The Clark Oxygen Sensor Through Mathematical Modelling

Introduction

The most common way to measure oxygen dissolved by electrochemical means is by the so-called Clark electrode. This sensor consists of two electrodes: a platinum working electrode (anode) and a silver counter/reference electrode (cathode). Platinum is catalytic towards oxygen reduction:

\[ \text{O}_2 + 4e^- + 4H^+ \xrightarrow{k^0} \text{H}_2\text{O} \]

However, platinum is not selective to oxygen and can reduce any electroactive species at the platinum-electrolyte interface. To ensure selectively, the electrodes are confined in a chamber filled with electrolyte solution and separated from the sample solution by a permeable membrane to oxygen (figure 1).

Figure 1. Schematic illustration of a Clark oxygen sensor.

This membrane is typically made of materials such as polytetrafluoroethylene (PTFE, known as Teflon), Silicon, polyethylene, or fluorinated ethylene propylene (FEP). These polymers possess high oxygen permeability. The membrane thickness is usually in the micrometre range, typically between 10 and 50 µm.

For the counter electrode to function as a stable reference electrode, it is usually coated with a layer of silver chloride (AgCl) or silver bromide (AgBr). Therefore, the camber must be filled an electrolyte solution that contains the corresponding anion to maintain the equilibrium, most commonly potassium chloride (KCl) or potassium bromide (KBr).

Modelling the system

For modelling, only the internal chamber of the sensor was considered. A two-dimensional (2D) model was constructed in which the lower boundary corresponds to the electrode domain. The working electrode is centred along this boundary and has a diameter of 100 µm. The upper boundary represents the oxygen-permeable membrane, through which oxygen transport occurs exclusively by diffusion (Figure 2).

Figure 2. Schematic representation of the mathematical model of the Clark oxygen sensor.

The flux of oxygen the membrane is described by:

\[ J_{\text{O}_2}=\left(\frac{\partial \text{O}_2}{\partial x}\right)_{x=b} = k_1[\text{O}_2]_s \]

Where k₁ is defined as the division between the membrane permeability coefficient (k_m) and the membrane thickness (δ), i.e. k₁=k_m/δ. This boundary condition assumes that oxygen transport across the membrane is proportional to the oxygen concentration at the membrane-solution interface ([O₂])_s.

At the electrode boundary (x=0), oxygen is detected under chronoamperometric conditions. Since a constant potential is applied, it is assumed that the rate of reduction is proportional to the oxygen concentration close to the electrode-electrolyte interface. The corresponding boundary condition is therefore expressed as:

\[ J_{\text{O}_2}=\left(\frac{\partial \text{O}_2}{\partial x}\right)_{x=0} = -k_2[\text{O}_2]_{x=0} \]

Here k₂ represents the heterogenous rate constant for oxygen reduction at the electrode surface. The mass transport of oxygen within the chamber is described by second Fick's law:

\[ \frac{\partial \text{O}_2}{\partial t} = D_{\text{O}_2}\left(\frac{\partial^2 [\text{O}_2]}{\partial x^2}\right) \]

To illustrate the system response when oxygen is removed from the external solution, two time domains were considered: (i) during the first 300 seconds, oxygen is present in the solution, and (ii) during the subsequent 300 seconds, oxygen is removed from the solution, such that [O₂]_s=0.

Finally, if oxygen reduction involves the transfer of four electrons per molecule, the resulting current is given by:

\[ i = -4 F A \frac{d[O_2]}{dx} \]

where A denotes the electrode area and F is Faraday’s constant.

Modelling Results

Animation 1. Effect of Oxygen concentration on the amperometric current of the Clark Sensor.

Animation 2. Heat map illustrating the evolution of oxygen concentration within the Clark sensor chamber.