Variational Models of Network Formation and Ion Transport: Applications to Perfluorosulfonate Ionomer Membranes

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.

[1]  S. Paddison,et al.  A statistical mechanical model for the calculation of the permittivity of water in hydrated polymer electrolyte membrane pores , 2001 .

[2]  Gérard Gebel,et al.  Fibrillar structure of Nafion: Matching fourier and real space studies of corresponding films and solutions , 2004 .

[3]  F. Beyer,et al.  Self-Assembling Nanomembranes Through Electrostatic Melt Processing of Copolymer Films , 2002 .

[4]  A. Weber,et al.  Transport in Polymer-Electrolyte Membranes II. Mathematical Model , 2004 .

[5]  S. Grot,et al.  SANS Study of the Effects of Water Vapor Sorption on the Nanoscale Structure of Perfluorinated Sulfonic Acid (NAFION) Membranes , 2006 .

[6]  H. Starkweather.  Crystallinity in perfluorosulfonic acid ionomers and related polymers , 1982 .

[7]  T. Gierke,et al.  Ion transport and clustering in nafion perfluorinated membranes , 1983 .

[8]  P. Canham.  The minimum energy of bending as a possible explanation of the biconcave shape of the human red blood cell. , 1970, Journal of theoretical biology.

[9]  Randall D. Kamien,et al.  The Geometry of Soft Materials: A Primer , 2002 .

[10]  H. Ohno,et al.  3D interconnected ionic nano-channels formed in polymer films: self-organization and polymerization of thermotropic bicontinuous cubic liquid crystals. , 2011, Journal of the American Chemical Society.

[11]  S. Hanna,et al.  Interpretation of the Small-Angle X-ray Scattering from Swollen and Oriented Perfluorinated Ionomer Membranes , 2000 .

[12]  U. Wiesner,et al.  A bicontinuous double gyroid hybrid solar cell. , 2009, Nano letters.

[13]  B. Balcom,et al.  Bi-modal water transport behavior across a simple Nafion membrane , 2011 .

[14]  S. J. Singer,et al.  The water-amorphous silica interface: analysis of the Stern layer and surface conduction. , 2011, The Journal of chemical physics.

[15]  Sia Nemat-Nassera,et al.  Micromechanics of actuation of ionic polymer-metal composites , 2014 .

[16]  Michael Eikerling,et al.  Poroelectroelastic theory of water sorption and swelling in polymer electrolyte membranes , 2011 .

[17]  Keith Promislow,et al.  PEM Fuel Cells: A Mathematical Overview , 2009, SIAM J. Appl. Math..

[18]  J. E. Hilliard,et al.  Free Energy of a Nonuniform System. I. Interfacial Free Energy , 1958 .

[19]  Noel J. Walkington,et al.  Digital Object Identifier (DOI) 10.1007/s002050100158 An Eulerian Description of Fluids Containing Visco-Elastic Particles , 2022 .

[20]  Gérard Gebel,et al.  Evidence of elongated polymeric aggregates in Nafion , 2002 .

[21]  Viatcheslav Freger,et al.  Hydration of ionomers and Schroeder's paradox in Nafion. , 2009, The journal of physical chemistry. B.

[22]  Frank S. Bates,et al.  Consequences of Nonergodicity in Aqueous Binary PEO-PB Micellar Dispersions , 2004 .

[23]  G. Voth,et al.  Probing selected morphological models of hydrated Nafion using large-scale molecular dynamics simulations. , 2010, The journal of physical chemistry. B.

[24]  S. Paddison.  The modeling of molecular structure and ion transport in sulfonic acid based ionomer membranes , 2001 .

[25]  Moo Hwan Kim,et al.  Role of water states on water uptake and proton transport in Nafion using molecular simulations and bimodal network , 2010 .

[26]  W. Helfrich.  Elastic Properties of Lipid Bilayers: Theory and Possible Experiments , 1973, Zeitschrift fur Naturforschung. Teil C: Biochemie, Biophysik, Biologie, Virologie.

[27]  A. B. Solov’eva,et al.  Structure alterations of perfluorinated sulfocationic membranes under the action of ethylene glycol (SAXS and WAXS studies) , 2003 .

[28]  Guillermo C Bazan,et al.  "Plastic" solar cells: self-assembly of bulk heterojunction nanomaterials by spontaneous phase separation. , 2009, Accounts of chemical research.

[29]  M. Eikerling,et al.  Defect structure for proton transport in a triflic acid monohydrate solid , 2003 .

[30]  Simon Sherman,et al.  Influence of the solvent structure on the electrostatic interactions in proteins. , 2004, Biophysical journal.

[31]  Stephen J. Paddison,et al.  Device and materials modeling in PEM fuel cells , 2009 .

[32]  E. D. Giorgi,et al.  Some remarks on Γ-convergence and least squares method , 1991 .

[33]  E. Vanden-Eijnden,et al.  A temperature accelerated method for sampling free energy and determining reaction pathways in rare events simulations , 2006 .

[34]  S. Paddison,et al.  Transport in proton conductors for fuel-cell applications: simulations, elementary reactions, and phenomenology. , 2004, Chemical reviews.

[35]  B. Améduri.  From vinylidene fluoride (VDF) to the applications of VDF-containing polymers and copolymers: recent developments and future trends. , 2009, Chemical reviews.

[36]  Robert B. Moore,et al.  Mechanical and transport property modifications of perfluorosulfonate ionomer membranes prepared with mixed organic and inorganic counterions , 2006 .

[37]  S. Paddison,et al.  Effect of Molecular Weight on Hydrated Morphologies of the Short-Side-Chain Perfluorosulfonic Acid Membrane , 2009 .

[38]  T. Springer,et al.  Water Uptake by and Transport Through Nafion® 117 Membranes , 1993 .

[39]  R. Caflisch.  The fluid dynamic limit of the nonlinear boltzmann equation , 1980 .

[40]  A. Kornyshev.  Nonlocal screening of ions in a structurized polar liquid — new aspects of solvent description in electrolyte theory , 1981 .

[41]  M. Bazant,et al.  Towards an understanding of induced-charge electrokinetics at large applied voltages in concentrated solutions. , 2009, Advances in colloid and interface science.

[42]  S. Paddison,et al.  Structure and dielectric saturation of water in hydrated polymer electrolyte membranes: Inclusion of the internal field energy , 2004 .

[43]  A. A. Kornyshev,et al.  Polar solvent structure in the theory of ionic solvation , 1974 .

[44]  C. R. Martin,et al.  Procedure for preparing solution-cast perfluorosulfonate ionomer films and membranes , 1986 .

[45]  Keith Promislow,et al.  Curvature driven flow of bi-layer interfaces , 2011 .

[46]  S. Paddison,et al.  A comparative study of the hydrated morphologies of perfluorosulfonic acid fuel cell membranes with mesoscopic simulations , 2008 .

[47]  Wataru Shinoda,et al.  Micellization Studied by GPU-Accelerated Coarse-Grained Molecular Dynamics. , 2011, Journal of chemical theory and computation.

[48]  Qiang Chen,et al.  Parallel cylindrical water nanochannels in Nafion fuel-cell membranes. , 2008, Nature materials.

[49]  Adam Z. Weber,et al.  Modeling Transport in Polymer‐Electrolyte Fuel Cells , 2004 .

[50]  K. Promislow,et al.  Critical points of functionalized Lagrangians , 2012 .

[51]  K. Ho,et al.  Dye-sensitized solar cells with a micro-porous TiO2 electrode and gel polymer electrolytes prepared by in situ cross-link reaction , 2009 .

[52]  Robert B. Moore,et al.  Chemical and morphological properties of solution-cast perfluorosulfonate ionomers , 1988 .

[53]  Marcus Müller,et al.  Concurrent coupling between a particle simulation and a continuum description , 2009 .

[54]  F. Golse.  Hydrodynamic Limits , 2005 .