Supercritical clusters

Figure 5 Comparison between simulated and experimental decay of MV growing clusters Ag ,cN including oxidation of subcritical oligomers by supercritical clusters by MV . The best fit yields Wc = 5. (From Ref 54.)...

Actually, the kinetics study of the redox potential of transient clusters (Section 20.3.2) has shown that beyond the critical nuclearity, they receive electrons without delay from an electron donor already present. The critical nuclearity depends on the donor potential and then the autocatalytic growth does not stop until the metal ions or the electron donor are not exhausted (Fig. 8c). An extreme case of the size development occurs, despite the presence of the polymer, when the nucleation induced by radiolytic reduction is followed by a chemical reduction. The donor D does not create new nuclei but allows the supercritical clusters to develop. This process may be used to select the cluster final size by the choice of the radiolytic/chemical reduction ratio. But it also occurs spontaneously any time when even a mild reducing agent is present during the radiolytic synthesis. The specificity of this method is to combine the ion reduction successively ... 

Equation 19.17 may be interpreted in a simple way. If the equilibrium concentration of critical clusters of size Afc were present, and if every critical cluster that grew beyond size Mc continued to grow without decaying back to a smaller size, the nucleation rate would be equal to J = (3CNexp[-AQc/(kT)]. However, the actual concentration of clusters of size Mc is smaller than the equilibrium concentration, and many supercritical clusters decay back to smaller sizes. The actual nucleation rate is therefore smaller and is given by Eq. 19.17, where the first term (Z) corrects for these effects. This dimensionless term is often called the Zeldovich factor and has a magnitude typically near 10-1. 

Instantaneous nucleation — This is the case when all nuclei of the new phase are formed within a short time period after supersaturating the parent phase. Then the nuclei only grow, which means that what we call instantaneous nucleation is the process of growth of a constant number N() of supercritical clusters. In this case the theoretical expression for the total current density j o (t) reads... 

Turner, C.H. and Gubbins, K.E. (2003). Effects of supercritical clustering and selective confinement on reaction equilibrium a molecular simulation study of the esterification reaction. J, Chem. Phys., 119, 6057-67. 

Above performed analysis leads to the consequence that clusters of critical sizes have properties which are widely different, in general, from the proproperties of the newly evolving macroscopic phases. By this reason, also the properties of sub- and supercritical clusters have to depend, in general, both on... 

A charged cluster may constitute an electron acceptor, but that depends on its own redox potential value, E (A -Agn) relative to the threshold imposed by the monitor potential, E°(Q -QH2). As the redox potential increases with cluster nuclearity (5, 6), a certain time after the pulse is required to allow the first supercritical clusters to be formed and their potential to reach the threshold value imposed by the hydroquinone. When time, t, is less than tc, where n < Uc, the transfer is not allowed. During this induction period, the kinetics at 380 nm correspond to pure coalescence of clusters (Figure 4), and hydroquinone is stable (the bleaching OD512 is constant). That means, obviously, that none of the silver species present at that time can react with hydroquinone, especially free Ag ions and Ag ions associated with the smallest clusters. 

The hydroquinone disappeai by reaction with silver ions adsorbed on supercritical clusters and is oxidized into the semiquinone. The semiquinone is also readily oxidized into naphtazarin, which strongly absorbs at 512 nm ... 

Electron Transfer Mechanism. Thus, the first oxidation step of the hydroquinone occurs, provided the nuclearity is supercritical, n> nc, which means when the potential °(Ag, + i-Ag +i) becomes higher than the threshold °(Q "-QH2). Then the supercritical cluster acts as a nucleus for its own growth through an autocatalytic electron transfer (reactions 5,6) according to the mechanism that was summarized by reaction 12 ... 

However, the semiquinone does not act as a redox probe, because its concentration is zero as long as hydroquinone itself has not started the transfer. Then it is always produced in the close vicinity of the supercritical cluster already selected by the hydroquinone, and it reacts readily with the same cluster before diffusion. It is worth noting that the semiquinone essentially amplifies the catalytic transfer. The overall hydroquinone oxidation into the quinone produces twice as many silver atoms as the initial QH2 concentration (reactions... 

Time > tc. Not only is tc longer at pH 3.9 than at pH 4.8, but the observed rate after tc is smaller (Figures 4 and 5). This rate would depend linearly on the supercritical cluster concentration. This confirms that, under conditions yielding identical initial amounts of silver atoms, larger clusters are required to accept electrons from a weaker donor and that the nuclei concentration is much lower at pH 3.9. 

We see from (11.64) that subcritical i-mers (/collision frequency. Critical size i-mers (i = / ) are in equilibrium with the surrounding vapor (y, = p,). Supercritical clusters (/ > f) grow since monomers tend to condense on them faster than monomers evaporate. 

The couple S/S is selected with a specific and intense optical absorption of S or S , so that the electron-transfer reaction can be observed directly. In the early stages of atom coalescence, the redox potentials of the atom and of the smallest clusters are generally far below that of the donor and the transfer from S to the oligomer does not occur. The ion reduction is caused exclusively by solvated electrons and alcohol radicals (Eqs. 2, 8, and 9). The nucleation and coalescence dynamics are thus the same as in the absence of (Eqs. 10 and 11). Beyond a certain critical time, tc, that is large enough to enable the growth of clusters and the increase of their potential above the threshold imposed by the electron donor S , electron transfer from this monitor to the supercritical clusters is allowed (Eq. 32) and detected by the absorbance decay of S (Fig. 6). For n > ny. 

While supercritical clusters ( > 6 1) (Table 5) accept electrons from MV+" with a progressive increase of their nuclearity (Eqs. 32-34), the subcritical clusters undergo a progressive oxidation by by means of the reactions depicted in... 

Eqs. 35-37). Eq. (35) n < nc) is the reverse of Eq. (32) n > nc). The reduced ions MV+ so produced also contribute to the growth of the supercritical clusters. Actually, they act as an electron relay favoring the growth of large clusters at the expense of the small ones. The coexistence of reduction by MV+ and oxidation by is observed because the coalescence in the presence of ligand CN" is, as... 

The MV+ formed can transfer an electron back to supercritical clusters (Eqs. 38, 41, and 42) so that it acts, as for silver, as an electron relay from subcritical to supercritical clusters3" ° ... 

As E is decreased one observes a change from the unimodal distribution for subcritical clusters to a bimodal form indicating growth of supercritical clusters. Because the system is adiabatic, the biomodal distributions also represent stationary states in which there are maximum supercritical cluster sizes, which, if exceeded, result in destruction of that supercritical cluster size new bonds formed in the system increase the cluster kinetic energy and decrease the pressure of the monomer gas. In the future it would be desirable to extract from the molecular-dynamics calculation accurate values for the free energy of formation of clusters. Such calculations would resolve the differences between the B - D theory and the Lothe-Pound theory. In the future, molecular-dynamics calculations should make possible development of correct mesoscopic and microscopic theories of homogeneous and even heterogeneous nucleation. 

After formation of ultrafine particles (ufp) by nucleation, subsequent growth of supercritical clusters occurs by condensation and coagulation. Coagulation is discussed in this section dealing with evolution of aerosols of ufp. The discussion is limited to Brownian coagulation which is the principal mode of coagulation for ufp. 

Doring [4] provided an analytical solution considering a system of constant composition in which supercritical clusters are reintroduced into the system as the equivalent amount of discrete units. A steady state expression of the nucleation rate follows, Jg = Zam Cm y where am is the net probability of addition of an atom per unit time from a critical cluster of size m, and Cm, the equilibrium concentration of critical clusters, is related to the monomer concentration through the Bolztmann equation, Cm = Cl exp(—AGm /kT) and the nondimensional Zeldovich factor Z, which accounts for the fact that the steady state concentration at m is only 1 /2 of the concentration at equilibrium, and that critical clusters may still decay [5]. 

As the cluster size increases towards a critical value, the point is approached where further addition of monomer results in the formation of a stable growth centre, and the centre is no longer involved in the steady state distribution. This can be accounted for by considering that supercritical clusters are removed and reintroduced as the equivalent amount of monomer, and solution of the set of steady state equations based on Equation (9.17) then leads to the steady state nucleation rate, /j, as... 

The first term of the right hand side of equation (2.124) gives the rate of birth of free active sites on the electrode surface and the second term gives the death rate of free active sites. As for equation (2.125), it gives the rate of appearance of supercritical clusters on the working electrode. We should... 

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