# Dielectric mismatch effects on polyelectrolyte solutions in electrified
nanopores: Insights from mean-field theory

Yury A. Budkov<EMAIL_ADDRESS>Laboratory for Computational Physics, HSE
University, Tallinskaya st. 34, 123458 Moscow, Russia  G.A. Krestov Institute
of Solution Chemistry of the Russian Academy of Sciences, 153045,
Akademicheskaya st. 1, Ivanovo, Russia  Nikolai N. Kalikin  Laboratory for
Computational Physics, HSE University, Tallinskaya st. 34, 123458 Moscow,
Russia  G.A. Krestov Institute of Solution Chemistry of the Russian Academy of
Sciences, 153045, Akademicheskaya st. 1, Ivanovo, Russia

###### Abstract

We utilize self-consistent field theory to explore the mechanical and
electrical properties of charged surfaces immersed in polyelectrolyte
solutions that could potentially be useful for electrochemical applications.
Our research focuses on how the dielectric heterogeneity of the solution could
affect the disjoining pressure and differential capacitance of the electric
double layer. Relying on the developed theoretical framework, based on the
Noether’s theorem, we calculate the stress tensor, containing the term,
arising from the conformational entropy of the polymer chains. With its help
we compute the disjoining pressure in polyelectrolyte solution confined
between two parallel charged surfaces and analyze its behavior as a function
of separation between the surfaces for different values of dielectric mismatch
parameter. We also calculate the differential capacitance of the electric
double layer and discuss how dielectric heterogeneity of the polyelectrolyte
solution influences its values.

self-consistent field theory, polyelectrolyte solution, disjoining pressure,
differential capacitance

##  I Introduction

The description of the polyelectrolyte solution behavior near the charged
surfaces is a challenging task due to a complex interplay of electrostatic,
steric, and entropic interactions. However, understanding the effects,
occurring in such systems may find a broad range of applications in a various
fields from material science to biophysics [1, 2].

Modern electrochemical devices, such as batteries and supercapacitors, make
extensive use of porous electrodes that are impregnated with low molecular
weight electrolyte solutions or room temperature ionic liquids [3, 4, 5].
However, considering long polyelectrolyte chains, one can expect a higher
electric double layer (EDL) charge. Recently, we proposed a theoretical model
that describes charged polymer chains near an electrified surface [6]. Our
theoretical findings show a significant increase of differential capacitance
values when these chains are in a solution with a polar organic solvent.

From the perspective of possible electrochemical applications [7], one should
emphasize not only the differential capacitance of the EDL but also essential
mechanical quantities such as disjoining pressure for discussing slit pores or
solvation pressure for pores of arbitrary geometries [8, 9].

The characteristic feature of the polyelectrolyte solution near the charged
surfaces is the presence of disjoining pressure oscillations with the change
of the distance between the walls. Such a behavior was observed experimentally
in ref. [10], where the authors studied thin liquid films formed from the
semidilute polyelectrolyte solution. They also observed the disappearance of
the oscillations with the addition of salt as a result of the additional
screening of the electrostatic forces. Similar behavior was studied in ref.
[11] for cationic polyelectrolyte solutions placed between cellulose surfaces.
It was shown that the surface forces exhibit oscillating behavior depending on
the separation between the surfaces, which is influenced by the molecular
weight and charge density of the polymers.

In paper [12] author provides a theoretical investigation of the forces
between hard walls immersed in the polyelectrolyte solution within the
integral equation framework. The theory predicts oscillatory behavior of the
surface force with a period of oscillation depending on the polymer
concentration. Studying how the amount of salt added to the solution affects
surface forces behavior revealed that it can cause a qualitative shift from
predominantly attractive to purely repulsive regimes.

A self-consistent field theory, modeling the depletion of the polyelectrolyte
macroions from the area between the charged with the same sign interacting
surfaces immersed in the solvent, which also contains electolyte ions, was
proposed in ref. [13]. The authors explain the ejection of the polymer from a
confined space into the bulk reservoir as a result of the competition between
the translational and conformational entropies, as well as a consequence of
the electrostatic repulsion caused by the polyelectrolyte and surfaces having
the same charge sign. The authors proposed an expression for grand potential
of the system, which was then used to compute the equilibrium ion
concentrations, disjoining pressure between the surfaces and effective Debye
length.

In a computer simulation of a system, consisting of two negatively charged
interacting surfaces immersed in a solution containing counterions and
positively charged polyelectrolytes, the authors of the ref. [14] discovered a
non-monotonic behavior of the force between the surfaces. The computed osmotic
pressure profile demonstrates a steep repulsion at very short separations,
followed by the attraction, which the authors attribute to the entropicaly-
driven effect of the formation of the polyelectrolyte bridges across the slit
from one surface to the other [15, 16].

It is worth noting that in the mentioned studies the disjoining pressure was
obtained via the differentiation of the grand potential with respect to the
separation between the walls. Recently [17] we proposed a self-consistent
field theory of macroscopic forces in spatially inhomogeneous equilibrium
polyelectrolyte solutions and derived the total stress tensor consistent with
obtained self-consistent field equations. Such tensor was discovered to
include the term, arising from the conformational entropy of flexible polymer
chains. Utilizing the expression for this tensor, we calculated the disjoining
pressure between two charged surfaces, immersed in the polyelectrolyte
solution.

On the other hand, strictly speaking, one can expect that the polyelectrolyte
solution has to display a rather significant dielectric heterogeneity due to
the difference in the dielectric permittivities of the polymer backbone,
solvent and counterions. One can also expect that such phenomenon can strongly
influence the properties of the EDL, namely differential capacitance. Thus, in
this paper we investigate the effect of the dielectric heterogeneity of the
polyelectrolyte solution, considering a dielectric mismatch effect within a
simple linear dependence on the concentration, on the disjoining pressure and
differential capacitance, basing on the previously developed theoretical
background [17].

##  II Theoretical background

###  II.1 Self-consistent field equations and stress tensor

We consider the case of the polyelectrolyte solution consisting of polymerized
flexible cations and low-molecular-weight anions, carrying positive $q>0$ and
negative $q<0$ charges, correspondingly. The grand thermodynamic potential
(GTP) of such system can be written within the developed [17] notations as
follows

$\Omega=\int d\mathbb{r}\omega(\mathbb{r}),$ (1)

where we have introduced the GTP density

$\omega=-\frac{\varepsilon(\nabla\psi)^{2}}{2}+\rho\psi+f+\frac{k_{B}Tb^{2}}{6}(\nabla
n_{p}^{1/2})^{2}-\mu_{p}n_{p}-\mu_{c}n_{c}.$ (2)

The first two terms in the expression are the electrostatic energy density in
the mean-field approximation with the local charge density
$\rho(\mathbb{r})=q\left(n_{p}(\mathbb{r})-n_{c}(\mathbb{r})\right)$ and
electrostatic potential $\psi(\mathbb{r})$, where $n_{p,c}(\mathbb{r})$ denote
the local concentrations of monomeric units and counterions; $\varepsilon$ is
the dielectric permittivity of the solution. Considering the difference in the
dielectric permittivities of the solvent $\varepsilon_{s}$, polymerized
cations $\varepsilon_{p}$ and monomeric anions $\varepsilon_{c}$, we introduce
the dielectric mismatch parameters
$\delta_{p}=(\varepsilon_{p}-\varepsilon_{s})/\varepsilon_{s}$ and
$\delta_{c}=(\varepsilon_{c}-\varepsilon_{s})/\varepsilon_{s}$. Thus, the
introduced above dielectric permittivity of the solution can be written as
[18, 19, 20, 21, 22]

$\varepsilon=\varepsilon_{s}\left(1+\delta_{p}\phi_{p}+\delta_{c}\phi_{c}\right),$
(3)

where $\phi_{p,c}=n_{p,c}v$ are the local volume fractions of the monomeric
units and counterions. The third term, $f=f(n_{p},n_{c})$, determines the
contribution of the volume interactions of monomeric units and counterions to
the total free energy density, which is described within the lattice model
[23, 4, 24] as
$f=k_{B}Tv^{-1}\left(\phi_{c}\ln\phi_{c}+\left(1-\phi_{c}-\phi_{p}\right)\ln\left(1-\phi_{c}-\phi_{p}\right)\right)$;
$v$ is the elementary cell volume; note that here we consider only the case of
very long polyelectrolyte chains, thereby, neglecting the contribution of
their translation entropy to the free energy density. The fourth term is the
density of the conformational free energy [25, 26, 27] of the flexible polymer
chains with a bond length $b$, which determines the volume of the elementary
cell in a standard way as $v=b^{3}$; $k_{B}$ is the Boltzmann constant, $T$ is
the temperature. The last terms in the GTP density expression contain the bulk
chemical potentials, $\mu_{p,c}$, of the monomeric units and counterions,
respectively.

The self-consistent field equations can be obtained as the Euler-Lagrange
equations for functional (1)

$\frac{\partial{\omega}}{\partial{\psi}}=\partial_{i}\frac{\partial\omega}{\partial(\partial_{i}\psi)},\leavevmode\nobreak\
\frac{\partial{\omega}}{\partial{n_{p}^{1/2}}}=\partial_{i}\frac{\partial\omega}{\partial(\partial_{i}n_{p}^{1/2})},\leavevmode\nobreak\
\frac{\partial{\omega}}{\partial{n_{c}}}=0,$ (4)

where $\partial_{i}=\partial/\partial{x}_{i}$ is the partial derivative with
respect to the Cartesian coordinates $x_{i}$ ($i=1,2,3$) and the standard
Einstein summation rule is implied. Using GTP density (2) and expression for
the dielectric permittivity (3), we obtain

$\displaystyle\bar{\mu}_{p}(\mathbb{r})-\frac{k_{B}Tb^{2}}{6n_{p}^{1/2}(\mathbb{r})}\nabla^{2}n_{p}^{1/2}(\mathbb{r})+q\psi(\mathbb{r})-\frac{\alpha_{p}(\nabla\psi(\mathbb{r}))^{2}}{2}=\mu_{p}$
$\displaystyle\bar{\mu}_{c}(\mathbb{r})-q\psi(\mathbb{r})-\frac{\alpha_{c}(\nabla\psi(\mathbb{r}))^{2}}{2}=\mu_{c}$
(5)
$\displaystyle\nabla(\varepsilon\nabla\psi(\mathbb{r}))=-q\left(n_{p}(\mathbb{r})-n_{c}(\mathbb{r})\right),$

where $\alpha_{p,c}=v\delta_{p,c}$ are the effective polarizabilities of the
species, and $\bar{\mu}_{c}={\partial f}/{\partial
n_{c}}=k_{B}T\ln\left({\phi_{c}}/{(1-\phi_{c}-\phi_{p}})\right)$,
$\bar{\mu}_{p}={\partial f}/{\partial
n_{p}}=-k_{B}T\ln\left(1-\phi_{c}-\phi_{p}\right)$ are the intrinsic chemical
potentials of monomeric units and counterions, respectively. The first and
second equations of the system (II.1) are the chemical equilibrium conditions
[28] for the monomeric units and counterions, respectively. In contrast to our
previous work [17], the left hand side of each chemical equilibrium conditions
contains additional term $-\alpha_{p,c}(\nabla\psi)^{2}/2$ that is nothing but
a potential energy of induced dipole
$\mathbb{p}_{p,c}=\alpha_{p,c}\boldsymbol{\mathcal{E}}$ in "external" field
$\boldsymbol{\mathcal{E}}=-\nabla\psi(\mathbb{r})$ [29]. The third equation is
the standard Poisson equation for the local electrostatic potential taking
into account the dielectric heterogeneity of the solution [30]. Taking into
account that in the bulk solution, where $\psi=0$, the local electroneutrality
condition, $n_{p}=n_{c}=n_{0}$, is fulfilled, we obtain the following
expressions for the bulk chemical potentials of the species
$\mu_{c}=k_{B}T\ln\left({\phi_{0}}/(1-2\phi_{0})\right)$,
$\mu_{p}=-k_{B}T\ln\left(1-2\phi_{0}\right)$, where $\phi_{0}=n_{0}v$ is the
bulk volume fraction of the monomeric units and counterions. The boundary
conditions for the polymer concentration and electrostatic potential are [2,
30] $n_{p}|_{s}=0$, $\psi|_{s}=\psi_{0}$, where the symbol $|_{s}$ means that
the variables are calculated at the surfaces of immersed macroscopic
conductors. These boundary conditions mean that near the surface of a
conductive wall (with a fixed surface potential, $\psi_{0}$) the monomeric
units are exposed to a strong repulsive force [2]. Note that for simplicity we
neglect the specific adsorption of the counterions. The latter can be directly
taken into account [31].

Referring to our recently developed theoretical framework, based on the
application of Noether’s theorem-based approach [32, 17, 33], we can write the
stress tensor as follows

$\sigma_{ik}=\omega\delta_{ik}-\partial_{i}n_{p}^{1/2}\frac{\partial\omega}{\partial(\partial_{k}n_{p}^{1/2})}-\partial_{i}\psi\frac{\partial\omega}{\partial(\partial_{k}\psi)},$
(6)

which after some algebra can be split into three contributions

$\sigma_{ik}=\sigma^{(h)}_{ik}+\sigma^{(M)}_{ik}+\sigma_{ik}^{(c)},$ (7)

where

$\sigma_{ik}^{(h)}=-P\delta_{ik}$ (8)

is the standard hydrostatic stress tensor with the local osmotic pressure
$P=n_{p}\bar{\mu}_{p}+n_{c}\bar{\mu}_{c}-f=-{k_{B}T}v^{-1}\left(\ln\left(1-\phi_{p}-\phi_{c}\right)+\phi_{p}\right),$

$\sigma_{ik}^{(M)}=\varepsilon\mathcal{E}_{i}\mathcal{E}_{k}-\frac{\varepsilon_{s}}{2}\mathcal{E}^{2}\delta_{ik}$
(9)

is the electrostatic contribution described by the standard Maxwell stress
tensor [30] for the case of continuous dielectric medium with the
electrostatic field components $\mathcal{E}_{i}=-\partial_{i}\psi$, and

$\sigma_{ik}^{(c)}=\frac{k_{B}Tb^{2}}{3}\left(\frac{1}{2}\nabla\cdot\left(n_{p}^{1/2}\nabla
n_{p}^{1/2}\right)\delta_{ik}-\partial_{i}n_{p}^{1/2}\partial_{k}n_{p}^{1/2}\right)$
(10)

is the contribution to the total stress tensor stemming from the
conformational entropy of the polymer chains (so-called conformational stress
tensor), which was first recently obtained in ref. [17].

We would like to note that discovered conformational stress tensor (10) can be
used for assessment of surface forces and deformations in various polymeric
systems with flexible chains, including polymer solutions, melts, gels,
membranes, etc.

###  II.2 Slit pore case. Disjoining pressure

For the case of the polyelectrolyte solution confined in a slit nano-sized
pore with identical charged conducting walls, placed at $z=0$ and $z=H$ with
$H$ being the pore size, and considering the equilibrium between the solution
phases in the confinement and in the bulk, we can reduce the self-consistent
field equations to the following form

$\displaystyle\bar{\mu}_{p}(z)-\frac{k_{B}Tb^{2}}{6\phi_{p}^{1/2}(z)}\frac{d^{2}\phi_{p}^{1/2}(z)}{dz^{2}}+q\psi(z)-\frac{\alpha_{p}}{2}\left(\frac{d\psi(z)}{dz}\right)^{2}=\mu_{p}$
$\displaystyle\frac{d}{dz}\left(\varepsilon\frac{d\psi(z)}{dz}\right)=-\frac{q}{v}\left(\phi_{p}(z)-\phi_{c}(z)\right)$
(11)

with the boundary conditions $\psi(0)=\psi(H)=\psi_{0}$,
$\phi_{p}(0)=\phi_{p}(H)=0$; we can also analytically express the volume
fraction of the counterions as

$\phi_{c}=\frac{(1-\phi_{p})e^{\frac{\mu_{c}+q\psi+\frac{\alpha_{c}\psi^{\prime
2}}{2}}{k_{B}T}}}{1+e^{\frac{\mu_{c}+q\psi+\frac{\alpha_{c}\psi^{\prime
2}}{2}}{k_{B}T}}},$ (12)

where ′ denotes the derivative with respect to z coordinate. In what follows,
we assume that the counterion-induced dielectric mismatch parameter is much
smaller than that associated with polymerized cations, which is the case for
the solutions of polymeric ionic liquids, i.e. $\alpha_{c}=0$.

Using the expression for the stress tensor (7), we can determine the
disjoining pressure in the pore. In the case of the slit pore, the mechanical
equilibrium condition can be written as

$\frac{d\sigma_{zz}}{dz}=0,$ (13)

thereby, leading to

$-\sigma_{zz}(z)=P_{b}+\Pi=const,$ (14)

where $\sigma_{zz}$ is the normal stress, $P_{b}$ is the bulk pressure and
$\Pi$ is the disjoining pressure [34]. Thus, determining $\sigma_{zz}$ at the
middle of the pore at $z=H/2$, where $n_{p}^{\prime}(H/2)=\mathcal{E}(H/2)=0$,
we obtain
$\Pi=-\sigma_{zz}\left({H}/{2}\right)-P_{b}=-{k_{B}Tb^{2}}n_{p}^{\prime\prime}\left({H}/{2}\right)/{12}+P_{m}-P_{b}$,
where we have introduced the pressure at the pore middle,
$P_{m}=P\left({H}/{2}\right)$. The latter equation can be rewritten in a form
that is more useful for applications without the second derivative of the
polymer concentration [17]. Using eq. (II.2) at $z=H/2$, i.e.
$-{k_{B}Tb^{2}}n_{p}^{\prime\prime}\left({H}/{2}\right)/{12}=n_{pm}\left(\mu_{p}-\bar{\mu}_{pm}-q\psi_{m}\right)$,
we arrive at

$\Pi=P_{m}-P_{b}+n_{pm}\left(\mu_{p}-\bar{\mu}_{pm}-q\psi_{m}\right),$ (15)

where $\psi_{m}=\psi\left({H}/{2}\right)$, $n_{pm}=n_{p}\left({H}/{2}\right)$,
$\bar{\mu}_{pm}=\bar{\mu}_{p}\left({H}/{2}\right)$.

##  III Numerical results

Now, we turn to the numerical results for the disjoining pressure and
differential capacitance. The main goal of the study is to investigate the
influence of the dielectric mismatch effect on the profiles of the disjoining
pressure and differential capacitance of the EDL. First, we calculate the
disjoining pressure. For this we need to solve self-consistent field equations
(II.2), calculate the respective variables at the midpoint of the pore and
then utilize the eq. (15) for different pore widths. Then, we examine the
influence of the dielectric mismatch effect on the behavior of differential
capacitance, $C=d\sigma/d\psi_{0}$, as a function of the surface potential,
where $\sigma=-\varepsilon\psi^{\prime}(0)$ is the surface charge density of
the pore walls [4, 35]. The following results were computed for temperature
$T=300$ K, elementary charge $q=1.6\times 10^{-19}$ C, segment length $b=0.5$
nm, bulk volume fraction $\phi_{0}=0.1$, which represents the case of the
polyelectrolyte solution, and the solvent dielectric permittivity
$\varepsilon_{s}=40\varepsilon_{0}$, which models polar solvents as dimethyl
sulfoxide or acetonitrile.

###  III.1 Disjoining pressure

Fig. 1(a) shows the disjoining pressure as a function of the surfaces
separation of the polyelectrolyte solution for rather high positive potential
$\psi=0.1\leavevmode\nobreak\ V$, with different values of the dimensionless
dielectric mismatch parameter $\delta_{p}=\alpha_{p}/v$. As is seen, the
disjoining pressure behaves non-monotonically and the position of the minimum
determines the width at which the polymer chains start to permeate the pore.
Moreover, we can see that disjoining pressure is not very sensitive to the
change in the mismatch parameter, leading to a practically unnoticeable shift
of the minimum.

The same nonmonotonic trend of the disjoining pressure on the pore width for
the case of the negative surface potential is shown in fig. 1(b). As is seen,
for the case of rather high values of negative surface potential, we observe
that as the $\delta_{p}$ increases, the minimum of the disjoining pressure
deepens. Such a behavior agrees with the results of the aforementioned Monte
Carlo simulation [14], where the oppositely charged polyelectrolytes and
surfaces stimulate the formation of the polymeric bridging [36, 37, 38],
displaying the attraction between the surfaces. One may conclude that the
change of $\delta_{p}$ from negative to positive values provokes an increase
in polymer concentration in the pores, which explains the deepening of the
minimum on the disjoining pressure curve, i.e. the growth of the attractive
bridging-induced interaction. It also should be noted that at such values of
the surface potential the model predicts that the polymerized cations enter
the pore at the width $H\approx 0.25$ nm, which is smaller than assumed bond
length $b$. Thus, we can consider that physically meaningful values of the
disjoining pressure start approximately from the distance, corresponding to
the location of the curve maximum. Considering this remark the obtained
disjoining pressure behavior is in agreement with the computer simulations
[14].

###  III.2 Differential capacitance

Now let us turn to the results of the electric differential capacitance
profiles calculation. As one can see at fig. 2(a) and 2(b) the values of the
differential capacitance are highly sensitive to the change in the parameter
$\delta_{p}$. The profiles differ only in the region of negative surface
potentials due to the fact that at the region of positive potentials
polymerized cations are practically absent in the pore due to the
electrostatic repulsion. At the increase in the negative surface potentials in
the dielectric mismatch parameter, $\delta_{p}$, results in an increase in the
differential capacitance (growth of the maximum differential capacitance).
Such behavior can be attributed to the fact that a larger values of
$\delta_{p}$ correspond to a stronger dielectrophoretic force[39, 40, 31, 41,
42], which in turn leads to a more effective "sucking" of the polarizable
macromolecules from the bulk solution into the charged pore. It should be
noted that this differential capacitance behavior resembles that discovered
recently within the theoretical framework for monomeric ionic liquids in ref.
[42].

Moreover, similar to our previous study [17], in the case of rather small pore
of 1 nm width (see fig. 2(a)) we obtain an abrupt jump in differential
capacitance values when turning to the region of positive surface potentials.
Such behavior is based on the same phenomenon as the ejection of the
polymerized cations from the charged slit pore. When the distance between the
surfaces increases, the differential capacitance profiles become smooth and
resemble those obtained for the case of isolated EDL, as demonstrated for the
case of pore of 2 nm width in Fig. 2(b).

##  IV Concluding remarks

To summarize, we used the developed self-consistent field theory to examine
how the mechanical and electrical properties of charged surfaces immersed in a
polyelectrolyte solution are affected by the dielectric heterogeneity of the
solution. Namely, we observed that the change of the dielectric mismatch
parameter $\delta_{p}$ from negative to positive values for the case of
polymerized cations and pore walls with opposite charges leads to a more
pronounced minimum on the disjoining pressure curve. Underlying such behavior
is the increase of the monomer concentration inside the pore, which leads to
the enhancement of the formation of a polyelectrolyte bridge between the walls
of the slit. We also obtained a significant increase in differential
capacitance values in the region of negative potentials for the same range of
values of mismatch parameter $\delta_{p}$. The explanation here is quite
intuitive: As the dielectric mismatch parameter increases, the total
dielectric permittivity also increases, leading to reduced screening of the
surface potential. This causes an increase in the monomer concentration in the
pore, which in turn leads to a higher charge accumulation and an observed
growth of the differential capacitance values.

We believe that our theoretical findings could be relevant for the design of
contemporary supercapacitors that make use of polymeric electrolytes, such as
polymeric ionic liquids and polyelectrolyte gels, as opposed to low-molecular
weight electrolyte solutions and room temperature ionic liquids.

Acknowledgements. Paper is dedicated to the memory of Igor Ya. Erukhimovich,
an outstanding theoretical physicist whose papers had a great influence on the
authors. This work is an output of a research project implemented as part of
the Basic Research Program at the National Research University Higher School
of Economics (HSE University). The numerical calculations were performed on
the supercomputer facilities provided by NRU HSE.

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(a)

(b)

Figure 1: Disjoining pressure as a function of the distance between the walls
for a set of dielectric mismatch parameter $\delta_{p}$ with positive
$\psi_{0}=0.1\leavevmode\nobreak\ V$ (1(a)) and negative
$\psi_{0}=-0.1\leavevmode\nobreak\ V$ (1(b)) surface potentials. The data are
shown for $\phi_{0}=0.1$, $\varepsilon_{s}=40\varepsilon_{0}$,
$b=v^{1/3}=0.5\leavevmode\nobreak\ nm$, $T=300\leavevmode\nobreak\ K$ and
$q=1.6\times 10^{-19}\leavevmode\nobreak\ C$.

(a)

(b)

Figure 2: Differential capacitance profiles for a pore of 1 nm (2(a)) and 2 nm
(2(b)) width, plotted for different values of the mismatch parameter
$\delta_{p}$. The data are shown for $\phi_{0}=0.1$,
$\varepsilon_{s}=40\varepsilon_{0}$, $b=v^{1/3}=0.5\leavevmode\nobreak\ nm$,
$T=300\leavevmode\nobreak\ K$ and $q=1.6\times 10^{-19}\leavevmode\nobreak\
C$.