Krieger-Dougherty Equation Research Articles

Rheological studies on concentrated polystyrene latex sterically stabilized by poly(ethylene oxide) chains

The rheological properties of aqueous concentrated polystyrene latex dispersions for three particles with core diameters of 155 nm, 612 nm, and 1004 nm were investigated using steady-state shear stress—shear rate and oscillatory measurements. The relative viscosity—effective volume fraction results were fitted to the Dougherty—Krieger equation for hard spheres for dispersions with various particle sizes by adjusting the value for the adsorbed layer thickness Δ. The latter was found to decrease with an increase in volume fraction of the dispersion which is caused by considerable compression of the chains when the volume fraction of the particles approaches close-packing. The value of Δ was also found to increase with increases in particle size. From the oscillatory measurements, the complex modulus G∗, the storage modulus G′, and the loss modulus G″ were obtained as a function of frequency at various latex volume fractions. The results showed that the dispersion changed from being more viscous ( G″ > G′) to more elastic ( G′ > G″) over a narrow range of volume fraction of the dispersion. Within this range of φ, it is likely that the chains undergo some compression and interpenetration of peripheries. When φ was increased significantly above a certain value, the system became predominantly elastic and significant interpenetration and compression of the chains occurred. If φ was further increased, the moduli increased by several orders of magnitude as did the dynamic viscosity. Under these conditions, the latex dispersion behaves as an elastic “gel” which can be fitted by a power law equation G′ = kφ m where m is related to the compressibility of particles which is determined by the ratio of adsorbed layer thickness to radius of particle, Δ R .

ドクターブレード法に用いるアルミナスラリーの流動性に及ぼす有機剤の添加効果

The fluidity of alumina slurry and the packing state of green sheets prepared by a doctor blade method were investigated with the variations of added organics, such as the ratio of binder to plasticizer, the degree of polymerization of binder (PVB), and the amount of organics in the alumina slurry. The homogeneous slurries were first prepared from alumina powder, organics, and solvent by ball milling. Then, the viscosity and relative viscosity of slurries and the relative density of green sheets were measured. The results are summarized as follows:(1) The viscosity of slurry was affected largely by the ratio of binder to plasticizer, the degree of polymerization of PVB, and the amount of organics.(2) The apparent hydrodynamic shape factor (α) of alumina particles calculated from Dougherty-Krieger equation was 6-8 in the slurry.(3) The packing density of alumina particles in the green sheet changed little with the variation of added organics.

Cement Viscosity As A Function Of Concentration

AbstractThe Krieger-Dougherty equation allows calculation of viscosity as a function of volume fraction for suspensions of noninteracting particles. For model suspensions (of spherical, monosized particles), it has been shown to provide excellent agreement between calculated and measured viscosities. In the present study, this equation was applied to portland cement pastes, also with good correlations between calculated and measured viscosities. Because cement has a broad particle size distribution and its particles are angular and elongated, the two constants in this equation (the maximum volume fraction and the intrinsic viscosity) were estimated using nonlinear optimization techniques. The equation provides an excellent fit to measured viscosity data. However, the nature of the equation makes the estimation somewhat difficult, and the solutions are not well-defined.

Rheological studies on inverse microlatices

The rheological properties of inverse microlatices prepared by polymerization in nonionic micro-emulsions have been investigated. These microlatices consist of sterically stabilized particles ( d < 130 nm) of polyacrylamide and polymethacrylate of trimethylaminoethyl chloride, swollen by high water contents (⋍50% water, 50% polymer) and dispersed in an organic medium. The rheological behavior of these materials follows quite closely that expected for hard-sphere suspensions. In particular, the data for the dependence on volume fraction of the high- and low-shear limiting viscosities are well fitted by the Krieger—Dougherty equation (I. M. Krieger and T.J. Dougherty, Trans. Soc. Rheol. 3, 137 (1959)). However, the study of viscosity versus shear stress shows that these microlatices have a Newtonian behavior extending up to a volume fraction of ca. 50%, in contrast to the limit of ca. 25% reported for conventional latices. Partition of the disperse phase into polymer particles and small surfactant micelles could account for this difference.

Effect of addition of hydroxyethyl cellulose on the viscoelastic properties of concentrated sterically stabilised latex dispersions

Concentrated sterically stabilised polystyrene latex suspensions were prepared by physical adsorption of a graft copolymer with poly (ethylene oxide) (PEO) side chains. The relative viscosity (ηr) -volume fraction (Φ) curves showed that these dispersions behave as hard spheres. By fitting the ηr - Φ to the Dougherty—Krieger equation, an adsorbed layer thickness of 4.0 nm was obtained and this showed a tendency to decrease (to 2.5 nm) with increase in Φ indicating some interpenetration and/or compression of the PEO chains. These concentrated sterically stabilised dispersions were used to study the effect of addition of hydroxyethyl cellulose (HEC) with three different molecular weights (70 000, 124 000 and 223 000) and a hydrophobically modified HEC. Rheological studies showed flocculation to occur above a range of HEC concentration (volume fraction Φ+p) which tended to decrease with increase in molecular weight of the polymer. From the extrapolated yield value (τβ) the energy of separation particles in the flocculated structure (ESep) was calculated assuming two values (4 and 8) for the average number of contacts between the particles. These values were compared with the theoretical calculations of the free energy of attraction due to depletion, (Gdep) based on Asakura and Oosawa as well as Fleer, Scheutjens and Vincent (FSV) models. The results showed considerable deviation between ESep and Gdep based on Asakura and Oosawa's theory. However, some agreement was obtained using the FSV model, particularly at low Φp values.

Viscoelastic properties of aqueous concentrated polystyrene latex dispersions containing grafted poly(ethylene oxide) chains

The viscoelastic properties of aqueous concentrated polystyrene latex dispersions containing grafted poly(ethylene oxide) (PEO) chains were investigated using oscillatory and steady-state shear stress-shear rate measurements. The relative viscosity-effective volume fraction results were fitted to the Dougherty-Krieger equation for hard spheres by adjusting the value for the adsorbed layer thickness Δ. The latter was found to decrease with an increase in volume fraction of the dispersion and near to close-packing considerable compression of the chains occurred. From the oscillatory measurements, the complex modules G ∗ , storage modules G′, and loss modulus G″ were obtained as a function of frequency at various latex volume fractions. The results showed that the dispersion changes from being more viscous ( G″ > G′) to more elastic ( G′ > G″) over a narrow range of volume fraction φ of the dispersion, i.e., when φ is increased from 0.465 to 0.5. Within the range of increase of φ, it is likely that the chains undergo some compression and interpenetration of the peripheries. When φ increased significantly above 0.5, the system becomes predominantly elastic and significant interpenetration and compression of the chains occur. Indeed when φ is further increased to 0.585 and 0.62, the moduli increase by several orders of magnitude and so does the dynamic viscosity. Under these conditions, the latex behaves as an elastic “gel.”

高濃度石炭水スラリー（ＣＷＭ）の流動性に及ぼす石炭性状の影響

Highly concentrated coal water mixtures (CWM's) were prepared using eight different coals and an anionic dispersant. Viscosity of the CWM's was investigated in relation to the coal properties. It was found that coals with lower water-holding capacity (WHC) would make more highly concentrated CWM's. Their viscosity was correlated well with the solid volume fraction and maximum packing density using Krieger-Dougherty equation. For these coals, the particle size distribution was found to influence the CWM viscosity only through the values of maximum packing density.CWM's made from coals with high WHC exhibited higher viscosity. This was attributed to the lower degree of dispersion of the coal particles. CWM's with coals having low WHC displayed similar viscosity-solid concentration relationship when electrolyte was added.

The sedimentation stability and viscosity of coal oil dispersions

Measurements of the sedimentation stability and viscosity of a number of suspensions of coal particles in heavy fuel oil known as Coal Oil Dispersions (COD) are described. It is shown that an estimate of the stability of a coal oil dispersion prepared by grinding coal in fuel oil can be made from simple viscosity measurements. The method is based on the use of the coal volume fraction and a parameter V which is characteristic of a given coal oil dispersion. V is determined from the variation of dispersion viscosity with coal volume fraction using the Krieger—Dougherty equation. The stability of an unknown dispersion is predicted by determining V and its coal content and using previously calibrated stability charts. The parameter V is a function of coal—coal interaction and particle size distribution. It is suggested that the method might be applicable to the quality control of other concentrated suspensions.