Letter pubs.acs.org/JPCL
Effect of Hydration of Ions on Double-Layer Repulsion and the Hofmeister Series Haohao Huang*,† and Eli Ruckenstein*,‡ †
School of Material Sciences and Engineering, South China University of Technology, Guangzhou 510640, China Department of Chemical and Biological Engineering, State University of New York at Buffalo, Buffalo, New York 14260, United States
‡
ABSTRACT: The repulsive force between two electrically charged parallel plates immersed in an electrolyte solution is calculated by taking into account the hydration of ions and the volume exclusion entropy thus generated. The ion specificity, ignored in the traditional DLVO theory, is thus accounted for. It is found that the hydration of ions affects, under some conditions, the repulsive force between the two plates and that the sequence of its strength follows closely the famous Hofmeister series for the salting out (precipitation) of proteins.
SECTION: Glasses, Colloids, Polymers, and Soft Matter
T
positions available to locate a hydrated ion is smaller than that when they are considered of zero size. As a result, the entropy of ions in the solution becomes smaller than that in the latter case. The fraction of sites occupied by hydrated ions is ∑j=1,2 cjτj/n, and the fraction that remains free is 1 − ∑j=1,2 cjτj/n, where cj is the concentration of ion j, τj is the number of sites occupied by the water molecules that hydrate ion j, and n is the total number of sites per unit volume (considered as the number of water molecules per unit volume). The probability Pi(x) of finding τi neighboring sites involved in a hydrated ion is approximately provided by (1 − ∑j=1,2 cjτj/n)τi, and the corresponding decrease in entropy can be evaluated as kB ln(1 − ∑j=1,2 cjτj/n)τi, where kB is the Boltzmann constant. This provides the ions with an excess chemical potential −kBT ln(1 − ∑j=1,2 cjτj/n)τi when compared with the chemical potentials of an ideal solution. One can therefore write that the excess free energy due to the decreased entropy is given by3
he double-layer interaction between colloidal particles is usually calculated on the basis of the traditional Derjaguin−Landau−Verwey−Overbeek (DLVO) theory.1 In this theory, the distributions of the electrostatic potential and ionic concentrations are obtained on the basis of the Poisson− Boltzmann equation, and the ions are assumed to be of vanishing sizes. The solvent is considered as a continuum, and its structure is neglected. In its improved version,2 the free energy of formation of the double layer contains three contributions, an electrostatic field energy, a surface chemical free energy, and the difference between the entropic free energies of ions and water in the double layer and under the conditions in the reservoir for the same ions and water molecules. The latter free energies were calculated as for an ideal mixture. Because the ion specificity is absent in the above contributions, this effect is disregarded in the previous two treatments of the double-layer interaction.1,2 However, in addition to the screening of the electrical field by ions, which is accounted for by the DLVO theory, their size due particularly to their hydration should also play a role in the free energy of formation of the double layer.3,4 It will be shown that the volume exclusion entropy generated by hydration can explain the ion specificity in the double-layer force and that the strength of the repulsive force follows closely the Hofmeister series. The Hofmeister series provides a sequence of ions in the order of their ability of salting out of proteins.5 Such ion specificity was also observed in a number of experiments6−9 regarding the colloidal repulsive force. Being hydrated, the ions acquire a relatively large size. In addition, a hydrated ion cannot occupy the sites already occupied by other hydrated ions. Consequently, the number of © 2013 American Chemical Society
Fv = −kBT
∫0
D
⎛ ∑ ci ln⎜⎜1 − ⎝ i = 1,2
cjτj ⎞ ⎟ dx n ⎟⎠ τi
∑ j = 1,2
(1)
where T is the temperature in K and D is the distance between the two parallel plates. By taking into account the volume exclusion entropy, the free energy F per unit area of the electrical double layer is given by Received: September 11, 2013 Accepted: October 17, 2013 Published: October 17, 2013 3725
dx.doi.org/10.1021/jz401948w | J. Phys. Chem. Lett. 2013, 4, 3725−3727
The Journal of Physical Chemistry Letters
Letter
σ (D) εε0 ⎛ dφ ⎞2 ⎜ ⎟ dx − 2[ φs dσ + φs(∞)σ(∞)] F= 0 σ (∞) 2 ⎝ dx ⎠ ⎞ D⎛ c ⎜⎜ci ln i − ci + ci ,b⎟⎟ dx + kBT ∑ ci ,b ⎠ i = 1,2 0 ⎝
∫
D
μwe = −
∫
− kBT
∑ i = 1,2
+ +
∫0
D
∑ i = 1,2
∫0
cjτj ⎡ ⎢ 1−∑ n ci ln⎢ cj ,bτj ⎢⎣ 1 − ∑ n
) ⎤⎥ dx τ⎥ ) ⎥⎦
( (
e c w(μwe − μw,b ) dx +
∑ i = 1,2
∫0
τi
∫0
∫μ (0) e 1
c1 dμ e − cw 1
μ2e (c 2)
∫μ (0) e 2
c2 dμ e cw 2
(3)
where μei is the excess chemical potential of hydrated ion i due to the volume exclusion. Calculations have shown that the term ∫ D0 cw(μew − μew,b) dx is negligible. The distribution of the electrical potential can be calculated using the Poisson equation
∫
D
μ1e (c1)
i
d2φ ρ =− εε0 dx 2
D
ci(αi − αi ,b) dx
(4)
where ρ is the charge density. The thermodynamic equilibrium of ions in the reservoir and the double layer provides the equation
D
ci Φi dx (2)
⎛ μi = μi + qiφ + kBT ln ci − kBT ln⎜⎜1 − ⎝
where φ is the electrical potential, ε the dielectric constant, ε0 the vacuum permittivity, φs the surface potential, σ the surface charge density, μew the excess chemical potential of water due to the excess chemical potentials of the anions and cations, αi the free energy of hydration of ion i, Φi the potential due to the other interactions between the plates and the hydrated ions, and subscript b indicates the values in the reservoir in which the two plates are immersed. In eq 2, the first term represents the electrical field energy, the second the surface chemical free energy,10 the third the difference between the entropic free energies of ideal solutions in the double layer and under the conditions in the reservoir for the same ions and water molecules, the fourth the difference between the free energies in the double layer and in the reservoir for the same ions due to the volume exclusion, the fifth the excess chemical potential of water, the sixth the difference between the free energies of hydration in the double layer and reservoir, and the seventh the other interactions between plates and hydrated ions. The first three terms in eq 2 were present in the traditional expression of the free energy of the double layer,2 but the other ones are new. At constant surface potential, the surface chemical free energy becomes −2σφs; at constant surface charge density, its contribution to the force is zero. Equation 2 also includes the difference between the free energies of formation of hydrated ions in the double layer and those in the reservoir. Over the distance D, the free energy of hydration is dominated by the interactions between the ions and water, and the difference αi − αi,b (hence, ∫ D0 ∑i=1,2 ci(αi − αi,b) dx) is zero. The interaction potential Φi depends on the nature of the surface. A hydrophilic surface will change very little the structure of water. In contrast, a hydrophobic surface will change its structure by decreasing the number of hydrogen bonds of each water molecule from 3.6 in the bulk to about 1.5 at the surface.11 The difference between the numbers of hydrogen bonds in a point near the hydrophobic surface and in the bulk generates a short-range attraction because the water molecules prefer the larger number of hydrogen bonds in the bulk. In addition, the van der Waals interactions may play a role in both cases. In the present Letter, we will consider only hydrophilic surfaces, and the van der Waals interactions (Φi) between the hydrated ions and the plates will be neglected. The change in the chemical potential of water due to the entropy decrease caused by the hydration of ions can be obtained using the Gibbs−Duhem equation
0
0
= μi + kBT ln ci ,b
⎛ − kBT ln⎜⎜1 − ⎝
∑ j = 1,2
∑ j = 1,2
cjτj ⎞ ⎟ n ⎟⎠
cj ,bτj ⎞ ⎟ n ⎟⎠
τi
τi
(5)
where μi is the chemical potential of ion i, μ0i its standard chemical potential, and qi its charge. Solving eqs 4 and 5 for appropriate boundary conditions, one obtains the distribution of the electrical potential. The repulsive force is the negative
Figure 1. Pressure between two parallel plates in solutions of sodium salts of F−, Cl−, Br−, I−, ClO4−, NO3−, Ac−, OH−, and H2PO4−. The electrolyte concentration is 0.1 M, and the temperature is T = 298 K. The values of τi are listed in Table 1. (a) At constant surface charge density for σ = 0.02C/m2; (b) at constant surface potential for φs = 0.05 V. 3726
dx.doi.org/10.1021/jz401948w | J. Phys. Chem. Lett. 2013, 4, 3725−3727
The Journal of Physical Chemistry Letters
Letter
Table 1. Ranges of the Hydration Number of Ions ions τ12,13
F−
Cl−
Br−
I−
Ac−
OH−
NO3−
H2PO4−
ClO4−
4−6
1.2−8.9
2.9−8
4.2−9.6
3−6.1
3−4
5.9−9
4.2−8.8
8−12
■
derivative of the free energy F with respect to the distance between the two plates. The above equations have been applied to the sodium salts of the anions F−, Cl−, Br−, I−, ClO4−, NO3−, Ac−, OH−, and H2PO4−. The repulsive force for the above electrolytes is plotted in Figure 1a and b. Unfortunately, the value of the hydration number of an ion depends on the experimental technique employed and was reported to vary in a wide range.12 Because no accurate hydration numbers are available, the arithmetic average values of the largest and the smallest acceptable reported values were employed in the calculations. The hydration number of the sodium cation considered in the calculations was taken as 4.12 The typical Hofmeister series for the ability of salting out of proteins decreases in the order H2PO4−, OH−, Ac−, F−, Cl−, Br−, NO3−, I−, ClO4−. The strength of the repulsive force for the distances calculated follows the order OH− < Ac− < F− ≈ Cl− < Br− < H2PO4− < I− < NO3− < ClO4−. The effects of ion specificity on the solubility of proteins and on the electrostatic repulsion between two plates are not completely the same. The difference might be due to the approximations employed in the calculations but may also be the result of the different quantities involved (salting out and repulsive force). Calculations carried out for numerous salt concentrations, surface charge densities, and surface potentials revealed that the order of anions remains the same. In addition, it was observed that for sufficiently large values of the above parameters, the specificity is strong (large differences between the repulsive forces for successive ions) but that at sufficiently small values, no specificity is present and the DLVO theory is recovered. Calculations have been also performed for monovalent cations, with the conclusion that their order almost coincides with the experimental one. One of the reviewers drew our attention to a group of papers by Levin et al. concerning the surface tension of electrolyte solution14,15 and ion specificity in colloid stability.16 In those papers, the authors concluded that ion specificity follows the Hofmeister series and obtained good agreement with experiments. There is a major difference between the present approach and that of Levin et al. In the present theory, the specificity is a result of the excess (negative) entropy generated by hydration of the ions. In Levin’s approach, short-range interactions between the plates and the hydrated ions (hardcore repulsion, dispersion interaction) as well as the cavitational energy provide the specificity.
■
REFERENCES
(1) Verwey, E. J.; Overbeek, J. Theory of the Stability of Lyophobic Colloids; Elsevier: Amsterdam, The Netherlands, 1948. (2) Overbeek, J. The Role of Energy and Entropy in the Electrical Double Layer. Colloids Surf. 1990, 51, 61−75. (3) Ruckenstein, E.; Schiby, D. Effect of the Excluded Volume of the Hydrated Ions on Double-Layer Forces. Langmuir 1985, 1, 612−615. (4) Manciu, M.; Ruckenstein, E. Lattice Site Exclusion Effect on the Double Layer Interaction. Langmuir 2002, 18, 5178−5185. (5) Hofmeister, F. Zur Lehre von der Wirkung der Salze. Zweite Mittheilung. Arch. Exp. Pathol. Pharmakol. 1888, 24, 247−260. (6) Marra, J. Effects of Counterion Specificity on the Interactions between Quaternary Ammonium Surfactants in Monolayers and Bilayers. J. Phys. Chem. 1986, 90, 2145−2150. (7) Petrache, H.; Zemb, T.; Belloni, L.; Parsegian, V. A. Salt Screening and Specific Ion Adsorption Determine Neutral−Lipid Membrane Interactions. Proc. Natl. Acad. Sci. U.S.A. 2006, 103, 7982− 7987. (8) Colic, M.; Fisher, M.; Franks, G. Influence of Ion Size on ShortRange Repulsive Forces between Silica Surfaces. Langmuir 1998, 14, 6107−6112. (9) Lutzenkirchen, J. Specific Ion Effects at Two Single-Crystal Planes of Sapphire. Langmuir 2013, 29, 7726−7734. (10) Manciu, M.; Ruckenstein, E. On the Chemical Free Energy of the Electrical Double Layer. Langmuir 2003, 19, 1114−1120. (11) Djikaev, Y. S.; Ruckenstein, E. Probabilistic Approach to the Length-Scale Dependence of the Effect of Water Hydrogen Bonding on Hydrophobic Hydration. J. Phys. Chem. B 2013, 117, 7015−7025. (12) Ohtaki, H.; Radnai, T. Structure and Dynamics of Hydrated Ions. Chem. Rev. 1993, 93, 1157−1204. (13) Asthagiri, D.; Pratt, L.; Kress, J.; Gomez, M. Hydration and Mobility of Ion HO−(aq). Proc. Natl. Acad. Sci. U.S.A. 2004, 101, 7229−7233. (14) Levin, Y.; dos Santos, A. P.; Diehl, A. Ions at the Air−Water Interface: An End to a Hundred-Year-Old Mystery? Phys. Rev. Lett. 2009, 103, 257802. (15) dos Santos, A. P.; Diehl, A.; Levin, Y. Surface Tensions, Surface Potentials, and the Hofmeister Series of Electrolyte Solutions. Langmuir 2010, 26, 10778−10783. (16) dos Santos, A. P.; Levin, Y. Ion Specificity and the Theory of Stability of Colloidal Suspensions. Phys. Rev. Lett. 2011, 106, 167801.
AUTHOR INFORMATION
Corresponding Authors
*E-mail:
[email protected], hhuang5@buffalo.edu (H.H.). *E-mail: feaeliru@buffalo.edu. Tel: (716) 645-1179. Fax:(716) 645-3822 (E.R.). Notes
The authors declare no competing financial interest. 3727
dx.doi.org/10.1021/jz401948w | J. Phys. Chem. Lett. 2013, 4, 3725−3727
