# Exploring Nonlinear Diffusion Equations for Modelling Dye-Sensitized Solar Cells

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## Abstract

**:**

## 1. Introduction

#### 1.1. Diffusion Equation

#### 1.2. Current-Voltage Characteristics

#### 1.3. Efficiency

## 2. Analytical Solution for the Linear Electron Diffusion Equation

#### 2.1. Short-Circuit Conditions

#### 2.2. Open-Circuit Model

## 3. Classical Lie Symmetry for the Nonlinear Electron Diffusion Equation

#### 3.1. Classical Lie Symmetry

- $D\left(n\right)=0$,
- $\frac{dD}{dn}=0$,
- $\frac{{d}^{3}D}{d{n}^{3}}\frac{dD}{dn}D\left(n\right)-2{\left(\frac{{d}^{2}D}{d{n}^{2}}\right)}^{2}D\left(n\right)+\frac{{d}^{2}D}{d{n}^{2}}{\left(\frac{dD}{dn}\right)}^{2}=0$,
- $x\frac{dG}{dx}+2G\left(x\right)+2R\left(n\right)=0$,
- The set $\{x\frac{dG}{dx},\frac{dG}{dx},G,1\}$ is linearly dependent, or
- $R\left(n\right)=0$.

#### 3.1.1. Constant Diffusion Coefficient

- The set $\{n\frac{\partial R}{\partial n},\frac{\partial R}{\partial n},R,n,1\}$ is linearly dependent or
- G is a constant.

- $R\left(n\right)={c}_{1}{(n+{c}_{2})}^{{c}_{3}}+{c}_{4}n+{c}_{5}$,
- $R\left(n\right)={c}_{1}{e}^{{c}_{2}n}+{c}_{3}n+{c}_{4}$,
- $R\left(n\right)={c}_{1}ln(n+{c}_{2})+{c}_{3}n+{c}_{4}$, or
- $R\left(n\right)={c}_{1}(n+{c}_{2})ln(n+{c}_{2})+{c}_{3}n+{c}_{4}$,

#### 3.1.2. Particular Diffusion Coefficient

- $G\left(x\right)=A{(x+B)}^{C}$,
- $G\left(x\right)=A{e}^{Bx}$, or
- $G\left(x\right)=Aln(x+B)$,

#### 3.1.3. Particular Source Term

#### 3.2. Summary

## 4. Nonclassical Lie Symmetry for the Nonlinear Electron Diffusion Equation

#### 4.1. Nonclassical Symmetry Analysis for $D\left(n\right)={n}^{m}$

#### 4.2. Nonclassical Symmetry Analysis for $D\left(n\right)={e}^{mn}$

## 5. Numerical Results and Discussion

#### Efficiency Calculations

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

DSSC | Dye-sensitized solar cell |

IPCE | Incident photon to current efficiency |

PDE | Partial differential equation |

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**Figure 8.**Comparison of the results with Cao et al. [6] (Reprinted with permission from (Cao, F., Oskam, G., Meyer, G.J. and Searson, P.C., 1996. Electron transport in porous nanocrystalline ${\mathrm{TiO}}_{2}$ photoelectrochemical cells. The Journal of Physical Chemistry, 100(42), pp.17021-17027.). Copyright 1996 American Chemical Society).

Parameter | Value | Reference |
---|---|---|

${D}_{0}$ | ${10}^{-11}\phantom{\rule{3.33333pt}{0ex}}{\mathrm{m}}^{2}\xb7{\mathrm{s}}^{-1}$ | [8] |

${\alpha}_{ab}$ | ${10}^{5}\phantom{\rule{3.33333pt}{0ex}}{\mathrm{m}}^{-1}$ | [12] |

d | $5\times {10}^{-5}\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}$ | [8] |

${k}_{R}$ | $4\times {10}^{-8}\phantom{\rule{3.33333pt}{0ex}}{\mathrm{s}}^{-1}$ | [8] |

${n}_{eq}$ | ${10}^{22}\phantom{\rule{3.33333pt}{0ex}}{\mathrm{m}}^{-3}$ | [16] |

$\phi $ | ${10}^{21}\phantom{\rule{3.33333pt}{0ex}}{\mathrm{m}}^{-2}\xb7{\mathrm{s}}^{-1}$ | [17] |

$\mu $ | $2.5\times {10}^{6}$ | |

$\nu $ | 5 | |

$\xi $ | ${10}^{-5}$ |

**Table 2.**Classical Lie symmetries found for special cases of Equation (4).

$\mathit{D}\left(\mathit{n}\right)$ | $\mathit{G}\left(\mathit{x}\right)$ | $\mathit{R}\left(\mathit{n}\right)$ | X | T | N |
---|---|---|---|---|---|

1 | $Ax+B$ | ${n}^{\frac{1}{3}}$ | x | $2t$ | $3n$ |

1 | $A{x}^{-2}$ | $\alpha {e}^{\beta n}$ | $\frac{x}{2}$ | t | $-\frac{n}{\beta}$ |

1 | $-2\alpha ln(x+B)+C$ | $\alpha ln(n+\beta )$ | $x+B$ | $2t$ | $2(n+\beta )$ |

1 | $A{e}^{B{x}^{2}+Cx}$ | $\alpha nln\left(n\right)+\gamma n$ | $\frac{1}{C{e}^{4Bt}}$ | 0 | $-\frac{(C+2Bt)}{C{e}^{4Bt}}$ |

1 | $A{x}^{-4}$ | $\alpha {n}^{2}$ | x | $2t$ | $-2n$ |

${n}^{m}$ | $A{(x+B)}^{C}$ | $\alpha {n}^{\frac{C}{(C+2)}}$ | $(m+1)(x+B)$ | $(2-Cm)t$ | $(C+2)n$ |

${n}^{m}$ | $A{e}^{Bx}$ | $\alpha {n}^{m+1}$ | $\frac{(m+1)}{Bm}$ | $-t$ | $\frac{n}{m}$ |

${n}^{m}$ | $Aln(x+B)+C$ | $-\frac{A(m+1)}{2}ln\left(n\right)$ | $(m+1)(x+B)$ | $2t$ | $2n$ |

${e}^{mn}$ | $A{x}^{B}$ | $\alpha {e}^{\frac{Bmn}{B+2}}$ | x | $-Bt$ | $\frac{B+2}{m}$ |

${e}^{mn}$ | $A{e}^{Bx}$ | $\alpha {e}^{mn}$ | 1 | $-Bt$ | $\frac{B}{m}$ |

${e}^{mn}$ | $Aln(x+B)+C$ | $-\frac{Amn}{2}$ | $x+B$ | 0 | $\frac{2}{m}$ |

$D\left(n\right)$ | $A{x}^{-2}$ | 0 | x | $2t$ | 0 |

$\mathit{D}\left(\mathit{n}\right)$ | $\mathit{\eta}$ | ${\mathit{J}}_{\mathbf{sc}}\left({\mathbf{Am}}^{-2}\right)$ | ${\mathit{V}}_{\mathbf{oc}}\left(\mathbf{V}\right)$ |
---|---|---|---|

1 | $7.0569$ | $133.8545$ | $0.6322$ |

$\sqrt{n}$ | $0.5054$ | $10.8597$ | $0.5673$ |

n | $0.1247$ | $2.8818$ | $0.5330$ |

${n}^{2}$ | $0.0298$ | $0.7452$ | $0.4980$ |

${e}^{\frac{ln\left({n}_{eq}\right)}{{n}_{eq}}n}$ | $7.0569$ | $133.8545$ | $0.6322$ |

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**MDPI and ACS Style**

Maldon, B.; Thamwattana, N.; Edwards, M.
Exploring Nonlinear Diffusion Equations for Modelling Dye-Sensitized Solar Cells. *Entropy* **2020**, *22*, 248.
https://doi.org/10.3390/e22020248

**AMA Style**

Maldon B, Thamwattana N, Edwards M.
Exploring Nonlinear Diffusion Equations for Modelling Dye-Sensitized Solar Cells. *Entropy*. 2020; 22(2):248.
https://doi.org/10.3390/e22020248

**Chicago/Turabian Style**

Maldon, Benjamin, Ngamta Thamwattana, and Maureen Edwards.
2020. "Exploring Nonlinear Diffusion Equations for Modelling Dye-Sensitized Solar Cells" *Entropy* 22, no. 2: 248.
https://doi.org/10.3390/e22020248