Eden cluster

Figure 1. A two-dimensional cluster grown by the Eden process. The clusters in this simulation Jill in because all surface sites have equal probability of growth whether or not they have access to the perimeter. This artifact is more unreal in two dimensions than in three. (Reproduced with permission from reference 19. Copyright 1986 Elsevier.)...

Figure 3. Two-dimensional clusters grown with the poisoned Eden model. The average functionality of the monomers in each cluster is three. The distribution of functional groups among the monomers causes the different structures.

Kazantzi V, Qin X, El-Halwagi M, Eljack F, Eden M. Simultaneous process and molecular design through property clustering — a visualization tool. Ind Eng Chem Res 2007 46 3400—9. 

For w 4, unhydrolyzed sites are incorporated into the growing cluster. The probability of condensation at these sites is less than at hydrolyzed sites. Under these conditions, growth is described by a poisoned Eden model. Depending on the number and distribution of the poisoned sites, mass fractals, surface fractals or uniformly porous objects can result. The addition of more water in a second hydrolysis step is expected to completely hydrolyze the clusters and further growth should be described by the Eden model. 

In order to convert property values from raw property data to cluster values, property operator mixing rules are required (Shelley El-Halwagi 2002 Eden et al. 2002). The property relationships can be described using the Kubelka-Munk theory (Biermann 1996). According to Brandon (1981), the mixing rules for objectionable material (OM) and absorption coefficient (k) are linear, while a non-linear empirical mixing rule for reflectivity has been developed (Willets 1958). 

Fig. 6 Internal structure of cluster obtained using Eden-like model of aggregation of charged particles (2 = 1, mq = 60). The cluster includes a core (crmtral dense charged part) and external branches [105]...

Consider, as an example, neutral pH. For r < 4, the condensation reaction initially proceeds between incompletely hydrolyzed species. Hydrolysis is rate-limiting, so the pattern of condensation reflects the pattern of hydrolysis. According to Eq. 37, the base-catalyzed condensation reaction occurs preferentially between acidic, deprotonated species and less acidic protonated species. Thus the hydrolyzed species are more likely to condense with a larger cluster than with themselves. Poisoned Eden growth should... 

The poisoned Eden mode) is relevant to silicate condensation under neutral conditions where the hydrolysis rate is minimized. Ether-forming (ROR) condensation is forbidden, and the rate of the alcohol-forming condensation reaction is lower than that of the water-forming condensation reaction [6,95]. Therefore, unhydrolyzed alkoxide sites, which are most abundant at neutral pH, act as poisons by inhibiting condensation (at least temporarily). Keefer argues that once unhydrolyzed sites are incorporated in the cluster, their hydrolysis may be impeded due to steric factors [136], If so, the poisoned Eden model would be physically and chemically relevant. 

K. A. Brueckner, R. J. Eden, and N. C. Francis, High-energy reactions and the evidence for correlations in the nuclear ground-state wavefunction, Phys. Rev. 98 1445 (1955) K. A. Brueckner, Many-body problems for strongly interacting particles. II. Linked cluster expansion, Phys. Rev. 100 36 (1955). 

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