For a binary electrolyte system, ions in the diffuse layer should abide by the Boltzmann distribution,

where (in units: mol· L^{− 1}) is the ionic concentration in bulk solution, φ (x) (in unit V) is the potential in the electric double layer, and Z_i is the ionic valence, F (in units: C· mol^{− 1}) is the Faraday constant, R (in unit: J· mol^{− 1}· K^{− 1}) is the gas constant, and T (in unit: K) is the absolute temperature.

The mean ion concentration in the diffuse layer equals

where N_i (mol· g^{− 1}) is the amount of the i ion that interacts with the charged particle surface, S (dm²· g^{− 1}) is the surface area of the charged particle, and κ (dm^{− 1}) is the Debye– Hü ckel parameter.

In our previous work, ^[36] the potential distribution φ (x) of the binary electrolyte system containing monovalent and divalent counterions (e.g., NaCl + CaCl₂) was derived on the basis of the nonlinear Poisson– Boltzmann (PB) equation,

where

In Eq. (4), φ ₀ represents the potential at the original plane of the diffuse layer (or Stern plane), and here is regarded as surface potential (see Fig. 1).

Introducing the potential distribution function into Eq. (1) and then combining Eq. (2), we arrived at

with

In Eq. (5), ε is the dielectric constant of water (8.9× 10^{− 10} C²· J^{− 1}· dm^{− 1}), F (C· mol^{− 1}) is the Faraday constant, m, also referred to as in some papers, ^[37] is written as , where and refer to the concentrations of univalent and bivalent ions, respectively.^[31]

Then the following expressions can be derived for the binary system (i and j)

As a result, the surface potential of the classical theory can be given as,

Equation (8) is the derivative of the original PB equation where only Coulomb interaction occurs between ions and surface.

It is apparent that ionic correlation affects the ion-exchange process and hence this factor should be taken into account:^{[38, 39]}

where μ _i0 is the standard chemical potential, while , , and are respectively the ionic concentration, ionic activity, and ionic activity coefficient in the bulk solution. That is, the RT ln term results from ionic correlation in a non-ideal system.^[38] The ionic activity is the effective ionic concentration in solution and there are several empirical formulae of calculating the ionic activity and ionic activity coefficient.^[40]

The chemical potential in the diffuse layer can be written as

where , ã _i, and are respectively the concentration, activity, and activity coefficient in the diffuse layer.

According to the previous studies, ^{[31, 37, 41]} ionic activity represents a function of the Coulomb energy , and the activity coefficient in the diffuse layer equals^[37]

When the ion exchange process reaches equilibrium, we have , and so the combination of Eqs. (9)– (11) leads to

In consideration of ionic correlation, the PB expression is modified into

Expressions similar to Eq. (7) can be derived as

where

As a result, surface potential with the inclusion of ionic correlation is

It can be seen from the above descriptions that the ionic activity in solution has been taken into account by considering ion size and ionic dispersion potentials of ion– ion and ion– interface interactions.^[28]

As is well known, strong electric fields (10⁸ V· m^{− 1} ∼ 10⁹ V· m^{− 1}) often exist at the surfaces of colloidal particles.^{[31, 32, 42– 50]} As reported previously, the strongly polarizable ions can be adsorbed on the air– water and oil– water interfaces thus affecting the interfacial tensions.^[23] For the negatively charged colloidal particles, the ions at their interfaces should be polarized more significantly, ^{[51, 52]} and the addition of polarization energy is absolutely needed. Therefore equation (14) is converted into the following equations:^[29]

The above equation can be further transformed into^[53]

with

where and Ẽ refer to the mean dipole moment in the diffuse layer and electric field strength, respectively; φ _1/κ is the potential at x = 1/κ (the upper limit of the electric double layer) and converged to the null point, as verified by the potential distributions in Section 3.

The exponential term in Eq. (18) equals

Then the coefficients β _i and β _j are

The coefficients β _i and β _j represent functions of the dipole moment difference of the i and j ions. Based on Eqs. (18) and (20), further correction to the surface potential by considering ion polarization turns Eq. (16) into

At low ionic concentrations, the effect of ionic activity can be neglected, that is, should be approximately equal to given below

The specific ion effects due to ion polarization are embodied with coefficients β _i and β _j, which are sensitive to the choice of ion pairs because ion polarizabilities in the diffuse layer are probably different.^[29] Note that refers to the surface potential corrected with ion polarization but not with ionic correlation. The PB equation cannot be resolved analytically for the mixed electrolyte solutions (e.g., NaCl + CaCl₂) in the presence of the polarization effect.^[54] As an alternative, we consider the polarization effect on the basis of the mean energy that is obtained directly from the PB equation.

The coefficients β _Na and β _Ca can be derived from Na⁺ – Ca²⁺ exchange equilibrium experiment, ^[43] and their empirical expressions are and , where I is the ionic strength.^[31] The difference between β _Na and β _Ca reflects the polarization difference between Ca²⁺ and Na⁺ . From the empirical expressions and Eq. (21), the experimental polarization difference between Ca²⁺ and Na⁺ in the electric double layer can be

On the other hand, according to the classical theory, the polarization effect for a given ion and the polarization difference between Ca²⁺ and Na⁺ are estimated by

where

ε ₀ is the dielectric constant in a vacuum and ε _i = 1 + 4π α _i/V_i, ^[55] is the effective ion polarizability in the solution, and ε _i and ε _w are the relative dielectric functions of the i ion and water, respectively, V_i is the ionic volume, and α _i is the static polarizability of the i ion, with α _Ca = 0.4692 Å ³ and α _Na = 0.139 Å ³, respectively.^{[55, 56]} According to the Ca²⁺ /Na⁺ exchange equilibrium results with different ionic strengths, ^[43]φ ₀ can be calculated by use of Eq. (22). The potential at the distance 1/κ (φ _1/κ ) is determined through the potential distribution of Eq. (3).^[36]

Figure 2 shows the polarization differences between Ca²⁺ and Na⁺ ions, obtained by the experimental electric double layer and classical theory. Owing to the neglect of ion polarization, the classical theory assumes that the dipole moments of Ca²⁺ and Na⁺ ions are close to each other and in addition, their difference does not show an observable change with the increase of ionic strength. These are apparently in discrepancy with the experimental observations. On the contrary, the electron configuration analysis is perfectly in line with the experimental observations in that there exists a remarkable difference between Ca²⁺ and Na⁺ dipole moments and the polarization effect of the former should be considerably larger (Fig. 2). As mentioned above, the strong electric fields (10⁸ V· m^{− 1} ∼ 10⁹ V· m^{− 1}) often exist at the interfaces of colloidal particles and in many other systems. Under such strong electric fields, the dipole moments of interacting counterions should be significantly affected and exhibit a considerable increase, and the increment differences between two different counterions can be substantial: e.g., Ca²⁺ and Na⁺ investigated in this work. As compared with Na⁺ , the electron clouds of Ca²⁺ are obviously softer and accordingly Ca²⁺ corresponds to a much stronger polarization effect when interacted with the negatively charged surface, which should be responsible for arousing the Hofmeister effects among the various ions. In addition, the polarization difference between Ca²⁺ and Na⁺ shows an increase with the decrease of ionic strength, and this trend becomes more conspicuous at relatively small ionic strengths (I ≤ 0.04 mol· L^{− 1}, see Fig. 2). Neither these experimental results nor electron-configuration results can be properly explained by the classical theory. The classical theory seriously underestimates the effective polarization of counterions. As a matter of fact, the various counterions near the charged surfaces should be polarized to a different degree, depending on their electron configurations.

Fig. 2.

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	Fig. 2. Polarization differences between the Ca2+ and Na+ ions (, in Debye), each as a function of the root square of ionic strength , in mol1/2· L− 1/2) in solution.

It is worth noting that the water should be also polarized by a strong electric field, ^[57] but which is obviously less than the charge dipole such as Ca²⁺ and Na⁺ ions; ^{[58– 60]} that is, the water solvent has a limited influence on the interactions between ions and surface charges that are mainly responsible for specific ion effects. Moreover, the influence of the ionic solution has already been included in the surface potential expression by ionic correlation (for the details, see Subsection 2.2.).

Si⁴⁺ and Al³⁺ are the principal constituents for most clay minerals, and isomorphous substitutions with metal ions of almost the same radii create permanent negative charges, e.g., usually the substitutions of Mg²⁺ /Al³⁺ for montmorillonite. It seems very difficult for lattice cations to be replaced by ions from the surfaces and bulk solutions, especially Ca²⁺ and Na⁺ that have substantially larger radii than Si⁴⁺ and Al³⁺ . Table 1 lists the surface potentials of montmorillonite (mean size: 1480 nm) and rutile titania (TiO₂, mean size: 25 nm), ^[32] that have been calculated by the various schemes in Section 2, i.e., the classical theory ( with only Coulomb interaction) as well as those with the addition of ion polarization , ionic correlation , and both effects . It shows that inclusion of only ion polarization can achieve comparable results to those with considering both ion polarization and ionic correlation ; instead, the ionic correlation in bulk solutions seems to play a minor role, because the values show severe deviations from those of whereas they are close to the data of the classical theory . A good example is montmorillonite with I (ionic strength) = 2.5 mmol· L^{− 1}, where , , , and are equal to − 201.1, − 207.5, − 106.3, and − 109.7 mV, respectively. It can be found that the inclusion of ion polarization in and causes a dramatic decrease to surface potential and results in only approximately half the value from the classical theory . That is, the ion polarization should be of vital importance in all the presented cases (Table 1). A careful analysis of ionic correlation will be performed later.

Polarization and gathering of counterions in proximity to the negatively charged surfaces cause an enhancement of the shielding effect on the surface charges, which further leads to a substantial decrease to surface potential. It is in good agreement with the predicted result of ion polarization. Instead, few researchers^{[55, 56, 61– 63]} insist that Hofmeister effects should result from the dispersion energy and ionic steric effect that are known to be significant only in the case of relatively high ion concentration. This seems to be a discrepancy with the fact that Hofmeister effects are more evident at low ionic concentrations due to the more pronounced enhancement of ion polarization in the electric double layer. With inclusion of ion polarization, the surface potentials ( and ) show an obvious decrease as compared with those from the classical theory (see Table 1).

Table 1.

Table 1.

Table 1. Values of ionic strength (I) of solution containing Na+ and Ca2+ ions and surface potentials (φ 0) for montmorillonite and rutile titania (TiO2) suspended in these solutions that are obtained by the various schemes in Section 2. Units of ionic strength and surface potential are mmol· L− 1 and mV, respectively. Ia Montmorillonite 2.52 − 201.1 − 207.5 − 106.3 − 109.7 2.72 − 183.2 − 193.1 − 97.0 − 102.3 5.35 − 140.7 − 154.4 − 75.3 − 82.7 Rutile titania 14.30 − 96.2 − 114.7 − 52.5 − 62.6 14.98 − 89.4 − 108.0 − 48.7 − 58.9 15.03 − 80.2 − 99.0 − 43.7 − 54.0 16.69 − 58.9 − 78.7 − 32.2 − 43.0 18.52 − 31.5 − 52.0 − 17.2 − 28.4 a Data of ionic strengths are cited from Ref. [32].

Table 1. Values of ionic strength (I) of solution containing Na+ and Ca2+ ions and surface potentials (φ 0) for montmorillonite and rutile titania (TiO2) suspended in these solutions that are obtained by the various schemes in Section 2. Units of ionic strength and surface potential are mmol· L− 1 and mV, respectively.

Figure 3 shows the variations of surface potentials for illite, i.e., , , , and with the root square of ionic strength , ^[43] corresponding to a much wider range of ionic concentrations than those shown in Table 1. Being in accordance with the results of montmorillonite and rutile titania (TiO₂), the surface potentials of illite decrease with the increase of ionic strengths, and especially, a more pronounced change rate on the surface potentials has been detected at relatively low ionic strengths. The logarithm functions can be established between the calculated surface potentials and root square of ionic strengths as follows:

The correlation coefficients (R²) given in Eqs. (27)– (30) indicate that these fitted functions describe reasonably the relationships between surface potentials (, , , and ) and ionic strengths (I). The correlation coefficients from the classical theory with ion polarization correction are close to 1.0, which demonstrates the fine prediction results for the two surface potentials ( and ), while those relating to the ion correlation correction ( and ) are somewhat less satisfactory, which may be caused by the charge oscillation at high ion concentration. Nonetheless, the changed trends of these two surface potentials (Eqs. (28) and (30)) are correctly described and also in agreement with the other two cases (Eqs. (27) and (29)). According to the versus results in Table 1 and Fig. 3, the surface potentials will be altered by 20.0% ∼ 50.0% when ion polarization is included. For surfaces with variable charges, charge and surface potential may be reversed at high ionic strengths; ^{[56, 64]} however, the montmorillonite and illite systems under study possess permanent charges resulting from isomorphous substitutions, and the surface charge densities will not be easily altered by ion strength, polarization, and other factors. Accordingly, charge oscillation may take place occasionally, while charge reversion is unlikely to occur.

The contribution of ion polarization can be evaluated by the difference between Eqs. (30) and (28) and expressed as

Therefore, ion polarization in the electric double layer strongly affects the surface potential and its contribution is quantitatively evaluated. The surface potential decays gradually within ionic solution, and then the potential distribution φ (x) can be described by the nonlinear PB equation.^[36]

Fig. 3.

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	Fig. 3. Variations of surface potentials (φ 0, in unit V) of illite in ionic solutions with root square of ionic strength (, in mol1/2· L− 1/2).

The surface potential decays gradually within ionic solution, and the potential distribution φ (x) does not need to additionally consider the ion polarization because this factor has already been accounted for during the calculation of surface potential. In order to investigate the relationship between potential distribution and ionic concentration, two ionic strengths (0.01 mol· L^{− 1} and 0.6 mol· L^{− 1}) are selected for the NaCl + CaCl₂ mixed solution . The various surface potentials are calculated by use of Eqs. (27)– (30), and the corresponding variations of potential distribution φ (x) with decay distance x which are derived from Eq. (3) are plotted in Fig. 4. The ϕ (x)^H and φ (x)^HI potentials descend quickly with the increase of decay distance x, because the counterions with a larger distance away from the charged surface are dramatically less polarized. As a result, Hofmeister effects due to ion polarization show a pronounced decrease with the increase of decay distance (x).

The φ (x)^H and φ (x)^HI potentials approach to the null point at x = 2.40 nm for I = 0.6 mol· L^{− 1} whereas at x = 13.3 nm for I = 0.01 mol· L^{− 1}, indicating an apparently faster decay rate at higher ion concentrations. In addition, at a given ionic strength, the φ (x)^H and φ (x)^HI potentials that include ion polarization vanish at the identical distances as the φ (x)^I potential. This demonstrates that the forces generated by the surface charges are independent of ion polarization. At low concentrations (e.g., I = 0.01 mol· L^{− 1} in Fig. 4(a)), ion polarization makes an important contribution to the surface potential and potential distribution, and the inclusion of only ion polarization in the electric double layer is already sufficient to obtain reasonable results as indicated by the comparable φ (x)^H and φ (x)^HI data (Fig. 4), which accord with the surface potential data shown in Table 1 and Fig. 3. However, at relatively high concentrations (e.g., I = 0.6 mol· L^{− 1} in Fig. 4(b)), the φ (x)^H results seem to deviate significantly from those of φ (x)^HI, and it suggests that only the ion polarization effect cannot correctly account for the surface potential nor potential distribution, which will be discussed in the following.

As indicated in Fig. 3, surface potentials including ionic correlation in bulk solution change inversely with the increase of ionic strength. Ionic correlation affects the distribution of counterions near the charged surface and further the surface potentials. At low ionic concentrations, surface potentials that include only ionic correlation have obvious deviations from those of but are rather close to the classical surface potentials . This indicates that at low ion concentrations, ionic correlation plays a minor role in determining the surface potential. However, when ion concentrations gradually increase, the values begin to diverge from those of , and instead the and results become gradually closer to each other, implying a strengthening role played by ionic correlation.^[35] In a manner similar to Eq. (31), the contribution of ionic correlation to surface potential can be assessed by the difference between Eqs. (30) and (29), and expressed mathematically as

It can be found that at high ion concentrations, ionic correlation affects surface potentials even more apparently than ion polarization (Fig. 3). For example, at I = 0.6 mol· L^{− 1}, and . The former neglects ionic correlation and results in an obviously larger deviation from . Accordingly, ionic correlation is a factor that greatly affects surface potential, especially at high ion concentrations. At the same time, it is found that even at such high ion concentrations, ion polarization cannot be ignored ( for I = 0.6 mol· L^{− 1}), which is quite different from the effect of ionic correlation at low ion concentrations. In addition, this gives a good explanation to the very slight influence of ionic correlation on the data shown in Table 1, for all the investigated cases therein are of rather low ion concentration.

Figure 4 shows that the potential distribution with considering the ionic correlation [φ (x)^I] becomes obviously less with the increase of decay distance (x), and the decay rates of φ (x)^I are more dramatic at higher ion concentrations. As discussed in Subsection 3.2, at a given ionic strength, the φ (x)^I, φ (x)^H, and φ (x)^HI potentials vanish at the same distances that are strongly concentration-dependent, that is, these potentials are equal to zero at x = 2.40 nm for I = 0.6 mol· L^{− 1} while x = 13.3 nm for I = 0.01 mol· L^{− 1}, respectively. This is assertive evidence for the association of concentration-associated ionic correlation for surface potential with the potential distribution. Accordingly, ionic correlation in bulk solution should be included when determining the surface potential and potential distribution.

At rather low ionic strengths (I ≤ 0.01 mol· L^{− 1}), the polarization effect predominates during the determination of surface potential and potential distribution, whereas the effect of ionic correlation can be almost neglected (Figs. 3 and 4). With the increase of ionic strength, each of these two effects plays a definite role and it is their synergy that results in the accurate surface potential and potential distribution. Compared with the referenced data, the polarization effect is underestimated whereas ionic correlation overestimates surface potential. Therefore, there should exist a critical point on the curve where these two factors can be exactly counteracted, and for the illite system, the critical ionic strength (I_c) equals 0.16 mol· L^{− 1} (see Fig. 5), that is, at I < 0.16 mol· L^{− 1}, ; at I = 0.16 mol· L^{− 1}, , and at I > 0.16 mol· L^{− 1}, . The presence of critical ionic strengths (I_c) can be safely extended to other systems albeit the exact values may change accordingly. The ion polarization and ionic correlation make a larger contribution to the surface potential at very low and high concentrations, respectively (Fig. 5). What should be noteworthy is that the effect of ionic correlation can be almost neglected at rather low ion concentrations, while ion polarization, although it makes less of a contribution to the ionic correlation at high concentrations, is still indispensable for the surface potential and potential distribution [φ (x)^HI] and hence should be considered across the entire concentration range (see Fig. 5).

Fig. 4.

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	Fig. 4. Variations of the potential distribution (∣ φ (x)∣ , in V) with decay distance (x, in nm) at two different values of ionic strength (I): (a) 0.01 mol· L− 1 and (b) 0.6 mol· L− 1.

Fig. 5.

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	Fig. 5. Illustration of region divisions of surface potentials (, in V) of illite according to the relative contributions of ion polarization and ionic correlation.