Dipolar depletion eﬀect on the diﬀerential capacitance of carbon based materials
Sahin Buyukdagli
1
∗
and T. AlaNissila
1
,
2
†
1
Department of Applied Physics and COMP center of Excellence,Aalto University School of Science, P.O. Box 11000, FI00076 Aalto, Espoo, Finland
2
Department of Physics, Brown University, Providence, Box 1843, RI 029121843, U.S.A.
(Dated: March 13, 2012)The remarkably low experimental values of the capacitance data of carbon based materials incontact with water solvent needs to be explained from a microscopic theory in order to optimizethe eﬃciency of these materials. We show that this experimental result can be explained by thedielectric screening deﬁciency of the electrostatic potential, which in turn results from the interfacialsolvent depletion eﬀect driven by image dipole interactions. We show this by deriving from themicroscopic system Hamiltonian a nonmeanﬁeld dipolar PoissonBoltzmann equation. This canaccount for the interaction of solvent molecules with their electrostatic image resulting from thedielectric discontinuity between the solvent medium and the substrate. The predictions of theextended dipolar PoissonBoltzmann equation for the diﬀerential capacitance are compared withexperimental data and good agreement is found without any ﬁtting parameters.
PACS numbers: 03.50.De,05.70.Np,87.16.D
I. INTRODUCTION
New generation supercapacitors are used for a broadrange of applications in nanoscopic scale technologies. Inwater puriﬁcation technology, capacitive desalination isan eﬃcient candidate that might replace the current leading technics such as reverse osmosis, a membrane basedpuriﬁcation method known to suﬀer from the membranefouling phenomenon [1]. Supercapacitors are also usedas low cost and long life energy storage devices with considerably higher energy densities than conventional electrolytic capacitors [2]. A through understanding of thedouble layer structure of these devices is thus necessaryto optimize their eﬃciency.The understanding of the double layer structure waslimited for several decades to the GouyChapmanSternmodel [3]. This model was later completed by considering additional eﬀects speciﬁc to electrolyte solutions, toname but a few, the steric layer associated with the size of solvent molecules as well as the dipolar alignment close tothe interface [4], nonlocal eﬀects in electrolytes at metallic interfaces [5], ionic crowding [6] and overscreening [7].
Supercapacitors are commonly fabricated from carbonbased materials with a dielectric permittivity
ε
m
≈
2
−
5much lower than the permittivity of the water solvent
ε
w
= 78. The polarization of the interface resulting fromthis large dielectric discontinuity can drastically changethe physics of the double layer. Image dipole interactions were considered in Ref. [8] for a metallic interface.However, the work accounted exclusively for the eﬀectof image interactions on the dipolar orientation withoutconsidering their role on the interfacial dipole density.Furthermore, it was recently shown in Ref. [9] that the
∗
email:
sahin buyukdagli@yahoo.fr
†
email:
Tapio.AlaNissila@aalto.fi
GouyChapman (GC) capacitance largely overestimatesthe experimental data obtained for carbon based materials. The failure of the GC capacitance was explainedby the unability of the Poisson Boltzmann formalism toaccount for nonlocal dielectric eﬀects.In order to gain insight about the contribution of surface polarization eﬀects on the capacitance of low dielectric substrates, we introduce in this work a ﬁrst microscopic modeling of solvent molecules beyond the MF levelapproximation. Namely, we derive an extended dipolarPB (EDPB) equation that can selfconsistently take intoaccount the interfacial solvent depletion. This depletionresults from the interaction of solvent molecules (modeledas dipoles) with their electrostatic image, an eﬀect absentin the meanﬁeld level DPB equation [11, 12]. The prediction of the EDPB equation is shown to agree well with experimental data for the diﬀerential capacitance of carbonbased materials. However, it is also shown that the DPBformalism yields the same result as the PB equation, thatis, it overestimates the experimental data by one orderof magnitude. These observations strongly suggests thatthe dielectric discontinuity between the substrate and thesolvent can solely explain the observed low values of thediﬀerential capacitance of carbon based materials, unlikethe conclusion of Ref. [9] where it was argued that thesurface polarity does not play a major role in the diﬀerential capacitance. Our results are also in agreement withthe experimental work in Ref. [10], where the surface hydrophobicity was actually shown to strongly reduce thecapacitance of carbone nanotubes.
II. EXTENDED DIPOLAR POISSONBOLTZMANN (EDPB) EQUATION
We will present in this part the derivation of an extended dipolar PoissonBoltzmann formalism. The ﬁeldtheoretic partition function of ions immersed in a dipo
a r X i v : 1 2 0 3 . 2 2 8 5 v 1 [ c o n d  m a t . s o f t ] 1 0 M a r 2 0 1 2
2lar liquid was derived in Ref. [11] as a functional integralover a ﬂuctuating electrostatic potential
φ
(
r
) in the form
Z
=
D
φ e
−
H
[
φ
]
, where the Hamiltonian functional isgiven by
H
[
φ
] =
d
r
[
∇
φ
(
r
)]
2
8
π
B
(
r
)
−
iσ
(
r
)
φ
(
r
)
(1)
−
d
r
d
Ω
4
π λ
d
e
E
d
−
V
w
(
r
)+
i
(
p
·∇
φ
)
−
i
λ
i
d
r
e
E
i
−
V
w
(
r
)+
i
[
q
i
φ
(
r
)]
.
The ﬁrst integral term of the Hamiltonian (1) is composed of the Maxwell tensor associated with a freely propagating electric ﬁeld
∇
φ
(
r
) in the air, and a second partthat couples the corresponding potential
φ
(
r
) to a ﬁxedsurface charge distribution
σ
(
r
). The second and thirdintegrals respectively account for the presence of solventmolecules (point dipoles) and ions of diﬀerent species denoted by the index
i
. Moreover,
r
= (
x,y,z
) is the conﬁgurational space and
Ω
= (
θ,ϕ
) stands for the solid anglecharacterizing the orientation of solvent molecules, with
θ
the angle between the dipole and the
z
axis. We notethat the external wall potential
V
w
(
r
) in Eq. (1) restrictsthe space accessible to the particles, and in the case of thesingle dielectric interface located at
z
= 0, it is of the form
V
w
(
z <
0) =
∞
and
V
w
(
z >
0) = 0. Furthermore,
λ
d
and
λ
i
are respectively the fugacity of dipoles and ions,
p
thedipole moment vector, and
q
i
stands for the valency of ions for the species
i
. The heterogeneous Bjerrum lengthis deﬁned as
B
(
r
) =
e
2
/
[4
πε
(
r
)
k
B
T
], where
e
is the elementary charge,
T
= 300 K is the ambient temperature,and
ε
(
r
) =
ε
0
θ
(
z
)+
ε
m
θ
(
−
z
) is the dielectric permittivityof the medium in the absence of solvent molecules for thesame single planar interface geometry. More precisely,
ε
0
and
ε
m
denote respectively the dielectric permittivity of the air (the subspace in
z >
0) and the low dielectricsubbstrate located at
z <
0. From now on, the dielectric permittivities will be expressed in units of
ε
0
. Wealso note that the Bjerrum length in the air is
B
≈
55nm. Finally, the self energy of ions and polar moleculesthat are substracted from the potential and electrostaticﬁeld respectively read
E
i
=
q
2
i
2
v
bc
(
r
−
r
)

r
=
r
and
E
d
=
12
(
p
·∇
r
)(
p
·∇
r
)
v
bc
(
r
−
r
), where the Coulomb operatorin the air is deﬁned as
v
bc
−
1
(
r
,
r
) =
−
k
B
Tε
0
e
2
∆
δ
(
r
−
r
).In this work, we aim at investigating the model of Eq. (1) beyond the MF approximation where surface polarization eﬀects are absent [11, 12]. One way to progress
consists in opting for a variational minimization procedure that aims at ﬁnding the upper boundary for thedimensionless Grand potential of the system Ω =
−
ln
Z
by minimizing the variational Grand potential deﬁned asΩ
v
= Ω
0
+
H
−
H
0
0
, where the reference Hamiltonianis a Gaussian functional of the form
H
0
= 12
r
,
r
[
φ
(
r
)
−
iφ
0
(
r
)]
v
−
10
(
r
,
r
)[
φ
(
r
)
−
iφ
0
(
r
)]
.
(2)Furthermore,
φ
0
(
z
) is a variational external potential andthe electrostatic trial kernel is chosen in the same formas in Refs. [13–15],
v
−
10
(
r
,
r
) =
k
B
T e
2
−∇
(
ε
v
(
r
)
∇
) +
ε
v
(
r
)
κ
2
c
(
r
)
δ
(
r
−
r
)
,
(3)where the piecewise variational dielectric permittivity isdeﬁned as
ε
v
(
r
) =
ε
w
θ
(
z
)+
ε
m
θ
(
−
z
) and the trial screening length is given by
κ
c
(
r
) =
κ
c
θ
(
z
). After performingthe functional integrals over
φ
(
r
), one getsΩ
v
= Ω
0
+
k
B
T
2
e
2
d
r
d
r
δ
(
r
−
r
)
×
[
ε
(
r
)
−
ε
v
(
r
)]
∇
r
·∇
r
−
ε
v
(
r
)
κ
2
c
(
r
)
v
0
(
r
,
r
)+
d
r
σ
(
r
)
φ
0
(
r
)
−
k
B
T
2
e
2
ε
(
r
)[
∇
φ
0
(
r
)]
2
(4)
−
i
d
r
ρ
i
(
r
)
−
d
r
d
Ω
4
π
¯
ρ
d
(
r
,
Ω
)
,
where the gaussian contribution reads Ω
0
=
−
ln
D
φ e
−
H
0
[
φ
]
. We also deﬁned above the localion density
ρ
i
(
r
) =
λ
i
e
E
i
−
V
w
(
r
)
e
−
q
i
φ
0
(
r
)
−
q
2
i
2
v
0
(
r
,
r
)
(5)and the local density of dipoles with orientation
Ω
¯
ρ
d
(
r
,
Ω
) =
λ
d
e
E
d
−
V
w
(
r
)
(6)
×
e
−
p
·∇
φ
0
(
r
)
−
12
(
p
·∇
r
)(
p
·∇
r
)
v
0
(
r
,
r
)

r
=
r
.
By taking the derivative of the variational Grand potential Eq. (4) with respect to
κ
c
and
ε
v
, one gets
κ
2
c
= 4
π
w
i
ρ
b,i
q
2
i
and
ε
v
=
B
/
w
=
ε
w
= 1 +
4
π
3
B
p
20
ρ
bd
. These two relations respectively introducethe DebyeHuckel screening parameter and the DebyeLangevin form for the bulk dielectric permittivity of thewater medium
ε
w
. The additional variational equationfor
φ
0
(
z
), i.e.
δ
Ω
v
/δφ
0
(
r
) = 0 yields
∂ ∂
z
˜
ε
(
z
)
∂φ
0
(
z
)
∂
z
+ 4
π
B
σ
(
z
) + 4
π
B
ρ
i
(
z
)
q
i
= 0
,
(7)where we took into account the translational symmetryof the electrostatic potential within the (
x,y
) plane. Wenote that the variational minimization left us in Eq. (7)with a spatially varying dielectric permittivity of theform˜
ε
(
z
) = 1
−
4
π
B
φ
0
(
z
)
d
Ω
4
π
¯
ρ
d
(
z,
Ω
)
p
z
,
(8)where
p
z
=
p
0
cos
θ
stands for the component of thedipolar moment vector
p
in the
z
direction. We willcall Eq. (7) the Extended Dipolar Poisson Boltzmann(EDPB) equation.The fugacity of dipoles and ions can be related to theirbulk density in the limit
z
→ ∞
of the equations (5)
3and (6). By injecting the obtained relations for the fugacities with the inverse of the kernel Eq. (3) [13] into
Eqs. (5) and (6), the local density functions take the
form
ρ
i
(
z
) =
ρ
b,i
e
−
V
w
(
z
)
e
−
q
i
φ
0
(
z
)
−
V
c
(
z
)
(9)¯
ρ
d
(
z,
Ω
) =
ρ
bd
e
−
V
w
(
z
)
e
−
p
·∇
φ
0
(
z
)
−
V
d
(
z,
Ω
)
,
(10)where we deﬁned the following ionic and dipolar potentials,
V
c
(
z
) =
q
2
w
2
∞
0
d
kk
ρ
c
∆
e
−
2
ρ
c
z
(11)
V
d
(
z,
Ω
) =
U
d
(
z
) +
T
d
(
z
)cos
2
θ
,
(12)with the functions
U
d
(
z
) =
w
p
20
4
d
kk
3
ρ
c
∆
e
−
2
ρ
c
z
(13)
T
d
(
z
) =
w
p
20
4
d
kk
ρ
c
(2
ρ
2
c
−
k
2
)∆
e
−
2
ρ
c
z
.
(14)and ∆ = (
ρ
c
−
η
k
)
/
(
ρ
c
+
η
k
),
η
=
ε
m
/
ε
w
, and
ρ
c
=
κ
2
c
+
k
2
. Carrying out the integral over
θ
inEq. (8) with the dipole density Eq. (10) and the potential
Eq. (12), the local dielectric permittivity takes the form˜
ε
(
z
) = 1 + 4
π
3
B
p
20
ρ
db
e
−
V
w
(
z
)
e
−
U
d
(
z
)
J
(
z
)
,
(15)where we deﬁned the function
J
(
z
) = 3
√
π
8
T
3
/
2
d
(
z
)
e
p
20
φ
20 (
z
)4
T d
(
z
)
{
Erf [Ψ
+
(
z
)] + Erf [Ψ
−
(
z
)]
}
.
−
3
e
−
T
d
(
z
)
2
T
d
(
z
)sinh[
p
0
φ
0
(
z
)]
p
0
φ
0
(
z
)
,
(16)and the potentialsΨ
±
(
z
) = 2
T
d
(
z
)
±
p
0
φ
0
(
z
)2
T
d
(
z
)
.
(17)The EDPB Eq. (7) has to be solved numerically withthe ionic density proﬁles of Eq. (9) and the dielectricpermittivity proﬁle of Eq. (15).The second order diﬀerential equation (7) should besolved with the boundary conditions
φ
0
(
z
→ ∞
) = 0and
φ
0
(
z
→
0
+
) = 2
ε
w
/µ
, where the second boundarycondition valid over the parameter domain 0
≤
ε
m
≤
ε
w
follows by integrating Eq. (7) in the close neighborhoodof the interface, and noting that according to the dipolar potentials of Eqs. (13) and (14), one has
ρ
d
(0) = 0and ˜
(0) = 1 on the boundary. We also note that in thelimit where the potentials
V
c
(
z
),
U
d
(
z
), and
T
d
(
z
) vanish, EDPB equation. (7) reduces to the mean ﬁeld DPBequation of Refs. [11, 12].We ﬁnally note that the orientation averaged densityof solvent molecules is obtained according to
ρ
d
(
r
) =
d
Ω
4
π
¯
ρ
d
(
r
,
Ω
). Evaluating the integral over
θ
, the solventdensity takes the form
ρ
d
(
z
) =
ρ
bd
√
π
4
√
T
d
(
z
)
e
−
V
w
(
z
)
e
−
U
d
(
z
)
e
p
20
φ
20 (
z
)4
T d
(
z
)
(18)
×{
Erf [Ψ
+
(
z
)] + Erf [Ψ
−
(
z
)]
}
.
III. NUMERICAL RESULTS
We will investigate in this part the EDPB Eq. (7) fora symmetric electrolyte composed of two ion species of bulk densities
ρ
b,i
=
ρ
bi
and valency
q
i
=
q
. All numerical results will be derived for monovalent ions (
q
= 1)in contact with a negatively charged planar surface, i.e.
σ
(
z
) =
−
σ
s
δ
(
z
) with
σ
s
>
0. We also note that withinthe convention adopted in this article, the surface charge
σ
s
is expressed in units of the elementary charge
e
. Moreover, the model parameters
ρ
db
and
ε
w
are taken thesame as in Ref. [9]. Namely, the bulk density of solvent molecules is
ρ
db
= 50
.
8 M, which yields with thedipole moment
p
0
= 1 ˚A the bulk dielectric permittivity
ε
w
= 71.The potential proﬁle obtained from the numerical solution of the EDPB Eq. (7) for the parameters
ρ
bi
= 0
.
1 M,
ε
m
= 1 and
σ
s
= 0
.
01 nm
−
2
is reported in Fig. 1.a. Onenotices that the potential proﬁle is composed of three regions, namely two successive layers close to the interfacewhere
φ
0
(
z
) behaves as a linear function of
z
, and a thirdlayer over which
φ
0
(
z
) exponentially decays.In order to understand the form of the potential proﬁle,we illustrate in Fig. 1 the form of the dielectric permittivity ˜
(
z
) and the screening parameter
κ
2
(
z
) =
κ
2
c
e
−
V
c
(
z
)
in the vanishing surface charge limit of Eq. (7). It isseen that with increasing distance from the surface, thedielectric permittivity increases from the air permittivity ˜
(
z
) = 1 to the bulk permittivity ˜
(
z
) =
ε
w
over adistance
h
≈
2 ˚A. We note that this dipolar exclusion effect is mainly due to the interaction of solvent moleculeswith their electrostatic images. Then, one sees that thissolvent depletion regime is followed by an ionic depletionregime of thickness
d
≈
6 ˚A, an eﬀect known to srcinatefrom image charge interactions [13].Inspired by the behaviour of ˜
(
z
) and
κ
(
z
) that results from the interfacial depletion of solvent moleculesand ions, we will introduce a restricted variational ansatzbased on a piecewise trial solution for the electrostaticpotential. We assume that
φ
0
(
z
) is the solution of Eq. (7) in the linear limit of weak surface charge, with˜
(
z
) =
θ
(
h
−
z
)+
ε
w
θ
(
z
−
h
) and
κ
(
z
) =
κ
c
θ
(
z
−
d
), wherethe dipolar and ionic depletion lengths
h
and
d
are trialparameters that will be obtained from a numerical optimization procedure of the Grand potential of Eq. (4).The solution of Eq. (7) with the above piecewise dielectric permittivity and ion density proﬁles, and satisfyingthe continuity of the potential
φ
0
(
z
) and the displacement ﬁeld
D
(
z
) = ˜
(
z
)
φ
0
(
z
) at
z
= 0,
z
=
h
, and
z
=
d
,
4
(a)
0 5 10 15 20 25 300.200.150.100.050.00
0
(
k
B
T
)
z(A)
0 1 2 30.50.40.30.20.10.0
0
(
k
B
T
)
z(A)
(b)
02460.00.20.40.60.81.0
z(A)
2
(z)
c2
(z)
w
eff
(z)
w
FIG. 1: (Color online) (a) Electrostatic potential proﬁle(
σ
s
= 0
.
01 nm
−
2
) and (b) renormalized density and dielectricpermittivity proﬁles for
ε
w
= 71 and
ρ
bi
= 0
.
1 M. The redline in (a) is from the restricted variational ansatz Eq. (19)and the dashed black line corresponds to the solution of theEDPB equation.
reads
φ
0
(
z
) =
−
2
µκ
c
[1 +
κ
c
(
d
−
h
)] + 2
ε
w
µ
(
z
−
h
)
,
0
< z
≤
h
φ
0
(
z
) =
−
2
µ
κ
c
+ 2
µ
(
z
−
d
)
, h
≤
z
≤
d
(19)
φ
0
(
z
) =
−
2
µ
κ
c
e
−
κ
c
(
z
−
d
)
, z
≥
d.
Numerical optimization yields
h
= 0
.
6 ˚A and
d
= 2
.
3˚A. We note that due to the piecewise nature of the trialpotential in Eqs. (19), these values correspond approximately to half saturation densities. Figure 1 shows thatthe potential proﬁle obtained from the numerical optimization agrees very well with the general form obtainedfrom the numerical solution of the EDPB equation. InEqs. (19), the ﬁrst linear regime at 0
< z
≤
h
correspondsto the solvent depletion layer resulting from image dipoleinteractions. This layer associated with dielectric screening deﬁciency is responsible for an ampliﬁcation of thePB prediction of the surface potential by a factor of ﬁve.The second and third intervals correspond respectivelyto the usual ionic depletion and diﬀuse layers [13]. The
(a)
1E3 0.01 0.1 1110100
C
(
F / c m
2
)
bi
(M)
PBMPBMDPBEq. 21DPB
(b)
1E3 0.01 0.1 110100
m
=1
bi
(M)
C
(
F / c m
2
)
PB
m
=60
m
=70
m
=
w
20 40 600.00.20.40.60.8
h ( A )
m
FIG. 2: (Color online) (a) Diﬀerential capacitance against thebulk ion concentration for
σ
s
= 0,
ε
m
= 1, and
ε
w
= 71.The black circles are the experimental data, the red solidcurve is the result of the EDPB equation, the red squaresare from Eq. (21), the dashed blue line is the MPB result,the dotted black line is the GC capacitance, and the blacksquares correspond to the prediction of the DPB equation.(b) The same plot as in (a) for various
ε
m
. The inset displaysthe evolution of the dipolar depletion length
h
(solid curve)and
l
d
(dashed curve) as a function of
ε
m
for
ρ
bi
= 0
.
1 M.
contribution of these layers to the diﬀerential capacitancewill be investigated below.The diﬀerential capacitance of the double layer is deﬁned as
C
d
=
qe
2
k
B
T
∂σ
s
∂φ
s
,
(20)where
φ
s
=
φ
0
(
z
= 0) is the surface potential. Fig. 2.acompares the diﬀerential capacitance computed withEq. (7) in the vanishing surface charge limit with experimental data obtained for several types of monovalent electrolytes at various concentrations (for details seeRef. [9] where the data were taken from). We also report in this ﬁgure the prediction of various formalisms.As stressed in Ref. [9], the PB result largely overestimates the experimental data. Furthermore, the result of the modiﬁed PB (MPB) equation (see Ref. [13]) that canexclusively take into account the ionic depletion eﬀect
5brings a very small correction to the PB result. However, the EDPB result that additionally contains the surface depletion eﬀect of solvent molecules exhibits a goodagreement with the experimental data. We ﬁnally notethat in the vanishing surface charge limit considered inthis part, the DPB equation yields the same result as thePB one (see black squares in Fig. 2.a).The physics of the EDPB prediction for the capacitance can be understood within the restricted selfconsistent scheme of Eq. (19), where the diﬀerential capacitance in Eq. (20) takes in the limit
σ
s
→
0 the simpleform
C
d
=
ε
w
κ
c
1 +
κ
c
(
d
−
h
) +
ε
w
κ
c
h
.
(21)We note that the prediction of this equation reportedin Fig. 2 (red squares) ﬁts very well the numerical result of the EDPB equation. The inverse capacitance of Eq. (21) is composed of three parts. The ﬁrst contribution from the diﬀuse layer is the inverse GC capacitance
C
−
1
d
1
= (
ε
w
κ
c
)
−
1
corresponding to the PB resultin Fig. 2.a. The second part
C
−
1
d
2
= (
d
−
h
)
/
ε
w
associated with the ionic depletion layer is shown to dropthe diﬀerential capacitance to the MPB curve. Finally,the third contribution from the solvent depletion layer
C
−
1
d
3
=
h
characterized by the dielectric screening deﬁciency brings the most important correction to the totalcapacitance by dropping the latter to the correct orderof magnitude.In order to estimate the order of magnitude of the dipolar depletion length
h
, we will compute the asymptoticlimit of ˜
(
z
) far from the interface, where the dipolar potentials in Eqs. (13) and (14) become weak, by expanding
Eq. (15) in
U
d
(
z
),
T
d
(
z
), and
p
0
φ
0
(
z
). Furthermore, wenote that in the limit
ε
m
= 0, the dipolar potentials aregiven by the closed form expressions
U
d
(
z
) =
w
p
20
1 + 2
κ
c
z
16
z
3
e
−
2
κ
c
z
(22)
T
d
(
z
) =
w
p
20
1 + 2
κ
c
z
(1 + 2
κ
c
z
)16
z
3
e
−
2
κ
c
z
.
(23)Renormalizing all lengths by the length scale
l
d
=(
w
p
20
/
10)
1
/
3
according to ¯
κ
c
=
κ
c
l
d
and ¯
z
=
z/l
d
, andtaking into account that
ε
w
1, the asymptotic form of Eq. (15) far from the dielectric interface reads˜
(
z
)
ε
w
≈
1
−
1 + 2¯
κ
c
¯
z
+ 3¯
κ
2
c
¯
z
2
/
2¯
z
3
e
−
2¯
κ
c
¯
z
.
(24)We now note that for a bulk permittivity
ε
w
= 71 andionic concentration
ρ
bi
= 10
−
1
M, one has
κ
c
l
d
= 8
.
10
−
2
.Hence, in the regime
z
∼
l
d
= 0
.
9 ˚A, the terms inEq. (24) that depend on the screening length becomenegligible. This simple calculation ﬁxes
l
d
as the characteristic length over which the local permittivity tends toits bulk value according to an inverse cubic power law,i.e. ˜
(
z
)
/
ε
w
≈
1
−
l
3
d
/z
3
. We note that an inverse cubiclaw for the dielectric permittivity proﬁle was derived inRef. [16] in the strict limit of a single dipole (i.e.
ε
w
= 1)and without salt. In the dilute salt limit
κ
c
→
0, onecan actually extend the estimation of
h
to ﬁnite values of
ε
m
by noting that the dipolar potentials possed the closeform expression
U
d
(
z
) =
T
d
(
z
) =
w
p
20
∆
0
/
(16
z
3
), where∆
0
= (
ε
w
−
ε
m
)
/
(
ε
w
+
ε
m
). Following the same steps asabove, one obtains for the characteristic dipolar depletionlength the more general expression
l
d
= (∆
0
w
p
20
/
10)
1
/
3
.Hence, for biological solvent concentrations and diluteelectrolytes with bulk density
ρ
bi
0
.
1 M, the dielectricscreening is mainly responsible for the decay of the imagedipole interactions and solely determines the region overwhich a reduced dielectric permittivity is observed. Forlarger ion concentrations, Eq. (24) shows that the screening of image dipole interactions by surrounding ions positively adds to the dielectric screening of these forces.We display in the inset of Fig. 2.b the evolution of
h
as a function of
ε
m
together with
l
d
, while the main plotshows
C
d
for various values of the membrane permittivity from
ε
m
= 1 to
ε
m
=
ε
w
. One notices that within therange 1
≤
ε
m
≤
60,
h
exhibits a slow linear decrease withincreasing
ε
m
while
C
d
remains within the same order of magnitude as the experimental capacitance data. However, with an increase of
ε
m
from 60 to
ε
w
, the dipolardepletion length quickly drops to zero and consequently,
C
d
approaches the PB result. One also sees in the insetthat although
l
d
is slightly higher than the dipolar depletion length
h
, it can reproduce the correct trend of thelatter as a function of
ε
m
. These observations suggestthat image dipole interactions are mainly responsible forthe low values of the experimental capacitance data inFig. 2.a. We emphasize that this result is in agreementwith the experimental observation of a strong reductionof the double layer capacitance with increasing surfacehydrophobicity [10].In addition to the dipolar depletion eﬀect, the orientation is also expected to play some role in the form of theinterfacial dielectric permittivity proﬁle. The measureof the dipolar orientation is deﬁned in the literature as
µ
m
(
z
) =
p
2
z
/
[
p
20
ρ
d
(
z
)], where
p
2
z
=
d
Ω
4
π
¯
ρ
d
(
z,
Ω
)
p
2
z
.The function
µ
m
was studied in Ref. [17] for multipolarions and it was found invariably below the free dipolevalue 1/3 in the SC limit (i.e. dipolar alignment parallelto the wall) and above this value in the WC limit (alignment along the electrostatic ﬁeld). We show in Fig. 3that for a neutral interface, one has
µ
m
(
z
)
<
1
/
3 as inthe SC limit, i.e. the solvent molecules exhibit a tendency to align parallel to the wall over a distance
≈
2 ˚A,that is, until image dipole forces vanish. We now deﬁnean eﬀective dielectric permittivity function of the form˜
ε
eff
(
z
) = 1+4
π
B
p
20
ρ
d
(
z
)
/
3 that solely accounts for thedipolar depletion. The comparison of ˜
ε
eff
(
z
) in Fig. 1.awith ˜
ε
(
z
) shows that the main eﬀect of the dipolar alignment close to the interface is a slight reduction of thelocal dielectric permittivity. However, it is seen that thiseﬀect is largely dominated by the solvent depletion.In the presence of a ﬁnite surface charge, Fig. (3) showsthat interestingly, the function
µ
m
(
z
) exhibits a non