Spatial distribution of clusters

Consider M clusters distributed in a spatial v -dimensional region Vy in such a way that the probability Pm to find m of them in the spatial element Oy of Vy is given by the Poisson law (3.1). In this case N denotes the expected average number of clusters in Oy. Following Hertz [3.27] Oy is presented as  

Equation (3.30) allows us to derive general expressions for two important physical quantities - the average and the most probable (r ) distance between nth neighbors. Thus  

A formula useful for comparison with experimental results can be obtained by introducing the non-dimensional distance / =r //v into equation 

An alternative is to reduce the probability distribution function with respect to the average distance F, between th neighbors [3.32]. In this case one obtains  

Having derived general theoretical formulae for the probability distribution function dPy,n (equations (3.30), (3.33) and (3.34)) in the following we present the results of a statistical analysis of the distances between clusters formed on a flat electrode surface. Several authors have performed such experimental studies in different experimental systems silver [3.36-3.38], lead [3.39-3.41], mercury [3.42] and gold [3.43] on glassy carbon, copper on evaporated silver [3.44] and mercury on platinum [3.31, 3.45]. Here we comment upon the data reported in Ref. [3.38]. 

---

Probabilistic approach to nucleation — Basic stochastic properties of assemblies of clusters randomly distributed in space or appearing in time as a nonstationary flux of random independent events can be examined in terms of the Poisson theory [i]. See - temporal distribution of clusters -> spatial distribution of clusters. 

Nucleation — Stochastic approach to nucleation — Spatial distribution of clusters — Figure. Experimental (histograms) and theoretical (lines) distribution of the distances between first (a), second (b) and third (c) neighbor silver crystals electro chemically deposited on a glassy carbon electrode [iii-v]... 

In addition to the local characterization by means of TEM in the previous section, the integral characterization of the prepared samples is described in the following. As a first step different sample types were probed to see if the sensitivity of the setup is sufficient to detect the low cluster catalyst amounts. Furthermore, a representative line scan for Ptes clusters was performed in order to get an idea of the spatial distribution of clusters on the sample materials. 

Fig. 5.8 Representative (a) Si 2p and (b) Pt 4/ peak and corresponding fits for 0.029e/nm Pte supported on a Si waver. The results of the presented fits, for each position of the fine scan are used to calculate the spatial distribution of clusters in Fig. 5.10b...

Fig. 5.10 Image plot of PtAf signals of 0.029 e/nm Pt(,% (corrected by the Pt to Si ratio) as a function of different spatial positions (a) and spatial distribution of clusters on the support, shown as ratio of the obtained areas from the fitted Si 2p and Pt 4/ signals (b) for each measured position from Fig. 5.9...

Figure 2.14. The molecular orbitals of gas phase carbon monoxide, (a) Energy diagram indicating how the molecular orbitals arise from the combination of atomic orbitals of carbon (C) and oxygen (O). Conventional arrows are used to indicate the spin orientations of electrons in the occupied orbitals. Asterisks denote antibonding molecular orbitals, (b) Spatial distributions of key orbitals involved in the chemisorption of carbon monoxide. Barring indicates empty orbitals.5 (c) Electronic configurations of CO and NO in vacuum as compared to the density of states of a Pt(lll) cluster.11 Reprinted from ref. 11 with permission from Elsevier Science.

Advection is important in fragmentation processes, and an initially homogeneous system may evolve spatial variations due to spatially dependent fragmentation rates. For example, Fig. 36 shows the spatial distribution of eroded clusters in the journal bearing flow operating under good mixing... 

We find Saurer 1 to be a member due to its Vgsr [2] did not correct this RV to Vgsr before incorrectly excluding Saurer 1 from the GASS member clusters. 2 It is noted from the spatial distribution of the clusters that GASS should have an elliptical orbit. If this change is made,... 

For example, the aggregated structures of the solutions containing polymer-metal complexes and the colloidal dispersions of metal nanoparticles stabilized by polymers have been analyzed quantitatively (64). SAXS analyses of colloidal dispersions of Pi, Rh, and Pt/Rh (1/1) nanoparticles stabilized by PVP have indicated that spatial distributions of metal nanoparticles in colloidal dispersions are different from each other. The superstructure (greater than 10.0 nm in diameter), with average size highly dependent on the metal element employed, is proposed. These superstructures are composed of several fundamental clusters with a diameter of 2.0-4.0 nm, as shown in Figure 9.1.13 for PVP-stabilized Pt nanoparticles. 

In the case of diffusion-controlled A + B —> 0 reaction distinctive spatial distributions of reactants observed in computer simulations (e.g., [21]) are qualitatively the same as were presented earlier in Figs 1.20 and 1.21. Quite similar aggregation of similar particles into loose clusters occurs in agreement with a distinctive block-structure characterized by the diffusion length Id = f Dt shown in Fig. 2.8. When the reaction is controlled by the particle diffusion, these clusters (domains) are less pronounced since diffusion is known to smooth nonuniform particle distribution created in a course of reaction. 

Fig. 10.16 The evolution of the spatial distribution of 10,000 particles initially clustered in 10 randomly placed clusters in the C-shaped chamber. The fraction or number of pitches denotes the axial advance of the material in the chamber due to the counterrotation. [Reprinted by permission from T. Li and lea Manas-Zloczower, A Study of Distributive Mixing in Counterrotating TSEs, Int. Polym. Process., 10, 314 (1995).]...

The theory reflects the solvent properties through the thermody-namic/hydrodynamic input parameters obtained from the accurate equations of state for the two solvents. However, the theory employs a hard sphere solute-solvent direct correlation function (C12), which is a measure of the spatial distribution of the particles. Therefore, the agreement between theory and experiment does not depend on a solute-solvent spatial distribution determined by attractive solute-solvent interactions. In particular, it is not necessary to invoke local density augmentation (solute-solvent clustering) (31,112,113) in the vicinity of the critical point arising from significant attractive solute-solvent interactions to theoretically replicate the data. 

---