Strong non-classical induction forces in ion-surface interactions: General origin of Hofmeister effects

Abstract

Hofmeister effects continue to defy all-encompassing theories and their origin is still a matter of debate. We observed strong Hofmeister effects in Ca²⁺/Na⁺ exchange on a permanently charged surface over a wide range of ionic strengths. They could not be attributed to dispersion forces, classical induction forces, ionic size, or hydration effects. We demonstrated that another stronger force was active in the ion-surface interactions and which would create Hofmeister effects in general. The strength of this force was up to 10⁴ times that of the classical induction force and could be comparable to the Coulomb force. Coulomb, dispersion and hydration effects appeared to be interwined to affect the force. The presence of the observed strong non-classical induction force implied that energies of non-valence electrons of ions/atoms at the interface might be heavily underestimated in current theories and possibly just those underestimated energies of non-valence electrons determined Hofmeister effects.

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 contested^{1,2,3,4,5,6,7} and has been recently brought to the forefront of research^{8,9,10,11,12,13,14,15,16,17,18,19}. 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 charges^1,2 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/m^{2 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/m^222,23,24,25, 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 surface^27,28,29, especially when considering the strong surface hydration force^30,31,32. At the shear plane, the charge density could reach 0.02–0.35 C/m^233,34,35, 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 field^37,38, 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 a_Na and a_Ca are the activities of Na⁺ and Ca²⁺, respectively, in bulk solution (mol/l) and N_Na and N_Ca 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 K_C 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; c_i(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 φ_1/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)

Using the data in Table 1, K_E = a_NaN_Ca/a_CaN_Na (K_E is the experimentally determined K) and K_C (Eq. 5) values could be calculated. The values of φ₀ in Table 1 indicated that the condition was satisfied for all K_C calculations. In Figure 1, it could be seen that the K_E were higher than the K_C 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.

Comparison between calculated () and experimental data ().

K_E is the experimental selectivity coefficient; K_C is the calculated selectivity coefficient based on the classical theory.

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 K_C+D is determined by both Coulomb and dispersion forces.

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

Figure 2 plots K_E/K_Cvs. φ₀, which is not constant. Therefore, the differences did not derive from the dispersion forces. The four red dots are K_E/K_C values with very large values of c⁰_Na/c⁰_Ca (red data in Table 1) and will be discussed below.

Figure 2.

K_E/K_Cvs. φ₀ (V).

Only under relatively high electrolyte concentrations is the dispersion force^{3,8,43,44,45,46} expected to become significant. Therefore, as the electrolyte concentration decreases, the difference between K_E and K_C decreases and at very low electrolyte concentrations, K_E/K_C will approach 1. However, Figure 2 showed that for decreasing electrolyte concentration (increasing surface potential), the difference between K_E and K_C 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 K_E/K_C < 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 K_E/K_C < 1. In Figure 2, however, K_E/K_C > 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 significant^{48,49,50,51,52,53,54,55,56,57,58}. Therefore, the difference between K_E and K_C 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 surface^1,2.

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 p_i(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 α_i* is the effective (excess) ionic polarizability in aqueous solution, ε_i and ε_w are the dielectric functions of the ith ion species and water, respectively, ε₀ is the dielectric constant in vacuum, V_i is the ionic volume, α_i is approximately equal to the intrinsic polarizability of the ith ion species, α_Ca = 0.4692 ų⁶⁰ and α_Na = 0.139 ų⁵⁹.

By substituting Eqs. 10 into Eq. 9, - can be calculated and by substituting the result into Eq. 8, K_C+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 Ca^2+/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.

Comparison between theoretical curves of K_C+I () and K_E ().

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 K_E and K_C 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 β_Ca and β_Na 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, β_Ca, β_Na 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 φ_0, φ(1/κ), β_Ca and β_Na are in Table 2.

Table 2.

The calculated φ₀, φ_1/κ, β_Ca and β_Na values based on the experimental data

From Table 2, β_Ca > 1 and β_Na < 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 β_Cavs. φ(1/κ)/φ₀ and β_Navs. 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/ε_water = 4.2 × 10⁸ V/m (ε_water = 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 β_Ca + β_Na = 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 K_C+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 K_C+NI ~ φ₀ relationship curve based on Eq. 17 is theoretical and not a fitting curve.

Figure 4.

Comparison between theoretical curves of K_C+NI () and K_E ().

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 Ca^2+/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 Ca^2+/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 α_i* and α_i 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

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 α_i* and α_i 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 ignored^1,2; 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.

vs. φ₀ (V) under different a⁰_Na/a⁰_Ca ratios (numbers aside dots are the values of a⁰_Na/a⁰_Ca).

Even though generally increased with φ₀, there was appreciable scatter in Figure 5. However, if the data were divided into three groups according to a⁰_Na/a⁰_Ca, the relationship of vs. φ₀ became more transparent: low values of a⁰_Na/a⁰_Ca over the range 1 –10; intermediate values over 10–100; and high values over 600–1000.

For low and intermediate values of a⁰_Na/a⁰_Ca, we found that = - sharply increased with φ₀. Since the electronic structures for Na⁺ and Ca²⁺ are, respectively, 1s²2s²2p⁶ (with a static polarizability α_Na = 0.139 ų⁵⁹) and 1s²2s²2p⁶3s²3p⁶ (with a static polarizability α_Ca = 0.4692 ų⁶⁰), 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 a⁰_Na/a⁰_Ca. A larger a⁰_Na/a⁰_Ca 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 a⁰_Na/a⁰_Ca, more Na⁺ might have random quantum fluctuations directionally enhanced. As a result, = - decreased.

Even with very high a⁰_Na/a⁰_Ca 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 kosmotropic^1,2). 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 preference^1,2.

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.

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

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.

Author Contributions

X.L. and H.L. performed the data analysis and modeling and wrote the main manuscript text. R.L. prepared Table 1–3. J.N. and D.X. supervised the project. L.W. made a thoroughgoing revision of the manuscript. X.L. and H.L. contributed equally to this work. All authors reviewed the manuscript.

Competing interests

The authors declare no competing financial interests.