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

Abstract

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.

Geometric evolution of bilayers under the functionalized Cahn–Hilliard equation

“…The identification of the FCH with a Canham-Helfrich-type sharp interface energy is predicated on the assumption that the underlying structures are of codimension 1 and free of defects, such as end-caps and junctions. Over R 3 , the FCH free energy supports co-dimension one bilayer interfaces, whose evolution we study here, as well as and a wide range of stable co-dimension 2 and co-dimension 3 morphologies [4,19] described below, in addition to many locally stable defect structures. The structure of the problem, and the physically motivating examples, change fundamentally and dramatically with the sign of η 2 .…”

“…There are many possible areas to investigate in both model development and model analysis. It is quite intriguing to consider extensions beyond binary mixtures; indeed, a preliminary discussion in this direction can be found in Gavish et al [4]. For example, bilayers need not only form a barrier between the same phase, but can also separate two distinct phases.…”

“…The phase u = b − with the higher self-energy is the majority phase, whereas the u = b + phase is the minority phase (amphiphilic surfactant or lipid). The well-tilt, or difference in self-energies W(b − ) − W(b + ) > 0 is a significant bifurcation parameter for network morphologies [4]. The single-layer interfaces of the CH free energy are natural minimizers of E for untilted wells W, i.e.…”

“…The end result is a 'spaghetti'-like structure with an O(ε) unscaled radius, but a spatially extended length. The FCH also possesses co-dimension three micellular structures in R 3 as well as defect states such as end-caps and multi-junctions that play an essential role in network formation, see [4,21]. This paper presents a formal reduction in the H −1 gradient flow of the functionalized energy F , called the FCH equation, …”

Resistor-Network Modeling of Ionic Conduction in Polymer Electrolytes

“…However, there is undoubtedly a mixture of modes, wherein the connectivity becomes increasingly important, and dominant transport pathways emerge. While the two mechanisms have been explored at the atomic scale through ab-initio and molecular dynamics [42][43][44][45], or at the coarse grain scale through dissipative particle dynamics and other methods [46][47][48], scaling up to a macroscopically measurable quantity across the membrane has been a challenge due to the connectivity issues and disparate length scales of the phenomena; a true, efficient multiscale representation of the conductivity does not exist.…”