Show Summary Details
More options …

# Journal of Non-Equilibrium Thermodynamics

Founded by Keller, Jürgen U.

Editor-in-Chief: Hoffmann, Karl Heinz

Managing Editor: Prehl, Janett / Schwalbe, Karsten

Ed. by Michaelides, Efstathios E. / Rubi, J. Miguel

4 Issues per year

IMPACT FACTOR 2017: 1.633
5-year IMPACT FACTOR: 1.642

CiteScore 2017: 1.70

SCImago Journal Rank (SJR) 2017: 0.591
Source Normalized Impact per Paper (SNIP) 2017: 1.160

Online
ISSN
1437-4358
See all formats and pricing
More options …
Volume 43, Issue 1

# Multiscale Transient and Steady-State Study of the Influence of Microstructure Degradation and Chromium Oxide Poisoning on Solid Oxide Fuel Cell Cathode Performance

Guanchen Li
• Corresponding author
• Mechanical Engineering Department, Center of Energy Systems Research, Virginia Tech, Blacksburg, USA
• Email
• Other articles by this author:
/ Michael R. von Spakovsky
/ Fengyu Shen
/ Kathy Lu
Published Online: 2017-11-14 | DOI: https://doi.org/10.1515/jnet-2017-0013

## Abstract

Oxygen reduction in a solid oxide fuel cell cathode involves a nonequilibrium process of coupled mass and heat diffusion and electrochemical and chemical reactions. These phenomena occur at multiple temporal and spatial scales, making the modeling, especially in the transient regime, very difficult. Nonetheless, multiscale models are needed to improve the understanding of oxygen reduction and guide cathode design. Of particular importance for long-term operation are microstructure degradation and chromium oxide poisoning both of which degrade cathode performance. Existing methods are phenomenological or empirical in nature and their application limited to the continuum realm with quantum effects not captured. In contrast, steepest-entropy-ascent quantum thermodynamics can be used to model nonequilibrium processes (even those far-from equilibrium) at all scales. The nonequilibrium relaxation is characterized by entropy generation, which can unify coupled phenomena into one framework to model transient and steady behavior. The results reveal the effects on performance of the different timescales of the varied phenomena involved and their coupling. Results are included here for the effects of chromium oxide concentrations on cathode output as is a parametric study of the effects of interconnect-three-phase-boundary length, oxygen mean free path, and adsorption site effectiveness. A qualitative comparison with experimental results is made.

This article offers supplementary material which is provided at the end of the article.

## 1 Introduction

Among the different types of fuel cells, solid oxide fuel cells (SOFCs) have intensively been studied. An SOFC is composed of two porous electrodes and one ionic conductive electrolyte. The most common electrolyte material is ion-conducting yttria-stabilized zirconia (YSZ), while perovskites such as strontium-doped lanthanum manganese (LSM) are used for the cathode and nickel/YSZ for the anode. Of the two electrodes, it is the oxygen reduction process at the cathode, which accounts for most of the performance losses. In order to improve cathode performance, new manufacturing techniques, as well as the synthesis of new materials, have been used to develop cathodes with greater stability and better electrical properties.

To help guide material development for improved oxygen reduction efficiencies, it is useful to study the oxygen reduction mechanism via both experiment and a theoretical model able to provide insight into the details of the phenomena, which take place at the cathode. There are two differing views of what these phenomena entail [1]. The first is that they involve a mixed chemical/electrochemical process and the other is that the process is purely electrochemical. The main difference in the two is the presence of the charged oxygen ion ${O}^{-}$. ${O}^{-}$ allows mass transfer via an additional charge transfer pathway other than that of neutral particle diffusion. There are many models based on one or the other of these two views. Gong et al. [2] have built a one-dimensional (1D) continuum model in order to study the interaction between ${O}^{-}$ formation and transport at the surface of the cathode material and ${O}^{X}$ (${V}_{O}^{..}$) diffusion in the bulk. Experimental observation is still needed to confirm the existence of the intermediate product ${O}^{-}$ [1]. Nonetheless, for generality, it is included in the nonequilibrium steepest-entropy-ascent quantum thermodynamic (SEAQT) model presented here, which includes three mass transfer pathways: ${O}_{2}$ diffusion in the pore, ${O}^{-}$ diffusion along the material surface, and ${O}^{X}$ (${V}_{O}^{..}$) diffusion in the bulk material.

Of course, improved material performance at the expense of material stability must also be dealt with since the development of La0.6Sr0.4Co0.2Fe0.8O3 (LSCF) cathodes has been hindered by their performance degradation due to the instability of the LSCF material [3, 4] and chromium poisoning from the interconnect [5, 6]. Thus, understanding how degradation mechanisms due to structural changes [7, 8, 9] and chromium oxide poisoning [10, 11] impact cathode lifetimes, as well as performance, is of great importance. The former has been studied experimentally, and results show that they significantly influence cathode performance, while some theoretical studies of the kinetics of poisoning and its coupling with the oxygen reduction process have been made at a microscale using kinetic theory and the lattice Boltzmann method (LBM) [12].

The goal of this paper is a cross-scale study of the influence of microstructural degradation and chromium oxide poisoning on performance. We study the macroscopic level cathode performance with respect to changes in reacting particle eigenstructure occupations and eigenstructure changes, which are atomistic level features. In other words, the model directly uses quantum mechanical energy eigenlevels (e.g., such as those which could be determined from the density of states of density functional theory (DFT) or from the potential energy surfaces of quantum chemistry) as input to the system description, a description which as a result contains the complete information of the particles (included in this case) and material (not included here) from an ab initio standpoint. The conventional approach for achieving a similar result requires several very different models to cooperate with each other, with each applicable to a given set of nonequilibrium phenomena at a particular time and length scale [2, 13, 14, 15, 16, 17, 18, 19, 20, 21] and each resulting in a large computational burden and requiring a range of expertise from the atomistic to the macroscopic. Clearly, the need for such cooperation is that very often microscopic and mesoscopic level properties not only influence performance at these levels but strongly do so at the macroscopic level as well. At times, the microscopic and/or mesoscopic model can be incorporated into the macroscopic one to form a multiphysics, multiscale model in which the phenomenological parameters of the macroscopic description can be updated in real time [21, 22, 23]. Of course, in doing so, the issue of dealing with huge differences in both the description of the system and the equations of motion at different scales must be dealt with. For example, a macroscopic continuum model may use concentration and temperature as state variables and balance equations resulting from the laws of thermodynamics as stand-ins for an equation of motion. In contrast, microscopic molecular dynamics lacks any thermodynamic information at all and uses particle position and momentum as state variables and the dynamics resulting from Newton’s laws as its equation of motion. As a result, there is no shared form for the microscopic and macroscopic variables or properties, and, thus, a need exists for some sort of link between scales to transform property information at one scale into that at another. This is an important feature of any multiphysics, multiscale model, and unfortunately comes with a large computational overhead. Furthermore, due to the complexity of such a process, it is not easy to parametrically optimize macroscopic performance based on atomistic energy eigenstructure information.

To address these issues of time and length scales and the influence of microscopic and mesoscopic parameters, a novel approach based on the SEAQT mathematical framework of intrinsic quantum thermodynamics (IQT) [24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36] is used here. The resulting model with a significantly smaller computational burden than that of the conventional approach links atomistic-level information with macroscopic, predicting both transport coefficients and reaction rate constants in the presence of coupled and uncoupled phenomena and doing so without the inclusion of all the intermediate models used in the more conventional approach. The SEAQT model has a range of applicability from that at equilibrium to that in the far-from-equilibrium realm and can in a single coherent description deal with a much wider range of scales than that of previous models since the model consists of a set of local systems with length scales in the mesoscopic range, while each local system is characterized by particle energy eigenstate properties, which provide information about atomistic or microscopic features. In the SEAQT framework, the basic system state variables are energy and entropy, which are well defined at both the microscopic and mesoscopic levels, can even be extended to the macroscopic level, and are used to describe the thermodynamic dependence of the coupled and uncoupled phenomena present. The mechanical dependence of the latter is taken into account with the proper choice of relaxation time, which, however, only represents timescale information. Using the same state variables at all scales of description permits the transfer of information between scales by simple scaling, which naturally extends the range of scales to which the SEAQT model can be applied. Furthermore, since the SEAQT framework uses the principle of steepest entropy ascent as the basis for its dynamics, its equation of motion is applicable from the atomistic to the macroscopic level, exhibiting the same basic features across different time and length scales. Thus, a consistent description and dynamics across scales makes this novel approach a good choice for studying microscopic parameter influences on mesoscopic and macroscopic system performance. It is also a good choice since it can predict system behavior in the presence of transport-reaction coupling both in the steady-state and transient regimes, even that far-from equilibrium where nonlinearities and entropy generation play a significant role. In this regard, it is particularly advantageous for studying the effects of chromium oxide poisoning on SOFC cathode performance since the presence of this oxide increases the complexity of the coupling, which exists between mass diffusion, chemical/electrochemical reactions, and heat diffusion phenomena.

Finally, the subsequent sections of the paper are organized as follows. Section 2 provides a brief description of the nonequilibrium SEAQT cathode model. Section 3 gives a presentation and discussion of results for the prediction of cathode performance relative to the coupled oxygen and chromium oxide reduction-diffusion pathways. Included is a parametric study of the degradation effects on performance of microstructural changes as well as a qualitative comparison of our theoretical predictions with some of our experimental results. Section 4 concludes with a few final remarks.

## 2.1 SOFC cathode model

The system studied is a 1D system from the interface of the interconnect with the cathode to the interface of the cathode with the electrolyte. There are three parallel regions in the system: the cathode pore, the surface of the cathode material, and the bulk of the cathode material. The ${O}_{2}$ molecule travels in the pore, the oxygen ion ${O}^{-}$ along the surface of the material, and the oxygen ion ${O}^{X}$ (or the vacancy ${V}_{O,MIEC}^{..}$) inside the bulk of the material. The oxygen molecule ${O}_{2}$ enters the system through the pore at the interface of the cathode with the interconnect, and the oxygen ion ${O}^{X}$ exits the system on the surface and from the bulk at the interface of the cathode with the electrolyte. Based on a multistep charge transfer model, three coupled pathways are considered enough for providing a sufficiently complete description of the oxygen reduction process. The three pathways are distinguished by where two steps of the reduction process occur. In the first pathway, ${O}_{2}$ diffuses through a nanopore of the porous cathode electrode material to a three-phase boundary (TPB) at or near the cathode–electrolyte interface after which a two-step charge transfer reaction reduces the oxygen molecule to $2{O}^{X}$ so that it can diffuse into the bulk electrolyte. A second pathway is for ${O}_{2}$ to be absorbed by the surface of the cathode material in the micropore after which it becomes $2{O}^{-}$ through a one-step charge transfer. This is followed by diffusion along the surface of the cathode material to a TPB where a one-step charge transfer transforms $2{O}^{-}$ into $2{O}^{X}$. This last species then diffuses into the bulk electrolyte. Finally, the third pathway is one where ${O}_{2}$ is absorbed by the surface of the cathode material and through a two-step charge transfer becomes $2{O}^{X}$ in the bulk of the cathode material where it diffuses through the bulk to arrive at the electrolyte. The design of our system and its pathways is inspired by the work of Gong et al. [2]. For the system results given in Sections 3.1, 3.2, and 3.3, adsorption sites are located at multiple positions from the interconnect to the electrolyte. The system for the chromium oxide pathway is composed of the same pore and cathode material surface that makes up the system for the oxygen reduction pathway. $Cr{O}_{3}$ enters the pore and then diffuses toward the TPB. Simultaneously, part of the $Cr{O}_{3}$ is reduced to $C{r}_{2}{O}_{3}$, which accumulates on the surface of the cathode material. Details of the reaction and diffusion pathways are listed in Table 1.

Table 1:

Model links and the relaxation times $\tau$ for interactions. Parameters to match for oxygen reduction are reaction exchange rate constants (mol${1}^{}$cm${-2}^{}$ s${-1}^{}$) and diffusion coefficients (cm${2}^{}$ s${-1}^{}$), which refer to Gong et al. 2012[2]. The relaxation time for chromium oxide reduction is estimated by the value of oxygen pathway for the order of magnitude.

## 2.2 SEAQT model for the SOFC cathode: Local system separation of pathways

Figure 1:

Diagram of pathways for the oxygen reduction model.

Figure 2:

Diagram of the pathway for the chromium oxide poisoning model.

The oxygen reduction pathways are depicted in Figure 1. Each of these three pathways, which corresponds to one region of the cathode, i.e. pore, surface, or bulk, is separated into $20$ locations where each constituent forms one of the local systems. There are $82$ local systems in total: $20$ for the oxygen (${O}_{2}$) in the pore region, $20$ for the absorbed oxygen atom (${O}_{ad}$) in the surface region, $20$ for the ionized oxygen (${O}^{-}$) in the surface region, $20$ for the oxygen ion inside the bulk material (${O}_{MIEC}^{X}$), and two for the oxygen ion inside the YSZ (${O}_{YSZ}^{X}$), which serves as the outlet. The diffusion and chemical/electrochemical reactions (see Table 1) are represented by the links between the local systems. Every two linked local systems can have diffusion or a reaction occurring, and there is mass and/or energy transfer between them. These mass and energy fluxes are modeled by the SEAQT model using information from the linked local system states. It is emphasized here that the SEAQT equation of motion can be applied to study local nonequilibrium systems and far-from-equilibrium interactions between local systems. Thus, it is not necessary for the local systems defined in our model to be small enough to support a local equilibrium assumption nor for the interactions between local systems to be constrained to the linear realm [33, 34]. The chromium oxide pathway is shown in Figure 2. For this pathway, there are $20$ local systems for the $Cr{O}_{3}$ in the pore and $20$ local systems for the $C{r}_{2}{O}_{3}$ on the surface of the cathode material. Two different kinds of links between local systems exist as shown in Table 1. These represent the diffusion of the $Cr{O}_{3}$ and the chemical reaction, which generates the $C{r}_{2}{O}_{3}$.

To describe the communication between the chromium oxide and oxygen reduction pathways, parameter $\alpha$, which describes the accumulation effectiveness of an adsorption site, can be defined as a function of the particle number of accumulated $C{r}_{2}{O}_{3}$ such that $\begin{array}{rl}\alpha & =\alpha \left({m}_{C{r}_{2}{O}_{3}}\right)\end{array}$(1) $\begin{array}{rl}{\left(\frac{d{m}_{{O}^{-}}}{dt}\right)}_{actual}& ={\alpha }_{0}^{{O}^{-}}\alpha \cdot {\left(\frac{d{m}_{{O}^{-}}}{dt}\right)}_{no\phantom{\rule{thinmathspace}{0ex}}loss}\end{array}$(2) $\begin{array}{rl}{\left(\frac{d{m}_{C{r}_{2}{O}_{3}}}{dt}\right)}_{actual}& ={\alpha }_{0}^{C{r}_{2}{O}_{3}}\alpha \cdot {\left(\frac{d{m}_{C{r}_{2}{O}_{3}}}{dt}\right)}_{no\phantom{\rule{thinmathspace}{0ex}}loss}\end{array}$(3)

where ${\alpha }_{0}$ (with superscript ${O}^{-}$ or $C{r}_{2}{O}_{3}$) is the concentration effectiveness of an adsorption site. The subscript “no loss” indicates that the rate of change in mass of the species (${O}^{-}$ or $C{r}_{2}{O}_{3}$) is acquired directly from the SEAQT equation of motion for the case when ${\alpha }_{0}^{{O}^{-}}={\alpha }_{0}^{C{r}_{2}{O}_{3}}=\alpha =1$, i.e., when there are no accumulation or concentration limitations relative to the site. The actual rates for oxygen adsorption in eq. (2) and for $C{r}_{2}{O}_{3}$ accumulation in eq. (3) are scaled by $\alpha$ and ${\alpha }_{0}$ to that actually absorbed and accumulated. The physical meaning for the dependence of $\alpha$ on ${m}_{C{r}_{2}{O}_{3}}$ is that as the accumulation occurs, $C{r}_{2}{O}_{3}$ covers the adsorption site and gradually reduces the ability of the site to absorb and ionize the oxygen as well as the $Cr{O}_{3}$. Thus, the greater the accumulation, the smaller the value of $\alpha$ is. In this paper, a linear form is chosen for $\alpha$, namely, $\alpha \left({m}_{C{r}_{2}{O}_{3}}\right)=1-{m}_{C{r}_{2}{O}_{3}}/{m}_{0}$(4)

where ${m}_{C{r}_{2}{O}_{3}}$ is the mass of $C{r}_{2}{O}_{3}$ accumulated and ${m}_{0}$ is the threshold value for the accumulated mass when the adsorption site is totally disabled. Another parameter in eq. (2) is the concentration effectiveness ${\alpha }_{0}^{{O}^{-}}$, which is the original adsorption site effectiveness without degradation for the oxygen pathway. ${\alpha }_{0}^{{O}^{-}}=1$ indicates that the ${O}_{2}$ molecules can find enough adsorption sites to react. However, since the number of adsorption sites to ${O}_{2}$ molecules present is very small in a real fuel cell, ${\alpha }_{0}^{{O}^{-}}$ is chosen to be a very small number, i.e., $0.0003$, in eq. (2) consistent with the current densities considered here. In contrast, due to the low concentration of $C{r}_{2}{O}_{3}$, the concentration effectiveness for the chromium oxide molecules ${\alpha }_{0}^{C{r}_{2}{O}_{3}}$ is chosen to be $1$ in eq. (3). More discussion about ${\alpha }_{0}^{{O}^{-}}$ is given in Section 3.4.

## 2.3 SEAQT chemical/electrochemical reaction and transport model: Interaction calculation

After establishing the local systems (see Figures 1 and 2), a model for the interactions between systems represented by the arrows in Figures 1 and 2 is needed to calculate the mass and heat fluxes as well as the reactions. These interactions result in entropy generation that is captured within the SEAQT framework by its equation of motion, which is used to model the kinetics and dynamics of the chemical and electrochemical reaction and diffusion pathways directly. This first-principles approach incorporates quantum as well as classical microscopic information directly into the model via the system energy eigenstructure, which plays a key role in the equation of motion.

The underlying theory of our approach is introduced here using an isolated, finite energy eigenlevel quantum system. A discussion of how the theory applies more broadly to other kinds of systems is included in [32, 33, 34]. A quantum system is well defined by a group of energy eigenstates and associated eigenvalues $\left\{{e}_{i},i=1,2,\cdots \right\}$. A given thermodynamic (nonequilibrium or equilibrium) state of the system is uniquely defined by a probability distribution among the energy eigenstates denoted by $\left\{{p}_{i},i=1,2,3,\cdots \right\}$, and all possible distributions form the state space of the system. In particular, the stable equilibrium state of an isolated system has a probability distribution described by the Maxwellian distribution. An evolution of state occurs when the system in a state of nonequilibrium evolves to one at stable equilibrium; and it is the given set of probability distributions predicted by the SEAQT equation of motion, which define the unique thermodynamic path that the system takes during the evolution.

For an isolated system, energy and mass conservations require $\sum {e}_{i}{p}_{i}=E$ and $\sum {p}_{i}=1$.

The SEAQT equation of motion is constructed so that it takes the direction in state space, which coincides with that of steepest entropy ascent constrained by energy and mass conservations. At this level of description, the entropy of the system takes the form $S=-{k}_{b}\sum {p}_{i}ln{p}_{i}$.

The equation of motion for an isolated system comprised of a single elementary constituent (i.e. a single particle, a single assembly of indistinguishable particles, or a single field) is then written for the case when the state operator commutes with the Hamiltonian operator of energy eigenstates as $\frac{d{p}_{j}}{dt}=-\frac{1}{\tau }D=-\frac{1}{\tau }\left|\begin{array}{ccc}{p}_{j}ln{p}_{j}& \phantom{\rule{thickmathspace}{0ex}}{p}_{j}& \phantom{\rule{thickmathspace}{0ex}}{e}_{j}{p}_{j}\\ \sum {p}_{i}ln{p}_{i}& \phantom{\rule{thickmathspace}{0ex}}1& \phantom{\rule{thickmathspace}{0ex}}\sum {e}_{i}{p}_{i}\\ \sum {e}_{i}{p}_{i}ln{p}_{i}& \phantom{\rule{thickmathspace}{0ex}}\sum {e}_{i}{p}_{i}& \phantom{\rule{thickmathspace}{0ex}}\sum {e}_{i}^{2}{p}_{i}\end{array}\right|/\left|\begin{array}{cc}1& \phantom{\rule{thickmathspace}{0ex}}\sum {e}_{i}{p}_{i}\\ \sum {e}_{i}{p}_{i}& \phantom{\rule{thickmathspace}{0ex}}\sum {e}_{i}^{2}{p}_{i}\end{array}\right|$(5)

where the only generators of the motion are the identity and Hamiltonian operators. Unlike traditional equations of motion such as, for example, the classical [37] and quantum [13, 38] versions of the Boltzmann equation, eq. (5) requires no near-equilibrium assumption. It can be used in the nonlinear, far-from-equilibrium realm without any theoretical ambiguities or violations of the laws of thermodynamics or of quantum mechanics. The only parameter, which must be estimated, is the relaxation time $\tau$ associated with the dissipation term $D$ in eq. (5). The relaxation time can be a constant or a function of the local system eigenstructure and state [39, 40]. The dissipation term predicts the rate of entropy generation for a given system evolving in time. The relaxation time can be realistically estimated using experimental data, a more microscopic model (e.g., from kinetic theory) such as that for a reaction rate constant [30], diffusion coefficient, and heat transfer coefficient [33, 34], or even ab initio [40].

The solution of the SEAQT equation of motion exhibits a number of convenient features, which lead to a complete description of each nonequilibrium state and a fundamental as opposed to phenomenological definition of nonequilibrium temperature. More discussion is presented in reference [32], and an example is provided below. The energy eigenlevels of the system $\left\{{e}_{i},i=1,2,\cdots \right\}$ can be divided into to two sets $\left\{{e}_{iA},i=1,2,\cdots \right\}$ and $\left\{{e}_{iB},i=1,2,\cdots \right\}$ so that the system state space (Hilbert space) ${H}$ can be expressed by ${H}={{H}}_{A}\oplus {{H}}_{B}$, which is the sum of the two subspaces ${{H}}_{A}$ and ${{H}}_{B}$.

The system state can then be represented by the distributions for the two subspaces’ energy eigenlevels $\left\{{p}_{iA},i=1,2,3,\cdots \right\}$ and $\left\{{p}_{iB},i=1,2,\cdots \right\}$. If the distribution in subspace A yields to the canonical distribution of temperature ${T}_{A}$, the temperature of subspace $A$ is defined to be ${T}_{A}$. Given the division of the state space, if the system probability distributions in the two subspaces are both canonical distributions, the state of the system is called a second-order hypoequilibrium state [32], which can be described uniquely by the total probability in each subspace (${p}^{A}=\sum _{i}{p}_{iA}$ and ${p}^{B}=\sum _{i}{p}_{iB}$) and the temperature of each subspace ( ${T}_{A}$ and ${T}_{B}$). If the initial state of the system is a second-order hypoequilibrium state, then $\begin{array}{rl}{p}_{iA\left(B\right)}^{initial}& =\frac{{p}^{A\left(B\right)}}{{Z}^{A\left(B\right)}\left({T}_{A\left(B\right)}\right)}{e}^{\frac{-{e}_{iA\left(B\right)}}{{k}_{b}{T}_{A\left(B\right)}}},\phantom{\rule{1em}{0ex}}i=1,2,\cdots ,{n}_{A\left(B\right)}\end{array}$(6)

where ${k}_{b}$ is Boltzmann’s constant, ${n}_{A}$ and ${n}_{B}$ are the number of energy eigenlevels in each subspace, respectively, ${Z}^{A}\left({T}_{A}\right)$ is the partition function of subspace $A$ at temperature ${T}_{A}$, and ${Z}^{B}\left({T}_{B}\right)$ the partition function of subspace $B$ at temperature ${T}_{B}$. When constant relaxation times are used for local systems, Li and von Spakovsky [32] have proven using eq. (5) that the system remains in a second-order hypoequilibrium state throughout the nonequilibrium state evolution so that $\begin{array}{rl}{p}_{iA\left(B\right)}\left(t\right)& =\frac{{p}^{A\left(B\right)}\left(t\right)}{{Z}^{A\left(B\right)}\left({T}_{A\left(B\right)}\left(t\right)\right)}{e}^{\frac{-{e}_{iA\left(B\right)}}{{k}_{b}{T}_{A\left(B\right)}\left(t\right)}},\phantom{\rule{1em}{0ex}}i=1,2,\cdots ,{n}_{A\left(B\right)}\end{array}$(7)

Thus, each subspace has temperature fundamentally (i.e., canonically) defined throughout the entire nonequilibrium process of relaxation since the hypoequilibrium concept assumes that the system in a nonequilibrium state (one even far-from-equilibrium) is composed of more than one subsystem (or subspace), each with a different temperature. This temperature is a “nonequilibrium temperature” because the system need not be in stable equilibrium. Furthermore, when relaxation times are not constant, temperature can still be fundamentally defined using a generalized form of hypoequilibrium state [39]. One further point is that it would be of interest to investigate how this concept and these temperatures relate to the “constraint potentials” [41, 42] of the Rate-Controlled Constrained-Equilibrium (RCCE) model in which the RCCE constraints slowly vary and hence are rate controlling, representing the bottlenecks of the dynamics.

In calculating the interaction between local systems, a combined system is set up for each link (Figure 3a for a reaction and Figure 3b for diffusion). The energy eigenlevels of the combined system are a combination of the energy eigenlevels of the two linked local systems, and the state space is the sum of the Hilbert spaces of the two linked local systems. If the distribution in each linked local system is canonical, the nonequilibrium state of the combined system is in a second-order hypoequilibrium state even when far-from equilibrium. Using the equation of motion, eq. (5), for the combined system, the nonequilibrium evolution of the combined system can be determined so that the nonequilibrium interactions between the two linked local systems can be modeled.

Figure 3:

Behavior of the mass flux of Figure 1 for (a: left) a chemical reaction, (b: middle) diffusion, and (c: right) more than one interaction.

The state space of the reactive system is spanned by the eigenstates of the reactants and those of the products. Intermediate states between reactants and products are represented by a mixture of reactant states and product states. This state space can be represented by the Hilbert space ${H}={{H}}_{1}\oplus {{H}}_{2}$, which is the sum of reactant and product subspaces.

Each subspace of the state is generated from single molecular state spaces by the cross product. For example, the reaction $A+BC↔AC+B$

is modeled so that the two subspaces are given by ${{H}}_{1}={{H}}_{BC}\otimes {{H}}_{A}$ and ${{H}}_{2}={{H}}_{AC}\otimes {{H}}_{B}$.

The eigenstate of each particle is generated from the one-particle eigenstates available to each molecule. For example, the state space of BC is expanded to include degrees of freedom for translation, rotation, and vibration such that ${{H}}_{BC}=\text{span}\left\{|{t}_{x}{t}_{y}{t}_{z}{〉}_{bc}\otimes |lm{〉}_{bc}\otimes |\nu {〉}_{bc}\right\}$(8)

where $|{t}_{x}{t}_{y}{t}_{z}〉$ represents the translational eigenstates, $|lm〉$ the rotational ones, and $|\nu 〉$ the vibrational ones. Inside each ket ($|\cdots 〉$) are the quantum numbers for the one-particle eigenstate. Starting from these one-particle eigenstates, one can then form the system-level eigenstate of the reactants and products, which spans the system state space. The thermodynamic state of the system at any given instant of time is then represented by a probability distribution among the system eigenstates. More detailed descriptions of this novel approach describing the chemical and electrochemical kinetics of reacting quantum systems can be found in [29, 30, 43]. Given the energy eigenlevels of the system and each nonequilibrium state represented by a probability distribution $\left\{{p}_{i},i=1,2,\cdots \right\}$, the SEAQT equation of motion is used to model the chemical and electrochemical reaction pathways. In the SEAQT SOFC model, the reactant subspace and the product subspace are formed by two local systems linked by a chemical reaction (see Figure 3b).

The state space of the diffusion system is spanned by the eigenstates of the particle at position $x$ and position $x+\mathrm{\Delta }x$. Strictly speaking, $\mathrm{\Delta }x$ needs to be much larger than the de Broglie wavelength of the particle, which is about ${10}^{-11}$ m for the problem studied here where $\mathrm{\Delta }x=1$ $\text{\hspace{0.17em}}\text{μ}$m. This state space can be represented by the Hilbert space ${H}={{H}}_{1}\oplus {{H}}_{2}$, which is the sum of

the subspaces of the particle at positions $x$ (${{H}}_{1}$) and $x+\mathrm{\Delta }x$ (${{H}}_{2}$). The eigenstate of the particle at $x$ and $x+\mathrm{\Delta }x$ can be generated in the same way as is done for the reactants and products described in the previous section. The set of system eigenstates is composed of the eigenstates of the particle at $x$ and $x+\mathrm{\Delta }x$ and is represented by a probability distribution among the system eigenstates. Given the energy eigenstates of the system and each nonequilibrium state represented by a probability distribution $\left\{{p}_{i},i=1,2,\cdots \right\}$, the SEAQT equation of motion is used to model the interaction between two neighboring particle systems via the diffusion process. In the SEAQT SOFC model, subspace 1 and subspace 2 are formed by two local systems linked by diffusion (see Figure 3b).

The equation of motion eq. (5) uses probability distributions to represent system state, and this can be generalized to systems with more than one particle. Consider the interaction (either mass diffusion or reaction) between two linked local systems. Local system $1$ has ${m}_{1}$ particles and has the distribution among its energy eigenstates of $\left\{{p}_{i,1},i=1,2,\cdots ,{n}_{1}\right\}$ where ${n}_{1}$ stands for the number of eigenstates. Local system $2$ has ${m}_{2}$ particles and has the distribution among its energy eigenstates of $\left\{{p}_{i,2},i=1,2,\cdots ,{n}_{2}\right\}$ where ${n}_{2}$ stands for the number of eigenstates. The combined system has ${m}_{1}+{m}_{2}$ particles with a new distribution $\left\{{p}_{i,1}^{new},i=1,2,\cdots ,{n}_{1},{p}_{i,2}^{new},i=1,2,\cdots ,{n}_{2}\right\}$ where $\begin{array}{rl}{p}_{i,1\left(2\right)}^{new}& ={m}_{1\left(2\right)}{p}_{i,1\left(2\right)}/\left({m}_{1}+{m}_{2}\right),\phantom{\rule{1em}{0ex}}i=1,2,\cdots ,{n}_{1\left(2\right)}\end{array}$(9)

Using these new distributions as input to the equation of motion, eq. (5) leads to a prediction of their time rate of change, i.e., $\left\{\frac{d{p}_{i,1}^{new}}{dt},\phantom{\rule{thinmathspace}{0ex}}i=1,2,\cdots ,{n}_{1}\right\}$ and $\left\{\frac{d{p}_{i,2}^{new}}{dt},\phantom{\rule{thinmathspace}{0ex}}i=1,2,\cdots ,{n}_{2}\right\}$.

The mass flow for one energy eigenlevel of the system then is $\begin{array}{rl}\frac{d{m}_{i,1\left(2\right)}^{new}}{dt}& =\left({m}_{1}+{m}_{2}\right)\frac{d{p}_{i,1\left(2\right)}^{new}}{dt},\phantom{\rule{1em}{0ex}}i=1,2,\cdots ,{n}_{1\left(2\right)}\end{array}$(10)

The mass flow rate for one local system from the interaction is then given by $\begin{array}{rl}\frac{d{m}_{1\left(2\right)}}{dt}& =\left({m}_{1}+{m}_{2}\right)\sum _{i}\frac{d{p}_{i,1\left(2\right)}^{new}}{dt}\end{array}$(11)

Thus, one can calculate the mass flow along a given link for every single eigenstate, as well as the total mass flow to a local system. If a local system has multiple links (e.g., as in Figure 3c), the total flow to one energy eigenlevel of the local system is the sum of the flows through all the links. Recall that each link maintains the local system’s distribution canonical. Thus, the total effect from multiple links is to as well keep the local system’s distribution canonical [44] so that the temperature is well defined for local systems even though it undergoes a nonequilibrium interaction.

A local system can also interact with a temperature reservoir with the result that only energy and no mass is transferred. The effect of the energy flow is a redistribution of particle number among the available energy eigenlevels. This heat diffusion between the reservoir and local system is modeled by forming a composite system of a given local system (local system $1$) and its duplicate (local system $2$) so that the systems have the same mass and energy eigenstructure but different temperatures. Unlike the case for mass diffusion and that for reaction, which yield to one probability conservation law and one energy conservation law, the equation of motion for the case of heat diffusion yields to two probability conservation laws and one energy conservation law. For more details, the reader is referred to [33, 44].

## 2.4 Numerical method

Since the description of system state uses a probability distribution among energy eigenlevels in the SEAQT framework, the equation of motion consists of a group of first-order ordinary differential equations (ODEs), one for each level. When the system is above room temperature, as many as ${10}^{130}$ or more energy eigenlevels and, thus, ODEs are required. Li and von Spakovsky [30, 32] have developed a numerical density-of-states method to deal with systems with large numbers of energy eigenlevels. For a detailed mathematical discussion, the reader is referred to [32]. The key idea here is that if the energy spectrum is divided into intervals with energy differences much smaller than ${k}_{b}T$, the ${N}_{j}$ energy levels in the $i\mathrm{t}\mathrm{h}$ interval can be combined into one energy level with average energy ${E}_{i}$ and degeneracy ${N}_{i}$. The state of the system is represented by the probability ${P}_{i}$ in each energy interval instead of that for each energy eigenlevel. The result is that the equation of motion is rewritten as $\frac{d{P}_{j}}{dt}=-\frac{1}{\tau }D=-\frac{1}{\tau }\left|\begin{array}{ccc}{P}_{j}ln\left({P}_{j}/{N}_{j}\right)& {P}_{j}& {E}_{j}{P}_{j}\\ \sum {P}_{i}ln\left({P}_{i}/{N}_{i}\right)& 1& \sum {E}_{i}{P}_{i}\\ \sum {E}_{i}{P}_{i}ln\left({P}_{i}/{N}_{i}\right)& \sum {E}_{i}{P}_{i}& \sum {E}_{i}^{2}{P}_{i}\end{array}\right|/\left|\begin{array}{cc}1& \sum {E}_{i}{P}_{i}\\ \sum {E}_{i}{P}_{i}& \sum {E}_{i}^{2}{P}_{i}\end{array}\right|$(12)

where ${P}_{i}$ is the probability in the $i$th interval and ${E}_{i}=\left(\sum {e}_{i}\right)/{N}_{i}$. The process of determining the energy intervals for the density-of-states method proceeds as follows. Based on the nonequilibrium temperature range of interest, a probability distribution (e.g., a gamma distribution or a linear combination of such distributions) over a subsystem’s energy eigenlevels decays exponentially at the upper limit of this temperature range, i.e. at the highest energy levels. The tail of this probability distribution can, thus, be truncated at a relatively high cutoff energy without losing appreciable probability. From a numerical integration standpoint, the truncated probability distribution can be approximated by a piece-wise step function. The energy intervals to use for the calculation of moment integrals can then be selected based on the desired accuracy. These intervals are used for the coarse graining of the energy eigenstructure. This selection of intervals provides very good accuracy for macroscopic properties such as the probability, energy, and entropy, all of which are lower moments of a probability function. Moreover, these macroscopic properties determine the system state evolution via the SEAQT equation of motion. The propagation of errors in time for this first-order ODE can easily be maintained below a desirable level with a sufficiently fine set of intervals. For the particular SEAQT SOFC model developed here, $750$ energy intervals are used to approximate about ${10}^{130}$ energy levels in each local system with excellent accuracy.

The SEAQT oxygen reduction pathway model includes interactions at different timescales from those for reactions (relaxation times of ${10}^{-9}$ s) to those for diffusion in the bulk (relaxation times of ${10}^{-2}$ s). Both mesoscopic information (local system separation) and microscopic information (energy eigenstructure) are included in the model. Heterogeneous multiscale methods (HMMs) [22, 23, 45] are, thus, needed to solve the coupling between the microscopic and mesoscopic models. The fast system, which is primarily microscopic, has a time step of $\delta t$ and comes to equilibrium after a time $\mathrm{\Delta }t$. In contrast, the slow system, which is mostly mesoscopic/macroscopic, has a time step of $\delta T$ and comes to equilibrium after a time $\mathrm{\Delta }T$. The four “times” have the relation that $\delta t\ll \mathrm{\Delta }t\ll \delta T\ll \mathrm{\Delta }T$. For each step of the slow system, the information provided by the microscopic (fast) system is assumed constant. In this time period of one step for the slow system, the fast system is solved to an equilibrium (or a steady) state using the information provided by the mesoscopic/macroscopic (slow) system, which is also assumed constant. The two systems then exchange information after $\mathrm{\Delta }T$ and update the information from both the fast and slow systems. Thus, a multiscale system can be solved coupled, taking into account mesoscopic/macroscopic and microscopic evolutions. Generally, in other approaches, the mesoscopic/macroscopic (slow) system and microscopic (fast) system have different models, governing equations, and state parameters. These differences make the communication between the two systems difficult, and the communication requires a major part of the computational resources. This is not the case in the SEAQT framework since slow and fast systems use the same framework. Thus, for example, for the case of the oxygen reduction pathway, the slow system is the local system network that includes diffusion interactions in the pore, on the surface, and in the bulk and whose relaxation time is on the order of ${10}^{-2}$ s. The fast system is the local system network that has the remaining interactions including reactions and diffusion in the pore and whose relaxation time varies between ${10}^{-9}$ s and ${10}^{-7}$ s. Slow and fast systems are solved coupled using HMM. Using SEAQT, the model, governing equations, and state parameters of the slow and the fast models are self-consistent, which makes the application of HMM very efficient. Each of the simulations presented in this paper requires only a few hours on a PC workstation.

## 2.5 Parameter estimation

System parameters include the mean free paths in three regions (pore, surface, and bulk), the distance between local systems, and the total length from the interconnect–cathode interface to the cathode–electrolyte interface. According to experimental observation, the thickness of the LSCF cathode is about $20$ $\text{\hspace{0.17em}}\text{μ}$m, the width of the pore is about $0.9$ $\text{\hspace{0.17em}}\text{μ}$m, and the grain size is about $1$ $\text{\hspace{0.17em}}\text{μ}$m. In our model, the mean free path of the particle in the pore is assumed to be $1$ $\text{\hspace{0.17em}}\text{μ}$m in each of the three directions, which is approximately the size of a grain in the bulk and the width of the pore. For a particle on the surface, the mean free path is chosen to be $0.1$ $\text{\hspace{0.17em}}\text{μ}$m in the direction perpendicular to the surface, which is one order of magnitude smaller than that for the free particle. The mean free paths in the other two directions are the same as for the free particle. For the particle inside the bulk, the mean free path is set to be $1$ $\text{\hspace{0.17em}}\text{μ}$m in the direction the particle travels, while those in the other two perpendicular directions are chosen to be $0.1$ $\text{\hspace{0.17em}}\text{μ}$m.

The length from the interconnect to the TPB is set to $20$ $\text{\hspace{0.17em}}\text{μ}$m. The system is separated into $20$ local systems for each region, and the distance between the center point of two local systems is chosen to be $1$ $\text{\hspace{0.17em}}\text{μ}$m, which just happens to be equal to the mean free path in the direction a particle travels toward the electrolyte boundary. The choice of mean free path length and that for the distance between two local systems are, however, independent of each other. The results of a parametric study on the effects of these lengths and distances are presented in Sections 3.2 and 3.3 in which the influence of pore size and grain size change on cathode performance is examined. The particular parameter values chosen are those which allow comparison with experimental results and are somewhat different (by less than an order of magnitude) from the ones used in the transient process results presented in Section 3.1.

The translational energy eigenstates of ${O}_{2}$, ${O}^{-}$, ${O}_{ad}$, and ${O}^{X}$ are calculated from the 3D infinite well. The three dimensions of the well are assumed to be the mean free paths in three directions. Furthermore, the average pore size and average grain size acquired from experiment is used as input to estimate the mean free path and to determine the eigenstructures of the components from the 3D infinite well potential. The rotational eigenenergies of ${O}_{2}$, $Cr{O}_{3}$, and $C{r}_{2}{O}_{3}$ are calculated from the rigid motor model, while the eigenstate of the electron is set to be the one-level Fermi energy ${E}_{f}$ with a density of states of ${D}_{f}$. The potential energy of ${O}_{2}$ is set to its dissociation energy ${D}_{{O}_{2}}$. The potential energy of ${O}_{ad}$ is assumed to be $0.5{D}_{{O}_{2}}$, and that of ${O}^{-}$ to be $0.5{D}_{{O}_{2}}+2{E}_{f}$. The potential energies of $Cr{O}_{3}$ and $C{r}_{2}{O}_{3}$ are set to their dissociation energies. The order of magnitude of the density of states ${D}_{f}$ for the electron is estimated based on the electrochemical reaction equilibrium concentration.

A detailed model for the eigenstructure of the cathode material is not included here since it is beyond the scope of the present work. Such a model would permit the prediction of microstructural changes due to specific degradation mechanisms, i.e., to conduct a detailed study of the morphology of the electrode. Instead, average pore size and average grain size are used here not as geometric parameters in a phenomenological equation such as that which would incorporate the tortuosity but instead as parameters to generate particle quantum eigenlevels that can then be parametrically varied to study their effects on performance. To achieve a detailed morphological study of the cathode material would require energy eigenlevel results for the material from DFT, which could then be directly used to replace the eigenlevel information generated from the pore and grain sizes used in our present model. Obviously, more experimental or theoretical information about the energy eigenstates and potential energy of a particle would improve the model. However, the estimates used above are reasonable and provide accuracies within the correct order of magnitude for the energy, entropy, and chemical reaction equilibrium concentration.

The mass flow due to an interaction is predicted via the equation of motion eqs. (5) and (11). The relaxation time for each interaction can then be set based on an initial mass flow determined from a phenomenological equation and experiment or, alternatively, from, for example, the ab initio approach outlined in [40]. For the results given here, the former approach is used. Furthermore, for each reaction, the relaxation time is chosen so that the initial reaction rate predicted coincides with a corresponding reaction rate found from experiment, although this as well could be done as outlined in [40]. Now, using Fick’s law, $\begin{array}{r}J=-{D}_{diff}\frac{dc}{dx}\end{array}$(13)

where $J$ is the flux (particle${1}^{}$s${-1}^{}$m${-2}^{}$), ${D}_{diff}$ the diffusion coefficient (s${-1}^{}$m${2}^{}$), and $c$ the concentration (particle${1}^{}$m${-3}^{}$). Using $m$ the particle number in the finite volume and $A$ the cross-sectional area perpendicular to the flux direction, the mass flow can in turn be written as $\frac{dm}{dt}=-{D}_{diff}A\frac{dc}{dx}=-{D}_{diff}\frac{A}{V}\frac{dm}{dx}$(14)

The discrete form of eq. (14) gives the mass flow between two local systems, namely, $\frac{d{m}_{1}}{dt}=-{D}_{diff}\frac{{m}_{1}-{m}_{2}}{\left(\mathrm{\Delta }x{\right)}^{2}}$(15)

where ${m}_{1}$ and ${m}_{2}$ are the mass of two neighboring local systems and $\mathrm{\Delta }x$=A/V is the distance between the local systems. Using experimental values for ${D}_{diff}$ in eq. (15) and comparing with, for example, eq. (11), the relaxation time for diffusion can be determined. All diffusion as well as reaction relaxation times are summarized in Table 1. For the chromium oxide poisoning pathway (Table 1), the relaxation time of diffusion in the pore (D4) uses the relaxation time of diffusion in the pore (D1) for oxygen as an approximation. The relaxation time of reaction R6 for the chromium oxide pathway uses the relaxation time for R3 of the oxygen pathway, since R3 is the slower step in the two-step reduction R2 and R3.

As already mentioned, the relaxation times could be calculated ab initio from the eigenstructure in our framework [39, 40] or even via mechanics (e.g., kinetic theory [32, 34]), although neither is used here since the main goal of our paper is to determine how energy eigenlevel changes and occupation probabilities influence the overall performance thermodynamically. Instead, they are chosen based on experimental data. Furthermore, the reaction rate constants and transport coefficients predicted here and used in the SEAQT model are not constant but changing in time, since the nonequilibrium thermodynamic effects are fully taken into account for the coupled chemical reaction and mass and heat diffusion phenomena via changes in the energy eigenlevels and their occupation probabilities which in turn cause changes in the specific entropy and energy of the particles studied.

Finally, the dissipation term of the equation of motion is what captures the cathode polarization effects due to irreversibilities present in the chemical/electrochemical reaction and diffusion processes modeled here. It does so via a continual redistribution of energy among the available energy eigenstates of the system’s eigenstructure guided by the principle of steepest entropy ascent (or maximum entropy production). Although it is true that additional polarization losses resulting from induced field effects are not taken into account since they are beyond the scope of the present paper, they could be. A brief outline of how this could be done follows. To begin with, this type of cathode polarization affects the electrochemical potential of the ions. From a quantum mechanical viewpoint, an electric potential difference between the material surface and the pore modifies the ground energy of the reactants and products of the heterogeneous interfacial electrochemical reactions, which in turn changes the chemical equilibrium of the reactions. To account for this, the eigenlevels of the reactants and products in the SEAQT model must be shifted by the ion charge times the overpotential to account for this effect on the equilibrium state. In addition, this type of cathode polarization alters the chemical kinetics. This effect can be captured by a more general form of the SEAQT equation of motion, in which the relaxation time $\tau$ is a function of the system density operator $\rho$ or probability distribution. With both effects incorporated into the model, the structure of the system’s state space as well as the relaxation time $\tau$ of its equation of motion changes throughout the nonequilibrium evolution, accounting intrinsically for the additional polarization. As indicated earlier, the functional form of $\tau$ can be derived ab initio as is done in [40] or from Fermi’s golden rule [46] via a classical mechanical scattering process.

## 3.1 Transient to steady-state process

Among the many strengths of the SEAQT framework is the fact that it is thermodynamically rigorous in the nonequilibrium region, which permits study of the transient process of fuel cell startup. The following section presents the results for the transient behavior of the cathode system considered here. This is followed in Sections 3.2 and 3.3 by results from parametric studies of the influence on cathode performance of distances to the TPB and of mean-free-path lengths, respectively. The time axis for all the figures is on the log scale.

Figure 4:

Particle number and temperature evolutions at positions 1, 5, 10, 15, 20 (see Figures 1 and 2 for the position index) for three pathways (a–f) and total system entropy evolution (g). (a) Particle number and (b) temperature evolutions in the pore pathway; (c) particle number and (d) temperature evolutions in the surface pathway; (e) particle number and (f) temperature evolutions in the bulk pathway.

Throughout the entire transient timescale studied (Figure 4), there exist two timescales due to the nature of the timescale separation of the different phenomena: (i) from ${10}^{-10}$ s to ${10}^{-7}$ s, the local systems at different positions $2$ to $20$ in each pathway have almost the same concentration and temperature evolution resulting from chemical reactions with little influence from diffusion given the same initial conditions; and (ii) from ${10}^{-7}$ s to ${10}^{-4}$ s, significant mass diffusion occurs, which makes the local systems at different positions exhibit a different behavior resulting from the coupled reactions and diffusion. A strong coupling effect is evident in the pore pathway (Figures 4a and 4b) due to the difference in the characteristic times of diffusion and reaction in the pore not being as large (the reaction speed here is slowed down by the adsorption effectiveness at the surface) as those happening at the surface and in the bulk. Mass diffusion produces different particle numbers for systems at different positions, leading to different chemical reaction equilibriums across locations and different and more developed temperature profiles. The coupling also includes the influence of the inlet and outlet as well as the dominant role of the pore pathway in the communication between pathways. For the inlet and outlet, one can observe that (i) the closer a local system is to the inlet, the earlier the temperature evolution of the system departs from that of the other systems since at this location, the influence of diffusion is greater (see Figure 4b) and (ii) due to the outlet, the steady temperatures of local systems at position $20$ on the surface and bulk depart from the others (see Figures 4d and 4f). One result of the communication of pathways is that the timescales of surface and bulk are the same as that in the pore (see Figures 4c and 4e). The change in mass at the surface and the bulk does not occur primarily due to surface or bulk diffusion but instead to the chemical reaction balance moving with the pore pathway since the characteristic time of diffusion at the surface is about two orders of magnitude smaller than that in the pore. Thus, it is diffusion to the surface and the bulk and not along it that is the driver here, i.e., the preferred path for mass transfer and subsequent reaction is that through the pore followed by adsorption at the surface or to the bulk as opposed to adsorption first and then movement along the surface and inside the bulk. The other result is that the steady-state temperature of all local systems is $700$ K except that at position $20$ at the surface and the bulk, because the particle number in the pore is much larger than that on the surface and in the bulk. Furthermore, the competition effect between heat diffusion in the pore and that at the outlet can be observed in the bulk pathway as evidenced by the peak in the transient temperature evolution at position $20$ and the flat temperature evolution from ${10}^{-6}$ s to ${10}^{-5}$ s at other positions (see Figure 4f).

Finally, to understand which interaction is the most important in terms of entropy generation, the evolution in the system entropy is depicted in Figure 4g. It is clear that the entropy generation is strongest in the time range from ${10}^{-7}$ s to ${10}^{-4}$ s, which is the same range in which the greatest diffusion in the pore occurs. This is reasonable, since mass transfer in the pore is larger than for the other pathways and the mass in the pore accounts for most of the total mass in the system. The system reaches steady state after ${10}^{-3.5}$ s, which is much faster than the characteristic time of surface and bulk diffusion, and this is a direct result of the dominant role of the pore pathway in mass transfer.

## 3.2 Parametric study on the effects of changes in cathode thickness

The nonequilibrium SEAQT model is used to study the effects of different interconnect–TPB distances ($L=10$ $\text{\hspace{0.17em}}\text{μ}$m, $20$ $\text{\hspace{0.17em}}\text{μ}$m, $40$ $\text{\hspace{0.17em}}\text{μ}$m, and $60$ $\text{\hspace{0.17em}}\text{μ}$m) on cathode performance. The total charge delivered to the electrolyte is determined, and the time evolution of charge flux calculated. At steady state, the charge flux in the bulk ${J}_{bulk}$ equals about $200$ mA cm${-2}^{}$, and that on the surface ${J}_{surface}$ equals about $20$ mA cm${-2}^{}$. The total current and the ratio of surface flux to bulk flux (${J}_{surface}/{J}_{bulk}$) are plotted in Figure 5 to study the influence of the interconnect–TPB distance $L$. Results shown in Figure 5 indicate that the greater $L$ is, the smaller the total current is. This result is consistent with the idea that longer interconnect–TPB distances cause larger resistances for oxygen ion diffusion. At the same time, as $L$ increases, the ratio of surface flux to bulk flux (${J}_{surface}/{J}_{bulk}$) stays the same because the local systems on the surface and in the bulk are linked by chemical/electrochemical reactions at every position. The ratio of local system surface and local system bulk concentrations at every position satisfies the same chemical equilibrium ratio. The decreasing trend of current (or current flux) when distance increases can be regarded as one possible explanation for the degradation in performance due to microstructural changes. However, the model shows that the influence of path length changes on performance is not very significant, which is consistent with the phenomena previously observed, namely, that the diffusion process is not the limiting step.

Figure 5:

Steady-state total current (black line) comparison and current flux ratio (red line, ${J}_{surface}/{J}_{bulk}$) for different interconnect–TPB distances resulting from microstructural changes due to grain-size growth.

## 3.3 Parametric study on the effects of particle mean free path

The nonequilibrium SEAQT model is also used to study the effects on performance of different mean free paths. The mean free path for all system dimensions is varied by the following factors: $1.1$, $1.05$, $0.95$, and $0.9$ relative to the mean free paths defined in Section 2.5. At steady state, the currents in the bulk and that on the surface are plotted in Figure 6a and 6b to study the influence of these lengths. The mean free path is related to the microscopic structure. A larger pore size (see Figure 7) in the cathode provides a larger mean free path in the pore, and a larger grain size of the material (see Figure 7) provides a larger mean free path on the surface and in the bulk. From an atomistic viewpoint, the mean free path affects the energy eigenstates. The smaller the mean free path, the lower the density of states at a given energy is and the lower the specific entropy for a given temperature. As a consequence, each eigenstate change affects the final cathode system steady state and the speed at which it is approached as well as the rate of entropy generation. In fact, for a given reaction, i.e., a given pair of linked local systems, if the specific entropy of the reactant increases, the reaction balance moves toward the reactant side, resulting in more reactant as the paired local systems approach chemical stable equilibrium. For diffusion, the larger mean free path results in a larger diffusion coefficient and faster diffusion; but the final stable equilibrium state approached by a given pair of linked local systems remains the same, since the diffusing component has the same energy eigenstructure and specific entropy at different positions. Thus, the dominant factor determining current at steady state is the influence of mean free path on the reactions. Neither diffusion in the pore nor the speed at which it occurs is limiting since as seen in Figure 4a, oxygen already has a relatively uniform distribution along the pore after time ${10}^{-4}$ s.

Results for different mean free paths in the pore (see Figure 6a) show that the greater the mean free path of the pore, the smaller the surface as well as bulk currents. With increasing mean free path in the pore, the specific entropy in the pore increases and the chemical reaction balance of the adsorption moves toward the oxygen molecule side. When the inlet particle number remains the same, there is less oxygen absorbed at the surface and into the bulk so that the currents on the surface and in the bulk both decrease.

Results for different mean free paths on the surface and in the bulk (see Figure 6b) show that the greater the mean free path of the surface and bulk, the larger the surface as well as bulk currents. The mean free path on the surface and in the bulk increases the specific entropy on the surface and in the bulk, while the chemical reaction balance of the adsorption moves toward the oxygen ion side. When the particle number of the inlet remains the same, there are more oxygen molecules absorbed to the surface and bulk so that the currents on the surface and in the bulk both increase. Another phenomena observed is that the currents are negative when the mean free path is $0.9$. This is because when the mean free path of the surface and bulk are too small, the equilibrium concentration in the bulk and on the surface can be even smaller than the concentration of the outlet, which is decided by other parts of the fuel cell.

Figure 6:

Steady-state current comparison for different mean free paths (a: left) in the pore; (b: right) on the surface and in the bulk.

Figure 7:

SEM image of the microstructure of the cathode material.

Table 2:

Ohmic resistance of the cathode sample after the heating process.

Figure 8:

SEM images of the microstructure of the cathode after  the heat processing for (a) 0 h, (b) 20 h, and (c) 40 h.

Long-term operation can change the microstructure of the cathode material. This process can be studied by thermal treatment under current load. (See the supplementary material for details about the experimental procedures.) The resistance of a sample of cathode material is measured for different durations of the thermal treatment, and the results are reported in Table 2. Scanning electron microscopy (SEM) images of the microstructure of the cathode sample for these different durations are shown in Figure 8. As is clearly seen, both grain and pore sizes and, thus, mean free path are significantly affected by the duration of the thermal treatment. Pore and grain sizes increase with duration, and as a consequence, the ohmic resistance increases, which, of course, means that the current decreases. This is consistent with the nonequilibrium SEAQT model results, which are used here to more deeply analyze the experimental results. A summary of this analysis is provided in Table 3. As pore size increases, fewer oxygen molecules are adsorbed leading to a decrease in current. In contrast, increasing grain size leads to more oxygen molecules being adsorbed and to increasing current. Thus, there are competing effects on current from microstructural changes. The experiment shows consistent with our model prediction that the net effect is a decrease in current, which means that the influence of pore size is dominant. After microstructural changes, the chemical reaction balance moves toward the oxygen molecule side. Since oxygen in the pore can be regarded as an ideal gas, the change of pore size has a significant influence on the mean free path in the pore. Even though the mean free paths on the surface and in the bulk are also changed by the grain size increase, other factors such as lattice structure and vacancy density limit the effects of these changes. Thus, decreasing pore size is a better way to improve cathode performance since it moves the chemical reaction balance toward the oxygen ion allowing more oxygen molecules to be adsorbed with the result that the current is higher. Of course, the caveat is a smaller diffusion speed. However, as discussed above, the limiting step is not the diffusion but rather the effects of mean free path on the reaction. Thus, the decrease in diffusion speed is easily offset by the gain relative to the reactions.

Table 3:

Analysis of the experimental results of the heating process using the nonequilibrium SEAQT model.

## 3.4 Chromium oxide accumulation and cathode material poisoning

The nonequilibrium SEAQT model for the chromium oxide pathway is used to study the kinetics of chromium (III) oxide ($C{r}_{2}{O}_{3}$) accumulation and chromium (VI) oxide gas ($Cr{O}_{3}$) distribution in the pore. Results provide insight into the influence on performance of chromium poisoning. The model in this section uses an initial concentration of $Cr{O}_{3}$ in the pore of $1×{10}^{-12}$ mol cm${-3}^{}$. Based on Kestell [12], the inlet concentration of $Cr{O}_{3}$ is chosen to be $2.33×{10}^{-10}$ mol cm${-3}^{}$, which is more than three orders of magnitude smaller than the concentration of oxygen. Thus, the competition of $Cr{O}_{3}$ reduction with that of oxygen is relatively small. In addition, heat diffusion occurs between the chromium oxide and other particles (e.g., ${O}_{2}$) in the pore and on the surface. Because the chromium oxide concentration is low, this diffusion can be modeled by an interaction with a reservoir at $700$ K, which is the steady-state temperature of the oxygen reduction pathway in the pore. Furthermore, since accumulation on the surface removes the $C{r}_{2}{O}_{3}$ product of the $Cr{O}_{3}$ reduction reaction pathway from the pore, the outlet boundary condition for the $C{r}_{2}{O}_{3}$ concentration is set to a very low value. For the results of the transient process, it is assumed that the accumulation does not slow down the reduction process either for the $Cr{O}_{3}$ or for the oxygen since this is a transient process and the timescale is smaller than the timescale for degradation. In a subsequent long-term study, the degradation effect is taken into account. Figure 9a shows the time evolution of $Cr{O}_{3}$ concentration in the pore, while Figure 9b shows the $C{r}_{2}{O}_{3}$ accumulation on the cathode surface. The results indicate that the diffusion process from the inlet concentration into the pore eventually results in a uniform distribution, which leads to a constant accumulation rate of $C{r}_{2}{O}_{3}$ on the surface. During the transient process and after the initial time, the inlet exhibits a lower accumulation rate of $C{r}_{2}{O}_{3}$ as indicated by the slight curvature to the right from top to bottom of the vertical colors in Figure 9b. This is because the diffusion and reduction reaction are in competition with each other, which contrasts with the last local system in the pore where all of the $Cr{O}_{3}$ is involved in the reduction reaction. Unlike the oxygen pathways whose transient process goes through multiple timescales, the transient process here only occurs over a single timescale, i.e., that from 0 s to 10${-4}^{}$ s. Thus, steady state and a uniform accumulation rate are reached very quickly.

Figure 9:

Time evolutions of (a) the pore $Cr{O}_{3}$ concentration distribution and (b) the surface $C{r}_{2}{O}_{3}$ accumulation.

The rate of $C{r}_{2}{O}_{3}$ accumulation is very slow relative to the timescales for reaction and diffusion. Thus, to assess if the chromium reduction process is delayed by the accumulation in the long term, the mass rate of change of the accumulated chromium (III) oxide expressed with eqs. (3) and (4) can be used. These equations are repeated here but with the subscript $C{r}_{2}{O}_{3}$ on $m$ dropped, i.e., $\begin{array}{rl}\alpha \left(m\right)& =1-\frac{m}{{m}_{0}}\end{array}$(16) $\begin{array}{rl}\frac{dm}{dt}& ={\alpha }_{0}^{C{r}_{2}{O}_{3}}\alpha \left(m\right){\left(\frac{dm}{dt}\right)}_{no\phantom{\rule{thinmathspace}{0ex}}loss}={\alpha }_{0}^{C{r}_{2}{O}_{3}}\left(1-\frac{m}{{m}_{0}}\right){r}_{0}\end{array}$(17)

In this last expression, ${r}_{0}=\left(dm/dt{\right)}_{no\phantom{\rule{thinmathspace}{0ex}}loss}$ is the initial reaction rate of $C{r}_{2}{O}_{3}$ accumulation equal to the steady-state rate acquired from the nonequilibrium SEAQT model for the chromium oxide pathway when the accumulating $C{r}_{2}{O}_{3}$ has almost no influence on the reaction rate. The solution of this ODE is $m\left(t\right)={m}_{0}\left(1-{e}^{-{\alpha }_{0}^{C{r}_{2}{O}_{3}}\frac{{r}_{0}}{{m}_{0}}t}\right)$(18)

or in terms of the accumulation site effectiveness $\alpha \left(t\right)=\alpha \left(m\left(t\right)\right)={e}^{-{\alpha }_{0}^{C{r}_{2}{O}_{3}}\frac{{r}_{0}}{{m}_{0}}t}$(19)

The total adsorption site effectiveness then becomes $\begin{array}{rl}{\alpha }_{total}^{{O}^{-}}\left(t\right)& ={\alpha }_{0}^{{O}^{-}}\alpha \left(m\left(t\right)\right)={\alpha }_{0}^{{O}^{-}}{e}^{-{\alpha }_{0}^{C{r}_{2}{O}_{3}}\frac{{r}_{0}}{{m}_{0}}t}\end{array}$(20) $\begin{array}{rl}{\alpha }_{total}^{C{r}_{2}{O}_{3}}\left(t\right)& ={\alpha }_{0}^{C{r}_{2}{O}_{3}}\alpha \left(m\left(t\right)\right)={\alpha }_{0}^{C{r}_{2}{O}_{3}}{e}^{-{\alpha }_{0}^{C{r}_{2}{O}_{3}}\frac{{r}_{0}}{{m}_{0}}t}\end{array}$(21)

where ${\alpha }_{0}^{{O}^{-}}$ is $0.0003$ for oxygen and ${\alpha }_{0}^{C{r}_{2}{O}_{3}}$ is $1$ for chromium oxide.

Figure 10:

Evolution of (a) the adsorption site effectiveness, (b) the surface and bulk currents for ${m}_{0}/{r}_{0}=5$ s and ${\alpha }_{0}^{{O}^{-}}=0.0003$, (c) the adsorption site effectiveness, and (d) the surface and bulk currents for ${m}_{0}/{r}_{0}=5$ s and ${\alpha }_{0}^{{O}^{-}}=1$.

Based on the results of the transient process, the chromium oxide accumulation process is quite uniform on the surface, which means that ${r}_{0}$ can be assumed to be constant across different positions on the cathode surface. Thus, the chromium oxide accumulation effectiveness and total adsorption site effectiveness at different positions all yield to the solutions given by eqs. (19) and (21). This result is used in the parametric study of total adsorption site effectiveness. Relatively long times are required for chromium poisoning to reduce cathode performance, which as already indicated above are much longer than the timescales of the transient processes for oxygen and chromium oxide reduction. At any instant of time $t$, the oxygen reduction pathway at steady state exhibits a constant total adsorption site effectiveness ${\alpha }_{total}^{{O}^{-}}\left(t\right)$. The steady-state surface and bulk currents for the oxygen reduction pathway for different ${\alpha }_{total}$ are calculated and plotted as a function of time in Figure 10b. Figure 10a provides the total adsorption site effectiveness evolution. Both figures are given for a particular value of the ratio ${m}_{0}/{r}_{0}$, which is the ratio between the threshold value for the accumulated mass when the adsorption site is totally disabled and the value for the steady-state reaction rate value for ${m}_{C{r}_{2}{O}_{3}}$ accumulation. In effect, this ratio is a characteristic time for chromium oxide poisoning. For different ${m}_{0}/{r}_{0}$, the shape of these curves is the same, with only the scaling of the time axis affected. As can be seen from Figure 10b, the performance of the oxygen pathway decreases due to the chromium poisoning process. Of course, the value of $5$ s used for ${m}_{0}/{r}_{0}$ in Figures 10a and 10b is somewhat arbitrary since a threshold value for ${m}_{0}$ should be determined from experimental data. This is not done here since the purpose is to simply demonstrate how the chromium poisoning process influences the oxygen reduction pathway. A more precise value could nonetheless be determined from, for example, experimental data such as that found in [11] or [10]. However, that is left for a future paper.

Finally, in order to illustrate which process limits the current generation, the focus is placed on the response of current generation to adsorption site degradation. This is done by setting ${\alpha }_{0}=1$, which implies that there are enough adsorption sites at the initial state to easily accommodate the adsorption of every oxygen molecule. In other words, for the cathode system at steady state, continuity requires that the speed of the adsorption process must be the same as that for the transport as long as there is a sufficient number of adsorption sites for the oxygen molecules and sufficient degradation of the sites has not yet occurred. Hence, to eliminate the first of these as a possible bottleneck or source of performance degradation, the value of ${\alpha }_{0}^{{O}^{-}}=1$ is chosen as indicated above for the results reported in Figures 10c and 10d. Thus, any degradation in performance seen in current in this figure must be due only to the degradation of the sites themselves. For example, in Figures 10d, there is little change in performance before 30 s (i.e., when $\alpha ={10}^{-6}$) even though chromium accumulation has led to a significant decrease of total adsorption site effectiveness as seen in Figures 10c. The reason is that before $30$ s (i.e. $\alpha >{10}^{-6}$), adsorption is not the limiting step for the mass transfer. On the other hand, after $30$ s (i.e. $\alpha <{10}^{-6}$), total adsorption site effectiveness has decreased sufficiently due to chromium oxide accumulation to exert a strong influence on performance and, thus, the ability to adsorb becomes the limiting factor.

## 4 Conclusions

Results presented here for the application of the SEAQT framework to a SOFC cathode include those for (i) the transient process of oxygen reduction at the cathode, (ii) the effects of changes in mean free path and interconnect–TPB distances on cathode performance due to microstructural changes, and (iii) the kinetics of the chromium (III) oxide accumulation and chromium oxide (VI) poisoning processes. In the transient study, the multi-timescale features of different nonequilibrium phenomena and how they communicate with each other are illustrated. With the parametric study on changes in mean free path and interconnect–TPB distances, a qualitative explanation of how degradation occurs via microstructural changes is provided so that a clearer understanding of the coupled phenomena underlying the experimental results are obtained. Model predictions and experimental results suggest that pore size reduction as opposed to grain size increases have a more profound effect on improving cathode performance since the former moves the chemical reaction balance toward the oxygen ion allowing more oxygen molecules to be adsorbed at a rate significantly higher than that achieved with the grain size increases. A clearer understanding of the degradation in cathode performance of chromium poisoning is also obtained from the model, underscoring its ability to distinguish between the effects on performance of reaction, transport, and adsorption site effectiveness. This ability makes possible the study in greater detail of the contributions that each of these phenomena individually make to cathode performance.

Finally, the SEAQT framework is able to effectively model coupled nonequilibrium phenomena, including mass and heat diffusion, and chemical and electrochemical reactions. A major benefit of this framework is that it accounts for both the quantum mechanical and thermodynamic features of the system and the nonequilibrium processes it undergoes. It also allows for a larger range of time and spatial scales to be used in a single model. In addition, it is thermodynamically rigorous even in the far-from-equilibrium realm and provides a consistent description at all scales using entropy and energy as state variables. The result is that particle number and temperature evolutions can be predicted with great efficiency (e.g., in only a few hours on a PC workstation for each of the simulations presented in this paper).

## Acknowledgements:

We acknowledge Advanced Research Computing at Virginia Tech for providing the necessary computational resources and technical support that have contributed to this work (http://www.arc.vt.edu). The authors would like to acknowledge the ASME as the original publisher of portions of the material in this paper in the proceedings of the 2015 ASME International Mechanical Engineering Congress and Exposition (IMECE2015) where this work was initially presented [35].

## References

• [1]

Li Y., Gemmen R. and Liu X., Oxygen reduction and transportation mechanisms in solid oxide fuel cell cathodes, J. Power Sources 195 (2010), 3345–3358.

• [2]

Gong M., Gemmen R. S. and Liu X., Modeling of oxygen reduction mechanism for 3pb and 2pb pathways at solid oxide fuel cell cathode from multi-step charge transfer, J. Power Sources 201 (2012), 204–218.

• [3]

Yokokawa H., Tu H., Iwanschitz B. and Mai A., Fundamental mechanisms limiting solid oxide fuel cell durability, J. Power Sources 182 (2008), 400–412.

• [4]

Liu Y., Chen J., Wang F., Chi B., Pu J. and Jian L., Performance stability of impregnated La0.6Sr0.4Co0.2Fe0.8O3–δ–Y2O3 stabilized ZrO2 cathodes of intermediate temperature solid oxide fuel cells, Int. J. Hydrogen Energy 39 (2014), 3404–3411.

• [5]

Lu K. and Shen F., Effect of stoichiometry on (La0.6Sr0.4)xCo0.2Fe0.8O3 cathode evolution in solid oxide fuel cells, J. Power Sources 267 (2014a), 421–429. Crossref

• [6]

Lu K. and Shen F., Long term behaviors of La0.8Sr0.2MnO3 and La0.6Sr0.4Co0.2Fe0.8O3 as cathodes for solid oxide fuel cells, Int. J. Hydrogen Energy 39 (2014b), 7963–7971.

• [7]

Asadi A. A., Behrouzifar A., Iravaninia M., Mohammadi T. and Pak A., Preparation and oxygen permeation of La0.6Sr0.4Co0.2Fe0.8O3-δ perovskite-type membranes: Experimental study and mathematical modeling, Ind. Eng. Chem. Res. 51 (2012), 3069–3080.

• [8]

Teraoka Y., Nobunaga T., Okamoto K., Miura N. and Yamazoe N., Influence of constituent metal cations in substituted lacoo3 on mixed conductivity and oxygen permeability, Solid State Ionics 48 (1991), 207–212.

• [9]

Mori M., Yamamoto T., Itoh H., Inaba H. and Tagawa H., Thermal expansion of nickel-zirconia anodes in solid oxide fuel cells during fabrication and operation, J. Electrochem. Soc. 145 (1998), 1374–1381.

• [10]

Paulson S. and Birss V., Chromium poisoning of LSM-YSZ SOFC cathodes i. detailed study of the distribution of chromium species at a porous, single-phase cathode, J. Electrochem. Soc. 151 (2004), A1961–A1968.

• [11]

Bentzen J. J., Høgh J. V. T., Barfod R. and Hagen A., Chromium poisoning of Chromium poisoning of LSM/YSZ and LSCF/CGO composite cathodes, Fuel Cells 9 (2009), 823–832.

• [12]

Kestell G. M., Model of chromium poisoning in the cathode of a solid oxide fuel cell using the lattice Boltzmann method, M.S. thesis, Virginia Tech, M.E. Department, 2010.

• [13]

J. M. Torres-Rincon, Hadronic transport coefficients from effective field theories, Springer, New York City, 2014. Google Scholar

• [14]

P. Vogl, T. Kubis, The non-equilibrium Green’s function method: an introduction, J. Comp. Elec. 9(2010), 237–242.

• [15]

C. H. Cheng, Y. W. Chang and C. W. Hong, Multiscale parametric studies on the transport phenomenon of a solid oxide fuel cell, J. Fuel Cell Sci. Technol. 2 (2005), 219.

• [16]

A. Modak and M. Lusk, Kinetic monte carlo simulation of a solid-oxide fuel cell: I. Open-circuit voltage and double layer structure, Solid State Ionics 176(2005), 2181–2191. Google Scholar

• [17]

A. S. Joshi, K. N. Grew, A. A. Peracchio and W. K. S. Chiu, Lattice Boltzmann modeling of 2d gas transport in a solid oxide fuel cell anode, J. Power Sources 164(2007), 631–638.

• [18]

A. Haghighat, Monte Carlo Methods for Particle Transport, CRC Press, Boca Raton, Florida, 2014. Google Scholar

• [19]

N. Autissier, D. Larrain, J. Van herle and D. Favrat, CFD simulation tool for solid oxide fuel cells, J. Power Sources 131(2004), 313–319. Google Scholar

• [20]

M. Andersson, J. Yuan and B. Sundén, Review on modeling development formultiscale chemical reactions coupled transport phenomena in solid oxide fuel cells, Appl. Energy 87(2010), 1461–1476.

• [21]

E. Weinan, Principles of Multiscale Modeling, Cambridge University Press, Cambridge, UK, 2011. Google Scholar

• [22]

E. Weinan, B. Engquist, X. Li, W. Ren and E. Vanden-Eijnden, Heterogeneous multiscale methods: A review, Commun. Comput. Phys. 2(2007), 367–450. Google Scholar

• [23]

E. Weinan, W. Ren and E. Vanden-Eijnden, A general strategy for designing seamless multiscale methods, J. Comput. Phys. 228 (2009), 5437–5453.

• [24]

G. P. Beretta, E. P. Gyftopoulos, J. L. Park and G. N. Hatsopoulos, Quantum thermodynamics. A new equation of motion for a single constituent of matter, Il Nuovo Cimento B Ser. 11. 82(1984), 169–191. Google Scholar

• [25]

G. P. Beretta, E. P. Gyftopoulos and J. L. Park, Quantum thermodynamics. A new equation of motion for a general quantum system, Il Nuovo Cimento B Ser. 11. 87(1985), 77–97. Google Scholar

• [26]

G. P. Beretta, Nonlinear model dynamics for closed-system, constrained, maximal-entropy-generation relaxation by energy redistribution, Phys. Rev. E 73 (2006), 026113.

• [27]

G. P. Beretta, Nonlinear quantum evolution equations to model irreversible adiabatic relaxation with maximal entropy production and other nonunitary processes, Rep. Math. Phys. 64(2009), 139–168.

• [28]

G. P. Beretta, Steepest entropy ascent model for far-nonequilibrium thermodynamics: Unified implementation of the maximum entropy production principle, Phys. Rev. E 90 (2014), 042113.

• [29]

G. P. Beretta, O. Al-Abbasi and M. R. von Spakovsky, Steepest-entropy-ascent quantum thermodynamic framework for describing the non-equilibrium behavior of a chemically reactive system at an atomistic level, Phys. Rev. E 95 (2017), 042139.

• [30]

G. Li, O. Al-Abbasi and M. R. von Spakovsky, Atomistic-level non-equilibrium model for chemically reactive systems based on steepest-entropy-ascent quantum thermodynamics, J. Phys.: Conf. Ser. 538 (2014), 012013. Google Scholar

• [31]

M. R. von Spakovsky and J. Gemmer, Some trends in quantum thermodynamics, Entropy 16 (2014), 3434.

• [32]

G. Li and M. R. von Spakovsky, Steepest-entropy-ascent quantum thermodynamic modeling of the relaxation process of isolated chemically reactive systems using density of states and the concept of hypoequilibrium state, Phys. Rev. E 93 (2016), 012137.

• [33]

G. Li and M. R. von Spakovsky, Generalized thermodynamic relations for a system experiencing heat and mass diffusion in the far-from-equilibrium realm based on steepest entropy ascent, Phys. Rev. E 94 (2016), 032117.

• [34]

G. Li and M. R. von Spakovsky, Modeling the nonequilibrium effects in a nonquasi-equilibrium thermodynamic cycle based on steepest entropy ascent and an isothermal-isobaric ensemble, Energy 115 (Part 1) (2016), 498–512. Google Scholar

• [35]

G. Li and M. R. von Spakovsky, Study of the transient behavior and microstructure degradation of a SOFC cathode using an oxygen reduction model based on steepest-entropy-ascent quantum thermodynamics, In: 2015 IMECE, ASME, 2015. No. IMECE2015-53726. Google Scholar

• [36]

G. Li and M. R. von Spakovsky, Study on nonequilibrium size and concentration effects on the heat and mass diffusion of indistinguishable particles using steepest-entropy-ascent quantum thermodynamics, J. Heat Transfer. 139 (2017), 122003. Google Scholar

• [37]

S. Harris, An Introduction to the Theory of the Boltzmann Equation, Dover Publications, Mineola, New York, 2004. Google Scholar

• [38]

A. Rossani and G. Kaniadakis, A generalized quasi-classical Boltzmann equation, Physica A: Stat. Mech. Appl. 277(2000), 349–358.

• [39]

G. Li, G. P. Beretta and M. R. von Spakovsky, Steepest-entropy-ascent thermodynamics under non-uniform state space geometry, (unpublished). Google Scholar

• [40]

I. Kim and M. R. von Spakovsky, Ab initio relaxation times and time-dependent hamiltonians within the steepest-entropyascent quantum thermodynamic framework, Phys. Rev. E 96 (2017), 022129.

• [41]

G. P. Beretta, J. C. Keck, M. Janbozorgi and H. Metghalchi, The rate-controlled constrained-equilibriumapproach to far-fromlocal- equilibrium thermodynamics, Entropy 14(2012), 92–130.

• [42]

G. P. Beretta, M. Janbozorgi and H. Metghalchi, Degree of disequilibrium analysis for automatic selection of kinetic constraints in the rate-controlled constrained-equilibrium method, Combust. Flame. 168(2016), 342–364.

• [43]

O. Al-Abbasi, Modeling the non-equilibrium behavior of chemically reactive atomistic level systems using steepest-entropyascent quantum thermodynamics, Ph.D. dissertation, Virginia Tech, M.E. Department, 2013. Google Scholar

• [44]

G. Li and M. R. von Spakovsky, Steepest-entropy-ascent quantum thermodynamic modeling of the far-from-equilibrium interactions between non-equilibrium systems of indistinguishable particle ensembles, 2016. arXiv:1601.02703. Google Scholar

• [45]

W. Ren, E.Weinan, Heterogeneous multiscale method for themodeling of complex fluids and micro-fluidics, J. Comput. Phys. 204 (2005) 1–26.

• [46]

G. Li, M. R. von Spakovsky and H. Celine, Steepest entropy ascent quantum thermodynamic model of electron and phonon transport, (unpublished). Google Scholar

## Supplemental Material

Revised: 2017-09-26

Accepted: 2017-10-13

Published Online: 2017-11-14

Published in Print: 2018-01-26

Funding for this research was provided by the US Office of Naval Research under Award No. N00014-11-1-0266.

Citation Information: Journal of Non-Equilibrium Thermodynamics, Volume 43, Issue 1, Pages 21–42, ISSN (Online) 1437-4358, ISSN (Print) 0340-0204,

Export Citation

© 2018 Walter de Gruyter GmbH, Berlin/Boston.