Article pubs.acs.org/JPCC
Evaluation of the Constant Wavenumber Cutoff Parameter for Modeling van der Waals Energy Nadia Shardt,† Subir Bhattacharjee,‡,§ and Janet A. W. Elliott*,† †
Department of Chemical and Materials Engineering, University of Alberta, Edmonton, AB, Canada T6G 2V4 Department of Mechanical Engineering, University of Alberta, Edmonton, AB, Canada T6G 2G8
‡
ABSTRACT: We examine the assumption that the wavenumber cutoff is constant for two parallel plates separated by a third material independent of which of the substances are interacting or forming the intervening medium. We take experimental values of interfacial tension from the literature to calculate wavenumber cutoffs for system configurations of decane, water, and air, and we find that these wavenumber cutoffs are not constant. The interaction energy of each system is plotted as a function of separation distance to compare the divergent Lifshitz, the constant wavenumber, and the systemspecific wavenumber models.
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INTRODUCTION van der Waals (vdW) intermolecular forces act over long ranges and are the result of fluctuating electric fields in every molecule. These forces can be categorized into three types: Keesom interactions between two dipoles, Debye interactions between a dipole and a nonpolar molecule, and dispersion (also called London) interactions between two nonpolar molecules.1 Historically, methods of modeling van der Waals interactions have struggled with a direct connection between the physical properties of materials and the near-contact interaction energy for a system of two parallel plates separated by a medium. A distance cutoff value related to interatomic spacing has traditionally been used, and the introduction of a wavenumber cutoff is the most recent advancement in the theory of van der Waals interactions.2 This wavenumber cutoff builds on the Lifshitz model, which assumes sharp interfaces, whereas other models such as those by Ninham3 and Podgornik and Parsegian4 incorporate diffuse interfaces. The classic Lifshitz equation is used to model the vdW interaction energy between two semi-infinite planar substrates separated by an intervening medium at nonretarded distances.5 To account for the divergence of this equation as the distance between two semi-infinite planes approaches zero, a distance cutoff parameter lc can be introduced. Physically, this is explained by the fact that two molecules cannot overlap due to the repulsion of their electron clouds, and this cutoff value varies on the basis of the substances’ atomic dimensions. The distance cutoff parameter does not, however, necessarily satisfy thermodynamic conditions pertaining to contact.2 A wavenumber cutoff, kc, was introduced by White6 to give a more natural link to the properties of materials (their dielectric permittivity) and to meet the thermodynamic restriction for the case of contact between two surfaces. In applying the wavenumber cutoff model for two materials, a and c, separated by a third intervening medium b (Figure 1), © 2014 American Chemical Society
Figure 1. Configuration of a system abc of materials a and c separated by material b at a distance L.
White made an assumption that the value of kc remains constant for all combinations of the three materials (i.e., abc, bac, acb). The interaction energy of the general system can be calculated by the Lifshitz method, which is then modified to remove divergence as the plates make contact. Because we are dealing with very close distances near contact (less than 2 nm apart), retardation effects can be disregarded.1 We take experimental values of interfacial tensions between material pairs to calculate the contact interaction energy, determine kc for each combination, and evaluate the validity of the constant kc assumption. The wavenumber cutoff values are calculated for the system composed of air, water, and decane utilizing interfacial tensions for these materials. Because hydrogen bonding interactions are not included in the Lifshitz formulation of vdW interactions,7 and because we are modeling systems consisting of water as one of the materials, the hydrogen bond contributions must be removed from our Received: October 15, 2013 Revised: December 31, 2013 Published: January 21, 2014 3539
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related to the material’s oscillator strength, ωj is the relaxation frequency, and gj represents the bandwidth of the relaxation frequency in the absorption spectrum. For the majority of common substances such as decane, ε(iξn) can be simplified further to include just one term from the ultraviolet frequency and one term from the infrared. The microwave term plays an insignificant role in the value of permittivity for these substances, and the bandwidth gj approaches zero. Thus,
calculations. In other words, we will focus solely on the dispersion component of vdW forces to accurately relate these to experimental values of interfacial tension. The hydrogen bond contribution cannot be completely isolated because it does still influence the dielectric spectrum that is used in the calculation of dispersion forces to follow.
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GOVERNING EQUATIONS The classic nonretarded Lifshitz equation for the interaction energy per unit area as a function of separation distance, L, between two parallel half-spaces a and c separated by an intervening medium b is given by8 A Eabc(L) = − abc2 (1) 12πL 8 where the Hamaker constant Aabc is defined by ∞
Aabc =
1+
(2)
Eabc(0) =
(3)
where ε(iξn) is the permittivity of a material at imaginary frequencies. The n = 0 term of the summation in eq 2 describes the Keesom and Debye contributions, whereas the remaining terms describe the London dispersion energy.1 We exclude the n = 0 term in our calculations because we are examining only the dispersion contribution to vdW interactions. Equation 1 can also be expressed in its standard form of2 kT Eabc(L) = B 2π
∞
∑′∫
∞
0
n=0
dk k ln(1 − ΔabΔcbe
−2kL
2 π
∫0
∞
ωε″(ω) dω ω 2 + ξn2
d + 1 + ξnτ
) (4)
kc(abc) =
j
1+
ωj 2
+
n=0
(9)
(10)
(11)
4π (γac − γab − γcb) ∞
kBT ∑n = 0 ′ ln(1 − ΔabΔcb)
(12)
PARAMETERS FOR THE WATER, DECANE, AND AIR SYSTEM We test the thermodynamically consistent wavenumber cutoff approach developed above for three materials: water (w), decane (d), and air (a). These fluids are studied extensively, and all pertinent parameters of interest are available in literature. Among these fluids, water is a polar molecule, and hence, we only consider the apolar (dispersion) contribution of the surface tension of water. The total water/air interfacial tension γwa can be considered as a sum of the apolar (γdwa) and polar (γhwa) components, where γhwa predominantly comes from the hydrogen bonding contribution.7,12,13 That is,
gjξn ωj 2
∞
∑ ′ ln(1 − ΔabΔcb)
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(6)
ξn 2
kBT kc(abc)2 4π
Because the values of γkj and Δkj are known, kc(abc) values can be calculated for every system containing the materials a, b, and c arranged in any combination.
Cj
∑
(8)
We are now in a position to determine the wavenumber cutoff, kc(abc), for any given combination of three materials by combining eqs 9 and 10, which yields
and ℏ = h/2π, where h is Planck’s constant. To increase the ease of calculation yet maintain accuracy, the main peaks of a material’s absorption spectrum can be isolated, and then the permittivity (eq 5) can be approximated by a form of the Ninham−Parsegian representation9 ε(iξn) = 1 +
ξn ωUV
Eabc (0) + Eacb(0) − Ebcb(0) = 0
(5)
(2πkBT ) for n = 0, 1, 2, ... ℏ
2
( )
where γkj is the interfacial tension between two materials k and j. This relationship between interaction energy and interfacial tension leads to the thermodynamic constraint equation2
where ε″(ω) is the absorption spectrum over frequencies denoted by ω, and ξn are defined by ξn = n
1+
Eabc(0) = γac − γab − γbc
To calculate the interaction energy of a system, the permittivity of a material is determined via the Kramers− Kronig relation8 ε(iξn) = 1 +
ξn ωIR
where kc(abc) is the wavenumber cutoff introduced by White.6 From the thermodynamic perspective, the equation for the contact interaction energy of a general system abc is given by11
εk(iξn) − εj(iξn) εk(iξn) + εj(iξn)
( )
C UV
+
where CIR and CUV are material-dependent constants. With the above theoretical foundation, it can be seen that the Lifshitz formula (eq 1) shows a divergence in interaction energy as the distance L approaches zero. To remove this divergence of the classic Lifshitz equation, a wavenumber cutoff model has been developed, and the contact interaction energy can be calculated with a wavenumber cutoff as follows
In eq 2, the prime on the outer summation indicates that the n = 0 term is multiplied by 0.5. Boltzmann’s constant is indicated by kB, T is the absolute temperature, and Δab and Δcb are defined by6 Δkj =
2
10
∞
(Δ Δ )s 3kBT ∑ ′ ∑ ab 3 cb 2 n=0 s=1 s
C IR
ε(iξn) = 1 +
(7)
d h γwa = γwa + γwa
where the term d/(1 + ξnτ) represents the microwave relaxation frequency (with d and τ being material dependent constants) and the summation represents frequencies in the infrared and ultraviolet range of frequencies. The variable Cj is
(13)
For water/air systems, the total interfacial tension is 72.78 mN/m,14 and the dispersion contribution is 21.8 mN/m.13,15 Using eq 13, water’s hydrogen bonding contribution is 3540
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determined to be 50.98 mN/m. The decane/air interfacial tension is γda = 23.83 mN/m.14 For this study, we are interested in the apolar (dispersion) component of the decane/water interfacial tension. To deduce this quantity, the specific case of the decane/water/air system is particularly useful to analyze because it has a negative Hamaker constant6 and, thus, a positive interaction energy. This means that the value of interfacial tension due to dispersion interactions for the decane/water interface must satisfy Edwa(0) = γda − γdw − γwa > 0. Therefore, the decane/water dispersion component of interfacial tension must be less than 2.03 mN/m. We assume that the hydrogen bond contribution to the interfacial tension of decane/water is similar in magnitude to the hydrogen bond contribution to the interfacial tension of water/air. By subtracting this 50.98 mN/m from the total interfacial tension of decane/water of 52.33,16 a value of 1.35 mN/m is obtained for the dispersion contribution to the decane/water interfacial tension. Such a low value is supported by the fact that interfacial tension does not have a large dispersion component when the substances are very different in polarity.7 Table 1 shows the contact energies of each system obtained by substituting the dispersion components of the interfacial tensions into eq 10.
h = 4.1356675 × 10−15 eV·s, and the constant Cj in eq 7 is calculated by Cj = f j/ωj2 with f j and ωj in units of eV2 and eV, respectively. The permittivity of water has since been more closely fitted by Roth and Lenhoff to experimental data. We have also used these newer constants17 to calculate resulting wavenumber cutoff values as a comparison to using the Parsegian values listed in Table 2. For the permittivity of decane, one can use the simplest approximation (eq 8) where CIR = 0.026, ωIR = 5.540 × 1014 rad/s, CUV = 0.965, and ωUV = 1.873 × 1016 rad/s.6 Note that our value of ωUV is greater than the listed value in Hough and White6 of 0.873 × 1016 because the lower value is inconsistent with the UV frequencies for similar alkanes (all vary around 1.85 × 1016). The parameter ωUV used in literature previous to Hough and White for the alkanes were all close in magnitude to 1.5 × 1016 rad/s.18
Table 1. Contact Interaction Energy Calculated from Values of Interfacial Tensiona
kc(abc) (×109 m−1)
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RESULTS AND DISCUSSION To evaluate the constant wavenumber cutoff assumption in modeling interaction energy, all possible combinations of the decane/water/air system are considered. Table 3 summarizes Table 3. Values of the System-Specific Wavenumber Cutoff and Its Inverse
system
equation
E(0) (mN/m)
system
Parsegian
Roth− Lenhoff
decane/air/decane decane/air/water water/decane/air water/air/water decane/water/decane decane/water/air
Edad(0) = γdd − γda − γda Edaw(0) = γdw − γda − γwa Ewda(0) = γaw − γwd − γad Ewaw(0) = γww − γwa − γwa Edwd(0) = γdd − γdw − γdw Edwa(0) = γda − γdw − γaw
−47.66 −44.28 −3.380 −43.60 −2.700 0.6800
decane/air/decane decane/air/water water/decane/air water/air/water decane/water/decane decane/water/air
4.274 4.453 2.999 4.763 6.490 1.477
4.274 4.157 8.486 4.149 12.465 4.797
Note, γkk = 0. Also, only the dispersion contribution is considered in these calculations (i.e., γwa = 21.8 mN/m, γda = 23.83 mN/m, γdw = 1.35 mN/m).
1/kc (nm) Parsegian
Roth− Lenhoff
0.2340 0.2246 0.3334 0.2099 0.1540 0.6770
0.2340 0.2406 0.1178 0.2410 0.0802 0.2085
a
the wavenumber cutoff values for each system as calculated using eq 12 along with eq 3 and either eq 7 or eq 8 as appropriate. The obtained system-specific wavenumber cutoff values vary within the same order of magnitude from 1.5 × 109 to 6.5 × 109 m−1 when the Parsegian representation is used for water’s permittivity, and from 4.1 × 109 to 1.2 × 1010 m−1 when the Roth−Lenhoff representation is used. The values of 1/kc
The data in Table 2 were used to calculate the permittivity of water using eq 7 and the Δkj value between water and decane or air using eq 3. Units of eV were converted to rad/s through the formula ω j (eV) × (2π/h) = ω j (rad/s), where
Table 2. Frequency, Oscillator Strength, and Bandwidth Values of Water19 Infrared Frequencies ωj (rad/s × 1014)
ωj (eV) −2
2.07 × 10 6.9 × 10−2 9.2 × 10−2 2 × 10−1 4.2 × 10−1
0.315 1.05 1.40 3.04 6.38
f j (eV2)
Cj −4
gj (eV)
6.25 × 10 3.5 × 10−3 1.28 × 10−3 5.69 × 10−3 1.35 × 10−2 UV Frequencies
1.46 0.735 0.151 0.142a 0.0765
1.5 3.8 2.8 2.5 5.6
× × × × ×
−2
10 10−2 10−2 10−2 10−2
gj (rad/s × 1013) 2.28 5.78 4.25 3.80 8.51
ωj (eV)
ωj (rad/s × 1016)
f j (eV2)
Cj
gj (eV)
gj (rad/s × 1015)
8.25 10 11.4 13 14.9 18.5
1.25 1.52 1.73 1.98 2.26 2.81
2.68 5.67 12 26.3 33.8 92.8
0.0394 0.0567 0.0923 0.156 0.152 0.271
0.51 0.88 1.54 2.05 2.96 6.26
0.775 1.34 2.34 3.11 4.50 9.51
As a point of interest, this value of 0.142 for Cj differs from the Gingell−Parsegian9 value by a factor of 10 (0.0136 for ωj = 3.065 × 1014) but agrees with Russel et al.20 This difference does not significantly change the value of εwater. a
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range from 0.15 to 0.68 nm using the Parsegian representation; inverse wavenumbers range from 0.080 to 0.24 nm using Roth−Lenhoff constants. Therefore, the assumption of a universally constant wavenumber cutoff across all combinations of materials may be questioned. System-specific wavenumber cutoffs generally lead to a greater magnitude in the interaction energy of analyzed systems, except for the case of water/decane/air where the cutoff leads to a lower magnitude. This trend is illustrated in Figures 2−6 for systems with attractive interaction energies.
Figure 5. Model comparison for the water/air/water system. N−P indicates the Ninham−Parsegian representation of water’s permittivity, whereas R−L indicates that of Roth−Lenhoff.
Figure 2. Model comparison for the decane/air/decane system. Note that the White and system-specific wavenumber cutoff values are identical for this system. N−P indicates the Ninham−Parsegian representation of water’s permittivity, whereas R−L indicates that of Roth−Lenhoff. Figure 6. Model comparison for the decane/water/decane system. N− P indicates the Ninham−Parsegian representation of water’s permittivity, whereas R−L indicates that of Roth−Lenhoff.
The system-specific wavenumber cutoff values for the attractive systems vary from 3.0 × 109 to 6.5 × 109 m−1 in the Parsegian representation. Using the more recent Roth−Lenhoff representation gives an even wider range of wavenumber cutoff values from 4.1 × 109 to 1.2 × 1010 m−1. In the repulsive decane/water/air system (Figure 7), a discrepancy also exists between the constant and systemspecific wavenumber models. When the Parsegian representation is used, an extreme of 1.5 × 109 m−1 for the system-specific wavenumber constant is calculated and gives a corresponding inverse wavenumber of 0.68 nm. Figure 8 shows the effect of changing the contact interaction energy calculated with eq 10
Figure 3. Model comparison for the decane/air/water system. N−P indicates the Ninham−Parsegian representation of water’s permittivity, whereas R−L indicates that of Roth−Lenhoff.
Figure 4. Model comparison for the water/decane/air system. N−P indicates the Ninham−Parsegian representation of water’s permittivity, whereas R−L indicates that of Roth−Lenhoff.
Figure 7. Model comparison between the classic Lifshitz, constant wavenumber cutoff, and system-specific wavenumber cutoff for the repulsive decane/water/air system. 3542
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tensions and IR spectra are known. Advancing the accuracy of this energy modeling with a system-specific cutoff may have key applications in developing adhesive and repellent surfaces.
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AUTHOR INFORMATION
Corresponding Author
*J. A. W. Elliott: e-mail,
[email protected]. Present Address §
Water Planet Engineering, 721 South Glasgow Avenue, Los Angeles, CA, 90301. E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
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Figure 8. Effect of changing γdw, and thus E(0) according to eq 10, on the wavenumber cutoff of the decane/water/air system. Both the Ninham−Parsegian representation and the Roth−Lenhoff representation are considered.
ACKNOWLEDGMENTS Funding from the Natural Sciences and Engineering Research Council (NSERC) of Canada is gratefully acknowledged. J.A.W.E. holds a Canada Research Chair (CRC) in Thermodynamics.
on the system-specific wavenumber cutoff for decane/water/air. The interaction energy was varied by increasing the value of γdw from zero up to a maximum possible 2.03 mN/m determined by the constraint of positive interaction energy. As the interfacial tension reaches its maximum, the energy approaches zero, and the wavenumber cutoff value accordingly goes to zero. A smaller discrepancy between the constant and system-specific wavenumber models is observed when the Roth−Lenhoff constants are used.
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REFERENCES
(1) Israelachvili, J. Intermolecular and Surface Forces; Academic Press: London, U.K., 1992. (2) White, L. R. van der Waals interaction energy and disjoining pressure at small separation. J. Colloid Interface Sci. 2010, 343, 338− 343. (3) Mahanty, J.; Ninham, B. W. Dispersion Forces; Academic Press Inc. (London) Ltd.: London, 1976. (4) Podgornik, R.; Parsegian, V. A. van der Waals interactions in a dielectric with continuously varying dielectric function. J. Chem. Phys. 2004, 121, 7467−7473. (5) Lifshitz, E. M. The theory of molecular attractive forces between solids. Sov. Phys. JETP-USSR 1956, 2, 73−83. (6) Hough, D. B.; White, L. R. The calculation of Hamaker constants from Lifshitz theory with applications to wetting phenomena. Adv. Colloid Interfac. 1980, 14, 3−41. (7) Wu, S. Polymer Interface and Adhesion; Marcel Dekker, Inc.: New York, 1982. (8) Hunter, R. J. Foundations of Colloid Science; Oxford University Press Inc.: New York, 1995; Vol. 1. (9) Gingell, D.; Parsegian, V. A. Computation of van der Waals interactions in aqueous systems using reflectivity data. J. Theor. Biol. 1972, 36, 41−52. (10) Bergstrom, L. Hamaker constants of inorganic materials. Adv. Colloid Interfac. 1997, 70, 125−169. (11) van Oss, C. K.; Chaudhury, M. K.; Good, R. J. Interfacial Lifshitz-van der Waals and polar interactions in macroscopic systems. Chem. Rev. 1988, 88, 927−941. (12) Kaelble, D. H. Dispersion-polar surface tension properties of organic solids. J. Adhes. 1970, 2, 66−81. (13) Fowkes, F. M. Contact Angle, Wettability, and Adhesion; American Chemical Society: Washington, DC, 1964. (14) Haynes, W. M. CRC Handbook of Chemistry and Physics, 92nd ed.; CRC Press: Boca Raton, FL, 2011. (15) van Oss, C. J. Interfacial Forces in Aqueous Media, 2nd ed.; CRC Press: Boca Raton, FL, 2006. (16) Zeppieri, S.; Rodriguez, J.; de Ramos, A. L. L. Interfacial tension of alkane and water systems. J. Chem. Eng. Data 2001, 46, 1086−1088. (17) Roth, C. M.; Lenhoff, A. M. Improved parametric representation of water dielectric data for Lifshitz theory calculations. J. Colloid Interface Sci. 1996, 179, 637−639. (18) Richmond, P.; Ninham, B.; Ottewill, R. A theoretical study of hydrocarbon adsorption on water surfaces using Lifshitz theory. J. Colloid Interface Sci. 1973, 45, 69−80. (19) Parsegian, V. A. In Physical Chemistry: Enriching Topics From Colloid and Surface Science; van Olphen, H., Mysels, K. J., Eds.; THEOREX: La Jolla, CA, 1975.
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CONCLUSIONS Experimental values of permittivity and interfacial tension were used to calculate the wavenumber cutoff specific to each system consisting of decane, air, and water. Because these wavenumber cutoffs varied from 1 × 109 to 6 × 109 m−1 (Parsegian) and 4 × 109 to 1.2 × 1010 m−1 (Roth−Lenhoff) over all systems, the assumption of a constant wavenumber cutoff can be considered accurate for this combination of materials to an order of magnitude. If more precision is needed in calculations, however, this assumption may not be made. Drummond and Chan21 performed a study relating the surface tensions of apolar substances to the contact energy based on Lifshitz theory and estimated the distance cutoff for various combinations of materials. They could reconcile the theoretical estimates of the contact energy with the experimental values for some of the hydrocarbon systems using a narrow range of distance cutoff values and combining rules. However, they observed a wider discrepancy between theory and experiment for PTFE/hydrocarbon systems. They also noted gaps related to consistent data for many of the materials. We note that, similar to the observation that the distance cutoff value is not unique for many material systems,21 we also observe a nonuniqueness of the wavenumber cutoff value in the decane/water/air systems. Whether this can be attributed to incompleteness of Lifshitz theory, various approximations inherent in the mathematical simplifications used, or errors in experimental values of the dielectric response or interfacial tension, still remain open questions. However, if we consider the fact that wavenumber cutoff values can be system-specific, we can establish a link between the van der Waals contact energies based on Lifshitz theory and the experimentally measured interfacial tensions. Such an approach is more thermodynamically consistent than the distance cutoff approach and can help create models of systems whose interfacial 3543
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(20) Russel, W. B.; Saville, D. A.; Schowalter, W. R. Colloidal Dispersions; Cambridge University Press: Cambridge, U.K., 1989. (21) Drummond, C. J.; Chan, D. Y. C. van der Waals interaction, surface free energies, and contact angles: dispersive polymers and liquids. Langmuir 1997, 13, 3890−3895.
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