Concentrated solution theory

The simplest practicable approach considers the membrane as a continuous, nonporous phase in which water of hydration is dissolved.In such a scenario, which is based on concentrated solution theory, the sole thermodynamic variable for specifying the local state of the membrane is the water activity the relevant mechanism of water back-transport is diffusion in an activity gradient. However, pure diffusion models provide an incomplete description of the membrane response to changing external operation conditions, as explained in Section 6.6.2. They cannot predict the net water flux across a saturated membrane that results from applying a difference in total gas pressures between cathodic and anodic gas compartments. 

The deviations and complications in the models arise from what function to use for the transport properties, K, I, and Z w, as well as the concentration of water in the membrane, Cw,2- To understand the differences in the models, a closer look at these functions is required, but first the models that use concentrated solution theory will be presented. 

Concentrated Solution "Theory. For an electrolyte with three species, it is as simple and more rigorous to use concentrated solution theory. Concentrated solution theory takes into account all binary interactions between all of the species. For membranes, this was initially done by Bennion ° and Pintauro and Bennion. ° They wrote out force balances for the three species, equating a thermodynamic driving force to a sum of frictional interactions for each species. As discussed by Fuller,Pintauro and Bennion also showed how to relate the interaction parameters to the transport parameters mentioned above. The resulting equations for the three-species system are... 

Membrane Water Content. Whether the dilute solution or concentrated solution theory equations are used to model the membrane system, functional forms for the transport parameters and the concentration of water are needed. The properties are functions of temperature and the water content, In the models, empirical fits are... 

Instead of the dilute solution approach above, concentrated solution theory can also be used to model liquid-equilibrated membranes. As done by Weber and Newman, the equations for concentrated solution theory are the same for both the one-phase and two-phase cases (eqs 32 and 33) except that chemical potential is replaced by hydraulic pressure and the transport coefficient is related to the permeability through comparison to Darcy s law. Thus, eq 33 becomes... 

The second term on the right hand side arises from concentrated solution theory. The sum of the currents of the two phases is equal to the applied current at any point in the electrode,... 

The water distribution within a polymer electrolyte fuel cell (PEFC) has been modeled at various levels of sophistication by several groups. Verbrugge and coworkers [83-85] have carried out extensive modeling of transport properties in immersed perfluorosulfonate ionomers based on dilute-solution theory. Fales et al. [109] reported an isothermal water map based on hydraulic permeability and electro-osmotic drag data. Though the model was relatively simple, some broad conclusions concerning membrane humidification conditions were reached. Fuller and Newman [104] applied concentrated-solution theory and employed limited earlier literature data on transport properties to produce a general description of water transport in fuel cell membranes. The last contribution emphasizes water distribution within the membrane. Boundary values were set rather arbitrarily. 

The application of Eq. (34) is limited to infinitely dilute systems (very dilute electrolytes). More details are given by Newman [15]. A more accurate representation of the flux is obtained when the gradient of the chemical potential is used for the derivation of the molar flux. This approach is known as concentrated solution theory [15]. For a binary electrolyte, the molar flux for the cations using concentrated solution theory becomes... 

The distinguishing feature of the classical diffusion model of Springer et al. [39] (hereafter SZG) is the consideration of variable conductivity. SZG relied on their own experimental data to determine model parameters, such as water sorption isotherms and membrane conductivity as a function of the water content. Alternative approaches include the use of concentrated solution theory to describe transport in the membrane [45], and invoking simplifying assumptions such as thin membrane with uniform hydration [46]. 

The physical model bridges the gap between the two types of mathematical models in the literature. Furthermore, it does so with a physically based description of the structure of the membrane. However, to put it to use in simulations a mathematical model and approach is required that describes the governing phenomena discussed above. In this section, the general governing equations based on the physical model are developed using concentrated-solution theory and the approach of having two transport modes is introduced. 

To account for all the relevant interactions between species in both transport modes concentrated-solution theory [11, 35] is used. Its use allows for the natural incorporation of coupled transport phenomena. The derivation starts with the equation of multicomponent transport... 

One comment should be made regarding the form of the transport equations. In the literature, two-phase flow has often been modeled using Schlogl s equation [50, 51]. This equation is similar in form to Eq. (5.9), but it is empirical and ignores the Onsager cross coefficients. Equations (5.8) and (5.9) stem from concentrated-solution theory and take into account all the relevant interactions. Furthermore, the equations for the liquid-equilibrated transport mode are almost identical to those for the vapor-equilibrated transport mode making it easier to compare the two with a single set of properties (i.e., it is not necessary to introduce another parameter, the elec-trokinetic permeability). 

Based on Newman s well-known modelling approach [8 10] the impedance of a commercial cell is described. This approach combines concentrated solution theory, porous electrode theory and Butler-Volmer kinetics to form a set of coupled partial differential equations. 

Thomas-Alyea KE, Newman J, Chen G, Richardson TJ (2004) A concentrated solution theory model of transport in solid-polymer-electrolyte fuel cells. J Electrochem Soc 151 A509... 

Concentrated solution theory allows the treatment of all species interacting with each other. We discuss this in more detail in the appendix to this chapter where we derive equation (8.22). There we show why the matrix relating electrochemical potential to flux is symmetric. If we know how the water flux is related to the proton flux, we can determine how the proton flux is affected by the water flux. In this equation, the Q are the concentrations of species H, B, and w. The Lij are unknown, only their symmetry property that CiCjL j = CjCiL i is known. 

Following the concentrated solution theory, the diHusional coefl cient, k, can be expressed for a binary electrolyte as [34, 49] ... 

The traditional approach to understanding both the steady-state and transient behavior of battery systems is based on the porous electrode models of Newman and Tobias (22), and Newman and Tiedermann (23). This is a macroscopic approach, in that no attempt is made to describe the microscopic details of the geometry. Volume-averaged properties are used to describe the electrode kinetics, species concentrations, etc. One-dimensional expressions are written for the fluxes of electroactive species in terms of concentration gradients, preferably using the concentrated solution theory of Newman (24). Expressions are also written for the species continuity conditions, which relate the time dependence of concentrations to interfacial current density and the spatial variation of the flux. These equations are combined with expressions for the interfacial current density (heterogeneous rate equation), electroneutrality condition, potential drop in the electrode, and potential drop in the electrolyte (which includes spatial variation of the electrolyte concentration). These coupled equations are linearized using finite-difference techniques and then solved numerically. 

Concentration solution theory can be used when an electrolyte is modeled with three species. This model is interpreted as the binary interactions between all of the species. The equations for the three species systen are... 

In this section we describe the equations required to simulate the electrochemical performance of porous electrodes with concentrated electrolytes. Extensions to this basic model are presented in Section 4. The basis of porous electrode theory and concentrated solution theory has been reviewed by Newman and Tiedemann [1]. In porous electrode theory, the exact positions and shapes of aU the particles and pores in the electrode are not specified. Instead, properties are averaged over a volume small with respect to the overall dimensions of the electrode but large with respect to the pore structure. The electrode is viewed as a superposition of active material, filler, and electrolyte, and these phases coexist at every point in the model. Particles of the active material generally can be treated as spheres. The electrode phase is coupled to the electrolyte phase via mass balances and via the reaction rate, which depends on the potential difference between the phases. AU phases are considered to be electrically neutral, which assumes that the volume of the double layer is smaU relative to the pore volume. Where pUcable, we also indicate boundary conditions that would be used if a Uthium foil electrode were used in place of a negative insertion electrode. 

Dilute solution theory is not often used in the treatment of lithium batteries, because most electrolytic solutions used in lithium batteries exhibit concentrated behavior. However, dilute solution theory becomes useful for cases such as the examination of side reactions such as redox shuttles for overcharge protection, because concentrated solution theory becomes more complicated when there are more than three species (anion, cation, and solvent) in solution. 

Concentrated solution theory includes interactions among aU species present in solution whereas dilute solution theory assumes that ions interact only with the solvent and not with other ions. In addition, dilute solution theory assumes that aU activity coefficients are unity. There is substantial evidence that both liquid and especially polymer electrolytes used in lithium batteries exhibit concentrated behavior [12,13,14,15]. 

The foundation of concentrated solution theory is the Stefan-Maxwell multicomponent diffusion equation [16,17],... 

For impurity species present in dilute concentrations, some may find it more convenient to treat the species using dilute solution theory, which accounts only for interactions of the dilute species with the solvent Rigorously, Equation 12 was derived for a binary electrolyte with no impurity species in the solution. While it is not completely rigorous to treat one species with dilute solution theory while treating the main electrolyte with equations derived from concentrated solution theory in the absence of the impurity species, the error may be small. The flux of the dilute species is given by Equation 5. The mass balance for the main electrolyte remains unchanged. If 2 is defined by Equation 3, then Us must be defined as a function of the concentration of the impurity species in order to include the concentration overpotential of the impurity species in the kinetic expression. Equation 53. The Nemst equation. Us = Uf + /JTln( Ci cf). is often used to account for the concentration overpotential of dilute species i. If Og is defined by Equation 6, then Us should not be defined as a function of solution composition. 

As mentioned earlier, electrolytes used in lithium batteries are usually concentrated, binary electrolytes that exhibit nonideal behavior. In addition, polymer and gel electrolytes are opaque, highly resistive, and sticky, and therefore their transference numbers are not easily measurable using traditional techniques such as the Hittorf or moving boundary methods. Recent theoretical studies have described the substantial error involved in measuring transference numbers with techniques that assume ideal behavior [14, 15], and have described how experimental data can be interpreted rigorously using concentrated-solution theory to obtain transference numbers. One method is the galvanostatic polarization technique [120,121,122] ... 

The discussion of concentrated solution models has indicated that, while the transport flux equations in their rigorous form (5) may be intractable, the use of the binary electrolyte approximation allows the convenient implementation of concentrated solution theory in pitting corrosion models. Engelhard et al. have shown that this approximation is valid over a surprisingly wide range of... 

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