Rate of coagulation

It is assumed that irreversible aggregation occurs on contact. The rate of coagulation is expressed as the aggregation flux J of particles towards a central particle. Using a steady-state approximation, the diffusive flux is derived to be... 

Reerink H and Overbeek J Th G 1954 The rate of coagulation as a measure of the stability of silver iodide sols Discuss. Faraday Soc. 18 74-84... 

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... 

The rate of coagulation of particles in a liquid depends on the frequency of collisions between particles due to their relative motion. When this motion is due to Brownian movement coagulation is termed perikinetic when the relative motion is caused by velocity gradients coagulation is termed orthokinetic. 

The rate of coagulation depends on the height of the potential energy barrier. In the absence of a barrier, the rate of disappearance of the primary particles is controlled by the number of collisions between particles, i. e. 

The mean radius of the sphere of action around each particle calculated from the observed rate of coagulation is 31 82 x 10 cm. or some one and a quarter times the actual radius of the particle. 

The rate of coagulation of night blue examined by Biltz and Steiner Zeit, Phys. Ohem, LXXili. 607, 1910) and of benzo-purpurin (Biltz and v, Vegesack, Zeit. Phys. Ghem. lxxiii. 500, 1910) likewise appear to conform reasonably closely to a reaction of the autocatalytic variety. 

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 ... 

Both secondary and primary minimum coagulation are observed in practice and the rate of coagulation is dependent on the height of the barrier. In general, coagulation into a primary minimum is difficult to reverse, whereas coagulation into a secondary minimum is often easily reversed, for example, by diluting the electrolyte. DLVO theory tells us... 

Using the approach developed in Example 13.3 and interaction energy expressions for spherical particles, it has been possible to predict how the stability ratio W varies with electrolyte concentration according to the DLVO theory. Since W can be measured by experimental studies of the rate of coagulation, this approach allows an even more stringent test of the DLVO theory than CCC values permit. We shall not bother with algebraic details, but instead go directly to the final result ... 

The presence of velocity gradients in the system may also increase the rate of coagulation above the value given by Equation (24) or (42). 

Exposure to light, including ultraviolet and infra-red, has little effect either on the cataphoretic speed or the rate of coagulation of arsenious sulphide sols.3 The absorption spectrum of the hydrosol has been studied4 and shows that there is simple absorption which increases with the size of the particles, and also a selective absorption in the region... 

The rate of coagulation depends upon the collision frequency, which is controlled by physical parameters describing perikinetic or ortho-kinetic particle transport (temperature, velocity gradient, number concentration and dimension of colloidal particles), and the collision efficiency factor a measuring the extent of the particle destabilization which is primarily controlled by chemical parameters. 

The existence of surface charge prevents coagulation in a low ionic strength suspension or sol of one solid component and thereby slows sedimentation (86, 87). The rate of coagulation and sedimentation can be increased by adjusting chemical conditions so that the surface charge is zero. Many examples can be found in the extensive literature of colloid and soil chemistry. 

Baxendale, Evans and coworkers reported in 1946 that the polymerization of methyl methacrylate (MMA) in aqueous solution was characterized by homogeneous solution kinetics, i.e. where mutual termination of free radicals occurred, in spite of the fact that the polymer precipitated as a separate phase. Increases in the rates of polymerization upon the addition of the surfactant cetyl trimethyl ammonium bromide (CTAB) were attributed to the retardation of the rate of coagulation of particles, which was manifested in a reduction in the effective rate constant for mutual termination,... 

The individual terms must be evaluated in order to obtain numerical values for N. In the simplest case the rate of coagulation, in the presence of adequate stabilizer, will be equal to zero. There is some question as to whether this can be accomplished in practice because of the extreme instability of primary particles which are 1 to 2 nm in size (16, 20). The rate of initiation may be obtained from literature values or from independent measurements. The initiator efficiency, f, must also be obtained. For example, in the thermal decomposition of persulfate ion the rate of initiation is given by... 

The rates of coagulation for the different types of association differ from each other. The EE aggregation is generally the fastest, since it involves the largest collision diameter (the platelet diameter). The FF aggregation is generally the slowest, since it involves the smallest collision diameter (the platelet thickness). Also, the three types of associations will not necessarily occur together or at the same rates when the day flocculates [49]. 

The rate of coagulation is considered to be dominated by a binary process involving collisions between two particles. The rate is given by bn,nj, where nl is the number of particles of z th size and b a collision parameter. For collision between i - and / -sized particles during Brownian motion, the physicist M. Smoluchowski derived the relation ... 

With the average size of the particle, the smaller the size, the faster the rate of coagulation. 

In order to evaluate the rate of coagulation, Eqs. [74], [77], and [78] must be solved for the constants A, B and the unknown dimensionless surface temperature 0. After considerable algebra, one can obtain the functional form of the expression for the constant A as... 

The rate of coagulation, of particles of mass mp is equal to the net rate of flow of particles of reduced mass mT across the sphere of influence of radius Rs, i.e.,... 

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