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**Modelling Diffusion Impedance in the Detection of Micron-sized Targets**

**Introduction**

The change in charge transfer resistance is the conventional indicator for analyte recognition in faradaic impedance biosensors, however, it is not sufficient in reflecting the influence of three-dimensional electrode geometry on the faradaic process. Therefore, an alternative model based on the diffusion impedance is to be developed. This chapter details the investigation of how the diffusion of electroactive molecules contributes to the change in electrode impedance upon analyte immobilisation. The effect of such process in biosensing applications has not been explored in research thus far. The model developed in this chapter would be used to characterise and study the influence of three-dimensional electrodes on faradaic mode impedance biosensors.

The model system selected for this purpose was a flat electrode in semi-infinite bulk medium. Micron-sized polystyrene (PS) bead was the target analyte of the model. Firstly, numerical simulation of the effects of surface immobilised beads on diffusion impedance is presented to provide a general understanding about the underlying process and impedance change. Subsequently, experimentally collected impedance data are analysed and compared to the simulation for model validation.

**Finite element simulation**

Theoretical investigation into the effects of blocked diffusion due to immobilised micro-particles was performed by analysing the change in impedance. This was done by solving Fick’s second law of diffusion to simulate the diffusion impedance in the presence of different densities of surface immobilised beads. The results were compared to the Warburg element, i.e. the diffusion impedance of bare electrode without any obstruction. A numerical approach was taken to solve for the diffusion impedance as the complex geometry involved in the model made it impossible for the solution to be obtained by analytical means. Finite element method (FEM) was selected for this purpose, which was useful for models with complex geometries such as the ones involved in this study.

COMSOL Multiphysics software package and its Electroanalysis module were used to facilitate the modelling process. The goal was to provide a qualitative understanding of how diffusion impedance changes in response to immobilised micro-particles on the electrode surface.

**FEM model setup**

**Calculating the diffusion impedance using FEM simulation**

FEM was used to calculate the diffusion impedance by solving Fick’s second law of diffusion (Equation 3.1). The numerical method solved for the concentration profile of chemical species k at user specified excitation frequencies. The excitation was setup as a harmonic voltage perturbation *v* of 10 mV amplitude, implemented as a boundary condition at the electrode surface.

∇ ∙ − _{k}∇ _{k}( , ) = ∇ ∙ _{k}( , )

(Equation 3.1)

*c*k: concentration of chemical species k (mol.m^{-3} ≡ mM)

*D*k: diffusion coefficient of chemical species k (m^{2}.s^{-1})

** J**k: molecule flux density vector of chemical species k (mol.m

^{-2}.s

^{-1})

**: position in model space**

*r**ω*: angular frequency of excitation signal (radians.s

^{-1})

^{ F }k

^{0}

^{(}

_{( ) =}

(Equation 3.2)

*J*k

^{0}: total flux of chemical species k at the electrode surface (mol.s

^{-1})

*i*: total current generated at the electrode boundary (A)

F: Faraday constant

*n*: number of electrons involved in the charge transfer reaction

νk: stoichiometric number of chemical species k in the redox reaction

( ) ()=

_{() }(Equation 3.3)

The calculated concentration profile allowed the molecular flux density

**k at any point in the model space to be determined from concentration gradient. The total flux through the electrode**

*J**J*k

^{0}could be obtained by integrating the flux through the electrode surface over the entire electrode area. This total flux was converted to an equivalent electrical current (Equation 3.2). Assuming there is only one redox reaction equilibrium involved in the system, the diffusion flux for one of the redox species only is needed to calculate the total current. This is because the simultaneous diffusion of all species (reactants and products) drives the transfer of

*n*electrons. This resulting current

*i*is the simulated current response upon the 10 mV amplitude sinusoidal voltage excitation. The impedance is therefore the ratio between the specified voltage excitation

*v*and the simulated current response

*i*(Equation 3.3).

**Model setup and parameters**

The electroactive chemical species considered in this study was the ferrocyanide/ferricyanide redox couple (n = 1, νR = -1, νO = 1). The diffusion coefficient for both molecules were approximated to 7.5×10^{-10} m^{2}.s^{-1} for simplicity [132]. The bulk concentration for both were specified as 5 mol.m^{-3} (equivalent of 5 mM). All instances of temperature values in equations or model parameters were approximated with a room temperature of 293.15 K (20˚C).

The target PS particle was modelled as a micron-sized sphere in the FEM simulation. The PS particles are dense and exclude most water, the electroactive molecules are not able to freely move through them. Therefore, the spheres were implemented as impenetrable to the movement of electroactive molecules. The spheres had a diameter of 0.7 μm.

The spherical particles were positioned evenly across the electrode surface as depicted in Figure 3.1A. The model assumed the condition of semi-infinite bulk volume above the electrode in the z direction. The ceiling boundary had fixed ferrocyanide and ferricyanide concentrations equal to their bulk concentrations to simulate the bulk solution. The lowest simulated frequency of excitation was 0.1 Hz, which corresponded to a characteristic distance of ~35 μm. Consequently, the modelled space extended 500 μm above the electrode surface to minimise any absorbing boundary effect imposed by the ceiling boundary. This was to uphold the assumption of semi-infinite diffusion.

Symmetry was exploited to simplify the modelled space. The diffusion pattern around each analyte particle was identical due to their regular distribution, therefore the simulation could be done for one unique section (unit cell) and the result extended to a greater area of the same particle coverage density. Figure 3.1B shows that there is no diffusion of electroactive molecules in the x and y directions at midpoints between each particle, as denoted by the dark blue colour in the figure. This means simulation could be performed as depicted in Figure 3.1C since the region represented a geometry that was repeated over the whole model. The newly created model boundaries (periodic boundaries) as a result of simplifying the model from Figure 3.1A to Figure 3.1C would be positioned at these planes that had no diffusion in the x or y directions. Therefore, the new boundaries could be implemented as blocking boundaries without changing the diffusion pattern after model reduction. The rotational symmetry around each particle allowed for further simplification down to a quadrant as shown in Figure 3.1D. The simplification significantly reduced the size of the model space, which allowed for more accurate solutions to be obtained with lower degrees of freedom and a shorter simulation time.

Simulation of diffusion impedance was performed with different percentages of particle coverage on the electrode surface (*X*%). The diffusion impedance for a specific coverage is symbolised as *Z*d *X*%. Coverage was defined as the ratio between the maximum horizontal cross-sectional area of the spherical analyte and the electrode area as depicted in Figure 3.1C. The model space dimension *L* was varied according to the percentage coverage modelled. A higher percentage coverage was equivalent to a higher density of bound particles.

The simulation also required configuration of parameters that defined the relationship between electrode potential and the rate of charge transfer. Since an arbitrary finite electron transfer rate constant (7×10^{-5} m/s) was used, this created a charge transfer resistance component that was included within the final simulation output. As the charge transfer resistance was not a part of this investigation, its value (7.6043×10^{-4} Ωm^{2}) was determined from the arbitrary reaction coefficient using Equation 2.9 and subtracted from the simulation output.

**Model verification**

The accuracy of finite element models is strongly influenced by its mesh [122]. Therefore, mesh refinement is of critical importance for obtaining reliable simulation results. Both h-refinement and p-refinement were performed. H-refinement involves improving solution accuracy by using finer mesh, while p-refinement uses higher order polynomials to more fittingly approximate the solution. Firstly, h-refinement study was performed with linear tetrahedral elements by halving the size of the mesh elements at the electrode surface until the changes in simulated impedance values were less than 0.5% at all frequencies between refinement steps. Next, p-refinement was attempted using quadratic tetrahedral elements. However, the simulation using higher order tetrahedral elements did not change the simulated result very much. Therefore, the final mesh setup after mesh refinement study consisted of linear tetrahedral elements of smaller than 10 nm in size on the electrode surface and grew at a rate of 1.03 towards the bulk solution.

The FEM generated numerical solutions were verified against the analytical solutions for semi-infinite planar diffusion pattern (Warburg element) and radial diffusion patterns (cylindrical and spherical [105]) to ensure the implemented model was accurate. The deviations between the numerical solutions and the analytical equations were less than 0.5% at all simulated frequencies.

**FEM simulation results**

*FEM simulation of diffusion impedance with micro-particle obstruction*

The simulated diffusion impedance *Z*d *X*% at three different percentages of electrode coverage (*X*%) are plotted with the diffusion impedance of the bare electrode (Warburg element) in Figure 3.2. The plot shows that the diffusion impedance does not change significantly at high frequencies (bottom left of the Nyquist plot), but shows a general increase in the real component with higher particle coverage at lower excitation frequencies.

The differences in the real (Δ*Z*d *X*%’) and imaginary (Δ*Z*d *X*% ») components of the diffusion impedance with particle obstructions from those of the Warburg element (*Z*Warburg’ and *Z*Warburg ») are shown separately in Figure 3.3. The differences are calculated according to Equation 3.4. The data are plotted for frequencies below 100 Hz, as this is the typical frequency range where the diffusion impedance is observable in experimentally measured impedance spectra. The real component appears to have a relatively even increase in magnitude within this frequency range for each percentage of coverage. In contrast, the increase in the magnitude of the imaginary component shows a positive correlation with frequency. The real component demonstrates greater difference than the imaginary component in general, however, the change in the imaginary component becomes more significant towards higher frequencies.

∆ _{d}^{′} _{ %}( ) = _{d}^{′} _{ %}( ) − _{Warburg}^{′}( )

∆ d » %( ) = d » %( ) − Warburg » ( )

(Equation 3.4)

Figure 3.4 shows the percentage change in the magnitude of the diffusion impedance in response to different percentages of particle coverage. The maximum relative change in the magnitude as seen from the plot is around 500 Hz. This frequency equates to a characteristic distance of around 0.49 μm, which is comparable to the size of the modelled PS bead with diameter of 0.7 μm. This is because a greater proportion of the diffusion pattern is obstructed when the diffusion distance matches with the dimensions of the analyte particle. However, diffusion impedance is normally unobservable at such high frequencies due to the dominating non-faradaic current path under such condition. Therefore, a more sensitive sensor signal in response to the presence of immobilised analyte is expected to be obtainable from analysing the electrode impedance at the highest frequency where the diffusion impedance can be observed reliably.

Figure 3.5A illustrates the diffusion pattern in the presence of particle obstructions at low excitation frequency. The long diffusion distance at low frequency means the majority of the diffusion occurs above the immobilised beads and are undisturbed. The increase in the magnitude of the real component dominates over the imaginary component in this situation. This increase in the real component of the diffusion impedance is analogous to the increase in charge transfer resistance used conventionally to interpret the data for faradaic mode impedance biosensors. The two models appear especially similar when the high frequency part of the diffusion impedance is unobservable due to the parallel non-faradaic process dominating in the sensor readout. Therefore, either the increase in charge transfer resistance or the change in diffusion impedance may be used to fit the same experiment impedance spectrum adequately when the higher frequency section of the faradaic impedance is obscured by the non-faradaic component.

In contrast, the shorter diffusion distance at higher excitation frequency means the majority of the diffusion occurs at a similar scale to the size of the analyte (Figure 3.5B). The diffusion impedance at this higher frequency range shows the more intricate changes, which reflect the disturbed diffusion pattern due to the presence of PS beads. The change in diffusion distance is no longer dominated by a simple increase in the real component at these higher frequencies, implying that the conventional model of increase in charge transfer resistance alone cannot be used to fully replace the diffusion impedance change suggested by the simulation. In reality, both components may contribute to the overall change in the electrode impedance in the presence of surface bound micro-particles.

The observations from the theoretical simulated diffusion impedance in response to micron-sized particle obstructions are compared to experimentally acquired impedance data for model validation in the next section.

**Experimental impedance measurement in the presence of ****polystyrene beads**

**Experiment procedure**

**Experiment design**

Sensing experiment was performed to validate the numerical model presented in the previous section. The gold working electrode was fabricated by sputter coating a gold layer onto flat polyethylene terephthalate glycol-modified (PETG) substrates using Nano36 sputtering system (Kurt J. Lesker Company). A fluid compartment was created above the working electrode by clamping down a Teflon well to the gold surface and sealed by a rubber O-ring in between as shown in Figure 3.6. The rubber O-ring had an inner diameter of 9 mm, which gave a circular working electrode with an area of ~64 mm^{2}. All electrochemical experiments were performed in the three-electrode configuration, involving the gold working electrode, a Ag/AgCl reference electrode (eDAQ), and a platinum counter electrode (eDAQ). All reported working electrode potentials are relative to the Ag/AgCl reference electrode.

Since the faradaic component was the focus of this study, a relatively faster faradaic process would allow the diffusion impedance component to be extracted from the total electrode impedance unambiguously up to higher frequencies. This would improve the confidence of the analysis on the diffusion impedance. Polypyrrole conducting polymer was chosen as the functional layer due to its high charge transfer rate across the surface-solution interface with the ferrocyanide/ferricyanide redox couple [37, 42, 133, 134]. Previous work has suggested that the charge transfer resistance of the electrodes reduced after deposition of polypyrrole conducting polymer on glassy carbon electrodes, possibly owing to its rough surface for providing a greater surface area for faradaic charge exchange to occur.

Biotin-coated PS micro-particle was the target analyte. Avidin-biotin interaction was used to immobilise the PS beads onto the electrode surface [135]. The gold working electrode was deposited with carboxylic acid group functionalised polypyrrole conducting polymer, which allowed for the anchoring of avidin probes on the polypyrrole polymer surface using carbodiimide crosslinking chemistry. This provided a surface for binding the biotin-coated PS beads.

**Preparation of polypyrrole conducting polymer**

The gold working electrode surface was rinsed with ethanol at the start of each experiment. Next, the electrode underwent electrochemical cleaning by potential cycling between -0.4 V and 1.8 V in 0.5 M nitric acid until a stable reduction peak was obtained [136].

Deposition of the polypyrrole conducting polymer on the cleaned gold electrode surface followed the reported process in [37]. The pyrrole monomer solution was made up with a mixture of 50 mM poly(styrene sulphonate) dopant, 5 mM distilled pyrrole, and 0.5 mM pyrrole-3-carboxylic acid (PCA) diluted in deionised water. This gives a pyrrole to PCA monomer ratio of 10:1. The monomer solution was then sparged with nitrogen gas to remove oxygen. Electropolymerisation of the monomers on the gold electrode surface was induced by applying a constant potential step of 0.8 V for 0.5 seconds while submerged in the monomer solution.

**Attachment of avidin probe and immobilisation of PS beads**

Avidin probes were covalently attached to the carboxylic acid functional groups of the conducting polymer layer using carbodiimide crosslinking chemistry. Activation of the carboxylic acid groups on the polypyrrole conducting polymer was achieved by incubating the polymer surface with 50 mM EDC (1-ethyl-3-(3-dimethylaminopropyl)-carbodiimide) and 25 mM NHS (N-hydroxysuccinimide) in phosphate-buffered saline (PBS) (Sigma-Aldrich) adjusted to pH 5.6 for 1 hour in room temperature. Next, the activated surface was rinsed with PBS and then incubated with avidin (Sigma-Aldrich) at a concentration of 0.5 mg/mL in PBS (pH 7.4) in room temperature for 1 hour. The unattached avidin was washed away with PBS afterwards. The immobilisation of analyte was performed by incubating the probed electrode surface with the biotin-coated PS beads (Spherotech, mean diameter 0.74 μm), suspended in PBS (pH 7.4) at the desired concentration, for 1 hour in room temperature. Impedances were recorded before and after incubation of the PS beads to assess how the presence of surface immobilised beads changed the electrode impedance.

Impedance drift over time creates problems with the reliability of data interpretation for electrochemical impedance sensors [74]. Therefore, the impedance data for each experiment were collected with a new electrode each time with the identical incubation steps, i.e. the same electrode was not incubated with incrementing beads concentrations to collect multiple data points. This minimises the contribution of impedance drifts over time when comparing between data points.

**Table of Contents**

**Abstract **

**Acknowledgement**

**Table of Contents**

**Common abbreviations **

**Common symbols **

**Published works **

**1. Bioanalytical Chemistry and Biosensors **

1.1. Bioanalytical chemistry and current limitations

1.2. Biosensors

1.3. Electrochemical biosensors and impedance spectroscopy

1.4. Contemporary bioanalytical chemistry methods and biosensing technologies

1.5. Research motivation and thesis outline

**2. Electrochemical Impedance Biosensors**

2.1. Principle of signal transduction in electrochemical impedance biosensor

2.2. Electrical models

2.3. Geometric optimisation of impedance biosensor

2.4. Faradaic current in sensing and the effects of three-dimensional electrode geometry

**3. Modelling Diffusion Impedance in the Detection of Micron-sized Targets**

3.1. Introduction

3.2. Finite element simulation

3.3. Experimental impedance measurement in the presence of polystyrene beads

3.4. Conclusion

**4. Fabrication of Three-dimensional Microelectrodes **

4.1. Introduction

4.2. Fabrication of the master template using photoresist reflow

4.3. Gold coating the master template

4.4. Nickel electroforming of the master template

4.5. Hot embossing

4.6. Defining electrodes with photolithography

4.7. Device setup

4.8. Conclusion

**5. Effects of Three-dimensional Electrode Geometry on Sensor Performance **

5.1. Introduction

5.2. Finite element model

5.3. Experimental validation

5.4. Conclusion

**6. Research Contributions and Future Works**

6.1. Significance of findings

6.2. Future research directions

6.3. Conclusion

Bibliography .

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Exploration of the Effects of Electrode Shape on the Performance of an Electrochemical Biosensor