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We present the functionalized Cahn-Hilliard (FCH) energy, a continuum characterization of interfacial energy whose minimizers describe the network morphology of solvated functionalized polymer membranes. With a small set of parameters the FCH characterizes bilayer, pore-like, and micelle network structures. The gradient flows derived from the FCH describe the interactions between these structures, including the merging and pinch-off of endcaps and formation of junctions central to the generation of network morphologies. We couple the FCH gradient flow to a model of ionic transport which incorporates entropic effects to localize counter-ions, yielding a flow which dissipates a total free energy, and an expression for the excess electrochemical potential which combines electrostatic and entropic effects. We present applications to network bifurcation and membrane casting.

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

SUMMARYIterative methods play an important role in solving large-scale systems of linear equations that arise in real-world applications. Due to numerous linear system properties that may affect the solution, it is rather difficult for a user to develop a good sparse linear system solver from scratch. Thus, various collections of solution methods are made available to the user. One such software package is SPARSKIT, which is well known in the scientific community. Written in FORTRAN77 and provided with a cumbersome interface, it is considered, however, a legacy code. Our objective is to enable its wider usage in modern applications and to facilitate further SPARSKIT enhancements.

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