Electro-neutrality local

An additional electrostatic component to the polymer interaction term is typically unimportant since the counterions strongly screen any Coulomb interactions [92]. Finally, an electrostatic interaction between polymers and counterions Tint occurs if the PE brush is not locally electro-neutral throughout the system, an example is depicted in Fig. 10a. This energy is given by... 

Among the basic concepts to be introduced are ionic equilibrium, local equilibrium, local electro-neutrality, etc. 

According to (1.9c) the smallness of suggests that the entire space is divided into the bulk where the local electro-neutrality relations... 

The above estimates suggest that, with the scaling (1.7) valid, it takes about time td for an ionic system to restore local electro-neutrality on the macroscopic length scale and to reach local equilibrium. By local equilibrium we merely imply a state with the normalized ionic fluxes ji, defined by... 

Summarizing, at equilibrium the entire ED cell is divided into the locally electro-neutral bulk solution at zero potential and the locally electroneutral bulk cat- (an-) ion-exchange membrane at ipm < 0 (> 0) potential. These bulk regions are connected via the interface (double) layers, whose width scales with the Debye length in the linear limit and contracts with the increase of nonlinearity. 

We will discuss next the ambipolar diffusion, that is, electro-diffusion of two oppositely charged ions in a solution of a univalent electrolyte with local electro-neutrality. Assume the dimensionless ionic diffusivities are constant. Then the relevant version of (1.9) is... 

Equation (1.57a) implies that in the locally electro-neutral ambipolar diffusion concentration of both ions evolves according to a single linear diffusion equation with an effective diffusivity given by (1.57b). Physically, the role of the electric field, determined from the elliptic current continuity equation... 

A somewhat similar situation occurs in one-dimensional multi-ionic systems with local electro-neutrality in the absence of electric current. It will be shown in Chapter 3 that in this case again the electric field can be excluded from consideration and the equations of electro-diffusion are reduced to a coupled set of nonlinear diffusion equations. 

Finally, we make a terminological remark. In a one-dimensional locally electro-neutral system the expression (1.11b) for the electric current density reduces to... 

The next level is that of one-dimensional electro-diffusion with local electro-neutrality in the absence of an electric current. This is the realm of nonlinear diffusion to be treated in Chapter 3. A still higher level of the same hierarchy is formed by the nonlinear effects of stationary electric current, passing in one-dimensional electro-diflFusion systems with local electro-neutrality. A few typical phenomena of this type will be studied in Chapter 4. The treatment of Chapter 4 will lay the foundation for the discussion of the effects of nonequilibrium space charge characteristic of the fourth level to be treated in Chapter 5. 

Locally Electro-Neutral Electro-Diffusion Without Electric Current... 

Preliminaries. In this chapter we shall address the simplest nonequilibrium situation—one-dimensional locally electro-neutral electrodiffusion of ions in the absence of an electric current. We shall deal with macroscopic objects, such as solution layers, ion-exchangers, ion-exchange membranes with a minimum linear size of the order of tens of microns. 

As pointed out in the Introduction, it is customary in the treatment of such systems to assume local electro-neutrality (LEN), that is, to omit the singularly perturbed higher-order term in the Poisson equation (1.9c). Such an omission is not always admissible. We shall address the appropriate situations at length in Chapter 5 and partly in Chapter 4. We defer therefore a detailed discussion of the contents of the local electro-neutrality assumption to these chapters and content ourselves here with stating only that this assumption is well suited for a treatment of the phenomena to be considered in this chapter. 

This concludes the proof of the above assertion regarding stability of the locally electro-neutral equilibrium and the way it is approached by the system. 

Here C is the concentration vector and D(C) is the diffusivity tensor defined by (3.1.15a). Thus, locally electro-neutral electro-diffusion without electric current is exactly equivalent to nonlinear multicomponent diffusion with a diffusivity tensor s being a rational function of concentrations of the charged species. 

Stationary Current with Local Electro-Neutrality... 

Preliminaries. In the previous chapter we dealt with locally electro-neutral time-dependent electro-diffusion under the condition of no electric current in a medium with a spatially constant fixed charge density (ion-exchangers). It was observed that under these circumstances electrodiffusion is equivalent to nonlinear diffusion with concentration-dependent diffusivities. 

For similar reasons ion diffusivities will be assumed piecewise constant, and we shall mostly limit ourselves to a discussion of one-dimensional situations, for which the combination of local electro-neutrality and piecewise constant fixed charge density yields instructive explicit solutions. 

We shall begin with a recapitulation of the conditions under which the local electro-neutrality approximation is expected to hold or, at least, to be consistent. Furthermore, we shall postulate using intuitive arguments the conditions of conjugation for the locally electro-neutral transport variables at the surfaces of discontinuity of N(x). (Some asymptotic justification for these conditions will be provided in the next chapter.)... 

In 4.4 the theory of 4.2 will be applied to study electro-diffusion of ions through a unipolar ion-exchange membrane, separating two electrolyte solutions. This will include the classical treatment of concentration polarization in a solution layer adjacent to an ion-exchange membrane under an electric current. The validity limits of this theory, set by the violations of local electro-neutrality and caused by the development of a macroscopic nonequilibrium space charge, will be indicated. (The effects of the nonequilibrium space charge are to be discussed at some length in Chapter 5.)... 

Circumstances in which this may occur are suggested by the following arguments. Local electro-neutrality implies that... 

In parallel, the simplicity of handling the locally electro-neutral case provides a convenient ground for studying the far less tractable one-dimensional version of the nonreduced system (4.1.1)-(4.1.2), asymptotically for small e. For an example of such a study we refer to [14] where the difficult question of multiplicity of steady solutions of the nonreduced system was approached through studying the multiplicity of solutions in the LEN approximation for a four layer (quadrupolar) arrangement. The theory of a bistable electronic device (thyristor) which resulted from this study will be presented in 4.3. 

We point out that the results of locally electro-neutral studies should be extrapolated upon the nonreduced systems with a certain caution even for e very small. This is so because, to the best of the author s knowledge, no asymptotic procedure for the singularly perturbed one-dimensional system (4.1.1), (4.1.2) has been developed so far that would be uniformly valid for the entire range of the operational parameters (e.g., for arbitrary voltages and fixed charge densities). 

Multiple steady states in one-dimensional electro-diffusion with local electro-neutrality [14]. This section is concerned with the construction and study of multiple steady states occurring in onedimensional ambipolar electro-diffusion with local electro-neutrality and... 

In this section we shall consider the simplest model problem for the locally electro-neutral stationary concentration polarization at an ideally permselective uniform interface. The main features of CP will be traced through this example, including the breakdown of the local electro-neutrality approximation. Furthermore, we shall apply the scheme of 4.2 to investigate the effect of CP upon the counterion selectivity of an ion-exchange membrane in a way that is typical of many membrane studies. Finally, at the end of this section we shall consider briefly CP at an electrically inhomogeneous interface (the case relevant for many synthetic membranes). It will be shown that the concentration and the electric potential fields, developing in the course of CP at such an interface, are incompatible with mechanical equilibrium in the liquid electrolyte, that is, a convection (electroconvection) is bound to arise. 

Locally electro-neutral concentration polarization of a binary electrolyte at an ideally cation-permselective homogeneous interface. Consider a unity thick unstirred layer of a univalent electrolyte adjacent to an ideally cation-permselective homogeneous flat interface. Let us direct the x-axis normally to this interface with the origin x = 0 coinciding with the outer (bulk) edge of the unstirred layer. Let a unity electrolyte concentration be maintained in the bulk. 

In order for the local electro-neutrality approximation to be consistent in the vicinity of the interface the following inequality must hold... 

Equation (4.4.17) corresponds to the ideal permselectivity of the membrane and (4.4.18) stands for local electro-neutrality within the solution (4.4.18a) and within the ideally permselective membrane (4.4.18b). 

Equation (5.2.42) and the treatment that leads to it remind us that in accordance with the description in the Introduction it takes about 1 je as long for an electro-diffusional system to get from an arbitrary initial state to a macroscopically locally electro-neutral one. 

Thus summarizing, we note that at the leading order the asymptotic solution constructed is merely a combination of the locally electro-neutral solution for the bulk of the domain and of the equilibrium solution for the boundary layer, the latter being identical with that given by the equilibrium electric double layer theory (recall (1.32b)). We stress here the equilibrium structure of the boundary layer. The equilibrium within the boundary layer implies constancy of the electrochemical potential pp = lnp + ip across the boundary layer. We shall see in a moment that this feature is preserved at least up to order 0(e2) of present asymptotics as well. This clarifies the contents of the assumption of local equilibrium as applied in the locally electro-neutral descriptions. Recall that by this assumption the electrochemical potential is continuous at the surfaces of discontinuity of the electric potential and ionic concentrations, present in the locally electro-neutral formulations (see the Introduction and Chapters 3, 4). An implication of the relation between the LEN and the local equilibrium assumptions is that the breakdown of the former parallel to that of the corresponding asymptotic procedure, to be described in the following paragraphs, implies the breakdown of the local equilibrium. 

A few remarks are due about this feature. The nonuniformity above is a formal expression of breakdown of the local electro-neutrality assumption in concentration polarization, described in the previous chapter. Essentially, this reflects the failure of a description based upon assuming the split of the physical region into a locally electro-neutral domain and an equilibrium double layer where all of the space charge is concentrated. The source of this failure, reflected in the nonuniformity of the corresponding matched asymptotic expansions, is that the local Debye length at the interface tends to infinity as the voltage increases. In parallel a whole new type of phenomena arises, which is not reflected in the simplistic picture above. The... 

For comparison, we present in Fig. 5.3.2 some numerical results for the following non-locally-electro-neutral generalization of the classical Teorell-Meyer-Sievers (TMS) model of membrane transport (see [11], [12] and 3.4 of this text). 

Anomalous rectification [3]. Our aim in this section is to show that under certain conditions development of a nonequilibrium space charge may yield, besides the punch through, some additional effects, unpredictable by the locally electro-neutral formulations. We shall exemplify this by considering two parallel formulations—the full space charge one and its locally electro-neutral counterpart. It will be observed that inclusion of the space charge into consideration enables us to account for the anomalous rectification effect that could not be predicted by the locally electro-neutral treatment. Physical motivation for this study is as follows. 

Fig. 5.4.1. (a) Calculated steady voltage-current curve for e=io-4, A=10. The dashed line marks the value of the limiting current in a locally electro-neutral model, (b) Calculated dependence of the relative rectification effect on the modulation frequency. 

Fig. 5.4.4. Time plots of voltage and minimal concentration in a locally electro-neutral model.

Generalized local Darcy s model of Teorell s oscillations (PDEs) [12]. In this section we formulate and study a local analogue of Teorell s model discussed previously. The main difference between the model to be discussed and the original one is the replacement of the ad hoc resistance relaxation equation (6.1.5) or (6.2.5) by a set of one-dimensional Nernst-Planck equations for locally electro-neutral convective electro-diffusion of ions across the filter (membrane). This filter is viewed as a homogenized aqueous porous medium, lacking any fixed charge and characterized... 

---