pubs.acs.org/Langmuir © 2009 American Chemical Society
Importance of Accurate Dynamic Polarizabilities for the Ionic Dispersion Interactions of Alkali Halides Drew F. Parsons* and Barry W. Ninham Research School of Physical Sciences and Engineering, Australian National University, Canberra, ACT 0200, Australia Received July 13, 2009. Revised Manuscript Received September 2, 2009 Ab initio quantum mechanical calculations of the dynamic polarizability of alkali metal and halide ions are performed as a function of imaginary frequency. Electron correlation is shown to provide a significant correction to ionic polarizabilities. Ab initio ion-surface dispersion coefficients are compared with single- and multimode London approximations. The commonly employed single-mode model with the characteristic frequency taken from the ionization potential of the ion is shown to be inadequate, underestimating dispersion forces with an average error around 40% or as high as 80% for halide ions. Decomposition of the polarizability data into five modes covers the major modes of each ion adequately (four modes for Li+). Illustrative calculations of surface potentials at the mica surface in aqueous alkali halide electrolytes are made. Charge reversal is obtained with the more polarizable cations, K+ and Rb+. The error in the single-mode ionization potential models is seen as a strong shift in the surface potential from negative toward positive values.
Introduction A great deal of recent work to do with Hofmeister effects focuses on the role of nonelectrostatic (dispersion) forces in ion-solvent, ion-ion, and ion-interface interactions. These ion-specific forces are missing from classical theories of physical chemistry and colloid science.1,2 A major inhibition to the development of a predictive theory that accounts for Hofmeister effects is a lack of knowledge of ionic polarizabilities as a function of frequency. Simple electrostatic theories, for instance, the DerjaguinLandau-Verwey-Overbeek (DLVO) theory of colloid interactions, characterize ions only by charge (apart from ionic radius). They embody no specific ion effects between ions of the same size and charge, contrary to experiment. It is generally accepted that the explanation for the Hofmeister series lies largely in quantum mechanical nonelectrostatic ionic interactions. These interactions, missing from classical theories, are highly specific. They involve the frequency-dependent polarizabilities of the ions and include induction (charge-induced dipole) forces3,4 as well as dispersion (van der Waals, induced dipole-induced dipole) forces.1,5-7 Ion specificity also appears due to the shape and size of the ion, with anisotropic interactions in molecular simulations playing a crucial role in describing solvation and complexation effects.4,8,9 In this paper, we focus on the role of dynamic polarizability in determining the strength of dispersion or van der Waals energies. (1) Kunz, W.; Lo Nostro, P.; Ninham, B. W. Curr. Opin. Colloid Interface Sci. 2004, 9, 1–18. (2) Ralston, J.; Fornasiero, D.; Mishchuk, N. Colloids Surf., A 2001, 192, 39–51. (3) Netz, R. R. Curr. Opin. Colloid Interface Sci. 2004, 9, 192–197. (4) Masia, M. J. Chem. Phys. 2008, 128, 184107. (5) Bostr€om, M.; Williams, D. R. M.; Stewart, P. R.; Ninham, B. W. Phys. Rev. E 2003, 68, 041902. (6) Bostrom, M.; Williams, D.; Ninham, B. Langmuir 2002, 18, 6010–6014. (7) Ninham, B.; Yaminsky, V. Langmuir 1997, 13, 2097–2108. (8) Gresh, N.; Cisneros, G.; Darden, T.; Piquemal, J. J. Chem. Theory Comput. 2007, 3, 1960–1986. (9) Rajter, R. F.; Podgornik, R.; Parsegian, V. A.; French, R. H.; Ching, W. Y. Phys. Rev. B: Condens. Matter Mater. Phys. 2007, 76, 045417.
1816 DOI: 10.1021/la902533x
van der Waals interactions have started appearing in solutions of the electrostatic Poisson-Boltzmann equation,10,11 represented in approximate form via the parameters for Lennard-Jones potentials. More fundamentally, they may be expressed via the dynamic polarizability R(iω) of an ion. As an example, the dispersion interaction between an ion in solution and a nearby interface is given by12-15 UðzÞ ¼
BhðzÞ z3
ð1Þ
where hðzÞ ¼ 1 ! " # " # 2z 2z2 z2 4z4 z 1 þ erfc -1 exp þ pffiffiffi a a2 a4 π a a2 ð2Þ and a is the Gaussian radius of the ion, corresponding to a Gaussian spatial spread of the ionic polarizability.15,16 Ion size is thereby included in the model, taking it beyond the limitations of point charge models. When the ion is sufficiently far from the interface (separated by more than several ion radii), then h(z) ≈ 1. At the surface, it reaches a finite limit, such that U(z=0) = 16B/[3(π)1/2a3], which is related to ion-surface binding energies. The dispersion B coefficient is calculated from (10) Baker, N. A.; Sept, D.; Joseph, S.; Holst, M. J.; McCammon, J. A. Proc. Natl. Acad. Sci. U.S.A. 2001, 98, 10037–10041. (11) Dzubiella, J.; Swanson, J.; McCammon, J. J. Chem. Phys. 2006, 124, 084905. (12) Mahanty, J.; Ninham, B. W. Dispersion Forces; Academic Press: London, 1976. (13) Mahanty, J.; Ninham, B. W. J. Chem. Phys. 1973, 59, 6157–6162. (14) Mahanty, J.; Ninham, B. J. Chem. Soc., Faraday Trans. 2 1974, 70, 637– 650. (15) Mahanty, J.; Ninham, B. W. Faraday Discuss. Chem. Soc. 1975, 59, 13–21. (16) Parsons, D. F.; Ninham, B. W. J. Phys. Chem. A 2009, 113, 1141–1150.
Published on Web 09/21/2009
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the excess polarizability R*(iω) of the ion in solution,3,17 mediated by the dielectric difference Δ(iω) between solvent and surface B ¼
kT X 0 Rðiωn ÞΔðiωn Þ 2 n εw ðiωn Þ
ð3Þ
where εw ðiωÞ -εs ðiωÞ ð4Þ εw ðiωÞ þ εs ðiωÞ P and ωn = 2πnkT/h; the summation prime ( 0 ) indicates that the zero-frequency term at n = 0 is taken with a factor of 1/2.18 The εw(iω) and εs(iω) are the dielectric functions of the solvent (water) and surface, respectively. The excess polarizability R*(iω) describes the effective polarizability of the ion in the presence of the depolarizing field of the surrounding solvent; further discussion of the meaning of R*(iω) is found in the Appendix. Thus, an accurate description of nonelectrostatic van der Waals interactions, and therefore an accurate and adequate explanation of ion-specific effects, is contingent upon an accurate description of the ionic dynamic polarizabilities. In general, the dynamic polarizability can be decomposed into a series of electronic transitions,19 each characterized by frequency ωj with oscillator strength fj (here, normalized with P j fj = 1) ΔðiωÞ ¼
RðiωÞ ¼ R0
X
fj
j
1 þ ðω=ωj Þ2
ð5Þ
R0 1 þ ðω=ωI Þ2
ð6Þ
where R0 is the static polarizability of the ion and ωI is the frequency corresponding to the ionization potential (IP) of the ion. We show here that this approximation is generally poor. Quantum chemical techniques for calculating dynamic polarizabililities are known24-27 but have been more commonly applied to neutral molecules. In this paper, we present the ab initio quantum mechanical calculation of the dynamic polarizability of a range of monatomic ions important to experimental ion-specific effects. The ab initio QM polarizabilities are fitted to modal decompositions (eq 5) with up to five modes, allowing the (17) Landau, L. D.; Lifshitz, E. M. Electrodynamics of Continuous Media; Pergamon Press: Oxford, U.K., 1960; Vol. 8. (18) Parsegian, V.; Ninham, B. Biophys. J. 1970, 10, 664–674. (19) Langhoff, P. W.; Gordon, R. G.; Karplus, M. J. Chem. Phys. 1971, 55, 2126–2145. (20) Malykhanov, Y.; Chadin, R. J. Appl. Spectrosc. 2005, 72, 1–8. (21) Bostr€om, M.; Ninham, B. W. Langmuir 2004, 20, 7569–7574. (22) Kunz, W.; Belloni, L.; Bernard, O.; Ninham, B. J. Phys. Chem. B 2004, 108, 2398–2404. (23) Tavares, F.; Bratko, D.; Blanch, H.; Prausnitz, J. J. Phys. Chem. B 2004, 108, 9228–9235. (24) Silvi, B.; Fourati, N. Mol. Phys. 1984, 52, 415–430. (25) Woon, D. E.; Thom, H.; Dunning, J. J. Chem. Phys. 1994, 100, 2975–2988. (26) Adamovic, I.; Gordon, M. Mol. Phys. 2005, 103, 379–387. (27) Bast, R.; Hesselmann, A.; Sazek, P.; Helgaker, T.; Saue, T. ChemPhysChem 2008, 9, 445–453.
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ion FClBrILi+ Na+ K+ Rb+ Cs+ Fr+
uncorr-R0 (A˚3) 1.218 4.220 4.350 8.967 0.0280 0.131 0.795 1.348 2.345 2.938
corr-R0 (A˚3) 1.913 4.861 6.490 9.649 0.0285 0.139 0.814 1.379 2.402 3.000
% corr. 36.3 13.2 33.0 7.1 1.5 5.6 2.4 2.3 2.4 2.1
nitrate 4.008 5.022 20.2 sulfate 6.132 8.056 23.9 hydroxide 2.746 4.753 42.2 formate 3.937 5.058 22.2 carbonate 6.043 9.835 38.6 phosphate 9.891 18.182 45.6 arsenate 11.430 21.116 45.9 a Electron correlation was performed by the CCSD method for all ions except carbonate, phosphate, and arsenate, for which DFT/PBE0 was used. The last column shows the correction due to electron correlation, given as the percent difference between uncorrelated and correlated values relative to the electron-correlated value.
validity of the single-mode IP approximation, in eq 6, to be tested. These fitted decompositions are intended for use in applications such as modified Poisson-Boltzmann studies which include dispersion forces calculated using Lifshitz theory.
Computational Details
where R0 is the static polarizability (an exact formula will also include other terms derived from complex characteristic frequencies20). However, since the full set of oscillator strengths and frequencies was not, in general, known, an approximation using a single-mode London approximation has been used,21-23 that is RðiωÞ≈
Table 1. Static Polarizabilities for Selected Ions Calculated by Ab Initio Quantum Calculations without (Hartree-Fock, headed “uncorr-r0”) and with Electron Correlations (headed “corr-r0”)a
Ab initio calculations of dynamic polarizabilities were performed using MOLPRO.28 Electron correlation was handled using the coupled cluster single and double (CCSD) excitation level of theory.29 The aug-cc-pVxZ family of basis sets30-32 was used for all but the heavier ions. The ECPnMDF family33,34 was used for larger ions (K+, Rb+, Cs+, Fr+, I-). It is worth commenting on the significance of electron correlation for ion polarizabilities. The static intrinsic polarizabilities of a sample of ions is shown in Table 1, comparing electron-correlated estimates to Hartree-Fock estimates with no electron correlation. The electron-correlated static polarizabilities for phosphate, arsenate, and carbonate were calculated using GAUSSIAN 03,35 (28) Werner, H.-J. et al. MOLPRO, version 2008.1, a package of ab initio programs; 2008, see http://www.molpro.net. (29) Hampel, C.; Peterson, K. A.; Werner, H.-J. Chem. Phys. Lett. 1992, 190, 1– 12. (30) Woon, D. E.; Thom, H.; Dunning, J. J. Chem. Phys. 1993, 98, 1358–1371. (31) Wilson, A. K.; Woon, D. E.; Peterson, K. A.; Thom, H.; Dunning, J. J. Chem. Phys. 1999, 110, 7667–7676. (32) Peterson, K. A.; Thom, H.; Dunning, J. J. Chem. Phys. 2002, 117, 10548– 10560. (33) Lim, I. S.; Schwerdtfeger, P.; Metz, B.; Stoll, H. J. Chem. Phys. 2005, 122, 104103. (34) Peterson, K. A.; Shepler, B. C.; Figgen, D.; Stoll, H. J. Phys. Chem. A 2006, 110, 13877–13883. (35) Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Scuseria, G. E.; Robb, M. A.; Cheeseman, J. R.; Montgomery, J. A., Jr.;Vreven, T.; Kudin, K. N.; Burant, J. C.; Millam, J. M.; Iyengar, S. S.; Tomasi, J.; Barone, V.; Mennucci, B.; Cossi, M.; Scalmani, G.; Rega, N.; Petersson, G. A.; Nakatsuji, H.; Hada, M.; Ehara, M.; Toyota, K.; Fukuda, R.; Hasegawa, J.; Ishida, M.; Nakajima, T.; Honda, Y.; Kitao, O.; Nakai, H.; Klene, M.; Li, X.; Knox, J. E.; Hratchian, H. P.; Cross, J. B.; Bakken, V.; Adamo, C.; Jaramillo, J.; Gomperts, R.; Stratmann, R. E.; Yazyev, O.; Austin, A. J.; Cammi, R.; Pomelli, C.; Ochterski, J. W.; Ayala, P. Y.; Morokuma, K.; Voth, G. A.; Salvador, P.; Dannenberg, J. J.; Zakrzewski, V. G.; Dapprich, S.; Daniels, A. D.; Strain, M. C.; Farkas, O.; Malick, D. K.; Rabuck, A. D.; Raghavachari, K.; Foresman, J. B.; Ortiz, J. V.; Cui, Q.; Baboul, A. G.; Clifford, S.; Cioslowski, J.; Stefanov, B. B.; Liu, G.; Liashenko, A.; Piskorz, P.; Komaromi, I.; Martin, R. L.; Fox, D. J.; Keith, T.; Al-Laham, M. A.; Peng, C. Y.; Nanayakkara, A.; Challacombe, M.; Gill, P. M. W.; Johnson, B.; Chen, W.; Wong, M. W.; Gonzalez, C.; Pople, J. A. Gaussian 03, revision D.01 ; Gaussian, Inc.: Wallingford, CT, 2004.
DOI: 10.1021/la902533x
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Table 2. Ab Initio Static Polarizabilities for Monatomic Ions Calculated by Various Electron Correlation Methodsa ion
CCSD
DFT/PBE0
MP2
-
1.913 1.917 1.902 F 4.861 5.035 4.860 Cl6.490 6.579 6.340 Br9.649 9.979 9.483 I+ 0.028 0.030 0.028 Li 0.139 0.144 0.139 Na+ 0.814 0.826 0.806 K+ 1.379 1.384 1.360 Rb+ + Cs 2.402 2.374 2.359 3.000 2.924 2.933 Fr+ avg % diff 2.3 1.2 a Comparison is made between CCSD, DFT/PBE0, and MP2 methods using the same basis sets in each method. The last row shows the average relative error in the latter two methods relative to CCSD.
using density functional theory (DFT) with the PBE0 functional,36 which generates polarizabilities very close to those found using CCSD.37 The correction due to electron correlation is vast for ions with delocalized electrons such as phosphate, carbonate, and formate, with the correction in phosphate exceeding 45%. The electron correlation correction is also seen to be large (42%) in hydroxide. This has significant implications for biological applications of the theory of dispersion forces involving carboxylate, hydroxyl, and phosphate groups and, no doubt, in other delocalized systems such as porphyrin rings; electron correlation cannot be ignored in these systems. In this paper, however, we focus on the monatomic alkali metal and halide ions. The correction due to electron correlation is less than 6% for the metal alkali cations but is larger for halide anions, exceeding 30% correction for Br-. Coupled cluster (CCSD) calculations represent the most accurate level of quantum chemistry available today and are computationally more expensive than other methods. The computational cost is affordable for the monatomic ions considered in this paper but quickly becomes prohibitive for more complex polyatomic ions. It is therefore worth comparing the output of various electron correlation methods, calculated using the same basis set. Table 2 compares static polarizabilities calculated under CCSD against those calculated under DFT/PBE0 (faster, less accurate) and under the M€ oller-Plesset second-order perturbation (MP2) method (middle speed and accuracy). The average relative error in DFT/PBE0 calculations is 2.3%, while the average relative error in MP2 is 1.2%. We would consider this level of accuracy sufficient to justify the use of these computationally cheaper methods for more complex ions and molecules. Note that one of the chief conclusions to be brought out of the current work is that accurate calculation of the dynamic polarizability at high frequencies is more significant than the value of the static polarizability (at zero frequency). We therefore repeat the comparison of CCSD against DFT/PBE0 in the context of dynamic polarizabilities (via surface dispersion B coefficients) below. Dynamic polarizabilities along imaginary frequencies are not currently available at the MP2 level of theory. Ab initio calculations determine the intrinsic polarizability of the ions in vacuum. Corresponding experimental values are not available. Published experimental polarizabilities38 for the alkali metal and halide ions are derived from crystal data and therefore represent excess polarizabilities in the condensed crystalline phase, rather than ion polarizabilities in vacuum. Our electroncorrelated ab initio methods were however validated by calculations for small neutral molecules, for which experimental data are
(36) Adamo, C.; Barone, V. J. Chem. Phys. 1999, 110, 6158–6170. (37) Adamo, C.; Cossi, M.; Scalmani, G.; Barone, V. Chem. Phys. Lett. 1999, 307, 265–271. (38) CRC Handbook of Chemistry and Physics, 84th ed.; Lide, D. R., Ed.; CRC Press: Boca Raton, FL, 2004.
1818 DOI: 10.1021/la902533x
Figure 1. Dynamic polarizability R(iω) for a chloride ion along the imaginary frequency axis. The ab initio curve is compared with the single-mode curve based on the ionization potential (IP, in vacuum and in water) of Cl- and with single- and two-mode fitted curves. The two-mode fitted curve is nearly congruent with the ab initio curve. available. Good agreement between experimental and ab initio calculation was found for these molecules.
Dynamic Polarizabilities Dynamic polarizabilities were calculated up to a frequency of 1017 Hz, corresponding to far-UV frequencies (around 5 nm in wavelength). In terms of Lifshitz sums over contributing frequencies for dispersion energies, for example, eq 3, this means summing up to ωn with n = 2100. An example is given for chloride in Figure 1. The simple single-frequency London model (eq 6) with a characteristic frequency corresponding to the ionization potential of the Cl- ion (that is, the electron affinity of the chlorine atom, not the ionization potential of the atom) is also given in the figure. We clearly see that the single-mode IP model matches the ab initio polarizability poorly, cutting out at far too low of a frequency and therefore grossly underestimating the full magnitude of the dispersion interaction. The ab initio polarizability may instead be fitted to a single mode, allowing the frequency to be fitted freely while constraining the weight of the mode to equal the static polarizability. This fitted single mode, shown in Figure 1, broadly follows the ab initio curve, slightly overestimating at lower frequencies and underestimating at higher frequencies. The fitting procedure followed a nonlinear least-squares } in eq 5 were approach. Each of the fitting parameters {fj, ωjP allowed to vary freely subject to the constraint j fj = 1. The fitted polarizability at frequency ωn due to the fitting parameters was given by R(ωn). If the ab initio polarizabilities at each frequency ωn (with n ranging from 0 to 2100) are given by Rn, then the least-squares method iteratively varies the fitting parameters {fj, ωP j} in order to minimize the sum of the squared residuals D = n [R(ωn) - Rn]2, with the aid of derivatives ∂R(ωn)/∂fj and ∂R(ωn)/∂ωj. Nonlinear least-squares routines were provided by the GNU Scientific Library (GSL). We continued decomposing the ab initio curve into a set of different modes, each with their own weight fj and characteristic fitted frequency ωj. Each extra mode successively improved the fit to the ab initio curve. The two-mode fitted curve is shown in Figure 1 (nearly congruent with the ab initio curve); higher modal fits are not shown since they essentially overlay the ab initio curve. We address the question of how many fitted modes are required to effectively reproduce the ab initio data by considering the Langmuir 2010, 26(3), 1816–1823
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Figure 2. Dynamic polarizability R(ω) for a chloride ion along the real axis. The frequencies of the major modes are marked. The values in parentheses are the frequencies found by fitting a five-mode function to R(iω) along the imaginary axis.
dynamic polarizability over the real axis. A sample for the chloride ion, calculated using GAUSSIAN, is given in Figure 2. This curve suggests that there are around five major modes, as suggested by the strength of the poles along the real axis, spread more or less evenly on a logarithmic scale between frequencies 0.1 and 10 au (where 1 au = 6.57968 105 Hz), with a number of additional minor modes with small weights. We note that five or four modes have previously been seen to sufficiently describe the dynamic polarizability of ions isoelectronic with the sodium atom.39 A five-mode fit to data along the imaginary axis yields modal frequencies at 0.21, 0.40, 0.86, 2.88, and 11.1 au, which matches reasonably well the real axis poles at 0.25, 0.44, 0.92, 2.68, and 11.6 au. We attribute the differences in the two sets of frequencies to an averaging effect where the fitted frequencies average out the neighboring minor modes. The close correspondence between fitted and physical frequencies indicates that the modal decomposition with five modes has satisfactorily captured the physics of the ion’s polarizability over the full frequency range. The light ion Li+ is the exception which, due to its small number of electrons, is modeled with only four major modes.
Surface Dispersion B Coefficients The quality of the fitted modes is best tested by applying the dynamic polarizability to the calculation of dispersion energies. In particular, ion-surface dispersion interactions may be characterized by dispersion B coefficients; see eq 3. The formula in eq 3 neglects retardation effects due to the finite speed of light. The effect of retardation is to dampen the contribution of high frequencies to the B coefficient.40 The dampening factor is given by exp(ω/ωs), where the dampening frequency ωs depends on the distance z of the ion from the surface, ωs = c/2z(εs)1/2, εs being the dielectric constant of the surface. Hence, retardation is only significant when z is large enough to bring ωs down to the optical/UV frequencies which provide most of the contribution to the B coefficient. At higher frequencies, X-ray and above, R*(iω) and Δ(iω) (see eq 3) are already close to 0 and therefore do not contribute to B. The ion-surface dispersion interaction is strongest within a few ionic radii, say around 5-10 A˚ from the surface. At z = 10 A˚, for instance, with ε ≈ 5 (mica), we have ωs ≈ 7 1016 Hz, (39) Kundu, B.; Ray, D.; Mukherjee, P. K. Phys. Rev. A 1986, 34, 62–70. (40) Ninham, B. W.; Parsegian, V. A. Biophys. J. 1970, 10, 646–663.
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corresponding to far-UV wavelengths of around 50 A˚. Higher frequencies, X-ray and above, will be dampened, but they, as just stated, already do not contribute significantly. Conversely, the contribution of optical/UV frequencies will be dampened when the ion is further than, say, 50 A˚ from the interface, but at this distance, the dispersion energy (≈B/z3) is already negligible. Retardation effects should therefore not be important for ionsurface interactions. Theoretical dispersion B coefficients at the mica-water interface, calculated for T = 298.15 K using ab initio dynamic polarizabilities, are presented in Table 3. We have applied the dielectric model of Dagastine, Prieve, and White for water41 and the Chan-Richmond model for mica.42 Values calculated using ab initio polarizabilities are compared against the B coefficients derived from the single-mode ionization potential model of the polarizability, using the ionization potential both in vacuum38 and in water.23 The polarizabilities of isolated ions were transformed into excess polarizabilities in water by the procedure described in the Appendix. Ion sizes (Gaussian radii a) were derived from ab initio estimates of ionic volumes.16 The error in the single-mode IP model relative to the ab initio values is also given in Table 4. Similarly, the average relative error in the modal decompositions is given in Table 4. Comparison for the single-mode model of the polarizability, eq 6, is made using both the ionic ionization potential (IP) for simple ions as well as a single fitted characteristic frequency. The ionization energy for halide anions is the atomic electron affinity, the second ionization potential for the singly charged group I cations, and the third ionization potential for the doubly charged group II cations. The error due to the ionization potential models is given in Tables 3 and 4. The performance of the single-mode ionization model is not too bad (≈5% error) for some of the metal cations such as K+ and Rb+ but is very poor (10-30% error) for other cations. For the halide anions, the ionization model is simply abysmal, missing three-quarters of the total dispersion energy. Moreover, the ionization potential in vacuum gets the sign wrong due to the impact of the error in the model on excess polarizabilities. Instead of bromide and iodide ions being attracted to the mica surface, in the vacuum ionization potential model, they are repelled! Magnitudes are not improved by using ionization potentials in water, although the sign is correct. The error in the halides is reduced slightly, from an average of 114 down to 69% error, but the performance of the metal cations is markedly degraded (from an average 10 to 19% error). In either case, whether taking ionization potentials of the ions in vacuum or in water, the average error over all ions is about 40%. Some previous studies21,22 have used atomic static polarizabilities and atomic ionization potentials for the single-mode IP model rather than the ionic values. The error due to using atomic values is omitted from Table 4 for the sake of brevity, but average errors are 48% for anions and 1430% for cations (that is, in error by an order of magnitude). The immense magnitude of the error can be readily understood from the static polarizabilities; ionic halide polarizabilities are twice as large as atomic halogen values, while the metal cation polarizabilities are at least an order of magnitude smaller than those for uncharged metal atoms. An additional minor source of error in ref 21 is that only the first 200 or so frequencies were used in that study (ωn up to n = 200, or 8 1015 Hz), losing higher frequencies. The current study uses (41) Dagastine, R. R.; Prieve, D. C.; White, L. R. J. Colloid Interface Sci. 2000, 231, 351–358. (42) Chan, D.; Richmond, P. Proc. R. Soc. London, Ser. A 1977, 353, 163–176.
DOI: 10.1021/la902533x
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Parsons and Ninham Table 3. Surface Dispersion B Coefficients (10-50 J m3) for Simple Ions at the Mica-Water Interfacea
ion
full ab initio (CCSD)
ab initio (DFT/PBE0)
one-mode IP (in vacuum)
one-mode IP (in water)
-
-1.210 -1.218 (0.70) -0.007 (-99.4) -0.517 (-57.3) Cl -1.301 -1.309 (0.56) 0.339 (-126.1) -0.312 (-76.0) Br-1.915 -1.891 (1.3) 0.288 (-115.0) -0.487 (-74.6) I -0.0057 -0.0067 (17.8) -0.0060 (5.7) -0.0075 (31.3) Li+ -0.040 -0.043 (7.9) -0.049 (24.5) -0.052 (30.2) Na+ + -0.336 -0.341 (1.3) -0.347 (3.0) -0.320 (-5.0) K + -0.559 -0.558 (0.10) -0.584 (4.6) -0.505 (-9.6) Rb -0.836 -0.817 (2.3) -0.921 (10.1) -0.675 (-19.2) Cs+ a Values calculated using ab initio polarizabilities are compared with the single-mode model with characteristic frequencies based on the ionic ionization potentials (IP) in vacuum38 and in water.23 The DFT/PBE0 ab initio method is also compared. Values in parentheses indicate the % error relative to the full ab initio (CCSD) value.
Table 4. Average Magnitude of Relative Error in Surface Dispersion B Coefficients for Simple Ions at the Mica-Water Interface, Calculated Using Different Modal Decompositions of the Polarizabilities Compared against Ab Initio Polarizabilitiesa model
avg. % rel. error
ab initio DFT/PBE0 3.99 (1.02) one-mode (IP in vacuum) 48.6 one-mode (IP in water) 37.9 one-mode (fitted) 4.8 two-mode 0.83 three-mode 0.35 four-mode 0.079 five-mode 0.022 a Three alternative single-mode models are compared, with the singlemode characteristic frequency based on the ion’s ionization potential (in vacuum38 and in water23) and on a single-mode fit. The DFT/PBE0 ab initio method is also compared (the value in parentheses excludes Li+ and Na+ from the average). Comparisons are expressed as the % error relative to the full ab initio (CCSD) value.
2100 points (up to n = 2099, or 8 1016 Hz), adding another 25% to the dispersion interactions. In the discussion of computational methods above, we remarked that the level of quantum chemical theory (CCSD) used to calculate ab initio dynamic polarizabilities is relatively computationally expensive and becomes prohibitive for larger polyatomic ions. Other, faster methods with electron correlation introduce errors in static polarizabilities less than 3% on average. Having seen in the preceding paragraphs that the dynamic behavior at high frequencies is extremely important for an accurate description of dispersion energies, we compare full ab initio (CCSD) dynamic polarizabilities against those calculated using the faster DFT/PBE0 method (dynamic polarizabilities along imaginary frequencies are not available under MP2). Comparison is again made via surface B values (listed in Table 3), with an average relative error of 4% shown in Table 4. Due to the small absolute value of the B values for Liþ and Naþ, the relative errors for these two ions are anomalously large. By excluding them from the average, the average error in DFT/PBE0 relative to CCSD is only 1%. We conclude that errors in dynamic polarizabilities calculated by the faster DFT/PBE0 method are consistent with the errors seen in static polarizabilities (Table 2). DFT/PBE0 should therefore be sufficient for calculating the dispersion forces of polyatomic ions. Finally, we compare the performance of the fitted modal decompositions with up to five modes (as discussed above, the ions have five major modes, or four for Liþ). The error in the dispersion B coefficients calculated using these decompositions is listed in Table 4. The single-mode fitted decompositions perform better than the single-mode ionization model but still have an overall 5% error on average. Unsurprisingly, the error decreases as the number of modes is increased, with the average error over all ions, relative to the ab initio values. At five modes, the average error over all ions is reduced to 0.02%. 1820 DOI: 10.1021/la902533x
We believe that any speed advantage in future applications of the dynamic polarizability due to applying a two- or three- or fourmode decomposition rather than the five-mode decomposition would not be significant and would be undesirable given the accompanying loss of physics (neglecting major modes). We therefore present the parameters of the five-mode (four-mode for Liþ) decompositions in Table 5. These decompositions of the dynamic polarizabilities are presented for the convenience of analytical studies or where otherwise useful; however, for numerical calculations we prefer to use the original ab initio polarizabilities. The necessity of accounting for multiple frequency modes enhances the ion-specific effect in that not only the static polarizability but also the frequency of several modes is specific to a particular ion. In fact, as shown in Table 5, the second lowestfrequency mode commonly has the largest weight, rather than the lowest frequency mode.
Application: Surface Potentials Surface potentials (zeta potentials) are measured experimentally by electrophoretic and other methods.43-46 We calculate theoretical surface potentials via a modified Poisson-Boltzmann approach.21,47 The standard Poisson-Boltzmann description of the electrostatic potential ψ(z) next to a flat plate is employed ε
X d2 ψ ¼ -4π qi ci ðzÞ 2 dz i
ð7Þ
with the boundary condition given by fixed surface charge σ, set to σ = -0.05 unit charges per nm2 (below full coverage, corresponding to slightly acidic conditions; the concentration of Hþ is not otherwise explicitly considered). The qi is the ionic charge, and ci(z) its concentration at a distance z from the surface. The modification due to ion-surface dispersion interactions, Ui(z) = Bif(z)/z3 (see eq 1), appears in the Boltzmann factor determining the ion concentrations ci ðzÞ ¼ ci0 expf -½qi ψðzÞþUi ðzÞ=kTg
ð8Þ
ci0 is the bulk concentration of the ion. The Bi coefficients (see eq 3) for the mica-water interface, giving the strength of the dispersion interactions, are listed in Table 3. (43) Lyons, J. S.; Furlong, D. N.; Healy, T. W. Aust. J. Chem. 1981, 34, 1177– 1187. € Dogan, M. Microporous Mesoporous Mater. (44) Alkan, M.; Demirbas, O.; 2005, 83, 51–59. (45) Lameiras, F. S.; Souza, A. L. d.; Melo, V. A. R. d.; Nunes, E. H. M.; Braga, I. D. Mater. Res. 2008, 11, 217–219. (46) Chassagne, C.; Mietta, F.; Winterwerp, J. J. Colloid Interface Sci. 2009, 336, 352–359. (47) Bostrom, M.; Lima, E. R. A.; Tavares, F. W.; Ninham, B. W. J. Chem. Phys. 2008, 128, 135104.
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Table 5. Static Polarizabilities, Weights, and Characteristic Frequencies for Five-Mode Decompositions of the Dynamic Polarizabilities of Simple Ions (see eq 5)a ioan
R0 (A˚3)
mode 1 f1
ω1 (au)
mode 2 f2
1.9133 0.4254 0.2525 0.4820 F4.8608 0.3200 0.2133 0.5335 Cl6.4902 0.3250 0.1969 0.5225 Br 9.6488 0.2937 0.1700 0.5421 I 0.0285 0.7160 2.3446 0.2814 Liþ 0.1390 0.2394 1.2812 0.4048 Naþ þ 0.8144 0.3165 0.7954 0.5885 K þ 1.3795 0.3174 0.6658 0.5458 Rb 2.4016 0.3644 0.5574 0.5249 Csþ 3.0001 0.3071 0.4892 0.5468 Frþ a Liþ has only four modes; 1 au = 6.57968 1015 Hz.
mode 3
mode 4
mode 5
ω2 (au)
f3
ω3 (au)
f4
ω4 (au)
f5
ω5 (au)
0.5906 0.4044 0.3677 0.3106 4.4211 1.9218 1.1088 0.9030 0.7395 0.6696
0.0175 0.1406 0.1429 0.1467 0.0004 0.3354 0.0890 0.1240 0.1054 0.1358
3.4706 0.8556 0.7904 0.6496 5.0299 3.8473 1.8043 1.3337 1.2041 1.0917
0.0743 0.0042 0.0062 0.0114 0.0022 0.0168 0.0058 0.0125 0.0050 0.0097
1.5314 2.8830 3.0964 2.3556 8.3180 8.4490 4.3207 2.3522 2.0469 1.9242
0.0007 0.0017 0.0033 0.0061
11.6356 11.0929 9.7500 4.8818
0.0036 0.0001 0.0003 0.0003 0.0005
18.9006 17.2179 7.5595 6.1339 5.8245
Table 6. Surface Potentials ζ0 (mV) of Various Electrolytes at the Mica Surfacea ζ0 (ab initio)
ζ0 (one-mode IP)
LiCl NaCl KCl RbCl
-10.2 -8.7 -0.9 0.8
-5.9 -3.7 2.3 3.1
42.0 57.6 353.3 275.4
LiBr NaBr KBr RbBr
-8.2 -6.7 1.1 2.9
-4.7 -2.4 3.6 4.4
43.3 63.8 222.4 54.1
salt
% err.
-9.4 -4.9 47.6 -7.9 -2.7 65.9 -0.1 3.3 4213.7 1.7 4.1 147.5 a Ion-surface dispersion interactions are determined from ab initio polarizabilities (second column) or by the single-mode IP model using ionization potentials in water23 (third column). The electrolyte concentration is 0.5 M, and the surface charge density is -0.05qe per nm2. The last column shows the % error in the single-mode model relative to the ab initio value.
LiI NaI KI RbI
Figure 3. Surface potentials of chloride electrolytes at the mica surface as a function of electrolyte concentration.
For simplicity, charges are permitted to approach up to the surface, that is, no Stern layer displacing the ions from the surface is invoked. The concept of the Stern layer has the effect of slightly reducing the magnitudes of zeta potentials by taking the potential at one ion radius out from the surface. The broad trends, including the appearance of a Hofmeister series across the ions, remain regardless. The impact of ion-surface dispersion interactions is relatively small at dilute electrolyte concentrations but starts to become significant above 0.1 M. Surface potentials calculated at a bulk electrolyte concentration 0.5 M are shown in Table 6. The table compares zeta potentials calculated using ab initio polarizabilities and those due to the single-mode IP model. The error in the singlemode IP model relative to the ab initio is found to be vast, averaging at 465%. As a rough trend, the error in the single-mode IP model increases as the polarizability and size of the ions increases. Concentration dependences of the surface potentials (zeta potentials, ζ0) in chloride electrolyte are displayed in Figure 3, compared to classical electrostatic DLVO theory without dispersion forces. Bromide and iodide salts, not shown here, yield similar curves, shifted slightly according to the ion-specific dispersion forces of the anions. To highlight the effect of dispersion interactions, concentrations below about 0.01 M are truncated in the figure (the curves converge at low concentration, reaching a surface potential at around -145 mV at 10-4 M). A strong deviation from DLVO theory is found, with cosmotropic Liþ and Naþ chlorides lying below the DLVO curve
(more strongly negative) while chaotropic Kþ and Rbþ chlorides lie above it (more positive). A Hofmeister series is found using ab initio polariabilities, with Liþ < Naþ < Kþ < Rbþ, Liþ showing the most negative surface potential. This is consistent with experimental measurements of the zeta potential at mica.43 It is interesting to find the two sets of cations lying on either side of the DLVO curve. At first glance, one might expect that once dispersion forces are switched on, all ion-specific curves would move in the same direction away from the DLVO result, differing only in their relative magnitudes. The result observed in Figure 3 may be understood by reference to the surface dispersion B values of the cations in Table 3, interpreted in conjunction with the B value for the anion. The surface dispersion interaction of all ions at the mica interface are attractive (the B values are negative), but the anions have the strongest dispersion interaction. Anions are therefore drawn toward the surface, despite the negative surface charge, effectively increasing the magnitude of the surface charge and driving the zeta potential toward a larger negative magnitude. In the case of the larger, more polarizable cations (Kþ and larger), the order of magnitude of their B values is around the same as that of the anion. That is, the larger cations are also attracted toward the surface by dispersion forces, thereby canceling the enhancement of the surface charge by the adsorbed anions. The combination of dispersion and electrostatic attraction of the more polarizable cations in fact brings an excess of cations over and beyond what is found in DLVO theory. Hence, zeta potentials are pushed further toward positive values by highly polarizable cations than is the case in DLVO. In the case of Liþ and Naþ, on the other hand, their B values are 2 orders of magnitude smaller than the anionic B values; they are nearly 0. There is therefore no dispersion attraction of these
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Figure 4. Surface potentials of chloride electrolytes at the mica surface as a function of electrolyte concentration. Calculations using ab initio polarizabilities are shown without symbols, and those using the single-mode model based on ionization potentials in water are shown with symbols.
less polarizable cations which might neutralize the dispersion enhancement of the effective surface charge by the anions. As result, more negative charge is found at the surface than is the case in DLVO theory; consequently, the zeta potentials are more strongly negative. Another interesting phenomenon is the appearance of charge reversal, with the surface potential of KCl passing from negative to positive at around 0.7 M. This is impossible in electrostatic DLVO theory. In that model, the surface charge may be merely negated by the electrolytic charge, with the surface potential reducing to zero at high concentration but never crossing zero. The ion-surface dispersion interactions, being attractive (for these particular ions at mica surface), cause the ions to adsorb more strongly to the surface. The negative mica surface charge draws in cations, inducing electrostatic cancellation of the charge at the surface. Then, the attractive cation-surface dispersion interaction draws in more cations, tipping the overall surface charge over to the opposite polarity. The charge reversal effect is stronger when the counterion has a greater polarizability, such that it is not seen in the less polarizable sodium and lithium cations, and kicks in at lower concentration in the case of the more polarizable rubidium cation. Charge reversal (at fixed pH) is readily observed in multivalent electrolytes.44,48,49 Fewer experimental reports of charge reversal in 1:1 electrolytes are found, partly because zeta potential studies frequently do not measure at concentrations higher than 0.7 M. Chassagne, Mietta, and Winterwerp46 measured zeta potentials at kaolinite surface up to 1 M concentrations. While kaolinite is not mica, they are both aluminosilicates, and it is not unreasonable to suppose that their dielectric spectra, and therefore ionsurface dispersion interactions, are comparable. In any case, the Chassagne zeta potential for KCl at kaolinite is indeed similar to our theoretical curve at mica in Figure 3, with charge reversal occurring near 0.7 M. Likewise, the Chassagne zeta potential of NaCl is more negative than that of KCl, and charge reversal is not seen, as is the case with our theoretical curves. Surface potentials are again shown in Figure 4, comparing calculations using ab initio polarizabilities to those using the single-mode IP model. The single-mode IP curves are shifted too strongly toward positive polarities, with charge reversal in KCl and RbCl kicking in at half of the concentration of the ab initio and Chassagne experimental curves. (48) Kekicheff, P.; Marcelja, S.; Senden, T. J.; Shubin, V. E. J. Chem. Phys. 1993, 99, 6098–6113. (49) Vane, L. M.; Zang, G. M. J. Hazard. Mater. 1997, 55, 1–22.
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When drawing a comparison between calculated and experimental zeta potentials, it should be noted that the raw experimental data is commonly interpreted and processed through classical Poisson-Boltzmann theory, accounting for electrostatic interactions while ignoring dispersion forces. There will consequently always be some discrepancy between reported experimental values and theory incorporating dispersion forces. The discrepancy will not be in the underlying experimental data but rather in its interpretation. Measurements based on theory that includes dispersion forces pose a major challenge for the interpretation of pH, pKas, membrane potentials. We close the discussion with a comment on the model of cosmotropic ions (namely, Liþ and Naþ). In this paper, we have used the polarizabilities and ion sizes of these ions on their own, without additional consideration of the polarizability of the hydration layer. Cosmotropic ions, which have tightly held hydration layers, may be differentiated from other, chaotropic ions with weakly held hydration layers.50,51 The Hofmeister series in bulk activity coefficients may only be reproduced by including the hydration layer of cosmotropic ions, giving an increased size to the hydrated ion.52 The correct Hofmeister series shown here in surface potentials, however, used unhydrated cosmotropic ions. The ion-surface dispersion interaction tends to become significant in the vicinity of the surface. The success of the unhydrated cosmotropic ions in reproducing the Hofmeister series suggests that a surface-induced dehydration effect is taking place. This is consistent with the literature.53-57 Our model is incomplete in that it continues to treat cosmotropic ions as “unhydrated” far from the surface, where they should be hydrated. The two regimes, with cosmotropic ions hydrated in the bulk, dehydrated near the surface, may be reconciled by including both hydrated and unhydrated ions alongside each other, with an equilibrium established between them. The equilibrium constant would be distance-dependent (and derived from the self-energies of hydrated and dehydrated ions, which change as the surface is approached), such that the hydrated ion is favored far from the surface while the dehydrated ion is favored close to the surface.
Conclusion The ab initio dynamic polarizabilities calculated in this paper have applicability in a wide range of areas where nonelectrostatic dispersion forces operating in ionic solutions are important. They provide the basis on which detailed theories of ion-specific effects can be developed and tested. The multimode decomposition now admits applications that go beyond the inadequately simple single-mode models used previously. That is, Hofmeister effects can now be explored with confidence, first in the continuum solvent approximation and then building in solvent effects. Calculations of surface potentials, for instance, including both ion-surface dispersion interactions and finite ion sizes, have successfully reproduced the Hofmeister series at mica surfaces. Charge reversal of the mica surface was found due to adsorption of the more polarizable cations. While this is rare for (50) Collins, K. Biophys. Chem. 2006, 119, 271–281. (51) Collins, K. D. Biophys. J. 1997, 72, 65–76. (52) Parsons, D. F.; Deniz, V.; Ninham, B. W. Colloids Surf., A 2009, 343, 57–63. (53) Zachara, J. M.; Cowan, C. E.; Resch, C. T. Geochim. Cosmochim. Acta 1991, 55, 1549–1562. (54) Brady, P. V.; Krumhansl, J. L.; Papenguth, H. W. Geochim. Cosmochim. Acta 1996, 60, 727–731. (55) Pandit, S. A.; Bostick, D.; Berkowitz, M. L. Biophys. J. 2003, 84, 3743– 3750. (56) Draper, D. E. RNA 2004, 10, 335–343. (57) Martı´ n-Molina, A.; Ibarra-Armenta, J. G.; Quesada-Perez, M. J. Phys. Chem. B 2009, 113, 2414–2421.
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1:1 electrolytes, it is commonly found with multivalent counterions. Ab initio studies of alkali earth and other multivalent ions will follow and will demonstrate that the charge reversal seen here continues to be consistent with experiment.
Article
Acknowledgment. We acknowledge the assistance of Rika Kobayashi and the staff at the Australian National University Supercomputing Facility.
Appendix Ongoing Developments: Hydration Effects The polarizabilities used here referred to the polarizabilities of the ions themselves, without explicit reference to the hydration shell. Solvation effects were included in the form of excess polarizabilities, accounting for the depolarization field of the solvent. An explicit hydration shell will be important for cosmotropic ions, which have tightly held hydration shells. The polarizability of the hydration shell will differ from that of bulk water due to, for instance, a different density of water molecules. To a first approximation, the total polarizability of the solvated ion may be taken to be the ionic polarizability plus the polarizability of the water molecules in the hydration shell. The number of water molecules may be taken from, for instance, experimental hydration numbers.58 The radius of the solvated ion may be increased by adding, for instance, a water radius to the intrinsic ion radius.16 The solvated ion may then be considered the fundamental ionic unit to be inserted into existing theories, with an accompanying change in polarizability and ion volume corresponding to the hydration complex rather than the bare ion. This should bring a nontrivial insight into ion-specific effects, with the outcome being a competition between two opposing trends, that of an increased total polarizability (strengthening dispersion interaction energies) and that of increased solvated ion size (weakening dispersion interaction energies). The self-energy (Born energy) of an ion in bulk solution, and therefore the transfer energy between media, is known to contain a large dispersion contribution.21,59 Accurate ion polarizabilities and hydration should have just as an important impact on these Born energies as they do on ion-surface interactions. Remaining theoretical developments to be considered include modeling anisotropic ions (particularly important for complex ions). Recent anisotropic polarizable force fields have been proposed for molecular modeling simulations with explicit water,4,8 but anisotropy has been largely neglected in continuum theories. Anisotropy provides a modest correction to self-energies of ions in bulk media.52 The correction should be much stronger at an interface, with the anisotropy of the medium introduced by the interface itself, and should have a significant impact on the surface adsorption of anisotropic ions such as the planar nitrate ion. Other developments include incorporating solvent spatial correlations,60-62 allowing the ionic volume to vary63 and unifying the electrostatic and dispersion components so that electrostatic interactions are treated by a quantum electrodynamic approach in the same way that the dispersion interactions have been. (58) (59) (60) (61) (62) (63)
Marcus, Y. Pure Appl. Chem. 1987, 59, 1093–1101. Bostr€om, M.; Ninham, B. W. Biophys. Chem. 2005, 114, 95–101. Basilevsky, M. V.; Parsons, D. F. J. Chem. Phys. 1998, 108, 9107–9113. Cherepanov, D. A. Phys. Rev. Lett. 2004, 93, 266104. Medvedev, I. G. Electrochim. Acta 2004, 49, 207–217. Basilevsky, M. V.; Parsons, D. F. J. Chem. Phys. 1998, 108, 9114–9123.
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Excess Polarizability. The polarizability of an ion determines the strength of the induced dipole formed when an external electric field distorts the electron cloud of the ion. In a medium such as water, however, it is not only the ion which gets polarized by an external electric field; the medium itself is also polarized. The response of the medium is described by its dielectric susceptibility. This means that an ion in solution does not simply experience the full external field; instead, it experiences a modified electric field formed from the original external electric field plus a “depolarization field” due to the polarized medium. By treating an ion as a simple dielectric sphere with volume Vi and ionic dielectric function εi(iω), Landau and Lifshitz17 deduced the resulting induced dipole, with the corresponding effective ionic polarizability being RðiωÞ ¼
3Vi ½εi ðiωÞ -εm ðiωÞ 4π ½εi ðiωÞþ2εm ðiωÞ
ð9Þ
εm(iω) is the dielectric function of the medium (i.e., water). The dielectric function of the ion is derived from its polarizability (its intrinsic polarizability, that is, its polarizability in vacuum). The most straightforward approach is to apply the simple model ε = 1 þ 4πnR, where n is the number density of the ion, that is, n = 1/V (one ion per single ionic volume). In other words, taking Ri to be the intrinsic polarizability of the ion (in vacuum) εi ðiωÞ ¼ 1þ4π
Ri ðiωÞ Vi
ð10Þ
A more sophisticated model may be considered,3,64 based on the Clausius-Mossotti relation between the polarizability and dielectric function. However, this model is highly sensitive to the ratio of ion polarizability to volume. If the ion volume has been slightly underestimated, then a negative dielectric value for the ion may be generated, which is physically impossible (violating causality). The simple formula offered in eq 10 is more robust to measurement errors in the ion volume. The effective polarizability R*(iω) of the ion in solution has come to be known as the “excess polarisability” since because of the “εi - εm” term, it describes the polarization response of the ion after subtracting out the polarization response of the medium. The term “excess polarisability” is, in a sense, misleading since the difference in the actual polarizabilities of the ion and solvent is not taken directly. The term “effective polarisability” seems more apt. (64) Boroudjerdi, H.; Kim, Y.-W.; Naji, A.; Netz, R.; Schlagberger, X.; Serr, A. Phys. Rep. 2005, 416, 129–199.
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