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1 the appropriate scale depends on N, shrinking to zero when N —> oo ((pm —> — 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. 0 (infinite polyelectrolyte dilution limit), coincides with the "structural" LCD a only for a values smaller than some critical is subharmonic in the vicinity of a. Possibility (a) is ruled out by the Hopf theorem which requires |^ > 0 which in turn contradicts (2.3.1c). This implies that possibility (b) is true; that is, f( , subharmonic in (6, c) so that -g^| _ > 0. But this contradicts the assumption that / decreases at r = c since sgn^ = sgn-^. / must thus remain of a constant sign in (6,1) which by the Hopf theorem makes it impossible for r = b to be a point of internal extremum. This in turn implies a monotonic decrease of (p in (a, 1). An identical argument yields nonnegativeness of f((f>) in (a, 1).
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
Here ji, jz = const are the ionic fluxes, subject to determination. For V -C 1 let us seek a solution of (1.37)-(1.42) as a slightly perturbed equilibrium (1.23b), (1.32) that is in the form
Here
CHAPTER 1
15
ji, J2 are also of order O(V). Linearization of (1.37)-(1.42) with respect to the perturbation around the equilibrium yields
Substitution of (1.23b) into (1.45), (1.46) yields
Integration of (1.51a,b) between x = 0 and x = 2 + A yields, taking into account (1.47), (1.48),
Here
16
INTRODUCTION
are termed ionic permeabilities of the layer 0 < x < 2 + A. Remembering that for e
with 1), (1.53c,d) yield
Equations (1.52), (1.53) are the quantitative expressions of the previous statements regarding permselectivity of an ion-exchange membrane. Of course, the treatment that led to (1.52), (1.53) is admissible only in the vicinity of equilibrium. Away from equilibrium treatment of the full nonlinear formulation of type (1.37)-(1.42) is required. Examples of such treatments with and without the LEN approximation will be presented in Chapters 4 and 5. We will discuss next the ambipolar diffusion, that is, electro-diffusion of two oppositely charged ions in a solution of a univalent electrolyte with local electro-neutrality. Assume the dimensionless ionic diffusivities are constant. Then the relevant version of (1.9) is
Denote
Addition of (1.54), (1.55), divided by QI and ct2> respectively, yields (taking into account (1.56), (1.57))
Here
CHAPTER 1
17
Furthermore, subtraction of (1.54), (1.55) yields (taking into account (1.56), (1.57)),
Equation (1.57a) implies that in the locally electro-neutral ambipolar diffusion concentration of both ions evolves according to a single linear diffusion equation with an effective diffusivity given by (1.57b). Physically, the role of the electric field, determined from the elliptic current continuity equation (1.58), is in preserving local electro-neutrality (1.56). A somewhat similar situation occurs in one-dimensional multi-ionic systems with local electro-neutrality in the absence of electric current. It will be shown in Chapter 3 that in this case again the electric field can be excluded from consideration and the equations of electro-diffusion are reduced to a coupled set of nonlinear diffusion equations. Finally, we make a terminological remark. In a one-dimensional locally electro-neutral system the expression (l.llb) for the electric current density reduces to
or, dividing by
By the current continuity (1.1 la) in a one-dimensional system / is spatially invariant, that is
(By |frr we denote independence of x.) From (1.59b), (1.60) the potential drop
18
INTRODUCTION
The first term in the right-hand side of (1.61a) is termed the Ohmic potential drop (J*2 [£)zfajCi] dx is the dimensionless solution resistance between x\ and #2) whereas the second term is termed diffusion potential. In the ambipolar case expression (1.61a) reduces to
The introductory information presented above allows us to outline a certain hierarchy of electro-diffusional phenomena and levels of description that form the backbone of this monograph. At the bottom, simplest level of this hierarchy lie the nonlinear equilibrium effects to be treated in Chapter 2. The entire treatment here will be based upon the Poisson-Boltzmann equation. The next level is that of one-dimensional electro-diffusion with local electro-neutrality in the absence of an electric current. This is the realm of nonlinear diffusion to be treated in Chapter 3. A still higher level of the same hierarchy is formed by the nonlinear effects of stationary electric current, passing in one-dimensional electro-diffusion systems with local electro-neutrality. A few typical phenomena of this type will be studied in Chapter 4. The treatment of Chapter 4 will lay the foundation for the discussion of the effects of nonequilibrium space charge characteristic of the fourth level to be treated in Chapter 5. The top level of the electro-diffusion hierarchy is formed by the electroconvection phenomena, of which electro-osmosis is in several respects the simplest one. Certain aspects of electro-osmosis will be treated in Chapter 6. The higher we climb the hierarchy outlined the less rigorous our mathematics will become and the more vague heuristic statements will appear. Before we proceed to the main topics of concern in this book two natural questions seem to be worth asking. First, since we are generally referring to nonstationary electric currents what happens to the associated magnetic field? In particular, why does the vector potential not show up in the above considerations; in other words why does the scalar potential satisfy the Poisson equation, rather than the wave equation common in nonstationary electrodynamics? The answer here is the one standard for slow processes. The maximum typical velocity in the ionic systems is that associated with the double layer relaxation r
CHAPTER 1
19
The second question concerns one particular aspect of general applicability of the simple "mean field" equations outlined above as opposed to more sophisticated statistical mechanical descriptions. In particular, the equilibrium Poisson-Boltzmann equation (1.24) is often used in treatments of some very short-scale phenomena, e.g., in the theory of polyelectrolytes, with a typical length scale below a few tens of angstroms (lA = 10~8 cm). On the other hand, the Poisson-Boltzmann equation implicitly relies on the assumption of a pointlike ion. Thus a natural question to ask is whether (1.24) could be generalized in a simple manner so as to account for a finite ionic size. The answer to this question is positive, with several "mean field" modifications of the Poisson-Boltzmanri equation to be found in [5], [6] and references therein. Another ultimately simple naive recipe is outlined below. Assume for simplicity that there is just one ionic species in the system with big ions, whose finite radius TO should be taken into consideration. Denote their concentration by C. Assume further that besides the electric force — z V acting on the ion that prevents its approaching another ion closer than 2ro- To account for this assume for the potential if) a steep smooth monotonic dependence on C of the sort
such that ij)(C} tends to zero rapidly with the decrease of C below some typical value C' and blows up upon C approaching another typical value C0 > C'. CQ is related to the ionic radius as
(C"1/3 is the average separation distance between the ions whereas (C*o)~ characterizes the range of "rigid" interaction.) The electrochemical potential of ions, modified by if), assumes the form (compare with (1.19b))
At equilibrium ionic electrochemical potential is constant throughout the system. Thus
Note that
20
INTRODUCTION
on the segment C G (0, Co) is a monotonic function, varying from —oo to +00. Thus (1.65) is resolvable uniquely with respect to C as
Substitution of (1.67) into the Poisson equation (1.9c), yields
Equation (1.68) is the sought analogue of the Poisson-Boltzmann equation. We know of no mathematical studies of (1.68) in the context of known or anticipated effects of finite ionic size. Finally, we point out that there is a close relation in description of ion electro-diffusion and the phenomenological theory of the electron and hole transport in semiconductors. In order to facilitate the reading we present below a brief ionics-semiconductor "vocabulary." electrolytes cations anions fixed charge density
semiconductors holes electrons
in an ion-exchanger
doping function
cation exchanger anion exchanger electrolyte solution bipolar membrane (two adjacent cat- and anion exchange membrane layers) quadrupolar membrane (four alternating mutually adjacent cat- and anion exchange layers)
^-semiconductor n-semiconductor undoped semiconductor
diode
thyristor
REFERENCES [1] J. O'M. Bockris and A. K. N. Reddy, Modern Electrochemistry, Vols. 1, 2, Plenum Press, New York, 1977. [2] Comprehensive Treatise of Electrochemistry, Vol. 6, Electrodics: Transport, E. Yeager, J. O'M. Bockris, B. E. Conway, and S. Sarangapani, eds., Plenum Press, New York, 1983. [3] F. Helfferich, Ion Exchange, McGraw-Hill, New York, 1962. [4] I. E. Tamm, Fundamentals of the Theory of Electricity, Mir, Moscow, 1979.
CHAPTER 1
21
[5] S. Levine and C. W. Outwaite, Comparison of theories of the aqueous electric double layers at a charged plane interface, J. Chem. Soc., Faraday Trans. II, 74 (1978), pp. 1670-1689. [6] G. A. Martynov, Statistical Theory of Electric Forces, Vol. 2, B. V. Derjagin, ed., Consultants Bureau, New York, 1966.
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Chapter 2
Nonlinear Effectsin Electro-Diffusional Equilibrium
2.1. The Poisson—Boltzmann equation. Equilibrium is a steady state without macroscopic fluxes. As we pointed out in the Introduction, under these conditions the equations of electro-diffusion reduce to
Recall that Ai are positive integration constants. For open systems Ai is equal to the known concentration of the charge carrier z, wherever tp vanishes (e.g., at infinity). For closed systems, in which only the total number of charge carriers may be known rather than their concentration somewhere, the Ai are subject to determination in the course of the solution. (The properties of the solutions for parallel open and closed system formulations may differ quite markedly, as was exemplified in [1].) Equation (2.1.2), the Poisson-Boltzmann equation, is a particular case of the nonlinear Poisson equations
Equation (2.1.3a) has been studied extensively in different mathematical and physical contexts ranging from differential geometry to reactiondiffusion, electrokinetics, colloid stability, theory of polyelectrolytes, etc. In 23
24
NONLINEAR EFFECTS
particular, it is well known that the conditions (2.1.3b,c) ensure the existence of a unique nonnegative solution to the Dirichlet problems for (2.1.3a) with a nonnegative boundary value of (f [2]-[6]. (Some condition of the sort (2.1.3c), or maybe weaker [2], is essential for existence-uniqueness since it keeps away from zero the eigenvalues of the linearization of (2.1.3a). On the other hand, (2.1.3b) together with (2.1.3c) ensures the nonnegativity of a solution for nonnegative boundary data [3].) The specific electro-diffusion phenomena, the field and force saturation and counterion condensation, as well a,s the corresponding features of the solutions to the Dirichlet problem for (2.1.2) to be addressed in this chapter, are closely related to those observed by Keller [7], [8] for the solutions of (2.1.3a) with /(y) positive definite, satisfying a certain growth condition. Keller considered /() > 0, satisfying the condition
In addition,
CHAPTER 2
25
the ionic systems even in the physically simplest equilibrium constellation. Note that, assuming a reasonable electrolyte concentration (10~3 — I N ) , the physical length scale of phenomena to be discussed in this chapter (10~ 7 — 10~ 5 cm) is somewhat shorter than in the rest of this text (typically 10~4- 1cm). Accordingly, our objects here will be colloid particles and polyelectrolyte molecules as compared with electrolyte diffusion layers, synthetic membranes, ion exchangers, etc., discussed in other chapters. This is so because the equilibrium phenomena of interest to us are related to the structure of the space charge, which makes all relevant distances to scale with the Debye length. Away from equilibrium both interesting nonspace-charge-related phenomena appear (Chapters 3 and 4) and the nonequilibrium space charge may reach a macroscopic size (Chapter 5). At the end of this chapter we propose a number of unresolved problems related to the subjects discussed herein.
2.2. Electric field and force saturation [10].
2.2.1. Recall that in vacuum or in dielectrics the electric field at a given distance from a charged particle increases indefinitely with the increase of the particle's charge. This is still true for the linearized version of (2.1.2), known as the Debye-Hiickel equation, although in this case the electric field is weaker compared to the vacuum or dielectric case (with the same dielectric constant as the electrolyte solution). This is due to the linear screening of particle charge by mobile ions, reflected in the Debye-Hiickel equation. It is shown below (§2.2.2) that a stronger nonlinear screening in the full Poisson-Boltzmann equation, results in a complete saturation of the electric field with the increase of the particle charge. The electric force, acting upon a particle is
where n is the outwards unit normal to the surface. Saturation of Fel with the increase of particle charge implies that
This result, which in the one-dimensional case (for parallel plates) follows from a direct computation (e.g., [9]), is shown below (Proposition 2.1), for parallel cylinders and for spheres. The developments of §§2.2.2 and 2.2.3 employ some elementary comparison results (Theorems 2.1, 2.2) of §2.2.2. These and much more general comparison results are well known for solutions of (2.1.3a) (e.g., [4]). They are presented in §2.2.2 for the sake of completeness. The same applies to the nonnegativeness part of Proposition 2.1, §2.2.2.
26
NONLINEAR EFFECTS
2.2.2. Field saturation. Consider a particle occupying a convex open domain u> C H3 (or T?2) with a smooth boundary duj, charged to the electric potential C > 0, at equilibrium with an infinite solution of a symmetric electrolyte of a given average concentration. (Properties described below are directly generalizable to an arbitrary electrolyte or electrolyte mixture.) The equilibrium electric potential (p in the space surrounding the particle is described by the following b.v.p.
(The electrolyte valency and "bulk" concentration have been normalized to unity by a proper rescaling of x and (p.) Existence and uniqueness of solutions to the b.v.p. analogous to (2.2.1) has been proved in numerous contexts (see, e.g., [2]-[6]) and can be easil inferred for (2.2.1). We shall not do it here. Instead we shall assume the existence and uniqueness for (2.2.1) and similar formulations and, based on this assumption, we shall discuss some simple properties of the appropriate solutions. These properties will follow from those of the solution of the one-dimensional Poisson-Boltzmann equation, combined with two ele mentary comparison theorems for the nonlinear Poisson equation. These theorems follow from the Green's function representation for the solution of the nonlinear Poisson equation with a monotonic right-hand side (or from the maximum principle arguments [20]) and may be stated as follows. THEOREM 2.1. Let u\ be the solution of b.v.p.
Here ft is an open (not necessarily bounded) domain with a boundary dfl, and f ( u ) is a monotonically increasing function o f u . Let further u% be the solution of
(For fi unbounded, assume that ui, oo.) Let gz(x) > gi(x) (x € d£l); then U2(x) > HI(X) (x £ fl).
CHAPTER 2
27
Proof. Write u\ and u-z in the form:
Here G(x, y) is the Green's function for the Dirichlet problem for the Laplace equation in $7 and n is the outward unit normal. It is well know that
Subtraction of (2.2.4a) from (2.2.4b) yields:
We observe from (2.2.5), taking into account (2.2.4c), that, for f(u] monotonically increasing, the inequality u^ < u\ cannot hold for all x in 12. Assume that u% < u\ for some connected subdomain S7i C fi. Let Oi be the closure of the domain fii. It is evident that for x e <9Oi, u\ = u^. Then by applying (2.2.5) to Oi we again arrive at a contradiction, which concludes the proof of Theorem 2.1. THEOREM 2.2. Let u\ be the solution of (2.2.2) and let u? be the solution of
Here O, Oft are as in Theorem 2.1 and
Proof of this theorem is completely analogous to that of Theorem 2.1. Besides Theorems 2.1 and 2.2, we shall employ the following properties of solution of the one-dimensional version of (2.2.la) for a half space. The appropriate b.v.p. is of the form:
28
NONLINEAR EFFECTS
Direct integration of (2.2.7) yields
Note that according to (2.2.8a) for x > 0 and £ —> oo, u ( x , £ ) tends to a finite upper limit: i
Below we shall make use of the following integral:
It follows from (2.2.7), (2.2.8) that
Another one-dimensional solution that we shall employ is that for (2.2.7a) in a segment [0, A], A > 0 with boundary conditions:
Integration of (2.2.7a) with (2.2.7c) yields u*(x, C) in terms of the Jacobian elliptic functions, with a property analogous to (2.2.8b), namely for A > 0,
Here again the limiting function u*(x, A) is finite for any 0 < x < A and blows up at the boundary x — 0. The following property of solutions of (2.2.1) is an immediate consequence of Theorem 2.1 and (2.2.8b).
CHAPTER 2
29
PROPOSITION 2.1. Let
Proof. Nonnegativeness of
By (2.2.1b) /(
and
which by (2.2.4a,c) represents a contradiction, implying that (p is nonnegative everywhere in 17. Boundedness of
Let us introduce a coordinate axis z, directed as np, with the origin at P. It follows from (2.2.12a) and Theorem 2.1 that for any (-}x with z > 0,
with u(z, C) being the solution of (2.2.7a,b), given by (2.2.8a). By (2.2.8b), (2.2.12b) implies for C -+ oo
Since (•) x was chosen arbitrarily on the convex surface du, (2.2.12c) implies uniform boundedness of
30
NONLINEAR EFFECTS
Remark 2.1. The estimate above for the rate of decay at infinity holds for an arbitrary system of TV bodies (wj> d^i-, 1 < i < N] confined to a bounded convex domain with a smooth boundary <9f2. Indeed let
By the arguments of Proposition 2.1, ip(x) is nonnegative for x € 7ln\ |J^ u^. This implies by Theorem 2.1 and the maximum principle for subharmonic functions that This equation, Theorem 2.1, and the arguments of Proposition 2.1 yield the estimate sought.
2.2.3. Saturation of the force of repulsion between two symmetrically charged parallel cylinders in an electrolyte solution. In
this section we describe one physical implication of Proposition 2.1, namely the saturation of the force of interaction for particles in an electrolyte solution upon the increase of their charge. The central result consists of the demonstration of this feature for parallel cylinders or spheres. We precede the formulation of the exact Proposition 2.2 and its proof by some heuristic discussion. The question to be discussed is whether saturation of the electric field (asserted by Proposition 2.1) implies saturation of the interparticle force of interaction. Consider for definiteness repulsion between two symmetrically charged particles in a symmetric electrolyte solution. In the onedimensional case (for parallel plates) the answer is known—the force of repulsion per unit area of the plates saturates. (This follows from a direct integration of the Poisson-Boltzmann equation carried out in numerous works, primarily in the colloid stability context, e.g., [9]. Recall that again in vacuum, dielectrics, or an ionic system with a linear screening, the appropriate force grows without bound with the charging of the particles.) In two or three dimensions the answer is not entirely obvious. On the one hand, according to Proposition 2.1 the field induced by each particle at the surface of its counterpart is expected to saturate as the charge increases. In parallel, the surface charge density increases indefinitely, so that the electric surface force density on the particle (proportional to the field intensity multiplied by the surface charge density) may seemingly increase indefinitely, too. On the other hand, the force in the one-dimensional case saturates, which seems to be suggestive, if only this is not specifically an accident of the one-dimensional arrangement. In fact, if the electric repulsion force for an arbitrary particle indeed saturates, this should imply severe requirements upon the structure of the electric field at the surface of a particle.
CHAPTER 2
31
Indeed, the total electric force F|], acting on a particle "fc" with a surface duk, charged to a constant potential £, is given by the following expression (see, e.g., [21]):
is the normal component of the electric field intensity at the surface of the particle. Let us express E\g in the form:
Here E^(Q is the electric field due to a single particle "fc" in an infinite domain and |E12 \s is the electric field induced by all other particles in the system (or external sources) at the surface of the particle "fc." Obviously, we expect that there should be no net electric force acting, in the absence of external sources, on a single charged particle in an infinite domain. We thus expect that
From Proposition 2.1 we expect that E\i dwk is bounded when £ —> oo. Thus, for the total electric force acting on a particle to remain bounded, we ought to have
which is likely to imply
In an attempt to clarify matters further by studying a specific example, let us evaluate the total force, acting per unit length of two parallel cylinders of radius R, symmetrically charged to a positive surface potential £ and
32
NONLINEAR EFFECTS
Fig. 2.2.1. Sketch for the b.v.p. (2.2.14)-(2.2.15) for the interaction of charged cylinders in an electrolyte solution.
fixed at a given distance 2A from each other in a symmetric electrolyte solution (see Fig. 2.2.1). The distribution of the electric potential in the ionic fluid surrounding the particles is described by the following b.v.p. in 7£2:
Here the x-axis has been directed through the centers of the particles, whereas the y-axis has been directed normal to x, with the origin x = 0, y = 0 coinciding with the center of particle I. In order to evaluate the total force acting on particle I, we note that this force is comprised of two components. The first component is the total pressure force exerted by the fluid. It is easily seen that at equilibrium this component vanishes. Indeed, for a fluid at equilibrium we have
Here, p is the pressure and p is the space charge density, which for the equilibrium ionic fluid under consideration assumes the form
CHAPTER2
33
For
Substitution of (2.2.17) into (2.2.16) with a subsequent integration yields, taking into account (2.2.18), (2.2.15c),
Equation (2.2.19) implies that at the surface of the particles pressure is constant and equal to PQ + e^ + e~^ — 2. This in turn implies that the total pressure force on the particle vanishes. The second force component is the electric force. As indicated in (2.2.13a), this component may be computed as follows (assuming independence of the dielectric constant on the density of the liquid [21]):
PROPOSITION 2.2. Let tp be the solution of the b.v.p. 2.2.14, 2.2.15. Then
Proof. In order to avoid cumbersome evaluation of the potential
Next, multiply (2.2.14) by V(p and integrate over {R < r < oo, x < A}. By Remark 2.1 to Proposition 2.1 the relevant integrals exist. The integration yields, taking into consideration the identity (2.2.21a), the boundary conditions (2.2.15) and the rate of decay at infinity, given by Remark 2.1,
34
NONLINEAR EFFECTS
Since the second term (2.2.21b) vanishes, it follows from (2.2.20),(2.2.21b) that the only nonvanishing (by symmetry) x component of the electric repulsion force F^, acting on the particle I may be computed as:
The evaluation of Fei for £ —> oc is thus reduced to an evaluation of two integrals I\ and /2 over the symmetry plane x = A in the right-hand side of (2.2.22). It is easily seen that both integrals in the right-hand side of (2.2.22) tend to a finite value when £ —»• oo. Note first that by arguments identical to those of Proposition 2.1, (p and accordingly the right-hand side of (2.2.14) are nonnegative. For evaluating /i, note that in the upper half plane y > 0, (p(x,y) is a monotonically decreasing function of y. Indeed, let
whereas, due to the nonnegativeness of
and, from the behaviour at infinity,
CHAPTER 2
35
We observe by differentiating (2.2.14) with respect to y that within fi+, v satisfies the equation
It follows from (2.2.25), (2.2.27) that
Indeed, assume the opposite, that is that there exists a maximal subdomain f2+ C ft+ such that
From (2.2.27), (2.2.29a) v is subharmonic in fi+; that is, by the maximum principle it has a boundary maximum, which by (2.2.29a,b) implies a contradiction. We conclude from here that (2.2.28) holds for all x,y € fi+, namely, that
Combining the property (2.2.8c) of the one-dimensional solution in a segment with the arguments of Proposition 2.1 yields boundedness of
which by (2.2.30) yields the boundedness of /i for (-+00. Boundedness of /2 at C —* oo is easily observed in a similar manner. Indeed, consider a straight line y = yo > R, —oo<x
Here
36
NONLINEAR, EFFECTS
and
Again, by arguments that led to (2.2.32a), 72 remains bounded when £ —» oo. In order to evaluate J2 , observe that for y = yo, is a smooth positive function of x, vanishing at infinity. Thus, tp(x, yo) achieves a maximum at some XQ. Denote this maximum value by y m (C)- According to an appropriate analogue of Proposition 2.1, (pm(Q remains bounded by some finite value (^ when £ —> oo. On the other hand, evidently,
where ip satisfies
The solution of (2.2.37) yields, according to (2.2.8a),
It follows from (2.2.36), (2.2.37), and (2.2.9b) that
Since (pm(() < (p1^ < oo when £ —> oo, we have according to (2.2.39)
which proves boundedness of 72 and thus of /2 at C —* oo. This completes the proof of boundedness of the repulsion force F*{ when £ —> oo. The above construction and accordingly the conclusion, obviously hold for an arbitrary electrolyte in Tln. "R.2 and symmetry of the electrolyte, resulting in the sinh
CHAPTER 2
37
2.3. Counterion condensation [17]. 2.3.1. This section is devoted to the transition occurring in the type of singularity, induced by a line charge in the solutions of the PoissonBoltzmann equation in a plane, at some critical value of the linear charge density. (See §2.3.2 for a more detailed motivation.) The basic b.v.p. under consideration is as follows:
Here f. 2, a, N are positive constants and
The central conjecture proved in this section is as follows. Conjecture 2.1. Let ip(r) be a solution of (2.3.1) and £ > z; then for a-» 0,
Similarly for z > 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
CHAPTER 2
Fig. 2.3.1. Scheme of the experimental dependence of the effective charge density on the structural linear charge density.
39
linear
charge density creff evaluated from the data on surface potential (obtained, e.g., via potentiometric titration), coincides with the structural, chemically determined linear charge density a only for the latter small enough. Upon increase of cr, a saturation in creff is rapidly reached with the limiting LCD determined by the highest valency of counterions present in the system in a nonvanishing concentration. Transition to saturation in the creff dependent on cr is sharper the higher the dilution so that asymptotically we end up with a sharp corner typical for second-order phase transitions. (It can be shown that discontinuity of the first derivative of creff as a function of cr implies a discontinuity of the second derivative of the free energy with respect to cr, characteristic for second-order phase transitions.) This limiting phase transition, known as counterion condensation, has received much attention in the polymer literature [24]-[30]. (See Fig. 2.3.1 for a schematic illustration.) As for the theory of this phenomenon, it was first observed by Onsager [27a] that, since in the limit a —> 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-
Here D\ is again a positive constant, satisfying
CHAPTER 2
43
Analogous to the above case it is observed from (2.3.5b) that for a/p small enough BI is uniquely determined by (2.3.5b) as
It is observed next that when a —> 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(
44
NONLINEAR EFFECTS
integrals Ac, ^4_, A+ fixed on the solution of the b.v.p. (2.3.1), is a monotonically increasing function of (p. Consider next an auxiliary b.v.p. of the form
with f((p) also monotonically increasing in
It can then be easily seen that for solutions v?(r), ip(r) monotonically decreasing in r,
Indeed, the solution of (2.3.1) satisfies the integral equation
Here
Analogously
Inequality (2.3.10) is then observed by subtracting (2.3.13) from (2.3.11) and employing (2.3.10) and the assumed monotonic decrease of
CHAPTER 2
45
with
provides an upper bound for
whenever (f>(r),
Recall further that sub- (super- ) harmonic functions have no inner maxima (minima) and that by the Hopf theorem the inward derivative of a sub(super-) harmonic function at the points of the boundary maxima (minima) are strictly negative (positive). Since
46
NONLINEAR EFFECTS
Indeed, since |^| . = 0 and (p is monotonically decreasing in (o, 1), /(v)| r=1 > 0 by the Hopf theorem. On the other hand, as we saw above, /(v)| r=0 > 0. J(
Due to the nonnegativeness of
(Ac, A-, A+ are determined by (2.3.1e-g).) Moreover, Ac remains bounded at a —» 0, since otherwise f(
for x £ (a, 1], which would further yield for the end point r = 1, according to (2.3.Id) and (2.3.19)
This would contradict in turn the nonnegativeness of f(
The solution of (2.3.22), as provided by (2.3.4), (2.3.5) is positive and monotonically decreasing. At the same time according to (2.3.Ib), (2.3.22b)
CHAPTER 2
47
and thus, according to (2.3.9), (2.3.10)
Formula (2.3.23b) in particular yields, according to (2.3.6), (2.3.7) for
Formula (2.3.24) provides the desired lower bound for the case ( > z under consideration. Upper bound. To obtain the upper bound, consider the following auxiliary b.v.p.
Here
<^o, defined by (2.3.25e), is such that for
whereas for
48
NONLINEAR EFFECTS
The solution of (2.3.25) is explicitly constructed as that of the following free boundary problem
Formula (2.3.27) determines (p!(r) (a < r < r0), identical to (2.3.4), (2.3.5) with p — r 0, (pp = ipo and ip11 — (tp0/\nr0) Inr (r0 < r < 1) with ra uniquely determined by (2.3.27f) to be such that 1/ro, 1/(1 — TO) = O(l) when a —» 0. It is then directly observed from (2.3.4), (2.3.5) that for a —» 0
On the other hand, (p as constructed via the solution of (2.3.27) is a monotonically decreasing function of r, and by construction
Thus according to (2.3.15) and (2.3.16)
CHAPTER 2
49
yielding by (2.3.28) for a -> 0
Formulae (2.3.24) and (2.3.31) finally yield for C > z, a -> 0
In the next section we treat the case C < z > which includes an interesting transition in the effective LCD determining counterions from those of the added electrolyte to the "proper" ones, upon a decrease of the added electrolyte concentration (N —> 0). 2.3.5. Limiting singularity for z > 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) = <^i) and match them continuously and with the continuous first derivative at r = 7*1, where r\ is determined by this construction. Treatment of this case requires certain caution, because r\ —> 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 /(
The solution of (2.3.38) is constructed analogously to that of (2.3.25) and (2.3.34), and is again by construction monotonically decreasing in r with the limiting singularity identical to that of (2.3.36). Together with the monotonicity of $ and inequalities (2.3.40), (2.3.15), (2.3.16) and (2.3.33), (2.3.36), (2.3.37), this finally yields for the case under consideration:
Formula (2.3.41) implies that in a system with added electrolyte of a higher valency 2, the magnitude of the "effective" linear charge density is determined by the latter. Upon reduction of the concentration of the added electrolyte (N —> 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 < <7^x, with (T^iax determined by Conjecture 2.1. This is schematically illustrated in Fig. 2.3.5b. Note that F as a function of creff is constructed in a single counterion case by solving (2.3.3a) with with the boundary a = conditions a/ f a e^r dr and (p(a) = -<jeff In a, |^ r=l = 0, and by going to the limit a —> 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,
56
NONLINEAR EFFECTS
5. Counterion condensation on a point charge in three dimensions for the excluded volume equation (1.68). It is expected that in contrast to the previous case (Problem 4) the limiting equilibrium solution will exist for the appropriate version of (1.68) and the corresponding limiting singularity in the electric potential should be studied.
REFERENCES [1] I. Rubinstein, An electrostatic model of cell junction, Zitologia, 16 (1974), p. 1117. (In Russian.) [2] N. Levinson, Dirichlet problem for Au = f(p,u), J. Math. Mech., 112 (1963), pp. 567-575. [3] S. V. Parter, Mildly nonlinear elliptic partial differential equations and their numerical solutions, I, Numero. Math., 7 (1965), pp. 113-128. [4] D. H. Sattinger, Monotone methods in nonlinear elliptic and parabolic boundary value problems, Indiana Univ. Math. J., 21 (1972), pp. 979-1000. [5] M. S. Mock, Analysis of Mathematical Models of Semiconductor Devices, Boole Press, Dublin, 1983. [6] P. A. Markovich, The Stationary Semiconductor Device Equations, Springer-Verlag, Vienna, New York, 1986. [7] J. B. Keller, Electrohydrodynamics I, The equilibrium of a charged gas in a container, J. Rational Mech. Anal., 5 (1956), pp. 715-724.
CHAPTER 2 [8] [9]
[10] [11] [12]
[13] [14] [15] [16] [17] [18] [19]
[20] [21] [22] [23]
[24]
[25] [26]
[27a]
57
, On solutions of Au = f(u), Comm. Pure Appl. Math., 10 (1957), pp. 503-510. B. Deryagin and L. D. Landau, A theory of the stability of strongly charged lyophobic sols and the coalescence of strongly charged particles in electrolytic solutions, Collected Papers by L. D. Landau, D. Ter-Haar, ed., Gordon and Breach, New York, 1967. I. Rubinstein, Field and force saturation in ionic equilibrium, SIAM J. Appl. Math., 48 (1988), pp. 1475-1486. A. D. MacGillivray, Upper bounds on solutions of the Poisson-Boltzmann equation near the limit of infinite dilution, J. Chem. Phys., 56 (1972), p. 80. , Analytic description of the condensation phenomenon near the limit of infinite dilution based on the Poisson-Boltzmann equation, J. Chem. Phys., 57 (1972), p. 4071. , Lower bounds on solution of the Poisson-Boltzmann equation near the limit of infinite dilution—the moderately charged case, J. Chem. Phys., 57 (1972), p. 4075. G. V. Ramanathan, Statistical mechanism of electrolytes and poly electrolytes. III. The cylindrical Poisson-Boltzmann equation, J. Chem. Phys., 78 (1983), p. 3223. T. Odijk, On the limiting solution of the cylindrical Poisson-Boltzmann equation for polyelectrolytes, Chem. Phys. Lett. 100 (1983), p. 145. G. V. Ramanathan and C. P. Woodbury Jr., The cell model for polyelectrolytes with added salts, J. Chem. Phys., 82 (1985), p. 1482. I. Rubinstein, Counterion condensation as an exact limiting property of the PoissonBoltzmann equation, SIAM J. Appl. Math., 46 (1986), pp. 1024-1038. A. Friedman and K. Tintarev, Boundary asymptotics for solutions of the PoissonBoltzmann equation, J. Differential Equations, 69 (1987), pp. 15-38. K. Tintarev, Fundamental solution of the Poisson-Boltzmann equation, in Differential Equations and Mathematical Physics, I. W. Knowles and Y. Saito, eds., Lecture Notes in Math. 1285, Springer-Verlag, Berlin, New York, 1987. M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Prentice-Hall, New York, 1967. I. E. Tamm, Fundamentals of the Theory of Electricity, Mir, Moscow, 1979. E. Hopf, A remark on linear elliptic differential equations of the second order, Proc. Amer. Math. Soc., 3 (1952), pp. 791-793. A. Katchalsky, Z. Alexandrovicz, and O. Kedem, Poly electrolyte Solutions in Chemical Physics of Ionic Solutions, B. E. Conway and R. G. Barradas, eds., John Wiley, New York, 1966, p. 295. M. Rinaudo, Comparison between experimental results obtained with hydroxylated poly acids and some theoretical models, in Polyelectrolytes, E. S&e'gny, M. Mandel, and U. P. Strauss, eds., D. Riedel, Dordrecht, the Netherlands, 1974, p. 157. F. Oosawa, Polyelectrolytes, Marcel Dekker, New York, 1971. G. S. Manning, Limiting laws for equilibrium and transport properties of polyelectrolyte solutions, in Polyelectrolytes, E. Selegny, M. Mandel, and U.P. Strauss, eds., D. Riedel, Dordrecht, the Netherlands, 1974, p. 9. , Limiting laws for equilibrium and counterion condensation in poly electrolyte solution. I. Colligative properties, J. Chem. Phys., 51 (1969), p. 923.
58 [27b] [27c]
[28] [29]
[30] [31] [32] [33]
[34] [35] [36]
[37]
[38]
NONLINEAR EFFECTS , Limiting laws for equilibrium and counierion condensation in polyelectrolyte solution. II. Self-diffusioof small ions, J. Chem. Phys., 51 (1969), p. 934. , Limiting laws for equilibrium and counterion condensation in polyelectrolyte solution. III. An analysis based on the Mayer ionic solution theory, J. Chem. Phys., 51 (1969), p. 3. J. L. Jackson and S. R. Corriell, On associated ions in poly electrolytes and trapped Brownian trajectories, J. Chem. Phys. 40 (1964), p. 1460. A. D. MacGillivray and J. J. Winklemari Jr., On an asymptotic solution of the Poisson-Boltzmann equation — The moderately charged cylinder, J. Chem. Phys., 45 (1966), p. 2184. H. Fixman, The Poisson-Boltzmann equation and its application to polyelectrolytes, J. Chem. Phys., 70 (1979), p. 4995. R. A. Marcus, Calculation of thermodynamic properties of polyelectrolytes, J. Chem. Phys., 23 (1955), p. 1057. T. Alfrey Jr., P. W. Berg, and H. Moravetz, The counterion distribution in solutions of rod-shaped polyelectrolytes, J. Polymer Sci., 7 (1951), p. 543. R. M. Fuoss, A. Katchalsky, and S. Lifson, The potential of an infinite rod-like molecule and the distribution of the counter ions, Proc. Nat'l. Acad. Sci. U.S., 37 (1951), p. 579. S. Lifson and A. Katchalsky, The electrostatic free energy of polyelectrolyte solutions, II. Fully stretched macromolecules, J. Polymer Sci. 8 (1954), p. 43. Z. Alexandrovicz, Calculation of the thermodynamic properties of polyelectrolytes in the presence of salt, J. Polymer Sci., 56 (1962), p. 97. L. Kotin and M. Nagasawa, Chain model for polyelectrolytes. VII. Potentiometric titration and ion binding in solution of linear polyelectrolytes, J. Chem. Phys., 36 (1962), p. 873. D. Dolar, Thermodynamic properties of polyelectrolyte solutions, in Polyelectrolytes, E. Selegny, M. Mandel, and U. P. Strauss, eels., D. Riedel, Dordrecht, the Netherlands, 1974, p. 98. L. D. Landau and E. M. Lifshitz, Statistical Physics, Pergamon Press, London, Paris, 1959.
Chapter 3
Locally Electro-Neutral Electro-DiffusionWithout Electric Current
3.1. Preliminaries. In this chapter we shall address the simplest nonequilibrium situation—one-dimensional locally electro-neutral electrodiffusion of ions in the absence of an electric current. We shall deal with macroscopic objects, such as solution layers, ion-exchangers, ion-exchange membranes with a minimum linear size of the order of tens of microns. As pointed out in the Introduction, it is customary in the treatment of such systems to assume local electro-neutrality (LEN), that is, to omit the singularly perturbed higher-order term in the Poisson equation (1.9c). Such an omission is not always admissible. We shall address the appropriate situations at length in Chapter 5 and partly in Chapter 4. We defer therefore a detailed discussion of the contents of the local electro-neutrality assumption to these chapters and content ourselves here with stating only that this assumption is well suited for a treatment of the phenomena to be considered in this chapter. In order to simplify matters as much as possible we shall restrict ourselves to one-dimensional situations only. The relevant LEN version of equations (1.9) thus reads
Recall that here Ci(x,t) is the dimensionless concentration of the ionic species z, scaled with some typical dimensional concentration Co, e.g., the initial or boundary value, typical concentration of the fixed charges, etc. 59
60
ELECTRO-NEUTRAL ELECTRO-DIFFUSION
on is the dimensionless diffusivity, scaled with some typical dimensional diffusivity DQ. (Further on o>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
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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)
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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
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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
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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:
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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
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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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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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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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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
This discontinuity of the diffusivity implies in turn a free boundary (front) at Ci = 0. The corresponding limiting problem is an instance of the one-phase Stefan problem of the form
or in terms of the total loading q
Here Zo(t) is the position of the free boundary. Problems (3.3.14) or (3.3.15) admit the well-known similarity solutions [48], [49]
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ELECTRO-NEUTRAL ELECTRO-DIFFUSION
with the front propagating as
Here £ is a constant determined from the equation
For e small but finite a sharp front smears out into a transition layer that appears in the solution of (3.3.8), (3.3.11) or (3.3.10), (3.3.12). The thickness and structure of this layer is determined by the asymptotic procedure outlined below. Note that the A, A terms in (3.3.8) and (3.3.10), respectively, are positive, bounded for any Ci, £ > 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
88
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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89
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,
90
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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91
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
92
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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93
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
94
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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95
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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97
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.
CHAPTER 3
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.
CHAPTER 3
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
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ELECTRO-NEUTRAL ELECTRO-DIFFUSION
[21] M. A. Herrero, A Limit Case in Nonlinear Diffusion, to appear. [22] , Singular Diffusion in the Line, to appear. [23] G. Rosen, Nonlinear heat conduction in solid H2, Phys. Rev. B., 19 (1979), pp. 23982399. [24] G. Bluman and S. Kumei, On the remarkable nonlinear diffusion equation d(a(u+b)~2 du/dx)/dx -du/dt=Q, J. Math. Phys., 21 (1980), pp. 1019-1023. [25] A. Kolmogoroff, I. Petrovsky, and N. Piscounoff, Etude de Vequation de la diffusion avec croissance de la quantite de matier et son application a un probleme biologique, Moscow Univ. Bull. Math., 1 (1937), pp. 1-25. [26] M. Friedlin, Functional integration and partial differential equations, Ann. of Math. Stud., Princeton University Press, Princeton, NJ, 1975. [27] P. S. Hagan, Travelling wave and multiple traveling wave solutions of parabolic equations, SIAM J. Math. Anal., 18 (1982), pp. 717-738. [28] G. W. Bluman and J. D. Cole, Similarity Methods for Differential Equations, SpringerVerlag, New York, 1974. [29] L. Dresner, Similarity Solutions of Nonlinear Partial Differential Equations, Pitman, Boston, 1983. [30] C. J. van Duijn, S. M. Gomes, and Z. Hongfei, On a class of similarity solutions of the equation ut=(\u\m-lux}x with 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:
CHAPTER 4
107
Here x, fi, c^, N are the dimensionless counterparts o f f , O, Q, N, respectively. The dimensionless electric potential
The sum of equations (4.1.1), multiplied by Zi for 1 < i < M, yields, employing (4.1.3),
(The continuity equation (4.1.4a,b) is a modification of Ohm's law for the steady state. For a general nonsteady LEN state the continuity of the electric current density reads
108
STATIONARY CURRENT
For a steady state the Nernst-Planck equations could be divided by £>i, yielding (4.1.1), whose summation leads to (4.1.4a,b) for constant N. The factor r in (4.1.4a) may be viewed as a modified steady state conductivity.) For N nonvanishing the factor r in (4.1.4a) may be evaluated as follows:
According to (4.1.4), (4.1.5) no 0. In other words, the LEN approximation is expected to hold on a macroscopic scale in any medium with a sufficiently high fixed charge density (ion-exchangers, doped semiconductors) unless the applied voltage is high enough for "punch through." On the other hand, for N ~ 0, (4.1.4a) suggests that y? ( Ay) become singular wherever all concentrations Ci vanish. In other words, in a noncharged medium (e.g., electrolyte solution), whenever the transport conditions are such that the conductivity factor r approaches zero, a macroscopic violation of local electro-neutrality is expected to occur due to a simultaneous growth of Ay? and decrease of all Cj. In contrast to the "punch through" case, this may already happen at a moderate voltage as in the case with concentration polarization. This is a prototype name for numerous effects occurring upon the passage of an electric current through an electrolyte solution adjacent to an ion-selective body (e.g., an ion-exchanger, electrode, etc.). Some LEN aspects of concentration polarization will be discussed in §4.4, whereas the relevant effects of the nonequilibrium space charge will be dealt with in Chapter 5. In particular, the LEN approximation will be considered there as a leading term of the appropriate asymptotic solution of the full problem for (4.1.1), (4.1.2). 4.2. Integration of the stationary electro-diffusion equations in one dimension. The integration of the stationary Nernst-Planck equations (4.1.1) with the LEN condition (4.1.3), in one dimension, for a medium with N constant for an arbitrary number of charged species of arbitrary valencies was first carried out by Schlogl [5]. A detailed account of Schlogl's procedure may be found in [6]. In this section we adopt a somewhat different, simpler integration procedure. One integration of the one-dimensional version of (4.1.1) yields
Here x is our only space variable and the constant ji (usually unknown) is a modified ionic flux, related to the true ionic flux j( as follows:
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109
Summation of (4.2.la) (directly after a multiplication by Zi) yields, employing (4.1.3),
Here a and r are, respectively, (compare with (4.1.4))
whereas
and B is another unknown integration constant. It follows from (4.2.3) that for / / 0, tp(x] is a monotonic function. (In the trivial case / = 0,
Substitution of
Equation (4.2.7) is a homogeneous system of linear differential equations. The search for a particular solution in the form
110
STATIONARY CURRENT
yields a characteristic equation for A
Assume for simplicity that all roots A^ of (4.2.9) are different (the case of multiple roots is handled in a standard fashion). Then according to (4.2.8)
Substitution of ip, expressed from (4.2.2a), into (4.2.10) yields:
Finally, summation of (4.2.11) for all i gives, using (4.2.2b),
Here
The integration of the nonlinear system (4.2.la), (4.1.3) is thus reduced to the solution of a single transcendental equation (4.2.12) which gives a (and by (4.2.2), (4.2.11) also ip and ci) as functions of x, j, and B for given initial values of ci0 (i = 1, • • • , k — 1, k + 1, • • • , M), ipo at some x — XQ. (c/co is then found from (4.1.3).) Existence and uniqueness of solutions to (4.2.12) and to the appropriate algebraic equations for j'j, B have to be studied separately in any specific electro-diffusional set-up. Different versions of the above calculations, carried out for particular ionic contexts, form the basis of numerous studies of the ion-selective membrane transport, starting with classical papers by Teorell [7], and Meyer and Sievers [8]. Without attempting to give a full or merely fair account of all these studies, we shall mention here just a typical few: Schlogl [5] (arbitrary number of ions in a monopolar membrane), Spiegler [9] (ambipolar ionic transport in a unipolar membrane and solution layers adjacent to it — concentration polarization), Oren and Litan [10], Brady and Turner [11], Rubinstein [12] (multipolar transport in a unipolar membrane and the adjacent solution layers—effects of concentration polarization upon
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the membrane selectivity for counterions; see §4.4), Sonin and Grossman [13], Rubinstein [14] (ionic transport in multipolar membranes; see §4.3). The principal value of these calculations is in the qualitative insight they provide into ionic membrane transport. In parallel, the simplicity of handling the locally electro-neutral case provides a convenient ground for studying the far less tractable one-dimensional version of the nonreduced system (4.1.1)-(4.1.2), asymptotically for small e. For an example of such a study we refer to [14] where the difficult question of multiplicity of steady solutions of the nonreduced system was approached through studying the multiplicity of solutions in the LEN approximation for a four layer (quadrupolar) arrangement. The theory of a bistable electronic device (thyristor) which resulted from this study will be presented in §4.3. We point out that the results of locally electro-neutral studies should be extrapolated upon the nonreduced systems with a certain caution even for e very small. This is so because, to the best of the author's knowledge, no asymptotic procedure for the singularly perturbed one-dimensional system (4.1.1), (4.1.2) has been developed so far that would be uniformly valid for the entire range of the operational parameters (e.g., for arbitrary voltages and fixed charge densities). Finally, the extreme simplicity of the procedure (4.2.2)-(4.2.12) makes us wonder if it could have any more-than-one-dimensional analogues. Such a generalization could be important in the semiconductor context (M = 2, zi= Z2 = —!)• For this case the multidimensional analogue of (4.2.2), (4.2.3) would read
Unfortunately, we do not know how to handle the nonlinear equation (4.12.14). Moreover, the harmonic function h is not continuous across the discontinuities of N (this follows from (4.1.9), (4.1.7), and (4.1.3)). The magnitude of the appropriate jumps in h is a nonlinear function of the local values of (p and h themselves, so that h cannot be computed separately from
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constant and equal charge carrier mobilities in a medium with a piecewise constant fixed charge density with four sign alterations, e.g., in a quadrupolar membrane or in a thyristor. (The term ambipolar electro-diffusion designates diffusion of two univalent charge species of opposite sign, combined with their migration in a self-consistent electric field.) The b.v.p. under consideration is as follows:
Here for the sake of convenience we have used p, n for the concentration of, respectively, positive (negative) univalent mobile charges, e.g., cations, holes (anions, electrons) instead of Ci of §§4.1 and 4.2. The piecewise constant fixed charge density N(x) is chosen of the form:
Furthermore, V > 0 is the voltage drop in the system (between x = 0 — 0 and x = L + 0); 0 < c0
of charge carriers, as well as the continuity of their electrochemical potentials (quasi-Fermi potentials)
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yielding conjugation conditions (see (4.1.9), (4.1.10)):
The b.v.p. (4.3.1)-(4.3.3) is solved explicitly, following the scheme of the previous section, and it is shown that for a certain range of the parameters {Aj}, {Ni}, V multiplicity of solutions of (4.3.1)-(4.3.3) occurs. The appropriate multiple solutions may be described as follows. There exists a "lower" solution branch, which is explicitly constructed in the asymptotic limit CQ—» 0. This lower solution appears right at equilibrium (V = 0) and whenever multiplicity occurs. The electric current /,
associated with this branch, saturates for V —* oo at some low value of order In parallel, when certain conditions on the parameters {Aj}, {N{} are satisfied, another two solution branches appear for V > V cr , with VCT appropriate for each {A^}, {Ni}, These two latter solution branches are connected via a turning point in the VI plane at V = VCT. The current Iu associated with the "upper" solution branch grows unboundedly as V —>• oo, while the current Im associated with the "middle" branch decreases with increasing V (negative differential resistance) to a limiting value 7^im (7m ^5°/yim). At some critical values of the system's parameters, solution multiplicity disappears through one of two mechanisms: either "blowup" of the low current solution, which connects to the high current solution, with the "negative differential resistance" branch disappearing, or, at a different critical parameter value, via the "run off' of the turning point to infinity, with only the low current branch of solution remaining in the IV plane. This situation is schematically depicted in Fig. 4.3.1. (Corresponding exact solutions will be presented in Figs. 4.3.3-4.3.6.) The existence of steady states described by the one-dimensional ambipolar version of (4.1.1), (4.1.2) of the form
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Fig. 4.3.1. Scheme of the computed voltage/current curves. — multiple steady states, "blowup" of the "low"current branch at a critical parameter value, — "run-off" of the turning point at another critical parameter value.
has been established under fairly broad conditions in several theoretical and numerical studies [1], [15], [16]. This is in sharp contrast to the question of uniqueness. Virtually no results are known, except for in the vicinity of equilibrium [1], [2]. Moreover, uniqueness away from equilibrium is not expected generally on physical grounds, in view of the existence of essentially one-dimensional multilayered semiconductor devices (p — n—p — n rectifiers or thyristors) exhibiting in a certain range of parameters a multiplicity of nonequilibrium steady states [17]. Until recently [14], the only known nonuniqueness result in a purely electro-diffusional formulation (no source terms, equal constant diffusivities) was due to Mock [18] who has produced a numerical example of steady state multiplicity in model (4.3.5) with N(x) piecewise constant, changing sign four times as prescribed by (4.3.2d) (the presence of at least four alterations of sign N(x) is likely to be necessary for the multiplicity of steady states). The above numerical example was obtained in [18] for one particular set of parameters (doping levels {Ni} and thicknesses of the doping layers {A,}), by employing elaborate finite difference methods for the solution of (4.3.5) with the appropriate boundary conditions. Computations of [18] resulted in a voltage current curve which is schematically reproduced in Fig. 4.3.2. (In actual computations the current / on the lower branch was about 15 orders of magnitude lower than the current on the middle and upper branches.) Recall that acquiring information about a multiplicity of electro-diffu-
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Fig. 4.3.2. Scheme of the voltage/current curve, computed in [18]. sional steady states is important for the following reasons. Electro-diffusion represents a paradigm of a conservative (sourceless, unlike most reactiondiffusional formulations), inertialess dissipative transport process with constant transport coefficients. In this sense, electro-diffusion is the simplest asymptotic limit of more complex transport phenomena, involving inertia, sources, or stronger nonlinearities. Besides, as was pointed out in the Preface, there exists a number of largely unexplained, practically relevant phenomena, occurring in purely electro-diffusional systems and potentially related to the multiplicity of steady states [19]-[21]. Finally, uniqueness results could be valuable for the numerical analysis of semiconductor models. Recall that considering a simpler reduced model (4.3.1) instead of (4.3.5) is motivated by the smallness of the parameter e and the appropriate results for a nowhere vanishing N(x) = 0(1) are expected to be valid for the nonreduced problem, unless very large voltages are applied, causing "punch through." We reiterate that although several studies were devoted to exploring (4.3.1) as an asymptotic limit (4.3.5) ([1], [2], [22]-[25]), no comprehensive uniformly valid analysis of the appropriate asymptotic transition exists for arbitrary N(x), to the best of our knowledge. 4.3.1. Integration, solution algorithm. Integration of b.v.p. (4.3.1)(4.3.3) proceeds along a simplified version of the scheme of §4.2.
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Assign to the variables p, n, y? in the regions Xi < x < £i+i, the superscript i (i — 1,2,3,4). Equation (4.3.1) yields, after one integration,
Here jp and jn are integration constants (the unknown fluxes of the charged species, corresponding to the applied voltage V) which have to be determined from the boundary and continuity conditions (4.3.2a-e), (4.3.3e-h). It follows from (4.3.3g-h) that jp, jn are independent of i. Now we introduce the following notation:
Recall that / is the total electric current to be found. By adding (4.3.6a) to (4.3.6b) and employing (4.3.6c), we obtain
For N(x) piecewise constant, subtraction of (4.3.6b) from (4.3.6a) yields, through (4.3.6c), (4.3.7)
Substitution of <^x, expressed from (4,3.8a), into (4.3.8b) with a subsequent integration yields
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Furthermore, we observe from (4.3.6c), (4.3.7a) that
and it follows from summing continuity conditions (4.3.3e) and (4.3.3f) that
In the light of the positivity of er, equations (4.3.9a-c) imply that
Here and in accordance with (4.3.7a) and boundary condition (4.3.2a), the following notation has been introduced for the sake of convenience:
Consideration of (4.3.8c) at x = Xj+i yields
Finally, integration of (4.3.8a) between Xi and x < xi+i gives us
Here again Once more, from consideration of (4.3.11) at x = £»+!, we have
where Introduce again the following convenient notation
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Equations(4.3.10a-c),(4.3.12a-b),togetherwith continuity relations (4.3.3e,f) and expressions (4.3.9a-b), employed to relate (pi to v'i-i? form a set, determining a\ > 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, <73, cr4 related to the above variables via (4.3.10a) with i = 2,3,4. The fourth equation (4.3.lOc) with i = 4 then serves to determine J. With a i, a i (i = 1, • • • , 4) and J determined in this fashion, <^,
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Fig. 4.3.3a. Voltage/current dependence for the low current branch. NI=I, JV 2 =-i,./V3 : =i.7,Ar4=-io,Ai=2,A2=3,A3=4,A4=i,Vo is defined in the text.
In Fig. 4.3.3b we present a V — I curve with a turning point and a negative differential resistance region with current saturation, computed for the same values of parameters {A^}, {Aj}, CQ as in Fig. 4.3.3a. This V—I curve corresponds to the "upper" and the "middle" solution branches. The range of parameters in which the high current solutions exist is again evaluated below, via an asymptotic treatment for /—> oo. We point out that current saturation at the "low" and the "middle" branches has to do with the vanishing of minority carrier concentrations at the middle (depletion) interface x = £3. (This is a general feature, common to all LEN formulations, to be discussed in detail in the next section in the context of concentration polarization.) Typical profiles of a, (p as functions of x, as computed from (4.3.8c), (4.3.11) for the same parameter values as in Fig. 4.3.3, are presented in Fig. 4.3.4a,b (V + V0 = 12.28; I£ = .5454 • 10~4, Jt = -.1051 • 1(T4; Im = .093, Jm = -.01685; Iu = 1.45, Ju = -.1662). The values of v, corresponding to the "lower" solution branch, represented in Fig. 4.3.4a by the point-dashed line, are hardly distinguishable from the appropriate equilibrium values, corresponding to V = 0, and are typified by a very low minority carrier invasion (crl — |JVj
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Fig. 4.3.3b. Voltage/current dependence for the "middle" and "high" current branches. Parameters as in Fig. 4.3.3a.
minority carrier invasion is appreciable at the enriched interfaces x = x2, x — x 4 (the appropriate a[, a'2 and <73', cr'4 are noticeably greater there than NI, \Ni and N3, |JV4|, respectively). Along the upper branch (the continuous line in Fig. 4.3.4a) minority carrier invasion is high everywhere in the system except at the end points, and this is the mechanism for the passage of high currents along this branch. This situation is carried to an extreme in the example presented in Fig. 4.3.6 (continuous line) where we depict the a profile at the high current branch for / = 1500, V + V0~ 225 (other parameters as in Figs. 4.3.3, 4.3.4). In accordance with these features of a, the if profiles presented in Fig. 4.3.4b provide evidence that practically the entire potential drop in the system along the lower branch falls upon the large potential jump at the depletion interface x 2 - This is still largely true along the "middle" branch, for which we already notice a potential drop within the bulk of charged regions i — 1, • • • , 4. The bulk potential drops form a major contribution towards the total potential drop at the "upper" branch. An example illustrating "appearance" and disappearance of multiplicity at the variation of the system's parameters, described in the Introduction, is presented in Fig. 4.3.5. Here parameters N\, N2, JV3, AI, A2, A3,
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Fig. 4.3.4a. a profiles, — "lower" solution branch within the current saturation region (/=.5454-io~ 4 , J=-.105MO~ 4 ),— "middle" branch within the current saturation region (/=.093, J=-.01685), — "upper" ( "high" current) branch (7=1.45, J=-.1662). V+V0=12.28, other parameters as in Fig. 4.3.3.
A4, CQ are kept as in Figs. 4.3.3, 4.3.4, whereas 7V4 is varied in the range — 10.5 < 7V4 < —8.5. It is observed that solution multiplicity occurs for N% < N4 < JVf with N% ~ -10.2 and JVf ~ -8.612. For N > TVf, N4 —> NT the "upper" limiting current I^m decreases until at AT4 = N^r it "clashes" with the "lower" limiting current I\im. As a result of this, solution multiplicity disappears via the "middle" current branch and the "lower" limiting part of the "low" current branch's eliminating each other, leaving behind an inflection point in the VI curve. For 7V4 < N% the "low" current solution "blows up", at a voltage somewhat lower than that corresponding to the turning point at 7V4 > N"', and the appropriate "low" current branch connects to the high current branch. Alternatively, with 7V4 —> 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.
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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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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
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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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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
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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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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,
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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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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 <& 1, thus provides us with full information about the parameter range in which the multiplicity of the steady state occurs. The conditions in terms of N± for the occurrence of the steady state multiplicity can be summarized as follows. According to (4.3.17b), for CQ -C 1, there exists a low current branch with current saturation as long as the following condition holds:
On this low current branch, concentration variations within the charged layers are small (of order CQ). The condition (4.3.17d) merely says that in order for the lower limiting current to exist, the total concentration gain a in the "enrichment" layers (i = 1,4), has to be smaller than the appropriate total concentration decrease in the "depletion" layers (i = 2,3). (According to (4.3.15), the total concentration variation within each layer is proportional to the passing current /, fixed charge density A^, and the thickness AJ of the layer.) In particular, (4.3.17d) implies the existence of some critical N± t QV£ r ) beyond which the low current branch with saturation ceases to exist. On the other hand, the high current solutions, whenever they exist, are characterized by a high minority carrier invasion and large total concentration variations, and these scale, for large currents /, as \/J. (In this high invasion state, the large total concentration within the bulk of the charged layers entirely "forgets" about its low outside value CQ and sees it instead as zero.) Nonetheless, as on the low current branch, the concentration variations within the charged layers are monotonic functions of Ni (this time, however, nonlinear). In particular, there exists a critical N±\ (N") such that for |-/V4 < JV"4 the interface concentration of the minority carriers at the central (depletion) interface would approach zero at some current, which is incompatible with the properties of the high current branch, implying the disappearance of the latter below IJVJ . It follows from this description -!
-h
that whenever \N^\ < N± t the low and the high current branches will coexist and thus the steady state multiplicity will occur for N± in the range:
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4.3.4. Existence of multiple solutions. Existence of solutions to (4.3.10a), (4.3.lOc), constructed numerically and described above, immediately follows from the classical Newton-Kantorovitch theorem (see, e.g., [26]) which asserts the following. If F: 7£n —» Kn is different iable on a convex set D, and
and there exists a:0 in D such that
then there exists a solution x* in S of F(x) = 0 and the Newton iterates
are well defined and converge to x* quadratically with
In our case, from (4.3.lOa), (4.3.10c), (4.3.13a,b),
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and
in accordance with (4.3.10b), (4.3.13b). Differentiation of (4.3.33a,b) yields for i = 1:
for
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It is easily checked that for F, F' given by (4.3.33a)-(4.3.45), with the accu racy employedin our computations, conditions of the Newton-Kantorovitct theorem are satisfied, unless we are in an extremely close vicinity of th "lower" and "middle" limiting currents. Indeed, even for the "stiff' ex ample of Fig. 4.3.4 (well within the range of current saturation at th "lower" and "middle" branches) with the accuracy employed, we have fron (4.3.33)-(4.3.45) the data shown in Table 4.3.1.
With data from Table 4.3.1, conditions (4.3.29), (4.3.30) of the NewtonKantorovitch theorem are clearly satisfied, which implies the existence of the appropriate solutions.
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4.4. Locally electro-neutral concentration polarization of an electrolyte solution under an electric current. Concentration polarization (CP) is an electrochemical and membranological nickname for a complex of effects. These are related to the formation of electrolyte concentration gradients resulting from the passage of an electric current through a solution adjacent to an ion-selective interface. This phenomenon forms a basic element of charge transfer from electrolyte solutions to electrodes and ion-selective membranes. Specifically, the situation to be discussed, as compared with that of the previous section, is that in one part of the system (solution) there are no fixed charges. This implies that somewhere in that region the total charge carriers concentration may approach zero at moderate voltages, causing a violation of the LEN approximation. Physically, this reflects a nonequilibrium broadening of the space charge, which may possibly reach a macroscopic size. (Recall that for noncharged systems at equilibrium or charged systems with \N\ = O(l) > 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.
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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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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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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 (<70 » 1) (4.4.14c) yields
i.e., the addition of a supporting electrolyte in a high concentration virtually eliminates the potential drop in solution. Accordingly, the voltage applied to the system practically entirely falls upon the interface potential jump. Simultaneously, the migrational component of the active cation's flux j becomes negligible as compared with its diffusional component.
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4.4.3. Effect of concentration polarization upon the valencyinduced counterion selectivity of ion-exchange membranes [12]. In this section we shall study the effect of CP upon the selectivity of an ion-exchange membrane with respect to counterions of different valencies. Near equilibrium, when the transmembrane fluxes are negligibly small, selectivity is largely determined by the ionic composition of the membrane. In particular, in the absence of any specific effects, with the ideal Donnan equilibrium, the membrane will take up multivalent counterions preferentially from a mixture with monovalent ones. If the multivalent ions are not immobilized near the fixed charges, their preferential uptake yields their preferential transfer by the membranes. On the other hand, as we move away from equilibrium, with the development of the concentration polarization, counterion selectivity is substantially reduced. An interesting feature here is that this happens at voltages that are considerably lower than those for which the concentration polarization manifests itself through a pronounced nonlinearity of the IV curves. Similar behaviour has been observed in membrane systems with a mechanism for counterion selectivity quite different from that considered here [33]. The effect of CP upon the selectivity due to differing counterion distribution coefficients in the membrane has been studied recently in [34]. For the sake of simplicity, we shall assume again an ideal permselectivity of the ion-exchange membrane. As mentioned before, this assumption is practically justified, since the co-ion transport numbers in modern monopolar ion-exchange membranes, employed, e.g., in electrodialysis, are limited to a few percent. The structure of the present subsection is as follows. First, the governing b.v.p. is formulated and reduced, following the scheme of §4.2, to a system of algebraic equations. Then, two important limit cases are discussed: counterion selectivity near equilibrium and selectivity at high concentration polarization. Finally, we present and discuss the results of a numerical solution of the above algebraic system for the intermediate range of deviations from equilibrium. 4.4.4. Formulation. Consider two unity thick diffusion layers of a mixture of 1, 1-, and 1, z-valent3 electrolytes with a common anion, adjacent to a planar ideally permselective cation-exchange membrane. Direct the axis x normally to the membrane; and let x = 0 coincide with the outer boundary of the diffusion layer. The diffusion layers will thus be located at 0 < x < 1 and 1 + A < £ < 2 - | - A , whereas the membrane will be at 1 < x < 1 + A. Here A is the thickness of the membrane. Steady electrodiffusional transfer of ions across the membrane and the diffusion layers is o
Consideration of a mixture of a symmetric and nonsymmetric electrolyte of arbitrary valencies ZQ, ZQ and ZQ, Z\ is reducible to the above case by means of a trivial rescaling, Z = ZI/ZQ.
140
STATIONARY CURRENT
described by the following b.v.p.:
Here pi (i = 1,2) are again the counterion concentrations, whereas n is the co-ion concentration. Finally, N is again the fixed charge concentration, and
where ji (i = 1,2) are the unknown ionic fluxes, Di is the dimensional diffusion constant of species i in the solution, D* is the appropriate diffusion constant within the membrane, and DO is the dimensional diffusion constant of the electrolyte, used for scaling. Equation (4.4.17) corresponds to the ideal permselectivity of the membrane and (4.4.18) stands for local electro-neutrality within the solution (4.4.18a) and within the ideally permselective membrane (4.4.18b). The appropriate standard set of boundary and conjugation conditions at the membrane/solution interfaces is as follows
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141
Here
V is the voltage applied to the system. Unit activity coefficients and zero standard counterion potentials in both phases are assumed for the sake of simplicity. As was pointed out in §4.3, the solution of (4.4.15)-(4.4.22) with voltage V given corresponds to potentiostatic experimental conditions and yields the concentration, electric potential fields, and the ionic fluxes as functions of V. We could alternatively fix the total current density
and calculate the total voltage and the ionic fluxes, j\ and j?, as functions of /, which would correspond to galvanostatic conditions. The latter approach is again mathematically easier as it will be commented on in due course. Integration of (4.4.15)-(4.4.17) yields for the depletion layer 0 < x < 1, taking into account conditions (4.4.18).
Here
with jt defined by expression (4.4.19).
142
STATIONARY CURRENT
Finally
A useful identity
has been employed here to simplify the algebra. Equation (4.4.24), together with (4.4.25) and (4.4.26), implicitly defines Pi as a function of x for 0 < x < 1. It is also easily found that for 0 < x < 1
and
For the enriched unstirred layer l + A < a : < 2 + A w e similarly have
Here
and a is again denned by (4.4.25d) and (4.4.26a). The analogue of (4.4.27), for pz is
Using (4.4.15) and (4.4.18) and boundary condition (4.4.22c), we then find the electric potential
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143
Furthermore, taking into account (4.4.18b), integration of the transport equations (4.4.15)-(4.4.17) within the membrane (1 < x < 1 + A) yields
Here
and Pi(i + 0), ¥?(! + 0) stand for the values of the appropriate quantities at the membrane/solution interface on the membrane side. Values of Pi(l + 0) and ip(l + 0) are easily related, through the Donnan relations (4.4.21), to the appropriate values pi(l — 0) and ip(l - 0) in the solution. In particular, for z = 2 (e.g., calcium) we have
where
144
STATIONARY CURRENT
The values pi (1 + A + 0),
This viplHs
where
Finally, substitution of (4.4.29) and (4.4.31) into, respectively, boundary equation (4.4.22a) and relation (4.4.38e) yields a set of two algebraic equations for the ionic fluxes ji, J2 at the given voltage V. These equations are further studied asymptotically or solved numerically; as indicated before, this corresponds to potentiostatic conditions. As in §4.3, in the galvanostatic formulation, the situation is even simpler. Indeed, in this case the total current density 7 is given and the ionic fluxes ji, j% and voltage V are sought. For one ionic flux, say j\ fixed at any particular value, the other flux j2 is immediately found from (4.4.23) as j? = ( I — j i ) / z . The transcendental equation for j\ does not contain V and thus splits apart from the system. The second equation for V is by construction resolved with respect to it. The advantage of the galvanostatic formulation, as compared with the potentiostatic one, thus lies in the possibility of reducing the former to a single equation for the unknown flux. Before turning to the appropriate numerical solution, we discuss next the two important limiting cases: equilibrium membrane selectivity and selectivity very far from equilibrium, near the saturation of the voltage against current curve. 4.4.5. Limiting cases. For an infinitesimal voltage V, the ionic composition in different parts of the system does not differ appreciably from the equilibrium one; i.e., for the solution layers we have
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145
whereas within the membrane
yielding, in particular for z = 2
It is easy to find the counterion flux ratio rj near the equilibrium:
If the membrane's permeability for counterions D* /A is much lower than that of the adjacent solution layers, which is normally the case, (4.4.42a) yields
For a convenient example, D* = D\, p2o — J'lo and N = 78pio, expression (4.4.42b) yields 7? = 12, thus implying quite a selective extraction o divalent counterions from a mixture with monovalent ones, near the equilibrium. Counterion selectivity at the other extreme, namely at the limiting current, is easily found from (4.4.24), describing the counterion concentration distribution within the depletion layer. To this end it is convenient to rewrite (4.4.24) at x = 1 as follows:
where
Since at the limiting current
146
STATIONARY CURRENT
it follows from (4.4.43) that in this limiting case, the last term in the lefthand side must vanish. This gives, using (4.4.25a),
Expressions (4.4.44a) and (4.4.45) yield for
the following relations
implying a residual selectivity at the limiting current. The physical content of this residual effect is easily clarified by the following argument. At the limiting current the concentration of all ions vanishes at the membrane/depletion layer interface. At this point counterion concentration profiles become linear and coincide throughout the Nernst layer. It then immediately follows from (4.4.15)-(4.4.18a), with pi = pi — p, that
and hence
yielding (4.4.47). Thus, the residual selectivity is due to a different contribution of migration into the limiting ionic fluxes for ions of different valencies, whereas the diffusional components of these fluxes are exactly equal. 4.4.6. Solution and discussion. In the intermediate range of currents the equations for j\ and V are solved numerically for an increasing sequence of currents, /. Some typical results of these computations, for z = 2, DI = £)0, Dl = D\ = O.OlDo, Pio = P20 and N = 1 - 0, are presented in Figs. 4.4.2 and 4.4.3. Figure 4.4.2a depicts the calculated voltage against current curve. The corresponding variation of the counterion flux ratio TJ with voltage is presented in Fig. 4.4.2b. We can see from these data that the bulk of selectivity reduction occurs before the saturation of current sets in. The mechanism for this is evident from the data presented in Fig. 4.4.3, depicting three typical counterion concentration profiles within the depletion layer and the membrane at, respectively, zero, medium, and limiting current. Near the equilibrium, counterion selectivity is largely determined by the membrane, while the membrane retains
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Fig. 4.4.2a. Calculated voltage against current curve for a mixture of 1,1and 1,2-valent electrolytes. Di=£>2=iOODj=iooDJ, W=iO,pi 0 =p2o (notation in the text). The dashed line indicates the limiting current.
its near-equilibrium ionic composition with a pronounced preference for multivalent counterions. In accordance with the above estimates, these two factors bring about high counterion selectivity. For a higher voltage, preferential transfer of multivalent counterions results in their enhanced depletion in the Nernst layer, so that the membrane is equilibrated with a solution containing an appreciably lower relative amount of the preferred ion than the bulk. These changes in the ionic contents of the membrane result in a rapid reduction of counterion selectivity, while the voltage against current curve still remains nearly linear. At still higher voltage, when the Nernst layer gets appreciably depleted of both ions, the limiting stage of counterion transfer shifts from the membrane to the unstirred layer. As a result, the selectivity approaches its low limiting value, while the voltage against current exhibits the typical nonlinearity with ultimate saturation.
4.4.7. CP of a binary electrolyte at an inhomogeneous permselective interface. As mentioned before, experiments on ion-exchange membranes point at an electric iiihomogeneity of their surface on a mi-
148
STATIONARY CURRENT
Fig. 4.4.2b. Calculated variation of counterion flux ratio with the applied voltage.
cron to tens of microns scale. The above surface may be viewed rather as an array of conductive and insulating spots. In order to evaluate the effect of this on the concentration polarization, let us consider the following three-dimensional modification of the b.v.p. (4.4.1). Let us view the membrane surface as consisting of a periodic two-dimensional array of circular conductive ideally perrnselective spots of radius R8, separated by an insulated space, with a center to center distance 2Rb between the spots. Consider again an "unstirred" layer (or thickness 6) of a symmetric zvalent electrolyte (of bulk concentration c 0 ), adjacent to the membrane. Let us introduce a rectangular system of coordinates j/i, y%, x, with the axis x directed normally to the membrane, through the center of the conductive spot, with the origin x = 0 coinciding with the outer edge of the unstirred layer. Let us introduce the scaled coordinates, denned this time as:
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149
Fig. 4.4.3. Counterion concentration profiles within the depletion layer and the membrane. Parameters of the system as in Fig. 4.4.2 (I) /—* 0, V —» 0; (II) 7=1.35, V = 10; (III) 7->/ lim , V-»oo.
along with the dimensionless parameters:
Due to symmetry, the description of the ionic transfer across the unstirred layer is equivalent to that in the rectangular cell
To simplify the description, let us replace the rectangular symmetry cell with a cylindrical one:
A stationary electro-diffusion in the unstirred layer is then described by the following b.v.p. (compare with (4.4.1)):
150
STATIONARY CURRENT
The boundary condition (4.4.52c) states that the normal component of the cationic flux at the membrane vanishes at the insulating portion of the membrane surface and is equal to a given constant i throughout the conducting site. The boundary condition (4.4.52d) asserts the vanishing of the normal component of the anionic flux at the membrane, corresponding to the ideal permselectivity of the latter. Finally, the boundary conditions (4.4.52e,f) state that the surface {r -= 0, 0 < x < 6} is that of symmetry. We observe that c(r,x),
with c(r, x) satisfying the following b.v.p.
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The b.v.p. (4.4.54) is easily solved by separation of variables, yielding
Here JQ and J\ are the zeroth- and first-order Bessel functions and \k are the roots of the equation
The first few A/t are:
Note that in the limit of a homogeneous membrane (R = 1), (4.4.55a) predicts
in accordance with (4.4.2.a). According to (4.4.55a) the interface concentration at x = 6 is given by the expression
It is observed from (4.4.55d) that c(r, 6\i) is a decreasing function of i and an increasing function of r, so that the concentration is minimal at the center of the conducting spot (r = 0) and is given by the expression:
152
STATIONARY CURRENT
The limiting current density i lim corresponds to c(0,<5) = 0, which yields, according to (4.4.55e)
For R —> 1 (4.4.56a) reduces to
in accordance with (4.4.3). According to (4.4.55d), (4.4.56a), the maximal interface concentration variation at the limiting current is:
For comparison with the homogeneous interface case it is convenient to define the effective (average) dimensionless current density at an inhomogeneous membrane as:
where i is a "true" current density at a conductive spot. In Table 4.4.1 we present some typical estimates for Ihm and the maximal interface concentration variation Ac for z = 2. Table 4.4.1 Concentration polarization at an inhomogeneous C-membrane.
6
R
zlim
Ac
200
.5
-995^
6-l(T 3
200
.1
-957^™
4-10~2
20
.5
.956tjj£
6-10- 2
20
.1
•692ihiS
3-1Q- 1
6 = b/Ri, — the relative thickness of the unstirred layer (8 — dimensional thickness of the unstirred layer, Rb — half typical separation between the centers of the conducting spots).
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Pig. 4.4.4. Interface concentration profiles for #=200. Curve 1: i=o, /3=io 2 . Curve 2: fi=.9, i/i"m=.9999, /3=103. Cwrue 3: R=.5, i/i linl =i.o, /3=io 2 . Cur-ue 4; R=.l, i/i lim = 1.0, /^lO 1 .
71 = Rs/Rb — dimensionless radius of a conductive spot (Rs — the corresponding dimensional quantity). z llm — the effective (average) limiting current density at the inhomogeneous membrane. zh™ — the limiting current density at a homogeneous membrane. Ac — the maximal interface concentration variation at the limiting current through an inhomogeneous membrane. Additional examples are presented in Figs. 4.4.4 and 4.4.5 where we depict the interface concentration profiles for 6 = 200 and 6 = 20 respectively, for different values of R at the limiting current. Furthermore, Figs. 4.4.6 and 4.4.7 depict the I dependence on V for different values of R and 6 = 200,20, respectively. When the thickness of the unstirred layer is much larger than the typical distance between the conductive inhomogeneities of the membrane surface (6 » 1), the deviation in the limiting current density at a nonhomogeneous membrane from the appropriate value at a homogeneous surface is hardly noticeable, unless the proportion of the conductive membrane surface area is very small (R -C 1). On the other hand, for thin unstirred layers (6 = 0(1)), the deviation
154
STATIONARY CURRENT
Fig. 4.4.5. Interface concentration profiles for S=W. Curve 1: t=0, 0=1. Curve 2: «=.9, i/i lim = 1.0, /3=102. Curve 3: R=.S, i/i lim =l.o, 0=10. Curved: H=.l, i/» Um =1.0, 0 = 1.
above is significant already for relatively large conductive spots (R = 0(1)). At the same time we notice that at the limiting current the variation of the interface concentration Ac is considerable in a very broad range of 6 and R. Let us show that no fluid equilibrium is compatible with the electrolyte concentration distribution given by (4.4.55d). Indeed, associated with this concentration distribution, there is an electric potential distribution, prescribed by (4.4.53). This latter is in turn inseparable from an electric space charge, whose density may be found from the Poisson equations as
Here d is the dielectric constant of the fluid. The presence of a space charge distributed in the fluid gives rise to a volume force:
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Fig. 4.4.6. Computed VC curves for 6=200. «— the effective (average) current density at the inhomogeneous membrane. i[j™— the limiting current density at a homogeneous membrane. Curve 1: /J=.9. Curve 1: R=A. Curve 3: R=.i
For a mechanical equilibrium in a Newtonian fluid to exist, the volume force / has to be balanced by a pressure gradient, according to the following condition:
Here p stands for pressure. Take the curl of both sides of (4.4.60). This yields, after a substitution of (4.4.59)
Condition (4.4.61) is necessary for the existence of a mechanical equilibrium, which can be rewritten as, employing (4.4.53) and some standard relations of the vector analysis,
Condition (4.4.62) implies that for a mechanical equilibrium to be possible the level lines of c and (Vc) 2 /c 2 have to coincide.
156
STATIONARY CURRENT
Fig. 4.4.7. Computed VC curves for 6=20. Curve 1: R=.9. Curve 1: R=.5. Curve 3: R=.i. Notation as in Fig. 4.3.6.
For a nonhomogeneous membrane, this condition is not generally satisfied, so that a convection due to the electric volume force should be expected in the solution layers adjacent to such a membrane. In particular, it is easily observed that the condition (4.4.62) is not satisfied by the concentration given by (4.4.55a). This is illustrated by Fig. 4.4.8 where we depict schematically the c and (Vc) 2 /c 2 dependencies on x and r. For instance, x = 0 is a level line for c and is not for (Vc) 2 /c 2 . Thus, generally Vc is not parallel to V [(Vc) 2 /c 2 ] and the condition (4.4.62) does not hold.
Fig. 4.4.8. Sketch of c and (Vc) 2 /c 2 dependence on X and r.
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In summary, the nonuniformities of the electric field, associated with those of concentration near an inhomogeneous membrane surface, give rise to a volume force that will set in motion the fluid in the diffusion layer. The corresponding convective pattern can be described as follows. The electric field is strongest at the center of the conducting spot and vanishes on the symmetry cell boundary at the membrane. A circulation is expected to result with a typical vortex size4 R(,. The appropriate electroconvective problem will be formulated in §6.5. 4.5. Open problems. (See §4.3.) 1. Stability of the described solution branches (the "middle" branch with negative differential resistance is expected to be unstable). 2. Effects of space charges (finite e). It is expected that the inclusion of space charges will eliminate current saturation (limiting currents) at the "lower" and "middle" branches. These branches are instead expected to meet at another turning point, due to "punch through" (in this purely electro-diffusional formulation without source terms).5 3. Uniqueness of electro-diffusional steady states is expected and should be proved for one-dimensional systems with less than three alterations of sign N. 4. The possibility should be considered of generalizing the present treatment with N(x) piecewise constant to the case of arbitrary sign varyiin.the asy 5. The precise nature of bifurcation at N^ = Nc^ (likewise, for other TVj, i = 1,2,3). It is expected that at AT4 = Nf the "lower" and "middle" limiting current branches fuse and annihilate each other, leaving an inflection point at the voltage-current curve for 7V4 < Nf.
4
In fact, due to an inevitable nonuniformity of the distribution of conductive spots over the membrane surface, a whole hierarchy of circulation on different length scales sets in the diffusion layer. Namely, this complex multiscale convection is expected to cause the mixing of the entire diffusion layer and the resulting "overlimiting" CP behaviour of the C-membranes. 5
The corresponding analysis has been carried out recently in [35].
158
STATIONARY CURRENT
REFERENCES [I] M. S. Mock, Analysis of Mathematical Models of Semiconductor Devices, Boole Press, Dublin, 1983. [2] P. A. Markowich, The Stationary Semiconductor Device Equations, Springer-Verlag,
New York, 1986. [3] S. Selberherr, Analysis and Simulation of Semiconductor Devices, Springer-Verlag, New York, 1984. [4] B. S. Polskii, Chislennoje Modelirovanije Poluprovodnikovikh Priborov, Zinatne, Riga, 1986. (In Russian.) [5] R. Schlogl, Electrodiffusion in freier losung und geladenen membranen, Z. Phys. Chem. (Frankfurt), 1 (1954) p. 305. [6] F. Helfferich, Ion Exchange, McGraw-Hill, New York, 1962. [7] T. Teorell, An attempt to formulate a quantitative theory of membrane permeability, Proc. Soc. Expt. Biol. Med., 33 (1935), p. 282. [8] 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. Chem.
Acta, 19 (1936) pp. 649-664, pp. 665-680. [9] K. S. Spiegler, Polarization at ion exchange membrane-solution interfaces, Desalination, 9 (1971), pp. 367. [10] Y. Oren and A. Litan, The state of the solution-membrane interface during ion transport across an ion-exchanger membrane, J. Phys. Chem., 78 (1974), p. 1805. [II] J. F. Brady and J. C. R. Turner, Coupled fluxes in electrochemistry. Concentration distributions near electrodialysis membranes, J. Chem. Soc., Faraday Trans. I, 74 (1978), p. 2839. [12] I. Rubinstein, Effect of concentration polarization upon the valency-induced counterion selectivity of ion-exchanger membranes, J. Chem. Soc., Faraday Trans. II, 80 (1984) p. 335. [13] A. A. Sonin and G. Grossman, Ion transport through layered ion-exchanger membranes, 3. Phys. Chem., 76 (1972), pp. 3996-4006. [14] I. Rubinstein, Multiply steady states in one-dimensional electrodiffusion with local elecroneutrality, SIAM J. Appl. Math. 47 (1987), pp. 1076-1093. [15] J. W. Jerome, Consistency of semiconductor modelling: an existence/stability analysis for the stationary van roosbroeck system, SIAM J. Appl. Math., 45 (1985), pp. 565-590. [16] I. Rubinstein, Effects of deviation from local electroneutrality upon electrodiffusional ionic transport across a cation-selective membrane, Reactive Polymers, 2 (1984), pp. 117-131. [17] F. E. Gentry, F. W. Gutzwiller, N. Holonyak, and E. E. Von Zastrow, Semiconductor Controlled Rectifiers: Principles and Applications of p—n—p—n Devices, PrenticeHall, Englewood Cliffs, NJ, 1964. [18] M.S. Mock, An example of nonuniqueness of stationary solutions in semiconductor device models, COMPEL — Int. J. for Comp. and Math, in EEE, 1 (1982), pp. 165174.
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[19] M. Seno and T. Yamabe, Anomalous conduction across ion-exchange Membranes, Bull. Chem. Soc. Japan, 36 (1963), pp. 877-878. [20] V. Shashoua, Electrical oscillatory phenomena in protein membranes, Symp. Faraday Soc., 9 (1974), pp. 174-181. [21] O. Kedem and I. Rubinstein, Polarization effects at charged membranes, Desalination, 46 (1983), pp. 185-189. [22] I. Rubinstein and L. Shtihnan. Voltage against current curves of cation exchange membranes, JCS Faraday Trans. II, 75 (1979), pp. 231-246. [23] C. P. Please, An analysis of semiconductor P-N junctions, IMA J. Appl. Math., 28 (1982), pp. 301-318. [24] F. Brezzi, A. Capelo, and L. Gastaldi, A Singular Perturbation for Semiconductor Device Equations, to appear. [25] M. J. Ward, Asymptotic Methods in Semiconductor Device Modeling, Ph.D. thesis, Caltech, Pasadena, CA, 1988. [26] J. M. Ortega, Numerical Analysis, Academic Press, New York, 1972. [27] H. Steinriick, A bifurcation analysis of the steady state semiconductor device equation, SIAM J. Appl. Math., 49 (1989), pp. 1102-1121. [28] V. G. Levich, Physicochemical Hydrodynamics, Prentice Hall, New York, 1962. [29] I. Tachi and T. Kambara, Bull. Chem. Soc. Japan, 28 (1955) p. 25. [30] M. Senda and I. Tachi, Bull. Chem. Soc. Japan, 28 (1955), p. 632. [31] V.K. Indusekhar and P. Mears, The effect of the diffusion layer on the ionic current from a solution into an ion-exchange membrane, in Physicochemical Hydrodynamics II, D. B. Spalding, ed., Advance Publications Limited, London, 1977, p. 1031. [32] I. Rubinstein, E. Staude, and 0. Kedem, Role of the membrane surface in concentration polarization at ion-exchange membrane, Desalination, 69 (1988), p. 101. [33] M. Peri, Hydrophobic solvent type charged membranes for selective electrodialysis, Ph.D. thesis, The Weizmann Institute of Science, Rehovot, Israel, 1980. [34] I. Rubinstein, Concentration polarization effects upon the counterion selectivity of an ion-exchange membrane with differing counterion distribution coefficients, J. Chem. Soc., Faraday Trans., 86 (1990), pp. 1857-1861. [35] H. Steinriick, Asymptotic analysis of the current-voltage curve of a pn pn semiconductor device, IMA J. Appl. Math., submitted.
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Chapter 5
Nonequilibrium Space Charge in One-Dimensional Electro-Diffusion
5.1. Preliminaries. This chapter is concerned with some aspects of the nonequilibrium space charge in a motionless ionic fluid. (Electro-osmosis as a particular instance of electro-convection is treated in the next chapter.) We shall begin (§5.2) with Jackson's theory [1] of the liquid junction— probably the simplest one-dimensional current free process at whose initial stage space charge is important. We hope that some important intuition can be gained from this example concerning the essence of the singular perturbation situation occurring in macroscopic nonequilibrium electrodiffusional systems. The insight gained from this example will be used in the further presentation built upon the case study of CP at a homogeneous permselective interface which began in the Introduction and was continued in the previous chapter. Technically, we shall consider from §5.3 the different modifications of the b.v.p. (4.4.1), with the singular terms preserved in the Poisson equation. Each particular modification will be chosen so as to provide the simplest object exhibiting a phenomenon described in the appropriate section. CP is a convenient foundation, first because it provides a simple context for a discussion of a number of nonequilibrium space charge effects induced by electric current, such as "punch through" (§5.3) and "anomalous rectification" (§5.4). Second, this framework permits an easy tracing of fundamental nonuniformity of direct matched asymptotic expansions in powers of the dimensionless Debye length (e of (1.9c), (4.1.2)) with respect to the applied voltage (§5.3). Physically, this is related to an essentially nonequilibrium phenomenon arising already at moderate voltages— formation of space charge fronts. Near the equilibrium or away from it in the absence of an electric current, as in the case of liquid junction, the fronts above do not exist and the space charge is entirely concentrated in the boundary (internal layers) at the permselective interface. A direct 161
162
NONEQUILIBRIUM SPACE CHARGE
matched asymptotic expansion is well suited for describing this situation. At the same time, this method fails to account for abrupt variations and possible nonmonotonicities of space charge distribution (fronts) which are formally reflected in the above-mentioned nommiformity of the corresponding expansions with respect to voltage. These events occur in concentration polarization in electrolyte solution at voltages that are much lower than those typical for the parallel events at the abrupt p — n junctions in semiconductor devices or in multipolar membranes. This is so because in the electrolyte solution, as opposed to its ion-exchanger or semiconductor counterparts, there is no fixed charge. The fixed charge, present in a finite concentration, confines the space charge to a narrow boundary (internal) layer for a range of voltages much broader than that for an electrolyte solution. Namely this development of a nonequilibrium macroscopic space charge at a moderate voltage makes the CP of an electrolyte solution a convenient object for the study of the space charge effects [2], [3]. Finally (§5.5), in order to illustrate an alternative asymptotic approach, available for systems with nowhere vanishing fixed charge density, we shall treat, following Please [4], a p — n junction (bipolar membrane). Presentation of this chapter is purely heuristic, based on numerical or formal asymptotic results. For some rigorous results concerning related matters, the reader is referred to [5]-[7]. 5.2. The space charge in the liquid junction [1]. By liquid junction or the liquid junction potential we mean the diffusion potential developing in an electrically insulated electrolyte solution with differing ionic diffusivities and an initial concentration discontinuity. Besides its conceptual importance as probably the simplest nonequilibrium electro-diffusional situation, the dynamics of liquid junction is important to understand for applications, such as salt bridges, etc. Consider two compartments, occupying the regions — 1 < x < 0 and 0 < x < 1, respectively, filled with solutions of the same univalent electrolyte at concentrations 1 and A, respectively. Let at some moment t — 0 the wall separating the compartments at x = 0 be removed. The solution within the compartments is assumed immobilized, say, with gelatin, so that the entire transport is due to electro-diffusion only. The initial values of the electrolyte concentration are maintained at the external walls x = ±1. These walls are electrically insulated so that no electric current can pass through them. The appropriate initial b.v.p. (i.b.v.p.) thus is
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Here the same scaling as in (4.1.1), (4.1.2) has been employed. Once again, £ is the square of the ratio of the Debye length to the dimensional thickness of the compartment (L). The time t has been scaled with the overall cation diffusion time tp = L2/DP, where Dp is the cation diffusivity. The dimensionless relative anion diffusivity D in (5.2.1b) is the ratio of the anion diffusivity Dn to Dp. Condition (5.2.3c) implies insulation, that is, vanishing of the total current density i, defined as we recall from the Introduction, as
Recall that i, as defined by (5.2.5), is comprised of three contributions—two ionic fluxes, as in (4.2.5), and the displacement current
Finally, recall that subtraction of (5.2.1b) from (5.2.la) yields, taking into account (5.2.1c) and (5.2.5a), ix = 0 (this corresponds to solenoidality of the total current density vector in more than one dimension). The classical Henderson treatment of the liquid junction consists of posing E ~ 0 in (5.2.Ic), (5.2.3c). The resulting locally electro-neutral i.b.v.p. is of the form
164 NONEQUILIBRIUM SPACE CHARGE
Here
Addition of (5.2.6a) to (5.2.6b), divided by D, yields
whereas subtraction of (5.2.6b) from (5.2.6a) gives
Integration of (5.2.11), (5.2.12) with the initial boundary conditions (5.2.7)(5.2.9) yields
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Here
is the effective (electrolyte) diffusivity, whereas
is the Green's function for the Dirichlet problem on the segment — 1 < x < 1. E(a,(3) in (5.2.16) is the fundamental solution of the diffusion equation
For x = 1, (5.2.13) yields)according to (5.2.8a),
Expressions (5.2.13), (5.2.14), (5.2.18) represent the classical Henderson solution to the liquid junction potential problem [8]. It will be shown below that in terms of the singular perturbation problem (5.2.1)-(5.2.4) the expressions above provide the leading term of the outer solution. This latter satisfies with corresponding accuracy all the initial and boundary conditions (5.2.2)-(5.2.4) of the original system (5.2.1)-(5.2.4). The higher-order corrections in e can be constructed by a straightforward regular perturbation procedure. The question then arises — what happened to the singular nature of the perturbation problem (5.2.1)-(5.2.4)? Where is the singularity hidden in the outer solution (5.2.13), (5.2.14)? The answer comes from looking in the solution for potential
166
NONEQUILIBRIUM SPACE CHARGE
Here u(x,t) is the solution vector (p, n,
(5.2.6)-
where e(x,t), pa(x,t) are given by (5,2.13) (5.2.14). The outer i.b.v.p. to order O(e) is
The initial conditions for the b.v.p. (5.2.21), (5.2.22) are to be worked out from matching with the initial layer solution. (It can be easily inferred that the latter yields identical zeros as the initial values to the O(e) order in the outer solution.) The solution of (5.2.21), (5.2.22) can be easily written out explicitly. Indeed, let us introduce the notation
With these notations, by adding and subtracting (5.2.21a) and (5.2.21b) and substituting (5.2.21c), the system (5.2.21) can be rewritten as
The integration of (5.2.24) is straightforward. We shall not carry it out here. Instead we shall turn to the construction of the initial layer solution.
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Introduce the fast time and space variables T and £ defined, respectively, as
In terms of T, £ the (5.2.1a-c) are rewritten as
For e —+ 0, the £ range becomes infinite — oo < £ < oo and the initial and boundary conditions for p(£, T), n(£,r) become
Seek a solution vector v(£, T) = (p, n, ip) of the i.b.v.p. (5.2.26)-(5.2.28) as an asymptotic series in e
The O(l) term of the expansion (5.2.29), satisfies the i.b.v.p. (5.2.26)(5.2.28). It is observed that <£,,(£, 0), defined by solving (5.2.26c) at T = 0, is C1 regular. On the other hand, it can be easily observed, following [1], that for T —> oo, H;O(£,T) matches with the outer 0(1) order term uo(x,t) for t —> 0. Indeed, in terms of the variables £, T, where
the i.b.v.p. (5.2.26)-(5.2.28) for UO(^T) is rewritten as
168
NONEQUILIBRIUM SPACE CHARGE
Equations (5.2.31a-c) suggest that as r —> oo
Accordingly, we seek the large r asymptotics of p 0 , n0,
By substituting (5.2.35) into (5.2.31) and equating powers of r, we obtain
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From (5.2,37), employing (5.2.36) and (5.2.32), we get
D in (5.2.39) is again the effective electrolyte diffusivity defined by (5.2.15). Since by (5.2.25), (5.2.30)
the leading term of the initial layer solution, with the fast time r asymptotics given by (5.2.39), (5.2.40), obviously matches with the leading term of the outer, slow time solution (5.2.20), (5.2.13), (5.2.14). The appropriate construction can be carried out to the next orders in e as well. We observe that the composite solution for the potential
withtp0(x,t), ^ 0 (^^)| 4 = x / v ^ v T = t / £ and^, 0) (C = £/V^) as described above, obviously is in (71 for t > 0. Equation (5.2.42) and the treatment that leads to it remind us that in accordance with the description in the Introduction it takes about 1/e as long for an electro-diffusional system to get from an arbitrary initial state to a macroscopically locally electro-neutral one. This concludes our discussion of the liquid junction problem. In the following sections we shall treat somewhat more complex nonequilibrium space charge phenomena, occurring under the passage of electric current.
170
NONEQUILIBRIUM SPACE CHARGE
5.3. The steady nonequilibrium space charge in concentration polarization at a permselective homogeneous interface [2]. It was shown in §4.4 that the locally electro-neutral description of CP at a homogeneous permselective interfa.ee becomes intrinsically contradictory already at moderate voltages. The essence of this contradiction is that the local normalized carrier concentration becomes the same order as the normalized space charge density as predicted by the locally electro-neutral description. In order to trace this phenomenon in more detail and to investigate some properties of the space charge that forms, consider the following "space charge" analogue of (4.4.1)
Here, again
Integration of (5.2.1)-(5.2.3) yields, taking into account (5.2.4b), (5.3.5b),
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Here j is again an unknown integration constant (dimensionless cation flux, electric current density) to be determined from the boundary conditions (5.3.8). Let us point out first that the system in (5.3.6), (5.3.7) can be integrated exactly in terms of the Painleve transcendents of the second kind. Indeed, by going back to (5.3.1)-(5.3.3) we observe that adding (5.3.1), (5.3.2) and substituting equation yields
Furthermore, after another integration we obtain, using (5.3.6b),
Here C is another integration constant that has to be determined from the boundary conditions (5.3.8). Substituting (5.3.9) into (5.3.7) and taking into account (5.3.6b) we find
By means of substitution
equation (5.3.10) is reduced to
which is a Painleve equation of the second kind. Because of the complex nature of the Painleve transcendents and of the resulting difficulties in satisfying the boundary conditions we shall not proceed with the exact analytical solution of b.v.p. (5.3.6)-(5.3.8) any further, but rather we turn to an asymptotic and numerical study of this singular perturbation problem. A direct matched asymptotics procedure as applied to the problem (5.3.1) (5.3.5) reads as follows. Rewrite (5.3.1)-(5.3.5), using (5.3.6b), as
172
NONEQUILIBRIUM SPACE CHARGE
We fixed the boundary value of p at x = 1 in (5.3.15a) up to O(e) in order to avoid consideration of the irrelevant boundary layer at x = 0 and to concentrate upon the crucial "double layer" at x = 1. Seek the outer solution of (5.3.13)-(5.3.14) in terms of the expansions
Expansions (5.3.16) are expected to give an asymptotic representation of the solution to (5.3.13)-(5.3.14) everywhere, except for a boundary layer of thickness ^/e, adjacent to x = 1, where an inner solution of the form
is to hold. The outer and inner solutions (5.3.16) and (5.3.17) are to be matched in each order of e in a standard fashion. The unknown integration constant j of (5.3.6a) is sought as
Substitution of (5.3.16), (5.3.17) into (5.3.13)-(5.3.15) yields the following order O(l) outer problem. O(l). outer expansion.
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Integration of (5.3.19), (5.3.20) yields the already familiar LEN solution (4.4.2)
The integration constant j° (ionic flux at order 0(1)), defined as
has yet to be determined from matching with the inner expansion (5.3.17) at the appropriate order. O(l). inner expansion. In terms of the inner variable £, defined by (5.3.17b) the basic equations (5.3.13), (5.3.14) assume the form
Substitution of (5.3.17) into (5.3.22) yields after one integration of (5.2.22a) at the leading order,
Here qo is an integration constant subject to determination from the boundary and matching conditions. Integration of (5.3.23a) yields, taking into account the boundary conditions (5.3.15c,d) at f = 0 (x = 1),
Matching for
174
NONBQUILIBR1UM SPACE CHARGE
By (5.3.24), (5.3.25) for j0 ^ 2, boundedness of p (0) (£) at £ ->• oo, necessary for the matching, implies
Hence, finally
Note that (5.3.27) asserts a Boltzmann distribution for cations in the boundary layer. (The Boltzmann distribution for anions holds everywhere by (5.3.6b).) The matching of p^(£) at £ —> oo with p^(x) at x —> 1 implies by (5.3.27), (5.3.25), (5i3.21a)
and hence yields
in accordance with (4.4.2d). Substitution of (5.3.27) into (5.3.23b) yields (taking into account (5.3.25), (5.3.29), and (5.3.15d)) the following b.v.p. for p ( 0 ) (f)
Integration of (5.3.30) with (5.3.31) yields
or, finally, taking into account (5.3.29), (5.3.5d),
Here
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The composite order O(l) term can thus be written, by combining (5-3.21b), (5.3.25), (5.3.32b) as
with jo given by (5.3.29). Thus summarizing, we note that at the leading order the asymptotic solution constructed is merely a combination of the locally electro-neutral solution for the bulk of the domain and of the equilibrium solution for the boundary layer, the latter being identical with that given by the equilibrium electric double layer theory (recall (1.32b)). We stress here the equilibrium structure of the boundary layer. The equilibrium within the boundary layer implies constancy of the electrochemical potential p,p = Inp +
176
NONEQUILIBRIUM SPACE CHARGE
After differentiation of (5.3.34b), this problem is rewritten, taking into account (5.3.19b), as
By subtracting (5.3.35b) from (5.3.35a), we get
Integration of (5.3.36) yields, taking into account (5.3.34b,c), (5.5.20)
The integration constant ji has to be determined from matching with the inner solution. O(e), inner expansion. The inner problem at this order is
Substitution of (5.3.27) into (5.3.38a) yields, after multiplying both sides byexp(£<°>),
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Integration of (5.3.39) yields, taking into account (5.3.38c,d),
Arguments, identical to those that lead to (5.3.26), yield again
in compliance with the above-mentioned constancy of the electrochemical potential across the boundary layer to this order also. From (5.3.40), matching with the outer order O(e) term yields, taking into account (5.3.41a),
Substitution of (5.3.37) into (5.3.41b) yields
We shall not go on with the construction of <£^(£) via a solution of (5.3.38b), with (5.3.38d) and matching with (^'(l). Instead, we emphasize the fundamental nonuniformity of the expansions obtained in voltage V at V ^oo. Indeed, from (5.3.29) for V -> 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
178
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),
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,
Proof. By Lemma 5.1 we have
Integration of (5.3.3) yields
Another integration of (5.3.55) yields, taking into account (5.3.4c), (5.3.54),
Equations (5.3.55) and (5.3.56) yield, taking into consideration (5.3.5c),
Substitution of (5.3.57) into (5.3.6a) yields
Integration of (5.3.58) yields, taking into account the boundary conditions (5.3.8a,c),
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NONEQUILIBRIUM SPACE CHARGE
in accordance with (5.3.53). An implication of Theorem 5.1 is that as voltage V —* oo the stationary voltage-current (VC) curve returns to its initial slope, corresponding to i = V. In parallel the entire unstirred layer is filled with a positive space charge. Some typical stationary voltage-current VC curves along with the ionic concentration, space charge density, and the electric field intensity profiles for an intermediate voltage range are presented in Fig. 5.3.1. The appropriate profiles are constructed using a numerical solution of the system (5.3.1), (5.3.5). The essence of the numerical procedure employed for this and similar problems discussed in due course is as follows. The stationary system, e.g., (5.3.1)-(5.3.5) is replaced by its time-dependent counterpart. In this counterpart, the Poisson equation is replaced by the total current continuity equation (1.5), obtained as a linear combination of the original equations. The resulting system is then solved by quasilinearization [9] with a simultaneous solution of quasilinearized equations and subsequent Newton's iterations at each time step. Integration is continued in time until the steady state is reached. This numerical procedure is a modification of that suggested by Mock in [10]. We note in Fig. 5.3.la the "punch through" — lack of saturation of the VC curves at V —» oo, as opposed to that observed in the locally electroneutral formulation (4.4.1), (4.4.2). Another feature worth noticing is the formation of nonmonotonicities of the space charge density at moderate voltages (Fig. 5.3.1e). These nonmonotonicities, which develop into fairly "sharp" charge fronts propagating towards the bulk of the diffusion layer with the increase of voltage, cannot be recovered from the direct matched asymptotic procedure outlined above and represent the physical source of its breakdown at moderate voltages. We reiterate that the asymptotic treatment above is thus suited only for nearly equilibrium situations, typified by a clear separation between the locally electro-neutral domains and the locally equilibrium electric double layers. For comparison, we present in Fig. 5.3.2 some numerical results for the following non-locally-electro-neutral generalization of the classical TeorellMeyer-Sievers (TMS) model of membrane transport (see [11], [12] and §3.4 of this text).
CHAPTER 5
Fig. 5.3.la. Calculated voltage against current curves for different parameter s.
181
values of
Here
is the fixed charge density, constant within the membrane 1 < x < 2, vanishing in the diffusion layers 0 < x < l , 2 < a ; < 3 adjacent to the membrane. In contrast to (5.2.1)-(5.2.5), the formulation (5.3.61)-(5.3.63) does not assume an ideal permselectivity of the membrane. Permselectivity of a noriideal membrane system is characterized by ionic transport numbers T;, defined as
182
NONEQUILIBRIUM SPACE CHARGE
Figs. 5.3.1b,c. Calculated ion concentration profiles at different voltages V for (b) e=iO~ 4 , (c) e=lO~ 8 . —, cation concentration at dimensionless voltage V; , anion concentration.
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Fig. 5.3.1d,e. (d) Calculated profiles of the electric field strength, E, at different voltages, V, for
184
NONEQUILIBRIUM SPACE CHARGE
Fig. 5.3.2a,b,c. (a) Polarization curves in a modifiedTMS model for different e (N=w). (b) Same for different N (£=io~ 4 ). (c) Co-ion transport numbers. rn dependence on voltage for different N (
In the bipolar case under consideration
It is observed from Fig. 5.3.2c that in the modified TMS model (5.3.61)(5.3.63) permselectivity (rn) remains essentially constant in a wide range of voltages. The corresponding VC curves (Fig. 5.3.2a,b) are similar in shape to those for an ideal permselectivity membrane. This is also true regarding the space charge density and profiles of the other fields in the depletion layer 0 < x < 1.
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The features illustrated by the two examples above are typical of all one-dimensional electro-diffusional systems composed of parts of differing charge selectivity. The specificity of the systems considered is that due to vanishing of the fixed charge density in one of their parts (electrolyte solution) the appropriate effects (and correspondingly the breakdown of the "single double layer" asymptotics) are set in closer to equilibrium than, for instance, for a p — n junction with a high fixed charge concentration at both sides of the charge-selective interface. An asymptotic treatment of p — n junctions similar to that presented here can be found in [13], [6], [7]. Moreover, a high nowhere vanishing fixed charge density, when present in the problem, allows for development of alternative asymptotic procedures, uniformly valid with respect to voltage of a suitable sign (§5.5 and [4], [14]). 5.4. Anomalous rectification [3], Our aim in this section is to show that under certain conditions development of a nonequilibrium space charge may yield, besides the "punch through," some additional effects, unpredictable by the locally electro-neutral formulations. We shall exemplify this by considering two parallel formulations—the "full" space charge one and its locally electro-neutral counterpart. It will be observed that inclusion of the space charge into consideration enables us to account for the "anomalous rectification" effect that could not be predicted by the locally electro-neutral treatment. Physical motivation for this study is as follows. We saw previously that concentration polarization results in the decrease of solute concentration near the permselective interface (right at the interface in the electro-neutral version) where most of the system's resistance thus concentrates, and where the space charge develops. The system is expected to be sensitive to the minimum concentration value, and because of nonlinearity nontrivial effects, could be anticipated in response to unsteady disturbances of this value (e.g., provided by harmonic modulation superimposed upon a constant voltage applied to the system). Since it is "easier" to increase the minimal concentration (close to zero at the limiting current) than to decrease it, we might expect a positive rectification effect for the direct current component, counterintuitive ("anomalous") in the present system with a convex stationary VC curve. Thus the topic of this section is the rectification effects that arise in the stationary concentration polarization in response to a harmonic voltage modulation. Consider the following time-dependent modification of the b.v.p. (5.3.1),
186
NONEQUILIBRIUM SPACE CHARGE
In (5.4.8) V is the direct voltage component, whereas A and u = 2?r/ = (27T/T) are, respectively, the amplitude and the cyclic frequency of a harmonic modulation superimposed upon V, above some critical voltage Vcr.
Fig. 5.4.1. (a) Calculated steady voltage-current curve for e=io~ 4 , JV=io. The dashed line marks the value of the limiting current in a locally electro-neutral model (b) Calculated dependence of the relative rectification effect on the. modulation frequency.
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The b.v.p. is solved numerically for a sequence of voltages from V = 0 to V = Vct until the steady state is reached at t —> 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.
190
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 <5, so that without loss of generality we may choose
On the other hand, matching with the "left" outer solution (5.5.15a) implies
Similarly, for the "right" transition layer we have
The inner region. In this region, located around x = 0, the exponentials in (5.5.17) are negligible, compared with the N(x) term. Thus to the leading order in A in this region
We have from (5.5.21a), (5.5.4a) for ^ continuously differentiable and vanishing at x = 0
Here C is another integration constant to be determined from matching with the transition layer solution (5.5.20). This matching may be achieved, following the standard prescription of matched asymptotics [15], by introducing the intermediate variable
200
NONEQUILIBRIUM SPACE CHARGE
with some
in such a way that
that is,
Choose for definiteness
we have to require that
Equations (5.5.23a,b) yield
An identical procedure yields the matching of the "right" transition layer solution VR(r) with #(z) for x > 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 /
VQ being the equilibrium solution constructed above. V\ is constructed via an asymptotic analysis essentially identical to that at equilibrium; it is shown that the expansion (5.5.25) is valid for
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By continuing his analysis Please [13] arrives at an asymptotic representation of the solution for the entire range of forward bias (7 < 0) currents. The following current subdomains are distinguished besides (5.5.26a)
In the current ranges (5.5.26a)-(5.5.26e), (5.5.26g) explicit asymptotic expressions for the solution are obtained. We reiterate that this construction of an asymptotic solution valid uniformly for arbitrary currents, became possible only due to the presence of a nowhere vanishing high fixed charge density. For a reverse biased p — n junction (/ > 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.
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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
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an oscillatory regime (the Hopf bifurcation of a steady state) appears in a seemingly purely dissipative inertialess system without chemical reactions. We begin in §6.2 with a bifurcation analysis of the transition to the oscillatory state in the simplest original Teorell model. The appearance of a stable limit cycle is inferred through simple phase plane arguments. The character of the appropriate Hopf bifurcation (sub- or supercritical) is found by a nonlinear perturbation analysis to depend on the properties of the postulated steady resistance function f(V/V0). This discussion lays the groundwork for the developments of §6.3 in which a local analogue of the Teorell model is formulated in terms of a system of partial differential equations (PDEs), which include the convective Nernst-Planck equations (1.9) for electro-diffusion of ions combined with a generalized Darcy's law for the flow rate. The resulting local model essentially represents a quantitative inscription of the intuitive ideas behind (6.1.5). This model is further treated by means of a formal weakly nonlinear asymptotic procedure. The purely supercritical nature of the appropriate Hopf bifurcation is established this time, in contrast to the original Teorell's (ODEs) case. Finally, in §6.4 the electro-osmosis in a capillary is studied in a more general electro-convection context based on the fundamental equations (1.6) treated by means of matched asymptotic expansions. In particular, we show that no mechanical equilibrium is possible with an electrolyte concentration gradient present along the capillary. We also observe that for the capillary diameters sufficiently large as compared with the thickness of the double layer the contribution of the intercapillary circulation to the solute transport is comparable to that of molecular electro-diffusion or may even dominate. In the concluding section (§6.5) some open questions related to electro-osmotic oscillations are formulated, and an unsolved problem is presented concerning electro-convection at an inhomogeneous ion-exchange membrane, which was referred to in §4.4. 6.2. yields
TeorelPs model (ODEs). Substitution of (6.1.1) into (6.1.2)
By elimination of V and F from (6.1.4), (6.1.5) via (6.1.3), (6.2.1a) we have
208
ELECTRO-OSMOTIC OSCILLATIONS
With a natural scaling (a detailed dimension analysis is deferred to subsequent sections), we get
RO is some typical value of the filter resistance. Equations (6.2.2), (6.2.3) are reduced to the dimensionless form
Here i = tiRoI/V0 is the dimensionless current —the control parameter of this model. Furthermore, f(pt) is the dimensionless stationary resistance function of a general shape sketched in Fig. 6.1.3 and k = k/sv ^> 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 <£ 1, to relaxation oscillations. We begin our analysis by rewriting (6.2.7) in terms of the phase variables u, p as a system of two first-order equations:
Recall that in physical terms u(t) is a suitably normalized flow rate.
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Consider f ( u ) of the general shape in Fig. 6.1.3. A simple qualitative analysis to follow yields sharp results for /(it) "symmetric" about the equilibrium u = 0 in the sense that
This will be the case upon which we shall mainly concentrate below. We shall also point out some important peculiarities implied by this very particular choice of f ( u ) as compared to the general nonsymmetric case. The two principal isoclines in the phase plane (u, p) are the p-axis
and the line
These two lines intersect at the only equilibrium point E(UQ,PQ) with coordinates
Due to our choice of the f ( u ) shape the equilibrium point E is also the point of inflection of the isocline F(u,p,i) = 0. Depending on the magnitude of the parameter z, the former is either monotonically decreasing in u (for i < ic) or nonmonotonic of "N-shape" (for i > 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
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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ELECTRO-OSMOTIC OSCILLATIONS
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
216
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 <^C 1). In contrast to this, in the nonsymmetric case with /"(O) ^ 0, ic > ix, and the difference UQ — UA is finite already at the bifurcation i = ic. Recall that this conclusion is valid only in the limit e —> 0. More precisely, in the "nonsymmetric" case the amplitude of the stable relaxation limit cycle becomes finite already in the very close vicinity of the critical value ic of the parameter i with this vicinity shrinking to nil when e —»• 0. This would be true for both a supercritical bifurcation, due to its relaxation character, and a subcritical bifurcation, due to the very meaning of subcriticality. To elaborate somewhat on the above issues and mainly to pave the ground for the treatment of the PDEs .version of the Teorell model in the next section we conclude here with the standard weakly nonlinear analysis of the vicinity of bifurcation i — ic in the model (6.2.7). To this end, address the equation (6.2.7) with e
as
with Pk (t), ik, wk,k = 12.... to be determined in due courase. Acordingly, f(Pt) expands as
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ELECTRO-OSMOTIC OSCILLATIONS
Substitution of (6.2.27a-c) into (6.2.7) yields to the leading 0(1) order in r? (6.2.26b). Thus
Furthermore, to order 0(r?) the equation for p i ( t ) is
or, taking into account (6.2.26a)
Integration of (6.2.28b) yields
Here @i is an integration constant to be determined. Similarly, to order 0(r/ 2 ) we have
or, due to (6.2.26a)
The solvability condition for (6.2.29b) (elimination of secular terms in j>2) implies
Equation (6.2.29b) is rewritten, taking into account (6.2.29c) and (6.2.28c), as
Here
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Integration of (6.2.29d) yields
Here 02 is another integration constant. Put
This can be done without loss of generality because of the arbitrariness in the choice of the small expansion parameter /?. One way to rationalize this would be to say that we expand the solution and the control parameter in the vicinity of bifurcation in powers of the amplitude of the main harmonics eir. This amplitude is thus assumed fixed, so that all corrections to it coming from the solution of homogeneous equations at higher orders are dropped. Furthermore, to order O(rf] we have, taking into account (6.2.29c,d),
or, according to (6.2.26a)
The solvability condition for (6.2.30b) implies, if we take into account (6.2.29g,h),
Let us rewrite (6.2.30c), using (6.2.26a), (6.2.28d), as
Recall that
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ELECTRO-OSMOTIC OSCILLATIONS
Here, according to (6.2.29f,g),
From (6.2.32a), (6.2.32b) and (6.2.31a), we have
that is, the amplitude of the limit cycle is proportional to the square root of deviation of i from its critical (bifurcation) value. The bifurcation is supercritical when 7 > 0, i.e., by (6.2.31b) when /'"(O) < 0. On the other hand, whenever 7 < 0, that is, whenever /"'(O) > 0, the bifurcation is subcritical. It is clear from Fig. 6.2.4d that in the "symmetric" case (/"(O) = 0), /'"(O) < 0, and thus the corresponding bifurcation is supercritical in accordance with the qualitative analysis above. In the "nonsymmetric" case (/"(O) ^ 0), /'"(O) can generally be of either sign, and correspondingly the bifurcation can be of either type. Finally, we point out that the present weakly nonlinear treatment of the relaxation regime (e
clearly incompatible with the essence of the above perturbation analysis. To overcome this difficulty and to recover the above-mentioned peculiarities of the near bifurcational behaviour (inferred qualitatively in the "symmetric" and "nonsymmetric" cases), a perturbation procedure uniformly valid in £ ought to be developed. For an account of such a procedure, designed for treatment of singular Hopf bifurcation to relaxation oscillations in an ODEs system in a plane, see [11].
6.3. Generalized local Darcy's model of Teorell's oscillations (PDEs) [12]. In this section we formulate and study a local analogue of Teorell's model discussed previously. The main difference between the model to be discussed and the original one is the replacement of the ad hoc resistance relaxation equation (6.1.5) or (6.2.5) by a set of one-dimensional Nernst-Planck equations for locally electro-neutral convective electro-diffusion of ions across the filter (membrane). This filter is viewed as a homogenized aqueous porous medium, lacking any fixed charge and characterized
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by some effective ionic diffusivity D which is assumed for the sake of briefness equal for the cat- and anions, e.g., for K + and Cl~. Let the filter occupy the layer 0 < x < L. The equations of locally electro-neutral convective electro-diffusion of ions yield for the dimensional electrolyte concentration C(x, t) and the electric potential (p(x, t) in the filter
or, after adding and subtracting (6.3.la), (6.3.1b),
Equation (6.3.2b) yields further
Here / is the electric current density in the filter—the time-independent control parameter in the galvanostatic regime. The flow velocity in the filter v is again related to the pressure and the electric potential gradients via a generalized Darcy's law of the form
Below we shall need some order of magnitude estimates for the hydraulic permeability v and the electro-osmotic coefficient u>. Such estimates are provided by the expressions
Equation (6.3.3b) results from assuming a Poiseille flow in a filter's pore of typical radius r, ^ is the dynamic viscosity of the fluid. Equation (6.3.3c) is a common expression for the electro-osmotic coefficient [13], with d and C, respectively, the dielectric constant of the fluid and the C-potential of the pore wall. For the time being, we shall assume u) constant (independent of C(x,t)). Furthermore, by incompressibility of the flow
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ELECTRO-OSMOTIC OSCILLATIONS
The system (6.3.2a,c), (6.3.3a), (6.3.4) for
The conditions (6.3.5a,b) merely state that at the outer edges of the filter the electrolyte concentrations are assumed fixed equal to the bulk concentrations in the vessels I and II. Conditions (6.3.5c,d) fix the electric potential and the pressure at the left I edge of the filter at zero reference level. Condition (6.3.5e) is just a combination of (6.1.1), (6.1.2) of §6.1, N is the average number of pores per unit cross-sectional area of the filter. Introduce the dimensionless variables
The normalization factors VQ, PQ, to are chosen as follows. Substitution of (6.3.3a-c) into (6.3.5e) yields, taking into account (6.3.6),
Choose to so as to make the factor at pt in (6.3.7a) unity. Thus
In accordance with (6.3.7a) define the dimensionless electro-osmotic coefficient (jj as
Furthermore, from (6.3.5e)
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Choose again VQ so as to make the factor at pt in (6.3.7d) unity. This yields, taking into account (6.3.7b), the following relation between VQ and po
Finally, in order to fix PQ rewrite (6.3.2a) in terms of the dimensionless variables (6.3.6) as
Normalizing the Peclet number voto/L in (6.3.7f) to unity yields, taking into account (6.3.7b)
Equations (6.3.7c), (6.3.7g) yield for u
Finally let us define the normalized diffusivity and the electric current density as respectively,
In terms of dimensionless variables (6.3.6) with (6.3.7), (6.3.8) the system (6.3.2a,c), (6.3.3a), (6.3.4), (6.3.5) can finally be rewritten in the following way. Equations.
Boundary conditions.
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ELECTRO-OSMOTIC OSCILLATIONS
HereC 2 = C2/Ci. In order to estimate the dimensionless parameters D and u in (6.3.9), (6.3.11) assume the following reasonable values for the parameters of the system (for the filter we adopt values characteristic for the Nuclepore membrane employed in [7]-[9]).
For these parameter values (6.3.8a), (6.3.7h) yield D ~ 26, u; ~ 1 (to ~ 260 sec). Let us emphasize that as a result of scaling i with the characteristic "hydrostatic" time to ~ 260 sec long as compared with that of diffusional relaxation time L2/D ~ 10 sec, D emerges in the system (6.3.9)-(6.3.15) as a large parameter. We shall not specify here the initial conditions on C(x, t] since in what follows we shall only be preoccupied with the limit state resulting from a Hopf bifurcation from the following stationary solution of the above system
Here
(The stationary solution (6.3.16) is in this context for the PDEs system (6.3.9)-(6.3.15) what the equilibrium solution (6.2.12) was for the ODEs system (6.2.8)-(6.2.9).) Let us begin by analyzing linear stability of solution (6.3.16) and showing that at some value of parameter / a Hopf bifurcation is indeed occurring. According to the common scheme, we look for a perturbed solution of the form
Here U_ is the solution vector (v, (7,
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Substitution of (6.3.17) into (6.3.9)-(6.3.15) yields, after a linearization with respect to the perturbation and integration of (6.3.11), an eigenvalue problem for u, a of the form
Integration of (6.3.18a) yields
Here
Furthermore, from (6.3.18b-e), (6.3.16c), we get
On the other hand, by eliminating p ( l ) from (6.3.18c) via (6.3.18f), we get
Comparison of (6.3.20a) with (6.3.20b) finally yields, taking into account (6.3.19a,b), a single algebraic equation for s of the form
Recalling that D > 1, consider the limit D —* oc, \s —> 0 such that |
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ELECTRO-OSMOTIC OSCILLATIONS
Here
Prom (6.3.22a) we have
Thus with a > 0, the equilibrium UQ is stable as long as b > 0 and unstable for b < 0. At the bifurcation
that is, the bifurcation is indeed of the Hopf type. From (6.3.23b), b > 0 when / < Ic and b < 0 when 7 > Ic where
Finally, let us evaluate Imcr at the bifurcation / = Ic. From (6.3.23c), (6.3.22c), (6.3.25) we have
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and thus by (6.3.24b) we have
Here
Thus, to summarize, a Hopf bifurcation of the stationary solution w 0 , given by (6.3.16), occurs at
with a characteristic frequency at the bifurcation Im a of the order
Moreover, according to (6.3.19a), C(x) at the bifurcation scales as
By (6.3.20a-c) we have, furthermore
This information is used for the following formal weakly nonlinear analysis of the bifurcating solution in the system (6.3.9)-(6.3.15). Let us begin by rewriting this system in a more convenient form. Integration of (6.3.10) yields, taking into account (6.3.14),
Furthermore, integration of (6.3.11), (taking into account (6.3.12), (6.3.14), (6.3.15a)) yields, after differentiation in time and use of (6.3.15b), (6.3.30),
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ELECTRO-OSMOTIC OSCILLATIONS
Equations (6.3.31) and (6.3.9) with the boundary condition (6.3.13) are to be addressed by the weakly nonlinear analysis. Define a small parameter e as Introduce the new variables f , v(f), C ( x , f ) and the parameters J related to t, v(t), C(x,t), I, as suggested by the scaling (6.3.29),
Rewritten in terms o f f , T, v, C, J, (6.3.9), (6.3.13), (6.3.31) read
Let us further rewrite (6.3.35), by expanding the integrand into powers of e 2 , as
Seek C(x, T) in the form of an asymptotic series in powers of e
Substitution of (6.3.37) into (6.3.34a) yields, after integrating at the corresponding order in e and satisfying (6.3.34b),
CHAPTER 6 etc Here p
229
10 are the following polynomials in x
Prom here on let us mean by "'" a derivative with respect to T. Let us introduce finally
fj is a yet undetermined constant of order 0(1). A substitution of (6.3.38)-(6.3.40) into (6.3.36) yields, after some rearrangement, up to the order 0(e4)
230
ELECTRO-OSMOTIC OSCILLATIONS
Here
Let us seek J, fi. V(T) in the form of the following asymptotic series in £
(The absence of odd powers in the expansions (6.3.45), (6.346) for thecontrol parameter J and the frequency parameter uis suggested by common
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experience with weakly nonlinear treatment of Hopf bifurcations. Recall in this connection (6.2.29c,d).) Substitution of (6.3.45)-(6.3.47) into (6.3.41) yields to O(l) order in e
which implies
Equation (6.3.48b) is an exact equivalent of (6.3.25). Furthermore, to order O(e) we have
Choose
Then from (6.3.49a), we have
The amplitude (3 is to be determined from the subsequent higher-order analysis. To order 0(e 2 ) we have, taking into account (6.3.49b),
With VQ given by (6.3.50) the solvability condition for (6.3.51) reads
Then from (6.3.51), we have
Here
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ELECTRO-OSMOTIC OSCILLATIONS
and (3i is another integration constant to be determined from further analysis. The equation to the next O(e 3 ) order reads
The corresponding solvability condition yields
Accordingly
Here fa is still another unknown integration constant and
Note that since p as given by (6.3.53b) is purely imaginary, r and q ar real. Finally, the order O(e4} equation is
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The solvability condition for (6.3.58) is as follows
Here A, B, C are real, defined by the following expressions
It may be assumed without loss of generality that both J3 and /?i are real. Then the real part of the equality (6.3.59a) implies
whereas the appropriate imaginary part yields
Equality (6.3.60b) in turn yields, taking into account (6.3.59b), (6.3.55),
Here
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ELECTRO-OSMOTIC OSCILLATIONS
Fig. 6.3.1. M dependence on A.
Equation (6.3.61) represents the main finding of this section. Recall, referring to (6.3.45), that (6.3.61a) implies that
with J0, J2, J* given by (6.3.48b), (6.3.52), is the critical value of the control parameter J. Accordingly, the bifurcation at J = Jc is subcritical when M < 0 and supercritical when M > 0. We observe from (6.3.61b), (6.3.48b), (6.3.49b), (6.3.57b), (6.3.54) that M is proportional to u. Furthermore, we observe from (6.3.61b,c), (6.3.57b), (6.3.54), (6.3.52), (6.3.49b), (6.3.48b), (6.3.42)-(6.3.44), (6.3.39)_that M,J0, J-z,J* —> oo when the other parameter of the basic system A = Ci -1 —> 0. We also observe that M grows linearly in A when A —> oo. For small and medium values of A, M is worked out numerically from (6.3.61b). In Fig. 6.3.1 we plot M as a function of A in the range 0 < A < 17 for w = 1. Due to proportionality, M corresponding to any other w is inferred from these data just by rescaling.
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We thus observe that M is positive for all w, A > 0 and, therefore, the bifurcation is supercritical. This stands in contrast with the result for the classical Teorell model discussed in the previous section. Recall that there the type of bifurcation depended on the properties of the postulated stationary resistance function f ( v ) , specifically on the sign of /'"(O) whenever /"(O) / 0. To stress this difference between the local and the classical Teorell's model let us work out the overall stationary resistance of the filter R(v, A, D} for a given time-independent flow rate v. We have analogously to (6.3.30)
Here C(x,v,D] is a solution of the stationary b.v.p.
Prom here we get
and thus by (6.3.63) we have
From (6.3.65a) we generally have
whereas
may be of either sign, depending on the value of A, as can be seen from the #£,(0, A, 1), #"™(0> A, 1) plots in Fig. 6.3.2. Thus with
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ELECTRO-OSMOTIC OSCILLATIONS
Fig. 6.3.2. R"v(o,A,i), #^ v (o,A,i) dependence on A.
the classical model would indeed predict the Hopf bifurcation of either type where the local model yields a strictly supercritical bifurcation. Another difference between the predictions of the two models is as follows. The classical model yields a finite value for the critical current when the relaxation parameter k tends to infinity (Ic — O(l), k —> oo). In contrast to this, the local model predicts
(Recall that D is the "local analogue" of k.) This discrepancy results from the oversimplification of the classical model which assumes the stationary
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resistance f ( v ) is independent of k. In fact, as reflected in particular in (6.3.64c), (6.3.65), with v = 0(1) and k —>• oc (D —* oo) the concentration profile within the filter becomes linear and thus independent of the flow rate, and so does the stationary resistance. This should make the transition to oscillations impossible for I — O(l). 6.4. Electro-osmosis in a capillary. In the local model of the previous section we assumed for the flow rate v the phenomenological generalized Darcy's law (6.3.3a) with constant coefficients. It is clear that at least one of them, the electro-osmotic factor, could not in fact be constant but rather should depend on local concentration C(x,i) through the respective dependence of the local Debye length in the filter. To elaborate somewhat on this point, namely, to work out the concentration dependence of u and to elucidate the actual contents of the phenomenological law (6.3.3a) let us consider in some detail convective electro-diffusion in a single capillary, .connecting two vessels with different concentrations C\ and C"2 of a univalent electrolyte and with a pressure drop p' and some voltage applied along the capillary. It will follow from our analysis that with a potential or concentration gradient present in the filter no mechanical equilibrium is possible within the pores. More precisely, even with a zero total flow discharge through the filter an intense electro-osmotic circulation is bound to be present in an individual pore. Thus, these are precisely the terms of total discharge in which a generalized Darcy's law of the type (6.3.3a) should be understood. This stands in complete accord with the commonly adopted view of electro-osmosis, so that the only novelty of the presentation to follow is in accents, such as stressing the concentration gradient-related effects, showing the relevance of the problematics involved to the performance of composite ion-exchange membranes, etc. For simplicity we restrict our discussion to a thin two-dimensional channel of a half width H which is much shorter than its length L, H -C L, and assume once again equal ionic diffusivities. Let the channel's wall be charged with a surface charge density a. Direct x along the symmetry axis of the capillary, with y directed across the channel and the origin at the middle of its left edge (see Fig. 6.4.1). Let us employ the following notation: w(i,y,i) = (u(x,y,t), w ( x , y , t ) J for the vector of fluid velocity in the capillary, C+(x, y, t), C~(x, y, t) for, respectively, the cation and anion concentration in the capillary andp(x, y, t),
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ELECTRO-OSMOTIC OSCILLATIONS
Fig. 6.4.1. Scheme of electro-osmotic circulation in a capillary.
Recall that in the Navier-Stokes equation (6.4.1) m is the fluid density, assumed constant further on, whereas p is the electric charge density related as usual to the ionic concentrations as
and to the electric potential
Boundary conditions at the channel walls are
The boundary conditions (6.4.7), (6.4.8) stand for the non-slip of the fluid at the walls and impermeability of the walls for ions, respectively. The electrostatic (6.4.9), exact for a metal wall, is adopted here as the simplest physically meaningful condition corresponding to a given surface charge density of the wall (n in (6.4.9) stands for a unity outer normal). It will
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be clear in a moment that the results of the analysis below depend very little on the precise form of the electrostatic boundary conditions (6.4.9). We shall not specify at this stage the boundary conditions at the ends of the channel. Using (6.4.6), equations (6.4.1), (6.4.2) can be rewritten for the velocity components as
Introduce the following natural scaling
The characteristic velocity v0 in (6.4.15) is chosen from the balance of the pressure and the viscous terms in (6.4.10) (the first and the third terms in the right-hand side of (6.4.10)) as
Accordingly, as a characteristic time to in (6.4.13c), we chose the residence time
For 10 < p' < 10 3 g/cm-sec 2 , L = 10-2cm, H = 10-4cm (6.4.16), (6.4.17) yield 10"3 < v0 < 10"1 cm/sec, 10"1
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ELECTRO-OSMOTIC OSCILLATIONS
Here
With p', H, L above 1(T7 < a < 1CT5, 1(T5 < Re < 1(T3, S = 1(T2, 10~2 < /3 < 1, 102 < D < 104, e ~ 10"4. Due to the smallness of a, Re, in what follows we shall omit the inertial terms in (6.4.18), (6.4.19).
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Since the electro-osmotic flow is induced by the interaction of the externally applied electric field with the space charge of the diffuse electric double layers at the channel walls, we shall concentrate in our further analysis on one of these O(e1/'2) thick boundary layers, say, for definiteness, at !/=-!. Let us fix the parameters 6, /?, D and define the inner variable z as
Denote by u(x, z), w_(x,z), p ( x , z ) , C_ ( x , z ) , (p(x,z) and u(x,y), w(x,y), p(x, y), C (x, y ) , ^(or, y), respectively, the leading order terms of the inner and outer asymptotic expansions of the appropriate variables in powers of e. Equations (6.4.18)-(6.4.26) yield for u, w_, p, C ± ,
The inner variables u(x,z), w(x,z), p(x,z),
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ELECTRO-OSMOTIC OSCILLATIONS
and thus from (6.4.31) that
or, after matching,
Substitution of (6.4.37b) into (6.4.31) yields to leading order
Equations (6.4.37), (6.4.38) imply physically that the flow in a "thin" electric double layer (e
Here
Equations (6.4.33), (6.4.39a,b) together with (6.4.36) and matching conditions yield for
or, after the integration
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Here
Finally, (6.4.41) is substituted into (6.4.38) and the integration is carried out with the boundary condition (6.4.34) and the matching condition
To simplify the calculation we shall do this in the Debye-Hiickel limit a —> 0. In this limit (6.4.41) yields
Equations (6.4.38), (6.4.41c) yield after the integration, taking into account (6.4.34),
and A is an integration constant to be chosen so as to enable matchin; (6.4.42). This matching yields
Equations (6.4.43a-c) yield the central result of this section—the following expression for the electro-osmotic slip velocity us under an applied potential and concentration gradient, in the Debye-Hiickel approximation for a "thin" double layer
The first term in the right-hand side of (6.4.44) represents the common electro-osmotic contribution whereas the second term stands for what could be termed the "negative" osmotic effect. Note that the pressure term is absent from (6.4.44). We are now in a position to write down the b.v.p. for the electro-osmosis in a channel in terms of the outer variables. Equations (6.4.18)-(6.4.26)
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ELECTRO-OSMOTIC OSCILLATIONS
yield to leading order in e, taking into account matching (hereon we omit overbars on the outer variables). Equations:
Boundary conditions at the channel's walls:
As a possible set of boundary conditions at the edges of the channel in a galvanostatic regime we may choose
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Once again we shall not specify the initial conditions, and in fact we shall not address the b.v.p. (6.4.45)^(6.4.55) at any length at all. The only point about it to be made here is as follows. It stems from (6.4.45)^(6.4.55) that for a given arbitrary concentration or voltage drop along the capillary no mechanical equilibrium is possible in it at any pressure drop. This is due to the absence of a pressure term from the expression for the slip velocity (6.4.51). Indeed, the slip induced by tangential concentration or potential gradients at the walls cannot be balanced by any pressure gradient acting in the bulk of the fluid. The best such pressure gradient can do is to provide for a zero total discharge through the channel. This would correspond to a circulation in the channel, with the fluid dragged along the wall by the electro-osmotic slip and returned along the central plane with a pressure driven backflow (see Fig. 6.4.1 for a schematic illustration). Note that, as is easily observed from b.v.p.(6.4.45)-(6.4.55), with a concentration gradient present no unidirectional developed Poiseuille-type channel flow is compatible with the boundary conditions (6.4.51). A fairly complicated two-dimensional flow pattern is thus generally expected even away from the edges of the channel. The appropriate rigorous flow calculation is still to be done. Here we shall content ourselves with the following crude order of magnitude estimate of the contribution of the above-mentioned circulation to the solute transport through the channel. Assume the following realistic values for the voltage V, concentration drop CilCi and the channel's wall (() potential
By (6.4.14), (6.4.41c) this yields in dimensionless terms, assuming roughly constant concentration and potential gradients along the channel 0yV 2Co ^ 4, V0x ~ 10, Cos/Co ~ 1 or by (6.4.44)
Assume, for the sake of this estimate only, a developed channel flow with the slip velocity (6.4.57). The corresponding velocity profile is
Choose 0(p') corresponding to a vanishing total discharge through the channel,
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ELECTRO-OSMOTIC OSCILLATIONS
From (6.4.58) we get
The velocity in the channel thus changes sign at
Estimate the total convective solute discharge Qc through the channel as
or, in dimensional terms, according to (6.4.16), (6.4.27d)
On the other hand, the total dimensional diffusional flux through the channel QD can be estimated as
Thus the relative contribution of the two mechanisms—convective-circulational and diffusional—may be finally estimated as
for the above reasonable choice of parameters. Two main conclusions are suggested by the above simple consideration. First, the solute transport in a porous filter (membrane) separating two solutions at different concentrations or electric potentials is likely to be dominated by electro-osmotic circulation as compared to molecular electrodiffusion in the pores. An accurate calculation of the circulation seems desirable. In particular, the observations upon the highly performing composite heterogeneous ion-exchange membrane [13], formed by casting a thin
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dense permselective layer upon a weakly charged porous support, make relevant a calculation of the above circulation in a dead-ended channel (dead-ended pore of the support). For a porous-capillary medium without bulk ionic concentration variations the transport (dispersion) effect of electro-osmotic circulation has been treated by Pismen and Babchin [14]. Second, we conclude that the generalized Darcy's law with constant coefficients represents a crude idealization in the following three respects. 1. No mechanical equilibrium is possible in a single pore with a thin double layer. The equilibrium of previous sections actually corresponds to an electro-osmotic circulation confined to a pore. In this case moving away from equilibrium corresponds to this circulation's breaking out of the capillary. 2. The "negative" osmosis term, proportional to the concentration gradient is expected to be of the same order of magnitude as the conventional electro-osmotic term and thus should be included in consideration. 3. The concentration dependence of the appropriate coefficients, as prescribed by (6.4.44), should be accounted for. This would make it a relevant task to rework the computations of §§6.2 and 6.3, taking into account the above factors. 6.5. Unsolved problems. 1. Bifurcation analysis in the local Teorell model accounting for "negative" osmosis and concentration dependence of the electro-osmotic factor as prescribed by (6.4.44). The analysis should be essentially identical to that of §6.3 with the generalized Darcy's law (6.3.11) replaced by the expression
with u> and a being positive constants. 2. Stationary electro-osmosis in a dead-ended pore adjacent to a dense permselective surface. The time-independent version of the b.v.p. (6.4.45)-(6.4.55) with the boundary conditions (6.4.55a-d) at x = 1 replaced by
Here the constant / once more stands for the DC current density, given in the galvanostatic regime. 3. Stationary electro-convection at an electrically inhomogeneous permselective membrane 7 Once again the time-independent version of (6.4.45)(6.4.49) with the boundary conditions (6.4.54a,b) at x = 0. 7 Asymptotic aspects of this problem have been treated recently in [15].
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ELECTRO-OSMOTIC OSCILLATIONS
The side wall conditions (6.4.51)-(6.4.53) are replaced by the symmetry conditions
The boundary conditions at x = I are replaced by those of the nonslip for the velocity and by the transport conditions at the electrically inhomogeneous surface for the electrolyte concentration and the electric potential of the form
This formulation is the electro-convective version of the transport problem (4.4.54a-d). A three-dimensional physically relevant generalization is straightforward. Problems 2 and 3 are of direct relevance for an adequate understanding of concentration polarization at, respectively, composite heterogeneous and "homogeneous" permselective membranes. The main difference between these formulations is that in Problem 2, relevant for a composite heterogeneous membrane, the motion in a pore of the support is induced by the electro-osmotic slip due to the interaction of the applied electric field with the space charge of the electric double layer which is present already at, equilibrium. In contrast to this, with a "homogeneous" membrane corresponding to Problem 3, the motion in a symmetry cell of the liquid boundary layer, adjacent to an electrically inhomogeneous membrane, is induced by the electric field interaction with an essentially nonequilibrium space charge, formed only in the course of the ionic transport itself. For an account of the corresponding transport problem see §4.4.
REFERENCES [1] T. Teorell, Transport processes in membranes in relation to the nerve mechanism, Exp. Cell Res., 5 (1958), pp. 83-100. [2] , Electrokinetic membrane of excitable tissues. I. Experiments on oscillatory transport phenomena in artificial membranes, J. Gen. Physiol., 42 (1959), pp. 831845. [3] , Electrokinetic membrane processes in relation to properties of excitable tissues. II. Some theoretical considerations, J. Gen. Physiol., 42 (1959), pp. 847-863. [4] , Oscillatory electrophoresis in ion exchange membranes, Ark. for Kemi, 18 (1961), pp. 401-408.
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[5] Y. Kobatake and H. Fujita, Flows through charged membranes. I. Flip-flop current vs voltage relation, J. Chem. Phys., 40 (1964), pp. 2212-2218. [6] , Flows through charged membranes. II. Oscillation phenomena, J. Chem. Phys., 40 (1964), pp. 2219-2222. [7] P. Meares and K. R. Page, Rapid force—flux transitions in highly porous membranes, Phil. Trans. Roy. Soc. London, 272 (1972), pp. 1-46. [8] P. Meares and K. R. Page, Oscillatory fluxes in highly porous membranes, Proc. Roy. Soc. London Sect. A, 339 (1974), pp. 513-532. [9] K. R. Page and P. Meares, Factors controlling the frequency and amplitude of the Teorell oscillator, Faraday Symp. Chem. Soc., 9 (1974), pp. 166-173. [10] A. A. Andronov, E. A. Leontovich, I. I. Gordon, and A. G. Mayer, Qualitative Theory of Dynamical Systems, Nauka, Moscow 1966, pp. 228(a), 238(b). (In Russian.) [11] S. M. Baer and T. Erneux, Singular Hopf bifurcation to relaxation oscillations, SIAM J. Appl. Math., 46 (1986), pp. 721-739. [12] T. Erneux and I. Rubinstein, Hopf bifurcation in a local model of Teorell oscillations, to appear. [13] I. Rubinstein, E. Staude and O. Kedem, Role of the membrane surface in concentration polarization at ion exchange membrane, Desalination, 69 (1988), p. 101. [14] L. M. Pismen and A. J. Babchin. The electro-osmotic enhancement of the neutral tracer transfer through a capillary membrane, J. Coll. Interf. Sci., 62 (1977), pp. 63-68. [15] I. Rubinstein, Electro-convection at an electrically inhomogeneous permselective interface, to appear.
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Index
Activity coefficient, 41, 141 Adsorption, 63 Backflow, 245 Backlund transformation, 65 Bendixon test, 211, 213 Bifurcation, 125, 157, 213, 220, 235 analysis, 125, 247 Hopf, 213, 224 singular, 220 sub critical, 216, 220 supercritical, 216, 220, 235, 236 Boltzmann distribution, 174 equilibrium, 38 Boundary layer, 86 Bulk concentration, 135 region, 14 C-membrane, 136 Cauchy problem, 65, 68, 81, 82, 100 Cell model, 40 Charge density effective, 51 linear, 24, 37-39, 51 structural, 54 surface, 30, 238 Circulation, 246, 247 Colloid stability, 23, 30 Comparison theorem, 26
251
Concentration loading, 80 profiles, 153 typical, 6 Concentration polarization, 133, 137, 139, 148, 152, 170, 177, 185, 201, 248 Conductivity, electric, 6 Convection, 7, 155 Convergence, quadratic, 97 Counterion, 2, 24, 39-41 capacity, 3 condensation, 24, 36, 38 selectivity, 139 Current continuity, 17 convection, 5 critical, 209 diffusion, 5 displacement, 5, 163 fluctuation, 136 limiting, 121, 132, 135, 136, 146, 152,153 saturation, 119, 125, 157
Darcy law, 237 Debye length, 3, 8, 13, 25, 107, 133, 161, 163, 177, 237 Debye-Hvickel equation, 13, 25 limit, 243 Depletion, 147
252 compartments, 2 interface, 119, 124 layer, 129, 141, 147 Diffusion ambipolar, 16 fast, 64 layer, 7, 139, 157 slow, 64 Diffusivity, 4 effective, 80, 81, 165 ionic, 4 tensor, 63 typical, 6 Diode, 20 Dirichlet problem, 27 Discharge, 246 Dissociation, 38 Donnan equilibrium, 13 potential, 13 Doping function, 20 layer, 114 level, 114, 194 Double layer, electric, 172, 241, 242, 248 thickness, 13 equilibrium, 175, 177 potential, 13
Electro-convection, 7, 18, 136, 203, 247 Electro-diffusion ambipolar, 111 convective, 237 Electro-neutrality approximation, 59, 133, 135 local, 1, 8, 10, 13, 60, 106, 107 Electro-osmotic, 7, 18 circulation, 237, 246, 247 coefficient, 221 factor, 247 flow, 241 oscillation, 203, 207 slip, 243, 245, 248 Electrodialysis cell, 3, 11 Electrolyte, 20 added, 49 binary, 98
INDEX low molecular, added, 41 solution, 1, 20 supporting, 136 symmetric, 26, 38 Electron, 20 Enrichment layer, 129 Equilibrium, 14, 23, 139, 209, 248 constant, 79 electro-diffusional, 23 ionic, 1, 98 local, 9, 13, 79, 175 mechanical, 136, 155, 237 point, 209, 213 solution, 200 potential, 13 electrochemical, 19 Excluded volume equation, 54
Fixed charge, 2, 5, 38, 60, 162 density, 5, 11, 13, 20, 106, 108, 112 Flux diffusional, 9 ionic, 3, 14 ionic, modified, 108 ionic, stationary, 14 ionic, true, 108 migration, 9 Force electric, 5, 6, 19, 25, 30 gravity, 6 potential, 19 saturation, 36 tangential, electric, 242 volume, 6, 154 Free energy, 52 Frequency, 186 Front formation, 78, 87 Gui-Chapman equation, 13 Henderson solution, 165 Hierarchy, electro-diffusional, 18 Hole, 20 Inhomogeneity, electric, 136, 147
253
INDEX Interface, 11 concentration, 152, 153 discontinuity, 10 potential jump, 13 Ion-exchange, 64 beads, 3 bed, 3 binary, 63, 78 column, 3 matrix, 3, 79 reactive, 60, 78 Ion-exchanger, 1, 20 anion, 20 cation, 20 Isocline, 209, 213
Modulation amplitude, 192 frequency, 186
Junction
Ohm's law, 5 Order parameter, 52, 53 Osmotic effect, 243
p-n, 162
liquid, 161, 163 liquid, potential, 162
Langmuir equilibrium, local, 79 isotherm, 63 Limit cycle, 211, 213, 220 stable, 216
Matrix, 2 polyanion, 2 Maximum principle, 26, 29, 45, 73 Mean field, 41 Membrane, 2, 98 anion-exchange, 3 anion-selective, 2 bipolar, 20, 162 cation-exchange, 3 cation-selective, 2 composite, 248 composite, heterogeneous, 246 homogeneous, 153, 248 inhomogeneous, 155 ion-exchange, 1, 3 multipolar, 105, 111 potential, 63, 98 quadrupolar, 20, 106 thickness, 3 Migration, 7, 8 Mobility, ionic, 4
Navier-Stokes equation, 4, 5 Nernst film, 7 layer, 147 Nernst-Planck equation, 108, 207 Newton iterates, 78, 96, 130 method, 96 Newton-Kantorovitch theorem, 118
Peclet number, 7, 223 Permeability hydraulic, 221 ionic, 16 Permselectivity, 14 ideal, 140, 150, 181 Perturbation, 61 singular, 86 Phase flow, 213 integral, 40 plane, 211, 213 transition, 38, 39, 51, 53 Picard's method, 97 Poiseille flow, 221 Poisson equation, 4, 154 nonlinear, 26 Poisson-Boltzmann equation, 18, 19, 23, 25, 37, 38, 40 Polarization curve, 136 Polarography, 135 Polyelectrolyte, 1, 37 core, 37 cross-linked, 2 linear, 40 molecule, 25 solution, 2, 38 Polymer
254 chain, 38 core, 38 film, 2 Pore aqueous, 3 radius, 3 Porous medium equation, 63, 64 Potential (0, 245 diffusion, 18 drop, Ohmic, 18 electric, 10 electrochemical, 10, 11, 98, 134 standard, 141 surface, 39 Pressure, 5, 32, 33, 221, 222, 237, 239 force, 33 hydrostatic, 203 Punch through, 108, 115, 178, 180
INDEX nonlinear, 25 Selectivity, residual, 146 Semiconductor, 20, 105 Shock, 86, 87 Similarity solution, 65, 67, 81 transformation, 68 variable, 65, 66, 68, 83 Singularity, 24, 37-39, 41 limiting, 46, 49 logarithmic, 38 Size, ionic, 19 Slip velocity, 245 Space charge, 4, 8, 25, 133, 154, 157, 162, 170, 241, 248 density, 32, 170, 180 front, 161 nonequilibrium, 161, 162, 169, 194, 248 Stefan problem diffusional, 78 one-phase, 81
Quasilinearization, 96, 180 Rankine-Hugoniot condition, 77 Rate constant, 206 Rayleigh equation, 208 Rectification, 185, 188, 193 anomalous, 161, 185 Relaxation law, 206 oscillation, 205 parameter, 208 Repulsion force, electric, 34 Resistance differential, negative, 113, 119 function, 208 instantaneous, 206 stationary, 206, 235
Saturation, 30, 39, 129, 146 field, 24 force, 24 Screening linear, 25, 30
Teorell oscillation, 207, 247 Teorell-Meyer-Sievers model, 98 Thyristor, 20, 111, 112, 114 Transition layer, 37, 82, 198, 199 solution, 199, 200 Turning point, 113, 114, 121 Unstirred layer, 147-149 thickness, 7 Valency, 40 Velocity characteristic, 7, 239 volumetric, 205, 206 Waiting time, 64 Wave equation, 18 monotonic, travelling, 65, 69 thermal, propagation, 64
