Article pubs.acs.org/Macromolecules
Theory of Lamellar Superstructure from a Mixture of Two Cylindrical PS−PMMA Block Copolymers John G. Spiro,† Nicolas Illy,† Mitchell A. Winnik,*,† Jeffrey D. Vavasour,‡ and Mark D. Whitmore‡ †
Department of Chemistry, University of Toronto, 80 St. George Street, Toronto, Ontario M5S 3H6, Canada Department of Physics and Astronomy, University of Manitoba, 186 Dysart Road, Winnipeg, Manitoba R2M 5C6, Canada
‡
ABSTRACT: The goal of this theoretical study was to examine whether formation of a lamellar superstructure was possible when blending relatively low-segregation cylindrical diblock copolymers (PSb-PMMA) of complementary composition. In the past, these kinds of experiments had only been carried out at high segregation levels. Our numerical self-consistent field (NSCF) simulations provided details of the morphology of the superstructure as well as of the components to be blended. For comparison, we also report our NSCF simulation giving the same detailsof a corresponding experiment with PS−PB copolymers.
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INTRODUCTION Block copolymer phase behavior has attracted intense interest for almost half a century, largely due to the self-assembling nature of many block copolymers. Considerable theoretical and experimental work has been done on the morphology, dimensions, and phase transitions of block copolymers particularly of diblock copolymersand in some cases it has been possible to reach broadly applicable and relatively simple generalizations. For instance, in the first approximation, phase behavior is controlled by only two parameters: (i) the χN product, where N is the total degree of polymerization of the diblock copolymer and χ is the Flory−Huggins χ parameter, and (ii) the volume fraction of either of the components. More recently, though, it was found necessary to add a third (conformational asymmetry) parameter, which incorporates differences in block coil statistics.1−3 Because of the great importance of block copolymer selfassemblies, as mentioned above, much effort has been expended on controlling the structure and repeat period. Apart from changing the block copolymer components and their volume fractions and molecular weights, adding a homopolymer compatible with one of the blocks has been explored as a means of influencing the structure. But the research of Hadziioannou and Skoulios4 suggested another approach to structure modification: mixing block copolymers with different compositions (component volume fractions and molecular weights), perhaps also of unlike morphologies. One of the early experimental and theoretical studies on blending block copolymers of different morphologies was a paper5 on blending two cylindrical polystyrene-b-polybutadiene (PS-b-PB) samples: one where the major component was PS and one where the major component was PB. The blend, with roughly 50 vol % of each PS and PB, gave a lamellar structure upon annealing. The Flory−Huggins χ parameter is quite high for PS-b-PB (we estimated 0.16 at 298 K6,7), and indeed most © 2012 American Chemical Society
experimentsas well as theorieshave dealt with strongly segregated block copolymers. Here, we discuss primarily a feasible experimental design and then theoretical analysis of blending 1:1 by volume cylindrical polystyrene-b-poly(methyl methacrylate) (PS-b-PMMA) samples of intermediate segregation level and complementary composition. We chose the latter system because cylindrical PS-b-PMMA is a copolymer of interest in its own right,8 our theoretical results are amenable to experimental confirmation, and the χ parameter is quite low and relatively well-known. We report details of how we arrived at the antisymmetric compositions and fairly low molecular weights of the cylindrical (hexagonally stacked cylinders) PS-b-PMMA components, showing the plausibility of our “design” by numerical selfconsistent field (NSCF) simulations, and prediction of the dimensions and the morphological details of the input and output structures. As well, for comparison and to demonstrate how the level of segregation affects matters, we also report our NSCF simulationgiving the same detailsof a corresponding experiment with PS−PB copolymers.
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COMPONENTS For low χN blends there was limited theoretical information in the literature, but Matsen and Bates reported applicable theoretical results9 for χN = 20. As for the morphology of χN = 20 diblock copolymers, information was available from published phase diagrams. We assumed an annealing temperature of 140 °C for films, and used the literature values ρPS10 = 5.795 nm−3, ρPMMA11 = 6.796 nm−3, bPS = 0.65 nm,12 and bPMMA = 0.652 nm12 and the Flory−Huggins χ parameter 0.02913−15(approximately), with the monomeric volume of PS as the Received: December 15, 2011 Revised: March 24, 2012 Published: April 30, 2012 4289
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the hexagon, via the relationship R = √[6 tan(π/6)/π]L (R ≈ 1.05L), giving the same cylinder cross section as the cross section of the hexagonal prism (L is half of the repeat distance). It is of interest that when we repeated the simulation18 for the 79 200 molecular weight PS-b-PMMA component using the full Wigner-Seitz cell rather than the cylindrical unit cell approximation, we obtained remarkably similar results. For instance, NSCF areas of the two kinds of unit cells differed by less than 1%. Morphologies of the Cylindrical Components. As mentioned earlier, the NSCF simulations did imply that our components 1 and 2 would have the desired morphology, i.e., cylindrical cores in hexagonal matrices. In addition, the computations provided segment density distributions, from which it was possible to calculate interface thicknesses (δ) (Figures 2 and 3). In these two figures R denotes the radius of
reference. The conformational asymmetry parameter,1,2 ε, defined as ε = (ρ0BbB2)/(ρ0AbA2), where ρ0A and ρ0B are the pure component number densities for block A and block B segments, respectively, and bA and bB are the Kuhn lengths of the components, was accordingly approximately 0.85 for PS-bPMMA. Phase diagrams had been published for ε = 11 and ε = 0.6.2 Each of them suggests volume fractions of the core components where the morphology is expected to be cylindrical, especially the middle of each range (see the illustration of Figure 1). Averaging led to the “design” volume
Figure 1. Partial phase diagram1 (conformational symmetry case) showing ranges of minor and major component volume fractions for χN = 20 that are expected to lead to cylindrical morphology. The centers of those ranges (marked) were considered the most favorable starting points toward obtaining a lamellar superstructure. However, as discussed in the text, the conformational asymmetry of PS-b-PMMA was also taken into consideration.
fractions of f PMMA1 = 0.32 for the core of “component 1” and f PS2 = 0.30 for the core of “component 2”. Those numbers were in reasonable agreement with the stipulations of Matsen and Bates’s theory,9 namely, that the diblock pair should be complementary: f PMMA1 = 1 − f PMMA2 (hence f PS1 = 1 − f PS2) and that these volume fractions should be in the range (0.3, 0.7). Matsen and Bates9 assumed conformational symmetry (ε = 1, see above) and specifically stated that the Kuhn lengths for the two blocks were assumed to be the same. This also implies equal densities. Accordingly, we employed effective degrees of polymerization (see, e.g., ref 16) ρPS eff eff = NPS; NPMMA = NPS NPMMA ρPMMA (1)
Figure 2. Component 1 PMMA cylinders (φPMMA1) and the PS matrix (φPS1) plot, also indicating the tangent lines used for interface thickness calculations.
Figure 3. Component 2 PS cylinders (φPS2) and the PMMA matrix (φPMMA2) plot. The tangent line used for the interface thickness calculation is also shown.
eff and the χN = 20 specification became χ(Neff PS + NPMMA) = 20, leading to the overall molecular molecular weight (Mn) 75 500 for component 1, with block degrees of polymerization NPS1 = 473, NPMMA1 = 262, and 79 200 for component 2, with block degrees of polymerization NPS2 = 209, NPMMA2 = 573. Our subsequent NSCF simulations did imply (via free energy minimization) that components 1 and 2 would be cylindrical and that their blend50 vol % of eachwould have the lamellar morphology.
the circularized unit cell (molecule): 16.31 nm for component 1 and 16.04 nm for component 2. The interface thicknesses were calculated essentially with Helfand’s formula for planar interfaces, based on the derivative of the volume fraction (φPS) profile,19 in the form quoted by Shull et al.:20
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δ=
NUMERICAL SELF-CONSISTENT FIELD SIMULATIONS We simulated the morphology and dimensions of both the cylindrical and lamellar structures using the methodology described by Vavasour and Whitmore.16 For the cylindrical (i.e., hexagonally stacked cylinders) samples we employed the unit cell approximation (UCA) first suggested by Helfand and Wasserman:17 the radius R of the approximated unit cell is related to the distance L from the center of the true, hexagonal unit cell to a point where the normal bisects one of the edges of
∂ϕPS ∂z
−1
ϕPS = 0.5
(2)
For Figures 2 and 3 z becomes the radial coordinate r and φPS = 0.5 is replaced by the intersection of the φPS and φPMMA curves. A small refinement is required, which follows from the geometric interpretation21 of eq 2. Namely, because of the low segregations, the minimum values of φPS and φPMMA are greater than zero, and their maximum values do not reach one. Therefore, the tangent lines that we work with per eq 2 must end at the minimum and maximum values of the volume 4290
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Helfand−Tagami (HT) theory,19 in conjunction with our NSCF results, also leads to another definiton of interface thickness for cylindrical diblock copolymers. In the past we used an approach similar to what follows to obtain junction distribution expressions applicable to spherical25 and cylindrical26 geometries. Namely, we can modify the usual HT expression for segment distributions in homopolymer blends or in A−B lamellar diblock copolymers:
fractions, and the interface thicknesses are the r distances between their end points. This is incorporated into Figures 2 and 3. Noolandi and Hong22 employed this approach for multicomponent systems, and the refinement is more significant in some of our calculations involving the lamellar superstructure. To our knowledge, there have been no interface thickness results published for cylindrical PS-b-PMMA samples; but we obtained values similar to those reported by Russell23 for lamellar PS-b-PMMA samples of a wide range of molecular weights. Russell,23 and previously Anastasiadis, Russell, and coworkers,24 also reported the volume fractions of different lamellar PS-b-PMMA samples occupied by the interfaces. Our ratios (34% for component 1 and 35% for component 2) are similar to what one would estimate by interpolation from the data of Anastasiadis et al.24 We will see that in the lamellar superstructure the separation of different blocks from one another is a more complex matter. In the subsection that follows we will also discuss a lamellar superstructure morphology studied by Vilesov and co-workers5 for a blend of cylindrical, antisymmetric PS-b-PB samples. To our knowledge, no interface thickness results are available for their samples, although they were much more highly segregatedas typical in this kind of research. (We calculated χN ≈ 79 for Vilesov sample 1 and χN ≈ 108 for Vilesov sample 2; details of the parameters we used for characterization and in NSCF simulations are given in the Appendix.) As shown in Figures 4 and 5 (volume fractions in these and all other figures are graphical representations of our NSCF
ϕB(z) = 0.5[1 + tanh(2z /δ)]
(3)
where z is distance along a horizontal axis perpendicular to the interface, δ is the interface thickness, and ϕB(z) is the segment density of component B, to ϕ(r ) = 0.5{1 + tanh[2(r − R 50)/δ]}
(4)
where ϕ(r) is now the segment density (volume fraction) of the corona component at distance r from the center of the cross section of a cylinder, R50 is the value of r where the volume fractions of the A and B components are equal, and δ is still the interface thickness. We can now compare the segment densities calculated by the NSCF method to the values from eq 4, only the latter values being dependent on δ. A reasonable “goodness of fit” measure would seem to be the sum of squares of deviations between corona component volumes at different values of r (from r/R = 0 to r/R = 1, in increments of 0.01); specifically, for Vilesov et al.’s sample 1: r/R=1
SS =
∑ r/R=0
4π 2r 2[ϕPS,NSCF − ϕPS,HT]2
(5)
This goodness of fit measure is very sensitive to the value of δ that one assumes in eq 4; minimizing SS leads to δ = 2.1 nm, almost identical to what we report in Figure 4 by the tangent line construction method. We also show there the good agreement between segment densities obtained from the modified Helfand−Tagami formula (eq 4, δ = 2.0 nm) and from the NSCF simulation. We note that unlike Vilesov et al.’s sample 1, which had 68 vol % PS and 32 vol % PB, their sample 2 is very asymmetric, with 19.5 vol % PS and 80.5 vol % PB. We speculate that it is for the latter reason that converting the HT eq 3 into a cylindrical form (eq 4) gave less satisfactory results. Nevertheless, the preceding discussion tends to confirm also the interface thickness result that we show in Figure 5 (δ = 2.1 nm) for sample 2. Lamellar Superstructures. In earlier sections we reported our goals, how Matsen and Bates’s theory9 provided a reasonable starting point, and how it led to a blend design. Our subsequent NSCF simulation did predict formation of a lamellar superstructure. (We compared the free energies of the lamellar morphology with free energies of cylindrical structures with either PS or PMMA core, and our conclusion is testable by experiment.) Expected segment distributions for the blocks of component 1 and component 2, as well as the total PS and PMMA volume fractions, are shown in Figure 6. One and onehalf periods are plotted there; the lamellar repeat distance is 31.38 nm. Averaging the input parameters for components 1 and 2 we did NSCF simulation for a lamellar diblock copolymer containing 49 vol % PS. For this structure the repeat distance was 29.60 nm. We will discuss this result below, in connection with high-segregation theory and experiments. We note that all four blocks overlap with each other significantly, presumably because of the low segregation. We
Figure 4. PB cylinders (φPB1) and the PS matrix (φPS1) plot for Vilesov et al.’s sample 1. The tangent line used for the interface thickness calculation is also shown, as well as the good agreement with the Helfand−Tagami PS segment density.
Figure 5. PS cylinders (φPS2) and the PB matrix (φPB2) plot for Vilesov et al.’s sample 2. The tangent line used for the interface thickness calculation is also shown.
simulation results), we obtained the circularized unit cell radius 19.0 nm and δ = 2.0 nm for sample 1 and circularized unit cell radius 17.3 nm with δ = 2.1 nm for sample 2. 4291
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distance of the lamellar superstructure. Our corresponding NSCF results were 36.24, 32.95, and 38.15 nm. The agreement is good for sample 2, quite acceptable for the superstructure; but there is a 36% devation for sample 1. We note that in their seminal paper on cylindrical block copolymers17 Helfand and Wasserman compared their theoretical results to experimental values for PS−PB cylindrical structures (among others) with either PS or PB core and also found appreciable deviations. Nevertheless, they considered the agreement between their theory and the experiments generally good, sinceas in our casethere were no adjustable parameters, the physical constants, especially χ were not well-known, and “the closeness of the samples to equilibrium is uncertain”.17 Localization of Blocks. Mayes et al.31 found by neutron reflectivity measurements on mixtures of low and high molecular weight lamellar PS-b-PMMA samples that there was a “substantial degree of interpenetration” between the PS and PMMA segments of a low molecular weight copolymer. Our NSCF results are entirely consistent with their findings. In Figure 8, we indicate a reduced distance between the short PS
Figure 6. Plot of the segment distributions of the individual blocks (φPS1, φPS2, φPMMA1, φPMMA2) and of the total PS and PMMA (φPS = φPS1 + φPS2; φPMMA = φPMMA1 + φPMMA2) in the lamellar superstructure.
analyze this matter by means of interface thickness and related calculations, but the question arose whether these overlaps are due only to the low segregation or whether they are caused, at least partially, by blending cylindrical block copolymers of antisymmetric composition to obtain a lamellar superstructureat any segregation level. Indeed, if we go to more strongly segregated (high χ parameter) cylindrical diblock copolymers, there is much more experimental and theoretical information available17 than on PS-b-PMMA and even on their antisymmetric blends5,27−29 of the type that we have designed for PS-b-PMMA. However, theoretical analyses always seem to have been based on the narrow interface approximation17 or on grafted chain theories that essentially assume zero interface thickness. Of the latter some of the Birshtein group’s papers30 seem the most relevant to our investigations. We thought that it would be informative if we applied our more rigorous NSCF analysis (again based on input parameters discussed in the Appendix) to the experiments of Vilesov et al.5 and compared the results to those we obtained for our designed PS-b-PMMA blend. Figure 7 (one
Figure 8. Localization of the short blocks in our designed blend of antisymmetric PS-b-PMMA components. Their interpenetration is indicated by the calculated “interface thickness”. The two tangent lines (shown) gave essentially identical results.
and PMMA blocks compared to the 5 nm typical23,24 for PS-bPMMA, whereas Figure 9a,b shows that the widths of the interfaces of the PS and PMMA blocks of both components 1 and 2 have increased in the superstructure. We have also calculated (Figure 10) an “interface thickness” between the total PS and total PMMA. The interpenetration and broadening effects seem to cancel out each other, and the total PS and PMMA distributions resemble what one would obtain for neat, lamellar PS-b-PMMA.23,24 The interpenetration of short blocks seems less significant (Figure 11) for the samples involved in the experiments of Vilesov et al.5 But there, too, distances between other blocks have increased considerably (Figure 12). According to our NSCF simulations, this also leads to an increase in the interdomain distanceto 38 nm from 35 nm for a single, symmetric diblock copolymer of equivalent composition. That kind of an effect was also reported by the Hashimoto group for experimental and theoretical studies29 involving binary mixtures composed of cylindrical PS-b-PI samples of complementary composition, thus forming lamellar superstructures. One of the reviewers mentioned that the lamellar period tends to decrease with increased interface thicknessunlike what we report for Vilesov et al.’s blend. We think that this reviewer was referring to the well-known scaling result D ∝ χ1/6N2/3, where D is the lamellar repeat period. And of course, increased δ and decreased χ go hand in hand. (The actual values of the exponents 1/6 and 2/3 depend on the degree of
Figure 7. Plot of the segment distributions of the individual blocks (φPS1, φPS2, φPB1, φPB2) and of the total PS and PB (φPS = φPS1 + φPS2; φPB = φPB1 + φPB2) in the lamellar superstructure of Vilesov et al.
and one-half periods plotted; the lamellar repeat distance is 38.15 nm) does indicate how the high segregation leads to narrower distributions for the short blocks than in the PS PMMA blend; we will report below whether there is significant interpenetration of these short blocks. One of the reviewers suggested that it would be of interest to compare the periods of the cylindrical and lamellar systems obtained by NSCF to the experimental (SAXS) data on Vilesov et al.’s PS−PB samples. They reported repeat distances of 26.7 and 30.5 nm, respectively, for their samples 1 and 2, and we interpreted their SAXS Q values as 33 nm for the repeat 4292
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Figure 12. (a) Vilesov sample 1 φPS (φPS1) and φPB (φPB1) in the lamellar superstructure. (b) Vilesov sample 2 φPS (φPS2) and φPB (φPB2) in the lamellar superstructure. The tangent lines used for the interface thickness calculations are also shown.
Figure 9. (a) Component 1 φPS (φPS1) and φPMMA (φPMMA1) in the lamellar superstructure. (b) Component 2 φPS (φPS2) and φPMMA (φPMMA2) in the lamellar superstructure. The tangent lines used for the interface thickness calculations are also shown.
Figure 13. Volume fraction profiles for the total PS and PB (φPS = φPS1 + φPS2; φPB = φPB1 + φPB2) in the lamellar superstructure of Vilesov et al. The tangent line used for the “interface thickness” calculation is also shown.
Figure 10. Volume fraction profiles for the total PS and PMMA (φPS = φPS1 + φPS2; φPMMA = φPMMA1 + φPMMA2) in the lamellar superstructure. The tangent line used for the interface thickness calculation is also shown.
segregation, but that is not the issue here.) However, even for the total PS−total PB “interface” we have calculated δ = 2.1 nm (eq 2 and Figure 13), whereas the corresponding number for the neat diblock copolymer would be only 1.9 nm. It was
incorporating the free energy of mixing of samples 1 and 2 into our model that led to our PS−PB simulation results.
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CONCLUSION
With theoretical work by Matsen and Bates9 as a starting point, we have developed a feasible experimental design for blending cylindrical PS-b-PMMA samples of complementary composition and fairly low molecular weight to obtain a lamellar superstructure. Our conclusions, based on numerical selfconsistent field (NSCF) simulations, are testable by experiment. However, to our knowledge, experiments of this kind have never been conducted; existing studies employed much more strongly segregated block copolymers. Our NSCF simulations also provided new information on the morphologies and dimensions of the input and output structures of our design as well as of corresponding PS−PB structuresmuch more strongly segregateddiscussed in ref 5.
Figure 11. Localization of the short blocks in the lamellar superstructure of Vilesov et al. The tangent lines used to calculate an “interface thickness” are also shown. 4293
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(15) Eitouni, H. B.; Balsara, N. P. Thermodynamics of Polymer Blends. In Physical Properties of Polymers Handbook, 2nd ed.; Mark, J. E. Ed.; Springer: Berlin, 2007; p 349. (16) Vavasour, J. D.; Whitmore, M. D. Macromolecules 2001, 34, 3471. (17) Helfand, E.; Wasserman, Z. R. Macromolecules 1980, 13, 994. (18) Whitmore, M. D.; Vavasour, J. D.; Spiro, J. G.; Winnik, M. A. On Cylindrical PS-b-PMMA, in preparation. (19) Helfand, E.; Tagami, Y. J. Chem. Phys. 1972, 56, 3592. (20) Shull, K. R.; Mayes, A. M.; Russell, T. P. Macromolecules 1993, 26, 3929. (21) Yekta, A.; Spiro, J. G.; Winnik, M. A. J. Phys. Chem. B 1998, 102, 7960. (22) Noolandi, J.; Hong, K. M. Macromolecules 1984, 17, 1531. (23) Russell, T. P. Mater. Sci. Rep. 1990, 5, 171. (24) Anastasiadis, S. H.; Russell, T. P.; Satija, S. K.; Majkrzak, C. F. J. Chem. Phys. 1990, 92, 5677. (25) Farinha, J. P. S.; Schillén, K.; Winnik, M. A. J. Phys. Chem. B 1999, 103, 2487. (26) Yang, J.; Lou, X.; Spiro, J. G.; Winnik, M. A. Macromolecules 2006, 39, 2405. (27) Sakurai, S.; Irie, H.; Umeda, H.; Nomura, S.; Lee, H. H.; Kim, J. K. Macromolecules 1998, 31, 336. (28) Noro, A.; Okuda, M.; Odamuki, F.; Kawaguchi, D.; Torikai, N.; Takano, A.; Matsushita, Y. Macromolecules 2006, 39, 7654. (29) Chen, F.; Kondo, Y.; Hashimoto, T. Macromolecules 2007, 40, 3714. (30) (a) Zhulina, E. B.; Birshtein, T. M. Polymer 1991, 32, 1299. (b) Zhulina, E. B.; Lyatskaya, Yu. V.; Birshtein, T. M. Polymer 1992, 33, 332. (c) Lyatskaya, Ju. V.; Zhulina, E. B.; Birshtein, T. M. Polymer 1992, 33, 343. (31) Mayes, A. M.; Russell, T. P.; Deline, V. R.; Satija, S. K.; Majkrzak, C. F. Macromolecules 1994, 27, 7447. (32) Banaszak, M.; Whitmore, M. D. Macromolecules 1992, 25, 3406.
APPENDIX. INPUT QUANTITIES AND PARAMETERS FOR NSCF SIMULATIONS RELATING TO THE EXPERIMENTS OF VILESOV ET AL.5 In a sense our simulations of Vilesov et al.’s samples and blend were extensions of results reported by Banaszak and Whitmore32 for blends of PS-b-PB with styrene solvent, and we continued to employ their Kuhn lengths for PS and PB segments, 0.68 nm, and their number density for PS segments, 6.07 nm−3. However, we accepted a more recent value, 9.90 nm−3, for the number density of PB segments, reported by Papadakis et al.7 (All values are for 298 K, since Vilesov’s SAXS measurements were made at room temperature.) The molecular weights and PS contents of “samples 1 and 2” (Figures 4 and 5) were taken from the Vilesov et al. paper as published,5 and we then calculated the concentrations of the four component blocks in the 1:1 blend to confirm and characterize the lamellar superstructure (Figure 7). Specifying the Flory−Huggins χ parameteras usualwas somewhat controversial. One difficulty is that the reference compoundif indeed χ was calculated for oneis frequently not specified in the literature. In our methodology1,2 we always use reference densities (the PS segment density here), with χ defined accordingly. Another consideration was that in this theoretical study we preferred a largish χ parameter, so that our simulations would deal with high segregation levels. We elected to base our calculations on two χ values from the literature 0.176 and 0.15,7 both when evaluated for PS as the reference and at 298 Kand averaged them.
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AUTHOR INFORMATION
Corresponding Author
*E-mail
[email protected]; Fax (416) 978-0541. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS We thank Dr. Gérald Guérin for his comments on the manuscript, especially on improving the graphics. REFERENCES
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