Kinetics of cluster growth by aggregation

Abstract

Abstract. The classical problem of the coalescence of isolated species to produce growing clusters/colloids/polymers by successive statistical encounters having the same rate constant, is revisited using numerical simulation for a maximum nuclearity value of a few 10 3 units. The evolution with time of the abundance of clusters of a given nuclearity and of the total population, and the distribution of sizes at a given time are obtained and compared with models from the literature. A remarkable feature of these curves is that they exhibit parity effects for the nuclearity, even clusters being systematically more abundant than odd ones. For easier comparison with experiments, some simulated curves are presented in the tbrm of an approximated analytical expression: kinetics of the total population, and of the monomer, dimer and higher oligomers populations, amplitudes at the maximum and delay for the maximum as functions of the nuclearity, size distribution at a given time. The validity of the approximations is discussed.

Growth and Reactivity of Silver Clusters in Cyanide Solution

“…The critical induction time, t c , depends strongly on the initial atom concentration, fixed by the dose per pulse, on the coalescence rate constant and on the critical nuclearity selected by the donor potential. At variable doses and at a given ratio [Ag(CN) 2 -] 0 /[MV 2+ ] 0 ) 1, yielding a constant initial ratio of silver atoms and donor molecules according to the scavenging competition for e aq -and radicals, it is observed (Figure 4) that t c -1 depends linearly in the range of uncertainty of measurements on the initial total absorbance at 700 nm (which is proportional to the dose), as has already been found in the kinetics model 1,24 and experimentally in the silver sulfate 1 or polyacrylate 4 systems.…”

“…21 Also, as in the latter case, the coalescence rate constant k d between two clusters may be supposed independent of the nuclearity n. Thus, the k d value can be derived from results on the growth process kinetics. In fact, from the general model of growth kinetics 1,24 resulting from all coalescence reactions between two clusters of any size, it was stated that the kinetics should be superimposed, provided the total cluster concentration Σx n is normalized relative to the initial concentration of atoms x 0 ) [Ag] t)0 as Σx n /x 0 and the time is expressed as t/τ with τ ) 1/k d x 0 .…”

“…The plateau absorbance is also lower with cyanide, due to a smaller yield, and therefore the noise is more pronounced. According to the model 1,24 and assuming the same variation of the normalized value (n)/ (nf∞) in both media, the superimposition of the 21 it follows from the signal comparison that k d ) (6 ( 2) × 10 6 L mol -1 s -1 in the presence of cyanide, i.e. more than 30 times slower than for the sulfate.…”

Development kinetics of silver clusters on silver halides

“…However, it is a much more complex system than in the previous study in free solution (Mostafavi et al, 1989;Sahyun et al, 1993;Khatouri et al, 1993Khatouri et al, , 1995a because it is twice microheterogeneous: first the silver ions belong to insoluble AgCl crystallites and second, after reduction, the silver atoms are clustering at the surface of these crystallites. The interest is to study how the environment is affecting the redox potentials and the kinetics processes compared to the previous study in free solution (Mostafavi et al, 1989).…”

“…The rate constant values of some of the above reactions in a homogeneous aqueous phase (k w ) are known from the literature (Buxton et al, 1988;Khatouri et al, 1993Khatouri et al, , 1995a: k 8;w ¼ 8:4 Â 10 10 M À1 s À1 ; k 10;w ¼ 5:5 Â 10 9 M À1 s À1 ; k 11;w ¼ 2:3 Â 10 10 M À1 s À1 ; k 13;w ¼ 3:5 Â 10 9 M À1 s À1 ;…”