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Minimising the viscosity of concentrated dispersions by using bimodal particle size distributions
Colloids and Surfaces A: Physicochemical and Engineering Aspects 144 (1998) 139–147
Minimising the viscosity of concentrated dispersions by using bimodal particle size distributions R. Greenwood a, P.F. Luckham b,*, T. Gregory c a Birchall Centre, Department of Chemistry, Keele University, Keele, Staffs. ST5 5BG, UK b Department of Chemical Engineering and Chemical Technology, Imperial College of Science, Technology and Medicine, Prince Consort Road, London, SW7 2BY, UK c 129 Orchard Road, Anerley, London, SE20 8DW, UK Received 2 November 1997; accepted 8 April 1998
Abstract The effect of composition (i.e. the volume of small particles compared to the volume of large particles) on the rheological properties of a bimodal dispersion has been studied in detail using two monodisperse polystyrene lattices. The diameter ratio of these lattices was 4.76. The relative high shear rate viscosity and dynamic viscosity of these bimodal dispersions were measured using the Bohlin VOR. Firstly, the rheology of the two monomodal dispersions was measured as a function of volume fraction, then 10 compositions ranging from 10% to 35% small particles by volume were prepared and again the rheology followed as a function of volume fraction. A minimum in the relative high shear rate limiting viscosity was found in the range 15–20% by volume of small particles and another minimum was also found at 20% by volume of small particles for the dynamic viscosity measurements. Thus, it can be concluded that reductions in viscosity can be achieved with a bimodal dispersion with the small particles occupying 20% of the total volume fraction. This is in agreement with other studies. However, if the adsorbed layer thickness is taken into account then the effective composition works out to be 27–36%; the lower limit of which is in excellent agreement with theory. The relative high shear rate viscosities of the bimodal dispersions were then compared to the theoretical viscosities calculated from an effective medium model. The results are in reasonable agreement with the measured data, but a rigorous analysis could not be performed due to the inability to calculate accurately the effective volume fraction, due to the presence of an adsorbed polymer layer on the particles. © 1998 Elsevier Science B.V. All rights reserved. Keywords: Bimodal dispersions; Polystyrene latex; Rheology; Viscosity minima
1. Introduction It is widely recognised that broadening the particle size distribution can increase the maximum packing fraction of a monodisperse system (i.e. all the same size). Thus, polydispersity can give a lower viscosity at the same volume fraction or * Corresponding author. Fax: +171 594 5604; e-mail: [email protected]
permit higher volume loading at the equivalent monodisperse viscosity. Despite this practical significance, until recently relatively few experiments and theoretical studies have been made. A systematic study of the effect of polydispersity can be achieved by investigating systems in which it is well-defined, i.e. bimodal dispersions. By introducing smaller particles, such that they could potentially fit in between the larger particles, it is possible to achieve much higher total volume fractions.
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Optimisation of the diameter ratio between these two sizes and the volume ratio (composition) leads to bimodal systems having greater volume fractions for the same viscosity as a monomodal system. This could produce a more environmentally sensitive product as there would be less polluting solvent. In other systems a lower viscosity would minimise pumping costs and much recent work in particle size distribution optimisation has been stimulated by research into high solids loading coal/water mixtures that can easily be pumped over long distances [1–3]. The earliest research into multimodal particle packing was carried out in the 1930s and then discontinued for a few decades. Two groups of researchers, White and Walton [4] and Westman and Hugill [5] calculated the optimum compositions for maximum packing of multimodal systems based on smaller particles fitting into the voids, using such diverse materials as lead shot, steel ball bearings, poppy seeds and sand. They suggested that 30% small particles by volume would be required for maximum packing fraction in a bimodal system. Further research [6,7] confirmed that the extent of the packing depends on the proportions of the two spheres, and is a maximum when the larger constituent is 73% of the total solids volume. But interest in the subject was not revived until 1968 when Farris [8] published optimum compositions for various maximum packing fractions in a bimodal system, see Table 1. With the advent of monodisperse polymer lattices other size scales and systems could be studied. Brodnyan [9] mixed two lattices, a factor of 11 apart in Table 1 Optimum compositions for maximum packing according to Farris Total volume (%)
Small (%)
Large (%)
70 72 74 76 78 80 82 84 86
35.5 34.5 33.5 33.0 32.0 31.0 30.0 28.5 27.5
64.5 65.5 66.5 67.0 68.0 69.0 70.0 71.5 72.5
diameter on an equal weight basis, and found that the viscosity of the mixture was lower than that of either component. Meanwhile Van Gilder [10] using optimised binary latex mixtures obtained improved coating properties complemented by higher solids loading. As well as the reductions in viscosity, other rheological improvements in bimodal systems have been observed. Wagstaff and Chaffey [11] reported that in a bimodal system the volume fraction at which shear thickening occurred increased. Similarly, Hoffman and coworkers [12,13] reported that dilatant behaviour could be avoided by broadening the size distribution of the particles. Sweeny and Geckler [14] carried out experiments with binary mixtures of glass spheres in an aqueous solution of zinc bromide and glycerol, and noted that the viscosity became less as the size ratio increased. Recently, Storms et al. [15] investigated the low shear rate viscosity of bimodal dispersions of PMMA in silicone fluids, by carefully choosing diameter ratios that would fill the interstices, and found similar results. Chong et al. [16 ] also noted the same trend in binary suspensions of glass beads, i.e. the viscosity decreases as the diameter ratio is increased at a fixed volume fraction and composition (25% small particles by volume). In a previous papers [17] of ours it has been shown that this picture is somewhat too simplistic. Bimodal polymer lattices, covering a wide range of diameter ratios, showed a series of peaks and troughs in the relative high shear rate viscosity against diameter ratio diagram. These can be explained by either packing arguments (if the particle is too large for the lattice hole one would not expect a reduction in viscosity), or a phase separation/superlattice formation argument. In this paper the effect of composition (i.e. number of smaller particles) at a fixed diameter ratio, 4.76 (a ratio where we observe a minimum in viscosity, and close to the size of the tetrahedral hole in a face centred cubic lattice, 4.44), is studied in an attempt to confirm the theoretical predictions and observed experimental values at which the maximum packing fraction and minimum viscosity occur. We note that this ratio is not too dissimilar to the ratio of 5 used in the diagram by Barnes et al. [18]. In their book if the viscosity of the
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bimodal suspension is plotted against the volume fraction of large particles as a function of total volume fraction then large reductions in viscosity can be seen when the fraction of large particles is above 0.60. This is called the Farris effect. Here many rheological experiments both in the steady shear and the oscillatory shear regimes have been performed, but only the relative high shear rate viscosity and dynamic viscosity are displayed for reasons of space. Similar trends in elastic modulus, dynamic viscosity, tan phase angle and yield values have also been found [19].
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2. Experimental
logical experiments. As theory and literature observations show that the position of maximum packing occurs at about 27% small particles by volume, a region around this value was selected for detailed study. A series of 10 samples was prepared from two polystyrene lattices (diameter 372 and 78 nm). The compositions were 10%, 15%, 20%, 22%, 24%, 26%, 28%, 30% and 35% by volume of small particles. The bimodals were all prepared at low volume fractions (approximately 0.15) to ensure good mixing and left for three to four days. The volume fraction was increased by placing the latex in dialysis tubing and applying a large hydrostatic pressure over a period of a few weeks.
2.1. Latex preparation
2.2. Rheological experiments
The large polystyrene particles were synthesised via a surfactant free dispersion polymerisation technique as described by Goodwin and coworkers [20,21]. The smaller sized polystyrene latex was made using a similar procedure, but with the addition of the surfactant sodium dodecylbenzenesulfonate (BDH Ltd, Upminster, Essex). The monomer was obtained from Aldrich Chemical Co. (Gillingham, Dorset) and was vacuum distilled before polymerisation to remove the inhibitor. Both the lattices were initiated using potassium persulfate (Aldrich). The lattices were filtered through glass wool and extensively dialysed against doubly distilled water for a period of four to five weeks with daily water changes. The latex particles were sized using a Malvern Zetasizer 3. Samples were diluted with 10−2 mol dm−3 sodium sulfate (Fisons, Loughborough, Leics.) until dilute enough to prevent multiple scattering. The stabilising polymer chosen for this study was Synperonic F127 (ICI Surfactants) which has a number average molecular weight of 9400, of which 8300 is PEO as reported previously [22]. This is an ABA block copolymer of ethyleneoxide (A) and propyleneoxide (B). This paper also showed that 1.60 mg/m2 is sufficient to stabilise the polystyrene particles. A final dialysis against 10−2 mol dm−3 sodium sulfate (Fisons), to screen out the effect of the particle charge, was carried out prior to the rheo-
The experiments were carried out using a Bohlin VOR rheometer (Metric Group, Cirencester, UK ) in one of two modes. Firstly the steady shear rate mode in which the outer cylinder is rotated at known shear rates c˙ . The resulting stress, s, is transmitted through the fluid to the inner cylinder. Hence, the viscosity (g) can be measured as a function of shear rate: g=
s c˙
(1)
One of the most common non-Newtonian behaviours is that of shear thinning, in which the viscosity decreases with increasing shear rate from a low shear rate limiting viscosity to a high shear rate limiting viscosity. It is this latter parameter that is recorded and plotted against volume fraction. The second mode of operation is oscillatory shear in which a sinusoidally varying strain of known amplitude (c ) is applied and the resulting o sinusoidal stress (amplitude s ) is measured. o Because of the viscoelasticity of the sample the stress is shifted out of phase by an angle h, but has the same frequency dependence. From these we can define the complex modulus G*: s G*= 0 c 0
(2)
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This complex modulus can be split into two parts, the elastic modulus G∞ and the viscous modulus G◊. These are related to the phase angle by: G∞=G* cos h
(3)
G◊=G* sin h
(4)
The complex modulus can then be expressed as: G*(v)=G∞(v)+iG◊(v)
(5)
where i is the square root of −1. It is important to note that the moduli are all frequency (v) dependent. For a Newtonian fluid h=p/2 and for a Hookean solid h=0, with viscoelastic materials having a phase angle between these two limits. The dynamic viscosity g∞ is then defined by: g∞=
G◊ v
(6)
These three moduli and the dynamic viscosity must be measured in the linear viscoelastic region (where the rheological parameters are independent of the amplitude). This can be located by a strain sweep experiment, where an oscillation experiment is performed as a function of strain amplitude (at a fixed frequency, typically 1 Hz) and the linear region established. Once this has been defined, measurements are carried out as a function of frequency (10−2–10 Hz) within it. By increasing the volume fraction (w) it is possible to follow changes from fluid behaviour to elastic behaviour.
3. Results and discussion 3.1. Maximum packing fractions and minimum viscosities A typical example of the relative high shear rate viscosity against volume fraction is displayed (for 15% small particles by volume) in Fig. 1. The data have been fitted with an exponential curve. Best curve fits are now used to compare the relative high shear rate viscosities plotted against volume fraction ( Fig. 2). Three curves can be identified that have a viscosity lower than the large polystyrene latex (shown as a solid curve): i.e. 10%, 15% and 20% small particles. In order to illustrate this
Fig. 1. Relative high shear rate viscosity for a bimodal suspension containing 15% small particles by volume.
effect more clearly, cuts have then been taken through this graph at several different volume fractions (0.45–0.65, in steps of 0.05) and the viscosity plotted against composition of small particles (Fig. 3). The viscosity data for 100% small particles (the 78 nm latex) has been omitted as it is two or three orders of magnitude higher; this is because the small particles have a much higher effective volume fraction than the large particles due to the adsorbed polymer layer. At medium volume fractions (0.45–0.50) the composition makes very little difference, but at higher volume fractions a pronounced minimum viscosity can be identified at a composition of 0.15–0.20 small particles by volume. Chang and Powell [23] from computer simula-
R. Greenwood et al. / Colloids Surfaces A: Physicochem. Eng. Aspects 144 (1998) 139–147
Fig. 2. Effect of composition of the bimodal suspensions on the relative high shear rate viscosity against volume fraction.
tions also noted that below a volume fraction of 0.4, the particle size distribution did not affect the viscosity, whereas above this bimodal suspensions had a lower viscosity than monomodals. This agreed with their rheological experiments on bimodal polystyrene lattices, i.e. a minimum viscosity occurring at 25% small particles and the relative viscosity decreasing with increasing size ratio. A typical example of the variation of dynamic viscosity (at 1 Hz) with volume fraction is shown in Fig. 4. Once again best fit curves are used to generate a similar curve to Fig. 3, which is plotted in Fig. 5. Some points have been omitted, especially the 100% small particles (78 nm), again as this is two or three orders of magnitude greater, but once again a definite minimum can be identified
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Fig. 3. Effect of total volume fraction on the relative high shear rate viscosity of the bimodal suspensions against composition of small particles.
at a composition of 0.20 small particles by volume. Again the composition has very little effect on the dynamic viscosity at low volume fractions, less than 0.40. Once again the lines are shown as a guide. This agrees very well with the previous result of 0.15–0.20 small particles from the high shear rate viscosity. Thus, a composition can be identified, namely 0.15–0.20 small particles by volume, which gives a minimum viscosity at a total volume fraction of 0.65. The question is why? Monodisperse systems of hard spheres can theoretically pack in several ways ranging from simple cubic (maximum packing fraction, w , 0.524) to max face centre cubic and hexagonal close packed (w , 0.740) to name just a few. max However, if steel balls are poured into a cylinder and gently shaken, Scott [24] observed that they
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Fig. 4. Dynamic viscosity of a bimodal suspension at 35% small particles by volume against total volume fraction.
pack to a well-defined limit called random close packing (r.c.p.): w =0.637. Similarly, Bernal and rcp Mason [25], by pouring paint over a random array of ball bearings and examining them for ‘‘dots’’ or ‘‘rings’’, obtained a value of 0.62 for random close packing. Computer simulations [26,27] reveal similar values. But here we shall take the estimate of Lee [28] of 0.639, which is an average of several publications over the last few decades. This means that the maximum volume fraction achieved by monodisperse latex is limited to that of random close packing. If smaller particles are added such that they randomly pack in between the randomly packed larger particles then higher volume fractions can be obtained. For bimodals this is given by the following: w
max
=w
rcp
+(1−w )w rcp rcp
(7)
Fig. 5. Effect of total volume fraction on the dynamic viscosity of the bimodal suspensions against composition.
which when the value of 0.639 is inserted yields: =0.639+(1−0.639)0.639=0.869 (8) max The theoretical volume ratio that the large particles should occupy to achieve this is given by the ratio of w to w , i.e. rcp max 0.639 =0.734 (9) 0.869 w
This is assuming an infinite diameter ratio. From simple geometry, if the ratio of the diameter of large spheres to the diameter of small spheres is too small, then the smaller spheres are too large to fit in the interstices created by the larger spheres. If the large particles form a random array then the limiting factor to allow the smaller particles
R. Greenwood et al. / Colloids Surfaces A: Physicochem. Eng. Aspects 144 (1998) 139–147
into the interstices is the triangular pore size, 6.46. Once above this critical ratio the small spheres are free to move through the labyrinth created by the larger spheres. This is the basis of McGeary’s [29] theoretical prediction of at least a seven-fold difference between successive particle sizes to allow movement of particles through the triangular pore. The composition of 0.15–0.20 small particles by volume is lower than the Farris prediction [8] for maximum packing, minimum viscosity. This discrepancy could be due to the inability to calculate the effective particle size and hence volume fraction. As the total volume fraction of the bimodal dispersion increases, the polymer layers begin to either compress or interpenetrate one another. As the exact structure of the dispersion is not known, it is difficult to accurately estimate the contribution the polymer layers make to the overall total volume fraction. The current results do agree with other experiments in the literature. Johnson and Kelsey [30] studied mixtures of styrene–butadiene lattices and reported a viscosity minimum with large particles consisting of 75% by volume. Similarly, Maron and Madow [31] reported maximum packing when the volume of large spheres is at 76–77%. Recently, Okubo [32] worked on binary mixtures of polystyrene and silica and reported a minimum viscosity with 25% small particles. Previous work in this laboratory by Kim and Luckham [33] obtained maximum packing, minimum viscosity, minimum elastic modulus and minimum osmotic pressure when the large particles were 80% of the total solids volume for a size ratio of seven. The problem is that the particles are coated with a thin polymer layer due to the adsorbed F127. Earlier studies of ours [34] have shown that the polymer layer thickness is dependent on particle size, a, and the volume fraction, w. If we recalculate the volume ratio based on the adsorbed layer thickness, d, whilst in dilute suspension, i.e. 20 nm on the large particles and 11 nm on the smaller particles using the following equation:
A B
d 3 w =w 1+ eff a then the effective volume ratio w
(10)
eff
works out as
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0.27–0.36. The lower limit of which is in excellent agreement with theory. 3.2. Predicting bimodal viscosities from the twomonomodal viscosities The form of the relative viscosity–volume fraction curve for monomodal spheres has been extensively studied over the years and numerous equations exist in the literature, the best known being the Krieger–Dougherty [35] equation:
A
B
w −[g]wm (11) g = 1− r w m where the relative viscosity (g ) is the viscosity of r the dispersion (g) compared to the viscosity of the suspending medium (g ). The intrinsic viscosity [g] s is taken to be 2.71 and w is taken to be 0.71. m These are the values reported by de Kruif et al. [36 ] for the limiting high shear rate viscosity. Fiderlis and Whitmore [37] and Eveson [38] have shown that the drag experienced by a large sphere moving in a suspension of small spheres is the same as a large sphere moving through a pure Newtonian liquid having the same viscosity and density as the suspension. This is an effective medium theory and basically assumes that the smaller particles act as a medium towards the larger particles. Using this concept, Farris was able to predict the viscosities of multimodal suspensions from monomodal data, i.e. the viscosity of the mixture of smaller fractions is the medium viscosity for the next largest fraction. This was assuming at least an order of magnitude difference in size between successive fractions: g (a )=g ×g ×g ×…×g (12) r r1 r2 r3 ri i So, an aqueous bimodal dispersion would have the relative viscosity: g g (13) = small × large g g∞ water small where g∞ is the viscosity of water thickened up small by the small component. So using the idea that, for a bimodal suspension, the mixture of small particles and suspending g
rdisp
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liquid is treated as one homogeneous fluid, if the volume of the suspending fluid is V and the liq volume of the small particles is V and that of small the large particles is V then two volume fraclarge tions can be defined [39]: w
large and
=v
large
(14)
v
large (15) 1−v small where v and v are the true volume fractions small large given by:
w
small
=
V large (16) +V +V large small liq V small v = (17) small V +V +V large small liq Using Eqs. (16) and (17) to generate the effective volume fractions, which are then fed into the Krieger–Dougherty equation, and using w as max 0.71, a theoretical curve can be calculated and compared with the experimental results. In Fig. 6 the relative high shear rate viscosity (at 20% small particles) is compared with the theoretical curve suggested by Farris. The correct trends are observed, but the data points lie above the curve as the adsorbed layers increase the effective volume fraction of the particles present and hence increase the viscosity. As mentioned previously it is not possible to accurately estimate this contribution to the volume fraction. At low volume fractions it is straightforward to estimate the effective volume fraction based on the thickness of the adsorbed layers, but as the volume fraction increases it is extremely difficult to calculate exactly at what rate each of the adsorbed layers is being deformed. Assuming the adsorbed layer thickness compresses at the same rate as in the monomodal systems is unrealistic, as both the large and the small particles will compress the polymer layers. Our data is the first attempt to fit the rheology of bimodal colloidal dispersions to that of the Farris theory. In the past the rheology of larger particles has been found to fit the Farris model well, for example Parkinson et al. [40] using v = large V
Fig. 6. Comparison of the high shear rate viscosity to the Farris theory at 22% small particles by volume.
PMMA spheres in Nujol fitted his data for bimodals, trimodals and tetramodals to the Farris theory and reported a viscosity minimum for bimodals at 25% small particles. More recently Gondret and Petit [41] have conducted similar experiments to those reported here for large noncolloidal spheres; broadly similar trends to those reported here were observed.
4. Conclusions At a diameter ratio of 4.76, a minimum viscosity can be achieved in a bimodal suspension with a composition of 0.15–0.20 small particles by volume. However, if the adsorbed layer thickness is taken into account then this recalculates as
R. Greenwood et al. / Colloids Surfaces A: Physicochem. Eng. Aspects 144 (1998) 139–147
0.27–0.36; the lower limit of which is in excellent agreement with the theory. The Farris theory seems to underestimate the viscosity due to an inability to calculate correctly the effective volume fraction. This of course can be minimised by the use of larger particles, but to study a wide range of diameter ratios would involve some particles that would have an appreciable settling rate.
Acknowledgment This research was sponsored by the EPSRC CASE studentship in conjunction with CoatesLorilleux, St Mary Cray, Orpington, Kent BR5 3PP.
References [1] R.F. Probstein, M.Z. Sengun, Proc. 7th Int. Symp on Coal Slurry Fuels Preparation and Utilisation, New Orleans, 21–24 May 1985, p. 115. [2] R.F. Probstein, M.Z. Sengun, Rheol. Acta 28 (1989) 382. [3] R.F. Probstein, M.Z. Sengun, Rheol. Acta 28 (1989) 394. [4] H.E. White, S.F. Walton, J. Am. Ceram. Soc. 20 (1937) 155. [5] A.E.R. Westman, H.R. Hugill, J. Am. Ceram. Soc. 13 (1930) 767. [6 ] C.C. Furnas, Ind. Eng. Chem. 23 (1931) 1052. [7] R. Fraser, J. Geol. 43 (1935) 910. [8] R.J. Farris, Trans. Soc. Rheol. 12 (1968) 281. [9] J.G. Brodnyan, Trans. Soc. Rheol. 12 (1968) 357. [10] R. Van Gilder, Paper and Timber 5 (1989) 537. [11] I. Wagstaff, C.E. Chaffey, J. Colloid Interface Sci. 59 (1977) 53. [12] R.L. Hoffman, Adv. Colloid Interface Sci. 17 (1982) 161.
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[13] E.A. Collins, D.J. Hoffmann, P.L. Soni, J. Colloid Interface Sci. 71 (1979) 21. [14] K.H. Sweeny, R.D. Geckler, J. Appl. Phys. 23 (1954) 1135. [15] R.F. Storms, B.V. Ramarao, R.H. Weiland, Powder Technol. 63 (1990) 247. [16 ] J.S. Chong, E.B. Christiansen, A.D. Baer, J. Appl. Polym. Sci. 15 (1971) 2007. [17] R. Greenwood, P.F. Luckham, T. Gregory, J. Colloid Interface Sci. 191 (1997) 11. [18] H.A. Barnes, J.F. Hutton, K. Walters, Introduction to Rheology, Elsevier, Amsterdam, 1993. [19] R. Greenwood, Ph.D. Thesis, Imperial College, University of London, 1995. [20] J.W. Goodwin, J. Hearn, C.C. Ho, R.H. Ottewill, Colloid Polym. Sci. 252 (1974) 464. [21] J.W. Goodwin, T. Gregory, J.A. Stile, Adv. Colloid Interface Sci. 17 (1982) 185. [22] I.T. Kim, P.F. Luckham, J. Colloid Interface Sci. 144 (1991) 174. [23] C. Chang, R.L. Powell, J. Fluid Mech. 253 (1993) 1. [24] G.D. Scott, Nature 188 (1960) 908. [25] J.D. Bernal, J. Mason, Nature 188 (1960) 190. [26 ] W.S. Jodrey, E.M. Tory, Phys. Rev. A 32 (1991) 2347. [27] J.G. Berryman, Phys. Rev. A 27 (1983) 1053. [28] D.I. Lee, J. Paint. Technol. 42 (1970) 579. [29] R.K. McGeary, J. Am. Ceram. Soc. 44 (1961) 513. [30] P. Johnson, R. Kelsey, Rubber World 138 (1958) 877. [31] S.H. Maron, B.P. Madow, J. Colloid Sci. 8 (1953) 300. [32] T. Okubo, J. Phys. Chem. 94 (1962) 1990. [33] I.T. Kim, P.F. Luckham, Powder Technol. 77 (1993) 31. [34] R. Greenwood, P.F. Luckham, T. Gregory, Colloids Surf. A 98 (1995) 117. [35] I.M. Krieger, T.J. Dougherty, Trans. Soc. Rheol. 3 (1959) 137. [36 ] C.G. de Kruif, E.M.F. van Iersel, A. Vrij, W.B. Russel, J. Chem. Phys. 83 (1985) 4717. [37] V. Fiderlis, R.L. Whitmore, Rheol. Acta 1 (1961) 573. [38] G.F. Eveson, in: C.C. Mill ( Ed.), Rheology of Disperse Systems, Pergamon Press, Oxford, 1959. [39] D.C.H. Cheng, A.P. Kruszewski, J.R. Senior, T.A. Roberts, J. Mater. Sci. 25 (1990) 353. [40] C. Parkinson, S. Matsumoto, P. Sherman, J. Colloid Interface Sci. 33 (1970) 150. [41] P. Gondret, L. Petit, J. Rheol. 41 (1997) 1261.
Minimising the viscosity of concentrated dispersions by using bimodal particle size distributions R. Greenwood a, P.F. Luckham b,*, T. Gregory c a Birchall Centre, Department of Chemistry, Keele University, Keele, Staffs. ST5 5BG, UK b Department of Chemical Engineering and Chemical Technology, Imperial College of Science, Technology and Medicine, Prince Consort Road, London, SW7 2BY, UK c 129 Orchard Road, Anerley, London, SE20 8DW, UK Received 2 November 1997; accepted 8 April 1998
Abstract The effect of composition (i.e. the volume of small particles compared to the volume of large particles) on the rheological properties of a bimodal dispersion has been studied in detail using two monodisperse polystyrene lattices. The diameter ratio of these lattices was 4.76. The relative high shear rate viscosity and dynamic viscosity of these bimodal dispersions were measured using the Bohlin VOR. Firstly, the rheology of the two monomodal dispersions was measured as a function of volume fraction, then 10 compositions ranging from 10% to 35% small particles by volume were prepared and again the rheology followed as a function of volume fraction. A minimum in the relative high shear rate limiting viscosity was found in the range 15–20% by volume of small particles and another minimum was also found at 20% by volume of small particles for the dynamic viscosity measurements. Thus, it can be concluded that reductions in viscosity can be achieved with a bimodal dispersion with the small particles occupying 20% of the total volume fraction. This is in agreement with other studies. However, if the adsorbed layer thickness is taken into account then the effective composition works out to be 27–36%; the lower limit of which is in excellent agreement with theory. The relative high shear rate viscosities of the bimodal dispersions were then compared to the theoretical viscosities calculated from an effective medium model. The results are in reasonable agreement with the measured data, but a rigorous analysis could not be performed due to the inability to calculate accurately the effective volume fraction, due to the presence of an adsorbed polymer layer on the particles. © 1998 Elsevier Science B.V. All rights reserved. Keywords: Bimodal dispersions; Polystyrene latex; Rheology; Viscosity minima
1. Introduction It is widely recognised that broadening the particle size distribution can increase the maximum packing fraction of a monodisperse system (i.e. all the same size). Thus, polydispersity can give a lower viscosity at the same volume fraction or * Corresponding author. Fax: +171 594 5604; e-mail: [email protected]
permit higher volume loading at the equivalent monodisperse viscosity. Despite this practical significance, until recently relatively few experiments and theoretical studies have been made. A systematic study of the effect of polydispersity can be achieved by investigating systems in which it is well-defined, i.e. bimodal dispersions. By introducing smaller particles, such that they could potentially fit in between the larger particles, it is possible to achieve much higher total volume fractions.
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R. Greenwood et al. / Colloids Surfaces A: Physicochem. Eng. Aspects 144 (1998) 139–147
Optimisation of the diameter ratio between these two sizes and the volume ratio (composition) leads to bimodal systems having greater volume fractions for the same viscosity as a monomodal system. This could produce a more environmentally sensitive product as there would be less polluting solvent. In other systems a lower viscosity would minimise pumping costs and much recent work in particle size distribution optimisation has been stimulated by research into high solids loading coal/water mixtures that can easily be pumped over long distances [1–3]. The earliest research into multimodal particle packing was carried out in the 1930s and then discontinued for a few decades. Two groups of researchers, White and Walton [4] and Westman and Hugill [5] calculated the optimum compositions for maximum packing of multimodal systems based on smaller particles fitting into the voids, using such diverse materials as lead shot, steel ball bearings, poppy seeds and sand. They suggested that 30% small particles by volume would be required for maximum packing fraction in a bimodal system. Further research [6,7] confirmed that the extent of the packing depends on the proportions of the two spheres, and is a maximum when the larger constituent is 73% of the total solids volume. But interest in the subject was not revived until 1968 when Farris [8] published optimum compositions for various maximum packing fractions in a bimodal system, see Table 1. With the advent of monodisperse polymer lattices other size scales and systems could be studied. Brodnyan [9] mixed two lattices, a factor of 11 apart in Table 1 Optimum compositions for maximum packing according to Farris Total volume (%)
Small (%)
Large (%)
70 72 74 76 78 80 82 84 86
35.5 34.5 33.5 33.0 32.0 31.0 30.0 28.5 27.5
64.5 65.5 66.5 67.0 68.0 69.0 70.0 71.5 72.5
diameter on an equal weight basis, and found that the viscosity of the mixture was lower than that of either component. Meanwhile Van Gilder [10] using optimised binary latex mixtures obtained improved coating properties complemented by higher solids loading. As well as the reductions in viscosity, other rheological improvements in bimodal systems have been observed. Wagstaff and Chaffey [11] reported that in a bimodal system the volume fraction at which shear thickening occurred increased. Similarly, Hoffman and coworkers [12,13] reported that dilatant behaviour could be avoided by broadening the size distribution of the particles. Sweeny and Geckler [14] carried out experiments with binary mixtures of glass spheres in an aqueous solution of zinc bromide and glycerol, and noted that the viscosity became less as the size ratio increased. Recently, Storms et al. [15] investigated the low shear rate viscosity of bimodal dispersions of PMMA in silicone fluids, by carefully choosing diameter ratios that would fill the interstices, and found similar results. Chong et al. [16 ] also noted the same trend in binary suspensions of glass beads, i.e. the viscosity decreases as the diameter ratio is increased at a fixed volume fraction and composition (25% small particles by volume). In a previous papers [17] of ours it has been shown that this picture is somewhat too simplistic. Bimodal polymer lattices, covering a wide range of diameter ratios, showed a series of peaks and troughs in the relative high shear rate viscosity against diameter ratio diagram. These can be explained by either packing arguments (if the particle is too large for the lattice hole one would not expect a reduction in viscosity), or a phase separation/superlattice formation argument. In this paper the effect of composition (i.e. number of smaller particles) at a fixed diameter ratio, 4.76 (a ratio where we observe a minimum in viscosity, and close to the size of the tetrahedral hole in a face centred cubic lattice, 4.44), is studied in an attempt to confirm the theoretical predictions and observed experimental values at which the maximum packing fraction and minimum viscosity occur. We note that this ratio is not too dissimilar to the ratio of 5 used in the diagram by Barnes et al. [18]. In their book if the viscosity of the
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bimodal suspension is plotted against the volume fraction of large particles as a function of total volume fraction then large reductions in viscosity can be seen when the fraction of large particles is above 0.60. This is called the Farris effect. Here many rheological experiments both in the steady shear and the oscillatory shear regimes have been performed, but only the relative high shear rate viscosity and dynamic viscosity are displayed for reasons of space. Similar trends in elastic modulus, dynamic viscosity, tan phase angle and yield values have also been found [19].
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2. Experimental
logical experiments. As theory and literature observations show that the position of maximum packing occurs at about 27% small particles by volume, a region around this value was selected for detailed study. A series of 10 samples was prepared from two polystyrene lattices (diameter 372 and 78 nm). The compositions were 10%, 15%, 20%, 22%, 24%, 26%, 28%, 30% and 35% by volume of small particles. The bimodals were all prepared at low volume fractions (approximately 0.15) to ensure good mixing and left for three to four days. The volume fraction was increased by placing the latex in dialysis tubing and applying a large hydrostatic pressure over a period of a few weeks.
2.1. Latex preparation
2.2. Rheological experiments
The large polystyrene particles were synthesised via a surfactant free dispersion polymerisation technique as described by Goodwin and coworkers [20,21]. The smaller sized polystyrene latex was made using a similar procedure, but with the addition of the surfactant sodium dodecylbenzenesulfonate (BDH Ltd, Upminster, Essex). The monomer was obtained from Aldrich Chemical Co. (Gillingham, Dorset) and was vacuum distilled before polymerisation to remove the inhibitor. Both the lattices were initiated using potassium persulfate (Aldrich). The lattices were filtered through glass wool and extensively dialysed against doubly distilled water for a period of four to five weeks with daily water changes. The latex particles were sized using a Malvern Zetasizer 3. Samples were diluted with 10−2 mol dm−3 sodium sulfate (Fisons, Loughborough, Leics.) until dilute enough to prevent multiple scattering. The stabilising polymer chosen for this study was Synperonic F127 (ICI Surfactants) which has a number average molecular weight of 9400, of which 8300 is PEO as reported previously [22]. This is an ABA block copolymer of ethyleneoxide (A) and propyleneoxide (B). This paper also showed that 1.60 mg/m2 is sufficient to stabilise the polystyrene particles. A final dialysis against 10−2 mol dm−3 sodium sulfate (Fisons), to screen out the effect of the particle charge, was carried out prior to the rheo-
The experiments were carried out using a Bohlin VOR rheometer (Metric Group, Cirencester, UK ) in one of two modes. Firstly the steady shear rate mode in which the outer cylinder is rotated at known shear rates c˙ . The resulting stress, s, is transmitted through the fluid to the inner cylinder. Hence, the viscosity (g) can be measured as a function of shear rate: g=
s c˙
(1)
One of the most common non-Newtonian behaviours is that of shear thinning, in which the viscosity decreases with increasing shear rate from a low shear rate limiting viscosity to a high shear rate limiting viscosity. It is this latter parameter that is recorded and plotted against volume fraction. The second mode of operation is oscillatory shear in which a sinusoidally varying strain of known amplitude (c ) is applied and the resulting o sinusoidal stress (amplitude s ) is measured. o Because of the viscoelasticity of the sample the stress is shifted out of phase by an angle h, but has the same frequency dependence. From these we can define the complex modulus G*: s G*= 0 c 0
(2)
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This complex modulus can be split into two parts, the elastic modulus G∞ and the viscous modulus G◊. These are related to the phase angle by: G∞=G* cos h
(3)
G◊=G* sin h
(4)
The complex modulus can then be expressed as: G*(v)=G∞(v)+iG◊(v)
(5)
where i is the square root of −1. It is important to note that the moduli are all frequency (v) dependent. For a Newtonian fluid h=p/2 and for a Hookean solid h=0, with viscoelastic materials having a phase angle between these two limits. The dynamic viscosity g∞ is then defined by: g∞=
G◊ v
(6)
These three moduli and the dynamic viscosity must be measured in the linear viscoelastic region (where the rheological parameters are independent of the amplitude). This can be located by a strain sweep experiment, where an oscillation experiment is performed as a function of strain amplitude (at a fixed frequency, typically 1 Hz) and the linear region established. Once this has been defined, measurements are carried out as a function of frequency (10−2–10 Hz) within it. By increasing the volume fraction (w) it is possible to follow changes from fluid behaviour to elastic behaviour.
3. Results and discussion 3.1. Maximum packing fractions and minimum viscosities A typical example of the relative high shear rate viscosity against volume fraction is displayed (for 15% small particles by volume) in Fig. 1. The data have been fitted with an exponential curve. Best curve fits are now used to compare the relative high shear rate viscosities plotted against volume fraction ( Fig. 2). Three curves can be identified that have a viscosity lower than the large polystyrene latex (shown as a solid curve): i.e. 10%, 15% and 20% small particles. In order to illustrate this
Fig. 1. Relative high shear rate viscosity for a bimodal suspension containing 15% small particles by volume.
effect more clearly, cuts have then been taken through this graph at several different volume fractions (0.45–0.65, in steps of 0.05) and the viscosity plotted against composition of small particles (Fig. 3). The viscosity data for 100% small particles (the 78 nm latex) has been omitted as it is two or three orders of magnitude higher; this is because the small particles have a much higher effective volume fraction than the large particles due to the adsorbed polymer layer. At medium volume fractions (0.45–0.50) the composition makes very little difference, but at higher volume fractions a pronounced minimum viscosity can be identified at a composition of 0.15–0.20 small particles by volume. Chang and Powell [23] from computer simula-
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Fig. 2. Effect of composition of the bimodal suspensions on the relative high shear rate viscosity against volume fraction.
tions also noted that below a volume fraction of 0.4, the particle size distribution did not affect the viscosity, whereas above this bimodal suspensions had a lower viscosity than monomodals. This agreed with their rheological experiments on bimodal polystyrene lattices, i.e. a minimum viscosity occurring at 25% small particles and the relative viscosity decreasing with increasing size ratio. A typical example of the variation of dynamic viscosity (at 1 Hz) with volume fraction is shown in Fig. 4. Once again best fit curves are used to generate a similar curve to Fig. 3, which is plotted in Fig. 5. Some points have been omitted, especially the 100% small particles (78 nm), again as this is two or three orders of magnitude greater, but once again a definite minimum can be identified
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Fig. 3. Effect of total volume fraction on the relative high shear rate viscosity of the bimodal suspensions against composition of small particles.
at a composition of 0.20 small particles by volume. Again the composition has very little effect on the dynamic viscosity at low volume fractions, less than 0.40. Once again the lines are shown as a guide. This agrees very well with the previous result of 0.15–0.20 small particles from the high shear rate viscosity. Thus, a composition can be identified, namely 0.15–0.20 small particles by volume, which gives a minimum viscosity at a total volume fraction of 0.65. The question is why? Monodisperse systems of hard spheres can theoretically pack in several ways ranging from simple cubic (maximum packing fraction, w , 0.524) to max face centre cubic and hexagonal close packed (w , 0.740) to name just a few. max However, if steel balls are poured into a cylinder and gently shaken, Scott [24] observed that they
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Fig. 4. Dynamic viscosity of a bimodal suspension at 35% small particles by volume against total volume fraction.
pack to a well-defined limit called random close packing (r.c.p.): w =0.637. Similarly, Bernal and rcp Mason [25], by pouring paint over a random array of ball bearings and examining them for ‘‘dots’’ or ‘‘rings’’, obtained a value of 0.62 for random close packing. Computer simulations [26,27] reveal similar values. But here we shall take the estimate of Lee [28] of 0.639, which is an average of several publications over the last few decades. This means that the maximum volume fraction achieved by monodisperse latex is limited to that of random close packing. If smaller particles are added such that they randomly pack in between the randomly packed larger particles then higher volume fractions can be obtained. For bimodals this is given by the following: w
max
=w
rcp
+(1−w )w rcp rcp
(7)
Fig. 5. Effect of total volume fraction on the dynamic viscosity of the bimodal suspensions against composition.
which when the value of 0.639 is inserted yields: =0.639+(1−0.639)0.639=0.869 (8) max The theoretical volume ratio that the large particles should occupy to achieve this is given by the ratio of w to w , i.e. rcp max 0.639 =0.734 (9) 0.869 w
This is assuming an infinite diameter ratio. From simple geometry, if the ratio of the diameter of large spheres to the diameter of small spheres is too small, then the smaller spheres are too large to fit in the interstices created by the larger spheres. If the large particles form a random array then the limiting factor to allow the smaller particles
R. Greenwood et al. / Colloids Surfaces A: Physicochem. Eng. Aspects 144 (1998) 139–147
into the interstices is the triangular pore size, 6.46. Once above this critical ratio the small spheres are free to move through the labyrinth created by the larger spheres. This is the basis of McGeary’s [29] theoretical prediction of at least a seven-fold difference between successive particle sizes to allow movement of particles through the triangular pore. The composition of 0.15–0.20 small particles by volume is lower than the Farris prediction [8] for maximum packing, minimum viscosity. This discrepancy could be due to the inability to calculate the effective particle size and hence volume fraction. As the total volume fraction of the bimodal dispersion increases, the polymer layers begin to either compress or interpenetrate one another. As the exact structure of the dispersion is not known, it is difficult to accurately estimate the contribution the polymer layers make to the overall total volume fraction. The current results do agree with other experiments in the literature. Johnson and Kelsey [30] studied mixtures of styrene–butadiene lattices and reported a viscosity minimum with large particles consisting of 75% by volume. Similarly, Maron and Madow [31] reported maximum packing when the volume of large spheres is at 76–77%. Recently, Okubo [32] worked on binary mixtures of polystyrene and silica and reported a minimum viscosity with 25% small particles. Previous work in this laboratory by Kim and Luckham [33] obtained maximum packing, minimum viscosity, minimum elastic modulus and minimum osmotic pressure when the large particles were 80% of the total solids volume for a size ratio of seven. The problem is that the particles are coated with a thin polymer layer due to the adsorbed F127. Earlier studies of ours [34] have shown that the polymer layer thickness is dependent on particle size, a, and the volume fraction, w. If we recalculate the volume ratio based on the adsorbed layer thickness, d, whilst in dilute suspension, i.e. 20 nm on the large particles and 11 nm on the smaller particles using the following equation:
A B
d 3 w =w 1+ eff a then the effective volume ratio w
(10)
eff
works out as
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0.27–0.36. The lower limit of which is in excellent agreement with theory. 3.2. Predicting bimodal viscosities from the twomonomodal viscosities The form of the relative viscosity–volume fraction curve for monomodal spheres has been extensively studied over the years and numerous equations exist in the literature, the best known being the Krieger–Dougherty [35] equation:
A
B
w −[g]wm (11) g = 1− r w m where the relative viscosity (g ) is the viscosity of r the dispersion (g) compared to the viscosity of the suspending medium (g ). The intrinsic viscosity [g] s is taken to be 2.71 and w is taken to be 0.71. m These are the values reported by de Kruif et al. [36 ] for the limiting high shear rate viscosity. Fiderlis and Whitmore [37] and Eveson [38] have shown that the drag experienced by a large sphere moving in a suspension of small spheres is the same as a large sphere moving through a pure Newtonian liquid having the same viscosity and density as the suspension. This is an effective medium theory and basically assumes that the smaller particles act as a medium towards the larger particles. Using this concept, Farris was able to predict the viscosities of multimodal suspensions from monomodal data, i.e. the viscosity of the mixture of smaller fractions is the medium viscosity for the next largest fraction. This was assuming at least an order of magnitude difference in size between successive fractions: g (a )=g ×g ×g ×…×g (12) r r1 r2 r3 ri i So, an aqueous bimodal dispersion would have the relative viscosity: g g (13) = small × large g g∞ water small where g∞ is the viscosity of water thickened up small by the small component. So using the idea that, for a bimodal suspension, the mixture of small particles and suspending g
rdisp
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liquid is treated as one homogeneous fluid, if the volume of the suspending fluid is V and the liq volume of the small particles is V and that of small the large particles is V then two volume fraclarge tions can be defined [39]: w
large and
=v
large
(14)
v
large (15) 1−v small where v and v are the true volume fractions small large given by:
w
small
=
V large (16) +V +V large small liq V small v = (17) small V +V +V large small liq Using Eqs. (16) and (17) to generate the effective volume fractions, which are then fed into the Krieger–Dougherty equation, and using w as max 0.71, a theoretical curve can be calculated and compared with the experimental results. In Fig. 6 the relative high shear rate viscosity (at 20% small particles) is compared with the theoretical curve suggested by Farris. The correct trends are observed, but the data points lie above the curve as the adsorbed layers increase the effective volume fraction of the particles present and hence increase the viscosity. As mentioned previously it is not possible to accurately estimate this contribution to the volume fraction. At low volume fractions it is straightforward to estimate the effective volume fraction based on the thickness of the adsorbed layers, but as the volume fraction increases it is extremely difficult to calculate exactly at what rate each of the adsorbed layers is being deformed. Assuming the adsorbed layer thickness compresses at the same rate as in the monomodal systems is unrealistic, as both the large and the small particles will compress the polymer layers. Our data is the first attempt to fit the rheology of bimodal colloidal dispersions to that of the Farris theory. In the past the rheology of larger particles has been found to fit the Farris model well, for example Parkinson et al. [40] using v = large V
Fig. 6. Comparison of the high shear rate viscosity to the Farris theory at 22% small particles by volume.
PMMA spheres in Nujol fitted his data for bimodals, trimodals and tetramodals to the Farris theory and reported a viscosity minimum for bimodals at 25% small particles. More recently Gondret and Petit [41] have conducted similar experiments to those reported here for large noncolloidal spheres; broadly similar trends to those reported here were observed.
4. Conclusions At a diameter ratio of 4.76, a minimum viscosity can be achieved in a bimodal suspension with a composition of 0.15–0.20 small particles by volume. However, if the adsorbed layer thickness is taken into account then this recalculates as
R. Greenwood et al. / Colloids Surfaces A: Physicochem. Eng. Aspects 144 (1998) 139–147
0.27–0.36; the lower limit of which is in excellent agreement with the theory. The Farris theory seems to underestimate the viscosity due to an inability to calculate correctly the effective volume fraction. This of course can be minimised by the use of larger particles, but to study a wide range of diameter ratios would involve some particles that would have an appreciable settling rate.
Acknowledgment This research was sponsored by the EPSRC CASE studentship in conjunction with CoatesLorilleux, St Mary Cray, Orpington, Kent BR5 3PP.
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