Computational interaction forces

Due to the very low volumetric concentration of the dispersed particles involved in the fluid flow for most cyclones, the presence of the particles does not have a significant effect on the fluid flow itself. In these circumstances, the fluid and the particle flows may be considered separately in the numerical simulation. A common approach is to first solve the fluid flow equations without considering the presence of particles, and then simulate the particle flow based on the solution of the fluid flow to compute the drag and other interactive forces that act on the particles. 

Fig. 3- Cose 1. Sherwood numbers computed for the convective-diffusion of particles of finite sine to the surface of a spherical collector by neglecting interaction forces. The dashed line is the Levich-LighthilJ equation (19) which is valid when a diffusion boundary-layer exists and the particles are infinitesimal.

Kinetic equations for reversible adsorption and reversible coagulation are established when the interaction potential has primary and secondary minima of comparable depths. The process is assumed to occur in two successive steps. First the particles move from the bulk of the fluid to the secondary minimum. A fraction of the particles which have arrived al the secondary minimum move further to the primary minimum. Quasi-steady state is assumed for each of the steps separately. Conditions are identified under which rates of reversible adsorption or coagulation at the primary minimum can be computed by neglecting the rate of accumulation at the secondary minimum. The interaction force boundary layer approach has been improved by introducing the tangential velocity of the particles near the surface of the collector into the kinetic equations. To account for reversibility a short-range repulsion term is included in the interaction potential. 

The direct measurement of the interaction force between two mica surfaces1 indicated a large repulsion at relatively short distances, which could not be accounted for by the DLVO theory. This force was associated with the structuring of water in the vicinity of the surface.2 Theoretical work and computer simulations8-5 indicated that, in the vicinity of a planar surface, the density of the liquid oscillates with the distance, with a periodicity of the order of molecular size. This reveals that, near the surface, the liquid is ordered in quasi-discrete layers. When two plan ar surfaces approach each other at sufficiently short distances, the molecules of the liquid reorder in discrete layers, generating an oscillatory force.6... 

Chan, B.K.C. Chan, D.Y.C. Electrical doublelayer interaction between spherical colloidal particles an exact solution. J. Colloid Interface Sci. 1983, 92 (1), 281-283 Palkar, S.A. Lenhoff, A.M. Energetic and entropic contributions to the interaction of unequal spherical double layers. J. Colloid Interface Sci. 1994, 165 (1), 177-194 Qian, Y. Bowen, W.R. Accuracy assessment of numerical solutions of the nonlinear Poisson-Boltzmann equation for charged colloidal particles. J. Colloid Interface Sci. 1998, 201 (1), 7-12 Carnie, S.L. Chan, D.Y.C. Stankovich, J. Computation of forces between spherical colloidal particles Nonlinear Poisson-Boltzmann theory. J. Colloid Interface Sci. 1994, 165 (1), 116-128 Stankovich, J. Carnie, S.L. Electrical double layer interaction between dissimilar spherical colloidal particles and between a sphere and a plate nonlinear Poisson-Boltzmann theory. Langmuir 1996,12 (6), 1453-61. 

To avoid using a predefined form for the interaction potential in molecular dynamics simulations, the quantum mechanical state of the many-electron system can be determined for a given nuclear configuration. From this quantum mechanical state, all properties of the system can be determined, in particular, the total electronic energy and the force on each of the nuclei. The quantum mechanically derived forces can then be used in place of the classically derived forces to propagate the atomic nuclei. This section describes the most widely used quantum mechanical method for computing these forces used in Car-Parrinello simulations. 

In principle, eqn (11.59) is all that we need to go about computing the interaction force between an obstacle and a dislocation. However, we will find it convenient to rewrite this equation in a more transparent form for the present discussion. Using the principle of virtual work, we may rewrite the interaction energy as... 

Intermolecular forces, sometimes called non-covalent interactions, are caused by Coulomb interactions between the electrons and nuclei of the molecules. Several contributions may be distinguished electrostatic, induction, dispersion, exchange that originate from different mechanisms by which the Coulomb interactions can lead to either repulsive or attractive forces between the molecules. This review deals with the ab initio calculation of complete intermolecular potential surfaces, or force fields, but we focus on dispersion forces since it turned out that this (relatively weak, but important) contribution took longest to understand and still is the most problematic in computations. Dispersion forces are the only attractive forces that play a role in the interaction between closed-shell ( 5) atoms. We will see how the understanding of these forces developed, from complete puzzlement about their origin, to a situation in which accurate quantitative predictions are possible. 

Although not yet obvious from the deceptively simple form of equation (37), this equation, the electrostatic Hellmann-Feynman theorem, allows one to use the electronic density and the simple internuclear Coulomb interactions to describe the forces acting on the nuclei of the molecule. A simple, classical interpretation of this theorem provides the key to the use of macromolecular electronic densities, such as those obtained within the MEDLA, ALDA, or ADMA methods, for the computation of forces within the macromolecule. 

It is certainly interesting to see here that the DFT treatment is able to provide us with a realistic description of these interaction forces which are likely to play a mayor role in the dynamics of cluster growth. It does so by markedly reducing the computational effort and thereby making amenable to computations ionic clusters of reahstic sizes. Furthermore, it shows very good agreement with the fully ab initio calculations which aheady exist on the smaller species and with which comparison has been possible. 

Iwata, H., A Six-Degree-of-Freedom Pen-Based Force Display, in Proceedings of the Fifth International Conference on Human-Computer Interaction (Orlando, FL, 1993), pp. 651-656. 

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