Article


Nature Chemistry 1, 80 - 86 (2009)
Published online: 8 March 2009 | Corrected online: 16 March 2009 | Corrected online: 27 March 2009 | doi:10.1038/nchem.125



There is a Corrigendum (May 2009) associated with this Article.

Subject Categories: Materials chemistry | Nanotechnology

Polymer nanofibre junctions of attolitre volume serve as zeptomole-scale chemical reactors

Pavel Anzenbacher, Jr & Manuel A. Palacios1

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Methods allowing chemical reactions to be carried out on ultra-small scales in a controllable fashion are potentially important for a number of disciplines, including molecular electronics, photonics and molecular biology, and may provide fundamental insight into chemistry in confined spaces. Ultra-small-scale reactions also circumvent potential problems associated with reagent and product toxicity, and reduce energy consumption and waste generation. Here, we report a technique for performing chemical reactions on a zeptomole (10-21 mol) scale. We show that electrospun polymer nanofibres with a diameter of 100–300 nm can be loaded with reactants, and that the junctions formed between crossed nanofibres can function as attolitre-volume reactors. Exposure to heat or solvent vapours fuses the fibres and initiates the reaction. The reaction products can be analysed directly within the nanofibre junctions by fluorescence measurements and mass spectrometry, and solvent extraction of multiple reactors allows product identification by common micromethods such as high-performance liquid chromatography–mass spectrometry.


From the perspectives of both fundamental and applied chemistry, the ability to carry out chemical reactions at a sub-attomole scale is highly desirable. If performed in a reliable and controlled manner, such methods would benefit numerous research and development efforts by providing data on relative reactivity alongside high-throughput liquid chromatography–mass spectrometry (LC-MS) analyses, while using small samples for preliminary analyses where reagent/product stability is a concern1, 2. Other desirable features of these methods include circumvention of potential problems associated with reagent and/or product toxicity and reduction of the amounts of reagents used, and energy consumption and waste during initial reaction testing. On a more fundamental level, methods for detection of chemical species reaching detection limits as small as individual molecules are available3, 4. These include methods based on metal nanoparticles5 or semiconductor quantum dots6, as well as methods using gas chromatography7 or capillary electrophoresis8 that allow the detection of ultra-low concentrations of individual components, which may be analysed spectroscopically9 or by mass spectrometry10.

Whereas analytical techniques have made strides towards sub-attomole detection limits, reactors capable of the preparation of chemical species on an ultra-small scale are not readily available, are expensive or do not allow effective control of chemical reactions11. This seems to be largely due to the complex methods of microreactor fabrication11, 12. In the case of sub-nanolitre-volume reactors, two main approaches have emerged: a top-down method of reactor fabrication using microfluidics13 and 'lab-on-a-chip' technologies14, 15.

The use of nanodroplets transported through microfluidic channels in a synchronized fashion allows reagent-carrying droplets to react in a controlled manner while eliminating dispersion16, 17, accelerating mixing18 and providing control over the chemistry19. This technique also allows screening of reactions, including nanoparticle synthesis20, protein crystallization21, DNA assays22, organic syntheses23 and combinatorial screening24.

The bottom-up approach is best illustrated by self-assembled molecular-scale reactors25. These can use inorganic materials such as zeolites26 and other mesoporous materials27. An alternative approach based on organic materials uses molecular reagent-sized cavities in self-assembled capsules28, imprinted polymers29 or supramolecular containers30. Some molecular-scale reactors exhibit catalytic activity, and in several cases impressive efficiencies have been observed31, 32. In spite of the indisputable achievements, most of these techniques seem to be far from easy to implement, and the attendant costs of synthesis and device preparation remain an obstacle.

We present here an easy-to-perform and versatile method enabling chemical reactions in femto- to attolitre volumes on a scale down to 1,000 molecules, that is, the zeptomole (1 zmol = 10-21 mol) scale. In theory, this method allows reactions to be carried out that potentially scale down to two single molecules. The method is based on the deposition of a rectangular grid of two nanofibre types, each nanofibre being doped with a different reagent (Fig. 1a). Heat or solvent vapour-welding of the softened polymer (for example polyurethane Tecoflex) nanofibres results in mixing of the contents of the fibres at the intersection, thereby establishing a mixed junction. In the example illustrated in Fig. 1b–d, polymer electrospinning33 was used to fabricate rectangular nanofibre grids34, 35, and the fibre fusion was performed by heating or simple exposure to solvent vapours. Solvent-vapour-mediated coalescence (similar to the 'welding' of polymer nanofibres demonstrated earlier by Sotzing and co-workers36 and to careful heat-welding of overlaid fibres in the form of nanofibre mats) does not result in collapse of the fibrous structure, and thus allows multiple junctions to be established in a small volume.

Figure 1: The principle of attolitre reactor formation.

Figure 1 : The principle of attolitre reactor formation.

Polymer nanofibres doped with reagents are overlaid to form a rectangular grid-like mat. Fusing the nanofibres at their intersections leads to mixing of their contents in the junctions to create an attoreactor containing zeptomole amounts of reagents and products. a, Schematic representation of the principle. b,c, Scanning electron microscope (SEM) images of electrospun polyurethane nanofibres overlaid in a rectangular grid. d, SEM image of the fused fibres.

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Figure 2 shows an example of a reaction carried out in a nanofibre junction. Neither of the starting materials, dansyl chloride (1-dimethylaminonaphthalene-5-sulfonyl chloride, 1Compound 1) and triethylenetetramine (2Compound 2), is fluorescent. The product, a dansylamide (3Compound 3), is brightly fluorescent37, which allows easy visualization of the reaction carried out in the attoreactor. After the nanofibre junction fusion, a combination of surface profilometry and scanning electron microscope (SEM) measurements indicates that the junctions are 90 nm high and 200 nm wide. The average volumes of the junctions, defined as the mixed polymer content, are approximately 5 plusminus 1 attolitres (al). The concentration of the reagents in the fibre after evaporation of the solvent is 0.5 mmol l-1, which in 5 al corresponds to about 2.5 zmol (approximately 1,500 molecules) of each reagent in an average attoreactor. However, the reagent concentration in the carrier polymer could easily be downscaled to achieve the formation of fewer product molecules, even just one molecule. In this study, we focused on proof of concept, and therefore clearly observable numbers of product molecules (about 1,000) were prepared to avoid the single-chromophore cycling between the excited and ground states resulting in photobleaching that occurs in most organic chromophores after the order of 105 cycles. Deposition of multiple-fibre mats containing different reactions, using a shadow mask to obtain well-defined reactor regions on a small substrate, is also possible (see Supplementary Information). This method will be tested for application in high-throughput screening.

Figure 2: Formation of reaction products at the junction of crossed polymer nanofibres.

Figure 2 : Formation of reaction products at the junction of crossed polymer nanofibres.

a, Reaction between two non-fluorescent reagents yields a fluorescent product; this may be used to visualize the attolitre reactor and the zeptomole amount of the product formed. Unlike the reagents, dansyl chloride (1Compound 1) and triethylenetetramine (2Compound 2), the product of the reaction, a dansylamide (3Compound 3), is fluorescent. b–e, Optical micrographs of attoreactors showing how a fluorescent product formed from two non-fluorescent reagents is used as a diagnostic signal. b, A single attoreactor comprising 500–1,000 molecules (bright field + ultraviolet image) separated by a wide margin allows harvesting of individual reactors. c, The same attoreactor on ultraviolet light excitation. d, Attoreactors deposited in a random mat. e, High-density rectangular network of nanofibres and reactors.

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The present approach is quite general. We were able to observe the diffusion of small donor–acceptor molecules between the fibres using fluorescence resonance energy transfer (FRET)38 between coumarin 521 (4Compound 4, donor) and rhodamine 6G (5Compound 5, acceptor) (Fig. 3). Because FRET efficiency decreases with the sixth power of the donor–acceptor distance, it is ideal to observe the donor and acceptor coming into close proximity in the attoreactor junction. The Förster distance (the distance at which FRET reaches 50% efficiency) for the above donor–acceptor pair is 5.11 nm. This experiment reveals that diffusion occurs from a distance of about 50–100 nm to 5.1 nm and less, that is, to distances where the chemical reaction might take place. The diffusion of this donor–acceptor pair in polyurethane fibres resulted in high FRET efficiency (>95%).

Figure 3: Reagent diffusion in nanofibre junctions.

Figure 3 : Reagent diffusion in nanofibre junctions.

Fluorescence resonance energy transfer (FRET) between a coumarin 521 donor in the horizontal fibre and a rhodamine 6G acceptor in the vertical fibre illuminates the diffusion of molecules within the nanofibre junction. a, Structures of the donor coumarin 521 (4Compound 4) and the acceptor rhodamine 6G (5Compound 5). b, Bright-field image of a four-reactor section. c, Image of the same section while both the donor and the acceptor are excited. d, The donor is excited, and donor emission is observed. In the junctions where the donor energy is transferred to the acceptor, quenched subsections corresponding to the mixed junctions within the fibre are observed. e, The donor is excited, energy is transferred to the acceptor, and acceptor emission is observed. The calculated FRET efficiency is >95%. Details of FRET efficiency calculation can be found in the Supplementary Information. Owing to the diffraction limit, the calibration bar cannot be used to estimate the diameter of the nanofibres, and serves merely to convey information about the distance of the reactors in the grid. See Supplementary Information for complete characterization of the fibres.

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The second set of experiments was focused on actual chemical reactions: two reactions using non-fluorescent reagents that produce fluorescent products were tested, with very similar results. These reactions were acylation of triethylenetetramine (2Compound 2) by dansyl chloride (1Compound 1) (shown in Fig. 2) and nucleophilic substitution of 4-chloro-7-nitro-2,1,3-benzoxadiazole (6Compound 6, NBD-chloride) with an amine (see Supplementary Information). In both cases, fluorescent products were formed in the junction. These experiments confirmed not only that the content of the fibre junctions is physically mixed, but also that the reagent molecules diffuse close enough to each other to undergo chemical reactions.

In practice, it is desirable that the reaction products be obtained in a form and amount that allows product purification, identification and potentially also quantitation. Indeed, in many instances, the polymer attoreactor method may be easily adapted to allow reaction scale-up and product isolation. We therefore chose an example reaction that could allow us to address these issues, and provide further insight into the factors that control reactions in polymer nanofibre junctions.

The diazo-coupling reaction39, 40 of activated aromatics such as resorcinol is an excellent model. Resorcinol (7Compound 7) reacts with 4-nitrobenzenediazonium tetrafluoroborate (8Compound 8) to yield chiefly products of multiple substitution. Even in the case of 1:1 reagent stoichiometry, when this reaction is carried out in fluid solution, the abundance of the individual products is as follows: disubstituted (10Compound 10) > trisubstituted (11Compound 11) > monosubstituted (9Compound 9) (Fig. 4a). This is because of the intramolecular hydrogen bond between the unshared electron pairs of the azo moiety and the proton of the phenolic group. This hydrogen bonding results in partial intramolecular charge transfer to the oxygen and the aromatic ring, which imparts the character of a phenolate anion, which is more reactive than the corresponding phenol.

Figure 4: The diffusion-limited nature of reactions in an attoreactor.

Figure 4 : The diffusion-limited nature of reactions in an attoreactor.

a, As a result of hydrogen-bonding-mediated activation, resorcinol (7Compound 7) reacts with 4-nitrobenzenediazonium tetrafluoroborate (8Compound 8) to yield multiple products of electrophilic substitution. b, In an attoreactor, where the reaction is diffusion-limited rather than activation-limited, only the monosubstituted product (9Compound 9) is observed. c, When the same reaction is performed in solution, disubstituted (10Compound 10) and trisubstituted (11Compound 11) products are also formed. All products 9Compound 911Compound 11 may be detected directly within the polymer matrix using MALDI-MS. Quenching the unreacted diazonium salt (8Compound 8) with dimethylamine (12Compound 12) before solvent extraction, resulting in the formation of a triazene (13Compound 13), allows the isolation of 50–500 microg amounts of products suitable for LC-MS analysis and NMR spectroscopy measurements using a nanoprobe. Ar = 4-nitrophenyl.

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The same reaction performed in a polymer attolitre reactor yields solely the monosubstituted product 9Compound 9 (Fig. 4b). When the nanofibres are deposited directly onto a matrix-assisted laser desorption/ionization mass spectrometry (MALDI-MS) target, the reactors can be sampled directly in the spectrometer without the use of an additional matrix. To demonstrate that the monosubstituted product is observed as the only compound volatilized from the polyurethane fibres and that the other products would have appeared in the mass spectrum if they had been formed, we electrospun the solution-synthesized mixture of products 9Compound 911Compound 11 onto a MALDI target and recorded the corresponding mass spectrum (Fig. 4c). As expected, all three diazo-coupling products were clearly observed. These experiments suggest strongly that the reactions taking place in the polymer nanofibre junctions are diffusion- rather than activation-limited, a feature that will be further explored to harness the utility of the reactions performed with this method.

The amount of products formed in single junctions is exceedingly small, but does allow product identification by ultrasensitive methods, such as capillary electrophoresis (CE) and gas chromatography coupled to mass spectrometry7, 10, both of which are capable of achieving zeptomole detection. Common laboratory micromethods require larger amounts of the sample (of the order of micrograms). To maintain the advantage of ultra-small reaction scales using this simple method, the scale-up is performed by fabricating dense nanofibre mats, in which the number of attolitre reactors may easily reach 106–109 per square centimetre. In the MALDI-MS experiment, the laser beam samples a number of junctions simultaneously, which allows a strong signal to be obtained (Fig. 4b).

The isolation and further characterization of the products present a unique set of problems, because it is necessary to prevent the reaction between the unreacted reagents taking place outside the junctions, for example during extraction. Hence, either the unreacted starting materials must be non-reactive at room temperature or the reagents must be quenched before removing the polymer. In such cases, the products may be easily obtained by solvent extraction of the nanofibre mat using a solvent or a solvent mixture that does not dissolve the polymer used to fabricate the fibres, but dissolves the products of the reaction or reactions. In our example, the unreacted diazonium salt was quenched by exposure of the fibres to dimethylamine (12Compound 12) vapours to yield the corresponding triazine (13Compound 13) in an instantaneous reaction41 (Fig. 4a). Here, care must be taken that the solvent or reagent vapours do not compromise the nanofibrous structure of the mat.

The poly(ether–urethane) used as a matrix is insoluble in both acetone and ethyl acetate, which were therefore used to dissolve the reaction products, namely the substitution products 9Compound 9, 10Compound 10 and 11Compound 11, the triazine 13Compound 13, and unreacted resorcinol. MS analysis using a direct insertion probe reveals the presence of the monosubstituted product and unreacted resorcinol. Contamination of the extracted sample by polymer fragments or shorter chains soluble in the extraction solvent can be minimized by treating the polymer with these solvents (before reagent doping and fibre fabrication) to remove soluble components such as short oligomers. It is conceivable that the product extraction from larger nanofibre mats followed by purification by high-performance liquid chromatography (HPLC) could also generate sufficient material for NMR spectroscopy, using a nanoprobe for recording spectra from less than 10–20 microg of the sample42.

This example illustrates that reactions performed in the attolitre polymer reactors may be conveniently sampled directly, that is, while still encased in the polymer by MALDI-MS, and that the reaction products may be isolated by extraction either immmediately or after the unreacted reagents have been quenched. The observation that the base used to quench one of the reagents can do so efficiently is also important, because it implies that breathable polymers43, 44 can be used to deliver a third component such as an acid or base catalyst to the reactors in the form of a vapour. Thus, the scope of the reactions may be extended to three-component reactions.

Large-molecule reagents may present a potential problem in diffusion-limiting environments such as polymer matrices, because their ability to diffuse through the matrix is limited. We have performed several reactions in which one of the reagents was a polymer, for example the reaction of poly-l-lysine (14) with dansyl chloride (1Compound 1) or fluorescamine (15Compound 15) and the reaction of a cyclopentadienone polymer with an acetylene (see below). In general, although they are not impossible, reactions requiring four or more components or more than one polymer reagent are expected to be significantly more difficult to carry out using this method, owing to difficulties in achieving balanced diffusion of the reagents or catalyst in an attolitre reactor. Our preliminary results suggest that this problem may be partly mitigated by the use of porous and breathable polymers or polymers with large interstitial space (free volume)45 to facilitate migration of the reagents through the matrix, because it is well known that the free volume in polymers has an important role in transport of small molecules and gases46.

The above reactions, despite their differences in mechanisms, all yielded a product, thereby suggesting the versatility of the method. In general, the reaction times are between 1 and 90 minutes, depending on the temperature and the nature of the reagent. Small-molecule reagents and reagents that are liquid at room temperature react more quickly. The reaction temperatures tested were 40–120 °C. Starting materials and products may be extracted from the polymer mats by an appropriate solvent that does not dissolve the polymer. Furthermore, the amount of products obtained is easy to scale up without compromising any of the advantages of this method. Dense or larger mats (with the order of 1012–1013 junctions per cm2) yield sufficient amounts of products for analysis using conventional LC-MS, infrared spectroscopy and 1H-NMR spectroscopy using a nanoprobe42.

A model Diels–Alder reaction of tetraphenylcyclopentadienone (16Compound 16) with dimethyl acetylenedicarboxylate (17Compound 17) to yield dimethyl tetraphenylphthalate (18Compound 18) was performed, and the product was extracted from the nanomat and analysed by gas chromatography-MS (see Supplementary Information). Also, because of the success of the experiments with small-molecule reagents, we attempted a similar experiment using polymeric reagents. We synthesized the material 19, a copolymer of 9,9-dialkylfluorene and tetraphenylcyclopentadienone (Fig. 5a), which was mixed with a polyurethane carrier before nanofibre deposition. Owing to the presence of the cyclopentanedienone (Cp) moiety, 19 is dark in colour, with only low fluorescence (Fig. 5b). A heat-induced Diels–Alder reaction with an activated alkyne such as 17Compound 17 followed by carbon monoxide extrusion at temperatures 100–120 °C results in the formation of a polyarylene-type material 20, in which the cyclopentanedienone are converted to arylene moieties (Fig. 5c). The elimination of the Cp quencher results in an order-of-magnitude increase in fluorescence intensity in the attoreactor junction (Fig. 5d,e), suggesting that the mixing of the junction content is effective even when one reagent is polymeric.

Figure 5: Reaction of a macromolecular substrate in an attoreactor.

Figure 5 : Reaction of a macromolecular substrate in an attoreactor.

a, Diels–Alder reaction of a cyclopentadienone–fluorene p(CpF) copolymer (19) with an activated acetylene, dimethyl acetylenedicarboxylate (17Compound 17): the reaction of a fibre composed of a blend of polyurethane and 19 with a second fibre containing 17Compound 17 results in the formation of a polyarene (20) at the junction. b,  A bright-field image of 400 nm wide (150 nm height) nanofibres shows the dark-coloured 19 in the horizontal fibre and 17Compound 17 in the vertical fibres. c, An image obtained with ultraviolet light excitation indicates the formation of the fluorescent polyarene product 20 at the junctions of the nanofibres. d, Quasi-3D (xy–intensity) representation of the attoreactors. e, False-colour intensity representation of the attoreactors. Note the residual fluorescence of the aromatic p(CpF) moieties in the horizontal fibre.

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In another example, poly-l-lysine (14), used as a model for proteins, was successfully labelled with dansyl chloride (1Compound 1), yielding essentially similar images to those in Figs 2d,e. This is an important finding, because it is possible to envisage a reaction in nanofibres where one of the fibres is doped with a protein to yield various derivatization and labelling schemes. Likewise, screening of nucleic acid sequences versus intercalators using this technique could easily be conceived.

Finally, the use of breathable or hydrophilic polymers allows rapid mass exchange between the attoreactors and the surrounding gases or liquids46, suggesting that reagents or catalysts could be delivered to the species formed in the attoreactor junctions. This feature could be used in molecular sensing, where the nanoscopic dimensions of sensors synthesized in the junctions would allow an instantaneous response. Thus, arrays of optical sensors could be fabricated in a combinatorial fashion to generate arrays of nanosensors. As a proof of principle, we used the dansylamide 3Compound 3 (Fig. 2a), which responds to the presence of heavy-metal ions such as Co2+ or Cu2+ with a marked decrease in fluorescence intensity37. As expected, a microdroplet of Co2+ solution in water resulted in instantaneous quenching of dansylamide fluorescence, suggesting the feasibility of in situ-prepared sensors within attoreactors as a basis for nanosensor arrays with fast analyte response.

In our investigations of reactions within polymer junction attoreactors, regardless of the existence of microdomains44 within the copolymer, we did not observe any uneven distribution of reagents or products within the fibres or attoreactors. Diffusion of small molecules within the nanofibres was also observed. We found that effective control of diffusion was possible with appropriate selection of the polymer matrix. Harder polymers suppress diffusion of the reagents, and softer matrices allow diffusion outside confinement by the junction. In this way, it is possible to use a large portion of the reagents in the deposited nanofibres. This approach has several further advantages. It is possible to limit the number of product molecules by using an extremely small reactor volume (in the attolitre range) rather than through dilution of the reagents. In fact, the concentration of reagents in the present attoreactors is relatively high (about 1–10 mmol), which allows fast diffusion and short reaction times, typically 1–90 minutes, depending on polymer hardness.

Attolitre reactors allowing chemical, analytical and biochemical reactions to be carried out on ultra-small scales in a controllable fashion have great potential in various fields of science and technology. This approach could allow reductions in reagent and energy consumption, and, in conjunction with MALDI-TOF or LC/FPLC-MS techniques, could provide data regarding the relative reactivities of ultra-small amounts of reagents. Current micro- and nanoreactors1, 11, 12 are far from inexpensive, and do not always allow chemical reactions to be performed with spatial and temporal definition under controlled conditions, nor do they allow delivery of a catalyst or quenching reagents. In an attempt to circumvent some of these drawbacks, we have developed a method in which chemical reactions can take place in attolitre volumes on a zeptomole scale (typically with the order of 1,000 molecules and less). This method uses polymer nanofibres with diameters 100–300 nm doped with reagents in such a way that fibres containing one reagent are overlaid with fibres doped with a second reagent, preferably at a 90 ° angle to form a rectangular nanofibre mat. Exposure of such mats to heat or solvent vapours results in coalescence of the fibres at their intersections, which results in mixing of the contents of the two fibres and in the two reagents coming into sufficiently close proximity to allow a physical interaction (FRET) or a chemical reaction within a semisolid immobile 'vessel', the attolitre reactor. The individual polymer attoreactors may be harvested individually from low-density mats (Fig. 2b,c) or manipulated collectively as a nanofibre mat.

This method is versatile, and various types of reactions have been carried out in polymer attoreactors, including labelling a polypeptide with fluorescent markers, and identification of reaction products by optical microscopy (when fluorescent), by direct detection in mats of nanofibres using MALDI-TOF or by LC-MS after product extraction. Both small-molecule and polymer reagents may be used in this method. This technique may yield methods to test new reactions, label proteins, explore nucleic acid intercalators, or fabricate optical nanosensor arrays. The use of attoscale reactors has several key advantages: adjustment of the reagent concentration used for electrospinning allows the reactions to be scaled up or down to a few molecules; reactors can easily be fabricated by techniques readily available for polymer fibre fabrication; and products and reagents are confined to the polymer matrix, but can be extracted after the reaction. A variety of polymers compatible with the reagents, including biopolymers, are available, adding versatility to this inexpensive and easy-to-perform method. On a more fundamental level, our results suggest that this method could also generate new composite materials with interesting photoactive entities formed in situ. Such host–guest chromophoric systems could provide information about the environments involved in dynamic processes within polymer matrices47.

Inspired by the work of Craighead and co-workers48 on identification of single biomacromolecules in electrospun fibres, our efforts in the near future will focus on scaling down the experiments to the ultimate frontier of single chromophores. Furthermore, as we have developed a method for deposition of multiple fibres carrying different reagents through a shadow mask onto small substrates (see Supplementary Information), we will use this method in high-throughput reaction and reagent screening.

* In the version of this Article originally published, in paragraph six of the main text, the concentration of reagents in the fibre after evaporation of the solvent was incorrectly given as 0.5 mol l-1, this should have read 0.5 mmol l-1. This has been corrected in the HTML and PDF versions of the Article.

** In the version of this Article originally published, the structure given in Fig. 3a for rhodamine 6G (compound 5) was incorrect. The error has now been corrected in the HTML and PDF versions.
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Acknowledgements

Financial support from the NSF (CHE No. 0750303, EXP-LA No. 0731153 to P.A.) and Bowling Green State University is gratefully acknowledged.

Author Contributions

P.A. and M.A.P. contributed equally to this work.

Received 24 September 2008; Accepted 28 January 2009; Published online 8 March 2009; Corrected 16 March 2009; Corrected 27 March 2009.

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  1. Center for Photochemical Sciences and Department of Chemistry, Bowling Green State University, Bowling Green, Ohio43403, USA

Correspondence to: Pavel Anzenbacher, Jr e-mail: pavel@bgsu.edu


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