Electrostatic energy upon cluster

The electrostatic energy per ion pair released upon cluster collapse is a function of the geometry of the cluster and the dielectric constant s of the medium. It is given by... 

Figure 6.11 shows the activity of an artificial enzyme can be controlled based on the phase behavior of a lipid bilayer. The catalytic site for hydrolysis was supplied by a monoalkyl azobenzene compound with a histidine residue which was buried in the hydrophobic environment of a hpid bilayer matrix formed using a dialkyl ammonium salt. Azobenzene compound association depended on the state of the matrix bilayer. The azobenzene catalyst aggregated into clusters when the bilayer matrix was in a gel state. In contrast, the azobenzene derivative can be dispersed into the liquid crystalhne phase of the bilayer matrix above its phase transition temperature. This bilayer-type artificial enzyme catalyzed the hydrolysis of a Z-phenylalanine p-nitrophenyl ester. The activation energy for this reaction in the gel state is twice as large as that observed in the hquid crystalline state. The clustering of the catalysts upon phase separation suppress their catalytic activity, probably due to the disadvantageous electrostatic environment around the catalysts and the suppressed substrate diffusion. This activity control is unique to such molecular assembhes. 

Here a and d are the number of atoms in the acceptor and the donor, respectively, Ry is the distance between atoms i and j and and are the van der Waals and electrostatic potentials, respectively. The van der Waals potential is often represented by a Lennard-Jones potential (Eq. 8) or by a Buckingham potential (Eq. 9). The parameters a, fi, y and o are obtained from solid-state crystal data. The leading term in the electrostatic potential is the Coulomb interaction (first term in Eq. 10), where D is the effective dielectric constant (usually < D <2). Other terms may be added to represent induced polarization, etc. [40]. The geometries of the two components of the cluster are obtained from microwave or electron diffraction data or from quantum chemical calculations. It is assumed that these geometries do not change upon adduct formation. An initial guess is made for the structure of the adduct, and then the relative positions of the two (or more) components are varied until a local energy minimum is obtained. 

What is found by experiment is that, as a general rule, at substitutional concentrations close to the maximum in the conductivity isotherms, there is a minimum in the activation energy. In an early (but very comprehensive) study of ceria solid solutions with the trivalent rare earths, Faber etal. [IS] showed that the depth of the minimum, and the concentration at which it occurs, depends upon the identity of the rare earth cation (Figure 9.1). The minima have been ascribed [19] to competitive defect interactions. Initially, the effect is a weakening ofthe association energy of the dimers caused by an electrostatic interaction between the cluster and the unassociated substitutionals having an opposite effective charge in the lattice note, however, that... 

Electrostatic and Entropic Terms. Just as charges coalesce into clusters in the dry state, upon hydration, the charges and water molecules are gathered into vesicles, which we assume to be spherical (radius R). Experiments show that the location of the ionomer peak varies continuously with the water content (v) Thus we assume that the sample is in thermodynamical equilibrium, and we have to look for its free energy. 

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