Doublet formation

Doublet formation is the first step of aggregate or cluster formation. When salt or pH is used to destabilize a colloidal suspension it is referred to as coagulation. When a polymer or surfactant is used to destabilize a colloidal suspension it is referred to as flocculation. The kinetics of doublet formation for both these methods of destabilizing a colloidal suspension is discussed in this section. 

The rate at which particles coagulate when the interaction energy between the particle is 0 was first investigated by von Smoluchowski 

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As mentioned above, in 1917 M. Smoluchowski applied the theory of diffusion to this situation to evaluate the rate of doublet formation. According to Fick s first law (Equation (2.22)) J, the number of particles crossing a unit area toward the reference particle per unit of time is given by... 

Fic. 36. Effect of the initial number density of polystyrene latices (2a = 0.7 /im) on the fraction of singlets remaining 1.5 hours after the addition of PEO at 0.34 mg/m2. The dashed line characterizes the doublet formation with relaxed chains and b indicates the transition to flocculation before reconformation (Pelssers, 1988). 

Problem 10.2. Determine the Half-Life for Doublet Formation for Various Initial Number Densities of Particles in Water... 

When Vy is 0 for all separations between the particles, W is equal to 1. In all practical cases, however, the interaction energy between the particles is not equal to 0, but is either repulsive or attractive or a mixture of both. The rate of collision between two particles is thus increased or decreased by the presence of an attractive or repulsive force between them. If we have only an attractive interaction (Vy is negative), the value of W is less than unity. W has been predicted (and measured [62]) to be as low as 0.5 for nonretarded attractive Hamaker interactions [3, p. 276 63]. If we have only a repulsive interaction (Vy is positive), W is larger than unity. Values of W can be obtained from doublet formation kinetic measurements as determined from photon... 

Studies of the doublet formation rate in polymer stabilized systems are far less numerous than those on electrostatically stabilized systems. Chang [68] has studied the doublet formation rate for 0.57 fim in diameter Si02 particles as a function of the amount of hydroxyl propyl cellulose (HPC) adsorbed onto their surface. Figure 10.32 is a plot of the colloid stability ratio as a function of the amount of HPC added to... 

Feigin et al. also calculated a second extreme case, that of doublet formation. They showed that the configurational free energy change per doublet is given by an expression closely analogous to the other limiting case... 

The singlet concentration decreases with time due to doublet formation, while the doublet concentration increases. As a result, the rates of aggregation and floe fragmentation will approach each other. Correspondingly, the change in the number of doublets [Pg.73]

Examine the spectrum of 2-methylbenzaldehyde shown below in Figure 4.6 and identify the aldehyde C-H stretching band. Describe the effect responsible for the doublet formation. 

The same type of kinetic equation as we have been using for doublet formation was put forward by von Smoluchowski in 1917 to describe the formation of any aggregate containing m particles. If is the number of m-particle aggregates per unit volume, their rate of formation is given by... 

Here, we have also assumed that the particles do not interact before they are in contact and all collisions lead to doublet formation. Moreover, hydrodynamic interactions have been neglected. An experimental verification of this formula showed, not unexpectedly, deviations. The coagulation was slower than that predicted by the rate constant given in equation (1.31). Derjaguin, in 1966, proposed the reason for this was that the particles interact hydrodynamically when they were sufficiently close to each other. The dispersion medium has to be removed from the space between the particles when they approach one another and the motion of the particles is retarded. The effect is in many cases quite large, i.e. about a factor of 2. This can be expressed as a reduction in the diffusion coefficient. Honig and co-workers have derived an approximate equation for how the diffusion coefficient D(H) varies with the interparticle surface-to-surface distance H. The expression is as follows ... 

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