Recent advances in liquid mixtures in electric fields

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by  Yael Katsir

Recentadvancesinliquidmixturesinelectricfields
Yael Katsir and Yoav Tsori
DepartmentofChemicalEngineeringandtheIlseKatzInstituteforNanoscaleScienceandTechnology,
Ben-Gurion University of the Negev, 84105 Beer-Sheva, Israel.
(Dated: January11,2017)
Whenimmiscibleliquidsaresubjecttoelectricfieldsinterfacialforcesariseduetoadifferencein
thepermittivityortheconductanceoftheliquids,andtheseforcesleadtoshapechangeindroplets
or to interfacial instabilities. In this Topical Review we discuss recent advances in the theory and
7 experiments of liquids in electric fields with an emphasis on liquids which are initially miscible
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and demix under the influence of an external field. In purely dielectric liquids demixing occurs if
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2 theelectrodegeometryleadstosufficientlylargefieldgradients. Inpolarliquidsfieldgradientsare

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prevalent due to screening by dissociated ions irrespective of the electrode geometry. We examine
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the conditions for these “electro prewetting” transitions and highlight few possible systems where
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CONTENTS
I. Introduction 2
II. Forceandstressinliquidsinelectricfields 4
A. Normalfieldinstabilityintwoimmiscibledielectricliquids 5
B. Theroleofasmallresidualconductivity 7
III. Changesintherelativemiscibilityofdielectricliquids 9
A. Landautheoryofcriticaleffectsofexternalfieldsonpartiallymiscibledielectricliquids 9
B. Experimentsthatfollowedinsimpleliquidsandinblockcopolymers 11
IV. Dielectricliquidsinelectricfieldgradients 13
A. Thepressuretensorandsurfacetensioninvapor-liquidcoexistence 14
B. Demixingdynamicsinliquidmixtures 16
V. Demixinginpolarsolutions 21
VI. Colloidalstabilizationbyadditionofsalt 23
VII. Porefillingtransitionsinmembranes 27
VIII. Outlook 29
IX. Acknowledgments 30
References 30
I. INTRODUCTION
Electrostatic forces are ubiquitous and their effect is important in many soft matter systems involving
liquidsboundedbyhardorsoftwalls. Theyariseonpurposeandareeasilycontrolledwhenwaterorother
solvents flow in microfluidics channels in contact with a metallic electrode whose potential is externally
controlled. Theyarelesseasilycontrolledwhenthesolventisnearbyachargednonmetallicsurfacewhich
can induce or impede the flow. In biological settings electrostatic forces determine whether proteins or
othermoleculesbindtoothermoleculesortocellularstructurewhichareoftencharged. Whencolloidsare
suspendedinsolventsthecompetitionbetweenentropicandelectrostaticforcesmayleadtointer-colloidal
3
attraction and eventually to coagulation and sedimentation of the colloids, or to repulsion between the
colloids and to stabilization of the suspension. The interplay between shear forces, surface tension and
electrostatic also plays a vital role in many industrial process where liquid droplets are transported and
ejectedviasmallorifices,asoccursforexampleinpesticidesprayinginagricultureorinink-jetprinting.
Thispapergivesaconciseoverviewofinterfacialinstabilitiesthatoccurwhenelectricfieldsareapplied
in a direction perpendicular to an initially flat interface between two liquids. Sec. IIA discusses this
normal-fieldinstabilityinpurelydielectricliquidswheretheelectrostaticforcesdestabilizingtheinterface
areproportionaltothedifferencebetweentheliquids’permittivitiessquared. Thesituationismorecomplex
when residual conductivity exists in the liquid phases and in this case mobile dissociated ions exert shear
forcesontheinterfaceandmodifyitsshape,Sec. IIB.
Section III then poses a more fundamental question: what if electric fields could affect the relative
miscibility of the two liquids? Namely, not only alter the interface but destroy it? Sec. IIIA shows the
Landautheorythataddressedthisquestionandprovedthatindeedsuchpossibilityexists. Theexperiments
supportingandcontradictingtheLandautheoryaresummarizedinSec. IIIB.
SectionIVgoesonestepfurtherandexaminessituationswhereelectricfieldgradientsactondielectric
liquids. In these systems a dielectrophoretic force acts on the liquids and, if strong enough, it may lead
to demixing of the liquids from each other. In those cases the shape, size and location of the electrodes
producing the fields are crucial for the understanding of the statics and dynamics of the phase transitions.
Peculiarly,aninterfacialinstabilityexistswheretheelectricfieldstabilizestheinterfacewhilesurfaceten-
siondestabilizesit,incontrasttothenormal-fieldinstabilityofSecs. IIAandIIB.Newexperimentalresults
ofphaseseparationdynamicsandequilibriumareshownandanalyzed.
Demixing occurs also in mixtures of polar solvents, but this time due to screening of the field which
alwaysexistirrespectiveoftheelectrodes. The“electro-prewetting”transitionsdescribedinSec. Vhavea
specific dependence on the salt content, temperature and relative composition of the mixture. The relative
miscibilityoftheionsinthesolventsplaysacrucialrole.
Aftersurveyingthebasicphysicalconceptstwo“applications”areconsidered: Sec. VIgivesanaccount
oftheelectrostaticandvanderWaalsforcesbetweentwocolloidsimmersedinapolarsolution. Thesection
details the complex interplay between these forces that depends on the relative adsorption of the liquids at
thesurfaceofthecolloids,inadditiontothetemperatureandmixturecomposition. Contrarytotheregular
Derjaguin,Landau,Verwey,andOverbeek(DLVO)behaviorinsimpleliquidsheretheadditionofionsleads
toarepulsionbetweenthecolloidsinacertainwindowofparameters. Sec. VIIconsidersanothersituation
where polar liquids are found in contact with hard surfaces: porous membranes. Pore gating between two
states can be achieved by controlling the surface potential of the membrane. This gating of membranes to
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small molecules by external potentials could be advantageous over other methods. Finally Sec. VIII is a
summaryandoutlook.
II. FORCEANDSTRESSINLIQUIDSINELECTRICFIELDS
WhenaliquidisplacedundertheinfluenceofanelectricfieldEstressdevelops. Thisstressoriginates
from the electrostatic free energy density −(1/2)E · D, where D = εE is the displacement field and ε
←→
is the local dielectric constant. Due to the vectorial nature of the field, the stress T is tensorial. For a
←→
unitsurfacewhosenormalisnˆ,theforceactingonthatsurfaceisgivenby−T ·nˆ (thei’thcomponentis
←→
−T n wherewehaveusedthesummationconventionontheindexj). Theelectricfieldhasdiagonaland
ij j
non-diagonalcontributionstothestresstensor[1–3]
(cid:18) (cid:18) (cid:19) (cid:19)
←→ 1 c ∂ε
T = −p (c,T)δ + εE2 −1+ δ +εE E (1)
ij 0 ij ij i j
2 ε ∂c
T
Herep (c,T)istheequationofstateoftheliquidintheabsenceoffield,wherecisthedensityandT isthe
0
←→
temperature. Influidsthe“regular”pressurehasadiagonalcontributionto T .
ij
←→
Thebodyforcef isgivenasadivergenceofthethisstress: f = ∂ T /∂x ,andisgivenby
i ij j
(cid:18) (cid:19)
1 ∂ε 1
f = −∇p + ∇ E2c − E2∇ε+ρE (2)
0
2 ∂c 2
T
where ρ is the charge density. The second and third terms describe electrostriction and dielectrophoretic
forceswhereasthelasttermreflectstheforcethatistransferredtotheliquidbyfreemovingcharges.
Thediscontinuityofthenormalfieldacrosstheinterfaceisobtainedas
D ·nˆ = σ (3)
(cid:74) (cid:75)
where D ≡ D(2) − D(1) is the discontinuity of the displacement field across the interface, σ is the
(cid:74) (cid:75)
surfacechargedensity,andthesurfaceunitvectornˆ pointsfromregion1toregion2. Thecontinuityofthe
tangentialfieldacrosstheinterfaceisgivenby
E ·tˆ = 0 (4)
i
(cid:74) (cid:75)
where tˆ (i = 1, 2) are the two orthogonal unit vector lying in the plane of the interface. At the interface
i
betweentworegionsofdifferentpermittivitytheforceisdiscontinuous. Thei’thcomponentofthenetforce
perunitareaoftheinterface,f ,isgivenby
s
←→
f = T n (5)
s,i ij j
(cid:74) (cid:75)
5
←→ ←→ ←→
(2) (1)
where T = T − T . When the isotropic parts of the force can be neglected [first and second
ij ij ij
(cid:74) (cid:75)
termsinEq. (2)],theelectricfieldbisectstheangelbetweennˆ andthedirectionoftheresultantforceacting
onthesurface. Thiscanbeseenbychoosingthex-axistobeparalleltoEandbynotingthatˆf ·Eˆ equals
s
ˆf ·nˆ [4].
s
The net force per unit area has three components: one in the direction perpendicular to the surface
(paralleltonˆ)andtwoindirectionsparalleltotˆ. Theyare[3]
i
←→T ·nˆ ·nˆ = 1 (E·nˆ)2−(cid:0)E·tˆ (cid:1)2−(cid:0)E·tˆ (cid:1)2−p +c∂εE2
1 2 0
2 ∂c
(cid:74) (cid:75) (cid:74) (cid:75)
←→
T ·nˆ ·tˆ = σE·tˆ , i = 1,2 (6)
i i
(cid:74) (cid:75)
InthesecondequationweusedEqs. (3)and(4).
A body force induces flow in the liquid. The Navier-Stokes equation for the the flow velocity u in
incompressibleliquidsis
(cid:20) (cid:21) (cid:18) (cid:19)
∂u 1 ∂ε 1
c +(u·∇)u = −∇p + ∇ E2c − E2∇ε+ρE+η∇2u. (7)
0
∂t 2 ∂c 2
T
Hereη isthefluid’sviscosityandtheithcomponentof∇2uis∇2u . Theterm(u·∇)uisavectorwhose
i
ithcomponentisu·∇u .
i
A. Normalfieldinstabilityintwoimmiscibledielectricliquids
Let us illustrate the force and stress in a simple example – a bilayer of two purely dielectric liquids,
1 and 2, with dielectric constants ε and ε , respectively, sandwiched inside a parallel-plate capacitor, see
1 2
Fig. 1a. The distance between the plates is L and the thickness of the first liquid is h. In this geometry
the electric fields E and E are oriented in the z-direction and are constant within the two regions. They
1 2
are found from the boundary conditions on the interface ε E = ε E (Eq. (3) with σ = 0) and from
1 1 2 2
E h+E (L−h) = E L,whereE istheaverageelectricfieldimposedbythecapacitor. Onethusfinds
1 2 0 0
that
ε E ε E
2 0 1 0
E = zˆ, E = zˆ. (8)
1 2
ε (1−h/L)+ε h/L ε (1−h/L)+ε h/L
1 2 1 2
←→ ←→
(1) (2)
From Eq. (1), when ∂ε/∂c = 0 the stresses just “below” and just “above” the interface, T and T ,
zz zz
respectively,arethengivenby
←→ 1 ε ε2E2 ←→ 1 ε ε2E2
T (1) = 1 2 0 , T (2) = 2 1 0 (9)
zz 2(ε (1−h/L)+ε h/L)2 zz 2(ε (1−h/L)+ε h/L)2
1 2 1 2
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Since these stresses are constant throughout the bulk of the liquids there is no body force of electrostatic
origin. Thedifference
←→ ←→ 1 ε ε ∆εE2
T (2)− T (1) = − 1 2 0 (10)
zz zz 2(ε +∆εh/L)2
1
givesthenetstressontheinterface. Hereweused∆ε ≡ ε −ε . If∆εispositivetheinterfaceispushed
2 1
downwardssoastodecreaseh,if∆εisnegativethentheinterfaceispushedupwards.
Undersufficientlylargeelectricfieldaninterfacialinstabilitymayoccurandthiscanbeseenasfollows.
Assume the bilayer divides into two parts, one with small value of h and one with a large value, as is
depictedinFig.1b. Inthisidealizedpictureallthreeinterfaces,markedby‘a’,‘b’,‘c’,areeitherparallelor
perpendiculartotheelectrodes. Farfrominterface‘b’thefringefieldcanbeignoredandthefieldisstillin
thez-direction. IneachdomaintheexpressionsforthefieldsstaythesameasinEq. (8).
AsEq. (10)showsthestressislargestwhenhissmallest; forincompressibleliquidsthismeansthatif
theinterface‘a’pushesdownwardsinterface‘c’will“cede”andwillmoveupwardstoconservethevolume
ofliquid1. TheconclusionisthattheinterfaceillustratedinFig.1bisnotstable; inthelongtimethefilm
willbedividedtotwoliquiddomains1and2withaninterfaceperpendiculartotheelectrodes(andparallel
tothefield). Inthisequilibriumstatethefieldsinbothliquidsareequal: E = E = E zˆ. Thestresstensor
1 2 0
←→
T from Eq. (1) is then diagonal and continuous across the interface, hence no net surface force acts to
ij
displacetheinterface.
At early times the destabilization of an initially flat interface is characterized by a fastest growing q-
modemodulationofthesurface. Leth(x,t)bethethicknessofthelayerofthefirstliquidandforsimplicity
assumethesecondliquidisgas. ForthinfilmsaPoiseuilleflowisassumedwherethex-componentofthe
flow velocity vanishes at z = 0 and is maximal at z = h. The integration of u in Eq. (7) along the z
coordinategivesaflux
h3 ∂p
− (11)
3η∂x
Thepressurehasthreecontributions[5]: oneisthedisjoiningpressuregivenbyA/6h3duetovanderWaals
forces, where A is the effective Hamaker constant of the system, the second occurs in curved interfaces
where surface tension plays a role: −γh(cid:48)(cid:48)(x), where γ is the surface tension between the two layers. The
thirdcontributiontothepressureiselectrostatic.
At the initial destabilization state the interface is only weakly perturbed and h(x) can be written as
h(x) = h +δh(x,t),whereh istheaveragefilmthicknessandδh (cid:28) h isthesmallspatially-dependent
0 0 0
perturbation growing in time. In the long wavelength approximation δh(cid:48) (cid:28) 1 and to lowest (linear) order
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inδhonecanwritethepressureas
A ε ε (∆ε)2E2
p(x) = − δh−γδh(cid:48)(cid:48)− 1 2 0 δh+const. (12)
2h4 L(ε +∆εh /L)3
0 1 0
ThesetofequationsforδhiscompletewhenoneusesEq. (12)andEq. (11)togetherwiththe“continuity”
equation for h: ∂h/∂t+∂(cid:0)−h3/(3η)∂p/∂x(cid:1)/∂x = 0. In this linear approximation one may substitute
a sinusoidal ansatz with q-number q and growth rate ω: δh = eiqx+ωt to obtain the dispersion relation
betweenω andq:
ω(q) = γh30 (cid:0)ξ−2q2−q4(cid:1) (13)
3η e
where
A ε ε (∆ε)2E2
ξ−2 = + 1 2 0 (14)
e 2γh4 γL(ε +∆εh /L)3
0 1 0
is the healing length having two contributions, from van der Waals and from electrostatics, both weighed
againstsurfacetension[5,6].
InEq. (13)thedependenceofωonqhasapositivecontributionscalingasq2andanegativecontribution
proportionalto−q4andthusforsmallqvaluesω(q)ispositiveandincreaseswithincreasingq. Thegrowth
rate ω(q) for all q’s smaller than ξ−1 is positive and they are unstable; modulations with large enough q’s,
e
q > ξ−1,arestableanddiminishexponentiallywithtime. Thefastestgrowingq-modeobeys∂ω(q)/∂q = 0
e
andhence
1 γh3
q = √ ξ−1 , ω = 0ξ−4 . (15)
fastest 2 e fastest 12η e
Which of the two forces is more dominant, the dispersion or electrostatic force? The van der Waals
pressurescalesasA/h3whereastheelectrostaticpressureis∼ εE2. IfwetakethefieldtobeE (cid:39) 1V/µm,
0 0 0
ε (cid:39) ε (ε is the vacuum permittivity), and A (cid:39) 10−20J we find that for film thicknesses h larger than
0 0 0
∼ 10nm the electrostatic force is the dominant force. For such relatively thick films the pattern period
2π/q observed in experiments scales as γ1/2/(∆εE ) and can be thus reduced if the surface tension
fastest 0
isdecreasedorifthe“dielectriccontrast”∆εorelectricfieldsareincreased[7–12].
B. Theroleofasmallresidualconductivity
In the classical experiments with liquid droplets embedded in an immiscible liquid under the influence
ofanexternalfieldthedropletselongatedinthedirectionofthefield,asexpected[13,14]. Insomecases,
however, droplets became oblate rather than prolate. Taylor and Melcher realized the importance of shear
stressduetoasmallnumberofdissolvedions[15,16]. Inperfectlyconductingliquidstheelectricfieldsare
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(a) (b)
z=L
z e c
L E2 e2 2 z=L-h
b
z=h
a
z=h
e
E e 1
1 1
z=0
FIG.1: Twoliquidsinelectricfield. (a)Schematicillustrationofabilayeroftwoliquids1and2withpermittivities
Fig. 1
ε andε ,respectively,confinedbyaparallel-platescapacitorwhoseplatesareatz =0andz =L. Thethicknessof
1 2
theliquidlayersarehandL−h. ThefieldsE andE areorientedinthez-direction. (b)Idealizedconfiguration
1 2
wherethebilayerbreaksintotwoparts,withsmall(leftside)andlarge(rightside)valuesofh. ‘a’,‘b’,and‘c’mark
thethreeinterfaces.
alwaysperpendiculartotheinterfacesandhencenoshearforceexists. Intheotherextreme,thatofperfect
dielectrics (σ = 0), the force density is perpendicular to the surface and the shear component vanishes as
well,ascanbeseenfromthecomponentoftheforceparalleltotheinterface,Eq. (6).
InTaylor’s“leakydielectric”modeltheMaxwellshearstressoftheresidualchargemustbebalancedby
a stress due to liquid flow inside and outside of the droplet. For a field alternating with angular frequency
ω much larger than the typical inverse ion relaxation time Σ/ε, where Σ is the electrical conductivity, the
behavioroftheliquidissimilartothatofapuredielectricsincetheionsmoveverylittleabouttheirplace.
When the frequency is reduced below this threshold, ω < Σ/ε, ions oscillation are large and they move
aboutmoresignificantlyasthefrequencyisfurtherreduced. InthelimitofaDCfield(ω → 0)clearlyeven
avanishinglysmallamountofionscanleadtoaverystrongresponse,recallingthatΣisproportionaltothe
ionnumberdensity.
TayloranalyzedtheelectrohydrodynamicsproblemandhisderivationleadtoafunctionΦgivenby
Φ = R(cid:0)D2+1(cid:1)−2+3(RD−1) 2M +3 (16)
5M +5
The parameters appearing M, R, and D are the ratios of the values of viscosity, resistivity, and dielectric
constant of the outer medium to that of the drop, respectively. Prolate drops are predicted when Φ > 0
whileoblatedropscorrespondtoΦ < 0. Sphericaldrops,Φ = 0,thusoccurasaspecialcase.
Based on this understanding one may ask how does residual conductivity affect the normal field insta-
bility described above? Namely how do the dispersion relation Eq. (13) and the fastest-growing q-mode
Eq. (15) change when conductivity is taken into account? It turns out that the existence of ions leaves the
general shape of the curve ω(q) intact but the maximum shifts to larger values – both the fastest-growing
q-modeanditsgrowthrateareincreased[6].
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Patterningoffilmsusingthenormal-fieldinstabilityisanappealingconceptfornanotechnologicalappli-
cationsbecauseofitssimplicityandsmallnumberofprocessingsteps[7,17,18]. Atypicalsetupinvolves
a polymer film of thickness ≈ 50-700nm placed on a substrate, and a gap with varying thickness between
the polymer and the mask. The idea is to quench the polymer structure at a specific time, at the onset of
the instability, where the most unstable mode is dominant, or at a later time, where nonlinear structures
with additional periodicities develop. Ideally one could increase the voltage and field across the substrate
and mask to decrease the period of unstable mode indefinitely. However, when the electric field increases
above∼ 100V/µm(dependingonthepolymerusedandtheoverallsamplegeometry)dielectricbreakdown
marked by a spark occurs, and current flows between the two electrodes. A possible route to decrease the
length-scale associated with the unstable mode, λ ∼ q−1 , is to decrease the surface tension γ, since
fastest
λ ∼ γ1/2 when van der Waals forces can be neglected. A smart strategy is to fill the air gap between the
polymerfilmandthemaskwithasecondpolymer,therebycreatingabilayerpolymersystem. Thesurface
tensionbetweenthetwopolymerswasindeedreducedthiswaybutthedielectriccontrast∆εwasreduced
too, and this had a detrimental effect. In addition, the leaky dielectric model has prompted researchers to
use polymers with a small conductivity or even to replace one of the polymers by an ionic liquid [8]. A
comparison between the theoretical and experimental values of the fastest growing wavelength is shown
in Fig. 2. This has led to a decrease in the feature size, though only down to a limit set by the dielectric
breakdownofthethinpolymerlayer.
III. CHANGESINTHERELATIVEMISCIBILITYOFDIELECTRICLIQUIDS
Theprecedingsectiondescribedtheinterfacialinstabilitythatoccurswhenthestressbytheelectricfield
opposes the stress by the surface tension between two existing phases. But a more fundamental question
arises: can the external field create or destroy an interface between two phases? That is, what is the effect
ofanelectricfieldontheliquid-vaporcoexistenceofapurecomponentortheliquid-liquidcoexistencefor
binarymixtures,andhowisthecriticalpointchanged? AtreatiseonthisproblemwasgivenbyLandau.
A. Landautheoryofcriticaleffectsofexternalfieldsonpartiallymiscibledielectricliquids
In the book of Landau and Lifshitz [1] the effect of a uniform electric field on the critical point was
given as a short solved problem. Unfortunately it appeared only in the first edition of the book and was
removed from the second edition by the Editors presumably because it was considered as “unimportant”.
Thisunimportantproblemhascaughtconsiderableattentioninrecentyears.
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2000
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m 1500
n
[
l

al
nt1000
e
m
ri
e
p
x 500
E
0
0 500 1000 1500 2000
Theoretical l [nm]
Fig. 3
FIG.2: Experimentalvstheoreticalfastestgrowingwavelengthinvariousexperimentswithvaryingfilm
thicknessesandvoltages. Theliquidsusedwerepolystyreneandanionicliquid. Theaveragefieldissmallerthan
139V/µm(filledsquares)andlargerthan158V/µm(opencircles). Thedottedlineistheexpectedλ =λ
th exp
relation. Thedeviationfromthislineoccursforhighfieldstrengths. Clearlythesizereductionstopsat(cid:39)400nm.
AdaptedfromRef. [8].
We illustrate Landau’s reasoning for a binary mixture of two liquids, A and B. The phase diagram is
given by T and φ, the volume fraction of A component (0 ≤ φ ≤ 1). The mixture’s free energy density
f (φ,T)includestheenthalpiccontributions,favoringseparation,andtheentropicforce,favoringmixing.
m
Theelectrostaticenergydensityisf = −(1/2)ε(φ)E2,whereε(φ)isaconstitutiveequationrelatingthe
es
localpermittivitywiththelocalcomposition. Closeenoughtothecriticalpoint(φ ,T )onecanexpandthe
c c
freeenergiesinaTaylorseriesinthesmalldeviationϕ ≡ φ−φ
c
∂f (φ ,T) 1∂2f (φ ,T)
f (cid:39) f (φ ,T)+ m c ϕ+ m c ϕ2+ ... (17)
m m c ∂φ 2 ∂φ2
1 1∂ε(φ ,T) 1∂2ε(φ ,T)
f (cid:39) − ε(φ ,T)E2− c ϕE2− c ϕ2E2 (18)
es 2 c 2 ∂φ 4 ∂φ2
Thetermslinearinϕareunimportanttothethermodynamicstatesincetheycanbeexpressedasachemical
potential. InLandau’sphenomenologicaltheoryofphasetransitions∂2f (φ ,T)/∂φ2 (cid:39) (k /v )(T−T ),
m c B 0 c
wherek istheBoltzmann’sconstantandv isamolecularvolume. Henceweseethatthequadraticterm
B 0
proportional to ϕ2 in the second line can be lumped into the first line as an effective critical temperature.
Theresultingfield-inducedshifttothecriticaltemperatureis
v ∂2ε
∆T = 0 E2 (19)
c 2k ∂φ2
B