New concepts in Emulsion Rheology

ELSEVIER Curr

New fundam

Corporate Research Science Lab

Abstract

The field of emulsion rheology is dedroplet size distributions. The droplesimulations. 0 1999 Elsevier Science

Keywords: Monodisperse emulsions; Drop

1. Introduction

Emulsions consist of droplets persed in another immiscible liqumicroemulsion phases, emulsionsdynamic states. Instead, emulsionspersions; external shear energy large droplets into smaller ones duSurfactants that provide a stabilizision are typically introduced to inhcence [l]. If the liquids are molecules of the dispersed phachanged between droplets, so droplet size distribution due to Onegligible. When coalescence andpressed, the emulsion can remaieven when osmotically compressedfoam.

Emulsions exhibit highly variedior that is useful and fascinating [

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nt Opinion in Colloid & Interface Science 4 (1999) 231-238 www.elsevier.nl/locate/cocis

ental concepts in emulsion rheology

T.G. Mason

ratory, Exxon Research and Engineering Co., Route 22 East, Annandale, NJ 08801, USA

eloping rapidly due to investigations involving monodisperse emulsions having narrow t uniformity facilitates meaningful comparisons between experiments, theories, and Ltd. All rights reserved.

t size distribution; Coalescence

f one liquid dis- id. By contrast to are not thermo- are metastable dis- used to rupture ing emulsification. g interfacial repul- ibit droplet coales- ighly immiscible, e cannot be ex- oarsening of the stwald ripening is ripening are sup- stable for years

to form a biliquid

rheological behav- ', 3-51. An emul-

sion's macroscopic constitutive relationships between the stress and strain depend strongly on its composition, microscopic droplet structure, and interfacial interactions. By controlling the droplet volume fraction, +, 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 further 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 distribution and hence packing. Additives such as polymers can also modify emulsion rheology by raising

Elsevier Science Ltd. All rights reserved.

232 T. G. Mason / Current Opinion in Colloid & Interface Science 4 (I 999) 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 & Interface Science 1

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 complexity arising from the interplay of bulk and interfacial elastic contributions [P I .

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, measurements using monodisperse emulsions have established 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 important in emulsion rheology. The monodispersity has facilitated comparisons between rheological experiments, theories, and simulations, and sparked a comparison 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 distributions 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 rupturing [19', 201, controlled coalescence [21"], membrane 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 emulsions 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 & Interface Scienci

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.

T. G. Mason / Current Opinion in Colloid & Interface Science 4 (I 999) 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 fraction, +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 monodisperse emulsions have been tested empirically at low enough shear rates that the shear stress, T, is less than IIL and 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 interfaces 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 hydrodynamic interactions 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 transition volume fraction, +g. Light scattering and rheology 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 transient 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 - k B T / V f , where V, is the translational free volume per droplet. Since V, - [a(+, - +)eff]3G1th would 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


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 microscopic 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 materials [38]. However, the connection between this theory’s parameters and the microscopic droplet structure and dynamics remains to be elucidated.

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Figure 4

6. Linear viscoelastic shear moduli of compressed emulsions

New developments in optical microrheology have enhanced our understanding of the frequency-dependent linear viscoelastic moduli of compressed emulsions. Diffusing wave spectroscopy (DWS) [39] has been used to measure the time-dependent mean square displacement, < A r 2 ( t ) > , of droplets in concentrated 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 loss moduli of a silicone oil-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) - o1I2 in 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 intermediate 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 monodisperse emulsion with ii = 0.5 y m and 4 = 0.8 based on mechanical oscillatory measurements at low w < lo2 rad/s and optical measurements 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 structure.

droplet caging. In other noteworthy experiments, 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 monodisperse 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-dimensional theory 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-


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', approaches a constant that lies within 10% of a prediction of GrP(l) = 0.5a/u [541 and 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 significantly increase 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 concentrated emulsion is shown in Fig. 5, along with a corresponding plot of ~(9 ) . At low +, the stress approaches a constant defined to be the yield stress, T ~ .

For higher y, the interplay of the fluid viscosities with the interfacial structures within the emulsion cause the shear stress to increase. For 7 < I I L , 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 studied. 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"]. Combined with G'p(+eff), this implies that T, varies nearly quadratically above + c :

T, = 0.5 (u/u)+eff(+eff - &I2. A new optical technique 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

Figure 5

A rupturing oooo.,

.coalescence

0) 0 -

zY - Current Opinion in Colloid & lntelface Science

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 emulsion. 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 interfacial properties, the droplets may also recombine through coalescence.

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 [611 predict T~ - j2 I3 , 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 compared to a theory of spherical inclusions of an isotropic viscoelastic material in an isotropic viscoelastic 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


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 rheology. 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, rupturing, and coalescence.

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