Electroosmosis of the second kind and current ... - Semantic Scholar
Colloids and Surfaces A: Physicochemical and Engineering Aspects 181 (2001) 131– 144 www.elsevier.nl/locate/colsurfa
Electroosmosis of the second kind and current through curved interface N. Mishchuk a,*, F. Gonzalez-Caballero b, P. Takhistov a a
Institute of Colloid and Water Chemistry of Ukrainian National Academy of Sciences, 42,Vernadsky pr., Kyi6, 03680, Ukraine b Department of Applied Physics, Uni6ersity Granada, Fuentenue6a s/n, ES-18071, Granada, Spain Received 17 March 2000; accepted 29 August 2000
Abstract The article focuses on the principles of strong concentration polarisation of flat and curved interfaces including the phenomenon of the limiting current. It is shown both theoretically and experimentally that electroosmosis of the second kind near the curved interface causes an important change of characteristics of concentration polarisation and a rapid growth of the current through an interface instead of current saturation in case of flat surface. At the same time, electroosmosis of the second kind can lead to such a strong decrease of concentration polarisation that the conductivity of ionexchange materials becomes a limiting factor. © 2001 Elsevier Science B.V. All rights reserved. Keywords: Concentration polarisation; Electroosmosis of the second kind; Limiting current; Desalination
Nomenclature List of symbols and abbreviations a b C0 CI CDL D0 Di E F
radius of fibre or ion exchange granule dimensionless parameter electrolyte concentration electrolyte concentration inside of ion exchange material convective-diffusion layer diffusion coefficient in electrolyte diffusion coefficient inside ion exchange material strength of electric field Faraday constant
* Corresponding author. Tel.: + 380-44-4504415; fax: + 380-44-4520276. E-mail address: [email protected] (N. Mishchuk). 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 7 4 1 - X
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h i IÞ I ilim L R t T V x l 0
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distance between membranes, granules or fibres density of the current current along the normal to the fibre current along the fibre limiting current length of fibre gas constant time temperature electroosmotic velocity of liquid distance along the fibre counted from the free end of the fibre thickness of convective-diffusion layer potential drop between the membranes, granules or fibres
1. Introduction Electrokinetic phenomena of the second kind were predicted theoretically [1 – 3] and have been investigated and corroborated experimentally for different types of dispersed particles [4 –7]. The theoretical prediction of electrokinetic phenomena of the second kind includes two aspects: (a) strong non-linearity of electroosmosis and electrophoresis; (b) unlimited growth of the current through a polarised interface. According to the first aspect of the theoretical prediction, electroosmotic and electrophoretic velocities are proportional to the particle size and approximately proportional to the squared value of electric field strength and, therefore, can reach values many times larger than Smoluchovski’s velocity. Measurement of electroosmotic and electrophoretic velocities has shown that the experiment corresponds well with the theory over a large range of voltage, electrolyte concentration, particle sizes and types of investigated particles [4–8]. According to the second aspect of the theoretical prediction, electroosmosis of the second kind is not only the result of concentration polarisation of an interface, but it also affects concentration polarisation and considerably changes its characteristics, including the current through a polarised region [1,9]. If, in the case of a flat
interface, concentration polarisation is accompanied by the phenomenon of the limiting current (an increase of voltage leads to saturation of the current: starting from a certain value of voltage its increase is very weak or impossible), in the case of a curved interface, due to electroosmosis of the second kind, the current growth is unlimited. This theoretical prediction [1,9] can be partially proved by the analysis of published experimental results in the field of electrodialysis intensification by the use of material with a curved surface [10,11]. However, these experimental works only indirectly confirm the idea discussed in the present paper. The objective of the present study is to develop the scheme of experimental investigation of the current under conditions of electroosmosis of the second kind and to obtain the main theoretical and experimental characteristics of the investigated processes. Experimental cells are developed for investigation of polarisation processes between: (a) two flat ion exchange surfaces (anion and cation exchange membrane); (b) two spherical surfaces (cation and anion exchange granules); (c) flat and spherical surfaces (a cation exchange granule and an anion exchange membrane; an anion exchange granule and a cation exchange membrane); (d) two cylindrical surfaces (cation and anion exchange fibres).
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2. Theory
2.1. Strong concentration polarisation of flat interface Limiting current Peculiarities of concentration polarisation of ion-exchange membranes (i.e. a change of electrolyte concentration near interface in electric field) have been examined in numerous theoretical and experimental papers [12 – 16]. This phenomenon is caused by the difference between ion transfer numbers in an electrolyte solution and in an ion exchange material. The electric current, passing through an ideal selective ion exchange material (ion exchange membrane) placed in an aqueous solution of electrolyte, is realised only by the migration of counterions (anions for anion exchange membrane; cations for cation exchange membrane). Thus, the transport properties of the ion exchange material and the electrolyte solution (where electric current is realised by both cations and anions) are not identical. The counterions move to and through the membrane, while the coions cannot move through it. The concentration of the coions decreases since their withdrawal is not compensated by the intake from the membrane. In a small field electroneutrality is preserved. The coions concentration decrease leads to the equivalent counterion concentration decrease, that is, the region of the low electrolyte concentration appears. However, the continuity condition for cation and anion fluxes during the transition from one medium into another cannot be satisfied only by means of their migration in the electric field. As a result, a diffusion flow from the electrolyte volume to the membrane surface occurs. In this way the diffusion layer is formed. At low voltage the picture of concentration polarisation is asymmetric: the change of electrolyte concentration near the opposite sides of ion exchange membranes has equal values and different signs (Fig. 1). However, these equal changes lead to different influence on the current density. This can be shown by the analysis of local values r1,2 of specific resistance near the opposite membrane surfaces, where the deviation of the concentration DC from the bulk value C0 is maximum.
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Without concentration polarisation, when the voltage is almost equal to zero, specific resistance is equal to r10 = r20 1/C0, while at larger voltage, with concentration polarisation, it takes the following values: r1 1/(C0 + DC) and r2 1/(C0 − DC). Thus, r1/r10 = C0/(C0 + DC) and r2/r20 =C0/ (C0 − DC). For example, at DC = 0.5C0, r1/r10 : 0.7 and r2/r20 : 2 or r2/r1 : 3. At larger voltage the difference between specific resistances sharply increases. For example, at DC = 0.9C0 and DC = 0.95C0 the ratio r2/r1 is equal to 20 and 40 respectively. Thus, the concentration decrease causes a considerable potential drop in the polarised regions and consequently limits the density of current. When the decrease of electrolyte concentration DC is close to C0, the density of the current is caused by the sum of diffusion and electromigration fluxes that is equal to i=
2FDC0 l
(1)
where F is Faraday constant, D the coefficient diffusion of ions, C0 the electrolyte concentration, l the thickness of region of concentration polarisation (convective-diffusion layer, CDL). In the case of the flat interface, the value d depends on external hydrodynamic flow or thermoconvection in the cell and is almost independent of voltage [12,14]. Thus, an increase of voltage does not result in a growth of the current. This is so-called ‘‘limiting current’’.
Fig. 1. Schematic representation of concentration polarisation of flat ion exchange membrane (1 — low voltage, 2 — high voltage).
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especially in the case of electroosmosis of the second kind, which will be discussed in the following section.
2.2. Strong concentration polarisation of cur6ed interface. Unlimited current under conditions of electroosmosis of the second kind
Fig. 2. Scheme of experimental cell with anion exchange and cation exchange membranes.
In real systems the growth of the current above the limiting value occurs. It is caused by water splitting [17,18] and the decrease of the thickness of convective-diffusion layer due to the appearance of the region of an induced space charge [2,3,12,14] and hydrodynamic flow [19]. However, the growth of the current for a flat interface is not very large. Taking into account the above analysis of specific resistance, one can see that the influence of the region of low concentration (desalination) on the general picture of polarisation is stronger than the influence of the opposite region of high concentration. Therefore, the present paper is devoted to the development of different schemes of experimental investigations that allow to study concentration polarisation related to desalination. The first scheme is traditional (Fig. 2), that is, a cell with two parallel anion and cation exchange membranes is used. Electrodes are connected with the power source in the regime of desalination. Thus, the concentration of electrolyte between the membranes decreases. To simplify the calculation of the thickness of CDL, a cell without hydrodynamic flow is used. In this case l = h/2, where h is the distance between membranes. The current is proportional to the membrane surface Sm and can be described as I = iSm =
2FDC0 4FDC0 Sm = Sm l h
The main feature of concentration polarisation of the ion exchange material in strong electric field [12,14,20] is that the concentration decrease leads to the deviation from electroneutrality. This property is independent of the interface profile or the homogeneity of conductivity of the ion exchange material. Yet, when the appearance of an induced space charge is combined with heterogeneous conductivity of the flat interface or the curved shape of the homogeneous surface of the solid phase, the current lines bend round non-conductive or low conductive parts of the flat surface or, otherwise, the curved surface of a particle. Thus, a component of the electric field, tangential to the interface, appears. Together with the induced space charge it causes a non-linear electroosmotic movement of liquid (Fig. 3) [1–3,5 –7,15,20] and non-linear electrophoresis [4,8]. Over a rather large range of parameters the velocities V of these phenomena (electroosmosis and electrophoresis of the second kind) are proportional to the particle radius a and squared value of electric field strength E 2. In particular, the velocity of electroosmosis V is approximately
(2)
The current/voltage dependence changes for the curved surface where the thickness of CDL is related to the electroosmotical mixing of liquid,
Fig. 3. Scheme of liquid flow near spherical particle under conditions of electroosmosis of the second kind.
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equal to mD(2FEa/RT)2/a, where for aqueous solution the parameter m =0.2. Taking into account that the thickness of CDL near the spherical particle depends on Peclet number Pe or the liquid velocity V [1 – 3] as l=
a
'
aD 1 V E
(3)
Pe one can see that electroosmosis of the second kind reduces the thickness of CDL inversely proportionally to the electric field strength. The current through the curved interface is related to the polarisation process and preserves its diffusion nature similarly to the current through the flat interface (1), but the thickness of CDL (3) decreases and causes the growth of the current. Considering Eqs. (1) and (3) it is clear that in the case of the spherical surface the density of the current is a linear function of the applied field i=
=
2FDC0 E l
(4)
However, it is difficult to measure the current through a particle and interpret the findings correctly. Therefore, in the following section we will introduce the scheme of the measurement of the current for the pair of particles that allows to overcome the obstacles.
Fig. 4. Scheme of experimental cell and electroosmotic flows with anion exchange and cation exchange granules.
namic fields that cannot be correctly described by the theoretical model. Another problem is created by the bipolar contacts that can cause water dissociation [17,18,21] and, as a result, change the overall picture of concentration polarisation. To eliminate these difficulties, another scheme of an experimental cell for the investigation of concentration polarisation between two curved surfaces is developed (Fig. 4 a). The electroosmotical mixing of liquid between granules determines the thickness of CDL and the density of the current. The theory in paper [9] was developed for polarisation of granules without the influence of membranes, therefore, the value of l derived in the paper [9] can be used for the purposes of the present study. The value decreases inversely proportionally to the applied voltage 0 10a ˜0
2.3. Electroosmosis of the second kind between two spherical surfaces. Unlimited current
l=
Electroosmotic movement of liquid between two ion exchange granules placed between two ion exchange membranes [9] is similar to the movement near a separate granule described in the Section 2.2 (Fig. 3). Here, the potential difference is fixed between membranes and non-homogeneity of the electric field is caused by the geometry of the space between membranes and granules. As a result, an electroosmotic flow near the surfaces of ion exchange granules occurs. However, it is difficult to undertake the analysis on the basis of the scheme presented in paper [9] because the space near the granules is limited by the presence of the membranes. This leads to the noticeable change of the electric and hydrody-
where ˜ 0=
(5)
F0 . RT Thus, similarly to the Section 2.2 we obtain the unlimited density of the current i=
2FD0C0 ˜0 10a
(6)
In the case, when the thickness of walls d that limit electrode chamber is small (d a), they do not affect electroosmosis, thus, the current can be calculated as I: iSg
(7)
where Sg = 2ya 2 is half the surface of the spherical particle.
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The comparison of the experimental results for flat (Section 2.2) and spherical (Section 2.3) ion exchangers with their theoretical models will enable better understanding of the peculiarities of polarisation processes for different shapes of an interface. Additional information about these processes can be obtained by the simultaneous investigation of flat and curved interfaces.
2.4. Electroosmosis of the second kind between spherical and flat surfaces. Unlimited current The scheme of the experimental cell with the flat and curved interfaces (Fig. 5 a, b) is similar to Fig. 2, and Fig. 4 presented above. The mathematical description of the processes in this cell is quite complicated. While in the case of two granules every granule is responsible for the structure of its own CDL, in the case of granule and membrane the electroosmotic flow of granule affects the CDL of the membrane. Due to the peculiarities of the electroosmotic flow between the granule and membrane, the thickness of CDL
Fig. 5. (a) Scheme of experimental cell and electroosmotic flows with cationexchange granule and anionexchange membrane. (b) Scheme of experimental cell and electroosmotic flows with anionexchange granule and cationexchange membrane.
Fig. 6. Scheme of experimental cell with cation exchange and anion exchange fibres.
of the membrane noticeably changes along its surface (Fig. 5a, b). The qualitative analysis of the processes in such systems will be undertaken in the Section 3.
2.5. Electroosmosis of the second kind between two ionexchange fibres. New factor that limits the current through interface The models for flat and curved surfaces presented above are based on the notion of the large conductivity of the ion exchange material and relatively low conductivity of the liquid. This corresponds to many systems, where the investigation of the current through the ion exchange material is both of scientific and practical interest. At the same time, it is well known that, due to low values of diffusion coefficient, the conductivity of ion exchange materials cannot be indefinitely large and considerably depends on the conductivity of the liquid that is used [22]. Therefore, the present paper aims at the combination of the analysis of the external and internal transport processes. This can be realised by the investigation of the current through ion exchange fibres connected to the power source as it is shown in the Fig. 6. Two cation and anion exchange fibres are parallel. On the one side their ends are inserted into electrode chambers, while on the other side the ends are held with the aid of a nonconducting material for fixing of the distance between the fibres. The potential difference 0, established by the external source, is distributed along and between the fibres. It has a complex nature because of the
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different electroconductivity of the electrolyte in the central and electrode chambers and in the ion exchange fibres. Owing to the action of electric field, the cations and anions move to the fibres and through them to the electrode chambers. As a result, the regions of low electrolyte concentration in the space between fibres near its surfaces can be observed. The thickness of region of concentration polarisation depends on the potential drop between the fibres and the velocity of the hydrodynamic flow near their surfaces. In the strong electric field the thickness of region of concentration polarisation is defined by electroosmosis of the second kind, whereas in the small electric field, when electroosmosis of the second kind is absent or very small, it is defined by the velocity of the external hydrodynamic flow or is equal to half the distance between the fibres in the absence of the external flow. For the mathematical description of the analysed system it is assumed that all the properties of the ion exchange fibres are identical and have an idealised nature, that is, the current through the fibres is provided only by counterions. The fibres are connected to the power source in the regime of desalination. The current will be analysed in the three models: (a) without concentration polarisation; (b) with concentration polarisation in the absence of electroosmotical mixing of electrolyte; (c) with concentration polarisation under conditions of electroosmosis of the second kind. (a) If there were no concentration polarisation, then the current outside the fibres would be determined by the electric field strength, i.e. it would be proportional to 0 I0 =iSf =2FDC0yaL ˜ 0/h
(8)
where Sf = yaL is half a fibre surface (the current is possible only through the part of fibre surfaces facing each other) and a,L are the fibre radius and the part of their length inside the chamber of desalination. (b) With the presence of concentration polarisation and in the absence of hydrodynamic flow, the diffusion layer thickness between the fibres l0 would be equal to h/2. This means that the liming current would be equal to
Ilim =
2FDC0 4FDC0 yaL yaL= l0 h
137
(9)
and it is smaller ˜ 0/2 times in comparison with the case when concentration polarisation is absent (8). For example, at 0 = 5V the relation I0/Ilim is equal to ˜ 0/2=100. (c) The curved surface of the fibres is similar to the surface of spherical granules. Thus, electroosmotic slip of the second kind appears and leads to significant compression of CDL: l h/2. Its thickness can be evaluated by expression (5) and the current can be described as Il =
FDC0 ˜ 0yaL 5a
(10)
i.e., it is smaller h/10a times than without concentration polarisation (8) and it is ˜ 0h/10a times larger than in the case of concentration polarisation without electroosmotic slip of the second kind (9). However, with such intense external diffusion transport (i.e. the transport of ions outside the fibres), the internal diffusion transport of ions becomes a new limiting factor. If conductivity of the fibres is not very large, they are not able to carry away all the current that can be provided by the ion fluxes to the fibre surface. The theory of concentration polarisation of fibres with arbitrary conductivity will be developed below. For the purposes of simplification of the theory a few approximations will be used. The object of the study is the long fibres placed in electrolyte with low conductivity with short distance h between them (hB B L). This allows us to describe the characteristics of concentration polarisation between the fibres by the solution of a local one-dimensional task and to divide the current into two components: the current in the electrolyte that is perpendicular to the fibre surface and the current in the fibre that is directed along the longitudinal axis. Hence, the current in the fibre can be presented as ( (x
III = FDi Ciya 2
(11)
where Di,Ci are the diffusion coefficient and the concentration of the current carriers (counterions) inside (i ) the fibre and l/lx is the gradient of
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the potential along the axis of the cylindrical fibre (measured from the free end of the fibre, see Fig. 6). The current along the normal to the fibre is equal to the product of the current density iÞ(x) by the surface area, through which the current flows, yaDx 2FDC0 IÞ = yaDx =JÞDx l(x)
(12)
where the current density is determined by the thickness of CDL l(x) specified by the electroosmotic flow that depends on x too. On the basis of Eq. (5) the local value of thickness of CDL is determined by the magnitude of the local potential difference ˜ (x) applied to each fibre 5a l(x)= ˜ (x)
(13)
The current along the fibre is replenished by the normal flux of ions with displacement from x=0 to x= L and reaches its maximum value at x= L. The current continuity condition should be satisfied in every point of the fibre surface. It can be presented as III (x +Dx)− III (x) = JÞDx
or
(III =JÞ (x (14)
Considering Eqs. (11) and (12) and normalisation of x on L: x˜ =x/L, Eq. (14) is reduced to the equation for the potential distribution along the fibre ˜ (x˜ ) ( 2 =b(x˜ ) (x˜ 2
(15)
where b= 2C0D0L 2/5DiCia 2 is a dimensionless parameter that determines the relative characteristics of the electrolyte and the fibre. The solution of this equation has the form of ˜ (x)= ˜ 1exp( − bx˜ ) + ˜ 2exp( bx˜ )
(16)
where the coefficients ˜ 1, ˜ 2 are determined from the boundary conditions, that is, the potential at the point where the fibre enters the electrode chamber ˜ (x˜ = 1) is equal to ˜ 0/2. Here, it is assumed that the property of the fibres are similar and, consequently, the potential difference is distributed uniformly between them. It is necessary
to take into account that the longitudinal current at the fibre’s free end is equal to zero III (x˜ =0)= 0. As a result, the solution takes the form of ˜=
˜ 0 exp(− bx˜ )+ exp( bx˜ ) 2 exp(− b)+ exp( b)
(17)
In the limiting case b 1 Eq. (17) can be rewritten as ˜=
˜0 2
(18)
Hence, the fibres are isopotential. In the opposite limiting case b 1, the following laws of potential variation near x˜ 0 and x˜ 1 are observed:
) )
˜0 exp(− b) 2 ˜ ˜ x˜ 1 $ 0[1− b(1− x˜ )] 2 ˜ x˜ 0 $
(19) (20)
i.e. in the first approximation the value of the potential near the free end of the fibre is constant (but significantly smaller than the value ˜ 0/2 that is given by the source of power) and the reduction of the potential near the point of the fibre attachment at the entrance to the electrode chamber is linear. It follows from Eqs. (19) and (20) that the potential decreases more rapidly, the larger the parameter b, i.e., the higher electrolyte concentration and the thinner and longer the fibre. But this means that the smaller is the current normal to the fibre surface, the less productively the fibre surface is utilised. Results of the numerical study of the potential (17) are shown in Fig. 7a. Using (17) and (11), we can represent the current as III (x) ˜0 exp( bx˜ )−exp(− bx˜ )
b 2L exp(− b)+exp( b)
= FD1C1ya 2
(21)
Thus, near the exit from the cell the current takes the value of III (x˜ =1)=
FD1C1 2 ya tan h( b) b ˜0 2L
which is measured in the experiment.
(22)
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It follows from the expression (22) that the current in the system is independent of the coefficient of diffusion and the concentration of the current carriers in the fibre under the condition that tanh( b) : b 1: III (x˜ =1)=yFD0C0L ˜ 0/5
(23)
that coincides with Eq.(Eq. (10)). In the opposite case b \ \1 Eq. (Eq. (22)) can be reduced to
'
III (x˜ =1)=F
D1C1D0C0 ya ˜0 10
(24)
The correlation h between the currents, calculated according to the model taking into account the internal resistance of fibres (22) and neglecting it (10)
h=
139
tanh b
(25)
b
is shown in Fig. 7b. The use of the scheme presented in Fig. 6 allows additional measurements that can be used for the analysis of the process. It is possible to use many pairs (n) of fibres, and, consequently, to increase the volume of the liquid, that the process of water desalination in a chamber with fibres can be analysed. As it was mentioned before, the fibres are connected to the electrodes in the desalination regime, i.e., the concentration of the electrolyte in the cell decreases. The magnitude of the reduction of electrolyte concentration is determined by the current III (x˜ =1), the desalination chamber volume Vc and the time t counted from the beginning of desalination dC(t) nIII (x˜ = 1,t) dt = FC0V C0
(26)
where the current III (x˜ =1,t) (22) is related with electrolyte concentration by the function b(C(t)) b(C(t))=
2D0C(t)L 2 = bC0 (t) 5D1C1a 2
(27)
where C0 (t)=C(t)/C0. As a result, the differential equation is obtained: dC0 (t) = nqtanh bC0 (t)· bC0 (t) dt
(28)
Di Ci ya 2 ˜ . C0Vc 2L 0 The solution of Eq.(Eq. (28)) is
where q=
1 C0 (t)= arcsinh 2(sinh bexp(− n q bt/2)) b
(29)
In the limiting cases this solution can be reduced to simpler forms
C(t)=C0 1− Fig. 7. (a) Distribution of potential along the fibre. The numbers of curves corresponds to following values of parameter b: 0.1 (curve 1); 0.3 (curve 2); 1 (curve 3); 3 (curve 4); 10 (curve 5); 30 (curve 6). (b) Correlation a between the theoretical values of the current calculated with and without internal conductivity of fibres as a function of parameter b.
ny ˜ 0at 2Vc
'
D0Di Ci 10C0
2
for b1 (30)
and C(t)=C0exp(−
y ˜ 0D0Lt ) 5V
for b 1
(31)
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140
It is necessary to stress that the approximation (30) is valid only for a small time interval because as t , the value b(C(t)) decreases to such extent that, instead of condition b \ \1, bB B 1 is obtained. Hence, it is necessary to use the solution (31). The obtained expressions for C(t) create an opportunity to evaluate the current in the system. The dependence of the current on the time is described by the following law:
'
III (t) =F
Di Ci D0C0 ny ˜ 0at ya ˜ 0 1− 2Vc 10
'
D0Di Ci 10C0
for b1 and
'
III (t)=F
2
(32)
Di Ci D0 ny ˜ 0D0Lt ya ˜ 0C0exp 10 5Vc
for b1
(33)
3. Experiment
3.1. The role of surface shape The experimental investigations of the current according to the different schemes presented in the Fig. 2, Fig. 4, and Fig. 5 were carried out with the use of membranes and granules placed at the same distance between them h =0.06 cm. The area of the membrane surface was 10 times larger than the area of the granule surface. This allowed to analyse the tenfold difference between the density of the current through the curved and flat surfaces. Considering that due to electroosmosis the density of the current through a granule is higher than through a membrane, the area of membranes is big enough to compensate the low density of the current through them. The space between ion exchangers was filled with the electrolyte of low conductivity (C0 = 0.001 and 0.002 mol/l), whereas in the electrode chambers the electrolyte with high conductivity (0.1 mol/l) was used. This means that the potential drop in the electrode chambers can be neglected. Moreover, in this case the relative role of
Fig. 8. Current as function of applied voltage in different schemes: 1 — two ion exchange flat membranes, 1% — theoretical model of limiting current, 2 — anion exchange granule/ cation exchange membrane, 3 — cation exchange granule/anion exchange membrane, 4-two ion exchange granules; 4% — theoretical dependency Eq. (7), 4%% — theoretical dependency with account peculiarity of electroosmosis near anion exchange granule Eq. (34), 4%%% — theoretical dependency with account of stagnant region.
concentration polarisation on the opposite sides of membranes (or granules) is lower than in the case of equal concentrations. Due to the high conductivity of the ion exchange material the potential drop in the membranes and the granules can be neglected, too. Thus, the potential drop is completely related to the space between the two membranes (two granules or membrane/granule). Taking into account this statement, an experimental analysis of polarisation related to the intermembrane (inter-granule) region can be undertaken. The results of such investigation are shown in Fig. 8.
3.1.1. Current through two membranes The current through two membranes is shown in Fig. 8 (curve 1). It is quite close to the theoretical value of the current (curve 1%). The deviation between the two curves is not very large for the traditional analysis of the current through membranes. The results for large field are not shown, because the current is accompanied by strong change of pH. As a result, the transport of H+ and OH− considerably increases the current.
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3.1.2. Current through two granules One can see that the current through the two granules (curve 4 in Fig. 8) is considerably higher than through the two membranes (curve 1). Nevertheless, the values of the current are lower than they should be according to the theoretical prediction (curve 4%) based on Eq. (7). This discrepancy can be explained by the action of the two factors: the property of the anion exchange material and the peculiarity of the experimental cell. The first factor is related to the difference between polarisation of cation and anion exchange materials. This phenomenon is well known in the field of polarisation of the ion exchange membrane and the experimental investigations of electrokinetic phenomena of the second kind. The anion exchange material is characterised by the lower velocity of electroosmosis and correspondingly the smaller size of electroosmotic whirl than the cation exchange material. This result was obtained by systematic investigation of electroosmosis for different types of anion exchange materials [5–7]. Similar results have been shown in electrophoresis: the velocity of anion exchange particles is considerably lower than the velocity of cationexchange ones [8]. This effect can be explained by the influence of two processes. The first of them, the catalysis of water dissociation by surface groups of the anion exchange material [17,18], leads to the decrease of the induced space charge and, as a result, to the decrease of electroosmosis velocity. However, it is necessary to stress that the change of pH was smaller than in the case of two membranes (The values of pH cannot be shown here, because the design of experimental cell does not allow to carry out these measurement). The second process that decreases velocity of electroosmosis is the concentration of ionised impurities (anions) in the region of the induced space charge [23]. As the anions play the part of coions, they reduce the induced space charge and electroosmosis velocity. Due to the both processes, the thickness of CDL layer is larger and the density of the current for the anion exchange granule is lower than the density for the cation exchange one. Taking into consideration that the experimental values of velocity is approximately 4 times lower than it is
141
predicted by the theory and that the thickness of CDL is the function of Peclet number Pe V/4, the thickness of the diffusion layer can be roughly evaluated as l
1
V/4
2
V
(34)
It is approximately twice larger than the thickness that was used in the calculations of the curve 4%. Thus, according to Eq. (4), the current should be roughly twice lower. This result is presented in the Fig. 8 as the curve 4%%. Notwithstanding the fact that the curve 4%% is near the experimental one, to analyse the processes more thoroughly, the evaluation of the current decrease caused by another factor related to the construction of the experimental cell (curve 4%%%) was introduced. In the case of a ‘‘free’’ granule the electroosmotic flow of the second kind is realised along more than half a particle surface (Fig. 3). But in the case of granules fixed by a special tube (Fig. 4, Fig. 5) the artificial factor that deteriorates the free liquid flow is received. In the space near the contact with the tube the so-called stagnant region appears. As a result, the electroosmotical flow cannot reduce the thickness of CDL in this place. It becomes many times larger than the thickness predicted for the ‘‘free’’ particle. Thus, this part of the granule surface is lost for the high density of the current. Taking into account the correlation between the thickness of the tube wall and the size of the granule in our experiment, the conclusion is made that due to the stagnant region about 30% of the current is lost. The curve 4%%% combines both factors, that is, the decrease of the current due to the peculiarities of the anion exchange material and the stagnant region, and appears below the experimental curve. Thus, it can be stated that the experimental results are in good agreement with the theoretical evaluations.
3.1.3. Current through the membrane and granule The values of the current (Fig. 8, curves 2,3) for the two versions (Fig. 5a,b) of the granule/membrane system are close to the values for two membranes (Fig. 8, curve 1). The current for the anion/exchange granule and cation exchange
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membrane (curve 2) is lower than for the opposite versions of the granule and the membrane (curve 3). Similarly to the case of 2 membranes, the results for large field are not shown, because the current is accompanied by strong change of pH. The change of pH for the case of the cation/exchange granule and anion exchange membrane is larger. The correlation between the curves 2 and 3 can be explained by the lower velocity of electroosmosis and correspondingly the lower current density through the anion exchange granule (see the explanation above). Due to lower velocity the electroosmotic whirl is smaller and, consequently, the influence of electroosmosis on CDL of a membrane is weaker. The qualitative picture of this process is presented in Fig. 5a,b. One can see the difference between the whirls and the thickness of CDL in both cases (we did not show here the thickness of the convective-diffusion layer of the granule because it is considerably lower and cannot be shown on the same scale). Therefore, the polarisation of the flat surface is a limiting factor. Even the area of the membrane, which is 10 times larger than the area of the granule surface, is not sufficient for the realisation of the current that can be provided by the spherical interface.
than the fibre diameter. Yet, it was small enough to produce the agitation of the majority of the electrolyte volume in the cell. As the investigation of the mass transfer through the fibres under the action of electroosmotic flow is of primary interest, the experiment was carried out at the conditions when the velocity of electroosmotic slip Veo was higher than the velocity of hydrodynamics flow Vg. The experimental data for the current through the ion exchange fibres are presented in the Fig. 9 and Fig. 10. It can be observed that the current is higher than in the case of the granules that is explained by the larger area of the fibres surface. However, the surface area of the fibre is approximately 300 times higher than the surface area of the granule, while the current, for instance, at 50 V is only 15 times higher. This considerable differ-
3.2. The role of internal conducti6ity In the experiment 5 pairs of antipolar fibres (radius 0.08 cm and length 3.5 cm) were used. The antipolar fibres were disposed at the distance of 0.15 cm and equipolar ones at the distance of 0,1 cm. Similarly to the scheme for granules and membranes (Fig. 2, Fig. 4, and Fig. 5), for the simplification of the analysis of the experimental results it is expedient to use an electrolyte with low conductivity in the central chamber and another one with high conductivity in the electrode chambers. The recirculation scheme of the electrolyte solution with the linear velocity of the flow Vg = 0.1= 0.1 cm/s was applied. The total volume of the liquid was equal to 32 cm3. The optimum size of a cell was used: to prevent distortion of the electroosmotic fluxes, it was several times larger
Fig. 9. (a) Current through 2 ion exchange fibres as function of applied voltage: curve 1 — experiment, curve, 2 – 4 theoretical curves for different value of diffusion coefficient inside the fibres: Di =10 − 8 cm2/sec (2), 2·10 − 8 cm2/sec (3) and 4·10 − 8 cm2/sec (4). (b) Time dependency of the current through two fibres. The numbers of curves are the same as in the Fig. 9a.
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In the present experiment the fibres are equilibrated by the low electrolyte concentration, and only a small part of their ends is immersed in the electrolyte with high conductivity. As a result, their internal conductivity is caused mainly by the low electrolyte concentration and therefore it is lower than the conductivity of the granules. Moreover, it changes along the fibres. Since the exact values of electroconductivity or diffusion coefficients are unknown, numerical calculations were carried out for a few values of the diffusion coefficient inside the fibres which, according to different papers (for example, [24]), are close to our experimental conditions. However, it is necessary to emphasize that for the analysis of the differences between the results for the granules and for the fibres the most important factor is the radius of the fibres, because the resistance of the fibres and parameter b are inversely proportional to the area of their cross section. Thus, there is a possibility of an increase of the current due to the larger surface area of the fibres (yaL for the fibres is noticeably larger than the surface area 2ya 2 for the granules), although, at the same time, there is a limiting factor related to the identical area of cross section ya 2. Fig. 10. Theoretical and experimental dependencies of desalination on the time for different electrolyte concentrations: (a) C =0.001 mol/l; (b) C =0.002 mol/l. The numbers of curves are the same as in the Fig. 8.
ence is caused by the low internal conductivity of the fibres. It is well known that the internal electroconductivity of the ion exchange material depends on the concentration of electrolyte in which it is placed [24]. This dependency is very strong and can lead to the alteration of the ion exchanger conductivity in tens or hundreds times. In the experimental scheme presented in Fig. 4 and Fig. 5 the granules are in equilibrium with the two electrolytes. In the electrode chamber the electrolyte concentration is considerably higher than in the central chamber between the granules. This leads to certain intermediate conductivity of the granules.
4. Conclusions On the basis of the developed theoretical and experimental investigations the conclusion is made that the main peculiarities of concentration polarisation of a curved interface considerably differ from the peculiarities of concentration polarisation of a flat one. Electroosmosis of the second kind is the important factor that affects the process of concentration polarisation of curved interface and leads to the decrease of thickness of convective-diffusion layer and to the considerable increase of the current density. The results that were obtained can be used for the intensification of electrodialysis, the efficiency of which is rather low because of the phenomenon of the limiting current.
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Acknowledgements The authors are grateful to the International Association for the Promotion of Cooperation with Scientists from the Commonwealth of Independent States for the financial support of this investigation in the framework of the projects INTAS 95-IN/UA-165.
References [1] S. Dukhin, N. Mishchuk, Colloid. J. (USSR) 51 (1989) 659. [2] S. Dukhin, N. Mishchuk, Colloid. J. (USSR) 52 (1990) 452– 456. [3] N. Mishchuk, S. Dukhin, Colloid. J. (USSR) 51 (1989) 497– 501. [4] A. Baran, Ja. Babich, A. Tarovsky, N. Mishchuk, Colloids and Surfaces. 68 (1992) 141–151. [5] N. Mishchuk, P. Takhistov, Chem.Tench. Water (Ukraine) 13 (1991) 106–110 Also pages 707– 715. [6] N. Mishchuk, P. Takhistov, Colloids and Surfaces A95 (1995) 119. [7] N. Mishchuk. Colloids and Surfaces, A (1999).
.
[8] N. Mishchuk, A. Barany, A. Tarovskij, F. Madai, Colloids and Surfaces. 140 (1998) 43. [9] S. Dukhin, N. Mishchuk, J.Membr.Sci. 79 (1993) 199. [10] O. Kedem, Desalination 16 (1971) 105– 118. [11] E. Korngold, Desalination 16 (1975) 225. [12] I. Rubinstein, L. Shtilman, J. Chem. Soc. Faraday Trans. II 75 (1979) 231. [13] I. Rubinstein, E. Staude, O. Kedem, Desalination 69 (1988) 101– 104. [14] V. Nikonenko, V. Zabolotski, N. Gnusin, Electrochemistry (Russia) 25 (1989) 180– 186. [15] I. Rubinstein, Phys.Fluids A3 (1991) 2301. [16] V. Zabolotsky, V. Nikonenko, Electrochemistry (Russia) 32 (1996) 246– 254. [17] R. Simons, Electrochim.Acta 30 (1985) 275. [18] Y. Tanaka, M. Seno, J. Chem. Soc. Faraday Trans. 82 (1986) 2065. [19] S. Dukhin, Chem. Techn. Water (Ukraine) 11 (1989) 675. [20] N.A. Mishchuk, Colloids and Surfaces 140 (1998) 75. [21] V.I. Zabolotskij, N.P. Gnusin, N.V. Sheldeshov, N.D. Pismenskaja, Electrochemistry (Russia) 21 (1985) 1059. [22] V.K. Varentsov, M.V. Pevnitskaja, Izvest. Sib. Ot. Akad. nauk SSSR. Ser. Khimiya 4 (1973) 134. [23] S.S. Dukhin, N.A. Mishchuk, N.A. Klimenko-Meshkova, V.N. Goncharuk, Chem. Techn. Water (Ukraine) 11 (1989) 885. [24] N.P. Gnusin, V.D. Grebenjuk, Electrochemistry of Granulated Ionites, Naukova Dumka, Kyiv, 1972.
Electroosmosis of the second kind and current through curved interface N. Mishchuk a,*, F. Gonzalez-Caballero b, P. Takhistov a a
Institute of Colloid and Water Chemistry of Ukrainian National Academy of Sciences, 42,Vernadsky pr., Kyi6, 03680, Ukraine b Department of Applied Physics, Uni6ersity Granada, Fuentenue6a s/n, ES-18071, Granada, Spain Received 17 March 2000; accepted 29 August 2000
Abstract The article focuses on the principles of strong concentration polarisation of flat and curved interfaces including the phenomenon of the limiting current. It is shown both theoretically and experimentally that electroosmosis of the second kind near the curved interface causes an important change of characteristics of concentration polarisation and a rapid growth of the current through an interface instead of current saturation in case of flat surface. At the same time, electroosmosis of the second kind can lead to such a strong decrease of concentration polarisation that the conductivity of ionexchange materials becomes a limiting factor. © 2001 Elsevier Science B.V. All rights reserved. Keywords: Concentration polarisation; Electroosmosis of the second kind; Limiting current; Desalination
Nomenclature List of symbols and abbreviations a b C0 CI CDL D0 Di E F
radius of fibre or ion exchange granule dimensionless parameter electrolyte concentration electrolyte concentration inside of ion exchange material convective-diffusion layer diffusion coefficient in electrolyte diffusion coefficient inside ion exchange material strength of electric field Faraday constant
* Corresponding author. Tel.: + 380-44-4504415; fax: + 380-44-4520276. E-mail address: [email protected] (N. Mishchuk). 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 7 4 1 - X
132
h i IÞ I ilim L R t T V x l 0
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distance between membranes, granules or fibres density of the current current along the normal to the fibre current along the fibre limiting current length of fibre gas constant time temperature electroosmotic velocity of liquid distance along the fibre counted from the free end of the fibre thickness of convective-diffusion layer potential drop between the membranes, granules or fibres
1. Introduction Electrokinetic phenomena of the second kind were predicted theoretically [1 – 3] and have been investigated and corroborated experimentally for different types of dispersed particles [4 –7]. The theoretical prediction of electrokinetic phenomena of the second kind includes two aspects: (a) strong non-linearity of electroosmosis and electrophoresis; (b) unlimited growth of the current through a polarised interface. According to the first aspect of the theoretical prediction, electroosmotic and electrophoretic velocities are proportional to the particle size and approximately proportional to the squared value of electric field strength and, therefore, can reach values many times larger than Smoluchovski’s velocity. Measurement of electroosmotic and electrophoretic velocities has shown that the experiment corresponds well with the theory over a large range of voltage, electrolyte concentration, particle sizes and types of investigated particles [4–8]. According to the second aspect of the theoretical prediction, electroosmosis of the second kind is not only the result of concentration polarisation of an interface, but it also affects concentration polarisation and considerably changes its characteristics, including the current through a polarised region [1,9]. If, in the case of a flat
interface, concentration polarisation is accompanied by the phenomenon of the limiting current (an increase of voltage leads to saturation of the current: starting from a certain value of voltage its increase is very weak or impossible), in the case of a curved interface, due to electroosmosis of the second kind, the current growth is unlimited. This theoretical prediction [1,9] can be partially proved by the analysis of published experimental results in the field of electrodialysis intensification by the use of material with a curved surface [10,11]. However, these experimental works only indirectly confirm the idea discussed in the present paper. The objective of the present study is to develop the scheme of experimental investigation of the current under conditions of electroosmosis of the second kind and to obtain the main theoretical and experimental characteristics of the investigated processes. Experimental cells are developed for investigation of polarisation processes between: (a) two flat ion exchange surfaces (anion and cation exchange membrane); (b) two spherical surfaces (cation and anion exchange granules); (c) flat and spherical surfaces (a cation exchange granule and an anion exchange membrane; an anion exchange granule and a cation exchange membrane); (d) two cylindrical surfaces (cation and anion exchange fibres).
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2. Theory
2.1. Strong concentration polarisation of flat interface Limiting current Peculiarities of concentration polarisation of ion-exchange membranes (i.e. a change of electrolyte concentration near interface in electric field) have been examined in numerous theoretical and experimental papers [12 – 16]. This phenomenon is caused by the difference between ion transfer numbers in an electrolyte solution and in an ion exchange material. The electric current, passing through an ideal selective ion exchange material (ion exchange membrane) placed in an aqueous solution of electrolyte, is realised only by the migration of counterions (anions for anion exchange membrane; cations for cation exchange membrane). Thus, the transport properties of the ion exchange material and the electrolyte solution (where electric current is realised by both cations and anions) are not identical. The counterions move to and through the membrane, while the coions cannot move through it. The concentration of the coions decreases since their withdrawal is not compensated by the intake from the membrane. In a small field electroneutrality is preserved. The coions concentration decrease leads to the equivalent counterion concentration decrease, that is, the region of the low electrolyte concentration appears. However, the continuity condition for cation and anion fluxes during the transition from one medium into another cannot be satisfied only by means of their migration in the electric field. As a result, a diffusion flow from the electrolyte volume to the membrane surface occurs. In this way the diffusion layer is formed. At low voltage the picture of concentration polarisation is asymmetric: the change of electrolyte concentration near the opposite sides of ion exchange membranes has equal values and different signs (Fig. 1). However, these equal changes lead to different influence on the current density. This can be shown by the analysis of local values r1,2 of specific resistance near the opposite membrane surfaces, where the deviation of the concentration DC from the bulk value C0 is maximum.
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Without concentration polarisation, when the voltage is almost equal to zero, specific resistance is equal to r10 = r20 1/C0, while at larger voltage, with concentration polarisation, it takes the following values: r1 1/(C0 + DC) and r2 1/(C0 − DC). Thus, r1/r10 = C0/(C0 + DC) and r2/r20 =C0/ (C0 − DC). For example, at DC = 0.5C0, r1/r10 : 0.7 and r2/r20 : 2 or r2/r1 : 3. At larger voltage the difference between specific resistances sharply increases. For example, at DC = 0.9C0 and DC = 0.95C0 the ratio r2/r1 is equal to 20 and 40 respectively. Thus, the concentration decrease causes a considerable potential drop in the polarised regions and consequently limits the density of current. When the decrease of electrolyte concentration DC is close to C0, the density of the current is caused by the sum of diffusion and electromigration fluxes that is equal to i=
2FDC0 l
(1)
where F is Faraday constant, D the coefficient diffusion of ions, C0 the electrolyte concentration, l the thickness of region of concentration polarisation (convective-diffusion layer, CDL). In the case of the flat interface, the value d depends on external hydrodynamic flow or thermoconvection in the cell and is almost independent of voltage [12,14]. Thus, an increase of voltage does not result in a growth of the current. This is so-called ‘‘limiting current’’.
Fig. 1. Schematic representation of concentration polarisation of flat ion exchange membrane (1 — low voltage, 2 — high voltage).
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especially in the case of electroosmosis of the second kind, which will be discussed in the following section.
2.2. Strong concentration polarisation of cur6ed interface. Unlimited current under conditions of electroosmosis of the second kind
Fig. 2. Scheme of experimental cell with anion exchange and cation exchange membranes.
In real systems the growth of the current above the limiting value occurs. It is caused by water splitting [17,18] and the decrease of the thickness of convective-diffusion layer due to the appearance of the region of an induced space charge [2,3,12,14] and hydrodynamic flow [19]. However, the growth of the current for a flat interface is not very large. Taking into account the above analysis of specific resistance, one can see that the influence of the region of low concentration (desalination) on the general picture of polarisation is stronger than the influence of the opposite region of high concentration. Therefore, the present paper is devoted to the development of different schemes of experimental investigations that allow to study concentration polarisation related to desalination. The first scheme is traditional (Fig. 2), that is, a cell with two parallel anion and cation exchange membranes is used. Electrodes are connected with the power source in the regime of desalination. Thus, the concentration of electrolyte between the membranes decreases. To simplify the calculation of the thickness of CDL, a cell without hydrodynamic flow is used. In this case l = h/2, where h is the distance between membranes. The current is proportional to the membrane surface Sm and can be described as I = iSm =
2FDC0 4FDC0 Sm = Sm l h
The main feature of concentration polarisation of the ion exchange material in strong electric field [12,14,20] is that the concentration decrease leads to the deviation from electroneutrality. This property is independent of the interface profile or the homogeneity of conductivity of the ion exchange material. Yet, when the appearance of an induced space charge is combined with heterogeneous conductivity of the flat interface or the curved shape of the homogeneous surface of the solid phase, the current lines bend round non-conductive or low conductive parts of the flat surface or, otherwise, the curved surface of a particle. Thus, a component of the electric field, tangential to the interface, appears. Together with the induced space charge it causes a non-linear electroosmotic movement of liquid (Fig. 3) [1–3,5 –7,15,20] and non-linear electrophoresis [4,8]. Over a rather large range of parameters the velocities V of these phenomena (electroosmosis and electrophoresis of the second kind) are proportional to the particle radius a and squared value of electric field strength E 2. In particular, the velocity of electroosmosis V is approximately
(2)
The current/voltage dependence changes for the curved surface where the thickness of CDL is related to the electroosmotical mixing of liquid,
Fig. 3. Scheme of liquid flow near spherical particle under conditions of electroosmosis of the second kind.
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equal to mD(2FEa/RT)2/a, where for aqueous solution the parameter m =0.2. Taking into account that the thickness of CDL near the spherical particle depends on Peclet number Pe or the liquid velocity V [1 – 3] as l=
a
'
aD 1 V E
(3)
Pe one can see that electroosmosis of the second kind reduces the thickness of CDL inversely proportionally to the electric field strength. The current through the curved interface is related to the polarisation process and preserves its diffusion nature similarly to the current through the flat interface (1), but the thickness of CDL (3) decreases and causes the growth of the current. Considering Eqs. (1) and (3) it is clear that in the case of the spherical surface the density of the current is a linear function of the applied field i=
=
2FDC0 E l
(4)
However, it is difficult to measure the current through a particle and interpret the findings correctly. Therefore, in the following section we will introduce the scheme of the measurement of the current for the pair of particles that allows to overcome the obstacles.
Fig. 4. Scheme of experimental cell and electroosmotic flows with anion exchange and cation exchange granules.
namic fields that cannot be correctly described by the theoretical model. Another problem is created by the bipolar contacts that can cause water dissociation [17,18,21] and, as a result, change the overall picture of concentration polarisation. To eliminate these difficulties, another scheme of an experimental cell for the investigation of concentration polarisation between two curved surfaces is developed (Fig. 4 a). The electroosmotical mixing of liquid between granules determines the thickness of CDL and the density of the current. The theory in paper [9] was developed for polarisation of granules without the influence of membranes, therefore, the value of l derived in the paper [9] can be used for the purposes of the present study. The value decreases inversely proportionally to the applied voltage 0 10a ˜0
2.3. Electroosmosis of the second kind between two spherical surfaces. Unlimited current
l=
Electroosmotic movement of liquid between two ion exchange granules placed between two ion exchange membranes [9] is similar to the movement near a separate granule described in the Section 2.2 (Fig. 3). Here, the potential difference is fixed between membranes and non-homogeneity of the electric field is caused by the geometry of the space between membranes and granules. As a result, an electroosmotic flow near the surfaces of ion exchange granules occurs. However, it is difficult to undertake the analysis on the basis of the scheme presented in paper [9] because the space near the granules is limited by the presence of the membranes. This leads to the noticeable change of the electric and hydrody-
where ˜ 0=
(5)
F0 . RT Thus, similarly to the Section 2.2 we obtain the unlimited density of the current i=
2FD0C0 ˜0 10a
(6)
In the case, when the thickness of walls d that limit electrode chamber is small (d a), they do not affect electroosmosis, thus, the current can be calculated as I: iSg
(7)
where Sg = 2ya 2 is half the surface of the spherical particle.
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The comparison of the experimental results for flat (Section 2.2) and spherical (Section 2.3) ion exchangers with their theoretical models will enable better understanding of the peculiarities of polarisation processes for different shapes of an interface. Additional information about these processes can be obtained by the simultaneous investigation of flat and curved interfaces.
2.4. Electroosmosis of the second kind between spherical and flat surfaces. Unlimited current The scheme of the experimental cell with the flat and curved interfaces (Fig. 5 a, b) is similar to Fig. 2, and Fig. 4 presented above. The mathematical description of the processes in this cell is quite complicated. While in the case of two granules every granule is responsible for the structure of its own CDL, in the case of granule and membrane the electroosmotic flow of granule affects the CDL of the membrane. Due to the peculiarities of the electroosmotic flow between the granule and membrane, the thickness of CDL
Fig. 5. (a) Scheme of experimental cell and electroosmotic flows with cationexchange granule and anionexchange membrane. (b) Scheme of experimental cell and electroosmotic flows with anionexchange granule and cationexchange membrane.
Fig. 6. Scheme of experimental cell with cation exchange and anion exchange fibres.
of the membrane noticeably changes along its surface (Fig. 5a, b). The qualitative analysis of the processes in such systems will be undertaken in the Section 3.
2.5. Electroosmosis of the second kind between two ionexchange fibres. New factor that limits the current through interface The models for flat and curved surfaces presented above are based on the notion of the large conductivity of the ion exchange material and relatively low conductivity of the liquid. This corresponds to many systems, where the investigation of the current through the ion exchange material is both of scientific and practical interest. At the same time, it is well known that, due to low values of diffusion coefficient, the conductivity of ion exchange materials cannot be indefinitely large and considerably depends on the conductivity of the liquid that is used [22]. Therefore, the present paper aims at the combination of the analysis of the external and internal transport processes. This can be realised by the investigation of the current through ion exchange fibres connected to the power source as it is shown in the Fig. 6. Two cation and anion exchange fibres are parallel. On the one side their ends are inserted into electrode chambers, while on the other side the ends are held with the aid of a nonconducting material for fixing of the distance between the fibres. The potential difference 0, established by the external source, is distributed along and between the fibres. It has a complex nature because of the
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different electroconductivity of the electrolyte in the central and electrode chambers and in the ion exchange fibres. Owing to the action of electric field, the cations and anions move to the fibres and through them to the electrode chambers. As a result, the regions of low electrolyte concentration in the space between fibres near its surfaces can be observed. The thickness of region of concentration polarisation depends on the potential drop between the fibres and the velocity of the hydrodynamic flow near their surfaces. In the strong electric field the thickness of region of concentration polarisation is defined by electroosmosis of the second kind, whereas in the small electric field, when electroosmosis of the second kind is absent or very small, it is defined by the velocity of the external hydrodynamic flow or is equal to half the distance between the fibres in the absence of the external flow. For the mathematical description of the analysed system it is assumed that all the properties of the ion exchange fibres are identical and have an idealised nature, that is, the current through the fibres is provided only by counterions. The fibres are connected to the power source in the regime of desalination. The current will be analysed in the three models: (a) without concentration polarisation; (b) with concentration polarisation in the absence of electroosmotical mixing of electrolyte; (c) with concentration polarisation under conditions of electroosmosis of the second kind. (a) If there were no concentration polarisation, then the current outside the fibres would be determined by the electric field strength, i.e. it would be proportional to 0 I0 =iSf =2FDC0yaL ˜ 0/h
(8)
where Sf = yaL is half a fibre surface (the current is possible only through the part of fibre surfaces facing each other) and a,L are the fibre radius and the part of their length inside the chamber of desalination. (b) With the presence of concentration polarisation and in the absence of hydrodynamic flow, the diffusion layer thickness between the fibres l0 would be equal to h/2. This means that the liming current would be equal to
Ilim =
2FDC0 4FDC0 yaL yaL= l0 h
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(9)
and it is smaller ˜ 0/2 times in comparison with the case when concentration polarisation is absent (8). For example, at 0 = 5V the relation I0/Ilim is equal to ˜ 0/2=100. (c) The curved surface of the fibres is similar to the surface of spherical granules. Thus, electroosmotic slip of the second kind appears and leads to significant compression of CDL: l h/2. Its thickness can be evaluated by expression (5) and the current can be described as Il =
FDC0 ˜ 0yaL 5a
(10)
i.e., it is smaller h/10a times than without concentration polarisation (8) and it is ˜ 0h/10a times larger than in the case of concentration polarisation without electroosmotic slip of the second kind (9). However, with such intense external diffusion transport (i.e. the transport of ions outside the fibres), the internal diffusion transport of ions becomes a new limiting factor. If conductivity of the fibres is not very large, they are not able to carry away all the current that can be provided by the ion fluxes to the fibre surface. The theory of concentration polarisation of fibres with arbitrary conductivity will be developed below. For the purposes of simplification of the theory a few approximations will be used. The object of the study is the long fibres placed in electrolyte with low conductivity with short distance h between them (hB B L). This allows us to describe the characteristics of concentration polarisation between the fibres by the solution of a local one-dimensional task and to divide the current into two components: the current in the electrolyte that is perpendicular to the fibre surface and the current in the fibre that is directed along the longitudinal axis. Hence, the current in the fibre can be presented as ( (x
III = FDi Ciya 2
(11)
where Di,Ci are the diffusion coefficient and the concentration of the current carriers (counterions) inside (i ) the fibre and l/lx is the gradient of
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the potential along the axis of the cylindrical fibre (measured from the free end of the fibre, see Fig. 6). The current along the normal to the fibre is equal to the product of the current density iÞ(x) by the surface area, through which the current flows, yaDx 2FDC0 IÞ = yaDx =JÞDx l(x)
(12)
where the current density is determined by the thickness of CDL l(x) specified by the electroosmotic flow that depends on x too. On the basis of Eq. (5) the local value of thickness of CDL is determined by the magnitude of the local potential difference ˜ (x) applied to each fibre 5a l(x)= ˜ (x)
(13)
The current along the fibre is replenished by the normal flux of ions with displacement from x=0 to x= L and reaches its maximum value at x= L. The current continuity condition should be satisfied in every point of the fibre surface. It can be presented as III (x +Dx)− III (x) = JÞDx
or
(III =JÞ (x (14)
Considering Eqs. (11) and (12) and normalisation of x on L: x˜ =x/L, Eq. (14) is reduced to the equation for the potential distribution along the fibre ˜ (x˜ ) ( 2 =b(x˜ ) (x˜ 2
(15)
where b= 2C0D0L 2/5DiCia 2 is a dimensionless parameter that determines the relative characteristics of the electrolyte and the fibre. The solution of this equation has the form of ˜ (x)= ˜ 1exp( − bx˜ ) + ˜ 2exp( bx˜ )
(16)
where the coefficients ˜ 1, ˜ 2 are determined from the boundary conditions, that is, the potential at the point where the fibre enters the electrode chamber ˜ (x˜ = 1) is equal to ˜ 0/2. Here, it is assumed that the property of the fibres are similar and, consequently, the potential difference is distributed uniformly between them. It is necessary
to take into account that the longitudinal current at the fibre’s free end is equal to zero III (x˜ =0)= 0. As a result, the solution takes the form of ˜=
˜ 0 exp(− bx˜ )+ exp( bx˜ ) 2 exp(− b)+ exp( b)
(17)
In the limiting case b 1 Eq. (17) can be rewritten as ˜=
˜0 2
(18)
Hence, the fibres are isopotential. In the opposite limiting case b 1, the following laws of potential variation near x˜ 0 and x˜ 1 are observed:
) )
˜0 exp(− b) 2 ˜ ˜ x˜ 1 $ 0[1− b(1− x˜ )] 2 ˜ x˜ 0 $
(19) (20)
i.e. in the first approximation the value of the potential near the free end of the fibre is constant (but significantly smaller than the value ˜ 0/2 that is given by the source of power) and the reduction of the potential near the point of the fibre attachment at the entrance to the electrode chamber is linear. It follows from Eqs. (19) and (20) that the potential decreases more rapidly, the larger the parameter b, i.e., the higher electrolyte concentration and the thinner and longer the fibre. But this means that the smaller is the current normal to the fibre surface, the less productively the fibre surface is utilised. Results of the numerical study of the potential (17) are shown in Fig. 7a. Using (17) and (11), we can represent the current as III (x) ˜0 exp( bx˜ )−exp(− bx˜ )
b 2L exp(− b)+exp( b)
= FD1C1ya 2
(21)
Thus, near the exit from the cell the current takes the value of III (x˜ =1)=
FD1C1 2 ya tan h( b) b ˜0 2L
which is measured in the experiment.
(22)
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It follows from the expression (22) that the current in the system is independent of the coefficient of diffusion and the concentration of the current carriers in the fibre under the condition that tanh( b) : b 1: III (x˜ =1)=yFD0C0L ˜ 0/5
(23)
that coincides with Eq.(Eq. (10)). In the opposite case b \ \1 Eq. (Eq. (22)) can be reduced to
'
III (x˜ =1)=F
D1C1D0C0 ya ˜0 10
(24)
The correlation h between the currents, calculated according to the model taking into account the internal resistance of fibres (22) and neglecting it (10)
h=
139
tanh b
(25)
b
is shown in Fig. 7b. The use of the scheme presented in Fig. 6 allows additional measurements that can be used for the analysis of the process. It is possible to use many pairs (n) of fibres, and, consequently, to increase the volume of the liquid, that the process of water desalination in a chamber with fibres can be analysed. As it was mentioned before, the fibres are connected to the electrodes in the desalination regime, i.e., the concentration of the electrolyte in the cell decreases. The magnitude of the reduction of electrolyte concentration is determined by the current III (x˜ =1), the desalination chamber volume Vc and the time t counted from the beginning of desalination dC(t) nIII (x˜ = 1,t) dt = FC0V C0
(26)
where the current III (x˜ =1,t) (22) is related with electrolyte concentration by the function b(C(t)) b(C(t))=
2D0C(t)L 2 = bC0 (t) 5D1C1a 2
(27)
where C0 (t)=C(t)/C0. As a result, the differential equation is obtained: dC0 (t) = nqtanh bC0 (t)· bC0 (t) dt
(28)
Di Ci ya 2 ˜ . C0Vc 2L 0 The solution of Eq.(Eq. (28)) is
where q=
1 C0 (t)= arcsinh 2(sinh bexp(− n q bt/2)) b
(29)
In the limiting cases this solution can be reduced to simpler forms
C(t)=C0 1− Fig. 7. (a) Distribution of potential along the fibre. The numbers of curves corresponds to following values of parameter b: 0.1 (curve 1); 0.3 (curve 2); 1 (curve 3); 3 (curve 4); 10 (curve 5); 30 (curve 6). (b) Correlation a between the theoretical values of the current calculated with and without internal conductivity of fibres as a function of parameter b.
ny ˜ 0at 2Vc
'
D0Di Ci 10C0
2
for b1 (30)
and C(t)=C0exp(−
y ˜ 0D0Lt ) 5V
for b 1
(31)
N. Mishchuk et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 181 (2001) 131–144
140
It is necessary to stress that the approximation (30) is valid only for a small time interval because as t , the value b(C(t)) decreases to such extent that, instead of condition b \ \1, bB B 1 is obtained. Hence, it is necessary to use the solution (31). The obtained expressions for C(t) create an opportunity to evaluate the current in the system. The dependence of the current on the time is described by the following law:
'
III (t) =F
Di Ci D0C0 ny ˜ 0at ya ˜ 0 1− 2Vc 10
'
D0Di Ci 10C0
for b1 and
'
III (t)=F
2
(32)
Di Ci D0 ny ˜ 0D0Lt ya ˜ 0C0exp 10 5Vc
for b1
(33)
3. Experiment
3.1. The role of surface shape The experimental investigations of the current according to the different schemes presented in the Fig. 2, Fig. 4, and Fig. 5 were carried out with the use of membranes and granules placed at the same distance between them h =0.06 cm. The area of the membrane surface was 10 times larger than the area of the granule surface. This allowed to analyse the tenfold difference between the density of the current through the curved and flat surfaces. Considering that due to electroosmosis the density of the current through a granule is higher than through a membrane, the area of membranes is big enough to compensate the low density of the current through them. The space between ion exchangers was filled with the electrolyte of low conductivity (C0 = 0.001 and 0.002 mol/l), whereas in the electrode chambers the electrolyte with high conductivity (0.1 mol/l) was used. This means that the potential drop in the electrode chambers can be neglected. Moreover, in this case the relative role of
Fig. 8. Current as function of applied voltage in different schemes: 1 — two ion exchange flat membranes, 1% — theoretical model of limiting current, 2 — anion exchange granule/ cation exchange membrane, 3 — cation exchange granule/anion exchange membrane, 4-two ion exchange granules; 4% — theoretical dependency Eq. (7), 4%% — theoretical dependency with account peculiarity of electroosmosis near anion exchange granule Eq. (34), 4%%% — theoretical dependency with account of stagnant region.
concentration polarisation on the opposite sides of membranes (or granules) is lower than in the case of equal concentrations. Due to the high conductivity of the ion exchange material the potential drop in the membranes and the granules can be neglected, too. Thus, the potential drop is completely related to the space between the two membranes (two granules or membrane/granule). Taking into account this statement, an experimental analysis of polarisation related to the intermembrane (inter-granule) region can be undertaken. The results of such investigation are shown in Fig. 8.
3.1.1. Current through two membranes The current through two membranes is shown in Fig. 8 (curve 1). It is quite close to the theoretical value of the current (curve 1%). The deviation between the two curves is not very large for the traditional analysis of the current through membranes. The results for large field are not shown, because the current is accompanied by strong change of pH. As a result, the transport of H+ and OH− considerably increases the current.
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3.1.2. Current through two granules One can see that the current through the two granules (curve 4 in Fig. 8) is considerably higher than through the two membranes (curve 1). Nevertheless, the values of the current are lower than they should be according to the theoretical prediction (curve 4%) based on Eq. (7). This discrepancy can be explained by the action of the two factors: the property of the anion exchange material and the peculiarity of the experimental cell. The first factor is related to the difference between polarisation of cation and anion exchange materials. This phenomenon is well known in the field of polarisation of the ion exchange membrane and the experimental investigations of electrokinetic phenomena of the second kind. The anion exchange material is characterised by the lower velocity of electroosmosis and correspondingly the smaller size of electroosmotic whirl than the cation exchange material. This result was obtained by systematic investigation of electroosmosis for different types of anion exchange materials [5–7]. Similar results have been shown in electrophoresis: the velocity of anion exchange particles is considerably lower than the velocity of cationexchange ones [8]. This effect can be explained by the influence of two processes. The first of them, the catalysis of water dissociation by surface groups of the anion exchange material [17,18], leads to the decrease of the induced space charge and, as a result, to the decrease of electroosmosis velocity. However, it is necessary to stress that the change of pH was smaller than in the case of two membranes (The values of pH cannot be shown here, because the design of experimental cell does not allow to carry out these measurement). The second process that decreases velocity of electroosmosis is the concentration of ionised impurities (anions) in the region of the induced space charge [23]. As the anions play the part of coions, they reduce the induced space charge and electroosmosis velocity. Due to the both processes, the thickness of CDL layer is larger and the density of the current for the anion exchange granule is lower than the density for the cation exchange one. Taking into consideration that the experimental values of velocity is approximately 4 times lower than it is
141
predicted by the theory and that the thickness of CDL is the function of Peclet number Pe V/4, the thickness of the diffusion layer can be roughly evaluated as l
1
V/4
2
V
(34)
It is approximately twice larger than the thickness that was used in the calculations of the curve 4%. Thus, according to Eq. (4), the current should be roughly twice lower. This result is presented in the Fig. 8 as the curve 4%%. Notwithstanding the fact that the curve 4%% is near the experimental one, to analyse the processes more thoroughly, the evaluation of the current decrease caused by another factor related to the construction of the experimental cell (curve 4%%%) was introduced. In the case of a ‘‘free’’ granule the electroosmotic flow of the second kind is realised along more than half a particle surface (Fig. 3). But in the case of granules fixed by a special tube (Fig. 4, Fig. 5) the artificial factor that deteriorates the free liquid flow is received. In the space near the contact with the tube the so-called stagnant region appears. As a result, the electroosmotical flow cannot reduce the thickness of CDL in this place. It becomes many times larger than the thickness predicted for the ‘‘free’’ particle. Thus, this part of the granule surface is lost for the high density of the current. Taking into account the correlation between the thickness of the tube wall and the size of the granule in our experiment, the conclusion is made that due to the stagnant region about 30% of the current is lost. The curve 4%%% combines both factors, that is, the decrease of the current due to the peculiarities of the anion exchange material and the stagnant region, and appears below the experimental curve. Thus, it can be stated that the experimental results are in good agreement with the theoretical evaluations.
3.1.3. Current through the membrane and granule The values of the current (Fig. 8, curves 2,3) for the two versions (Fig. 5a,b) of the granule/membrane system are close to the values for two membranes (Fig. 8, curve 1). The current for the anion/exchange granule and cation exchange
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membrane (curve 2) is lower than for the opposite versions of the granule and the membrane (curve 3). Similarly to the case of 2 membranes, the results for large field are not shown, because the current is accompanied by strong change of pH. The change of pH for the case of the cation/exchange granule and anion exchange membrane is larger. The correlation between the curves 2 and 3 can be explained by the lower velocity of electroosmosis and correspondingly the lower current density through the anion exchange granule (see the explanation above). Due to lower velocity the electroosmotic whirl is smaller and, consequently, the influence of electroosmosis on CDL of a membrane is weaker. The qualitative picture of this process is presented in Fig. 5a,b. One can see the difference between the whirls and the thickness of CDL in both cases (we did not show here the thickness of the convective-diffusion layer of the granule because it is considerably lower and cannot be shown on the same scale). Therefore, the polarisation of the flat surface is a limiting factor. Even the area of the membrane, which is 10 times larger than the area of the granule surface, is not sufficient for the realisation of the current that can be provided by the spherical interface.
than the fibre diameter. Yet, it was small enough to produce the agitation of the majority of the electrolyte volume in the cell. As the investigation of the mass transfer through the fibres under the action of electroosmotic flow is of primary interest, the experiment was carried out at the conditions when the velocity of electroosmotic slip Veo was higher than the velocity of hydrodynamics flow Vg. The experimental data for the current through the ion exchange fibres are presented in the Fig. 9 and Fig. 10. It can be observed that the current is higher than in the case of the granules that is explained by the larger area of the fibres surface. However, the surface area of the fibre is approximately 300 times higher than the surface area of the granule, while the current, for instance, at 50 V is only 15 times higher. This considerable differ-
3.2. The role of internal conducti6ity In the experiment 5 pairs of antipolar fibres (radius 0.08 cm and length 3.5 cm) were used. The antipolar fibres were disposed at the distance of 0.15 cm and equipolar ones at the distance of 0,1 cm. Similarly to the scheme for granules and membranes (Fig. 2, Fig. 4, and Fig. 5), for the simplification of the analysis of the experimental results it is expedient to use an electrolyte with low conductivity in the central chamber and another one with high conductivity in the electrode chambers. The recirculation scheme of the electrolyte solution with the linear velocity of the flow Vg = 0.1= 0.1 cm/s was applied. The total volume of the liquid was equal to 32 cm3. The optimum size of a cell was used: to prevent distortion of the electroosmotic fluxes, it was several times larger
Fig. 9. (a) Current through 2 ion exchange fibres as function of applied voltage: curve 1 — experiment, curve, 2 – 4 theoretical curves for different value of diffusion coefficient inside the fibres: Di =10 − 8 cm2/sec (2), 2·10 − 8 cm2/sec (3) and 4·10 − 8 cm2/sec (4). (b) Time dependency of the current through two fibres. The numbers of curves are the same as in the Fig. 9a.
N. Mishchuk et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 181 (2001) 131–144
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In the present experiment the fibres are equilibrated by the low electrolyte concentration, and only a small part of their ends is immersed in the electrolyte with high conductivity. As a result, their internal conductivity is caused mainly by the low electrolyte concentration and therefore it is lower than the conductivity of the granules. Moreover, it changes along the fibres. Since the exact values of electroconductivity or diffusion coefficients are unknown, numerical calculations were carried out for a few values of the diffusion coefficient inside the fibres which, according to different papers (for example, [24]), are close to our experimental conditions. However, it is necessary to emphasize that for the analysis of the differences between the results for the granules and for the fibres the most important factor is the radius of the fibres, because the resistance of the fibres and parameter b are inversely proportional to the area of their cross section. Thus, there is a possibility of an increase of the current due to the larger surface area of the fibres (yaL for the fibres is noticeably larger than the surface area 2ya 2 for the granules), although, at the same time, there is a limiting factor related to the identical area of cross section ya 2. Fig. 10. Theoretical and experimental dependencies of desalination on the time for different electrolyte concentrations: (a) C =0.001 mol/l; (b) C =0.002 mol/l. The numbers of curves are the same as in the Fig. 8.
ence is caused by the low internal conductivity of the fibres. It is well known that the internal electroconductivity of the ion exchange material depends on the concentration of electrolyte in which it is placed [24]. This dependency is very strong and can lead to the alteration of the ion exchanger conductivity in tens or hundreds times. In the experimental scheme presented in Fig. 4 and Fig. 5 the granules are in equilibrium with the two electrolytes. In the electrode chamber the electrolyte concentration is considerably higher than in the central chamber between the granules. This leads to certain intermediate conductivity of the granules.
4. Conclusions On the basis of the developed theoretical and experimental investigations the conclusion is made that the main peculiarities of concentration polarisation of a curved interface considerably differ from the peculiarities of concentration polarisation of a flat one. Electroosmosis of the second kind is the important factor that affects the process of concentration polarisation of curved interface and leads to the decrease of thickness of convective-diffusion layer and to the considerable increase of the current density. The results that were obtained can be used for the intensification of electrodialysis, the efficiency of which is rather low because of the phenomenon of the limiting current.
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Acknowledgements The authors are grateful to the International Association for the Promotion of Cooperation with Scientists from the Commonwealth of Independent States for the financial support of this investigation in the framework of the projects INTAS 95-IN/UA-165.
References [1] S. Dukhin, N. Mishchuk, Colloid. J. (USSR) 51 (1989) 659. [2] S. Dukhin, N. Mishchuk, Colloid. J. (USSR) 52 (1990) 452– 456. [3] N. Mishchuk, S. Dukhin, Colloid. J. (USSR) 51 (1989) 497– 501. [4] A. Baran, Ja. Babich, A. Tarovsky, N. Mishchuk, Colloids and Surfaces. 68 (1992) 141–151. [5] N. Mishchuk, P. Takhistov, Chem.Tench. Water (Ukraine) 13 (1991) 106–110 Also pages 707– 715. [6] N. Mishchuk, P. Takhistov, Colloids and Surfaces A95 (1995) 119. [7] N. Mishchuk. Colloids and Surfaces, A (1999).
.
[8] N. Mishchuk, A. Barany, A. Tarovskij, F. Madai, Colloids and Surfaces. 140 (1998) 43. [9] S. Dukhin, N. Mishchuk, J.Membr.Sci. 79 (1993) 199. [10] O. Kedem, Desalination 16 (1971) 105– 118. [11] E. Korngold, Desalination 16 (1975) 225. [12] I. Rubinstein, L. Shtilman, J. Chem. Soc. Faraday Trans. II 75 (1979) 231. [13] I. Rubinstein, E. Staude, O. Kedem, Desalination 69 (1988) 101– 104. [14] V. Nikonenko, V. Zabolotski, N. Gnusin, Electrochemistry (Russia) 25 (1989) 180– 186. [15] I. Rubinstein, Phys.Fluids A3 (1991) 2301. [16] V. Zabolotsky, V. Nikonenko, Electrochemistry (Russia) 32 (1996) 246– 254. [17] R. Simons, Electrochim.Acta 30 (1985) 275. [18] Y. Tanaka, M. Seno, J. Chem. Soc. Faraday Trans. 82 (1986) 2065. [19] S. Dukhin, Chem. Techn. Water (Ukraine) 11 (1989) 675. [20] N.A. Mishchuk, Colloids and Surfaces 140 (1998) 75. [21] V.I. Zabolotskij, N.P. Gnusin, N.V. Sheldeshov, N.D. Pismenskaja, Electrochemistry (Russia) 21 (1985) 1059. [22] V.K. Varentsov, M.V. Pevnitskaja, Izvest. Sib. Ot. Akad. nauk SSSR. Ser. Khimiya 4 (1973) 134. [23] S.S. Dukhin, N.A. Mishchuk, N.A. Klimenko-Meshkova, V.N. Goncharuk, Chem. Techn. Water (Ukraine) 11 (1989) 885. [24] N.P. Gnusin, V.D. Grebenjuk, Electrochemistry of Granulated Ionites, Naukova Dumka, Kyiv, 1972.
