Modeling the effects of polydispersity on the viscosity of noncolloidal hard sphere suspensions. Paul M. Mwasame, Norman J. Wagner, Antony N.

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1 Submitte to the Journal of Rheology Moeling the effects of polyispersity on the viscosity of noncolloial har sphere suspensions Paul M. Mwasame, Norman J. Wagner, Antony N. Beris a) epartment of Chemical an Biomolecular Engineering, University of elaware, Newark, E Corresponing Author. Tel , E mail aress: beris@uel.eu Synopsis The present stuy evelops an extension of the approach pioneere by Farris [T. Soc. Rheol., 12, , (1968)] to moel the viscosity in polyisperse suspensions. Each smaller particle size class is assume to contribute to the suspension viscosity through a weighting function in two ways. First, inirectly, by altering the backgroun viscosity, an secon, irectly, by increasing the contribution of the larger particles to the suspension viscosity. The weighting functions are evelope in a consistent fashion as a power law with the exponent, h, a function of the relative volume fraction ratios an the base, g, a function of the soli particle size ratio. The moel is fit to available theoretical an experimental results for the viscosity of several binary suspensions an shows goo to excellent agreement epening on the functions g an h chosen. Once parameterize using binary suspension viscosity ata, the preictive capability to moel the viscosity of arbitrary continuous size istributions is realize by representing such istributions with equivalent ternary approximations selecte to match the first six moments of the actual size istribution. Moel preictions of the viscosity of polyisperse suspensions are presente an compare against experimental ata. Keywors: Viscosity moel, Polyispersity, Noncolloial suspension, Huggins coefficient, Moment approximation a) Author to whom corresponence shoul be aresse; beris@uel.eu 1

2 I. INTROUCTION The theoretical stuy of the rheology of suspensions has a long history. It ates back to the seminal work by Einstein who first escribe theoretically the relationship between viscosity an the solis volume fraction in ilute suspensions of soli spheres [Einstein (1906)]. Subsequent works on unerstaning higher orer effects in more concentrate systems inclue the work of Batchelor (1970) an Happel an Brenner (1983). It is clear from experimental evience that the total solis fraction is not the only factor affecting the rheological properties of suspensions [Jeffrey an Acrivos (1976)]. One key factor is particle size polyispersity which has immeiate consequences in numerous inustrial processes incluing the processing of slurries an in the foo inustry. In many applications it is important to etermine the particle size istribution that minimize the viscosity of a given particulate formulation. Therefore, it is not surprising that attempts to moel the limit of no flow (maximum packing) in particulate systems [Furnas (1931), Ouchiyama an Tanaka (1984)] an more generally polyispersity effects on rheology [Mooney (1951), Farris (1968), Chong et al. (1971)] comprise a longstaning area of inquiry. Over the past few years, there has been renewe interest in moeling the effects of the particle size istribution on suspension viscosity. In particular, there have been a number of recent publications that have propose alternative moels to escribe the effects of polyispersity on viscosity, incluing works by Qi an Tanner (2011), orr an Saiki (2013) an Farr (2014). Experimental measurements [Chong et al. (1971), Sweeny (1959), Poslinski et al. (1988), Chang an Powell (1994)] an simulation results [Chang an Powell (1993)] show that polyispersity inuces significant changes in suspension rheology, when compare to monoisperse systems. For example, computer simulations by Chang an Powell (1993) starting from initially ranom configurations inicate that clusters of large an small particles are a plausible microstructural explanation for the changes in rheology observe in biisperse systems. At the moment, no first principles theory is available to preict the effects of polyispersity except for certain limiting cases. More specifically, Wagner an Woutersen 2

3 (1994) presente exact calculations for the effects of polyispersity on the viscosity of Brownian suspensions in the ilute regime. However, the evelopment of first principles theories for concentrate suspensions remains an open question espite being of great practical relevance. In the absence of such theories, there are three main phenomenological approaches that have been use to unerstan an moel the effects of polyispersity on suspension rheology. These are the maximum packing fraction approach, the Mooney approach [Mooney, (1951)] an the Farris approach [Farris (1968)]. From the theoretical point of view, there has always been an interest to evaluate the limit of flowability in isperse systems [Furnas (1931), Ouchiyama an Tanaka (1984)] leaing to the evelopment of moels to preict the maximum packing fraction of particulate systems. Base on these moels, rheologists have use the maximum packing fraction to moel the viscosity of binary suspensions [Chang an Powell (1994), Chong et al. (1971), Poslinski et al. (1988)]. In the first variant of this approach, the viscosity is assume to obey any one of the empirical or semi empirical viscosity correlations for monoisperse har sphere suspensions [Eilers (1941), Maron an Pierce (1956), Krieger an ougherty (1959)]. A review of these an other viscosity correlations is presente by Faroughi an Huber (2015). The auxiliary information on the particle size istribution (PS) only enters into the calculation via a moification of the maximum packing fraction base on various moels [Furnas (1931), Ouchiyama an Tanaka (1984), Suuth (1993)]. Qi an Tanner (2011), ientifie shortcomings in the maximum packing moel by Ouchiyama an Tanaka (1984). In particular, this moel yiels unphysical preictions in the limit of vanishing values of the small particle volume fraction in a binary suspension. On the other han, the moel by Furnas (1931) is base on geometrical arguments an is inherently limite to large size ifferences only. In the case of binary suspensions, this moel preicts a single maximum packing fraction that correspons to the theoretical maximum attainable packing. Qi an Tanner (2011, 2012) evelope a moel that provies a metho to irectly calculate the suspension viscosity in binary an multimoal suspensions by consecutively accounting for the effects of 3

4 the ifferent size classes on the overall viscosity in a multiplicative fashion. Starting from the larger size particles, where the maximum packing fraction is assume to be ranom close packing, each subsequent smaller particle size is assume to have an ajuste maximum packing fraction that epens on the volume fractions an the particle sizes present in the system. The suspension viscosity is then calculate as a multiplicative prouct of the contributions of each iniviual size class to the relative viscosity of the suspension. Relate to the Qi an Tanner approach is the work of örr et al. (2013) who have evelope an effective meium approach that consiers the contribution of each size class to suspensions viscosity explicitly. In their moel the suspension viscosity is compute recursively base on the aition of particles of a larger size class to an effective suspension of smaller size particles, unlike Qi an Tanner (2011,2012) who use a multiplicative rule. Their moel also uses a moifie maximum packing fraction that is associate with the stepwise aition of each size class an is compute base on exclue volume arguments. However, like the Furnas maximum packing moel [Furnas (1931)], this moel is currently limite to suspensions with large size ifferences between successive classes. A unique aitional contribution of this work is the matche asymptotic expansion to a generalize viscosity correlation that allows one to arbitrarily choose the secon orer Taylor coefficient (Huggins coefficient), in aition to satisfying the Einstein limit. This provies aitional flexibility to account for ifferent inter particle interactions present in real systems an introuces an interesting paraigm to systematize the analysis of viscosity measurements across a wie variety of systems. Mooney (1951) presente an alternative to the maximum packing theoretical moel for viscosity in suspensions by using a crowing factor. Using symmetry arguments as constraints, he erive an expression to escribe the suspension viscosity taking into account explicitly the contributions of each size class. This was essentially a renormalization of the Einstein ilute limit result to escribe the relative viscosity in a monoisperse suspension through an exponential function of the volume fraction. However, instea of the 2.5 epenence, the effective volume fraction is increase 4

5 up to /1 k, where k is a crowing factor that is selecte phenomenologically to represent concentration effects in monoisperse suspensions. For polyisperse suspensions, Mooney postulate a epenence of the crowing factor on the relative particle sizes. Finally, the total suspension viscosity is calculate from multiplicative contributions from each size class. Even though Mooney allue to the extension of his approach to polyisperse suspensions, he never completely aresse this aspect in the original publication. Following Mooney s work, Farr (2014) extene the original approach with two main contributions. First, he propose a moel for how the crowing factor epens on size ratios for arbitrary size istributions, completing an important aspect of Mooney s original iea. Secon, Farr allowe for further aitional complexity in the moeling of the suspension viscosity by incluing a ispersity effect to account for the heterogeneity of particle interactions in a polyisperse suspension, therefore introucing more flexibility to allow for better fits to experimental ata. Faroughi an Huber (2014) have also recently escribe a theoretical argument for a crowing base rheological moel for binary suspensions. They establish the crowing effect as the reuction in the ea flui volume that is associate with a given level of packing an are able to show goo agreement with experimental ata. Farris (1968) escribe an alternative theoretical approach towars calculating the effect of polyispersity on the suspension viscosity. His moel was motivate by previous work on seimentation in binary suspensions [Fileris an Whitmore (1961)], where the large particles in the presence of much smaller particles are observe to behave in a manner that suggests that they are interacting with an effective Newtonian viscosity corresponing to a suspension of the smaller particles. Uner such conitions, he assume there are no interactions between particles of ifferent sizes. The total suspension viscosity is compute as a prouct of the relative viscosity of the large particles multiplie by an effective viscosity of the renormalize meium. The attractive feature of this approach is that it provies a methoology towars constructing the viscosity of a polyisperse suspension of particles by explicitly consiering the effect of each particle size class uring the viscosity calculation. However, it is 5

6 currently limite to suspensions with large size ifferences between successive particle size classes. Although Farris emonstrate the possibility of introucing crowing factors to exten his moel applicability to systems with arbitrary size istributions, no systematic methoology was provie to achieve this, leaving this moeling approach incomplete. In our work, a moification an extension of the Farris approach is presente. As in the original approach, we start by requiring the relative viscosity of the monoisperse suspension to be a function of volume fraction represente by one of the many empirical formulae available, epening on the nature of the suspension. Next, a systematic methoology is evelope to weight contributions from ifferent size classes. On aition of a larger size particle to a suspension, the weighting functions relegate a fraction of the alreay present smaller particles to interact irectly with the ae particle volume fraction, with the remaining contributing towars increasing the effective backgroun viscosity. A key element of our approach is to use formulae that by construction preserve the consistency of the moel for all possible limiting cases. Still these constraints are not enough to uniquely efine the weighting function. Thus by necessity, the weighting function involves fitting parameters that nee to be etermine empirically. This can be achieve using theoretical results as well as experimental an/or simulation ata on binary suspensions. Once parameterize base on binary viscosity ata, our moel can preict the viscosity of suspensions containing particles with multiple sizes e.g. ternary suspensions. This is analogous to the approach of Renon an Prausnitz (1968) to preict the excess Gibbs free energy of multicomponent mixtures from binary ata alone. The success of such an approach here inicates the power of linking the complex behavior of systems to that of known limiting cases an of systematically interpolating it in a consistent fashion base on a limite set of empirical parameters. This is an especially useful first approach when face with complex systems for which there is little guiance from first principles theory on how to evelop a comprehensive moel of the phenomena to be escribe. 6

7 The rest of the paper is organize as follows: In Section II, we present the relevant theory an moel evelopment for binary an ternary suspensions, as well as the etails of its implementation to polyisperse suspensions. In Section III we escribe the parametrization of the moel base on theoretical calculations of the Huggins coefficient by Wagner an Woutersen (1994). In the same section we also show how the moel can be parameterize base on either numerical simulations of spherical particles on a monolayer or experimental viscosity ata of a binary suspension. In Section IV, we valiate the moel by comparing its preictions of binary suspension viscosities against those obtaine with recent alternative moels from the literature. In Section V, we evelop preictions of the viscosity of polyisperse suspensions, comparing them against available experimental ata, an we emonstrate their insensitivity to the etails of the implementation. Finally, our conclusions follow in Section VI. II. MOEL EVELOPMENT A. Propose approach an unerlying physical picture The general framework within which we have evelope our viscosity moel assumes the presence within the suspension of multiple particle size classes with the viscous effects attributable to hyroynamic interactions only. Therefore, the moel is currently limite to noncolloial suspensions as well as colloial suspensions in the plateau viscosity region at high shear rates (or Peclet numbers). Similar to the maximum packing fraction moels, the starting point of the propose constitutive equation is provie by the relative viscosity relationship of a mononisperse suspension. A number of such empirical an semi empirical relationships exist [Eilers (1941), Maron an Pierce (1956), Morris an Boulay (1999), Zarraga et al. (2000)]. For example, Singh an Nott (2003) fit the volume fraction ( ) epenence of the shear viscosity ( ) measurements in r noncolloial suspensions using Eiler s equation 7

8 r max 1 2, (1) with max The moel for the viscosity of polyisperse suspensions evelope in this work allows for flexibility in the choice of the functional form of r as neee for practical applications. The particular relative viscosity moel chosen (e.g. Eq. (1)) can then be use to efine a hyroynamic function for a single size particle suspension, f u, as exp( f ). (2) r Through this efinition, the Farris moel [Farris (1968)] can now be recast using the formalism of hyroynamic functions. The original approach use by Farris (1968) to calculate binary suspension viscosity assume that the smallest particles act to change the effective suspension meium in which the larger particles exist. Mathematically, we can efine a binary hyroynamic function, f, approach as where u bi, Farris u L u S, bi Farris, to represent Farris f f f (3) (1 w) S (1 w ) w L., (4) The subscripts an enote the small an large particle iameters, respectively, in the suspension an w is a suitable weight function that epens on the relative size ratio, /. The renormalize volume fractions of the respective particle sizes appearing in the arguments of the hyroynamic functions are labelle as an for small an large particles respectively. The total volume fraction is compute 8

9 as the sum of these two quantities. L an S are intermeiate variables that represent ajuste volume fractions of large (L) an small (S) particle contributions to the effective viscosity of the suspension. In Eq. (3), f, bi Farris escribes the combine effective hyroynamic effect in a binary suspension. Farris (1968) primarily iscusse the case where w=0. This represents the limit of large ifferences between the two particle size classes i.e. /<<1. The separation of length scales in such a suspension implies that the large an small particles are non interacting. Therefore, the viscosity in such a suspension can be escribe base on exclue volume arguments alone [Farris (1968)]. The equivalent calculation of the viscosity of a binary suspension as escribe by Farris (1968), in terms of the binary hyroynamic function is given by exp( f ). (5) r, Farris bi, Farris Although Farris (1968) postulate that the weighting function w appearing in Eq. (4) coul be generalize to represent suspensions in which the separation of length scales is not very large, he i not provie a methoology to o so. In this work, we present an alternative formulation to systematically evelop a viscosity moel that is applicable to suspensions with arbitrary separation of scales, as well as polyisperse an continuous particle size istributions. The viscosity in such suspensions may be calculate in a recursive fashion. We start from the consieration of the smallest particle fraction, which is assume to obey one of the monoisperse suspension viscosity rules such as provie by Eq. (1). The effect of the next larger in size particle fraction is then consiere. uring this construction step, a fraction ( ) of the alreay existing small particles is assume to interact with the ae volume fraction of larger particles, resulting in an effective volume fraction of larger particles ( ). These are then assume to exist in an effective meium whose hyroynamic function is a fraction (1 ) of the hyroynamic function evaluate consiering only the smaller particles. This construction can be carrie out repeately, aing larger an larger particle fractions to the suspension, 9

10 while renormalizing the volume fractions at each construction step with appropriate weighting function. The essence of this approach is summarize in Fig. 1 for a binary suspension, an can be represente quantitatively by a binary hyroynamic function given by fbi fu + fu 1, (6) A B where, is a weighting function that is assume to epen on both the relative size ratio (/) an composition / of the binary suspension. The viscosity of the binary suspension is then calculate from rbi, exp( fbi). (7) At the heart of this moel is the weighting function,, that accounts for the two fol effect of the smaller particles in (a) increasing the effective volume fraction of the larger particles (term A in Eq. (6)) an in (b) enhancing the overall backgroun viscosity (term B in Eq. (6)). Therefore, the evelopment of an appropriate form of the weighting function is the focus of the following section. FIG. 1: An illustration of the construction steps to renormalize a binary suspension allowing for the calculation of the viscosity. The goal is to express the viscosity of the real binary suspension (far left) in terms of an effective renormalize suspension (far right) consisting of an effective meium (shae backgroun) an effective volume fraction (large circles). 10

11 B. Moel evelopment for binary suspensions The success of the propose moel epens on careful selection of the weighting function. This selection is guie by observations from experimental measurements of the viscosity in binary suspensions from literature [Chong et al. (1971), Poslinksi et al. (1988), Gonret an Petit (1997)]. The only constraints explicitly consiere in the selection of the weighting function are those originating from the ability to recover characteristic limiting behaviors. More specifically, a binary suspension behaves like a monoisperse suspension uner certain limiting conitions of relative size an composition of the constituent particles. Therefore, the binary hyroynamic function f bi, escribe in Eq. (6) must fulfill the 4 key limits outline below lim f f lim f f bi u bi u 0 1 lim f f lim f f bi u bi u 0 0. (8) Consequently, from the efinition of fbi in Eq. (6), the weighting function,,, for a binary suspension consisting of small particles ( ) an large particles ( ) must obey the following limits,, lim, 1 lim, 0 / 1 / 0 lim, 1 lim, constant 0. (9) A robust an useful form of the weighting function that satisfies the above limits, is given by a power law g h g h, (10) 11

12 where the exponent, h, is assume to epen only on the particle volume fraction ratio, /, an the base, g, on the particle size ratio, /. These can therefore be interprete to represent an effective volume fraction ratio an an effective size ratio respectively. For consistency with the limiting behavior of the weighting function given in Eq. (9), the functional forms of g an h / must, at a minimum, obey the following limiting behaviors: lim g 0 0 lim g 1 1 lim h 1 0 lim h 0. (11) It shoul be note that the thir limit in Eq. (11) can be any constant, but by selecting it to be a specific fixe value (1 is chosen for simplicity) we can uniquely efine the g an h functions. The particular functional form of the weighting function,, in Eq. (10) is chosen for convenience, in orer to facilitate the extension of the moel to multimoal an polyisperse suspensions (see Section II C an ). In aition, the form of Eq. (10) allows us to separate the effects of relative size ( / ) an composition ( / ) on the viscosity of a binary suspension. This assertion will be justifie later on in this section. For the effective volume fraction ratio, we further shall assume h 1, (12) where the parameter plays a similar role to the crowing factor in Mooney s (1951) viscosity expression. More sophisticate mixing rules are possible by allowing aitional complexity (more 12

13 parameters) in the functional form of / h. On the other han, the yet to be etermine, effective particle size ratio, g, accounts for all the epenence of the viscosity on the relative size ratio, /, an is an increasing function of it. A parametric stuy of the weighting function escribe by Eqs. (10) an (12), as a function of the effective particle size ratio, g, 0 g 1, is presente in Fig. 2 for various values of / an. This result verifies that the limiting behaviors of the weighting function outline in Eq. (11) are preserve for the choice of h escribe by Eq. (12). FIG 2: A parametric stuy showing the epenence of the weighting function, efine in Eq. (10), on the effective size ratio, g, for ifferent values of the parameter an the volume fraction ratio, /. All the limits of Eq. (9) are fulfille for the functional form of the weighting function in Eqs. (10) an (12) irrespective of the values of the parameter. A key, implie property, of the form of the weighting function in Eq. (10) is the ecoupling of the effects of relative size an composition on the overall viscosity of a binary suspension. Therefore, it is important to show that the parameter appearing in Eq. (12) primarily controls the occurrence of the 13

14 viscosity minimum. This is emonstrate by assuming using a Krieger oherty viscosity relationship for a monoisperse suspension given by r 1 max 2.5max, (13) where the maximum particle volume for flow, max, is typically assume to be ranom close packing limit (0.64). This expression is use to efine the monoisperse hyroynamic function efine by Eq. (2). The occurrence of the viscosity minimum in a binary suspension can be calculate from the first erivative of the binary hyroynamic function in Eq. (6) with respect to the fraction (by volume) of small particles in the suspension,. This is expresse as f bi min 0, (14) where min represents the soli volume fraction of small particles in the total solis loaing at which the viscosity minimum is observe. Using Eqs. (2), (10), (12) an (13) to efine f bi in Eq. (6), the extremum conition represente by Eq. (14) can be written explicitly as, 1 1 min min min ln min max max, (15) where, an ln g 1 1 min min 2, (16) 1 min min 1 min g. (17) 14

15 The behavior of Eq. (15) is now stuie parametrically for two scenarios. In the first scenario, we consier the relationship between min an g for various values of while holing the total volume fraction ( ) fixe. The results in Fig. 3 suggest that the position of the viscosity minimum is a strong function of the choice of an epens only weakly on the relative size ratio (/), which is implicit in the g function. In the secon scenario the relationship between min an g at fixe values of for various values of is examine. The calculations, summarize in Fig. 3, suggest that the occurrence of the viscosity minimum is only weakly epenent on. Furthermore, for 6 we observe that the value of min at which the viscosity minimum is seen lies between 0.25 an This choice of is consistent with empirical observations where, for a fixe size ratio (/) an total solis loaing ( ), the viscosity minimum is seen to occur when 25 35% of the total soli particles by volume are small [Mewis an Wagner (2012)]. 15

16 FIG 3: A parametric stuy of the relationship between the minimum viscosity soli particle volume fraction, min, an the effective size ratio, g, efine by Eq. (15), at two ifferent values of the parameter an at various total solis volume fractions,. The ability of the moel to ecouple the size an composition effects in a binary suspension is emonstrate. This means that two pieces of information are neee to parametrize the moel. The parameter may be chosen such that the viscosity minimum preicte by the moel occurs over the esire composition range that is observe experimentally. On the other han, to etermine the functional form of the effective size ratio, g, one nees measurements of binary suspensions viscosity as a function of size ratio (see Section III). While the analysis presente in Eqs. (14) (17) can be applie to any choice of the monoisperse viscosity correlation, the final close forms solution may be more complex epening on the particular choices of r an / h. Therefore, in subsequent sections, will be treate as an aitional fitting parameter whenever a Krieger ougherty viscosity relationship is not use for r an/or no ata on the viscosity minimum are available. 16

17 C. Moel extension to ternary suspensions The ability to calculate the viscosity of a ternary suspension requires the evelopment of an appropriate hyroynamic function, f tri, such that the viscosity of the ternary suspension is given by rtri, exp( ftri). (18) We achieve this by consiering the effect of aing another larger size particle at volume fraction to an existing binary suspension of particle sizes an an corresponing volume fractions an. To be consistent, ftri must reuce to the relevant binary an monoisperse limits, as the ternary suspension egenerates into either a binary an monoisperse one. The trimoal hyroynamic function must therefore obey the following limits lim f f lim f f lim f f Tri bi Tri bi Tri u 1 1, 1 lim f f lim f f lim f f Tri bi Tri bi Tri bi lim f f lim f f lim Tri u Tri u, 0, 0, 0 f Tri f u. (19) Consiering these limits, the binary hyroynamic function from Eq. (6) is extene to evelop an analogous expression for a ternary suspension of successive particle iameters << an respective volume fractions, an given by f f f f. (20) tri u u u The corresponing extene weighting functions,, are given by i h1 1, g, * * (21) an 17

18 * h2 * * * 2, g, (22) where, 1, *,, 1, (23) *, (24) 1 h1 1, (25) an * * h2 1. * (26) The functional form for 1 an 2 appearing in Eqs. (21) an (22) is similar to that efine for binary suspensions in Eq. (10). The set of expressions in Eqs. (21) (26) appear to be more complex, but are strictly consistent with all known limits of the monoisperse an binary suspensions. For example, by setting equal to 0 in these equations, we recover from Eq. (20) the binary hyroynamic function in Eq. (6). Note that the formulae above only require information that can be obtaine from bimoal ata. The moel can be further extene to quaternary suspensions in a straightforwar fashion as shown in Appenix A. Similarly one can procee in a recursive fashion, to exten the moel to arbitrary multi n ary suspensions. However, the formulae are complex, an as will be argue shortly, unnecessary. Finally, it is noteworthy that the extene weighting functions escribe by Eqs. (21) (26) (as well as Eqs. (49) (51) for quaternary suspensions in Appenix A) incorporate the same parameters appearing in the weighting function efine for binary suspensions in Eqs. (10) an (12). This means that 18

19 the ternary suspension moel only requires binary suspension ata to specify the form of the weighting function. This feature gives the moel preictive power with respect to estimating the viscosity of ternary an polyisperse suspensions (see Section V).. Implementation of the moel to polyisperse suspensions Suspensions of practical relevance such as coal slurries [Rosin an Rammler (1933)] are comprise of continuous size istributions 1 such that is esirable to be able to preict the viscosity of such suspensions. One approach is to iscretize the continuous size istribution an procee with moeling it as a multi n ary istribution following the metho outline above. However, as we shall show, this is not necessary. Instea, it suffices to simply iscretize the continuous particle size istribution with a small but sufficient number of finite size classes in such a way as to fit the first few moments of the actual istribution. Wagner an Woutersen (1994) propose that just three particle size classes (or fitting the first 6 moments) are aequate to represent the rheological properties of a suspension with a continuous particle size istribution. This means that a continuous size istribution can be escribe by an equivalent ternary suspension. Therefore, the ternary hyroynamic function evelope in Eq. (20) can be use to preict the viscosity in polyisperse suspensions. The information to etermine the equivalent ternary suspension first nees to be extracte from the volume weighte continuous size istribution. The moments of the continuous istribution are efine by where f v k mk R fvr R 0, (27) R R represents the normalize volume weighte number ensity of non colloial particles with raii between sizes R an R R an mk is the k th moment of the istribution. The first 6 1 Continuous size istributions as iscusse here refers to single peake istributions with relatively short tails. 19

20 moments of the continuous volume weighte size istribution ( m ; 0 i 5) are then use to generate an equivalent ternary approximation base on the following equation i m 3 k k ili i1, (28) where i an L i are the i th weight an sizes of the equivalent ternary system respectively. The relevant moeling information is then obtaine from these weights an sizes using the following relationships: i i i i1 i1 i1, (29) L1 L2 L3. (30) In Eq. (29), is the total solis volume fraction an, an represent the small, meium an large particles respectively in the system with, an the respective volume fractions. Using the equivalent, but approximate, ternary representation of a continuous size istribution, all the variables appearing in Eqs. (20) (26) can be efine base on Eqs. (27) (30). The sensitivity of the moel to the number of iscrete size classes is examine in Section V to justify the 6 moment approximation propose for continuous size istributions in the context of the moel evelope in this work. III. BINARY SUSPENSIONS: COMPARISONS AGAINST EXISTING THEORY AN EXPERIMENTS TO ETERMINE MOEL PARAMETERS To evelop the weighting function outline in Eq. (10), the constituent functions representing the effective size ratio an effective volume ratio, g an h respectively, must be specifie. For the effective volume ratio ( h ) specifie in Eq. (12), only the parameter nees to be etermine. On the other han, the effective size ratio ( g ) is an unknown function whose epenence on the size ratio 20

21 (/) must be etermine. In this section, we present a theoretical evelopment of the effective size ratio ( g ) as well as an alternative empirical evelopment base on simulation or experimental ata of the viscosity of binary suspensions. A. etermining moel parameters from the Huggins coefficient in ilute limit In ilute har sphere colloial suspensions, the Huggins coefficient, k H, accounts for pair interactions an therefore provies information on colloial interactions [Russel (1984)]. In practice, the 2 Huggins coefficient is relate to the coefficient in a Taylor expansion of the viscosity with respect to the volume fraction. (31) r c 2... For the Krieger oherty relationship in Eq. (13), c 2 turns out to be equal to 5.0 for monoisperse particles if we assume max This is exactly the value compute by Wagner an Woutersen (1994) for a ranom configuration of hyroynamically interacting spheres. The corresponing measure for binary an polyisperse suspensions has also been etermine for a ranom binary suspension of hyroynamically interacting unequal spheres [Wagner an Woutersen (1994)] as c i jih ij. (32) 2 j1 i1 where 1 an 2 represent the composition of small an large particles in the suspension with an / H ij ij i j I accounts for the hyroynamic pair interactions between spheres compute from previous theoretical results of Jeffrey s resistivities [Jeffrey (1992), Wagner an 21

22 Woutersen (1994)] an is efine to be equivalent upon 1/ substitution. The calculations of ij ij c 2 from Eq. (32) are presente schematically in Fig. 4, for ifferent compositions an size ratios. FIG 4: Calculate values of 2 c using Eq. (32) an the values of / I provie by Wagner an Woutersen (1994) of as a function of the solis fraction of small particles for ifferent size ratios (/). H Using our moeling approach, we calculate the binary suspension viscosity (, rbi ), as escribe in Section II, by assuming the Krieger oherty relationship in Eq. (13) as the moel for the monoisperse suspension viscosity. The Taylor expansion of, rbi efine by Eq. (7), is then given by 2 2 g, 11 1 g, rbi, O 2 max, (33) where g, g, (34) 22

23 an max is assume to be 0.67 to preserve the consistency with the Wagner an Woutersen (1994) relation in the monoisperse limit. By equating the expression for c 2 provie by Eq. (32) an the 2 coefficient in Eq. (33), the relationship between the relative size ratio ( g ) an the size ratio (/) can be etermine once the parameter is etermine. Following the proceure escribe in Section II B, is compute inepenently using Eqs. (15) (17) together with the observation that the minimum in c 2 always occurs at equal to 0.5 (see Fig 4). From this proceure, the value of is etermine to be 1,min Next, by enforcing equality of Eq. (32) an the coefficient in Eq. (33), the effective size ratio values ( g ) that best parametrize the results of Wagner an Woutersen (1994) (see Fig. 4) are extracte an are presente in Fig. 5. These can be fit to a power law given by g (35) 23

24 FIG. 5: The empirically etermine g function values an their epenence on (/) obtaine by 2 enforcing equality of Eq. (32) an the coefficient in Eq. (33). Soli points represent g function values applie to generate the fits at the ifferent size ratios (/) shown in Fig. 5. The soli line is a parameterization of the g function values obtaine by fitting to a power law. The full weighting function efine in Eqs. (10) an (12) corresponing to the effective size ratio provie by Eq. (35) is then given as 0.54 h g. (36) The corresponing comparison for c 2 of the moel preictions base on this weighting function an the theoretical results from the calculations by Wagner an Woutersen (1994) using Eq. (32) is then shown in Fig

25 2 FIG. 6. Comparison of calculations of the coefficient following the theoretical results from Wagner an Woutersen (1994) (symbols) an the calculations from the moel (lines) using Eqs. (33) an (36). The approach outline in this section represents one way to etermine the components of the weighting function, g an, that appear in the efinition of the binary hyroynamic function. The ability of the moel to capture the semi ilute behavior (the 2 coefficient) as provie by an alternative, first principles, approach, provies some justification for the form of the weighting function use, as well as the efinition of the bimoal hyroynamic function. However, it shoul be note that the theoretical results from Wagner an Woutersen (1994) assume that the particle configurations are ranom. This is not necessarily true for suspensions uner flow, even if one starts from a ranom configuration [Batchelor an Green (1972), Chang an Powell (1993), Mewis an Wagner (2012)]. For example Chang an Powell (1993) reporte formation of clusters of large an small particles in simulations of non Brownian binary sphere suspensions espite starting from a ranom configuration of particles. Therefore, alternative, necessarily empirical, weighting functions must be evelope to reflect the actual microstructure that evelops uner flow, especially for non colloial suspensions. In the 25

26 following section, we iscuss how to evelop such empirical weighting functions by comparison against either numerical simulations or experimental ata on the viscosity of binary suspensions. B. Estimating weighting function using concentrate binary suspensions ata The evaluation of the weighting function efine by Eq. (10) can procee in two ways epening on the available ata. Like the previous section, if the relative viscosity as a function of the composition of small particles ( 1 ) is known an isplays a viscosity minimum at 1,min, then an approach analogous to that escribe in Eqs. (13) (17) (Section II B) may be use to inepenently etermine. Subsequently, the relationship between g an / can be estimate by fitting the viscosity moel to binary suspension viscosity at ifferent / ratios. Otherwise, if such ata is not available, shoul be consiere as a fitting parameter, together with g, to be etermine from the binary suspension viscosity ata. Therefore at a minimum, one must have binary suspension viscosity ata taken over multiple / ratios to parametrize the moel. In this section, the latter approach is use to etermine the weighting function. The following methoology was use to fit the parameters. First, an appropriate monoisperse viscosity relationship, such as Eq. (1), specific to the particular system, nees to be selecte such that it best escribes the monoisperse suspension viscosity. The next step is to etermine the empirical weighting function. For this section, we limit ourselves to the simple expression for h that involves a single parameter (see Eq. (12)). As such, for a selecte value we fit the corresponing g for any given / ratio so that the error is minimize. This proceure is repeate iteratively until the overall relative error between the calculate viscosities from Eq. (7) an the experimental binary viscosities is minimize. At the en of the fitting process, there are as many g values as there are / ratios an a 26

27 single value. Finally, a parametrization for g vs. / is evelope, allowing us to interpolate for all possible effective size ratios. This parametrization must fulfil the limits outline in Eq. (11) for / equal to 0 an 1. This approach to evelop the weighting function in Eq. (10) is now emonstrate using simulation ata as well as experimental measurements on binary suspension viscosity. 1. Simulations of monolayers of non colloial suspensions The various components of the weighting function, g an, can be etermine by fitting the bimoal suspension viscosity moel to the Stokesian ynamics simulation results by Chang an Powell (1993). These simulations emulate binary suspensions of spherical particles in a monolayer. Owing to their two imensional nature, these simulations o not incorporate all the features of a real suspension. Nevertheless they provie a goo test be to evelop insights on the form of the weighting function. The monoisperse viscosity relation for the simulation ata use is given by r 1 0< <0.6 (1 / 0.78), (37) where is the areal fraction with a maximum packing of Using Eqs. (6) an (7) to efine the binary hyroynamic function, the simulate binary suspension viscosity results are fit to the viscosity moel. The best fit value of is 3.13 an the best representation of the relationship between effective size ratio ( g ) an the true size ratio ( / ), shown in Fig. 7, is given by g (38) The corresponing weighting function is given by 3.13 h g 11, (39) an the resulting moel fits are compare to the simulate binary viscosity ata in Figs. 8 an 9. 27

28 Chang an Powell (1993) reporte formation of clusters in their 2 simulations. This eparture of the 2 suspension from ieal binary suspension behavior can be quantifie by comparing the effective size ratio, g, in Eq. (38) to that in Eq. (35) (which is assume to represent an ieal ranom binary suspension). This comparison is summarize in Fig. 7. The effective size ratio from Eq. (38) is consistently smaller than that of Eq. (35) as well as the monoisperse value of g 1. This suggests that the effect of polyispersity in mitigating hyroynamic interactions is far larger in the non ranom binary suspensions simulate by Chang an Powell (1993), when compare against the calculations by Wagner an Woutersen (1994) on ranom binary suspensions. FIG. 7: The empirical relative size ratio ( g ) as a function of the true size ratio (/) etermine from the 2 monolayer simulations of Chang an Powell (1994). The soli points represent the empirically etermine relative size ratios fit to the simulation ata shown in Fig. 8 an 9. The continuous line represents (Eq. (38)) the parameterization that is most consistent with the fit values of the relative size ratios. For comparison, the otte line (Eq. (35)) correspons to the parametrization base on the effective Huggins coefficient of ranom suspensions (see Fig. 5). 28

29 FIG. 8. Moel fit compare to simulation ata from Chang an Powell (1993) showing the epenence of the relative viscosity on the particle size ratio (/) with the fraction of small spheres is fixe at 0.27 for binary all cases. The weighting functions for the fits were calculate using the effective size ratio values ( g ) in Fig. 7 an Monoisperse /=1, Binary /=0.5 an /=0.25. FIG. 9. Relative viscosity of binary suspensions at ifferent compositions an particle size ratios. Total areal fraction is fixe at 0.5. Moel fit (soli lines) compare to simulation ata from Chang an Powell (1993) showing the epenence of relative viscosity on fraction of small particles in the suspension for ifferent particle size ratios (/). The weighting functions for the fits were calculate using the g function values in Fig. 7 an Monoisperse, /=1; Binary /=0.5 an /=

30 In this case, the binary viscosity moel shows its ability to represent the behavior of the simulate two imensional binary suspensions proviing an analytical parameterization of all the simulation ata. Nevertheless, for practical applications, it may be necessary to evelop an alternative parameterization of the moel base on more realistic, three imensional binary experimental ata. The penalty incurre is that the resulting moel lacks full preictive capability regaring the estimation of the viscosity of real binary suspensions. However, it will be shown that this shortcoming is mitigate by the fact that once the moel is parametrize, it can be extene to truly multimoal or polyisperse noncolloial suspensions without the nee to a any new parameters as iscusse in Section II C. 2. Experimental measurements of concentrate non colloial suspensions The experiments by Chong et al. (1971) on binary suspensions of glass spheres are taken as a moel system to evelop an alternative set of moel parameters. Chong et al. (1971) reporte that the relative viscosities of the monoisperse systems that were ultimately blene to form the binary suspensions isplaye relative viscosities inepenent of size an temperature, epening only the total solis fraction. This suggests that the system is a reasonable representation of an ieal noncolloial suspension. A viscosity correlation from Morris an Boulay (1999) is aopte to efine the monoisperse viscosity of experiments by Chong et al. (1971): m max max max / / 1 /. (40) r The parameters m 0.41 an max are etermine by fitting the monoisperse viscosity ata from Chong et al. (1971). Subsequently, the binary hyroynamic function is efine an the parameter an relative size ratio g are obtaine by fitting the moel to the experimental ata using Eqs. (2), (6), (7), (10) an (12). The resulting fitting parameter is estimate to be 2.46 an the effective size ratio, g, that best escribes the experimental ata, shown in Fig. 10, is given by 30

31 The overall weighting function is then given by g (41) h g 11, (42) The associate moel fit an comparison to the binary suspension viscosity ata of Chong et al. (1971) are presente in Fig. 11 for comparison. FIG 10: The effective size ratio ( g ) as a function of true relative size ratio (/) from fitting the viscosity moel to experimental ata by Chong et al. (1971). The soli points represent the g values that were use to fit the experimental results in Fig. 11. The soli line represents the parameterization of the g values obtaine by fitting to the ata in Fig. 11. For comparison, the otte line is the effective size ratio previously extracte from the 2 simulations of Chang an Powell (1994) (see Fig 7). 31

32 FIG 11. Relative viscosity as a function of total volume fraction. Viscosity moel fit (soli lines) are compare to simulation ata from Chong et al. (1971) for ifferent particle size ratio (/). The fraction of small spheres ( 1 ) is fixe at 0.25 for binary all cases. Monoisperse, /=1; Binary /=0.477, /=0.313 an /= The effective size ratio ( g ) an parameter ( ) etermine here are similar to the parametrization of the 2 simulation ata in section III B 1. In Fig. (10), the two ifferent parametrizations of g are seen to be closely relate. In aition, the values from these two parametrizations are 2.46 an 3.13 respectively. At this point, the ifferences in parameters arising from the fits to the simulation an experimental ata sets can be attribute to the ifferent microstructures that govern the rheological behavior of the two systems, reflecting the 2 an 3 nature of the two ata sets. Therefore, aitional flexibility in fitting the weighting function shoul be allowe in the choice of weight function to reflect such effects as well as the aitional complexity that is encountere in real systems, cause by particle interactions an non universal particle configurations. IV. COMPARISON TO EXISTING MOELS IN LITERATURE In this section, the binary suspension viscosity moel evelope from the parametrization base on experiments by Chong et al. (1971) (see Section III B 2) is compare to a moel evelope by Qi an 32

33 Tanner (2011). The latter moel is parameterize base on experiments by Chang an Powell (1994) on binary suspensions. Also inclue is a comparison against the results obtaine from a moel for binary suspension recently evelope by Farr (2014). For consistency with the work of Qi an Tanner (2011), we aopte here the same viscosity expression for the monoisperse suspension that was use in their work where c is given by r ( ) 1 1 c, (43) c. (44) These expressions together with the weighting function evelope in Section III B 2 is use to calculate the binary suspension s viscosity. The moel evelope by Farr (2014) has its own viscosity relationship an is use as is. A comparison of these three moels is presente in Fig. 12. In the same figure, the three moels are also compare against experimental ata by Chang an Powell (1994) an Chong et al. (1971). Fig. 12 shows that our moel preictions capture the experimental trens but lack the asymmetry seen in the experimental ata. It shoul be note that the moel of Tanner an Qi (2011) has a total of 7 parameters an it explicitly enforces a viscosity minimum at a fraction of small particles of By amitting extra complexity in the interpolating functions, especially for the form of / h, an equally goo fit may also be obtaine from our moel as well (see Appenix B). On the other han, the moel by Farr (2014) that has 2 parameters has the poorest fit to the ata seen in Fig. 12. Therefore, our moel compares favorably to existing works in the literature, in relation to the esire complexity. 33

34 FIG 12. Relative viscosity as a function of the fraction of small particles in the suspension. Comparison of calculate viscosity from moel of Qi an Tanner (2011) (otte line) to preictions from our moel (soli lines) an Farr s moel (ashe lines). Circle symbols correspon to ata from Chang an Powell (1994). Square symbols correspon to ata from Chong et al. (1971). (1), (2) an (3) represent the viscosities calculate corresponing to the experimental conitions given by the circle, triangle an square symbols respectively for the three moels. V. PREICTING THE VISCOSITY OF POLYISPERSE SUSPENSIONS: RESULTS AN ISCUSSION In this section, the ternary viscosity moel evelope in Sections II C an is applie to preict the viscosity of several suspensions characterize by continuous polyispersity using the framework outline in Sections II C an. The moel preictions in this section are base on the weighting function evelope in Section III B 2 base on the fit to the binary ata set by Chong et al. (1971). The results in this section are base on the assumption that a continuous size istribution may be represente by an equivalent ternary suspension as explaine in Section II. This assumption is also 34

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