THE JOURNAL OF CHEMICAL PHYSICS 147, 134704 (2017)

Ionic liquids-mediated interactions between nanorodsZhou Yu,1 Fei Zhang,1 Jingsong Huang,2 Bobby G. Sumpter,2 and Rui Qiao1,a)1Department of Mechanical Engineering, Virginia Tech, Blacksburg, Virginia 24061, USA2Center for Nanophase Materials Sciences and Computational Sciences and Engineering Division,Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA

(Received 19 May 2017; accepted 1 August 2017; published online 6 October 2017)

Surface forces mediated by room-temperature ionic liquids (RTILs) play an essential role in diverseapplications including self-assembly, lubrication, and electrochemical energy storage. Therefore, theirfundamental understanding is critical. Using molecular simulations, we study the interactions betweentwo nanorods immersed in model RTILs at rod-rod separations where both structural and doublelayer forces are important. The interaction force between neutral rods oscillates as the two rodsapproach each other, similar to the classical structural forces. Such oscillatory force originates fromthe density oscillation of RTILs near each rod and is affected by the packing constraints imposedby the neighboring rods. The oscillation period and decay length of the oscillatory force are mainlydictated by the ion density distribution near isolated nanorods. When charges are introduced on therods, the interaction force remains short-range and oscillatory, similar to the interactions betweenplanar walls mediated by some protic RTILs reported earlier. Nevertheless, introducing net chargesto the rods greatly changes the rod-rod interactions, e.g., by delaying the appearance of the first forcetrough and increasing the oscillation period and decay length of the interaction force. The oscillationperiod and decay length of the oscillatory force and free energy are commensurate with those of thespace charge density near an isolated, charged rod. The free energy of rod-rod interactions reacheslocal minima (maxima) at rod-rod separations when the space charges near the two rods interfereconstructively (destructively). The insight on the short-range interactions between nanorods in RTILshelps guide the design of novel materials, e.g., ionic composites based on rigid-rod polyanions andRTILs. Published by AIP Publishing. https://doi.org/10.1063/1.5005541

I. INTRODUCTION

The past decades have witnessed extensive interest andattention to room-temperature ionic liquids (RTILs) due totheir unique combination of various properties such as lowvolatility, wide electrochemical windows, and excellent ther-mal stability.1–3 Because of these properties, RTILs have foundmany potential and even practical applications in diverse areasincluding electrochemical energy storage, green solvents, anddrug delivery, to name a few.1,4–6 In many of these applications,RTILs are confined in nanostructures or between extendedsurfaces, e.g., they are confined between nanorods in macro-molecular ionic composites and ion gels7 or in nanopores inporous electrodes of supercapacitors2,4,8 or between extendedsurfaces when used as lubricants.9–13 Because the structureof the interfacial RTILs and the surface forces mediated byRTILs play an essential role in the aforementioned appli-cations, they have been intensively investigated in recentyears.

The interactions between surfaces mediated by RTILsare often investigated experimentally using surface forcesapparatus (SFA) and atomic force microscopy (AFM).14,15

The forces between two negatively charged mica surfacesseparated by ethylammonium nitrate were measured using

a)Author to whom correspondence should be addressed: ruiqiao@vt.edu

SFA as early as 1988.16 Systematic experimental studiesusing various surfaces (e.g., mica and silica) and RTILs(e.g., 1-ethyl-3-methylimidazolium ethylsulfate, 1-butyl-3-methylimidazolium bis(trifluoromethanesulfonyl)amide, etc.)have been reported recently.11,17–20 These studies revealedthat, for small surface separations of typically less than 10 iondiameters, the surface force resembles the classical structuralforce, i.e., the force is oscillatory with its period commensuratewith the estimated diameter of an ion pair and it decays rapidlyas the surface separation increases. While some experimentssuggest that surface forces mediated by some RTILs, espe-cially protic ones, are short-range, a growing body of exper-iments shows that such surface force can also be long-range,especially when aprotic RTILs are involved.20–23 Specifically,the surface force decays exponentially as the surface separationincreases and decay lengths of 5-10 nm have been reported.15

Such long-range interactions are not anticipated from the clas-sical DLVO theory. Although their physical origins are notfully understood, it has been suggested that they are elec-trostatic in nature based on the observation that they dependgreatly on the ion density and the dielectric constant of theRTILs.15,21,24

Since the inter-surface forces, especially their short-rangeoscillations, are closely related to the organization and distri-bution of RTILs confined between surfaces, the structure ofRTILs can be inferred from the force-distance curves mea-sured experimentally. For instance, based on the oscillation of

0021-9606/2017/147(13)/134704/10/$30.00 147, 134704-1 Published by AIP Publishing.

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the interaction force, it has been suggested that eight to nineordered layers of ions form in the RTILs that are confinedbetween mica surfaces,16 although a more elaborate assign-ment of the RTIL structure is also possible.25 The layeredstructure of RTILs near solid surfaces has been confirmedexperimentally using high-energy x-ray reflectivity studies.26

However, the direct measurement of the RTIL structure, espe-cially in confined space, is usually challenging. This limitation,however, can be addressed partially using theoretical or sim-ulation studies in which the structure of RTILs is resolvedexplicitly at molecular scales. Indeed, the structure of RTILsconfined between surfaces (usually planar walls) has beeninvestigated extensively in the past decade using analyticaltheories, molecular simulations, and classical DFT calcula-tions.27–40 The findings of these studies are largely in reason-able agreement with those derived from experimental studies,and interested readers are referred to two recent reviews forfurther details.2,3

While the theoretical and computational studies of thestructure of confined RTILs are fairly advanced by now, sim-ilar studies of the surface forces mediated by RTILs are stillat a relatively early stage. The classical DLVO theory is notexpected to describe the interactions between charged surfacesseparated by RTILs because the excluded volume effects asso-ciated with the finite size of ions and the correlations betweenions, both important for RTILs, are not taken into account inthis theory.10 In addition, the Debye length, the characteris-tics length scale for the electrostatic inter-surface interactionsin the DLVO theory, is usually smaller than the ion size inRTILs. This suggests that other length scales, e.g., the ion-ioncorrelation length,31,41 are involved in the long-range electro-static interactions revealed in recent experiments. The limi-tations of the classical DLVO theory can be addressed usingmore advanced theories or molecular simulations, which havebeen successful in predicting the structure of simple liquids,aqueous electrolytes, and polymers confined between surfacesand inter-surface interactions.42–45 Nevertheless, only a lim-ited number of analytical theories and simulations have beenreported for inter-surface interactions mediated by RTILs.Using the Coulomb gas model, Lee et al. showed that inter-surface force between two planar walls separated by RTILsdecays in an oscillatory manner as the inter-surface separationincreases and the force maxima correspond to the insertionof new ion layers into the gap formed between the planarwalls, which agree qualitatively with available experimen-tal data.46 A key insight from the model is that the inter-surface force is controlled strongly by the fugacity of bulkRTILs. More recently, using models built upon the Ginzburg-Landau theory and molecular simulations, Vaikuntanathan,Talapin, and their colleagues studied the interactions of pla-nar surfaces and nanocrystals separated by a high-temperaturemolten inorganic salt KCl.47 Their analytical model and sim-ulations captured the oscillatory nature of the inter-surfaceforces well. A key insight from their study is that the inter-ference between the charge density oscillations near opposingsurfaces greatly affects the sign and magnitude of the inter-surface interactions.47 The fact that theories and simulationscan capture key aspects of the inter-surface forces mediatedby ionic liquids revealed in the experiments suggests that

theories and simulations are promising tools for studying theseforces.

The prior studies on the interfacial structure of RTILs andsurface forces mediated by RTILs have greatly advanced ourunderstanding of these problems. Nevertheless, some impor-tant knowledge gaps still remain. For instance, few simu-lations have been performed to delineate the surface forcesand interfacial RTIL structure concurrently to provide insightinto their correlation. Furthermore, most of the available stud-ies deal with extended surfaces like planar walls. Situationsin which the critical dimension of the objects is compara-ble to the ion size, which are often encountered in molecu-lar self-assembly, are rarely explored. It is not yet clear towhat extent the interactions between these objects resem-ble the interactions between extended surfaces and thereforemany questions remain open. For example, how does surfacecharge affect the interactions between these objects? Whatis the range of these interactions? How do the distributionsof ions around the objects determine these interactions? Inthis work, we investigate the interactions between nanorodsmediated by RTILs using MD simulations to address thesequestions. We focus on the situation in which nanorods arefully aligned because of its relevance to the recently synthe-sized macromolecular ionic composites, in which polyanionrods are highly aligned.7 The insight gained here will helplay a foundation for understanding the interactions betweentilted rods and surfaces.47,48 The rest of the manuscript is orga-nized as follows. The simulation system, molecular model, andmethods are presented in Sec. II. The simulation results onthe interactions between neutral or charged nanorods and theirrelation with the underlying RTILs structure near the nanorodsare presented in Sec. III. Finally, conclusions are drawn inSec. IV.

II. SIMULATION SYSTEM, MOLECULAR MODELS,AND METHODS

Figure 1 shows a schematic of the simulation system,which features two aligned nanorods immersed in modelRTILs. The rod-rod distance D is defined as the distance mea-sured between the axes of the two rods. We performed twosets of studies, one with neutral nanorods and the other with

FIG. 1. A schematic of the simulation system. Two aligned nanorods areimmersed in model ionic liquids. The system is periodic in all three direc-tions. The cations and anions have geometry identical to the BF4

� ions butonly their central atoms carry a unit charge.

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charged nanorods. For each set of study, the distance D wasvaried systematically to study the rod-rod interactions medi-ated by the RTILs. The number of ions inside the system wasfixed in all simulations. Due to the large size of the simulationbox (see Fig. 1 for the dimension of the simulation system),the nanorods occupy less than 0.9% of its total volume. Con-sequently, a bulk like behavior is recovered at positions awayfrom the rods and ions in the system have approximatelythe same chemical potential in systems with different rod-rodspacings.

In the spirit of the coarse-grained models used in manyprior studies of RTILs, which have been successful in captur-ing key features of the structure and dynamics of bulk andinterfacial ions,32,33,36,49 herein we adopted similar coarse-grained models for both the nanorod and model RTILs. Eachrigid nanorod consists of 33 layers of carbon atoms dis-tributed evenly in the rod length direction. Each layer includesseven carbon atoms arranged as a regular heptagon with aside length of 0.17 nm. For charged nanorods, small partialcharges are assigned equally to each atom in the nanorodto give a line charge density of �2.38 e/nm. The struc-ture and charge of the rod thus constructed are relevant tothe rigid-rod polyanion poly(2,2′-disulfonyl-4,4′-benzidineterephthalamide), or PBDT, used in the synthesis of somemacromolecular ionic composites.7 Cations and anions in ourmodel RTILs, both rigid, have geometries identical to theBF4

� ions (see Fig. 1) and only their central atoms carrya unit positive/negative charge. To account for ions’ elec-tronic polarizability, a background dielectric constant of ε r =2.5 was used in the calculation of electrostatic interactions.All atoms in the nanorod have Lennard-Jones (LJ) param-eters of σLJ = 0.355 nm and εLJ = 0.293 kJ/mol. Atomsin the cation and anion have LJ parameters of σLJ = 0.336nm and εLJ = 0.360 kJ/mol. More details of the force fieldfor RTILs can be found in Table S1 in the supplementarymaterial.

With the above force fields, the bulk density of the RTILsis ρ+ = ρ

�

= 4.11 nm�3 (P = 1 atm; T = 400 K). The radiusof the cation/anion is ∼0.22 nm based on the radial distribu-tion function of bulk RTILs (see Fig. S1 in the supplementarymaterial). The effective radius of each nanorod is ∼0.35 nmbased on the ion density profile around isolated nanorods (seeFig. S2 in the supplementary material). The coarse-grainedmodels adopted here do not take into account many of the

subtle details of real polyanions and RTILs, e.g., the discrete-ness of polyanions’ surface charge groups and the complexshape of the ions. With these simplifications, we seek toreveal the most essential features of the interactions betweennanorods mediated by RTILs, unobscured by the chemicalcomplexity of RTILs and nanorods. The potential limitationsof the coarse-grained model are discussed in Sec. IV.

Simulations were performed using the Gromacs code.50

All simulations were executed in the NVT ensemble(T = 400 K). Similar to many prior studies on RTILs, an ele-vated temperature was used to ensure that the phase space isexplored effectively within the time scale accessible to oursimulations (∼50 ns). The temperature of the system wasmaintained using the velocity rescaling thermostat.51 The non-electrostatic interactions were computed by direct summationwith a cut-off length of 1.5 nm. The electrostatic interac-tions were computed using the Particle Mesh Ewald (PME)method. The real space cutoff and FFT spacing were 1.5and 0.13 nm, respectively. For each system, an equilibriumrun of 20 ns was performed, followed by a 30 ns productionrun.

During the production run, the force acting on each atomof the rod was recorded and summed to give the total forceacting on each rod. The free energy (potential of mean force,or PMF) for rod-rod interactions was calculated using E (D)= − ∫

DD0

f (x)dx, where f (x) is the force at a rod-rod separationof x, D0 = 4 nm is the largest rod-rod separation exploredin our simulations, and the free energy at rod-rod separationof D0 is taken as zero. Hereafter, energy is measured in kBTwith T = 400 K for convenience, where kB is the Boltzmannconstant.

III. RESULTS AND DISCUSSIONA. Interactions between neutral nanorods

Figure 2(a) shows the rod-rod interaction force as a func-tion of the distance between two neutral nanorods. As the tworods initially in physical contact are separated from each other,they first attract each other (0.7 nm < D < 1.0 nm). As theseparation further increases, their interaction force exhibitsstrongly damped oscillations, which nearly vanish at separa-tions larger than∼2.0 nm. The force profile f (D) at D > 0.9 nmcan be fitted to a damped sine wave function

FIG. 2. Interactions between two neu-tral nanorods immersed in ionic liq-uids. (a) Interaction force. (b) Interac-tion energy.

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f (D) = ae−D/Lfd sin *

,

2π

Lfp

(D − b)+-

, (1)

where Lfd is the decay length, Lf

p is the oscillation period,and a and b are fitting constants. Using the data shown inFig. 2(a), Lf

d and Lfp are determined to be 0.50 nm and 0.48 nm,

respectively.The integration of the force shown in Fig. 2(a) leads to

the interaction free energy E(D) between the nanorods pre-sented in Fig. 2(b). E(D) is also oscillatory and decays withD rapidly. Its decay length (LE

d = 0.44 nm) and oscillationperiod (LE

p = 0.51 nm) are determined by fitting E(D) to theform of Eq. (1), and they are found to be comparable to thoseof the interaction force. It can be seen that separating twonanorods initially in contact must overcome an energy bar-rier of ∼10 kBT /nm. Such a large energy barrier is attributedmostly to the attractive rod-rod interactions at separations of0.7-1.0 nm, which are caused by the van der Waals attrac-tions between the two rods and the capillary effects. For D≤ 1.0 nm, one cannot even fit one ion between the closestsurface points of the two rods (see Fig. S3 in the supplemen-tary material). As such, a vacuum is developed between thetwo rods and a vacuum-RTIL interface is formed. To lowerthe free energy associated with the vacuum-RTIL interface,an attractive force between the rods, which pulls the rodscloser to reduce the area of the vacuum-RTIL interface, isdeveloped.

The short-range, oscillatory forces between two neutralrods observed here resemble the structural forces betweenextended surfaces separated by simple fluids. The latter isusually understood using the contact value theorem and thesuperposition approximation method.10,14 Following the con-tact value theorem, the interaction force P between two planarwalls separated by a distance D is given by P(D) = kBT [ρD(0)� ρ∞(0)], where ρD(0) is the contact fluid density (i.e., fluiddensity on a wall’s surface) when the walls are separatedby a distance D. Therefore, repulsive (attractive) forces areaccompanied by the highly (weakly) ordered fluid structurebetween the walls and accordingly the high (low) fluid con-tact density on the walls.14 The fluid structure between thetwo walls, embodied in the contact fluid density and affectedby the geometrical constraints imposed by the two walls, canbe qualitatively inferred from the fluid structure near iso-lated walls by superposition:52 the fluid density between twowalls separated by a distance of h varies as ρh(z) ∼ ρ∞(z)+ ρ∞(h � z), where z is the distance from one wall andρ∞ is the fluid density near an isolated wall. It follows thatP (D) ∼ kBT

[ρ∞ (D) − ρb

]∼ kBT ρex

∞(D), where ρb is thedensity of bulk fluids and ρex

∞ (D) = ρ∞ (D) − ρb is the excessfluid density at a distance of D from the wall. This relationhas been used successfully to understand structural forces.43,53

Importantly, it suggests that the oscillation period of struc-tural forces between two planar walls is close to the oscillationperiod of the fluid density near an isolated planar wall, whichin turn is comparable to the size of the fluid molecules. Thestructural forces between curved surfaces are often understoodfrom those between planar walls with the help of the Derjaguinapproximation, and it can be shown that key characteristics of

structural forces, e.g., their oscillation period, are not affectedby the curvature of the surfaces.14

For the interactions between molecularly thin, neutralnanorods in RTILs considered here, the Derjaguin approxima-tion is no longer rigorous since the radius of nanorods, the sizeof ions, and rod-rod separations are all comparable. Neverthe-less, we shall examine to what extent the rod-rod interactionscan be understood using the above framework establishedfor the structural forces between extended surfaces. Assum-ing that the rod-rod interaction is controlled primarily by thestructure of RTILs along the centerline connecting the tworods’ axes, we have f (D) ∼ (ρ∞(D) � ρb). Because the cationsand anions are symmetric in our RTILs and the nanorods areneutral, the fluid densities ρ∞(D) and ρb are taken as the sum-mation of cation and anion density. Figure 3 compares therod-rod interaction force f (D) and the excess fluid densityρex∞ (D). Excellent commensuration between f (D) and ρex

∞ (D)is observed. Importantly, the oscillation period of the rod-rodinteractions (0.48 nm) agrees with that of the fluid density nearisolated rods (0.48 nm), and both oscillation periods are closeto the ion diameter (∼0.44 nm). These results indicate that thestructural forces between molecularly thin nanorods immersedin RTILs can be understood qualitatively using the establishedmethod for the structural forces between planar walls mediatedby simple fluids.

The above discussion suggests that the liquid structurearound nanorods, which is controlled by the geometrical con-straints imposed by the nanorods, greatly affects the rod-rodinteractions. Nevertheless, the superposition method for infer-ring the liquid structure near the rods does not take into accountthe rods’ small radius and thus cannot provide a full picture ofthe liquid structure near them. To address this limitation, nextwe examine the distribution of RTILs around the nanorods atrod-rod separations corresponding to the three extrema of f (D)in Fig. 2(a).

The left column of Fig. 4 shows the fluid density distri-bution (ρf = ρ+ + ρ

�

) near the two nanorods corresponding topoints A, B, and C in Fig. 2(a). Clearly, each nanorod affects thefluid structure near the other rod, especially in the gap betweenthem. The schematics in the right column of Fig. 4 highlight theevolution of the fluid structure as the rods are gradually sepa-rated from each other and the areas in which strong interferenceof fluid structure near neighboring nanorods is enclosed by the

FIG. 3. Correlation between the rod-rod interaction force f (D) with the excessfluid density near an isolated, neutral nanorod ρex

∞ (D).

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FIG. 4. Distribution of ionic liquidsnear neutral nanorods. The three rowsare for three rod-rod separations cor-responding to the force extrema inFig. 2(a) {[(a1) and (a2)] → point A;[(b1) and (b2)] → point B; [(c1) and(c2)]→ point C}. Fluid density is com-puted as the summation of cation andanion densities. Fluid density is color-coded in panels [(a1), (b1), and (c1)](unit: nm�3) and illustrated schemati-cally in panels [(a2), (b2), and (c2)]to highlight the interference betweenfluid distributions near the two rods (theschematics are drawn to scale). Theinner solid circle, the dashed circle, andthe outer solid circle around each roddenote the first peak, first valley, andsecond peak of the fluid density arounda rod if it is isolated in ionic liquids(see Fig. S2 in the supplementary mate-rial). The filled green and blue circlesdenote the nanorods and ions, respec-tively. Notable interference of fluid dis-tribution occurs in areas enclosed by reddashed lines. Positions of the rod sur-face where ion density is modified bythe interference between ion distribu-tions around the two rods are markedusing magenta lines.

red dashed lines. At a rod-rod separation of 1.10 nm, the stronginterference of the first ion density peaks centering on the tworods leads to greatly enhanced ion density along a portion ofthe rod surface [see the high density spots in Fig. 4(a1) andthe magenta lines in Fig. 4(a2)], and consequently a strongrepulsion between the rods as shown in Fig. 2(a). Figure 4(b2)shows that as the rod-rod separation is increased to 1.30 nm,two kinds of interferences set in: a constructive interferencebetween the first ion density peak centering on each rod anda destructive interference between the first ion density peakcentering on one rod and the first ion density valley centeringon the other rod. The constructive interference enhances theion density in the space between the two rods [see Fig. 4(b1)and the grey patch between the two rods in Fig. 4(b2)] but thedestructive interference decreases the ion density along a por-tion of each rod’s surface [see the magenta lines in Fig. 4(b2)].Following the contact value theorem, the latter interferencereduces the repulsion between the rods, which is consistentwith the force valley at point B in Fig. 2(a). As the rods movefurther apart to a rod-rod separation of 1.6 nm, the geometri-cal constraints provided by one rod to the ions near the otherrod become quite weak. However, a weak constructive inter-ference between the first ion density peak centering on onerod and the second ion density peak centering on the other rodleads to a slight enhancement of the ion density along a por-tion of each rod’s surface [see the magenta lines in Fig. 4(c2)],and consequently a peak in the rod-rod repulsion force inFig. 2(a).

B. Interactions between charged nanorods

Next we turn to the rod-rod interactions when both rods arenegatively charged. Here, the double layer forces are expectedto become important. Figure 5(a) shows the rod-rod interac-tion force as a function of the distance between two chargednanorods. The interaction force between charged rods showsdamped oscillations similar to those between neutral rods.However, some differences are also notable. First, comparedto that between neutral rods, the interactions between chargedrods have a longer range (∼3 nm) and the period of force oscil-lation is larger. Indeed, fitting the force curve in Fig. 5(a) toEq. (1) gives a period of Lf

p = 0.58 nm and a decay length of

Lfd = 0.80 nm, compared to Lf

p = 0.48 nm and Lfd = 0.50 nm

for the neutral rods. Second, the first valley of the force curveappears at a rod-rod separation ∼0.30 nm farther comparedto that for neutral rods and it is shallower than the first val-ley in the force curve for neutral rods. This observation iscaused by the electrostatic repulsion between the two negativecharged rods. Finally, a kink is observed in the force curveat a rod-rod separation of 1.15 nm (point A in Fig. 5). Thisobservation can be attributed to the structural force. At a rod-rod separation of ∼1.15 nm, the first counter-ion layer aroundboth rods merges in the gap between them to enhance the iondensity there (see Fig. S4 in the supplementary material). Thehigh density of this counter-ion layer helps reduce the elec-trostatic repulsion between the rods. Meanwhile, since theseions also serve as the liquid medium for the two interacting

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FIG. 5. Interactions between twocharged nanorods immersed in ionicliquids. (a) Interaction force. (b)Interaction energy.

rods, their high local density in the gap between the rods con-tributes to a repulsive force peak. This force peak is similarto the repulsive, structural (entropic) force peak for the inter-actions between two neutral rods (cf. the force peak at pointA in Fig. 2), which occurs at essentially the same rod-rodseparation.

Figure 5(b) shows the interaction free energy between thetwo charged rods as a function of the rod-rod separation. Thefree energy curve shows oscillation period (LE

p = 0.59 nm)and decay length (LE

d = 0.87 nm) similar to those for theforce curve in Fig. 5(a). Despite that both rods carry the samecharge, a primary free energy minimum occurs at a rod-rodseparation of 1.00 nm. Separating two rods initially posi-tioned at this separation must overcome an energy barrierof ∼4.6 kBT /nm. Such a barrier can help stabilize nanorodsin RTILs and may contribute to the mechanical strength

of the polyanion nanorod-based ionic composites developedrecently.7

Since the current understanding of the interactionsbetween charged rods mediated by RTILs is quite limited,hereinafter we focus on understanding how the interactionsoscillate and decay with the rod-rod separation. Because thedecay length and oscillation period are similar for the interac-tion force and free energy and it is customary to focus on thefree energy in the analysis of colloidal interactions,14 below weexamined the interaction free energy between charged rods. Inparticular, we focused on how the structure of RTILs near tworods evolves as their separation changes and how the structurechange affects the oscillation and decay of the free energy withrod separation. Given that the interactions between chargedcolloids are most closely tied to the distribution of the spacecharge around the objects, the structure of the RTILs near

FIG. 6. Interference between ion distri-butions near two nanorods. The threerows are for three rod-rod separa-tions corresponding to the three energyextrema in Fig. 5(b) {[(a1) and (a2)]→ point B; [(b1) and (b2)] → point D;[(c1) and (c2)] → point F}. The localspace charge density is color-coded inpanels [(a1), (b1), and (c1)] (unit: enm�3). The charge density profiles mea-sured along the rod-rod centerline arecompared with that computed by simplesuperposition of charge density profilesnear isolated rods in panels [(a2), (b2),and (c2)].

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the rods will be characterized using the local space chargeρe = e(ρ+ � ρ

�

), where ρ± is the local cation/anion numberdensity.

The left column of Fig. 6 shows the space charge distribu-tion near the two rods at three rod-rod separations correspond-ing to the three local free energy maxima in Fig. 5(b) (pointsB, D, and F). The interference of the space charge distribu-tion near the two rods is evident in the gap between the tworods. Prior studies have shown that, for space charges enclosedbetween two charged planes, their distribution can be under-stood from the simple superposition of the space charge near anisolated plane.47 Here we check whether this simple superpo-sition is also reasonable for the space charge near rods whoseradius and separation from each other are both comparable tothe ion size. To this end, we computed the space charge alongthe centerline connecting the axes of the two rods by super-posing the space charge near isolated nanorods and the resultsare compared to that measured directly from MD simulationsin the right column of Fig. 6. For all three rod-rod separa-tions, the space charge along the rod-rod centerline is capturedquite well by the superposition method. A key observation ofFigs. 6(a2), 6(b2), and 6(c2) is that the interference of the spacecharge from the opposing rods is “destructive” in nature, i.e., aspace charge peak (valley) associated with one rod is cancelledby a space charge valley (peak) associated with the other rod.Such a destructive interference of the space charges near tworods reduces the positive space charge along the centerline ofthe two negatively charged rods, thus causing less effective

screening of the rods’ negative surface charges by the counter-charges in the gap between them. The weakened screening ofthe electrostatic repulsion between the like-charged rods drivesup the overall interaction energy between them energetically,which helps us to explain why free energy is maximized locallyat points B, D, and F in Fig. 5(b).

The interference of the space charge near two rods withseparations corresponding to the local minima of the freeenergy [points C, E, and G in Fig. 5(b)] is shown in Fig. 7.A key difference from the interference shown in Fig. 6 is thatthe interference is “constructive” in nature here, i.e., a spacecharge peak (valley) associated with one rod is enhanced byanother space charge peak (valley) near the other rod. Such aconstructive interference tends to increase the positive spacecharge density in the gap between the two rods and thus leadsto more effective screening of the surface charge of the rods.Consequently, the free energy is minimized locally at pointsC, E, and G in Fig. 5(b).

In the above discussion (in particular the right columnsof Figs. 6 and 7), only the interference of the space chargealong the centerline of the two rods is explicitly addressed.Consequently, the effect of the finite radius of the rods onspace charge interference is not considered. To examine theinterference more accurately, we computed the net spacecharge in the gap between the two rods for rod-rod separationscorresponding to points B to G in Fig. 5. The gap is defined asa rectangle box with a height of 1.0 nm and its two sides arebounded by the axes of the two rods (see Figs. 6 and 7). Table I

FIG. 7. Interference between ion distri-butions near two nanorods. The threerows are for three rod-rod separa-tions corresponding to the three energyextrema in Fig. 5(b) {[(a1) and (a2)]→ point C; [(b1) and (b2)] → point E;[(c1) and (c2)] → point G}. The localspace charge density is color-coded inpanels [(a1), (b1), and (c1)] (unit: enm�3). The charge density profiles mea-sured along the rod-rod centerline arecompared with that computed by simplesuperposition of charge density profilesnear isolated rods in panels [(a2), (b2),and (c2)].

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TABLE I. The net charges confined between two charged nanorods, mea-sured in a 1.0 nm-wide box between the two rods (cf. dashed boxes in Figs. 6and 7).

Maximal free energy positions Minimal free energy positions

Point B Point D Point F Point C Point E Point G+1.24 e/nm +1.37 e/nm +1.71 e/nm +2.61 e/nm +2.29 e/nm +2.12 e/nm

shows that the net space charge in the gap at points B, D, andF is much smaller than at points C, E, and G. This is in goodagreement with the difference expected from the destructiveinterference and constructive interference for these two sets ofpoints.

The above results suggest that the interference of spacecharge near molecularly thin rods can be understood reason-ably well by superposition of the space charge near isolatedrods. Together with the understanding that the destructive(constructive) interference of the space charge around twoneighboring rods tends to increase (decrease) their interactionfree energy, it is reasonable to conclude that the interactionsbetween thin rods in RTILs can be qualitatively understoodfrom the superposition of the space charge around isolatedrods. In a recent study of the interactions between planar sur-faces mediated by molten salts, it has been analytically shownthat using the superposition method for evaluating the spacecharge between surfaces, the oscillation period and decaylength of the interactions between two surfaces are the same asthose for the space charge profile near the isolated surfaces.47

Here we checked whether the same would be true for the thinrods considered in this work. Figure 8 shows the space chargedensity profile near an isolated rod, where the damped oscil-lation of the space charge is evident. Fitting the space chargecurve to the form given by Eq. (1) gives an oscillation periodof Le

p = 0.61 nm and a decay length of Led = 0.80 nm. The

computed Lep agrees with the free energy oscillation period

of LEp = 0.59 nm extracted from Fig. 5(b). Physically, the

oscillation period of the space charge profile near an isolatedrod is similar to the thickness of a counter-ion and co-ionlayer positioned next to each other shown in Fig. S5 in thesupplementary material. Since earlier experiments showedthat each period of interaction force (and consequentlyinteraction energy) oscillation corresponds to a neighboring

FIG. 8. The space charge density profile near an isolated, charged nanorod.

pair of counter- and co-ion layer being removed between twointeracting surfaces,10 the agreement of Le

p with LEp is physi-

cally sound. The computed decay length of the space chargenear isolated rods Le

d agrees with the rod-rod free energy decaylength of LE

d = 0.87 nm extracted from Fig. 5(b). These resultsthus support the method of inferring rod-rod interactions fromthe space charge profiles near isolated rods.

The results in Figs. 2 and 5 show that the wavelength(Lp) and decay length (Ld) of the damped, oscillatory rod-rodinteractions mediated by RTILs depend apparently on the sur-face charge of the rods. Fundamentally, however, Lp and Ld

are mainly the properties of the RTILs,15,54 and to a lesserextent, they are also affected by the geometry of the inter-acting surfaces. Two important types of length scales existin RTILs. One is the size of the ions and the other is Lp

and Ld of the space charge density wave within them (forbulk RTILs, a charge density wave is observed near individ-ual ions; for RTILs near electrified surfaces, a charge densitywave is also observed as one moves from the surfaces towardbulk liquids). Lp and Ld depend on the size and the interac-tions between ions and are thus primarily the properties ofthe RTILs. For RTILs near electrified surfaces, Lp and Ld arealso affected by the geometry of the interacting surfaces tosome extent because the distribution of space charge near anelectrified surface is known to be affected somewhat by its cur-vature.55 For example, Ld is 1.25 nm in our bulk RTILs, but it isreduced by∼20% in the same RTILs near planar walls (surfacecharge density: 0.75 e/nm2) and is reduced further by ∼15%in the same RTILs near the charged rods considered in thiswork.

The principal role of the surface charge is to control whichlength scales in RTILs are manifested more prominently inrod-rod interactions. The interactions between neutral rodsare dominated by structural forces, which are mainly con-trolled by the packing of ions in the gaps between the rods.Consequently, the wavelength and decay length of these inter-actions are controlled mostly by the ion size. In comparison,the interactions between highly charged rods are dominatedby double layer forces, which depend strongly on the inter-ferences of the space charge near individual rods. As such,the wavelength and decay length of these interactions are con-trolled primarily by Lp and Ld of the space charge densitywaves within RTILs. One can expect the wavelength and decaylength of the rod-rod interaction force/energy to evolve con-tinuously as the surface charge on the rods deviates from zero.A systematic investigation of such an evolution is beyondthe scope of the present work but is worth pursuing in thefuture.

The rod-rod interactions revealed in our simulations arerather short-range. This seems to be in agreement with therange of protic RTIL-mediated surface forces measured exper-imentally,12,15 but different from the long-range interactionsreported in many other experiments (e.g., a decay length of∼5-10 nm has been reported).15,54 The absence of long-rangerod-rod interactions in our simulations can potentially becaused by the limitations of the MD simulations and mayalso be related to the nature of the model RTILs and thenanorods used here. First, in our simulations, the periodicboundary conditions are used. Therefore, long-range rod-rod

134704-9 Yu et al. J. Chem. Phys. 147, 134704 (2017)

interactions may be smeared out because of the interactionsbetween the rods and their images. Second, since only verystrong forces can be reliably measured in MD simulations,it is possible that the weak long-range interactions are sim-ply buried in statistical noises. Finally, some recent theoreticaland experimental studies suggested that the decay length ofthe inter-surface interactions in RTILs is likely a propertyof bulk RTILs.54 Given that the model RTILs adopted herehave a small ion diameter (∼0.44 nm) and the nanorods aremolecularly narrow, it is possible that inter-surface interac-tions mediated by our model RTILs are truly short-range.In light of the uncertainties associated with these consider-ations, we caution that the present study of rod-rod interac-tions mediated by RTILs does not provide a definitive answeron the decay length scale of RTIL-mediated interactions.Instead, it mainly provides insight into the interactions atshort-range when both structural and double layer forces arestrong, which are important in applications such as synthe-sis of macromolecular ionic composites using polyanions andRTILs.7

IV. CONCLUSIONS

Using molecular simulations, we have studied the interac-tions between nanorods mediated by RTILs when both struc-tural and double layer forces are important. When the rodsare neutral, the force between them shows damped oscillationas a function of increasing separation, and both the oscillationperiod and decay length of the interaction force are close to theion diameter. Such oscillatory force is essentially the classicalstructural force and is controlled by the packing constraintsimposed by the rods to their neighboring RTILs. When therods are charged, the force between them still shows dampedoscillation as their separation increases. However, both theoscillation period and the decay length are larger than thosefor the interactions between neutral rods. In addition, the firstvalley of their interaction force becomes shallower and shifts toa larger rod-rod separation. Nevertheless, separating two rodsinitially positioned at the first force valley must overcome anenergy barrier of ∼5 kBT /nm.

We have shown that, using the simple superpositionmethod for estimating the ion density profile between rods andthe classical contact value theorem, the oscillation pattern anddecay length for interactions between neutral rods can be esti-mated using the ion density profile despite the diameter of therod being comparable to the ion size. Likewise, the interactionsbetween charged rods can be interpreted qualitatively by con-sidering the interference of the space charge around differentrods. It is found that, even for molecularly thin rods, the inter-ference of space charge can be estimated from the space chargenear isolated rods using the simple superposition method. Thedestructive (constructive) interference of the space charge nearneighboring rods leads to the reduced (enhanced) screening ofthe surface charge of the rods and thus higher (lower) rod-rodinteraction energy. Because of such interferences, the oscil-lation period of the rod-rod interaction energy is close to thethickness of a pair of counter- and co-ion layers, in good agree-ment with that found experimentally. The rod-rod interactionsrevealed by our simulations are short-range. However, for

reasons such as the nature of the model RTILs used and noisein the measurement of interaction forces, we caution that thisresult does not preclude the existence of long-range forcebetween charged surfaces separated by RTILs.

We note that the insight gained in this work is derivedfrom simulations based on RTILs featuring cations and anionswith identical size. In practice, many RTILs have cations andanions with different sizes (e.g., [BMIM][Cl]). Nevertheless,the basic features of the rod-rod interactions mediated by theseasymmetric RTILs can be inferred from the present study.Specifically, the periodicity and decay length (Lp and Ld) ofthe interactions between neutral rods should be determinedprimarily by the size of the larger ions. This is because thedistribution of ions near the rods, which governs the structuralforce between them, is constrained mostly by the packing ofthe larger ions. On the other hand, Lp and Ld of the interac-tions between highly charged rods should still depend mostlyon Lp and Ld of the charge density wave in RTILs. SinceLp and Ld of the charge density wave in RTILs are affectedgreatly by both the small and the large ions, we expect thesmall ions to have a stronger effect on Lp and Ld of the inter-actions between highly charged rods than between neutralrods. A systematic study of how asymmetric ions affect RTIL-mediated interactions between nanorods will be pursued in thefuture.

SUPPLEMENTARY MATERIAL

See supplementary material for the force fields of nanorodand RTILs, the radial distribution function of ions in bulkRTILs, the ion distribution near isolated neutral rod, the iondistribution near neutral rods with a separation of 1.0 nm, theion distribution near charged rods corresponding to point A inFig. 5, and ion distribution near isolated charged rod.

ACKNOWLEDGMENTS

We thank the ARC at Virginia Tech for generous alloca-tions of computer time on the BlueRidge and NewRiver cluster.R.Q. gratefully acknowledges the support from NSF (CBET-1461842). R.Q. was partially supported by an appointmentto the HERE program for faculty at the Oak Ridge NationalLaboratory (ORNL) administered by Oak Ridge Institute ofScience and Education. J.H. and B.G.S. acknowledge workdone at the Center for Nanophase Materials Sciences, whichis a DOE Office of Science User Facility. We thank Dr. AlphaLee for insightful comments.

1N. V. Plechkova and K. R. Seddon, Chem. Soc. Rev. 37, 123 (2008).2M. V. Fedorov and A. A. Kornyshev, Chem. Rev. 114, 2978 (2014).3R. Hayes, G. G. Warr, and R. Atkin, Chem. Rev. 115, 6357 (2015).4H. Ohno, Electrochemical Aspects of Ionic Liquids (John Wiley & Sons,2005).

5M. Koel, Crit. Rev. Anal. Chem. 35, 177 (2005).6T. L. Greaves and C. J. Drummond, Chem. Rev. 108, 206 (2008).7Y. Wang, Y. Chen, J. Gao, H. G. Yoon, L. Jin, M. Forsyth, T. J. Dingemans,and L. A. Madsen, Adv. Mater. 28, 2571 (2016).

8C. Merlet, C. Pean, B. Rotenberg, P. A. Madden, B. Daffos, P.-L. Taberna,P. Simon, and M. Salanne, Nat. Commun. 4, 2701 (2013).

9F. Zhou, Y. Liang, and W. Liu, Chem. Soc. Rev. 38, 2590 (2009).10S. Perkin, Phys. Chem. Chem. Phys. 14, 5052 (2012).

134704-10 Yu et al. J. Chem. Phys. 147, 134704 (2017)

11S. Perkin, T. Albrecht, and J. Klein, Phys. Chem. Chem. Phys. 12, 1243(2010).

12H. Li, M. W. Rutland, and R. Atkin, Phys. Chem. Chem. Phys. 15, 14616(2013).

13F. B. O. Y. Fajardo, A. A. Kornyshev, and M. Urbakh, J. Phys. Chem. Lett.6, 3998 (2015).

14J. N. Israelachvili, Intermolecular and Surface Forces (Academic Press,2011).

15M. A. Gebbie, A. M. Smith, H. A. Dobbs, G. G. Warr, X. Banquy,M. Valtiner, M. W. Rutland, J. N. Israelachvili, S. Perkin, and R. Atkin,Chem. Commun. 53, 1214 (2017).

16R. Horn, D. Evans, and B. Ninham, J. Phys. Chem. 92, 3531 (1988).17S. Perkin, L. Crowhurst, H. Niedermeyer, T. Welton, A. M. Smith, and

N. N. Gosvami, Chem. Commun. 47, 6572 (2011).18K. Ueno, M. Kasuya, M. Watanabe, M. Mizukami, and K. Kurihara, Phys.

Chem. Chem. Phys. 12, 4066 (2010).19I. Bou-Malham and L. Bureau, Soft Matter 6, 4062 (2010).20M. A. Gebbie, M. Valtiner, X. Banquy, E. T. Fox, W. A. Henderson, and

J. N. Israelachvili, Proc. Natl. Acad. Sci. U. S. A. 110, 9674 (2013).21A. M. Smith, A. A. Lee, and S. Perkin, J. Phys. Chem. Lett. 7, 2157 (2016).22H. W. Cheng, P. Stock, B. Moeremans, T. Baimpos, X. Banquy, F. U. Renner,

and M. Valtiner, Adv. Mater. Interfaces 2, 1500159 (2015).23R. Espinosa-Marzal, A. Arcifa, A. Rossi, and N. Spencer, J. Phys. Chem.

Lett. 5, 179 (2013).24A. A. Lee and S. Perkin, J. Phys. Chem. Lett. 7, 2753 (2016).25K.-i. Amano, Y. Liang, K. Miyazawa, K. Kobayashi, K. Hashimoto,

K. Fukami, N. Nishi, T. Sakka, H. Onishi, and T. Fukuma, Phys. Chem.Chem. Phys. 18, 15534 (2016).

26M. Mezger, H. Schroder, H. Reichert, S. Schramm, J. S. Okasinski,S. Schoder, V. Honkimaki, M. Deutsch, B. M. Ocko, and J. Ralston, Science322, 424 (2008).

27Y. Shim, Y. Jung, and H. J. Kim, J. Phys. Chem. C 115, 23574 (2011).28S. A. Kislenko, I. S. Samoylov, and R. H. Amirov, Phys. Chem. Chem.

Phys. 11, 5584 (2009).29C. Pinilla, M. Del Popolo, J. Kohanoff, and R. M. Lynden-Bell, J. Phys.

Chem. B 111, 4877 (2007).30S. Wang, S. Li, Z. Cao, and T. Yan, J. Phys. Chem. C 114, 990 (2009).31M. Z. Bazant, B. D. Storey, and A. A. Kornyshev, Phys. Rev. Lett. 106,

046102 (2011).

32D.-e. Jiang, Z. Jin, and J. Wu, Nano Lett. 11, 5373 (2011).33J. Wu, T. Jiang, D.-e. Jiang, Z. Jin, and D. Henderson, Soft Matter 7, 11222

(2011).34L. B. Bhuiyan, S. Lamperski, J. Wu, and D. Henderson, J. Phys. Chem. B

116, 10364 (2012).35S. Li, K. S. Han, G. Feng, E. W. Hagaman, L. Vlcek, and P. T. Cummings,

Langmuir 29, 9744 (2013).36M. V. Fedorov and A. A. Kornyshev, J. Phys. Chem. B 112, 11868

(2008).37N. N. Rajput, J. Monk, R. Singh, and F. R. Hung, J. Phys. Chem. C 116,

5169 (2012).38W. Silvestre-Alcantara, L. B. Bhuiyan, J. Jiang, J. Wu, and D. Henderson,

Mol. Phys. 112, 3144 (2014).39C. Merlet, B. Rotenberg, P. A. Madden, P.-L. Taberna, P. Simon, Y. Gogotsi,

and M. Salanne, Nat. Mater. 11, 306 (2012).40M. V. Fedorov and A. A. Kornyshev, Electrochim. Acta 53, 6835 (2008).41A. A. Lee, S. Kondrat, D. Vella, and A. Goriely, Phys. Rev. Lett. 115, 106101

(2015).42P. Attard and J. L. Parker, J. Phys. Chem. 96, 5086 (1992).43A. Yethiraj and C. K. Hall, Macromolecules 23, 1865 (1990).44A. Trokhymchuk, D. Henderson, A. Nikolov, and D. T. Wasan, J. Phys.

Chem. B 107, 3927 (2003).45J. Gao, W. Luedtke, and U. Landman, Phys. Rev. Lett. 79, 705 (1997).46A. A. Lee, D. Vella, S. Perkin, and A. Goriely, J. Chem. Phys. 141, 094904

(2014).47H. Zhang, K. Dasbiswas, N. B. Ludwig, G. Han, B. Lee, S. Vaikuntanathan,

and D. V. Talapin, Nature 542, 328 (2017).48A. Kornyshev and S. Leikin, Phys. Rev. E 62, 2576 (2000).49Y. He, J. Huang, B. G. Sumpter, A. A. Kornyshev, and R. Qiao, J. Phys.

Chem. Lett. 6, 22 (2014).50B. Hess, C. Kutzner, D. Van Der Spoel, and E. Lindahl, J. Chem. Theory

Comput. 4, 435 (2008).51G. Bussi, D. Donadio, and M. Parrinello, J. Chem. Phys. 126, 014101 (2007).52W. van Megen and I. Snook, J. Chem. Soc., Faraday Trans. 2 75, 1095

(1979).53H. Eslami and F. Muller-Plathe, J. Phys. Chem. B 113, 5568 (2009).54A. M. Smith, A. A. Lee, and S. Perkin, Phys. Rev. Lett. 118, 096002 (2017).55G. Feng, R. Qiao, J. Huang, S. Dai, B. G. Sumpter, and V. Meunier, Phys.

Chem. Chem. Phys. 13, 1152 (2011).