Electrolyte continuum treatment

Integration of KMC simulations with larger-scale models such as finite element continuum approaches can enhance the impact of a KMC simulation. In many cases, certain components of an electrochemical device can be effectively approximated by a bulk continuum treatment (e.g., ionic diffusion through the bulk of an electrolyte) while other components of a device that involve interfacial transport require discrete treatments like either Butler-Volmer empiricism or an atomistic treatment to capture important details related to the distribution of charge in the double layer. 

Note that the MCT treatment presented above is quite general and can be extended to describe relaxation in many different systems, such as orientational relaxation in dipolar liquids [54]. This approach can also be extended to multicomponent systems, in particular to describe transport properties of electrolyte solutions [55]. The usefulness and the simplicity of the expressions lies in the separation between the single particle and collective dynamics (as in Eq. 98). Actually this sepration allows one to make connections with hydrodynamic (or continuum frameowrk) models where only the collective dynamics is included but the single particle motion is ignored. However, the same separation is also the reason for the failure... 

Aside from the energetics and dynamics of solvent reorganization, the roles of dissolved electrolyte on ET processes carried out in solution with finite ionic strengths have been the subject of recent experimental [56] and theoretical [57] study. The various analyses suggest that continuum-based treatments or Debye-Htickel descriptions of ionic atmospheres are inadequate and point to the importance of specific ion-pairing effects, including dynamic as well as energetic factors. 

In this chapter some aspects of the present state of the concept of ion association in the theory of electrolyte solutions will be reviewed. For simplification our consideration will be restricted to a symmetrical electrolyte. It will be demonstrated that the concept of ion association is useful not only to describe such properties as osmotic and activity coefficients, electroconductivity and dielectric constant of nonaqueous electrolyte solutions, which traditionally are explained using the ion association ideas, but also for the treatment of electrolyte contributions to the intramolecular electron transfer in weakly polar solvents [21, 22] and for the interpretation of specific anomalous properties of electrical double layer in low temperature region [23, 24], The majority of these properties can be described within the McMillan-Mayer or ion approach when the solvent is considered as a dielectric continuum and only ions are treated explicitly. However, the description of dielectric properties also requires the solvent molecules being explicitly taken into account which can be done at the Born-Oppenheimer or ion-molecular approach. This approach also leads to the correct description of different solvation effects. We should also note that effects of ion association require a different treatment of the thermodynamic and electrical properties. For the thermodynamic properties such as the osmotic and activity coefficients or the adsorption coefficient of electrical double layer, the ion pairs give a direct contribution and these properties are described correctly in the framework of AMSA theory. Since the ion pairs have no free electric charges, they give polarization effects only for such electrical properties as electroconductivity, dielectric constant or capacitance of electrical double layer. Hence, to describe the electrical properties, it is more convenient to modify MSA-MAL approach by including the ion pairs as new polar entities. 

Macroscale cell-level models are able to provide a great amount of insight into the operation and performance of SOFCs. With the newer mesoscale electrochemistry models, information about the conditions within the SOFC electrodes and electrolytes can even be resolved. However, due to the continuum-scale treatment of the SOFC, these models stiU rely on effective parameters, which need to be determined through smaller scale modehng or by fitting the models to experimental data. 

The MSA was first introduced by Percus and Yevick [9] as an approximate way of introducing hard-core effects rm the distribution of charged particles. Then, Waisman and Lebowitz applied this approximation to electrolytes [10]. They obtained the solution to the Omstein-Zenuke equation [8] in the case of ions having the same diameter, in the primitive model of solutions in which ions are regarded as charged spheres immersed in a continuum (the solvent) of relative permittivity e. The treatment was later extended to the case of ions of different diameters by Blum [11] and Blum and Hpye [12]. 

The porous structure filled with electrolyte is considered as the superposition of two continua, the electrode matrix and the solution in the unoccupied spaces within the matrix. The two phases, which complement one another, are supposed to be homogeneous and isotropic. Effective parameters rather than actual parameters are used for the description of the properties like pore size, conductivity, diffusion coefficient etc. The problem is treated as a one-dimensional one. This is equivalent to the assumption that the penetration depth of the current is larger than the size of the structural units (grains, holes) of the porous electrode. Continuum models were analyzed under different assumptions in references 1,4,9,12 to 14,16 to 20. Comprehensive treatments with strict derivations were published by Tobias and coworkers [21, 22] and by Micka [23]. 

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