Deposition of Colloids on Fibers Subjected to Shear

As soon as polymer has adsorbed on the fibers, creating patches of polymer on the fibers, the colloidal particles can deposit on these polymer patches. The kinetics of colloid deposition on fibers are very similar to those of polymer adsorption on fibers, discussed above. It can be described by [13]  

Here 6 is the fractional coverage of fibers by colloids, the initial colloid concentra- 

For fibers the collision efficiency y depends on the size ratio of colloid and fiber diameter and on the ratio of colloidal (van der Waals) to hydrodynamic forces, but does not depend on fiber length [15]. The term a is the deposition efficiency, given according to the classical polymer bridging model [22] by  

Assumptions made in writing Equation 1.7 are that collisions between a bare patch and a polymer patch lead to deposition and those between polymer patches or between bare patches do not. This is plausible when, during a collision, the particle trajectory is determined mainly by the attractive force between a bare patch and a polymer patch. Collisions in shear are more complicated, as rotating particles are surrounded by fluid that rotates with it, leading to trajectories that can orbit a fiber several times before colloid-fiber contact occurs [23]. When particles are partially coated by polymer, they will experience fluctuating colloidal interaction forces, depending on where the patches are located. Nevertheless, also in such cases a = 0 when polymer is absent and a = 1 when fibers are fully coated (and colloids not). Thus, also in this case Equation 1.7 is probably not a bad approximation. 

The dimensionless colloid concentration can be expressed, for random deposition of spheres of radius a, or disks of thickness h, on fibers, as  

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