Variational Models of Pore Networks in Ionomer Membranes: The Role of Electrostatics

Abstract

We present a self-consistent, dynamic model for the morphology of the pore networks which form in Nafion and other ionomer membranes when they imbibe solvent. The model incorporates the interactions of the tethered ionic groups with the solvent and the entropic energy of the counter ions. Numerical simulations are presented which display hysteresis, and the development of pearled pore structures. A framework for extensions of the model that couple the electrostatic double layer at the interface of the pore wall and the transport equations for the counterions is presented. The framework derives the ionic transport and generalized Poisson equation from an action whose gradient flows dissipate a related free energy, in agreement with the second law of thermodynamics. Network Formation in Ionomer MembranesWhile ionomer materials play a central role in Polymer Electrolyte Membrane (PEM) fuel cells, may basic questions about the morphology of their pore networks remain unanswered. Nafion is the industry standard for membrane separators in PEM fuel cells due to its high ionic conductivity, mechanical robustness, and durability. It is a peruorosulfonate ionomer membrane with tethered SO 3 H acid groups which disassociate in the presence of polar solvent. The chemical energy behind the disassociation leads to one of Nafion's key properties: the imbibition of solvent, principally water, to produce a pore network on the 2-4 nanometer length scale. The network facilitates the conduction of counter-ions, chiefly protons in PEM fuel cell applications, and generates the ionic selectivity of the membrane.There have been many proposed forms for the ionic-cluster morphology, i.e., the solventfilled pore network, of Nafion. Early SAXS experiments, (1-2), lead Hsu and Gierke to hypothesize that a balance between the elastic energy of the interface and the hydrophilic surface interactions among the charged functional groups and the solvent drive Nafion to generate a water-filled, "pearled" pore network comprised of small 4-5 nanometer balls interconnected by thin 1-2 nanometer cylindrical pores, see Figure 4 (bottom-right). Nanoscale pearling has also be studied in functionalized diblock polymers, (3-4)The morphologies of Hsu and Gierke are somewhat controversial, as numerical and experimental investigation of the microstructure of Nafion and related perfluorinated membanes have 10.1149/05002.0161ecst ©The Electrochemical Society ECS Transactions, 50 (2) 161-173 (2012) 161

Systematic interpretation of differential capacitance data

“…The same Hamiltonian describes electrolyte behavior in a system with more complex geometries, e.g., near a curved charged body or inside a narrow channel. In addition, it is possible to use the Hamiltonian to describe dynamics, such as arises under ionic current or time-dependent applied voltages [39].…”

New Insights into Perfluorinated Sulfonic-Acid Ionomers

“…Simulated morphology of PFSAs at various length scales as determined from (a) ab initio models (Paddison et al), (c and d) mesoscale (coarse-grained) models, and (e) phase-field models. , Panel a demonstrates the impact of backbone length on min number of water molecules required to bridge SO 3 – sites. Panel b illustrates the concept of coarse-graining.…”

“…Also see Table for the computed interaction parameters for these structures. (e) Possible morphologies obtained from variational model for low high and low mixing energies corresponding to the strength of segregation of phases (from Promislow , ). ((a) Reproduced from ref .…”

“…Other approaches that can be used to predict the morphology are energy-balance-variational and phase-field models , that utilize a functionalized Cahn–Hilliard (FCH) energy, as discussed by Promislow et al In these simulations, an overall expression for the free-energy of the system is minimized among possible geometries to derive a description of the hydrophilic domain microstructure as introduced in section . , In this fashion, they can predict the impact of environment on water uptake and provide a domain distribution that can be used to run transport simulations as shown in Figure . The figure shows the structures for various mixing energies, η mix , which reflect the entropic penalty for high densities of interspersed ions and counterions, with larger values leading to segregation of charge groups on the length scale of the domain . At higher water contents, the appearance of the network starts to mimic that observed using the cryo-TEM (Figure ), composed of flat, worm-like small channels.…”

“…443,449,451,452 In fact, substantial rearrangement of SO 3 − groups and conformational changes in polymer chain was suggested to result in nonaffine swelling correlations, even in the absence of mesoscale organization of domains of lower dimensionality. 71,73,368 Other approaches that can be used to predict the morphology are energy-balance-variational and phase-field models 353,468 that utilize a functionalized Cahn−Hilliard (FCH) energy, as discussed by Promislow et al In these simulations, an overall expression for the free-energy of the system is minimized among possible geometries to derive a description of the hydrophilic domain microstructure as introduced in section 2.6. 353,468 In this fashion, they can predict the impact of environment on water uptake and provide a domain distribution that can be used to run transport simulations as shown in Figure 25.…”