Percolation model of water

3 Reason for large isobaric specific heat of water 

The specific heat of water (4.2 kJ-kg -K ) is typically two to three times greater than the specific heat of other common liquids such as acetonitrile (2.23 kJ-kg -K ) or ethanol (2.44 kJ kg K ), even when the water molecule is much smaller, with fewer degrees of freedom. The large value of specific heat can be attributed to the existence of the local quasi-stable low-frequency oscillatory modes. Examples include hindered translation around 50 cm , intermolecular vibration around 200 cm and librational modes around 585 cm . In addition, HB breaking and re-formation also contribute to the specific heat as all of them contribute to fluctuation in the enthalpy. Note that 

More detailed explanation of the thermodynamic and structural anomalies requires the formulation of theories and models with predictive power. This model was proposed for liquid water in 1970s by Julian Gibbs and co-workers [2]. It was obtained by consideration of the two transitions (melting and boiling) which define the liquid phase. These transitions were discussed with the aid of two analogies to well-known phenomena in polymer physical chemistry. In analogy to the helix-coil 

These concepts also lead naturally to an interpretation of the triple point and sublimation. This random gel model is seen to be consistent with most of the known properties of liquid water, in particular the radial distribution function, infrared and Raman spectra, dielectric properties, density maximum, and anomalous properties in the supercooled region. The difficulty of such analogies is the quantification, as the order parameters are all collective many-body quantities which are not always easy to measure, even in simulations. 

Stanley and Teixeira have introduced a new polychromatic correlated-site percolation model [4], which has the novel feature that the partitioning of the sites into different species arises from a purely random process - that of random bond occupancy. By polychromatic one thus means that each lattice site is differently colored according to bond occupancy. 

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The earliest fully atomistic molecular dynamic (MD) studies of a simplified Nation model using polyelectrolyte analogs showed the formation of a percolating structure of water-filled channels, which is consistent with the basic ideas of the cluster-network model of Hsu and Gierke. The first MD... 

One can picture the percolation process detected in the dielectric response as proton transfer along a thread of hydrogen-bonded water molecules adsorbed on the protein surface (Careri et al., 1986). The water molecules are formally equivalent to the conducting elements of the familiar percolation model of a conducting network. Above the thresh-... 

On the other hand, the nature of the microstructure and the physics of concentrated electrolytes in the context of these systems have also been considered. Hsu(46) has formulated a theoretical percolation model of ion transport that considers ionic clusters that conduct water but which cannot contribute to long-range transport at low water contents where no connectivity of clusters is expected. As the water content Increases, an insulator-to-conductor transition occurs at a cluster volume-fraction percolation threshold. 

By changing the site site Mayer function, this formalism can be applied to a number of different physical situations. For example, if the site-site Mayer functions are chosen as those of an interaction site model of water, then the formalism could be used to study hydrogen bonding as a percolation problem, as has been done in the lattice context by Stanley and Texeira. Another possibility is to choose the Mayer functions to model polyfunctional chemical bonding. The formalism could then form the basis of treatment of gelation in polymer networks. 

Effectiveness factor approaches Macrohomogeneousmodel of ionomer-bound CL Structural (percolation) model of ionomer-bound CL Structural model coupled with water balance in pores Thin-film morphology of ionomer in CL Hierarchical Model, coupling of meso-and macroscale... 

Below we show how the appearance of spanning water networks may be detected in computer simulations. In particular, a percolation transition of water upon hydration was studied by simulations in model lysozyme powders and on the surface of a single lysozyme molecule. In protein crystals, increase in hydration of a biomolecular surface may be achieved by applying pressure. In some hydration range, pressurization leads to the formation of spanning water networks enveloping the surface of each biomolecule. Finally, the formation of the spanning water network is shown for the DNA molecule at various conformations and for different forms of DNA. 

In low-humidity tetragonal crystal with the partial density of lysozyme of about 0.80 g/cm, approximately 120 water molecules are in the first hydration shell of lysozyme molecule. In order to explore a wide range of hydration level up to monolayer coverage (about 300 water molecules), partial density of lysozyme in powder should be < 0.80 g/cm. In Ref. [401], two models for protein powder were studied densely packed powder with the density of dry protein 0.66 g/cm and loosely packed powder with a density 0.44 g/cm. In loosely packed powder, the percolation transition of water was noticeably (by a factor of two) shifted to higher hydration levels compared with experiment. The fractal dimension of the water network at the percolation threshold as well as other properties evidenced that the percolation transition of water in this model was not two dimensional. The spanning water network consists of the 2D sheets at the protein surface as well as of the 3D water domains, formed due to the capiUaiy condensation of water in hydrophilic cavities. The latter effect causes essential distortion of various distribution functions of water clusters in loosely packed powder. Therefore, below we present an overview of the results obtained for the densely packed model powder. 

Spanning probability R, defined as a probability to observe a water cluster that crosses the model system at least in one dimension, shows sigmoid dependence on the mass fraction C of water (Fig. 98, upper panel). At ambient temperature (T = 300 K), the inflection point of this dependence corresponding to R = 50% is located at about C = 0.122. This hydration level is close to that where the mean cluster size Smean passes through a maximum (Fig. 98, middle panel). Fractal dimension of the largest water cluster achieves the value at C 0.155 (Fig. 98, lower panel). Summarizing, the percolation transition of water may be attributed to the hydration level C 0.155. The cluster size distribution ns supports this conclusion [401]. 

Note that a correct comparison of the absolute values of the temperatures of the percolation transitions of water in the hydration shells of ELP and Snase, obtained in simulations, with the real temperature scale needs special consideration, as the phase diagrams of the available water models differ noticeably from the phase diagram of real water (see [5, 6] for a comparative analysis of the phase diagrams of various water models). There are two main characteristic temperatures that can be used for estimating the temperature shift of the phase diagram of model water with the behavior of real water the critical temperature of the hquid-vapor phase transition and the temperature of the liquid density maximum. The latter temperature is the most important parameter for studies carried on close to ambient conditions. For example, the phase diagram of TIP3P water model is shifted downward by at least 35 K with respect to real water. 

A kinetic study of the basic hydrolysis in a water/AOT/decane system has shown a change in the reactivity of p-nitrophenyl ethyl chloromethyl phosphonate above the percolation threshold. The applicability of the pseudophase model of micellar catalysis, below and above the percolation threshold, was also shown [285],... 

Fig. 2.9.3 Proton spin density diffusometry in a two-dimensional percolation model object [31]. The object was initially filled with heavy water and then brought into contact with an H2O gel reservoir, (a) Schematic drawing ofthe experimental set-up. The pore space is represented in white, (b) Maps ofthe proton spin density that were recorded after diffusion times t varying from 1.5 to 116 h. Projections of the...

Fig. 2.9.13 Qu asi two-dimensional random ofthe percolation model object, (bl) Simulated site percolation cluster with a nominal porosity map of the current density magnitude relative p = 0.65. The left-hand column refers to simu- to the maximum value, j/jmaK. (b2) Expedited data and the right-hand column shows mental current density map. (cl) Simulated NMR experiments in this sample-spanning map of the velocity magnitude relative to the cluster (6x6 cm2), (al) Computer model maximum value, v/vmax. (c2) Experimental (template) for the fabrication ofthe percolation velocity map. The potential and pressure object. (a2) Proton spin density map of an gradients are aligned along the y axis, electrolyte (water + salt) filling the pore space...

As a second example, we construct a simple model of how minerals might dissolve and precipitate as rainwater percolates through a soil (Bethke, 1997). The soil, 1 m thick, is composed initially of 50% quartz by volume, 5% potassium feldspar (KSiAEOfO, and 5% albite (sodium feldspar, NaSiAEOx). The remaining 40% of the soil s volume is taken up by soil gas (15% of the bulk) and water (25%). 

The physical mechanism of membrane water balance and the formal structure of modeling approaches are straightforward. Under stationary operation, the inevitable electro-osmotic flux has to be compensated by a back flux of water from cathode to anode, driven by gradients in concentration, activity, or liquid pressure of water. The water distribution in PEMs that is generated in response to these driving forces decreases from cathode to anode. With increasing/o, the water distribution becomes more nonuniform. the water content near the anode falls below the percolation threshold of proton conduction, X < X. This leaves only a small conductivity due to surface transport of water. As a consequence, increases dramatically this can lead to failure of the complete cell. 

Of course, this equation and its theoretical underpinnings does not constitute a model as such and certainly does not address the structural specifics of Nafion, so that it is of no predictive value, as experimental data must be collected beforehand. On the other hand, the results of this study clearly elucidate the percolative nature of the ensemble of contiguous ion-conductive clusters. Since the time of this study, the notion of extended water structures or aggregated clusters has been reinforced to a degree by the morphological studies mentioned above. 

This model, when applied to Nation as a function of water content, indicated a so-called quasi-percolation effect, which was verified by electrical impedance measurements. Quasi-percolation refers to the fact that the percolation threshold calculated using the single bond effective medium approximation (namely, x = 0.58, or 58% blue pore content) is quite larger than that issuing from a more accurate computer simulation. This number does not compare well with the threshold volume fraction calculated by Barkely and Meakin using their percolation approach, namely 0.10, which is less than the value for... 

Careri et al. (1986), using the framework of percolation theory, analyzed the explosive growth of the capacitance with increasing hydration above a critical water content (Fig. 14). The threshold for onset of the dielectric response was found to he 0.15 h for free lysozyme and 0.23 h for the lysozyme—substrate complex. In the percolation model the thresh-... 

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