Revisiting the Theory of Coagulation of Colloidal Dispersions: An Improved Expression for the Stability Ratio

Abstract

The stability of a colloidal dispersion has long been expressed in terms of the stability ratio. Based on the available theories of coagulation of colloidal dispersions, a novel expression, complying with the classical definition, is developed for the stability ratio. It accounts for the contributions of both primary and secondary minimum coagulations to the overall rate of coagulations. In addition, it can also be regarded as the result of a combination of the kinetic theory of an ideal gas and the Smoluchowski theory with Fuchs' correction, considering the interaction between identical spherical particles and their surfaces immersed in a symmetrical electrolyte solution. The agreement with experimental data suggested that it is superior to the classical ones in describing the weak dependence of the stability ratio on the particle size and the valence of the counterion, by emphasizing the importance of the secondary minimum coagulation in dispersion stability and the complementation between the two modes of coagulation.

Update to the Newly Developed Expression for the Stability Ratio of Colloidal Dispersions

“…However, as pointed out by Liu et al, such a definition of f P and f S works well only if ΔΦ > (−Φ min ), since then the combination of particle speeds which leads to the fact that one type of coagulation is not effective for the other type . In otherwise cases, f P + f S > 1 would result, which is certainly not plausible.…”

“…To account for the mechanism that PMC and SMC may occur simultaneously in dispersed systems and that not every collision between particles leads to permanent attachment, it is necessary to introduce efficiency factors or fractions of successful collisions to correct Fuchs’ formula for the stability ratio. This motivates Liu et al to write the total rate constant, k P&S , for coagulation of an initially monodispersed colloidal system as k normalP & normalS = 8 k normalB T 3 μ f normalP 2 ∫ 0 ∞ exp ( Φ false( u false) k B T ) normald u false( 2 + u false) 2 + 8 k normalB T 3 μ f normalS 2 ∫ u min ∞ exp ( Φ false( u false) k B T ) normald u false( 2 + u false) 2 where f P is the fraction of particles which overcome the potential barrier and adhere at the primary minimum, and f S is the fraction of particles which arrive at the secondary minimum and remain to form agglomerates.…”

“…The expression given in eq is a variant form of the one proposed by Liu et al It makes, however, the contributions of PMC and SMC to the stability ratio, W P&S , much clearer. In addition, it should be noted that W PMC reduces to McGown–Parfitt’s formula for W PMC in the case of f P = 1, and that W SMC becomes Bagchi’s formula for W SMC in the case of unprotected particles (absence of any surface charge). Therefore, one can envisage the new expression of W P&S as a generalization/combination of the studies of McGown and Parfitt and Bagchi by accounting for the efficiency of successful collisions that lead to coagulation at both the primary and the secondary minimum.…”

“…Since adhesion at the primary minimum is mostly irreversible, while adhesion at the secondary minimum is likely to be reversible, a pair of particles once get chance to move from the secondary minimum to the primary minimum will stuck into the deep well forever. This happens whenever the total kinetic energy Φ of a single particle or two interacting particles be greater than ΔΦ, according to the imagination of Marmur . Hence, we should keep f P as it was originally defined, but modify f S as f normals = f ( min ( Δ Φ , false| normalΦ min false| ) / k B T ) …”

“…10 In otherwise cases, f P + f S > 1 would result, which is certainly not plausible. To tackle this problem, Liu et al 2 proposed to keep f S as it is but replace f p by f P (1 − f S ). The consequence is, then, that the sum of f P and f S never exceeds unity.…”