pubs.acs.org/Langmuir © 2010 American Chemical Society
Characterization of a Reverse Micellar System by 1H NMR Jean-Luc Lemyre and Anna M. Ritcey* D epartement de chimie and CERMA, Universit e Laval, Qu ebec, Canada Universit e Laval, Pavillon A.-Vachon, 2240-C, G1 V 7P4 Received October 23, 2009. Revised Manuscript Received December 16, 2009 The 1H NMR spectrum of IgepalCO520 in ternary mixtures containing water and cyclohexane shows a complex dependence on water content. This is in part because of rapid exchange between surfactant molecules within the micelles and free surfactant dissolved in the continuous phase. The analysis of this two-state system is further complicated by the fact that the chemical shifts of both free and micellar surfactants vary with micelle size. We demonstrate that the relative quantities of free and micellar surfactants can be determined from the NMR spectra if the data are compared within sample sets of constant micelle size but differing global composition. By fixing micelle size, the spectra of both surfactant states remain constant within a given series and only the relative populations of the free and micellar species change with overall composition. This method of analysis allows for the determination of free surfactant concentration as a function of micelle size. Results are presented for the water/IgepalCO520/cyclohexane system and indicate that the free surfactant concentration is far from negligible and strongly dependent on micelle size. The free surfactant concentration increases with decreasing micelle size, reflecting the lower stability of the smaller micelles. Similar behavior can be expected for other reverse micellar systems based on non-ionic surfactants.
I. Introduction Reverse microemulsions are dispersions of surfactant-stabilized water in a nonpolar solvent. These systems are thermodynamically stable and optically transparent and present complex phase diagrams.1,2 This paper focuses on the specific region of the phase diagram corresponding to reverse micelles in equilibrium with free surfactant dissolved in the continuous oil phase. The concentration of free surfactant, while often negligible for ionic surfactants, can be considerable in the case of non-ionic surfactants. Reverse micelles formed from non-ionic surfactants can be viewed as spherical self-assembled aggregates in which the hydrophilic segment of the surfactant is co-mixed with water in the core and the hydrophobic tail extends into the organic phase. Reverse micelles are of great interest for both fundamental science and technological applications. They have been employed, for example, as enzymatic reaction media3-5 and models for biological systems6 and for the synthesis of inorganic nanoparticles,7-10 the extraction and purification of biomolecules,11,12 and drug delivery.13 All of these applications could benefit from a better understanding and a more complete characterization of reverse microemulsions. *Phone: 418-656-2368. Fax: 418-656-7916. E-mail
[email protected]. (1) Hellweg, T. Curr. Opin. Colloid Interface Sci. 2002, 7(1-2), 50. (2) Sager, W. F. C. Microemulsion Templating. In Nanostructured Soft Matter, Zvelindovsky, A. V., Ed.; Springer: The Netherlands, 2007. (3) Biasutti, M. A.; Abuin, E. B.; Silber, J. J.; Correa, N. M.; Lissi, E. A. Adv. Colloid Interface Sci. 2008, 136(1-2), 1. (4) Miyake, Y. Colloids Surf., A 1996, 109, 255. (5) Volkov, A. G. Interfacial Catalysis; Marcel Dekker: New York, 2003. (6) Van Horn, W. D.; Ogilvie, M. E.; Flynn, P. F. J. Am. Chem. Soc. 2009, 131 (23), 8030. (7) Lemyre, J.-L.; Ritcey, A. M. Chem. Mater. 2005, 17(11), 3040. (8) Eastoe, J.; Hollamby, M. J.; Hudson, L. Adv. Colloid Interface Sci. 2006, 128-130, 5. (9) Pileni, M.-P. Nat. Mater. 2003, 2(3), 145. (10) Destree, C.; Nagy, J. B. Adv. Colloid Interface Sci. 2006, 123-126, 353. (11) Mathew, D. S.; Juang, R.-S. Sep. Purif. Technol. 2007, 53(3), 199. (12) Krishna, S.; Srinivas, N.; Raghavarao, K.; Karanth, N., Reverse Micellar Extraction for Downstream Processing of Proteins/Enzymes. In History and Trends in Bioprocessing and Biotransformation, Springer-Verlag: Berlin, 2002; p 119. (13) M€uller-Goymann, C. C. Eur. J. Pharm. Biopharm. 2004, 58(2), 343.
6250 DOI: 10.1021/la904033e
One of the attractive features of reverse micelles is the ease with which micelle size can be controlled through the water-to-surfactant ratio.14 The relationship between composition and micelle size is, however, somewhat more complicated when the free surfactant concentration cannot be neglected. In such cases, micelle size, aggregation number, free surfactant concentration, the number of micelles, and the molecular area of surfactant at the water/organic interface can all simultaneously change as the global composition of the system is modified.15 This variability leads to difficulty in the interpretation of data obtained from characterization techniques such as NMR spectroscopy. Because of the presence of both free and aggregated surfactant in rapid exchange, the NMR spectrum will appear as the weighted average of the spectrum corresponding to each of the two species.16-18 Each of these contributing spectra, however, also depends on composition. This is because the conformation and environment of the micellar surfactant molecules and the concentration of free surfactant all vary with micelle size. NMR spectra recorded as a function of composition will therefore be modified by both changes in the chemical shift and in the relative populations of free and micellar surfactant. In this paper, we demonstrate that the interpretation of surfactant spectra can be greatly simplified by comparing samples of differing global composition but fixed micelle size. With this approach, all parameters affecting the chemical shifts of the free and aggregated species (aggregation number, surfactant hydration and conformation, and free surfactant concentration) are fixed, and systematic changes in the overall spectrum can be attributed solely to changes in the relative (14) Pileni, M. P. J. Phys. Chem. 1993, 97(27), 6961. (15) Lemyre, J.-L.; Lamarre, S.; Beaupre, A.; Ritcey, A. M., Langmuir 2009, Submitted. (16) Du, Y. R.; Zhao, S.; Shen, L. F. Nuclear magnetic resonance studies of micelles. In Annual Reports on NMR Spectroscopy; Academic Press: 2002; Vol 48, p 145. (17) Llor, A.; Zemb, T. What can be expected from NMR in reversed micelles. In Structure and reactivity in reverse micelles, Pileni, M. P., Ed.; Elsevier: Amsterdam, 1989; Vol 65, p 54. (18) Fendler, J. H.; Fendler, E. J. Catalysis in Micellar and Macromolecular Systems; Academic Press: New York, 1975.
Published on Web 01/25/2010
Langmuir 2010, 26(9), 6250–6255
Lemyre and Ritcey
quantities of free and micellar surfactant. In this way, the free surfactant concentration can be determined as a function of micelle size. Our particular interest in reverse micelles is for their use in the synthesis of inorganic nanoparticles. Although the extraordinary control over particle size offered by this method of nanoparticle synthesis is well-documented,7,14,19,20 the detailed mechanism of particle nucleation and growth remains unknown. Monte Carlo simulations have been used to model the mass distribution of the micelle contents during random collisions and particle growth.21,22 The testing of such models, as well as their use for predicting the appropriate conditions for obtaining a desired particle size, however, requires knowledge of characteristic parameters such as the aggregation number and the quantity of free surfactant. Relatively few methods have been developed to measure the free surfactant concentration in reverse microemulsions. Optical spectroscopic techniques are typically of little use because the spectra of free and micellar surfactant are indistinguishable. Recently, we have demonstrated that DLS measurements, in conjunction with an appropriate geometric model, can lead to a number of important system parameters, including the concentration of free surfactant as a function of reverse micelle size.15 The present article employs a complementary method for the determination of free surfactant concentration based on 1H NMR spectroscopy, assisted by DLS. The method is applied to the specific system composed of polyoxyethylene(5)nonylphenyl ether, better known under the commercial name of Igepal CO520 in cyclohexene, but may be extended to other non-ionic systems.
Article Table 1. Examples of Global Compositionsa Resulting in Reverse Micelles of Equal Size reverse micelle size (nm)
surfactant mass (mg)
water volume (μL)
5
75 96 110 128 138 62 80 94 111 119 47 62 74 91 96 36 47 58 73 76
6.7 16.7 33.3 50.0 66.7 6.7 16.7 33.3 50.0 66.7 6.7 16.7 33.3 50.0 66.7 6.7 16.7 33.3 50.0 66.7
6
7.7
10
a
per mL of cyclohexane.
II. Experimental Section All chemicals were supplied by Aldrich and used as received without further purification. Water was first deionized and then purified to 18.2 MΩ 3 cm by a Barnstead NanoPure II purification system. Microemulsions were prepared by mixing varying amounts of water, the surfactant Igepal CO520 (polyoxyethylene(5)nonylphenyl ether), and cyclohexane. Homogeneous microemulsions were obtained by mixing with a magnetic stirrer followed by 10 min in an ultrasonic bath. Global compositions were selected to yield sample sets of constant micelle size. Table 1 presents the exact global compositions employed for four of the numerous micelle sizes considered. Global compositions resulting in identical micelle size were identified by dynamic light scattering measurements (DLS),15 performed at 25 °C using by a Malvern instrument (Zetasizer Nano ZS). Measurements were carried out without sample dilution. The dynamic viscosity of each sample, required for the calculation of particle size from the StokesEinstein equation, was obtained from kinematic viscosities measured with an Ubbelohde viscometer and densities determined with a pycnometer. 1 H nuclear resonance magnetic measurements were performed at 25 °C on a 400 MHz Varian Inova spectrometer. Samples were prepared using an 80:20 mixture of cyclohexane and cyclohexaned12, and the cyclohexane signal was presaturated for each acquisition. Tetramethylsilane served as the internal standard for chemical shift calibration. The uncertainty associated with the chemical shifts evaluated from these spectra is estimated to be 0.0025 ppm. (19) Boutonnet, M.; Kizling, J.; Stenius, P.; Maire, G. Colloids Surf. 1982, 5(3), 209. (20) Capek, I. Adv. Colloid Interface Sci. 2004, 110(1-2), 49. (21) Ethayaraja, M.; Dutta, K.; Bandyopadhyaya, R. J. Phys. Chem. B 2006, 110(33), 16471. (22) Ethayaraja, M.; Dutta, K.; Muthukumaran, D.; Bandyopadhyaya, R. Langmuir 2007, 23(6), 3418.
Langmuir 2010, 26(9), 6250–6255
Figure 1. 1H NMR spectrum of Igepal CO520 in cyclohexane. Resonances are assigned as indicated on the molecular structure provided in the inset. An expansion of the region of the spectrum assigned to the oxyethylene protons is also provided.
III. Results and Discussion H NMR Spectrum of Igepal CO520. The 1H NMR spectra of Igepal CO520 in cyclohexane is presented in Figure 1. Resonances associated with the hydrophobic alkyl chain (A) appear between 0.5 and 1.6 ppm, those assigned to the aromatic protons (B) around 6.7 to 7.1 ppm and the oxyethylene (C, D, and E) protons between 3.4 and 4.0 ppm. As illustrated in Figure 2, creation of a reverse microemulsion by the addition of water significantly modifies the spectrum. The most notable alterations are observed in the oxyethylene region of the spectrum, reflecting the change in environment and conformation that occur as the hydrophilic segment of the surfactant is sequestered into the aqueous phase. This region is expanded in the inset of Figure 1. The oxyethylene protons that are the furthest from the aromatic ring (C) appear as relatively large unresolved signals between 3.4 and 3.6 ppm. Protons D and E appear as resolved triplets centered at 3.7 and 4.0 ppm, respectively. It is this latter signal that is the prime focus of this study. As shown in Figure 2, the resonance 1
DOI: 10.1021/la904033e
6251
Article
Lemyre and Ritcey
1
Figure 2. H NMR spectra of Igepal CO520 in cyclohexane in the absence (top) and presence (bottom) of water, at similar surfactant concentrations. Resonances are assigned as indicated on the molecular structure provided in the inset.
Figure 4. Chemical shift of peak E of IgepalCO520 as a function of the mass of surfactant dissolved per milliliter of cyclohexane in the absence of water.
are in rapid dynamic equilibrium with free molecules dissolved in the continuous phase, the observed spectrum is an average of the signals arising from the two species.16-18 The effective chemical shift can be expressed as the weighted mean of the chemical shifts associated with all possible surfactant states, as given by δeff ¼ Σχi δi
ð1Þ
where χi and δi are, respectively, the mole fraction and the chemical shift of the ith state (Σχi = 1). In the case of the current system, it can be reasonably assumed that each surfactant molecule resides in one of two limiting states: either as free surfactant in cyclohexane solution or within a micelle. Using eq 1, it should then be trivial to determine the free surfactant concentration from δeff ¼ χfs δfs þ χms δms
Figure 3. Peak E from the 1H NMR spectra of reverse microemulsions at constant surfactant and increasing water concentrations. From left to right: 0, 6.7, 16.7, 33.3, 66.7, 100 μL of water with 133 mg surfactant per milliliter of cyclohexane.
shifts to lower frequency and broadens upon the addition of water. The presence of water can also be clearly seen in the lower spectrum of Figure 2, appearing as an intense signal centered at 4.67 ppm. The chemical shift of this peak is strongly influenced by the exact composition of the system, systematically shifting from 4.1 to 4.8 ppm as the water content is increased, as is anticipated for increased hydrogen bonding. Attempts to utilize this shift to deduce information about the system failed, however, because the water protons are in rapid exchange with the surfactant alcohol proton, which is present in both free and micellar forms. The detailed changes in the 1H NMR signal of proton E that accompany the progressive addition of water in the system are illustrated in Figure 3. The resonance clearly shifts to lower frequency and becomes less well resolved as the water content of the microemulsion is increased. The change in chemical shift over the entire range of compositions is relatively small, on the order of 0.03 ppm, but is sufficient to be measured with accuracy. Variation of Chemical Shift with Composition. Figure 3 clearly indicates that the chemical shift is sensitive to surfactant environment. Since the surfactant molecules within the micelles 6252 DOI: 10.1021/la904033e
ð2Þ
where χfs, χms, δfs, and δms are the mole fractions and chemical shifts for free and micellar surfactant, respectively. However, the true situation is not so simple. This is because the chemical shifts characteristic of the surfactant in each of the limiting environments (δfs and δms) also vary with the exact global composition of the system. Composition Dependence of the Chemical Shift of Free Surfactant. In the case of the free surfactant, even in the absence of water, the chemical shift is found to vary with concentration. In cyclohexane solution, Igepal has been shown to exist as a mixture of monomers, dimers, and, to a lesser degree, higher aggregates.23 The exact distribution of surfactant among these states varies with concentration, with the mole fraction of monomers decreasing at higher concentration. It can be expected that the chemical shift of each of these states of aggregation is slightly different, therefore resulting in an effective chemical shift that varies as a function of total concentration. This is indeed the case, as illustrated in Figure 4, which shows the chemical shift of resonance E as function of surfactant concentration in cyclohexane in the absence of water. The free surfactant chemical shift used in calculations that follow is derived from the data of Figure 4, which can be fitted linearly to the reciprocal surfactant concentration, according to δ = a/nsurf þ b. The reciprocal form of Figure 4 is available in the Supporting Information. Composition Dependence of the Chemical Shift of Micellar Surfactant. In the case of surfactant molecules incorporated (23) Sheih, P. S.; Fendler, J. H. J. Chem. Soc., Faraday Trans. I 1977, 73, 1480.
Langmuir 2010, 26(9), 6250–6255
Lemyre and Ritcey
Article
in reverse micelles, the chemical shift is anticipated to depend on the specific size and composition of the micelle. This dependence can be predicted to be particularly important for the hydrophilic part of the surfactant, reflecting changes in oxyethylene chain conformation, intermolecular interactions, as well as changes in environmental parameters such as polarity and viscosity. Unfortunately, it is impossible to independently directly measure the NMR spectrum of micellar surfactant because of the unavoidable simultaneous presence of free surfactant. As described below, however, the chemical shift characteristic of a surfactant molecule in a micelle can be determined by comparing samples of differing global composition but fixed micelle size. The following analysis is possible because reverse micelles are relatively monodisperse in size. Furthermore, the micellar composition, defined as the ratio of the number of surfactant molecules to the number of water molecules within a micelle, is constant for any given micelle size.15 For a given size, surfactant molecules may exist in a variety of conformations, but for a large number of molecules, these variations will be averaged on the time scale of the NMR measurement. Therefore, each reverse micelle size presents a single micellar surfactant chemical shift for each proton within the molecule. A final key element of the analysis is the fact that only a single free surfactant concentration is possible for each micelle size. This constraint arises from the thermodynamic equilibrium between free surfactant and surfactant within the micelles. Since all micelles of a given size are identical, the chemical potential of the surfactant within them is also identical. The chemical potential, and thus the concentration, of free surfactant in equilibrium with the micelles is therefore also constant for a given micelle size, regardless of the exact global composition. Through dynamic light scattering experiments, we recently demonstrated that a given micelle size can be generated from many different overall system compositions of the water/Igepal/ cyclohexane system.15 Relevant to the present discussion is the consequence that, within a series of micelles of constant size and shape, the chemical shifts of both free and micellar surfactants will be independent of the global composition. This is because neither micellar composition nor free surfactant concentration vary within a given series. It is important to note that hydrodynamic diameters evaluated from DLS measurements are calculated from diffusion coefficients through the assumption of a spherical shape for the scattering body. This means that the observation of identical hydrodynamic diameters only implies the presence of identical micelles if micelle shape remains constant. In particular, the possible formation of cylindrical micelles must be considered.24,25 As discussed in detail elsewhere,15 the dependence of both the hydrodynamic diameter and the viscosity on global composition supports the assumption of spherical reverse micelles in the case of the water/Igepal/cyclohexane system studied here. Since neither micellar composition nor free surfactant concentration vary within a given series of a fixed micelle size, it is the number of micelles that changes with global composition. The effective chemical shift will therefore be modified because of a change in the ratio of free to micellar surfactant. Solution of eq 2 for this situation leads to
δeff ¼
nsurf;free ðδfs - δms Þ þ δms nsurf;total
ð3Þ
where nsurf,free is the amount of free surfactant, nsurf,total is the total amount of surfactant, δfs* and δ*ms are the chemical shifts for free (24) Strey, R.; Glatter, O.; Schubert, K.-V.; Kaler, E. W. J. Chem. Phys. 1996, 105(3), 1175. (25) Blokhuis, E. M.; Sager, W. F. C. J. Chem. Phys. 2001, 115(2), 1073.
Langmuir 2010, 26(9), 6250–6255
Figure 5. Resonance E of the 1H NMR spectrum of Igepal CO520 recorded for 4 nm reverse micelles prepared with various global compositions. From left to right: 96/6.7, 121/16.7, 135/33.3, 150/ 50, 164/66.7 mg of surfactant/μL of water per milliliter of cyclohexane.
and micellar surfactant at a fixed reverse micelle size. The complete derivation of eq 3 is provided in the Supporting Information. Since the free surfactant concentration and the chemical shifts δfs* and δ*ms are all constant for a fixed micelle size, eq 3 represents a linear relationship between the effective chemical shift and the reciprocal of the total surfactant in the system. Furthermore, the intercept of the corresponding linear plot provides the desired value of δ*ms and the slope is equal to nsurf, * * free(δfs - δms). Thus, the free surfactant concentration present at various micelle sizes can be evaluated from simple linear regressions. Since δfs* changes with free surfactant concentration, this dependence must be included, leading to the following equation nsurf;free ¼
slope - a
b - δms
ð4Þ
with the slope being that obtained from linear plots of the effective chemical shift as a function of reciprocal surfactant concentration at fixed reverse micelle size and a and b being the fitting parameters evaluated for the data of Figure 4. Derivation of this equation can be found in the Supporting Information. Figure 5 presents the evolution of the NMR signal of resonance E for a series of samples of identical micelle size but different global compositions. Under these conditions, an increase in the amount of surfactant and water leads to an increase in the number of micelles. Since the free surfactant concentration remains constant, the molar ratio of free to micellar surfactant decreases leading to a downshift of the chemical shift, as expected from eq 2, knowing that δfs > δms. This experiment has been conducted for about twenty different micelle sizes, ranging from 4 to 12 nm, with five samples of differing global composition for each size. Figure 6 presents examples of the linear plots based on eq 3. From a practical point of view, the number of overall system compositions used for each size is only limited by the number of DLS experiments one chooses to perform. As predicted, a linear relationship is found at each micelle size, with correlation coefficients above 0.98 for most of the regressions. The linearity of the plots presented in Figure 6 supports the hypothesis that the micellar composition remains constant within a given series of global compositions resulting in constant micelle size. DOI: 10.1021/la904033e
6253
Article
Figure 6. Examples of linear regressions based on eq 3. The effective chemical shift of peak E is plotted as a function of the reciprocal surfactant mass, per milliliter of cyclohexane, for global compositions from Table 1 that produce reverse micelles with diameters of (b) 10, (O) 7.7, (1) 6, and (4) 5 nm. Error bars correspond to the estimated uncertainty associated with evaluation of the chemical shift from the NMR spectra.
Figure 7. Micellar surfactant chemical shift of peak E, obtained from the intercept at the origin of linear regressions based on eq 3, plotted as a function of the reverse micelle size.
Figure 7 shows the micellar surfactant chemical shift, evaluated from the intercepts of the plots of Figure 6, as a function of micelle size. Although the data are somewhat scattered, a clear trend is evident, with the chemical shift moving to higher frequencies for the larger micelles. This shift is consistent with the increased deshielding that would be anticipated in the more polar environment experienced by the oxyethylene chain as the relative quantity of water increases within the micelle.26,27 Finally, the free surfactant concentration can be evaluated from eq 4 for each micelle size. The 1H NMR results are presented in Figure 8 and compared with previously reported values obtained from DLS measurements.15 Excellent agreement is found between the results of the two methods, particularly for smaller micelles. As anticipated, the free surfactant concentration is found to vary with micelle size, rising sharply for smaller micelles. For larger micelles, the concentration of free surfactant tends toward a constant value that, as discussed elsewhere,15 can be indentified as (26) Kumar, C.; Balasubramanian, D. J. Colloid Interface Sci. 1980, 74(1), 64. (27) Podo, F.; Ray, A.; Nemethy, G. J. Am. Chem. Soc. 2002, 95(19), 6164.
6254 DOI: 10.1021/la904033e
Lemyre and Ritcey
Figure 8. Mass of free surfactant in the system, per milliliter of cyclohexane, as a function of reverse micelle diameter. Results obtained by 1H NMR in this study (circles) are compared with those obtained by DLS from ref 15 (line). Error bars correspond to the combined uncertainty associated with the slopes of the linear regressions of Figure 6 and the parameters a and b of eq 4.
Figure 9. Schematic representation of (a) a small non-ionic reverse micelle with confined hydrophilic chains and (b) a non-ionic reverse micelle large enough to permit full chain extension.
the critical micellar concentration. The increase in free surfactant concentration with decreasing micelle size corresponds to an increase in surfactant chemical potential and thus indicates that smaller micelles are less stable than larger ones. This difference in stability can be attributed to conformational changes imposed on the hydrophilic oxyethylene chain upon confinement in small micelles. In large micelles, particularly when the polar core is at least twice the length of the surfactant hydrophilic segment, the chain has sufficient space to extend into the water phase, as illustrated in Figure 9b. As the reverse micelle becomes smaller, the hydrophilic chains are forced into coiled conformations (see Figure 9a) that result in increasing steric repulsions, destabilizing the micelle. This interpretation is supported by the corresponding increase in the area occupied by each surfactant molecule at the water/oil interface, as revealed by DLS measurements.15
IV. Conclusion The 1H NMR spectrum of Igepal CO520 in ternary mixtures containing water and cyclohexane shows a complex dependence on water content. This is in part because experimental spectra correspond to the weighted averages of spectra arising from free and micellar surfactants, which undergo rapid exchange on the NMR time scale. The analysis of this two-state system is further Langmuir 2010, 26(9), 6250–6255
Lemyre and Ritcey
complicated by the fact that the chemical shifts of both free and micellar surfactant vary with micelle size. The distribution of surfactant between the free and micellar states can, however, be determined if spectra are recorded for a series of samples of constant micelle size but differing global composition. Within a given series, the spectrum of both surfactant states remains constant and only the relative populations of the free and micellar species change with overall composition. This method of analysis allows for the determination of free surfactant concentration as a function of micelle size. The results presented for the water/ Igepal CO520/cyclohexane system are in excellent agreement with those previously determined by DLS. In this system, the free surfactant concentration is found to be far from negligible and strongly dependent on micelle size. The free surfactant concentration increases with decreasing micelle size, reflecting
Langmuir 2010, 26(9), 6250–6255
Article
the lower stability of the smaller micelles. Similar behavior can be expected for other reverse micellar systems based on non-ionic surfactants. Acknowledgment. The authors would like to acknowledge NanoQuebec, le Fonds Quebecois de la recherche sur la nature et les technologies (FQRNT) and the National Sciences and Engineering Research Council of Canada (NSERC) for financial support. Michele Auger is also acknowledged for helpful discussions. Supporting Information Available: The definition of all symbols, the reciprocal form of Figure 4 and detailed demonstrations of eqs 3 and 4. This material is available free of charge via the Internet at http://pubs.acs.org.
DOI: 10.1021/la904033e
6255