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= 0 is normalized to unity.) Equation (1.23b) is the equilibrium Boltzmann distribution in a potential field. Substitution of (1.23b) into (1.9c) yields the Poisson-Boltzmann equation for equilibrium electric potential of the form
Consider potential distribution in the right half cell sufficiently far from edges z = 0, L where (1.24) may be replaced by its one-dimensional version
Here A is the dimensionless thickness of the membrane A = A//. Supplement (1.25) by the boundary conditions
12
INTRODUCTION
Equation (1.26a) fixes the potential at the reference zero level at the middle plane of the cell, whereas (1.26b) asserts zero normal derivative at the symmetry middle plane of the membrane. Let us work out an approximate solution of the b.v.p. (1.25), (1-26) for e ^ 1. Introduce a new variable
The b.v.p. (1.25), (1.26) is rewritten in terms of y to leading order in e as
It follows from (1.28), (1.29b) that for y -> oo, p -> — oo). This illustrates the previously made statement concerning the meaningfulness of the presented interpretation of r^ only as long as the scaling of the electric potential <jj with RT/F is uniformly valid in the system; that is naturally not the case when TV —> oo ( 0, y(oo) —> —oo). 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 (pm < 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. A terminological remark is due. An equilibrium between two media with different fixed charge density (e.g., an ion-exchanger in contact with an electrolyte solution) is occasionally termed the Donnan equilibrium. The corresponding potential drop between the bulks of the respective media is then termed the Donnan potential By the same token, we speak of the local Donnan equilibrium and the local Donnan potential, referring, respectively, to the local equilibrium and the interface potential jump at the surface of discontinuity of the fixed charge density, considered in the framework of the LEN approximation.
14
INTRODUCTION
We have now introduced equilibrium (some particular problems of ionic equilibrium will be treated in Chapter 2). Let us trace now how the equilibrium picture above projects into the major nonequilibrium property of an ion-exchange membrane—its permselectivity. In particular, let us clarify the quantitative meaning of the previous statement concerning the cat(an-) ion-exchange membrane as a potential barrier for an- (cat-) ions. Let us concentrate for definiteness on the cation-exchange membrane. Consider a cation-exchange membrane of an electrodialysis cell with two adjacent solution layers of unity width. Assume that a unity bulk (feed) concentration is maintained at the outer edges of the solution layer and that a low voltage V C, N = 0(1), a -> 0,
The problem under consideration arises in the theory of polyelectrolytes. (p in (2.3.1) is the electric potential, produced by a central cylindrical polyelectrolyte core of radius a, charged with a normalized "structural" linear
38
NONLINEAR EFFECTS
charge density (LCD) cr, determined by the chemical structure of the polyelectrolyte; and small mobile ions of valencies £ and 2, distributed with the equilibrium Boltzmann distribution in a cylindrical cell of unit radius, surrounding the central core. Mobile ions of valency (" are the "proper" counterions produced by dissociation of the polyelectrolyte, whereas ions of valency z are those of a low molecular symmetric electrolyte of normalized bulk concentration N added to the polyelectrolyte solution. The physical meaning of Conjecture 2.1 is that the "effective" linear charge density, measurable as the proportionality coefficient at the logarithmic singularity in
0 (infinite polyelectrolyte dilution limit), coincides with the "structural" LCD a only for a values smaller than some critical 0 an LCD a is expected to yield a singularity of the type —cr In a in the surface potential, the statistical-mechanical phase integral for counterions should diverge for a greater than some critical value, characteristic of a given valency. Indeed consider a counterion (for definiteness anion) of valency z. The appropriate phase integral is of the form
Here (r, 9) are polar coordinates, v?(r, 8) is the normalized local electric potential, and integration is carried Os-er the region accessible for counterions. If the singularity in (p induced by the positive line charge indeed were of
40
NONLINEAR EFFECTS
Fig. 2.3.2. The cell model.
the type — crlna, the integral would diverge in the limit a —» 0 for a > 2/z. Divergence of the statistical sum implies thermodynamic instability of the ionic system. It was further postulated by Manning [26] that whenever the structural LCD a is greater than 2/£, where £ is the greatest valency of counterions present in the system, a counterion "condensation" occurs on the line charge which reduces LCD to the critical value 2/£. Based on this postulate, Manning developed his approach to thermodynamics of linear polyelectrolyte systems with one or several types of counterions, in the presence or absence of an added low molecular electrolyte. As far as we know, attempts at experimentally proving the existence of "condensed" counterions bound at the polyelectrolyte core in a state different from that in the bulk have so far been unsuccessful. We reiterate that Onsager and Manning's consideration is entirely based upon their apparently natural assumption about the type of limiting surface potential singularity. On the other hand, this singularity can be easily evaluated directly in the mean field approximation. Probably the simplest way for doing so is provided by the classical Katchalsky cell model [23]. In terms of this model three-dimensional space is viewed as filled with a regular array of infinitely long cylindrical polyelectrolyte cores (see Fig. 2.3.2). Each of these cores is surrounded by a square-cylindrical solution cell whose surface is that of the symmetry. At the next step the square-cylindrical outer shell is replaced by a circular-cylindrical one. The resulting geometrical model is reasonable as long as the length of each linear polyelectrolyte core is large compared to the individual core thickness. The electric field within each cell is determined in the mean field approximation from the Poisson-Boltzmann equation (2.3.1), written for the prototypical case of a symmetric low molecular electrolyte of valency z added to a polyelectrolyte with a single type of "proper" counterion of va-
CHAPTER 2
41
lency £ and LCD a. (Cell radius in (2.3.1) is normalized to unity.) We point out that the form (2.3.1) of the Poisson-Boltzmann equation, introduced in the polyelectrolyte context by Marcus [31], contains as denominators on the right-hand side the normalizing integrals AC, A~, A+ identical in the mean field approximation with the phase integrals referred to by Manning. As will be demonstrated in due course, boundedness of these integrals is necessary for the existence of a solution of the appropriate b.v.p. In the case of a single exponent in the right-hand side of (2.3.1) (physically corresponding to a single counterion without a low molecular electrolyte added) the analogue of (2.3.1) can be explicitly integrated (see §2.3.2). The appropriate solutions, first introduced in the polyelectrolyte context by Alfrey, Berg, and Moravetz and Fuoss, Katchalsky, and Lifson in their classical papers [32], [33], formed the basis for numerous later studies [23], [24], [34]-[37]. In particular, it was observed [24], [26], [37] that in the case of a single counterion and no added electrolyte the above explicit solutions yield in the infinite dilution limit the same predictions for several thermodynamic properties of a polyelectrolyte system (osmotic coefficients, counterions activity coefficients, etc.) as Manning's counterion condensation model. It is thus the purpose of this section to show that a "sharp counterion condensation," as postulated by Manning and as expressed by Conjecture 2.1, is an exact limiting property of solutions of the b.v.p. (2.3.1) for the Poisson-Boltzmann equation. Part (C3) of Conjecture 2.1 refers to a particular case when the valency of "proper" counterions is lower than that of the added electrolyte, whereas concentration of the latter may become vanishingly small. As a result, the potential determining role is transferred from the counterions of the added electrolyte to the "proper" counterions. The prototypical example (2.3.1), treated in this section, concerns the case of a single symmetric low molecular electrolyte of valency z added to a linear polyelectrolyte with a single "proper" counterion with valency £. The results presented here are generalizable in a straightforward manner to the case of any number of low molecular ionic species present in the system. A less straightforward generalization of these results, carried out by Friedman and Tintarev [18], [19], concerns lifting the restriction of axial symmetry of the cell model (obviously irrelevant for the type of singularity at a line charge). Another possible generalization which has not yet been carried out concerns replacing the straight charged cylinder with an arbitrary cylindrical manifold in ~R? without self-crossings. 2.3.3. Methodology. The explicit solution for the analogue of (2.3.1), in a ring a < r < p, with a single exponent of the form
42
NONLINEAR EFFECTS
is constructed with the aid of a substitution For a < 2/£ the appropriate solution is
Here B is a positive constant, satisfying
It can be directly shown that a solution to (2.3.4b) exists and is unique. This is particularly easily observed for a/p sufficiently small, i.e., in the limit we shall employ in due course. Indeed, in this case the right-hand side of (2.3.4b) may be fixed at an arbitrary large value. On the other hand, for the definition range of the left-hand side 0 < B < (2- 0, (2.3.4) predicts for the singularity
whereas (2.3.5) predicts
and
in accordance with the postulates of Manning and Conjecture 2.1. The particular form of estimate (2.3.7b) is due to the fact that for a > 2/£ the function u(r), defined by (2.3.3d), possesses a minimum of order —2 lnln(p/a) — In ap2 at the distance of order p(ajp)a = ME/Lre of the polymer.(p/a)
from the origin. When cr > 2/£ the above distance is of order p(ajp) ' , whereas for a = 2/C it becomes of order a, i.e., the minimum of u shifts to the left end point r = a. (We have allowed for a to become small and for the outer ring radius and the appropriate boundary value to be arbitrary positive numbers—possibilities we shall employ in due course.) The heuristic idea behind generalizing this fact for additional kinds of ions present in the system is similar to that employed by Alexandrovicz in [35]. Namely, we expect that the electric field near the polyelectrolyte core (and in particular the type of the limiting singularity) is largely determined by counterions of the highest valency, whose proportion increases wherever the electric field is higher. It is thus expected that the appropriate field singularity can be evaluated through upper and lower bounds, obtained by replacing the right-hand side of (2.3.1) in the vicinity of the polyelectrolyte core by an appropriate single exponent. The formal basis for obtaining such bounds is a straightforward analogue of Theorems 2.1 and 2.2, and Proposition 2.1. Notice first, that f( C- The limit of vanishing added electrolyte concentration. Without loss of generality, concentrate upon the case a > 2/£. (For a < 2/£ no counterion condensation occurs and the limiting singularity is accordingly of the type — a In a. Treatment of the intermediate case 2/z < cr < 2/£ is analogous to the one presented below.) By directly repeating arguments of the previous section we arrive at the following crude estimate for a —> 0:
In order to refine the rough estimate (2.3.33) we employ constructions similar to those of the previous section. Let us start by refining the lower bound. To this end we consider the following auxiliary b.v.p.:
50
NONLINEAR EFFECTS
Here
is defined in such a way that for ^ > (p\
whereas for
It is thus obvious by construction that
(p is next explicitly constructed analogously to the solution of (2.3.25) and (2.3.27). We employ solutions of the type (2.3.4) and (2.3.5) within the rings a < r < TI and TI < r < 1 (^(ri) = 0 when N —>• 0, a —> 0. Yet it follows from a straightforward calculation that the resulting t/>(r) is monotonically decreasing in r with the property:
It follows from the monotonicity of (r) and inequalities (2.3.9), (2.3.10), (2.3.35) that
which together with (2.3.36) provides the sought lower bound for the limiting singularity in y?(a) for a —> 0. For refining the upper bound in (2.3.33), it is sufficient to consider the auxiliary problem:
CHAPTER 2
51
Here
is again defined so that for -0 > (^2?
Bearing in mind the positiveness of /( 0) a transition occurs in the way prescribed by (2.3.41) to the "effective" linear charge as determined by the "proper" counterions of valency ( in a polyelectrolyte solution free from added low molecular electrolyte.
52
NONLINEAR EFFECTS
2.3.6. Counterion condensation as a nonbifurcational secondorder phase transition. In order to put the mathematical phenomenon described above into a clearer physical context, let us point out the following. As can be easily observed from the explicit expressions for the electrostatic free energy in the "no added electrolyte" case [34] the described switch in the type of singularity, induced by a line charge (Fig. 2.3.3), implies a discontinuity of the second derivative of the free energy with respect to the structural charge density a. According to the common phenomenological classification, this in turn implies a second-order phase transition.
Fig. 2.3.3. Theoretical dependence of the effective linear charge density a on the structural linear charge density a.
A wide class of "analytic" second-order phase transitions is characterized by their Landau bifurcational mechanism [38]. According to this mechanism, a system characterized by order parameter 77, possesses a single stable equilibrium solution (rje = 0) for a range of the external parameter T (T > Tcr; see a schematic illustration in Fig. 2.3.4a). This single solution corresponds to an absolute internal minimum of the system's free energy F as a function of the order parameter (Fig. 2.3.4b, Curve 1). As the external parameter T decreases, at a critical value T = Tcr, the solution with r)e = 0 becomes unstable with two more stable solutions with r)e^Q (for T < Tcr) bifurcating fro~n it (Fig. 2.3.4a). In the (F, rj) plane this corresponds to the appearance of two new local free energy minima that flank the former one, which now turns into a local maximum (Fig. 2.3.4b, Curve 2). The situation is completely different with counterion condensation, considered in this section. A natural order parameter here would be
CHAPTER 2
53
Fig. 2.3.4a. Schematic dependence of the equilibrium order parameter r)cq on the external parameter T in the Landau mechanism for a second-order phase transition.
Fig. 2.3.4b. Schematic dependence of the system's free energy F on the order parameter TJ in the Landau mechanism for a second-order phase transition.
54
NONLINEAR EFFECTS
Fig. 2.3.5a. Schematic dependence of the equilibrium order parameter on the structural linear charge density a.
A plot of equilibrium r\e as a function of the external parameter a is schematically presented in Fig. 2.3.5a. The plot in Fig. 2.3.5a is markedly different from that in Fig. 2.3.4a by its lack of bifurcation. (Uniqueness of the appropriate solutions of the Poisson-Boltzmann equation for any values of a is proved in [18].) In the (F, rj) or (F, creff) plane this corresponds to the existence of solutions of the Poisson-Boltzmann equations with finite F (bounded norm of the appropriate solution with a subtracted singular part due to the effective line charge) only for creff < 0. For a < ff^ax, F possesses an absolute internal minimum at creff = a. At a — (r^ftx, this minimum hits the vertical asymptote and thereafter (ff — "'max) staYs there. The nonbifurcational scheme outlined herein is thus responsible for the appearance of a weak singularity in the free energy dependence on the external parameter a, phenomenologically reminiscent of a nonanalytic second-order phase transition. 2.4. Unsolved problems. 1. An actual calculation of the maximal force of repulsion between two symmetrically charged parallel cylinders in an electrolyte solution (see §2-2).
CHAPTER 2
55
Fig. 2.3.5b. Schematic dependence of the system's free energy F on the effective linear charge density
2. Proof of boundedness of the force of interaction between two charged particles of an arbitrary shape in 7£3, held at a given distance from each other in an electrolyte solution, upon an infinite increase of the particle's charge. (It was shown in §2.2 that the repulsion force between parallel symmetrically charged cylinders saturates upon an infinite increase of the particle's charge. This is also true for infinite parallel charged plane interaction [9]. The appropriate result is expected to be true for particles of an arbitrary shape.) 3. Study of counterion condensation as a limiting property of the solutions of the Poisson-Boltzmann equation for arbitrary, charged cylindri cal manifolds in "ft3 (see §2.3). 4. Study of the singularity induced by a point charge to the solutions of the Poisson-Boltzmann equation in 11? (see §2.3 and [19]). It is expected that no solutions of the Poisson-Boltzmann equation exist with a singularity of the type 0, i is assumed to be a known positive constant.) Zi is the ionic charge (valency); (p is the dimensionless potential scaled with RT/F; N(x) is the known dimensionless concentration of fixed charges (present in ion-exchangers or ion-exchange membranes), scaled with CQ and assumed further on piecewise constant. In reactive ionic systems, a source term has to be added to the righthand side of (3.1.1). We shall relate to this in due course when discussing reactive ion-exchange (see §3.3). Before we turn to particular instances of system (3.1.1), (3.1.2) let us make a few general observations. First, note that the equilibrium in (3.1.1), (3.1.2) is linearly stable and is approached monotonically in time. Indeed, consider (3.1.1), (3.1.2) on the segment —L < x < L. Assume N = const. The boundary conditions for concentrations compatible with equilibrium are
and
with constants Cf satisfying the electro-neutrality relation
Any combination of conditions (3.1.3), one for each end of the segment, fits equilibrium. These conditions are to be supplemented by those for the electric potential of the form
It is observed from the definition of the ionic fluxes
that use of the conditions (3.1.3a), (3.1.3c), (3.1.4) is equivalent to requiring explicitly the vanishing of the ionic fluxes at the boundaries. The equilibrium solution to (3.1.1)-(3.1.4) is
CHAPTER 3
61
Here C® is defined by the boundary value in the case of the Dirichlet conditions (3.1.3b), (3.1.3d) at one of the end points or by the space averages of the initial concentrations in the case of the Neumann conditions (3.1.3a), (3.1.3c) at both ends. In the spirit of a standard linear stability analysis consider a small perturbation of the equilibrium of the form
A substitution of (3.1.7) into (3.1.1)-(3.1.4) yields, after a linearization with respect to the perturbation u^, ?/>, the eigenvalue problem
The complex conjugates cr, Wj(x), i^(x) naturally also satisfy the system (3.1.8)-(3.1.11). With this in mind multiplication of (3.1.8) by u i? followed by integration over the segment — L < x < L, summation over 1 < i < M, and integration by parts in the first term of the right-hand side, yields (taking into account (3.1.10), (3.1.11))
From here, taking into account (3.1.9), we get
62
ELECTRO-NEUTRAL ELECTRO-DIFFUSION
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. Our second observation concerns an important integral of the system (3.1.1), (3.1.2). Multiply (3.1.1) by z{ and sum over all i. Equation (3.1.2) then yields
Here
is an integral of the system (3.1.1), (3.1.2) termed electric current density, already mentioned in the Introduction. It follows from (3.1.13a) that in one-dimensional systems / is spatially invariant. Let us observe that when the boundary conditions are such that / vanishes (the system is electrically isolated), the system (3.1.1), (3.1.2) is reduced to a set of coupled nonlinear diffusion equations. Indeed, by (3.1.13b)
implies
Substitution of (3.1.14) into (3.1.1) yields finally
Here
CHAPTER 3
63
Equation (3.1.15a) is the aforementioned system of coupled quasilinear diffusion equations for Cj, 1 < i < n — 1 (Cn is eliminated via (3.1.2)) of the form
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. In this chapter we shall treat some particular instances of the system (3.1.15) and the related phenomena. Thus in §3.2, we shall concentrate upon binary ion-exchange and discuss the relevant single nonlinear diffusion equation. It will be seen that in a certain range of parameters this equation reduces to the "porous medium" equation with diffusivity proportional to concentration. Furthermore, it turns out that in another parameter range the binary ion-exchange is described by the "fast" diffusion equation with diffusivity inversely proportional to concentration. It will be shown that in the latter case some monotonic travelling concentration waves may arise. Furthermore, in §3.3 we turn to reactive binary ion-exchange. An equilibrium binding reaction (adsorption) with a Langmuir-type isotherm is considered. Formation of sharp propagating concentration fronts is studied via an unconventional asymptotic procedure [1]. Finally, in §3.4 we present a calculation of membrane potential in term of the classical Teorell-Meyer-Sievers (TMS) [2], [3] model of a charged permselective membrane. In spite of its extreme simplicity, this calculation yields a practically useful result and is typical for numerous membrane computations, some more of which will be touched upon subsequently in Chapter 4. 3.2. Slow and fast diffusion in ion-exchange. 3.2.1. Consider a particular case of (3.1.15) with n = 2, signal = sign z-2. This corresponds to the exchange of two counterions in an ideal ionexchanger (complete co-ion exclusion). Accordingly, (3.1.15) is rewritten as
Here
Note that in the limit a —> 0, aC = 1> (3.2.la) reduces to
64
ELECTRO-NEUTRAL ELECTRO-DIFFUSION
Equation (3.2.2b) with m > 1 is known as the "porous medium" equation. Historically it was indeed first Inferred for transport in a porous medium [4] and combustion (propagation of strong thermal waves) [5], [6]. Somewhat more recently this equation became a subject for extensive mathematical studies (see [7]-[9] and references therein). The central feature of the porous medium equation is that diffusivity vanishes wherever u does. The two major resulting peculiarities of the corresponding solutions are the compact supports and the waiting time. The essence of the first feature is that a solution that initially vanished identically outside some space domain (support) continues to do so at all later times, with the support evolving (propagating) with a finite speed. In particular an initially compact support remains such at all finite times. This stands of course in complete contrast to the corresponding feature of diffusion with a nonvanishing diffusivity as in the case of conventional linear diffusion. The other peculiarity concerns the beginning of movement of the support's boundary. For an arbitrary initial condition, the former does not start propagating right away; rather it takes some finite "waiting" time to build the boundary concentration gradient that is necessary for the support's propagation to begin. In this section we shall concentrate on another somewhat less explored limit case of (3.2.la). For a —> oo, £ = 1 equation (3.2.la) yields
The "fast" diffusion equation (3.2.3) is yet another version of (3.2.2a), this time with ra = 0. In contrast to the "porous medium" case, diffusivity here blows up when u —+• 0. This equation, or more generally (3.2.2a) with m < 1, has attracted much less attention compared to its "porous medium" counterpart, in spite of the fact that it occurs in numerous physical situations. Thus the case m = 0 to be discussed here was inferred previously in the plasma containment context [10]-[12] , in the thermalized electron cloud expansion [13], and in the central limit approximation to the Carleman's model of the Boltzmanri equation [14]-[16j. In the context of electro-diffusion of ions in an ion-exchanger this equation was inferred in [17]. Quite a few mathematical studies were devoted to this equation with many interesting features of the solutions to the corresponding b.v.p.s and Cauchy problems found in [10]-[12], [18]-[22]. They include the results concerning the separable time asymptotics for a solution of (3.2.3) on a segment [10], existence-uniqueness results for the Cauchy problem in 7£n for 0 < m < 1 [19] , and existence-uniqueness for the solution of the Cauchy problem in K1 with the "total mass" conserved for —1 < m < 0 [18]. The requirement of conservation of the total mass is important in the latter case because it is the one that guarantees uniqueness. Along with the maximal conserved mass solution, additional noriconserved solutions vanishing in a
CHAPTER 3
65
finite time [20], [21] may exist. For m < —1 no finite-mass solution exists in any strip Kl x (0, T), 0 < T < oo [22]. The appropriate study is carried out with the aid of the Backlund transformation which maps the case m = — into that of linear heat diffusion (m = 1) [14], [23], [24] , whereas the range m < — 1 is mapped into that of the porous medium equation (m > 1) [22]. In this section we shall study still another peculiarity of (3.2.3)—the occurrence of uniformly bounded, monotonic travelling waves. These waves, very common in reaction-diffusion (see, e.g., [25]-[27]), seem fairly unexpected in the reactionless diffusion under discussion. Their occurrence here is directly related to the singularity of diffusivity in (3.2.3) and thus can be viewed as the "fast" diffusional counterpart of the aforementioned peculiarities of the "porous medium" equation. 3.2.2. Before we turn to this issue, we would like to substantiate the above discussion of basic features of nonlinear diffusion with some examples based upon the well-known similarity solutions of the Cauchy problems for the relevant diffusion equations. Similarity solutions are particularly instructive because they express the intrinsic symmetry features of the equation [6], [28], [29]. Recall that those are the shape-preserving solutions in the sense that they are composed of some function of time only, multiplied by another function of a product of some powers of the time and space coordinates, termed the similarity variable. This latter can usually be constructed from dimensional arguments. Accordingly, a similarity solution may only be available when the Cauchy problem under consideration lacks an explicit length scale. Thus, the two types of initial conditions compatible with the similarity requirement are those corresponding to an instantaneous point source and to a piecewise constant initial profile, respectively, of the form
Here 0 is the total "initial mass" ("amount of heat" etc.) and UQ is the concentration at oo. (The concentration at — oo has been assumed to be zero to take into consideration the singular value of interest in (3.2.2a).) Consider the Cauchy problem for (3.2.2a) with m arbitrary, on the real line —oo < x < oo with the initial condition (3.2.4). In order to infer the similarity variable, pass to the dimensional form of (3.2.2a), (3.2.4)
66
ELECTRO-NEUTRAL ELECTRO-DIFFUSION
Note that
Equation (3.2.7b) suggests for the dimensionless similarity variable £,
Equations (3.2.7a,b) suggest the power of time dependence in the scaling function for C(x,i) and thus, finally, we seek a solution of (3.2.5)-(3.2.6) of the form
Here f m (£) is a dimensionless function of £ only. Substitution of (3.2.9) into (3.2.5), (3.2.6) yields the following b.v.p. for
Integration of (3.2.10a) with boundary conditions (3.2.10b,c) yields
Here
CHAPTER 3
67
and £o is a positive integration constant determined from the condition (3.2.10(1). Equation (3.2.1 la) reflects that which was said above concerning the structure of a solution for different m. Thus for m < 1 for all £, vm(^) > 0, that is, for all x, for all i > 0, C(x, t) > 0; in other words, the solution's support is noncompact. On the other hand, for m > 1, for all £ > £o v(£) = 0, that is,
i.e., the solution possesses a compact support that grows in time by the law
Another group of similarity solutions corresponds to the initial discontinuity condition (3.2.4b), which might be rewritten in dimensional terms as
Here, the similarity variables, the same for any m, are
or more conveniently,
(Note that the power structures of the similarity variables (3.2.8) and (3.2.13a) for the initial conditions (3.2.4a) and (3.2.4b) coincide only in the case of linear diffusion, m = 1.) The b.v.p. for w is
68
ELECTRO-NEUTRAL ELECTRO-DIFFUSION
A closed form analytic solution to b.v.p. (3.2.15)
is known only for the linear case m = 1. For m > 1 there exists a solution to (3.2.15) with the property
In terms of physical coordinates, (3.2.16) corresponds to a support whose boundary propagates to the right according to the law
This is an analogue of the evolving compact support (3.2.12) for (3.2.3), (3.2.4a) with m> 1. Existence of a class of similarity solutions to (3.2.2a) with -1 < m < 1 and a step function initial condition (3.2.4b) has been established recently [30]. Furthermore, it is known that in the case m > 1 the similarity solutions represent the longtime asymptotics for the solution of the Cauchy problem with initial conditions compatible respectively with (3.2.4a) and (3.2.4b) at x = ±00 [9], [31]. We do not know in what sense, if any, this could also be the case for m < 1. Let us reiterate that whenever a similarity solution to (3.2.5), (3.2.4c) exists, the physical space coordinate of any given value of concentration between 0 and CQ propagates as const • Vt. The shape of the solution (the concentration profile) evolves accordingly in terms of the physical space variable i, whereas it is preserved unchanged (either after a rescaling with some function of time only as in (3,2.9) or without it as in (3.2.13b)) in terms of the similarity variable. Another type of shape-preserving behaviour, the one we shall be preoccupied with in the rest of this section, is characteristic for travelling wave solutions of the form
Here c is a constant speed of wave propagation to be determined. For solutions of this type the spatial distributions of properties at different times are obtained from one another by a spatial shift rather than through a power law similarity transformation, as discussed previously. Note that a travelling wave solution is related to a similarity solution via the following known transformation:
CHAPTER 3
69
In terms of the variables u, C7, £, r the expression (3.2.17) is put into the similarity form
(It seems curious to ask what sort of a travelling wave is obtained when a transformation inverse to (3.2.18) is applied for m > 1; in particular, we ask what is the wave parallel of the "analogue" of compact support.) 3.2.3. Observe that a monotonic travelling wave solution to (3.2.2a) with boundary conditions
exists for ra < 0. Indeed, let us seek u(x, t) in the form
Here c is some constant, still to be specified. Substitution of (3.2.19) into (3.2.2a,d,e) yields the following b.v.p. for «(0
Integration of (3.2.20a) yields, taking into account (3.2.20c),
For a solution of (3.2.20a), satisfying the boundary condition (3.2.20b), to exist, the condition
70
ELECTRO-NEUTRAL ELECTRO-DIFFUSION
must hold for all v G (0,1). Prom (3.2.22) a travelling wave solution to (3.2.2a,d) exists if
In particular for m = 0, integration of (3.2.21b) yields
or in terms of physical coordinates
Equation (3.2.24b) (or more generally for ra < 0, the appropriate integral of (3.2.21b), satisfying the boundary condition (3.2.20b)), represents a monotonic wave, travelling from left to right with speed c. In order to specify c the boundary condition (3.2.20b) has to be modified to
with c given. This specifies the flux at the left infinity, where the concentration vanishes. (Fulfillment of (3.2.25) implies vanishing of concentration at x = —oo, whereas (3.2.2d) alone leaves the flux there undefined together with the propagation speed.) Thus (3.2.25) specifies a travelling wave solution to (3.2.2a,e), (3.2.25) uniquely up to a shift
Moreover, it will be shown in due course that for m = 0 it also ensures uniqueness of a global solution to the corresponding Cauchy problem. The Cauchy problem is obtained by supplementing (3.2.2a,e), (3.2.25) by the initial condition
with u0(x) satisfying (3.2.25), (3.2.2e). Note finally that by rescaling x the condition (3.2.25) is reduced to the form
In due course we shall restrict our analysis to the case m = 0.
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The Cauchy problem to be studied thus consists of a search for uniformly bounded function u(x,t), such that
where UQ(X) is assumed to be an entire function such that
Note that there exists no less than one point of inflection of UQ(X) and assume that there is no more than a finite number of points of extrema of UQ(X).
The results for the Cauchy problem (3.2.29)-(3.2.33) are presented in §§3.2.4-3.2.6. Thus in §3.2.4 we introduce and study an auxiliary problem on a finite large interval. In §3.3.5 we employ the results of §3.2.4 to infer existence-uniqueness of a global classical solution to the main Cauchy problem considered. Section 3.2.6 contains a remark on the stability of the travelling wave (3.2.24). 3.2.4. Consider the following auxiliary problem. Find u(x,t\N) such that
Under the assumptions of §3.2.3 this problem obviously satisfies Gevrey's conditions [32]. Hence the local classical solution of this auxiliary problem exists on the time interval (0, T) whatever N > 0 is. Here
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ELECTRO-NEUTRAL ELECTRO-DIFFUSION
Moreover, whatever N is, this solution is an analytic function of x in (—N,N) and an analytic function of i for 0 < t < T, whatever T > 0 is. Assume that a solution of the auxiliary problem is obtained for 0 < t < T*. Then the solution of this problem may be continued onto the time interval (T*, T* + AT*), where
In what follows we call the operation of such a continuation "a step." Perform a countable number of steps. Let
be the sequence of their lengths. If the series
diverges, then a classical global solution u(x, t \ N) of the auxiliary problem exists and is unique, and moreover this solution is an analytical function of a; in (—N, N) and an analytical function of t at any finite interval of the t variation. Divergence of (3.2.41) follows from the following estimates. Estimate 3.1.
Equation (3.2.29) is a nondegenerate parabolic equation at any finite interval -N < x < N. Hence, by the maximum principle [33] max(u) is on one of the lines x = -TV, x = N or at t = 0. Due to (3.2.35) for all e > 0 there exists N so large that
Assume that the maximum of u is at the boundary x — -N at t0. Then according to the Hopf theorem for parabolic functions [34a,b],
so that (3.2.43b) implies
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which by the maximum principle contradicts (3.2.33). Thus, max(w) is positive and lies on x = JV, or at t = 0. Likewise, a minimum at x = —N could not be nonpositive which together with (3.2.32a), (3.2.33) yields (3.2.42). Estimate 3.2.
To obtain (3.2.46) let us introduce the notation
Then (3.2.34) implies
and consequently
Hence p is a parabolic function in —N < x < N, t > 0 and therefore its extrema are at x = —N, x = N or at t = 0. We have
Due to (3.2.48) and (3.2.50)
Due to the Hopf theorem extrema of p cannot be at x = N. Hence these extrema are at x = —N or at t — 0, so that
which implies (3.2.46). Estimate 3.3. We shall obtain here some estimates for the time and the higher space derivatives and infer the existence-uniqueness of a global classical solution for the auxiliary problem (3.2.34)-(3.2.36). Denote
Differentiation of (3.2.54) with respect to time yields
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ELECTRO-NEUTRAL ELECTRO-DIFFUSION
Hence Z is subparabolic for —TV < x < N which implies that its maximum is at x — —N, x = N or at t = 0. This means that
and, moreover,
in any domain of the u(x,t N) convexity. Hence it only remains to estimate the minimum of Z. Let D be any subregion of the region where u(x, t) is a function concave with respect to x so that
Hence due to (3.2.29)
Applying the estimate (3.2.52), we find that such that Now let D be the subregion of u convexity so that both terms in the righthand side of the equality
are of the same sign. The positiveness of u together with the negativity of uxx within this domain yields
Thus we may choose TO > 0, sufficiently large, so that
On the other hand,
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Let us define
With this definition, we have
which is the required estimate on ^ from below. This together with Estimates 3.1 and 3.2 also provides the existence of uniform bounds for uxx in all such domains for 0 < t < oo. Analogous arguments demonstrate the uniform boundedness of uxxx which is sufficient for a reference to Gevrey's results [32]. All stated above proves that the length of any interval of the solution continuation depends only on the initial data, so that ATn is independent of N. Hence the series (3.2.41) diverges, which proves the existence and uniqueness of a global classical solution to the auxiliary problem under consideration and, moreover, con firms our assertion that this solution may be constructed by the method of continuation. Moreover, analyticity of u(x, t \ N] with respect to x for all N > 0 shows that
is an entire function of x for all t > 0. 3.2.5. We may prove now the existence of a solution to the main problem (3.2.29)-(3.2.31). Indeed, let G(x,£,t-r) be Green's function of the b.v.p.
It is known [35], [36] that for all k > 0 and for all ra > 0 there exists Kkm > 0 such that
where E is the fundamental solution of the heat equation
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ELECTRO-NEUTRAL ELECTRO-DIFFUSION
Taking this into account, we rewrite equation (3.2.29) as
Hence a solution of the auxiliary problem (3.2.29)- (3.2.31) admits the integral representation
Fix x € (-N, N), t > 0 and pass to the limit N -> oo. Condition (3.2.32), inequalities (3.2.71), and definition (3,2.72) yield1
We find from here that u(x,t) is a solution of the integrodifferential equation
which implies the existence and uniqueness of the global solution to the main problem (3.2.29)-(3.2.31). 3.2.6. Unfortunately, nothing much is known presently about stability of travelling waves (3.2.21), (3.2.24). Nothing is known also about these waves as the time asymptotics of the solutions of the Cauchy problems with the initial conditions compatible with the corresponding waves at infinity. 1
Analyticity of u(x, t \ N) for all N > 0 justifies the passage to the limit under the sign of the double integral in (3.2.74).
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The best we can say now concerning stability is that if the initial conditions can be majorized and minorized uniformly by two shifts of the same wave, then the solution of the Cauchy problem with that initial condition will remain bounded by the appropriate two shifted propagating waves as the upper and the lower solutions. This follows immediately from the maximum principle. Thus the following theorem is true. THEOREM 3.1. Letu(x,t] be a solution of the Cauchy problem (3.2.29)(3.2.32), and let there exist d = const > 0 such that
where
Denote
Then
Naturally, this is also true for the solution of the Cauchy problems and the corresponding travelling waves for (3.2.2a) with boundary conditions (3.2.28) and m < 0. From here a sufficiently small disturbance of an initial wave profile develops into a state that remains close to the appropriate propagating wave; that is, the waves above are at least "marginally stable."2 3.3. Asymptotic front formation in reactive ion-exchange [1], [51]. 3.3.1. As illustrated by (3.2.11), for m > 2 the first derivative of concentration at the boundary of support is discontinuous; that is, a weak shock is formed at the zero concentration front. This stands in accord with the classical Rankine-Hugoniot condition that prescribes for any moving interface Xi(i)
Equation (3.3.1) implies that with a boundary of the support moving at a finite speed, the derivative at the boundary is finite, discontinuous for m = 2, and blows up one-sidedly for m > 2. Asymptotic L stability of these waves has been announced recently by Takac [37].
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ELECTRO-NEUTRAL ELECTRO-DIFFUSION
It was observed in the previous section that a certain limit case of nonreactive binary ion-exchange is described by the porous medium equation with m = 2; in other words, a weak shock is to be expected at the boundary of the support. Recall that this shock results from a specific interplay of ion migration in a self-consistent electric field with diffusion. Another source of shocks (weak or even strong in the sense to be elaborated upon below) may be fast reactions of ion binding by the ion-exchanger. The possibility of occurrence of concentration shocks in reactive ionexchange was first suggested by HelfFerich in [38]. These predictions were further experimentally supported by several investigators [39]-[45] whose measurements have provided information about the parameter range in which sharp fronts are formed. In particular, it was demonstrated that upon a certain alteration of systems' parameters (e.g., reduction of the externa'l concentration of the penetrating species [42]) the sharp front gets smeared out and thus transition occurs from shell progressive kinetics to continuous reaction kinetics (following the terminology of [46], [47]). Theoretical computations of the above authors [39]-[41] mainly concentrated upon tracing the propagation of an ideally sharp front under the a priori assumption of its presence. Technically, these treatments amounted to solutions, under different specific conditions, of the so-called diffusional Stefan problem [48], [49]. Exceptions were papers by Weisz [46], H611 and Sontheimer [42], H611 and Geiselhart [43], H611 and Kirch [44], and Weisz and Hicks [50]; they, without presupposing the presence of a sharp front, numerically treated ion diffusion in an ion-exchanger, accompanied by fast reversible binding of ions to the matrix. Local reaction equilibrium of Langmuir type was assumed. Upon the increase of the external "sorbtive" concentration, these solutions exhibited the formation of a sharp propagating concentration front. A good fit of the limiting front propagating rates, with the appropriate results stemming from the discontinuous treatment [42], [39], was observed, thus demonstrating the close relation between the two approaches. The numerical nature of the solutions in the references cited above did not permit inference of the explicit dependence of the front properties and structure on the system's parameters. In this section we address formation of concentration shocks in reactive ion-exchange as an asymptotic phenomenon. The prototypical case of local reaction equilibrium of Langmuir type will be treated in detail, following [1], [51]. For a treatment of the effects of deviation from local equilibrium the reader is referred to [51]. The methodological point of this section consists of presentation of a somewhat unconventional asymptotic procedure well suited for singular perturbation problems with a nonlinear degeneration at higher-order derivatives. The essence of the method proposed is the use of Newton iterates for the construction of an asymptotic sequence. Consider the exchange of two univalent counterions with concentrations Ci(x,t) (i = 1,2), in an infinite ion-exchange slab — oc < x < oo, electri-
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cally insulated at infinity. Let the ion C\ participate in a fast reversible binding reaction
with electrically neutral reactive sites F of the ion-exchange matrix. Denote by a the ratio of the immobilized reaction product A concentration [A] to that of reactive sites 7 < N, N being again a constant concentration of fixed charges in the ion-exchanger. (All variables introduced are dimensionless, normalized in some natural fashion.) With this notation conservation of mobile species reads
(For simplicity diffusivities of both counterions have been assumed equal.) By electro-neutrality
Assume that typical time scales of both direct and reverse reactions in (3.3.2) are much shorter than any other time scale in the system. Then the reaction (3.3.2) yields the Langmuir's local equilibrium relation between Ci and a of the form
Equilibrium constant e is defined as
(For the contents of local equilibrium approximation (3.3.6) in transport context, see [51], [52].) Similarly to the derivation of (3.1.13), summation of (3.3.3), (3.3.4) yields, using the local electro-neutrality condition (3.3.5) and insulation at infinity,
Elimination of C2, a, ip from (3.3.3), (3.3.4), using (3.3.5)-(3.3.7), finally yields for C\(x,t] the equation
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ELECTRO-NEUTRAL ELECTRO-DIFFUSION
Here
Equation (3.3.8) can be rewritten in terms of total (loading) concentration
as
Here
with Ci(q) defined by the inverse of (3.3.9) as
Equation (3.3.10a) represents a proper nonlinear diffusion equation with effective diffusivity £>eff5 defined by (3.3.10b) We shall trace the formation of a shock in the system above ((3.3.8) or (3.3.10)) by considering the evolution of an initial discontinuity of the ionic concentration.
or, respectively, in terms of total loading
Recall that physically the Cauchy problems (3.3.8), (3.3.11) or (3.3.10), (3.3.12) correspond to the exchange of the nonreactive counterion 62, initially loading the right half space, to the reactive counterion Ci, initially loading the left half of the ion-exchanger.
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The heuristic mechanism for persistence and propagation of an initially existing front in a solution to the Cauchy problem (3.3.8), (3.3.11) (or (3.3.10), (3.3.12)) in the limit e —> 0 is as follows. According to the Langmuir isotherm (3.3.6a), for e 0 with their particular structure irrelevant for the essence of the asymptotic phenomenon to be studied. Thus to maximally simplify the presentation in what follows we shall omit the A, A terms from (3.3.8) and (3.3.10), limiting ourselves to consideration of the Cauchy problem
After a shift
and introduction of the similarity variable x = z/2^/t, the Cauchy problem (3.3.17) is rewritten in terms of M, x as
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Below we discuss some alternative approaches to constructing an asymptotic solution to the b.v.p. (3.3.19) for e < 1 and outline the idea of the asymptotic procedure proposed (§3.3.2). This latter is carried out in §3.3.3 and is put into the context of Newton's method as an asymptotic procedure. 3.3.2. The heuristic mechanism for the creation of a weak shock in the solution of (3.3.19) in the limit e —> 0 is as follows. Start the integration of (3.3.19a) from the right end (+00). Since at x —> oc, u —> e, the coefficient of the second-order term in (3.3.19a) is of order e. yielding in the limit e^O,
Equation (3.3.20) suggests that the boundary value u = e will be transferred in the limit e —> 0 from +00 to some finite point £ (whose location has to be determined). On the other hand, with the integration started from the left (—oc), we have u( — oc) = ! + £ , £ = o(u), implying in the limit e —> 0,
Condition (3.3.21c) follows from the requirement of continuity of u at the point £. The presence of a weak shock at £ is now obvious, since (as is easily seen from (3.3.21)) the solution approaches £ from the left with a finite slope, whereas to the right of £ the limiting solution is identically zero. Observe with the aid of the scaling transformation
that the above heuristic image corresponds to the macroscopic (outer) scale with d = 0, a = 0 (u(x) = O(l), x < f) to the left of the front £ and d = 0, a = 1 (UR(X) = eui(x), x > £) to the right of £. (Here a is determined from the boundary conditions (3.3.19b,c).) The corresponding outer equations are
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ELECTRO-NEUTRAL ELECTRO-DIFFUSION
The outer solutions WL(#O), x < £; UR(X], x > £ are sought as asymptotic series of the form
Likewise the location of the front £ is sought as
Substitution of (3.3.24), (3.3.25) into (3.3.23) yields to leading order in e
Integration of (3.3.26a) yields
The outer solutions UL(X,S), UR(X,E) have to be smoothly matched with the aid of the appropriate inner solution. Moreover, the matching procedure must specify the location of the front £. The inner scale is easily found from (3.3.19), (3.3.22) to satisfy
which yields an inner equation of the form
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The inner solution
is again sought in the form of an asymptotic series
Substitution of (3.3.31) into (3.3.29) and the subsequent integration gives to leading order
Here A, B are integration constants. Matching (3.3.32) with (3.3.26b) yields
and hence, according to (3.3.28)
Matching (3.3.32) with (3.3.27) requires, according to (3.3.33a,c), that
Prom here on u\ stands for Ua with a = 1 and Z for Zd with d = 1. Matching conditions (3.3.34), according to (3.3.32) and (3.3.27), imply
which serves for determining £oWe point out here that since, according to (3.3.32),
conditions (3.3.34a,b) are identical with the Stefan conditions (3.3.14d,e), rewritten in terms of the similarity variable x, and (3.3.35) is identical to (3.3.16d). Determining £o from (3.3.35) concludes the construction of the leading term in the direct procedure of singular perturbation. Unfortunately, an essential difficulty already arises at the construction of the first correction.
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ELECTRO-NEUTRAL ELECTRO-DIFFUSION
This difficulty stems from the nonlinear degeneration of the highest order term in the outer equation (3.3.23a). Substituting (3.3.24a) into (3.3.23a) and equating terms proportional to s yields the following equation for the first correction in the outer expansion:
Equation (3.3.36) has to be solved within the interval (—00, £o)- The requirement that UL vanishes at the right end of this interval implies that there is an unlimited growth of UL . This difficulty is of a fundamental nature and cannot be avoided by refining the direct procedure (say, by allowing fractional powers of e in the expansion). This would only push the singularity into higher approximations. Reiss [53] has expressed the view that problems with jumps could hardly be treated with the aid of singular perturbations. Reiss suggests in this connection an alternative approach based on the rational function approximation of the jump and illustrates this method through a number of Cauchy problems for simple first-order model equations. The cumbersomeness of the realization of the Reiss method in the context of problem (3.3.19), likewise in the "weak shock" or in the equivalent "strong shock" formulation, makes it hardly applicable in this case. Kassoy, in his debate with Reiss [54] concerning the applicability of singular perturbations to jump phenomena, points out the possibility of overcoming the difficulties of the above-mentioned type by introducing a number of imbedded boundary layers, each of which carries a corresponding solution, singular at the inner side of the appropriate boundary layer. A similar, though simpler situation, is described in [55] in the context of applying the singular perturbation procedure to a linear problem with nonanalytic coefficients. It seems to us that an approximation of a smooth, regular exact solution by a sequence of singular functions is somewhat artificial and does not correspond to the physical essence of the matter. Moreover, realization of this procedure in our case is quite cumbersome. This motivated the search for an alternative approach to the asymptotic solution of the problem (3.3.19) [1]. The idea of the proposed method is as follows. Recall that the above formal leading; approximation appeared to be fairly reasonable, except for the inappropriate vanishing of the left outer term at the front, instead of its just being small there. Equality
caused the blowup of the first correction, which reflected the lack of uniform validity in omitting the e order term in the original equation
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on the way to (3.3.23a). With the leading order approximation small but finite at x = £o the unlimited growth of the correction term would probably be avoided; the smallness of u at x = £o would then just imply, according to (**), a large curvature, perfectly appropriate around the "weak shock." Summarizing these observations, we find it appealing to construct a procedure based upon a direct regular perturbation of a smooth, matched multiscale "starting" approximation (e.g. the composite term of the standard singular perturbation procedure [55]). Perturbations around such a nowhere-vanishing term would not blow up and it would then suffice to demonstrate the uniform asymptotic smallness of the correction to ensure that the desired asymptotic procedure is indeed found. This program is realized in the next subsection. 3.3.3. Let UQ(X, e) be a twice continuously differentiable starting approximation of the sort described above, satisfying the boundary conditions
Let us seek a solution of problem (3.3.19) in the form
Substitution of (3.3.38) into (3.3.19) yields for ui(x, e) after dividing through
byt/ 0 2 (z,e)
Here the upper dot stands for an z-derivative. Assume that the correction Vi(x, e) is of a smaller order in t than Uo(x, e), uniformly for all x, that is,
Then omission in (3.3.39a) of the v\/Uo terms as compared to unity reduces problem (3.3.39) at leading order in £ to
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ELECTRO-NEUTRAL ELECTRO-DIFFUSION
Here
Equation (3.3.40a) is linear of reducible order, and its direct integration with boundary conditions (3.3.40b,c) yields
Here
The correction VI(X,E), given by (3.3.42a), has to be evaluated next, in order to show that relation (* * *) holds, but before this the starting approximation U$(x,e} has to be specified. A natural candidate for UQ(X, e) is the composite "leading" term of the above singular perturbation procedure, with the outer and inner parts defined by (3.3.27), (3.3.32). This composite term is of the form
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Unfortunately, such a function UQ(X, e), with UL (x), u[ (Z) satisfying th matching conditions (3.3.34), does not suit our purpose. The inner term e • ul°}(Z) (as defined by (3.3.42), (3.3.43)) as Z -> -oo does have a term UL • (x — £o)5 in common with the outer term UL (x}. Unfortunate it also contains a term proportional to e In Z. This violates the boundary condition at — oo and brings about a divergence of the integral in the second term of (3.3.42a). Alternatively, we can try to construct UQ(X,E) from the solution of the leading order outer equation (3.3.26a), used to the left of £0 and EU\ (Z), used to the right of £cb with both parts matched at x = £o with continuous curvature. We thus consider
where u stands for the solution of (3.3.26a) with the boundary conditions
eQ stands here for the value of UQ at the matching point £0? that is,
with Q to be fixed by matching.
Finally, u(Z) in (3.3.44) stands for wL 0) , denned by (3.3.32), (3.3.33) with B fixed by (3.3.45c). This yields for u
Integration of (3.3.26a) with (3.3.45a) yields
Twice continuously differentiate matching of (3.3.46a), (3.3.46b) at x = £o implies for Q, £0,
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ELECTRO-NEUTRAL ELECTRO-DIFFUSION
The starting approximation UQ(X,S) thus finally becomes
with u given by (3.3.46a), (3.3.47a). It is easily seen that with UQ(X,E] defined by (3.3.48) all the integrals of (3.3.42a) converge and the whole expression (3.3.42a) is meaningful. We now proceed to demonstrate the uniform smallness in e of the correction vi(x, e), defined by (3.3.42), as compared to the leading term UQ(X, e), defined by (3.3.48), (3.3.47), (3.3.46a). Rewrite expression (3.3.42a) as
where R(x,e), P(x,e] are defined by (3.3.41), (3.3.42b). First let us evaluate the correction v\ at x = £o- According to simple estimates whose detailed derivation may be found in [1] the following equalities hold
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6 in (3.3.50a) satisfies e1/2 < O(6) < I (e.g., 6 = £l/3). Substitution of (3.3.50) into (3.3.49a) yields, to leading order
i.e., the correction v\ is by order y^ smaller at £o than the leading term U0. Consider further some x < £o- According to (3.3.49d,e), (3.3.50b,d)
We introduce the notation
Expression (3.3.49) can be rewritten, employing (3.3.52), (3.3.53) as
Evaluation similar to that leading to (3.3.50a,b) yields the estimates
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ELECTRO-NEUTRAL ELECTRO-DIFFUSION
Substitution of (3.3.50), (3.3.55) into (3.3.54) yields to leading order
According to (3.3.48), (3.3.47),
Thus for x < £o the correction vi(x, E) is uniformly smaller than the leading term UQ(X,E). Finally, for x > £o the following equalities hold to leading order in e (see again [1] for details).
with u(Z(x)} denned by (3.3.46a), (3.3.47a). Substitution of (3.3.57) into (3.3.49a) thus yields for x > £0
Here, employing (3.3.52), (3.3.46a), (3.3.47a),
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Finally
Expression (3.3.59a) can be rewritten as
Here employing again (3.3.46a), (3.3.47a),
Furthermore, it is easily seen from (3.3.46a), (3.3.47a) that
where T.S.T. denotes a transcendentally small term. Estimates (3.3.52c), (3.3.52d) yield
which together with (3.3.58b,c) finally give
This completes the proof of the uniform asymptotic smallness of the correction ^i(x, e), given by (3.3.42), as compared with the leading term UQ(X, e), given by (3.3.48). For illustration, we present in Fig. 3.3.1a,b the results of a numerical solution of the original system (3.3.19) (Curve 1) for e = 10~2, 10~3, 7 = 1 together with a plot of the leading term (3.3.48) (Curve 2). We also present for comparison a plot of | erfcx (Curve 3), the similarity solution for the linear diffusion equation with the boundary and initial conditions analogous
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ELECTRO-NEUTRAL ELECTRO-DIFFUSION
to those employed in our study. A comparison of Curve 3 with Curves 1, 2 illustrates the role of nonlinearity in the asymptotic production of the weak shock. It is finally seen from Fig. 3.3.1 that the exact solution of the nonlinear problem (3.3.19) (Curve 1) is fairly well approximated for these parameter values by the leading term (3.3.48) (Curve 2). In Fig. 3.3.2 we present the corresponding results for the concentration a(x, t} of bound ions, related to the data of Fig 3.3.1 through (3.3.6a). So far we have dealt with the construction of the leading term UQ(X,£] and of the first asymptotic correction vi(x, e). The next correction v%(x, e) is constructed in a fashion completely analogous to that leading to (3.3.42a). To this end u(x, e] is sought as
Here
Substitution of (3.3.61) into (3.3.19) and assumption of a uniform smallness in order e of v-2(x,e) compared with Ui(x,e) yields an equation for vz(x,£) that is identical to (3.3.40a), with UQ replaced by Ui in (3.3.41), and homogeneous boundary conditions. Integration of this b.v.p. yields for v%(x,e) an expression analogous to (3.3.42a), without the last term (which came from the inhomogeneity of (3.3.40b)). The same is true, regarding a correction vn(x, e) of any order.
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Fig. 3.3.Ib. Same as Fig. 3.3.1a for £ = io~ 3 .
The cumbersome part will always consist of proving the uniform smallness of the appropriate order correction v n (x, e) as compared with the total previous approximation
In practice this test could be performed numerically without much difficulty due to the identical structure of all the corrections (transition from n to n+1 amounts to replacing Un-i by Un in the expressions for P(x, e), R(x, e] that appear in the integrals in (3.3.42)). Of course, for a conventional asymptotic power expansion there is no need to compare the successive terms as long as the coefficient functions remain bounded. For a "starting" approximation in a more general situation than that considered here, a composite term of the last singularity-free approximation of a conventional singular perturbation procedure seems to be a suitable candidate, whenever it exists. When it does not, due to reasons similar to the one described ("wrong" asymptotic behaviour of the inner solution), a possibility for a twice continuously differentiate patching at a finite point is always available. The above smoothness is required by the proposed procedure for a second-order equation, whereas the possibility of a patching is guaranteed by the presence of three free constants, provided by the integration of the second-order inner and outer equations and by the unknown position of the shock. The procedure outlined thus allows a simple construction of a uniformly valid formal asymptotic solution, free of the difficulties invoked by the use of the standard asymptotic methods when applied to a situation similar
96
ELECTRO-NEUTRAL ELECTRO-DIFFUSION
Fig. 3.3.2. Profiles of bound ion concentration a(x). solution, Leading order asymptotic term.
Numerical
to that discussed here, with a nonlinear degeneration at the highest order derivative. The method discussed is nothing but the employment of the Newton iteration for constructing an asymptotic sequence. Whenever the starting approximation tends to the limiting solution as e —* 0 and the Newton method converges, the procedure proposed ceases to be formal and becomes a generator of a rigorous asymptotic sequence. Indeed, let C/(x, e) be an exact solution of some nonlinear b.v.p. and let UQ(X, e} be the starting approximation of the functional Newton's (quasilinearization) process, converging to t/(x, e) for any e > 0. Assume also that
Denote by Un(x,e) the nth Newton's iterate of UQ(X,E). Recall that the
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expression
is said to be an asymptotic representation of U(x, e] if the following equality holds for any n:
Observe that for a converging Newton's process (3.3.66a,b) hold. Indeed, due to quadratic convergence of Newton's method, we have and
Here k(e] is a positive bounded function of e only; for briefness of notation the supremum norm has been introduced as By (3.3.67a) we have
Substitution of (3.3.69), (3.3.67b) into (3.3.66b) yields
Prom (3.3.67b) we have
Here Equations (3.3.70), (3.3.71) together with (3.3.64) yield (3.3.66). Finally, let us stress that the obtained asymptotic feature is entirely due to quadratic convergence characteristic of Newton's method. Thus no process with a linear convergence, e.g., Picard's method, would generate an asymptotic sequence.
98
ELECTRO-NEUTRAL ELECTRO-DIFFUSION
3.4. Membrane potential of a binary electrolyte. A standard characteristic of a permselective membrane is the so-called membrane potential. It is measured between two well-stirred compartments, containing some binary electrolyte at different concentrations (Ci, 62), separated by the tested membrane. Potential drop between the compartments quickly arrives at a quasi-stationary value (stationary, assuming the concentrations strictly fixed in time), termed the membrane potential of the tested membrane for given concentrations. The higher the permselectivity of a membrane, the closer is its membrane potential to the "ideal'' equilibrium value corresponding to a complete impermeability of the membrane for co-ions. This equilibrium value
or
in dimensional units, is obtained by equating the electrochemical potentials (1.19a,b) of the penetrating ion in both compartments. (For a cation(anion-) selective membrane the potential is lower (higher) in the high concentration compartment). The measured value is normally lower than (3.4.1) due to the nonvanishing permeability at a real membrane to co-ions. Below we present a well-known calculation of membrane potential based on the classical Teorell-Meyer-Sievers (TMS) membrane model [2], [3]. The essence of this model is in treating the ion-selective membrane as a homogeneous layer of electrolyte solution with constant fixed charge density and with local ionic equilibrium at the membrane/solution interfaces. In spite of the obvious idealization involved in the first assumption the TMS model often yields useful results and represents in fact the main tool for practical membrane calculations. We shall return to TMS once again in §4.4 when discussing the electric current effects upon membrane selectivity. In the case of our present interest, the simplest TMS model of membrane potential for a l,zvalent electrolyte reads
Here p, n, ji, ji are, respectively, the dimensionless cat- and anion concentrations in the membrane and the ionic fluxes subject to determination.
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99
The boundary and the conjugation conditions for p, n, (f> are
Here Ci, Ci are the cation concentrations in both compartments. The conjugation conditions (3.4.6) express the continuity of the ionic electrochemical potentials at the membrane/solution interfaces. Finally, the electric insulation condition reads
The solution of the b.v.p. (3.4.2)-(3.4.7) is straightforward. Indeed, elimination of (px from (3.4.2), (3.4.3) yields, taking into account (3.4.4), (3.4.7),
Here
Integration of (3.4.8) in the range 0 < x < I yields
Here
100
ELECTRO-NEUTRAL ELECTRO-DIFFUSION
By (3.4.6), (3.4.5), and (3.4.4), pi, p 2 , ¥>' are related to Ci, C2 through the equations
Elimination of pi, p2 from (3.4.9b) via (3.4.10a) yields J. This in turn yields through (3.4.10b), (3.4.9a), (3.4.6c) the membrane potential
Below we present the appropriate explicit expressions for the case of a univalent electrolyte, z = 1
For N —> oo (3.4.13) yields the ideal equilibrium potential (3.4.la), independent of the relative ionic diffusivity a. In the opposite limit N —»• 0 membrane permselectivity is lost and the potential drop Ay> is reduced to the diffusion potential
3.5. Open questions. 1. Stability of the travelling wave (3.2.21)-(3.2.24). 2. Longtime asymptotics for the solution of a Cauchy problem for (3.2.2) with 0 < m < 1 and an initial distribution compatible with (3.2.4b) at x = ±00.
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101
REFERENCES [1] I. Rubinstein, Asymptotics of propagating front formation in diffusion kinetics, SIAM J. Appl. Math., 45 (1985), pp. 403-419. [2] T. Teorell, An attempt to formulate a quantitative theory of membrane permeability, Proc. Soc. Expt. Biol. Med., 33 (1935), p. 282. [3] K. H. Meyer and J. F. Sievers, La permeabilite des membranes. I. Theorie de la permeabilite ionique. II. Essais avec des membranes selectives artificielles, Helv. Chim. Acta, 19 (1936) pp. 649-664, pp. 665-680. [4] L. S. Leibenzon, Flow of Natural Fluids and Gases in Porous Medium, Gostechizdat, Moscow, 1947. (In Russian.) [5] Y. B. Zel'dovich and A. S. Kompaneets, On the theory of propagation of heat with thermal conductivity depending on temperature, in Collection of Papers Dedicated to the 70th Birthday of A.F. Yoffe, Izd. Akad. Nauk SSSR, Moscow (1947), pp. 61-71. [6] G. I. Barenblatt, Similarity, Self-Similarity and Intermediate Asymptotics, Consultants Bureau, New York, 1979. [7] O. A. Oleinik, A. S. Kalaschnikov, and Y. L. Czhou, The Cauchy problems for equations of the type of nonstationary filtration, Izv. Akad. Nauk SSSR, Ser. Math., 22 (1958), pp. 667-704. [8] L. A. Peletier, The Porous Media Equation in Application of Nonlinear Analysis in the Physical Sciences, H. Amann, N. Bazley, and K. Kirchgassner, eds., Pitman, Boston, 1981, pp. 229-241. [9] S. Kamin, The asymptotic behaviour of the solution of the filtration equation, Israel J. Math., 14 (1973), pp. 76-78. [10] J. G. Berryman and C. J. Holland, Stability of the separable solution for fast diffusion, Arch. Rational Mech. Anal., 74 (1980), pp. 379-388. [11] , Asymptotic behaviour of the nonlinear diffusion equation n t — ( n - 1 n x ) , J. Math. Phys., 2 (1982), pp. 983-987. [12] J. G. Berryman, Evolution of stable profile for a class of nonlinear diffusion equations with fixed boundaries, J. Math. Phys., 18 (1977), pp. 2108-2115. [13] K. E. Lonngren and A. Hirose, Expansion of an electron cloud, Phys. Lett. A, 59 (1976), pp. 285-286. [14] T. G. Kurtz, Convergence of semigroups of nonlinear operators with an application to gas kinetics, Trans. Amer. Math. Soc., 186 (1973), pp. 259-272. [15] H. P. McKean, The central limit theorem for Carleman's equations, Israel J. Math., 21 (1975), pp. 54-92. [16] T. Carleman, Problemes mathematiques dans la theorie cinetique de gas, AlmquistWiksells, Upsala, Sweden, 1957. [17] F. Helfferich and M. S. Plesset, Ion exchange kinetics. A nonlinear diffusion problem, J. Chem. Phys., 28 (1958), pp. 667-704. [18] J. R. Esteban, A. Rodriges, and J. L. Vazquez, Heat Equation with Singular Diffusivity, to appear. [19] M. A. Herrero, and M. Pierre, The Cauchy problem for w t =A(u m ) when 0<m-l, IMA J. Appl. Math., 41 (1988), pp. 147-163. [31] A. Friedman and S. Kamin, The asymptotic behaviour of a gas in an n-dimensional porous medium, Trans. Amer. Math. Soc., 262 (1980), pp. 551-563. [32] M. Gevrey, Sur les equations aux derivees partielles du type parabolique, J. Math. Pure Appl., 9 (1913), pp. 394-406, p. 435. [33] M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Partial Differential Equations Series, Prentice-Hall, Englewood Cliffs, NJ, 1967. [34a] A. Friedman, Remarks on the maximum principle for parabolic equations and its applications, Pacific J. Math., 8 (1958) 201-211. [34b] R. O. Vyborny, Properties of the solution of certain boundary problems of parabolic type, Dokl. Akad. Nauk SSSR, 114b (1954), pp. 563-565. [35] A. N. Tichonov, Sur I 'equation de la Shaleur a plusier variables, Bull. Univ. de Moscqou. Ser. Inter., Sect. Al (1938), p. 9. [36] L. Rubinstein, Free boundary problem for a nonlinear system of parabolic equations, including one with reversed time, Ann. Mat. Pura Appl., 135 (1983), pp. 29-42. [37] P. Takac, A fast diffusion equation which generates a monotonic local semiflow. II. Global existence and asymptotic behaviour, preprint. [38] F. Helfferich, Ion-exchange kinetics. V. Ion exchange accompanied by reactions, J. Phys. Chem., 69 (1965), p. 1178. [39] M. Sami Selim and R. C. Seqgrave, Solution of moving-boundary transport problems in finite media by integral transforms. I. Problems with a plane moving boundary, Industr. Engrg. Chem. Fundam., 12 (1973), p. 12. [40] P. R. Dana and T. D. Wheelock, Kinetics of a moving boundary ion-exchange process, Industr. Engrg. Chem. Fundam., 13 (1974), p. 20. [41] M. Nativ, S. Goldstein, and G. Schmuckler, Kinetics of ion-exchange processes accompanied by chemical reactions, J. Inorganic Nuclear Chem., 37 (1975), p. 1951.
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[42] W. H611 and H. Sontheimer, Ion-exchange kinetics of the protonation of weak acid ion exchange resins, Chem. Engrg. Sci., 32 (1977), p. 755. [43] W. H611 and G. Geiselhart, Kinetics of the neutralization of weak acid ion exchange resins with different solutions, Desalination, 25 (1978), p. 217. [44] W. H611 and R. Kirch, Regeneration of weak base ion exchange resins, Desalination, 26 (1978), p. 153. [45] W. H611, Optical verification of ion exchange mechanisms in weak electrolyte resins, Reactive Polymers, 2 (1984), p. 93. [46] P. B. Weisz, Sorption-diffusion in heterogeneous systems, Trans. Far. Soc., 63 (1967), pp. 1801. [47] F. G. Helfferich, Ion Exchange Kinetics — Evolution of a Theory in Mass Transfer and Kinetics of Ion Exchange, L. Liberti and F. G. HelfFerich, eds., NATO ASI Ser., E71, Nijhoff, Hague, the Netherlands, 1983, p. 157. [48] J. Crank, The Mathematics of Diffusion, Oxford University Press, Oxford, (1956), p. 102. [49] L. I. Rubinstein, The Stefan Problem, Trans, of Math. Monographs, 27 (1971), p. 18. [50] P. B. Weisz and J. S. Hicks, Sorption-diffusion in heterogeneous systems, Trans. Far. Soc., 63 ( 1967), pp. 1807. [51] I. Rubinstein, Asymptotics front formation in heterogeneous reaction diffusion kinetics, Phys. Chem. Hydrodyn., 6 (1985), pp. 879-899. [52] J. B. Keller, Liesegang rings and theory of fast reaction and slow diffusion, in Dynamics and Modelling of Reactive Systems, W. E. Stewart, W. H. Ray and C. C. Couley, eds., Academic Press, New York, 1980. [53] E. D. Reiss, A new asymptotic method for jump phenomena, SIAM J. Appl. Math., 39 (1980), pp. 440-455. [54] D. R. Kassoy, A note on asymptotic methods for jump phenomena, SIAM J. Appl. Math., 42 (1982), pp. 926-932. [55] J. Kevorkian and J. D. Cole, Perturbation Methods in Applied Mathematics, SpringerVerlag, Berlin, New York, 1981, p. 36.
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Chapter 4
Stationary Current with Local Electro-Neutrality
4.1. 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. In this chapter we shall treat the stationary passage of an electric current in locally electro-neutral systems with piecewise constant fixed charge density N(x) (ion-exchangers in contact, ion-exchange membranes—uni- and multipolar, semiconductors with piecewise constant doping, etc.). Assuming N(x) piecewise constant greatly simplifies the discussion as compared with the case of arbitrary N(x), while still providing the necessary insight into the effects of variable fixed charge density. (For a computational treatment of the variable doping case, see [l]-[4].) 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.) The equations of stationary one-dimensional electro-diffusion will be further integrated (§4.2) for an arbitrary number of the transferred charged species of arbitrary valencies. This result will be applied next to a num105
106
STATIONARY CURRENT
her of prototypical situations. Thus in §4.3 we will treat ionic transport through a multipolar membrane, consisting of adjacent ion-exchange layers with an alternating sign of the fixed charge. The case of a quadrupolar membrane (a thyristor, in semiconductor context) will be treated in detail. In particular it will be shown that a multiplicity of steady states, associated with the ionic transport through such a membrane may occur in a certain parameter range. The appropriate solution branches will be studied both numerically and analytically via suitable asymptotic procedures. 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.) Furthermore, the practically important effect of concentration polarization upon the counterion selectivity of ion-exchange membranes will be treated. Finally, some simple estimates will be presented for the three-dimensional electrolyte concentration and electric potential fields resulting from concentration polarization in a diffusion layer adjacent to a spatially inhomogeneous ion-selective interface (membrane). It will be shown that the appropriate fields are incompatible with mechanical equilibrium in an ionic fluid, so that a related (nongravitational) convection is expected to arise at an inhomogeneous ion-exchange membrane upon the passage of an electric current. Let us recapitulate the main notions involved in the local electro-neutrality (LEN) approximation. Consider a stationary electro-diffusion of n charged species (i = 1, • • • , M) of valencies Z{ in an open bounded macroscopic domain J7 c iln (n=l, 2, or 3) with a single macroscopic length scale L. Let the piecewise constant dimensional fixed charge density N(x) and the ionic concentration Ci within the ion-exchanger (doped semiconductor) scale with N. In the case of N ~ 0 (weakly charged ion-exchangers, electrolyte solution) let the ionic concentrations scale with the typical concentration CQ or TV, whichever is the larger. For typical "macroscopic" systems (electrolyte solutions, ion-exchangers, semiconductor devices) 10~3 < L(cm) < 1, 1(T8< AT(mol/cm3) < 10~3, 1(T6 < c0(mol/cm3) < 1(T3. As was pointed out in the Introduction, with the above scaling the equations of stationary electro-diffusion assume the form:
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107
Here x, fi, c^, N are the dimensionless counterparts o f f , O, Q, N, respectively. The dimensionless electric potential
0, (p'i, (p'i (i = 1,2, 3,4), /, J as functions of V. Instead of solving this system as it stands for different values of V, we shall consider V as a dependent variable, to be determined as a function of the electric current /, taken to be as known. This corresponds, in electrochemical terms, to replacing "potentiostatic" conditions with "galvanostatic" ones. With such an approach the a^ ai , J part of system (4.3.10), (4.3.12), given by (4.3.10a-c), splits from the rest of the equations, and greatly simplifies the treatment. Indeed, fr l 5 cr4 are immediately found from (4.3.10a,b) for i = 1,5 to be
We are then left with three equations for cr1, cr 2 , er3 of identical form (4.3.10c) with i = 1,2,3 and cr2, Nf, the turning point of the "upper" and "middle" branches shifts to infinity (V = oo, / = oo) with the multiplicity again disappearing and only the "low" current branch remaining in the V I plane. 4.3.3. Asymptotics. We begin with construction of the "low" current branch for CQ —> 0.
122
Fig. 4.3.4b. (p profiles. parameters as in Fig. 4.3.4a.
STATIONARY CURRENT
equilibrium potential profiles, other notation, and
Consider / of the order
It is observed from (4.3.8c) that
Furthermore, it is found from (4.3.8c) that to first order in /
Equations (4.3.15a) and (4.3.10a), satisfied to first order in / and CQ, yield
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123
Fig. 4.3.5. "Blow up" of the "low" current branch at N4=N^=-io.2 and "run off" of the turning point at AT4-+ #"=-8.612. Other parameters as in Figs. 4.3.3 and 4.3.4.
On the other hand, (4.3.13b) requires, to the first order in CQ, that
124
STATIONARY CURRENT
Fig. 4.3.6. cr profile at the "upper" branch at a high current, — numerical solution of the exact system (4.3.10a), (4.3.10c), — solution of the asymptotic system (4.3.22) (7=1500, J=-7.no, other parameters as in Figs. 4.3.3 and 4.3.4).
Then (4.3.15e) and (4.3.15f) yield for ae
Finally, (4.3.16), (4.3.14b) together with (4.3.15a-f) and (4.3.10a), taken to first order in / and CQ, accomplish the construction of the leading term in.the asymptotic expansion of crl(x) in CQ. (pl(x] and V to the same order are found from (4.3.11), (4.3.12a,b), (4.3.3e,f), (4.3.9a,b). It is observed that the minority carrier concentration, as predicted by the above construction, vanishes at the "depletion" interface #3 (cr2 = |N2|, cr3 = |-/V3|) at some value of current /, which is found from (4.3.15c), (4.3.16) to be
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125
Equations (4.3.17a,b) provide an expression for the "lower" branch limiting current to leading order in CQ. It is observed from (4.3.17a,b), (4.3.16) that 7]im "blows up" when {JV;}, {A;} are such that Dt = 0. Condition
together with (4.3.17a,b), (4.3.16) provides an asymptotic bound for the parameter range, in which the "lower" branch with current saturation exists, to leading order in CQ. For parameter values of Figs. 4.3.3, 4.3.4, the above asymptotic development predicts a "lower" branch voltage current curve indistinguishable from the exact one in Fig. 4.3.3a. The corresponding asymptotic value for the "lower" limiting current is 5.450 • 10~5 (the exact value computed numerically is 5.453 • 10~5). For JV^, defined in the previous subsection, (4.3.17), (4.3.16) predict the value —10.2, coinciding with that found via a numerical solution of the exact system. The bifurcation at 7V4 = jV%r may be studied by expanding cr'(x), cr-in (4.3.8c) to next (O(/ 2 )) order, as compared with 0(7) in (4.3.15), and again considering the vanishing of the minority carrier concentration at 0:3. Bifurcation analysis of this type has been carried out in [27]. Turning to the asymptotic analysis of the "high" current branch, we observe from (4.3.8c) that for 7 —> oo, crl(x) (i = 1, • • • , 4, 0 < x < L) and J scale with
as follows:
with a* (a;), j = O(l). With the above scaling, (4.3.10a) and (4.3.13a,b) yield to leading order in
126
STATIONARY CURRENT
and (4.3.10c) for i = 1, • • • , 4 can be rewritten to the same order as
The above equations (4.3.20a-d) represent the asymptotic version of (4.3.10c), (4.3.10a) to leading order in 7a. In order to outline the solution of system (4.3.20a-d) with respect to the unknowns Si (i = 1, 2,3) and j, and to infer the range of parameters {A^}, {Ai} in which the "high" current solution exists, we introduce the function
With this notation, (4.3.21a-d) can be rewritten as follows:
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127
Fig. 4.3.7. Graph of f(t)=t-\n(i+t) and scheme of solution of the asymptotic equilibrium (4.3.23).
The graph of f ( t ) is presented in Fig. 4.3.7 (continuous line). Denote the inverse of the left branch of f ( t ) (t < 0) by fl1 and the inverse of the right branch (t > 0) by /+1. With this notation, a consecutive exclusion of Si (i — 1,2,3) from (4.3.22ad) yields a single equation for j of the form:
Here i = —j. A chart of flow, as prescribed by the left-hand side of (4.3.23), for some initially picked value of z, is depicted in Fig. 4.3.7 (dashed line with arrows). A numerical realization of this scheme yields the value i = .03374 ({A^}, {A} as in Figs. 4.3.3, 4.3.4), coinciding with the value obtained via the numerical solution of the exact system (4.3.10c), (4.3.10a), (4.3.13a,b) for / = 1500 (J = -7.1141, parameters {JVJ, {AJ, c0 as in Figs. 4.3.3,
128
STATIONARY CURRENT
4.3.4). The appropriate asymptotic profile of a(x] (rescaled back from s with T = -7/J, / = 1500, J = -7.1141), presented in Fig. 4.3.6 (dashed line), is seen to coincide with the exact profile everywhere, except, naturally, in the immediate vicinity of the end point x = L. We shall illustrate the use of the above asymptotic treatment in order to evaluate the range of parameters {Afj}, {Aj} in which the "high" current branch exists, upon calculating JV|r of the previous subsection. We point out first that the "high" current solution exists as long as «2> as determined by (4.3.22a-d), is positive. Accordingly, define N%T so that for
From (4.3.22a,b), we have
On the other hand, it follows from (4.3.22c,d) that
It is easily observed from (4.3.24c) that for j < 0 fixed, 82 is a monotonically decreasing function of AT4. Taking into consideration that
equations (4.3.24a-c) yield:
Here jCT stands for j corresponding to N± = N". The solution of (4.3.25a) for jcr, with a subsequent substitution of the latter into (4.3.25b), yields a simple single equation for N". The solution of (4.3.25a) for jcr and (4.3.25b) for AT|r is trivially accomplished, say, graphically, from graphs of f ( t } . Thus, for Ni (i = 1,2,3), AJ (i = 1, • • • ,4) as in Fig. 4.3.5, such a crude graphic construction yields N" = -8.5 instead of N™ = -8.612, as is found via a numerical solution of the full exact system (4.3.10c), (4.3.13a,b). The appropriate critical values
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129
of the parameters {Ni} (i = 1,2,3), {Aj| (i = 1, • • • , 4) can be found in a completely analogous manner. The condition
together with (4.3.16), (4.3.17a-c), valid for CQ 0 at moderate voltages the space charge is confined to regions with a size of the order of the Debye length.) The nonequilibrium space charge effects in CP will be considered in the next chapter. 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. 4.4.1. 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. The stationary ionic transport across the unstirred layer is described by the following b.v.p.
134
STATIONARY CURRENT
Equation (4.4.1b) expresses impermeability of the ideally cation-permselective interface under consideration for anions; j is the unknown cationic flux (electric current density). Furthermore, (4.4.Id) asserts continuity of the electrochemical potential of cations at the interface, whereas (4.4.1g) states electro-neutrality of the "interior" of the interface, impenetrable for anions. Here N is a known positive constant, e.g., concentration of the fixed charges in an ion-exchanger (membrane), concentration of metal in an electrode, etc. E in (4.4.1h) is the equilibrium potential jump from the solution to the "interior" of the interface, given by the expression:
Finally, V in (4.4.1h) is the bias voltage, applied to the system. Integration of (4.4.1) yields
The following observation can be made about (4.4.1), (4.4.2). From (4.4.2d) j is a monotonic function of V, bounded from above by jiim, termed the limiting current density, such that
From (4.4.1a-c), we get
that is, the diffusional flux component is equal to the migrational component which implies that no steady current can be passed through the system without creating concentration gradients. Equation (4.4.2b) implies that
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135
that is, the applied bias voltage splits in half between the potential drop in the solution and the interfacial jump. Finally, we notice from (4.4.2a,b), (4.4.3) that the steady passage of the limiting current implies a vanishing interface electrolyte concentration.
According to (4.4.4) this corresponds to the greatest achievable constant concentration gradient within the unstirred layer. Recall that (4.4.1c) is a formal limit of (4.1.2) when e —> 0. For reasonable values of parameters e lies in the range 10~9 < E < 10~4. The second derivative of the electric potential at the interface, as given by (4.4.2c-d), is
In order for the local electro-neutrality approximation to be consistent in the vicinity of the interface the following inequality must hold
According to (4.4.2a,d) the inequality (4.4.8a) can be rewritten as
We observe that even for e extremely small of order 10 10 the condition (4.4.8b) (and accordingly the LEN approximation) is violated already for V as low as 16 (corresponding to the physical voltage of about .4 volts). This motivates in part the study of the space charge effects undertaken in the next chapter. In Fig. 4.4.1 (Curve 1) we present the IV plot prescribed by (4.4.2d). A perfect prototype of an ideally cation-permselective interface is a cathode upon which the cations of a dissolved salt are reduced. Experimental polarization curves measured on metal electrodes fit the theory very closely. Since in dimensional units the limiting current is proportional to the bulk concentration, the polarization measurements on electrodes may serve for determining the former. This is the essence of the electrochemical analytical method named polarography. (For the theory of polarographical methods see [28]-[30].) Another prototype of an ideally cation-permselective interface would be a cation-exchange membrane (C-membrane). Most practically employed Cmembranes are extremely permselective, so that their polarization curves would be expected to coincide with those at electrodes (given the same
136
STATIONARY CURRENT
Fig. 4.4.1. Curve 1 — IV curve prescribed by (4.4.2d). Curve 2 — a typical IV curve of a cation-exchange membrane.
geometry, thickness of the unstirred layer, etc.). Surprisingly, this is not the case. A prototypical polarization curve of a C-membrane is given in Fig. 4.4.1, Curve 2. The limiting current at a C-membrane is typically about twice as low as that at an electrode [31], [32]. Moreover, the fairly short "plateau" of the IV curve is followed by a sharp "second rise" of the current characterized by strong current fluctuations growing witgh voltage current The source of this behaviour probably lies in the electric inhomo (on the micron or tens of microns scale) of most synthetic ion-exchange membranes [32]. As will be shown at the end of this section, such an inhomogeneity may cause an appreciable reduction of the limiting current as compared with the homogeneous interface case. Moreover it will be shown that the electric potential and concentration fields developing in the course of CP at an inhomogeneous interface are incompatible with mechanical equilibrium of the liquid solution. The resulting electro-convection (see also §6.5) causes mechanical mixing, which, when it has grown strong enough with the increase of the applied voltage, probably results in the noisy second rise of the IV curves. 4.4.2. CP with an added supporting electrolyte. Quite often in electrochemistry a passive supporting electrolyte (with ions not taking part
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137
in the electrode reactions) is added to the system. This is done in order to increase the conductivity of the solution and thus to reduce the drop of potential in it. Let us trace how the addition of a univalent supporting electrolyte (with an anion identical and a cation different from those of the active electrolyte) will affect the above scheme of concentration polarization. The modification of the b.v.p. (4.4.1) to be considered is as follows:
Here c is the concentration of cations of the added electrolyte. Integration of (4.4.9), (4.4.10) yields
Here
138
STATIONARY CURRENT
For x = 1, (4.4.lib) yields
Prom (4.4.12), p ( l ] is a decreasing positive function of j for
For
equation (4.4.12) predicts p(l) < 0 and for j > j*im, according to (4.4.lla), a is negative. It follows from this consideration that with a supporting electrolyte present, the limiting current density is:
or, according to (4.4.lie) for CQ ^> 1
i.e., half of the value (4.4.3) which is found in the case of no supporting electrolyte. Accordingly, from (4.4.lie) for x = 1, j = jf im , we get
For c0 > 1 ( oc
whereas from (5.3.4)
It can easily be inferred that the subsequent terms grow in V even faster. 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
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NONEQUILIBRIUM SPACE CHARGE
essence of these phenomena, termed a "punch through" in the semiconductor context, is the proliferation of the space charge at high voltage upon the entire physical region. This may be accompanied by additional effects, some of which will be traced below for the b.v.p. (5.3.1)-(5.3.5) or related settings. Let us begin by considering the asymptotics of the solution to the b.v.p. above for V —> oo and e fixed. LEMMA 5.2. Let p(x,), n(x, V), oo. Proof. Assume the opposite, i.e., that, say, p (the proof for n is identical) increases unboundedly somewhere within the interval 0 < x < I when V —* oo. Since p(0) = 1, p ( l ) = N there exists M > 0, such that for all V > M there will be a maximum of p at some XQ € (0,1). For x = XQ we have
Equations (5.3.1), (5.3.3), (5.3.45a,b) imply
Equation (5.3.46) implies that n grows unboundedly too, when V —> oo. Since n(0) = 1, n(l) = e~^E~v\ n has a maximum for x — yo € (0,1). Thus,
From (5.3.2), (5.3.3)
that is, by (5.3.47a,b)
Equations (5.3.46) and (5.3.49) yield
From (5.3.6), we have
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By (5.3.45a), (5.2.47a), (5.3.50) equations (5.3.51a,b) yield
Equation (5.3.52b) implies
which is incompatible with p ( l ) = N prescribed by (5.3.5a), that is, a contradiction has been arrived at. A simple asymptotic estimate based on Lemma 5.1 yields the following theorem. THEOREM 5.1. Let p, ra,
oo. As an initial condition we employ the steady concentration fields, computed for the previous voltage value, starting from the known equilibrium fields at V = 0. For V < VCT the appropriate solution for t —> oo coincides with those for (5.3.1), (5.3.5).
Fig. 5.4.2a,b. Same as Fig. 5.4.1, for e=iQ~6.
In Figs. 5.4.la and 5.4.2a, we present the steady VC curves, corresponding to Vcr —> oo, computed, respectively, for £ = 10~~4, 10~6. For Vcr finite the above procedure is carried out until VCI is reached. At this point the modulation is switched on and the unsteady computation is performed for a few tens of periods T until the transients die out. The computed current density
is then averaged over one period, yielding the DC current component / :
188
NONEQUILIBRIUM SPACE CHARGE
Here t'» T is the time moment when the averaging is started. In Figs. 5.4.1b and 5.4.2b we present some plots of
as a function of modulation frequency /, computed for e = 10 4 (Vcr = 15, A = 10) and e= 10~6 (Vcr = 45, A = 35), respectively. Here 7£ is the steady current, corresponding to Vct.
Fig. 5.4.3. Conventional scheme of rectification for a convex VC curve.
It is observed from Figs. 5.4.Ib and 5.4.2b that for a given e there exists a frequency range wherein the AV modulation, applied at the "plateau" of the steady VC curve, yields an increase of the DC current beyond the corresponding steady value. In order to interpret this result, recall first that a straightforward quasisteady reasoning, predicts a negative rectification effect for convex VC curves. Indeed we expect that a harmonic voltage modulation, applied at a convex region of the VC curve, will yield, in the limit u —> 0, a greater decrease in the current in the negative half-period than increase in the positive half (see Fig. 5.4.3 for a schematic illustration). The net rectification effect is thus expected to be a decrease of the direct (DC) current component as compared to the steady value. For a finite u, we expect an intermediate effect between a maximal value for u —» 0 and nil for di —> oo. According to this naive scheme no effect at all is expected at the linear part of the VC curve, particularly at the plateau. This scheme is in accord with the results of the detailed electro-neutral calculation,
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based upon the model (5.4.1)-(5.4.8) with the Poisson equation replaced by the local electro-neutrality condition and the boundary conditions (5.4.6), (5.4.8) replaced by a condition of continuity of the electrochemical potential of cations crossing the interface. A direct numerical integration in this locally electro-neutral i.b.v.p.6 shows that, e.g., for V cr = 15, A= 10, / =
Fig. 5.4.4. Time plots of voltage and minimal concentration in a locally electro-neutral model.
6
A straightforward calculation in the electro-neutral model yields a single nonlinear integral equation for the direct component of the current with a harmonic voltage modulation at a given direct bias of the type, known in the polarographical literature. Study of this equation appears more cumbersome than a direct numerical solution of the corresponding i.b.v.p.
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NONEQUILIBRIUM SPACE CHARGE
1, the negative rectification effect is about 0.05% of the steady current value and rapidly decreases further until it ultimately vanishes with increasing /. The immediate question is then how is this compatible with the arguments concerning sensitivity of the system to the value of concentration at the minimum and the expected related positive rectification? To answer, we have to examine the detailed time evolution of the minimal electrolyte concentration Cm\n(t) (the interface concentration in the electro-neutral picture) during one period. Bear in mind that, since at the plateau of the VC curve practically all of the system's resistance is concentrated at the location where the concentration is at its minimum, the electric current in the system is proportional to C m i n (t) • V(t). In Fig. 5.4.4 we present the calculated time plots of Cm-in = C(t, 1), V(t) during one period for / = 1, A= 10, Vcr= 15. It is observed that in one half-period the minimal concentration indeed increases considerably more than it decreases in the other half-period. However, since the voltage is applied right where the concentration is at its minimum, the latter is always exactly in counterphase with the former. As a result there is no positive current rectification, despite the net increase of the minimal concentration during one period, as compared to the steady value. The situation becomes essentially different if an account is taken of the space charge near the interface. In the steady picture, the location of minimum cation concentration is then shifted towards the bulk by a distance A, dependent on e and V. This is illustrated in Fig. 5,4.5, where the steady
Fig. 5.4.5. Steady ionic concentration profiles for e=io~ 4 . —, cationic concentration, , anionic concentration.
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ionic profiles are plotted for e = 10 4 and increasing values of the applied voltage. With AV modulation applied, concentration at the minimum is periodically effected. However, this time, as opposed to the previous case, the spatial separation between the position of the concentration minimum and the interface determines a diffusional time lag of order r ~ A2/D between the voltage and the minimal concentration variation, with a corresponding frequency dependent phase shift. In a certain frequency range this
Fig. 5.4.6. (a) Time plots of voltage and minimal cation concentration for e=io- 4 , /=125. (b) Same for /=iooo.
phase shift becomes such that the alternating voltage gets into phase with the increase of the minimal concentration in the appropriate half-period. As in the previous electro-neutral case, this increase of the minimal concentration is appreciably bigger than the respective decrease in the other half-period, which together with the "favorable" phase adjustment yields the positive rectification effect of Figs. 5.4.1b, 5.4.2b. These points are illustrated in Figs. 5.4.6a,b where the time plots of the minimal concentration and voltage are presented for e = 10~4 and two
192
NONEQUILIBRIUM SPACE CHARGE
different frequencies (/= 125, 1000), the second well above the optimum. When the frequency is too high, the phase shift between the voltage and the minimal concentration becomes again "unfavorable" for positive rectification. Besides, upon the increase of modulation frequency the amplitude of concentration disturbance decreases, eventually resulting in the disappearance of the rectification effect. An experimental test of the "anomalous" rectification has been carried out upon a cathodic reduction of copper ions from a CuS04 solution. The measurements were carried out in a standard three electrode setup, with a stationary vertical copper disk (0.15cm in diameter) employed as the working cathode, and another copper disk (2.5cm in diameter) as the counterelectrode. The reference electrode was a copper wire. The working electrolyte was aqueous CuSC>4 in the concentration range from 0.002'10~3M/cm3 to O.MO~ 3 M/cm 3 . The steady direct voltage, maintained between the small working disk cathode and the reference electrode with the aid of a potentiostat, was modulated with a high frequency volt-
Fig. 5.4.7. (a) • - Measured steady voltage-current curve for O.oo2-io~3 M/cm3 CuSot solution. A -Absolute rectification effect for A=.1V, /=iMHz. (b) Relative rectification effect in the same solution for: • - A-.2V, /=.5MHz. o - A=.3V, /=.5MHz. A - A=.1V, / = l M H z .
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Fig. 5.4.8. (a) Measured steady VC curve for o.oi-icr 3 M/cm 3 CuSOt solution. (b) Relative rectification effect for /=iMHz: x - A=.iv, • - A=.1V, o - A=.275V. (c) Relative rectification effect jor /=.5MHz.- o - A=.275V.
age of a given amplitude. The DC current and the direct and alternating AV voltage components were recorded. The steady polarization curves in this arrangement exhibited a virtually perfect saturation. AV modulation experiments were carried out in the frequency range from 10 kHz to 1 MHz with modulation amplitude ranging from 0.1 V to 0.3 V. With AV modulation away from the plateau region, no change in the DC current as compared with the steady value was observed at any frequency. Similarly, no effect was observed upon applying the modulation at the plateau in the frequency range below 100 kHz. On the other hand, upon applying an AV modulation of frequency above several hundred kHz at the plateau of the polarization curve, a marked steady increase of the DC current was seen. The magnitude of this increase was strongly dependent on the modulation frequency, the modulation amplitude, and the electrolyte concentration. In Figs. 5.4.7 and 5.4.8 we present some typical results for C = 0.002 • 10~3 M/cm 3 and C = 0.01 • 10"3 M/cm3. It was observed that upon increasing the working electrolyte concentration, there was an increase in the characteristic frequency at which the
19 4
NONEQUILIBRIUM SPACE CHARGE
rectification effect appeared. In other words, for a given frequency and modulation amplitude, an increase in the concentration yielded reduction and an eventual elimination of the effect. Thus, increase of concentration from 0.01 • 10~3 M/cm3 to 0.05 • lO^M/cm 3 , for a_modulation frequency of 1 MHz and a modulation amplitude of 0.2V at V = .3V, resulted in a reduction of the rectification effect from 15% to virtually nil. A similar result was obtained by the addition of a supporting electrolyte. Thus addition of 0.1- 10~ 3 M/cm 3 (V = .3 V, A = .2V) completely eliminated the rectification effect. Recall that increase of the working electrolyte concentration yields contraction of the extended nonequilibrium space charge, whereas the addition of a supporting electrolyte virtually eliminates the former. Reduction of the rectification effect upon increasing the working electrolyte concentration and its complete disappearance upon adding a supporting electrolyte thus stand in agreement with the mechanism outlined. Typical frequencies at which the rectification effect was observed correspond to a diffusional length beyond 300 A in accord with the characteristic thickness of the space charge region, as evaluated from steady state calculations (§5.3). 5.5. A uniform asymptotics for the nonequilibrium space charge in a bipolar membrane under a steady electric current [4]. It is our purpose in this section to outline an asymptotic procedure, alternative to that of the §§5.2, 5.3, that was based upon a matched asymptotics expansion in powers of the relative squared equilibrium Debye length. We recall that these expansions, valid for arbitrary fixed charge densities or bulk electrolyte concentrations, broke down for moderate voltages, due to their intrinsic nonuniformity with respect to the latter. The asymptotics to be outlined here briefly is a version of that developed in great detail by Please [4], [13] for a semiconductor p — n junction. The specificity of this asymptotics is that it is valid uniformly for any voltage. On the other hand the applicability of this asymptotics is limited to systems with high nowhere vanishing fixed charge density, such as unipolar and multipolar membranes or semiconductors with high doping levels. Accordingly the asymptotic procedure to be outlined is not suitable for treating systems containing parts with zero fixed charge density, e.g., an electrolyte solution layer. Consider the following simplest prototype problem for stationary electrodiffusion of a univalent symmetric electrolyte through a bipolar ion-exchange membrane with an antisymmetric piecewise constant fixed charge density \N(x).
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Here
and jp, jn are, respectively, the constant unknown cationic and anionic fluxes, to be determined from the solution of (5.5.1)-(5.5.3). Conditions (5.5.2a,b) and (5.5.3a,b) imply that the end points are assumed to be locally electro-neutral at equilibrium with an external solution of unit normalized concentration. The latter is in turn assumed to be much lower than the relative fixed charge density XN, which implies by (5.5.4a),
The b.v.p. (5.5.1)-(5.5.3) again stands for galvanostatic conditions, with the electric current density
in (5.5.3c) regarded as known. Please [4] begins his construction of an asymptotic solution to (5.5.1)(5.5.3) for A ^> 1 by considering the zero current limit
In this limit the problem (5.5.1)-(5.5.3) is easily reduced to a problem for the Poisson-Boltzmann equation similar to the b.v.p. (1.25)-(1.26) of the Introduction. The corresponding developments are as follows.
196
NONEQUILIBRIUM SPACE CHARGE
5.5.1. 7 = 0, equilibrium.
Basic b.v.p. First let us show that zero current in the present stationary, nonlocally electro-neutral constellation implies a true equilibrium, that is, the vanishing of jp and jn separately. Indeed, from (5.5.5a,b) we have
and, using (5.5.11b), we get
Substitution of (5.5.6b). into (5.5.la) yields
or,using the boundry conditions (5.5.2a)
It follows from (5.5.7b), taking into account the nonnegativity of p and n, that the boundary condition (5.5.3a) at x = I can be satisfied if and only if
which completes the proof. Integration of (5.5.1a,b) with (5.5.8), (5.5.6a) yields, taking into account the boundary conditions (5.5.2c)
The boundary values p(—1), n(—1) are determined from (5.5.2a,b) as
Similarly, by (5.5.3a,b), we have
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and thus from (5.5.9)
Substitution of (5.5.9) into (5.5.1) yields
By means of the shift,
equations (5.5.12a), (5.5.lie), (5.5.2c) are reduced to a more symmetric final form
of the equilibrium b.v.p. (In what follows we use (5.5.13b) as the boundary value for (p(—l), which is always possible due to the arbitrariness of (p up to a constant.) The outer problem. Equations (5.5.13b,c) yield
198
NONEQUILIBRIUM SPACE CHARGE
The balance between the exponentials and the XN term in (5.5.13a) yields up to the O(^) order a piecewise constant outer solution of the form
This outer solution, discontinuous at x = 0, has to be smoothed out via an internal layer solution around this point. In this internal layer we distinguish the inner region around x — 0 in which the potential is close to zero and the derivatives term is balanced by N, flanked by two transition layers. In those layers, three terms balance—the derivative, the N term, and one of the two exponents (the positive one for x < 0 and the negative one for x > 0). To realize this program, we introduce the inner ~x and \t variables as
In terms of x, * (5.5.13a) is rewritten as
The transition layer, Seek
in the transition layer as
to hold for
and as
to hold for
Here s, r > 0 are the transition layer coordinates and § is a yet unknown constant, to be determined from matching. 6 determines the position of
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the transition layer of "thickness" A"1/2 within the entire internal layer of "thickness" (InA/A) 1 / 2 . Substitution of (5.5.18) into (5.5.13a) yields to main order in A for x < 0
which, in turn, yields after two integrations
The integration constant A can be incorporated into 0. This completes the construction of the equilibrium solution to leading order in A. From here Please [4] goes on to the analysis of the nonequilibrium case starting from the low current limit. 5.5.2. Nonzero current. Please begins with low current / 0) an asymptotic analysis for the intermediate-high range of voltages V — O(e~1) (see the notation of §5.3) has recently been carried out by Schmeiser [16]. His method, combined with those of §5.3 is likely to yield an asymptotic solution of the CP problem of §5.3, valid uniformly for any voltage. 5.6. Open questions. 1. Construction of an asymptotic solution to the stationary concentration polarization problem of §5.3, uniformly valid for all voltages (see [16]). 2. Application of the uniform asymptotics above to the study of nonstationary effects occurring at high concentration polarization, such as the anomalous rectification in §5.4.
202
NONEQUILIBRIUM SPACE CHARGE
REFERENCES [1] J. L. Jackson, Charge neutrality in electrolytic solutions and the liquid junction potential, J. Phys. Chem., 78 (1974), pp. 2060-2064. [2] I. Rubinstein and L. Shtilman, Voltage against current curves of cation exchange membranes, J. Chem. Soc., Trans. Faraday Trans., II, 75 (1979), pp. 231-246. [3] Isaak Rubinstein, Israel Rubinstein, and E. Staude, High frequency rectification in concentration polarization, PCH Phys. Chem. Hydrodynamics, 6 (1985), pp. 789802. [4] C. P. Please, An analysis of semiconductor P—N junction, IMA J. Appl. Math., 28 (1982), pp. 301-318. [5] M. S. Mock, Analysis of Mathematical Models of Semiconductor Devices, Boole Press, Dublin, 1983. [6] P. A. Markowich and C. A. Ringhofer, A singularly perturbed boundary value problem modelling a semiconductor device, SIAM J. Appl. Math., 44 (1984), pp. 231-256. [7] C. Ringhofer, An asymptotic analysis of a transient p—n junction model, SIAM J. Appl. Math., 47 (1987), pp. 624-642. [8a] P. Henderson, Zur Thermodynamik der Fliissig Keitsketten, Z. Phys. Chem., 59 (1907), p. 118. [8b] , Zur Thermodynamik der Fliissig Keitsketten, Z. Phys. Chem., 63 (1908), p. 325. [9] R. E. Bellman and R. E. Kalaba, Quasilinearization and Nonlinear Boundary Value Problems, Elsevier, New York, 1965. [10] M. S. Mock, A time dependent numerical model of the insulated gate field effect transistor, Solid State Electronics, 24 (1981), pp. 959-966. [11] T. Teorell, An attempt to formulate a quantitative theory of membrane permeability, Proc. Soc. Expt. Biol. Med., 33 (1935), p. 282. [12] K. H. Meyer and J. F. Sievers, La permeabilite des membranes. I. Theorie de la permeabilite ionique. II. Essais avec des membranes selectives artificielles, Helv. Chim. Acta, 19 (1936), pp. 649-680. [13] P. A. Markowich, The Stationary Semiconductor Device Equations, Springer-Verlag, New York, 1986. [14] C. P. Please, An Analysis of Semiconductor p—n Junctions, D. Phil, thesis, Oxford University, Oxford, 1978. [15] J. Kevorkian and J. D. Cole, Perturbation Methods in Applied Mathematics, SpringerVerlag, New York, 1981. [16] C. Schmeiser, On strongly reverse biased semiconductor diodes, to appear.
Chapter 6
A Prototypical Connective Electro-Diffusional Phenomenon—Electro- Osmotic Oscillations
6.1. Preliminaries. This entire chapter is devoted to one physical phenomenon—electro-osmotic (Teorell) oscillations. As opposed to phenomena discussed in previous chapters, electro-convection will be of importance here in its interaction with electro-diffusion. Electro-osmotic oscillation (first observed by Teorell [l]-[4] in a laboratory set-up devised to mimic nerve excitation) may likely represent a common source of oscillations in various natural or synthetic electrokinetic systems such as solid microporous filters, synthetic ion-exchange membranes or their biological counterparts. The original experimental set-up, which contained all essential elements to look for when the electro-osmotic oscillations are suspected in a natural system, is schematically as follows. Two vessels (I and II) are connected through a microporous glass sinter filter (see Fig. 6.1.1) with a weak negative charge on the matrix while in an aqueous solution. The vessels contain an electrolyte solution, e.g., KC1, at different concentrations C\ and Ci, respectively, (C\ < C^}. Vessel I is open through a large orifice whereas vessel II possesses a manometer tube, so that liquid elevation h in it measures the hydrostatic pressure difference between the vessels. Both vessels contain large working electrodes that allow a prolonged galvanostatic passage of a DC current / through the system in the direction from I to II. Both vessels are supplied with small test electrodes to measure the voltage drop E = $1 — $2 on the filter. Concentration equilibration between the vessels is sufficiently slow for the concentrations in each vessel to be viewed as constant in the course of the experiment. Permselectivity of the filter is very low so that at zero current 7 = 0 the voltage drop on it is practically zero (membrane potential is negligible as is the diffusion potential for practically equal ionic diffusivities in the KC1 case). 203
204
ELECTRO-OSMOTIC OSCILLATIONS
Fig. 6.1.1. Scheme of Teorell's cell.
The osmotic pressure drop on the filter also is negligible, so that at zero current the liquid level in both vessels is equal (h = 0). The experiment consists of passing an increasing sequence of DC currents with a small increment between the subsequent current values, while observing the evolution in time of the voltage E(t) and of the hydrostatic pressure drop P(t) related to the elevation h(t) as
Here m is the liquid density in vessel II and g is the gravitational constant. For each new current value sufficient time is allowed for the establishment of the steady state if possible. Phenomenology observed by Teorell was roughly as follows. For the current range between zero and some lower threshold value J°, the system approached the new voltage monotonically in time. For the current in the range 7° < / < 7C, Ic being another threshold value, the system approached the steady state, this time through a decaying oscillation. For I > Ic the system did not approach any steady state at all but rather oscillated with p and E amplitudes dependent on the increment of the current above the critical value . With the current increasing above /c, the stabilized shape of the oscillations soon received a relaxation character, schematically depicted in Fig. 6.1.2.
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Fig. 6.1.2. Sketch of typical Teorell's oscillations.
Period of oscillations varied from a few seconds to a few tens of minutes, depending on the system parameters such as filter surface area Ss, manometer tube cross section area Sm, etc. To rationalize his observations Teorell suggested the following model in terms of ordinary differential and algebraic equations for the average dynamic characteristics of the system concerned. Let V(t) be the average volumetric velocity of the liquid in the filter and R(t) be its instantaneous electric resistance. By mass conservation
with h(t) related to P(t) through (6.1.1). Furthermore, by Ohm's law The flow rate V(t) is related to E(i) and P(i) via a linear phenomenological relation of the form Equation (6.1.4) asserts that the volumetric flow rate is a superposition of two components. They are the electro-osmotic component proportional to the electric field intensity (voltage) with the proportionality factor u> and the filtrational Darcy's component proportional to — P with the hydraulic permeability factor v. Teorell assumed both Hi and v constant. Finally another equation, crucial for Teorell's model, was postulated for the dynamics of instantaneous electric resistance of the filter R(t}. Teorell assumed a relaxation law of the type
206
ELECTRO-OSMOTIC OSCILLATIONS
Fig. 6.1.3. Typical stationary resistance f as a function of the flow rate v.
Equation (6.1.5) is motivated by recognizing that the instantaneous resistance R(t) tends to relax quickly by electro-diffusion with some effective rate constant k to a steady value f(V/Vo), corresponding to the instantaneous flow rate V(i) and depending upon the concentrations C\ and C^. Here VQ is some typical flow rate picked from a calculated or experimental stationary resistance curve of the sort depicted schematically in Fig. 6.1.3. This type of dependence merely stands for the filter resistance's being high when the solution flows from the vessel I to II (V > 0). In this case the filter ionic composition is dominated by the low concentration C-\_. In the opposite case, when V < 0, the stationary resistance is low due to the high concentration solution's (Cj) being brought into the filter by the flow from the vessel II. Teorell studied the system (6.1.1)-(6.1.5) graphically, by the isocline method, and also numerically. He recovered most of the features observed experimentally. This study was further elaborated by several investigators. Thus, Kobatake and Pujita [5], [6] criticized the original model for invoking the "ad hoc" equation (6.1.5). These authors assumed instead instantaneous relaxation of the resistance to its stationary value while preserving the overall order of the relevant ordinary differential equation (ODE) system by including consideration of the mechanical inertia of the liquid column in the manometer tube. In addition, the simple phenomenological relation (6.1.4), with a constant electro-osmotic coefficient ui, was replaced by a more elaborate one, accounting for the w dependence on the flow rate and the concentrations Ci, C 1 is the dimensionless relaxation parameter, assumed large in accordance with Teorell's intuitive ideas. By expressing R(t) through p(t) from (6.2.4) and substituting the result into (6.2.5) we arrive at a single equation for p(t) of the form
Here
To the leading order in e, (6.2.6), known as the Rayleigh's equation, can be rewritten as
Equation (6.2.7) is the one to be treated in this section. We shall observe that in a suitable range of i the solutions of (6.2.7) tend to a limit cycle that corresponds, for e ic). The critical current value ic is determined from the condition
which yields, according to (6.2.11)
A standard stability analysis of the equilibrium E(UQ,PQ) implies that the latter is linearly stable for i < ic and unstable for i > i°. Indeed, for e fixed the eigenvalues cr li2 of the appropriate linearized problem are given by the expression
210
ELECTRO-OSMOTIC OSCILLATIONS
Thus
whereas
The equilibrium point E(0,po) is thus either a node for \i - i°\ > 2-\/£//'(0) (stable for i < ic and unstable for i > i°) or a focus for |f - ic\ < 2 A /e//'(0) (stable for i < i° and unstable for i > ic). This is illustrated schematically in Fig. 6.2.1 where the flow of a with variation of i is depicted in the complex plane.
Fig. 6.2.1. Scheme of a flow in a complex plane at varying i.
At i =
we have
i.e., at this value ofi the focus at E(0,po) loses its stability; that is, a Hopf bifurcation takes place. As a result, a limit cycle appears around E(Q,po) and stays on for all i > ic', whereas no limit cycles in the phase plane are possible for i < ic. This stems from combining the fact that E(Q,po) is the only equilibrium point in the phase plane, with the instability of the infinity arguments to be
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211
discussed below and Bendixon's test [lOa] for nonexistence of limit cycles in a plane. This test applies to systems of the form
The test asserts that if the expression
preserves its sign in a simply connected domain in the x, y plane, then there are no limit cycles of the system (6.2.16) in this domain. In our case
and accordingly, remembering that by (6.2.10), (6.2.13b) sup|/'(w)| < |/'(0)| = l/t c ,
This implies that for i < ic there are no limit cycles in the phase plane (w,p). Note that this conclusion relies on (6.2.10) and thus is only true for a "symmetric" f ( u ) . For a non-symmetric f ( u } by the above argument there are no limit cycles for i < ix = l/f'(ux) < i°, ux being the coordinate of the point of inflection of/(u), generally different from zero. On the other hand, some limit cycles in this case still might exist in the current range ix < ic. Furthermore, instability of the infinity in our case may be inferred from the following simple argument due to Andronov et al. [10b] Let the system (6.2.18) be such that
and thus preserves its sign.
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ELECTRO-OSMOTIC OSCILLATIONS
From here we observe that if (x = x ( t ] , y = y(t)} is a trajectory, then
i.e., all the circles with their center at the origin are "noncontact loops"; that is, all the trajectories cross them transversally with increasing t, going outwards if J > 0 or going inwards if J < 0. Accordingly, the infinity is stable when J> 0 and unstable when J< 0. In order to enable us to employ this criterion to the system (6.2.18) let us re normalize the variables as follows
a, /3, 7 are some constants to be determined. In terms of the new variables the system (6.2.18) assumes the form
For the system (6.2.22), in accordance with the definition (6.2.20),
Choose
With this choice, J(p, u) assumes the form
From (6.2.24), since sup/(y) < oo, for all i there exists r0 > 0 such that for \u\ > TO and for all p that is, the infinity is unstable. This conclusion is obviously independent of the scaling transformation (6.2.21), as well as of the symmetry of f ( u ) . Since the only equilibrium point E(Q,po) in the phase plane becomes unstable for i > i° and the infinity is unstable for any i, we conclude that limit cycles must exist around J3(0,po) for i > ic. At the same time, the proven nonexistence of the limit cycles for i < ic implies the supercritical nature of the Hopf bifurcation at i — ic in the "symmetric" case /"(O) = 0.
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Fig. 6.2.2. Schematic bifurcation diagrams in a super- (a) and subcritical (b) case.
Since, as we saw previously, for a general "nonsymmetric" f ( u ) (/"(O) ^ 0) the limit cycles may exist already in the current range
or rather are not excluded there by Bendixon's test, the bifurcation at i = ic in this case does not necessarily have to be supercritical. (In fact it will be shown below that the appropriate bifurcation is subcritical if /"'(O) > 0.) Recall that a Hopf bifurcation is termed supercritical if its bifurcation diagram is as shown schematically in Fig. 6.2.2a. Correspondingly, in this case a stable limit cycle is born around the equilibrium, unstable hereon, only at a critical (bifurcation) value of the control parameter A = A c . In contrast, in the subcritical case (Fig. 6.2.2b), the equilibrium is surrounded by limit cycles already for A < A c , with an unstable limit cycle separating the stable one from the still stable equilibrium. At the bifurcation A = Ac the unstable limit cycle "dies" out with the equilibrium, unstable hereon, surrounded by a stable limit cycle. Thus the main feature of the subcritical case (as opposed to the supercritical one) is that a stable equilibrium and a stable limit cycle coexist in a certain parameter range, with a possibility to reach the limit cycle through a sufficiently strong perturbation of the equilibrium. The information acquired so far is reflected in Fig. 6.2.3a,b for two choices of f ( u ) — symmetric (a) and nonsymmetric (b). The phase plane diagrams in these figures contain a scheme of phase flow for the overcritical value of z = 2 > ic. For the relaxation oscillation regime, implied by smallness of parameter e, the explicit asymptotic (e —» 0) shape of the limit cycle ABCD is inferred directly from the isocline F(u,p,i)=Q (Fig. 6.2.3a or b). Indeed for e < 1 the system is rapidly "thrown" along a nearly horizontal path from any "initial" point S in the phase plane towards one of the stable branches EA or CF of the above isocline. After that the phase point of the system moves "slowly" along the isocline towards one of the holding points A or C, where it then "jumps" again along a nearly horizontal path AB or
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Fig. 6.2.3a. Phase plane diagrams for a symmetric case (/"(o)=o).
CD, moves slowly towards C or A jumps and thus the pattern continues forever. The location of holding points A, C can be determined from the condition
Here
is given by the isocline equation (6.2,lib) resolved with respect to p.
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Fig. 6.2.3b. Phase plane diagrams for a nonsymmetric case (/"(0)^0).
Substitution of (6.2.25b) into (6.2.25a) yields for WA,B the equation
In Fig. 6.2.4 we present once more a schematic plot of f ( u ) and its first three derivatives which we shall need in due course. It is observed from Fig. 6.2.4b that for i < ix — l / f ' ( u x ) < ic there is no real solution to (6.2.25), whereas for i > ix there are two of them, which fuse together at i = ix. This stresses once again the difference between the
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ELECTRO-OSMOTIC OSCILLATIONS
Fig. 6.2.4. Scheme of dependence of f and its first three derivatives with respect to u on the latter.
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symmetric case (/"(O) = 0) and the nonsymmetric one (/"(O) ^ 0). In the first case, the amplitude of the limit cycle, a magnitude of the order of difference UQ — WA, is zero at the bifurcation at i = i° and grows gradually as i is increased above ic, in accordance with the common pattern of a supercritical bifurcation, essentially unaffected by the relaxation regime (e
