Diffusion-controlled coagulation

The potential energy barrier to coagulation can be reduced to zero by the addition of excess electrolyte, which creates a situation in which every encounter between the particles leads to permanent contact. The theory of rapid (diffusion-controlled) coagulation was developed by Smoluchowski110. For a monodispersed sol containing... 

For a hydrosol at room temperature, the time ty2 in which the number of particles is halved by diffusion-controlled coagulation is calculated from the above equations to be of the order of 10u//io seconds, if n0 is expressed in the unit, particles cm-3. In a typical dilute hydrosol, the number of particles per cm3 may be about 10lw-10u, and so, on this basis, tm should be of the order of a few seconds. 

By mistake, a much too high salt concentration is used in the coating and diffusion-controlled coagulation takes place. How long time does it take before 90% of aU the colloids are coagulated ... 

Diffusion-controlled coagulation can be calculated using Equation 11.2 ... 

It can be seen that Equation 11.4.22 is independent of particle size and all particles and droplets can be pooled in one calculation. The term k2° is the rate constant for diffusion-controlled coagulation. 

Smoluchowski, who worked on the rate of coagulation of colloidal particles, was a pioneer in the development of the theory of diffusion-controlled reactions. His theory is based on the assumption that the probability of reaction is equal to 1 when A and B are at the distance of closest approach (Rc) ( absorbing boundary condition ), which corresponds to an infinite value of the intrinsic rate constant kR. The rate constant k for the dissociation of the encounter pair can thus be ignored. As a result of this boundary condition, the concentration of B is equal to zero on the surface of a sphere of radius Rc, and consequently, there is a concentration gradient of B. The rate constant for reaction k (t) can be obtained from the flux of B, in the concentration gradient, through the surface of contact with A. This flux depends on the radial distribution function of B, p(r, t), which is a solution of Fick s equation... 

Colloidal solutions are characterized by the degree of stability or instability. This is related to the fact that both kinds of properties in everyday phenomena need to be understood. The kinetics of coagulation is studied using different methods. The number of particles, Np, at a given time is dependent on the diffusion-controlled process. The rate is given by... 

If the electrostatic barrier is removed either by specific ion adsorption or by addition of electrolyte, the rate of coagulation (often followed by measuring changes in turbidity) can be described fairly well from simple diffusion-controlled kinetics and the assumption that all collisions lead to adhesion and particle growth. Overbeek (1952) has derived a simple equation to relate the rate of coagulation to the magnitude of the repulsive barrier. The equation is written in terms of the stability ratio ... 

Calculate a second-order rate constant, k2, and compare it with the value, 2, calculated on the assumption that coagulation is a diffusion-controlled process. 

The classical treatment of such processes derives from the consideration of the coagulation of colloids (Smoluchowski, 1917), but many accounts have been given of how the same approach can be used for diffusion-controlled reactions (Noyes, 1961 North, 1964 Moelwyn-Hughes, 1971). The starting point is the assumption of a random distribution of the two reactants (here given the symbols X and B) in the solution. Then, if B is capable of reacting on encounter with a number of molecules of X, it follows that such reactions deplete the concentration of X in the neighbourhood of B and therefore set up a... 

If the energy barrier to aggregation is removed (e.g., by adding excess electrolyte) then aggregation is diffusion controlled only Brownian motion of independent droplets or particles is present. For a monodisperse suspension of spheres, Smoluchowski developed an equation for this rapid coagulation ... 

Suppose the aerosol contained in a large chamber is composed of particles larger than the mean free path of the gas. The surface-to-volume ratio of the chamber is sufficiently small to neglect deposition on the walls, and the composition of the system is uniform. Coagulation takes place, and at the same time the particles grow as a result of diffusion-controlled condensation but sedimentation can be neglected. Homogeneous nucleation does not occur and the system is isothermal. A system of this type has been used to model aerosol formation in photochemical air pollution. 

The electrolyte induced rapid coagulation of polydisperse mixtures of polystyrene microspheres, titania and kaolinite was studied by Photon Correlation Spectroscopy (PCS). A method has been developed that enables the rate constant to be calculated without precise knowledge of the particle size distribution (7). For Brownian Diffusion-controlled Aggregation ... 

Strength of the aqueous medium [149], At high electrolyte concentrations, the repulsion between the particles vanishes and the coagulation of the particles is fully diffusion controlled [150, 151]. 

This treatment of the rate of diffusion-controlled encounters in solution was developed originally by Smoluchowski [13,a] for the rates of coagulation of colloidal solutions, and was later applied to reactions between molecules. It assumes that the diffusive motions of molecules can be treated like those of macroscopic particles in a continuous viscous fluid. A simplified version is as follows. 

Use of implies that the supposed process of heterocoalescence between antifoam drops and bubbles is diffusion controlled. In practice, it seems likely, however, that convection will be important under the conditions assumed for this model. The adjustable coagulation efficiency factor presumably accounts for the probability of antifoam drops overcoming colloidal repulsion forces to adhere to bubbles to which... 

Solidification of the aramid filaments is called coagulation but is in fact a simple freezing-in of the solution. It is not a speed-limiting step. The subsequent removal of sulfuric acid from the solid filaments in the washing process is diffusion-controlled, however, and hence relatively slow. 

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