# Non-Equilibrium Thermodynamics and Stochastic Dynamics of a Bistable Catalytic Surface Reaction

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Bistable Catalytic Reaction Model

## 3. Theoretical Framework

#### 3.1. Mean-Field Stochastic Description

#### Stochastic Entropy Production Rate

#### 3.2. Deterministic Mean-Field Description

#### 3.3. Macroscopic Entropy Production Rate

## 4. Results and Discussion

#### 4.1. Deterministic Analysis

#### 4.2. Stochastic Analysis

## 5. Overall CO_{2} Production Rate

## 6. Summary and Conclusions

_{2}production rate, while the state of low entropy production rate corresponds to a NEES characterised by a low CO

_{2}production rate. In the stochastic description, there is a unique entropy production rate. Inside the bistable region, the stochastic entropy production rate exhibits a first-order phase transition at a critical point, which becomes sharper as ${N}_{L}$ increases. For ${k}_{co}^{ads}$ below the critical point (for ${k}_{co}^{ads}$ above the critical point), a state with high (low) entropy production is selected. These observations suggest that, in our model, the entropy production rate is neither maximised nor minimised for all non-equilibrium states.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

NESS | Non-equilibrium steady state |

ODEs | Ordinary differential equations |

CO | Carbon monoxide |

CO_{2} | Carbon dioxide |

O_{2} | Oxygen |

LH | Langmuir-Hinshelwood |

CME | Chemical master equations |

2D | two dimensional |

EP | Macroscopic entropy production rate |

sn | Saddle node |

UR | Upper rate |

LR | Lower rate |

MinEPP | Minimium entropy production principle |

MaxEPP | Maximium entropy production principle |

## Appendix A. Transition Rates and Detailed Balance

#### Appendix A.1. CO(gas) Adsorption and CO(ads) Desorption.

#### Appendix A.2. Dissociative O_{2}(gas) Adsorption and Associative O(ads) Desorption.

#### Appendix A.3. CO(ads)+O(ads) Reaction and CO_{2}(gas) Dissociative Adsorption

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**Figure 1.**Steady state bifurcation diagram in the parameter space (${k}_{co}^{ads}$, ${k}_{co}^{des}$) from Equations (19) and (20). Inside black region the system exhibits the bistable phenomenon. Its boundaries are given by the upper saddle node ($s{n}_{h}$) line and the lower saddle node ($s{n}_{l}$) line meeting each other in a cusp. Along the dashed-red line (${k}_{co}^{ads}\approx 0.05{k}_{co}^{des}$), the system relaxes to thermodynamic equilibrium. Other parameters are ${k}_{{o}_{2}}^{ads}=0.2$, ${k}_{o}^{des}=20$, ${k}_{c{o}_{2}}=0.5$, ${k}_{r}=50$, and $\zeta =4$ (square lattice).

**Figure 2.**(

**a**) and (

**b**) Steady state bifurcation diagrams of the CO and oxygen coverages as a function of ${k}_{co}^{ads}$, for ${k}_{co}^{des}=0.15$. The figures clearly show a bistable region in which two NESS stable branches (blue lines) coexist with an unstable or saddle one (red dots). In all cases, the boundaries of the bistable region are characterised by saddle node bifurcations at the $s{n}_{l}$ and $s{n}_{h}$ points. Other parameters are ${k}_{{o}_{2}}^{ads}=0.2$, ${k}_{o}^{des}=20$, ${k}_{c{o}_{2}}=0.5$, ${k}_{r}=50$, and $\zeta =4$ (square lattice)

**Figure 3.**Time evolution of the system inside the bistable region and on the plane (u, $\nu $), for two different initial conditions. In this case, ${k}_{co}^{ads}=0.57$ and ${k}_{co}^{des}=0.15$. Depending on the initial condition, the system converges to one of the two NESS branches of the bistable region. Other parameters are ${k}_{{o}_{2}}^{ads}=0.2$, ${k}_{o}^{des}=20$, ${k}_{c{o}_{2}}=0.5$, ${k}_{r}=50$, and $\zeta =4$ (square lattice)

**Figure 4.**(

**a**) Macroscopic entropy production rate, $EP$ (Equation (22)), corresponding to the steady-state solutions of Equations (19) and (20) as a function of ${k}_{co}^{ads}$, for ${k}_{co}^{des}=0.15$. (

**b**) The same $EP$ but only for values of ${k}_{co}^{ads}$ around the bistable region. Full blue lines are the $EP$ associated with the (${u}_{-}$, ${\nu}_{+}$) and (${u}_{+}$, ${\nu}_{-}$) stable branches and dotted line to the one of the unstable or saddle branch or (${u}_{saddle}$, ${\nu}_{saddle}$). Other parameters are ${k}_{{o}_{2}}^{ads}=0.2$, ${k}_{o}^{des}=20$, ${k}_{c{o}_{2}}=0.5$, ${k}_{r}=50$, and $\zeta =4$ (square lattice).

**Figure 5.**Time series of CO coverage from the deterministic and stochastic approaches. Blue dashed lines correspond to the deterministic prediction and full black lines to the stochastic one. (

**a**) Stochastic simulations for a surface area of ${N}_{L}=1500$. Note as the stochastic trajectories follow on average the trajectories predicted by the deterministic approach; (

**b**) Stochastic simulations with a surface area of ${N}_{L}=100$. Note the phenomenon of fluctuation-induced transitions. Other parameters are ${k}_{co}^{ads}=0.57$, ${k}_{co}^{des}=0.15$, ${k}_{{o}_{2}}^{ads}=0.2$, ${k}_{o}^{des}=20$, ${k}_{c{o}_{2}}=0.5$, ${k}_{r}=50$, and $\zeta =4$ (square lattice).

**Figure 6.**Normalised steady state probability distribution, ${P}_{st}\left(\mathbf{Z}\right)$, of finding a population vector $\mathbf{Z}=\{{N}_{CO},{N}_{O}\}$, for ${N}_{L}=200$. (

**a**) ${k}_{co}^{ads}=0.55$; (

**b**) ${k}_{co}^{ads}=0.56$; (

**c**) ${k}_{co}^{ads}=0.57$. Other parameters are ${k}_{co}^{des}=0.15$, ${k}_{{o}_{2}}^{ads}=0.2$, ${k}_{o}^{des}=20$, ${k}_{c{o}_{2}}=0.5$, ${k}_{r}=50$, and $\zeta =4$ (square lattice). The steady state probability distribution was obtained after averaging over an ensemble (20 independent realisations sampled at a fixed time).

**Figure 7.**(

**a**) and (

**b**) Steady state bifurcation diagrams of the average CO and oxygen coverages as a function of ${k}_{co}^{ads}$, for ${k}_{co}^{des}=0.15$. Other parameters are ${k}_{{o}_{2}}^{ads}=0.2$, ${k}_{o}^{des}=20$, ${k}_{c{o}_{2}}=0.5$, ${k}_{r}=50$, and $\zeta =4$ (square lattice). The steady state probability distribution was obtained after averaging over an ensemble (20 independent realisations sampled at a fixed time). This figure should be compared with Figure 2.

**Figure 8.**(

**a**) Stochastic entropy production rate measured as ${N}_{L}^{-1}d{S}_{i}/dt$ versus ${k}_{co}^{ads}$ for three different system sizes, and ${k}_{co}^{des}=0.15$; (

**b**) The same as in (

**a**) but only for values of ${k}_{co}^{ads}$ around the bistable region. Other parameters are ${k}_{{o}_{2}}^{ads}=0.2$, ${k}_{o}^{des}=20$, ${k}_{c{o}_{2}}=0.5$, ${k}_{r}=50$, and $\zeta =4$ (square lattice). To calculate this entropy production rate, the steady state probability distribution was obtained after averaging over an ensemble (20 independent realisations sampled at a fixed time). This figure should be compared with Figure 4.

**Figure 9.**(

**a**) Deterministic overall CO

_{2}production rate; (

**b**) Stochastic overall CO

_{2}production rate at stationary state. In both cases, ${k}_{co}^{ads}$ was variated and ${k}_{co}^{des}$ was fixed at $0.15$. Other parameters are ${k}_{{o}_{2}}^{ads}=0.2$, ${k}_{o}^{des}=20$, ${k}_{c{o}_{2}}=0.5$, ${k}_{r}=50$, and $\zeta =4$ (square lattice).

**Table 1.**Processes, population changes, and transition rates for our well-mixed CME treatment of the dynamics of $\mathbf{Z}=\{{N}_{CO},{N}_{O}\}$ for a surface with ${N}_{L}$ available sites. Parameters ${k}_{co}^{ads}$ and ${k}_{co}^{des}$ represent the rate constants for CO(gas) adsorption and CO(ads) desorption, respectively. ${k}_{{o}_{2}}^{ads}$ and ${k}_{o}^{des}$ are the rate constants for O${}_{2}$(gas) dissociative adsorption and O(ads) associative desorption. The parameter ${k}_{c{o}_{2}}$ is the rate constant for CO${}_{2}$(gas) dissociative adsorption, and ${k}_{r}$ is the rate constant for CO(ads) + O(ads) reaction. $\zeta $ is the coordination number or the number of nearest neighbours of a site. In a 2D regular lattice $\zeta $ can take one of the following values: 3 for honeycomb-type lattice, 4 for a square lattice, and 6 for a hexagonal lattice [31,32]. Please note that the number of free sites on the surface is ${N}_{\ast}={N}_{L}-{N}_{CO}-{N}_{O}$. In Appendix A, we demonstrate the consistency of these transition rates.

Process | Population Change | Transition Rate |
---|---|---|

CO(gas) adsorption | $({N}_{CO},{N}_{O})\to ({N}_{CO}+1,{N}_{O})$ | ${W}_{+1}={k}_{co}^{ads}\left({N}_{L}-{N}_{CO}-{N}_{O}\right)$ |

CO(ads) desorption | $({N}_{CO},{N}_{O})\to ({N}_{CO}-1,{N}_{O})$ | ${W}_{-1}={k}_{co}^{des}{N}_{CO}$ |

O${}_{2}$(gas) dissociative adsorption | $({N}_{CO},{N}_{O})\to ({N}_{CO},{N}_{O}+2)$ | ${W}_{+2}=\frac{\zeta {k}_{{o}_{2}}^{ads}}{2({N}_{L}-1)}{N}_{\ast}\left({N}_{\ast}-1\right)$ |

O(ads) associative desorption | $({N}_{CO},{N}_{O})\to ({N}_{CO},{N}_{O}-2)$ | ${W}_{-2}=\frac{\zeta {k}_{o}^{des}}{2({N}_{L}-1)}{N}_{O}\left({N}_{O}-1\right)$ |

CO${}_{2}$(gas) dissociative adsorption | $({N}_{CO},{N}_{O})\to ({N}_{CO}+1,{N}_{O}+1)$ | ${W}_{+3}=\frac{\zeta {k}_{c{o}_{2}}}{2({N}_{L}-1)}{N}_{\ast}\left({N}_{\ast}-1\right)$ |

CO(ads) + O(ads) reaction | $({N}_{CO},{N}_{O})\to ({N}_{CO}-1,{N}_{O}-1)$ | ${W}_{-3}=\frac{\zeta {k}_{r}}{{N}_{L}-1}{N}_{CO}{N}_{O}$ |

**Table 2.**Reaction rates or fluxes pertaining to Equations (19) and (20). Please note that the macroscopic rate constants are ${K}_{co}^{ads}={k}_{co}^{ads}$, ${K}_{co}^{des}={k}_{co}^{des}$, ${K}_{{o}_{2}}^{ads}=\zeta {k}_{{o}_{2}}^{ads}/2$, ${K}_{o}^{des}=\zeta {k}_{o}^{des}/2$, ${K}_{c{o}_{2}}=\zeta {k}_{c{o}_{2}}/2$, and ${K}_{r}=\zeta {k}_{r}$.

Process | Reaction Rates or Fluxes |
---|---|

CO(gas) adsorption | ${w}_{+1}={K}_{co}^{ads}\left(1-u-\nu \right)$ |

CO(ads) desorption | ${w}_{-1}={K}_{co}^{des}u$ |

O${}_{2}$(gas) dissociative adsorption | ${w}_{+2}=2{K}_{{o}_{2}}^{ads}{\left(1-u-\nu \right)}^{2}$ |

O(ads) associative desorption | ${w}_{-2}=2{K}_{o}^{des}{\nu}^{2}$ |

CO${}_{2}$(gas) dissociative adsorption | ${w}_{+3}={K}_{c{o}_{2}}{\left(1-u-\nu \right)}^{2}$ |

CO(ads) + O(ads) reaction | ${w}_{-3}={K}_{r}u\nu $ |

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**MDPI and ACS Style**

Pineda, M.; Stamatakis, M. Non-Equilibrium Thermodynamics and Stochastic Dynamics of a Bistable Catalytic Surface Reaction. *Entropy* **2018**, *20*, 811.
https://doi.org/10.3390/e20110811

**AMA Style**

Pineda M, Stamatakis M. Non-Equilibrium Thermodynamics and Stochastic Dynamics of a Bistable Catalytic Surface Reaction. *Entropy*. 2018; 20(11):811.
https://doi.org/10.3390/e20110811

**Chicago/Turabian Style**

Pineda, Miguel, and Michail Stamatakis. 2018. "Non-Equilibrium Thermodynamics and Stochastic Dynamics of a Bistable Catalytic Surface Reaction" *Entropy* 20, no. 11: 811.
https://doi.org/10.3390/e20110811