# A Three-Scale Model of Basic Mechanical Properties of Nafion

- 109 Downloads

The mechanical properties of Nafion are explained and modeled on the basis of Kafka’s general mesomechanical model and confronted with experimental results. In this approach, Nafion is looked upon as a composite consisting of three constituents: a crystalline Nafion, amorphous Nafion, and water. Taking into account the degree of hydration, its elastic, elastic-plastic, and hysteretic properties are discussed and modeled. It is shown how the interaction between the three constituents manifests itself on the macroscale.

## Keywords

Nafion mechanical properties mesomechanics material structure hydration## Nomenclature

## General mechanics

*σ*_{ij}stress tensor

*δ*_{ij}*σ*isotropic part of

*σ*_{ ij }(*σ*= σ_{ ii }*/*3)*s*_{ij}deviatoric part of

*σ*_{ ij }(*σ*_{ ij }− δ_{ ij }*σ*)*ε*_{ij}strain tensor

- δ
_{ij}*ε* isotropic part of

*ε*_{ ij }(*ε*=*ε*_{ ii }*/*3)*e*_{ij}deviatoric part of

*ε*_{ ij }(*ε*_{ ij }−*δ*_{ ij }*ε*)*δ*_{ij}Kronecker’s delta

*E*Young’s modulus

- ν
Poisson’s ratio

*μ*= (1+ν)/*E*deviatoric elastic compliance

*ρ*= (1 − 2ν) /*E*isotropic elastic compliance.

## Specific symbols

- \( \overline{\Big|} \)
overbar that relates the symbol | to its macroscopic value — the average in the representative volume element RVE

- |
^{ω}or |_{ω} index that relates the symbol | to the

*ω*-constituent — the average in the subvolume of RVE that is filled in by the*ω*-constituent*ω*=*e*elastic constituent in the general two-phase model

*ω*=*n*inelastic constituent in the general two-phase model

*ω*=*a*amorphous constituent in Nafion

*ω*=*c*crystalline constituent in Nafion

*ω*=*w*water comprised in Nafion

*ω*=*wa*aggregate of two constituents in Nafion: of the amorphous Nafion with water

- |
_{ij} Einstein’s notation

\( {\varepsilon}_{ij}^{\prime }={\varepsilon}_{ij}-{\overline{\varepsilon}}_{ij}; \)

*δ*_{ij}*ε*′isotropic part of

*ε*_{ ij }^{′}*e*_{ij}^{′}deviatoric part of

*ε*_{ ij }^{′}*σ*_{ij}^{′}stress related to

*ε*_{ ij }^{′}similarly as*σ*_{ ij }is related to*ε*_{ ij }*δ*_{ij}*σ*′isotropic part of

*σ*_{ ij }^{′}*s*_{ij}^{′}deviatoric part of

*σ*_{ ij }^{′}*ν*_{e}{*ν*_{n}}volume fraction of the elastic {inelastic} constituent in the general two-phase model

*ν*_{a}{*ν*_{w}}volume fraction of the amorphous Nafion {of water} in the aggregate of amorphous Nafion with water

*V*_{ω}volume fraction of the

*ω*- constituent in the total Nafion (*ω*=*a*,*c*,*w*,*wa*);\( {R}_a^c=\frac{V_c}{V_a}; \)

*c*_{a}elastic limit of

*s*_{11}^{ a }in the*a*-constituent*C*_{wa}elastic limit of

*s*_{11}^{ wa }in the*wa*-constituent*n*_{ω}structural parameter (

*ω*=*a*,*c*,*w*,*wa*)- |
^{Ω} superscript that relates the symbol | to the respective Ω -specimen of Nafion

- Ω =
*H* hydrated specimen

- Ω =
*D* dry specimen

*p*=*V*_{ c }^{ H }*η*_{ c }*μ*_{ c }+*V*_{ wa }^{ H }*η*_{ wa }*μ*_{ wa }^{ H }\( q=\frac{\mu_c}{V_c^H}\left(p\left|+{\eta}_c{\eta}_{wa}{\mu}_{wa}^H\right.\right) \).

- \( \overset{\cdot }{h} \)
formal variable equal to 0 in elasticity and to \( \overset{\cdot }{\lambda } \) in plasticity

- |
_{L} value of | at the elastic limit.

## Notes

### Acknowledgements

This work was supported by the Czech Science Foundation within projects P108/10/1296 and 103/09/2101. Acknowledged is also the support through the Institutional Project RVO: 68378297. D. Vokoun would like to thank Dr. M. Paidar for fruitful discussions.

## References

- 1.K. Schmidt-Rohr and Q. Chen, “Parallel cylindrical water nanochannels in Nafion fuel-cell membranes,” Nature Mater.,
**7**, 75-83 (2008).CrossRefGoogle Scholar - 2.R. Knake, P. Jacquinot, A. W. E. Hodgson, and P. C. Hauser, “Amperometric sensing in the gas phase,” Analytica Chimica Acta,
**549**, 1-9, (2005).CrossRefGoogle Scholar - 3.F. Opekar and K. Stulik, “Electrochemical sensors with polymer electrolytes,” Analytica Chimica Acta,
**385**, 151-162, (1999).CrossRefGoogle Scholar - 4.V. Mehta and J. S. Cooper, “Review and analysis of PEM fuel cell design and manufacturing,” J. of Power Sources,
**114**, 32-53, (2003).CrossRefGoogle Scholar - 5.V. Antonuccia, A. Di Blasi, V. Baglioa, R. Ornelasb, F. Matteuccib, J. Ledesma-Garciac, L. G. Arriagac, and A. S. Arico, “High-temperature operation of a composite membrane-based solid polymer electrolyte water electrolyser,” Electrochimica Acta,
**53**, 7350-7356, (2008).CrossRefGoogle Scholar - 6.A. A. Gronowski and H. L. Yeager, “Factors which affect the permselectivity of Nafion membranes in chloralkali electrolysis,” J. of the Electrochemical Soc.,
**138**, 2690-2697, (1991).CrossRefGoogle Scholar - 7.M. Shahinpoor, Y. Bar-Cohen, J. O. Simpson, and J. Smith, “Ionic polymer-metal composites (IPMCs) as biomimetic sensors, actuators and artificial muscles — a review,” Smart Mater. Struct.,
**7**, R15-R30, (1998).CrossRefGoogle Scholar - 8.J. Brufau-Penella, M. Puig-Vidal, P. Giannone, S. Graziani, and S. Strazzeri, “Characterization of the harvesting capabilities of an ionic polymer metal composite device,” Smart Mater. Struc.,
**17**, 015009, (2008).CrossRefGoogle Scholar - 9.Y. Tang, A. M. Karlsson, M. H. Santare, M. Gilbert, S. Cleghorn, and W. B. Johnson, “An experimental investigation of humidity and temperature effects on the mechanical properties of perfluorosulfonic acid membrane,” Mater. Sci. Eng., A,
**425**, 297-304, (2006).CrossRefGoogle Scholar - 10.M. B. Satterfield, P. W. Majsztrik, H. Ota, J. B. Benziger, and A. B. Bocarsly, “Mechanical properties of Nafion and titania/Nafion composite membranes for polymer electrolyte membrane fuel cells,” J. Polym. Sci., Part B, Polymer Physics,
**44**, 2327-2345, (2006).CrossRefGoogle Scholar - 11.M. N. Silberstein and M. C. Boyce, “Constitutive modeling of the rate-, temperature-, and hydration-dependent deformation response of Nafion to monotonic and cyclic loading,” J. of Power Sources,
**195**, 5692-5706, (2010).CrossRefGoogle Scholar - 12.G. Gebel, “Structural evolution of water-swollen perfluorosulfonated ionomers from dry membrane to solution,” Polymer,
**41**, 5829-5838, (2000).CrossRefGoogle Scholar - 13.D. Liu, S. Kyriakides, S. W. Case, J. J. Lesko, Y. Li, and J. E. McGrath, “Tensile behavior of Nafion and sulfonated poly(arylene ether sulfone) copolymer membranes and its morphological correlations,” J. Polym. Sci., Part B, Polymer Physics,
**44**, 1453-1465, (2006).CrossRefGoogle Scholar - 14.A. Kusoglu, A. M. Karlsson, and M. H. Santare, “Structure–property relationship in ionomer membranes,” Polymer,
**51**, 1457-1464, (2010).CrossRefGoogle Scholar - 15.Y. Qi and Y. H. Lai, “Mesoscale modeling of the influence of morphology on the mechanical properties of proton exchange membranes,” Polymer,
**52**, 201-210, (2011).CrossRefGoogle Scholar - 16.V. Freger, “Hydration of ionomers and Schroeder’s paradox in Nafion,” J. Phys Chem. B,
**113**, 24-36, (2009).CrossRefGoogle Scholar - 17.M. N. Silberstein, P. V. Pillai, and M. C. Boyce, “Biaxial elastic-viscoplastic behavior of Nafion membranes,” Polymer
**52**, 529-539, (2010).CrossRefGoogle Scholar - 18.M. N. Silberstein and M. C. Boyce, “Hygro-thermal mechanical behavior of Nafion during constrained swelling,” J. of Power Sources,
**196**, 3452-3460, (2011).CrossRefGoogle Scholar - 19.K. J. Kim and M. Shahinpoor, “Ionic polymer-metal composites: II. Manufacturing techniques,” Smart Mater. Struct.,
**12**, 65-79, (2003).CrossRefGoogle Scholar - 20.R. Tiwari and K. J. Kim, “Disc-shaped ionic polymer metal composites for use in mechano-electrical applications,” Smart Mater. Struct.,
**19**, 065016, (2010).CrossRefGoogle Scholar - 21.D. Pugal, K. J. Kim, A. Punning, H. Kasemagi, M. Kruusmaa, and A. Aabloo, “A self-oscillating ionic polymer-metal composite bending actuator,” J. of Appl. Phys.,
**103**, 084908, (2008).CrossRefGoogle Scholar - 22.S. Nemat-Nasser, “Micromechanics of actuation of ionic polymer-metal composites”, J. of Applied Physics,
**92**, 2899-2915, (2002).CrossRefGoogle Scholar - 23.S. Nemat–Nasser and S. Zamani, “Modeling of electrochemomechanical response of ionic polymer-metal composites with various solvents,” J. of Appl. Phys.,
**100**, 064310, (2006).CrossRefGoogle Scholar - 24.G. Alberti, R. Narducci, and M. Sganappa, “Effects of hydrothermal/thermal treatments on the water-uptake of Nafion membranes and relations with changes of conformation, counter-elastic force and tensile modulus of the matrix,” J. of Power Sources,
**178**, 575-583, (2008).CrossRefGoogle Scholar - 25.V. Kafka, Mesomechanical Constitutive Modeling, World Scientific, Singapore (2001).Google Scholar
- 26.A. Eisenberg and J. S. Kim, Introduction to Ionomers, Wiley, New York (1998).Google Scholar