Percolation viscoelastic properties

Verification of the values of the exponents in Equations 6.14 and 6.15 continue to be the subjects of considerable debate between proponents of the two theories. The viscoelastic properties of the system that characterize the sol-gel transition are also important features of the percolation model. 

Stauffer, D., Introduction to Percolation Theory, Taylor Friends, London, 1985. Yanez, J.A., Laarz, E., and Bergstrom, L., Viscoelastic properties of particle gels, J. Colloid Interface ScL, 209, 162, 1999. 

As an example of the viscoelastic properties of an actual medium, we consider the viscoelastic properties of a charged polymeric material. Percolation properties will be exhibited by the charged polymeric composite, if the stiffness of the agglomerates of particles is greater by some orders of magnitude than the stiffness of the unperturbed polymetric compound [49]. This can occur, for example, if the polymetric compound in the vicinity of a boundary (e.g., the surface of a particle) attains the superstrong state [171-173]. 

Another approach to determining the viscoelastic properties of dense microemulsions at high frequencies is to conduct ultrasonic absorption experiments. In such experiments it has been found that the percolation process is correlated to a shift of the ultrasonic dynamics from a single relaxation time to a distribution of relaxation times [121]. Other experiments showed an increase in the hypersonic velocity for samples at and beyond the percolation threshold. The complex longitudinal modulus deduced from such experiments is also correlated with the occurrence of the percolation phenomenon, which suggests that the velocity dispersion is clearly correlated with structural transformations [122]. 

Non-linear viscoelastic properties were observed for fumed silica-poly(vinyl acetate) (PVAc) composites, with varying PVAc molar mass and including a PVAc copolymer with vinyl alcohol. Dynamic mechanical moduli were measured at low strains and found to decrease with strain depending on surface treatment of the silica. The loss modulus decreased significantly with filler surface treatment and more so with lower molar mass polymer. Copolymers with vinyl alcohol presumably increased interactions with silica and decreased non-linearity. Percolation network formation or agglomeration by silica were less important than silica-polymer interactions. Silica-polymer interactions were proposed to form trapped entanglements. The reinforcement and nonlinear viscoelastic characteristics of PVAc and its vinyl alcohol copolymer were similar to observations of the Payne effect in filled elastomers, characteristic of conformations and constraints of macromolecules. ... 

Khalkhal and Carreau (2011) examined the linear viscoelastic properties as well as the evolution of the stmcture in multiwall carbon nanotube-epoxy suspensions at different concentration under the influence of flow history and temperature. Initially, based on the frequency sweep measurements, the critical concentration in which the storage and loss moduli shows a transition from liquid-like to solid-like behavior at low angular frequencies was found to be about 2 wt%. This transition indicates the formation of a percolated carbon nanotube network. Consequently, 2 wt% was considered as the rheological percolation threshold. The appearance of an apparent yield stress, at about 2 wt% and higher concentration in the steady shear measurements performed from the low shear of 0.01 s to high shear of 100 s confirmed the formation of a percolated network (Fig. 7.9). The authors used the Herschel-Bulkley model to estimate the apparent yield stress. As a result they showed that the apparent yield stress scales with concentration as Xy (Khalkhal and Carreau 2011). 

Arbabi, S Sahimi, M, Critical Properties of Viscoelasticity of Gels and Elastic Percolation Networks, Physical Review Letters 65, 725, 1990. 

In the present case, the foam density relates perfectly with the previously observed rheological properties, as a transition in the flow behavior was detected at approximately 20 wt% of PPE (Fig. 13). In the viscoelastic case (below the percolation limit), the PPE content neither significantly influences the foamability nor the blend rheology. At elevated contents (beyond percolation), however, the PPE content strongly affects the rheological response of the blend and, subsequently, degrades the foaming behavior, which is verified by a reduced expandability. 

The power laws for viscoelastic spectra near the gel point presumably arise from the fractal scaling properties of gel clusters. Adolf and Martin (1990) have attempted to derive a value for the scaling exponent n from the universal scaling properties of percolation fractal aggregates near the gel point. Using Rouse theory for the dependence of the relaxation time on cluster molecular weight, they obtain n = D/ 2- - Df ) = 2/3, where Df = 2.5 is the fractal dimensionality of the clusters (see Table 5-1), and D = 3 is the dimensionality of... 

Recent work (Altmann, 2002) has focussed on combining a dynamic Monte Carlo percolation-grid simulation for reaction kinetics and an enthalpy-based group-interaction viscoelastic model to develop a model for the chemorheological and network properties of reactive systems. More emphasis will be placed on this model in Chapter 6. 

Here, the discussion of the viscoelastic exponent n in relation to the assumed gelation model (e.g. electrical analogy percolation or Rouse model), as well as the fractal dimension dfof the critical gel (Muthukumar and Winter 1986 Muthukumar 1989) will be ignored. The reader is referred e.g. to (Adam and Lairez 1996 Martin and Adolf 1991). It has been also shown, that stoichiometry, molecular weight and concentration have an impact on the critical gel properties (Winter and Mours... 

The co-continuous structure and the final rheological properties of an immiscible polymer blend are generally controlled by not only the viscoelastic and interfacial properties of the constituent polymers but also by the processing parameters. For example, the effect of plasticizer on co-continuity development in blends based on polypropylene and ethylene-propylene-diene-terpolymer (PP/EPDM), at various compositions, was studied using solvent extraction. The results showed more rapid percolation of the elastomeric component in the presence of plasticizer. However, the same fuUy co-continuous composition range was maintained, as for the non-plasticized counterparts (Shahbikian et al. 2011). It was also shown that the presence of nanoclay narrows the co-continuity composition range for non-plasticized thermoplastic elastomeric materials (TPEs) based on polypropylene and ethylene-propylene-diene-terpolymer and influences their symmetry. This effect was more pronounced in intercalated nanocomposites than in partially exfoliated nanocomposites with improved clay dispersion. It seems that the smaller, well-dispersed particles interfere less with thermoplastic phase continuity (Mirzadeh et al. 2010). A blend of polyamide 6 (PA6) and a co-polyester of... 

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