Summary
A theory is developed for the rate of doublet formation in dilute dispersions of spheres subjected to a simple shear flow of gradient G when Brownian motion can be neglected but taking account of both hydrodynamic and interparticle interactions. When Coulombic repulsive forces are negligible, the capture frequency is found to be approximately proportional toG 0.82; when they dominate the van der Waals attractive forces, the capture frequency is zero. When both attractive and repulsive forces are important no simple relationship exists, but the capture frequency can be calculated numerically. Experiments are described which are in qualitative agreement with the theory. Hamaker constants for polystyrene in water calculated from the measured doublet formation agree with those calculated by means of the present theory from experimental data of other workers, but are somewhat higher than those obtained from perikinetic coagulation rates.
Zusammenfassung
Eine Theorie für Doppelkugelformationen in verdünnten Dispersionen, welche einer einfachen Scherströmung mit dem GradientenG ausgesetzt sind, bei dem die Brown'sche Molekularbewegung vernachlässigt werden kann, unter Berücksichtigung aber von hydrodynamischen und Zwischenkörper-Kräften, wird entwickelt. Wenn Coulomb'sche Abstoßungskräfte vernachlässigt werden können, ergibt sich die Einfanghäufigkeit mit ca.G 0.82; wenn deren Größe aber die der Van der Waals Kräfte überschreitet, ist die Einfanghäufigkeit null. Bei Auftreten von anziehenden und abstoßenden Kräften existiert keine einfache Beziehung, sondern es kann nur eine numerische Berechnung durchgeführt werden.
Experimente werden beschrieben, die in qualitativer Übereinstimmung mit der Theorie sind. HamakerKonstanten für Polystyrol in Wasser, errechnet von den gemessenen Doppelkugelformationen, stimmen mit den, mit Hilfe der bestehenden Theorie von anderen Forschungsergebnissen errechneten Werte überein, sind aber etwas höher als jene, die von perikinetischen Koagulationsmaßen erhalten wurden.
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Abbreviations
- A :
-
Hamaker constant
- A(r*), B(r*), C(r*) :
-
functions introduced in [7]
- b :
-
sphere radius
- c, c t :
-
volumefraction ofsingle spheres, total volume fraction
- C A ,C R :
-
ratio of attractive and repulsive forces to hydrodynamic forces, resp.
- d :
-
distance between top and bottom of tube
- \(f\left( {\bar \lambda } \right)\) :
-
function introduced in [7]
- F attr,F rep :
-
attractive and repulsive forces, resp
- G, G c :
-
shear rate, critical value
- J; J 8 :
-
capture frequency per particle; according toSmoluchowski
- K :
-
dielectric constant of suspending medium
- l; l* :
-
semi-axis ofcapture crosssection inX 2-direction;l/b
- N o ;N t :
-
number concentration of particles att=0; at timet
- p :
-
(r* - 2)r
- r;r*:
-
distance between sphere centers;r/b
- R, R o :
-
radial distance from tube center, tube radius
- S;S 1; S2 :
-
ratio of singlets to doublets; at top of tube; at bottom of tube
- t,t*;t*:
-
real and dimensionless time (Gt); average ofGt
- u 3 :
-
undisturbed fluid velocity in X3-direction
- x i ;X i ;X i :
-
Cartesian coordinates;xi/b; coordinate axes
- Z(x 2);Z* (x 2):
-
boundary of capture crosssection; dimensionless value (Z(x 2)/b)
- α0 :
-
orthokinetic capture efficiency, defined in [6]
- εo :
-
permittivity of free space
- n :
-
suspending fluid viscosity
- 0,φ :
-
polar coordinates
- x :
-
reciprocal double layer thickness
- λ; xxx :
-
London wavelength; λ/2πb
- σ :
-
ratio of capture to collision cross-section
- τ; τ c :
-
ratio of sphere radius and double layer thickness; critical value
- ψ 0;ψ 0 c :
-
surface potential of sphere; critical value
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van de Ven, T.G.M., Mason, S.G. The microrheology of colloidal dispersions VII. Orthokinetic doublet formation of spheres. Colloid & Polymer Sci 255, 468–479 (1977). https://doi.org/10.1007/BF01536463
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DOI: https://doi.org/10.1007/BF01536463