General network theory

One approach to extend such theories to more complex media is network theory. This approach utihzes solutions for transport in single pores, usually in one dimension, and couples these solutions through a network of nodes to mimic the general structure of the porous media [341], The complete set of equations for aU pores and nodes is then solved to determine overall transport behavior. Such models are computationally intense and are somewhat heuristic in nature. 

Nonlinear Theory. It is straightforward to generalize the above linear dynamics to cases with general network deformations [12]. This is necessary for description of dynamics in deformed gels and phase separation. First, the relative velocity should be written as... 

In general, it appears that the fraction of configurations in the various topological classes can be determined for models in which one of the elements is a fixed curve and the other is a random coil. The detailed calculations are intricate and difficult, however, and some simple generalizations are needed which could be used as a step towards building classification effects into the network theories. Classification for the case of two random coils and for self-entanglement are unsolved problems at the present time. 

Ziabicki,A., Takserman-Krozer.R. General dynamic theory of macromolecular networks. I. Definitions and classification. J. Polymer Sci. Part A-2 7,2005-2018 (1969). 

A discussion of Lodge s original network theory (4) can be kept very short, since the general approach has already been sketched in Section2.2. This theory is of importance, as it gives support to the expectation that... 

Both these models find their basis in network theories. The stress, as a response to flow, is assiimed to find its origin in the existence of a temporary network of junctions that may be destroyed by both time and strain effects. Though the physics of time effects might be complex, it is supposed to be correctly described by a generalized Maxwell model. This enables the recovery of a representative discrete time spectrum which can be easily calculated from experiments in linear viscoelasticity. 

Due to the changes in the dynamics, a general relationship for stochastic dynamics is not available like it is for deterministic dynamics. However, for mesoscopic systems, a mesoscopic FR is useful. Therefore, there has been much work on developing stochastic models with different conditions. Andrieux and Gaspard developed a stochastic fluctuation relation for nonequilibrium systems whose dynamics can be described by Schnakenberg s network theory (e.g. mesoscopic electron transport, biophysical models of ion transport and some chemical reactions). Due to early experimental work on protein unfolding and related molecular motors, and their ready treatment by stochastic dynamics, a number of papers have appeared that model these systems and test the or JE for these. FR... 

A Bethe-tree is a particular case of more general networks considered in percolation theory, which is used to a growing extent to describe transport inside catalysts, as evidenced by a recent review by Sahimi et al [ref 26] Sahimi and Tsotsis [ref. 27] applied percolation theory and Honte Carlo simulation to deactivation in zeolites, represented by a simple cubic lattice. 

Meixner, J. (1966). Network theory and its relation to thermodynamics. In Proceedings of the Symposium on Generalized Networks. Polytechnic Press of the Polytechnic Institute of Brooklyn, New York. pp.13-25. 

Today the network theory is generally accepted for ordinary glass. It appears, however, that some complex multicomponent types of glass may also consist to some extent of very small ordered zones in an amorphous network matrix. This is especially true after heat treatment, which can induce phase separation and crystallisation. 

In order to understand ionic distributions in the EDL a realistic description of hydration or, more generally, solvation phenomena is necessary, which in protic liquids implies an adequate description of the hydrogen-bond network. Theory and computer simulation of bulk liquids showed that the most efficient way to include these properties into the models is via distributed charge models in which the intramolecular charge distribution is represented by several point charges. The point charges are adjusted to reproduce experimental dipole and/or quadrupole moments of the molecule, the bulk structure... 

KKTs are tools brought to network theory by the work of Kramers (1926) and Kronig (1929) on X-ray optics. Just as the reciprocity theorem, they are purely mathematical rules of general validity in any passive, linear, reciprocal network of a minimum phase shift type. By minimum-phase networks, we mean ladder networks that do not have poles in the right half plane of the Wessel diagram. A ladder network is of minimum phase type a bridge where signal can come from more than one ladder is not necessarily of the minimum-phase type. The transforms are only possible when the functions are finite-valued at all frequencies. With impedance Z = R- -jX the transforms are ... 

The primary goal of a general statistical theory is to derive an equation of state for the elastomeric molecular network which will hold for any deformation including swelling. Since the major contribution to the elasticity is entropic the molecular interpretation depends on how the stress affects the conformational distribution of an assembly of chains. The successful statistical model will provide predictive relationships between the molecular structure and topology of the network and its macroscopic behavior, e.g., mechanical and swelling responses. 

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