How Clark Oxygen Sensor works: Mathematical Model and simulations

Key Concepts

• Oxygen Diffusion: Describes mass transport from the membrane to the electrode and determines the sensor current.
• Chamber Geometry: Describes the diffusion path length and influences both current magnitude and response time.
• Membrane Permeability: Determines how efficiently oxygen enters the sensor through the membrane.

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 shows how the current signal changes as the oxygen concentration increases. During the first time domain, the current increases until steady-state conditions are reached. When oxygen is subsequently removed, the current decreases and becomes almost zero.

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

The steady state is achieved when the rate of oxygen consumption at the electrode interface equals the rate at which oxygen diffuses through the membrane towards the electrode surface. Animation 2 illustrates this behaviour: a concentration gradient is formed from the membrane towards the electrode, and a diffusion layer is formed around the electrode. Due to the small size of the electrode, radial diffusion is important in the mass transport process.

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

Once the oxygen concentration in the external solution becomes zero, oxygen diffuses from the chamber thought the membrane until oxygen concentration is almost null. Hence the oxygen consumption depends on several physical and geometrical parameters:

• Membrane thickness (δ).
• Membrane material, through its permeability coefficient (k_m).
• Distance between the electrode and the membrane
• Electrode size, since smaller electrodes enhance radial diffusion

Membrane Permeability

Animation 3 shows how the sensor current changes with membrane permeability. As the membrane permeability increases, the steady-state current also increases due to the enhanced transport of oxygen through the membrane.

Animation 3. Effect of membrane permeability, k_m, on the amperometric current of the Clark Sensor.

At the same time, the sensor response time, defined as the time required to reach 90% of the steady-state current (t₉₀), decreases exponentially as the membrane permeability coefficient (k_m) increases (Figure 3). This behaviour reflects how critical is the selection of membrane material.

Figure 3. Dependence of the sensor response time (t₉₀) on the membrane permeability, k_m.

Distance between Electrode and Membrane

Animation 4 shows that the current decreases significantly as the distance between the membrane and the electrode increases. A larger separation makes the diffusion path longer within the chamber. Consequently, the flux of oxygen is reduced towards the electrode surface.

Animation 4. Effect of the distance between the electrode and the chamber , L, on the amperometric current of the Clark Sensor.

In addition, the sensor response time (t₉₀) increases as the electrode –membrane distance becomes larger (Figure 4). As mentioned previously, this is due to the reduction of oxygen flow within the chamber. Hence, it is desired to diminish this distance as much as possible, so sensor response becomes faster and enhanced.

Figure 3. Dependence of the sensor response time (t₉₀) on the distance between the electrode and the membrane, L.

Clark Sensor Simulator

This interactive app simulates the dynamic response of a Clark oxygen sensor. Explore how key parameters affect the current signal and optimize the sensor performance by adjusting:

• Gap length (\(L\)): Distance between the electrode and the membrane.
• Membrane Thickness (\(δ\)): Select how thick is the membrane in micrometres.
• Catalytic rate constant (\(k_3\)): Convertion of glucose to gluconolactone; Logarithmic scale.
• Membrane Permeability: Select between different polymer materials (PDMS, PE, and PTFE), each with distinct oxygen permeability.
• Global charge transfer rate (\(k\)): Control the rate at which the oxygen is reduced. Parameter in logarithmic scale.
• Oxygen Concentration: Bulk concentration of oxygen in the sample.
• Oxygen concentration gradient: Visualize the oxygen concentration profile at the desired time in seconds.

By adjusting these parameters, you can gain deeper understanding of how the oxygen flow is within the sensor chamber. In addition, plots can be exported as CSV files for further analysis.

If you want to cite this blog post, use:
Robayo, I. (2026). The Clark Oxygen Sensor Through Mathematical Modelling. Available at: https://electrochemeisbasics.blogspot.com/2026/03/the-clark-oxygen-sensor-through.html [Accessed Date Accessed].