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The membrane transport matrix: Relationship between concentration dependence, matrix transformation, and accuracy of flux representation
Journal of Membrane Science, 2 (1977) 269-287 0 Elsevier Scientific Publishing Company, Amsterdam -
Printed in The Netherlands
269
THE MEMBRANE TRANSPORT MATRIX: RELATIONSHIP BETWEEN CONCENTRATION DEPENDENCE, MATRIX TRANSFORMATION, AND ACCURACY OF FLUX REPRESENTATION
A. ZELMAN,
J. COHEN, and D. GISSER*
Center for Biomedical Rensselaer Polytechnic
Engineering (*Department of Electrical Institute, Troy, N. Y., 12181 (U.S.A.)
and Systems
Engineering),
(Received March 8, 1977; revised version May 18, 1977)
Summary Completely unambiguous characterization of the membrane transport matrix is impossible. The problem lies in that a second state of the membrane-solution system is necessary to generate sufficient data to determine the complete matrix of coefficients. The transport coefficients are not obtained experimentally from a single state of the membrane-solution system, but they are functions of at least two different states. The magnitude of the coefficients will depend on the method of averaging the coefficients over these different thermodynamic states as well as the particular states over which the coefficients have been arbitrarily chosen to be evaluated. The principal purposes of this research are (a) to provide a format for accurate representation of membrane transport coefficients, (b) to show that Fa is a “natural” variable with which to catalog the concentration dependence of transport coefficients and (c) to demonstrate the general validity and utility of this method of representation by developing matrix transformations to several matrix forms, and by comparing the accuracy of the transformed coefficients to generate the experimental fluxes to the actual experimental fluxes in each reference frame.
Staverman [ 1, 21 initiated a revolution in membranology by forming the first membrane transport coefficient matrix from the linear theory of nonequilibrium thermodynamics developed by Onsager [ 3, 41. Since that time a huge number of experimental and theoretical membrane transport papers have ensued. Several research groups have developed specific apparatus and techniques for evaluating three or more membrane transport coefficients using, for example, true steady state [ 5, 61, quasi-steady state [ 7-141, and nonsteady [15-171 techniques, Zelman has attempted to combine and refine the various experimental methods into a single apparatus for non-steady state characterization of membrane transport phenomena and has catalogued extensive data on cellulose membranes [18]. His studies have led to the conclusion that completely unambiguous characterization of membrane transport coefficients is impossible. The following argument illustrates the general problem. Consider, for example, the “practical” transport equations written
for a single permeable solute and a single state of the membrane-solution system indicated by the prime: J; = -I,,
(AP’-
uA&)
(1)
A~I;
J, = ‘F-,’ (1-o)J;-o
(2)
There are two equations and three transport coefficients. Thus, either An, or AP or both must be altered to develop sufficient data to evaluate the transport coefficients (Lp, (Tand w). Should AP be changed, the altered membrane tension or compression will alter the transport coefficients. If AU, is changed, the altered solute concentration profile and membrane hydration will affect the transport coefficients. Consider the case where only An, is altered, Then J;‘= 4,
(A&“-uAIl,“)
(3)
J;’ = F”(1-o)&-wArI;
(4)
Any three equations from eqns. l-4 can be used to evaluate the three independent coefficients. The important aspect, however, is that a second state of the chemical system is necessary to generate sufficient data to determine the complete matrix of coefficients. The transport coefficients, then, are not obtained experimentally from a single state of the membrane--solution system, but they are functions of at least two different states. The magnitude of the coefficients will depend on the method of averaging the coefficients over these different thermodynamic states as well as the particular states over which the coefficients have been arbitrarily chosen to be evaluated. This intrinsic problem in accurate characterization of transport processes can be ameliorated only by careful choice of experimental protocol and precise statements for what is meant by “accuracy”. The following methods are suggested by our laboratory: (1) The transmembrane hydrostatic pressure drop is maintained constant during each experiment and the membrane support is well specified; thus the tension or compression of the membrane can be characterized. (2) The concentration variation during each experiment is maintained small according to: (a)
AC,& << 1
(b)
cs N
and
= constant during each experiment.
During each experiment, the average state of membrane hydration remains constant and can be specified by c. Perturbations of membrane hydration due to an altering concentration profile will be small since AC,/& is small. We predict that this choice of experimental conditions will provide transport coefficients independent of AC, and allow their concentration dependence to be accurately catalogued as a function of G.
271
(3) The accuracy of each experiment is verified by evaluating the reliability coefficient
(5) and the fractional reliability coefficient
(6) where Xi may be either the flux or its conjugate thermodynamic force. Superscripts e and c refer to experimental values and the value calculated from the transport equation after evaluating the transport coefficients, respectively. The reliability coefficient is a direct measure of how accurately the experimental fluxes Or forces Can be reproduced from the determined matrix of transport CO efficients. The fractional reliability coefficient allows comparison of data representation between different matrices using different coordinate systems. (More will be said of the use of the reliability coefficients in the discussion.) A series of experiments at increasing rates of solution agitation will allow extrapolation of the matrix to infinite solution agitation. With Fs, AP, Rx,, and R$, reported for each coefficient matrix, the accuracy of representation will be komplete. We recommend that transport coefficients and experimental conditions be catalogued together as in Table 1. The principal purposes of this paper are (a) to provide a format for accurate representation of membrane transport coefficients, (b) to show that FS is a “natural” variable with which to catalog the concentration dependence of transport coefficients, and (c) to demonstrate the general validity and utility of this method of representation by developing matrix transformations to several matrix forms and by comparing the accuracy of the transformed coefficients to generate the experimental fluxes to the actual experimental fluxes in each reference frame. Also discussed are the “real” vs. “artifactu~” COncentration dependence of transport coefficients and the unreliability of catdoting transport coefficients solely by the standard error or standard deviation of each coefficient. Experimental and analytical procedure h apparatus and analytical method have been constructed for non-steadystate characterization of membrane transport phenomena and are described in detail elsewhere [ 181. This system is unique in that an entire matrix Of transport coefficients can be obtained from a single experiment. Every necessary experimental parameter can be determined, with precision, in each half-cell simultaneously.
1
3.15 f 0.03
_____._
w x lop
_.-I___
Range of experiments 1.18 ?; 0.04 3.12 + 0.06
Single experiment 1.12 f 0.02
L, x lo5
0.0452
0.0446 f 0.0030
+ 0.0030
_~_
Q 0.200
< 0.043
--------
(dimensionless)
0.197, 0.417, 0.478, 1.18
25
1.80
1.37
-.._
~ _ __ 25
(cm/set)
(Torr )
AP
--
_
0.058
0.128
___~
~.
(dimensionless)
1.40
1.15
.._
0.0 600
0.0649
_____-_
(dimensionless)
for one experiment and a series of experiments on the same
1.180 t 0.005
(5)
Es x lo4
An accurate display for characterizing the membrane transport coefficients piece of membrane
TABLE
tQ 2
273
Temperature, solution concentration, and solution volume were monitored
continuously in each half-cell, Experiments were designedsuch that the solute concentration in each half-cell varied with time while the solution temperatures and transmembrane hydrostatic pressure difference were maintained constant. The “practical” transport equations [ 191 were chosen for evaluation because L,, u, and w had been predicted by Spiegler and Kedem [ 201 to be reasonably independent of concentration. Linearized, time-dependent transport equations were written J,(t) = -L,AP
Js (t) G (t) Jv (t)
+ L,uA&
= (l-u)
(t)
(7)
(t) qt) J,(t)
-wAIl
Least squares linear functions were developed from which I&, U,and o could be determined from the Y intercepts and slopes together with the standard deviation of each coefficient [21]. The linear fit for all data reported here was r > 0.999 with p < 0.001. The practical coefficients were evaluated for C, between 0.0197 and 0.345 M NaCl. The membrane used was cellophane (No. 70160-3, 3-7/16 flat cellophane tubing, Central Scientific Co., Chicago, Ill.). These experiments were designed so that as A& decays toward zero, the volume flow will decay to zero and then reverse direction. The reported coefficients are equally accurate for volume flow in either direction relative to AII,. The zero volume flow method for calculating the transport coefficients was used as a check on the least squares method. Detailed experimental comparison between the two analytical methods can be found in [18]. The values of L,, u, and w were shown to remain essentially constant over a wide range of values for q. However, the non-linear transformations from the “practical coefficients” to any other transport matrix involves &. We shall use an average value of L,, u, and u to calculate transport coefficients of other reference frames as a function of Fs. The accuracy of the average L,, u, and w to characterize J, and Js at different values of q follows. Table 1 contains experimental values for L,, IS, and w from our four best experiments on a single membrane. An arithmetic mean value of L,, u, and w were obtained from these experiments. The mean values of L,, u, and w were used to calculale the fluxes for 7 experiments where c = (0.197, 0.417, 0.478, 1.18, 2.89, 3.15 and 3.45) X 10m4mole/cm3 NaCI. The reliability and fractional reliability coefficients are tabulated in Tables 1 and 3 and will be used as indices for accuracy throughout this paper. The average values of L,, u, and w will now be used to generate the transport coefficients in various frames of reference. Since the fluxes in these frames of reference can be calculated directly from the experimental data, the reliability of the transformation in each case can be accurately evaluated with
274
the concentration dependence of the coefficient matrix determined as a function of c. Fluxes used in this paper The following fluxes appear in the equations of the reference frames used in this paper: (1) Volume Flux: The volume flux, J,, is not conservative because the partial molar volume is concentration dependent. However, with AC, I& C-C 1, Jv is conservative within experimental error. The experimental formula for Jv is : 1
Jv (t) = ~ A
dV(t) ~ -dt
,
(9)
where A is the membrane area, V is the half-cell volume, and t is time. (2) Salt Flux: The salt flux, J,, is a conservative flux taken relative to the laboratory. The experimental formula for J, is: 1 dn, (t) J, (t> =- ~ Adt
(10) ’
where “T is the salt increase in one half-cell. (3) Water Flux: The water flux, J w, is a conservative flux taken relative to the laboratory. The experimental formula for J, is: & (t) = (J’“(t)
-T,’ (t) J: (t))/i&
(11)
where the prime indicates measurements taken in one half-cell. (4) Diffusion Flux: The diffusion flux, JD, is not a conservative flux. The experimental formula for JD is:
Js (t) JD (t) =-
-
G (tl
GJw (t) .
(12)
Results The average L,, CT,and o have been transformed to the conductance coefficients of the reference frames below:
-Js = +L Us + Lsw &xv -J,
= +L,,
ApS + L,,
AP,
(13) .
(14)
275
Zelman [22] has shown that the straightforward transformation from the L coefficients to the “practical” coefficients of necessity produced two different reflection coefficients, one for the solute flow equation, usvI and one for the volume flow equation, uv. This apparent discrepancy results from not making the dilute solution approximation and it has been verified by Mikulecky [ 231 using the Kedem-Katchalsky approach without approximations. Numerical evaluation below indicates that the difference between usv and crv is calculable, but very small. Onsager reciprocity is assumed in these calculations. Zelman’s equations were used to derive the following transformation of L,, uv,and w into L,,, LSw, and L, w (see Appendix A for derivation).*
-L,
2iqwu,
(
-VW2
(uq2
zr;,
uv
-
FS
--1
4
+o
(a2
(
-3 VW”,
“WV,
-I1 &
(17)
4
L ww = ---
2i?$i&
-
---+v$$
F; +(Q2
cs
The second set of three equations from Zelman is the transformation CJSV,and w into LfSs, LISw, and Ltww (see Appendix B for derivation).
-L,
(
-2~.oSV
+
F; + “$
(QSV)
of L,,
2)-_(;) S
(18)
L’,, = -3 “w -I_
iX-& -__
(
) (GA)
G
*The two appendices to this paper are lengthy and have not been included. They may be obtained by writing to the authors.
276
G &l
-~$&IJSv
-
-
+ FWir,
(oSv)2
+
( c, Gw
=
s
-
--__-~
(19)
-
WI_ 1
k 6%‘)”
&VW=
S
-3VW us
iT3 W
L,
Gusv Q-)+cd(F)
G
2iz,iQF -“,IJs -2 (@V)” +-------
(
ix-xi& if& - __ +-G
c,
-ij-,z)-
-2
(C)
--iT3 W
-3UWUS
-__-__ (
) (Cl2
cs
The difference between the computed values of coefficient sets (L,,, Law, Lww) and (LlsS, LISw, LrWw) were not experimentally significant as can be seen for two experiments in Table 2. Therefore, set (LSS,LSw, Lw w) was arbitrarily chosen to evaluate the fluxes in further computations. The plots of the transport coefficients versus ?SSare virtually the same for both coefficient sets. Table 3 contains the reliability and fractional reliability coefficients for the coefficient set (L,,, LSw, L,, ). Figs. 1, 2, and 3 are the plots of Z&, Lsw, and L,, versus E& respectively. One may form the resistance matrix from the conductance matrix eqns. (13) and (14) by forming an inverse matrix: -4~s
= RssJs + RswJw
(21)
-Aclw = RswJS + RwwJw .
(22)
Figs. 4, 5, and 6 are the plots of Rss, Rsw, and Rww versus Fs, respectively. Table 3 contains the reliability and fractional reliability coefficients for this matrix. The frictional type coefficients, rik, are defined by the following matrix: -A& = rsw (J,-J,)
-bw
+ rssJs
= rsw(J,- Js) + rwwJw
(23) (24)
Figs. 7, 8, and 9 are the plots of r,, rSw, and rww versus cs, respectively. The reliability coefficients for the frictional equations are given in Table 3. Kedem and Katchalsky [24] derived the following transformation for the transport coefficients of reference frame (I&-,,u, w), into the transport coef-
277
TABLE
2
Values of coefficient
sets (L,,, L,,
L,,
Fs x lo4 Exp.
L ss
L’ss
L SW
L’SW
L ww
1.180 * 0.005
5.12 x lo-l3 2.02 x 10-12
5.12 x 1o-‘s 2.02 x lo-‘*
7.14 x lo-” 1.90 x lo-lo
7.16 x lo-” 1.91 x 10 -lo
3.54 x 1o-a 3.51 x lo-*
3.150 * 0.006
TABLE
) and (L&, L&, L&)
L’ww
3.54 x lo-* 3.51 x lo-’
3
The reliability and fractional reliability coefficients for various matrices, all of which were calculated from average values of L,, o, and w. Coefficients applicable to 0.0197 a Cs Q 0.315M NaCl, 25 Q AP < 125 Torr. Coefficient values at any < can be obtained from the graphs.
&?3 a, WI
(See Table 1 for values)
RJ, = 1.80
x lo-*
[RJs = 1.40 x lo-‘O
RJW
f = 0.058 RJ”
cm set-’ mole see-’ cm-’
= 9.40 X 10m9 molesec-’
cm-’
RJ~ = 2.98 x 10-l a mole see-’ cmea
R&&I
(a,,, a,, a,,, all)
RJ~ = 1.75 x lo-’
R&s
RJ, = 1.75 x lo-’ RJ~ = 2.94
U&v R,,, %,I
x lO_“’
mole set“
_ cs
= 9.72
RJ, = 2.74 x lo-’
RJ~ = 8.80 x lo-’
= 0.710
f RJ~ = 0.138
R;”
cm-’
cm set-’
R An = 7.64 cm3 atm mole-’ R(A~_A~)
tLP. L,D, ‘$I,)
cm set-’
X 10-l ’ mole set-’
&
= 0.151
f Raps = 0.133
= 764 cm3 atm mole-’
RJ~ = 4.42
s
f B&V
= 13.1 cm’ atm moles’
(R,,, R,, Rww1and P 39 rsw. rww1
RfT = 0.060
f RJ~ = 0.270
R;”
cm-’
= 0.346
= 0.346
R:, = 0.138
R’,n=0.093
x 10e2 atm
REP_,,,=
cm see-’
R;,
cm set-’
= 0.256
= 0.058
0.151
cgxto’ (vimole Fig.1. Transport coefficient ficients
Lssversus CS from eqn. 15.
of (all, u12,azl, a22 ). The equations are given by
where
(27)
atI = L,
(28) al2 = 5 (I-U)Lp a21=
&
(l-u)
az2 = c, w
(29)
(30) Table 3 contains the reliability and fractional reliability coefficients for the fluxes of eqns. 28,29, and 30. It may be noted that alI is independent of concentration. Also from Kedem and Katchalsky [ 241, the following reference frame was formulated:
2?9
L
csxlo4(*,) Fig.2. Transport coefficient L,
Fig. 3. Transport coefficient L,,
versus CS from eqn. 16.
versus Cs from eqn. 17.
280
. .
Fig.4. Transport coefflclent
R,, versus -cS from eqn. 21.
An -J, = L,, (AP- AiI) + L,, _ c,
( 1
(31)
(32) We may express &I, LIZ, Lzl, and Lz2in terms of L,, U, and Ll
O:
(33)
= L,
L1*=_d (I-a)Lp
(34)
L21 = 5
(35)
(1-o)L,
L22=(cg)2(1LqLp +&cd
(36)
Table 3 contains the reliability and fractional reliability coefficients for the fluxes of this reference frame. Inverse Transport Coefficient Matrix: We may write eqns. 31 and 32 with the following inverse matrix: (37) -(AP-An) = RI& + RI&
281
-547-
- 5.48 -
E2:N ii_? “E 67 g-5.490 ol ‘0 X 1
-5.50-
IL”
-5.51
0
I
I
I
I
I
2
3
4
CS
Fig. 5. Transport coefficient R,
x lO4y-J
versus CS from eqn. 2 1.
- - - = R21Jv + Rz2Js cs
(38)
Table 3 contains the reliability and fractional reliability coefficients for (AP-AII) and (An&). Katchalsky and Curran [19] wrote a set of transport coefficients as: -Jv = LpAP + L~DAII
(39)
-JD=LD,AP+LDAFI
(40)
The transformation,
(Lp, u, w) + (Lp, L,D,
LD), is:
Lp = Lp
(41)
L pD = QLp
(42)
LD
==+t,‘Lp
.
(43)
7s
Table 3 contains the reliability and fractional reliability coefficients for eqns. 42 and 43. Inverse Transport Coefficient Matrix: We may write eqns. 39 and 40 in the following inverse matrix notation:
282
6.0
-
E Gc.4 z-”
*
0
E
5.0-
-E 0
bIO x ;
40-
cc
I
3.01
0
Fig.6. Transport
-AI’=
I
coefficient
RpJv + R,DJD
-AH = RDPJ,
+ RDJD
2
3
4
R,, versus ES from eqn. 22.
(44) (45)
Table 3 contains the reliability and fractional reliability coefficients for this matrix. Discussion This report has tried to point out the intrinsic problems in the efforts to accurately and unambiguously catalog the membrane transport matrix, and to offer a methodology by which the membrane transport matrix can be catalogued with its range of uncertainty sufficiently well characterized that the accuracy with which the matrix can be used to approximate the experimental fluxes is easily understood. The “practical” coefficients are clearly the easiest coefficients to evaluate experimentally; however, the transformation from the “practical coefficients” to other reference frames is non-linear. The high accuracy with which (Lp, u, and o with C,) can predict the correct concentration dependence of other transport coefficients clarifies the usefulness of the practical equations. Their accuracy is sufficiently great that any theoretical treatment can be tested on the basis of its ability to duplicate the concentration dependence formulated here for cellulose membranes.
283
Fig.7. Transportcoefficient rss versusCs from eqn. 23. In determining the validity of any theoretical formulation one must not forget that any matrix of membrane transport coefficients can only be experimentally evaluated for a range of concentration values, not concentration per se. The practical coefficients offer C, as a reasonable and n&Ural method of cataloging the particular range for which the coefficients are valid. In order to more clearly delineate the concentration state of the membrane the additional stipulations that AC& << 1 and Fs NCC,> seem like natural qualifiers. These experiments also indicate that perhaps (at least for cellulose membranes) artifactual concentration dependence will appear in transport experiments where & is allowed to vary during the experiment. Since all of the transport equations can be derived from (Lp, u, o , and G), it is clear that by maintaining Fs constant during any experiment, the transport coefficients of any reference frame will be independent of concentration during that experiment. However, if Cs alters significantly during any experiment, then most transport equations will behave non-linearly. This non-linear behavior could be interpreted as a violation of the linear assumption, but, in reality, it would be nothing more than particular experimental conditions producing the natural concentration dependence of the transport coefficients in that particular frame of reference. me choice of an experimental reference frame has been primarily the experimenter’s expediency. This research should clearly indicate that the choice of reference frame cannot be arbitrary. Because the experimental data must
Fig.8. ‘Transport coefficient rsw versus G from eqn. 23
be averaged over a range of concentrations, there undoubtedly will be one reference frame more suited to accurate averaging than any other. To find the “best fit” reference frame one should construct a table similar to Table 3, but where each coefficient matrix has been evaluated directly from the data, then choose the most accurate reference frame by locating the matrix with the smallest fractional reliability, Rf: _. The accuracy of representation within the selected reference frame will be given by the reliability coefficient, Rxi.This selection process can be easily automated with the computer programmed to determine the matrix with the best fit. Once the best fit reference frame has been determined, matrix transformations can be made to any other, perhaps more convenient set of transport equations. The accuracy of transformations of this kind is shown in Table 3. The most accurate transformation can be determined by choosing the minimum fractional reliability coefficient. The uncertainty value for each transport coefficient specifies a limited range of validity for that coefficient; some accuracy will be lost in matrix transformations where uncertainty regions overlap. Values determined here indicate that for reasonably precise work, this effect will not be a limiting factor. The concentration dependence of the transport coefficients reported here is real. This is tested by using the concentration-dependent coefficients evaluated from (Lp, u, o, and &) and the thermodynamic forces from the various reference frames to evaluate the conjugate fluxes. Even though some coefficients are highly concentration dependent, the reliability coefficients indicate that the accuracy of the concentration-dependent representation given
285
Fig.9.
Transport
coefficient
rww versus Cs from
eqn. 24.
here is the correct form. Interestingly, Figs.1 to 9 show that within a single matrix both very linear and very non-linear coefficients are generated. A close examination of these properties will be forthcoming. This research has tried to indicate the inadequacy in reporting only the accuracy or standard deviation of individual transport coefficients. The coefficient accuracy is not sufficient information to determine the overall accuracy of the matrix to describe the fluxes. Within each concentration range for which a matrix of coefficients has been evaluated, the precision of the fluxes will vary as the relative importance of the product of the transport coefficient and its thermodynamic force, i.e., _I&Apk ~ changes. The reliability coefficient and the fractional reliability coefficient depict the accuracy of the transport coefficients to reproduce the experimental fluxes. This includes both the error in the coefficient determination and the relative importance of the individual force-flux terms, &k Apk I in determining the actual flux. Use of the reliability and Ifractional reliability coefficients together with the experimental data representation of Table 1 will result in a far more understandable and less ambiguous cataloging of transport coefficients. Acknowledgement. This research was supported in part by Public Health Service Grant No. AM 19198 and contract No. l-AM-7-2206.
286
Table of symbols Area of membrane, cm’ Arithmetic mean concentration, mole cmm3. Transmembrane concentration difference, mole cmm3, Logarithmic mean concentration, mole cmm3. Diffusion flux, cm set-‘. Solute and water flux, respectively, mole set-’ cm-?. Volume flux through membrane, cm see-‘. Hydraulic permeability, cm set-’ atm.-‘. Total transmembrane chemical potential drop of the salt and water, respectively, cm3 atm mole-‘. Salt transport, mole. Solute permeability coefficient, mole set-’ cmW2atm-‘. Transmembrane hydrostatic pressure drop, atm. Transmembrane osmotic pressure drop, atm. F’ractional Reliability Coefficient, see eq. (6). Reliability Coefficient, see eq. (5). Reflection coefficient, dimensionless. Time, set Partial molar volume of solute and water respectively cm3 mole-’
(bs, Liw, Lww) (Rss,Raw, Rww) Ok, rsw, rww) (%I,
a12,
(Ll,
b2I
a219
a22)
L22)
m11,
J&t,
(-$a
LpD,
R22)
LD)
@p,
R,D>
ED)
See eqns. See eqns. See eqns. See eqns. See eqns. See eqns. See eqns. See eqns.
13 21 23 25 31 37 39 44
and and and and and and and and
14. 22. 24. 26. 32. 38. 40. 45.
References 1 2 3 4 5 6 7 a 9
A.J. Staverman, Rec. Trav. Chim. Pays-Ras, 70 (1951) 344. AJ. Staverman, Trans. Faraday Sot., 48 (1952) 176. L Onsager, Phys. Rev., 37 (1931) 405. L Onsager, Phys. Rev., 38 (1931) 2265. A Zelman, J.C.T. Kwak, J. Liebovitz, and K.S. Spiegler, Experimentia Suppl., 18 (1971) 679. M. Demarty and E. Selegny, Compt. Rend., C276 (1973) 1549. T.G. Kaufman and E.F. Leonard, Amer. Inst. Chem. Engrs. J., 14 (1968) 110. P. Meares and A.H. Sutton, J. Colloid Interface Sci., 28 (1968) 118. W. J, McHardy, P. Meares, A.H. Sutton, and J.F. Thain, J. Colloid Interface Sci., 29 (1969) 116.
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Printed in The Netherlands
269
THE MEMBRANE TRANSPORT MATRIX: RELATIONSHIP BETWEEN CONCENTRATION DEPENDENCE, MATRIX TRANSFORMATION, AND ACCURACY OF FLUX REPRESENTATION
A. ZELMAN,
J. COHEN, and D. GISSER*
Center for Biomedical Rensselaer Polytechnic
Engineering (*Department of Electrical Institute, Troy, N. Y., 12181 (U.S.A.)
and Systems
Engineering),
(Received March 8, 1977; revised version May 18, 1977)
Summary Completely unambiguous characterization of the membrane transport matrix is impossible. The problem lies in that a second state of the membrane-solution system is necessary to generate sufficient data to determine the complete matrix of coefficients. The transport coefficients are not obtained experimentally from a single state of the membrane-solution system, but they are functions of at least two different states. The magnitude of the coefficients will depend on the method of averaging the coefficients over these different thermodynamic states as well as the particular states over which the coefficients have been arbitrarily chosen to be evaluated. The principal purposes of this research are (a) to provide a format for accurate representation of membrane transport coefficients, (b) to show that Fa is a “natural” variable with which to catalog the concentration dependence of transport coefficients and (c) to demonstrate the general validity and utility of this method of representation by developing matrix transformations to several matrix forms, and by comparing the accuracy of the transformed coefficients to generate the experimental fluxes to the actual experimental fluxes in each reference frame.
Staverman [ 1, 21 initiated a revolution in membranology by forming the first membrane transport coefficient matrix from the linear theory of nonequilibrium thermodynamics developed by Onsager [ 3, 41. Since that time a huge number of experimental and theoretical membrane transport papers have ensued. Several research groups have developed specific apparatus and techniques for evaluating three or more membrane transport coefficients using, for example, true steady state [ 5, 61, quasi-steady state [ 7-141, and nonsteady [15-171 techniques, Zelman has attempted to combine and refine the various experimental methods into a single apparatus for non-steady state characterization of membrane transport phenomena and has catalogued extensive data on cellulose membranes [18]. His studies have led to the conclusion that completely unambiguous characterization of membrane transport coefficients is impossible. The following argument illustrates the general problem. Consider, for example, the “practical” transport equations written
for a single permeable solute and a single state of the membrane-solution system indicated by the prime: J; = -I,,
(AP’-
uA&)
(1)
A~I;
J, = ‘F-,’ (1-o)J;-o
(2)
There are two equations and three transport coefficients. Thus, either An, or AP or both must be altered to develop sufficient data to evaluate the transport coefficients (Lp, (Tand w). Should AP be changed, the altered membrane tension or compression will alter the transport coefficients. If AU, is changed, the altered solute concentration profile and membrane hydration will affect the transport coefficients. Consider the case where only An, is altered, Then J;‘= 4,
(A&“-uAIl,“)
(3)
J;’ = F”(1-o)&-wArI;
(4)
Any three equations from eqns. l-4 can be used to evaluate the three independent coefficients. The important aspect, however, is that a second state of the chemical system is necessary to generate sufficient data to determine the complete matrix of coefficients. The transport coefficients, then, are not obtained experimentally from a single state of the membrane--solution system, but they are functions of at least two different states. The magnitude of the coefficients will depend on the method of averaging the coefficients over these different thermodynamic states as well as the particular states over which the coefficients have been arbitrarily chosen to be evaluated. This intrinsic problem in accurate characterization of transport processes can be ameliorated only by careful choice of experimental protocol and precise statements for what is meant by “accuracy”. The following methods are suggested by our laboratory: (1) The transmembrane hydrostatic pressure drop is maintained constant during each experiment and the membrane support is well specified; thus the tension or compression of the membrane can be characterized. (2) The concentration variation during each experiment is maintained small according to: (a)
AC,& << 1
(b)
cs N
and
= constant during each experiment.
During each experiment, the average state of membrane hydration remains constant and can be specified by c. Perturbations of membrane hydration due to an altering concentration profile will be small since AC,/& is small. We predict that this choice of experimental conditions will provide transport coefficients independent of AC, and allow their concentration dependence to be accurately catalogued as a function of G.
271
(3) The accuracy of each experiment is verified by evaluating the reliability coefficient
(5) and the fractional reliability coefficient
(6) where Xi may be either the flux or its conjugate thermodynamic force. Superscripts e and c refer to experimental values and the value calculated from the transport equation after evaluating the transport coefficients, respectively. The reliability coefficient is a direct measure of how accurately the experimental fluxes Or forces Can be reproduced from the determined matrix of transport CO efficients. The fractional reliability coefficient allows comparison of data representation between different matrices using different coordinate systems. (More will be said of the use of the reliability coefficients in the discussion.) A series of experiments at increasing rates of solution agitation will allow extrapolation of the matrix to infinite solution agitation. With Fs, AP, Rx,, and R$, reported for each coefficient matrix, the accuracy of representation will be komplete. We recommend that transport coefficients and experimental conditions be catalogued together as in Table 1. The principal purposes of this paper are (a) to provide a format for accurate representation of membrane transport coefficients, (b) to show that FS is a “natural” variable with which to catalog the concentration dependence of transport coefficients, and (c) to demonstrate the general validity and utility of this method of representation by developing matrix transformations to several matrix forms and by comparing the accuracy of the transformed coefficients to generate the experimental fluxes to the actual experimental fluxes in each reference frame. Also discussed are the “real” vs. “artifactu~” COncentration dependence of transport coefficients and the unreliability of catdoting transport coefficients solely by the standard error or standard deviation of each coefficient. Experimental and analytical procedure h apparatus and analytical method have been constructed for non-steadystate characterization of membrane transport phenomena and are described in detail elsewhere [ 181. This system is unique in that an entire matrix Of transport coefficients can be obtained from a single experiment. Every necessary experimental parameter can be determined, with precision, in each half-cell simultaneously.
1
3.15 f 0.03
_____._
w x lop
_.-I___
Range of experiments 1.18 ?; 0.04 3.12 + 0.06
Single experiment 1.12 f 0.02
L, x lo5
0.0452
0.0446 f 0.0030
+ 0.0030
_~_
Q 0.200
< 0.043
--------
(dimensionless)
0.197, 0.417, 0.478, 1.18
25
1.80
1.37
-.._
~ _ __ 25
(cm/set)
(Torr )
AP
--
_
0.058
0.128
___~
~.
(dimensionless)
1.40
1.15
.._
0.0 600
0.0649
_____-_
(dimensionless)
for one experiment and a series of experiments on the same
1.180 t 0.005
(5)
Es x lo4
An accurate display for characterizing the membrane transport coefficients piece of membrane
TABLE
tQ 2
273
Temperature, solution concentration, and solution volume were monitored
continuously in each half-cell, Experiments were designedsuch that the solute concentration in each half-cell varied with time while the solution temperatures and transmembrane hydrostatic pressure difference were maintained constant. The “practical” transport equations [ 191 were chosen for evaluation because L,, u, and w had been predicted by Spiegler and Kedem [ 201 to be reasonably independent of concentration. Linearized, time-dependent transport equations were written J,(t) = -L,AP
Js (t) G (t) Jv (t)
+ L,uA&
= (l-u)
(t)
(7)
(t) qt) J,(t)
-wAIl
Least squares linear functions were developed from which I&, U,and o could be determined from the Y intercepts and slopes together with the standard deviation of each coefficient [21]. The linear fit for all data reported here was r > 0.999 with p < 0.001. The practical coefficients were evaluated for C, between 0.0197 and 0.345 M NaCl. The membrane used was cellophane (No. 70160-3, 3-7/16 flat cellophane tubing, Central Scientific Co., Chicago, Ill.). These experiments were designed so that as A& decays toward zero, the volume flow will decay to zero and then reverse direction. The reported coefficients are equally accurate for volume flow in either direction relative to AII,. The zero volume flow method for calculating the transport coefficients was used as a check on the least squares method. Detailed experimental comparison between the two analytical methods can be found in [18]. The values of L,, u, and w were shown to remain essentially constant over a wide range of values for q. However, the non-linear transformations from the “practical coefficients” to any other transport matrix involves &. We shall use an average value of L,, u, and u to calculate transport coefficients of other reference frames as a function of Fs. The accuracy of the average L,, u, and w to characterize J, and Js at different values of q follows. Table 1 contains experimental values for L,, IS, and w from our four best experiments on a single membrane. An arithmetic mean value of L,, u, and w were obtained from these experiments. The mean values of L,, u, and w were used to calculale the fluxes for 7 experiments where c = (0.197, 0.417, 0.478, 1.18, 2.89, 3.15 and 3.45) X 10m4mole/cm3 NaCI. The reliability and fractional reliability coefficients are tabulated in Tables 1 and 3 and will be used as indices for accuracy throughout this paper. The average values of L,, u, and w will now be used to generate the transport coefficients in various frames of reference. Since the fluxes in these frames of reference can be calculated directly from the experimental data, the reliability of the transformation in each case can be accurately evaluated with
274
the concentration dependence of the coefficient matrix determined as a function of c. Fluxes used in this paper The following fluxes appear in the equations of the reference frames used in this paper: (1) Volume Flux: The volume flux, J,, is not conservative because the partial molar volume is concentration dependent. However, with AC, I& C-C 1, Jv is conservative within experimental error. The experimental formula for Jv is : 1
Jv (t) = ~ A
dV(t) ~ -dt
,
(9)
where A is the membrane area, V is the half-cell volume, and t is time. (2) Salt Flux: The salt flux, J,, is a conservative flux taken relative to the laboratory. The experimental formula for J, is: 1 dn, (t) J, (t> =- ~ Adt
(10) ’
where “T is the salt increase in one half-cell. (3) Water Flux: The water flux, J w, is a conservative flux taken relative to the laboratory. The experimental formula for J, is: & (t) = (J’“(t)
-T,’ (t) J: (t))/i&
(11)
where the prime indicates measurements taken in one half-cell. (4) Diffusion Flux: The diffusion flux, JD, is not a conservative flux. The experimental formula for JD is:
Js (t) JD (t) =-
-
G (tl
GJw (t) .
(12)
Results The average L,, CT,and o have been transformed to the conductance coefficients of the reference frames below:
-Js = +L Us + Lsw &xv -J,
= +L,,
ApS + L,,
AP,
(13) .
(14)
275
Zelman [22] has shown that the straightforward transformation from the L coefficients to the “practical” coefficients of necessity produced two different reflection coefficients, one for the solute flow equation, usvI and one for the volume flow equation, uv. This apparent discrepancy results from not making the dilute solution approximation and it has been verified by Mikulecky [ 231 using the Kedem-Katchalsky approach without approximations. Numerical evaluation below indicates that the difference between usv and crv is calculable, but very small. Onsager reciprocity is assumed in these calculations. Zelman’s equations were used to derive the following transformation of L,, uv,and w into L,,, LSw, and L, w (see Appendix A for derivation).*
-L,
2iqwu,
(
-VW2
(uq2
zr;,
uv
-
FS
--1
4
+o
(a2
(
-3 VW”,
“WV,
-I1 &
(17)
4
L ww = ---
2i?$i&
-
---+v$$
F; +(Q2
cs
The second set of three equations from Zelman is the transformation CJSV,and w into LfSs, LISw, and Ltww (see Appendix B for derivation).
-L,
(
-2~.oSV
+
F; + “$
(QSV)
of L,,
2)-_(;) S
(18)
L’,, = -3 “w -I_
iX-& -__
(
) (GA)
G
*The two appendices to this paper are lengthy and have not been included. They may be obtained by writing to the authors.
276
G &l
-~$&IJSv
-
-
+ FWir,
(oSv)2
+
( c, Gw
=
s
-
--__-~
(19)
-
WI_ 1
k 6%‘)”
&VW=
S
-3VW us
iT3 W
L,
Gusv Q-)+cd(F)
G
2iz,iQF -“,IJs -2 (@V)” +-------
(
ix-xi& if& - __ +-G
c,
-ij-,z)-
-2
(C)
--iT3 W
-3UWUS
-__-__ (
) (Cl2
cs
The difference between the computed values of coefficient sets (L,,, Law, Lww) and (LlsS, LISw, LrWw) were not experimentally significant as can be seen for two experiments in Table 2. Therefore, set (LSS,LSw, Lw w) was arbitrarily chosen to evaluate the fluxes in further computations. The plots of the transport coefficients versus ?SSare virtually the same for both coefficient sets. Table 3 contains the reliability and fractional reliability coefficients for the coefficient set (L,,, LSw, L,, ). Figs. 1, 2, and 3 are the plots of Z&, Lsw, and L,, versus E& respectively. One may form the resistance matrix from the conductance matrix eqns. (13) and (14) by forming an inverse matrix: -4~s
= RssJs + RswJw
(21)
-Aclw = RswJS + RwwJw .
(22)
Figs. 4, 5, and 6 are the plots of Rss, Rsw, and Rww versus Fs, respectively. Table 3 contains the reliability and fractional reliability coefficients for this matrix. The frictional type coefficients, rik, are defined by the following matrix: -A& = rsw (J,-J,)
-bw
+ rssJs
= rsw(J,- Js) + rwwJw
(23) (24)
Figs. 7, 8, and 9 are the plots of r,, rSw, and rww versus cs, respectively. The reliability coefficients for the frictional equations are given in Table 3. Kedem and Katchalsky [24] derived the following transformation for the transport coefficients of reference frame (I&-,,u, w), into the transport coef-
277
TABLE
2
Values of coefficient
sets (L,,, L,,
L,,
Fs x lo4 Exp.
L ss
L’ss
L SW
L’SW
L ww
1.180 * 0.005
5.12 x lo-l3 2.02 x 10-12
5.12 x 1o-‘s 2.02 x lo-‘*
7.14 x lo-” 1.90 x lo-lo
7.16 x lo-” 1.91 x 10 -lo
3.54 x 1o-a 3.51 x lo-*
3.150 * 0.006
TABLE
) and (L&, L&, L&)
L’ww
3.54 x lo-* 3.51 x lo-’
3
The reliability and fractional reliability coefficients for various matrices, all of which were calculated from average values of L,, o, and w. Coefficients applicable to 0.0197 a Cs Q 0.315M NaCl, 25 Q AP < 125 Torr. Coefficient values at any < can be obtained from the graphs.
&?3 a, WI
(See Table 1 for values)
RJ, = 1.80
x lo-*
[RJs = 1.40 x lo-‘O
RJW
f = 0.058 RJ”
cm set-’ mole see-’ cm-’
= 9.40 X 10m9 molesec-’
cm-’
RJ~ = 2.98 x 10-l a mole see-’ cmea
R&&I
(a,,, a,, a,,, all)
RJ~ = 1.75 x lo-’
R&s
RJ, = 1.75 x lo-’ RJ~ = 2.94
U&v R,,, %,I
x lO_“’
mole set“
_ cs
= 9.72
RJ, = 2.74 x lo-’
RJ~ = 8.80 x lo-’
= 0.710
f RJ~ = 0.138
R;”
cm-’
cm set-’
R An = 7.64 cm3 atm mole-’ R(A~_A~)
tLP. L,D, ‘$I,)
cm set-’
X 10-l ’ mole set-’
&
= 0.151
f Raps = 0.133
= 764 cm3 atm mole-’
RJ~ = 4.42
s
f B&V
= 13.1 cm’ atm moles’
(R,,, R,, Rww1and P 39 rsw. rww1
RfT = 0.060
f RJ~ = 0.270
R;”
cm-’
= 0.346
= 0.346
R:, = 0.138
R’,n=0.093
x 10e2 atm
REP_,,,=
cm see-’
R;,
cm set-’
= 0.256
= 0.058
0.151
cgxto’ (vimole Fig.1. Transport coefficient ficients
Lssversus CS from eqn. 15.
of (all, u12,azl, a22 ). The equations are given by
where
(27)
atI = L,
(28) al2 = 5 (I-U)Lp a21=
&
(l-u)
az2 = c, w
(29)
(30) Table 3 contains the reliability and fractional reliability coefficients for the fluxes of eqns. 28,29, and 30. It may be noted that alI is independent of concentration. Also from Kedem and Katchalsky [ 241, the following reference frame was formulated:
2?9
L
csxlo4(*,) Fig.2. Transport coefficient L,
Fig. 3. Transport coefficient L,,
versus CS from eqn. 16.
versus Cs from eqn. 17.
280
. .
Fig.4. Transport coefflclent
R,, versus -cS from eqn. 21.
An -J, = L,, (AP- AiI) + L,, _ c,
( 1
(31)
(32) We may express &I, LIZ, Lzl, and Lz2in terms of L,, U, and Ll
O:
(33)
= L,
L1*=_d (I-a)Lp
(34)
L21 = 5
(35)
(1-o)L,
L22=(cg)2(1LqLp +&cd
(36)
Table 3 contains the reliability and fractional reliability coefficients for the fluxes of this reference frame. Inverse Transport Coefficient Matrix: We may write eqns. 31 and 32 with the following inverse matrix: (37) -(AP-An) = RI& + RI&
281
-547-
- 5.48 -
E2:N ii_? “E 67 g-5.490 ol ‘0 X 1
-5.50-
IL”
-5.51
0
I
I
I
I
I
2
3
4
CS
Fig. 5. Transport coefficient R,
x lO4y-J
versus CS from eqn. 2 1.
- - - = R21Jv + Rz2Js cs
(38)
Table 3 contains the reliability and fractional reliability coefficients for (AP-AII) and (An&). Katchalsky and Curran [19] wrote a set of transport coefficients as: -Jv = LpAP + L~DAII
(39)
-JD=LD,AP+LDAFI
(40)
The transformation,
(Lp, u, w) + (Lp, L,D,
LD), is:
Lp = Lp
(41)
L pD = QLp
(42)
LD
==+t,‘Lp
.
(43)
7s
Table 3 contains the reliability and fractional reliability coefficients for eqns. 42 and 43. Inverse Transport Coefficient Matrix: We may write eqns. 39 and 40 in the following inverse matrix notation:
282
6.0
-
E Gc.4 z-”
*
0
E
5.0-
-E 0
bIO x ;
40-
cc
I
3.01
0
Fig.6. Transport
-AI’=
I
coefficient
RpJv + R,DJD
-AH = RDPJ,
+ RDJD
2
3
4
R,, versus ES from eqn. 22.
(44) (45)
Table 3 contains the reliability and fractional reliability coefficients for this matrix. Discussion This report has tried to point out the intrinsic problems in the efforts to accurately and unambiguously catalog the membrane transport matrix, and to offer a methodology by which the membrane transport matrix can be catalogued with its range of uncertainty sufficiently well characterized that the accuracy with which the matrix can be used to approximate the experimental fluxes is easily understood. The “practical” coefficients are clearly the easiest coefficients to evaluate experimentally; however, the transformation from the “practical coefficients” to other reference frames is non-linear. The high accuracy with which (Lp, u, and o with C,) can predict the correct concentration dependence of other transport coefficients clarifies the usefulness of the practical equations. Their accuracy is sufficiently great that any theoretical treatment can be tested on the basis of its ability to duplicate the concentration dependence formulated here for cellulose membranes.
283
Fig.7. Transportcoefficient rss versusCs from eqn. 23. In determining the validity of any theoretical formulation one must not forget that any matrix of membrane transport coefficients can only be experimentally evaluated for a range of concentration values, not concentration per se. The practical coefficients offer C, as a reasonable and n&Ural method of cataloging the particular range for which the coefficients are valid. In order to more clearly delineate the concentration state of the membrane the additional stipulations that AC& << 1 and Fs NCC,> seem like natural qualifiers. These experiments also indicate that perhaps (at least for cellulose membranes) artifactual concentration dependence will appear in transport experiments where & is allowed to vary during the experiment. Since all of the transport equations can be derived from (Lp, u, o , and G), it is clear that by maintaining Fs constant during any experiment, the transport coefficients of any reference frame will be independent of concentration during that experiment. However, if Cs alters significantly during any experiment, then most transport equations will behave non-linearly. This non-linear behavior could be interpreted as a violation of the linear assumption, but, in reality, it would be nothing more than particular experimental conditions producing the natural concentration dependence of the transport coefficients in that particular frame of reference. me choice of an experimental reference frame has been primarily the experimenter’s expediency. This research should clearly indicate that the choice of reference frame cannot be arbitrary. Because the experimental data must
Fig.8. ‘Transport coefficient rsw versus G from eqn. 23
be averaged over a range of concentrations, there undoubtedly will be one reference frame more suited to accurate averaging than any other. To find the “best fit” reference frame one should construct a table similar to Table 3, but where each coefficient matrix has been evaluated directly from the data, then choose the most accurate reference frame by locating the matrix with the smallest fractional reliability, Rf: _. The accuracy of representation within the selected reference frame will be given by the reliability coefficient, Rxi.This selection process can be easily automated with the computer programmed to determine the matrix with the best fit. Once the best fit reference frame has been determined, matrix transformations can be made to any other, perhaps more convenient set of transport equations. The accuracy of transformations of this kind is shown in Table 3. The most accurate transformation can be determined by choosing the minimum fractional reliability coefficient. The uncertainty value for each transport coefficient specifies a limited range of validity for that coefficient; some accuracy will be lost in matrix transformations where uncertainty regions overlap. Values determined here indicate that for reasonably precise work, this effect will not be a limiting factor. The concentration dependence of the transport coefficients reported here is real. This is tested by using the concentration-dependent coefficients evaluated from (Lp, u, o, and &) and the thermodynamic forces from the various reference frames to evaluate the conjugate fluxes. Even though some coefficients are highly concentration dependent, the reliability coefficients indicate that the accuracy of the concentration-dependent representation given
285
Fig.9.
Transport
coefficient
rww versus Cs from
eqn. 24.
here is the correct form. Interestingly, Figs.1 to 9 show that within a single matrix both very linear and very non-linear coefficients are generated. A close examination of these properties will be forthcoming. This research has tried to indicate the inadequacy in reporting only the accuracy or standard deviation of individual transport coefficients. The coefficient accuracy is not sufficient information to determine the overall accuracy of the matrix to describe the fluxes. Within each concentration range for which a matrix of coefficients has been evaluated, the precision of the fluxes will vary as the relative importance of the product of the transport coefficient and its thermodynamic force, i.e., _I&Apk ~ changes. The reliability coefficient and the fractional reliability coefficient depict the accuracy of the transport coefficients to reproduce the experimental fluxes. This includes both the error in the coefficient determination and the relative importance of the individual force-flux terms, &k Apk I in determining the actual flux. Use of the reliability and Ifractional reliability coefficients together with the experimental data representation of Table 1 will result in a far more understandable and less ambiguous cataloging of transport coefficients. Acknowledgement. This research was supported in part by Public Health Service Grant No. AM 19198 and contract No. l-AM-7-2206.
286
Table of symbols Area of membrane, cm’ Arithmetic mean concentration, mole cmm3. Transmembrane concentration difference, mole cmm3, Logarithmic mean concentration, mole cmm3. Diffusion flux, cm set-‘. Solute and water flux, respectively, mole set-’ cm-?. Volume flux through membrane, cm see-‘. Hydraulic permeability, cm set-’ atm.-‘. Total transmembrane chemical potential drop of the salt and water, respectively, cm3 atm mole-‘. Salt transport, mole. Solute permeability coefficient, mole set-’ cmW2atm-‘. Transmembrane hydrostatic pressure drop, atm. Transmembrane osmotic pressure drop, atm. F’ractional Reliability Coefficient, see eq. (6). Reliability Coefficient, see eq. (5). Reflection coefficient, dimensionless. Time, set Partial molar volume of solute and water respectively cm3 mole-’
(bs, Liw, Lww) (Rss,Raw, Rww) Ok, rsw, rww) (%I,
a12,
(Ll,
b2I
a219
a22)
L22)
m11,
J&t,
(-$a
LpD,
R22)
LD)
@p,
R,D>
ED)
See eqns. See eqns. See eqns. See eqns. See eqns. See eqns. See eqns. See eqns.
13 21 23 25 31 37 39 44
and and and and and and and and
14. 22. 24. 26. 32. 38. 40. 45.
References 1 2 3 4 5 6 7 a 9
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