# Structural and electrostatic properties between pH-responsive
polyelectrolyte brushes studied by augmented strong stretching theory

Jun-Sik Sin<EMAIL_ADDRESS>Natural Science Center, Kim Il Sung
University, Taesong District, Pyongyang, Democratic People’s Republic of Korea

###### Abstract

In this paper, we study electrostatic and structural properties between pH-
responsive polyelectrolyte brushes by using a strong stretching theory
accounting for excluded volume interactions, the density of polyelectrolyte
chargeable sites and the Born energy difference between the inside and outside
of the brush layer.

In a free energy framework, we obtain self-consistent field equations to
determine electrostatic properties between two pH-responsive polyelectrolyte
brushes. We elucidate that in the region between two pH-responsive
polyelectrolyte brushes, electrostatic potential at the centerline and osmotic
pressure increase not only with excluded volume interaction, but also with
density of chargeable sites on a polyelectrolyte molecule.

Importantly, we clarify that when two pH-responsive polyelectrolyte brushes
approach each other, the brush thickness becomes short and that a large
excluded volume interaction and a large density of chargeable sites yield the
enhanced contract of polyelectrolyte brushes. In addition, we also demonstrate
how the influence of such quantities as pH, the number of Kuhn monomers, the
density of charged sites, the lateral separation between adjacent
polyelectrolyte brushes, Kuhn length on the electrostatic and structural
properties between the two polyelectrolyte brushes is affected by the
exclusion volume interaction.

Finally, we investigate the influence of Born energy difference on the
thickness of polyelectrolyte brushes and the osmotic pressure between two pH-
responsive polyelectrolyte brushes.

Electrostatic interaction, Strong Stretching Theory, Electric double layer,
Polyelectrolyte brush, Osmotic Pressure

###### pacs:

82.45.Gj

††preprint: AIP/123-QED

##  I Introduction

Grafting polyelectrolyte brushes onto solid-liquid interfaces is a promising
strategy to fabricate functionalized nanochannels and nanoparticles because it
is capable to regulate many physicochemical properties of interfaces formed
between hard surfaces and ionic solutions. ali_jacs_2008 ; vilozny_ns_2013 ;
sadeghi_aca_2020 ; ali_jacs_2009 ; sadeghi_pccp_2021 ; kreer_sm_2016 ;
nap_jcp_2018 ; marins_acs_2018 ; kegler_prl_2007 ; sachar_matter_2020 ;
zhao_lang_2018

Ionization and charging of strong polyelectrolytes depends on only the ionic
strength of the electrolyte solution, whereas weak (or annealed)
polyelectrolytes involve pH-dependent ionization and charging and hence are
widely employed to receive a variation in pH or ionic concentration.

For weak polyelectrolyte brushes, a change in solution pH or ionic strength
causes noticeable changes in its conformation or its brush thickness which is
the reason why weak polyelectrolyte brushes not only is called pH-responsive
brushes but also are widely employed for industrial applications such as
biosensingali_jacs_2008 , current rectifiers vilozny_ns_2013 ;
sadeghi_aca_2020 ; ali_jacs_2009 and power generation in nanochannels
sadeghi_pccp_2021 , polymer-brush lubrication kreer_sm_2016 , nanoparticles
for biomedical applications nap_jcp_2018 ; marins_acs_2018 ; kegler_prl_2007 ,
nanofiltration sachar_matter_2020 and many more zhao_lang_2018 ;
zhulina_jcp_2002

Some studies to describe pH-responsive PE brushespincus_macromolecules_1991 ;
ross_macromolecules_1992 ; borisov_jp_1991 ; joanny_macromolecules_1993 ;
borisov_macromolecules_1994 ; zhulina_macromolecules_1995 ; zhulina_sm_2012 ;
zhulina_macromolecules_1994a aimed to obtain scaling laws by considering the
balance between the different energies (elastic, electrostatic, and excluded
volume) to yield the brush thickness as functions of quantities such as the
grafting density and charge density of the brushes, number of monomers, and
the concentration of the added salt. For establishing a more complete picture
of pH-responsive polyelectrolyte brushes, the Poisson-Boltzmann equation
das_jpcb_2015 ; das_rsc_2015 ; das_jpcb_2016 ; das_jpcb_2017 ; das_sm_2018 ;
das_jpcb_2018 ; biesheuvel_macromolecules_2008 was widely studied for the
electrostatics of the induced electric double layer (EDL)by using the monomer
interactions such as the Alexander-de-Gennes model and the parabolic model.
Moreover, a more innovative approach for pH-responsive polyelectrolyte brushes
is the strong stretching theory zhulina_jcp_1997 ;
zhulina_macromolecules_1995a ; zhulina_jcp_2017b ; zhulina_macromolecules_2000
; zhulina_lang_2011 presented to describe the polyelectrolyte brushes.

The self-consistent strong stretching theory and the Poisson-Boltzmann
equation have also been employed to study the configuration (or brush
thickness) of two interacting pH-responsive polyelectrolyte brushes and
proposed that when approaching two interacting polyelectrolyte brushes, the
thickness of polyelectrolyte brush can be reduced.

Unfortunately, the papers zhulina_jcp_1997 ; zhulina_macromolecules_1995a ;
zhulina_jcp_2017b ; zhulina_macromolecules_2000 ; zhulina_lang_2011 discard
the excluded volume interaction between monomer units in a polyelectrolyte
chain and hence considers the polyelectrolyte brushes being in a $\theta$
-solvent. In many realistic experiments kumar_jps_2016 ;
wu_macromolecules_2007 ; ito_jacs_1997 , physicochemical and structural
properties of pH-responsive PE brushes were performed by using good solvents
and, accordingly, the theory of zhulina_jcp_1997 dealing with $\theta$
-solvent doesn’t apply to such cases. On the other hand, since the theory does
not account for the density of chargeable sites of the polyelectrolyte
brushes, the theory is unable to study the variation of polyelectrolyte brush
thickness with the density of chargeable sites of the polyelectrolyte brush,
as reported in zhulina_macromolecules_2014 ; kumar_macromolecules_2005 ;
naji_epje_2003 .

In order to account for such effects, the authors of das_sm_2019a proposed a
self-consistent field approach to describe the behavior of the pH-responsive
brush by considering (a) the excluded volume interactions between the
polyelectrolyte segments and (b) a more expanded form of the mass action law
valid for $\gamma a^{3}$. The study unravels an increase of the brush
thickness due to the excluded volume interaction which is originated from
polyelectrolyte inter-segmental repulsion, as well as a variation in the brush
thickness for $\gamma a^{3}$, attributed to counterion-induced osmotic
swelling of the brushes. Furthermore, with the help of the augmented strong
stretching theory, the authors provided the fascinating theoretical analysis
das_sm_2019b ; das_ep_2020 ; das_pf_2020 ; das_pre_2020 ; das_jfm_2021 of
liquid transport driven by an axial temperature gradient, concentration
gradient, electric potential gradient in polyelectrolyte-brush-grafted
nanofluidic channel. However, to the best of our knowledge, there is no
researcher which discusses how the excluded volume interaction and the
expanded form of the mass action law affect the electrostatic interaction and
structure between pH-responsive polyelectrolyte brushes.

To unravel the issue, we generalize the augmented strong stretching theory for
a pH-responsive polyelectrolyte brush presented in das_sm_2019a to the case of
two interacting pH-responsive polyelectrolyte brushes and investigate the
influence of the excluded volume interaction and the expanded form of the mass
action law on interaction and structural properties between two pH-responsive
brushes in connection with other quantities.

Here, it is worthwhile to provide a brief overview of the modeling studies of
electrostatic interactions between two polymer-grafted-surfaces. In fact,
although many works duval_lang_2011 ; duval_lang_2015 ; ohshima_lang_2006 ;
Das_csb_2018 ; dahnert_jcis_1994 ; ohshima_jcis_2002 ; richmond_jcsf_1974 ;
richmond_jcsf_1975 ; nelson_ijtb_1975 ; micklavic_jpc_1994 ;
mickavic_jcis_1999 ; ohshima_cps_2014 ; das_rsc_2015b ; duval_pccp_2011 ;
sin_csa_2021 have dealt with the electrostatic interaction between the
polyelectrolyte grafted particles in electrolyte solutions, most of them are
based on the assumption that lengths of chains in a polyelectrolyte layer are
equal and constant and are independent of external stimuli such as
temperature, pH and salt concentration in electrolyte solutions. Consequently,
such an oversimplification allows the theories not to predict the influence of
a polyelectrolyte chain on properties of polyelectrolyte grafted particles.
Moreover, recent works sin_csa_2021 ; horno_jcis_2003 ; sadeghi_eleccom_2017 ;
gopmandal_pre_2018 ; sadeghi_csb_2018 ; gopmandal_pre_2020 ; gopmandal_ep_2021
; poddar_sm_2016 ; pandey_csa_2021 anticipate that physicochemical properties
of polyelectrolyte grafted nanochannels or nanoparticles is significantly
affected by the Born energy difference between the inside and outside of
polyelectrolyte layer.

Thus we aim at investigating the electrostatic and structural properties
between two interacting pH-responsive polyelectrolyte brushes. The rest of the
paper is organized as follows: We will first consider thermodynamics of the
system consisting of two interacting planar polyelectrolyte brushes in an
electrolyte solution and then derive the electrostatic potential distribution
and structural properties of pH-responsive polyelectrolyte brushes tethered on
neutral surfaces. Finally, the influence of various quantities on brush
thickness, electrostatic potential and osmotic pressure are demonstrated in
detail.

##  II Theory

We consider two parallel pH-responsive polyelectrolyte brushes separated by a
distance $2h\left(-h<x<h\right)$. Theoretical description of the brushes is
based on the free energy formalism using the augmented strong stretching
theory accounting for a more generic mass action law and the effects of the
polyelectrolyte excluded volume interactions.

The present theory will present the equilibrium configuration of the
polyelectrolyte brushes and the equilibrium electrostatics of the brush-
induced electric double layer by minimizing the total free energy of a given
brush molecule that consists of the elastic, excluded volume, electrostatic,
and ionization energies of the brush and the electrostatic energy of the
brush-induced electric double layer.

Figure 1: (Color online) Schematic of two pH-responsive polyelectrolyte
brushes in an electrolyte solution. Here the polyelectrolyte ions are
represented in green, electrolytic anions and cations are represented in red
and blue, respectively. Hydrogen ions and hydroxyl ions are represented in
cyan and orange.

The total energy functional $F$ of the PE brush system consists of the elastic
$F_{els}$, excluded volume $F_{EV}$, electrostatic $F_{elec}$, and ionization
$F_{ion}$ free energies of a polyelectrolyte brush molecule, the electrostatic
energy of the EDL ($F_{EDL}$) induced by the brush and the free energy due to
Born energy difference($F_{Born}$) .

$\displaystyle F=F_{els}+F_{EV}+F_{elec}+F_{ion}+F_{EDL}+F_{Born}.$ (1)

Due to axial symmetry of the present system, we shall only consider the left
half of the region between two interacting pH-responsive brushes in our
subsequent calculation. In the same way as in zhulina_jcp_1997 ; das_sm_2019a
, we can express

$\displaystyle
F_{els}=\frac{3k_{B}T}{{2pa^{2}}}\int_{-h}^{-h+H}{g\left({x^{\prime}}\right)dx^{\prime}}\int_{-h}^{x^{\prime}}{E\left({x,x^{\prime}}\right)dx},$
(2) $\displaystyle F_{EV}=\frac{k_{B}T}{{\sigma
a^{3}}}\int_{-h}^{-h+H}{f_{EV}\left[{\phi\left(x\right)}\right]dx}.$ (3)

where $k_{B},T,p,a,H,\sigma$ and $l_{0}$ denote the Boltzmann constant, the
temperature, chain rigidity, Kuhn length, brush thickness, polyelectrolyte
brush grafting density, and the lateral separation between the adjacent
polyelectrolyte brushes, respectively. Here
$\sigma\propto\frac{1}{l_{0}^{2}}$. $\phi\left(x\right)$ and
$f_{EV}\left[{\phi\left(x\right)}\right]\left({\approx
v\phi^{2}+\omega\phi^{3}}\right)$ represent the dimensionless monomer
distribution profile of a polyelectrolyte chain and the non-dimensionalized
per unit volume free energy associated with the excluded volume interactions,
respectively. $E\left({x,x^{\prime}}\right)=\frac{{dx}}{{dn}}$ expresses chain
stretching (for a chain whose end is at $x^{\prime}$) at a location $x$ and
$g\left(x^{\prime}\right)$(the normalized chain end distribution function) is
expressed as

$\displaystyle\int_{-h}^{-h+H}{g\left({x^{\prime}}\right)dx^{\prime}}=1.$ (4)

The Born energy difference of the $i$-th ionic species , $\Delta W_{i}$, is
expressed as follows

$\Delta
W_{i}=\frac{{e}^{2}}{8\pi\varepsilon_{0}r_{i}}\left(\frac{1}{\varepsilon_{p}}-\frac{1}{\varepsilon_{r}}\right)=\frac{{e}^{2}}{8\pi\varepsilon_{0}\varepsilon_{r}r_{i}}\left(\frac{1}{\zeta}-1\right)$
(5)

where $r_{i}\left(i=1,2\right)$ is the radius of ion having the elementary
charge $e$. $\varepsilon_{0}$ is the absolute permittivity of vaccum,
$\varepsilon_{p}$ and $\varepsilon_{r}$ are the relative permittivities of the
inside and outside of polyelectrolyte brush layer, respectively.
$\zeta=\frac{\varepsilon_{p}}{\varepsilon_{r}}$. For simplicity, we assume
that anions and cations have an equal value of hydrated radius, $r$. Eq. (5)
imply that the energy difference decreases with the permittivity of
polyelectrolyte layer. If the polyelectrolyte brush layer contains less water,
the difference in permittivity between inside and outside of polyelectrolyte
brush layer is enhanced and finally the Born energy difference will increase.

$F_{Born}=\int\limits_{-h}^{-h+H}{\Delta
W\left({n_{+}+n_{-}+n_{H^{+}}+n_{OH^{-}}}\right)dx}$ (6)

Next, following das_sm_2019a , $F_{elec}+F_{EDL}$ can be expressed as

$\displaystyle\begin{array}[]{l}F_{elec}+F_{EDL}=\frac{1}{{\sigma
k_{B}T}}\int_{-h}^{{\rm{-}}h+H}{\left[{-\frac{{\varepsilon_{0}\varepsilon_{p}}}{2}\left|{\frac{{d\psi}}{{dx}}}\right|^{2}+e\psi\left({n_{+}-n_{-}+n_{H^{+}}-n_{OH^{-}}}\right)}\right]dx}\\\
+\frac{1}{{\sigma
k_{B}T}}\int_{-h+H}^{0}{\left[{-\frac{{\varepsilon_{0}\varepsilon_{r}}}{2}\left|{\frac{{d\psi}}{{dx}}}\right|^{2}+e\psi\left({n_{+}-n_{-}+n_{H^{+}}-n_{OH^{-}}}\right)}\right]dx}-\frac{1}{{\sigma}}\int_{-h}^{-h+H}{\left[{e\psi
n_{A^{-}}\phi}\right]dx}\\\ +\frac{1}{{\sigma
b^{3}}}\int_{-h}^{0}{\left[{n_{+}b^{3}\ln\left({n_{+}b^{3}}\right)+n_{-}b^{3}\ln\left({n_{-}b^{3}}\right)+n_{H^{+}}b^{3}\ln\left({n_{H^{+}}b^{3}}\right)+n_{OH^{-}}b^{3}\ln\left({n_{OH^{-}}b^{3}}\right)}\right]}dx\\\
+\frac{1}{{\sigma
a^{3}}}\int_{-h}^{0}{\left[{\left({1-n_{+}b^{3}-n_{-}b^{3}-n_{H^{+}}b^{3}-n_{OH^{-}}b^{3}}\right)\ln\left({1-n_{+}b^{3}-n_{-}b^{3}-n_{H^{+}}b^{3}-n_{OH^{-}}b^{3}}\right)}\right]}dx\\\
+\frac{1}{{\sigma
a^{3}}}\int_{-h}^{0}{\left[{-\mu_{+}n_{+}-\mu_{-}n_{-}-\mu_{H^{+}}n_{H^{+}}-\mu_{OH^{-}}n_{OH^{-}}}\right]}dx\end{array}$
(12)

where $\psi$ is the electrostatic potential, $n_{i}$ and $n_{i,\infty}$ are
the number density and bulk number density for ion $i$, where
$i=\pm,H^{+},OH^{-}$. Here $n_{i,\infty}=10^{3}N_{A}c_{i,\infty}$, where
$c_{i,\infty}$ and $N_{A}$ are the bulk ionic concentration, and Avogardro
number, respectively.

In order to determine the translational entropy of electrolyte ions, we use
the same lattice statistics as in andelman_prl_1997 ; iglic_jp_1996 . In the
statistics, each electrolyte ion occupies a cell, the length of each side of a
cell is $b$. $\mu_{-},\mu_{+},\mu_{H^{+}},\mu_{OH^{-}}$ are the chemical
potentials of cations, anions, hydrogen ions and hydroxyl ions, respectively.

As in das_sm_2019a , a pH-responsive polyelectrolyte brush is charged by an
acidlike dissociation of HA producing $H^{+}$ and $A^{-}$ ions. $K_{a}$ is the
ionization constant for the acidlike dissociation process. The number density
of these $A^{-}$ ions is expressed as $n_{A^{-}}$. Consequently, $F_{ion}$ can
be written as

$F_{ion}=\frac{k_{B}T}{{\sigma
a^{3}}}\int_{-h}^{-h+H}{\phi\left[{\left({1-\frac{{n_{A^{-}}}}{\gamma}}\right)\ln\left({1-\frac{{n_{A^{-}}}}{\gamma}}\right)+\frac{{n_{A^{-}}}}{\gamma}\ln\left({\frac{{n_{A^{-}}}}{\gamma}}\right)+\frac{{n_{A^{-}}}}{\gamma}\ln\left({\frac{n_{H^{+,\infty}}}{{K_{a}^{\prime}}}}\right)}\right]dx}.$
(13)

where
$K_{a}^{\prime}=10^{3}N_{A}K_{a},n_{H^{+},\infty}=10^{3}N_{A}c_{H^{+},\infty}$.
$c_{H^{+},\infty}$ is the bulk concentration of hydrogen ions, which can be
related with bulk $pH$ or $pH_{\infty}$ as
$c_{H^{+},\infty}=10^{-pH_{\infty}}$, $pK$ as $K_{a}=10^{-pK}$ and $\gamma$ is
the density of polyelectrolyte chargeable sites. Substituting Eqs. (2), (3),
(12), (13) into Eq. (1), we can finally get the whole expression for the total
free energy as

$\displaystyle\begin{array}[]{l}\frac{F}{{k_{B}T}}=\frac{3}{{2pa^{2}}}\int_{-h}^{-h+H}{g\left({x^{\prime}}\right)dx^{\prime}}\int_{-h}^{x^{\prime}}{g\left({x,x^{\prime}}\right)dx}+\frac{1}{{\sigma
a^{3}}}\int_{-h}^{-h+H}{f_{EV}\left[{\phi\left(x\right)}\right]dx}\\\
+\frac{1}{{\sigma
k_{B}T}}\int_{-h}^{{\rm{-}}h+H}{\left[{-\frac{{\varepsilon_{0}\varepsilon_{p}}}{2}\left|{\frac{{d\psi}}{{dx}}}\right|^{2}}\right]dx}+\frac{1}{{\sigma
k_{B}T}}\int_{-h+H}^{0}{\left[{-\frac{{\varepsilon_{0}\varepsilon_{r}}}{2}\left|{\frac{{d\psi}}{{dx}}}\right|^{2}}\right]dx}\\\
+\frac{1}{{\sigma k_{B}T}}\int_{-h}^{-h+H}{\left[{\Delta
W\left({n_{+}+n_{-}+n_{H^{+}}+n_{OH^{-}}}\right)}\right]dx}\\\
+\frac{1}{{\sigma
k_{B}T}}\int_{-h}^{0}{\left[{e\psi\left({n_{+}-n_{-}+n_{H^{+}}-n_{OH^{-}}}\right)}\right]dx}-\frac{1}{{\sigma
k_{B}T}}\int_{-h}^{-h+H}{\left[{e\psi n_{A^{-}}\phi}\right]dx}\\\
+\frac{1}{{\sigma
b^{3}}}\int_{-h}^{0}{\left\\{{n_{+}b^{3}\ln\left({n_{+}b^{3}}\right)+n_{-}b^{3}\ln\left({n_{-}b^{3}}\right)+n_{H^{+}}b^{3}\ln\left({n_{H^{+}}b^{3}}\right)+n_{OH^{-}}b^{3}\ln\left({n_{OH^{-}}b^{3}}\right)}\right\\}}dx\\\
+\frac{1}{{\sigma
a^{3}}}\int_{-h}^{0}{\left\\{{\left({1-n_{+}b^{3}-n_{-}b^{3}-n_{H^{+}}b^{3}-n_{OH^{-}}b^{3}}\right)\ln\left({1-n_{+}b^{3}-n_{-}b^{3}-n_{H^{+}}b^{3}-n_{OH^{-}}b^{3}}\right)}\right\\}}dx\\\
+\frac{1}{{\sigma
a^{3}}}\int_{-h}^{-h+H}{\phi\left[{\left({1-\frac{{n_{A^{-}}}}{\gamma}}\right)\ln\left({1-\frac{{n_{A^{-}}}}{\gamma}}\right)+\frac{{n_{A^{-}}}}{\gamma}\ln\left({\frac{{n_{A^{-}}}}{\gamma}}\right)+\frac{{n_{A^{-}}}}{\gamma}\ln\left({\frac{{n_{H^{+},\infty}}}{{{K_{a}}^{\prime}}}}\right)}\right]dx}\\\
+\frac{1}{{\sigma
a^{3}}}\int_{-h}^{0}{\left[{-\mu_{+}n_{+}-\mu_{-}n_{-}-\mu_{H^{+}}n_{H^{+}}-\mu_{OH^{-}}n_{OH^{-}}}\right]}dx\\\
\end{array}$ (22)

Eq. (22) should be minimized using the variational formalism in the presence
of the following constraints:

$N=\int_{-h}^{x^{\prime}}{\frac{{dx}}{{E\left({x,x^{\prime}}\right)}}},$ (23)
$N=\frac{1}{{\sigma a^{3}}}\int_{-h}^{-h+H}{\phi\left(x\right)dx},$ (24)

where $N$ is the number of Kuhn monomer in every polyelectrolyte brush
molecule. Additionally, we have $\phi\left(x\right)$ related to the functions
$g$ and $E$ as

$\phi\left(x\right)=\sigma
a^{3}\int_{x}^{-h+H}{\frac{{g\left({x^{\prime}}\right)dx^{\prime}}}{{E\left({x,x^{\prime}}\right)}}}.$
(25)

This minimization procedure eventually yields the final set of equations
dictating the equilibrium of the system. We have only considered $p=1$
corresponding to the case of the fully flexible polyelectrolyte chains.

The Euler-Lagrange equations for
$n_{A^{-}},\psi,n_{\pm},n_{H^{+}},n_{OH^{-}},E\left(x,x^{\prime}\right),g\left(x\right)$
lead to the following equations:

$n_{A^{-}}=\frac{{K_{a}^{\prime}\gamma}}{{K_{a}^{\prime}+n_{H^{+},\infty}\exp\left({-\gamma
a^{3}\frac{{e\psi}}{{k_{B}T}}}\right)}},$ (26)
$\varepsilon_{0}\varepsilon_{r}\left({\frac{{d^{2}\psi}}{{dx^{2}}}}\right)+e\left({n_{+}-n_{-}+n_{H^{+}}-n_{OH^{-}}-n_{A^{-}}\phi}\right)=0,-h+H\leq
x\leq 0,$ (27)
$\varepsilon_{0}\varepsilon_{r}\left({\frac{{d^{2}\psi}}{{dx^{2}}}}\right)+e\left({n_{+}-n_{-}+n_{H^{+}}-n_{OH^{-}}}\right)=0,-h+H\leq
x\leq 0,$ (28)

In the region of $-h\leq x\leq-h+H$,

$n_{\pm}=n_{\pm,\infty}\exp\left({\mp\frac{{e\psi}}{{k_{B}T}}}-\frac{\Delta
W}{k_{B}T}\right)/D_{in},$ (29)
$n_{H^{+}}=n_{H^{+},\infty}\exp\left({-\frac{{e\psi}}{{k_{B}T}}}-\frac{\Delta
W}{k_{B}T}\right)/D_{in},$ (30)
$n_{OH^{-}}=n_{OH^{-},\infty}\exp\left({\frac{{e\psi}}{{k_{B}T}}}-\frac{\Delta
W}{k_{B}T}\right)/D_{in},$ (31)

where
$D_{in}=1-\left({n_{+,\infty}+n_{H^{+},\infty}+n_{-,\infty}+n_{OH^{-},\infty}}\right)b^{3}\\\
+b^{3}\exp\left(-\frac{\Delta
W}{k_{B}T}\right)\left({\left({n_{+,\infty}+n_{H^{+},\infty}}\right)\exp\left({-\frac{{e\psi}}{{k_{B}T}}}\right)+\left({n_{-,\infty}+n_{OH^{-},\infty}}\right)\exp\left({\frac{{e\psi}}{{k_{B}T}}}\right)}\right)$.

In the region of $-h+H\leq x\leq 0$,

$n_{\pm}=n_{\pm,\infty}\exp\left({\mp\frac{{e\psi}}{{k_{B}T}}}\right)/D_{out},$
(32)
$n_{H^{+}}=n_{H^{+},\infty}\exp\left({-\frac{{e\psi}}{{k_{B}T}}}\right)/D_{out},$
(33)
$n_{OH^{-}}=n_{OH^{-},\infty}\exp\left({\frac{{e\psi}}{{k_{B}T}}}\right)/D_{out},$
(34)

where
$D_{out}=1-\left({n_{+,\infty}+n_{H^{+},\infty}+n_{-,\infty}+n_{OH^{-},\infty}}\right)b^{3}\\\
+b^{3}\left({\left({n_{+,\infty}+n_{H^{+},\infty}}\right)\exp\left({-\frac{{e\psi}}{{k_{B}T}}}\right)+\left({n_{-,\infty}+n_{OH^{-},\infty}}\right)\exp\left({\frac{{e\psi}}{{k_{B}T}}}\right)}\right)$.

if $\nu\neq 0$ and $\omega\neq 0$, the equation for monomer distribution
profile is

$\begin{array}[]{l}\phi\left(x\right)=\frac{v}{3\omega}\left\\{\left[1+\kappa^{2}f\left(x\right)\right]^{1/2}-1\right\\},\end{array}$
(35)

where $f\left(x\right)=\lambda-\left(x+h\right)^{2}+\beta
n_{A^{-}}\psi-\rho\left(1-\frac{n_{A^{-}}}{\gamma}\right)\ln\left(1-\frac{n_{A^{-}}}{\gamma}\right)-\rho\frac{n_{A^{-}}}{\gamma}\ln\left(\frac{n_{A^{-}}}{\gamma}\frac{{n_{H^{+},\infty}}}{{K_{a}^{\prime}}}\right)$.

if $\nu\neq 0$ and $\omega=0$,

$\phi\left(x\right)=\frac{1}{{2\nu\rho}}f\left(x\right)$ (36)

if $\nu=0$ and $\omega\neq 0$,

$\phi\left(x\right)=\frac{1}{{\sqrt{3\omega\rho}}}{{f\left(x\right)}^{1/2}}$
(37)

We can be easily aware of that if $\nu=0.0$ and $\omega=0.0$, there exists a
division by zero in Eqs. (35), (36) and (37), it is impossible to calculate
the monomer distribution profile.

To solve the issue for $\nu=0.0,\omega=0.0$, in the similar way in
zhulina_lang_2011 , we can derive a new formula for
$\phi\left(x\right)$.(refer Appendix)

$E\left({x,x^{\prime}}\right)=\frac{\pi}{{2N}}\sqrt{\left({x^{\prime}+h}\right)^{2}-\left({x+h}\right)^{2}},$
(38) $\left({q_{tot}}\right)_{H=H_{0}}=0,$ (39)
$\left({q_{tot}}\right)_{H=H_{0}}=\frac{e}{\sigma}\int_{-h}^{0}{\left({n_{+}-n_{-}+n_{H^{+}}-n_{OH^{-}}-\phi
n_{A^{-}}}\right)dx},$ (40)
$g\left(x\right)=\frac{{\left({x+h}\right)}}{{\sigma
Na^{3}}}\left[{\frac{{\phi\left({-h+H}\right)}}{{\sqrt{H^{2}-\left({x+h}\right)^{2}}}}-\int_{x}^{-h+H}{\frac{{d\phi\left({x^{\prime}}\right)}}{{dx^{\prime}}}\frac{{dx^{\prime}}}{{\sqrt{\left({x^{\prime}+h}\right)^{2}-\left({x+h}\right)^{2}}}}}}\right].$
(41)

Eq. (26) denotes the expanded form of the mass action law. Eq. (27),(28)
provides the electrostatic potential distribution in both $-h\leq x\leq-h+H$
and $-h+H\leq x$.

Eqs. (29-31), Eqs. (32-34) relate the ion number densities to electrostatic
potential and the corresponding bulk number density $n_{i,\infty}$ through the
Boltzmann distribution.

Eq. (35) provides the monomer distribution profile using virial coefficients
$\omega$ and $\nu$ , parameters $\rho=\frac{{8a^{2}N^{2}}}{{3\pi^{2}}}$,
$\lambda=-\lambda_{1}\rho$ ($\lambda_{1}$ is the Lagrange multiplier
attributed to the constraint condition Eq. (24)).
$\beta=\frac{{8N^{2}ea^{5}}}{{3\pi^{2}k_{B}T}}$.

While Eq. (38) quantifies the local stretching of the polyelectrolyte brush,
Eq. (39) is used as the condition for determining the equilibrium brush
thickness $H_{0}$.

Finally, Eq. (41) expresses the normalized chain end distribution $g(x)$.

Summarizing, the electrostatic potential and the brush thickness for the
system consisting of two interaction pH-responsive polyelectrolyte brushes in
a good solvent is self-consistently solved by combining Eqs. (27-41) and the
following boundary condition:

$\displaystyle\begin{array}[]{l}\left(\psi\right)_{x=\left({-h+H}\right)^{-}}=\left(\psi\right)_{x=\left({-h+H}\right)^{+}},\\\
\varepsilon_{p}\left({\frac{{d\psi}}{{dx}}}\right)_{x=\left({-h+H}\right)^{-}}=\varepsilon_{r}\left({\frac{{d\psi}}{{dx}}}\right)_{x=\left({-h+H}\right)^{+}},\\\
\left({\frac{{d\psi}}{{dx}}}\right)_{x=-h}=0,\\\
\left({\frac{{d\psi}}{{dx}}}\right)_{x=0}=0.\\\ \end{array}$ (46)

In order to combine and solve the equations obtained above, we apply the
folloiwng procedure.

Firstly, we give a guess of $H$ and then provide a initial value of $\lambda$.

Secondly, we can determine $\psi$ by solving Eq. (27) and Eq. (28).

Then, $\lambda$ is self-consistently determined with the help of the
normalization condition Eq. (25).

Next, to obtain
$\phi\left(x\right),\psi\left(x\right),n_{A^{-}}\left(x\right),n_{\pm},n_{H^{+}},n_{OH^{-}}$,
we use Eq. (26), Eq. (29-31) and Eq. (32-38). Finally, the equilibrium brush
height $H_{0}$ will be self-consistently determined by solving Eq. (39) and
Eq. (40).

The solution provides
$\phi,\psi,g\left(x\right),H_{0},n_{A^{-}},n_{i}\left({i=\pm,H^{+},OH^{-}}\right)$
and therefore provides the complete equilibrium description of the system.

Multiplying both sides of Eq. (28) by $\frac{d\psi}{dx}$ and rearranging the
expression, the expression for the electrostatic interaction $P$ is obtained
as follows.

${\rm{P=}}\Pi{\rm{+P}}_{{\rm{bulk}}}=-\frac{{\varepsilon_{0}\varepsilon_{r}}}{2}\left({\frac{d\psi}{dx}}\right)^{2}+2n_{0}k_{B}T\cosh\left({\frac{{e\psi}}{{k_{B}T}}}\right),$
(47)

where $n_{0}=n_{H^{+},\infty}+n_{+,\infty}=n_{OH^{-},\infty}+n_{-,\infty}$ and
$\Pi$ and ${\rm{P}}_{{\rm{bulk}}}$ are the osmotic pressure and the
electrostatic interaction for infinite distance between two polyelectrolyte
brushes, respectively. As the separation between the two polyelectrolyte
brushes reaches to the positive infinity, the electrostatic interaction
${\rm{P=P}}_{{\rm{bulk}}}$. Substituting ${\rm{P}}_{{\rm{bulk}}}=2n_{0}k_{B}T$
into Eq. (47), we get the following expression

$\Pi=-\frac{{\varepsilon_{0}\varepsilon_{r}}}{2}\left({\frac{d\psi}{dx}}\right)^{2}+2n_{0}k_{B}T\left[{\cosh\left({\frac{{e_{0}\psi}}{{k_{B}T}}}\right)-1}\right].$
(48)

For the sake of ease, if we consider at the centerline, the expression for
osmotic pressure can be represented more concisely

$\Pi=2n_{0}k_{B}T\left[{\cosh\left({\frac{{e_{0}\psi\left({x=0}\right)}}{{k_{B}T}}}\right)-1}\right].$
(49)

It is important to emphasize that we rely on the assumption that two brushes
aren’t interpenetrate. Therefore, we consider only the cases when the distance
between the two polyelectrolyte brushes is larger than twice the thickness of
polyelectrolyte brushes.

##  III Results and Discussion

Figure 2: (Color online) For two pH-responsive polyelectrolyte brushes,
variation of electrostatic potential (a), number density of polyelectrolyte
ions (b), monomer distribution profiles (c) , non-dimensional chain end
distribution profiles (d) with $x$-coordinate. The case of zhulina_jcp_1997 is
the one where $\nu=0$ and $\omega=0$. $a=1nm,pH=3,pK=3.5,\gamma
a^{3}=1,c_{b}=0.01mol/L,N=400,l_{0}=60nm,\zeta=1,b=0,h=30nm$.

We account for the influence of the excluded volume interaction and the
expanded form of the mass action law on pH-responsive brushes to be in a good
solvent. To compare our results with previous studies zhulina_jcp_1997 ;
das_sm_2019a , we use the same values of parameters $\nu$ and $\omega$ as in
the papers.

Fig. 2(a) shows the electrostatic potential as a function of $x$-coordinate by
considering both finite excluded volume interactions and no excluded volume
interactions. As in ref das_sm_2019a , near the surface the magnitude of
electrostatic potential increases with excluded volume parameter $\nu$,
whereas at the intermediate position ($x=0$) between two polyelectrolyte
brushes the electrostatic potential decrease with excluded volume parameter
$\nu$. This is explained by the fact that a higher excluded volume interaction
leads to a longer thickness of polyelectrolyte brush (see Fig. 2(b)) and
causes a slow change in the electrostatic potential. Considering the spatial
behavior of electrostatic potential, Eqs. (32-34) provide the fact that the
number densities of positive ions and hydrogen ions decrease with the distance
from the supporting surface. In contrast, it is clear that the number
densities of negative ions and hydroxyl ions increase with the distance from
supporting surfaces.

Figure 3: (Color online) For two pH-responsive polyelectrolyte brushes,
variation of the thickness of polyelectrolyte brush (a), the relative change
of thickness of polyelectrolyte brush (b), the electrostatic potential at the
centerlinel (c), the osmotic pressure (d) with the half distance between two
pH-responsive polyelectrolyte brushes. The other parameters are the same in
Fig. 2.

Fig. 2(b) depicts the number density of polyelectrolyte ions as a function of
$x$-coordinate. We confirm that a smaller $\nu$ factor provides a shorter
polyelectrolyte brush thickness, attributed to excluded volume interaction.
Moreover, from the insets we can know that near the end of polyelectrolyte
brush, a small $\nu$ provides a larger number density of polyelectrolyte ions.
This can be understood by combining Eq. (26) and the fact that a small $\nu$
provides a smaller magnitude of electrostatic potential as shown in Fig. 2(a).

Fig. 2(c) displays the dimensionless monomer distribution profile of a
polyelectrolyte chain as a function of $x$-coordinate. In the same way in
das_sm_2019a , a small $\nu$ yields a denser monomer concentration near to the
wall, and accordingly, in order to ensure a constant N, results in a smaller
monomer concentration away from the wall.

Fig. 2(d) shows the non-dimensional chain end distribution profiles as a
function of the $x$-coordinate. A smaller excluded volume effect provides a
higher density of the monomers in the close region from the surface. On the
other hand, a large excluded volume effects leads to a flatter distribution of
$g$ and ensures a larger $g$ value at farther positions from the surface.

Figure 4: (Color online) For two pH-responsive polyelectrolyte brushes,
variation of the thickness of polyelectrolyte brush (a), the electrostatic
potential at the centerline (b), the osmotic pressure (c) with $N$, the number
of Kuhn monomer in every polyelectrolyte brush molecule. While for $\nu=0.0$,
$\omega=0.0$, for the other values of $\nu$, $\omega=0.1$. The other
parameters are the same as in Fig. 2

Fig. 3(a) shows the brush thickness as a function of the half separation
between two pH-responsive polyelectrolyte brushes for different values of the
excluded volume interactions (quantified by different values of
$\nu=0.0,\omega=0.0;\nu=0.1,0.5,1,\omega=0.1$). We should first indicate that
when two pH-responsive polyelectrolyte brushes get close to each other, the
equilibrium thickness of polyelectrolyte brush is diminished. i.e. the
polyelectrolyte brush contracts. Moreover, such a trend is enhanced with
increasing the excluded volume interaction($\nu$). This is attributed to the
fact that a larger osmotic pressure due to a higher excluded volume parameter
allows the polyelectrolyte brush to contract more strongly.

Fig. 3(b) shows the relative variation of the brush thickness as a function of
the half separation between two pH-responsive polyelectrolyte brushes. The
variation is defined as $\Delta H=H-H_{\infty}$,where $H_{\infty}$ is the
brush thickness when the distance between two pH-polyelectrolyte brushes is
infinite. The figure displays that at any distance, a large $\nu$ produces a
larger relative decrease in brush thickness. The reason is the same as that in
Fig. 3(a). In other words, a large excluded volume interaction results in not
only a larger absolute contract in brush thickness but also a larger relative
contract in brush thickness. This is the first key finding of the present
study.

Fig. 3(c) depicts the electrostatic potential at the centerline between two
pH-responsive polyelectrolyte brushes as a function of the half separation
between the brushes for different values of excluded volume parameter
($\nu=0.0,\omega=0.00:\nu=0.1,0.5,1,\omega=0.01$). It is shown that a decrease
in the distance between two pH-responsive polyelectrolyte brushes results in
enhancement in the magnitude of the centerline potential, attributed to the
nature of the electrostatic interaction which decreases with the distance
between two charged objects. Moreover, a higher excluded volume parameter
provides a larger magnitude of electrostatic potential at the centerline
between two charged surfaces, while the difference in electrostatic potential
for the cases with different excluded volume parameters increases with
decreasing the distance between two charged surfaces. This can be explained by
following the same reason as in Fig. 2(a).

Fig. 3(d) displays the osmotic pressure between two pH-responsive
polyelectrolyte brushes as a function of the half separation between the
brushes for different values of excluded volume parameter. From the formula
for osmotic pressure, Eq. (49), we can be aware of the fact that the osmotic
pressure between two polyelectrolyte brushes decreases with the distance
between two surfaces and is enhanced with increasing the excluded volume
parameter in the same way as the electrostatic potential at the centerline
between the two brushes.

Fig. 4(a) shows the thickness of polyelectrolyte brush as a function of the
number of Kuhn monomer in every polyelectrolyte brush molecule. It displays
that the thickness of polyelectrolyte brush increases nearly linearly with the
number of Kuhn monomer. Such a behavior can be understood by the fact that the
osmotic pressure is very weak and thus the change in the thickness of
polyelectrolyte brush due to the osmotic pressure is negligible.

Fig. 4(b) depicts the electrostatic potential at the centerline between two
charged plates as a function of the number of Kuhn monomer in every
polyelectrolyte brush molecule. Increasing the number of Kuhn monomer in a
polyelectrolyte brush means an increment of the thickness of polyelectrolyte
layer and hence yields an enhancement in the magnitude of electrostatic
potential at the centerline between two pH-responsive polyelectrolyte brushes.
Due to the same reason as in Fig. 3(b), an increase in the excluded volume
interaction provides an enhancement of the electrostatic interaction.

Fig. 4(c) displays the osmotic pressure between the two pH-responsive
polyelectrolyte brushes as a function of the number of Kuhn monomer in every
polyelectrolyte brush molecule. From the formula for the osmotic pressure, the
osmotic pressure increases with the number of Kuhn monomer in a
polyelectrolyte brush. In addition, the difference in osmotic pressure between
the cases with different values of excluded volume parameter is enhanced with
the number with Kuhn monomer in a polyelectrolyte brush.

Figure 5: (Color online) For two pH-responsive polyelectrolyte brushes,
variation of the thickness of polyelectrolyte brush (a), the electrostatic
potential at the centerline (b), the osmotic pressure (c) with Kuhn length.
While for $\nu=0.0$, $\omega=0.0$, for the other values of $\nu$,
$\omega=0.1$. The other parameters are the same as in Fig. 2.

Fig. 5(a) shows the thickness of polyelectrolyte brush at the centerline
between two pH-responsive polyelectrolyte brushes as a function of Kuhn length
in every polyelectrolyte brush molecule. An increase of Kuhn length represents
an expansion of every polyelectrolyte chain and hence gives an enhancement in
the thickness of polyelectrolyte brush. It is noticeable that a larger
excluded volume parameter provides a steeper change in the thickness of
polyelectrolyte brush. This can be understood by combining the facts that a
larger excluded volume parameter provides a stronger excluded volume
interaction and that a larger Kuhn length induces a larger deformation of
every segment of a polyelectrolyte chain under a given tension.

Fig. 5(b) depicts the electrostatic potential at the centerline between two
pH-responsive polyelectrolyte brushes as a function of Kuhn length in every
polyelectrolyte brush molecule. The figure shows that the magnitude of
electrostatic potential increases with Kuhn length and the difference in the
electrostatic potential between different values of excluded volume parameter
is enhanced with Kuhn length. This is attributed to the fact that a large Kuhn
length yields a longer thickness of polyelectrolyte brushes (see Fig. 5(a))
and that a long thickness of polyelectrolyte brush allows the centerline
potential to be larger(see Fig. 2(a)). Fig. 5(c) displays the osmotic pressure
between two pH-responsive polyelectrolyte brushes as a function of Kuhn length
in every polyelectrolyte brush molecule. The osmotic pressure increases with
Kuhn length and the difference in the osmotic pressure between different
values of excluded volume parameter is enhanced with Kuhn length. The fact can
be understood by combining the fact that from the definition for osmotic
pressure, osmotic pressure increases with the centerline potential and the
reason explained in Fig. 5(b)

Figure 6: (Color online) For two pH-responsive polyelectrolyte brushes,
variation of the thickness of polyelectrolyte brushes (a), the electrostatic
potential at the centerline (b), the osmotic pressure (c) with pH.
$c_{b}=0.1mol/L,h=40nm$. The other parameters are the same as in Fig. 2.

Fig. 6(a) shows the thickness of polyelectrolyte brush as a function of pH. A
larger pH (a smaller H+ ion concentration) provides enhanced charging of the
polyelectrolyte brushes and hence ensures enhanced counterion-induced osmotic
expansion of the brushes causing a larger brush thickness for all values of
$\nu$. It should be emphasized that as pH increases, the difference in brush
thickness between the cases having different values of excluded volume
parameter is reduced. This can be explained by the fact that as pH increases,
counterion-induced osmotic swelling is enhanced and plays more and more
important role in stretching of polyelectrolyte chains compared to excluded
volume interaction between monomers in polyelectrolyte chains.

Fig. 6(b) depicts the electrostatic potential at the centerline between two
pH-responsive polyelectrolyte brushes as a function of pH. The figure shows
that the magnitude of electrostatic potential increases with pH and the
difference in the electrostatic potential between different values of excluded
volume parameter is enhanced with pH. Such behavior can be explained by
considering the fact that a large pH allows the thickness of polyelectrolyte
brushes to be larger (see Fig. 6(a)) and a long polyelectrolyte brush yields a
lager centerline potential (see Fig. 2(a)). Fig. 6(c) displays the osmotic
pressure between two pH-responsive polyelectrolyte brushes as a function of
pH. The osmotic pressure increases with pH and the difference in osmotic
pressure between different values of excluded volume parameter is enhanced
with pH. Considering the properties of centerline potential, this behavior can
be understood in the similar way as Fig. 5(c)

Figure 7: (Color online) For two pH-responsive polyelectrolyte brushes,
variation of the thickness of polyelectrolyte brush (a), the electrostatic
potential at the centerline (b), the osmotic pressure (c) with length $l_{0}$.
$h=50nm$. The other parameters are the same as in Fig. 6.

Fig. 7(a) shows the thickness of polyelectrolyte layer as a function of the
lateral separation between the adjacent polyelectrolyte brushes. One can first
know that the thickness of polyelectrolyte brush decreases with the lateral
separation between the adjacent polyelectrolyte brushes. In fact, from the
definition of $l_{0}$, a larger value of $l_{0}$ provides a smaller
polyelectrolyte brush grafting density and hence a smaller polyelectrolyte
ions concentration. Such a smaller polyelectrolyte ion concentration will
cause a smaller stretching of polyelectrolyte brushes and provide a smaller
thickness of polyelectrolyte brush.

Fig. 7(b) depicts the electrostatic potential at the centerline between two
pH-responsive polyelectrolyte brushes as a function of the lateral separation
between the adjacent polyelectrolyte brushes. The figure shows that the
magnitude of electrostatic potential decreases with $l_{0}$ and the difference
in the electrostatic potential between different values of excluded volume
parameter is reduced with $l_{0}$. Fig. 7(c) displays the osmotic
pressurebetween two pH-responsive polyelectrolyte brushes as a function of the
lateral separation between the adjacent polyelectrolyte brushes. The figure
shows that the magnitude of electrostatic potential decreases with $l_{0}$ and
the difference in the osmotic pressure between different values of excluded
volume parameter is reduced with $l_{0}$. The reason for this phenomenon is
the same as in Fig. 7(b).

Figure 8: (Color online) For two pH-responsive polyelectrolyte brushes,
variation of the thickness of polyelectrolyte brush (a), the electrostatic
potential at the centerline(b), the osmotic pressure (c) with pK. $h=40nm$.
The other parameters are the same as in Fig. 6.

Fig. 8(a) shows the thickness of polyelectrolyte brush as a function of pK. A
larger pK (a smaller polyelectrolyte ion concentration) provides weakened
charging of the polyelectrolyte brushes and hence ensures weakened counterion-
induced osmotic swelling of the brushes yielding a smaller brush thickness for
all values of $\nu$. It should be emphasized that an increase in pK enhances
the difference in brush thickness between the cases with different values of
excluded volume parameter. This can be understood by the fact that when pK
decreases, counterion-induced osmotic swelling increases and plays more and
more important role in stretching of polyelectrolyte chains compared to
excluded volume interaction between monomers in polyelectrolyte chains.

Fig. 8(b) depicts the electrostatic potential at the centerline between two
pH-responsive polyelectrolyte brushes as a function of pK. The figure shows
that the magnitude of electrostatic potential decreases with pK and the
difference in the electrostatic potential between different values of excluded
volume parameter is reduced with pK. Fig. 8(c) displays the osmotic pressure
between two pH-responsive polyelectrolyte brushes as a function of pK. The
osmotic pressure decreases with pK and the difference in osmotic pressure
between different values of excluded volume parameter is reduced with pK. The
reason for this phenomenon is the same as in Fig. 8(b).

Figure 9: (Color online) For two pH-responsive polyelectrolyte brushes,
variation of the thickness of polyelectrolyte brush (a), the electrostatic
potential at the centerline(b), the osmotic pressure (c) with $\gamma a^{3}$.
While for $\nu=0.0$, $\omega=0.0$, for the other values of $\nu$,
$\omega=0.1$. The other parameters are the same as in Fig. 2.

Fig. 9(a) shows thickness of polyelectrolyte brush as a function of $\gamma
a^{3}$. Under a given Kuhn length, a larger $\gamma a^{3}$ means a stronger
density of polyelectrolyte chargeable sites and hence ensures a stronger
counterion-induced osmotic swelling of the brushes implying a longer brush
thickness for all values of $\nu$. It should be emphasized that an increase in
$\gamma a^{3}$ diminishes the difference in brush thickness between the cases
with different values of excluded volume parameter. This can be proved by
understanding that when $\gamma a^{3}$ increases, counterion-induced osmotic
swelling increases and plays more and more important role in stretching of
polyelectrolyte chains compared to excluded volume interaction between
monomers in polyelectrolyte chains. The reason for this fact is identical to
the corresponding one of Fig. 8(b).

Fig. 9(b) depicts the electrostatic potential at the centerline between two
pH-responsive polyelectrolyte brushes as a function of $\gamma a^{3}$. The
figure shows that the magnitude of electrostatic potential increases with
$\gamma a^{3}$ and the difference in the electrostatic potential between
different values of excluded volume parameter enhances with $\gamma a^{3}$.
Fig. 9(c) displays the osmotic pressure between two pH-responsive
polyelectrolyte brushes as a function of $\gamma a^{3}$. The osmotic pressure
increases with $\gamma a^{3}$ and the difference in the electrostatic
potential between different values of excluded volume parameter increases with
$\gamma a^{3}$. This phenomenon is attirbuted to the same reason as in Fig.
9(b).

Figure 10: (Color online) For two pH-responsive polyelectrolyte brushes,
variation of the electrostatic potential (a), concentration of polyelectrolyte
ions (b), dimensionless monomer distribution profile (c), non-dimensional
chain end distribution profile (d) with $x$-coordinate.
$pH=3,c_{b}=0.01mol/L,pK=3.5,a=1nm,N=400,l_{0}=60nm,b=0,\nu=0.5,\omega=0.1,r_{0}=3.3\times
10^{-10}m$.  Figure 11: (Color online) For two pH-responsive polyelectrolyte
brushes, variation of the electrostatic potential at the centerline (b), the
osmotic pressure (c) with the half distance between two pH-responsive
polyelectrolyte brushes, $h$. The other parameters are the same as in Fig. 10.
Figure 12: (Color online) For two pH-responsive polyelectrolyte brushes,
variation of the thickness of polyelectrolyte brush (a), the electrostatic
potential at the centerline(b), the osmotic pressure (c) with $\nu$. $h=55nm$.
The other parameters are the same as in Fig. 10.

Fig. 10(a) shows the electrostatic potential as a function of $x$-coordinate
for different values of Born energy difference inside polyelectrolyte brush
layers. It turns out from the figure that an increase in Born energy
difference (i.e. a decrease in $\zeta$) results in an increase of magnitude of
electrostatic potential at all locations. As pointed out in sin_csa_2021 ;
horno_jcis_2003 ; sadeghi_eleccom_2017 ; gopmandal_pre_2018 ; sadeghi_csb_2018
; gopmandal_pre_2020 ; gopmandal_ep_2021 ; poddar_sm_2016 ; pandey_csa_2021 ,
a small values of electric permittivity of polyelectrolyte layer allows Born
energy difference to be high. Therefore, overcoming such a region needs more
external energy. As a consequence, a smaller permittivity ratio yields a
higher magnitude of electrostatic potential.

Fig. 10(b) depicts the concentration of polyelectrolyte ions as a function of
$x$-coordinate for different values of Born energy difference inside
polyelectrolyte brush layers. Noticeably, we find that when Born energy
difference is enhanced, the thickness of polyelectrolyte brush layer is
expanded. This can be explained as follows. Firstly, an increase in Born
energy difference provokes a decrease in number densities of all kinds of
electrolyte ions. Thus, the concentration of hydrogen ions becomes lower and
accoring to the mass action law, dissociation reaction of polyelectrolyte
molecules is enhanced and hence polyelectrolyte ions are more produced. As a
results, the stronger the electric repulsion between polyelectrolyte ions, the
longer the lengths of polyelectrolyte chains.

Fig. 10(c) displays the monomer distribtuion profile with $x$-coordinate for
different values of Born energy difference inside polyelectrolyte brush
layers. From the normalization condition of monomer distribution profiles,
near the a core surface the monomer density lowers with increasing Born energy
difference(i.e. decrasing $\zeta$), whereas at distant positions from the
surface, the monomer density increases with Born energy difference.

Fig. 10(d) shows non-dimensional chain end distribution profiles with
$x$-coordinate for different values of Born energy difference inside
polyelectrolyte brush layers. It should be pointed out that inside the brush
layer, a large value of Born energy difference(i.e. a small value of $\zeta$)
provides a small value of chain end distribution at any positions inside
polyelectrolyte brush layer.

Fig. 11(a) shows the variation of the electrostatic potential at the
centerline between two pH-responsive polyelectrolyte brushes with half
separation between the two brushes for different values of Born energy
difference inside polyelectrolyte brush layers. From it, we can see that a
decrease in Born energy difference (i.e. an increase in $\zeta$) results in a
decrease of magnitude of electrostatic potential at the centerline. This can
be understood by following the same reason as in Fig. 10(a). Combining Eq.
(49) and the above fact, we can understand that as shown in Fig. 11(b), an
increase in Born energy difference provides an increase in osmotic pressure
between two pH-responsive polyelectrolyte brushes.

Fig. 12(a) shows the thickness of pH-responsive polyelectrolyte brush as a
function of excluded volume interaction factor $\nu$ for different values of
Born energy difference inside polyelectrolyte brush layers. Combining the the
results from Fig. 3(a) and Fig. 10(b), both a large excluded volume
interaction and a large Born energy difference provide a longer thickness of
pH-responsive polyelectrolyte layer.

We can expect that the electrostatic potential at the centerline and osmotic
pressure between two pH-responsive polyelectrolyte brushes increase with both
Born energy difference and excluded volume interaction factor(Fig. 12(b) and
Fig. 12(c)).

Now we should consider ion size effect on the structure and interaction
between two pH-responsive polyelectrolyte brushes. Although we have probed ion
size effect concretely, it turns out that in weak polyelectrolyte brush, the
effect becomes insignificant because the magnitude of the electrostatic
potential is not high.

Summarizing, it is elucidated that a large excluded volume interaction yields
a long pH-responsive polyelectrolyte brush and a centerline potential and
osmotic pressure between two charged surfaces is enhanced for the case with a
large excluded volume interaction than a smaller excluded volume interaction.
In particular, as two pH-responsive brushes approach, the absolute and
relative decrease in brush thickness increases with excluded volume
interaction. The result distinguishes our theory from the previous results
zhulina_jcp_1997 ; zhulina_lang_2011 ; das_sm_2019a .

On the other hand, the consideration of density of chargeable sites on a
polyelectrolyte molecule involves a change in brush thickness, centerline
potential and osmotic pressure. As a result, increasing density of chargeable
site on a polyelectrolyte molecule reasonably not only enhances brush
thickness, centerline potential and osmotic pressure but also relative
decrease of brush thickness when two brushes approach.

However, the present study does not consider the influence of solvent
polarization iglic_bioelec_2012 ; sin_EA_2015 ; sin_jcp_2017 , ionic
correlation bazant_pre_2012 and surface charge distribution bohinc_jcp_2016 on
the interaction between two pH-responsive polyelectrolyte brushes. In fact,
because in the paper, the bulk ionic concentration and the magnitude of
electrostatic potential is not high, the two effects become trivial. The
justification for using the present theory is that we study weak
polyelectrolyte brushes, not strong polyelectrolyte brushes. If we consider
the case of high salt concentration or high magnitude of electrostatic
potential, the ion size, solvent polarization and ionic correlation will be
significant. In the future, the effects should be considered by the authors in
more detail, in combination with molecular simulations, which will require
very expensive cost of computational efforts.

##  IV Conclusions

In this paper, we predict the behavior of two interacting pH-responsive brush
system by using an augmented strong stretching theory considering not only the
excluded volume interactions between the polyelectrolyte segments but also a
more expanded form of the mass action law valid for $\gamma a^{3}$.

We unravel that when two interacting pH-responsive polyelectrolyte brushes
approach, contraction of the brush thickness is enhanced with the excluded
volume interaction and with $\gamma a^{3}$.

We also find that centerline potential and osmotic pressure between two
interacting pH-responsive polyelectrolyte brushes decreases with distance,
increases with Kuhn length, $\gamma a^{3}$, pH and number of Kuhn monomer and
decreases with pK and the lateral separation between polyelectrolyte chain
molecules $l_{0}$.

Moreover, we demonstrate that a consideration of Born energy difference
enhances thickness of polyelectrolyte brush layer and osmotic pressure between
two pH-responsive polyelectrolyte brushes.

We conclude that simultaneous considerations of excluded volume interaction,
density of polyelectrolyte chargeable sites and the Born energy difference are
necessary for explaining experiments.

Our study will be useful for explaining physicochemical properties of
nanoparticles grafted with pH-responsive brushes to be in a ”good” solvent and
will be further improved to a more advanced theory dealing with various kinds
of polyelectrolytes.

##  V Conflicts of interest

The authors have no conflicts to disclose.

##  VI Data Availavility

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

##  Appendix A Variation of Free energy for $\nu=0.0$ and $\omega=0.0$

As already mentioned in Section. II, for case of $\nu=0.0$ and $\omega=0.0$,
the formulae for the monomer distribution profile( i.e. Eqs. (35, 36, 37))
diverge.

To avoid the difficulty, here, we derive a different solving method for the
case without excluded volume interaction($\nu=0.0$ and $\omega=0.0$). We
consider the case accounting for the expanded form of the mass action law and
Born energy difference, but not excluded volume interaction between monomers
on polyelectrolyte chains.

If we neglect the two effects, the theory leads to ones of zhulina_lang_2011 .

The total free energy functional ($F$) of the PE brush system consists of the
elastic ($F_{els}$), electrostatic ($F_{elec}$), and ionization ($F_{ion}$)
free energies of polyelectrolyte brush, the electrostatic energy of the EDL
($F_{EDL}$) induced by this brush and the Born energy difference($F_{Born}$).

$F=F_{els}+F_{elec}+F_{ion}+F_{EDL}+F_{Born}.$ (50)

While $F_{els}$ is given by Eq. (4), $F_{elec}+F_{EDL}$ is written as

$\displaystyle\begin{array}[]{l}\frac{{F_{elec}+F_{EDL}}}{{k_{B}T}}=\frac{1}{{\sigma
k_{B}T}}\int_{-h}^{{\rm{-}}h+H}{\left[{-\frac{{\varepsilon_{0}\varepsilon_{p}}}{2}\left|{\frac{{d\psi}}{{dx}}}\right|^{2}+e\psi\left({n_{+}-n_{-}+n_{H^{+}}-n_{OH^{-}}}\right)+\Delta
W\left({n_{+}+n_{-}+n_{H^{+}}+n_{OH^{-}}}\right)}\right]dx}\\\
+\frac{1}{{\sigma
k_{B}T}}\int_{-h+H}^{0}{\left[{-\frac{{\varepsilon_{0}\varepsilon_{r}}}{2}\left|{\frac{{d\psi}}{{dx}}}\right|^{2}+e\psi\left({n_{+}-n_{-}+n_{H^{+}}-n_{OH^{-}}}\right)}\right]dx}-\frac{1}{{\sigma
k_{B}T}}\int_{-h}^{-h+H}{\left[{e\psi n_{A^{-}}\phi}\right]dx}\\\
+\frac{1}{\sigma}\int_{-h}^{0}{\left\\{{n_{+}\left[{\ln\left({\frac{{n_{+}}}{{n_{+,\infty}}}}\right)-1}\right]+n_{-}\left[{\ln\left({\frac{{n_{-}}}{{n_{-,\infty}}}}\right)-1}\right]}\right\\}}dx\\\
+\frac{1}{\sigma}\int_{-h}^{0}{\left\\{{n_{H^{+}}\left[{\ln\left({\frac{{n_{H^{+}}}}{{n_{H^{+},\infty}}}}\right)-1}\right]+n_{OH^{+}}\left[{\ln\left({\frac{{n_{OH^{+}}}}{{n_{OH^{+},\infty}}}}\right)-1}\right]}+\left({n_{+,\infty}+n_{-,\infty}+n_{H^{+},\infty}+n_{OH^{-},\infty}}\right)\right\\}dx}\\\
\end{array}$ (55)

$F_{ion}$ and $F_{Born}$ are written as Eq. (22) and Eq. (6), respectively.
Substituting Eqs.(2), Eq. (55) and Eq. (6) into Eq. (50), the free energy is
obtained as follows.

$\begin{array}[]{l}\frac{F}{{k_{B}T}}=\frac{3}{{2pa^{2}}}\int_{-h}^{-h+H}{g\left({x^{\prime}}\right)dx^{\prime}}\int_{-h}^{x^{\prime}}{g\left({x,x^{\prime}}\right)dx}+\\\
\frac{1}{{\sigma
k_{B}T}}\int_{-h}^{{\rm{-}}h+H}{\left[{-\frac{{\varepsilon_{0}\varepsilon_{p}}}{2}\left|{\frac{{d\psi}}{{dx}}}\right|^{2}}\right]dx}+\frac{1}{{\sigma
k_{B}T}}\int_{-h+H}^{0}{\left[{-\frac{{\varepsilon_{0}\varepsilon_{r}}}{2}\left|{\frac{{d\psi}}{{dx}}}\right|^{2}}\right]dx}\\\
+\frac{1}{{\sigma k_{B}T}}\int_{-h}^{-h+H}{\left[{\Delta
W\left({n_{+}+n_{-}+n_{H^{+}}+n_{OH^{-}}}\right)}\right]dx}\\\
+\frac{1}{{\sigma
k_{B}T}}\int_{-h}^{0}{\left[{e\psi\left({n_{+}-n_{-}+n_{H^{+}}-n_{OH^{-}}}\right)}\right]dx}-\frac{1}{{\sigma
k_{B}T}}\int_{-h}^{-h+H}{\left[{e\psi n_{A^{-}}\phi}\right]dx}\\\
+\frac{1}{\sigma}\int_{-h}^{0}{\left\\{{n_{+}\left[{\ln\left({\frac{{n_{+}}}{{n_{+,\infty}}}}\right)-1}\right]+n_{-}\left[{\ln\left({\frac{{n_{-}}}{{n_{-,\infty}}}}\right)-1}\right]}\right\\}}dx\\\
+\frac{1}{\sigma}\int_{-h}^{0}{\left\\{{n_{H^{+}}\left[{\ln\left({\frac{{n_{H^{+}}}}{{n_{H^{+},\infty}}}}\right)-1}\right]+n_{OH^{+}}\left[{\ln\left({\frac{{n_{OH^{+}}}}{{n_{OH^{+},\infty}}}}\right)-1}\right]}\right\\}dx}\\\
+\frac{1}{{\sigma
a^{3}}}\int\limits_{-h}^{-h+H}{\phi\left(x\right)\left[{\left({1-\frac{{n_{A^{-}}}}{\gamma}}\right)\ln\left({1-\frac{{n_{A^{-}}}}{\gamma}}\right)+\frac{{n_{A^{-}}}}{\gamma}\left({\ln\frac{{n_{A^{-}}}}{\gamma}-\ln\frac{{K_{a}^{\prime}}}{{n_{H^{+},\infty}}}}\right)}\right]}\\\
\end{array}$ (56)

Eq. (56) should be minimized by performing the variation under the constraints
of Eq. (24, 25,26).

$\begin{array}[]{l}\delta\left({\frac{{F_{els}}}{{k_{B}T}}}\right)=\frac{3}{{2a^{2}}}\int\limits_{-h}^{-h+x}{\delta
g\left({x^{\prime}}\right)dx^{\prime}\int\limits_{-h}^{x^{\prime}}{E\left({x,x^{\prime}}\right)dx}}\\\
+\frac{3}{{2a^{2}}}\int\limits_{-h}^{-h+x}{g\left({x^{\prime}}\right)dx^{\prime}\int\limits_{-h}^{x^{\prime}}{\delta
E\left({x,x^{\prime}}\right)dx}}+\lambda\int\limits_{-h}^{-h+H}{\delta\phi\left(x\right)dx}\\\
-\int\limits_{-h}^{-h+H}{\lambda_{1}\left(x\right)dx\int\limits_{0}^{x}{\frac{{\delta
E\left({x,x^{\prime}}\right)dx^{\prime}}}{{E^{2}\left({x,x^{\prime}}\right)}}}}\\\
\end{array}$ (57)

where $\delta\phi\left(x\right)=\sigma
a^{3}\int\limits_{x}^{-h+H}{dx^{\prime}\left({\frac{{\delta
g\left({x^{\prime}}\right)}}{{E\left({x,x^{\prime}}\right)}}-\frac{{g\left({x^{\prime}}\right)\delta
E\left({x,x^{\prime}}\right)}}{{E^{2}\left({x,x^{\prime}}\right)}}}\right)}$.

$\begin{array}[]{l}\delta\left({\frac{{F_{ion}}}{{k_{B}T}}}\right)=\frac{1}{{\sigma
a^{3}}}\int\limits_{-h}^{-h+H}{\delta\phi\left(x\right)\left[{\ln\left({1-\frac{{n_{A^{-}}}}{\gamma}}\right)+\frac{{n_{A^{-}}}}{\gamma}\left({\ln\frac{{\frac{{n_{A^{-}}}}{\gamma}n_{H^{+},\infty}}}{{1-\frac{{n_{A^{-}}}}{\gamma}}}-\ln
K_{a}^{\prime}}\right)}\right]}dx\\\ +\frac{1}{{\sigma
a^{3}}}\int\limits_{-h}^{-h+H}{\frac{{\phi\left(x\right)}}{\gamma}\left[{\ln\left({\frac{{\frac{{n_{A^{-}}}}{\gamma}}}{{1-\frac{{n_{A^{-}}}}{\gamma}}}}\right)-\ln\frac{{K_{a}^{\prime}}}{{n_{H^{+},\infty}}}}\right]}dx\delta
n_{A^{-}}.\end{array}$ (58)

The Euler-Lagrange equation for $n_{A^{-}}$ leads to the following equation.

$\begin{array}[]{l}-\frac{{e\psi\phi}}{{k_{B}T}}+\frac{{\phi\left(x\right)}}{{\gamma
a^{3}}}\left[{\ln\left({\frac{{\frac{{n_{A^{-}}}}{\gamma}}}{{1-\frac{{n_{A^{-}}}}{\gamma}}}}\right)-\ln\frac{{K_{a}^{\prime}}}{{n_{H^{+},\infty}}}}\right]=0,\\\
\ln\left({\frac{{\frac{{n_{A^{-}}}}{\gamma}}}{{1-\frac{{n_{A^{-}}}}{\gamma}}}}\right)-\ln\frac{{K_{a}^{\prime}}}{{n_{H^{+},\infty}}}=\gamma
a^{3}\frac{{e\psi}}{{k_{B}T}},\\\
n_{A^{-}}=\frac{{K_{a}^{\prime}\gamma}}{{K_{a}^{\prime}+n_{H^{+},\infty}\exp\left({-\gamma
a^{3}\frac{{e\psi}}{{k_{B}T}}}\right)}}.\end{array}$ (59)

The Euler-Lagrange equation for $\psi$ provides the Poisson equations.

$\displaystyle\varepsilon_{0}\varepsilon_{p}\left({\frac{{d^{2}\psi}}{{dx^{2}}}}\right)+e\left({n_{+}-n_{-}+n_{H^{+}}-n_{OH^{-}}-n_{A^{-}}\phi}\right)=0,\left(-h\leq
x\leq-h+H\right)$ (60)
$\displaystyle\varepsilon_{0}\varepsilon_{r}\left({\frac{{d^{2}\psi}}{{dx^{2}}}}\right)+e\left({n_{+}-n_{-}+n_{H^{+}}-n_{OH^{-}}}\right)=0,\left(-h+H\leq
x\leq 0\right)$ (61)

The Euler-Lagrange equations for $n_{\pm},n_{H^{+}},n_{OH^{-}}$ yield the
following number densites of cations, anions, hydrogen ions and hydroxyl ions.

In the region of $-h\leq x\leq-h+H$

$\displaystyle
n_{\pm}=n_{\pm,\infty}\exp\left({\mp\frac{{e\psi}}{{k_{B}T}}-\frac{{\Delta
W}}{{k_{B}T}}}\right),$ (62) $\displaystyle
n_{H^{+}}=n_{H^{+},\infty}\exp\left({-\frac{{e\psi}}{{k_{B}T}}-\frac{{\Delta
W}}{{k_{B}T}}}\right),$ (63) $\displaystyle
n_{OH^{-}}=n_{OH^{-},\infty}\exp\left({\frac{{e\psi}}{{k_{B}T}}{\rm{-}}\frac{{\Delta
W}}{{k_{B}T}}}\right).$ (64)

In the region of $-h+H<x\leq 0$

$\displaystyle
n_{\pm}=n_{\pm,\infty}\exp\left(\mp\frac{{e\psi}}{{k_{B}T}}\right),$ (65)
$\displaystyle
n_{H^{+}}=n_{H^{+},\infty}\exp\left(-\frac{{e\psi}}{{k_{B}T}}\right),$ (66)
$\displaystyle
n_{OH^{-}}=n_{OH^{-},\infty}\exp\left(\frac{{e\psi}}{{k_{B}T}}\right).$ (67)

The Euler-Lagrange equation for $E\left(x,x^{\prime}\right)$ yields the
following expression.

$\frac{{3g\left({x^{\prime}}\right)}}{{2a^{2}}}-\left({\lambda+\ln\left({1-\frac{{n_{A-}}}{\gamma}}\right)}\right)\frac{{g\left({x^{\prime}}\right)}}{{E^{2}\left({x,x^{\prime}}\right)}}-\frac{{\lambda_{1}\left({x^{\prime}}\right)}}{{E^{2}\left({x,x^{\prime}}\right)}}=0.$
(68)

After a brief rearrangement, we get
$E\left({x,x^{\prime}}\right)=\sqrt{U_{1}\left({x^{\prime}}\right)-U_{2}\left({x^{\prime}}\right)},U_{1}\left({x^{\prime}}\right)=\frac{{2a^{2}\lambda_{1}\left({x^{\prime}}\right)}}{{3g\left({x^{\prime}}\right)}},U_{2}\left(x\right)=-\frac{{2a^{2}}}{3}\left[{\lambda+\ln\left({1-\frac{{n_{A^{-}}}}{\gamma}}\right)}\right]$.

Tension should be vanished at the free end of the brush

$E\left({x^{\prime},x^{\prime}}\right)=0,x^{\prime}>0.$ (69)

one finds $U_{1}\left(x\right)\equiv U_{2}\left(x\right)\equiv
U\left(x\right)$.

By using the normalization condition Eq. (24, 25, 26), we obtain a well-known
result for the Gaussian elasticity of tethered chains.

$U\left(x\right)=\frac{{\pi^{2}}}{{4N^{2}}}\left({x+h}\right)^{2}.$ (70)

Hence, the function of local chain stretching is given by
$E\left({x,^{\prime}x}\right)=\frac{\pi}{{2N}}\sqrt{\left({x^{\prime}+h}\right)^{2}-\left({x+h}\right)^{2}}$.

Combining Eq. (68), Eq. (69) and Eq. (70) and rearranging, we lead to the
equation for determining the local number density of the polyelectrolyte ion
$A^{-}$

$\begin{array}[]{l}\ln\left({1-\frac{{n_{A^{-}}}}{\gamma}}\right)=-\lambda-\frac{3}{{2a^{2}}}U\left(x\right)=-\lambda-\frac{{3\pi^{2}}}{{8N^{2}a^{2}}}\left({x+h}\right)^{2}=-\lambda-m^{2}\left({x+h}\right)^{2}\\\
\frac{{n_{A^{-}}}}{\gamma}=1-\exp\left({-\lambda-m^{2}\left({x+h}\right)^{2}}\right),\end{array}$
(71)

where $\lambda$ is a local Lagrange parameter and
$m^{2}=\frac{{3\pi^{2}}}{{8N^{2}a^{2}}}$.

We introduce the number density of polyelectrolyte ions at the edge of the
brush, $n_{H}=n_{A^{-}}\left({x=-h+H}\right)$, and specify the value of the
Lagrange multiplier as

$\begin{array}[]{l}\ln\left({1-\frac{{n_{H}}}{\gamma}}\right)=-\lambda-m^{2}H^{2}\\\
\lambda=-\ln\left({1-\frac{{n_{H}}}{\gamma}}\right)-m^{2}H^{2}.\end{array}$
(72)

Substituting Eq. (72) into Eq. (71), we obtain the following equation

$\frac{{n_{A^{-}}}}{\gamma}=1-\exp\left({-\lambda-m^{2}\left({x+h}\right)^{2}}\right)=1-\left({1-\frac{{n_{H}}}{\gamma}}\right)\exp\left({m^{2}H^{2}-m^{2}\left({x+h}\right)^{2}}\right).$
(73)

To obtain the electrostatic potential inside the pH-responsive polyelectrolyte
brush, we combine Eq. (73) with Eq. (59).

$\begin{array}[]{l}\frac{{n_{A^{-}}}}{\gamma}=\frac{{K_{a}^{\prime}}}{{K_{a}^{\prime}+n_{H^{+},\infty}\exp\left({-\gamma
a^{3}\frac{{e\psi}}{{k_{B}T}}}\right)}}=1-\left({1-\frac{{n_{H}}}{\gamma}}\right)\exp\left({m^{2}H^{2}-m^{2}\left({x+h}\right)^{2}}\right)\\\
\frac{{e\psi}}{{k_{B}T}}=\frac{1}{{\gamma
a^{3}}}\left[{m^{2}\left({\left({x+h}\right)^{2}-H^{2}}\right)+\ln\frac{{n_{H^{+},\infty}}}{{K_{a}^{\prime}\left({1-\frac{{n_{H}}}{\gamma}}\right)}}\left({1-\left({1-\frac{{n_{H}}}{\gamma}}\right)\exp\left({m^{2}H^{2}-m^{2}\left({x+h}\right)^{2}}\right)}\right)}\right].\end{array}$
(74)

Combining Eq. (74), Eq. (61) and Eq. (64) lead to the following equations for
monomer density distribution

$\begin{array}[]{l}\phi\left(x\right)=\left({\frac{{\varepsilon_{0}\varepsilon_{p}}}{e}\frac{{d^{2}\psi_{1}\left(x\right)}}{{dx^{2}}}+\left({n_{+}+n_{H^{+}}-n_{-}-n_{OH^{-}}}\right)}\right)/n_{A^{-}}=\\\
=\frac{{2\varepsilon_{0}\varepsilon_{p}}}{{e^{2}}}\frac{{m^{2}k_{B}T}}{{\gamma
a^{3}}}\left[{\frac{{1-\left({1-\frac{{n_{H}}}{\gamma}}\right)\exp\left({m^{2}H^{2}-m^{2}\left({x+h}\right)^{2}}\right)\left({1+2m^{2}\left({x+h}\right)^{2}}\right)}}{{\left[{1-\left({1-\frac{{n_{H}}}{\gamma}}\right)\exp\left({m^{2}H^{2}-m^{2}\left({x+h}\right)^{2}}\right)}\right]^{2}}}}\right]\\\
+\exp\left({-\frac{{\Delta
W}}{{k_{B}T}}}\right)\left[{\left({n_{H^{+},\infty}+n_{+,\infty}}\right)\exp\left({-\frac{{e\psi_{1}}}{{k_{B}T}}}\right)-\left({n_{OH^{-},\infty}+n_{-,\infty}}\right)\exp\left({\frac{{e\psi_{1}}}{{k_{B}T}}}\right)}\right].\end{array}$
(75)

Here, the following equations should be used

$\exp\left({\frac{{e\psi_{1}}}{{k_{B}T}}}\right)=\frac{{n_{H^{+},\infty}}}{{K_{a}^{\prime}\left({1-\frac{{n_{H}}}{\gamma}}\right)}}\left({1-\left({1-\frac{{n_{H}}}{\gamma}}\right)\exp\left({m^{2}H^{2}-m^{2}\left({x+h}\right)^{2}}\right)}\right)\exp\left({\frac{{m^{2}\left({\left({x+h}\right)^{2}-H^{2}}\right)}}{{\gamma
a^{3}}}}\right).$ (76)

When we don’t consider the Born energy difference and density of
polyelectrolyte chargeable sites (i.e. $\Delta W=0,\gamma a^{3}=1$), Eq. (75)
and Eq. (59) reach to ones of zhulina_lang_2011 .

## References

  * (1) M. Ali, B. Yameen, R. Neumann, W. Ensinger, W. Knoll and O. Azzaroni, Biosensing and Supramolecular Bioconjugation in Single Conical Polymer Nanochannels. Facile Incorporationof Biorecognition Elements into Nanoconfined Geometries, J. Am. Chem. Soc. 130, 16351-16357 (2008),
  * (2) B. Vilozny, A. L. Wollenberg, P. Actis, D. Hwang, B. Singaram and N. Pourmand, Carbohydrate-Actuated Nanofluidic Diode: Switchable Current Rectification in a Nanopipette, Nanoscale 5, 9214-9221 (2013).
  * (3) M. Khatibi, S.N. Ashrafizadeh, A. Sadeghi, Covering the conical nanochannels with dense polyelectrolyte layers significantly improves the ionic current rectification, Anal. Chim. Acta 1122 48-60 (2020).
  * (4) B. Yameen, M. Ali, R. Neumann, W. Ensinger, W. Knoll and O. Azzaroni, Single Conical Nanopores Displaying pH-Tunable Rectifying Characteristics. Manipulating Ionic Transport with Zwitterionic Polymer Brushes, J. Am. Chem. Soc. 131, 2070-2071 (2009).
  * (5) M. Khatibi, A. Sadeghi, S.N. Ashrafizadeh, Tripling the reverse electrodialysis power generation in conical nanochannels utilizing soft surfaces, Phys. Chem. Chem. Phys. 23, 2211-2221 (2021).
  * (6) T. Kreer, Polymer-brush lubrication: a review of recent theoretical advances, Soft Matter 16, 3479-3501 (2016).
  * (7) RJ. Nap and I. Szleifer, Effect of calcium ions on the interaction between surfaces end-grafted with weak polyelectrolytes, J. Chem. Phys. 149, 163309 (2018).
  * (8) J. A. Marins, T. Montagnon, H.Ezzaier, Ch. Hurel and P. Kuzhir, Colloidal stability of aqueous suspensions of polymer-coated iron oxide nanorods: Implifications of biomedical applications, ACS Appl. Nano Mater, 12, 6760-6772 (2018).
  * (9) K. Kegler, M. Salomo and F. Kremer, Forces of interaction between DNA-grafted colloids: an optical tweezer measurement, Phys. Rev. Lett. 98, 058304 (2007).
  * (10) H. S. Sachar, T. H. Pial, P. R. Desai, S. A. Etha, Y. Wang, P. W. Chung and S. Das, Densely grafted polyelectrolyte brushes trigger ”Water-in-Salt like scenarios and ultraconfinement effect, Matter, 2, 1509-1521 (2020).
  * (11) B. Zhao, G. Yuna, X. Chu, J. Yang, and J. Zhao, Response of a permanently charged polyelectrolyte brush to external ions: the aspects of structure and dynamics, Langmuir 23, 6757-6765 (2018).
  * (12) M. Biesalski, D. Johannsmann and J. Rühe, Synthesis and swelling behavior of a weak polyacid brush, J. Chem. Phys. 117, 4988 (2002).
  * (13) P. A. Pincus, Colloid Stabilization with Grafted Polyelectrolytes, Macromolecules 24, 2912-2919 (1991).
  * (14) R. Ross and P. A. Pincus, The polyelectrolyte brush: poor solvent, Macromolecules 25, 2177-2183 (1992).
  * (15) O. V. Borisov, T. M. Birshtein and E. B. Zhulina, Collapse of Grafted Polyelectrolyte layer, J. Phys. II 1, 521-526 (1991).
  * (16) J. Wittmer and J. F. Joanny, Charged Diblock Copolymers at Interfaces, Macromolecules 26, 2691-2697 (1993).
  * (17) O. V. Borisov, E. B. Zhulina and T. M. Birshtein, Diagram of the States of a Grafted Polyelectrolyte Layer, Macromolecules 27, 4795-4803 (1994).
  * (18) E. B. Zhulina, T. M. Birshtein and O. V. Borisov, Theory of Ionizable Polymer Brushes, Macromolecules 28, 1491-1499 (1995).
  * (19) E. B. Zhulina and M. Rubinstein, Ionic Strength Dependence of Polyelectrolyte Brush Thickness, Soft Matter 8, 9376-9383 (2012).
  * (20) R. Israel̈s, F. A. M. Leermakers, G. J. Fleer and E. B. Zhulina, Charged Polymeric Brushes: Structure and Scaling Relations, Macromolecules 27, 3249-3261 (1994).
  * (21) G. Chen and S. Das, Scaling Laws and Ionic Current Inversion in Polyelectrolyte-Grafted Nanochannels, J. Phys. Chem. B 119, 12714-12726 (2015).
  * (22) G. Chen and S. Das, Electrostatics of Soft Charged Interfaces with pH-Dependent Charge Density:Effect of Consideration of Appropriate Hydrogen Ion Concentration Distribution, RSC Adv. 5, 4493-4501 (2015).
  * (23) G. Chen and S. Das, Anomalous Shrinking-Swelling of Nanoconfined End-Charged Polyelectrolyte Brushes: Interplay of Confinement and Electrostatic Effects, J. Phys. Chem. B 120, 6848-6857 (2016).
  * (24) G. Chen and S. Das, Massively Enhanced Electroosmotic Transport in Nanochannels Grafted with End-Charged Polyelectrolyte Brushes, J. Phys. Chem. B 121, 3130-3141 (2017).
  * (25) G. Chen, H. S. Sachar and S. Das, Efficient Electrochemomechanical Energy Conversion in Nanochannels Grafted with End-Charged Polyelectrolyte Brushes at Medium and High Salt Concentration, Soft Matter 14, 5246-5255 (2018)
  * (26) R. S. Maheedhara, H. S. Sachar, H. Jing and S. Das, Ionic Diffusoosmosis in Nanochannels Grafted with End-Charged Polyelectrolyte Brushes, J. Phys. Chem. B 122, 7450-7461 (2018)
  * (27) P. M. Biesheuvel, W. M. de Vos and V. M. Amoskov, Semianalytical Continuum Model for Nondilute Neutral and Charged Brushes Including Finite Stretching, Macromolecules 41, 6254-6259 (2008).
  * (28) E. B. Zhulina and O. V. Borisov, Structure and Interaction of Weakly Charged Polyelectrolyte Brushes: Self-Consistent Field Theory, J. Chem. Phys. 107, 5952-5967 (1997).
  * (29) Y. V. Lyatskaya, F. A. M. Leermakers, G. J. Fleer,E. B. Zhulina and T. M. Birshtein, Analytical Self-Consistent-Field Model of Weak Polyacid Brushes, Macromolecules 28, 3562-3569 (1995).
  * (30) I. O. Lebedeva, E. B. Zhulina and O. V. Borisov, Self-Consistent Field Theory of Polyelectrolyte Brushes with Finite Chain Extensibility, J. Chem. Phys. 146, 214901 (2017).
  * (31) E. B. Zhulina, J. K. Wolterink and O. V. Borisov, Screening Effects in a Polyelectrolyte Brush: Self-Consistent-Field Theory, Macromolecules 33, 4945-4953 (2000).
  * (32) E. B. Zhulina and O. V. Borisov, Poisson-Boltzmann Theory of pH-Sensitive (Annealing) Polyelectrolyte Brush, Langmuir 27, 10615-10633 (2011).
  * (33) J. P. Mahalik, Y. Yang, C. Deodhar, J. F. Ankner, B. S. Lokitz, S. M. Kilbey II, B. G. Sumpter and R. Kumar, Monomer Volume Fraction Profiles in pH Responsive Planar Polyelectrolyte Brushes, J. Polym. Sci., Part B: Polym. Phys. 54, 956-964 (2016).
  * (34) T. Wu, P. Gong, I. Szleifer, P. Vlcek, V. Subr and J. Genzer, Behavior of Surface-Anchored Poly(acrylic acid) Brushes with Grafting Density Gradients on Solid Substrates? 1. Experiment, Macromolecules 40, 8756-8764 (2007).
  * (35) Y. Ito, Y. S. Park and Y. Imanishi, Visualization of Critical pH Controlled Gating of a Porous Membrane Grafted with Polyelectrolyte Brushes, J. Am. Chem. Soc. 119, 2739-2740 (1997).
  * (36) E. B. Zhulina and M. Rubinstein, Lubrication by Polyelectrolyte Brushes, Macromolecules 47, 5825-5838 (2014).
  * (37) N. A. Kumar and C. Seidel, Polyelectrolyte Brushes with Added Salt, Macromolecules 38, 9348-9350 (2005)
  * (38) A. Naji, R. R. Netz and C. Seidel, Non-linear Osmotic Brush Regime: Simulations and Mean-field theory, Eur. Phys. J. E: Soft Matter Biol. Phys. 12, 223-237 (2003).
  * (39) H. S. Sachar, V.S. Sivasankar, and S. Das, Revisiting the strong stretching theory for pH-responsive polyelectrolyte brushes: effects of consideration of excluded volume interactions and an expanded form of the mass action law. Soft Matter 15, 559-574 (2019).
  * (40) H. S. Sachar, V.S. Sivasankar, and S. Das, Electrokinetic energy conversion in nanochannels grafted with pH-responsive polyelectrolyte brushes modelled using augmented strong stretching theory. Soft Matter 15, 5973-5986 (2019).
  * (41) H. S. Sachar, V.S. Sivasankar, S.A. Etha, G. Chen, and S. Das, Ionic current in nanochannels grafted with pH-responsive polyelectrolyte brushes modeled using augmented strong stretching theory. Electrophoresis 41, 554-561 (2020).
  * (42) V. S. Sivasankar, S.A. Etha, H.S. Sachar, and S. Das, Ionic diffusioosmotic transport in nanochannels grafted with pH-responsive polyelectrolyte brushes modeled using augmented strong stretching theory. Phys. Fluid. 32, 042003 (2020).
  * (43) V.S. Sivasankar, S.A. Etha, H.S. Sachar and S. Das, Theoretical study on the massively augmented electro-osmotic water transport in polyelectrolyte brush functionalized nanoslits. Phys. Rev. E 102, 013103 (2020).
  * (44) V. S. Sivasankar, S.A.Etha, H.S. Sachar and S. Das, Thermo-osmotic transport in nanochannels grafted with pH-responsive polyelectrolyte brushes modelled using augmented strong stretching theory,J. Fluid Mech. 917 (2021).
  * (45) P. Richmond, Electrical forces between particles with arbitrary fixed surface charge distributions in ionic solution, J. Chem. Soc. Faraday 70, 1066-1073 (1974).
  * (46) P. Richmond, Electrical forces between particles with discrete periodic surface charge distributions in ionic solution, J. Chem. Soc. Faraday 71, 1154-1163 (1975).
  * (47) A.P. Nelson, D.A. McQuarie, The effect of discrete charges on the electrical properties of a membrane, Int. J. Theor. Biol. 55, 13-27 (1975).
  * (48) S.J. Micklavic, C.D.Y. Chan, L.R. White, T.W. White, Double layer forces between heterogeneous charged surfaces, J. Phys. Chem. 98, 9022-9032 (1994).
  * (49) S.J. Micklavic, S.L. Carnie, Interactions between two spherical particles with non-uniform surface potentials: the linearized Poisson-Boltzmann theory, J. Colloid Interface Sci. 216, 329-347 (1999).
  * (50) K. Dähnert, M. Rödenbeck, Exact solution of bisphere Debye-Hückel problem, J. Colloid Interface Sci. 163, 229-233 (1994).
  * (51) H. Ohshima, Surface charge density/surface potential relationship for a spherical colloidal particle in a salt-free medium, J. Colloid Interface Sci. 24, 18-23 (2002).
  * (52) J.F.L. Duval, J. Merlin, Pal Narayana, Electrostatic interactions between diffuse soft multi-layered (bio)particles: beyond Debye-Hückel approximation and Deryagin formulation, Phys. Chem. Chem. Phys. 13, 1037-1053 (2011).
  * (53) JFL. Duval, D. Kuttner, C. Werner, R. Zimmermann, Electrohydrodynamics of soft polyelectrolyte multilayers: point of zero-streaming current, Langmuir 27, 10739-10759 (2011).
  * (54) M. Moussa, C. Caillet, R.M. Town, JFL. Duval, Remarkable electrokinetic features of charge-stratified soft nanoparticles: mobility reversal in monovalent aqueous electrolyte, Langmuir 31, 5656-5666 (2015).
  * (55) JFL. Duval, H. Ohshima, Electrophoresis of diffuse soft particles, Langmuir 22 (2006) 3533-3546.
  * (56) H. Jing, S. Das, Electric double layer electrostatics of lipid-bilayer-encapsulated nanoparticles: toward a better understanding of protocell electrostatics, Electrophoresis 39, 752-759 (2018).
  * (57) H. Ohshima, Discrete charge effects on the Donnan potential and surface potential of a soft particle, Colloid Polym. Sci. 292, 749-756 (2014).
  * (58) J. Andrews, S. Das, Effect of finite ion sizes in electric double layer mediated interaction force between two soft charged plates, RSC Adv. 5, 46873-46880 (2015).
  * (59) J.-S. Sin, Ion partitioning effect on the electrostatic interaction between two charged soft surfaces, Colloid Surf. A, 628, 127296:1-7 (2021).
  * (60) J.J. Lopez-Garcia, J. Horno, C. Grosse, Suspended particles surrounded by an inhomogeneously charged permeable membrane. Solution of the Poisson-Boltzmann equation by means of the network method, J. Colloid Interface Sci. 268, 371-379 (2003).
  * (61) A. Ganjizade, S.N. Ashrafizadeh, A. Sadeghi, Effect of ion partitioning on the electrostatics of soft particles with a volumetrically charged core, Electrochem. Commun. 84, 19-23 (2017).
  * (62) S.K. Maurya, P.P. Gopmandal, S. Bhattachryya, H. Ohshima, Ion partitioning effect on the electrophoresis of a soft particle with hydrophobic core, Phys. Rev. E 98, 023103 (2018).
  * (63) A. Ganjizade, A. Sadeghi, S.N. Ashrafizadeh, Effect of ion partitioning on electrostatics of soft particles with volumetrically charged inner core coated with pH-regulated polyelectrolyte layer, Colloids Surf. B 170, 129–135 (2018).
  * (64) P.P. Gopmandal, S. De, S. Bhattacharyya, H. Ohshima, Impact of ion-steric and ion partitioning effects on electrophoresis of soft particles, Phys. Rev. E 102, 032601 (2020).
  * (65) P. Mahapatra, P.P. Gopmandal, JFL. Duval, Effect of dielectric gradients-mediated ions partitioning on the electrophoresis of composite soft particles: an analytical theory, Electrophoresis 42, 153-162 (2021).
  * (66) A. Poddar, D. Maity, A. Bandopadhyya, S. Chakraborty, Electrokinetics in polyelectrolyte grafted nanofluidic channels modulated by the ion partitioning effect, Soft Matter 12, 5968-5978 (2016).
  * (67) D. Pandey, S. Bhattacharyya, Impact of finite ion size, Born energy difference and dielectric decrement on the electroosmosis of multivalent ionic mixtures in a nanotube, Colloids Surf. A 610, 125905-125919 (2021).
  * (68) I. Borukhov, D. Andelman, H. Orland, Steric effects in electrolytes: a modified Poisson-Boltzmann equation, Phys. Rev. Lett. 79, 435-438 (1997).
  * (69) V. Kralj-Iglič, A. Iglič, A simple statistical mechanical approach to the free energyof the electric double layer including the excluded volume effect, J. Phys. France 6, 477-491 (1996).
  * (70) E. Gongadze, A. Igliˇc, Decrease of permittivity of an electrolyte solution neara charged surface due to saturation and excluded volume effects, Bioelectrochemistry 87, 199-203 (2012).
  * (71) J.-S. Sin, S.-J. Im, K.-I. Kim, Asymmetric electrostatic properties of an electric double layer: a generalized Poisson-Boltzmann approach taking into account non-uniform size effects and water polarization, Electrochim. Acta 153, 531-539 (2015).
  * (72) J.-S. Sin, Influence of solvent polarization and non-uniform ion size on electrostatic properties between charged surfaces in an electrolyte, J. Chem. Phys. 147, 214702 (2017)
  * (73) B.D. Storey, M.Z. Bazant, Effects of electrostatic correlations on electrokinetic phenomena, Phys. Rev. E 86, 056303 (2012).
  * (74) K. Bohinc, G. V. Bossa, S. Gavryushov and S. May, Poisson-Boltzmann model of electrolytes containing uniformly charged spherical nanoparticles, J. Chem. Phys. 145, 234901 (2016).