Exact solutions of kinetic equations in an autocatalytic growth model

Abstract

Kinetic equations are introduced for the transition-metal nanocluster nucleation and growth mechanism, as proposed by Watzky and Finke. 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 (ξ1) ∝ ξ k 1 /k was obtained, while in the latter, the functional form of s k (ξ1) is more complicated. In both cases, ξ1(t) = hµ(Mµ(t)) is a function of the moments of the mass distribution. Both functions, s k (ξ1) and hµ(Mµ), depend on the assumed mechanism of autocatalytic growth and monomer production, and not on other chemical reactions present in a system. PACS numbers: Keywords:Nucleation and growth phenomena, resulting in occurrence of a new phase from a homogeneous host phase, are ubiquitous in nature [1,2]. In many cases, apart from processes of coagulation and fragmentation, chemical reactions are also present in a system. Frequently, the chemical reactions account for phenomena studied by polymer and colloidal science.Nucleation and subsequent growth of metal nanoclusters in aqueous solution is a topic of considerable current interest, since solution route synthesis is one of the most convenient methods of producing transition-metal nanoparticles [3]. However, applicability of this method depends on the ability to control size and shape of the produced nanoparticles, which determine their unique optical, electronic and catalytic properties. For that reason, theoretical models capable of predicting the cluster size distribution, as well as its dependence on the experimentally controllable parameters of the system, are required.A mechanism of transition-metal colloidal nanoparticle formation has been proposed by Watzky and Finke [4], cf. [5][6][7][8]. The WF mechanism consists of (i) slow monomer, i.e., the zerovalent transition-metal atom (B 1 ) production due to reduction reaction A → B 1 of a metal precursor (A), usually a transition-metal complex coordination compound, (ii) fast autocatalytic reduction reaction A+B i → B i+1 taking place on the surface of growing metal nanoparticles, consisting of i atoms (B i ), and (iii) process of coagulation B i + B j ⇄ B i+j , reversible or otherwise. In the original WF scheme [4], step (iii) had not been considered. Irreversible coagulation was first introduced in [5].For the transition metals in which higher oxidation states are present, e.g. Au, at least one additional preliminary step of the form (iv) P → A is needed [9][10][11][12][13].As an excess of the reducing agent is usually used in the reactions (i), (ii), (iv), its concentration is fairly timeindependent, and all chemical reactions may be treated as irreversible. Furthermore, (i) and (iv) may be treated as reactions of pseudo-first order, while (ii) as a reaction of pseudo-second order.The WF mechani...

Cluster-size distribution in the autocatalytic growth model

“…The remaining (apart from B i ) products of reactions (1) and (2) are collectively denoted X 1 and X 2 . In contrast to our previous treatment [19] of the WF mechanism in its original formulation [20][21][22][23], here the presence of the reducing agent has been explicitly taken into account in both (1) and (2). Usually, as an excess of the reducing agent is used, we may assume that its concentration is time-independent.…”

“…[21]. In addition, although chemical reactions (1) and (2) are assumed to be irreversible due to the presence of large amount of the reducing agent, this does not need to be the case for the physical processes, and (3) is generalized to include fragmentation [19]. Various extensions of the original WF scheme (1)-(3) are possible, and frequently required, depending on the experimental situation at hand.…”

“…Again, (73) corresponds to initial conditions (19) for the ξ k variables. As in thek α = 0, e 0 = 0 case, we first consider k < n [cf.…”

“…III we provide time evolution rate equations of the model, being generalization of those introduced and analyzed in Ref. [19]. In Sec.…”

“…In a recent paper [19], we have introduced reactionaggregation model, which reduces to the model of autocatalytic reaction in absence of coagulation. In such a situation, we were able to find analytical solution of the model equations in two nontrivial cases.…”