Molecular dynamics for the charging behavior of nanostructured … · 2014-09-05 · Molecular...

Nano Res

1

Molecular dynamics for the charging behavior of

nanostructured electric double layer capacitors

containing room temperature ionic liquids

Xian Kong1,2, Diannan Lu2(), Zheng Liu2(), and Jianzhong Wu1()

Nano Res., Just Accepted Manuscript • DOI: 10.1007/s12274-014-0574-0

http://www.thenanoresearch.com on September 1, 2014

© Tsinghua University Press 2014

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DOI 10.1007/s12274-014-0574-0

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Xian Kong1,2, Diannan Lu2*, Zheng Liu2*, and

Jianzhong Wu1*

1 University of California, Riverside, USA

2 Tsinghua University, Beijing, China

Strong electrostatic correlations in a room temperature ionic

liquid are responsible for the oscillatory variation of the surface

charge density during the constant-potential charging of electric

double layer capacitors. The legends denote different

separations between two parallel electrodes.

Provide the authors’ website if possible.

Diannan Lu, http://www.chemeng.tsinghua.edu.cn/scholars/ludn/index.htm

Zheng Liu, http://www.chemeng.tsinghua.edu.cn/scholars/liuzheng/liuzheng_e.htm

Jianzhong Wu, http://www.cee.ucr.edu/jwu/index.html

0 5 10 150.00

0.05

0.10

0.15

0.20

Q (

e/n

m2)

t (ps)

2.1 nm

2.3 nm

2.8 nm

3.1 nm

1.0 nm

1.4 nm

1.7 nm





Received: day month year

Revised: day month year

Accepted: day month year

(automatically inserted by the

publisher)

© Tsinghua University Press and

Springer-Verlag Berlin

Heidelberg 2014

KEYWORDS

electric double layer,

room temperature ionic

liquids,

nanostructured capacitor,

charging dynamics

ABSTRACT

The charging kinetics of electric double layers (EDLs) is closely related to the

performance of a wide variety of nanostructured devices including

supercapacitors, electro-actuators, and electrolyte-gated transistors. While room

temperature ionic liquids (RTIL) are often used as the charge carrier in these

new applications, the theoretical analyses are mostly hinged on conventional

electrokinetic theories suitable for macroscopic electrochemical phenomena in

aqueous solutions. In this work, we study the charging behavior of RTIL-EDLs

using a coarse-grained molecular model and constant-potential molecular

dynamics (MD) simulations. In stark contrast to the predictions of conventional

theories, the MD results show oscillatory variations of ionic distributions and

electrochemical properties in response to the separation between electrodes.

The rate of EDL charging exhibits non-monotonic behavior revealing strong

electrostatic correlations in RTIL under confinement.

1 Introduction

Electric double layer capacitors (EDLC), also

known as supercapacitors, have received a great

deal of attention in recent years for their

outstanding performance as an efficient energy

storage device with large power density, high

capacitance, and long lasting cycle life[1-4]. Recent

developments in EDLC research have greatly

benefitted from rapid advances in the fabrication of

nanostructured electrodes with ultra-high specific

surface area and microporous dimension

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2 Nano Res.

comparable to the ionic size[5-8]. The EDLC

performance, especially the energy density and

thermal stability, can be further enhanced by the

adoption of room-temperature ionic liquids (RTIL)

or organic electrolytes as the charge carrier [9-12].

The non-aqueous electrolytes allow for EDLC

operation at a wider voltage window, thereby

providing higher energy density.

Recent experimental investigations of EDLC have

inspired considerable theoretical interests in

examining the interfacial structure and the

electrokinetic behavior of EDLs consisting of ionic

liquids [13, 14]. The theoretical and simulation

studies help to elucidate novel electrochemical

phenomena such as the microscopic mechanisms

responsible for the drastic increase of capacitance in

nanoporous electrodes and predict an oscillatory

behavior in response to the changes in the pore

size[15-18]. In stark contrast to numerous reports

on the equilibrium properties of RTIL-EDLs,

relatively little investigations have been devoted to

understanding the dynamics and transport

behavior of RTIL in confined geometry, especially

on their connections with the unique microscopic

details of EDLs. While the electrokinetics in RTIL

systems has been speculated to be distinctively

different from that corresponding to aqueous

electrolyte solutions, theoretical interpretation of

the voltammetry data from experiments are mostly

hinged on conventional equivalent-circuit (EC)

models or microscopic electrokinetic theories

established for systems containing ions in dilute

aqueous solutions[19-22]. Because the conventional

methods entail drastic approximations suitable only

for macroscopic electrochemical phenomena, their

applicability to RTIL-EDLs is questionable and has

to be validated from a microscopic perspective[23,

24]. Towards that end, Péan et al studied the

charging dynamics of supercapacitors based on

constant-voltage molecular dynamics (MD)

simulations for a realistic model of an ionic liquid

and nanoporous carbon[25]. It was found that ion

transport in nanoporous materials is not much

affected by the confinement and that the charging

kinetics can be nicely fitted with an EC model.

However, simulation on simpler models of ionic

liquids and porous electrodes indicates that ion

diffusion in ionophilic nanopores could be an order

of magnitude faster than that in the bulk[26]. In

addition, it was found that the EDL charging in

RTIL typically follows a diffusive process with

“overfilling” at short time and “defiling” at the late

stage. Such charging behavior cannot be faithfully

described with conventional EC models[26].

In this work, we investigate the charging kinetics

of EDLC using MD simulation for a realistic model

of RTIL but with a deliberately simplified

configuration for the electrodes. While the model

system is not intended to mimic any experimental

setup for EDLCs, it allows for fast equilibrium and

better control of the temperature without

comprising the essential features of EDL charging.

We show that the surface layering of counterions

and coions near the electrodes have profound

influences on the kinetics of EDL charging in ionic

liquids. Unlike the predictions of conventional

electrokinetic models, RTIL-EDL charging is highly

sensitive to the pore size and may lead to a

non-monotonic variation of the surface charge

density.

2 Molecular Model and Methods

Consider a model electrochemical cell consisting

of two metal electrodes and an ionic liquid. As

shown schematically in Figure 1, the ionic liquid is

described in terms of a coarse-grained model where

each cation is represented by 3 partially charged,

spherical particles, and each anion is one spherical

particle with a charge of -0.78e. The coarse-grained

model has been used in a number of earlier

simulation studies of ionic liquids (e.g.,[25, 27, 28]).

Whereas the molecular parameters had been fitted

with various static and dynamic properties of a

specific ionic liquid, namely 1-butyl-3-methyl-

www.theNanoResearch.com∣www.Springer.com/journal/12274 | Nano Research

3 Nano Res.

imidazolium exafluorophosphate (BMI-PF6) in the

bulk, the coarse-grained model was intended to be

generic, representing typical RTILs used in

experiments[28].

For computing electrostatic interactions and

confinement effects on mobile ions, the charged

surface is modeled as a single layer of

Lennard-Jones (LJ) particles arranged in a

configuration identical to that of carbon atoms in a

perfect graphite sheet[29]. For convenience, the size

and energy parameters of the LJ particles are

selected to be identical to those corresponding to

individual carbon atoms, i.e., σ=0.337 nm and ε=0.23

kJ/mol, respectively.

Figure 1 A snapshot of the ionic distributions in the model

electrochemical cell from MD simulation. The electrode atoms

are colored gray, each cation (red) consists of 3 coarse-grained

beads of different sizes and partial charges, and each green

sphere represents an anion with a charge of -0.78e. The

electrode polarizability is accounted for by the image charges of

the mobile ions shown in semitransparent colors.

We assume that each electrode is made of a

perfect metal, i.e., the electrode charges can be

accumulated only at the interface and the dielectric

constant in the bulk electrode is infinite. To account

for the effects of electrode polarizability, we adopt

the image charge method proposed by Petersen et al

[30]. Briefly, the polarization of each electrode is

described by the primary image charges of the

mobile ions with higher-order terms approximated

by a uniformly charged surface. The charge of each

particle at the electrode surface is allowed to

fluctuate such that the difference between the

electrical potentials of the electrodes, i.e., the voltage

of the electrochemical cell, is kept as a constant.

Because the image charge method avoids iterative

minimization of the total energy at each simulation

step, its computational efficiency is superior to an

alternative protocol proposed by Reed et al., which

was based on an earlier simulation method

developed by Siepmann and Sprik[31, 32].

We implemented MD simulations for the above

model ionic system using Gromacs 4.6.5 simulation

package[33]. The x-y directions of the simulation

box were fixed at 3.118nm×3.600nm with periodic

boundary conditions; the z-direction is confined

between two parallel electrodes, each containing 96

LJ particles to represent the surface atoms. At a

given separation between the symmetric planes of

the surface atoms, the simulation cell is filled with

an equal number of cations and anions such that the

average density of the confined ionic liquid is the

same as that of the bulk electrolyte. An equal

number of image charges are placed on each side

out of the electrode for both cations and anions.

Throughout this work, all MD simulations were

conducted in the NVT ensemble. For all cases, the

temperature was kept at 400 K using the

Nose-Hoover thermostat with 1.0 ps coupling

coefficient[34, 35]. The integration step was fixed at

5 fs. The slab particle mesh Ewald (PME) method[36,

37] was used to compute electrostatic interactions. A

slab of vacuum was left on each side of the

simulation box to ensure the accuracy of

electrostatic force calculation. A cutoff distance of

1.2 nm was applied to both electrostatic interactions

in the real space and non-electrostatic interactions.

During each step of MD simulation, we propagate

only the positions of the ionic liquid particles. The

positions of primary image charges at each

electrode are updated to make sure that a primary

image charge and its real particles form mirror

symmetry with the corresponding electrode as the

symmetry plane.

For each atomic configuration of the ionic liquid,

we can calculate the surface charge of the electrode


4 Nano Res.

analytically[30],

0

1

ni i

E E

i

q dn q q

D

(1)

where nE is the number of carbon atoms from the

electrode, qE is the charge of each electrode atom, n

is the total number of ionic liquid atoms between

the two parallel electrodes, qi is the charge for atom

i, di is the distance of atom i to the left electrode, D is

the separation between the two parallel electrodes,

and 0q is the total charge for the same electrode

without the ionic liquid. It can be shown that, in the

absence of ionic liquid,

q

0= DV

0Ae

0/ D (2)

where DV

0 is the difference in electrical potential

between the two electrodes, e

0is the vacuum

electrical permittivity, and A is the cross-section

area of the electrodes. For each ionic configuration,

the charge density for each electrode is calculated

from /E EQ n q A .

For simulating the equilibrium properties of

EDLs, we carried MD production runs with a target

electric potential drop after the system was subject

to energy minimization and 1 ns equilibration with

no electrical potential bias to the electrodes. Table 1

lists all simulation setups used in this work. The

charging kinetics of the model electrochemical cell

was studied by conducting 500 parallel

constant-potential simulations. For each MD

trajectory, the starting configuration was generated

after energy minimization and 50 ns equilibration

with no electrical potential bias to the electrodes. At

t = 0 , the applied voltage was subject to a sudden

increase from 0 to 1 V. All dynamic properties

reported in this work were obtained by averaging

over 500 MD trajectories.

Table 1 Simulation parameters for systems with different

distances between parallel electrodes (D)

D (nm)

# of

atoms in

electrode

# of

IL pairs

# of

image

IL pairs

# of

total

atoms

1.0 96 13 26 348

1.1 96 16 32 384

1.2 96 19 38 420

1.3 96 21 42 444

1.4 96 24 48 480

1.5 96 27 54 516

1.6 96 30 60 552

1.7 96 32 64 576

1.8 96 35 70 612

1.9 96 38 76 648

2.0 96 41 82 684

2.1 96 43 86 708

2.2 96 46 92 744

2.3 96 49 98 780

2.4 96 51 102 804

2.5 96 54 108 840

2.6 96 57 114 876

2.7 96 60 120 912

2.8 96 62 124 936

2.9 96 65 130 972

3.0 96 68 136 1008

3.1 96 71 142 1044

3.2 96 73 146 1068

3.3 96 76 152 1104

3.4 96 79 158 1140

3.5 96 82 164 1176

3.6 96 84 168 1200

3.7 96 87 174 1236

3.8 96 90 180 1272

3.9 96 93 186 1308

4.0 96 95 190 1332

4.1 96 98 196 1368

4.2 96 101 202 1404

4.3 96 103 206 1428

4.4 96 106 212 1464

4.5 96 109 218 1500

4.6 96 112 224 1536

4.7 96 114 228 1560

4.8 96 117 234 1596

4.9 96 120 240 1632

5.0 96 123 246 1668

3 Results and Discussions

3.1 RTIL-EDL at equilibrium

To ensure that the customized simulation

protocol generates reliable results, we have first

performed MD simulations for the model system at

various constant electrode voltages and also at


5 Nano Res.

constant surface charge densities. For the latter case,

the simulation can be easily implemented with the

original Gromacs 4.6 simulation package. For

equilibrium systems, the constant potential and

constant charge simulations should lead to the same

results. Figure S1 presents the Q-V curves generated

from the two different methods. The excellent

agreement between these two curves indicates that

the image charge method can be successfully used

to control the electrode electrical potential.

Figure 2 presents the charge density of the

cathode as a function of the separation between the

parallel electrodes for the model electrochemical

cell at equilibrium. Here the applied voltage on the

electrodes was fixed at DV

0= 1.0V . Because of

electrostatic neutrality, the cathode and anode bear

the same total amount of charge but with opposite

signs.

Figure 2 The surface charge density for the model EDLC at

equilibrium versus the separation between the electrodes. Here

the voltage between two the parallel electrodes is fixed at

ΔV0=1.0 V. The symbols are simulation results; the solid line is

for the guide to the eye.

While the equilibrium charge of a macroscopic

capacitor is inversely proportional to the dielectric

thickness, our simulation results indicate that the

capacitor charge of RTIL-EDL exhibits an oscillatory

decay as the separation between the electrodes

increases. The distance between two neighboring

peaks (or valleys) is about 0.7nm, approximately

equal to two times the average ion diameter. Similar

oscillatory behavior has been reported before using

classical density functional theory (DFT)

calculations and all-atom MD simulations for an

ionic liquid confined between two surfaces of the

same charge[15-17]. The amplitude of the oscillation

falls as D increases and remains significant beyond

the maximum distance (5nm) studied in this work.

(a) (b)

Figure 3 (a) The local densities of cations and anions based on

the center of mass; (b) the local charge density and the local

mean electrical potential inside the cell. In both cases, the

separation between electrodes is D=3 nm, the cell voltage is

fixed at ΔV0=1.0 V.

The oscillatory charge variation is closely

affiliated with the layering structures of ionic

liquids within the electrical cell. Qualitatively, the

variation in the surface charge density corroborates

recent investigations on surface forces in ionic

liquid systems[38]. To illustrate, we show in Figure

3 the local densities of cations and anions, and the

local charge density and electrical potential within

the cell when the separation between the parallel

electrodes is fixed at 3D nm. Because of the

differences in ion size and shape, the ionic and

charge distributions are not symmetric as revealed

in recent DFT calculations[15]. Nevertheless, the

layer-by-layer distributions of cations and anions

between the electrodes remain evident. The

thickness of each ionic layer is about 0.7 nm, close

to the average diameter for a pair of cations and

anions. It is worthwhile mentioning that the cation

orientations are distinctively different near positive

and negative electrodes. As shown in Figure 3(a),

the density profile for cations ([BMI+]) exhibits

1 2 3 4 50.14

0.16

0.18

Q (

e/n

m2)

D (nm)

1.0 V

0 1 2 3

0

400

800

1200

Den

sity

(k

g/m

3)

z (nm)

BMI

PF6

0 1 2 3-3.0

-1.5

0.0

1.5

3.0

z (nm)

Ch

arg

e (e

)

-1

0

Po

tential (V

)


6 Nano Res.

double peaks near the cathode, while there is only a

single peak near the anode. Approximately, double

and single peaks correspond to the perpendicular

and parallel alignments of the trimeric cations with

the electrode, respectively.

Figure 3b) indicates that the layer-by-layer

distributions of cations and anions are accompanied

by near sinusoidal variations of the local electrical

charge and potential. The phase shift between the

two sinusoidal curves is consistent with the

prediction of the Poisson equation

Ñ2Y(z) = -q(z) / e

0 (3)

where Y(z) and

q(z) = qin

i(z)

i

å stand for the

local electrical potential and charge density,

respectively, and e

0is permittivity in the vacuum.

Figure 4 Distribution of the surface charge density versus the

separation between two neutral electrodes. The color bar shows

the probability of the surface charge at each electrode.

At a given surface potential, the total charge of

the electrode fluctuates with the ionic

configurations due to the electrode polarizability

effect. To calculate the charge fluctuations, we have

conducted a series of MD simulations with the

voltage bias changing from -0.5 to 1.0 V at an

interval of 0.1 V. The weighted histogram method

was used to obtain the surface charge fluctuations.

Figure 4 shows a “free-energy landscape” of the

system according to the distribution of electrode

charges at zero voltage bias. We see that the charge

fluctuation depends on the distance between

electrodes, especially at low D values.

We have calculated the differential capacitor Cd

from the surface charge fluctuations according to an

analytical method proposed by Limmer et al[39].

Briefly, this method relates the differential capacitor

to the fluctuation of the charges on the electrodes,

C

d= b dQ2 (4)

where dQ = Q - Q .

Figure 5 Differential capacitance versus separation at zero

electrode voltage calculated from the fluctuations of the surface

charge density Q. Black circles are simulation points. Red line

is a spline fitting of the simulation data to guide the eye.

Figure 5 presents the differential capacitance at

zero voltage bias versus the separation between the

electrodes. As demonstrated in our previous work

for porous electrodes[15-18], the differential

capacitance oscillates with the separation between

electrodes. The period of oscillation is about 0.7nm,

the same as that for the variation of the electrode

charges. The amplitude of the capacitance

oscillation falls as the separation between the

electrodes increases, although the trend is not as

obvious as that for the variation of the electrode

charges. The oscillatory behavior of the capacitance

again indicates that the separation between

electrodes may have tremendous effects on the

performance of the nanostructured EDLCs.

1 2 3 4 52.0

2.5

3.0

3.5

4.0

Cd(

F/c

m2)

D (nm)


7 Nano Res.

3.2 Charging kinetics

We have studied the charging kinetics of the

model electrochemical cell with separation between

electrodes D=1.0nm, 1.4nm, 1.7nm, 2.1nm, 2.3nm,

2.8nm, 3.1nm. These conditions are selected because

they correspond to either the peak or valley of the

charge and capacitance oscillations shown in

Figures 2 and 5, respectively.

Previous research indicated that during the

charging process, the temperature of ILs in the

nanopores would increase due to the Joule heat

effect [26, 40]. We studied the temperature change

in two typical cases with D=2.0 nm and 5.0 nm and

the results are shown in Figure S2. The Joule heat

effect is insignificant in our simulations because the

increase of temperature is proportional to the

squared voltage drop[40] (ΔV2), and we used a

relatively low voltage value, ΔV=1.0 V, at which the

temperature increase rate is only about 0.1K∙ps-1. By

contrast, significant temperature increase was

reported in earlier studies [26, 40].

(a) (b)

Figure 6 (a) Evolution of the cathode charge density after the

electrodes are exerted with a 1.0 V voltage bias. Simulation

results are give for the separation between electrodes fixed at

1.0 nm, 1.4 nm, 1.7 nm, 2.1 nm, 2.3 nm, 2.8 nm and 3.0 nm; (b)

The RC values using equivalent circuit (EC) model. Symbols

are fitted RC values from (a), and the red line is for the guide of

the eye.

As shown in Figure 6 (a), the charging kinetics is

very fast for the model electrochemical cell; the

equilibrium state could be reached within about 500

ps except for the case of D=1.4 nm. Note that in our

simulations, we used a high temperature (about 400

K) to accelerate the dynamics of ionic motions and

thus reducing the simulation cost. For real systems,

the charging may be many times slower due to

much larger system size.

Since the same voltage drop of 1.0 V was used for

all the charging processes studied in this work, the

driving force of the electrical charging, namely the

total electric field across the cell, E=ΔV/D, decreases

with the increasing the cell width. Thus we would

expect a slow charging kinetics with the increase of

separation between the electrodes. However, Figure

6(a) shows that the charging kinetics at D= 1.4 nm

and 2.1 nm is much slower than other cell

configurations; they are even slower than those

with much larger electrode separations (e.g., D=3.1

nm). The nonmonotonic charging behavior is

closely related to the formation of layer-by-layer

structures[41]. For most cases, the charging process

completes within about 200 ps, while for D=2.1 nm,

the charging process lasts more than 400 ps, and the

duration of charging is even longer than 1000 ps for

D=1.4 nm.

For all the cases studied, the variation of the

surface charge density versus time shows a rough

exponential decay as predicted by the equivalent

circuit (EC) model. In the EC model, the total

charge density of a parallel capacitor follows an

exponential decay towards the asymptotic value at

equilibrium

Q(t) = Q

e(1- e-t/RC ) (5)

where Q

e stands for the equilibrium charge

density at the electrode surface, C and R

represent the differential capacitance and the

effective resistance for each EDL, respectively. The

quantity RC provides a characteristic time

reflecting the kinetics of EDL charging/discharging.

The fitted values of RC are plotted in Figure 6(b).

The RC values also show an oscillation behavior

with the increase of D, indicating that the

separation between electrodes not only influences

0 200 400 600 800 1000

0.08

0.12

0.16

0.20

2.1 nm

2.3 nm

2.8 nm

3.1 nm

1.0 nm

1.4 nm

1.7 nm

Q (

e/n

m2)

t (ps)

1 2 3

0

10

20

30

40

RC

(p

s)

D (nm)


8 Nano Res.

the equilibrium properties of the capacitors, but

also the charging kinetics of the nanostructured

capacitors.

A careful inspection of the first picoseconds of the

charging process reveals interesting “overfilling”

and “defiling” behavior as identified by earlier

simulations[26]. Figure 7 indicates that, right after

the exertion of the voltage bias to two originally

neutral electrodes, the charge density first increases

rapidly. However, the increase is not always

exponentially as predicted by EC model. At about 1

~2 ps, there will be an oscillatory variation of the

surface charge density, in particular for cases with

small electrode separations (D=1.0 nm, 1.4 nm), the

electrode charge could even decrease, before

increasing again till reaching the equilibrium values.

Recently, the nonmonotonic variation of the surface

charge density during EDL charging has been

examined in detail with the time-dependent density

functional theory (TDDFT) [41].

Figure 7 The initial stage of the charging dynamics for the

model electrochemical system after the application of 1.0 V

voltage bias at t=0. The different curves correspond to electrode

separation D=1.0 nm, 1.4 nm, 1.7 nm, 2.1 nm, 2.3 nm, 2.8 nm

and 3.0 nm, respectively.

Because the electrode charge is induced by ion

displacements, we expect that the nonmonotonic

charging kinetics should be closely correlated with

the ion motions. Figure 8 shows the evolution of the

ion numbers in the left half-cell of the EDLC during

the charging process. The total number of cations

increases sharply after applying the voltage bias for

all cases. Whereas anions move out of the left

half-cell at the initial stage of charging, they may

flow back into the left half-cell at late stages. As

predicted by TDDFT, the non-monatomic charging

behavior is closely related to layer-by-layer

formation and strong charge correlations between

cations and anions in the IL. The oscillatory flow of

anions is not captured by the conventional EC

models.

Figure 8 Evolutions of the numbers of cations and anions in the

left half-cell of the model electrochemical system at different

electrode separations during the charging process. In each panel,

the black line is for the number of BMI cations and the red line

is for the number of PF6 anions.

4 Conclusions

In this work, we have studied the electrochemical

properties and the charging behavior of a model

electric double layer (EDL) capacitor composed of a

room temperature ionic liquid (RTIL) sandwiched

between two planar electrodes. Using a

coarse-grained model for the RTIL and

constant-potential molecular dynamics simulations,

we investigated the influences of the separation

between the electrodes on the ionic and charge

distributions, the differential capacitance, the total

stored charge, and their variations with time during

the constant-potential charging processes. We

identified oscillatory variations of both the total electrode

0 5 10 150.00

0.05

0.10

0.15

0.20

Q (

e/n

m2)

t (ps)

2.1 nm

2.3 nm

2.8 nm

3.1 nm

1.0 nm

1.4 nm

1.7 nm

0 10 20 30

-2

0

2

4

D = 3.1 nmD = 2.8 nm

D = 1.4 nm

BMI

PF6

# o

f io

ns

D = 1.0 nm

0 20 40 60 80

0.0

0.5

1.0

1.5

2.0

# o

f io

ns

0 20 40 60 80 100-0.4

0.0

0.4

0.8

1.2

# o

f io

ns

t (ps)

0 20 40 60 80 100-0.4

0.0

0.4

0.8

# o

f io

ns

t (ps)


9 Nano Res.

charge and differential capacitance as functions of the

separation between the electrodes similar to previous

results obtained from the classical density functional

theory and all atom simulations for porous electrodes.

The EDL charging kinetics is highly sensitive to the

separation between electrodes; it is most sluggish when

the separation coincides with the valley points of the

oscillatory profile for the electrode charge or the

differential capacitance. While the variation of electrode

charge can be approximately correlated with the

equivalent circuit model, the charging kinetics exhibits

strong nonmonotonic behavior at the beginning stage of

EDL charging, in particular at small electrode separations.

The nonmonotonic charging behavior arises from the

strong correlations between cations and anions of

confined ionic liquids that are ignored in conventional

theories.

Acknowledgements

This work was supported as part of the Fluid

Interface Reactions, Structures and Transport (FIRST)

Center, an Energy Frontier Research Center funded

by the U.S. Department of Energy, Office of Science,

Office of Basic Energy Sciences. K.X. is grateful to the

Chinese Scholarship Council for the visiting

fellowship. Additional support is provided by

National Natural Science foundation of China, No.

21276138 and Tsinghua University Foundation, No.

2013108930. The numerical calculations were

performed at the National Energy Research Scientific

Computing Center (NERSC).

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Nano Res.

Electronic Supplementary Material





erside, CA 92521,USA

2 Department of Chemical Engineering, Tsinghua University, Beijing 100084, China

Supporting information to DOI 10.1007/s12274-****-****-* (automatically inserted by the publisher)

Figure S1 (a) Potential profile of between two electrodes separated by 12.3 nm under different voltage biases.

This demonstrates the viability of our simulation technique. (b) Q-V relation from constant voltage and

constant charge simulations. The collapse of two curves demonstrates the reliability of our simulation method.

0 2 4 6 8 10 12

-8

-6

-4

-2

0

-8

-6

-4

-2

0

Po

ten

tial

(V

)

z (nm)

0V

2V

3V

4V

6V

9V

0 2 4 6 8 10

0.0

0.4

0.8

1.2 constant V

constant Q

Q

e/n

m2

V (V)


Nano Res.

Figure S2 Temperature change during the course of MD simulation for the model electrochemical system at

two electrode separations.

0 200 400 600300

360

420

480

540

T (

K)

Time (ps)

2.0 nm

5.0 nm

Address correspondence to [email protected]; [email protected]; [email protected]



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