## Abstract

Many properties of the interfacial layer of water at surfaces differ significantly from those of bulk water. The consequences are most significant for the double-layer capacitance and the electrokinetic properties. We model the interfacial hydration layer by a modified dielectric constant and a modified local viscosity over a single interfacial width. Analytic expressions in the low-charge Debye–Hückel approximation are derived and shown to describe experimental surface capacitance and electro-osmotic data in a unified framework.

## 1. Introduction

The presence of a solid surface in an aqueous solution has a strong influence on the local water structure and dynamics [1,2]. Although the spatial range over which the water structure deviates from its bulk configuration is of the order of several ångströms only, the interfacial layer has a profound influence on surface properties measured at large distances from the interface. In fact, the local water structure dominates ion distributions [3,4], double-layer capacitance [5,6], electrokinetic mobility and surface conductivity [7–10]. The changing structure and dynamics of the water close to the interface manifest themselves primarily in the form of two quantities: the viscosity profile and the dielectric profile. In their seminal work, Lyklema & Overbeek [8] consider a decrease in the interfacial dielectric constant and an increase in the interfacial viscosity due to the strong electric field in the interfacial layer. They argue that, although electric field effects on the dielectric constant may be neglected, the viscosity is significantly enhanced due to the electric field. Apart from the electric field, however, both the viscosity and the dielectric properties of the interfacial layer are strongly affected by short-range (Lennard–Jones) interactions, as well as by collective effects and electric multipole correlations between water molecules. Direct investigation of the effects of this molecular water structure has become possible in recent years through the use of molecular dynamics simulations. The dielectric profile, calculated using molecular dynamics simulations, shows strong oscillations as a function of both the distance to the interface [5] and the wavevector [11]. The viscosity profile shows an increased viscosity at hydrophilic surfaces. At hydrophobic surfaces, however, a depletion layer and a finite slip length are observed [12], contrary to the effect expected from the strong electric field alone.

The viscosity and dielectric profiles can be incorporated in the Stokes–Poisson–Boltzmann equations and the resulting integro-differential equations have been solved numerically [3–7]. However, an analytical solution showing the effect of the interfacial layer has been lacking so far. In this paper, we derive analytical expressions for the double-layer capacitance and the electrokinetic mobility of hydrophilic and hydrophobic surfaces in the linearized (Debye–Hückel) limit.

## 2. Modified Poisson–Boltzmann equation

We deal with the simple case of a planar interface, in which the electrostatic potential and all density distributions only depend on the perpendicular coordinate *z*. In the presence of an inhomogeneous dielectric profile, the Poisson equation reads
2.1where ∇=*d*/*dz*, *ε*_{0} is the vacuum permittivity, *ψ*(*z*) is the electrostatic potential and *ρ*(*z*) is the charge density due to ions. At an interface, the dielectric response function is in fact a tensor; for laterally homogeneous charge distributions only the perpendicular component *ε*_{⊥}(*z*) is relevant. Introducing the perpendicular electrostatic field *E*_{⊥}(*z*)=−∇*ψ*(*z*) and the approximate relation
2.2which becomes virtually exact in the long-wavelength limit for constant vertical displacement field *D*_{⊥}(*z*), we can rewrite the Poisson equation as
2.3According to the Boltzmann equation, the charge density is determined by the electrostatic potential via
2.4where *μ*_{+}(*z*) and *μ*_{−}(*z*) are the non-electrostatic contributions to the chemical potential of positive and negative ions, respectively. In the following, we set *μ*_{+}(*z*)=*μ*_{−}(*z*)=0. Note that the vertical displacement is determined by the charge density via
2.5In fact, the system of equations (2.3)–(2.5) constitutes a nonlinear integro-differential equation. In [3–7], we have solved this differential equation numerically, and in the present communication we obtain approximate analytic solutions. For this we choose a simplified box profile for the dielectric profile in the form of
2.6The interfacial width *z** is constructed by integrating over *z* both according to the box profile of equation (2.6) and the inverse dielectric profile obtained from simulations, and requiring the integrals to be equal to each other for large *z* [6]. We choose the value of *ε*_{⊥}(*z*) for *z*<*z**—which affects the value of *z** in this construction—equal to 1. This value matches the simulated dielectric profiles, which do not account for atomic polarizability [6]. From equation (2.6), we immediately obtain
2.7From the differential equation (2.3), we conclude that the electrostatic potential satisfies the boundary condition
2.8at the location of the discontinuity in the dielectric profile at *z*=*z**, in addition to the continuity boundary condition
2.9The Poisson–Boltzmann equation splits into two separate equations and reads
2.10We assume the surface charge to be located at *z*=0, so that the charge boundary condition reads
2.11

### (a) Debye–Hückel approximation

The classical Debye–Hückel approximation consists of linearizing the Boltzmann equation (2.4) according to
2.12so that we obtain from equation (2.10)
2.13where the screening length *κ*^{−1} is defined by
2.14The general solution of the linear differential equation (2.10) reads
2.15Using the three conditional equations (2.8), (2.9) and (2.11), we obtain
2.16
2.17
2.18

### (b) Surface capacitance

The surface capacitance is defined by
2.19and depends on the surface charge density *σ*_{0}. On the linearized level the derivative yields the capacitance at the point of zero charge defined by *σ*_{0}=0. From the potential in equation (2.15) and the coefficients in equations (2.16)–(2.18), we obtain
2.20It is instructive to consider the asymptotic limits of this expression. For , i.e. a for a uniform bulk dielectric constant, we obtain
2.21which is the standard capacitance for a dielectric constant *ε* and capacitor thickness *κ*^{−1}. In the opposite limit , i.e. if the double-layer width *κ*^{−1} is much smaller than the interfacial layer width *z**, we obtain
2.22which corresponds to a capacitor of plate separation characterized by vacuum permittivity *ε*_{0}.

To linear order in *z***κ*, we obtain
2.23which can be inverted and rewritten again to linear order as
2.24This expression can be interpreted as the capacitance of two capacitors that are connected in series, namely an interfacial capacitor of thickness *z** with vacuum permittivity *ε*_{0} and the outer double layer of thickness *κ*^{−1}−*z** characterized by bulk water permittivity *ε*_{0}*ε*. In figure 1, we compare the exact expression equation (2.20) and the asymptotic limits equations (2.21) and (2.22), as well as the linearized equation (2.24), with experimental data.

### (c) Electro-osmotic velocity

The hydrodynamic equation that describes the tangential flow of an electrolyte solution at a charged planar surface reads
2.25where *u*(*z*) denotes the tangential flow velocity profile and *η*_{⊥}(*z*) is the tangential viscosity profile. The term on the right-hand side is the force density, which is proportional to the externally applied tangential electric field *E*_{∥} and the local charge density *ρ*(*z*), which is determined by the Poisson–Boltzmann equation discussed in the previous sections. Integrating the equation once we obtain
2.26where we used the absence of stress infinitely far away from the surface and the definition of the displacement field in equation (2.5). After a second integration, we obtain
2.27where we used the stick boundary condition at the surface *u*(*z*=0)=0. The displacement is given by the constitutive relation equation (2.2), leading to
2.28

For the viscosity profile, we choose a step profile in the same manner as for the dielectric constant 2.29

The remaining integral in equation (2.27) can be performed, leading to the explicit expression for the solvent velocity infinitely far away from the surface
2.30In the limit we obtain the standard Helmholtz–Smoluchowski result for the electro-osmotic flow
2.31Based on equation (2.31), we define the electrokinetic surface charge density *σ*_{ek} in the Debye–Hückel limit as
2.32Expanding equation (2.30) in powers of *z***κ*, we obtain to leading order
2.33Using the definition of the slip length
2.34this can be rewritten as
2.35The correction linear in the interfacial width *z** in equation (2.33) thus corresponds to the slip length enhancement factor *κb* derived earlier [13–15]. The reduced dielectric constant inside the interfacial layer comes in only at quadratic order in *z** and reduces the electro-osmotic flow velocity. We conclude that, on the linearized level of the Poisson–Boltzmann equation, viscosity effects are dominant over dielectric effects, and a low interfacial viscosity or positive slip length increases the flow, while an increased interfacial viscosity or negative slip length decreases the electro-osmotic flow. The reduced dielectric constant in the interfacial layer reduces the electro-osmotic flow, but this effect is second order in *z***κ* and thus becomes important only at large salt concentrations.

## 3. Results and discussion

In figure 1, we show the double-layer capacitance of equation (2.20) as a function of the bulk salt concentration *c*_{0} as a solid line. We use *z**=0.1 nm, which is appropriate for both hydrophilic and hydrophobic surfaces [5]. At low salt concentrations, the capacitance follows the asymptotics of equation (2.21) (shown as a dash-dotted line), which coincides with the double-layer capacitance of an electrolyte having a homogeneous dielectric constant *ε* throughout the entire fluid (corresponding to a vanishing interfacial layer thickness *z**=0). At higher salt concentrations, the capacitance switches to a different asymptotic (equation (2.22), shown as a dash-double-dotted line), still linear in *c*_{0}, but reduced by with respect to the result for *z**=0. This reduced double-layer capacitance partly explains why the experimental data (shown as triangles in figure 1) typically lie one to two orders of magnitude below the standard capacitance of equation (2.21). We also show the low-concentration prediction of equation (2.24) by a dashed line which is seen to correctly describe the exact result up to salt concentrations of approximately *c*_{0}=100 mmol l^{−1}. In our previous work, we have shown that the remaining deviation between experimental data and theory can be accounted for by additional ion–surface repulsive interactions, which can be conveniently incorporated into the theory by the non-electrostatic contributions to the chemical potential of positive and negative ions, *μ*_{+}(*z*) and *μ*_{−}(*z*) [6].

In figure 2, we show the electrokinetic velocity of equation (2.30), normalized with respect to the Helmholtz–Smoluchowski velocity of equation (2.31). We use three different values for the viscosity in the interfacial layer: *η*_{int}=3*η* to model water at hydrophilic surfaces, *η*_{int}=*η* for a fluid with bulk viscosity everywhere and *η*_{int}=*η*/15 for water at hydrophobic surfaces. The values for the viscosity are taken from [7]. For *η*_{int}=*η*, the electrokinetic mobility is reduced with respect to the Helmholtz–Smoluchowski equation for large *z***κ*, caused by the low dielectric constant of the interfacial layer. The reduction of the mobility is more pronounced for hydrophilic surfaces, as noted earlier [8], but the situation is reversed for hydrophobic surfaces.

In figure 3, we plot the electrokinetic surface charge density *σ*_{ek} (equation (2.32)) as a function of the bare surface charge density *σ*_{0}, together with the experimental data from [9,10] as presented in [7]. The maximum Debye length (at *c*_{0}=1 mmol l^{−1}) is 10 nm, whereas the colloids used experimentally are typically significantly larger^{1} [16–19]. Therefore, the experiments are treated in terms of planar electrokinetics. Two striking properties of the experimental data are well reproduced by the Debye–Hückel theory presented here. First, the mobility of the hydrophilic surface is reduced with respect to the Helmholtz–Smoluchowski line *σ*_{ek}=*σ*_{0}, whereas at the hydrophobic surface the mobility is enhanced, in agreement with the experimental data for low *σ*_{0}. Second, while the mobility at hydrophilic surfaces decreases with increasing *c*_{0}, the mobility at hydrophobic surfaces increases (up to *z***κ*∼0.1; figure 2), in agreement with the experimental data. Nevertheless, the saturation of the mobility for high *σ*_{0}, which is accurately reproduced by the full nonlinear Poisson–Boltzmann equation [7], is not captured within the Debye–Hückel approximation, as expected.

## 4. Conclusion

We derive analytical expressions for the double-layer capacitance and the electrokinetic mobility in the limit of low surface potential, using block functions for the viscosity profile and the dielectric profile. The double-layer capacitance can be approximated as the capacitance of two capacitors in series, one of thickness *z** and vacuum permittivity and one of thickness *κ*^{−1}−*z** and bulk water permittivity. The electrokinetic mobility is reduced due to the dielectric profile at high *z***κ*, but this effect is overshadowed by the effect of the viscosity profile. The viscosity profile enhances the electrokinetic mobility at hydrophobic surfaces, where the interfacial viscosity is lower than the bulk viscosity, and it reduces the electrokinetic mobility at hydrophilic surfaces, where the interfacial viscosity is higher than the bulk viscosity. This effect of the hydrophobicity as well as the change in mobility with increasing salt concentration (increasing mobility at hydrophobic surfaces and decreasing mobility at hydrophilic surfaces) are in agreement with experimental data. The experimentally well-established saturation at high surface charge density, however, is only reproduced in the nonlinear Poisson–Boltzmann theory. It follows that, in order to theoretically model the experimentally measurable electrostreaming potentials and conductivities in single nanotubes [20], nonlinear electrostatic effects have to be included in addition to effects due to the cylindrical geometry.

## Authors' contributions

All authors equally carried out the calculations, performed the analysis and drafted the manuscript.

## Competing interests

We have no competing interests.

## Funding

D.J.B. acknowledges the Glasstone Benefaction and Linacre College for funding. Y.U. is supported by the JSPS Core-to-Core Program Non-equilibrium Dynamics of Soft Matter and Information and a Grant-in-Aid for JSPS fellowship. R.R.N. acknowledges funding by the Deutsche Forschungsgemeinschaft via SFB 1078.

## Footnotes

One contribution of 11 to a Theo Murphy meeting issue ‘Nanostructured carbon membranes for breakthrough filtration applications: advancing the science, engineering and design’.

↵1 Although the colloid size used in these experiments is not reported, the same authors use colloids of radius 30–300 nm in similar experiments.

- Accepted September 7, 2015.

- © 2015 The Author(s)

Published by the Royal Society. All rights reserved.