The operator, K, is known as the coagulation kernel and describes the rate at which particles of size coagulate with particles of size. Analytic solutions to the equation exist when the kernel takes one of three simple forms: known as the constant, additive, and multiplicative kernels respectively. For the case it could be mathematically proven that the solution of Smoluchowski coagulation equations have asymptotically the dynamic scaling property. This self-similar behaviour is closely related to scale invariance which can be a characteristic feature of a phase transition. However, in most practical applications the kernel takes on a significantly more complex form. For example, the free-molecular kernel which describes collisions in a dilute gas-phase system, Some coagulation kernels account for a specific fractal dimension of the clusters, as in the diffusion-limited aggregation: or Reaction-limited aggregation: where are fractal dimensions of the clusters, is the Boltzmann constant, is the temperature, is the Fuchs stability ratio, is the continuous phase viscosity, and is the exponent of the product kernel, usually considered a fitting parameter. Generally the coagulation equations that result from such physically realistic kernels are not solvable, and as such, it is necessary to appeal tonumerical methods. Most of deterministic methods can be used when there is only one particle property of interest, the two principal ones being the method of moments and sectional methods. In the multi-variate case, however, when two or more properties are introduced, one has to seek special approximation methods that suffer less from curse of dimensionality. Approximation based on Gaussian radial basis functions has been successfully applied to the coagulation equation in more than one dimension. When the accuracy of the solution is not of primary importance, stochastic particle methods are an attractive alternative.
Condensation-driven aggregation
In addition to aggregation, particles may also grow in size by condensation, deposition or by accretion. Hassan and Hassan recently proposed a condensation-driven aggregation model in which aggregating particles keep growing continuously between merging upon collision. The CDA model can be understood by the following reaction scheme where denotes the aggregate of size at time and is the elapsed time. This reaction scheme can be described by the following generalized Smoluchowski equation Considering that a particle of size grows due to condensation between collision time equal to inverse of by an amount i.e. One can solve the generalized Smoluchowski equation for constant kernel to give which exhibits dynamic scaling. A simple fractal analysis reveals that the condensation-driven aggregation can be best described fractal of dimension The th moment of is always a conserved quantity which is responsible for fixing all the exponents of the dynamic scaling. Such conservation law has also been found in Cantor set too.
