Calculations with cross-linked network model

For the nonlinear step growth case above, eiTg, the crosslink density must be related to p. A relevant model, based on calculating the probabilities of finite chains being formed, has been published For the reaction of A -1- 2B2 (e.g., tetra-functional amine -b difunctional epoxy), A4 is considered to be an effective cross-linking site if three or more of its arms lead out to the infinite network. The probability of finding an effective crosslink is related to one minus the probability of a randomly chosen A leading to the start of a finite chain, which in turn is related to the extent of reaction. Application of this procedure to the system of Fig. 15 has been presented in detail The more complicated reaction of a tetrafunctional amine with a trifunctional epoxy was also considered. ... 

As a result, the concentration of the elastically effective subchains (i.e., subchains contributing to the elasticity modulus) differs from the concentration of subchains Vch calculated from the amount of cross-linker used in the gel synthesis under the assumption of the formation of an ideal network, where all the cross-linker molecules are incorporated in such a way that they connect elastically effective network subchains. To describe this difference, the elasticity modulus G is represented as a sum of two components Gch and Gg associated with chemical cross-links and entanglements, respectively. This approach was called a two-network model. " °... 

The molecular model of an elastomeric network with local intermolecular correlations, given by Flory, is used to calculate the components of the molecular deformation tensor and molecular orientation. Effects of molecTilar parameters such as severity of entanglements, network inhomogeneities and conditions during cross-linking are discussed. Components of molecular deformation and orientation are calculated for a network under uniaxial stress. 

The above picture of the network structure of vulcanized rubber is supported by -the success of the kinetic theory of rubberlike elasticity (see part 4, page 14) calculations based on this model agree well with experimental measurements of stress-strain curves and other properties (James and Guth, 1943 Flory, 1944). Excellent evidence that the swollen gel contains the same network as the unswollen rubber has been presented by Flory (1944, 1946), based on studies of butyl rubber. Using the network model, the number of cross-links in the structure can be calculated in three ways (o) from measurements of the proportions of insoluble (network) and soluble (unattached) material in samples of different initial molecular lengths (b) from the elastic modulus of the unswollen rubber (c) from the maximum amount of liquid imbibed by the gel when swollen in equilibrium with pure solvent. The results of these three calculations for butyl rubber samples were in good agreement. 

Chains in Networks. One of the first studies of chains in networks is by Gao and Weiner (235) where they performed extended simulations of short chains with fixed (affinely moving) end-to-end vectors. The first extensive molecular dynamics simulations of realistic networks were performed by Kremer and collaborators (236). These calculations were based on a molecular dynamics method that has been applied to study entanglement effects in polymer melts (237). The networks obtained by cross-linking the melts were then used to study the effect of entanglements on the motion of the cross-links and the moduU of the networks. The moduli calculated without any adjustable parameters were close to the phantom network model for short chains, and supported the Edwards tube model for long ones. Similar molecular dynamics analyses were used to understand the role of entanglements in deformed networks in subsequent studies (238-240). 

To use the elastic floe model in a quantitative manner, it is necessary to have a model for the flocculated structure. For the present case of particles with adsorbed PVA chains, the most likely structure is that produced by interpenetration of the tails, under worse tha d-conditions for the chains. A typical floe may be assumed to consist of strings of particles linked together in a more-or-less three dimensional network. The floe density (as measured by Cpp) is related to its strength by the number of chains, which pass through unit cross sectional area of floe [9,10,13]. /ic can be calculated from the total number of bonds per floe [10], i.e. 

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