Fig. 8. Cake layer volume fraction in relation to
observed permeate flux with varying particle radius. The following parameters
(in addition to those listedinTable1)wereusedinthesimulations:zetapotential=−30mV,ionic strength=10−_{3 }M.

in Eqs. (10)–(12). This model relies on using a linear superposition approximation with constant potential. Rigorous derivation and defense of this method are given in Bell et al. [25] and Chew and Sen [26].

Otherexpressionsforelectrostaticdoublelayerinteraction are available. Among them are the fundamental expressions for interactions at constant surface potential, constant surface charge, and many more other models [3,38]. A thorough discussion of the various options available is given in Chen and Kim [8].

The choice of models bears significant consequences. To
further illustrate this point, a parameter *R *is introduced to
gaugetherelativemagnitudeofthedownwardhydrodynamic drag versus the
inter-particle electrostatic potential:

(20)

where *V*_{s
}is the difference in inter-particle potentials measured at a particle
separation of 2.5 times particle radius, with a displacement *r *of 0.1
times particle radius. Eq. (20) essentially
measures the ratio of the hydrodynamic force that the particle experiences
while displacing 0.1 times particle radius in the negative *z*-direction
towards another particle 2.5 times particle radius away versus the resistance
against this displacement from the inter-particle repulsion.

Fig. 9
shows the particle volume fraction as a function of the logarithm of *R*.
The curves exhibit two distinct regions. In region I, the particle volume
fraction is an increasing function of log*R*. This is an expected trend
and signifies that a greater hydrodynamic drag will compress the particle configuration
and generate a higher volume fraction. The model implemented in the current
study according to Eqs. (10)–(12) does in
fact fall within region I as supported by results given in Figs. 4, 5 and 8. However, in region II, the
volume fraction becomes a decreasing function of log*R*. In this case, the
hydrodynamic drag has surpassed a threshold beyond which it is so strong
relative to the inter-particle potential that particles coalesce rapidly with
less opportunity to reorder or re-

Fig. 9. Sensitivity of cake layer volume fraction to
the relative magnitudes of hydrodynamic drag force vs. inter-particle
potential. The following parameters (in addition to those listed in Table 1) were used in the simulations: (a)
ionic strength=10−^{1},
(b) ionic strength=10−^{2}.

structure their configuration. As a result, the volume fraction decreases with increasing hydrodynamic drag. Switching to a different model for the inter-particle potential or varying the magnitude of the physical parameters inputted into the model may cause the model to shift from region I to region II or vice versa.

Fig.
9(a) and (b) also shows a comparison of the behavior of the ratio log*R
*in response to adjustments in the ionic strength. With a lower ionic
strength of 10−^{2
}M in Fig. 9(b), the values of log*R *decrease
significantly from that of Fig. 9(a). This
is because at a lower ionic strength, inter-particle repulsive interactions are
stronger and therefore greatly reduces the ratio of hydrodynamic force versus
inter-particle repulsion. Nevertheless, even when the interparticle repulsions
are stronger, a threshold still exists where the particle deposition dynamics
will transition into a region II behavior as shown in Fig.
9(b).

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