# Mixing-Demixing Phase Diagram for Simple Liquids in Non-Uniform Electric Fields

## Abstract

We deduce the mixing-demixing phase diagram for binary liquid mixtures in an electric
field for various electrode geometries and arbitrary constitutive relation for
the dielectric constant. By focusing on the behavior of the
liquid-liquid interface, we produce simple analytic expressions for the dependence of the
interface location on experimental parameters. We also show that the phase diagram
contains regions where liquid separation cannot occur under *any* applied field. The
analytic expression for the boundary “electrostatic binodal” line reveals that the
regions’ size and shape depend strongly on the dielectric relation between the liquids.
Moreover, we predict the existence of an “electrostatic spinodal” line that identifies
conditions where the liquids are in a metastable state. We finally construct the phase
diagram for closed systems by mapping solutions onto those of an open system via an
effective liquid composition. For closed systems at a fixed temperature and mixture
composition, liquid separation occurs in a finite “window” of surface potential (or
charge density). Larger potentials or charge densities counterintuitively destroy the
interface, leading to liquid mixing. These results give valuable guides for experiments by
providing easily testable predictions for how liquids behave in non-uniform electric
fields.

Phase transitions describe fundamental transformations in substances, where material properties, such as viscosity, refractive index, etc., often dramatically change. These changes are not only mediated by intrinsic thermodynamic variables (temperature, pressure, etc.), but also by external forces (gravitational (1), magnetic (2) and electric (3) fields, and shear flows (4); (5)). Scientific interest in using electric fields to alter the phase behavior spans over half a century and resulted in theories and experiments devoted to the application of uniform fields in dielectric liquid mixtures (3); (6); (7); (8); (9); (10); (11); (12); (13). Unfortunately, the liquid-field coupling in uniform electric fields is very weak since in such cases variations in the field strength occur as a result of variations in the permittivity of the liquid. As a consequence, theories predict that even minuscule changes to the phase diagram require enormous applied voltages (3); (6); (9).

In contrast, recent theoretical and experimental results reveal that nonuniform fields can effectuate large changes in phase diagrams (14); (15); (16). The externally produced spatial variations in field strength occur even in homogeneous materials, and lead to liquid rearrangement that can potentially induce liquid-liquid separation. High-gradient fields readily emerge from a modest potential or surface charge on misaligned plate capacitors as well as from small objects with high surface curvature, like nanowires and colloids (14); (15); (16). Thus the relative ease for creating nonuniform fields underscores the potential to profoundly influence the behavior of complex liquids.

The challenge of non-uniform fields, however, resides in distinguishing true liquid-liquid phase separation from mere concentration gradients. In the more common case of uniform fields, the free energy has a double-well form with two coexisting minima. Since the system possesses translational invariance, the two liquids can replace each other in space without changing the total energy. This does not hold for nonuniform fields where translational invariance is broken. Here, the spatial location of the liquids directly ties to the free energy, and as a consequence, the total free energy can have a single minimum even with two-phase coexistence. We point out that not all spatially nonuniform fields display this property, as for example in the case of random-field (17) and periodic-field (18) Ising models.

To overcome the difficulty in determining a transition, we defined phase separation by
observing a *local* property—the behavior of the interface.
Using this perspective, we derived analytic expressions for predicting the location of the
interface from experimental parameters.
We additionally adapted the standard methods used in creating phase diagrams and found the
electrostatic-equivalent of binodal and spinodal lines as well as critical points.
The methods presented here can, in principle, apply to any geometry, and we
explicitly give results for three basic electrode shapes: wedge, cylinder, and sphere.
Furthermore, these methods can incorporate an arbitrarily complicated dielectric relation
for the liquid composition, provided that derivatives to the expression exist.

The manuscript is arranged as follows. We describe the theory for liquid mixtures with electric fields in Sec. I and briefly review general properties of phase diagrams in the absence of external fields in Sec. II. In Sec. III, we introduce a useful definition of phase separation in an electric field that is essential for simplifying theoretical expressions. In Sec. IV, we assume phase separation exists and derive simple expressions for the location of the liquid-liquid interface. The mixing-demixing regions of the phase diagram as well as the dividing “electrostatic binodal” line are discussed in Sec. V, while the theoretical stable-metastable states and dividing “electrostatic spinodal” line are presented in Sec. VI. Finally, we discuss important differences between open and closed systems in Sec. VII.

## I Theory

Using a mean-field approach, we consider a binary mixture of two liquids, and , in an electric field , and write the total free energy for a volume as

(1) |

where , and are the free energy densities for mixing and electrostatics, respectively.

The liquids, in the absence of an electric field, can mix or demix due to a competition between entropy and enthalpy, where temperature adjusts the relative balance. For concreteness, we use the following Landau free energy of mixing where the expansion is performed around the critical volume fraction

(2) |

where is Boltzmann’s constant, such that is the volume fraction of component , and is the Flory interaction parameter (19). Without loss of generality, we set , and , where is the critical temperature. Simple liquids have , while polymers are composed of monomers with volume . Here, we consider the symmetric simple liquid . Real interfaces consist of a gradual change in composition. In contrast, generates an interface marked by a discontinuity in composition. We will find, however, that the discontinuity greatly simplifies the analysis to follow.

For electrostatics, the free energy is given by

(3) |

where is the vacuum permittivity, and is the electrostatic potential (). The positive (negative) sign corresponds to constant charge (potential) boundary conditions.

The dielectric permittivity at zero frequency depends on the relative liquid-liquid composition. For clarity in the discussions, we mainly consider a linear relation, , where and are the dielectric constants for pure liquids and , respectively. Excluding the possibility of critical behavior in in the immediate vicinity of the liquid’s critical point (20); (21), the measured often approximates a quadratic function for various liquid combinations (9); (10). We, therefore, highlight some significant changes in the results that occur with higher order relations.

To determine the equilibrium state in the presence of a field, we minimize with respect to and using calculus of variations and obtain the following Euler-Lagrange equations

(4) | |||||

(5) |

where the “prime” represents the derivative with respect to . The first equation is Laplace’s equation for the potential , while the second equation gives the composition distribution . Both and couple the two equations.

The Lagrange multiplier in eq. 5 differentiates between open and closed systems. For a closed system (canonical ensemble), is adjusted to satisfy the mass conservation constraint: , where is the average composition. When the system under consideration is coupled to an infinite reservoir at composition , is the chemical potential of the reservoir.

We conduct detailed investigations of the phase transition with three simple yet fundamental shapes—cylinder, sphere, and wedge. A closed system with cylindrical geometry consists of two concentric cylinders with radii and , where produces an open system, Fig. a. We impose cylindrical symmetry such that and , where is the distance from the inner cylinder’s center. Furthermore, the prescribed charge density on the inner cylinder allows integration of Gauss’s law to obtain an explicit expression for the electric field. By using a similar construction for spherical geometry we find that the electric field for both cylindrical and spherical configurations is , where and for cylinders and spheres, respectively. Combining this result with in eq. 5, we obtain a single equation determining the composition profile :

(6) |

The wedge geometry consists of two “misaligned” flat plates with an opening angle , Fig. b. Using a constant potential boundary condition, we obtain an electric field , where is the potential difference across the electrodes, is the distance from the imaginary intersection of the two plates, and is the azimuthal angle. Combining this result with eq. 5, we obtain

(7) |

In this manuscript we mainly present results for cylindrical geometry. This geometry intrinsically presents a mathematically unsavory dependence of on (via ), and therefore creates more complicated solutions than, for example, in the wedge. Also, the difference in equational form between the cylinder and sphere does not present new information for discussion. The methods presented here can easily be adapted to both wedge and sphere geometries.

As will become evident, the precise surface charge density (surface potential) necessary to induce a transition depends on experimental parameters like the size and relative concentration of the liquid molecules, size of the charged material, temperature, etc. We consider a wide range of surface charges , from approximately zero up to (equivalent to ). For comparison, colloidal particles immersed in the non-polar phase of an inverse-micelle liquid have been measured to have large surface potentials, with an estimate of to charges (22). This amount of charge on a colloid could induce phase separation in a binary mixture if its composition is close enough to the demixing curve. The demixed liquid layer surrounding the colloid is predicted to be several tens to hundreds of nanometers thick, thereby altering the local environment of the colloid in an otherwise mixed liquid suspension. Of course, having the ability to externally apply a field, for example via an electrode, can be useful in some applications.

## Ii Phase Diagram without an Electric Field

We briefly discuss some features of the mixing-demixing phase diagram in the absence of electric fields that are essential in the derivations below. A “double well” function (for example eq. 2 when ) possesses two local minima, one local maximum, and two inflection points located between the maximum and each minimum.

To ascertain the minimum of at constant , we find the solution to , Fig. c, that also satisfies , Fig. d, where the derivatives are taken with respect to . The two solutions for each create the binodal curve, thick solid line in Fig. a. Fluids demix if the initial conditions are under the binodal curve, and mix if they are above this curve. There in fact exists a third solution to , Fig. c—the local maximum at concentration , dashed line in Fig. a. Even though this solution is physically unstable [, Fig. d], it will be useful in subsequent sections. If the local minima satisfy and the local maximum satisfies , then there must exist inflection points between the extrema that satisfy , Fig. d. These solutions for each create the spinodal line, dash-dotted line in Fig. a, and describe liquid behavior dynamically. If the initial point is located below the spinodal curve, then the liquids demix spontaneously. If, however, exists between and , then the liquid can be “stuck” in a local minimum, resulting in a metastable mixed state.

At the critical point the shape of changes from having a single to double minima. As increases to , the two minima , the two inflection points , and maximum converge and convert into a single minimum . To meet these requirements the critical point must satisfy and . Figures c and d display two of the four requirements. Finally, the light solid line in Fig. a shows the single solution to above the critical point.

## Iii Defining Phase Separation

Nonuniform electric fields impose a nonuniform “pull” on the liquid mixture, manifesting as an -dependent total free energy density . The behavior of can be conceptualized as a competition between mixing and electrostatic energies. As , the electric field is weak, , and governs liquid behavior. The solid line in Fig. a shows a typical example of at a large value of using , , and in an open cylinder system. The minimum of , marked by a symbol, gives the value of as , which in this case is . At the other distance extreme, , the electric field is the strongest, and the dashed line in Fig. a shows the resulting . Note the dramatic difference in the value of when the value of is small () versus large.

By finding the minimum of for all values of , it is possible to construct the full
concentration profile , where the solid line in Fig. b corresponds
to the data from Fig. a. Whether or not a phase transition occurs in the equilibrium solution resides in how
the minimized changes as varies between the two distance extremes.
Specifically, if there exists an where and
contains *two* minima [see dash-dotted line in Fig. a], then is
an interface between the two liquids. Figure b illustrates how the two
minima in translate into a discontinuity at , thereby creating a
distinct boundary between the two phases.

A closer inspection of at reveals important mathematical features similar to those in discussed in the previous section. The similarity is not surprising, since is a component of . The dash-dotted lines in Figs. a, c, and d show that possesses two local minima we call and , one local maximum , and two inflection points we call and . In addition, can have critical behavior. We will demonstrate that all these features at behave analogously to those in Fig. a and show how to use this information to construct the mixing-demixing phase diagram with an electric field.

## Iv Composition Profiles and Location of the Interface

Not all applied fields induce liquid demixing, and based on our definition of a phase
transition, there are two possible causes. First, exists in “virtual” (
or ) rather than “real” space, dash-dotted line in Fig. b.
Second, contains a single minimum for *all* , including , dashed line in
Fig. b.

We begin with the first cause. For a constant and , Fig. b shows that certain values of induce a transition, whereas others do not. In fact, there exists a transition that marks the lowest necessary for liquid-liquid separation. Figure a also shows how increasing moves the interface to larger , using an open cylinder system as an example. Noting that mathematical solutions exist for all (including those distances in non-physical space), the vertical dashed line in Fig. a at marks the surface of the cylinder. To the right of this line is real (physical) space, while to the left is the virtual space inside the electrode (or not between the plates as defined in Fig. b for wedge geometries). This observation inspires an alternative definition: the surface charge density is the that places exactly at . We stress that profiles at constant and in open systems with varying values of all collapse to a single curve when plotted versus a scaled distance , Fig. b.

Varying (holding and constant) and (holding and
constant) reveals the second cause for no phase separation, illustrated in
Fig. c and d, respectively. In these cases,
both the interface location *and* the size of the discontinuity change.
Importantly, the discontinuity can even vanish, as the high and low concentrations
and at the interface merge to the same value at certain
or . Notice the remarkable similarity between the behavior of the discontinuity at
, dash-dotted lines in Figs. c and d, and
the binodal curve, Fig. a.

The location of the interface, once it exists, is controlled by , , and . In general, increases with increasing [Fig. a and a], increasing [Fig. c and b], decreasing [Fig. d and c], and increasing [discussed in Sect. VII, Fig. b).

Besides solving the full profile, a quicker method for determining the location of the interface consists of solving these three equations (16); (23)

(8) |

for three unknowns: and the high and low concentrations and , respectively, at . The plus (minus) sign in the third equation is for constant charge (potential) boundary conditions. The first two equations find extrema points and are simply eq. 6 or 7, depending on system geometry. The third equation ensures that the free energy for the high concentration is as favorable as the low concentration .

An even simpler method for finding consists in recalling that there exists a third solution to —the local maxima , Fig. c. For cylindrical () and spherical () geometries, the explicit equation for is

(9) | |||||

If is known, then can, in principle, be deduced from experimental parameters (, , etc.). For now, we will borrow ideas from the binodal curve and make the assumption , but will see later that indeed approximately equals under many conditions. Rearranging eq. 9, we now have the useful relation

(10) |

In open systems, this equation is further simplified by substituting . The lines in Figs. a, b, and c use eq. 10 to solve in an open cylinder system, and reveal an excellent agreement to the solutions from eqs. 8 (symbols). Figure d combines all data from Figs. a, b, and c, revealing that the agreement spans many orders of magnitude.

The analogous equation for finding in a wedge geometry is

(11) |

## V Stability Diagram and Electrostatic Binodal

If an electric field can cause phase separation in a region of space
*above* the binodal curve, a natural question arises: what is the new stability
diagram for a particular value of surface charge density ?
This can be constructed by holding constant
and probing space for liquid-liquid demixing. Since the electric field breaks
the symmetry of the free energy with respect to composition (), the
stability diagram is asymmetric with respect to .
Figure a compares a typical stability curve for an open cylindrical
system, solid line, to the binodal curve, dashed line. Clearly, nonuniform fields can
produce large changes to the phase diagram.

Figure b shows the superposition of stability diagrams from a wide
range of in an open cylindrical system, where the color indicates the transition
for each point . (Points beneath the binodal curve are omitted since
phase separation occurs there without an electric field.) Figure b
clearly illustrates two distinct regions in the plane. In the “demixed”
region, there exists a for each (,) such that any
results in liquid demixing. In the “mixed” region, there does not
exist *any* that results in liquid demixing. Notice how the mixed region
extends well below , indicating that simply setting is not sufficient for
producing a phase transition with an electric field. We will call the curve that divides
these two regions the “electrostatic binodal”.

To derive the electrostatic binodal, we draw inspiration from the “regular” binodal curve. The convergence of the interface concentrations and in Figs. c and d suggest the existence of a critical point at . If there exists a critical point at , then the two minima and , the local maximum , and the two inflection points and converge to a single point , resulting in and . We call the coordinates in the plane that produce a critical point at the critical and critical .

We now show one method for finding and , using an open cylinder system as an example and begin with :

(12) |

The derivation of eq. 10 depends on finding that satisfies , but does not specify as a local maximum or minimum. In particular, also satisfies eq. 10. We therefore substitute eq. 10 for into eq. 12, use for an open system, and rearrange to obtain

(13) | |||||

Notice that as , we recover the solution to , and when equals the critical composition .

Proceeding, must also satisfy at . Figure c shows the solutions to for a wide range of , where the curves from left to right are low to high . The values of (solid lines), (symbols), and (dashed lines) form a continuous variation with , Fig. c, analogous to , , and with , Fig. a. Since , Fig. c, we can simplify eq. 13 to obtain the expression for the electrostatic binodal in an open cylinder system

(14) |

Interestingly, this equation only depends on and the functional form of , but is independent of and . This finding is a consequence of the self-similarity of solutions in open systems for a constant and , described in Sect. IV and shown in Fig. b. Moreover, the geometry difference between cylinders and spheres does not influence the electrostatic binodal. Equation 14 is, in fact, the same equation for the electrostatic binodal in an open sphere system, using the same assumptions.

The thick solid line in Fig. b shows the results from eq. 14, as it accurately divides the plane into mixed and demixed regions. With each point , there is an associated critical : the that places exactly at . It’s important to recognize that is not constant along the electrostatic binodal— is at and increases as , Fig. d, and/or decrease. Figure d compares from calculations (symbols) versus derived from eq. 10 using , , and (line). Equation 14 predicts that the electrostatic binodal also exists for , thin solid line in Fig. b; however, the associated values of are imaginary and not possible in real physical systems.

The electrostatic binodal is a line of critical points, or simply a “critical line”.
This finding explains some curious observations found previously (16): If
and/or is changed such that the stability diagram for a constant is
crossed on the boundary between the kink and , for example the arrow in
Fig. a, then emerges at some distance *greater* than
, Fig. d. The kink marks . The
boundary of the stability diagram to the right of the kink is exactly the electrostatic
binodal. On this boundary, is now *larger* than . In other words,
is no longer the minimum surface charge that induces the transition; therefore,
necessarily emerges at some distance greater than .

The open wedge system produces analogous results; however, we will use the simplicity of the equations in this geometry to demonstrate the effects of quadratic relations, Fig. e. Following the same reasoning as for an open cylinder system, we find the electrostatic binodal for an open wedge

(15) | |||||

We add that , , and exactly equal if and higher derivatives vanish. Notice the similarity between eqs. 14 and 15, where the main difference is that higher derivatives of control the electrostatic binodal in the wedge geometry. Figure f shows how the electrostatic binodal for the wedge curves downwards to upwards as changes from negative to positive. And if , then for the electrostatic binodal simply equals for all . By comparing Fig. e to the results in Fig. f, it is evident that small amounts of curvature in can create large changes in the electrostatic binodal, in agreement with previous findings (16).

We briefly discuss an alternate derivation presented in Ref. (16) to emphasize that we have not exhausted all possible relations between parameters. Beginning with for and in the wedge geometry, we obtain

(16) |

Interestingly, using the Flory-Huggins approximation for results in exactly the same relation, eq. 16, as the Landau approximation. The differences between the two approximations instead arise when determining , where the biggest discrepancies occur, as expected, for values of that are far from , Fig. f.

## Vi Electrostatic Spinodal

We now turn the discussion to possible metastable states, recalling the meaning of the spinodal curve in the mean-field theory (24). Earlier in the manuscript, we rationalized the existence of inflection points at through the presence of a maximum and minima , . Both high and low values , satisfy and exist at all interfaces. Figure a, for example, explicitly shows the mathematical features [, (solid line), , (dash-dotted line), (dashed line), and critical point] occurring at with changing in an open cylinder system. For comparison, the dotted lines display the behavior of the binodal points with .

Despite the ubiquitous presence of and , only carries physical meaning in open systems, and only in a limited region of the stability diagram. To see how this occurs, we return to the solutions of . Thus far, we focused on , the location of the interface for the minimized ; however, there can be many that posses the same mathematical features. Figure b shows all possible solutions to , where the thin solid, dash-dotted, and dotted lines are the “lower”, “upper”, and “unstable” solutions, respectively. The heavy solid line depicts the solution that actually minimizes , and the two dashed lines denote and found at .

We start from a homogeneous mixture at composition and perform the thought experiment of turning on an electric field. Considering diffusive liquid movement in the absence of other factors (ex. liquid convection, noise), this experimental setup implies that the profile initially develops along the free energy “well” created by the lower solution. If the electric field can sufficiently “pull” the higher dielectric material such that there is at least one distance where , then the liquid can escape the metastable (mixed) state at the local free energy minimum to find the global minimum (demixed). We call the distance where and find by solving at . For a cylindrical geometry we have

(17) |

Knowing that the highest value of occurs closest to the electrode at , we seek the conditions where . These conditions, therefore, mark the electrostatic spinodal: If at a particular , then demixing occurs spontaneously. If at a particular , for example Fig. b, then the liquids can be metastabaly mixed. The long time solution for dynamics in these cases therefore resides along the thin solid curve, Fig. b.

Figure a shows the location of the electrostatic spinodal for a particular value of . The curve begins at the critical point and travels down, on the right side of the stability diagram boundary. Similar to the “regular” spinodal curve, demixing occurs spontaneously (non-spontaneously) for to the right (left) of the electrostatic spinodal. Since the electrostatic spinodal cuts inside the stability diagram, the location of the interface emerges at distances greater than , with only at . Figure c displays the behavior of along the spinodal in Fig. a. Finally, the electrostatic spinodal exists for all . Figure d shows the superposition of the electrostatic spinodal curves, where the color indicates the associated .

## Vii Closed Systems

Up until now, we focused on liquid behavior in open systems, where we considered the location of the second boundary as . A closed system with a finite markedly alters the phase diagram (16); however, we will show that these alterations naturally arise from the solutions of open systems.

We begin as previously, with variations in the concentration profiles , and
identify interesting changes with two parameters, and . Both
Figs. a and b clearly reveal
that the discontinuity at the interface decreases and vanishes with increasing
and decreasing , respectively, in closed cylinder systems. Intriguingly, the profiles
in Fig. a stand in sharp contrast to the self-similar
solutions found in open systems, Figs. a
and b. Closer inspection of
Fig. a also reveals that the parabolic-like shape in the
discontinuity with various opens to the left, rather than to the right as in
Figs. c, d,
and b. An important consequence is that for closed systems
there are *two* transition surface charge densities
: the first is the that places
exactly at , while the second is the where the interface
discontinuity vanishes.
Therefore, the interface between the liquids in closed systems only exists when
satisfies , shaded region in Fig. c.

Material conservation drives all differences between closed and open systems, thus, the key to understanding these differences resides in understanding . Recall that in open systems, while is adjusted to account for material conservation in closed systems. Mathematically, the adjusted for a closed system at exactly matches the for an open system with a different “effective” concentration in the bath. Consequently, the profile between and at in a closed system exactly matches the profile at in an open system. In other words, the behavior of a closed system maps onto that of an open system via .

We can explain the variation of with in closed systems using this
construct. Intuitively, the higher dielectric material is pulled closer to the electrode
as the value of increases. In order to conserve material in a closed system,
necessarily decreases near , Fig. a. This
shift in liquid concentration translates as a decrease in , hence increasing
in a closed system maps as increasing *and* decreasing in
an open system. Recall that the interface discontinuity becomes smaller with lower
in an open cylinder system, Fig c, and eventually vanishes
when crosses the electrostatic binodal. The same principles apply to closed
systems, where the second transition marks this crossing.

Now that we understand how experimental parameters change , we focus on how these
changes affect the stability diagram. Figure a shows a
typical stability diagram for a constant and
in a closed cylinder system. One striking difference between open,
Fig. a, and closed, Fig. a, systems
is that liquid separation can now occur for . Experimentally, this
manifests as an interface emerging close to , rather than . A second more subtle
difference is that the stability diagram for closed systems occupies a slightly smaller
region of space for compared to open systems with the same
. Finally, the upper boundary for the closed system stability curve travels
*below* to *exclude* a portion of the binodal curve. Closed systems,
therefore, provide the interesting possibility of an electric field mixing
liquids that normally demix.

We can use the mapping construct to not only comprehend these changes but also to produce the stability diagram of closed system. Open systems link to closed systems via integration. Specifically, integrating between and in an open system at gives the corresponding for the closed system. We begin with the left boundary of the stability diagram for an open system, label in Fig. a, and integrate between and to determine the location of the left boundary in a closed system, label in Fig. a. The difference between and along this boundary is small. If we look at an example profile, Fig. b, we see that the interface location equals and that the electric field for produces only small variations in . Truncating the integration at , therefore, only minimally alters the liquid concentration.

Next, we consider the upper boundary of the open system stability diagram, label in Fig. a, and integrate from to to obtain the upper boundary for the closed system stability diagram, label in Fig. a. Here, large differences between and can occur. This boundary for open systems is the electrostatic binodal. As previously described in Sect. V, , which causes the location of the interface to emerge at distances greater than . The inclusion of high dielectric material from to can substantially increase when integration stops at .

The upper boundary of the stability diagram for a closed system ends when . And to form the right boundary in a closed system, we must find the conditions where places at in an open system. There are two methods by which to proceed. First, we present the simple straightforward approach. We use eq. 10 with , , and various to determine the appropriate , label in Fig. a, and then integrate profiles from to to create the right boundary for the closed system, label in Fig. a. The second method relies on the self-similarity of the solutions in open systems. We recognize that the line labeled in Fig. a is the stability line (where ) for a rescaled surface charge, namely for cylindrical geometry. The ability to shift the interface and rescale the solution with a modified will prove useful in creating the closed system electrostatic binodal.

The superposition of the stability diagrams from many produces
Figs. c and d, where color
indicates and , respectively, for .
These figures reveal striking asymmetry with respect to in the values of both
and . Notably, higher are necessary to both create
() and eventually destroy () the interface when .
The outer bounding line in
Figs. c and d represents the
electrostatic binodal for a closed system. This line is also asymmetric with respect to
. And due to the structure of the stability diagram in closed systems,
is both *and* for all , see Fig. c.

In order to find this electrostatic binodal, we follow the same methods we used for finding the stability diagram of the closed system. We begin with the open system solutions at , and integrate between and to determine (the for the closed system). Notice that this procedure accounts for interfaces emerging at ; however, closed systems can also have interfaces emerging from . Therefore, we rescale the open system solutions by increasing so that [precisely for cylindrical geometry], and integrate . This rescaling links and , as evident in Fig. e. Consequently, phase separation for concentrations greater than technically exist for open systems, and requires infinitely large to produce an interface at . Practically speaking however, even closed systems with a “large enough” would need unreasonably high values of to induce a transition in this region of space. Under these conditions, other events, such as heating, liquid ionization, bubble formation, and electrical breakdown of the liquids would need to be considered (25); (26); (27).

Figure f shows how the electrostatic binodal changes with , where the curve surrounds a smaller region of space as decreases. This change, however, is relatively minor, unless becomes sufficiently “small”.

Material conservation produces two spinodal lines in a closed system—one line
associated with each boundary. Finding the spinodal line associated with consists of
finding on the lower solution of and ensuring , similar
to open systems. However, the lower solution from in an open system does not
fulfill the material conservation requirement. Instead, the lower solution from yet
another open system concentration must be used.
Figure a shows example profiles associated
with . The heavy line corresponds to the profile that satisfies the free
energy minimum of , the dotted line is the lower solution for the open system, and the
thin line is the lower solution for the closed system with . In
Fig. a, the open system could be in a metastable state
(compare thick solid and dotted lines), while the closed system would not be metastable
(compare thick and thin solid lines). Similar behavior applies for the location of the
spinodal line at ; however, this line consists of finding on
the *upper* solution of . The line styles in
Fig. b are as those in
Fig. a. In Fig. b, the
closed system could be metastable, while the open system would not be metastable (recall
that the upper solutions have no meaning in open systems).

Finally, Fig c shows the location of the electrostatic spinodal lines in a closed system for particular values of and . Each line begins at the critical points on either side of and travel down “inside” the stability diagram.

## Viii Conclusion

In summary, we describe the mixing-demixing phase diagram for two dielectric liquids in an electric field. By focusing on the liquid-liquid interface and adapting standard methods for determining phase diagrams, we found the electrostatic-equivalent of binodal lines, spinodal lines, and critical points. Given this new perspective, the dynamics of phase separation with non-uniform electric fields requires reinvestigation, with an emphases on validating predicted metastable states and uncovering critical dynamic behavior. Perhaps similar adaptations of existing theory for dynamics will uncover new features in the electric-field modified liquid-liquid phase diagram.

In addition, we restricted our analysis to solutions with radial symmetry, enforcing one dimensional solutions that only depend on the distance . This constraint, however, might not satisfactorily apply to all experimental conditions, and allowing for full two- or three-dimensional theoretical investigations could uncover non-radially symmetric solutions. For example, interfacial energies, both liquid-liquid and liquid-surface energies, dominate the liquid patterning for phase separation beneath the regular binodal curve in the absence of a field. And in the case where both liquids have an equal preference for the surface, liquid-liquid interfaces emerge normal to a surface. This configuration, however, can be electrostatically unfavorable since the low dielectric material is adjacent to the charge. It will be interesting to determine if, when, and how instabilities in the interface develop and if these instabilities modify the phase diagram.

Also, highly confined cylindrical geometries do not show a true liquid-liquid phase transition (28). Here, the system can be approximated as one dimensional, with the expectation that correlations diverge as the length of the cylinder goes to infinity. It is unknown if the addition of a non-uniform electric field is sufficient to induce a true transition in this case. An appropriate investigation on this topic would, of course, require theories that go beyond the mean-field approach.

Finally, we have not considered the fluid wetting behavior on the electrode surfaces. In the wedge geometry, for example, these phenomena include wedge filling, where a liquid transitions between partial and complete filling (29); (30); (31). This transition can be either first or second order and depends on factors like the wedge opening angle, liquid contact angle, and temperature. Since our results show that the interface location directly ties with the electric field, it currently remains unclear if the electric field enhances or diminishes the effects of wetting, or possibly both (depending on experimental conditions).

### Acknowledgements

This work was supported by the Israel Science Foundation under grant No. 11/10, the COST European program MP1106 “Smart and green interfaces - from single bubbles and drops to industrial, environmental and biomedical applications”, and the European Research Council “Starting Grant” No. 259205.