Schroeder s paradox

The subsequently presented model of water sorption in PEMs reconciles vapor sorption and porosity data. At sufficiently large water contents exceeding the amount of surface water, T > equilibrium water uptake is controlled by capillary forces. Deviations from capillary equilibrium arising at A < can be investigated by explicit ab initio calculations of water at dense interfacial arrays of protogenic surface groups. ° In the presented model, the problem of Schroeder s paradox does not arise and there is no need to invoke vapor in pores or hydrophobicity of internal channels. Here, we will present a general outline... 

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. 

The authors discuss Schroeder s paradox, referred to elsewhere in this review, and the fact that liquid water uptake increases but saturated water uptake decreases with temperature. And, at low temperature, the water uptake by membranes in contact with saturated vapor is greater than that by membranes in contact with liquid water, which suggests a fundamental difference in membrane microstructure for the two situations. An energy level diagram of thermodynamic states versus temperature was proposed, based on this Flory—Huggins-based model. 

Different models determine A in different ways. Nation exhibits a water-uptake isotherm as shown in Figure 7. The dashed line in the figure shows the effects of Schroeder s paradox, where there is a discontinuous jump in the value of A. Furthermore, the transport properties have different values and functional forms at that point. Most models used correlate A with the water-vapor activity, since it is an easily calculated quantity. An exception to this is the model of Siegel et al., ° which assumes a simple mass-transfer relationship. There are also models that model the isotherm either by Flory—Huggins theory" or equilibrium between water and hydrated protons in the membrane and water vapor... 

Figure 7. Equilibrium water-uptake or isotherm curve at 30 °C. The dashed line signifies the effect of Schroeder s paradox, a change in water uptake at the same chemical potential depending on the phase of water next to the membrane liquid is at A = 22.

Schroeder s paradox is an observed phenomenon that needs to be considered in any model where the membrane is not either fully hydrated or dehydrated. There are various methods to account for Schroeder s paradox. The easiest way is to ignore it (i.e., either only vapor filled or fully hydrated), which a majority of the models do. Next, it can be treated as a discontinuity or by assuming a functional form of the water content such that A and continue to in-... 

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. 

On the other hand, when the membrane is saturated, transport still occurs. This transport must be due to a hydraulic-pressure gradient because oversaturated activities are nonphysical. In addition, Buechi and Scherer found that only a hydraulic model can explain the experimentally observed sharp drying front in the membrane. Overall, both types of macroscopic models describe part of the transport that is occurring, but the correct model is some kind of superposition between them. - The two types of models are seen as operating fully at the limits of water concentration and must somehow be averaged between those limits. As mentioned, the hydraulic-diffusive models try to do this, but from a nonphysical and inconsistent standpoint that ignores Schroeder s paradox and its effects on the transport properties. 

Comparison of water uptake by PFSA membranes from the liquid and from the vapor phase reveals an interesting apparent paradox the water content of the membrane in equilibrium with saturated water vapor (aw = 1) is lower than the water content of a similarly prepared membrane in contact with liquid water (aw = 1). Under isopiestic conditions (Fig. 9(b)), 14 waters per sulfonate group are sorbed from vapor at unit activity (saturated vapor), whereas 22 are sorbed from liquid water at the same temperature. The phenomenon was first reported in 1903 by Schroeder [24] and is named after him as Schroeder s paradox. The explanation suggested for the lower water uptake from the saturated vapor is that sorption of water from the vapor... 

Both membranes in Figure 4.4 exhibit the so-called Schroeder s paradox, an observed difference in the amount of water sorbed by a liquid-equilibrated membrane and a saturated vapour-equilibrated membrane, with both reservoirs at the same temperature and pressure [27, 35, 36]. This difference leads to the jump in lambda when the membrane is water equilibrated (activity = 1), as shown in Figure 4.4. The underlying mechanisms for this behaviour are not completely resolved, but Choi and Datta [29] proposed a good explanation, arguing that an additional capillary pressure causes the vapour-equilibrated membrane to sorb less water than the liquid-equilibrated membrane from an external solvent with the same activity. 

We have presented a review of experimental and macroscopic modelling aspects of transport phenomena in polymer electrolyte membranes. This included examination of the connection between the hydration scheme and the behaviour of the membrane, a discussion of the so-called Schroeder s paradox, and the influence of the membrane phase on transport mechanisms. We also provided a critical examination of various approaches to modelling transport phenomena in membranes, and established that binary friction model provide a correct and rational framework for modelling membrane transport. 

P. Choi and R. Datta, Sorption in Proton-Exchange Membranes. An Explanation of Schroeder s Paradox, Journal of the Electrochemical Society, 150, E601... 

The observed scatter in the data is in part due to differences in measurement temperature, which was not well controlled in some of the cases, but it is mainly due to different membrane treatment previous to the alcohol uptake. As in the case of water uptake, methanol uptake from the liquid phase is higher than from the vapor phase when compared for the same membrane under the same treatment. This phenomenon, known as Schroeder s paradox, is related to the thermal history of the adsorbing polymer [90]. 

Water sorption in physically crosslinked poly(vinyl alcohol) membranes an experimental investigation of Schroeder s paradox. J. Memhr. Sci., 337, 291-296. 

Onishi, L.M., Prausnitz, J.M., Newman, J. 2007. Water Nafion equilibria. Absence of Schroeder s paradox. The Journal of Physical Chemistry B 111 (34) 10166-10173. 

Choi, R, and Datta, R. 2003. Sorption in proton-exchange membranes—an explanation of Schroeder s paradox. LElectrochenL SoCi, 150(12), E601—E607. 

Freger, V. 2009. Hydration of ionomers and Schroeder s paradox in Nafion. 

A surprising result also arises from the comparison of the water uptake values obtained after membrane equihbration in pure water and in a saturated vapor atmosphere. The obtaining of different values in these conditions where the water activity is supposed to be identical is called Schroeder s paradox. This effect was mainly observed at elevated temperature for Nation membranes [139] and was shown to be related to the membrane pretreatment [3,140]. In the case of SPIs, a wide difference close to a factor of 2 (X = 19 in liquid water and 11 at 100% RH) was observed at room temperature [141]. Moreover, this effect was shown to be reversible. A membrane equilibrated in liquid water and placed in a saturated atmosphere will slowly lose some weight and reach the equilibrium value determined at 100% RH in around 15 days. 

Bass M, Freger V (2006) An experimental study of Schroeder s paradox in Nation and Dowex polymer electrolytes. Desalination 199(l-3) 277-279... 

The sharp difference in water uptake between a membrane equilibrated with vapor and liquid water is known as Schroeder s paradox. This important phenomenon is of... 

Example 5.1 Water Uptake in Nafion Plot the expected water uptake in Nafion as a function of RH and include Schroeder s paradox uptake value of 22 at 80° C and a water activity of one for liquid water. 

The uptake of Nafion when equilibrated with liquid water illustrates Schroeder s paradox, whereby a water content of up to A = 22 has been measured in elevated temperature environments [7]. 

For gas-phase contact of vapor with the membrane, the water activity in the gas phase is equivalent to the gas-phase relative humidity at the cell operating temperature, as discussed in Chapter 5. For contact with liquid water, X increases to 22, despite the same thermodynamic activity as water vapor due to Schroeder s paradox. 

As a result of Schroeder s paradox, there is a very high electro-osmotic drag coefficient of 2-5 H2O/H+ in applications where the anode is in contact with liquid-phase water such as the DMFC. This and the lack of back diffusion result in a very positive net drag coefficient and more severe cathode flooding without special engineering of the DM to provide a strong capillary pressure gradient toward the anode. 

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