Electrolytes concentrated solution theory

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... 

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... 

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... 

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]. 

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... 

A finite time is required to reestabUsh the ion atmosphere at any new location. Thus the ion atmosphere produces a drag on the ions in motion and restricts their freedom of movement. This is termed a relaxation effect. When a negative ion moves under the influence of an electric field, it travels against the flow of positive ions and solvent moving in the opposite direction. This is termed an electrophoretic effect. The Debye-Huckel theory combines both effects to calculate the behavior of electrolytes. The theory predicts the behavior of dilute (<0.05 molal) solutions but does not portray accurately the behavior of concentrated solutions found in practical batteries. 

Battery electrolytes are concentrated solutions of strong electrolytes and the Debye-Huckel theory of dilute solutions is only an approximation. Typical values for the resistivity of battery electrolytes range from about 1 ohmcm for sulfuric acid [7664-93-9] H2SO4, in lead—acid batteries and for potassium hydroxide [1310-58-3] KOH, in alkaline cells to about 100 ohmcm for organic electrolytes in lithium [7439-93-2] Li, batteries. 

The physical picture in concentrated electrolytes is more apdy described by the theory of ionic association (18,19). It was pointed out that as the solutions become more concentrated, the opportunity to form ion pairs held by electrostatic attraction increases (18). This tendency increases for ions with smaller ionic radius and in the lower dielectric constant solvents used for lithium batteries. A significant amount of ion-pairing and triple-ion formation exists in the high concentration electrolytes used in batteries. The ions are solvated, causing solvent molecules to be highly oriented and polarized. In concentrated solutions the ions are close together and the attraction between them increases ion-pairing of the electrolyte. Solvation can tie up a considerable amount of solvent and increase the viscosity of concentrated solutions. 

According to modem views, the basic points of the theory of electrolytic dissociation are correct and were of exceptional importance for the development of solution theory. However, there are a number of defects. The quantitative relations of the theory are applicable only to dilute solutions of weak electrolytes (up to 10 to 10 M). Deviations are observed at higher concentrations the values of a calculated with Eqs. (7.5) and (7.6) do not coincide the dissociation constant calculated with Eq. (7.9) varies with concentration and so on. For strong electrolytes the quantitative relations of the theory are altogether inapplicable, even in extremely dilute solutions. 

It is assumed that the quantity Cc is not a function of the electrolyte concentration c, and changes only with the charge cr, while Cd depends both on o and on c, according to the diffuse layer theory (see below). The validity of this relationship is a necessary condition for the case where the adsorption of ions in the double layer is purely electrostatic in nature. Experiments have demonstrated that the concept of the electrical double layer without specific adsorption is applicable to a very limited number of systems. Specific adsorption apparently does not occur in LiF, NaF and KF solutions (except at high concentrations, where anomalous phenomena occur). At potentials that are appropriately more negative than Epzc, where adsorption of anions is absent, no specific adsorption occurs for the salts of... 

This theory of the diffuse layer is satisfactory up to a symmetrical electrolyte concentration of 0.1 mol dm-3, as the Poisson-Boltzmann equation is valid only for dilute solutions. Similarly to the theory of strong electrolytes, the Gouy-Chapman theory of the diffuse layer is more readily applicable to symmetrical rather than unsymmetrical electrolytes. 

At low electrolyte concentrations, up to about a 10 3 M solution, the Gouy-Chapman theory agrees quite well with experimental values of... 

The Gouy-Chapman theory treats the electrolyte as consisting of point ions in a dielectric continuum. This is reasonable when the concentration of the ions is low, and the space charge is so far from the metal surface that the discrete molecular nature of the solution is not important. This is not true at higher electrolyte concentrations, and better models must be used in this case. Improvements on the Gouy-Chapman theory should explain the origin of the Helmholtz capacity. In the last section we have seen that the metal makes a contribution to the Helmholtz capacity other contributions are expected to arise from the molecular structure of the solution. 

For water at room temperature, A 2.65. The natural interpretation of Eq. (17.20) is this The structure of the solution at the interface causes deviations from the Gouy-Chapman theory. The leading correction term is independent of the electrolyte concentration and therefore contributes to the Helmholtz capacity for water (s 3 A) one obtains a contribution of about 7.1 A (0.64 cm2//F-1). At very high concentrations terms of order k and higher become significant. These should cause deviations from a straight line in a Parsons and Zobel plot, which have indeed been observed [10]. 

Use schematic diagrams to describe the influence of electrolyte concentration, type of electrolyte, magnitude of surface electrostatic potential and strength of the Hamaker constant on the interaction energy between two colloidal-sized spherical particles in aqueous solution. What theory did you use to obtain your description Briefly describe the main features of this theory. 

---