# Charged rotating dilaton black strings in AdS spaces

Ahmad Sheykhi<EMAIL_ADDRESS>Department of Physics, Shahid Bahonar
University, P.O. Box 76175, Kerman, Iran
Research Institute for Astronomy and Astrophysics of Maragha (RIAAM), Maragha,
Iran

###### Abstract

We derive a class of charged rotating dilaton black string solutions in the
background of anti-de-Sitter spaces with an appropriate combination of three
Liouville-type dilaton potentials. We also present the suitable counterterm
which removes the divergences of the action in the presence of dilaton
potential. The solutions are analyzed and their thermodynamics is discussed by
using the counterterm method.

There has been considerable attention in the past years in dilaton gravity. It
is of great importance to investigate the effect of the dilaton field on the
properties of the solutions. It was found that the dilaton field changes the
causal structure of the black hole and leads to the curvature singularities at
finite radii. In the absence of dilaton potential, exact solutions of charged
dilaton black holes have been constructed by many authors CDB1 ; CDB2 . These
black holes are all asymptotically flat. The presence of Liouville-type
dilaton potential, which is regarded as the generalization of the cosmological
constant, changes the asymptotic behavior of the solutions to be neither
asymtotically flat nor (anti)-de Sitter [(A)dS]. Indeed, it has been shown
that with the exception of a pure cosmological constant, no dilaton de-Sitter
or anti-de-Sitter black hole solution exists with the presence of only one
Liouville-type dilaton potential MW . In the presence of one or two Liouville-
type potential, black hole spacetimes which are neither asymptotically flat
nor (A)dS have been explored by many authors (see e.g. CHM ; Cai ; Clem ;
Shey0 ). Although these kind of solutions may shed some light on the possible
extensions of AdS/CFT correspondence, they are physically less interesting due
to their unusual asymptotic behavior.

On the other side, the construction and analysis of black hole solutions in
(A)dS space is a subject of much recent interest. This is primarily due to
their relevance for the AdS/CFT correspondence. It was argued that the
thermodynamics of black holes in AdS spaces can be identified with that of a
certain dual CFT in the high temperature limit Witt . Having the AdS/CFT
correspondence idea at hand, one can gain some insights into thermodynamic
properties and phase structures of strong ’t Hooft coupling CFTs by studying
thermodynamics of AdS black holes. Recently, the dilaton potential leading to
(anti)-de Sitter-like solutions of dilaton gravity has been found Gao1  (see
also Gao2 ). It was shown that the cosmological constant is coupled to the
dilaton in a very nontrivial way. With the combination of three Liouville-type
dilaton potentials, a class of static dilaton black hole solutions in (A)dS
spaces has been obtained by using a coordinates transformation which recast
the solution in the schwarzschild coordinates system Gao1 . The purpose of the
present Letter is to construct a class of charged rotating dilaton black
string solutions in the background of AdS spacetime. We will also present the
suitable counterterm which removes the divergences of the action. Finally, we
analyze the solutions and calculate their conserved and thermodynamic
quantities by using the counterterm method inspired by AdS/CFT correspondence.

Our starting point is the four-dimensional Einstein-Maxwell-dilaton action

$\displaystyle I_{G}$ $\displaystyle=$
$\displaystyle-\frac{1}{16\pi}\int_{\mathcal{M}}d^{4}x\sqrt{-g}\left(R\text{
}-2\partial_{\mu}\Phi\partial^{\mu}\Phi-V(\Phi)-e^{-2\alpha\Phi}F_{\mu\nu}F^{\mu\nu}\right)$
(1)
$\displaystyle-\frac{1}{8\pi}\int_{\partial\mathcal{M}}d^{3}x\sqrt{-\gamma}\Theta(\gamma),$

where ${R}$ is the Ricci scalar curvature, $\Phi$ is the dilaton field, and
$V(\Phi)$ is a potential for $\Phi$. $\alpha$ is a constant determining the
strength of coupling of the scalar and electromagnetic field,
$F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ is the
electromagnetic field tensor and $A_{\mu}$ is the electromagnetic potential.
The last term in Eq. (1) is the Gibbons-Hawking surface term. It is required
for the variational principle to be well-defined. The factor $\Theta$
represents the trace of the extrinsic curvature for the boundary
${\partial\mathcal{M}}$ and $\gamma$ is the induced metric on the boundary.
While $\alpha=0$ corresponds to the usual Einstein-Maxwell-scalar theory,
$\alpha=1$ indicates the dilaton-electromagnetic coupling that appears in the
low energy string action in Einstein’s frame. In this Letter, we examine
action (1) with three Liouville-type dilaton potentials Gao1

$V(\Phi)=\frac{2\Lambda}{3(\alpha^{2}+1)^{2}}\left[{\alpha}^{2}\left(3\,{\alpha}^{2}-1\right){e^{-2\Phi/\alpha}}+\left(3-{\alpha}^{2}\right){e^{2\,\alpha\,\Phi}}+8\,{\alpha}^{2}{e^{\Phi(\alpha-1/\alpha)}}\right],$
(2)

where $\Lambda$ is the cosmological constant. It is clear the cosmological
constant is coupled to the dilaton in a very nontrivial way. This type of
dilaton potential can be obtained when a higher dimensional theory is
compactified to four dimensions, including various supergravity models Gid .
The equations of motion can be obtained by varying the action (1) with respect
to the gravitational field $g_{\mu\nu}$, the dilaton field $\Phi$, and the
gauge field $A_{\mu}$ which yields the following field equations

${R}_{\mu\nu}=2\partial_{\mu}\Phi\partial_{\nu}\Phi+\frac{1}{2}g_{\mu\nu}V(\Phi)+2e^{-2\alpha\Phi}\left(F_{\mu\eta}F_{\nu}^{\text{
}\eta}-\frac{1}{4}g_{\mu\nu}F_{\lambda\eta}F^{\lambda\eta}\right),$ (3)
$\nabla^{2}\Phi=\frac{1}{4}\frac{\partial
V}{\partial\Phi}-\frac{\alpha}{2}e^{-2\alpha\Phi}F_{\lambda\eta}F^{\lambda\eta},$
(4) $\partial_{\mu}\left(\sqrt{-g}e^{-2\alpha\Phi}F^{\mu\nu}\right)=0.$ (5)

Our aim here is to construct charged rotating black string solutions of the
field equations (3)-(5) and investigate their properties. The metric of four-
dimensional rotating solution with cylindrical or toroidal horizons can be
written as Lem

$\displaystyle ds^{2}$ $\displaystyle=$ $\displaystyle-f(r)\left(\Xi dt-
ad\phi\right)^{2}+r^{2}R^{2}(r)\left(\frac{a}{l^{2}}dt-\Xi
d\phi\right)^{2}+\frac{dr^{2}}{f(r)}+\frac{r^{2}}{l^{2}}R^{2}(r)dz^{2},$
$\displaystyle\Xi^{2}$ $\displaystyle=$ $\displaystyle 1+\frac{a^{2}}{l^{2}},$
(6)

where $a$ is the rotation parameter. The functions $f(r)$ and $R(r)$ should be
determined and $l$ has the dimension of length which is related to the
constant $\Lambda$ by the relation $l^{2}=-3/\Lambda$. The two dimensional
space, $t$=constant and $r$ =constant, can be (i) the flat torus model $T^{2}$
with topology $S^{1}\times S^{1}$, and $0\leq\phi<2\pi$, $0\leq z<2\pi l$,
(ii) the standard cylindrical model with topology $R\times S^{1}$, and
$0\leq\phi<2\pi$, $-\infty<z<\infty$, and (iii) the infinite plane $R^{2}$
with $-\infty<\phi<\infty$ and $-\infty<z<\infty$. We will focus upon (i) and
(ii). The Maxwell equation (5) can be integrated immediately to give

$\displaystyle F_{tr}$ $\displaystyle=$ $\displaystyle\frac{q\Xi
e^{2\alpha\Phi}}{r^{2}R^{2}(r)},$ $\displaystyle F_{\phi r}$ $\displaystyle=$
$\displaystyle-\frac{a}{\Xi}F_{tr},$ (7)

where $q$, an integration constant, is related to the electric charge of black
string. Inserting the Maxwell fields (Charged rotating dilaton black strings
in AdS spaces) and the metric (6) in the field equations (3) and (4), we can
write these equation for $a=0$ as

$\displaystyle
2r^{4}R^{3}R^{\prime}f^{\prime}+r^{4}R^{4}f^{\prime\prime}+2r^{3}R^{4}f^{\prime}+r^{4}R^{4}V(\Phi)-2q^{2}e^{2\alpha\Phi}=0,$
(8) $\displaystyle
2r^{3}R^{4}f^{\prime}+r^{4}R^{4}f^{\prime\prime}+8r^{3}R^{3}fR^{\prime}+4r^{4}R^{3}fR^{\prime\prime}+2r^{4}R^{3}R^{\prime}f^{\prime}$
$\displaystyle+4r^{4}R^{4}f\Phi^{\prime
2}+r^{4}R^{4}V\left(\Phi\right)-2q^{2}e^{2\alpha\Phi}=0,$ (9) $\displaystyle
8r^{3}R^{3}fR^{\prime}+2r^{3}R^{4}f^{\prime}+2r^{4}R^{3}f^{\prime}R^{\prime}+2r^{2}R^{4}f+2r^{4}R^{2}fR^{\prime
2}$
$\displaystyle+2r^{4}R^{3}fR^{\prime\prime}+r^{4}R^{4}V\left(\Phi\right)+2q^{2}e^{2\alpha\Phi}=0,$
(10) $\displaystyle
r^{4}R^{4}{\Phi^{\prime}}{f^{\prime}}+r^{4}R^{4}{\Phi^{\prime\prime}}{f}+2r^{3}R^{4}\Phi^{\prime}f+2r^{4}R^{3}R^{\prime}{\Phi^{\prime}}{f}-r^{4}R^{4}\frac{\partial{V}}{4\partial{\Phi}}-\alpha
q^{2}e^{2\alpha\Phi}=0,$ (11)

where the “prime” denotes differentiation with respect to $r$. Subtracting Eq.
(9) from Eq. (8) gives

$\displaystyle 2R^{\prime}+rR^{\prime\prime}+rR\Phi^{\prime 2}=0.$ (12)

Then we make the ansatz

$R(r)=e^{\alpha\Phi}.$ (13)

Substituting this ansatz in Eq. (12), it reduces to

$\displaystyle
r\alpha\Phi^{\prime\prime}+2\alpha\Phi^{\prime}+r(1+{\alpha}^{2})\Phi^{\prime
2}=0,$ (14)

which has a solution of the form

$\Phi(r)=\frac{\alpha}{\alpha^{2}+1}\ln(1-\frac{b}{r}),$ (15)

where $b$ is an integration constants. Inserting (15), the ansatz (13), and
the dilaton potential (2) into the field equations (8)-(11), one can show that
these equations have the following solution

$f(r)=-\frac{c}{r}\left(1-{\frac{b}{r}}\right)^{{\frac{1-{\alpha}^{2}}{1+{\alpha}^{2}}}}-\frac{\Lambda}{3}\,{r}^{2}\left(1-{\frac{b}{r}}\right)^{{\frac{2{\alpha}^{2}}{{\alpha}^{2}+1}}},$
(16)

where $c$ is an integration constant. The above solutions will fully satisfy
the system of equations (8)-(11) provided we have $q^{2}(1+\alpha^{2})=bc$.
One can also check that these solutions satisfy equations (3)-(5) in the
rotating case where $a\neq 0$. It is apparent that this spacetime is
asymptotically (anti)-de-Sitter. The Kretschmann scalar
$R_{\mu\nu\lambda\kappa}R^{\mu\nu\lambda\kappa}$ and the Ricci scalar $R$
diverge at $r=0$ and therefore there is an essential singularity located at
$r=0$. The explicit form of the Kretschmann scalar is complicated and we do
not present it here, however, the Ricci scalar curvature of the metric has a
simpler form and can be written as

$R=-\frac{2\,{q}^{2}{\alpha}^{2}b\left(1-{\frac{b}{r}}\right)^{{\frac{1-\alpha^{2}}{{1+\alpha}^{2}}}}}{\left(r-b\right)^{2}\left({\alpha}^{2}+1\right){r}^{3}}+\frac{2\Lambda\,\left(1-{\frac{b}{r}}\right)^{{\frac{2{\alpha}^{2}}{{\alpha}^{2}+1}}}\left(2\left(r-b\right)^{2}+4\,{\alpha}^{2}{r}(r-b)+2\,{\alpha}^{4}{r}^{2}-{\alpha}^{2}{b}^{2}\right)}{\left({\alpha}^{2}+1\right)^{2}\left(r-b\right)^{2}}.$
(17)

One can easily check that for the arbitrary values of the dilaton coupling
constant, $R\rightarrow 4\Lambda$ as $r\rightarrow\infty$. We also find out
that for arbitrary $\alpha$ the Kretschmann scalar approach $8\Lambda^{2}/3$
as $r\rightarrow\infty$. This confirm our above discussion that our solution
is asymptotically anti-de-Sitter. In the absence of a nontrivial dilaton
($\alpha=0$), the solution reduces to the asymptotically (anti)-de-Sitter
charged rotating black string Lem . However, for $\alpha\neq 0$ the solution
is qualitatively different. As one can see from Eq. (17), the surface $r=b$ is
a curvature singularity except for the case $\alpha=0$ when it is a
nonsingular inner horizon. This is consistent with the idea that the inner
horizon is unstable in the Einstein-Maxwell theory. Therefore, our solutions
describe black strings in the case $b<r_{+}$, where $r_{+}$ is the outer
horizon of the black string (the root of Eq. $f(r)=0$). The above discussions
will become more clear if we look on the figures 1 and 2, where we have
plotted the function $f(r)$ versus $r$ for different values of the dilaton
coupling $\alpha$ and the charge parameter $q$. In the case $r=b$, it is clear
from (Charged rotating dilaton black strings in AdS spaces) and (13) that for
$\alpha\neq 0$ the area of the event horizon goes to zero (since
$R(r)\rightarrow 0$ in this case).

In summary, compared to the charged black holes/strings in (A)dS universe, the
dilaton version has some remarkable properties. In the first place, there may
be three horizons in the Reissner Nordstr m de Sitter spacetime, i.e., black
hole event horizon, black hole Cauchy horizon and cosmic event horizon.
However, the charged dilaton black holes/strings in (A)dS spaces has at most
two horizons. Here the inner Cauchy horizon disappears. This is due to the
fact that the inner horizon is unstable, as pointed by Garfinkle CDB2 .

Figure 1: The function $f(r)$ versus $r$ for $\alpha=1$, $b=2$ and
$\Lambda=-1$. $q=0.5$ (bold line), $q=1$ (continuous line) and $q=1.5$ (dashed
line).

Figure 2: The function $f(r)$ versus $r$ for $b=1$, $q=2$ and $\Lambda=-1$.
$\alpha=0$ (bold line), $\alpha=1$ (dashed line).

Next, we calculate the conserved quantities of the solutions. For
asymptotically (anti)-de-Sitter solutions, the way that one can calculate
these quantities and obtain finite values for them is through the use of the
counterterm method inspired by (A)dS/CFT correspondence Mal . In this paper we
deal with the spacetimes with zero curvature boundary, $R_{abcd}(\gamma)=0$,
and therefore the counterterm for the stress energy tensor should be
proportional to $\gamma^{ab}$. We find the suitable counterterm which removes
the divergences in the form

$I_{ct}=-\frac{1}{8\pi}\int_{\partial\mathcal{M}}d^{3}x\sqrt{-\gamma}\left(-\frac{1}{l}+\frac{\sqrt{-6V(\Phi)}}{2}\right).$
(18)

One may note that in the absence of a dilaton field where we have
$V(\Phi)=2\Lambda=-6/l^{2}$, the above counterterm has the same form as in the
case of asymptotically (A)dS solutions with zero-curvature boundary. Having
the total finite action $I=I_{G}+I_{\mathrm{ct}}$ at hand, one can use the
quasilocal definition to construct a divergence free stress-energy tensor BY .
Thus the finite stress-energy tensor in four dimensional Einstein-dilaton
gravity with three Liouville-type dilaton potentials (2) can be written as

$T^{ab}=\frac{1}{8\pi}\left[\Theta^{ab}-\Theta\gamma^{ab}+\left(-\frac{1}{l}+\frac{\sqrt{-6V(\Phi)}}{2}\right)\gamma^{ab}\right],$
(19)

The first two terms in Eq. (19) are the variation of the action (1) with
respect to $\gamma_{ab}$, and the last two terms are the variation of the
boundary counterterm (18) with respect to $\gamma_{ab}$. To compute the
conserved charges of the spacetime, one should choose a spacelike surface
$\mathcal{B}$ in $\partial\mathcal{M}$ with metric $\sigma_{ij}$, and write
the boundary metric in ADM (Arnowitt-Deser-Misner) form:

$\gamma_{ab}dx^{a}dx^{a}=-N^{2}dt^{2}+\sigma_{ij}\left(d\varphi^{i}+V^{i}dt\right)\left(d\varphi^{j}+V^{j}dt\right),$

where the coordinates $\varphi^{i}$ are the angular variables parameterizing
the hypersurface of constant $r$ around the origin, and $N$ and $V^{i}$ are
the lapse and shift functions respectively. When there is a Killing vector
field $\mathcal{\xi}$ on the boundary, then the quasilocal conserved
quantities associated with the stress tensors of Eq. (19) can be written as

$Q(\mathcal{\xi)}=\int_{\mathcal{B}}d^{2}x\sqrt{\sigma}T_{ab}n^{a}\mathcal{\xi}^{b},$
(20)

where $\sigma$ is the determinant of the metric $\sigma_{ij}$, $\mathcal{\xi}$
and $n^{a}$ are, respectively, the Killing vector field and the unit normal
vector on the boundary $\mathcal{B}$. For boundaries with timelike
($\xi=\partial/\partial t$) and rotational
($\varsigma=\partial/\partial\varphi$) Killing vector fields, one obtains the
quasilocal mass and angular momentum

$\displaystyle M$ $\displaystyle=$
$\displaystyle\int_{\mathcal{B}}d^{2}x\sqrt{\sigma}T_{ab}n^{a}\xi^{b},$ (21)
$\displaystyle J$ $\displaystyle=$
$\displaystyle\int_{\mathcal{B}}d^{2}x\sqrt{\sigma}T_{ab}n^{a}\varsigma^{b}.$
(22)

These quantities are, respectively, the conserved mass and angular momenta of
the system enclosed by the boundary $\mathcal{B}$. Note that they will both
depend on the location of the boundary $\mathcal{B}$ in the spacetime,
although each is independent of the particular choice of foliation
$\mathcal{B}$ within the surface $\partial\mathcal{M}$. The mass and angular
momentum per unit length of the string when the boundary $\mathcal{B}$ goes to
infinity can be calculated through the use of Eqs. (21) and (22). We find

$\displaystyle{M}$ $\displaystyle=$
$\displaystyle\frac{\alpha^{2}(\alpha^{2}-1)b^{3}}{24\pi
l^{3}(\alpha^{2}+1)^{3}}+\frac{(3\Xi^{2}-1)c}{16\pi l},$ (23)
$\displaystyle{J}$ $\displaystyle=$ $\displaystyle\frac{3\Xi
c\sqrt{\Xi^{2}-1}}{16\pi}.$ (24)

For $a=0$ ($\Xi=1$), the angular momentum per unit volume vanishes, and
therefore $a$ is the rotational parameters of the spacetime. The entropy of
the dilaton black string still obeys the so called area law of the entropy
which states that the entropy of the black hole is a quarter of the event
horizon area Beck . This near universal law applies to almost all kinds of
black holes, including dilaton black holes, in Einstein gravity hunt . It is a
matter of calculation to show that the entropy per unit length of the string
is

${S}=\frac{r_{+}^{2}\Xi(1-\frac{b}{r_{+}})^{\frac{2\alpha^{2}}{\alpha^{2}+1}}}{4l}.$
(25)

By analytic continuation of the metric we can obtain the temperature and
angular velocity of the horizon. The analytical continuation of the Lorentzian
metric by $t\rightarrow i\tau$ and $a\rightarrow ia$ yields the Euclidean
section, whose regularity at $r=r_{+}$ requires that we should identify
$\tau\sim\tau+\beta_{+}$ and $\phi\sim\phi+i\beta_{+}\Omega_{+}$ where
$\beta_{+}$ and $\Omega_{+}$ are the inverse Hawking temperature and the
angular velocity of the horizon. We find

$\displaystyle T_{+}$ $\displaystyle=$ $\displaystyle\frac{f^{\text{
}^{\prime}}(r_{+})}{4\pi\Xi}=\frac{\left(1-{\frac{b}{r_{+}}}\right)^{{\frac{{\alpha}^{2}-1}{{\alpha}^{2}+1}}}}{2\pi
r_{+}^{2}\Xi({\alpha}^{2}+1)}\left(-\frac{\Lambda}{3}r_{+}^{2}\left({\alpha}^{2}r_{+}+r_{+}-b\right)+\frac{c\left({\alpha}^{2}r_{+}+r_{+}-2\,b\right)}{2r_{+}}\left(1-{\frac{b}{r_{+}}}\right)^{{\frac{1-3\,{\alpha}^{2}}{{\alpha}^{2}+1}}}\right)$
$\displaystyle\Omega_{+}$ $\displaystyle=$ $\displaystyle\frac{a}{\Xi l^{2}}.$
(26)

The next quantity we are going to calculate is the electric charge of the
string. To determine the electric field we should consider the projections of
the electromagnetic field tensor on special hypersurface. The normal vectors
to such hypersurface are

$u^{0}=\frac{1}{N},\text{ \ }u^{r}=0,\text{ \ }u^{i}=-\frac{V^{i}}{N},$ (27)

where $N$ and $V^{i}$ are the lapse function and shift vector. Then the
electric field is $E^{\mu}=g^{\mu\rho}e^{-2\alpha\phi}F_{\rho\nu}u^{\nu}$, and
the electric charge per unit length of the string can be found by calculating
the flux of the electric field at infinity, yielding

${Q}=\frac{\Xi q}{4\pi l}.$ (28)

The electric potential $U$, measured at infinity with respect to the horizon,
is defined by Cvetic

$U=A_{\mu}\chi^{\mu}\left|{}_{r\rightarrow\infty}-A_{\mu}\chi^{\mu}\right|_{r=r_{+}},$
(29)

where $\chi=\partial_{t}+\Omega\partial_{\phi},$ is the null generator of the
event horizon. One can easily show that the vector potential $A_{\mu}$
corresponding to the electromagnetic tensor (Charged rotating dilaton black
strings in AdS spaces) can be written as

$A_{\mu}=-\frac{q}{r}\left(\Xi\delta_{\mu}^{t}-a\delta_{\mu}^{\phi}\right).$
(30)

Substituting (30) in (29) we obtain the electric potential as

$U=\frac{q}{\Xi r_{+}}.$ (31)

Having the conserved and thermodynamic quantities of the rotating dilaton
black string at hand, we are in a position to check the first law of black
hole thermodynamics. Numerical calculations show that the conserved quantities
calculated above satisfy the first law of thermodynamics

$dM=TdS+\Omega d{J}+Ud{Q}.$ (32)

In conclusion, we constructed a new class of charged rotating solutions in
four-dimensional Einstein-Maxwell-dilaton gravity with cylindrical or toroidal
horizons in the presence of dilaton potentials and investigate their
properties. These solutions are asymptotically anti-de Sitter. The
cosmological constant couples the dilaton field in a nontrivial way. The
dilaton potential with respect to the cosmological constant includes three
Liouville-type potentials. This is consistent with the arguments in MW  that
no (anti)-de-Sitter version of dilaton black holes exist with only one
Liouville-type dilaton potential. We found the suitable counterterm which
removes the divergences of the action in the presence of three Liouville-type
dilaton potential. We also computed the conserved and thermodynamic quantities
of the solutions by using the conterterm method and verified numerically that
these quantities satisfy the first law of black hole thermodynamics.

###### Acknowledgements.

This work has been supported financially by Research Institute for Astronomy
and Astrophysics of Maragha, Iran.

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11institutetext:  Department of Physics and Astronomy, University of
California, Los Angeles - California 90095, USA

# The Debye layer as a transmission line in the 4 Hz - 100 kHz frequency range

Thanh-Trị Châu 11 Giovanni Zocchi 1111

###### Abstract

We report measurements on the dynamics of the Debye layer at a gold electrode
in several electrolytes. In the experiments, the Debye layer transmits a
damped voltage wave along the electrode, which we use to probe the dynamics.
We compare the measurements with traditional impedance models, which
schematize the Debye layer as a capacitance. We find good agreement for very
dilute electrolytes, but also for an ionic liquid. However, the same model
fails for the concentrated electrolyte, as ion - solvent - ion interactions
become important.

##  1 Introduction

A charged surface in an electrolyte drives the formation of a cloud of
counterions within a thin boundary layer (Debye layer) next to the surface.
This electric double layer (EDL) has a profound effect on macromolecular and
colloidal interactions. Ionic screening transforms the long range Coulomb
interaction into a short range force. The large electric field within the
Debye layer (typically of order $100\,\mathrm{mV}/1\,\mathrm{nm}$) can affect
a variety of chemical and physical processes. For example, the binding
dynamics of biological macromolecules involves the disruption of the EDL at
the surface of contact. Action potential transmission is accompanied by a
large perturbation of the EDL at the cell membrane [10]. Another motivation
for the study of EDL dynamics lies in the development of capacitive energy
storage devices (supercapacitors) and electrochemical processes for energy
conversion [4, 6].
Direct experimental measurements of the Debye layer are challenging, due to
the $\mathrm{nm}$ scale. The static properties, such as the concentration
profile of ions in the diffuse layer near a charged surface, can be deduced
from measurements of the force between such surfaces [7, 15]. The
corresponding theoretical framework is based on the Poisson-Boltzmann equation
and known as the Gouy-Chapman theory. Measurements of the dynamics are mostly
obtained from electrochemical cells. There are three common methods [4]. In
Electrochemical Impedance Spectroscopy (EIS) a voltage consisting of a DC
component plus a sine wave is imposed; the measured quantity is the current.
The measurements are often presented in terms of a complex impedance vs
frequency (a Nyquist plot). Cyclic Voltammetry (CV) is essentially the same
measurement with a different voltage protocol (a triangular wave). In
Galvanometric Cycling the electrode current is imposed and the measured
quantity is the voltage. Physical quantites of interest, such as the EDL
capacitance, are extracted by comparing to a model [4, 6]. Increasingly,
models are compared with MD simulations, which provide quantities (such as the
structure of the EDL) only indirectly accessible to experiments [11, 12].
To a first approximation, and for non-Fradaic processes, the EDL behaves like
a capacitor, charging and discharging as charges on the two sides of the
electrode-electrolyte interface accumulate and disperse without an actual
charge transfer across the interface. To introduce the characteristic scales,
let us consider a planar electrode in a 1:1 electrolyte (such as NaCl in
water). From the linearized Debye-Falkenhagen equation for the charge density
$\rho$:

$\frac{\partial\rho}{\partial t}-\chi
k_{\mathrm{B}}T\,\nabla^{2}\rho+\frac{2\chi}{\epsilon}|e|^{2}n_{0}\,\rho=0$
(1)

where $\chi$ is the mobility of the ions (for simplicity assumed the same for
the two species e.g. Na+ and Cl-), $n_{0}$ the bulk concentration of NaCl,
$|e|$ the proton charge and $\epsilon$ the dielectric constant of the medium,
one obtains the characteristic length scale $\ell=\sqrt{(\epsilon
k_{\mathrm{B}}T)/(2|e|^{2}n_{0})}$ (the Debye length) and time scale
$\tau=\ell^{2}/(\chi
k_{\mathrm{B}}T)=\epsilon/(2|e|^{2}n_{0}\chi)=\epsilon/\sigma$ ($\sigma$ is
the ionic conductivity of the solution). Below we write the equations in
dimensionless form using these characteristic scales and scaling $\rho$ by
$2|e|n_{0}$. In an axis-symmetric geometry with the $z$ axis perpendicular to
the electrode, (1) becomes:

$\frac{\partial\rho}{\partial t}-\frac{\partial^{2}\rho}{\partial
z^{2}}+\rho=0\,.$ (2)

The electrostatic potential $\phi$ is related to $\rho$ through the Poisson
equation; scaling $\phi$ by $k_{\mathrm{B}}T/|e|$, it reads:

$\frac{\partial^{2}\phi}{\partial z^{2}}=-\rho\,.$ (3)

With sinusoidal forcing $\phi(z=0,t)=\exp(\mathrm{i}\omega t)$ and
$\phi,\,\rho\rightarrow 0$ for $z\rightarrow\infty$ (in the bulk), the
solution of (2) and (3) is:

$\displaystyle\phi(z,t)$ $\displaystyle=$ $\displaystyle
e^{-kz}e^{\mathrm{i}\omega t}\,,$ (4) $\displaystyle\ \rho(z,t)$
$\displaystyle=$ $\displaystyle-k^{2}e^{-kz}e^{\mathrm{i}\omega t}\,,$ (5)
with $\displaystyle k^{2}=1+\mathrm{i}\omega\,.$

We are concerned only with the regime $\omega\ll 1$ , in which case $k\approx
1+\mathrm{i}\omega/2$ and (5) describes the formation of a Debye layer of size
one (size $\ell$ in dimensional variables), independent of frequency (apart
from a slight phase lag). The reason is that, taking as an example a
$150\,\mathrm{mM}$ (“physiological”) NaCl solution, $\sigma\approx
10\,\mathrm{mS/cm}$, $n_{0}\approx 10^{20}\,\mathrm{cm}^{-3}$ so that
$\tau\approx 1\,\mathrm{ns}$ and $\ell\approx 1\,\mathrm{nm}$ .
The capacitive current at the electrode is obtained by considering the
relation of the surface charge to the electric field: $\epsilon E_{n}=Q/A$
where $Q/A$ is the charge per unit area and $E_{n}$ the normal component of
the electric field at the surface. The capacitive current density $j$ is then
$j=(\partial/\partial t)(Q/A)=\epsilon(\partial/\partial
t)E_{n}=-\epsilon(\partial/\partial
t)(\vecfont{\nabla}\phi\cdot\vecfont{n})_{z=0}$ with $\vecfont{n}$ the unit
normal. Using (4) in the regime $\omega\ll 1$ we then find
$|j|\approx\epsilon\,\omega(1+\omega^{2}/8)$, to be compared with a driven RC
circuit, where, in the same approximation, the current is $|I|\approx
C\omega(1-\omega^{2}/2)$. We see that the frequency behavior of the capacitive
current at the electrode departs from that of an RC circuit only for
$\omega\sim 1$. For this reason one associates to the Debye layer a
capacitance per unit area
$c=|j|/|\dot{\phi}(z=0)|\approx\epsilon(1+\omega^{2}/8)$ which, for $\omega\ll
1$, is the constant $\epsilon$ (or $\epsilon/l$ in dimensional variables).
However, this simple theory neglects hydrodynamic effects, as well as
interactions between ions, and between ions and the electrode.
Here we probe Debye layer dynamics through an experimental configuration where
the Debye layer forms part of an RC transmission line. We measure voltage
transmission at the end of the line, given a sinusoidal input at the other
end. The Debye layer at a planar gold electrode provides the capacitive part
of the line, while the resistive part is provided by the thin film gold layer
which forms the electrode. It is helpful to reason in terms of the continuum
limit of the equivalent discrete elements circuit. As a first approximation,
Fig. 1 is a schematic of the (1D) transmission line if the EDL can be
considered as a distributed capacitance with capacitance per unit length $c$ ;
$r$ is the resistance per unit length of the gold electrode. The potential
along the line satisfies the diffusion equation

$\frac{\partial V(x,t)}{\partial
t}-\frac{1}{rc}\frac{\partial^{2}V(x,t)}{\partial x^{2}}=0\,.$ (6)

For AC driving voltage $V(x=0,t)=V_{\mathrm{in}}e^{\mathrm{i}\omega t}$ with
the Ansatz

$V(x,t)=\exp(-kx)\exp(\mathrm{i}\omega t)$ (7)

[width=1]plots/RC_schematic.png

Hình 1: Schematic of the Debye layer at the gold electrode as a distributed RC
transmission line in 1D. The resistive path in the gold is of resistance $r$
per unit length and the electrolyte within the Debye layer is considered as a
capacitor with capacitance $c$ per unit length. $R$ is a trailing resistance,
present in the experiments, between the input AC source and the starting point
$x=0$ of the line. In the experiments, we measure the amplitude and phase of
$V_{\mathrm{out}}$ relative to $V_{\mathrm{in}}$, at $x=L$.

one obtains a complex wave vector

$k=(1+\mathrm{i})\sqrt{\frac{\omega rc}{2}}\,.$ (8)

In this simple model, for increasing frequency the output voltage
$V_{\mathrm{out}}$ has an amplitude going exponentially to zero and a
monotonously decreasing phase (modulo $2\pi$).

##  2 Materials, sample preparation and experimental procedure

The electrolyte solutions under study are NaCl aqueous solutions of
$10\,\mathrm{mM}$, $100\,\mathrm{mM}$ and $1\,\mathrm{M}$ concentrations in
$10\,\mathrm{mM}$ NaH2PO4/Na2HPO4 buffer with pH $7$ at $25$ ℃. We choose
phosphate buffer because its $\mathrm{pK_{a}}$ at the second step is close to
the neutral $\mathrm{pH}$ and has modest variation with
temperature.111$\mathrm{pK_{a}2=7.21}$ with a variation of approximately
$-0.0028/\text{\textcelsius}$. The ionic liquid is 1-Butyl-3-methylimidazolium
hexafluorophosphate (C8H15F6N2P). All chemicals are from Sigma-Aldrich.
The ionic liquid or salt solution is sealed with a UV-activated epoxy (Loctite
AA 3525) between two gold-coated glass slides arranged perpendicular to one
another and separated by two spacer strips $120\,\mathrm{\mu m}$ thick (see
Fig. 2). The top plate provides a resistive path for the voltage signal and is
prepared by depositing $3\,\mathrm{nm}$ of Cr followed by $20\,\mathrm{nm}$ of
Au on a glass slide using electron-beam evaporation method (CHA Mark 40). The
bottom plate, having an extra $10\,\mathrm{nm}$ of Au compared to the top,
serves as the ground electrode. The contacts are made by soldering gauge-28
wires on a buffering layer of electrically conductive silver paint (MG
Chemicals 842AR-P) applied on the gold layer to prevent the latter from being
peeled off.
We use a lock-in amplifier (SR850) for the measurements. The reference signal
from the lock-in drives an in-house built voltage clamp circuit which applies
the input voltage between $V_{\mathrm{in}}$ and ground, as in Fig. 2. By
switching the input to the detection path of the lock-in between
$V_{\mathrm{in}}$ and $V_{\mathrm{out}}$, we measure both their amplitude and
phase relative to the reference. We deduce from these measurements their
relative amplitude and phase difference. For each experimental condition, we
keep the temperature of the sample stable within $\pm 0.01\text{\textcelsius}$
in an in-house built thermoelectrically controlled chamber.

[width=0.9]plots/sample_schematic.png

Hình 2: Schematic of an experimental sample. The electrolyte is sealed in a
$120\,\mathrm{\mu m}$ thick cell obtained from two microscope slides separated
by spacers. The inner surface of the slides is conductive through a thin layer
of gold evaporated on it. One electrode serves as the ground, while the other
provides a resistive path for the transmission line.

##  3 Results

Fig. 3 shows the ratio of the rms amplitudes of the voltage at the end of the
electrolytic cell, $V_{\mathrm{out}}$, to the input voltage $V_{\mathrm{in}}$,
as a function of the square root of the driving frequency (blue symbols). Also
shown is the phase of $V_{\mathrm{out}}$ relative to $V_{\mathrm{in}}$ (red
symbols). Measurements are displayed for 3 different NaCl concentrations of
the electrolyte: 10, 100, and 1000 $\mathrm{mM}$. In the text and in the
figures we refer to the above NaCl concentrations to identify the samples,
however all these solutions also contain $10\,\mathrm{mM}$ phosphate buffer,
so for example the total ionic strength of the “$10\,\mathrm{mM}$ salt”
samples is actually $20\,\mathrm{mM}$. We notice immediately that at high
enough frequencies the behavior of both the amplitude and phase of the output
signal is non monotonic with salt concentration; this feature cannot be
explained by a model based on the mean field theory of the Introduction.
Equation (8) predicts that the amplitudes in Fig. 3 should decrease linearly
on this log-linear scale above some small frequency222The behavior at low
frequencies reflects the finite size of the transmission line, among other
things..

[width=]plots/Fig_3_P.png

Hình 3: Measured amplitude (blue) and phase (red) of
$V_{\mathrm{out}}/V_{\mathrm{in}}$ vs the square root of the driving frequency
$f=\omega/2\pi$ for three concentrations of buffered NaCl solution:
$10\,\mathrm{mM}$ (filled squares), $100\,\mathrm{mM}$ (empty squares), and
$1\,\mathrm{M}$ (stars). The amplitude is shown on a log scale, the phase on a
linear scale. The driving amplitude was
$V_{\mathrm{in}}=24\,\mathrm{mV}\,\mathrm{rms}$, and the temperature $25$ ℃.

Similarly the phase should decrease linearly on the same scale with jumps from
$-\pi$ to $\pi$ due to its periodic nature. Up to some moderate frequency,
such as $110^{2}=12.1\,\mathrm{kHz}$ for $100\,\mathrm{mM}$, this is indeed
the case. However at higher frequency, the amplitude saturates to some
constant level while the phase goes back up to zero. The reason is that in the
experiment there are two Debye layers, one at the “live” and one at the ground
electrode, connected by a resistive path through the electrolyte. A more
appropriate transmission line model is therefore as shown in Fig. 4, where the
ground electrode is endowed with its own (capacitive) Debye layer and the
conductance (per unit length) $\sigma$ refers to conduction through the
electrolyte in the direction orthogonal to the plates.

[width=0.9]plots/RCR_schematic.png

Hình 4: Schematic of the 1D transmission line model where the electrolyte is
considered as two capacitors sandwiching a series resistor. Each capacitor has
a value of $2c$ per unit length, whereas the resistor corresponds to a
conductance of $\sigma$ per unit length. The transmission line electrode has
resistance per unit length $r$, whereas the ground electrode is an
equipotential, due to the thicker gold layer on it.

We now consider this 1D transmission line, of finite length $L$, and solve
analytically for the output voltage $V_{\mathrm{out}}\equiv V(L)$, under
harmonic driving. Ohm’s law relates the current $I(x)$ through the resistive
gold layer to the voltage $V(x)$ at an arbitrary point along the transmission
line as

$I(x)=-\frac{1}{r}\frac{\partial V(x)}{\partial x}\,.$ (9)

We atttribute the spatial variation of this current to a capacitive current
per unit length $I_{\mathrm{c}}(x)$, related to the local voltage $V(x)$
through the impedance of the electrolyte strip of width $\mathrm{d}x$:

$-\frac{\partial I(x)}{\partial
x}=I_{\mathrm{c}}(x)=V(x)\left(\frac{1}{\mathrm{i}\omega
c}+\frac{1}{\sigma}\right)^{-1}\,.$ (10)

As a result, the voltage $V(x)$ satisfies

$\frac{\partial^{2}V(x)}{\partial x^{2}}=\frac{\mathrm{i}\omega
rc\sigma}{\mathrm{i}\omega c+\sigma}V(x)$ (11)

where $\omega$ is the forcing frequency. We solve eq. (11) under boundary
conditions that account for a vanishing current at $x=L$

$V^{\prime}(L)\sim I(L)=0$

and a voltage drop across the trailing resistance $R$

$V_{\mathrm{in}}+\frac{R}{r}V^{\prime}(0)=V(0)\,.$

The result for the complex output $V_{\mathrm{out}}$ is

$\frac{V_{\mathrm{out}}}{V_{\mathrm{in}}}=\frac{1}{\cosh(kL)+\alpha
kL\sinh(kL)},$ (12)

with

$\alpha=\frac{R}{rL}$ (13)

being the ratio between the trailing resistance $R$ and the resistance of the
metal layer in direct contact with the electrolyte.
The real and imaginary parts of the wave vector $k$ are:

$\begin{split}k^{\prime}L&=\left[\frac{\omega/\omega_{rc}\left(\sqrt{1+\omega^{2}/\omega_{c\sigma}^{2}}+\omega/\omega_{c\sigma}\right)}{2\left(1+\omega^{2}/\omega_{c\sigma}^{2}\right)}\right]^{1/2}\\\
k^{\prime\prime}L&=\left[\frac{\omega/\omega_{rc}\left(\sqrt{1+\omega^{2}/\omega_{c\sigma}^{2}}-\omega/\omega_{c\sigma}\right)}{2\left(1+\omega^{2}/\omega_{c\sigma}^{2}\right)}\right]^{1/2}\,.\end{split}$
(14)

The frequencies $\omega_{rc}$ and $\omega_{c\sigma}$ are set by the charging
time of the capacitors, and limited by the (longitudinal) resistance of the
gold layer and the (transverse) resistance of the electrolyte, respectively:

$\displaystyle\omega_{rc}$ $\displaystyle=$ $\displaystyle\frac{1}{(rL)(cL)}$
(15) $\displaystyle\omega_{c\sigma}$ $\displaystyle=$
$\displaystyle\frac{\sigma}{c}\,.$ (16)

$\alpha$ is not a fitting parameter as we can deduce it from measurements of
the resistive gold layer’s geometry and surface resistance. For a rectangular
gold layer $2.98\pm 0.30\,\mathrm{cm}$ long and $0.93\pm 0.10\,\mathrm{cm}$
wide, we measure a resistance of $9.3\pm 0.1\,\mathrm{\Omega}$. This
corresponds to a surface resistance of $2.9\pm
0.4\,\mathrm{\Omega}/\mathrm{sq}$ ($\mathrm{sq}$ refers to any square patch of
surface). We are left with two fitting parameters: $\omega_{rc}$ and
$\omega_{c\sigma}$.

[width=]plots/Fig_5_P.png

Hình 5: Amplitude and phase of $V_{\mathrm{out}}/V_{\mathrm{in}}$ for sample
A: $100\,\mathrm{mM}$ NaCl in $10\,\mathrm{mM}$ phosphate buffer, pH 7 at $25$
℃ (same data as in Fig. 3). The frequency sweep is carried out at $25$ ℃ and
$24\,\mathrm{mV}$ rms driving voltage. The solid lines show the global two-
parameter fit according to Eq. (12), using a measured $\alpha=1.69$.

[width=]plots/Fig_6_P.png

Hình 6: Amplitude (blue) and phase (red) of $V_{\mathrm{out}}/V_{\mathrm{in}}$
for the ionic liquid C8H15F6N2P, at $25$ ℃ and $24\,\mathrm{mV}$ rms driving
voltage. Solid lines show the two parameter fit to the model Eq. (12), using a
measured $\alpha=1.29$.

The two parameters fit to the model of Eq. (12) is displayed in Fig. 5 for the
$100\,\mathrm{mM}$ salt solution (same data as in Fig. 3). The fit for the
$10\,\mathrm{mM}$ salt is noticeably better than the case shown, while for the
$1\,\mathrm{M}$ case it is noticeably worse (data not shown). The conclusion
we draw is that the model of Fig. 4, which schematizes the Debye layer as a
fixed (frequency independent) capacitance and the bulk electrolyte as a
resistance, describes the frequency dependence fairly well. However we also
see systematic departures from the data, at the level of $10\,\%$ for the
$100\,\mathrm{mM}$ salt and $20\,\%$ for the $1\,\mathrm{M}$ concentration.
The discrepancy between model and experiments is significant as the stability
and reproducibility of the experiment (for the same sample) is of order the
size of the dots in the plots. It cannot be attributed to nonlinearities
arising from voltage dependent processes (such as chemical reactions): the
measurements of Fig. 3, obtained for $V_{\mathrm{in}}=24\,\mathrm{mV}$, were
repeated (on the same samples) for $V_{\mathrm{in}}=50\,\mathrm{mV}$, and
identical results (overlapping symbols, not shown) were obtained.
Intriguingly, the measurements on the ionic liquid, shown in Fig. 6, show no
discrepancy at all with the model (11).
However, the virtues of the RC transmission line model are somewhat tarnished
if one tries to directly relate its effective parameters to the physical
properties of the electrolytes. Table 1 summarizes the parameters obtained
from the fits, for the 3 salt concentrations and the ionic liquid. The
$100\,\mathrm{mM}$ salt condition was measured for two independent samples A
and B. The two fitting parameters are $\omega_{rc}$ and $\omega_{c\sigma}$
(Eq. (14)); $\alpha$ is obtained from the measured conductivity of the gold
layer and the geometry of the sample. The capacitance per unit area of one
EDL, $C$, is obtained from the value of $\omega_{rc}$ using (15) and the
measured geometry (length and width) of the cell. For comparison, the next
column in the Table lists the corresponding capacitance value $\epsilon/\ell$
obtained from the (calculated) Debye length $\ell$. Note that for the $1\,M$
salt and more so for the ionic liquid, the calculated $\ell$ is smaller than
the size of the ions. The last two columns display the bulk conductivity of
the electrolyte obtained from $\omega_{c\sigma}$ using (16) and the cell
geometry, and for comparison the literature values. The quantity
$\epsilon/\ell$, which is proportional to $\sqrt{\epsilon}$, is calculated
using $\epsilon=80$ (relative to the permittivity of free space) for the salt
solutions and $\epsilon=11$ for the ionic liquid [13]. Focusing on the
capacitance $C$, we see that in all cases the EDL capacitance measured in the
experiment is more than an order of magnitude lower than $\epsilon/\ell$,
which is the value calculated assuming a parallel plates capacitor of
thickness $\ell$ (the Debye length). This well known phenomenon is usually
attributed to the existence of a Stern layer of immobilized water molecules
and counterions at the metal surface. In terms of the model of Fig. 4 the
effect is to add a “Stern layer capacitance”
$C_{\mathrm{S}}=\epsilon_{\mathrm{S}}/\delta_{\mathrm{S}}$ in series to the
Debye layer capacitance $C$; $\epsilon_{\mathrm{S}}$ is the dielectric
constant of the Stern layer, $\delta_{\mathrm{S}}$ its thickness [4]. The
composite EDL capacitance is then
$C_{\mathrm{EDL}}=(CC_{\mathrm{S}})/(C+C_{\mathrm{S}})<C_{\mathrm{S}}$ i.e. it
is bounded by $C_{\mathrm{S}}$. With representative values
$\epsilon_{\mathrm{S}}\approx 2$ and $\delta_{\mathrm{S}}\approx
4\mathrm{\AA}$ for the Stern layer (see for example the detailed analysis in
[12]) one obtains $C_{\mathrm{S}}\approx 4.4\,\mathrm{\mu F/cm^{2}}$, not
inconsistent with our measured values for the NaCl electrolytes.
For the conductivity $\sigma$, at low ionic strength (the $10\,\mathrm{mM}$
sample) there is rough agreement between the value obtained from the
experiment (which assumes that the resistors labeled $1/(\sigma\Delta x)$ in
Fig. 4 correspond to the conductivity of the bulk electrolyte) and the actual
bulk conductivity of the electrolyte. But for high ionic strength there is a
discrepancy of more than an order of magnitude. On the other hand, the ionic
liquid again seems to display “ideal” behavior in that there is rough
agreement between the experimental value and the actual bulk conductivity.
Next to ionic strength, temperature is another thermodynamic control parameter
in the experiments, affecting the Debye layer, the Stern layer, and the bulk
conductivity, among other factors. Without providing a comprehensive analysis
in this Letter, we show in Fig. 7 a series of measurements at different
temperatures, for the $100\,\mathrm{mM}$ salt. The data for $25$, $15$, $5$,
and $0$ ${}^{\circ}\mathrm{C}$ are from the same sample, while $-10$ and $-15$
${}^{\circ}\mathrm{C}$ are from another. The monotonic rise of the high
frequency amplitude plateau with decreasing temperature, and corresponding
behavior of the phase, are due to the decrease in the conductivity of the
electrolyte (due to the decreased ion mobility) with decreasing temperature.
At $-15$ ${}^{\circ}\mathrm{C}$ the sample is frozen; there is no Debye layer
(the conductivity $\sigma$ and capacitance per unit length $c$ are essentially
zero) and we obtain a featureless response
($V_{\mathrm{out}}/V_{\mathrm{in}}\approx 1$ and $\Delta\theta\approx 0$).
Note that under our conditions the $100\,\mathrm{mM}$ samples freeze at about
$-10$ ${}^{\circ}\mathrm{C}$, because the cell keeps the sample at
approximately constant volume and therefore under pressure when frozen.

[width=]plots/Fig_7_P.png

Hình 7: Effect of temperature on the amplitude and phase of $V_{\mathrm{out}}/V_{\mathrm{in}}$ , measured for the $100\,\mathrm{mM}$ NaCl solution; drive amplitude $V_{\mathrm{in}}=24\,\mathrm{mV}$. The behavior is monotonic with temperature; the different plots correspond to $T=25^{\circ}\mathrm{C}$ (filled squares), and $15,5,0,-10,-15\,^{\circ}\mathrm{C}$. At $-15\,^{\circ}\mathrm{C}$ (open diamonds) the sample is frozen (whereas at $-10\,^{\circ}\mathrm{C}$ (open circles) it is still liquid, due to the increased pressure).  Bảng 1: The parameters $\omega_{rc}$ and $\omega_{c\sigma}$ (see (12), (14)) obtained from fitting the data to the model Eq. (12) , for the $10\,\mathrm{mM}$ , $100\,\mathrm{mM}$ (two different samples), and $1\,\mathrm{M}$ NaCl electrolytes, and the ionic liquid. The parameter $\alpha$ is measured from the conductivity of the gold electrodes and the geometry of the cells (see (13)). The capacitance per unit area $C$ is derived from $\omega_{rc}$ (see (15)), the ionic conductivity $\sigma$ from $\omega_{c\sigma}$ (see (16)). For comparison, the column $\epsilon/\ell$ displays the capacitance per unit area derived from the calculated Debye length $\ell$, while the column $\sigma_{ref}$ displays literature values of the ionic conductivity. Note that in the Table, C is the capacitance of one Debye layer, corresponding to $2c\,\Delta x$ in Fig. 4. | $\alpha$ | $\omega_{rc}\,[\mathrm{kHz}]$ | $\omega_{c\sigma}\,[\mathrm{kHz}]$ |  $C$ [$\mathrm{\mu F/cm^{2}}$] | $\epsilon/\ell$ | $\sigma\,[\mathrm{mS/cm}]$ | $\sigma_{ref}$
---|---|---|---|---|---|---|---
10 mM | 1.79 | 33.7 | 47.4 | 3.32 | 33 | 0.94 | 1.2
100 mM (A) | 1.69 | 28.8 | 88.4 | 3.90 | 76 | 2.07 | 10.7
100 mM (B) | 1.44 | 22.9 | 82.1 | 4.90 |  | 2.41 |
1 M | 2.11 | 22.2 | 62.5 | 5.08 | 230 | 1.90 |  $\sim$ 90
Ionic Liquid | 1.29 | 20.3 | 31.4 | 5.52 | 192 | 1.04 | 1.5

##  4 Summary and Discussion

We have introduced a setup in which the Debye layer at the electrode-
electrolyte interface forms part of a transmission line along the electrode.
Measurements of the voltage along the electrode, alternative to traditional
current measurements with equipotential electrodes, reflect the dynamics of
the Debye layer. We present measurements for buffered NaCl solutions of
different ionic strengths, in the low voltage regime where hydrolysis and
other chemical reactions are unimportant. We also present a set of
measurements for an ionic liquid. We find that the traditional view of the EDL
as a (frequency independent) capacitance describes the dynamics fairly well in
some cases, but not others. Specifically, for the low salt concentration
electrolyte ($10\,\mathrm{mM}$ NaCl), and, at the other end, for the ionic
liquid, the frequency dependence of the measurements is well described by the
model of Fig. 4. However, for the more concentrated $100\,\mathrm{mM}$ salt
electrolyte, and more so for the $1\,\mathrm{M}$ concentration, there are
large discrepancies between the measurements and the model. The two parameters
measured from the fits, $\omega_{rc}$ and $\omega_{c\sigma}$, can be
converted, within this model, into an effective capacitance for the EDL and an
effective ionic conductivity for the electrolyte. In all cases, the
capacitance values thus measured are an order of magnitude or more smaller
than the values calculated for a parallel plates capacitor of thickness a
Debye length $\ell$ , and using the bulk static dielectric constant of the
medium. This behavior is attributed to the existence of a Stern layer of
immobilized electrolyte molecules at the electrode [14, 12]. The conductivity
values measured from the fits are roughly consistent with the actual bulk
ionic conductivity of the electrolyte for the very dilute salt solution
($10\,\mathrm{mM}$) and for the ionic liquid. However for the more
concentrated salt solutions the conductivity from the fits clearly does not
represent the bulk conductivity. Phenomenologically one could possibly invoke
a reduced mobility of the ions in the region of the Stern / Debye layer. The
more fundamental conclusion is however that the impedance model Fig. 4 misses
part of the physics.
The hydrodynamics of interacting ions close to a surface forms indeed an
interesting mathematical problem, due to the range of scales: the Debye layer
at the $\mathrm{nm}$ scale, the bulk electrolyte at the $\mathrm{\mu m}$ or
$\mathrm{mm}$ scale. The method of matched asymptotic expansions has been used
in this context [1, 5]. The impedance elements models (such as Fig. 4)
circumvent the mathematical difficulties by placing a resistance ($1/\sigma$
in Fig. 4) for the bulk electrolyte, but this is unsatisfactory in general.
The transmission line model based on impedances was introduced in the 60’s to
describe electrochemistry at a porous electrode [2], a subject of renewed
interest today [3]. Remarkably, this simple approach seems to work well for
the ionic liquid. Ion - solvent interactions are absent in this case, a
situation analogous to the ideal behavior of a polymer melt. Moreover,
measurements with the surface force apparatus indicate that ionic liquids
behave as dilute electrolytes in terms of the static properties of the diffuse
layer [16, 17], apparently for the reason that only a small fraction of the
charges are effectively dissociated.
From a purely experimental point of view, looking at the columns $C$ and
$\sigma$ in Table 1, one would think that what is measured are properties of
the electrode rather than the different electrolytes. In fact it is well known
that even in the static case, the structure of the EDL is more complicated
than the result of the mean field Gouy-Chapman-Stern theory suggests [6]. For
high ionic strength, the ion density profile away from the electrode is
oscillatory rather than monotonic [12], while the Debye length may increase
with salt at high enough concentrations [8]. The non-monotonicity with
increasing salt concentration visible in the plots of Fig. 3 may be related to
this latter phenomenon; further measurements with this setup at higher salt
concentrations could be informative. Similarly it should be interesting to
compare our measurements with the predictions from continuum theories which
take into account the finite size of the ions and ion-ion interactions [18,
5].

Measurements with this setup may be extended in a number of ways. A DC bias
can be added to the drive, in order to probe the dynamics with different
electrode potentials. This is routinely done in EIS, where a third (reference)
electrode is normally used to standardize the measurements. The high voltage
regime ($|e|V/k_{\mathrm{B}}T>1$) will introduce nonlinearities and eventually
new processes (water hydrolysis, potentially surface remodelling), and remains
to be studied in this system. Coupling this transmission line configuration to
redox chemical reactions in the electrolyte will generate reaction-diffusion
systems where voltage is one phase space coordinate. These should be
interesting as pattern forming systems.

###### Acknowledgements.

We thank Anastassia Alexandrova for suggesting the ionic liquid measurements.

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