C A R B O N 6 6 ( 2 0 1 4 ) 3 9 – 4 7

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Negatively curved carbon as the anode for lithiumion batteries

0008-6223/$ - see front matter � 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.carbon.2013.08.033

* Corresponding author: Fax: +82 52 217 3208.E-mail address: noejung@unist.ac.kr (N. Park).

Dorj Odkhuu a,b, Dong Hyun Jung a, Hosik Lee c, Sang Soo Han d, Seung-Hoon Choi a,Rodney S. Ruoff e, Noejung Park b,*

a Insilicotech Co., Ltd., C602, Korea Bio Park, 694-1, Sampyeong-dong, Seongnam-si, Gyeonggi-do 463-400, Republic of Koreab Interdisciplinary School of Green Energy and Low Dimensional Carbon Materials Center, Ulsan National Institute of Science and Technology

(UNIST), Ulsan 689-798, Republic of Koreac School of Mechanical and Advanced Materials Engineering, Ulsan National Institute of Science and Technology (UNIST), Ulsan 689-798,

Republic of Koread Institute for Multidisciplinary Convergence of Matter, Korea Institute of Science and Technology, 14-Gil 5, Hwarang-Ro, Seoungbuk-Gu,

Seoul, Republic of Koreae Department of Mechanical Engineering and the Materials Science and Engineering Program, The University of Texas, Austin, TX 78712,

United States

A R T I C L E I N F O

Article history:

Received 11 June 2013

Accepted 18 August 2013

Available online 28 August 2013

A B S T R A C T

The inclusion of heptagonal, octagonal, or larger rings in an sp2-bonded carbon network

introduces negative Gaussian curvature that can lead to a high network porosity. Here

we investigated a particular negatively curved nonplanar sp2-carbon structure namely

688P schwarzite, with a view toward the possible use of negatively curved carbons as lith-

ium ion battery anodes. Our first principles calculations show that the presence of pores in

schwarzites can lead to three-dimensional Li ion diffusion paths with relatively small

energy barriers. We calculated the binding energy of Li (which donates 1 electron to the

schwarzite) in different positions in the schwarzite structure, and the open-circuit voltage

(OCV) with respect to Li metal and found that this schwarzite has a positive OCV for a Li

concentration as high as LiC4. The advantages of the particular schwarzite studied here

for use as an anode are expected to be present in other sp2-bonded carbon networks that

feature large polygonal rings.

� 2013 Elsevier Ltd. All rights reserved.

1. Introduction

Allotropes of trivalent (sp2-bonded) carbon structures have

fascinated the scientific community over the last several dec-

ades. Such allotropes including graphene, fullerenes, and car-

bon nanotubes (CNTs), can be classified by the Gaussian

curvature of the overall surface [1–3]. Fullerenes include a

bonding network of pentagonal and hexagonal carbon rings

that form a surface with a positive Gaussian curvature. CNTs

and graphene consist of only hexagons whose surfaces have

zero Gaussian curvature. On the other hand, negatively

curved surfaces can be constructed by including, e.g., hepta-

gons and/or octagons (or larger n-gons) in the hexagonal ring

network. Periodic three-dimensional structures of any

sp2-bonded carbon atoms that form a surface with a negative

Gaussian curvature are called schwarzites, after the mathe-

matician Schwartz who studied a related set of topologies

[4]; they are also referred to as ‘negative curvature carbon’

40 C A R B O N 6 6 ( 2 0 1 4 ) 3 9 – 4 7

or NCC. To date no practical synthetic approach to the pure

periodic schwarzites has been reported, and scalable meth-

ods to make NCCs would be of great interest [5–10]. First prin-

ciples calculations have predicted that schwarzites can be

thermodynamically more stable than certain carbon nano-

structures having positive-curvature [11–14]. The mechanical,

electronic, and magnetic properties of schwarzites are ex-

pected to be distinct from the corresponding properties of po-

sitive- or zero-curvature structures. For example, the band

gap in such a structure is predicted to vary from metallic to

insulating character depending on the particular arrange-

ment of carbon polygons [5,14–19]. In particular, the NCC

structures have large pores and exceptionally high specific

surface areas that could offer notable advantages in energy

storage applications [20].

Today’s demand for electric vehicle and energy storage

applications requires breakthroughs that lead to a drastic in-

crease in the storage capacity of the anodes and cathodes in

lithium ion batteries (LIBs), and improvements in charging

rate and lifetime are also called for. The current LIB anode

is typically based on graphite and it has a fundamental Li

ion storage limit of LiC6, which corresponds to a specific

capacity of 372 mAh/g. To realize a full-scale commercial elec-

tric vehicle, the anode and cathode capacities should be

boosted beyond the current limit. To this end, a variety of can-

didate materials, including silicon, germanium, and tin, and

various metal alloys, have been evaluated when comprised

in composites as high-capacity anode active materials. Some

partial successes have been reported, including storage

capacities several times larger than the capacity of graphite

[21,22]. The large increase in the storage capacity is usually

accompanied by large volume changes in the Si (Ge, Sn, etc.)

during the lithiation/delithiation processes, and these volume

changes often leads to poor cyclability including through frac-

ture of the Si (Ge, Sn, etc.) particles. In this regard, a recent

first principles study suggested that the two-dimensional sil-

icon within one or two layers thick has small volume changes

during the lithiation and delithiation cycle, and also preserves

the Li ion capacity and diffusion affinities found in bulk and

nanowire structures [23].

With respect to carbon-based materials, single- and multi-

walled CNTs and their hybrid materials involving fullerenes

have also been proposed as candidate high-capacity anode

materials [24–27]. In these materials, the Li ion mobility,

which is crucial to achieving favorable kinetics during charg-

ing and discharging, is restricted to one- or two-dimensional

diffusion channels. Recent theoretical studies have explored

the potential utility of novel carbon allotropes, including the

layered graphyne and graphdiyne structures, which contain

alternating sp and sp2 hybrid bonds [28,29]. Graphyne and

graphdiyne structures include larger holes, therefore, they

are expected to yield better Li ion diffusion properties than

graphite or perhaps than the aforementioned CNTor CNT/ful-

lerene hybrid materials. One of the major drawbacks of

graphite-like carbon materials is that the layered structures

are vulnerable to exfoliation as a result of cycling charging

and discharging. Layer exfoliation reduces the battery capac-

ity over time. Other carbonaceous structures, including amor-

phous, mesoporous, and disordered carbon, showed

somewhat improved stability and diffusion properties [30,31].

In the present study, we examined the effects of the sur-

face topology on the storage and mobility of Li ions in pure

sp2-bonded carbon structures having atom-thick walls. We

compared the properties of Li binding and diffusion in partic-

ular sp2-bonded carbon structures having zero-, positive-, and

negative-curvature surfaces. Our first principles calculations

further suggest that the presence of larger polygons (hepta-

gons and octagons), either in the form of periodic or aperiodic

schwarzites, could lead to high storage capacities and isotro-

pic three-dimensional diffusion paths, offering the possibility

of large performance improvements over the currently used

graphitic anode materials.

2. Computation

The first principles density functional theory calculations

using the projector augmented wave (PAW) pseudopotential

method [32] were performed by implementation in the Vienna

ab initio Simulation Package (VASP) [33,34]. Exchange correla-

tion interactions between electrons were described using the

generalized gradient approximation (GGA) formulated by Per-

dew, Burke, and Ernzerhof (PBE) [35]. The convergence of the to-

tal energy with respect to the energy cutoff for the plane-wave

basis set and the k-point sampling for Brillouin zone integra-

tion was tested in all calculations. The energy barrier for Li

ion diffusion was estimated using the nudged elastic band

method [36].

3. Results and discussion

The finite fragments of curved sp2-bonded surface structures pre-

sented in Fig. 1 lend insight into the topology of the structures. A

zero-curvature structure, graphene, is shown in Fig. 1(a) for refer-

ence. The inclusion of heptagons and octagons in the hexagonal

network produces a negative Gaussian curvature, as illustrated in

Fig. 1(b) and (c), respectively. The addition of a five-membered

ring gives rise to the positive Gaussian curvature (Fig. 1(d)). Note

that the inclusion of twelve pentagons in the hexagonal network

closes the surfaces, as is observed in the fullerenes. On the other

hand, regular periodic structures comprising hexagons and high-

er polygons can lead to a variety of three-dimensional structures

that as mentioned have been collectively called schwarzites, and

these will be discussed later.

Ionic interaction is the main mechanism that underlies

the interaction of Li with sp2 carbon structures; however we

would like to note that the most extensive experimental stud-

ies have been done on the layered material graphite, which

fundamentally differs in structure from sp2 carbon structures

having atom thick walls, such as single walled CNTs, mono-

layer graphene, and schwarzites. (Thus for monolayer graph-

ene one might discuss ‘decoration’ rather than intercalation

by Li, and the terminology for formation of ‘Li-decorated’ sch-

warzites of given stoichiometry will likely be a topic for future

discussion within the community). We find that Li cations are

stabilized at the centers of hexagons or polygons of a sp2 car-

bon material. Hereafter, PEN will be used to indicate the

center of a pentagon, HEX the center of a hexagon, HEP the

center of a heptagon, and OCT the center of an octagon. The

adsorption of Li onto the inner or outer surfaces of a ring

Fig. 1 – Side and top views of surfaces with different curvatures: (a) graphene; the surfaces containing (b) a heptagon, (c) an

octagon, and (d) a pentagon. With the exception of the periodic graphene structure, the finite fragment structures were

modeled with H atom terminations, as represented by the blue ends of the stick models. The energetics of the Li ion diffusion

pathways between neighboring polygons in each structure are shown in (e)–(h). The positions of the initial and final states of

the Li atom are labeled in (a)–(d), as described in the text. A colour version of this figure can be viewed online.

C A R B O N 6 6 ( 2 0 1 4 ) 3 9 – 4 7 41

yields an endohedral or exohedral complex, respectively. The

distinction between the inner and outer surfaces of PEN is

obvious; however, HEP and OCT form saddle structures, thus,

there is no obvious distinction between the inner and outer

surfaces. The hexagons adjacent to HEP and OCT structures

have different local curvatures that define the ‘endo-hexagon’

(HEXendo) and ‘exo-hexagon’ (HEXexo), depending on the local

curvature, as shown in Fig. 1(b) and (c). The geometric and

energetic features of a structure comprising a single Li atom

bound to the center of each type of polygon are summarized

in Table 1.

Table 1 – The binding energy of a Li atom on the centers of polygLi–C distance indicates the closest bond length in the equilibriu

Surface type Binding site

Graphene HEXHeptagon-containing HEP

HEXendo

HEXexo

Octagon-containing OCTHEXendo

HEXexo

Pentagon-containing PENendo

HEXendo

PENexo

HEXexo

The binding energy calculated for a single Li ion bound to

graphene (1.2 eV/Li) is consistent with the value reported in

previous studies [37,38]. As shown in Table 1, the curved

structures have stronger binding affinities to Li ions than

the zero-curvature graphene; the positively curved surface

studied here has the largest binding energies (typically more

than 2 eV/Li) in both the endo and exohedral configurations.

The negatively curved surfaces also lead to largely increased

binding affinity with the Li adatom. The octagon-included

surface has stronger Li binding strength by about 0.2 eV/Li

compared with the case of the pentagon-included one. For

ons in the sp2-bonded carbon surfaces defined in Fig. 1. Them geometry.

Binding energy (eV/Li) Li–C distance (A)

1.20 2.2161.66 2.1841.54 2.1821.42 2.1481.93 2.1551.75 2.1261.68 2.1612.47 2.1742.38 2.1752.49 2.1742.34 2.182

42 C A R B O N 6 6 ( 2 0 1 4 ) 3 9 – 4 7

each curved structure, the center of non-hexagon polygons

shows larger Li binding energy compared to that at the center

of the hexagons. In addition, the Li binding energies onto

HEXendo and HEXexo neighboring to non-hexagon polygons

are higher than that of graphene, indicating that the presence

of higher polygons in graphene enhances greatly the overall Li

binding affinity. Previous studies demonstrated that the pres-

ence of carbon vacancies in graphene could increase the bind-

ing strength of Li on the carbon surface [30]. The outer surface

of C60 has a stronger calculated binding affinity toward Li (as

Li+) than graphene [27,39,40]. These results are consistent

with the finite fragment results described here. In all the cal-

culations, we simulate charge neutral super cells in which the

Li and carbon walls form cation/anion pairs as result of spon-

taneous electron transfer.

We calculated the barrier for Li ion diffusion across a C–C

bond that bridges two neighboring polygon centers. Fig. 1(e)

shows that the barrier for diffusion from a HEX to a neighbor-

ing HEX in graphene is about 0.5 eV, in agreement with previ-

ously reported results [38,41,42]. The diffusion barriers from

HEP, OCT, or PEN to a neighboring HEX on a curved surface

are presented in Fig. 1(f)–(h), respectively. The barrier for dif-

fusion to an endohedral configuration is always much smaller

than the barrier for diffusion to an exohedral configuration

because the diffusion path of the former case is shorter. Li+

binds to the HEP and OCT structures slightly off-center; thus,

the diffusion from HEP to the HEXexo can proceed through

either of two paths. The barrier to diffusion onto the adjacent

HEXexo (solid triangles in Fig. 1(f)) is lower than the barrier to

diffusion onto the HEXexo site positioned across from the hep-

tagon center (solid squares in Fig. 1(f)). The surface containing

an octagon yields the energy barrier shown in Fig. 1(g), similar

to those on the heptagon-containing surface.

Binding energies were calculated for the adsorption of two

Li adatoms onto the neighbor and next-neighbor polygon

pairs. As expected, in the cases of graphene (shown in

Fig. 2(a)), the binding energy (0.89 eV/Li) of two Li ions on

two neighboring HEX’s is much weaker than that of the single

isolated Li adsorption onto a HEX. This energy cost arises

from the Li–Li Coulombic repulsion. If Li ions occupy every

HEX position in graphite, the stoichiometry would be LiC2;

however, the small distance between such Li ions requires

too large an energy cost mainly incurred by the Coulombic

repulsion. When the second neighbor HEX’s are occupied by

Li ions (Fig. 2(a); the Li–Li distance in this case corresponds

to the case of the maximally intercalated LiC6), the binding

energy per Li ion tends to reach the optimum value (1.2 eV/

Li). It is noteworthy that, as shown in Fig. 2(b) and (c), the

binding energies per Li of the pair, on the nearest neighbor

HEP–HEX and OCT–HEX, are not much reduced compared to

the cases of single Li ion adsorption the on HEP, OCT, and HEX-

endo/exo (Table 1). Such enhanced binding can be understood in

terms of the larger Li–Li distances due to the presence of lar-

ger polygons. For the second neighbor HEP–HEX and OCT–HEX

pairs, we consider mostly the Li adsorption sites on HEXendo,

in which the pairs have slightly more and less binding ener-

gies depending on the local curvature (see Fig. 2). Importantly,

the binding energies of both negative curvature structures

(heptagon- and octagon-included) do not deviate much from

those of the nearest neighbor pairs. A similar trend as

observed in graphene can be seen for binding of 2 Li ions to

the outer surface of a pentagon-containing structure

(Fig. 2(d)). The binding energies of the second neighbor PEN–

HEX pairs are greatly enhanced with respect to the nearest

neighbor pair, and those are much smaller than the binding

energies for single Li adatom adsorption. These results sug-

gested that some schwarzites may host a larger density of

Li ions than graphite (as we will show later). In addition, the

Li–Li distances in Fig. 2(b) and (c) are shorter than or compa-

rable to the Li–Li distance in LiC6. This indicates that the re-

peated structures of hexagons and higher polygons permit

both a higher gravimetric density and a higher volumetric

density of the adsorbed Li atoms relative to that of graphite.

We now consider several types of schwarzites. The three-

dimensional all-carbon sp2-bonded networks with particular

unit cells that are repeated, have been grouped as P, G, or D

surfaces, depending on the overall surface topology [16]. Pre-

vious theoretical studies have reported that some of the sch-

warzites are energetically more stable than some of the

positively curved structures, i.e., some of the fullerenes

[5,11,16]. The likelihood of forming randomly pored Schwarz-

ite-like structures has been discussed in previous experi-

ments [7–9], but the synthesis of periodic Schwarzite

remains as a great challenge. In the present work, we focus

on 688P schwarzite, illustrated in Fig. 3(a), which is composed

of only octagons and hexagons. This structure has cubic sym-

metry and 24 carbon atoms per primitive unit cell. All hexa-

gons are surrounded by octagons, and each octagon has

four hexagon and octagon neighbors. In the 688P schwarzite,

the octagon rings have C–C bond lengths of 1.38 A (octagon

edge) and 1.46 A (hexagon edge), which are smaller than

those (1.42 and 1.48 A) of the single octagon surrounded by

hexagons in the finite fragment shown in Fig. 1(c). In addition,

the C–C–C bond angles (114.5� for hexagonal and 122.5� for

octagonal rings) of the 688P schwarzite deviate more from

the ideal sp2 network compared with those of the finite frag-

ment. Note that the C–C–C bond angles are 123.7� and 129.5�for the octagon, and 116.5� and 117.8� for the hexagon in

the finite fragment.

As presented in Fig. 3(a), five different Li adsorption sites

were identified and labeled as A–E in the order of Li binding

energy. The Li binding energies and configurations are sum-

marized in Table 2. The binding energies associated with Li

binding onto either side of the OCT, denoted by A and B, are

2.26 eV/Li for the A site and 2.18 eV/Li for the B site, respec-

tively. These energies are roughly comparable but somewhat

larger than the calculated binding energy (1.93 eV/Li) of the

Li ion on OCT of the octagon-included finite fragment (see Ta-

ble 1). The C and D adsorption sites, which are the corre-

sponding sites of HEX in the finite fragment (Fig. 1(c)), have

still stronger binding (about 2.05 eV/Li) affinity with Li adatom

compared to the HEXendo of the octagon-containing fragment

(1.75 eV/Li). These enhanced binding energetics can mostly be

attributed to the increased curvature of 688P schwarzite com-

pared with the fragment structure shown in Fig. 1(c). Note

that, in Fig. 1(c), the isolated central octagon is surrounded

by hexagonal networks, while the 688P schwarzite has

alternating hexagons and octagons. Therefore the 688P sch-

warzite has increased local curvature compared to the finite

fragment, leading to a larger deviation of the C–C bond

Fig. 2 – Top view of the two-lithium nearest neighbor exo pair configurations (represented by purple balls) for (a) graphene, (b)

heptagon-included, (c) octagon-included, and (d) pentagon-included curvature surfaces. The Li–Li distance denoted by d and

the binding energies Eb of the Li ions are shown at the bottom of each panel. The numbers indicate the binding energies (in

the unit of eV/Li) of the two Li ions positioned on the next-neighbored polygon pairs. For (b), (c), and (d), one Li atom is fixed

onto the polygon (heptagon, octagon, and pentagon, respectively) and the second Li is placed on one of the hexagon centers

for which the binding energy of the pair is shown. Atomic symbols are the same as used in Fig. 1. A colour version of this

figure can be viewed online.

C A R B O N 6 6 ( 2 0 1 4 ) 3 9 – 4 7 43

lengths and bond angles from the perfect graphitic network

as mentioned above. The schwarzite cavity center, denoted

by E, has the weakest Li binding (1.75 eV). Bader charge anal-

ysis showed that the Li atoms are fully ionized to Li+, irrespec-

tive of the Li binding site, see Table 2. The carbon atoms

Fig. 3 – (a) The geometry of the negatively curved schwarzite 68

Energy barriers for the different Li ion diffusion paths from the

energies in (b) are reported with respect to the binding energy of

diffusion from HEX to HEX in graphite is shown as open circles

neighboring the Li primarily share one extra electron from

the Li atom.

The calculated barriers for diffusion of Li+ from the most

stable A site to the neighboring stable binding sites are shown

in Fig. 3(b). Compared to the barrier for diffusion of Li+ in

8P structure. A–E indicate the five distinct binding sites. (b)

initial state at the A site to the B–E sites. The potential

the A site for this schwarzite. For comparison, the barrier to

. A colour version of this figure can be viewed online.

Table 2 – The calculated binding energy of Li on five different binding sites of 688P schwarzite. The Li–C distance is from theequilibrium geometry. The values for graphite are presented for reference.

Binding site Binding energy (eV/Li) Li–C distance (A) Lithium charge (e/Li)

A 2.26 2.29 +1B 2.18 2.24 +1C 2.07 2.38 +1D 2.02 2.30 +1E 1.75 3.49 +0.98Graphite 1.77 2.36 +1

44 C A R B O N 6 6 ( 2 0 1 4 ) 3 9 – 4 7

graphite (0.45 eV), all barriers are quite low, analogous to the

finite curvature fragments (Fig. 1). Such low barriers, in con-

junction with a low energy difference between the locally sta-

ble binding sites (less than 0.4 eV/Li), should lead to faster

transport of Li+ through the pores. The diffusion pathways

in the 688P schwarzite are considerably more ‘accessible’ than

are the interlamellar spaces in graphite. Thus schwarzites

might offer (much) better charging and discharging rates than

graphite-based anodes. Graphitic or CNT-based anodes are

typically treated chemically and/or mechanically to provide

more diffusion paths than only the interlamellar spaces pres-

ent in perfect graphite [30,38,43]. The sp2 bonding network of

schwarzites may also provide much better structural stability

than graphitic anodes during lithiation and delithiation, as

graphite can fragment (exfoliate) during the swelling (lithia-

tion) and contraction (delithiation) occurring during charging

and discharging of the anode.

Fig. 4(a) and (b) present the electronic band characteristics

and density of states (DOS) for graphite and the 688P schwarz-

ite. Unlike the semi-metallic graphite that has a Dirac cone

structure at the k-point, the 688P schwarzite is metallic with

a finite DOS at the Fermi level. The p bands of the schwarzite

are more concentrated just below the Fermi level (compare

the starred regions in Fig. 4(a) and (b)), which is due to the re-

duced overlap between two neighboring p orbitals on the

curved surface (which in turn reduces the dispersion). The

atom-projected DOS and band structures of the Li interca-

lated 688P schwarzite for the stoichiometries Li0.25C6 and

LiC6 are shown in Fig. 4(c) and (d), respectively. The intercala-

tion of Li atoms into schwarzite results in transfer of one elec-

tron per Li atom to yield Li+, thereby filling the conduction

band states of the schwarzite. A comparison to the pristine

schwarzite (Fig. 4(b)) shows that the carbon band structures

are rigid upon intercalation of the Li atoms, but the Fermi le-

vel shifted up to accommodate the electrons transferred from

the Li atoms. The LiC6 stoichiometry illustrated in Fig. 4(d)

shows that although the overall carbon band structure re-

mained nearly rigid, the Li conduction bands broadened, con-

tributing substantially to the Fermi level DOS. Overall, this

implies that the Li–C bonds are mainly ionic in character,

and the p electronic structures of the sp2-bonded carbon net-

works are mostly unperturbed.

We further calculated the maximum storage capacity of the

688P schwarzite. Fig. 5(a) presents the average intercalation

energy per Li atom as a function of the Li concentration for this

schwarzite and graphite. Several configurations were found to

be locally stable for a given Li content, and the most stable con-

figuration was selected for the plot in Fig. 5(a). The Li-graphite

intercalation energy decreases monotonically with increasing

Li concentration but exceeds the cohesive energy of bulk Li

(1.6 eV) up to stoichiometry LiC6. As the Li concentration

increases further, the intercalation energy per Li atom

becomes smaller than the cohesive energy of bulk Li. The Li

intercalated 688P schwarzite has larger binding energy/Li com-

pared to graphite for the same stoichiometries up to LiC6, and

furthermore was found to be stable relative to disproportion-

ation to Li(s) even at stoichiometry LiC4.

The open-circuit voltage (OCV) is calculated for a given

concentration (x) as follows [44]:

OCV ¼ �GðLix2C6Þ � GðLix1

C6Þ � ðx2 � x1Þ � GðLiÞðx2 � x1Þ � F

; ð1Þ

where G is the Gibbs free energy of the compound and F is the

Faraday’s constant. In the present work we took G as the total

energy of the compound without considering the entropy.

G(Li) is the total energy of a Li atom in the form of the bulk

body-centered cubic lattice.

The OCV, as defined by Eq. (1), can be evaluated from the

derivative of the intercalation energy with respect to the con-

centration x. In the present work, the OCV calculated at the

interval between x1 and x2 was considered the OCV at the

midpoint x, i.e. (x1+x2)/2. The existence of various configura-

tions with marginal energy differences were considered for

the calculations of OCV with a selected Li concentration for

both 688P schwarzite and graphite, and the calculated OCV’s

are shown in Fig. 5(b). The published experimental data for

Li intercalated graphite [45] are presented in Fig. 5(b). The

experimental OCV of graphite decreases as x increases, dis-

playing a sequence of plateaus, and eventually approached

0 V vs. Li+/Li at LiC6. Although a stage-dependent OCV was

not identified thoroughly in the theoretical calculations, the

overall trend from the low concentration (about 0.5 V vs. Li+/

Li near x = 0.2) to the maximum concentration (about 0 V vs.

Li+/Li near x = 1) is roughly consistent with the experimental

data. As shown in Fig. 5(b), an OCV profile similar to that of

graphite is obtained for 688P schwarzite, but the crossover

point from the positive to the negative OCV occurred at

around Li1.5C6. Furthermore, the volumetric capacity of the

Li1.5C6 configuration for schwarzite (about 39 A3 per Li that

is 1233 mAh/cc) corresponds to 150% of the capacity of max-

imally lithiated graphite (LiC6). In our PBE calculation, the

volumetric density of graphitic LiC6 is 58 A3/Li or 825 mAh/

cc in good agreement with the experimentally observed

820 mAh/cc. This indicates that the schwarzite could offer

Fig. 4 – Electronic band structure and density of states for pure (a) graphite and (b) schwarzite 688P. The Li intercalated

schwarzites, with concentrations of (c) Li0.25C6 or (d) LiC6. The Fermi level is set to zero. The blue and orange shaded areas

represent the density of states projected onto the carbon and Li atoms, respectively. A colour version of this figure can be

viewed online.

Fig. 5 – (a) The calculated binding energy Eb per Li (b) OCV for the 688P schwarzite and graphite as a function of the Li

concentration. (c) The volume changes with respect to the concentration of Li. (d) Atomic structure of the Li intercalated 688P

schwarzite for the stoichiometry Li1.5C6. In all figures, squares and circles indicate the calculated results for the schwarzite

and graphite, respectively. The lines serve as guides to the eye. In (a), the horizontal dashed line indicates the calculated

cohesive energy of Li(s). In (b), the experimental OCV values taken from Ref. [43] are shown in triangles for comparison. In (d),

all Li atoms are depicted as pink balls positioned in the most stable A site, as shown in Fig. 3(a). A colour version of this figure

can be viewed online.

C A R B O N 6 6 ( 2 0 1 4 ) 3 9 – 4 7 45

46 C A R B O N 6 6 ( 2 0 1 4 ) 3 9 – 4 7

both a better gravimetric capacity and a higher volumetric

density capacity than graphite.

Fig. 5(c) shows the calculated volume expansion of the

schwarzite and graphite with reference to the fully delithiated

structures. Graphite reveals a sharp increase at early stage of

lithiation, which is due to the layered nature, and reaches 10%

increase at the maximum lithiation (LiC6). The volume

changes observed in the schwarzite structure are smaller

and about 5% at Li1.5C6. The optimized atomic structure for

Li1.5C6 is shown in Fig. 5(d). The smaller volume change in

the lithiated schwarzite suggests another advantage of this

interlinked three-dimensional network.

4. Conclusion

We have calculated Li binding and diffusion in a three-dimen-

sional sp2-bonded carbon network with negative Gaussian

curvature. Li ions have smaller energy barriers for diffusion

in the 688P schwarzite studied here than in Li intercalated

graphite. The graphite anode in LIBs has maximum gravimet-

ric density of 372 mAh/g and volumetric density of 820 mAh/

cc (LiC6) but the Li intercalated schwarzite is calculated as sta-

ble for stoichiometry Li1.5C6. The open porous structure and

atom thick walls of schwarzites suggest their use as the an-

ode in LIBs and could result in extremely fast charging and

discharging, high specific gravimetric and volumetric capaci-

ties, and perhaps much longer lifetimes. Finally, we expect

that such advantages of negatively curved structures are not

limited to the specific schwarzite structure studied here but

may be expected in other sp2-bonded carbon containing lar-

ger polygons.

Acknowledgements

This study was supported by Grant No. 10041589 from the

Industrial Strategic Technology Development Program funded

by the Ministry of Knowledge Economy, Republic of Korea. HL

was supported by the National Research Foundation of Korea

(NRF) (Grant No. 2012R1A1A2043431). RSR appreciates support

from his Cockrell Family Endowed Regents Chair. NP was sup-

ported by Basic Science Research Program through the Na-

tional Research Foundation of Korea (NRF) funded by the

Ministry of Education (NRF-2013R1A1A2007910). We appreci-

ate M. Stoller for commenting on the manuscript.

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