Stability of Binary Mixtures in Electric Field Gradients
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] N ov Stability of Binary Mixtures in Electric Field Gradients
Sela Samin and Yoav Tsori
Department of Chemical Engineering and The Ilse Katz Institute for Nanoscale Science and Technology,Ben-Gurion University of the Negev, 84105 Beer-Sheva, Israel. (Dated: 19/11/2009)We consider the influence of electric field gradients on the phase behavior of nonpolar binarymixtures. Small fields give rise to smooth composition profiles, whereas large enough fields leadto a phase-separation transition. The critical field for demixing as well as the equilibrium phaseseparation interface are given as a function of the various system parameters. We show how thephase diagram in the temperature-composition plane is affected by electric fields, assuming a linearor nonlinear constitutive relations for the dielectric constant. Finally, we discuss the unusual casewhere the interface appears far from any bounding surface.
I. INTRODUCTION
The effect of gravitational and magnetic fields on thephase diagram of liquid mixtures is quite small in gen-eral due to the weak coupling of the field with the mix-ture’s composition. The influence of electric fields wasstudied extensively, extensively in geometries where thefield is uniform. Theoretical work predicted that for abinary mixture the upper critical solution temperature T c is shifted upward by a small amount, of the order ofmilikelvins . Experiments in low molecular weight liq-uids predominantly showed an opposite shift of the samemagnitude . Two exceptions are Reich and Gordon who measured the cloud point temperature of a polymermixture and G´abor and Szalai who predicted a down-ward shift in the critical pressure of dipolar fluid mixturesin uniform electric fields.In a uniform field the shift in T c is proportional tothe square of electric field E and d ε/dφ – the secondderivative of the dielectric constant with respect to themixture composition . However, in realis-tic systems, such as colloidal suspensions and microflu-idic devices, the electric field varies in space due to thecomplex geometry. In such systems, electric fields on theorder of 10 V/m naturally occur due to the small length-scales involved. Recently, we have shown that a homo-geneous mixture confined by curved charged surfaces un-dergoes a phase-separation transition and two distinctdomains of high and low compositions appear . Thetransition occurs when the surface charge (or voltage)exceeds a critical value. In this paper we examine indetail numerically and analytically the location of thetransition. We also show how this transition affects thetemperature-composition phase diagram. We find thatnear T c the spatial variation of the field leads to a non-trivial modification of the stability lines. Our analysisshows that in nonuniform electric fields even a linearconstitutive relation can lead to a substantial change ofthe transition (binodal) temperature. Lastly, we demon-strate that the phase separation interface can appear farfrom any of the surfaces bounding the mixture. II. THEORY
Consider an A/B binary mixture confined by chargedconducting surfaces giving rise to an electric field. Thefree energy of the mixture is F = Z [ f m ( φ, T ) + f es ( φ, E )] d r , (1)where f m is the bistable mixing free energy density, givenin terms of the A-component composition φ (0 < φ < T . f es is the electrostatic free energy density due to theelectric field E , given by f es = − ε ( φ )( ∇ ψ ) . (2) ψ is the electrostatic potential. The negative sign cor-responds to cases where the potential is prescribed onthe bounding surfaces. When the charge is given on thebounding surfaces the sign should be replaced by a pos-itive one . ε is the mixture’s permittivity, and is afunction of the composition φ . Note that in order to iso-late the electric field effect we do not include any directshort- or long-range interactions between the liquid andthe confining surfaces. The equilibrium composition pro-file φ ( r ) and electrostatic potential ψ ( r ) are given by theextremization of the free energy with respect to φ and ψ . The resulting Euler-Lagrange equations are δFδφ = δf m δφ − dε ( φ ) dφ ( ∇ ψ ) − µ = 0 , (3) δFδψ = ∇ · ( ε ( φ ) ∇ ψ ) = 0 . (4)Eq. (4) is simply Gauss’s law. Notice that the relation ε ( φ ) couples these nonlinear equations. In the canonicalensemble µ is a Lagrange multiplier adjusted to satisfymass conservation: Z [ φ ( r ) − φ ] d r = 0 , (5)where φ is the average composition. In the case of a sys-tem in contact with a matter reservoir (grand-canonical FIG. 1: The three model systems. (a) A single charged spher-ical colloid with surface charge density σ and radius R . (b)A charged wire with surface charge density σ and radius R ,or two concentric cylinders with radii R and R . (c) A wedgecomprised of two flat electrodes with an opening angle β andpotential difference V . R and R are the minimal and maxi-mal values of the distance r from the imaginary meeting pointof the electrodes. ensemble), the chemical potential is set by the reservoir, µ = µ ( φ ) where φ is the reservoir composition.In order to simplify the solution of Eqs. (3)–(5), weconsider the three simple model systems shown schemat-ically in Fig. 1. The first one is a charged isolatedspherical colloid of radius R and surface charge den-sity σ , immersed in an infinite mixture bath. In thissystem, spherical symmetry dictates that φ = φ ( r ) and E = E ( r ) where r is the distance from the colloid’s center.Since the colloid has a prescribed charge we can integrateGauss’s law and obtain an explicit expression for the elec-tric field: E ( r ) = σR / ( ε ( φ ) r )ˆ r . The second geometry isa charged wire of radius R and surface charge density σ ,coupled to a reservoir at r → ∞ . Alternatively, we mayconsider a closed condenser made up of two concentriccylinders of radii R and R . In both cases, we readilyobtain the electric field E ( r ) = σR / ( ε ( φ ) r )ˆ r , where r isthe distance from the inner cylinder’s center. The lastsystem is the wedge condenser, made up from two flatelectrodes with an opening angle β and a potential dif-ference V across them. Solution of the Laplace equationgives E ( r ) = ( V /βr )ˆ θ , where r is the distance from theimaginary meeting point of the electrodes and θ is the az-imuthal angle. In this geometry φ = φ ( r ) and therefore E · ∇ ε = 0. The explicit expressions for the electric fieldin all three systems outlined above decouple equations(3) and (4).We will show that Eq. (3) leads, under certain con-ditions, to a phase-separation transition. This transi-tion is independent of the exact form of f m , and canbe realized as long as f m is bistable and the dielectricconstant ε depends on the composition φ . In order tobe specific, we will consider the mixing free energy de-rived from the Flory-Huggins lattice theory, with latticesite volume v . We consider the simple symmetric casewhere each component occupies N successive lattice cells.Simple liquids have N = 1, while polymers have N > f m = k B T ˜ f m /N v , where˜ f m = φ log( φ ) + (1 − φ ) log(1 − φ ) + N χφ (1 − φ ) . (6) k B is the Boltzmann constant, and χ ∼ /T is the Floryinteraction parameter . We limit ourselves to the case where χ >
0, leading to an Upper Critical Solution Tem-perature type phase diagram in the φ − T plane. Inthe absence of electric field, the mixture is homogeneousabove the binodal curve φ t ( T ), and phase separates intotwo phases having the binodal compositions φ t belowit. Below the binodal curve, but above the spinodal,given by φ s ( T ) = (1 / ± p − / ( N χ )], the mixtureis metastable. The binodal and spinodal curves meet atthe critical point ( φ c , ( N χ ) c ) = (1 / , T t for a given composition is givenby T t ( φ ) = ( N χ ) c T c [log( φ/ (1 − φ )) / (2 φ − − .Using the expressions given above for the electric field,we write the generalized composition equation valid forcylindrical and spherical geometries:˜ f ′ m ( φ ) − N χM sc d ˜ ε/dφ ˜ ε ( φ ) ˜ r − n − ˜ µ = 0 . (7)Here, ˜ r ≡ r/R is the scaled distance and ˜ ε = ε/ε , with ε the vacuum permittivity. Where M sc ≡ σ N v k B T c ε (8)is the dimensionless field, and n is an exponent charac-terizing the decay of E : n = 2 for concentric cylinders,and n = 4 for spherical colloid. For the wedge geometrywe find: ˜ f ′ m ( φ ) − N χM w d ˜ εdφ ˜ r − n − ˜ µ = 0 , (9)where M w ≡ V N v ε β k B T c R , (10)and n = 2. M sc and M w are dimensionless quantitiesmeasuring the magnitude of the maximal electrostaticenergy stored in a molecular volume compared to thethermal energy. The second term in Eqs. (7) and (9) isthe variation of the electrostatic free energy with respectto φ , and is only present when the mixture componentshave different permittivities.The constitutive relation ˜ ε ( φ ) is a smooth function of φ . Experiments show that the curve can be slightly con-vex or concave, and is dominantly linear . Theyare mostly in agreement with Clausius-Mossotti andOnsager-based theories for the dielectric constant .Thus, for a mixture of liquids A and B with dielectricconstants ˜ ε a and ˜ ε b , respectively, the experiments yielda polynomial relation in the form˜ ε ( φ ) ≃ ˜ ε b + ˜ ε ′ φ + ˜ ε ′′ φ + . . . . (11)We start by focusing on a linear relation, namely ˜ ε ′′ = 0;in this case ˜ ε ′ = ∆˜ ε ≡ ˜ ε a − ˜ ε b . Even in such a simple caseit turns out that a phase separation transition occurs,in contrast to the Landau mechanism which relies on a ˜ f ′ m φ φ s φ (2) s φ (˜ r ) ˜ µ ˜ µ (˜ r )˜ µ a ˜ µ b FIG. 2: Graphical solution of Eq. (12) for an open wedge at
T < T c and a symmetric mixture. Solid curve is ˜ f ′ m ( φ ). Itsroots are the transition (binodal) compositions. The intersec-tion between ˜ f ′ m ( φ ) and the horizontal dash-dotted line ˜ µ (˜ r )gives the composition φ (˜ r ). If ˜ µ ( ˜ R ) is at ˜ µ a , the profile φ (˜ r )varies smoothly, but if ˜ µ ( ˜ R ) = ˜ µ b , φ (˜ r ) has a discontinuity. nonvanishing ˜ ε ′′ . After investigating linear relations weexamine how our results change when ˜ ε ( φ ) has a positiveor negative curvature, by allowing for ˜ ε ′′ = 0. In thiscase ˜ ε ′ is different from ∆˜ ε . Higher order terms in theexpansion Eq. (11) are not expected to change the resultsqualitatively, since they do not affect much the curvatureof ˜ ε ( φ ) . III. RESULTS AND DISCUSSION
Before we present the numerical solutions of Eqs. (7)and (9), it is illustrative to consider a graphical solutionfor a wedge condenser. Recall that in the absence of fieldit is assumed that T is above the binodal temperature.We rewrite Eq. (9) as:˜ f ′ m ( φ ) = ˜ µ (˜ r )˜ µ (˜ r ) ≡ N χ ∆˜ εM w ˜ r + ˜ µ (12)where ˜ µ is the dimensionless reservoir chemical poten-tial. At a given temperature, the intersection of ˜ f ′ m ( φ )and the horizontal line ˜ µ (˜ r ) gives the local composition φ (˜ r ) (Fig. 2). When ˜ r → ∞ , ˜ µ (˜ r ) → ˜ µ and the compo-sition is φ = φ , corresponding to a homogeneous phase.For simplicity we consider φ < φ c . As ˜ r decreases, ˜ µ (˜ r )(and hence φ (˜ r )) increase until they attain their maximalvalue at ˜ R . Above T c , the free energy is always convex,˜ f ′ m is a monotonic function of φ , and the compositionprofile φ (˜ r ) is hence continuous. However, below T c , ˜ f m is bistable and ˜ f ′ m is sigmoidal. In this case there are ˜ R φ ˜ r FIG. 3: The three types of equilibrium profiles φ (˜ r ) for a sys-tem of two concentric cylinders. Dash-dot line: T = 1 . T c and M sc = 0 .
04: above T c the profile is smoothly varying.Dashed line: T = 0 . T c , M sc = 0 .
01 smaller than the crit-ical value for demixing. Solid line: same T , but M sc = 0 . r = ˜ R . We took an average composition φ = 0 . R = 1, ˜ R = 5, ˜ ε a = 5, and ˜ ε b = 3. two possible scenarios shown in Fig. 2. If, for example,˜ µ ( ˜ R ) = ˜ µ a , the composition profile varies smoothly. If,on the other hand, ˜ µ ( ˜ R ) = ˜ µ b , there is a discontinuityin the profile since there is a radius ˜ R > ˜ R where thevalue of φ can “jump” from high to low values. Below T c , there is a range of radii, or compositions, where thediscontinuity in φ (˜ r ) can occur. The equilibrium profile φ (˜ r ; ˜ R ) is the one that minimizes the total free energyintegral F = R f [ φ (˜ r ; ˜ R )]d r . These conclusions also holdfor Eq. (7).In Fig. 2 we show the composition φ (2) s defined by therelation ˜ f ′ m ( φ (2) s ) = ˜ f ′ m ( φ s ). Clearly, this is the minimalcomposition for which exist more than one solution toEq. (9). The role of this special composition will bediscussed later.The three typical composition profiles are shown inFig. 3. Above T c (dash-dot line), φ (˜ r ) varies smoothlydue to the dielectrophoretic force, whereby the high-˜ ε liq-uid is drawn into the strong electric field region. Below T c (dashed line), at a temperature where the field-freemixture is homogeneous, if M is small the profile φ (˜ r )is again smoothly decaying, exhibiting the same dielec-trophoretic behavior. However, if at the same temper-ature, M is increased, either by adding charge to thesurface or by increasing the curvature (smaller R ), wearrive at a critical value, denoted M ∗ . Above it, a phase-separation transition occurs. This is shown in the solidline of Fig. 3, where the mixture consists of two coexist-ing domains separated by an interface at ˜ r = ˜ R .The typical value of charge/voltage required for demix-ing can be estimated from the value of M being in the ˜ R M sc (a) φ = 0 . φ = 0 . φ = 0 . ˜ R M w (b) φ = 0 . φ = 0 . φ = 0 . FIG. 4: Location of the demixing interface ˜ R as a functionof M , (a) for three average compositions φ in a closed cylin-drical system (lines with symbols) and for φ = 0 .
28 (dashedline) and φ = 0 . R = 1. (b) The same, in the closed ”wedge”geometry. We took T = 0 . T c . range M ∼ . − . . Consider a colloid of radius R ∼ µ m placed in a mixture having a molecular vol-ume N v ≃ × − m and T c ≃ − e charges(surface voltage is 1 − V ). It scales linearly with 1 /N :in a polymer mixture the confining surfaces require N times smaller charge compared to molecular liquids inorder to induce phase separation. A. The phase-separation interface
At the critical value of M , a sharp interface first ap-pears separating coexisting regions of high- and low- φ value. If the average composition φ is smaller than φ c ,the interface appears at ˜ r = ˜ R . When we further in-crease M and supply more electrostatic energy to thesystem, dielectrophoresis leads to an increase in the size of the high composition domain. Thus, the location ofthe separation interface ˜ R increases. Fig. 4 shows how˜ R varies with M at a constant temperature in the con-centric cylinders and wedge systems. Notice that as φ approaches the binodal composition ( φ t ≃ . R islarger at the same M . It also grows more rapidly withincreasing φ . Indeed, when the binodal is approached,the mixing free energy barrier is smaller. Secondly, ˜ R grows faster in an open system than in a closed one.This is because in a closed system the energy penaltyin ˜ f m grows faster than the energy gain in ˜ f es . Materialconservation gives the maximum value of ˜ R , ˜ R ∞ , givenby ˜ R ∞ = φ ( ˜ R − ˜ R ) + ˜ R . (13)This M → ∞ limit is physically unattainable and is pre-empted by dielectric breakdown.Note that the typical values of M in the cylindrical andspherical cases are an order of magnitude larger than inthe wedge condenser, see Fig. 4 (a) and (b). Indeed,in the spherical and cylindrical symmetries, E is paral-lel to ∇ φ : the dielectrophoretic force, (proportional ∆ ε ),has to be large enough to overcome the energy penaltyassociated with dielectric interfaces parallel to E (pro-portional to (∆ ε ) ) . On the other hand, in the wedgecondenser E is perpendicular to ∇ φ , and the requireddielectrophoretic force for demixing is correspondinglysmaller, leading to smaller values of M w . This could beseen by comparing the electrostatic terms in Eqs. (3)and (4), differing by a factor of ˜ ε ( φ ) − ∼ .
1, which M sc has to compensate for in order for the values of ˜ f es to beequal.Alternatively, an increase in T at constant electric fielddecreases χM ∼ M/T and decreases ˜ R . Fig. 5 showshow an increase in T shrinks the high- φ domain and de-creases ˜ R in a closed and open cylindrical system. InFig. 5, when ˜ R = 1 the temperature is that for which M = M ∗ . The effect of temperature is much more pro-nounced in an open system: ˜ R tends to infinity when ap-proaching the binodal temperature (not shown). On theother hand, ˜ R is finite when approaching the binodal in aclosed system. Its maximal value is larger when | φ − φ c | is smaller, because then the mixing free energy differencebetween low- and high- φ values is reduced. In Fig. 5 (a),˜ R appears to be linear simply because T changes over asmall interval.One can estimate the value of φ at the demixing in-terface in an open system. At the interface there is a“jump” in φ (˜ r ) from φ to φ > φ . Let us denote by φ ( ˜ R ) the upper interface composition – φ ( ˜ R ) = φ . Theconditions for a “jump” in the wedge geometry are:˜ f ′ m ( φ ) − N χ ∆˜ εM w ˜ R − ˜ µ = 0 , (14)˜ f ′ m ( φ ) − N χ ∆˜ εM w ˜ R − ˜ µ = 0 , (15)˜ f m ( φ ) − N χ ˜ ε ( φ ) M w ˜ R ≥ ˜ f m ( φ ) − N χ ˜ ε ( φ ) M w ˜ R . (16) ˜ R T /T c (a) ˜ R T /T c (b) FIG. 5: Location of the demixing interface ˜ R as a functionof T /T c , (a) for three average compositions in a closed cylin-drical system: φ = 0 .
28 (squares), 0 . . M sc = 0 .
04. When ˜ R = ˜ R , the tempera-ture corresponds to M sc = M ∗ sc = 0 .
04. ˜ R grows as T /T c isreduced until it attains it maximal value at the binodal tem-perature (dashed line for each value of φ ) (b) Same, in anopen cylindrical system with M sc = 0 . R → ∞ atthe binodal. The first two equations define the local solutions of Eq.(12), and the third one is the condition that a high com-position is favorable: ˜ f ( φ ) < ˜ f ( φ ). The true valueof ˜ R is the one that gives the global minimum of thefree energy. Putting Eq. (14) in Eq. (16) and using˜ µ = ˜ f ′ m ( φ ), we get˜ f ′ m ( φ ) − ˜ f ′ m ( φ ) ≥ ˜ f m ( φ ) − ˜ f m ( φ ) φ − φ . (17)The right-hand side of Eq. (17) is maximal when thetransition occurs from φ = φ (2) s to φ = φ s ( φ (2) s < φ s ).We therefore denote ∆ f w , max by∆ f w , max = ˜ f m ( φ s ) − ˜ f m ( φ (2) s ) φ s − φ (2) s . (18) T / T c φ (a) φ t φ s φ (2) s φ ˜ r (b) φ t φ s φ (2) s φ ˜ r (c) (c)(b) FIG. 6: (a) The “differentiating” curve – φ † (dashed line) foran open wedge. Above it, when phase separation occurs, thecomposition at the interface jumps from φ (2) s to φ s . Below φ † the front composition jumps from φ (2) s < φ < φ t to φ > φ s .Examples of this behavior are shown in (b) and (c), wherecomposition profiles for points above and below φ † with twovalues of M are given, showing this behavior is independentof M . The inset in (a) is a blowup showing the locationof (b) and (c) in the φ − T plane. In (b) φ = 0 .
38 with M w = 4 × − and M w = 6 × − , and in (c) φ = 0 . M w = 1 × − and M w = 1 . × − . In (b) and (c) wetook T = 0 . T c . If the inequality˜ f ′ m ( φ ) − ˜ f ′ m ( φ ) ≥ ∆ f w , max (19)holds, the transition must be at φ = φ (2) s , since for largervalues of φ the right-hand side of Eq. (17) is smallerwhile the left-hand side is larger, so a higher compositionis surely favored. The equality sign in Eq. (19) corre-sponds to the maximal average composition φ for whichthis equation holds.The locus of such compositions is the “differentiatingcurve” – φ † , shown in the dashed curve of Fig. 6 (a).When the zero-field point in the phase diagram ( φ , T )is above the φ † ( T ) curve, Eq. (19) holds, and the com-position at the interface jumps from φ (2) s to φ s . When( φ , T ) is below φ † ( T ) the upper interface composition φ ( ˜ R ) is between φ s and φ t . Eq. (17) shows this result isindependent of M . An example of this situation is givenin Fig. 6 (b), where the compositions φ and φ are thesame for two values of M . Fig. 6 (c) shows the inter-face compositions are independent of M also when φ islarger than φ † . M w , M sc φ ( ˜ R ) φ t FIG. 7: Composition at the demixing interface ˜ R as a functionof M for a closed wedge (crosses) and concentric cylinders(circles). The dashed line is the binodal composition. Here φ = 0 . T = 0 . T c . When the discontinuity in the composition profile oc-curs at φ = φ s , we can invert Eq. (9) to get ˜ R ∝ M / for the open wedge system. In particular, above the dif-ferentiating curve we get in the Flory-Huggins model˜ R = (cid:20) N χ ∆˜ εM w ˜ f ′ m ( φ s ) − ˜ µ (cid:21) / . (20)Thus, ˜ R varies linearly with the wedge potential V .In the other geometries ˜ f ′ es depends on φ , and thisinfluences the value of pairs φ and φ . However, a verygood approximation, valid when M sc is not too large and T is not too close to T c , is that the transition remainsfrom φ = φ (2) s to φ = φ s . One can then repeat a similarderivation and obtain˜ f ′ m ( φ ) − ˜ f ′ m ( φ ) ≥ ∆ f sc , max , (21)∆ f sc , max = ˜ f m ( φ s ) − ˜ f m ( φ (2) s ) φ s − φ (2) s ˜ ε ( φ s )˜ ε ( φ (2) s ) . (22)Using these equations one can determine the differenti-ating curve for cylindrical and spherical geometries.The situation is different in a closed system, as Fig.7 shows. In the wedge, the demixing interface occurs atthe binodal composition, φ t , irrespective of M . In thecylindrical and spherical systems, φ ( ˜ R ) is lower than butclose to φ t , and decreases when M grows. The qualita-tive explanation is as follows. In the wedge geometry ˜ f ′ es only adds a constant to ˜ f ′ , and the binodal compositionsremain the only pair of solutions of Eq. (9) that have thesame mixing free energy ˜ f m . Hence, the mixing free en-ergy penalty is minimized when the transition is at thebinodal compositions . This also explains why in thewedge geometry φ ( ˜ R ) is independent of M . In the cylin-drical and spherical geometries on the other hand, ˜ f ′ es affects φ ( ˜ R ), resulting in a value of φ ( ˜ R ) smaller than T / T c φ (a) T t T / T c φ (b) T t kinkkink FIG. 8: (a) Stability diagram of a spherical colloid with M sc =0 .
04 (thick solid line) and M sc = 0 .
08 (dashed line). Dash-dotcurve is φ † [see Fig. 6(a)] and thin solid line is the binodal T t ( φ ) . (b) Stability diagram showing φ ∗ ( T ) for concentriccylinders with M sc = 0 .
04 (thick solid line) and M sc = 0 . φ t . This reflects the fact that dielectrophoresis favorshigh values of φ . Since ˜ f es ∝ M , larger values of M leadto lower values of φ ( ˜ R ). B. Stability diagrams
One can also set M constant and for a given electricfield draw the a stability curve φ ∗ ( T ) in the φ − T plane,see Fig. 8. φ ∗ ( T ) is defined such that below it phaseseparation occurs, while above it composition profiles aresmooth. Fig. 8 (a) and (b) show the stability diagram of
1 1.5 2 0.450.50.55 ˜ r φ T / T c φ T t φ ∗ A B C
CBA
FIG. 9: Composition profiles above the kink temperature foran isolated spherical colloid. The profiles change from smooth(dash-dot and dashed lines) to discontinuous (solid line) witha discontinuity at finite value of ˜ R : ˜ R ≤ ˜ R ≤ ˜ R , when φ increases at constant T and M (see inset). a spherical colloid and concentric cylinders, respectively.In both diagrams, an increase in M increases the unstableregion. For the same value of M , in a closed systemthe phase separation region is smaller than in an openone, because the mixing energy penalty makes it moredifficult to induce phase separation. The range of valuesof φ that are unstable in nonuniform electric fields growswhen T increases (but still T < T c ). For low valuesof T , there is a significant difference between open andclosed systems: in open systems, if M is large enoughthe stability curve tends to φ † [see Fig. 6(a)], whereasfor closed systems the stability curve tends to φ t wheredemixing is spontaneous.In both parts of Fig. 8, there is a kink in all the curves φ ∗ ( T ) at a temperature we denote T k, . In the Flory-Huggins model and for spherical colloids and concentriccylinders, the second derivative of the free energy is ∂ ˜ f∂φ = ∂ ˜ f m ∂φ + 2 N χM sc ( d ˜ ε/dφ ) ˜ ε ( φ ) ˜ r − n . (23)The second term in this equation is the positive electro-static contribution f ′′ es . It is clear that when an electricfield is present, even at T < T c , the electrostatic contri-bution can lead to a positive value of ˜ f ′′ ( φ, ˜ r ) and phaseseparation cannot occur.When T < T k, , ˜ f ′′ is negative for all values of ˜ r , andthe phase separation interface appears first at ˜ R = ˜ R .However, when T > T k, , for a given value of φ , a specialradius ˜ R c ( T, φ ) exists. This is the largest value of ˜ r forwhich ˜ f ′′ in Eq. (23) can be negative. In this case, thedemixing interface appears first at ˜ R = ˜ R c . An exampleof this behavior is shown in Fig. 9: at constant T c >T > T k, , at points A and B ( φ < φ ∗ ) φ (˜ r ) is smooth, φ φ ( ˜ R ) (a) T/T c = 0 . T/T c = 0 . T/T c = 0 . φ φ ( ˜ R ) (b) ~ T/T c = 0 . T/T c = 0 . T/T c = 0 . FIG. 10: Surface composition φ ( ˜ R ) when approaching thebinodal, (a) for three temperatures in a closed cylindrical sys-tem. (b) Same, for an isolated charged cylinder. We took M sc = 0 . similar to φ ( r ) above T c . However, at point C ( φ > φ ∗ ) φ ( r ) has a discontinuity at ˜ R > ˜ R c . As the critical pointis approached, ˜ R c → ∞ and φ ∗ approaches the criticalcomposition.The kink temperature T k, is given by setting ˜ R c = ˜ R and can be obtained from solution of: ∂ ˜ f∂φ ( ˜ R ) = 0 , ∂ ˜ f∂φ ( ˜ R ) = 0 . (24)We stress that this result is independent of the exactform of mixing free energy. Notice that for a wedge, theelectric field has no effect on the convexity of the freeenergy, and one finds T k, = T c (see Fig. 11).The surface composition φ ( ˜ R ) when approaching thebinodal at constant T and M is given in Fig. 10. When T < T k, the surface composition has a discontinuity at φ = φ ∗ (dashed and dash-dot lines) and the value of φ ( ˜ R ) becomes larger than φ c . When T > T k, the sur-face composition varies smoothly (solid lines). The dis-continuity in φ ( ˜ R ) occurs at lower values of φ in open T / T c φ (a) T / T c φ (b) A CBA CB D D T t T t ˜ ε ′′ < ε ′′ > E FIG. 11: (a) Stability diagram for an open wedge with M w = 4 × − and three constitutive relations Eq. (11)with ˜ ε a = 5 and ˜ ε b = 3. Thick solid line is the transition(binodal) curve T t ( φ ). When ˜ ε ′′ = 0 (dash-dot line) there isno phase separation above T c . For ˜ ε ′′ = 1 . T c . In contrast, when ˜ ε ′′ = − . φ (˜ r ) changes from smoothto discontinuous at ˜ R along lines A and B, and at ˜ R c > ˜ R along C and D. (b) The same for a closed wedge system. Notethat the stability curve extends to compositions larger than φ c . φ (˜ r ) changes from smooth to discontinuous at ˜ R alongA and B, at ˜ R < ˜ R c < ˜ R along C and D, and at ˜ R alongE. Here T k, /T c = 1 . systems compared to closed ones; open systems are lessstable than closed systems. When T increases at a givenvalue of φ , the surface composition decreases since mix-ing is favored at high temperatures. C. Quadratic constitutive relation
We now examine how the stability diagram changesif the dielectric constant has a quadratic dependence oncomposition: ˜ ε ′′ = 0 in Eq. (11). For simplicity, wetreat the wedge system where a linear constitutive rela-tion means that T k, = T c . In the Flory-Huggins model,the conditions in Eq. (24) give T k, T c = 1 + M w ˜ ε ′′ , (25)where we used ˜ R = 1.Note that ˜ ε ′′ can be positive or negative. When ˜ ε ′′ < f ′′ es is positive and we return to the same behavior we sawfor spherical and cylindrical systems with linear relation˜ ε ( φ ). On the other hand, if ˜ ε ′′ > f ′′ es is negativeand phase separation is possible above T c . The stabilitydiagram of a wedge with a quadratic constitutive rela-tion is shown in Fig. 11. In this figure, arrows labeledA–E indicate the variation of φ at constant T in differ-ent areas of the stability diagrams. For each arrow thelocation for which the interface first appears is given inthe caption of Fig. 11. For the data in Fig. 11, thekink temperature is given by T k /T c = 1 ± . ε ′′ . Taking T c ≈ T ( φ ∗ ( T k )) = T k − T c ≈ ± T c istwo orders of magnitude larger than the correspondingchange in uniform electric fields. Note that in most ofthe phase space the displacement of the transition tem-perature due to a nonvanishing ˜ ε ′′ is much smaller thanthat due to ˜ ε ′ ; in spatially nonuniform fields far fromthe critical composition, the demixing transition is welldescribed by a linear constitutive relation ˜ ε ( φ ). D. Demixing transitions for φ > φ c Since the electric field breaks the symmetry of the freeenergy with respect to composition ( φ → − φ ), the fullstability diagram is asymmetric with respect to φ − φ c .Fig. 11 (a) shows that in an open system, phase separa-tion does not occur when φ > φ c . In Fig. 11 (b) thereare unstable compositions φ such that φ > φ c . This isan important feature of the stability diagram in closedsystems. Here, the dielectrophoretic force that pulls thehigh dielectric constant liquid toward the electrode cre-ates a depletion in the region where electric field is low,and phase separation will occur if the composition at thisregion is close to the binodal composition. The stabilitycurves in Fig. 11 (b) show that higher values of potentialor charge are required for demixing when φ > φ c . In-deed, the ratio of electrostatic energies at the inner andouter radii is ≃ ˜ R / ˜ R .Examples of this behavior are shown in Fig. 12.When φ > φ c , and M is small (dashed line) φ (˜ r ) variessmoothly, exhibiting the usual dielectrophoretic behav-ior. However, if M is sufficiently large, that is M > M ∗ , φ ˜ r FIG. 12: Equilibrium profiles φ (˜ r ) for a closed wedge sys-tem with an average composition φ = 0 .
58 larger than φ c .Dashed line: M w = 1 × − smaller than M ∗ w , and φ (˜ r ) issmoothly varying. Solid line: M w = 4 × − is large enoughto induce phase separation. Dash-dot line: M w = 8 × − ,the separation interface moves to a smaller radius. We used T = 0 . T c . the long range dielectrophoretic force gives rise to aphase-separation transition near ˜ R (solid line). If M is further increased the phase-separation interface movesto smaller radii (dash-dot line).Fig. 13 shows how ˜ R varies with M and T in the wedgegeometry for φ > φ c . At M ∗ , the interface appears wherethe field is minimal, i.e at ˜ R = 5, see Fig. 13 (a). Anincrease in M decreases ˜ R and the volume of the high- φ domain. Fig. 13 (b) shows that at fixed M , ˜ R decreaseswith T and attains its minimal value at T t . The minimalvalue of ˜ R is ˜ R ∞ given by Eq. (13).When ˜ ε ′′ = 0, the stability curves in Fig. 11 (b) haveanother kink at φ > φ c at a temperature we denote T k, .From the previous discussion, T k, is obtained by replac-ing ˜ R by ˜ R in Eq. (24). In the wedge geometry wethus find T k, T c = 1 + M w ˜ ε ′′ ˜ R . (26)Hence, T c − T k, is smaller than T c − T k, by a factor of˜ R . IV. CONCLUSIONS
We present a systematic study of electric field inducedphase-separation transitions in binary mixtures on themean field level. The behavior described by us comple-ments the findings of Onuki and co-workers andBen-Yaakov et al. who mainly focused on effects abovethe critical temperature or on miscible liquids. The tran- . −3 ˜ R M w (a) ˜ R T /T c (b) FIG. 13: (a) Interface location ˜ R for a closed wedge systemwith an average composition φ = 0 . > φ c as a function of M w at T = 0 . T c . The minimal value of ˜ R is ˜ R ∞ = 3 . R as a function of T at M w = 4 × − .Dashed line is T t . sitions should occur in any bistable system with a dielec-tric mismatch sufficiently close to the transition tempera-ture, e.g a vapor phase close to coexistence with its liquid.The differences between closed and open systems and be-tween the three geometries are highlighted. The stabil-ity diagrams in the temperature-composition plane aregiven. These diagrams show that the change in the tran-sition temperature is much larger in nonuniform fieldsthan in uniform fields. We describe the special temper-atures T k, and T k, where the stability diagram has a“kink”. It should be emphasized that the location of thephase separation interface is not restricted to the vicinityof the confining surfaces, and can appear at a finite loca-tion ˜ R in the range of temperatures T k, < T < T k, asdescribed above in Fig. 11. In other geometries, too, thedemixing interface can be created far from any surface,e.g in quadrupolar electrode array.In order to test our predictions we suggest the follow-ing experiments. Consider a wedge capacitor partially0immersed in a binary liquid mixture near the coexis-tence temperature. Neutron reflectometry may be usedto probe directly the composition profile and to deter-mine the critical voltage for demixing. Here we expect˜ R to be proportional to the voltage difference across theelectrodes. A different experiment can be realized bysuspending a conducting wire in a dilute vapor phase ofa pure substance near the coexistence temperature withthe liquid. When a potential V is applied to the wire,its fundamental frequency of vibration perpendicular toits length changes from f to f V because a liquid layercondenses around the wire. We expect that the ratio( f /f V ) should be a linear function of V . Alterna-tively, one can measure directly the change in the wiremass ∆ m using a microbalance – ∆ m should depend lin-early on V . In these experiments, care must be taken tosubtract wetting or confinement effects due to the elec-trodes.Capillary condensation has been described for colloidal suspensions in binary mixtures. Accordingto our work, charged colloidal suspensions can floccu-late or be stabilized due to the formation of liquid layersaround the colloids. The liquid layer may also influencethe interaction between a colloid and a flat surface bothfar and close to the critical point .Our results may also be measurable in Surface ForceBalance and Atomic Force Microscope experiments. Insuch apparatus, a bridging transition has been ob-served in binary mixtures and explained by capillaryforces . Since in most cases the surfaces are charged, a capillary bridge could be the result of the merging oftwo field-induced layers, leading to long range attractiveforces between the surfaces.In the current work we neglect a ( ∇ φ ) in the mixingfree energy f m , since the system size was relatively largeand the electrostatic energy acts throughout the wholesystem volume, and therefore is dominant . We haveverified that inclusion of such term leads to smoothing ofcomposition profiles but otherwise to no other noticeablechanges in the figures presented. In order to isolate theelectric field effect, we have also not considered any di-rect short- or long-range interactions with the confiningsurfaces. In a future study it would be very interestingto relax these assumptions and to look at ever smallersystems. Here we are intrigued by the possibility to findqualitative, and not just quantitative, differences fromour current profiles and diagrams. ACKNOWLEDGMENTS
We acknowledge useful discussions with L. Leibler, D.Andelman, H. Diamant, and J. Klein, and critical com-ments from K. Binder. This research was supported bythe Israel Science foundation (ISF) grant no. 284/05, bythe German Israeli Foundation (GIF) grant no. 2144-1636.10/2006, and by the COST European program P21“The Physics of Drops”. L. D. Landau and E. M. Lifshitz,
Elektrodinamika Splosh-nykh Sred Chap. II, Sec. 18, Problem 1 (Nauka, Moscow,1957). A. Onuki, Europhys. Lett. , 611 (1995). P. Debye and K. Kleboth, J. Chem. Phys. , 3155 (1965). D. Beaglehole, J. Chem. Phys. , 5251 (1981). K. Orzechowski, Chem. Phys. , 275 (1999). D. Wirtz and G. G. Fuller, Phys. Rev. Lett. , 2236(1993). S. Reich and J. M. Gordon, J. Pol. Sci.: Pol. Phys. , 371(1979). A. G´abor and I. Szalai, Molecular Physics , 801 (2008),ISSN 0026-8976. A. Onuki, Phys. Rev. E , 021506 (2006). H. G. Schoberth, K. Schmidt, K. A. Schindler, andA. Boker, Macromolecules , 3433 (2009). Y. Tsori, D. Andelman, C.-Y. Lin, and M. Schick, Macro-molecules , 289 (2006). Y. Tsori, Rev. Mod. Phys. , 1471 (2009). It has been argued by Stepanow and co-workers thatfluctuation effects near the critical point change thisbehavior . I. Gunkel, S. Stepanow, T. Thurn-Albrecht, andS. Trimper, Macromolecules , 2186 (2007). S. Stepanow and T. Thurn-Albrecht, Phys. Rev. E ,041104 (2009). Y. Tsori, F. Tournilhac, and L. Leibler, Nature , 544 (2004). G. Marcus, S. Samin, and Y. Tsori, J. Chem. Phys. ,061101 (2008). L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii,
Electro-dynamics of Continuous Media (Butterworth-Heinemann,Amsterdam, 1984), 2nd ed. A. Onuki,
Nonlinear dielectric phenomena in complex liq-uids (Kluwer Academic, Dordrecht, 2004). A. Onuki and H. Kitamura, J. Chem. Phys. , 3143(2004). A. Onuki,
Phase transition dynamics (Cambridge Univer-sity Press, 2004). M. Doi,
Introduction to polymer physics (Oxford Univer-sity Press, Oxford, UK, 1996). Y. Y. Akhadov,
Dielectric properties of binary solutions (Oxford : Pergamon Press, 1981). A. D. Sen, V. G. Anicich, and T. Arakelian, J. Phys. D:Appl. Phys. , 616 (1992). Strictly speaking, the spinodal compositions φ s are “for-bidden”, because the spinodal is the locus of critical points,and thus critical fluctuations are expected to be important.The current mean-field treaties therefore is not expectedto hold when φ ≃ φ s . L. D. Landau and E. M. Lifshitz,
Statistical Physics Part1 § (Butterworth-Heinemann, Amsterdam, 1980), 3rded. A. Onuki and R. Okamoto, J. Phys. Chem. B , 3988 (2009). D. Ben-Yaakov, D. Andelman, D. Harries, and R. Pod-gornik, J. Phys. Chem. B , 6001 (2009). J. Bowers, A. Zarbakhsh, I. A. McLure, J. R. P. Webster,R. Steitz, and H. K. Christenson, J. Phys. Chem. C ,5568 (2007). H. T. Dobbs and J. M. Yeomans, J. Phys.: Condens. Mat-ter , 10133 (1992). C. Bauer, T. Bieker, and S. Dietrich, Phys. Rev. E ,5324 (2000). D. Beysens and D. Est`eve, Phys. Rev. Lett. , 2123(1985). T. Narayanan, A. Kumar, E. S. R. Gopal, D. Beysens,P. Guenoun, and G. Zalczer, Phys. Rev. E , 1989 (1993). R. D. Koehler and E. W. Kaler, Langmuir , 2463 (1997). C. Hertlein, L. Helden, A. Gambassi, S. Dietrich, andC. Bechinger, Nature , 172 (2008). H. Wennerstrom, K. Thuresson, P. Linse, andE. Freyssingeas, Langmuir , 5664 (1998). D. Andrienko, P. Patricio, and O. I. Vinogradova, J. Chem.Phys. , 4414 (2004). M. Olsson, P. Linse, and L. Piculell, Langmuir20