Vilgis models

Below we consider practical aspects of the estimation of the fractal dimension D of a macromolecule section between chemical crosslinking points with allowance for these statements as well as for those considered in the preceding Sections. This approach differs fundamentally from that of the Cates [56] and Vilgis models [61, 62]. All the foregoing is valid not only for a network of chemical bonds but also for a network of macromolecular entanglements in linear polymers. 

Initial modulus from stress-strain curves was as well used for determining the percolation threshold of CNT in IR. Initial modulus values were elaborated through the Huber and Vilgis model, reporting in a double logarithmic plot the excess of initial modulus (E — Eo)/Eq, versus the CNT content. Experimental data were interpolated with two different straight lines with slope 0.9 and 2.1, respectively, whose intersection gave the percolation threshold at... 

Figure 8.19 Percolation threshold predicted from Huber-Vilgis model, (a) NR-OMt system, (b) NR-EOMt system.

Fig. 3. Visualization of the viscoelastic Zener model within the Huber/Vilgis approach...

Data on the fractal forms of macromolecules, the existence of which is predetermined by thermodynamic nonequilibrium and by the presence of deterministic order, are considered. The limitations of the concept of polymer fractal (macromolecular coil), of the Vilgis concept and of the possibility of modelling in terms of the percolation theory and diffusion-limited irreversible aggregation are discussed. It is noted that not only macromolecular coils but also the segments of macromolecules between topological fixing points (crosslinks, entanglements) are stochastic fractals this is confirmed by the model of structure formation in a network polymer. 

A crosslinked polymer is formally one giant macromolecule (fractal cluster) therefore, the models considered above provide the dimension d( of exactly this cluster. The Vilgis concept [61, 62] assumes the existence of several clusters of this type and linear macromolecules between them. In practice, parameters used more widely are the density of the network of nodes (chemical crosslinks) or the molecular mass of the macromolecular section between the crosslinks M, which are related to each other in the following way ... 

Extrapolation of the SANS data in [4] to the isotropic state confirms, indirectly, the presence of a diffuse PS-PI transition layer between filler and rubbery matrix with thickness A 0.5 nm around the PS domain with a mean filler radius of about 84 A. Excellent agreement between measured reinforcing factor and corresponding model predictions could be realized within a very recent approach of Huber and Vilgis [5] for the hydrodynamic reinforcement of rubbers filled with spherical fillers of core-shell structures [6]. 

Vilgis et al. commented that the disadvantages of this model are the small range of application and the idealizations which they introduced in order to make the calculations tractable. The advantages are the successful derivation of a structure-property relationship, the possibility of explicitly including the fractal filler structure, and the universality (transfer to all types of branched aggregates). Refinements of the present model require the inclusion of local properties, such as particle-particle binding between the primary filler particles. 

This theory was introduced by Vilgis el al The simplest model consists of randomly dispersed uniform soft spheres. There are two limiting cases if the modulus of the soft filler particles is zero, the matrix contains holes (resembling a Swiss cheese) and thus becomes softer. On the other hand, in the case of a very large modulus of the filler particles, the Einstein-Smallwood formula (Equation (3.1)) will be reproduced. For uniform soft filler particles with elastic modulus Gf > Gm the intrinsic modulus [/r], is given by... 

Vilgis and Erman that the constraint models and slip-link models have much in common, (iv) elucidating the effects of cross-link functionality and degree of cross linking, (v) exploring a variety of elastomeric polymers, particularly those having very different values of the plateau modulus, and (vi) generalizing rubber-elasticity models to include viscoelastic effects as well. 

Vilgis, T. A. Erman, B., Comparison of the Constrained Junction and the Slip-Link Models of Rubber Elasticity. Macromolecules 1993,26(24), 6657-6659. 

The mutual steric restrictions of entangled chains at deformation are accotmted for in a tube model considering the reptation motion of network subchains. This approach was proposed by Edwards and Vilgis ° and Heinrich et al. Later the tube model was further developed (see References 42 and 43). 

The scaling model of Vilgis and Halperin (VH) provides a theoretical Ifame-woik to help us understand aspects of our observations. VH consider a chain-folded crystalline core in which each chain experiences an integral number of chain... 

Abstract The present chapter is written as an introduction towards this book on nonlinear viscoelasticity of rubber composites and nanocomposites. Rather than introducing the concept of the book to the readers this chapter reveals the basics behind rubber viscoelasticity and explains both linearity and nonlinearity from this behavior. Various filler reinforced rubbers are introduced emphasising the flow behavior of such nanocomposites. Major mathematical models proposed by Kraus, Huber and Vilgis and Maier and Goritz for the Payne Effect are briefly addressed based on the filler matrix interactions existing in the composite systems. 

Of the several mechanisms investigated, the most commonly adopted is based on the filler network breakage [48, 49]. Kraus [7, 50] proposed a phenomenological model of the Payne effect based on this interpretation. In this model, under dynamic deformation, filler-filler contacts are continuously broken and reformed. The Kraus model considers filler-filler interactions but the loss modulus and effect of temperature were not taken into account. In the model of Huber and Vilgis [9, 50, 51] the existence of dynamic processes of breakage and reformation of the filler network is explained. In this model, the Payne effect is related to the fractal nature of the filler surface. At sufficiently high volume fractions of filler, percolation occurs and a continuous filler network is formed, characterized by its fractal dimension and its... 

Usual microscopic models of chemical reaction assumes the existence of ordered energy barriers. Chemical reaction is used to be considered as diffusion in the phase space, Kramers (1940). It is not clear, whether how to switch such kinds of microscopic models to traditional CDS models. One of the main open problems in chemical kinetic is to derive CDS model from microscopic picture. It seems to be credible, that ordered energy barriers can lead to traditional Markov models. Random fluctuations in the energy barrier can have ultrametric structure (e.g. Zumofen et al 1986) and may lead to much more complex, i.e. hierarchical dynamic models (Vilgis 1987). Chemical reaction can be interpreted as anomalous, i.e. ultradiffusion (Huberman and Kerszberg 1985).While normal diffusion is characterized by the relation... 

Vilgis T, Benmouna M, Benoit H (1991) Static scattering from multicomponent polymer systems theoretical models. Macromolecules 24 4481 488... 

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