Junction constrained diffusion

II. The Constrained Diffusion Jimction.—The assumption made by Planck in order to integrate the equation for the liquid junction potential is equivalent to what has been called a constrained diffusion junction this is supposed to consist of two solutions of definite concentration separated by a layer of constant thickness in which a steady state is reached as a result of diffusion of the two solutions from opposite sides. The Planck type of junction could be set up by employing a membrane whose two surfaces are in contact with the two electrolytes which are continuously renewed in this way the concentrations at the interfaces and the thickness of the intermediate layer are kept constant, and a steady state is maintained within the layer. The mathematical treatment of the constrained diffusion junction is complicated for electrolytes consisting entirely of univalent ions, the result is the Planck equation,... 

The number of monomers in virtual chains is assumed to change with deformation according to Eq. (7.60), similar to the constrained-junction and diffused-constraints models. If one virtual chain is attached to every entanglement strand of monomers, it contains of order virtual monomers in the undeformed state of the network. The number of monomers in each virtual chain changes as the network is deformed... 

The affine and the phantom models derive the behavior of the network from the statistical properties of the individual molecules (single chain models). In the more advanced constrained junction fluctuation model the properties of these two classical models are bridged and interchain interactions are taken into account. We remark for completeness that other molecular models for rubber networks have been proposed [32,57,75-87], however, these are not nearly as widely used and remain the subject of much debate. Here we briefly summarize the basic concepts of the affine, phantom, constrained junction fluctuation, diffused constraint, tube and slip-tube models. 

As already described, the upper three portions of Figure 2 summarize the differences in the way the constraints are applied in the constrained-junction theory, constrained-chain theory, and the diffused-constraints theory, respectively [4], Additional comparisons between theory and experiment for a variety of elastomeric properties should be very helpful [20], Also, neutron-scattering measurements conducted on series of networks having different values of the junction functionality , which is the number of chains emanating from a junction (cross-link), would be extremely useful in suggesting how to position the constraints along a chain in refining such models, since should have a pronounced effect on the... 

FIGURE 29.4. Effect of constraints on the fluctuations of network junctions, (a) Phantom model and (b) constrained junction fluctuation model. Note that the domain boundaries (circles in the figures) are diffuse rather than rigid. The action of domain constraint is assumed to be a Gaussian function of the distance of the junction from B similar to the action of the phantom network being a Gaussian function of AR from the mean position A. 

In summary, the common feature of all constrained chain models is that they impose only limited constraints on chain fluctuations. [101] The constrained-junction fluctuation model restricts fluctuations of junctions and of the center of mass of network chains. The diffused constraint model restricts fluctuations of a single randomly chosen monomer for each network strand. Consequently, all these models can only represent the crossover between the phantom and afflne limits. [101] The phantom limit corresponds to a weak constraining case, while the affine limit corresponds to a very strong constraining potential. 

Kloczkowski, Mark, and Erman [95] compared the prediction of the diffused constraint model with the results of the Flory constrained-junction fluctuation theory [36] and the Erman-Monnerie constrained chain theory [94]. They found that the shapes of the [/ ] vs. a curves for all three theories were very similar. Rubinstein and Panyukov [101] reanalyzed the data of Pak and Flory [118] obtained for uniaxially deformed crosslinked PDMS samples. They concluded that the fit of the experimental data by the diffused... 

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