Particle Capture with Surface Forces

The basic difference between particle capture in the case of non-zero molecular and hydrodynamic forces (Fig. 10.9) from particle capture in the case when these forces are ignored (Fig. 10.7) is in the limiting trajectory dividing the family of trajectories into those that lead the particles to collide with the cylinder, and those that lead the particles away from the cylinder. In the first case the limiting trajectory ends at the back critical point of the cylinder Op = tt, and in the second case -at the point of contact between particle Up and the cylinder surface at Op = njl. 

At the times when DLVO theory was developed, the direct measurement of forces between colloidal particles and surfaces in solution was not possible, and the macroscopic observation of colloidal stability was the only experimental reference data. With increasing technological advancement, setups have been developed for the direct observation of such forces The surface force apparatus (SFA) allows for the measurements of forces between surfaces in solution [6], and with an atomic force microscope (AFM), forces on a colloidal particle can be detected [7]. It is a major success that DLVO theory predicts forces that agree nicely with the measured forces for large particle separations (more than 3-10 nm), but at the same time, it is obvious that in the regime of short particle separations, not aU effects are captured by DLVO. When the barrier for coagulation occurs at such low separations, the DLVO prediction for colloidal stability is not accurate (Fig. 2). 

The total number of independent variables appearing in Fq. (4.32) is thus quite large, and in fact too large for practical applications. However, as mentioned earlier, by coupling Eq. (4.32) with the Navier-Stokes equation to find the forces on the particles due to the fluid, the Ap-particle system is completely determined. Although not written out explicitly, the reader should keep in mind that the mesoscale models for the phase-space fluxes and the collision term depend on the complete set of independent variables. For example, the surface terms depend on all of the state variables A[p ( x ", ", j/p" j, V ", j/p" ). The only known way to determine these functions is to perform direct numerical simulations of the microscale fluid-particle system using all possible sets of initial conditions. Obviously, such an approach is intractable. We are thus led to reduce the number of independent variables and to introduce mesoscale models that attempt to capture the average effect of multi-particle interactions. 

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