Random heterogeneous media

At the mesoscopic scale, interactions between molecular components in membranes and catalyst layers control the self-organization into nanophase-segregated media, structural correlations, and adhesion properties of phase domains. Such complex processes can be studied by various theoretical tools and simulation techniques (e.g., by coarse-grained molecular dynamics simulations). Complex morphologies of the emerging media can be related to effective physicochemical properties that characterize transport and reaction at the macroscopic scale, using concepts from the theory of random heterogeneous media and percolation theory. 

Torquato, S., Random heterogeneous media Microstructure and improved bounds on effective properties. Appl. Meeh. Rev. 44, 37 (1991). 

After all our efforts, membrane research is still challenging and in need of fresh and innovative ideas. It is a highly interdisciphnary field, based on molecular chemistry, polymer physics, interfacial science and the science of random heterogeneous media. Could it be possible that the future lies in ordered nanostructured materials such as, for example, ordered polyelectrolyte brushes In such materials, studying the role of the sidechains (length, separation, controlled flexibility, hydrophobicity) and mechanisms of self-assembly, which will determine proton distribution at the mesoscopic scale, will be central for design and optimization. 

A further approach stems from the Debye-Bueche description of scattering from random heterogeneous media, which gives, for spherically symmetrical systems... 

The representative elementary volume approach in continuum modeling of random heterogeneous media is similar to unit cell approaches in the theory of solids with crystalline ordering. 

PEM research is a multidisciplinary, hierarchical exercise that spans scales from Angstrom to meters. It needs to address challenges related to (i) to ionomer chemistry, (ii) physics of self-organization in ionomer solution, (iii) water sorption equilibria in nanoporous media, (iv) proton transport phenomena in aqueous media and at charged interfaces, (v) percolation effects in random heterogeneous media, and (vi) engineering optimization of coupled water and proton fluxes under operation. Figme 1.13 illustrates the three main levels of the hierarchical structure and phenomena in PEMs. 

Percolation theory represents the most advanced and most widely used statistical framework to describe structural correlations and effective transport properties of random heterogeneous media (Sahimi, 2003 Torquato, 2002). Here, briefly described are the basic concepts of this theory (Sahimi, 2003 Stauffer and Aharony, 1994) and its application to catalyst layers in PEFCs. 

The water content is the state variable of PEMs. Water uptake from a vapor or liquid water reservoir results in a characteristic vapor sorption isotherm. This isotherm can be described theoretically under a premise that the mechanism of water uptake is sufficiently understood. The main assumption is a distinction between surface water and bulk water. The former is chemisorbed at pore walls and it strongly interacts with sulfonate anions. Weakly bound bulk-like water equilibrates with the nanoporous PEM through the interplay of capillary, osmotic, and elastic forces, as discussed in the section Water Sorption and Swelling of PEMs in Chapter 2. Given the amounts and random distribution of water, effective transport properties of the PEM can be calculated. Applicable approaches in theory and simulation are rooted in the theory of random heterogeneous media. They involve, for instance, effective medium theory, percolation theory, or random network simulations. 

Patelli, E. Schueller, G. On optimization techniques to reconstruct microstructures of random heterogeneous media. Comput. Mater. Sci. 45 (2009), pp. 536-549. 

Fig. 3 Structural evolution of PEMs from primary chemical architectme to aggregate formation at mesoscale to random heterogeneous medium at macroscale...

The effects of two types of randomness on the behaviour of directed polymers are discussed in this chapter. The first part deals with the effect of randomness in medium so that a directed polymer feels a random external potential. The second part deals with the RANI model of two directed polymers with heterogeneity along the chain such that the interaction is random. The random medium problem is better understood compared to the RANI model. 

Heterogeneity, nonuniformity and anisotropy are based on the probability density distribution of permeability of random macroscopic elemental volumes selected from the medium, where the permeability is expressed by the one-dimensional form of Darcy s law. 

The situation becomes quite different in heterogeneous systems, such as a fluid filling a porous medium. Restrictions by pore walls and the pore space microstructure become relevant if the root mean squared displacement approaches the pore dimension. The fact that spatial restrictions affect the echo attenuation curves permits one to derive structural information about the pore space [18]. This was demonstrated in the form of diffraction-like patterns in samples with micrometer pores [19]. Moreover, subdiffusive mean squared displacement laws [20], (r2) oc tY with y < 1, can be expected in random percolation clusters in the so-called scaling window,... 

Compared to rivers and lakes, transport in porous media is generally slow, three-dimensional, and spatially variable due to heterogeneities in the medium. The velocity of transport differs by orders of magnitude among the phases of air, water, colloids, and solids. Due to the small size of the pores, transport is seldom turbulent. Molecular diffusion and dispersion along the flow are the main producers of randomness in the mass flux of chemical compounds. 

The above considerations referred to the practically important examples of more or less ordered heterogeneities. If we face random distribution, usually effective medium and percolation theory have to be referred to in order to evaluate the inhomogeneous situations properly. However, attention has to be paid to the fact that they often require nonrealistic approximations. For more details see Ref.300 In such cases numerical calculations, e.g., via finite element methods are more reliable. 

In all previous dissolution models described in Sections 5.1 and 5.2, the variability of the particles (or media) is not directly taken into account. In all cases, a unique constant (cf. Sections 5.1, 5.1.1, and 5.1.2) or a certain type of time dependency in the dissolution rate constant (cf. Sections 5.1.3, 5.2.1, and 5.2.2) is determined at the commencement of the process and fixed throughout the entire course of dissolution. Thus, in essence, all these models are deterministic. However, one can also assume that the above variation in time of the rate or the rate coefficient can take place randomly due to unspecified fluctuations in the heterogeneous properties of drug particles or the structure/function of the dissolution medium. Lansky and Weiss have proposed [130] such a model assuming that the rate of dissolution k (t) is stochastic and is described by the following equation ... 

In all the cellular models described, the cells throughout the medium were initially uniform. A microheterogeneous cell model has recently been developed, where the cells have varying initial properties (Hwang et ai, 1997). In such a model, the structure of the reaction medium is stochastic, and the heterogeneity of the reactant medium is considered explicitly. For example, the structure can be described as a regular two-dimensional matrix of circular cylinders (with nominally flat sides) in contact, with a number of cylinders randomly removed until the ratio of the voids to total volume equals the porosity (Fig. 23a). The cylinder diameter is assumed to be similar to the particle diameter. 

Equation (6.12) faces one main difficulty, namely the determination of the explicit functional form of D(s). It is important to stress that D s) does not have exactly the same properties as the classical diffusion coefficient, and we refer to it here as the conductivity. Likewise, we define the resistivity of the medium as R = l/D. We expect the resistivity to be proportional to the number of steps of the particle, and arguments from random walks on fractals should be useful to determine R. Walks on fractals are characterized by the existence of two scales. Divide the medium into small blocks of size, such that the diffusion is normal within the small blocks, f. At scales larger than f, the effect of the heterogeneities becomes important, and motion depends on the fractal parameters. The self-invariance properties of the fractal are not valid at short distances. Similarly, the idealized concept of self-similarity at all scales does not hold for fractals in practice. 

So far we have been considering the problem of random medium. A different situation arises if there is randomness in the interaction of polymers. This is the RANI model [36,37]. Consider the problem of two directed polymers interacting with a short range interaction as in Eq. (51) but take v to be random. Such problems are of interest, especially in the context of DNA where the base sequence provides heterogeneity along the chain. In this DNA context, the randomness can be taken to be dependent only on the z coordinate and not on others like the transverse position r. It can be written as... 

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