Intensiﬁcation of electrodialysis by applying a non-stationary …hera.ugr.es/doi/15090802.pdf · 2006. 7. 18. · Colloids and Surfaces A: Physicochemical and Engineering Aspects

Colloids and Surfaces

A: Physicochemical and Engineering Aspects 176 (2001) 195–212

Intensification of electrodialysis by applying anon-stationary electric field

N.A. Mishchuk a, L.K. Koopal b,* , F. Gonzalez-Caballero c

a Institute of Colloid and Water Chemistry of Ukrainian National Academy of Sciences, 42, Vernadsky pr., Kyi6,252680, Ukraine

b Laboratory of Physical Chemistry and Colloid Science, Wageningen Agricultural Uni6ersity, Dreijenplein 6,6703 HB Wageningen, The Netherlands

c Department of Applied Physics, Uni6ersity Granada, Fuentenue6a s/n, ES-18071, Granada, Spain

Received 13 December 1997; accepted 14 March 2000

Abstract

Electrodialysis is an effective method of water desalination. However, the efficiency of the method is limited by theconcentration polarization of the ion-exchange membranes. Applying an electric field with special pulse characteristicscan diminish the effect of the concentration polarization. A theoretical analysis is carried out for galvanostatic andpotentiostatic pulse regimes. The time dependencies of the extent of the concentration polarization near themembrane surface during the pulse are described theoretically for both pulse regimes and a qualitative discussion ofthe pause duration is presented. The main characteristic of the non-stationary process is the transition time betweenthe state without polarization and the state with stationary polarization. In principle, the electrodialysis process canbe intensified significantly when the applied pulse is sufficiently smaller than this transition time. Experimental resultsare quoted that qualitatively support the model predictions. It is shown that the desalination can be intensified severaltimes, depending on the pulse-pause characteristics. © 2001 Elsevier Science B.V. All rights reserved.

Keywords: Concentration polarization modeling; Desalination; Electrodialysis; Galvanostatic regime; Intensification of electrodialy-sis; Limiting current; Non-stationary electric field; Overlimiting current; Potentiostatic regime; Sublimiting current

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1. Introduction

Electrodialysis as a method of demineralisationof natural and wastewater is ecologically expedi-ent and economically profitable. However, the useof ion-selective membranes is complicated by the

phenomenon of concentration polarization, whichis caused by a difference between the ion transfernumbers in the solution and in the membranes [1].The effect is connected with a decrease of theelectrolyte concentration in a thin layer near themembrane surface in the desalination cell, and itcan be observed in the form of a retardation ofthe current growth when the applied voltage isincreased. In the absence of thermal and hydrody-namic turbulence of the liquid flow, and of the

* Corresponding author. Tel.: +31-317-482278; fax: +31-317-483777.

E-mail address: [email protected] (L.K. Koopal)

0927-7757/01/$ - see front matter © 2001 Elsevier Science B.V. All rights reserved.

PII: S 0 9 2 7 - 7 7 5 7 ( 0 0 ) 0 0 5 6 8 - 9

N.A. Mishchuk et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 176 (2001) 195–212196

dissociation of water, the limiting current ilim canbe expressed as follows [1–6]:

ilim=2FD+C0

L(1)

where F is Faraday’s constant, D+ is the diffusioncoefficient of counterions (for example, cations),C0 is the electrolyte concentration and L is thethickness of the concentration polarization region,that is called the Nernst layer. The limiting valueof the current is rather low, especially when elec-trolytes with millimolar and lower concentrationsare used.

Mass transfer through a membrane, and there-fore the electrodialyser capacity, are directly pro-portional to the current density in anelectromembrane channel. Consequently the limit-ing current restricts an electrodialyser capacity.When the voltage drop over the membrane risesabove the value providing the limiting current,unproductive expenditure of power takes place.Therefore, an investigation of the possibilities toincrease the current density above the limitingvalue is important.

Traditional methods to prevent current satura-tion are based on extending the membrane surfaceand on promotion of turbulence in the electrodi-alyser channel. The latter can be achieved bymeans of membrane profiling [7,8], turbulencepromoters [9], conducting spacers [10] and airsparing [11]. However, all these methods are char-acterized by a common imperfection: due to vis-cous tension the velocity of liquid motiondecreases sharply on approaching a solid surface.Therefore, these methods lead only to moderateincreases in current density.

Investigations of polarization processes in sta-tionary and non-stationary regimes [12–16] sug-gest that a non-stationary regime can lead to aweakening of concentration polarization and toan intensification of electrodialysis. One way torealize a non-stationary regime is to use pulses ofcurrent or voltage that alternate with pauses. Thistype of non-stationarity will be explored in detailin this paper.

For the analysis of the optimal conditions of apulse regime it is important to consider both thepulse and the pause time. The characteristic time

of a pulse has to be smaller than the transitiontime. This is the time required to build up theconcentration polarization layer near a mem-brane. The pause duration has to be comparablewith the duration of pulse in order to enable theconcentration profiles to be restored to their ini-tial state (without polarization). However, thepause should not to be too large, otherwise diffu-sion and osmotic processes through the mem-brane counteract the desalination.

The theoretical background of both stationaryand non-stationary electrodialysis processes willbe considered first in order to gain insight into thepulse and pause duration. In the second part ofthe paper the theoretical predictions will be com-pared with experimental results, obtained by elec-tro-filtration and electrodialysis.

2. Theory

2.1. General considerations

The concentration polarization of a membranewill be investigated for an ideally selective mem-brane, for which the flux of coions through amembrane is completely absent. The treatment ofthe stationary regime of concentration polariza-tion has been analyzed theoretically in Refs. [2–5]. That of the non-stationary concentrationregime will be based on the Nernst model [17–20].In the Nernst model a practically motionless layerof liquid, through which the ions diffuse to thesurface, is situated near the electrode or mem-brane surface.

The treatment will be limited to low electrolyteconcentrations (C0B10−2 mol l−1) because athigher electrolyte concentrations, heat processeslead to thermoconvection and modification of thestructure of the polarized layer [21].

Water dissociation that usually accompaniesthe electrodialysis at low electrolyte concentration[16,21–26] can, however, be neglected because itdecreases strongly when a pulse regime is applied.

The processes will be analyzed only at the sideof a membrane where in a stationary regime theelectrolyte concentration decreases. On this sidethe main potential drop occurs that limits the

N.A. Mishchuk et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 176 (2001) 195–212 197

electrodialysis capacity [2–6]. An exact descrip-tion of the polarization processes is very compli-cated. It would be necessary to take into account:(1) the non-stationarity of Nernst layer in thedesalination channel, (2) the non-stationarity inthe concentration channel, (3) the concentrationprofiles inside the membrane, (4) the interfacialpotential drop (Donnan potential) and (5) thepolarization of the electrodes. For the moment itwill be assumed that an analysis of the non-sta-tionarity of the Nernst layer in the desalinationchannel is sufficient to obtain the correct order ofmagnitude of the time scale of the polarizationprocesses.

2.2. Stationary regime

In order to be able to analyze qualitatively andquantitatively the differences between stationaryand non-stationary concentration polarization,first the structure of the stationary Nernst layerand its characteristics in weak, moderate andstrong electric fields will be discussed.

In Fig. 1, the structure of a stationary Nernstlayer with thickness L is shown schematically for

three values of the electric field. Switching on theelectric field leads to a decrease of the electrolyteconcentration, C(x), from the value C0 in thebulk of the electrolyte solution to some value C(0)near the membrane surface (more exactly, on theexternal boundary of electrical double layer). Atlow values of the electric field, the value C(0) isessentially greater than zero and the concentrationprofiles of both ions of the 1-1 electrolyte coincide(Fig. 1, curves 1). Under this condition, the cur-rent increases linearly with the potential growth.This regime is usually called ‘sublimiting’.

When the value of C(0) is close to zero, bothion concentrations in the Nernst layer decreaselinearly to zero (Fig. 1, curves 2) and the currentdensity approximates its limiting value ilim. Theelectric field range corresponding with this situa-tion is called the ‘limiting’ regime. For both thesublimiting and the limiting regime the electroneu-trality behind the electrical double layer of themembrane is conserved.

Further growth of the electric field leads to theappearance of an extended region of low ionconcentrations and to a deviation from the elec-troneutrality, because the concentration of coun-terions, Cs, is substantially higher than that of thecoions, which is negligibly small (Fig. 1, curves 3).This range of the electric field is called ‘overlimit-ing’. The polarization processes in these strongelectric fields have been analyzed for flat [2–6]and for curved [27,28] surfaces. It has been shownthat the charge density or the counterion concen-tration Cs and the thickness S0 of the inducedcharge layer depend on the bulk electrolyte con-centration, the current or the voltage and thethickness of the Nernst layer. The conductivity inthis region is Cs/C0 times lower than in the bulksolution (or inside this layer without concentra-tion polarization). The thickness of the diffusionpart of the Nernst layer d=L−S0 decreases withthe expansion of the induced space charge.

The term ‘overlimiting’ for this regime pointstowards to two different aspects. On the one handit takes into account that the electrodialyserworks in the regime where the limiting current isreached. So, a voltage is applied above a valuethat can lead to an increase of the current. Thiscauses an excessive expenditure of power. On the

Fig. 1. Schematic representation of the concentration polariza-tion of an ion-exchange membrane in the sublimiting (1), thelimiting (2) and the overlimiting (3) regime. The drawn curvesrepresent the counterion concentrations, the dashed curves thecoions, x is the distance from the membrane surface, S0 and C0 s

the thickness and density of induced space charge.


Fig. 2. The current in the stationary regime as a function ofthe potential drop across the Nernst layer for different magni-tudes of the concentration polarization as expressed by theparameter g =k−1/L. Without concentration polarization (1);limiting current for g�0 (2); overlimiting regime at g=10−4

(3); 10−3 (4); 10−2 (5).

be higher than 10−4–10−3 which means that thecurrent is very close to the limiting value.

The current value in the overlimiting regime isconnected to the structure of the polarizationregion. When the diffusion layer is formed inweak or moderate electric fields (Fig. 1, curves 1and 2), the current through the Nernst layer isgiven by diffusion flow and the nature of currentleads to Eq. (1) for the limiting current. When theinduced space charge appears, the nature of thecurrent is more complicated. In the region of theinduced space charge the counterion concentra-tion is approximately constant and the current isprovided by electromigration (almost withoutdiffusion).

In the diffusion part of the Nernst layer, thecurrent maintains its diffusion nature. Owing tothe continuity of the ion flux, the current shouldhave the same value in all sections of the channeland can be described as [27,28]

ist=2FD+C0

d=

2FD+C0

L−S0

= ilimL

L−S0

(2a)

or

i0 st=ist

ilim=

1d0 =

11−S0 0

(2b)

where d0 =d/L, and S0 0=S0/L. According to Eqs.(2a) and (2b) the current density is L/(L−S0)times higher than the limiting one. For example,the concentration profiles shown by curves 3 inFig. 1 correspond to the situation that i0 st= ist/ilim=2.

However, the current density cannot be muchlarger than ilim, because the decrease of the diffu-sion layer thickness is accompanied by a verystrong growth of the potential drop over theregion of the low electrolyte concentration. Forexample, the limiting current i0 st=1 can bereached by the voltage 8:0.25 V or 8̃:10,whereas the overlimiting value i0 st=2 for g=10−3

is realized at 8:10.5 V or 8̃:420.In the hypothetical absence of the concentra-

tion polarization the current growth is propor-tional to the voltage and i0 st=2 can be reached bya voltage of only 8:0.5 V or 8̃:10. So, fori0 st=2 only 0.5 V (4.7% of 10.5 V) is used effec-tively. The situation changes for the worse withan increase of i0 st and a decrease of g.

other hand, the term takes into account that thecurrent in the overlimiting regime is slightlyhigher than the limiting current.

Different possibilities for the dependence of thecurrent, ist, in the stationary regime (subscript ‘st’indicates the stationary regime) on the voltage, 8,are presented in the Fig. 2. The current ist and thevoltage 8 are expressed in the dimensionlessquantities i0 st= ist/ilim; 8̃=F8/RT, respectively(where R is the gas constant and T is the absolutetemperature). In the hypothetical absence of con-centration polarization the current grows linearlywith the voltage increase (curve 1). In reality,owing to the strong concentration polarization,the voltage growth leads only to an increase of theresistance of the Nernst layer. Thus, the currentdensity (curves 2–5) is essentially lower than with-out concentration polarization (curve 1). The cur-rent density depends on the ratio g between thethickness of double layer k−1 and the thickness ofNernst layer L [2–5]. Under real conditions ofelectrodialysis the thickness of the convective–dif-fusion layer of a membrane is such that g cannot


2.3. Non-stationary processes

The analytical expressions describing the situa-tion in the stationary regime have been obtainedafter various simplifications [5]. For the analysisof a non-stationary regime, further simplifyingassumptions have to be made.

After switching on an electric field, the elec-trolyte concentration near the membrane surfacegradually decreases. In the sublimiting and limit-ing cases where the electroneutrality is conserved,the non-stationary process reduces to a solutionof the diffusion problem in the Nernst layer, aswill be shown below.

In the overlimiting case, directly after switchingon the electric field, the electroneutrality is alsomaintained and the analysis is similar to that ofthe sublimiting regime. However, for somewhatlonger times the decrease of the electrolyte con-centration near the membrane is accompanied bya violation of the electroneutrality and an inducedspace charge appears. In this case it is impossibleto reduce the problem to a solution of the diffu-sion equation. To avoid this problem we restrictthe present investigation to the very short timeinterval in which the region of induced spacecharge has not yet been formed. In this limitingcase the problem reduces to solving the diffusionproblem.

Although this seems a rather strong simplifica-tion, the present treatment will show that it is stilla valuable theoretical exercise. Moreover, whenthe region of induced space charge is not formed,the potential drop within the polarization regionis not very large and the unproductive expenditureof electric power is small. Therefore this regime isalso interesting from a practical point of view.

In general, the ion fluxes j 9 of either the cation(+ ) or the anion (− ) in the non-stationaryregime are connected with the changes of thecation and anion concentrations C9 by the conti-nuity equations

di6 j 9(x,t)= −(C9(x,t)(t

(3)

with

j 9(x,t)

= −D9(C9(x,t)(x

9D9C9(x,t)F

RT(8(x,t)(x

(4)

where x is a distance from the membrane surfaceand t is the time. Eqs. (3) and (4) lead to thefollowing expressions for the rate of change of thecation and anion concentration

(C+(x,t)(t

=D+ (2C+(x,t)(x2 +D

+(

(x�

C+(x,t)F

RT(8(x,t)(x

�(5a)

(C−(x,t)(t

=D− (2C−(x,t)(x2 +D

−(

(x�

C−(x,t)F

RT(8(x,t)(x

�(5b)

The solution of Eqs. (5a) and (5b) can benoticeably simplified by assuming that the diffu-sion coefficients for cations D+ and anions D−

are equal:

D+=D−=D (6)

Furthermore, the electroneutrality conditionC+(x, t)=C−(x, t)=C(x, t) applies for any timein the sublimiting and limiting regime and for theinitial stage of the process at the overlimitingregime. Under these conditions and using reducedvariables, Eqs. (5a) and (5b) may be written as:

(C0 (x̃,t0 )(t0 =

(2C0 (x̃,t0 )(x̃2 (7)

and the differential equation for potential distri-bution becomes

(

(x̃�

C0 (x̃,th ) (8̃(x̃,t0 )(x̃

�=0

or C0 (x̃,t0 ) (8̃(x̃,t0 )(x̃

=const (8)

where x̃=x/L, t0 = t/2td and C0 =C/C0. The time t0is normalized with respect to the characteristictime of diffusion, td, through the Nernst layer

td=L2

2D(9)

In Eq. (9), L appears rather than d because, forthe present approximation, the thickness of the


diffusion layer d is equal to the thickness of theNernst layer L.

The constant in Eq. (8) is independent of thetime and can be obtained from the transition tothe stationary regime. According to Eq. (4), onefinds for t��

2DC(x)F

RT(8(x)(x

= j(x,t)�t��= ist (10)

or

C0 (x̃)(8̃(x̃)(x̃

= i0 st (11)

Thus, by using the simplifications describedabove, one can analyze the peculiarities of non-stationary polarization near a membrane surfaceby solving the diffusion Eq. (7) using Eq. (11) forthe potential distribution.

The boundary and initial conditions for Eq. (7),and correspondingly, the peculiarities of the tran-sition processes that occur, essentially depend onthe electrodialysis regime. Two regimes can bedistinguished: (1) using a fixed current throughthe membrane (galvanostatic regime) and (2) us-ing a fixed voltage (potentiostatic regime). Belowthese two regimes will be analyzed.

2.4. Concentration polarization and pulse time atthe gal6anostatic regime

To solve Eq. (7) at the galvanostatic regime(i= ist=const), use can be made of (1) theboundary and initial condition for the electrolyteconcentration

C0 (1,t0 )=1 and C0 (x̃,0)=1 (12)

and (2) the boundary condition for the gradient ofelectrolyte concentration

(C0 (x̃,t0 )(x̃

)x̃=0

= i0 st (13)

In the stationary regime, Eq. (12) is fulfilled forall values of x̃, and it leads to a linear change ofthe concentration in the diffusion layer. In thenon-stationary regime the gradient of electrolyteconcentration is not constant in the entire diffu-sion layer. Switching on an electric field leads to adecrease of the electrolyte concentration first of

all near the membrane surface. At slightly largerdistances the electrolyte concentration duringsome time is still equal to the bulk concentrationC0. Thus, the diffusion layer appears near thesurface and gradually moves to the externalboundary of the Nernst layer. So, Eq. (13) iswritten for x̃=0.

The solution of Eq. (7) with the conditions(Eqs. (12) and (13)) may be presented in thefollowing form

C0 (x̃,t0 )=1− i0 st!

(1− x̃)

−2 %�

n=1

1Qn

2 exp(−Qn2t0 ) cos(Qnx̃)

"(14)

with

Qn=2n−1

2p.

In spite of the restriction of the time interval tovery short times, this expression leads for thelimiting case, t��, to the stationary distributionof the electrolyte concentration in the diffusionlayer for all possible regimes: sublimiting, limitingand overlimiting.

Results of the numerical calculation of the con-centration profiles in the Nernst layer as functionof time are presented in Fig. 3a,b. Fig. 3a showsthe concentration as a function of the distance forfive values of the time and i0 st=1. Owing to thegalvanostatic regime the decrease of the elec-trolyte concentration is uniform with time. Fig. 3bshows the concentration profiles in the Nernstlayer for five different values of the current den-sity at a fixed time t0 =0.1. One can see that thenon-stationary concentration distribution essen-tially depends on the value of current. For exam-ple, in the sublimiting (i0 st=0.5, curve 1) orlimiting (i0 st=1, curve 2) regimes, the decrease ofthe concentration near the membrane is verysmall. However, in the overlimiting regime (i0 st=4,curve 5), the electrolyte concentration near themembrane reaches zero. Starting with Eqs. (8)and (14), an expression can be found for thepotential drop within the region of the concentra-tion polarisation:


8̃=F8

RT= i0 st

& 1

0

dx̃

C0 (x̃,t0 )

= i0 st

& 1

0

dx̃

1− i0 st

!(1− x̃)−2 %

�

n=1

1Qn

2 exp(−Qn2t0 ) cos(Qnx̃)

"(15)

Fig. 4. (a) The time dependence of the potential growth D8̃

over the Nernst layer at the galvanostatic regime for differentvalues of the current i0 st: i0 st=0.5 (1); 0.75 (2); 1 (3); 1.25 (4);1.5 (5); 2 (6). (b) The time dependence of the normalizedpotential growth D8̃/D8̃st (curves 1 to 5) and electrolyteconcentration C0 (0) (curves 1% to 5%) for different values of thecurrent i0 st: i0 st=2 (1); 3 (2); 4 (3); 5 (4); 6 (5). Curves 6 and 6%show D8̃/D8̃st=0.9 and C0 (0)=0.1.

Fig. 3. The electrolyte concentration in the Nernst layer as afunction of the distance from a membrane at the galvanostaticregime. Panel (a) shows the behavior at i0 st=1 and differentvalues of the time t0 : t0 =0.1 (1); 0.15 (2); 0.2 (3); 0.25 (4); �(5). Panel (b) shows the behavior at t0 =0.1 and differentvalues of the current i0 st: i0 st=0.5 (1); 1 (2); 2 (3); 3 (4); 4 (5).

The electrolyte concentration at x̃=0 decreasesin time, so the potential drop increases. The dete-rioration of the electrodialysis characteristics withthe development of the concentration polarizationcan be demonstrated by the calculation of thepotential growth (the potential difference betweenthe non-polarized state at t=0 and the potentialat time t) for different values of the currentdensity

D8̃= i0 st& 1

0

dx̃C0 (x̃,t0 )− i0 st

& 1

0

dx̃1

= i0 st�& 1

0

dx̃C0 (x̃,t0 )−1

�(16)

Fig. 4a,b shows some results. In the sublimitingregime (Fig. 4a, curves 1 and 2) the potentialgrowth D8̃ over the Nernst layer is increasingslowly with time, but in the limiting (curve 3) andoverlimiting (curves 4–6) regimes, D8̃ increases


strongly. For example, for t0 =0.25 at the sublimit-ing regime (i0 st=0.5) D8̃:0.068, at the limitingregime (i0 st=1) D8̃:0.34 and at the overlimitingregime (i0 st=1.5) D8̃:1.27. So, at i0 st=1.5, D8̃ is3.7 times larger than at i0 st=1 and almost 20 timeslarger than at i0 st=0.5. Hence, at the appearanceof a substantial region of induced space charge thepotential drop is expected be 10–100 times higherthan that at the sublimiting regime and the unpro-ductive power consumption is large. The pulse ofcurrent has to end before this situation occurs.

To estimate the pulse duration of the non-sta-tionary process Eq. (14) can be used. On the basisof this equation one can obtain the maximumtime, t0 *, up to which the region of induced spacecharge is not yet formed. In the time interval0B t0B t0 *, the concentration polarization is weak,but in the time interval t0\ t0 * it strongly increases.The pulse duration should therefore be smallerthan t0 *.

The time t0 * is connected with the period todecrease the initial electrolyte concentration atx=0 to the value of Cs at the beginning of theappearance of region of induced space charge. Thecharge density Cs depends on the electrolyte con-centration C0, the current density and the thick-ness of the Nernst layer [3–5]. Thus, t0 * has to bea function of these parameters too. However, forpractical purposes it is enough to evaluate t0 * for aconcentration decrease at x̃=0 to some value C*,which is connected only with the bulk concentra-tion of electrolyte C0. Let us suppose that C*=C0/10 or C0 *=0.1 for the overlimiting regime andC0 *=1.1C0 st for sublimiting regime.

For the calculation of t0 *, we have to take intoaccount that the time dependence of the concen-tration decrease is provided by the exponentialseries in Eq. (14). The main contribution to theseries is its first term (n=1). By neglecting thenext terms one finds for the relation between C0 *and t0 *

C0 *=1− i0 st+8

p2 i0 st exp�

−p2

4t0 *� (17)

So that the characteristic time or the maximumpulse time is found as

t0 i*=4

p2 ln� 8i0 st

p2(i0 st−1+C0 *)�

(18a)

or

t i*=8td

p2 ln� 8i0 st

p2(i0 st−1+C0 *)�

(18b)

where the subscript i indicates the galvanostaticregime. Note that these expression are only ade-quate to provide a rough evaluation of the charac-teristic time, for more exact values, it is necessaryto use the full Eq. (14).

For a comparison of the concentration decreaseand the potential growth in overlimiting regimethe concentration C0 (x̃=0,t0 ) and the potentialgrowth D8̃ are shown in the same picture (Fig.4b). The value of potential growth D8̃ is normal-ized on the stationary potential growth D8̃*st=D8̃st (i0 st=0.9), because the value of currenti0 st=0.9 corresponds to the stationary value of theconcentration C0 (0,t0��)=C0 *=0.1. One can see(Fig. 4b) that the reduced electrolyte concentra-tion C0 (0,t0 ) (curves 1%–5%) reaches the value C0 *=0.1 (curve 6%) at approximately the same time asthe ratio D8̃/D8̃*st (curves 1–5) reaches the value0.9.

Some values of the characteristic times as afunction of i0 st, calculated with Eqs. (18a) and(18b), are shown in Fig. 5 for C0 *=0.1 and thethickness of the Nernst layer ranging from 0.025(curve 4) to 0.1 (curve 1). The traditional electro-dialysis regimes are represented by curve 4.

It follows that the characteristic time can beequal to several minutes in the case of a largeNernst layer. However, t* sharply decreases to lessthan 1 min with a growth of current density anda decrease of the Nernst layer thickness.

2.5. Concentration polarization and pulse time atthe potentiostatic regime

The description of the transition processes atthe potentiostatic regime (8(t)=const) is morecomplicated than that at galvanostatic conditions,even in the case of the idealized model of a fixedpotential drop over only one Nernst layer. Al-though it is impossible to realize this situation inpractice, the simple model can still be used toobtain a reasonable estimate of the transitiontime.


Fig. 5. The characteristic time t i* as a function of the currentdensity i0 st for several values of the Nernst layer thickness L : L(cm)=0.1 (1); 0.075 (2); 0.05 (3); 0.025 (4).

8̃= i0 & 1

0

dx̃C0 (x̃,t0 )=8̃st (20)

where C0 (x̃,t0 ) is an unknown function. However,at all times the potential drop 8̃ is equal to thepotential drop in the stationary regime 8̃st. Thevalue of 8̃st in the sublimiting regime is related tothe stationary current i0 st by the following expres-sion [20,29]

8̃st= ln(1− i0 st) (21)

Although the potentiostatic regime is analyzed,the parameter i0 st indicates, as before, the dimen-sionless stationary current density. For both thegalvanostatic and potentiostatic regime this choiceallows a convenient comparison of the time de-pendencies of the electrolyte concentration in theNernst layer.

Unfortunately, the quick decrease of the elec-trolyte concentration at x̃=0 makes it is impossi-ble to use the diffusion Eq. (7) for the overlimitingregime. Even at very short times after switchingon the power, the electrolyte concentration nearthe membrane decreases so sharply that first of allthe region of the induced space charge forms andonly thereafter the concentration of electrolytedecreases at larger distance from the membranesurface. So, in order to simplify the situation weare forced to limit the investigation to currentsi0 st51.

The solution of Eq. (7) using the complicatedconditions expressed by Eqs. (19)–(21) can beobtained only in a numerical way. The calculatedconcentration profiles in the Nernst layer at thepotentiostatic regime as a function of time arepresented in Fig. 6 for 0.005B t0B� and twovalues of i0 st. The parameter i0 st accounts for thefact that the potential drop over the Nernst layercorresponds to the one in the stationary regime atthe indicated value of the current. This allows usto compare the two non-stationary regimes on anequal basis. In Fig. 6, a gradual increase in con-centration polarization with time is observed. Foran impression of the effect of the current densityi0 st on the concentration profiles in the Nernstlayer the results of Fig. 6a,b can be compared.

One can see, first of all, a steep decrease of theelectrolyte concentration in a narrow region nearthe surface (curve 1). The value of Cs becomes

In the potentiostatic regime a growth of theconcentration polarization with time leads to adecrease of the number of current carriers (ions)in the polarization layer and to an increase of theresistance. Consequently, the current through themembrane decreases with time. But right afterswitching on the power the current is very large,especially in the case of the overlimiting regime,when the concentration of electrolyte near themembrane reaches zero almost instantaneously.

Taking into account the similarity of the transi-tion processes for a membrane and an electrode,the following conditions [20,29] can be used forthe solution of Eq. (7):

C0 (1,t0 )=1 C0 (x̃\0,0)=1 (19)

The first condition is the same as for the gal-vanostatic regime, the second one indicates that att0 =0, the electrolyte concentration on an infinitelysmall distance x̃\0 is still equal to the initialvalue C0.

Furthermore, instead of Eq. (13), we now haveto use the condition that the potential drop isconstant. On the basis of Eqs. (8), (11) and (15),this condition can be expressed as


Fig. 6. The electrolyte concentration in the Nernst layer as afunction of the distance from the membrane at potentiostaticregime. Panels (a) and (b) show the behavior at i0 st=0.4 and0.8, respectively and different values of the pulse time t0 :t0 =0.0025 (1); 0.025 (2); 0.1 (3); 0.25 (4); � (5). Curve 4 canbe compared with curve 4% obtained for the galvanostaticregime at t0 =0.25 (note the y-scale difference in panel (a) and(b)).

polarisation extends over the entire Nernst layerand then reaches the stationary regime (curve 5).

In Fig. 6a,b one curve (4%) is also shown for thegalvanostatic regime (t0 =0.25). One can see thatthe non-stationary distribution of the electrolyteconcentration essentially depends on the regimeused: at the same value of i0 st the decrease of theelectrolyte concentration at a potential regime(curve 4) is stronger than at a galvanostatic one(curve 4%).

The change of the concentration profile leads tothe change of the current. At the beginning of thepulse, when the steep change of concentrationoccurs in a narrow region, the diffusion flux tothe membrane is essentially higher than for longerpulse times, when the concentration decrease oc-curs over the entire Nernst layer and its localvalue near the surface, Cs, increases.

The time dependence of the real currentthrough the membrane in the potentiostatic casecan be obtained with an equation similar to Eq.(10). Similarly as at the galvanostatic regime, thediffusion layer is engendered near the membranesurface and the current density at time t0 can beobtained from the electrolyte concentration gradi-ent at x̃=0:

i0 (t0 )=(C0 (x̃,t0 )(x̃

)x̃=0

(22)

In general, the current density decreases withtime and at t0�� it reaches the stationary valuei0 st.

To arrive at the pulse time range for an effec-tive electrodialysis in the potentiostatic regime itis necessary to analyze the mass transfer (integraltransport of ions) through the membrane as afunction of time t0 . The mass transfer can beobtained by the integration of the current over thetime as expressed by Eq. (23)

M0 (t0 )=1t0& t0

0

i0 (t0 ) dt0 (23)

where M0 (t0 ) is normalized on the limiting value ofmass transfer Mlim= ilimt.

Some results of numerical calculations of bothi0 (t0 ) (curves 1 and 1%) and M0 (t0 ) (curves 2 and 2%)are shown in Fig. 7. The normalized value of themass transfer at t0�0 coincides with the value of

Fig. 7. The normalized current density (curves 1 and 1%) andthe mass transfer (curves 2 and 2%) as function of time at thepotentiostatic regime at i0 st=0.4 (curves 1 and 2) and i0 st=0.8(curves 1% and 2%). Curves 3 and 3% demonstrate the stationaryvalues of the current: i0 st=0.4 and 0.8.

very low. At longer pulse times (curves 2, 3 and 4)the decrease of concentration spreads over alarger region. Near the surface the local value ofconcentration Cs increases, but the concentrationat larger distances decreases. Thus, concentration


the current. Later, at a fixed time t0 1\0 the masstransfer is always higher than the current becauseit is calculated as the integral of the current overthe time till t0 1 is realized (0B t0B t0 1) and for t0B t0 1the current is higher than its value at time t0 1.

On the basis of the results presented in Fig. 7,one can see that the current and the mass transferthrough a membrane (or the electrodialyser pro-ductivity) both quickly decrease with time. Forexample, at t0 =0.005 the current (curves 1 and 1%)is 10 times stronger than at the stationary regime(curves 3 and 3%), whereas at t0 =0.05 and 0.075,the current is, respectively three and two timesstronger.

Because the electrolyte concentration C0 (x̃=0)in the non-stationary regime is lower than in thestationary one, the characteristic time for thepulse duration, t0 *, cannot be calculated for thepotentiostatic regime in the same way as for thegalvanostatic regime. Instead, it is convenient touse for this purpose the current density. Twoexpressions can be used for the characteristic timet0 *8 (subscript 8 indicates the potentiostaticregime). In the first place this time can be re-garded as the time to increase the current from itsstationary value by 10%:

i0 *=1.1i0 st (24)

Some values of the characteristic time t0 *8 basedon Eq. (24) are presented in Fig. 8 (curve 1) asfunction of parameter i0 st.

Alternatively, one can consider t0 *8 as the timerequired for a 10-fold decrease of the initialcurrent:

i0 (t0 = t0 *8)/i0 (t0 =0)=0.1 (25)

Fig. 8, curve 1% shows that t0 *8 based on Eq. (25)is essentially lower than the time based on Eq.(24) (curve 1). As we want to increase the masstransfer, we have to terminate the pulse at a valueof the current that is sufficiently large. For practi-cal purposes the time based on Eq. (25) is there-fore more relevant than t0 *8 based on Eq. (24).

For the sake of comparison, the time t0 *i at thegalvanostatic regime is shown in Fig. 8 too(curves 2 and 2%). Curve 2 is calculated on thebasis of the condition

8̃(t0 *i )=0.98̃*st (26)

Alternatively, curve 2% describes the characteris-tic time t0 *i , when

C0 (0,t0 *i )=1.1C0 st(0) (27)

These two times are comparable and muchlarger than those for the potentiostatic regime.The transition process at the potentiostatic regimeis quicker than that at the galvanostatic one. Thedifference may be compared with the difference inthe time dependence of the electrolyte concentra-tion for both conditions, see Fig. 3 (galvanostatic)and Fig. 6 (potentiostatic).

2.6. Comparison of the potentiostatic and thegal6anostatic regimes

The desalination of water or mass transferthrough the membrane is caused by the currentdensity. A higher current density means a higherion flux through the membrane and a higherdegree of desalination. In the stationary case thepeculiarities of the structure of the Nernst layerand the power expenditure are the same in thegalvanostatic and potentiostatic cases: at a fixedcurrent density i0 a certain potential drop 8̃ occursand at a fixed potential drop 8̃ the certain currentdensity i0 occurs. Consequently, for i0 =const,8̃st=const and the power consumption of electro-dialyser, i0 st8̃st, is constant too.

Fig. 8. The characteristic time t0 * as a function of the currentdensity i0 st for the potentiostatic t0 *8 (curves 1 and 1%) andgalvanostatic t0 i* (curves 2 and 2%) regime. The calculations arecarried out according to the conditions: Eq. (24), curve 1; Eq.(25), curve 1%; Eq. (26), curve 2; Eq. (27), curve 2%.


The two processes lose their identity in non-stationary conditions. At the galvanostaticregime, where i0 (t0 )= i0 st=const, the removal ofions away from the membrane surface is inde-pendent of the time. Therefore, the non-station-arity is related to the growth of the potential.With the growth of the potential the power ex-penditure increases (i0 8̃(t0 )� i0 st8̃st), but the desali-nation remains the same.

At the potentiostatic regime the potential 8̃ isconstant and the non-stationarity is related tothe change of the current density i0 (t0 ). The cur-rent is maximal when the power is switched onand decreases with time. It reaches its minimumvalue at the stationary regime where i0 (t0 )� i0 st(=const). Hence, the desalination in the initialstage of the potentiostatic regime corresponds toa high value of the current. This means that fora given value of the stationary current (i0 st=const) the desalination in the potentiostaticregime is quicker than in the galvanostaticregime. Consequently, the concentration decreasein the Nernst layer is stronger and the transitiontime to the stationary regime or characteristictime t0 *8 is shorter than t0 i*. The power expendi-ture at the potentiostatic regime decreases withtime i0 (t0 )8̃� i0 st8̃st, but this decrease is not usefulas it is caused by a decrease of the current andthis decrease leads to a decrease of the desalina-tion.

It should be noted that forced water dissocia-tion is unimportant in the pulse regime. Thiswater dissociation is due to the fact that thepolarization processes lead to local electrolyteconcentrations below 10−7 eq. l−1 [23]. Thus,the concentration of electrolyte in the region ofthe induced space charge reaches a value corre-sponding to the local equilibrium of the waterdissociation reaction (CHCOH=10−14 eq. l−1).If the process of the concentration decrease islimited to times t0B t0 *, the electrolyte concentra-tion is higher than 10−7 eq. l−1 in every pointof the Nernst layer and forced water dissociationdoes not occur.

It is necessary to stress that the theory devel-oped is based on the notion of electroneutrality.However, it is very possible that non-stationarypolarization is combined with a deviation from

electroneutrality. Due to non-stationarity the in-duced space charge can appear in the sublimit-ing regime and can increase in the overlimitingregime.

2.7. Pause duration

The choice of a characteristic time of thepause is also important. The optimal time forthe intensification of the electrodialysis is thetime required to restore the initial state of theelectrolyte concentration near the membrane sur-face (without any concentration polarization). Atpresent it is not possible to analyze the restora-tion time quantitatively, but a qualitative analy-sis will already give an indication of the lengthof the restoration time compared to that of thepulse duration time.

The transition time to the initial state dependsstrongly on the degree of the concentration po-larization, so, on the duration of the pulse andon the value of the current or the voltage. Bytaking into account only the diffusion processthrough the Nernst layer, one can show [14,15]that the transition time is of the same order ofmagnitude as the pulse time for the sublimitingregime. However, the transition time quickly in-creases above this value with current growth inthe limiting and overlimiting regimes.

Several other processes also influence therestoration time. First of all, the return to theinitial state is connected with diffusion and os-motic ion transport through the membrane.Both are caused by the different electrolyte con-centrations at the two sides of the membrane.

Secondly, the restoration is connected withliquid movement near the membrane, as it iswell known that in the stationary regime a slightelectro-osmotic flow occurs through the mem-brane [30]. After switching off the power, thelocal electric field does not disappear absolutely.The existence of a diffusion layer in the absenceof the external electric field means that a poten-tial difference is preserved. Therefore, a slightmovement of liquid directed to the membranesurface still continues.

Thirdly, because in the non-stationary regimethe electro-osmotic flows inside and outside the


membrane cannot be compensated, a micro tur-bulence of liquid near the membrane results, thatspeeds up the restoration process.

In the fourth place, partial restoration of theinitial state of the electrolyte concentration duringthe pause is possible through hydrodynamic trans-port of salt along the membrane surface. Theeffect of hydrodynamic transport on the concen-tration polarization is not important when theexternal electric field is applied, but it is importantin the absence of the field.

Finally, the pulse regime can stimulate the in-stability of the induced space charge at the over-limiting regime, as has been predicted byRubinstein [1–4] for the stationary regime. Thisinstability also speeds up the restoration.

Based on these arguments one may expect thatthe return to the state without concentration po-larization is significantly quicker than a commondiffusion process. Therefore, it may be concludedthat the pause duration has to be about 10–50%of the pulse duration.

The argumentation presented above is relatedonly to the restoration of the initial state of theelectrolyte near the membrane surface. However,on the basis of a general notion about electroos-mosis of the second kind, we can predict a newform of this process related to the deviation fromelectroneutrality in the non-stationary regime.Similarly to electroosmosis of the second kindnear a flat heterogeneous membrane [1,12] thisphenomenon can be responsible for the intensifi-cation of electrodialysis. Moreover, due to inertialproperties this electroosmotical mixing of liquidcan exist not only in the period of the pulse, butalso in the period of pause. This complicatedproblem will be analyzed in a future paper.

3. Comparison with experimental results

In the present section experimental results willbe compared with theoretical predictions based onthe equations derived above. All results are ob-tained with a traditional electrodialyser cellequipped with an anion-exchange membrane MA-40 and a cation-exchange membrane MK-40 pro-duced in Russia.

3.1. In6estigation of the transition processes bytransport of macroions in electrofiltration

The processes of electrofiltration and electrodi-alysis are similar. The difference between the twois that the liquid in the electrofilter contains notonly the salt ions, but, for instance, alsomacroions. If the concentration of macroions isnot very large they will not change the membranepolarization and this is especially true in the be-ginning of process. As we can analyze the concen-tration of macroions in the membrane channelindependently of the salt concentration, addi-tional information can be obtained about thedistribution of the electric field in the channel.Electrofiltration is therefore a suitable method toillustrate the transition processes near the mem-brane surface.

For the present analysis use can be made of theresults presented in Ref. [16], on the electrofiltra-tion of a polyelectrolyte solution (1.6 mg l−1) indistilled water. The electrofiltration was carriedout at constant pH in a channel with a lengthl=4 cm and a thickness h=1 cm. The currentdensity during filtration was 1.5 mA cm2. Thepotential drop 8 over the channel in the station-ary regime was 180 V. The polyelectrolyte concen-tration at the exit of the channel was determinedspectrophotometrically. The resulting poly-elec-trolyte concentration as a function of time ispresented in Fig. 9 (curve 1).

For the analysis of the results it should berealized that in the beginning of the electrofiltra-tion all macroions sediment on the membranesurface and form an immobile layer. Under theseconditions, the decrease of the concentration ofmacroions in the liquid flow with time can beexpressed by the following equation [16,30]

Ci(t)Ci0

=3�h−Dh(t)

h�2

−2�h−Dh(t)

h�3

(28)

where Ci(t) is the measured polyelectrolyte con-centration at the exit of the channel at time t ; Ci0

its initial concentration; h the thickness of thechannel and

Dh=UefEt (29)


Fig. 9. The experimentally observed polyelectrolyte concentra-tion after electrofiltration as a function of time (curve 1) [16].For sake of comparison the calculated values in the absence ofthe concentration polarization (curve 2) and at stationaryconcentration polarization (curve 3) are also shown.

is the distance passed by the polyelectrolyte in thedirection normal to the membrane surface in thetime t. In Eq. (29), Uef is the electrophoreticmobility of the impurity and E is the electric fieldstrength in the channel.

If we suppose that the concentration polariza-tion is absent, the electric field strength can becalculated as E=80/h=180 V cm−1. On theother hand, an electric field strength in the core ofthe channel of about E=2 V cm−1 has beencalculated in the presence of concentration polar-ization [24]. For these two extreme values of thefield strength the decrease of the polyelectrolyteconcentration can be calculated with Eqs. (28) and(29) using the experimental value of the polyelec-trolyte mobility Uef=5×10−4 cm2 V−1 s−1. Theresults are presented in Fig. 9 as curves 2 (180 Vcm−1) and 3 (2 V cm−1). One can see that theexperimental curve (1) is located between the twotheoretical curves. This shows that the process ofbuilding up the concentration polarization and itsinfluence on the transport properties inside themembrane channel governs the actual electrofiltra-tion.

3.2. Transition time at the gal6anostatic regime

Further proof of the correctness of the galvano-static model can be obtained by a comparison ofthe experimental transition times and the calcu-lated characteristic times for the process of waterdesalination. Experimental results reported byKarlin and Kropotov [31] can be used for thispurpose.

The experiment has been carried out with anelectrodialyser channel with a length l=10 cm anda thickness h=1 cm. The liquid velocity was 18 cms−1. The measured transition times to the standardstate are plotted in Fig. 10 (symbols and curve 1)together with the calculated characteristic times t0 *i(Eq. (18b)) for the same conditions (curve 2).

One can see that at i0 st=1, the theoretical (t0 *i )and the experimental (t0 st) results coincide. This isevident, because a 10 times decrease of the elec-trolyte concentration C0 (0) at i0 st=1 is almostequivalent to reaching the stationary state. Atlarger values of i0 st, the difference between thestationary and the intermediate states increases.

Fig. 10. Comparison of the experimentally observed depen-dence of the transition time on the current density [31] (curve1) and the theoretically calculated (Eq. (18b)) characteristictime t0 i* (curve 2). �, C0=1.2×10−2 mol l−1; �, C0=1.18×10−3 mol l−1.


So, the difference between the theoretical (t0 *i ) andthe experimental (t0 st) times increases too. Forexample, for i0 st=3 the values of t0 *i and t0 st differby a factor of two, for i0 st=4 this differenceincreases by another factor of two and so on. In

general, Fig. 10 shows that the rate of the transi-tion processes is rather fast and that the charac-teristic time of the concentration polarizationdepends strongly on the current.

According to Eqs. (18a) and (18b), t0 *i is approx-imately proportional to 1/i0 st

2 for i0 st\1. The resultsof Karlin and Kropotov [31] show a similar de-pendency for the experimentally observed transi-tion times t0 st. Both this and the results of Fig. 10confirm that the derived model is qualitativelycorrect.

3.3. Transition process at the potentiostaticregime and water desalination at a pulse regimewith a fixed potential drop

Finally the transition process and the desalina-tion are investigated at potentiostatic conditionsin a specifically designed experiment. An electro-dialyser channel with a thickness h=0.15 cm, alength l=4 cm and a flow regime with a smallliquid velocity of V1=0.1 cm s−1 with a recircu-lation scheme (liquid is returned in the channel ina closed circle) has been used. The electrolyteconcentration as a function of time has beencalculated on the basis of the measured electrolyteconductivity. The investigations were carried outat a starting electrolyte concentration C0=10−3

mol l−1 (NaCl) and voltages ranging from 5 to 55V. Numerical calculations of the structure of theNernst layer and the potential drop over this layer(using the calculation scheme presented in Ref.[16]) show that the dimensionless current i0 st underthe conditions of the present experiment changesfrom 0.2 to 1.4.

The investigation of transition process was car-ried out for 10 pulses. The frequency of the pulseregime, v, was 1 Hz (duration of the pulsest1=0.8 s and the duration of the pauses t2=0.2 s). The transition time to the stationaryregime is measured with the condition that thedeviation of the measured current i(t) from thestationary value i0 st equals 10%: i(t0 )= i0 st=1.1.

The measured transition time as a function ofthe applied voltage for the first, the second andtenth voltage pulse are shown in Fig. 11. Thedecay of the current density as a function of timeis shown in Fig. 12 for an applied potential of 30

Fig. 11. Experimental dependence of the transition time on thevoltage at the first (1); second (2) and tenth (3) pulse.

Fig. 12. Experimental current density as a function of the timefor the first (1), the second (2) and the tenth (3) pulse of thevoltage (8=30 V).


Fig. 13. Experimental dependence of the water desalination onthe time (80=50 V; C0=10−3 mol l−1) under stationary(curve 1) and non-stationary (curves 2–5) conditions. Pulseregime: v=1 Hz; t1/(t1+ t2)=0.9 (2); v=10 Hz; t1/(t1+t2)=0.8 (3); v=10 Hz; t1/(t1+ t2)=0.9 (4); v=10 Hz; t1/(t1+ t2)=0.75 (5).

C0) and a subsequent lowering of the desalinationwith time.

Comparison of curves 1 and 2 shows that theintensification of the desalination is not very large(5–50%) at the low frequency (v=1 Hz) of theapplied pulse. Therefore the investigation was ex-tended to a higher frequency (10 Hz) and therelative duration of the pulse was varied. Theexperimental results show (Fig. 13, curves 3–5)that the efficiency of the pulse regime stronglydepends on both the frequency and the durationof pause. It is gratifying to see that at well chosennon-stationary conditions the desalination can beimproved some three times. For the present set upthe optimal result was received at t1/(t1+ t2)=0.8.

Future theoretical and experimental investiga-tions in this area will have to provide the back-ground for this optimum and its dependence onthe cell characteristics and the peculiarities of theelectrodialysis. In particular, it is important todevelop the theory of non-stationary processeswith account of deviation from electroneutrality.

4. Conclusions

Theoretical and experimental investigationshave shown that the non-stationary concentrationpolarization of membranes increases with time.Applying a non-stationary regime with short cur-rent or voltage pulses can intensify the process ofelectrodialysis.

The intensification of the electrodialysis will bemost noticeable if the duration of the current orthe voltage pulses is considerably shorter than thecalculated characteristic time to build up the po-larization layer.

The experimental investigation of the desalina-tion process in the pulse regime shows that thepulse–pause duration ratio and the pulse fre-quency are important parameters for the opti-mization of the desalination. Under optimalconditions an increase of the desalination ofabout three times could be achieved.

5. Nomenclature

C0 initial electrolyte concentration

V and the same pulse numbers as in Fig. 11. Onecan see that for the higher pulse number thetransition time to the stationary regime is shorter(Fig. 11) and the current is smaller (Fig. 12). Thereason for this behavior is that the return to theinitial state without the concentration polarizationdoes not occur, because the pause is too short.Moreover, Fig. 11 shows that the distinction be-tween the transition times of the different pulsenumbers increases with increasing voltage.

In Fig. 13, the relative concentration decreasein the desalination channel is shown as a functionof the time for different pulse–pause regimes.Curve 1 is obtained in the stationary state. Theconditions for curve 2 are comparable to those ofFigs. 11 and 12. The investigations show that thedesalination strongly depends on the time andthat it reaches an approximately constant value inabout 1 h. This result is related to the use of arecirculation scheme in the experimental cell. Dueto the desalination and the recirculation of liquidthe concentration of electrolyte C0 at the entranceto the membrane channel decreases in time. Thisleads to a lowering of the current (proportional to


concentration distribution in theC(x,t)Nernst layer

Ci(t) concentration of impurityCi0 initial concentration of impurityCs density of induced space charge

ion distribution in the Nernst layerC9(x,t)critical value of the electrolyteC*

concentrationD9=D diffusion coefficients of the ionsE electric field strength

Faraday constantFlimiting currentilim

i0 st stationary current densitycurrent densityi(t)thickness of channelhlength of channellthickness of Nernst layerL

R gas constantS0 thickness of induced space charge

layerT absolute temperaturet timet i* characteristic time for galvanostatic

regimecharacteristic time for potentiostatict*8

regimediffusion (transition) timetd

pulse durationt1

pause durationt2

electrophoretic mobility of macroionUef

x distance from the surface ofmembrane

d thickness of diffusion layerk−1 thickness of electrical double layer

(Debye layer)8 (x, t) potential distribution

potential drop over the Nernst layer8

potential drop within the electrodi-8o

alyser channelfrequency of pulsesv

Acknowledgements

Thanks are due to the International Associa-tion for the Promotion of Cooperation with Sci-entists from the Commonwealth of Independent

States for financial support of this investigation(projects INTAS 95-IN/UA-165 and INTAS 93-3372-Ext). P.V. Takhistov is kindly acknowledgedfor the experimental investigations at the poten-tiostatic regime.

References

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