evaluation of methods for molecular dynamics simulation of
TRANSCRIPT
Evaluation of Methods for Molecular Dynamics Simulation of Ionic LiquidElectric Double Layers
Justin B. Haskins1 and John W. Lawson21)
AMA Inc., Thermal Protection Materials Branch, NASA Ames Research Center, MS N234-1, Mo↵ett Field,
California 94035, USA
a)
2)
Thermal Protection Materials Branch, NASA Ames Research Center, MS N234-1, Mo↵ett Field, California 94035,
USA
We investigate how systematically increasing the accuracy of various molecular dynamics modeling techniquesinfluences the structure and capacitance of ionic liquid electric double layers (EDLs). The techniques probedconcern long-range electrostatic interactions, electrode charging (constant charge versus constant potentialconditions), and electrolyte polarizability. Our simulations are performed on a quasi-two-dimensional, or slab-like, model capacitor, which is composed of a polarizable ionic liquid electrolyte, [EMIM][BF4], interfacedbetween two graphite electrodes. To ensure an accurate representation of EDL di↵erential capacitance, wederive new fluctuation formulas that resolve the di↵erential capacitance as a function of electrode charge orelectrode potential. The magnitude of di↵erential capacitance shows sensitivity to di↵erent long-range elec-trostatic summation techniques, while the shape of di↵erential capacitance is a↵ected by charging techniqueand the polarizability of the electrolyte. For long-range summation techniques, errors in magnitude can bemitigated by employing two-dimensional or corrected three dimensional electrostatic summations, which leadto electric fields that conform to those of a classical electrostatic parallel plate capacitor. With respect tocharging, the changes in shape are a result of ions in the Stern layer (i.e. ions at the electrode surface) havinga higher electrostatic a�nity to constant potential electrodes than to constant charge electrodes. For elec-trolyte polarizability, shape changes originate from induced dipoles that soften the interaction of Stern layerions with the electrode. The softening is traced to ion correlations vertical to the electrode surface that inducedipoles that oppose double layer formation. In general, our analysis indicates an accuracy dependent di↵er-ential capacitance profile that transitions from the characteristic camel shape with coarser representations toa more di↵use profile with finer representations.
I. INTRODUCTION
In response to a charged electrode, ionic liquids formelectric double layers (EDLs). The EDL consists of al-ternating layers of cations and anions that screen thesurface charge. Ionic liquid EDLs can have large values(⇠10-20 µF/cm2) of di↵erential capacitance, Cdl, whichis the rate of change of surface charge density (h�i) withrespect to surface potential (h i) on a given electrode,
Cdl =@h�i@h i , (1)
making them well suited for non-Faradaic energy stor-age. Recent experimental work has delineated the sensi-tive relationship of the Cdl of ionic liquids to atomic-leveldetails of the EDL, including electrolyte molecular size1
as well as electrode surface structure.2–7 The key to im-proved energy storage lies in the optimization of theseproperties, which will require careful control of molecu-lar level features. In this regard, theoretical techniques,especially those that capture the full atomic detail of theinterface, will be a useful guide for future development.
Molecular dynamics (MD) has been broadly and suc-cessfully employed to understand both the structure of
a)Electronic mail: [email protected]
ionic liquids at electrified interfaces8–17 as well as to es-timate the electrode potential dependent di↵erential ca-pacitance.18–28 A variety of ionic liquid interatomic po-tentials have been applied to the study of EDLs, includ-ing coarse grained,19,29–31 all atom,8–15,17,18,20–25,28,32,33
and polarizable models.34–39 Coarse grain simulations ofionic liquids19,29–31 have provided important insight intohow ionic liquid size and shape influence the EDL struc-ture and capacitance. For instance, coarse grained mod-els19,40 have been used extensively to corroborate meanfield predictions that describe how disparities in ion sizeand packing at the electrode surface lead to the experi-mentally observed camel- and bell-shaped profiles of Cdl
with respect to h i. All atom interatomic potentials pro-vide a richer representation of EDLs and have been usedto detail both the electrode potential dependent structureand the di↵erential capacitance of specific ionic liquidcompounds.8–15,17,18,20–25,28,32,33 The predictive power ofthese methods has been recently highlighted though theexcellent agreement of MD derived EDL structures withatomic force microscopy measurements41 and the abilityto capture the magnitude and camel shaped profile ofCdl.22–26,28 Adding a level of refinement beyond the allatom models, polarizable models of ionic liquids, whereeach atom has a dynamic dipole moment, have producedhighly accurate measures of bulk structure and trans-port;34–39 however, they have had limited application tothe study of EDLs, with the only application at presentbeing to organic electrolyte systems.25
Several methodologies have been developed for apply-
2
ing charge or, alternatively, setting the potential of elec-trodes in a manner consistent with a metal in MD simu-lations. The most direct of these methods applies a con-stant charge to each electrode surface atom, referred to asconstant-� ensemble. Electrodes treated in this mannerhave produced favorable comparisons of Cdl with cyclicvoltammetry measures on some carbon nanostructures.18
A more complex approach is to redistribute the electrodesurface charge over the course of the simulation suchthat the potential di↵erence, � , between two electrodesmeets a pre-defined value. This approach, referred to asthe constant-� ensemble,42,43 has been employed to ob-tain the di↵erential capacitance of ionic liquids at bothideal and roughened graphitic electrodes,22–26,28 wherethe application of the constant-� condition is not intu-itive. Notably, the extensive simulations of Vatamanuand coworkers23 using this approach have provided a de-tailed description of how the electrode surface, includingspacing, curvature, and disorder, influences the magni-tude and shape of Cdl.
In addition, numerical issues related to simulationmethodology can a↵ect the accuracy of EDL computa-tions. Since slab-like, quasi-two-dimensional (2D) geom-etry is often employed in these computations (see Fig. 1),the appropriate technique for the long range electrostaticsummation must be considered. Typically, such summa-tions assume three dimensional (3D) periodicity, whichconflicts with the quasi-2D symmetry of a slab in an infi-nite vacuum and introduces spurious electrostatic inter-actions between periodic images. To correct for this, thecomputationally costly, though exact, 2D Ewald summa-tion44 is often utilized. Alternatively, less expensive 2Dcorrections the the 3D Ewald summation have also beenconsidered.45 The comparative accuracy of these options,however, has yet to be rigorously determined.
Another source of potential error arises in the determi-nation of Cdl, which is often computed as the numericalderivative of h�i with respect to h i on a given elec-trode. Due to the slow dynamics of ionic liquids and thecomplexity of interfacial interactions, however, the noisepresent in the numerical derivative is often large. Reduc-ing the noise requires fitting the h�i data over many simu-lations using a fine voltage mesh and taking the derivativeof the curve fit.24 Alternatively, fluctuation formulas forCdl have recently been derived,30,31,46 which remove theambiguity of a noisy numerical derivative and do not re-quire many simulations over a dense potential grid. Thus,Cdl can be determined at particular points of interest.At present, the fluctuation formulas have not been ex-tensively tested and are available for only two electrodesystems treated with the constant-� approach. Fur-thermore, the available fluctuation expressions define thequantity @h�i/@� , which is the total capacitance of twoelectrodes, a positive electrode and oppositely chargednegative electrode, together as opposed to the capaci-tance of an individual electrode. Fluctuation expressionsfor the capacitance of individual electrodes, which arecommonly measured in experiments, are necessary to de-
scribe di↵erences in the EDL capacitance at negative andpositive electrodes, leading, for example, to asymmetric,camel-shaped Cdl profiles.Despite the considerable amount of work using MD
simulations to study EDL structure and capacitance, is-sues related to simulation methodology have only begunto be examined recently.29,47–49 In the present work, weevaluate the impact of di↵erent modeling approaches onthe computed structure and capacitance of the EDL. Webegin with an evaluation of the numerical accuracy ofvarious long-range summation techniques on electrodepotential and EDL structure. We then investigate theevaluation of di↵erential capacitance using constant-�and constant-� electrodes. For both electrode ap-proaches, we derive, benchmark, and evaluate new fluc-tuation expressions for Cdl that extend previously de-rived expressions and enable the analysis of single elec-trodes. Di↵erences in capacitance are then rationalizedthough an evaluation of the potential dependent densityand the configuration of the surface ion layer, or Sternlayer. Finally, we comment on the role of polarizationon EDL structure and capacitance. To perform thesesimulations, we employ a model system consisting of anelectrolyte, 1-ethyl-3-methylimidazolium boron tetraflu-oride ([EMIM][BF4]) as represented by the polarizableforce field, APPLE&P,50–52 interfaced between two idealgraphite electrodes (see Fig. 1).
II. METHODS
A. Long-Range Electrostatics
Our capacitor system has a slab-like, or quasi-2D,geometry, with the rk dimensions being fully periodicand the z dimension, normal to the surface, being fi-nite. While the covalent interactions are local and easilyamenable to such a geometry, the 3D periodic techniquestypically employed to perform long-range summation ofelectrostatic interactions do not easily conform to slab-like geometry and can lead to spurious interactions be-tween electrode images. There are various approachesthat can be used to perform such a summation in slabgeometry, which include the exact 2-D Ewald summa-tion, the 3-D Ewald summation with a slab correction,and the 3-D Ewald summation with no correction.The 2-D Ewald summation technique, as developed by
Parry,53 Heyes and coworkers,54 and Leeuw and cowork-ers55 and further streamlined by Kawata and Mikami,44
is an exact method for the evaluation of the electrostaticinteractions in a system having slab geometry. Similarto 3-D Ewald summation, one partitions the electrostaticenergy as UES = Ur +Uk,2D, where Ur is the real spacecontribution and Uk,2D is the reciprocal space contribu-tion. The real space contribution is given by
Ur =X
i
X
j 6=i
qiqj|rij |
erfc(↵|rij |), (2)
3
Lz = 10 nm!
vacuum!
- 12Lz
12Lz0
vacuum!
z
r||
FIG. 1. Representation of dual graphite electrodes interfaced with [EMIM][BF4
]. The system has slab geometry and is periodicin the directions parallel to the electrode surface (rk) and non-periodic in the direction perpendicular to the electrode surface(z).Throughout this work, z = 0 is defined as the point equidistant to both electrodes. The spacing between the electrodes isdefined as Lz, and beyond the electrodes an infinite vacuum is assumed. The large purple and blue atoms are F and N atoms,respectively, while the small grey and white atoms are C and H, respectively.
which represents the mutual interaction between atomiccharges as well as the interaction of atomic charges withGaussian distributed charges (the half maximum widthof these distributions is 2↵�1
pln(2) A which is stan-
dard to the Ewald procedure). Each Gaussian is cen-tered on a given atom and bears opposite charge to thatatom, screening the real space interaction and allowingthe use of a cuto↵. The long range interactions are han-dled through the reciprocal space summation over a gridof k-point in the rk directions according to
Uk,2Dkk 6=0 =
1
Ak
X
k 6=0
Z 1
�1dh
1
|kk|2 + h2⇥
exp
✓�|kk|2 + h2
4↵2
◆|S2D(kk, h)|2,
(3)
where Ak is the area of the simulation cell in the rk di-rections, kk are reciprocal space points the rk directions,h is a real space integration variable in the direction ofz, and S2D(kk, h) =
PNj=1 qjexp(ikk ·rk,j + ihzj). Unlike
the 3-D Ewald summation, the infinite contribution atkk = 0 is dependent on the z direction according to
Uk,2Dkk=0
= � 1
Ak
NX
i=1
NX
j=1
qiqj⇥
⇢p⇡
↵exp(�[↵zij ]
2) + ⇡zijerf(↵zij)
�.
(4)
Such an N2 contribution is not easily computed for largesystems without the use of basis splines, which reducethis term to an order N operation. In the present work,we compute this term exactly at 100 values of z andemploy 4th order basis splines to evaluate this quantityat all atomic positions.
Alternatively, the relatively less computationally ex-pensive 3-D Ewald summation with an approximate cor-rection for slab-like geometry can be employed. In this
approach, a large amount of vacuum space is includedbetween the capacitor periodic images in the z directionsuch that the electrostatic interactions between the im-ages is considered small. The correction to the 3D-Ewaldsummation, as given by Yeh and coworkers,45 may beimplemented to remove the remaining long-range inter-action between the electrode images in the z direction.Though originally developed for systems of interactingpoint charges, we have extended this correction to ac-count for the long-range interaction of point charges withthe atomic dipoles produced from the polarizable forcefield according to
U2Dcorr =
1
2✏0M2
z , (5)
where ✏0 is the permittivity of free space and Mz is thenet dipole moment in the z direction,
PNi=1 (qizi + µz).
We use a vacuum three times larger than the electrodeseparation.
B. Constant Potential versus Constant Charge Electrodes
In the present work, our model capacitor consists ofelectrodes composed of three layers of graphite, as shownin Fig. 1. The position of an atom, r, is decomposed intothe direction normal to the electrode surface, z, and thedirections parallel to the electrode surface, rk. The elec-trodes are separated by a distance Lz, and the regionbeyond the electrodes in the z direction is consideredinfinite vacuum. To characterize the double layers ateach surface accurately, it is important that they do notinteract. Accordingly, we find Lz = 10 nm meets thiscriteria; however, for computational ease, select bench-marking simulations are performed with smaller Lz =4 nm systems. The electrode atoms interact with theionic liquids through both repulsive-dispersive Lennard-Jones interactions as well as electrostatic interactions,
4
though the atomic positions are held fixed throughoutthe dynamics. To approximate a metallic surface, thecharges on the electrode atoms are taken to be Gaussiandistributed with a width at half maximum of
pln(2) A.
Two approaches are employed in the present workto induce a potential on the electrodes: the constant-� emsemble and the constant-� ensemble. For theconstant-� ensemble, the charges on the electrodes aredetermined through a modified version of the constantpotential formalism first implemented by Siepmann42,43
that ensures all atoms on a given electrode are at anidentical electrostatic potential and that the potentialdi↵erence between the two electrodes is at the definedtarget value, � . Similar approaches have found use insimulations of ionic liquids in the presence of electrifiedinterfaces.24,27,48 In this method the system Hamiltonian,H, for our dual electrode, model capacitor is given by
H = UK + URD + UES �A|�|� , (6)
where UK is the kinetic energy, URD is the totalrepulsive-dispersive energy, UES is the total electrostaticenergy, A is the electrode cross sectional area, |�| is theabsolute value of the electrode surface charge density onthe positive (�+) or negative (��) electrode, and � is the imposed potential di↵erence between positive andnegative electrodes, given by +� �. This Hamiltonianresults from a constrained optimization that distributescharges on the electrode atoms such that �+ = ���.Thus, all atoms on a given electrode are at an identi-cal electrostatic potential, and the potential di↵erencebetween the electrodes is the defined value of � . Forclarity, throughout the majority of this work the super-scripts + and � are not explicitly given, but are indicatedthrough the sign of the surface potential or the charge.
An alternative to treating the electrodes in theconstant-� ensemble is to simply define a static chargedistribution on the electrode, referred to as the constant-� ensemble. For our model capacitor, we distribute equalcharge to all surface atoms (i.e. those on the graphiteplane nearest the liquid) on a given electrode and en-sure �+ = ���. The Hamiltonian in this case is simply,H = UK +URD +UES . Care must be taken when defin-ing the electrode potential, however, as
=@H
@�=
1
N
X
i
@H
@qi
@qi@�
, (7)
which implies any electrode atom, “i,” not participat-ing in the charging process (@qi/@� = 0) should not beincluded in the potential computation. For the presentwork, because we smear charge only on the graphite planenearest the electrolyte, we tabulate the potential on onlyatoms in this plane. This method is clearly more straight-forward than the constant-� formalism; however, asthe surface structure grows in complexity, the assignmentof charges in this manner may not be trivial.
C. Capacitance Fluctuation Formulas
For the constant-� ensemble, the Helmholtz free en-ergy of our model capacitor can be written as F =���1ln(⌦), where � is the inverse product of Boltz-mann’s constant and temperature and ⌦ the partitionfunction of our system. The partition function is equal toRe��HdRdP , where
R...dRdP represents a phase space
integration and H is given by Eq. 6. The derivative of Fwith respect to � may then be written as,
@F
@� = �Ah|�|i �A�
⌧@|�|@�
�+
⌧@UES
@�
�, (8)
where h...i represents the ensemble average of a givenquantity. The final two terms that describe the ensembleaverage of the derivatives of |�| and UES with respectto � arise from the implicit, and instantaneous, depen-dence of the electrode charge on � introduced throughour constant-� formalism. The integral of Eq. 8 withrespect to � gives a direct means to obtain di↵erencesin free energy from molecular dynamics simulations. Weinclude this expression here to provide a complete outlineof the thermodynamics of our model capacitor system.Further evaluation and implementation of this expressioncan be found in an alternate work.56
Expressions for the di↵erential capacitance of ourmodel system can also be derived.31,46,57,58 The aver-age surface charge of our constant-� electrode is givenby h�i = ⌦�1
R�e��HdRdP . As described in recent
work,31,46 the derivative of h�i with respect to � yieldsthe combined di↵erential capacitance of the two elec-trodes in our model capacitor. This quantity can bewritten as
@h�i@�
=�A h|�|��i+⌧
@�
@�
�
+ �A�
⌧@|�|@�
��
�� �
⌧@UES
@� ��
�,
(9)
where � is the inverse product of Boltzmann’s constantand temperature, UES is the electrostatic energy, and�� = � � h�i. Eq. 9 is the full fluctuation formula forthe total di↵erential capacitance of two electrodes in theconstant-� ensemble. The final three terms in Eq. 9,which have not been included in previous derivations of@h�i/@� , account for the implicit dependence of elec-trode charge on the value of � . For the system studiedhere, the first term on the right side of Eq. 9 is by andlarge the leading contribution to the total di↵erential ca-pacitance. Included in the second and third terms, h@�/@� i represents the self-capacitance of the electrodesunder the influence of the electrolyte structure. Simi-larly, in the fourth term, h@UES/@� i is dominated bychanges in the self-energy of the electrode. The value ofh@�/@� i, being the capacitance of the electrode only,is an order of magnitude smaller than �A h|�|��i, thoughnon-neglegible. We find that h@�/@� i and hUES/
5
@� i are relatively constant for the model system con-sidered in the present work. As such, the final two termsin Eq. 9 that include fluctuations of these values are van-ishingly small. We thereby truncate the expression afterthe first two terms. For other systems, however, theseterms may not be negligible.
Of experimental interest, the resolution of capacitancefor a single electrode, @h�i/@h i, yields information onthe comparative behavior of the EDL at positive and neg-ative potential conditions. An expression for @h�i/@h i,which has not previously been reported, can be derivedfor use with our model capacitor system by starting withan expansion of di↵erential capacitance,
@h�i@h i =
@h�i@�
@�
@h i , (10)
where the ensemble averages, h�i and h i, are taken tobe on the same electrode. The average potential on oneof our electrodes is given by h i = ⌦�1
R e��HdRdP .
As with surface charge previously, the derivative of thisquantity with respect to � may be taken to yield
@�
@h i
��1
= �A h|�|� i+⌧
@
@�
�
+ �A�
⌧@|�|@�
�
�� �
⌧@U
@� �
�,
(11)
where � = � h i. For the systems of interest in thepresent work, we have found, numerically, that
⌦@ @�
↵,
representing the instantaneous change of electrode po-tential with a change in potential di↵erence, is 0.5 and-0.5 for the positive and negative electrodes, respectively,to within the nearest thousandth. For reasons previouslydescribed, the final two terms in Eq. 11 are vanishinglysmall and are not included in the current computations.The total expression for di↵erential capacitance for eitherthe positive or negative electrodes is then given as
C� =@h�i@h i =
�A h|�|��i+
⌧@�
@�
��⇥
�A h|�|� i+
⌧@
@�
���1
,
(12)
where C� indicates capacitance in the constant-� en-semble.
Complementary thermodynamic expressions can bederived for our model capacitor in the constant-� ensem-ble. The Helmholtz free energy can again be determinedfrom F = ���1ln(⌦); however, the simpler form of theconstant-� Hamiltonian when compared to that of theconstant-� case leads to
@F
@�=
X
i
@F
@qi
@qi@�
. (13)
As a constant value is used for each surface charge atom,@F@� further reduces to h i, the average potential on
the electrode. Concerning capacitance using constant-�electrodes, the quantities of interest are @�/@h� i and@�/@h i, where the potential di↵erence between the elec-trode and the potential of each electrode are ensembleaverages as the potential is free to fluctuate in time. Em-ploying a procedure similar to that used for the constant-� ensemble, the fluctuation expressions can be writtenas
@|�|@h� i =
⌧@�
@|�|
�� �A h� �� i
��1
(14)
and
C� =@�
@h i =
⌧@
@�
�� �A h| |�� i
��1
, (15)
where �� = � � h� i and C� indicates capacitancein the constant-� ensemble.
D. Polarizable Force Field
We employ the atomistic polarizable potential for liq-uids, electrolytes, and polymers (APPLE&P)50–52 to rep-resent [EMIM][BF4]. In this potential, each atom has apolarizability, ↵, which induces an atomic dipole, µ, ac-cording to
µi = ↵i ·Ei, (16)
where Ei is the electric field due to charge-charge andcharge-dipole interactions at atomic site i. The dipolesare iteratively evaluated at each step of the simulationuntil the polarization energy converges to within 1E-9kcal/mol. The polarization energy, Upol, can be given as
Upol = �1
2
X
i
µ ·E0i , (17)
where E0i is the charge contribution to the electric field.
Because the dipole-charge interactions are long range, wehave extended both the 2D and 3D Ewald summationsto include contributions from charge-dipole interactions.The dipole-dipole interactions are alternatively treatedwith a reaction field to approximate their long-range in-teractions during the iterative update procedure. For alllong-range techniques, the k-space grid is chosen to pro-duce a 0.01% error in the long-range energy.
E. Molecular Dynamics Simulations
All simulations are performed in the canonical ensem-ble using the Nose-Hoover formalism having a tempera-ture of 363 K. The rRESPA formalism is used for timeintegrate, with the time step being 3 fs. We employthree rRESPA levels, with bonds, angles, and dihedralsbeing determined on the first level; impropers and non-bonded interactions between atoms no further than 6 A
6
apart being determined on the second level; and all non-bonded interactions, long-range interactions, dipole up-dates, and surface charge updates being performed onthe third level. There are 9 first level integrations and3 second level integrations for every 3 fs outer integra-tion. For the constant-� simulations, the potential isupdated in the 3rd rRESPA step, and the charges aredetermined such that the variance of on each atom iswithin 0.025 V of the electrode average. As previouslymentioned, our model capacitor is composed of two elec-trodes, a positive and negative electrode, between whichis the ionic liquid electrolyte. The electrodes are com-posed of three layers of graphite having the basal planeexposed to the electrolyte. For our investigation intothe influence of method on EDL structure and capaci-tance, we use a large capacitor system with Lz = 10 nmand 216 [EMIM][BF4] pairs, which, based on average iondensity profiles, is free of size errors related to the in-teraction of the EDLs of the positive and negative elec-trodes. In a limited number of numerical tests, however,we also employ a small capacitor with an electrode spac-ing, Lz = 4 nm and 140 [EMIM][BF4] pairs as an e�cientmeans of comparing the accuracy of various methods.
III. RESULTS
A. Long-Range Electrostatics
To begin our study on di↵erential capacitance, we ad-dress the issue of the performance of electrostatic summa-tion techniques with respect to accuracy of the electrodepotential and EDL structure. To do this, we use a smallcapacitor system consisting of [EMIM][BF4] interfacedwith electrodes Lz = 4 nm apart at 363 K, which allowsthe e�cient comparison of our most expensive, and accu-rate, numerical methods to those having approximationsthat reduce computational expense.
Ideally, our model capacitor should electrostaticallyconform to the slab condition, periodic in the plane ofthe electrode and having infinite vacuum exterior to theelectrode. To understand the implications of this condi-tion on the electrostatic potential, we refer to the classicalcase of parallel plate capacitors separated by a distanceLz and having uniform charge densities of ±�. For sucha system, the magnitude of the electric field between theplates is �/✏0, while the field is zero external to the plates.The lack of electric field external to the plates indicatesthat the potential is constant in these regions, and a pathintegral yields a potential di↵erence of �Lz/✏0 betweenthe plates. A direct extension of this model to that of aparallel plate capacitor filled with a medium of a given di-electric constant yields similar results. Therefore, for ourmodel capacitor to be consistent with the classical elec-trostatic system, the potential external to the electrodesshould be constant. Some deviation from this behavior,however, could be expected near the electrodes becausethe charges in our system are localized to atoms.
To compare the behavior of the multiple electrostaticsummation techniques in this respect, we provide a mea-sure of the electrostatic potential averaged in the rk di-rections, given as a function of z in Fig. 2 for our smallcapacitor in the constant-� ensemble. We immediatelynote a distinct di↵erence between the 3D-Ewald and theother summation techniques. This is, in part, becausethe 3D Ewald summation necessitates the use of a pe-riodic boundary condition in the z direction, which hasbeen applied to allow a vacuum of 3Lz to separate theelectrode periodic images, leading to a periodic bound-ary at ±80 A. Because of periodicity in the perpendic-ular direction in the 3D-Ewald approach, the potentialmust necessarily cross zero in the vacuum region, whichis seen at the intersection of the potential from the pos-itive and negative electrodes at the periodic boundary.According to our assumption of an infinite medium be-yond the boundary of the electrodes, the 2D-Ewald and3D/slab Ewald techniques conform with our expectationsfrom the parallel plate capacitor model, having constantpotential for |z| > 1
2Lz. Interestingly, the internal struc-ture of the ionic liquid appears similar between the threesummation techniques.
0 20 40 60 80|z| (Å)
-4-2024
�^� || (V
)
2D-Ewald3D-Ewald3D/slab-Ewald
-80 -40 0 40 80z (Å)
-4
-2
0
2
4�^
� (V)
2D-Ewald3D/slab-Ewald3D-Ewald
0 1 2 3 4 5 6 7z (Å)
05
101520
^ (V
)
(c)
Vacuum
IL
Electrodes
(a)
(b)
Vacuum
FIG. 2. Electrostatic potential profiles obtained from anLz = 4 nm capacitor having a constant � of 6 V. The pro-files are obtained from simulations employing the 3D-Ewald(dashed line), the 2D-Ewald (dotted line), and the 3D/slab-Ewald (solid line) summation techniques. The profiles areshown from the center of the capacitor to z = 80 A, whichrepresents the periodic boundary for the 3D and 3D/slab-Ewald techniques.
To better understand the implications of the variouspotential profiles, we examine the relationship between �and � using the di↵erent electrostatic summation tech-niques for our model capacitor systems in the constant-� and constant-� ensembles, given in Fig. 3. In bothelectrode ensembles, we find the 3D/slab-Ewald approachto be commensurate with the 2D-Ewald correction. Be-tween the constant-� and constant-� values using thesesummation techniques, we see close agreement, whichsuggests the average �-� relationship is weakly depen-dent on electrode conditions for the ideal graphite system
7
used here. For the 3D-Ewald summation, we find di↵er-ent behaviors that depend on the specific electrode condi-tions. For constant-� electrodes, shown in Fig. 3a, wefind that average surface charge, h�i, is overestimated by5-10%, which is a result of the interaction between oppo-sitely charged electrodes across the vacuum. The reducedpotential is compensated by a larger h�i. For constant-�electrodes, shown in Fig. 3b, the average potential drop,h� i, is an underestimate of the 3D/slab and 2D Ewaldtechniques, again due to interaction across the vacuum.Across the range of explored values, the overestimationof h�i and the underestimation of h� i for constant-� and � electrodes, respectively, by the 3D-Ewald summa-tion is ⇠5-10%. The error in the estimates of h�i andh� i using the 3D-Ewald approach directly translatesto a 10-20% error in di↵erential capacitance of a singleelectrode, which is approximately twice the derivate ofsurface charge with respect to the potential di↵erencebetween the two electrodes.
0 1 2 3 4 5 6�6^� (V)
0
5
10
15
|m| (µ
C/c
m2 )
2D-Ewald3D-Ewald3D/slab-Ewald
0 1 2 3 4 5 66^ (V)
0
5
10
15
�|m|�
(µC
/cm
2 )
2D-Ewald3D-Ewald3D/slab-Ewald
0 1 2 3 4 5 6 7z (Å)
05
101520
^ (V
)
constant-6^
constant-m
(a)
(b)
FIG. 3. Influence of long-range electrostatic summation (2D,3D, and 3D/slab Ewald summation) on the �- relationshipof the Lz = 4 nm model capacitor for both (a) constant-� and (b) constant-� conditions.
While the potential on the electrode is a↵ected by thechoice of long range summation technique, the associatedinfluence on structure is not clear. To probe structural ef-fects, we have computed the ion density profile (⇢) alongz using the three summation techniques as well as thedeviation of ⇢ obtained by the 3D Ewald and 3D/slabEwald from that of the 2D-Ewald technique (�⇢) for theconstant-� and � conditions. The density profiles, asshown in Fig. S1a and b of the supplemental material,59
indicate that all three summation techniques for bothelectrode conditions result in almost identical measuresof ⇢. An analysis of the �⇢, given in Fig. S1c and d,59 in-dicates that the resulting deviation in density when using3D/slab-Ewald and 3D-Ewald summations is only a few
percent. The similarity in the magnitude of �⇢ betweenthe two techniques indicates the structure is relativelyinsensitive to the up to 10% error in � or � inherentto the 3D-Ewald summation.Our results suggests that the 3D/slab-Ewald summa-
tion technique is appropriate for the system of inter-est here, providing an accuracy comparable to the 2D-Ewald summation with the lower computational expenseof the 3D-Ewald summation technique. For other moregeneral systems, we conjecture that the 3D/slab-Ewaldwill be a good approximation to the exact 2D-Ewald ap-proach if the periodic slab images are su�ciently sep-arated such that their leading interaction is dipolar innature. In the remainder of this work, therefore, we ap-ply the 3D/slab-Ewald summation technique. With re-gards to the constant-� and constant-� ensembles, ourresults suggest both methods behave in a qualitativelysimilar manner. As shown in Fig. 3, the �-� relation-ship is similar for both surface conditions, leading to ourexpectation of similar magnitudes of di↵erential capaci-tance.
B. Electrode Ensemble: Capacitance Sensitivity
We now turn to an investigation of the comparativebehavior of di↵erential capacitance in the constant-� and constant-� ensembles. We first describe the generalbehavior and function of the terms present in the fluctu-ation expressions used to compute Cdl. For the expres-sion for constant-� electrodes in Eq. 12, the dominantterms include �Ah|�|��i, h@�/@� i, and �Ah|�|� i.The value of �Ah|�|��i and h@�/@� i on the positiveand negative electrodes are of equal magnitude but op-posite sign as �+ = ���. Alternatively, the value of�Ah|�|� i is of equal magnitude and sign for both elec-trodes because � = +� � is constant. The functionof �Ah|�|��i is to include the capacitance that is pri-marily attributed to electrolyte interaction with the elec-trode, while h@�/@� i encapsulates the self-capacitanceof the electrodes. Any asymmetry in C� related to thedi↵erent screening properties of the EDL at the positiveand negative electrodes is manifested through �Ah|�|� i.From Eq. 12, we then see that from a single simulationat a given value of � , two values of C� are obtainedusing @h|�|i/@�
h@�/@� i±�Ah|�|� i , where the addition or subtrac-tion in the denominator yields the capacitance of the pos-itive and negative electrode, respectively. With regardto the simpler fluctuation expression used for constant-�electrodes, Eq. 15, both h@ /@�i and �Ah� �| |i arepositive for both electrodes. As with the constant-� expression, the capacitance contribution from the liquidand self-capacitance of the electrode are primarily con-tained in the h@ /@�i and �Ah� �| |i terms, respec-tively.We now perform a convergence study on @h�i/@h i, as
defined in Eq. 12, on our prototype Lz = 4 nm capacitorusing the constant-� ensemble for potential di↵erences
8
of 0, 2, and 4 V. Regarding the value of �Ah|�|��i, shownin Fig. 4a, we find the magnitude reaches to within 10%of its long-time average after ⇠10 ns. Longer simula-tions lead to subtle variations, which can be seen in thecrossing of the value of �Ah|�|��i for 0 and 2 V poten-tial di↵erences near 45 ns. These long time processesthat a↵ect capacitance are di�cult to characterize andare likely related to minor changes in the EDL struc-ture, such as the ion density at the electrode surface. Wehave also provided a measure of the instantaneous deriva-tive of surface charge with respect to surface potential,h@�/@ i, as a function of simulation time. As given inFig. 4b, this term converges within ⇠ 10 ns and is nearlyconstant and for multiple values of � , exhibiting varia-tions on the order of 10�3, which are orders of magnitudesmaller than the total value of capacitance. We thereforeconclude that this term is e↵ectively constant across ourpotential di↵erence range and here further determine itsvalue from a single simulation at � = 0 V. The thirdterm, �Ah|�|� i, in Fig. 4c also shows similar conver-gence trends with the prior two terms. It is important tonote that at � = 0 V, this term is nearly zero, which isnecessary as the positive and negative electrodes are iden-tical in this case. Furthermore, the value of this term at2 and 4 V is small, and this indicates that in our specificsystem, there is little di↵erence between the capacitanceof the positive and negative electrodes.
The convergence behavior of @�/@h i for constant-�electrodes is qualitatively similar to that of the constant-� electrodes. The value of �Ah| |�� i, shown inFig. 4d, shows reasonable convergence within ⇠25 ns,which is longer than required for constant-� electrodes.The value of fluctuations on constant-� electrodes mayconverge more slowly due to harder electrostatic interac-tions between the electrode and surface ions when com-pared to the constant-� electrodes, where the chargeredistributes on the electrode in response to the surfaceion environment and could lead to a more favorable ioninteractions. As given in Fig. 4e, the value of h@ /@�iappears to be converged again within 25 ns, though thespread in values is relatively small, on the order of 10�3.Based on our convergence tests, all further measures ofdi↵erential capacitance in the present work are obtainedfrom 50-100 ns simulations.
To better understand the e↵ect of electrode ensemble,we compare the di↵erential capacitance of our prototypeLz = 4 nm system using constant-� and � electrodesas obtained from both fluctuation expressions and nu-merical derivatives of � with respect to . To provide a� profile su�ciently dense enough in space to takea numerical derivative, we have performed 11 simula-tions equally spaced between 0 and 2 V for the constant-� condition and between 0 and 4.44 µC/cm2 for theconstant-� condition. Beyond these limits, we have per-formed additional simulations at 3 V (6.58 µC/cm2) and4 V (8.78 µC/cm2) for the constant-� (�) electrodes.Following previous approaches,27 � is fit to a high orderpolynomial, the derivative of which provides a smooth,
well-behaved measure of Cdl. Furthermore, we fit the re-sults of our fluctuation formulas to a high order polyno-mial to highlight the general trends in di↵erential capac-itance. The exact values resulting from the fluctuationformula as well as the derivative of surface charge as ob-tained from a central di↵erences approach of raw data areshown in Fig. S2 of the supplemental material.59 The re-sult of this procedure, shown in Fig. 5, indicates that thefluctuation expressions provide close agreement with thenumerical derivative approach. This validates our fluc-tuation expressions for di↵erent electrode ensembles andenables the treatment of larger systems, where obtaininga dense �( ) curve is prohibitively expensive.While there are minor variances between capacitances
obtained with di↵erent electrode conditions, both themagnitude and global, camel-shaped profile of Cdl aresimilar. However, the prototype system used to obtainthese results necessarily introduces confinement e↵ectsinto the measure of Cdl. We therefore perform compu-tations on larger Lz = 10 nm systems to remove sizee↵ects, the results of which are shown in Fig. 6. Follow-ing our previous convention, we show a fit to the fluc-tuation formula results to highlight trends in the data,while the exact values can be found in Fig. S3 of thesupplemental material.59 As with the smaller capacitor,the magnitude of di↵erential capacitance is similar be-tween the constant-� and constant-� ensembles, eachresulting in averaged capacitances of 4.8 µF/cm2. Wealso note that for both electrode ensembles the value ofcapacitance is larger at negative potentials that at pos-itive potentials. A few di↵erences, however, are appar-ent between the ensembles. Constant-� electrodes pro-duce a highly resolved camel-shaped profile (akin to ex-pectations from mean field theory), with the peaks be-ing centered on ±0.5 V, while constant-� electrodeslead to a more di↵use capacitance with lower maximumvalues. The value of surface potential at � = 0 and� = 0, or the potential of zero charge, is virtually zerofor our system in both electrode ensembles, which some-what appears to be nearer to the maximum in C� atnegative potentils on constant-� electrodes and in awell between the dual peaks of the camel-shaped C� onconstant-� electrodes. At larger magnitudes of potential(> |1| V), constant-� electrodes have larger values ofdi↵erential than constant-� electrodes. The error asso-ciated with both measures of capacitance, shown in Fig.S3, are obtained from the standard deviation of Cdl as afunction of simulation length and are similar in magni-tude for both electrode systems, ⇠0.1-0.2 µF/cm2.
C. Electrode Ensemble: Structural Sensitivity
To understand the origin of the noted di↵erences inCdl with respect to electrode ensemble, we now investi-gate the potential dependent structural behavior of theEDL from the Lz = 10 nm capacitor. As a first step inthis direction, we have evaluated the ion density (⇢) as
9
0 25 50Simulation Length (ns)
0.37
0.375
0.38
�,m/,6^� (µ
F/cm
2 ) 0 V2 V4 V
0 25 50Simulation Length (ns)
1
2
3
`A�|m|bm� (µ
F/cm
2 ) 0 V2 V4 V
0 25 50Simulation Length (ns)
-0.01
0
0.01
`A�|m|b^�
0 V2 V4 V
0 25 50SimulationLength (ns)
1.1
1.2
`A�|^
|b6^� (
cm2 /µ
F) 0 µC/cm2
4.39 µC/cm2
8.78 µC/cm2
0 25 50Simulation Length (ns)
1.346
1.348
�,^/,m� (
cm2 /µ
F) 0 µC/cm2
4.39 µC/cm2
8.78 µC/cm2
(a) (b) (c)6^ =6^ = 6^ =
(d)
(d) (e)m = m =
FIG. 4. Convergence of terms in Eqs. 12 and 15, which is used to compute Cdl
with constant-� and constant-� electrodes,respectively, as a function of total simulation time. Results are obtained from an Lz = 4 nm model capacitor and are shownfor constant � values of 0, 2, and 4 V and for constant � values that produce similar values of potential di↵erence, 0, 4.39,and 8.78 µC/cm2.
-1.5 -1 -0.5 0 0.5 1 1.5�^� (V)
4
4.5
5
5.5
Cm (µ
F/cm
2 )
FluctuationDerivative
4
4.5
5
5.5
C6^
(µF/
cm2 )
FluctuationDerivative
0 1 2 3 4 5 6 7z (Å)
05
101520
^ (V
)
(c)
(b)
constant-6^
constant-m
(a)
(b)
FIG. 5. Comparison of the di↵erential capacitance obtainedfrom Lz = 4 nm systems having (a) constant-� and (b)constant-� electrodes. Capacitance from both the fluctuationformulas (solid lines) as well as the numerical derivative ofthe charge density (dashed lines) are provided.
a function of distance (�z) from both constant-� andconstant-� electrodes, shown for neutral, negative, andpositive electrodes in Fig. S4 of the supplemental mate-rial. For the case of neutral electrodes displayed in Fig.4a and b, the total density, which includes both cations
-2 -1 0 1 2�^� (V)
3.5
4
4.5
5
5.5C
dl (µ
F/cm
2 )
C6^
(constant-6^)Cm (constant-m)
FIG. 6. Di↵erential capacitance of an Lz = 10 nm capacitoras obtained from the use of constant-� (dashed lines) andconstant-� (solid lines) electrodes.
and anions, shows a sharp signature within �z < 5 A,indicating a dense layer of adsorbed ions at the electrodesurface. The electrode surface ion layer shares similari-ties with the Stern layer, which in the classical picture ofthe EDL is the layer closest to the surface. Less promi-nent accumulations in ⇢ occur at larger values of �z,5 < �z < 15 A, and represent the liquid response tothe ordering and possible imbalance of cations and an-ions in the surface ion layer. Further decomposing ⇢ intoits ionic components, the surface ion layer shows a sin-gle, strong cation signature near the surface and two pri-mary anion signatures, one overlapping the cation sig-nature and one more distance from the surface. This
10
suggests a finer structure within the surface ion layer,where anions assume district configurations. Compar-ing ⇢ between the two electrode conditions, the surfaceion layer at the constant-� electrodes (Fig. S4a) hasa higher initial peak than the one at the constant-��electrodes (Fig. S4b). This implies constant-� elec-trodes lead to a denser surface ion layer. Such a result isplausible considering the electrode charges respond to thesurface ion environment in the constant-� approach,allowing for more energetically favorable charge distri-butions than that of the constant-� approach and thusstronger ion binding at the surface.
Extending our analysis of the EDL density, we pro-vide ⇢ profiles for electrodes having a constant � of4.2 V or, equivalent in terms of potential, a constant-|�| of 9.63 µC/cm2. The density as a function of �zfrom the positive electrodes is shown in Fig. S4c andd, while density from the negative electrodes is shownin Fig. S4e and f. Beyond the expected di↵erences ofa cation and anion dominated surface ion layer at thenegative and positive electrodes, respectively, the samegeneral structure described for the neutral electrode ispresent at the charged surfaces. This includes a strongsignature at small �z, representing the surface ion layer,and less prominent signatures at larger �z, represent-ing ion accumulations that compensate for an imbalanceof cations and anions in surface ion layer. The surfaceion layer occurs at �z < 7 A at the positive electrodeand �z < 4 A at the negative electrode. The close ap-proach of the cations near the negative electrode suggests[EMIM] assumes a highly aligned conformation with re-spect to the surface, which is somewhat surprising con-sidering it is considerably larger than [BF4]. We againnote that di↵erent ion distributions at the constant-� and constant-� electrode surfaces imply a denser surfaceion layer at the constant-� electrode.
A simple comparison of the ion density profiles indi-cates that constant-� electrodes lead to a denser sur-face ion layer than constant-� electrodes. To quantifythis, we have explicitly evaluated the number of ions inthe surface layer per unit area, N, by simply integratingover the initial peak found in the ⇢(�z) profile. The re-sults of this procedure are shown in Fig. 7a for constant-� and constant-� electrodes. For both electrode con-ditions, we find a slight tendency toward more anionsthan cations in the surface layer at h i = 0 V, whichis likely a result of the small radius of [BF4]. With anincrease in h i, we see an increase in the number of an-ions and a decrease in the number of cations, leading tomore negatively charged surface ion layer; naturally, theopposite e↵ect is noted when decreasing potential. Wenote that the number of [EMIM] cations is generally largeand weakly varying, changing by only ⇠0.25 nm�2 acrossthe investigated range of potential. On the other hand,[BF4] exhibits large changes with potential, varying by⇠1.25 nm�2 from the lowest to highest surface poten-tials. This implies [EMIM] interacts more favorably withthe surface than [BF4].
-3 -2 -1 0 1 2 3�^� (V)
0
25
50
75
W (%
)
0.5
1
1.5
2
N (n
m-2
)
0 1 2 3 4 5 6 705101520
^ (V
)
constant-6^
constant-m
(a)
(b)
[EMIM]
[BF4]
FIG. 7. Properties of the electrode surface ion layer as afunction of surface potential h i for constant-� (solid lines)and constant-� (dashed lines) electrodes of an Lz = 10 nmmodel capacitor. Both the (a) surface density, N , and (b)alignment with respect to the electrode surface, ⇧, are givenfor cations and anions.
Contrasting both electrode ensembles, we see thatthere are, indeed, more cations and anions in the sur-face layer at constant-� electrodes than the one atconstant-� electrodes. At the largest magnitude of sur-face potential, | ± 2.1| V, the ion density is on the or-der of 10-20% higher at constant-� electrodes. Cor-relating this observation with the previously noted dif-ferences in Cdl, we suggest that the energetic favorabil-ity of having more of both ionic species in the surfaceion layer on constant-� electrodes makes their sepa-ration, and thus the polarization of the electrode, moredi�cult. Essentially, this suggests that to obtain a givenvalue of � requires larger potentials on constant-� elec-trodes than on constant-� electrodes. This is consis-tent with the larger values of Cdl at low potentials onconstant-� electrodes and larger values at high potentialson constant-� electrodes. The average capacitanceon both electrodes are practically identical because thecharge accumulation attributed to higher capacitances atlow potentials obtained from constant-� electrodes is es-sentially spread over a larger potential range in the caseof constant-� electrodes.
In addition to surface ion density, we also compute thepotential-dependent ion configuration at the electrodesurface. The possible alignments are shown in Fig. S5of the supplemental material,59 with [EMIM] having a
11
planar configuration, where the charge center is highlyexposed to the surface, and a perpendicular configura-tion, where the interactions are primarily with the chainand the charge center is more removed from the surface.Similarly, [BF4] can assume highly interacting configura-tions with three F atoms close to the surface and otherswith fewer F atoms, one or two, in proximity to the sur-face. We define the percent of highly interacting planar[EMIM] cations and [BF4] having three F atoms near thesurface with ⇧. The value of ⇧ is determined from theangular distribution of the ring normal in [EMIM] or theB-F bond in [BF4] with respect to the electrode surface.Values of ⇧ as a function of surface potential for constant-� and constant-� surfaces are given in Fig. 7b.
The value of ⇧ is also influenced by electrode ensemble,with constant-� electrodes leading to a higher value of⇧ for [BF4] at h i > 0 V than constant-� electrodes.The higher level of alignment of [BF4] with the electrodesurface, and thus more compact packing, naturally ac-companies the higher density noted at constant-� elec-trodes. There is no apparent, systematic di↵erence in thevalue of ⇧ for [EMIM] between the two electrodes, thoughalignment appears to be higher at positive potentialsfor constant-� electrodes when compared to constant-� electrodes. These trends are consistent with interac-tions of surface ions being more favorable with constant-� electrodes than constant-� electrodes, which leadsto more ion alignment at the surface. This further addsveracity to the argument that a higher density, and thusmore aligned, neutral surface ion layer on constant-� electrodes resists the formation of an EDL at low val-ues of � . This washes out the marked camel-type Cdl
profile that result from surface ions being less stronglybound, and thus having a lower � barrier to ion segre-gation and EDL formation, on constant-� electrodes.
Common to both electrode ensembles, the alignment of[BF4] shows a strong dependence on h i, with ⇧ increas-ing from ⇠25% to ⇠75% across the 4.2 V potential range.At neutral surfaces, [EMIM] cations predominantly as-sume the highly interacting planar configuration at theelectrode surface, having ⇧ of ⇠70 %. The value of ⇧for [EMIM] changes little as potential decreases, suggest-ing the surface is saturated with planar cations even atneutral conditions. Such a fact agrees with the previousobservation of a closely packed surface ion layer at thenegative electrode. At the positive electrode, the value of⇧ for [EMIM] decreases, reaching 42% at a surface poten-tial of 2.1 V. Overall, this agrees with the previous asser-tion, based on the behavior of N in Fig. 7a, that [EMIM]has more conformational freedom at the electrode sur-face than [BF4]. Planar conformations of [EMIM] canexist at the negative electrode through chain interactingconfigurations where the charge center is moved form thesurface.
D. Electrolyte Polarizability
The inclusion of atomic polarizability into interatomicpotentials has been shown to be important to accuratelyrepresent the dynamics of ionic liquids. Ionic liquidEDLs, on the other hand, are commonly studied throughnon-polarizable potentials, and the e↵ect of polarizationon such interfaces has not been systematically charac-terized. As an initial means of understanding the influ-ence of polarization on EDL properties, we have com-puted the average induced dipole moment perpendicularto the electrode surface (µz) as obtained from constant-� simulations where � = 0 V, given in Fig. 8a. Thevalue of µz is provided as a function of distance from theelectrode, �z. Negative and positive values of µz implyatomic dipoles pointing toward and away from the sur-face, respectively. The µz profile shares similarities withthe ion density profile, shown in Fig. S4, specifically inthat the atomic dipoles oscillate with the same period in�z as ⇢. The resulting oscillations dampen to µz ⇠ 0after the first few ion layers, which can be seen from acomparison with ⇢ in Fig. S4.
0 5 10 156z (Å)
-5
0
5µ
z x 1
03 (eÅ)
Total[EMIM][BF4]
Total[EMIM]+
[BF4]-
0 5 10 156c (Å)
-15
0
15
lQ (e
·nm
-3)
[EMIM][BF4]
Total[EMIM]+
[BF4]-
-150
15
latom
(nm
-3)
CNHBF
0 5 10 15 20 256z (Å)
Total[EMIM]-
[BF4]-
-50-25 0 25 50z (Å)
(a)
(b)
(c)0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
more cations
more anions
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
more cations
more anions
0 5 10 156z (Å)
-10
0
10
r || (Å)
Total[EMIM]+
[BF4]-
0 5 10 156c (Å)
-15
0
15
lQ (e
·nm
-3)
[EMIM][BF4]
Total[EMIM]+
[BF4]-
-150
15
latom
(nm
-3)
CNHBF
0 5 10 15 20 256z (Å)
Total[EMIM]-
[BF4]-
-50-25 0 25 50z (Å)
(b)
(b)
(c)
g(r,z)
0 5 10 156z (Å)
-10
0
10
r || (Å)
Total[EMIM]+
[BF4]-
0 5 10 156c (Å)
-15
0
15
lQ (e
·nm
-3)
[EMIM][BF4]
Total[EMIM]+
[BF4]-
-150
15
latom
(nm
-3)
CNHBF
0 5 10 15 20 256z (Å)
Total[EMIM]-
[BF4]-
-50-25 0 25 50z (Å)
(b)
(b)
(c)
g(r,z)
FIG. 8. Comparison of (a) the average induced dipole nor-mal to the electrode, µz, for [EMIM] and [BF
4
] and (b) thenet ion distribution, �g = gcations � ganions, as a functionof distance from a surface cation in the rk direction and theelectrode surface, �z. Results are shown for our Lz = 10 nmmodel capacitor having � = 0 V. More cation concentra-tion is represented by blue regions while anion concentrationis represented by red regions. The direction of the averageinduced dipole in a given accumulation of ions is indicated byblack arrows.
The ion resolved induced dipole moments at � =0 V, again provided in Fig. 8 a, further show surface
12
cations to have dipoles directed away from the electrode,while surface anion induced dipoles are directed towardthe electrode. As such, the induced dipoles are ener-getically unfavorable to EDL formation. For instance,at negative values of h i, there is an electric field di-rected toward the electrode. While this attracts cations,it opposes the cation dipole moment, which is directedaway from the electrode. The same mechanism is alsoat play for anions. Thus, the induced dipoles will softenthe electrostatic interaction between the surface ion layerand the electrode surface. For instance, as the electrodebecomes negatively charged, the electric field originat-ing from the electrode opposes the alignment of induceddipoles on cations, while it is aligned with the induceddipoles of anions. This leads to a lower and higher inter-action between the negative electrode with cations andanions, respectively. A similar behavior is noted as theelectrode becomes positively charged, with cations beingless repulsed and anions being less attracted to the elec-trode surface. The net e↵ect of induced dipoles, then, isto create an energetic barrier to EDL formation.
We rationalize the behavior of the induced dipolesthrough an analysis of surface structure using a rk andz dependent net ion distribution function, �g(rk,�z),for surface cations, given in Fig. 8b. (The valueof �g(rk,�z) is computed as the di↵erence betweenthe cation and anion distribution surrounding surfacecations, where the distribution is converted to distancefrom the cation in the rk direction and from the electrodesurface in the z direction for ease of comparison with thez-dependent induced dipole profile.) In e↵ect this pro-vides a map of ion probability as a function of distancefrom a surface cation, with a net cation accumulationrepresented by blue regions and net anion accumulationrepresented by red regions. Within the surface layer ofions, which occurs as �z < 5 A, we see surface anionsat rk = ±5 A crowding the surface cation at rk = 0 A.Beyond these surface anions are aggregations of cationsoccurring at rk = ±8 A, which suggests electrostatic or-dering at the surface. Interestingly, the second molecu-lar layer, occurring at 5 A < �z < 10 A shows corre-lation with the surface ions. Specifically, for a surfaceion at a given rk, we see a corresponding accumulationof counterions at the same rk in the second layer. Thisis in agreement with the noted induced dipoles, surfacecations have dipoles which point away from the surfacetoward their correlated anions in the second layer, whilesurface anions point toward the surface and away fromtheir correlated cations in the second layer. This furtherunderlines the three-dimensional nature of the EDL.
To further elucidate the implications of the induceddipoles, we have performed a suite of simulations inthe constant-� ensemble using the APPLE&P elec-trolyte potential without atomic polarization. Fromthese simulations, we have computed the di↵erence incapacitance between polarized and non-polarized com-putations, given by �Cdl and shown in Fig. 9 a. Wesee negative regions of �Cdl around ±0.5 V and posi-
tive region for 1 V < h i < -1 V. Overall, this leads toa net decrease in capacitance. The regions of negative�Cdl occur where highly resolved maxima in Cdl werenoted for constant-� electrodes. Having higher capacityat lower surface potentials suggests less resistance in thenon-polarizable force field to formation of the EDL.
-1
-0.5
0
0.5
bCdl (µ
F/cm
-2)
Total[EMIM]+
[BF4]-
-0.1
-0.05
0
0.05
bN (n
m-2
)
[BF4][EMIM]
Total[EMIM]+
[BF4]-
-2 -1 0 1 2�^� (V)
-20
-10
0
10
bW (%
)[BF4][EMIM]
0 5 10 15 20 256z (Å)
Total[EMIM]-
[BF4]-
-50-25 0 25 50z (Å)
(a)
(b)
(c)
FIG. 9. Potential dependent influence of electrolyte polar-ization on (a) di↵erential capacitance (�C
dl
) and (b) surfaceion alignment (�⇧) for a Lz = 10 nm model capacitor usingthe constant-� formalism.
To evaluate the structural origins of the trends in�Cdl, we compute the di↵erence in surface ion alignment,�⇧, between polarized and non-polarized simulations, asgiven in Fig. 9 b. We see significant di↵erences in surfaceion alignment, with the anion layer at the positive elec-trode showing 10% lower alignment, which allows the in-corporation of more cations into the surface and results ina 10-15% increase in alignment of cations at the positiveelectrode. A similar trend, though less marked, is seen atthe negative electrode. The alignment of surface cationsis decreased by up to 5%, while anions are 5-10% morealigned with the surface. The relatively small change inthe value of ⇧ for [EMIM] at the negative electrode is anadditional indication that the surface is saturated withplanar cations, which was previously suggested from thesmall changes in [EMIM] density and configuration inthe polarizable system (see Fig. 7). The increased degreeof configurational disorder in the polarizable EDL corre-sponds to the noted negative regions of �Cdl at relativelylow potentials, which as previously noted results fromthe induced dipole softening of electrostatic interactionswith the electrode.
13
IV. CONCLUSIONS
The present work is an investigation of various ap-proaches for the MD modeling of ionic liquid EDLs. Thesystem of focus is a model capacitor of slab-like geom-etry that is composed of two graphite electrodes inter-faced with a polarizable electrolyte, [EMIM][BF4]. Onthis system, we have quantified the influence of electro-static summation technique, the method of charging theelectrode, and the electrolyte atomic polarizability onthe EDL structure and di↵erential capacitance. Further-more, we have compared and contrasted di↵erent meth-ods of computing di↵erential capacitance.
To ensure the validity of our model capacitor, we haveevaluated the interplay between electrostatic summationand the assumption of quasi-2D system geometry. Inthis regard, our model capacitor should conform to theclassical, quasi-2D parallel plate capacitor and have noelectric field external to the electrodes. We find both the3D Ewald summation corrected for slab geometry andthe more expensive, though exact, 2D Ewald summa-tion yield model capacitors that are e↵ectively quasi-2D.Alternatively, the use of an uncorrected 3D Ewald sum-mation is not compatible with quasi-2D geometry, whichleads to errors in the electrode potential and charge.
We have additionally investigated how di↵erent tech-niques for charging the electrode, namely constant-� and constant-� ensembles, influence di↵erential capaci-tance. To e�ciently perform this analysis, we derive fluc-tuation expressions for Cdl for both constant-� (C� )and constant-� (C�) ensembles. These expressions pro-vide di↵erential capacitance on a per electrode basis andare validated through a comparison to the direct deriva-tive of � with respect to . Di↵erential capacitance com-putations performed for both constant-� and constant-� electrodes show di↵erences in overall shape, but pro-vide similar averages over �2 V < h i < 2 V. Constant-� electrodes produce capacitance profiles with highlyresolved peaks conforming to the camel-shaped profile,while constant-� electrodes yield a capacitance that ismore di↵use and does not as clearly exhibit camel-typecharacter.
The di↵erences in Cdl with respect to electrode ensem-ble originate in the behavior of the surface ion layer, orStern layer. An analysis of the h i-dependent densityof this layer shows more ions, both cations and anions,are present at the constant-� electrode than at theconstant-� electrode. Such an e↵ect likely arises fromthe variable surface charge on the constant-� electrode,which, unlike the constant-� approach, responds to sur-face ions to create a more favorable electrostatic interac-tion. The denser surface ion layer on constant-� elec-trodes, then, requires a higher value of h i to achievethe same polarization as the layer on the constant-�electrode. Thus, large values of Cdl present between�0.75 V < h i < 0.75 V on constant-� electrodes aredistributed over a larger potential range on constant-� electrodes, leading to a more di↵use profile with lower
maximum values of Cdl.The role of electrolyte polarization has been ana-
lyzed through comparisons with non-polarizable elec-trolyte models on constant-� electrodes. We find thatthe average value of capacitance increases slightly andthat the camel shaped profile becomes more resolvedwhen polarization is removed. This e↵ect has been re-lated the fact that induced dipoles oppose the field gener-ated by the electrode, thereby softening the electrostaticinteraction between ions and the electrified surface. Thiscan be further traced to ion correlation e↵ects betweensurface ions and those in the second layer, which un-derscore the three dimensional nature of the EDL. Forcharged electrodes, there is a competition between theelectrode surface charge and the second molecular layer,whose electric fields oppose, to determine the directionof induced dipoles on the surface ions.We have shown, then, how details of MD simulation
can a↵ect the resulting h i-dependent profile of Cdl. Ourresults suggest that simpler models, such as electrolytesinteracting through charge alone and surfaces having aconstant �, lead to highly idealized camel-type di↵er-ential capacitance profiles that more clearly reflect thepredictions of mean field models of ionic liquid behaviorat ideal metallic electrodes. As additional atomic detailis incorporated into the simulation, such as electrolytepolarization and charge redistribution on the electrodesurface, the h i-dependent profile of Cdl becomes morecomplex and, in our case, loses some of its camel-typecharacter. This reflects the sensitive balance of forcesthat influence the density and configuration of ions inthe Stern layer and underscores potential challenges incomparing the experimentally obtained values of capaci-tance with those obtained from theory.
ACKNOWLEDGMENTS
This work was supported by funding from the NASAAeronautics Research Institute Seedling program.
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SUPPLEMENTARY MATERIAL
Contents
-20 -10 0 10 20z (Å)
3D-Ewald3D/slab-Ewald
-20 -10 0 10 20z (Å)
0
0.2
0.4
δρ (n
m-3
) 3D-Ewald3D/slab-Ewald
2D-Ewald3D-Ewald3D/slab-Ewald
0
10
20
ρ (n
m-3
)
2D-Ewald3D-Ewald3D/slab-Ewald
(d)
(b)
constant-∆Ψ constant-σ
(a)
(c)
constant-∆Ψconstant-σ
+∆Ψ/2-∆Ψ/2 +|σ|-|σ|
Figure S1. Influence of long-range electrostatic summation technique(2D, 3D, and 3D/slab Ewald summation) on structure. Both the (a,b)total ion density and (c,d) deviation of density obtained from simulationsusing the 3D and 3D/slab summation techniques from that given by the2D Ewald summation technique are shown. The simulations are performedusing (a,c) constant-∆Ψ electrodes having a potential drop of 6 V and (b,d)constant-σ electrodes having a surface charge density of 13.18 µC/cm2.
1
SUPPLEMENTARY MATERIAL 2
-1.5 -1 -0.5 0 0.5 1 1.5⟨Ψ⟩ (V)
4
4.5
5
5.5
Cdl (µ
F/cm
2 )
FluctuationDerivative
4
4.5
5
5.5
Cdl (µ
F/cm
2 )
FluctuationDerivative
0 1 2 3 4 5 6 7z (Å)
05
101520
Ψ (V
)
(c)
(b)
constant-∆Ψ
constant-σ
(a)
(b)
Figure S2. Comparison of the differential capacitance obtained from Lz
systems having (a) constant-∆Ψ and (b) constant-σ electrodes. Capacitancefrom both the fluctuation formulas (solid lines) as well as the numericalderivative of the charge density (dashed lines) are provided.
SUPPLEMENTARY MATERIAL 3
-2 -1 0 1 2�^� (V)
3.5
4
4.5
5
5.5
Cdl (µ
F/cm
2 )
constant-6^constant-m
Figure S3. Differential capacitance of an Lz = 10 nm capacitor as ob-tained from the use of constant-∆Ψ (dashed lines) and constant-σ (solidlines) electrodes. Error bars are obtained from the standard deviation ofCdl as a function of simulation time are included.
SUPPLEMENTARY MATERIAL 4
0
4
8
12
16
l(6
z) (n
m-3
) Total[EMIM]+
[BF4]-
0
4
8
12
16
l(6
z) (n
m-3
) Total[EMIM]+
[BF4]-
0 5 10 15 20 256z (Å)
0
5
10
15
20
l(6
z) (n
m-3
)
0 5 10 15 20 256z (Å)
Total[EMIM]-
[BF4]-
-50-25 0 25 50z (Å)
(a)
(c)
(e)
(b)
(d)
(f)
constant-6^ constant-m
Figure S4. Ion density profiles as a function of distance, ∆z, from (a,b)neutral, (c,d) negative, and (e,f) positive electrodes. Results are obtainedusing both (a,c,e) constant-∆Ψ and (b,d,f) constant-σ conditions.
SUPPLEMENTARY MATERIAL 5
Figure S5. Cation and anion alignment with the electrode surface rep-resenting high and low values of Π.