# Phase lines in mean-field models with nonuniform external forces

Roni Kroll and Yoav Tsori<EMAIL_ADDRESS>Department of Chemical Engineering,
Ben-Gurion University of the Negev, Beersheba, Israel.

###### Abstract

We look at the influence of external fields on systems described by generic
free energy functional of the order parameter. The external force may have
arbitrary spatial dependence and the coupling to the order parameter may be
nonlinear. The treatment generalizes seemingly disparate works, such as pure
fluids, liquid and polymer mixtures, lipid monolayers, and colloidal
suspensions in electric fields, fluids and nematics in gravity, solutions in
an ultracentrifuge, and liquid mixtures in laser radiation. The phase lines
and thermodynamic behavior are calculated on the mean-field level. We find a
“surface” critical point that can be shifted to higher or lower temperatures
compared to the bulk critical point. Below this point, the transition from a
“gas” phase to a “liquid” phase is first-order, while above it the transition
is second-order. Moreover, the susceptibility may diverge at a finite location
${\bf r}$. Several analytical expressions are given in the limit where a
Landau expansion of the free energy is valid.

##  I Introduction

The influence of external fields, such as electric Landau and Lifshitz (1957)
or magnetic fields Kittel and Kroemer (1998), gravity Moldover  _et al._
(1979), shear Larson (1999), compression and deformation fields Landau  _et
al._ (1986), on given systems have been studied extensively over the decades
both theoretically and experimentally. A microscopic picture of local and
global forces enables one to obtain the dynamical and thermodynamic behavior
on a sufficiently large scale using a coarse-grained order-parameter Brini
_et al._ (2013); Müller-Plathe (2002). While a small field may change the
system’s state slightly, in the thermodynamic context, the interesting cases
occur when the field is large enough to induce a phase transition. In most
studies, the external field is assumed to be (i) constant in space, and (ii)
the coupling between field and order-parameter is bilinear, e.g., a term $mh$
in classical magnetic (Ising) models Huang (2008) ($m$ is magnetization and
$h$ is the field amplitude), or a term $\rho gh$ in gravity ($\rho$ is a
fluid’s density, $g$ the acceleration on Earth, and $h$ a height above an
arbitrary reference) Moldover  _et al._ (1979).

The influence of Earth’s gravity on pure fluids and liquid mixtures has been
studied thoroughly, and gravity was shown to cause small changes in the
behavior close to the critical point Hohenberg and Barmatz (1972); Dickinson
_et al._ (1975); Greer  _et al._ (1975); Leung and Miller (1975); Moldover
_et al._ (1979). Electric fields have been the subject of equally extensive
study. Strain due to electrostriction is well documented in solids and fluids
Zimmerli  _et al._ (1999a, b). Even in the absence of electrostriction, Landau
and Lifshitz showed that when a simple fluid is subject to a uniform electric
field $E$, if the constitutive relation between permittivity and density is
nonlinear, the critical temperature may shift by a small amount, proportional
to $E^{2}~{}$ Landau and Lifshitz (1957). Debye and Kleboth Debye and Kleboth
(1965) and later Orzechowski Orzechowski (1999) measured the field effect in a
mixture of isooctane and nitrobeneze, and found a small negative shift to
$T_{c}$ of about $-15$ mK, with a sign opposite to that expected from the
Landau’s prediction. Beaglehole worked with cyclohexane and aniline and found
a negative shift Beaglehole (1981). Early worked on the same system but found
no shift to $T_{c}$ Early (1992). Wirtz and Fuller measured the effect on a
mixture of n-hexane and nitroethane and also found a comparable small negative
shift Wirtz and Fuller (1993). This increased miscibility was later explained
by the increase of free energy caused by permittivity fluctuations, not
included in Landau’s treatise Amundson  _et al._ (1991); Onuki and Kitamura
(2004); Onuki (1995); Tsori (2009), Orzechowski  _et al._ (2014); Kriisa and
Roth (2014, 2019). The combined effect of both electric and gravitation or
centrifugal fields on colloidal suspensions have been studied as well, leading
to nontrivial density profiles van Roij (2003); Rasa  _et al._ (2005); Royall
_et al._ (2005) Kuijk  _et al._ (2014).

The field affects the system’s state as it couples to its mass or charge
densities, magnetic or electrical dipoles, etc. In equilibrium, the system’s
state is given by an equation expressing energy minimization. In a rigorous
treatment, though, the field obeys a second equation, and changes in the
system affect the field itself. In most models, this back-action on the field,
manifest in the electrostatic case by the existence of Laplace or Poisson’s
equations, is absent, and the field is assumed to be unaffected by the
material on which it acts.

Here we investigate the thermodynamic equilibrium of a system exposed to a
spatially nonuniform external field. The system is described by a generic
phenomenological coarse-grained order-parameter $\phi$. The expression for the
field is quite general: the spatial dependence can be arbitrary and the
coupling to the order-parameter is nonlinear. The current framework thus
unifies previous works on different systems, such as pure fluids Samin and
Tsori (2011); Vink and Archer (2012), liquid and polymer mixtures Tsori  _et
al._ (2004), colloidal suspensions Sullivan  _et al._ (2006), and lipid
monolayers Lee  _et al._ (1994); McConnell  _et al._ (1984) in electric
fields, mixtures in laser radiation Delville  _et al._ (1999); Nishioka  _et
al._ (2000); Tran-Cong-Miyata  _et al._ (2004); Walton and Wynne (2018),
nematic rigid rods in gravity Baulin and Khokhlov (1999), and molecular
solutions in an ultracentrifuge Rossen  _et al._ (1986).

##  II Model

The starting point of our treatment is the following mean-field free-energy
density

$\displaystyle f=f_{b}(\phi,T)+\frac{1}{2}c^{2}(\nabla\phi)^{2}+H(\phi,r),$
(1)

where $f_{b}$ is a bulk energy depending on the order-parameter $\phi$ and on
temperature $T$; it has specific forms for liquid-liquid, liquid-vapour, or
magnetic systems but we leave it here in a general form. We are interested in
looking at the shift to the phase lines at all values of $T$ and especially
close to the bulk critical temperature $T_{c}$. $f_{b}(\phi)$ is assumed to be
convex at all values of $\phi$ when $T>T_{c}$ and concave at a certain
“window” of $\phi$ when $T<T_{c}$. $H(\phi,r)$ is a function that contains the
influence of the external field. The coordinate $r$ is a distance in effective
one-dimensional systems, i.e., in systems with Cartesian, cylindrical, or
spherical symmetries. There is a location where the external force attains its
maximum. For the monotonic forces considered below, this point can be thought
of as being a “surface” even in the absence of a physical surface.

The bulk critical point $(\phi_{c},T_{c})$ is classically obtained as a
solution to the two coupled equations:
$\partial^{2}f_{b}(\phi_{c},T_{c})/\partial\phi^{2}=\partial^{3}f_{b}(\phi_{c},T_{c})/\partial\phi^{3}=0$
for the two variables $\phi_{c}$ and $T_{c}$. A nonuniform force leads to a
nonuniform profile $\phi(r)$. One may then ask: what is the bulk point
$(\phi_{0},T)$ for which the composition and temperature at the “surface” obey
the critical-point conditions?

We focus on external fields that can be written in the form

$\displaystyle H(\phi,r)=h(\phi)q(r).$ (2)

For simplicity in the analysis, we assume that $h(\phi)$ is a monotonously
decreasing or increasing function of $\phi$ and is independent on $T$. In
addition, $q(r)$ decreases in absolute value, e.g., $q(r)=r^{-n}$ with $n>0$.
The dimensionless distance $r$ is in the range $1\leq r\leq\infty$ ($r=1$ is
the “surface”). Without loss of generality $q(r=1)=1$ and $|q(r)|\leq 1$.

##  III Three equations for surface critical point and phase diagram

The equilibrium profile $\phi(r)$ obeys the Euler-Lagrange equation

$\displaystyle\frac{\delta f}{\delta\phi}=\frac{\partial
f_{b}}{\partial\phi}-c^{2}\nabla^{2}\phi+\frac{\partial
h(\phi)}{\partial\phi}q(r)-\mu_{0}=0.$ (3)

At large distances, $q(r\to\infty)=0$, the external force vanishes and we
retrieve a bulk system at composition $\phi_{0}$; hence the bulk chemical
potential is $\mu_{0}=f_{b}^{\prime}(\phi_{0},T)$. In contrast to classical
wetting phenomena Cahn (1977), this equation cannot be integrated because of
the $r$-dependence. It is instructive to consider first cases where $q(r)$ is
characterized by a spatial scale much larger than the typical interfacial
length $\sim c$. In this sharp-kink approximation, the gradient-squared term
in $f$ can be neglected and the free energy is governed by bulk effects. We
denote by $\phi_{s}$ the surface composition, i.e. $\phi_{s}\equiv\phi(r=1)$.
When $\phi_{s}$ satisfies the critical-point conditions it obeys the following
three equations:

$\displaystyle f_{b}^{\prime}(\phi_{s},T)+h^{\prime}(\phi_{s})q(r)-\mu_{0}=0,$
(4) $\displaystyle
f_{b}^{\prime\prime}(\phi_{s},T)+h^{\prime\prime}(\phi_{s})q(r)=0,$ (5)
$\displaystyle
f_{b}^{\prime\prime\prime}(\phi_{s},T)+h^{\prime\prime\prime}(\phi_{s})q(r)=0,$
(6)

We now solve these equations with $q(r)=1$ as follows: We solve Eqs. (4) and
(5) simultaneously: Eq. (5) gives $\phi_{s}$; when substituted in Eq. (4), the
bulk value $\phi_{0}$ for the given temperature can be found from the
definition of $\mu_{0}$. Similarly, we solve Eqs. (4) and (6) simultaneously:
$\phi_{s}$ that solves Eq. (6) can be used in Eq. (4) to give the
corresponding bulk value $\phi_{0}$.

###  III.1 Solutions of Eqs. (4)-(6)

Figure 1: Graphical solution of Eq. (5). When the temperature is small enough,
the intersection of the blue curve ($f_{b}^{\prime\prime}$) with the red cure
($-h^{\prime\prime}$, assumed negative) gives two solutions,
$\phi_{s,a}^{(2)}$ and $\phi_{s,b}^{(2)}$, marked by circles. If the
temperature is too high, $T_{c}-T$ is small, and there is no solution to Eq.
(5). At $T=T^{*}$ (dashed-dot line), $\phi_{s,a}^{(2)}=\phi_{s,b}^{(2)}$ and
there is one solution to the equation. Green curve shows $-h^{\prime\prime}$
when it is positive.

Direct solution of equations (4)-(6) is straightforward; however, to emphasize
the general aspects of the solution and to increase the physical insight, we
present here a graphical construction.

Graphical solution of Eq. (5) and definition of $T^{*}$. Figure 1 shows
$f^{\prime\prime}_{b}(\phi)$ when $T$ is sufficiently small (large $T_{c}-T$,
dashed blue line) and $-h^{\prime\prime}(\phi)$ (red line). The intersection
of the two curves is the solution to Eq. (5). The curves intersect at two
compositions, denoted $\phi_{s,a}^{(2)}$ and $\phi_{s,b}^{(2)}$. We define
$\phi_{s,a}^{(2)}>\phi_{s,b}^{(2)}$. The subscript a or b denotes either
solution; the superscript $2$ signifies that they are solutions of the second
derivative of $f$. Each of the two solutions $\phi_{s,i}^{(2)}$ (i=a,b) has a
different bulk value of $\phi_{0}$, obtained when it is substituted in Eq.
(4). We call these $\phi_{0,a}^{(2)}$ and $\phi_{0,b}^{(2)}$.

A different behavior occurs if the temperature is not low enough
(alternatively, if the strength of the external force $|h^{\prime\prime}|$, is
too large), as indicated by the dashed blue line – in this case the curve do
not intersect, hence there is no solution to Eq. (5). We now define $T^{*}$ as
the temperature at which $\phi_{s,a}^{(2)}$ and $\phi_{s,b}^{(2)}$ are equal,
see dashed-dot line in Fig. 1.

Figure 2: Graphical solution of Eq. (6). The $f_{b}^{\prime\prime\prime}$
curve looks typically like the blue curve. Two curves are shown for
$h^{\prime\prime\prime}$ – for the two cases where $h^{\prime\prime\prime}$ is
either positive or negative. In both cases there is always a solution
(circles). Figure 3: Graphical solution of Eq. (4). Blue curve is
$f_{b}^{\prime}(\phi)+h^{\prime}(\phi)$ and green curve is $\mu_{0}(\phi)$.
The circles mark the solutions of Eq. (5). The values of $\phi_{0,a}^{(2)}$
and $\phi_{0,b}^{(2)}$ can be obtained as the intersection of the horizontal
lines with the $\mu_{0}(\phi)$ curve. Top and bottom panels are for low and
high temperatures, respectively (both lower than $T^{*}$).

Graphical solution of Eq. (6). The graphical solution is shown in Fig. 2. Here
it is easy to see that one solution $\phi_{s}^{(3)}$ always exists. Once this
root is known, it can be substituted in Eq. (4) to yield the corresponding
bulk value $\phi_{0}^{(3)}$.

Figure 4: Phase diagram in the $\phi-T$ plane when $-h^{\prime\prime}$ is
negative (as the red curve in Fig. 1). The bulk critical point (BCP) is at
$(\phi_{c},T_{c})$ (square). The bulk binodal and spinodal curves are in black
and red, respectively. The curves $\phi_{s,a}^{(2)}(T)$,
$\phi_{s,b}^{(2)}(T)$, and $\phi_{s}^{(3)}(T)$ are in solid green, blue and
purple, while $\phi_{0,a}^{(2)}(T)$, $\phi_{0,b}^{(2)}(T)$, and
$\phi_{0}^{(3)}(T)$ are in identical colors and dashed lines.
$\phi_{0}^{(3)}(T)$ is a first-order transition line (dashed purple curve).
The surface critical point (SCP), marked with a circle, is at a temperature
below the critical point ($T^{*}<T_{c}$). The second-order line is the dashed
thick black curve.

Graphical solution of Eq. (4). Figure 3 shows the graphical construction for
finding the value of $\phi_{0,a}^{(2)}$ and $\phi_{0,b}^{(2)}$. The figure has
two parts, for two different temperatures – top panel is for low $T$ and the
bottom is for higher $T$ (both smaller than $T^{*}$). There are two curves in
each part, representing two parts of Eq. (4). The blue curve is
$f_{b}^{\prime}(\phi)+h^{\prime}(\phi)$ and the green curve is
$\mu_{0}(\phi)$. The two circles mark the values of $\phi_{s,a}^{(2)}$ and
$\phi_{s,b}^{(2)}$. The solution to Eq. (4) is obtained as the intersection of
the horizontal line at each of these points with the green curve. At the low
temperature, there are three possible values of $\phi_{0,b}^{(2)}$. The
correct solutions are marked with squares, the two other solutions are marked
with “*”. At the higher temperature (bottom panel), there is only one possible
value of $\phi_{0,b}^{(2)}$.

Phase diagram. The resulting phase diagram is shown in Fig. 4. For
sufficiently low temperatures, two solutions to Eq. (5) exist,
$\phi_{s,a}^{(2)}(T)$ and $\phi_{s,b}^{(2)}(T)$ (green and blue,
respectively), and usually they are below the spinodal curve. As the
temperature is elevated, the two solutions approach each other. At $T=T^{*}$,
they merge. The corresponding bulk composition curves $\phi_{0,a}^{(2)}(T)$
and $\phi_{0,b}^{(2)}(T)$ (dashed green and blue) behave similarly, and merge
at $T^{*}$ at a “surface critical point” (SCP), marked by a large circle. At
this point, the bulk values $\phi_{0}$ and $T$ are such that at the surface,
the second and third derivative of the energy with respect to $\phi$ both
vanish.

The curve $\phi_{0}^{(3)}$ is a first-order phase line; crossing it from low
to high values of $\phi_{0}$ at constant $T$ leads to a discontinuous change
in the profile $\phi(r)$.

Figure 5: The same as in Fig. 4 but this time $-h^{\prime\prime}$ is positive
(green curve in Fig. (1). The surface critical point is displaced upwards and
$T^{*}>T_{c}$.

The SCP point depends on the amplitude of the external forcing. The point
displaces when the amplitude of the external force reduces toward zero. The
locus of such points is indicated by the dashed black curve. This is a line of
second-order phase transition; crossing it leads to a continuous change in the
profile $\phi(r)$. Equivalently, one can hold the amplitude of external force
fixed and look for solutions of Eqs. (4)-(6) with values of $r$ increasing
from $1$ to $\infty$ (see a specific example in Section V.1). Note that on the
second-order line, Eq. (5) is satisfied not on the surface ($r=1$) but at a
finite distance from it, at $r=r_{0}>1$ (see below).

One can now see that the SCP point and the phase transition lines would not
appreciably change even if the $(\nabla\phi)^{2}$ term in the energy were
included. Indeed, if Eq. (4) has a $\phi^{\prime\prime}$ term, just before the
transition, the profiles are smoothly varying and this second derivative is
small. In addition, by definition, the interface separating high and low
values of $\phi$ is an inflection point in the profile so
$\phi^{\prime\prime}=0$.

Figure 6: Plots of the function $g(\phi)$ given in Eq. (7) for several values
of $r$. Top: when the external force is small, the bulk composition $\phi_{0}$
is outside of the stability line (e.g. point “$\Diamond$” in Fig. 4). As $r$
decreases from $\infty$ to $1$ ($r$ values are in the legend), the minimum of
$g(\phi)$, indicated by circles, shifts to larger values. But the minimum
stays close to the bulk value, $\phi_{0}$. The resulting profile $\phi(r)$ in
shown in Fig. 7 (top). Bottom: when the external forcing is large, $\phi_{0}$
is inside the stability line (e.g. point “$\star$” in Fig. 4). As $r$
decreases, the minimum of $g(\phi)$ jumps from small to large values of
$\phi$. See profile in Fig. 7 (bottom).

##  IV Construction of profiles

Figure 7: Composition profiles at a temperature below $T^{*}$ in the sharp
kink approximation. Top and bottom panels correspond to bulk points marked by
“$\Diamond$” and “$\star$”, respectively, in Fig. 4. While at point
“$\Diamond$” $\phi(r)$ is smoothly varying, at “$\star$” it exhibits a
discontinuity. The transition from the top curve to the bottom one is
discontinuous (first-order). At any given value of $r$, there are two values
of $\phi$ that obey $f^{\prime\prime}(\phi,r)=0$ [Eq. (8)]. The dashed lines
show these values vs. $r$. The horizontal dash-dot lines are the bulk spinodal
values at this temperature (i.e. solutions of
$f_{b}^{\prime\prime}(\phi,T)=0$).

To construct the composition profiles, we look at the function $g$ defined by

$\displaystyle g(\phi)\equiv f_{b}(\phi)+h(\phi)q(r)-\mu_{0}\phi-p_{0},$ (7)

where $p_{0}=f_{b}(\phi_{0})-\mu_{0}\phi_{0}$ is the bulk pressure. The
“correct” value of composition at a specific $r$ is the value of $\phi$ that
minimizes $g(\phi)$. Next, we concentrate on two points, marked in Fig. 4: the
star is “inside” the first-order line, and the diamond is outside. The grand-
potential $g(\phi)$ plots for the two points are shown in Fig. 6 for several
values of $r$. The top panel shows the $g(\phi)$ curves when the bulk
composition is outside of the first-order line (diamond). When $r$ is large,
the minimum is at $\phi_{0}$ (dashed vertical line). As $r$ decreases, the
minimum gradually shifts to larger values of $\phi$. Yet, even at the smallest
possible value of $r$ ($r=1$) the minimum stays rather close to $\phi_{0}$.
The behavior is different at the “star” point, found below the first-order
line. The bottom panel shows the $g(\phi)$ curves in this case. Here, as $r$
decreases from large to small values, a discontinuous transition occurs
between low and high values of $\phi$.

Figure 8: Composition profiles at a temperature above $T^{*}$. Top and bottom
panels correspond to bulk points marked by “+” and “*”, respectively, in Fig.
4. While at the point “+”, $\phi(r)$ is smoothly varying, at “*” it exhibits a
discontinuity. But the transition from the top curve to the bottom one is
continuous (second-order). At any given value of $r$, there are two values of
$\phi$ that obey $f^{\prime\prime}(\phi,r)=0$ [Eq. (8)]. The dashed lines show
these values vs. $r$. The two curves merge at $r=r_{0}$. The horizontal dash-
dot lines are the bulk spinodal values at this temperature (i.e. solutions of
$f_{b}^{\prime\prime}(\phi,T)=0$).

The corresponding profiles $\phi(r)$ are shown in Fig. 7.

We now look at two points at a temperature above $T^{*}$, marked by “+” and
“*” in Fig. 4. Across the dashed line in the figure, the transition is second-
order. Indeed, Fig. 8 shows the corresponding $\phi(r)$ curves. When the point
is to the left (“+”), the profile is continuous (though it might have a
relatively steep domain). When the point is to the right (“*”), the profile
becomes discontinuous. The difference between this scenario and the previous
one is that here the discontinuous profile emerges from the continuous one as
a second-order phase transition. The second-order line is shown as a dashed
black curve in Fig. 4.

In Figs. 7 and 8 we show two dashed black lines. They correspond to the two
possible solutions of $f^{\prime\prime}(\phi)=0$, that is, they are solutions
of

$\displaystyle f_{b}^{\prime\prime}(\phi)+h^{\prime\prime}(\phi)q(r)=0$ (8)

for a specific value of $r$. For low temperatures, $T<T^{*}$, the two curves
only meet at some $r$ smaller than unity, which is not valid. For high
temperatures, $T>T^{*}$, they meet at a finite value of $r>1$, denoted
$r_{0}$. When the jump in the profile occurs exactly at $r=r_{0}$, one is on
the second-order phase transition line. In Fig. 8 (bottom) the jump is for
$r>r_{0}$ and therefore in this example we are already past the phase
transition line (point “*”). For a given $T$ and amplitude of external force,
the second-order line can thus be obtained when $\phi_{s}$ is the single
solution to Eq. (8), with $r$ varying from $1$ to $\infty$, and $\phi_{0}$
obtained from Eq. (4). The SCP is retrieved as the special case when
$r_{0}=1$.

The susceptibility $\chi$. There are several ways to define the susceptibility
in spatially-varying fields. We rewrite the sharp-kink profile equation as

$\displaystyle f_{b}^{\prime}(\phi)+h^{\prime}(\phi)q(r)-\mu_{0}=0.$ (9)

and interpret $-h^{\prime}(\phi)q(r)$ as the field. A derivative of the
profile equation with respect to the field then gives

$\displaystyle\chi=\frac{1}{f_{b}^{\prime\prime}(\phi(r))}.$ (10)

Importantly, $\chi$ diverges at a point $r$ if $\phi(r)$ equals one of the
bulk spinodal values.

Figure 9: Susceptibility $\chi$ defined by Eq. (10) and applied to the
profiles $\phi(r)$ from Figs. 7 and 8. $\chi$ stays finite for the profile at
point “$\diamond$” in the phase-diagram Fig. 4 and diverges at the points
“$\star$”, “+”, and “*”.

Figure 9 shows $\chi$ calculated by Eq. (10) for the profiles shown in Figs. 7
and 8. $\chi$ diverges for the profiles $\phi(r)$ that cross one of the two
spinodal values (points “$\star$”, “+”, and “*”) and remains finite otherwise
(point “$\diamond$”).

Note that the susceptibility can tend to $+\infty$ at one side of a point $r$
and to $-\infty$ at the other side. This happens when the curve $\phi(r)$
crosses the bulk spinodal value. In uniform fields this does not happen.

##  V Landau expansion of the free energy

To get concise explicit expressions for the quantities presented in the
previous section, we now cast the above insight onto a standard (symmetric)
Landau expansion of the free energy and external field:

$\displaystyle f_{b}$ $\displaystyle=$ $\displaystyle
f_{c}+\frac{1}{2}t\varphi^{2}+\frac{1}{24}d\varphi^{4},$ $\displaystyle h$
$\displaystyle=$ $\displaystyle
h_{c}+h_{1}\varphi+\frac{1}{2}h_{2}\varphi^{2}+\frac{1}{6}h_{3}\varphi^{3},$
(11)

with $t\sim(T-T_{c})/T_{c}$, and $d$ a positive constant independent of $T$.
Quantities are further expressed in terms of $\varphi=\phi-\phi_{c}$ instead
of $\phi$. The bulk chemical potential is

$\displaystyle\mu_{0}=t\varphi_{0}+\frac{1}{6}d\varphi_{0}^{3}.$ (12)

Equations (4)-(6) can now be written for the surface ($q(r)=1$) as

$\displaystyle
t\varphi_{s}+\frac{1}{6}d\varphi_{s}^{3}+h_{1}+h_{2}\varphi_{s}+\frac{1}{2}h_{3}\varphi_{s}^{2}-\mu_{0}=0,$
(13) $\displaystyle t+\frac{1}{2}d\varphi_{s}^{2}+h_{2}+h_{3}\varphi_{s}=0,$
(14) $\displaystyle d\varphi_{s}+h_{3}=0.$ (15)

The solutions to these equations give the qualitative phase diagram in Fig. 4
if $h^{\prime\prime}=h_{2}+h_{3}\varphi>0$ and
$h^{\prime\prime\prime}=h_{3}>0$ or Fig. 5 if
$h^{\prime\prime}=h_{2}+h_{3}\varphi<0$ and $h_{3}<0$.

To find the SCP point, we use
$\varphi_{0,a}^{(2)}=\varphi_{0,b}^{(2)}=\varphi_{0}^{(3)}$ at the SCP, and at
the SCP temperature, $t^{*}$,
$\varphi_{s,a}^{(2)}=\varphi_{s,b}^{(2)}=\varphi_{s}^{(3)}$, so this
composition can be simply called $\varphi_{s}$. From Eq. (15) we find that

$\displaystyle\varphi_{s}=-h_{3}/d.$ (16)

$\varphi_{s}$ is independent on temperature. Hence the line
$\varphi_{s}^{(3)}$ is entirely to the left of $\phi_{c}$ if $h_{3}>0$ and to
the right otherwise. Substitution in Eq. (14) gives for $t=t^{*}$

$\displaystyle t^{*}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\frac{h_{3}^{2}}{d}-h_{2}.$ (17)

Interestingly, $t^{*}$ depends on the amplitude of the external forcing
(expressed via $h_{2}$ and $h_{3}$) both linearly and quadratically. $T^{*}$
is above the critical point temperature if

$\displaystyle\frac{1}{2}\frac{h_{3}^{2}}{d}-h_{2}>0$ (18)

and below otherwise.

The first-order line $\varphi_{0}^{(3)}(t)$ is obtained from Eq. (13) as the
solution of the third-order polynomial

$\displaystyle
t\varphi_{s}+\frac{1}{6}d\varphi_{s}^{3}+h_{1}+h_{2}\varphi_{s}+\frac{1}{2}h_{3}\varphi_{s}^{2}-t\varphi_{0}-\frac{1}{6}d\varphi_{0}^{3}=0$

with $\varphi_{s}=-h_{3}/d$. The composition of the SCP point,
$\varphi_{0}^{*}$, is a special case with $t=t^{*}$, simplifying to

$\displaystyle
h_{1}-\frac{1}{6}\frac{h_{3}^{3}}{d^{2}}-t^{*}\varphi_{0}^{*}-\frac{1}{6}d\varphi_{0}^{*3}=0.$
(20)

The lines $\varphi_{s,a}^{(2)}(t)$ and $\varphi_{s,b}^{(2)}(t)$ in the phase
diagram are the solutions to the quadratic equation
$f^{\prime\prime}(\varphi,r=1)=0$ [Eq. (14)]. The dashed curves in Fig. 8 are
the solution to $f^{\prime\prime}(\varphi_{s},r)=0$, namely

$\displaystyle
t+\frac{1}{2}d\varphi_{s}^{2}+\left(h_{2}+h_{3}\varphi_{s}\right)q(r)=0.$ (21)

The value of $r_{0}$ is defined such that the two solutions $\varphi_{s}$ are
the same, i.e., when the discriminant vanishes: $\Delta^{2}=0$. This yields a
quadratic equation for $q(r_{0})$:

$\displaystyle\Delta^{2}=h_{3}^{2}q^{2}(r_{0})-2dh_{2}q(r_{0})-2dt=0.$ (22)

(if $r=1$ then $t=t^{*}$). At $t<t^{*}$, there is no solution with $r_{0}>1$
because $|q(r_{0})|<1$, while at $t>t^{*}$ there is one. The value of the
single solution $\varphi_{s}$ is then

$\displaystyle\varphi_{s}=-h_{3}q(r_{0})/d.$ (23)

How to obtain the second-order transition line? This line is obtained by
increasing $r$ from $1$ to $\infty$ as a parameter, with $t$ given from Eq.
(22), $\varphi_{s}$ from Eq. (23) (replace $r_{0}\to r$), and both substituted
in

$\displaystyle
t\varphi_{s}+\frac{1}{6}d\varphi_{s}^{3}+h_{1}+h_{s}\varphi_{s}+\frac{1}{2}h_{3}\varphi_{s}^{2}-t\varphi_{0}-\frac{1}{6}d\varphi_{0}^{3}=0$

to find $\varphi_{0}$.

Figure 10: Non-monotonous profiles $\phi(r)$ that occur when $h_{1}<0$ and
$h_{2}>0$ is sufficiently large. In the limit $r\to\infty$, the free energy
$f(\phi)$ has a shape of a double well, with the global minimum at the “gas”
phase. When $r\to 1$, the external field dominate and $f(\phi)$ is
approximately the parabola $h_{1}\varphi+(1/2)h_{2}\varphi^{2}$, whose minimum
is at $-h_{1}/h_{2}$. Hence the decrease in $\phi$ from its value at the
liquid-gas interface to its value at $r=1$.

Non-monotonous profiles. The profiles $\phi(r)$ usually look like the ones in
Figs. 7 and 8: $\phi(r)$ is a monotonously decreasing function, with
$\phi(r=\infty)=\phi_{0}$. If $\phi_{0}$ is within the unstable area of the
phase diagram, $\phi(r)$ exhibits a sharp drop at the “liquid”-“gas”
interface. If, however, $h_{1}<0$ and $h_{2}>0$ are sufficiently large, close
to the surface $(r\gtrsim 1$) the external field dominates over the bulk free
energy and $\phi(r)$ can increase with $r$ inside the liquid phase. To see
this, assume that $h_{1}$ and $h_{2}$ are very large, then at $r=1$ the free
energy is $f\approx h_{1}\varphi+(1/2)h_{2}\varphi^{2}$ and in this parabola
the linear and quadratic terms are negative and positive, respectively. The
minimum value, $-h_{1}/h_{2}$, can have any positive value and, in particular,
it can be smaller than the (upper) spinodal value. Figure 10 shows such a
profile.

The free-energy picture as $r$ decreases is this: at $r\to\infty$, $f(\phi)$
has a double minima, with the most stable one being the bulk “gas”, at
$\phi_{0}$. As $r$ decreases, the “liquid” minimum lowers until, at the
interface, the two minima are in equal depth. As $r$ further decreases, the
right minimum becomes the global minimum; it continuously shifts with $r$. If
$h_{2}=0$ then it shifts to the right. If $h_{2}$ is sufficiently large, it
shifts to the left, while the $f(\phi)$ curve loses the “gas” minimum and
becomes the parabola mentioned above.

###  V.1 Two examples

We give here two “applications” from nonuniform fields in dielectric fluids.
The first is the wedge capacitor and the second is a charged spherical
particle. In both cases the fluid is purely dielectric and its relative
permittivity depends on composition via an arbitrary constitutive relation
$\varepsilon(\phi)$. In the vicinity of the critical point, one writes it as a
power series around the critical composition to third order in $\varphi$:

$\displaystyle\varepsilon(\varphi)=\varepsilon_{c}+\varepsilon_{1}\varphi+\frac{1}{2}\varepsilon_{2}\varphi^{2}+\frac{1}{6}\varepsilon_{3}\varphi^{3}.$
(25)

Wedge capacitor. The wedge is made up of two semi-infinite planes that have an
opening angle $\beta$ between them Tsori  _et al._ (2004) and the potential
difference between the plates is $V$. The fluid is placed between these
planes. In dimensionless units, $h(\varphi)$ and $q(r)$ are given by

$\displaystyle h(\varphi)$ $\displaystyle=$
$\displaystyle-\frac{1}{2}M\varepsilon(\varphi)~{}~{}~{}~{}~{}~{}~{}~{}q(r)=r^{-2},$
(26) $\displaystyle M$ $\displaystyle=$
$\displaystyle\frac{V^{2}v_{0}\varepsilon_{0}}{\beta^{2}R_{1}^{2}k_{B}T}.$
(27)

Here $v_{0}$ is a molecular volume, $\varepsilon_{0}$ is the vacuum
permittivity, $r$ is scaled by $R_{1}$, the minimal value of $r$, and $k_{B}T$
is the thermal energy. $M$ denotes the amplitude of the external field, which
is proportional to $V^{2}$ because the force acting on the fluid is
dielectrophoretic in nature.

Thus, the general results from the preceding sections are obtained with

$\displaystyle h_{c}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}M\varepsilon_{c}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}h_{1}=-\frac{1}{2}M\varepsilon_{1}$
(28) $\displaystyle h_{2}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}M\varepsilon_{2}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}h_{3}=-\frac{1}{2}M\varepsilon_{3}.$

From Eq. (17) we find $t^{*}$ to be

$\displaystyle
t^{*}=\frac{1}{8}\frac{M^{2}\varepsilon_{3}^{2}}{d}+\frac{1}{2}M\varepsilon_{2},$
(29)

where $d$ is the coefficient of the fourth order term in the Landau expansion
of the bulk energy $f_{b}$. As denoted above, $t^{*}$ depends on the amplitude
of the external field, $M$, both linearly and quadratically.

We can estimate $t^{*}$ by using the values of $\varepsilon(\phi)$ measured in
the experiment by Debye and Kleboth Debye and Kleboth (1965). They found
$\varepsilon_{2}=28.6$ and $\varepsilon_{3}=-116.4$. Since $M$ is typically
much smaller than unity, the $M^{2}$ term in $t^{*}$ can be safely neglected
and one finds that $t^{*}\approx 14M$. If $M=10^{-3}$ and $T_{c}=350^{\circ}$K
we find $T^{*}-T_{c}\approx 5^{\circ}$K.

Charged colloid. The second example is a spherical particle with surface
charge density $\sigma$ and radius $R$ embedded in the fluid. Here

$\displaystyle h(\phi)$ $\displaystyle=$
$\displaystyle\frac{1}{2}M\frac{1}{\varepsilon(\phi)}~{}~{}~{}~{}~{}~{}~{}~{}q(r)=r^{-4},$
(30) $\displaystyle M$ $\displaystyle=$
$\displaystyle\frac{v_{0}\sigma^{2}}{\varepsilon_{0}k_{B}T}.$ (31)

If one assumes a linear constitutive relation
$\varepsilon(\phi)=\varepsilon_{c}+\varepsilon_{1}\varphi$ with
$\varepsilon_{1}\varphi\ll\varepsilon_{c}$

$\displaystyle h(\varphi)$ $\displaystyle\simeq$
$\displaystyle\frac{1}{2}\frac{M}{\varepsilon_{c}}\left[1-\left(\frac{\varepsilon_{1}}{\varepsilon_{c}}\varphi\right)+\left(\frac{\varepsilon_{1}}{\varepsilon_{c}}\varphi\right)^{2}-\left(\frac{\varepsilon_{1}}{\varepsilon_{c}}\varphi\right)^{3}\right],$

and it is easy to make the following identification

$\displaystyle h_{c}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\frac{M}{\varepsilon_{c}}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}h_{1}=-\frac{1}{2}M\frac{\varepsilon_{1}}{\varepsilon_{c}^{2}}$
(33) $\displaystyle h_{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2}M\frac{\varepsilon_{1}^{2}}{\varepsilon_{c}^{3}}~{}~{}~{}~{}~{}~{}~{}h_{3}=-\frac{1}{3}M\frac{\varepsilon_{1}^{3}}{\varepsilon_{c}^{4}}.$

The value of $t^{*}$ is now [Eq. (17)]

$\displaystyle
t^{*}=\frac{1}{18}M^{2}\frac{\varepsilon_{1}^{6}}{d\varepsilon_{c}^{8}}-\frac{1}{2}M\frac{\varepsilon_{1}^{2}}{\varepsilon_{c}^{3}}.$
(34)

For binary mixtures of two nonpolar solvents, $\varepsilon_{1}\sim 1$ and
$\varepsilon_{c}\sim 1$. Neglecting $M^{2}$ compared to $M$, we find that
$t^{*}\sim M$ and is negative.

##  VI Conclusions

We study a generic mean-field model of systems in external fields. The field
may have arbitrary dependence on the order parameter and on the spatial
coordinate. This treatment generalizes numerous works on vastly different
systems. The graphical construction shown in Figs. 1, 2, 3, and 6 provides a
simple presentation of the governing equations Eqs. (4)-(6) and leads to the
phase diagrams in Figs. 4 and 5. Monotonous fields have a point in space where
they are maximal and this point is analogous to a surface in wetting
phenomena. We find that the bulk critical point expands to a line of second-
order phase transition that ends in a “surface critical point”. A line of
first-order transition meets the second-order line at this point. This line
resembles the prewetting line in wetting phenomena and the adsorption diverges
on it (not shown) Cahn (1977); De Gennes (1985); Tarazona and Evans (1983,
1984). If some conditions apply on $h(\phi)$, non-monotonous profiles can form
even when external forcing $\sim q(r)$ is monotonous. It would be interesting
to investigate how the order-parameter evolves dynamically to the equilibrium
profiles found here. We predict an interfacial instability as a result of
competition between the long-range nature of the force and the local diffusive
transport and surface-tension related stress.

Acknowledgement

This work was supported by the Israel Science Foundation (ISF) grant No.
274/19.

## References

  * Landau and Lifshitz (1957) L. D. Landau and E. M. Lifshitz,  _Electrodynamics of continuous media_ (Nauka (Moscow), 1957).
  * Kittel and Kroemer (1998) C. Kittel and H. Kroemer,  _Thermal physics_ (American Association of Physics Teachers, 1998).
  * Moldover  _et al._ (1979) M. R. Moldover, J. V. Sengers, R. V. Gammon, and R. J. Hocken, Rev. Mod. Phys. 51, 79 (1979).
  * Larson (1999) R. G. Larson,  _The structure and rheology of complex fluids_ , Vol. 150 (Oxford university press New York, 1999).
  * Landau  _et al._ (1986) L. D. Landau, E. M. Lifshitz, A. M. Kosevich, and L. P. Pitaevskii,  _Theory of elasticity_ , Vol. 7 (Elsevier, 1986).
  * Brini  _et al._ (2013) E. Brini, E. A. Algaer, P. Ganguly, C. Li, F. Rodriguez-Ropero, and N. F. van der Vegt, Soft Matter 9, 2108 (2013).
  * Müller-Plathe (2002) F. Müller-Plathe, ChemPhysChem 3, 754 (2002).
  * Huang (2008) K. Huang,  _Statistical mechanics_ (John Wiley & Sons, 2008).
  * Hohenberg and Barmatz (1972) P. Hohenberg and M. Barmatz, Phys. Rev. A 6, 289 (1972).
  * Dickinson  _et al._ (1975) E. Dickinson, C. M. Knobler, V. N. Schumaker, and R. L. Scott, Phys. Rev. Lett. 34, 180 (1975).
  * Greer  _et al._ (1975) S. C. Greer, T. E. Block, and C. M. Knobler, Phys. Rev. Lett. 34, 250 (1975).
  * Leung and Miller (1975) H. Leung and B. N. Miller, Phys. Rev. A 12, 2162 (1975).
  * Zimmerli  _et al._ (1999a) G. A. Zimmerli, R. A. Wilkinson, R. A. Ferrell, and M. R. Moldover, Phys. Rev. Lett. 82, 5253 (1999a).
  * Zimmerli  _et al._ (1999b) G. A. Zimmerli, R. A. Wilkinson, R. A. Ferrell, and M. R. Moldover, Phys. Rev. E 59, 5862 (1999b).
  * Debye and Kleboth (1965) P. Debye and K. Kleboth, J. Chem. Phys. 42, 3155 (1965).
  * Orzechowski (1999) K. Orzechowski, Chem. Phys. 240, 275 (1999).
  * Beaglehole (1981) D. Beaglehole, J. Chem. Phys. 74, 5251 (1981).
  * Early (1992) M. D. Early, J. Chem. Phys. 96, 641 (1992).
  * Wirtz and Fuller (1993) D. Wirtz and G. G. Fuller, Phys. Rev. Lett. 71, 2236 (1993).
  * Amundson  _et al._ (1991) K. Amundson, E. Helfand, D. D. Davis, X. Quan, S. S. Patel, and S. D. Smith, Macromolecules 24, 6546 (1991).
  * Onuki and Kitamura (2004) A. Onuki and H. Kitamura, J. Chem. Phys. 121, 3143 (2004).
  * Onuki (1995) A. Onuki, EPL 29, 611 (1995).
  * Tsori (2009) Y. Tsori, Rev. Mod. Phys. 81, 1471 (2009).
  * Orzechowski  _et al._ (2014) K. Orzechowski, M. Adamczyk, A. Wolny, and Y. Tsori, J. Phys. Chem. B 118, 7187 (2014).
  * Kriisa and Roth (2014) A. Kriisa and C. B. Roth, J. Chem. Phys. 141, 134908 (2014).
  * Kriisa and Roth (2019) A. Kriisa and C. B. Roth, ACS Macro Lett. 8, 188 (2019).
  * van Roij (2003) R. van Roij, J. Phys.: Condens. Matter 15, S3569 (2003).
  * Rasa  _et al._ (2005) M. Rasa, B. H. Erné, B. Zoetekouw, R. van Roij, and A. P. Philipse, J. Phys.: Condens. Matter 17, 2293 (2005).
  * Royall  _et al._ (2005) C. Royall, R. van Roij, and A. van Blaaderen, J. Phys.: Condens. Matter 17, 2315 (2005).
  * Kuijk  _et al._ (2014) A. Kuijk, T. Troppenz, L. Filion, A. Imhof, R. Van Roij, M. Dijkstra, and A. Van Blaaderen, Soft Matter 10, 6249 (2014).
  * Samin and Tsori (2011) S. Samin and Y. Tsori, J. Phys. Chem. B 115, 75 (2011).
  * Vink and Archer (2012) R. L. Vink and A. J. Archer, Phys. Rev. E 85, 031505 (2012).
  * Tsori  _et al._ (2004) Y. Tsori, F. Tournilhac, and L. Leibler, Nature (London) 430, 544 (2004).
  * Sullivan  _et al._ (2006) M. Sullivan, K. Zhao, A. Hollingsworth, R. H. Austin, W. Russel, and P. Chaikin, Phys. Rev. Lett. 96, 015703 (2006).
  * Lee  _et al._ (1994) K. Y. C. Lee, J. F. Klingler, and H. M. McConnell, Science 263, 655 (1994).
  * McConnell  _et al._ (1984) H. M. McConnell, L. K. Tamm, and R. M. Weis, Proc. Natl. Acad. Sci. U.S.A. 81, 3249 (1984).
  * Delville  _et al._ (1999) J. P. Delville, C. Lalaude, and A. Ducasse, Physica A 262, 40 (1999).
  * Nishioka  _et al._ (2000) H. Nishioka, K. Kida, O. Yano, and Q. Tran-Cong, Macromolecules 33, 4301 (2000).
  * Tran-Cong-Miyata  _et al._ (2004) Q. Tran-Cong-Miyata, S. Nishigami, T. Ito, S. Komatsu, and T. Norisuye, Nature Mater. 3, 448 (2004).
  * Walton and Wynne (2018) F. Walton and K. Wynne, Nature chemistry 10, 506 (2018).
  * Baulin and Khokhlov (1999) V. A. Baulin and A. R. Khokhlov, Phys. Rev. E 60, 2973 (1999).
  * Rossen  _et al._ (1986) W. R. Rossen, H. T. Davis, and L. E. Scriven, J. Colloid. Interf. Science 113, 248 (1986).
  * Cahn (1977) J. W. Cahn, J. Chem. Phys. 66, 3667 (1977).
  * De Gennes (1985) P.-G. De Gennes, Rev. Mod. Phys. 57, 827 (1985).
  * Tarazona and Evans (1983) P. Tarazona and R. Evans, Mol. Phys. 48, 799 (1983).
  * Tarazona and Evans (1984) P. Tarazona and R. Evans, Mol. Phys. 52, 847 (1984).