Ion exchange diffusion in electromembranes and its description using the Maxwell-Stefan formalism

journal of MEMBRANE SCIENCE ELSEVIER

Journal of Membrane Science 137 (1997) 121-132

Ion exchange diffusion in electromembranes and its description using the Maxwell-Stefan formalism A. Heintz*, E. Wiedemann

~, J. Z i e g l e r ~

Department o "Physical Chemistry, University ~/'Rostock, Hermannstr 14, D-18051 Rostock. Germany Received 9 June 1997: received in revised form 18 July 1997: accepted 22 July 1997

Abstract Exchange diffusion experiments of the exchanging ion pairs C1 /Br and C1 /SO42- in an anion exchange membrane and of Na'/H ~ in three cation exchange membranes have been carried out using a dialysis cell. The time dependent concentrations in the cell are measured with precision densimetry based on the principle of the vibrating tube. Thermodynamic ion exchange equilibria between the membrane phase and the electrolyte solution have also been measured for the three systems CI /Br , CI /SOl , and Na+/H" using atomic absorption spectrometry and ion chromatography as analytical methods. The experimental data of exchange diffusion and exchange equilibria are used to study the diffusion velocity of the mobile ions in the membrane on the basL,;of the Maxwell-Stefan formalism of multicomponent diffusion processes. In frame of the solutiondiffusion model equations are derived for the ion exchange diffusion process which allow to determine diffusion coefficients of the exchanging ions in lhe membrane and which account also for frictional effects between the ions expressed by a coupling diffusion coefficient. The method developed can generally be applied to study transport processes in electromembrane system~. ~' 1997 Elsevie~~ Science B.V. Kevword~s: Ion exchange membranes; Ionic diffusion coefficients; Maxwell-Stefan theory: Ion exchange equilibrium

1. I n t r o d u c t i o n Mobilities and diffusion coefficients of single ions in charged membranes are of interest in various membrane processes where the transport of ions across cation or anion exchange membranes plays an important role such as electrodialysis, fuel batteries, redox flow batteries and others [1-8]. In principle, mobilities *Corresponding author. Fax: +49 381 498 1854. tPresent address: Physikalisch Chemisches Institut, Universit~it Heidelberg, Im Neuenheimer Feld 253, D-69120 Heidelberg.

Germany. 0376-7388/97/$17.00 t~, 1997 Elsevier Science B,V. All rights reserved. P l l SI)3 7 6 - 7 3 8 8 ( 9 7 ) 0 ~ ) 1 8 5 - 3

of ions in charged membranes can be obtained by measuring the electrical membrane resistance under the condition that only one kind of mobile ions acts as counter ion in the membrane. This procedure has an advantage that simple experimental equipment can be used. However, the experimental error is quite large and the method of deriving diffusion coefficients of the mobile counter ions is complicated and involves ambiguous approximations [9]. Another procedure which is preferable due to its experimental simplicity and its direct relationship with diffusion coefficients is the exchange diffusion of two mobile ions in a given electromembrane. The process is sketched schemati-

122

A. Heintz et al./Journal of Membrane Science 137 (1997) 121-132

2. Exchange diffusion measurements

Mere ~rane Na+ CI-

Na+ <

t3r" Br-

Na+

Na+

Na+ CI"

~ Na+

el

,~

Br-

Fig. 1. Schematics of an ion exchange experiment,

cally in Fig. 1 for the example of the exchange diffusion of C1- and B r - ions across an anion exchange membrane with equal sodium ion concentration but different concentrations of each of the two anions in the aqueous solutions being in contact with the two outer surfaces of the membrane. Due to the different concentrations of C1 and B r - in the solutions on both sides of the membrane exchange diffusion will occur as function of time until the concentrations of the anions are identical in the two solutions. Since there is no concentration gradient of cations (Na +) no transport of Na + will be observed. If the electrolyte concentrations are not too high, the thermodynamic activity of water is constant and identical on both sides of the membrane at any time and no osmotic water flow through the membrane will take place. Exchange processes of ions with equal sign of electrical charge passing the membrane have already been studied previously [ 10,11 ]. In this paper we report ion exchange diffusion measurements of the system N a + / H + in three cation exchange membranes and of C1 /Br- and C1-/SO 2- in an anion exchange membrane using a dialysis cell. Working equations for the quantitative description of the exchange process have been developed based on the MaxwellStefan formalism of diffusion processes, which also accounts for diffusion coupling of the interchanging ions and allow to determine single ion diffusion coefficients from the experimental data.

Fig. 2 shows the dialysis apparatus which was used to measure the exchange diffusion of ions. A similar technique has already been applied for determining diffusion coefficients of Br2 in cation exchange membranes [12]. The two compartments of the cell are filled with the solutions under study. They are separated by the membrane and each compartment is stirred using two independently working magnetic stirrers. The whole cell is surrounded by a water jacket which is kept at constant temperature by means of the thermostat TH. In the upper-half cell, the concentration of the exchanging ions is measured as function of time using precision densimetry. The solution of the upper cell is pumped continuously through a vibrating tube densimeter (A. Paar, DMA 60). The frequency of the vibrating tube is a measure for the concentration which is sensitive enough to detect density changes of :[:10 -5 g cm -3. For example, the accuracy of the determined concentration of Brions is =t=0.005tool 1 l (and vice versa for CI-) in a NaCI+NaBr solution with constant Na + concentration

I

[ oo

l

] .r'--:7:--._ V13 ~

~]

~

[ . ~

II

iiC~ ~

~

-'=11

"] /

/ M

C~

2

J

[ MS I Fig. 2. Dialysis apparatus for ion exchange measurements. M membrane, VB=vibrating tube densiometer, MS=magnetic stirrer, P=liquid pump, TH=thermostats, Vu=volume of upper cell, VL=volume of lower cell.

A. Heintz et al./Journal qf Membrane Sciem'e I.¢7 (1997) 121 132

123

0.]-.

I

I

L ]

!

]

o:

~i

•

"

c> ~

I

c':

i

!

............................................. 2 J c n i1,, d ~lclmlce t,} [, ] M ix ~:{ ] - I [ x~ N a B r l amd 0,l M NaC] solution [ ~ m ]

:

Fig. 3. Calibration curve for the determination of Br lions.

z

4::,,

6o0

s00

concentra-

10c

!,,; . . . . .

,0

,,,,,~1..... Fig. 4. Dialysis measurements No. I (see Table 11 with the AMS membrane. Br concentration in the upper-half cell at 298 K. •

o f 0.1 tool 1 I. This is demonstrated by the calibration

experimental data, - - - -

curve shown in Fig. 3. An almost linear relationship is obtained for the plot of Br concentration versus the density difference b e l w e e n the m i x e d electrolyte 0.1 ( x N a C l + (1 x)NaBr)M (0
(g.~7, ~.

fore, the final concentration,

membrane. SOl concentration in the upper-half cell at 298 K. • experimental data, - - .... Eq. (2), ,5"~ equilibrium concentration

which is established ~t.

i. . . . .... [ . . ...... ] " 7 .... : > "i ~ , i ' ~:~~! • ~ .... ! ,,'" ~ l. JL"" 0 03a .

=c

al.

-} t : ~" )

=

VL¢{t 0 ) - -

V ~-.w U (t=0)

VU - - V L

•

.

I ! '

" _

.

.

/ ill , ~.

:,, .

l)

m e a s u r e d concentration o f e x c h a n g i n g ions in the upper-half cell as function of time. The curves a r e denoted by an identification n u m b e r which corre--

. i.

Fig. 5. Dialysis experiment No. 9 (see Table 11 v~iih the AMS

after

where g,, tT -,L > 0 / and c(r=o ! are the concentrations of the e x c h a n g i n g ion at the beginning o f the e x c h a n g e e x p e r i m e n t for the upper- and the l o w e r - h a l f cell, respectively. In the used dialysis cell Vu 338 ml and Vt = 2 6 3 ml, the rnembrane area A 28 cm 2. . Figs. 4 - 6 show three selected e x a m p l e s o f the

. .

.............................. ......~"<

the ion e x c h a n g e process is finished ( ( t ~ i and g,tr c, ~. ,), has to be calculated according to the f o l l o w i n g relation obtained from the balance of moles:

~.L ~i \

Eq. (21. C equilibrium concentration

.

.

.

.

,. . . . . . . . . .

7: .

.

.

.

.

.

• .................

.

7 ':" ..- ...-* ~ ......... • . . ~ ..... f ! ~ ........ . : ...... . . .

" °

"

~h~

Fig. 6. Dialysis expemnent No. 13 (see Table 1) w.ith the CMV membrane. Na ~ concentration in the upper-half cell at 298 K. • experimental data. - -

(~'I~:, I.

Eq. (2). C equilibrium concentration

0.1 M NaBr 0.02 M NaCI+0.08 0.07 M NaCI+0.03 0.05 M Na2SO4 0.04 M NaCI+0.03 0.05 M Na2SO4 0.02 M NaCl+0.04 0.1 M NazSO4 0.06 M NaCI+0.07 0.01 M Na2SO4 0.1 M NaCI 0.1 M NaC1 0.1 M NaCI 0.1 M NaCI 0.08 M NaCI+0.02 0.08 M NaCI+0.02 M HCI M HCI

M Na2SO4

M Na2SO4

M NaeSO4

M NaBr M NaBr

(t=0)

c(t=0)

0.1 M NaCI 0.06 M NaCI+0.04 M NaBr 0.08 M NaCI+0.02 M NaBr 0.1 M NaCI 0.08 M NaCl+0.01 M NazSO4 0.08 M NaCI+0.01 M NazSO4 0.06 M NaCI+0.02 M NazSO4 0.2 M NaC1 0.14 M NaCI+0.03 M Na2SO4 0.08 M NaCI+0.06 M NazSO4 0.1 M HCI 0.015 M NaCl+0.085 M HCI 0.03 M NaCI+0.07 M HC1 0.04 M NaCI+0.06 M HCI 0.02 M NaCI÷0.08 M HCI 0.04 M NaCI+0.06 M HCI

Starting concentration in the lower-half cell

Starting concentration in the upper-half cell

~ Calculated with Eq. (1). b Measured at 750 rev min ~ for speed of stirring. T=supplied by Tokuyama Soda; A supplied by Asahi Glass.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

System no.b

Table 1 Mixed electrolytes and membranes studied with the dialysis cell (see Fig. 2)

0.056 0.043 0.076 0.056 0.062 0.044 0.042 0.114 0.104 0.044 0.056 0.048 0.039 0.034 0.054 0.042

M M M M M M M M M M M M M M M M

NaCl+0.044 M NaBr NaCl+0.057 M NaBr NaCl+0.024 M NaBr NaCI+0.022 M Na2SO 4 NaCI+0.019 M Na2SO4 NaCI+0.028 M Na2SO4 NaCI+0.029 M Na2SO4 NaCI÷0.043 M NazSO4 NaCI+0.048 M Na2SO4 NaCI+0.078 M NazSO4 HC1+0.044 M NaC1 HC1+0.052 M NaC1 HCI+0.061 M NaCI HC1+0.066 M NaCI HC1+0.046 M NaC1 HC1+0.058 M NaC1

c(t=oc)c(t=~/

Final concentration in both half cells a

AMS AMS AMS AMS AMS AMS AMS AMS AMS AMS CMX CMX CMX CMX CMX CMX

(T) (T) (T) (T) (T) (T) (T) (T) (T) (T) (T), (T), (T), (T), (T), (T),

CMV CMV CMV CMV CMV CMV

Membranec

(A), (A), (A), (A), (A), (A),

CMS CMS CMS CMS CMS CMS

(T) (T) (T) (T) (T) (T)

"q

~-

t~

~.

~"

4~

A. Heintz et al./Journal of Membrane Science 137 (1997) 121-132 sponds to the systems characterized in Table 1. Measurements of several test systems have been carried out with different speeds of magnetic stirring in order to study a possible influence of a laminar boundary layer adjacent to the solution sides of the membrane on the diffusion process. Stirring speeds of 250, 500, 750 and 1000 rev rain ~ have been applied to the 0.1 NaCI+0. I M NaBr system (No. 1). No significant effect on the results has been observed. 750 rev rain i has been used for all systems studied, It turned out that the following simple empirical equation fits all curves within the experimental error: ~.u -.u ( c~:~ = ~ , =0~ ÷ ~!v

-U ) e(~t) ) - c/i=0) (1 -

t2)

where ~,~ is the concentration at any time t between t = 0 and t=oc. The corresponding concentration Z'~/in the lower cell can be calculated from Eq. (2) using the molar balance of the exchanging ion: _ c'~ = c~=l,i

Vtj {g.tJ -u ) (1 - e '~') V-~-\ I,=--!-c':t=~l:

(3)

~u and b L are identical a n d g i v e n b y Eq. (1). ~ (:=~; ~;=~/ is a parameter adjustable to each curve. The concentration of the second exchanging ion is simply calculated using the condition of electroneutrality which couples the fluxes of the two exchanging ions. The amount of fluxes is identical, but their directions are opposite,

3. M e a s u r e m e n t of ion e x c h a n g e

equilibria

In order to evaluate the exchange diffusion measurements in frame of the so-called solution-diffusion model on the basis of the Maxwell-Stefan formalism, experimental data of ion exchange equilibria have to be known, The quantitative description of ion exchange equilibria is the plot of counter ion concentration in the membrane versus the ion concentration in the electrolyte solution being in electrochemical equilibrium with the ion exchange membrane. To have the cornplete information, it is sufficient to present the concentmtions of only one of the two exchanging ions in the membrane and solution phases, the concentration of the second ion in the two phases is given by the requirement of electroneutrality in each phase,

125

3.1. Cation exchange equilibria We have measured exchange equilibrium curves of the ion pair Na VH + in three kinds of cation exchange membranes (see Table 1) which were immersed into 0.1 M (x H C I + ( I - x ) NaC1) solutions with varying values of x (0 < x < 1) and a constant concentration of 0.1 M C1-. The procedure was as follows. A weighted sample of membrane material (area ca. 5 cm 2, thickhess 0.015 cml was swollen in 0.1 M (HCI+NaCI) solution of given H + and N a ' concentration. The volume of the solution was 50 ml. Since the W and Na ~ ions are present in large excess compared to the amounts of these ions in the membrane sample, the known composition in the solution remains constant after the exchange equilibrium has been established. The membrane sample is then removed from the solution and rinsed careflflly with pure water. Afterwards, the sample is swollen in a large excess of 0. l M HC1 solution until the new equilibrium is reached and all N a ' ions in the membrane are exchanged by H + ions. The N a ions, previously solved in the membrane phase are now present in the HC1 solution at a very low concentration. The solution is filled up to a known volume with water and from this solution a sample is withdrawn whose concentration is analysed using an atomic absorption spectrometer IAAS Varian Spectra 6001. Knowing this concentration and the total volume of the solution, the amount of Na + ions can be calculated which was previously solved in the membrane phase. The H ~ concentration in the membrane is simply the difference of the fixed anion concentration CM in the cation exchange membrane and the calculated Na" concentration, cM is determined by a separate experiment where the membrane in its Na ~ form is completely exchanged by H ~ ions and the desorbed Na ~ ions are again determined in the described manner. 3.2. Anion eachange equilibria Corresponding experiments have been performed with the exchanging anion pairs CI /Br and CI / SO~ using an anion exchange membrane (AMS). The procedure is similar as in case with cation exchange membranes. Low concentrations of Br and C1 ions have to be analysed after their complete extraction from the membrane phase. The suitable analytical

126

A. Heintz et al./Journal of Membrane Science 137 (1997) 121-132

i n s t r u m e n t for this p r o b l e m is ion chromatography. 3We have used an ion chromatograph (Dionex DX . . . . 3000) with an electrical conductivity detector for ! a n a l y s i n g C1- in presence of SO42- and an UV-photo2- 21 metric detector for B r - in presence of C I - . ~ : Selected experimental results of exchange equili¢. . . bria are presented in Figs. 7-9. ~ Ion exchange e q u i l i b r i u m data can be described by equations derived from the t h e r m o d y n a m i c equili . . . . 1 b r i u m condition which requires the equality of the 0 electrochemical potential of each ion in the solution . . and m e m b r a n e phases. We consider two kinds of v

e x c h a n g i n g ions (i=1 or 2): #o i + R T In ai + ziF& = fi~ + R T In gli + ziF(h

(4)

1 T

- ~. .

--7 " . . . . . . . i /. . . . . . . */ : . . . . ~ : ; ! i | t ___.L _ _ _ :, /i[ __ :' _-] _ ~ ~. . . . . . . . . . . . . ~ ...... ~B~So,o,,o,)Lmo~,'j k

- "

i

__

~

i

-

]

Fig. 9. Ion exchange equilibrium isotherm oftheCl /SO4 system in the AMS membrane at 298 K. • experiments, - - Eqs. (11) and (13) with parameters from Table 2.

where the quantities on the right side refer to the solution phase (superscript ~ ) and the u n m a r k e d .... I .... ] z .... i i --~ I o~o~ . . . . ~ I

-- - ~ .

0,00

.

0,02

.-

,

_ _*Z . . . . . . .

I

. . . . •. _. - . __ o,oo * :::

.

--

i

.

.

.

, ~ ~--~__

l,] .

.

0,04

" .

.

quantities on the left side to the m e m b r a n e phase. #i° and/~.0 are the standard chemical potentials, ai and di are the t h e r m o d y n a m i c activities, 0 and ~b are the electrical potentials, zi is the electrical charge n u m b e r and F is F a r a d a y ' s constant. In case of m o n o v a l e n t ions (]all = IzzJ = 1), substraction of Eq. (4) for i=1 and i = 2 gives:

; g_

~-_ i ! {.

-

-

:

#10 __ /~1 __#2+0 ]~0= R T I n 2

~

0.06

0,08

o IO

~

~,

--

: '

; U . .,,. . . . . . .

(5)

a p p r o x i m a t e l y by the concentrations c; and g'i. Taking into account the electroneutrality in both the phases we have CM = CI q- C2

----

~ic2 ~C I C2

where the activities a i and di have been substituted

~,~.~sot~,o.)i~o~;']

Fig. 7. Ion exchange equilibrium isotherm of the Na+/H + system in the CMX membrane at 298 K. • experiments, - Eq. (8) with parameters from Table 2.

:~i~

•

and

"

~'s = ~'l + ~'2 ~

t oo_..Z . . . oo. . . . . . oo~ . . . . 0o~ . . .

(6)

,,,

Z cf (Solulion) [moll "l]

Fig. 8. Ion exchange equilibrium isotherm of the CI-/Br system in the AMS membrane at 298 K. • experiments, Eq. (8) with parameters from Table 2.

(7)

where cM and ?s are the concentration of the m o n o valent ions fixed to the m e m b r a n e network and the concentration of the electrolyte in the solution, respectively. Using Eqs. (5)-(7), we can write:

KCM?I Cl = _ Cs + ( K - 1 ) ~ ,

(8)

where K = e x p [ - ( # ° - rio _ #o2 + f i O ) / R T ] is the ion exchange e q u i l i b r i u m constant. Eq. (8) gives cl as function of {], the corresponding c o n n e c t i o n of c2 with ~'2 is obtained from Eq. (8) by substituting cl and ?] by c 2 and ?2, using Eqs. (6) and (7). Eq. (8) has been used

A. Heintz et al./Journal of'Membrane Science 137 (1997) 121 132

for describing the measured ion exchange equilibria of the Na t/H+ system ( q ==CNa, C2=CH ~ ) with the cation

Table 2

exchange membranes CMX, CMV and CMS and of the C1 /Br- system with the AMS membrane. K has been adjusted to the experimental data. Examples are shown in Figs. 7 and 8. The condition for the ion exchange equilibrium between a monovalent ion and a bivalent ion such as the C1 //SO] system is obtained using Eq. (4) for q col withzl - 1 andforc2 = Cso2 w i t h z 2 = - 2 . Combining both equations by eliminating the terms with =iFO and 2 i F o gives the following equilibrium:

Ion pair

2p~, I

~ ~o

o

-o

-P.Cl - / t s o ] ['~7 Cso~

4- IZso~

~ ~ RTInK (9) Ccl Cs°~ Using the corresponding conditions for the electroneutrality in tlhe aqueous and membrane phases: = RTIn

I b~ - c ' c i Cso~ = ~(

)

I

Cso ~ = 5 ( c v - c c l

(10) )

Parametersof ion exchange equilibria at 298 K Membrane

K

cM (moll

Na '/H ~

CMX

(I.618

1,35

Na'/H ~ Na/H ~ Cl /Br cI /SO~

(TMS CMV .\MS AMS

0.902 0.565 2.89 10.52

1.56 1.61 2.23 2.58

I)

these parameters for the ion exchange equilibria studied. We have still to justify why concentrations have been used instead of activities. Activity coefficients are difficult to estimate, particularly inside the membrane. It can be assumed that their values in Eqs. (5) and (12) cancel each other essentially, so concentrationscan be written instead of activities with sufficient degree of approximation. Calculated curves using Eqs. (8) and (13) turn out to describe the experiments accurately enough.

(11)

4. Application of the Maxwell-Stefan (MS) equations and evaluation of diffusion coefficients

we obtain K =

127

(~'~ -~'c, )c~-i

~, ( 121 (CM -CCI )C(, 1 The solution of this quadratic equation for co is: ~ ccl ...... K C c ~ I - 2(/'~ g'cl

The MS-equations describe diffusional processes in multicomponent systems including the influence of external fields such as the electrical field [13-16]. The equation system for n diffusing species is given by [ 171: tZ

_,

KQ: 1

-F 2(~', -/"el

[7_

i 1 4Cm (~'~ -- CCl ) t -~ K~2

__1 V I i i (13)

Eq. (13) is the relationship between the concentration of C1 in the membrane ( c c l ) and in the aqueous solution ( b c l ) for the case that the exchanging ion is bivalent (SO 4 ). The corresponding concentrations Cso] and ~:so] are obtained from Eqs. (10) and (11) using Eq. (13). The equilibrium constant K has to be adjusted to the experimental data of the C I - / S O 4 system. Results and comparison with experimental data are shown in Fig. 9. Parameters for describing the ion exchange equilibria are the fixed ion concentration in the membrane CM which is experimentally available and the equilibrium constant K which has to be adjusted. Table 2 presents

__ ~

~

RT

vpj(ll i -- Hj) Dii

( 14 t

where V~i is the gradient of the electrochemical potential, ui and Ug are the individual velocities of the diffusing species i and j, ,:, is the volume fraction of i and D o are MS-diffusion coefficients. 9; is related to the molar concentration c, by :2, -

ciVi

~15J

where Vi is the partial molar volume of species i in the mixture. Defining a frame of reference for the movement of the ions through an ion exchange membrane being at rest, we get for the molar fluxes .I,: J, :(:'illi CMlt M where the index M denotes the membrane. -

-

(16)

A. Heintz et al./Journal of Membrane Science 137 (1997) 121-132

128

Using the definition of the electrochemical potential in Eq. (4) together with Eq. (15), Eq. (14) can be

Substituting Eqs. (24) and (25) into Eqs. (22) and (23) and solving for J1 gives:

rewritten:

J1 = -[[[z2 IIZM[ - Izl I(Iz21 - Iz, I ) ( c , / c M )]

V-fii = -- ~ Ct JiXj---- Jjxi Ci R T j=, Dij/V,j

(17)

x(DIMD2M/VM)VC']/[]ZII2CID1M +IZeI(IZMICM -- ]Zl ICl)D2M

where c t = ~ c i is the total concentration of all s p e c i e s and xi, x i are mole fractions defined as: ci xi = --

(18)

Ct

Following a general procedure suggested by Taylor and Krishna [15], Eq. (17) can be rewritten: V~ i

~n, = - 2.., BijJj ci R T j=t

.4_[ZMI2CM(D,MD2M/DI2)(VI/VM)]

VM in Eq. (26) is the average molar volume of the fixed ions in the membrane. Approximately V , ~ V 2 is the molar volume of the mobile counterion 1 or 2 in the membrane. The correct relation between ci and V~ in the membrane phase is given by c , V 1 -.-c2V2-}-CMV M =9.9, q- 992 q- 99M = 1

(19)

By equating coefficients in Eqs. (17) and (19) and simultaneous elimination of n--I

(26)

(27)

where the definition of ~2i of Eq. (15) is used. Considering the case Iz, l = Iz21 = IZMI = 1, we have according to Eq. (25): (28)

CM = C1 -~ C2

and, therefore,

J,, = - ~) ~ J~ i we find:

l V~ V2 ~MM= CM + Cl ~MM+ C2 ~MM

(29)

with V1~V2 it follows: Bii =

Dik/(Vkct)

(20)

k=l i¢k

Bij --

1

(30)

CM -- VM q- V~

In case obtain

xi D,a/(V;ct)

of

I z l l = 2, Ig21 = 1 and IZMI = 1 we

(21)

2Cl -? C2 = CM Applying Eq. (19) to a system of two diffusing ions and neglecting activity coefficients, one obtains:

and again setting V1.-~V2 we have

F -(gTc, q- ClZl ~ V ~ b ) = B,1JI + B,2J2

(22)

CM -- - VM + Vl

(23)

Since normally VI<
1 -

F

- - ( V C 2 if- C2Z2 ~

~76~) = B21J1 + B22J2

There is no electrical current present in the membrane:

IZl IJ1 q- ]z2[J2 = 0 is:

(24)

The condition of electroneutrality in the membrane

]ZMICM = ]Z, ]el q- Ig21c2

(25)

CM ~ -

c,V,

(31)

1

(32) VM Application of Eq. (26) requires values of VM and VI~V2. We have estimated these data from VM=I/CM, where CM was measured as described in the previous section. For V1~-,V2 averaged values calculated by 71 = V2 = NL 5 7r

(33)

A. Heintzet al./Journal (~fMembrane Science 137 (1997) 121-132 have been used with rl and r2 being the van der Waals radii of the corresponding ions taken from literature [18]• Nt=6.022×1023 mol i is Avogadro's number. VM.AMS=0.39 I mol - I , VM,CMX=0.741 tool i VM.CMS=0.64 1 tool-1 and VM,CMV=0.621 mol I have been obtained. According to Eq.(33).

Ja~- = 1 ((VM q- VNa)/VM)DH.MDNaMCM h (DH.M -- DNa.M) × lni(DN~,.M -- DH.M)CNa. q t'M(DHM

÷(DNa,MDuM/DNa.H)(VNa/VM)}]/ [(DN~,M--DH.M)C~.~+cM(DuM

@(DNa.MDH,M/DNaH)(VN,a/VM))]

VB~, ~ V2 ~7 VSO,~_,= 215 ix 10-21mo1-1 and V N a VH. Z.O) X It1 "mol . Obviously Vio, is only 3-5% of the VM value. For the CI /Br system, the flux is according to Eq. (26):

1 ( c~ q) ~ d 2 - c ~- -\

Jso~

JBr = -[(Dc1,MDBr.M/'VM)VCBr ]/

q-d ('so:

[CBr DBrM + (CM -- CBr )DcI,M

129

(38)

+2, ;-.0 3Cso~

l n - -

C;o:

)]

(39)

with

+ON ( Vm / VM) (De l.MDBr.g/DCLBr)]

(34)

Ocl .~IDso,.M c~: -

140)

where loll = ]gBr] = I and ]zcll = 1. For the CI /SO] system we have

3

Jso~ =- I(I+2Cso~ /':'M)(Dsoa,MDcI,M/VM)VCso~ ]/ ,

~/ =CM(DcI.M + DCI'MDSOaM VSO'~ , Dso4 cl ~ J

(42)

~

(43)

dcso] DSO~M F (CM -- 2Cso ] )DcI,M

VM 2(2Dso~M -- DCLM)

2DcI.MDsoa.M

CMVM

]

cM(VsQ/VM)(DcI.MDsoa,M/DCl SOd)/ (35) where I:1{ [=so~{ 2 and Iz.21 = [Z.cll = 1. For the Na~/H ~ system in the cation exchange membranes, the flux is identical with Eq. (34), only indices have to be changed:

JNa = - [(DH.MDNa.M/VM)VCNa ] /

c°~ and c ~ . c ° , and c';a., ,Cs°~ .0 .a and ~so~ are the concentrations of Br , N a and SOl runs inside the membrane on the boundary to the solution of the lower cell (index 0) and the upper cell (index h). respectively. The left-hand sides of Eqs. (37)-t39) are related to the concentrations C~r • bU :.u N~,'' or ~sol in the upper cell solution by

[CNa DNa.M + ("M -- CNa )DH.M

4 CM(VNa/VM)(DH,MDNaM/DNa.H)]

(36)

,

Since the gradients VCB,- VCso 4 and VCya are unidirectional along the x-axis and perpendicular to the membrane surface, Eqs. (34)-(36) can be integrated from x = 0 to x=~5 when b is the membrane thickness. The results are:

J~r

dB, JNa --

VU df'~,. A dt

(44)

Vu d~ if

(45)

"~a A dt Vu di. so~ U

Jso::, - A

dt

(46)

where A is the membrane area• In Eqs. (37)-(39) the concentrations CBr ' c'~~~ , and c ~SO~ depend on the concentrations

1 ( ( V M q- VBr)/FM)DcI,MDBr,MCM =h (DcI.~I- DBr.M)

~U

x lnI(Du,.M - DC,.M)C~ + CM(Dc,.M

+(DBr.MDcI,M/ OBr,CI)( gBr/VM ))] / [(DBr,M --DCl.M)cO~ + CM(DcI,M ~-(DBrMDCLM/DBr,CO(VB~/VM))]

(41

(37)

Ci in the upper cell and c°~ , c°~, . and C~o5 on the concentrations i..L in the lower cell by the ion exchange equilibria, this means, local thermodynamic phase equilibrium is assumed to exist at the phase boundaries. Eq. (21 is substituted into the right-hand side of

130

A. Heintz et al./Journal of Membrane Science 137 (1997) 121-132

Eq. (8) to obtain cl --- c~_ or cl = C~a+ and Eq. (3) is substituted into Eq. (8) to obtain cl -- c°~ or 0 The corresponding procedure holds for C1 ~ CNa+. the C1 /SO 2- system. Here Eq. (2) for ?sUo~ is substituted into Eq. (1 1) with ccl- taken from Eq. (13) to obtain C~o2 and Eq. (3) for C~o: is substituted into Eq. (1 1) w~th ccl taken from Eq~ (13) to obtain C°o2 . Using this procedure, all unknown concentratiohs + "~ c~ and c o (i = Br , Na , SO5 ) appearing in Eqs. (37)-(39) are now functions of the known time dependent concentrations of ion i in the upper and lower cells, respectively. Therefore, Eqs. ( 3 7 ) - ( 3 9 ) e x h i b i t differential equations of the following structure: dCBUr dt -F(~'~r,~'~r I

(47)

where DB~,M, DC~,M and DBr,CI are parameters. -u dCNa+ dt - G(e~, ' ' CLa4 )

4 . 6 × 1 0 ~ m 2 s t and DC~,M=8.4× 10-11 m 2 s l have been found, the average coupling diffusion (48) Table 3 Diffusion coefficients D~,i (10-l~ m2s ~) in the AMS anion exchange membrane

where DNa,M, OH, M and DNa,H are parameters. d?sUo2_ dt

4

- H(z.v 2 , CSO ,-.L~ ) SO4

(49)

where DSO4,M, DCI,M and Dso4~c~ are parameters. F, G, and H are functions of the variables as indicated. Eqs. (47)-(49) can be integrated humerically and the adjustable parameters, which are the three specific diffusion coefficients appearing in each system, have to be chosen in such a way that an optimized agreement is obtained with the experimental concentration curves in the upper-half cell (see examples of Figs. 4-6). Different starting concentra-

Table 4 Diffusion coefficients Dij NO.

(10-

tions of the same ion exchange system (see Table 1) should lead to identical diffusion coefficients within the experimental error independent of the starting concentrations. This was the reason why different starting concentrations of a given ion exchange pair have been studied. The numerical procedure of fitting the diffusion coefficients to each of the three systems C I - / B r - , Na+/H +, and C1-/SO 4- has been performed, using a least square fitting method (Levenberg-Marquardt Algorithm) applied simultaneously to all measurements belonging to each of the systems. According to Table 1, there are three curves (No. 1-3) for the C1 / B r - system, seven curves (No. 4 10) for the C1-/SO 4 system and six curves (No. 1116) for the Na+/H + system. The results for the diffusion coefficients are presented in Tables 3 and 4. For the A M S membrane averaged values for DBr, M=

No. 1

2 3 Averaged:(1-3) 4 5 o 7 Averaged: (4-7)

DBr,M

DCI,M

DBr,CI

4.6 3.9

5.2

9.2 8.3 7.6

10.7 9.0 10.4

4.6

8.4

10.0

DSO~.M

DCt,M

DcI.so~

1.8

6.3

1.0 1.7 1.8 1.6

7.8 8.0 9.7 8.0

2.3 1.7 2.2 3.2 2.4

l i m2 s- i ) in the cation exchange membranes CMX, CMS, CMV DNa.M

DH, M

CMX

CMS

11

11

55

12

8

31

13

11

64

14 15 16 Averaged: (11-16)

9 12 9 10

33 34 39 43

CMV

DNa,H

CMX

CMS

11

3.6

13

3.5

10

2.6

4

3,4

4

2.0

13

1.7

0.5

2.5

7 13 9 9

6.1 1.8 1.8 3.0

2.3 3.5 4.0 3.1

1.0 0.4 0.4 0.6

2.3 1.5 2.2 1.9

7 5 9 8.5

CMV

CMX

CMS

CMV

0.6

1.5

0.9

0.4

1.4

0.4 0.2 0.3 0.8 0.9 0.6

A. Heint~ et al./Journal of Membrane Science 137 (1997) 121-1.{2

coefficient Dm, c~=10× 10 J i m 2 s-~ is in the same order of magnitude. The diffusion coefficient of S O ~in the A M S membrane is smaller due to the larger size

13 I

Table 5 Diffusion coefficients DNa,M and l)mM {10 if m-'s +) in cation exchange membranes without coupling coefficien!

DN:,,M

DH.M

1.3±4597 1.5±30';} 2.1±359;

5.1±30c/~ 4.9-609, 7.74459~

of SO] compared to B r - and C1 and the higher negative electrical charge. It is reasonable that DC|.M>DBr,M, because the e l -ion is smaller and more mobile than the Br -ion. The average coupling coefficient D(.l so~=2.4× l0 11 m 2 s 1. The value of DCLM

Membrane CMS CMV CMX

calculated from results of the C l - / B r - system (8.4x 10 i~ tn 2 s - i ) agrees very well with the value obtained for DCI.M from the C1-/SO24 - system (8.())< l0 I I m 2 S 1), This confirms the consistency' of the calculation procedure. Standard deviations from the averaged values of the diffusion coefficients are 20~40%. This is not surprising due to the limited sensitivily of the evaluation method including experimental error sources concerning the measurements o17 the exchange equilibria and the diffusional fluxes and due to the membrane material, which is not completel.,,' uniform for each membrane sample,

which is identical with the exchange diffusion equation already derived in literature [10] with 1/VM=CM. If Eq. (50) is used for fitting DN~,M and DH,M to the experimental data of the diffusion curves (No. 1 1-16 in Table l), the results listed in Table 5 are obtained. Comparison of the data in Tables 3 and 4 shows that the values of I)N,,.M and DH.M are distinctly snmller if the coupling term is neglected. This is a reasonable result. In case of the frictional coupling, the fluxes must be identical with those in case without coupling. This can only be obtained if higher mobilities are assumed in case of coupling, since the influence of

5. Discussion and conclusion

friction of opposite fluxes must be compensated by. higher D,M values in order to reach the same diffusion velocity, as in case where no frictional effects occur.

The M a x w e l l - S t e f a n formalism offers a suitable way for describing the ion exchange process in electromembranes providing diffusion coefficients D~,Mof the exchanging ions i in the membrane, Also coupling diffusion coefficients L)ij of the exchanging ions i and j are obtained. Their physical meaning is the frictional influence which the two ions exert to each other by slowing down their opposite diffusion velocity. It is interesting to see what happens if no diffusion coupiing is considered. We discuss this case for the Na~/ H~ system in the three cation exchange membranes, W i t h o u t c o u p l i n g e f f e c t DNa,H g o e s to i n f i n i t y and Eq. (38~ b e c o m e s

I ((VM ÷ VNa)/VM)DN,,~,MDH,MCM JNa == (DNa.M -- DH,M) ca In (DNa,M -- DH,M) "t
(50)

Eq. (50) is the integrated form of the differential ( DNa,MDH,M / VM ) :

II [ H. Cnobloch, W. Kellennann, H. Nischik, K. Pantel. G. Siemensen. Redox ion flow cell for solar energy storage. Siemens Forsch. Entwickl. Bet. 12 t1983) 79-84. 12l E. Korngold, Electrodialysis Membranes and Mass Transport, Synthetic Membrane Process, Academic Press, New York, 1984.

[31 H. Ohya, M. Kuromoto, H. Matsumoto, Y. Negishi, Electrical resistivities and pemaeabilities of composite membranes based on a cation exchange membrane for a redox flo~ battery,, J. Membr. Sci. 51 (19901 201 214. [41 K. Dorfner (Ed.), Ion Exchangers, de Gruyter, Berlin, 1990. 151 R. Kohler, R.E. Brunner, Das Umkehrelektrodialyse-Verfahten zur Wasser- und Abwasserentsalzung,Preprints Aachener Membran Knlloquium. GVC-VDI Gesellschaft Verfahrenstechnik, 1991, pp. 163-191. [~1 T. Sata, Studies nn ion exchange membranes with perm selectivity for specific ions in electrodialysis, J. Membr. Sci. 93 (1994) 117-135.

[71 H. Strathmann, Electrodialysis and related processes, in: R. Noble, A. Stern (Eds.), Membrane Separation Technology, Principles and Applications. Chap. 6, Elsevier, Amslerdam.

equation: JNa

References

--

VCNa+ CNa,MDNaM ÷ (CM -- CNa-)DH,M (51)

1995. [81 1.I. Schoeman, Models for selectiviD in electrodialysls, in: Water Treatment Membrane Processes, (7hap. 7. McGra,a Hill. New York. 1996.

132

A. Heintz et al./Journal of Membrane Science 137 (1997) 121-132

[9] J. Ziegler, Diploma Thesis, University of Heidelberg, Heidelberg, 1995. [10] N. Lakshminarayanaiah, Transport Phenomena in Merebranes, Academic Press, New York, 1969. [11] T. Zugi, Z. Yuqing, Y. Genliang, C. Zheng, O. Mingan, C. Xingqu, Binary and ternary interdiffusion of counter ions across cation exchange membranes, J. Membr. Sci. 52 (1990) 143-156. [12] A, Heintz, C. Illenberger, Diffusion coefficients of Br2 in cation exchange membranes. J. Membr. Sci. 113 (1996) 175-181. [13] E. Scattergood, E.N. Lightfoot, Diffusional interaction in an ion-exchange membrane, Trans. Faraday Soc. 64 (1968) 1135-1143.

[14] J.A. Wesselingh, R. Krishna, Mass Transfer, Ellis Horwood, New York, 1990. [15] R. Taylor, R. Krishna, Multicomponent Mass Transfer, Wiley, New York, 1993. [16] R. Krishna, J.A. Wesselingh, The Maxwell-Stefan approach to mass transfer, Chem. Eng. Sci. 52 (1997) 861-911. [17] A. Heintz, W. Stephan, A generalized solution-diffusion model of the pervaporation process through composite membranes. Part lI. Concentration polarization, coupled diffusion and the influence of the porous support layer, J. Membr. Sci. 89 (1994) 153-169. [18J C.H. Hamann, W. Vielstich, Elektrochemie, Verlag Chemie, Weinheim, 1985.