Korea-Australia Rheology Journal Vol. 23, No. 2, June 2011 pp. 105-111 DOI: 10.1007/s13367-011-0013-7
Relative viscosity of bimodal suspensions Fuzhong Qi and Roger I. Tanner* School of Aerospace, Mechanical and Mechatronic Engineering, University of Sydney, NSW 2006, Australia (Received February 13, 2011; final revision received April 11, 2011; accepted April 26, 2011)
Abstract A new differential (or multi-scale, mean field approach) model for the relative viscosity of bimodal suspensions is discussed in this paper. Solid spherical particles with a bimodal size distribution in a Newtonian solvent are considered. The problem of random close packing for a bidisperse system is studied. The bounds on volume fractions are given by 0.639 < ϕbm < 0.869 , where the random close packing volume fraction for a monodisperse system, ϕrcp = 0.639 , is assumed. We propose that the bimodal suspension has a dominant large particle composition and that the small particles fill the empty spaces between the large particles. The model can therefore be based on the theory of monodisperse suspensions. The predictions of the relative viscosity for several bimodal suspensions given by the model are compared to experimental measurements. A reasonably good agreement is observed. Keywords : relative viscosity, bimodal suspension, monomodal suspension, random close packing, volume fraction
1. Introduction Suspensions are involved in many industrial processes and our daily life, for example, in mine exploitation and transport, concrete casting and painting. It is thus important to study the rheological properties for suspensions and this problem has received attention for more than a century. Much research on suspensions has been done and many models have been given, but most of these works are on monodisperse suspensions. Here we ignore Brownian motion, so all particles are greater than about 1 micron in size, and the suspension is regarded as being non-colloidal. Actually, in practice, the solid phase of the suspension is a multimodal size distribution in most cases. The effect of the particle size distribution and particle size ratio on the rheological behaviour of multimodal suspensions is investigated here. To predict the relative viscosity of bimodal suspensions, the influence of the particle size distribution and particle size ratio has to be taken into account. For bimodal suspensions, there are some published papers investigating rheological behaviour by experimental tests and theories, for example reviews by Farris (1968), Chong et al. (1971), Poslinski et al. (1988), Shapiro et al. (1992), Chang et al. (1994), Barnes (2003) and Tanner et al. (2010). We also note the work of Pishvaei et al. (2006) which has a review of earlier work. However, the rheological behaviour of bimodal suspensions is not clearly *Corresponding author:
[email protected] © 2011 The Korean Society of Rheology and Springer
Korea-Australia Rheology Journal
understood. It is still a challenging problem, and to complete the theory, there is much more work to be done. To study the rheological properties of bimodal suspensions, a major factor involved is the random close packing of bimodal systems of particulates (with a maximum volume fraction ϕbm ). The relative viscosity will be expressed here in terms of the volume fraction ϕ and the parameter ϕbm . The value of the relative viscosity is very sensitive to the particle size distribution, especially at high volume fractions. Thus, it is essential to set up a model to estimate the value of the maximum volume fraction for bidisperse systems. Two theoretical models have been reported by Yerazunis et al. (1962 and 1965) and Ouchiyama et al. (1984), but both models are invalid in some cases since the value of the maximum packing fraction parameter tends to unity. Recently, Bournonville et al. (2004) developed the model given by de Larrard (1999). The model indeed gives a better estimation than that given by the models of Yerazunis et al. (1962 and 1965) and Ouchiyama et al. (1984). However, there is a singular point at the maximum value when ϕbm is expressed as a function of the fraction of small particles with a constant particle size ratio. Therefore, the aim of this work is to set up a simple model to predict the relative viscosity of bimodal suspensions. A new model to estimate the maximum volume fraction for bidisperse system is suggested, and a new differential-type model is used to compute the viscosity. Comparisons have been made between the calculated results of relative viscosity by using the model and experimental data.
Fuzhong Qi and Roger I. Tanner
ϕrcp = 0.639 into Eqn. (2), one finds
2. Random close packing model The random close packing fraction is an important parameter in describing the rheological and mechanical properties of suspensions. In the present work, we discuss suspensions which consist of rigid spherical particles with a Newtonian matrix. For a bidisperse system of spherical particles, two parameters are defined here; (a), particle size ratio, λ = Dl ⁄ Ds , where Dl is the diameter of the large particles and Ds is the diameter of the small particles, and (b), the volume fraction of the small particles is kϕ . Hence the volume fractions of the small particles ( ϕs ) and that of the large particles ( ϕl ) can be written as ϕs = kϕ and ϕl = ( 1 – k) ϕ
(1)
respectively, in which ϕ is the total volume fraction of the solid phase. These parameters will be used in later discussions. a) Bounds of random close packing for bimodal systems The random close packing fraction, namely the maximum volume fraction of particles, has been widely studied by previous researchers. Close packing of monodisperse systems of spheres can be theoretically described in many ways. As we know, the range of this packing fraction is from 0.524 (simple cubic lattice) to 0.74 (face centered cubic lattice). However, in random close packing for a monodisperse system the value is not so clear. Much work has been done by previous researchers, for example experimental tests (Scott 1960, Bernal and Mason 1960) and by numerical simulations (Jodrey et al. 1991 and Berryman 1983). A wide range of maximum volume fractions has been suggested and used in studies (Nielsen, 1977, Chong et al.1971, Kitano et al. 1980 and Maron and Kreiger 1960). Most of the numbers suggested are in the range from 0.55 to 0.68. Greenwood et al. (1997) suggested the random close packing fraction of a monodisperse system to be ϕrcp = 0.639 , which is estimated by an average of several publications (Lee 1970). This value will be used in this work. For bimodal suspensions, the maximum volume fraction ϕbm can be obtained if the small particles are randomly packed in between the randomly packed large particles. If the volume fraction of the large particles reaches the random close packing value ϕl = ϕrcp and the diameter of the small particle tends to infinitesimally small ( λ → ∞ ), the upper bound of maximum volume fraction ϕbm is determined and it can be written in terms of ϕrcp (Greenwood et al., 1997 and Yerazunis et al., 1962) ϕubm = ϕrcp + ϕrcp( 1 – ϕrcp)
(2)
Clearly, the upper bound depends on the random packing fraction of the monodisperse system. By substituting 106
ϕubm = 0.869
(3)
In this case, the fraction of the large particles is given by 0.639/0.869 (=1-k = 0.73) and so the fraction of the small particles is given by k = 0.27 (Greenwood et al., 1997). The lower bound is simply given by ϕlbm = ϕrcp = 0.639
(4)
Therefore, the random close packing fraction of the solid phase must satisfy 0.639 < ϕbm < 0.869 for bimodal suspensions ( λ > 1 and 0 < k < 1 ). b) Review For a bidisperse system of spherical particles, the random close packing fraction is expressed as a function of the particle size ratio λ and the fraction of the small particles k. To recall the history of the development and compare the models, we review some models in the following text. In the 1960s, research on the random close packing of a bidisperse system of spheres was made by Yerazunis et al. (1962 and 1965) following Scott’s (1960) experimental work. Chang et al. (1994) used this model in their research work. In this model, the random close packing fraction as a function of the size ratio λ and the fraction of small particle k is given by ϕrcp - (5) ϕbm = -------------------------------------------------------------------------------------------------------------------------–0.706 –4 2 1 – ( 1 – ϕrcp – 0.315 λ ) ( 1 – k ) + 0.955 λ ( 1 – k) ⁄ k
which, when the value of 0.639 is inserted, becomes 0.639
- (6) ϕbm = ----------------------------------------------------------------------------------------------------------------------–0.706 –4 2 )( 1 – k ) + 0.955 λ ( 1 – k ) ⁄ k 1 – ( 0.361 – 0.315λ
We now check if the result given by Eqn. (6) satisfies the bound conditions. First, if the size ratio λ is not a relatively large number, say < 4, and k → 0 , the maximum volume fraction ϕbm tends to zero (because of the 1 – k factor in the denominator), which is lower than the lower bound of ϕbm = 0.639 ; second, if the size ratio λ → ∞ ( Ds → 0 ), Eqn. (6) simplifies to 0.639 1 – 0361( 1 – k )
ϕbm = ----------------------------------
(7)
When k → 0 , Eqn. (7) gives a result of ϕbm → 1 . This value is higher than the upper bound of ϕbm = 0.869 . Fig. 1 shows the results of Eqn. (6). It is clear that the model is invalid at least at λ < ~4 , k → 0 and λ > ~10 , k→0 . Ouchiyama et al. (1984) investigated the porosity problem for random packing of a multimodal system of spherical particles by using statistical theory. The equation given in their work can be converted to calculate the maximum random packing fraction of the solid phase and has been used by Gupta et al. (1986) and Poslinski et al. (1988). The Korea-Australia Rheology Journal
Relative viscosity of bimodal suspensions
Fig. 1. Maximum volume fraction for bidisperse system as a function of fraction of small particle k for various particle size ratios. Solid lines: predictions given by Eqn. (6) (Yerazunis et al., 1962 and 1965).
Fig. 2. Maximum volume fraction for bidisperse system as a function of fraction of small particle k for various particle size ratios. Solid lines: predictions given by Eqn. (8) (Ouchiyama et al., 1984).
maximum random packing is given by (see below for the meaning of the notation “~”),
Bournonville et al. (2004) have slightly modified the model given by de Larrard (1999) to calculate the maximum volume fraction of bidisperse spherical systems. The modified model has been used in research work by Vu et al. (2010). In this model, two different configurations of the bidisperse system are recognized: dominant small particle configurations (large particles are embedded in a small particle matrix) and dominant large particle configurations (small particles were inserted in the empty space between the large particles). The volume fraction of dominant small particle ϕsm and dominant large particle ϕlm are given by
3
∑D n
i i ϕbm = ----------------------------------------------------------------------------------------------------------3 –1 3 3 ∑ ( Di ∼ Da ) ni + β ∑ [ ( Di + Da ) – ( Di ∼ Da ) ]ni
(8)
in which 3Da ⁄ 8 2 - n ∑ ( Di + Da ) 1 – -------------------( D i + Da ) i 4 β = 1 + ------ ( 8ϕrcp – 1 )Da ----------------------------------------------------------------3 3 13 ∑ [ Di – ( Di ∼ Da ) ]ni
(9)
and (10)
Da = ∑ Di ni
where Da is the average diameter of the different sizes of the spheres, Di is the diameter of the i-th component (in this work, it is Dl or Ds ), the notation Di ∼ Da is defined as a step function: ⎧0 ( D i ∼ Da ) = ⎨ ⎩ Di – D a
D i ≤ Da Di > D a
(11)
and ni is the number fraction of the i-th component. It is given by 3
ϕi ⁄ Di -3 ni = ----------------∑ ϕi ⁄ Di
(12)
in which ϕi denote the volume fraction of the i-th component. Results given by Eqn. (8) are shown in Fig. 2. It is clear that this model gives better results than that given by the Yerazunis et al. (1965) model. At least the results given by this model satisfy the lower bound condition (> 0.639). But when the size ratio λ > ~12 and the fraction of the small particles k < ~0.3 , this model is still over the upper bound (0.869) and tends to unity if the size ratio λ → ∞ ( Ds → 0 ). It is therefore invalid in this region. Korea-Australia Rheology Journal
ϕrcp ϕsm = -------------------------------------------------------------------------1 – (1 – k )[ 1 – ϕrcp + bsl ( ϕrcp – 1 ) ]
(13)
and ϕrcp l -, ϕm = ---------------------------
(14)
1 – k( 1 – als )
in which bsl and als are two functions describing the wall effect and loosening effect respectively. They only depend on the particle size ratio λ and were determined experimentally by Bournonville et al. (2004) 1 1.79 bsl = 1 – ⎛ 1 – ---⎞ ⎝ λ⎠
0.82
1 1.13 als = 1 – ⎛ 1 – ---⎞ ⎝ λ⎠
0.57
(15) (16)
Finally, the maximum volume fraction of the bidisperse system is defined as ϕbm = min( ϕlm, ϕsm)
(17)
The calculated results from Eqn. (13) to (17) are shown in Fig. 3. The predictions given by this model indeed sat-
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Fuzhong Qi and Roger I. Tanner
Fig. 3. Maximum volume fraction for bidisperse system as a function of fraction of small particles k for various particle size ratios. Solid lines: predictions given by Eqn. (17) (Bournonville et al. 2004).
Fig. 4. Maximum volume fraction for bidisperse system as a function of fraction of small particle k for various particle size ratios. Solid lines: predictions given by Eqn. (19) (21).
isfy the boundary conditions discussed above, but the model does not give a smooth curve. It has a discontinuous slope (due to the min( ϕlm, ϕsm ) condition (17)) at the maximum value when keeping the particle size ratio constant.
Predictions given by the model are shown in Fig. 4. These results satisfy the boundary conditions, 0.639 < ϕbm < 0.869 and give smooth curves.
c) A new model According to the discussion in Section a), we know that the maximum volume fraction of a bidisperse system has to satisfy ϕrcp < ϕbm < [ ϕrcp + ϕrcp( 1 – ϕrcp) ] .
(18)
Therefore, it might be written as ϕbm = ϕrcp + ξkξλ ϕrcp( 1 – ϕrcp )
(19)
where ξk is a function of the fraction of small particles k and ξλ is a function of the particle size ratio λ. The range of the function, ξk , is 0 ≤ ξk ≤ 1 . When k = 1 or k = 0, the bidisperse system reduces to a monodisperse system and the function ξk should be zero and ξk should be unity when k = 0.27. The range of the function, ξλ , is also 0 ≤ ξk ≤ 1 . It is clear that the function ξλ should be unity and zero when λ → ∞ (the size of the small particles tends to infinitesimally small) and λ = 1 respectively. These functions can be determined by fitting the experimental viscosity data of Chang et al. (1994). We find 1⁄3
ξλ = 1 – e
(λ – 1) – ---------------------6
(20)
and 1 2 ⎧ 1 – --------------( k – 0.27 ) k ≤ 0.27 ⎪ 0.0729 ξk = ⎨ 1 4 ⎪ 1 – -----------( k – 0.27 ) k > 0.27 ⎩ 0.284 108
(21)
3. Relative Viscosity For the relative viscosity of monodisperse spherical system, many models have been proposed by previous researchers. In the present work, the model given by Mendoza and Santamaria-Holek (2009 and 2010) is adopted. We assume that there is a unique, smooth relation between relative viscosity, particle size ratio and composition. Mendoza and Santamaria-Holek (2009) found that the relative viscosity is given by ϕ ηr = ⎛⎝ 1 – -------------⎞⎠ 1 – cϕ
5 – --2
(22)
where c is a constant defined as 1 – ϕrcp c = ---------------
(23)
ϕrcp
The ratio ϕ ⁄ ( 1 – cϕ ) is called the effective volume fraction in their work. To investigate the rheological properties of a bimodal suspension, a number of attempts can be seen in the literature (Farris 1968, Chong et al. 1971, Poslinski et al. 1988, Shapiro et al. 1992, Chang et al. 1994 and Tanner et al. 2010). The relative viscosity is defined as a product of two mean field functions: η ( ϕ l, ϕ s ) η ( ϕ l ) η ( ϕ l, ϕ s ) - = ------------- -------------------- = H( l ⁄ 0 )H( s ⁄ l) ηr = ------------------η0 η0 η( ϕl )
(24)
in which η( ϕl, ϕs ) is the viscosity of the bimodal suspension, η( ϕl) is the viscosity of a suspension consisting of Korea-Australia Rheology Journal
Relative viscosity of bimodal suspensions
large particles and solvent, η0 is the viscosity of the solvent. From Eqn. (24), we know that the relative viscosity of the bimodal suspension can be expressed as the product of the relative viscosity of two different unimodal suspensions, H( l ⁄ 0 ) and H( s ⁄ l ) . H( l ⁄ 0 ) is the relative viscosity of the unimodal suspension consisting of large particles and solvent. Hence, we have ϕl ⎞ H ( l ⁄ 0 ) = ⎛ 1 – --------------⎝ 1 – clϕl⎠
5 – --2
(25)
with 1 – ϕrcp -. cl = ---------------
(26)
ϕrcp
When ϕl → ϕrcp or ϕ → ϕbm , the relative viscosity ηr → ∞ as expected. The bimodal suspension reduces to an unimodal suspension if ϕs → ϕrcp . Clearly, Eqn. (29) is back to the well known Einstein equation at low volume fractions 5 2
ηr = 1 + --- ϕ + O( ϕ2 ) .
(30)
The results calculated by using Eqns. (19) and (29) are compared to experimental data given by Chang et al. (1994), Chong et al. (1971) and Poslinski et al. (1988) as shown in Fig. 5 and Fig. 6. The agreement is reasonably good. For comparison, we also use the model of the maximum volume fraction given by Bournonville et al. (2004), Eqn. (17), to fit these experimental data. The results are shown in
H ( s ⁄ l ) is the relative viscosity of a unimodal suspension consisted of small particles and matrix. It is given by
ϕs ⎞ H ( s ⁄ l ) = ⎛ 1 – ---------------⎝ 1 – cs ϕs⎠
5 – --2
(27)
Note, the matrix here is not a solvent in itself, instead it is the suspension consisting of large particles and solvent. The small particles are embedded in the suspension. Therefore, the random close packing fraction of the small particle is no longer ϕrcp = 0.639 in Eqn. (23). It should be ϕbm – ϕl due to the existence of the large particles. We find 1 – ( ϕbm – ϕl ) 1 – ϕbm + ϕl - = ------------------------- . cs = ---------------------------ϕbm – ϕl ϕbm – ϕl
(28)
By substituting Eqns. (25) and (27) into Eqn. (24), the relative viscosity of a bimodal suspension is given by ϕl ⎞ ⎛ ϕs ⎞ - 1 – ---------------ηr = ⎛⎝ 1 – --------------⎠ ⎝ 1 – cl ϕl 1 – csϕs⎠
5 – --2
(29)
Fig. 5. The calculated results compared to experimental data. Lines: calculated results by using Eqns. (19) and (29); Symbols: experimental measurements – circles and squares given by Chang et al. (1994) and triangles given by Chong et al. (1971). Korea-Australia Rheology Journal
Fig. 6. The calculated results compared to experimental data. Lines: calculated results by using Eqns. (19) and (29); Symbols: experimental measurements given by Poslinski et al. (1988).
Fig. 7. The calculated results compared to experimental data. Lines: calculated results by using Eqns.(17) and (29); Symbols: experimental measurements - circles and squares given by Chang et al. (1994) and triangles given by Chong et al. (1971).
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Fuzhong Qi and Roger I. Tanner Table 1. Comparisons of fluidity limits of bimodal suspensions, ϕbm , between Shapiro and Probstein’s results and Eqn. (19) Volume fraction of Small particle k
Fig. 8. The calculated results compared to experimental data. Lines: calculated results by using Eqns. (17) and (29); Symbols: experimental measurements given by Poslinski et al. (1988).
λ=2 Shapiro and Probstein’s results
0.25
λ=4
Eqn. (19)
Shapiro and Probstein’s results
Eqn. (19)
0.546
0.5580
0.588
0.5730
0.35
0.550
0.5580
0.597
0.5733
0.50
0.550
0.5579
0.597
0.5728
0.75
0.536
0.5512
0.558
0.5633
Table 1 shows the comparison between their results and the one calculated by using Eqn. (19). The agreement is quite good; the errors are under 4%.
4. Conclusion
Fig. 9. Calculated results compared to experimental data. Lines: calculated results by using Eqns. (19) and (29) with ϕrcp = 0.52 ; Symbols: experimental measurements given by Shapiro and Probstein (1992).
Fig. 7 and Fig. 8. It is clear that reasonably good agreement can be observed at lower volume fractions, but at high volume fractions, the experimental data cannot be fitted. Shapiro and Probstein (1992) studied the problem of the fluidity limits for monodisperse and bidisperse suspensions. In their work, the model given by Sengun and Probstein (1989) was employed to calculate the relative viscosity. The parameters in the model, the fluidity limit and a constant, were determined by best fitting of the experimental data. For the monodisperse suspension, they found ϕrcp = 0.524 . It is interesting that their experimental data can be also reasonably well fitted by using our model (Eqn. (29)) with ϕrcp = 0.52 even though this value is quite low. The results are shown in Fig. 9. Fluidity limits of several bidisperse suspensions were also found in their study. 110
The rheological properties of bimodal suspensions with Newtonian matrices have been discussed in this paper. The effects of the particle size ratio and the fraction of different size particles are shown. We focused our attention to find a simple theoretical model to predict the relative viscosity. In summary: • To estimate the maximum volume fraction of bidisperse spherical system, a new formulation (Eqns. (18) (21)), which is expressed in terms of the particle size ratio and the fraction of small particle, is proposed. • The bounds of the maximum volume fraction are pointed out. They are given by ϕlbm = ϕrcp , and ϕubm = ϕrcp + ϕrcp( 1 – ϕrcp) . When ϕrcp = 0.639 , one has 0.639 < ϕbm < 0.869 . • A new model for the relative viscosity has been developed. The model is based on two different monomodal suspensions and uses the concept of effective packing fraction Mendoza and Santamaria-Holek (2009 and 2010). The model is quite simple. There are many factors which we have ignored: possible shear thinning for example, ambiguity about the maximum volume fraction (Buscall et al., 1994). Comparisons have been made between the calculated results given by the model and experimental measurements of viscosity given by previous researchers with reasonable success. We believe that the model will be useful in practice. Acknowledgements We thank the Australian Research Council for support of this work via Grant DP110103414. Korea-Australia Rheology Journal
Relative viscosity of bimodal suspensions
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