The eects of ion adsorption on the potential of zerocharge and the dierential capacitance of charged
aqueous interfaces
Yuki Uematsu1,2 , Roland R. Netz2 , and Douwe Jan
Bonthuis21 Department of Chemistry, Kyushu University, Fukuoka 819-0395, Japan
2 Fachbereich Physik, Freie Universität Berlin, 14195 Berlin, Germany

E-mail: uematsu@chem.kyushu-univ.jp
Using dierent box prole approximations for the non-electrostatic
surface adsorption potentials of anions and cations, we calculate the dierential
capacitance of aqueous electrolyte interfaces from a numerical solution of the
Poisson-Boltzmann equation, including steric interactions between the ions and an
inhomogeneous dielectric prole. Preferential adsorption of the positive (negative)
ion shifts the minimum of the dierential capacitance to positive (negative)
surface potential values. The trends are similar for the potential of zero charge,
however, the potential of zero charge does not correspond to the minimum of
the dierential capacitance in the case of asymmetric ion adsorption, contrary to
the assumption commonly used to determine the potential of zero charge. Ourmodel can be used to obtain more accurate estimates of ion adsorption properties
from dierential capacitance or electrocapillary measurements. Asymmetric ion
adsorption also aects the relative heights of the characteristic maxima in thedierential capacitance curves as a function of the surface potential, but even forstrong adsorption potentials the eect is small, making it dicult to determine
the adsorption properties from the peak heights.
Abstract.

Keywords : electrolyte solution, ion adsorption, capacitance
Submitted to: J. Phys.: Condens. Matter

The eects of ion adsorption on the potential of zero charge and the dierential capacitance

1. Introduction
The adsorption of ions on electrodes and solutes isimportant for wetting phenomena, colloid stability
and electrochemical kinetics [13]. Apart from thedirect Coulomb interaction between the ions and thesurface charges, the ion adsorption potential is strongly
inuenced by short-ranged Lennard-Jones interactions
and dielectric eects, as well as by steric and image
charge interactions. Clearly, the consequences of ion
adsorption are especially large when anions and cations
adsorb asymmetrically  which is typically the case rendering a net charge to the surface even in theabsence of charged surface groups. The potential ofzero charge, dened as the surface potential at which
the surface charge density vanishes, is therefore aparticularly important quantity in the characterizationof solid-electrolyte interfaces.
In the absence ofasymmetric ion adsorption, the potential of zero charge
coincides with zero surface potential.
Therefore,
deviations of this correspondence can be used to study
ion adsorption.
There are various ways of measuring the potentialof zero charge, often giving disparate results [4].
The most popular experimental method is based on
measurements of the dierential capacitance of theelectric double layer, which characterizes how muchcharge is stored at an interface between a solid andan electrolyte [57]. The capacitance depends on theapplied potential in a non-linear fashion, typicallyshowing a minimum between two peaks of diering
heights which appear at larger absolute potentials.
According to the Gouy-Chapman theory, the minimumof the dierential capacitance corresponds to thepotential of zero charge. However, the Gouy-Chapman
theory does not include surface eects such as specic
ion adsorption, and the dependence of the potential ofzero charge on the potential of minimum capacitancein the presence of surface eects is yet unclear.
The peaks in the dierential capacitance curves
have been attributed to steric eects, causing
saturation of the double-layer charge, as expressedby the Bikerman equation [814].
However, theexperimental data show a shift of the potential at
the capacitance minimum with increasing salinity,
as well as an asymmetric shape with respect tothe minimum [57], both of which are beyond theassumptions underlying the Bikerman equation. These
asymmetric experimental characteristics have been

2

explained by the eect of the dierent sizes of cations
and anions [12] and by the dielectric decrement in
the double layer [13, 14]. Ionic adsorption also aects
the asymmetry of the peak heights, which has beenused instead of the capacitance minimum to estimate
the adsorption potentials [6, 7, 15]. Despite the longstanding interest, a unied model for the analysis of
the interfacial capacitance at varying potentials andsalt concentrations has been lacking so far.
The ion adsorption potential  comprising allterms except for the direct Coulomb interaction 
has been calculated using molecular dynamics (MD)simulations [1618].
For the studied monovalentions, these potentials of mean force (PMFs) show
adsorption or desorption potentials with magnitudes
of up to ∼ 2kB T at several Ångstroms from thesurface [18, 19]. In an alternative approach, thestructure of water near interfaces has been studied
using MD simulations [1922], revealing that thedielectric constant is highly inhomogeneous near theinterface, oscillating and reversing sign repeatedly.
This inhomogeneous dielectric prole decreases thedierential capacitance because the spatial average ofthe inverse dielectric prole near the surface is higher
than the inverse of the bulk dielectric constant. The
dierential capacitance at the point of zero charge(PZC) has also been analyzed using this framework forcarbons [20,21], but an analysis for nite surface charge
density with dierent surface anities for cations andanions, as well as a theory for the determination of the
PZC, is still lacking.
In this paper, we describe the development ofa model of the dierential capacitance based ona modied Poisson-Boltzmann equation, where wetake ion adsorption into account via step-function
approximations of the PMFs. We also take the stericinteractions between ions and a dielectric prole into
account. With these components, our model contains
the essential parameters necessary to study the eects
of ion adsorption on the interfacial capacitance andthe potential of zero charge. Of course, this simplicity
comes at a cost. In particular, interfacial ion densities
will not be reproduced accurately, and the model
will not be able to precisely reproduce experimental
capacitance curves. However, our model shows how
the dierent features observed in experimental data are
related to the order of magnitude, the typical lengthscale and the extent of asymmetry of the ion adsorption
and desorption potentials.
We nd that in theThe eects of ion adsorption on the potential of zero charge and the dierential capacitancepresence of asymmetric ion adsorption, the potential
at the capacitance minimum diers from the potentialof zero charge. Even for mild adsorption potentials of
2kB T , the potentials dier by tens of mV, necessitating
a more careful analysis of the experimental data.
Asymmetric ion adsorption potentials also aect theasymmetry of the saturation peaks, but the eect issmall, making this an inconvenient way to estimate theextent of ion adsorption.

2. The model
2.1. Extended Poisson-Boltzmann equation
We consider a planar interface (z = 0) between an
aqueous electrolyte solution (z > 0) and a conducting
solid or liquid (z < 0). All quantities are laterally
homogeneous, so the Poisson equation reads [20, 21]ddε⊥ (z)ε0 ψ(z) = −ρ(z),
(1)
dz
dz
where ε⊥ (z) is the local perpendicular dielectric
constant, ε0 is the electric permittivity of vacuum, ψ(z)is the local electrostatic potential, and the ionic charge
density ρ(z) is given byρ(z) = e c+ (z) − c− (z) ,(2)where c± (z) denote the local concentrations ofmonovalent cations (+) and anions (−) and e isthe elementary charge. Including steric interactions
between the ions, the ionic concentration can be
written as [8, 11]c± (z) =cbsalt e∓Ψ(z)−U± (z),(3)
1 + (ν/2)(e−Ψ(z)−U+ (z) + eΨ(z)−U− (z) − 2)

where cbsalt is the bulk concentration of electrolyte,
Ψ(z) = eψ(z)/kB T is the dimensionless potential
with kB T being the thermal energy, and U± (z) is
the non-electrostatic interaction between the ions andthe surface. We set T = 298 K throughout the
paper. The ionic packing parameter ν is√related tothe eective ionic diameter a via ν = 2cbsalt a3 (<1). This√ determines the maximum ion concentrationc± = 2/a3 , which is the maximum density of closepacked (face-centered cubic or hexagonal close-packed)
spheres of diameter a. In the following, models withoutsteric eect are based on (3) with ν = 0, and models
with steric eects included have a nite ν based on an
estimate for the ionic radius a. As the approach of (3)is not applicable to the case of dierent ion diameters
for cations and anions, we use identical diameters inthis work. The boundary conditions for (1) are given
by

dψ
= −σ0
ε⊥ (z)ε0
dz z=0and ψ(z)

= 0,
z→+∞

(4)

3

with σ0 being the surface charge density.
The
additional non-electrostatic ion-surface interaction,
which is the primary subject of our present work, ismodeled asα± for 0 < z < zα± ,U± (z) =(5)0
for zα± < z,where zα± is the width of the adsorption layer and α±
is the ion-specic surface interaction parameter which
determines whether the ion preferentially adsorbs ordesorbs from the surface. The non-uniform dielectric
constant is also modeled as a box prole [1924]
εint for 0 < z < zε ,ε⊥ (z) =(6)εfor zε < z,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. We set ε = 78 throughout the paper. The
extended Poisson-Boltzmann equation in combination
with the dielectric prole of (6) has been shown toaccurately reproduce the dierential capacitance atthe potential of zero charge as a function of salt
concentration [20, 21, 23].

2.2. Parameterization of the box models
The dielectric prole ε⊥ (z) and the ionic potentialof mean force U± (z) have been obtained previously
by MD simulations [17, 20, 21]. The full dielectric
prole ε⊥ (z) is an oscillating function at diamondsurfaces [20, 21] and between polar soft surfaces [25].
The potential of mean force U± (z) exhibits either aminimum or a monotonic repulsion, depending on ion
and surface type [19]. From the full dielectric prole,we determine zε and εint as follows [20, 21]. Using thatin the absence of free charges the displacement eldis constant along the z direction (perpendicular to theinterface), the dierential capacitance C between z = 0
and z = zb (in the aqueous bulk phase far from thesurface) is given in terms of ε⊥ (z) by
Z zb
dz
1
=
.
(7)
C
ε
(z)ε
⊥
0
0
If we use the box prole of (6) for ε⊥ (z), we obtain1zεzb − zε=+.
(8)Cεint ε0εε0Equating (7) and (8) leads toZ zb
ε⊥ (z)−1 − ε−1
zε =
dz.
(9)
−1
ε−1
0
int − εSimilarly, the potential of mean force U± (z) determines
the excess number of ions per unit surface area Γ± ,denedZ as
zGDS

Γ± =

cbsalt e∓Ψ(z)−U± (z) dz

0+

Z zb 
zGDS

cbsalt e∓Ψ(z)−U± (z) − cbsalt dz,(10)

The eects of ion adsorption on the potential of zero charge and the dierential capacitance
where zGDS is the Gibbs dividing surface of water.
For a neutral surface and at innite dilution, theelectrostatic contribution Ψ(z) in (10) vanishes, andwhen using the box prole of (5) for the potential ofmean force, we obtain

Γ± = cbsalt (e−α± − 1)zα± + cbsalt zGDS .
Again, by equating (10) and (11), we obtain
Z zb −U± (z)
e
−1
dz.
zα± =
−α± − 1
e
0

(11)

(12)

Obviously, (9) and (12) are insucient to determine
all free parameters zε , εint , zα± and α± . Moreover,
we do not have the full dielectric proles and thepotentials of mean force for all pairs of surfaces andelectrolytes. In this paper, therefore, we consider
only the case of zα± = zε to reduce the number ofparameters, and we set zε = 0.1 nm, which has been
extracted from MD simulations as the width of thelayer of low eective dielectric constant for uncharged
hydrophilic surfaces [20, 21]. Note that these MD
simulations have been performed for diamond surfaces
with ε = 1, whereas capacitance experiments are
performed for conducting electrodes. The diamondvalues still provide a good estimate, however, because
similar values of the interfacial capacitance εint /zεhave been found in simulations of metallic surfaces
including image charge eects [26]. Moreover, theeects of εint and zε on the shape of the capacitancecurves are minor. Finally, image charge eects, which
are dierent at conducting and isolating surfaces, areincluded in our adsorption parameters α± , which we
set independently.
We assume these interfacial
parameters to remain constant when we vary thetitrated surface charge density σ0 and the bulk salt
concentration cbsalt . This means that we neglect thedielectric decrement due to the presence of salt [27, 28]
which has been shown to slightly aect the nonlinear
capacitance [14].
In the presence of steric eects and setting zα± =zε , (1) yields
εκ2eΨ−α− − e−Ψ−α+−Ψ−α+ + eΨ−α− − 2)
 2εint 1 + (ν/2) (ed2 Ψ 
for 0 < z < zε ,
=
(13)

dz 2


2

κ sinh Ψ
for z > zε ,1 + ν (cosh Ψ − 1)qwhere κ−1 = εε0 kB T /2e2 cbsalt is the Debye length.

3. Dierential capacitance
From the numerical solutions of (13), we calculate
the dierential capacitance.
Apart from the ion
distribution in the aqueous phase, the potential

4variation across the interface between a conducting
solid or liquid and an electrolyte is aected by
the electron distribution in the conductor. If weassume the contribution of the conductor to thetotal capacitance to be minor, which is equivalent toassuming a high dielectric constant in the conductor,
the dierential capacitance is given by
dψ0
1
=
,
(14)
C
dσ0
with ψ0 being the surface potential ψ(z = 0). Note,however, that the capacitance of a solid substrate isoften nite, and its contribution cannot always beneglected [2935].

3.1. Relation between the point of zero charge and theminimum of the dierential capacitance
Before calculating the dierential capacitance, we
study the relation between the point of zero charge andthe minimum of the dierential capacitance. In theliterature, it is typically assumed that these two points
are identical [57], but this is not true in general.
From the Gibbs adsorption isotherm, the increment of the surface tension γ with the surface potential
ψ0 and the chemical potentials µi of the componentsobeysXdγ = −σ0 dψ0 −Γi dµi ,(15)
i=±where µ± are the chemical potentials of cations andanions. We do not consider the variation of thechemical potential of the components, and thus γ is afunction of the potential dierence across the interface
denoted by ψ0 . The condition determining the PZC is
dγ
= 0,
(16)
σ0 |PZC = −dψ0 PZC
which corresponds to the zero of the rst derivativeof γ . The subscript PZC means the derivative atψ0 = ψPZC , where ψPZC denotes the potential at
the PZC. Note that if the cations or anions reactwith the electrode involving charge transfer, dµi is notindependent of dψ0 in (15). In this case, the potentialof zero total charge is dened by dγ/dψ0 = 0 anddistinguished from the potential of zero free chargedened by σ0 = 0 [36]. Since we do not consider
reactions of ions here, the potentials of zero total
charge and zero free charge are identical, and we refer
to it as the potential of zero charge. Now, we do a
Taylor expansion of γ with respect to ψPZC , yielding
1 d2 γ(ψ0 − ψPZC )2γ(ψ0 ) = γ(ψPZC ) +2 dψ0 2 PZC+

1 d3 γ(ψ0 − ψPZC )36 dψ0 3 PZC+

1 d4 γ(ψ0 − ψPZC )4 + · · · .
24 dψ0 4 PZC(17)

Note that the term linear in ψ0 −ψPZC in (17) does notappear by virtue of (16). The dierential capacitanceis by denition

d2 γ
dσ0
=−
,
C(ψ0 ) =
dψ0
dψ0 2

(18)

C [ µFcm-2]

The eects of ion adsorption on the potential of zero charge and the dierential capacitanceand using (17), we have

−

d3 γ
d2 γ
−
(ψ0 − ψPZC )dψ0 2 PZCdψ0 3 PZC
1 d4 γ(ψ0 − ψPZC )2 + · · · .
2 dψ0 4 PZC

30

30

5

5

-0.6 -0.3(19)Equation (19) is the Taylor expansion of C(ψ0 )with respect to ψPZC .
The potential of theminimum or maximum capacitance close to the PZC
is approximately equal tod3 γ/dψ0 3,
ψm ≈ ψPZC − 4d γ/dψ0 4 PZC

180

(a)

(20)

which shows that the minimum of the dierential
capacitance generally does not correspond to thepotential at the PZC.

3.2. Dierential capacitance at nite surface chargeFirst, we calculate the dierential capacitance forsymmetric ion adsorption potentials (α+ = α− ). We
use z ∗ = 0.1 nm, and εint = 1 which are extracted
from molecular dynamics simulations of hydrophilic
surfaces [20, 21]. The results for α± = 0, 2, and −2
are shown in gure 1. Without adsorption potentials(α+ = α− = 0), the capacitance curves show two
maxima for low salt concentration, whereas for large
salt concentration, the two maxima move close to eachother and nally form a single maximum. As shownin gure 1d, the height of the maxima for cbsalt ≤100 mM is almost independent of the salt concentration(black triangles). Because of the symmetric adsorption(α+ = α− ), the potentials at the minimum or the
single maximum of the capacitance curves (dened byψm ) are identical to the potentials of zero charge ψPZC
for all cases (α± = 0, 2, −2). When both cations
and anions are repelled (α+ = α− = 2), as shown ingure 1b, the shape of the capacitance curves becomes
broader compared to the case (α+ = α− = 0). By
contrast, when both cations and anions are adsorbed(α+ = α− = −2), as shown in gure 1c, the parabolic
shape of the capacitance curves becomes narrower
compared to the case (α+ = α− = 0). As shown ingure 1d, we nd that the repulsion (α+ = α− = 2)
decreases and the adsorption (α+ = α− =2) increases
the height of the capacitance maxima on both sides ofthe minimum, but only by a negligibly small amountfor small concentration (cbsalt ≤ 100 mM).
Second, we study the case of asymmetricadsorption, α+ 6= α− .
In gure 2a, we show

00.3
ψ0 [ V ]0.6-0.6 -0.3(b)α+ = α- = 000.3
ψ0 [ V ]0.6α+ = α- = 2

180
C [ µFcm-2]

C(ψ0 ) = −

180

5

90
85

30
80
5

75
70

-0.6 -0.3

(c)

0
0.3
ψ0 [ V ]α+ = α- = - 2
0.1 mM
1 mM
10 mM
100 mM

1M
10 M
20 M

0.6

(d)

10-4

10-2
100
b
csalt [ M ](α+, α-) = (0, 0)(2, 2)(-2, -2)(a,b,c) Dierential capacitance as a function of thesurface potential. We use zε = 0.1 nm, εint = 1, and a = 0.3 nm.
The anity parameters are (a) α+ = α− = 0, (b) α+ = α− = 2,
and (c) α+ = α− = −2. (d) The peak values of the dierential
capacitance as a function of salt concentration. The broken lines
are guides to eyes.
Figure 1.

the dierential capacitance for α+ = 2, α− = 0(cation more repulsive than anion), calculated using
a numerical solution to (13). The position of thecapacitance minimum ψm moves to more negativevalues, as expected. As shown in gure 2b, theheights of the maxima for negative ψ0 (open squares)
slightly decrease compared to those for ψ0 > 0 (lled
squares), although for cbsalt > 10 M the dierence is
indistinguishable. In gure 2c, we plot ψm and ψPZC ,both of which are decreasing as a function of salt
concentration. However, the ψm and ψPZC dier by
several tens of mV in the range of 1 mM< cbsalt < 1 M,
which shows that the dierence derived in (20) is nonnegligible in this case. The red broken lines in gure
2a,c denote the potential
kB T α+ − α−,(21)ψ0 = −e2
which is a special surface potential where the charge
density (2) near the surface (z → +0) vanishes.
Equation (21) has the same sign as ψm and ψPZC ,and both ψm and ψPZC approach (21) at large salt
concentration. In gure 2d, we plot the case α+ = −2

The eects of ion adsorption on the potential of zero charge and the dierential capacitance

2

30

5

85
80
75
70

-0.6 -0.3

(a)

0
0.3
ψ0 [ V ]0.04
ψPZC, ψm [ V ]90
C [ µF/cm ]C [ µFcm-2]

180

0.6

(b)

6

10-4

0.02
0
-0.02
-0.04

10-2
100
cbsalt [ M ]

(c)

10-4

10-2
100
cbsalt [ M ]

10-4

10-2
100
b
csalt [ M ]α+ = 2, α- = 0

2

30

5

85
80
75
70

-0.6 -0.3

(d)

0
0.3
ψ0 [ V ]α+ = - 2, α- = 0
0.1 mM
1 mM
10 mM
100 mM

1M
10 M
20 M

0.04
ψPZC, ψm [ V ]90
C [ µF/cm ]C [ µFcm-2]

180

0.6

(e)

10-4

10-2
100
b
csalt [ M ]max in negative side
max in positive side

0.02
0
-0.02
-0.04

(d)

PZC
min or single max-(kBT/e)(α+-α-)/2

Figure 2. (a) Dierential capacitance as a function of the surface potential ψ0 . We use zε = 0.1 nm, εint = 1, and a = 0.3 nm. Theanity parameters are α+ = 2 and α− = 0. (b) The values of the peaks in the capacitance curves as a function of salt concentration
for α+ = 2 and α− = 0. (c) ψm and ψPZC are plotted as a function of salt concentration for α+ = 2 and α− = 0. (d) Dierential
capacitance as a function of the surface potential for α+ = −2 and α− = 0. (e) The values of the peaks in the capacitance curves
as a function of salt concentration. The anity parameters are α+ = −2 and α− = 0. (f) ψm and ψPZC are plotted as a functionof salt concentration for α+ = −2 and α− = 0.

and α− = 0 (cation more adsorbing than anion).
The position of the minimum shifts to more positive
values with increasing salt concentration, reecting
the cation adsorption. In this case, the heights ofthe maxima for ψ0 < 0 are decreased, compared tothose for ψ0 > 0 (see gure 2e). Figure 2f showsthat ψm and ψPZC are both increasing functions of
cbsalt , but again, their values are dierent and both
approach (21) at large salt concentration. The absolutedeviations between ψm and ψPZC show a similar
trend for cation desorption (gure 2a-c) and cation
adsorption (gure 2d-f), but their magnitudes depend
intricately on the salt concentration, as well as on theadsorption asymmetry (α+ − α− ) and even on theindividual values of α+ and α− . Therefore, over a
very large range of salt concentrations, an accurate
determination of the potential of zero charge from thedierential capacitance requires explicit knowledge ofthe ion adsorption potentials in the interfacial model.
Because the height of the maxima are not aected
so much by weak anities αi = ±2, we also examine
the eects of strong anities αi = ±10, which
is signicantly stronger than those extracted frompotentials mean force of simple monovalent ions in MD
simulations [1619]. Figure 3a shows the dierential
capacitance for α+ = 10 and α− = 0. Like ingure 2a, ψm moves to the negative side. Furthermore,
the maximum on the positive side becomes higher
than on the negative side (see gure 3b). However,the dierence of ∼ 5 µF/cm2 , which is induced here
only by the asymmetry in the adsorption potentials
of cations and anions, is negligible compared to thevariation of the capacitance found in experiments fordierent salt types [57]. This implies that the large
asymmetry of the capacitance curves in experiments
is caused by other eects, for example by asymmetricsteric interactions between the ions or by changes of thedielectric constant due to added ions [13,14], which are
not studied in this paper. In gure 3d, the dierential
capacitance for α+ = −10 and α− = 0 is plotted, andψm moves to the positive side. The increase of theheights of the maxima for ψ0 < 0 are small, but thetwo maxima move closer to each other and the heightincreases for large salt concentration. In gure 3c,f,
we plot ψm and ψPZC , showing qualitatively similarfeatures to those in gure 2c,f.

The eects of ion adsorption on the potential of zero charge and the dierential capacitance

2

30

5

85
80
75
70

-0.8 -0.4

(a)

0
0.4
ψ0 [ V ]0.2
ψPZC, ψm [ V ]90
C [ µF/cm ]C [ µFcm-2]

180

0.8

(b)

7

10-4

0.1
0
-0.1
-0.2

10-2
100
cbsalt [ M ]

(c)

10-4

10-2
100
cbsalt [ M ]

10-4

10-2
100
b
csalt [ M ]α+ = 10, α- = 0

2

30

5

85
80
75
70

-0.8 -0.4

(d)

0
0.4
ψ0 [ V ]0.8α+ = - 10, α- = 0
0.1 mM
1 mM
10 mM
100 mM

1M
10 M
20 M

0.2
ψPZC, ψm [ V ]90
C [ µF/cm ]C [ µFcm-2]

180

(e)

10-4

10-2
100
b
csalt [ M ]max in negative side
max in positive side

0.1
0
-0.1
-0.2

(f)

PZC
min or single max-(kBT/e)(α+-α-)/2(a) Dierential capacitance as a function of the surface potential. We use zε = 0.1 nm, εint = 1, and a = 0.3 nm. Theanity parameters are α+ = 10 and α− = 0. (b) The values of the peaks in the capacitance curves as a function of salt concentration
for α+ = 10 and α− = 0. (c) ψm and ψPZC are plotted as a function of salt concentration for α+ = 10 and α− = 0. (d) Dierential
capacitance as a function of the surface potential for α+ = −10 and α− = 0. (e) The values of the peaks in the capacitance curves
as a function of salt concentration. The anity parameters are α+ = −10 and α− = 0. (f) ψm and ψPZC are plotted as a functionof salt concentration for α+ = −10 and α− = 0.

Figure 3.

4. Conclusions
When anions and cations have dierent surfaceanities, the potential of zero charge and the potential
at which the capacitance has a minimum, ψm , have thesame sign. In the limit of low salt concentration bothpotentials are zero whereas in the limit of large salt
concentration both approach −(kB T /e)(α+ − α− )/2.
However, in the intermediate concentration range
1 mM to 1 M, they have dierent values. Therefore,
the common method to determine of the potentialof zero charge from the minimum in the dierential
capacitance is inaccurate in the presence of asymmetricion adsorption. The deviation between ψm and ψPZC
depends intricately on the salt concentration and theion adsorption potentials. Numerical solutions of
our simple model can be used to extract a moreaccurate estimate of the ion adsorption potentials fromdierential capacitance curves, ideally combined withelectrocapillary curves γ(ψ0 ).
Coion-repulsion decreases the height of thecapacitance maxima, but for weak repulsion (αi = 2)
the change is negligible, and for strong repulsion (αi =10) it goes down by approximately 5 µF/cm2 . Coionadsorption increases the height of the maxima a little
bit for both weak and strong adsorption (αi = −2 and−10), but for large salt concentration (cbsalt ≥ 1 M)
the two maxima move close to each other, their heightincreases, and nally the capacitance curve shows asingle maximum. Overall, the change of the peakheights for small concentration (cbsalt ≤ 100 mM) is
not very sensitive to the underlying ion adsorption
potentials, and negligible in comparison to the eects
typically found in experiments. This suggests thatthe large dierence of the two maximum capacitances
found in experiments is most likely caused by othereects such as asymmetric ion steric interactions orvariations of the dielectric constant due to added ions.
Direct comparison of our curves with experimental
data is not straightforward because we neglect thecontributions of the substrate capacitance, dierent
ionic sizes of cations and anions, and the dielectricdecrement with salt concentration. In this paper
we concentrate on the eect of ion adsorption andrepulsion on the potential of zero charge and thedierential capacitance only.

The eects of ion adsorption on the potential of zero charge and the dierential capacitanceAcknowledgments
R.R.N. and D.J.B acknowledge support from the DFG
project NE 810/10-1. Y.U. acknowledges support fromGrant-in-Aid for JSPS Fellows 16J00042.

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