Orthokinetic collisions

Although reduction or elimination of the repulsion barrier is a necessary prerequisite of successful flocculation, the actual flocculation in such a destabilized suspension is effected by particle—particle collisions. Depending on the mechanism that induces the collisions, the flocculation process may be either perikinetic or orthokinetic. 

Studies on orthokinetic flocculation (shear flow dominating over Brownian motion) show a more ambiguous picture. Both rate increases (9,10) and decreases (11,12) compared with orthokinetic coagulation have been observed. Gregory (12) treated polymer adsorption as a collision process and used Smoluchowski theory to predict that the adsorption step may become rate limiting in orthokinetic flocculation. Qualitative evidence to this effect was found for flocculation of polystyrene latex, particle diameter 1.68 pm, in laminar tube flow. Furthermore, pretreatment of half of the latex with polymer resulted in collision efficiencies that were more than twice as high as for coagulation. 

In summary, polymeric flocculants generally increase peri-kinetic flocculation rates compared with perikinetic coagulation rates. This is not necessarily true for orthokinetic flocculation, and experimental results in the literature are seemingly in conflict. Collision rate theory predicts that the polymer adsorption step may become rate limiting in orthokinetic flocculation. The present study was designed to elucidate the relationship between polymer adsorption rates and particle flocculation rates under orthokinetic conditions. 

The rate of coagulation of particles in a liquid depends on the frequency of collisions between particles due to their relative motion. When this motion is due to Brownian movement coagulation is termed perikinetic when the relative motion is caused by velocity gradients coagulation is termed orthokinetic. 

Analyses of the orthokinetic encounters between equi-sized spheres 30-1 have shown that, as with perikinetic encounters, equation 5.28 can be modified to include a ratio a0 to give the collision frequency J as ... 

The ratio of the probability of a collision induced by a fluid velocity gradient (dv/dx) (i.e., orthokinetic coagulation) to the collision probability under the influence of Brownian motion (perikinetic coagulation—what we have considered so far) has been shown to be (Probstein 1994)... 

TJhe aggregation of particles in a colloidal dispersion proceeds in two distinct reaction steps. Particle transport leads to collisions between suspended colloids, and particle destabilization causes permanent contact between particles upon collision. Consequently, the rate of agglomeration is the product of the collision frequency as determined by conditions of the transport and the collision efficiency factor, the fraction of collisions leading to permanent contact, which is determined by conditions of the destabilization step (2). Particle transport occurs either by Brownian motion (perikinetic) or because of velocity gradients in the suspending medium (orthokinetic). Transport is characterized by physical parame-... 

The rate constant k0 for orthokinetic coagulation is determined by physical parameters (velocity gradient du/dz, floe volume ratio of the dispersed phase, = sum over the product of particle number and volume), and the collision efficiency factor a0 observed under orthokinetic transport conditions ... 

From a comparison of the two collision frequency terms, described in detail in the Equations 3 and 6, one obtains the relative contributions of the perikinetic and orthokinetic transport to the total particle agglomeration. The ratio is a function of the radius of the colloid, r, and the absolute value of the velocity gradient du/dz ... 

The collision efficiency factors, describing the extent of the colloid destabilization, within certain limits are equal under perikinetic and orthokinetic conditions (3). 

Collision efficiency factor, measured under orthokinetic conditions... 

For a given, low applied shear rate, perikinetic collision rates tend to be higher than orthokinetic rates when the dispersed species are quite small, of the order of 100 nm or less. The orthokinetic collision rates tend to dominate for larger-sized dispersed species, of the order of several pm or more. For more information on aggregation kinetics see Refs. [27,292,318,319]. 

Conceptually similar results were demonstrated by Krutzer et al. [14], who measured the orthokinetic coagulation rate under laminar Couette flow and isotropic turbulent flow (as well as other flow conditions). Despite equal particle collision rates, significance differences were observed in the overall rates indicating different collision efficiencies (higher collision efficiencies were found under a turbulent flow regime). Thus, identical chemical properties of a dispersion do not determine a single collision efficiency the collision efficiency is indeed dependent upon the physical transport occurring in the system. 

The mechanism of orthokinetic particle collisions is a subject that has continued to receive a significant amount of attention. In 1952, Manley and Mason verified the rectilinear model given above for relatively large glass... 

Acoustic agglomeration is a process in which acoustic forces cause particles to interact and, eventually, to collide. The complex mechanisms behind this process involve orthoki-netic and hydrodynamic interactions. The orthokinetic interaction is founded on the hypothesis that collisions are produced due to the different acoustic entrainments experienced by particles of different size and weight. In order to describe this mechanism, an agglomeration volume is defined around each particle as a volume where another particle can be captured [49], However, this mechanism, which constitutes the basis for most existing interaction models, can explain neither the agglomeration of monodispersed aerosols nor the way in which the agglomeration volume is refilled once the initial particles are captured. 

Among the primary collision mechanisms is Brownian flocculation, also termed perikinetic flocculation, which dominates for submicrometer particles at relatively high number densities. The second principal collision mechanism is that of velocity gradient flocculation, also termed orthokinetic flocculation, which dominates for particles of micrometer size and larger. Evidently, the presence of any stabilizer in the solution will reduce the number of particle encounters and subsequent floccing, as discussed in the last section, resulting in slow flocculation. In our discussion we shall separate the transport and stability problems by assuming that the suspension is completely destabilized, so flocculation occurs on encounter rapid flocculation). Our concern here is with the effect of the particle motion alone on the number of encounters between the suspended 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. 

When the dispersion is stirred or when it flows, diffusion is not the only mechanism determining the collision probability. Under such conditions, the shear rate, that is, the velocity gradient, normal to the particle surface, dv(x)/dx (see Section 17.1.3) enhances the collision frequency, so that the orthokinetic aggregation rate is faster than the perikinetic aggregation rate. Under conditions of rapid aggregation, it can be derived that... 

---