Equilibrium Electro-Convective Instability in Concentration ...leonid/Ramadan_et_al2017.pdf ·...

903

ISSN 1023-1935, Russian Journal of Electrochemistry, 2017, Vol. 53, No. 9, pp. 903–918. © Pleiades Publishing, Ltd., 2017.Published in Russian in Elektrokhimiya, 2017, Vol. 53, No. 9, pp. 1014–1031.

Equilibrium Electro-Convective Instabilityin Concentration Polarization: The Effect

of Non-Equal Ionic Diffusivities and Longitudinal Flow1, 2Ramadan Abu-Rjal, Leonid Prigozhin, Isaak Rubinstein*, and Boris Zaltzman**

Jacob Blaustein Institutes for Desert Research, Ben-Gurion University of the Negev, Sede Boqer Campus, 84990 Israel*e-mail: [email protected]**e-mail: [email protected]

Received October 12, 2016; in final form, November 17, 2016

Abstract—For a long time, based on the analysis pertaining to a perfectly charge selective interface, electro-convective instability in concentration polarization was attributed to a nonequilibrium mechanism related tothe extended space charge which forms next to that of the electric double layer near the limiting current. Morerecently, it was shown that imperfect charge selectivity of the interface makes equilibrium instability possible,driven by either equilibrium electro-osmosis or bulk electro-convection, or both. In that study, addressingstability of a quiescent binary electrolyte, equal ionic diffusivities were assumed. Here we study the effect ofnon-equal ionic diffusivities and imposed longitudinal f low upon the onset and further nonlinear develop-ment of the equilibrium electro-convective instability at a non-perfectly permselective interface. It isobserved through a suitable analytical and numerical study that the imposed f low along the perm-selectiveinterface does not affect fundamentally the equilibrium electro-convective instability in concentrationpolarization either in terms of the temporal instability threshold or the resulting nonlinear f low. For theformer, the critical voltage is practically identical with that in quiescent concentration polarization. For thelatter, with non-slip interface conditions, the resulting nonlinear f low, with high accuracy, may be repre-sented as a superposition of the imposed Poiseuille f low and the vortices of the quiescent instability. Dif-fering ionic diffusivities may have a considerable effect upon the onset of the electro-convective instability.In particular, co-ionic diffusivity appreciably lower than the counter-ionic one may yield an appreciableincrease of the critical voltage. This is explained by the stabilizing effect of the diffusion potential’s contri-bution to the electric potential f luctuations.

Keywords: interfacial instability, electro-kinetic effects, electro-osmosis, membrane processesDOI: 10.1134/S1023193517090026

LIST OF NOTATIONS

Acronyms

EC Electro-convectionED electrodialysisEO electro-osmosisEDL electric double layerESC extended space chargeCP concentration polarizationVC voltage–currentOLC “over-limiting” conductanceECI electro-convective instability

Symbols tangential and normal coordinates

dimensional and dimensionless concen-trations of cations

dimensional and dimensionless concen-trations of anions

dimensional and dimensionless averageionic concentration

stirred bulk salt concentration dimensional and dimensionless fixed

charge density in the membrane dimensional and dimensionless electric

potential dimensional and dimensionless voltage

dimensional and dimensionless velocityvector

1 This paper is the authors’ contribution to the special issue ofRussian Journal of Electrochemistry dedicated to the100th anniversary of the birth of the outstanding Soviet electro-chemist Veniamin G. Levich.

2 The article is published in the original.

,x y+ +� ,c c

− −� ,c c

� , C C

0c� ,N N

ϕ ϕ�,

�, V Vv v�,

904

RUSSIAN JOURNAL OF ELECTROCHEMISTRY Vol. 53 No. 9 2017

RAMADAN ABU-RJAL et al.

dimensionless horizontal and verticalvelocity components

dimensional and dimensionless pressure dimensional and dimensionless time

variable dimensionless electric current

dimensionless cationic and anionicfluxes

dimensionless cationic and anionic elec-trochemical potential

diffusion layer width membrane thickness universal gas constant absolute temperature Faraday constant dielectric constant dynamic viscosity of the f luid dimensionless Zeta potential

dimensional cations and anions diffusivity dimensionless relative anionic diffusivity dimensionless Peclet number

dimensionless discharge perturbation parameter

wave number spectral parameter quiescent 1D steady-state concentration quiescent 1D steady-state electric potential quiescent 1D steady-state pressure

leading order perturbation of the concen-tration

leading order perturbation of the electricpotential

leading order perturbation of the velocity leading order perturbation of the vertical

velocity component leading order perturbation of the pressure velocity vector of the test vortex

horizontal and vertical velocity compo-nents of the test vortex

leading order perturbation of the concen-tration (test vortex)

leading order perturbation of the electricpotential (test vortex)

leading order perturbation of the velocity(test vortex)

leading order perturbation of the verticalvelocity components (test vortex)

, u w

�, p p�, t t

I

+ −j j,

+ −μ μ,

LlRTFdηζ

+ −,D DDPe

Qε � 1kλ

0Cϕ0

0pξ

Φ

V

W

Pv0

0 0,u w

1C

ϕ1

v1

1w

instantaneous velocity responseWDO, WEO instantaneous velocity response of the

‘diffusion-osmotic’ (‘buoyant’) and elec-tro-osmotic driving forces

, diffusion and Ohmic components of theelectric potential disturbance

residual space charge

INTRODUCTIONElectro-convection (EC) [flow of a liquid electro-

lyte induced by the action of Coulombic forces] is oneof the most promising ways for enhancing the masstransfer rate and reducing diffusion limitations in elec-trodialysis (ED) [1–7]. It is also pertinent in micro-nano-fluidic devices [8, 9], which are, over the pastdecade, being used for variety of applications, such asDNA analysis [10], micro-pumps used in fuel cells[11], electrophoresis [12] and others.

Two modes of EC may be distinguished in strongelectrolytes. The first is the relatively recentlydescribed ‘bulk’ EC, due to the volume electric forcesarising from the action on a macroscopic scale of theelectric field upon the residual space charge in thestoichiometrically electro-neutral electrolyte [13–21].The second is the common electro-osmosis (EO)induced by electro-osmotic slip resulting from theaction of a tangential electric field upon the spacecharge of the electric double layer (EDL). There aretwo regimes of EO that correspond to different statesof the EDL and are controlled by the non-equilibriumvoltage drop (over-voltage) across it [22–29]: equilib-rium EO [27–29] and non-equilibrium EO, or EO ofthe second kind [22–24, 26]. While both regimesresult from the action of a tangential electric fieldupon the space charge of the EDL, equilibrium EOpertains to the charge of the equilibrium EDL,whereas non-equilibrium EO pertains to the extendedspace charge (ESC) of the non-equilibrium EDL,which develops in the course of concentration polar-ization (CP) near the limiting current [24, 30–33].

The term ionic CP pertains to a complex of effectsrelated to the formation of concentration gradients inan electrolyte solution adjacent to a perm-selectiveinterface (an electrode, an ion-exchange membrane oran array of nano-channels in a micro-fluidic system)upon the passage of a DC current. CP is expression ofdiffusion limitation of ion transfer: solution is depletedupon the current entering the membrane (andenriched upon its exit) so that the diffusion in-flux ofco-ions into the vicinity of the membrane would partlycompensate (fully, for a perfectly charge selectivemembrane) their electro-migration out-flux in theelectric field. This is a basic electrochemical transportphenomenon which has been traditionally studied as akey element of the ion transfer through an electrode oran ion exchange membrane [27, 34, 35]. Morerecently, there has been a surge in the interest in CP

W

Φ1 Φ2

ρ


EQUILIBRIUM ELECTRO-CONVECTIVE INSTABILITY 905

related to the developments in micro-nano fluidics.This renewed interest roots in that fact that a nano-channel with overlapping EDLs at the channel sidewalls closely mimics an ion exchange membrane, sothat most CP related membrane phenomena havetheir exact analogs in micro-nano-fluidics and viceversa [4, 36–39]. Thus, the ‘desalination shock’ – anexpanding zone of low electrolyte concentration in ashallow micro-channel adjacent to a nano-channelunder the passage of a DC current – has as its exactanalog the ‘salt exclusion’ from a weakly charged ionexchange membrane adjacent to a highly charged oneof the same kind in the course of CP [4, 38–41].Moreover, the electrolyte concentration gradientsresulting from CP combined with the electric fieldmay induce several electro-kinetic effects, such as,electro-diffusio-osmosis and electro-diffusio-phore-sis, potentially useful for particle separation (e.g.,DNA sequencing) in micro-nano-fluidic systems [42]and electrode and membrane coating [43, 44].

In practice, a common expression of CP is thecharacteristic non-linear shape of experimentallymeasured steady-state voltage-current (VC) curve:low current linear (Ohmic) region is followed by cur-rent saturation at the “limiting” value, correspondingto the vanishing interface concentration, which is inturn followed by an inflection of the VC curve and thetransition to the “over-limiting” conductance (OLC)regime. The mechanism of OLC remained unclear fora long time, and still is a subject of intense research[1, 27, 45–53]. Only recently it was shown that, inopen systems, OLC is due to the destruction of the dif-fusion layer by a microscale vortical f low (the afore-mentioned ‘bulk’ EC or EO) which spontaneouslydevelops as a result of instability of CP near the limit-ing current and provides an additional ionic transportmechanism yielding OLC [1, 22, 40, 54–59].

For a long time, hydrodynamic instability in CPhas been attributed exclusively to non-equilibrium EOrelated to the ESC [24, 26, 30, 60]. This attributionstemmed from recognizing that, for a perfectlycharge-selective interface, neither equilibrium EO nor‘bulk’ EC can yield instability, [61], whereas non-equilibrium EO can [1, 54–56]. In the early theoreti-cal studies, the assumption of perfect charge selectivitywas employed for the sake of simplicity [1, 26, 54–56].Also, the subsequent studies of various time-depen-dent and non-linear features of EO instability contin-ued to use this assumption [23, 57, 62, 63], recogniz-ing that the ESC lying at the basis of non-equilibriumEO is essentially identical for a perfect and non-per-fect interface [64, 65]. Recently, Rubinstein andZaltzman found [66], that equilibrium EC (eitherequilibrium EO or bulk EC, or both) can yield insta-bility if the assumption of perfect perm-selectivity isrelaxed. The physical reason for this stems from thenon-constancy of the counter-ion electrochemicalpotential in the lateral direction at the outer edge ofEDL. Such non-constancy may result either fromnon-ideal perm-selectivity of the interface [66], or

from a finite rate of electrode reactions (e.g., incathodic deposition). In the case of a perfectly perm-selective membrane, the electrochemical potential ofcounter-ion does not vary in the membrane in the lon-gitudinal direction, and this creates stabilizing effectthrough the Donnan contribution to the electricpotential variation [66]. If the membrane is not per-fectly perm-selective, the electrochemical potential ofcounter-ions can vary along the interface if the con-centration and the electric potential in the enricheddiffusion layer vary. In general, any deviations fromconstancy of the electrochemical potential of counter-ions at the outer edge of EDL at the depleted interfacecan induce equilibrium instability, driven by eitherequilibrium EO or bulk EC, or both.

As often done for simplicity, equal ionic diffusivi-ties were assumed in [66]. This assumption, reason-able for potassium chloride (

versus ), is clearly inaccuratefor most electrolytes. Moreover, for low interface con-centration, at the limiting current, the related diffu-sion potential can be important, in particular, becauseof its possible effect on the electro-convective instabil-ity (ECI); this issue deserves a study. In addition, [66]and many other theoretical works studying perm-selective membranes or micro-nano-fluidic systems inOLC regime [1, 4, 24, 57, 67–71], did not address theimposed pressure-driven f low along the interface;such flow is typical of many practical electrochemicalsystems, e.g., ED channels. Recently, the influence ofthis f low was studied, experimentally and theoreti-cally, for an ideally perm-selective interface with a spec-ified constant electrochemical potential of counter-ions[5, 72, 73]. The novel features, related to the imposedflow and observed in these works, included the break ofvortex symmetry and occurrence of a particular vortexscaling. However, the ECI studied in these works was ofa strictly non-equilibrium type.

In this work we investigate the effects of imposedlongitudinal f low in the depleted diffusion layer and ofthe non-equal ionic diffusivities upon the onset andfurther non-linear development of the equilibriumECI at a non-perfectly perm-selective interface. Wepoint out that our study is of a purely qualitativenature, based, on the model in Ref. [66], in which theequilibrium ECI was announced. In particular, thisimplies that the current study is aimed at investigatingthe effect of the two aforementioned factors upon theonset and development equilibrium ECI in the verysame artificial framework which was employed in [66].In this framework, the ion exchange membrane ismodelled as merely a quiescent solution layer withfixed charges and ionic diffusivities identical withthose in the surrounding aqueous solution, no f low isassumed in the membrane and the enriched diffusionlayer, creeping f low is assumed in the depleted diffu-sion layer of constant width, diffusion layer boundaryconditions are postulated at the external edges of thediffusion layers, etc. All these assumptions are made

cm+−∼ × 5 2K 1.96 10 sD

cm−−∼ × 5 2Cl 2.03 10 sD

906



for maximal simplifications in order to distill the veryessence of the phenomenon addressed. Developmentof a more realistic model valid for engineering applica-tions and free from the excessive simplifications of thecurrent treatment is a natural issue for future studies.

ELECTROCONVECTION MODEL

Following [66], we consider a segment of an infinite 2D cation-exchange membrane,

, , f lanked by two univalentbinary electrolyte diffusion layers of constant width,

, , and ,, with equal fixed concentrations and a

given drop of electric potential maintained in the res-ervoirs at the outer boundary of diffusion layers. Inaddition, we consider an imposed pressure-drivenPoiseuille f low along the membrane in ,(see Fig. 1). This three-layer system is modeled by thefollowing dimensionless mass conservation (Nernst–Planck) equations,

, (1)

where the dimensionless ionic f luxes and are

,

. (2)

Assuming local electroneutrality in the diffusion lay-ers and the membrane, we obtain

, (3)

. (4)

<



the cross f lows whose velocity does not surpass by theorder of magnitude the maximal linear velocity of theemerging instability vortices. In particular, we wishour velocity scaling to be valid for the no imposed flowcase, for which we analyze the effect of differing ionicdiffusivities. This implies that we cannot use theimposed flow rate for the velocity scaling. Instead, ascommonly done in analyzing free convection, and asdone in all aforementioned studies of EC, the typicalvelocity and pressure are inferred from the force bal-ance in the dimensional version of the momentumEq. (6). This yields:

, (10)

(11)

Here, is the dielectric constant and thedynamic viscosity of the f luid. Finally, is the dimen-sionless time,

, (12)

where is the cations diffusivity.Correspondingly, the resulting Peclet number

defined as

, (13)

or, using Eq. (10)

(14)

is not the usual one, based on a prescribed(imposed) velocity, but rather an intrinsic materialcharacteristic of the electrolyte independent of itsconcentration and the typical length scale of thesystem [15]. Other names for this dimensionlessmaterial characteristic which have been used in dis-cussing EC are Electric Rayleigh number, ElectricHartman number, Levich number [14], etc. As indi-cated in [15], for a typical aqueous low molecularelectrolyte is of the order of unity (more pre-cisely, ).

The relative anionic diffusivity in the secondequation of (2) is the ratio of the dimensional anionand cation diffusivities, and , respectively,

. (15)

As mentioned previously, for simplicity, and areassumed the same for the membrane and solution,which implies that the membrane is treated merely asa quiescent solution layer with fixed charges. For real-

( )=πη

2

4sd RT F

lv

η= .ssp lv

d ηt

+=�

2tDtl

+D

Pe

+= sl

Dv

( )Pe+

=πη

2

4d RT

D F

0cl

Pe

Pe ≈ 0.5D

−D +D

−

+= DD

D

+D −D

istic aqueous electrolyte solutions D may vary by twoorders of magnitude in the range .

The system (1)–(7) is supplemented by the follow-ing boundary and interface conditions:

(16)

(17)

(18)

(19)

(20)

(21)

(22)

(23)

(24)

Condition (16) prescribes the concentration andthe electric potential at the outer edge of the enrichedlayer . Here is the voltage (normalized bythe thermal potential ). At the outer edges of thedepleted layer , condition (17) prescribes abulk solution that is experiencing no shear stress andzero inflow into the system, along with zero electricpotential. Condition (18) prescribes the slip conditionfor the tangential velocity [54], along with zero f lowinto the membrane, where is the dimensionless electricpotential drop between the interface and the outer edge of

the EDL, . Condition (19)defines the f luid f low along the membrane in thedepleted layer, with being the given discharge. In theabsence of EC, this f low has the parabolic Poiseuillevelocity profile. Equalities (20) prescribe periodicboundary conditions in the longitudinal direction.(Recall that the length of the segment isassumed short enough, comparable to , to make thevariation of the width of the diffusion layer induced bythe imposed cross f low negligible.) Conditions (21)and (22) impose the continuity of f luxes at the electro-lyte-membrane interfaces and , respec-tively, whereas (23) and (24) impose the continuity ofthe electrochemical potentials at these interfaces.

<

908



The bulk EC-no-slip setup corresponds to replac-ing the first equation of (18) [bulk EC-EO setup] bythe no-slip condition, . The peculiarity of thefirst condition in (18) is that, for an ideally perm-selective cation exchange membrane maintained at afixed electric potential, the electrochemical potentialof counterions in the membrane, , isconstant, and so, in equilibrium conditions, the sameis true for the outer edge of the EDL. In other words,

, and for , this conditionyields

. (25)

Hydrodynamic stability of the quiescent CP, with alimiting equilibrium EO slip condition (25), was stud-ied by Zholkovskij et al. [61], who found that 1D CP isstable. They concluded that with a perfectly perm-selective interface no bulk EC instability is feasible fora low molecular electrolyte [26]. In brief, the physicalreason for this is that for an ideal interface, the stabi-lizing Donnan contribution to the electric potentialperturbation, resulting from the concentration pertur-bation by the f low, dominates the correspondingdestabilizing Ohmic contribution. Recognizing thisbalance has motivated the reexamination of the role ofperfect charge selectivity of the interface [66]. On theother hand, it was shown that the non-equilibrium sliprelated to the ESC does yield instability. Because ofthis reason, since its prediction until now the hydrody-namic instability in CP was attributed to and studiedsolely for non-equilibrium EO.

By adding the mass conservation equations (1) forthe two ionic species, upon using (2) and applying theelectro-neutrality conditions (3) and (4), we arrive tothe following equations:

(26)

(27)

(28)

Furthermore, by subtracting we obtain

(29)

(30)

Equations (26)–(30) together with (5) and (6) formthe final set describing macroscopic EC in the localstoichiometric electro-neutrality approximation. Thethree-layer models of this type constitute a standardstudy tool in membrane electrochemistry and theoryof electro-kinetic phenomena at charge-selectiveinterfaces [65, 66, 74–76].

= 0u

ln c++=μ + ϕ

∂ ∂ = − ∂ϕ ∂C x C x ζ → −∞

( )= − ϕ4 ln 2 xu

= ∇+

− <



(16)–(24), (26)–(30), and linearization with respect to yield the following Sturm–Liouville problem.

Enriched layer, :

(32)

(33)

Membrane, :

, (34)

(35)

Depleted layer, :

, (36)

(37)

(38)

ε− < < 0L y

( )λξ = ξ − ξ+2 ,2

1 yykD

D

( )( ) ( )( )Φ −

− ξ − ξ+

+ Φ + ξϕ + Φ =20 0

2

0

11

0.yy y yy

y

y

y

C k C

D kD

<

910



(43)

(44)

( )( )( )

( )( )( )

0 0

0 0

0 0

0 0

1

1

1

1

2,

2,

y y y

y y y

y y y

y y y

y

y

y

y

C N

N

C

C

C

= −

= +

= −

= +

ξ + ξϕ + Φ

= ξ + ξϕ + Φ

ξ − ξϕ − Φ

= ξ − ξϕ − Φ

+

−

= + = +

ζ

⎛ ⎞ξ= = ζ Φ +⎜ ⎟⎝ ⎠

ξ +−

21 1

02

20

0,

14 ln ,2

yy yW W k

C

ekC

(45)

(46)

For the boundary value problem (32)–(46)determines the marginal stability relation between thethreshold value of the control parameter and thewave number . This problem has been solved by theshooting method using the Maple ODE solver. Fortwo setups, the bulk EC–EO and the bulk EC –no-slip, the results of the linear stability analysis of thequiescent 1D steady state are shown in Figs. 3–5. InFig. 3 we present the typical neutral-stability curves

in the V–k plane computed for three realistic

= == =2 2 0,yyy yW W

2 2| 0, | 0.y y= =ξ = Φ =

λ = 0

Vk

( )λ = 0

Fig. 3. Neutral stability curves in the voltage V(V*), wave number k plane (instability–above the curve): (a) and (b) curves inscaled voltage V* for, respectively, N = 1 and N = 2: D = 0.5 (1), D = 1 (2) and D = 5 (3); L = 1, Pe = 0.5 and Q = 0. Solid line standsfor the bulk EC–EO setup and dashed line stands for the bulk EC–no-slip setup; (c) and (d) curves same for the unscaled voltage V.

3.0

3.9

4.2

3.6

3.3

(а)

1 1

22

33

156 9 1230k

V *

3.9

5.1

5.4

4.8

4.5

4.2

(b)

1

12

2

3

3

156 9 1230k

V *

25

31

33

29

27

(d)

1

1

2

2

3

3

156 9 1230k

V

36

48

44

40

(c)

11

22

3

3

156 9 1230k

V



values of D, D = 0.5, 1 and 5, and two different mem-brane perm-selectivities, ; the region abovethese curves corresponds to instability. The criticalvalues, , for each pair correspondto the curve minimum. We note that both scaled and

unscaled voltages, and , decrease with theincrease of . Whereas for thisdecrease is slow ( nearly constant), for

, sharply increases with the decrease of, Fig. 4. (Physically, this effect is explained in section

Instability Mechanism) An inverse dependence isobserved for the critical wave number, , variationwith the variation of , Fig. 5.

NUMERICAL SIMULATIONS

The problem (5), (6), (16)–(24), (26)–(30) wassolved numerically by the method of lines [77–79].First, the finite element method was employed toapproximate the problem in space on a non-uniformmesh. The piece-wise linear conforming element wasused for both the concentration and the potential, thenonconforming linear element (Crouzeix–Raviartelement) was used for the velocity components and thepiece-wise constant element was employed for thepressure. We note that our choice of finite elements forthe velocity and pressure provides for the simplest

= 1, 2N

= crk k = crV V ,D N

crV cr*VD > 1D ( )+ −D D crVD

crkD

convergent approximation of the Stokes equation (theinf-sup condition [80] is satisfied). To approximatethe Laplacian in the electric body force term

of this equation using only the piece-wise lin-∇ ϕ2

∇ ϕ ϕ2 ∇

Fig. 4. Critical voltage dependence on D. (a) Scaled critical voltage versus D for N = 1 and N = 2; here L = 1, Pe = 0.5, and

Q = 0. Solid line stands for bulk EC–EO setup and dashed line for bulk EC–no-slip setup. Inset: versus logD. (b) Same forthe unscaled critical voltage, Vcr.

3

5

6

4

(а)

D

N = 1

N = 1

N = 2

N = 2

104 6 820

3

4

5

6

1.0–0.5 0 0.5–1.0log D

*crV

* crV

24

48

56

40

32

(b)V cr

D

N = 1

N = 1

N = 2

N = 2

104 6 820

25

35

45

1.0–0.5 0 0.5–1.0log D

* crV

cr*V

cr*V

Fig. 5. Critical wave number kcr dependence on D forN = 1 and N = 2. Inset: kcr versus logD. Here, L = 1, Pe =0.5 and Q = 0.

1.0

3.0

2.5

3.5

2.0

1.5

k cr

k cr

D

N = 1

N = 1

N = 2

N = 2

104 6 820

1.25

2.00

2.75

3.50

1.0–0.5 0 0.5–1.0log D

912



ear representation of the potential , we computedfirst the area-weighted average vertex values of ,then used them to find the piece-wise constantapproximation of the Laplacian. Test examples con-firmed that such an approximation preserved thedesired second order approximation of the velocitycomponents. All other variables except pressure,which is only an auxiliary variable in this problem,were also approximated with the second order inspace. We note that significant mesh refinement wasnecessary in the depleted layer near the interface

to accurately represent strong potential andconcentration variations in this area.

To integrate the obtained semi-discretized equa-tions in time we employed a standard solver for stiffODE problems, ode15s, from the Matlab ODE Suite[81]; this solver enabled us to control the accuracy ofintegration.

ϕϕ∇

( )= 1x

Numerical simulations, carried out for in therange and several values in both set-ups (bulk EC–EO and bulk EC–no-slip), Fig. 6, arein good correspondence with the linear stability resultspresented in Figs. 3–5. In Fig. 6 we present the steadyhydrodynamic vortex pair (in one spatial period)developed in the vicinity of the instability (Fig. 6a) andfor (Fig. 6b) for a poorly selective mem-brane, , and three values of . We note thedevelopment of the steady vortex pair that has twostagnation points located at the vortices’ core.

In our study of the effect of imposed longitudinalflow, , we limit ourselves to the case of equalionic diffusivities, . For an ideally perm-selectiveinterface with a specified constant electrochemicalpotential of counterions this type of effect was recentlyexperimentally and theoretically studied in [5, 72, 73];

D<



in this case the ECI is of a non-equilibrium type. Itwas found that the forced f low breaks symmetry of theemerging bidirectional vortex pairs; for a sufficientlystrong f low there develops an asymmetric unidirec-tional vortex structure [5, 72, 73]. For the equilibriuminstability studied here, the situation is similar: thesymmetry of vortex pairs is broken by the imposed

flow with a tendency to form unidirectional vortices,Fig. 7. The essence here is the enhancement of the‘favorite’ EC vortices by the imposed shear f low andsuppression of the ‘un-favorite’ ones. In addition, thevortex pairs are being advected by the imposed flowwith its mean velocity, Fig. 8. Numerical simulations(Fig. 9) showed, and this is our main observation, that

Fig. 7. Concentration distribution (in color) and flow streamlines (black lines) computed in bulk EC –no-slip setup for L = 1,N = 2, Pe = 0.5, D = 1, and V = 27.22 ≈ Vcr for various discharges Q: 0 (1), 0.5 (2), 1 (3) and 2 (4). The white lines are equi-concentration contours and the white arrows point in the velocities’ direction.

(1)

x

2.0

1.5y

1.00 0.5 1.0 1.5 2.0

(2)

x

2.0

1.5y

1.00 0.5 1.0 1.5 2.0

(3)

x

2.0 1.0

0.5

0

1.5y

1.00 0.5 1.0 1.5 2.0

(4)

x

2.0

1.5y1.0

0 0.5 1.0 1.5 2.0

Fig. 8. Shift of the two vortices, (a) right vortex and (b) left vortex, by the imposed flow for V = 1.08Vcr, N = 1, D = 1, Pe = 0.5,L = 1 and various discharges Q: 0.5 (1), 1 (2), and Q = 2 (3); dashed lines are the linear fits of the numerical experiment dots andis the coefficient of determination.

4

2

6

(а)

x = 1.051t + 0.46R2 = 0.998

x = 1.031t + 0.46R2 = 0.999

x = 1.97t + 0.48R2 = 0.999

51 2 3 40t

1Cen

ter o

f vor

tex

1 in

x

2

34

2

6

8(b)

x = 1.51t + 1.45R2 = 0.997

x = 1.03t + 1.47R2 = 0.999

x = 1.97t + 1.47R2 = 0.999

51 2 3 40t

1

Cen

ter o

f vor

tex

2 in

x

23

914



in the parameters’ range studied the resulting non-lin-ear f low is merely a superposition of the emerging ECvortices, identical with those in quiescent CP, with theimposed Poiseuille f low. The critical voltage for theonset of instability remains unaffected by the imposedflow.

INSTABILITY MECHANISM

To discuss the mechanism of the equilibrium ECIwe employ the ‘test vortex’ approach. The essence ofthis approach is as follows. One considers the system’sresponse to an accidental test vortex. When thisresponse yields vortex’ acceleration, a positive feed-back is identified and further analyzed as a possiblephysical cause of instability. Test vortex deceleration isviewed as evidence of negative feedback and stability.

For qualitative interpretation of non-equilibrium elec-tro-osmotic instability, this approach has beenemployed in Ref. [26]. Analytical and numericalimplementation of this approach has been recentlypresented in [82]. In spite of its intuitive plausibilitythe relation of the test vortex approach to the linearstability analysis has never been rigorously exploredand, at this stage, it should be viewed as purely heuris-tic means of physical interpretation. In terms of thisapproach, the physical mechanism of equilibrium ECIfor equal ionic diffusivities is as follows. In CP at a cat-ion selective interface, the space charge near thedepleted interface is positive. Consider an accidentaltest vortex near this interface. The f low towards theinterface in this vortex brings with it the high bulk con-centration, that is induces a positive conductivity per-turbation. This, in its turn, increases the electric poten-

Fig. 9. Flow streamlines (black lines) computed in bulk EC–no-slip setup for L = 1, N = 2, Pe = 0.5, D = 1, and V = Vcr = 27.22for various discharges Q: 0.5 (1), 1 (2), 2 (3), 4 (4), 10 (5) and 30 (6).

(1)2.0

1.5y

1.00 0.5 1.0

x

1.5 2.0

(2)2.0

1.5y

1.00 0.5 1.0

x

1.5 2.0

(3)2.0

1.5y

1.00 0.5 1.0 1.5 2.0

(4)2.0

1.5y1.0

0 0.5 1.0 1.5 2.0

(5)2.0

1.5y

1.00 0.5 1.0 1.5 2.0

(6)2.0

1.5y

1.00 0.5 1.0 1.5 2.0



tial, inducing the electric field disturbance along theinterface accelerating the vortex rotation (either due tocommon equilibrium EO or its equivalent bulk electricforce) that is resulting in a positive feedback [82].

To implement this approach for elucidation of theobserved effect of differing ionic diffusivities on thestability of 1D conduction, let us consider an asym-metric test vortex (reminiscent of the marginally stablevelocity mode of the linear stability analysis) with thevelocity

(47)

super-imposed upon the basic 1D quiescent CP.Hereon, for brevity, a unity diffusion layer width,

, is assumed.

The time-dependent problem for this vortex’induced electrolyte concentration and electric poten-tial disturbances reads:

(48)

, (49)

(50)

(51)

The velocity response corresponding to these dis-turbances is obtained as a solution of the equations

, (52)

(with the non-slip boundary conditions at the mem-brane/solution interface and reservoir boundary con-ditions at the depleted diffusion layer/bulk interface).

Let us define the instantaneous velocity response,, in the depleted diffusion layer as

. (53)

( ) ( )

( ) ( )

− −= + −

+ − − < <

v i j = i

j

5

0 0 0

6

8 7 2

1 2 , 1 2,

ikx

ikx

y yu w e

iky y e y

= 1L

( ) ( )( )Pe+ + + += ⋅ + ϕ − ⋅

∂∂

v + v0 ,c c c ct

∇ ∇ ∇ ∇

( ) ( )( )Pe− − − −= ⋅ − ϕ − ⋅

∂∂

v + v0c D c c ct

∇ ∇ ∇ ∇

+ −= =+ −= +

− < < < <

= + < <

1 0, 1 2;

,

,2

0 1,

y yC

C yNN

c c

c c

( ) ( )( ) ( )

, 1, 1, ,2, 1, , 1, , ,2, 0.

c x t c x tx t V x t

± ±− = =ϕ − = − ϕ =

2 2 0, 0,1 2

py

∇ ⋅ =∇ − ∇ + ∇ ϕ∇ϕ =< <

vv

W

( )

( )

21

12

0

1

, ,0

,

tw x y

W

w x y

+

=∫

∫

To evaluate W, we seek a solution of (48)–(52) inthe following form:

(54)

Here and are 1D steady-state solutions to theproblem (48)–(52). Keeping the leading order termsin Eqs. (48)–(52), we find

(55)

(56)

, (57)

with

(58)

In the r.h.s of (57), the first term is the analog ofdiffusion-osmosis (or buoyancy term in the Rayleigh–Bénard instability), whereas the second term is theanalog of EO. Here, is the tangentialvariation of the residual space charge of the distur-bance, , is the normal unperturbed field compo-

nent, is the minus normal derivative of the unper-

turbed residual space charge, , and ϕ1 is propor-tional to the tangential component of the electric fieldof the disturbance. Let us represent the solution of(55)–(58) as a superposition of terms proportional tothe normalized ionic diffusivities difference, D* = (1 –D)/(1 + D) and those independent of it of the form:

(59)

(60)Here Φ1 and Φ2 may be interpreted as the diffusionand Ohmic components of the electric potential dis-turbance, respectively, satisfying the equations:

(61)

(62)

( )( )

( )( )( )

( ) ( ) ( )= ==

⎛ ⎞ ⎛ ⎞⎛ ⎞⎜ ⎟ ⎜ ⎟⎜ ⎟ϕ = +ϕ ϕ⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠

∂ϕ∂ ∂= ϕ = =∂ ∂ ∂

vv v

vv

0 1

0 1

0 1

1 1 1

0 00

;

, , , , , .

ikx

t tt

C y C yCt ey y

yCC x y x y x yt t t

0C ϕ0

<

916



Furthermore, WA,B,C,D stand for the correspondingcontributions to the velocity response found from thesolution of the equations:

(63)

Thus, D*WA is the ‘diffusion-osmosis’ (‘buoyant’)contribution of the diffusion potential component ofthe electric potential disturbance, WB is the ‘electro-osmotic’ contribution of this component, whereas WCand WD are the respective contributions of the Ohmiccomponent of the electric potential disturbance. In addi-tion, let us denote the overall ‘diffusion-osmotic’ (‘buoy-ant’) contribution to the velocity response as WDO,

(64)

and the corresponding ‘electro-osmotic’ contribution as

(65)

In accord with Eqs. (59)–(63), these contributionssatisfy the equations

(66)

( )

( )

2 2 2 01 1

2 2 0

2 2 2 02 2

2 2

1

20

Φ

Φ

,

,

,

.

A yy y

B yyy

C yy y

D yyy

W k k

W k

W k k

W k

Δ = − Φ ϕ

Δ = − ϕΔ = − Φ ϕ

Δ − ϕ

Φ

Φ=

= +* , DO A CW W WD

= +* .EO B DW W WD

( )2 2 1 2 1 0,DO yy yW k kΔ = ϕ − ϕ ϕ

and

(67)

respectively.In Fig. 10a we depict the ‘diffusion-osmotic’

(‘buoyant’) and ‘electro-osmotic’ driving forces—theright hand sides in the equations (66), (67)—as func-tions of the relative co-ions’ (anions’) diffusivity, D.We note that whereas the first is negative (stabilizing)and nearly constant for the entire range of (and so isits velocity response, WDO), the second, positive(destabilizing) for large D, sharply decreases andbecomes negative (stabilizing) with D decreasingbelow 1 and approaching zero, causing the sharpincrease of the critical voltage for, observed in Fig. 4.This effect is due to the diffusion potential componentin the electric potential disturbance, as illustrated inFig. 10b. In this figure we depict various contributions,D*WA, D*WB, WC and WD, to the overall velocityresponse W as functions of D (positive values corre-spond to positive feedback and destabilization). Wenote that WDO is stabilizing and nearly constant for theentire range of D. The contribution of its diffusionpotential component, D*WA, slightly destabilizing forD < 1 and stabilizing for D > 1, is, on the whole, neg-ligible compared to the Ohmic one, WC. The situationis different for the destabilizing ‘electro-osmotic’ con-tribution, WEO. Here, for D > 1, both diffusion and

= ϕ ϕΔ − 22 1 0 ,EO yyyW k

Fig. 10. (a) Dependence of the average, in the depleted diffusion layer, of the two ‘diffusion-osmotic’ (‘buoyant’),

, and electro-osmotic, , driving forces on the relative co-ion diffusivity, D. (b) Dependence of the con-

tribution of the two driving factors, WDO and WEO, and its components, , , WC and WD, to the total instan-

taneous velocity response W on the diffusivity D. N = 2, , and L = 1.

–250

125

0

–125

250(а)

–ϕ0yyyϕ1

102 4 6 80D

1 2 1 0( – )yy ykϕ ϕ ϕ

–200

200

0

400 (b)

102 4 6 80D

W

WD

WC

WDO

D*WA

D*WB

WEO

( )ϕ − ϕ ϕ1 2 1 0yy yk −ϕ ϕ0 1yyy−+

11 A

DWD

−+

11 B

DWD

( )= =cr 1V V D ( )= =cr 1k k D



Ohmic components of the electric potential distur-bance are of the same sign, both contributing to thepositive destabilizing feedback; whereas, for D < 1, thestabilizing diffusion potential component of the elec-tric potential disturbance partly compensates/over-takes the destabilizing Ohmic one.

CONCLUSIONS(1) The imposed flow along the perm-selective

interface does not affect fundamentally the equilib-rium ECI in concentration polarization either in termsof the temporal instability threshold or the resultingnonlinear f low. For the former, the critical voltage ispractically identical with that in quiescent concentra-tion polarization. For the latter, within the frameworkof the current qualitative model, for non-slip interfaceconditions, the resulting non-linear f low may be rep-resented with high accuracy as a superposition of theimposed Poiseuille f low and the suitable vortices ofthe quiescent instability.

(2) Differing ionic diffusivities may have a consid-erable effect upon the onset of the ECI. In particular,co-ionic diffusivity significantly lower than thecounter-ionic one may yield an appreciable increaseof the critical voltage due to the stabilizing effect of thediffusion potential contribution to electric potentialdisturbance by the emerging vortical f low.

ACKNOWLEDGMENTThe authors appreciate the most helpful discussion

of the numerical approximation with J.W. Barrett.

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