Irreversible aggregation kinetics: Power-law exponents from series

Cluster-size distribution in the autocatalytic growth model

We generalize the model of transition-metal nanocluster growth in aqueous solution, proposed recently [J. Jȩdrak, Phys. Rev. E 87, 022132 (2013)], by introducing a more complete description of chemical reactions. In order to model time evolution of the system, equations describing chemical reaction kinetics are combined with the Smoluchowski coagulation equation. In the absence of coagulation and fragmentation processes, the model equations are solved in two steps. First, we obtain the explicit analytical form of the i-mer concentration, ξ(i), as a function of ξ(1). This result allows us to reduce considerably the number of time-evolution equations. In the simplest situation, the remaining single kinetic equation for ξ(1)(t) is solved in quadratures. In a general case, we obtain a small system of time-evolution equations, which, although rarely analytically tractable, can be relatively easily solved numerically.

Exact solutions of kinetic equations in an autocatalytic growth model

Kinetic equations are introduced for the transition-metal nanocluster nucleation and growth mechanism, as proposed by Watzky and Finke [J. Am. Chem. Soc. 119, 10382 (1997)]. Equations of this type take the form of Smoluchowski coagulation equations supplemented with the terms responsible for the chemical reactions. In the absence of coagulation, we find complete analytical solutions of the model equations for the autocatalytic rate constant both proportional to the cluster mass, and the mass-independent one. In the former case, ξ(k)=s(k)(ξ(1))[proportionality]ξ(1)(k)/k was obtained, while in the latter, the functional form of s(k)(ξ(1)) is more complicated. In both cases, ξ(1)(t)=h(μ)(M(μ)(t)) is a function of the moments of the mass distribution. Both functions, s(k)(ξ(1)) and h(μ)(M(μ)), depend on the assumed mechanism of autocatalytic growth and monomer production, and not on other chemical reactions present in a system.

Kinetics of migration-driven aggregation processes on scale-free networks

We propose a solvable model for the migration-driven aggregate growth on completely connected scale-free networks. A reversible migration system is considered with the produce rate kernel K(k;l|i;j) approximately k(u)i(upsilon)(lj)(nu) or the generalized kernel K(k;l|i;j) approximately (k(upsilon)i(omega)+k(omega)i(upsilon)(lj)(nu), at which an i-mer aggregate locating on the node with j links gains one monomer from a k-mer aggregate locating on the node with l links. It is found that the evolution behavior of the system depends crucially on the details of the rate kernel. In some cases, the aggregate size distribution approaches a scaling form and the typical size S(t,l) of the aggregates locating on the nodes with l links grows infinitely with time; while in other cases, a gelation transition may emerge in the system at a finite critical time. We also introduce a simplified model, in which the aggregates independently gain or lose one monomer at the rate I(1)(k;l)=I(2)(k;l) proportional to k(omega)l(nu), and find the similar results. Most intriguingly, these models exhibit that the evolution behavior of the total distribution of the aggregates with the same size is drastically different from that for the corresponding system in normal space. We test our analytical results with the population data of all counties in the U.S. during the past century and find good agreement between the theoretical predictions and the realistic data.

Kinetics of a migration-driven aggregation process with birth and death

We propose an irreversible aggregation model driven by migration and birth-death processes with the symmetric migration rate kernel K(k;j)=K'(k;j)=Ikj(upsilon), and the birth rate J(1)k and death rate J(2)k proportional to the aggregate's size k. Based on the mean-field theory, we investigate the evolution behavior of the system through developing the scaling theory. The total mass M1 is reserved in the J(1)=J(2) case and increases exponentially with time in the J1>J2 case. In these cases, the long-time asymptotic behavior of the aggregate size distribution a(k)(t) always obeys the scaling law for the upsilon<or=2 case. This model may provide a more natural description for diverse aggregation processes such as the evolution of the distribution of city population and individual wealth.

Kinetics of the catalysis-driven aggregation processes

We study a catalysis-driven aggregation model in which irreversible growth of A aggregates occurs only with the help of the catalyst. The results show that kinetics of the system depends strongly on whether the catalyst coagulates by itself or not. The mass distribution of A clusters obeys a conventional scaling law in the case without self-coagulation of the catalyst, while for the reverse case the evolution of the system falls in a peculiar scaling regime. Our theory applies to diverse phenomena such as the cluster-size distribution in a chemical system.

Kinetics of migration-driven aggregation processes

We study the kinetic behavior of the growth of aggregates driven by reversible migration between any two aggregates. For the simple system with the migration rate kernel K(k;j)=K(')(k;j) proportional, variant kj(upsilon) at which the monomers migrate between the aggregates of size k and those of size j, we find that for the upsilon< or =2 case the evolution of the system always obeys a scaling law. Moreover, the typical aggregate size grows as exp(2IA(0)t) in the case of upsilon=2 and as t(1/(2-upsilon)) in the case of -1<upsilon<2. In particular, when upsilon< or =-2, the typical aggregate size always grows as t(1/3) and the aggregate-size distribution approaches a similar scaling form.

Solvable n-species aggregation processes with joint annihilation

We study the kinetic behavior of the aggregation-annihilation processes of an n-species (n> or =3) system, in which an irreversible aggregation reaction occurs between any two clusters of the same species and an irreversible complete annihilation reaction occurs only between one certain A(n) species and each of the other A(m) species (m=1,2,...n-1). Based on the mean-field theory, we investigate the rate equations of the processes to obtain the asymptotic solutions of the cluster-mass distributions in several different cases. The results show that the evolution behavior of the system depends crucially on the ratios of the equivalent aggregation rate of A(m) species and the aggregation rate of A(n) species to the annihilation rate. The cluster-mass distribution of each species always obeys a conventional scaling law or a modified one, and the scaling exponents depend only on the reaction rates for most cases. However, when both the equivalent aggregation rate of A(m) species and the aggregation rate of A(n) species are twice as large as the annihilation rate, the scaling exponents depend on the reaction rates as well as the initial concentrations.

Kinetic behavior of aggregation processes with complete annihilation

The kinetic behavior of an aggregation-annihilation process of an n-species (n> or =2) system is studied. In this model, an irreversible aggregation reaction occurs between any two clusters of the same species and an irreversible complete annihilation reaction occurs between any two different species. Based on the mean-field theory, we investigate the rate equations of the process with constant reaction rates to obtain the asymptotic solutions for the cluster-mass distributions. We find that the cluster-mass distribution of each species satisfies a modified scaling law, which reduces to the standard scaling law in some special cases. The scaling exponents of the system may strongly depend on the reaction rates for most cases; however, for the case with all the aggregation rates twice the annihilation rate, these exponents depend only on the initial concentrations. All the species annihilate each other completely except in the case in which at least one aggregation rate is less than twice the annihilation rate.