Theory of Phase Separation and Polarization for Pure Ionic LiquidsNir Gavish*,† and Arik Yochelis*,‡

†Department of Mathematics, Technion - IIT, Haifa, 3200003, Israel‡Department of Solar Energy and Environmental Physics, Swiss Institute for Dryland Environmental and Energy Research, BlausteinInstitutes for Desert Research (BIDR), Ben-Gurion University of the Negev, Sede Boqer Campus, Midreshet Ben-Gurion, 84990,Israel

ABSTRACT: Room temperature ionic liquids are attractive to numerousapplications and particularly, to renewable energy devices. As solvent freeelectrolytes, they demonstrate a paramount connection between the materialmorphology and Coulombic interactions: the electrode/RTIL interface is believedto be a product of both polarization and spatiotemporal bulk properties. Yet,theoretical studies have dealt almost exclusively with independent models ofmorphology and electrokinetics. Introduction of a distinct Cahn−Hilliard−Poisson type mean-field framework for pure molten salts (i.e., in the absence ofany neutral component), allows a systematic coupling between morphologicalevolution and the electrokinetic phenomena, such as transient currents.Specifically, linear analysis shows that spatially periodic patterns form via a finite wavenumber instability and numericalsimulations demonstrate that while labyrinthine type patterns develop in the bulk, lamellar structures are favored near chargedsurfaces. The results demonstrate a qualitative phenomenology that is observed empirically and thus, provide a physicallyconsistent methodology to incorporate phase separation properties into an electrochemical framework.

High energy consumption stresses the need for thedevelopment of renewable and clean devices that will

ease the transition from reliance on fossil fuels towardindependent alternatives. Room temperature ionic liquids(RTILs) are attractive to several technological applicationsand in particular to renewable energy devices,1−5 due to theirhigh charge density and tunable anion/cation design, low vaporpressure, and wide electrochemical windows. RTILs are moltensalts and thus propose superior properties to electrolytesolutions when incorporated into devices such as batteries,supercapacitors, dye-sensitized solar cells. Even though RTILsresemble highly concentrated electrolytes, they also exhibitfundamental physicochemical differences: (i) electrical doublelayer (EDL) structure exhibits in many cases charge layering,crowding, and overscreening effects that do not arise intraditional (dilute) electrolytes;6−10 (ii) important classes ofRTILs also exhibit a nanostructure that displays lamellar,bicontinuous, and sponge-like morphologies with polar andapolar domains of order of nm size in the bulk.11−16 Theseordered nanodomains apparently arise from electrostaticinteraction between the oppositely charged ions and/or frominteractions between the large functional groups, e.g., hydrogenbonding or solvophobic/solvophillic interactions.13,14,17,18

The bulk and interfacial nanostructuring of the RTIL isfundamental for device efficiency as it controls the time scalesof ion transport (e.g., conductivity) and charge transfer19 orcapacitance.20 The current theoretical studies of RTIL mostlydevote their efforts to either interfacial nanostructuring byelectrode polarization21−23 or to bulk nanostructuring.24

However, experimental data shows that interfacial RTILnanostructure can be also a consequence of both surface-

specific and bulk liquid interactions.25 Thus, together withadditional empirical and theoretical evidence, a combinedtheory of RTIL bulk and interfacial nanostructuring is required(cf. ref 26 and the references therein). Atomistic (force field)methods such as molecular dynamics, allow access to relativelyrealistic properties of RTILs. These methods, however, arelimited to relatively small systems due to a finite number ofmolecules that can be traced simultaneously.27,28 On the otherhand, although mean−field formulations do not offer atomisticinsights, they are amenable to extended analytical andnumerical computations and thus may provide fundamentalunderstanding of the system behavior at larger scales.Respectively, mean-field modeling of bulk nanostructure was

advanced by formulating the problem via a gradient flowapproach, such as Flory−Huggins.29,30 These models predictamphiphilic-type bulk structures, in qualitative agreement withempirical data.11,12,15,16 Nevertheless, these models do notaccount for an external electrical field and thus do not provideinsights to the interfacial nanostructure. On the other hand,charge layering of the EDL was captured via incorporation ofthe over−screening effect31,32 through the so-called Bazant-Story-Kornyshev (BSK) framework, but while assuming astructureless bulk. These equations allowed insights into theion-pairing evolutions puzzle33,34 and moreover, stressed theneed in connecting the EDL structure with the bulk nature.35,36

In this Letter, we develop a mean-field theory that combinesphase separation and ionic transport via an Onsager frame-

Received: February 18, 2016Accepted: March 8, 2016

Letter

pubs.acs.org/JPCL

© XXXX American Chemical Society 1121 DOI: 10.1021/acs.jpclett.6b00370J. Phys. Chem. Lett. 2016, 7, 1121−1126

work,37,38 while incorporating finite size (a.k.a steric) effectsand Coulombic interactions. The resulting model bearssimilarity to the Ohta-Kawasaki formulation for morphologydevelopment driven by competing short-range, and long-rangenonlocal interactions,39−41 and in fact is complementary to theBSK approach31,32 in the limit of absence of solvent or otherneutral subsets, such as ion pairs.34 Using linear stabilityanalysis, we show that the bulk morphology emerges via a finitewavenumber instability type.42 Below the onset and uponapplied potential, we observe also the crowding effect, which isconsistent with the BSK model. Moreover, numericalsimulations show that the same phenomenology persists inhigher space dimensions. Finally, we address implications toempirical observations and the outlook for other electro-chemical systems.Coupling Phase Separation and Polarization. Pure RTILs (i.e.,

fully dissociated molten salts) are fundamentally distinct fromdilute electrolytes, which obey the Poisson−Nernst−Planckdescription, due to the absence of a solvent, such as water forthe KCl⇌ K+ + Cl− salt. Consequently, we start by consideringa system of a symmetric RTIL of monovalent anions (n) andcations (p), confined in between two flat parallel electrodes.The salt ions are assumed be fully dissociated and thus, theirlocal volume fractions, 0 ≤ (p, n) ≤ 1, preserve the uniformdensity constraint:

+ =n px x( ) ( ) 1 (1)

where ∈ x 3.In the absence of a solvent (namely, large concentration

gradients) and since the molecules are charged, the drivingforces for ionic transport and structural evolution of the RTILare attributed to short-range interactions and to long-rangeCoulombic interactions. Thus, the mean-field free energy of thesystem is given by

= +E E ECH C (2a)

where

∫ κ= + |∇ | + |∇ |E c f p n

Ep n x( , )

4( ) dmCH

02

2 2

(2b)

∫ ϕ ϕ= − − ϵ|∇ |E qc p n x( )12

dC2

(2c)

and

β= + +⎡⎣⎢

⎤⎦⎥f p n k T p

pn

nnp( , ) ln

2ln

2m B (2d)

Here, c is the concentration of the anions and of the cations inthe mixture, kB is the Boltzmann constant, T is the temperature,β is interaction parameter for the anion/cation mixture and hasunits of energy, and E0κ

2/4 is the gradient energy coefficientwhere E0 has units of energy and κ has units of length. We notethat entropic terms describing the solvent, voids, or neutralsubsets for concentrated electrolytes, such as the Bikermanterm31,43−45 ∝ (1 − p − n) ln(1 − p − n), are commonly usedto account for steric effects in BSK and other mean-fieldmodels, but are not appropriate for pure molten salts due tocomplete absence of any subsets besides charged ions (see eq1). Therefore, higher order terms are required to describe ion−ion steric effects. Indeed, in (2a), the first term ECH stands forthe Cahn−Hilliard energy and accounts for the energetic costof short-range interactions including anion/cation mixing and

for composition inhomogeneities.46−48 When β > βc = 2kBT,the function fm(p, n ≡ 1 − p) takes the form of a double wellpotential, thus driving phase separation. The secondcomponent, EC of eq 2a, accounts for long-range Coulombicinteractions where ϕ is the electric potential, q is theelementary charge, and ϵ is the permittivity. Requiring that ϕis a critical point of the action yields Poisson’s equation,

δδϕ

ϕ= − + ϵ∇ =Eqc p n( ) 02

(3)

Notably, eqs 2b and (3), comprise the Ohta−Kawasaki freeenergy, a nonlocal Cahn−Hilliard model for the structure ofdiblock copolymers mixtures.39−41

The uniform density constraint (1) for RTILs, implies thation transport is governed by an interdiffusion process and notby a standard diffusion, i.e., not by a random walk of isolatedions in a solvent that leads to Einstein-Stokes relations.Following Onsagers’ framework,37 the Cahn−Hilliard−Poisson(CHP) equations of motion for cations and anions read:

δδ

δδ

∂ = ∇· ∇ ∇⎡⎣⎢

⎤⎦⎥

⎛⎝⎜⎜

⎡⎣⎢

⎤⎦⎥

⎞⎠⎟⎟

pn

Ep

En

L x( ) ,t

T

(4a)

where L(x) is the 2 × 2 matrix of coefficients and thesuperscript T stands for transpose. Onsagers’ reciprocalrelations combined with (1), lead to the choice of the Onsagermatrix38

=

−−

⎡⎣⎢

⎤⎦⎥

Mp nc

L xx x

( )( ) ( ) 1 1

1 1 (4b)

where M is the mobility coefficient.49,50 Introduction of thestandard dimensionless variables

ϕ ϕλ

λτ

τ λ

= = =ϵ

=

=

qk T

xk Tq c

tt

Mk T

x, ,

2, ,

B

B2

2

B

leads, after omitting the tildes, to the dimensionless form of eqs3 and 4:

ϕ∇ = − p1 22(5a)

and

σ χ ϕ

∂ = −∇· ∂ = −∂

= − ∇ − ∇ +−

+ − +⎛⎝⎜

⎞⎠⎟

p n p

p p pp

pp

J

J

, ,

( 1) log1

(1 2 ) 2

t t t

2

(5b)

Here, σ = κλ

Ek T

0

B

2

2 controls the competition between short- and

long-range interactions, and χ = β/(kBT) is the Floryparameter. For consistency with traditional dimensionlessanalysis, we have chosen to scale x by the Debye-like scale λwhile noting that this choice does not reflect a typical electricscreening length, as for dilute electrolytes. We further note thatthe resulting system (eqs 5) is distinct from the Otha−Kawasaki model by its dissipation mechanism.Equations 5 are supplemented with boundary conditions of

fixed potential at the electrodes and Neumann (i.e., no-flux) forp and n:

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1122

ϕ ϕ= = − = = | =∂Ωx V x d V J( 0) /2, ( ) /2, 0(6)

where ∂Ω is the volume boundary and d is the distancebetween electrodes; at the boundaries of the rest of the volume(that is in the y−z planes), we take a Neumann boundarycondition, i.e., ∇ϕ|∂Ωy−z = 0.Bulk Stability, Polarization, and Electrokinetics. For the sake of

analysis, we first consider an infinite one-space dimensionaldomain for which the electrode polarization effects vanish.Under such conditions, eq 5b reduces to

σ χ ϕ∂ = ∂ − − ∂ − ∂ + ∂ + ∂p p p p p p[(1 ) ( 2 2 )]x x x x xt3 2

(7)

Next, we substitute the expansions

∑ ∑ε ϕ ε ϕ= + ==

∞

=

∞

p p p ,i

ii

i

ii0

1 1 (8)

into eq 7 and collect terms up to the first order in ε ≪ 1.Noting that terms of the type (∂x p) (∂x ϕ) ∼ o(ε2), we identifyusing

= ++p e c.c.st ikx1 (9)

the stability properties of the equilibrium uniform state (p0, ϕ0)= (1/2, 0); s is the temporal growth rate of perturbationsassociated with wavenumbers k.42 The resulting dispersionrelation reads:

χ σ= − + − −⎜ ⎟⎛⎝

⎞⎠s k k1

21

42 4

(10)

The uniform state is linearly stable if for all k, s < 0, while theinstability corresponds to a band of wavenumbers for which thegrowth rate becomes s > 0; the dispersion relation here isalways real. The instability onset is obtained by seeking for acritical wavenumber for which s(k = kc) = 0, s(k ≠ kc) < 0 andds/dk = 0. Accounting for these conditions, we find that for χ >2 and σ = σc, the instability is of finite wavenumber type with

σ χχ

= − =−

k( 2)

4and

22c

2

c(11)

as demonstrated in Figure 1. In fact, finite wavenumberinstability is known to arise in the Otha−Kawasaki typemodels,51 but it is distinct from generalized Poisson−Boltzmann type systems, which are characterized by a stable

uniform bulk.36 In the latter case, structure formation may bedriven by boundary conditions or external forces, e.g., theformation of the double layer structure near a charged wall inthe Poisson−Boltzmann model.Next, we consider a finite domain and apply boundary

conditions (6). Indeed, numerical integrations of eq 7 showthat, in addition to the EDL crowding (plateau region near theboundaries) and over-screening effects (spatially decayingoscillations), as shown in Figure 2a, there is a bifurcation to a

periodic structure, as shown in Figure 2b. The bifurcation is of asupercritical type, i.e., the amplitude of the emerging solutionsscales as σ σ∼ −p c (details will be given elsewhere), andemerges also without any potential difference at the walls.42

Notably, finite wavenumber bifurcation cannot mathematicallyarise in the BSK model31,32,36 (which belongs to the Poisson−Boltzmann class), namely, the bulk will preserve spatialsymmetry. The typical length-scale of the spatial oscillationsis attributed to the molecular characteristic sizes andinteractions that are being introduced to the free energy (seeeq 2b). Consequently, the lower bound that results here isabout angstroms, where upon increasing, for example χ, thespatial oscillations become of the order of nanometer, namely,the spatial oscillations (decaying or persisting) near theinterface have the same quantitative characteristics as for theBSK model equations.31 More generally, at large appliedvoltages, the BSK model is only capable of describing interfaces(heteroclinic-type connections) between the plateau region

Figure 1. Dispersion relations (eq 10), showing the finite wavenumberinstability onset at σ = σc = 1/4 (solid line), below the onset σ = 0.255> σc (bottom dashed line), and above the onset σ = 0.24 < σc (topdashed line); in all three simulations, χ = 3.

Figure 2. Asymptotic numerical solutions of model eqs 5 in 1D, below(a) and above (b) the instability onset, showing the asymptotic statesfor χ = 3 and (a) σ = 0.255 > σc = 1/4, (b) σ = 0.24 < σc; the initialstate was taken as spatially uniform (p, ϕ) = (0.5, 0). The boundaryconditions are no-flux for p and a fixed potential difference of 20 (V =20) for ϕ; see eq 6). Both cases demonstrate, in addition, crowdingand over−screening effects near the boundaries. The integration wasperformed using the commercial software COMSOL 5.2 with anequally discretized grid Δx = 0.05.

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(crowding) and the uniform bulk,36 while the CHP equationsdescribe in addition a transition between heteroclinicconnections to fixed points below the onset and once theonset is exceeded, forming heteroclinic connections between afixed point (crowding) and a (spatial) limit cycle (bulk region).The latter property implies that the interface width dictates theextent of the electrode polarization impact.Morphology existence within the bulk also has a distinct

impact on the dynamical properties, such as charging/discharging of the EDL. To demonstrate the differences, wecompute the transient currents via voltage reversal below andabove the instability onsets, as shown in Figure 3(a) and (b),

respectively. The currents are computed via numericalintegration of the flux over the entire domain35

∫ σ χ ϕ= | − − ∂ − ∂ + ∂ + ∂ |−

I tL

p p p p p x( )1

[(1 ) ( 2 2 ) ] dL

L

x x x x/2

/23

(12)

using profiles that are also obtained numerically at each timestep from eq 7. Below the instability onset (σ > σc), the currentshows a standard exponential decay, as shown by the semilogplot in Figure 3a. Above the onset (σ < σc), the primaryexponential decay is followed by a slow relaxation; see times160 < t < 1000. This additional relaxation is not exponentialand is due to weak modulations of the bulk structure beforereaching an equilibrium. In particular, in the absence of bulkinstability, such additional relaxations cannot appear and thus,this prediction suggests a new empirical method for detectingthe presence of bulk nanostructure.Finally, we note that, as expected, the finite wavenumber

instability persists also in higher space dimensions (with d ≫λ). Here, however, the 1D periodic pattern becomes a lamellarone and thus is subjected to a transverse secondary phaseinstability of zigzag, similar to the Ohta−Kawasaki model.51Consequently, since the equations are derived for a symmetriccase, random initial conditions may result in labyrinhtinepatterns depending on the distance from the primary onset, i.e.,σ = σc. Namely, the stripes become sensitive to zigzag instabilitywith an increasing distance from onset σ = σc, where thedependence can be obtained via the construction of the Busseballoon,42 and will be detailed elsewhere. Figure 4 shows threeasymptotic lameller (Figure 4a), mixed (Figure 4b), andlabyrinthine (Figure 4c) patterns, which dominate even in theEDL region, at three distances from the onset σ < σc(χ = 4) =1. We further note that, integration of p(x, y) over y will resultin a vanishing charge density (e.g., Figure 4c), so that theresulting p x( ) profile will by resemblance be assumed to be a1D asymptotic solution below the instability onset with nomorphology, e.g., Figure 2a. However, even though theasymptotic solutions may resemble the 1D case and also theprofiles obtained via BSK theory, it is a distinct solution withdistinct properties; for example, the respective time scales uponapproaching them will differ. Recovering these distinct timescales in eqs 5 is a prerequisite to a systematic comparison with

Figure 3. Transient currents evaluated numerically by integrating eq 7at (a) σ = 0.3, below the onset and (b) σ = 0.2 above the onset. Theinitial conditions are similar to the asymptotic solutions in Figure 2,respectively, and for both cases the applied voltage was reversed, V =−20. Other parameters and numerical conditions are as in Figure 2.

Figure 4. Numerical solution of eqs 5 in 2D at increasing distance from the onset σc = 1. Graphs present p(t,x, y) at t = 500 000 for χ = 4 withpotential ϕ(±50, y) = ∓V/2 = ∓10: (a) σ = 0.9, (b) σ = 0.6, and (c) σ = 0.4. All three solutions arise from the same random initial data, and havereached steady state. Light and dark colors mark the upper (p = 1) and the lower (p = 0) limits, respectively. The integration was performed usingthe commercial software COMSOL 5.2 with minimum finite triangular element size of 0.1 on a physical domain size of 100 × 100.

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empirical observations and for determining strategies to tailorby-demand RTIL compositions.Discussion. Electrical diffuse layer in RTILs displays a puzzling

charge layering near the liquid/solid interface. This uniqueproperty stimulated recently intensive studies and distinctapproaches ranging from electrode polarization effects to bulkmorphology based descriptions.21,24 However, since under-standing of EDL properties is required to control and optimizecharge transport and transfer in numerous energy conversiondevices, a study of the device-wise mechanisms governing RTILstructure was moved to a spotlight26 and the references therein.To capture both the polarization effects and the bulkproperties, we consider a mean-field framework that couplesshort-range interactions and long-range Coulombic interactionsbetween the ions. The approach keeps a qualitative fidelity tothe physicochemical empirical observations, within thethermodynamic Onsager system framework. Specifically, weuse phase separation of Cahn−Hilliard-type coupled to Poissonequation to unfold the origin of the emerged spatially periodicand isotropic bulk morphology, which then becomes ordered(anisotropic) near the electrode surface (see Figure 4; cf. refs24 and 26). As expected, the results show tight couplingbetween bulk-nanostructure and EDL structure. For example,zigzag instability of the bulk nanostructure impacts and mayeven dominate the EDL structure itself (see Figure 4c).Our approach considers the limit of pure molten salts, which

is complementary to the BSK approach,31,32 that is apparentlyonly appropriate in the presence of additional neutralsubsets,31,43,44 e.g., ion pairing or dilution by organic solvents.Second, the model mechanistically describes the emergence ofbulk nanostructure. Notably, as shown in refs 35 and 36, bulknanostructure cannot arise in the BSK model. It is possible tofurther generalize the framework by merging the approachpresented here with the BSK model to address, for example, ionintercalation.52 However, the short-range electrostatic correla-tion length scale that was introduced as a fourth-orderderivative in the Poisson’s equation, contributes only at higherorders and thus will have no qualitative impact on the results ofthe CHP model. Third, while in the existing literature mean-field models of RTILs the ions are assumed to incorporate theEinstein−Stokes relations,31,32 in purely molten salts, theabsence of “solvent” inherently relates ion transport tointerdiffusion while Coulombic interactions enter naturallythrough Poissons’ equation. However, the detailed effects ofinterdiffusion and bulk nanostructure on bulk transport andmolecular properties in RTILs are beyond the scope of thisstudy and will be addressed elsewhere. Fourthly, we havesuggested a mean field formulation; however, as also in othercontexts, our model equations miss atomistic interpretations,which are essential to quantitative estimation of parameters ineqs 2 and a detailed comparison to available experiments.24 Webelieve that designing molecular dynamics force fields will allowbridging and better understanding of the nature of the ion−ioninteractions in RTIL and the mechanisms that dominate in pureRTILs, such as chemical reactions that were advanced byBazant.52

In general, our formulation bears similarity to the well-studied Ohta−Kawasaki energy for diblock copolymermixtures. Specifically, a similar instability have been alsostudied for the Ohta−Kawasaki model and Newell-White-head-Segel equation together with the respective Busse balloonhave been indeed derived.51 The presence of the zigzaginstability is also responsible for labyrinthine bulk morphology

in 1:1 symmetric RTILs. Once this symmetry is broken, e.g.,due to different ion sizes, we expect that also othermorphologies that were observed in the Ohta−Kawasakimodel will form,53 such as spheres, circular tubes, andbicontinuous gyroids patterns. In particular, since RTIL ionsare not chemically bonded like the diblock copolymers, theinterplay between morphology and long-range interactions(electrokinetics in the case of RTILs) is richer and will bestudied in detail elsewhere. As such, we expect that the platformdeveloped here, can be extended to a much wider range ofmaterial science and energy conversion applications thatinvolve coupling between material nanostructure, electrostatics,and electro-diffusion, such as packed colloidal media.54

■ AUTHOR INFORMATION

Corresponding Authors*E-mail: ngavish@tx.technion.ac.il; Phone: +972 (4)8294181;Fax: +972 (4)8294181 (N.G.).*E-mail: yochelis@bgu.ac.il; Phone: +972 (8)6596794; Fax:+972 (8)6596736 (A.Y.).

NotesThe authors declare no competing financial interest.

■ ACKNOWLEDGMENTS

We thank Martin Bazant for helpful comments and discussions.This research was done in the framework of the GrandTechnion Energy Program (GTEP) and of the BGU EnergyInitiative Program, and supported by the Adelis Foundation forrenewable energy research. N.G. acknowledges the supportfrom the Technion VPR fund and from EU Marie−Curie CIGGrant 2018620.

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