Diffused-constraint model

The interactions between long overlapping network strands suppress fluctuations not only of the network junctions, but of all monomers in every network strand. In an attempt to capture this effect, Kloczkowski, Mark, and Erman proposed a diffused-constraints model. Instead of the... 

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... 

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... 

Urayama et al. [119-121] tested the diffused constraint model using both uniaxial compression and equibiaxial elongation data for end-linked PDMS networks in which trapped entanglements were dominant in number relative to chemical crosslinks. The parameter k was used as an empirical fitting parameter, and the best-fit procedure yielded k = 2.9. The structural parameters (v, jj., /)... 

This study was carried out to simulate the 3D temperature field in and around the large steam reforming catalyst particles at the wall of a reformer tube, under various conditions (Dixon et al., 2003). We wanted to use this study with spherical catalyst particles to find an approach to incorporate thermal effects into the pellets, within reasonable constraints of computational effort and realism. This was our first look at the problem of bringing together CFD and heterogeneously catalyzed reactions. To have included species transport in the particles would have required a 3D diffusion-reaction model for each particle to be included in the flow simulation. The computational burden of this approach would have been very large. For the purposes of this first study, therefore, species transport was not incorporated in the model, and diffusion and mass transfer limitations were not directly represented. 

Rieckmann and Volker fitted their kinetic and mass transport data with simultaneous evaluation of experiments under different reaction conditions according to the multivariate regression technique [116], The multivariate regression enforces the identity of kinetics and diffusivities for all experiments included in the evaluation. With this constraint, model selection is facilitated and the evaluation results in one set of parameters which are valid for all of the conditions investigated. Therefore, kinetic and mass transfer data determined by multivariate regression should provide a more reliable data basis for design and scale-up. 

Constraints may be introduced either into the classical mechanical equations of motion (i.e., Newton s or Hamilton s equations, or the corresponding inertial Langevin equations), which attempt to resolve the ballistic motion observed over short time scales, or into a theory of Brownian motion, which describes only the diffusive motion observed over longer time scales. We focus here on the latter case, in which constraints are introduced directly into the theory of Brownian motion, as described by either a diffusion equation or an inertialess stochastic differential equation. Although the analysis given here is phrased in quite general terms, it is motivated primarily by the use of constrained mechanical models to describe the dynamics of polymers in solution, for which the slowest internal motions are accurately described by a purely diffusive dynamical model. 

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... 

Serious efforts have been made to explain the atypical lithium transport behavior using modified diffusion control models. In these models the boundary conditions -that is, "real potentiostatic constraint at the electrode/electrolyte interface and impermeable constraint at the back of the electrode - remain valid, while lithium transport is strongly influenced by, for example (i) the geometry of the electrode surface [53-55] (ii) growth of a new phase in the electrode [56-63] and (iii) the electric field through the electrode [48, 56]. 

Here k is a parameter which measures the strength of the constraints. For k = 0 we obtain the phantom network limit, and for infinitely strong constraints (k = oo) the affine limit is obtained. Erman and Monnerie [27] developed the constrained chain model, where constraints effect fluctuations of the centers of the mass of chains in the network. Kloczkowski, Mark, and Erman [28] proposed a diffused-constraint theory with continuous placement of constraints along the network chains. 

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. 

Comparison of results between GEOSECS and TTO/SAVE shows that the bomb radiocarbon inventory has increased by 36% for the region north of 10° N, by 69% for the equatorial region and by 71% for the region south of 10° S. These data reflect the radiocarbon uptake for the Atlantic Ocean between 1973 (GEOSECS) and 1985 (TTO/SAVE). Along with global bomb radiocarbon distribution, this information provides crucial constraints for the carbon cycle in the ocean. Preliminary results from CGC-91, one of the WOCE cruises, show that the observed increase in bomb radiocarbon inventory from 1974 to 1991 in the northern Pacific Ocean is consistent with the first-order prediction from a box-diffusion ocean model. 

Experimental methods which yield precise and accurate data are essential in studying diffusion-based systems of pharmaceutical interest. Typically the investigator identifies a mechanism and associated mass transport model to be studied and then constructs an experiment which is consistent with the hypothesis being tested. When mass transport models are explicitly involved, experimental conditions must be physically consistent with the initial and boundary conditions specified for the model. Model testing also involves recognition of the assumptions and constraints and their effect on experimental conditions. Experimental conditions in turn affect the maintenance of sink conditions, constant surface area for mass transport, and constant and known hydrodynamic conditions. 

With respect to SCF models that focus on the tail properties only (typically densely packed layers of end-grafted chains), the molecularly realistic SCF model exemplified in this review needs many interaction parameters. These parameters are necessary to obtain colloid-chemically stable free-floating bilayers. A historical note of interest is that it was only after the first SCF results [92] showed that it was not necessary to graft the lipid tails to a plane, that MD simulations with head-and-tail properties were performed. In the early MD simulations (i.e. before 1983) the chains were grafted (by a spring) to a plane it was believed that without the grafting constraints the molecules would diffuse away and the membrane would disintegrate. Of course, the MD simulations that include the full head-and-tails problem feature many more interactions than the early ones. 

As a result of steric constraints imposed by the channel structure of ZSM-5, new or improved aromatics conversion processes have emerged. They show greater product selectivities and reaction paths that are shifted significantly from those obtained with constraint-free catalysts. In xylene isomerization, a high selectivity for isomerization versus disproportionation is shown to be related to zeolite structure rather than composition. The disproportionation of toluene to benzene and xylene can be directed to produce para-xylene in high selectivity by proper catalyst modification. The para-xylene selectivity can be quantitatively described in terms of three key catalyst properties, i.e., activity, crystal size, and diffusivity, supporting the diffusion model of para-selectivity. 

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