Hard phase percolation

It is important to note that Eq. (2.7) should be used only in the vicinity of the hard phase percolation. Indeed, the meaning of the percolation exponent, 6, is to eiccomit for the fact that the percolated pathways are not linear but have some complicated morphology thus, the efficiency of the reinforcement ( fraction of elastically active hard phase elements ) is much less than 100%. 

If the hard phase in a polymer blend is of much higher modulus than the soft phase so that the soft phase contribution to the composite modulus can be neglected, and if co-continuity of the hard phase sets in at a threshold volume fraction (percolation threshold pc) beyond which the hard phase behaves like percolating randomly-placed objects, simple phenomenological models based on percolation theory (see Equation 20.2 for an example) often work quite well both for E and for ay [12-16]. The power law exponent [3, whose theoretical value is... 

Figure 20.3. Comparison of the predicted Young s moduli of binary multiphase materials with morphologies best described by the aligned lamellar fiber-reinforced matrix model (Equation 20.1), the blend percolation model (Equation 20.2), and Davies model for materials with fully interpenetrating co-continuous phases (Equation 20.3). The filler Young s modulus in Equation 20.1 was assumed to be 100 times that of the matrix, and calculations were performed at Af=10, At-=100 and Af=l()00 to compare the effects of discrete filler particles with differing levels of anisotropy. It was assumed that E(hard phase)=100, pc=0.156 and (3=1.8 in Equation 20.2. For...

Based on various experimental studies, one can schematically represent the morphology of segmented polyurethane (elastomer or flexible foam polymer) on the nano- and micro-scale as shown in Figm-e 2.1. For the range of hard segments volume fraction less than 50%, much of the space is occupied by the soft phase matrix. Microphase-separated nano-domains of the hard phase are dispersed in this matrix they can be individual islands or can form percolated networks. Finally, there could also be some larger (micron-sized) macrophase-separated domains of hard phase, where hard phase domains are ordered at the macro-scale (this is especially true in the case of flexible foams). The relative amounts of all these elements depend on the formulation and processing history. 

Figure 2.1. Complexity in polyurethanes. On a nano- to micro-scalet ywethanes contam various components, including macrophase separated (micron-siz ) pbase domams, disperaed nanometer-sized hard phase islands , and percolated hard phase ano-network, all dispersed in the soft phase matrix ...

Here, <5 is the percolation exponent (we take 6 = 2.5), and is the percolation threshold for the hard phase. The percolation exponent, 6, typically ranges between 1.5 and 2 (see, e.g., ref. [52]), depending on the type of the system and the property described by a percolation model (modulus, conductivity, etc.). There are instances, however, when the percolation exponent could be larger than 2 (see, e.g., ref. [53]). Various models (e.., double percolation -see ref. [54]) have been proposed to explain these high percolation exponents. In our analysis, we refrain firom ascribmg any specific meaning to exponent 5 = 2.5, and treat it simply as an adjustable parameter that is found fi-om the best fit to experimental data. 

An important point is that the percolation threshold, depends on the morphology of the percolating clustei-s, and thus on the incompatibility between the two phases. We postulate that = fgQ, so that once hard phase spheres aggregate into infinite cylinders, percolated hard phase is formed. This assumption is a crucial link that relates thermodynamic information about the hard segment (its cohesive energy density or solubility parameter) to mechanical properties of polyurethanes based on that hard segment. 

The above discussion was mainly dealing with the percolated hard phase. The remaining portion of the material is the filled soft phase which consists of the soft phase matrix and hard phase spheres. Modulus enhancement provided by the spheres can be estimated by [39] ... 

It should be mentioned that in general, hard phase clusters can be non-spherical, as discussed in various earlier papers. In this case, the modulus increase could strongly depend on the aspect ratio the effect of the aspect ratio can be modeled through the micromechanical models of Halpin and Tsai [55] or Mori and Tanaka [56]. However, as we already commented above, below the spherical-to-cylindrical transition, most of the hard phase nano-domains have an aspect ratio close to 1. Above the spherical-to-cylindiical transition that is in our model associated with percolation threshold, most of the cylinders participate in the formation of the percolated hard phase, while the soft phase primarily contains hard phase islands with smaller aspect ratios. Therefore, in our analysis we assume that all the fillers dispersed within the soft phase are spherical (or have aspect ratios close to one). 

Stress should have contributions from a (hyper)elastic soft phase and elasto-plastic (or visco-elasto-plastic) percolated hard phase ... 

While the existing approaches (such as the model of Qi and Boyce) often provide a good description of polyurethane tensile curves, th typically treat hard and soft phase volume fractions as adjustable (fitting) parameters. In a fully predictive theory, one needs to combine the Qi-Boyce or similar framework with a thermodynamic model to predict hard and soft phase volume fractions, as we discussed in the previous section. Below, we illustrate how one can build such a theory and obtain a qualitative, if not quantitative, agreement with experiment. We start from a micromechanical model of Figure 2.7. The initial value of Vfj (volume fi action of the elastically active regions of the percolated hard phase) is determined on the basis of thermodynamic considerations and the percolation model, as described in the previous section. We assume that each elastically active region of the har d phase can be described as an elasto-plastic material ... 

We now have to determine how the number of the elastically active hard phase elements evolves with time (or, assuming the constant strain rate, s, as a function of strain, e). It is important to note that initially, the fraction of elastically active elements, % = [(/—/5c)/(l /5c)] is much smaller than the total volmne fraction of the percolated hai-d phase, E = f—fgQ. It is then natural to assume (in a spirit similar to the Qi-Boyce model) that many inactive hard segments would become activated during deformation, primarily due... 

Figure 2.17. Typical DMA curves of polyurethane elastomers tensile storage modulus, E, (left) and loss tangent, tan 6 = E"/E, (right). Measurements were performed at a frequency of 1 s The PEUU curves are typical for weakly phase-segregated elastomer, while PEU curves are typical for strongly phase-separated elastomer with percolated hard phase [27]...

Lagues et al. [17] found that the percolation theory for hard spheres could be used to describe dramatic increases in electrical conductivity in reverse microemulsions as the volume fraction of water was increased. They also showed how certain scaling theoretical tools were applicable to the analysis of such percolation phenomena. Cazabat et al. [18] also examined percolation in reverse microemulsions with increasing disperse phase volume fraction. They reasoned the percolation came about as a result of formation of clusters of reverse microemulsion droplets. They envisioned increased transport as arising from a transformation of linear droplet clusters to tubular microstructures, to form wormlike reverse microemulsion tubules. 

Sheth JP, Wilkes GL, Fornof AR, Long TE, Yilgor I. Probing the hard segment phase coimec-tivity and percolation in model segmented poly(urethane urea) copolymers. Mactomolecules 2005 38 5681-5685. 

Figure 7.4 Phase diagram for adhesive hard spheres as a function of Baxter temperature rg. The solid line is the spinodal line for liquid-liquid phase separation (the dense liquid phase is probably metastable), the dot-dashed line is the freezing line for appearance of an ordered packing of spheres, and the dashed line is the percolation transition. (Adapted from Grant and Russel 1993, reprinted with permission from the American Physical Society.)...

Therefore, we conclude that the minimum crystallite size together with the concomitant maximum hardness of the ncTiN/aSi3N4 composites is a result of a (relative ) thermodynamic stability of such a nanostructure at the percolation threshold (see remark added in proof and ref. [170]). We recall that at this percolation threshold the HR-TEM and XRD data show that the nanostructure consists of isolated TiN nanocrystals with a nearly spherical shape and only very few nanocrystals touching each other [61,62], that is, there is no indication of bicontinous or an interwoven bi-phase systems. Such a nanostructure with the minimum crystallite size has a maximum specific area of the interface. If, as experimentally observed, the system adjusts the minimum crystallite size at the percolation threshold resulting in the maximum specific area of the interface, this interface must possess an unusual stability. As such a behavior was also found with the other systems (ncW2N/aSi3N4... 

The threshold volume fraction of percolation (( >,) is guided by the amphiphile shell length and the overall volume fraction of the dispersed phase. For zero shell length and no interparticle attractive interaction, according to the randomly close-packed hard sphere model, 4, = 0.65 systems with strong attractive interactions end up with (f), being lowered from 0.65 to 0.10. 

Based on the discussion above, it is clear that the amount and distribution of the CB particles on the surface of microfibrils are very critical factors to determine the percolation threshold of i-CB/PET/PE composite. When the CB content in the PET phase is just beyond the percolation threshold of CB/PET compound, the continuous network of CB particles may be formed inside the microfibrils. However, there are hardly any CB particles on the surface of the CB/PET microfibrils and there exists a pure polymer layer below the surface of CB/PET microfibrils. This results in a high contact resistance among the microfibrils. The whole system exhibits an insulator state though the electrically conductive microfibrils may form a network. As the CB content in the CB/PET microfibrils reaches max> fhe number of CB paxticles on the surface evidently increases. Conduction pathways axe formed between some contact points in the microfibril network. With a further increase of CB content, the amount of CB particles on the microfibril surface increased significantly, and the electrically conductive contact points also increased. When the number of contact points is large enough to form a network to sustain the electron transmission in the whole... 

The actual compaction takes place in the final phase and can hardly be differentiated from the previous molding. As in the case of hot-compaction, matrix percolation and melt-based squeeze flow of separate fiber and matrix areas determine the interlaminary adhesion as well as the fiber impregnation in the composite. 

Figure 2.7. Schematic representation of Kolafik rnicromechanical model used to calculate Young s modulus of segmented poljmrethaue elastomers. Solid black rectangles represent hal d phase domains. Percolated hai d phase has effective volume fraction, and the rest of the hard domains ai e dispersed in the soft phase matrix [33]...

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