Percolation transition of water

Figure 4 Liquid-vapor coexistence curve of water (thick solid line) and various specific lines emanating from the critical point maximum heat capacity Cp [3] (dashed line), minimum speed of sound [22] (circles), and percolation transition of water clusters [24] (star).

Electrical conductivity of various sulfonated proton-conducting membranes, such as Nation membranes, is strongly sensitive to the water content and develops stepwise, clearly indicating percolation transition of water [411-415]. In accordance with the percolation theory, above the percolation threshold he, conductivity a varies as ... 

Formation of the hydrogen-bonded water networks may affect conductivity of a system in a drastic way, as these networks provide the paths for the conduction of protons, ions, or other charges in the system. So, the qualitative changes in the conductivity may be expected at hydrations, close to the percolation transition of water. Surface conductivity of quartz increases relatively slowly with increasing hydration level until the completion of the adsorbed water monolayer, but much faster at higher hydrations [582]. The hydration dependence of the dielectric losses of hydrated collagen... 

The conductivity exponent of about 1.23 indicates 2D character of the percolation transition. Similar values of the conductivity exponent were obtained for the hydration dependence of the conductivity of embryo and endosperm of maize seeds [595, 596], where the percolation threshold is /t = 0.082 and 0.127 g/g, respectively. In hydrated bakers yeast, protonic conductivity evidences 2D percolation transition of water at h = 0.163 g/g, and the value of the conductivity exponent is about 1.08 [597]. In this system, increase in conductivity due to 3D water percolation is observed at essentially higher hydration level h= 1.47 g/g, where conductivity exponent is about 1.94, i.e., close to the 3D value t = 2.0. Conductivity measurements of Anemia cysts at various hydrations show strong increase in conductivity starting from the threshold hydration h = 0.35 g/g [598] (see Fig. 97). The conductivity exponent in this system is 1.635, which is in between the values expected for 2D and 3D systems. DC conductivity of lichens, evaluated from the dielectric studies at frequencies between 100 Hz and 1 MHz [599], shows strong enhancement at some hydration level. Fit of the conductance-hydration dependence to equation (24) gave the following parameters he = 0.0990 g/g, t = 1.46 for Himantormia lugubris and he = 0.0926 g/g, t = 1.18 for Cladonia... 

The above experimental studies of the conductivity of hydrated biosystems directly evidence the formation of a spanning network of hydration water via percolation transition. The charge transfer itself may play a crucial role in biofunction [607]. In most of the cases, described above, the percolation transition of water occurs at the hydration level, where various forms of biological activity develop in a stepwise manner (see Section 6). In particular, the following biological processes starts close... 

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]. 

Figure 98 2D percolation transition of water in the hydrated densely packed powder of lysozyme at two temperatures. Spanning probability R (upper panel), mean cluster size 5 mean (middle panel), and fractal dimension of the largest cluster d (lower panel) are shown as functions of a mass fraction of water C. The dashed lines are guides for eyes only. Vertical hnes indicate the 2D percolation threshold. Reprinted, with permission, from [401]. 

Effect of hydration on the properties of biosystems was extensively studied both experimentally and by computer simulations. We have already considered how biological activity and conformational dynamics of hydrated biomolecules (Section 6) as well as conductivity of biosystems (Section 7.1) develop upon hydration. Now we analyze some other physical properties of hydrated biosystems (first, their dynamical properties) in relation to the percolation transition of water. Typical biomolecular surface is characterized by heterogeneity (presence of strongly hydrophilic and strongly hydrophobic groups), roughness, and finite size (closed surface of a single biomolecule). These features determine several steps in the process of hydration of biomolecules. 

This picture agrees with the available experimental analysis of the effect of hydration on lysozyme dynamics [473, 508-510, 512, 513]. Namely, internal dynamics of lysozyme molecule is restored when it is covered by some minimal amount of hydration water. Note that correlation between the percolation transition of water and pressure-induced dynamic transition of protein molecules was also observed in simulations of crystalline Snase [612]. 

At any temperature, a true percolation transition of water upon increasing hydration shell may be located based on the cluster size distribution ns, whereas a midpoint of this transition may be estimated based on... 

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. 

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