Abstract
A large amount of data from the literature on viscosity of concentrated suspensions of rigid spherical particles are analyzed to support the new concept that the maximum packing fraction (ϕ M ) is shear-dependent. Incorporation of this behavior in a rheological model for viscosity (η) as a function of particle volume fraction (ϕ) succeeds in describing virtually all non-Newtonian effects over the entire concentration range and also accounts for a yield stress. The most successful model is one proposed by Krieger and Dougherty for Newtonian viscosities,η (ϕ, ϕ M ), but withϕ M varying from a low-shear limitϕ M0 to a high-shear limitϕ M∞. Microstructural interpretations of this behavior are advanced, with arguments suggesting that similar rheological models should apply to suspensions of nonspherical and irregular particles.
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Abbreviations
- a :
-
particle size scale (for spheres, the diameter)
- A :
-
lumped kinetic parameter in eqs. (23) and (24)
- BS:
-
butadiene-styrene
- C :
-
coefficient in Arrhenius model, eq. (2)
- D :
-
coefficient in Mooney model, eq. (3)
- e i :
-
parameter representing one of the three electroviscous effects (i = 1, 2, or 3)
- f :
-
fraction of total particulates that exist in the dispersed phase, eq. (22)
- h :
-
solution factor, in Arrhenius model, eq. (2)
- k :
-
crowding factor, in Mooney model, eq. (3)
- k D ,k F :
-
kinetic rate coefficient for producing particles of dispersed or flocculated type, respectively
- K :
-
Einstein coefficient for particles of any shape, eq. (1); equal to [η]
- KD:
-
Krieger-Dougherty model, eq. (6)
- m :
-
exponent to characterize shear-dependence in viscosity models of Cross, eq. (10), and eq. (23), and also in yield stress prediction eq. (24)
- N :
-
number of monodisperse components in a blend of spheres with different diameters
- PD:
-
polydispersity (in size) parameter
- S :
-
generalized shape parameter
- T :
-
temperature
- V c :
-
volume of “chamber” in figure 6, representing the entire volume of the sample
- V P :
-
total volume of particles in the sample
- V D ,V F :
-
sample volumes in which dispersed particles or flocculated particles, respectively, prevail; volumes of the “dispersed phase” or “flocculated phase”, containing both particles and carrier fluid
- V PD ,V PF :
-
particle volume within the phase volumeV D orV F , respectively
- α :
-
coefficient in definition ofτ c in eq. (8); of order unity
- β :
-
coefficient regulating\(\dot \gamma \)-sensitivity in eq. (10)
- \(\dot \gamma \) :
-
shear rate,dv 1/dx 2 in simple shear
- η :
-
shear viscosity of the suspension
- η 0,η ∞ :
-
low-shear and high-shear limiting values ofη
- η s :
-
viscosity of the suspending fluid
- [η]:
-
intrinsic viscosity,\(\mathop {\lim }\limits_{\phi \to 0} (\eta - \eta _s )/\phi \eta _s \)
- η r :
-
reduced viscosity,η/η s
- ϰ:
-
Boltzmann's constant; inτ c
- τ :
-
shear stress
- τ c :
-
parameter characterizing sensitivity of viscosity to stress, in eq. (8)
- τ B :
-
dynamic yield stress in the floc model
- τ y :
-
yield stress
- ϕ :
-
volume fraction occupied by solids in a suspension
- ϕ M :
-
maximum value ofϕ attainable by a given collection of particles under given conditions of flow
- ϕ M0,ϕ M∞ :
-
limiting values ofϕ M at the low-τ and high-τ conditions, respectively
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Wildemuth CR, Williams MC, Rheol Acta, in press
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Wildemuth, C.R., Williams, M.C. Viscosity of suspensions modeled with a shear-dependent maximum packing fraction. Rheol Acta 23, 627–635 (1984). https://doi.org/10.1007/BF01438803
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DOI: https://doi.org/10.1007/BF01438803