Basics of Electrochemical Impedance Spectroscopy（四）

We will calculate two examples to illustrate a point about combining circuit elements. Suppose we have a 1 Ω and a 4 Ω resistor in series. The impedance of a resistor is the same as its resistance (see Table 1). We thus calculate the total impedance as:

(12)

Resistance and impedance both go up when resistors are combined in series.

Now suppose that we connect two 2 μF capacitors in series. The total capacitance of the combined capacitors is 1 μF.

(13)

Impedance goes up, but capacitance goes down when capacitors are connected in series. This is a consequence of the inverse relationship between capacitance and impedance.

Physical Electrochemistry and Equivalent Circuit Elements

Electrolyte Resistance

Solution resistance is often a significant factor in the impedance of an electrochemical cell. A modern three electrode potentiostat compensates for the solution resistance between the counter and reference electrodes. However, any solution resistance between the reference electrode and the working electrode must be considered when you model your cell.

The resistance of an ionic solution depends on the ionic concentration, type of ions, temperature, and the geometry of the area in which current is carried. In a bounded area with area, A, and length, l, carrying a uniform current, the resistance is defined as,

(14)

ρ is the solution resistivity. The reciprocal of ρ (κ) is more commonly used. κ is called the conductivity of the solution and its relationship with solution resistance is:

(15)

Standard chemical handbooks will often list κ values for specific solutions. For other solutions, you can calculate κ from specific ion conductances. The units of κ is Siemens per meter (S/m). The Siemen is the reciprocal of the ohm, so 1 S = 1/ohm.

Unfortunately, most electrochemical cells do not have uniform current distribution through a definite electrolyte area. The major problem in calculating solution resistance therefore concerns determination of the current flow path and the geometry of the electrolyte that carries the current. A comprehensive discussion of the approaches used to calculate practical resistances from ionic conductances is well beyond the scope of this application note.

Fortunately, you usually don't calculate solution resistance from ionic conductances. Instead, you calculate it when you fit experimental EIS data to a model.

Double Layer Capacitance

An electrical double layer exists on the interface between an electrode and its surrounding electrolyte. This double layer is formed as ions from the solution adsorb onto the electrode surface. The charged electrode is separated from the charged ions by an insulating space, often on the order of angstroms.  Charges separated by an insulator form a capacitor so a bare metal immersed in an electrolyte will be have like a capacitor.  You can estimate that there will be 20 to 60 μF of capacitance for every 1 cm² of electrode area though the value of the double layer capacitance depends on many variables. Electrode potential, temperature, ionic concentrations, types of ions, oxide layers, electrode roughness, impurity adsorption, etc. are all factors.

Polarization Resistance 

Whenever the potential of an electrode is forced away from its value at open-circuit, that is referred to as “polarizing” the electrode. When an electrode is polarized, it can cause current to flow through electrochemical reactions that occur at the electrode surface. The amount of current is controlled by the kinetics of the reactions and the diffusion of reactants both towards and away from the electrode.

In cells where an electrode undergoes uniform corrosion at open circuit, the open circuit potential is controlled by the equilibrium between two different electrochemical reactions. One of the reactions generates cathodic current and the other generates anodic current. The open circuit potential equilibrates at the potential where the cathodic and anodic currents are equal. It is referred to as a mixed potential. If the electrode is actively corroding, the value of the current for either of the reactions is known as the corrosion current.

Mixed potential control also occurs in cells where the electrode is not corroding. While this section discusses corrosion reactions, modification of the terminology makes it applicable in non-corrosion cases as well as seen in the next section.

When there are two, simple, kinetically-controlled reactions occurring, the potential of the cell is related to the current by the following equation.

(16)

where,

• I = the measured cell current in amps,

• I_corr = the corrosion current in amps,

• E_oc = the open circuit potential in volts,

• β_a = the anodic Beta coefficient in volts/decade,

• β_c = the cathodic Beta coefficient in volts/decade.

If we apply a small signal approximation to equation 16, we get the following:

(17)

which introduces a new parameter, R_p, the polarization resistance. As you might guess from its name, the polarization resistance behaves like a resistor.

If the Beta coefficients, also known as Tafel constants, are known you can calculate the I_corr from R_p using equation 17. I_corr in turn can be used to calculate a corrosion rate.

We will discuss the R_p parameter in more detail when we discuss cell models.

Charge Transfer Resistance

A similar resistance is formed by a single, kinetically-controlled electrochemical reaction. In this case we do not have a mixed potential, but rather a single reaction at equilibrium.

Consider a metal substrate in contact with an electrolyte. The metal can electrolytically dissolve into the electrolyte, according to,

(18)

or more generally

 (19)

In the forward reaction in the first equation, electrons enter the metal and metal ions diffuse into the electrolyte. Charge is being transferred.

This charge transfer reaction has a certain speed. The speed depends on the kind of reaction, the temperature, the concentration of the reaction products and the potential.

The general relation between the potential and the current (which is directly related with the amount of electrons and so the charge transfer via Faradays law) is:

(20)

with,

When the concentration in the bulk is the same as at the electrode surface, C_O=C_O* and C_R=C_R*. This simplifies equation 20 into:

(21)

This equation is called the Butler-Volmer equation. It is applicable when the polarization depends only on the charge-transfer kinetics. Stirring the solution to minimize the diffusion layer thickness can help minimize concentration polarization.

When the overpotential, η, is very small and the electrochemical system is at equilibrium, the expression for the charge-transfer resistance changes to:

(22)