# Correlation decoupling of Casimir interaction in an electrolyte driven by
external electric fields

Guangle Du1, David S. Dean2,3, Bing Miao4, and Rudolf Podgornik1,3,5,6 1
School of Physical Sciences, University of Chinese Academy of Sciences (UCAS),
Beijing 100049, China.  2Univerité Bordeaux, CNRS, LOMA, UMR 5798, F-33400
Talence, France.  3 Kavli Institute for Theoretical Sciences, University of
Chinese Academy of Sciences (UCAS), Beijing 100049, China.  4Center of
Materials Science and Optoelectronics Engineering, College of Materials
Science and Opto-Electronic Technology, University of Chinese Academy of
Sciences (UCAS), Beijing 100049, China.  5Wenzhou Institute, University of
Chinese Academy of Sciences, Wenzhou, Zhejiang 325001, China.  6Institute of
Physics, Chinese Academy of Sciences, Beijing 100190, China.

###### Abstract

It has been established for a long time that the long range van der Waals or
thermal Casimir interaction between two semi-infinite dielectrics separated by
a distance $H$ is screened by an intervening electrolyte. Here we show how
this interaction is modified when an electric field of strength $E$ is applied
parallel to the dielectric boundaries, leading to a non-equilibrium steady
state with a current. The presence of the field induces a long range thermal
repulsive interaction, scaling just like the thermal Casimir interaction
between dielectrics without the intervening electrolyte, i.e. as $1/H^{3}$. At
small $E$ the effect is of order $E^{2}$ while at large fields it saturates to
an $E$ independent value. We explain the results in terms of a decoupling
mechanism between the charge density fluctuations of cations and anions at
large applied fields.

Introduction– Fluctuation induced interactions are ubiquitous both from the
fundamental as well as applied point of view and can arise due to both quantum
and thermal fluctuations par05 ; woo16 ; dan23 . These interactions can be
varied by changing material properties in the case of van der Waals
interactions woo16  or by chemical variations of surface properties in the
case of the critical Casimir effect RevModPhys.90.045001 . Another way to
modify these interactions is to apply external fields, for example electric or
magnetic fields dea16 ; mah21 . The application of an electric field generates
a current in systems which can conduct, and the presence of a current implies
that the system is out of equilibrium. The out of equilibrium nature of the
problem means that standard equilibrium methods, for example the Lifshitz
formulation of van der Waals forces cannot be applied directly Advances , it
can however be extended to compute van der Waals forces between dielectrics
held at different temperatures Antezza2 ; Antezza1 ; Kardar . Out of
equilibrium thermal or classical fluctuation induced forces can be studied via
the Langevin dynamics of dipole fields in dielectrics and/or ionic Brownian
dynamics in electrolytes UOELangevin ; UOELiving ; BSLu .

Figure 1: Geometry of the confined electrolyte of dielectric constant
$\epsilon$ between two dielectric semispaces of dielectric constant
$\epsilon^{\prime}$, driven by an external electric filed ${\bf E}$. $H$ is
the separation between the dielectrics, $z$ is the normal coordinate with the
bottom surface located at $z=0$.

An illuminating model in the theory of the thermal Casimir effect is the
living conductor model where two plates containing an electrolyte interact
across a dielectric gap jan04 ; UOELiving ; YLevin . In equilibrium this model
exhibits the universal long range attractive Casimir interaction for free
scalar field theories Advances . Recently dea16 , it was shown that when an
electric field is applied parallel to one plate this attractive Casimir
interaction is reduced by the fact that the driving electric field destroys or
scrambles the charge-charge correlations which yield the attraction in the
equilibrium case. More recently mah21 , the force between two planar
dielectrics with an electrolyte in between, and driven by an electric field
has been studied as a function of the field strength (see the schematic in
Fig. (1)). Here, it was found that driving the system leads to an effective
long range interaction which can be attractive or repulsive. This is a
particularly interesting result as, in equilibrium, the long range van der
Waals interaction between two dielectric slabs with an intervening electrolyte
is established to be screened, i.e., it decays exponentially with the plate
separation at distances larger than the Debye screening length nin76 ; net01 .
The Casimir interaction in the presence of a driving electrostatic field was
analyzed by using an approximation scheme for the linearized stochastic
density functional theory (SDFT) that describes the classical charge
fluctuations in the system mah21 . This leads to an effective theory
reminiscent of statistical field theories where the introduction of anisotropy
(due to the applied field) is mismatched with the thermal noise and leads to
long-range correlations gar90 ; gri90  which in turn generate a long-range
fluctuation induced interaction Sengers1 ; Sengers2 . It furthermore leads to
a limiting result at small driving electric field magnitude $E$, with the
force scaling as $E^{4}$, while at large fields it scales as $E^{2}$. These
results are somewhat surprising, first one might expect that the small field
force should scale as $E^{2}$ unless there is some fine tuning arising
naturally in the problem, secondly it is not clear how a laterally imposed
large electric field could lead to arbitrarily large forces.

Here we reconsider the problem studied in Ref. mah21  in the limit where the
dielectric constant of the surrounding (outer) dielectrics,
$\epsilon^{\prime}$, is much smaller than the dielectric constant of the
intervening solvent, $\epsilon$. This case can be solved without any
approximation beyond the linearization of the SDFT for the electrolyte. We
find that an electric field does indeed induce a long range interaction and
this interaction gives a total, or net, repulsive force per unit area

$f_{\rm t}(H)=\frac{\zeta(3)}{8\pi\beta
H^{3}}\left[1-\left({1+\frac{\beta^{2}q^{2}E^{2}}{m^{2}}}\right)^{-{\textstyle\frac{1}{2}}}\right].$
(1)

as well as exponentially screened terms which are thus subdominant for large
$H$. In Eq. (1) $\beta=1/(k_{\mathrm{B}}T)$ where $k_{\mathrm{B}}$ is
Boltzmann’s constant and $T$ is the temperature of the system (assumed
constant), $\pm q$ is the charge of the cations/anions, and
$m=\sqrt{2\overline{\rho}q^{2}/(\epsilon k_{\mathrm{B}}T)}$ is the inverse
Debye screening length where $\overline{\rho}$ is the mean density of
cations/anions. We thus predict that the long range force in this dielectric
set up is always repulsive, scales as $E^{2}$ for small applied electric
fields and saturates for large applied fields. We also identify the origin of
this limiting form of the repulsion at large electric fields which is due to
the decoupling of the cationic and anionic charge fluctuations, which then
become independent of one another for large fields. For $E=0$, we recover the
well known result for the screened thermal Casimir interaction nin76 ; net01 ,
from our purely dynamical approach.

The model– We consider (see Fig. (1)) two semi-infinite dielectrics of
dielectric constant $\epsilon^{\prime}$ separated by a slab of thickness $H$
in the direction $z$, containing an electrolyte solution of dielectric
constant $\epsilon$ and ionic densities $\overline{\rho}$ of cations and
anions of charges $\pm q$, but with otherwise identical properties, notably
they have the same friction coefficient $\gamma$. These conditions of symmetry
can be relaxed but the resulting formulas become much more complicated. This
symmetric case is sufficient to highlight the underlying physics of the
problem. The over-damped Langevin equation for the anions and cations (labeled
by the index $i$) is given by

$\gamma\frac{\mathrm{d}{\bf x}_{i}}{\mathrm{d}t}=q_{i}({\bf E}-\nabla\phi({\bf
x}_{i}))+\sqrt{2k_{\mathrm{B}}T\gamma}~{}\bm{\eta}_{i}(t),$ (2)

which corresponds to force balance between the frictional force on the left
hand side and the electrical and thermal noise forces on the right hand side.
The first term on the right is the force due to the applied electric field
parallel to the slabs (taken to be in the direction $x$) ${\bf E}=E{\bf
e}_{x}$, and the second one is the force due to the electric field generated
by the ions themselves:

$\phi({\bf x})=q\int\mathrm{d}{\bf x}^{\prime}G({\bf x},{\bf
x}^{\prime})[\rho_{+}({\bf x}^{\prime})-\rho_{-}({\bf x}^{\prime})].$ (3)

In Eq. (3) $\rho_{\pm}({\bf x})=\sum_{i\pm}\delta({\bf x}-{\bf x}_{i})$
denotes the density of ions/cations and $G({\bf x},{\bf x}^{\prime})$ is the
Green’s function for the slab geometry where

$\nabla\cdot\epsilon({\bf x})\nabla G({\bf x},{\bf x}^{\prime})=-\delta({\bf
x}-{\bf x}^{\prime}).$ (4)

Finally, the term $\bm{\eta}_{i}(t)$ denotes a Gaussian white noise of zero
mean with
$\langle\eta_{i\alpha}(t)\eta_{j\beta}(t^{\prime})\rangle=\delta_{ij}\delta_{\alpha\beta}\delta(t-t^{\prime})$,
where $\alpha$ and $\beta$ label the spatial components $(x,\,y,\,z)$.
Starting from the Langevin equations we can formally write down the SDFT for
the densities $\rho_{\pm}({\bf x})$, this non-linear theory becomes solvable
if we expand about a background of constant density $\rho_{\pm}({\bf
x})=\overline{\rho}+n_{\pm}({\bf x})$. Note that when the electric field is
parallel to the bounding surfaces it does not generate a surface charge and so
the expansion of the density about its average bulk value is justified. The
analysis of the linearized SDFT is simplified by using the fluctuations of the
total density $n_{\mathrm{t}}({\bf x})=n_{+}({\bf x})+n_{-}({\bf x})$ and the
fluctuations of the charge density $\rho_{\mathrm{c}}({\bf x})=q[n_{+}({\bf
x})-n_{-}({\bf x})]=qn_{\mathrm{d}}({\bf x})$. One thus finds SM

$\displaystyle\frac{\partial n_{\mathrm{t}}({\bf x})}{\partial t}$
$\displaystyle=$ $\displaystyle\nabla\cdot D[\nabla n_{\mathrm{t}}({\bf
x})-\beta qn_{\mathrm{d}}({\bf x}){\bf E}]+\xi_{\mathrm{t}}({\bf x},t),$ (5)
$\displaystyle\frac{\partial n_{\mathrm{d}}({\bf x})}{\partial t}$
$\displaystyle=$ $\displaystyle\nabla\cdot D[\nabla n_{\mathrm{d}}({\bf
x})-\beta qn_{\mathrm{t}}({\bf x}){\bf
E}+2q\beta\overline{\rho}\nabla\phi({\bf x})]$ (6)
$\displaystyle+\xi_{\mathrm{d}}({\bf x},t).$

where the spatio-temporal white noise terms are independent and have
correlation functions $\langle\xi_{\mu}({\bf x},t)\xi_{\nu}({\bf
x}^{\prime},t^{\prime})\rangle=-4\overline{\rho}D\delta_{\mu\nu}\delta(t-t^{\prime})\nabla^{2}\delta({\bf
x}-{\bf x}^{\prime}),\,\mu,\nu\in\\{\mathrm{t},\mathrm{d}\\}$. The term $D$ is
the diffusion constant of the cations and anions, given by
$D=k_{\mathrm{B}}T/\gamma$. The boundary conditions on the linearized SDFT are
no-flux boundary conditions, so $\nabla_{z}n_{\mathrm{t}}({\bf x})=0$ and
$\nabla_{z}n_{\mathrm{d}}({\bf x})+2q\beta\overline{\rho}\nabla_{z}\phi({\bf
x})=0$ at the boundaries $z=0$ and $z=H$. No flux boundary conditions are also
imposed independently on the noise terms, as the noise is independent of the
ionic distribution at the time when it acts. The second boundary condition (on
$n_{\mathrm{d}}({\bf x})$) is non-local due to the appearance of the
electrostatic potential which depends on the whole charge density in the
problem. This non-locality can be dealt with Future , but here we consider the
simplifying case where we assume that $\epsilon^{\prime}\ll\epsilon$. This
means that both the density and the electrostatic potential have the same,
Neumann boundary conditions at the bounding surfaces. Consequently, both
fields, $n_{\mathrm{t}}({\bf x})$ and $n_{\mathrm{d}}({\bf x})$, as well as
the noise fields, can be expanded in terms of a Fourier cosine expansion SM ,
yielding

$\displaystyle n_{\mu}({\bf x}_{\|},z,t)$ $\displaystyle=$
$\displaystyle\int\frac{\mathrm{d}{\bf
k}}{(2\pi)^{2}}\sum_{n=0}^{\infty}\frac{1}{\sqrt{N_{n}}}\tilde{n}_{\mu n}({\bf
k},t)\exp(\mathrm{i}{\bf k}\cdot{\bf
x}_{\|})\cos(p_{n}z),\,\mu\in\\{\mathrm{t},\mathrm{d}\\},$ (7)

where $p_{n}=n\pi/H$ enforces the Neumann conditions at $z=0$ and $z=H$, while
$N_{n}$ normalizes the corresponding eigenfunction, with $N_{n}=H/2$ for
$n\geq 1$ and $N_{0}=H$. The Fourier transform is taken in the plane
$\mathbf{x}_{\|}=(x,y)$ parallel the bounding dielectric surfaces. In the
steady state the resulting Lyapunov equation SM  can be solved analytically
for the equal time correlation functions $\langle\tilde{n}_{\mu m}({\bf
k},t)\tilde{n}_{\nu n}(({\bf
k}^{\prime},t)\rangle=(2\pi)^{2}\delta_{mn}\delta({\bf k}+{\bf
k^{\prime}})\tilde{C}_{\mu\nu n}$, $\mu,\nu\in\\{\mathrm{t},\mathrm{d}\\}$,
where

$\displaystyle\tilde{C}_{\mathrm{tt}n}({\bf k})$ $\displaystyle=$
$\displaystyle
2\overline{\rho}\frac{(k^{2}+p_{n}^{2})\left[(K^{2}+p_{n}^{2})(K^{2}+k^{2}+2p_{n}^{2})+2\beta^{2}q^{2}E^{2}k_{x}^{2}\right]}{(K^{2}+k^{2}+2p_{n}^{2})\left[(K^{2}+p_{n}^{2})(k^{2}+p_{n}^{2})+\beta^{2}q^{2}E^{2}k_{x}^{2}\right]},$
(8a) $\displaystyle\tilde{C}_{\mathrm{td}n}({\bf k})$ $\displaystyle=$
$\displaystyle-\tilde{C}_{\mathrm{dt}n}({\bf
k})=\mathrm{i}2\overline{\rho}\frac{\beta
qEk_{x}m^{2}(k^{2}+p_{n}^{2})}{(K^{2}+k^{2}+2p_{n}^{2})\left[(K^{2}+p_{n}^{2})(k^{2}+p_{n}^{2})+\beta^{2}q^{2}E^{2}k_{x}^{2}\right]},$
(8b) $\displaystyle\tilde{C}_{\mathrm{dd}n}({\bf k})$ $\displaystyle=$
$\displaystyle
2\overline{\rho}\frac{(k^{2}+p_{n}^{2})\left[(k^{2}+p_{n}^{2})(K^{2}+k^{2}+2p_{n}^{2})+2\beta^{2}q^{2}E^{2}k_{x}^{2}\right]}{(K^{2}+k^{2}+2p_{n}^{2})\left[(K^{2}+p_{n}^{2})(k^{2}+p_{n}^{2})+\beta^{2}q^{2}E^{2}k_{x}^{2}\right]}$
(8c)

with $K^{2}=k^{2}+m^{2}$ and $k_{x}={\bf e}_{x}\cdot{\bf k}$.

We now turn to the computation of the force between the two dielectrics. In
writing Eq. (4) we have implicitly assumed that the dipole degrees of freedom
in the system adjust instantaneously to generate image charges, and are thus
fully equilibrated with respect to a given configuration of charge density
(dynamics of dipoles being governed by rapid electronic degrees of freedom
orders of magnitude faster than the diffusive dynamics of the ions of the
electrolyte). However, in principle, potentially slower dynamics of the
dipoles in the dielectrics can be taken into account UOELangevin . The dipole
fluctuations also generate their own thermal component of the Casimir force
between two dielectrics, which would be present in the absence of any
electrolyte, i.e., when $\overline{\rho}=0$. There is also an ideal gas
component coming from the ions when they are uncharged or due to the constant
non-fluctuating part of the densities which is by definition also assumed to
be equilibrated. This means that the average total force per unit area is
given by

$f_{\rm t}=f_{\rm vdW}+f_{\rm ion}+2\,\overline{\rho}\,k_{\mathrm{B}}T,$ (9)

where $f_{\rm vdW}$ is the van der Waals interaction between the two
dielectrics in the absence of ions (including contributions from the zero and
non-zero Matsubara frequencies), which are assumed to be instantaneously
equilibrated, and $2\,\overline{\rho}\,k_{\mathrm{B}}T$ is the bulk ideal
van’t Hoff pressure term, which is also assumed to be equilibrated as it is
not affected by the external field, and will be ignored subsequently since it
is independent of separation $H$. The second term $f_{\rm ion}$ is the
contribution due to the ions which can be deduced by constructing a stress
tensor from the local body force, which can then be written down in terms of
the density fluctuations, and is given by SM ; kru18

$\displaystyle\sigma_{\alpha\beta}({\bf x})=$
$\displaystyle-\frac{k_{\mathrm{B}}T}{4\,\overline{\rho}}\delta_{\alpha\beta}(n^{2}_{\mathrm{t}}({\bf
x})+n^{2}_{\mathrm{d}}({\bf x}))+$ (10) $\displaystyle\epsilon({\bf
x})(\nabla_{\alpha}\phi({\bf x})\nabla_{\beta}\phi({\bf
x})-\frac{1}{2}\delta_{\alpha\beta}[\nabla\phi({\bf x})]^{2}).$

The first term on the right hand side is generated by the density
fluctuations, while the second term is the contribution from the standard
Maxwell stress tensor. Furthermore, in the limit
$\epsilon^{\prime}\ll\epsilon$ only the part of the stress tensor inside the
electrolyte slab yields a non-zero contribution to the force.

The outward acting surface force density due to the ions at $z=0$ is given by
$f_{\rm
ion}=-\sum_{\beta}\langle\sigma_{z\,\beta}(0,0,0^{+})-\sigma_{z\,\beta}(0,0,0^{-})\rangle$
(by translational invariance in the $(x,y)$ plane). The average is taken over
the spatio-temporal noise, the minus sign comes from the downward normal and
$z=0^{\pm}$ denotes the position just above and just below the bounding
surface at $z=0$, respectively. In the limit $\epsilon^{\prime}\ll\epsilon$
the stress tensor is diagonal and we find

$f_{\rm ion}=\frac{k_{\mathrm{B}}T}{4\overline{\rho}}\langle
n^{2}_{\mathrm{t}}({\bf 0})+n^{2}_{\mathrm{d}}({\bf
0})\rangle+\frac{\epsilon}{2}\langle\nabla_{\|}\phi({\bf 0})\rangle^{2},$ (11)

where $\nabla_{\|}$ denotes the gradient in the $(x,y)$ plane. One can
explicitly extract the terms which give the total $H$ dependent force
(including the contribution from the dielectric slabs $f_{\rm vdW}$ in Eq.
(9)) SM , these are

$f_{\rm t}(H)=-\frac{1}{\beta}\int\frac{\mathrm{d}{\bf
k}}{(2\pi)^{2}}\sum_{n}\frac{1}{2N_{n}}\frac{K^{2}(k^{2}+p_{n}^{2})^{2}(K^{2}+k^{2}+2p_{n}^{2})+\beta^{2}q^{2}E^{2}k_{x}^{2}\left[2k^{2}(k^{2}+p_{n}^{2})+m^{2}(k^{2}+2p_{n}^{2})\right]}{(k^{2}+p_{n}^{2})(K^{2}+k^{2}+2p_{n}^{2})\left[(k^{2}+p_{n}^{2})(K^{2}+p_{n}^{2})+\beta^{2}q^{2}E^{2}k_{x}^{2}\right]}.$
(12)

Figure 2:  Dependence of the interaction force
$\bar{f}_{\mathrm{t}}=f_{\mathrm{t}}/[\zeta(3)/8\pi\beta H^{3}]$ on (a) the
separation $\bar{H}=Hm$ with fixed coupling constant $g=\beta qE/m$ and (b)
$g$ with fixed $\bar{H}$. The asymptotic large $\bar{H}$ and $g$ results are
shown as the dashed lines $\bar{f}_{\mathrm{t}}=1$.

For $E=0$, the above equation yields the well known equilibrium result for the
screened thermal Casimir interaction which can be written in the same limit
$\epsilon^{\prime}\ll\epsilon$ as nin76

$\displaystyle f_{\rm t}(H)$ $\displaystyle=$
$\displaystyle-\frac{1}{4\pi\beta
H^{3}}\int_{mH}^{\infty}\mathrm{d}K\,K^{2}\left(\coth K-1\right)$ (13)

which is rather remarkable viewed in the context of the formalism adopted
here. The charge fluctuations of the electrolyte obviously produce a repulsive
thermal Casimir interaction which cancels exactly the attractive thermal
Casimir interaction between the two bounding dielectrics. This effect was
already pointed out explicitly for an equilibrium system in jan04  and was
also discussed previously by a number of other authors att87 ; att88 . In the
$E=0$ limiting case this attractive interaction is

$f_{\rm vdW}(H)=-\frac{\zeta(3)}{8\pi\beta H^{3}}.$ (14)

The residual interaction after this cancellation then turns out to be
attractive and screened.

The integral and sum in Eq. (12) can be evaluated analytically SM  but the
result is rather unwieldy and does not have an intuitive interpretation.
However, the large distance behavior of the interaction, more precisely in the
limit where $Hm\gg 1$, can be simply extracted from the small $k$ and $p_{n}$
components in Eq. (12). This then gives for large $H$ the particularly simple
formula in Eq. (1) for the dominant large distance component of $f_{\rm
t}(H)$, which is exact up to terms which decay as $\exp(-2mH)$. We emphasize
that the long distance repulsion is only present for non zero $E$ and is a
purely non-equilibrium effect. For small $E$ the force behaves as $f_{\rm
t}(H)=\frac{\zeta(3)}{16\pi\beta H^{3}}\frac{\beta^{2}E^{2}q^{2}}{m^{2}}$. On
the other hand, for strong electric field driving as $E\to\infty$ in the same
limit of $\epsilon^{\prime}\ll\epsilon$, we find

$f_{\rm t}(H)\\!=\\!\frac{\zeta(3)}{8\pi\beta H^{3}}-\frac{2}{4\pi\beta
H^{3}}\\!\\!\int_{\frac{mH}{\sqrt{2}}}^{\infty}\\!\\!\mathrm{d}KK^{2}\\!\left(\coth
K-1\right).$ (15)

The first term is a repulsive thermal Casimir force of exactly the same
absolute magnitude as the equilibrium attraction between two dielectrics with
no intervening electrolyte. This result can be understood as a correlation
decoupling between the cations and anions due to the strong external electric
field SM , indicated by Eq. (8b). If only the cationic type was present then
the effect of the electric field would be to drive all the cations at the same
average velocity $v=\beta DqE$. In the rest frame of the cations the system
would then be in an effective equilibrium. Charge fluctuations in this
equilibrium state would then generate a repulsive thermal Casimir interaction
as given by Eq. (15). However, the same argument is equally valid for the
anions that move with velocity $v=-\beta DqE$. The crucial point is that due
to the external electrostatic field driving the dynamics of the cations and
anions becomes decoupled in the limit of $E\to\infty$, and so they both
generate a repulsive force of the form in Eq. (15), subtracting off the
attractive thermal Casimir interaction of the dielectrics then giving the net
repulsive force Eq. (15). Notice that the $E$ dependence in the decouplimg
effect occurs via a coupling constant $g=\beta Eq/m$ which can be written as
$g=v/v^{*}$, where $v^{*}=Dm$ is an intrinsic velocity scale for the non-
driven electrolyte system, corresponding exactly to the same non-equilibrium
parameter found in Ref. dea16  for the Casimir interaction between two plates
containing Brownian electrolytes, where one is driven by an electric field.

The second term in Eq. (15), which again exhibits a form of the screened
thermal Casimir force, is also consistent with the above interpretation in
terms of the correlation decoupling SM . In fact the inverse screening length
of $m/\sqrt{2}$ indicates that the ions of either type are screening
independently. Since in the rest frame of the anions/cations they are in an
effective equilibrium at half the total density (see above), the Casimir
interaction is exactly twice (explicit factor 2 in the second term of Eq.
(15)) the screened Casimir force at $m/\sqrt{2}$ screening length, giving the
strong electric field driving limiting law a consistent intuitive
interpretation.

One can express the full interaction $f_{\rm t}$ at arbitrary separation $H$
and external electric field strength $E$ in terms of integrals which need to
be evaluated numerically. In Fig. (2) we have shown $f_{\rm t}$ as functions
of $\bar{H}=Hm$ (i.e. with $H$ measured in units of the Debye length) and $g$,
where $f_{t}$ are normalized by the absolute value of van der Waals attraction
in Eq. (14). In Fig. (2a) we see the behavior of $f_{\rm t}$ at fixed values
of $g$ as $\bar{H}$ is varied. The lowest curve corresponds to $g=0$ the
screened thermal Casimir force given in Eq. (13) which is always attractive.
The cases $g=1,\ 10$ and $g\to\infty$ are also shown, at short distances all
these forces are attractive but on increasing $\bar{H}$ they become repulsive,
the crossover value of $\bar{H}$ from attraction to repulsion decreases on
increasing $g$. The large $g$ limits agree with the asymptotic result Eq.
(15). In Fig (2b) we show $f_{\rm{t}}$ at fixed values of $\bar{H}$ as a
function of $g$. At all separations increasing $g$ increases the repulsive
component of the force, at short distances the force remains net attractive
but at large distances it eventually becomes net repulsive. The large
$\bar{H}$ limits agree with the asymptotic result Eq. (1). $f_{t}$ saturates
to the repulsion $\zeta(3)/(8\pi\beta H^{3})$ at either large separation $H$
or large coupling $g$.

Conclusions– Using non-equilibrium SDFT for an electrolyte confined between
two dielectrics in an external driving electric field, we first of all recover
the well known result for the screened thermal Casimir interaction from a
purely dynamical approach. For asymptotically large separations we find a
quadratic scaling of the repulsive Casimir interaction for small electric
fields, and show that for large electric field it saturates. The exponentially
screened terms correspond to the standard screened thermal Casimir force with
the Debye screening length for vanishing electric field, and to a Debye
screening length at half the ion density for large electric fields, this
effect being due to decoupled ionic correlations in the presence of strong
external driving. The non-equilibrium effects predicted here could be tested
either in the surface force apparatus ric20  or colloidal probe atomic force
microscopy interaction geometry Trefalt20 .

Acknowledgments– G.D., B.M. and R.P. acknowledge funding from the Key Project
No. 12034019 of the National Natural Science Foundation of China and the
support by the Fundamental Research Funds for the Central Universities (Grant
No. E2EG0204). D.S.D. acknowledges support from the grant No.
ANR-23-CE30-0020-01 EDIPS, and by the European Union through the European
Research Council by the EMet-Brown (ERC-CoG-101039103) grant.

## References

  * (1) L.M. Woods, D.A.R. Dalvit, A. Tkatchenko, P. Rodriguez-Lopez, A.W. Rodriguez and R. Podgornik, Materials perspective on Casimir and van der Waals interactions, Rev. Mod. Phys. 88, 045003 (2016).
  * (2) D.M. Dantchev and S. Dietrich, Critical Casimir Effect: Exact Results, Physics Reports, 1005, 1 (2023).
  * (3) V.A. Parsegian, Van der Waals Forces: A Handbook for Biologists, Chemists, Engineers, and Physicists, Cambridge University Press (Cambridge) (2005).
  * (4) A. Maciołek and S. Dietrich, Collective behavior of colloids due to critical Casimir interactions, Rev. Mod. Phys., 90, 045001 (2018).
  * (5) D. S. Dean, B.-S. Lu, A. C. Maggs, and R. Podgornik, Nonequilibrium Tuning of the Thermal Casimir Effect, Phys. Rev. Lett. 116, 240602 (2016).
  * (6) S. Mahdisoltani and R. Golestanian, Long-Range Fluctuation-Induced Forces in Driven Electrolytes, Phys. Rev. Lett. 126, 158002 (2021).
  * (7) M. Bordag, G. L. Klimchitskaya, U. Mohideen, and V. M. Mostepanenko, Advances in the Casimir Effect (Oxford University Press, New York, 2009).
  * (8) M. Antezza, L.P. Pitaevskii, S. Stringari, V.B. Svetovoy, Casimir-Lifshitz force out of thermal equilibrium and asymptotic nonadditivity, PRL 97, 223203 (2006).
  * (9) M. Antezza, L. P. Pitaevskii, S. Stringari, and V. B. Svetovoy, Casimir-Lifshitz force out of thermal equilibrium, Phys. Rev. A 77, 022901 (2008).
  * (10) M. Krüger, T. Emig, and M. Kardar, Nonequilibrium Electromagnetic Fluctuations: Heat Transfer and Interactions, Phys. Rev. Lett. 106, 210404 (2011).
  * (11) D. S. Dean, V. Démery, V. A. Parsegian, and R. Podgornik, Out-of-equilibrium relaxation of the thermal Casimir effect in a model polarizable material, Phys. Rev. E 85, 031108 (2012).
  * (12) D. S. Dean and R. Podgornik, Relaxation of the thermal Casimir force between net neutral plates containing Brownian charges, Phys. Rev. E 89, 032117 (2014).
  * (13) B.-S. Lu, D. S. Dean and R. Podgornik, Out of equilibrium thermal Casimir effect between Brownian conducting plates, EPL, 112, 20001, (2015).
  * (14) B.W. Ninham and J. Mahanty, Dispersion forces, London Academic Press (1976).
  * (15) R.R. Netz, Static van der Waals interactions in electrolytes, Eur. Phys. J. E 5, 189 (2001).
  * (16) P. L. Garrido, J. L. Lebowitz, C. Maes, and H. Spohn, Long-range correlations for conservative dynamics, Phys. Rev. A 42, 1954 (1990).
  * (17) G. Grinstein, D.-H. Lee, and S. Sachdev, Conservation Laws, Anisotropy, and Self-Organized Criticality in Noisy Non- equilibrium Systems, Phys. Rev. Lett. 64, 1927 (1990).
  * (18) T. R. Kirkpatrick, J. M. Ortiz de Zárate, and J. V. Sengers, Giant Casimir Effect in Fluids in Nonequilibrium Steady States, PRL 110, 235902 (2013).
  * (19) T. R. Kirkpatrick, J. M. Ortiz de Zárate, and J. V. Sengers, Fluctuation-induced pressures in fluids in thermal nonequilibrium steady states, Phys. Rev. E 89, 022145 (2014).
  * (20) See Supplementary Material for more detailed explanation of the calculation.
  * (21) Guangle Du, D.S. Dean, Bing Miao, R. Podgornik, in preparation.
  * (22) M. Krüger, A. Solon, V. Démery, C. M. Rohwer, and D. S. Dean, Stresses in non-equilibrium fluids: Exact formulation and coarse-grained theory, J. Chem. Phys. 148, 084503 (2018).
  * (23) V. Démery and D. S. Dean, The conductivity of strong electrolytes from stochastic density functional theory, J. Stat. Mech. 023106 (2016).
  * (24) D. S. Dean, Langevin equation for the density of a system of interacting Langevin processes, J. Phys. A 29, L613 (1996).
  * (25) B. Jancovici and L. Šamaj, Screening of classical Casimir forces by electrolytes in semi-infinite geometries, J. Stat. Mech. P08006 (2004).
  * (26) P. Attard, R. Kjellander and D. J. Mitchell, Interactions between electroneutral surfaces nearing mobile charges, Chem. Phys. Lett. 139, 219 (1987).
  * (27) P. Attard, D. J. Mitchell, and B. W. Ninham, Beyond Poisson–Boltzmann: Images and correlations in the electric double layer. I. Counterions only, J. Chem. Phys. 88, 4987 (1988).
  * (28) D.S. Dean and R.R. Horgan, Electrostatic fluctuations in soap films, Phys. Rev. E 65, 061603 (2002).
  * (29) I.M. Telles, Y. Levin, A. dos Santos, Reversal of Electroosmotic Flow in Charged Nanopores with Multivalent Electrolyte, Langmuir 38, 3817-3823,(2022).
  * (30) L. Richter, P. J. Żuk, P. Szymczak, J. Paczesny, K. M. Bkak, T. Szymborski, P. Garstecki, H. A. Stone, R. Hołyst, and C.Drummond, Ions in an AC Electric Field: Strong Long- Range Repulsion Between Oppositely Charged Surfaces, Phys. Rev. Lett. 125, 056001 (2020).
  * (31) A. Smith, M. Borkovec, G. Trefalt, Forces between solid surfaces in aqueous electrolyte solutions, Advances in Colloid and Interface Science, 275, 102078 (2020).