Total Internal Reflectance-Infrared Structural Studies on Tensile Water

Apr 2, 2008 - Reference is made to the solar powered water ascent in trees, where this phenomenon of tensile water has evolved into a working technolo...
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J. Phys. Chem. C 2008, 112, 6313-6318

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Total Internal Reflectance-Infrared Structural Studies on Tensile Water Formation during Evaporation from Nanopores Nadine Szabo´ , Christian To1 tzke,* and Helmut Tributsch Hahn-Meitner Institute, Department of Solar Energetics, 14109 Berlin, Germany ReceiVed: December 7, 2007; In Final Form: January 23, 2008

Attenuated total reflectance (ATR) infrared spectroscopy of water evaporating from within a TiO2 nanolayer with pores of a mean diameter of 23 nm is applied to detect the transition to the tensile state where water molecules are stretched. It is seen how the concentration of the strongly H-bonded (network) water population, measured within the OH-stretching band, is strongly increasing at the expense of the medium (intermediate) H-bonded population. This observation suggests that water molecules exposed to tensile stress (in nanoenvironments) get involved in additional H-bonding, generating a sigmoid (autocatalytic) growth of the network water population. Such a feedback mechanism is related to known molecular electronic properties and to irreversible thermodynamic concepts of water. Reference is made to the solar powered water ascent in trees, where this phenomenon of tensile water has evolved into a working technology.

1. Introduction Water is of paramount importance for life. Many of its properties have consequently been studied thoroughly (see, e.g, ref 1 and references herein). Although the significance of water is generally recognized, there are still undeveloped water-based technologies that are fundamental for biology and the environment but unexploited by man (e.g., solar water splitting for fuel production, tensile water technology of green plants). In these cases and generally often water is not involved in the form of a homogeneous liquid but instead attached to substrates or embedded in nanostructured environments. This is, for example, the case in the interior of cells and in the vicinity of cell membranes. In inorganic systems, water can be absorbed into nanoporous materials such as clay minerals or into porous silica. Water existing in such restricted geometries is typically referred to as “confined”, “vicinal”, or “interfacial” water.2 A biological phenomenon in which confined water plays a very prominent role is the solar-mediated water ascent in trees.3 The water, lifted by the energy turned over during evaporation in the leaves and needles, is thereby experiencing tensile stress; this is why physiologists also use the term “tensile water” in this context. In fact, trees employ a solar-powered pumping mechanism in which evaporation is used to convert solar radiation into mechanical pumping energy. This is a physicochemical solar energy conversion process, which functions parallel to and independent of photosynthesis. Within the leaves, evaporation takes place from nanoporous evaporation sites. Menisci are formed while the water retreats into hydrophilic nanopores of the cell walls in the mesophyll. Thereby, strong capillary forces are activated and tension within the xylem, that is, within the water conducting tissue of trees below the bark, is built up. Taking advantage of these effects, trees, such as Sequoia and Eucalyptus, are able to pull water up to heights of over 100 m. Xylem pressures in the range of -20 to -30 bar (-2 to -3 MPa) have been reported by numerous plant physiologists.4-6 In the West Australian bush, tensile stresses of 4-5 MPa have been measured reliably in Eucalyptus species.7 * Corresponding author. E-mail: [email protected].

Such high tensions can only be transmitted along water columns when a significant increase in hydrogen bonding is assumed to occur in water within the nanometer and micrometer large dimension of the conductive plant tissues (typical effective diameters of xylem conduits are between 5 and 50 µm, those of cell wall pores of evaporative tissues are on the order of 10 nm4). Expanding stretched water within an adapted nanoenvironment should, therefore, be associated with an increasing concentration of hydrogen bonds. This may be intuitively understandable on the basis of the known properties of water. Even though ice modifications under atmospheric pressure are less dense than liquid water, they are known to be solid bodies held together by hydrogen bonds, which are more numerous as well as stronger than those in liquid water. An appropriate method to study H-bonding structures is infrared (IR) spectroscopy. But the interior location of evaporation sites in leaves creates great difficulties in applying any kind of IR spectroscopy. Therefore, in vivo IR measurements are not performed straightforwardly. It is, however, in a first step, possible to simulate evaporation processes that may lead to the buildup of water tension in appropriate model structures. In addition, exploration of stable, inorganic nanostructures for solar evaporation experiments is a stepping stone toward artificial devices using such bioanalog energy technology. During evaporation, water confined in inorganic model structures should behave similarly to water in cell wall nanopores. In the present study, a hydrophilic nanoporous structure was prepared in order to model the tensile water and also the evaporation environment of leaves on the basis of a simple, inorganic nanomaterial. When water confined in such structures evaporates, changes in the nature, quantity, and distribution of H bonding are expected to occur. These changes can be detected by means of ATR-IR spectroscopy. There are two main absorption bands of water in the mid-IR region: the H-O-H bending band located at 1640 cm-1 and the OH stretching band at a wave number of about 3400 cm-1. In contrast to the bending band, the stretching band is very sensitive to different degrees of connectivity of water molecules and provides, therefore, insight into changes of the H-bonding structure. The more

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Figure 2. ATR crystal prepared with the sample and placed into the divided sample chamber.

Figure 1. SEM image of the TiO2 layer prepared from P25 nanoparticles. Mean pore size is about 23 nm.

hydrogen bonds a molecule establishes with its neighbor molecules, the weaker the OH oscillation strength becomes.8 Consequently, the OH band shifts toward lower frequencies if additional H bonds are activated. However, the OH band is not a homogeneous vibration but exhibits several substructures. This is because molecules taking part in H-bonded water clusters vibrate differently according to their degree of interconnectivity. H-bond coordination numbers can range from 0 to 4.9 Following these considerations, water can be thought of as a mixture of molecule populations with different mean numbers of active hydrogen bonds.8 Understanding the structural dynamics of water in nanostructures is equally important for oxygen reduction catalysis research in fuel cells, where water is turned over in and evaporated from nanostructures (e.g., ref 10). It is also relevant for the understanding of proton conduction via amino acids and water bound to oxide nanoparticles in nanowires within fuel cell membranes.11 2. Experimental Section The designed model system was investigated using ATR-IR spectroscopy. This type of spectroscopy is based on the total reflectance of an IR beam at the surface of an ATR crystal, which is in direct contact with the sample. The reflection at the surface induces an evanescent wave, which extends into the sample, typically a few micrometers. Consequently, the IR beam will be attenuated while passing the waveguide before it eventually meets the detector. A specific requirement of ATRIR spectroscopy is the direct contact between the crystal and the water-containing sample.12 For that reason, the TiO2 layer was directly coated onto the upper surface of the silicon ATRcrystal. For the preparation of the porous layer structure, we used hydrophilic TiO2 nanoparticles (P25 Degussa, grain size 25 nm, 70% anatase, 30% rutile). In a beaker, 2 g of TiO2 (previously heated to 450 °C) were treated with 80 wt % ethanol. The resulting thick paste was stirred for 2 h and subsequently homogenized for 30 s employing ultrasound. The area to be covered by TiO2 was limited by a tape of the desired thickness (5 µm). Then the paste was applied using a glass rod and the layer was prepared using the doctor blade technique.13 Subsequently, the layer was shortly dried and compressed applying a pressure of 12 kN/cm2. This pressure treatment replaced a heat treatment, which was avoided because the TiO2 layer had to be prepared on top of the delicate ATR crystal. The nanostructure of the TiO2 layer obtained is shown in Figure 1. The rigid nanostructure formed was very hydrophilic14-16 and produced pores with a mean diameter of approximately 23 nm (according

to our measurements using the gas sorption (BET) technique). This corresponds to the order of magnitude of the biological pore size. Preliminary experiments showed that the water storage capacity of this TiO2 layer was smaller than required. Evaporation would proceed too fast; that is, the layer was dry before the tensile state of water within the pores could be detected accurately by the spectrometer. Therefore, it was necessary to decelerate the drying procedure. On one hand, the storage capacity was extended by covering the porous layer with a water-saturated paper filter that served as a water reservoir. On the other hand, a microscope glass slide on top of the reservoir reduced the evaporation rate. Spectra were recorded in the mid-infrared region (1400-4000 cm-1) using a FT-IR spectrometer (model Bruker IFS113v). Thirty-six scans per spectrum were taken with a spectral resolution of 1 cm-1. The experiments were conducted at atmospheric pressure and room temperature (22 °C). In order to keep the course of IR beams clear of water vapor released by the sample, the sample chamber was divided horizontally into two sections. The ATR crystal carrying the watered sample was placed in the upper section, whereas the IR beam was passed through the lower section of the sample chamber. The whole sample chamber was flushed with a constant flow of nitrogen, which ensured low carbon dioxide and water vapor concentrations in the lower partition of the chamber and stable evaporation conditions in the upper one. The principal configuration is shown schematically in Figure 2. At the beginning of the measurements, the sample was saturated with distilled water. Spectra were recorded every 30 s until evaporation ceased. As a background spectrum of all measured spectra, the coated but dry ATR-crystal under dry nitrogen atmosphere was subtracted. 3. Results Figure 3 shows the OH stretching band of a water sample that has been absorbed by the nanoporous TiO2 layer on the ATR crystal. The signatures of different molecule populations in the OH stretching band can be described well by a superposition of several Gaussian curves. Deconvolution of the band into 3 Gaussians, as done in Figure 3, delivers quite a satisfactory fit of measured data. The position of the Gaussian was deduced by calculation of the second derivation of the spectra that exhibit local minima at the center of the absorbance band of each component. The positions of Gaussians were found to remain constant throughout the measurements. The Gaussian with the lowest wavenumber (3230 cm-1) represents the strongly bonded population. Molecules belonging to this group have a coordination number close to 4 and are likely to be involved in extended transient water networks. In the literature this population is often referred to as “network water”.8,17,18 In contrast,

Internal Reflectance-Infrared Structural Studies

Figure 3. Deconvolution of the OH band of water confined in a porous TiO2 layer. The lower-energy Gaussian is ascribed to “network water”, the medium energy Gaussian to “intermediate water”, and the higher energy Gaussian to “multimer water” molecules. The spectrum was recorded after 40 min of wetting the TiO2 layer.

Figure 4. Water distribution within the TiO2 nanolayer showing changes in water configuration (drawings 1-3) in the course of evaporation: (A) bulk water, (B) TiO2 particle, (C) adsorbed water, (D) Meniscus.

the Gaussian with the highest wavenumber (3590 cm-1) represents the molecule population that is rather poorly interconnected standing as free monomers or as dimers or trimers.18 It is, therefore, referred to as “multimer water”. Such molecules can particularly be found at gas-liquid interfaces.19 In between lies the third Gaussian (3390 cm-1) that is ascribed to the medium bonded population, the so-called “intermediate water” having a mean coordination number larger than that of multimer but smaller than that of network water (i.e., between 2 and 3).8 In this study, we observed the evolution of the OH band of water that had been absorbed by a nanoporous hydrophilic TiO2 layer. As mentioned in the Introduction, this model system was designed to simulate the water situation within leaves. Spectra were taken while the absorbed water evaporated from this layer expecting tensile stress to develop. At a certain point during the drying, water should retreat into the TiO2 layer thereby forming menisci. The resulting capillary forces then stretch the remaining water within the pores, which, in turn, should change the intermolecular bonding structure. However, previous experiments with uncovered TiO2 layers have shown that menisci could only persist for a very short time because evaporation proceeded too rapidly. In order to detect the formation of menisci and the related stretching of water accurately it is, therefore, useful to decelerate the drying process (as described above). The desiccation of the water-saturated TiO2 layer is expected to take place in three stages (Figure 4). During the first stage (Figure 4-1) the TiO2 layer remains saturated with water as the water reservoir fully compensates the evaporational losses. With progressive evaporation, however, the reservoir depletes gradually and is eventually not able to compensate for evaporation any more. At this point, the second stage begins: water retreats into the pores (Figure 4-2) of the TiO2 layer where capillary

J. Phys. Chem. C, Vol. 112, No. 16, 2008 6315 forces start to act on the remaining water and cause a pressure drop. Finally, the layer runs successively dry in stage 3, starting with the largest pores until the situation depicted in Figure 4-3 is reached. All bulk water has been evaporated and only a thin adhesive layer of water remains on the surface of the TiO2 particles. A typical run of such an experiment is presented in the following. Figure 5 shows the evolution of the OH band during the experiment. All plots in Figure 5a are recorded during the first stage of drying the porous layer still being saturated with water. At the beginning, the TiO2 layer needs some time (minutes) for complete wetting before stable conditions are established. During this time, a shoulder at about 3600 cm-1 is forming and the band intensity rises slightly. Once the layer is wetted, the conditions stabilize so that no drastic change of the OH band occurs for about 110 min. However, after 60 min the band maximum shifts very moderately toward lower wave numbers. After 112 min, the situation changes drastically (compare Figure 5b). The OH band swells over its left wing and the band maximum shifts toward lower wavenumbers while the overall intensity remains nearly constant for another 10 min (see also Figure 6). Finally, the band intensity decreases, indicating a dropping of water concentration within the sample. After 135 min, the porous layer has run dry so that the OH band almost vanishes. As from 113 min, one can also observe the formation of an additional shoulder at about 3050 cm-1. This shoulder could indicate the emergence of another subcomponent within the strongly bonded network water population. Molecules belonging to this subcomponent can be expected to have even stronger H-bonding than the rest of the network water population. However, for simplicity, this potential subcomponent was not analyzed separately but integrated into the network water molecule population. All spectra have been deconvoluted into 3 Gaussians, respectively, to allow for an interpretation of the OH band evolution in terms of changing intermolecular bonding structures (as discussed further above). In Figure 6, relative areas of Gaussians are plotted against elapsed time of the experiment. The relative area of each Gaussian complies with the fraction of the corresponding molecule population, respectively. In addition, the integral of absorbance is shown to detect the changing water content of the porous layer at the interface of the ATR crystal. After the initial phase of wetting, the TiO2 layer reaches quasi-steady-state conditions; that is, the distribution of molecule populations remain quite stable for about 1 h. During this time evaporated water is replaced by the water reservoir, this way keeping the porous layer water saturated. At this stage, the absorbed water is composed of mainly intermediate water and network water whereby the intermediate water is the dominating population. In contrast, the weakly bonded population of multimer water contributes only a minority of about 10% of water molecules. The slight shift of the band maximum starting at t ) 60 min is reflected in a moderate rise of the network water population. Conversely, the intermediate water population shrinks slightly. This trend may already indicate some water stress in the TiO2 layer but it still takes almost another hour before the second stage starts. Within the second stage, changes in the intermolecular bonding structure can be detected by a shift of the OH band. The interesting process starts when the water reservoir is depleted so that the porous layer itself begins to lose water. Consequently, water retreats into the pores of the layer and forms tiny menisci. The magnitude of pressure that can develop inside a cylindrical pore of radius R is the result of surface tension, σ,

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Figure 5. Evolution in time of the OH band of water within a TiO2 nanolayer during evaporation. (a) Initial 112 min, (b) after 112 min.

Figure 6. Evolution of the three water populations as derived from the Gaussian fit of the OH band: upright triangles, multimer water; squares, network water; circles, intermediate water. The integral of absorbance of the OH band (downward triangles) serves as indicator for changing total water content (see text).

and adhesion of water molecules to the pore wall that determines the contact angle, R. The resulting pressure difference across the gas-liquid interface can be expressed by the well-known equation3

∆p)

-2σ cosR R

The pores of the used TiO2 layer have a mean radius of about 11-12 nm. In the case of hydrophilic TiO2 surfaces, it is reasonable to assume a contact angle near 0°. Considering these conditions, a negative pressure of over 10 MPa could theoretically develop inside these pores. The resulting water stress leads to a significant change of the bonding structure. This could be detected at t ) 112 min as a significant downshift of the dominating wave numbers (Figure 5). At the same time, the overall IR-band intensity does not decrease (but increases) for another 10 min (see Figure 6, the plot of the integrated absorbance of the OH band), which indicates that all pores located at the interface of the ATR crystal are still water-filled and that apparently additional water is pulled within the reach of the evanescent IR wave. Similar effects of increasing signal intensity were reported for the phase transition from water to ice.8,20

Figure 7. Evolution of the three water populations during the buildup of tensile state. A sigmoid-shaped curve is fitted to the increase of network water population.

Figure 6 reveals the structural transition of the water in those pores: the network water population grows drastically at the expense of the medium bonded population eventually becoming the dominating population. In contrast, the fraction of multimer water remains nearly constant. A possible explanation is that latter contribution involves water at the water-silicon interface of the ATR crystal or at the gas-liquid interface. The OHband shift is consequently a result of the activation of additional H bonds. It may be concluded that water molecules experiencing tensile stress tend to establish additional H bonds. This apparently gives rise to the organization of intensively connected transient water clusters with the correspondingly changed water property. Because the transition to strongly H-bonded water in Figure 6 is quite rapid, the transition is better resolved in Figure 7. It can be observed that a sigmoid-type of growth kinetics is generated. Sigmoid-shaped curves typically describe population dynamics21 or autocatalytic chemical reactions.22 This supports the conclusion that we are dealing with a self-organized (autocatalytic or feedback-controlled) mechanism of network water formation toward the development of extended water clusters or tensile water conditions.

Internal Reflectance-Infrared Structural Studies

Figure 8. OH-stretch spectrum of tensile water formation normalized against water reference band at 110 min (see Figure 5). An isosbestic point is recognized near 3290 cm-1. The insert shows the corresponding normalized spectrum from the phase transition of water to ice (from ref 23).

In the Introduction, the formation of stretched, tensile water has been compared with the formation of ice from water during cooling. In both cases, water assumes a less dense state and simultaneously increases its concentration of H bonds. If OH bands are measured at various temperatures, then they approximately cross each other in an isosbestic point.8 Cooling of water and normalizing the observed IR changes within the OH-stretch band against a reference band leads to an isosbestic point at 3280 cm-1 (compare Figure 8, insert23). A similar normalization can be performed with our data on tensile water (Figure 8). It can be seen that a similar isosbestic point can be identified near 3290 cm-1, which separates the increasingly H-bonded water (left) from the decreasingly H-bonded water (right). The difference, however, is that the increase in water H-bonding during the water-ice transition occurs between +8 and -5 °C, in our case of tensile water formation, however, at a temperature near 20 °C. This confrontation (in Figure 8) indeed justifies the comparison of stretching water with the formation of an ice-like structure. In the present study the effect of menisci formation, favoring tensile water, could be observed for about 10 min. After that, an obvious reduction of the overall intensity of the band indicated the beginning of the third and last stage of drying, which is related to a falling water concentration within the TiO2 layer. The proceeding evaporation forced the pores to run successively dry. It started with the largest pores at the top of the layer. Then the drying front proceeded downward until it reached the interface of the ATR crystal. At this point, all water had evaporated. 4. Discussion and Conclusions The present ATR-infrared study shows the possibility of building up tension in water, which is confined in a nanoporous hydrophilic structure (here nanoporous TiO2) by means of evaporation. Tensile stress originates from cohesive and adhesive forces that are activated as soon as water menisci are formed inside the pores. The energy required for the transition to the tensile state originates here from the evaporation process. It is energy needed to shift the system away from equilibrium. This shift facilitates additional hydrogen bonding, which is associated

J. Phys. Chem. C, Vol. 112, No. 16, 2008 6317 with a release of bonding energy in the form of heat. Consequently, the tensile water state is, compared to ordinary water, characterized by a lower free energy. Similar to supercooled water or ice, more energy is required to transfer tensile water into the vapor state. To a first approximation, this free energy decrease reflects the number of hydrogen bonds additionally formed as well as the increased structural order (lower entropy) attained. Altogether the process of tensile water formation and maintenance requires an input and throughput of energy from outside (evaporation process) so that it has to be considered to be a phenomenon of irreversible thermodynamics. On a molecular scale, the resulting tension alters vibrational spectra of water molecules in the mid-IR, which was experimentally verified by monitoring the OH-stretch band. Deconvolution of the band into 3 Gaussians revealed the establishment of additional H bonds during the time of menisci formation. Consequently, a significant fraction of the medium bonded water population (intermediate water) switches over to the strongly bonded population (network water). This way they contribute to the extension of transient water networks. Expressed in a simplified way, the structure of liquid water evolves toward that of ice as the fraction of network water strongly increases (Figure 7). This result is in agreement with experimental findings along similar lines presented by other authors: A trend for H-bond activation was reported for bulk water upon cooling8 and is evident from the paper of Raichlin.23 With decreasing temperature, more and more H bonds were established before crystallization occurred. Interestingly, experiments with 25% hydrated Vycor samples (nanoporous glass) indicated that, at room temperature, interfacial water has a structure similar to that of bulk supercooled water at a temperature of about 0 °C, which corresponds to a shift of about 25 K. The author characterized the structure of confined water by an increase of the long-range correlations, which corresponds to the building of the H-bond network as it appears in low density amorphous ice.2 That suggests that the water structure could be similarly affected by a decrease in temperature and pressure, respectively. It seems that the lower the internal energy the more hydrogen bonding can withstand thermal excitation. Water experiencing tensile stress thus undergoes a similar structural transition as it does upon cooling below 8 °C (Figure 8). A higher degree of structural order within the H-bonded water network is reached; that is, the structure of liquid water evolves toward that of less dense ice. This includes higher mean coordination numbers as well as a higher tensile strength in spite of decreasing water density. To understand this phenomenon of increasing H-bond formation upon water stretching better, it is useful to refer to a known molecular electronic property of water: From molecular electronic considerations, it can be deduced that the formation of one hydrogen bond may induce such electronic changes that an additional H-bond formation is favored.1,24 Such an interaction has the autocatalytic character needed for self-organization processes. The kinetics of hydrogen-bond formation (formation of network clusters) could indeed be fitted by a sigmoid growth curve (Figure 7), which is typical for autocatalytic, selforganized phenomena.22 On the basis of such a molecular feedback process for H-bond formation, a nonlinear kinetic model has been derived for the interaction between water molecules.25 It turned out to be consistent and analog with the empirical Van der Waals equation describing a realistic watervapor system considering interactive forces between water

6318 J. Phys. Chem. C, Vol. 112, No. 16, 2008 molecules. This means that it describes the buildup of negative pressure (or positive tension) when the water expands. But the same model also describes chaotic cavitation and oscillations, which are sometimes observed with sap water in living trees.25 Tensile water formation itself, as we have observed here, may thus be considered to be a dynamic self-organization phenomenon, which occurs when energy is converted by the system. Water molecules in nanoenvironments are behaving as microcanonical ensembles and are thus not subject to reversible thermodynamic laws but to irreversible thermodynamic phenomena. Nature has, in the tensile water supply system for trees, transferred a simple property of H-bond-interacting water into a highly elaborate and well-functioning technology. Without the ability of trees to extract water from soils, to pump it, and to desalinate it, our ecosystem and climate would be entirely different. An interesting future challenge of physical chemical research would be to follow evolution, building up knowledge from simple model experiments, as demonstrated here, toward a fully grown new water technology. References and Notes (1) Chaplin, M. Water Structure and Science 2007; Vol. 2007. (2) Bellissent-Funel, M.-C. J. Mol. Liq. 1998, 78, 19. (3) Physiochemical & EnVironmental Plant Physiology, 2nd ed.; Nobel, P. S., Ed.; Academic Press: Oxford, 1999; p 474. (4) Xylem Structure and the Ascent of Sap, 2nd ed.; Tyree, M. T., Zimmermann, M. H., Eds.; Springer: New York, 2002; p 278. (5) Pockman, W. T.; Sperry, J. S.; O’Leary, J. W. Nature 1995, 378, 715.

Szabo´ et al. (6) Bauerle, W. L.; Hinckley, T. M.; Cermak, J.; Kucera, J. Trees 1999, 13, 211. (7) Poot, P.; Venaklaas, E. In 6th International Workshop on Xylem Sap Flow and its Application to Plant Sciences; Perth, Western Australia, 2006. (8) Brubach, J.-B.; Mermet, A.; Filabozzi, A.; Gerschel, A.; Roy, P. J. Chem. Phys. 2005, 122, p.184509. (9) Cabane, B. C. R. Geoscience 2005, 337, 159. (10) Fiechter, S.; Dorbandt, I.; Bogdanoff, P.; Zehl, G.; Schulenburg, H.; Tributsch, H.; Bron, M.; Radnik, J.; Fieber -Erdmann, M. J. Phys. Chem. C 2007, 111, 477. (11) Leem, H.-J.; Dorbandt, I.; Rojas-Chapana, J.; Fiechter, S.; Tributsch, H., submitted to J. Phys. Chem. C. (12) Internal Reflection Spectroscopy (Practical Spectroscopy); Mirabella, F. M., Ed.; Marcel Dekker Inc.: New York, 1992; p 384. (13) Lindstrom, H.; Magnusson, E.; Holmberg, A.; Sodergren, S.; Lindquist, S. E.; Hagfeldt, A. Sol. Energy Mater. Sol. Cells 2002, 73, 91. (14) Diebold. Appl. Phys. A 2003, 76, 681. (15) Henderson, A. M. Surf. Sci. Rep. 2002, 46, 1. (16) Wang, R.; Hashimoto, K.; Fujishima, A.; Chikuni, M.; Kojima, E.; Kitamura, A.; Shimohigoshi, M.; Watanabe, T. Nature 1997, 388, 431. (17) Boisse`re, C.; Brubach, J. B.; Mermet, A.; de Marzi, G.; Bourgaux, C.; Prouzet, E.; Roy, P. J. Phys. Chem. B 2002, 106, 1032. (18) Brubach, J.-B.; Mermet, A.; Filabozzi, A.; Gerschel, A.; Lairez, D.; Krafft, M. P.; Roy, P. J. Phys. Chem. B 2001, 105, 430. (19) Ludwig, R. Chemie unserer Zeit 2005, 39, 164. (20) Ewing, G. J. Phys. Chem. C 2004, 108, 15953. (21) Mathematical Biology I: An Introduction, 3rd ed.; Murray, J. D., Ed.; Springer: New York, 2002; p 541. (22) Physikalische Chemie; Atkins, P. W., Ed.; VCH: Weinheim, 1996. (23) Raichlin, Y. Phys. ReV. Lett. 2004, 93, 185703 1. (24) Bartha, F.; Kapuy, O.; Kozmutza, C.; Van Alsenoy, C. J. Mol. Struct.: THEOCHEM 2003, 666-667, 117. (25) Tributsch, H.; Cermak, J.; Nadezhdina, N. J. Phys. Chem. B 2005, 109, 17693.