Composite properties percolation theory

The formulation plays a very important role in composite properties. Percolation theory predicts a maximum enhancement in composite properties when there are just enough nanoparticles properly dispersed in the matrix material to form a continuous structure. However, beyond the optimum filler loading, the properties generally decrease [84]. Uniform dispersion of fillers in the matrix is a major concern when it comes to the composite properties [85, 86]. Inferior mechanical property due to poor distribution of cellulosic nano elements has been reported by many researchers [87]. The relationship between quality of dispersion of the nano element and the resulting effect on the mechanical properties is well described in the review by Schaefer and Justice [88]. They came to the conclusion that almost all so-called nanocomposites have fallen far short of the most optimistic expectations, and the reason for their relatively poor performance can be attributed to large-scale agglomeration of filler loading. 

Hsu and Berzins used effective medium theories to model transport and elastic properties of these ionomers, with a view toward their composite nature, and compared this approach to that of percolation theory. ... 

Webman and Jortner [68] used the following formulas obtained by combining the formulas of effective medium theory with the formulas of percolation theory to calculate the effective Hall properties of a composite ... 

Percolation theory [53] is also used to calculate the effective properties such as the ionic conductivity in the SOFC electrodes. The effective conductivity of a composite electrode is less than that of the pure material due to the composite structure and porosity of the electrode. Percolation theory calculates an effective ionic conductivity that accounts for the tortuous path of the electrolyte phase in the electrodes and is based on the probability of finding a percolated chain of the electrolyte phase through the electrode [53]. 

Percolation theory to predict effective properties of solid oxide fuel-cell composite electrodes. J. Power Sources, 191 (2), 240-252. 

For materials with random morphologies, the effect of microscopic composition, i.e., of size, shape, and random distribution of phase domains and of their coimectedness, on its macroscopic properties is the topic addressed by percolation theory [129, 130], Here, we briefly describe the basic concepts of this theory and outline their application to determine the morphology and effective properties of random composite materials. We refer the interested reader to Stauffer and Aharony [46] and Sahimi [129] for detailed discussion of percolation theory and its applications. 

Percolation phenomena deal with the effect of clustering and coimectivity of microscopic elements in a disordered medium [129], Percolation theory represents a random composite material as a network or lattice structure of two or more distinct types of microscopic elements or phase domains, the so-called percolation sites. These elements represent mutually exclusive physical properties, e.g., electrically conducting vs. isolating phase domains, pore space vs. solid matrix, atoms with spin up vs. spin down states. Here, we will refer to black and white elements for definiteness. The network onto which black and white elements of the composite medium are distributed could be continuous (continuum percolation) or discrete (discrete or lattice percolation) it could be a disordered or regular network. With a probability p a randomly chosen percolation site will be... 

Percolation theory deals with the size and distribution of connected black and white domains and the effects on macroscopic observable properties, e.g., eleetrie conductivity of a random composite or diffusion coefficient of a porous roek. A percolation cluster is defined by a set of connected sites of one color (e.g., white ) surrounded by sites of the complementary color (i.e., black ). If p is sufficiently small, the size of any connected cluster is likely to be small compared to the size of the sample. There will be no continuously connected path between the opposite faces of the sample. On the other hand, the network should be entirely eonnected if is close to 1. Therefore at some well-defined intermediate value,... 

Combination of the macrohomogeneous approach for porous electrodes with a statistical description of effective properties of random composite media rests upon concepts of percolation theory (Broadbent and Hammersley, 1957 Isichenko, 1992 Stauffer and Aharony, 1994). Involving these concepts significantly enhanced capabilities of CL models in view of a systematic optimization of thickness, composition, and porous structure (Eikerling and Komyshev, 1998 Eikerling et al., 2004). The resulting stmcture-based model correlates the performance of the CCL with volumetric amounts of Pt, C, ionomer, and pores. The basis for the percolation approach is that a catalyst particle can take part in reaction only if it is connected simultaneously to percolating clusters of carbon/Pt, electrolyte phase, and pore space. Initially, the electrolyte phase was assumed to consist of ionomer only. However, in order to properly describe local reaction conditions and reaction rate distributions, it is necessary to account for water-filled pores and ionomer-phase domains as media for proton transport. 

At a critical value of the fraction of objects of one type, these objects would form an extended cluster that connects the opposite external faces of the sample. At this so-called percolation threshold, the corresponding physical property represented by the connected objects would start to increase above zero. Thereby percolation theory establishes constitutive relations between composition and structure of heterogeneous media and their physical properties of interest. For porous electrodes or catalyst layers in PEFC, these properties are electrical conductivities of electrons and protons, diffiisivities of gaseous reactants and water vapor, and liquid water permeability. 

Simulations of physical properties of realistic Pt/support nanoparticle systems can provide interaction parameters that are used by molecular-level simulations of self-organization in CL inks. Coarse-grained MD studies presented in the section Mesoscale Model of Self-Organization in Catalyst Layer Inks provide vital insights on structure formation. Information on agglomerate formation, pore space morphology, ionomer structure and distribution, and wettability of pores serves as input for parameterizations of structure-dependent physical properties, discussed in the section Effective Catalyst Layer Properties From Percolation Theory. CGMD studies can be applied to study the impact of modifications in chemical properties of materials and ink composition on physical properties and stability of CLs. 

Figure 4.6b compares calculated plots of Eceii versus jo with experimental data of Uchida et al. (1995a,b) for CCL with different ionomer content, as specified in the legend. Composition-dependent properties were parameterized using the functions proposed in the section Effective Catalyst Layer Properties from Percolation Theory in Chapter 3. The fuel cell voltage was assumed to be of the form... 

As discussed in the section Ionomer Structure in Catalyst Layers Redefined in Chapter 3, a theory of composition-dependent effective properties that incorporates recent insights into stmcture formation in CCLs is yet to be developed. At present, the relations presented in the section Effective Catalyst Layer Properties from Percolation Theory in Chapter 3 do not account for agglomerate formation and skin-type morphology of the ionomer film at the agglomerate surface. Qualitative trends predicted by the simple structure-based catalyst layer theory should be correct, as confirmed by the results discussed in this section. 

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