Nanochannel Design by Molecular Imprinting on a Free-Standing

Jun 3, 2009 - A free-standing ultrathin film of a poly(vinyl alcohol) (PVA)/titania (TiO2) composite was prepared by spin coating. The thickness of th...
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Nanochannel Design by Molecular Imprinting on a Free-Standing Ultrathin Titania Membrane Shigenori Fujikawa,*,†,‡,§ Emi Muto,† and Toyoki Kunitake† †

Topochemical Design Laboratory and ‡Innovative Nanopatterning Laboratory, RIKEN, Wako, Saitama 351-0198, Japan. §Current address: Nanocompartment Engineering Laboratory, ASI, RIKEN, Wako, Saitama 351-0198, Japan Received April 27, 2009. Revised Manuscript Received May 7, 2009 A free-standing ultrathin film of a poly(vinyl alcohol) (PVA)/titania (TiO2) composite was prepared by spin coating. The thickness of the film was adjusted to 30-50 nm by changing the spin-coating speed and the concentrations of PVA and the TiO2 precursor. A template molecule, (4-phenylazo)benzoic acid (4PABA), was introduced into the film as a mixture in the TiO2 precursor and was removed after film formation by dipping the film in aq NH3 (1%). Aqueous solutions of tetraphenylporphyrin tetrasulfonic acid (TPPS), 4PABA, and sodium benzoate (SB) were filtered through this film, and the concentrations of these compounds in the filtered solution were monitored by UV-vis absorption measurements. The filtered TPPS solution was colorless, and its absorbance at 413 nm was 8% that of the original solution. In contrast, almost 100% of SB and 28.4% of 4PABA were passed through the film. The ultrathin TiO2/PVA film obtained without imprinting with 4PABA had no cavities, and aqueous solutions did not pass through this film. Therefore, it was concluded that the nonimprinted film was defect free and that imprinting of the template molecule in the film resulted in the formation of a size-selective channel across a 40 nm thickness.

1. Introduction In modern separation science, highly selective and efficient filtration techniques are desired. Typical cases where selective filtration is required include separation of proteins from blood in an artificial kidney (hemodialysis), desalination of seawater, and wastewater treatment. In these cases, the thickness and cavity size of the membrane used for filtration determine the filtration efficiency. Commercially available membranes are very thick, and their cavities are considerably larger than the target molecules; the use of such membranes results in poor selectivity, filtrate loss, and low transport rates. Although thin membranes with small cavity sizes are preferred for efficient and selective filtration,1,2 ultrathin films are not commonly used in conventional filtration processes because they are highly fragile and difficult to fabricate. Hence, the thickness of most commercially available membranes is at the least of the order of a few micrometers.3-8 If mechanically strengthened artificial nanomembranes are fabricated, these artificial nanomembranes would have great potential for achieving more precise and efficient separation membranes.

*Corresponding author. (1) Tong, H. D.; Jansen, H. V.; Gadgil, V. J.; Bostan, C. G.; Berenschot, E.; van Rijn, C. J. M.; Elwenspoek, M. Nano Lett. 2004, 4, 283–287. (2) Kuiper, S.; van Rijn, C. J. M.; Nijdam, W.; Elwenspoek, M. J. Membr. Sci. 1998, 150, 1–8. (3) Yan, F.; Goedel, W. A. Adv. Mater. 2004, 16, 911–915. (4) Martin, F.; Walczak, R.; Boiarski, A.; Cohen, M.; West, T.; Cosentino, C.; Shapiro, J.; Ferrari, M. J. Controlled Release 2005, 102, 123–133. (5) Mulder, M. Basic Principles of Membrane Technology, 2nd ed.; Kluwer Academic Publishers B. V.: The Netherlands, 1996. (6) Bhave, R. R. Inorganic Membranes: Synthesis, Characteristics, and Applications; Chapman & Hall: New York, 1991. (7) Yamaguchi, A.; Uejo, F.; Yoda, T.; Uchida, T.; Tanamura, Y.; Yamashita, T.; Teramae, N. Nat. Mater. 2004, 3, 337–341. (8) Lee, S. B.; Martin, C. R. J. Am. Chem. Soc. 2002, 124, 11850–11851. (9) Striemer, C. C.; Gaborski, T. R.; McGrath, J. L.; Fauchet, P. M. Nature 2007, 445, 749–753. (10) Hashizume, M; Kunitake, T. Langmuir 2003, 19, 10172–10178.

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We have recently developed robust, free-standing ultrathin membranes of metal oxides by a simple spin-coating process.10-12 The thickness of these membranes is of the order of a few tens of nanometers. We could also introduce small organic molecules and proteins into such nanometer-thick membranes. Removal of these organic components resulted in cavities that retained the shape of the template molecules. Using a transmission electron microscope (TEM), we observed the cavities formed on the membrane when ferritin protein was used as the template.13 In this paper, we report the channel formation in free-standing ultrathin films of metal oxides via molecular imprinting and cavity clustering.

2. Experimental Section The schematic illustration of the film fabrication and filtration processes is shown in Figure 1. An ethanol-soluble polymer (Tokyo Ohka Kogyo, TDUR-P015) was used as the underlayer in this study. A Si wafer coated with a 500 nm layer of this polymer was supplied by Tokyo Ohka Kogyo (Figure 1-1). An aqueous solution of poly(vinyl alcohol) (PVA, Polysciences, 98 mol % hydrolyzed, Mw ∼ 78 000, 5 mg/mL, 250 μL) was spin-coated on this wafer at 3000 rpm for 2 min (Figure 1-2), and the substrate was left under ambient conditions for 1 h (Figure 1-3). In order to prepare macroscopically manipulable ultrathin films, the dependence of the thickness of the TiO2/PVA film on the concentration of the TiO2 precursor and the spin-coating speed were investigated. In order to create cavities by imprinting in the TiO2 layer, the template (4-phenylazo)benzoic acid (4PABA, Aldrich) was dissolved in a solution of titanium tetra-n-butoxide (Gelest, 100 mM in CHCl3). The molar percentages of 4PABA with respect to titanium n-butoxide were varied from 0 to 20 mol %. This precursor mixture (25 μL) was then spin-coated on the underlayer at 4000 rpm for 2 min and air-dried for 1 h (Figure 1-4, 1-5). (11) Vendamme, R.; Onoue, S.; Nakao, A.; Kunitake, T. Nat. Mater. 2006, 5, 494–501. (12) Watanabe, H.; Kunitake, T. Adv. Mater. 2007, 19, 909–912. (13) Fujikawa, S.; Muto, E.; Kunitake, T. Langmuir 2007, 23, 4629–4633.

Published on Web 06/03/2009

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Figure 1. Schematic representation of film preparation and filtration experiment.

Figure 2. Molecular structures of the template and filtrate molecules. (1) SB, λmax = 223 nm; (2) 4PABA, λmax = 325 nm in aq NH3 (1%); and (3) TPPS, λmax = 413 nm. The underlayer was dissolved by immersing the modified Si wafer in ethanol in order to detach the PVA/TiO2 film from the wafer (Figure 1-6). The film floating in ethanol was then transferred onto a porous alumina membrane (Whatman Anodisc; pore size, 0.1 μm; diameter, 25 mm; thickness 60 μm) (Figure 1-7) for scanning electron microscopy (SEM) observations and filtration experiments. The thickness of the films was determined by cross-sectional SEM observations. The 4PABA template molecule was then removed from the film by immersing the film in aq NH3 (1%) for 1 h and rinsing it thoroughly with ionexchanged water (Figure 1-8). For the filtration experiment, the TiO2/PVA film on the alumina membrane was placed on a glass filtration disk. A Si rubber ring supporter was positioned over the TiO2/PVA film to attach it to the glass reservoir containing the solution to be filtrated. This filtration apparatus was connected to a suction flask, and the pressure in the flask was reduced to 0.07 MPa in order to aspirate the solution through the filter membrane (Figure 1-9). Filtration was stopped when 1 mL of the filtrate was collected in the receiving flask. For UV-vis measurements (Shimadzu, UV3100), the filtered solution was diluted to 10 times its original concentration using ion-exchanged water. Aqueous solutions of sodium tetraphenylporphyrin tetrasulfonate (TPPS, Dojindo), 4PABA, and sodium benzoate (SB, Wako Pure Chem.) were used for the filtration experiment.

3. Results A. Thickness of the TiO2/PVA Film. The PVA underlayer apparently induces pseudo-two-dimensional cross-linking of the metal oxide network to give mechanically stable and free-standing metal oxide films, as we had discussed in our previous report.10 In order to obtain a uniform PVA underlayer, an aqueous solution of PVA (5 mg/mL, 250 μL) was directly spin-coated on a 11564 DOI: 10.1021/la9014916

Si wafer at a spin-coating speed of 3000 rpm, and the thickness of the PVA layer formed on the substrate was determined using a DEKTAK instrument. This instrument can be used to measure the height profile of a sample surface with sub-nanometer precision by line scanning using the probe tip. The maximum scanning length of the instrument was 30 mm. The maximum and minimum thicknesses of the PVA layer were measured by scanning the surface along three parallel lines separated by 20 mm (Figure 3-1). From these three measurements, the average thickness of the PVA layer was determined to be 5.8 nm. To obtain fairly robust and macroscopically manipulable ultrathin films, the preparation conditions of the TiO2/PVA film were investigated. First, the dependence of the film thickness on the concentration of the TiO2 precursor was examined. Chloroform solutions (5-100 mM, 250 μL) of the TiO2 precursor were subsequently spin-coated at 3000 rpm on the PVA-coated substrate. With an increase in the concentration of the precursor, the average thickness of the TiO2/PVA film increased from 8 to 30 nm (Figure 3-2) from DEKTAK measurements. Second, the effect of spin-coating speed on the thickness of the TiO2/PVA film was investigated and was clearly noticed only at high spinning rates, as shown in Figure 3-3. Finally, the TiO2/PVA film was detached from the substrate by dissolving the sacrificial underlayer and transferred onto an alumina membrane in ethanol. Figure 4-1 shows the digital camera image of the film being detached from the Si wafer. The detached film was found to remain intact in all experiments. Cross-sectional SEM images of the films prepared at different TiO2 concentrations are shown in Figure 4-2-4-4. The thicknesses of the films prepared from 10, 50, and 100 mM TiO2 solutions were estimated to be 15, 23, and 40 nm, respectively. Langmuir 2009, 25(19), 11563–11568

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Figure 3. Dependence of film thickness on the concentration of the TiO2 precursor and spin-coating speed. (1) Schematic illustration of the scanning process followed for determination of the thickness (DEKTAK measurements). Thickness measurements are carried out along three parallel scanning lines. (2) Effect of the TiO2 concentration on film thickness (spin-coating speed = 3000 rpm). (3) Effect of spin-coating speed on the film thickness ([Ti] = 100 mM). In all experiments, 250 μL of TiO2 solution is used for spin coating. Experiments are repeated three times under identical conditions, and the values obtained in the three cases are averaged. The thicknesses of the films are determined by DEKTAK measurements.

Figure 4. Digital camera image showing the detachment of the film from the substrate and cross-sectional SEM images of the films transferred on to the alumina membrane. (1) Digital camera image of the film during the detachment process in ethanol. (2-4) Images of the films transferred onto the alumina membrane; PVA/TiO2 composite film prepared by spin coating at 3000 rpm. The concentration of the TiO2 precursor is indicated below each image. Scale bars correspond to 200 nm.

These SEM thicknesses were similar to those obtained in the corresponding TAK measurements. The TiO2/PVA films prepared from precursor solutions of low concentrations (less than 50 mM) were fragile, and it was very difficult to manipulate the films in ethanol. Fairly robust and macroscopically manipulable films were obtained from precursor solutions whose concentrations were more than 70 mM (the thickness of the films thus obtained was approximately 30 nm or greater). From the above-mentioned observations, we concluded that a sufficiently robust ultrathin TiO2/PVA membrane, which could serve as a matrix for molecular imprinting, could be obtained by spin coating 250 μL of 100 mM precursor solution on the substrate at 4000 rpm. B. Molecular Imprinting on a Free-Standing TiO2/PVA Membrane. A mixture of 4PABA (20 mM) and the TiO2 precursor (100 mM) was spin-coated at 4000 rpm on a PVAcoated quartz substrate, which was then analyzed by UV-vis spectroscopy to confirm the existence of 4PABA molecules in the film. Figure 5-1 shows the results of this analysis. The template molecule showed an absorption peak centered at 325.5 nm. After the sample was immersed in aq NH3 solution for 1 h, the absorbance at 350 nm decreased and remained constant after 2 h. This indicated the removal of the template molecules from the film within 1 h of soaking it in aq NH3. Elemental analysis of the film was conducted by X-ray photoelectron spectroscopy Langmuir 2009, 25(19), 11563–11568

(XPS) (Figure 5-2). A small amount of nitrogen species was detected in the nonimprinted TiO2/PVA film, probably because of contamination from the XPS chamber. The N/Ti atomic ratio in the 4PABA-containing the TiO2/PVA film was 0.54. After correction for contamination, the Ni/Ti ratio is 0.43, which is close to the N/Ti molar ratio (0.4) in the original spin-coating solution. After immersion in aq NH3, the N/Ti ratio essentially reduced to zero after correction for contamination. Thus, it was confirmed that the incorporation and removal of the 4PABA template are quantitative. C. Channel Formation. We had previously reported cavity formation in a metal oxide film via imprinting of template molecules.14 The nanocavities formed could selectively rebind to the template molecules after their removal; these nanocavities could also be connected to form a channel. The amount of template required for channel formation was decided by varying the molar ratio of the template (4PABA) and TiO2 from 0/100 to 1/5. SB was used as the solute, since solute molecules smaller than the template would readily pass through the channel, if formed. In the reference experiment, a nonimprinted TiO2/PVA film ([4PABA/Ti] = 0/100) was set on the filtration apparatus. An SB solution (1.0  10-4 M) was filtrated by aspiration of the flask which was connected to the filtration apparatus. The pressure in the flask was maintained at 0.07 MPa throughout the aspiration period (1 h). No leakage of the solution was observed during the aspiration in the case of the nonimprinted TiO2/PVA film. Further, no leakage was observed when the 4PABA/Ti molar ratio was increased from 0/100 to 1/20. When the 4PABA/Ti molar ratio was 1/10, the solution passed through the film very slowly; the filtration speed was approximately 1 mL/h. The filtration speed drastically increased with the template concentration; rapid filtration occurred through the film with a 4PABA/Ti molar ratio of 1/5 (approximately 0.2 mL/min). In these experiments, the films remained undamaged, and no holes or cracks were observed during SEM observation of the films after filtration. These results strongly support the fact that imprinting of the 4PABA template on the film results in the formation of rather robust channels. At low 4PABA/Ti molar ratios, however, the number of cavities formed was not sufficient for the formation of channels across the film. D. Selective Filtration. A TiO2/PVA membrane containing 20 mol % of the template (4PABA/Ti = 1/5) was used for the selective filtration experiment. Solutions of SB (1.0  10-4 M), (14) Lee, S.-W.; Ichinose, I.; Kunitake, T. Langmuir 1998, 14, 2857–2863. (15) He, J.; Ichinose, I.; Kunitake, T. Chem. Lett. 2001, 850–851.

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Figure 5. UV-vis and XPS results obtained for the film before and after template removal. (1) UV-vis spectroscopy results obtained for the 4PABA-imprinted TiO2/PVA film transferred on a quartz plate before and after immersion ion aq NH3 (1 and 2 h). (2) XPS analysis results obtained for the TiO2 and 4PABA/TiO2 composite film before and after template removal. (*) In our XPS system, nitrogen species were detected in the nonimprinted TiO2 sample, probably due to contamination from the XPS chamber. Thus, the calculated N/Ti ratios in theTiO2/4PABA composite film before and after template removal were corrected for this contamination present in the pure TiO2 /PVA film (a).

Figure 6. UV-vis spectroscopy results obtained during the filtration experiment. (1-3) UV-vis spectroscopy data of SB, 4PABA, and TPPS, respectively. The dotted and solid lines in each spectrum correspond to the original and filtered solutions, respectively. (4) Summary of the absorbance data (column marked by “Abs”) and the absorbance ratio (column marked by “Af/Ao”) obtained at λmax of each molecule.

4PABA (1.0  10-5 M), and TPPS (5.0  10-6 M) were separately filtered in the same manner as in the leakage test. The filtration efficiency (Af/Ao) in each case was determined from the UV-vis absorbances of the filtered (Af) and original (Ao) solutions; the results are shown in Figure 6. In the case of SB, Af was close to Ao, indicating that 86% of the solute had passed through the filter membrane. In the case of 4PABA, Af was 28% of Ao. The TPPS solution was deep green in color before filtration but turned colorless after filtration; in this case, Af was less than 10% of Ao, indicating that TPPS molecules essentially did not pass through the filter membrane. E. Filtration of a Binary Mixture. In the preceding experiments, the relative filtration efficiency was examined separately for individual guest molecules. It is equally important to assess if these filtration efficiencies are determined independently from each other. A mixture solution of SB and TPPS solutions was used in the filtration experiment. The concentration of each component in the mixture was 100 mM, and absorption peaks of SB and TPS were observed at 223 and 413 nm, respectively. In spectrum of the 11566 DOI: 10.1021/la9014916

aforementioned mixture, the absorption edge of TPPS slightly overlapped with that of SB. Thus, the contribution of TPPS to the absorption at 223 nm was corrected as follows. In a singlecomponent solution consisting of TPPS only, the ratio of the absorbances at 223 and 413 nm was 0.09, and the contribution of TPPS to the absorbance at 223 nm could be given as 0.09Aobs-413. Thus, the absorbance of the SB molecules at 223 nm (ASB-223) in the mixture of TPPS and SB could be calculated by subtracting the observed absorbance of the solution at 223 nm for 0.09Aobs-413 from Aobs-413: ASB223 ¼ Aobs223 -0:09  Aobs413 After filtration of the binary mixture, Aobs-413 of the filtrate significantly decreased to 14% of that of the original solution, though Aobs-223 decreased only to 19% (Figure 7-1). The difference between the Aobs-223 values of the original solution and the filtrate was 0.021, and the absorbance of the TPPS molecules in the original solution at 223 nm was calculated to be 0.019 (Aobs-413  0.09), which was close to the above-mentioned Langmuir 2009, 25(19), 11563–11568

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Figure 7. UV-vis spectroscopy results obtained in the filtration experiments. (1) UV-vis spectroscopy data recorded before and after filtration of the SB/4PABA mixture. The dotted and solid lines in each spectrum correspond to the original and filtered solutions, respectively. (2) Summary of the absorbance observed at λ = 223.5 and 413 nm.

difference in the absorbance at 223 nm. The Af/Ao values at 223 and 413 nm were calculated to be 0.95 (0.086/0.091) and 0.14 (0.030/0.208), respectively; these values were averages of the ratios obtained in three filtration experiments. From these results, it could be stated that most of the SB molecules (95%) penetrated the membrane film, whereas only 14% of the TPPS molecules penetrated the film. Prior to this filtration experiment, it could not be confirmed whether a given guest molecule, particularly TPPS, obstructed the filtration of smaller molecules by clogging the channel. However, the results of the filtration experiment suggested that this did not happen, since the SB molecules in the binary mixture were efficiently filtered through the membrane film.

4. Discussion A. Channel Formation by Molecular Imprinting. Molecular imprinting is a well-known method for the formation of cavities that are complementary to the shape and (often) functional properties of a template molecule. Amorphous networks of metal oxides are highly flexible and can adjust their network morphology according to the shape of the template. Thus, metal oxides are considered as superior materials for molecular imprinting. In the channel formation via molecular imprinting, the molar ratio of the template molecules to the matrix molecules plays a crucial role in controlling the porosity of the imprinted material. Suppose a template molecule is surrounded by an oligomer ring (TiO2 oligomer) as shown in Figure 8-1. A 12-atom ring consisting of 6 -Ti-O- units has an external diameter of approximately 6 and 8 A˚ along the short and long axes, respectively (Figure 8-1b). This is the minimum size of the ring required to accommodate the azobenzene moiety, as can be seen in Figure 8-1. The N/Ti atomic ratio required for obtaining a ring of the above-mentioned size is less than 1/3 (2N/6Ti). Therefore, the template molecule may be isolated at lower concentrations of the template molecule in the film, and the corresponding cavities formed after removal of the template are not connected to one another (Figure 8-2a). Under the filtration conditions used in the present experiment, the molar ratio of 4PABA and TiO2 is greater than this minimal requirement. Hence, the template molecules in the film are expected to come in contact with one another, as shown in Figure 8-2b. Removal of the template molecules results in the formation of channels across the film. From the 4PABA/TiO2 molar ratio in our experiment, the TiO2/PVA film should be highly porous. The van der Waals volume of 4PABA is calculated to be 692.29 A˚3 per molecule, while that of the O-Ti-O unit involved in the tetrahedral TiO2 Langmuir 2009, 25(19), 11563–11568

Figure 8. Schematic representations of the template molecule surrounded by TiO2 and cavity formation in the film. (1) (a, b) Side and front views of a 4PABA molecule surrounded by TiO2. Color index: Ti, pale gray; C, dark gray; N, blue; O, red. (2) (a, b) Schematic illustrations of cavity formation and filtration process at low and high concentrations of 4PABA in the film, respectively.

molecule is 169.56 A˚3 per unit. Assuming that the film contains only 4PABA and TiO2 molecules, the space occupied by the 4PABA molecules is roughly estimated to be 50.5 vol % from the feed molar ratio of 20:80. This rough estimation suggests that random incorporation of the template molecule is not very efficient for the channel formation. At the same time, it is noteworthy that the highly porous ultrathin film has a mechanical robustness to maintain its macroscopic morphology during suction filtration. B. Channel Property on the Selective Molecular Filtration. The cavities imprinted in the ultrathin TiO2/PVA film provide a multifunctional wall that is complementary to the shape and function of the template molecules.14 Thus, surfaces of channels prepared under identical conditions are expected to show identical physical and chemical interactions with the guest molecules; hence, the size and chemical nature of the channel surface decide the selectivity of the channel toward the guest DOI: 10.1021/la9014916

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molecules. The guest molecules we investigated here are negatively charged under the current experimental conditions, and the electrostatic interaction between the guest and the channel surface may give a similar effect in filtration selectivity. In general, π-conjugated compounds tend to self-aggregate at high concentrations, and this aggregation leads to enhancement of apparent molecular sizes. However, in the present case, significant molecular aggregation is not expected to occur, because the solute concentrations in each solution are lower than the aggregation concentration. The examined filtrates are planar molecules, and the lengths of the long axis of SB, 4PABA, and TPPS are ca. 0.6, 1.2, and 2.0 nm, respectively. Further, the lengths of the short axis in SB, 4PABA, and TPPS are ca. 0.4, 0.5, and 1.6 nm, respectively (Figure 2). In the case of the 4PABA-imprinted film, the concentration of the filtrated molecules decreases with an increase in the lengths of their long and short axes. Apparently, the channel can recognize the size of the filtrate molecules rather precisely. In the imprinting process, the template molecules are expected to be randomly oriented in the film. If the imprinted molecules are randomly aggregated in the film, the channel formed would have a broad size distribution, losing its precise size selectivity. It should be emphasized that small size differences of less than 1 nm were distinguished in the filtration experiment. Therefore, the channel diameter is determined by the size of individual template molecule. Striemer and co-workers recently reported the fabrication of an ultrathin porous nanocrystalline Si membrane using Si fabrication techniques for the filtration and separation of proteins.9 Since protein molecules have diameters greater than several nanometers, they are separated on the basis of their charge and size. In contrast, our system is based on the morphology of small organic molecules of a few nanometer size.

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5. Conclusion and Prospects An ultrathin TiO2/PVA film with a molecular sieving property has been successfully fabricated by a combination of spin coating and molecular imprinting. It is noteworthy that molecular channels are created through connection of molecularly imprinted cavities. The membrane is free-standing and shape selective, in spite of its high porosity. Since the template molecule used in this study is simple, the structure of the channel formed in the current study is rather rudimentary, and the channel is probably formed by random linking of individual cavities. There is considerable scope for improvement of the channel morphology. Incorporation of a molecular transport machine in the molecularly thin membrane is an exciting challenge task in this area. In biological cell membranes, selective molecular separation and transport are achieved across the membrane at a distance of 5-10 nm. The thickness of lipid bilayer is sufficient for separation of molecules and ions between the inside and outside of the membrane. The transport of molecules is driven by molecular mechanisms (molecular machinery) and is highly sophisticated. In the filtration system used in the current study, filtration selectivity is derived from connected cavities formed during molecular imprinting. When filtration is carried out using a membrane with a thickness of 30 nm, molecules may migrate in the channel via random collision. With a thinner membrane, the length of the channel is reduced, and this facilitates highly efficient filtration. An additional potential expected for a thinner membrane is that molecular mechanisms could be operative at around 10 nm membrane thickness. This mechanism would provide more selective transport without relying on random molecular walk and extend possibilities to active and facilitated transports.

Langmuir 2009, 25(19), 11563–11568