The role of size distribution of spherical shaped aggregates on relative viscosity was examined. An approximate log normal distribution was used and sphere size distributions were characterized by the mean squared deviation of sphere radii, normalized to the average sphere radius, σrms, that is given by
where fi is the volume fraction of spheres with radius
ai (normalized by the total volume of spheres) and
is a similarly weighted average sphere radius. We allowed σrms to range
from 0 to 1 (Figs. 7 and
8). For σrms = 0, the spheres are
monosize and when σrms = 1 the spheres size varied by a factor of about
30. In this study, the focus was more on the role of size variation and it was decided to
not include lubrication forces in the simulation as very small times steps would be needed,
making the simulation too time consuming. Hence, only a moderate Pe
≈ 10 was considered.
Figure 7. Suspension of polydisperse spheres with φ = 0.55 and φ= 0.2.
Figure 8. Suspension of polydisperse spheres with φ = 0.55 and σ = 0.8.
In these simulations the suspension was sheared using the Lees–Edwards boundary condition. The stresses in the system were then calculated and the viscosity determined using Eqs. 12, 13. Figure 9 shows the relative viscosity as a function of solid fraction for different values of σ. Note that at low solid fractions the data did not appear very sensitive to the value of σrms. This is, in part, an artifact of plotting our data on a log scale, although it was not expected that there would be a large difference in this regime. However, as the solid fraction increases the relative viscosity, at the same φ, clearly decreased with increasing φrms. This can be understood as a consequence of the maximum packing φc of the sphere system increasing as the particle size distribution becomes wider.
Figure 9. Relative viscosity of polydisperse suspensions. Shown are simulation data for σrms = 0, 0.2, 0.4, 0.6, 0.8, and 1.0. Solid lines are fits of data to Krieger–Doughtery equation. Curves offset to the right correspond to increasing σrms. Statistical uncertainties in the simulation data were approximately 10% or smaller.
One of the most well known equations for fitting relative viscosity data, for a broad range of φ, is the Krieger–Dougherty (KD) equation [Krieger and Dougherty (1959)]. The KD equation equation is based on effective medium theory arguments. Here, incremental changes to the solid volume fraction of a suspension increases the viscosity as if small particles were being added to the suspension, which is treated as a homogeneous viscous medium. In addition, a correction is needed to allow the viscosity to diverge at φc. The KD equation has the following form:
Fits to the KD equation were reasonable and are shown in Fig. 9. Consider the expansion of the KD equation in terms of φ:
By construction, the KD equation obtains the correct intrinsic viscosity. For polydisperse sphere systems, the KD equation would predict that the Huggins coefficient varies from approximately 5.08 to 4.375 as φc increases from 0.64 to 1. Theoretical work [Wagner and Woutersen (1994)] shows KH weakly depends on the polydispersivity of spheres in a suspension. For example, it was found that KH was reduced by about 13% for suspensions where the ratio of maximum to minimum radii was about 10. One might find troubling the increase of the exponent, 1.6 ≤ ηoφc ≤ 2.5, describing the divergence of viscosity as φc is approached from below. Some experimental results point to a divergence of viscosity with a critical exponent of 2 [de Kruif et al. (1985)]. Regardless, the KD equation captures the main trends correctly although the value of KH and the critical exponent are not exact.
Bicerano et al. (1999) suggested the following equation, which was intended to describe a suspension with uniform sized spheres and a maximum random packing fraction φc = 0.64:
By construction, it recovers the Einstein intrinsic viscosity and a Huggins coefficient of KH = 6.2 while fixing the critical exponent to the value 2. Equation 20 can be generalized for suspensions composed of particles with arbitrary shape and size distributions
where n is the critical exponent describing the divergence of the viscosity as
the critical packing is approached and K1 = φc
ηo–n and K2 = φ
KH
–nφcηo + n
(n–1/2) are chosen to match the intrinsic viscosity and Huggins coefficient
for that suspension, respectively [note, the generalization of Eq.
20 was done in collaboration with Flatt]. In principle, such
terms can also be generalized to account for a shear dependence and interparticle
interactions. The intrinsic viscosity is known for many shapes [Douglas and Garboczi (1995)] and KH is
predicted as a function of polydispersivity of sphere systems
[Wagner and Woutersen (1994)]. For arbitrarily shaped
objects, one could determine ηo and KH
by simulations in the regime where 0<φ<0.15. Higher order terms proportional
to φ3 and so forth may become important as the volume fraction is
increased although at some point the singular term should dominate. Also, it is not
clear if the critical exponent n is truly universal. The value of 2 used in Eq.
20 is based on a formal hydrodynamic-electrostatic analogy of
suspensions. In this analogy, n is equal to the percolation theory insulator
exponent, which has a value close to 2 in three dimensions. Off lattice models can give
rise to different values of n so that the universality of this exponent may only
be approximate [Feng et al. (1987)].
Regardless, precise determination of a critical exponent from simulation would require a
finite size scaling study that is beyond our current computational capabilities. Figure
10 shows the same data as in Fig.
9 but fit with Eq.
21 with the constraint that the critical exponent n = 2.
Figure 10. Fig. 10. Fit of same data in Fig. 8 to Eq. 21 with n = 2 and terms up to K2 retained. Statistical uncertainties in the simulation data were approximately 10% or smaller.