Network structure topology

Equation (29) shows that the modulus is proportional to the cycle rank , and that no other structural parameters contribute to the modulus. The number of entanglements trapped in the network structure does not change the cycle rank. Possible contributions of these trapped entanglements to the modulus therefore cannot originate from the topology of the phantom network. 

The division by two denotes the topological requirement (6) of two inter-crosslink chains per DVB, in the present model of a closed X-type network structure. (Consider two X units with their chain ends joined together. There are four chain lines and two vertices.)... 

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. 

Computer simulations were performed taking into account the component ratio P, the relative reactivity of primary and secondary amines, and different probabilities of monocycle formation. The simulations showed the structural features of networks (both topology and defects) at all stages of the cure process. Two examples of network structure received from computer simulation are shown in Fig. 2. Here... 

The glass transition temperature Tg is one of the most important structural and technical characteristics of amorphous solids. The correlations of Tg of linear polymers with their chemical composition, molecular weight, rigidity and symmetry of chains, as well as some other characteristics of macromolecules are well documented 57,58) Thg information on networks is much poorer. At present, for networks there exists mainly one parameter in structure-Tg correlations. It is the concentration of crosslinks — a parameter which is very insufficient, since in networks there are chemical crosslinks of different functionality (connectivity) which are distinguished by their molecular mobility. This means that the topological aspect of the network structure should be taken into account in the Tg analysis. Another difficulty connected with Tg determination of polymers lies in vitrification occurring during polymer formation (Sect. 6). 

In the era of systems biology, great attention is paid to the structures of networks of reactions and interacting molecules (i.e., the topological connectivities). In some ways network structures have replaced molecular structures as the central object of biological attention. 

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. 

Solid state NMR offers several advantages for the investigation of filled rubbers since molecular properties of elastomer chains can be measured selectively by NMR e>q)eriments. The method is very sensitive to the molecular scale heterogeneity in a sample. The network structure which is composed of chemical, physical and topological junctions can also be andyzed by NMR relaxation experiments [11,12,14,15],... 

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. 

It follows from this Fig. that the amount of chemical junctions in silicon rubber increases with increasing fractions of Aerosil. The chemical junctions are apparently formed by scission of PDMS chains under the mechanical forces during milling. However, the fraction of these junctions is the lowest. The fraction of adsorption junctions increases proportionally to the filler content as shown in Fig. 11. The major contribution to the network structure is provided by topological hindrances near the filler surface as shown in Fig. 11. 

To sum it up it can be said, that solid state NMR is a very sensitive tool for the study of chain dynamics at the elastomer-filler interface as well as the network structure resulting from chemical Junctions, adsorption junctions and topological hindrances from the filler particles. The method is of interest for establishing structure-property relations for filled elastomers. 

In order to raise the stability of nonmetal nitride network structures the element phosphoms has to be exchanged against silicon. In phosphoms nitrides the relative amount of (N ) never seems to exceed a value of 2/5 and in most examples it is zero. In contrast, in silicon nitrides even all of the nitrogen atoms each may connect three silicon atoms (N ). This topological situation is realized in the two modifications of binary silicon nitride Si3N4 (Fig. 2). 

Sr2SisN8 and Ba2SisNg are isotypic and have a network structure of corner-sharing SiN4 tetrahedra similar to Ca2SisN8. However both structure types are topologically different [10] as the... 

There are two directions of study. One is with regard to the static aspect of reaction network structure (e.g., topology). The other is the number distribution of chemical species and their dynamics. Of course, one needs to combine the two aspects to fully understand the condition for recursive production of a cell. 

The primary structure of macromolecules is defined as the sequential order of monomers connected via covalent chemical bonds. This structural level includes features such as chain length, order of monomer attachment in homopolymers (head-to-head, head-to-tail placement), order of monomer attachment in various copolymers (block copolymers, statistical and graft copolymers, chemical composition of co-monomers), stereoregularity, isomers, and molecular topology in different branched macromolecules and molecular networks. Structure at this primary level can be manipulated by polymer synthesis [4]. With AFM it is possible to visualize, under certain conditions, single macromolecules (Fig. 3.2) and it is even possible to manipulate these (i.e. push with AFM tips). Characteristics of chain-internal... 

Another two key concepts in metabolic engineering are metabolic pathway analysis and metabolic pathway modeling. The former is used for assessing inherent network properties in the complete biochemical reaction networks. It involves identification of the metabolic network structure (or pathway topology), quantification of the fluxes through the branches of the metabolic network, and identification of the control structures within the metabolic network. 

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