Parallel surface method , mean

Now that we have determined the 3D structure, which reflects truly the real structures from global-to-local scale, and that we have obtained a sound physical basis of this structure, we proceed to analyze the structure in terms of interface curvatures, based on differential geometry. We used the parallel-surface method to determine the area-averaged mean curvature = <(/ci + k2)/2> and Gaussian curvature = ... 

The rapid rise in computer speed over recent years has led to atom-based simulations of liquid crystals becoming an important new area of research. Molecular mechanics and Monte Carlo studies of isolated liquid crystal molecules are now routine. However, care must be taken to model properly the influence of a nematic mean field if information about molecular structure in a mesophase is required. The current state-of-the-art consists of studies of (in the order of) 100 molecules in the bulk, in contact with a surface, or in a bilayer in contact with a solvent. Current simulation times can extend to around 10 ns and are sufficient to observe the growth of mesophases from an isotropic liquid. The results from a number of studies look very promising, and a wealth of structural and dynamic data now exists for bulk phases, monolayers and bilayers. Continued development of force fields for liquid crystals will be particularly important in the next few years, and particular emphasis must be placed on the development of all-atom force fields that are able to reproduce liquid phase densities for small molecules. Without these it will be difficult to obtain accurate phase transition temperatures. It will also be necessary to extend atomistic models to several thousand molecules to remove major system size effects which are present in all current work. This will be greatly facilitated by modern parallel simulation methods that allow molecular dynamics simulations to be carried out in parallel on multi-processor systems [115]. 

The reaction coordinate that describes the adsorption process is the vibration between the atom and the surface. Strictly speaking, the adsorbed atom has three vibrational modes, one perpendicular to the surface, corresponding to the reaction coordinate, and two parallel to the surface. Usually the latter two vibrations - also called frustrated translational modes - are very soft, meaning that k T hv. Associative (nondissociative) adsorption furthermore usually occurs without an energy barrier, and we will therefore assume that A = 0. Hence we can now write the transition state expression for the rate of direct adsorption of an atom via this transition state, applying the same method as used above for the indirect adsorption. 

An alternative method of reducing the resistance to filtration is to recirculate the slurry and thereby maintain a high velocity of flow parallel to the surface of the filter medium. Typical recirculation rates may be 10-20 times the filtration rate. By this means the cake is prevented from forming during the early stages of filtration. This can be particularly beneficial when the slurry is flocculated and exhibits shear-thinning non-Newtonian properties. This method of operation is discussed by Mackley and Sherman(21) and by Holdich, Cumming and Ismail(22). 

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