Network density topological structure

It is well known that there are numerous factors that influence the gel collapse (charge density, topological structure of a network, medium composition, etc.). The same factors are effective in the case under study as well (see... 

The investigation of the collapse phenomenon have shown that the topological structure of the network plays an essential role in the process of gel collapse [42, 43]. In order to check the influence of the topology of a network on the equilibrium properties of the network-surfactant complexes, a set of experiments with gels differing in the number of crosslinks or in the conditions of synthesis have been performed. It has been shown that tie decrease of crosslink density or concentration of monomers in the polymerization mixture results in a sharper gel collapse. 

The mean average molecular mass of the network chains is determined for the elastomer matrix outside the adsorption layer. Contributions to the network structure fi om different types of junctions (chemical junctions, adsorption junctions, and topological hindrances due to confining of chains in the restricted geometry (entropy constraints or elastomer-filler entanglements) are estimated. The major contributions to the total network density are provided by the topological hindrances near the filler surface and by the adsorption junctions. The apparent number of the elementary chain units between the topological hindrances is estimated to be approximately 40-80 elementary chain units. 

For the observed distinetions explanation it is necessary to point out, that the Eqs. (2.8) and (2.12) take into consideration only molecular characteristic, namely, maeromolecule flexibility, characterized by the value C. Although the Eq. (2.12) takes into account additionally topological factor (traditional macromolecular binary hooking network density v ), but this factor is also a function of [40, 42], The Eqs. (2.16) and (2.5) take into account, besides C, the structural organization of HDPE noncrystalline regions within the frameworks of cluster model of polymers amorphous state structure [5] or fractal analysis with the aid of the value [22], Hence, HDPE noncrystalline regions structure appreciation changes sharply the dependence DJJ). 

Let us show how the value of (i.e., at the molecular and topological levels) influences the suprasegmental level of the structure. Within the frameworks of the cluster model [5,6] the number of densely packed segments in clusters per volume unit is approximately equal to the cluster network density and is connected with v by Equation 2.7, in which the numerical constant accounts for the necessary molecular constants of polymers. 

Now, if we apply the model approach proposed above, we shall see that the main effect on the effective cross-linking density in semi-IPNs is produced by the kinetic conditions and the nature of the components. Changing reaction conditions leads to the formation of the more defective structures of networks as compared with the pme network. Thus, the effective network density is determined not only by the theoretical topology of the network but also by the reaction conditions. It is worth noting that the formation of IPNs imder phase-separated conditions enhances the formation of the defective network because phase separation leads to the formation of two phases and of a transitional region, which may be considered as an independent IPN with its own composition and cross-linking density. 

The chemical bonding and the possible existence of non-nuclear maxima (NNM) in the EDDs of simple metals has recently been much debated [13,27-31]. The question of NNM in simple metals is a diverse topic, and the research on the topic has basically addressed three issues. First, what are the topological features of simple metals This question is interesting from a purely mathematical point of view because the number and types of critical points in the EDD have to satisfy the constraints of the crystal symmetry [32], In the case of the hexagonal-close-packed (hep) structure, a critical point network has not yet been theoretically established [28]. The second topic of interest is that if NNM exist in metals what do they mean, and are they important for the physical properties of the material The third and most heavily debated issue is about numerical methods used in the experimental determination of EDDs from Bragg X-ray diffraction data. It is in this respect that the presence of NNM in metals has been intimately tied to the reliability of MEM densities. 

The mathematical theory of topology is the basis of other approaches to understanding inorganic structure. As mentioned in Section 1.4 above, a topological analysis of the electron density in a crystal allows one to define both atoms and the paths that link them, and any description of structure that links pairs of atoms by bonds or bond paths gives rise to a network which can profitably be studied using graph theory. 

As in carbon-black-filled EPDM and NR rubbers, the physical network in silica-filled PDMS has a bimodal structure [61]. A loosely bound PDMS fraction has a high density of adsorption junctions and topological constraints. Extractable or free rubber does virtually not interact with the silica particles. It was found that the density of adsorption junctions and the strength of the adsorption interaction, which depends largely on the temperature and the type of silica surface, largely determine the modulus of elasticity and ultimate stress-strain properties of filled silicon rubbers [113]. 

The description of a network structure is based on such parameters as chemical crosslink density and functionality, average chain length between crosslinks and length distribution of these chains, concentration of elastically active chains and structural defects like unreacted ends and elastically inactive cycles. However, many properties of a network depend not only on the above-mentioned characteristics but also on the order of the chemical crosslink connection — the network topology. So, the complete description of a network structure should include all these parameters. It is difficult to measure many of these characteristics experimentally and we must have an appropriate theory which could describe all these structural parameters on the basis of a physical model of network formation. At present, there are only two types of theoretical approaches which can describe the growth of network structures up to late post-gel stages of cure. One is based on tree-like models as developed by Dusek7 I0-26,1 The other uses computer-simulation of network structure on a lattice this model was developed by Topolkaraev, Berlin, Oshmyan 9,3l) (a review of the theoretical models may be found in Ref.7) and in this volume by Dusek). Both approaches are statistical and correlate well with experiments 6,7 9 10 13,26,31). They differ mainly mathematically. However, each of them emphasizes some different details of a network structure. 

Fracture processes become very different in the glassy state as compared with the rubbery state. In the glassy state, the sensitivity of fracture to network topology is lost. The chemical structure of the network, crosslink density and the type of bond overloading do not play a key role and defects of glassy samples become very important. 

The elastic properties of rubbers are primarily governed by the density of netw ork junctions and their ability to fluctuate [35]. Therefore, knowledge of the network structure composed of chemical, adsorption and topological junctions in filled elastomers as well as their relative weight is of a great interest. The H T2 NMR relaxation experiment is a well established method for the quantitative determination of the network structure in the elastomer matrix outside the adsorption layer [14, 36]. The method is especially attractive for the analysis of the network structure in filled elastomers since filler particles are "invisible" in this experiment due to the low fraction of protons at the Aerosil surface as compared with those in the host matrix. 

To sum up this rather technical analysis, it seems indisputable that the fitting of classical "monolayer" zeolites to IPMS is not just an elegant mathematical curiosity. The quantitative analysis that follows from this description allows predictions of framework densities as a function of the network topology. This understanding of structure, which views the (Euclidean) three-dimensional structure in terms of its intrinsic two-dimensional hyperbolic geometry opens up a predictive understanding of structure. 

A full description is beyond the scope of this review, but it is noted that the topological method identifies other chemical features in the electron density. The union of all bond paths gives a bond path network that is normally in a 1 1 correspondence with the chemical bond network drawn by chemists. The bond paths for bonds in strained rings are curved, reflecting their bent nature. In Figure 6, we show the gradient paths in the molecular plane of cyclopropane. The C—C bond paths are distinctly bent outward. The value of the Laplacian at the bond critical point discriminates between ionic and covalent bonding." Maps of the Laplacian field reveal atomic shell structure, lone pairs, and sites of electrophilic and nucleophilic attack. The ellipticity of a bond measures the buildup of density in one direction perpendicular to the... 

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