Noname manuscript No. (will be inserted by the editor)
A generalized equation for rheology of emulsions and suspensions of deformable particles subjected to simple shear at low Reynolds number
Salah Aldin Faroughi Christian Huber ·
Received: date / Accepted: date
Abstract We present analyses to provide a general- when extrapolated to high capillary numbers (Ca 1). ized rheological equation for suspensions and emulsions We also predict the existence of two dimensionless num- of non-Brownian particles. These multiparticle systems bers; a critical viscosity ratio and critical capillary num- are subjected to a steady straining flow at low Reynolds bers that characterize transitions in the macroscopic number. We first consider the e↵ect of a single de- rheological behaviour of emulsions. Finally, we present a formable fluid particle on the ambient velocity and stress regime diagram in terms of the viscosity ratio and capil- fields to constrain the rheological behaviour of dilute lary number that constrains conditions where emulsions mixtures. In the homogenization process, we introduce behave like Newtonian or Non-Newtonian fluids. a first volume correction by considering a finite do- Keywords Emulsion Rheology Suspension Rheol- main for the incompressible matrix. We then extend · ogy particle deformation Relative viscosity regime the solution for the rheology of concentrated system · · diagram using an incremental di↵erential method operating in a fixed and finite volume, where we account for the e↵ective volume of particles through a crowding fac- 1 Introduction tor. This approach provides a self-consistent method to approximate hydrodynamic interactions between bub- Suspensions and emulsions of a Newtonian fluid in- bles, droplets or solid particles in concentrated sys- cluding dispersed non-Brownian particles are ubiqui- tems. The resultant non-linear model predicts the rel- tous in nature, and have many applications in indus- ative viscosity over particle volume fractions ranging try. When the Brownian motion due thermal energy from dilute to the the random close packing in the is neglected, the dynamics of suspensions/emulsions is limit of small deformation (capillary or Weissenberg mainly governed by external body forces, interparticle numbers) for any viscosity ratio between the dispersed forces and long range hydrodynamic interactions due and continuous phases. The predictions from our model to the presence of other particles (Brady & Bossis , are tested against published datasets and other consti- 1988). It is known that the existence of a cloud of par- tutive equations over di↵erent ranges of viscosity ra- ticles in a Newtonian fluid at low Reynolds number dra- tio, volume fraction and shear rate. These comparisons matically changes the mechanism by which momentum show that our model, is in excellent agreement with is exchanged between particles and the ambient fluid. published datasets. Moreover, comparisons with exper- Moreover, the macroscopic rheological behaviour of sus- imental data show that the model performs very well pensions and emulsions, which are heterogeneous mi- croscopically, depends mainly on the particle size distri- S. A. Faroughi Department of Civil and Environmental Engineering, Geor- bution, particle concentration, shear dynamic viscosity gia Institute of Technology, Atlanta, GA, 30332, USA of the matrix (continuous phase) and dispersed phase, E-mail: [email protected] the order of the particle deformation, and the rate of C. Huber deformation. Department of Earth and Atmospheric Sciences, Georgia In- In the last century, since the calculation conducted stitute of Technology, Atlanta, GA, 30332, USA independently by Sutherland (1905) and Einstein (1906,1911) 2 Salah Aldin Faroughi, Christian Huber to obtain the viscosity of a very dilute suspension of quantify the shear dynamic viscosity of suspensions of non-deformable solid spheres, the macroscopic rheologi- rigid and spherical bimodal-sized particles with inter- cal behaviour of multiparticle systems has received a re- fering size ratios. markable attention. Solutions are mainly based on con- These theoretical models have been tested and com- ceptual models that account for the change in hydro- plemented with several numerical and experimental stud- dynamic interactions based on the particle concentra- ies. For instance, Brady & Bossis (1988) used numer- tion and deformation. Einstein-Sutherland theory was ical modelling based on Stokesian dynamics and took first extended to very dilute emulsions by Taylor (1932) into account lubrication forces at high particle density where he assumed that fluid particles remain spherical, to study the rheology of monosize suspensions. They i.e. the dimensionless capillary number, Ca, (the ratio showed that microstructures can form in sheared sus- of viscous force to the force associated with the sur- pension, and outlined the role of particle clusters on face tension) is assumed Ca 1. These models pre- the rheological behaviour of concentrated suspensions. ⌧ dict an increase in the macroscopic shear viscosity of Schaink et al. (2000) extended the Stokesian dynamics the system that is linearly proportional to the particle method to study the rheology of suspensions of rigid concentration, with a greater e↵ect for solid spheres. spheres suspended in viscous and viscoelastic matrices. The Einstein-Sutherland law for a dilute suspension of The rheological behaviour of suspensions of rigid par- identical rigid spheres was then extended to second- ticles has also been investigated using other numerical order in volume fraction by Batchelor & Green (1972). techniques (for example Aidun & Lu (1995); Ladd & Other investigations by Mackenzie (1950), Ducamp & Verberg (2001) who used Lattice Boltzmann method Raj (1989) and Bagdassarov & Dingwell (1992) have and studies of Koelman & Hoogerbrugge (1993) and constrained expressions for the rheology of dilute emul- Strating (1999) for Brownian dynamics method, see sions including highly deformable fluid particles (Ca also the recent numerical studies by Rexha & Minale 1). More generalized constitutive equations for the rhe- (2011); D’Avino et al. (2013) and Villone et al. (2014)). ology of dilute systems were derived by Oldroyd (1959) Experimental studies also have provided a great in- for emulsions of two immiscible Newtonian fluids, by sight into the role of particles on the suspension rheol- Goddard & Miller (1967) for suspensions including ogy (e.g. Rutgers (1962); Thomas (1965); Chan & Pow- deformable Hookian solid sphere, and by Frankel & ell (1984); Rodriguez et al. (1992); Segre et al. (1995); Acrivos (1970) for emulsions consisting of deformable Cheng et al. (2002); Pasquino et al. (2008); Mueller et fluid particles up to the first order of the particle defor- al. (2009); Boyer et al. (2011); Dai et al. (2013)) and mation. These constitutive equations predict the macro- bubbly emulsion rheology (e.g. Stein & Spera (2002); scopic viscosity of relatively dilute systems over a wide Manga & Loewenberg (2001); Rust & Manga (2002) range of deformation rates (capillary number) and vis- and Llewellin & Manga (2005)). Additionally, the role cosity ratios. of the viscosity ratio and capillary number on the vis- For concentrated systems, Pal (2003-a, 2004) em- coelastic properties and rheology of dilute and concen- ployed the Di↵erential E↵ective Medium (DEM) the- trated emulsions has been studied extensively by Pal & ory Norris (1985) to determine phenomenologically the Rhodes (1989); Pal (1992, 1996, 2001, 2003-b). These relative viscosity for elastic solid particle suspension studies provided many experimental data on the viscos- ( ) and bubbly emulsion ( 0). Pal (2003- ity of emulsions which will be used here to validate the !1 ! b) developed a more general model for concentrated accuracy of our new theoretical model for predicting emulsions with di↵erent viscosity ratio and deformable the shear viscosity in multiparticle systems. particle using the analogy between shear modulus and We summarize the contribution and applicability of shear viscosity. In these studies di↵erent interpretations several studies (non-exhaustive) for the rheology of sus- and definitions are used for the change in the volume pensions and emulsions as function of the particle vol- available for adding particles (termed ”free volume” by ume fraction, , viscosity ratio, , and capillary num- Robinson (1949)), which leads to di↵erent sub-models. ber, Ca in Fig. 1. For example, the model proposed by More recently, new rheological equations for concen- Lim et al. (2004) is suitable for <0.2, 0 and ! trated suspensions of rigid solid particles have been Ca 1, while models of Pal (2003, 2004) cover the en- ⌧ proposed by Mendoza (2011) using DEM theory, and tire range of and capillary number within the limit of by Brouwers (2010) who matched the viscosity of bi- 0. This diagram serves to clearly identify regions ! modal suspensions with identically sized particles to of the volume fraction, viscosity ratio, capillary num- yield a closed form solution for the relative viscosity ber parameter space that need to be further explored. of monomodal suspensions. Faroughi & Huber (2014) It also points to the lack of unified model valid over the also proposed a crowding-based rheological model to entire space. A more complete list of published equa- Title Suppressed Due to Excessive Length 3
Fig. 1 Summary of some of published rheological models and their range of applicability with respect to particle volume fraction, ,viscosityratio, ,andcapillarynumber,Ca.Seetables1and2formoredetailsaboutpublishedmodels. tions developed for solid particle suspensions and emul- the second volume correction accounts for the volume sions along with the range over which they are deemed of matrix trapped in interstices formed by particles applicable is reported in Tables 1 and 2, respectively. through a crowding factor). The second objective of the present work is to provide a robust and general equa- The present study is undertaken for three reasons. tion for the macroscopic rheology of emulsions applica- The first goal is to present a complete derivation of ble for a wide range of viscosity ratio, capillary number the macroscopic rheology for both dilute and concen- and particle concentration which is missing in the liter- trated monosized suspensions/emulsions under simple ature (see figure 1). This general equation shall reduce steady shearing flow conditions. The e↵ective viscosity to the well-known relative viscosity law developed by is determined from the knowledge of the influence of Sutherland-Einstein (1906,1911) and Taylor (1932) in individual particles on the fluid flow and the pressure the limiting cases when 1 along with either field by taking two steps of volume correction into con- ⌧ !1 or 0, respectively. The third objective is to provide sideration. The first volume correction serves to build a ! a regime diagram which illustrates how the relative vis- general rheological model for dilute systems. With this cosity for emulsions depends on the viscosity ratio and correction, each particle inside the finite volume can capillary number. We find di↵erent regimes that are de- interact with all particles added simultaneously to the limited by two critical dimensionless numbers; a critical system through a decrease in the volume of the ambi- viscosity ratio and a critical capillary number. These ent fluid. We then introduce the second volume correc- regimes constrain the influence of di↵erent parameters tion to extend the model phenomenologically to highly on the deformation of particles, and provide insight on concentrated systems (up to the random close pack- transitions in the rheology of non-Brownian emulsions ing). This correction accounts for the interaction of par- from Newtonian to shear thinning due to the particle ticles added during the Di↵erential E↵ective Medium deformation (the e↵ect of microstructure changes such procedure with particles already present in the system. as shear-induced migration, wall-slip and heterogeneity Therefore, the second volume correction includes a term is not considered in this study). that carries the e↵ect of the particle shape and size dis- tribution as a geometrical constraint on the amount of In the following sections, we present a brief physical volume that can be eventually filled by particles (i.e. description of the perturbation in the flow fields due 4 Salah Aldin Faroughi, Christian Huber to the presence of a single fluid particle. Next, we dis- surface of particle controls the final shape of the par- cuss the homogenization process and the application of ticle, and leads to the definition of the dimensionless the first volume correction to obtain the macroscopic deformation number; the capillary number, property of dilute systems. Then, we explain the pro- ⌥µ R Ca = m d , (3) cedure of the fixed volume di↵erential e↵ective method including the second volume correction to extend the for the case of fluid particles and the Weissenberg num- relative viscosity model to concentrated systems. The ber, model is then tested against a number of experimental ⌥µ data and published constitutive equations. Finally, the Wi = m , (4) ability of the model to approximate the relative viscos- G ity for polydisperse systems including non-deformable for the case of solid particles. The required sets of bound- particles is discussed. ary conditions directly depends on the order of particle deformation considered. Here, we shall consider a ho- mogeneous straining flow at a large distance from the 2 Physical description center of the fluid particle, (1), along with the continu- ity of tangential velocity and tangential components of We shall consider two incompressible and immiscible the stress tensor at the surface of the particle in order Newtonian fluids forming a matrix (the continuous phase) to find the zeroth order of deformation solution (assum- and the dispersed phase (a single fluid particle at this ing the particle remains spherical). We can also obtain stage). The fluid flow at large distances from the fluid the first order of deformation solution by using the dis- particle satisfies the conditions of a simple steady strain- continuity in normal components of the stress tensor ing flow: across the particle surface based on Laplace’s equation. u (x)=⌥ x. (1) Overall, the velocity, pressure and stress fields outside · the particle are decomposed up to the second order of Here, x denotes the position vector with respect to the particle deformation O(D2) as follows the origin located at the center of the fluid particle, ut = u + u0,d + Du1,d + O(D2), (5) and ⌥ is a given velocity gradient tensor for which the incompressibility of the matrix imposes tr⌥ = 0. pt = p + p0,d + Dp1,d + O(D2), We shall assume that inertial forces can be neglected t = + 0,d + D 1,d + O(D2). (small Reynolds number, Re 1), and the density of ⌧ Here, D is a dimensionless parameter which spec- dispersed particles is the same as that of the matrix. ifies the amount of deformation (departure from the The force balance which governs the equation of mo- spherical shape), and it is proportional to either Ca tion is characterized by Stokes creeping equations or Wi number respectively for fluid particle and solid m m m m m T =0, = p I + µm u +( u ) , (2) particle. p is an arbitrary constant pressure at a large r· r r distance from the particle which is normally assumed where sub/superscript m refers to⇥ properties associated⇤ with the matrix, m is the total stress tensor, pm is the to be zero. At large distances from the particle, the ze- roth and first order correction terms (parameterized by dynamic pressure, I is the unit tensor, and µm denotes the shear dynamic viscosity of the matrix. um is the superscript 0,d and 1,d) vanish. velocity vector that satisfies the continuity equation, um = 0. Similar expressions can be formulated for r· 2.1 Zeroth order deformation the fluid flow inside the particle just by changing the superscript m to d which refers to the dispersed phase. Using the general solution for Stokes equations formu- We assume that the particle deforms due to the shear- lated by Lamb, and considering appropriate solid spher- ing. To the first order, the stress that acts to elongate ical harmonics of degree j (pj & j) for the exterior the particle is proportional to µ ⌥ where ⌥ = ⌥ is m | | fluid, one can express both velocity and pressure fields. the magnitude of the velocity gradient (or the shear rate The zeroth order deformation solution was first pro- 1 magnitude) with unit [t ]. The resisting stress on the vided by Taylor (1932) who used the following solid surface of the particle opposing the induced shear stress spherical harmonic functions of degree 3: is of order /Rd where is the surface tension, and Rd 3 is the radius of the particle. For the case of a deformable 0,d m 0,d Rd p 3 = µ A 3 2 (⌥ S : xx), r elastic solid particle, the resisting stress will be propor- ✓ ◆ 5 tional to the shear modulus G. The equilibrium state 0,d 0,d Rd 3 = B 3 (⌥ S : xx), (6) between these two counteracting surface stresses on the r ✓ ◆ Title Suppressed Due to Excessive Length 5 where ⌥ S is the normalized pure shear rate tensor (the all unknown fields (velocity and pressure) and the par- rate of deformation tensor as the symmetric part of the ticle shape. According to Greco (2002), one arrives to velocity gradient normalized by the magnitude of the the following function for the fluid particle shape up to shear flow). By applying the aforementioned boundary the third order of the deformation (O(D3)), conditions for the zeroth order, Taylor (1932) arrived 2 Ca 19 + 16 2 at the following expression for the constants in Eq. (6): r R 1+ (⌥ S : xx)+Ca d r2 16 +1 2 0,d + 5 0,d ⌥ S1( ) S2( ) A 3 = 5⌥ ,B3 = , (7) (⌥ S ⌥ S :: xxx)+ (⌥ S ⌥ S : xx) +1 2 +1 · r3 r2 ✓ ◆ ✓ ◆ ✓ in which is the viscosity ratio defined as, S4( ) +S3( )(⌥ S : ⌥ S )+ (⇧ : xx) =0, (11) µ r2 = d . (8) ◆ µ m where coe cients S1 through S4 depend only on the viscosity ratio and are listed in the Appendix F of Greco (2002). In Eq. (11), ⇧ is the second Rivlin-Ericksen 2.2 First order deformation tensor (Astarita & Marrucci , 1974) that, under simple The solution for a first order deformation can be ob- shear flow conditions, reduces to tained with the same method, and by using Laplace’s ⇧ = 2(⌥ S ⌥ A ⌥ A ⌥ S )+4⌥ S ⌥ S , (12) equation as a proper boundary condition to define the · · · stress jump at the boundary of the particle (see Frankel where ⌥ A is the normalized spin tensor (skew-symmetric & Acrivos (1967, 1970) for more details). The solid part of the velocity gradient normalized by the magni- tude of the shear flow). spherical harmonics p 3 are the only functions needed for the integration of the stress components over a large While Greco (2002) provides a starting point to volume, owing to the fact that other solid spherical har- further develop our model to higher orders of parti- monics vanish. The final result for the solid spherical cle deformation, we restrict our derivation to the first 2 harmonic reduces to the following expression Schowal- order of the deformation (up to O(D ), therefore the ter et al. (1968): model is theoretically applicable only for small particle deformations). Interestingly, we show below that our 2 3 1,d 20 25 + 41 +4 Rd model predictions for emulsions sheared at high Ca are p 3 = µm⌥ 2 7 25( + 1) r in good agreement with experiments which suggest that ✓ ◆ 3 2 19 + 16 the second order truncation with respect to particle de- (⌥ S : ⌥ S ) (⌥ S x) + 12µ ⌥ · r2 · m 15( + 1) formation does not introduce significant errors when 3 extrapolated to high Ca. Rd u ( ⌥ S x) (⌥ S x) . (9) The harmonic functions for the case of a matrix in- · r2 ⌥ · · · ✓ ◆ h i cluding a Hookean elastic solid particle are obtained Furthermore, the shape of the particle up to the with the same methodology to the zeroth order of defor- second order of the deformation (O(D2)) is calculated mation. For first order deformations, we replace Laplace’s as: equation with another stress boundary condition at the fluid-elastic solid interface (Goddard & Miller , 1967). 2 Ca 19 + 16 This is the case even for a fluid particle which has a in- r Rd 21+ (⌥ S : xx)3 =0. (10) r2 16 +1 finite viscosity where the spherical shape of the particle 6 7 is not maintained by surface tension forces, but rather 6 D 7 4 5 by its shear modulus G. | {z } 2.3 Second order deformation 3 Relative viscosity of a dilute system To proceed to higher orders deformation, for instance to Ca2, one needs to derive the complete second-order so- According to Batchelor (1967), the rate of energy dis- lutions of the spherical harmonics for the pressure and sipation per unit volume inside a suspension (or emul- velocity fields and an expression for the particle shape. sion) increases when more solid particles (or fluid par- Deriving these solutions following the same methodol- ticles possessing high surface tension or shear viscos- ogy is complex and tedious (Cha↵ey & Brenner, 1967; ity) are fed to the system. Therefore, a multiparticle Greco , 2002). Alternatively, Greco (2002) presented systems can be treated as a homogeneous Newtonian an analysis that calls for rotational invariance to find fluid of the same average density in a fixed volume of 6 Salah Aldin Faroughi, Christian Huber
r t t V = V + V (in which V is the total volume occu- m p p ⌥ik(2µ ⌥ij)xknjdA = ⌥ik(2µm⌥ij)xknjdA r pied by particles, and Vm is the remaining volume of ZA ZA the matrix) and with viscosity µ . The stress tensor at + ⌥ x n dA, (17) any point of the system (outside particles), t, is given ik ij0 k j A by Z where n is a outward unit vector normal to the sur- t m T = p I +2µm⌥ p0I + µm u0 +( u0) , (13) face. We proceed by transforming the first two surface r r integrals into integrals over the boundary of the remain- ⇥ 0 ⇤ ing ambient fluid, A , (a surface enclosing the matrix). where the primed terms are associated with the distur- m | {z } Thus, by applying the divergence theorem, Eq. (17) can bance in stress tensor, velocity and pressure fields due be recast as to the presence of particles, and consequently they in- clude both perturbation arising from zeroth and first @ui 2µ ⌥ij dV + 2µ ⌥ij⌥ikxknjdA = order of deformation. Namely, V V r @xj A Z m Z m 0,d 1,d @ui 0 = + D . (14) 2µm⌥ij dV + 2µm⌥ij⌥ikxknjdA + r @x V Vm j Am Besides, the stress tensor for the homogeneous equiv- Z Z alent fluid at any point can be calculated as ⌥ik ij0 xknjdA. (18) ZA = p I +2µ ⌥ . (15) Assuming that equations governing the perturba- The equivalence assumption implies the equality of tion in the fluid flow caused by particles are in Stokes the total rate of work done on the boundary of the regime, we obtain emulsion/suspension, A , in both structures character- @( ij0 xk) ized with the stress tensors defined in Eqs. (13) and = 0 . (19) @x ik (15). j In previous models (Einstein, 1906,1911; Taylor, 1932; Using Eq. (19), the third integral in the right hand Batchelor, 1967; Goddard & Miller , 1967; Frankel & side of Eq. (18) can be expressed as Acrivos , 1970; Schowalter et al., 1968; Landau & Lif- shitz , 1987), the matrix is considered unbounded (in- ⌥ik ij0 xknjdA = finite volume). Therefore, the excluded volume taken ZA by particles has been overlooked which results in par- ⌥ik ik0 dV + N ⌥ik ij0 xknjdA. (20) ticles being represented as mass points. These models V NVp Ap Z Z provide valuable results only in cases where the parti- N 2⌥ikµmu0 nkdA cle concentration is low (less than 5%). In this study, Ap i R a finite volume for the matrix is considered, however | N is the{z number} of particles fed to the system, and it is assumed large enough to satisfy the fact that the Ap is the surface of a particle. Integrals in Eq. (20) perturbation of single particles on the flow fields are are treated in such a way that it is assumed particles independent of each other (no hydrodynamic interac- are far apart and the disturbance they generate does tions in the dilute limit). The model, thus, takes into not a↵ect the flow field around other particles. There- account the excluded volume of the matrix replaced by fore, the averaged rate of energy dissipation per unit of particles using a first volume correction. Using this cor- volume is calculated only for one particle and then gen- rection, particles added simultaneously interact by de- eralized (linearly summed) to account for the e↵ect of creasing the volume available in the ambient fluid. We other particles on the rate of dissipation. This assump- note that the consideration of a finite volume is physi- tion is true only for very dilute suspensions/emulsions cally more consistent when the model is tested against where V V r . As a result, the following equation can ! m experiments. be retrieved from Eq. (18) by a simple integration, Owing to the fact that the rate of work associated 2⌥ ⌥ (µ µ )V r = with the isotropic component of the stress tensors stated ij ij m m in Eqs. (13) and (15) are the same on the boundary of N (⌥ 0 x n 2⌥ µ u0 n )dA. (21) ik ij k j ik m i k the physical domain (far from particles), the equality of ZAp the rate of work exerted by the deviatoric components, The integral in the right hand side of Eq. (21) indi- represented by ⌧ ,yields cates the average additional rate of energy dissipation caused by a single particle (Batchelor, 1967; Happel & u ⌧ n dA = u (⌧ + 0) n dA, (16) · · Brenner , 1983). To calculate this integral, we can use ZA ZA Title Suppressed Due to Excessive Length 7 the reciprocal theorem developed by Happel & Bren- and ner (1983) or simply replace Ap by an arbitrary large 1 19 + 16 2 surface, A , enclosing a single particle at its center. For B = ⌥ A ⌥ S ⌥ S ⌥ A . (29) a 40 +1 · · the latter method, the ambient stress and velocity fields ✓ ◆ of the fluid disturbed by the presence of this particle Equation (27) is a special case of simple fluids fam- should be considered as well, namely ily of constitutive equations (Schowalter et al., 1968). We note that the deformation introduces a non-linear relationship between the stress and the rate of strain. u00 = u + u0 Thus, emulsions/suspensions behave as non-Newtonian 00 = + 0. (22) fluids. Following Frankel & Acrivos (1970), we can ap- Therefore, we shall restate Eq. (21) as follows ply the operator t S Vp D ⌧ik =2µm⌥⌥ik + t 1+⇤ , (30) V V Dt p 1 2 D on both sides of Eq. (27). In Eq. (30), t denotes the ij00 xknjdA µmui00nk dA ,(23) D · Vp Vp Jaumann derivative (Goddard & Miller , 1967), which ✓ ZAa ZAa ◆ t where Vp = NVp, Using Lamb’s general solution we is defined as follows for an arbitrary tensor ↵, have D d ↵ = ↵ + u ↵ + ⌥ ⌥ A ↵ ↵ ⌥ A . (31) Dt dt ·r · · 1 u00 = u + xp 3, (24) In Eq. (30), ⇤ is determined as 2µm (2 + 3)(19 + 16) µ R and ⇤ = m d , (32) T 40( + 1) 00 = µm u +( u ) ✓ ◆ r r 1 T has unit of time and is defined as the relaxation time + ⇥ (xp 3)+( (xp⇤ 3) Ip 3, (25) 2 r r (Oldroyd, 1959) that characterizes the time-dependency with ⇥ ⇤ of the flow response to deformation (the time required 0,d ⌥µmRd 19 + 16 1,d for a slightly deformed particle to relaxes exponentially p 3 = p 3 + p 3. (26) 16 +1 to its spherical equilibrium shape). The value of the re- ✓ ◆✓ ◆ In Eq. (23), the first correction that accounts for the laxation time diverges as the viscosity ratio approaches volume taken by particles appears in the homogeniza- to infinity or as surface tension approaches zero. tion process. Models which overlook this correction un- In a steady and laminar simple straining flow, when derpredict the shear dynamic viscosity of the equivalent ↵ = ⌥ S , the material derivative part of the Jaumann fluid in a finite system. Therefore, if a set of particles derivative, first two terms of the RHS of Eq. (31), van- are added to the matrix (forming a dilute system), the ishes. This simplification is valid even when the Jau- position of each particles is restricted by the presence mann derivative is applied to the stress tensor associ- of other particles. ated with dilute systems subjected to a steady simple The detailed solution to integrals in Eq. (23) can be shear. For these systems, fluctuations caused by varia- found in Landau & Lifshitz (1987); Batchelor (1967) for tion in particle arrangement and deformation far away system of non-deformable particles and in Goddard & from the considered particle remain relatively small, Miller (1967); Frankel & Acrivos (1970); Schowalter et therefore, we expect this simplification does not a↵ect al. (1968) for systems composed of deformable particles. our model under steady conditions. Finally, By applying the operator defined in Eq. (30) to Eq. (27), we obtain Eq. (33), in which A and B are defined 1+2.5 in Eqs. (28) and (29), respectively. We drop the last ⌧ =2µ ⌥ 1+ ⌥ S m 1 1+ term in Eq. (33) to maintain the order of deformation ✓ ◆ 2 2 in Eq. (33) similar to that of Eq. (27) (second order µ ⌥ Rd + m (A B), (27) with respect to ⌥ ). 1 ✓ ◆ For a simple steady straining flow with the following NV where p is the particle volume fraction and dimensionless symmetric and skew-symmetric part of ⌘ V 3(19 + 16)(25 2 + 41 + 4) the velocity gradient A = 140( + 1)3 010 0 10 1 1 1 ⌥ S = 100 , ⌥ A = 100 , ⌥ S ⌥ S I ⌥ S : ⌥ S , (28) 2 2 3 2 2 3 · · 3 000 000 4 5 4 5 8 Salah Aldin Faroughi, Christian Huber
1+2.5 ⌧ + ⇤⌥ ⌥ A ⌧ ⌧ ⌥ A =2µ ⌥ 1+ ⌥ S + ⇤⌥ ⌥ A ⌥ S ⌥ S ⌥ A + · · m 1 1+ ⇥ · · ✓ ◆ ⇥ ⇤ 2 2 2 3 µm⌥ Rd µm⌥ Rd (A B)+ ⇤ ⌥ A (A B) (A B) ⌥ A , (33) 1 1 · · ✓ ◆ ✓ ◆
⌧ ⇤⌥ ⌧ ⌧ + ⇤⌥ (⌧ ⌧ )0 ⇤⌥ 10 11 12 12 2 11 22 1+2.5 ⌧ + ⇤⌥ (⌧ ⌧ ) ⌧ + ⇤⌥ ⌧ 0 = µ ⌥ 1+ ( ) 1 ⇤⌥ 0 + 2 12 2 11 22 22 12 3 m 1 1+ 2 3 00⌧33 000 4 (158 2 + 286 +5 116) 04 0 5 µ ⌥ 2⇤ m ( ) 0 (108 2 + 204 + 108) 0 , (34) 14(2 + 3)( + 1)2 1 ⇥ 2 3 ✓ ◆ 00(50 2 + 82 + 8) 4 5
70 3 + 45 2 111 74 ⌧ = ⇤⌥ ⌧ µ ⇤⌥ 2 1+ ( ) , 11 12 m 14(2 + 3)( + 1)2 1 ✓ ◆ 70 3 + 95 2 29 66 ⌧ = ⇤⌥ ⌧ + µ ⇤⌥ 2 1+ ( ) , 22 12 m 14(2 + 3)( + 1)2 1 ✓ ◆ 1 1+2.5 ⌧ = ⇤⌥ (⌧ ⌧ )+µ ⌥ 1+ ( ) . (35) 12 2 11 22 m 1+ 1 ✓ ◆
we can restate Eq. (33) in a matrix form, see Eq. (34). ratios and capillary number. Note for the case of infi- The deviatoric stress components in the direction nite viscosity ratio, the fluid particle acts like a Hookian of the first and second principal axes are obtained in solid particle and remains spherical because of the large Eq. (35). A simple manipulation of Eq. (35) yields an shear dynamic viscosity, not surface tension. Therefore, expression for ⌧12 which can be used to find the macro- another proper set of boundary conditions for the nor- scopic viscosity of the emulsion, µ = ⌧12/⌥ . Finally, mal components of the stresses on the surface of the we obtain deformed particle should be used (Goddard & Miller , µ 1 1967). Applying the boundary condition of Goddard & =1+ ( ) µ 1+⇤2⌥ 2 1 Miller (1967) and introducing the first volume correc- m 1+2.5 140( 3 + 2 1) tion in the homogenization process, we find + ⇤2⌥ 2 . (36) · 1+ 28(2 + 3)( + 1)2 µ 2.5 3 Now by substituting ⇤ from Eq. (32) into Eq. (36), =1+ 1 Wi2 , (39) µ 1+ 9 Wi2 1 2 and using the definition of the capillary number in Eq. m 4 ✓ ◆ (3), we can recast Eq. (36) into the following general for suspensions of elastically deformable solid particles. equation for any capillary number and finite viscosity We note that at low particle volume fraction, where ratio = + O( 2), Eq. (37) recovers the equation of µ 1 1 =1+ ( ) Taylor (1932) using 0 and Ca 1, the equation µ 1+Ca2 1 ! ⌧ m of Mackenzie (1950) using 0 and Ca 1 and that 1+2.5 140( 3 + 2 1) ! + Ca2 , (37) of Oldroyd (1959) using 0 (see table 1). Similarly, · 1+ 28(2 + 3)( + 1)2 ! at low solid particle volume fraction, Eq. (39) reduces to where the well-known Einstein-Sutherland law when Wi 1 2 ⌧ (2 + 3)(19 + 16) (see table 2). = . (38) 40( + 1) An extension of rheological model to concentrated ✓ ◆ It should be noted again that the model stated in systems requires a self consistent approach to account Eq. (37) is only valid for a dilute system up to the sec- for particle hydrodynamic interactions. Additionally, for ond order of particle deformation for any finite viscosity a multiparticle system of rigid solid or non-deformable Title Suppressed Due to Excessive Length 9
fluid particles, the relative viscosity should satisfy erations. We should account for the fact that this pro-
µ cedure cannot be followed until the entire volume of lim , (40) the matrix is replaced by particles ( 1). This re- M µm '1 9 ! striction arises because of the geometrical constraint where is the threshold packing (commonly known M dictated by the shape and size distribution of particles. as the maximum random close packing, RCP ) fraction Firstly, we will extend the model for the relative vis- for spherical particles. We note that in the case of emul- cosity of a dense system in the case of zeroth order of sions including deformable fluid particles, the relative particle deformation. Then, we can find an expression viscosity at = exhibits considerable increase but M for the geometrical constraint by utilizing the packing does not diverge (Pal, 2000), however to the first order limit condition of Eq. (40). deformation for emulsions of slightly deformable fluid particles, we will assume that (40) still holds. We note that M depends strongly on the particle size distribu- 4.1 Relative viscosity for a dense system of tion, particle shape and deformation, and the protocol non-deformable particles employed to produce the random packing (Faroughi & Huber , 2014). This quantity is also defined as the max- To start the procedure, we can rewrite Eq. (37) in the imally random jammed state by Torquato et al. (2000), following form (assuming Ca 1) who argued that the concept of the RCP as the high- ⌧ µ 1+2.5 est possible density that a random sphere packing can =1+ , = . (41) µ c 1+ c 1 attain is ill-defined. For these reasons, in the literature, m ✓ ◆ the value of M for mono-disperse spheres is found in Here c is called the corrected volume fraction of the range of 0.56 < <0.74 which is related to the me- particles for the dilute system (first volume correction). chanically stability of packing starting from the random In other words, this volume correction considers the fi- loose packing to face-centered cubic structure, respec- nite space taken by other particles of the same gen- tively (Rust & Manga , 2002; Song et al., 2008; Boyer eration. Based on the fixed volume DEM theory, the et al., 2011). Under static conditions, the value of 0.637 homogenization process is characterized by taking a is reported for M in classical studies (e.g. Scott & Kil- portion of the ambient fluid out and replacing it with gour (1969)), and it is assumed to be an acceptable particles at each step. We define the particle fraction value for the remaining of this study. We keep this pa- added to the system during each step as d i, and the rameter constant, however, our model allows to modify corresponding corrected particle fraction (e↵ective con- it freely, if necessary, to account for di↵erent packing i centration) added to be c. Therefore, the viscosity protocols in experiments, especially when high shear change of the homogenized equivalent fluid during step stresses (or shear rates) are imposed. i +1is 1+2.5 µ µ = µ i+1 , (42) i+1 i i c 1+ 4 Extension to concentrated suspension ✓ ◆ where the current value µi represents the matrix vis- We use a phenomenological approach based on the Dif- cosity µm and the next value µi+1 denotes the e↵ective ferential E↵ective Medium (DEM) method (Norris, 1985) suspension viscosity µ . The e↵ective concentration in operating in a fixed volume to extend our model to high Eq. (42) is defined as, concentration systems. The DEM approach is an incre- d i+1 mental method in which, at each conceptual step, a few i+1 = , (43) c 1 ⌦ i particles are introduced into the suspension/emulsion and interact with particles present in the medium. The where i is the total volume fraction of particles inside homogenized macroscopic property (e↵ective viscosity) the medium at step i. In Eq. (43), the e↵ective con- i+1 is then computed for the whole system. It should be centration at step i + 1, c , introduces the second noted that the Di↵erential E↵ective Medium theory is volume correction combining the first volume correc- physically appropriate only in the case where the in- tion c and a self-crowding factor parameter denoted cremental addition is su ciently sparse that it does by ⌦. This parameter, ⌦, is a positive constant that not form a preferential connected network throughout accounts for the fact that particles cannot fill all the the system. Due to the first volume correction, parti- volume of the suspension/emulsion (a geometrical con- cles added simultaneously can interact with each other. straint). Theoretically, this parameter takes the e↵ec- Therefore, we only need to account for interactions be- tive volume of particles into consideration knowing that tween a new generation of particles and previous gen- some fluid located in interstices formed by particles is 10 Salah Aldin Faroughi, Christian Huber no longer available to suspend particles. ⌦ is called the self-crowding factor because we assume that all par- ticles have the same size (volume). In general, we ar- gue that ⌦ is related to the size distribution (assuming small deformation) through the maximum random close packing concentration, M . Substituting Eq. (43) into Eq. (42) yields µ µ d i+1 1+2.5 i+1 i = . (44) µ 1 ⌦ i 1+ i ✓ ◆ Upon integrating Eq. (44) from a system with zero particle and shear dynamic viscosity µm to a desired particle volume fraction c and shear dynamic viscosity µ , µ 1 c 1 1+2.5 dµ = d , (45) µ 1 ⌦ 1+ Zµm Z0 ✓ ◆ Eq. (41) becomes
1+2.5 µ ⌦(1+ ) = 1 ⌦ . (46) µ 1 m ✓ ◆ The model described by Eq. (46) predicts the rela- Fig. 2 Rheology of suspension of rigid solid particles ( tive viscosity for a dense system at any finite viscosity and G ). Comparison of the model in Eq. (49) with! 1 !1 ratio to the zeroth order of particle deformation. The previous published models (see Table 1) and experimental self-crowding factor ⌦ is determined by applying the data from dilute up to the intermediate particle concentra- tion. The shaded area highlights the region that regroups constraint stated in Eq. (40) most of the experimental data. ⌦ 1 lim =1 ⌦ = M . (47) M 1 ! ! M Based on Eq. (47), we find that ⌦<1if M > 0.5. data. Our model agrees very well with published experi- This implies that the added particle volume fraction, ments for suspensions. One can observe that commonly say = a, practically occupies an e↵ective volume of used models for concentrated suspensions, like (Krieger a/⌦. This is also equivalent to argue that the volume & Dougherty , 1959), (Barnea & Mizrahi , 1973) and a(1/⌦ 1) of the matrix is trapped in interstitial spaces (Eilers, 1943), deviate from the experimental data as between particles. the particle concentration increases. As mentioned ear- Substituting Eq. (47) into Eq. (46) yields the fol- lier, ignoring the first volume correction (e.g. (Einstein, lowing equation for the relative viscosity of emulsions 1906,1911)) results in underpredicting the shear vis- of non-deformable fluid particles cosity of even dilute suspensions. It is also interesting
M (1+2.5 ) to stress that our model closely follows the empirical (1 )(1+ ) µ M M model proposed by Mooney (1951) when the free pa- = . (48) µ (1 ) rameter in his model is set to 1.35 (see Table 1). In Fig. m ✓ M ◆ For the particular case of a suspension of rigid solid 3, the relative viscosity predicted with our model Eq. particles ( and G ), Eq. (48) reduces to (49) is plotted for dense systems up to the packing limit !1 !1 M , and shows an excellent agreement with ex- 2.5 M 1 ! µ M perimental data. Here again the shaded area indicates = M . (49) µ (1 ) the range observed in experiments. Over this range of m ✓ M ◆ particle concentration (0.35 < <0.6), we observe Equation (49) is plotted for intermediate and high that models that do not include the volume corrections volume fractions of particles respectively in Figs. 2 and discussed above underpredict the relative viscosity by 3 where it is compared to published experimental data up to two orders of magnitude. As a note, we emphasize and well-known equations listed in Table 1. Figure 2 here that the model does not include free parameters shows a monotonically increasing relative viscosity with to fit the data and that we used M =0.637 which cor- particle concentration. The shaded area in Fig. 2 high- responds to the volume fraction for the random close lights the region that regroups most of the experimental packing of spherical particles under static conditions. Title Suppressed Due to Excessive Length 11
Fig. 3 Rheology of suspension of rigid solid particles ( Fig. 4 Rheology of emulsion of non deformable inviscid fluid and G ). Comparison of the model in equation (49)! particles (bubbly emulsion where 0andCa 1). Com- with1 previous!1 published models (see table 1) and experimen- parison of the model in equation (48)! with existing⌧ models tal data at intermediate to high particle volume fraction. (see table 2) and published experimental data.
4.2 Relative viscosity for a dense system of deformable particles
To extend our model to concentrated systems with de- formable particles, we shall first rewrite Eq. (37) as µ 1 =1+ ( ) + 2µ2 , (50) We compare the model in Eq. (48) with published µ 1+ 2µ2 1 N M L m m L m data for dense emulsions in the limit of Ca 0. The ⇥ ⇤ ! in which = ⌥R / and predicted value of the relative viscosity as function of L d particle volume fraction is depicted in Fig. 4 for an 1+2.5 140( 3 + 2 1) emulsion of non-deformable fluid particles where 0 = , = 2 . (51) ! N 1+ M 28(2 + 3)( + 1) (bubbly emulsion). Similarly to the results for solid sus- pensions in Fig. 2, we observe that the relative viscosity Applying the same procedure (fixed volume DEM increases monotonically with volume fraction, but with theory) to Eq. (50) leads to the following ordinary dif- a smaller rate than for solid particles. Based on Fig. 4, ferential equation one can see that our model performs very well to repro- 2 2 4 3 1 µ 2 µ duce experimental data in the limit of 0. Predicted M L 2 2 + µ M L 2 2 dµ ! µ + µ L + µ results from other well-known models reported in Ta- ✓ N M L N M L ◆ d ble 2 that are applicable to this range of , Ca 1 = .(52) ⌧ N 1 ⌦ and 0 are also depicted in Fig. 4 for comparison. ! Figures 2-4 clearly show the importance of consider- Upon integrating this equation with respect to the ing a finite volume (the influences of the first volume volume fraction from zero to c, with corresponding correction in the range of dilute emulsions <0.15) viscosity of µm and µ , we can find the following non- and defining an appropriate self-crowding factor (sec- linear relation for the relative viscosity ond volume correction) to improve the model’s ability 1 2 ( N 1) N M 2 2 M 1 to describe interparticle hydrodynamic interactions at µ + µ M M N M L = (53). high volume fraction. µ + 2µ2 (1 ) m N M L m ! ✓ M ◆ 12 Salah Aldin Faroughi, Christian Huber
µ 3 Alternatively, defining f ( , , Ca)=µ /µ and values of capillary numbers (Ca 10 ), ii) the relative m Ca = µm, Eq. (53) can be restated in the following viscosity decreases over intermediate values of capillary L 3 general form numbers (10 Fig. 5 Rheology of emulsion of deformable inviscid fluid particles ( 0) versus capillary number. Comparison between our model (solid line), containing no fitting parameter, and experimental data! for four di↵erent measured particle concentrations. µ Fig. 6 Relative viscosity, f = µ /µm,asafunctionofcapillarynumberfordi↵erentparticleconcentrationscalculatedwith (54) for a system where (a) the viscosity ratio is zero (bubbly emulsion) and, (b) the viscosity ratio is =1.1. Our model predicts that a critical capillary number exists only when the viscosity ratio <1, while for emulsions with >1, the e↵ective viscosity is greater than that of the ambient fluid for all Ca. the greatest influence on the viscosity of the emulsion is no longer controlled by surface tension (and conse- (maximum possible shear thinning). For viscosity ra- quently the capillary number). Under these conditions, tios >1, the e↵ect of the capillary number decreases. the physical property that acts to keep the particle’s At high viscosity ratio ( >103), the capillary number shape spherical is the shear viscosity of the dispersed plays a negligible role on the rheology of emulsions (Fig. phase. 7(a-b)). In addition, at >103, the relative viscosity For solid suspensions, , of elastic particles !1 of emulsions for a given particle volume fraction does (Hookian particles), the deformation is controlled by no longer depend on the viscosity ratio. This e↵ect is the Weissenberg number. The rheological behaviour of clearly depicted in Fig. 7(b). It suggests that when the these concentrated suspensions is also obtained by ap- shear viscosity of the dispersed phase is much greater plying the fixed volume DEM theory along with the than that of the matrix, the deformation of the particle second volume correction to account for the geometri- 14 Salah Aldin Faroughi, Christian Huber Fig. 7 The e↵ect of the capillary number and viscosity ratio on the relative viscosity at a constant particle volume fraction =0.4; (a) shows that increasing the viscosity ratio results in an increase in relative viscosity up to a point ( >103)beyond which increasing viscosity ratio does not a↵ect the relative viscosity. Also the sensitivity of the shear thinning behaviour to viscosity ratio first increases from zero to unity, and then as viscosity ratio increases the shear thinning behaviour decreases. The shear thinning behaviour vanishes for systems where >103.(b)showsthee↵ectofthecapillarynumberindi↵erent viscosity ratios. We observe that the e↵ect of the capillary number on the relative viscosity is maximum around the critical viscosity ratio, and decreases as the viscosity ratio increases. cal constraint described by Eq. (39). This leads to Equation (56) has a real and physical root only in 5 2.5 M 3 2 µ 2 4 cases where < 0, owing to the fact that and are 1 M 1 Wi (f ) M M N f µ 2 = ,(55) always positive. For a bubbly emulsion, 0, the crit- 3 2 1 Wi M (1 ) ! ✓ 2 ◆ ✓ ◆ ical capillary number is found to be 0.645 (Fig. 6(a)), which is a non-linear equation in terms of the relative while for a system where 1, the relative viscosity is µ viscosity, f . As expected, the relative viscosity of a always a function of the particle concentration. In other suspension including Hookian solid particles increases words, there is no critical capillary number for such sys- as more particles are fed to the system. At a fixed par- tem (because > 0) as depicted in Fig. 6(b). Thus, M ticle volume fraction, when the Weissenberg number in- the presence of the critical capillary number strongly creases (i.e. particles deform), shear thinning behaviour depends on the viscosity ratio between the two phases occurs. The shear thinning behaviour decreases as the which controls the sign of the parameter . Following particle shear modulus increases, and at G the M !1 this assertion, we define the critical viscosity ratio cr suspension behaves like a Newtonian fluid. It is also such that ( ) = 0. Therefore, the critical viscosity M cr worth mentioning that Wi =0.816 calculated based on ratio satisfies Eq. (55) is a critical Weissenberg number at which the 3 2 shear viscosity of the deformable solid suspensions is + cr 1=0, (57) cr cr independent of the particle volume fraction, and it is which has only one real physical root, = 1. For equivalent to the shear viscosity of the matrix. cr < cr a critical capillary number exists and conse- quently a behaviour similar to that shown in Fig. 6(a) 5 Regime diagram will be expected. For a viscosity ratio greater than cr, the behaviour of the system (the relation between the The macroscopic responses of emulsions to shear, and relative viscosity, capillary number and the particle con- more specifically the role of and Ca observed in Figs. centration) will be similar to that displayed in Fig. 6(b). 6 and 7, suggest that there should be a set of critical Note that the critical viscosity ratio and capillary num- numbers that controls transitions in the behaviour of bers are independent of the particle volume fraction. the relative viscosity. Based on Eq. (54), we can de- We summarize the prediction of our rheological model duce that a critical capillary number, Cacr, at which for the relative viscosity of emulsions as function of the the viscosity of the emulsion is identical to that of the viscosity ratio and capillary number Ca in a regime matrix and hence is independent of the particle volume diagram in Fig. 8. This regime diagram includes three fraction satisfies regions A, B & C, which are delimited by the criti- + Ca2 =0. (56) cal numbers discussed above. In the region where the N M cr Title Suppressed Due to Excessive Length 15 Fig. 8 Aregimediagramconstrainedbythecriticalviscosityratio( cr)andcriticalcapillarynumber(Cacr)shownby solid lines. In region A, where the viscosity ratio is bigger than cr, the relative viscosity is always greater than unity. While at smaller viscosity ratio ( < cr), regions B & C, there is a critical capillary number determined by Eq. (56) at which atransitioninthemacroscopicrheologicalbehaviouroccurs.RightatCacr,therelativeviscosityisalwaysunityandis independent of the particle volume fraction. In region C, where Ca < Cacr the relative viscosity is greater than unity, whereas at Ca > Cacr the shear viscosity of the emulsion becomes lower than that of matrix (region B). The dashed line separates regions where di↵erent parameters control the stress partitioning between the matrix and particles, and the shape of the fluid particles (deformation), surface tension in region a and shear viscosity of the dispersed phase in region b . viscosity of the dispersed phase is greater than the vis- ering a viscosity ratio smaller than cr and a capillary cosity of the matrix (region A in Fig. 8 where > cr), number equals to that of determined by Eq. (56) for the viscosity of the emulsion is always greater than the that specific viscosity ratio (Ca = Ca ), the viscos- cr | viscosity of the matrix for any given particle concentra- ity of the emulsion is identical to the viscosity of the tion. In this region of the diagram, the high viscosity matrix regardless of particle concentration. From Fig. of the dispersed phase generates a resisting stress that 8, we can also observe the limiting behaviour of the rel- balances the shear applied on the surface of the par- ative viscosity at = and 0wherethetwo cr ! ticles. We note that increasing the volume fraction of phase fluid flow actually reduces to a single phase flow. particles results in an increase in the e↵ective viscosity In this scenario, the shear viscosity of the emulsion is of the emulsion. independent of the particle concentration and morphol- ogy, and hence the critical capillary number approaches As the viscosity ratio of the emulsion decreases be- infinity. low the critical viscosity ratio cr = 1, two opposite scenarios emerge for the relative viscosity depending This regime diagram can be interpreted di↵erently on the capillary number. When the capillary number by highlighting four regions as shown in Fig. 9. The is smaller than its critical value Cacr (region C in Fig. solid lines that delimit these regions represent transi- 8where < cr and Ca < Cacr), the viscosity of the tions in the general rheological behaviour (Newtonian emulsion is greater than the viscosity of the matrix. The or non-Newtonian) of emulsions. In region 1 ,there- large capillary stresses between the two phases strongly sistance to deformation of fluid particles is controlled oppose particle deformation. On the other hand, when by surface tension, . According to Fig. 6, at Ca 1 ⌧ the capillary number increases beyond its critical value where the capillary stresses are important, an increase (region B in Fig. 8 where < cr and Ca > Cacr), in the viscosity of the emulsion is expected. Therefore, the viscosity of the emulsion is lower than the viscos- at low capillary numbers (region 1 ), emulsions behave ity of the matrix. This reduction is more pronounced like a Newtonian fluid, i.e. the shear dynamic viscos- as more particle are fed to the system (see Figs. 6(a) ity of the emulsion is independent of the strain rate for and 8). This shear thinning behaviour results from the a given particle volume fraction. When the capillary accommodation of most of the induced shear stress by number increases beyond Cacr, the emulsion displays low viscosity and deformable particles. Finally, consid- a non-Newtonian behaviour indicated by region 2 in 16 Salah Aldin Faroughi, Christian Huber Fig. 9 Same regime diagram highlighting four regions that distinguish di↵erent rheological responses for emulsions. In region 1 ,wherethecapillarystressesbetweentwophasesareimportant,anincreaseintheviscosityoftheemulsionisexpected, and emulsions behave like Newtonian fluid. Region 2 illustrates the shear thinning behaviour that occurs when the capillary number at a given viscosity ratio is increased. The largest shear thinning occurs at =1.Region 3 ,wheretheresistance force against deformation is dominated by the shear dynamic viscosity of the dispersed phase, is characterized by a Newtonian behaviour for emulsions. However, it has lower shear viscosity than that obtained for the region 1 .Thehatchedarea(region 4 )representsaregionwherethevalueoftherelativeviscosityisindependentofbothcapillarynumberandviscosityratio and behaves like a Newtonian fluid. Fig. 9. This implies that an increase in the strain rate form and align with the flow direction. Owing to the leads to a decrease in the viscosity of the emulsion). We smaller viscosity of the dispersed phase, as the particle observe that the range of capillary number over which volume fraction increases, the shear viscosity of emul- shear thinning occurs is the broadest at = 0, and de- sions decreases (see Fig. 6(a)). Thus in this portion of creases as the viscosity ratio increases. However, it is the regime diagram, the relative (Newtonian) viscosity worth mentioning that for a given particle volume frac- is smaller than unity. However, for parts of the region tion the amount of viscosity reduction due the shear 3 that belong to region A in Fig. 8 ( > ), the cr thinning is the greatest at = cr. This range shrinks relative (Newtonian) viscosity is greater than unity. gradually when the viscosity ratio decreases, and it also The hatched area (region 4 in Fig. 9) represents decreases significantly as the viscosity ratio increases a region where the value of the relative viscosity is in- beyond cr as illustrated in Fig. 7. dependent of both capillary number and viscosity ratio In regions 3 and 4 of Fig. 9, the resisting force (Fig. 7(b)), and emulsions behave macroscopically like against deformation is dominated by the shear viscosity Newtonian fluids. Beyond a viscosity ratio of O(103), of the dispersed phase which tends to keep the particle the viscosity of emulsions is only controlled by the vol- spherical regardless of surface tension. Therefore, the ume fraction of particles (e.g. it will be roughly 8.2 and viscosity of emulsions in these region strongly depends 3.4 times greater than the viscosity of the matrix for on the viscosity ratio and particle concentration. Region fluid particle concentrations of =0.4 and =0.3, 3 is characterized by a Newtonian behaviour for emul- respectively). Mathematically, for the hatched region sions, however it has lower shear viscosity than that we have obtained in region 1 . In the portion of region 3 where the viscosity ratio is smaller than cr (which belongs to lim lim 2.5, (58) O(103) M' O(103) N! the region B in Fig. 8) the resisting forces imposed by ! ! both surface tension and the viscosity of the dispersed phase are small compared to the external shear force consequently, the non-linear part of Eq. (54) vanishes, exerted by the ambient fluid. As a results, particles de- and the expression to estimate the relative viscosity of Title Suppressed Due to Excessive Length 17 mostly constant. We test this hypothesis by compar- ing the predictions from our model, Eq. (54) with the experimental and numerical data published in Lejeune et al. (1999); Stein & Spera (2002); Pal (2004) and Manga & Loewenberg (2001) in Fig. 10 . These datasets for bubbly emulsions ( 0) provide useful test for ! 4 our model in the limit of non-deformable (Ca = 10 ) and highly deformable bubbles (Ca = 104). As shown in Fig. 10, both limits are successfully modelled by our model Eq. (54) using a fixed maximum packing M =0.637, corresponding to the random close pack- ing of uniform spherical particles. The model developed by Pal (2003) (model 4) fits these datasets with a vary- ing maximum random packing limit that increases sig- nificantly with Ca ( M is changed from 0.54 to 0.7). In Fig. 10, we also show the relative viscosity of bub- bly emulsion predicted by our model at Cacr to high- light the transition in the rheological behaviour across Ca = Cacr, and the fact that at Ca = Cacr the relative Fig. 10 Comparison between our model and published ex- viscosity is independent of particle concentration. perimental and numerical data for the rheology of bubbly emulsion ( 0) over a range of volume fraction. The com- parison is performed! for two bounding values of the capil- 6 Polydisperse systems lary number representing emulsions including non-deformable fluid particles (small capillary number) and deformable fluid Providing a theoretical value for M in polydisperse particles (large capillary number) using Eq. (54). The dashed emulsions/suspensions is more challenging because the line represents the relative viscosity versus particle volume fraction for a bubbly emulsion at Ca = Cacr. void space between large particles can be filled by smaller particles (Faroughi & Huber , 2014). This causes M to reach a higher limit in polydisperse systems. Therefore, emulsions becomes we expect to observe lower e↵ective viscosities in poly- 2.5 M 1 disperse suspensions/ emulsions compared to monodis- M f µ M (59) perse systems for a given volume fraction (below ). ⇡ (1 ) M ✓ M ◆ For a multimodal system (with a wide range of parti- which is similar to the relative viscosity of a suspension cle sizes), the particle size distribution and the parti- of rigid spherical particles in (49). cle size ratio have significant impact on the rheologi- It is important to stress that for an emulsion pos- cal behaviour of the system.The greatest e↵ect of poly- sessing a viscosity ratio exactly equal to cr,themodel dispersity occurs when the modality is changed from presented here diverges because ( ) = 0. Thus, we monomodal to bimodal, subsequent modality changes M cr consider two emulsions of viscosity ratios = cr + " have lesser influences on rheology (Farris, 1968; Faroughi and = ",where" is an arbitrary small number. & Huber , 2014). Experiments on bimodal suspensions cr We then determine the relative viscosity of the emulsion reported by Chong et al. (1971) at constant fraction of by matching solutions in the limit where " 0. smaller size particles revealed that the e↵ective viscos- ! Our model for the rheology of emulsions ( < ) or ity decreases as the size ratio of spheres (small to large) 1 suspension ( = ) relies on our choice for the maxi- decreases. They also showed a negligible reduction in 1 mum random close packing M . For slightly deformable the shear viscosity when the particle size ratio decreases particles (up to the first order of deformation) we expect below 0.1 (Chong et al., 1971; Stickel & Powell , 2005). M to deviate slightly from its value for the random For bimodal suspensions of any size ratio, the largest close packing of spherical particles. In the majority of fraction for the random close packing (and hence the published models for monosized systems, M has been minimum relative shear viscosity) in a fixed volume of used as a fitting parameter. M must be a function of total particles occurs when suspensions consist of 65% particle shape, size distribution and dynamical condi- to 80% large particles, or in other words, 20% to 35% of tions (order of deformation). However, for monosized the total particle volume fraction is made of small par- spherical particles undergoing no or small deformation, ticles (Santiso & Muller , 2002; Stickel & Powell , 2005; which is assumed to be the case here, it should remain Faroughi & Huber , 2014). Quemada (1977) discussed 18 Salah Aldin Faroughi, Christian Huber Fig. 11 Relative viscosity versus solid phase concentrations and particle size ratio (PSR) for bimodal systems obtained from Eq. (49). The maximum packing is computed from Eq. (60) where it is assumed that the fraction of the small size particles is 25%. that for a highly polydisperse systems M approaches spaces between larger particles. In Fig. 12, we use our unity, because the broad distribution of particle sizes rheological model for monosized particles Eq. (54) and decreases the void ratio to a negligible value. It is impor- test it against experimental data provided for polydis- tant to stress that, using multimodal size distribution, persed emulsions at Ca 1. For this comparison, we ⌧ we can produce an emulsion/suspension possessing a use M =0.9 in Eq. (54) and plot the results from the fixed shear viscosity but with various amount of parti- model against experimental data over a wide range of cles. For example, in a bimodal suspension, the particle viscosity ratio and particle volume fraction. For exam- concentration can be increased while maintaining the ple, the experiments associated with =5.52 (inset shear viscosity fixed by varying the size ratio. This is of Fig. 12), was conducted with particle sizes ranging illustrated in Fig. 11 where we compare the relative vis- from 1 µm to 24 µm (Pal, 2001). While our model, in cosity for two bimodal suspensions possessing particle the limit of Ca 1, provides satisfying approximation ⌧ size ratio of PSR =0.35 and PSR =0.1 and contain- of the rheology of multimodal emulsions by considering ing 25% of small particles with a monodispersed sus- a corrected maximum packing limit, a more rigorous ac- pension (PSR = 1). The maximum packing for these count of polydisperse dynamics is required to find more bimodal systems are estimated with the experimental accurate results, and extend the model to predict the correlation (Chong et al., 1971; Costa et al., 2009) relative viscosity of multimodal emulsions at (Ca 1). s 0.104 R b = p , (60) M M Rl ✓ p ◆ 7 Conclusion b where M denotes the estimated critical fraction for a s l bimodal system, and Rp and Rp are the radius of the The primary goal of this paper is to provide a gener- smaller and larger particles respectively. Other models alized equation to determine the relative viscosity of for the maximum close packing of bimodal systems with both dilute and concentrated emulsions made of two di↵erent size ratios and fraction of sizes have been pub- Newtonian incompressible and immiscible fluids under lished recently and can be used alternatively (Boumonville a simple straining flow. First, we obtain a constitu- et al., 2005; Qi & Tanner , 2011; Brouwers, 2013; Faroughi tive equation in the dilute limit using the perturbation & Huber , 2014). The maximum attainable packing for of the flow field caused by a single fluid particle. The bimodal systems is around 0.869 (Faroughi & Huber , model is then extended to concentrated systems using 2014), which occurs when the size ratio (small to large) the di↵erential e↵ective medium theory operating in a approaches zero. Therefore, we expect that the max- fixed volume. In our derivation, two volume corrections imum random close packing can be even higher than are introduced. The first correction accounts for a finite 0.869 for multimodal systems involving a broad range spatial domain where the addition of a particle requires of particle sizes where the smaller particles fill void the removal of the same volume of the matrix. The sec- Title Suppressed Due to Excessive Length 19 viscosity is unity (i.e. the relative viscosity is indepen- dent of the particle volume fraction). At Ca < Cacr, the relative viscosity is greater than unity, while the viscosity of emulsions is smaller than that of the ma- trix when Ca > Cacr. In addition, the regime diagram provides informa- tion regarding parameters that control the stress parti- tioning between the matrix and particles, and regions where the emulsion behaves like a Newtonian or non- Newtonian fluid. We find that beyond a viscosity ratio of order 103, the e↵ect of capillary number and viscosity ratio on the relative viscosity of emulsion is negligible, and the relative viscosity is only function of the parti- cle volume fraction. In addition, we consider the case of suspensions that contain either deformable Hookian or rigid solid particles. We derive a model for these sus- pensions following the same approach as for emulsions. For both emulsions ( < ) and suspensions ( ), 1 !1 we find an excellent agreement between our model and experimental data over a wide range of viscosity ratio (0 ) particle volume fraction (0 < ) 1 M and capillary number (0 Ca < ) or finite Weis- 1 Fig. 12 Comparison between experimental data and our senberg number for Hookian solid suspensions. Finally, model in (54) at =0.5, Ca 0andassuming =0.9for ! M we discuss the application of the proposed model to the maximum packing of multimodal emulsions. The two in- predict the relative viscosity of multimodal emulsions sets also show how the relative viscosity of multimodal emul- in the limit of Ca 1. sions varies as function of volume fraction for two di↵erent ⌧ viscosity ratios at Ca 0. ! ond volume correction accounts for the amount of the matrix inaccessible to other particles and trapped in the interstices formed by particles through a self-crowding factor. The resulting general equation is a function of the viscosity ratio, capillary number, particle volume fraction and the maximum fraction for random close packing of particles. The maximum packing is expected to depend on dynamical conditions (deformation) that a↵ect particles. However, to the first order, assuming small deformations, we use the static packing limit for spheres ( M =0.637). The model is then tested against published experimental data, and we find an excellent agreement with these data sets. The proposed model provides a generalized framework to accurately predict the viscosity of emulsions over a wide range of capillary number, volume fraction and viscosity ratios. Our theoretical model allows us to construct a regime diagram to highlight the transition in rheological be- haviour in emulsions as a function of two critical dimen- sionless numbers. The critical viscosity ratio, cr = 1, determines whether the system has a critical capillary number or not. The existence of a critical capillary num- ber requires a viscosity ratio smaller than cr.Thecriti- cal capillary number defines a regime where the relative 20 Salah Aldin Faroughi, Christian Huber Table 1 Selected published models to predict the relative viscosity of suspensions of rigid solid particles and the range of volume fraction over which they intend to be applied. Reference Equation Specific Parameter Concentration Einstein (1906,1911) and µ Sutherland (1905) =1+K K =2.5lim 0 µm ! Hatschek (1913) µ =1+K K =4.5Upto40% µm µ K 2 Eilers (1943) =(1+ ) K =2.5Allrange µm 2(1 ) M 2 µ 2.5 +2.7 Vand (1948) µ = exp( 1 K ) K =0.609 Up to 60% m Mooney (1951) µ = exp( 2.5 )1.35 µ M 2 Chong et al. (1971) =(1+K ) K =0.75 All range µm (1 ) M µ K1 5 Barnea & Mizrahi (1973) = exp( ) K1 = ,K2 =1 Upto10% µm 1 K2 3 µ 1 2 Quemada (1977) =(1 K0 ) 2.54 K 3.71 Up to 50% µm 2 µ K 2 Lewis & Nielsen (1968) =(1+ ) K =1.5Allrange µm 1 M Cichocki & Felderhof (1991) µ =1+2.5 + K 2 K =5.00 Up to 15% µm Verberg et al. (1997) µ =1+2.5 + K 2 K =6.03 Up to 20% µm µ 1 Morris & Boulay (1999) =1+2.5 (1 ) —Upto20% µm M 2 2 +0.1( ) (1 ) M M µ 3 Zarraga et al. (2000) = exp(K )(1 ) K = 2.34 Up to 30% µm M Cheng et al. (2002) µ = 1+1.5(1+K) K = + 2 2.3 3 Up to 56% µm 1 (1+K) K M µ 1 5 1 M Brouwers (2010) =( ) K = All range µm 1 2 M µ 2.5 1 M Mendoza (2011) µ =(1 1 K ) K = All range m M This study Equation (49) — All range Title Suppressed Due to Excessive Length 21 ,and 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ! ! ! ! ! ! ! ! ! ! ! ! ! ! lim lim lim 100 lim 25 lim . 1Allrange 1Allrange 1lim 1Allrange 1lim 1lim 1lim 1lim 0 1 1 1 1 1 1 ⌧ ⌧ ⌧ ⌧ lim All range lim Up to 20% All range lim — Up to 17% Up to 15% All range All range lim All range All range All range Up to 50% Up to 70% 2 M M 8 . 16 9 15 8 5 ) ) M M 2 ) 14 4Upto45% 2 5 2 3 3 5 . 5 =100 . 5 5 1 . . 1 Ca 8415 1+ 1 1+ m 4Upto70% 1Upto5%1 1+ . . . µ µ 9 1+ 1+ 0 16 1+2 K 1+2 1+2 5 3 = = =(1 =(1 1 2 = 1 2 =1 = = = = =22 =13 =( 5 K K K K . A K K —Upto10%Allrangelim —Upto20%Allrangelim —Upto5% K K —Upto40% K Minimum modelMaximum model — Up to 7% 8 < : ( ⇢ ) 2 ) K 2 ) ) 2 2 2 5 ) Ca 6 5 Ca ( 6 Ca Ca 2 ) ( 5 5 6 5 2 12 . 202 1 K 2 K 0 ) M ) 1 ) 2 K 43 ) 2 . 1 K 5 1 K 1+( Ca ) 6 1+ 5 Ca 2 2 ( ) 6 1+ 2 1 2 ) ) 2 ( ( Ca ) 5 ) M 2 1 . K ) 9+412 ) Ca M 2 1 1+369 )2 . 5 Ca 72 K . . 3 ) . 5 6 5 1 1 M 12 K / 4 5 . 1+ (1 ) ) ) ) 1+ 3(1 K 1 K 1 5 1+( )+(1+ +(22 1+(0 1+(23 M K 3+2 k A (1 + 2 2(1 1 K ( K K K K K K ( 2 5 3 5 1+ 1 (1 (1 1+9 (1 + 22 22+ . 1+ 1 2(1+2 1 exp ⇢ A ⇢ ⇢ =1+ =(1+ =1 = =1+ =1+ = = =(1+ =(1+ =1+ = = = =(1 = = m m m m m m m m m m m m m m m m m µ µ µ µ µ µ µ µ µ µ µ µ µ µ µ µ µ µ µ µ µ µ µ µ µ µ µ µ µ µ µ µ µ µ ,overwhichtheyintendtobeapplied. Ca Selected published models to predict the relative viscosity of emulsions of deformable fluid particles and the range of volume fraction, viscosity ratio, ReferenceTaylor (1932) Eilers (1943) Mackenzie (1950) Oldroyd (1959) Frankel Equation & AcrivosChoi (1970) & Schowalter (1975) Scherer (1979) Ducamp & RajPal (1989) & Rhodes (1989) Bagdassarov & Dingwell (1992) Stein & Spera (1992) Manga & Loewenberg (2001) Specific ParameterRust & Manga (2002) Lim et al. (2004) ConcentrationPal (2004) DeformationLlewellin Viscosity & ratio Manga (2005) This study Equation (54) Table 2 capillary number, 22 Salah Aldin Faroughi, Christian Huber References Chan, D., & Powell, R. L. (1984). Rheology of suspen- sions of spherical particles in a Newtonian and a non- Aidun, C. K., & Lu, Y. (1995). Lattice Boltzmann sim- Newtonian fluid. Journal of non-Newtonian fluid me- ulation of solid particles suspended in fluid. Journal chanics, 15(2), 165-179. of statistical physics, 81(1-2), 49-61. 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