1752
Y.KOBATAKE, R4. YUASA,AND H. FUJITA
Studies of Membrane Phenomena. VI.
Further Study of Volume Flow
by Y. Kobatake, M. Yuasa, and 11. Fujita Department of Polymer Science, Osaka University, Toyonaka, J a p a n
(Received November 16, 1967)
Volume flows (Jy)of aqueous solutions of KC1 and LiCl through oxidized collodion membranes were measured under the condition in which either pressure difference A p or electric potential difference A t waq applied eaternally across the membrane, and for each system the electroosmotic coefficient wiz defined by - L ( J v / A { ) ~ ~ and the hydraulic coefficient wZ2defined by -L(J,/Aplo)~~-o(where L is the effectivethickness of the membrane) were calculated from these data as functions of the concentration of the external electrolyte solution. Expressions for these phenomenological coefficients were derived on the basis of the thermodynamics of irreversible processes in conjunction with an approximate hydrodynamic equation for the movement of the local center of mass. They showed that both w12 and ~ 2 are % composed of two parts, one associated with the movement of individual ions relative to the local center of m a s and the other with the mass flow. The former term in each coefficient was estimated approximately using our previous assumptions for the mobilities and activities of mobile ions in charged membranes. Comparison with the experimentally determined values of w12 and w Z 2 indicated that the contribution of this term to either coefficient was essentially negligible. Thus for the membrane-electrolyte systems studied the observed volume flows can be argued to have consisted mainly of the mass flow. The observed values of the electroosmotic coefficient approached a nonzero value even at the limit of high electrolyte concentration and, on the average, accorded well with those derived from osmosis data in a previous part of this series.
Introduction Part V of this series1 dealt with the volume flow of liquid which occurs in the system where an ionizable membrane separates bulk solutions of a 1:l electrolyte of different concentrations. The consideration was confined to the case where neither hydrostatic pressure nor electric field is applied externally across the membrane. lloreover, it was assumed that the system is maintained a t a constant temperature. The present paper considers the volume flow occurring under the condition that there are no gradients of concentration and temperature between the bulk solutions but they are subject to externally applied differences in pressure and electric potential. I n doing this, emphasis is pIaced on taking the movement of the local center of mass explicitly into account, with the interest in obtaining relations which allow the contributions of mass flow to electroosmotic and pressure flows to be evaluated. With respect to this point, the present study may be differentiated from those of previous authors2-6 mho concerned themselves with these flow processes. Application of the theory is made to data which are obtained with oxidized collodion membranes over wide ranges of concentration of aqueous KC1 and LiC1.
water (denoted by subscript w) and those of cation and and -) occur, on anion (designated by subscripts the average, only in the direction ( 2 ) perpendicular to the membrane surfaces. Considerations below are restricted to the system which is in the steady state. Then, according to the thermodynamics of irreversible processes, the electric current density I and the volume flow J,, both relative to the frame fixed to the membrane, are represented by7
+
+,
Here ( L j J r n( k , j = -) are the mass-fixed phenomenological coefficients and obey the Onsager reciprocal relation (A+-), = (L-+),; e j , Pj,Cj, and ill5 are the
Theoretical Section
(1) Y . Toyoshima, Y . Kobatake, and H. Fujita, Trans. Faraday Soc., 63, 2828 (1967). (2) D. Mackay and P. Meares, ibid., 5 5 , 1221 (1959).
The system considered is the same as in part V, except that here the concentrations of the bulk solutions on both sides of the membrane are taken to be equal (denoted by C in moles/cc). The electric potential and pressure differences applied externally across the membrane are denoted by A{ and A p , respectively. It is assumed, as before, that the flow of
(3) C. W. Carr, R, McClintock, and K. Sollner, J . Electrochem. Soc., 109, 251 (1962). (4) N. Lakshminarayanaiah and V. Subrahmanyan, J. Polymer Sei., A2,4491 (1964). (5) J. H. B. George and R. A. Courant, J . Phys. Chem., 71, 246 (1967). (6) J. Greyson, ibid., 71, 259 (1967). (7) S. R. deGroot and P. Xazur, "Non-Equilibrium Thermodynamics," North-Holland Publishing Co., Amsterdam, 1962, p 423.
T h e Journal of Physical Chemistry
-
0
STUDIES OF MEMBRANE PHENOMENA
1753
molar electric charge, partial molar volume, concentration in moles/cc, and molar weight of species j (w, -); Urn is the velocity of the local center of mass; { is the local electric potential; and p is the local pressure. Keglecting, as before, interacting flows between different ion species and introducing the mass-fixed ionic mobilities, u+ and u-,in terms of the coefficients (L++),and (L&,, eq 1 and 2 are transformed to give
+,
I
=
-(u+C+
-t u-C-)F- d l -
ones which can migrate with mass flow. If this is the e-C- in eq 6 may also be case, the term e+C+ written F # X , with the dimensionless parameter 9 introduced above. With these assignments to e+C+ e L - in eq 3 and 6, substitution of eq 6 into eq 3 and 4, followed by integration of the resulting equations over the membrane thickness ( L ) under the steady-state condition (dI/dx = 0 and dJv/dx = 0), gives
+
+
I = -wii(Al/L) - wiz(Ap/L)
(7)
J, = -wzi(Al/L) - wzz(Ap/L)
(8)
dx
(V+*u+C+- V-*u-C-)-
dP dx
+ (e+C+ + e-C-)U,
(3)
where the coefficients wt5 (i,j = by
where VTC* (k
==
+, -)
+
dP V-*2u-C-)dx
+ Urn
(4)
a22
stands for
v,* = v, - Mvw W A4Tk
(5)
and F is the Faraday constant. I n part V we set the last term on the right-hand side of eq 3 equal to FXU,, since formally the condie-C- = F X . tion of electric neutrality requires e+C+ Here X represents the density of electric charges fixed on the polyelectrolyte molecules constituting the membrane and is expressed in equiv/cc of membrane phase. However, this assignment assumes that all excess counterions, L e . , those dissociated from the membrane skeletons, are carried by mass flow. In reality, only a fraction of them may migrate with mass flow and contribute to the flow of electricity, since the electrostatic interaction of the membrane polyions tends to immobilize part of the counterions in their vicinity. Thus it is more appropriate to write the last term in eq 3 as F$XUrn,where $ is a dimensionless parameter to represent the fraction of X that can be conveyed by mass flow. Following the argument given in part V, the steadystate value of U , may be represented by the approximate hydrodynamic equation
+
where k is a constant which may depend on the fluidity of solvent as w'ell as the compactness and other structural characteristics of the membrane. Formal application of the condition of electric neutrality again requires setting the term e+C+ e-C- in this equation equal to F X . However, physically, this term represents the number density of excess counterions which can transmit the external electric force acting on them to their surrounding liquid medium. We assume here that such "effective" counterions are
+
~ 1 = 2 w21
=
are represented
+ u-C-)F + kF2($X)' = (V+*u+C+ - V-*u-C-) + kF$X (V+*Zu+C++ V_*%-C-)(l/F) + k
~ 1 = 1
1 -(V+*%+C+ F
+, -)
(u+C+
(9) (10) (11)
It should be noted that, in deriving eq 7 and 8, the parameter $ may not be treated as independent of ion concentrations. The result that w12 = w21 indicated by eq 10 is the familiar Saxen relation.* Tracing back the above development, we find that this relation is a consequence of our assumption made above for the term e+C+ e-C- in eq 3 and 6. The ample experimental evidence so far reporteds on the Saxen relation may be taken as a justification of this assumption. All wtl are composed of two terms. The first term in each coefficient refers to the movement of individual ions relative to the local center of mass and may not vanish even when the charge density of the membrane is diminished. On the other hand, the second term is associated with the mass flow caused by gradients of pressure and electric potential, and depends on the charge density and structural details of the membrane as well as the fluidity of solvent. From eq 7 it follows that
+
=
-L(I/Ay)A,=o
~
2
L/r
(12) where r is the electrical resistance of the membrane per unit area in the absence of pressure gradient (and of concentration gradient as well). Since this quantity has been studied separately in part II1,lono further comment on wll is made below. From eq 8, we obtain 011
/
= ~
(Jv/I) 1
~p
=
-o
(13)
which may be combined with eq 12 to give ~ 1 = 2
L(Jv/Ir)~p-o = [Jv/(&/L)l~~-o
(14)
(8) U. SaxBn, Ann, Physik, 47, 46 (1892). (9) D. G. Miller, Chem. Rev., 60, 15 (1960). (10) Y. Toyoshima, M. Yuasa, Y . Kobatake, and H. Fujita, Trans. Faraday Soc., 63, 2803 (1967).
Volume 7'3, Number 6 M a y 1068
Y. KOBATAKE, $1. YUASA,AND H. FUJITA
1754 This indicates that w12 is the volume flow produced by an electric field of unit strength in the absence of pressure gradient (and of concentration gradient as well). Thus this coefficient may be termed the electroosmotic coefficient of a given system. Finally, it also follows from eq 8 that w22
=
- [JV/(AP/L)Iar-0
(15)
which indicates that wZ2 is the volume flow produced when the membrane is subject to unit pressure gradient in the absence of electric potential gradient (and of concentration gradient as well). Thus with the larger wzg, the membrane is more permeable to liquid under a pressure gradient. Hence we may term w22 the hydraulic permeability of a given membrane. Equations 12, 14, and 15 tell how the coefficients (~11, WE, and w22 can be determined from experiment. For instance, w12 can be obtained by measuring J v (volume of liquid transferred through 1 cm2 of the membrane in 1 sec) as a function of A { / L (electric potential gradient) under the condition where there exists no difference of hydrostatic pressure (and of concentration) across the membrane, followed by evaluating the dope of J , against Ap/L. I n practice, first J, may be measured as a function of electric current density I , and then the slope of J , vs. I may be multiplied by the value of L / r which can be determined from a separate resistance measurement conducted under the same experimental conditions. Once data for wt5 are obtained in this way, we can proceed to examine the relative contributions of the two terms comprising each coefficient. The experiment described below was undertaken to obtain data relevant for this purpose.
Experimental Section Two oxidized collodion membranes numbered 2 and 3 in part V, together with one newly oxidized collodion membrane numbered 4, were chosen for the present measurements. For each of these membranes, data Table I: Some Characteristics of Membrane-Electrolyte Pairs Studied @ X x 103
Mernbrane
Electrolyte
2
KC1 LiCl
1.20" 1.31
0.50 0.37
3
KC1 LiCl
1.91" 1.97
0.50 0.37
4
KC1 LiCl
1.7OC 1.70
0.50 0.37
equiv/l.
ab
+
LY = u+O/(u+O U-O), a Taken from Table I of part 111. taken from Table I of part IV. ' Actually, the values of +'.X for membrane 4 determined from membrane potential data by the method described in part IV.
The Journal of Physical Chemistry
Figure 1. Schematic diagram of the cell used for measurements of the volume flows, (J,/Z)ap-o and ( J / A P ) ~ = O : A, membrane; B, Ag-AgC1 plate electrode; C, stirrer; D, Teflon tubing; E, capillary.
were taken over wide ranges of C of KC1 and LiC1. Table I summarizes pertinent characteristics of these membrane-elec troly te pairs. Before use, each membrane was thoroughly conditioned with a concentrated solution of a given electrolyte, following the procedure described in part 1 1 1 . 1 0 Figure 1 shows a schematic diagram of the cell used. The bulk solutions of equal concentration C on both sides of the membrane were stirred vigorously by a pair of magnetic stirrers in order to maintain then1 uniform in composition. The cell was immersed in a water bath thermostatted at 30 f 0.02", and moreover, the entire system was placed in an air bath of 30". Great care was taken to prevent the solution from leaking out of the cell and joint portions. The solution in the cell was led to two horizontally mounted capillaries of uniform bore through Teflon tubing, and the movement of the liquid meniscus in one of them was followed as a function of time by a travelling microscope. I n every case treated, plots of the distance travelled by the liquid meniscus against time became accurately linear after some interval of time from the start of a particular run. The slope of this linear portion multiplied by a factor A-' ( = S,/X,) was taken as the steady-state value of J , under given experimental conditions. Here 8, denotes the cross section of either capillary and X, the effective area of the membrane. Electric current was delivered to the cell from a battery through a pair of Ag-AgC1 electrodes mounted as shown in Figure 1, and its strength was measured with a microammeter connected in series. During the measurement at a given electric current, the two capillaries were held at an equal level so as to obtain the condition A p = 0. At any concentration of the external solution, plots of J v against I were accurately linear over the range of I studied (0-20 mA/cm2). Pressure difference A p was applied across the membrane by shifting one of the capillaries relative to the other in the vertical direction. The distance between them was measured accurately by means of a cathetometer. Ap was varied in the range from 0 to 20 cm in
1755
STUDIESOF MEMBRANE PHENOMENA water, and in this range J, increased linearly with Ap. The measurements were made under the condition of no electric current being delivered to the cell. Therefore, the resulting data for J , did not exactly correspond to the absence of potential gradient across the membrane, but must have contained a contribution from the so-called streaming potential. However, as shown in the Appendix, this effect was negligibly small in the membrane-electrolyte systems studied. Thus, in the presentation below, data obtained under I = 0 are simply taken as those corresponding to the condition AT = 0.
Results and Discussion Hydraulic Permeability. Figure 2 plots data for ozz/L (= - (Jv/Ap),,,o) on membranes 3 and 4 as a function of log C. One can conclude from this graph that wzz/L is essentially independent not only of the species and concentration of electrolyte but also of the charge density of the membrane. The average value of the plotted data is indicated by a horizontal line and yields 8.2 x 10-1' cm3 dynedi sec-l. All membranes exa,mined had essentially the same thickness of about 0.05 cm. Hence wZ2 for the systems em4 dyne-' sec-l on the studied was about 4.1 X average. Our problem now is to estimate the contributions of the first and second terms of eq 11 to this experimental value of wZ2. To this end, we invoke our previous assumptions1 for the mobilities and activity coefficients of mobile ions in charged membranes, together with the Donnan equilibrium between the membrane phase and the external salt solution. Then it is shown that the first term in question is represented bY
+ P_*'u-C- = (A~/~)[P.+*'cY + V-*'(l
V+*'u+C+
- a>]Z(C,4X)+ (AO/~)[V+*~CY - V-*'(l - C Y ) ] ( ~ X(16) )
where
+ l#PX')''t + +
Z(C,C+X) = (4C' A0
a =
=
u+O
U+O/(U+O
u-O
u-0)
(17)
(18) (19)
and u+O and u.-O are the mobilities of cation and anion species in polyelectrolyte-free solution. I n deriving eq 16, it has been assumed, in conformity to our previous finding with oxidized collodion membranes, that the thermodynamically effective charge density of the membrane, rpX, and the hydrodynamically effective one, +'X, have the same va1ue;'OJl for the meaning of these effective charges the reader is referred to part I11 of this series. Actually, this assumption need not be introduced, but it is pertinent for the present purpose.
log c Figure 2. Plots of w z z / L against log C for membranes 3 and 4 with KCI, LiCI, and distilled water: 0 , membrane 3 with KCI; 0, membrane 3 with LiC1; @, membrane 4 with KCl; membrane 4 with LiCI; C is expressed in moles/l.
For KCl,
P+*'u+C+
CY
= 0.5, so eq 16 is simplified to
+ P-*'U_CP-*"Z(C,q5X)
(Ao/4)[( V+*'
+
+ (P+*, - P-"($X) ]
(20)
I n order to evaluate the right-hand side of this equation, the values of 'i5 for K + and C1- must be known. Unfortunately, these are not experimentally accessible quantities. Therefore, we estimate them in terms of the Stokes radii of individual ions using the relation
P,
(4nN*/3)rt3 (21) u-here N A is the Avogadro number and rt is the Stokes radius of ion species i. Using literature values for r K t and rc1-,12 we obtain PK+= 19.5 cc/mole and PCI-= 18.2 cc/mole. Hence, with Vw being taken as 18.1 cc/mole
PK+*=
=
-19.6 cc/mole;
V C ~ -= * -17.3 cc/mole
Introduction of these values, together with AoF = 122 cm2 ohm-l equiv-' for KC1 in water a t 3O0l3and c$X = 2 X equiv/cc for membrane 2-KC1 (see Table I), into eq 20 gives 4.4 X em4 dyne-' V-*2u-C- at C = 1 mole/l. sec-1 for P+*zu+C+ This value is negligibly small compared with the experimental value of w2'. Similar results are deduced at other values of C as well as for other pairs of membrane and electrolyte treated. Therefore we can conclude that for the present systems the contribution of the first term of eq 11 to the observed hydraulic permeability was negligible. Thus Figure 2 may be taken as representing the values of k / L for these systems, and a value of 8.2 X lo-" cm8 dyne-' sec-l can be assigned to their average. Electroosmotic Coeflcient. Figure 3 shows that ex-
+
(11) Y. Toyoshima, Y. Kobatake, and H. Fujita, Tram. Faraday Soc., 63, 2814 (1967).
(12) H. S. Harned and B. B. Owen, "The Physical Chemistry of Electrolyte Solutions," Reinhold Publishing Gorp., New York, N. Y., 1957, p 217. (13) .'%andolt-Bornstein Tabellen," 11-7, 6th ed, Springer, Berlin, Gottingen, and Heidelberg, 1960, pp 54, 88.
Volume 78, Number 6 May 1968
Y. KOBATAKE, 111. YUASA,AND H. FUJITA
1756
I
-3 Figure 3. Plots of F(J,/I)4p-,, against the reduced concentration C/+X for membranes 2, 3, and 4 with KC1 and LiCl: 0 , membrane 2 with KCl; 0, membrane 3 with KCI; (3, membrane 4 with KCl; 8, membrane 2 with LiC1; 0 membrane 3 with LiCl; 0, membrane 4 with LiCl.
I
I
-2 log c
-I
I
0
Figure 5. w12/L as a function of log C for membrane 2 with KCI. M-2 and M-3 refer to membranes 2 and 3. C is expressed in moles/l.
P . .e
4-
1
I
I
1
1
1
4-
t------
.I
'k-----
$11 1 0
0
2 4 I/C x 16' (crna~equlv9
-3
-2
~
log c
-I
1 0
Figure 4. Plots of F(J,/l)ap-o against 1/C for membrane 3 with KC1 and LiCl: 0, KC1; 0, LiC1.
Figure 6. w12/L as a function of log C for membranes 2 and 3 with LiC1. Dashed lines represent values of w l 2 / L for the same membrane-electrolyte systems calculated from osmosis data obtained in part V. C is expressed in moles/l.
perimental values of F ( J v / I )4p=,0 for membranes of different charge density in solutions of either KC1 or LiCl collect to a single composite curve when plotted against a reduced concentration defined by C / 4 X . Here the necessary values for 4 X have been taken from Table I. I n Figure 4 is shown the behavior of F ( J v / I ) A p = 0 in the region of high salt concentration for membrane 3 in KCI and LiCl. It is interesting to see the plotted points for a given system follow a straight line over a remarkably wide range of C-l and the line nearly passes through the coordinate origin. The data of Figure 3 can be transformed to those of -(JV/Af)ap=~, Le., w l z / L according to eq 14, provided that electric resistance data r for these membrane-electrolyte systems are available. Actually, such data have been obtained in part I11 for membranes 2 and 3, but not yet for membrane 4. Therefore, in this paper, we present w l z / L only for the former two membranes. The results are shown in Figures 5 and 6. I n deriving these curves, use was made of smoothed data for both ( J , / I ) e P - o and r , since the measurements of these quantities had not been performed a t exactly identical values of C. The order of magnitude of the first term on the right-hand side of eq 10 may be estimated by the same
method as has been used for the first term of eq 11. It can be shown that for the systems studied here this term is also quite small compared with the observed values of w12, but the difference is not as extreme as has been found in the case of w22. For example, for membrane 2 in KC1 at 30" this term is about 1.7 X cm2 sec-l V-' at C = 1 mole/l. and thus amounts to be nearly 10% of the observed value of w12. However, its contribution diminishes monotonically as C is lowered. Thus at C = 0.1 mole/l. it becomes only about 2% of the observed w12. From this consideration one may conclude that for the present systems the electroosmotic coefficient w12 is practically equal to k F # X , the second term of eq 10, except in the region of high salt concentration. I n other words, the curves shown in Figures 5 and 6 may be taken as representing the dependence of #X on concentration C except in such a region of C. Two features of them may be pointed out here. One is that the indicated dependence of #X on C is not appreciable in any of the four systems shown, and the other is that # X appears to level off a nonzero value in the limit of high concentration. This latter behavior does not accord with a recent theory of Kobatake and Fujita14 and of Manning,I6
The Journal of Physical Chemistry
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STUDIESOF MEMBRANE PHENOMENA
1757
which predicts that a quantity equivalent to our kF+X approaches zero with C-'/z as C is increased. Comparison with Previous Results. As has been demonstrated in part V, it is possible to evaluate a parameter K , defined by
K,
=
akRT/L
(22)
from measurements of volume flows caused under concentration gradients. Here R is the gas constant, T is the absolute temperature, u is the reflection factor which represents permeability of the membrane surface to small ions, and k and L have the same meaning as those defined above. The values of K , determined in part V with membranes 2 and 3 were nearly independent of electrolyte species and the membrane's charge density and, on the average, yielded a value of 8.7 X lo-' cm/sec. Introducing this, with the k / L value determined above (8.2 X lo-'' cm3 dyne-l sec-l) and T = 303°K) into eq 22 gives u N 0.4. This indicates that our membrane surfaces were partially leaky to small ions. It was also shown in part V that if is assumed to be constant, a parameter K , defined by
+
K,
=
kRT+X/L
(23)
can be evaluated from volume flow data obtained in the presence of concentration gradients. We insert the K , values determined in part V for membrane 2-LiC1 and membrane 3-1,321 into this equation, together with the k/L value indicated above. The results are
+X
=
2.5 X
equiv/cc
(for membrane 2-LiC1)
+X
= 3.9 X
equiv/cc
(for membrane 3-LiC1)
which are only about 1 / 6 the corresponding values of CpX (see Table I). The values of kF+XjL recalculated from these +X and k / L are shown by dashed lines in Figure 6. It can be seen that they are quite consistent with the corresponding da,ta (solid curves) derived from the present electroosmotic measurements. No such comparison between the present and previous results was attempted with respect to KC1 solutions, since K , for membranes 2 and 3 with KC1 had to be left undetermined in our previous experiment.
Concluding Remarks This study has demonstrated for oxidized collodion
mernbrane-1 :1 electrolyte systems that the volume flow under either pressure gradient or electric potential gradient is mainly brought about by the movement of the local center of mass, i.e., the so-called mass flow. Specifically, in the case of pressure flow, the mass flow almost entirely dominated the transport of liquid, whereas, in the case of electroosmotic flow, a small portion of the volume flow stemmed from the opposite movement of cation and anion species relative to the local center of mass when the concentration of the external solution was relatively high. Mathematically expressed, the first term on the right-hand side of eq 11 was negligible, but the first term of eq 10 was not necessarily negligible at high concentrations. As far as we are aware, none of the previous authors has estimated the relative contributions of the first and second terms in the expression for either hydraulic coefficient or electroosmotic coefficient. Therefore, it is too early to generalize the above arguments on our particular membrane-electrolyte systems. Nevertheless, we wish to point out that, in general, no satisfactory interpretation of volume flow data might be made unless the effect of mass flow is taken properly into consideration.
Appendix Referring to eq 7 and 8, ( J v / A p ) ~ = 0can be expressed by (J,/Ap),o
=
[(WIZ~/WII)
- ~z2IL-l
(AI)
Introducing eq 13, 14, and 15 transforms eq A1 to (Jv/Ap)I-O
=
- [(Jv/I)(Jv/Ir)
lAp=O
+
( J v / A P >Ar 3 0 (A21
As found from Figure 3, the maximum value of ( J v / I ) A p = is o 7.5 X cm3coulomb-' for membranes 2 and 3 in KC1, while the maximum value of ( J y j 1 r ) A p read off Figure 5 is 3.75 X cm sec-l V-l. Therefore, the first term on the right-hand side of eq A2 is about 3 X cm4 sec-I coulomb-l V-' = 3 X lo-'* cm3dyne-l sec-l, which is less than 1% of the observed value of - (Jv/Ap)I,o (8.2 X lo-" cm3 dyne-' sec-I). Thus ( J v / A p ) , = , may be equated to ( J y / A p ) A r , o , as far as the present systems are concerned. (14) Y. Kobatake and H. Fujita, J. Chem. Phys., 40, 2212 (1964). s. Manning, ibia., 46, 4976 (1967).
(16) G.
volume 78, Number 6 May 1968
