An experimental route to Hofmeister

Abstract

The puzzling effects of the Hofmeister series is the result of a subtle balance of several competing evenly matched interactions. The complex interplay of electrostatics, dispersion forces, thermal motion, fluctuations, hydration, ion size effects and the impact of interfacial water structure make it hard or even impossible to identify an universal law. The diversity of specific ion effects as manifested by the Hofmeister series of ions is a direct consequence of the subtle interplay of forces. The decisive information is completely contained in the ion distribution. We discuss a simple approach developed in our laboratory providing first insights in the prevailing ion distribution at a charged interface based on ellipsometry. The distribution of ions is transformed in a layer with effective surface charge density and a Poisson–Boltzmann distribution within the diffuse layer. Under these assumptions Fresnel's equations can be simplified to an analytical expression, which directly relates the effective surface charge density and the ellipsometric signal. The formula does not contain free parameters and can be used to interpret ellipsometric isotherms at charged interfaces.

Introduction

The distribution of ions at a charged surface is a fundamental problem of colloid and interface science [1], [2]. Gouy and Chapman were the first who treated this problem in a quantitative fashion [3], [4]. The ions were considered as point charges embedded in a continuum of a constant dielectric constants. The chemical potential μ_i of the ion i with charge eZ_i is then given by μ_i=μ_oi+k_BTln (c_i)+eZ_iΨ, μ_o, represents the standard chemical potential of the ion of the species ‘i’, k_b is the Boltzmann constant, T the temperature, c_i stands for the ion concentration, e the elementary charge, Z_i is the ion valence and Ψ a ‘mean potential’. In thermal equilibrium μ_i remains constant throughout the system. Consequently, each sort of ion i obeys a Boltzmann distribution in the solution. The combination of the Boltzmann distribution with the Poisson equation leads to a nonlinear second order differential equation for the electric potential Ψ:

$$\text{∂}{}^2\text{Ψ}\text{z}\text{∂z}{}^2\text{=−}\text{4πe}\text{ε}{}_{\mathrm{w}}\text{Z}\text{exp}\text{−eZ}\text{Ψ}\text{z}\text{k}{}_{\mathrm{B}}\text{T}$$

where ε_w is the dielectric constant of the aqueous solution and z is the distance to the interface.

The solution of Eq. (1) yields the number density of the counter-ions as a function of the distance to the interface:

$$\text{n}\text{z}\text{=}\text{1}\text{2πZ}{}^2\text{l}{}_{\mathrm{B}}\text{1}\text{z+b}{}^2$$

The number density decays to zero with the characteristic Gouy–Chapman (GC) length $\text{b=e/2π}\text{σ}\text{l}{}_{\mathrm{B}}\text{=ε}{}_{\mathrm{w}}\text{k}{}_{\mathrm{B}}\text{T/2πe}\text{σ}$, where l_B=e²/ε_wk_BT≈7 Å is the Bjerrum length in water at room temperature. The GC length defines a distance to the interface at which half of the surface charge is screened by the counter-ions and is effectively a function of the surface charge density at constant temperature. The Bjerrum length defines the distance where the thermal energy is balanced by the electrostatic interaction.

The shortcoming of the Gouy–Chapman theory were obvious from the beginning and Stern was the first who noticed that this theory predicts unrealistic high concentrations of counter-ions in the vicinity of the interface due to a neglect of the geometrical dimensions of the ions [5]. He bridged this discrepancy by the division of the interfacial architecture in two distinct regions, a layer of directly bound counter-ions and a diffuse layer where the counter-ions can move freely. The directly bound counter-ions effectively reduce the surface charge density σ₀:

$$\text{σ}{}_{\mathrm{<mtext>net</mtext>}}\text{=σ}{}_0\text{=en}{}_{\mathrm{c}}$$

where n_c is the number density of the condensed counter-ions. The net surface charge denstity σ_net effectively determines the counter-ion distribution in the diffuse layer according to the PB theory. This approach appears to be rather artificial, however, it turns out that profiles of sophisticated molecular dynamic simulation can be mapped in a corresponding Stern–Gouy Chapman profile [6], [7]

The classical theory accounts only for electrostatics and thermal motion and neglects several important effects such as dispersion forces, fluctuation, hydration, ion size effects and the impact of interfacial water. Several extensions of the theory are aiming to include this in the framework of mean-field PB theory [8]. The chemical potential is extended by a further additive term W_i(z). The interaction free energy W_i includes all other interactions of ion ‘i’ with the medium and may depend on the electric potential Ψ, the concentration c_i and further parameters.

$$\text{μ}{}_{\mathrm{i}}\text{=μ}{}_{\mathrm{oi}}\text{+k}{}_{\mathrm{B}}\text{T}\text{ln}\text{(c}{}_{\mathrm{i}}\text{)+eZ}{}_{\mathrm{i}}\text{+W}{}_{\mathrm{i}}\text{(z)}$$

The distribution of the ion i is given by Boltzmann

$$\text{c}{}_{\mathrm{i}}\text{=c}{}_{\mathrm{0i}}\text{exp}\text{−Z}{}_{\mathrm{i}}\text{e}\text{Ψ}\text{z}\text{+W}{}_{\mathrm{i}}\text{z}\text{k}{}_{\mathrm{b}}\text{T}$$

In the context of specific ion interactions dispersion forces become important. Ninham and Yaminsky pointed out that the van der Waals interaction is not screened by the electrolyte at the interface and may play a decisive role [9], [10]. The van der Waals attraction between the ion and the interface of the two dielectric media has a leading term as calculated by Lifschitz theory:

$$\text{W}\text{z}\text{=}\text{B}\text{z}{}^3$$

where B_i is the interaction coefficient.

This term is specific for each ion and has an inherent potential to account for specific ion effects. However, we would like to stress that several important effects cannot be captured on a mean field level such as for instance fluctuations. The sound treatment requires pair correlation functions in the general framework of statistical mechanics [11].

The explicit consideration of fluctuation can give raise to new phenomena such as a phase transition towards a condensed state of ions with increasing surface charge. Lau et al. predicts a fluctuation driven counter-ion condensation [12], [13]. For a system consisting of a single charged surface and its oppositely charged counter-ions, Netz and Orland showed that a perturbative expansion about the mean-field PB solution breaks down if the surface charge is sufficiently high [14]. Thus, in this limit, fluctuation and correlation corrections are important and the mean-field picture must be modified [15]. To overcome this difficulty, a two-fluid model was proposed in Ref. [12], in which the counter-ions are divided into a free and a condensed fraction. The free counter-ions have the usual three-dimensional mean-field spatial distribution, while the condensed counter-ions are confined to move only on the charged surface and thus effectively reduce its surface charge density. The number of condensed counter-ions is determined self-consistently, by minimizing the total free energy which includes fluctuation contributions. This theory predicts that if the surface charge density of the plate is sufficiently high, a large fraction of counter-ions is condensed. The result resembles features of a phase transition similar to the liquid–gas transitions. However, this theory lacks ion-specific effects.

For experimentalists the situation appears as follows: The PB theory is getting more complex with a zoo of parameters accounting for the various effects which have been neglected in the original version of PB theory. There are also sound molecular dynamic simulations and several theories, which go beyond the mean field level predicting new phenomena. The assessment of these theories requires reliable experimental data. The decisive information is contained in the prevailing ion distribution. Therefore experiments are required which directly assess the ion distribution.

Such an experiment is outlined in the next section. We demonstrate that the optical reflection technique ellipsometry gives valuable insights in the prevailing ion distribution. We are able to condense the Fresnel formalism to an equation that directly relates the effective surface charge density to the measured ellipsometric signal. This equation does not contain any free parameter.

Ellipsometry is an optical technique which measures and analyses changes in the state of polarization of light upon reflection. Light with a well-defined state of polarization is incident on the sample. Upon reflection the state of polarization is changed. These changes can be expressed by two measurable quantities, ψ and Δ, which are related to the ratio of the complex reflectivity coefficients for ŝ and p̂ polarization by the basic equation of ellipsometry [17]:

$$\text{tan}\text{ψe}{}^{\mathrm{<mtext>i</mtext><mtext>\Delta </mtext>}}\text{=}\text{r}{}_{\mathrm{p}}\text{r}{}_{\mathrm{s}}$$

This equation outlines the strategy for ellipsometric data analysis. The task requires the translation of the interfacial architecture in the corresponding refractive index profile. The reflectivity can then be calculated by a numerical Fresnel algorithm as outlined in Ref. [18]. The correct treatment of charged interfaces requires the explicit consideration of the counter-ion distribution and their corresponding refractive index profile.

In the thin film limit ellipsometry measures only a single parameter that is proportional to the following integral of the dielectric function ε across the interfacial region [19]

$$\text{η=∫}\text{ε−ε}{}_0\text{ε−ε}{}_2\text{ε}\text{d}\text{x}$$

and

$$\text{Δ}\text{≈}\text{4}\text{ε}{}_0\text{ε}{}_2\text{π}\text{cos}\text{ϕ}\text{sin}{}^2\text{ϕ}\text{ε}{}_0\text{−ε}{}_2\text{ε}{}_0\text{+ε}{}_2\text{cos}{}^2\text{ϕ−ε}{}_0\text{·}\text{1}\text{λ}\text{·η}$$

where ε₀ and ε₂ are the dielectric constants of the adjacent bulk phases, in our case air and water, respectively, ϕ is the angle of incidence and λ the wavelength of light. This equation is used for a further analytical simplification as outlined in the next section. Ellipsometric measurements are fast, simple and do not disturb the system [16]. Because of this it is easy to carry out concentration series of different species. The ellipsometric angle Δ monitors film properties, whereas the second ellipsometric angle ψ records bulk properties.

The condensed counter-ions in the compact or so-called Stern layer reduce the surface charge density and influence the counter-ion distribution in the diffuse layer. The distribution of the counter-ions is effectively a function of net surface charge density σ_net and can be described by Eq. (2). Since there is a linear relation between the dielectric function and the concentration in the diffuse layer (Fig. 1) we can translate the ion distribution in a refractive index profile:

$$\text{ε}\text{z}\text{=ε}{}_2\text{+c}\text{z}\text{d}\text{ε}\text{d}\text{c}\text{=ε}{}_2\text{+}\text{1}\text{2πl}{}_{\mathrm{B}}\text{N}{}_{\mathrm{A}}\text{z+b}{}^2\text{d}\text{ε}\text{d}\text{c}$$

where N_A is Avogadros constant. To quantify the impact of the counter-ion distribution in the diffuse layer on the ellipsometric angle Δ we separate the integral over the interface in Eq. (8):

$$\text{η=∫}{}_0{}^{\mathrm{d}}\text{ε−ε}{}_0\text{ε−ε}{}_2\text{ε}\text{d}\text{z+∫}{}_{\mathrm{d}}{}^{\mathrm{a}}\text{ε−ε}{}_0\text{ε−ε}{}_2\text{ε}\text{d}\text{z}$$$$\text{=η}{}_{\mathrm{SL}}\text{+η}{}_{\mathrm{DL}}$$

where the second term η_DL describes the influence of the counter-ions in the diffuse layer on the ellipsometric angle Δ. The length d is the thickness of the compact layer and a resembles a sufficient high distance to the interface, where the dielectric function reaches the value of ε₂. Ellipsometry probes the system off-resonant at optical frequencies (10¹⁵ H_z). If the adsorption process does not form new species with new electronic signatures the polarizability at optical frequencies remains unchanged. Hence η_SL is dominated by the adsorbed surfactants and not significantly changed upon ion condensation. Note that the dielectric constant of the layer consisting of the surfactants is typically about 2.2. Inserting Eq. (10) in η_DL and solving the integral leads to:

$$\text{η}{}_{\mathrm{DL}}\text{=−}\text{dε}\text{dc}\text{·}\text{σ}{}_{\mathrm{<mtext>net</mtext>}}\text{eN}{}_{\mathrm{A}}\text{+kε}{}_0\text{·}\text{π}\text{2}\text{−}\text{arctan}\text{e}\text{σ}{}_{\mathrm{<mtext>net</mtext>}}\text{ε}{}_2\text{N}{}_{\mathrm{A}}\text{2πl}{}_{\mathrm{b}}\text{dε}\text{dc}$$

with

$$\text{k=}\text{d}\text{ε}\text{d}\text{c}\text{2πl}{}_{\mathrm{B}}\text{N}{}_{\mathrm{A}}\text{ε}{}_2$$

This equation gives the desired relation between the measurable quantity Δ and the net surface charge density of the compact layer σ_net. Eq. (12) contains only direct measurable quantities. Experimentalist can use these formula for a straight forward interpretation of measured ellipsometric isotherms for ionic surfactants at the air–water interface [20]. The situation is modelled in Fig. 2.

Eq. (12) contains an ion-specific parameter $\text{d}\text{ε}\text{d}\text{c}$ which can also be directly measured by ellipsometry. The ellipsometric angle ψ is not influenced by the features of the adsorption layer and is proportional to the dielectric function of the bulk for known angle of incidence [21]. Fig. 3 shows the corresponding measurement using sodium and potassium as cations and the ions of the Hofmeister series as anions. The quantity ψ changes in a linear fashion and the slope yields the unknown refractive index increment for each ion.