (disjoining pressure) that prevents droplet
coalescence. The net consequence of these repulsions
is depicted in Fig. 3 by the rise in both lines for the
droplet pair interaction potential, U, near and below
Figure 2
Current Opinion in Colloid& InterfaceScienci
Optical micrograph of a concentrated monodisperse emulsion of
uniformly sized droplets having an average radius Z = 0.5 wm,
polydispersity Pa = 0.1, and volume fraction q5 = 0.6. Some droplet
ordering has been induced by the shear when the microscope slide
is prepared.

3. T.G.Mason / Current Opinion in Colloid & Interface Science 4 (I999)231-238 233
the separation r = 2a. Describing how the droplets’
interfaces deform as they are forced together is com-
plicated, so the surfactant’s repulsive contribution is
usually crudely represented by a thickness, h, of the
film between the droplets [25]. Since h must be con-
sidered when droplets pack, the effective volume frac-
tion, +eff, is slightly larger than +:+e,, = +[1+3h/
(2a)], valid for h <<a and weakly deformed droplets.
Repulsive emulsions do not have potentials which
exhibit a deep potential well relative to k,T (dashed
line - Fig. 31, where k, is Boltzmann’s constant and
T is the temperature, but attractive emulsions do
(solid line - Fig. 3). Droplets in attractive emulsions
flocculate or gel. Depletion attractions can arise from
surfactant micelles [181, polymers [71, or even smaller
droplets [26]. Other attractions can be induced by
adding excess salt to emulsions stabilized by ionic
surfactants [27] or changing the solvent quality [28].
However, even a small density difference between the
continuous and dispersed phases can lead to rapid
gravity-driven creaming of flocs or aggregates, so
measuring the rheology of attractive emulsions can be
problematic. We focus on the rheology of repulsive
emulsions and comment about attractive emulsions
when appropriate.
Figure 3
I L
$4
, Repulsive 00
00
Schematic diagram of the pair potential, U,as a function of
separation, r, between the centers of two identical interacting
droplets. The dashed line depicts a repulsive positive potential, and
the solid line depicts an attractive potential with a well that is
significantly deeper than the thermal energy, k,T, so that droplets
can flocculate or aggregate. Both potentials rise toward low r
because of the short-range stabilizing repulsion of the surfactant
and the resistance of the droplets to deformation due to surface
tension.
or aggregates as i, is increased. For strong attractions,
tenuous gels of droplets [27]even exhibit weak elastic
shear moduli.
4. Dilute emulsion rheology
5. Glass transition in colloidal emulsions
Predictions of the viscosity, q, of dilute monodis-
perse emulsions have been tested empirically at low
enough shear rates that the shear stress, T, is less
than IILand there is little droplet deformation and
no rupturing. Steady shear viscosity measurements for
+e,, <0.4 [15]agree with simulations of monodisperse
HS suspensions [29] at large Peclet numbers, Pe =
q?/(kBT/U3)>> 1, where convection dominates dif-
fusion, yet at small Capillary numbers, Ca =
qi,/(a/a) a 1, where the droplets are not greatly
deformed. By contrast to Taylor’s theory for emulsion
viscosity [30], q(+) is well described by HS predictions
[29,31] even when the external viscosity, qe,is larger
than the internal droplet viscosity, qi.From this, one
can infer that the Gibbs elasticity opposing gradients
in the surfactant concentration on the droplet inter-
faces through the Marangoni effect, is typically large
enough to decouple external flow from that within the
droplets. However, polydisperse emulsion viscosities
can depart from the monodisperse HS prediction,
especially at higher +,because hydrodynamicinterac-
tions between droplets depend upon the distribution
and especially +c. As Ca + 1,a recent simulation [32]
predicts that emulsions with += 0.3 may exhibit a
pronounced shear thinning behavior (q decreasing as
i, increases). Finally, attractive emulsions can be shear
thinning even at dilute + due to the breakup of flocs
The identification of features of the colloidal glass
transition [33,34] in emulsion rheology is one of the
most important recent conceptual advances [14”1.
For hard spheres, the colloidal glass transition occurs
when the spheres become sufficiently concentrated
that a given droplet becomes caged by its neighbors
indefinitely. Thermal excitations are insufficient to
destroy these cages when + exceeds the glass transi-
tion volume fraction, +g. Light scattering and rhe-
ology measurements for HS are consistent with the
mode coupling theory prediction of +g =0.58 [35”1,
[36’] (see Fig. 1).For <+g, the cages are tran-
sient and break up over time scales that diverge as
+e,, + +g. By analogy to HS, an emulsion’s low-
frequency linear shear response for +e,, near +g
should be dominated by a plateau elastic modulus,
GIP, that is entropic in origin and scales with the
thermal energy density: GIth-kBT/Vf,where V, is
the translational free volume per droplet. Since V, -[a(+, - +)eff]3G1thwould diverge at +c for hard
spheres (or for emulsions if (T + a). For deformable
droplets, G’, does not diverge but instead approaches
IIL.Because Gth-a-3,the entropic elasticity and the
glass transition dynamics are most noticeable for
emulsions with sub-micron radii. For +<+g and IIL
zz=- k,T/l/f, the emulsion’s frequency-dependent

4. 234 T.G. Mason / CurrentOpinionin Colloid& Interface Science 4 (I999)231-238
lo6
lo5
lo4
lo3-
storage modulus, G’(o) and loss modulus, G”(o),
resemble those of a glassy HS suspension [37’] and
can be described using mode coupling theory. By
contrast, compressed emulsions with +eff significantly
larger than +c still exhibit slow relaxation resulting in
G”(o)> G’(o) as o + 0 due to droplet deformability
and finite h. The relationship between the emulsion’s
macroscopic rheology and the these slow glassy mi-
croscopic relaxations in the droplets’ positional and
interfacial structures is a subject of current interest. A
modified mode coupling theory has been proposed to
describe the glassy dynamics of disordered soft mate-
rials [38]. However, the connection between this the-
ory’s parameters and the microscopic droplet struc-
ture and dynamics remains to be elucidated.
I
- /-
-
G’ Ghe/ /
- /
/
/
/
G” /
/
I /
- - _ * .I-
---
, I, , , ‘ , , , , , -
Figure4
6. Linear viscoelastic shear moduli of compressed
emulsions
New developments in optical microrheology have
enhanced our understanding of the frequency-depen-
dent linear viscoelastic moduli of compressed emul-
sions. Diffusing wave spectroscopy (DWS) [39] has
been used to measure the time-dependent mean
square displacement, <Ar2(t)> ,of droplets in con-
centrated turbid monodisperse emulsions, and G’(0)
and G”(w) are obtained using a generalized
Stokes-Einstein relation [40”,41]. This method is
approximate because it treats the emulsion as an
isotropic viscoelastic continuum. By using DWS to
probe high w and mechanical rheometry to probe low
o,the storage and lossmoduli of a siliconeoil-in-water
emulsion with I+= 0.8 and a =0.5 km have been
measured over nine decades in o,as is shown by the
solid (G’) and dashed (G”) lines in Fig. 4. At low w,
G’(o) dominates G”(w),exhibits a plateau, and rises
at high frequencies as G’(o) -o1I2.This scaling and
a corresponding ‘anomalous viscous loss’ in G”(0)
implied by the Kramers-Kronig relations has been
predicted based on a theory of the collective slipping
motion of clusters of droplets in random directions
due to the disorder [42’]. The persistence of G’(o) -o1I2in measurements for +< + c may be due to a
crossover between this collective slipping motion and
the simple diffusive entropic relaxation of the un-
packed droplet structures, as in HS predictions [43,44].
By contrast, G”(o)exhibits a minimum at intermedi-
ate frequencies and rises rapidly at high frequencies
as G”(o)-o where it dominates G’(o). The rise in
G”(o) toward low o reflects droplet rearrangements
that slowly relax the emulsion’s quenched-in glassy
structure. Although HS mode coupling theory cannot
predict an emulsion’s viscoelastic spectra, it provides
a conceptual basis for explaining the development of
the plateau in G’(o) and minimum in G”(o) through
Frequency-dependent linear storage modulus, G’(w ) (solid line)
and loss modulus, G ” ( w )(dashed line) of a concentrated monodis-
perse emulsion with ii = 0.5 ym and 4 = 0.8 based on mechanical
oscillatory measurements at low w < lo2 rad/s and optical mea-
surements using Diffusing wave spectroscopy (DWS) at high w.
The low frequency plateau modulus, GI,, given by the inflection
point in G ’ ( w ) of the DWS measurements has been rescaled to
G’, of the mechanical measurements in order to correct for order
unity errors introduced by the non-spherical shape of the droplets
and the continuum approximation in the generalized Stokes-Ein-
stein equation. At high w, G ’ ( w ) scales as wl/*. The minimum in
G “ ( w ) is indicative of slow glassy relaxations in the droplet struc-
ture.
droplet caging. In other noteworthyexperiments,DWS
has been used to probe thermally-induced droplet
shape fluctuations [45’1 and foam film dynamics [461
and coarsening [471.
7. Elasticity of concentrated emulsions
The universal +dependence of the linear plateau
elasticity of disordered concentrated monodisperse
emulsions has been established. Measurements on
four emulsions having different a are described by:
Grp(+eff)= 1.5(o/a)(~+,~- + c ) [14”1 where 4, has
been identified as random close packing of monodis-
perse spheres, +c = +RCP =0.64 [48]. Although a
quasi-linear rise in Grp(+eff had been previously
measured [lo], little insight into the reported +c =
0.715 could be offered due to polydispersity. The
quasi-linear rise contrasts with a two-dimensionalthe-
ory of ordered droplets in which Grp(+eff)jumps
discontinuously from zero to the Laplace pressure
scale at +eff = +c [49]. Recent simulations of the
shape of three-dimensional droplets deformed by
plates [50”] using surface evolver software [511 have
demonstrated an anharmonic repulsion between
droplets that depend on the coordination number, 2,
of neighboring droplets; this anharmonicity is in ac-

5. T.G. Mason / Current Opinionin Colloid & Interface Science 4 (I999)231-238 235
cord with an earlier theory [52] and leads to a more
gradual increase in G'p(+eff)above +c. By combining
the average z-dependent anharmonic potentials with
a disordered three-dimensional droplet positional
structure and applying a small shear strain, Grp(+eff)
has been calculated [53"] and agrees well with the
measurements. These simulations also show the non-
affine motion of the disordered droplets. Measure-
ments and simulations of the osmotic equation of
state, II(+eff)[14"] exhibit a remarkable similarity to
Grp(+eff)for +eff immediately above + c . However, as
+eff + 1, II diverges and the measured G', ap-
proaches a constant that lies within 10% of a predic-
tion of GrP(l)= 0.5a/u [541and simulations that con-
sider different droplet structures [55]. Attractions do
not usually affect compressed emulsion elasticity
strongly because droplet deformation dominates the
rheology, but attractions can significantlyincrease G',
for +eff near and below +c by comparison to repulsive
emulsions [2,561.
8. Non-linear rheology of concentrated emulsions
Basic concepts for understanding yielding, fracture
flow, and emulsification are beginning to appear. A
schematic illustration of these phenomena for a con-
centrated emulsion is shown in Fig. 5, along with a
corresponding plot of ~(9).At low +, the stress ap-
proaches a constant defined to be the yield stress, T ~ .
For higher y, the interplay of the fluid viscositieswith
the interfacial structures within the emulsion cause
the shear stress to increase. For 7<IIL, droplet
rearrangements occur, but for T = IIL the droplets
can stretch, rupture, and, possibly even coalesce.
Given these complex phenomena at large 9 yielding
just beyond the linear regime has mostly been stud-
ied. Mechanical oscillatory measurements of the yield
strain, y, = TJG',, show that yy is much less than
unity and rises linearly: yy(+eff = 0.3 (+eff - + c ) for
+eff >+c = +RCP [15"]. Combinedwith G'p(+eff),this
implies that T, varies nearly quadratically above + c :
T, = 0.5 (u/u)+eff(+eff - &I2. A new optical tech-
nique has provided microscopic insight into yielding.
DWS has been applied to concentrated emulsions
[57"1, hard sphere suspensions [%I, and foams [591
that are sheared between two transparent plates at a
controlled strain amplitude and frequency. The strain
induces periodic echoes in the intensity autocorrela-
tion function that are used to deduce the proportion
of droplets that rearrange irreversibly. A comparison
of DWS echo to mechanical measurements implies
that yielding occurs when only approximately 5% of
the droplets rearrange irreversibly [57"]. Beyond the
yield regime, mechanical rheometry has been used to
Figure5
A rupturing
oooo.,
.coalescence
0)
0-
zY
-
Current Opinion in Colloid& lntelfaceScience
log y
Schematic log-log diagram of the steady shear stress, T, as a
function of the shear rate, j~ (solid line) for a concentrated emul-
sion. As y increases, T rises above the elastic yield stress, T,,, as
viscous contributions become important. As T approaches the
Laplace pressure scale, u/a (dashed line) the droplets can deform,
stretch, and rupture, as shown at right. Depending upon the inter-
facial properties, the droplets may also recombine through coales-
cence.
measure the steady-shear viscous stress: T~ = T - 7,.
For (beff <0.7, the flow is uniform, and T~ -+",where
x = 1/2 at (Peff = 0.63 to x = 2/3 at +eff = 0.58. A
theory [60] and a simulation for incompressible foams
[611predict T~ -j2I3,but no general prediction exists
for x(+eff). For +eE >0.7, the emulsion can fracture
[15,62] and + is not uniform throughout the rheome-
ter's gap. However, fracturing can be suppressed if
the gap is very small. Shear rupturing viscoelastic
polydisperse emulsion in a thin gap can lead to a
monodisperse emulsion of smaller droplets [19'1, [631.
Extensions of theories on the capillary instability
modified by membrane curvature elasticity [64] and
on the stability of cylindrical domains in phasesepa-
rating binary fluids in a shear flow [65] may provide
future insight into emulsification. Another interesting
instability occurs when draining foams are driven by
viscous flows of the continuous phase [66'].
9. Emulsions of viscoelastic materials
Emulsions need not be comprised solely of isotropic
viscous liquids, but may include viscoelastic or
anisotropic liquids such as polymers [ P I or liquid
crystals [67]. Bulk and interfacial energy storage com-
bine to provide a wide range of rheological behavior
[68',69-71'1. The measured G'(o) and G"(o)of
copolymer blends [71'] have been successfully com-
pared to a theory of spherical inclusionsof an isotropic
viscoelastic material in an isotropicviscoelastic matrix
[72"]. In the non-linear regime, droplets in blends
have been stretched by an elongational shear and can
form ellipsoids or long needles [691; such shears can
lead to cusped ends and tip streaming modes of

6. 236 T.G. Mason / CurrentOpinionin Colloid& Interface Science 4 (I999)231-238
droplet breakup [73]. Finally, a theoretical picture of
how compatibilizers inhibit droplet collisions during
copolymer emulsification has been developed [741.
10. Conclusions
Monodisperse emulsions have provided much new
insight into emulsion rheology, including the notion
of colloidal glasses of deformable droplets, yet many
challenges remain. Perhaps the most important is to
understand how polydispersity affects emulsion rhe-
ology. This could be studied by combining different
monodisperse emulsions to control the polydispersity.
Other rheological frontiers lie in crystalline emulsions
with ordered droplet structures, binary emulsions,
emulsions of liquid crystals, multiple emulsions,
inverse emulsions, attractive emulsions, and in shear-
induced droplet rearrangements, deformation, rupt-
uring, and coalescence.
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of special interest
w of outstanding interest
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