# Modeling and Simulation of Enzymatic Biofuel Cells with Three-Dimensional Microelectrodes

^{*}

## Abstract

**:**

## 1. Introduction

**Figure 1.**(

**a**) A miniaturized EBFC with 3D interdigitated microelectrode arrays; (

**b**) schematic depiction of EBFC reaction mechanism.

## 2. Mechanism of EBFCs

- Anode:
- Cathode:

_{2}, which can be oxidized by the electrode back to GDH-FAD shown in the following reactions:

_{2}+ Gluconolactone

_{2}↔ GDH-FAD + 2H

^{+}+ 2e

^{−}

_{1}, k

_{−1}are the rate constants for the association and breakdown, respectively of the complex and k

_{2}denote the rate constants for the breakdown of the complex. The steady state kinetics of the enzyme reaction (v) is expressed by:

_{cat}(=k

_{2}) is catalytic rate constant, K

_{M}(=(k

_{2}+ k

_{−1})/k

_{1}) is the Michaelis-Menten constant of the enzyme. [E] and [S] are the concentration of the enzyme and the substrate.

## 3. Simulation Modeling

^{−3}. In the diffusion module, the diffusion of substrate with enzyme kinetics is solved based on reaction-diffusion equations:

_{2}for GDH and type 1 copper for laccase.

Boundary | Diffusion | Potential |
---|---|---|

Top boundary of bulk domain | $c={c}_{0}$ | $n\xb7J=0$ |

Bulk-enzyme interface | $-n\left({N}_{1}-{N}_{2}\right)=0$ | $V={V}_{0}$ |

Enzyme-electrode interface | $-n\left({N}_{1}-{N}_{2}\right)=0$ | $n\left({J}_{1}-{J}_{2}\right)=0$ |

Side and bottom boundaries of bulk domain | $-n\left(-D\nabla c\right)=0$ | $\text{}n\xb7J=0$ |

Constant | Ref. Value | Reference |
---|---|---|

R | 8.314 J·mol·K^{−1} | - |

T | 300 K | - |

F | 96,485 C·mol^{−1} | - |

D_{glucose} | 7 × 10^{−10} m^{2}·s^{−1} | [42,43,44] |

D_{oxygen} | 1.74 × 10^{−9} m^{2}·s^{−1} | [45,46] |

K_{M_GDH} | 17.4 mM | [47] |

K_{M_laccase} | 133.4 mM | [48] |

k_{cat_GDH} | 360 s^{−1} | [47] |

k_{cat_laccase} | 117 s^{−1} | [48] |

ø ^{o}_{A} | −0.32 V | [49] |

ø ^{o}_{C} | 0.585 V | [49] |

σ_{carbon} | 8000 S·m^{−1} | [50] |

σ_{substrate} | 4 S·m^{−1} | - |

- (1)
- 2-D simulation is used to simplify the 3-D microelectrode design.
- (2)
- The DET between enzyme and electrode is assumed.
- (3)
- The enzyme kinetics constant is obtained from the literatures based on immobilized enzymes.
- (4)
- The enzyme is uniformly distributed in the enzyme layer.
- (5)
- Negligible change in heat transfer is assumed between enzyme layer and electrode interface.
- (6)
- Temperature distribution around the EBFCs is assumed to be uniform.

## 4. Results and Discussions

#### 4.1. The Steady State Response

**Figure 2.**Response time to reach steady state for electrode at height of 200 µm with well width of 100 µm.

#### 4.2. Impact of Mass Transport and Reaction Rate

**Figure 3.**Glucose concentration from the bottom to the top of electrode at electrode height of 200 µm with different well width (ww) (

**a**) 40 µm; (

**b**) 100 µm respectively; (

**c**) gluocse concentration from the bottom to the top of electrode at different height from 100 µm to 200 µm.

**Figure 4.**Change in reaction rate along the surface of anode and cathode at electrode height of 200 µm.

#### 4.3. The Cell Performance of EBFCs

^{2}. Ease of fabrication is an important consideration and since it is difficult to manufacture carbon microelectrodes higher than 200 µm by C-MEMS. Therefore, based on the results and practical application, to design the microelectrode arrays within 15 mm × 15 mm foot print, the optimum configuration is height and well width keeping as 200 µm and 100 µm.

^{2}when the voltage is approximately 0.44 V. This performance of EBFC is adequate for operation of low-voltage CMOS integrated circuits.

**Figure 5.**(

**a**,

**b**) Current density profile for one row of microelectrodes at 100 µm with different well width 50 µm and 100 µm; (

**c**) current density at the surface of electrode along the vertical direction inside the well: 50 µm (dash line); 100 µm (solid line); (

**d**) change of line current density with respect to electrode height and well width; (

**e**) power density vs. cell voltage at different electrode heights. The height to well width ratio of electrode is 2:1.

#### 4.4. Geometry of the Electrodes

**Figure 6.**Electric field (surface, contour and arrow plot) profile surrounding the rectangle electrodes with constant potential of 0.585 V on cathode and −0.32 V on anode.

_{2}would get reduced on the carbon and then decrease the current, voltage and faradic efficiency [51]. In order to simplify the simulation, we don’t take into account any putative O

_{2}reduction on the anode. The current density and resistive heating profiles for: (a) rectangular; (b) triangular; (c) tapered; and (d) semi-elliptical electrodes with height of 100 µm, well width of 20 µm at bottom and diameter of 20 µm are shown in Figure 7a–d, respectively. In the figures, the left Y-axis and the right Y-axis show the values for current density and resistive heating, respectively. The current density and resistive heating profiles follow almost the same trend for all the four geometries since heat dissipation is proportional to current density. The values for current density and resistive heating, for all the geometries, are summarized in Table 3.

**Table 3.**Statistical analysis of current density (CD) and resistive heating (RH) for rectangular, triangular, tapered and semi-elliptical geometry of electrodes.

Geometry | Side Edges | Top Edge | Corners | |||
---|---|---|---|---|---|---|

Property | CD (μA·cm^{−2}) | RH (μW·cm^{−3}) | CD (μA·cm^{−2}) | RH (μW·cm^{−3}) | CD (μA·cm^{−2}) | RH (μW·cm^{−3}) |

Rectangular | 170 | 4 | 100 | 1.5 | 410 | 23 |

Triangular | 90–225 | 1–8 | - | - | 270(top) 230(bottom) | 12(top) 7(bottom) |

Tapered | 130–210 | 2.5–5 | 125–135 | 1–3 | 250 | 8.5 |

Semi-elliptical | 45–90 | 0.05–0.17 | - | - | 105(top) 47(bottom) | 0.27(top) 0.05(bottom) |

**Figure 7.**Current density and resistive heating distribution plots for four different geometries of electrodes: (

**a**) rectangular; (

**b**) triangular; (

**c**) tapered; and (

**d**) semi-elliptical.

## 5. Conclusions

^{2}at 0.44 V in voltage when the dimension of electrodes is keeping height as 200 µm and well width as 100 µm. From current density and resistive heating distribution analysis for different geometries of electrodes, we highly recommend that semi-elliptical shaped electrode is more favorable to deliver uniform current density along the electrode.

## Author Contributions

## Nomenclature

[i] | concentration of component i (mol·m ^{−3}) |

c | concentration |

D | diffusion coefficient (m ^{2}·s^{−1}) |

F | Faraday’s constant (C·mol ^{−1}) |

JJ | current density (mA/cm ^{2}) |

k | rate constant for enzyme complex (s ^{−1}) |

k_{cat} | catalytic rate constant (s ^{−1}) |

K_{M} | Michaelis Mention constant (mM) |

N | flux (mol·m ^{−2}·s^{−1}) |

R | universal gas constant (J·mol·K ^{−1}) |

T | room temperature (K) |

v | enzyme reaction (mol·m ^{−3}·s^{−1}) |

z | number of electron transferred |

## Greek

σ | electric conductivity (S·m ^{−1}) |

ø | electric potential (V) |

## Conflicts of Interest

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**MDPI and ACS Style**

Song, Y.; Penmatsa, V.; Wang, C.
Modeling and Simulation of Enzymatic Biofuel Cells with Three-Dimensional Microelectrodes. *Energies* **2014**, *7*, 4694-4709.
https://doi.org/10.3390/en7074694

**AMA Style**

Song Y, Penmatsa V, Wang C.
Modeling and Simulation of Enzymatic Biofuel Cells with Three-Dimensional Microelectrodes. *Energies*. 2014; 7(7):4694-4709.
https://doi.org/10.3390/en7074694

**Chicago/Turabian Style**

Song, Yin, Varun Penmatsa, and Chunlei Wang.
2014. "Modeling and Simulation of Enzymatic Biofuel Cells with Three-Dimensional Microelectrodes" *Energies* 7, no. 7: 4694-4709.
https://doi.org/10.3390/en7074694