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Transport numbers from initial time and stationary state measurements of EMF in non-ionic polysulphonic membranes
'
ELSEVIER
journalof MEMBRANE SCIENCE
Journal of MembraneScience 123 (1997) 293-302
Transport numbers from initial time and stationary state measurements of EMF in non-ionic polysulphonic membranes V. Compafi a,* T.S. Scrensen b A. Andfio a L. L6pez a, j. de Abajo c a Dpto. Ciencies Experimentals, Universitat Jaume L 12080 Castell6, Spain b Physical Chemistry, Modelling and Thermodynamics, DTH, NOrager Plads 3, DK2720 VanlOsse, Denmark c Instftuto de polfmeros, Consejo Superior de Investigaciones Cient[ficas (CSIC), 28006-Madrid, Spain
Received 22 May 1996; revised26 July 1996; accepted 26 July 1996
Abstract Initial membrane potentials of two non-ionic polysulphonic (PS) membranes, used in hyperfiltration experiments, have been measured in an electrochemical cell filled with NaC1 solution using Ag/AgC1 electrodes. The concentration in one chamber (left chamber of the cell) was fixed to be 0.01 M. In the other chamber of the cell the concentration was varied between 10 -4 and 0.5 M. The two membranes were (1) a dense, almost uniform membrane, and (2) a porous, asymmetric membrane. It is possible from these initial time EMF measurements to evaluate the transport numbers of the sodium ion corresponding to each of the two surface layers of the membrane. From these transport numbers, an electrochemical characterization of the asymmetry in hyperfiltration membranes can be made. A comparison with the mean transport number obtained by stationary state EMF measurements is also performed. This transport number is always greater than the transport numbers obtained by the initial state method. The increased asymmetry of the membrane No. 2 is reflected in the initial time transport numbers (especially at low salt concentrations) but only to a minor extent in the stationary state mean transport numbers. Keywords: EMF; Polysulphonicmembranes;Transport numbers
1. Introduction Electrochemical properties of asymmetric membranes have been discussed in several papers [1-7]. In these systems, peculiar effects are found with respect to the direction of diffusion in the membrane. In an earlier paper [3], it was expressed that membranes may transitorily exhibit asymmetric EMF values even in cases with the same salt concentration at the two faces of the membrane, when there is a
* Correspondingauthor.
non-zero gradient of the concentration in at least some part of the membrane, and when the transport number is a function of the position in the membrane in the parts of the membrane where concentration gradients exist. The Maxwell electric potential is immeasurable by electrochemical methods in such systems of electrodes, solutions and membranes. Instead an 'observable electric potential' (OEP) linked to measurements with for example a reversible A g / A g C 1 electrode has been defined in previous papers [3,4,7-10]. The difference in the OEP between two electrodes is
0376-7388/97/$17.00 Copyright © 1997 Elsevier Science B.V. All rights reserved. PH S0376-73 8 8(96)00228- 1
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V. Compah et al. / Journal of Membrane Science 123 (1997) 293-302
then equal to the EMF. The application of the concept of OEP, in the framework of the thermodynamics of irreversible processes, produces a profitable formalism which can describe the system in a complete manner, where the true electric potential, not accessible by experiment, is replaced by something we can measure. Heterogeneities in ion-exchange membranes have also been studied more or less directly by means of radiotracer autoradiography experiments [11], by electron microprobe/X-ray spectroscopy, small-angle X-ray and neutron scattering, electron spin resonance, and Mtissbauer spectroscopy [12]. Furthermore, membrane heterogeneities have been studied by electric impedance spectroscopy [13-15]. In the present paper, we investigate two different non-ionic polysulphonic (PS) membranes using the initial time method to evaluate the asymmetry of the two faces. For these two faces, we calculate the transport number of the cation (Na ÷) from the initial values of EMF. The two membranes are (1) an almost homogeneous, dense membrane, and (2) an asymmetric, porous membrane. We have also measured the EMF values in the stationary state of diffusion with the membrane turned in one or the other direction. From such measurements, we may calculate mean transport numbers in the membrane, which can be compared to the transport numbers determined by the initial time method. In order to facilitate the understanding of the effects of membrane asymmetry on measured EMF values, also in cases more general than studied here, we shall incorporate in the theoretical section a short derivation of the most general formula for the calculation of EMF in asymmetric, ion-exchange membranes with a subsequent specialisation to the case of homogeneous membranes.
2. Theory Our analysis is based on a formulation of Thermodynamics of Irreversible Processes where all the fluxes and forces are observable variables. The system studied in the present work is a membrane placed between NaCI solutions in thermal equilibrium and with uniform pressure.
The local dissipation for this system is given by: To- = -j0V~0 - jlV/21 - j2~7/~2
( 1)
where o- is the source strength of entropy production, J0, Jl and J2 are the matter flux densities of the solvent, anion and cation, respectively, in the x-direction in a membrane fixed frame of reference, and V~1, V/22, and V/2o are the gradients of the electrochemical potentials of the anion and the cation, and the gradient of the chemical potential of the solvent, respectively. The electrochemical potentials of the constituent ions are related to the electrochemical potential of the solution, Vixs by: viva, + v2V~2 = V/~s
(2)
We introduce an 'observable electric potential' (OEP or ~/'obs) [4] related to the electrochemical potential of the anion (No. 1). The differences in OEP between two points are simply the EMF measured between two electrodes reversible to the anion. For the gradient in OEP we have:
1 V0obs = - - V ~ l zlF
(3)
If we use matrix notation the new and old forces are:
(x) T = Ivy0;
v 2];
(Xt) T = [V/.L0 ; V/tZs ; V~Yobs]
(4)
The fluxes and forces are related by the following congruent transformation, where T signifies the transpose of a matrix: X = M T x ' ; Y' = MY
(5)
where n is the transformation matrix. In our particular case, from Eqs. (2)-(4): 1 0
0 0 1
0
-1.12
MT=
0 ] zlF (6) z2F
From Eq. (5), we deduce the new fluxes of our formulation from the old ones. The old fluxes were:
(y)T -----[J0; Jl; J2]
(7)
V. Compafi et al. / Journal of Membrane Science 123 (1997) 293-302 and the new fluxes deduced from Eqs. (5) and (6) are:
(y,)X__
[']
J0;--J2; i
(S)
112
the same chemical potential of the salt /z~ as is found in the membrane at the position considered. In the case of zero current density the last line in Eq. (11) may be written (using Eq. (12)): t2(x,c
where i is the electric current density. Notice that only when i = 0, the new flux JJ112 is equal to Jl/111 ( = the common salt flux). We write the local dissipation in the form:
FV0obs = -to(X,C~)VtXo
(9)
122
1 F × EMF
/22 Z2
In the new variables the phenomenological equations are given by:
JolVlool°llo2 =1/10
- V/z 0
/11
(10)
- V/z,
/12 / " -
re,oh
s
The matrix of phenomenological coefficients is symmetric, since a congruent transformation preserves the Onsager symmetry. We may now perform a partial inversion:
( J2/112 0/i =/&o 00 021 -V~Oob~
IX2°
~'ll
~'12/"
A21
AeeJ
A02
F
A20...A12-
_ _ 112 Z2 F
~d
0/Zs
)o t2(x,c,) --~xdX
-d
0/z0
0
0X
(14)
O/zs = 0
(15)
Using this equation we can write:
-
- -
1 fod(
112Z2
× Ol~Sdx
t2(x,c~) -
(13)
F × EMF
The coefficients h02 and A12 are expressed through the usual definitions of the transport numbers of solvent and the cation (No. 2):
to(X,Cs)
Vtx s
The external solutions yield no contribution to EMF and the membrane-solution interfaces also do not contribute since the OEP is continuous through the interfaces. Since all the quantities in Eq. (14) are characterized by the external solutions with salt concentration c~ with which each membrane point is in electrochemical equilibrium, we have the GibbsDuhem relation: O/zo
(11)
112 Z2
- ] i t°( X ' C s ) - - d x
Co- ~x + cs Ox
V/z~J
),
- -
Since the EMF is the difference between the OEP it can be found simply by integrating the above equation through the membrane from x = 0 to x = d:
1
To"= -joVtZo - - - jzV tXs - i~7~bobs
JJ/11
295
t2(x,cs)
112Z2Cs to(X,Cs)} C0 ) (16)
Ox /~21
(12)
The relations to the coefficients A20 and A2~ follow from the fact that a partial inversion makes the phenomenological coefficients between non-inverted and inverted variables antisymmetric. In general, the transport numbers depend on the position x in the membrane and on the concentration of the salt c s in a fictitious external solution in electrochemical (Donnan) equilibrium with the membrane at the point x. This external solution is characterized by having
This is the most general EMF expression one can write for a general, asymmetric ion-exchange membrane with a single diffusing electrolyte under isothermal and isobaric conditions. The EMF is according to this equation, a complicated functional of the 'external' salt concentration profile. The last term in the integrand may be neglected in dilute solutions (Cs/Co<< 1) a n d / o r in non-charged or weakly charged membranes, where the transport number of water is small. In the case of a homogeneous membrane, the expression (16) may be simplified. Then, the explicit
V. Compahet al./ Journalof MembraneScience123 (1997)293-302
296
x-dependence of the transport numbers disappears, and we write: F × EMF I
-
o~{
1"12Z2 fc L tdOs)
-P2Z2CSto(Cs) } -3/xs -dc
CO
s
OCs
(17) Since /xs is a function of c s only, we have: F x EMF 1
tZRI
f /22 Z2 ~L
t2(Ides)
OP2Z2Cs c--to(
} ILl,s) d[d~s
(18) In this case the EMF is completely independent of the salt concentration profile (and therefore also independent of time). The integration through the membrane in Eq. (18) may be replaced by an integration spacing up the difference between the two solutions in the electrode chambers with salt concentrations c L in the left chamber and c R in the right. The trick is to differentiate the expression (18) with respect to tZR: 0EMF
1
3/d,R
t22 Z2
F
t2(]ZR)
-I"2ZzCR -to(uR)
}
CO
(19) Integrating again between /.% and /~R we recover Eq. (18), but with a somewhat different interpretation. Thus, Eq. (18) may be interpreted in two ways: (1) As an integration through the fictitious 'external' solutions which are in equilibrium with the membrane in each point. (2) As an integration over variable solutions in the right electrode chamber from c L to the final value of c R using the whole time transport numbers determined in experiments with electric current, but no salt concentration gradients, and with the membrane in contact with an external salt solution equal to the actual integration value. In this way, the EMF expression has been transformed to a fully 'operational' form using only quantities which can be determined from 'outside'. The latter transformations are not possible with inhomogeneous membranes, where there is no alternative to the full integration of Eq. (16) through the membrane. Recently, we have performed such calculations in the case of ideal 1:1 and 2:1 electrolytes
[16,17]. Also in the case of more than two diffusing ions, there is no alternative to the consideration of the complete ion concentrations profiles and also in that case we in general obtain time dependent EMF values. In Refs. [3,5,6], however, a method was devised for two ion systems in order to circumvent the requirement that the membranes should be homogeneous. We have called this method the 'initial time EMF method'. The basis of this method is the following. Consider Eq. (16). If there is a gradient OtG/Ox different from zero only in one small layer of the membrane (from x 0 - 6 to x 0 + 6, say), then only this layer will contribute to the EMF. The two transport numbers can then be replaced by the transport numbers at the position x 0' and the transport numbers will only be functions of the salt concentration. Thus, we have a 'pseudo-homogeneous' case and all the transformations [17-19] will be valid. Furthermore, Eq. (18) can be interpreted in the two different ways referred to above. If the membrane is equilibrated in an external solution with one salt concentration and then brought into contact at the right hand side with a solution of another concentration, the whole diffusion zone is very narrow and the diffusion zone is limited to the right surface layer of the membrane. Taking the derivative with respect to the salt chemical potential at the right hand side we obtain the equation: 0EMF Fu2' - 3/ZR
t2(/XR) V2CR - + to(P~R) z2 Co -
~'2(CR)
(20)
We have introduced the so-called 'reduced transport number' T2 of the cation at the right interface by the last identity in Eq. (20). The reduced transport number may be considered to be the real transport number of the cation in the membrane corrected for the motion of water. If the diffusion had been in free solution, the reduced transport number would have been the usual Hittorf transport number. In that case, the reduced transport number is the 'real' transport number, since the solvent is the frame of reference. In the presence of a third component (the membrane), however, the physical interpretation of the reduced transport number is less clear. It is a combination of the two measurable quantities t 2 and t 0, measurable
297
V. Compafi et al. / Journal of Membrane Science 123 (1997) 293-302
from the slope of the EMF vs. lna +,variable curve. For the present membranes - carrying no fixed charge the transport number of water will be close to zero, and r 2 will be practically identical to t 2. An equation completely analogous to Eq. (20) may be obtained for the left hand interface. It should be noticed that the variation in the salt chemical potential is given by
dl~ = vRTdln a+= vRTdln(a~/~a~ 2/~)
IxR ~mbmne IlZ
1-8
(21)
and we thus have for the right hand reduced transport number:
IlL IViem'bxane
b'2C R =
-
-
-
Z2 Fv 2
-
Ill,
-
CO 3EMF
RTv Oln a _+,R
(22)
Obviously, if the reduced transport numbers for the right hand face is different from that of the left hand face, the membrane is asymmetric (inhomogeneous). On the other hand, even if the two interfacial transport numbers are practically identical, the transport number in the interior of the membrane might be different. We have also measured the EMF in the stationary state of diffusion with the membrane turned in one or the other direction with respect to the concentration gradient. Experimental measurements [2,18,19] as well as theoretical calculations [16,17] show that the EMF in such cases may easily differ by 5-10 mV at great concentration differences between the two sides.
3. E x p e r i m e n t a l
The measurements were performed with differences in the concentrations at both sides of the membrane. We always first equilibrate the membrane with a solution at a concentration equal to 0.01 M NaC1, which is kept in a glass cell with two electrode chambers filled primarily with this solution. Thereafter, we replace the solution in one of the chambers by another with a different concentration, and the EMF is measured immediately with Ag/AgC1 electrodes already equilibrated in the respective solutions. We repeat the same procedure
Fig. 1. Salt chemical potential profiles in the membrane corresponding to initial time EMF characterisation of the right and the left hand face of the membrane.
with the other electrode chamber to obtain the reduced transport number for the other membrane face. Electric potential differences were measured by means of a pH-meter of high impedance. In all measurements, the temperature was within 25 + I°C. If we follow the experimental procedure described before, the EMF is given under potentiometric conditions by Eq. (18). Fig. 1 shows the concentration profile when the diffusion process starts. The upper picture corresponds to measurements of the EMF at the fight hand side and the lower one to measurements of the EMF at the left hand side. When we are doing initial time measurements we can consider the transport number as a constant in an interval 6 because the concentration, as we have shown in Fig. 1, varies only in a thin layer inside the membrane. EMFright =
RT F %(nght)ln
(23)
EMFleft---
~- r 2 ( l e f t ) l n
(24)
•
These expressions permit a characterization of each of the surface layers of the membrane and make
298
V. Compah et a l . / Journal of Membrane Science 123 (1997) 293-302
possible a determination of the two reduced transport numbers corresponding to the near interfacial layers of an asymmetric membrane. The asymmetry is given by the difference between the surface layer transport numbers. The 'initial' EMF values have all been taken after 15 s of contact of the membrane face with the new solution. The 'mean reduced transport numbers' under stationary state conditions are calculated by formulas similar to Eqs. (23) and (24) for the two orientations of the membrane. The stationary EMF values were read after the observation of a constant voltage during one half hour, controlling also that the conductivities in the electrode chambers did not change during the measurements. Stirring, which is most important for the stationary state EMF measurements [6], were performed in each electrode chamber by means of high rpm magnetic stirrers. The rotation speed was taken as high as possible, and no changes in the EMF values could be seen when the rpm was lowered a little.
4. Preparation and electron micrography of the membranes used Technical polysulfone (UDEL P3500) was used as polymer material for all the membranes. The chemical structure of the repeating unit of the polymer can be seen in Fig. 2. It appears that the polymer is non-ionic. The dense (pore free) membrane was prepared by casting polymer solutions (8% w / v in N,Ndimethylformamide) on glass plates. The solvent was evaporated in an air-circulating oven at 110°C for 6 h, and at 100°C under vacuum for 8 h. Residual solvent was removed by rinsing the polymer film in methanol for 24 h, and by subsequent drying in vacuum at 80°C until constant weight. The preparation of asymmetric, porous mem-
SO2
0
Repeatunit ofpoiym~'one Fig. 2. Chemical structure of the repeating unit of the polysulfone membranes.
Fig. 3. SEM micrograph of a cross section of the dense polysullone membrane. Magnification 1600. At the top we have the membrane face A (turning towards the air during casting). At the bottom membrane face B (in contact with the glass plate during casting). No pore structure is seen at this magnification.
branes with a skin layer was done by casting polymer solutions (16% w / v in N,N-dimethylacetamide) on a flat glass plate and subsequently quenching the cast film in water at 10°C. The membrane was kept in the coagulation bath for 20 min and then transferred to a running water bath, where it was washed for 2 h. Membrane samples for SEM micrographs were prepared from strips approximately 8 mm wide, fracturing the samples under liquid nitrogen. The fractured samples were covered with a thin gold coating (approximately 40 nm) by sputtering in a Balzers SCD004 sputtering coater. The cross sections were photographed on a Jeol JSM 6400 microscope under an accelerating voltage of 20 kV. In Figs. 3 and 4 the SEM micrographs of (1) the dense, and (2) the porous, asymmetric membrane are shown. The cross section of the dense membrane shown in Fig. 3 does not exhibit any difference between the face turning towards the air (at the top) and the face turning towards the glass (bottom) during the process of casting. There are no visible pores or voids at this level of magnification. The asymmetric, porous membrane (Fig. 4) ex-
V. Compafi et al. / Journal of Membrane Science 123 (1997) 293-302
299
barriers for the diffusion. In the bottom of the memb r a n e , e l o n g a t e d m a c r o v o i d s are i n t e r d i s p e r s e d in t h e f o a m - l i k e s t r u c t u r e . T h e r e are n o v i s i b l e ' p o r e s ' in t h e u s u a l s e n s e o f t h e w o r d , as is t h e c a s e w i t h m a n y other membranes for desalination.
5. Results and discussion I n T a b l e s 1 a n d 2, t h e E M F v a l u e s m e a s u r e d f o r t h e t w o m e m b r a n e s u n d e r initial a n d s t a t i o n a r y state c o n d i t i o n s (in t h e t w o d i r e c t i o n s : r i g h t a n d left) are g i v e n . W e h a v e f i t t e d t h e E M F vs. ln(a,,ariable/afi,,) d a t a to l e a s t s q u a r e 3 r d d e g r e e p o l y n o m i a l s p a s s i n g t h r o u g h ( 0 , 0 ) in all f o u r c a s e s (initial left, initial r i g h t , s t a t i o n a r y in o n e d i r e c t i o n , s t a t i o n a r y in t h e o t h e r d i r e c t i o n ) . F o r t h e initial t i m e E M F v a l u e s , t h e third Fig. 4. SEM micrograph of a cross section of the asymmetric, porous polysulfone membrane. Magnification 1600. At the top we have the membrane face A (turning towards the air during casting and quenched by cold water). At the bottom membrane face B (in contact with the glass plate during casting). The structure is evidently a quite open 'frozen foam' structure with large elongated voids near one boundary. The foam lamellae acts as diffusion barriers, but the microscopic diffusion channels in the lamellae cannot be seen at this magnification.
degree
polynomial
has
been
used
for easy
c a l c u l a t i o n o f t h e d e r i v a t i v e ~(EMF)/~lnavari~ble n e e d e d to c a l c u l a t e t h e s u r f a c e t r a n s p o r t n u m b e r f o r each variable concentration. In the case of the stationary state measurements, we define a 'reduced mean transport number' by the relation 'r2,mean ~ ( F / 2 R T ) × IEMF(polynomial)/ln[a
vari~b,e/afix ][ (25)
h i b i t s a d e n s e s t r u c t u r e in t h e t o p l a y e r ( t u r n i n g t o w a r d s t h e air) a n d b e l o w a s t r u c t u r e l o o k i n g like
T h e r e is n o t m u c h s e n s e in d e f i n i n g t h e m e a n v a l u e
'frozen foam'
in t h e d i f f e r e n t i a l w a y a n a l o g o u s to t h e initial t i m e
w h e r e t h e f o a m l a m e l l a e w i l l a c t as
Table 1 Dense membrane c (tool/l)
a
0.5 0.2 0.1 0.07 0.05 0.02 0.01 0.005 0.002 0.001 0.0005 0.0001
0.34036 0.1479 0.077857 0.05586 0.041089 0.01720 0.008896 0.0046376 0.001885 0.00096508 0.00046376 0.00009885
(EMF)A (mY)
-
66.1 56.1 41.0 35.8 26.1 11.7 3.2 16.0 18.5 43.1 62.2 86.0
(EMF) B (mV)
(EMF)st I (mY)
(EMF)st2 (mY)
65.1 54.8 39.2 27.2 23.0 14.2 - 0.2 - 13.1 - 23.2 - 40.1 - 59.9 - 80.1
71.6 51.0 37.6 0.1 - 23.0 - 54.1 - 77.2 - 117.0
71.0 46.5 23.5 0.1 - 22.1 - 53.6 - 81.0 - 107.0
The fixed concentration was c = 0.01 M, a = 0.0090293. The mean ionic activity coefficients ( y ± ) in the external NaC1 solutions were evaluated by the empirical ASPEV formula given in Ref. [20]. The activity is a = y + c / c o where c o = 1 mol/l.
30O
V. Compafi et al. /Journal of Membrane Science 123 (1997) 293-302
Table 2 A s y m m e t r i c porous m e m b r a n e c (mol/1)
a
(EMF) A ( m V )
(EMF) B ( m V )
(EMF)~t I (mV)
(EMF)st 2 ( m V )
0.5 0.2 0.1 0.07
0.34036 0.1479 0.077857 0.05586
66.0 55.1 50.1 32.1
67.0 40.0 41.0 38.1
88.3 61.2 -
81.1 53.7 -
0.05 0.02 0.01
0.041089 0.01720 0.008896
31.0 6.3 -0.9
32.0 7.0 - 1.4
45.1 0.2
38.3 0.1
0.005 0.002 0.001 0.0005
0.0046376 0.001885 0.00096508 0.00046376
- 20.0 - 31.0 - 44.0 -62.0
- 20.0 - 29.0 - 45.0 -65.0
- 44.0 - 91.0 - 124.0
- 40.1 - 85.0 - 117.0
0.0001
0.00009885
- 68.0
- 83.0
- 139.0
- 134.0
The fixed concentration was c = 0.01 M, a = 0.0090293. The mean ionic activity coefficients ( y ± ) in the external NaC1 solutions were evaluated by the empirical A S P E V formula given in Ref. [20]. The activity is a = y ± c / c o where c o = 1 m o l / 1 .
value, since the stationary EMF is a functional of the whole concentration profile. Fig. 5 shows the two initial value EMF transport numbers for the dense membrane as a function of the logarithm of the salt concentration. The variation of the two reduced transport numbers with concentration is very small or rather absent if the level of uncertainty is taken into account. The reduced transport numbers for the two interfaces are 0.38 _ 0.01 and 0.36+0.01, respectively, and the systematic variation shown are the ones induced by the 3rd degree polynomial. The transport numbers for the two membrane faces are quite close to each other if
0"
/
0.40
IIIJIE
Reduced transport number(I~)
I l lllll)l
] I ITI~IIT
not statistically identical. The transport numbers found are for some reason quite close to the transport number of Na + in dilute NaC1 in free aqueous solution at 25°C (ca. 0.40). Thus, the diffusion of the two ions in the molecular channels of the dense membrane seems to be equally restricted. Fig. 6 shows the interracial transport numbers for the porous, asymmetric membrane. The concentration dependence is much more pronounced, especially for one of the interfaces, and the difference between the two transport numbers seems to be greater at the lower concentrations. The transport number of the dense interface (coagulated by water)
tLg
1
T
IHIT number(17)
0.30
0.0001
Concentration moV L
1
Fig. 5. Reduced initial time transport numbers for the Na + ion for the two m e m b r a n e faces as a function of the salt concentration with which the face is in contact. Results for the dense membrane. The rectangles correspond to face A (towards air) and the crosses to face B (towards glass).
~
o.oHII 0.0001
I
IConcentration
1
mol / L
Fig. 6. Reduced initial time transport numbers for the N a + ion for the two m e m b r a n e faces as a function of the salt concentration with which the face is in contact. Results for the asymmetric, porous membrane. The rectangles correspond to face A (towards air) and the crosses to face B (towards glass).
V. Compafi et al. / Journal of Membrane Science 123 (1997) 293-302 0,6
Reduced transport number ( 1 : ) ~
0.30
.............
o.oool
"
'
'"
Concentration mol / L
1
Fig. 7. Reduced mean transport numbers for the Na + ion as a functionof the variable salt concentrationcalculatedfrom stationary state EMF values for the two directions of diffusion through the membrane. Results for the dense membrane. The rectangles correspond to variable concentrationsat the A face and a fixed concentration (0.01 mol/1) at the B face of the membrane.The crosses correspond to variable concentrationsat the B interface and fixed (0.01 mol/1) in face A.
shows less variation with the concentration, and the values are approximately in the same range as for the dense membrane (Fig. 5). Figs. 7 and 8 show the 'reduced mean transport numbers' calculated from stationary state measurements with the membrane in one or the other orientation relative to the concentration drop. The dependence of the mean transport numbers on the variable
0.90
Reduced transport number (I:)
g
301
concentration for the dense membrane is shown in Fig. 7. The two curves for the two directions of diffusion are quite close to each other and show a similar variation with concentration. The mean transport number is quite constant around 0.5 in the range of variable concentrations between 1 0 - 4 m o l / 1 and 0.01 mol/1, Above 0.01 m o l / 1 Z2,mean decreases somewhat to ca. 0.38 at 0.5 mol/1. (The fixed concentration is in all cases equal to 0.01 mol/1). Thus, except for the highest concentration, the mean transport numbers appear to be somewhat higher than the initial time transport numbers. Fig. 8 shows the reduced stationary state mean transport number for Na + in the asymmetric, porous membrane as a function of the variable concentration. The values exhibit a great dispersion with this concentration increasing from ca. 0.6 for the most dilute variable concentration through a maximum value (ca. 0.8) situated around 0.008 mol/1. At higher variable concentrations, the mean transport number decreased to ca. 0.45 at 0.5 reel/1. Thus, in all cases, the stationary state reduced mean transport numbers are higher than the initial time values of the reduced transport numbers (cf., Fig. 6). Also, we observe that the difference between the stationary state mean transport numbers for the two directions of diffusion is generally larger in the case of the asymmetric, porous membrane than in the case of the dense membrane (cf., Fig. 7). Thus, the conclusion is that for the dense membrane the transport numbers are practically independent of concentration and also of position in the membrane, so that the dense membrane is effectively homogeneous. However, the stationary state values of the transport numbers seem to be somewhat higher than the initial time values.
I
Acknowledgements
0.30 0.000
Concentration mol / L
:L
Fig. 8. Reduced mean transport numbers for the Na + ion as a function of the variable salt concentrationcalculatedfrom stationary state EMF values for the two directions of diffusion through the membrane.Results for the asymmetric,porous membrane.The rectangles correspond to variable concentrationsat the A face and a fixed concentration(0.01 mol/1) at the B face of the membrane. The crosses correspond to variable concentrationsat the B interface and fixed (0.01 mol/1) in face A.
The authors wish to acknowledge the financial support provided from the Direcci6n General de Investigaci6n Ciencia y Tecnolog~a, D G I C Y T (Mininistry of Education and Science of Spain) under project PB92-0773-C03-03 and also to Bancaja under project P1B95-04. One of the authors (Torben Smith SCrensen) is indebted to DGICYT for grants for research at Universitat Jaume I, Castell6n, Spain.
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References [1] N. Kamo and Kobatake, Interpretation of asymmetric membrane potential, J. Colloid Interface Sci., 86 (1988) 85. [2] T.S, Serensen, J.B. Jensen and B. Malgren-Hansen, Electrochemical characterisation of cellulose acetate membranes. 1. Influence of hydrogen and calcium ions on the emf of LiC1 concentration cells with a CA-membrane as separator, J. Non-Eq. Therm., 13 (1988) 57. [3] J. Garrido and V. Compafi, Asymmetry potential in inhomogeneous membranes, J. Phys. Chem., 96 (1992) 2721. [4] J. Garrido, V. Compafi and M.L. L6pez, Observable electric potential in nonequilibrium electrolyte solutions with a common ion, J. Phys. Chem., 98 (1994) 6003. [5] V. Compafi, M.L. L6pez, T.S. S0rensen and J. Garrido, Transport numbers in the surface layers of asymmetric membranes from initial time measurements, J. Phys. Chem., 98 (1994) 9013. [6] V. Compafi, T.S. S0rensen and S.R. Rivera, Comparison of initial time and stationary state measurements of the emf of concentration cells using phenolsulfonic acid membrane separators, J. Phys. Chem., 99 (1995) 12553. [7] J. Garrido, S. Maf6 and V.M. Aguilella, Observable variables of the transport processes in discontinuous systems, Electrochim. Acta, 33 (1988) 1151. [8] J. Garrido, V. Compafi, V.M. Aguilella, S. Maf6, J. Garrido and V. Compafi, Thermodynamics of electrokinetic processes. I. Formulations, Electrochim. Acta, 35 (1990) 705. [9] J. Garrido and V. Compafi, Thermodynamics of electrokinetic processes. II. Systems with different kinds of electrodes, Electrochim. Acta, 35 (1990) 711. [10] J. Garrido, V. Compafi and M.L. L6pez, Electrical magnitudes in gravitational and centrifugal systems, Electrochim. Acta, 38 (1993) 877. [ 11 ] R. W6dski, A. Narebska and J. Ceynova, Nonuniform distribution of the ionogenic groups in permselective membranes, J. Angew. Makromol. Chem., 78 (1979) 145.
[12] M. Pineri, Coulombic interactions in macromolecular systems, ACS Symp. Set., 302 (1986). [13] B. Malmgren-Hansen, T.S. S0rensen, B. Jensen and M. Hennenberg, Electric impedance of cellulose actetate membranes and a composite membrane at different salt concentrations, J. Colloid Interface Sci., 130 (1989) 359. [14] T.S. Serensen, Capillarity today, Lect. Notes Phys., 386 (1991); A. Eisenberg and F.E. Bailey (Eds.), ACS. Symp. Set., 302 (1986). [15] I.W. Plesner, B. Malmgren-Hansen and T.S. S0rensen, Distribution of electrolytes between membraneous and bulk phases and the dielectric properties of membraneous water studied by impedance spectroscopy measurements on dense cellulose acetate membranes, J. Chem. Soc., Faraday Trans., 90 (1994) 2381. [16] T.S. Serensen and V. Compafi, Nemst Planck model simulating the electromotive force measured over aymmetric membranes with special reference to the initial time method for investigation of surface layers, J. Phys. Chem., 100 (1996) 7623. [17] T.S. S0rensen and V. Compafi, Salt flux and EMF in concentration cells with asymmetric ion exchange membranes and ideal 2:1 electrolytes, J. Phys. Chem., 100 (1996) 15261. [18] T.S. S~rensen and J.B. Jensen, Membrane charge and Donnan distribution of ions by electromotive force (EMF) measurements on a homogeneous cellulose acetate membrane, J. Non-Eq. Therm., 9 (1984) 1. [19] T.S. Serensen, J.B. Jensen and B. Malgren-Hansen, Electrochemical characterisation of cellulose acetate membranes. 2. Influence of casting conditions and of thermal curing of CA-membranes on the EMF of concentration cells with HC1, LiCI, NaCI and KCI solutions, J. Non-Eq. Therm., 13 (1988) 193. [20] T.S. S0rensen, P. Sloth and M. Schreder, Ionic radii from experimental activities and simple statistical-mechanical theories for strong electrolytes with small Bjerrum parameters, Acta Chem. Scand. A, 38 (1984) 735.
ELSEVIER
journalof MEMBRANE SCIENCE
Journal of MembraneScience 123 (1997) 293-302
Transport numbers from initial time and stationary state measurements of EMF in non-ionic polysulphonic membranes V. Compafi a,* T.S. Scrensen b A. Andfio a L. L6pez a, j. de Abajo c a Dpto. Ciencies Experimentals, Universitat Jaume L 12080 Castell6, Spain b Physical Chemistry, Modelling and Thermodynamics, DTH, NOrager Plads 3, DK2720 VanlOsse, Denmark c Instftuto de polfmeros, Consejo Superior de Investigaciones Cient[ficas (CSIC), 28006-Madrid, Spain
Received 22 May 1996; revised26 July 1996; accepted 26 July 1996
Abstract Initial membrane potentials of two non-ionic polysulphonic (PS) membranes, used in hyperfiltration experiments, have been measured in an electrochemical cell filled with NaC1 solution using Ag/AgC1 electrodes. The concentration in one chamber (left chamber of the cell) was fixed to be 0.01 M. In the other chamber of the cell the concentration was varied between 10 -4 and 0.5 M. The two membranes were (1) a dense, almost uniform membrane, and (2) a porous, asymmetric membrane. It is possible from these initial time EMF measurements to evaluate the transport numbers of the sodium ion corresponding to each of the two surface layers of the membrane. From these transport numbers, an electrochemical characterization of the asymmetry in hyperfiltration membranes can be made. A comparison with the mean transport number obtained by stationary state EMF measurements is also performed. This transport number is always greater than the transport numbers obtained by the initial state method. The increased asymmetry of the membrane No. 2 is reflected in the initial time transport numbers (especially at low salt concentrations) but only to a minor extent in the stationary state mean transport numbers. Keywords: EMF; Polysulphonicmembranes;Transport numbers
1. Introduction Electrochemical properties of asymmetric membranes have been discussed in several papers [1-7]. In these systems, peculiar effects are found with respect to the direction of diffusion in the membrane. In an earlier paper [3], it was expressed that membranes may transitorily exhibit asymmetric EMF values even in cases with the same salt concentration at the two faces of the membrane, when there is a
* Correspondingauthor.
non-zero gradient of the concentration in at least some part of the membrane, and when the transport number is a function of the position in the membrane in the parts of the membrane where concentration gradients exist. The Maxwell electric potential is immeasurable by electrochemical methods in such systems of electrodes, solutions and membranes. Instead an 'observable electric potential' (OEP) linked to measurements with for example a reversible A g / A g C 1 electrode has been defined in previous papers [3,4,7-10]. The difference in the OEP between two electrodes is
0376-7388/97/$17.00 Copyright © 1997 Elsevier Science B.V. All rights reserved. PH S0376-73 8 8(96)00228- 1
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V. Compah et al. / Journal of Membrane Science 123 (1997) 293-302
then equal to the EMF. The application of the concept of OEP, in the framework of the thermodynamics of irreversible processes, produces a profitable formalism which can describe the system in a complete manner, where the true electric potential, not accessible by experiment, is replaced by something we can measure. Heterogeneities in ion-exchange membranes have also been studied more or less directly by means of radiotracer autoradiography experiments [11], by electron microprobe/X-ray spectroscopy, small-angle X-ray and neutron scattering, electron spin resonance, and Mtissbauer spectroscopy [12]. Furthermore, membrane heterogeneities have been studied by electric impedance spectroscopy [13-15]. In the present paper, we investigate two different non-ionic polysulphonic (PS) membranes using the initial time method to evaluate the asymmetry of the two faces. For these two faces, we calculate the transport number of the cation (Na ÷) from the initial values of EMF. The two membranes are (1) an almost homogeneous, dense membrane, and (2) an asymmetric, porous membrane. We have also measured the EMF values in the stationary state of diffusion with the membrane turned in one or the other direction. From such measurements, we may calculate mean transport numbers in the membrane, which can be compared to the transport numbers determined by the initial time method. In order to facilitate the understanding of the effects of membrane asymmetry on measured EMF values, also in cases more general than studied here, we shall incorporate in the theoretical section a short derivation of the most general formula for the calculation of EMF in asymmetric, ion-exchange membranes with a subsequent specialisation to the case of homogeneous membranes.
2. Theory Our analysis is based on a formulation of Thermodynamics of Irreversible Processes where all the fluxes and forces are observable variables. The system studied in the present work is a membrane placed between NaCI solutions in thermal equilibrium and with uniform pressure.
The local dissipation for this system is given by: To- = -j0V~0 - jlV/21 - j2~7/~2
( 1)
where o- is the source strength of entropy production, J0, Jl and J2 are the matter flux densities of the solvent, anion and cation, respectively, in the x-direction in a membrane fixed frame of reference, and V~1, V/22, and V/2o are the gradients of the electrochemical potentials of the anion and the cation, and the gradient of the chemical potential of the solvent, respectively. The electrochemical potentials of the constituent ions are related to the electrochemical potential of the solution, Vixs by: viva, + v2V~2 = V/~s
(2)
We introduce an 'observable electric potential' (OEP or ~/'obs) [4] related to the electrochemical potential of the anion (No. 1). The differences in OEP between two points are simply the EMF measured between two electrodes reversible to the anion. For the gradient in OEP we have:
1 V0obs = - - V ~ l zlF
(3)
If we use matrix notation the new and old forces are:
(x) T = Ivy0;
v 2];
(Xt) T = [V/.L0 ; V/tZs ; V~Yobs]
(4)
The fluxes and forces are related by the following congruent transformation, where T signifies the transpose of a matrix: X = M T x ' ; Y' = MY
(5)
where n is the transformation matrix. In our particular case, from Eqs. (2)-(4): 1 0
0 0 1
0
-1.12
MT=
0 ] zlF (6) z2F
From Eq. (5), we deduce the new fluxes of our formulation from the old ones. The old fluxes were:
(y)T -----[J0; Jl; J2]
(7)
V. Compafi et al. / Journal of Membrane Science 123 (1997) 293-302 and the new fluxes deduced from Eqs. (5) and (6) are:
(y,)X__
[']
J0;--J2; i
(S)
112
the same chemical potential of the salt /z~ as is found in the membrane at the position considered. In the case of zero current density the last line in Eq. (11) may be written (using Eq. (12)): t2(x,c
where i is the electric current density. Notice that only when i = 0, the new flux JJ112 is equal to Jl/111 ( = the common salt flux). We write the local dissipation in the form:
FV0obs = -to(X,C~)VtXo
(9)
122
1 F × EMF
/22 Z2
In the new variables the phenomenological equations are given by:
JolVlool°llo2 =1/10
- V/z 0
/11
(10)
- V/z,
/12 / " -
re,oh
s
The matrix of phenomenological coefficients is symmetric, since a congruent transformation preserves the Onsager symmetry. We may now perform a partial inversion:
( J2/112 0/i =/&o 00 021 -V~Oob~
IX2°
~'ll
~'12/"
A21
AeeJ
A02
F
A20...A12-
_ _ 112 Z2 F
~d
0/Zs
)o t2(x,c,) --~xdX
-d
0/z0
0
0X
(14)
O/zs = 0
(15)
Using this equation we can write:
-
- -
1 fod(
112Z2
× Ol~Sdx
t2(x,c~) -
(13)
F × EMF
The coefficients h02 and A12 are expressed through the usual definitions of the transport numbers of solvent and the cation (No. 2):
to(X,Cs)
Vtx s
The external solutions yield no contribution to EMF and the membrane-solution interfaces also do not contribute since the OEP is continuous through the interfaces. Since all the quantities in Eq. (14) are characterized by the external solutions with salt concentration c~ with which each membrane point is in electrochemical equilibrium, we have the GibbsDuhem relation: O/zo
(11)
112 Z2
- ] i t°( X ' C s ) - - d x
Co- ~x + cs Ox
V/z~J
),
- -
Since the EMF is the difference between the OEP it can be found simply by integrating the above equation through the membrane from x = 0 to x = d:
1
To"= -joVtZo - - - jzV tXs - i~7~bobs
JJ/11
295
t2(x,cs)
112Z2Cs to(X,Cs)} C0 ) (16)
Ox /~21
(12)
The relations to the coefficients A20 and A2~ follow from the fact that a partial inversion makes the phenomenological coefficients between non-inverted and inverted variables antisymmetric. In general, the transport numbers depend on the position x in the membrane and on the concentration of the salt c s in a fictitious external solution in electrochemical (Donnan) equilibrium with the membrane at the point x. This external solution is characterized by having
This is the most general EMF expression one can write for a general, asymmetric ion-exchange membrane with a single diffusing electrolyte under isothermal and isobaric conditions. The EMF is according to this equation, a complicated functional of the 'external' salt concentration profile. The last term in the integrand may be neglected in dilute solutions (Cs/Co<< 1) a n d / o r in non-charged or weakly charged membranes, where the transport number of water is small. In the case of a homogeneous membrane, the expression (16) may be simplified. Then, the explicit
V. Compahet al./ Journalof MembraneScience123 (1997)293-302
296
x-dependence of the transport numbers disappears, and we write: F × EMF I
-
o~{
1"12Z2 fc L tdOs)
-P2Z2CSto(Cs) } -3/xs -dc
CO
s
OCs
(17) Since /xs is a function of c s only, we have: F x EMF 1
tZRI
f /22 Z2 ~L
t2(Ides)
OP2Z2Cs c--to(
} ILl,s) d[d~s
(18) In this case the EMF is completely independent of the salt concentration profile (and therefore also independent of time). The integration through the membrane in Eq. (18) may be replaced by an integration spacing up the difference between the two solutions in the electrode chambers with salt concentrations c L in the left chamber and c R in the right. The trick is to differentiate the expression (18) with respect to tZR: 0EMF
1
3/d,R
t22 Z2
F
t2(]ZR)
-I"2ZzCR -to(uR)
}
CO
(19) Integrating again between /.% and /~R we recover Eq. (18), but with a somewhat different interpretation. Thus, Eq. (18) may be interpreted in two ways: (1) As an integration through the fictitious 'external' solutions which are in equilibrium with the membrane in each point. (2) As an integration over variable solutions in the right electrode chamber from c L to the final value of c R using the whole time transport numbers determined in experiments with electric current, but no salt concentration gradients, and with the membrane in contact with an external salt solution equal to the actual integration value. In this way, the EMF expression has been transformed to a fully 'operational' form using only quantities which can be determined from 'outside'. The latter transformations are not possible with inhomogeneous membranes, where there is no alternative to the full integration of Eq. (16) through the membrane. Recently, we have performed such calculations in the case of ideal 1:1 and 2:1 electrolytes
[16,17]. Also in the case of more than two diffusing ions, there is no alternative to the consideration of the complete ion concentrations profiles and also in that case we in general obtain time dependent EMF values. In Refs. [3,5,6], however, a method was devised for two ion systems in order to circumvent the requirement that the membranes should be homogeneous. We have called this method the 'initial time EMF method'. The basis of this method is the following. Consider Eq. (16). If there is a gradient OtG/Ox different from zero only in one small layer of the membrane (from x 0 - 6 to x 0 + 6, say), then only this layer will contribute to the EMF. The two transport numbers can then be replaced by the transport numbers at the position x 0' and the transport numbers will only be functions of the salt concentration. Thus, we have a 'pseudo-homogeneous' case and all the transformations [17-19] will be valid. Furthermore, Eq. (18) can be interpreted in the two different ways referred to above. If the membrane is equilibrated in an external solution with one salt concentration and then brought into contact at the right hand side with a solution of another concentration, the whole diffusion zone is very narrow and the diffusion zone is limited to the right surface layer of the membrane. Taking the derivative with respect to the salt chemical potential at the right hand side we obtain the equation: 0EMF Fu2' - 3/ZR
t2(/XR) V2CR - + to(P~R) z2 Co -
~'2(CR)
(20)
We have introduced the so-called 'reduced transport number' T2 of the cation at the right interface by the last identity in Eq. (20). The reduced transport number may be considered to be the real transport number of the cation in the membrane corrected for the motion of water. If the diffusion had been in free solution, the reduced transport number would have been the usual Hittorf transport number. In that case, the reduced transport number is the 'real' transport number, since the solvent is the frame of reference. In the presence of a third component (the membrane), however, the physical interpretation of the reduced transport number is less clear. It is a combination of the two measurable quantities t 2 and t 0, measurable
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V. Compafi et al. / Journal of Membrane Science 123 (1997) 293-302
from the slope of the EMF vs. lna +,variable curve. For the present membranes - carrying no fixed charge the transport number of water will be close to zero, and r 2 will be practically identical to t 2. An equation completely analogous to Eq. (20) may be obtained for the left hand interface. It should be noticed that the variation in the salt chemical potential is given by
dl~ = vRTdln a+= vRTdln(a~/~a~ 2/~)
IxR ~mbmne IlZ
1-8
(21)
and we thus have for the right hand reduced transport number:
IlL IViem'bxane
b'2C R =
-
-
-
Z2 Fv 2
-
Ill,
-
CO 3EMF
RTv Oln a _+,R
(22)
Obviously, if the reduced transport numbers for the right hand face is different from that of the left hand face, the membrane is asymmetric (inhomogeneous). On the other hand, even if the two interfacial transport numbers are practically identical, the transport number in the interior of the membrane might be different. We have also measured the EMF in the stationary state of diffusion with the membrane turned in one or the other direction with respect to the concentration gradient. Experimental measurements [2,18,19] as well as theoretical calculations [16,17] show that the EMF in such cases may easily differ by 5-10 mV at great concentration differences between the two sides.
3. E x p e r i m e n t a l
The measurements were performed with differences in the concentrations at both sides of the membrane. We always first equilibrate the membrane with a solution at a concentration equal to 0.01 M NaC1, which is kept in a glass cell with two electrode chambers filled primarily with this solution. Thereafter, we replace the solution in one of the chambers by another with a different concentration, and the EMF is measured immediately with Ag/AgC1 electrodes already equilibrated in the respective solutions. We repeat the same procedure
Fig. 1. Salt chemical potential profiles in the membrane corresponding to initial time EMF characterisation of the right and the left hand face of the membrane.
with the other electrode chamber to obtain the reduced transport number for the other membrane face. Electric potential differences were measured by means of a pH-meter of high impedance. In all measurements, the temperature was within 25 + I°C. If we follow the experimental procedure described before, the EMF is given under potentiometric conditions by Eq. (18). Fig. 1 shows the concentration profile when the diffusion process starts. The upper picture corresponds to measurements of the EMF at the fight hand side and the lower one to measurements of the EMF at the left hand side. When we are doing initial time measurements we can consider the transport number as a constant in an interval 6 because the concentration, as we have shown in Fig. 1, varies only in a thin layer inside the membrane. EMFright =
RT F %(nght)ln
(23)
EMFleft---
~- r 2 ( l e f t ) l n
(24)
•
These expressions permit a characterization of each of the surface layers of the membrane and make
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V. Compah et a l . / Journal of Membrane Science 123 (1997) 293-302
possible a determination of the two reduced transport numbers corresponding to the near interfacial layers of an asymmetric membrane. The asymmetry is given by the difference between the surface layer transport numbers. The 'initial' EMF values have all been taken after 15 s of contact of the membrane face with the new solution. The 'mean reduced transport numbers' under stationary state conditions are calculated by formulas similar to Eqs. (23) and (24) for the two orientations of the membrane. The stationary EMF values were read after the observation of a constant voltage during one half hour, controlling also that the conductivities in the electrode chambers did not change during the measurements. Stirring, which is most important for the stationary state EMF measurements [6], were performed in each electrode chamber by means of high rpm magnetic stirrers. The rotation speed was taken as high as possible, and no changes in the EMF values could be seen when the rpm was lowered a little.
4. Preparation and electron micrography of the membranes used Technical polysulfone (UDEL P3500) was used as polymer material for all the membranes. The chemical structure of the repeating unit of the polymer can be seen in Fig. 2. It appears that the polymer is non-ionic. The dense (pore free) membrane was prepared by casting polymer solutions (8% w / v in N,Ndimethylformamide) on glass plates. The solvent was evaporated in an air-circulating oven at 110°C for 6 h, and at 100°C under vacuum for 8 h. Residual solvent was removed by rinsing the polymer film in methanol for 24 h, and by subsequent drying in vacuum at 80°C until constant weight. The preparation of asymmetric, porous mem-
SO2
0
Repeatunit ofpoiym~'one Fig. 2. Chemical structure of the repeating unit of the polysulfone membranes.
Fig. 3. SEM micrograph of a cross section of the dense polysullone membrane. Magnification 1600. At the top we have the membrane face A (turning towards the air during casting). At the bottom membrane face B (in contact with the glass plate during casting). No pore structure is seen at this magnification.
branes with a skin layer was done by casting polymer solutions (16% w / v in N,N-dimethylacetamide) on a flat glass plate and subsequently quenching the cast film in water at 10°C. The membrane was kept in the coagulation bath for 20 min and then transferred to a running water bath, where it was washed for 2 h. Membrane samples for SEM micrographs were prepared from strips approximately 8 mm wide, fracturing the samples under liquid nitrogen. The fractured samples were covered with a thin gold coating (approximately 40 nm) by sputtering in a Balzers SCD004 sputtering coater. The cross sections were photographed on a Jeol JSM 6400 microscope under an accelerating voltage of 20 kV. In Figs. 3 and 4 the SEM micrographs of (1) the dense, and (2) the porous, asymmetric membrane are shown. The cross section of the dense membrane shown in Fig. 3 does not exhibit any difference between the face turning towards the air (at the top) and the face turning towards the glass (bottom) during the process of casting. There are no visible pores or voids at this level of magnification. The asymmetric, porous membrane (Fig. 4) ex-
V. Compafi et al. / Journal of Membrane Science 123 (1997) 293-302
299
barriers for the diffusion. In the bottom of the memb r a n e , e l o n g a t e d m a c r o v o i d s are i n t e r d i s p e r s e d in t h e f o a m - l i k e s t r u c t u r e . T h e r e are n o v i s i b l e ' p o r e s ' in t h e u s u a l s e n s e o f t h e w o r d , as is t h e c a s e w i t h m a n y other membranes for desalination.
5. Results and discussion I n T a b l e s 1 a n d 2, t h e E M F v a l u e s m e a s u r e d f o r t h e t w o m e m b r a n e s u n d e r initial a n d s t a t i o n a r y state c o n d i t i o n s (in t h e t w o d i r e c t i o n s : r i g h t a n d left) are g i v e n . W e h a v e f i t t e d t h e E M F vs. ln(a,,ariable/afi,,) d a t a to l e a s t s q u a r e 3 r d d e g r e e p o l y n o m i a l s p a s s i n g t h r o u g h ( 0 , 0 ) in all f o u r c a s e s (initial left, initial r i g h t , s t a t i o n a r y in o n e d i r e c t i o n , s t a t i o n a r y in t h e o t h e r d i r e c t i o n ) . F o r t h e initial t i m e E M F v a l u e s , t h e third Fig. 4. SEM micrograph of a cross section of the asymmetric, porous polysulfone membrane. Magnification 1600. At the top we have the membrane face A (turning towards the air during casting and quenched by cold water). At the bottom membrane face B (in contact with the glass plate during casting). The structure is evidently a quite open 'frozen foam' structure with large elongated voids near one boundary. The foam lamellae acts as diffusion barriers, but the microscopic diffusion channels in the lamellae cannot be seen at this magnification.
degree
polynomial
has
been
used
for easy
c a l c u l a t i o n o f t h e d e r i v a t i v e ~(EMF)/~lnavari~ble n e e d e d to c a l c u l a t e t h e s u r f a c e t r a n s p o r t n u m b e r f o r each variable concentration. In the case of the stationary state measurements, we define a 'reduced mean transport number' by the relation 'r2,mean ~ ( F / 2 R T ) × IEMF(polynomial)/ln[a
vari~b,e/afix ][ (25)
h i b i t s a d e n s e s t r u c t u r e in t h e t o p l a y e r ( t u r n i n g t o w a r d s t h e air) a n d b e l o w a s t r u c t u r e l o o k i n g like
T h e r e is n o t m u c h s e n s e in d e f i n i n g t h e m e a n v a l u e
'frozen foam'
in t h e d i f f e r e n t i a l w a y a n a l o g o u s to t h e initial t i m e
w h e r e t h e f o a m l a m e l l a e w i l l a c t as
Table 1 Dense membrane c (tool/l)
a
0.5 0.2 0.1 0.07 0.05 0.02 0.01 0.005 0.002 0.001 0.0005 0.0001
0.34036 0.1479 0.077857 0.05586 0.041089 0.01720 0.008896 0.0046376 0.001885 0.00096508 0.00046376 0.00009885
(EMF)A (mY)
-
66.1 56.1 41.0 35.8 26.1 11.7 3.2 16.0 18.5 43.1 62.2 86.0
(EMF) B (mV)
(EMF)st I (mY)
(EMF)st2 (mY)
65.1 54.8 39.2 27.2 23.0 14.2 - 0.2 - 13.1 - 23.2 - 40.1 - 59.9 - 80.1
71.6 51.0 37.6 0.1 - 23.0 - 54.1 - 77.2 - 117.0
71.0 46.5 23.5 0.1 - 22.1 - 53.6 - 81.0 - 107.0
The fixed concentration was c = 0.01 M, a = 0.0090293. The mean ionic activity coefficients ( y ± ) in the external NaC1 solutions were evaluated by the empirical ASPEV formula given in Ref. [20]. The activity is a = y + c / c o where c o = 1 mol/l.
30O
V. Compafi et al. /Journal of Membrane Science 123 (1997) 293-302
Table 2 A s y m m e t r i c porous m e m b r a n e c (mol/1)
a
(EMF) A ( m V )
(EMF) B ( m V )
(EMF)~t I (mV)
(EMF)st 2 ( m V )
0.5 0.2 0.1 0.07
0.34036 0.1479 0.077857 0.05586
66.0 55.1 50.1 32.1
67.0 40.0 41.0 38.1
88.3 61.2 -
81.1 53.7 -
0.05 0.02 0.01
0.041089 0.01720 0.008896
31.0 6.3 -0.9
32.0 7.0 - 1.4
45.1 0.2
38.3 0.1
0.005 0.002 0.001 0.0005
0.0046376 0.001885 0.00096508 0.00046376
- 20.0 - 31.0 - 44.0 -62.0
- 20.0 - 29.0 - 45.0 -65.0
- 44.0 - 91.0 - 124.0
- 40.1 - 85.0 - 117.0
0.0001
0.00009885
- 68.0
- 83.0
- 139.0
- 134.0
The fixed concentration was c = 0.01 M, a = 0.0090293. The mean ionic activity coefficients ( y ± ) in the external NaC1 solutions were evaluated by the empirical A S P E V formula given in Ref. [20]. The activity is a = y ± c / c o where c o = 1 m o l / 1 .
value, since the stationary EMF is a functional of the whole concentration profile. Fig. 5 shows the two initial value EMF transport numbers for the dense membrane as a function of the logarithm of the salt concentration. The variation of the two reduced transport numbers with concentration is very small or rather absent if the level of uncertainty is taken into account. The reduced transport numbers for the two interfaces are 0.38 _ 0.01 and 0.36+0.01, respectively, and the systematic variation shown are the ones induced by the 3rd degree polynomial. The transport numbers for the two membrane faces are quite close to each other if
0"
/
0.40
IIIJIE
Reduced transport number(I~)
I l lllll)l
] I ITI~IIT
not statistically identical. The transport numbers found are for some reason quite close to the transport number of Na + in dilute NaC1 in free aqueous solution at 25°C (ca. 0.40). Thus, the diffusion of the two ions in the molecular channels of the dense membrane seems to be equally restricted. Fig. 6 shows the interracial transport numbers for the porous, asymmetric membrane. The concentration dependence is much more pronounced, especially for one of the interfaces, and the difference between the two transport numbers seems to be greater at the lower concentrations. The transport number of the dense interface (coagulated by water)
tLg
1
T
IHIT number(17)
0.30
0.0001
Concentration moV L
1
Fig. 5. Reduced initial time transport numbers for the Na + ion for the two m e m b r a n e faces as a function of the salt concentration with which the face is in contact. Results for the dense membrane. The rectangles correspond to face A (towards air) and the crosses to face B (towards glass).
~
o.oHII 0.0001
I
IConcentration
1
mol / L
Fig. 6. Reduced initial time transport numbers for the N a + ion for the two m e m b r a n e faces as a function of the salt concentration with which the face is in contact. Results for the asymmetric, porous membrane. The rectangles correspond to face A (towards air) and the crosses to face B (towards glass).
V. Compafi et al. / Journal of Membrane Science 123 (1997) 293-302 0,6
Reduced transport number ( 1 : ) ~
0.30
.............
o.oool
"
'
'"
Concentration mol / L
1
Fig. 7. Reduced mean transport numbers for the Na + ion as a functionof the variable salt concentrationcalculatedfrom stationary state EMF values for the two directions of diffusion through the membrane. Results for the dense membrane. The rectangles correspond to variable concentrationsat the A face and a fixed concentration (0.01 mol/1) at the B face of the membrane.The crosses correspond to variable concentrationsat the B interface and fixed (0.01 mol/1) in face A.
shows less variation with the concentration, and the values are approximately in the same range as for the dense membrane (Fig. 5). Figs. 7 and 8 show the 'reduced mean transport numbers' calculated from stationary state measurements with the membrane in one or the other orientation relative to the concentration drop. The dependence of the mean transport numbers on the variable
0.90
Reduced transport number (I:)
g
301
concentration for the dense membrane is shown in Fig. 7. The two curves for the two directions of diffusion are quite close to each other and show a similar variation with concentration. The mean transport number is quite constant around 0.5 in the range of variable concentrations between 1 0 - 4 m o l / 1 and 0.01 mol/1, Above 0.01 m o l / 1 Z2,mean decreases somewhat to ca. 0.38 at 0.5 mol/1. (The fixed concentration is in all cases equal to 0.01 mol/1). Thus, except for the highest concentration, the mean transport numbers appear to be somewhat higher than the initial time transport numbers. Fig. 8 shows the reduced stationary state mean transport number for Na + in the asymmetric, porous membrane as a function of the variable concentration. The values exhibit a great dispersion with this concentration increasing from ca. 0.6 for the most dilute variable concentration through a maximum value (ca. 0.8) situated around 0.008 mol/1. At higher variable concentrations, the mean transport number decreased to ca. 0.45 at 0.5 reel/1. Thus, in all cases, the stationary state reduced mean transport numbers are higher than the initial time values of the reduced transport numbers (cf., Fig. 6). Also, we observe that the difference between the stationary state mean transport numbers for the two directions of diffusion is generally larger in the case of the asymmetric, porous membrane than in the case of the dense membrane (cf., Fig. 7). Thus, the conclusion is that for the dense membrane the transport numbers are practically independent of concentration and also of position in the membrane, so that the dense membrane is effectively homogeneous. However, the stationary state values of the transport numbers seem to be somewhat higher than the initial time values.
I
Acknowledgements
0.30 0.000
Concentration mol / L
:L
Fig. 8. Reduced mean transport numbers for the Na + ion as a function of the variable salt concentrationcalculatedfrom stationary state EMF values for the two directions of diffusion through the membrane.Results for the asymmetric,porous membrane.The rectangles correspond to variable concentrationsat the A face and a fixed concentration(0.01 mol/1) at the B face of the membrane. The crosses correspond to variable concentrationsat the B interface and fixed (0.01 mol/1) in face A.
The authors wish to acknowledge the financial support provided from the Direcci6n General de Investigaci6n Ciencia y Tecnolog~a, D G I C Y T (Mininistry of Education and Science of Spain) under project PB92-0773-C03-03 and also to Bancaja under project P1B95-04. One of the authors (Torben Smith SCrensen) is indebted to DGICYT for grants for research at Universitat Jaume I, Castell6n, Spain.
302
V. Compafi et al. / Journal of Membrane Science 123 (1997) 293-302
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