Interface model

In the interface model, the gradients of concentration, potential and current inside the catalyst layer are neglected. In order to apply the interface model, the model electrodes must satisfy its main assumptions. Two types of electrodes that we utilized largely met these conditions. The first one was based on a macroscopically flat carbon support with immobilized enzymes (this type has already been described with respect to fluorescence microscopy measurements in section 2.3.1) and the second one was prepared by following the intermixed PVDF procedure but where the porous layer had a very low thickness. Both electrodes were studied in a rotating disc electrode assembly, which ensured defined mass transport conditions. To further reduce the number of model parameters, steady state conditions were applied.

3.2.1. Steady state solution

Steady state solution is illustrated on the basis of Model 3 (Table 3.2). By assuming the steady state conditions, the following overall current density expression, here shown in its reciprocal form, can be derived:

1

𝑖electrode_M3_ss= 1 2𝐹_t^[

𝐾_m+ 𝑐_H2O2(0, ss) 𝑘_cat𝑐_H2O2(0, ss) +

1

𝑐_H⁺(0, ss)𝑘_e𝑒^{−𝑅𝑇𝐹}electrode_ss

]

(3.33) Eq. 3.33 demonstrates the series connection between enzymatic and electrochemical steps as also shown by Andreu et al ⁴⁹. In eq. 3.33, ke stands for an effective electrochemical rate constant defined as:

e2 e1

e2 e e1

k k

k k k

+

=  (3.34)

where ke1 and ke2 are defined as follows:

𝑘_e1= 𝑘_e10𝑒^{𝑅𝑇1𝐹10} (3.35) 𝑘_e2= 𝑘_e20𝑒^{𝑅𝑇2𝐹20} (3.36) The Km is the Michaelis-Menten constant defined as usual:

31

1 cat 1 m

-k k

K = k + (3.37)

To calculate the overall current density based on eq.3.33, concentrations of substrate and protons at the electrode surface are also needed. These concentrations are obtained by solving the mass balance equations (eq.3.5-i) with appropriate boundary conditions. Under steady state conditions, the number of original parameters is significantly reduced. Instead of two electrochemical rate constants only one lumped constant ke appears. Similarly, from 3 enzymatic rate constants only kcat and Km can be determined independently, while k1 and k-1 will be lumped together in the Km value. An additional parameter is the total enzyme surface concentration t.

The overall rate expressions for two other model variants M1 and M2 mentioned previously (Table 3.2), can be derived in a similar way (not shown here). In these two models the number of model parameters is even fewer. For example, only one enzymatic rate constant K1 and one electrochemical rate constant ke were involved.

The difference between M1 and M2 was only in the concentrations of hydrogen peroxide / protons which are considered to be the same as in bulk (M1), or their concentration on the electrode surface was considered (M2).

In the next steps the goal was to identify the most appropriate model for the description of the experimental electrode responses by comparing experimental data with the predictions of the models M1, M2 and M3. First estimations of the kinetic parameters were obtained from experimental data by using a linearized form of the overall rate expressions (e.g. eq.3.33) and by assuming no mass transfer resistance effects (see for example ¹⁴), while their final values were a result of the global optimization ⁴⁵. Finally, the steady state solutions for the three different models are plotted in Figure 3.1.

As can be seen, all 3 models can qualitatively describe the experimental data, with M3 having the best quantitative agreement. No model discrimination was possible. We have shown previously that dynamic models have a higher capability for model discrimination ⁵⁰. Therefore, the dynamic formulation of the models was reconsidered. From different dynamic methods, electrochemical impedance spectroscopy was selected as the method of choice.

32 Figure 3.1: Steady state polarization curves at pH 6 for hydrogen peroxide reduction on HRP modified SPG RDE; symbols – experimental data and lines – simulated curves (M1 - blue dashed dot line, M2 – green dashed line and M3 – black solid line). Conditions: fixed delay of 1 min, hydrogen peroxide concentrations from 10 to 160 M, pH 6, room temperature, rotation rate 400 rpm.

3.2.2. Dynamic response

At first we adopted an approach for deriving the theoretical impedance of enzyme/electrode systems based on the reaction mechanisms of enzyme catalyzed electrochemical reactions ⁴⁵. In this way mechanistic details relating to bioelectrochemical reactions including all relevant kinetic parameters were obtained.

This approach overcomes the limitations of the classical equivalent circuit approach, since it does not rely on phenomenological elements, but on the particular reaction mechanism. It is based on the more general theory of nonlinear frequency response (NLFR) ^{51, 52}, which in the present case was restricted to its linear part. This method has a generic mathematical background that allows the analysis of any weakly nonlinear system with a polynomial nonlinearity. It was initially developed for the analysis of nonlinear electrical circuits and in chemical engineering for the investigation of the adsorption equilibrium and kinetics ^{52, 53}. We first demonstrated the application of this method in theory and experiment using the example of simple electrochemical reaction ferrocyanide oxidation kinetics ^{54, 55}. For all details of the derivations related to the enzymatic electrode case, please refer to our paper ⁴⁵.

For the weakly nonlinear system, its input–output behavior around the steady state (Δ𝜑electrode,SS, 𝑖electrode,SS) can be described with a Volterra series. For a harmonic input signal:

𝐸̃(𝑡) = 𝐴𝑐𝑜𝑠(_{𝑡) =}^𝐴

2 (𝑒^𝑗𝑡+ 𝑒^{−𝑗𝑡}) (3.38)

it follows that:

0.2 0.3 0.4 0.5 0.6 0.7 0.8

-0.4 -0.35 -0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0

Electrode potential / V Current density / A m-2 geo

exp 10M exp 20M exp 30M exp 40M exp 80M exp 160M

33 𝑖̃(𝑡) = ∑ (𝐴

2)

 𝑛 𝑛=1

∫ ℎ_𝑛



−

(_{1, … .}_𝑛)(𝑒^{𝑗(𝑡−1)}_{+ 𝑒}^{−𝑗(𝑡−1)}_{) … (𝑒}^{𝑗(𝑡−𝑛)} + 𝑒^{−𝑗(𝑡−𝑛)})𝑑_{1… 𝑑}_𝑛

(3.39) where 𝐸̃(𝑡) =_electrode(𝑡) −electrode,SS and 𝑖̃(𝑡) = 𝑖electrode(𝑡) − 𝑖electrode,SS.

After applying the n-dimensional Fourier transformation the system output can be expressed as:

𝑖̃(𝑡) = (𝐴

2) [𝐻¹(_)𝑒^𝑗𝑡_{+ 𝐻1(−}_)𝑒^{−𝑗𝑡}_] + (𝐴

2)

2

[𝐻2(_,_)𝑒^2𝑗𝑡_{+ 2𝐻2(}_{, −}_)𝑒⁰ + 𝐻₂(−_{, −}_)𝑒^{−2𝑗𝑡}_]

+ (𝐴 2)

3

[𝐻3(_,_,_)𝑒^3𝑗𝑡_{+ 3𝐻3(}_,_{, −}_)𝑒^𝑗𝑡 + 3𝐻3(_{, −}_{, −}_)𝑒^{−𝑗𝑡}

+ 𝐻₃(−_{, −}_{, −}_)𝑒^{−3𝑗𝑡}_{]+. ..}

(3.40)

The function 𝐻₁() is the so-called first-order frequency response function of the system. It can be shown that 𝐻1(), is identical to the reciprocal of the electrochemical impedance. The remaining functions 𝐻_𝑛(, … …) are called higher order frequency response functions. They contain the nonlinear fingerprint of the system. The higher order frequency response functions can be derived analytically from the model equations ^{45, 54}. Additionally, they can be determined experimentally.

This can be briefly demonstrated by rearranging eq.3.40 sorted by the power of 𝑒^𝑗𝑡:

𝑖̃(𝑡) = 𝑒⁰[𝐴²

2 𝐻²(_{, −}₎

⏟

𝐻_{𝑞,𝐷𝐶}(,𝐴)

+. . ]

+1

2 {𝑒^𝑗𝑡[𝐴𝐻₁(_{) +}³

4 𝐴³𝐻₃(_,_{, −}_{) + ⋯ ]}

⏟

𝐻_𝑞,𝐼(,𝐴)

+ 𝑒^{−𝑗𝑡}[𝐴𝐻₁(−_{) +}³

4 𝐻³(−_{, −}_{, −}_{) + ⋯ ]}

⏟

𝐻_𝑞,𝐼(−,𝐴)

}

+1 2

{

𝑒^2𝑗𝑡[𝐴²

2 𝐻²(_,_{) + ⋯ ]}

⏟

𝐻𝑞,𝐼𝐼(,𝐴)

+ 𝑒^{−2𝑗𝑡}[𝐴²

2 𝐻²(−_{, −}_{) + ⋯ ]}

⏟

𝐻_{𝑞,𝐼𝐼}(−,𝐴) } +. ..

(3.41)

The individual higher order frequency response functions 𝐻𝑛(, … …) are related to the amplitude-dependent quasi-response functions of the harmonics 𝐻𝑞,𝑖(, 𝐴). The quasi-response functions of the harmonics can be obtained with an experimental setup similar to EIS. This was demonstrated taking ferrocyanide

34 oxidation as an example (Figure 3.2) ⁵⁵. The results are shown here as an illustration of a response of a simple mass transfer limited reversible electrochemical reaction.

In contrast to EIS the excitation amplitude in NLFR is considerably larger in order to exceed the quasi-linear range (EIS) and to raise higher harmonics in the output signal (to determine the linear part of the response a 10 mV amplitude is typically used, for the second order frequency response function an amplitude of 50 mV was selected as an optimal value ⁵⁵). The comparison of analytically derived and experimentally determined higher order frequency response functions can be used for model discrimination and to determine the nonlinear system parameters.

Figure 3.2: Amplitude and phase shift frequency spectra of the first (H1()) and second order (H2(,)) frequency response functions at different potentials (in mV); Symbols: experimental data; lines: theoretic curves (input amplitude: 50 mV r.m.s.; electrolyte: 20 mM [Fe(CN)6]^4–/3–, 1.0 M KCl; rotation speed: 4500 rpm; t = 25.0 °C). Adopted from ⁵⁵ with permission from the American Chemical Society.

Since the method described is generic, it was further used to derive the analytical expressions of electrochemical admittances based on the models M1, M2, M3 already introduced above⁴⁵. The analysis was restricted to the linear part of the response. The results show that only M3 can predict the correct order of the reaction resistance change with potential, while models M1 and M2 fail in this aspect (Figure 3.3). Therefore, M3 was further implemented into the porous electrode model and used to simulate electrode responses. The kinetic parameters determined here were

35 used for simulations in the framework of the porous electrode model. The only remaining parameters were related to the electrode structure (porosity, internal surface area and the electrode thickness).

Figure 3.3: Experimental (symbols) vs. simulated (lines) EIS data with optimized parameters at three steady state potentials a) M1 (dashed line) and b) M3. Conditions: pH 5, 80 μM hydrogen peroxide concentration, room temperature and rotation rate 400 rpm. Adopted from

45 with permission from Elsevier.

Im Dokument Bioelectrochemical systems for energy and materials conversion (Seite 30-35)