Mechanics of Composite Materials

, Volume 50, Issue 6, pp 763–776 | Cite as

A Three-Scale Model of Basic Mechanical Properties of Nafion

  • V. Kafka
  • D. Vokoun

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.


Nafion mechanical properties mesomechanics material structure hydration 


General mechanics


stress tensor


isotropic part of σ ij (σ = σ ii /3)


deviatoric part of σ ij (σ ij − δ ij σ)


strain tensor


isotropic part of ε ij (ε = ε ii /3)


deviatoric part of ε ij (ε ij δ ij ε)


Kronecker’s delta


Young’s modulus


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


Einstein’s notation

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


isotropic part of ε ij


deviatoric part of ε ij


stress related to ε ij similarly as σ ij is related to ε ij


isotropic part of σ ij


deviatoric part of σ ij


volume fraction of the elastic {inelastic} constituent in the general two-phase model


volume fraction of the amorphous Nafion {of water} in the aggregate of amorphous Nafion with water


volume fraction of the ω - constituent in the total Nafion (ω = a, c, w, wa);

\( {R}_a^c=\frac{V_c}{V_a}; \)


elastic limit of s 11 a in the a -constituent


elastic limit of s 11 wa in the wa -constituent


structural parameter (ω = a, c, w, wa)


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


value of | at the elastic limit.



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.


  1. 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. 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. 3.
    F. Opekar and K. Stulik, “Electrochemical sensors with polymer electrolytes,” Analytica Chimica Acta, 385, 151-162, (1999).CrossRefGoogle Scholar
  4. 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. 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. 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. 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. 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. 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. 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. 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. 12.
    G. Gebel, “Structural evolution of water-swollen perfluorosulfonated ionomers from dry membrane to solution,” Polymer, 41, 5829-5838, (2000).CrossRefGoogle Scholar
  13. 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. 14.
    A. Kusoglu, A. M. Karlsson, and M. H. Santare, “Structure–property relationship in ionomer membranes,” Polymer, 51, 1457-1464, (2010).CrossRefGoogle Scholar
  15. 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. 16.
    V. Freger, “Hydration of ionomers and Schroeder’s paradox in Nafion,” J. Phys Chem. B, 113, 24-36, (2009).CrossRefGoogle Scholar
  17. 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. 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. 19.
    K. J. Kim and M. Shahinpoor, “Ionic polymer-metal composites: II. Manufacturing techniques,” Smart Mater. Struct., 12, 65-79, (2003).CrossRefGoogle Scholar
  20. 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. 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. 22.
    S. Nemat-Nasser, “Micromechanics of actuation of ionic polymer-metal composites”, J. of Applied Physics, 92, 2899-2915, (2002).CrossRefGoogle Scholar
  23. 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. 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. 25.
    V. Kafka, Mesomechanical Constitutive Modeling, World Scientific, Singapore (2001).Google Scholar
  26. 26.
    A. Eisenberg and J. S. Kim, Introduction to Ionomers, Wiley, New York (1998).Google Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Institute of Theoretical and Applied Mechanics ASCRPragueCzech Republic
  2. 2.Institute of Physics ASCRPragueCzech Republic

Personalised recommendations