Vapor-equilibrated membrane

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. 

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]. 

Figure 5.7. Plots of the gas permeation coefficients in 1100 equivalent weight Nafion . (a) Hydrogen and oxygen permeation coefficients at 30°C as a function of water content. The dotted lines signify the transition-region values, (b) Arrhenius plot of the oxygen permeation coefficient as a function of temperature for a liquid-equilibrated membrane, a vapor-equilibrated membrane, and a dry membrane. Also plotted are the oxygen permeation coefficients in water [79] and Teflon [80, 81]. (Figure b is reproduced from Ref. [39] with permission of The Electrochemical Society, Inc.)...

To examine the transport-mode-transition region in more detail, simulations were run at different current densities [71]. The resultant membrane water profiles are shown in Figure 5.11 where a vapor-equilibrated membrane at unit activity has a water content of L = 8.8 as calculated by the modified chemical model (see Section 5.5.1). The profiles in the figure demonstrate that the higher the current density the sharper the transition from the liquid-equilibrated to the vapor-equilibrated mode as well as the lower the value of the water content at the anode GDL/membrane interface. The reason why the transition occurs at the same point in the membrane is that the electro-osmotic flow and the water-gradient flow are both proportional to the current... 

Thus, water transport by electro-osmotic drag is always from the anode to cathode and is two to three times greater than the water generation at the cathode for vapor-equilibrated membranes In terms of flooding at the cathode, the electro-osmotic drag to the cathode results in more water than the electrochemical generation. 

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. 

Unlike the cases of the single-phase models above, the transport properties are constant because the water content does not vary, and thus, one can expect a linear gradient in pressure. However, due to Schroeder s paradox, different functional forms might be expected for the vapor- and liquid-equilibrated membranes. 

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... 

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. 

In summary, when both the liquid- and vapor-equilibrated transport modes occur in the membrane they are assumed to occur in parallel. In other words, there are two separate contiguous pathways through the membrane, one with liquid-filled channels and another that is a one-phase-type region with collapsed channels. To determine how much of the overall water flux is distributed between the two transport modes, the fraction of expanded channels is used. As a final note, at the limits of S = 1 and S = 0, Eqs. (5.17) and (5.18) or their effective property analogs collapse to the respective equations for the single transport mode, as expected. 

As discussed in the physical model, the conductivity is caused by both the vehicle and Grotthuss mechanisms. The main difference between the conductivity of the vapor and liquid-equilibrated membranes stems from their respective water contents. A percolation-type equation is used for the conductivity although, at higher water contents, the conductivity should level out and ideally approach that of liquid water at infinite dilution (i.e., the polymer is dissolved in an infinite amount of water). The temperature dependence of the conductivity is due to the change in the equilibrium constant for the dissociation of the sulfonic acid sites and the activation energies for the Grotthuss mechanism and vehicle mechanisms [39]. Finally, the expressions given in Table 5.2 are valid for pure water and the... 

The permeation coefficients, like the other transport properties, are expected to depend on water content, temperature, and the state of the membrane (i.e., collapsed or expanded channels). Fitting the experimental data [39] yields the following expressions for vapor- and liquid-equilibrated membranes respectively... 

Figure 5.7(b) except that all the values are increased by a factor of about 1.5. As can be seen in the figure, the permeation-coefficient values are basically bounded with the liquid-equilibrated values higher than the vapor-equilibrated ones. The values are bounded because at higher water contents the gases mainly move through the bulk-like liquid water, and under dry conditions the membrane is very similar to Teflon (see Figure 5.2). 

In the 1 + ID model the MEA can be in one of three distinct states, as indicated in Figure 7.1. Either the cathode GDL is entirely single phase, and the membrane water content is based upon disequilibrium from the vapor equilibrated phase, or the cathode GDL is partially or entirely two-phase, and the membrane water content is based upon disequilibrium from the liquid equilibrated phase. The switching between these states drives the hysteresis however, lateral motion of liquid water within the membrane or within the GDL is not permitted within this model. Such motion could lead to slow transients, as found in the STR fuel cell of Section 7.3.1. 

Equilibrating membranes over salt solutions that set the vapor pressure have provided membrane water activity versus water content data In... 

FIGURE 2.21 Pressure equilibrium between fluid pressure (solid and dashed lines), pA, and elastic pressure (dash-dotted line), at the pore wall for a = 0.5 in the case of anisotropic in-plane expansion in Figure 2.20. The pore fluid pressure of both vapor-equilibrated (dashed) and liquid-equilibrated (solid) membranes increases with the wall charge density (here, —0.05, —0.1, —0.2, —0.4C m ). (Eikerling, M. and Berg, P. 2011. Soft Matter, 7(13), 5976-5990, Figures 1-7. Reproduced by permission of The Royal Society of Chemistry.)... 

Two major groups of performance models have been proposed. The first group considers the membrane as a homogeneous mixture of ionomer and water. The second group involves approaches that consider the membrane as a porous medium. Water vapor equilibrates with this medium by means of capillary forces, osmotic forces resulting from solvated protons and fixed ions, hydration forces, and elastic forces. In this scenario, the thermodynamic state of water in the membrane should be specified by (at least) two independent thermodynamic variables, namely, chemical potential and pressure, subdued to independent conditions of chemical and mechanical equilibrium, respectively. The homogeneous mixture model is the basis of the so-called... 

Romero, T. and Merida, W. 2009. Water transport in liquid and vapor equilibrated Nafion membranes. 338, 135-144. 

K) 2 is the electron transfer number for each H2O molecule produced by a 4-electron ORR F is the Faraday constant (96,487 A s moP ) and m is the amount of water osmotically dragged per proton transfer from the anode to the cathode (m is also called the osmotic drag coefficient). For a water-vapor-equilibrated system, the value of m is likely to be a constant number over a broad range of water content values, from a nearly dry membrane (--2H2O per SO3) to a fully hydrated membrane (with water vapor -14H20 per SO. The... 

Regarding the membrane as a single phase and assuming that the membrane is vapor equilibrated, where the water and protons dissolve and move by diffusion under the condition of dilute and concentrated solution Assuming the membrane layer has two parts, which are the membrane and liquid water Considering both diffusion and pressure-driven flow Combining all the aforementioned types and being suitable for the cases when membrane is either saturated or dehydrated with water... 

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