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Numerical Prediction of Multiscale ElectronicConductivity of Lithium-Ion Battery Positive Electrodes

François Cadiou, A. Etiemble, T. Douillard, François Willot, O. Valentin,Jean-Claude Badot, Bernard Lestriez, E. Maire

To cite this version:François Cadiou, A. Etiemble, T. Douillard, François Willot, O. Valentin, et al.. Numerical Predictionof Multiscale Electronic Conductivity of Lithium-Ion Battery Positive Electrodes. Journal of The Elec-trochemical Society, Electrochemical Society, 2019, 166 (8), pp.A1692-A1703. �10.1149/2.1221908jes�.�hal-02137415v2�

Numeri al predi tion of multis ale ele troni ondu tivity of

Lithium-ion battery positive ele trodes

F. Cadiou

1

, A. Etiemble

2

, T. Douillard

1

, F. Willot

3

, O. Valentin

4

, J-C. Badot

5

, B. Lestriez

∗,6, and

E. Maire

1

1

MATEIS, INSA Lyon, Villeurbanne, Fran e

2

P�le Matériaux et Stru tures, ECAM Lyon, Lyon, Fran e

3

Center for Mathemati al Morphology, Mines Pariste h, Fontainebleau, Fran e

4

Auro k, Albi, Fran e

5

Institut de Re her he de Chimie Paris, Chimie-Pariste h CNRS, Paris, Fran e

6

IMN Institut des Matériaux Jean Rouxel, Universté de Nantes CNRS, Nantes, Fran e,

Referen e: Journal of The Ele tro hemi al So iety, 2019, 166 (8), pp. A1692�A1703.

DOI: 10.1149/2.1221908jes. This is a �post-print� version. A few typos orre ted;

units in Figs. 3, 7 and pp. 19, 21 have been amended ( hanges highlighted in blue).

Results un hanged.

Abstra t

The ele troni ondu tivity, at the multis ale, of lithium-ion positive omposite ele trodes based on

LiNi1/3Mn1/3Co1/3O2 and/or arbon- oated LiFePO4, arbon bla k and poly(vinylidene �uoride) mix-

ture is modelled. The ele trode mi rostru tures are a quired numeri ally in 3D by X-ray tomography

and FIB/SEM nanotomography and numeri ally segmented to perform ele trostati simulations using

Fast Fourier Transform (FFT) method. Su h simulations are easy and qui k to perform be ause they are

dire tly omputed on the grid represented by the voxels in the 3D volumes. Numeri al results are om-

pared with experimental measurements of the multis ale ele troni ondu tivity by broadband diele tri

spe tros opy (BDS). A good predi tion is realised for the bulk ondu tivities of the C/LiFePO4 phase and

the CB/PVdF mixture. The ombination of X-ray and FIB/SEM tomography, FFT simulation method,

and BDS is thus well adapted to understand the in�uen e of the ele trode omposition and mi rostru ture

on its ele troni ondu tivity.

∗

Corresponding author: Bernard.Lestriez� nrs-imn.fr

1

1 Introdu tion

Lithium-ion batteries (LiB) are nowadays widely used to store energy in nomad devi es su h as laptops and

smartphones. Another �eld of appli ation for LiB is the automotive industry, in luding ele tri ars, hybrid

ele tri vehi le, plug-in hybrid ele tri vehi les [1, 2℄. In those ases, both high power and high energy density

LiB are required to extend the vehi les autonomy range and to redu e the harge duration.

Ele trodes for LiB are omplex omposite materials obtained by mixing the a tive material (AM) pow-

der with an ele troni ondu tive additive and a polymeri binder in a solvent. The ele trode slurry is then

oated on a metalli foil that will serve as urrent olle tor. After solvent evaporation, the ele trode �lms

are pressed and assembled with a separator to form a battery ell. The ele trodes and separator porosity

are impregnated by a liquid ele trolyte to allow the battery ell to work. It is well established that battery

performan e depends signi� antly on the ele trons and ions transport properties of the omposite ele trodes,

whi h are riti ally a�e ted by their nano-, meso- and ma rostru ture through the interfa ial areas, material

onne tivities, and the transport length s ales [3, 4, 5, 6, 7, 8℄.

In the past years, implementation of X-ray omputed tomography (XRCT) and fo used ion beam

ombined with s anning ele tron mi ros opy (FIB-SEM) te hniques has enabled the quanti� ation of the

omposite ele trodes ar hite ture. These novel three dimensional hara terisation methods allow one to

interpret transport properties and eventually assess the limiting fa tors in ele tro hemi al performan e

[9, 10, 11, 12, 13, 14, 15, 16, 17, 18℄. The re onstru ted 3D geometries, obtained with these imaging

methods, an subsequently be used as inputs to model the ele tro hemi al behaviour at the mi rostru -

ture s ale [19, 20, 21, 22, 23℄. Combined with experiments, these methods redu e the time and ost resour es

needed to design new batteries, when ompared to empiri al approa hes. The present work fo uses on the

modelling of ele troni transport properties, whi h are ru ial with respe t to ele tro hemi al performan e

[24, 25, 26, 27, 6, 28, 29℄, and are often less studied than ioni mass transport properties. Indeed, it is often

assumed ioni transport is a limiting fa tor, espe ially under fast harge, while ele troni transport is not,

due to the presen e of ondu tive additive.

Simulation methods, on this topi , have evolved from numeri ally-generated mi rostru tures (ideal par-

ti le geometries as in Chen et al. [30℄ or Awarke et al. [31℄, to a tual mi rostru tures a quired by XRCT or

FIB/SEM. For omputation at the parti le s ale, Chen et al. and Awarke et al. used pa kings of spheri al or

ellipsoidal elements to generate a �nite element mesh representing the ele trode mi rostru ture. Chen et al.

a ounted for the presen e of the mixture of arbon bla k and polymer binder by in luding a uniform oat-

2

ing surrounding a tive material parti les. In Awarke et al., the a tive and the ondu tive additive parti les

were randomly mixed; while the binder phase was not modelled. However, its volume was a ounted for by

varying the void volume fra tions [32℄. Both studies aimed at al ulating the per olation transition. Above

the per olation threshold an ele tri ally onne ted ontinuous ondu tive network is formed and an abrupt

hange leading to a mu h enhan ed transport behaviour o urs. For Awarke et al., per olation threshold

is between 7.81 and 11.31% (volume fra tion) depending on the AM parti le size. A ording to Chen et

al., the per olation is dependent on the AM volume fra tions as these authors are using a uniform oating

on the AM parti le. As AM volume fra tion is over 30% (theoreti al threshold of 29% for spheres) per-

olation is a hieved. They also have tried ellipsoidal graphite �bers and platelets and they have found a

per olation threshold ranging from 7.5% to 10% (volume fra tion). Both studies gave interesting trends to

be followed for optimising the e�e tive ondu tivity of omposite ele trodes with respe t to the parti le size

ratio or shape fa tor. However, they fell short in taking into a ount the a tual distribution of the mixture

of ondu tive additive and binder that riti ally depends on the ele trode pro essing route [33, 6℄. Re ently,

Usseglio-Viretta et al. [34℄ have quanti�ed the per olation of the Carbon binder additive domain for positive

NMC and negative graphite ele trodes, based upon a physi s-based numeri al generation of the additive

domain, within real three-dimensional ele trode geometries re onstru ted from X-ray omputed tomography,

des ribed in [35℄. While no per olation path has been found for the arbon-binder embedded within the

graphite (4.6-6.0vol% arbon-binder), the authors found a onne ted luster for the NMC ele trode (10.6-

14.6%vol arbon-binder). When ompared to Awarke et al. and Chen et al. the threshold values are are

di�erent be ause the latter studies used numeri ally generated mi rostru tures with ideal parti le shapes on

the ontrary to [34℄, and also be ause di�erent AM parti le sizes and shapes were onsidered.

Imaging te hniques in 3D provide a wealth of information on the mi rostru ture. One of the main di� ul-

ties, nevertheless, is to a ount for the presen e of widely separated length s ales. The development of XRCT

allows one to obtain 3D maps of a tual mi rostru tures but only of the a tive material (AM) phase. This is

be ause, with this te hnique, the mixture of the binder with the ondu tive additive parti les (referred to as

the ondu ting mixture for simpli ity in the following) is di� ult to separate from the porosity in terms of

attenuation ontrast. XRCT based simulation was used to al ulate ondu tivity using the thermal/ele tri

analogy and �nite element method in Cooper et al. [18℄. This kind of simulations provides a fairly good

insight on the ondu tivity in the a tive material phase but ignores the role of the ondu ting mixture,

whi h is a strong approximation as it is the main ele tron ondu ting phase. Grillet et al. [25℄ used XRCT

as a basis to generate realisti AM mi rostru tures and, in their simulations, assumed that the ondu ting

mixture uniformly oated the surfa e of the AM parti les. The authors also investigated the e�e t of the

3

deterioration of the ondu tion of this ondu ting mixture upon ageing on the variation of the ele trode

e�e tive ondu tivity upon y ling. However, su h uniform oating assumption may not be representative

of the a tual mixture spatial distribution, as it may form some aggregates of various size and shape more

or less randomly spatially distributed as well as some dendriti , fra tal-like, stru tures onstituting a 3D

onne ted (or not) network through the ele trode thi kness. For instan e, di�erent arbon-binder mixture

morphology, ranging from �lmlike to tenta lelike, an be generated through a physi s-based pro ess related

with interfa ial energy [35℄. Kashkooli et al. [20℄ tried to a ount for this morphologi al hara teristi of

the ondu ting mixture, whi h however was not onsidered separately from the a tive material phase, but

rather as random bridges between neighbouring a tive parti les. An ele tri al ondu tivity was a ordingly

postulated for the obtained re onstru ted mi rostru ture and used to simulate the ele tro hemi al ( harge

and dis harge) behaviour. Wieser et al. [36℄ used the higher resolution of FIB/SEM to image in 3D the

ondu ting mixture. The authors studied the ontribution of this nanoporous phase in the mass transport

restri tion of the ele trolyte salt spe ies within the ele trode. Inoue et al. [37℄ studied realisti LiCoO2 and

graphite ele trode mi rostru tures obtained from FIB/SEM re onstru tions and evaluated their ele troni

ondu tivities. The ele troni ondu tivity was al ulated assuming that the ele tron transport through the

ele trode ould be modelled with homogenisation theory. This simply allowed them to al ulate the e�e -

tive ele troni ondu tivity of a omposite from the bulk ondu tivity in the di�erent phases, their volume

fra tions and the tortuosity of the transport path, whi h were all obtained from the FIB/SEM re onstru tions.

The ele troni ondu tivity of omposite ele trodes for lithium batteries has also been predi ted using

full-�eld numeri al omputations with �nite element method [38, 39℄ or else [40℄. Contrarily to e�e tive

medium theories, this approa h allows one to estimate the �elds at the lo al s ale in the mi rostru ture

while taking into a ount the �exa t� morphology of the ele trode. In the present paper, ele trode mi rostru -

tures designed for ele tri vehi le appli ation, in luding LiNi1/3Mn1/3Co1/3O2 (NMC) and/or arbon oated

LiFePO4 (LFP) as a tive material, arbon bla k (CB) as ele troni ondu tive additive and poly(vinylidene

�uoride) (PVdF/CB) as binder, were a quired by XRCT and FIB/SEM, and numeri ally segmented to

perform ele trostati simulations using Fast Fourier Transform (FFT) methods. Simulations are omputed

dire tly on the regular grid represented by the voxels in the 3D volumes without meshing. This approa h

was originally developed for me hani s [41℄ and extended to ondu tivity [42, 43℄. This method is applied

here for the �rst time to Lithium-ion battery omposite ele trodes. The simulation results are ompared

to experimental measurements of the ele troni ondu tivity done on the same ele trodes or on the on-

stituents taken separately. This allows us to establish relationships between the ele troni ondu tivity of

the di�erent phases in the omposite ele trode, its a tual ar hite ture and its e�e tive ele troni ondu tivity.

4

2 Digitalisation of the ele trode mi rostru ture

The ele trodes onsidered in the present arti le have been studied in previous works. Two ele trodes denoted

F and H are based on a blend of LFP and NMC, while a third one, denoted J, is solely based on LFP. The

names F, H and J are the same as those used in [3, 12℄. The ele trode volumes were a quired using the porous

non-impregnated ele trodes with XRCT and FIB/SEM te hniques at a voxel resolution of 320×320×320nm3

and 10×10×10nm3respe tively, as des ribed in [12℄.

XRCT volumes were a quired in phase ontrast mode in order to be able to separate the NMC lusters

from the other omponents in the ele trodes. At this s ale (∼ s ale 1), the ele trode thi kness is entirely

in luded in the �eld of view. A simple segmentation pro ess, based on grey level thresholding, allows us

to obtain segmented volumes ( f. Figure 3) and thus the NMC volume fra tions in these volumes (Table

1). X-Ray tomography volumes were segmented using a ombination of simple grey level thresholdings.

First the NMC parti les were extra ted by thresolding. Then another thresholding allowed us to extra t the

homogenised phase. The ombination of these two thresholds enables the volume re onstru tions. Internal

porosity (along with some ra ks reated by the sample preparation method) is the omplementary part of

these two solid phases. No arti� ial intermediate layer has been noti ed through volume inspe tion. In order

to get segmented volumes without ra ks, we used the same �rst threshold to obtain NMC parti les. The

omplementary being the homogenised phase with internal NMC porosity. To distinguish both features, the

internal NMC porosities are dete ted with a ��ll holes� type algortihm (Fiji software) on the NMC phase

and the di�eren e with the non �lled phase. This gives a good result only be ause the NMC lusters are

quite well separated, thus only internal porosities are losed and not possible losed spa es between NMC

lusters. This last part was veri�ed by areful eye inspe tion of the NMC-�lled volumes.

For FIB/SEM volumes, after a quisition, the image pro essing was performed with a home-made Fiji

ma ro [12℄ whi h is based on the algorithm developed in [44℄ for the shine through artefa t orre tion of

FIB/SEM images. Brie�y, the shine through artefa ts stem from the fa t that SEM imaging retrieves data

from inside the pores. This reates a gradient, by adding data from matter below the urrent observation

plane, that is perturbing grey levels ommonly used for segmentation. This is dis ussed and explained further

in [12, 44℄. As explained in [12℄, the Fiji ma ro is used to distinguish the solid phase from porosity. Then,

a tive materials (NMC and LFP) and PVdF/CB mixture were learly identi�ed by thresholding the grey

levels of BSE data. This methodology is illustrated in the Supporting Information. FIB-SEM re onstru tions

5

of ele trodes F, H and J are shown in Figure 1 together with an example of an X-Ray volume re onstru -

tion for sample F. The raw data are available upon request to the authors for resear hers eventually interested.

Figure 1: 3D entire segmented volumes a quired by FIB/SEM a), b) and ) and X-Ray tomography (d)

and used in simulations, a) Sample H, b) and d) Sample F, ) Sample J. For FIB/SEM samples, white is

LFP, light grey is NMC, dark grey is the CB/PVdF mixture and the porosity is transparent for all three

volumes. For X-Ray samples light grey is also the NMC lusters and the o�-white/lighter grey olour is the

homogenised phase (LFP, CB/PVdF, and porosity). Dimensions reported on the di�erent axes are in µm.

LFP (in white) presents the shape of nanometri parti les, from few tens of nanometres to around 800nm,

with some of them aggregated as lusters of parti les of a few mi rometers. LiFePO4 parti les are overed

by a nanometre thin layer of amorphous arbon, whi h is invisible at the FIB/SEM resolution. The arbon

ontent is about 2wt% in the LFP a tive material. NMC (in light grey) is in the form of large quasi-spheri al

lusters (∼10µm) of smaller grains. Parti le size distribution for NMC and LFP phases omposing the ele -

trodes studied here have been previously reported in [12℄. The PVdF and CB appear as one indis ernible

mixture (in dark grey). CB is a �ne graphite powder with high degree of rystallinity. The elementary

parti les are spheres made up of the sta king of broken, quasi-graphiti , layers. These elementary parti les

(mean diameter ∼40nm) are fused together by hemi al bounds in various forms of aggregations, alled

primary aggregates with average diameter of 100-300nm [45℄. Attra tive intera tions between CB and the

PVdF in the ele trolyte slurry favour their oagulation [46, 47℄. Thus, in omposite ele trodes PVdF and CB

are generally intimately mixed and asso iated into one indis ernible PVdF/CB ondu ting phase [48℄. We

believe that, with our 10nm voxel size, we an a ess pores of size above 20nm. Therefore, the smaller pores

6

within the PVdF/CB mixture might be invisible for us. Vierrath et al. [49℄ used FIB-SEM tomography to

re onstru t the arbon-binder domain of a LiCoO2 ele trode with ontrast enhan ement by ZnO in�ltration

via atomi layer deposition. They found the porosity inside the arbon domain is about 58% with the pore-

size distribution entred at 64nm. A ording to their results, we should have a ess to around 75% of the

pores within the PVdF-CB domains. The last phase present in the studied samples is a mostly open porosity

(intra onne tivity above 96% in all ases), whi h is �lled with a liquid ele trolyte in fun tional onditions to

ondu t the lithium ions.

Table 1 gives the volume fra tion φv (%) of the LFP, NMC, PVdF/CB and porosity phases, and intra on-

ne tivity of the LFP and PVdF/CB phases, at various ele trode s ales. These di�erent s ales are depi ted

in Figure 2. S ale 0 is the entire ma ros opi ele trode. The values reported in the table for this s ale are

average values measured ma ros opi ally from the ele trode thi knesses, areas, weights and the material den-

sities. S ale 1 is the XRCT volume and s ale 2 is the FIB/SEM volume for whi h the values were measured

in [12℄ using standard 3D image analysis. S ale 3 represents a small subdomain within the FIB/SEM volume.

The pe uliarity of this sub-volume is that it does not ontain the large NMC parti les. Su h subdomains

will be used in the multis ale simulations dis ussed in this work. Simulation results are shown in se tion 5.

Starting from the experimentally measured ele troni ondu tivity for ea h ele trodes, the e�e tive ondu -

tivity of the LFP/PVdF/CB/porosity homogenised phase will be determined with the XRCT volumes. Then

this ondu tivity will be onsidered as the e�e tive one for the FIB/SEM volumes, and the ondu tivity of

the LFP and the PVdF/CB phases will be then determined. These ones will be ompared to referen e values.

In Table 1, we quantify the intra onne tivity as the volume fra tion of the most voluminous luster of

one material/phase in the analysed volume. A value of 100% of this parameter means that all voxels of

this material/phase are part of a same unique per olated obje t in the analysed volume. F and H are made

with NMC/LFP (50:50 mass ratio) as a tive material and J only ontains LFP. The PVdF/CB phase vol-

ume fra tions in ele trodes F, H and J are respe tively 16.4, 11 and 9.1%. The PVdF/CB phase is quasi

fully per olated within the FIB/SEM volumes of F and H (intra onne tivity at s ale 3 equals to 97% and

96% respe tively), while it is rather not in J (intra onne tivity at s ale 3 equals to 22%). The LFP phase

fully per olates within the FIB/SEM volumes. One an noti e some di�eren es between volume fra tions

measured from tomographi data and standard measurements (s ale 0). These di�eren es an ome from a

representativity issue (as will be dis ussed after) and have also been addressed in [12℄ from a manufa turing

point of view.

7

The tortuosities given in Table 1 are geometri ones and these are measured along the y axis of Figure

1 (through the ele trode thi kness) to omply with the dire tion of the imposed potential di�eren e used

in the simulation developed in Se tions 4 and 5. By de�nition, the geometri tortuosity is the length of

the shortest path between two lo ations, going only through the phase of interest, divided by the straight

distan e between these two lo ations.

τ =shortest path length

straight distance(1)

It is then always superior to 1. In this work, the tortuosity is al ulated by the following pro edure. A

seed plane is initially de�ned in the middle of the binarised tomographi volume, perpendi ular to the y

axis. Then, a front is propagated (∼pixel agglomeration through dilation) only in the phase of interest,

materialising the length of the tortuous path. Then, at the end of the propagation pro edure, ea h rea hable

voxel in the phase of interest is labelled with its distan e from the seed plane. A measurement of the average

distan e from the seed plane an then easily be al ulated in ea h plane perpendi ular to the seed plane. The

so- al ulated distan e is however not equal to the true eu lidean distan e and depends on the riterion used

to propagate the front. A riterion based on six neighbours (the voxel is added if it shares a ommon fa e

with one of the front voxels) overestimates the distan e while a riterion based on 26 neighbours (the voxel

is agglomerated if it shares a ommon fa e, edge, or summit with one of the front voxels) underestimates

the distan e. In the present study, we make the al ulation twi e (with 6 and 26 neighbours). Then, with

respe ts to the straight distan e from the seed plane, a linear interpolation allows us to al ulate a mean

tortuosity in the phase (one for 6 neighbours and one for 26 neighbours). And, as mentioned, we estimate

the geometri al tortuosity as the average of both measurements.

8

Table 1: Properties of the volumes used for the simulation. φv stands for volume fra tion. Sour e 0

orresponds to (real) ma ros opi properties from the manufa turer. Other sour es refer to the imaging

te hniques used to a quire the volume. Corresponding dimensions in µm are reported in the Size olumn

whi h has been formatted like A×B×C with C the dimension along the ele trode thi kness

.

Sample Sour e Size (µm3)∗

Volume fra tion φv (%)

Intra onne tivity (%) /

Geometri tortuosity

NMC LFP PVdF/CB Pores LFP PVdF/CB

F

0 � 24 30.6 16.4 29 � �

XRCT 128×160×72 18.5 81.5 � �

FIB/SEM 10×4×7 20.4 36.1 10.0 33.5 99.8 / 1.17 87 / 1.25

FIB/SEM 2.3×2.3×2.3 � 37.8 18.5 43.7 98.5 / 1.36 96 / 1.35

H

0 � 25.5 32.5 11 31 � �

XRCT 128×160×67.8 21.1 78.9 � �

FIB/SEM 10×6×5.4 15.4 49.4 10.6 24.6 99.7 / 1.11 90.8 / 1.39

FIB/SEM 4×4×4 � 54.3 16.3 37 99.6 / 1.1 95 / 1.34

J

0 � � 53.9 9.1 37.0 � �

FIB/SEM 10×8×5.5 � 58.5 8.1 33.4 99.8 / 1.09 23 / -

FIB/SEM 5×5×5 � 64.9 4.7 30.4 99.9 / 1.08

17.6 / -

∗ F, H and J thi kness are respe tively 77.5, 72 and 91.5 µm (in luding the urrent olle tor).

One an noti e that no more NMC enter the omposition of F and H volumes for the last sour e from

FIB/SEM. This is explained in Figure 2 and serves the purpose of multis ale simulation dis ussed in this

work.

The sign � means that the property is not measurable.

9

Figure 2: Multis ale view for the mi rostru tures studied. The homogenised phase in the X-Ray tomography

pi ture is omposed of the PVdF/CB, LFP and porosity phases.

3 Ele tri al ondu tivity measurements

The ele troni ondu tion properties were a quired with Broadband diele tri spe tros opy (BDS). The theo-

reti al ba kground and experimental setup to implement this te hnique are well des ribed in previous papers.

See for example Badot et al. [50℄. With the BDS te hnique, it is possible to measure the ele tri al properties

at di�erent s ales (sample, luster, parti le and atom) of an ele trode material. When a time-dependent ele -

tri �eld is applied to the material, polarisations ( harge-density �u tuations) are produ ed. The times ale

(or relaxation time) of the polarisation depends on the s ale at whi h it o urs and on the ondu tivity

at this s ale. In ele troni ondu tors, di�erent relaxation me hanisms generally result from their rystal

stru ture, nanostru ture and mi rostru ture. Be ause they have distin t hara teristi frequen ies, they an

be separated and treated individually. All details about the BDS measurements are given in the Supporting

Information as most of them are already published elsewhere. Results for the ondu tivity measurements are

thus summarised in Table 2 and dis ussed hereafter. The interested reader is en ouraged to refer to se tion

5.4 for omparisons of ondu tivity values with analyti al estimates.

10

Table 2: Room temperature ondu tivity at di�erent s ales for NMC and LFP powders (pressed pellets),

PVdF/CB mixture, ele trodes F, H and J.

Material

Volume fra tions (%)

Condu tivity (S.m

−1) Referen e

NMC LFP PVdF/CB

NMC pellet

(7% PVdF)

74 � �

Sample 0.000035

[53℄Cluster

0.0001

(e�e tive)

Grain in a

luster

0.2

(e�e tive)

Interatomi

19

(e�e tive)

LFP pellet

(2% PVdF)

� 65 �

Sample 0.26

This work

Coating

0.4

(e�e tive)

Intra sp

2

domains

> 20

(e�e tive)

PVdF/CB

�lms

� � 100 Sample 50 to 400 [55℄

Ele trode F

24 30.6 16.4

Sample 4

[3℄ and this work

PVdF/CB

lusters

9.1

(e�e tive)

Ele trode H

25.5 32.5 11

Sample 2.94

[3℄ and this work

PVdF/CB

lusters

4.26

(e�e tive)

Ele trode J

� 53.9 9.1

Sample 0.3

[3℄ and this work

PVdF/CB

lusters

4.1

(e�e tive)

The bare LiFePO4 ompound has very low intrinsi ondu tivity. The ondu tivity of the arbon oated

ompound is an e�e t of the arbon oating in whi h LiFePO4 parti les are embedded. It ontains short-order

sp

2- oordinated arbon domains (graphite nano rystallites) bonded by sp

3- oordinated arbon atoms. At

11

the mi ros opi (parti le) s ale, the ele tri al ondu tivity is due to ele trons hopping between sp

2domains

within the arbon oating around the LiFePO4 parti le. At the ma ros opi (sample) s ale, the ele tri al

ondu tivity within this phase depends on the onta t resistan e between the LFP parti les and on the void

volume fra tion [51℄. The ondu tivity at the di�erent s ales of the LFP phase is given in Table 2. It is

very important to note that the measured values are e�e tive. True ondu tivity ould only be measured for

dense, non-porous samples. For example, the ondu tivity of graphite is about 10

5S.m

−1[52℄. The e�e tive

ondu tivity measured for graphite nano rystallites in the arbon oating of C-LFP is measured to be in the

20-100S.m

−1range, be ause the arbon volume fra tion in the sample is very low, about 3.8% in C-LFP and

2.5% in the porous pellet.

The ondu tivity of the NMC ompound stems from the presen e of Ni

3+ions, whi h are balan ed by

negatively harged M va an ies. This enables a hole 2D ondu tion in the ni kel layer. The ondu tivity

within the NMC lusters is however limited due to blo kage of the harge arriers (holes) at the interstitial

grain-boundary amorphous region between NMC rystallites (grains). Furthermore, at the ma ros opi (sam-

ple) s ale, the ele tri al ondu tivity is hindered due to a onstri tion resistan e at inter- lusters onta ts [53℄.

The ondu tivity of the PVdF/CB phase is provided by the arbon bla k nanoparti les. In this mixture,

CB aggregates are generally not in dire t onta t but are separated by an interfa ial PVdF layer. The ombi-

nation of per olation theory and interparti le tunnelling ondu tion a ounts well for the ele tri al behaviour

of su h polymer/CB mixture [54, 26℄. At the mi ros opi s ale (a few aggregates), the ele tri al ondu tivity

is due to the tunnelling of ele trons through the thin polymer layers adsorbed between the CB aggregates.

This tunnel resistivity in reases exponentially with the polymer gap thi kness. At the ma ros opi (sample)

s ale, an ele tri al ondu tivity is only observed if the CB aggregates per olates through the PVdF matrix.

This ondition is ful�lled for the PVdF to CB mass ratios (5:1 to 1:1) typi ally used in omposite ele trodes

[48℄. The typi al ondu tivity of PVdF/CB mixture was measured in [55℄ on dense PVdF/CB �lms. The

ondu tivity values range from 50 to 400S.m

−1for PVdF to CB ratios from 5:1 to 5:4.

The ondu tivity of the three ele trodes studied here was measured by BDS and some values were already

reported in [3℄. These values however were here re�ned. Resistivity Nyquist plots for the three ele trodes are

given and interpreted in the Supporting Information. By having a se ond look at these data, it was possible

to measure three parameters: (i) the onta t resistan e at the ele trode/aluminium urrent olle tor, RC , (ii)

the ele trode layer ondu tivity, here named sample ondu tivity, σS, and (iii) the e�e tive ondu tivity of

the PVdF/CB lusters at high frequen y (in the 1-100 MHz range), σC . The sample ondu tivity of J is lose

12

to that of a pressed LFP pellet, 0.3 and 0.26S.m

−1, respe tively. These two ondu tivities show Arrhenius-

like temperature dependen e with similar a tivation energy, 0.07 and 0.09eV. These omparisons strongly

suggest that the ele trode layer ondu tivity of J is dominated by the ontribution of the LFP phase, this one

being per olated ontrarily to the PVdF/CB mixture. The sample ondu tivity of J is however higher and

its a tivation energy is lower than that of a pure LFP pellet as J ontains isolated PVdF/CB lusters, whi h

temperature independent e�e tive ondu tivity ould be determined to be equal to 4.1S.m

−1as explained in

the Supporting Information (Figure S2). The temperature independen e of this high frequen y ondu tivity,

whi h is typi al of the tunnelling me hanism, allows its unambiguous attribution to the PVdF/CB unper o-

lated lusters.

The sample ondu tivity of F and H is one order of magnitude higher than that of J. A ording to

FIB/SEM re onstru tions, these high ondu tivities an be explained by the per olation of the PVdF/CB

mixture that likely short- ir uits the less ondu tive LFP and NMC parti les in the ele trode. The e�e tive

ondu tivity of the PVdF/CB lusters is higher in F than in H, while it is similar in H and J, depending on

their di�erent volume fra tions.

4 Modelling and omputer simulation

The simulations des ribed here use the FFT method by solving the ele trostati equations with the Mor-

phhom software, making use of the �dis rete� Green operator introdu ed in [43℄. This software has already

been used to perform simulations on the ele tri al ondu tivity of fuel ells [40℄ and for the homogenisation

of me hani al properties [56, 57℄ or opti al response [58℄. The interest of this method is that simulations are

omputed dire tly on the regular grid represented by the voxels in the 3D re onstru ted FIB/SEM volumes.

The ele trostati equations solved here are:

div(J(−→x )) = 0, E(−→x ) = −

−−−−−→

grad(φ), J(−→x ) = σE(−→x ) (2)

Where φ is the ele tri potential, E the ele tri �eld, J the urrent �eld and σ is the lo al ondu tivity

at point

−→x . A strong hypothesis is that all onta ts are supposed to be perfe t, i.e. no onta t resistan e

nor onstri tion resistan e between the di�erent phases are onsidered. Phase ontinuity is assumed to o ur

when two voxels pertaining to the same phase are tou hing by their fa es. The impa t of the mi rostru ture

is then essentially re�e ted in the phase volume fra tion, intra onne tivity (or extent of per olation) and

13

tortuosity.

5 Results and dis ussion

5.1 Simulations based on X-Ray tomography volumes

In this subse tion, we fo us on the �rst simulation s ale performed on X-Ray tomography volumes.

One an observe in Figure 3a) and f) that some ra ks are present in the X-Ray volumes. These ra ks

are most ertainly due to the sample preparation for tomography. In order to be imaged at su h a small

resolution, the sample has to be very thin (around 0.8 or 1mm thi k). Thus, utting from the mother ele -

trode is very likely to indu e ra ks in these fragile samples. The X-Ray samples, for H and F, were then

segmented in two di�erent ways, with and without ra ks, to observe their impa t on the urrent density

�eld. The dis ernible internal voids inside the NMC lusters are onsidered as porosity in both ases. Their

volume fra tion is however negligible with less than 0.01%. Internal voids plus ra ks together ount for up

to 2.9% volume fra tion in the volumes. The segmentation results are shown in Figure 3b), d), g) and i). In

these images, the white phase indi ates the NMC lusters and the bla k one the porosities. The light grey

areas symbolises an homogenised phase omposed of LFP, PVdF/CB mixture and porosity at a smaller s ale

a essible with the FIB/SEM resolution.

14

Figure 3: From a) to e) is sample F and from f) to j) is sample H. a) and f) are X-Ray original data. b)

and g) segmentation with ra ks and d) i) segmentation without ra ks. ), h) and e), j) are respe tively 2D

maps of the urrent density �eld with and without ra ks. These maps are se tions from the 3D FFT data

results. A 1V/m [instead of 1V ℄ di�eren e potential is imposed between the left and right fa es (thi kness

dire tion of the ele trode sample), olour s ale is expressed in A.m

−2[instead of A.m

−1℄. In the segmented

images, white is for NMC, bla k for porosity and the light grey for an homogenised phase (LFP, PVdF/CB

and porosity at the FIB/SEM smaller s ale).

The FFT simulations were then arried out on both types of segmented volumes for H and F. The simula-

tions were performed by �xing the ondu tivity of the low ondu ting NMC phase to 0.0001S.m

−1, a ording

to experimental data [53℄ (see Table 2 and 3), and varying the ondu tivity of the homogenised phase, σvar

(bulk ondu tivity), from 2S.m

−1to 8S.m

−1. The e�e tive volume ondu tivity, predi ted by the FFT

method, varies linearly with respe t to σvar. We a ordingly determine for ea h sample the ondu tivity of

the homogenised phase whi h allows one to re over the experimentally measured e�e tive ondu tivity. This

homogenised phase ondu tivity will then be used to determine the ondu tivities of the individualised LFP

and PVdF/CB phases, by hange of s ale, using the FIB/SEM volumes ( f. Se tion 5.2). The simulation

results and input parameters are gathered in Table 3. Figure 4 shows the variation of the al ulated volume

ondu tivity as a fun tion of the homogenised phase ondu tivity σvar. The horizontal lines orrespond to

experimentally measured sample ondu tivities. The homogenised phase ondu tivity is higher for F than

for H, in agreement with the higher e�e tive ondu tivity in F. As expe ted, when ra ks are onsidered, a

higher ondu tivity is obtained for the homogenised phase for both ele trodes. This is mostly a onsequen e

of the more pronoun ed tortuosity and onstri tivity of the homogenised phase, as illustrated in Figure 3.

15

Table 3: Simulation results and input/target parameters for the FFT simulations on XRCT volumes

Sample

Input: NMC

phase (S.m

−1)

Target:

experimental

sample

ondu tivity

(S.m

−1)

Result:

LFP/PVdF/CB/porosity

homogenised phase

ondu tivity determined by

inverse method (S.m

−1)

F

With ra ks

0.0001 4

5.88

Without ra ks 5.51

H

With ra ks

0.0001 2.94

4.71

Without ra ks

4.26

Figure 4: E�e tive volume ondu tivity of the XR volumes for samples F and H as a fun tion of the

homogenised phase ondu tivity. Dashed lines represent the experimental results. The text �n � refers to

volumes without ra ks, otherwise the plot on erns volumes with ra ks. Arrows indi ate the values of the

homogenized phase orresponding to experimental measurements.

The �eld maps of the lo al urrent density are shown in Figure 3 ), e), h) and j). These �elds are repre-

16

sented along a 2D se tion. The urrent is enhan ed in regions where the ondu ting phase is onstri ted (at

bottlene ks between NMC lusters) or in ra ked zones. The ra ks strongly deviate the urrent density �ow

leading to highly on entrated and depleted regions, as illustrated in Figure 3h). The �eld �u tuations are

signi� antly smaller in the absen e of ra ks (see Figure 3j). In the same way, the presen e of more NMC

lusters (21.06% instead of 18.54%) seems to also have a onstri tion e�e t in H ompared with F when

examining the slopes in Figure 4.

5.2 FFT simulations on the FIB/SEM sub-volumes

Results presented hereafter on ern the sub-volumes a quired by FIB/SEM (s ale 3) whi h are exempt of

ra ks ontrarily to the XRCT volumes. The simulation results and input parameters are gathered in Table

4. Sample J is made of the homogenised phase only, within whi h the PVdF/CB mixture has not per o-

lated. This makes it the best ase to ompute the LFP phase ondu tivity. It is not not possible to do so

from samples F and H be ause of the PVdF/CB phase per olation that hinder the e�e t of the LFP phase

ondu tivity. Volume fra tions of the di�erent onstituents are re alled in Table 1. FFT omputations are

performed �xing the ondu tivity of the CB/PVdF mixture to 250S.m

−1, following [55℄ (see also Table 2),

and varying the ondu tivity of the LFP phase σLFP . Figure 5 illustrates the variation of the sub-volume

ondu tivity (J, 5µm edge size) as we vary the LFP phase ondu tivity. This is also done for two other

�xed values of the CB/PVdF phase ondu tivity. The e�e tive volume ondu tivity varies linearly with

σLFP in the range of interest (between 0.1 and 0.6S.m

−1, see Figure 5). Experimental data for the sample

ondu tivity, i.e. 0.3S.m

−1(dashed line in Figure 5), is re overed for one parti ular value for σLFP . We

determine a LFP phase ondu tivity σLFP of 0.38S.m

−1using the ondu tivity value of 250S.m

−1for the

PVdF/CB mixture. This value seems reasonable sin e it is in the same order of magnitude than the e�e tive

ondu tivity of the arbon oating that was measured by BDS (0.40S.m

−1, Table 2). The ondu tivity of

the LFP phase is indeed due to the arbon oating.

17

Figure 5: Variation of the e�e tive volume ondu tivity for J×500 as a fun tion of the LFP phase ondu -

tivity for three �xed values of the CB/PVdF phase ondu tivity (170, 250 and 300S.m

−1) along with the

experimental sample ondu tivity value to �t (horizontal dashed line).

In all FFT omputations hereafter, we assume that σLFP is 0.38S.m

−1. We let the CB/PVdF phase on-

du tivity vary and ompute the resulting e�e tive ondu tivity for samples F and H at s ale 3. As observed

previously, this results in quasi linear evolutions (see Figure 6). One an noti e that these two evolutions

are quite similar despite the di�eren es in ar hite ture and omposition of the two ele trodes. This an be

explained by the fa t that the volume fra tion of the highly ondu tive phase (PVdF/CB mixture) is quite

similar at this s ale (18.5 and 16.3 for F and H respe tively). The geometri al tortuosities are also really

lose, 1.34 for F and 1.35 for H. In terms of e�e tive volume ondu tivity, the slightly higher PVdF/CB

volume fra tion in F is �nally possibly ompensated by the mu h higher LFP volume fra tion in H and/or

lower porosity in H. The horizontal dashed lines orrespond to the homogenised phase ondu tivity values

al ulated using X-Ray volumes for F and H ( f. Se tion 5.1). The omputed sub-volume ondu tivity

mat hes the homogenised phase ondu tivity of ele trode F and H for a PVdF/CB phase ondu tivity equal

to 305S.m

−1and 210S.m

−1respe tively. This gives a mean value of 258S.m

−1for the CB/PVdF phase.

These values are remarkably lose to the experimental ondu tivity value (250S.m

−1) measured in [55℄ on

ma ros opi PVdF/CB �lm exhibiting the same weight ratio as those in the ele trodes studied here. The

ratios measured in [55℄ are all displayed in the x axis of Figure 6a. The ratio that was used on the ele trodes

18

(F, H and J) is 5:3 (in bold hara ters in Figure 6a). This suggests that this PVdF/CB mixture shows similar

ele tri al properties when forming a bulky ma ros opi �lm or when it is distributed as mi rometri lusters

within a omposite ele trode (as in our study). This apparent very good agreement between experimental

and al ulated PVdF/CB ondu tivity values ould indi ate (even though our subvolumes are not represen-

tative of the large NMC lusters) that the spatial distribution of the CB/PVdF is representative. It may

also imply that the spatial distribution of the large NMC lusters does not impa t the e�e tive ondu tivity

enough to be fully taken into onsideration. That may be why we do not need to apture the NMC luster

RVE (as dis ussed in se tion 5.3) to obtain relevant results as those we are presenting here.

Figure 6b displays the umulative volume fra tion of the CB/PVdF phase vs. the lo al urrent density

�eld in this phase for F and H. This lo al urrent density �eld is extra ted from the FFT omputation

results. It an be seen that the urrent is distributed over two de ades as it varies typi ally between 5 and

100A.m

−2[instead of 100A.m

−1℄ when the umulative volume fra tion in reases from about 10 to 80%. In

sub-volume F, a less intense urrent streams through the CB/PVdF phase ompared to sub-volume H. This

is onsistent with the two sub-volumes having the same tortuosity but more PVdF/CB in F than in H, with

also a slightly higher intra onne tivity for this phase in sub-volume F.

Figure 6: a) Computed volume ondu tivity for the sub-volumes from samples F and H as a fun tion of

the PVdF/CB phase ondu tivity. Dashed horizontal lines are the homogenised phase ondu tivity dedu ed

from the previous simulations on the XRCT volumes. Verti al lines are experimental measurements for the

PVdF/CB phase from [55℄, ratios are PVdF/CB weight ratios. The ratio used in our ele trodes F, H and J

is the one written in bold font (5:3). b) Cumulative volume fra tion of the PVdF/CB phase as a fun tion of

the urrent value in this phase. Only the results for the lowest urrent density values are displayed.

Table 4: Simulation results and input/target parameters for the FFT simulations on FIB/SEM sub volumes.

19

Sample

Target: LFP, PVdF/CB

and porosity homogenised

phase ondu tivity

(S.m

−1)

Input Result

Experimentally

measured

al ulated

with XRCT

volume

PVdF/CB

phase

ondu tivity

experimen-

tally

measured

(S.m

−1)

LFP phase

ondu tivity

al ulated

with

FIB/SEM

sub-volume

from J

(S.m

−1)∗

LFP phase

ondu tivity

(S.m

−1)

PVdF/CB

phase

ondu tivity

(S.m

−1)

F � 5.51 � 0.38 � 305

H � 4.26 � 0.38 � 210

F 0.3 � 250 � 0.38

�

∗ Note that the LFP phase ondu tivity al ulated with FIB/SEM sub-volume from J is very lose to that

experimentally measured on a LFP pellet (see in Table 2 the � arbon oating� ondu tivity value, whi h

gives the e�e tive ondu tivity of the arbon oated LFP material).

Finally, 2D maps of the urrent �eld, visualised for one sli e of volumes F and J (at s ale 2), are shown in

Figure 7. A olour map is used to distinguish areas in whi h the appli ation of a potential di�eren e auses

an enhan ed ele tri urrent (in yellow and white) from areas where it is moderate or inexistent (in blue and

bla k). The olour s ale is di�erent for F, from 0 to 126A.m

−2[instead of 126A.m

−1℄, than for J, from 0 to

6A.m

−2. It is learly visible in these images that the urrent mainly streams through the PVdF/CB phase

when it is per olated ( f. Sample F). This also learly illustrates that the ondu tivity of the homogenised

phase onsidered in the previous subse tion is dominated by the per olated PVdF/CB mixture. In ele trode

J, it is also lear that the urrent is more intense in the PVdF/CB lusters even if it is not per olating in

this ase.

20

Figure 7: Segmented sli e (left olumn) and 2D urrent maps (right olumn, FFT al ulations) from one

sli e of the whole volumes, from top to bottom: Sample F and J. Colour s ales for the urrent density are in

A.m

−2[instead of A.m

−1℄. The left arrow indi ates the dire tion of the imposed potential (1V/m) [instead

of 1V℄.

5.3 Impa t of the volume size on representativity

Given its mi rostru ture, the positive ele trode an be seen as a random medium from a mathemati al point

of view. It means the heterogeneous ele trode mi rostru ture an be represented with an homogeneous

medium hara terised by a set of e�e tive (i.e. ma ros opi ) parameters. Thus, the question of the rep-

resentativeness of the analysed volumes is relevant, espe ially for the FIB/SEM volumes and sub-volumes.

The simulated volumes must be large enough to hara terise properly the e�e tive behaviour of the global

ele trode. This is veri�ed a ording to two di�erent riteria in the following se tions.

5.3.1 Empiri al riterion

The more intuitive way to address this question is an assessment on the geometri al dimensions involved.

As a rule of thumb (see for example [59, 31℄), the volume is onsidered to be representative if its size is �ve

to ten times larger than the so alled �Elements of Interest� (EI) in the mi rostru ture. Here, the EI are

the NMC lusters at s ale 1 and the LFP parti les at s ale 3. This size (mean diameter) has been measured

on the three studied ele trodes by image analysis using a lo al thi kness algorithm within the Fiji software

[60℄. This rule of thumb statement tends to be nuan ed by other onsiderations su h as the size distribution

heterogeneity as stated in [61℄. Results are gathered in Table 5. The ratio between the lateral size of the EI

21

and the lateral size of the analysed volume is also al ulated in this table. In the three studied ases, this

ratio is nearly equal or higher than 15. These results might provide a �rst look at the representativeness

of its simulation volumes. However, it is hard to �rmly on lude on su h a riterion if there is not a lot

of volumes from the same sour e to onverge on a statisti al evaluation. This is why another riterion is

developed in the next subse tion in order to be more systemati and sele tive when only one volume, or a

few, is available.

One an noti e that, in Table 5, we do not take the thi kness (the minimal dimension) of the XRCT

volumes as the lateral size. This is be ause we have aptured the entire ele trode thi kness in the XRCT

volumes. This dimension is thus inherently exa t and representative. Only the two other dimensions an

possibly be non-representative. However, the dimension that is of primary importan e in our simulation on

XRCT volumes is the thi kness, making all other analysis of their representativeness of little interest. That

is why the mathemati al evaluation dis ussed in the next paragraph (5.3.2) only on erns the FIB/SEM

volumes and s ale 3.

Table 5: Geometri al analysis of the representativity of the simulation volumes. Comparison between the

lateral size of the simulation volume and the �Element of Interest� (EI) in the mi rostru ture, depending on

the s ale.

Sample

Lateral size of the simulation

volume (µm)

Chara teristi length of the

EI (µm)

Ratio

S ale 1 S ale 2 S ale 1 S ale 2 S ale 1 S ale 2

F 128 2.3 8.29 0.16 15.44 14.4

H 128 4 7.27 0.15 17.61 26.7

J 128 5 � 0.22 �

22.7

5.3.2 Mathemati al evaluation

It is possible to evaluate the representativity of a volume by appealing to the theory originally developed by

Matheron [62℄ and Lantuéjoul [63℄. In this theory, �representativity� is estimated by examining how the vari-

an e of the mean of a spatial �eld over a volume de ays with the volume size, as the later be omes very large.

In the present ontext, the �eld is the lo al urrent and the mean is the apparent ondu tivity. Matheron's

main result is that the varian e de ays proportionally to the inverse of V, where V is the volume size, ex ept

in parti ular ases where the �eld presents orrelations at in�nite length s ales. Matheron's theory, however,

22

assumes that the �eld is stationary (statisti ally independent of translations). In all rigour, this ondition

is not ful�lled here, be ause of the e�e t of boundary onditions. Although Matheron's theorems do not

rigorously apply to the present problem, this theory nevertheless provides a useful riterion for quantifying

the representativity (see e.g. [64℄).

Hereafter, we use sample J at s ale 2 be ause its mi rostru ture is similar to the one of the other samples

and be ause it is the largest volume available ontaining only LFP, CB/PVdF and porosity. The whole

simulation volume is divided in height equal parts, themselves divided in height equal parts and so on over

several levels. Then, we ompute the e�e tive ondu tivity for ea h volume by using the same phase on-

du tivities as in se tion 5.1 (LFP = 0.38S.m

−1, PVdF/CB = 250S.m

−1). The result of these al ulations

is shown in Figure 8). The varian e D

2σ(V) is then al ulated for ea h size level. A ording to [65, 62℄, to

obtain a relative error εrel on the estimate of the e�e tive volume ondu tivity, the omputation volume V

must be larger than about:

V =4D2

σA3

Z2ε2rel(3)

with Z, the e�e tive ondu tivity value for the whole volume. D

2σ is the point varian e al ulated a ross

the whole volume, i.e. the varian e of the lo al �eld σ. A3 is the integral range, a onstant linked to the

ovarian e fun tion of the �eld [65℄. In pra ti e, A3 is determined by a �t of the urve D

2σ(V)/ D

2σ vs. 1/V

(see Figure 8b). We obtain A3 ∼ 2.46×105(in voxels) for the sample studied here.

23

Figure 8: a) Evolution of the normalised omputed e�e tive ondu tivity (referen e value is the omputed

e�e tive ondu tivity for level 1) with the volume size. b) Evolution of the varian e with the inverse of the

volume size. Level 1 orresponds to ele trode J at s ale 2.

A ording to FFT numeri al results, the size of the RVE is 100×100×100µm3when εrel equals 10%.

Hen e, the simulation volumes used in Se tion 5.2 are too small to be representative a ording to this sta-

tisti al riterion with a relative error superior to 120%. Equation 3 also allows us to al ulate that for a

volume similar in size to the one of ele trode J (s ale 2), we should use 15 di�erent volumes in order to

statisti ally extra t the omputed e�e tive ondu tivity value for a relative tolerated error εrel of 10% and

4 for a εrel of 20%. Although not fully satisfa tory from a statisti al point of view, we however managed

to ompute realisti properties when omparing to experimental ones (see Table 2). We an also point out

that the volume fra tions of the di�erent phases in the numeri al volumes are lose to experimental ones

espe ially for samples J and F. Although our volume is smaller than the theoreti al statisti al RVE, the

geometri al riterion is not totally on lusive but the results mat h the experimental measurements. This

may be the result of the CB/PVdF spatial distribution at the FIB/SEM s ale being representative of the

one at the other s ales. Clearly, we would need several FIB-SEM imaging to assess representativity.

5.4 Analyti al estimates

In this part, we use the results of our simulations, more parti ularly the ondu tivity values of the di�erent

phases, to evaluate lassi al analyti al approa hes for al ulating the e�e tive ondu tivity of omposite

materials. Indeed, it is not always possible to make use of multiple experimental te hniques as sophisti ated

as XR and FIB/SEM tomography, and to ouple them with numeri al simulations, to estimate the e�e tive

ondu tivity of a omposite material from the values of its di�erent phases. Simple analyti al approa hes

24

an therefore be useful for the engineer.

Among all the literature, we �rst onsider the Voigt and Reuss bounds, whi h are parallel and series

resistan e ir uit analogy respe tively. We have then the following equations 4 and 5, where σvol is the

ondu tivity of the omposite volume and φv the phase volume fra tion.

σvol,V oigt = φv,NMCσNMC + φv,homogσhomog (4)

σvol,Reuss =σNMCσhomog

φv,NMCσhomog + φv,homogσNMC(5)

More narrow bounds are obtained for isotropi two-phase media, whi h lead to equations 6 and 7 [66℄.

σvol,HSmin = σNMC

1 +φv,homog

σNMC

σhomog−σNMC+

φv,NMC

3

(6)

σvol,HSmax = σhomog

1 +φv,NMC

σhomog

σNMC−σhomog+

φv,homog

3

(7)

We also make use of the Self Consistent (SC) approximation [67℄. In the limiting ase where the NMC

ondu tivity vanishes, it gives equation 8.

σSC =

(

1− φv,NMC −

φv,NMC

2

)

σhomog (8)

We onsider here s ale 1 where the volumes are modelled as a mixture of the NMC phase and a se ond

homogeneous one ontaining, at a lower s ale, the LFP, PVdF/CB and porosity phases. We use 0.0001S.m

−1

as the phase ondu tivity for NMC ( σNMC , f. Table 2). As in Se tion 5.1, we let the ondu tivity

of the homogenised phase vary and determine the value of σhomog so that analyti al predi tions mat h

experimentally-measured values. We fo us here on sample F, as an example. Results for the analyti al

simulations are plotted in Figure 9 along with the experimental value to �t. Results are similar to the FFT

ones for the upper bounds (Voigt and Hashin-Shtrikman) and the SC estimate. As expe ted, the lower

bounds (Reuss and Hashin-Shtrikman) are lose to zero. The numeri al values al ulated using a �t of the

experimental values and that of the FTT predi tions on the same sample are ompared in Table 6. We

observe ex ellent agreement between SC estimates and FFT predi tions.

25

Figure 9: Variation of the analyti al models with the ondu tivity of the homogenised phase for XRCT

sample F along with the measured experimental global ondu tivity value

Table 6: Analyti al and FFT simulation results for the homogenised phase ondu tivity in order to �t the

experimentally measured e�e tive ondu tivity of sample F.

Sample

Voigt

(S.m

−1)

Hashin-

Shtrikman

upper

(S.m

−1)

Reuss

(S.m

−1)

Hashin-

Shtrikman

lower

(S.m

−1)

SC model

(S.m

−1)

FFT

simulation

(S.m

−1)

F 4.88 5.32 2.53×108

2.81×107

5.48

5.51

At the large �eld of view ontaining the NMC lusters (noted s ale 1), the mi rostru ture an be assumed

as nearly non ondu ting spheri al parti les embedded in a ondu ting matrix. In this ase, the upper bounds

and the SC estimate lead to good predi tions as we remain lose to the hypothesis on whi h they are based.

If we take into onsideration the real mi rostru ture at the �ner s ales (s ale 2 and 3) this is no longer the

ase. As examples, let us onsider ele trode J and F, at s ale 3, in order to ompare ases where there is

26

per olation (F) or not (J). At these s ales, the ele trodes are omposed by three phases, the LFP a tive

material, the PVdF/CB mixture and the porosity. Only the Voigt and Reuss bounds and the SC estimate

an be extended to a three phase material, the Hashin and Shtrikman bounds are thus not al ulated in

the following. From these analyti al tools, we try to predi t the e�e tive volume ondu tivity based on the

phase properties we determined earlier (i.e. 0.38S.m

−1for the LFP, 250S.m

−1for the PVdF/CB mixture

and 0S.m

−1for the porosity). Results are gathered in Table 7 with the FFT predi tions and experimental

values to be ompared with. As one an see, results show not so good agreement for ele trode J and really

bad agreement for ele trode F. This an simply be explained by the fa t that real mi rostru tures no longer

�t, at these �ner s ales, into the assumptions that underlie analyti al models.

Table 7: Analyti al and FFT predi tion results or experimental measurements for the e�e tive volume

ondu tivity at s ale 3 for ele trodes J and F.

Sample

Voigt

(S.m

−1)

Reuss

(S.m

−1)

SC model

(S.m

−1)

FFT

simulation

(S.m

−1)

Experimental

measurements

(S.m

−1)

F 46.4 0 0.29 5.51 �

J 12 0 0.24 0.3

0.3

6 Con lusion

In this work, we have used an FFT based numeri al method oupled with experimental measurements, to

hara terise the in�uen e of an ele trode mi rostru ture on its ele troni ondu tion properties. As shown

here, one may predi t the ele troni ondu tivity of battery omposite ele trodes through FFT simulations

by using a ombination of X-Ray and FIB/SEM segmented volumes. The a ura y of this te hnique depends

on the di�erent s ales in the mi rostru ture. Based on the experimental measurement of an ele trode on-

du tivity and its segmented X-Ray and FIB/SEM volume, it was possible to ba k- al ulate the ondu tivity

of the PVdF/CB mixture and the LFP phase. Considering the variability of the mi rostru ture, the volumes

analysed using FIB/SEM are not fully representative but the values obtained are in good agreement with the

experimental measurements made on these materials taken separately. This suggests the CB/PVdF spatial

distribution at the FIB/SEM small s ale is a tually representative of the spatial distribution of this ondu t-

ing phase at the more global s ales (like X-Ray tomography). The opposite approa h is also possible, namely

to predi t the ondu tivity of an ele trode starting from its segmented FIB/SEM volume and experimental

27

measurements of the ondu tivities of the various materials that ompose it.

This method allows one to predi t lo al urrent paths in the mi rostru ture. It is however ru ial to ad-

dress the representativity of the mi rostru ture as the numeri al results are based on it. In this regard, tests

to hara terise the representativeness of the volumes were presented. A ording to our statisti al analysis,

the FIB/SEM simulation volumes used are too small to be fully representative. Nevertheless, even with a

rough pre ision, FFT predi tions are in good agreement with experimental values.

Analyti al bounds and the lassi self- onsistent estimate also give a urate results for the e�e tive prop-

erties at the XRCT s ale. These tools may then be of some use to monitor the a ura y of numeri al

simulations or to simply a ess or assess some values in a multi-s ale approa h. However, at �ner s ales and

for more omplexe mi rostru tures, these methods have shown their limitations.

This study opens several new perspe tives, su h as the al ulation of the e�e tive ondu tivity of an ele -

trode from its real mi rostru ture and the ondu tivity of its di�erent phases, in order to better understand

the e�e t of the ele trode formulation, or of various me hani al or physi o- hemi al stimuli applied to the

ele trode during its manufa ture or battery operation.

A knowledgement

We are grateful to the ANR for the funding of the Pepite proje t (ANR-15-CE05-0001).

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