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theory:electrochemistry:electrochemistry7_electrolyte_conductivity

# Methods for Electrolyte Conductivity

## A cell filled with electrolyte

The measurement of the the electrolyte conductivity can be done by placing the liquid between two capacitor plates. Due to the conductive elements in the electrolyte, there is a resistive element placed in the capacitor. The electrical equivalent circuit is given in figure 1. Fig. 1: Electric equivalent circuit for a conductivity cell

We are interested in the value of the resistance $R_{Cel}$ which represents the total ion concentration and is related to the conductivity $\kappa_{sol}$ by means of the cell-constant $K_{Cell}$:

\begin{equation} R_{Cell} = \frac{K_{Cell}}{\kappa_{sol}} \end{equation}

which is a geometrical constant only. For a parallel plate electrode setup the cell constant can be calculated easily:

\begin{equation} K_{Cell} = \frac{d}{A} \end{equation}

with $d$ the distance bewteen the plates and $A$ the surface area size of the plates.

The capacitors in figure 1 represent the interfering effects. The perallel capacitance $C_{Cell}$ is the result of the direct AC-coupling between the electrodes an is here equal to

\begin{equation} C_{Cell} = \frac{\epsilon A}{d} = \frac{\epsilon}{K_{Cell}} \end{equation}

with $\epsilon$ the dielectric constant of the electrolyte. The series capacitors are interface polarization effects of the electrode-electrolyte surfaces that can be simplified to

\begin{equation} C_{interface} = A \cdot C_{dl} \end{equation}

with again $A$ the surface area size of the electrode and $C_{interface}$ the double-layer capacitance as described before.

## The planar interdigitated finger electrode

As an example of a practical implementationn of a conductivity cell, a planar construction will be calculated. Alternative set-ups which are commercially used are round sticks with two or four metal rings around them. What is needed for modelling such a configuration is just the cell constant and the surface area, because then all modelleing components can be calculated. The surface area is normally not so difficult to calculate, but the cell constant may involve some mathematical complexity. Fig. 2: Planar interdigitated finger structure

The cellconstant can be found by means of a conformal mapping transformation. A two dimensional evaluation is in most cases sufficient. Two dimensional conformal mapping is described in books and readers about complex function theory and in specific papers about calculating cell constants1). A conformal mapping transformation is transforming a space in such a way that the Maxwell equations remain valid. As a result when we can transform the electrical fieldlines conform Maxwell, we can transform the electrode geometry accordingly. In that case, we can transform a known configuration (for example the parallele plate capacitor) to the complex geometry of interest and transform the value of the cell constant as well.

For the interdigitated finger electrode we find

\begin{equation} K_{Cell-finger} = \frac{2}{\left ( N-1 \right ) L} \cdot \frac{K \left ( k_{1} \right )}{K \left ( k_{2} \right )} \end{equation}

with

\begin{equation} K \left ( k \right ) = \int_{0}^{1} \frac{1}{ \sqrt{\left ( 1-t^{2} \right ) \left ( 1-k^{2}t^{2} \right ) } } dt\\ k_{1}=\cos \left( \frac{\pi}{2} \cdot \frac{w}{s+w}\right )\\ k_{2}=\cos \left( \frac{\pi}{2} \cdot \frac{s}{s+w}\right )\\ \end{equation}

and $S$ the spacing between the fingers, $W$ the width of the fingers, $L$ the length of the fingers and $N$ the number of fingers. To choose these geometries ($S$, $W$, $L$ and $N$) we have to optimse in such a way that the effect of the paracitic capacitances in the frequency range of interest are as small as possible.

## Impedance measurements

The simulation of figure 3 is easily made from the electric equivalent circuit of figure 1. The numerical values $S = 4 \mu m$, $W = 200 \mu m$, $N = 5$ and $L = 1 mm$ were taken.  Fig. 3: Simulation of the spectrum of a conductivity cell - Bode plot and polar plot

In the first graph we can see the modulus as a function of frequency. The working region is easily recognized: it is the ferquency range in which the modulus is only dependent on concentration. The lower boundary of the sensitive region is about $10 kHz$ and is determined by the interface capacitances and the electrolyte conductivity. The upper boundary is the result of the cell-capacitance.

In the second graph, the data is plotted as a polar plot (imaginary part and real part as a parametric plot with the frequency as the independent parameter). In such a plot, the RC-couples are easily recognized as semi-circles. Polar plots can be used to identify electrode phenomena. This method is referred to as impedance spectrostopy. A real measurement picture conform figure 3 can be made with a gain-phase analyser, for example the HP4184.

When the user is only interested in a single frequency, a Phase-Locked-Loop (PLL) can be used. In its simples form, the PLL consists of an oscillator with a DC voltage dependent frequency, the Voltage-Controlled-Oscillator (VCO), and a phase detector, which can be a simple XOR port. The XOR port compares the phase of the signal of interest with the PLL internal phase, and adjust until these are equal. Fig. 4: Principle of a Phase-Locked-Loop

The signals do not have to be sinewave-shaped, there are complete integrated circuits that have VCO, phase detector and filters in a single TTL component (LM565).

# Electrochemistry TOC

1)
P. Jacobs, A. Varlan, W. Sansen, Design optimisation of planar electrolytic conductivity sensors, Medical & Biological Engineering & Computing, November 1995 