Observation of significant steric, valence and polarization effects and their interplay: a modified theory for electric double layers

Abstract

Presented electric double layer theory achieves comparable results with experiments by including steric, polarization and valence effects and is applicable to all ions contrasting the classical theory only to hydrogen ions, which thus capably describes the selective adsorption on charged surfaces and is expected to clarify specific ion effects.

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The array of charged particles and oriented dipoles can be called electric double layer (EDL),¹ which is believed to exist at every interface and represents a focusing topic in a wide range of subjects such as electrochemistry, polymer, biophysics as well as colloid, surface and soil science.^1,2 Since the beginning of the 20^th century, a number of EDL theories have been developed successively,^3–7 and because of being able to give a quantitative description, the Gouy–Chapman model that was developed in 1910s has become one of the most widely used EDL theories.^3,4

The Gouy–Chapman model with the Poisson–Boltzmann equation as the mathematical basis was born with significant flaws that can result in flawed results, however. Only the ion-surface Coulomb interactions have been taken into account in this model.⁸ As indicated by the charge density analyses, soil particles,⁹ proteins,¹⁰ membranes¹¹ and a variety of other systems^12–14 often establish strong electric fields (usually 10⁸–10⁹ V m⁻¹) at their surfaces. As a result, the adsorbed ions, especially those with relatively large sizes, will be strongly polarized,^15,16 and the polarization effect should have been an integral part of interaction forces.¹⁷ The reason of neglecting polarization effect is due to that all ions in the Gouy–Chapman model are assumed as point-like charges albeit with consideration of the sign for these charges (+/−).^18–20 Whereas the ion exchange experiments¹⁷ clearly showed that different ions are selectively adsorbed on the charged surface, which seems contradictory to the point-like charge approximation that ignores the difference of the various ions. Another consequence of this approximation is that the amount of ions adsorbed on the charged surface can be overwhelmingly more than that observed experimentally.¹⁹

Owing to the aforementioned deficiencies, the Gouy–Chapman model has been subjected to a number of revisions and improvements.^19–23 The correction by inclusion of the steric effect was made by Borukhov et al.,^19,20 choosing a lattice-gas version where each lattice site was occupied by at most one ion. This provides a good solution to the adsorption of large ions (e.g., H₃PW₁₂O₄₀ with an estimated size of 10 Å). Apparently, the ion sizes we are usually concerned about are substantially smaller. The polarization effect was also considered by Boström et al.;²³ nonetheless, the ion polarizabilities that are significantly dependent on the electric field strength have to be derived from ab initio calculations. In this study, we proposed a modified EDL theory accounting for the steric, polarization and valence effects, which was testified to show obvious improvement as compared to the classical theory and good agreement with experimental observations. We then evaluated the respective contributions of the three effects for ion adsorption on the charged surface. Thus, we demonstrated that the steric effect is indispensible to describe all ions with outer electrons, and it is the synergistic interplay of the steric, polarization and valence effects that distinguishes the various electron-inclusive ions. Using this modified theory, the selective adsorption process of the various ions can be quantitatively characterized.

The adsorption of two different ions on the charged surface is a competitive process, which can be described by selectivity coefficient (K_i/j),

where a is the activity of ions in bulk solution, N is the amount of ions adsorbed on the charged interface, 1/κ is the thickness of EDL, S is the surface area of the charged particle, and f(x) is the distribution of the ion concentration. Here the selectivity coefficient (K_i/j) is defined similarly to that of the Vanselow coefficient (K_V) specific for univalent ions.²⁴K_i/j > 1 means that the i ion has a stronger adsorption capability than the j ion; vice versa.

According to the classical theory,²⁵ the Coulomb force is the sole source to drive the adsorption process, and thus the interaction energy w(x) is written to be,

w(x) = ZFφ(x) (2)

In eqn (2), Z is the ion valence and F is Faraday constant, whereas φ(x) represents the potential distribution in the EDL (x = 0 − 1/κ), with φ₀ corresponding to the surface potential (x = 0). w̄₀ is the Coulomb energy required to carry one mole ion from x = 1/κ to x = 0, and according to the previous results^9,26,27 it equals,

w̄_0i = Z_iFφ₀ = 2RT ln(ma_iS/N_iκ) (3)

Accordingly, the concentration distribution function (f(x)) and selectivity coefficient (K_i/j) of the classical theory can be expressed as,²⁸

The analytical solution of the Poisson–Boltzmann equation demonstrates m = 2 for 1 : 1 + 1 : 1 mixed electrolytes and for 1 : 1 + 2 : 1 mixed electrolytes,^9,17 wherein f⁰₊ and f⁰₂₊ refer to the concentrations in bulk solution for the univalent and divalent cations, respectively.

With use of the surface potential φ₀ derived from eqn (3) and (5) can be further transformed to,

where Ω = mSa_i/N_iκ.

The classical theory (eqn (6)) is tested for describing the binary exchange reactions on the montmorillonite surface. It can be seen from Fig. 1 that the selectivity coefficients (K_i/j) predicted by the classical theory are far less than those of experimental observations.²⁹ This is due to that the Coulomb force alone cannot account for the adsorption process of electron-inclusive ions.

Fig. 1 The selectivity coefficients (K_i/j) of the classical theory as function of Ω (eqn (6)) for (a) Na–Ca and (b) Na–Mg exchange processes on the montmorillonite surface (S = 103.0 m² g⁻¹).

It is known that the surface charges can establish a strong electric field (usually 10⁸–10⁹ V m⁻¹) towards any common electrolyte solutions (e.g., NaCl or NaNO₃ + Mg(NO₃)₂ mixed solutions) with a depth of several nm, and this will cause serious polarization to adsorbed ions. As a result, the electron cloud configurations of these ions are altered to a degree depending on the distance to the charged surface,^15,16 see the Na⁺ ion in Fig. 2 for an example. That is, the polarization effect should be included in the calculations of interaction energy,

where p̄ and Ē stand for the ionic dipole moment and electric field in diffusion layer, respectively.

Fig. 2 Schematic representation of the competitive adsorption of the hydrogen (H⁺) and sodium (Na⁺) ions on the charged surface, where the approximate radii of H⁺ and Na⁺ are given (unit in Å).

With consideration of the steric and polarization effects, the interaction energy of ions with the charged surface equals,

where φ_r corresponds to the potential with a distance of r (radius of the ion) from the charged surface. That is, the steric effect has been accounted for by use of φ_r instead of φ₀ as in the classical theory.

Then the selectivity coefficient (K_i/j) of the modified theory can be given as,

Below is an equivalent to eqn (9), and this transformation is similar to that of eqn (6),

In eqn (10), , where ε and I (I = 0.5Σa_iZ²_i) refer to the dielectric constant and ion strength of the electrolyte solution, respectively.

In contrast to the classical theory, the modified theory (eqn (9) and (10)) includes the polarization and steric effects and therefore is expected to be able to describe and distinguish the competitive adsorption of the various ions. As is known to us, the classical theory is applicable to the hydrogen ion (H⁺).³⁰ H⁺ consists of merely one proton in the nucleus with no outer electrons, and hence no polarization will be resulted in (p̄ = 0). Compared with other ions, the radius of H⁺ is so slight that can almost be neglected; that is, φ_r(H⁺) ≈ φ₀ (Fig. 2), and as a result of eqn (8), we can derive that Π_H = 1. The activities (a) and adsorption quantities (N) of Na⁺, Ca²⁺ and H⁺ listed in Table 1 are available from the ion exchange equilibrium experiments.^17,31 When the i ion is selected as H⁺, eqn (10) can be used to calculate the Π_Na and Π_Ca values, equal to 0.19 and 0.27, respectively (Table 1). Both of them are far less than the Π_H value of H⁺. This indicates that steric effect should be the most significant factor for describing the electron-inclusive ions. The Π values of other ions can be determined in a similar manner.

Table 1 Π _Na and Π_Ca values calculated on basis of ion activities (a) in solution and amounts (N) of ions adsorbed on the montmorillonite surface^a

a Na	a Ca	a H	N Na	N Ca	N H	Π Na	Π Ca
a Units of a and N are mmol L^−1 and mmol g^−1, respectively. b The average ΠNa and ΠCa values to be used in the text are given in parentheses.
2.31	0.07	0.002	0.16	0.21	0.55	0.20	0.28
2.21	0.05	0.001	0.17	0.21	0.56	0.19	0.28
3.78	0.30	0.01	0.13	0.21	0.59	0.18	0.26
						(0.19)^b	(0.27)^b

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The selectivity coefficient for Na⁺ and Ca²⁺ adsorbed on the charged surface (K_Ca/Na) is then calculated using the obtained Π_Na and Π_Ca values,

K_Ca/Na = Ω^−0.65 (11)

Fig. 3a clearly shows that the modified theory is in good agreement with experimental results. The exponential term equals −0.65 and almost reproduces the experimental value (−0.66).¹⁷ In addition, the correlation coefficient 0.96 implies that the modified theory describes satisfactorily the selective adsorption of different ions on the charged surface. On the other hand, the modified theory can be used to obtain the Π values by fitting the experimental data. For the selective adsorption of Na⁺ and Mg²⁺ (Fig. 3b), the fitted equation is given as,

K_Mg/Na = 1.00Ω^−0.62 (R²= 0.95) (12)

Fig. 3 The selectivity coefficient corresponding to the modified theory as a function of Ω (eqn (10)) for (a) Na–Ca and (b) Na–Mg exchange processes on the montmorillonite surface (S = 103.0 m² g⁻¹).

The Π_Mg value is calculated to be 0.25. The correlation coefficient equals 0.95 and this further validates the modified theory for describing the selective adsorption on the charged surface.

According to the Boltzmann distribution function of the modified theory:

the corresponding Poisson–Boltzmann equation is expressed as,

On basis of the Π values obtained thus far, the Boltzmann distribution functions for the discussed ions have been derived:

f_Na(x) = a_Nae^{−0.19Fφ(x)/RT} (15)

f_Mg(x) = a_Mge^{−0.50Fφ(x)/RT} (16)

f_Ca(x) = a_Cae^{−0.54Fφ(x)/RT} (17)

f_H(x) = a_He^−Fφ(x)/RT (18)

Among these ions, the hydrogen ion (H⁺) is also applicable to the classical theory. It can be seen that H⁺ is significantly different from the ions that have outer electrons. This further evidences the indispensability of steric effect for the electron-inclusive ions, in satisfactory agreement with the above discussions.

In addition, the adsorption process of the various ions on the charged surface can be distinguished by the modified theory. The size difference between two electron-inclusive ions is not so obvious as compared to H⁺, and accordingly the steric difference among them is less dramatically. For example, the valences of Ca²⁺ and Mg²⁺ are identical, and the difference between their exponential terms (−0.04, see eqn (16) and (17)) is a coupling result of the steric and polarization effects. Ca²⁺ (r ≈ 0.99 Å) has a larger steric hindrance than Mg²⁺ (r ≈ 0.66 Å) while its adsorption is facilitated by larger polarization owing to a more apt deformation of electron clouds. The slightly larger interaction energy for Ca²⁺ indicates that the polarization rather than steric effect is slightly preferred as compared to Mg²⁺. eqn (13) shows that the interaction energy is exponentially proportional to the valence. Ca²⁺ (r ≈ 0.99 Å) and Na⁺ (r ≈ 0.98 Å) have very close radii and their steric effects can be considered identical. As seen from eqn (15) and (17), the exponential term of Ca²⁺ is more than twice as that of Na⁺ by −0.16. The additional energy (−0.16) should be caused totally by polarization difference. Accordingly, for the case of Ca²⁺vs. Na⁺ the difference of polarization effect is almost equivalent to, or even larger than, that of valence effect, because Na⁺ is also polarized under such strong electric fields; that is, the interaction energy of Na⁺ (−0.19, eqn (15)) should also include the contribution from polarization effect but not exclusively from valence effect. Accordingly, for the electron-inclusive ions each of the steric, polarization and valence effects plays a definite role during the adsorption on the charged surface. It is the synergistic interplay of the three effects that distinguishes the various ions and makes the theoretical results comparable to experimental observations. Because of the efficient differentiation of the various electron-inclusive ions, the modified theory is expected to helpfully clarify specific ion effects.

Acknowledgements

This work was supported by the Major Science and Technology Program for Water Pollution Control and Treatment (2012ZX07104-003), Natural Science Foundation Project of CQ CSTC (2011BA7001), 973 Program (2010CB134511), the Fundamental Research Funds for the Central Universities (XDJK2011C008) and Special Fund for Basic Scientific Research of Central Colleges (SWU113049).

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