Space-filling condition

Due to the characteristic self-similar structure of the CCA-clusters with fractal dimension df 1.8 [3-8, 12], the cluster growth in a space-filling configuration above the gel point O is limited by the solid fraction Oa of the clusters. The cluster size is determined by a space-filling condition, stating that, up to a geometrical factor, the local solid fraction Oa equals the overall solid concentration O ... 

The CCA-model considers the filler network as a result of kinetically cluster-cluster-aggregation, where the size of the fractal network heterogeneity is given by a space-filling condition for the filler clusters [60,63,64,92]. We will summarize the basic assumptions of this approach and extend it by adding additional considerations as well as experimental results. Thereby, we will apply the CCA-model to rubber composites filled with carbon black as well as polymeric filler particles (microgels) of spherical shape and almost mono-disperse size distribution that allow for a better understanding of the mechanisms of rubber reinforcement. 

Equation (70) is a scaling invariant relation for the concentration-dependency of the elastic modulus of highly filled rubbers, i.e., the relation is independent of filler particle size. The invariant relation results from the special invariant form of the space-filling condition at Eq. (67) together with the scaling invariance of Eqs. (68) and (69), where the particle size d enters as a normalization factor for the cluster size only. This scaling invariance disappears if the action of the immobilized rubber layer is considered. The effect of a hard, glassy layer of immobilized polymer on the elastic modulus of CCA-clusters can be de-... 

The mechanical action of the immobilized rubber layer on spherical filler particles, that are assumed to form a CCA-filler network in a rubber matrix for

cp, is obtained if the mechanically effective solid fraction q>A (Eq. (71)), is applied in Eq. (69) instead of cpA and the space-filling condition [Pg.35]

A realistic model of a solution requires at least several hundred solvent molecules. To prevent the outer solvent molecules from boiling off into space, and minimizing surface effects, periodic boundary conditions are normally employed. The solvent molecules are placed in a suitable box, often (but not necessarily) having a cubic geometry (it has been shown that simulation results using any of the five types of space filling polyhedra are equivalent ). This box is then duplicated in all directions, i.e. the central box is suiTounded by 26 identical cubes, which again is surrounded by 98 boxes etc. If a... 

Figure 11.43 Space-filling models of the ligands C32 (high activity of Fe(ll) complex) and A32 (low activity of Fe(ll) complex). Structure of the metal ligand complex under polymerization conditions.

Lack of steady flow of a liquid-bearing colloidal solution requires the existence of a space-filling, three-dimensional structure. As we might select a perfect crystal as a csuionical solid, or liquid argon as a prototypical liquid, we csui choose the covalently crosslinked network, without any entanglements, to represent the ideal gel state. Then an appropriate time scale for reversible gels would be the lifetime of a typical crosslink bond if subjected to conditions that would cause flow in a pure... 

The violation of electroneutrality can be avoided by using a space-filling model. Such a model must fulfill the condition... 

The effects mentioned in the last sub-section are all local ones and do not depend upon the macromolecular nature of the material. A more important group of properties comprises those that are affected by the changes that LCB produces in the space-filling behaviour of polymer molecules under conditions in which they are free or relatively free from conformational constraints, that is in solution or in the melt, or in the amorphous state below 7. ... 

The effect of solid particles. In his studies of the detonation products of explosives stemmed in different ways, Audibert [29, 32] paid attention to the fact that in charges insufficiently stemmed a certain amount of the explosive remains in the form of small particles which may undergo further explosive decomposition according to conditions, i.e., temperature and ambient pressure. If these particles are ejected into a space filled with a methane-air mixture they may lead to the explosion of this mixture. The possibility of the existence of particles of undecomposed explosive in ejected detonation products has been disputed by some authors (Segay [33]), but many others have proved that it can occur. T. Urbanski [34] found that a thin layer on the periphery of a cylindrical charge of ammonium nitrate explosive is... 

As discussed by Andersen [9, 10] for muffin-tin orbitals, the locally regular components y defined in each muffin-tin sphere are cancelled exactly if expansion coefficients satisfy the MST equations (the tail-cancellation condition) [9, 384], The standard MST equations for space-filling cells can be derived by shrinking the interstitial volume to a honeycomb lattice surface that forms a common boundary for all cells. The wave function and its normal gradient evaluated on this honeycomb interface define a global matching function %(cr). 

The Schlosser-Marcus variational principle is derived for a single surface a that subdivides coordinate space 9i3 into two subvolumes rm and rout. This generalizes immediately to a model of space-filling atomic cells, enclosed for a molecule by an external cell extending to infinity. The continuity conditions for the orbital Hilbert space require i>out =a i>in This implies a vanishing Wronskian surface integral... 

The same principle of formation of a spherical foam film is used in the device (Fig. 2.23) for direct measurement of the film tension y in static [126] and dynamic [127] conditions. The lower part of the capillary is placed in a closed space filled up with a gas, whose pressure can be measured precisely and by means of a special pump can be varied at different rates. The manometer registers the difference with the atmospheric pressure Ap, which is equal to the capillary pressure pa of the spherical foam film... 

The barium bis(dnunod) complex is readily prepared by a redox reaction combining elemental barium with Hdnunod (equation 43). Purified by distillation of the excess Hdnunod, Ba(dmmod)2 was identified to be authentic by elemental analysis, and found to be monomeric in benzene solution. Examination of the literature reports currently available indicates that this represents the first observation of an ambient condition liquid barium compound that is based on )3-diketonate ligands. Utihzing the data, it would appear that the metal center is six coordinate. Examination of space-filling molecular models lends some credence to this suggestion. 

Portions of polymer molecules which are in crystalline regions have overall dimensions and space-filling characteristics that arc determined by the particular crystal habit which the macromoleculc adopts. Here, however, we are concerned with the sizes and shapes of flexible polymers in the amorphous (uncrystallized) condition. It will be seen that the computation of such quantities provides valuable insights into the molecular nature of rubber elasticity. 

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