# Ions and dipoles in electric field: Nonlinear polarization and field-
dependent chemical reaction

Akira Onuki111e-mail<EMAIL_ADDRESS>(corresponding author)
Department of Physics, Kyoto University, Kyoto 606-8502, Japan

###### Abstract

We investigate electric-field effects in dilute electrolytes with nonlinear
polarization. As a first example of such systems, we add a dipolar component
with a relatively large dipole moment $\mu_{0}$ to an aqueous electrolyte. As
a second example, the solvent itself exhibits nonlinear polarization near
charged objects. For such systems, we present a Ginzburg-Landau free energy
and introduce field-dependent chemical potentials, entropy density, and stress
tensor, which satisfy general thermodynamic relations. In the first example,
the dipoles accumulate in high-field regions, as predicted by Abrashikin et
al.$[$Phys.Rev.Lett. 99, 077801 (2007)$]$. Finally, we consider the case,
where Bjerrum ion pairs form a dipolar component with nonlinear polarization.
The Bjerrum dipoles accumulate in high-field regions, while field-induced
dissociation was predicted by Onsager $[$J. Chem. Phys.2, 599 (1934)$]$. We
present an expression for the field-dependent association constant $K(E)$,
which depends on the field strength nonmonotonically.

##  I Introduction

The electrostatic interactions among electric charges and dipoles in a solvent
are of central importance in various situations in soft matter physics Russel
; Stokes ; Is . In this paper, we consider dilute electrolytes composed of a
waterlike liquid solvent, cations, anions, and a dipolar component with a
dipole moment $\mu_{0}$. Andelman, Orland, and their coworkers (AO) An1 ; An2
; An3 ; An4  proposed a dipolar Poisson-Boltzmann (dPB) equation, where the
dipoles can respond to electric field nonlinearly and their polarization
density ${\mbox{\boldmath$p$}}_{d}$ yields the effective charge density
$-4\pi\nabla\cdot{\mbox{\boldmath$p$}}_{d}$. In their papers the dipolar
component is the solvent itself. Theoretically, it can be different from the
solvent with some changes of the equations. The dPB equation is a
generalization of the classical Poisson-Boltzmann equation and is convenient
to investigate the nonlinear polarization around charged objects. In
particular, AO group concluded that the dipole density is increased by the
factor $\sinh h/h$ with $h=\mu_{0}E/k_{B}T$ An1 , where
$E=|{\mbox{\boldmath$E$}}|$ is the field strength, $T$ is the temperature, and
$k_{B}$ is the Boltzmann constant. In this paper we show the following. The
dipole accumulation in high-field regions occurs if the dipolar component is a
dilute solute in a solvent. On the other hand, a nearly incompressible polar
solvent is hardly enriched in high-field regions.

The physics of ion-dipole systems is even more intriguing if associated ion
pairs, Bjerrum dipoles Bje ; Ma ; Fuoss ; Roij ; Dill ; Vegt , are treated to
form a dipolar component. In electrolytes, association of free ions and
dissociation of bounded ion pairs balance on the average in equilibrium, while
strong acids in full dissociation have long been studied since the seminal
work by Debye-Hückel Russel ; Is ; Debye ; McQ ; Stokes . Although not well
known, OnsagerOnsager34  predicted that applied electric field enhances
breakage of the ion bonding to increase dissociation. He intended to explain
conductance increases in weak electrolytes under electric field (the Wien
effect) Wien ; Eck ; Kaiser ; Korea . In his theory, the effect is significant
for $E\gtrsim 2k_{B}T/e\ell_{B}(=0.08$V$/$nm for ambient water), where $e$ is
the elementary charge and $\ell_{B}$ is the Bjerrum length. Therefore, in
high-field regions, some fraction of the Bjerrum dipoles dissociate in the
Onsager theory, while dipoles of a dilute dipolar component accumulate without
chemical reactions in the AO theory.

In treating the Coulombic and dipolar interactions, AO group used a Hubbard-
Stratonovich transformation of the grand-canonical partition function. In this
paper, we propose a Ginzburg-Landau free energy of electrolytes, where the
polarization can be nonlinear in inhomogeneous electric field. With this
approach we can draw unambiguous conclutions systematically. Because the
problem is complex, we divide this paper into three parts: (i) A dipolar
component is added as a dilute solute, (ii) the solvent exhibits nonlinear
polarization in Sec.III, and (iii) Bjerrum ion pairs form a dipolar component
in Sec.IV. There is no chemical reaction in the first two parts.

In our problem, we need to develop a thermodynamic theory of inhomogeneous
electrolytes. In their book, Landau and Lifshits Landau-e  examined
thermodynamics of dielectric fluids in electric field, providing explanations
of the electrostriction effect Hakim ; Stell  and the Maxwell stress tensor.
It shows that the stress tensor is not diagonal and the scalar pressure is not
well-defined in electric field in such fluids. In this paper, we use the
Maxwell stress tensor for electrolytes and introduce field-dependent chemical
potentials $\mu_{i}$ and entropy density $S$. We find that these variables
satisfy general thermodynamic relations. For Bjerrum dipoles, we determine the
field-dependent association constant $K(E)$ using the field-dependent
$\mu_{i}$. In the literature Bje ; Ma ; Fuoss ; Roij ; Dill ; Vegt ; Kirk ;
Landau-s , the zero-field association constant $K(0)$ has been discussed. We
can then examine the distributions of ions and dipoles near charged surfaces,
incorporating the AO and Onsager theories.

The organization of this paper is as follows. In Sec.II, we will present a
theromodynamics theory of dilute electrolytes consisting of four-components in
electric field. In Sec.III, we will examine dilute electrolytes consisting of
a polar solvent and a strong acid, where the solvent exhibits nonlinear
polarization. In Sec.IV, we will investigate association and dissociation in
electrolytes in electric field, where Bjerrum dipoles constitute the fourth
component. In Appendix A, we will derive the electric free energy density
using the Onsager theory of rodlike molecules Onsager49 .

##  II Thermodynamics of dilute electrolytes in electric field

In this section, we treat a four component system composed of a single-
component polar solvent (1), cations with charge $Ze$ (2), anions with charge
$-Ze$ (3), and a dipolar component with a molecular dipole moment $\mu_{0}$
(4). Here, $Z=1$ for monovalent salts (NaCl) and $Z=2$ for divalent salts
(MgSO4). We assume that the solvent dipole moment is considerably smaller than
$\mu_{0}$ and the ions have no dipole moment, so the solvent exhibits linear
polarization. As a typical solvent, we can suppose ambient liquid water at
$T=300$ K and $p=1$ atm. In this paper, we use the cgs units.

###  II.1 Polarization of dipolar component

In our continuum theory, the densities $n_{i}({\mbox{\boldmath$r$}})$ of the
four components are coarse-grained ones with spatial scales longer than the
Bjerrum length $\ell_{B}=e^{2}/\epsilon_{\rm w}k_{B}T$, where $\epsilon_{\rm
w}$ is the pure-solvent dielectric constant. The electric potential $\Phi$ and
the electric field ${\mbox{\boldmath$E$}}=-\nabla\Phi$ also vary smoothly. To
apply electric field, we suppose parallel charged plates separated by a
distance $L$, within which an electrolyte is inserted in the region $0<z<L$.
The lateral dimensions of the container much exceed $L$ and the edge effect is
negligible.

The charge density $\rho$ and the polarization ${\mbox{\boldmath$p$}}_{d}$ of
the dipolar component are written as

$\rho=Ze(n_{2}-n_{3}),~{}~{}~{}{\mbox{\boldmath$p$}}_{d}=n_{4}{\mbox{\boldmath$\mu$}}_{d},$
(1)

where ${\mbox{\boldmath$\mu$}}_{d}({\mbox{\boldmath$r$}})$ is the average
dipole moment vector of the dipolar component per particle, as will be defined
in Eq.(7) below. The polarization of the solvent molecules is expressed in the
linear response form,

${\mbox{\boldmath$p$}}_{\rm w}=(4\pi)^{-1}(\epsilon-1){\mbox{\boldmath$E$}}.$
(2)

We then introduce the electric induction,

${\mbox{\boldmath$D$}}={\mbox{\boldmath$E$}}+4\pi({\mbox{\boldmath$p$}}_{\rm
w}+{\mbox{\boldmath$p$}}_{d})=\epsilon{\mbox{\boldmath$E$}}+4\pi{\mbox{\boldmath$p$}}_{d},$
(3)

which is related to $\rho$ by the Poisson equation,

$\nabla\cdot{\mbox{\boldmath$D$}}=\nabla\cdot(\epsilon{\mbox{\boldmath$E$}}+4\pi{\mbox{\boldmath$p$}}_{d})=4\pi\rho.$
(4)

As shown in Eqs.(2) and (3), $\epsilon$ is the linear dielectric constant of
the solvent molecules in the static limit. In dielectric experiments of 1:1
small salts in liquid water Co ; Wei ; Bu , $\epsilon$ decreased as
$\Delta\epsilon=\epsilon-\epsilon_{{\rm w}}\cong-A_{1}c$ for the salt
concentration $c$ below 1M$/\ell$, where $A_{1}$ is a constant. Here,
$\Delta\epsilon$ ranged from $-8$ to $-20$ at $c\sim 1$M$/\ell$ depending on
the salt species. This large decrease arises from the formation of the
hydration shells of water around ions. That is, the water molecules inside the
shells do not freely rotate due to the ion-dipole interaction and their
contribution to $\epsilon$ is largely suppressed. For small $n_{i}$ ($i\geq
2$) we thus express $\epsilon$ as

$\epsilon=\epsilon_{{\rm w}}(n_{1},T)+{\sum}_{i}^{\prime}\epsilon_{i}^{{\rm
w}}(n_{1},T)n_{i}+\cdots,$ (5)

where the coefficients $\epsilon_{i}^{{\rm w}}$ are independent of $n_{i}$
$(i\geq 2$). Hereafter, ${\sum}_{i}^{\prime}$ denotes summation over the
solute species. Since $n_{2}=n_{3}$ on the average, the degree of hydration is
indicated by the dimensionless coefficient,

${\hat{\gamma}}=-n_{1}(\epsilon_{2}^{\rm w}+\epsilon_{3}^{\rm
w})/2\epsilon_{\rm w},$ (6)

which is positive in the range $2.5\lesssim{\hat{\gamma}}\lesssim 6$ (even
divided by $2\epsilon_{\rm w}=160$). Thus, $\Delta\epsilon/\epsilon_{{\rm
w}}\cong-\hat{\gamma}(n_{2}+n_{3})/n_{1}$ for 1:1 salts with $c\lesssim
1$M$/\ell$ in water. For larger $c$, Buchner et al. Bu  obtained
$\Delta\epsilon\cong-A_{1}c+A_{2}c^{3/2}$ with $A_{1}$ and $A_{2}$ being
positive constants. For NaCl they argued that the degradation of $\epsilon$ is
mostly due to Na cations. In Eq.(5) we add the term $\epsilon_{4}^{\rm
w}n_{4}$ in case the dipolar particles affect the solvent polarization.

The dipole moment vector ${\mbox{\boldmath$\mu$}}_{d}$ in Eq.(1) is the
statistical average defined by

${\mbox{\boldmath$\mu$}}_{d}=\mu_{0}\langle{{\mbox{\boldmath$n$}}_{d}}\rangle=\frac{\mu_{0}}{4\pi}\int
d\Omega
P({\mbox{\boldmath$n$}}_{d},{\mbox{\boldmath$r$}}){\mbox{\boldmath$n$}}_{d},$
(7)

where ${\mbox{\boldmath$n$}}_{d}$ is the microscopic unit vector along the
dipole direction, $\int d\Omega$ is the angle integration of
${\mbox{\boldmath$n$}}_{d}$, and
$P({\mbox{\boldmath$n$}}_{d},{\mbox{\boldmath$r$}})$ is the angle distribution
function at position $r$ with $\int d\Omega
P({\mbox{\boldmath$n$}}_{d},{\mbox{\boldmath$r$}})=4\pi$. In Appendix A, we
shall see that the equilibrium form of
$P({\mbox{\boldmath$n$}}_{d},{\mbox{\boldmath$r$}})$ in the dilute reime is
written in terms of the local electric field
${\mbox{\boldmath$E$}}({\mbox{\boldmath$r$}})$ as

$P({\mbox{\boldmath$n$}}_{d},{\mbox{\boldmath$r$}})={\cal N}_{\rm
d}^{-1}\exp[\mu_{0}{\mbox{\boldmath$n$}}_{d}\cdot{\mbox{\boldmath$E$}}({\mbox{\boldmath$r$}})/k_{B}T],$
(8)

Here, ${\cal N}_{\rm d}$ is the normalization factor,

${\cal N}_{\rm d}=\frac{1}{4\pi}\int
d{\Omega}\exp[\mu_{0}{\mbox{\boldmath$n$}}_{d}\cdot{\mbox{\boldmath$E$}}/k_{B}T]=\frac{\sinh
h}{h}.$ (9)

which depends on the dimensionless field strength,

$h=\mu_{0}E/k_{B}T,$ (10)

with $E=|{\mbox{\boldmath$E$}}|$. Here, we introduce $W(h)=\ln{\cal N}_{\rm
d}$ and $W^{\prime}(h)=dW(h)/dh$ expressed as

$W=\ln(\sinh h/h),~{}~{}~{}W^{\prime}={\cal L}(h)=\coth(h)-1/h,$ (11)

where ${\cal L}(h)$ is the Langevin function with ${\cal L}\cong h/3$ for
$h\ll 1$ and ${\cal L}\cong 1$ for $h\gg 1$. Then, Eqs.(7)-(9) give

${\mbox{\boldmath$p$}}_{d}=n_{4}\mu_{0}{\cal
L}(h)E^{-1}{{\mbox{\boldmath$E$}}}={\chi}_{d}{\mbox{\boldmath$E$}}.$ (12)

Here, ${\chi}_{d}(h)=n_{4}\mu_{0}{\cal L}(h)/E$ is the nonlinear
susceptibility of the dipolar component. Setting
${\mbox{\boldmath$D$}}=\epsilon_{\rm eff}{\mbox{\boldmath$E$}}$, we obtain the
total nonlinear dielectric constant,

$\epsilon_{\rm eff}=\epsilon+4\pi\chi_{d}=\epsilon+4\pi n_{4}\mu_{0}{\cal
L}(h)/E.$ (13)

As $h\to 0$ it follows the total linear dielectric constant,

$\epsilon_{\rm eff}^{0}={\lim}_{h\to 0}\epsilon_{\rm
eff}=\epsilon+4\pi(\mu_{0}^{2}/3k_{B}T)n_{4}.$ (14)

For $h\gg 1$, the dipolar particles tend to fully align along $E$ with
${\chi}_{d}\cong n_{4}\mu_{0}/E$.

AO group An2 ; An4  calculated $\epsilon_{2}^{\rm w}$ and $\epsilon_{3}^{\rm
w}$ using their dPB equation around ions, where the solvent (water) itself is
the dipolar component. In our scheme, we assume their presence with large
sizes from the beginning. Frydel Frydel  constructed the dPB equation for ions
with polarizabilities $\alpha_{i}$. The ionic polarization can be accounted
for if we change $\epsilon_{i}^{\rm w}$ to $\epsilon_{i}^{\rm
w}+4\pi\alpha_{i}$ in Eq.(5). However, $|\epsilon_{2}^{\rm
w}+\epsilon_{3}^{\rm w}|$ is much larger than $4\pi\alpha_{i}$ for small
ionsMadden .

###  II.2 Free energy functional and chemical potentials

We set up the free energy functional ${\cal F}=\int d{\mbox{\boldmath$r$}}f$,
where the integration is within the cell confining the electrolyte solution.
We neglect the van der Waals fluid-solid interaction and the image potential,
which can be important near the surfaces, however. For $n_{i}\ll n_{1}$
($i\geq 2$), the Helmholtz free energy density $f$ is written as Onukipolar ;
anta ; Ben ; Oka1 ; Oka2

$f=f_{\rm
w}+k_{B}T{\sum}^{\prime}_{i}n_{i}[\ln(n_{i}\lambda_{i}^{3})+\nu_{i}-1]+f_{e},$
(15)

where $f_{\rm w}(n_{1},T)$ is the pure-solvent free energy density without
applied field. The first two terms constitute the non-electric part. For each
$i(\geq 2)$, $\lambda_{i}(\propto T^{-1/2})$ is the thermal de Broglie length
and $\nu_{i}(n_{1},T)$ consists of ideal-gas and interaction parts. The former
is written as Landau-s ,

$\nu_{i}^{0}(T)=-(k_{i}-3/2)\ln(T/\Theta_{i}^{0}),$ (16)

where $2k_{i}-3$ is the degree of vibration-rotation freedom and
$\Theta_{i}^{0}$ is a constant temperature. The difference
$\Delta\nu_{i}=\nu_{i}-\nu_{i}^{0}$ is determined by the solute-solvent
interactions at infinite dilution. Although not written in Eq.(15), the Debye-
Hückel free energy density $f_{\rm DH}=-k_{B}T\kappa^{3}/12\pi$ Debye  and the
short-range ion-ion interaction terms are important to explain experimental
data for not very dilute electrolytes McQ ; Stokes ; Oka2 , where
$\kappa=Z[4\pi(n_{2}+n_{3})\ell_{B}]^{1/2}$ is the Debye wave number.

As will be shown in Appendix A, we use Onsager’s theory of rodlike molecules
Onsager49  to obtain the electric free energy density $f_{e}$. Neglecting the
short-range dipole-dipole interactions, we obtain

$\displaystyle f_{e}=\epsilon
E^{2}/8\pi+{\mbox{\boldmath$p$}}_{d}\cdot{\mbox{\boldmath$E$}}-k_{B}Tn_{4}W$
$\displaystyle=\epsilon E^{2}/8\pi+k_{B}Tn_{4}(hW^{\prime}-W).$ (17)

where $W(h)$ and $W^{\prime}(h)$ are defined in Eq.(11). For $h\ll 1$, we have
$W\cong h^{2}/6$ and $hW^{\prime}\cong h^{2}/3$, so $f_{e}$ tends to the
linear response form,

$f_{e}^{0}=\epsilon_{\rm eff}^{0}E^{2}/8\pi=(\epsilon+4\pi
n_{4}\mu_{0}^{2}/3k_{B}T)E^{2}/8\pi.$ (18)

Next, we slightly change $n_{i}$ and $T$ by $\delta n_{i}$ and $\delta T$.
Then, from $\delta(\epsilon
E^{2})=2{\mbox{\boldmath$E$}}\cdot\delta{\mbox{\boldmath$D$}}-8\pi{\mbox{\boldmath$E$}}\cdot\delta{\mbox{\boldmath$p$}}_{d}-E^{2}\delta\epsilon$
and
${\mbox{\boldmath$p$}}_{d}\cdot\delta{\mbox{\boldmath$E$}}=n_{4}\mu_{0}W^{\prime}\delta
E$, $f_{e}$ is changed by

$\displaystyle\delta
f_{e}={\mbox{\boldmath$E$}}\cdot\delta{\mbox{\boldmath$D$}}/4\pi-E^{2}\delta\epsilon/8\pi$
$\displaystyle-(k_{B}TW)\delta n_{4}+k_{B}n_{4}(hW^{\prime}-W)\delta T.$ (19)

Here, integration of the first term within the cell becomes

$\frac{1}{4\pi}\int
d{\mbox{\boldmath$r$}}{\mbox{\boldmath$E$}}\cdot\delta{\mbox{\boldmath$D$}}=\int
d{\mbox{\boldmath$r$}}\Phi\delta\rho-(\Phi_{0}\delta Q_{0}+\Phi_{L}\delta
Q_{L}),$ (20)

where $(\Phi_{0},Q_{0})$ and $(\Phi_{L},Q_{L})$ are the surface potential and
the surface charge at $z=0$ and $z=L$, respectively. If we assume the overall
charge neutrality, we require

$\int d{\mbox{\boldmath$r$}}\rho=Q_{0}+Q_{L},$ (21)

under which we can replace $\Phi$ by $\Phi-C_{0}$ with $C_{0}$ being an
arbitrary constant. The surface term in Eq.(20) vanishes at fixed surface
charges $(\delta Q_{0}=\delta Q_{L}=0$).

We define the chemical potentials $\mu_{i}$ by the functional derivatives
$\mu_{i}=\delta{\cal F}/\delta n_{i}$ at fixed $T$ and $n_{j}$ with $j\neq i$.
Then, Eqs.(15), (19), and (20) give

$\displaystyle\mu_{1}/k_{B}T=\mu_{\rm
w}/k_{B}T+{\sum}^{\prime}_{i}\nu^{\prime}_{i}n_{i}-\zeta_{1},$ (22)
$\displaystyle{\mu}_{2}/k_{B}T=\ln(n_{2}\lambda_{2}^{3})+\nu_{2}+U-\zeta_{2},$
(23)
$\displaystyle{\mu}_{3}/k_{B}T=\ln(n_{3}\lambda_{3}^{3})+\nu_{3}-U-\zeta_{3},$
(24)
$\displaystyle\mu_{4}/k_{B}T=\ln(n_{4}\lambda_{4}^{3})+\nu_{4}-W-\zeta_{4},$
(25)

where $\mu_{\rm w}(n_{1},T)=\partial f_{\rm w}/\partial n_{1}$ is the chemical
potential of pure solvent. For each $i\geq 1$ we define

$\displaystyle\zeta_{i}=\epsilon_{i}^{\prime}E^{2}/8\pi k_{B}T,$ (26)
$\displaystyle U=Ze\Phi/k_{B}T.$ (27)

We write the density derivatives of $\nu_{i}$ and $\epsilon$ at fixed $T$ as

$\displaystyle\nu^{\prime}_{i}=\partial\nu_{i}/\partial n_{1}~{}(i\geq 2),$
(28) $\displaystyle\epsilon_{i}^{\prime}=\partial\epsilon/\partial
n_{i}~{}(i\geq 1),$ (29)

so $\epsilon_{1}^{\prime}\cong\partial\epsilon_{\rm w}/\partial n_{1}$ and
$\epsilon_{i}^{\prime}\cong\epsilon_{i}^{\rm w}$ for dilute solutes. In
Eq.(22), the last term $-\zeta_{1}$ gives rise to the electrostriction (see
Eq.(50)) Landau-e ; Stell ; Hakim ; Kirk . In Eq.(23) and (24), the terms
$-\zeta_{i}$ serve to increase $\mu_{i}$ in $E$ for $\epsilon_{i}^{{\rm w}}<0$
(see below Eq.(5)). In Eqs.(23)-(25), $\nu_{i}$ can be treated to be
homogeneous constants in one-phase states. However, from their strong
dependence on $n_{i}$, they exhibit discontinuities across interfaces in two-
phase states, giving rise to solute density differences in the two phases.

We also define the entropy density as the functional derivative
$S=-\delta{\cal F}/\delta T$ at fixed densities $\\{n\\}$ and fixed surface
charges. The integral of
${\mbox{\boldmath$E$}}\cdot\delta{\mbox{\boldmath$D$}}/4\pi$ does not
contribute to $S$ from Eq.(20) so that

$\displaystyle S=S_{\rm
w}-k_{B}{\sum}^{\prime}_{i}n_{i}\big{[}\ln(n_{i}\lambda_{i}^{3})-\frac{5}{2}+\nabla_{T}(T\nu_{i})\big{]}$
$\displaystyle+\frac{1}{8\pi}E^{2}\nabla_{T}{\epsilon}-k_{B}n_{4}(hW^{\prime}-W),$
(30)

where $S_{\rm w}(n_{1},T)=-\partial f_{\rm w}/\partial T$ is the pure-solvent
part and the last term is the entropy density of dipole orientation (see
Appendix A). Here, $\nabla_{T}(\cdots)=\partial(\cdots)/\partial T$ denotes
the temperature derivative at fixed densities $\\{n\\}$.

###  II.3 Free energy variations

As in Eq.(19) we consider small changes in $n_{i}$ and $T$. Then, the free
energy density $f$ in Eq.(15) is changed by

$\delta f={\sum}_{i}\mu_{i}\delta n_{i}-S\delta
T-\nabla\cdot(\Phi\delta{\mbox{\boldmath$D$}})/4\pi,$ (31)

where we use the field-dependent $\mu_{i}$ in Eqs.(22)-(25) and $S$ in
Eq.(30). If the cell volume $V$ is fixed, the free energy increment
$\delta{\cal F}=\int d{\mbox{\boldmath$r$}}\delta f$ is expressed as

$\delta{\cal F}=\int d{\mbox{\boldmath$r$}}\Big{[}{\sum}_{i}\mu_{i}\delta
n_{i}-S\delta T\Big{]}-\Phi_{0}\delta Q_{0}-\Phi_{L}\delta Q_{L},$ (32)

Thus, ${\cal F}$ is used at fixed surface charges $\delta Q_{0}=\delta
Q_{L}=0$.

If the potential difference $\Delta\Phi=\Phi_{0}-\Phi_{L}$ is fixed, we
perform the Legendre transformation Landau-e ,

$\tilde{\cal F}={\cal F}+\Phi_{0}Q_{0}+\Phi_{L}Q_{L}={\cal F}+\int
d{\mbox{\boldmath$r$}}\big{[}\Phi\rho-\frac{1}{4\pi}{\mbox{\boldmath$E$}}\cdot{\mbox{\boldmath$D$}}\big{]}.$
(33)

Setting $\tilde{\cal F}=\int d{\mbox{\boldmath$r$}}{\tilde{f}}$, we can define
the electric free energy density ${\tilde{f}}_{e}$ by

$\displaystyle{\tilde{f}}_{e}=f_{e}+\Phi\rho-{{\mbox{\boldmath$E$}}\cdot{\mbox{\boldmath$D$}}}/{4\pi}$
$\displaystyle=\Phi\rho-{\epsilon E^{2}}/{8\pi}-k_{B}Tn_{4}W.$ (34)

The incremental changes in $\tilde{f}$ and $\tilde{\cal F}$ are given by

$\displaystyle\delta{\tilde{f}}={\sum}_{i}\mu_{i}\delta n_{i}-S\delta
T+\nabla\cdot(\delta\Phi{\mbox{\boldmath$D$}})/4\pi,$ (35)
$\displaystyle\delta\tilde{\cal F}=\int
d{\mbox{\boldmath$r$}}\Big{[}{\sum}_{i}\mu_{i}\delta n_{i}-S\delta
T\Big{]}+Q_{0}\delta\Phi_{0}+Q_{L}\delta\Phi_{L}.$ (36)

For $h\ll 1$, ${\tilde{f}}_{e}$ tends to $\Phi\rho-f_{e}^{0}$ with $f_{e}^{0}$
being given by Eq.(18), as it should be the case.

Previously, Ben-Yaakov et al. Ben  used the electrostatic free energy density
composed of the first two terms in the second line of Eq.(34), while we set it
equal to $\epsilon E^{2}/8\pi$, the first term in Eq.(17) Onukipolar ; anta ;
Oka2 . In these papers, the polarization was assumed to be linear.

###  II.4 Maxwell stress tensor

The solution stress tensor $\Pi_{\alpha\beta}$ consists of two parts,

$\Pi_{\alpha\beta}=p_{\rm n}\delta_{\alpha\beta}+M_{\alpha\beta},$ (37)

where $\alpha$ and $\beta$ stand for ($x,y,z)$ and

$\displaystyle p_{\rm n}=p_{\rm
w}+k_{B}T{\sum}^{\prime}_{i}(1+n_{1}\nu^{\prime}_{i})n_{i},$ (38)
$\displaystyle
M_{\alpha\beta}=\frac{1}{8\pi}\big{(}\epsilon-{\sum}_{i}n_{i}\epsilon_{i}^{\prime}\big{)}E^{2}\delta_{\alpha\beta}-\frac{1}{4\pi}E_{\alpha}D_{\beta},$
(39)

The $p_{\rm n}$ is the non-electric solution pressure arising from the first
two terms in $f$ in Eq.(15) Oka2  with $p_{\rm w}(n_{1},T)=\partial f_{\rm
w}/\partial n_{1}-f_{\rm w}$ being the pure-solvent pressure. The derivatives
$\nu_{i}^{\prime}$ are related to the solute partial volumes (see Eq.(51)).
The $M_{\alpha\beta}$ is the Maxwell stress tenor of the solution Landau-e
with $D_{\beta}=\epsilon_{\rm eff}E_{\beta}$ and
$M_{\alpha\beta}=M_{\beta\alpha}$, where $\epsilon_{i}^{\prime}$ are defined
in Eq.(29). In its first term, $\epsilon-{\sum}_{i}n_{i}\epsilon_{i}^{\prime}$
is equal to $\epsilon_{\rm w}-n_{1}\epsilon_{1}^{\prime}$ to leading order in
$n_{i}$ ($i\geq 2$) from $\epsilon$ in Eq.(5). It is also equal to
$\epsilon_{\rm eff}^{0}-{\sum}_{i}n_{i}\partial\epsilon_{\rm eff}^{0}/\partial
n_{i}$ in terms of $\epsilon_{\rm eff}^{0}$ in Eq.(14), so Eq.(39) surely
tends to the linear polarization limit $\epsilon_{\rm eff}\to\epsilon_{\rm
eff}^{0}$ in the Landau-Lifshits book. These authors presented
$M_{\alpha\beta}$ for one-component dielectric fluids without ions in applied
electric field (see Eq.(15.9) in their book Landau-e ). In this paper, we use
it for dilute electrolytes containing dipoles with nonlinear polarization.

The reversible (non-dissipative) force density acting on the fluid is given by
$-{\sum}_{\beta}\nabla_{\beta}\Pi_{\alpha\beta}$. From Eqs.(4), (12), and (39)
its electric part is calculated as

$\displaystyle-{\sum}_{\beta}\nabla_{\beta}M_{\alpha\beta}=\rho
E_{\alpha}+{\mbox{\boldmath$p$}}_{d}\cdot\nabla E_{\alpha}$
$\displaystyle+{\sum}_{i}\nabla_{\alpha}(n_{i}\epsilon_{i}^{\prime}E^{2})/8\pi-{E^{2}}\nabla_{\alpha}\epsilon/8\pi,$
(40)

where $\nabla_{\alpha}$ and $\nabla_{\beta}$ stand for the components of
$\nabla=(\partial/\partial x,\partial/\partial y,\partial/\partial z)$. The
second term can also be written as $\chi_{d}\nabla_{\alpha}E^{2}/2$ from
$\nabla_{\beta}E_{\alpha}=\nabla_{\alpha}E_{\beta}$. In
${\sum}_{\beta}\nabla_{\beta}\Pi_{\alpha\beta}$, the pressure gradient
$\nabla_{\alpha}p_{\rm n}$ is equal to the non-electric part of the
combination ${\sum}_{i}n_{i}\nabla_{\alpha}\mu_{i}+S\nabla_{\alpha}T$ from the
Gibbs-Duhem relation. We further confirm that
${\sum}_{\beta}\nabla_{\beta}M_{\alpha\beta}$ in Eq.(40) coincides with the
electric part of this combination from Eqs.(22)-(25) and (30). Thus, we obtain

${\sum}_{\beta}\nabla_{\beta}\Pi_{\alpha\beta}={\sum}_{i}n_{i}\nabla_{\alpha}\mu_{i}+S\nabla_{\alpha}T,$
(41)

including the electric parts. The mechanical equililbrium
${\sum}_{\beta}\nabla_{\beta}\Pi_{\alpha\beta}=0$ is attained for homogeneous
$\mu_{i}$ and $T$. We note that Eq.(41) can be used in the hydrodynamic
equations of nonequilibrium electrolytes.

We can calculate the pressure contribution from the Debye-Hückel free energy
$f_{\rm DH}=-k_{B}T\kappa^{3}/12\pi$ in the form $p_{\rm
DH}=(1-3n_{1}\partial\ln\epsilon_{\rm w}/\partial n_{1})f_{\rm DH}/2$ Oka2 ,
where $\kappa$ is the Debye wave number. We can also derive this $p_{\rm DH}$
from the thermal average of $M_{ij}$ using the Debye-Hückel structure factor
of the charge density McQ ; Stokes ; Debye . We note that $M_{ij}$ can be used
in its nonlinear form for the thermal charge density fluctuations at small
wave numbers in agreement with the Debye-Hückel theory.

###  II.5 Equilibrium solute densities

In equilibrium, the chemical potentials $\mu_{i}$ and the temperature $T$ are
homogeneous constants. If we neglect the wall and image potentials,
Eqs.(23)-(25) yield

$\displaystyle n_{2}=n_{2}^{0}\exp(-\nu_{2}-U+\zeta_{2}),$ (42) $\displaystyle
n_{3}=n_{3}^{0}\exp(-\nu_{3}+U+\zeta_{3}),$ (43) $\displaystyle
n_{4}=n_{4}^{0}{\cal N}_{\rm d}\exp(-\nu_{4}+\zeta_{4}),$ (44)

where $n_{i}^{0}$ ($i\geq 2)$ are constants independent of space, $\zeta_{i}$
are defined in Eq.(26). In Eqs.(42) and (43), $\zeta_{i}$ decrease $n_{i}$ for
$\epsilon_{i}^{\rm w}<0$. Indeed, we find $\exp(\zeta_{i})=\exp(-0.05h^{2})$
($i=2,3$) in ambient liquid water, where we set ${\hat{\gamma}}=3.5$ and
$\epsilon_{2}^{\rm w}=\epsilon_{3}^{\rm w}$ in Eqs.(5) and (6) and
$\mu_{0}=e\ell_{B}/2$ in Eq.(10). Despite this factor, $n_{3}$ ($n_{2})$
should increase near positively (negatively) charged surfaces, owing to the
factor $e^{U}$ ($e^{-U}$). In Eq.(44), the factor ${\cal N}_{\rm d}(>1)$ in
Eq.(9) comes from $-W$ in Eq.(25) amplifying $n_{4}$. AO group An1  found that
the dipole density is multiplied by ${\cal N}_{d}$ (without $\nu_{i}$ and
$\zeta_{i}$ in their expressions).

From Eqs.(12) and (44) the polarization ${\mbox{\boldmath$p$}}_{d}$ becomes

${\mbox{\boldmath$p$}}_{d}=\mu_{0}n_{4}^{0}e^{-\nu_{4}+\zeta_{4}}{\cal
G}(h)E^{-1}{\mbox{\boldmath$E$}},$ (45)

where ${\cal G}(h)$ is defined by An1 ; An2 ; An4 ; An3

${\cal G}(h)=\partial{\cal N}_{d}/\partial h={\cal N}_{d}{\cal L}(h)=\cosh
h/h-\sinh h/h^{2}.$ (46)

Now, Eq.(4) gives the four-component dPB equation,

$\nabla\cdot\epsilon{\mbox{\boldmath$E$}}+4\pi\nabla\cdot{\mbox{\boldmath$p$}}_{d}=4\pi
Ze(n_{2}-n_{3}),$ (47)

where $n_{i}$ and ${\mbox{\boldmath$p$}}_{d}$ are given in Eqs.(42)-(45). We
use the linear dielectric constant $\epsilon$ in Eq.(5), which depends on
$n_{2}$ and $n_{3}$. The decrease of $\epsilon$ due to accumulation of cations
or anions is crucial in solving Eq.(47) Roij1 , though not studied in this
paper.

###  II.6 Density change of nearly incompressible solvent and osmotic stress

Next, we examine the equilibrium solvent density $n_{1}$ in a nearly
incompressible solvent in a one-phase state. Its deviation is small if the
solvent isothermal compressibility $\kappa_{T}^{\rm w}=(\partial
n_{1}/\partial p)_{T}/n_{1}$ is much smaller than $1/k_{B}Tn_{1}$. For
example, $k_{B}Tn_{1}\kappa_{T}^{\rm w}\cong 0.06$ in ambient liquid water. We
can then expand $\mu_{\rm w}(n_{1},T)$ in Eq.(22) as

$\delta\mu_{\rm w}=\mu_{\rm w}(n_{1},T)-\mu_{\rm
w}(n_{1}^{0},T)=(n_{1}-n_{1}^{0})/n_{1}^{2}\kappa_{T}^{\rm w},\\\ $ (48)

where $n_{1}^{0}$ is a constant reference density.

For simplicity, let the electrolyte be in equilibrium with a large pure
solvent without applied field, for which $n_{1}^{0}$ is the solvent density in
the reservoir. Here, the solvent chemical potential is common in the two
regions as

$\mu_{1}(\\{n\\},T)=\mu_{\rm w}(n_{1}^{0},T),$ (49)

so Eq.(22) yields the solvent density deviation,

$\displaystyle n_{1}/n_{1}^{0}-1=-n_{1}\kappa_{T}^{\rm
w}k_{B}T[{\sum}_{i}^{\prime}\nu_{i}^{\prime}n_{i}-\zeta_{i}]$
$\displaystyle=-{\sum}_{i}^{\prime}v_{i}^{*}n_{i}+n_{1}\kappa_{T}^{\rm
w}{\epsilon_{1}^{\prime}}E^{2}/8\pi.$ (50)

For each $i\geq 2$, $v_{i}^{*}$ is a volume related to the partial volume
${\bar{v}}_{i}$ at infinite dilution Buff ; Oc ; Oka1 ; Oka2  as

$v_{i}^{*}=n_{1}k_{B}T\kappa_{T}\nu_{i}^{\prime}={\bar{v}}_{i}-k_{B}T\kappa_{T}.$
(51)

The difference ${\bar{v}}_{i}-v_{i}^{*}=k_{B}T\kappa_{T}^{\rm w}$ stems from
the partial pressure $k_{B}Tn_{i}$ and is small, which is $0.06/n_{1}$ in
ambient liquid water. In Eq.(50), the first term represents the steric effect
Oka2  and the second term the electrostriction Landau-e ; Stell ; Hakim . Note
that $v_{i}^{*}$ can be negative due to the hydration (see the last paragraph
in this subsection). On the other hand, in the lattice theory of fluid
mixtures, all the particles have a common volume $v_{0}$ and the space-filling
relation ${\sum}_{i}v_{0}n_{i}=1$ is assumed PG ; Iglic ; An5 .

In the above situation we also calculate the equilibrium stress component
$\Pi_{zz}$ in the one-dimensional geometry, where the electric field $E$
changes along the $z$ axis. Here, $p_{\rm
w}(n_{1},T)=p_{0}+n_{1}\delta\mu_{\rm w}$, with $p_{0}$ being the reservoir
pressure. From Eqs.(37)-(39), (48), and (50) we thus find Is

$\Pi_{zz}=p_{0}+k_{B}T{\sum}_{i}^{\prime}n_{i}+(\epsilon_{\rm
w}-2\epsilon_{\rm eff})E_{z}^{2}/8\pi.$ (52)

From Eqs.(42)-(44) we confirm that the above $\Pi_{zz}$ is independent of $z$
(or $d\Pi_{zz}/dz=0$) in accord with Eq.(41).

We comment on the partial volumes of salts in ambient liquid water. In
experiments with the overall charge neutrality, the sum
${\bar{v}}_{s}={\bar{v}}_{2}+{\bar{v}}_{3}$ has been measured. From data of
${\bar{v}}_{s}$ Mil ; Craig , $n_{1}({v}^{*}_{2}+{v}^{*}_{3})$ was $-0.20$,
$-0.55$, $0.93$, and $2.0$ for LiF, MgSO4, NaCl, and NaI, respectively. Then,
$n_{1}(\nu_{2}^{\prime}+\nu_{3}^{\prime})$ is $-3.4$, $-10$, $15$, and $32$,
respectively, for these salts. For small and/or multivalent ions, $v_{i}^{*}$
and $\nu_{i}^{\prime}$ are negative, while $\nu_{i}^{\prime}\gg 1$ for large
ions.

##  III Solvent nonlinear polarization

In this section we consider equilibrium three-component electrolytes composed
of a solvent, cations, and anions, where the solvent exhibits nonlinear
polarization and is nearly incompressible. Indeed, AO group treated such
electrolytes An1 ; An2 ; An3 ; An4 . In this case, the linear dielectric
constant $\epsilon_{\rm w}$ of the solvent is given by

$\epsilon_{\rm w}=1+4\pi\mu_{0}^{2}n_{1}/3k_{B}T.$ (53)

Since the linear dielectric constant $\epsilon$ of the solution is given in
Eq.(5), the nonlinear dielectric constant $\epsilon_{\rm eff}$ of the solution
is expressed as

$\epsilon_{\rm eff}=1+\epsilon-\epsilon_{\rm w}+n_{1}{\mu_{0}}E^{-1}{\cal
L}(h).$ (54)

where $\epsilon-\epsilon_{\rm w}\cong\epsilon_{2}^{\rm
w}n_{2}+\epsilon_{3}^{\rm w}n_{3}$. See $\epsilon_{\rm eff}$ in Eq.(13) also.
The electric free energy density is then given by

$f_{e}=(1+\epsilon-\epsilon_{\rm w})E^{2}/8\pi+k_{B}Tn_{1}(hW^{\prime}-W),$
(55)

which tends to $\epsilon E^{2}/8\pi$ as $h\to 0$ from Eq.(53). The increment
$\delta f_{e}$ is obtained if we replace $\delta\epsilon$ by
$\delta(\epsilon-\epsilon_{\rm w})$, $\delta n_{4}$ by $\delta n_{1}$, and
$n_{4}$ by $n_{1}$ in Eq.(19). Then, it follows the solvent chemical
potential,

$\mu_{1}=\mu_{\rm
w}(n_{1},T)+k_{B}T(\nu^{\prime}_{2}n_{2}+\nu^{\prime}_{3}n_{3})-k_{B}TW,$ (56)

which should be compared with $\mu_{1}$ in Eq.(22). The expressions for
$\mu_{2}$ and $\mu_{3}$ are still given by Eqs.(23) and (24). Notice that the
last term $-k_{B}TW$ in Eq.(56) tends to the electrostriction term
$-k_{B}T\zeta_{1}$ as $h\to 0$ from

$k_{B}TW\cong k_{B}Th^{2}/6=(\partial\epsilon_{\rm w}/\partial
n_{1})E^{2}/8\pi~{}~{}(h\ll 1).$ (57)

We should consider the deviation of $n_{1}$. Let us determine the reference
solvent density $n_{1}^{0}$ from the osmotic condition (49). As in Eq.(50), we
find

$n_{1}/n_{1}^{0}-1=-v_{2}^{*}n_{2}-v_{3}^{*}n_{3}+(k_{B}Tn_{1}\kappa_{T}^{\rm
w})W.$ (58)

The last term is small from $k_{B}Tn_{1}\kappa_{T}^{\rm w}\ll 1$ and $W\cong
h$ for $h\gg 1$ (see the first paragraph in Sec.IIE). That is, the
polarization-induced increase in $n_{1}$ is negligible for small solvent
compressibility, while that in $n_{4}$ can be significant as found in Sec.II.
Thus, regarding $n_{1}$ as a constant, we now set up the three-component dPB
equation as

$\nabla\cdot\big{[}1+\epsilon-\epsilon_{\rm w}+n_{1}{\mu_{0}}{E}^{-1}{\cal
L}(h)\big{]}{\mbox{\boldmath$E$}}=4\pi Ze(n_{2}-n_{3}),$ (59)

where $n_{4}^{0}e^{-\nu_{4}+\zeta_{4}}$ is replaced by $n_{1}$ and ${\cal
G}={\cal N}_{d}{\cal L}$ by ${\cal L}$ in Eq.(45). Here, we neglect the steric
effect due to the first two terms in the right hand side of Eq.(58).

In their three-component dPB equation, AO group used ${\cal G}$ instead of
$\cal L$, where the polarization contribution is larger than in Eq.(59) by
${\cal N}_{d}$. See Eq.(5) in Ref.An1  and Eq.(27) in Ref.An3 . They further
accounted for the steric effect dividing the polarization contribution and the
charge density by ${\cal D}=(1-\phi_{0}){\cal N}_{d}+\phi_{0}\cosh(U)$, where
$\phi_{0}$ is the bulk ion volume fraction. See Eq.(9) in Ref.An1  and Eq.(13)
in Ref.An4 . However, as indicated by Eq.(58), we should divide them by An5 ;
Iglic

${\cal D}_{0}=1-\phi_{0}+\phi_{0}\cosh(U),$ (60)

where all the particles are assumed to have the same volume (see below
Eq.(51)). The steric effect is crucial in high-potential regions with
$\phi_{0}\cosh(U)\gtrsim 1$.

##  IV Ion association and dissociation

In this section, we examine the equilibrium electric-field effect for Bjerrum
dipoles composed of monovalent ions Bje ; Fuoss ; Roij ; Ma ; Dill ; Vegt .
For simplicity, we assume that they have a fixed dipole length $\ell_{d}$ with
a constant dipole moment,

$\mu_{0}=e\ell_{d},$ (61)

though $\ell_{d}$ should thermally fluctuate due to rather weak bonding. In
the original theories of ion pairing Ma ; Bje ; Fuoss , $\ell_{d}$ is of order
$\ell_{B}$. We then use the Onsager theory of the orientation entropy in
Appendix A. For relatively high salt concentrations ($\gtrsim 1$M$/\ell$),
clustering of ions becomes conspicuous in simulations Vegt ; Ma ; Deg ; Hassan
.

On symmetric multivalent ion pairs (A+ZB-Z), we make comments here and in the
last paragraph of this section. The dipole moment of such ion pairs is of
order $Z^{3}e\ell_{B}$ if there separation length is of order $Z^{2}\ell_{B}$.
However, this picture is not justified because a number of water molecules are
nonlinearly oriented between them. Nevertheless, the electric field effect
should be enhanced for multivalent ion pairs. It is worth noting that the
chemical reaction of MgSO4 is the main origin of low-frequency sound
attenuation in sea water Yeager1 ; Sim ; Eigen , where the dissociation time
is of order $10^{-6}$sec and is very long.

###  IV.1 Chemical equilibrium

In chemical thermodynamics Landau-s ; Kirk , the condition of chemical
equilibrium is given by $\mu_{2}+\mu_{3}=\mu_{4}$, where dissociation and
association balance on the average. This condition is rewritten as

$\mu_{2}+\mu_{3}-\mu_{4}=k_{B}T\ln[Kn_{2}n_{3}/n_{1}n_{4}]=0,$ (62)

where $K$ is the equilibrium association constant in the dimensionless form.
From Eqs.(23)-(25) we have

$\ln
K=\ln[n_{1}(\lambda_{2}\lambda_{3}/\lambda_{4})^{3}]+\nu_{r}-W-\epsilon_{r}^{\prime}E^{2}/(8\pi
k_{B}T),$ (63)

where $\nu_{r}$ and $\epsilon_{r}^{\prime}$ are defined by

$\displaystyle\nu_{r}=\nu_{2}+\nu_{3}-\nu_{4},$ (64)
$\displaystyle\epsilon_{r}^{\prime}=\epsilon_{2}^{\prime}+\epsilon_{3}^{\prime}-\epsilon_{4}^{\prime}\cong\epsilon_{2}^{\rm
w}+\epsilon_{3}^{\rm w}-\epsilon_{4}^{\rm w}.$ (65)

In experiments, $K$ is determined in equilibrium by

$K=n_{1}n_{4}/n_{2}n_{3}=c_{4}/c_{2}c_{3},$ (66)

where $c_{i}=n_{i}/n_{1}$ $(i\geq 2$) are the solute concentrations.
Typically, $K=10^{2}$ for $Z=1$ and $K=10^{4}$ for $Z=2$ without applied field
Ma ; Rie .

In $f$ in Eq.(15), $\nu_{4}$ is the parameter for the Bjerrum dipoles. If
there is no applied field, its non-ideal part $\nu_{4}-\nu_{4}^{0}$ is
determined by the Coulombic attraction between an associating ion pair under
the influence of the solvent. For small 1:1 ion pairs, the mutual potential is
given by $-(k_{B}T\ell_{B})/r$ if the separation distance $r$ is shorter than
$\ell_{B}$ Bje ; Fuoss ; Ma ; Dill ; Vegt ; Roij . Bjerrum Bje  and Fuoss
Fuoss  estimated $K$ as

$K_{\rm B}\sim a^{3}\int_{a}^{\ell_{B}/2}drr^{2}e^{\ell_{B}/r},~{}~{}K_{\rm
F}\sim e^{\ell_{B}/a},$ (67)

respectively. Here, $a$ is the closest center-to-center distance of an ion
pair. In Bjerrum’s theory, the bounded pairs are those with $a<r<\ell_{B}/2$.
If we use their estimations, the association contribution to
$\nu_{4}-\nu_{4}^{0}$ is given by $-\ln K_{\rm B}$ or $-\ln K_{\rm F}$ in the
absence of applied field from Eqs.(63) and (64).

###  IV.2 Field-dependent association constant

In applied electric field $E$, the ion bonding tends to be partially broken
when the dipole direction ${\mbox{\boldmath$n$}}_{d}$ is parallel to $E$,
while the bonding is stronger for
${\mbox{\boldmath$n$}}_{d}\cdot\mbox{\boldmath$E$}<0$. The Wien effect for
weak electrolytes Wien ; Eck ; Kaiser ; Korea  indicates that the dissociation
constant $K_{d}$ should increase with increasing $E=|\mbox{\boldmath$E$}|$,
where $K_{d}$ is the inverse of the association constant $K$. For 1:1 salts,
Onsager Onsager49  calculated the $E$-dependence of $K_{d}$ in the form,

$K_{d}(E)/K_{d}(0)=F_{\rm O}(b)=I_{1}(\sqrt{8b})/\sqrt{2b}.$ (68)

where $I_{1}(x)=\pi^{-1}\int_{0}^{\pi}d\theta\cos\theta\exp({x\cos\theta})$ is
the modified Bessel function of the first kind and $b$ is given by

$b=\ell_{B}eE/2k_{B}T=(\ell_{B}/2\ell_{d})h,$ (69)

with $\ell_{d}$ being defined in Eq.(61). Here, $F_{\rm O}(b)\cong 1+b$ for
$b\ll 1$ and $F_{\rm O}(b)\cong\sqrt{2/\pi}(8b)^{-3/4}\exp(\sqrt{8b})$ for
$b\gg 1$. The characteristic field at $b=1$ is
$E=2k_{B}T/e\ell_{B}=0.08$V$/$nm. In particular, $b=h$ for
$\ell_{d}=\ell_{B}/2$. Onsager derived Eq.(68) kinetically from the equations
for Brownian motion in the combined Coulomb and external fields. However,
$K_{d}(=1/K)$ is determined by Eq.(62) in equilibrium, so it can be calculated
by a static theory.

To understand the Onsager formula (68), let us assume that
${\mbox{\boldmath$n$}}_{d}$ is parallel to $E$. Then, the sum of the Coulombic
potential and the field potential is written as

$\psi_{B}(r)=-k_{B}T\ell_{B}/r-eEr.$ (70)

which assumes a maximum $\psi_{B}^{\rm max}$ at $r=\ell_{B}/\sqrt{2b}$ with

$\psi_{B}^{\rm max}/k_{B}T=-\sqrt{8b}.$ (71)

Thus, we find $\sqrt{8b}$ in Eq.(68). However, in the above argument (and in
Osager’s calculation also), the thermal random rotations of the dipoles are
neglected, which should be important particularly for weak applied field.

In this paper, we treat the field-induced dissociation as a consequence of the
field-induced potential change, as argued in the above paragraph. That is, we
assume that $\nu_{4}$ in the chemical potential $\mu_{4}$ is field-dependent
as

$\nu_{4}(E)=\nu_{4}(0)+\ln F_{\rm O}(b),$ (72)

where we use Onsager’s $F_{O}$. Then, $\mu_{4}$ in Eq.(25) becomes

$\mu_{4}(E)=\mu_{4}(0)-k_{B}T\ln({\cal N}_{d}/F_{\rm
O})-\epsilon_{4}^{\prime}E^{2}/8\pi.$ (73)

Thus, in the presence of electric field, $\mu_{4}/k_{B}T$ decreases by
$W=\ln{\cal N}_{d}$ due to the dipole alignment but increases by $\ln F_{\rm
O}$ due to the potential change. Here, $W\propto h^{2}$ and $\ln F_{O}\propto
h$ for small $h$. In Fig.1(a), we plot $\ln({\cal N}_{d}/F_{\rm O})$ for $b=h$
using Onsager’s $F_{O}$, where $F_{\rm O}$ is larger (smaller) than ${\cal
N}_{d}$ for $h\lesssim 7$ (for $h\gtrsim 7$).

For 1:1 salts, Eqs.(63)-(65) and (72) yield the field-dependent association
constant $K(E)$ in the form,

$K(E)/K(0)=\exp(A_{r}h^{2}){\cal N}_{d}/F_{\rm O}(b).$ (74)

If we use $\hat{\gamma}$ in Eq.(6) and set $\epsilon_{4}^{\prime}=0$, $A_{r}$
is written as

$A_{r}=-(\epsilon_{2}^{\rm w}+\epsilon_{3}^{\rm
w})k_{B}T/8\pi\mu_{0}^{2}={\hat{\gamma}}/(4\pi\ell_{B}\ell_{d}^{2}n_{1}).$
(75)

In Fig.1(a), $K(E)/K(0)$ is plotted for $b=h$ and $A_{r}=0.10$, where the
factor $\exp(A_{r}h^{2})$ is important for $h\gtrsim 3$. Thus, $K(E)$
decreases due to Onsager’s dissociation mechanism for relatively small
$h(\lesssim 3)$, but it increases due to the dipole accumulation and the ion-
induced dielectric degradation for larger $h$.

###  IV.3 Electric double layer with Bjerrum dipoles

Figure 1: (a) Field-dependence of the association constant $K(E)/K(0)$ in
Eq.(74) and the dipole density factor $c_{4}/c_{d}^{0}={\cal N}_{d}/F_{O}$ in
Eq.(79) as functions of the normalized field strength $h=\mu_{0}E/k_{B}T$ in
Eq.(10) for 1:1 salts on a semi-logarithmic scale. In $\mu_{4}$ in Eq.(73)
$\ln({\cal N}_{d}/F_{O})$ also appears. (b) Near-wall profiles of $U(z)$
(electric potential), $c_{2}(z)/c_{0}$ (cation), and $c_{3}(z)/c_{0}$ (anion)
vs $\kappa_{0}z$ on a semi-logarithmic scale, where the bulk concentrations
are $c_{0}=c_{d}^{0}=10^{-2}$ and $\kappa_{0}$ is the bulk Debye wave number.

We consider an electric double layer with a charged surface at $z=0$ in the
semi-infinite limit ($0<z<\infty$). For large $z$, $c_{2}(z)=n_{2}(z)/n_{1}$
and $c_{3}(z)=n_{3}(z)/n_{1}$ tend to $c_{0}$, while $c_{4}(z)=n_{4}(z)/n_{1}$
tends to $c_{d}^{0}$. From Eq.(66) the bulk concentrations $c_{0}$ and
$c_{d}^{0}$ satisfy

$c_{d}^{0}=K(0)c_{0}^{2}.$ (76)

Neglecting the steric effect we write the solute concentration profiles
$c_{i}(z)=n_{i}(z)/n_{1}$ as

$\displaystyle c_{2}(z)=c_{0}\exp(-U-A_{2}h^{2}),$ (77) $\displaystyle
c_{3}(z)=c_{0}\exp(U-A_{3}h^{2}),$ (78) $\displaystyle c_{4}(z)=c_{d}^{0}{\cal
N}_{d}/F_{\rm O},$ (79)

where $A_{i}=-\epsilon_{i}^{\rm w}k_{B}T/8\pi\mu_{0}^{2}$ ($i=2,3$) with
$A_{2}+A_{3}=A_{r}$ for $\epsilon_{4}^{\prime}=0$. The $U(z)$ and $h(z)$ tend
to 0 as $z\to\infty$. See Fig.1(a) for $c_{4}/c_{d}^{0}={\cal N}_{d}/F_{O}$ vs
$h$.

In Fig.1(b), we plot $U(z)$ $c_{2}(z)/c_{0}$, and $c_{3}(z)/c_{0}$ vs
$\kappa_{0}z$ for a 1:1 salt, where $c_{0}=c_{d}^{0}=10^{-2}$, $K(0)=10^{2}$,
and $A_{2}=A_{3}=0.05$ with $\kappa_{0}$ being the bulk Debye wave number. In
this example, $U$ exhibits a nearly exponential decay. The parameter values
are common to those in (a).

Finally, we examine the dielectric constant $\epsilon_{\rm eff}$ in the bulk
region for $Z:Z$ salts. From Eq.(14) it is written as

$\epsilon_{\rm eff}/\epsilon_{\rm w}\cong
1-2{\hat{\gamma}}c_{0}+A_{d}c_{d}^{0},$ (80)

where we use Eq.(6) with $\epsilon_{4}^{\rm w}=0$ in the second term. From
Eqs.(14) and (57) the coefficient $A_{d}$ is given by

$A_{d}=4\pi Z^{2}\ell_{B}\ell_{d}^{2}n_{1}/3.$ (81)

For $Z=1$ and $\ell_{d}=\ell_{B}/2$, $A_{d}$ is about $12$. For multivalent
ion pairs, $A_{d}$ is larger (see below Eq.(60)). Thus, the third dipolar term
in Eq.(80) can exceed 1 for $c_{d}^{0}>A_{d}^{-1}$ or for
$c_{0}>[A_{d}K(0)]^{-1/2}$, where the lower bound of $c_{0}$ is 0.03 for $Z=1$
and $1.5(\ell_{B}/2\ell_{d})\times 10^{-3}$ for $Z=2$. Here, we set
$K(0)=10^{2}$ for $Z=1$ and $K(0)=10^{4}$ for $Z=2$ Ma ; Rie . If $c_{0}$ much
exceeds the lower bound, we have $\epsilon_{\rm eff}\propto c_{0}^{2}$ and the
Debye wave number becomes proportional to $[c_{0}/\epsilon_{\rm
eff}]^{1/2}\propto c_{0}^{-1/2}$, as predicted by Zwanikken and Roij Roij . We
also note that the second term in Eq.(80) serves to decrease $\epsilon_{\rm
eff}$, which can also be important with increasing the $c_{0}$.

##  V Summary and remarks

We have studied the effects of electric field in dilute electrolytes, which
contain dipoles exhibiting nonlinear polarization. Our main results are
summarized below.

In Sec.II, we have discussed thermodynamics of electrolytes in inhomogeneous
electric field, where the dipolar component is a dilute solute. The electric
free energy density $f_{e}$ has been presented in Eq.(17) using Onsager’s
theory Onsager49 . We have then defined the field-dependent chemical
potentials $\mu_{i}$ in Eqs.(22)-(25) and the entropy density $S$ in Eq.(28).
We have calculated the equilibrium density profiles of the solutes and the
solvent in Eqs.(42)-(45) and (49). In Sec.III, we have considered three-
component electrolytes (solvent+ strong acid), where the solvent polarization
can be nonlinear. In this case, if the solvent is nearly incompressible, the
solvent density increase due to nonlinear polarization is negligible, which is
not in accord with the theory by the Andelman-Orland group. In Sec.IV, we have
examined Bjerrum dipoles created by association and dissociation of 1:1 salts,
which have a large dipole moment. We have calculated the density profiles in
Eqs.(71)-(73) and the field-dependent association constant $K(E)$ in Eq.(68).

We make some remarks below. (i) The Debye-Hückel free energy density and the
short-range ion-ion interaction terms are needed for not very dilute
electrolytes McQ ; Stokes ; Oka2 . We should further include the steric effect
in our theory to account for the accumulation of ions and dipoles near charged
surfaces Iglic ; An5 ; Oka2 . We have also neglected the short-range solute-
wall interactions and the image potentials. (ii) The decrement of the
dielectric constant $\epsilon$ due to ions in Eq.(5) should greatly affect the
ion densities near charged surfaces Roij1 , so its influence should be
examined in detail. (iii) The Onsager theory of field-induced dissociation
Onsager34  has not attracted enough attention, so his formula (68) should be
checked in detail (see our remark below Eq.(71)). (iv) We should study the
effect of electric field on electrolytes with multivalent salts such as MgSO4
and MgCl2. (v) Recently, the screening length has been shown to be longer than
the Debye length $\kappa^{-1}$ in ionic liquids Is1  and concentrated
electrolytes ($>1$M$/\ell$ for NaCl) Perkin1 . At present, we cannot decide
whether or not the last paragraph in Sec.IV is related to this issue. (vi)
Phase separation in electrolytes with mixture solvents are also of great
interest Onukipolar ; Tsori ; elec ; anta , where electric field appears
across two-phase interfaces and is particularly strong in the presence of
hydrophobic and hydrophilic (antagonistic) ions anta . The role of Bjerrum
dipoles remains unknown in mixture solvents.

Data availability: The data that supports the findings of this study are
available within the article.

Appendix A: Polarization free energy
In this appendix, we consider four-component electrolytes, where the solvent
polarization ${\mbox{\boldmath$p$}}_{\rm w}$ and the orientational
distribution $P({\mbox{\boldmath$n$}}_{d},{\mbox{\boldmath$r$}})$ for the
dipolar component are unknown variables to be determined below. Here, the
dipole polarization ${\mbox{\boldmath$p$}}_{d}$ is given by Eqs.(1) and (7) in
terms of $P({\mbox{\boldmath$n$}}_{d},{\mbox{\boldmath$r$}})$ with
$\int{d\Omega}P=4\pi$.

We minimize ${\cal F}$ at fixed surface charges. The electric free energy
density is given by

$f_{e}=\frac{1}{8\pi}E^{2}+\frac{1}{2\chi}p_{\rm w}^{2}-Tn_{4}s_{\rm d}.$ (A1)

where $\chi=(\epsilon-1)/4\pi$ and $p_{\rm w}=|{\mbox{\boldmath$p$}}_{\rm
w}|$. If we neglect the short-range interactions among the dipolar particles,
$s_{d}$ is the orientation entropy per dipolar particle Onsager49 ,

$s_{\rm d}=-\frac{1}{4\pi}k_{B}\int{d\Omega}P\ln P,$ (A2)

where $s_{\rm d}=0$ for the isotropic distribution $P=1$.

We then superimpose small variations on the physical quantities, where
${\mbox{\boldmath$p$}}_{\rm w}$ is changed by
$\delta{\mbox{\boldmath$p$}}_{\rm w}$ and $P$ by $\delta P$. Since
$\delta{\mbox{\boldmath$\mu$}}_{d}=\mu_{0}\int
d\Omega{\mbox{\boldmath$n$}}_{d}\delta P/4\pi$, we obtain

$\displaystyle\delta
f_{e}=\frac{{\mbox{\boldmath$E$}}\cdot\delta{\mbox{\boldmath$D$}}}{4\pi}-{p}_{\rm
w}^{2}\frac{\delta\chi}{2\chi^{2}}+\Big{(}\frac{{\mbox{\boldmath$p$}}_{\rm
w}}{\chi}-{\mbox{\boldmath$E$}}\Big{)}\cdot\delta{\mbox{\boldmath$p$}}_{\rm
w}-s_{\rm d}\delta(Tn_{4})$
$\displaystyle-{\mbox{\boldmath$E$}}\cdot{\mbox{\boldmath$\mu$}}_{d}\delta
n_{4}+\frac{n_{4}}{4\pi}\int{d\Omega}\Big{(}k_{B}T\ln
P-\mu_{0}{\mbox{\boldmath$E$}}\cdot{\mbox{\boldmath$n$}}_{d}\Big{)}\delta P.$
(A3)

Thus, to minimize $f_{e}$, we obtain ${\mbox{\boldmath$p$}}_{\rm
w}=\chi{\mbox{\boldmath$E$}}$ and
$P=\exp(\mu_{0}{\mbox{\boldmath$E$}}\cdot{\mbox{\boldmath$n$}}_{d}/k_{B}T)/{\cal
N}_{d}$, under which the third and last terms vanish and the second term
becomes $-E^{2}\delta\epsilon/8\pi$ in Eq.(A3). Then, we have Eqs.(2) and (8).
For this equilibrium $P$ we find

$Ts_{\rm
d}=-{\mbox{\boldmath$E$}}\cdot{\mbox{\boldmath$\mu$}}_{d}+k_{B}T\ln{\cal
N}_{\rm d}=k_{B}T(W-hW^{\prime}),$ (A4)

which leads to Eq.(17). We can see $s_{\rm d}\leq 0$ and $ds_{\rm
d}/dh=-k_{B}h(d{\cal L}(h)/dh)\leq 0$, In the three-component case in Sec.III,
$f_{e}$ is given by Eq.(55).

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