1. Impedance Spectroscopy (Or, how a sinusoidally varying voltage is used to probe multiple electrical properties of materials) Yun-Ju (Alex) Lee June 13, 2002

2. Introduction IS measures the resistance of a circuit to an applied voltage Z(t) = E(t)/I(t), Z(  )=E(  )/I(  ) When E is applied as a sinusoidal function in a linear system, I response can be represented by a sum of sinusoidal fuctions with phase shifts. If an equivalent circuit for the system being probed can be constructed, then the resistance or capacitance values for each circuit element can be backed out from Z. Especially in electrochemical impendance spectroscopy (EIS), the the voltage probe signal is small (1-10mV) so the current response is pseudolinear (side benefit: it doesn’t perturb the sample very much) A Bard and L Faulkner, Electrochemical Methods: Fundamentals and Applications (2000) http://www.gamry.com/G2/Appnotes/Reference/eistheory/theory/EIS_Theory.htm

3. Background – Impedance and Circuit Elements E(t) = E 0 cos(  t),  =2  f I(t) = I 0 cos(  t-  ) Or, if one writes in complex notation: E(t) = E 0 exp(i  t) I(t) = I 0 exp(i  t - i  ) Z(t) = Z 0 exp(i  ) = Z 0 (cos  + isin  ) E/I response for a resistor (  =0) E/I response for a capacitor (  =-90) E/I response for an inductor (  =90) http://www.gamry.com/G2/Appnotes/Reference/eistheory/theory/EIS_Theory.htm http://www.bath.ac.uk/~chsacf/solartron/circp/html/acrci.htm

4. =-j X c = j X l http://www.bath.ac.uk/~chsacf/solartron/circp/html/acrci.htm R For more complex circuits, use Kirchoff’s rules

5. Plotting Impedance A Cole-Cole (or Nyquist) plot of the real and imaginary components of Z for a circuit containing a capacitor and resistor in parallel (real world example: a charged layer at an interface).  = 1/  = 1/RC Named after KS Cole and RH Cole, J Chem Phys 9, 341 (1941)

6. Plotting Impedance A Bode plot of a circuit containing a capacitor and resistor in parallel. log |Z| and  vs. log  are plotted, showing explicitly the impact of  impedance. http://www.gamry.com/G2/Appnotes/Reference/eistheory/theory/EIS_Theory.htm

7. Circuit Elements from Electrochemistry 1.Electrolyte resistance 2.Double layer capacitance (~30  F/cm 2 for aqueous systems) 3.Polarization resistance 4.Charge transfer resistance 5.Coating capacitance 6.Constant phase element (a fudge factor) 7.Virtual inductor (adsorption?) And… l A WE, RE, CE R = l /  A http://www.gamry.com/G2/Appnotes/Reference/eistheory/theory/EIS_Theory.htm

8. Diffusion (aka Warburg Impedance) Warburg impedance = diffusion of chemical species to a large planar electrode Infinite diffusion layer thickness Diffusion layer thickness  Also, in the Warburg region,  = 45º http://www.consultrsr.com/resources/eis/diffusion.htm

9. Equivalent Circuit Model for Electrochemical Cells Mixed Kinetic + Diffusion model Cole-Cole plot Bode plot http://www.gamry.com/G2/Appnotes/Reference/eistheory/theory/EIS_Theory.htm

10. Some Applications of IS Determine corrosion rates of materials* Measure capacitance and resistance of SAMs to ascertain thickness and packing quality Probe linearity of electrical/electrochemical reactions Characterize electrical activity across interfaces to determine carrier concentration, etc. Determine diffusion rate of counter ions in conducting polymer/carbon nanotube composites ASL Castela et al., Prog Org Coat 38, 1-7 (2000) S Flink et al., Adv Mater 12, 1315-1327 (2000) K Krischer et al., Angew Chem Int Ed 40, 850-869 (2001) GZ Chen et al., Adv Mater 12, 522 (2000)