A Simple Model for the Viscosity of Pickering Emulsions Rajinder Pal

fluids Article

AArticle Simple Model for the Viscosity of Pickering EmulsionsA Simple Model for the Viscosity of Pickering Emulsions Rajinder Pal

DepartmentRajinder Pal of Chemical Engineering, University of Waterloo, Waterloo, ON N2L 3G1, Canada; [email protected]; Tel.: +1-519-888-4567 (ext. 32985) Department of Chemical Engineering, University of Waterloo, Waterloo, ON N2L 3G1, Canada; Received:[email protected]; 23 November Tel.: 2017; +1-519-888-4567 Accepted: 22 (ext. December 32985) 2017; Published: 23 December 2017 Received: 23 November 2017; Accepted: 22 December 2017; Published: 23 December 2017 Abstract: A new model is proposed for the viscosity of Pickering emulsions at low shear rates. The modelAbstract: takesA into new consideration model is proposed the increase for the in the viscosity effective of volume Pickering fraction emulsions of droplets, at low due shear to rates.the presenceThe model of an takes interfacial into consideration layer of solid the nanoparticles increase in the at effective the oil-water volume interface. fraction The of droplets,model also due considersto the presence aggregation of an interfacialof droplets layer and ofeventual solid nanoparticles jamming of Pickering at the oil-water emulsion interface. at high The volume model fractionalso considers of dispersed aggregation phase. ofAccording droplets to and the eventual proposed jamming model, of the Pickering relative emulsionviscosity of at higha Pickering volume emulsionfraction ofat low dispersed shear rates phase. is Accordingdependent toon thethree proposed factors: model,contact theangle, relative ratio viscosityof bare droplet of a Pickering radius toemulsion solid nanoparticle at low shear radius, rates is dependentand the volume on three fraction factors: of contact bare angle,droplets. ratio For of bare a given droplet radius radius of to nanoparticles,solid nanoparticle the relati radius,ve andviscosity the volume of a Pickering fraction of emulsion bare droplets. increases For a givenwith radiusthe decrease of nanoparticles, in bare dropletthe relative radius. viscosity For O/W of aPickering Pickering emulsions, emulsion increasesthe relative with viscosity the decrease decreases in bare with droplet the increase radius. Forin contactO/W Pickeringangle. The emulsions, W/O Pickering the relative emulsion viscosity exhibits decreases an opposite with the behavior increase in contactthat the angle. relative The viscosityW/O Pickering increases emulsion with exhibitsthe increase an opposite in co behaviorntact angle. in that The the relativeproposed viscosity model increases describes with the the experimentalincrease in contact viscosity angle. data The for proposed Pickering model emulsions describes reasonably the experimental well. viscosity data for Pickering emulsions reasonably well. Keywords: Pickering emulsions; contact angle; nanoparticles; interface; rheology; viscosity Keywords: Pickering emulsions; contact angle; nanoparticles; interface; rheology; viscosity

1. Introduction 1. Introduction Pickering emulsions, also referred to as Ramsden emulsions, are dispersions of two immiscible liquidsPickering (oil and water) emulsions, where also the referred dispersed toas droplets Ramsden areemulsions, stabilized against are dispersions coalescence of two solely immiscible by the presenceliquids (oilof andsolid water) particles, where usually the dispersed nanoparticles, droplets at are the stabilized oil-water against interface. coalescence Figure solely1 shows by thea schematicpresence ofdiagram solid particles, of a droplet usually of Pickering nanoparticles, emul atsion. the oil-waterThe droplet interface. has a “core-shell” Figure1 shows morphology, a schematic wherediagram the of“core” a droplet is composed of Pickering of liquid emulsion. phase and The the droplet “shell” has consists a “core-shell” of an interfacial morphology, layer where of solid the nanoparticles.“core” is composed of liquid phase and the “shell” consists of an interfacial layer of solid nanoparticles.

FigureFigure 1. 1.DropletDroplet of of a aPickering Pickering emulsion. emulsion.

There has been a rapid growth of interest in nanoparticles-stabilized Pickering emulsions in There has been a rapid growth of interest in nanoparticles-stabilized Pickering emulsions in recent recent years due to their many potential applications in industries such as: food, pharmaceutical, years due to their many potential applications in industries such as: food, pharmaceutical, petroleum, petroleum, biomedical, environment, cosmetics, and catalysis [1–18]. A recent article by Binks [16] biomedical, environment, cosmetics, and catalysis [1–18]. A recent article by Binks [16] discusses many discusses many potential applications of Pickering emulsions. The biomedical and pharmaceutical potential applications of Pickering emulsions. The biomedical and pharmaceutical applications are applications are discussed at length in a paper by Wu and Ma [1]. In a series of articles by Leclercq discussed at length in a paper by Wu and Ma [1]. In a series of articles by Leclercq and Nardello-Rataj and Nardello-Rataj group [19–22], the applications of Pickering emulsions in interfacial catalysis for group [19–22], the applications of Pickering emulsions in interfacial catalysis for biphasic systems are Fluids 2017, 3, 2; doi: 10.3390/fluids3010002 www.mdpi.com/journal/fluids

Fluids 2018, 3, 2; doi:10.3390/ﬂuids3010002 www.mdpi.com/journal/ﬂuids Fluids 20172018,, 33,, 2 2 2 of 12 13 biphasic systems are discussed. For example, reactions can be carried out simultaneously in both discussed.aqueous and For non-aqueous example, reactions phases can by be placing carried solid out simultaneously catalyst nanoparticles in both aqueousat the oil-water and non-aqueous interface phases[18–22]. by The placing solid solidnanoparticles catalyst nanoparticles placed at the at oil- thewater oil-water interface interface serve [18 the–22 ].dual The purpose solid nanoparticles of catalyst placedand emulsion at the oil-water stabilizer. interface serve the dual purpose of catalyst and emulsion stabilizer. For solid particles to be adsorbed at the oil-water interface, it is necessary that the particles are partially wetted by both oil and water phases. When solid particles adsorb at the oil-water interface, they reduce the interfacial area of the high energy oil-wateroil-water interface. Thus Thus,, the driving force for the transfer of particles at the interface is the reductionreduction of the interfacial area. Depending on the relative wettability of the liquid phases for the solid nanoparticles, two types of emulsions could be formed: oil-in-water (O/W) emulsion, emulsion, where where oil oil forms forms the the dr dropletsoplets and and water is the continuous phase, and water-in-oil (W/O) emulsion, emulsion, where where water water forms forms the the droplets and oil is the continuouscontinuous phase. The liquid phase, which preferentially wets the solid particles tends to become the continuouscontinuous phase of the emulsion and the liquid phase, which is relativelyrelatively less wetting of the solid particles becomes the dispersed phase. The relative wettability of the liqu liquidid phases for the solid particles is determined by the three-phase contactcontact angleangleθ θ.. If If the the contact contact angle angle measured measured through through the the aqueous aqueous phase, phase, as as shown shown in Figurein Figure2, is 2, greater is greater than than 90 ◦ ,90°, then then the solidthe solid particles particles are relatively are relatively more more wetted wetted by the by oil the phase, oil phase, and in suchand in situations, such situations, W/O emulsions W/O emulsions are generally are generally formed. formed. When theWhen contact the contact angle θ angleis less θ than is less 90◦ than, the solid90°, the particles solid particles are preferentially are preferentially wetted by wetted the aqueous by the phase aqueous and phase in such and situations, in such O/Wsituations, emulsions O/W areemulsions favored. are When favored.θ = 90When◦, both θ = liquid 90°, both phases liquid wet phases the solid wet particles the solid equally, particles and equally, in such and situations, in such theresituations, is no preferredthere is no emulsion—O/W preferred emulsion—O/W or W/O. or W/O.

Figure 2. DifferentDifferent wetting conditions descr describedibed in terms of contact angle θ..

The energy required to desorb a single solid nanoparticle from the oil-water interface is given as The energy required to desorb a single solid nanoparticle from the oil-water interface is given [8]: as [8]: E = =ππR22γγ (()1±± cosθθ2)2 (1) E Rnpnp OWOW 1 cos (1) where E is the energy required to desorb a single solid nanoparticle of radius Rnp from the oil-water where E is the energy required to desorb a single solid nanoparticle of radius Rnp from the oil- interface with interfacial tension γOW. The plus sign in the brackets gives the energy that is required γ forwater desorption interface of with a nanoparticle interfacial fromtension oil-water OW . interfaceThe plusinto sign the in oilthe phase, brackets and gives the negative the energy sign that gives is energyrequired for for desorption desorption of of a a nanoparticle nanoparticle into from the oil- aqueouswater interface phase. Equationinto the oil (1) phase, could and be re-writtenthe negative in dimensionlesssign gives energy form for as: desorption of a nanoparticle into the aqueous phase. Equation (1) could be re- 2 2 written in dimensionless form as:E /kT = πRnpγOW (1 ± cos θ) /kT (2) where k is the Boltzmann constant and T=isπ the2 γ absolute()± temperature.θ 2 This Equation (2) gives the E/kT Rnp OW 1 cos /kT (2) desorption energy relative to thermal energy of a particle. whereFigure k is3 the shows Boltzmann the plots constant of relative and desorption T is the energyabsolute of atemperature. nanoparticle This from Equation the oil-water (2) gives interface the asdesorption a function energy of contact relative angle. to thermal The plots energy are shown of a particle. for different values of nanoparticle radius (ranging fromFigure 5 nm to 3 100 shows nm). the The plots interfacial of relative tension desorptionγOW is ﬁxed energy at 10 mN/m.of a nanoparticle The relative from desorption the oil-water energy neededinterface to as desorb a function a nanoparticle of contact angle. from the The oil-water plots are interface shown for to different the aqueous values phase of nanoparticle decreases with radius the decrease in contact angle. The relative desorption energy neededγ to desorb a nanoparticle from the (ranging from 5 nm to 100 nm). The interfacial tension OW is fixed at 10 mN/m. The relative oil-water interface to the oil phase decreases with the increase in contact angle. desorption energy needed to desorb a nanoparticle from the oil-water interface to the aqueous phase decreases with the decrease in contact angle. The relative desorption energy needed to desorb a nanoparticle from the oil-water interface to the oil phase decreases with the increase in contact angle. Fluids 2018, 3, 2 3 of 13 Fluids 2017, 3, 2 3 of 12 Fluids 2017, 3, 2 3 of 12

Figure 3. Relative desorption energy of a single solid nanoparticle as a function of contact angle θ for Figure 3. Relative desorption energy of a single solid nanoparticle as a function of contact angle θ for Figuredifferent 3. Relativevalues of desorption nanoparticle energy radius. of a single solid nanoparticle as a function of contact angle θ for differentdifferent values of nanoparticle radius. γ The desorption energy of a nanoparticle also depends on the interfacial tension OW . Figure 4 TheThe desorption desorption energy energy of of a a nanoparticle nanoparticle also also depends depends on on the the interfacial interfacial tension tension γγ . Figure Figure 44 shows the plots of relative desorption energy for different values of interfacial tension. OWTheOW radius of shows the plots of relative desorption energy for different values of interfacial tension. The radius of showsthe nanoparticle the plots of is relative fixed at desorption 25 nm. As energythe interfac for diffeial tensionrent values increases, of interfacial the relative tension. desorption The radius energy of the nanoparticle is ﬁxed at 25 nm. As the interfacial tension increases, the relative desorption energy theincreases. nanoparticle The effect is fixed of interfacial at 25 nm. tensionAs the interfac on the ialdesorption tension increases, energy is theless relativesevere in desorption comparison energy with increases. The effect of interfacial tension on the desorption energy is less severe in comparison with increases.the effect Theof nanoparticle effect of interfacial size, as expectedtension on from the desorptionEquation (2). energy is less severe in comparison with thethe effect effect of of nanoparticle nanoparticle size, as expected from Equation (2).

Figure 4. Relative desorption energy of a single solid nanoparticle as a function of contact angle θ for FigureFiguredifferent 4. 4. RelativevaluesRelative of desorption desorption interfacial energy energytension. of of a a single single solid solid nanoparticle nanoparticle as as a a func functiontion of of contact contact angle angle θθ forfor differentdifferent values of interfacial tension. The desorption energy should be significantly larger than the thermal energy in order to produce The desorption energy should be significantly larger than the thermal energy in order to produce a stableThe emulsion. desorption In energy general, should the contact be signiﬁcantly angle for larger the stabilization than the thermal of O/W energy emulsions in order should to produce be in athe stable range emulsion. 15oo<<θ In general, 90 [23]. the For contact stable angle W/O foremulsions the stabilization to be produced, of O/W theemulsions contact shouldangle should be in a stable emulsion.oo In general, the contact angle for the stabilization of O/W emulsions should be in 15<<θ oo 90 o thebe inrange the range ◦ 90<<◦θ [23]. 165 For [23]. stable If the W/O solid emulsions nanoparticles to be areproduced, too hydrophobic, the contact θ angle≈ 180 should, then the range 15 < θ surfactant or nanoparticles), the rheological constitutive equation for a dilute emulsion of two immiscible Newtonian liquids is given as:

2 + 5λ σ = −P δ + 2η 1 + ϕ E (3) c 2 + 2λ where σ is the bulk stress tensor, P is the average pressure, δ is the unit tensor, E is the bulk rate of strain tensor, ηc is the viscosity of the continuous phase, λ is the ratio of dispersed-phase viscosity to continuous-phase viscosity, and ϕ is the volume fraction of the dispersed phase. From Equation (3), it follows that the effective viscosity of a dilute emulsion is:

2 + 5λ η = η 1 + ϕ (4) c 2 + 2λ

Equation (4) is referred to as the Taylor emulsion viscosity equation [26]. It could also be expressed as: 2 + 5λ η = η/η = 1 + ϕ (5) r c 2 + 2λ where ηr is the relative viscosity deﬁned as the ratio of emulsion viscosity to continuous phase viscosity. From Equation (5), the intrinsic viscosity [η] of an emulsion is as follows:

η − 1 2 + 5λ [η] = Lim r = (6) ϕ→0 ϕ 2 + 2λ

For dilute suspensions of rigid hard spheres, Equation (5) reduces to the well-known Einstein suspension viscosity equation [27,28]:

ηr = η/ηc = [1 + 2.5ϕ] (7)

Note that the intrinsic viscosity of suspension of hard spheres has a constant value of 2.5 whereas the intrinsic viscosity of emulsion is dependent on the viscosity ratio λ (see Equation (6)). The Einstein suspension viscosity equation as well as the Taylor emulsion viscosity equation are valid only for very dilute systems. Several authors [29–33] have developed viscosity equations for non-dilute dispersed systems taking into account the packing limits of particles and droplets. The following equation proposed by Krieger and Dougherty [31] is quite popular in the literature available on the viscosity equations for concentrated suspensions of rigid particles:

ϕ −2.5ϕm ηr = 1 − (8) ϕm where ϕm is the maximum packing volume fraction of particles where the relative viscosity of suspension diverges. Pal [30] has extended the Taylor emulsion viscosity equation for dilute systems to concentrated emulsions using the differential effective medium approach and taking into consideration the packing limits of droplets. He proposed the following equation for the viscosity of concentrated emulsions at low shear rates: 3/2 −2.5ϕm 2ηr + 5λ ϕ ηr = 1 − (9) 2 + 5λ ϕm Fluids 2018, 3, 2 5 of 13

In the limit λ → ∞ (rigid particles), Equation (9) reduces to the Krieger-Dougherty Equation (8). In the limit ϕ → 0, Equation (9) reduces to the Taylor emulsion viscosity equation for dilute emulsions. This equation is found to adequately describe a large pool of available viscosity data on unstable (without interfacial additives) and surfactant-stabilized emulsions.

3. Rheology of Pickering Emulsions—Brief Literature Review As noted in the preceding section, the rheology of traditional emulsions (surfactant-stabilized emulsions and emulsions without any interfacial additives) is well understood. Models are available to predict a priori the rheological properties of such emulsions. However, the rheology of Pickering emulsions stabilized by solid amphiphilic nanoparticles has received a limited attention. Binks et al. [34] experimentally studied the rheological behavior of W/O emulsions stabilized by hydrophobic bentonite clay particles. The emulsions exhibited shear-thinning and yield stresses due to the formation of a network structure of clay particles and water droplets. Wolf et al. [35] investigated the rheology of Pickering O/W emulsions stabilized by hydrophilic silica nanoparticles. The droplets were coated by a monolayer of closely packed silica nanoparticles. At high volume fractions of oil droplets, the emulsions exhibited shear-thickening behavior at intermediate shear rates. The shear-thickening behavior is frequently observed in suspensions of rigid particles, but not in traditional emulsions. Thus, the authors concluded that the droplets of oil coated with monolayers of solid nanoparticles behave more like rigid spheres. Katepalli et al. [36] investigated the effects of nanoparticles shape and inter-particle interactions on the rheology of Pickering O/W emulsions at a ﬁxed volume fraction of oil of 0.50. The spherical silica nanoparticles stabilized the emulsion by forming a hexagonally packed monolayer of particles at the oil-water interface. The non-spherical fumed silica nanoparticles stabilized emulsion droplets by forming an interlocked network of particles at the oil-water interface. At high salt concentrations, the interparticle interactions were attractive, resulting in a network of nanoparticles and emulsion droplets, and consequently, non-Newtonian rheology. Braisch et al. [37] studied the viscosity behavior of O/W emulsions stabilized by hydrophilic silica nanoparticles. However, the matrix of emulsions was thickened by a polymer (polyethylene glycol, PEG 6000). The droplets were fully covered by a monolayer of silica nanoparticles. The O/W Pickering emulsions thickened by PEG exhibited shear-thinning behavior. Ganley and Van duijneveldt [38] investigated the rheology of O/W Pickering emulsions that were stabilized by montmorillonite clay platelets. The clay was present in excess of what was needed to fully coat the oil droplets. The excess clay platelets formed a network structure in the matrix phase imparting non-Newtonian rheology to emulsions. Hermes and Clegg [39] studied the rheology, including yielding behavior, of highly concentrated O/W emulsions (also referred to as emulsion gels) stabilized by silica nanoparticles. The Pickering O/W emulsion gels were viscoelastic in nature and exhibited yield stress. In summary, the rheology of Pickering emulsions stabilized by solid nanoparticles is not as well understood as that of traditional surfactant-stabilized emulsions. Little or no work has been reported on modelling of the rheological behavior of Pickering emulsions. The experimental studies that are reported in the literature on the rheology of Pickering emulsions often deal with complicated systems where the non-Newtonian behavior is caused by factors, such as: network formation of droplets and nanoparticles; network of ﬁlms of matrix phase in highly concentrated emulsion gels; and incorporation of thickeners (clays, polymers) in the matrix phase.

4. Modelling of the Viscosity of Pickering Emulsions

4.1. Dilute Pickering Emulsions Consider a single “core-shell” droplet of a Pickering emulsion, as shown in Figure1. While the core of the composite droplet is mobile, the presence of a rigid interfacial layer of solid nanoparticles at the surface of the droplet disrupts the transmission of tangential and normal stresses from the Fluids 2018, 3, 2 6 of 13

Fluidscontinuous 2017, 3, 2 phase liquid to the core liquid. Thus, the composite “core-shell” droplets of a Pickering6 of 12 Fluids 2017, 3, 2 6 of 12 emulsion can be treated as rigid particles of effective radius Re f f given as: = + Reff R=d h+ (10) RReeff f f = RRdd +hh (10)(10) where Rd is the radius of the core droplet and h is the thickness of the solid nanoparticle layer where RRd isis thethe radiusradius ofof thethe corecore dropletdroplet and hh isis thethe thicknessthickness ofof thethe solidsolid nanoparticlenanoparticle layer protruding into the continuous phase (see Figure 5). protruding into the continuous phase (see(see FigureFigure5 5).).

Figure 5. Composite “core-shell” droplet of a Pickering emulsion with core radius Rd and interfacial Figure 5. Composite “core-shell” droplet of a Pickering emulsion with core radius Rd and interfacial layerFigure thickness 5. Composite h protruding “core-shell” into dropletthe continuous of a Pickering phase. emulsion with core radius Rd and interfacial layer thickness hhprotruding protruding into into the th continuouse continuous phase. phase. It should be noted that the thickness of the rigid interfacial layer h protruding into the It should be noted that the thickness of theθ rigid interfacial layer h protruding into the continuousIt should phase be is noted governed that the by thicknessthe contact of theangle rigid interfacial and radius layer of interfacialh protruding nanoparticles into the continuous Rnp . continuous phase is governed by the contact angle θ and radius of interfacial nanoparticles R . R np Considerphase is a governed single solid by nanoparticle the contact angle presentθ and at the radius oil-water of interfacial interface, nanoparticles as shown in Figurenp. Consider 6. The asolid single Consider a single solid nanoparticle present at the oil-water interface, as shown in Figure 6. The solid nanoparticlesolid nanoparticle is assumed present to be at present the oil-water at the interface interface, of asan shown oil droplet. in Figure From6. th Thee figure, solid nanoparticleit is clear that is nanoparticle is assumed to be present at the interface of an oil droplet. From the figure, it is clear that theassumed thickness to of be the present interfacial at the interfacelayer protruding of an oil into droplet. the continuous From the ﬁgure, phase it (aqueous is clear that phase the thicknessin Figure of the thickness of the interfacial layer protruding into the continuous phase (aqueous phase in Figure the interfacial layer protruding into the continuousθ phase (aqueous phase in Figure6) can be expressedR 6) can be expressed in terms of the contact angle θ and radius of interfacial nanoparticles np as 6)in can terms be ofexpressed the contact in angleterms θofand the radiuscontact of angle interfacial and nanoparticles radius of interfacialRnp as follows: nanoparticles Rnp as follows: follows: h== Rnp(+1 + cosθ θ) (11) h R=np(1 cos+ )θ (11) h Rnp(1 cos ) (11)

Figure 6. Determination of interfacial thickness of solid nanoparticles protruding into the continuous Figure 6. Determination of interfacial thickness of solid nanoparticles protruding into the continuous phase of an emulsion. phase of an emulsion. Equation(11) is valid for O/W emulsions. For W/O emulsions, the thickness of the interfacial layer Equation(11)Equation(11) is valid is valid for O/W for O/Wemulsions. emulsions. For W/O For W/Oemulsions, emulsions, the thickness the thickness of the ofinterfacial the interfacial layer protruding into the continuous phase is: protrudinglayer protruding into the into continuous the continuous phase phase is: is: = − θ h R=np(1 cos− )θ (12) h Rnp(1 cos ) (12) h = Rnp(1 − cos θ) (12) The increase in the effective volume fraction of droplets due to the presence of an interfacial The increase in the effective volume fraction of droplets due to the presence of an interfacial monolayer of solid nanoparticles at the oil-water interface can be expressed in terms of the layer monolayer of solid nanoparticles at the oil-water interface can be expressed in terms of the layer thickness as: thickness as: Fluids 2018, 3, 2 7 of 13

The increase in the effective volume fraction of droplets due to the presence of an interfacial monolayer of solid nanoparticles at the oil-water interface can be expressed in terms of the layer thickness as: h 3 ϕS = ϕ 1 + (13) Rd where ϕS is the effective volume fraction of composite core-shell droplets and ϕ is the volume fraction of bare droplets without interfacial layer of solid nanoparticles. As composite droplets are assumed to behave as rigid particles, the relative viscosity of dilute Pickering emulsions can be calculated from the Einstein viscosity equation:

ηr = η/ηc = [1 + 2.5ϕS] (14) where ϕS is given by Equation (13).

4.2. Concentrated Pickering Emulsions

The modiﬁed Einstein equation, Equation (14), is valid only in the limit ϕS → 0. We can re-write Equation (5) as: Volume of composite droplets η = 1 + 2.5 (15) r Total volume of emulsion Pal [40] has argued that it is the free volume available to the inclusions that is more relevant in determining the viscosity of non-dilute dispersed systems. Accordingly, the following equation is applicable to concentrated Pickering emulsions:

Volume of composite droplets η = 1 + 2.5 (16) r Free volume available to emulsion

From Equation (16), it follows that: ϕS ηr = 1 + 2.5 (17) 1 − ϕS

This equation is applicable at low to moderate volume fractions of composite droplets. At high concentrations of dispersed phase, clustering of droplets dominates. When clusters are formed, a significant amount of continuous-phase fluid is trapped within the clusters resulting in an increase in the effective dispersed phase concentration. At glass transition concentration of composite droplets ϕg, the entire emulsion is jammed into a single large cluster. In the jammed or arrested state of the emulsion, the emulsion is expected to behave as a semi-solid with significant yield-stress. Figure7 schematically shows the change in emulsion morphology, with the increase in volume fraction of composite droplets. In order to take into account the aggregation of droplets in Pickering emulsions, Equation (17) is modified as follows: ! ϕe f f ηr = 1 + 2.5 (18) 1 − ϕe f f where ϕe f f is the effective volume fraction of the dispersed phase. The relationship between ϕe f f and ϕS can be developed on the following basis: (a) in the dilute limit ϕS → 0, ϕe f f = ϕS; (b) in the jammed state limit ϕS → ϕg , ϕe f f = 1; (c) ∂(ϕe f f /ϕS)/∂ϕS ≥ 0; and (d) in the limit ϕS → ϕg , ∂(ϕe f f /ϕS)/∂ϕS = 0. Fluids 2017, 3, 2 7 of 12

3  h  φ = φ +  S 1  (13)  Rd  φ where S is the effective volume fraction of composite core-shell droplets and φ is the volume fraction of bare droplets without interfacial layer of solid nanoparticles. As composite droplets are assumed to behave as rigid particles, the relative viscosity of dilute Pickering emulsions can be calculated from the Einstein viscosity equation: η =η η =[]+ φ r / c 1 2.5 S (14) φ where S is given by Equation (13).

4.2. Concentrated Pickering Emulsions φ → The modified Einstein equation, Equation (14), is valid only in the limit S 0. We can re-write Equation (5) as:

η = + Volume of composite droplets r 1 2.5  (15)  Total volume of emulsion 

Pal [40] has argued that it is the free volume available to the inclusions that is more relevant in determining the viscosity of non-dilute dispersed systems. Accordingly, the following equation is applicable to concentrated Pickering emulsions:  Volume of composite droplets  η = +   r 1 2.5  (16)  Free volume available to emulsion

From Equation (16), it follows that:  φ  Fluids 2018, 3, 2 η =1+ 2.5 S  8 of 13 r  −φ  (17) 1 S 

ThisThe followingequation is empirical applicable expression at low to moderate meets all ofvolume the conditions fractions of just composite mentioned, droplets. and has At beenhigh concentrationsfound to describe of aggregationdispersed phase, behavior clustering of a variety of droplets of dispersed dominates. systems When [40– 42clusters]: are formed, a significant amount of continuous-phase fluid is trapped within the clusters resulting in an increase  s  2 in the effective dispersed phase concentration.1 − Atϕg glass transitionϕg concentration− ϕS of composite droplets φ ϕe f f /ϕS = 1 +  1 −  (19) g , the entire emulsion is jammed into a singleϕg large cluster. In ϕtheg jammed or arrested state of the emulsion, the emulsion is expected to behave as a semi-solid with significant yield-stress. Figure 7 schematicallyIn summary, shows the modelthe change proposed in emulsion in this work morphology, for the viscosity with the of diluteincrease to concentratedin volume fraction Pickering of compositeemulsions droplets. at low shear rates is Equation (18) in conjunction with Equations (11)–(13) and (19).

Figure 7. 7. ChangeChange in morphology in morphology of Pickering of Pickering emulsion emulsion with the with increase the increasein composite in composite droplets concentration.droplets concentration.

4.3. Limitations of the Proposed Model The proposed model has the following restrictions: the solid nanoparticles are spherical in shape; the solid nanoparticles coat the surface of the emulsion droplets by forming a monolayer of closely packed nanoparticles; the excess nanoparticles, over and above the amount that is needed to form a closely packed monolayer on the surface of the emulsion droplets, simply enhance the viscosity of the matrix phase without any network structure formation; the nanoparticles-coated droplets behave as rigid hard spheres; and, the composite nanoparticles-coated droplets are reasonably monodisperse. Some of these assumptions such as hard-sphere interactions and mono-dispersity of droplets can be relaxed if the glass-transition concentration ϕg is taken to be an adjustable parameter. The jamming concentration of emulsion droplets is clearly dependent on the type of interactions and the size distribution of emulsion droplets.

5. Predictions of the Proposed Viscosity Model for Pickering Emulsions According to the proposed model, the relative viscosity of a Pickering emulsion (O/W or W/O) is a function of the following factors: (a) volume fraction of bare droplets (without interfacial nanoparticle layers); (b) the relative size of nanoparticles to bare droplets; and, (c) contact angle. Figure8 shows the model predictions for the relative viscosity O/W Pickering emulsion as a function of volume fraction of ◦ bare droplets for different values of Rd/Rnp, ranging from 10 to 40. The contact angle is taken to be 45 and the glass transition volume fraction of composite droplets where jamming of Pickering emulsion is expected is taken as 0.58. Clearly, the relative viscosity is strongly dependent on the relative sizes of the nanoparticles and bare droplets. For a given bare droplet radius, the relative viscosity increases with the increase in the radius of the nanoparticles at any given volume fraction of bare droplets. The effect of contact angle θ on the relative viscosity of O/W and W/O emulsions, as predicted by the model, is shown in Figure9. The Rd/Rnp ratio is ﬁxed at 10 and the glass transition volume fraction of composite droplets is 0.58. For O/W emulsions, the relative viscosity at any given volume fraction of bare droplets decreases with the increase in contact angle. A reverse behavior is observed in the case of W/O emulsions where the relative viscosity increases with the increase in contact angle. FluidsFluids 20182017,, 33,, 22 99 of of 13 12 Fluids 2017, 3, 2 9 of 12

Figure 8. Model predictions of relative viscosity versus dispersed-phase (bare droplets) concentration Figure 8. Model predictions of relative viscosity versus dispersed-phase (bare droplets) concentration Figure(vol. 8. Model %) of predictionsoil-in-water (O/W)of relative Pickering viscosity emulsions versus for dispersed-phase different ratios (bareof bare droplets) droplet concentrationradius to (vol.(vol. %)nanoparticle%) ofof oil-in-wateroil-in-water radius. (O/W)(O/W) Pickering emulsions for different ratiosratios ofof barebare dropletdroplet radiusradius toto nanoparticlenanoparticle radius. radius.

Figure 9. Model predictions of relative viscosity versus contact angle of oil-in-water (O/W) and water- in-oil (W/O) Pickering emulsions for different volume fractions of bare droplets. Figure 9. Model predictions of relative viscosity versus contact angle of oil-in-water (O/W) and water- Figure6. Comparison 9. Model of predictions Model Predictions of relative with viscosity Experimental versus contactData angle of oil-in-water (O/W) and in-oil (W/O) Pickering emulsions for different volume fractions of bare droplets. water-in-oilThe experimental (W/O) Pickering viscosity emulsions data for for Pickering different volumeO/W emulsions fractions were of bare collected droplets. using a co-axial 6. Comparisoncylinder viscometer. of Model The Predictions emulsions withwere preparedExperimental using hydrophilicData starch nanoparticles at 2 wt % 6. Comparisonconcentration of Modelin the aqueous Predictions phase. with The Experimentaloil used was a Datawhite mineral oil, which was supplied by ThePetro-Canada. experimental The oil viscosity viscosity data was 21for mPa Pickering s at 28 °C. O/W The emulsionsaverage diameter were ofcollected the nanoparticles using a wasco-axial The experimental viscosity data for Pickering O/W emulsions were collected using a co-axial cylinder20.9 viscometer. nm and the average The emulsions droplet diameter were prepared was 91.7 µm.using The hydrophilic contact angle starch was 46°, nanoparticles as reported inat our 2 wt % cylinder viscometer. The emulsions were prepared using hydrophilic starch nanoparticles at 2 wt % concentrationearlier study in [43].the Theaqueous viscosity phase. measurements The oil usedwere carriedwas a outwhite at a temperaturemineral oil, of which about 28was °C. supplied by concentrationFigure in 10 the compares aqueous the phase. low shear The relative oil used viscos wasity a data white of Pickering mineral oil,O/W which emulsions was stabilized supplied by Petro-Canada. The oil viscosity was 21 mPa s at 28◦ °C. The average diameter of the nanoparticles was Petro-Canada.by starch Thenanoparticles oil viscosity with was the 21 proposed mPa s at model. 28 C. TheThe averageexperimental diameter relative of theviscosity nanoparticles data for was 20.9 nm and the average droplet diameter was 91.7 µm. The contact angle was 46°,◦ as reported in our 20.9 nmPickering and the O/W average emulsions droplet can diameter be described was reasonably 91.7 µm. Thewell contactby the model angle using was 46 the, glass as reported transition in our earlier study [43]. The viscosity measurements were carried out at a temperature of about 28 °C. earliervolume study [fraction43]. The of viscosity0.52. This measurementsvolume fraction wereof droplets carried where out viscosity at a temperature divergence of occurs about is 28 lower◦C. Figure 10 compares the low shear relative viscosity data of Pickering O/W emulsions stabilized Figurethan the 10 reportedcompares literature the low value shear of relative 0.58 correspo viscositynding data to of glass Pickering transition O/W concentration emulsions of stabilized mono- by by starch nanoparticles with the proposed model. The experimental relative viscosity data for starchsized nanoparticles hard spheres. with theThis proposed is not unexpected, model. The given experimental that the emulsions relative viscosity are not datamonodisperse. for Pickering Pickering O/W emulsions can be described reasonably well by the model using the glass transition O/WFurthermore, emulsions can due be to described the large reasonablysize of the welldroplets, by the the model droplets using also theprobably glass transitionundergo some volume volumedeformation fraction ofwhen 0.52. they This are volume subjected fraction to a shear of dropletsfield. where viscosity divergence occurs is lower fraction of 0.52. This volume fraction of droplets where viscosity divergence occurs is lower than the than the reported literature value of 0.58 corresponding to glass transition concentration of mono- reported literature value of 0.58 corresponding to glass transition concentration of mono-sized hard sized hard spheres. This is not unexpected, given that the emulsions are not monodisperse. spheres. This is not unexpected, given that the emulsions are not monodisperse. Furthermore, due to Furthermore, due to the large size of the droplets, the droplets also probably undergo some the large size of the droplets, the droplets also probably undergo some deformation when they are deformation when they are subjected to a shear field. subjected to a shear ﬁeld. Fluids 2018, 3, 2 10 of 13 Fluids 2017, 3, 2 10 of 12 Fluids 2017, 3, 2 10 of 12

Figure 10. Comparison of the experimental relative viscosity data for Pickering O/W emulsions with FigureFigure 10.10. ComparisonComparison ofof thethe experimentalexperimental relativerelative viscosityviscosity datadata forfor PickeringPickering O/WO/W emulsions withwith the predictions of the proposed model. thethe predictionspredictions ofof thethe proposedproposed model.model. Figure 11 compares the experimental low-shear viscosity data of Wolf et al. [35] with the Figure 11 compares the experimental low-shear viscosity data of Wolf et al. [35] with the proposedFigure model. 11 compares The experimental the experimental data low-shearwere obta viscosityined for dataO/W of emulsions Wolf et al. that [35] were with thestabilized proposed by proposed model. The experimental data were obtained for O/W emulsions that were stabilized by model.hydrophilic The experimental silica nanoparticles. data were The obtained Sauter mean for O/W diameter emulsions of emulsion that were droplets stabilized was by 3.15 hydrophilic µm. The hydrophilic silica nanoparticles. The Sauter mean diameter of emulsion droplets wasµ 3.15 µm. The silicasilica nanoparticles.nanoparticles had The an Sauteraverage mean diameter diameter of 8 nm. of emulsionAlthough dropletsthe contact was angle 3.15 wasm. not The reported, silica silica nanoparticles had an average diameter of 8 nm. Although the contact angle was not reported, nanoparticles had an average diameter of 8 nm. Although the contact angle was not reported,R / R it it is assumed to be 45°.◦ Due to small sized nanoparticles relative to the droplet size (large Rd / Rnp isit is assumed assumed to to be be 45 45°.. Due Due to to small small sizedsized nanoparticlesnanoparticles relativerelative toto the droplet size size (large (large Rd/Rnpnp ratio of approximately 394), the predictions of the proposed model are insensitive to contact angle. ratioratio ofof approximatelyapproximately 394), the predictionspredictions of the proposed model areare insensitive to contact angle. Interestingly, the model prediction using the glass transition concentration of monodisperse hard Interestingly,Interestingly, thethe modelmodel predictionprediction usingusing thethe glassglass transitiontransition concentrationconcentration ofof monodispersemonodisperse hardhard spheres of 0.58 is in excellent agreement with the experimental viscosity data of Wolf et al. [35], as spheresspheres ofof 0.580.58 isis inin excellentexcellent agreementagreement withwith thethe experimentalexperimental viscosity data of Wolf et al.al. [[35],35], asas obtained for Pickering O/W emulsions at a low shear stress of 0.2 Pa. Note that the Pickering obtainedobtained forfor Pickering Pickering O/W O/W emulsions emulsions at aat low a shearlow sh stressear ofstress 0.2 Pa.of Note0.2 Pa. that Note the Pickering that the emulsionsPickering emulsions that were prepared by Wolf et al. consisted of relatively much smaller droplets as thatemulsions were preparedthat were by prepared Wolf et al. by consisted Wolf et ofal. relatively consisted much of relatively smaller dropletsmuch smaller as compared droplets with as compared with the emulsions of Figure 10. The droplets of Figure 10 emulsions were nearly 30 times thecompared emulsions with of the Figure emulsions 10. The of droplets Figure 10. of The Figure drople 10 emulsionsts of Figure were 10 emulsions nearly 30 timeswere nearly larger than30 times the larger than the droplets of emulsions prepared by Wolf et al. Thus, the Pickering emulsions prepared dropletslarger than of emulsionsthe droplets prepared of emulsions by Wolf prepared et al. Thus, by Wolf the et Pickering al. Thus, emulsionsthe Pickering prepared emulsions by Wolf prepared et al. by Wolf et al. are closer to the assumptions made in the proposed model. areby Wolf closer et to al. the are assumptions closer to the made assumptions in the proposed made in model.the proposed model.

Figure 11. Comparison of the literature relative viscosity data for Pickering O/W emulsions with the Figure 11. Comparison of the literature relative viscosity data for Pickering O/W emulsions with the Figurepredictions 11. Comparison of the proposed of the model. literature relative viscosity data for Pickering O/W emulsions with the predictionspredictions ofof thethe proposedproposed model.model. 7. Conclusions 7. Conclusions 7. Conclusions The viscosity behavior of nanoparticles-stabilized Pickering emulsions is studied. A new model The viscosity behavior of nanoparticles-stabilized Pickering emulsions is studied. A new model is developedThe viscosity to describe behavior and of to nanoparticles-stabilized predict the viscosity of Pickering Pickering emulsions emulsion is as studied. a function A new of volume model is developed to describe and to predict the viscosity of Pickering emulsion as a function of volume isfraction developed of bare to describedroplets. and The to droplets predict of the Picker viscositying emulsion of Pickering are emulsionmodelled asas a“core-shell” function of droplets volume fraction of bare droplets. The droplets of Pickering emulsion are modelled as “core-shell” droplets fractionwith liquid of bare core droplets. and interfacial The droplets layer ofof Pickeringsolid nanopa emulsionrticles are as shell. modelled As the as “core-shell”presence of dropletsan interfacial with with liquid core and interfacial layer of solid nanoparticles as shell. As the presence of an interfacial liquidrigid layer core of and solid interfacial nanoparticles layer of at solid the oil-water nanoparticles interface as shell. disrupts As the the presencetransmission of an of interfacial tangential rigid and rigid layer of solid nanoparticles at the oil-water interface disrupts the transmission of tangential and normal stresses from the continuous phase to the inner core liquid, the composite “core-shell” normal stresses from the continuous phase to the inner core liquid, the composite “core-shell” droplets behave as rigid particles of effective radius larger than the radius of bare droplets. The droplets behave as rigid particles of effective radius larger than the radius of bare droplets. The Fluids 2018, 3, 2 11 of 13 layer of solid nanoparticles at the oil-water interface disrupts the transmission of tangential and normal stresses from the continuous phase to the inner core liquid, the composite “core-shell” droplets behave as rigid particles of effective radius larger than the radius of bare droplets. The effective radius of composite droplets is dependent on the contact angle and the size of the interfacial solid nanoparticles. The viscosity model that is proposed in this study takes into consideration of clustering of composite droplets and eventual jamming of Pickering emulsion at high volume fraction of dispersed droplets. For a given size nanoparticles, the viscosity of Pickering emulsion increases with the decrease in bare droplet size. The contact angle also has a signiﬁcant effect on the viscosity of Pickering emulsion. For O/W type Pickering emulsion, the viscosity decreases with the increase in contact angle, whereas the W/O type Pickering emulsion exhibits an opposite behavior, that is, the viscosity increases with the increase in contact angle. The proposed model describes the experimental viscosity data for Pickering emulsions reasonably well. More experimental work is needed on the low-shear viscosity behavior of well-characterized Pickering emulsions to validate the proposed model.

Acknowledgments: Financial support from the Natural Sciences and Engineering Research Council of Canada (NSERC) is appreciated. Upinder Bains assisted in the measurement of the viscosity of starch-stabilized emulsions. Conﬂicts of Interest: The author declares no conﬂict of interest.

References

1. Wu, J.; Ma, G.H. Recent studies of Pickering emulsions: Particles make the difference. Small 2006, 12, 4633–4648. [CrossRef][PubMed] 2. Yang, Y.; Fang, Z.; Chen, X.; Zhang, W.; Xie, Y.; Chen, Y.; Liu, Z.; Yuan, W. An overview of Pickering emulsions: Solid-particle materials, classification, morphology, and applications. Front. Pharmacol. 2017, 3, 287. [CrossRef][PubMed] 3. Berton-Carabin, C.C.; Schroen, K. Pickering emulsions for food applications: Background, trends, and challenges. Annu. Rev. Food Sci. Technol. 2015, 6, 263–297. [CrossRef][PubMed] 4. Chevalier, Y.; Bolzinger, M.A. Emulsions stabilized with solid nanoparticles: Pickering emulsions. Colloids Surf. A 2013, 439, 23–34. [CrossRef] 5. Timgren, A.; Rayner, M.; Dejmek, P.; Marku, D.; Sjoo, M. Emulsion stabilizing capacity of intact starch granules modified by heat treatment or octenyl succinic anhydride. Food Sci. Nutr. 2013, 1, 157–171. [CrossRef][PubMed] 6. Zoppe, J.O.; Venditti, R.A.; Rojas, O.J. Pickering emulsions stabilized by cellulose nanocrystals grafted with thermos-responsive polymer brushes. J. Colloid Interface Sci. 2012, 369, 202–209. [CrossRef][PubMed] 7. Chen, X.; Song, X.; Huang, J.; Wu, C.; Ma, D.; Tian, M.; Jiang, H.; Huang, P. Phase behavior of Pickering emulsions stabilized by graphene oxide sheets and resins. Energy Fuels 2017, 31, 13439–13447. [CrossRef] 8. Binks, B.P.; Fletcher, P.D.I.; Holt, B.L.; Beaussoubre, P. Phase inversion of particle-stabilized perfume oil-water emulsions: Experiment and theory. Phys. Chem. Chem. Phys. 2010, 12, 11954–11966. [CrossRef][PubMed] 9. Fournier, C.O.; Fradette, L.; Tanguy, P.A. Effect of dispersed phase viscosity on solid-stabilized emulsions. Chem. Eng. Res. Des. 2009, 87, 499–506. [CrossRef] 10. Tsabet, E.; Fradette, L. Effect of the properties of oil, particles, and water on the production of Pickering emulsions. Chem. Eng. Res. Des. 2015, 97, 9–17. [CrossRef] 11. Simon, S.; Theiler, S.; Knudsen, A.; Oye, G.; Sjoblom, J. Rheological properties of particle-stabilized emulsions. J. Dispers. Sci. Tech. 2010, 31, 632–640. [CrossRef] 12. Li, C.; Liu, Q.; Mei, Z.; Wang, J.; Xu, J.; Sun, D. Pickering emulsions stabilized by paraffin wax and laponite clay particles. J. Colloid Interface Sci. 2009, 336, 314–321. [CrossRef][PubMed] 13. Tarimala, S.; Wu, C.Y.; Dai, L.L. Pickering emulsions—A paradigm shift. In Proceedings of the 05AIChE: 2005 AIChE Annual Meeting and Fall Showcase, Cincinnati, OH, USA, 30 October–4 November 2005. 14. Dickinson, E. Use of nanoparticles and microparticles in the formation and stabilization of food emulsions. Trends Food Sci. Technol. 2012, 24, 4–12. [CrossRef] 15. Dickinson, E. Food emulsions and foams: Stabilization by particles. Curr. Opin. Colloid Interface Sci. 2010, 15, 40–49. [CrossRef] Fluids 2018, 3, 2 12 of 13

16. Binks, B.P. Colloidal particles at a range of fluid-fluid interfaces. Langmuir 2017, 33, 6947–6963. [CrossRef] [PubMed] 17. Leclercq, L.; Nardello-Rataj, V. Pickering emulsions based on cyclodextrinx: A smart solution for antifungal azole derivatives topical delivery. Eup. J. Pharm. Sci. 2016, 82, 126–137. [CrossRef][PubMed] 18. Crossley, S.; Faria, J.; Shen, M.; Resasco, D.E. Solid nanoparticles that catalyze biofuel upgrade reactions at the water/oil interface. Science 2010, 327, 68–72. [CrossRef][PubMed] 19. Yang, B.; Leclercq, L.; Clacens, J.M.; Nardello-Rataj, V. Acidic/amphiphilic silica nanoparticles: New eco-friendly Pickering interfacial catalysis for biodiesel production. Green Chem. 2017, 19, 4552–4562. [CrossRef] 20. Pera-Titus, M.; Leclercq, L.; Clacens, J.M.; De Campo, F.; Nardello-Rataj, V. Pickering interfacial catalysis for biphasic systems: From emulsion design to green reactions. Angew. Chem. Int. Ed. 2015, 54, 2006–2021. [CrossRef][PubMed] 21. Leclercq, L.; Company, R.; Muhlbauer, A.; Mouret, A.; Aubry, J.M.; Nardello-Rataj, V. Versatile eco-friendly Pickering emulsions based on substrate/native cyclodextrin complexes: A winning approach for solvent-free oxidations. ChemSusChem 2013, 6, 1533–1540. [CrossRef][PubMed] 22. Leclercq, L.; Mouret, A.; Proust, A.; Schmitt, V.; Bauduin, P.; Aubry, J.M.; Nardello-Rataj, V. Pickering emulsions stabilized by catalytic polyoxometalate nanoparticles: A new effective medium for oxidation reactions. Chem. Eur. J. 2012, 18, 14352–14358. [CrossRef][PubMed] 23. Kaptay, G. On the equation of the maximum capillary pressure induced by solid particles to stabilize emulsions and foams and on the emulsion stability diagrams. Colloids Surf. A 2006, 282/283, 387–401. [CrossRef] 24. Pal, R. Rheology of Particulate Dispersions and Composites; CRC P: Boca Raton, FL, USA, 2007. 25. Pal, R. Fundamental rheology of disperse systems based on single-particle mechanics. Fluids 2016, 1, 40. [CrossRef] 26. Taylor, G.I. The viscosity of a fluid containing small drops of another liquid. Proc. R. Soc. Lond. A 1932, 138, 41–48. [CrossRef] 27. Einstein, A. Eine neue Bestimmung der Molekuldimension. Ann. Phys. 1906, 19, 289–306. [CrossRef] 28. Einstein, A. Berichtigung zu meiner Arbeit: Eine neue Bestimmung der Molekuldimension. Ann. Phys. 1911, 34, 591–592. [CrossRef] 29. Pal, R. Single parameter and two parameter rheological equations of state for nondilute emulsions. Ind. Eng. Chem. Res. 2001, 40, 5666–5674. [CrossRef] 30. Pal, R. Novel viscosity equations for emulsions of two immiscible liquids. J. Rheol. 2001, 45, 509–520. [CrossRef] 31. Krieger, I.M.; Dougherty, T.J. Mechanism for non-Newtonian flow in suspensions of rigid particles. Trans. Soc. Rheol. 1959, 3, 137–152. [CrossRef] 32. Mooney, M. The viscosity of a concentrated suspension of spherical particles. J. Colloid Sci. 1951, 6, 162–170. [CrossRef] 33. Pal, R.; Yan, Y.; Masliyah, J. Rheology of emulsions. In Emulsions: Fundamentals and Applications in the Petroleum Industry. Adv. Chem. Ser. 1992, 231, 131–170. 34. Binks, B.P.; Clint, J.H.; Whitby, C.P. Rheological behavior of water-in-oil emulsions stabilized by hydrophobic bentonite particles. Langmuir 2005, 21, 5307–5316. [CrossRef][PubMed] 35. Wolf, B.; Lam, S.; Kirkland, M.; Frith, W.J. Shear-thickening of an emulsion stabilized with hydrophilic silica particles. J. Rheol. 2007, 51, 465–478. [CrossRef] 36. Katepalli, H.; John, V.T.; Tripathi, A.; Bose, A. Microstructure and rheology of particle stabilized emulsions: Effects of particle shape and inter-particle interactions. J. Colloid Interface Sci. 2017, 485, 11–17. [CrossRef] [PubMed] 37. Braisch, B.; Kohler, K.; Schuchmann, H.P.; Wolf, B. Preparation and flow behavior of oil-in-water emulsions stabilized by hydrophilic silica particles. Chem. Eng. Technol. 2009, 32, 1107–1112. [CrossRef] 38. Ganley, W.J.; Van Duijneveldt, J.S. Controlling the rheology of montmorillonite stabilized oil-in-water emulsions. Langmuir 2017, 33, 1679–1686. [CrossRef][PubMed] 39. Hermes, M.; Clegg, P.S. Yielding and flow of concentrated Pickering emulsions. Soft Matter 2013, 9, 7568–7575. [CrossRef] Fluids 2018, 3, 2 13 of 13

40. Pal, R. Viscosity-concentration relationships for nanodispersions based on glass transition point. Can. J. Chem. Eng. 2017, 95, 1605–1614. [CrossRef] 41. Pal, R. Modeling the viscosity of concentrated nanoemulsions and nanosuspensions. Fluids 2016, 1, 11. [CrossRef] 42. Pal, R. Modeling and scaling of the viscosity of suspensions of asphaltene nanoaggregates. Energies 2007, 10, 767. [CrossRef] 43. Ogunlaja, S.B.; Pal, R.; Sarikhani, K. Effects of starch nanoparticles on phase inversion of Pickering emulsions. Can. J. Chem. Eng. 2017.[CrossRef]

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