Exact kinetics of sol-gel transition in a coagulating mixture

Cyclization in bipartite random graphs

In this paper the time evolution of a finite bipartite graph initially comprising two sorts of isolated vertices is considered. The graph is assumed to evolve by adding edges, one at a time. Each new edge connects either two linked components and forms a new component of a larger order (coalescence of graphs) or increases (by one) the number of edges in a given linked component (cycling). Any state of the graph is thus characterized by the set of occupation numbers (the numbers of linked components comprising a given numbers of vertexes of the both sorts and a given number of edges. Once the rate of appearance of an extra edge in the graph being known, the master equation governing the time evolution of the probability to find the random graph in a given state is reformulated in terms of the functional generating the probability to find the evolving graph in a given state. The exact solution of the evolution equation for the generating functional applies for analyzing the average population numbers of linked components. In the limit of large order of the graph the distribution factorizes into two multipliers, one of which is just the spectrum of linked components in the infinite bipartite graph, The second multiplier includes the dependence on the total size of the graph. Both these multipliers contain information on the emergence of the giant component that forms at a critical time.

Kinetic model of two-monomer polymerization

We propose a kinetic gelation model of polymer growth with two monomeric types that have distinct functionalities (reaction sites), and can polymerize using different reaction types. The heterotypic aggregation of two monomer types is modeled using a moment generating function approach by tracking the temporal evolution of a closed system of moment equations up until gelation. We investigate several scenarios of polymerization with two distinct monomers that differ in the types of reactions that can occur. We determine numerical and analytical conditions for finite time blow-up (the emergence of an oligomer of infinite size) that depend on initial conditions, reaction rates, and number of reaction sites per monomer.

Composition distributions of particles in a gelling mixture

Gelation in a two component disperse system wherein binary coagulation governs the temporal changes of particle composition spectra is studied for the crossproduct coagulation kernel proportional to m1n2+m2n1, with m,n being the numbers of monomers of the first and the second component in the coalescing pair of particles. This model reveals the sol-gel transition, i.e., after a finite interval of time the conservation of the total particle mass concentration violates because of the formation of giant particles (the gel). This paper reports on the exact solution of this model for arbitrary initial particle composition spectra. Exact expressions for the particle composition spectrum, the gel mass, and the second moments of the composition distribution are derived. Two scenarios of gelation, where the gel is either active or passive, are considered.

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.

Two-dimensional colloidal aggregation mediated by the range of repulsive interactions

We study the effect of the interaction's range on the structural and kinetic properties of a computer-simulated two-dimensional aggregating colloidal system. For this purpose, we considered that the particles of the system interact through a repulsive Yukawa potential which depends on two parameters: the value of the interaction potential between particles in contact V0 and the range of the interaction kappa(d) . We observed that the increase of the interaction range or V0 provokes the arrangement of the small aggregates in linear structures. The repulsive interactions have also a strong influence on the kinetic behavior of the coagulation process. Indeed, they induce the formation of three different time-separated aggregation regimes. In the first regime (at early states) the aggregation is dominated by the range of the repulsive forces, and the cluster-cluster repulsion increases with the cluster size. The second regime (at intermediate times) is reached when the average cluster size is larger than the interaction range. Here, the cluster-cluster repulsions do not grow anymore with the cluster size, so the probability of overcoming the repulsive barrier is the same for all clusters. This corresponds with the so-called reaction-limited-cluster-aggregation regime, where more than one collision between the clusters is needed to form a bond. The third aggregation regime is found at long aggregation times. In this region the coagulation is mainly determined by the diffusion time and the kinetics becomes diffusion controlled. A physical interpretation for the transition between chain structures and the typical fractals aggregates from the point of view of the range of the interactions is discussed. Moreover, a method has been developed in order to obtain the effect of the interactions with a non-negligible range over the aggregation rates directly from the simulations. The relation between these different regions with the parameters of the interaction potential V0 and kappa(d) is analyzed.