Behavior of AF r at v const

Let us examine the behavior of AF(r) when v = const. This situation reflects the process of a stepwise (virtual) grinding of the initial volume v into identical spherical particles whose size decreases with every step of the grinding. Such an approach was used in both the original and the subsequent studies [61,66-68]. In this case, the number of particles, n, is inversely proportional to the third power of the particle size, that is, n = vK l icr ), and Equations 4.6 and 4.7 acquire a simplified form, namely. 

Both expressions show that as the grinding yields smaller and smaller particles, the value of AF grows slowly at first as +l/r from zero up to some maximum value, AF, , corresponding to and then rapidly decays as -1/r. The function then passes through zero at r = and reaches negative values. For the critical value of r = one can write 

Both C=vlV and C=n/N approaches show that a positive maximum of the AF function is observed when dAF/dr = 0, which corresponds to the particle radius Such a maximum is of a 

Physical-Chemical Mechanics of Disperse Systems and Materials 

FIGURE 4.33 Schematic illustration of the behavior of AF/kT as a function of the particle radius, r, on a system approaching molecular dimensions, b a monotonous decrease when there is no factor that prevents dispersion down to molecular dimensions (1) the trend in the presence of a factor resisting dispersion (2) the appearance of the minimum in the AF function positive minimum (3) and negative minimum (4), corresponding to the formation of a stable lyophilic colloidal system. (From Shchukin, E.D., J. Dispers. Sci. TechnoL, 25, 875, 2004.) 

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