Gas and water liquid transport through nanoporous block copolymer membranes

Gas and water liquid transport through nanoporous block copolymer membranes

Journal of Membrane Science 286 (2006) 144–152 Gas and water liquid transport through nanoporous block copolymer membranes William A. Phillip a , Jav...

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Journal of Membrane Science 286 (2006) 144–152

Gas and water liquid transport through nanoporous block copolymer membranes William A. Phillip a , Javid Rzayev b , Marc A. Hillmyer a,b , E.L. Cussler a,∗ a

Department of Chemical Engineering & Materials Science, University of Minnesota, Minneapolis, MN 55455, United States b Department of Chemistry, University of Minnesota, Minneapolis, MN 55455, United States Received 28 February 2006; received in revised form 11 September 2006; accepted 19 September 2006 Available online 23 September 2006

Abstract ABC triblock terpolymers containing polylactide, poly(dimethylacrylamide) and polystyrene, which can self-assemble as aligned cylinders, are etched to give nanoscopic pores. The diffusion of gases through these pores is quantitatively consistent with Knudsen diffusion. The flow of water liquid in the pores is consistent with the Hagen–Poiseuille law. If these polymers can be produced as commercial ultrafiltration membranes, they should give a larger hydraulic permeability and a sharper molecular weight cut-off. © 2006 Elsevier B.V. All rights reserved. Keywords: Nanopore; Gas diffusion; Block copolymers

Water purification seems poised for rapid growth [1,2]. The public water system in North America is over 100 years old and seems increasingly strained by the large demand caused by higher population. The domestic water market also seems ready for growth, as consumers hesitate to drink tap water but resent the cost and effort required to stock bottled water. Moreover, all water systems are subject to terrorist attack, an additional source of anxiety. These stresses will probably cause both increased public and private efforts to insure water quality. Ultrafiltration will play a large role in this renewed interest in pure water [3,4]. In ultrafiltration, a feed water at high pressure is forced across a nanoporous membrane filter to give a pure permeate. A typical pressure drop is about 50 kPa; a typical flux is 1 m3 /m2 day. Such a porous membrane will normally retain both bacteria and viruses; it will normally pass dissolved ions and salts, including Cl− and Ca2+ . The key to ultrafiltration is the membrane, which should allow a large flux and show a sharp molecular weight cut-off. Many current ultrafiltration membranes are made by phase inversion, in which a solution of membrane polymer is spread on a moving

∗ Corresponding author at: Department of Chemical Engineering & Materials Science, University of Minnesota, Minneapolis, MN 55455, United States. Tel.: +1 612 625 1596; fax: +1 612 626 7246. E-mail address: [email protected] (E.L. Cussler).

0376-7388/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.memsci.2006.09.028

web, allowed to dry slightly, and then plunged into a bath of nonsolvent. The result is a tangle of polymer fibers like those shown at the left of Fig. 1. By changing the process conditions, the size and packing of the fibers can be controlled. Not surprisingly, the flux can be adjusted without a lot of trouble, but realizing a sharp molecular weight cut-off is more challenging. An alternative ultrafiltration membrane structure [5,6], shown in the middle panel of Fig. 1, begins with a thin, pore-free polycarbonate film. The film is then exposed to ionizing radiation, which weakens regions of the film. If the weakened film is then etched with caustic, it develops small near-cylindrical pores like those in the figure. Because the pores are nearly monodisperse, this membrane has a sharp molecular weight cut-off. However, because the pore density is low, the flux is usually dramatically lower than that of more conventional ultrafiltration membranes. (We understand that the picture of the track-etched membrane in the center of Fig. 1 shows a smaller population of pores which do not look especially monodisperse. Manufacturers of these membranes often publish more impressive pictures. The picture we show is typical of those which we obtain for membranes which we purchased.) Recently, several groups have made membranes which have the potential of both high flux and sharp molecular weight cut-off [7,8] and so have the promise of improved ultrafiltration. In some cases, these membranes are made of aligned carbon nanotubes [9–11]. In other cases, they are made of self-assembled block

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Fig. 1. Three types of membranes. That on the left is a conventional ultrafiltration membrane formed by phase inversion, and that in the center is a track-etched polycarbonate film. The membrane on the left is that made here.

copolymers. When their molecular weight and composition are within carefully specified limits these materials spontaneously self-assemble into cylinders [12–14] that can be aligned perpendicularly to the membrane surface. One segment of the block forms the cylinders; the other forms the surrounding continuum. After this structure is formed, the cylindrical blocks are removed by etching with, for example, UV light or ozone. The particular system studied here is typical of these efforts. It consists of a triblock copolymer of polylactide– poly(dimethylacrylamide)–polystyrene (PLA–PDMA–PS) as shown:

When n/p is about 0.1 the system can form small spherical domains of PLA coated with PDMA and immersed in a continuum of PS. When n/p is about 1, the system forms lamellae of PLA and PS; the interfaces between the lamellae are coated with PDMA. When n/p equals 0.44, the system assembles as cylinders of PLA coated with PDMA in a continuum of PS. The PLA blocks are removed by etching with aqueous base to form the desired pores. The result is a new ultrafiltration membrane like that shown on the right of Fig. 1. The large concentration of pores (ca. 1011 cm−2 ) means that this membrane will have a large flux: if we can make an 0.3 ␮m selective layer on a support, we can easily obtain fluxes more than 10 times larger than conventional membranes. The monodisperse pores promise a much sharper molecular weight cut-off. A membrane like this could really represent a step change in the capability of ultrafiltration. However, the promised advantages of this new ultrafiltration membrane have not been demonstrated. To be sure, the apparent pores shown at the right of Fig. 1 are an attractive structure. But these pores are often found not to go all of the way through the membranes: they are not so much pores as pits [15]. Moreover,

details about the internal pore architecture are unknown: even if they do span the membrane, the pores may be branched, with bends or twists. Any of these complexities could compromise their performance. In this paper, we begin to measure the properties of nanoporous membranes made by etching block copolymers. We decided to start by measuring the diffusion of gases and the hydraulic permeability of water liquid. Later, we plan to study ultrafiltration across these porous membranes. 1. Theory The mechanism of gas diffusion in small pores depends on the Knudsen number [16], i.e. the ratio of the mean-free path to the pore diameter. When the mean-free path is much smaller than the pore size, the Knudsen number is small and diffusion is the result of random collisions between different molecules. In this case, the diffusion coefficient D can be estimated from the Chapman–Enskog kinetic theory:  √  (kB T )3/2 4 2 (1) D= 3π3/2 pσ 2 m ˜ 1/2 Ω where kB is the Boltzmann’s constant, T the temperature, p the pressure, σ the collision diameter, m ˜ the mass of one molecule, and Ω is a dimensionless function of temperature which is of order 1. Note that for small Knudsen number, kinetic theory predicts that the diffusion coefficient varies inversely with pressure but is independent of pore size. The situation when the mean-free path is much larger than the pore diameter is different. Now, when the Knudsen number is large, diffusion is the result of collisions between the diffusing gas and the walls of the pores. The diffusion coefficient D for this case of “Knudsen diffusion” is now given by √    2 kB T 1/2 D= (2) d 3 m ˜


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where d is the pore diameter. This equation is more approximate than Eq. (1); more accurate theories of Knudsen diffusion include such features as a sticking coefficient to correct for nonelastic collisions between the diffusing gas molecule and the pore wall [17]. For our purposes, the important feature is the prediction that the diffusion coefficient does not vary with the pressure but is proportional to the pore diameter. The results for liquid flowing through the pores in these membranes is different again. As in the first case, the Knudsen number is small, but the molecular collusions are so frequent and the fluid density is so high that the liquid behaves as a continuum. Because the pores are so small, the fluid velocity v is laminar and described by the Hagen–Poiseuille law: v=

pd 2 32μl


where p is the pressure drop across a membrane of thickness l, and μ is the liquid viscosity. Of course, we can rewrite this expression as an apparent diffusion coefficient, but this obscures the physical mechanism. We use these equations as a comparison for our results for diffusion and flow through our new membranes in the following sections. First, we describe how the membranes are made and how the diffusion coefficients and the flow are measured. We then compare these measurements with those predicted. These measurements suggest pore diameters within 3 nm of those observed by microscopy or calculated using small-angle X-ray scattering data. We conclude with a discussion about the possibilities for ultrafiltration. 2. Experimental 2.1. Fabrication of the nanoporous monoliths Chemically incompatible block copolymers undergo phase separation into different morphologies on the nanometer scale. A block copolymer precursor that self assembles into cylinders can be used as a precursor to the nanoporous membrane shown at the right of Fig. 1. The cylindrical domains in this precursor are first shear-aligned and then removed via selective etching [8,18]. Here, we utilize a polylactide–poly(dimethylacrylamide)–polystyrene (PLA– PDMA–PS) triblock copolymer precursor where polystyrene constitutes the continuous matrix, PLA is the etchable component, and PDMA remains as a coating the nanochannel walls [19,20]. This PLA–PDMA–PS triblock copolymer is prepared by a combination of controlled ring-opening and freeradical polymerization protocols. First, the polylactide segment is synthesized by an aluminum-catalyzed ring-opening polymerization of d,l-lactide. The reaction, conducted in toluene at 90 ◦ C, produces PLA with Mn = 11 × 103 (NMR) and Mw /Mn = 1.05 (GPC). The native hydroxyl end-group of PLA is then linked to S-1-dodecyl-S -(␣,␣ -dimethyl-␣ acetic acid)trithiocarbonate [21] via an acid chloride intermediate to provide a chain-transfer agent (PLA-TC) in quan-

titative yield, as determined by NMR. Subsequently, PLATC is employed under the reversible addition-fragmentation chain-transfer (RAFT) polymerization to grow PDMA and PS blocks sequentially. The analysis by NMR and GPC after each step corroborates the efficient re-initiation, and the final PLA–PDMA–PS triblock copolymer obtained has block molecular weights of 11 × 103 − 2.2 × 103 − 25 × 103 , respectively, and Mw /Mn = 1.14. Powdery samples of PLA–PDMA–PS are pressed in a rectangular mold, and then forced through a channel die (length × area = 50 mm × 4 mm2 ) at 160 ◦ C to provide 1–2 mm thick monoliths [22,23]. Following annealing in a vacuum oven at 190 ◦ C for 16 h, the monoliths are analyzed by smallangle X-ray scattering, which confirms the formation of the aligned cylindrical morphology with hexagonal symmetry in the direction of flow. The degree of alignment, quantified using the second-order orientation factor F2 [8,24,25] is 0.94– 0.96. 2.2. Membrane preparation We use two different membranes in our experiments. The first is based on the etched block copolymer monoliths whose synthesis is described above. The second uses commercially available track-etched polycarbonate membranes as a benchmark. The preparation of these two membranes differs. Extruded, aligned monoliths of PLA–PDMA–PS are cut into 2 mm lengths. A 3 mm length of 5 mm i.d. PTFE tubing is filled with epoxy (DP-460 Off White, 3M, St. Paul, MN) and the PLA–PDMA–PS chunk is submerged in the epoxy with the extruded direction (i.e. the pores) parallel to the axis of the tubing. The epoxy is cured in an oven at 75 ◦ C for 1 h. Upon removal from the oven, the cured epoxy and PLA–PDMA–PS are removed from the PTFE tubing and the epoxy ends are whittled away with a razor blade to expose the PLA–PDMA–PS chunk. The final sample is approximately 2 mm thick and 5 mm in diameter, with an exposed area of PLA–PDMA–PS approximately 1.5 mm × 2 mm. The cylindrical epoxy PLA–PDMA–PS sample is inserted into a gasket made by punching a hole in a GC septum. This assembly is then inserted into a metal disk. The disk is clamped between two metal sieve plates, causing the gasket to compress in the vertical direction and expand in the horizontal direction. This pressure creates a good seal between the epoxy PLA–PDMA–PS and the metal disks. At this point, the polymer monolith has not been etched. The preparation of the commercially obtained track-etched membranes is simpler. A typical membrane (Steriltech, Kent, WA), made from polycarbonate, is mounted between two sheets of aluminum foil, using the same epoxy to avoid contamination of the membrane by the sealing gaskets. A sheet of aluminum foil is folded in half and a 1.1 cm diameter hole is punched through both layers. The PCTE membrane, which has a diameter of 1.3 cm, is glued between the two layers using epoxy. The epoxy is cured in an oven at 75 ◦ C for 1 h. This membrane is now ready for mounting.

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2.3. Diffusion measurements The membranes prepared as above are used in the gas diffusion cell [26] shown in Fig. 2. The cell consists of two compartments separated by the membrane. The donating compartment has a volume of 22.1 cm3 and the receiving compartment has a volume of 21.9 cm3 . In some experiments, both are connected to 18 L tanks to allow for experiments with longer times. Receiving and donating compartment pressures are measured by electronic transducers (Cole Parmer Model 97356-61, Vernon Hills, IL) and the temperature is monitored with a thermocouple (National Semiconductor, Santa Clara, CA). Both pressures and the temperature were recorded in Microsoft Excel using a Super Logics 8017 (Waltham, MA) interface. To prepare for an experiment, a membrane is clamped between the two compartments in this cell. The procedure for a track-etched film, clad with aluminum foil, is simple: it is put between a sieve plate and a metal ring with gaskets and clamped in place. The procedure with the PLA–PDMA–PS films is more elaborate because each film is tested before etching of the PLA block to ensure that a good seal is formed. This shows that the only flux through the membrane will be due to the pores and not to leaks at the epoxy PLA–PDMA–PS interface. Earlier attempts to pot the entire extruded PLA–PDMA–PS sample and then cut-off 2 mm lengths resulted in leaks, probably because the adhesion between the epoxy and the nanoporous PS–PDMA is not sufficient to counteract the stress caused by the razor blade. To eliminate this problem, we found it necessary to cut the PLA–PDMA–PS before potting or etching it. If the unetched PLA–PDMA–PS membrane is found to be impermeable, it is removed and immersed in a stirred 0.5 M NaOH, 60:40 (v/v) methanol/water solution at room temperature.


Earlier experiments at 70 ◦ C were not successful because the methanol swelled the epoxy, destroying the adhesion between the epoxy and PS. Etching at room temperature takes about 8 days. The sample is then removed from the solution and rinsed thoroughly with 60:40 (v/v) methanol/water. It is dried under vacuum for approximately 8 h. The resulting membrane does have the highly aligned nanoporous structure with hexagonal symmetry, as verified by SAXS and scanning electron microscopy (Hitachi S-900 FE-SEM). In the dry state, the pore diameter calculated from the volume fractions of the PLA–PDMA–PS precursor and the SAXS data is 16.8 nm. This membrane can now be clamped in the diffusion cell. Once the membrane is clamped in the cell, both tanks and the gas cell are thoroughly flushed with the gas being studied. Both vents are opened and the donating tank is kept at about 20 psig while the receiving tank is kept at about 10 psig. After 2 h, the feed to the receiving tank is cut-off, so its pressure decreases quickly to atmospheric. At this point, the donating feed valve and receiving vent valve are closed and the data acquisition program is started. Experiments generally take about 1 h. Duplicate runs are performed after about 20 min of additional flushing. The pressure data are adjusted for temperature fluctuations using the ideal gas law. The permeability is calculated from    p PAt 1 1 = exp − +  p0 l V V


where l is the membrane thickness, A the membrane area, t the time, V and V the receiving and donating compartment volumes, and p0 and p are the pressure differences between the donating and receiving compartments initially and at time t. Normally the data are plotted as ln(p0 /p) versus t/l, and P is determined from the slope. We expect that the permeability in

Fig. 2. The gas diffusion cell. The membrane is clamped between two volumes at different pressures, and the pressure difference is measured vs. time.


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Eq. (4) is related simply to the diffusion coefficient: P=

Dε τ


where ε is the void fraction of the membrane, τ the tortuosity of any pores and D is the diffusion coefficient within the pore itself. This is the coefficient appearing in Eqs. (1) and (2), and so is the key to comparing experiments and theory. In our experiments, we estimate the void fraction from the volume fraction of PLA. We expect that the pores are straight and so assume that the tortuosity is one. Because we can calculate the diffusion coefficients from Eqs. (1) and (2), we can compare these calculations with the experimental values found from Eqs. (4) and (5). These experimental values are reported in the next section. Our early experiments were prey to artifacts due to the silicone grease used in sealing the cell. Commonly, these artifacts appeared as a permeability which kept dropping as the experiments were repeated, apparently because the grease slowly wet the membrane and blocked some of the pores. To avoid this, we used a minimum amount of grease, and then found that permeability was constant.

partment is attached to a large reservoir and the receiving compartment is connected to a precision bore capillary (Friedrich & Dimmock Inc., Millville, NJ). Once the membrane is clamped in the cell, it is prewet with the 60% methanol/40% water solution. Then the large reservoir is completely filled with pure water, while the capillary is partly filled. Because the water height in the reservoir is greater than that in the capillary, a hydrostatic pressure difference drives the flow through the membrane. The height difference between the donating and receiving volumes is recorded as a function of time using a cathetometer (Gaertner Scientific Corporation, Chicago, IL). The Hagen–Poiseuille law in Eq. (3) is used to derive an equation which allows determining the pore diameter from volumetric flow rate:     h d 4 ρg 1 1 = exp −NA t (6) + h0 128μl R 2 R 2 where h0 and h are the height differences between the donating and receiving compartments initially and at the time t, N the membrane pore density per area, ρ and μ the density and viscosity of water, g the acceleration due to gravity, and R and R are the reservoir and capillary radii, respectively. When the data are plotted as ln(h0 /h) versus t, d may be determined from the slope.

2.4. Liquid flow measurements 3. Results The preparation of the monoliths for liquid flux measurements varies from that for diffusion measurements in only one detail. The PFTE tubing used as a mold for the membrane for the diffusion measurements is replaced with a cylindrical die bored out of a Teflon block. Using this die, we make a sample approximately 2 mm thick and 31 mm in diameter with an exposed area of copolymer around 1.5 mm × 2 mm. These dimensions allow the sample to be directly inserted into the liquid flow cell. The preparation of the track-etched membranes is the same as that for gas diffusion. A membrane prepared as above is now clamped into the diaphragm cell shown in Fig. 3. This cell also consists of two compartments separated by the membrane. The donating com-

In this section we first show typical data for our diffusion experiments and then compare the results of these experiments with the predictions summarized above. We conclude with our measurements of liquid flows. Typical data for nitrogen diffusion across the etched PLA–PDMA–PS triblock copolymer membranes are shown in Fig. 4. The data, plotted as suggested by Eq. (4), show that the logarithm of the ratio of pressure differences is linear in time. The slope of this plot is proportional to the permeability: a larger slope means a larger permeability. The slope for the unetched membrane is near zero because it has no pores, and diffusion through the polymer itself is so small. The slope of the two different etched membranes is much

Fig. 3. The liquid flow cell. The diameter of the membrane’s pores is found by measuring the height of water in the capillary vs. time.

Fig. 4. Diffusion data vs. etching. The pressure with time is reproducible for etched films, but very small for nonetched films.

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Fig. 5. Diffusion data for different gases. As expected, permeabilities are higher for gases with smaller molecular weights.

larger and equal within 5%. We believe that this is typical of the reproducibility of our experiments. As expected, different gases have different permeabilities, as shown in Fig. 5. The permeability of helium is greater than oxygen, which is in turn greater than argon. This inverse dependence on molecular weight is that expected for either conventional or Knudsen diffusion, described by Eqs. (1) and (2), respectively. This dependence reflects the larger velocity of the smaller species, which is a basis of both models of diffusion. The transport mechanism in these membranes is Knudsen diffusion, as shown by the summary of results in Table 1. The upper part of this table gives the results for the PLA–PDMA–PS etched block copolymer membranes, and the lower part is for the track-etched membranes studied as a benchmark. The organization of the results within these two parts is the same. The first column gives diffusing gas, and the second column gives


the experimentally determined diffusion coefficient. These values depend on the void fraction occupied by the pores, which is 0.26 for the block copolymer membrane (based on the PLA volume fraction) and 0.0060 for the track-etched membrane. The next two columns give the diffusion coefficients calculated from Eqs. (2) and (1), respectively. The last two columns give the membrane selectivities relative to helium found from experiment and calculated for Knudsen diffusion. The results in Table 1 show conclusively that transport through the pores etched in the block copolymer occurs by Knudsen diffusion. The measured coefficients are within 5% of those calculated from the Knudsen mechanism. They are about six times smaller than the values expected from kinetic theory. This is expected because estimates of the mean-free path of these gases average about 10 times larger than the pores themselves. In addition, the selectivities measured relative to helium are close to those calculated from Eq. (2). They are a function only of the inverse square root of the molecular weight and are not dependent on factors like the collision diameter, which alter kinetic theory estimations. Finally, the experiments with the track-etched membranes used as a benchmark give very similar results, justifying our procedure. The results in Table 1 begin to realize the promise of etched block copolymer membranes for fast ultrafiltration with sharp molecular weight cut-off. However, the predicted values of Knudsen diffusion coefficients are based on a pore diameter of 13.7 nm, somewhat less than the value of 16.8 nm found from both electron microscopy and SAXS. We believe that this difference reflects both limitations in the theory summarized by Eq. (2) and consequences of the detailed chemistry, especially of the PDMA left at the pore walls. We will discuss this in detail later in the paper. We did explore this discrepancy via experiment as well, by measuring flow of water liquid through the etched pores. The results are shown in Fig. 6 where the data, plotted as the

Table 1 Experimental and predicted diffusion coefficients for etched PLA–PDMA–PS block copolymer membranes and for commercial track-etched polycarbonate membranes Gas

D Experimentala

Etched PLA–PDMA–PS He 0.051 0.019 N2 O2 0.017 Ar 0.017 Gas

Track etched He Ar


Kinetic theory

0.051 0.019 0.018 0.016

1.12 0.135 0.137 0.122

D Experimentalc


Kinetic theory

0.128 0.040

0.124 0.039

1.12 0.122

Diffusion coefficients are given in cm2 /s. The selectivity is relative to helium. a Assuming void fraction ε = 0.26. b Assuming 13.7 nm pores. c Assuming void fraction ε = 0.006. d Assuming 29.5 nm pores.

Experimental selectivity

Predicted selectivity

1.00 2.59 2.90 3.08

1.00 2.65 2.83 3.16 Experimental α

Predicted α

1.00 3.21

1.00 3.16


W.A. Phillip et al. / Journal of Membrane Science 286 (2006) 144–152 Table 2 Predicted permeances Gas

Permeance Experimentalb


Kinetic theory

442 165 156 139

9700 1060 1170 1190



Kinetic theory

2.57 0.80

2.48 0.78

22.3 2.43


442 165 149 144


Track He Ar

Fig. 6. Liquid water flow through nanoetched pores. The solid line is based on a pore diameter of 13 nm, close to the value of 14 nm found from the data in Table 1.

logarithm of the ratio of height differences, are linear in time, as predicted by Eq. (6). The average pore diameter as calculated from the slope in Fig. 6 is 13.0 nm, in close agreement with the diffusion results. We have not yet measured changes in flux as a function of pH or solute concentration. We understand that these changes may be major [27]. 4. Discussion The results given above begin to realize the promise of ultrafiltration membranes based on etched block copolymers. The diffusion coefficients that are observed are consistent with those expected for Knudsen diffusion. The nearly monodisperse pores within these films should give sharp molecular weight cut-offs like those observed using track-etched membranes. We can make this comparison with track-etched films more explicit by comparing the permeances across a 0.3 ␮m film, as shown in Table 2. The permeance of the block copolymer films is over 100 times faster than that of a track-etched film, even though the block copolymer film has significantly smaller pores. If the pores were the same size, the permeance of the block copolymer films would be 4000 times larger. The close agreement between theory and experiment implies that the pore geometry we have obtained is close to that of right circular cylinders with an aspect ratio of over 100,000:1. Any branching or constrictions in the pores would mean that the agreement between theory and experiment would be weaker. At the same time, the diffusion data are consistent with a large body of earlier literature that explores whether these simple theories of membrane transport are valid for very small pores [28–30]. Our data, which seem reliable, do support these simple theories, both for gases and liquids [16,30]. Our data are also consistent with earlier efforts using etched block copolymers, though these make no comparisons with predictions [18]. However, our results and most earlier data are inconsistent with results for water liquid transport in aligned carbon nanotubes, where a new, different transport mechanism suggested by quantum mechanics may be



The values for the copolymer and marked “experimental” are based on the pore diameters inferred from Table 1. Permeances are in cm/s. a Assuming 300 nm thickness. b Assuming void fraction ε = 0.26. c Assuming 13.7 nm pores. d Assuming 5 ␮m thickness. e Assuming void fraction ε = 0.006. f Assuming 29.5 nm pores.

operating [9–11]. We find no evidence whatsoever of any new transport mechanism in our work. However, the pore diameter of 13.7 nm, which is consistent with our diffusion and flow results, is less than 16.8 nm which is observed in our SEM and SAXS measurements. The cause of this discrepancy is not known. The obvious explanation is that the experimental procedure is somehow flawed. However, the experiments with track-etched films, which use the same procedure, give pore diameters from diffusion measurements that do agree with those from the SEM measurements. Thus the procedure seems correct. Another possible explanation for this discrepancy is that the pore geometry implied by the picture at the right of Fig. 1 is misleading. That picture implies straight monodisperse cylindrical pores. Perhaps the pores have big mouths and narrow bodies within the film. Perhaps they are branched, or have many dead ends. Perhaps they are somehow polydisperse. In other words, perhaps their geometry is not be what it seems. However, SEM images from related monoliths are consistent with well-aligned straight pores with uninterrupted lengths over several microns. Moreover, the data do not support this explanation. If the pores had wide mouths and narrow bodies, the SEM results would suggest larger diameters than the SAXS results. They do not. If the pore geometry were complex and polydisperse, then the gas diffusion and liquid flow results would be inconsistent. They are not. Thus the pore geometry is probably close to right circular cylinders. The most likely reason for this discrepancy is based on the chemistry in the pores. Once the poly(lactide) blocks are removed by etching with base, we expect that the pores will be coated with poly(dimethylacrylamide). This will create a polymer brush on the inside of the pores that will extend into the fluid inside the pores. If a PDMA coating of 0.8 nm were to swell to 1.4 nm in both gas and liquid, it would reduce the 16.8 nm

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measured from SAXS to the 13.7 nm measured by diffusion. Other changes in pore chemistry may include adsorption and desorption kinetics, which are often summarized as a “sticking coefficient.” Analyzing the data in these terms introduces a new parameter that cannot be independently measured. However, any chemical interactions like those postulated in these analyses should be different for the different gases, and we find the same 13.7 nm diameter explains data for He, N2 , O2 , and Ar. We recognize that many will be interested in the results for diffusion of gases, but that many more will be interested in the results for ultrafiltration of liquid solutions. After all, while the gas results are interesting scientifically, the liquid results may have considerable practical value. We decided to begin our results with gases for two reasons. First, the diffusion of gases is chemically simple. It is unaffected by solvation, by solute charge, or by polymer polarity. It provides a chance to see if the geometry of etched pores shown in the right panel of Fig. 1 is really as simple as it looks. The result is a definitive yes. The second reason that we began by studying gases rather than liquid solutions is the difficulty of the experiments themselves. Once the basic synthesis and the etching of the block copolymers is established, it is relatively easy to get micrographs and X-ray scattering indicating that the aligned pore geometry exists. It is considerably more difficult to show that these materials can show the transport properties that are expected. We personally know of three groups who have failed to date, and we expect that there are more. Because of this, we wanted to establish a strong baseline before moving on to experiments with liquid solutions. This paper establishes that baseline. We do want to speculate about what the performance of these membranes will be when they are used for ultrafiltration. To do so, we consider the case of bovine serum albumin, because the separation of this protein has been so carefully studied, as summarized by Mehta and Zydney [31]. One way to evaluate this case is to plot the separation factor versus the hydraulic permeability. The separation factor is a measure of the membrane’s selectivity; the hydraulic permeability is inversely proportional to the membrane’s resistance to flow. We seek a high separation factor and a high hydraulic permeability. Not surprisingly, we get either a membrane with a high separation factor or a membrane with a high hydraulic permeability, as the data in Fig. 7 show for commercially available membranes [31]. This figure also shows as the solid line the results expected for etched block copolymer membranes. These data were calculated by assuming a pore diameter, and then calculating the hydraulic permeability from Eq. (3) and the separation factor α from [32]: 1 2 = (1 − λ)2 [2 − (1 − λ)2 ] e−0.71λ α


where λ is the ratio of the diameter of BSA (7.3 nm) to the diameter of the pores. The estimates for membranes like that used here have both a faster flow and a more abrupt increase in separation factor. We look forward to finding out if this expected improvement can be achieved experimentally for aqueous ultrafiltration. This paper, showing that gas diffusion does behave as anticipated, is the first step in this effort.


Fig. 7. Separation factor vs. hydraulic permeability. The data reviewed in Ref. [31] are for commercially available membranes. The solid curve is for the membranes being developed here.

Acknowledgments The authors benefited from suggestions of J.P. DeRocher and G.D. Moggridge of Cambridge University. This work was largely supported by the Air Force Office of Scientific Research (grant F49620-01-1-0333). Other support came from the Department of Energy (grant DE-FG-02ER63509), the Petroleum Research Fund (grant 39083-AC9), and the Initiative for Renewable Energy and the Environment at the University of Minnesota.

Nomenclature A d D g h kB l m ˜ N p P R , R t T V , V

membrane area pore diameter diffusion coefficient gravitational acceleration liquid height Boltzmann’s constant membrane thickness molecular mass pore density (per area) pressure permeability reservoir and capillary radii, respectively time temperature volumes in the diffusion cell

Greek letters α separation factor ε void fraction μ liquid viscosity ρ liquid density σ collusion diameter τ tortuosity Ω correction factor for bulk diffusion


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