Analytical interfacial layer model for thecapacitance and electrokinetics of charged
aqueous interfaces
Yuki Uematsu,∗,†,‡ Roland R. Netz,∗,‡ and Douwe Jan Bonthuis‡†Department of Chemistry, Kyushu University, Fukuoka 819-0395, Japan‡Fachbereich Physik, Freie Universität Berlin, 14195 Berlin, Germany
E-mail: uematsu@chem.kyushu-univ.jp; rnetz@physik.fu-berlin.de

Abstract
We construct an analytical model to account for the influence of the subnanometerwide interfacial layer on the differential capacitance and the electro-osmotic mobilityof solid-electrolyte interfaces. The interfacial layer is incorporated into the PoissonBoltzmann and Stokes equations using a box model for the dielectric properties, theviscosity and the ionic potential of mean force. We calculate the differential capacitanceand the electro-osmotic mobility as a function of the surface charge density and the salt
concentration, both with and without steric interactions between the ions. We compare
the results from our theoretical model with experimental data on a variety of systems(graphite and metallic silver for capacitance and titanium oxide and silver iodide forelectro-osmotic data). The differential capacitance of silver as a function of salinity
and surface charge density is well reproduced by our theory, using either the width of
the interfacial layer or the ionic potential of mean force as the only fitting parameter.
The differential capacitance of graphite, however, needs an additional carbon capacitance to explain the experimental data. Our theory yields a power-law dependence of1the electro-osmotic mobility on the surface charge density for high surface charges, reproducing the experimental data both using the interfacial parameters extracted frommolecular dynamics simulations, as well as using fitted interfacial parameters. Finally,
we examine different types of hydrodynamic boundary conditions for the power-law
behavior of the electro-osmotic mobility, showing that a finite-viscosity layer explains
the experimental data better than the usual hydrodynamic slip boundary condition.
Our analytical model thus allows us to extract the properties of the subnanometer-wide
interfacial layer by fitting to macroscopic experimental data.

Introduction
The electric double layer, formed by ions which are electrostatically attracted or otherwise adsorbed at interfaces, plays an essential role in electrochemistry, colloidal physics,
and interfacial science. 1–3 In particular, a charged surface and its counterions constitute ananometer-scale capacitor, the capacitance of which depends sensitively on the properties of
the interfacial layer. When a lateral electric field or pressure gradient is applied, the double layer gives rise to electrokinetic transport phenomena. Both the interfacial capacitanceand the electrokinetic transport are of notable interest in many industrial and biologicalprocesses. 1–3 The traditional model describing both the double layer capacitance and electrokinetics consists of the Poisson-Boltzmann (PB) and hydrodynamic (Stokes) equations 1–3
and is based on point charges in a homogeneous viscous fluid. However, the classical theory cannot consistently reproduce some experimental observations, such as the significantly
reduced differential capacitance, 4,5 the saturation of the electro-osmotic flow with rising surface charge density, 6,7 ion-specific effects, 8,9 overscreening in ionic liquids 10 , and interactions
between surfaces in water when they are close together. 11,12
Seeking to capture these effects, many modifications of the PB equation have been published over the past decades. It has been recognized early in the 20th century that thedielectric environment of water is not spatially homogeneous, and several modified PB equa2

tions incorporate the dielectric decrement due to the presence of ions. 13,14 Attempts to also
include water-water and water-surface interactions on the dielectric environment has led tothe development of the dipolar PB equation. 15 Another key approximation in the tradition
model, the point-charge nature of the ions, has been circumvented through the introductionof a finite ion size. 16–19 Other models for the ion’s steric effects have also been suggested, suchas one based on the virial expansion of the equation of state of a hard-sphere liquid. 20,21 The
differences between these different steric interaction models for the predicted electrophoretic
mobility of spherical particles with fixed charge density have been found to be minor. 22 As
an alternative to these adjustments, the PB equation has been modified by incorporating
a potential of mean force in addition to the electrostatic potential. 9,23,24 For example, thepotential of mean force between surfaces and hydrated ions has been extracted from classical molecular dynamics (MD) simulations, 24,25 taking into account the combined effects ofion-specific interactions, dielectric profile, steric interactions and image charge effects.
More recently, the interfacial structure of water itself has been studied using MD simulations, 26–30 revealing that the dielectric constant of pure water is highly non-uniform near theinterface, oscillating and reversing sign repeatedly. The same behavior has been observed forwater confined between polar soft surfaces. 31 This inhomogeneous dielectric profile reduces
the differential capacitance because the spatial average of the inverse dielectric profile near
the surface is higher than the inverse of the bulk dielectric constant at both hydrophilic
and hydrophobic surfaces. Apart from the dielectric profile, the viscosity near the interface
is also non-uniform, and strongly depends on the affinity between the surface and the solvent. Results obtained for uncharged surfaces show an increased viscosity at hydrophilic
surfaces, 26,27 which can be modeled by a stagnant layer of fluid. At hydrophobic surfaces,
however, a water-depleted layer is observed, modeled by a layer of decreased viscosity or
by a finite slip length. 26,27 The slip length depends sensitively on the presence of a surface
charge, 32 but the relation between the interfacial structure and charge density, surface slip
and the interfacial viscosity is not clear yet. The PB and Stokes equations including the3effects of these inhomogeneous profiles for the dielectric constant and the viscosity, as well
as ion-specific interactions, have been solved numerically, and the resultant electrokinetic
flow (expressed as an effective surface charge density) as a function of the titrated surface
charge density is in excellent agreement with experimental data without using viscoelectriceffects. 27,30 The differential capacitance at the point of zero charge (PZC) has also been
analyzed using this framework for carbon surfaces, 28,29 but an analysis for metals is stilllacking.
With these modified profiles, the PB equation cannot be solved analytically, but ananalytical solution exists if the profiles are approximated by step functions. In contrast to
the numerical solutions, analytical expressions for the electrokinetic flow and the interfacial
capacitance allow the effective parameters characterizing the double layer to be deduced
directly from experimental data by fitting. The primary challenge now is to develop ananalytically tractable model capable of explaining the multitude of experimental observations
simultaneously. In this work, we focus on a simultaneous description of the differential
capacitance and the electrokinetic mobility.
The differential capacitance, which characterizes how much charge is stored at an interface between a solid and an electrolyte, has been measured for many different combinationsof conducting materials and aqueous electrolytes, 5,33,34 typically by measuring the current
in response to an applied alternating current (AC) voltage, see figure 1a. It depends on theapplied potential in a nonlinear fashion, as well as on the type of ions in the electrolyte. 33,34
For the capacitance of metal-electrolyte interfaces at the PZC, the standard PB equation
predicts higher values than those measured experimentally in the case of high salinity. At low
salinity, however, the experimental capacitance of metal-electrolyte interfaces mostly agrees
with, and sometimes exceeds, the prediction of the standard PB equation. In addition to
the standard PB equation, the phenomenological framework available to describe this dataconsists of an excess interfacial capacitance and a finite roughness of the metallic surfaces,
both of which are extracted from the experimental data via Parsons-Zobel plots. 33–35 For

4carbonic materials like graphites 36,37 and graphenes, 38 the experimental differential capacitance is generally much lower than both the capacitance of metal-electrolyte interfaces andthe standard PB prediction. To address these discrepancies, Stern considered the effectsof the dielectric properties of the interfacial layer, as well as ion adsorption at large potentials 39 in his phenomenological model of the interfacial capacitance. 4 This Stern layer
effect is also important for electrification of colloidal surfaces, 40 electro-osmosis 41 and electrophoresis. 42 At large potentials, steric interactions between the ions have been identified as
the cause of the experimentally observed maximum in the capacitance. 16,17,19,22,43–50 Despite
the long-standing interest in the interfacial capacitance from both theoretical and applied
perspectives, however, a unified model for this phenomenon at varying potentials and salt
concentrations is lacking so far.
The dynamics of the double layer determine the electrophoretic mobility of charged solutes in an electrolyte and the electro-osmotic flow at charged surfaces. The electrophoretic
mobility of a spherical particle in solution has been intensely studied by the standard PB
and Stokes equations, and the dependence of the electrophoretic mobility on the salt concentration, the particle radius and the surface potential is known. 51 Specifically, in terms of thesurface potential ψ0 , the dielectric constant ε and the viscosity η, the mobility equals εε0 ψ0 /η
for arbitrary surface potential and large radius, known as the Helmholtz-Smoluchowski formula, 52 and (2/3)εε0 ψ0 /η for arbitrary surface potential and small radius. 53,54 For intermediate radii, the electrophoretic mobility becomes nonlinear with respect to the surface
potential, known as the relaxation effect or double layer polarization effect. 51,55 In experiments, the electrokinetic surface charge density or the associated zeta potential are definedby the measured electro-osmotic or electrophoretic mobility via these standard equations, see
figure 1b. 3,6,56 Generally, however, even if the double layer polarization effect is eliminatedfrom the electrophoretic mobility, the electrokinetic surface charge density deviates fromthe bare surface charge density measured by titration. 3,6,56–58 In fact, for strongly charged
surfaces, the electrokinetic surface charge density is much smaller than the titrated surface

5charge density. One way to explain the difference between the electrokinetic and titrated
surface charge densities is to assume that the no-slip boundary is not located at the solidliquid interface, but at a finite distance away from the interface. 56,58 The layer between thesolid surface and the no-slip boundary is called the stagnant layer and, according to the simplest model, has infinite viscosity. This way, however, the predicted saturated (high surface
charge) zeta potential is much larger than the experimental one, and the predicted doublelayer conductivity contradicts experimental measurements. 27,30 Another scenario to explain
the difference between the electrokinetic and titrated surface charge densities is the viscoelectric effect, which refers to the enhancement of the local viscosity due to the double-layerelectric field near the surface. 56 A microscopic foundation of the viscoelectric effect is stilllacking, however. Finally, for strong electric fields, the dielectric constant has been foundto decrease as a function of applied field, 59 which affects the electrostatic potential in theelectric double layer. Even so, the latter can typically be neglected, since experimentally
applied electric fields are small and because the effect of the interfacial structure dominates
the effect of the applied field. 29
In this paper, we explore the analytical solution of the PB equation with interfacial inhomogeneities beyond the previously reported power-law scaling of the electrokinetic mobility
with surface charge. 60 In particular, we study the differential capacitance within the same
framework, both at the PZC and at a finite surface charge. We also examine the effects ofsteric interactions on the differential capacitance and on the electrokinetic power law, andstudy different descriptions of the finite-viscosity layer and the interfacial slip length. We
fit our theory to the relevant experimental data sets which are available in the literature.
Because to our knowledge no complete data set of electrokinetic flow, titrated surface charge
density and interfacial capacitance for one surface exists, we are forced to use different data
sets for the different aspects of our model. We hope that our analysis will encourage further
experimental studies on the electric double layer.
This paper is organized as follows. In the second section, we describe the PB equation

6

including inhomogeneous dielectric profiles and nonelectrostatic interactions between ionsand surfaces, also accounting for the finite ion size. In the third section, we derive the exact
solution of the PB equation with inhomogeneous dielectric profile and ion-specific interactions. In the fourth section we discuss the differential capacitance and in the fifth section
we discuss the electrokinetic power law with different inhomogeneous viscosity profiles. We
summarize the results in the last section.
(a) Capacitance(b) Electrokinetics
u

A

E

~

Figure 1: Illustration of the two types of experiments that we use as a reference in this work.
(a) The interfacial capacitance is typically measured by measuring the electric current inresponse to an applied AC voltage. (b) The electrokinetic mobility is measured by measuring
the velocity difference u between the substrate and the fluid induced by an applied lateralelectric field Ek .

The model
Extended Poisson-Boltzmann equations
We consider a planar interface (z = 0) between an aqueous electrolyte solution (z > 0) andanother liquid, solid or gas (z < 0). All quantities are laterally homogeneous, so the Poisson
equation reads 28,29ddε⊥ (z)ε0 ψ(z) = −ρ(z),
dz
dz

(1)

where ε0 is the electric permittivity of vacuum and ψ(z) is the local electrostatic potential.
The local perpendicular dielectric constant ε⊥ (z) can be derived from the general nonlocal
inverse dielectric response function in the limit of a slowly varying displacement field. 28 The

7ionic charge density ρ(z) is given byρ(z) = e c+ (z) − c− (z) ,(2)where c± (z) is the local concentration of monovalent cations and anions and e is the absolute value of the elementary charge. If we neglect the steric interactions between ions, theconcentrations of cations and anions obey the Boltzmann distribution±

c± (z) = c0 e∓Ψ(z)−µ0 (z) ,(3)where c0 is the bulk concentration of electrolyte, Ψ(z) = eψ(z)/kB T is the dimensionlesspotential with kB T being the thermal energy, and µ±
0 (z) is the nonelectrostatic interaction
potential between the ions and the surface. If, instead, we include the steric interaction
between ions, the ionic concentration can be written as 16,19,43±

c0 e∓Ψ(z)−µ0 (z)±c (z) =+−

1 + (ν/2)(e−Ψ(z)−µ0 (z) + eΨ(z)−µ0 (z) − 2),(4)

where ν is the ionic packing parameter. The packing parameter has a relation with the√effective ionic diameter a via ν = 2c0 a3 (< 1). This determines the maximum ion concen√
tration c± = 2/a3 , which is the maximum density of close-packed (face-centered cubic or
hexagonal close-packed) spheres of diameter a. In the following, models without steric effects
are based on eq 3 and models with steric effects are based on eq 4. The boundary conditions
for eq 1 are determined by the surface charge density σ0 and a vanishing potential far fromthe surface,ε⊥ (z)ε0dψ= −σ0dz z=0and ψ(z)

= 0.

(5)

z→+∞

At the surface, we do not consider any chemical reaction involving protons, ions and functional groups. Therefore, when studying the differential capacitance, the electrodes are assumed to be ideally polarizable and the surface charge density σ0 is a given parameter which8can be controlled by applying an electrostatic potential difference between the electrode andthe bulk electrolyte. In the case of electro-osmosis, the surface charge density σ0 can becontrolled by adjusting the concentration of protons or potential-determining ions. In particular, when the proton or hydroxide concentration is well below the bulk salt concentration(c0  10−pH or c0  10pH−14 ), any σ0 can be obtained by adjusting the pH and we may consider σ0 as a given parameter without considering the chemical equilibrium. For analyticaltractability, the non-uniform dielectric constant is modeled as a box profile 27–30,60,61 εint for 0 < z < zε ,ε⊥ (z) = εfor zε < z,(6)where ε is the dielectric constant of the bulk solution, εint is the dielectric constant in the
dielectric interfacial layer and zε is the width of the dielectric interfacial layer. The additional
non-electrostatic ion-surface interaction is modeled asµ±
0 (z) =

 α± for 0 < z < z ± ,
α

 0

for

(7)

zα± < z,where zα± is the width of the adsorption layer and α± is the ion-specific surface interaction
parameter. If we consider the symmetric case α+ = α− , we simply denote the ion-specificsurface interaction parameter as α = α+ = α− . In general, zα± 6= zε , but in the following we
consider only the case of zα± = zε for analytical tractability. In the presence of steric effects,
eq 1 yieldsεκ2eΨ−α− − e−Ψ−α+2εint 1 + (ν/2) (e−Ψ−α+ + eΨ−α− − 2)





for 0 < z < zε ,

d2 Ψ
=

dz 2










κ2 sinh Ψ
for z > zε ,1 + ν (cosh Ψ − 1)9(8)

where Ψ = eψ/kB T and κ−1 =p
εε0 kB T /2e2 c0 is the Debye length. Without steric effects(ν = 0), eq 8 simplifies toεκ2 Ψ−α−−Ψ−α+
for 0 < z < zε ,
−
e
e


 2εint

d2 Ψ
=

dz 2



 κ2 sinh Ψ(9)for z > zε .

Parameterization of the box models
The introduction of the simplified interfacial box models (eqs 6 and 7) enables the analyticalsolution of the extended PB equation. The parameters of the box profiles can be obtainedby fitting the model results to experimental data, as will be explained in detail below.
As a check of the fitting results, we also estimate the parameters based on the interfacialdielectric profile and the ionic potentials of mean force which have been obtained previously
by MD simulations. 24,28,29 We use this second independent parameterization to guarantee theconsistency of our approach, and to facilitate the interpretation of the fitted box profiles interms of the molecular interfacial structure. The full dielectric profile ε⊥ (z) is an oscillating
function for diamond surfaces. 28,29 The potential of mean force exhibits either an attractive
minimum or a monotonic repulsive decrease, depending on ion and surface type. 27,62 From
the full dielectric profile, we determine zε and εint as follows. 28,29 Using that the displacement
field is constant along the z direction (perpendicular to the interface) in the absence of freecharges, the differential capacitance C between z = 0 and z = zb (in the aqueous bulk phasefar from the surface) is given in terms of ε⊥ (z) by
1
=
C

Z zb
0

dz
.
ε⊥ (z)ε0(10)

If we use the box profile (eq 6) for ε⊥ (z), we obtain1zεzb − zε=+.
Cεint ε0
εε0
10

(11)

Equating eqs 10 and 11 leads to
Z zb
zε =
0

ε⊥ (z)−1 − ε−1
dz.
−1
ε−1
int − ε(12)

Similarly, the potential of mean force determines the excess number of ions per unit surface
area Γ± , defined as±Γ =Z zb 

±c0 e∓Ψ(z)−µ0 (z) − c0 dz.

(13)

0

For a neutral surface in contact with an infinitely dilute electrolyte, the electrostatic contribution Ψ(z) in eq 13 vanishes, and with the box profile of eq 7 for µ±
0 (z) we obtain
Γ± = c0 (e−α± − 1)zα± .

(14)Equating eqs 13 and 14 yields

zα± =

Z zb
0

±

e−µ0 (z) − 1dz.
e−α± − 1(15)

Obviously, eqs 12 and 15 are insufficient to determine all free parameters zε , εint , zα± and
α± . Moreover, we do not have the full dielectric profiles nor the potentials of mean force
for all pairs of surfaces and electrolytes. In this paper, therefore, we consider only thecase of zα± = zε to reduce the number of parameters, and we set zε = 0.1 nm, which has
been extracted from MD simulations as the width of the low-dielectric layer for hydrophilic
surfaces. 29,29 Furthermore, we consider these interfacial parameters to remain constant when
we vary the titrated surface charge density σ0 and the bulk salt concentration c0 . This meansthat we neglect the dielectric decrement due to a finite salt concentration 63,64 which has beenshown to affect the nonlinear capacitance. 46,47

11

Atomistic model(a)Without
steric effectStandard PB
equationInterfacial
model

(b)

Ψ∗

α±

εint

ηint

With
steric effect
(c)

a

Ψ0

α±

ε

εint

η

(d)

ε

η

ηint
(e)

a

ε

η

ε

η

Figure 2: Illustration of the different models studied, including sketches of the electrostatic
potential Ψ(z) in panels (b-e). The surface always carries a surface charge density σ0 . A
representative snapshot of a molecular simulation of water and ions at a solid surface isshown in (a). The interfacial layer (0 < z < zε = zα± ) is indicated by the shaded purple area
in panels (b-c). The ion diameter is denoted by a, the parameters εint , α± and ηint denotethe dielectric constant, the nonelectrostatic contribution to the potential and the viscosity inthe interfacial layer, respectively. The standard Poisson-Boltzmann (PB) model with bulk
dielectric constant ε and bulk viscosity η is shown in panels (d-e).

12

Exact solution and the effects of steric interactions
In this section, we present an exact solution to the extended PB equation introduced in
the previous section, and study the influence of steric interactions between the ions on theelectrostatic potential. First, we give an overview of the different models we discuss in
the remainder of this paper, illustrations of which are shown in figure 2. A snapshot of
the molecular structure of a solid-electrolyte interface is shown in figure 2a. The model
introduced in the previous section with εint 6= ε and finite interfacial affinities α± 6= 0 we callthe interfacial box model (figure 2bc), 60 for which we will eventually also include interfacial
viscosity effects. We call the model with εint = ε and α± = 0 the standard PB equation(figure 2de). For each of these models, we consider the cases with and without steric effects.

Exact solution without steric effects
We derive the exact solution of the interfacial box model without steric effects (figure 2c).
The solution of eq 9 in the range z > zε is 2

Ψ(z) = 2 ln

1 + e−κ(z−zε ) tanh(Ψ∗ /4)
for z > zε ,1 − e−κ(z−zε ) tanh(Ψ∗ /4)(16)

where Ψ∗ = Ψ(zε ) is the interfacial potential (see figure 2c). What remains to be solved is
eq 9 in the interfacial layer (0 < z < zε ). Multiplying by dΨ/dz and integrating once, we
obtain"  #2d 1 dΨ
κ2 ε Ψ−α−−e+ e−Ψ−α+ = 0,
dz 2 dz
2εint

(17)

which means that the part in square brackets equals an integration constant. Using this fact,
we define p by


 2

2κ2 εχ+11 dΨ
κ2 ε Ψ−α−p−=−e+ e−Ψ−α+ ,
εint
2
2 dz
2εint

13

(18)

where the symmetric affinity parameter is defined as χ+ = exp[−(α+ + α− )/2]. Using theexact solution for z > zε (eq 16), we know that Ψ∗ satisfies (dΨ/dz)2z=zε +0 = 2κ2 (cosh Ψ∗ −1).
Inserting this expression into the boundary condition εint dΨ/dz|z=zε −0 = εdΨ/dz|z=zε +0 , wearrive at


dΨ
dz

2


=2

zε −0

ε
εint

2

κ2 (cosh Ψ∗ − 1).

(19)

Thus, the constant p can be written as a function of Ψ∗ ,p=ε1 −1 Ψ∗∗(cosh Ψ∗ − 1) −χ− e + χ− e−Ψ − 2 .
2χ+ εint
4(20)

where the antisymmetric affinity parameter is defined as χ− = exp[−(α+ − α− )/2].
Our aim is now to obtain a general solution of eq 18 and to determine Ψ∗ based on theboundary condition at z = 0, given by dΨ/dz|z=0 = −eσ0 /εint ε0 kB T (charge neutrality).
Using Jacobian elliptic functions, 65 the general solution of eq 18 for 0 < z < zε equals
1 − φχ− dn (xp |mp ) α+ − α−



ln
−
, for p < 0,


1
+
φ
dn
(x
|m
)
2
χ
p
p

−







 1 − cn (x |m ) α − α+−ppΨ(z) =−, for 0 ≤ p ≤ 1,
ln

1
+
cn
(x
|m
)
2
p
p










1 − sn (xp |mp ) α+ − α−−, for 1 < p,
 ln
1 + sn (xp |mp )2(21)

where φχ− = sgn(ln χ− ), and the index 0 ≤ mp ≤ 1 of the Jacobian elliptic functions isdefined by
(1 − p)−1 , for



mp =
1 − p,for 1 − p−1 ,for

p < 0,
0 ≤ p ≤ 1,

(22)

1 ≤ p.

A detailed derivation is given in the Supporting Information. The index mp of the Jacobian
elliptic functions will be omitted in the remainder of this paper. The arguments xp (z) of the14

Jacobian elliptic functions in eq 21 are defined byx (z) =p−φχ− φΨ∗ κp (z − zε ) + dn−1 (φχ− ω ∗ ), for p < 0,−φΨ∗ κp (z − zε ) + cn−1 (ω ∗ ), for 0 ≤ p ≤ 1,





φΨ∗ κp (z − zε ) + sn−1 (ω ∗ ), for 1 < p,
∗

(23)

∗

where ω ∗ = (χ− − eΨ )/(χ− + eΨ ), φΨ∗ = sgn(Ψ∗ ), and κp is given by

p
p


1
−
pκ
χ+ ε/εint , for p < 0,


 p
κp =
κ χ+ ε/εint ,for 0 ≤ p ≤ 1,


p


 √pκ χ+ ε/εint ,for 1 < p.

(24)

The values of Ψ∗ and p are determined by the electroneutrality boundary condition as
−2φΨ∗ κp cs (xp (0)) , for p < 0,



eσ0
=
−
−2φΨ∗ κp ds (xp (0)) , for 0 ≤ p ≤ 1,εint ε0 kB T
 −2φΨ∗ κp dc (xp (0)) , for 1 < p.

(25)

The electrostatic potential of eq 21 forms the basis of our treatment of the interfacial box
model without steric effects in the following sections. General solutions of the PB equationin terms of Jacobian elliptic functions have been reported previously in the literature, 66,67
but the combined effects of an interfacial low-dielectric layer and interfacial ion adsorption
have not been treated analytically before except in ref 61.

Linear analysis including steric effects
In order to include steric effects in the model while keeping the model analytically tractable,
we linearize the interfacial box model (figure 2c). Note that linearization of the standard
PB equation (figure 2e) eliminates the effect of steric interactions, but in the presence of ion

15

adsorption the steric effect survives even in the linear limit. If we assume |Ψ(z)|  1 forall z, we can linearize e±Ψ . Such a situation is realized in the case of |α+ − α− |  1. After
linearization, eq 9 in the presence of steric effects becomes
κ2 (Ψ − Ψint ) , for 0 < z < zε ,


 int

d2 Ψ
=

dz 2



 κ2 Ψ, for z > zε ,(26)

where κint (> 0) is defined asεκ2 (e−α+ + e−α− ) (1 − ν) + 2νe−α+ −α−,2εint [1 + (ν/2) (e−α+ + e−α− − 2)]2(27)

(e−α+ − e−α− ) [1 + (ν/2) (e−α+ + e−α− − 2)].
(e−α+ + e−α− ) (1 − ν) + 2νe−α+ −α−(28)κ2int =

and
Ψint =The solution of eq 26 can be written as a linear combination of exponential functions, Ae−zκint + Bezκint + Ψint , for 0 < z < zε ,Ψ(z) =
 De−zκ ,for z > z .

(29)

ε

With the boundary conditions εint dΨ/dz|z=zε −0 = εdΨ/dz|z=zε +0 , Ψ|z=zε −0 = Ψ|z=zε +0 , anddΨ/dz|z=0 = −eσ0 /εint ε0 kB T , we determine the coefficients A, B, and D to be


1
eσ0
εκe−zε κint Ψint
A =−,1 + G εint ε0 kB T κintεκ + εint κint


G
eσ0
εκezε κint Ψint
B =
−
−
,
1+G
εint ε0 kB T κint εκ − εint κint
D = Aezε (κ−κint ) + Bezε (κ+κint ) + ezε κ Ψint ,

(30)
(31)
(32)

where
G=

εκ − εint κint −2zε κint.
eεκ + εint κint

16

(33)

Eqs 29–33 constitute an analytical solution to the extended PB equation in the presence ofsteric interactions, valid for small electrostatic potentials. However, for medium-sized ions,a = 0.3 nm, the steric effects become important only when νe−α± ≈ 1, which happens forlarge adsorption potentials, for example, α± ≈ −3 for c0 = 1 M. Because the case of large
adsorption potentials, while keeping |α+ − α− |  1, is unrealistic for common electrolytes,
we will study the effects of steric interactions numerically for the nonlinear PB equation
only, at the end of this section. Now, we will focus instead on the low salinity limit κzε  1,where steric effects can be neglected (ν = 0). Expanding the surface potential Ψ0 = Ψ(0)with respect to κzε , we obtain to first order in κzε

Ψ0




ε
e−α+ + e−α−eσ0=
1 + zε κ−εε0 kB T κ
εint
2
−α+
−α−
e
−e
+ O(κ2 zε2 ),
+zε κ
2

(34)

and the interfacial boundary potential becomes

∗

Ψ



eσ0
e−α+ + e−α−=1 − zε κεε0 kB T κ
2
−α−
−α+
−e
e
+zε κ
+ O(κ2 zε2 ).
2(35)

The last terms in eqs 34 and 35 contain the increase of the potential due to asymmetricspecific ion adsorption.

Asymptotic analysis for high surface charge
We now consider the asymptotic behavior of the nonlinear PB equation in the limit of
σ0 → ±∞ for the interfacial box model without steric effects (figure 2b). In this limit, Ψ∗(determined from eqs 23, 24 and 25) approaches a constant value for any specific adsorption
strength, independent of the salt concentration. This means that a given charge incrementon the surface is totally compensated for by the increase of the charge density within the17interfacial layer (z < zε ), because any other distribution of the extra counterions wouldchange Ψ∗ . We restrict our discussion to the cases in which p, defined in eq 18, is muchhigher than unity. The detailed derivation of the asymptotic equations is given in the
Supporting Information. Further analysis of eqs 21 and 25 yields the boundary potential in
the limit of high surface charge and low salinity#p2
α∓ ε/εe4arccoshint,Ψ∗ ≈ ± ln(ε/εint )zε2 κ2 (ε/εint − e−α∓ )"(36)and the surface potential in the limit of high surface charge density"Ψ0eα∓ εint≈ ± lnκ2 εeσ0εint ε0 kB T

2 #
,

(37)

where the ± in front of the logarithm correspond to the limits of σ0 → ∞ and σ0 → −∞,respectively. Note that eqs 36 and 37 depend on the ion-specific interaction parameter α±of the counterion only. For example, in the limit of σ0 → ∞, Ψ∗ and Ψ0 depend only on α− ,not on α+ . This is a direct consequence of the exclusion of the coions from the interfacial
layer. Finally, we remark that eqs 36 and 37 are also valid for the homogeneous dielectric√√case (εint = ε). Using the relation arccosh x ≈ x − 1 in the limit x → 1, eq 36 with
εint = ε and α± = 0 becomes
2
,
κzε

(38)

eσ0
.
εε0 kB T κ(39)Ψ∗ ≈ ±2 ln
and eq 37 becomesΨ0 ≈ ±2 ln

Eqs 38 and 39 correctly recover the expressions which follow from the exact solution of thestandard PB equation. 2
Figure 3 shows Ψ∗ for different salt concentrations as a function of the surface charge
density σ0 without steric effects. In figure 3a, the solid lines denote the full analytic solution
given by eq 25, and the dotted lines denote the linearized solution given by eq 29 61 for
18

101

5
4

100

-Ψ*

3

10-1

2
1

1mM
10mM
100mM
1M

-2

10

0

-3

-1
0

2

4
2

(a)

linear behavior

10

10-1

(b)

-σ0 [µC/cm ]

100

101
2

102

-σ0 [µC/cm ]Figure 3: Interfacial boundary potential Ψ∗ as a function of the titrated surface charge σ0 ,eq 25, (solid lines) on (a) linear and (b) double-logarithmic scales. We use the parameters
ε = 78, 68 T = 298 K, εint = 1, zε = 0.1 nm, and α± = 0 without steric effects. Dotted lines in(a) are calculated by linear analysis eq 29, 61 and broken lines in (b) denote the asymptotic
result eq 36 for 1 mM and 10 mM.

20

20

-Ψ0

15

15

1mM
10mM
100mM
1M

10
ln 10

10
5
5

0
-5

0
0

(a)

2

4
2

(b)

-σ0 [µC/cm ]

10-1

100

101
2

102

-σ0 [µC/cm ]Figure 4: The surface potential Ψ0 as a function of the titrated surface charge density σ0(solid lines) on (a) linear and (b) semi-logarithmic scales. We use the parameters ε = 78, 68
T = 298 K, εint = 1, zε = 0.1 nm, and α± = 0 without steric effects. Dotted lines in (a) arecalculated by linear analysis eq 29, 61 and broken lines in (b) denote eq 37. The broken linesin (b) increase by ln 10 when the salinity increases by a factor 10, indicating that ψ0 depends
logarithmically on c0 .

19parameters ε = 78, 68 T = 298 K, εint = 1, zε = 0.1 nm, and α± = 0. For weakly charged
surfaces, Ψ∗ increases linearly with the surface charge density. However, above a specific
σ0 depending on the salinity, Ψ∗ shows a nonlinear dependence. Plotting the same curves
on a double-logarithmic scale, figure 3b, clearly shows the saturation of Ψ∗ in the limit ofstrongly charged surfaces. Horizontal broken lines indicate eq 36. The saturation of Ψ∗ isnot specific for the interfacial box model without steric effects, figure 2b, but also occurs
for the standard PB equation without steric effects, figure 2d. 2 Figure 4 shows the surface
potential Ψ0 at a given salt concentration without steric effects. In figure 4a (linear scale),
the solid lines denote the numerical result obtained by solving eqs 21 and 25, and the dottedlines denote the linearized solution eq 29. 61 The broken lines in figure 4b, which show Ψ0
on a semi-logarithmic scale, indicate the logarithmic dependence |Ψ0 | ∼ ln(σ02 /c0 ) for large|σ0 |, which follows from eq 37.
Finally, we examine the effects of steric interactions on the asymptotic behavior of Ψ∗ andΨ0 by numerically solving eq 8. We use the effective ion diameter a = 0.3 nm, corresponding
to the ionic diameter of potassium. 69 The solid lines in figure 5 show Ψ∗ and Ψ0 , includingsteric effects, as a function of σ0 for different ion concentrations. Comparing figure 5ab withthe corresponding curves without steric effects, figure 3b and 4b, respectively, it becomes
clear that steric effects induce strong deviations of both Ψ∗ and Ψ0 from the asymptotic
limit only at high surface charge density (|σ0 |  10 µC/cm2 ). For clarity, the asymptotic
equations without steric effects, eqs 36 and 37, are reproduced in figure 5 as broken lines.
This comparison shows that for ions with the effective size of potassium, there still is a large
range of medium surface charge densities in which the asymptotic behavior predicted by the
analytically solvable model is valid, and steric effects are negligible.

20

10

1

20

0

15

-Ψ0

-Ψ*

10

-1

10

1 mM
10 mM
100 mM
1M

-2

10

10-3

-1

10

(a)

0

10

10
5

1

10
2

2

0

10

-σ0 [µC/cm ]

-1

10

(b)

0

10

1

10
2

2

10

-σ0 [µC/cm ]Figure 5: Ψ∗ and Ψ0 as a function of the titrated surface charge density σ0 (solid lines)
including steric effects. We use parameters ε = 78, 68 T = 298 K, εint = 1, zε = 0.1 nm, and
α± = 0. The effective ion diameter is a = 0.3 nm. Broken lines denote eq 36 (a) and eq 37(b). Panel (a) corresponds to figure 3b and panel (b) corresponds to figure 4b when stericeffects are included.

Differential capacitance
From the surface potential Ψ0 derived in the previous section, we calculate the differential capacitance, and compare it with experimental measurements on carbon- and metal-electrolyte
interfaces. The potential variation across a solid-liquid interface, which is caused in the solidby the electron distribution and in the liquid by the ionic distribution and the water polarization, is shown schematically in figure 6. In general, the definition of the total capacitanceisdψ0 dψsub
1
1
1
=
+
≡
+
,
C
dσ0
dσ0
Cliq Csub(40)

where ψsub is the potential difference between the bulk solid phase and the interface (atz = 0), ψ0 and σ0 (defined in the previous section) are the surface potential and the surface
charge density, respectively, and Cliq and Csub denote the capacitances of the liquid and thesolid substrate, respectively. Note that we use the potential ψ(z) = kB T Ψ(z)/e in units ofVolt in this section.
To compare with the extended PB equation developed in the previous sections, we firstsummarize the results of the standard PB equation. From the standard PB equation, 70,71

21

ψ

ψ=0

z

Solid

ψ0

Liquidψ0 + ψsubFigure 6: Schematic picture of the typical profile of the electrostatic potential at the solidliquid interface.
the surface charge density in terms of ψ0 equals
2εε0 kB T κ
σ0 =
sinh
e



eψ0
2kB T


,

(41)

which is the equivalent of eq 25 in the absence of interfacial effects, i.e. for εint = ε and
α± = 0. The capacitance of the liquid phase equals
dσ0
Cliq == εε0 κ cosh
dψ0



eψ0
2kB T


.

(42)

We refer to eq 42 as the capacitance of the diffuse layer. At the PZC (σ0 = 0 and therefore
ψ0 = 0), the capacitance of the liquid phase, eq 42, becomes

CPZC = εε0 κ.

(43)

The contribution of the solid phase to the capacitance, 1/Csub , has been considered tobe minor for a long time, which is equivalent to assuming perfect conduction in the solid.
However, it has been known since the 1980s that the capacitance of the solid phase istypically finite, and its contribution cannot always be neglected. 72–78 For graphite, the solid
phase capacitance is positive (Csub > 0), decreasing the total capacitance. Our interfacial
box model considers only the capacitance of the liquid, and therefore, we add the additional
contribution from the solid phase when we compare with the experimental data of carbon,
as described below.
22Capacitance at the point of zero charge
For a surface in contact with an electrolyte, the salt type and concentration, the surface
potential and the pH determine the surface charge density σ0 . The point where σ0 = 0 isdefined as the point of zero charge (PZC), which is typically expressed as a pH value. In
experiments, the PZC is usually defined by the minimum of the capacitance as a functionof the interface potential ψ0 , based on the standard PB equation (eqs 41 and 42). Unlike
eqs 41 and 42, however, the PZC does not correspond to ψ0 = 0 for most combinationsof surfaces and electrolytes, partly due to asymmetric ion-surface affinities. Including ion
affinities using the interfacial box model, eqs 8 and 9 are symmetric with respect to thetransformation ψ0 → −ψ0 only in the case of symmetric ion affinities, α+ = α− . In this case,therefore, the minimum of the differential capacitance as a function of ψ0 exactly corresponds
to the PZC indeed. However, in the case of α+ 6= α− , the minimum is generally not identicalto the PZC. 79 Further discussion is given in the Supporting Information. Here, we consider
α+ = α− ≡ α, in which case we can calculate the capacitance at the PZC from the linear
PB equation (eq 29–33) as1=
CPZC
εint ε0 κint
1



1−G
1+G


+

1
,
Csub

(44)

where G is defined in eq 33. For CPZC , the effects of steric interactions are minor except at
high salt concentration (ν . 1) and in the case of strong adsorption (−α  −1). Therefore,
we neglect steric interactions at the PZC, setting ν = 0. Assuming a conducting surface(Csub → ∞), eq 44 is approximated in the low salinity regime as 611zεe−α zεCPZC ≈+−
εε0 κ εint ε0
εε0


−1
, for κzε  1,(45)and in the high salinity regime as

CPZC ≈√

εεint ε0 κe−α/2 , for κzε  1.
23

(46)

Eq 45 shows that in the low salinity regime, the total capacitance is equivalent to a series
circuit of two capacitors: the diffuse layer capacitance εε0 κ and the interfacial layer capacitance Cint = (zε /εint ε0 − e−α zε /εε0 )−1 , which depends on zε and εint . In the high salinity
regime, by contrast, eq 46 cannot be understood as a series of capacitors. Instead, it follows from the standard PB equation by a transformation of ε → εint and c0 → c0 e−α , sop
κ → κ ε/εint e−α/2 . That means that the ions are predominantly located inside the interfacial layer, where the system behaves as a standard PB system with a dielectric constantεint and a reference concentration c0 e−α . For strong attraction (α < − ln(ε/εint )), the capacitance is larger than the result of the standard PB equation, eq 43, whereas for weak
attraction the interfacial layer makes CPZC smaller, for any value of κ.
Figure 7 shows the differential capacitance at the PZC as a function of the salt concentration. The black broken line in figure 7 denotes the capacitance of the standard PB
equation, eq 43. We compare the results from the standard PB equation and from our interfacial box model with experimental data for the capacitance of metals (blue symbols),
planar carbon (graphite, graphene, and diamond) (red symbols), and porous carbon (black
symbols), plotted in figure 7 and summarized in the Supporting Information. The capacitance of metals almost coincides with the result of the standard PB equation, eq 43 (black
broken line), and in some cases slightly exceeds the result of the standard PB equation, which
has been rationalized by the roughness of the metallic surfaces. 33,80 In contrast, the planar
non-porous carbon data at high salinity fall short of the standard PB prediction. Surprisingly though, the planar carbon data exceed the standard PB equation at low salinity. Dueto the indirect measurement method, the capacitance per unit mass of porous carbon does
not scale proportional to the surface area per mass, 38 which is discussed in the Supporting
Information.

24

Variation of the ion-surface interaction α
In our interfacial box model, the capacitance of the interface is decreased by the low-dielectricinterfacial layer, depending on its width. The black solid line in figure 7 denotes the case of
the interfacial box model with α = 0, εint = 1 and zε = 0.1 nm, eq 44, showing a decrease of
the capacitance with respect to the standard PB equation of about one order of magnitude athigh salt concentration. In addition, the ion-surface interaction can increase the capacitance,
for attractive potentials (α < 0), or decrease the capacitance, for repulsive potentials (α > 0).
These tendencies can be inferred directly from eqs 45 and 46. The colored lines in figure 7
denote the results of our interfacial box model without steric effects (figure 2b) for different
values of α, calculated using eq 44, εint = 1 and zε = 0.1 nm. To reproduce the experimental
data, we vary the ion-surface interaction α in the interfacial box model according to α = −4,−2, 2, and 4. The prediction according to strong attraction (α = −4, blue solid line) agrees
well with the experimental data of metals, and the results of ion-specific repulsion (α = 2,
orange broken line) agree well with the experimental data of porous carbons, 28,29 suggestingthat the nonelectrostatic interaction is attractive for metals and repulsive for porous carbons.
However, even using the strong repulsive interaction α = 4, the model does not reproduce thecapacitance of planar carbons (red data points), such as graphenes and graphites. This factindicates the necessity of assuming a finite Csub for planar non-porous carbons, which we will
discuss later. The difference between porous and non-porous carbons can be understood byrealizing that the electronic structure of porous carbons is more metallic than that of planar
non-porous carbons, 81,82 and thus the capacitance of the liquid phase might dominate Csub
as discussed in the Supporting Information.

Displacement of the image plane – silver surfaces
An alternative explanation for the experimental data for metals is based on the spill-overof electrons at metallic surfaces. As shown in figure 7, the predicted capacitance CPZC withzε = 0.1 nm, εint = 1, and α = 0 is an order of magnitude smaller than the one measured
25

CPZC [µF/cm2]

100
Standard PB eq.
εint = 1, zε = 0.1 nm, α = 0
α=-4
α=-2

10

α=2
α=4
Metal exp.
Planar carbon exp.
Porous carbon exp.

1

10-5 10-4 10-3 10-2 10-1 100 101

c0 [M]

Figure 7: Differential capacitance at the PZC CPZC as a function of the salt concentration
c0 . The black solid line denotes the interfacial box model with α = 0, εint = 1, zε = 0.1 nm,
and ν = 0 and the black broken line denotes the standard PB equation. Symbols denotethe experimental data for metals (blue), planar carbons (red), and porous carbons (black).
Colored solid and broken lines denote the interfacial box model for varying surface affinity
α.
at metal surfaces. 28,29,83,84 Additional contributions to the capacitance originating from theexcess charge on metallic surfaces have been considered in the literature. 73–75,78,85 Here, we
include the spill-over of electrons, which causes a displacement of the image plane toward
the liquid, by adjusting the width of the interfacial layer zε to fit the experimental data. As
the reference experimental data set, we use the capacitance of a silver (110) surface with a
KPF6 (potassium hexafluoro-phosphate) solution. 33 The blue solid line in figure 8a denotes
the interfacial box model result when we adjust zε to fit the experimental data (black points)
with εint = 1 and α = 0 fixed. The resultant interfacial layer width is zε = 0.007 nm, which ismuch thinner than zε = 0.1 nm extracted for diamond surfaces from classical MD simulations
with constant surface charge 28,29 and used in figure 7. As a possible explanation for the originof this discrepancy, we note that the difference between the fitted and predicted widths,
0.093 nm, is of the same order as the displacement of the image plane of metals obtainedby quantum-mechanical calculations. 73–75,78 Therefore, the reduced width of the interfacial
layer could be caused by the spill-over of electrons from the metal up to the image plane,
which is known to be displaced toward the liquid.

26

Substrate capacitance – graphite surfaces
Using the strong repulsive interaction α = 4 in figure 7, the interfacial box model stillshows a discrepancy with the experimental data for the capacitance of planar carbon (redsymbols in figure 7). One explanation for the very small capacitance of planar carbon isprovided by the low density of electronic states at the Fermi level, 72,76,78,86,87 which causes
deep penetration of the external electric field into the substrate. As a result, the surface
charge forms a diffuse-layer-like structure on the carbon side, and the image plane is locatedinside the carbon. Here, we include the contribution from the surface charge distribution
in the carbon substrate by adding an effective carbon capacitance Csub in the series circuit(eq 40). Based on the results of density functional theory simulations, 78 figure S2b in the
Supporting Information, we use the following heuristic expression for the additional effective
carbon capacitance
Csub =

ε0 |σ0 |/Σ
e
,
zim

(47)

where zim > 0 is the displacement of the image plane from the surface of the carbon and Σ isan adjustable parameter necessary for finite surface charge. We discuss the validity of eq 47
in the Supporting Information. As experimental reference, we consider the capacitance ofstress-annealed pyrolytic graphite, which is essentially non-porous, against a NaF solution
plotted in figure 8b. 37 Because the experimental capacitance is larger than the standard PB
result in figure 8b for low salinity, eq 47 does not explain the carbon capacitance in the
low salinity limit. Therefore, we adjust both α and zim to fit the experimental data with
εint = 1 and zε = 0.1 nm fixed. The red solid line in figure 8b is the result of the interfacial
box model with the adsorption energy α = −8.5, and zim = 0.32 nm (corresponding toCsub = 2.8 µF/cm2 ). At about half the value, Csub is in reasonable agreement with a density
functional theory estimate. 78 Note that Σ is not needed since σ0 = 0. Agreement with theexperimental data is excellent for both low and high salinity. However, the resultant ionspecific parameter α = −8.5 is strongly attractive, whereas the potentials of mean force at27

different surface types calculated by MD simulations are either repulsive (fluoride) or neverexceed −0.5kB T (sodium). 27 A strong surface affinity is typically observed, however, fororganic surfactants. 88,89 Therefore, the presence of charged organic impurities might explain
the good agreement found in figure 8b.

2

CPZC [µF/cm ]

50

3

40

2.5

30
2

20
10
10-3

(a)

10-2
10-1
c0 [M]

100

Standard PB eq.
Interfacial box model
Silver ( 110 ) + KPF6 exp.

1.5
10-5 10-4 10-3 10-2 10-1 100(b)c0 [M]
Standard PB eq.
Interfacial box model
Graphite + NaF exp.

Figure 8: Differential capacitance at the PZC as a function of the salt concentration. The
black broken line denotes the standard PB equation eq 43. Black symbols in (a) denotethe experimental data for a silver (110) surface in contact with a KPF6 solution. 33 The
solid line in (a) denotes the interfacial box model without steric effects, using the adjustedzε = 0.007 nm in eq 44. Black symbols in (b) denote the experimental data for stressannealed pyrolytic graphite surface in contact with a NaF solution. 37 The solid line in (b)denotes the interfacial box model without steric effects, using the adjusted zim = 0.32 nm
and α = −8.5 in eqs (44), (40) and (47).
Summarizing, the capacitance of metal surfaces can be explained by the interfacial box
model assuming either an attractive ion-surface potential (figure 7), or a reduced width of
the interfacial layer due to the spill-over of electrons, figure 8a. Which of these phenomena
dominates the behavior of the metal-electrolyte interface remains to be determined, forexample by explicitly calculating the ionic surface affinity at metallic surfaces. The lowdielectric interfacial layer also explains the capacitance of porous carbon surfaces using theparameters extracted from simulations, as noted earlier, 28 whereas it does not explain thecapacitance of planar non-porous carbon, for which we have to assume a finite substratecapacitance. This fact might be caused by the different electronic structures of porous andplanar carbon. 81,82

28

Differential capacitance at finite surface charge
Comparison with experimental data of silver
The dependence of the interfacial capacitance on the surface potential is highly nonlinear
due to the specific interactions between the ions, the water and the solid substrate. It is
well-known that steric effects are essential to understand the differential capacitance at finite charge because of the increased ion concentration near the interface. 16,17,19,43–47,49 Here,
we study the same systems as the ones we considered in the previous section, silver andgraphite, away from the PZC. In figure 9a, we show the differential capacitance (the capacitance as a function of the surface potential measured by AC voltammetry or impedance
spectroscopy) of the interface between silver (110) and a KPF6 solution as a function of thesurface potential for various salt concentrations. The experimental data shows two maxima
which correspond to the voltages at which the counterion concentration reaches its maximum
packing density near the surface. 44 The absolute voltages of the maxima on the positive andnegative sides are almost equal, whereas the height of the maximum is slightly larger on
the positive side compared to the negative side. The near symmetry is remarkable because
PF6 − is a polyatomic anion which is very different from the mono-atomic cation K+ . We
calculate the capacitance of the interfacial box model including steric effects (figure 2c). To
fit the location of the maxima in the capacitance curves, we adjust the effective ion diameter.
Note that we use a single effective diameter for both ions. In the Supporting Information,
we discuss the effect of varying the ion diameter in detail. The other parameters are thesame as those used in figure 8a. We find quantitative agreement with the experimental data
regarding both the voltage dependence (being parabolic at low voltages, and reproducing the
maximum and decay at high voltages) and the salinity dependence. The quantitative agreement is extraordinary: although the qualitative behavior has been discussed intensely, very
few studies have considered a quantitative comparison with experiments. 45,49 The effective
diameter a = 1.2 nm resulting from the fit is larger than the largest diameter a = 0.59 nm

29

of the octahedral shape of PF6 − . 90 This large effective diameter shows that the definition of
an ionic radius from the ionic crystal structure is not meaningful for the highly concentrated
counterion situation near the surface. 63,64 The decrement of the local dielectric constant due
to the local counterion concentration also contributes to the differential capacitance, affecting
the resulting estimates of the ion diameter. 46,47

Comparison with experimental data of graphite
The capacitance of graphite in contact with a NaF solution, shown in figure 9b, lacks apeak and its increase with absolute surface potential is slower than for silver. We calculate
the capacitance again using the interfacial box model with steric effects, now including theadditional carbon capacitance defined in eq 47, as we did for the graphite capacitance atthe PZC in figure 8b. Because of the additional capacitance, the experimentally measured
potential is not ψ0 , but ψ0 + ψsub , where ψsub is the contribution from the carbon capacitance(figure 6). We calculate the potential difference across the carbon ψsub by integrating eq 47

ψsub (σ0 ) = sgn(σ0 )zim Σ
1 − e−|σ0 |/Σ .
ε0

(48)

We fit the capacitance curves by adjusting Σ and using a = 0.24 nm for the effective diameter
expected from the structure of the NaF ionic crystal, 69 yielding Σ = 6 µC/cm2 . All otherparameters are the same as in figure 8b. In the Supporting Information, we explain thedetails of the fitting procedure and compare different ion diameters. The colored solidlines in figure 9b show the results from the fitting procedure as a function of ψ0 + ψsub fordifferent salt concentrations, showing a quantitative agreement with the experimental data.
At high voltages, the agreement with the experimental capacitance is better for higher salt
concentrations. At concentrations above 10 mM, however, the theoretical capacitance curves
collapse onto a single curve because the additional carbon capacitance, which is independentof salinity, dominates the total capacitance. Interestingly, however, previous theories on the30

graphite capacitance completely neglect the contribution of the diffuse-layer capacitance,
which is important at low salinity, and only consider the case of high salinity. 72,76,78 Therefore,
our result is the first theoretical calculation for the finite-voltage capacitance of graphite withvarying salt concentration.
Summarizing, the capacitance of silver at finite surface charge can be explained by theinterfacial box model with steric effects. The effective ion diameter is a = 1.2 nm, which islarger than those of K+ and PF−
6 found in ionic crystals. The capacitance of graphite canbe explained by a σ0 -dependent capacitance of the substrate, eq 47. The agreement withexperimental data of graphite indicates strong nonelectrostatic attraction of charged species,

C [µFcm-2]

α = −8.5, the origin of which is yet unclear.
60

4

50

3.5

40

3

30

2.5

20

2

10
-0.4 -0.2

(a)

0
0.2
ψ0 [V]

0.4

zε = 0.007 nm, α = 0, a = 1.2 nm

1.5-0.4 -0.2
00.2(b)ψ0 + ψsub [V]0.4zε = 0.1 nm, α = - 8.5, a = 0.24 nm
0.01 mM
0.1 mM
1 mM
10 mM
100 mM
900 mM

2.5 mM
5 mM
10 mM
20 mM
40 mM
100 mM

Figure 9: (a) Differential capacitance of a silver (110) surface in contact with a KPF6 solution
as a function of the surface potential. The symbols are extracted from the continuous
experimental curve. 33 The experimental data shows a PZC at low salinity of −0.975 V relative
to the saturated calomel electrode (SCE), 33 which we have shifted to ψ0 = 0 for clarity.
The parameters used are ε = 78, T = 298 K, zε = 0.007 nm, εint = 1, a = 1.2 nm, and
α = 0. (b) Differential capacitance of stress-annealed pyrolytic graphite in contact with a
NaF solution as a function of the total potential difference. The symbols are extracted fromthe experiment, 37 where the voltage is measured relative to the normal hydrogen electrode(NHE), 37 so we have not shifted the potential. The parameters used are ε = 78, T = 298 K,zε = 0.1 nm, εint = 1, a = 0.24 nm, α = −8.5, zim = 0.32 nm, and Σ = 6 µC/cm2 .

31

Electrokinetic power law
The ion distribution determining the interfacial capacitance also affects the electro-osmotic
velocity of the liquid when a tangential electric field is applied. In this section, we use theinterfacial box model (figure 2bc) to model the electro-osmotic flow velocity, expressed interms of the electrokinetic surface charge density. For an inhomogeneous viscosity profile,the hydrodynamics equation reads 27,30
du (z)dη⊥ (z)+ ρ (z) Ek = 0,
dz
dz

(49)

where η⊥ (z) is the viscosity profile, u(z) is the tangential velocity profile and Ek is the applied
tangential electric field. We define the zeta potential ζ by the solvent velocity infinitely far
away from the surface u(z)|z→∞ , which is the experimentally observed quantity 27,30,61ζ=−ηu|z→∞.
εε0 Ek(50)

Combining eqs 1 and 49, and using the boundary conditions du(z)/dz|z→∞ = 0 and u(z =0) = 0, the zeta potential follows as 27,30,60
η
ζ=−
ε

Z ∞
0

ε⊥ (z) dψdz.
η⊥ (z) dz(51)

Consistent with the interfacial box model, we also approximate the viscosity profile by a box
function, ηint for 0 < z < zηη⊥ (z) = η

for z > zη ,(52)

where ηint is the viscosity in the interfacial layer. The viscosity profile can be obtained by
MD simulations. 26,27 For hydrophilic surfaces, the viscosity increases when approaching theinterface, whereas for hydrophobic surfaces, there is a depletion layer, and the definition ofa local viscosity is not straightforward. However, for both cases, it is easy to evaluate the32slip length in a shear situation by extrapolating the bulk velocity gradient to the surface.
Therefore, to determine zη and ηint , we define the slip length according to 26
b = zη


η
−1 .
ηint

(53)

Similar to εint and α± , we assume ηint and zη to remain constant when varying σ0 and c0 ,which means that we neglect the coupling of surface charge and slip, even though a finite
coupling has been found in simulations. 32 The width of the viscous interfacial layer generally
differs from the dielectric one, 27,30 depending on the assumed values of the interfacial dielectric constant εint and viscosity ηint . For analytical tractability, however, we first examine thesame width for both dielectric and viscosity profiles (zη = zε ), allowing εint and ηint to vary.
The case of a finite-viscosity layer with a width different from the dielectric layer is discussed
later. From eqs 6, 51 and 52 the zeta potential follows as 60,91

ζ = γψ0 + (1 − γ) ψ ∗ ,(54)

where ψ ∗ = ψ(zε ), and where we defineγ=εint η
.
εηint

(55)

Based on the Grahame equation (eq 41), the electrokinetic surface charge density is definedas 27,30peζ
σek = 8εε0 kB T c0 sinh
.
2kB T

(56)

For homogeneous systems, that means for ηint = η and εint = ε, we obtain σek = σ0 , i.e.
the surface charge density estimated from electrokinetic experiments σek equals the surface
charge measured by titration σ0 . Therefore, deviations from σek = σ0 are a measure forinterfacial effects. 27,30
We derive the asymptotic behavior of the electrokinetic surface charge density for high33

titrated surface charge density σ0 . Substituting eqs 36 and 37 into eq 54, and substituting
eq 54 into eq 56 leads to 60r|σek | ≈

εeα∓
εint

!1−γ
p
2εint ε0 kB T arccosh εeα∓ /εintp|σ0 |γ .
ezε εeα∓ /εint − 1(57)

Eq. (57) shows that the high-|σ0 | behavior of |σek |, which has traditionally been interpreted
as saturation, obeys a power law as a function of σ0 instead. The exponent of the power law
is analytically given by γ = εint η/εηint , which is the ratio of the bulk and interfacial viscousand dielectric properties. Furthermore, the prefactor of the power law in eq 57 is also given
analytically. In contrast to the universal power laws found e.g. in phase transitions, theexponent and the prefactor of the electrokinetic power law are not universal, but depend on
the interfacial material properties. Therefore, the interfacial parameters εint , ηint , zε and α±
can be extracted from the standard electrokinetic experiments via eq 57.

Simplified qualitative model using a uniform dielectric constant
The surprising emergence of a power law can be easily explained qualitatively by the saturation of Ψ∗ and the inhomogeneity of the viscosity. As a minimal model, we consider the
standard PB equation (εint = ε and α± = 0, see figure 2d). Then the surface and interfacialpotentials, Ψ0 and Ψ∗ , becomeeσ0,= 2arcsinh
εε0 kB T κ
1 + e−κzε tanh(Ψ0 /4)= 2 ln,1 − e−κzε tanh(Ψ0 /4)


Ψ0
Ψ∗

(58)
(59)

and in the limit of σ0 → ±∞,eσ0,εε0 kB T κ
1 + e−κzε≈ ±2 ln.
1 − e−κzεΨ0 ≈ ±2 ln

(60)

Ψ∗

(61)

34

We substitute eqs 60 and 61 in eq 54, and take the limit of low salinity to arrive at|σek | ≈2εε0 kB T
ezε

1−γ

|σ0 |γ ,(62)

where γ = η/ηint is the ratio of the viscosities only. Eq. (62) can also be obtained bysubstituting α± = 0 and εint = ε in eq 57, so the simple derivation using the asymptotic
eqs 60 and 61 is correct. This means that a power law is obtained even for the standard
PB equation with uniform dielectric constant. When we consider a stagnant interfacial layer(ηint → ∞), ζ is identical to ψ ∗ . In this limiting case, γ = 0 and σek is completely saturated
and equal toσek =2εε0 kB T κ,e sinh κzε(63)which is a decreasing function of salinity. Note that experimental values for σek for hydrophilic surfaces indeed decrease with increasing salinity. 3,6 In fact, the model with εint = ε,α = 0, and ηint → ∞ has been studied by Lyklema and Overbeek, 56 but it has not beenused to explain the saturation of σek because the model predicts much larger saturated zeta
potentials than the experimentally measured ones. In summary, eq 62 provides an intuitive
qualitative explanation for the electrokinetic saturation and the emergence of a power law,
but quantitatively, this simplified model predicts relatively large values of ηint because of thelack of the εint /ε factor in the exponent. Both the interfacial dielectric and viscosity effects
are needed in order to explain the experimental data quantitatively.

Comparison with experiments
In this section, we compare our exact solution of σek given by eqs 21, 25, 54, and 56, as well as
the asymptotic eq 57, with experiments. In typical experiments, the electrophoretic mobilityof finite-sized particles is measured rather than the electro-osmotic mobility, so we compare
our theory with electrophoresis measurements of colloidal solutions. For the colloids withfinite radius, the relaxation or the double layer polarization effect is crucial to determine the35

20
γ = 0.237-σek [µC/cm2]

3.2
2
0.8
0.2

γ = 0.015

0.02
0.02

(a)

0.2
0.2
2
2
-σ0 [µC/cm ]full (hydrophilic, MD)
full (hydrophobic, MD)
full (hydrophilic, Fit)
full (hydrophobic, Fit)

20

0.2

(b)

0.8
3.2
-σ0 [µC/cm2]

TiO2 exp.
AgI exp.
power law (hydrophilic, Fit)
power law (hydrophobic, Fit)

Figure 10: The electrokinetic surface charge density σek for a c0 = 1 mM electrolyte solution
as a function of the titrated surface charge density σ0 (double-logarithmic). The points
are the experimental data extracted from refs 92 and 93. The colored lines show the fullanalytical solutions (eqs 21, 25, 54, and 56) with parameters obtained in MD simulations(solid lines, denoted MD) and with fitted parameters (broken lines, denoted Fit). The colored
dotted lines denote eq 57 with fitted parameters. The black broken line denotes σek = σ0 .
We set ε = 78 68 and T = 298 K. The interfacial parameters are summarized in table 1. (b)is a magnification of (a).
relation between the surface potential and the electrophoretic mobility. 51,55 Therefore, we
eliminate this effect from the experimental electrophoretic mobility data by the proceduredescribed in detail in the Supporting Information, instead of using eq 50, which is only valid
for planar surfaces or very large radii. Figure 10 shows a double-logarithmic plot of σek .
Without interfacial effects we obtain the black broken line σek = σ0 . The strong deviationof the experimental data from the black broken line reveals the importance of interfacial
effects. The experimental data (symbols) have been obtained with TiO2 (rutile) and AgI in
KNO3 aqueous solutions (the details of the σek conversion are explained in the Supporting
Information) 92,93 . At the hydrophobic surface, the data points 2 µC/cm2 < |σek | < 4 µC/cm2
have large uncertainties because they are close to the maximum point of the conversion
function from the electrophoretic mobility to ζ (see Supporting Information). Note thathydrophobic surfaces are possibly affected by small amounts of impurities. 94 The surface
charge of TiO2 is governed by the pH, 95 whereas the surface charge of AgI is governedby the bulk concentration of Ag+ . 96–98 In general, TiO2 (rutile) is hydrophilic and AgI is36

hydrophobic. 99–101 The colored solid and broken lines denote the model prediction definedby the exact solution of the interfacial box model without steric effects (eqs 21, 25, 54, and56), and the colored dotted lines denote the asymptotic solution for large |σ0 | (eq 57).
The interfacial parameters of the model are obtained by two different approaches. In
the first approach, we take the parameters obtained from MD simulations of generic hydrophilic and hydrophobic surfaces, which have been independently verified by comparison
with multiple sets of experimental data on surface capacitance, 28,29 surface conduction andelectrokinetics. 27,30 Even though the simulations were performed for diamond surfaces instead of the surface types used in the experiments that we compare with, the advantage ofthis approach is the total absence of fit parameters. The specific adsorption parameter α is
expected to depend sensitively on the combination of ions and surfaces used, but is set toα± = 0 due to the lack of literature data. For hydrophilic surfaces, the slip length b and ηint
have been measured in MD simulations. 26 Therefore, we use zε = zη = b(η/ηint − 1)−1 as thewidth for both the dielectric and viscosity layers, and εint is determined from the simulations
by the value of the surface capacitance εint /zε . 27–30 For hydrophobic surfaces, slip is caused
by a low-density depletion layer, which can be modeled by a slip boundary condition or
by a low-viscosity layer. Either description constitutes an approximation that captures thelarge-scale hydrodynamic properties well, but at the molecular scale, both approximations
are imperfect. For hydrophobic surfaces, we choose εint = 1, fix zε through the capacitance,and determine ηint from the simulated value for b by ηint = ηzε /(b + zε ). For both surface
types we observe a fair agreement between the experimental data and our analytical theory
without fit parameters (figure 10, solid lines). However, especially for high |σ0 | the amplitudes of the experimental data are different from the ones predicted by the theory without
fit for both hydrophilic and hydrophobic surfaces (colored solid lines).
In the second approach, we extract the ion-surface interaction parameter α from a fit
to the experimental data. At both surface types, excellent agreement is achieved by fitting
only α (figure 10, red broken lines), resulting in the attractive interaction of α = −1.7

37for the hydrophilic surface, and the repulsive interaction of α = 2.0 for the hydrophobic
surface. Other parameters are the same as those extracted from MD simulations and are
given in table 1. Details of the fitting procedure are given in the Supporting Information.
The stronger repulsion of ions from the interfacial layer at hydrophobic surfaces (α = 2.0)
than at hydrophilic surfaces (α = −1.7) is qualitatively consistent with ionic potentials ofmean force calculated by MD simulations of hydrophobic and hydrophilic self-assembled
monolayers. 27,62 The fitted and MD parameters used in figure 10 are summarized in table 1.
In summary, our theory and the comparison with experimental data reveal that theelectrokinetic surface charge σek exhibits a power law behavior with a small but non-zero
exponent (γ = 0.237 for the hydrophobic and γ = 0.015 for the hydrophilic surface), determined by the subnanometer-wide interfacial structure. Therefore, if accurate experimental
data for highly charged surfaces are available, eq 57 provides a method to measure theparameters characterizing the interfacial structure, and in particular the elusive interfacial
viscosity, by standard electrokinetic experiments.
Table 1: Interfacial parameters from molecular dynamics simulations (MD) 26,28 and fromfitting (Fit). The interfacial capacitance εint /zε and the slip length b are shown for comparing
the difference between MD and Fit.
εinthydrophilic MD(TiO2 )
Fithydrophobic MD
(AgI)
Fit

4.4
4.4
1.0
1.0

zε
ηint /η
(nm)0.44 3.7 26
0.44
3.7
0.12 0.054
0.12 0.054

α
0.0
-1.7
0.0
2.0

εint /zεb−1(nm ) (nm)
10 28
-0.32 26
10
-0.32
28
8.3
2.1 26
8.3
2.1

Different widths of the interfacial viscosity and dielectric layers
Now we examine the more general case where zη 6= zε . Note that this more general model
contains both the model used previously in this paper but also the pure slip boundary
condition (zη → 0) as limiting cases. We calculate σek for various values of zη /zε while38

keeping the slip length fixed at b = zη (η/ηint − 1). The zeta potential follows from eq 51 as
ζ = γψ0 + (γ 0 − γ)ψ1 + (1 − γ 0 )ψ2 ,(64)

where γ 0 , ψ1 , and ψ2 are given byγ 0 = εint /ε, ψ1 = ψ(zη ), ψ2 = ψ(zε ),forzη < zε ,(65)

γ 0 = η/ηint , ψ1 = ψ(zε ), ψ2 = ψ(zη ),forzη > zε .
(66)

Note that the exact solution of the PB equation does not change when we set zη 6= zε .
However, ζ and its dependence on γ change because the values of ψ1 and ψ2 depend on zη .
In the case of zη > zε , ψ1 and ψ2 are saturated in the limit of high surface charge density,and therefore a power law |σek | ∼ |σ0 |γ holds, similar to the zη = zε case. In contrast, ψ1 is
not saturated in the high surface charge limit in the case of zη < zε , and σek no longer follows
the power law. The limit zη → 0 corresponds to the usual pure slip boundary condition,bdu= u(0),
dz z=0(67)

and the zeta potential is equal toζ = γ 0 ψ0 + (1 − γ 0 )ψ ∗ +

bσ0
,
εε0

(68)

where γ 0 = εint /ε is the ratio of the dielectric constants only. Eq. (68) shows that a positive slip length enhances the zeta potential whereas a negative slip length decreases it, inagreement with previous results. 61 Note that eq 67 is typically used with a positive b forhydrophobic surfaces, but not for hydrophilic surfaces, where b is negative. We call eq 67 the39

pure slip boundary condition even for hydrophilic surfaces, though it is “physically no-slip”.
Figure 11 shows σek as a function of σ0 for varying zη /zε while keeping the slip length
b fixed. All the parameters used are the same as the fitted ones in table 1. The reason forvarying zη while fixing b and εint /zε in our comparison (figure 11) is that b and εint /zε aremeasurable by macroscopic experiments, whereas measuring zη is difficult. Even for the caseof zη = 10zε , curves of σek as a function of σ0 are similar to the case of zη = zε for bothhydrophilic and hydrophobic surfaces. However, for zη = 0.1zε , the change from the resultsof zη = zε , and thus the deviation from the experimental data, is large. In the case of zη → 0,
the hydrophilic σek curve (red dotted-dashed line) is lower than the result of zη = zε , whereas
the hydrophobic curve (blue dotted-dashed line) is higher, in agreement with the findings ofeq 68. We conclude that the results produced by a pure-slip description, which is equivalent
to choosing a vanishing thickness of the viscosity layer zη , in combination with an interfacialdielectric layer are in strong disagreement with the experimental evidence. However, a good
match with experimental data is obtained when the value of the viscosity dividing surface
position zη is of the order of the dielectric width zε .

-σek [µC/cm2]

20

TiO2 exp.
zη = zεzη = 10 zεzη = 0.1 zεzη = 02

AgI exp.
zη = zεzη = 10 zεzη = 0.1 zεzη = 0

0.2

0.02
0.02

0.2

2

20

-σ0 [µC/cm2]

Figure 11: Electrokinetic surface charge density σek for varying viscous dividing surface
positions zη while keeping the slip length fixed. Other parameters are fitted and describedin table 1 and the salt concentration is 1 mM. The black broken line denotes σek = σ0 .

40Steric effects
Finally, we examine the effect of steric interactions on the electrokinetic power law. It has
been shown previously that the electrophoretic mobility is strongly enhanced when stericeffects are dominant. 48 Figure 12 shows σek as a function of σ0 with steric effects included.
We use a = 0.3 nm which corresponds to the ionic diameter of potassium. 69 All the otherparameters used in figure 12 are the same as the fitted parameters in table 1. The stericeffect significantly enhances σek in the range of |σ0 | ≥ 20 µC/cm2 . However, there is still awide range of 2–20 µC/cm2 in which the asymptotic power law is valid. Moreover, the surface
charge density in colloidal systems is typically the result of the dissociation of surface groups,
and has an upper bound given by the surface density of dissociable groups of the order of|σ0 | = 10 µC/cm2 . Therefore, the high surface charges where steric effects enhance σek are
typically only reached with specially designed ions.
20-σek [µC/cm2]

TiO2 exp.
AgI exp.

2

philicphobic0.2

philic + stericphobic + steric

0.02
0.02 0.2

2

20

200

2

-σ0 [µC/cm ]Figure 12: The electrokinetic surface charge density σek for hydrophilic and hydrophobic
surfaces in contact with a 1 mM electrolyte solution, including steric effects (solid lines)
based on eqs 1, 2, 4, 54, and 56, compared with the case without steric effects (broken lines)
based on eqs 21, 25, 54, and 56. The effective diameter of the ions is a = 0.3 nm. Other
parameters are the fitted ones described in table 1. The black broken line denotes σek = σ0 .

Conclusions
The modified dielectric constant, viscosity and ion adsorption potential in the subnanometerwide interfacial layer dominate the surface capacitance and the electrokinetic properties of41

aqueous solutes, leading to deviations from the standard PB and Stokes equations. Even
though the underlying molecular interactions are complex and involve a multitude of specific parameters, we show that the effects of the interfacial layer can be accurately andeffectively modeled by an interfacial box model. The primary advantage of our interfacial
box model is that the exact solution is available (in the case without steric effects) enabling
the analysis of experimental data by simple fitting procedures, thus giving direct access tothe effective properties of the subnanometer-wide interfacial layer. Moreover, the model isgeneral and flexible enough to analyze a multitude of phenomena originating in the electric
double layer within the same framework. For the simultaneous extraction of all parametersof the interfacial box model, however, concurrent measurements of the capacitance and theelectro-osmotic mobility for identical surface types are important. 102
We apply our interfacial box model to different sets of literature experimental data.
First, we show that the capacitance of a silver (110) surface in contact with a KPF6 solutionagrees with the result of the interfacial box model, giving a dielectric interfacial layer widthof zε = 0.007 nm. We attribute this narrow width of the interfacial layer compared to thewidth found in classical MD simulations (zε = 0.1 nm) to the spill-over of electrons from themetal. Fitting the interfacial box model to the capacitance of planar, non-porous pyrolytic
graphites in contact with NaF solutions shows the presence of strongly adsorbing charged
species (α = −8.5). MD simulations of simple ions on different types of surfaces show that
their adsorption potentials are typically smaller, 27 suggesting that the adsorbing species in
the capacitance measurements are, for example, charged organic surfactants instead. 88,89
For pyrolytic graphites, the capacitance needs an additional carbon capacitance of Csub =2.8 µF/cm2 in quantitative agreement with density functional calculations. 78 This substratecapacitance corresponds to a shift of the image plane of zim = 0.32 nm toward the solid
phase, which can be rationalized by the formation of a diffuse-layer like structure of thesurface charge on the carbon. The calculated differential capacitance at finite potentials is
also in good agreement with the experimental data for both the silver and graphite surfaces.

42Second, we derive that the electro-osmotic flow velocity, quantified by the electrokinetic
surface charge density, obeys a power law with increasing surface charge density, with an
exponent equal to εint η/εηint . We examine the robustness of this power law by using different widths for the interfacial dielectric and viscosity layers and by including steric ion-ion
interactions. For both hydrophilic and hydrophobic surfaces, we show that a model with afinite-viscosity layer describes the experimental data better than a model with a pure slip
boundary condition, which corresponds to the limit of a vanishing interfacial viscosity layer.
Steric interactions enhance the electro-osmosis, eventually obscuring the power law at highcharge density, but for medium-sized ions the onset of the steric effects starts beyond theexperimentally accessible surface charge density.
The success of our analytically solvable model at capturing a wide range of experimental
results provides a promising route to extract the effective interfacial properties of aqueous solutes from standard electrostatic and electrokinetic experiments, and to relate the results toa molecular model of the interface. Although complete experimental surveys of electrostaticand electrokinetic properties on a single surface type are lacking at present, recent electrokinetic experiments using carbon and boron-nitride nanotubes 103–105 provide an exciting newarea for the future application of our theoretical framework.

Acknowledgement
This work was supported by Grant-in-Aid for JSPS Research Fellow Number 16J00042 andfunding from the Deutsche Forschungsgemeinschaft (project NE 810/10-1).

Supporting Information
Derivation of the analytic solution of extended Poisson-Boltzmann equation, capacitancedata, discussion of capacitance, conversion methods of the electrokinetic surface charge density, and fitting procedures of the ion-surface interaction parameters in the electrokinetic
43

power law.

Author biography
Yuki Uematsu:

Yuki Uematsu studied physics at Kyoto University and received his Ph.D. in theoretical soft
matter physics, especially electrokinetics of polymer solutions. Since 2016, he has a JSPS
Research Fellowship and spent his first postdoc position at the Free University of Berlin.
Currently, he holds a postdoc position at the department of chemistry, Kyushu University.
Roland R. Netz:

44

Roland R. Netz studied physics at the Technical University of Berlin and at MIT and received
his Ph.D. in 1994 from the University of Cologne. After postdocs at Tel-Aviv University,
UCSB, Seattle, Institute Charles Sadron in Strasbourg, and CEA in Paris, he was a research associate at the Max-Planck Institute for Colloids and Interfaces in Potsdam. He wasappointed associate professor of physics at the LMU Munich in 2002 and full professor ofphysics at the TU Munich in 2004. Since 2011, he has held a chair in theoretical bio-soft
matter physics at the Free University of Berlin.
Douwe Jan Bonthuis:

Douwe Jan Bonthuis studied physics at Delft University of Technology and Harvard University. He received his Ph.D. in theoretical physics in 2012 from the Technical University of
Munich. From 2012, he held a position as a Glasstone Fellow at the Rudolf Peierls Centre
for Theoretical Physics, University of Oxford. Since 2016, he works at the Free Universityof Berlin.

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