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An electrical model for the relaxation of streaming potential across a charged membrane
Journal of Membrane Science, 84 (1993) 29-36 Elsevier Science Publishers B.V.. Amsterdam
29
An electrical model for the relaxation of streaming potential across a charged membrane M.A. Islam, N.D. Nikolov* and J.D. Nikolova Department of General Chemical Technology, Department of Applied Mechanics, Higher Institute of Chemical Technology, 8010 Bourgas (B&aria) (Beceived December 3,1992; accepted in revised form April 30,1993)
Abstract An electrical model is presented to describe the streaming potential relaxation in charged membranes. The model consists of three elements: two active resistances and one capacity resistance. It describes satisfactorily the relaxation phenomenon irrespective of the applied law of pressure-variation. Finally the values of the electrical parameters of the model are evaluated for a polymethacrylic acid grafted polyethylene membrane and compared with those for a Nuclepore polycarbonate membrane previously used by some authors. Key words: ion-exchange
membranes;
streaming potential; electrical model; electrokinetic
Introduction In recent years, a lot of attention has been focussed on the electrokinetic effects of ion-exchange membranes [l-9]. A potential difference is observed across a membrane separating two electrolyte solutions with the same concentration but at different pressure. This potential difference is called the streaming potential. The dependence of streaming potential on time was observed when polymeric membranes were used as a Some authors interpreted the phenomenon of time dependence being a of the occurrence of a potential due to the concentration difference developed across the membrane. Kumar and Singh [6] considered that the time dependence of the streaming % whom correspondence should be addressed at: k-c Izgrev, bl. 38, bx. 3, ap. 6,8008 Bourgas, Bulgaria.
0376-7388/93/$06.00
phenomena
potential is due to the relaxation phenomenon associated with the establishment of an electric double layer. Tasaka et al. [ 71 investigated the streaming potential relaxation by a sudden jump of pressure from zero to a higher value. They observed that immediately after pressurization, the streaming potential changes instantaneously upto a certain value and then increases gradually with time until it reaches an equilibrium value. The authors also showed that the streaming potential relaxation follows is exponential. They explained the observed relaxation phenomenon with the assumption that the polyacrylic acid grafted membrane is not entirely rigid and its structure alters partially after the driving forces are applied as the polymer chains dissolved in the liquid phase are forced to move in the direction of the resultant volume flow. Thus the authors considered that the phenomenological coefficients in the equa-
0 1993 Elsevier Science Publishers B.V. All rights reserved.
30
tions of irreversible thermodynamics are not constants, but functions of time. In their next investigation [ 81, the authors reported that the number of components of relaxation could be assigned to the grafted components of the membrane. In practice, streaming potential relaxation is observed even in the absence of dissolved grafted chains as observed by Ibaiiez et al. [ 91. They investigated the streaming potential relaxation across a Nuclepore polycarbonate membrane by a gradual change of pressure from a higher to a lower value. They observed that the measured streaming potential is always higher than that of the equilibrium one. In the present paper, we have made an attempt to unify the apparently different approaches of the previous authors [7,9] by considering that the charged membrane acts as a condenser with a definite capacity. When a pressure difference is applied between the two surfaces of the membrane, the observed streaming potential is lower than its equilibrium value during the transition period, as the membrane condenser is being charged. In the same way, if the pressure is decreased gradually, the observed streaming potential is higher than its equilibrium value, as the membrane condenser is being discharged. Finally an electrical model consisting of two active elements and one capacity element is proposed to describe the streaming potential relaxation across a charged membrane. The model satisfactorily explains the relaxation process of the streaming potential observed by the previous authors [7,9]. The values of the parameters of the electrical model have been evaluated for a polymethacrylic acid grafted polyethylene membrane and compared with those for the Nuclepore polycarbonate membrane used by Ibafiez et al. [ 9 1. Theory Applying the theory of thermodynamics of irreversible processes, the volume flow J and
M.A. Islam et al/J. Membrane Sci. 84 (1993) 29-36
electric current I through an ion-exchange membrane subjected to the simultaneous action of a hydrostatic pressure difference AZ’ and an electric potential difference Acy are expressed by [ 3 ] : -J=k,1APfk,2A.ry
(la)
-I=~1AP+~2A(y
(lb)
where k, depends on the solution and membrane parameters such as ion mobility, membrane geometry, fixed charge density in the membrane phase, etc. [ 3 1. If the potential difference Aw is measured under the condition I=O, then it is called the streaming potential. Thus, from eqn. (lb ) , we obtain
& [
1
&I
(2)
A.P I=o=-K2
Under unsteady state, the coefficients kzl and kzz are functions of time [ 71. The observation of the relaxation phenomenon of the streaming potential is performed mainly by two methods: (a) Applying a sudden change of pressure difference across the membrane from 0 to AP and keeping it constant. Such a method is applied by Tasaka et al. [ 71. AV.mV
’
a) Q%(t)
I
b)
I,
i,mln
Fig. 1. Streaming potential Ay/ aa a function of time t. The discrete line indicates the equilibrium streaming potential Awp. (a) The relationship observed by Tasaka et al. [7] after a sharp change of pressure from 0 to AP. (b) The same relationship observed by Ibafiez et al. (91 after a gradual decrease of pressure from AP to 0.
31
M.A. Islam et al./J. Membrane Sci. 84 (1993) 29-36
(b) Following the decrease of the streamingpotential from a non-zero equilibrium value to zero by a gradual decrease of pressure-difference. Such a method is applied by Ibaiiez et al.
PI.
The experimental observation of streaming potential relaxation by Tasaka et al. [7] and Ibatiez et al. [9] is shown qualitatively in Fig. 1. The streaming potential relaxation may be described by an equivalent electric circuit as shown in Fig. 2. The model consists of two active resistances R1 R2, one capacity element C,. The electric current I through the circuit may be described by eqn. (3) [lo] I
AYZ =F+c,-2
d&z dt
difference is equivalent to the electric current
I. From eqn. (lb) we have
I=kAP
for Av=O
(4)
att+co
(4a)
with k= -b,
Substituting the current with the equivalent pressure, eqn. (3) may be rewritten as
kAP++c,~
kAP=$
(aa)
The total potential drop Avis expressed by eqn. (6). A(I/= Ary, +
=-WI Rl
(5b)
1
and I
(54
2
(3b)
If a potential difference Acyis applied, a current with strength I will flow through the circuit. In other words, if there is a current I flowing through the circuit, potential drops Aly, and Av2 will he observed as shown in Fig. 2. The streaming potential arises due to the difference in the hydrostatic pressure applied across the membrane. Under such a condition, the pressure
(6)
Av2
Case 1 There is a sharp change of pressure from 0 to AP and the pressure is then kept constant. Then solving eqn. (5) for AP=const, we obtain
Aw2=Ay2,,[1-exp(-t/72)l
(7d
with ~2= R2C2 and Av2,, = kAPR2
ml
WI = AYO, with A% =kAPR1 Combining eqns. (6) and (7), we have
Aw= Ayol + 4,
[ 1- exp( - t/r2 )I
(8)
at t-rco Fig. 2. An electrical model representating streaming potential relaxation across an ion-exchange membrane.
Acy, =kAP(R,
+R,)
Equation (8) entirely describe the relaxation
M.A. Islam et al.fJ. Membrane Sci. 84 (1993) 29-36
32
phenomenon observed by Tasaka et al. [ 71. Equation (8) may be rewritten as eqn. (9) which is used as the working equation to determine the electric relaxation time. In” ‘,
-*w(t) *%X
= -t/q
.hp=Pg(Yr -Y) where p is the solution density (kg/m 3) and g is the acceleration due to gravity ( m/sec2). From eqns. (2) and (ll), we have
(9)
(12)
Case II
with
The pressure difference decreases gradually as observed by Ibafiez et al. [ 91. The mechanism of pressure regulation is shown schematically in Fig. 3. The liquid level yr on the right-hand side is kept constant, while in the other tube it decreases gradually from yi to yr. To interprete the observation by Ibafiez et al. [ 91 in the light of the electrical model, the equations deduced by the authors are reproduced. The volume flux J is described by the following equation
A= kll
J= -s
dy
(10)
where s is the cross-sectional area of the tube at the left-hand side (m’) and t is time (set). From eqns. (la) and (lo), we have
dy
-J=sdt=kl#‘+klzA~
-k!
h2 k,, 22
where ;1 is the hydraulic permeability of the membrane ( m3/sec-Pa). 1 is a function of time under unsteady state. It may be considered to be constant, if the contribution of AI,U to flux J is negligible compared with that of AP (which is valid, of course, in most practical case), and kll is independent of time. Considering I to be constant, Ibafiez et al. [9] solved eqn. (12)) obtaining eqn. (13). Y=Yr-
(Y,-YibP(-t/z,)
with z1= s/pgA, where r1 is the time for pressure relaxation. The hydrostatic pressure difference AP( t) is given by eqn. (14). U(t)
(11)
=PdYr
-Y)
(14)
=p&-yi)exp(-t/r,) or
with
A.P(t) = AP(O)exp( - t/q)
Yl
(13)
h
mbmne
Fig. 3. Two tubes mounted near the membrane to regulate the hydrostatic pressure difference across it.
(15)
with AP(O)=pg(y,--yi), where AZ’(O) is the initial pressure difference. Thus if the contribution of Av to the flux is negligible compared to that of AP, the pressure relaxation obeys the relationship described by eqn. (15). If the streaming potential follows instantaneously the change in pressure difference, only then the potential difference at every moment
MA.
Islam et al./J. Membrane
33
Sci. 84 (1993) 29-36
coincides with the equilibrium potential difference Ay/, (eqn. 16). (16)
Ay/,(t)=A%exp(-t/r,)
Equation (16) is used by Ibaiiez et al. [ 91 to determine the electric relaxation time. But in practice, the measured streaming potential in the transient period is higher than its equilibrium value. As the membrane acts as a condenser, which is being discharged during the pressure relaxation, the measured streaming potential at every moment is higher than its equilibrium value as was also experimentally observed by Ibaiiez et al. [9]. Equation (16) gives simply the equilibrium value of the streaming potential at the corresponding pressure. It should be noted here that, by definition, the relaxation phenomenon consists of the fact that the equilibrium value of the “effect” is achieved a certain time after the “cause” of the effect has come into action. Thus if eqn. (16) describes the change in streaming potential with time, the phenomenon should not be considered to be a process of streaming potential relaxation in the true sense. solving Now eqn. for (5) AP=AP(O)exp( -t/rl), we have
Experimental The membrane used was based on polymethacrylic acid grafted polyethylene and had a working area of 3.14 cm’. The method for preparation of the membrane was described in Befs. [ 11,121. The degree of grafting was 8.5% with the vacuum dried polyethylene membrane as a reference. The degree of swelling was 45%. In principle, the membrane cell was similar to that used by lbafiez et al. [9]. The electrolytic solution used had a concentration of 10e4 mol/ 1 KCl. The streaming potential was measured with two Ag/AgCl electrodes. The registration of the potential across the electrodes was performed with a digital voltmeter.
Determination of phenomendogical coefficients kij (a) A constant pressure difference was maintained across the membrane. The electrodes were grounded to ensure Avu= 0 and k,, was determined from the eqn. (la): k 11=G-4~)&/4 (b) Under steady state, the volume flow and streaming potential are measured under the condition I=0 and at constant pressure. Knowing the value of kll, k,, was determined from eqn. (la). Considering that the Onsager reciprocal relation is valid, the value of b2 was determined from eqn. (2).
G--z, 1 (174 exp( -t/rz)
--%-
Ary,= Avolexp( - t/z11
(1%)
with ALO1=kAP(0)R1 andAWoz=KAP(0)Rz. Thus
-exp( -t/q)
----
”
~1-72
exp(-t/z2)
1
(18)
Results and discussion Figure 4 shows the dependence of the streaming potential across membrane Ml on time when the pressure difference was changed suddenly from 0 to 70 cm H,O. As can be seen from the figure, the streaming potential increases instantaneously to Avol = 2.3 mV and then gradually increases to a stationary value of Ay/,=4.2 mV. The relaxation time ~~~3.1
M.A. Islam et al./J. Membrane
I
.
0
4
8
12t.min
Fig. 4. The dependence of streaming potential Ay/ across membrane MI on time after an abrupt change in pressure from 0 to 70 cm HzO.
Sci. 84 (1993) 29-36
investigation and those of membrane M2 used by Ibafiez et al. [9] are represented in Table 1. The electrical parameters of membrane Ml were calculated from eqns. (7a) and (7b) and those of membrane M2 were determined by a trial and error method from eqn. (18). It should be noted that the relaxation time r1= 2.95 set from eqn. (16) was considered by Ibafiez et al. [9] to be the electric one. According to the proposed model in our investigation, r1 is the mechanical relaxation time. Table 1 shows that the coupling coefficient & of membrane M, is about 14 times lower than that of membrane M,. The latter had a negative charge of the order of mC/m’ [ 91. However due to the specific method of preparation of the Nuclepore membranes, the ions are located mainly on the porewalls and contribute to the coupling coefficient as well as to the electrical parameters of the membrane. On the other hand, it is very difficult to assess what part of the grafted chains in membrane M, is located on the pore-walls and what part of the -COOH groups is dissociated to contribute to the coupling coefficient as well as to the electrical parameters. Most probably TABLE 1
1
0
5
10
15
20 t,mln
The phenomenological coefficients and electrical parameters of membrane Ml used in our investigation and those of membrane M2 used by Ibaiiez et al. [ 9 ]
Fig. 5. The dependence of streaming potential Ay/ across membrane Ml on time at a mechanical relaxation time r, = 8 min.
min was evaluated from the slope of the curve of eqn. (9). Figure 5 shows the dependence of the streaming potential across membrane Ml on time when the pressure difference was changed gradually from 70 cm Hz0 to 0 with a mechanical relaxation time r1= 8 min. The dense curves corresponds to the values calculated by eqn. (18) for 7z= 3.1 min. The phenomenological coefficients and electrical parameters of membrane Ml used in our
Membrane Ml Phenomenological coefficients, k, 1.2 k,, X 10” (m3/Pa-see) k,, X lo6 ( m3/V-set) 3.3 3.3 kzlx lo6(A/Pa) 5.4 k,, (a-‘) Electrical parameters R,xlO* (a)
R2x102(fi, C2x lo-’
(F)
Electric relaxation time T2 (min)
10.3 8.5 22.0
3.1
Membrane M2
24 45 45
7.4 3.1 32.3
1.67
35
M.A. Islam et al/J. Membrane Sci. 84 (1993) 29-36
Fig. 6. (a) The theoretical Ay=f( t) relationship (eqn. 8) for membrane M2 for a sudden change of pressure from 0 to 675 Pa. (b) Streaming potential Ay/ as a function of time t as observed by Ibafrez et al. [9 ] after a gradual decrease of pressure from 675 Pa to 0. (1) The equilibrium streaming potential, (2) experimental curve, (3) the theoretical curve calculated from the electrical model.
the grafted membrane used has a very low concentration of ions on the pore-walls and correspondingly the electrical capacitance of membrane M, is lower than that of membrane M,. It should be noted here that the coupling coefficient of a given membrane is not a constant quantity but a complex function of the ionic strength of the solution, mobility of the solute ions, degree of dissociation of the grafted chains, forms of the counter-ions, etc. [3]. Therefore the electrical parameters of a given membrane are subject to variation depending on the mentioned parameters. Whatever might be the variation in the absolute values of the electrical parameters, the proposed model entirely describes the behaviour of streaming potential relaxation across a charged membrane. Figure 6(a) shows the theoretical Au/=f(t) (eqn. 8) relationship for membrane M2 for a sudden change of pressure from 0 to AP and Fig. 6(b) shows the same relationship for a gradual decrease of pressure from AP to 0 as observed by lbafiez et al. [9] as well as the relationship calculated by eqn. (18). Thus the proposed model satisfactorily describes the re-
laxation phenomenon of the streaming potential. It may be concluded that an electrical model with three elements (two active and one capacity resistance) may be used to describe the relaxation phenomenon of streaming potential irrespective of the law of change in pressure. List of symbols phenomenological coefficients in the equations of the thermodynamics of irreversible processes streaming potential (mV) equilibrium streaming potential (mV) pressure difference applied between the two surfaces of a charged membrane (Pa) cross-sectional area of the tube (m’) fluid density ( kg/m3) acceleration due to gravity ( m/sec2) hydraulic permeability ( m3/sec-Pa) time of pressure relaxation (min) time of electric relaxation (min )
36
M.A. Islam et al./J. Membrane
I
RipCi
electric current (A) respectively the active (0) & capacity resistance (F) of the proposed electric circuit equivalent to the charged membrane.
References H.-U. Demisch and W. Pusch, Electrical and electroosmotic transport behaviour of asymmetric cellulose acetate membranes, I & II, J. Colloid Interface Sci., 76 (1980) 445,464. S. Tamura, T. Yamazaki and M. Tasaka, Electro-osmosis and streaming potential across charged membranes, Kobunshi Ronbonshu, 34 (1977) 719. Y. Toyoshima and H. Nozaki, Streaming potential across a charged membrane, J. Phys. Chem., 73 (1969) 2134. M. Tasaka, S. Tamura, N. Takemura and K. Morimoto, Concentration dependence of electro-osmosis and streaming potential across charged membranes, J. Membrane Sci., 12 (1982) 169. T.S. Brun and D. Vaula, Correlation of measurements of electro-osmosis and streaming potential in ion-exchange membranes, Ber. Bunsenges. Phys. Chem., 71 (1967) 825.
6
7
8
9
10
11
12
Sci. 84 (1993) 29-36
R. Kumar and K. Singh, Relaxation times for buildup and decay of electroosmotic pressure and for buildup of streaming potential, Indian J. Chem., Sect. A, 19A (1980) 511. M. Tasaka, 0. Sekiguchi, M. Urahama, T. Matsubara, R. Kiyono and S. Suzuki, Relaxation of polymer chains dissolved in the liquid phase of membranes under a pressure gradient, J. Membrane Sci., 48 (1990) 91. 0. Sekiguchi, R. Kiyono, T. Matsubara and M. Tasaka, Relaxation of polymer chains dissolved in the liquid phase of membranes under a pressure gradient. II. Correlation with membrane composition, J. Membrane Sci., 48 (1990) 309. J.A. Ibafiez, J. Forte, A. Hernandez and F. Tejerina, Streaming potential and phenomenological coefficients in Nuclepore membranes, J. Membrane Sci., 36 (1988) 45. G.A. Nenov, Electrotechniques and Electronics, Chemical Institute Publishing House, Bourgas, 1989. (in Bulgarian). A. Dimov and M.A. Islam, Preparation conditions affecting the permeability of polyethylene microfiltration membranes, J. Membrane Sci., 52 (1990) 109. M.A. Islam, A. Dimov and A.L. Malinova, Environment sensitive properties of polymethacrylic acid grafted polyethylene membranes, J. Membrane Sci., 66 (1992) 69.
29
An electrical model for the relaxation of streaming potential across a charged membrane M.A. Islam, N.D. Nikolov* and J.D. Nikolova Department of General Chemical Technology, Department of Applied Mechanics, Higher Institute of Chemical Technology, 8010 Bourgas (B&aria) (Beceived December 3,1992; accepted in revised form April 30,1993)
Abstract An electrical model is presented to describe the streaming potential relaxation in charged membranes. The model consists of three elements: two active resistances and one capacity resistance. It describes satisfactorily the relaxation phenomenon irrespective of the applied law of pressure-variation. Finally the values of the electrical parameters of the model are evaluated for a polymethacrylic acid grafted polyethylene membrane and compared with those for a Nuclepore polycarbonate membrane previously used by some authors. Key words: ion-exchange
membranes;
streaming potential; electrical model; electrokinetic
Introduction In recent years, a lot of attention has been focussed on the electrokinetic effects of ion-exchange membranes [l-9]. A potential difference is observed across a membrane separating two electrolyte solutions with the same concentration but at different pressure. This potential difference is called the streaming potential. The dependence of streaming potential on time was observed when polymeric membranes were used as a Some authors interpreted the phenomenon of time dependence being a of the occurrence of a potential due to the concentration difference developed across the membrane. Kumar and Singh [6] considered that the time dependence of the streaming % whom correspondence should be addressed at: k-c Izgrev, bl. 38, bx. 3, ap. 6,8008 Bourgas, Bulgaria.
0376-7388/93/$06.00
phenomena
potential is due to the relaxation phenomenon associated with the establishment of an electric double layer. Tasaka et al. [ 71 investigated the streaming potential relaxation by a sudden jump of pressure from zero to a higher value. They observed that immediately after pressurization, the streaming potential changes instantaneously upto a certain value and then increases gradually with time until it reaches an equilibrium value. The authors also showed that the streaming potential relaxation follows is exponential. They explained the observed relaxation phenomenon with the assumption that the polyacrylic acid grafted membrane is not entirely rigid and its structure alters partially after the driving forces are applied as the polymer chains dissolved in the liquid phase are forced to move in the direction of the resultant volume flow. Thus the authors considered that the phenomenological coefficients in the equa-
0 1993 Elsevier Science Publishers B.V. All rights reserved.
30
tions of irreversible thermodynamics are not constants, but functions of time. In their next investigation [ 81, the authors reported that the number of components of relaxation could be assigned to the grafted components of the membrane. In practice, streaming potential relaxation is observed even in the absence of dissolved grafted chains as observed by Ibaiiez et al. [ 91. They investigated the streaming potential relaxation across a Nuclepore polycarbonate membrane by a gradual change of pressure from a higher to a lower value. They observed that the measured streaming potential is always higher than that of the equilibrium one. In the present paper, we have made an attempt to unify the apparently different approaches of the previous authors [7,9] by considering that the charged membrane acts as a condenser with a definite capacity. When a pressure difference is applied between the two surfaces of the membrane, the observed streaming potential is lower than its equilibrium value during the transition period, as the membrane condenser is being charged. In the same way, if the pressure is decreased gradually, the observed streaming potential is higher than its equilibrium value, as the membrane condenser is being discharged. Finally an electrical model consisting of two active elements and one capacity element is proposed to describe the streaming potential relaxation across a charged membrane. The model satisfactorily explains the relaxation process of the streaming potential observed by the previous authors [7,9]. The values of the parameters of the electrical model have been evaluated for a polymethacrylic acid grafted polyethylene membrane and compared with those for the Nuclepore polycarbonate membrane used by Ibafiez et al. [ 9 1. Theory Applying the theory of thermodynamics of irreversible processes, the volume flow J and
M.A. Islam et al/J. Membrane Sci. 84 (1993) 29-36
electric current I through an ion-exchange membrane subjected to the simultaneous action of a hydrostatic pressure difference AZ’ and an electric potential difference Acy are expressed by [ 3 ] : -J=k,1APfk,2A.ry
(la)
-I=~1AP+~2A(y
(lb)
where k, depends on the solution and membrane parameters such as ion mobility, membrane geometry, fixed charge density in the membrane phase, etc. [ 3 1. If the potential difference Aw is measured under the condition I=O, then it is called the streaming potential. Thus, from eqn. (lb ) , we obtain
& [
1
&I
(2)
A.P I=o=-K2
Under unsteady state, the coefficients kzl and kzz are functions of time [ 71. The observation of the relaxation phenomenon of the streaming potential is performed mainly by two methods: (a) Applying a sudden change of pressure difference across the membrane from 0 to AP and keeping it constant. Such a method is applied by Tasaka et al. [ 71. AV.mV
’
a) Q%(t)
I
b)
I,
i,mln
Fig. 1. Streaming potential Ay/ aa a function of time t. The discrete line indicates the equilibrium streaming potential Awp. (a) The relationship observed by Tasaka et al. [7] after a sharp change of pressure from 0 to AP. (b) The same relationship observed by Ibafiez et al. (91 after a gradual decrease of pressure from AP to 0.
31
M.A. Islam et al./J. Membrane Sci. 84 (1993) 29-36
(b) Following the decrease of the streamingpotential from a non-zero equilibrium value to zero by a gradual decrease of pressure-difference. Such a method is applied by Ibaiiez et al.
PI.
The experimental observation of streaming potential relaxation by Tasaka et al. [7] and Ibatiez et al. [9] is shown qualitatively in Fig. 1. The streaming potential relaxation may be described by an equivalent electric circuit as shown in Fig. 2. The model consists of two active resistances R1 R2, one capacity element C,. The electric current I through the circuit may be described by eqn. (3) [lo] I
AYZ =F+c,-2
d&z dt
difference is equivalent to the electric current
I. From eqn. (lb) we have
I=kAP
for Av=O
(4)
att+co
(4a)
with k= -b,
Substituting the current with the equivalent pressure, eqn. (3) may be rewritten as
kAP++c,~
kAP=$
(aa)
The total potential drop Avis expressed by eqn. (6). A(I/= Ary, +
=-WI Rl
(5b)
1
and I
(54
2
(3b)
If a potential difference Acyis applied, a current with strength I will flow through the circuit. In other words, if there is a current I flowing through the circuit, potential drops Aly, and Av2 will he observed as shown in Fig. 2. The streaming potential arises due to the difference in the hydrostatic pressure applied across the membrane. Under such a condition, the pressure
(6)
Av2
Case 1 There is a sharp change of pressure from 0 to AP and the pressure is then kept constant. Then solving eqn. (5) for AP=const, we obtain
Aw2=Ay2,,[1-exp(-t/72)l
(7d
with ~2= R2C2 and Av2,, = kAPR2
ml
WI = AYO, with A% =kAPR1 Combining eqns. (6) and (7), we have
Aw= Ayol + 4,
[ 1- exp( - t/r2 )I
(8)
at t-rco Fig. 2. An electrical model representating streaming potential relaxation across an ion-exchange membrane.
Acy, =kAP(R,
+R,)
Equation (8) entirely describe the relaxation
M.A. Islam et al.fJ. Membrane Sci. 84 (1993) 29-36
32
phenomenon observed by Tasaka et al. [ 71. Equation (8) may be rewritten as eqn. (9) which is used as the working equation to determine the electric relaxation time. In” ‘,
-*w(t) *%X
= -t/q
.hp=Pg(Yr -Y) where p is the solution density (kg/m 3) and g is the acceleration due to gravity ( m/sec2). From eqns. (2) and (ll), we have
(9)
(12)
Case II
with
The pressure difference decreases gradually as observed by Ibafiez et al. [ 91. The mechanism of pressure regulation is shown schematically in Fig. 3. The liquid level yr on the right-hand side is kept constant, while in the other tube it decreases gradually from yi to yr. To interprete the observation by Ibafiez et al. [ 91 in the light of the electrical model, the equations deduced by the authors are reproduced. The volume flux J is described by the following equation
A= kll
J= -s
dy
(10)
where s is the cross-sectional area of the tube at the left-hand side (m’) and t is time (set). From eqns. (la) and (lo), we have
dy
-J=sdt=kl#‘+klzA~
-k!
h2 k,, 22
where ;1 is the hydraulic permeability of the membrane ( m3/sec-Pa). 1 is a function of time under unsteady state. It may be considered to be constant, if the contribution of AI,U to flux J is negligible compared with that of AP (which is valid, of course, in most practical case), and kll is independent of time. Considering I to be constant, Ibafiez et al. [9] solved eqn. (12)) obtaining eqn. (13). Y=Yr-
(Y,-YibP(-t/z,)
with z1= s/pgA, where r1 is the time for pressure relaxation. The hydrostatic pressure difference AP( t) is given by eqn. (14). U(t)
(11)
=PdYr
-Y)
(14)
=p&-yi)exp(-t/r,) or
with
A.P(t) = AP(O)exp( - t/q)
Yl
(13)
h
mbmne
Fig. 3. Two tubes mounted near the membrane to regulate the hydrostatic pressure difference across it.
(15)
with AP(O)=pg(y,--yi), where AZ’(O) is the initial pressure difference. Thus if the contribution of Av to the flux is negligible compared to that of AP, the pressure relaxation obeys the relationship described by eqn. (15). If the streaming potential follows instantaneously the change in pressure difference, only then the potential difference at every moment
MA.
Islam et al./J. Membrane
33
Sci. 84 (1993) 29-36
coincides with the equilibrium potential difference Ay/, (eqn. 16). (16)
Ay/,(t)=A%exp(-t/r,)
Equation (16) is used by Ibaiiez et al. [ 91 to determine the electric relaxation time. But in practice, the measured streaming potential in the transient period is higher than its equilibrium value. As the membrane acts as a condenser, which is being discharged during the pressure relaxation, the measured streaming potential at every moment is higher than its equilibrium value as was also experimentally observed by Ibaiiez et al. [9]. Equation (16) gives simply the equilibrium value of the streaming potential at the corresponding pressure. It should be noted here that, by definition, the relaxation phenomenon consists of the fact that the equilibrium value of the “effect” is achieved a certain time after the “cause” of the effect has come into action. Thus if eqn. (16) describes the change in streaming potential with time, the phenomenon should not be considered to be a process of streaming potential relaxation in the true sense. solving Now eqn. for (5) AP=AP(O)exp( -t/rl), we have
Experimental The membrane used was based on polymethacrylic acid grafted polyethylene and had a working area of 3.14 cm’. The method for preparation of the membrane was described in Befs. [ 11,121. The degree of grafting was 8.5% with the vacuum dried polyethylene membrane as a reference. The degree of swelling was 45%. In principle, the membrane cell was similar to that used by lbafiez et al. [9]. The electrolytic solution used had a concentration of 10e4 mol/ 1 KCl. The streaming potential was measured with two Ag/AgCl electrodes. The registration of the potential across the electrodes was performed with a digital voltmeter.
Determination of phenomendogical coefficients kij (a) A constant pressure difference was maintained across the membrane. The electrodes were grounded to ensure Avu= 0 and k,, was determined from the eqn. (la): k 11=G-4~)&/4 (b) Under steady state, the volume flow and streaming potential are measured under the condition I=0 and at constant pressure. Knowing the value of kll, k,, was determined from eqn. (la). Considering that the Onsager reciprocal relation is valid, the value of b2 was determined from eqn. (2).
G--z, 1 (174 exp( -t/rz)
--%-
Ary,= Avolexp( - t/z11
(1%)
with ALO1=kAP(0)R1 andAWoz=KAP(0)Rz. Thus
-exp( -t/q)
----
”
~1-72
exp(-t/z2)
1
(18)
Results and discussion Figure 4 shows the dependence of the streaming potential across membrane Ml on time when the pressure difference was changed suddenly from 0 to 70 cm H,O. As can be seen from the figure, the streaming potential increases instantaneously to Avol = 2.3 mV and then gradually increases to a stationary value of Ay/,=4.2 mV. The relaxation time ~~~3.1
M.A. Islam et al./J. Membrane
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.
0
4
8
12t.min
Fig. 4. The dependence of streaming potential Ay/ across membrane MI on time after an abrupt change in pressure from 0 to 70 cm HzO.
Sci. 84 (1993) 29-36
investigation and those of membrane M2 used by Ibafiez et al. [9] are represented in Table 1. The electrical parameters of membrane Ml were calculated from eqns. (7a) and (7b) and those of membrane M2 were determined by a trial and error method from eqn. (18). It should be noted that the relaxation time r1= 2.95 set from eqn. (16) was considered by Ibafiez et al. [9] to be the electric one. According to the proposed model in our investigation, r1 is the mechanical relaxation time. Table 1 shows that the coupling coefficient & of membrane M, is about 14 times lower than that of membrane M,. The latter had a negative charge of the order of mC/m’ [ 91. However due to the specific method of preparation of the Nuclepore membranes, the ions are located mainly on the porewalls and contribute to the coupling coefficient as well as to the electrical parameters of the membrane. On the other hand, it is very difficult to assess what part of the grafted chains in membrane M, is located on the pore-walls and what part of the -COOH groups is dissociated to contribute to the coupling coefficient as well as to the electrical parameters. Most probably TABLE 1
1
0
5
10
15
20 t,mln
The phenomenological coefficients and electrical parameters of membrane Ml used in our investigation and those of membrane M2 used by Ibaiiez et al. [ 9 ]
Fig. 5. The dependence of streaming potential Ay/ across membrane Ml on time at a mechanical relaxation time r, = 8 min.
min was evaluated from the slope of the curve of eqn. (9). Figure 5 shows the dependence of the streaming potential across membrane Ml on time when the pressure difference was changed gradually from 70 cm Hz0 to 0 with a mechanical relaxation time r1= 8 min. The dense curves corresponds to the values calculated by eqn. (18) for 7z= 3.1 min. The phenomenological coefficients and electrical parameters of membrane Ml used in our
Membrane Ml Phenomenological coefficients, k, 1.2 k,, X 10” (m3/Pa-see) k,, X lo6 ( m3/V-set) 3.3 3.3 kzlx lo6(A/Pa) 5.4 k,, (a-‘) Electrical parameters R,xlO* (a)
R2x102(fi, C2x lo-’
(F)
Electric relaxation time T2 (min)
10.3 8.5 22.0
3.1
Membrane M2
24 45 45
7.4 3.1 32.3
1.67
35
M.A. Islam et al/J. Membrane Sci. 84 (1993) 29-36
Fig. 6. (a) The theoretical Ay=f( t) relationship (eqn. 8) for membrane M2 for a sudden change of pressure from 0 to 675 Pa. (b) Streaming potential Ay/ as a function of time t as observed by Ibafrez et al. [9 ] after a gradual decrease of pressure from 675 Pa to 0. (1) The equilibrium streaming potential, (2) experimental curve, (3) the theoretical curve calculated from the electrical model.
the grafted membrane used has a very low concentration of ions on the pore-walls and correspondingly the electrical capacitance of membrane M, is lower than that of membrane M,. It should be noted here that the coupling coefficient of a given membrane is not a constant quantity but a complex function of the ionic strength of the solution, mobility of the solute ions, degree of dissociation of the grafted chains, forms of the counter-ions, etc. [3]. Therefore the electrical parameters of a given membrane are subject to variation depending on the mentioned parameters. Whatever might be the variation in the absolute values of the electrical parameters, the proposed model entirely describes the behaviour of streaming potential relaxation across a charged membrane. Figure 6(a) shows the theoretical Au/=f(t) (eqn. 8) relationship for membrane M2 for a sudden change of pressure from 0 to AP and Fig. 6(b) shows the same relationship for a gradual decrease of pressure from AP to 0 as observed by lbafiez et al. [9] as well as the relationship calculated by eqn. (18). Thus the proposed model satisfactorily describes the re-
laxation phenomenon of the streaming potential. It may be concluded that an electrical model with three elements (two active and one capacity resistance) may be used to describe the relaxation phenomenon of streaming potential irrespective of the law of change in pressure. List of symbols phenomenological coefficients in the equations of the thermodynamics of irreversible processes streaming potential (mV) equilibrium streaming potential (mV) pressure difference applied between the two surfaces of a charged membrane (Pa) cross-sectional area of the tube (m’) fluid density ( kg/m3) acceleration due to gravity ( m/sec2) hydraulic permeability ( m3/sec-Pa) time of pressure relaxation (min) time of electric relaxation (min )
36
M.A. Islam et al./J. Membrane
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RipCi
electric current (A) respectively the active (0) & capacity resistance (F) of the proposed electric circuit equivalent to the charged membrane.
References H.-U. Demisch and W. Pusch, Electrical and electroosmotic transport behaviour of asymmetric cellulose acetate membranes, I & II, J. Colloid Interface Sci., 76 (1980) 445,464. S. Tamura, T. Yamazaki and M. Tasaka, Electro-osmosis and streaming potential across charged membranes, Kobunshi Ronbonshu, 34 (1977) 719. Y. Toyoshima and H. Nozaki, Streaming potential across a charged membrane, J. Phys. Chem., 73 (1969) 2134. M. Tasaka, S. Tamura, N. Takemura and K. Morimoto, Concentration dependence of electro-osmosis and streaming potential across charged membranes, J. Membrane Sci., 12 (1982) 169. T.S. Brun and D. Vaula, Correlation of measurements of electro-osmosis and streaming potential in ion-exchange membranes, Ber. Bunsenges. Phys. Chem., 71 (1967) 825.
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7
8
9
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11
12
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R. Kumar and K. Singh, Relaxation times for buildup and decay of electroosmotic pressure and for buildup of streaming potential, Indian J. Chem., Sect. A, 19A (1980) 511. M. Tasaka, 0. Sekiguchi, M. Urahama, T. Matsubara, R. Kiyono and S. Suzuki, Relaxation of polymer chains dissolved in the liquid phase of membranes under a pressure gradient, J. Membrane Sci., 48 (1990) 91. 0. Sekiguchi, R. Kiyono, T. Matsubara and M. Tasaka, Relaxation of polymer chains dissolved in the liquid phase of membranes under a pressure gradient. II. Correlation with membrane composition, J. Membrane Sci., 48 (1990) 309. J.A. Ibafiez, J. Forte, A. Hernandez and F. Tejerina, Streaming potential and phenomenological coefficients in Nuclepore membranes, J. Membrane Sci., 36 (1988) 45. G.A. Nenov, Electrotechniques and Electronics, Chemical Institute Publishing House, Bourgas, 1989. (in Bulgarian). A. Dimov and M.A. Islam, Preparation conditions affecting the permeability of polyethylene microfiltration membranes, J. Membrane Sci., 52 (1990) 109. M.A. Islam, A. Dimov and A.L. Malinova, Environment sensitive properties of polymethacrylic acid grafted polyethylene membranes, J. Membrane Sci., 66 (1992) 69.