Article pubs.acs.org/Langmuir
A Theory for the Surface Tensions and Contact Angles of HydrogenBonding Liquids Robert David*,† and A. Wilhelm Neumann‡ †
1145 Boundary Rd. W, Pembroke, Ontario K8A 7T9, Canada Department of Mechanical & Industrial Engineering, University of Toronto, 5 King’s College Rd., Toronto, Ontario M5S 3G8, Canada
‡
S Supporting Information *
ABSTRACT: The surface tensions of non-hydrogen-bonding, organic liquids can be accurately calculated from their electromagnetic properties, using an approximate form of the Lifshitz theory. A simple extension of this approach to the calculation of the surface tensions of hydrogen-bonding liquids is proposed. It is shown that the higher surface tensions of hydrogen-bonding liquids can be accounted for, with reasonable accuracy, by the increase in dispersion due to the shortened distance of approach between hydrogen-bonded atoms. Similar considerations allow calculations of contact angles on several low-energy solid surfaces in terms of molecular and electromagnetic properties. In accordance with well-known experimental observations, the calculated contact angles of both hydrogen-bonding and non-hydrogen-bonding liquids on the same lowenergy surface nearly follow a single, smooth pattern.
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angle θ is determined by the balance of liquid−liquid and solid−liquid intermolecular forces. Further progress in understanding the origins of surface tensions and contact angles requires knowledge of the intermolecular forces. For organic liquids and solids, the intermolecular forces are usually divided into three categories: dispersive forces, polar forces, and hydrogen bonds. The dispersive, or London, force is the induced dipole−induced dipole force. It arises from the instantaneous polarization of molecules due to the quantum mechanical zero-point energy of the electrons and is present in all materials. The polar forces are the Debye force and the Keesom force. The Debye force is the dipole−induced dipole force and results from the polarization of a nonpolar molecule by a polar molecule. The Keesom force is the dipole−dipole force; it causes the mutual orientation of two polar molecules. Finally, the hydrogen bond is a special dipole−dipole interaction involving a hydrogen atom situated between two electronegative atoms. The London, Debye, and Keesom intermolecular forces are collectively referred to as van der Waals forces. Liquid and solid surface tensions have been formally split into components, with each component accounting for the contribution of a different intermolecular force.1 For purely dispersive (nonpolar) liquids, Israelachvili2,3 showed that the values of surface tensions could be predicted within a few percent from other macroscopic and molecular properties, via a simple model based on the Lifshitz theory. The surface tension
INTRODUCTION Liquid surface tension γlv plays a central role in many natural and engineered systems that involve liquid surfaces or small amounts of liquid. Examples include water-walking insects, lung surfactant, condensing vapors, digital microfluidic devices, and superhydrophobic surfaces. When the liquid in such a system contacts a solid surface, the contact angle θ is also of importance. Surface tension and contact angle are linked by Young’s equation: γsv = γlv cos θ + γsl
(1)
where γsv is the solid surface tension and γsl is the solid−liquid interfacial tension. Surface tension can be defined as the reversible work required to create a unit area of surface. Thus, if hypothetically two half-spaces of a liquid are separated from initial contact and Wll is the work per unit area required to achieve this, then Wll = 2γlv. Wll is called the work of cohesion and represents the work needed to overcome the liquid intermolecular forces that initially held the half-spaces together. Analogous definitions can be made for γsv and Wss. The interfacial tension γsl between a solid and a liquid can be expressed by the Dupré equation: γsl = γlv + γsv − Wsl
(2)
where Wsl, the work of adhesion, is the work performed to separate unit areas of the solid and the liquid from contact. Wsl represents the solid−liquid intermolecular forces. By combining the Young and Dupré equations, it can be seen that the contact © 2014 American Chemical Society
Received: July 12, 2014 Revised: September 7, 2014 Published: September 9, 2014 11634
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is always significant, and for a pair of nonpolar molecules, it is the only nonzero term; but, if both molecules are polar, then all four terms are nonzero, and the Keesom interaction can be dominant.12 The work of adhesion between two half-spaces of material can be obtained by integrating eq 3 over all pairs of molecules on opposite sides of the interface:
of water has been calculated from intermolecular forces (e.g., see ref 4), but to our knowledge, more general calculations for hydrogen-bonding liquids have been performed using only a highly complex model.5 The contact angles of purely dispersive liquids on a solid surface have also been related to intermolecular forces. Hough and White6 used the Lifshitz theory to compute the contact angles of a series of liquid alkanes on solid polytetrafluoroethylene (PTFE) with reasonable accuracy. Their results showed a regular increase of the contact angle θ with the surface tension γlv of the alkane, a well-known pattern that was originally observed experimentally by Fox and Zisman.7 The same pattern of contact angles is in fact observed on a wide range of low-energy solid surfaces and spans not only dispersive liquids but also polar and hydrogen-bonding liquids.8,9 The latter point is surprising in that, despite the direct connection between intermolecular forces and interfacial properties, the same empirical relationship between γlv and θ is shared by liquids with apparently quite dissimilar intermolecular forces. It is not yet understood why this is the case. In a computational study, van Giessen et al.10 used density functional theory to calculate the contact angles of a range of dispersive, polar, and hydrogen-bonding liquids on a solid surface. While the computed contact angles of liquids of all types did fall close to a single curve, as in experiments, the model for molecular interactions contained only a Lennard− Jones potential and did not account for polar forces or hydrogen bonds. In previous work,11 we showed that the density depletions computed by van Giessen et al.10 could be approximately derived analytically, but we also did not account for differences in intermolecular forces. Thus, the current understanding of how hydrogen bonds contribute to both the surface tensions and contact angles of hydrogen-bonding substances is incomplete. In this paper, a simple model, based on the work of Israelachvili,3 is developed for the contribution of hydrogen bonding to the surface tensions of hydrogen-bonding liquids. Surface tensions calculated from the model are compared with experimental values. The model is then applied to predict the contact angles of a range of hydrogen-bonding and non-hydrogen-bonding liquids on four low-energy surfaces, and the results are again compared to experimental observations. We conclude with a discussion.
W=
A12 12πd 2
(4)
where d is the separation between the half-spaces and all the molecular properties from eq 3 are bundled into A12, the Hamaker coefficient between materials 1 and 2. However, the van der Waals interactions are in fact not pairwise additive because the polarization of a molecule (i.e., how it interacts with a third molecule) is altered by its electromagnetic interaction with a second molecule. There are therefore multibody interactions that are significant in condensed media and are not taken into account in the integration of eq 3 to produce eq 4. The Lifshitz theory13,14 avoids this difficulty by calculating the energy of electromagnetic interaction between two halfspaces using only macroscopic rather than molecular parameters. The effect of multibody interaction is automatically included in the macrosopic parameters. For two half-spaces separated by a distance d, eq 4 can still be used, but the Hamaker coefficient is now expressed, after some approximations to be discussed below, as A12 = ×
⎛ε − 3 kBT ⎜ 1 4 ⎝ ε1 +
1 ⎞⎛ ε2 − 1 ⎞ 3hν ⎟+ ⎟⎜ 1 ⎠⎝ ε2 + 1 ⎠ 8 2
(η12 − 1)(η2 2 − 1) η12 + 1 η2 2 + 1 ( η12 + 1 +
η2 2 + 1 )
(5)
where ε is a dielectric constant, η is an index of refraction, and the subscripts 1 and 2 again refer to the two materials.3 In eq 5, the absorption frequency ν is assumed equal for both materials, which is often a good approximation. The full Lifshitz theory calculates the total interaction between the two materials as the sum of interactions at all frequencies of the fluctuating electromagnetic field.13 In the simplified form of eq 5, the first term is the interaction at zero frequency and contains both the Debye and Keesom interactions. The second term is the interaction at the main ultraviolet frequency ν corresponding to electronic absorption. It accounts for the London force. The zero-frequency term in eq 5, representing the polar interactions, typically contributes only a few percent of the total value of A12 (at room temperature and with the materials separated by vacuum).3 The unimportance of the polar interactions in the Lifshitz theory contrasts sharply with the situation between individual molecules (eq 3), in which the Keesom interaction can dominate, as mentioned above. The reason for the difference is that the Keesom interaction is directional; it results from the favorable mutual orientation of two isolated dipolar molecules. However, in a condensed medium, if a polar molecule is favorably oriented with respect to one neighbor, it will be unfavorably oriented with respect to others.12 The Keesom interaction is therefore much less important in a condensed medium than it is between isolated molecules. In the Lifshitz theory, this issue is automatically taken into account by using the macroscopic dielectric response
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MOLECULAR INTERACTIONS The total energy of van der Waals interaction between two molecules separated by a distance r is 1 E=− (4πε0)2 u 2u 2 ⎤ νν 1 ⎡ 3h × 6 ⎢ α1α2 1 2 + u12α2 + u 2 2α1 + 1 2 ⎥ 3kBT ⎦ ν1 + ν2 r ⎣2 (3)
where ε0 is the permittivity of free space, h is Planck’s constant, α is a molecular polarizability, ν is an absorption frequency, u is a dipole moment, kB is Boltzmann’s constant, T is absolute temperature, and the subscripts 1 and 2 refer to the two molecules.3 The intermolecular force is the derivative of E with respect to r. The first term in the brackets in eq 3 is the London interaction, the second and third are the Debye interaction, and the fourth is the Keesom interaction. The London term in eq 3 11635
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(ε and η). These properties encode the ability of the molecules in a condensed medium to find favorable mutual orientations at a given frequency. While the first term of eq 5 accounts for zero-frequency polar interactions, a polar molecule can also rotate in order to interact with more partners. However, the frequency of rotation of an entire molecule is much slower than the frequency of electronic (London) “rotations”, covered by the second term of eq 5. For example, the absorption frequency of water associated with molecular rotation is in the microwave region, while electronic interactions occur, as mentioned, at ultraviolet frequencies. The energy associated with interaction at a given frequency is proportional to that frequency, ultimately due to Planck’s relation for a photon (E = hν). Thus, eq 5, in which the second term neglects all frequency-dependent interactions other than those at the highest frequency, namely, the London interactions, gives a good approximation for the Hamaker coefficient, even for water.3 In summary, between two condensed media with vacuum intervening, directional (zerofrequency) electromagnetic interactions are minor, and interactions at nonzero frequencies are dominated by the London force. Thus far, we have not discussed hydrogen bonds. In a hydrogen bond X−H···A, the electronegative atom X creates a partial positive charge on the hydrogen atom H, which is then attracted to the partial negative charge on the electronegative atom A. Atoms X and A are often oxygen, nitrogen, or fluorine. While many varieties of the hydrogen bond exist, the common type considered here is dominated by this electrostatic interaction between polar molecules.15 In other words, it is essentially an especially strong Keesom interaction. However, one important distinction is that, because the H atom has only a single electron, which is pulled toward the electronegative X atom, the H nucleus is exposed and the A atom can approach it closely. A typical measured distance between H and A in a hydrogen bond is ∼0.18 nm, whereas the sum of van der Waals radii (i.e., the approach distance with no hydrogen bond) is ∼0.27 nm.15
enhance the dispersive interaction. Therefore, Israelachvili’s method can be simply extended by using, in eq 4, an effective separation d that is a weighted sum of the non-hydrogenbonding and hydrogen-bonding contacts across the interface: d=
nHB n ⎞ ⎛ dHB + ⎜1 − HB ⎟d0 ⎝ n n ⎠
(6)
where n is the number of contacts or nearest neighbors of a molecule, nHB is the number of those that are hydrogen bonds, dHB is the effective distance to be used for hydrogen-bonded contacts, and d0 = 0.165 nm is the effective distance for nonhydrogen-bonded contacts. Table 1 shows the surface tensions γcalc of a number of liquids at room temperature calculated with this method. Table 1. Calculated Surface Tensions liquid heptane decane hexadecane 2,2,2trifluoroethanol methanol ethanol formic acid methylformamide formamide glycerol water a
nHB
n
γcalc*a (mJ/m2)
γcalc (mJ/m2)
γexp (mJ/m2)
ref
0 0 0 1.62
16 22 34 14
22 25 27 14
22 25 27 16
20 24 27 21
− − − 16
1.77 1.8 2.07 2.00 2.0 5.68 3.58
6 8 6 7 5 14 4
18 21 22 28 30 33 18
26 27 34 40 50 55 76
22 22 37 39 57 63 73
17 18 19 20 21 22 23
Calculated without accounting for hydrogen bonds (see text).
Values of ε and η for the liquids were taken from handbooks,24,25 and a common value of ν = 3 × 1015 Hz was used.3 For nHB, measurements from neutron or X-ray diffraction were employed (the hydrogen-bonding liquids were chosen based on the availability of these data), with the sources given in the last column of the table. Values of n are unavailable and so were estimated (see the Supporting Information) by counting the exposed atoms in each molecule, with O counted twice (for two lone pairs of electrons) and F thrice (for three lone pairs). The value of dHB, the effective Lifshitz separation for a hydrogen bond, was optimized so as to minimize the sum of squared errors for the calculated surface tensions γcalc. The optimal value was dHB = 0.071 nm. This value is similar to what would be obtained by taking 0.4 times the interatomic centerto-center distance of ∼0.18 nm, as argued by Israelachvili,2,3 and is therefore physically reasonable. For the last eight (hydrogen-bonding) liquids in Table 1, the average error for the calculated surface tension relative to the experimental value, γexp, was 13%. This is a reasonable level of error, considering the simplicity of the calculation. A column listing the calculated surface tensions, γcalc*, with hydrogen bonds neglected (i.e., with d = 0.165 nm instead of eq 6) is also included for comparison. In the next section, the model is applied to the calculation of the contact angles of both dispersive and hydrogen-bonding liquids on solid surfaces.
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CALCULATION OF SURFACE TENSIONS The surface tension of a liquid can be estimated from intermolecular forces using the relation γlv = Wll/2, with Wll from eq 4 and All from eq 5.3 The required inputs are the dielectric constant ε, the UV absorption frequency ν, the index of refraction η, and the distance d. The first three of these properties are available for many liquids, leaving the choice of an appropriate value for d to be made. The value of d represents the perpendicular separation of two half-spaces of the liquid that are “in contact” and should be of the order of the size of an atom. Israelachvili3 argued from geometry that d should equal 0.4 times the interatomic center-to-center distance, which he took as ∼0.4 nm. He showed that a value d = 0.165 nm gave accurate results for the surface tensions of a number of dispersive and polar materials, both liquid and solid. For most hydrogenbonding liquids, predictions of γlv using the same formula were too low. As discussed above, the polar nature of a hydrogen bond is not expected to contribute strongly to the molecular interaction in a medium. However, a distinguishing feature of the hydrogen bond is that the bonded atoms (H and A) are separated by much less distance than they would be in the absence of a hydrogen bond. This shorter distance of approach must 11636
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CALCULATION OF CONTACT ANGLES Contact angles can be estimated by first computing the surface tensions γlv, γsv, and γsl and then using Young’s equation (eq 1). The solid surface tension γsv can be calculated in the same way as was shown for γlv in the previous section. The solid−liquid interfacial tension γsl can then be found by calculating the work of adhesion Wsl and using the Dupré equation (eq 2). There is one further detail regarding γsl. For an interface between a solid and a liquid for which γsv < γlv (as is the case for the systems considered here, with contact angle θ > 0), the liquid molecules are depleted adjacent to the solid surface, because of the weaker attraction they experience from the solid molecules compared to that from the other liquid molecules.26 Taking this density depletion into account in eq 4 gives Wsl ≈
A sl 12πd
e 2
2 − δ γsl kBT
from eq 8 and eliminating Wsl with the aid of eq 2) and then using γsl, γlv, and γsv to obtain the contact angle θ from Young’s equation. As above, values of d0 = 0.165 nm and dHB = 0.071 nm were used in eq 8, ν = 3 × 1015 Hz in eq 5, and as mentioned, δ = 0.185 nm in eq 7. The results are shown in Figure 1. The calculated cosines of contact angles of both the hydrogen-bonding and non-
(7)
where δ2 is the area in the interface occupied by an atom. An average value of δ = 0.185 nm was found to fit contact angle data,11 excluding data for two solids to be considered below. The present calculations are not very sensitive to the value of δ; for example, using δ = 0.2 nm changes calculated contact angles below by only a few degrees. In order to estimate Wsl, an appropriate value for the liquid− solid contact distance d in eq 7 must also be determined. The same approach is used as that described above for the calculation of surface tensions (eq 6). We consider only nonhydrogen-bonding solids, because no values of nHB/n are available for hydrogen-bonding solid surfaces. For simplicity, it is assumed that the lower-energy solid does not affect the arrangement of the higher-energy liquid molecules, so that the liquid packing adjacent to the solid “wall” is equivalent to its packing in the bulk (this might not be the case for a hydrogenbonding solid). Then, for an interface between a non-hydrogenbonding liquid and a non-hydrogen-bonding solid, d = d0 = 0.165 nm. For an interface between a hydrogen-bonding liquid and a non-hydrogen-bonding solid, the contact distance between the solid and a hydrogen-donating site on a liquid molecule is dHB (as before, due to the exposure of the hydrogen nucleus); between the solid and a hydrogen-accepting site on a liquid molecule, it is d0; and between the solid and all other, non-hydrogen-bonding sites on a liquid molecule, it is d0. With the assumption of unaltered packing of the liquid molecules, the number of hydrogen-donating and hydrogen-accepting sites per liquid molecule that contact the solid must each equal nHB/ 2. Thus, the following averaged value for the effective contact distance between liquid and solid is obtained: 11
d=
nHB n ⎞ ⎛ dHB + ⎜1 − HB ⎟d0 ⎝ 2n 2n ⎠
Figure 1. Calculated contact angles for Teflon AF 1600.
hydrogen-bonding liquids fell nearly along a single, smooth curve. Calculated values also lay close to the pattern of experimental values for various liquids (both non-hydrogenbonding and hydrogen-bonding) measured on Teflon AF 1600.28 Note that for consistency, calculated contact angles are plotted versus calculated surface tensions, while experimental contact angles are plotted versus experimental surface tensions. The experimental values are advancing contact angles, which on low-energy solid surfaces, reflect the surface energy of the main constituent of the solid. Figure 2 shows analogous results for contact angles at room temperature on polystyrene. Because of its aromatic content, the ultraviolet absorption frequency of polystyrene is 2.3 × 1015 Hz.3 For the calculations of Asl (eq 5), an average between the liquid and solid absorption frequencies (i.e., ν = 2.65 × 1015 Hz) was used. This approximation produces only about 1% error (see the Supporting Information). The calculated contact
(8)
Equation 8 amounts to an arithmetic mean of the solid−solid contact distance (d0) and the liquid−liquid contact distance (eq 6), which is the classical Lorentz combining rule. Contact angles for the 11 liquids of Table 1 were calculated on four solid surfaces chosen for the availability of required input parameters and contact angle data. For the solid polymer Teflon AF 1600, with ε = 1.93 and η = 1.31,27 using eqs 4 and 5 with ν = 3 × 1015 Hz and d = 0.165 nm gives γsv = 15 mJ/m2. With this value of γsv as well as those calculated for γlv (see the previous section) in hand, contact angles for liquids at room temperature on Teflon AF 1600 can be calculated as outlined above: first by finding γsl from eq 7 (using Asl from eq 5 and d
Figure 2. Calculated contact angles for polystyrene. 11637
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there is cross-interaction between the components for dispersion and hydrogen bonding because both are ultimately dispersive. Indeed, without a cross-interaction, the observed near-independence of contact angles from molecular propertiesthat is, the observation that the contact angles of hydrogen-bonding and non-hydrogen-bonding liquids fall nearly on a single, smooth curvewould not be possible. The cross-interaction between energetic components takes a mathematical form very similar to a geometric mean, or Berthelot, combining rule (eq 5). It was shown previously11 that experimental patterns of contact angles on low-energy surfaces could be obtained from two ingredients: density depletion of liquid molecules adjacent to the solid surface, as described by eq 7, and a then-assumed Berthelot combining rule for interactions between any pair of unlike molecules. The Berthelot-like behavior is now seen to arise from the ultimately mainly dispersive nature of the molecular interactions. Thus, the near-conformity to the same pattern of contact angles by liquids ranging from nonpolar to highly polar (Figures 1 and 2), as well as the simple shift of this pattern from one solid surface to another, both first noted experimentally by the Zisman group,8 are due to the root commonality of molecular interactions in all the liquids and solids involved. Contact angle patterns and related issues have been discussed at greater length elsewhere.9,11,28 Besides the dispersive and hydrogen-bonding terms, the calculations above also included a third “component” for zerofrequency, polar interactions (the first term of eq 5); this component does not cross-interact with the others, but its magnitude is small. This is consistent with experimental observations that have shown a measurable but minor effect of polar liquid−solid interactions on contact angles.33 In summary, the contribution of hydrogen bonding to interfacial properties is primarily dispersive, via the reduction in molecular separation. A simple model based on this principle leads to reasonable estimates for the surface tensions of organic liquids and their contact angles on low-energy solids. In addition, it provides an explanation for the experimental observation that, on the same low-energy solid surface, the contact angles of both hydrogen-bonding and non-hydrogenbonding liquids fall close to the same, smooth curve.
angles were again reasonably close to the pattern of experimental values for various liquids on the same solid,29 with no apparent distinction between non-hydrogen-bonding and hydrogen-bonding liquids. Note that calculated values of cos θ that were greater than 1, indicating complete wetting, were set equal to 1 in Figure 2. Similar results were obtained from calculations of contact angles on two other low-energy solid surfaces: Teflon FEP and poly(methyl methacrylate) (see the Supporting Information). For those 11 liquid/solid pairs for which both calculated and experimental contact angles were available, the average error in the calculated values was 12%.
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DISCUSSION It is generally understood that hydrogen bonding leads to higher liquid surface tensions. Nevertheless, a simple, quantitative link between surface tensions and molecular properties has been available only for dispersive liquids, via the Lifshitz theory. One of the well-known defining features of the hydrogen bond is that the two involved molecules approach each other more closely than they would in the absence of a hydrogen bond; this fact must be taken into account in any consideration of the intermolecular forces. It has been shown in the above, using a simple model with physically reasonable parameters, that taking only this closer distance of approach into account is enough to produce reasonable estimates for the surface tensions of hydrogen-bonding liquids within the Lifshitz theory. This implies that, for interactions between extended surfaces (as opposed to individual molecules), the principal energetic effect of the hydrogen bond is not polar but dispersive. Such a conclusion is consistent with the Lifshitz theory, as discussed above. The Lifshitz theory shows (eq 5) that the relative importance of omnidirectional dispersive forces, compared to unidirectional polar forces, is greatly amplified in a condensed medium relative to a rarified medium. In a rarified medium, a molecule usually has only a single neighbor with which to interact, while in a condensed medium, it is surrounded by neighbors. Of course, directional forces cannot be wholly negligible in condensed media, as they are responsible for the preferred orientations of molecules in associated liquids and solids. The model developed here is highly simplified. Strictly speaking, the Lifshitz theory is applicable only to continuous media, and the separation distances d0 and dHB are equivalent classical representations of complex quantum interactions. The limitations of Israelachvili’s models2,3 apply here as well. Our model is intended as support for the main argument about the nature of the contribution of the hydrogen bond to interfacial properties, rather than as an accurate prediction tool. As such, it does capture the essential physics of the situation, and the approximate calculations give reasonably accurate results and reproduce the patterns of experimental observations. Classical models of molecular interaction are also commonly used in molecular dynamics simulations of liquid water.30,31 Unlike the present work, such models often consider only the O atom as a source of dispersion and treat hydrogen bonding as a Coulombic interaction. In the calculations of both surface tensions (eq 6) and contact angles (eq 8), the molecular interactions were essentially broken into components for dispersion and hydrogen bonding, as in a theory of surface tension components.32 The difference is that, in the present theory,
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ASSOCIATED CONTENT
S Supporting Information *
A spreadsheet containing all calculations for surface tensions and contact angles (including on Teflon FEP and PMMA solid surfaces), a check of the approximation used for the absorption frequency in liquid/polystyrene systems, and details of the calculation of n. This material is available free of charge via the Internet at http://pubs.acs.org.
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AUTHOR INFORMATION
Corresponding Author
*Phone: 613-639-0730; e-mail:
[email protected]. Notes
The authors declare no competing financial interest.
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REFERENCES
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