Sphere-plate model

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.

When bounding walls exist, the particles confined within them not only collide with each other, but also collide with the walls. With the decrease of wall spacing, the frequency of particle-particle collisions will decrease, while the particle-wall collision frequency will increase. This can be demonstrated by calculation of collisions of particles in two parallel plates with the DSMC method. In Fig. 5 the result of such a simulation is shown. In the simulation [18], 2,000 representative nitrogen gas molecules with 50 cells were employed. Other parameters used here were viscosity /r= 1.656 X 10 Pa-s, molecular mass m =4.65 X 10 kg, and the ambient temperature 7 ref=273 K. Instead of the hard-sphere (HS) model, the variable hard-sphere (VHS) model was adopted in the simulation, which gives a better prediction of the viscosity-temperature dependence than the HS model. For the VHS model, the mean free path becomes ... 

In Eq. (15), the electrostatic potential, iJ/, is for the overlapping electric double layer of the interacting particles. Numerous models have been created to predict the overlapping field electrostatic potential between parallel plates. However, calculation of the EDL interaction for the common geometry of two spheres has not been satisfactorily resolved, due mainly to the nonlinear partial differential terms in Eq. (13) arising because of the three-dimensional geometry of the system. As a consequence, a number of approximate and numerical models have been developed for the calculation of the EDL interaction between two spheres. These models are briefly described below. 

The circuits discussed in Sect. 2 contain discrete mechanical elements. They predict the sign, the n-dependence, and the relative magnitude of A/ and AT, but they make no suggestion of how to assign a physical meaning to the model parameters, once they have been determined from experiment. Continuum models evidently are more complicated. On the other hand, they are not only more realistic, they also provide quantitative guidelines for the interpretation of experimentally derived parameters. Two situations have been analyzed, which are the sphere-plate contact and the sheet contact. 

Both (i) and (ii) necessitate recourse to a model of pore shape. By far the commonest, chosen on grounds of simplicity, is the cylinder but the slit model is being increasingly used where the primary particles are plate-like, and the model where the pore is the cavity between touching spheres is beginning to receive attention. 

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. 

Lin, J. R., Squeeze Film Characteristics Between a Sphere and [40] a Flat Plate Couple Stress Fluid Model, Camputers Structures, Vol. 75,2000, pp. 73-80. 

FIG. 2 Model for the calculation of electrostatic forces. The tip-lever system (top) is approximated by a sphere of radius equal to the apex tip radius R and (bottom) a flat plate of area S equal to that of the support lever. 

The theory has been developed for two special cases, the interaction between parallel plates of infinite area and thickness, and the interaction between two spheres. The original calculations of dispersion forces employed a model due to Hamaker although more precise treatments now exist [194],... 

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 ... 

A plate is a model for membranes, films and coatings. Cylinders can be models for fibres, but also for pores. We often regard particles, drops and bubbles as if they are little spheres. 

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... 

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