B. Jachimska, Z Adamczyk /Joumal of the European Ceramic Society 27(2007)2209-2215 expression for p, valid for spherical particles8 seems, therefore, that the non-spherical shape of zirconia parti- regates is not responsible for the deviation of the slope 22(1+ka2F(ka) (5) of ln] versus v dependence from the from Einsteins formula, which was, as previously estimated, about 180%0 where E is the dielectric constant of water(relative permittivity), The most likely explanation is that the zirconia aggregates s the zeta potential of the particle forming the suspension, D, form a loose structure, characterized by a high degree of poros the ith ion diffusion coefficient, F(xa) the function of the dimen- ity, as suggested by the SEM picture shown in Fig. 2. As a result, sionless ka parameter, K-=(e kT/8Ten 2 the double-layer the effective hydrodynamic volume of particles in the suspension thickness,k the Boltzmann constant, T the absolute tempera-(at fixed mass fraction )becomes larger than the net volume of ture, e the elementary charge, I the ionic strength of solution isolated primary particles, which results in a decreased appar and a=dn2 is the particle radius. ent density of particles. A similar model was effectively used For a ionic strength of l=3 x 10-M, Kaequals 56 for the zir- efore to explain the anomalous viscosity behavior of silica conia suspension(assuming 0.6 um as the averaged diameter of suspensions 8 It was postulated that upon contact with water, a the aggregate).As calculated in for this value of Ka, the value gel-like layer was formed on the surface of silica, having lower of the function(1+Ka) F(xa)is about 3 x 10-4. Then, by tak- apparent density than the core volume of particles ing S=-50mV and D=2 x 10-cm-/s one can estimate from Since the primary electroviscous effect remains negligible as Eq (5)thatp=7x 10-4. As can be seen, this nis value is more than previously estimated, the apparent density of the aggregates can three orders of magnitude smaller than experimentally found be calculated from the dependence& for zirconia, equal 0.8. The comparison proves quite unequivo- cally, that the primary electroviscous effect is not responsible for the deviation of zirconia suspension viscosity from the Einstein model By knowing the apparent density, one can calculated the aver- Therefore, the most probable reason of this deviation can age porosity of aggregates Ep from the formula be sought in the presence of aggregates in the suspension. For Pp=p aggregates of non-spherical shape, the increase in the intrinsic"PPp-Pe viscosity of suspensions occurs because of the increase in the fuid velocity gradients due to particle rotation. This effect was By substituting into described quantitatively by Brenner for spheroidal particles, iments, i.e., Pp=5.47 g/cm, p=3.03 g/cm, Pe=1.01 g/cm3 having both a prolate and an oblate shape. The parameter of one obtains Ep=0.55. Hence, the apparent volume fraction of primary significance governing suspension viscosity in this case solid in the aggregate equals 1-Ep=0.45. This seems a quite is the Peclet number( for rotary Brownian motion) defined as the reasonable estimate in view of the fact that the maximum packing ratio of the characteristic rotation velocity (G)d to the diffusion of spheres in three dimensions is 0.62.20 Our value is smaller velocity D/d because the aggregate structure is more loose than under the maximum packing when, be definition, all particles must con- 6) tact with each other Having established the basic relationship for the pure zir- where(G)is the averaged shear rate in the capillary, D=kT/3nd conia suspension, we have studied the effect of the anionic the diffusion coefficient of the aggregate and d is the typical polyelectrolyte PSS on the viscosity of zirconia suspension. This diameter of the aggregate. polyelectrolyte, bearing sulfonic groups in chain was thoroughly In the case of the capillary viscometer the averaged shear rate characterized in our previous work. The extended length of the can be calculated as polyelectrolyte chain, having the molecular weight of 70,000 is 91 nm and the length to diameter ratio of the chain equal 3TR (7) 78. One can draw quantitative conclusions about the ionization degree of the Pss chain from the electrophoretic mobility mea By taking the typical experimental data: R=0.035 cm,Usus cm, t=50s, one obtains(G)=4 x 10s-l. Using this value surements as a function of the pH and ionic strength. The data compiled in Table 2 show that the zeta potential of PSs increased and taking d=0.66 um as the averaged size of the zirconia aggre- from -90-+5.0 mV for 1=1x 10-3M to-55+5.0mV for gate one can calculate from Eq. (7)that Pe=6.6x 10. Such a 1=0.15. that suggests a significant reduction in the number of large value of Pe indicates that the suspension viscosity was dominated by shear rather than diffusion. Thus, in the limit Table 2 of such large Peclet numbers, the intrinsic viscosity of parti- Zeta potential 5, of PSS (mv) at pH cle suspensions attains the limiting, high-shear values, tabulated by Brenner. For prolate spheroids having the length to width (M) Zeta potential (MV) aspect ratio of 3 the limiting value of [n]=2684, for aspect 1 x 10-3 ratio 4, [n]=2.801 and for aspect ratio 5, [n]=2.918. As can 5×10- be noticed, even for very elongated spheroids, i.e. for aspect 0.15 55±5.0 ratio 5. the correction to the einsteins formula is about 12%o. ItB. Jachimska, Z. Adamczyk / Journal of the European Ceramic Society 27 (2007) 2209–2215 2213 expression for p, valid for spherical particles18: p = 2εζ2(1 + κa) 2F(κa) 3πηDi (5) where ε is the dielectric constant of water (relative permittivity), ζ the zeta potential of the particle forming the suspension, Di the ith ion diffusion coefficient, F(κa) the function of the dimen￾sionless κa parameter, κ−1 =(ε kT/8πe2I) 1/2 the double-layer thickness, k the Boltzmann constant, T the absolute tempera￾ture, e the elementary charge, I the ionic strength of solution and a = d/2 is the particle radius. For a ionic strength ofI = 3 × 10−3 M, κa equals 56 for the zir￾conia suspensioun (assuming 0.6 m as the averaged diameter of the aggregate). As calculated in18 for this value of κa, the value of the function (1+κa) 2F(κa) is about 3 × 10−4. Then, by tak￾ing ζ = −50 mV and D = 2 × 10−5 cm2/s one can estimate from Eq. (5) that p = 7 × 10−4. As can be seen, this value is more than three orders of magnitude smaller than experimentally found for zirconia, equal 0.8. The comparison proves quite unequivo￾cally, that the primary electroviscous effect is not responsible for the deviation of zirconia suspension viscosity from the Einstein model. Therefore, the most probable reason of this deviation can be sought in the presence of aggregates in the suspension. For aggregates of non-spherical shape, the increase in the intrinsic viscosity of suspensions occurs because of the increase in the fluid velocity gradients due to particle rotation. This effect was described quantitatively by Brenner19 for spheroidal particles, having both a prolate and an oblate shape. The parameter of primary significance governing suspension viscosity in this case is the Peclet number (for rotary Brownian motion) defined as the ratio of the characteristic rotation velocity Gd to the diffusion velocity D/d: Pe = G d2 4D (6) where Gis the averaged shear rate in the capillary, D = kT/3πηd the diffusion coefficient of the aggregate and d is the typical diameter of the aggregate. In the case of the capillary viscometer the averaged shear rate can be calculated as G = 8vsus 3πR3t (7) By taking the typical experimental data: R = 0.035 cm, vsus = 10 cm3, t = 50 s, one obtains G = 4 × 103 s−1. Using this value and taking d = 0.66m as the averaged size of the zirconia aggre￾gate one can calculate from Eq. (7) that Pe = 6.6 × 102. Such a large value of Pe indicates that the suspension viscosity was dominated by shear rather than diffusion. Thus, in the limit of such large Peclet numbers, the intrinsic viscosity of parti￾cle suspensions attains the limiting, high-shear values, tabulated by Brenner.19 For prolate spheroids having the length to width aspect ratio of 3 the limiting value of [η] = 2.684, for aspect ratio 4, [η] = 2.801 and for aspect ratio 5, [η] = 2.918. As can be noticed, even for very elongated spheroids, i.e., for aspect ratio 5, the correction to the Einstein’s formula is about 12%. It seems, therefore, that the non-spherical shape of zirconia parti￾cle aggregates is not responsible for the deviation of the slope of [η] versus ΦV dependence from the from Einstein’s formula, which was, as previously estimated, about 180%. The most likely explanation is that the zirconia aggregates form a loose structure, characterized by a high degree of poros￾ity, as suggested by the SEM picture shown in Fig. 2. As a result, the effective hydrodynamic volume of particles in the suspension (at fixed mass fraction) becomes larger than the net volume of isolated primary particles, which results in a decreased appar￾ent density of particles. A similar model was effectively used before to explain the anomalous viscosity behavior of silica suspensions.8 It was postulated that upon contact with water, a gel-like layer was formed on the surface of silica, having lower apparent density than the core volume of particles. Since the primary electroviscous effect remains negligible as previously estimated, the apparent density of the aggregates can be calculated from the dependence8: ρ∗ = 2.5ρp CV (8) By knowing the apparent density, one can calculated the aver￾age porosity of aggregates εp from the formula8: εp = ρp − ρ∗ ρp − ρe (9) By substituting into Eq. (9) the data obtained in our exper￾iments, i.e., ρp = 5.47 g/cm3, ρ* = 3.03 g/cm3, ρe = 1.01 g/cm3 one obtains εp = 0.55. Hence, the apparent volume fraction of solid in the aggregate equals 1 − εp = 0.45. This seems a quite reasonable estimate in view of the fact that the maximum packing of spheres in three dimensions is 0.62.20 Our value is smaller because the aggregate structure is more loose than under the maximum packing when, be definition, all particles must con￾tact with each other. Having established the basic relationship for the pure zir￾conia suspension, we have studied the effect of the anionic polyelectrolyte PSS on the viscosity of zirconia suspension. This polyelectrolyte, bearing sulfonic groups in chain was thoroughly characterized in our previous work.11 The extended length of the polyelectrolyte chain, having the molecular weight of 70,000 is 91 nm and the length to diameter ratio of the chain equals 78. One can draw quantitative conclusions about the ionization degree of the PSS chain from the electrophoretic mobility mea￾surements as a function of the pH and ionic strength. The data compiled in Table 2 show that the zeta potential of PSS increased from −90 ± 5.0 mV for I = 1 × 10−3 M to −55 ± 5.0 mV for I = 0.15, that suggests a significant reduction in the number of Table 2 Zeta potential ζp of PSS (mV) at pH 10 I (M) Zeta potential (MV) 1 × 10−3 −90 ± 5.0 5 × 10−3 −83 ± 5.0 1 × 10−2 −74 ± 5.0 0.15 −55 ± 5.0