Relaxation, cross equilibrium properties

It was proved experimentally, that expressions (4) and (5) are effective only for hyperelastic state of cross-linked polymers. To describe change of strain electromagnetic anisotropy in all cross-linked polymers physical states will consider strain electromagnetic susceptibility s C (MPa" ) and electromagnetic susceptibility elastic coefficient s relaxation operators. Suppose that they are submitted by such appropriateness, by which equilibrium properties in equation (5)... 

Eq. (14), which was originally postulated by Zimmerman and Brittin (1957), assumes fast exchange between all hydration states (i) and neglects the complexities of cross-relaxation and proton exchange. Equation (15) is consistent with the Ergodic theorem of statistical thermodynamics, which states that at equilibrium, a time-averaged property of an individual water molecule, as it diffuses between different states in a system, is equal to a... 

Different equilibrium, hydrodynamic, and dynamic properties are subsequently obtained. Thus, the time-correlation function of the stress tensor (corresponding to any crossed-coordinates component of the stress tensor) is obtained as a sum over all the exponential decays of the Rouse modes. Similarly, M[rj] is shown to be proportional to the sum of all the Rouse relaxation times. In the ZK formulation [83], the connectivity matrix A is built to describe a uniform star chain. An (f-l)-fold degeneration is found in this case for the f-inde-pendent odd modes. Viscosity results from the ZK method have been described already in the present text. 

Operator in glassy and hyperelastic states of cross-linked polymers is equal to from 0 to 1, respectively, and in transition region between these conditions from 0 to 1. Therefore Equations (1) and (2) reproduce change of concerned cross-linked polymers hyperelastic properties in all their physical states in hyperelastic, where is being momentary a-process, shear pliability s relaxation operator is equal to equilibrium shear pliability in glassy, where is only local conformational mobility of polymeric mesh s cross-site chains, shear phabihty s relaxation operator is equal to shear pliability of glassy state in transition region between these states, where both... 

Rouse chains. The fundamental feature here is the appearance of cooperative interchain motions due to cross-linking. First approaches to evaluating the dynamical properties of such networks started from the intrachain relaxation, and accounted for the connectivity between chains only in simplified, effective ways. For instance, the dynamics of Rouse chains that have fixed (constant) end-to-end distances were studied [60]. Alternatively, Mooney considered Rouse chains with fixed (immobile) ends as a model for a polymer network [3,61]. In particular, he found that the relaxation modulus of such a chain coincides with that of a Rouse chain with free ends, except for a constant contribution, which can be considered as being the nonvanishing, equilibrium modulus of the network However, the idea of the GGS formalism is to take the connectivity exactly into account, see Eqs. 1 and 2. In order to gradually increase the complexity of the networks, one can start by first considering chains cross-linked into regular spatial structures. This is the subject of the present section. 

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