Network junction model development

This chapter is devoted to the molecular rheology of transient networks made up of associating polymers in which the network junctions break and recombine. After an introduction to theoretical description of the model networks, the linear response of the network to oscillatory deformations is studied in detail. The analysis is then developed to the nonlinear regime. Stationary nonhnear viscosity, and first and second normal stresses, are calculated and compared with the experiments. The criterion for thickening and thinning of the flows is presented in terms of the molecular parameters. Transient flows such as nonhnear relaxation, start-up flow, etc., are studied within the same theoretical framework. Macroscopic properties such as strain hardening and stress overshoot are related to the tension-elongation curve of the constituent network polymers. 

The pore connectivity aspect has been neglected in the DFT analysis. It is possible that the adsorption in an individual pore is affected by the adsorption in an adjacent or a networked pore, which can complicate the adsorption integral. Better insight into the connectivity phenomenon has been provided by Seaton and co-workers using Monte Carlo simulations [32,33] and efforts to develop pore-junction models are on. 

According to the importance of the cross-links, various models have been used to develop a microscopic theory of rubber elasticity [78-83], These models mainly differ with respect to the space accessible for the junctions to fluctuate around their average positions. Maximum spatial freedom is warranted in the so-called phantom network model [78,79,83], Here, freely intersecting chains and forces acting only on pairs of junctions are assumed. Under stress the average positions of the junctions are affinely deformed without changing the extent of the spatial fluctuations. The width of their Gaussian distribution is predicted to be... 

Early Development of the Drosophila Neuromuscular Junction A Model for Studying Neuronal Networks in Development Akira Chiba... 

Little information has been published on the question of how filler network structure actually affects the energy dissipation process during dynamic strain cycles. The NJ-model focuses on modeling of carbon black network structure and examination of the energy dissipation process in junction points between filler aggregates. This model was further developed to describe the strain amplification phenomenon to provide a filler network interpretation for modulus increase with increasing filler content. 

The theory of affine networks was developed by Kuhn and improved by Treloar, and is based on the assumption that the network consists of v freely-jointed Gaussian chains and the mean-square end-to-end vector of network chains in the undeformed network is the same as of chains in the uncross-linked state. This assumption is supported by experimental data. It is also assumed that there is no change in volume on deformation and the junctions displace affinily with macroscopic deformation. The intermolecular interactions in the model are neglected, i.e., the system is similar to the ideal gas. 

A molecular model for the deformation is required. The Flory-Rehner Eqs. (4) and (5) were developed for a network deforming affinely, i.e. a network without junction fluctuations. A more general treatment by Flory and Erman which includes such fluctuations is described in this Section. 

Stress-strain measurements in uniaxial extension have revealed that real networks have a behavior closest to the affine limit at small deformations and approach the phantom limit at large deformations. The recent molecular theory developed by Flory and Erman accounts for this transition. In this model, the restrictions on junction... 

The idea of entanglements acting as physical junctions was originally developed in the literature to explain deviations from the predictions for affine networks (185,213). The possibility of a contribution at equilibrium caused by trapped entanglements has been tested with model networks, ie, those prepared in such a way that the niunber and functionality of the cross-links are known. A... 

In recent years, with the development of model networks it has been possible to prepare networks of controlled and junction functionality These are prepared by endlinking functionalized prepolymers with cross-linking agents of known functionality. Therefore, by choosing the appropriate molecular weight distribution of the prepolymers it is possible to prepare unimodal and bimodal networks. Mark and coworkers (5-11) have performed extensive studies on model networks to test the various theories of rubber elasticity. In the case of unimodal networks they find that the macroscopic properties such as stress or swelling ratios can be described reasonably well by the Flory-Erman theory (12,13). 

In the case of a highly crosslinked network, the assumption that the chain lengths between junctions follows a Gaussian distribution may be incorrect. Therefore, the Peppas and Lucht [12] model (equation (17)) was developed to describe the equilibrium swelling behavior for a polymeric gel where the network was formed in the solid state, but with a non-Gaussian chain length distribution. 

The Affine Network Model. One of the earlier assumptions regarding microscopic deformation in networks is that the junction points in the networks move affinely with macroscopic deformation. The affine network model was developed by Kuhn, Wall, and Flory (1,175,176). According to the model, chain end-to-end vectors deform affinely, which gives... 

Theoretical developments after 1975 have not supported the affine network model (82,177-182), mainly because the assumption of embedding each junction securely into the continuum of the network volume was judged to be unrealistic. Even more important, neutron scattering results (5,183-185) provide clear evidence for junction fiuctuations of the nature assumed in the phantom network model described below. 

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