Exact solution for stochastic degradation and fragmentation processes in arbitrary chain and ring aggregates with multiple bonds

This paper presents a statistical theory of stochastic evaporation and degradation processes in complex polymerlike ring and chain aggregates with multiple degrading bonds between the primary particles (monomers). The exact kinetic solution fully describing fragmentation processes is obtained for such aggregates with arbitrary number of primary particles (monomers) and bonds between them. The effects of additional interaction of multiple bonds with each other is shown to have a drastic impact on the predicted kinetic processes and time-dependent particle size distributions during aggregate degradation. Structural effects associated with different distributions of multiple bonds and bonding configurations in the aggregates are also investigated and shown to have a significant impact on typical fragmentation time and accumulation of fragmenting aggregates in intermediate modes. The developed theory and its results will be important for degradation of multistranded polymers, polymer networks, self-assembling structures, surface nanoclusters and nanotechnology, and formation and evolution of aerosol aggregates resulting from transport and industry emissions.

A study of the collisional fragmentation problem using the Gamma distribution approximation

The nonlinear fragmentation population balance formulation has been elevated in recent years from a prototype for studying nonlinear integro-differential equations to a vehicle for analyzing and understanding several physicochemical processes of technological interest. The so-called pure collisional fragmentation, which is the particular mode of nonlinear fragmentation induced by collisions between particles, is studied here. It is shown that the corresponding population balance equation admits large time asymptotic (self-similarity) solutions for homogeneous fragmentation and collision functions (kernels). The self-similar solutions are given in closed form for some simple kernels. Based on the shape of the self-similar solutions the method of moments with Gamma distribution approximation is employed for transient solution (from initial state to establishment of the asymptotic shape) of the collisional fragmentation equation. These solutions are presented for several sets of parameters and their behavior is discussed rather extensively. The present study is similar to the one has already been performed for the case of the much simpler linear fragmentation equation [G. Madras, B.J. McCoy, AIChE J. 44 (1998) 647].

The equal-size binary breakage problem: evolution toward a steady shape or periodic behavior?

During the last few years, the self-similar particle size distribution for a particle population undergoing breakage in equal size fragments has been derived using approximating, numerical, and analytical means. But very recently it was shown [N.V. Mantzaris, J. Phys. A Math. Gen. 38 (2005) 5111] through transient simulation of the breakage process that the particle size distribution in case of breakage in two equal fragments, never attains a steady shape, i.e., a self-similar form. The new results give rise to questions about the real meaning and utility of the previously derived self-similar distributions for these systems. The scope of the present work is to answer these questions and it is attempted using only analytical (exact) means for the solution of the transient breakage problem. In doing so, the very interesting and rich underlying structure and properties of the solutions of the equal size breakage problem (seemingly, very simple) are revealed. It appears that the utility of the known self-similar distributions for this particular problem has to be redefined but yet not entirely abandoned.

On the self-similar solution of fragmentation equation: Numerical evaluation with implications for the inverse problem

It is well known that the fragmentation equation admits self-similar solutions for evolving particle-size distributions (PSD); i.e., if the shape of PSD is independent of time after an initial transient period. Although an analytical derivations of the self-similar PSD cases have been studied extensively, results for cases requiring numerical solutions are rare. The aim of the present work is to fill this gap for the case of homogeneous breakage functions. The known analytical and approximate solutions for the self-similar PSD are reviewed and a general algorithm for the numerical solution is proposed. Results for a broad range of breakage functions (kernel and rate) are presented. Further, the work is focused on the sensitivity of the relation between self-similar PSD and breakage kernel and its influence on the inverse breakage problem, i.e., that of estimating the breakage kernel from experimental self-similar PSDs. Useful suggestions are made for tackling the inverse problem.

Numerical and Similarity Solutions for Reversible Population Balance Equations with Size-Dependent Rates

Population balance equations (PBEs) for reversible aggregation-fragmentation processes are important to particle agglomeration and dissolution, polymerization and degradation, liquid droplet coalescence and breakup, and floc coagulation and disintegration. Moment solutions provide convenient solutions to the PBEs, including steady state and similarity solutions, but may not be feasible for complex forms of size-dependent rate coefficients and stoichiometric kernels. Numeric solutions are thus necessary not only for applications, but also for the study of the mathematics of PBEs. Here we propose a numerical method to solve PBEs and compare the results to moment solutions. The numeric results are consistent with known steady state and asymptotic long-time similarity solutions and show how processes can be approximated by self-similar formulations.