Gradient Shear Coagulation

The second mechanism of coagulation is coagulation of particles suspended in a laminar flow of Uquid whose velocity is a regular function of position. As the characteristic distance of particle interaction is small, the structure of the velocity field on such linear scales can be considered as linear. In other words, the flow may be regarded as a shear flow, hence the name of the coagulation mechanism. This problem was also studied by SmolukhowsM [59]. 

Consider a test particle of radius ai whose center coincides with the origin of coordinates, and a particle of radius a2 moving relative to the test particle in a linear velocity field (Fig. 10.5). 

One can see from Fig. 10.5 that collision of particle U2 with the test particle ai occurs when the distance over %-axis between them is less or equal to (ai + U2) sin 0. The velocity of particle a2 relative to ai is 

Consider a strip of width dy. The number of particles U2 crossing the strip dy in a unit time is 

Multiplying (10.48) by the number of test particles rii and integrating, we obtain the collision frequency of particles U2 with particles Ui  

---

Fig. 10.5 The model of gradient (shear) coagulation of particles according to Smoluchowski. Here y denotes the shear rate.

From (10.54), it follows that increase of the particle size and the shear rate makes the frequency of shear coagulation larger in comparison with that of Brownian one. The particle radius has an especially strong influence ( a ) on the ratio (10.54). It is worth to note that the parameter (10.54) is just the Peclet number, equal to the ratio of the characteristic time of Brownian coagulation to the characteristic time of gradient coagulation. 

Illustrative Cases. Three cases are illustrated in Figure 9, marked by the circles labeled A, B, and C. Case A refers to classical experiments by Swift and Friedlander (27) on the coagulation of monodisperse latex particles (diameter = 0.871 pm) in shear flow and in the absence of repulsive chemical interactions. Considering a velocity gradient of 20 s 1, HA is 0.0535, log HA is — 1.27, and dfdj is 1.0 for these experimental conditions. The circle labeled A in Figure 9 marks these conditions and indicates that the hydrodynamic corrections to Smoluchowsla s model predict a reduction of about 40% in the aggregation rate by fluid shear. The experimental measurements by Swift and Friedlander showed a reduction of 64%. This observed reduction from Smoluchowski s rectilinear model was therefore primarily physical or hydro-dynamic and consistent with the curvilinear model. 

Particles in a fluid in which a velocity gradient du/dy exists have a relative motion that may bring them into contact and cause coagulation (Figure 13.A.1). Smoluchowski in 1916 first studied this coagulation type assuming a uniform shear flow, no fluid dynamic interactions between the particles, and no Brownian motion. This is a simplification of the actual physics since the particles affect the shear flow and the streamlines have a curvature around the particles. 

An important effect in Equation (3.34) is that the collision radius enters as Rl and since we have approximated Rc = 2R it becomes 8R consequently this indicates that shearing is very sensitive to particle size. For this reason small particles are rather insensitive to shearing forces, whereas larger particles, for instance with R > 0.5 pm, can often be flocculated by stirring or shaking particularly at electrolyte concentrations close to the ccc. The sensitized coagulation which occurs in the presence of a velocity gradient is known as orthokinetic flocculation. 

A helpful presentation which illustrates the effect of shear rate on dispersions is to use dimensionless quantities as the axes. In this way the colloidal forces can be represented as the ratio of electrostatic repulsive terms to the attractive term, i.e. ereo i R/A and the hydrodynamic terms as the ratio of the shear term to the attractive term, i.e. (mr)R y/A. Following Zeichner and Schowalter [90] this is illustrated in Figure 3.30. Thus as illustrated by the arrow the impact of a shear gradient can move particles out of the secondary minimum association into a region of stability. However, as the shear rate increases still further primary minimum coagulation can occur until at even higher shear rates the particles are redispersed again. 

It is a general observation that gentle stirring promotes coagulation. The reason for this is that velocity gradients in the flow field create relative particle movements and therefore an increased collision frequency. The simplest case to treat is that of a uniform shear field. 

When the stirring speed increases, there is a transition from laminar to turbulent flow. Treatment of this kind of orthokinetic coagulation is much more difficult than the case of laminar flow. An approximate way to treat this situation is to use the Smoluchowski equation for orthokinetic coagulation in laminar flow and to employ an average shear gradient in the turbulent flow. Various estimates of this average have been derived. One of these... 

---