Coupled convection of solution near the (1)

Coupled Convection of Solution near the Surface of Ion-Exchange Membranes in Intensive Current Regimes

N. D. Pismenskaya a , z , V. V. Nikonenko a , E. I. Belova a , G. Yu. Lopatkova a ,

Ph. Sistat b , G. Pourcelly b , and K. Larshe c

a Kuban State University, ul. Stavropol’skaya 149, Krasnodar, 350040 Russia b Institut Europeen des Membranes, Montpellier, France c Université Paris XII, Avenue du Général de Gaulle, Creteil, 94010 France Received May 16, 2006

Abstract —Mechanisms responsible for the overlimiting ion transfer in membranes systems are discussed. The overlimiting transfer is shown to be due largely to the action of four effects coupled with the concentration polarization of the system. Two of these are connected with the water dissociation near the membrane/solution interface: the emergence of additional charge carriers (ions H + or OH – ) in the depleted solution layer and the exaltation of transfer of salt counterions. The latter effect is connected with the perturbation of electric field caused by the water dissociation products. The other two effects are two versions of coupled convection, which leads to partial destruction of the depleted diffusion layer. These include gravitational convection and electro- convection. The former is caused by the emergence of the solution’s density gradient. The latter develops via a mechanism of electroosmotic slip. In this work, methods of voltammetry and chronopotentiometry and pH measurements are used to study the transfer of ions through homogeneous membranes Nafion-117 and AMX as a function of the concentration of sodium chloride solutions in the underlimiting and overlimiting current regimes. In a 0.1 M NaCl solution, gravitational convection makes a considerable contribution to the transfer of salt ions near the membrane surface in intensive current regimes. The influence of this effect on the electro- chemical behavior of membrane systems weakens with the solution dilution and with increasing relative trans- fer of the H + and OH – ions that are generated at the membrane/solution interface. In conditions where gravita- tional convection is suppressed and the water dissociation near the membrane/solution interface is not great, the major contribution to the overlimiting growth of current is made by electroconvection. Topics for discussion in the paper include the mutual influence of effects on one another, in particular, the effect the rate of generation of the H + and OH – ions exerts on the gravitational convection and electroconvection and the reasons for the dif- ferent behavior of cation- and anion-exchange membranes in intensive current regimes.

DOI: 10.1134/S102319350703010X

Key words : electrodialysis, overlimiting mass transfer, coupled convection, water dissociation

1. INTRODUCTION plex of phenomena caused by the flowing of a current, With an electric current flowing through an ion- including the formation of concentration gradients and

exchange membrane, due to the difference between secondary convective flows and other effects [6] (in a transport numbers for ionic components in the mem- broad meaning of the term). According to notions of brane and solution, there arises a concentration gradient classic electrochemistry, the formation of concentration of electrolyte in boundary layers of solution. In turn, gradients near the membrane (electrode)/solution inter- this concentration gradient is the reason for the poten- face leads to a restriction that is imposed on the current tial near the membrane surface shifting away from its density ( i ) by the so-called limiting current density equilibrium value [1]. A similar phenomenon takes ( i lim ). Once a zero concentration of electrolyte is place in electrode systems as well [2]. This shift of reached near a surface, i tends to limiting value i lim , potential is usually denoted by the term kontsentra- whereas a potential drop tends to infinity [2]. In real tsionnaya polyaritsatsiya (concentration polarization) membrane and electrode systems, however, the density in the Russian language literature [2] and overpotential of a limiting current may be exceeded by several times or overvoltage in English [3, 4]. In the last case, the at the expense of the emergence, near the surface of a concentration polarization is understood to mean the membrane (electrode), of a complex of effects that are current-induced formation of concentration gradients caused by concerted action of the flowing current and [5] (in a narrow meaning of the term) or the entire com- concentration variations in the system. These effects

may be united by the term “coupled effects of concen-

z Corresponding author, email: pism@chem.kubsu.ru

tration polarization” (CECP). Understanding these

PISMENSKAYA et al.

effects, which are responsible for the overlimiting mass tion only. Other coupled effects must make a substan- transfer, opens additional possibilities for intensifying tial contribution to the overlimiting transfer of salt electrolysis, electrodialysis, and electrodeionization of counterions as well. Specifically, we speak here of two liquid media and makes it possible to broaden areas of types of coupled convection, which provides for addi- application of these methods. In addition, exploring the tional, as compared with forced convection, agitation of coupled convection, which enters CECP, is of interest solution. This particular agitation is caused by local for perfecting electrokinetic micropumps [7], processes vortexes that result from the action of bulk forces which of electrophoresis [8, 9], electrodeposition [10], layer- are engendered by the flowing of an electric current. ing colloidal crystals onto the surface of electrodes The first type of the coupled convection is a gravita- [11], and so on.

tional convection. It emerges owing to nonequilibrium Four effects that explain the phenomenon of the distribution of density of solution, which is the reason

overlimiting mass transfer are under discussion in the for the emergence of a volume buoyancy force [21–23]. literature at present. Two of these are connected with The second type is an electroconvection. It emerges as the water dissociation at the membrane/solution inter-

a result of the action of electric field on a space electric face. The emergence of additional charge carriers, charge in the depleted solution adjacent to the mem-

namely the H + and OH – ions that are generated during brane [6, 24–28]. the water dissociation in membrane systems [12–15],

The emergence of a bulk force in the case of gravi- for a long time was considered as the principal and, fre- tational convection is caused by gradients of concentra- quently enough, sole, reason for the overlimiting con- tion and/or temperature [21–23]. This phenomenon is duction [16]. At the same time, generation of the H + and more probable in relatively concentrated solutions, for OH – ions gives rise to another, less obvious, mechanism there take place in such solutions a stronger solution of the overlimiting transfer. This is the effect of exalta- heating and a larger gradient of concentrations [22, 23], tion of a limiting current [2], which was explored as which are caused by a larger value of limiting density applied to electromembrane systems (EMS) for the first of an electric current, which is proportional, to a first time by Yu.I. Kharkats [17]. The emergence of the H + approximation, to the solution concentration. In the and OH – ions in the vicinity of the surface of a mem- case of a vertical solution/electrode (membrane) inter- brane perturbs electric field and is capable of increasing face and the horizontal solution density gradient, the (exalting) transfer of counterions of a salt. Consider, for gravitational convection generally emerges in a thresh- example, the OH – ions, which are charged negatively. oldless regime, i.e. it emerges always and gradually

When generated at the interface between a depleted dif- increases with the current (potential) [21]. The inter- fusion layer and the surface of a cation-exchange mem- face may be horizontal and the density of solution con- brane, these pull salt cations out of the solution depth fined in between two parallel horizontal planes may towards the interface. With the effect of exaltation alter along the perpendicular coordinate. Two cases are taken into account, the density of the flux of salt coun- possible in such a situation. The lighter layer of solu-

terions ( j 1 ) is described by the equation [17] tion (depleted diffusion layer) may be positioned under

0 the membrane and the heavier layer of solution

j = --------------- + ------- j (enriched diffusion layer), above the membrane. In that

2D 1 c D 1

1 w , δ (1)

case no convection emerges near the membrane. The lighter layer of solution may be positioned above the

where D 1 and Ò 0 are the diffusion coefficient for salt membrane. Then, a certain threshold takes place in the counterions and concentration (for simplicity, 1 : 1) of development of gravitational convection. The threshold electrolyte in the solution depth, δ is the thickness of a in question is defined by the value of the Rayleigh num-

depleted diffusion layer (DDL), D w and j w are the diffu- ber [21–23]:

sion coefficient and the density of a flux inside a diffu-

3 sion layer of the water dissociation products that are 3 ∆ρ gX ν ∆ρ gX generated near the surface of a membrane (in the case

Ra = Gr Sc = ------- --------- 2 ---- = ------- ---------. ρ (2) of a cation-exchange membrane, these are the OH –

ρ ν D νD ions). As follows from the Kharkats equation, i.e. equa-

∆ρ gX

tion (1), the increment of the flux of salt counterions in

0 Here, Gr = ------- --------- 2 is the Grashof number; Sc = ν / D excess of the limiting value is j 1 lim =2 D i c / δ is propor- ρ ν tional to the flux of generated OH – H ( + ) ions, with a

a Schmidt number; ∆ρ is the change in the solution den-

proportionality coefficient being equal to D 1 / D w .

sity ρ , which occurs between the upper and lower por- The increase in the flux of salt counterions at the tions of a layer of thickness X , in which the change in expense of the effect of exaltation in EMS is relatively the solution density occurs; g is the gravitational accel-

not great. For example, it amounts to about 0.2 j 1 lim in eration; ν is the solution viscosity; and D is the diffu- the case where the flux of the OH – ions reaches values sion coefficient of electrolyte. A system is stable (i.e. no that are equal to j 1 lim . In practice, the increment of salt convection arises) if Ra < Ra cr = 1708. In such a case, counterions is considerably greater [18–20] and, conse- the characteristic time that is required for the diffusion quently, it cannot be explained by the effect of exalta- dissipation of the density fluctuation in a small volume

RUSSIAN JOURNAL OF ELECTROCHEMISTRY Vol. 43 No. 3 2007

COUPLED CONVECTION OF SOLUTION NEAR THE SURFACE

of solution is shorter than the characteristic time required for this volume to float up. At Ra > Ra

cr

, the

volume with a negative density gradient floats up with acceleration, because the density inside a volume that is floating up increases slower than in the solution sur- rounding it. The amplitude of a small perturbation in this case increases with time, and the solution that is confined in between two planes achieves a certain state. This state is characterized by a periodic cellular vortex structure, where the liquid in two neighboring cells (Bénard convection cells) rotates in the opposite direc- tions [22, 23, 29]. Heat and mass transfer theory main- tains that gravitational convection in an “empty” (con- taining no spacer) rectangular channel is not suppressed by forced convection, provided Ri = Gr

Re

> 1. Here, Ri

is the Richardson number and Re =

VX

is the Rey-

nolds number, where

is the average linear velocity of

forced flow of solution [30, 31]. Research into the hydrodynamic instability due to

gravitational convection in electrode systems was reviewed by V.M. Volgin and D.A. Davydov [23]. Results of mathematical modeling of this phenomenon in EMS were reported in [9, 19, 32, 33], for example. Gravitational convection plays a role in boosting the mass transfer and reducing the DDL thickness in inten- sive current regimes. This role was estimated experi- mentally using voltammetry and measuring partial transport numbers for ions [19, 20]. Its influence on the concentration polarization of EMS was probed with the aid of the chronopotentiometry [34] and laser interfer- ometry [5, 35] methods. Laser interferometry [5, 35] and microphotography with background laser illumina- tion were employed for observing convective motion of liquid visually. In parallel, the structure of the Bénard convection cells was examined by the chronopotenti- ometry method [36, 37] with use made of a Fourier analysis of obtained curves. The sequence of events involved in the development of coupled convection near the surface of a homogeneous cation-exchange membrane was described in [38]. Using the Fourier and wavelet analyses, the authors of [38] showed that the frequency of revolutions of vortexes in a stationary state of EMS is equal to 0.1–0.4 Hz. This result con- forms to the data reported in [37], where it was estab- lished that the potential oscillations in chronopotentio- grams correlate with the life cycle of vortexes near the electrode/solution interface. The authors of [37] also demonstrated that the coupled convection may manifest itself at Ra < Ra

cr

= 1708 and explained this phenome- non by the action of electrostatic forces, i.e. electrocon- vection.

According to modern theoretical notions, which were reviewed in [6, 25–28], the major mechanism governing the development of electroconvection in membrane systems is electroosmotic slip of second kind or, in terms of the authors of [24], who were the first to look into this phenomenon, “electroosmosis II.” Electroosmosis of second kind arises in membrane sys-

tems due to an electric field interacting with a space charge induced by this electric field. The space charge in question appears in DDL near the interface. As its extension increases in more dilute solutions [25, 39, 40], the electroconvection contribution to overlimiting mass transfer would presumably increase with decreas- ing salt concentration.

The space charge extension and density and the strength of the applied electric field are not the only fac- tors affecting the electroconvection intensity. The latter is also likely to depend on the Stokes radius of ions that make up the space-charge region. Indeed, the larger the Stokes radius of ions, the more efficient should be the involvement of the liquid into convective motion. The authors of [41] were probably the first to notice this cir- cumstance. Having compared current–voltage curves (CVC) recorded for a CMX membrane in various elec- trolytic solutions, they discovered that the plateau in CVC shrinks with increasing the Stokes radius of coun- terions. The shorter the plateau, the lower the voltage at which intensive coupled convection arose. They also observed that the EMS resistance decreased in overlim- iting regimes. The plateau length was maximum in a hydrochloric acid solution, for the H

ions are trans- ferred in solution via a relay mechanism, rather than a hydrodynamic mechanism [2]. Electroosmosis of sec- ond kind near granules of an ion-exchange resin, which were deployed in between two polarizing electrodes in

a dilute solution, was confirmed experimentally in [25, 42].

The aim of this work is to experimentally investigate how the coupled convection influences electrochemical behavior of ion-exchange membranes and expose fac- tors that define its type and intensity in EMS. To do this, we will consider CVC and chronopotentiometric char- acteristics of homogeneous membranes under condi- tions favorable and not favorable for the development of coupled convection. We are also going to compare characteristics of cation- and anion-exchange mem- branes that have similar surface morphology but differ- ent capability to generate the H

The objects under study in this work are cation- exchange membrane Nafion-117 (Du Pont, United States) and anion-exchange membrane AMX (Tokuyama Soda, Japan). The fixed groups in homoge- neous membrane AMX are quaternary ammonium bases [43]. The catalytic activity of these groups rela- tive to the water dissociation reaction is not great [12, 14]. Nevertheless, the water dissociation reaction pro- ceeds on the AMX membrane at a relatively high rate owing to the appearance on its surface of secondary and tertiary amines that form during its storage and exploi- tation in overlimiting current regimes [44]. The cata- lytic activity of sulfo acid groups of homogeneous

PISMENSKAYA et al.

cles, etc.) are 1–3 µm. The dimensions of microirregu- larities on the surface of membranes and in their bulk are connected with the technique used for manufactur- ing membranes. In the case of an AMX manufactured by a paste method [46], they lie in the limits from 100 to 300 nm. In the case of Nafion-117 produced by a method of polymerization [45], they do not exceed

30 nm. Basic characteristics of both membranes appear in Table 1. Values of the exchange capacity and mois- ture content were borrowed from [43, 45, 47].

The experiments were run at a temperature of (25 ± 1)°ë in 0.005, 0.02, 0.05, or 0.1 M solutions of NaCl, which were prepared from the reactant of analytical grade and distilled water with a resistance of no smaller than 0.5 Mohm and pH 5.5–6.2. Prior to experiment,

membranes were equilibrated with a solution of a spec- (‡) ified concentration. In accord with recommendations of

3 µm

the firm-manufacturer, membranes Nafion-117 had first been subjected to a boiling procedure in distilled water for the duration of half an hour.

2.2. Investigation of the Surface of Membranes When obtaining micrographs of the surface and

cross-section of membranes under study, we used a scanning electron microscope LEO (Leica, Cam- bridge), type S260. Prior to taking micrographs, mem- branes 24 h were kept in an exsiccator with a moisture absorber. The procedure used for preparing specimens was described in more detail in [34, 48].

2.3. Investigation of the Electrochemical Behavior (b)

3 µm

of Membrane Systems

2.3.1. Experimental setup. The current–voltage curves, the chronopotentiograms, and the dependences

Fig. 1. Photographs of surfaces of ion-exchange membranes (a) AMX and (b) Nafion-117.

of pH of near-membrane layers on the potential drop were obtained in a four-chamber cell of the flow- through type (Fig. 2a) that was described in [48]. The

membrane Nafion-117 [45] relative to the water disso- flow-through chambers of the flow-through cell are ciation reaction is low [14]. The generation rate of the formed by membranes 1 as well as by plastic (Plexi-

glas) frames 2 and rubber gaskets 3 and 4 with square and éç ions on Nafion-117 is low, indeed. It is holes of area S = 2 time 2 cm 2 5. The thickness of the substantially lower than that on AMX. The surface of Plexiglas frames had been fixed and equaled 4 or 5 mm.

both membranes is practically homogeneous and The distance between membranes can be varied by smooth (Fig. 1). The linear dimensions of small irregu- varying the thickness of the rubber gaskets 3 and 4. The larities on it (protuberances, pits, bacteria, dirt parti- chambers that are abutting the membrane under inves-

Table 1. Principal characteristics of membranes under investigation Membrane Thickness,

Moisture content, Electroconductivity in 0.05 M NaCl, µm

Ionogenic

Exchange capacity,

groups

mg-equiv ml –1 *

mS cm 2 –1 /g*

AMX + 170 ± 10 –N(CH

Nafion-117 212 ±4 –SO 3 H 1.09 [45]

* For swollen membrane in the Cl – form (AMX) of the Na + form (Nafion-117).

RUSSIAN JOURNAL OF ELECTROCHEMISTRY Vol. 43 No. 3 2007

COUPLED CONVECTION OF SOLUTION NEAR THE SURFACE 311

Waste solution

Autolab 16

Vol. 43 No. 3 2007

PISMENSKAYA et al.

tigation A* on the side facing a flat platinum cathode 6 hydrostatic pressure in the course of experiment, the are separated by cation-exchange C and on the side of exit tubes of cell 12 are fixed in the lid of auxiliary res-

a flat platinum anode 7 by anion-exchange A mem- ervoir 24. Values of the flow velocity are regulated by branes CMX and AMX, respectively. To take solution valves 19, which are positioned on bendable tubes from near-surface layers of solution in order to monitor before the entrance into chambers of the cell. its pH, tips of two plastic capillaries 8 with an external

Upon leaving capillaries 8, solution enters flow- diameter of about 0.8 mm are brought to the center of through cells 13, into which combined glass and silver

either side of the membrane under investigation. The chloride electrodes WTW SenTix 97T 14 are placed. end faces of the capillaries are situated in the immediate These electrodes are connected with the pH meters vicinity of the membrane at an angle of 45° to its sur- (pH-320 WTW) 15, which allow us to measure pH in face. Two silver wires 9 with a diameter of 0.25 mm, flow-through cells 13. The rate of collecting solution which are covered with polyterefluoroethylene for measuring pH in flow-through cells 13 is regulated 0.024 mm thick squeezed in between rubber gaskets 3 by valves 25 that are deployed on short (8 cm) tubes, and 4 from both sides of the membrane under investiga- which connect capillaries 8 and flow-through cells 13. tion. The end faces of these silver wires are fixed in the In order to ensure a specified hydrodynamic regime in vicinity of capillaries 8 at a distance of approximately chambers of cell 12, this rate must not exceed 5% of the

0.8 mm from every side of the membrane under inves- rate of supply of solution into chambers of cell 12 that tigation. They are covered with AgCl by means of are abutting the membrane under investigation. When polarization in the role of an anode in a 0.1 M solution obtaining galvanodynamic CVC the time of “lag” of of HCl for the duration of one hour at a current of registered values of pH relative to the current values of

0.1 mA and serve for measuring the potential drop. The the current t lag is defined by the volume of the connect- supply and removal of the feeding solution in each ing tube W and the bulk velocity of solution flow w out

chamber is realized through connecting tubes 10 that of flow-through cells 13 (t = W/w). are inserted into plastic frames. The devices for the

lag

solution insertion into a chamber and its removal out of By rotating cell 12 it was possible to preset any it are performed in plastic frames in the form of comb angle between the membrane under investigation and

11 in order to provide for a laminar uniform flow of the direction of the gravitational field of Earth. In this solution between membranes. The latter is necessary particular work we present results of experiments in for alleviation of a mathematical description of transfer which the membrane under investigation was situated of ions in the cell, in particular, for calculation of the in vertical or horizontal positions at an average velocity limiting current density and the diffusion layer thick- of the flow of solution. In the cases where the mem- ness (Section 2.3.2.).

brane under investigation is situated in a horizontal position, the current is directed in such a manner that

Figure 2b shows a schematic depiction of the exper- the lighter depleted diffusion layer is situated below the imental setup. The flow of liquid between membranes membrane under investigation and the gravitational is ensured by the difference between hydrostatic pres- convection in the immediate vicinity of the membrane sures between the tube that removes solution out of cell under investigation is absent (Section 1). If a cell is in a

12 and intermediate reservoir 18 with the feeding solu- vertical position, solution is supplied into it from below tion. Reservoir 18 with the aid of pump 17 is replen- upward. ished by solution from reservoir 23 containing a con- siderably larger volume of the feeding solution. A con-

In order to register potentials and specify currents, stant level of solution in reservoir 18 is ensured by the we used electrochemical complex 16 Autolab (Eco

hole 22, which is connected by means of a bendable Chemie B.V., The Netherlands). The complex in ques- tion was equipped with a potentiostat PGSTAT 100 hose with reservoir 23. In order to maintain a constant (Institut Europeen des Membranes, Montpellier,

France) or a programmable power source Programma-

Fig. 2. Schematics of (a) experimental cell and (b) experi-

ble Current Source 220, Keithley, and a high-resistance

mental setup for recording CVC and chronopotentiograms and measuring solution pH near the surface of the mem-

voltmeter Multimeter-3478A, Hewlett Packard (Uni-

brane under investigation: (1) membranes, (2) plastic

versity Paris XII, France) or potentiostat PI-50.1.1, and

frames, (3, 4) rubber gaskets, (5) square holes, (6) cathode,

voltmeter V7-65/5 (Kuban State University, Russia).

(7) anode, (8) plastic capillaries, (9) silver–silver chloride

The specially designed software allowed us to realize

electrodes, (10) holes, (11) special comb, (12) experimental cell, (13) cells for measuring solution pH and temperature,

digital recording of the measured potential drop with a

(14) combined glass and silver–silver chloride electrodes,

minimum interval between signals equal to 0.01 s

(15) pH meters, (16) either complex Autolab-100 or poten-

(Autolab) or 0.1 s (Multimeter-3478A) or 0.2 s

tiostat and high-resistance voltmeter, (17) pump, (18) inter-

(V7-65/5).

mediate vessel with feeding solution, (19) valve, (20– 22 ) holes in intermediate vessel 18, (23) another vessel with

2.3.2. The treatment of experimental data: the

feeding solution, (24) auxiliary vessel, and (25) valve; (A*)

limiting current density and the diffusion layer

membrane under investigation, (C) auxiliary cation-

thickness. As it was established previously in [33], in

exchange membrane, and (A) auxiliary anion-exchange

the case of low currents, the mass transfer in the desali-

membrane.

nation channel of the cell under investigation is well

Vol. 43 No. 3 2007

313 Table 2. Principal characteristics of electromembrane systems under investigation System 0 Membrane c

COUPLED CONVECTION OF SOLUTION NEAR THE SURFACE

Gr/Re 1 AMX

2800 1.8 3a AMX

700 0.4 3b Nafion-117

described by a convective–diffusion model [40], pro- point of intersection of tangents drawn to the initial seg- vided the desalination channel is formed by smooth ment at i = 0 and to the segment of slanted “plateau” in homogeneous membranes. The model assumes a sta- CVC. tionary laminar forced convection of solution with a

The average value of the thickness of the depleted parabolic profile of velocity. To satisfy these conditions diffusion layer δ may be calculated after finding the

we used devices for supply and removal of solution 10 limiting current density with the aid of equation (3) and 11 that were described in the foregoing (Fig. 2a). from the know equation According to this model [40, 49], the limiting current

density in a cell that is formed by smooth homogeneous 0

0 FDc

ion-exchange membranes with a small value of desali-

δT ( 1 – t 1 ) nation length Y = LD/Vh (on the order of 10 , as is the

i lim = -----------------------. (4)

case in this work) is well described by the Leveck equa- In the small-length electrodialysis channel under tion

consideration we can presume that the concentration of

electrolyte in the solution depth is identical to the con-

i lim = 1.47 ----------------------- ⎛ --------- ⎞ ,

0 FDc

h V centration c 0 at the entrance into the cell. Equation (4)

is valid in the case where generation of the ç and éç ions at the membrane/solution interface is absent. Oth-

where Ò 0 is the concentration of electrolyte at the erwise one should make use of the known Kharkats entrance to the desalination channel, L is the length of equation [17], i.e. equation (1), which takes into the active surface of the membrane under investigation, account the effect of exaltation of the current of salt

h is the intermembrane distance, V is the linear velocity counterions by the water dissociation products.

of the flow of solution, T 1 is the effective transport num-

For calculations, the results of which are presented ber of the salt counterion in the membrane, t 1 is the below, in all cases we used the following parameters:

C D A = 1.61 × 10 cm s , T Na = T Cl = 0.998, t Na = 0.396, solution, and F is Faraday’s constant. The superscript

electromigration number of the salt counterion in the

0 and t Cl = 0.604 [50], where the superscript “C” refers to “0” in the designation for the current density i lim

a cation-exchange membrane and the superscript “A” implies that we mean by it a quantity that would have refers to an anion-exchange membrane. Other charac- taken place should there have been no coupled convec- teristics of the EMS under investigation are presented tion.

in Table 2 together with average values of the diffusion Note that equation (3) has an approximate character. layer thickness. These values were calculated with the

A more correct equation has a two-member right-hand aid of equation (4) for the case where there takes place part [40]. However, for simplicity we will use here the solely the forced convection. one-member equation (3) while keeping it in mind that

The ratio between the salt concentration near the the coefficient of proportionality in the right-hand part membrane surface and the salt concentration in the bulk of (3) insignificantly varies with Y. Specifically, at Y = solution in underlimiting current regimes is defined by

lim [40, 49], see equation (A5) in Appendix. equation (3); at Y = 10 –2 , however, its value is close to

10 –4 , this coefficient is equal to 1.47, as it is written in

the ratio 0 i

1.43. In the overlimiting current regimes, however, the ratio

i /i lim defines the extension of the space-charge region Another observation: the Poiseuille distribution of [39, 40]. Bearing this in mind, we deem it possible to

velocity may be violated if coupled convection of solu- state that the magnitude of the i/i lim ratio characterizes tion arises in the system under investigation. In that the degree of development of the concentration polar- case one may expect that the experimental value of the ization. It is suitable to perform normalization of cur- limiting current density could be greater than the quan-

tity calculated with the aid of the convective–diffusion 0 rent density to quantity i lim . The latter is easy to calcu- model. Note that the experimental value of the limiting late with the aid of equation (3). Such a normalization current density is determined approximately from the allows us to compare the behavior of different mem-

Vol. 43 No. 3 2007

PISMENSKAYA et al.

brane systems at similar conditions of the development the separation between the measuring electrodes, the of coupled effects. It also makes it possible for us to membrane thickness, and some other parameters. estimate the influence of one effect or another on their These parameters are frequently not that important for electrochemical behavior.

the behavior of a membrane but are difficult to take into The treatment of experimental data (continued): the account upon going from one membrane system to

potential drop. In the experimental cell we described in another. the foregoing, the total potential drop between two

To compare results obtained during chronopotenti- measuring electrodes ( ∆ϕ tot ) consists of the sum of sev- ometry of different membrane systems, the difference

eral potential drops as follows: between potentials ∆ϕ tot and ∆ϕ Ohm is used. The ohmic

I II constituent ∆ϕ Ohm = iR ohm is found as the potential drop ∆ϕ tot = ∆ϕ DL + ∆ϕ DL + ∆ϕ Don + ∆ϕ m + ∆ϕ sol . (5) between measuring electrodes that is caused by switch-

The first two addends in the sum are the potential ing a current on under conditions where concentration drops that occur in the depleted (designated with super- gradients are absent. In practice, the value of ∆ϕ Ohm is script “I”) and enriched (superscript “II”) diffusion lay- found from a chronopotentiogram by means of extrap- ers (DL). The third addend comprises the Donnan drops olation of ∆ϕ tot at t 0. The ohmic constituent ∆ϕ Ohm of potential on both membrane/solution interfaces comprises the ohmic drops of potential in all layers of ( ∆ϕ Don ). The last two addends refer to the ohmic drops

a system that consists of a membrane, two diffusion of potential on the membrane ( ∆ϕ m ) and in solution lay- layers, and two layers of solution in between the mea- ers that are situated in between the measuring elec- suring electrodes and external boundaries of the diffu-

trodes and the external boundaries of diffusion layers sion layers. The difference between the ohmic resis- ( ∆ϕ sol ). The ∆ϕ m and ∆ϕ sol quantities are in essence tance R ohm and the effective resistance R ef of a system is ohmic drops of potential. The point here is that, in defined by equation (A8). The difference in question dilute solutions, the diffusion drop of potential in the consists of that the effective resistance R ef , apart from membrane may be ignored [1] in view of the smallness the ohmic resistance R ohm , includes diffusion resis- of the concentration gradients. Moreover, in solution tance, which arises upon establishing a concentration layers that are situated in between the measuring elec- profile in EMS [3, 4]. trodes and the external boundaries of diffusion layers,

The treatment of experimental data (continued): concentration gradients are absent altogether.

chronopotentiometry. At current densities below or When performing a comparison of the electrochem- ical behavior of different membrane systems with use equal to i lim , chronopotentiograms for the diffusion-

made of voltammetry, it is convenient to use, instead of controlled EMS with a smooth mass exchange surface the total drop of potential ∆ϕ tot , the quantity ∆ϕ', which have the shape presented in Fig. 3 [34, 46, 48]. The ini- is defined by the equation

tial portion of the curve contains three segments. Seg- ment ‡, confined by point 1 in Fig. 3, is practically ver-

⎛ ∂∆ϕ tot --------------- ⎞ ∆ϕ' tical. Its height is defined by the value of the total ohmic = ∆ϕ tot – i ⎝

∂i ⎠

= ∆ϕ tot – iR ef . (6)

i = 0 resistance of the membrane and two layers of solution that are confined in between the measuring electrodes

Here, R ef =( ∂∆ϕ tot / ∂i) i=0 is the effective resistance of

I II

the membrane system at low current densities, i.e. at and the membrane at c s = c s = c 0 . The slope of this

0 depends on the capacitance of the iⰆ i lim . This effective resistance includes the ohmic electrical double layer (EDL) at the membrane/solution

0 segment at t

resistance of the space (membrane–solution) in interface [52]. Segment b corresponds to a slow build- between the measuring electrodes and the diffusion ing up of the potential until inflection 2. This magnifi- resistance of the depleted and enriched diffusion layers cation of the potential is caused by a decrease of con- [3]. The value of this resistance is found from the slope centration in the solution layer near the membrane that of the initial segment of a CVC.

undergoes desalination. At point 2 the potential rises at The ∆ϕ' quantity had been introduced probably by

a maximum rate. Thereafter the potential growth rate Maletzky et al. [51]. It was called a “reduced polariza- diminishes and the potential drop ∆ϕ reaches a certain tion voltage” or a “reduced voltage of polarization” steady-state value or a quasi-steady-state value (seg- (“corrected polarization voltage” in [51]). We will call ment d). In the quasi-steady state, the potential drop ∆ϕ it by the name “reduced drop of potential.” The ∆ϕ' slowly alters or periodicaly oscillates. The fact that the quantity shows the excess of a drop of potential in a sys- growth of ∆ϕ slows down testifies to the amplification tem over the quantity that would have taken place upon of coupled effects, in the first place, of the coupled con- retaining a linear growth of potential that is observed at vection. These effects reduce the degree of concentra-

i 0. The physical meaning of the ∆ϕ' quantity is tion polarization of the system. Note that in segment d close to overvoltage η [2–4]. Expressions for ∆ϕ' and η the potential drop ∆ϕ is equal to the potential drop that are derived in Appendix. Resorting to the reduced drop is observed in galvanostatic or galvanodynamic CVC of potential ∆ϕ' allows us to exclude from consideration that are recorded at a low rate of the current scan the initial ohmic resistance. The latter is dependent on (~0.01 mA s –1 ). The shape of the time dependences of

Vol. 43 No. 3 2007

COUPLED CONVECTION OF SOLUTION NEAR THE SURFACE

25 mA/cm 2

a 10 mA/cm 2

f 400 t ,s

Fig. 3. Chronopotentiograms for membrane AMX placed in the vertical position in a 0.1 M NaCl solution; h = 7.0 mm, V =

0.32 cm s –1 .

∆ϕ and the character of oscillation of this potential drop concentration in the vicinity of the membrane/solution in segments c and d yield information concerning the interface to drop to such values c s Ⰶc 0 at which cou- type and the scenario of the development of coupled pled effects start manifesting themselves. The action of effects of concentration polarization. The last segment, these effects is manifested in the slowing-down of the denoted as f, is obtained after switching the current off. magnification of the potential drop ∆ϕ. Consequently, It describes the process of diffusion relaxation of the the transition time may be approximately determined system.

from the inflection point 2.

The simplest mathematical model for semiinfinite The potential difference between points 4 and 5 in diffusion toward a flat mass exchange surface is the Fig. 3 (segment e) is equal to the ohmic drop of poten-

Sand model [52]. It takes no account of the coupled tial of a polarized membrane system. The potential dif- effects of concentration polarization. The model has

ference 1 – ∆ϕ corresponds to the change in the parameter τ that is called a transition time. This param- ohmic drop of potential in EMS that is caused by the eter is defined by the following expression [46]:

development of concentration polarization. The major

0 2 πD contribution to this quantity is made by the increment

τ = ⎛ ------- ⎞c ⎛ i z i F ------------- ⎞ 1 --- .

(7) of the resistance of the depleted diffusion layer:

– 1 ∆ϕ )/I. (8) In the framework of this model, parameter τ corre- sponds to the time instant when the electrolyte concen-

∆R δ ≈ (∆ϕ 4–5

tration near the surface of a membrane turns zero and

3. RESULTS AND DISCUSSION the potential drop tends to infinity. The concentration of

3.1. Effect of Concentration Polarization counterions 0 c

i in the solution depth in the systems we on the Gravitational Convection are investigating is presumably none other than the con-

As we have already mentioned in Introduction, a centration c 0 of a 1 : 1 electrolyte at the entrance into the rather simplistic analysis demonstrates that the gravita- cell. As follows from equation (7), the magnitude of the tional convection weakens with decreasing salt concen- product i τ 1/2 is independent of the current density in the tration in solution. The degree, to which the gravita- overlimiting current regimes. This statement is true if tional convection has developed, may be prognosti- the mass transfer in the electrochemical system is con- cated with the aid of values of the Rayleigh number Ra

trolled by electrodiffusion not complicated by coupled and the Richardson criterion Ri = Gr/Re 2 . Calculations effects of concentration polarization. From the view- of these quantities for conditions that were used in this point of some modern notions, the quantity τ may be work were performed with the aid of formula (2) for a defined as the time period required for the electrolyte temperature of 25°ë. The calculations were conducted

Vol. 43 No. 3 2007

PISMENSKAYA et al.

∆ϕ', V

∆pH

proportionally to concentration c 0 . The viscosity ν was

0 set equal to 0.009 cm 2 s –1 . The solution flow velocities, which are required for calculating the Reynolds num-

1 pH ber, appear in Table 2. The estimate was conducted 4 pH

without allowance for possible heating of solution near 2

the interface.

As expected, values of the Rayleigh number and the Richardson criterion diminish with the solution dilution

(Table 2). In the case of a 0.1 M NaCl solution

2 V (system 1 in Table 2), Ra > Ra cr and Gr/Re 2 Ⰷ 1. This implies that the conditions that are required for the

1 emergence of gravitational convection are created if the

depleted diffusion layer is not situated under the mem- brane. In the 0.05 and 0.02 M NaCl solutions (systems 2 and 3 in Table 2), values of Ra and Ri are

2 pH

almost critical. This points to a weak contribution of

4 V gravitational convection. The contribution may be

1 V increased or decreased due to heat evolution or con- sumption near the membrane surface, which is ignored

0 1 2 3 4 by calculations of the Rayleigh and Grashof numbers.

i /i lim 0 The development of gravitational convection in a 0.005 M NaCl solution (system 4 in Table 2) is hardly

probable.

Fig. 4. Dependences of the corrected potential drop of con-

centration polarization and variations in the pH value of a

The obtained experimental data (Figs. 4–6) corrob-

near-membrane layer of depleted solution on the current

0 orate the validity of the estimates we made and corre-

density normalized to the limiting current i lim , which was

late with the results reported in [19, 20]. In the case of

calculated with the aid of equation (3) for the AMX mem-

a 0.1 M NaCl solution (system 1 in Table 2), the shape

brane at h = 7.0 mm and V = 0.32 cm s –1 . The calculations

of the current–voltage curves (Figs. 4, 5) and the chro-

are performed for the following different values of the NaCl

nopotentiograms (Fig. 6) that were recorded at current

solutions: (1 G ,1 V ,1 pH ) 0.1, (2 G ,2 V ,2 pH ) 0.5, and (4 G ,4 V ,

4 pH ) 0.05 M. Points represent data obtained in galvanostatic

densities in excess of i lim substantially depends on the

regime and curves refer to data obtained in galvanodynamic

membrane’s position: vertical or horizontal. (In this

regime with the current scanned at a rate of 0.01 mA s –1 .

case and in what follows, for a membrane in the hori-

Subscripts “h,” “v,” and “pH” denote, respectively, horizon-

zontal position, the electric field is oriented so that the

tal and vertical positions of the membrane under investiga-

lighter depleted diffusion layer is situated under the

tion and the solution pH. Numbers identifying the curves correspond to numbers of systems in Table 2.

membrane under investigation, which practically rules out the emergence of gravitational convection in the vicinity of this membrane.)

under the assumption that characteristic distance X may Let us compare curves 1 G and 1 V in Fig. 5a, which

be estimated as the total thickness of a diffusion layer were obtained for AMX deployed in, respectively, hor- δ tot . Here, the characteristic distance X is the size of the izontal and vertical positions in a 0.1 M NaCl solution. region where the solution density gradient is other than The slope of the portion of the “plateau” in the CVC for zero. As to the total diffusion layer thickness δ tot , it is the AMX membrane positioned horizontally is very determined from the point where salt concentration small and the length of the plateau is large enough,

c x = δ tot differs by a mere 1% from the value of c , which testifies to considerable diffusion limitations that hamper the mass transfer in this portion. The value of which is the salt concentration in the solution depth. In the limiting current density i lim that was found from the

the cases of forced [49] and natural [53] convection of intersection of tangents as described in Subsection solution, the total diffusion layer thickness is ~1.7 times the Nernst layer thickness δ [40]. Then, δ tot =

2.3.2 is very close to the value of i lim that was calcu- lated with the aid of equation (3). The evaluations made

1.7 × 0.028 ≈ 0.05 (cm) for systems 1, 2, and 4 in using formula (A9) (Appendix) suggest that the salt Table 2 and δ tot = 1.7 × 0.025 ≈ 0.04 (cm) for systems concentration c

s near the membrane surface decreases 3a and 3b in the same table. When performing calcula- tions we assumed that the difference between densities at i = i lim by four orders of magnitude as compared

of a 0.1 M NaCl solution (external boundary of a total with the concentration in the depth of solution (the cal- depleted diffusion layer) and pure water (near the inter- culation was made on the basis of the known potential

0 drop at this current density). Upon going from the hor- face, at i ≥ i lim ) is equal to 0.004 g cm –3 and decreases izontal position to the vertical position, which is favor-

Vol. 43 No. 3 2007

317 able for the development of gravitational convection

COUPLED CONVECTION OF SOLUTION NEAR THE SURFACE

i /i 0 lim

(Fig. 5a, curve 1 V ), similar values of the salt concentra- 4 (‡)

tion c 0 s occur solely at i ≈1.55 i lim . The value of the lim- 4 G ,4 V iting current density that is determined from the inter-

section of tangents is approximately 1.5 times the value of 0 i

lim . A subsequent build-up of the preset current 3 2 V 2 G

leads to the appearance of a plateau in curve 1 V in

Fig. 5a. However, the slope of this plateau is smaller than that for the membrane in the horizontal position. These data point to the removal of a part of the diffusion 2 limitations in the case of the membrane positioned ver-

1 V 1 G tically, due to the development of gravitational convec-

tion near its surface.

A twofold decrease in the concentration of the feed- 1 ing solution (system 2 in Table 2) leads to a substantial shrinkage of differences in the behavior of AMX

deployed in a horizontal position (Fig. 5a, curve 2 G ) and in a vertical position (Fig. 5a, curve 2 V ). For the

same membrane in a 0.005 M NaCl solution (system 4 in Table 2), these differences are practically absent 0 1 2

(Fig. 5a; curves 4 G , 4 V ). The absence of these differ-

(b)

ences testifies to a negligibly small influence the gravi- 3 3 ‡ V

tational convection exerts on the mass transfer in this

particular system.

Note that the gravitational convection in the AMX −0.1 M NaCl system starts manifesting itself as early as at current densities below the limiting current 2

3 density. However, the influence exerted by it is not b G

great. At i = 10 mA cm 0 (i/ i

lim = 0.7), for example, the

difference ∆ϕ tot – ∆ϕ Ohm for the AMX membrane in the horizontal and vertical positions does not exceed

0.025 V in the quasi-steady state (Fig. 6; curves 1 G ,1 V ). 1

At higher current densities, this difference is greater. The difference is maximum at such current densities that are limiting or overlimiting in the horizontal posi- tion but are still underlimiting in the vertical position. The case where i = 20 mA cm –2 is a good example. A

calculation with (3) for this case yields i/ 0 i lim = 1.4. The 0 1 2 3 ∆ϕ', V clearly pronounced portion of the transition time (Fig. 6, curve 2 G ) testifies to the reaching of the limiting

Fig. 5. Dependences of current density normalized to the

state by the membrane positioned horizontally. At the

limiting current density 0 i

lim on the corrected potential drop

same time, for the vertical position, the portion of the transition time is absent and the steady-state drop of of concentration polarization for membranes (a, b) AMX

and (b) Nafion-117. Curves in panel a are obtained at NaCl

potential reaches a mere ~0.100 V (Fig. 6, curve 2 V ).

concentrations equal to (1 G ,1 V ) 0.1, (2 G ,2 V ) 0.5, and (4 G ,

As follows from Fig. 6b, within the first seven sec-

4 V –1 ) 0.05 M and h = 7.0 mm, V = 0.32 cm s . Panel b:

onds after switching the current on, the chronopotentio-

0.02 M NaCl, h = 5.8 mm, V = 0.39 cm s –1 . Numbers iden-

gram that was recorded at i = 20 mA cm –2 in the vertical

tifying the curves correspond to numbers of systems in

position of AMX repeats the run of the curve for the

Table 2.

same membrane that was obtained for the horizontal position. This time period may be referred to the time period required for the development of gravitational down, and low-frequency oscillations of potential convection after switching the current on. After the appear in the chronopotentiogram. The period of these “splitting” of curves, the difference ∆ϕ tot – ∆ϕ Ohm oscillations is equal to 7–8 s and their amplitude is low. sharply increases for the membrane in the horizontal Emulating the authors of [37], we deem it possible to position. Conversely, for the membrane in the vertical assume that the period of oscillation of potential corre- position, the growth of this difference sharply slows sponds to the time period that elapses from the instant

Vol. 43 No. 3 2007

PISMENSKAYA et al.

0.15 2 G 2 V

Fig. 6. (a) Chronopotentiograms for the AMX membrane in horizontal (subscript “h”) and vertical (subscript “v”) positions in a

0.1 M NaCl solution ( 0 c NaCl = 0.1 å, h = 7.0 mm, V = 0.32 cm s –1 ), obtained at the following current densities: (1 G ,1 V ) 10 and (2 ,2 ) 20 mA cm –2 G V . Panel b: initial segments of the same chronopotentiograms in an enlarged scale.