Percolation hypothesis

The integrated intensities of PHB and NHB water show an opposite temperature behavior for T > 300K. Although the intensities of NHB increase with increasing T, those of PHB decrease. The classification of these contributions reflects that used in the percolation hypothesis for water (/ species of water, with i indicating the number of bonds) [21]. Thus, HB component I is /4, NHB components IV and V are /o, and PHB components II and III are /i, fz, and fs. According to water polymorphism, the HDL phase is represented by both the NHB and PHB components. 

A quantitative study of the influence of particle size on the percolation threshold employing inert matrix tablets prepared with KC1 and Eudragit RS-PM as matrix-forming material [44-46] showed that experimental data are in agreement with this hypothesis. 

The formation of spanning H-bonded water networks on the surface of biomolecules has been connected with the widely accepted view that a certain amount of hydration water is necessary for the dynamics and function of proteins. Its percolative nature had been suggested first by Careri et al. (59) on the basis of proton conductivity measurements on lysozyme this hypothesis was later supported by extensive computer simulations on the hydration of proteins like lysozyme and SNase, elastine like peptides, and DNA fragments (53). The extremely interesting... 

Employing Eq. (24), we may assume that the percolation probability for the sublattice of voids with r > rp is the same as the universal percolation probability for the bond problem (Section II). The latter probability was originally calculated only for regular lattices. The sublattice of voids with r > rp is not regular. In addition, some of the voids with r > rp are not connected with the other voids having r > rp. However, the numerical results obtained by Yanuka (33) for a randomized cubic lattice (see also the discussion in Section II) support the hypothesis on the universality of the percolation probability for both regular and irregular lattices. 

As for the linear properties, numerous approaches have been proposed to predict and explain the nonlinear optical response of nanocomposite materials beyond the hypothesis leading to the simple model presented above ( 3.2.2). Especially, Eq. (27) does not hold as soon as metal concentration is large and, a fortiori, reaches the percolation threshold. Several EMT or topological methods have then been developed to account for such regimes and for different types of material morphology, using different calculation methods [38, 81, 83, 88, 96-116]. Let us mention works devoted to ellipsoidal [99, 100, 109] or cylindrical [97] inclusions, effect of a shape distribution [110, 115], core-shell particles [114, 116], layered composites [103], nonlinear inclusions in a nonlinear host medium [88], linear inclusions in a nonlinear host medium [108], percolated media and fractals [101, 104-106, 108]. Attempts to simulate in a nonlinear EMT the influence of temperature have also been reported [107, 113]. 

In the cases where the percolating networks of both A-rich and B-rich domains persist throughout the last stages, the data comply with the scaling hypothesis of Binder and Stauffer [178, 179]... 

The percolation model next assumes that molecules with a larger number of HBs occupy a larger volume than those with a smaller number of bonds. In other words, one may associate a molecular volume related to the family f to which the moleeule belongs. The volume increases with i. This rather strong hypothesis is confirmed by molecular simulations and by X-ray experiments. 

The dielectric measurements performed for the AOT/water/decane and AOT/water/hexane micro emulsions at the volume fraction of the dispersed phase of (p= 0.13 demonstrate the significant shift of the percolation region to the direction of high temperatures when the oil chain lengfli decreased (Fig. 18) (118, 119). However, the values of s for bofli die microemulsions are die same at low temperatures, i.e., below die percolation onset. Thus, diose results do not support die hypothesis that the clustering can be responsible for the temperature behavior of die static dielectric permittivity at F < and it must be the internal processes within a droplet diat determine the behavior of the dielectric polarization in the system. 

It is obvious that the dynamical scaling hypothesis is not valid through the dynamical percolation-to-cluster transition. This is because the pattern changes from the bicontinuous periodic structure to the cluster of spheres. In fact the scaled structure factor was found to substantially broaden before and after the transition. 

Into these relations one may introduce specific values (s,z) from percolation theory or from branching theory and determine the corresponding values for Wc- The wide range of values for the relaxation exponent 0 < c < 1 lets us expect that the dynamic exponents s and z are nommiversal. Since s and z can be predicted from theory (47), Uc values can be calculated from equation 13. This result, however, relies on the symmetry hypothesis, which does not seem to be generally valid, at least not for highly entangled polybutadienes (48). 

In this study, the percolation threshold of HPMC and NaCMC binary matrices were estimated analyzing the water uptake and the release behavior of the matrices. Ihe HPMC percolation threshold was situated between 29 and 41% (v/v) HPMC for the binary KCl-HPMC matrices, while the excipient percolation threshold for the binary KCl-NaCMC hydrophihc matrices was found between 39 and 54% (v/v) NaCMC. As it can be appreciated, there is a narrow range of overlapping concentrations for the polymers percolation threshold. Thus it suggests that there is a small possibility that these two polymers can show a combined percolation threshold. This hypothesis is confirmed since considerable differences in the Higuchi s slope values (1.416 and 0.49, respectively) were found in two batches of matrices prepared employing very similar volume percentages of both polymers. 

Previous works have explained the low values obtained for the polymer percolation threshold in hydrophilic matrices based on the contribution of the initial porosity of the tablet in order to form the gel layer that controls the drug release [13,94,95]. The results obtained by Aguilar-de-Leyva et al. do not contradict this hypothesis, since the porosity values of the previous studies correspond to the lower level (5-10%), and this study proposes that it could be an involvement of the initial porosity. Nevertheless, this new study concluded that the contribution of initial porosity to establish the gel layer would be restricted to a low range of tablet porosity [101]. Above this range, an increase in the tablet initial porosity does not affect the polymer percolation threshold. 

The high aspect ratio of nanorods can facilitate charge transport, while the handgap can he tuned by vaiying the nanorod radius. This enables the absorption spectmm of the devices to be tailored to overlap with the solar emission spectmm, whereas traditionally polymer absorption has been limited to only a small fraction of the incident solar irradiation. At present, the nanorods in polymer solar cells are typically incorporated into a homopolymer matrix. An alternative to this approach is to incorporate the nanorods into either a polymer blend or diblock copolymer system. The photovoltaic properties of nanorod polymer composites could potentially be improved due to the percolation of nanorods, and the presence of continual electrical pathways, from the DA interfaces to the electrodes. To test this hypothesis, we use the distribution of nanorods from the self-assembled stmcture in Figure 1(b) as the input into a drift-diffusion model of polymer photovoltaics. 

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