Aggregation kernel constant

In the case of a constant aggregation kernel it is possible to obtain... 

A quick demonstration may be made of the constant aggregation kernel, which allows exact calculation of the self-similar solution. We let a x, x ) = a, and recognize it to be homogeneous as described by (5.2.14) with m = 0 so that the scaling size h t) is proportional to t. Equation (5.2.16) may be solved by Laplace transform, which is left as an exercise to the reader. However, the reader may more readily verify that for this case by... 

Smoluchowski also presented a simple theory of aggregation kinetics assuming collisions of perfect collection efficiency to predict spherical particle size distributions in a uniform liquid shear field of constant velocity gradient. The aggregation kernel is then expressed as... 

The time evolution of the cluster size distribution, arising in an aggregating system may be obtained in the framework of Smoluchowski s equation once the aggregation kernel, that is, the set of aggregation rate constants is known. However, no valid kernel for the description of aggregation processes of functionalized particle suspensions is known [54]. In order to overcome this difficulty, we use the kernel... 

The Kij in Eqs. (1)—(3) is the rate constant of aggregation. The set of rate constants for all i and j, often called the coagulation kernel, is an infinite symmetric matrix with nonnegative elements. In order to keep the form of Kij as simple as possible we shall assume that... 

A constant reaction kernel is to be expected in the absence of enzyme reaction if the aggregation rate is determined by the Brownian motion of spherical particles which coalesce to form larger spheres. To a first approximation, the increased collisional cross-section is then compensated for by the decrease in diffusion rate (von Smoluchowski,... 

X is the crossover time when the balance between the aggregation processes is established. The validity of this approach was confirmed by Montecarlo simulations of the full equation using a constant coagulation kernel and a breakup probability equal to k(i+j) where a and k were adjustable parameters [15]. Spatial fluctuations were compensated by cluster breakup and the generalized Smoluchowski equation had a critical dimension dc < 1. 

We have shown how this set of differential equations is closed in some simple cases (i.e. for nucleation at zero size, with constant growth rate, and constant aggregation and breakage kernels). In general, when the system is not closed, closure must be sought through a stable numerical method. 

As an example, consider a monodisperse population of particles characterized by mass as internal coordinate and moments m = mjt(O) = 1 with k = 0,..., 2N 1. This population of particles is continuously fed to a system wherein particles undergo aggregation and symmetric binary breakage with constant kernels. The equations describing the evolution of the moments are... 

For this specific example, consider the case of constant aggregation and breakage kernels, namelyPa,y,s,[Pg.326]

The studies reviewed here show that precipitation of uniform particles requires two conditions to be met. First, at a point early in the reaction, a constant number of colloidally stable particles must be established. Second, these particles must remain stable to mutual coagulation throughout the subsequent reaction. If the primary particles formed are unstable with respect to larger particles, a short nucleation period is not a prerequisite of a narrow final particle-size distribution. Indeed, continuous slow nucleation with the correct aggregation rate kernels can produce very uniform particles. If the nuclei are unstable and aggregation is very fast, a broad particle-size distribution may result. For colloidally stable nuclei, a short nucleation period will be important in establishing uniform precipitates. However, nuclei with 1-15-nm radii are difficult to stabilize under typical precipitation conditions. 

A discrete version of the master density equations (7.3.10), without particle growth, has been solved by Bayewitz et al (1974), and later by Williams (1979), to examine the dynamic average particle size distribution in an aggregating system with a constant kernel. When the population is small EN < 50) their predictions reveal significant variations from those predicted by the population balance equation. However, the solution of such master density equations is extremely difficult even for the small populations of interest for nonconstant kernels. It is from this point of view that a suitably closed set of product density equations presents a much better alternative for analysis of such aggregating systems. We take up this issue of closure again in Section 7.4. 

The case of constant kernel is similar to the situation that was analyzed in Sect. 4.1. The fast aggregation problem with the constant kernel was exactly solved by Smoluchowski in 1917 [73]. This model is based on the more complicated kernel for 3D Brownian aggregation ... 

It was assumed that the stability ratio is constant in the course of aggregation, Wij Ik"". The initial value, Ik"", was experimentally determined for polymer latex particles in aqueous suspensions. The aggregation rate was measured at the very initial stage, where the presence of triplets was negligible [151]. The kernels in Eqs. 60, 62 were found to be appropriate for simulation of experimental results subject to proper tuning of the exponent X. Moreover, the kernels in Eq. 62 were suitable for description of the continuous transition from DLA-like aggregation at IT = 1 to Rl.A-like aggregation at IT oo [152]. 

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