I am confused about the equation used for calculating the proton conductivity from the impedance data
The first equation is
Sigma = d / (A* RMBR)
where d = the sample thickness, A = membrane area and RMBR= The high-frequency intercept with real impedance axis in Nyquist plot
The previous equation that I had known from the literature. But then I have faced another equation also in the literature
the second equation:
Sigma = L / (W*T*R)
Where: L is the length between the voltage measuring probes, R is the measured resistance, W is the membrane width, and T is the membrane thickness.
As you noticed the two equations are extremely different and they are both mentioned in references
So I am confused about which equation I can use for my calculations or what are the differences of application of these two equations.
The reference for the 1st equation is
Lvovich, V. F. (2012). Selected Examples of Impedance-Analysis Applications Impedance Spectroscopy (pp. 281-318): John Wiley & Sons, Inc.
The reference for the 2nd equation is
Yuan, X.-Z., Song, C., Wang, H., & Zhang, J. (2010). EIS Diagnosis for PEM Fuel Cell Performance Electrochemical Impedance Spectroscopy in PEM Fuel Cells: Fundamentals and Applications (pp. 193-262). London: Springer London.
I did not check in literature (your task) and it is just a guess, but it might have something to do with in-plane and through-plane conductivity of membranes. Equation one seems to be the setup in a fuel cell test bench and might be through-plane. Equation two seems to be an in-plane measurement with a 4 electrode system. If conductivity would be isotropic in space there would be no difference, but this is usually not the case with membranes.
Kind regards
Peter
Start with the 1st method (equation, in your representation). I think that, it is more simple, for your case[1]. You can verify the method, if you form an explanatory/diagnostic plot[2] : Sigma versus 1/N.
Also, you can verify the (1st method), if you form an alternative diagnostic plot[3] : Sigma versus 1/A.
1. Proton conductivity percolation laws (and critical exponents) in membranes: a) Percolation theory http://www.wow.com/wiki/Percolation_theory b) Proton conduction in crystalline and porous covalent organic frameworks http://www.nature.com/nmat/journal/vaop/ncurrent/full/nmat4611.html?foxtrotcallback=true c) Proton conducting ceramic membranes for hydrogen separation US 8012380 B2 https://www.google.com/patents/US8012380
2. Check (and verify) that Sigma scales with 1/N, where N is the Number of the (same) membranes, in series.
3. Where A is the adjacent membrane Area, in Sigma = d / (A* R). As an example, you might smoothly modify A (Area), by filling more and more electrolyte in your (regularly horizontal) cell, having a vertical membrane, etc.
How you carried out your measurement, in-plane or through-plane?
Please, consider any of these literature to answer your question. I did many in-plane (membrane as single material) and through-plane (in the fuel cell) measurements during my Ph.D:
1) A novel titanium PBI-based composite membrane for high temperature PEMFCs. Journal of Membrane Science, Volume 369, Issues 1–2, 1 March 2011, Pages 105-111
2) Enhancement of the fuel cell performance of a high temperature proton exchange membrane fuel cell running with titanium composite polybenzimidazole-based membranes. Volume 196, Issue 20, 15 October 2011, Pages 8265-8271.
3) Titanium composite PBI-based membranes for high temperature polymer electrolyte membrane fuel cells. Effect on titanium dioxide amount. RSC Advances, vol. 2, issue 4 (2012)
4) Promising TiOSO4 composite polybenzimidazole-based membranes for high temperature PEMFCs. ChemSusChem, vol. 4, issue 10 (2011)
Hope that helps you!
The first equation is in agreement with the definition of a material conductivity https://en.wikipedia.org/wiki/Electrical_resistivity_and_conductivity . The resistance of a material should increase as the material length increase and decease when the cross-section area of the material increase. The second equation may be the same if W and T defines the cross-section (cross-section is perpendicular to the current path). Coming back to the first equation, keep in mind that the high frequency intercept of the real axis also contains the ohmic resistance of the plates, electrolyte and connections not only the membrane resistance. The assumption should be that all these other resistances are negligible regarding to the membrane resistance. A 4 point connection is mandatory with the sensing wires directly connected to the fuel cell plates. Otherwise you will measure extra resistance coming from the wiring.
- Badges
- Science topic
- Similar topics
- Physical Sciences
- Materials Science
- Materials Chemistry
- Polymer Chemistry