Analysis and Prediction of Sieving Curves for Ultrafiltration Membranes A Universal Correlation

ISSN: 0149-6395 (Print) 1520-5754 (Online) Journal homepage: http://www.tandfonline.com/loi/lsst20Analysis and Prediction of Sieving Curvesfor Ultrafiltration Membranes: A UniversalCorrelation?

Alan S. Michaels

To cite this article: Alan S. Michaels (1980) Analysis and Prediction of Sieving Curves for

Ultrafiltration Membranes: A Universal Correlation?, Separation Science and Technology, 15:6,1305-1322, DOI: 10.1080/01496398008068507To link to this article: http://dx.doi.org/10.1080/01496398008068507Published online: 05 Dec 2006.

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SEPARATION SCIENCE AND TECHNOLOGY, 15(6), pp. 1305-1322, 1980 Analysis and Prediction of Sieving Curves for Ultrafiltration Membranes: A Universal Correlation? Downloaded by [The UC Irvine Libraries] at 08:21 27 January 2016 ALAN S. MICHAELS DEPARTMENT OF CHEMICAL ENGINEERING STANFORD UNIVERSITY STANFORD, CALIFORNIA 94305 Abstract Sieving curves [variations in sieving coefficient (6) with Einstein-Stokes radius (a) of the permeating macromolecule] of a number of synthetic ultra- filtration membranes, and of a variety of mammalian glomerular membranes studied in vivo, conform surprisingly closely to a log-normal-probability relationship between 6 and a which allows determination of the complete sieving curve from experimental measurement of only two sieving coefficients for two macrosolutes of differing ESR. Even more striking is the finding that, for all membranes examined, the value of a corresponding to 6 = 0.5 (the inflection point in the sieving curve) varies only between 17 and 34& and geometric standard deviation about the mean macrosolute radius h), which is inversely related to the “sharpness” of the sieving curve, lies between 1.2 and 1.7. It is concluded that not only is the log-probability correlation a reason- able and convenient means for interpreting and predicting membrane sieving data, but that most natural and synthetic ultrafiltration membranes have very closely related matrix morphologies. In 1976, Green et al. (I) reported data on the sieving coefficients of two hemodialysis membranes [Cuprophan and Rhone-Poulenc’s RP 69 poly(acrylonitri1e) membrane] when operated as ultrafilters, and found that the sieving coefficients for the two membranes varied with the Einstein-Stokes radius (ESR) of the permeating macrosolutes in a manner which yielded straight (and parallel) lines on normal probability coordi- nates. While their mathematical representation of the consequence of this correlation was in error, and their inferences about the relations between I305 Copyright 0 1980 by Marcel Dekker, Inc. I306 MICHAELS Downloaded by [The UC Irvine Libraries] at 08:21 27 January 2016 a sieving curve and a membrane pore-size distribution incorrect, their observations seemed to us to be sufficiently provocative to merit closer scrutiny as a basis for correlation of the sieving (rejection) characteristics of a variety of natural and synthetic ultrafiltration membranes. There is a large body of data in the literature describing the macrosolute sieving behavior of commercially-available polymeric ultrafiltration and dialysis membranes, and of mammalian glomerular membranes. The most commonly employed “probe” solutes for these measurements are mono- disperse purified proteins, and polydisperse, linear, water-soluble macro- molecules exemplified by the dextrans and their derivatives, poly(viny1 pyrrolidone) and poly(ethy1ene glycol). Dextran has been extensively employed in human and animal kidney-clearance studies because of its nontoxicity, stability, availability in widely differing molecular weight ranges, and relative ease of analysis. The ESR is the traditional parameter used to characterize a macrosolute in ultrafiltration studies; it is the “apparent equivalent spherical radius” of the macromolecule as computed from the measured diffusivity of the molecule in free solution by using the Stokes-Einstein equation: D = kT/6qa where D is the measured diffusion coefficient, is the solvent viscosity, and a is the ESR. The ESR has unequivocal physical meaning only for a rigid, spherical particle in a fluid continuum, and its significance for asymmetric, solvated, free-draining, or compliant-chain macromolecules is ambiguous. For polydisperse polymer mixtures, gel-permeation chromatography has become well established as a technique for determining molecular size and size-distribution. For aqueous-phase GPC, it is customary to calibrate the chromatographic column with a series of monodisperse proteins of known ESR, and to ascribe to the chromatographic fractions eluted from the column the same ESR as that of a protein displaced at the corresponding elution-volume. ESR values determined by GPC are, therefore, empirical characterization parameters related in some complex way to “effective molecular size.” For a homologous series of polymer molecules, however, the ESRs determined by GPC undoubtedly correlate monotonically with the “true” molecular radii important in determining ultrafiltrative rejection, but to equate such a parameter to a genuine molecular dimension would be ill-advised. The sieving coefficient, 0, of an ultrafiltration membrane for a particular solute is the fraction of that solute present in the solution upstream of the membrane which is delivered in the ultrafiltrate. This coefficient (or its complement, the rejection coefficient R = 1 - 0) varies between zero SIEVING CURVES FOR ULTRAFILTRATION MEMBRANES I307 Downloaded by [The UC Irvine Libraries] at 08:21 27 January 2016 and unity with changing solute “molecular size” to yield a characteristic S-shaped “sieving curve.” The similarity of appearance of these sieving curves to cumulative-particle-size-distribution curves for particulate solids has undoubtedly been responsible for suggesting a probabilistic approach to correlating membrane rejection data. The use of the normal probability (Gaussian distribution) function to correlate sieving coefficients with ESR, as proposed and tested by Green et al., is functionally inadmissible, since this distribution function is finite and symmetrical about all positive and negative values of its argument. Since negative values of molecular radius (or pore radius) are impossible, a more acceptable and reasonable correlation-basis is the log-normal distribution function; the cumulative probability function is then given by @ = 1 - erf (2) (1) where x is the value of a particular population in the distribution, K is the geometric mean, and og is the geometric standard deviation about the mean. If the sieving coefficient 8 correlates with Einstein-Stokes radius a via the log-normal probability function, then the analytical relationship between 8 and a will be and a is the ESR of the permeating molecule, Z is the ESR of the “mean” molecule (for which 8 = OS), and o, is the geometric standard deviation about the mean ESR. On log-probability coordinates, Eqs. (4) and (5) linearize in the form qe) = A + Briog whence, if F(0.5) = 0, loga log ci F(e) = - - = T logo, logo, (6) - (7) I308 MICHAELS Downloaded by [The UC Irvine Libraries] at 08:21 27 January 2016 Thus, when log (a/ii)/log na = 1.0 (or a/C = no), 8 = 0.159; hence no can be determined from the ratio of a at 8 = 0.159 (or 8 = 0.841) and 5 at 8 = 0.5. Sieving curves which linearize on log-probability coordinates, therefore, are completely defined by two characteristic constants: (1) the “mean ESR,” a, for which 8 = 0.5; and (2) the geometric standard devia- tion, no. Evidence of the validity of this basis for correlating the sieving coefficients as functions of solute molecular weight for ultrafiltration membranes has recently been provided by Cooper (2) for a polydisperse neutral dextran and a series of asymmetric hollow-fiber membranes. Since the ESR is expected to be a simple power function of molecular weight for a homologous series of macromolecules, a log-normal relation between sieving coefficient and ESR of the form of Eq. (7) is expected for this system. We have reexamined a number of recent publications reporting experi- mentally determined values of the ultrafiltration sieving coefficients (and their molecular-size dependency) for the glomerular membranes of the kidneys of the rat, the dog, and man, wherein the probe macrosolutes studied have included neutral dextrans, dextran sulfate, diethylaminoethyl dextran, Ficoll (an epichlorhydrin crosslinked polymer of sucrose), and poly(viny1 pyrrolidone) for the purpose of testing the applicability of this log-normal probability correlation. In addition, published sieving- coefficient data for several synthetic polymeric membranes, as determined with dextrans, PVP, and other probe macrosolutes of known ESR, have been similarly evaluated. The results of this exercise are summarized below. GROUP I: NORMAL MAMMALIAN GLOMERULI; UNCHARGED MACROSOLUTES Table 1 and Fig. 1 present sieving coefficient data for the normal TABLE 1 Sieving Curves for Normal Mammalian Curve no. Species 1 2 3 4 5 6 7 Source Probe solute Chang (9) Dextran Chang (3) Dextran Bohrer (4) Dextran Bohrer (5) Ficoll Lambert (6) PVP Verniory (7) PVP Myers (8) Dextran 18 20 22 24 0.82 0.92 0.73 0.86 0.50 0.74 0:80 26 0.69 0.83 0.60 0.74 0.39 0.63 0.74 Rat Rat Rat Rat Dog Dog Man 0.99 0.96 0.92 0.99 0.99 0.97 1.00 0.97 0.87 0.96 0.96 0.93 -_ 0.68 __0.82 - 0.97 0.90 SIEVING CURVES FOR U LTRAFl LTRATION MEMBRANES I309 Downloaded by [The UC Irvine Libraries] at 08:21 27 January 2016 (Wistar) rat glomerulus with respect to neutral dextrans [Chang (.?), Bohrer (41 and Ficoll [Bohrer (5)]; for the normal dog glomerulus to polydisperse Verniory (7)]; and for the normal poly(viny1 pyrrolidone) [Lambert (6), human glomerulus to dextrans [Myers (S)]. As will be seen from Fig. 1, the log-normal probability relationship against ESR is closely obeyed for all the data sets. Linear regression analysis of the data has been performed [recognizing that F(0) is the inverse error function] ; the appro- priate mean values of ii and a, are tabulated in Table 1 for each data set, along with the coefficient of correlation, rz. The results are significant on at least three counts: The coefficients of correlation for the seven independent data sets vary between a minimum of -0.97 and a maximum of >0.99. This, of course, indicates that the log-normal-probability relation- ship between sieving coefficient and macrosolute ESR is an extraordinarily accurate representation of the sieving curves for these membranes. The “mean macrosolute ESRs,” ii, for the glomeruli of the three species studied-corresponding to a 0 of 0.50-differ surprisingly little. In man, the value appears to be slightly over 30 A; for the rat, perhaps 28 to 29 A; and for the dog, around 25 A. The geometric standard deviation about the mean (a,) for all three species is very nearly identical, lying between 1.2 and 1.3. This quantity is a direct measure of the “sharpness” of the rejection spectrum of the membrane, indicating that, functionally, glom- erular membranes of these three mammalian species must be morphologically almost identical. Another rather interesting and provocative observation is the finding that the rat glomerular sieving curve with respect to polydisperse Ficoll Glomeruli and Neutral Macrosolutes e vs a, Einstein-Stokes radius (A) 28 0.56 0.69 0.45 0.60 30 32 0.33 0.42 0.22 0.28 0.14 0.30 0.44 34 0.23 0.30 0.15 0.16 0.09 0.22 0.34 36 0.15 0.19 0.09 0.09 0.06 0.15 0.26 38 0.08 0.11 0.045 0.04 - 0.10 0.20 40 d(A) - 42 a. r2 0.44 0.55 0.32 0.44 0.28. 0.23 0.50 0.40 0.62 0.55 0.05 0.03 28.9 0.06 0.03 30.4 27.4 0.02 28.0 0.02 24.5 26.1 30.3 0.15 1.22 1.20 1.20 1.20 1.29 1.30 1.30 0.999 0.987 0.996 0.969 0.995 0.998 0.986 1310 MICHAELS 0.005 0.01 Downloaded by [The UC Irvine Libraries] at 08:21 27 January 2016 18 I 20 22 I I I 24 I IIIII 26 28 30 3234363640 I I I EINSTEIN-STOKES RADIUS, 1 FIG. 1. Sieving curves for normal mammalian glomeruli and uncharged macrosolutes. (Numbers on lines correspond to entries in Table 1 ; points on graph are experimental data points corresponding to Line 1.) is substantially identical to that measured with polydisperse neutral dextran. Inasmuch as it is known that the conformation of Ficoll mole- cules in solution is substantially different from (more spheroidal than) that of dextran, this result might not have been expected. Since, however, the putative ESRs of both polymeric species were assigned on the basis of their elution times from the samc precalibrated Sephadex gel-permeation chromatographic column, one is tempted to postulate that the molecular conformational parameters which govern gel-permeation dynamics within hydrogels such as Sephadex are very closely related to those which SIEVING CURVES FOR ULTRAFILTRATION MEMBRANES 131 I govern macrosolute transport through ultrafiltration membranes. A comparison of the apparent molecular dimensions of dextran and Ficoll species displaying the same apparent ESR by GPC, using some other “size”-measuring technique such as laser-Doppler autocorrelation spectroscopy, is likely to clarify this important question. GROUP II: NORMAL VS NEPHROTIC RAT KIDNEYS; NEUTRAL VS IONOGENIC MACROSOLUTES Downloaded by [The UC Irvine Libraries] at 08:21 27 January 2016 In the absence of electrostatic interactions between a permeating macromolecule and an ultrafiltration membrane [a condition which should obtain if either membrane or macromolecule (or both) were uncharged], and if specific van der Waals force-interactions between membrane and macrosolute are not significant, then the principal determinant of the sieving coefficient should be the size and conformation of the penetrating molecule, and the morphology of the membrane matrix. This indeed seems to be borne out with “normal” glomeruli and neutral macromolecules. Since, however, the glomerular membrane is known to be normally negatively charged, and since pathological changes in the membrane are frequently accompanied by changes in membrane microstructure and charge density, use of this log-normal-probability-correlation to charac- terize the response of normal and pathological glomeruli to neutral, anionic, and cationic macromolecules was expected to be informative. Data are presented for polydisperse sodium dextran sulfate [Chang (9. ZO), Bohrer (41, and for polydisperse diethylaminoethyl dextran [Bohrer (I])] with normal rat glomeruli, and also for neutral and charged dextrans with rat glomeruli displaying pathologies induced by chemical treatments known to cause irreversible glomerulonephrosis. Results are summarized in Table 2 and Fig. 2. For data-sets 8, 9, and 10, which constitute sieving curves for ionically charged dextrans through the normal rat glomerulus, the log-probability correlation continues to describe the experimental data quite accurately, with coefficients of correlation in the range -0.99. It is noteworthy that, for dextran sulfate, the “mean” ESR, ii, is depressed to 20 A from nearly 30A for neutral dextran, whereas ii for DEAE dextran is elevated to about 34 A. Such a result is qualitatively consistent with electrostatic retardation of penetration of the negatively charged macromolecule through a negatively charged membrane, and enhancement of penetration of a positively charged macromolecule. The concomitant observation that a, values for charged solutes are also elevated above those observed for neutral dextran (1.25 to 1.29 vs 1.20) lends credibility to this hypothesis : a, is a measure of the “sharpness-of-cutoff’’ of the membrane (for a, = - - 1312 MICHAELS TABLE 2 Sieving Curves for Charged Macrosolutes and Curve no. Condition 8 9 10 11 12 13 14 15 16 17 Normal Normal Normal A-11\" A-I1 NSNb NSN NSN PAN' PAN Source Probe solute 18 0.74 0.56 1.00 1.00 0.74 0.90 0.90 0.83 0.88 20 22 0.42 0.19 0.97 0.95 0.37 0.81 0.81 0.72 0.69 0.90 24 0.29 0.11 0.93 0.85 0.25 0.73 0.75 0.66 0.56 0.80 26 0.19 0.06 0.87 0.74 0.16 0.63 0.66 0.58 0.43 0.74 Downloaded by [The UC Irvine Libraries] at 08:21 27 January 2016 Chang (3) Dextran sulfate Bohrer (4) Dextran sulfate Bohrer (11) DEAE dextran Bohrer (4) Dextran Bohrer (4) Dextran sulfate Bohrer (11) Dextran Bohrer (11) Dextran sulfate Bohrer (11) DEAE dextran Bohrer (4) Dextran Bohrer (4) Dextran sulfate 0.58 0.35 0.99 0.99 0.55 0.86 0.86 0.78 0.82 - 0.97 transglomerular leakage of albumin 'Treatment with angiotension 11. Causes irreversible bInduced nephrotoxic serum nephritis. 'Treatment with puromycin aminonucleoside. Causes lipoid nephrosis and disorganiza- 1.0, a membrane would display a step change in sieving coefficient from 0.0 to 1.0 for a macrosolute of ESR = a). An increase in u. indicates a reduction of membrane separation capacity on a basis of macrosolute size, and it is logical that electrostatic interactions should (if repulsive) tend to cause preferential retardation of transport of small molecules or (if attractive) cause relatively enhanced transport of large ones. If, however, the log-normal probability correlation for the sieving coefficient were to have some fundamental physical significance for describing the macrosolute rejection behavior of neutral membranes toward neutral macrosolutes, it seems unlikely that such a correlation would be equally applicable to membrane/macrosolute systems wherein electrostatic or electrokinetic phenomena are also involved in the sieving process. Accordingly, we do not believe there is any rational basis for ascribing a physicochemical explanation to the correlation; nonetheless, the correlation is an extremely good approximation to the experimental observations. Examination of the sieving coefficient/ESR data (Table 2, Fig. 2, Entries 11 to 17) for rat glomeruli subjected to chemical or biochemical insults known to induce irreversible changes in glomerular membrane-structure characteristic of glomerular nephroses (such changes resulting in partial collapse of the membrane-matrix and loss of much of the normal negative charge-functionality of the membrane) shows that the log-probability correlation continues to be an accurate representation of the sieving SIEVING CURVES FOR ULTRAFILTRATION MEMBRANES 1313 Normal and Pathologic Rat Glomeruli 8 vs a, Einstein-Stokes radius (A) 28 0.13 0.03 0.80 0.59 0.10 0.50 0.56 0.49 0.30 0.62 30 0.09 0.02 0.74 0.46 0.06 0.38 0.46 0.39 0.21 0.55 32 0.05 0.01 0.66 0.32 0.04 0.28 0.39 0.31 0.14 0.44 34 0.03 0.01 0.56 0.22 0.02 0.20 0.31 0.22 0.07 0.34 36 38 40 42 d(A) a. r2 Downloaded by [The UC Irvine Libraries] at 08:21 27 January 2016 0.015 0.01 - - 0.44 0.32 0.14 0.075 0.01 - 0.14 0.09 0.24 0.18 0.15 0.09 0.05 0.02 0.26 0.20 - 18.1 0.20 0.11 34.0 0.04 0.02 19.3 - 20.7 0.05 0.025 27.1 0.11 0.05 28.5 0.05 0.03 30.3 0.01 0.01 26.2 0.15 - 24.5 - - 21.0 1.29 1.28 1.25 1.19 1.25 1.29 1.35 1.27 1.34 1.26 0.998 0.989 0.987 0.998 0.992 0.980 0.973 0.986 0.960 0.996 and other plasma proteins. tion of the glomerular epithelium. curves (rZ values between 0.96 and 0.99). For neutral dextran, these membrane changes (Entries 11, 13, and 16) are accompanied by a reduction in ii and usually an increase in a,; this is an anticipated consequence of a shrinkage-induced reduction in “mean pore size” and a corresponding widening of the “pore size distribution.” For dextran sulfate these same membrane changes (Entries 12, 14, and 17) lead to an increase in a; this observation is consistent with a decline in negative-charge-density within the membrane, with consequent loss in Donnan-exclusion of polyanions. For DEAE dextran (a cationic polymer), the trend with glomerular membrane-damage (Entry 15) is toward reduction in ii, a result again consistent with loss in electrostatic augmentation of polycation transport with loss of membrane negative charge. It is thus apparent that the log- probability correlation, and the two characteristic parameters ii and a, which are derived from that correlation, can facilitate interpretation of sieving curves in terms of membrane matrix structural features. GROUP 111 : SY NTH ETlC ULTRAFILTRATION MEMBRANES The sieving coefficient data obtained by Green et al. (I) for Cuprophan and the Rhone-Poulenc AN 69 dialysis membrane (an asymmetric, microporous polyacrylonitrile structure) are reported in Table 3. These data also linearize quite well on log-probability coordinates (see Fig. 3), 1314 MICHAELS Downloaded by [The UC Irvine Libraries] at 08:21 27 January 2016 18 I 20 I 22 I 24 I 26 28 30 32 34 36 38 40 I I I IIIIII EINSTEIN- STOKES RADIUS, A FIG. 2. Sieving curves for normal and pathologic rat glomeruli, and un- charged and ionogenic dextrans. (Numbers on lines correspond to entries in Table 2; points on graph are experimental data points corresponding to Lines 8, 9, and 10.) SIEVING CURVES FOR ULTRAFILTRATION MEMBRANES 1315 Downloaded by [The UC Irvine Libraries] at 08:21 27 January 2016 TABLE 3 Correction of Sieving Coefficients Reported by DuBois (12) for Transport of Polydisperse PVP through XM-50 Membranes Calculated mass transfer coefficient,’ k (cm/sec x lo5) exp (JJR) 7.6 6.6 5.8 5.2 4.8 4.4 4.1 3.9 3.7 7.6 6.6 5.8 5.2 4.8 4.4 4.1 3.9 3.7 1.07 1.08 1.09 1.10 1.11 1.12 1.13 1.14 1.14 1.48 1.58 1.68 1.78 1.87 1.98 2.08 2.16 2.25 Measured True sieving sieving coefficient, coefficient, 0. e 0.96 0.86 0.79 0.66 0.46 0.24 0.12 0.06 0.03 0.82 0.59 0.45 0.38 0.20 0.16 0.06 0.04 0.02 0.96 0.85 0.78 0.64 0.43 0.22 0.11 0.05 0.03 0.75 0.48 0.33 0.26 0.12 0.09 0.03 0.02 0.01 Operating pressure ESR (cmH,O) (A) 5 16 20 24 28 32 36 40 44 48 16 20 24 28 32 36 (cm/sec) x lo5 0.5 J” 30 3.0 40 44 48 ‘Via method of Smith et al. (13). 1316 MICHAELS 0 Line 18 Line 19 A Line 20 A Line 21 12 16 20 24 28 32 36 404448 EINSTEIN-STOKES RADIUS, A FIG. 3. Sieving curves for synthetic ultrafiltration membranes. (Numbers on lines correspond to entries in Table 3.) Downloaded by [The UC Irvine Libraries] at 08:21 27 January 2016 SIEVING CURVES FOR ULTRAFILTRATION MEMBRANES 1317 yielding correlation coefficients between 0.97 and 0.98. The slopes of the lines for these two membranes are nearly identical (a, z 1.5), although the values of ii (corresponding to 8 = 0.5) differ substantially. DuBois et al. (12) measured sieving coefficients, using a polydisperse polyvinyl pyrrolidone, for several synthetic, asymmetric ultrafiltration membranes (Diaflo) manufactured by Amicon Corp. Their experiments were conducted in a stirred, batch ultrafiltration cell which the authors assumed provided sufficient agitation to eliminate concentration polariza- tion; our computations, however, indicate that polarization may have been significant in most of their determinations. Their sieving data obtained with XM-50 membranes at low pressures were amenable to correction for polarization using the mass-transfer-coefficient correlation and for stirred cells developed by Smith et al. (13) and Colton et al. (14, the estimated trans-membrane water flux (J,), to compute the Sherwood Number (J,/k) for each PVP molecular radius. The results of these computations are summarized in Table 4 where the measured and “true” sieving coefficients are tabulated as functions of ESR for two different operating pressures. As expected, the polarization correction is rather small for the low-pressure (lower flux) run, but is significant for the higher-flux run. As shown in Table 3 and Fig. 3, the corrected sieving curves obey the log-probability correlation quite well (r2 between 0.97 and 0.99). The observed decrease in slope (or increase in a,) and ii with increasing J, is consistent with the Kedem-Spiegler theory of coupled solute/solvent transport through ultrafiltration and hyperfiltration With increasing water-flux, the relative diffusive contribution membranes : of solute-leakage to total solute transport (most marked for smaller solute molecules) is reduced, rendering the sieving curve flatter, and lowering the value of ii corresponding to 8 = 0.5. Finally, we have taken the sieving data reported by Cooper et al. (2) for polydisperse dextran through Amicon PM-5 and PM-10 hollow fiber ultrafiltration membranes (which relate sieving coefficients to dextran molecular weight), computed the ESR values for the various molecular weight dextrans using the correlation proposal by Granath (15,16) based on dextran diffusion measurements: u = 0.33(M)0.463 Downloaded by [The UC Irvine Libraries] at 08:21 27 January 2016 (8) and retabulated and replotted their data with 8 a function of a. As shown in Table 3, the log-probability correlation is a very accurate representation of the function (r2 2 0.99) for both membranes. Interestingly, the PM-5 (nominal 5000 MW cutoff) and PM-10 (nominal 10,000 MW cutoff) display nearly identical values of 5, but the former has a smaller a, than the latter. Thus the PM-5 membrane may be characterized as having a MICHAELS TABLE 4 Sieving Curves for Membrane water permeability [mL/(min) Equip- AP (cm2) Probe curve no. Membrane ment (cmH20) (cmH,O)] solute 18 Source - 12 Downloaded by [The UC Irvine Libraries] at 08:21 27 January 2016 19 20 21 22 23 Cuprophan Dialysis 2,100 cell Dialysis 1,033 RP 69 cell Amicon Stirred UF 5 cell XM-50 Amicon StirredUF 30 XM-50 cell Amicon PM-10 Amicon PM-5 Hollow fiber module Hollow fiber module ? ? 2.8 x Various Green (I) 0.72 species 3.7 x 10-s Various Green (I) 0.97 species 6 x PVP DuBois (12)” 6 x lo-’ PVP DuBois (12)” Neutral Cooper (2) dextran Neutral Cooper (2) dextran Corrected via method of Smith et al. (13). “narrower pore-size distribution” than does the PM-10, although the “mean pore size” of the two membranes is nearly the same. These observa- tions emphasize the inadequacy of a singre parameter (such as the “molecular weight cutoff”) as a useful descriptor of the sieving charac- teristics of an ultrafiltration membrane. GE N ERAL OBSERVATIONS Perhaps the most surprising and astonishing observation to be drawn from the foregoing analysis is not the apparent universality of the log- probability correlation to describe ultrafiltration membrane sieving curves (which is surely operationally convenient, but of doubtful theoretical significance), but the extraordinary similarity of all the sieving curves. Consider that the data analyzed include (1) normal and diseased glomerular membranes of three mammalian species; (2) synthetic, sub- and (3) synthetic, polymeric, stantially homogeneous, dialysis membranes ; asymmetric ultrafiltration membranes, the range of ii values falls between 17 and 34 A, and of a, only between 1.2 and 1.7. This suggests that virtually all membrane ultrafilters, irrespective of their origin, are quite similar in their microstructure. On balance, glomerular membranes display signifi- values lie between 1.2 and 1.3) cantly narrower pore-size distributions (a,, SIEVING CURVES FOR U LTRAFl LTRATION MEMBRANES 1319 Synthetic Membranes 0 vs a, Einstein-Stokes radius (A) 16 0.55 20 0.40 24 0.26 0.70 28 0.14 0.55 32 0.09 0.43 36 0.03 0.25 40 0.01 44 - 48 ci(& rz - - 17.0 1.50 0.968 27.7 1.49 0.978 Downloaded by [The UC Irvine Libraries] at 08:21 27 January 2016 0.92 0.83 0.15 0.08 (0.96) (0.86) (0.79) (0.66) (0.46) (0.24) (0.12) (0.06) (0.03) 0.96 0.85 0.78 0.64 0.43 0.22 0.11 0.05 0.02 28.2 1.33 0.974 (0.82) (0.59) (0.45) (0.38) (0.20) (0.16) (0.06) (0.04) (0.02) 0.75 0.48 0.33 0.26 0.12 0.09 0.03 0.02 0.01 20.5 1.45 0.987 MW ESR, A MW ESR,A 6 e 7,000 11,000 18,000 28,000 43,000 19.8 24.5 30.8 37.8 46.1 0.47 0.26 0.14 0.07 0.04 3,200 4,800 7,500 12,000 18,000 28,000 13.8 16.7 20.5 25.5 30.8 37.8 0.74 0.53 0.38 0.16 0.08 0.03 18.3 1.66 0.988 17.6 1.49 0.997 than do synthetic ultrafilters (0, values between 1.3 and 1.7), although their “mean pore size” (represented by ii) may span as broad a range as those of synthetic polymeric membranes. Reexamination of Eqs. (4) and (5) indicates that the “universal” sieving coefficient vs solute ESR correlation can be reduced to dimensionless form by defining 8 in terms of a “reduced ESR” (a/ii) and the (already dimensionless) geometric standard deviation about the mean ESR, 0,. By plotting 8 against a/Z for various parametric values of b,, a family of generalized sieving curves can be generated upon which experimentally determined sieving coefficients for a specific membrane can be located to obtain complete sieving curves for that membrane. Such a generalized correlation is presented in Fig. 4. If, for a given membrane, one has determined the sieving coefficients for two macrosolutes of differing ESR, the ratio of the values of a corresponding to those two sieving coefficients uniquely determines the valu of o,, and thus also of ii, for that membrane. For example, if a membrane displays a sieving coefficient of 0.75 for a solute of ESR = 21 A, and 0.35 for a solute of ESR = 30 A, these points fall on the line for o, 1.4, and lead to a value of ii E 26 A. Thus Fig. 4 can be used to estimate the entire sieving curve for a membrane from a knowledge of only two sieving coefficients for two different macrosolutes, without recourse to algebraic manipulation of Eqs. (4)-(7). I 320 MICHAELS I 0.OoI I I tI I I I I I1111 -1 o.m[ 0.0 I a - Downloaded by [The UC Irvine Libraries] at 08:21 27 January 2016 O.0.10 7 I I 1 t LI IIIIIII 1.1 1.2 1.3 1.4 1.5 1.6 18 20 I FIG. 4. Generalized sieving curves for membranes of varying a,, values. (Dotted lines illustrate means for determining entire sieving curve from two arbitrary data-points: e.g., 0 = 0.35 when a = 30A; 0 = 0.75 when a = 21 A. In this case, the sieving curve lies on the line corresponding to na = 1.4.) It is concluded that the log-normal probability function is a generally accurate means for describing sieving curves for ultrafiltration membranes, and can be used quite reliably to construct a complete sieving curve for a given membrane from only two experimental sieving-coefficient values for two differing solutes of known ESR. Of particular note is the observa- tion that a wide variety of synthetic and biological membranes falls within a quite narrow range of apparent mean pore size and size- distribution, suggesting that all such membranes have very similar morphologies. Further experimental tests of this correlation, via experi- SIEVING CURVES FOR ULTRAFILTRATION MEMBRANES 1321 mental determination of sieving coefficients of a number of synthetic ultrafiltration membranes toward a variety of macrosolutes of known Einstein-Stokes radii, are currently in progress and will be the subject of a subsequent paper. SYMBOLS Downloaded by [The UC Irvine Libraries] at 08:21 27 January 2016 Einstein-Stokes radius, A, lo-* cm geometric mean ESR, A, lo-* cm diffusivity, cmZ/sec Boltzman constant, ergs/(molecule) (“K) molecular weight of a dextran fraction K absolute temperature, Oarbitrary value of a population of a distribution geometric mean value of the distribution generalized argument of the error function, dimensionless solvent viscosity, dyn-sec/cm2 sieving coefficient, dimensionless inverse error function of 8, dimensionless geometric standard deviation of x, dimensionless geometric standard deviation of a, dimensionless argument of the error function in terms of a, Z, and ua, dimensionless Acknowledgments This study was conducted under the auspices and support of the Office of Water Research and Technology, U.S. Department of the Interior, and the Satellite Dialysis Center of San Jose, California. REFERENCES I. D. M. Green, G. D. Antwiler, J. W. Mancrief, J. F. Decherd, and R. P. 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