Flocculation models

It is demonstrated in Figure 22.11 that the quasi-static stress-strain cycles at different prestrains of silica-filled rubbers can be well described in the scope of the above-mentioned dynamic flocculation model of stress softening and filler-induced hysteresis up to large strain. Thereby, the size distribution < ( ) has been chosen as an isotropic logarithmic normal distribution (< ( i) = 4> ) = A( 3)) ... 

In particular it can be shown that the dynamic flocculation model of stress softening and hysteresis fulfils a plausibility criterion, important, e.g., for finite element (FE) apphcations. Accordingly, any deformation mode can be predicted based solely on uniaxial stress-strain measurements, which can be carried out relatively easily. From the simulations of stress-strain cycles at medium and large strain it can be concluded that the model of cluster breakdown and reaggregation for prestrained samples represents a fundamental micromechanical basis for the description of nonlinear viscoelasticity of filler-reinforced rubbers. Thereby, the mechanisms of energy storage and dissipation are traced back to the elastic response of tender but fragile filler clusters [24]. 

The various flocculation models which are valid in the different regimes described above allow one to compute the particle/ particle collision rate for any given particle sizes, chemical and physical condition. From the magnitude of this collision rate, one can estimate a colloidal system s stability in cases (iv) and (v). However, in cases (ii) and (iii), both flocculation and creaming will be important in the colloidal breaking process. Consequently, in order to determine whether a colloidal system will be stable in these two cases, we have to determine the net rate of particle loss due to both creaming and flocculation. 

Particle concentration and size distribution in raw water have extensive and complex effects on the performance of individual treatment units (flocculator, sedimentation tank, and filter) and on the overall performance of water treatment plants. Mathematical models of each treatment unit were developed to evaluate the effects of various raw water characteristics and design parameters on plant performance. The flocculation and sedimentation models allow wide particle size distributions to be considered. The filtration model is restricted to homogeneous suspensions but does permit evaluation of filter ripening. The flocculation model is formulated to include simultaneous flocculation by Brownian diffusion and fluid shear, and the sedimentation model is constructed to consider simultaneous contacts by Brownian diffusion and differential settling. The predictions of the model are consistent with results in water treatment practice. 

Figure 5. Flow diagram of computer program for flocculation model...

With four vertical boxes and 21 particle sizes, there are 84 equations of the form of Equation 16 that must be integrated simultaneously. The format of the computer program is similar to that described previously for the flocculation model. The use of more particle sizes or vertical boxes does not significantly alter the results developed here, but does increase the computer time necessary to obtain a solution. With the 84 equations, the simulation of 2 hr of settling is accomplished with approximately 1 min of computer time. 

The first clear-cut demonstration that unattached water soluble polymers can flocculate model sterically stabilized dispersions was provided by Li-in-on et al. (1975). Subsequently, Vincent, with these and other coworkers, has further elaborated the phenomenology of flocculation by free polymer in aqueous dispersions (Cowell et al., 1978 Vincent et al., 1980 Cowell and Vincent, 1982a). 

A more refined Monte Carlo flocculation model was considered, based on energetic considerations rather than sticking probability factors, and included charged polymers (polyelectrolytes), the effect of solution pH, and the rearrangement of biopolymers at the particle surface. This model is presented in the next section, where the formation of a simple polymer-particle entity is examined, due to the required computational time. 

THE DYNAMIC FLOCCULATION MODEL STRESS SOFTENING AND FILLER INDUCED HYSTERESIS... 

The dynamic flocculation model of stress softening and filler induced hysteresis assumes that the breakdown of filler clusters during the first deformation of the virgin samples is totally reversible, though the initial virgin state of filler-... 

A consideration of the dispersion/flocculation model suggests that during flocculation parts of the polymer matrix can no longer be adsorbed on the surface of the conductivity phase but act as a connecting sleeve that surrounds the particles, like the skin of a snake that has swallowed a number of golf balls (as shown later in Figs. 19.39 and 19.40). 

With respect to Equation 5.37, the aggregates flocculation model allows thus, to consider that the low strain modulus of a CB filler compounds... 

Comparing the aggregate flocculation model (Equation 5.46) with the Guth and Gold equation data used Np(D)=200 particles, d=30nm, D=200 nmandfl=2 nm, G(0)=1 MPa,... 

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