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The Einstein coefficient of suspensions in,generalized Newtonian liquids. Jozua Laven and Hans N Stein, Laboratory of Colloid Chemistry and Thermodynamics. Technical University of Eindhoven P O Box 513 5600 MB. Eindhoven The Netherlands,Received 1 7 August 1990 accepted 10 June 1991. A new theoretically more satisfactory definition for relative viscosities of sus. pensions in generalized Newtonian media is presented which according to the. viscosities of suspensions and pure liquids should be compared to equal aver. aged squared strain rates 3 in the liquid phases Comparison of this quantity. 17 is made with two definitions currently in use in which viscosities are. compared either at equal macroscopic stresses r or at equal macroscopic. squared strain rates Interrelations between the three different relative vis. cosities have been derived in the case of dilute suspensions of particles of arbi. trary shape in generalized Newtonian liquids Values for limiting viscosity num. bers Einstein coefficients KE have been derived from experimental data on. suspension viscosities both as obtained by the authors and as obtained from. other sources The data cover suspensions of spherical particles in liquids with. power law exponents n between 0 07 and 1 0 According to the new definition. the Einstein viscosity coefficient KE lim u d In 7 p is constant with a. value of 2 5 over the whole range of n According to the other definitions KE is. 2 5 only for n 1 with n decreasing to zero KE decreases linearly to 0 75. n under equal strain rate or rises exponentially to m 7 under equal strain. stress The values of KE as deduced from experiments obey the theoretical. interrelations derived There are strong indications that the value of n has no. influence on the inhomogeneity in rate of strain in dilute suspensions. I INTRODUCTION, The rheology of suspensions in non Newtonian media is from a. technological point of view even more interesting than its counterpart. a Dedicatedto Prof Dr H Janeschitz Kriegl on the occasion of his 67th birthday. 1991by The Society of Rheology Inc, J Rheol 35 8 November 1991 0148 6055 91 081523 27 04 00 1523.

1524 LAVEN AND STEIN, with Newtonian media Important industrial applications are in the. processing of filled polymers technical ceramic pastes and drilling. muds for oil recovery Theoretical analyses be it nonrigorous of these. flow problems are only available for high volume fractions 12 where. particle particle lubrication hydrodynamics dominate the particle fluid. interactions, One of the major problems in developing such rigorous theories is the. nonlinearity of the viscous contribution in the Navier Stokes equation. Apparently research efforts in the area of suspension rheology at present. are focused mainly in the rheology of highly filled suspensions in New. tonian media 3 rather accurate predictions are now available for sim. ple shear flow of hard spheres both at high and low P clet numbers. Pe This complements the numerous publications on the rheology of. dilute suspensions in Newtonian media First Einstein published his. classical result 8 KE 2 5 with KE limq o qr 11 a where v is the. relative viscosity and 9 is the volume fraction of spherical particles as. dispersed phase In the last three decades many studies have been pub. lished on deviations of KE from 2 5 e g due to electrostatic interac. Experimental results of the suspension rheology with non Newtonian. media is mainly restricted to high Pe numbers i e with almost neg. ligible Brownian motion This is due to the fact that in most industrial. applications the particles are relatively large and because mixing of. small particles 1 pm with pseudoplastic liquids is impossible in most. cases without degrading the liquid excessive shear rates and viscous. A difficulty in evaluating relative viscosities qr qsu qliq in such. systems is in defining what circumstances in suspension and in pure. liquid should be kept equal Experimental data in this field are mainly. interpreted in terms of relative viscosities while taking the macroscopic. shear rates or shear stressesequal We will denote these quantities with. 7 and The quantity 7 was inspired by the time temperature su. perposition method employed in polymer rheology when comparing. data taken at different temperatures 13 In such a way these pseudoplas. tic viscosities can be better compared A logical but until now unem. ployed method compares viscosities of suspension and pure liquid while. their liquid phase viscosities are equal This method will be evaluated in. the present paper, The precise definition of 7r has a large influence on its value as is. evident e g from the value of the Einstein viscosity coefficient KE. limq O rl 1 p where p is the volume fraction of the filler par. DILUTE SHEAR THINNING SUSPENSIONS, titles Published experimental results1 6 indicate that Kb i e KE. using 7 is approximately zn where n d lnr d In F Ir is equal or. slightly larger than 2 5, The Einstein coefficient is a tool for studying the effects of the defi.

nition of vr on its quantitative value A disadvantage of this tool is that. experimental data are only useful if they are very accurate see also Sec. IV However it has an important advantage as well which is related to. the difficulty in obtaining monosized particles of a size large enough. both to neglect the influence of electrical double layers on the suspen. sion viscosity and to be able to mix the particles into the fluid The. almost inevitable polydispersity of such particles has considerable influ. ence on the suspension viscosity at high volume fractions of solids. However it has no influence on the value of the Einstein coefficient. because particle interactions are absent under q 0 conditions and be. cause KE is in principle independent of the particle radius. In Sec II A we will propose a new definition of q which is much. more satisfactory from a theoretical point of view than the former two. In Sec II B we calculate the relationship between the macroscopically. applied strain rate and the local one in the Newtonian liquid phase of a. dilute suspension we also derive to what extent the latter one is different. from the strain rate in the pure liquid when evaluating qr according to. the three definitions In Sec II C we show how to calculate these three. relative viscosities for dilute suspensions in generalized Newtonian me. dia In Sec II D we indicate how to cross section a diagram of a col. lection of flow curves in order to calculate the three types of relative. viscosity We demonstrate the inhomogeneity of strain rates in the liq. uid phase of a dispersion to be independent of the degree of pseudoplas. ticity Experimental results with suspension systems described in Sec. III and as taken from the literature are analyzed Einstein coefficients. according to the different definitions are evaluated in Sec IV. II THEORY OF THE RELATIVE VISCOSITY OF SUSPENSIONS. For the derivation of Einstein s law of the relative viscosity of sus. pensions two approaches appear to exist in literature the original one by. Einstein see also his book on this subject I7 and the one described by. Landau and Lifshitz I8 The original one best suits our purpose. The starting point of this section is the notion that a comparison of. 1526 LAVEN AND STEIN, the viscosities of a liquid and a suspension of that liquid can best be. made under circumstances that the liquid viscosity in both systems is. In this study we describe suspensions of particles in non Newtonian. nonelastic liquids generalized Newtonian liquids t9 These liquids. where T and are the deviatoric stress and the rate of strain tensors and. where the viscosity function 7 is a function of the second invariant or. the magnitude p of the rate of strain tensor only, where Vv Vv Vv being the dyadic product of the liquid veloc. ity v and of V Bei J i, In this study we exclude special suspension flow effects arisin from. Brownian motion of the particles inducing pseudoplasticity 2 and. from a competition of hydrodynamic forces with long range repulsive. forces on the particles inducing viscosity dilatancy2. B Dilute suspensions in Newtonian media, If one sphere volume P is placed in a very extended Newtonian. liquid total volume V which is subjected to simple shear then the. average rate of dissipation per volume of the dispersion will be. where p is the undisturbed shear rate at infinite distance to the. sphere and v is the viscosity of the liquid phase Becausein very dilute. suspensions the areas of flow disturbances of individual spheres are far. apart their effects on are additive giving,li2 ql I 4.

For suspensions it is also possible to write, where qSand i are the macroscopically measurable suspension viscos. ity and suspension strain rate respectively,It can be shown17 that. DILUTE SHEAR THINNING SUSPENSIONS 1527, in which q denotes the volume fraction of the dispersed phase Substi. tution of Eq 6 into Eq 4 and comparing the result with Eq 5. gives Einstein s law for suspension viscosity first order in p. qr 77s qo 1 p 7, In reality dissipation in a suspension is confined to the volume oc. cupied by the liquid thus the intensity of the rate of dissipation Li in a. suspension volume V can be written as,w v Jileply i MUr 770 1 91Z3.

where pi stands for the square of the strain rate as averaged over the. liquid phase Comparing Eqs 8 and 5 and eliminating r r with the. aid of Eq 7 gives, Equation 9 indicates that the determination of v7r while keeping the. macroscopic strain rates in suspension s and in pure liquid p equal. involves different average liquid strain rates according to. The alternative used by Kataoka et a1 14i e keeping the shear stresses. equal in a determination of q means that, In view of our starting point Sec II A the latter method is debatable. as an objective definition of 7 of non Newtonian suspensions. C Dilute suspensions in generalized Newtonian liquid media. 1 Relative viscosity at equal averaged squared strain. rates in the liquid phases, Also with generalized Newtonian liquids a single sphere in an ex. tended liquid phase will disturb the local flow field increasing the aver. aged strain rate in the liquid phase if the macroscopic deformation rate. is kept constant For a specific flow rate indicated with index 1 in the. dilute suspension in a generalized Newtonian liquid we can in similarity. to Eq 4 formally write,1528 LAVEN AND STEIN, where I is a specific average value of the strain rates in the liquid. phase the exact way of averaging to be defined later Parentheses. indicate that the quantity concerned e g 0 is a function of the pa. rameter concerned e g vo p Note that 70 is the local viscosity of the. liquid phase its value being defined see Eqs 1 and 2 by the local. squared strain rate p However the viscosity of a suspension can be. defined as a function of either of two types of squared strain rates F or. ii We will use either of them where appropriate, In Eq 12 any deviation off from unity accounts for the fact that.

the flow profile around the sphere is different from that in the Newton. ian l uid case as well as for the fact that the liquid viscosity is not. vc J at any location In the case of a power law liquid we could. write f n where n is the power law exponent The exact value off will. of course depend on how is averaged, In an analogous way Eq 6 can be adjusted to the present case. where g may be unequal to unity accounting for the fact that IZq 6. may be invalid in caseof a non Newtonian liquid medium Additionally. Eq 5 in the non Newtonian case reads,Combination of Eqs 12 13 and 14 gives. c l l 2g f 2lq l fhp 15, The left hand side of Eq 15 can be interpreted as the relative viscosity. of a dilute suspension while keeping the strain rates in the liquid phases. averaged in a way still to be defined equal, Because in the remaining of the present paper f and g would appear. always in the same combination we introduce h 2g f 2 The quan. tity h can be seen as the Einstein coefficient at that average squared. liquid strain rate K h,DILUTE SHEAR THINNING SUSPENSIONS 1529.

The dissipation is confined to the liquid phase The intensity of dis. sipation as averaged over the whole suspension can be written in anal. ogy to Eq 8,770 1 2 r lj2 rMUr 1 ITo,v I ll PIV, where the bar indicates averaging over the liquid phase In a way anal. ogous to that employed by Fowler and Guggenheim22in the description. of strictly regular solutions we define introduced in Eq 12 accord. In this way Eq 17 can be converted into, Equating Eqs 14 and 19 with 77 defined in Eq 16 we arrive at. Inspection of Eqs 15 and 20 shows that calculationrf the Ein. stein coefficient Ki requires knowledge of 1 in terms of J 1 For this. purpose we define n as the local slope of the non Newtonian flow curve. log T versus log p We assume that T i is a continuous function of p. which can be differentiated to first order This implies that the non. Newtonian behavior of the suspension can in a small range of strain. rates be represented with a power law behavior r kf where in. principle the exponent n may still be a function of 9 and of i This. linearization is allowed because in determining Kg we are only inter. ested in p Q which only small variations in mean strain rates occur. The Einstein coefficient of suspensions in generalized Newtonian liquids Jozua Laven and Hans N Stein Laboratory of Colloid Chemistry and Thermodynamics Technical University of Eindhoven P O Box 513 5600 MB Eindhoven The Netherlands

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