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The fol lowing article has been published in Journal of Rheology 57 (3), 743-765, 2013. DOI: 10.1122/1.4795746

A discrete model for the apparent viscosity of polydisperse suspensions including

maximum packing fraction

Aaron DrrInstitute of Nano- and Microfluidics, Technische Universitt Darmstadt,

Alarich-Weiss-Str. 10, 64287 Darmstadt, Germany andCenter of Smart Interfaces, Petersenstr. 32, 64287 Darmstadt, Germany

Amsini SadikiInstitute of Energy and Power Plant Technology, Technische Universitt Darmstadt,

Jovanka-Bontschits-Str. 2, 64287 Darmstadt, Germany andCenter of Smart Interfaces, Petersenstr. 32, 64287 Darmstadt, Germany

Amirfarhang MehdizadehInstitute of Energy and Power Plant Technology, Technische Universitt Darmstadt,

Jovanka-Bontschits-Str. 2, 64287 Darmstadt, Germany(Dated: February 28, 2014)

Based on the notion of a construction process consisting of the stepwise addition of particles to the

pure fluid, a discrete model for the apparent viscosity as well as for the maximum packing fractionof p olydisperse suspensions of spherical, non-colloidal particles is derived. The model connects theapproaches by Bruggeman and Farris and is valid for large size ratios of consecutive particle classesduring the construction process, appearing to be the first model consistently describing polydispersevolume fractions and maximum packing fraction within a single approach. In that context, theconsistent inclusion of the maximum packing fraction into effective medium models is discussed.Furthermore, new generalized forms of the well-known Quemada and Krieger-Dougherty equationsallowing for the choice of a second-order Taylor coefficient for the volume fraction (2-coefficient),found by asymptotic matching, are proposed. The model for the maximum packing fraction as wellas the complete viscosity model are compared to experimental data from the literature showing goodagreement. As a result, the new model is shown to replace the empirical Sudduth model for largediameter ratios. The extension of the model to the case of small size ratios is left for future work.

I. INTRODUCTION

Multiphase flow is a very important but unexploitedfield of research according to the variety of unsolved ques-tions. In both nature and technology multiphase flowis rather the rule than the exception. The field of ap-plications includes sprays, bubbly flows, process and en-vironmental engineering, combustion, rheology of bloodand suspension systems as well as electro- and magne-torheological fluids. At this point of time, researchersagree neither on the cause of various effects nor on theirtheoretical description. In this paper, focus is put on

suspensions of non-colloidal, non-Brownian hard sphereswith a multimodal size distribution. Therefore, the P-clet number (Pe= 6sa3/(kT), where s is the fluidviscosity,athe particle radius, the shear rate, k Boltz-manns constant and T the temperature), is assumedto be large due to a large particle radius. At the sametime, shear rate variations at small shear rate ( [6, 8, 35])result in an apparent viscosity that is independent ofshear rate (this limit is confusingly called high shear limitby[35]). Depending on the length scale of observation aswell as on the effects to be described, different approachesmay be chosen. On the scale of individual particles in a

microscopic approach, besides the strongly restricted pos-sibilities for exact calculation ([13]), Stokesian dynamics([4, 34]) and Lattice-Boltzmann methods ([20, 21]) areemployed, for instance. These methods provide a meansto investigate the mechanisms occurring in suspensionsand to explain the origin of macroscopically observableeffects. However, resolving individual particle requests alarge computational effort and is therefore not applica-ble for engineering purposes. Increasing the observationlength scale thus leads to reduced computational effort butalso to the loss of information because of coarse-graining.The Euler-Lagrange method uses groups of particlesso-called parcelsto represent the particle phase within the

fluid carrier phase whereas the Euler-Euler method con-siders both phases as interacting continuous media (seee.g.[9]). Both methods are frequently used to investigatetransport, dispersion and reaction processes in dilute anddense suspension systems. In case a pure macroscopicdescription of the suspension is sufficient, one may modelcertain flow parameters of the suspension as a whole sothat only, for instance, the volume fraction of the particlephase has to be determined during the computation toprovide a basis for the calculation of macroscopic sus-pension properties such as the apparent viscosity. Here,micromechanics models ([29, 39]) are well-suited, espe-

arXiv:1202.3804v2

[physics.flu-dyn]27Feb2014

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cially the notion of construction processes employed inthe present work. In the effective medium approach, thesuspension is constructed from successively added particlesize classes, where the newly added particles interact withthe present particles as with an effective medium due tolarge size ratios.

In the following, we intend to describe the apparentviscosity as well as the maximum packing fraction of

disperse systems by means of the volume fractions of theparticle phase, which accounts for the excluded-volumeeffect governing the apparent viscosity of hard-spheresuspensions ([41]). This restriction of the parameter spaceby excluding the shear rate corresponds to the asymptoticlimit of large Pclet numbers. Especially, we considerpolydisperse suspensions as they are the most generalcase of dispersions. The apparent viscosity has first beendescribed by[13] in the dilute limit, that is small volumefractions of the particle phase. Later, various attemptsto extend the validity of the viscosity relation to highervolume fractions were made, see for instance[8, 24, 29,36,37].

The paper is organized as follows. In sectionII somebasics on apparent viscosity are provided along with anoverview of recent work. Section III is dedicated to ageneralization of the viscosity correlation for monodis-perse suspensions. In sectionIV a construction processapproach used to describe polydisperse suspension viscos-ity by monodisperse viscosity correlations is presented.Accordingly, a model for the maximum packing fractionof polydisperse systems is developed. The resulting modelis then compared to experimental data from the literature.SectionV is devoted to conclusions.

II. BASICS OF APPARENT VISCOSITY AND

REVIEW OF RECENT WORKS

Throughout this work, we will disregard the existenceof single particles but represent the particles summarily bythe so-calledparticle size classes. The presence of particleswithin the flow increases the viscous dissipation comparedwith the pure fluid phase with viscosity 0 which leadsto the measurability of the apparent viscosity app. Inorder to isolate the influence of the particle phase on theapparent viscosity one defines the relative viscosityr as

r :=

app

0 (1)

The apparent viscosity is mainly dependent on the volumefraction

:= Vparticle

Vfluid+ Vparticle(2)

of the particle phase, where Vparticle and Vfluid denote thevolumes of the particle and fluid phase, respectively. Thevolume fraction is sometimes called packing fraction aswell. From experiments it is well known that the relative

viscosity monotonously increases with increasing volumefraction and exhibits a singular behavior at a value i(compare Tab.III). If we succeed in calculating the volumefractions k, we are in a position to calculate the volumefractions of the individual size classes ik and the total

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After Step Volume Fraction of Particle

Phase SC 1 SC 2 SC 3 . . . SC n

1 1 11

2 2 21

22

3 3 31

32

33

......

......

.... . .

n n n

1 n

2 n

3 . . . n

n

Table III. Scheme for the volume fractions during the construction process (SC = size class)

volume fractions after each step i from Eqs. (30), (31)and from

i =

ik=1

ik (32)

(consider Eq. (26) as an example). Writing down thevolume fractions of the last size classes in the completesuspension and using definition (28) as well as Eqs. (30)and(31) reveals the possibility to calculate the volumefractionsk:

n =

n (33)

n1=

n1(1

n) (34)

n2=

n2(1

n1)(1

n) (35)

n3= . . .

So we find by recursive insertion that

n = n (36)

n1= n11n

(37)

n2= n2

1n n1(38)

n3= . . .

This may be generalized in the form

k = k

1n

m=k+1m(39)

Inserting Eq. (39) into Eq. (30) yields

ik = k

1nm=i+1m

(40)

So we have represented all of the quantities occurring inthe construction process by the volume fractions in thecomplete suspension.

D. A discrete model for the relative viscosity

In this section, we apply the volume fraction relationsderived above to the viscosity change during the con-struction process. As we have already noted, the dif-ferential Bruggeman model lacks any information about

the volume fraction of the individual particle size classes.For that reason we have described the construction pro-cess of the suspension in a discrete form in Sec. IVC.Preparing the development of the viscosity model, we firstneed to introduce the maximum packing fraction into theBruggeman model.

The Roscoe equation (17)diverges as the total volumefraction approaches unity. In a real suspension, the

achievable value of is limited by the maximum packingfraction. In order to formally introduce the maximumpacking fraction into the differential Bruggeman modelwe proceed in a way proposed in Hsueh and Wei [19]. Asimilar way can be found in Mendoza and Santamara-Holek[24], where hard-sphere scaling is used (Quin andZaman [32]). In both publications, it emerges that thenotion of maximum packing fraction is introduced underlittle convincing considerations.

The approach followed in Hsueh and Wei [19] consistsof modifying Eq. (14) by using the maximum packingfraction T in the description of the volume fractionof newly added particles . Therefore, it is supposed

without derivation that

= d

1 T

(41)

In combination with Eq.(15) and after integration underthe condition app( = 0) = 0 one finds the Krieger-Dougherty relation (6)

app = 0

1

T

[]1T

(42)

It would formally be possible to transfer the modifica-

tion (41) to the discrete construction process, that is thevolume fractions (39), and apply the result to the vis-cosity calculation. In the following it will be explainedwhy this approach cannot be valid in general. Partiallyanticipating the later viscosity calculation, we raise twopoints.

Firstly, the construction process described in Sec.IV Cis by no means dependent on the particle geometry. Thisis emphasized by the notion of homogenization betweentwo construction steps. In contrast, the maximum packingfraction is strongly influenced by the particle geometry.So it would be artificial to introduce this quantity into the

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description of volume fractions during the constructionprocess.

Secondly, the consideration of the volume fractionsduring the construction process is independent of thephysical quantity that is calculated (here: the viscosity).It does not make any difference whether one calculates theviscosity or, for instance, the electric conductivity (or bothat the same time). In both cases the construction process

is constituted by the same volume fractions. Only throughthe employed relation between the volume fractions andthe change in the physical quantity of interest parameterslike the maximum packing fraction are included. Thiswill be the approach followed during the later viscositycalculation.

The above considerations imply that the differentialBruggemanmodel only allows for the derivation of theRoscoe equation (17) because as a consequence of thisdifferential approach the right-hand side of Eq. (16)mayonly consist of a linear expression (the Einstein relation)that cannot contain the maximum packing fraction. Sothe approaches presented in Hsueh and Wei [19]and Men-doza and Santamara-Holek [24]are formally possible butphysically questionable. This disadvantage of the differen-tial approach can be avoided in case of a discrete model,where finite volume fractions of particles are added duringthe construction process. The viscosity change is then de-scribed by nonlinear expressions containing the maximumpacking fraction.

At this point it is necessary to introduce a distinctnotation for the maximum packing fraction in order toavoid misinterpretations. The calculation of the maximumpacking fraction will be conducted in Sec.IV E. We choosethe following notations partly referring to Brouwers[5]:iTk

: We denote as iTk

the maximum packing fractionof a polydisperse suspension consisting ofk sizeclasses after theith construction step (ith line inTab.III).

T : The maximum packing fraction of the completesuspension is denoted as T :=

nTn. Hence, it is

obvious why this notation has already been used inthe correlations in Tab.I and in the introduction.

c : We writec :=iT1 for the monodisperse packing

fraction. The monodisperse maximum packingfractionc is constant throughout the construc-tion process and thus carries no upper index i.

ik : The volume fraction of the kth size class after the

ith construction step in the state of maximumpacking fraction is denoted by ik.

This notation is visualized in Tab.IV (compare Tab.III).After each construction step the maximum packing

fraction is newly calculated according to the new com-position of the suspension. This is conducted within arecursive process (arrows in Tab.IV). Starting from themonodisperse maximum packing fractioncall previouslyadded size classes are taken into account and so for eachstepi the value iT i is calculated. This value is needed inEq. (46), which is still to derive.

We now build on the description of the discrete con-

struction process given in Sec. IV Cderiving the viscosityrelations. When during the construction process the(i+ 1)th size class is added, the total volume fractionof the particle phase changes from i to i+1. This isassociated with a change in apparent viscosity from ito i+1. To emphasize the analogy to the differentialBruggemanmodel, we temporarily confine ourselves tothe linear Einsteinrelation as a description of the viscos-

ity change and afterwards we will extend the model tohigher orders.

According to Sec.IV Cthe volume fraction of the newlyadded (i + 1)th size class is denoted as i+1, so the newapparent viscosity is given by

i+1= i

1 + []1

i+1

(43)

The explicit form of Eq. (43) is

i+1= 0

i+1m=1

(1 + []1

m) (44)

Fori + 1 =n we find the apparent viscosity = n of thecomplete suspension (it is well known that

nl=n+1 x= 0

for all x).The differential Eq.(16)contains only the first-order

Einstein relation because of the infinitesimal characterofd. In the case of the discrete model represented byEq. (44) it is not necessary to confine oneself to linearterms. Therefore, we are allowed to describe the modifica-tion of the apparent viscosity more accurately by higherorder terms ofk. So Eq.(44)can be extended accordingto

n

= 0

n

m=1

1 + []1 m+ []2(m)2 + (45)In this way, one can establish a connection betweenEq. (45)and models existing in the literature. Selectingthe coefficients from the Taylor expansion of a viscosityrelation ((12), for instance) at the point = 0 for thecoefficients[]k in Eq.(45) and formally considering aninfinite number of terms, we may substitute the bracketedterm in Eq. (45)with the viscosity relation itself. Thisstep requires the introduction of the maximum packingfraction. Using a viscosity relation in the sense of Eq.(12)we find

n = 0n

m=1

r(Pm, mTm) (46)

In Eq.(46) mTm refers to the maximum packing fractionafter the mth construction step that will be calculated inSec.IVE.Writing down Eq. (46)we make the importantassumption that the viscosity change in each step m de-pends on the actual maximum packing fraction value mTmand not only on the final valueT =nTn.

Despite the use of the variable maximum packing frac-tion Eq. (46) corresponds to the so-called Farris model

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After Step Volume Fraction of Particle

Phase SC 1 SC 2 SC 3 . . . SC n

1 1 c

2 2 c 2T2

3 3 c 3T2

3T3

......

......

...

n n c n

T2 n

T3 . . . n

Tn

Table IV. Notation for the maximum packing fraction during the construction process (SC = size class); the arrows indicatethat the far right values are calculated recursively from the previous values (for details of the calculation see Sec. IVE)

(Farris [16]) also referred to in Brouwers [5], Cheng et al.[7], Sudduth[36]. So we have found an interesting connec-tion between the differential Bruggeman model and theFarris model which is drawn by the discrete constructionprocess.

It is important to note that in extending Eq. (45)to(46)we have considered the complete Taylor expansion of

relation (12), suggesting the admissibility of arbitraryvolume fractions of the individual size classes which isuncertain regarding the notion of the construction process.Since it is not possible to state a limiting value ofm forthe validity of the result (46), we will use this equation atleast as a reasonable approximation also for higher valuesofm.

E. Determination of the maximum packing fraction

In the following we review two models for computing

the polydisperse maximum packing fraction reported inthe literature. The first model has been proposed byFurnas [17] (see also Brouwers [5], Sudduth [37]) and thesecond one, using Furnas results, by Sudduth [37]. Wethen derive a model which modifies the Furnas approachand serves as an alternative to the Sudduth model forlarge diameter ratios. During the calculation we willuse the quantities occurring in the construction processaccording to Tab.IIIandIV. The new model for the max-imum packing fraction can be easily introduced into theviscosity model (as explained in the context of Eq. (46))because it is conceptually consistent with the construc-tion process. This is an advantage over models that use

a maximum packing fraction that has been derived ormeasured entirely independent of the viscosity relationin which it is to be used (e.g. Chong et al. [8], Dameset al. [11] or Poslinski et al. [30] using a model takenfrom Ouchiyama and Tanaka [28]). The same appliesfor the model proposed by Farr and Groot [15], which isbased upon a mapping of the 3D packing problem ontoa 1D problem allowing for inexpensive estimation of themaximum packing fraction. This model is very promis-ing even for finite size ratios but still needs a non-trivialsorting-like algorithm. However, a close look at that algo-rithm reveals that it is very similar to the construction

process used in the present work and therefore offers thepossibility to be used as an alternative, especially whenit comes to small size ratios.

1. The Furnas model

The Furnas model estimates the highest possible valueof the maximum packing fraction for a given number ofsize classes with large diameter ratios (cf. Sec.IV B). It isbased on the assumption that the state of maximum pack-ing fraction is constructed successively from size classeswith decreasing diameter. At first, the larger spheres fillthe entire available suspension volume with the monodis-perse maximum packing fraction c. So the total volumefraction is

iT1,Furnas= i1= c (47)

Subsequently, the remaining volume fraction (1 c)is filled by the spheres with the next smaller diameter,also to a realizable part ofc. So the additional volumefraction of small spheresi2 is given by

iT2= i1+

i2= c+ (1 c)c = 1 (1 c)

2 (48)

Generalizing these considerations to a number ofn sizeclasses, one finds (Sudduth[37])

nTn,Furnas = 1 (1 c)n (49)

In the Furnas model the maximum packing fraction iscalculated under the assumption that the size distributionallows that each size class fills the entire interstitial space

to a fraction of c. This corresponds to a certain sizedistribution for a given number of size classes McGeary[23].

2. The Sudduth model

To account explicitly for the size distribution, Sudduth[37] proceeded as follows. Starting from experimentaldata by McGeary [23] for bidisperse suspensions, Sud-duth found that the maximum packing fraction of poly-disperse suspensions can be parameterized using certain

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moments (Dx for the xth moment) of the particle sizedistribution

Dx =

ni=1 NiD

xin

i=1 NiDx1i

(50)

whereDi, as before, is the particle diameter and Ni thenumber density of the ith size class, respectively. In con-

junction with the maximum packing fraction value(49)from the Furnas model, Sudduth ended up with the ap-proximation

nTn,Sudduth = nTn,Furnas

nTn,Furnas c

e0.27(1D5/D1) (51)

for the complete suspension, where D5, D1 are given byEq. (50). We will compare the Sudduth model to ourresults in Sec. IV F. Note that, despite the model (51)can possibly cover all kinds of size distribution, it lacks atheoretical foundation. Therefore, we attempt to propose

an alternative model for large size ratios consistent withthe construction process and the viscosity model (46).

3. A new model for the maximum packing fraction

We will derive the new model in a way that first theinstructive case of bidisperse suspensions is treated explic-itly in order to prepare the subsequent general derivation.It should be noted that the bidisperse model presentedbelow has already been proposed by several authors (forinstance Gondret and Petit [18], Yu and Standish [42]),but it is helpful to discuss it in detail to prepare the

polydisperse derivation (which appears to be novel).A consequent calculation of the maximum packing frac-tion can be achieved by retaining the particle size distri-bution existing in the fluent suspension state during thecomposition of the maximum packing fraction. This mainassumption can be expressed in the form

ik+1ik

=ik+1

ik(52)

whereik in the state of maximum packing fraction cor-responds to ik in the fluent state, that is the volumefraction of the kth size class.

Derivation for bidisperse systems Bidisperse (orbimodal) suspensions contain two different particle sizes.Given the ratio

i2i1

=i2i1

(53)

we ask how the state of maximum packing fraction can beachieved retaining the volume fraction ratio (53). Thereare two possibilities differing with respect to the valueofi2. Both situations are visualized in Fig. 1.

(a)

(b)

Figure 1. Schematic representation of the two possible situa-tions for the bidisperse maximum packing fraction: (a)Situa-tion : The fraction of large particles lies below c while thesmall particles fill the interstices with a fraction ofc. (b)Sit-uation : The large particles occupy a fraction of c whilethe small particles are present in the interstices according tothe volume fraction ratio.

a. Situation The volume fractioni2 of the largespheres lies below the monodisperse packing fraction cand the remaining interstice 1 i2 is filled to an amountofc with small spheres, compare Fig.1(a). So the totalparticle volume fraction is

iT2= i1+

i2=

i2+ (1

i2)c (54)

(compare Eq. (48), where restrictively i2 = c). UsingEq. (53) we find from Eq. (54)

i2= c

c+i

1

i2

(55)

and furthermore

iT2= c(i

2+ i

2)ci2+

i1

(56)

The limitation for the validity of Eq. (56) follows fromthe realizability condition i2< c (arbitrarily, we as-sign the equal sign to the situation in order to avoidan ambiguous definition of the case i2 = c) which incombination with Eq. (55) yields

i2< i11 c

(57)

as the condition for the validity of situation .

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b. Situation In this situation the volume frac-tion i2 of the large spheres equals the monodispersemaximum packing fractionc (it is not possible to exceedthe value of c in a monodisperse loading) and so wehave i2= c, see Fig.1(b). Using Eq. (53)we immedi-ately find

i

1=

i

2

i1

i2 and (58)

iT2= i1+

i2= c

1 +

i1i2

(59)

It follows from the fundamental requirementiT2< 1 andEq. (59) that

i2>c

i1

1 c(60)

whereas the realizability condition i1 (1i2)c under

consideration ofi2= c and Eq. (58) yields

i2 i11 c

(61)

The condition (61) includes the inequality (60). SoEq. (61) serves as a condition for the validity of situ-ation .

Generalization for polydisperse systems In thefollowing, we will generalize the model for polydispersesystems which corresponds to deriving expressions for thetotal volume fractioniTk+1in the situations and and

for the limiting value ofik+1. Regardless of the number

of size classes, there are always two situations. Thisrelies on the geometric fact that no particle class exceptthe one with the largest particle diameter can reach themonodisperse packing fraction. If hypothetically a particleclass with a smaller diameter reached the monodispersemaximum packing fraction, the larger particles wouldnot fit into the interstices between the smaller particlesand could thus not contribute to this state of maximumpacking fraction.

a. Situation In the polydisperse case the inter-stices between the largest particles are filled by the smallerparticle size classes with the packing fractionkTi which isdetermined by the volume fractions i1 to

ik. So the

volume fraction of the largest particles is

i

k+1 and theone of all the smaller particles is (1ik+1)iTk. The sum

of both fractions yields the maximum packing fraction,that is

iTk+1= ik+1+ (1

ik+1)

iTk (62)

So we may express the volume fractions of all the smallparticles by

km=1

im = (1 ik+1)

iTk (63)

Eq. (52)allows for representing all the occurring volumefractions by the fractionik+1 of the largest particles, sowe can reformulate Eq. (63):

km=1

im= ik+1

1

ik+1

km=1

im= (1ik+1)

iTk (64)

Solving Eq. (64) for ik+1 and inserting the result intoEq. (62) yields the maximum packing fraction in situa-tion :

iTk+1= iTk

k+1m=1

im

iTkik+1+

km=1

im

(65)

The limiting value forik+1 follows analogously to the

bi- and tridisperse cases from the requirement ik+1< cand is thus given by

ik+1< ckm=1im

iTk(1 c) (66)

b. Situation In situation the largest particlesoccupy a volume fraction equal to the monodisperse max-imum packing fraction, so ik+1 = c. Therefore, bymeans of Eq. (53) we may write

iTk+1= c

k+1m=1

im

ik+1(67)

for the maximum packing fraction in situation . Thelimiting value ofik+1 in this situation is determined bythe condition

km=1

im (1 c)iTk (68)

and combined with (63) and(67) takes the form

ik+1 ck

m=1im

iTk(1 c) (69)

c. Properties of the bounds The bounds(66)and(69)can alternatively be derived by equaling the prescrip-tions(65) and (67). This underlines the consistency be-tween the situations and because there is a continuoustransition. This is equivalent to choosing the smallestvalue out of the two calculated maximum packing frac-tions because for all values ofik+1 the situation thatis present is always the one with the smaller maximumpacking fraction. Thus we are lead to the prescription

iTk+1= min

iTk+1, iTk+1

(70)

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F. Comparison with experiments and models from

the literature

In this section, we compare our model to experimentaldata and models reported in the literature. McGeary[23] conducted experiments to determine the maximumpacking fraction of polydisperse compositions of spheres.To achieve high packing fractions, he investigated systemsof up to four particle size classes mostly with a largediameter ratio (cf. Sec.IV B). Note that we consider hisdata without applying the manipulation used by Sudduth[37], which would not change the result of our compari-son. Fig. 2compares the bidisperse maximum packingfraction calculated from Eqs. (56)and (59), depending onthe criteria(57)and (61), respectively, with experimentaldata for three different large diameter ratios. All exper-

0 0.2 0.4 0.6 0.8 10.6

0.65

0.7

0.75

0.8

0.85

0.9

fraction of large particles L

maximum

packingfraction

2T2

from Eqs. (56) and (59)

2T2

from Eq. (49)

1 = 11.3

1 = 16.5

1 = 19.1

Figure 2. Comparison of Eqs. (56) and (59) for the maxi-mum packing fraction of bidisperse suspension with the datafrom McGeary [23] using c = 0.64, accordingly; the Furnasmodel (48) gives the maximum value of2T2,Furnas = 0.8704;1 is the diameter ratio (18)

imental values lie below the curve given by the model,indicating that the latter correctly predicts the bidisperselarge size ratio limit of the maximum packing fraction.In addition, we calculate the maximum packing fractionfor the quaternary system from McGeary [23], as wellas for the corresponding bi- and tridisperse composition,

and compare it to the experimental values in Tab. V.Therein, also the limiting values provided by the Furnasmodel (Sec.IV E 1) are given. The fact that our modelrepresents the experimental data more accurately thanthe Furnas model shows that the theoretical limit doesnot coincide with McGearys densest packing but liesvery close to it. Also the model by Sudduth [37] deviatesmore strongly from the experimental data than our model.Within the experimental uncertainties of the data, thisshows that our model for the maximum packing fractionis at least equivalent to the Sudduth model for large sizeratios. Note that the smallest size ratio of= 5.4 dies

not fulfill the condition from Sec. IV B. However, thishas a negligible effect in the considered case because thetwo corresponding size classes (2 and 3 in Tab. V) to-gether occupy less than one third of the total solid volumefraction.

For comparison of the complete viscosity model withexperimental data we choose the case of bidisperse sus-pensions. In Poslinski et al. [30], experimental datafrom Chong et al. [8], Sweeny [38] as well as originaldata for the relative viscosity of bidisperse suspensionswith large size ratios (1 = 5.2, 7.2, 20.8) for severaltotal volume fractions is provided. In order to apply the

Start i = 1

ik from Eq.(40)for k = 1, . . . , i

Viscosityrelation depen-

dent on iTi

No

Yes

iT1 = c

iT2 fromEq. (65) or (67)

. . .

i

Ti fromEq. (65) or (67)

i fromEq.(46)

i fromEq. (45)

i = nNo

Yes

End

i i+ 1

Figure 3. Scheme for the calculation of maximum packingfraction and relative viscosity after the ith construction stepfor n particle size classes with references to the respectiveequations

discrete viscosity model as presented above to the exper-imental data, we proceed as follows (cf. Fig.3). First,we determine the parameters of the underlying monodis-

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i 1

2

3

4

nTnMcGeary

nTnthis work

nTn Sudduth nTn Furnas

48.35.4

7

1.000 0.580 0.580 0.580 0.5803 0.274 0.726 0.800 0.799 0.784 0.8242 0.109 0.244 0.647 0.898 0.896 0.926 0.9261 0.061 0.102 0.230 0.607 0.951 0.956 0.969 0.969

Table V. Maximum packing fraction Tn (n= 1, 2, 3, 4) from McGeary [23] compared against three different models: This work

(Eqs. (65) and (67), respectively), the model by Sudduth [37], and the Furnas model (Eq. (49)); i denotes the index of a sizeclass, is the diameter ratio between two consecutive classes

perse viscosity relation, for which we choose Eq. (12)(this choice is due to consistency with Poslinski et al.[30], where the Quemada relation (9) is used, alternativechoices are given in Tab.I). We set []1= 2.5, []2= 5.2(according to Batchelor and Green [2]) and c = 0.64.Second, we evaluate the model Eqs. (46), (56) and (59)for bidisperse systems (n= 2), thereby introducing thefraction of large particles

L:= 22

21+ 22

=: 22

(71)

with the total volume fraction. The viscosity Eq.(46)can thus be written as

2= 0

2m=1

[]1

2

mTm

m

+

[]2

3

(mTm)2

(m)

2 +

1

mmTm

2 (72)

where 1 = 1/(1 2) = L/(1 L) and2 = 1 = L are calculated from Eq. (39) (recallthat21= 1 and

22= 2 for bidisperse systems,

cf. Eq. (40)). The maximum packing fraction in eachof the situations and (see Sec. IV E) according toEqs. (56) and (59), respectively, may be written as

2T2= c

c2+ 1=

ccL+ 1 L

(73)

2T2= c

2=

cL

(74)

Situations or are chosen from condition (70)

2T2= min

2T2, 2T2

(75)

Fig.4shows good agreement between theory and experi-ment especially for the highest volume fractions, but anunderestimation of the effect of bidispersity, that is theviscosity minimum, for the two smallest volume fractions.Note that, despite the small size ratio of 1 = 5.2 inPoslinskis data, our model can be applied as a good ap-proximation. We may conclude that the model presentedin this work correctly displays the behavior of bimodalsuspensions with large size ratio, which implies that the

0 0.2 0.4 0.6 0.8 110

0

101

102

103

fraction of large particles L

relat

ive

viscosity

r

Poslinski et al. [30] = 0.60 = 0.30 = 0.50 = 0.20 = 0.40 = 0.10This workChong et al. [8]Sweeny [38]

Figure 4. Comparison of the model consisting of Eqs. (46)for n = 2, (56) and (59) with experimental data from Chonget al. [8], Poslinski et al. [30], Sweeny [38] for various totalvolume fractions ([]1 = 2.5, []2 = 5.2, c = 0.64); missing

total volume fractions are = 0.60 (Chong et al.[8]) and =0.55 (Sweeny [38]); diameter ratios are 1 = 5.2 (Poslinskiet al. [30]), 1 = 7.2 (Chong et al. [8]) and 1= 20.8 (Sweeny[38]), respectively

way in which the maximum packing fraction has beenintroduced into the construction process is reasonable.It must be emphasized that the viscosity reduction withrespect to a monodisperse suspension, as shown in Fig.4by our model, is not only due to the increased maxi-mum packing fraction for 0 <

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As a starting point, we proposed generalized forms ofthe well-known Quemada and Krieger-Dougherty equa-tions that allow for the choice of the second order intrinsicviscosity[]2. These modified equations are more suitableto describe the relative viscosity over the whole range ofconcentrations than the original relations.

Later, we described the construction process in detailapplying a dimensionless way of description based on

volume fractions. This rigorous description served as abasis for the calculation of the relative viscosity duringthe construction process. Starting from theBruggemanmodel, we finally arrived at the Farris model, connectingtwo approaches commonly regarded as uncorrelated.

As an entirely new component, we introduced the poly-disperse maximum packing fraction into the Farris model.The way of introducing the maximum packing fractioninto the effective medium approach has been justifiedby conceptual considerations regarding the constructionprocess. Consistently with the relative viscosity calcula-tion and therefore in contrast to most approaches in theliterature, we derived a formalism to determine the poly-

disperse maximum packing fraction by means of the same

construction process. It is assumed that the maximumpacking fraction affects the viscosity in each step of theconstruction process.

Comparing separately the maximum packing fractionmodel and the complete viscosity model to experimen-tal data from the literature, good agreement has beenachieved. We showed that for hard-sphere suspensionswith large diameter ratios the empirical Sudduth model

can be replaced by a maximum packing fraction modelbased on an intuitive geometrical argumentation. Thiscould offer the possibility to modify the Sudduth modelusing the model presented here as a starting point.

Additionally, we revealed a possible approach for inte-grating particle deformability, represented by a particlephase viscosity, into the viscosity model using a resultfrom the literature.

So far, our model is only valid for large diameter ratiosof consecutive size classes during the construction process.An attempt to generalize the present model, which fullyaccounts for the size distribution in the limiting case oflarge size ratios, to the case of small size ratios will be

presented in a future work.

[1] G.K. Batchelor. The effect of Brownian motion on thebulk stress in a suspension of spherical particles. J. FluidMech., 83(1):97117, 1977.

[2] G.K. Batchelor and J.T. Green. The determination ofthe bulk stress in a suspension of spherical particles toorder c2. J. Fluid Mech., 56(3):401427, 1972.

[3] D. Bedeaux, R. Kapral, and P. Mazur. The effective shearviscosity of a uniform suspension of spheres. PhysicaA: Statistical Mechanics and Applications, 88(1):88121,

1977.[4] J.F. Brady. The rheological behavior of concentrated

colloidal dispersions. J. Chemical Physics, 99(1):567581,1993.

[5] H.J.H. Brouwers. The viscosity of a concentrated suspen-sion of rigid monosized particles. Phys. Rev. E, 81:051402,2010.

[6] C. Chang and R.L. Powell. Dynamic simulation of bimodalsuspensions of hydrodynamically interacting sphericalparticles. J. Fluid Mech., 253:125, 1993.

[7] D.C-H. Cheng, A.P. Kruszewski, J.R. Senior, and T.A.Roberts. The effect of particle size distribution on therheology of an industrial suspension.J. Materials Science,25:353373, 1990.

[8] J.S. Chong, E.B. Christiansen, and A.D. Baer. Rheologyof concentrated suspensions. J. Appl. Polym. Sc., 15:20072021, 1971.

[9] M. Chrigui. Eulerian-Lagrangian approach for modelingand simulations of turbulent reactive multi-phase flows

under gas turbine combustor conditions. PhD thesis, Tech-nische Universitt Darmstadt, 2005.

[10] B. Cichocki and B.U. Felderhof. Linear viscoelasticity ofsemidilute hard-sphere suspensions. Phys. Rev. A, 43(10):54055411, 1991.

[11] B. Dames, B.R. Morrison, and N. Willenbacher. An empir-ical model predicting the viscosity of highly concentrated,

bimodal dispersions with colloidal interactions. RheologicaActa, 40:434440, 2001.

[12] H. Eilers. Die Viskositt von Emulsionen hochviskoserStoffe als Funktion der Konzentration. Kolloid-Zeitschrift,97(3):313321, 1941.

[13] A. Einstein. Eine neue Bestimmung der Molekldimen-sionen. Annalen der Physik, 19:289306, 1906.

[14] A. Einstein. Berichtigung zu meiner Arbeit: Eine neueBestimmung der Molekldimensionen. Annalen der

Physik, 34:591592, 1911.[15] R.S. Farr and R.D. Groot. Close packing density of

polydisperse hard spheres. J. Chem. Phys., 131:244104,2009.

[16] R.J. Farris. Prediction of the viscosity of multimodalsuspensions from unimodal viscosity data. Trans. Soc.Rheol., 12:281301, 1968.

[17] C.C. Furnas. Grading aggregates - I. - Mathematicalrelations for beds of broken solids of maximum density.Ind. Eng. Chem., 23:1052, 1931.

[18] P. Gondret and L. Petit. Dynamic viscosity of macroscopicsuspensions of bimodal sized solid spheres. J. Rheology,41(6):12611274, 1997.

[19] C.H. Hsueh and W.C.J. Wei. Analyses of effective viscosity

of suspensions with deformable polydispersed spheres. J.Phys. D: Appl. Phys., 42:17, 2009.

[20] J. Hyvluoma, P. Raiskinmki, and A. Koponen. Lattice-boltzmann simulation of particle suspensions in shear flow.J. Statistical Physics, 121(1&2):149161, 2005.

[21] D. Kehrwald. Lattice boltzmann simulation of shear-thinning fluids. J. Statistical Physics, 121(1&2):223237,2005.

[22] I.M. Krieger and T.J. Dougherty. A mechanism for non-Newtonian flow in suspensions of rigid spheres. Trans.Soc. Rheol., III:137152, 1959.

[23] R.K. McGeary. Mechanical packing of spherical particles.

8/11/2019 1202.3804f

14/14

14

J. Am. Ceramic Soc., 44(10):513522, 1961.[24] C.I. Mendoza and I. Santamara-Holek. The rheology of

hard sphere suspensions at arbitrary volume fractions:An improved differential viscosity model. J. Chem. Phys.,130:17, 2009.

[25] M. Mooney. The viscosity of a concentrated suspensionof spherical particles. J. Colloid and Interface Science, 6(2):162170, 1951.

[26] L. n. Krishnamurthy and N.J. Wagner. The influence ofweak attractive forces on the microstructure and rheologyof colloidal dispersions. J. Rheology, 49(2):475499, 2005.

[27] A.N. Norris. A differential scheme for the effective moduliof composites. Mechanics of Materials, 4:116, 1985.

[28] N. Ouchiyama and T. Tanaka. Porosity estimation forrandom packings of spherical particles. Ind. Eng. Chem.Fundam., 23:490493, 1984.

[29] W. Pabst. Fundamental considerations on suspensionrheology. Ceramics Silikty, 48:613, 2004.

[30] A.J. Poslinski, M.E. Ryan, R.K. Gupta, S.G. Seshadri,and F.J. Frechette. Rheological behavior of filled poly-meric systems II. The effect of a bimodal size distributionof particulates. J. Rheology, 32(8):751771, 1988.

[31] D. Quemada. Rheology of concentrated disperse systems

and minimum energy dissipation principle I. Viscosity-concentration relationship. Rheologica Acta, 16:8294,1977.

[32] K. Quin and A.A. Zaman. Viscosity of concentratedcolloidal suspensions: Comparison of bidisperse models.J. Colloid and Interface Science, 266:461467, 2003.

[33] J.V. Robinson. The viscosity of suspensions of spheres.J. Phys. Chem., 53(7):10421056, 1949.

[34] W.B. Russel, D.A. Saville, and W.R. Schowalter. Colloidaldispersions. Cambridge University Press, 1989.

[35] A.P. Shapiro and R.F. Probstein. Random packings ofspheres and fluidity limits of monodisperse and bidispersesuspensions. Phys. Rev. Lett., 68(9):14221425, 1992.

[36] R.D. Sudduth. A generalized model to predict the viscos-ity of solutions with suspended particles. J. Appl. Polym.Sc., 48:2536, 1993.

[37] R.D. Sudduth. A new method to predict the maximumpacking fraction and the viscosity of solutions with a sizedistribution of suspended particles. II. J. Appl. Polym.Sc., 48:3755, 1993.

[38] K.H. Sweeny. Paper presented at the Symposium on HighEnergy Fuel, 135th National Meeting, Am. Chem. Soc.,Boston, Mass., 1959. as cited in[8, 30].

[39] S. Torquato. Random heterogeneous materials Mi-crostructure and macroscopic properties. Springer, 2002.

[40] M. van Dyke. Perturbation methods in fluid mechanics.Parabolic Press, 1975.

[41] N.J. Wagner and A.T.J.M. Woutersen. The viscosity ofbimodal and polydisperse suspensions of hard spheres in

the dilute limit. J. Fluid Mech., 278:267287, 1994.[42] A.B. Yu and N. Standish. Porosity calculations of multi-

component mixtures of spherical particles. Powder Tech-nology, 52:233241, 1987.