2836 | Phys. Chem. Chem. Phys., 2014, 16, 2836--2841 This journal is© the Owner Societies 2014

Cite this:Phys.Chem.Chem.Phys.,

2014, 16, 2836

Transition from non-monotonic to monotonicelectrical diffuse layers: impact of confinementon ionic liquids

Arik Yochelis

Intense investigations of room temperature ionic liquids have

revealed not only their advantages in a wide range of technological

applications but also triggered scientific debates about charge

distribution properties within the bulk and near the solid–liquid

interfaces. While many observations report on an alternating

charge layering (i.e., spatially extended decaying charge density

oscillations), there are recent conjectures that ionic liquids bear

similarity to dilute electrolytes. Using a modified Poisson–Nernst–

Planck model for ionic liquids (after Bazant et al., Phys. Rev. Lett.

2011, 106, 046102), we show that both behaviors are fundamental

properties of ionic liquids. The transition from the non-monotonic

(oscillatory) to the monotonic structure of electrical diffuse layers

appears to non-trivially depend on ionic density in the bulk, electro-

static correlation length, confinement and surface properties.

Consequently, the results not only reconcile the empirical results

but also provide a powerful methodology to gain insights into the

nonlinear aspects of concentrated electrolytes.

Room temperature ionic liquids (RTILs) have become an increasingsubject of investigations due to their unique physicochemicalproperties and their advantages in numerous technologicalapplications.1–11 Fundamental and yet unclear characteristicsof RTILs concern the spatial organization of anions and cationsin the bulk and at liquid–solid interfaces, both significant toelectrochemical applications.1–7,12 While many studies have reportedon novel self-assembled charge layering near surfaces,13–23 it wasrecently conjectured that nevertheless RTILs bear similarity to diluteelectrolytes,24 where decrease in conductivity and monotonic decayof the electrical diffuse layer (EDL) was explained by ion association.The latter observation implies fundamental controversy25,26

regarding the basic nature of RTILs and thus the subject thatwe attempt to advance in this study.

The charge layering phenomenon at the liquid–solid inter-face can be viewed as a non-monotonic (spatially oscillatory)EDL displaying an alternating segregation of cations and anionsthat decay toward the bulk region.27–35 For metal surfaces thislayering was attributed to ‘overscreening’ of the electrodepotential.19,36,37 However, a similar behavior is also found inexperiments employing confined RTILs with surface charge andno applied voltage,33 suggesting that electrical ordering is ageneric meso-scale self-assembly property of RTILs.37 In general,the physicochemical origin of such phenomena is a combinationof many interactions (electrostatic, van der Waals, hydrogen-bonding, etc.) that result between two confined solid surfaces38

making it difficult to construct an intuitive theoretical frame-work as the macroscopic Poisson–Nernst–Planck (PNP) typeformulation.26,39 In this direction, Bazant et al.36 have proposeda modification for the PNP equations by taking into accountboth steric effects40–43 and electrostatic correlations.37,44,45 Theinclusion of the latter showed that the EDL can exhibit spatialoscillations (due to overscreening) under both high and lowapplied voltages and as such, these oscillations are independentof the condensed layer that forms near the surface at highapplied voltages due to steric effects (crowding).36

To investigate (and reconcile) the RTIL spatial properties, weimplement spatial dynamics methods46,47 and show that inconfined ionic liquids electrostatic correlations are not the onlydriving force and that the EDL structure also depends on ionicdensity in the bulk. The analysis exploits the methodology developedfor reaction–diffusion–migration systems, i.e., the selectionmechanism of spatial patterns using boundary conditions.46

Through a semi-phenomenological modified PNP model,36,48,49

we show the existence of spatially extended oscillations alreadyin the bulk even in the absence of any confinement. On infinitedomains these oscillations are temporally unstable and beingstabilized for example, under applied voltages. In addition, underapplied voltages, the charges are attracted and accumulated nearthe surfaces and thus reduce the ion density within the bulk.50

Consequently, depletion of ions within the bulk and the extent ofelectrostatic correlations may lead to monotonic EDL structure.

Department of Solar Energy and Environmental Physics and Ben-Gurion National

Solar Energy Center, Swiss Institute for Dryland Environmental and Energy

Research, Jacob Blaustein Institutes for Desert Research (BIDR), Ben-Gurion

University of the Negev, Sede Boqer Campus, Midreshet Ben-Gurion 84990, Israel.

E-mail: yochelis@bgu.ac.il

Received 27th November 2013,Accepted 16th December 2013

DOI: 10.1039/c3cp55002h

www.rsc.org/pccp

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The two cases are schematically described in Fig. 1. Temporalimplications near the transition onset, in the form of transientcurrents (a.k.a. chronoamperometry), are also discussed. Impor-tantly, this approach enables consideration of both applied voltagecases, symmetrically charged surfaces and relatively small confine-ments, which are of interest in most applications.

Model equations

We start with a brief description of the modified PNP model for abinary (1 : 1) electrolyte (with identical anion–cation mobilities)that showed capture of the non-monotonic EDL structure,36,48,49

i.e., additionally to the standard transport fluxes for positive (C+)and negative (C�) subsets, the model incorporates steric effectsand short-range electrostatic correlations. In the dimensionlessform, the model reads as:

@c

@t¼ @

@xJc ¼ @

@x

@c

@xþ r

@j@xþ nc1� nc

@c

@x

� �; (1a)

@r@t¼ @

@xJr ¼ @

@x

@r@xþ c

@j@xþ nr1� nc

@c

@x

� �; (1b)

lc2 @

2

@x2� 1

� �@2j@x2¼ r; (1c)

where c(x,t) is the local ion density, r(x,t) is the localcharge density, j(x) is the electric potential, lc is the ratiobetween the electrostatic length scale and Debye length

lD ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffieRT= 2F2C0ð Þ

p� �, R is the ideal gas constant, T is

the absolute temperature, F is the Faraday constant, e is thedielectric permittivity and n characterizes the ability of theliquid to compress (we use here n = 0.5), i.e. molecular packing;for details and physical interpretations of the model, we referthe reader to ref. 49 and the references therein. Eqn (1) wereobtained from the original equations by rescaling the spatialscale by lD, time by lD

2/D (where D is the diffusion coefficientthat is taken here to be equal for anions and cations), potentialby RT/F and defining c = (C+ + C�)/(2C0), r = (C+ � C�)/(2C0),where C0 = C� is the ion concentration of each in the absence ofthe electrical field. Eqn (1) is supplemented by boundaryconditions (BCs) that correspond to inert (no charge transfer)electrodes and fixed potentials.

Numerical solutions of the above model showed that thespatially oscillatory nature of the EDL is independent ofthe applied voltages.36 This observation in turn, impliesan inherited oscillatory structure that is already ‘hidden’within the bulk, as has been shown in reaction–diffusion–migration systems, for example.46,47 Thus, the knowledgeregarding the spatial bulk structure is an important missingpiece in the mechanism behind EDL emergence. To under-stand the spatial structure of the EDL, we first consider aninfinite bulk while the impact of the boundary conditionsfollows at later stages.

Spatially oscillatory structure in aninfinite bulk

To study the nonuniform spatial properties within the bulk weemploy the spatial dynamics methods, i.e., due to flux con-servation; we rewrite (after some algebra) eqn (1) as a set ofordinary differential equations:

c0 = (1 � nc)Er, (2a)

r0 = (c � nr2)E, (2b)

E0 = u, (2c)

u0 = w, (2d)

w0 = lc�2(u � r), (2e)

where E = �qxj and primes denote derivatives with respect tothe spatial coordinate.

In the absence of applied potential, eqn (1) admit a uniformsteady state (c,r,f) = (c*,0,0). With this respect, weak spatial

Fig. 1 Schematic representation of alternating charge layering (top andbottom) and monotonic decay (middle) for an RTIL under oppositely (topand middle) and equally (bottom) applied external potential. Grey colormarks approach electroneutrality (vanishing net charge density) and L isthe separation distance.

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deviations about the uniform state can be viewed as a spatial‘instability’46,54 that obeys the following form:

c

r

E

u

w

0BBBBBBBBBB@

1CCCCCCCCCCA�

c�

0

0

0

0

0BBBBBBBBBB@

1CCCCCCCCCCA/ delx þO d2

� �; (3)

where |d| { 1 is an auxiliary parameter. By inserting (3) into (2)and solving for l, we obtain five eigenvalues:

l0 = 0, (4a)

l�2 ¼1

2lc21�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� 4lc2c�

p� �: (4b)

The trivial eigenvalue corresponds to the translational invarianceof the concentration (c*). The other four eigenvalues are eitherreal or complex conjugated, depending on the relation betweenthe electrostatic correlation length and the ionic concentrationaccording to:

lc� ¼ 1

2ffiffiffiffiffic�p : (5)

Notably, eqn (4b) is identical to an anzats that was usedfor low applied voltages (V o RT/F) and electro-neutrality r = 0,c* = 1.49 Under these assumptions the transition between weakand strong electrolytes occurs at lc* = 1/2, which is in agreementwith (5).

The distinct configurations of the spatial eigenvalues arepresented in Fig. 2. At the onset lc = lc* (right inset), there isa double multiplicity at the real axis, for which l+ = �l�. Forlc o lc* (left inset), the four eigenvalues are real while for lc > lc*(top inset), the eigenvalues become complex conjugated. Underan applied voltage, since Iml� corresponds to a wavenumberand the potential is antisymmetric [j(x) = �j(�x)], the formercase corresponds to a monotonic approach toward x - 0� whilethe latter case corresponds to spatially decaying oscillations.

This behavior can be numerically verified (including the resultingperiod of the oscillations, 2p/Iml�) by assuming a large domain,L c 1 (in dimensional units L c lD and c* (x - 0�) - 1), asshown in Fig. 3 for lc = lc* = 0.5 and lc = 10 > lc* [the oscillationsamplitude p<el�

�1 increases with lc, as given by (4b)]. Note thatwithout any applied voltages (as BCs), these deviations decay intime to a uniform state.

The lower limit for electrostatic correlations is of order of theBjerrum length which due to large ion sizes is of order of nm. Now,for fully dissociated ions of RTILs lD B Å for which lc > 1 andthus the criterion (5) is always fulfilled. The latter leads to theemergence of an oscillatory EDL structure, as has been alsoobserved by more refined computations.37 However, followingthe recent observations, ions in RTILs may not obey fulldissociation.24 Namely, under ‘‘dilution’’ of RTILs lD increasestowards the nm size and thus lc significantly decreases,51,52

i.e., lc r 1, and thus depletion of ion density, contributessignificantly to the spatial structure of the EDL. For largeseparation distances between electrodes, the bulk impact isnegligible; however, bulk ion density does become crucialunder small confinements, as will be demonstrated next.

Diffuse layer emergence underconfinement

Here we consider confinements of moderate separation distances[L/2 B O(10–100)] (in dimensional units L/2 B O(10–100 lD)) andapplied voltages V B O(10) (in dimensional units V B O(10 RT/F).To show that the above analysis also holds under these condi-tions, we perform numerical integrations and supplementeqn (1) with boundary conditions at x = �L/2:36,48,49

J c = J r = 0, (6a)

j = �V, (6b)

@3j@x3¼ 0; (6c)

which corresponds to inert electrodes (i.e., no charge transfer) andconstant potential with no electrostatic correlations at solid surfaces.The initial conditions c(x,0),r(x,0),j(x) are (c,r,f) = (1,0,2Vx/L).

Fig. 2 Spatial ‘instability’ onset (5) that marks the transition from non-monotonic to monotonic EDLs. At the onset, the spatial eigenvaluescorrespond to one trivial and a double multiplicity at the real axis(right inset). Above the onset, the eigenvalues are complex (top inset)implying spatial oscillations while below the onset all the eigenvalues arereal (left inset) implying spatially monotonic decay. The eigenvalues aregiven by eqn (4).

Fig. 3 Demonstration of typical solutions in the bulk region above and atthe onset, i.e., non-monotonic and monotonic behaviors. Eqn (1) wereintegrated numerically on a large domain, with c* = 1 fixed at x = 0 andlc as marked in the figure. The inset zooms into the x - 0 region todemonstrate that the oscillations persist all the way.

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Asymptotic solutions of eqn (1) show that near the electrodesthere is a mass/charge saturation due to steric constraints,36,40,41 aso called ‘crowding’ effect, according to c = r = n�1. Naturally, thelarger the applied potential the wider is the saturation plateau36

since more ions are being attracted towards the surface.50 On theother hand, widening of the plateau regions (near the electrodes)reduces c in the bulk. This enables the densities and potential thatdecay toward the bulk to be either monotonic or non-monotonic(oscillatory) depending on the value of lc, as shown in Fig. 4.The transition from non-monotonic to monotonic EDLs isfound to correspond well to the criterion given by (5): thereexist lc for which the ion density in the bulk becomes lower thanthe respective critical value, i.e., around x = 0: c > c*(lc) for non-monotonic and c o c*(lc) for monotonic.

The behavior described above is rather general and can bealso obtained by keeping lc constant and varying either domainsize, applied voltage or molecular packing (n).53 Due to ourinterest in a confined medium, we chose to demonstrate theeffect via domain size since in most applications electrolytesare desirably to be confined at meso-scales (10–100 nm). Insuch a situation, the bulk approaches quasi-electroneutralityand yet, the criterion for the transition from non-monotonic tomonotonic diffuse layer can also be efficiently deduced even forlarge potentials and for relatively small separation distances(as shown in Fig. 5). However, once the separation distance

becomes of the order of 1, screening of the electrical field isimpossible and electroneutrality cannot be approached.

Existence of spatially extended bulk oscillations also explainscharge layering observed in RTILs under equal surfaces,37,55,56 asschematically described in Fig. 1. Importantly, the RTILs in thissetup are not subjected to any applied voltage (but rather to asurface charge) and yet exhibit layering (spatial oscillations). Thisphenomenon is consistent with the analysis discussed here sinceunder symmetric surfaces the solution obeys even symmetry inspace (unlike odd symmetry as in the previous case):

cðxÞrðxÞjðxÞ

24

35 ¼ cð�xÞ

rð�xÞjð�xÞ

24

35: (7)

As such, once the separation distance is increased, addi-tional charge layers are added to the bulk, making the charge atthe mid-plane change signs exactly as described by Perkin.33

Dynamical properties of the emergentdiffuse layer

Another significant property of non-monotonic diffuse layers isrelated to a transient current that is developed under appliedconstant voltage.48,57 Thus, we compare between transient currentsthat are developed for monotonic and non-monotonic EDLs, asshown in Fig. 6. The current is evaluated via integration of thecharge flux over the domain:48,57

I ¼ 1

L

ðL=2�L=2

@r@xþ c

@j@xþ nr1� nc

@c

@x

dx; (8)

where at each time step the profiles are computed numericallyfrom eqn (1), as shown through representative profiles in Fig. 7.For large L c 1 or small L B O(10) domains, the transientcurrents are expected to be dominated by the formation ofthe condensed layer near the electrode (i.e., the crowdingeffect). At moderate distances, L B O(100), spatially extendedoscillations occupy a significant portion of the domain andthus their formation contributes significantly to the transientcurrent. The current decay can be qualitatively divided intothree temporal stages (Fig. 7):

(A) Initial formation of space charge [representative profilesin Fig. 7 at t = 20 in (a) and t = 50 in (b)], which also marks a‘breaking’ point as shown in the inset of Fig. 6;

(B) Development of the plateau near the electrode [representa-tive profiles in Fig. 7 at t = 50 in (a) and t = 120 in (b)]. Note that theprofile in (b) already exhibits weak spatial oscillations;

(C) Approach to quasi-electroneutrality (I - 0 as t -N) andenhancement of spatial oscillations [representative profile inFig. 7 at t = 200 in (b) while in (a) there is no significant change].

Although at later times both monotonic and nonmonotonicbehaviors exhibit exponential decay (see inset in Fig. 6), spatialoscillations play the role of ‘rate determining step’ since alreadyin stage (B) they dominate the decay rate even though they arejust being formed. Moreover, since these oscillations areextended all over the bulk, the system will not reach ultimate

Fig. 4 Ion density and charge (inset) profiles for distinct values of lc: lc > lc*(thin line), lc slightly above lc* (thick line) and lc o lc* (dashed line). Thecritical ion density for lc = 0.6 is marked by the dotted line while for all othervalues of lc the critical density is way below the bulk density. Eqn (1) werenumerically integrated with V = 10 and L = 22.

Fig. 5 Distinct ion density and charge (inset) profiles for lc = 1, above(solid line) and below (dashed line) the onset. Eqn (1) were numericallyintegrated with V = 10 and L = 22 (solid line) and L = 20 (dashed line).

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electroneutrality under meso-scale confinement (i.e. a uniformbulk state for which r = E = 0).

Conclusions

To conclude, studies of RTILs demonstrate not only theiradvantages in technological applications but also debate25,26

about the physicochemical properties of highly concentratedelectrolytes, such as spatial distributions of ions near the solidsurfaces (the EDL structure). While many studies have reportedon alternating layers of opposite ions near the surfaces13–23,28–35

it was recently conjectured through a revision of empiricalmethods24 that RTILs are not much different from dilute electro-lytes. Using spatial dynamics methods, we have demonstratedthe nature of decaying charge distributions near the surfaces arein fact enslaved by the bulk and thus, EDLs in RTILs can exhibitboth ion layering and a standard decaying behavior as in diluteelectrolytes. Since the EDL structure is very basic to technologicalapplications,1–11 we hope that the approach and results providedhere will allow new vistas and advances.

To resolve the emergence of spatiotemporal ion distributions,we have analyzed a modified PNP model36 which captures semi-phenomenologically the critical physicochemical basis of RTILsand it is ‘simple’ enough to allow analytic explorations.48,49

Particularly, we showed that non-monotonic (spatially oscillatorycharge layering) and monotonic EDL behaviors are both proper-ties of confined RTILs. This unusual emergence of the EDLs stemsfrom the missing puzzle piece that is related to the existence ofspatial oscillations already in the bulk. Consequently, the overallEDL behavior (transition from non-monotonic to monotonic) isdetermined by electrostatic correlations and ion density in thebulk. The decrease in ionic density, however, depends implicitlyon applied voltages, molecular packing, domain size,53 and ionicdissociation level (the latter was very recently conjectured byGebbie et al.24). The results appear to be general and apply equallyto systems with metal electrodes (under applied voltage) and toconfined RTILs under identical surfaces (no applied voltage).In addition, we have performed a temporal analysis (chrono-amperometry) and showed that development of spatial oscilla-tions results in an exponential current decay which indeedbears qualitative similarity to dilute electrolytes but with a

distinct time scale, as has already been conjectured for lowapplied voltages48 (V o RT/F). Thus, our results should allowconduction and control of specifically targeted experimentstoward realization of the basic EDL properties.

Acknowledgements

We are grateful to Susan Perkin (Oxford) and Jacob Klein(Weizmann Institute of Science) for stimulating discussionsand elaborations.

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