Sphere-plate model surface

To evaluate DLVO-Lifshitz potentials we approximate the highly complex interaction of a spherical-icosahedral, deformable virus adsorbing to a real surface (Figure 1) with sphere-plate models (Figure 5). The complex real interactions, even if they were well defined, cannot be quantified by present ab initio quantum mechanical procedures. 

Figure 5. Sphere-plate model of virus adsorbing to a flat surface, showing inner (Stern) layer and outer (Gouy) double layers.

Two types of model surfaces were used in this study printing plates etched with a screen pattern and glass micro-sphere surfaces. 

The pores in question can represent only a small fraction of the pore system since the amount of enhanced adsorption is invariably small. Plausible models are solids composed of packed spheres, or of plate-like particles. In the former model, pendulate rings of liquid remain around points of contact of the spheres after evaporation of the majority of the condensate if the spheres are small enough this liquid will lie wholly within the range of the surface forces of the solid. In wedge-shaped pores, which are associated with plate-like particles, the residual liquid held in the apex of the wedge will also be under the influence of surface forces. 

The molecule is often represented as a polarizable point dipole. A few attempts have been performed with finite size models, such as dielectric spheres [64], To the best of our knowledge, the first model that joined a quantum mechanical description of the molecule with a continuum description of the metal was that by Hilton and Oxtoby [72], They considered an hydrogen atom in front of a perfect conductor plate, and they calculated the static polarizability aeff to demonstrate that the effect of the image potential on aeff could not justify SERS enhancement. In recent years, PCM has been extended to systems composed of a molecule, a metal specimen and possibly a solvent or a matrix embedding the metal-molecule system in a molecularly shaped cavity [62,73-78], In particular, the molecule was treated at the Hartree-Fock, DFT or ZINDO level, while for the metal different models have been explored for SERS and luminescence calculations, metal aggregates composed of several spherical particles, characterized by the experimental frequency-dependent dielectric constant. For luminescence, the effects of the surface roughness and the nonlocal response of the metal (at the Lindhard level) for planar metal surfaces have been also explored. The calculation of static and dynamic electrostatic interactions between the molecule, the complex shaped metal body and the solvent or matrix was done by using a BEM coupled, in some versions of the model, with an IEF approach. 

The potential energy of interaction 4> between a sphere and a plate, which serve in this approach as a model for the cell and the contacted surface, can be obtained by summing up the individual contributions of the above mentioned forces ... 

As in the case of two interacting soft plates, when the thicknesses of the surface charge layers on soft spheres 1 and 2 are very large compared with the Debye length 1/k, the potential deep inside the surface charge layer is practically equal to the Donnan potential (Eqs. (15.51) and (15.52)), independent of the particle separation H. In contrast to the usual electrostatic interaction models assuming constant surface potential or constant surface... 

The current model is a step closer toward a reliable working description of the sulfite oxidation rate in scrubber slurries. By incorporating a boundary layer description of the film around each particle, this model predicts the conditions at the particle surface which drive the mass transfer. The interfacial area for mass transfer was discovered to be more closely represented by a sphere than by a plate-like shape. From the model, using highly catalyzed experiments, a mass transfer coefficient of 0.015 cm sec-1 was found - quite close to literature correlation predictions. 

Constant capacitance model (CCM) was proposed in 1972 by Schindler and Stumm (Schindler, R. W. et at, 1976 Stumm, W. et at, 1980) mostly for the surface of oxides. It is based on the very first model of the dual electric layer developed by Helmholtz. Its core concept is an assumption that only inner-sphere ion complexes form, which are positioned as an individual layer at some distance from the surface, and the diffusion layer is absent. It is believed that Na+, K+, Cb and NO ", as well as inert, do not form bond with the surface and affect only the ion force of the solution. For this reason the model is viewed as two parallel capacitor plates surface of the mineral with charge a, on the one hand, and adsorbed H+, OH and other ions (Figure 2.18, A) with charge + a. on the other. At that, the electric potential value on the surface of the mineral is equal to... 

For the entropic repulsion calculations, the model system considered is that of a spherical particle, radius a, separated by a distance H from a plate, with both surfaces coated with adsorbed polymer chains of root-mean-square (rms) height /r at a surface coverage 0. If pairs of chains of height /i on opposing surfaces just touch when those on the sphere lie on a circle which subtends, at the centre of the sphere, a semi-angle

entropic repulsion Kr for the system is given in kT units by Equation 9.9, where is the... 

The simplest model is that of a plate capacitor developed very early by Helmholtz. The idea is that the ions of the electrolyte, which form the excess charge there, can approach the metal surface only up to the distance of the radius which includes the irmer solvation sphere in liquid solutions. Measurements of the differential capacity of smooth electrodes yielded values for the Helmholtz double-layer capacity, Ch, on tlie order of 20 to 30 pF cm . The model of a plate capacitor gives for the differential capacity... 

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