Phase separation transition in liquids and polymers induced by electric field gradients

0.18 MB·English

by  G. Marcus

Phase separation transition in liquids and polymers induced by electric field
gradients
Gilad Marcus1 and Yoav Tsori2
1Max-Planck-Institut fu¨r Quantenoptik, Hans-Kopfermann-Str. 1, D-85748 Garching, Germany.
2Department of Chemical Engineering, Ben-Gurion University of the Negev, 84105 Beer-Sheva, Israel.
Spatially uniform electric fields have been used to induce instabilities in liquids and polymers,
9 and to orient and deform ordered phases of block-copolymers. Herewe discuss the demixing phase
0 transition occurringinliquidmixtureswhentheyaresubject tospatially nonuniformfields. Above
0 thecriticalvalueofpotential,aphase-separationtransitionoccurs,andtwocoexistingphasesappear
2
separated by a sharp interface. Analytical and numerical composition profiles are given, and the

n interfacelocation as afunction ofchargeorvoltage isfound. Thepossible influenceofdemixingon
a thestability of suspensions and on inter-colloid interaction is discussed.
J

4
1 INTRODUCTION givencompositionisgivenbydfm(Tt,φ)/dφ=0,thatis,
T /T =2(2φ−1)/log(φ/(1−φ))[2]. Inthe absenceof
t c
h] Electric fields influence the structure and thermody- electricfield,themixtureishomogeneousifT >Tt,and
c namic behavior of charged as well as neutral matter. unstable otherwise. The field-induced phase-transition
e Their effect is strong, they can be switched on or off, discussed below does not depend on the exact form of
m
and they are easily scalable to the sub-micron regime fm; it occurs also in a Landau series expansion of Eq.
t- [1]. Therearetwomaindistinctionswithrespecttothe (2) around the critical composition φc, and in other
a field: spatially uniform vs. nonuniform fields. There forms having a “double-well” shape.
t
s are also two broad classes of material properties: pure The dielectric constant ε depends on φ via a consti-
t. dielectric vs. conducting media. All four combinations tutive relation. A variation of φ from its critical value,
a
m are relevant to phase transitions in liquid and polymer φc, induces a variation of ε from the critical permittiv-
mixturesandtoliquid-vaporcoexistenceinpureliquids. ity εc. When the composition deviation ϕ ≡ φ− φc
d- Perfect dielectrics. The electrostatic energy of di- is small enough, |ϕ|≪1, the constitutive relation ε(φ)
n electric materials is given by the expression canbewrittenasaTaylorseriesexpansiontoquadratic
o order:
c 1
[ Fes =−2Z εE2d3r (1) ε(φ)=εc+∆εϕ+ 12ε′′ϕ2 (3)
1
where ε is the dielectric constant and E is the electric
v
0 field. Thenegativesignbeforetheintegralisapplicable iTfhεe′′“vdaineilsehctersi.c contrast”∆ε is simply equal to ε2−ε1,
9 to situationswherethe electricpotentialψ (E=−∇ψ)
9 is given on the bounding surfaces; in cases where the Theelectricfielddependsontheimposedexternalpo-
1 tentials or charges and on the local dielectric constant.
chargeisprescribed,E is givenasafunction ofthe dis-
1. placementfieldD,andtheLegendretransformreverses Let us denote by E0 the electric field corresponding to
0 the sign [1]. the system with uniform composition φc everywhere.
9 Compositionchangesinφinducechangesinε,andsince
The phase-transition described below occurs in sys-
0 ε and E are coupledvia Laplace’s equation∇(εE)=0,
: tems described by bistable free-energy functionals
v one has variations in electric field.
giving rise to a phase-diagram in the composition-
i We may thus write to quadratic order in ϕ
X temperature plane divided into two regions: homo-
r geneous mixture and a phase-separated state. For 1
a concreteness, we consider the symmetric mixture free- E=E0+E1ϕ+ E2ϕ2 . (4)
2
energy density f given by
m
NotethatE0 isconstantinspaceonlyinsideaparallel-
v0 f = (2) plate capacitor, or if the sources of the field are very
m
k T
B far from the system under investigation. Clearly, even
[φlog(φ)+(1−φ)log(1−φ)]+2kBTcφ(1−φ) if E0 is uniform, composition variations lead to field
nonuniformities.
This free-energy is given in terms of the dimensionless
One can expand the electrostatic energy density in
composition φ (0≤φ≤1). In a binary mixture of two
Eq. (1) in powers of ϕ:
liquids 1and2,withdielectricconstantsε1 andε2,φis
the relativecompositionof(say)liquid2. InA/Bpoly-
1
merblends,itistherelativevolumefractionofpolymer fes = const.−(cid:18)εcE0·E1+ 2∆εE20(cid:19)ϕ (5)
A, and similarly for an A/B diblock-copolymer melt.
1 1
v0 is a molecular volume, kB is the Boltzmann con- − 2(cid:18)2ε′′E20+εcE21+2∆εE0·E1+εcE2·E0(cid:19)ϕ2
stant and T is the critical temperature. In symmetric
c
mixtures, the transition (binodal) temperature T at a + O(ϕ3) .
t
2
The unimportant constant corresponds to the electro- MIXTURES OF NONPOLAR LIQUIDS IN
static energy of the system with uniform composition, FIELDS GRADIENTS
anditservesasareferenceenergy. IfthefieldE0 isuni-
forminspace,the twotermsinlinearorderofϕsimply Fieldgradientsaregeneral,andoccurinallelectrodes
add a constantto the chemicalpotential, and therefore unless special care is taken to eliminate them (super-
are inconsequential for the thermodynamic state of the flat and parallel conducting surfaces). When mixtures
system [3]. of pure dielectric liquids are subjected to a spatially
Landau and Lifshitz showed that the existence of a nonuniform field, the situation is very different. The
ε′′ϕ2 term in Eq. (5) is responsible to a shift of the direct coupling between field variations and composi-
critical temperature Tc [1, 4]. They found that Tc is tionfluctuationsthenleadstoadielectrophoreticforce,
increased by ∆Tc given by depending on ∆ε in Eq. (3), which tends to “suck”the
component with large ε to regions with high electric
∆T = v0ε′′E02 , (6) field,asinthecaseofthewell-knownriseofadielectric
c
2k
B liquid in a capacitor [1].
T andthewholebinodalcurveclosetothecriticalpoint
c
are increased if ε′′ > 0 (field-induced demixing) or de- (a) V (b) (c)
creased if ε′′ < 0 (field-induced mixing). A similar ex-
b r r
pression exists for a pure liquid in coexistence with its r R s
vapor. R1 1 R1
The experiments, starting with P. Debye and Kle- R2
R
both [5], are in contradiction with this prediction. De- 2
bye and Kleboth investigated the critical temperature
of a Isooctane-Nitrobenzene mixture (relative permit-
FIG. 1: Three model systems where field gradients lead to
tivities 2.0 and 34.2, respectively) They observed re- demixing. (a)Awedgecomprisedoftwoflatelectrodeswith
duction of Tc by 15 mK in a field of 4.5 V/µm. Their an opening angle β and potential difference V. R1 and R2
measurements were later verified by Orzechowski [6]. are theminimal and maximal values of the distance r from
BeagleholeworkedwithaCyclohexane-Anilinemixture theimaginarymeetingpoint. (b)Achargedwirewithradius
(relative permittivities 2 and 7.8, respectively), and he R1,ortwoconcentriccylinderswithradiiR1 andR2. (c)A
measuredreductionofT byasmuchas80mKina 0.3 single charged colloid of radius R1 and surface charge σ.
c
V/µm dc field [7]. Early worked on the same mixture
but in 1 V/µm ac field, and found no change in T . He
c
attributed the results ofBeagleholeto spuriousheating Statics
[8]. Wirtz andFuller performedsimilar experiments on Three “canonical” geometries with electric field gradi-
n-hexane-Nitroethanemixture(relativepermittivities2 ents are presented in Fig. 1. The first is the “wedge”
and 19.7, respectively), and found a reduction of T by capacitor, made up from two flat and nonparallel sur-
c
20 mK in a 5 V/µm field [9]. In all cases, ε′′ was pos- faces with potential difference V, and opening angle β.
itive but still mixing was observed. In addition, the The electric field is then azimuthal, E(r) = V/(βr)θˆ,
observed change in T is quite small, typically in the where r is the distance from the imaginary meeting
c
10-20 mK range. The only exception is the work of point of the surface. r is bounded by the smallest
Gordon and Reich [10]. They worked on polymer mix- and largest radii R1 and R2, respectively. The sec-
tures of poly(vinyl methyl ether) (PVME)-polystyrene ond model system consists of a charged wire of radius
(PS) system (relative permittivities 2.15 and 2.6, re- R1, or two concentric metallic cylinders with radii R1
spectively), and observed changes significantly larger and R2 > R1. In this case the azimuthally-symmetric
than 1 K. Their strong effect can be attributed to the field is E(r)=σR1/(rε(r))ˆr, where σ is the charge per
largemolecularweightofthepolymer(14,000–30,000 unit area on the inner cylinder. Lastly, for a charged
gr/mol)andtotheir reducedentropycomparedtothat spherical colloid of radius R1 and surface charge σ,
of simple liquids. one readily finds the spherically-symmetric field to be
One is inclined to explain the experimental findings E(r)=σR1/(r2ε(r))ˆr.
bythe secondandthirdtermsonthe secondline ofEq. In all three cases a general scenario occurs: when
(5) (proportional to ϕ2). The third term is twice as T is above Tc, the composition profile φ(r) is smooth,
large as the second one and opposite in sign, and the and its gradients increase as the charge on the objects
twosumtogiveafreeenergycontributionproportional increases. However,belowTc the behaviorisdifferent–
to the dielectric contrast squared +(∆ε)2. This is a φ(r) is smooth as long as the charge (voltage) is small,
free energypenalty for dielectric interfacesperpendicu- and becomes discontinuous when the charge (voltage)
lar to the externalfield. Indeed, these additional terms attainsacriticalvalue. Atthischarge,asharpinterface
are responsible to the normal field instability in liquids appearsbetweenthecoexistingdomains[19,20]. Asthe
[4], and to orientation of ordered phases (e.g. block- charge further increases, the interface location and the
copolymers)inexternalfields[11,12,13,14,15,16,17]. compositions of the coexisting domains change.
In liquid mixtures they favor mixing (lowering of T ). To see this, consider the wedge capacitor, for which
c
3
(a) The right hand side of the equation is independent
f' (j ) of ϕ, and is indicated by the horizontal lines in Fig. 2
m (a). Suppose the mixture composition in the absence
b' of field corresponds to a point above and to the left of
the binodal curve (homogeneous mixture). A graphical
j (r)
solution of the governing equation is obtained by the
j
intersection of the horizontal line, whose location de-
b
pends on the field, and therefore on r, with the curve
f′ (ϕ). If T is above T , f is convex, and therefore
m c m
a the intersection of the two curves changes smoothly as
r decreases (E increases). The resulting composition
profile is shown in Fig. 2 (b).
(b)
However, the situation is different below T : here
c
1 f′ (ϕ) behaveslike −ϕ+ϕ3. Whenthe appliedvoltage
m
is small enough such that the maximum value of the
0.8
right-hand side of Eq. (8) occurs at line b of Fig. 2
f (r) 0.6 V>V* (a),asonegoesfromlargetosmallvaluesofr (increas-
ing E), the composition increases, but ϕ(r) is always
0.4 V<V* continuous. There is a criticalvalue ofthe voltage,V∗,
where this is not true: above the critical potential, the
0.2
maximumvalueofthehorizontallinecanbeatb’inthe
0 figure. Therefore, ϕ increases with decreasing r until,
R R R at a certain location r = R, there are three solutions.
1 r 2
The middle one is an unstable while the other two are
stable. Atthispoint,thecomposition“jumps”between
FIG. 2: (a) Graphical solution to Eq. (8). Solid curve is the two stable values and a discontinuity appears. At
f′ (ϕ). Its roots are the transition (binodal) compositions. suchvoltages,theprofilesarediscontinuousandtheco-
m
The intersection between f′ (ϕ) and the horizontal dashed existence between two distinct phases occurs.
m
linegivesthesolutionϕ(r)toEq. (8). ForvoltagesV below Assuming that the jump in φ occurs at the binodal
the critical value V∗, the dashed line is bounded by lines a
values, one obtains the stability criterion [19]
andb,correspondingtothethemaximalandminimalvalues
opfrotfihleerϕig(hrt)-.haAntdVsid>eVof∗E,qli.ne(8b),igsivdiinspglraicseedtotoabc’o,natninduothues ∆T = v0 ∆ε E2 . (9)
intersection is at ϕ < 0 for large r’s and at ϕ> 0 at small 2kB (cid:12)(cid:12)φc−φ0(cid:12)(cid:12)
(cid:12) (cid:12)
r’s. (b) Qualitative composition profiles φ(r). Horizontal Here E = V/(βR1) is the(cid:12)largest v(cid:12)alue of the field. A
dfiaelsdh.edφ(lirn)eviasritehsesamvoeroatghelycowmhpenosVitio<nVφ0∗,inantdhehaasbsaenschearopf mixtureofinitialhomogeneouscompositionφ0isunsta-
ble and demixes into two coexisting domains under the
jump at r=R when V >V∗.
given field if the temperature is below T +∆T, where
t
Tt(φ0)isthezero-fieldtransition(binodal)temperature
the electrostatic energy density is at composition φ0. In contrast to uniform fields, where
2 fieldvariationsresultfromcompositionvariations,here
1 V
fes =− (εc+∆εϕ) (7) field gradient are due to the non-flat geometry of elec-
2 (cid:18)βr(cid:19)
trodes. Hence,∆T aboveistypically2-100timeslarger
Note that we have used a linear constitutive relation. than ∆T in uniform fields [Eq. (6)]. Note that simi-
In uniform electric fields, such a linear relation would lar demixing is also expected to occur in a rapidly ro-
mean that the electrostatic energy is simply a constant tating centrifuge. In that case (ωr)2 is the analogue of
independent of the composition profile. In addition, in thespatially-dependentfieldE2,whereristhedistance
the wedge geometry the electrostatic energy does not fromtherotationaxisandωtheangularfrequency. The
have a term proportional to (∆ε)2 because the electric density difference ∆ρ ≡ ρ2−ρ1 replaces the dielectric
field is parallel to the dielectric interfaces (both are in contrast ∆ε [21].
the θˆdirection). Eq. (9) may be inverted to give the critical volt-
The equation that governs the composition profile age for demixing V∗ as a function of φ0 and tem-
ϕ(r), derivedfromthe Euler-Lagrangeequationδ(fm+ perature. One finds that V∗ ∝ (T − Tt)1/2. In
fes)/δϕ=0, is the following: the experiments of the Leibler group, conducted us-
ing sharp “razor-blade” electrodes, the measured ex-
2
1 V
f′ (ϕ)= ∆ε +µ . (8) ponent was 0.7±0.15, larger than the value 1/2 cited
m 2 (cid:18)βr(cid:19) here. One may write the dimensionless potential as
Hereµisthechemicalpotentialofthelargereservoirat Uw ≡ V v0ε0/(4β2kBTcR12) 1/2, where ε0 is the vac-
infinity. Note that Eq. (8) gives an analytical expres- uum perm(cid:2)ittivity. The critica(cid:3)l value of Uw for a closed
sion for r as a function of ϕ. wedge, U∗, is obtained by an approximation similar to
w
4
that of Eq. (9), namely [22] Consider, for example, the simplifications of the phase-
U∗2 = v0 φt−φ0 d2fm(φt)g(x) , (10) otrridcerciynlginedqeursa.tioInnstohcicsuarnrinnuglainr ctahpeascyitsoter,mthoef cnoon-cfleonw-
w kBTc4|∆ε|/ε0 dφ2 conditions on the inner and outer cylinders lead to
where φt is the transition composition, x = R2/R1, a vanishing flow velocity: u ≡ 0 everywhere. One
and the dimensionless function g is given by g(x) = is therefore left with only a single equation to solve,
2(x2−1)/(x2−1−2lnx). ∂φ/∂t = L∇2δf/δφ. This equation can be viewed as
a continuity equation ∂φ/∂t = −∇·J, with a current
Dynamics densityJ =−L∇δf/δφ. Inaclosedsystem,Jvanishes
The phase ordering dynamics of mixtures in electric at R1 and R2, and the integral R22πrφ(r,t)dr is kept
field is quite different from the no-field case, since the constantthroughoutthesystemRdRy1namics. Gauss’slaw
electric field introduces a preferred direction and thus readily gives the electric field in the concentric capac-
breaks the initial system symmetry. The phase transi- itor when the charge is given. In the explicit scheme
tionstudiedhereisevenmoredifficult,becausespatially we used, φ(t) is given by a successive summation of
nonuniform fields also break the translational symme- −(∇·J)dt, calculated for each time interval dt. The
try. initial condition for the calculationwas a homogeneous
In this phase transition, droplets nucleate every- distribution φ0.
where,notonlyinregionsofhighelectricfield. Asthey When the charge on the capacitor is larger than a
grow, they move under the external force. The viscos- thresholdcharge,we observefastcreationofadisconti-
ity plays an important role, in addition to the field’s nuity nearthe innercylinder whichthen startstomove
amplitude, location in the phase diagram and distance outsides. Thelocationofthisfront,separatingtheinner
from the binodal, and dielectric constant mismatch andouterregions,R(t),isshownagainsttime inFig. 3
∆ε. Clearly,the spatialdependence of the electric field (a). Inclosedsystems,Rcannotgrowindefinitely,since
meanstheinitialdestabilizationandphaseorderingdy- mass conservationdictates anupper bound Rmax given
namicsarequitedifferentfromthewell-studiednormal- by
fieldinstabilityinthinliquidfilms[4,23,24,25,26]and
the regular coarsening dynamics [27, 28]. Rm2ax = R22−R12 φ0+R12 . (15)
For salt-free mixtures, the starting point for the dy- (cid:0) (cid:1)
namics is the following set of equations [28, 29, 30]: In the numerical calculation, we find a match to an
exponential relaxation with a single time constant τ:
∂φ δf
+u·∇φ = L∇2 , (11)
∂t δφ R(t)=Rt=0+(R∞−Rt=0)(1−e−t/τ) . (16)
∇·(ε(φ)∇ψ) = 0 , (12)
∇·u = 0 , (13) R∞ correspondstothesteady-statesolution;ittendsto
∂u δf Rmax when the voltage or charge tend to infinity. The
ρ +(u·∇)u = η∇2u−∇P −φ∇ .(14) time constant τ depends on the external potential (or
(cid:20)∂t (cid:21) δφ
charge)and onthe temperature andcomposition. Part
u is the velocity field corresponding to hydrodynamic (b) plots τ as a function of temperature for different
flowandηistheliquidviscosity. Equation(11)isacon- average compositions. The calculations indicate faster
tinuity equation for φ, where −L∇(δf/δφ) is the diffu- dynamics (smaller τ) when the average composition is
sive currentdue to inhomogeneitiesof the chemicalpo- farther from the criticalvalue (large|φ0−φc|) or when
tential,andListhetransportcoefficient(assumedcon- the cylinder’s charge is large, see Fig. 3 (b).
stant). Equation (12) is Laplace’s equation, Eq. (13)
implies incompressible flow, and Eq. (14) is Navier-
Stokes equation with a force term −φ∇δf/δφ [27, 28]. MIXTURES CONTAINING SALT
It should be noted that similar models have been pro-
posed in the literature; the main differences here are Mixtures of polar liquids (e.g. aqueous solutions)
thebistabilityoff andthenonuniformfieldsderivable contain some amount of charge carriers. In such mix-
m
from the potential ψ. The presence of salt is naturally tures, the physics is rich and quite different from the
incorporated into the model by adding two continuity simple dielectric case. The most important feature is
equations for the two ionic species, and by using Pois- due to screening, occurring when dissociated ions ac-
son’sequationinsteadofLaplace’s. Asastartingpoint, cumulate at the charged surfaces. This means that
weassumethereisnonetflowduetopressuregradients the electric field is substantial only close to the sur-
ormovingsolidsurfaces–flowwillbe purelyaresultof faces, within the screening distance λ. Field gradi-
the forces exerted by the electric field. In addition, the ents thus originate from both geometry and screen-
liquid viscosity η is taken as a simple constant scalar, ing, and the phase transition depends on at least two
independent of mixture composition. lengths. The ionic screening therefore adds to the di-
Since the equations are coupled and nonlinear, it is electrophoreticforcewhichseparatestheliquidscompo-
useful to first study the demixing in one of the ge- nents fromeachother. Since screening is omni-present,
ometries mentionedabove (sphere, cylinder, or wedge). phase-separation may occur even near parallel and flat
5
1.5 (a) addition, the parameters ∆u+ and ∆u− measure the
Simulation results
Exponent fit affinity of the positive and negative ions toward the
1.45
liquid-1environment,respectively[33]. ∆u+,forexam-
1.4 ple, measures how much a positive ions prefers liquid-2
environmentoverthatofliquid1. λ± andµaretheLa-
1.35
grange multipliers (chemical potentials) of the positive
and negative ions and liquid composition, respectively,
1.3
and e is the electron charge.
R(t) 1.25 The free energy is extremized with respect to the
fields φ, ψ, and n±:
1.2
δf δf 1 δε
1.15 = m − (∇ψ)2−∆u+n+−∆u−n−−µ=0
δφ δφ 2δφ
1.1
(18)
1.05 δf = ∇(ε(φ)∇ψ)+e n+−n− =0 (19)
δψ
1 (cid:0) (cid:1)
0 50 100 150 200 250 300 350 400 450 δf
± ± ±
time δn± = ±eψ+kBT ln(v0n )+1 −∆u φ−λ =0
(cid:0) (cid:1)
(20)
(b)
τ inkeepingwith afixedmixture andionconcentrations:
V−1 φ(r)d3r =φ0 (21)
Z
T/Tc V−1 n±(r)d3r =n0 (22)
Z
Here V is the total volume and n0 the average ion
FIG. 3: (a) Plot of theR(t),thedynamicsof front location concentration. The Poisson-Boltzmann equation is ob-
between coexisting phases, for a mixture confined by two
tained from substitution of Eq. (20) in the Poisson
concentric cylinders when the charge on the inner cylinder
equation, Eq. (19).
isabovethedemixingthreshold. φ0 =0.3,T/Tc =0.95,and
Due to these forces, the phase transition is expected
thedimensionlesschargeoftheinnercylinderisU =0.445,
c
where Uc ≡ σ[v0/(4ε0kBTc)]1/2, σ is the charge per unit to be greatly enhanced compared to the no-ions case:
area of the cylinder, and R1 is its radius. The exponential it should occur at elevated temperature above the bin-
time constant τ in Eq. (16) is plotted in (b) for different odal,andleadto averythindemixinglayeraroundthe
values of φ0 and Uc. Blue circles: φ0 = 0.2, Uc = 0.252. charged object [33]. Consider a mixture in the semi-
Green stars: φ0 = 0.3, Uc = 0.199. Red squares: φ0 = infinite space x > 0 confined by one wall at x = 0
0.3, Uc = 0.252. R is scaled by R1, and time is scaled by charged at potential V. An approximate formula for
LkBT/(v0R12). thetemperaturebelowwhichaphasetransitionoccurs,
T +∆T, can be obtained by performing a first loop in
t
charged surfaces, i.e. in one dimension. But ions have a perturbative solution of the equations, namely using
another effect besides increasing the dielectrophoretic a uniform dielectric constantin Eqs. (19) and (20) and
force. Ions have in general different solubilities in the substitutioninEq. (18). Theexpressionfor∆T isthen
different liquids. As an ion drifts toward the electrode, found to be [33]:
it might “drag” with it the preferred liquid component
[31, 32]. Thus, the solubility introduces a force of elec- ∆T = |∆ε| + ∆u n0v0 exp − eV (23)
trophoretic origin, proportional to the ions’ charge. Tc (cid:18) εc kBTc(cid:19)φ0−φc (cid:18) kBTc(cid:19)
We use the following free energy density to describe
Here ∆u=|∆u±|. In most cases, ∆ε/ε ∼∆u/k T ∼
the system: c B
1, and therefore the dielectrophoretic and solubility
f = f (φ)− 1ε(φ)(∇ψ)2+ n+−n− eψ (17) forces have the same magnitudes. The numerator n0v0
m 2 is quite small: if we take v0 = 8×10−27 m3 and av-
+ kBT n+ln v0n+ +n−(cid:0)ln v0n− (cid:1)−µφ erage ion density n0 = 6×1019 m−3 (10−7M) we get
− ∆u+(cid:2)n++(cid:0)∆u−n(cid:1)− φ−λ+(cid:0)n+−(cid:1)λ(cid:3)−n−+const. n0v0 ≃5×10−7. However,∆T isusuallylarge. Evenif
we ignorethe denominator|φ −φ |−1, the exponential
(cid:0) (cid:1) c c
Thefreeenergydependsonthreefields: theelectricpo- factorcanbehuge: ifthesurfacepotentialisonly1Volt
tential ψ(r), and the two number densities of positive and T is the roomtemperature, we get eV/k T ≃40.
c B c
and negative ions: n(r)±. The new terms added here This shows us that demixing should be observed even
aretheinteractionofionswiththepotential(n±ψ)and if the surface potential V or the charge density n± are
0
the ideal-gas entropy of ions (logarithmic terms). In much smaller.
6
We would like to note that for homogeneous dielec- Acknowledgment
tric liquids, the equation ∇(ε∇ψ) = 0 means that an
increase of the potential on the bounding surfaces sim-
WethankL.LeiblerandF.Tournilhacforhelpinde-
ply increases the potential ψ proportionally, but that
velopingtheideaspresentedinthiswork. Thisresearch
for ion-containing mixtures this is not true: due to the
was supported by the Israel Science foundation (ISF)
nonlinearityoftheproblem,increaseoftheexternalpo-
grant no. 284/05, and by the German Israeli Founda-
tentialleadstoachangeinthewholedistributionψ(r).
tion (GIF) grant no. 2144-1636.10/2006.
The composition difference between coexisting phases
increases with V, and the frontseparatingthe domains
may move to larger or smaller radii.
[1] L. D. Landau and E. M. Lifshitz: Elektrodinamika
CONCLUSIONS Sploshnykh Sred, Nauka, Moscow (1957) Ch. II, Sect.
18, problem 1.
[2] M. Doi: Introduction to Polymer Physics, Oxford Uni-
The steady-state and dynamics of phase transitions
versity Press, Oxford,UK (1996).
due to inhomogeneous electric field are discussed. In
[3] One may write E1 as a sum: E1 = αkE0 +α⊥E1⊥,
nonpolarmixtures,thecompositionprofileofamixture whereE1⊥ isthefield’scomponentinthedirectionper-
isgivenforthreegeometrieswithazimuthalorspherical pendiculartoE0.IfE0 isuniform,itcanbeshownthat
symmetries. Above Tc, the profile is smooth, while be- αk isindependentofr,andhenceR εcE0·E1ϕd3r=0.
lowT itbecomesdiscontinuousifthesurfacechargeor [4] Y.Tsori: Rev.Mod. Phys.(2009).
c
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