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A Simple Model for the Viscosity of Pickering Emulsions Rajinder Pal

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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 angleangleq 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 q 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.q = 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 θq.. 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 = =pπR22gγ (()1±± cosθq2)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 gOW. 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
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