Theory of Multi-Ion Transport and Thermodynamic Phenomena in Solvent-Filled Membranes
Authors: Crothers Andrew Robert , Kushner Douglas I. , Darling Robert M. , Kusoglu Ahmet , Radke Clayton J. , Weber Adam Z. 
Published: February 2020

Solvent-filled membranes, such as Nafion, are an important class of polymer electrolytes because the imbibed solvent imbues the material with high ionic conductivity, while still allowing the membrane to maintain its mechanical stability. Both of these attributes are essential for membrane electrolytes in numerous energy-storage and -conversion devices including redox-flow batteries. The transport phenomena is complicated due to the multiple ions in the electrolytes employed in these applications and the higher concentrations desired for energy density. Furthermore, while there is understanding on the single-ion transport in these materials, there is a need to develop a self-consistent theory for describing the ion transport and ion partitioning and thermodynamics when interacting with the charged polymer membrane. Moreover, there are few quantitative material design guides for polymers that balance the movement of certain species that are desirable, such as current-carrying ions, against transport of contaminants, additives, or redox-active species, which decrease device performance. To address the above gaps and provide material design insights, this work describes the theory of multi-ion transport in polymer-electrolyte membranes.
To specify the thermodynamics of the system, we formulate the membrane and electrolyte solution free energies. The model accounts for long-range electrostatic and short-range solvation and physical interactions between ions. Externally validated solution and membrane properties parameterize the model. The model calculations agree with measured water and ion uptake in dilute and concentrated binary and ternary salt electrolytes, with the uptake being a balance between osmotic and elastic pressures that are impacted by ion specificity and membrane pretreatment.1-2

To model the transport phenomena, we use Stefan-Maxwell-Onsager theory. To connect performance to the molecular-level structure of the polymer, hydrodynamic theory provides constitutive relations for the involved ion/solvent and ion/ and solvent/polymer friction coefficients, with classical porous-media theories scaling tortuosity. The model compares favorably to a number of published measured membrane transport properties1-2 (i.e. conductivity, transference numbers, electroosmosis, and water-transport coefficient) in dilute and concentrated binary and ternary electrolytes. Importantly, the model isolates the contribution of each type of molecular-scale to the measured properties.

Acknowledgements:

This study was funded with support of the Fuel Cell Performance and Durability Consortium (FC-PAD) funded by the Energy Efficiency and Renewable Energy, Fuel Cell Technologies Office, of the U.S. Department of Energy under contract number DE-AC02- 05CH11231 and, in part, by the Advanced Research Projects Agency - Energy (ARPA-E), U.S. Department of Energy (DOE) under Award Number DEAR0000149 .

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