Chinese Journal of Chemical Physics  2018, Vol. 31 Issue (1): 61-65

The article information

Tie-jun Xiao, Yun Zhou
肖铁军, 周匀
Stochastic Thermodynamics of Mesoscopic Electrochemical Reactions
介观电化学反应的随机热力学
Chinese Journal of Chemical Physics, 2018, 31(1): 61-65
化学物理学报, 2018, 31(1): 61-65
http://dx.doi.org/10.1063/1674-0068/31/cjcp1705110

Article history

Received on: May 27, 2017
Accepted on: June 26, 2017
Stochastic Thermodynamics of Mesoscopic Electrochemical Reactions
Tie-jun Xiao, Yun Zhou     
Dated: Received on May 27, 2017; Accepted on June 26, 2017
Guizhou Provincial Key Laboratory of Computational Nano-Material Science, Guizhou Synergetic Innovation Center of Scientific Big Data for Advanced Manufacturing Technology, Guizhou Education University, Guiyang 550018, China
*Author to whom correspondence should be addressed. Tie-jun Xiao, E-mail:tjxiao@gznc.edu.cn
Abstract: In this work, we discussed the stochastic thermodynamics of mesoscopic electron transfer reactions between ions and electrodes. With a relationship between the reaction rate constant and the electrode potential, we find that the heat dissipation βq equals to the dynamic irreversibility of the reaction system minus an internal entropy change term. The total entropy change Δst is defined as the summation of the system entropy change Δs and the heat dissipation βq such that Δsts+βq. Even though the heat dissipation depends linearly on the electrode potential, the total entropy change is found to satisfy the fluctuation theorem < (e)st>=1, and hence a second law-like inequality reads <Δst>≥0. Our study provides a practical methodology for the stochastic thermodynamics of electrochemical reactions, which may find applications in biochemical and electrochemical reaction systems.
Key words: Mesoscopic chemical reaction    Electron transfer    Electrode potential    Entropy production    Fluctuation theorem    
Ⅰ. INTRODUCTION

Stochastic thermodynamics (ST) has been a very active field in the modern statistic physics [1-4]. The fluctuations could be significant in mesoscopic systems, and hence various thermodynamic properties such as work, heat, and system entropy become stochastic variables [5-8]. One can introduce the system entropy $s$ and the heat dissipation $q$ or equivalently the medium entropy change $\Delta s_\textrm{m}$=$\beta q$ to characterize a stochastic system. A very important result from ST is that the total entropy change defined as $\Delta s_\textrm{t}$=$\Delta s$+$\Delta s_\textrm{m}$ may satisfy the fluctuation theorem $\langle \textrm{e}^{-\Delta s_\textrm{t}} \rangle$=1, which leads to a second-law-like inequality $\langle \Delta s_\textrm{t} \rangle$$\geq$0. Such a physical picture has been revealed in dissipative systems with Newtonian dynamics [9-12], the underdamped or overdamped Brownian particles [13-17] and the quantum systems [18, 19].

As a special case of the birth-death stochastic processes, chemical reactions have gained considerable attentions. In general, the state of a reaction system is described by the number vector $\textbf{X}$ of active species X, which could vary according to the reaction events. When the transition rates $W_{\pm \rho} (\textbf{X})$ for a forward and a backward reaction are identified, the probability $p(\textbf{X}; t)$ of variable $\textbf{X}$ evolves according to a master equation. Note that the heat dissipation of a reaction system is known to have very subtle relationship with the transition rates [8, 20], a careful treatment of the heat dissipation is necessary. Previous studies on reaction network with elementary reactions $\displaystyle\sum_{i} r^i_{\rho} X_i$$\rightleftharpoons$$\displaystyle\sum_{i} p^i_{\rho} X_i $ demonstrate that the heat dissipation $\beta q$ can be related to the quantity $\Sigma$=$\displaystyle\sum_i \ln { W_{\rho_i } \over W_{-\rho_i }}$, so that the total entropy change $\Delta s_\textrm{t}$=$\Delta s$+$\beta q$ would satisfy the fluctuation theorem [21, 22]. A key intermediate step to prove this relation is to use the relationship between the equilibrium constant and the chemical potential of the reactant and product species, which can either be found from the rate theory of gas phase reactions [23] or from the self-consistent thermodynamics constraints [22]. ST of simple reactions has been successfully applied to chemical reactions which are far away from equilibrium [24-29].

In this study, we focus on the electrochemical reactions where the electrons are transferred between ions and electrodes, and hence the electrode potential $\psi$ plays a role. One may note that the electron transfer reactions have wide applications in sensing devices [30-32]. To the best of our knowledge, the effect of electric potential is not explicitly addressed in the previous studies of ST, so it would be useful to see how to incorporate the electrode potential into the ST of electrochemical reactions. Specifically, based on the constraints from thermodynamic self-consistency or the Marcus' theory of electron tranfer reactions [33-36], a relation between the equilibrium constant and the electrode potential $\psi$ is found. With this relation, we demonstrate onstrate that $\Sigma$=$\displaystyle\sum_i\ln { W_{\rho_i } \over W_{-\rho_i }}$ equals to the heat dissipation $\beta q$ minus an entropy change of the active species, so that the total entropy change defined as $\Delta s_\textrm{t}$=$\Delta s$+$\beta q$ satisfies the fluctuation theorem $\langle \textrm{e}^{-\Delta s_\textrm{t}}\rangle$=1 and the second law reads $\langle \Delta s_\textrm{t} \rangle$$\geq$0. For a reaction system with given propensity functions $ \{ W_{\pm \rho} \}$, the heat dissipation $\beta q$ depends explicitly on the electrode potential as well as the internal entropy of the reactive species. When the electrode potential and the internal entropy of the system are not specified, $\beta q$ is not well defined while the total entropy change $\Delta s_\textrm{t}$ can be uniquely determined. Our finding implies that the ST of electron transfer reactions can be treated in the same way as ST of reactions without electron transfer.

In this work, a model electrochemical reaction system is first introduced. ST of electrochemical reactions is shown in a thermodynamic relationship between the reaction rate constant and the electrode potential is presented and the fluctuation theorem is discussed.

Ⅱ. MODEL OF THE MESOSCOPIC ELECTROCHEMICAL SYSTEMS

Consider a model electrochemical reaction system with fixed volume $V$. The reaction system contacts to a heat bath with temperature $T$, and then the reduced inverse temperature can be defined as $\beta$=$1/k_\textrm{B}T$. As an illustrative example, we consider an electrochemical reaction between an ion species ${Y}$ and an electrode with electric potential $\psi$

$ \begin{eqnarray} {Y} \rightleftharpoons {X} + e^-(\textrm{electrode}) \end{eqnarray} $ (1)

where ${Y}$=$A^{n+}$ and ${X}$=$A^{(n+1)+}$ are the active species. Denote $\textbf{X}$ and $\textbf{Y}$ as the particle numbers of the ${X}$ species and $ {Y}$ species. In the dilute limit, the reaction rate for elemental reactions satisfies the mass action law, where the reaction rates for the forward and backward reactions are given by [23]

$ \begin{array}{*{20}c} W_{+}( \textbf{X} ) = k_{+} \textbf{Y}\\ W_{-}( \textbf{X} ) = k_{-} \textbf{X}\\ \end{array} $ (2)

where $k_{+}$ and $k_{-}$ are the reaction constant for the forward and backward reaction. The rate constant $K$=$k_+/k_-$ for electron transfer reactions is known to depend on the chemical potential of $\textbf{X}$ and $\textbf{Y}$ as well as electrode potential $\psi$ [33, 36]. In the following section we will show that a simple relationship between the equilibrium constant $K$ and the electrode potential $\psi$ can be worked out.

The $ {Y}$ species is assumed to contact to a particle reservoir with chemical potential $\mu_{Y}$ and hence keeps its particle number fixed. Denote the probability distribution of the intermediate species $X$ as $p(\textbf{X}; t)$. The evolution of $p(\textbf{X}; t)$ is governed by the master equation as

$ \begin{eqnarray} \partial_t p(\textbf{X};t) &=& \sum\limits_{\rho = \pm } [W_{\rho}(\textbf{X}-v_{\rho}) p(\textbf{X}-v_{\rho};t)-\nonumber\\ &&{}W_{\rho}(\textbf{X}) p(\textbf{X};t)] \end{eqnarray} $ (3)

where $v_{+}$=1 and $ v_{-}$=$-$1 are the stoichiometric coefficients of X species for the forward and backward reactions, i.e., the particle number changes from $\textbf{X}$ to $\textbf{X}$+$v_{+}$ or $\textbf{X}$+$v_{-}$ given a forward or a backward reaction happens. In the long time limit, the system could reach a stationary distribution $ p_s(\textbf{X})$=$p(\textbf{X}; t$$\rightarrow$$\infty)$, such that $\partial_t p_s(\textbf{X})$=0.

Ⅲ. STOCHASTIC THERMODYNAMICS IN THE SINGLE TRAJECTORY LEVEL A. Single trajectory and dynamic irreversibility

For the reaction system, the reaction events change the state variable of the system and hence generate a stochastic trajectory. Denote $\textbf{X}_{j}$$\equiv$$\textbf{X}(t''$=$t_j)$ as the particle number of species $X$ at time $t''$=$t_j$. A forward trajectory $\chi(t)$ consisted of $n$ concessive reactions is defined as

$ \begin{eqnarray} \chi(t)&=&\{ \textbf{X}_0(t''=t_0)\rightarrow\textbf{X}_1(t''=t_1)\rightarrow... \rightarrow\nonumber\\[2ex] &&{}\textbf{X}_{n-1}(t''=t_{n-1})\rightarrow\textbf{X}_n(t''=t_n) \rightarrow\nonumber\\[2ex]&&{}\textbf{X}_n(t''=t)\}\nonumber \end{eqnarray} $

its conjugate backward trajectory $\chi^R(t)$ is defined as

$ \begin{eqnarray} \chi^R(t)&=&\{\textbf{X}_n(t''={t'}_0)\rightarrow\textbf{X}_{n-1}(t''={t'}_1)\rightarrow...\rightarrow\nonumber\\[2ex] &&{} \textbf{X}_{1}(t''={t'}_{n-1})\rightarrow\textbf{X}_0(t''={t'}_n)\rightarrow\textbf{X}_0(t''=t)\}\nonumber \end{eqnarray} $

$\tau_j$=$t_j$$-$$t_{j-1}$ and $\tau'_j$=$t'_j$$-$$t'_{j-1}$ are two time interval parameters. The conjugate condition requires that $\tau_j$=$\tau'_{n+1-j} $. According to the Gillespie's method [37, 38], the conditional probability to find such a trajectory or its conjugated one reads

$ \begin{eqnarray} P[\chi(t)| \textbf{X}_0]&=& \Pi_{j=1}^n \textrm{e}^{- r( \textbf{X}_{j-1}) \tau_j } \displaystyle{W_{\rho_j}( \textbf{X}_{j-1}) \over r( \textbf{X}_{j-1})}\nonumber\\ &&{} \textrm{e}^{-r( \textbf{X}_n)(t- t_n)} \end{eqnarray} $ (4)
$ \begin{eqnarray} P[\chi^R(t)| \textbf{X}_t]&=& \Pi_{j=1}^n \textrm{e}^{- r( \textbf{X}_{j-1}) \tau_j } \displaystyle {W_{-\rho_j}( \textbf{X}_j) \over r( \textbf{X}_{j-1})}\nonumber\\ &&{} \textrm{e}^{-r( \textbf{X}_n)(t- t_n)} \end{eqnarray} $ (5)

where $r(\textbf{X}_i)$=$\displaystyle\sum_{\rho = \pm } W_{ \rho }(\textbf{X}_i) $ is the total propensity function. With the log ratio of those two probabilities, one can introduce a dynamic irreversibility term $\Sigma$ as

$ \begin{eqnarray} \label{Sigm} \Sigma &\equiv& \ln {P[\chi(t)| \textbf{X}_0] \over P[{\chi^R(t)}| \textbf{X}_t]}\nonumber\\ &=& \sum\limits_{j=1}^n \ln {W_{\rho_j}( \textbf{X}_{j-1}) \over W_{-\rho_j}( \textbf{X}_j)} \end{eqnarray} $ (6)

For the simple reaction systems, the dynamic irreversibility $\Sigma $ differs from $ \beta q$ by a term which depends on the degeneracy of particles [22], and the averaged change rate $ \langle \dot{\Sigma} \rangle$ can be related to the entropy production defined as the product of the thermodynamic flux and the thermodynamic force [21]. In the following part, we will show that such relations are also valid for the reaction systems with electron transfer.

B. Energy balance in the single trajectory level

In order to study the energetics of the reaction system, we consider the thermodynamic properties. The chemical potential $\mu_i$ for the i-th active species with particle number $N_i$ can be expressed as $\beta \mu_{i}(N_i)$=$\beta \mu_{i}^*$+$\ln N_i $, where $\beta \mu_{i}^* $ is the chemical potential of a reference state [23]. The entropy $s_i$ and the internal energy $\epsilon_i$ of the species i are connected via the thermodynamic relation $ \beta \epsilon_i$=$\beta \mu_i$+$s_i$ [23].

Denote $A(\textbf{X})$ as the free energy of the reaction system in state $ \textbf{X} $. After a reaction of the $\rho_j$-th kind, the state varies from $ \textbf{X}_{j-1}$ to $\textbf{X}_{j}$=$\textbf{X}_{j-1}$+${v}_{\rho_j}$, then the free energy change $\beta \Delta A^j$, the internal entropy change $\Delta S_0^j$ and the internal energy change $\Delta U^j$ for the reaction system reads

$ \beta \Delta A^j = v_{\rho_j} [\beta \mu_{X}^* + \ln {(X_{j-1} + v_{\rho_j})}] $ (7)
$ \Delta S_0^j = v_{\rho_j} s_X $ (8)
$ \beta \Delta U^j = \beta \Delta A^j + \Delta S_0^j $ (9)

In order to keep the particle number of $Y$ of the reaction system unchanged, the amount of chemical work done by particle reservoirs is as the following

$ \begin{eqnarray} \beta w_{\textrm{chem}}^j &=& - u_{\rho_j} \beta \mu_{Y}\nonumber\\ &=& - u_{\rho_j} (\beta \mu_{Y}^* + \ln \textbf{Y}) \end{eqnarray} $ (10)

where $u_{+}$=$-$1 and $ u_{-}$=1 are the stoichiometric coefficients of Y species for the forward and backward reactions. The electric work by the electrode reads

$ \begin{eqnarray} \beta w_{\textrm{ele}}^j= v_{\rho_j} e_0 \psi \end{eqnarray} $ (11)

where $e_0$ is the elemental charge.

Since the temperature $T$ of the reaction system is fixed during the reaction process, the amount of energy $\beta w_{\textrm{chem}}^j$+$\beta w_{\textrm{ele}}^j$$-$$\Delta U^j$ should be released to the heat bath as heat dissipation $\beta q^j$, so the energy balance equation can be written as

$ \begin{eqnarray} \beta q^j = \beta w_{\textrm{chem}}^j +\beta w_{\textrm{ele}}^j- \beta \Delta U^j \end{eqnarray} $ (12)

Along a single trajectory which consists of $n$ reactions, the heat dissipation $ \beta q$$\equiv$$\displaystyle\sum_{j=1}^n \beta q^j$ reads

$ \begin{eqnarray} \beta q &=& - \sum\limits_{j=1}^n [u_{\rho_j} \beta \mu_{Y}^* + v_{\rho_j} \beta \mu_{X}^* + v_{\rho_j} \textrm{e}_0 \psi]\nonumber \\ &&{}\hspace{-0.2cm}-\sum\limits_{j=1}^n [u_{\rho_j} \ln \textbf{Y} + v_{\rho_j} \ln {(\textbf{X}_{j-1} + v_{\rho_j}) }]\nonumber \\ &&{}\hspace{-0.2cm}- \sum\limits_{j=1}^n \Delta S_0^j \end{eqnarray} $ (13)

which implies that the heat dissipation has a linear dependence on the electrode potential.

C. Dynamic-thermodynamic relation

Herein we demonstrate that the heat dissipation $\beta q$ could be related to the dynamic irreversibility $\Sigma$ based on the constraints from macroscopic equilibrium system. Denote the equilibrium concentration of species $X$ and $Y$ as $\textbf{x}_s$ and $\textbf{y}$:

$ \textbf{x}_s\equiv\lim\limits_{V\rightarrow \infty} \frac{\textbf{X}}{V} $ (14)
$ \textbf{y}\equiv\lim\limits_{V\rightarrow \infty} \frac{\textbf{Y}}{V} $ (15)

The zero current condition in the equilibrium system is:

$ \begin{eqnarray} k_{+}\textbf{y}-k_{-} \textbf{x}_s=0\end{eqnarray} $ (16)

or equivalently:

$ \begin{eqnarray} \ln K= \ln\frac{k_{+}}{ k_{-}}= \ln\frac{\textbf{x}_s}{\textbf{y}} \end{eqnarray} $ (17)

The electric potential equilibrium condition reads:

$ \begin{eqnarray} \beta \mu_X(\textbf{x}_s V)-\beta \mu_Y(\textbf{y} V)+e_0 \psi=0\end{eqnarray} $ (18)

or equivalently:

$ \begin{eqnarray} u_{+} \beta \mu_Y^*+v_{+} \beta \mu_X^*+v_{+} e_0\psi=-\ln{\frac{\textbf{x}_s}{\textbf{y}}} \end{eqnarray} $ (19)

By comparing these two constraints, one can find that

$ \begin{eqnarray} \ln K &\equiv& \ln \frac{k_{+ }}{k_{- }} \nonumber\\ &=& -( u_{+} \beta \mu_{Y}^* + v_{+} \beta \mu_X^* + v_{+}e_0 \psi) \end{eqnarray} $ (20)

One may note that it is possible to derive this equation in a different way, e.g., one can use the reaction rate constant $k_{\pm}$ from Marcus' theory of electron transfer [33, 36] to derive Eq.(14). With Eq.(2), Eq.(6), Eq.(13) and Eq.(20), it is straightforward to prove that

$ \begin{eqnarray} \beta q&=&\displaystyle \sum\limits_{j=1}^n \ln\displaystyle \frac{W_{\rho_j}( \textbf{X}_{j-1})}{ W_{-\rho_j}( \textbf{X}_j)} - \sum\limits_{j=1}^n \Delta S_0^j\\ &=& \Sigma - \Delta S \end{eqnarray} $ (21)
$ \Delta S_0\equiv\displaystyle\sum\limits_j \Delta S_0^j $ (22)

where $\Delta S_0$ is the internal entropy change along the trajectory. Eq.(21) implies that the heat dissipation equals to the dynamic irreversibility minus the internal entropy change of the reaction system.

D. Entropy balance and the fluctuation theorem

For the reaction system along the single trajectory $\chi(t)$, one should note that the stochastic system entropy $s(t)$ consists of an internal entropy contribution $S_0$ and the Shannon entropy contribution $-\ln [p(\textbf{X}_t; t)] $ [20, 21], i.e.,

$ \begin{eqnarray} \label{entropy} s = S_0 - \ln [p(\textbf{X}_t;t)] \end{eqnarray} $ (23)

The system entropy change can then be evaluated as

$ \begin{eqnarray} \label{entropy-change} \Delta s = \Delta S_0 + \ln {p(\textbf{X}_0, t_0)\over p(\textbf{X}_t;t)} \end{eqnarray} $ (24)

When the internal entropy term is evaluated analytically from the grand canonical ensemble, one can find that $\Delta S_0$ reduces to the degeneracy entropy term introduced by Schmiedl and Seifert [20, 22]. As one can see, the reactive species which contact with particle reservoir would not contribute to the system entropy change as their particle numbers are constants during the reactions.

Using Eq.(21) and Eq.(24), the total entropy change $\Delta s_\textrm{t}$ defined as $\Delta s_\textrm{t}$$\equiv$$\Delta s$+$\beta q$ can be rewritten as

$\begin{eqnarray} \label{1Law-c} \Delta s_\textrm{t} &=& \Sigma + \ln{ p(\textbf{X}_0, t_0) \over p(\textbf{X}_t, t)}\nonumber\\ &=& \ln\frac{P(\chi)}{P(\chi^R)} \end{eqnarray} $ (25)

According to Eq.(25), the total entropy change is equal to the log ratio of the probability of a forward trajectory to that of a conjugate backward trajectory. Then following the standard procedure in the stochastic thermodynamics [3, 22], it is straightforward to prove that $\Delta s_\textrm{t}$ satisfies the fluctuation theorem as

$ \begin{eqnarray} \langle e^{-\Delta s_\textrm{t}}\rangle =1 \end{eqnarray} $ (26)

and then the second law of thermodynamics follows as $\langle \Delta s_\textrm{t} \rangle$$\geq$0.

As one can see, the propensity function $\{ W_{\rho}(X) \}$ is insufficient to define a chemical reaction system with unique thermodynamic properties, i.e., chemical reaction systems with fixed $\{ W_{\rho}(X) \}$ may have different internal entropy $\{ s_i \}$ and different chemical potential $\{ \mu_i^* \} $. According to Eq.(21) and Eq.(25), these systems would have different heat dissipation $\beta q$ and the same total dissipation $\Delta s_\textrm{t}$. Even though the heat dissipation $\beta q$ does depend on the electrode potential $\psi$, the dynamic-thermodynamic relations $\beta q$=$\Sigma$$-$$\Delta S_0$ and the fluctuation theorem $\langle e^{-\Delta s_\textrm{t}}\rangle$=1 are still valid. Our study implies that the ST of electrochemical reactions can be treated in the same way as ST of simple reactions. Note that the mass action law is the starting point of this study and is known to be valid for dilute reactants, one can expect that our theory is at least applicable to dilute solutions. For concentrated solutions where the mass action law fails, the ST of reactions remains an open question and deserves further investigations.

Ⅳ. CONCLUSIONS

In this study, we discussed the stochastic thermodynamics of electrochemical reactions, where the electrode potential plays a role. We find a relationship between the reaction constant $K$=$k_+/k_-$ and the electrode potential $\psi$. With such a relationship, one can discuss the energy balance in stochastic trajectory level, where the heat dissipation $\beta q$ is found to be equal to the log ratio of the conditional probabilities for a forward and a backward trajectory minus an internal entropy change term. Even though the heat dissipation $\beta q$ has a linear dependence on the electrode potential $\psi$, the total entropy change $\Delta s_\textrm{t}$=$\Delta s$+$\beta q$ satisfies the fluctuation theorem $\langle e^{-\Delta s_\textrm{t}}\rangle$=1 and is similar to the case of simple reactions without electron transfer. Our study provides a practical methodology for the stochastic thermodynamics of mesoscopic electron transfer reactions, which may find applications in small biochemical and electrochemical reaction systems.

Ⅴ. ACKNOWLEDGMENTS

This work was supported by the National Natural Science Foundation of China (No.21403041 and No.21503048) and the Startup Packages from Guizhou Education University.

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介观电化学反应的随机热力学
肖铁军, 周匀     
贵州师范学院, 贵州省纳米材料模拟与计算重点实验室, 贵州省科学云大数据开发及其产业应用协同创新中心, 贵阳 550018
摘要: 本文讨论了离子和电极之间发生的介观电子转移反应的随机热力学.基于反应速率常数与电极电势的关系,发现反应体系的热耗散βq等于反应系统的动力学不可逆性减去体系的内部熵变项.尽管体系的热耗散对于电极电势存在线性依赖,由体系熵变Δst以及热耗散βq的加和定义的体系总熵变Δsts+βq仍满足传统的涨落定理 < (e)st>=1,从而热力学第二定律可表述为<Δst>≥0.
关键词: 介观化学反应    电子转移    电极电势    熵产生    涨落定理