ELSEVIER Journal of Membrane Science 118 (1996) 199-212
Asymmetric solute transport and solvent flux in dual-skinned hollow fiber membranes
Paul J. Soltys a,*, Norma J. Ofsthun b, Andrew L. Zydney c a Renal Division, Baxter Healthcare Corp., 1620 Waukegan Road, Mail code MPR-D1, McGaw Park, IL 60085, USA
b Corporate Research, Baxter Healthcare Corp., Route 120 and Wilson Road, Mail code RLT-12, Round Lake, IL 60073, USA c 227 Colburn Laboratory, Department of Chemical Engineering, University of Delaware, Newark, DE 19716, USA
Received 14 December 1995; revised 24 January 1996; accepted 20 March 1996
Abstract
Dual-skinned hollow fiber membranes were produced with thin skins on both the inner (lumenal) and outer surfaces of an annular macroporous matrix. These dual-skinned membranes demonstrated a clear directional selectivity with sieving coefficients that were dramatically different in the two flow directions (shell-to-lumen and lumen-to-shell). Unlike the incidental directional selectivity previously reported with single-skinned reverse osmosis membranes, sieving coefficients in both directions are controllable by varying the properties of the two skin layers. The directional sieving behavior of these membranes is a result of the directional nature of the convective solute transport across a membrane having two skin layers with different effective pore sizes. Internal concentration polarization significantly increases solute sieving coefficients when flow occurs through the more open skin layer first, but is largely absent when the flow is in the other direction. The internal concentration polarization also has a strong effect on the solvent flux through these membranes causing large directional differences in flux during filtration of a dilute macromolecular solution. The unique transport characteristics of these dual-skinned membranes thus provide an extraordinarily powerful tool for the design and development of novel membrane devices and processes that exploit the directional selectivity of these new membrane structures.
ment of various metabolic or immunologic disorders, the immobilization of enzymes or cells in membrane bioreactors, and the immunoisolation of transplanted cells in hybrid artificial organs. Membranes are at- tractive for such a broad range of applications be- cause it is possible to tailor the membrane pore size, and thus control the membrane selectivity, to obtain the desired rates of solute transport for the different molecular weight species that determine the overall performance of the given system.
200 P.J. Soltys et al. / Journal of Membrane Science 118 (1996) 199-212
However, in a number of these applications, it would actually be desirable to employ membranes which also provide a "directional" selectivity, i.e. a greater rate of solute transport across the membrane in one direction than in the other. For example, an ideal hemodialysis membrane would provide rapid removal of both small and middle molecular weight solutes from the blood, but would not allow any significant transport of similarly-sized contaminants (e.g. endotoxin fragments) from the dialysate back across the membrane and into the bloodstream [1]. Likewise, the performance of many membrane biore- actors and artificial organs could be significantly enhanced if the rate of nutrient transport from the feed solution to the immobilized cells could be inde- pendently controlled relative to the rate of product (or by-product) transport back across the membrane.
One way of achieving this directional selectivity is to employ a membrane with a multi-layer struc- ture, with each layer having its own (different) pore size. The unusual transport characteristics of these multi-layer membrane structures was first described over 30 years ago in the context of biological mem- branes by Curran [2], Durbin [3], Katchalsky and Kedem [4] and Patlak et al. [5]. The effects of such a multi-layer structure on the performance of reverse osmosis membranes has been discussed by Jonsson [6,7]. However, there has been no effort to date to exploit the unique properties of these multi-layer structures through the development of novel syn- thetic membranes with desired directional selectivi- ties.
The overall objective of this study was to produce and characterize new polymeric hollow fiber mem- branes that would have significant directional selec- tivity. These membranes were designed to have se- lective skin layers on both the inner and outer sur- faces of a porous (annular) matrix, with the transport characteristics of these dual-skinned membranes de- termined by independently controlling the pore size of these two skin layers. We first present a theoreti- cal analysis of the transport characteristics of these dual skinned hollow fiber membranes, including a brief discussion of the origin of their directional selectivity. A general description of the methods used to fabricate these dual skinned membranes is then presented along with scanning electron micro- graphs of the resulting membrane structure. Experi-
mental data for dextran transport and solvent flow were obtained for several of these dual skinned membranes, including membranes with tight inner skins as well as those with tight outer skins. All of these fibers exhibited a distinct directional selectiv- ity, with the results in good agreement with predic- tions from the theoretical model for solute transport through these dual skinned membranes.
2. Theoretical analysis
A schematic representation of the dual skinned hollow fiber membrane is shown in Fig. 1. The lumenal (or inner) radius of the fiber is R L, and the outer radius is R o. The bulk of the membrane struc- ture is occupied by the macroporous support matrix, with thin skin layers located on both the inner and outer surfaces of this annular matrix. Our theoretical analysis of solute transport across this multi-layer" membrane structure is analogous to that used by Patlak et al. [5] and Jonsson [6,7], although these earlier analyses were performed for flat sheet mem- branes and thus neglected the effects of membrane curvature in the hollow fiber system.
The local solute flux (N i) through a given region i of the multi-layer membrane is expressed as the sum of the convective and diffusive contributions [8]:
oci Ni = V iK c iC i -D ~K di Or (1)
where V i is the local filtration velocity, C i is the local solute concentration, Do is the free solution diffusion coefficient, and r is the radial position in the hollow fiber geometry. The coefficients Kci and Kdi represent the hindrance factors for convective
lumen axis r=O
fiber lumen / r = R L inner skin
] r Porous Support Matrix I / r = R 0 - ~2
outer skin \
Outer shell " r = R o
F i g . ] . S c h e m a t i c d i a g r a m o f d u a l - s k i n n e d h o l l o w f i b e r m e m -
P.J. Soltys et al. / Journal of Membrane Science 118 (1996) 199-212 201
and diffusive transport in region i, respectively. These hindrance factors arise from the additional drag on the solute molecules due to the presence of the pore walls, and are thus functions of the solute to pore size ratio as well as any long range (e.g. electro- static) interactions between the solute and the pore wall (see Appendix).
Eq. (1) can be integrated over each (homoge- neous) region of the membrane using the continuity equations for the filtration velocity and solute flux in the hollow fiber geometry:
1 0 -- - - (rVi) = 0 , r ar
1 0 r Or (rNi) = 0 (2)
yielding:
RL VKci ln(VKciC i - N ) ~ In ( r ) + A 1 (3)
where V and N are the filtration velocity and solute flux in the bulk solution at the lumen-membrane interface ( r = RL). Note that in developing Eq. (3) we have implicitly assumed that the system is at quasi-steady state and that there is no chemical reaction (or adsorption) of the solute of interest. The integration constant A 1 is evaluated by matching the flux expressions in the different regions using the appropriate continuity constraints:
= < + , v i + , ,
EiNi = Ei+ iNi+l (4)
and an expression for the thermodynamic (equi- librium) partitioning of the solute between adjacent regions of the membrane:
4)i+1 Ci+l - - = - - ( 5 )
4'i Ci
where e i is the membrane porosity and thi is the equilibrium partition coefficient for region i. Eq. (5) can also be used to describe solute partitioning be- tween the external solutions and the inner and outer skins by setting the partition coefficient in the free (external) solution equal to one. Eqs. (4) and (5) are
substituted into Eq. (3), which is then solved explic- itly for the solute flux yielding:
N = VS~lS~2S~3( 0/10/2 ~3CL -- Co)
/[S~lgm2(o' 3 - 1 ) -['- SmlS~3( 0/2 - 1)0/3
+ S~2 S~3( a , - 1) a 2 a3] (6)
where C L and C O are the lumen and exterior solute concentrations, respectively. S< is the asymptotic sieving coefficient for region i [9]:
S< = ~i Kci (7)
which is the ratio of the concentration at the down- stream surface of region i to the concentration at the upstream surface of region i in the limit as V ap- proaches infinity. The parameter 0/~ provides a mea- sure of the relative importance of convection and diffusion:
= ( Ri+ l ) Pei 0/i R i ( 8 )
where R i is the radius of region i and Pei is the corresponding membrane Peclet number:
RL V] S~i
The solute flux is thus determined by both the geometry (R i) and the transport properties (S< and ei (~i Kdi) of the three regions of the membrane.
Eq. (6) is clearly asymmetric with respect to the properties of the two skin layers, i.e. the expression for the solute flux varies upon interchanging the properties of regions 1 and 3. This asymmetry is not simply due to the curvature of the hollow fiber system, although the change in cross-sectional area with increasing r does affect the final result. Instead, this asymmetry arises because of the vectorial nature of the convective solute transport, with the direc- tional character of the flow causing the detailed solute concentration profiles to be a strong function of the order in which the fluid passes through the two skins. For example, if the fluid passes through the tighter of the two skin layers first, the solute concentration throughout the hollow fiber membrane will be relatively low. This is in sharp contrast to the behavior that would be seen if the flow were in the
202 P.J. Soltys et a l . / Journal of Membrane Science 118 (1996) 199-212
reverse direction (i.e. if the fluid were to pass through the more open skin first). This effect can be most easily seen by examining the behavior of Eq. (6) at large (absolute) values of the filtration velocity, i.e. under conditions where the solute flux is governed by convection. For high positive filtration velocities (high flow rates out from the lumen), the solute flux across the dual-skinned membrane reduces to:
N -- S~lVC L (10)
Thus under these conditions, the flux is governed entirely by the sieving characteristics of the inner skin (region 1), irrespective of the properties of the rest of the membrane (as long as the matrix and outer skin are at least partially permeable to the solute of interest). In the limit of large negative filtration velocities (high flow rates back into the lumen), the Peclet numbers all approach negative infinity and the solute flux depends only on the sieving coefficient of the outer skin (region 3):
N = S~3VC o (11)
Note that even under conditions where C L = C o, the rate of solute transport out from the lumen for a given (absolute) value of the filtration velocity will be different than that in the reverse direction unless the asymptotic sieving coefficients of the two skin layers are equal.
The actual sieving coefficient of the dual-skinned membrane, defined as the ratio of the concentration at the downstream wall of the membrane to the concentration at the upstream wall, can be evaluated from an overall solute mass balance on the solution immediately downstream of the multi-layer mem- brane, yielding:
N C e = - - (12)
V
where Cf is the filtrate concentration (more specifi- cally, Cf = C o when the flow is from lumen to shell and Cf = C L when the flow is from shell to lumen). Substitution of Eq. (12) into Eq. (9) gives, upon rearrangement:
Sa + = S~l 8~2 8~3 o/113/213/3//[ S~l 8~2 8~3
+ & 2 & 3 ~ 2 1 3 ( ~ , - 1)
+S~l&2(~3- 1) + &,S~3~3(~2- 0 ] (13)
S a =- S~I S~2S~3/ / [S~1S~2S~3 ol 1 ol 2 ol 3
- &2&3~2c~3(~, - 1)
-- S~1S~2 ( Og 3 -- 1) - SmlS~e3 0¢3( ~2 -- l ) ]
(14)
where S + is the actual sieving coefficient when flow is from lumen to shell (Sa + = Co/CL) , and S~ is the actual sieving coefficient when flow is from shell to lumen (S 5 = CL/Co) . Note that in the limit of zero filtration velocity (corresponding to c~ i --* 1), Eqs. (13) and (14) both predict that the actual sieving coefficient is equal to one. Under these conditions, solute diffusion tends to equalize the solute concen- trations on the two sides of the membrane, causing S a ~ 1 at very low filtration velocities. The actual membrane sieving coefficient decreases with increas- ing filtration velocity, attaining its asymptotic value (S< when V is positive and S~ 3 when V is negative) at very large absolute values of V. These effects are discussed in more detail subsequently.
3. Experimental
3.1. Materials
Membranes were formed from polysulfone (Udel 1700, purchased from Union Carbide, Danbury, CT) and polyetherimide (Ultem 1000 purchased from General Electric, Pittsfield, MA). Additional polysul- fone hollow fibers were fabricated per our specifica- tions by A / G Technology (Needham, MA). Epoxy resin, obtained from Devcon (Wood Dale, IL), was used to pot the fibers in small-scale cartridges. Ace- tone, isopropyl alcohol (IPA), and N-methylpyrroli- done (NMP) were obtained from Burdick and Jack- son (Muskegon, MI). Dextran 4 and dextran FPI were purchased from Crescent Chemical (Hauppage, NY); dextran T40 and T10 were from Pharmacia LKB (Uppsala, Sweden).
3.2. Membrane Fabrication
The three types of hollow fibers fabricated in- house (designated polysulfone A, polyetherimide X, and polyetherimide Y) were formed using a phase inversion process. In each case, the polymer (20-25%
P.J. Soltys et al. /Journal of Membrane Science 118 (1996) 199-212 203
by weight) was dissolved in a solvent (0-10% ace- tone in NMP) and extruded through an annular die. Development of the appropriate skin structure was controlled by adjusting the composition of the core solution (0-50% IPA or NMP in water) and by varying the die height above the bath. (The mecha- nisms by which these parameters affect membrane structure, e.g. through differences in viscosity, diffu- sivity, and solubility, have been examined in the literature [10] and will not be discussed here.) The extruded fibers were directed through a number of water baths, collected with a multi-segment winder, and allowed to air dry.
To prepare specimens for scanning electron mi- croscopy (SEM), hollow fibers were frozen in liquid nitrogen and cut with a disposable microtome blade either perpendicular to the fiber axis (cross-sectional views) or longitudinally (radial views). The samples were then mounted on aluminum stubs and sputtered with platinum (70 A) in a VWR ion sputterer (VWR, San Francisco, CA). The specimens were examined using a JEOL 6300 low voltage scanning electron microscope (JEOL USA, Peabody, MA) operated at 2 kV and a magnification of 250-100000 X.
Approximately 100 dried polysulfone fibers were carefully selected, cut to length, and sealed in a cylindrical polycarbonate device using an epoxy resin. Approximately ! cm of the fiber length at each end of the device was encased in epoxy. Because the size of the different fibers varied, the number of fibers and resulting effective surface area of the devices varied as listed in Table 1. The device was 22 cm in length and 0.6 cm in diameter. Fluid ports (0.25 cm in diameter) were located 2 cm from each end. Polycarbonate end caps, also having 0.25 cm diameter fluid ports, were sealed to each end of the device with hot melt adhesive.
The fluid circuit for the transport experiments
consisted of a bulk solution reservoir, a peristaltic pump (Flo-gard ~ 6300, Baxter Healthcare, Deer- field, IL), the test device, and appropriate tubing. Transmembrane pressure was adjusted by flow re- striction distal to the test device. Pressure transducers (PPM-15, Ohmic Instruments Co., Easton, MD) were located anterior and posterior to the device. An additional pressure transducer was used to monitor the filtrate pressure for experiments in which the filtrate exit was not level with the test system.
For filtration in the lumen-to-shell direction, fluid from the reservoir was pumped directly into the fiber lumen at a flow rate of 3-5 ml /min. A small fraction was removed as filtrate. For filtration in the shell-to-lumen direction, fluid from the reservoir was directed into the shell space around the fibers through one of the integral side ports present in the device. Filtrate was collected out the lumen ports, with the retentate collected from the second side port. Filtrate and retentate samples were collected at specified time intervals. All experiments were single pass.
Transport measurements were made using a mix- ture of polydisperse dextrans containing: 1.0 g/1 Dextran 4, 1.0 g/1 Dextran T40, 0.3 g/1 Dextran T10, and 0.2 g/1 Dextran FP1 in a phosphate buffered saline solution. Filtrate flux was determined gravi- metrically using timed collection. All experiments were performed at ambient temperature.
Dextran concentrations were determined by high performance liquid chromatography using the proce- dure developed by Frigon et al. [11]. The column was a G3000PWXL TSK Gel (Phenomenex, Tor- rance, CA). The chromatography system consisted of a model 600E solvent delivery system, model 712 WISP sample injector, and model 410 refractive index detector (Waters Associates, Milford, MA). The mobile phase was phosphate buffered saline (Sigma Chemical Co., Saint Louis, MO) at a flow
Table 1 Dimensions of hollow fiber modules
Module type Fiber size
i.d. ( /zm) o.d. (/xm)
Device dimensions
Number of fibers Surface area (cm 2)
Polysulfone A 203 290 100 126 Polysulfone B 160 240 100 100 Polyetberimide X 2500 2960 4 69 Polyetherimide Y 2300 2760 4 63
204 P.J. Soltys et al. / Journal of Membrane Science 118 (1996) 199-212
rate of 1 ml /min. The column was calibrated with dextran standards of low polydispersity (American Polymer Standards Corporation, Mentor, OH). Data acquisition and analysis were performed with Waters Expert Ease GPC software option.
Each chromatogram was divided into 50 equal fractional slices. The average dextran molecular weight for each slice was calculated from the reten- tion time using the previously developed calibration curve. The dextran concentration for each slice was determined by integrating the area under the curve. Dextran concentrations in the bulk solution were determined from samples taken at the housing inlet. The observed dextran sieving coefficients, S o, were then evaluated for each slice using the bulk dextran concentration determined from samples taken at the housing inlet.
Cdextra n in filtrate So = ( i s )
Cdextr~ . in bulk
4. Results
Scanning electron micrographs of three hollow fibers produced in a single Polysulfone A spinning run are shown in Fig. 2 (panels A and B), Fig. 3 (panels A and B) and Fig. 3 (panels C and D). As seen in Fig. 2, the spinning conditions used here resulted in a porous support matrix containing both macrovoids and spongy regions. Fig. 3B shows that the cells within the spongy regions are greater than 0.2 /xm in diameter. The high magnification (10000 X ) cross-sectional views in Figs. 3A and 3B show the dense structure of the inner and outer skins. Very high magnification (100000 × ) views of the inner and outer surfaces such as those seen in Figs. 3C and 3D reveal mean pore sizes of approximately 20 nm and 100 nm, respectively. The 70 A (7 nm) platinum coating may have obscured any pores smaller than about 14 nm in diameter. Note that the SEM results were obtained after completion of the pore size estimation by mathematical modeling, preventing un- intentional bias in the modeling.
Typical experimental data for the dextran sieving coefficients for a cartridge employing polysulfone fiber A are shown in Fig. 4. The squares represent
Fig. 2. Scanning electron micrographs depicting a dual-skinned polysulfone membrane in cross-sectional view of entire hollow fiber (panel A; 250 X, scale bar = 100 /xm), and cross-sectional view of membrane wall (panel B; 1000 × , scale bar = 10 #m).
data obtained with flow in the shell-to-lumen direc- tion, while the circles are for flow from lumen-to- shell. Both sets of data were obtained at a transmem- brane pressure drop of 75 mmHg, corresponding to filtration velocities (evaluated using the lumenal sur- face area) of V = 1.3 X 10 -4 c m / s for flow in the shell-to-lumen direction and V = 2.4 X 10 - 4 c m / s
for flow from lumen-to-shell. The directional depen- dence of the filtration velocity is discussed in more detail subsequently. Results obtained at other trans- membrane pressure drops were similar to those shown in Fig. 4, with the large difference in the dextran sieving coefficients in the two flow directions also seen for data sets obtained at the same filtration velocity (rather than the same transmembrane pres- sure drop). The curves in Fig. 4 are model calcula- tions as discussed below. The fiber examined in Fig.
P,J. Solrys et aL / Journal of Membrane Science 118 (1996) 199-212 205
4 displays a very large directional sieving behavior. For example, the observed sieving coefficient for an 80000 molecular weight dextran in the shell-to-lu- men direction is about S o = 0.9, while the observed sieving coefficient in the opposite flow direction is less than 0.1. This membrane thus behaves as a highly retentive membrane (for an 80K dextran) when operated with flow from lumen-to-shell, but is highly permeable to this same dextran when operated in the opposite flow direction.
The curves in Fig. 4 represent the calculated values of the dextran sieving coefficients determined using the theoretical analysis for solute transport through a dual-skinned membrane [Eqs. (13) and (14)]. The membrane transport parameters for the inner and outer skin layers were evaluated using the hydrodynamic model described in the Appendix, which implicitly accounts for the presence of a pore
O' O' ® = 0.8 ~ Shell-to-Lumen o 0.6
0.4
0.2
0
O 0 , I , I , I , I ,
20 40 60 80 1 O 0
Dextran Molecular Weight (kD)
Fig. 4. Directional sieving coefficients for polydisperse dextrans in polysulfone fiber A, which has a tighter inner skin. The squares represent filtration in the shell-to-lumen direction, while the cir- cles represent filtration in the lumen-to-shell direction. Lines represent model predictions with s I = 26 A and s 3 = 130 A.
Fig. 3. Scanning electron micrographs of the inner surface (panels A and C) and outer surface (panels B and D) of a dual-skinned polysulfone membrane in cross-sectional view (panels A and B; 10000 X, scale bar = 1 /xm) and perpendicular view (panels C and D; 100000 x , scale bar = 100 nm).
206 P.J. Soltys et aL / Journal of Membrane Science 118 (1996) 199-212
size distribution in the two skin layers. The hin- drance factors and equilibrium partition coefficients for the porous matrix were all assumed to be equal to one due to the very large pore size ( > 0.2 /xm) in this annular region. The observed dextran sieving coefficients were then calculated from the values for S, + and S~ using a stagnant film model to account for bulk mass transfer limitations [12]:
s: so+ = (1 - S + ) e x p ( - V / k +) + S + (16 )
where So + is the observed sieving coefficient for flow in the lumen-to-shell direction. An analogous expression was used for S o . The mass transfer coef- ficients, k + (for lumen-side feed) and k- (for shell- side feed), were determined from the Leveque solu- tion using the appropriate cross-flow velocities and channel geometries for the two flow configurations. This external concentration polarization was rela- tively small at the filtration velocities used in these experiments. The best fit values for the pore size of the inner and outer skin layers (s I = 26 ,~ and s 3 = 130 A) were determined by minimizing the sum of the squared residuals between the model calcula- tions and the data. The model was in excellent agreement with the dextran sieving data obtained in both the shell-to-lumen and the lumen-to-shell flow directions over the entire range of dextran molecular weights. The slight increase in S o for the intermedi- ate molecular weight dextrans predicted in the shell- to-lumen direction arises from the increase in the extent of both external and internal (discussed be- low) concentration polarization for the larger molec- ular weight (i.e. smaller diffusion coefficient) dex- trans.
The origin of the directional sieving behavior seen in Fig. 4 can be most easily understood by examin- ing the concentration profiles within the dual-skinned membrane for the two different flow directions. Fig. 5 shows the predicted profiles for a 50K dextran through the polysulfone membrane examined in Fig. 4. Calculations were performed using the best fit values for s l and s 3 with R L = 100 /xm, E i = 0.5, skin thickness of 0.25 /xm, and total membrane thickness of 45 /xm (determined from the scanning electron micrographs). When the flow is from shell- to-lumen, the dextran passes through the relatively
g-
o
e~
102 I ' ' I ' ' I ' '
Shell-to-Lumen Filtration Velociyly,
V (cm/s x 10 ~)
I 1,0
100
10 "1
Filtration Velocity, V (crn/s x 1 0 " )
0.4
4.0 Lumen-to-Shell
10 -2 I I 100 115 130 145
R a d i a l Pos i t i on , r (p ro )
Fig. 5. Solute concentration profiles within the matrix of a dual- skinned membrane. Calculations are performed for a 50K dextran in fiber A as discussed in the text.
open outer skin first. Only a slight drop in the dextran concentration occurs across this outer skin since the actual sieving coefficient of this layer is quite large. In contrast, the actual sieving coefficient of the inner skin of fiber A is very small (Sal = 0.045 at V = 4 × 10 -5 c m / s and Sal = 0.027 at the high- est V), causing the 50K dextran to accumulate at the interface between the matrix and the inner skin.
This "internal concentration polarization" can be quite dramatic; the dextran concentration at the inter- face between the matrix and the inner skin is pre- dicted to be as much as 40 times greater than that in the bulk (external) solution at the high filtration velocities. This internal polarization is much more significant than the external concentration polariza- tion in this system. For example, the ratio of the concentration at the wall to the bulk concentration for the shell-to-lumen configuration is less than 1.5 over the entire velocity range examined in Fig. 5. The net result of this large internal concentration polarization is that the observed dextran sieving co- efficient for flow in the shell-to-lumen direction is quite high (S O = 0.9) even though the inner skin is very retentive to the 50K dextran. The behavior when the flow is from lumen to shell is very differ- ent. In this case, the dextran passes through the tight (inner) skin first, causing a significant drop in the dextran concentration just inside the porous matrix. There is essentially no internal concentration polar- ization under these conditions since the second (outer)
P.J. Soltys et al. / Journal of Membrane Science 118 (1996) 199-212 207
skin layer does not retain a significant fraction of the 50K dextran. The net result is that the dextran siev- ing coefficient in the lumen-to-shell configuration is determined almost entirely by the properties of the tight inner skin. The extent of external concentration polarization is somewhat greater in the lumen-to-shell configuration than was seen in the shell-to-lumen direction due to the much larger solute retention in the lumen-to-shell direction. In this particular case, the ratio of the concentration at the wall to the bulk concentration is predicted to increase from 1.2 to 8.1 as V increases from 4 X 10 -5 c m / s to 4 × 10 -4
cm/s , which causes the observed dextran sieving coefficient to increase from 0.06 to 0.22 over this flux range.
Comparison of calculated values of s with pore diameters observed by scanning electron microscopy requires knowledge of pore geometry. Assuming straight-through cylindrical pores, the pore diameter is equal to 4s. Thus, pore diameters for inner and outer skins are 104 and 520 A, or 10 and 52 nm, respectively. These values are within a factor of two of the pore diameters observed by SEM -about 20 and 100 nm for the inner and outer skins, respec- tively.
Fig. 6 shows data for a second polysulfone fiber (fiber B), which was fabricated by A / G Technology. In this case, the data were obtained at a transmem- brane pressure of 80 mmHg, with V= 5.0 x 10 -5 c m / s for flow from lumen-to-shell (circles) and V = 6.9 × 10 5 c m / s for shell-to-lumen (squares). The curves are model calculations using the best fit values of s 1 (21 A) and s 3 (14 ,~). The model calculations for fiber B are in good qualitative agree- ment with the data, although the model does predict somewhat steeper sieving curves under these condi- tions. Fiber B also displays a distinct directional
1 . 0 g i
,g c 0.8
o 0.6 0
- > Shell 0.4
"0
0.2
S h e l l - t o - L u m
0 i I • ~ i i
10 20 30
Dextran Molecular Weight (kD)
Fig. 6. Directional sieving coefficients for polydisperse dextrans in polysulfone fiber B, which has a tighter outer skin. The squares represent filtration in the shell-to-lumen direction, while the cir- cles represent filtration in the lumen-to-shell direction. Lines represent model predictions with s 1 = 21 ik and s 3 = 14 ~,.
selectivity, but in this case the dextran sieving coeffi- cients are smaller in the shell-to-lumen configura- tion, which is exactly the opposite of the behavior seen for fiber A (Fig. 4). For example, So + for a 15K dextran is about a factor of 4 greater than S o. This dramatic difference arises from the very different pore size characteristics of the inner and outer skin layers of these two polysulfone fibers. Fiber B has a tighter outer skin (s 3 = 14 A vs. s I = 21 A), while fiber A had a tighter inner skin.
The results for the different fibers examined in this study are summarized in Table 2. The nominal molecular weight cut-off in the two flow directions was determined from the smallest molecular weight dextran which has greater than 90% rejection (i.e. S o < 0.1). The s-values for the inner and outer skin layers were determined from comparison of the ob-
Table 2 Effective pore size for the two skin layers and nominal molecular weight cut-offs in the two flow directions for the different hollow fiber membranes
Membrane type S 1 (inner) (A) s 3 (outer) (A) Nominal molecular weight cut-off (kD)
Lumen-to-shell Shell-to-lumen
Polysulfone A 26 130 55 > 100 Polysulfone B 21 14 20 13 Polyetherimide X 16 30 33 77 Polyetherimide Y 18 13 > 60 20
208 P.J. Soltys et aL / Journal of Membrane Science 118 (1996) 199-212
, , , i , , i
~ " " Increasing Time (rain)
30
o~ 2 3
0 ,8
8 0 . 6
._~ ~ 0 . 4 " 0
0 . 2
0 i i i i i i i 2 0 4 0 6 0 8 0 1 O0 120 140 160
Dextran Molecular Weight (kD)
Fig. 7. Time dependence of observed sieving coefficient through the polysulfone A membrane for filtration in the shell-to-lumen direction.
served sieving coefficient data with the model calcu- lations as discussed in Figs. 4 and 6. All 4 mem- branes exhibit a very clear directional sieving. Poly- sulfone A and polyetherimide X have much larger nominal molecular weight cut-offs for flow in the shell-to-lumen configuration than for flow from lu- men-to-shell, with exactly the opposite behavior seen for polysulfone B and polyetherimide Y. These dif- ferences are also apparent in the calculated values of the effective pore size for the different skin layers. The pore size of the inner skins (s l) of polysulfone A and polyetherimide X are much smaller than those of the outer skins (s3), with the reverse behavior seen for the other two membranes. Note that there is not a direct, one-to-one correspondence between s i and the nominal molecular weight cut-off due to the different geometries (R E and matrix thickness) and different ultrafiltration rates for the various fibers.
All of the data presented in Figs. 4 and 6 (and Table 2) were for the steady-state dextran sieving coefficients (determined from samples obtained after 60 min of filtration). Fig. 7 examines the transient development of the sieving coefficients for the poly- sulfone A hollow fibers in the shell-to-lumen config- uration. The dextran sieving coefficients increase with increasing filtration time, varying by as much as a factor of 8 over the 30 min filtration. This time
dependence arises from the transient development of the internal concentration polarization within the porous matrix. At very short filtration times, the dextrans have not had an opportunity to accumulate at the interface between the porous matrix and the inner skin (for flow in the shell-to-lumen direction). Under these conditions, the solute concentration pro- file within the matrix is relatively flat, with the sieving coefficient determined by the large drop in concentration as the solute crosses the tight inner skin. In fact, the dextran sieving coefficients at t = 1 rain in the shell-toqumen configuration (Fig. 7) are very similar to the steady-state values obtained when the flow is in the opposite direction (Fig. 4). The sieving coefficients in both of these situations are determined almost entirely by the properties of the tight (inner) skin. As filtration continues, the dextran concentration at the interface between the matrix and the inner skin begins to increase due to solute reten- tion by the tight inner skin. This internal polarization causes the large increase in S O seen with increasing filtration time in Fig. 7. The largest variation in S o occurs for the largest molecular weight dextrans, since these solutes have the smallest diffusion coeffi- cients and thus polarize to the greatest extent. The characteristic time for the development of the steady-state concentration profiles in this system can be estimated as 12/Dm where 1 is the membrane thickness (diffusion distance) and D m is the effective dextran diffusion coefficient in the porous matrix. This gives tss = 2 min for the 20K dextran, which is in good agreement with the results in Fig. 7. The corresponding analysis for a 100K dextran gives tss = 5 min, which is significantly less than the time seen experimentally. This discrepancy is due to the time required to reach the high dextran concentra- tions that accumulate within the matrix under these conditions. A simple mass balance analysis over the porous matrix yields t~ = 30 rain for the 100K dextran, which is in good agreement with the experi- mental data.
As mentioned previously, these dual-skinned membranes also exhibit a distinct directionality with respect to the solvent (filtration) flow. This is shown explicitly in Fig. 8 for filtration of the mixed dextran solution through polysulfone fiber A (top panel) and polysulfone fiber B (bottom panel). The flow rates for the pure solvent (in the absence of any dextrans)
P.J. Soltys et al. / Journal of Membrane Science 118 (1996) 199-212 209
125
Polysulfone A Solvent I " / .
100 O n l y ~ . /
75 ~ Lumen-to-Shell
o X 5O t~ ^ ~'//y/' - Shell-to-Lumen _
2s
v i i i
"O(:~ 125
0~ Polysulfone B ~> Solvent t " ~"
1 O0 Only f . t 0 i
iz 50
i . I f I - - . o
, 8
0 ~ ~ l 200 400 800 800
Transmembrane Pressure (mm Hg)
Fig. 8. Solvent flux as a function of applied hydrostatic pressure during filtration of a polydisperse dextran solution. Flux of PBS, shown by the broken line, was independent of filtration direction. Squares represent filtration in the shell-to-lumen direction, while circles represent filtration in the lumen-to-shell direction. The top panel shows data for polysulfone A, while the bottom panel shows data for polysulfone B.
through a given fiber were identical in the two flow directions. In contrast, the filtration velocity for the dextran solution through fiber A in the lumen-to-shell direction was considerably larger than that for flow in the shell-to-lumen direction, with the exact oppo- site behavior seen for fiber B. The difference in the values of the filtration velocity for the two flow directions increased significantly with increasing transmembrane pressure. For example, at A p = 750 mmHg, the filtration velocity for fiber A in the lumen-to-shell direction ( V = 8.5 × 10 -4 c m / s ) is more than a factor of 4 greater than that for flow from shell-to-lumen (V = 2.0 × 10 -4 cm/s ) .
The directional dependence of the filtration veloc- ity is again due to the development of an internal concentration polarization within the annular matrix. The filtration velocity across each region of the membrane can be written in terms of the effective pressure driving force as:
V = Lp[ Ap -- o -A/ / ] (17)
where Lp is the hydraulic permeability, tr is the osmotic reflection coefficient, and A H is the os-
motic pressure difference across that region of the membrane. Eq. (17) is valid for both the inner and outer skin layers, using the appropriate values for the transport properties (Lp and ~r) and the hydrostatic and osmotic pressure differences ( A p and A H ) for that particular skin. Let us consider the behavior of polysulfone fiber A (top panel in Fig. 8). The hy- draulic permeability of the outer skin layer in this fiber is quite large, while the osmotic reflection coefficient is quite small, due to the relatively large effective pore size of this skin (s 3 = 130 /~,). Thus, the filtration velocity through fiber A is determined almost entirely by the solvent flux across the (tight) inner skin. When the filtration is from shell to lu- men, the internal concentration polarization causes a very large concentration difference to develop across the inner skin (Fig. 5). This generates a large os- motic pressure difference, which reduces the effec- tive pressure driving force and in turn the filtration velocity [Eq. (17)]. This large concentration differ- ence is absent when the flow is from lumen to shell since there is no significant internal concentration polarization under these conditions (Fig. 5). The filtration velocity for flow from lumen to shell should thus be much larger than that in the reverse direction, and this is exactly what is seen in the top panel of Fig. 8. This same physical reasoning can be used to explain the results for fiber B (lower panel in Fig. 8). In this case, the large osmotic pressure effect occurs for flow from lumen to shell since the internal concentration polarization develops from solute re- tention by the tight outer skin of fiber B. It is not possible to quantify the polarization effects in this system due to the polydisperse nature of the dex- trans, with each of the different molecular weight dextrans having a different degree of polarization and a corresponding contribution to the overall os- motic pressure difference.
5. Discussion
The polysulfone and polyetherimide hollow fiber membranes produced in this study had thin skin layers on both the inner (lumenal) and outer surfaces of an annular macroporous matrix. Most signifi- cantly, these dual-skinned membranes exhibited a clear directional selectivity. The sieving coefficients
210 P.J. Soltys et al. / Journal of Membrane Science 118 (1996) 199-212
for these membranes were dramatically different in the two flow directions (shell-to-lumen and lumen- to-shell), with the magnitude of this effect deter- mined by the effective pore size of the inner and outer skins. This makes it possible to produce mem- branes with greater sieving coefficients (or nominal molecular weight cut-offs) in either the lumen-to- shell or the shell-to-lumen configuration, demonstrat- ing the extraordinary flexibility of this new mem- brane technology.
The directional sieving behavior of the dual- skinned membrane arises from the directional char- acter of the convective flow across the membrane. When the fluid flow passes through the more open skin first, the solute accumulates within the porous matrix since it is retained by the tight (second) skin. This internal concentration polarization significantly increases the solute sieving coefficients, analogous to the external (bulk) polarization effects seen for con- ventional membrane systems. This internal polariza- tion is largely absent when the fluid passes through the tighter skin first, since the second (more open) skin does not retain a significant fraction of the solute. These effects were described using a multi- layer membrane transport model, with the calculated sieving coefficients in excellent agreement with those observed experimentally.
The internal concentration polarization also has a dramatic effect on the solvent flux through these dual-skinned membranes. In this case, the accumula- tion of retained solute immediately upstream of the tighter skin (i.e. at the interface between the mem- brane matrix and the tight inner skin for fluid flow from shell-to-lumen) causes a large osmotic pressure difference to develop across the tight skin layer. This significantly reduces the effective pressure driving force, causing a dramatic reduction in the filtration velocity compared to that obtained with the pure solvent. This effect is largely absent when the flow passes through the tighter skin first since there is no significant internal concentration polarization under these conditions.
The directional transport characteristics of these dual-skinned membranes can have important impli- cations in a variety of membrane systems. For exam- ple, the performance of many membrane reactors is strongly influenced by the Starling recirculation flow caused by the axial pressure drop due to flow through
the fiber lumens [13]. This recirculation flow causes a convective solute transport from lumen-to-shell in the first part of the reactor with a reverse flux (shell-to-lumen) occurring near the outlet. The use of a dual-skinned membrane would make it possible to independently control the rate of solute transport in these two flow directions. Such a reactor could have significantly better performance characteristics, with the dual-skinned membrane enhancing reactant trans- port into the shell and /or improving product (or unwanted by-product) removal from the annular shell space. Similar improvements could be achieved in bioartificial organs using immobilized (transplanted) cells. In this case, the directional selectivity could be used to improve the overall nutrient status of the cells (cultured in the external shell region), enhance delivery of secreted products back out the lumen, and/or reduce the transport of immunological agents that might illicit an undesirable immune response.
In the field of hemodialysis, improvements in hollow fiber membrane design have provided greater clearance of waste products from blood. At the same time, these membranes may also increase the passage of harmful bacterial endotoxin fragments from the dialysate (flowing in the shell space) back into the blood. These bacterial fragments can illicit an unde- sirable immune response in the dialysis patient. The use of a dual-skinned membrane would make it possible to maintain high rates of solute transport from blood to dialysate while minimizing the solute transport back into the blood, providing better over- all clinical performance.
The unique transport characteristics of these dual-skinned membranes thus provide an extraordi- narily powerful tool for the design and development of novel membrane devices and processes that ex- ploit the directional selectivity of these new mem- brane structures.
6. List of symbols
Cf filtrate concentration (mo l / cm 3) C i local solute concentration (mo l / cm 3) D~ free solute diffusion coefficient (cm2/s) D m effective dextran diffusion coefficient (cm2/s) k + mass transfer coefficient (for lumen side feed)
(cm/s)
P.J. Soltys et al. / Journal of Membrane Science 118 (1996) 199-212 211
Kci hindrance factor for convective transport in region i ( - )
Kdi hindrance factor for diffusive transport in re- gion i ( - )
1 membrane thickness (cm) Lp hydraulic permeability N solute flux at the lumen-membrane interface
(mol /cm 2 s) N i local solute flux in region i (mol /cm 2 s) Pe i membrane Peclet number in region i ( - ) r radial position in hollow fiber geometry (cm) r s solute radius (cm) R L lumen (inner) radius (cm) R o outer radius (cm) s effective pore size in transport model (A) S + actual sieving coefficient when flow is in the
lumen-to-shell direction ( - ) S 2- actual sieving coefficient when flow is in the
shell-to-lumen direction ( - ) S~ i asymptotic sieving coefficient for region i ( - ) S O observed sieving coefficient ( - ) V filtration velocity at the lumen-membrane in-
terface (cm/s) V i local filtration velocity in region i (cm/s) A H osmotic pressure difference (mmHg) A p transmembrane pressure (mmHg) E i membrane porosity of region i ( - ) th~ equilibrium partition coefficient for region i
(-) /~eff effective solute to pore size ratio ( - ) o- osmotic reflection coefficient ( - )
Acknowledgements
The authors gratefully acknowledge the technical contributions of other scientists to this work: Dan Boggs, Marsha Kray, and Robin Pauley produced the polysulfone membranes, Steve Briana produced the polyetherimide membranes, Fazal Khan performed the solvent flux experiments, and Stanley Westphal prepared the scanning electron micrographs.
Appendix A
The transport parameters ( l~i , Kci , and Kdi) for spherical solutes in cylindrical pores in the absence
of any long-range (e.g. electrostatic) interactions can be evaluated in terms of the solute to pore size ratio (A i) using the analytical expressions developed by Bungay and Brenner [14]:
~b i = (1 - Ai) 2 (A1)
(2 - ~bi) K s Kci = (A2)
2K t
67r Kdi = - - (A3)
Kt
where the hydrodynamic functions K S and K t a r e
both expressed as expansions in Ai:
4
+ __~o\bn+3 A~
Zydney and co-workers [9,15] have shown that the transport characteristics of a variety of polymeric ultrafiltration membranes can be effectively de- scribed using Eqs. (A1)-(A4) by defining an effec- tive solute to pore size ratio (A~ff) using a theoretical expression for the partitioning of a rigid solute in an isotropic porous media formed by a random array of parallel planes:
(~i = exp( - rs/S ) (m5)
where r s is the solute radius and s is an effective pore size, equal to the ratio of the pore volume to the pore surface area. Aef f is then evaluated from Eqs. (A1) and (A5) as:
"~eff = l -- exp( - rs /2S ) ( a 6 )
Eq. (A6) implicitly accounts for the presence of a pore size distribution in these polymeric membranes through the definition of Aee f. Unlike classical mod- els for membrane transport that consider only a single pore size, the membrane transport properties (~b i, K~i, and Koi) given by the above analysis decay to zero only as r s ~ ~ due to the presence of a small number of very large pores in the pore size distribu- tion.
212 P.J. Soltys et al. / Journal of Membrane Science 118 (1996) 199-212
Mochizuk i and Zydney [9] have shown that Eqs.
( A 1 ) - ( A 5 ) can also be used to descr ibe the transport
o f different molecu la r weight dextrans. In this case,
the characterist ic dextran radius is evaluated f rom
the dextran diffusion coeff ic ient (D~) using the
S tokes -E ins t e in equat ion as:
k B T
r S 6~'tzD~ ( A 7 )
where k B is B o l t z m a n n ' s constant, T is the absolute
temperature, /z is the fluid phase viscosi ty, and
log D~ = - 4 . 1 1 5 4 - 0 . 4 7 7 5 2 1 o g ( M W ) ( A S )
as g iven by Granath [16]. Recent ly Boggs [17] ex-
tended this correlat ion by employ ing the Tay lor dis-
pers ion technique to obtain diffusivi t ies o f nearly
monodisperse dextran fractions in the molecu la r
weight range o f 1 0 0 0 - 5 0 0 0 0 daltons. L imi ted re-
analysis of our data using his sl ightly different val-
ues o f the coeff ic ients in A8 ( - 4 . 0 0 9 and - 0 . 4 6 5 ,
respect ively) did not affect any of the calculated
(best fit) values o f s.
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