Langmuir 1997, 13, 2097-2108
2097
Ion Binding and Ion Specificity: The Hofmeister Effect and Onsager and Lifshitz Theories Barry W. Ninham*,† and Vassili Yaminsky Department of Applied Mathematics, Research School of Physical Sciences and Engineering, Institute of Advanced Studies, Australian National University, Canberra, Australia 0200 Received October 7, 1996. In Final Form: December 31, 1996X Present theories of colloid science do not account for specific ion effects as exemplified by the Hofmeister effect, surface tension of electrolyte solution interfaces, binding to micelles, effective charge in double layer interactions, and attractive interactions in low Hamaker constant systems. It is argued that specificity emerges naturally and can be rationalized if dispersion interactions acting on ions are included in the theory. These are in principle accessible from bulk solution properties. Specific ion adsorption due to dispersion interactions can be dominant even at charged interfaces especially at high salt concentrations (∼0.1Μ). The effects can be qualitatively different at air-water and oil-water surfaces. That part of extended Lifshitz theory for low Hamaker constant systems, in which the forces are mainly due to temperature and salt dependent interaction, is re-examined. It is shown to be at the same level of approximation as, and precisely equivalent to, the Onsager limiting law for the interfacial tension change due to dissolved salt at a single interface, i.e., to linearization of the Poisson-Boltzmann distribution, and restriction to electrostatic potentials as the sole determinant of adsorption excesses. The usual description of interactions into separate electrostatic, and dispersion interaction, is invalid, even at the level of continuum (primitive model) theories.
1. Introduction A large number of phenomena in colloid, polymer, and interfacial science that involve electrolytes are not explained by present theories. When theory fails it is usual to ascribe that failure to “ion specificity” or “ion binding”. From one point of view this is hardly surprising. All of chemistry is specific. From another point of view the situation is most unsatisfactory, because the accounting of events that do not fit into a supposedly predictive theory by invoking a specificity that varies from case to case means that the theory is inadequate and needs revision. To make clear what we have in mind, we first list and discuss a number of phenomena involving electrolytes at interfaces where a breakdown of theory is evident. We will then explore present theories to see where and in what manner they become inadequate. We then argue the case that ion specificity can emerge from a more general theory and how this might be accomplished. Examples of Ion Specificity. For over a hundred years the Hofmeister series has been unexplained.1 Salting out of proteins depends on counterion and co-ion, with a particularly strong dependence on anion type. The same applies for hydrophobic chromatography. Again, the conformation and solubility of polyelectrolytes exhibit similar systematic dependence on counterion and salt type.2 For example chitosan, a highly insoluble hydrophobic polycationic material with Cl- as counterion, unravels and is solubilized with Ac- as counterion.3 Another striking example is provided by flocs of hydrophobic particles formed with salt solutions.4 At the same salt concentration floc volumes are dramatically different depending on anion. A large floc volume is indicative of † Present address: Physical Chemistry 1, Center for Chemistry and Chemical Engineering, University of Lund, P.O. Box 124, S-211 00 Lund, Sweden. X Abstract published in Advance ACS Abstracts, February 15, 1997.
(1) Collins, K. D.; Washabaugh, M. W. Q. Rev. Biophys. 1985, 18, 323-422. (2) Tomas, S.; Sarkar, M.; Ratilainen, T.; Wittung, P.; Nielsen, P.; Norde´n, B.; Gra¨slund, A. J. Am. Chem. Soc. 1996, 118, 5544. (3) Claesson, P.; Ninham, B. W. Langmuir 1992, 8, 1406. (4) Yaminksy, V. V.; Pchelin, V. A. Dokl. Akad. Nauk SSSR 1973, 310, 154.
S0743-7463(96)00974-2 CCC: $14.00
strong interparticle forces; a smaller, closer packed volume indicates weak forces between the particles. “Ion binding” parameters for micelles depend on counterion, an effect that is especially marked for anions. The classic example is cetyltrimethylammonium salts, CTAClvs CTABr-, the one forming spherical micelles and the other rodlike structures at the critical micelle concentration (cmc). With such cationic surfactants, if the counterion is changed to Ac- or other carboxylates, OH- or F-, cmc’s are twice as large and aggregation numbers half those with Br- as counterion.5 For micelles and polyelectrolytes the self-organization phenomena are characterized in terms of the ion-binding model for micellization6 and the Manning theory of counterion condensation7 for polyelectrolytes. Both of these phenomenological theories can be put on a firm statistical mechanical basis6,8,9 that uses the Poisson-Boltzmann equation. Ion binding models emerge as a special asymptotic approximation to the more general theory valid in the limit of “tight binding”. There is in fact no real binding, and the “binding” that occurs in ion binding models is related at a deeper level to physisorption excesses of counterion about the micelle or polyelectrolyte.6,9 These theories seem to work very well in the most studied systems and appear to be confirmed by neutron scattering9 as a function of both surfactant chain length and added salt and as a function of temperature10 up to 130 °C. Given cmc’s and aggregation numbers, the effective fractions of bound charge interpreted as an electrostatic adsorption excess all come out quantitatively with only another single apparently reasonable parameter, the “hydrated” counterion radius. Typically, the fraction of bound charge is 70-80%. But the ion binding model, or a more general theory of self-assembly based on electrostatic forces, fails for systems with different counterions like the acetates (5) Brady, J. E.; Evans, D. F.; Warr, G.; Griesser, F.; Ninham, B. W. J. Phys. Chem. 1986, 90, 1853. (6) Evans, D. F.; Mitchell, D. J.; Ninham, B. W. J. Phys. Chem. 1984, 88, 6344. Jo¨nsson, B.; Wennerstro¨m, H. J. Phys. Chem. 1987, 91, 338. (7) Manning, G. S. Q. Rev. Biophys. 1978, 11, 179. (8) Mohanty, U.; Ninham, B. W.; Oppenheim, I. Proc. Natl. Acad. Sci. U.S.A. 1996, 93, 4342. (9) Hayter, J. B. Langmuir 1992, 8, 2873. (10) Evans, D. F.; Allen, M.; Ninham, B. W.; Fonda, A. J. Solution Chem. 1984, 13, 87.
© 1997 American Chemical Society
2098 Langmuir, Vol. 13, No. 7, 1997
or carboxylates already mentioned. This can be argued away by invoking water structure, Stern layers, extravagant hydrated ion sizes, local pH, and so on. At first sight this is reasonable. After all, we know that the primitive model of electrolytes is not always successful in accounting for activity coefficients11,12 of bulk solutions. (Hard core (hydration) radii that fit the data are indeed additive for most alkali halides but are not so successful for nitrates and sulfates, cesium, and other ions.11) We are forced to conclude that the theory of micellization based on the same primitive model description is valid only for particular charged surfactant head group, counterion (and co-ion) pairs. The theory cannot claim a generality that we would wish. But even if exceptional cases be accepted as a necessary limitation, other related phenomena cannot be so reconciled. Consider lamellar phases of double chained cationic and anionic analogues of the single-chained micelleforming surfactants already considered. Take in particular, didodecyl- or dihexadecyldimethylammonium bromides13 (DDAB, DHAB) or corresponding sodium dialkyl sulfosuccinates.14 Both are quite insoluble. If the bromide of DDAB is ion exchanged for acetate13,15 or hydroxide16,17 or the sodium ion for lithium, the lamellar phases swell enormously, and the solubility changes. The phenomenon is not simply a reflection of the change in Krafft temperature. The forces between bilayers are increased by counterion exchange and are highly ion specific. Direct force measurements between bilayers of DHABr- or DHAAc- adsorbed on mica have been made18 as a function of added salts(NaBr) or (NaAc). For the acetates the measured forces fit precisely to PoissonBoltzmann double layer theory to give the effective charge which is the bare charge of the fully ionized surfactant. The Debye length is as expected for given salt concentration. By contrast, with NaBr, the effective charge of the interacting surfaces that fits the interaction corresponds to about 80% “bound counterion”. Now the PoissonBoltzmann (P-B) (point ion) model of the double layer interaction is known to provide an upper bound to the predicted force. Refinements due to introduction of finite counterion size reduce the predicted force, by a maximum of about 20-30% at close separations. This should reduce correspondingly19,20 the effective surface charge deduced from a fit of experiments to the P-B theory of the double layer. But that reduction is compensated by occurrence of a short-range hydration force.21 For acetate as counterion, the agreement of the measured double layer force with the prediction of Poisson-Boltzmann theory is coincidence. However even with such refinements it is the bare surface charge which is the effective charge, and this is as it should be. (11) Pailthorpe, B. A.; Mitchell, D. J.; Ninham, B. W. J. Chem. Soc., Faraday Trans. 2 1984, 80, 115. (12) Knackstedt, M. A.; Ninham, B. W. J. Phys. Chem. 1996, 100, 1330. (13) Ninham, B. W.; Evans, D. F. Faraday Discuss. Chem. Soc. 1986, 81, 1. (14) Karaman, M. E.; Ninham, B. W.; Pashley, R. M. J. Phys. Chem. 1994, 98, 11512. (15) Radlinska, E. Z.; Zemb, T. N.; Dalbiez, J.-P.; Ninham, B. W. Langmuir 1993, 9, 2844. (16) Brady, J. E.; Evans, D. F.; Kachar, B.; Ninham, B. W. J. Am. Chem. Soc. 1984, 106, 4279. (17) Talmon, Y.; Evans, D. F.; Ninham, B. W. Science 1983, 221, 1047. (18) Pashley, R. M.; McGuiggan, P. M.; Ninham, B. W.; Evans, D. F.; Brady, J. J. Phys. Chem. 1986, 90, 1637. (19) Kjellander, R.; Marcelja, S. J. Phys. (Paris) 1988, 49, 1009. (20) Kjellander, R.; Marcelja, S.; Pashley, R. M.; Quirk, J. P. J. Chem. Phys. 1990, 92, 4399. (21) Parsegian, V. A.; Rand, R. P. Proc. Natl. Acad. Sci. U.S.A. 1991, 95, 4779.
Ninham and Yaminsky
By contrast, with bromide as counterion as already noted the effective charge from experimental force measurements that fits the Poisson-Boltzmann description or its generalizations is reduced by about 80%, and the forces are correspondingly weaker. This cannot be reconciled with the behavior of micelles of cetyltrimethylammonium bromide (CTAB), for which the effective bound charge, related to a physisorption excess, is apparently predicted correctly, as a function of temperature and salt, from exactly the same theory.7,9 The double layer force, due to overlap of the inhomogeneous counterion and co-ion distributions, ought to be described by the bare surface charge, as it appears to be for acetate. There is here a clear inconsistency. Either the theory of micellization is incorrect, the double layer theory is incorrect, or both are (or else the experiments). More subtle effects with “ion binding” are seen with the cloud points of nonionic surfactants in water.22 For a given salt concentration, the cloud points change with ion pair. Much more dramatic effects can be seen in the phase diagrams of dialkyldimethylammonium/water/alkane mixtures. With bromide as counterion, the L2 phase region is very wide23 and its microstructure predictable from simple packing arguments.24 But with chloride as counterion,25 the L2 phase region is extremely narrow. Curvature at the interface and interaggregate interactions that set microstructure and phase separation are critically dependent on “binding” not accessible to electrostatic theories. With sulfate as counterion26 the phase behavior is reversed. Here the microemulsion region corresponds with oil-in-water structures, normal curvature, whereas bromide and choride give water-in-oil structures. Admixture of very dilute NaBr to the sulfate microemulsion system with [Br-] , [SO42-] induces a transition back to the reversed phase water-in-oil system. If one thinks in terms of an electrostatic model for counterion adsorption at the water-surfactant-oil interface, the divalent sulfate ion ought to win out over a small concentration of univalent bromide ions, not vice versa. This effect can be exploited to advantage in synthesis of alkylsulfonates.27 Even more startling effects occur at the air-water interface. With bubble-bubble interactions,28 for some ion pairs, and as a function of added salt, bubbles coalesce below 0.175 M. Above this concentration bubble coalescence is totally inhibited. The inhibition of bubble coalescence effect scales with Debye length. (Curiously this is almost the concentration of salt in the blood of all land animals.) This shows that the hindrance of the coalescence occurs when electrostatic interactions are effectively screened. However, for different ion pairs, there is no effect on coalescence up to molar concentrations. No classical theory of colloid science can yet explain this phenomenon quantitatively. What is even more surpris(22) Kabalnov, A.; Olsson, U.; Wennerstro¨m, H. J. Phys. Chem. 1995, 99, 6220. (23) Evans, D. F.; Mitchell, D. J.; Ninham, B. W. J. Phys. Chem. 1986, 90, 2817. Fontell, K.; Ceglie, A.; Lindman, B.; Ninham, B. W. Acta Chem. Scand. 1986, A40, 247. (24) Barnes, I. S.; Hyde, S. T.; Ninham, B. W.; Derian, P.-J.; Drifford, M.; Warr, G. G.; Zemb, T. Prog. Colloid Polym. Sci. 1988, 76, 1. See also: Knackstedt, M. A.; Ninham, B. W.; Monduzzi, M. Phys. Rev. Lett. 1995, 75, 653. (25) Chen, V.; Evans, D. F.; Ninham, B. W. J. Phys. Chem. 1987, 91, 1823. (26) Nyde´n, M.; So¨derman, O. Langmuir 1995, 11, 1537. (27) Holmberg, K.; Oh, S.-G.; Kizling, J. Prog. Colloid Polym. Sci. 1996, 100, 281. See also: Gudfeldt, S.; Kizling, J.; Holmberg, K. Colloids Surf., in press. (28) Craig, V. S. J.; Ninham, B. W.; Pashley, R. M. J. Phys. Chem. 1993, 97, 10192.
Ion Binding and Ion Specificity
ing28 is that if each cation and anion is assigned a “flavor”, R or β, an apparently universal rule emerges: RR and ββ pairs exhibit the phenomenon of inhibition of bubble coalescence at a critical concentration; βR and Rβ pairs have no effect. The phenomenon is due to a GibbsMarangoni effect for which the only substantial prediction is that it should scale as [dγ/dc]-2, where c is the concentration of electrolyte and γ the interfacial tension at the air-water interface. In fact that expectation almost follows from dimensional arguments. Two interfaces are involved in the interaction that gives rise to bubble coalescence, and the main interfacial parameter involved, namely, the surface activity dγ/dc, is related to the adsorption excess Γ ) -(dγ/dc)c/kT. That has been borne out by measurements of interfacial tension29 induced by the different ion pairs. The surface tension isotherms are essentially linear in concentration. dγ/dc can be positive or negative, and broadly speaking, the isotherms do fall into two classes. The RR and ββ electrolyte pairs exhibit large absolute values of dγ/dc; Rβ and βR pairs have much lower values. Thus for (RR) pairs like Na+Brand (CH3)4N+Ac-, dγ/dc takes on larger values than do the values for (Rβ) pairs like NaAc- and (CH3)4N+Br-. The first two correspond to large absolute values of adsorption, the second two to small values. That a correlation exists between the viscous Marangoni drag and the magnitude of adsorption excesses is reassuring.30 The combination rule shows that surface activity is additive for the ions of an electrolyte. However the fundamental question remains. Why should those adsorption excesses be so qualitatively different in magnitude for different ion pairs? Given this, why are ionic contributions additive? And why should some give positive adsorption, some negative adsorption? Electrostatics alone cannot account for the first or even for the second or the third, at least quantitatively.31,32 To restate our problem in other words: A theory of micellization for sodium dodecyl sulfate (SDS) and CTAB based on the Poisson-Boltzmann equation appears to fit experiment very well, so too does the primitive model description of the activity coefficients at the level of DebyeHu¨ckel theory plus hydrated hard core interaction for the corresponding electrolytes (Na+2SO42-, (CH3)4N+Br-) at least up to 0.1-0.3 M concentrations. If so, then why does the same level of theory applied to the air-aqueous electrolyte interface give such dramatically different adsorption effects as revealed by surface tension data29 and bubble-bubble interactions? We here argue the case that the apparent inconsistencies can be traced to an omission in present continuum theories of interfacial tension and of surface forces. Ions must experience not just an electrostatic potential near an interface but also a potential due to dispersion forces. That potential, depending on atomic number, is highly specific, less significant, and negligible at low electrolyte concentration but can dominate when electrostatic forces become screened (g0.1 M). The effects are much amplified in a complete nonlinear theory. For interactions between interfaces or micelles, lamellar phases, etc., the usual Lifshitz theory, being linear in nature, quite misses the specificity of interactions. The standard decomposition of forces in colloid science into separate electrostatic and dispersion contributions is therefore generally invalid. Stairs32 and others dating back to Onsager33 have already (29) Weissenborn, P. K.; Pugh R. J. Langmuir 1995, 11, 1422. (30) Christenson, H. K.; Yaminsky, V. V. J. Phys. Chem. 1995, 99, 10420. (31) Bhuiyan, L. B.; Bratko, D.; Outhwaite, C. W. J. Phys. Chem. 1991, 95, 336. (32) Stairs, R. A. Can. J. Phys. 1995, 73, 781.
Langmuir, Vol. 13, No. 7, 1997 2099
argued cogently for the inclusion of ion-dipole polarizability effects, and the dispersion effects that we discuss include these as well as ultraviolet correlation contributions. 2. The Onsager-Samaris Limiting Law To make our thesis explicit, we need to rehearse the derivation due to the Onsager-Samaris limiting law for the change in interfacial tension γ due to a dissolved electrolyte at concentration c ) ∑jνjczj ) ∑jcj. The Gibbs adsorption equation gives
∑j Γj dµ j,
dγ ) -
µ j ) kT ln cj + ψj
(1)
where Γj is the adsorption excess of the jth species. Hence
dγ ) -kT
Γj ) cj
∑j
Γj
dcj cj
∫0∞[exp(-wj(x) + zjeφ)/kT - 1] dx
(2)
The distance of the center of an ion from the interface is x, and the image potential wj(x) (with κD the inverse Debye length) is
wj(x) =
zj2e2 4x
∆e-2κDx,
∆)
-1 +1
(3)
A self-consistent potential φ(x) is set up near the interface because the size of ions is different and therefore so is their distance of closest approach. In writing down eq 2, we have ignored the contributions to adsorption excesses of the solvent, which can be shown to be small. Much more elaborate expressions for the potential of mean force wj(x) can be written down using extended Debye-Hu¨ckel theory31 or more sophisticated primitive model statistical mechanics, and for the chemical potentials. (These complications inevitably take on the nature of additional fitting parameters for “hydrated” ion size and obscure the basic physics of the problem, at least for our purpose. And in any event the hard core parameters necessary to fit the adsorption data for real electrolytes are inconsistent with those required to fit bulk activity coefficients, even for those salts for which the primitive model works.) If this be accepted, we remark that the expression for Γj in eq 2 does not diverge, no cut-off distance is required, and the only length scale is the Bjerrum length, e2/kT, which is independent of ion size (and electrolyte). We can anticipate only a weak dependence of Γ on ion size. If eq 2 is linearized, the term in φ(x) vanishes by electroneutrality because ∑jzj dcj ) 0 and we are left with the approximation
dγ =
e2
∑j d(zj2cj) ∫a
4
∆
∞ j
dx
e-2κx x
(4)
Here a cut-off distance aj for the jth species is necessary to guarantee convergence of the integral, unnecessary as already remarked if the complete (nonlinear) expression, eq 2, for the adsorption excess is used. Then the change ∆γ in interfacial tension follows as (33) Onsager, L.; Samaris, N. N. T. J. Chem. Phys. 1934, 2, 529. (34) Mahanty, J.; Ninham, B. W. Dispersion Forces; Academic Press: London, 1976.
2100 Langmuir, Vol. 13, No. 7, 1997
∆γ )
e2
∆
4
∑j ∫0
cj
Ninham and Yaminsky
d(zjc′j) I(2κ′Daj)
∫2κ∞ a ex
(5)
-x
I(2κDaj) )
D j
dx
Using the electroneutrality condition we have for a twocomponent electrolyte
4πe2 2 (z c + z22c2), kT 1 1
κD2 )
e2
( )
∆γ )
4 e2 ) 4
∆
4πe2 kT
∑i ∫0
R(0)
κD
(
∑j cj zj2∑i -
∆
z1c1 + z2c2 ) 0
2κ′ dκ′ I(ajκ′)
1 (2κaj)
2
∑i
2κai
{
-ln(2κai) +
R(iξ) )
e-2κai +
∞ ∫2κa
( )
e-x i
x
)
dx (6)
4 - γ + κai + 2 3 7 (2κai)2 + ... 8
[
1
}
x2(x1 + n )(w1 + w2′′) (7)
3. The Contribution of Dispersion Forces 32
3.1. Air-Water Interfaces. Stairs has taken up the issue of ion-induced permanent dipole terms and their role in affecting the image potential. The results certainly point to the importance of such effects. However there is a further, so far neglected, contribution whose role we now explore. Ions should experience an additional potential due to dispersion interactions. The dispersion33,35,36 potential corresponding to the electrostatic image potential eq 3 has the asymptotic form
∆
(8)
(10)
with
w2′′ )
w2
x2
x1 + n2
(11)
We note the limiting forms
{
π I ) R(0)(n2 - 1) × 2
[
w1
[ [
1
1 +
n2
(1 + n2)
]
w1 , ww
]
w1
1 1 + n(1 + n) x 1 + n2 x2 x1 + n2
w1 ) wω
w2
1 + n
w1 . ww
1
]
x2 (x1 + n2)
(12)
Then in any case we have
wDISP(x)/kT )
1 π p$ R*(0)(n2 - 1) 3 2 kT 4πx
(13)
where w j is some average frequency characteristic of a given ion. We compare this dispersion potential with the electrostatic image potential
2
wes(x)/kT )
where
2 - 1 , 2 + 1
]
1
w2′ ) w2n,
21 ∫0∞dξ Rj*(iξ) (iξ)
(1 + (ξ2/w22)) (9)
2
where γ ) 0.5772... is Euler’s constant. The last asymptotic expansion is restricted in validity to (2κai) , 1 and what is germane to our thesis, this (linear) theory is clearly a truly limiting law. At high salt concentrations, say g0.1 M where ∆γ values are experimentally significant, a calculation of ∆γ based on the nonlinear expression for Γ (eq 2) with or without decorations that include hard core interactions must be used. In this region where “specific ion effects” do show up strongly, as evidenced, e.g., by the bubble-bubble interaction problem, electrostatics is not dominant and something more is involved.32
p 4πx3
(n2 - 1)
1 π + I ) R(0)(n2 - 1)(w1w2) 2 n(w1 + w2′)
aκD , 1
wjd(x) )
(1 + (ξ2/w12))
w(iξ) ) 1 +
,
where R(0) is the static effective polarizability of the ion, pw1, its unknown electron affinity, n the refractive index of water, and w2 a typical ultraviolet relaxation potential. The integral of eq 8 is then
(e-2κaj - 1) -
1
= cj zj2
water, and medium 1 vacuum. This continuum theory expression holds to a very good approximation for x g aj. At smaller distances or at the interface, the more complete expression is finite and does not diverge.34-36 It can be extended in principle to include induced permanent dipole interactions. Essentially the potential equation (8) is the change in dispersion self-energy34 of the ion in medium 2 due to its interaction with the interface. The difference in dispersion self-energy of the ion in medium 1 or 2 is a measure of solubility differences in the media. To obtain some estimate of how large this potential might be, take34,37-39
1 - 1 e2 e-2κx 4 + 1 kT x
(
)
(14)
Rj*(iξ) is the excess polarizability of the ion, medium 2 is
To do this, measure w j in units of a typical UV adsorption j ) γww, and frequency for water, ww ) 1016 rad/s, write w measure all distances units of Å. Since e2/kT ≈ 7 Å, ( - 1)/( + 1) ≈ 1. Then wDISP/kT > wes/kt if
(35) Mahanty, J.; Ninham, B. W. J. Chem. Phys. 1973, 59, 6157. (36) Mahanty, J.; Ninham, B. W. J. Chem. Soc., Faraday Trans. 2 1974, 70, 637.
(37) Ninham, B. W.; Parsegian, V. A. Biophys. J. 1970, 10, 646. (38) Parsegian, V. A.; Ninham, B. W. Biophys. J . 1970, 10, 664. (39) Ninham, B. W.; Parsegian, V. A. J. Chem. Phys. 1970, 52, 4578.
∆21 )
2 ) 2(iξ),
1 ) 1
Ion Binding and Ion Specificity
Langmuir, Vol. 13, No. 7, 1997 2101
( )
pww R*(0) 7e-2κx (n2 - 1) γ > kT x 2x3
(15)
since
pww 103 ∼ , kT 4
(n2 - 1) ) (4/3)2 - 1 ) 7/9
the inequality is satisfied if
5 ye-y e x5xRγκD, 3
y ≡ κDx
(16)
The maximum value of ye-y is e-1 ∼ 1/3 at y ) 1, and it follows that wDISP > wes if
5x5xRγκD > 1
reflect water structure around the ions. The notion of hydrophobic hydration can be rationalized on these lines. 3.2. Oil-Water Interfaces. The Hofmeister Series. We use the term “oil” loosely to mean any surfacesphospholipids, Teflon, silica, etc.sdifferent from air. The integral which occurs in eq 8 for the dispersion potential can now change sign. It depends on the dielectric properties of the two media, and their adsorption frequencies in the UV. To see this, consider the integral corresponding to eqs 8 and 16. It is now
I)
(
)
R(iξ) w - 0 w w + 0
(18)
It can be estimated by taking in the denominator of eq 18
(17)
If we take a polarizability as small as 1 Å3, γ ) 1 corresponding to w j in the UV ) 1016 rad/s, then the dispersion potential is always larger than the electrostatic potential for κD-1 e 12 Å, i.e., salt concentrations >0.1 M. This means that the Onsager-Samaris theory, even in its full nonlinear form, cannot be used in the range of concentrations where the ∆γ are experimentally significant. If ω j is an infrared frequency ∼1015 rad/s, with the same small value for excess polarisability, wDISP(x) still dominates above about 0.3 M salt. Even at quite low salt concentrations the dispersion potential may be important in dictating adsorption excesses. Thus if κD-1 = 30 Å corresponding to 10-2 M, from eq 16 we see that the dispersion potential can dominate at very small distances (∼5 Å) or very large distances (100 Å). The arguments are qualitative. However they do suggest that dispersion forces can play a very strong role in setting ion specificity. That specificity should be accessible from bulk solution properties like partial molal volumes and refractive indices. The excess polarizabilities can be much greater than the value 1 Å3 chosen for the illustrative estimates above. Thus Br-, with 80 electrons, can be expected to have a much higher polarizability, whereas the Ac- ion with roughly the same electron density as water can be expected to have essentially zero excess polarizability. In general, we can expect excess polarizabilities and electron affinities (in solution) to change through the Hofmeister series of alkali metal halides and halide cations. The way that they do so change should be experimentally accessible, but in fine detail is not so obvious as the differences between Br- and carboxylates. The inference is that current theories of interfacial tensions for which adsorption excesses are at the airwater interface and are based on electrostatic potentials alone must be inadequate. We note two further matters: (1) Provided the excess polarizability is positive, the sign of the dispersion potential is the same as that of the electrostatic potential at the air-water interface. Both give rise to negative adsorption. We can expect much enhanced negative adsorption, larger ∆γ(c), for NaBr than for NaAc. Awareness of this effect may explain the ion pair correlation effects and its additivity for surface activity as evidenced by, e.g., bubble-bubble interactions.28 (The qualitative differences for different ion pairs occur at concentrations g0.1 M, where dispersion effects dominate.) (2) Positive adsorption, i.e., reduction in γ with added salt can occur if the excess polarizability is negative. This can occur and depends on partial molal volumes which
∫0∞dξ
w + 0 ) 2 +
nw2 - 1 (1 + ξ2/ww2)
n02 - 1 +
(1 + ξ2/w02)
≈ 2 (19)
Then we have
I∼
[
]
(nw2 - 1)ww (n02 - 1)w0 π R(0) w1 4 (w1 + ww) (w1 + w0)
(20)
with limiting forms
I∼ I∼
π R(0) w1(nw2 - n02); 4
w1 < ww, w0
π R(0) [(nw2 - 1)ww - (n02 - 1)w0] 4 w1 > ww, w0 (21)
Clearly, and exactly as for the problem of spreading of alkanes on water,40 the sign of the dispersion potential is delicately poised. It depends on the interface. It can be repulsive, so enhancing electrostatic image effects. Or it can be attractive, leading to positive or reduced negative adsorption excesses. Thus for KI, dγ/dc changes from positive (Γ < 0) at the water-air interface to negative (Γ > 0) at the water-octane interface.41 The expression corresponding to Γ of eq 2 is now schematically
Γj ) cj
∫0∞(e-[((A/x)e
-2κx((B/x3))+z
jeφ]/kT
- 1) dx
(22)
(1) There can now be a significant potential set up at the interface with, e.g., positive adsorption for the anion and negative for the cation because the magnitude of the dispersion potentials acting on cation and anion will be very different. (2) The potential B/x3 can be expected to be much larger for halide ions than for say carboxylates or acetates with low electron density. Note that |ΓKCl| > |ΓKBr| > |ΓKI| are all negative at the air-water interfaces, and this apparently is a convenient way to systematize the Hofmeister series. The maximum effect at Na+ in the cation series should be reflected in bulk data. (3) What may be more of significance is that it would appear that no inferences of ion binding to say proteins at the air-water interface can be carried over to binding at an oil or phospholipid water interface. (4) There is no reason to expect that the dispersion potential will not act at a charged interface and even (40) Richmond, P.; Ninham, B. W.; Ottewill, R. H. J. Colloid Interface Sci. 1973, 45, 69. (41) Aveyard, R.; Saleem, S. M. J. Chem. Soc., Faraday Trans. 1 1976, 72, 1609.
2102 Langmuir, Vol. 13, No. 7, 1997
Ninham and Yaminsky
dominate when local potentials felt by an ion are highly screened, and this would explain a number of puzzling effects, outlined in the introduction, to do with counterion dependence of double layer forces. Ion exchange effects occur at concentrations orders or magnitude lower than for interfaces of water with air, octane, or Teflon. Nonspecific electrostatic interactions here are modulated by dispersion effects which vary between ion pairs and interfaces. (5) The hydronium ion, with no dispersion electrons, is clearly very different. Pure paraffins and air bubbles charge in water. With an acid we may expect enhanced adsorption of a halide ion due to dispersion effects and no such effect on H+ (a reverse situation occurs at the waterair interface). This sets up a potential which may bear on the matter of what we mean by pH measured by electrodes. 4. The Limitations of Lifshitz Theory 4.1. We have discussed above some consequences of inclusion of dispersion forces into the ion-surface potential. We turn now to interactions between surfaces. In colloid science generally it has been usual to decompose interactions between surfaces into double layer (repulsive) forces, purely electrostatic in origin (due to ions), and quantum mechanical (attractive) forces (due to solvent, and the bodies themselves). Lifshitz42,37-39 theory extended the theory of DLVO and Hamaker to include all many body forces with ultraviolet, infrared, microwave (classical, zero frequency), temperature and salt dependent contributions including retardation. And in low Hamaker “constant” systems like microemulsions, phospholipid membranes, micelles, oil-in-water, the temperature and salt dependent contributions to the net potential of interaction dominate.34,37,38 These ideas concerning interactions are at the core of analysis of experiments on microstructured fluids. The interpretation and theories of microstructure from scattering experiments in terms of the fluctuation forces of Helfrich do depend on the assignment of and magnitude of the direct double layer and attractive forces. With well-ordered mesophases like uncharged phospholipids, disagreement with theory can sometimes be associated with co-operative surface dipole correlation interactions.43 With micelles and microemulsions, microstructure depends critically on both counterion and co-ion. We next explore the nature of the extended Lifshitz theory of interactions and show that it is precisely equivalent to the Onsager limiting law for a single interface. That is, it is precisely equivalent to a pointion, electrostatic image correlation model alone, with a linearized approximation to adsorption excesses. The theory therefore suffers precisely the same limitations as the Onsager limiting law at a single interface. 4.2. Lifshitz Theory. Explicit Forms. In the absence of explicit conduction processes due to salt, the Lifshitz interaction free energy between two uncharged bodies can be derived in several ways.34 All are equivalent. The simplest is the normal mode approach. In the nonretarded limit we have the well-known result for the Helmholtz free energy of interaction F(l) between two planar surfaces a distance l apart
F(l) )
kBT
∞′
k dk ln(1 - ∆2e-2kl) ∫ ∑ 0 2π n)0 ∞
(23)
(42) Dzyaloshinskii, I. E.; Lifshitz, E. M.; Pitaevskic, L. P. Adv. Phys. 1961, 10, 165. (43) Attard, P.; Mitchell, D. J.; Ninham, B. W. Biophys. J. 1988, 53, 457.
Here kB is Boltzmann’s constant, the prime indicates that the term in n ) 0 is to be taken with a factor 1/2, and
∆(ξn) )
2(iξn) - 1(iξn) 2(iξn) + 1(iξn)
(24)
The dielectric susceptibilities of the interacting media are evaluated at imaginary frequencies iξn, ξn ) 2πn kBT/ p. (All many body effects are built into the measured dielectric properties.) The theory is exact, subject to the limitations that the interaction media are continnua. (This is a subtle point. Although the free energy expression (eq 23) is derived by the normal mode approach with the assumption that the media have continuum bulk properties up to the interface, it does not mean that the concentration profile is constant. This is a consequence of the fact that thermodynamic perturbation theory for the free energy to first order requires the distribution function to zeroth order only.) That limitation is more, or less, severe depending on circumstance. The manner of derivation of this result is hardly transparent, and its connection to the Gibbs equation is not obvious. In fact by definition interaction free energies are changes in interfacial energies:
F(l) ) γ(l) - 2γ∞ )
∑i ∫-∞[Γi(l) - 2Γ∞] dµi µ
Here γ(∞) ) 2γ∞ and Γ(∞) ) 2Γ∞. From the Gibbs adsorption equation if the intervening medium is a liquid at constant chemical potential, dF(l)/dµ measures the change in adsorption excess Γ(l) - Γ(∞) at an interface induced by a neighboring interface. In the theory the intervening liquid is considered as a continuum of constant dielectric properties, so that the theory is in fact a pertrubation theory, deviations from which are taken into account by invoking “hydration”, surface induced liquid structure, or Deryaguin’s disjoining pressure depending on taste. (Deryaguin and Obuchov45) first used the term “disjoining pressure” long before the DLVO and Hamaker results were firmly established, later as a substitute for interaction force (dF(l)/dl) of arbitrary nature that may occur in liquid films. Some of these effects have been suspected and are either implicit or explicit in the works of Hall and Pethica and of Ash, Everett, and Radke.46-48) For low Hamaker constant systems (oil-in-water, phospholipids) given these restrictions, the expression eq 23 loosely splits into two components. Terms in the sum for which n * 0 are essentially quantum mechanical and depend only weakly on temperature. They involve London (ultraviolet induced dipole-dipole correlations) and Debye (dipole-induced dipole correlations). The term n ) 0 is explicitly temperature dependent and classical in origin. This term usually dominates in low Hamaker constant systems. In the presence of salt, the same technique can be used34 to derive a modified expression (44) Landau, L. D.; Lifshitz, E. M. Statistical Physics; Pergamon Press Ltd.: London, 1958; Chapter VII, p 74. (45) Derjaguin, B. V.; Obuchov, E. V. Colloid J. (USSR) 1935, 1, 385. (46) Hall, D. G. J. Chem. Soc., Faraday Trans, 2 1972, 68, 2169. (47) Ash, S. G.; Everett, D. H.; Radke, C. J. Chem. Soc., Faraday Trans, 2 1973, 69, 1256. (48) Hall, D. G.; Pethica, B. A. Proc. R. Soc. London 1978, A364, 457.
Ion Binding and Ion Specificity
Fn)0(l) ≡ F0(l) )
Langmuir, Vol. 13, No. 7, 1997 2103
kBT 4π
∫0∞k dk ln[1 - ∆122(s)e-2sl]
s ) xk2 + κD2,
2s - 1k 2s + 1k
∆12(s) )
(25)
F0(l) )
∫
kT k dk ln(1 - ∆2e-2lk) 4π kT κD2 dp Γ(p) ζ(p + 1) (8πi)
(26)
∫c
(2κDl)p(p - 2)
(33)
0 < Rep < 2 4.3. Small Distance Behavior. We shall explore the genesis of this contribution at a more fundamental level later. For the moment we wish to exhibit its explicit connection to the Onsager limiting law. To do this consider the case κDl f 0. Change variables to x ) 3Dk/κD and expand the logarithm of the integrand of eq 25. Then the representation
1 2πi
e-y )
c+i∞ ∫c-i∞
Γ(p) dp yp
where ζ(p + 1) is the Riemann zeta function, whose only pole is at p ) 0. The gamma function also has a pole at p ) 0, and evaluating the residue at the double pole, we get
F0(l)(κDlf0) )
kT
∞ k dk ln(1 - ∆2e-2lk) + ∫ 0 4π
(27)
e2
(∑c z ) ln(2κ l) + ... 2 2
j j
where Γ(p) is a gamma function and the contour of integration in the complex p plane is parallel to the imaginary axis and allows us to write
F0(l) ) -
kT κD2 1
∞
Γ(p) dp
c+i∞ ∫ c-i∞ ∑ 2πi
4π
∫0∞
(
n)1(2κ
p Dl)
np+1
(2x1 - x2 - 1x)
(2x1 + x - 1x) 2
)
×
2n
x dx (x2 + 1)p/2
(28)
The integral in x and sum over n converges and is an analytic function of p for Rep > 2, so that
F0(l) ) -
-kT κD2 1
∫ 2πi
4π
Γ(p) dp (2κDl)
∞
∑ p n)1
1
I(p) (29)
np+1
c ) Rep > 2 The analytic structure of I(p) is determined by the behavior of the integral at x f ∞ and to leading order in 2n
∫0∞(x2 x+dx1)p/2 ) (p∆- 2)
I(p) ) ∆2n
(30)
whence
F0(l) ∼
-kT κD2 4π 2πi
∫c-i∞
c+i∞
∞
Γ(p) dp
∑
∆2n
(2κDl)p(p - 2) n)1 np+1
(31)
The first role of the integrand is at p ) 2, and evaluating the residue at this pole ∞
F0(l) ∼
-kT
∑
-
4π(2πi)
(34)
By comparison with eq 7, we see that at small distances κDl f 0; eq 25 gives the Lifshitz (salt free) interaction energy plus twice the Onsager limiting change in interfacial free energy due to electrostatics, which cancels out. This exactly matches the general thermodynamic limit F(lf0) f 2γ∞ for which the surface tension of the solvent includes the surface pressure (∆γ) of the ions. 4.4. Large Distance Behavior. Ion Depletion. At large distances, the interaction free energy takes a different form. Take 2 . 1, ∆(s) = 1 in eq 25. Then expansion of the logarithm and integration gives
F0(l) ≈ -
[
]
kT 2 -2lκD 1 1 + 0(e-4κDl) (35) κ e + 4π D 2κDl (2κ l)2 D
This large distance behavior holds for κDl . 1. The Onsager-Samaris law and the corresponding interaction free energy for low Hamaker constant systems are genuinely limiting laws holding only when 2κDl f 0 and l is a molecular dimension, unlike the Debye-Hu¨ckel theory which holds over a much wider range of concentrations. The attractive force underlying eq 35 can be considered to be due to an osmotic pressure due to depletion on ions inside the gap of distance l between the inteacting systems. To see this, consider for simplicity a 1:1 electrolyte. For the bulk solution the electrolyte has a chemical potential43
µB ) kT ln FB -
( )
e3 πFB 3/2 kT
1/2
Ni
FB )
∑i V ) ∑i ci
Ni is the total number of electrolyte ions of species i, in volume V, and FB the density. For the electrolyte in the gap
∆2n
16πl2 n)1 n3 kT κD2
D
2
∫c
Γ(p) dp p
(2κDl) (p - 2)
∞
∑ n)1
1 n
∆2n (32)
p+1
0 < Rep < 2 i.e., the leading term in the electrolyte free case. In the second term we can take ∆ ) 1 to first approximation to get
µG ) kT ln FG µint )
e3 π FG 3/2 kT
(
1/2
)
+ µint
∂F (∂N )V,T
For the gap of volume V ) Al, where A is the area of the plates, we have from (35)
2104 Langmuir, Vol. 13, No. 7, 1997
Ninham and Yaminsky
{
Fint 1 kT 2 -2lκD 1 + )κ e A 4π D 2lκD (2lκ )2 D
}
() ( )
(
In terms of extensive variables N, V, T
Fint
e )-
A
Ni
2
∑l
{
1
e-2lκD
1 + (2lκD)2
(2lκD)
}
4π kT
We require
Fint )
Ni e2 ν
∑
)
1/2
1 δ 3/2 - Pint (37) 3/2 FB
-e3 el
∑Ni(gap)e-2lκ
(
1 1 + 2lκG (2lκ )2 D
G
)
Since
κD ∼
∂F
∑ 2 i ∂N
µint )
(FB3/2 - FG3/2) - Pint
Now
( )
1
1/2
( )
δ e3 π FB 3 kT
) kTFB
where we have used
κD2 )
e3 π δ FB 3 kT
(PB - PG) ) kTFB
() N V
( )
2lκD ∂ (2lκD) ∼ ∂V 2lV
1/2
,
i l,Nj
So and use
( )
2 ∂ 4πNie
∂κD
∑∂N ) ∑∂N i
i
1/2
Pint ) -
1
)
kTν
κD 2N
e2
∂F
∑i ∂N
)el
i
e2 el
{
1
e-2lκD
1 +
[ ( )]
(lκD)
∂
∂x
e-x
}
1
1 e2
1
)
+
x
x2
x)2lκD
2 l
ln FG ) ln FB +
(
FG1/2 = FB1/2 1 +
(
δ e-2lκG ) e-2lκB(Bulk) 1 - (lκB) FB
)
1 δ 2 FB
)
( )
1/2
=-
e2 e-2lκD 2l kT
This gives the difference in density between the bulk electrolyte and that in the gap. The bulk electrolyte pressure is
FB -
( )( )
e3 π 3 kT
1/2
∑ci 3/2 3/2
and the pressure between the plates
∑
PG ) kT Hence
( )( )
e3 π FG 3 kT
1/2
∑FG 1/2 + P 3/2
-
-
[( )
int
(36)
∑
(
)]
Fie2e-2lκD ∂F kT 1 ∂ 4π 1+ )∂l ∂l 4π kT 2lκD 2lκD )
Equating chemical potentials, we have after a little algebra
e2 -2lκB e δ 2l ) FB e3 πFB kT - 3/2 kT 2
3/2
But from eq 35 the force per unit area from the free energy expression follows directly as (κD measures the bulk electrolyte concentration)
δ , FB
-
FB3/2
1/2
2 e2 -2lκDFB e2 2 = FGe-2κGl 1 + + e 2l 2κGl (2lκ )2 el G
e-2lκD
Now put
FG ) FB + δ;
(38)
[ () ] ( )
δ e3 π (PB - PG) ) kT FB FB 2 kT
+
(2lκD)2
(2lκD)
)
where the Debye length is evaluated for the electrolyte density in the gap. Taking eqs 36-38 together we get
Hence
µint )
(
∂F 2 2 e2 + ) F e-2lκD 1 + ∂V 2l G 2lκD (2lκ )2 D
( ( ))
e2 -2lκD 1 FB 1 + 0 3 e l l
We have thus shown explicitly that the attractive force indeed can be considered to be an osmotic effect, as it must be as long as the µi are everywhere constant throughout the system. 4.5. Genesis of the Electrostatic Free Energy. The meaning of the salt dependent extension of Lifshitz theory embodied in eqs 23 and 25 and its connection to the Onsager limiting law for an isolated surface is by now explicit. But its derivation via the normal mode analysis obscures its connection to, and genesis from, the Gibbs adsorption isotherm and statistical mechanics. To understand where “ion binding” comes into the picture, we now proceed to a more fundamental derivation. This will reveal the nature of and limitation of the approximations involved in the theory. 4.5.1. Surface and Interaction Energies. Preliminary Considerations. Consider again surface and interaction energies when the electrolyte separates two planar surfaces. The interaction energy is just the change in surface energy. Consider the geometry of Figure 1.
Ion Binding and Ion Specificity
Langmuir, Vol. 13, No. 7, 1997 2105
1 d2κ iκF -iκF′ -|z-z′|κ e + a1(κ) e-|z|κ] e [e 2π1 κ (43b)
∫
φ(r,r′) )
z1
Figure 1. Geometry for calculation of interaction energies.
The condition that φ and ∂φ/∂z be continuous across the boundaries gives [write a1, a2, a3, a4 e-iκF′ a1′, a2′, a3′, a4′ and drop the prime for notational convenience]
1 -|z′|κ 1 -|l-z′|s + a2 e-ls + a4] ) + a1] (44) [e [e 2s 1κ
We compute the self-energy of an ion at a distance d from the interface as shown. The potential satisfies
1 -|l-z′|s 1 -|l-z′|κ + a2 e-ls + a4] ) + a3] [e [e 2s 1κ
(∇2 - κD2) φ(r,r′) ) 4π - [δ(r,r′) + A2 δ(z) + A4 δ(l - z)] (39a) 2
e-|z′|s - a2 + a4 e-ls ) e-|z′|κ + a1
0
