The consolidation of the
particles is visually apparent in Fig. 6.
Note that the point of view in Fig. 6(a)–(c)
spans from the membrane surface to the bulk in the positive *z*-direction.
Take, for instance, the case of particle zeta potential −30mV. With an observed permeate flux of 6.9m/s,
the particles remain sparsely concentrated (volume fraction=0.423), and an
appreciable average center-to-center nearest pair interparticle spacing of
2.250 times particle radius is clearly discernible. Upon increasing the
observed permeate flux to

27.6m/s, the volume fraction rises accordingly to a value of 0.527 with average inter-particle spacing of 2.075 times radius.Theparticlesareconsolidatingbutstillnotyetincontact with one another. A comparison of Fig. 6(a) and (b) reveals the noticeable compression of the particles and reduction in inter-particle spacing due to heightened hydrodynamic drag. Finally, the particles contact one another when the observed permeate flux reaches 82.8m/s as seen in Fig. 6(c), with average center-to-center inter-particle spacing of 2.002 times particle radius (i.e., particle contact). The volume fraction of 0.608 at this point suggests that the particles are configured in random packing [33]. The illustration given in Fig. 6 captures well the randomness of the particle configuration.

The role of inter-particle repulsion can be seen for the case of −90mV in Fig. 7. In this case, the cake layer volume fraction at permeate flux 6.9m/s is 0.331 with sparse average center-to-center inter-particle spacing of 2.450 times particle radius. The volume fraction compresses to only 0.456 with average inter-particle spacing 2.175 times particle radius at

82.8m/s. The particles do not come into contact despite the relatively high permeate velocity. The more porous cake structureformedatthehigherzetapotentialisvisiblyevident.

Besides
showing the effects of increasing hydrodynamic drag force, the progression
depicted in Fig. 6 provides one
possibleanswertothequestionofphasetransition.*Thepolarized layer of colloidal
particles near the membrane surface progressively consolidates under increasing
hydrodynamic drag force to reach a point of contact among the particles. The
point of particle contact delineates the onset of a solid phase. The volume
fraction of particles need not reach ordered maximum packing but only random
maximum packing*.

The definition of phase transition given above proceeds directly from inspection of the inter-particle spacing. It is straightforward and simplistic, but does not include practical considerations that will furnish it to be more pertinent to the operations of membrane filtration systems. The definition of phase transition will be addressed with greater refinement and sophistication in Section 5, with inclusion of concepts of critical flux and cake reversibility.

The prominent roles that the hydrodynamic drag force and inter-particle electrostatic interaction play in cake layer formation are definitely evident in Figs. 6 and 7. Therefore, it becomes imperative for any simulation model to accurately characterize these two quantities. This crucial step in the development of the model will be discussed in greater detail in Section 4.3.

Fig. 8 shows the model results for simulations of different sizes of particles. As the hydrodynamic drag force is a function of the particle radius according to Eq. (13), it would follow that a smaller particle radius would then produce particle configurations of lower volume fraction. For instance at 82.8m/s permeate flux, the particle volume fraction drops off from a value of 0.659 for 200nm particles to 0.489 for 50nm particles. So, the particle radius proves to be another important model parameter. The need to accurately size the model particles stands without question. However, implications of the sensitivity of the particle configuration to the radius pervades to other deeper issues.

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