Impedance Spectroscopy

Specimen conductivity can be most accurately determined using the principles of IS [24]. In an IS measurement, the specimen is subjected to an AC potential over a range of frequencies, and the phase (with respect to the applied potential) and amplitude of the current are measured at each frequency. In order to interpret the IS results, the impedance response of a specimen is approximated by an equivalent circuit. The components of the circuit have a physical correspondence to components of the specimen. For metal electrodes against a saturated porous material composed of an insulating solid framework and a pore space filled with electrolyte, the impedance response can be approximated by an equivalent circuit composed of resistors (R) and capacitors (C)  [24, 25]. A simple equivalent circuit for the RCT is shown in Fig. 1. The subscripts "E" and "B" represent the electrode and the bulk components, respectively. The term bulk represents the porous solid and electrolyte pore solution composite, and is interchangeable in meaning with specimen in this context. The resistor R_B in Fig. 1 represents the DC resistance of the RCT specimen. The capacitor C_B represents the specimen capacitance due to the electrolyte in the pore space. The electrode elements R_E and C_E represent the impedance response of the electrodes due to polarization and charging effects.

Figure 1: A simplified equivalent circuit used to model the electrode (E) and bulk (B) impedance of the RCT cell.

Figure 2: Impedance plane plot of the circuit in Fig. 1 showing the real (Z ' ) and negative imaginary (-Z " ) components of the impedance.

Figure 3: Typical impedance spectroscopy result from measurements made on the RCT setup. Solid circles represent data at decade frequencies.

The equivalent circuit in Fig. 1 is composed of parallel resistors and capacitors connected in series. The impedance of a resistor ( Z_R ) and a capacitor ( Z_C ) are complex quantities ($i=\sqrt{-1}$) that are parametrized by the AC angular frequency $\omega$ [26]:

$$Z_{\mbox{\tiny R}}= R \hspace{1.0in} Z_{\mbox{\tiny C}}= \frac{1}{i\omega C}$$ (1)

The complex nature of the impedance corresponds to a phase difference between the current and the voltage through these devices; a resistor, having a pure real impedance, does not contribute to a change in the phase.

The circuit in Fig. 1 can exhibit both a capacitive and resistive response. For some values of ω, the current and the voltage are nearly in phase. In this case, the impedance has no complex component, and the entire system behaves like a purely resistive element. These values of ω for which the system is purely resistive can be determined from the total impedance of the equivalent circuit in Fig. 1:

$$Z(\omega) \left(\frac{2R_{\mbox{\tiny E}}}{1+\left(\frac{w}{w_{\mbox{\tiny ... ...\mbox{\tiny B}}}\right)} {1+\left(\frac{w}{w_{\mbox{\tiny B}}}\right)^2}\right)$$ =

$$Z^{\prime}(\omega) + i Z^{\prime\prime}(\omega)$$ = (2)

The quantities Z' and Z" represent the real and imaginary components of Z, respectively. The constants $\omega_{\mbox{\tiny E}}$ _E and ω _B are equal to (R _E C _E ) ⁻¹ and (R _B C _B )⁻¹, respectively, and have the same dimensional units as ω. In a system like saturated concrete, the quantity ω_E may be several orders of magnitude smaller than ω _B ( $\omega_{\mbox{\tiny E}}<\!\!<\!\!<\omega_{\mbox{\tiny B}}$). Given this information, there are three ranges of values for ω that are of interest:

Z(w  → 0)	=	2RE + R B
Z(w  → ∞)	=	0
Z(ωE   ω   wB )	=	RB

The third relationship expresses mathematically the fact that for intermediate values of ω, orders of magnitude from either ω_E or $\omega_{\mbox{\tiny B}}$_B, the entire system becomes purely resistive; the difference between the applied voltage and the resultant current is negligible. Most importantly, this value of Z is equal to the bulk specimen resistance R_B that is used to calculate the specimen conductivity.

A schematic representation of Z( ω ) is shown in Fig. 2 for $\omega_{\mbox{\tiny E}}<\!\!<\!\!<\omega_{\mbox{\tiny B}}$. The figure is an impedance plane plot, typically referred to as a Nyquist plot, and is parametrized by ω, where ω  = 0 is at the right hand side of the curve, and ω  = ∞ is at the left. The values of ω at the maximum values of −Z" are shown. Experimentally, impedance analyzers can only produce a finite range of frequencies, and typically only the portion of the curve near Z ' = R_B is measured. For an estimate of bulk conductivity, this is all that is required. However, only under ideal conditions does the imaginary component of Z go to zero at Z ' = R_B. In practice, the response of a specimen is more like that shown in Fig. 3, where the bulk resistance R_B must be estimated from the value of Z ' at the minimum of −Z". The data collected for Fig. 3 consist of 10 data points per decade of frequency. The datum at each decade is shown as a filled circle along the curve. Therefore, the value of R_B in Fig. 3 was determined at a frequency between 10 kHz and 100 kHz.

Although a more complete equivalent circuit for the bulk and electrode response of the RCT cell would be more complicated than that shown in Fig. 1[27,28], this simple circuit captures the major behavior. However, there is an additional component of the bulk impedance that is not represented in Fig. 1. A schematic cross section of the RCT with sample and holders is shown in Fig. 4. Typically, between the specimen and each brass electrode, there is a 1-5 mm gap that is filled with aqueous electrolyte: either 3 % by mass NaCl or 0.3 mol/L NaOH.

Figure 4: A schematic cross section of the RCT cell showing the relative positions of the sample and the electrodes.

The contribution of this resistive component to the total resistance can be calculated from the solution conductivities found in published tables[29]: σ_NaC1  = 4.4 mS/mm and σ_NaOH  = 5.7 mS/mm. For the geometry of the RCT cell, the resistances per unit length are 0.029 Ω/mm and 0.022 Ω/mm for the NaCl and the NaOH solutions, respectively. Therefore, a gap of 10 mm between each electrode and the specimen contributes less than 1 Ω to the bulk resistance. Since the bulk resistance for concrete is typically in the range of 100 Ω to 1000 Ω, the contribution by the electrolyte between the electrodes and the sample can be neglected.