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For developing high-performance devices, it is important to understand the processes involved in the system. The processes are described as follows: After the oxidized or reduced species is produced on a base electrode, the concentration distribution of the species changes constantly not only by the diffusion of the redox species but also by the following chemical reactions. Hence, the analysis used in a homogeneous solution cannot be applied to the electrode system except in a few cases. Such a situation can be expressed by partial differential equations, which can be solved not analytically but numerically. However, a computer with a high process capability is required to solve the equations. In 2002, the capability of a personal computer, e.g., the Pentium III machine, becomes more than six times as high as that of a first commercial supercomputer, CRAY-1 (80MFLOPS), which would be enough to solve the equations.

In the present study, we have developed a program, called PLEC-1, which can simulate electrochemical measurements with a polymer-coated electrode, and analyze cyclic voltammograms by the Gauss-Newton method. PLEC-1 can simultaneously treat seven materials and three reactions in both simulation and analysis.

Figure 1. Illustration of a polymer-coated electrode soaked in an electrolyte solution.

The program can treat four kinds of chemical reactions as follows:

where S is a substrate, P a product, E an enzyme, and B a by-product. *k*_{f} (s^{-1} or mol^{-1}cm^{3}s^{-1}) and *k*_{b} are rate constants of a forward reaction and backward reaction, respectively.

However, when a material is localized in the polymer matrix because of chemical reactions, the molar flux of electrons by electron hopping is not always represented by Fick's law. Such a situation causes the reversal of electric potential calculated by Nernstian equation in the direction of electron propagation. Hence, we assumed that the molar flux of electrons by electron hopping,

where

where

Assuming that Material-

where

where

Assuming that the concentration ratio on the electrode obeys a Nernstian equation, the concentration of the oxidant on the electrode is represented as

where

The concentration of the reductant,

Total material balance of the species in the immediate neighborhood of the electrode is expressed as

where

where

where

When the material-

Implicit types allow using larger values of D

When an electron exchange reaction between a base electrode and a material is slow or the chemical reactions are included in the electrode system, the Crank-Nicolson method cannot be applied to solve PDEs. Thus, we adopted an "iterative" Crank-Nicolson method in the program as mentioned below. The following equations do not include the charge hopping mechanism to explain the iterative Crank-Nicolson method concisely.

When the electron exchange reaction is slow, the finite differential equation of the oxidant evaluated as an average at the time

where

The diffusion equation of a reductant is represented as

Eqs. 19 and 21 have the other unknown quality; eq. 19 has

One is the method where we must solve the large matrix containing the two species. In this method, it is complicated to accelerate the Gauss method for solving the simultaneous equations, and the second-order chemical reactions cannot be treated by the method.

The other is an "iterative" Crank-Nicolson method. Although the amount of calculation is higher than the above method, it is easy to introduce the calculation of the chemical reactions into the program. However, this method may not have the stability and accuracy of the Crank-Nicolson method.

In the present study, the "iterative" Crank-Nicolson method was adopted for solving PDEs including the second-order chemical reactions.

As mentioned above, eq. 19 which is the finite difference equation of the oxidant contains the unknown reductant concentration,

Figure 2. Simplified flow chart of the estimation of the concentration distribution under the condition where the electron exchange reaction is the rate-determining step on the electrode. *C*_{m,ox,tmp}(*i*) and *C*_{m,rd,tmp}(*i*) are temporary variables, respectively.

**(b)Following chemical reaction with reversible reaction on base electrode**

We obtained a finite differential equation of a substrate in a bulk polymer by evaluating an average at the time *t* = *j* and *j* - 1 as

where *D*_{S} (cm^{2} s^{-1}) is the diffusion coefficient of the substrate, *C*_{S} (mol cm^{-3}) is the concentration of the substrate, *R*_{b} (mol cm^{-3} s^{-1}) is the increasing concentration caused by the reverse reaction, and the relation between *Q*_{f} (s^{-1}) and *R*_{f} (mol cm^{-3} s^{-1}) is expressed by

In the same manner as above, the finite difference equation of a product in the bulk polymer is represented as

where *D*_{P} (cm^{2} s^{-1}) is the diffusion coefficient of the product and *C*_{P} (mol cm^{-3}) is the concentration of the product. The relation between *Q*_{b} (s^{-1}) and *R*_{b} (mol cm^{-3} s^{-1}) is given by

For example, when the chemical reaction expressed by eq. 1 takes place in the polymer layer of the electrode, eq. 22 contains an unknown quantity, *C*_{P}(*i*, *j*), because *R*_{b} is defined by eq. 25. Eq. 22 cannot be solved independently, so the following procedures were adopted for solving equations: *C*_{P}(*i*, *j*-1) is substituted for an initial value of *C*_{P}(*i*, *j*). Until tolerance was satisfied, eq. 22 and eq. 24 were solved alternately as the electrode reaction was irreversible.

In short, the concentrations that should be simultaneously estimated are processed in a loop, and the loop is repeated until tolerance is satisfied, which is the algorithm adopted in PLEC-1.

Figure 3. Screen shot of PLEC-1. (a) Control panel. (b) Numerical values of a simulated cyclic voltammogram. (c) Potentiostat PLEC-1. (d) Graph window of cyclic voltammogram. (e) Concentration distribution of material species.

Menu/Submenu | Description | |
---|---|---|

File | ||

Open parameters | Load the parameters from the PLEC-1 format file | |

Save parameters | Save the parameters as PLEC-1 format | |

Quit | Quit PLEC-1 | |

Edit | ||

Paste parameters | Paste parameters from Microsoft Excel | |

Analysis | ||

CV analysis (one material) | Analysis of the cyclic voltammogram with Gauss-Newton method | |

Concentration Analysis | Concentration analysis using a cyclic voltammogram with Gauss-Newton method | |

Reaction Analysis | Reaction analysis using a cyclic voltammogram with Gauss-Newton method | |

Settings | ||

AVI Setting | Settings for the animation of the concentration distribution graph | |

Reaction Setting | Settings for the reactions among materials | |

Gauss-Newton Setting | Settings for the Gauss-Newton method | |

Help | ||

Help | Show help file. (English and Japanese) | |

About | Show information about PLEC-1 |

Figure 4. Cyclic voltammogram of a virtual material with three methods. Scan rate is 20mVs^{-1}. -, the forward difference method; ---, Crank-Nicolson method; ..., Iterative Crank-Nicolson method. (D*t* = 0.01 s, Div. Num. = 40, *l* = 1 × 10^{-4} cm) The properties of the material are as follows: *n* = 1, *E*° = 1 V vs. SE, *D*_{phys} = 1 × 10^{-11} cm^{2} s^{-1}, *k*_{ex} = 0 mol^{-1}cm^{5}s^{-1}, *C*_{T} = 2 × 10^{-4} mol cm^{-3}, *i*_{0} = 1 × 10^{-2}Acm^{-2}mol^{-1}cm^{3}, a= 0.5.

[ 2] Y. Kurimura, E. Tsuchida and M. Kaneko,

[ 3] M. Kaneko, S. Nemoto, A. Yamada and Y. Kurimura,

[ 4] P. Mariaulle , F. Sinapi, L. Lamberts, A. Walcarius,

[ 5] M. Yagi, K. Kinoshita and M. Kaneko,

[ 6] T. Abe, T. Yoshida, S. Tokita, F. Taguchi, H. Imaya and M. Kaneko,

[ 7] A. Merz, A. J. Bard,

[ 8] N. Oyama, F. C. Anson,

[ 9] J. S. Facci, R. H. Schmehl, and R. W. Murray,

[10] W. J. Vining and T. J. Meyer,

[11] A. M. Hodges, O. Johansen, J. W. Loder, A. W. H. Mau, J. Rabani, W. H. F. Sasse,

[12] A. Aoki, R. Rajagopalan and A. Heller,

[13] J. W. Long, C. S. Velazquez, and R. W. Murray,

[14] J. Zhang, M. Yagi, and M. Kaneko,

[15] M. Yagi, H. Fukiya, T. Kaneko, T. Aoki, E. Oikawa, and M. Kaneko,

[16] H. Shiroishi, K. Ishikawa, K. Hirano, M. Kaneko,

[17] H. Shiroishi, T. Nomura, K. Ishikawa, S. Tokita, and M. Kaneko,

[18] H. Shiroishi, T. Shoji, T. Nomura, S. Tokita, M. Kaneko,

[19] T. Saito,

[20] J. Crank, P. Nicolson,

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