Clustering cluster-network model

Inhibition of anion transport in Nation was attributed to the inhomogeneous structure of the ion exchange sites in the polymer network (Gierke cluster network model). It was found that Nation contains (even in... 

The earliest fully atomistic molecular dynamic (MD) studies of a simplified Nation model using polyelectrolyte analogs showed the formation of a percolating structure of water-filled channels, which is consistent with the basic ideas of the cluster-network model of Hsu and Gierke. The first MD... 

The effective conductivity of the membrane depends on its random heterogeneous morphology—namely, the size distribution and connectivity of fhe proton-bearing aqueous pafhways. On fhe basis of the cluster network model, a random network model of microporous PEMs was developed in Eikerling ef al. If included effecfs of varying connectivity of the pore network and of swelling of pores upon water uptake. The model was applied to exploring the dependence of membrane conductivity on water content and... 

An external gas pressure gradient applied between anode and cathode sides of the fuel cell may be superimposed on the internal gradient in liquid pressure. This provides a means to control the water distribution in PEMs under fuel cell operation. This picture forms the basis for the hydraulic permeation model of membrane operation that has been proposed by Eikerling et al. This basic structural approach can be rationalized on the basis of the cluster network model. It can also be adapted to include the pertinent structural pictures of Gebel et and Schmidt-Rohr et al. ... 

Figure 1. Cluster-network model for the morphology of hydrated Nafion. (Adapted with permission from ref 16. Copyright 1983 Elsevier.)...

Based on the fiber diffraction data and the morphological constraints imposed by Geirke s cluster-network model,Starkweather developed a model for the crystalline structure of perfluorosulfonate iono-mers. " Given a 1 nm wall space between clusters. 

The original cluster-network model proposed by Gierke et al. (also referred to as the cluster-channel model) has been the most widely referenced model in the history of perfluorosulfonate ionomers. Despite the very large number of papers and reports that have strictly relied on this model to explain a wide variety of physical properties and other characteristics of Nafion, this model was never meant to be a definitive description of the actual morphology of Nafion, and the authors recognized that further experimental work would be required to completely define the nature of ionic clustering in these iono-mers. For example, the paracrystalline, cubic lattice... 

The cluster-network model of Gierke et al. has already been discussed in the Introduction as being the first realistic model for rationalizing a number of properties of Nafion membranes. 

Fig. 15. Cluster network model for highly cation-permselective Nafion membranes126). Counterions are largely concentrated in the high-charge shaded regions which provide somewhat tortuous, but continuous (low activation energy), diffusion pathways. Coions are largely confined to the central cluster regions and must, therefore, overcome a high electrical barrier, in order to diffuse from one cluster to the next...

Figure 3 Schematic representation of the two-phase cluster-network model for Nafion membrane.

Gierke, T., Hsu, W. (1982). The cluster-network model of ion clustering in perfluoro-sulfonated membranes. In "Perfluorinated lonomer Membranes", American Chemical Society Symp. Series 180, Washington, DC. 

Gierke also considered that these clusters are interconnected by short, narrow channels in the fluorocarbon backbone network. The diameter of these channels is about 1 mm estimated from hydraulic permeability data. He further considered that the Bragg spacing ( 5nm from SAXS data) can represent the distance between clusters. The cluster-network model is a phenomenological description. Recently, Hsu and Gierke " have derived a semi-phenomenological expression to correlate the variation of cluster diameter with water content, equivalent weight, and cation form of the membrane. They have shown that the short channels are thermodynamically stable. 

Since the conductivity of electrolytes and the cross section and thickness of the membrane are known, a can be determined from the voltage drops across the three pairs of probe electrodes 1-2, 3-4 and 5-6. The sodium current efficiency (CE) can also be determined by titrating the amount of caustic soda generated over a given period of time. The confinement chambers around the working electrodes are used to eliminate free bubbles near the membrane. Our normalized transport data for sulfonate, carboxylate and sulfonamide ionomers are plotted In Figure 5 the universal percolative nature of perfluorinated ionomers can be clearly eeij. The prefactor sulfonate ionomers. The exponent t is 1.5 0.1 in reasonable agreement with theory and the thresholds are between 8 to 10 vol. %, which are consistent with the bimodal distribution in cluster size postulated by the cluster-network model (5.18). This theory has also been applied recently to delineate sodium selectivity of perfluorinated ionomers (20). 

The Cluster-Network Model of Ion Clustering in Perfluorosulfonated Membranes... 

A model for ionic clustering in "Nafion" (registered trademark of E. I. du Pont de Nemours and Co.) perfluorinated membranes is proposed. This "cluster-network" model suggests that the solvent and ion exchange sites phase separate from the fluorocarbon matrix into inverted micellar structures which are connected by short narrow channels. This model is used to describe ion transport and hydroxyl rejection in "Nafion" membrane products. We also demonstrate that transport processes occurring in "Nafion" are well described by percolation theory. 

In this work we propose a model for ionic clustering, which we have called the cluster-network model (2), to account for hydroxyl rejection in nNafionM perfluorinated membranes. In developing this model we have been guided by two requirements 1. the model should be consistent with the available data on the microscopic structure of the polymer (1-5) 2. the model should... 

In the next section we will present the data and arguments on which the cluster-network model is based. We will also discuss the effects of equivalent weight, ion form, and water content on the dimensions and composition of the clusters. In the third section we will present a formalism, which follows from the cluster-network model, based on absolute reaction rate theory (2) and hydroxyl rejection in "Nation perfluorinated membranes. Finally we will outline the concepts of percolation theory and demonstrate that ion transport trough "Nation" is well described by percolation. 

Combining the results of these water transport experiments with the inverted micellar structure proposed for the clusters, we arrive at the cluster-network model shown in Figure 5. In this model the clusters are connected by short narrow channels whose dimensions are derived from the water transport measurements. 

Figure 5. Cluster-network model for Nafion perfluorinated membranes. The polymeric ions and absorbed electrolyte phase separate from the fluorocarbon backbone into approximately spherical clusters connected by short, narrow channels. The polymeric charges are most likely embedded in the solution near the interface between the electrolyte and fluorocarbon backbone. This configuration minimizes both the hydrophobic interaction of water with the backbone and the electrostatic repulsion of proximate sulfonate groups. The dimensions shown were deduced from experiments. The shaded areas around the interface and inside a channel are the double layer regions from which the hydroxyl ions are excluded electrostatically.

Table V. Cluster-Network Model Calculations of NaOH Apparent Diffusion Coefficients.

To test this theory, the room temperature conductivity of "Nafion" perfluorinated resins was measured as a function of electrolyte uptake by a standard a.c. technique for liquid electrolytes (15). The data obey the percolation prediction very well. Figure 9 is a log-log plot of the measured conductivity against the excell volume fraction of electrolyte (c-c ). The principal experimental uncertainty was in the determination of c as shown by the horizontal error bars. The dashed line is a non-linear least square law to the data points. The best fit value for the threshold c is 10% which is less than the ideal value of 15% for a completely random system. This observation is consistent with a bimodal cluster distribution required by the cluster-network model. In accord with the theoretical prediction, the critical exponent n as determined from the slope of... 

Summary. We have shown that ion transport in "Nafion" per-fluorinated membrane is controlled by percolation, which means that the connectivity of ion clusters is critical. This basically reflects the heterogeneous nature of a wet membrane. Although transport across a membrane is usually perceived as a one-dimensional process, our analysis suggests that it is distinctly three-dimensional in "Nafion". (Compare the experimental values of c and n with those listed in Table 7.) This is not totally unexpected since ion clusters are typically 5.0 nm, whereas a membrane is normally several mils thick. We have also uncovered an ionic insulator-to-conductor transition at 10 volume % of electrolyte uptake. Similar transitions are expected in other ion-containing polymers, and the Cluster-Network model may find useful application to ion transport in other ion containing polymers. Finally, our transport and current efficiency data are consistent with the Cluster-Network model, but not the conventional Donnan equilibrium. 

Figure 4.21 Cluster network model of perfluorocarbon cation exchange membrane (Nafion ).

FIGURE 8. Cluster-network model proposed by Gierke. Reprinted with permission from T. D. Gierke and W. Y. Hsu, in Perfluorinated Ionomer Membranes (Eds. A. Eisenberg and H. L. Yeager), Chap. 13, ACS Symp. Ser. No. 180, 1980, p. 286. Copyright (1980) American Chemical Society. 

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... 

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