Vapor-equilibrated

Fig. 9. Schematic of water vapor equilibration apparatus featuring carousels that hold up to 49 samples in crimped silver cups (from Sauer et al., 2009).

The experimental basis of sorption studies includes structural data (SANS, SAXS, USAXS), isopiestic vapor sorption isotherms,i and capillary isotherms, measured by the method of standard porosimetry. i 2-i44 Thermodynamic models for water uptake by vapor-equilibrated PEMs have been suggested by various groupThe models account for interfacial energies, elastic energies, and entropic contributions. They usually treat rate constants of interfacial water exchange and of bulk transport of water by diffusion and hydraulic permeation as empirical functions of temperature. 

This implies that vapor equilibration of PEMs corresponds to large negative liquid pressures inside the membrane or that Tj, would increase from zero to the saturation value for P very close to P. Moreover, the effect of the total gas pressure on water uptake should be insignificant at normal values of Ps 1 atm. Heuristic solutions out of this dilemma would be to recalibrate the value of V or to normalize P to a reference value of a... 

Equation (6.20) determines the maximum degree of swelling and the maximum pore radius of a liquid-equilibrated membrane. This relation suggests that the external gas pressure over the bulk water phase, which is in direct contact with the membrane, controls membrane swelling. The observa-hon of different water uptake by vapor-equilibrated and by liquid water-equilibrated PEMs, denoted as Schroeder s paradox, is thus not paradoxical because an obvious disparity in the external conditions that control water uptake and swelling lies at its root cause. 

In the physical model, there are two separate structures for the membrane depending on whether the water at the boundary is vapor or liquid these are termed the vapor- or liquid-equilibrated membrane, respectively. The main difference between the two is that, in the vapor-equilibrated membrane, panel c, the channels are collapsed, while, in the liquid-equilibrated case, panel d, they are expanded and filled with water. These two structures form the basis for the two types of macroscopic models of the membrane. 

The diffusive models treat the membrane system as a single phase. They correspond more-or-less to the vapor-equilibrated membrane (panel c of Figure 6). Because the collapsed channels fluctuate and there are no true pores, it is easiest to treat the system as a single, homogeneous phase in which water and protons dissolve and move by diffusion. Many membrane models, including some of the earliest ones, treat the system in such a manner. 

Weber and Newman do the averaging by using a capillary framework. They assume that the two transport modes (diffusive for a vapor-equilibrated membrane and hydraulic for a liquid-equilibrated one) are assumed to occur in parallel and are switched between in a continuous fashion using the fraction of channels that are expanded by the liquid water. Their model is macroscopic but takes into account microscopic effects such as the channel-size distribution and the surface energy of the pores. Furthermore, they showed excellent agreement with experimental data from various sources and different operating conditions for values of the net water flux per proton flux through the membrane. 

The term plate height comes from distillation theory. Some high-performance distillation columns contain discrete units called plates, in which liquid and vapor equilibrate with each other. As a teenager, A. J. R Martin, coinventor of partition chromatography, built distillation columns in discrete sections from coffee cans. (We don t know what he was distilling ) When he formulated the theory of partition chromatography, he adopted terms from distillation theory. 

The Vario-KS chamber (Camag, Fig. If) is used for optimization of developing conditions for 10 x 10 cm plates by simultaneously testing of up to six different mobile phases and vapor equilibration conditions with N or S chamber conditions. Also, the Vario-KS chamber may be used for optimization of developing conditions for 20 X 20 cm TLC plates. The thin-layer plate (1) is laid down on the support (2). The eluent in the reservoir is connected to the thin layer of adsorbent by means of a filter paper strip. Under the thin-layer plate, there are many troughs (3) filled with solvents for preconditioning of the adsorbent layer. The chamber is tightly sealed by two clamps (4). 

Since polyethylene glycol) solutions are not volatile, this precipitant must be used like salt and equilibrated with the protein by slow mixing or vapor equilibration. This latter approach, utilizing either hanging drops over 0.5 ml reservoirs, or sitting drops in plastic plates, has proved the most popular. When the reservoir concentration is in the range 5% to 20%, the protein solution to be equilibrated should be at an initial concentration of about half of that, which is conveniently obtained by adding an equal volume of the reservoir to that of the protein solution. 

Figure 7. Formaldehyde liberation from particleboards and CH20-sorbed wood at 27°C and 33 percent relative humidity (RH) weighing bottle test with -80 mesh materials (o Southern pine impregnated with pH 2 tartaric acid and vapor-equilibrated with CH20/salt solution at pet RH O as before except heated 4 min. 16O°C after CH2O sorption 0 urea-formaldehyde particle board (b) phenol-formaldehyde particleboard, values approximate P = Perforator value at indicated moisture content (MC)).

Figure 12. The effect of NaCl on the hydrogen isotope fractionation (10 InF) obtained from experiments involving liquid-vapor equilibration (solid bold curves) compared to those obtained from brucite-water (solid curves, Horita et al., in press) and epidote-water (dashed curves, Graham and Sheppard 1980) partial exchange results. Data from Horita et al. (in press) Solid circles = 1 molal NaCl, Sohd squares = 3 molal NaCl, Sohd triangles = 5 molal NaCl. Data from Graham and Sheppard (1980) Open, inverted triangles = 4 molal NaCl, Open circles = 1 molal NaCl.

The negative values of In S indicate that the D/H ratio of water vapor equilibrated with aqueous solutions is higher than that of water vapor equilibrated with pure water. 

In terms of the structure within the membrane, the idealized Hsu and Gierke cluster-network model is used as a picture where the pathways between the clusters are interfacial regions. These pathways are termed collapsed channels since they can be expanded by liquid water to form a liquid-filled channel. In essence, the collapsed channels are sulfonic acid sites surrounded by the polymer matrix having a low enough concentration such that the overall pathway between two clusters remains hydrophobic. In other words, they are composed of bridging ionic sites [31] and the electrostatic energy density is too low compared to the polymer elasticity to allow for a bulk-like water phase to form and expand the channels. In all, for a vapor-equilibrated membrane the structure is that of ionic domains that are hydrophilic and contain some bulk-like water. These clusters are connected by... 

For a vapor-equilibrated membrane (i.e., one that is in contact with water vapor only), the physical model proposes that there is water in the ionic domains but none in the collapsed channels except for the bound water hydrating the few sulfonic acid sites present. Furthermore, the sulfonic acid sites that make up the collapsed channels are always fluctuating, but the elusters are elose enough together to form a transport pathway after the pereolation threshold has been reached. Due to the nature of the collapsed ehaimels, the membrane is treated as a homogenous single-phase system. In this sense, the water vapor does not penetrate into the cluster-network, but instead dissolves into the membrane. Thus, the vapor-equilibrated membrane transport mechanism is similar to the single-phase transport models mentioned previously. 

In summary, the transport mode of a vapor-equilibrated membrane is that of a single membrane phase in which protons and water are dissolved. The chemical-potential gradient is used directly since it precludes the necessity of separating it into pressure and activity terms. Thus, Eqs. (5.8-5.11) are used directly without any modifications. Although it makes sense to use the chemical-potential driving force, most of the experimental data are a function of water content or X. Thus, a way is needed to relate X to the chemical potential. 

Chemical potential and water content, X, can be related through an uptake isotherm. Uptake isotherms of k as a function of water-vapor activity or relative humidity, such as that given in Figure 5.1, are prevalent in the literature [4, 6, 42, 43]. They have been used in almost every model that deals with vapor-equilibrated membranes and treats the membrane as a single phase [1]. As discussed in the proposed physical model, the water uptake is described by the hydration of the sulfonic acid sites in the membrane clusters and a balance between osmotic, elastic, and electrostatic forces. The approach taken here is to calculate the isotherms using the chemical model of Meyers and Newman [5] with some modifications [39]. 

In the chemical model, equilibrium is assumed between protons and water with a hydronium ion. This equilibrium considers the tightly bound water in the membrane [13, 19, 44] and agrees with the vapor-equilibrated transport picture of a hydronium ion being the dominant proton-transfer species in the membrane. The equilibrium relates the electrochemical potentials of the species, and at the boundary the water in the membrane is in equilibrium... 

One comment should be made regarding the form of the transport equations. In the literature, two-phase flow has often been modeled using Schlogl s equation [50, 51]. This equation is similar in form to Eq. (5.9), but it is empirical and ignores the Onsager cross coefficients. Equations (5.8) and (5.9) stem from concentrated-solution theory and take into account all the relevant interactions. Furthermore, the equations for the liquid-equilibrated transport mode are almost identical to those for the vapor-equilibrated transport mode making it easier to compare the two with a single set of properties (i.e., it is not necessary to introduce another parameter, the elec-trokinetic permeability). 

As in the case for the vapor-equilibrated transport mode, the properties of the liquid-equilibrated transport mode depend on the water content and temperature of the membrane. For a fully liquid-equilibrated membrane, the properties are uniform at the given temperature. This is because the water content remains constant for the liquid-equilibrated mode unlike in the vapor-equilibrate one. From experimental data, the value of A, for liquid-equilibrated Nafion is around 22, assuming the membrane has been pretreated correctly [6, 7, 52]. In agreement with the physical model, the water content is only a very weak function of temperature for extended (E)-form membranes (as assumed in our analysis) and can be ignored [6]. For other membrane forms, this dependence is much stronger and cannot be ignored, as discussed in the Section 5.10.1. 

When the membrane is in contact with liquid water on one side and vapor on the other (i.e., it is neither fully liquid nor vapor equilibrated), as can often occur during fuel-cell operation, both the liquid- and vapor-equilibrated transport modes will occur. This results in a transition between modes that exists in the membrane. As discussed in the physical model, a continuous transition between the two transport modes is assumed. Thus, transport in the transition region is a superposition between the two transport modes they are treated as separate transport mechanisms occurring in parallel (i.e., the middle region in Figure 5.3). In this section, an approach to modeling the transition region is introduced followed by a discussion of its limitations, other approaches, and points to consider. 

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