Introduction

Hofmeister effects, also known as specific ion effects, were observed over 120 years ago. Even though they are ubiquitous in the physical, chemical and biological literature, their origin is still contested1,2,3,4,5,6,7 and has been recently brought to the forefront of research8,9,10,11,12,13,14,15,16,17,18,19. New work in this area may break down barriers between physics and biology9.

Ionic sizes, hydration, quantum fluctuations (or dispersion forces)3,4,8 and surface charges1,2 are crucial in Hofmeister effects. Nano-scale surfaces and colloidal particles (e.g., DNA, proteins, cells, bacteria, metal oxides and clay) are usually strongly charged in aqueous solution and the sign of the charge or the charge number for biological macromolecules will be dependent on pH and ionic strength20. In classical theory, the surface charges can set up a strong electric field extending from the surface to several nanometers in solution. Typical surface charge densities and their corresponding electric field strengths include: (1) membrane with surface charge densities of 0.28 C/m2 20 and electric fields of 4 × 108 V/m (assuming a planar surface); (2) for natural clay (e.g., illite), it is ~0.2895 C/m221, with an electric field of ~4.2 × 108 V/m at the surface; (3) for metal oxides it is 0.3–1.0 C/m222,23,24,25, with a surface electric field of 108 ~ 109 V/m; (4) for artificially synthesized nano-TiO2, it is 0.25 C/m2 and 3 × 108 V/m, respectively26. For proteins, surface charge density measurements are primarily based on zeta potentials; therefore, they would be equal to the charge density at the shear plane, which is possibly much lower than the charge density at the surface27,28,29, especially when considering the strong surface hydration force30,31,32. At the shear plane, the charge density could reach 0.02–0.35 C/m233,34,35, with an electric field of 107–108 V/m. Therefore, it is reasonable to expect electric field strengths >108 V/m at protein surfaces in aqueous solutions. However, in those calculations, adsorbed counter-ions near the surface are treated as point charges. If their ionic size were taken into account, then the electric field near the surface would be much greater than 108 V/m. This is because the finite size of counter ions could weaken their screening effect, as compared with point charges.

Noah-Vanhoucke and Geissler found that, the persistence of electric field in the space near liquid-vapor interface shapes the sensitivity of solute distributions to ion polarizability and the electric field often plays central role in the influence of ionic polarizability on ion density profiles36. Surely, the distribution of the polarized ions would reversely influence the electric field itself. Therefore, at the interface, ionic polarization, ionic distribution and the electric field are interwined. At the solid/liquid interface, however, we often meet strong electric field greater than 108 V/m, the effects observed by Noah-Vanhoucke and Geissler might be much stronger and much more complex.

In classical theory, induction forces are much weaker than dispersion forces; thus the contribution of induction forces to Hofmeister effects can be neglected. In strong electric fields of 108–109 V/m, however, we must question the validity of classic induction theory for three reasons. First, if an atom or ion is strongly polarized, the binding force of the nucleus to the extranuclear electrons decreases, and, simultaneously, the contribution of the high external electric field to the energy of the electrons is greatly enhanced. The exact additional potential energy of the electrons from the high external electric field should be included in the Hamilton operator. Second, the range of electrostatic forces from surface charges is longer than that from a single ion20, which means that at the surface the additional force on an ion/atom from the external electric field and polarization effects will be long range. Third, there are many counter-ions around the surface and if they are all strongly polarized in the high external electric field37,38, then they will reversibly and strongly influence the external electric field itself. All of these effects must be correctly evaluated.

Because the selectivity coefficient in cation exchange equilibrium varies exponentially with cation-surface interaction energies for the two cation species involved, a slight difference in their respective interaction energies could result in significant differences in selectivity coefficient. Therefore, the ion selectivity coefficient is a useful parameter in quantitative evaluation of Hofmeister effects. For quantitatively evaluate the Hofmeister effects, we should firstly evaluate the contribution of Coulomb interaction to the selectivity coefficient accurately. We recently derived exact analytic solutions for the non-linear Poisson-Boltzmann equation for cation exchange in mixed electrolyte solutions39. This enables accurate evaluations of Coulomb interaction effects on the selectivity coefficient, which in turn allows experimental evaluation of Hofmeister effects on the selectivity coefficient from cation exchange equilibrium data. As a result, a comparison can be made between experimental results and calculated Hofmeister effects that take into account ionic size, hydration, dispersion forces and classical induction forces.

By analyzing published Ca2+/Na+ exchange equilibrium data under a wide range of electrolyte concentrations and ionic strengths (0.00075 –0.354 mol/l21), we found that the observed Hofmeister effects could not be understood in terms of any classical interaction effects noted above, but instead requires a strong new force between cation and surface.

Results

Selectivity coefficient of cation exchange by classical Coulomb force

For Ca2+/Na+ exchange, the selectivity coefficient may be defined as:

where aNa and aCa are the activities of Na+ and Ca2+, respectively, in bulk solution (mol/l) and NNa and NCa are the adsorbed quantities (mmol/g) of Na+ and Ca2+, respectively, because of the Coulomb force.

An advantage of Eq. 1 is that it can quantitatively evaluate the relative preference between Ca2+ and Na+. Thus, when K > 1, there is a preference of Ca2+ over Na+ in exchange; and vice versa for K < 1.

If cation adsorption forces were Coulomb, Eq. 1 gives40:

where KC is the selectivity coefficient determined by the Coulomb forces for Ca2+ and Na+ in cation-surface interactions; R is the gas constant (J/mol·K); T is temperature (K), F is the Faraday constant, Z is the charge number of cation species; φ(x) is the potential at position x in the diffuse layer; S is the specific surface area; ci(x) is the concentration of the ith cation species at x; and 1/κ is the effective thickness of the diffuse layer (or Debye length).

Liu et al. obtained the exact analytical solution of the non-linear Poission-Boltzmann equation for 2:1 and 1:1 mixed electrolyte solutions39. From the analytical solution of φ(x), we can get the following simple relationships under relatively high surface potential conditions ():

Thus Eq. 2 yields:

Taking x = 1/κ to be the upper limit of the integrations in Eq. 2 (meaning φ1/κ → 0), we thus have , where is the mean electric field strength in the diffuse layer and is the mean Coulomb potential energy of the ith cation species in the diffuse layer. Thus Eq. 3 can be expressed as:

According to Eq. 5, it was the mean potential energies of cation species in the diffuse layer that determine the selectivity coefficient K.

Strong Hofmeister effects in Ca2+/Na+ exchange

If we consider Bolt's21 experimental results for Ca2+/Na+-illite exchange equilibria in a solution of NaCl and CaCl2, there are four relevant aspects: (1) the Ca2+/Na+ exchange was determined under a wide range of electrolyte concentrations and ionic strengths (0.00075–0.354 mol/l); (2) illite particle surfaces can be considered planer; (3) illite surface charges are constant and therefore the surface charge density is independent of pH and ionic strength; (4) the ionic radii of Ca2+ (0.099 nm) and Na+ (0.095 nm) are almost the same, but the hydration diameter of Ca2+ (0.52 nm) is much larger than that of Na+ (0.356 nm)41.

Bolt independently determined the surface charge density of illite (0.2895 C/m2) by the negative adsorption method21. With the Poisson-Boltzmann equation of the mean force, the surface potentials can be estimated from the classical σ0 ~ φ0 relationship, where σ0 is surface charge density (C/m2) and φ0 is surface potential (V). The results were shown in Table 1. We note that ionic interaction energies in bulk solution would influence the distribution of cations in the diffuse double layer; therefore, we used activity instead of concentration for cation species in bulk solution, by using the modified Debye-Hückel equation42.

Table 1 Surface potentials of illite particles for each exchange equilibrium21 (In the calculation of surface potential, the dielectric constant of water is 8.9 × 10−9 C2/Jm)
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Using the data in Table 1, KE = aNaNCa/aCaNNa (KE is the experimentally determined K) and KC (Eq. 5) values could be calculated. The values of φ0 in Table 1 indicated that the condition was satisfied for all KC calculations. In Figure 1, it could be seen that the KE were higher than the KC and the difference increased with surface potential. The difference clearly revealed Hofmeister effects and implied that there are additional adsorption forces other than Coulomb for cation adsorption in Ca2+/Na+ exchange. Thus there are strong Hofmeister effects present in Ca2+/Na+ exchange, which are strengthened by the increasing surface potential of illite particles.

Figure 1
figure 1

Comparison between calculated () and experimental data ().

KE is the experimental selectivity coefficient; KC is the calculated selectivity coefficient based on the classical theory.

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Strong Hofmeister effects in Ca2+/Na+ exchange based on classical interaction forces

The effect of dispersion forces

Equation 5 shows that selectivity coefficient K is a function of mean potential energy of cation species in the diffuse layer. If dispersion forces were present, Eq. 5 becomes:

where i(D) was the mean dispersion energy of cation species i in the diffuse layer; the selectivity coefficient KC+D is determined by both Coulomb and dispersion forces.

Considering i(D) is constant (independent from surface potential)3 and if the differences between KC and KE in Figure 1 come from dispersion forces, then KC+D = KE. From Eqs. 1, 5 and 6, we obtain:

Figure 2 plots KE/KCvs. φ0, which is not constant. Therefore, the differences did not derive from the dispersion forces. The four red dots are KE/KC values with very large values of c0Na/c0Ca (red data in Table 1) and will be discussed below.

Figure 2
figure 2

KE/KCvs. φ0 (V).

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Only under relatively high electrolyte concentrations is the dispersion force3,8,43,44,45,46 expected to become significant. Therefore, as the electrolyte concentration decreases, the difference between KE and KC decreases and at very low electrolyte concentrations, KE/KC will approach 1. However, Figure 2 showed that for decreasing electrolyte concentration (increasing surface potential), the difference between KE and KC increased. Therefore, we conclude that strong Hofmeister effects in Ca2+/Na+ exchange did not derive from dispersion forces.

The same phenomena are also observed for enzyme activities in <0.2 mol/L solutions of LiCl, NaCl and CsCl47 and for K+/Na+, K+/Li+, Na+/Li+ and Mg2+/Na+ cation exchanges40.

The effects of ionic size and hydration

If the hydration effect became important in exchange, it would certainly decrease the preference of Ca2+ over Na+ in the exchange since the hydration diameter for Ca2+ (0.52 nm) is much larger than that for Na+ (0.356 nm)41 and then KE/KC < 1. According to Yan Levin et al.1,2, the chaotropic Na+ cation would be more likely adsorbed at particle surfaces, whereas the kosmotropic Ca2+ cation would be more likely present in solution. This would decrease the preference of Ca2+ over Na+ in the exchange because of the stronger dispersion and electrostatic forces per charge for Na+ than that for Ca2+ and KE/KC < 1. In Figure 2, however, KE/KC > 1 and is not constant vs. surface potential. Therefore, the hydration effect could not explain the Ca2+/Na+ exchange. One possible reason was that hydration and a hydration-dispersion combination2 may be correct only in relatively weak external electric field conditions of air/water and solid/liquid interfaces. In the dos Santo and Yan Levin study1, the surface charge density was merely 0.04–0.06 C/m2. Moreover, only under relatively high electrolyte concentrations would the ionic size (including hydration diameter) effect become significant48,49,50,51,52,53,54,55,56,57,58. Therefore, the difference between KE and KC should have decreased with electrolyte concentration, in contrast to Figure 1. Hence, the strong Hofmeister effects in Ca2+/Na+ exchange did not come from differences in the ionic size, ionic hydration volume, or hydration-adjusted dispersion interactions between the ions and the surface1,2.

The effect of ionic induction force

Because increasing surface potential strengthened Hofmeister effects, they appear to be independent on dispersion forces, ionic sizes and hydration effects. If it is the induction force in the electric field of the diffuse layer that was responsible for the large Hofmeister effects, then we have:

where is the mean induction energy of cation species i in the diffuse layer.

Assuming that the dipole orientation near the particle surface is co-directional and parallel with the field direction, then the classical mean induction energy (J/mol) of Ca2+ and Na+ dipoles can be calculated by the classical equations, and:

where pi(I) is the mean dipole moment of the ith cation that results from classical induction theory. is the mean electric field strength in the diffuse layer and . The φ0 values are from Table 1 and the potential at x = 1/κ φ(1/κ) can also be obtained from the analytical solutions of the non-linear Poisson-Boltzmann equation for the 1:1 and 2:1 electrolyte mixtures39.

The can be estimated from:

in which59 and .

where αi* is the effective (excess) ionic polarizability in aqueous solution, εi and εw are the dielectric functions of the ith ion species and water, respectively, ε0 is the dielectric constant in vacuum, Vi is the ionic volume, αi is approximately equal to the intrinsic polarizability of the ith ion species, αCa = 0.4692 Å360 and αNa = 0.139 Å359.

By substituting Eqs. 10 into Eq. 9, - can be calculated and by substituting the result into Eq. 8, KC+I can be calculated. The results plotted in Figure 3 indicate that the classical induction force could not explain the observed strong Hofmeister effects in Ca2+/Na+ exchange. This is because the classical induction potential energies are very small relative to the Coulomb potential energies and can be completely neglected. Actually, the excess polarizabilities for Ca2+ and Na+ are negative, which means that the classical induction forces between the surface and the cations are repulsive.

Figure 3
figure 3

Comparison between theoretical curves of KC+I () and KE ().

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We have now shown that the strong Hofmeister effects in Ca2+/Na+ exchange did not derive from classic dispersion forces, induction forces, ionic sizes, or hydration effects. The London-Lifshiz theories on dispersion forces, induction forces and hydration effects might be valid only when the external electric field is weak. In high electric fields >4.2 × 108 V/m, however, it is possible that new interaction forces are present.

The general origin of Hofmeister effects at the interface

Because the differences between KE and KC sharply increase with surface potential, the unknown interaction force between the cations and the surface must be a function of the potential. Thus two parameters βCa and βNa are introduced to modify the Coulomb interaction potential energies (ZFφ0) of Ca2+ and Na+. Eq. 4 can be changed to61:

Using the iteration approach suggested by Li et al.61, βCa, βNa and φ0 can be calculated. If φ0 is known, the potential at x = 1/κ can also be calculated from the analytical solutions of the non-linear Poisson-Boltzmann equation for 1:1 and 2:1 electrolyte solutions. The calculated values of φ0, φ(1/κ), βCa and βNa are in Table 2.

Table 2 The calculated φ0, φ1/κ, βCa and βNa values based on the experimental data
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From Table 2, βCa > 1 and βNa < 1, which implied that the unknown force for Ca2+ in cation-surface interactions was stronger than that for Na+. A comparison of surface potentials in Tables 1 and 2 also indicated that the unknown force decreased the surface potential significantly, perhaps because Ca2+ was more effective in screening than Na+.

We used Table 2 to examine βCavs. φ(1/κ)/φ0 and βNavs. φ(1/κ)/φ0. It was surprising that these were unusual linear relationships (with relative error <0.3%):

They were unusual because the slopes were equal to the “1-intercept” for the two cation species. If a new induction force in cation-surface interactions was introduced, Eqs. 12 could be theoretically derived. We tentatively refer to the new surface-potential-dependent force as a non-classic induction force, because it can be explained by the enhanced polarizability in strong external electric field from surface charges of illite particle.

An illite surface charge density of 0.2895 C/m2 corresponds to an electric field strength of 4πσ0/εwater = 4.2 × 108 V/m (εwater = 8.9 × 10−9 C2/J·m) at the surface. The high external electric field may non-classically and greatly enhance the dipole moments of the two cation species. The dipole moment of a cation species will be more strongly enhanced than others if it has a softer electron cloud and/or prefers to stay near surface of stronger electric field (e.g., a cation with a stronger electrostatic or dispersion force, and/or a chaotropic cation).

If the additional energies do come from the strong non-classic induction force, then:

where and are the strong non-classical induction energies (J/mol) in the adsorption phase of Ca2+ and Na+, respectively, which come from the strong polarization of the cations in the high electric field at the surface.

If the dipole orientation of the particle surface was co-directional and parallel with the high external electric field, the mean induction energies of Ca2+ and Na+ dipoles in the field are:

where and are the mean dipole moments (dm·C/mol) of Ca2+ and Na+ in the diffuse layer, respectively.

Introducing Eq. 14 into Eq. 13:

in which

where a is constant and ; = -.

Substituting Eq. 16 into Eq. 15, one obtains:

Comparing Eqs. 17 and 11, we have:

It is very interesting that the fitting equation Eq. 12 for the experimental data could be explained by the theoretical equation Eq. 18. There are several reasons why this is very interesting. The expressions Eq. 18 are the same as Eq. 12; Eq. 18 verifies that βCa + βNa = 2; Eq. 18 verifies the “slope = 1-intercept” in Eq. 12; and from Eqs. 18 and 12, aΔ = 0.357.

By including the strong induction force, the relationship of KC+NI ~ φ0 based on Eq. 17 (where ) fitted the experimental data very well, as shown in Figure 4. It is important to note that if we did not have the specific mathematical form of Eq. 18, the comparison with Eq. 12 could not have been done and the value of = 0.357 would not have been obtained. Therefore the KC+NI ~ φ0 relationship curve based on Eq. 17 is theoretical and not a fitting curve.

Figure 4
figure 4

Comparison between theoretical curves of KC+NI () and KE ().

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In summary, we have shown that a strong non-classic induction force of adsorbed ions in the electric field is the origin of the strong Hofmeister effects in Ca2+/Na+ exchange.

Comparison of the strong non-classical and the classical induction energies

From = 0.357, we have = 0.357 × 3 × F = 103337 C for Ca2+ and Na+. The differences in the non-classical induction energies between Ca2+ and Na+ could be calculated from the Ca2+/Na+ exchange experiments. From Eq. 14, we have:

Thus the experimental values of can be calculated with Eq. 19.

On the other hand, the surface-cation Coulomb force in such a high electric field would be much stronger than that for surface-water (surface hydration) and cation-water (cation hydration) interactions. As a result, the adsorbed cation near the surface might be to some degree dehydrated3 in the high electric field near the surface. In the extreme case of complete dehydration near the surface, we can use the intrinsic cation polarizability to calculate the dipole moment. Table 3 shows comparisons of and among the calculated results from cationic excess (Eqs. 9 and 10), the intrinsic polarizabilities and the observed non-classic results. There are several observations: (1) The observed non-classical induction energies are comparable with the Coulomb energies in a wide range of electrolyte concentrations and ionic strengths over 0.00075–0.354 mol/l. (2) Even at very high electrolyte concentrations, the observed non-classic induction energies are large. (3) The classical induction energies (applying both excess and intrinsic polarizabilities) are so low relative to the observed high non-classic induction and Coulomb energies that they could be completely ignored. (4) From classical induction theories, or decrease with an increase in external electric field (low electrolyte concentration corresponding to high external electric field); and this is not correct because in the classic theories we used constant ionic polarizabilities αi* and αi in different electric fields.

Table 3 Comparison of mean potential energies of cations in diffusion layer producing from Coulomb force, classical induction and non-classical forces respectively
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The comparison of the observed large non-classical induction energy and the calculated classical induction energy indicated that in a strong external electric field of 108–109 V/m, the ionic polarizabilities αi* and αi would be heavily underestimated by the classical induction theories and that they would sharply increase with electric field strength (Table 3). Therefore, a new theory to calculate ionic polarizability in strong electric fields is required; a successful theory for Hofmeister effects at solid/liquid interfaces should address the statistical mechanics of density fluctuations and their impact on solvent polarization36.

There are high electric fields >108–109 V/m at the solid/liquid interfaces of clay, oxides, proteins, nanomaterials and cell membranes. Therefore, the observed strong non-classical induction force will generally be the origin of Hofmeister effects at interface surfaces. In current treatments of Hofmeister effects, however, this important force has been ignored1,2; thus they may be correct only for weak electric fields.

It should be noted that the dielectric constant of water for bulk solution was used in all of the calculations. If it is actually lower near a surface relative to that in the bulk, then the values in Table 3 might be overestimated. Even if this was the case, it is unlikely to change the general conclusions of this study.

An extended analysis of the strong non-classic induction force

Table 3 had very large values, indicating that adsorbed ions in the electric field of the diffuse layer were strongly polarized. It also indicated that the polarization strength sharply increased with decreasing cation activity and that it was sensitive to the electric field strength in diffuse layer.

To demonstrate the effect of electric field strength in the diffuse layer on the polarization of adsorbed ions, we plotted vs. φ0 in Figure 5.

Figure 5
figure 5

vs. φ0 (V) under different a0Na/a0Ca ratios (numbers aside dots are the values of a0Na/a0Ca).

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Even though generally increased with φ0, there was appreciable scatter in Figure 5. However, if the data were divided into three groups according to a0Na/a0Ca, the relationship of vs. φ0 became more transparent: low values of a0Na/a0Ca over the range 1 –10; intermediate values over 10–100; and high values over 600–1000.

For low and intermediate values of a0Na/a0Ca, we found that = - sharply increased with φ0. Since the electronic structures for Na+ and Ca2+ are, respectively, 1s22s22p6 (with a static polarizability αNa = 0.139 Å359) and 1s22s22p63s23p6 (with a static polarizability αCa = 0.4692 Å360), the electronic cloud for Ca2+ is “softer” than that for Na+. Therefore, quantum fluctuations for Ca2+ should be higher than those for Na+. The observation that = - sharply increased with φ0 indicated that the difference in random quantum fluctuation between Ca2+ and Na+ was strongly and directionally enhanced by the high external electric field.

At a constant surface potential φ0 (e.g., −0.12 V), = - sharply decreased with increased a0Na/a0Ca. A larger a0Na/a0Ca means more Na+ could be distributed in the inner space of diffuse layer. The electric field was higher in the inner diffuse layer; thus, for larger a0Na/a0Ca, more Na+ might have random quantum fluctuations directionally enhanced. As a result, = - decreased.

Even with very high a0Na/a0Ca ratios (e.g., 604–991), = - > 0, which implies a strong preference of the surface for Ca2+ over Na+. Three key factors will influence that preference: cation charge number (Coulomb force), cation flexibility (dispersion force) and hydration (chaotropic or kosmotropic1,2). Even though Ca2+ is kosmotropic and Na+ is chaotropic, the charge number and flexibility of Ca2+ are higher than that of Na+, so the surface prefers Ca2+ over Na+. Therefore, Coulomb forces, dispersion forces and hydration effects appeared to be interwined to influence cation distribution at the interface. In Levin theory, only hydration determines that preference1,2.

The difference in quantum fluctuations between Ca2+ and Na+ could be strongly and directionally enhanced by the high external electric field that created enormous values of - = .The complex combination of Coulomb, dispersion and hydration effects would also greatly affect the enhancement. In addition, the enhancement would decrease the external electric field because the additional strong non-classical induction force significantly concentrated the counter-ions in the near-surface region.

Similar to London-Lifshiz forces, quantum fluctuations might be the essential origin of the new force. However, unlike London-Lifshiz forces, the random quantum fluctuations might be strongly and directionally enhanced by the high external electric field. Therefore if the quantum fluctuation was the essential cause of the new force, the high electric field would be its external cause. Here, we tentatively refer to this new force as a non-classical induction force, but the induction concept could not exactly describe it. Therefore, we suggest that the new force might derive from the coupling of ionic quantum fluctuations and the high external electric field. The classic Coulomb forces, dispersion forces, ionic size and hydration effects combine to determine the preference for ion to stay at a surface over others, thus affecting the coupling effect. This concept is schematically illustrated in Figure 6. We emphasize we are simply illustrating a possible explanation of the new force; a complete description about the nature of the new force is still required.

Figure 6
figure 6

The schematic diagram of a cation that is strongly polarized in an external electric field and the field is weakened by the polarization.

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Conclusions

We have observed a strong non-classical induction force in cation-surface interactions. Hofmeister effects in general may derive from this force and the results presented here may fundamentally challenge all related theories. Currently, it is believed that Derjaguin-Landau-Verwey-Overbeek theory and the double-layer theory are exact at low electrolyte concentrations; and at high concentrations, dispersion forces, ionic sizes and hydration effects must be taken into account for the explanation of Hofmeister effects. In contrast, our results indicate that the most important forces are not those classic interactions. Instead, they are the strong non-classical induction forces at high and especially at low concentrationsfor the explanation of Hofmeister effects.The classic Coulomb forces, dispersion forces, ionic size and hydration effects appeared to be interwined in determining the preference for an ion species to stay at a surface over others, thus affecting the strong non-classical induction forces.

The strong non-classical induction force implies that the energies of non-valence electrons of ion species at the interface might be substantially understated and that they may profoundly influence the physical and chemical properties of ions, atoms and molecules. Therefore, to describe Hofmeister effects occurring at interfaces with high electric fields, a combined solution of Schrodinger's equation and the Poisson-Boltzmann equation would be required and that the energy produced by the field near the interface should be exactly included in the Hamilton operator.