Fractal elastic moduli

The behaviour and magnitude of the storage and loss moduli and yield stress as a function of applied stress or oscillatory frequency and concentration can be modelled mathematically and leads to conclusions about the structure of the material.3 For supramolecular gels, for example, their structure is not simple and may be described as cellular solids, fractal/colloidal systems or soft glassy materials. In order to be considered as gels (which are solid-like) the elastic modulus (O ) should be invariant with frequency up to a particular yield point, and should exceed G" by at least an order of magnitude (Figure 14.2). 

Accordingly, we expect a power law behavior G,0 (O/Op)3 5 of the small strain elastic modulus for 0>0. Thereby, the exponent (3+df [j)/(3—df)w3.5 reflects the characteristic structure of the fractal heterogeneity of the filler network, i.e., the CCA-clusters. The strong dependency of G 0 on the solid fraction Op of primary aggregates reflects the effect of structure on the storage modulus. 

The established concepts predict some features of the Payne effect, that are independent of the specific types of filler. These features are in good agreement with experimental studies. For example, the Kraus-exponent m of the G drop with increasing deformation is entirely determined by the structure of the cluster network [58, 59]. Another example is the scaling relation at Eq. (70) predicting a specific power law behavior of the elastic modulus as a function of the filler volume fraction. The exponent reflects the characteristic structure of the fractal heterogeneity of the CCA-cluster network. 

The strong-, weak- and intermediate regimes are all a product of the elastic constant of the basic mechanical unit (the floe, the links between the floes, or a combination of both) and the number of these units present in the direction of the externally applied force (Shih et al. 1990). Therefore, the fractal dimension defines to the size of the clusters. A large fractal dimension represents a large cluster that translates to less cluster-cluster interactions per unit volume and a decreased elastic modulus. At high volume fractions, cluster size decreases and the number of cluster-cluster interactions increases, and thus the elastic constant also increases. 

In 1992, Vreeker et al. presented rheological data for aggregated fat networks in the framework of previous fractal theories. These authors indicated that the elastic modulus varied with particle concentration according to a power law, in keeping with the proposed models for the elasticity of colloidal gels. 

Filler Particles. We will consider gels containing particles that are much larger than the pores in the gel network. The primary gel (i.e., without particles) may be a polymer gel, a fractal particle gel as obtained with casein, or a heat-set protein gel. We will consider the effect of filler particles on gel properties, especially the elastic modulus, which has been studied best. The following factors are known to affect the properties. 

The dependence of the elastic modulus on protein concentration has been used to establish the framework of fractal geometry. Bremer et al. (1990) indicated that a cluster of protein molecules would possess a fractal nature if the power-law dependence exists between the amount of floes in the cluster and the radius of the cluster. In addition, the magnitude of this power, signified as the fractal dimension, D, would be below 3. The elastic constant of protein aggregates could be described as a function of the aggregate volume fraction ... 

Let us fulfill the value theoretical estimation according to the two methods and compare these results with the ones obtained experimentally. The first method simulates interfacial layer in polymer composites as a result of interaction of two fractals— polymer matrix and nanofiller surface [19,20]. In this case, there is a sole linear scale /, which defines these fractals interpenetration distance [21]. As nanofiller elasticity modulus is essentially higher than the corresponding parameter for rubber (in the considered case—in 11 times, see Figure 6.1), then the indicated interaction... 

The results observed in this ehapter emphasize that the microcomposites rheology description models do not give adequate treatment of melt viscosity for particulate-filled nanocomposites. The correct description of the nanocomposites rheological properties can be obtained within the frameworks of viscous liquid flow fractal models. It is significant, that such an approach differs principally from the used ones to describe microcomposites. So, nanofiller particles aggregation reduces both melt viscosity and elastic modulus of nanocomposites in the solid-phase state. For microcomposites, melt viscosity enhancement is accompanied by elastic modulus increase. 

Hence, the results stated above confirmed that the models, developed for the microcomposites rheology description, did not give melt viscosity adequate treatment for nanocomposites polymer/oiganoclay as well. And as earlier, the indieated nanoeomposites iheologieal properties deseription ean be obtained within the fiamewoik of a viseous liquid flow fractal model. Na -montmorillonite plates aggregation in paekets (tactoids) simultaneously reduces both melt viscosity and elasticity modulus in solid-phase state of nanocomposites. 

Kozlov G.V Novikov V.U. Mikitaev A.K. Fractal analysis of structure elements connectivity with elasticity modulus for cross-linked polymers. Materialovedenie, 1997 4,2-5. 

The viscoelastic properties, in particular how the elastic modulus scales with the volume fraction, have been modelled based on the assumption that the particle network consists of close-packed fractal floes. Although differing in details, these models all predict a power-law behaviour, which can be written as follows ... 

The interrelation of elasticity modulus and amorphous chain s tightness characterized by fractal dimension of chain part between its fixation points for nanocomposites based on the polypropylene is shown. This assumes the polymeric matrix stmcture change in comparison with initial polymer the role of densely-packed regions for it is played by interphase areas. An offered fractal model allows estimation of elasticity modulus limiting values. 

Keywords Nanocomposite polypropylene elasticity modulus amorphous chain tightness fractal analysis. 

Figure. 1. The elasticity modulus vs. fractal dimension of chain part between interfacial areas D at testing temperatures 293 (1) and 373 K (2) for nanocomposites based on the polypropylene...

In Fig. 9.1, the temperature dependences of elasticity modulus E for the studied HDPE have been adduced. As one can see, at comparable testing temperatures E value in case of quasistatic tests is about twice smaller, than in impact ones. Let us note that this distinction is not due to tests type. As it has been noted above, for HDPE with the same ciystalUnity degree at r=293 K E value can reach 1252 MPa [10]. Let us consider the physical grounds of this discrepancy. The value of fractal dimension of polymer stiucture, which is its main characteristic, can be determined by several methods appUcation. The first from them uses the following equation [11] ... 

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