Electrochemical Impedance Spectroscopy EIS Ref

In contrast to the EIS method, the Tafel-extrapolation, Tafel-curve-modeling and polarization-resistance methods are conducted under essentially dc conditions. In these cases, in generating the appropriate Eexp versus log iex or iex curve, the potentiodynamic potential scan rate is very slow, or the time between potentiostatic potential steps is very long. The common practice is a potential scan rate of 600 mV/h or an equivalent step rate of 50 mV every 5 min. Underthese conditions, it is assumed that a steady-state, extemal-current-density results at every discrete potential. Consequently, every element in the electrical circuit is purely resistive in nature, and therefore, the applied potential and resultant extemal-current-density are exactly in phase. Since the impedance (normalized with respect to specimen area) is dEexp/diex, under these conditions, the impedance, Z, at Ecorr is given by (see Eq 6.29)  

Some Basic Relationships in ac Circuit Analysis. Consider an ac voltage, V, applied to the circuit shown in Fig. 6.14(a), which consists of a resistor and a capacitor in parallel. For the resistor, the voltage, V, and current, Il3 as a function of time, t(s), at a given frequency, (O radi-ans/s), are illustrated in Fig. 6.14(b) and are given by the equations  

The current and voltage are exactly in phase, and therefore, the phase angle of the current relative to the voltage, 0, is zero. Next, consider the capacitor, where the same ac voltage, V, is applied at the same frequency, cOj. The capacitance, C, is given by  

I2 vectors in Fig. 6.15(a) to obtain Ir, then recognizing that the vertical component of Ir is the value of Ir at tj that is, Ir = Ir max sin (ajjtj + 0r), where 9r is the phase angle between the resultant current and the voltage. It is noted that the vector addition easily provides both the maximum value and the phase angle for the resultant current these quantities are all that are needed to fully describe the resultant current. 

90° out of phase with V. For example, consider an arbitrary current vector, I, with a phase angle, 0, relative to V. The real component is Imax cos 0, the imaginary component is Imax sin 0, and the complex-notation description is I = Imax (cos 0 + j sin 0), where j = V-T. Therefore, with reference to Fig. 6.16, the vectors V, I1 and I2 may be expressed as  

---