The intrinsic electrostatic dielectric behaviour of graphite anodes in Li-ion batteries across the entire functional range of charge

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Highlights

The intrinsic electrostatic dielectric behaviour of graphite anodes in Li-ion batteries – across the entire

functional range of charge

Simon Anni´es, Christoph Scheurer, Chiara Panosetti

•The relative permittivity of lithium interca-

lated graphite anodes is linearly dependent on

the state of charge, with approximate values of

around 7 at 0% and around 25 at 100%.

•Sampling the Coulomb interaction between

two charge carriers within the material is

enabled by our recently published DFTB

parametrization for lithium intercalated

graphite.

•The dielectric screening in graphite is signiﬁ-

cantly larger in the direction perpendicular to

the graphene-sheets, than in the direction par-

allel to them. This is important when compar-

ing experimental results for graphite powder

and for perfect graphite crystals.

arXiv:2210.14641v1 [cond-mat.mtrl-sci] 26 Oct 2022

The intrinsic electrostatic dielectric behaviour of graphite anodes in Li-ion

batteries – across the entire functional range of charge

Simon Anni´esa,b, Christoph Scheurera, Chiara Panosettia

aFritz-Haber-Institute of the Max-Planck-Society, Faradayweg 4-6, 14195, Berlin, Germany

bChair for Theoretical Chemistry, TU Munich, Lichtenbergstr. 4, 85747, Garching b. M¨unchen, Germany

Abstract

Lithium-graphite intercalation compounds (Li-GICs) are by far the most common anode material for modern

Li-ion batteries. However, the dielectric response of this material in the electrostatic limit (and its variation

depending on the state of charge (SOC)) has not been investigated to a satisfactory degree – neither by

means of theory nor by experiment – and especially not for the higher range of SOC. Nevertheless, said

dielectric behaviour is a highly desired property, particularly as an input parameter for charged kinetic Monte

Carlo simulations – one of the most promising modeling techniques for energy materials. In this work, we

make use of our recently published DFTB parametrization for Li-GICs based on a machine-learned repulsive

potential in order to overcome the computational hurdles of sampling the long-ranged Coulomb interactions

within this material – as experienced by the charge carriers within. This approach is rather novel due to

computational cost, but best suited for investigating our speciﬁc property of interest. For the ﬁrst time, we

discover a mostly linear dependency of the relative permittivity ron the SOC, from ≈7at SOC 0% to ≈25

at SOC 100%. In doing so, we also present a straightforward approach that can be used in future research

for other intercalation compounds – once suﬃciently fast and long-ranged computational methods – such as

linear-scaling DFT, a good DFTB parametrization, or atomic potentials with inbuilt electrostatics – become

available. However, while the presented qualitative behaviour is robust and our results compare favourably

with the very few experimental studies available, we do stress that the quantitative results are strongly

dependent on our estimation of the partial charge transfer from the intercalated Li-ion to the carbon host

structure, and need to be veriﬁed by further experiments and other calculations. Yet, our research shows

that in principle, two measurements – one at low and one at high SOC – should suﬃce for that purpose.

Keywords: relative permittivity, dielectric response, graphite anodes, lithium intercalated graphite,

multiscale modeling

PACS: 0000, 1111

2000 MSC: 0000, 1111

1. INTRODUCTION:

The relative permittivity (RP) is one of the deﬁn-

ing properties of many materials, as it describes the

degree to which the Coulomb interactions between

charge carriers are screened within the material

compared to vacuum. Especially in older literature,

this property is also known as “dielectric constant”,

even though it is far from being constant, but de-

pendent on temperature, the frequency of probing

electric ﬁelds and even the underlying mechanisms

in terms of polarizability, conductivity and others.

Preprint submitted to Electrochimica Acta October 27, 2022

Figure 1: Illustration of lithium-ions (purple spheres) in-

tercalated into a graphite host structure (grey lines) at

50% state of charge, based on the Daumas-Herold domain

model [3]. Li-ions tend to ﬁll up every second layer com-

pletely (staging, left), before starting to intercalate into the

other half of the layers. However, this behaviour is not

global, but occurs in ﬁnite-sized domains (right) and is not

expected to be perfect in real systems. During the inter-

calation process, the distance between the graphene sheets

(interlayer distance) is increased by around 10%.

Traditionally, the RP has primarily been of in-

terest for insulators, but in recent times it has also

been increasingly investigated for conducting ma-

terials [1], where it stems from the interaction be-

tween a small fraction of the charge carriers and the

atoms. The fact that these charge carriers are mo-

bile in conducting materials fundamentally changes

the way the property and its dependencies need

to be understood within these materials, compared

with insulators.

One group of materials of great interest

are lithium-graphite intercalation compounds (Li-

GICs), which constitute by far the most common

anode material in modern lithium ion batteries.

During the charging and discharging cycles of the

battery, Li-ions are stored between the layers of the

graphite host structures, up to a stoichiometry of

LiC6, which traditionally translates to a state of

charge (SOC) of 100% and corresponds to one Li-

ion above every third C6ring of the hexagonal base

lattice – even though recent studies have shown that

overlithiation beyond that point is possible at am-

bient conditions [2]. The distribution of the Li-ions

for intermediate SOCs is not uniform, but ordered

as shown in ﬁg. 1, as ﬁrst explained in [3].

Within the scope of this speciﬁc material, we de-

ﬁne the RP we are investigating as “the damping of

the electrostatic interaction between two Li-ions or

Li-ion vacancies embedded in the material caused

by the surrounding charge-carrier density”. As of

note, this is the electrostatic, low-frequency RP,

as opposed to what is measured in many experi-

ments, which make use of alternating AC-ﬁelds at

a vast variety of frequencies [1, 4]. Furthermore,

we point out that this speciﬁc property is direc-

tionally separated – its contribution in the xy-plane

(in this work deﬁned as parallel to the graphene

sheets) is expected to be diﬀerent from the contri-

bution in z-direction (orthogonal to the graphene

sheets). This makes direct comparison with exper-

iments performed on graphite powder as opposed

to a “perfect” crystal non-trivial – a problem we

address in section 3.4.

One of the primary motivations for investigating

the RP of Li-GICs is the fact that it is a required in-

put parameter for including charge in kinetic Monte

Carlo simulations (kMCs), which in turn are a cru-

cial method for studying charge carrier dynamics in

functional energy materials [5, 6, 7, 8]. Long-ranged

Coulomb interactions are a necessity when perform-

ing kMC on systems, which include charged or par-

tially charged particles. For example, Casalegno et

al. [9] have shown that not including such interac-

tions (as it would be the case when using e.g. force

ﬁeld approaches with ﬁnite-size descriptors) causes

an error of 14% in the protonic diﬀusion coeﬃcients

in doped perovskites. The situation becomes sig-

niﬁcantly more complicated when looking at anode

and cathode materials, but also electrolytes, per-

ovskites in solar cells, and any other type of func-

tional materials that involves changes in the den-

sity and/or local ordering of charge carriers as part

of their intended function. This is due to the fact

that the local relative permittivity then is not only

inﬂuenced by the “host” material, but also by the

charge carriers close by. Therefore, the RP of Li-

GICs changes signiﬁcantly depending on the SOC

(vide infra).

In this work, we put forward a systematic ap-

proach to investigating this crucial property. Based

on this, we determine for the ﬁrst the relative per-

mittivity of Li-GICs, as a function of the SOC, for

the entire functional range of the material.

Beyond the previously outlined interest for

charge-kMC, we believe there to be many more

valuable applications for the relative permittivity

of Li-GICs also at higher levels of the multiscale

simulation hierarchy: one such motivation is un-

derstanding charge gradients, as they occur dur-

ing the fast charging of modern batteries in electric

vehicles, and the chemical pressure which leads to

plating and dendrite formation inside the batter-

ies under certain operational conditions. The latter

phenomenon is typically investigated by means of

continuum simulations like e.g. by Hein et al. [10],

which also rely at least implicitly on knowledge

of the dielectric response. Furthermore, a sim-

ple model of the charge carrier electrostatics could

be used as a physical baseline for otherwise short-

ranged machine learning models or cluster expan-

sions. Another related ﬁeld is the development of

functional materials based on doped graphite [11].

The dielectric behaviour of Li-GICs (and solid

materials in general) is signiﬁcantly more com-

plicated than the expression “dielectric constant”

would suggest, and is governed by vastly diﬀer-

ent physics at diﬀerent frequencies of a probing

ﬁeld. In the static limit (which this work aims

to investigate – the probing ﬁeld is essentially the

electrostatic ﬁeld of the intercalated ion itself), no

periodic movement (beyond thermal ﬂuctuations)

of the electrons is induced. However, this pic-

ture changes in the kHZ range, where the entire

charge carrier density oscillates with the probing

ﬁeld, causing large polarization and large dielectric

screening. For example, Chung et al. [1] measured

an RP of r=2100 for highly oriented pyrolytic

graphite (HOPG) and even higher ones for other

carbon structures, at 2-10KHz. Moving on to the

GHz regime, a balance is reached where the ﬁeld

oscillations are too fast for macroscopic bulk cur-

rents to build up, and a situation occurs that is

arguably similar to the static limit and may serve

as comparison for our research. Hotta et al. [4] put

forward a dielectric constant for graphite powder of

r≈15, at 6GHz. Finally, at even higher frequen-

cies beyond THz, the electric ﬁeld becomes high

enough in energy to excite a signiﬁcant number of

electrons, again creating a physically diﬀerent sit-

uation with much lower dielectric screening, which

converges to transparency in the ω→ ∞ limit. A

study by Jellison et al. [12] in the frequency regime

of visible light ﬁnds an RP in xy-plane for HOPG

of r=4.21, which, for the previously mentioned

reasons, cannot be used as comparison either and

is expected to serve as a lower bound in the follow-

ing.

It is apparent that there is a glaring lack of

studies investigating the exact property of inter-

est to us, which is – again – the electrostatic di-

electric response of a perfect graphite crystal in

xy-plane, i.e. parallel to the graphene sheets, as

experienced by some internal charge carriers (in

this case Li-ions and vacancies). There are some

studies available on graphene, either on some sub-

strate or quasi-freestanding, with results ranging

from r=2.2−5.0by Elias et al. [13] to r=15.4

by Reed et al. [11], and another study by Bostwick

et al. ﬁnding r≈4.4[14], none of which can serve

as direct comparison to our research either. How-

ever, there is a study on bilayer graphene (which

according to our calculations can be compared with

graphite quite well) by Bessler et al. [15], putting

forward an RP of r=6±2. This is likely the most

reliable direct experimental comparison currently

available to us.

In terms of theoretical approaches to determin-

ing the dielectric response of materials, substantial

work has been done on water [16, 17, 18, 19]. There

is also some promising work by Gigli et al. [20] in

the development of an integrated machine learning

model predicting the dielectric response of BaT iO3.

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HighlightsTheintrinsicelectrostaticdielectricbehaviourofgraphiteanodesinLi-ionbatteriesacrosstheentirefunctionalrangeofchargeSimonAnnies,ChristophScheurer,ChiaraPanosettiˆTherelativepermittivityoflithiuminterca-latedgraphiteanodesislinearlydependentonthestateofcharge,withapproximatevaluesofaround7...

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