Egon Heuson

Franck Charmantray

Jean-Louis Petit

Véronique De Berardinis

Thierry Gefflaut
email: thierry.gefflaut@uca.fr

Enantioselective Synthesis of D-and L-Amino Acids by Enzymatic Transamination Using Glutamine as Smart Amine Donor

Keywords: Amination, Amino acids, Biocatalysis, Transferases

Enzymatic transamination is a useful method for the green and highly enantioselective synthesis of chiral amines and non-canonical amino acids which are of major importance as intermediates in medicinal chemistry. However, transamination reactions are usually reversible and synthetic applications of transaminases often require the implementation of an equilibrium shift strategy. Herein, we report a highly effective approach using glutamine as smart amine donor. This amino acid is converted upon transamination into 2-oxoglutaramate which undergoes a fast cyclisation displacing the transamination equilibrium.

We have developed a new activity assay in order to identify transaminases from biodiversity able to convert various αketo acids into valuable amino acids of L-or D-series in the presence of glutamine as amine donor. Discovered transaminases were then used to prepare in high yield and with high enantioselectivity three amino acids of pharmaceutical importance, homophenylalanine, homoalanine and tert-leucine by simply using a nearly stoichiometric amount of glutamine as amine donor.

Introduction

The tremendous structural diversity of biomolecules notably relies on the combination of 20 standard amino acids in proteins. Moreover, the chemists ability of preparing an unlimited variety of nonproteinogenic (also named "non-canonical" or "unusual") amino acids still expand the chemical space of peptides and peptidomimetics which are of increasing importance in modern drug discovery. [1] Among the numerous synthetic routes towards amino acids, [START_REF] Blaskovich | Handbook on Syntheses of Amino Acids: General Routes to Amino Acids[END_REF] bioprocesses and biocatalytic methods benefit from the high enantioselectivity of enzymes while meeting the criteria of green and sustainable chemistry. [3] Among various enzyme families used for amino acids synthesis, transaminases (TA) have been extensively studied in the large-scale preparation of unusual L-or D-α-amino acids as well as a variety of chiral amines. [4] TA catalyze the pyridoxal 5'-phosphate-dependent reversible transfer of the amino group from an amine donor substrate onto an acceptor carbonyle substrate. The TA family is highly diverse, comprising many ubiquitous L-α-TA involved in proteinogenic L-amino acids metabolism. Moreover, some microbial D-α-TA are specific for amino acids of the D-series whereas amine-TA (ATA also called ω-TA) are active with a variety of amines lacking the α-carboxylic acid moiety. [5] All L-or D-α-TA studied to date, proved to be highly enantioselective catalysts and most of them displayed relaxed substrate specificity. High turnovers and no requirement for external cofactor recycling are additional features making TA valuable biocatalysts to access a variety of unusual amino acids.

However, the major drawback of transamination reactions lies in their equilibrium constants, often close to unity and resulting in limited yields and purification issues. [6] To overcome these thermodynamic limitations, an equilibrium shift strategy is needed. It usually consists in the removal or transformation of the carbonyl coproduct formed from the amine donor substrate. One striking example is the removal under reduced pressure of acetone formed from isopropylamine used as amine donor for particular ATA. [7] This approach was notably adopted in a synthetic process for the antidiabetic drug sitagliptine. [8] However only few ATA are active with isopropylamine and this strategy is not applicable to α-TA which are highly specific for α-amino acids.

Alternatively, equilibrium shift was often achieved through auxiliary enzymatic reactions as exemplified by the lactic-dehydrogenase catalyzed reduction, [9] acetolactate-synthase catalyzed autocondensation [10] or pyruvate-decarboxylase catalyzed decarboxylation [11] of pyruvic acid (Pyr) formed from alanine (Ala) used as amine donor.

In a simpler way, a transamination reaction can be displaced through the use of smart amine donors. These so-called substrates are indeed converted upon transamination into unstable carbonyl coproducts that undergo a spontaneous transformation affording the needed equilibrium shift. For example, aspartic acid (Asp) or its analogue cysteine sulfinic acid (CSA) used as amine donors are converted into β-keto carboxylic or sulfinic acids which lose carbon or sulfur dioxide to give pyruvic acid (Pyr). This strategy was however limited to particular TA like Aspartate-TA or Tyrosine-TA, that are active with Asp or CSA as amine donor and unable to convert Pyr, formed from these substrates, into Ala. [12] Another class of smart amine donors is defined by compounds bearing an extra amino group able to react with the carbonyl generated upon transamination to give a cyclic imine. Therefore, the α,ε-diamino acid lysine was used as irreversible amine donor with a genetically modified AspTA to produce L-homophenylalanine (Hfe) [START_REF] Lo | [END_REF] whereas ornithine was used in an equilibrium shifted process using an ornithine-ω-TA as auxiliary enzyme. [14] More recently, several diamines proved efficient irreversible amine donors with various ATA. Indeed, the amino-carbonyl by-product formed from oxylylenediamine [15] and but-2-ene-1,4-diamine [16] are readily cyclized and subjected to further aromatization and polymerization. Interestingly, several bio-based diamine including putrescine, cadaverine and spermidine were efficiently used as smart amine donors with ATA. [17] Besides cyclic imine formation from diamino substrates, other transamination by-product formed from polyfunctional amine donors are prone to cyclisation. Noteworthy, reactions catalyzed by glutamine-TA (GlnTA, EC 2.6.1.15) yield 2oxoglutaramate (OGA) which was shown to exist almost exclusively (99,7%) in a cyclic hydroxylactame form [18] and some natural transaminations involving Gln as donor were reported to be displaced towards OGA formation. [18], [19] Herein, we report the design of a new activity assay for TA using L-or D-Gln as amine donor, the implementation of this assay to identify L-and D-α-TA from biodiversity active with Gln, and application examples of these newly discovered TA for the efficient synthesis of valuable unusual amino acids. Hence, we demonstrate that Gln can truly be considered as a smart amine donor for equilibriumshifted enzymatic transaminations.

Results and Discussion

New TA activity assay with Gln as amine donor.

Assuming that Gln could be a substrate for various TA from biodiversity, [20] we firstly developed a continuous spectrophotometric assay in a 96-well microplate format in order to precisely quantify transamination activities with Gln as amine donor and various keto acid precursors of amino acid targets. As shown in Scheme 1, our assay was based on 2 auxiliary enzymes: an ω-amidase from Arthrobacter nicotinovorans (ωA, Uniprot ID: Q93NG1) catalyzing the hydrolysis of OGA amide into 2oxoglutaric acid (OG) [21] and a 2-oxoglutarate reductase from Acidaminococcus fermentans (OGR, Uniprot ID: D2RJU7) which catalyzes the reduction of OG by NADH, thus allowing its spectrophotometric monitoring at 340 nm (ε = 6220 M -1 .cm -1 ). [22] One NADH-based assay of OGA was already described using ωA and glutamicdehydrogenase (GluDH). [23] However reductive amination catalyzed by GluDH requires the addition of ammonium ions which could impair TA activity. That is why OGR was preferred to GluDH for OG titration.

Scheme 1. Enzymatic assay for transaminase with Gln as amine donor.

Both auxiliary enzymes, ωA and OGR, were cloned and overexpressed in E. coli with addition of an His-Tag for easy purification (see table S1). Approximately 150,000 U of purified OGR were obtained from 1 L culture of the over-expressing strain. OGR proved to be highly active with a specific activity close to 1600 U.mg -1 . . In the case of ωA, a total activity of approximately 500 U was obtained from 1 L microbial culture, corresponding to a modest specific activity of 2.4 U.mg -1 . However, it should be pointed out that ωA activity was measured with OGR as auxiliary enzyme in non-optimal conditions, using a sample of OGA that was shown by 1 H NMR to be almost exclusively cyclized in aqueous solution. Considering that the acyclic form of OGA is the true substrate of ωA, [18] it is likely that the activity was underestimated in this assay and it was expected that ωA should display a much higher specific activity in the multienzymatic assay described in scheme 1 where OGA hydrolysis can compete efficiently with cyclisation. Moreover Both enzymes showed good stability and could be stored for few months without noticeable loss of activity.

In order to adjust the parameters and evaluate the performances of the new Gln-TA assay, we used as model enzyme, a D-amino acid-TA from Enterococcus italicus DSM 15952 (D-TA Ei , Uniprot ID : E6LHY8) which was previously identified as a highly active broad spectrum TA. [20], [24] The activity of D-TA Ei towards D-Gln (40 mM) was first evaluated in the presence of Pyr (10 mM) as acceptor substrate by monitoring Pyr consumption using lactic dehydrogenase (LDH). In these conditions, a specific activity of 45 U.mg -1 was measured. D-TA Ei (20 mU) was then combined with various amounts of ωA and OGR in the Gln-TA assay. We could observe that 0.4 U OGR and 40 mU ωA are sufficient to observe nonlimiting auxiliary reactions and these conditions were fixed for the following experiments. The apparently low needed amount of ωA confirmed the assumption that ωA specific activity measured with cyclic OGA was largely underestimated.

As shown in figure S1, the calibration curve using various amounts of D-TA Ei showed a broad dynamic range with a good linearity up to 20 mU/well and a calculated limit of detection of 0.3 mU/well. Finally, we considered that possible hydrolytic activity of ωA towards Gln could generate a false positive signal. Indeed, hydrolysis of Gln would give Glu which is a standard amine donor for most L-α-TA. In order to rule out this possibility, Branched-Chainaminotransferase from Escherichia coli (BCAT Ec ) was used as a negative control enzyme. BCAT Ec activity was measured in our assay with L-Gln as donor and its natural substrate 4-methyl-2oxopentanoic acid (MOP) as acceptor. At the same time, BCAT Ec activity was also measured in the same conditions with L-Glu as amine donor. Specific activities of 0.02 and 11.5 mg.mL -1 were measured with Gln and Glu respectively, confirming the very low activity of BCAT towards Gln and indicating that no false positive signal comes from Gln hydrolysis.

Hence, this new Gln-TA assay appeared well adapted to perform the screening of TA collections in order to highlight transamination activities with L-or D-Gln as amine donor. Screening of a TA collection. Screening experiments were performed on a set of 22 L-TA from various classes and 2 D-TA including D-TA Ei (see table S1) that were identified in previous experiments for their synthetic potential with various keto acid acceptors and Glu as standard amine donor. [20] A series of 9 keto acids were used as acceptor substrates, including precursors of canonical amino acids (glyoxylic acid (GL), Pyr, MOP, hydroxypyruvic acid (HPA) and phenylpyruvic acid (PhP)) as well as unusual amino acids (2-oxobutyric acid (OB), trimethylpyruvic acid (TMP), phenylglyoxylic acid (PhG) and 2-oxo-4phenylbutyric acid (OPB)). Recombinant E. coli strains overexpressing each of the 24 TA were cultured in 96-well microplates and cell-lysates were directly used in the Gln-TA assay. For each substrate, a cell lysate from host E coli without overexpressed TA was used as control in order to discard false positives coming from side reactions catalysed by the host-cell enzymes. The average protein content of cell lysates was 1 mg.mL -1 and 0.5-5 μg TA were used in each assay. In these conditions, the 2 D-TA and 5 L-TA among 22 showed activities > 0.1 U/mL of lysate with at least one keto acid substrate (table 1). The best activities in D-and L-series were measured with D-TA Ei and L-TA from Megasphaera elsdenii DSM 20460 (L-TA Me , Uniprot ID: G0VQA2) respectively. Although L-TA Me displayed lower activity levels than those of D-TA Ei , a broad substrate spectrum was evidenced for both enzymes.

GL

Pyr OB MOP TMP HPA PhG PhP OPB L-TA1 G0VQA2 0,10 0,12 0,15 0,19 0,03 0,01 0,18 0,09 0,06 L-TA2 B2JE86 0,04 0,15 0,13 0,15 0,01 0,15 0,06 0,03 L-TA3 B7WY27 0,03 0,02 0,03 L-TA4 DOMDHO 0,02 0,07 0,04 0,03 0,02 0,03 L-TA5 G8R8R4 0,11 0,03 0,01 L-TA6 E1VWL8 0,02 0,05 0,02 0,03 0,01 0,01 L-TA7 C1CZ11 0,02 0,04 0,06 0,02 0,03 0,02 L-TA8 Q1J263 0,01 0,03 L-TA9 Q9RRM7 0,02 0,07 0,04 0,03 0,02 0,03 L-TA10 C0GI63 0,05 0,06 0,01 0,05 0,01 L-TA11 C4LDS0 0,02 0,03 0,04 L-TA12 Q0AV15 0,01 L-TA13 Q0AT08

L-TA14 Q3SH06 0,08 0,08 0,01 0,01 0,06 0,01 L-TA15 B7X3V3 0,03 0,04 0,05 0,01 L-TA16 Q72ES5 0,03 0,06 0,05 0,01 0,02 L-TA17 E1Q1V7 0,02 0,01 0,01 0,01 L-TA18 A0A0H3K2P4 0,06 0,04 0,07 0,09 0,01 0,04 0,07 L-TA19 C5CIR2 0,02 L-TA20 A1HUC4 0,09 0,02 0,11 0,01 L-TA21 F6B3V8 0,03 0,17 0,01 0,09 0,01 0,01 L-TA22 F6CII1 0,01 0,08 0,03 0,02 D-TA1 P54694 0,50 0,78 0,60 0,04 0,01 0,25 0,07 0,18 29.9 ± 1.8 39 ± 2 1.3 ± 0.1 0.9 ± 0.1 In the case of D-TA Ei , activities ≥ 1 U.mL -1 were measured with all substrates except TMP and PhG. In the case of L-TA Me activities were always ≥ 0.1 U.mL - 1 except with TMP and HPA. However L-TA Me activity with TMP was significantly higher than the control indicating that L-TA Me could be used for Ltert-leucine (Tle) synthesis from TMP and Gln.

Kinetic parameters. In order to better characterize both enzymes and to confirm screening results, the Gln-TA assay was used with purified D-TA Ei and L-TA Me , to determine kinetic parameters of transamination reactions with Pyr, KB, TMP or OPB as acceptor and L-or D-Gln as amine donor (table 2).

Enzyme kinetics clearly confirmed the greater activity of D-TA Ei over L-TA Me due to a better affinity towards the amine donor D-Gln as well as a 10 to 75 fold increased k cat with the various substrates. The inactivity of D-TA Ei with TMP was confirmed whereas a modest activity was evidenced for L-TA Me with this substrate. Moreover, L-TA Me showed lower K M and higher k cat with all other acceptor substrates, and the best specific activity of this enzyme was measured with OPB.

In order to better characterize the substrate spectrum of both TA, activities with OPB and D/L-Gln were compared to those measured with D/L-Glu and D/L-Ala as standard amine donors. Results reported in figure 1 indicated that although D-TA Ei showed comparable activities with the 3 tested donors, L-TA Me displayed a marked preference for Glu with a ten-time activity reduction when Gln was used as amine donor. Nevertheless, L-TA Me was considered as a suitable enzyme to demonstrate the relevance of the equilibrium shift strategy using Gln as amine donor.

Synthetic experiments. As a first model reaction, the transamination reaction between Pyr and L-Gln was studied using L-TA Me as catalyst. In a first analytical experiment, Pyr (50 mM) was reacted with L-Gln at various concentrations (0-60 mM). Residual Pyr was titrated over time with LDH and NADH until equilibration (stabilization of Pyr concentration). In order to highlight the interest of Gln as amine donor, the same experiment was run using L-Glu as amine donor. Final Pyr concentrations measured at various initial donor/Pyr ratios are reported in Figure 2. As expected, we could observe a linear correlation between residual [Pyr] at equilibrium and the initial donor/Pyr ratio indicating an apparently complete reaction and therefore a very efficient equilibrium shift afforded by OGA cyclisation. Considering that the conversion obtained with one molar equivalent of Gln was above 98%, indicated an equilibrium constant over 2000 for the transamination reaction between Pyr and Gln. Conversely, when using a stoichiometric amount of Glu as amine donor, the conversion was less than 60%, thus indicating an equilibrium constant lower than 2.25. A larger-scale reaction between Pyr (50 mM, 0.7 mmol) and Gln (60 mM) was then run using L-TA Me (9.3 μM) as catalyst. In those conditions, the conversion rate reached 95% in less than 1.5 h and Pyr titration after 3 h indicated a completely displaced reaction (> 99% conversion). Amino acids were removed from the reaction mixture by selective adsorption on a sulfonic acid containing ion exchange resin and the lithium salt of OGA was quantitatively isolated after neutralization with lithium hydroxide and lyophilisation. NMR analyses showed that OGA was the only keto acid present in the reaction mixture and confirmed that it exists almost exclusively in its cyclic form in aqueous solution (see SI). It should be pointed out that this transamination reaction between Pyr and Gln thus constitutes a highly efficient method to prepare OGA, which is an important biomarker of hyperammonemia. [25] In light of these first encouraging results, L-TA Me and D-TA Ei were evaluated as catalysts for the synthesis of valuable unusual L-or D-amino acids using L-or D-Gln as amine donor (scheme 2). In accordance with the substrate spectra evaluated through screening and kinetic experiments, L-TA Me was assayed for the synthesis of L-homo-alanine (Hal), L-tert-leucine (Tle) and L-homophenylalanine (Hfe). These three amino acids are included in the peptidomimetic structures of widely used drugs [1] and their stereoselective synthesis has been the subject of many studies. [START_REF] Blaskovich | Handbook on Syntheses of Amino Acids: General Routes to Amino Acids[END_REF], [26] Furthermore, the potential of D-TA Ei was assessed through the synthesis of D-Hal and D-Hfe which also constitute important building blocks for biologically active compounds. [27] Synthesis experiments were run on 0.1 mmol scale and 100 mM Gln and 120 mM OB or TMP were first chosen as standard substrate concentrations. A slight excess of acceptor was used in order to achieve a complete conversion of Gln allowing the isolation of pure amino acid products by selective adsorption on a cation exchange resin. Enzymes specific activities were measured in these conditions in order to address possible inhibition at high substrate concentrations. In the case of D-TA Ei , the results were very close from those reported in table 1 (36 and 28 U.mg -1 for OB and OPB respectively). Specific activities of L-TA Me with 120 mM OB or TMP and 100 mM L-Gln (1,8 and 0.7 U.mg -1 respectively) were found higher than those measured previously at 40 mM L-Gln (table 1). These results were in accordance with the high K M value determined for L-Gln and L-TA Me (table 1) and indicated that no significant substrate inhibition was observed for L-TA Me with these 2 substrates. In the case of the lithium salt of OPB, a maximum solubility of 14 mM was determined in water at pH 7.6. Consequently, reactions were run in heterogeneous conditions with a suspension of OPB (18.4 g.L -1 , theoretically 100 mM). As Hfe is even less soluble in water (maximum solubility < 3 mM), it was simply isolated by filtration at the end of the reaction. For that reason, a slight excess of Gln (120 mM) was used in that case to ensure a complete conversion of OPB. Syntheses results are summarized in table 3. Amino acids of L-or D-series were isolated with excellent enantiopurity, confirming again that TA are highly enantioselective catalysts. All reactions showed conversions over 90%, clearly demonstrating that using Gln as amine donor is an effective and Scheme 2. Synthesis of unsual L-and D-amino acids using L-or D-Gln as amine donor. general method to shift the equilibrium of transamination reactions.

It might be argued however that Gln is maybe not commonly accepted as a good substrate by L-α-TA and that finding one enzyme showing a good activity for both Gln and the keto acid precursor of a valuable target might be a cumbersome task. Nevertheless, a Gln-TA could also be used as an auxiliary enzyme to shift the equilibrium of another transamination reaction catalyzed by an α-TA or an Amine-TA devoid of activity towards Gln. In this bienzymatic system, Ala or Glu could be used as standard primary amine donor ensuring the coupling between both transamination reactions. In order to illustrate this approach, we designed the new synthetic process of L-Tle described in scheme 3, by coupling BCAT Ec and a previously assayed serine-glyoxylate-TA from Thermosinus carboxydivorans Nor1 (L-TA Tc , Uniprot ID: A1HUC4). BCAT Ec was previously used for Tle synthesis [14], [28] whereas thermostable L-TA Tc was recently combined with a transketolase for in situ production of HPA from L-Serine. [29] This TA couple was chosen because although neither of the two TA showed any activity with TMP and Gln (< 0.02 U.mg - 1 ), BCAT Ec proved active with TMP and Glu (1.1 U.mg -1 ) whereas L-TA Tc showed significant activity with Gln or Glu using Pyr as acceptor (0.6 or 1.1 U.mg -1 respectively). The process of scheme 3 was implemented on 0.1 mmol scale using TMP (110 mM), Gln (100 mM), Glu (10 mM), and 100 μM of each TA. Under these conditions, a conversion of 94% was reached in 48 h, thus demonstrating the applicability of this Gln-TA coupled approach.

Conclusion

In summary, we have developed an efficient spectrophotometric assay, which is suitable not only for screening enzyme collections but also for kinetic studies of TA which are active with L-or D-Gln as amine donor. The screening of a small collection of 24 TA highlighted several L-or D-TA showing significant activity with L-or D-Gln. The 2 most active enzymes in L-and D-series were used for high yielding and enantioselective synthesis of three valuable unusual amino acids, thus demonstrating that Gln, a readily available and bio-based substrate, can be used as a smart amine donor in order to shift the equilibrium of transamination reactions. Further large scale screening experiments will undoubtedly allow the discovery of highly active and robust new Gln-TA for use as catalyst or auxiliary enzyme in large scale and highly enantioselective syntheses of a variety of chiral amines.

Experimental Section

General procedure for amino acid synthesis using L-TA Me or D-TA Ei . Reactions were carried out in Eppendorf tubes (1,5 mL). To a 1 mL aqueous solution containing L-or D-Gln (0.1 or 0.12 mmol), OPB (0.1 mmol), OB (0.12 mmol) or TMP (0.12 mmol) and PLP (0.1 μmol) at pH 8 was added L-TA Me (6.5-13 nmol) or D-TA Ei (0.84 nmol): a suspension of the TA in 3 M (NH 4 ) 2 SO 4 was centrifuged (14000 rpm, 5 min.), the supernatant was discarded and the pellet diluted with the solution of substrates and cofactor. The reaction mixture was orbitally stirred at room temperature and controlled by thin layer chromatography. After 7 to 48 h, L-or D-Hfe was isolated by filtration on a 0.22 μm membrane. The white solid was washed with H 2 O (4x0.5 mL) and dried under vacuum (0.5 mm Hg) for 24 h at 40 °C. In the case of Hal and Tle, the reaction mixture was poured on a small column of dowex® 50WX8 (H + form, 1 mL). The column was washed with H 2 O (10 mL) before elution with 1 M NH 3 . Ninhydrin positive fractions were pooled and concentrated under reduced pressure. Amino acids were isolated as white solids. Conversions, chemical and enantiomeric purity were assessed by NMR and HPLC analyses (See SI).

Table 1 .

 1 Transamination activities (U.mL -1 ) of 22 L-TA and 2 D-TA a) measured in cell lysates with L-or D-Gln as amine donor and various acceptor substrates b) .a) Activities of L-TA and D-TA are highlighted in blue and red respectively and are given after subtraction of the control activities measured for each substrate without overexpressed TA. b) Glyoxylic acid (GL), pyruvic acid (Pyr), 2-oxobutyric acid (OB), 4-methyl-2-oxopentanoic acid (MOP), trimethylpyruvic acid (TMP), hydroxypyruvic acid (HPA), phenylglyoxylic acid (PhG), phenylpyruvic acid (PhP), 4-phenyl-2-oxobutanoic acid (OPB).

  concentrations of D-or L-Gln were used in combination with 5 mM Pyr. b) variable concentrations of keto acid were used in combination with 40 mM D-or L-Gln.

Figure 1 .

 1 Figure 1. Activities of D-TA Ei and L-TA Me with OPB (1 mM) and various amine donors (40 mM): OGR or LDH were used as auxiliary enzymes for the reduction by NADH of OG or Pyr formed from D/L-Glu and D/L-Ala respectively.

Figure 2 .

 2 Figure 2. Final concentrations of Pyr measured in transaminations with various initial [donor]/[Pyr] ratios.

Table 2 .

 2 Apparent kinetic parameters of transaminations catalysed by D-TA Ei and L-TA Me

	TA	Substrate	K M (mM)	k cat (min -1 )	k cat / K M (min -1 .mM -1 )	Specific activity (U.mg -1 )
	D-TA Ei	D-Gln	a)	9.0 ± 0.1	1904 ± 28	211 ± 5	57 ± 1
		Pyr b)		0.9 ± 0.1	1631 ± 16	1812 ± 219	49 ± 1
		OB b)		0.6 ± 0.1	1266 ± 14	2110 ± 375	38 ± 1
		OPB b)	0.5 ± 0.1	836 ± 31	1672 ± 396	25 ± 1
	L-TA Me	L-Gln	a)

Table 3 .

 3 Synthesis of unsual amino acids catalysed by L-TA Me and D-TA Ei with L-or D-Gln as amine donor Conversions of the limiting substrate were calculated by NMR analysis of reaction media and were always inferior or equal to yields calculated from isolated products. b) Enantiomeric excess determined by HPLC. Synthesis of L-Tle by coupling BCAT Ec with L-TA Tc using Gln as smart amine donor

	TA (μM)	Acceptor (mM)	Donor (mM)	Time (h)	Product	Conversion a) (%)	ee b) (%)
	L-TA Me (65)	OB (120)	L-Gln (100)	24	L-Hal	98	> 99
	L-TA Me (130)	TMP (120)	L-Gln (100)	48	L-Tle	92	> 99
	L-TA Me (90)	OPB (100)	L-Gln (120)	24	L-Hfe	93	> 99
	D-TA Ei (8.4)	OB (120)	D-Gln (100)	7	D-Hal	99	> 99
	D-TA Ei (8.4)	OPB (100)	D-Gln (120)	24	D-Hfe	96	> 99
	a)

Acknowledgements

We thank the French National Center for Scientific Research (CNRS) and University Clermont-Auvergne for financial support. We also thank Martine Sancelme for the enzyme production and Aurélie Job for HPLC analyses.

FULL PAPER Enantioselective Synthesis of D-and L-Amino Acids by Enzymatic Transamination Using Glutamine as Smart Amine Donor

Adv. Synth. Catal. Year, Volume, Page -Page Egon Heuson, Franck Charmantray, Jean-Louis Petit, Véronique de Berardinis, and Thierry Gefflaut a *


E Lestriez
email: bernard.lestriez@cnrs-imn.fr

Maire

F Cadiou

T Douillard

F Willot

J-C Badot

B Lestriez

E Maire

Effective Electronic and Ionic Conductivities of Dense EV-Designed NMC-Based Positive Electrodes

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Introduction

The electronic and ionic transport properties of battery electrodes significantly influence their electrochemical behaviour and performance [START_REF] Robert | Multiscale phase mapping of LiFePO 4 -based electrodes by transmission electron microscopy and electron forward scattering diffraction[END_REF][START_REF] Katayama | X-ray absorption fine structure imaging of inhomogeneous electrode reaction in LiFePO 4 lithium-ion battery cathode[END_REF][START_REF] Zhang | Direct observation of li-ion transport in electrodes under nonequilibrium conditions using neutron depth profiling[END_REF][START_REF] Orikasa | Ionic conduction in lithium ion battery composite electrode governs crosssectional reaction distribution[END_REF][START_REF] Liu | Quantifying reaction and rate heterogeneity in battery electrodes in 3d through operando x-ray diffraction computed tomography[END_REF]. These properties strongly depend on the electrode compositions and their microstructures. Having a good knowledge and understanding of the relationships between these properties and these electrode parameters is very useful for experimenters and engineers for a good interpretation of the electrochemical performance limitations of their electrodes [START_REF] Grillet | Conductivity degradation of polyvinylidene fluoride composite binder during cycling: measurements and simulations for lithium-ion batteries[END_REF][START_REF] Ogihara | Impedance spectroscopy characterization of porous electrodes under different electrode thickness using a symmetric cell for highperformance lithium-ion batteries[END_REF][START_REF] Besnard | Multiscale morphological and electrical characterization of charge transport limitations to the power performance of positive electrode blends for lithium-ion batteries[END_REF]. It is also useful for modellers who, in a complementary way, develop electrochemical models which make it possible moreover to carry out predictive calculations and to specify, at lower cost, the design of more powerful electrodes and batteries [START_REF] Doyle | Modeling of galvanostatic charge and discharge of the lithium/polymer/insertion cell[END_REF][START_REF] Fuller | Simulation and optimization of the dual lithium ion insertion cell[END_REF][START_REF] Higa | Comparing macroscale and microscale simulations of porous battery electrodes[END_REF].

The measurement of the electronic and ionic transport properties of composite electrodes is not trivial.

While it is easy to determine the overall electronic conductivity of the electrode film, quantitatively relating this macroscopic value to the microstructure and to the electrode composition actually requires being able to measure the electronic transport properties at different scales of the electrode to identify the most limiting factors [START_REF] Badot | A multiscale description of the electronic transport within the hierarchical architecture of a composite electrode for lithium batteries[END_REF][START_REF] Seid | Multiscale electronic transport mechanism and true conductivities in amorphous carbon-LiFePO 4 nanocomposites[END_REF][START_REF] Seid | Multiscale electronic transport in Li 1+x Ni 1/3-u Co 1/3-v Mn 1/3-w O 2 : a broadband dielectric study from 40 Hz to 10 GHz[END_REF]. To our knowledge, only broadband dielectric spectroscopy is capable of reaching this level of information, but it is a complex and very uncommon technique [START_REF] Badot | Interest in broadband dielectric spectroscopy to study the electronic transport in materials for lithium batteries[END_REF]. The measurement of ionic conductivity also requires specific experimental approaches, and the results which have been acquired in the few studies on this subject are invaluable [START_REF] Orikasa | Ionic conduction in lithium ion battery composite electrode governs crosssectional reaction distribution[END_REF][START_REF] Ogihara | Impedance spectroscopy characterization of porous electrodes under different electrode thickness using a symmetric cell for highperformance lithium-ion batteries[END_REF][START_REF] Thorat | Quantifying tortuosity in porous li-ion battery materials[END_REF][START_REF] Zacharias | Direct measurements of effective ionic transport in porous Li-ion electrodes[END_REF][START_REF] Seid | An in situ multiscale study of ion and electron motion in a lithium-ion battery composite electrode[END_REF][START_REF] Landesfeind | Tortuosity determination of battery electrodes and separators by impedance spectroscopy[END_REF][START_REF] Pouraghajan | Quantifying tortuosity of porous li-ion battery electrodes: Comparing polarization-interrupt and blocking-electrolyte methods[END_REF].

The measurement interpretations of the electronic and ion transport properties of composite electrodes has greatly benefited in recent years from the development of characterisation techniques, by XR and FIB/SEM tomography, of the microstructure of the electrodes [START_REF] Stephenson | Modeling 3d microstructure and ion transport in porous li-ion battery electrodes[END_REF][START_REF] Inoue | Numerical and experimental evaluation of the relationship between porous electrode structure and effective conductivity of ions and electrons in lithium-ion batteries[END_REF]. It also greatly benefits from the development of numerical simulations. Initially carried out on simplified virtual microstructures [START_REF] Chen | Selection of conductive additives in Li-ion battery cathodes[END_REF][START_REF] Awarke | Percolation-tunneling modeling for the study of the electric conductivity in LiFePO4 based Li-ion battery cathodes[END_REF], they have recently been boosted by the development of aforementioned characterisation techniques, which offer the possibility of carrying out these simulations on real or semi-real microstructures. The FIB/SEM resolution makes it possible to distinguish with good precision all the phases present (the active material (AM), the porosity, and the mixture formed by the binder and the conductive carbon), but on a limited volume which may not be representative of the corresponding electrode [START_REF] Grillet | Conductivity degradation of polyvinylidene fluoride composite binder during cycling: measurements and simulations for lithium-ion batteries[END_REF][START_REF] Hutzenlaub | Threedimensional electrochemical Li-ion battery modelling featuring a focused ion-beam/scanning electron microscopy based three-phase reconstruction of a LiCoO 2 cathode[END_REF][START_REF] Mendoza | Mechanical and electrochemical response of a LiCoO 2 cathode using reconstructed microstructures[END_REF]. In order to perform simulations on larger and therefore more representative electrode volume, it is possible to consider microstructures acquired by XR tomography, which however hardly distinguish the porosity from the carbon and binder domains (CBD). Thus the pore/CBD mesostructure must be computationally generated in the empty space of the AM skeleton, either from FIB/SEM data [START_REF] Zielke | Three-phase multiscale modeling of a LiCoO 2 cathode: Combining the advantages of FIB-SEM imaging and X-ray tomography[END_REF][START_REF] Usseglio-Viretta | Li-ion battery electrodes through micro-macro modeling and experiment[END_REF][START_REF] Wieser | Multiscale simulation process and application to additives in porous composite battery electrodes[END_REF], or by various mathematical models to reproduce the various morphologies of the CBD observed experimentally [START_REF] Trembacki | Editors-choice-mesoscale analysis of conductive binder domain morphology in lithium-ion battery electrodes[END_REF][START_REF] Ferraro | Electrode mesoscale as a collection of particles: Coupled electrochemical and mechanical analysis of NMC cathodes[END_REF]. The generation of entirely virtual microstructure remains used for the simulation of the transport properties because this makes it possible for example to simulate the influence of the compactness of the electrodes or of the physico-chemical processes during the implementation of the electrode slurries [START_REF] Mistry | Secondary-phase stochastics in lithium-ion battery electrodes[END_REF][START_REF] Laue | Joint structural and electrochemical modeling: Impact of porosity on lithium-ion battery performance[END_REF][START_REF] Ngandjong | Multiscale simulation platform linking lithium ion battery electrode fabrication process with performance at the cell level[END_REF].

Several studies have investigated how virtual microstructure models, coupled with numerical homogenisation tools, may be used to simulate the microstructure and transport properties of porous materials similar to Li-ion electrodes such as fuel cell materials and more particularly their anode layers (see e.g. [START_REF] Neumann | Stochastic 3D modeling of three-phase microstructures for the prediction of transport properties in solid oxide fuel cells[END_REF][START_REF] Gaiselmann | Quantitative relationships between microstructure and effective transport properties based on virtual materials testing[END_REF][START_REF] Abdallah | Morphological modeling of three-phase microstructures of anode layers using SEM images[END_REF]).

The use of mathematically-rigorous random structure models allows one to quantify and decorrelate geometrical features of interest, and assess their effect on transport properties. Neumann et al. [START_REF] Neumann | Stochastic 3D modeling of three-phase microstructures for the prediction of transport properties in solid oxide fuel cells[END_REF] highlighted the role of phase connectivity and "constrictivity", and that of "dead-end" branches on the performance of materials with complex, random microstructures (see also [START_REF] Nguyen | The electrode tortuosity factor: why the conventional tortuosity factor is not well suited for quantifying transport in porous li-ion battery electrodes and what to use instead[END_REF]). Such studies suggest how microstructures may be tailored for increased performance, based on numerical computations, carried out on tomography images and numerical models. Nevertheless they do not address whether tomography images are sufficient to predict the behaviour of multiscale materials (we refer to [START_REF] Geers | Multiscale modeling of microstructure-property relations[END_REF] for a review of similar question in mechanics).

The simulated effective transport properties can be used as inputs for electrochemical modelling to improve the prediction accuracy of models regarding electrode microstructure effects and in particular the tortuosity coefficient [START_REF] Thorat | Quantifying tortuosity in porous li-ion battery materials[END_REF][START_REF] Laue | Joint structural and electrochemical modeling: Impact of porosity on lithium-ion battery performance[END_REF][START_REF] Tjaden | Tortuosity in electrochemical devices: a review of calculation approaches[END_REF]. This way, the most recent and advanced approaches consist in performing electrochemical modelling directly in these microstructures [START_REF] Hutzenlaub | Threedimensional electrochemical Li-ion battery modelling featuring a focused ion-beam/scanning electron microscopy based three-phase reconstruction of a LiCoO 2 cathode[END_REF][START_REF] Wieser | Multiscale simulation process and application to additives in porous composite battery electrodes[END_REF][START_REF] Ferraro | Electrode mesoscale as a collection of particles: Coupled electrochemical and mechanical analysis of NMC cathodes[END_REF][START_REF] Wu | Three-dimensional finite element study on stress generation in synchrotron x-ray tomography reconstructed nickel-manganese-cobalt based half cell[END_REF][START_REF] Danner | Thick electrodes for Li-ion batteries: A model based analysis[END_REF]. Beyond predicting macroscale quantities such as half-cell voltage or charge/discharge capacity, it can provide a spatially mesoscale (particlescale) resolved description of the electrochemical reaction intensity. This modeling at the mesoscopic scale is extremely rich in new information, which is however difficult to verify because it requires for this purpose the use of operando and/or spatially resolved techniques for characterising localised electrochemical phenomena [START_REF] Robert | Multiscale phase mapping of LiFePO 4 -based electrodes by transmission electron microscopy and electron forward scattering diffraction[END_REF][START_REF] Katayama | X-ray absorption fine structure imaging of inhomogeneous electrode reaction in LiFePO 4 lithium-ion battery cathode[END_REF][START_REF] Zhang | Direct observation of li-ion transport in electrodes under nonequilibrium conditions using neutron depth profiling[END_REF][START_REF] Orikasa | Ionic conduction in lithium ion battery composite electrode governs crosssectional reaction distribution[END_REF][START_REF] Liu | Quantifying reaction and rate heterogeneity in battery electrodes in 3d through operando x-ray diffraction computed tomography[END_REF][START_REF] Harris | Direct in situ measurements of Li transport in Li-ion battery negative electrodes[END_REF][START_REF] Liu | Visualization of charge distribution in a lithium battery electrode[END_REF][START_REF] Roberts | Direct observation of active material concentration gradients and crystallinity breakdown in LiFePO 4 electrodes during charge/discharge cycling of lithium batteries[END_REF][START_REF] Kitada | Factors determining the packing-limitation of active materials in the composite electrode of lithium-ion batteries[END_REF][START_REF] Nakamura | Visualization of inhomogeneous reaction distribution in the model LiCoO 2 composite electrode of lithium ion batteries[END_REF].

In the present work, experimental conductivity measurements, obtained on NMC 532 -based electrodes with markedly different connectivity properties, made of percolating and non-percolating solid phases, are compared with full-field numerical predictions. Our numerical predictions are based on segmented nanotomography images and phase bulk properties and contain no tunable parameter. The trade-off between representativity and resolution is investigated, in an effort to quantify the reliability and accuracy of numerical predictions. A good agreement between the calculated and measured transport properties is observed.

The analysis of the current density distribution in the electrode microstructures allows us to deepen our understanding of the influence of this microstructure. Two results can be highlighted. For electronic transport, minimisation of contact resistances between NMC clusters thanks to CBDs which form gateways between these clusters, even in the absence of percolation of the CB/PVdF phase. For ionic transport, the existence of dead zones in the porosity which participate very little in ionic conduction.

Electrode materials and electrical measurements

The electrodes studied here have an EV-design. They are fairly dense (porosity is lower than 30 %) and contain a low amount of CB and PVdF additives (total is lower than 6 % by weight). Their microstructures were characterised and digitised in a previous work where the imaging conditions and segmentation process are detailed [START_REF] Cadiou | Multiscale characterisation of composite electrode microstructures for high density lithium-ion batteries guided by the specificities of their electronic and ionic transport mechanisms[END_REF]. Table 1 gives the electrodes mass and volume composition, thickness and density. NA3 and NA2 electrodes have similar composition and density (porosity) at the electrode scale as they only differ by their thickness. NA7 has a higher density and thus lower porosity than NA3 and NA2. NA2b and NA7b contain more CB/PVdF than NA3, NA2 and NA7. NA7b has a higher density and thus lower porosity than NA2b. The NMC volume fractions are close for NA3, NA2 and NA2b on the one hand, and for NA7 and NA7b on the other hand. As a consequence of their higher CB/PVdF content, NA2b has lower porosity than NA3 and NA2, and NA7b than NA7. Table 2 gives the corresponding FIB/SEM volumes compositions and densities.

The electrical properties were acquired with Broadband dielectric spectroscopy (BDS). The theoretical background and experimental setup to implement this technique are well described in previous papers.

Electronic conductivity measurements are made on dry electrodes [START_REF] Badot | A multiscale description of the electronic transport within the hierarchical architecture of a composite electrode for lithium batteries[END_REF] or active material pellet [START_REF] Seid | Multiscale electronic transport mechanism and true conductivities in amorphous carbon-LiFePO 4 nanocomposites[END_REF][START_REF] Seid | Multiscale electronic transport in Li 1+x Ni 1/3-u Co 1/3-v Mn 1/3-w O 2 : a broadband dielectric study from 40 Hz to 10 GHz[END_REF][START_REF] Badot | Interest in broadband dielectric spectroscopy to study the electronic transport in materials for lithium batteries[END_REF] with the classic ex-situ cell, while ionic conductivity measurements are made from the same materials but soaked in the electrolyte with the in-situ cell [START_REF] Seid | An in situ multiscale study of ion and electron motion in a lithium-ion battery composite electrode[END_REF].

Fast Fourier Transform numerical simulation method

The simulations were conducted using a Fourier-based method. The starting point of these methods is an integral equation involving a convolution product of the local fields (e.g. the strain tensor in mechanics) with the Green operator associated to a homogeneous reference medium [START_REF] Moulinec | A fast numerical method for computing the linear and nonlinear properties of composites[END_REF], and referred to as the "Lippmann-Schwinger" equation by analogy with scattering theory [START_REF] Kröner | Statistical continuum mechanics[END_REF]. In linear conductivity problems, the unknown is the local electric field E(x), expressed as the sum of a mean applied field E and contributions at all points x = x, proportional to the deviation [σ(x) -σ 0 ]E(x ) of the local current field with respect to the reference conductivity σ 0 :

E(x) = E -d d x G 0 (x -x ) • (σ(x) -σ 0 )E(x ), (1)
and G 0 is the Green operator associated to σ 0 with symbolically G 0 = graddiv/(σ 0 div•grad) [START_REF] Willot | Fourier-based schemes with modified Green operator for computing the electrical response of heterogeneous media with accurate local fields[END_REF]. The convolution product is advantageously computed in Fourier space, whereas the local constitutive laws are evaluated in the direct space, in regularly-spaced voxel grids [START_REF] Eyre | A fast numerical scheme for computing the response of composites using grid refinement[END_REF] allowing for the use of complex microstructure derived from microtomography images [START_REF] Cadiou | Numerical prediction of multiscale electronic conductivity of lithium-ion battery positive electrodes[END_REF]. Recent works have focused on reinterpreting the Fourier-Lippmann-conductivity-type problems [START_REF] Zeman | A finite element perspective on nonlinear FFT-based micromechanical simulations[END_REF], establishing links between Fourier and Finite Element methods [START_REF] Schneider | An FFT-based fast gradient method for elastic and inelastic unit cell homogenization problems[END_REF][START_REF] Schneider | FFT-based homogenization for microstructures discretized by linear hexahedral elements[END_REF][START_REF] Vondřejc | Energy-based comparison between the Fourier-Galerkin method and the finite element method[END_REF],

and introducing new numerical schemes inspired by those employed in minimization problems such as the conjugate gradient or Barzilai and Borwein methods [START_REF] Schneider | On the barzilai-borwein basic scheme in FFT-based computational homogenization[END_REF], to name a few. Studies have highlighted the improvement achieved by Fourier methods in terms of computational and memory requirements over finite element techniques [START_REF] Gasnier | Thermoelastic properties of microcracked polycrystals. Part I: Adequacy of Fourier-based methods for cracked elastic bodies[END_REF][START_REF] Vondřejc | Energy-based comparison between the Fourier-Galerkin method and the finite element method[END_REF]. Fourier-based methods, it has been argued, benefit from a better conditioning of the linear system provided by the Lippmann-Schwinger equation. In certain situations, Finite element methods, however, require a lower number of degrees of freedom to achieve comparable accuracy [START_REF] Vondřejc | Energy-based comparison between the Fourier-Galerkin method and the finite element method[END_REF].

Fourier simulations are here performed using the homogenised Ohm's law. The derivation for the effective electric conductivity and the effective ionic conductivity are mathematically identical and based on the following equations:

div(J( - → x )) = 0, E( - → x ) = - -----→ grad(φ), J( - → x ) = σE( - → x ) (2)
Where φ is the local potential, E the electric field, J the current field and σ is the local conductivity at point -→ x . The effective electronic conductivity, σ ef f , is calculated by applying a voltage across the domain in the z direction (perpendicular to the current collector) and then normalising the mean current density by the applied mean electric field. Note that this approach assumes that transport across both CBD-AM interfaces and AM-AM particle contacts is ideal (no contact resistance). A calculation on a volume of around 10 9 voxels takes a few hours (on a standard computing station using 14 threads) depending on the microstructure and the input parameters.

An important point must be emphasised here. As in [START_REF] Gully | Effective transport properties of porous electrochemical materials -A homogenization approach[END_REF], but contrarily to many other works, we have simply used the ohm's law to describe the ionic current density. From the perspective of the solver, Ohm's law, is identical to steady state diffusion. However, during battery operation, the ionic current density within the pore space filled with electrolyte is not only governed by migration but also by diffusion and thus Ohm's law must be modified to take into account the effect of diffusivity and its dependence on the lithium/salt concentration [START_REF] Hutzenlaub | Threedimensional electrochemical Li-ion battery modelling featuring a focused ion-beam/scanning electron microscopy based three-phase reconstruction of a LiCoO 2 cathode[END_REF]. Using the simple ohm's law amounts to considering a situation without dynamic salt concentration gradient. Our simulation results appear however in good agreement with our measurements as well as with the main trends in the literature, as we will see. An important aspect in our work is the study of the geometric distribution of current densities in the microstructure. A key result is that the electrode effective porosity is reduced to its percolated network of micrometric pores from the point of view of ionic conductivity. This should also apply when a salt concentration gradient is established during battery operation.

Effective electronic conductivity simulation

This section focuses on electronic conductivity in the electrodes and hence the electrolyte conductivity will be considered equal to 0 Sm -1 . Table 2 gives the overall (sample) and local (FIB/SEM volume) compositions of the electrode materials studied here.

In order to calibrate the NMC phase conductivity for the Fourier based simulations we have used measurements made by BDS (Broadband Dielectric Spectroscopy) on a dry stacking made of NMC particles only (NA0a sample).

As the NMC clusters used in this stacking are the same as for the electrodes, it is then possible to draw direct comparisons between this stacking conductivity and the one of the NMC skeleton in the electrodes NAX and NAXb, as long as the volume fractions are comparable.

In order to estimate by reverse calculation the NMC phase conductivity, NA2 and NA3 FIB/SEM volumes were then numerically striped of their CB/PVdF to leave only NMC and porosity to mimic NA0a type microstructure (cf. Figure 1 for an example). Indeed, no 3D volume FIB/SEM acquisition has been performed on NA0a sample. In this one, NMC volume fraction is 65 % while it is 70.5 % and 61 % respectively for NA2 and NA3, which frames the NA0a one. These modified volumes are called NA2-0 and NA3-0 hereafter. Fourier based simulations are then performed on a binary microstructure of NMC (conductive) and porosity (insulating) using equations 2.

NA0a conductivity at its different scales was measured by BDS. The sample conductivity was determined at 0.00136 Sm -1 , the cluster one at 0.48 Sm -1 and the grain one at 120 Sm -1 . The conductivity drop from grain to cluster scale is due to the resistive grain boundaries, while the drop from cluster to sample scale is due to the constriction and contact resistances between the clusters. Several simulations were then conducted by varying the NMC phase conductivity until one value was found to give the right effective conductivity (equal to the experimental one) for both volumes. We respectively obtained an NMC phase conductivity of 0.00417 Sm -1 and 0.0068 Sm -1 for the microstructures from NA2-0 and NA3-0. This discrepancy can be explained by the difference in NMC volume fraction between the two microstructures.

In fact, NA3-0 has a lower NMC volume fraction than NA2-0 (respectively 61 and 70.4 %) which implies that its NMC conductivity must be higher than in NA2-0 for the two volumes to show the same effective conductivity. Furthermore, both values are supposed to frame the one in NA0a as its NMC volume fraction is intermediate. By using a simple linear interpolation we could estimate the NMC phase conductivity in NA0a to be 0.0057 Sm -1 for its 65 % NMC volume fraction.

This NMC phase conductivity shows a quite different value than the one measured experimentally by BDS at the NMC cluster scale which is two orders of magnitude higher (0.48 Sm -1 ). It is thus worth mentioning that the numerically determined NMC phase conductivity considers here the NMC phase as a continuous skeleton and is then homogenised. This means the resistive contacts between NMC clusters are directly integrated in the phase and then not geometrically discriminated. This value of 0.0057 Sm -1 is then the consequence of this homogenisation and of geometrical features such as phase intraconnectivity, volume fraction and tortuosity.

To explain further this discrepancy between experimental and numerical results spotted in the previous paragraph we can stress more the fact that they are not directly comparable. Contact areas are indeed present between NMC clusters. Some of them can be very narrow thus creating a so called bottleneck effect.

Above all, these contact areas are sources of discontinuities at the nanometer scale, such as insulating air gaps, that act as energy barriers to the electron transfer between two touching clusters. Moreover, the structure at the NMC cluster surface itself can also act as a conductivity barrier if, as an example, the outer shell has not exactly the same composition as the cluster "core" [START_REF] Badot | Dielectric and conductivity spectroscopy of Li 1-x Ni 1+x O 2 in the range of 10-10 10 Hz: polaron hopping[END_REF]. Such kind of data are not accessible from the FIB/SEM volumes due to the limited resolution and the nature of the acquired information. To image these defects at the electron hopping phenomenon scale we should be able to image chemical maps at a 1 nm size resolution as this phenomenon is typically at the range of a few nanometers. It then makes sense that the NMC conductivity numerically measured is the NMC skeleton one, thus including contact resistances between clusters, and not the NMC cluster one. To sum up, such numerical conductivity estimation leads to a "pragmatic" value corresponding to an homogenised NMC network. Once the question of the NMC phase conductivity is solved, it is possible to compute the effective electronic conductivity in the FIB/SEM volumes from the real electrode microstructures. To do so, we used a CB/PVdF phase conductivity of 350 Sm -1 as experimentally determined in [START_REF] Zheng | Cooperation between active material, polymeric binder and conductive carbon additive in lithium ion battery cathode[END_REF]. The use of such experimental value is justified in another work [START_REF] Cadiou | Numerical prediction of multiscale electronic conductivity of lithium-ion battery positive electrodes[END_REF] for a different CB/PVdF binder composition hence the difference in conductivity between this work and the previous one [START_REF] Cadiou | Numerical prediction of multiscale electronic conductivity of lithium-ion battery positive electrodes[END_REF].

Simulation results are gathered in Table 3 alongside experimental measurements made by BDS. The data ranges in the experimental values in Table 3 is the result of several measurements. Numerical values show overall a good agreement with the experimental ones except for the NA7 sample which exhibits considerably higher simulated effective conductivity. This difference can be explained by the presence of a localised CB/PVdF percolation in the FIB/SEM volume in the simulation direction (see Figure S1), while experimental measurement indicates that the CB/PVdF phase does not percolate in the whole electrode.

This is explainable by a statistical variation in the captured microstructure due to the relatively small FIB/SEM volume size. Note that the electrodes contain a CB/PVdF volume fraction significantly higher than the one found in the FIB/SEM volumes (Table 2). This difference is due to the presence of some very bulky CB/PVdF agglomerates resulting from insufficient mixing, or from a lack of stability, in the ink used to manufacture the electrodes [START_REF] Cadiou | Multiscale characterisation of composite electrode microstructures for high density lithium-ion batteries guided by the specificities of their electronic and ionic transport mechanisms[END_REF]. These bulky CB/PVdF agglomerates are few in number and clearly identifiable in the electrode cross-sections and care has been taken to make the FIB/SEM acquisitions away from them. The good agreement between the values simulated from the FIB/SEM volumes and the measurements carried out on the electrodes confirm that these very bulky CB/PVdF agglomerates form islands, isolated from each other, inside the electrode and therefore contribute very little to the conductivity of the whole.

Once one has segmented volumes from actual microstructures, it is possible to numerically modify these volumes to assess the impact of microstructural / composition modifications on effective properties. Such modifications were here applied to the CB/PVdF phase particularly to assess the percolation effect.

Erosion and dilatation operations were simply applied to modify the CB/PVdF phase morphology. The simplest process was to increase the CB/PVdF volume fraction using dilatation operations. The CB/PVdF phase was isotropically dilated on several distances (1, 2 or 3 voxels) and then subsequently re-added to the unmodified NMC phase. We then obtained a microstructure similar to an electrode with the same NMC loading but a virtually increased CB/PVdF volume fraction. This CB/PVdF addition was only performed in the porosity and did not affect the NMC phase.

Mimicking a CB/PVdF volume fraction decrease was slightly more complicated as a simple erosion on the CB/PVdF phase automatically led to a complete contact loss with the NMC phase. We then had to erode the CB/PVdF phase only at its contacts with the porosity. These interfaces could be identified by dilating the porosity once and then performing a boolean "and" operation between the dilated porosity and the CB/PVdF phase which gave us the CB/PVdF surface in contact with the porosity. This interface could be subtracted from the CB/PVdF phase to decrease its volume fraction. By dilating the interface, the decrease process could be magnified to further erode the CB/PVdF phase and its volume fraction. By applying this method we could obtain microstructures with the same NMC loading but less CB/PVdF. Such method naturally erases the smallest CB/PVdF regions first. These methods, graphically summarised in Figure 2, have however the disadvantage to act on the CB/PVdF internal porosity by increasing it with the erosions and closing it with the dilatation process. This microstructural feature is then altered regarding what is observable in the raw FIB/SEM volumes. However, keeping it would lead to a more complex process and we can consider that this internal porosity has little effect on the electronic conductivity within the CB/PVdF mixture.

Simulated effective electronic conductivity evolution from the modified FIB/SEM volumes is shown in Figure 3 where the CB/PVdF volume fraction is either increased by dilatation for all electrodes or decreased by erosion for NA2b and NA7b electrodes only. We can notice that this curve set seems to draw a master curve at the exception of NA2 (from 4.8 % to 12 %) and NA3 (from 4.7 % to 8 %). This tendency towards a master curve existence is certainly a picture of microstructural common characteristics in the considered electrodes. In particular in the way the CB/PVdF domains are arranged on the NMC stacking surface constraining their distribution within the electrode. As the NMC clusters have the same dimensions in all the electrodes, and same way of stacking, it seems logical to find similar electronic conductivity values at similar CB/PVdF volume fractions. This so-called master curve does not exhibit a clear percolation threshold. Having a second less conductive and percolated phase (the NMC one) can hinder this threshold a little by smoothing the transition in effective conductivity, which has been experimentally observed on LiV 3 O 8

based electrodes [START_REF] Guy | Critical role of polymeric binders on the electronic transport properties of composites electrode[END_REF]. NA2 and NA3 volumes however clearly stand aside from this so-called master curve within the 5 to 8 % range (CB/PVdF volume fraction) and show a clear increase in their effective electronic conductivity around 5 to 6 % CB/PVdF volume fraction illustrating then the existence of a percolation threshold for this phase. The NA2 and NA3 FIB/SEM volumes appear more representative than NA7, as it is mentioned above, in which partial percolation of the CB/PVdF phase is observed in the direction of the simulation (i.e. the direction of the applied electrical potential difference). The difference between effective conductivities for NA2b and NA7b when the CB/PVdF volume fraction is reduced by erosion and the effec-tive conductivities for NA2 and NA3 therefore indicates that the erosion process creates a microstructure that differs from reality. For NA2b and NA7b, starting from a CB/PVdF percolated network, the erosion process at the interface with porosity tends to retain the percolated state. Whereas an actual electrode is produced from an ink in which the CB/PVdF phase tends to agglomerate with itself under the effect of attractive interactions and thus will rather tend to form domains isolated from each other at low volume fractions.

Figure 4 provides a comparison of the current density maps in both the NMC (Figure 4a and4c) and CB/PVdF phases (Figure 4b and4d) for NA3 (Figure 4a and4b) and NA7b (Figure 4c and4d). The current density distributions are also reported in the NMC phase Figure 4e. First note that the cumulative current density flowing through these two volumes, in response to the application of a potential difference of 1 V, is 5 Amm -2 for NA3 and 275 Amm -2 for NA7b, which are distributed as follows: for NA3, 0.30 Amm -2 (6 %) in the NMC phase and 4.7 Amm -2 (94 %) in the CB/PVdF phase; for NA7b, 0.63 Amm -2 (0.2 %) in the NMC phase and 274 Amm -2 (99.8 %) in the CB/PVdF phase. The higher cumulative current density in NA7b is related to its higher effective conductivity stemming from the CB/PVdF phase percolation.

Proportionally, the current uses more the CB/PVdF phase in a microstructure in which it is percolated compared to a microstructure in which it is not (respectively 99.8 vs. 94 % of the cumulative current density).

However, the NMC phase in NA7b is, on the whole, crossed by a higher total current density than in NA3

(respectively 0.63 and 0.30 Amm -2 ). This is due to the CB/PVdF phase percolation which distributes this current more widely, and at higher levels, throughout the electrode volume and thus maximise the NMC phase use. This feature is illustrated by the current density distribution, which is wider in NA7b. The most frequent current density is lower for NA7b compared to NA3, but a fraction of the NMC phase sees a higher current density in NA7b than in NA3. The current density distribution in NA3-0 is also plotted in Figure 4e. This distribution is shifted toward lower current densities, in relation with its lower effective conductivity. Compared to the two others, the cumulative current density flowing through NA3-0 is only 0.12 Amm -2 .

In addition, the current density maps allows to see that near areas where the CB/PVdF phase is crossed by a high current density, the NMC phase is very little if not used. Conversely, the NMC phase appears more used where the CB/PVdF phase percolation is interrupted and thus where the NMC phase must take over from the CB/PVdF mixture for carrying the current. The comparison with the current density maps from NA2-0 volume (without CB/PVdF) shows that in NA3 and NA7b zones of current constriction at the junctions between NMC clusters are not witnessed. In fact, this current is distributed for the most part by the CB/PVdF phase and not the NMC one. Resistive constrictions between the NMC clusters are thus avoided thanks to the CB/PVdF aggregates.

Effective ionic conductivity simulation

FIB/SEM volumes can also be used to compute effective ionic conductivities. To achieve that, NMC and CB/PVdF were considered as complete insulators. The same electrostatic equations were solved considering the electrolyte local ionic conductivity equal to 1 Sm -1 as measured in [START_REF] Panabière | Electronic and ionic dynamics coupled at solid-liquid electrolyte interfaces in porous nanocomposites of carbon black, poly(vinylidene fluoride), and gamma-alumina[END_REF]. Results are gathered in Table 4 and plotted in Figure 5a. An experimental value could be determined by BDS on a stacking made of NMC clusters (binded with 5 %wt of PVdF) with porosity 26 % and wetted by a liquid electrolyte, LP30

(conductivity at room temperature about 1 Sm -1 ). The experimentally measured ionic effective conductivity for this sample (named NA0-b) is 0.0625 Sm -1 which is very close to the simulated one (see Figure 5a). It suggests that the Fourier based simulations undertaken here on FIB/SEM volumes can perform a fair prediction of the electrode effective ionic conductivities. No further experimental data is however available for comparison as it is difficult to discriminate the ionic from the electronic contributions to the total conductivity. This part thus needs further investigation to assess the Fourier based method reliability to precisely predict the effective ionic conductivity in composite electrodes wetted by a liquid electrolyte.

It shows nevertheless the interest of numerical estimations to overcome some experimental difficulties in measuring this property. Simulation results are also well fitted by an Archie's law [START_REF] Glover | A generalized Archie's law for n phases[END_REF].

σ ef f = σ ion φ m poro (3)
With σ ef f is the effective conductivity, σ ion the electrolyte conductivity and φ poro the porosity volume fraction wetted by the electrolyte and m is an exponent that depends on the conducting phase topology and is thus related to the porous phase tortuosity. The fit results here in a m = 2 exponent. When fitted from experimental ionic conductivity from NA0-b this Archie's law gives a m = 2.05 exponent; previous measurements done on another NMC stacking wetted by the same electrolyte gave m = 2.5 [START_REF] Seid | An in situ multiscale study of ion and electron motion in a lithium-ion battery composite electrode[END_REF]. Simulation results are also compared in Figure 5a with effective ionic conductivities inferred through the tortuosity factor equation.

σ ef f = σ ion φ poro T au (4)
Where T au is the tortuosity factor. This tortuosity factor accounts for all aspects that are hindering the ion transport through the electrolyte within the electrode porosity. One of these hindering aspects, at a first order, is the porosity geometric tortuosity τ . For a porous medium with L pore as the actual path length through pores, and L straight as the straight (Euclidian) distance, the strict geometrical definition of tortuosity is:

τ = L pores L straight (5)
Epstein [START_REF] Epstein | On tortuosity and the tortuosity factor in flow and diffusion through porous media[END_REF] showed that for a bundle sinuous but parallel pores, if the geometric tortuosity τ is the sole aspect influencing the ion diffusion, the tortuosity is the square root of the tortuosity factor T au as presented in equation 6.

T au = τ 2 (6)
Other factors such as interactions at the solid/liquid interfaces between the active material or the CB/PVdF mixture and the electrolyte [START_REF] Panabière | Electronic and ionic dynamics coupled at solid-liquid electrolyte interfaces in porous nanocomposites of carbon black, poly(vinylidene fluoride), and gamma-alumina[END_REF][START_REF] Seid | An in situ multiscale study of ion and electron motion in a lithium-ion battery composite electrode[END_REF] can influence the ion transport. It is generally considered that only taking into account the geometric tortuosity is insufficient and leads to overestimating the effective ionic conductivity. The simulation results are well fitted by equation 4 considering tortuosity factors between 3.1 (for NA3) and 6.2 (for NA7b) (Table 4).

These values are in good agreement with the estimations from Usseglio-Viretta et al [START_REF] Usseglio-Viretta | Li-ion battery electrodes through micro-macro modeling and experiment[END_REF] as shown in Figure 5b. The lower and upper bound lines give limiting values that were also determined through numerical simulation but by solving the Laplace steady-state diffusion equation. This was done within the pore network of 3D NMC-based electrode geometries reconstructed from X-ray computed tomography data for the active material phase and mathematically generated carbon/binder domains in the empty spaces of the macrostructures. The two bounds correspond to two different arrangements of the carbon/binder domains [START_REF] Usseglio-Viretta | Li-ion battery electrodes through micro-macro modeling and experiment[END_REF]. The lower bound corresponds to a film-like morphology where the carbon/binder phase is preferentially layered at the NMC cluster surface, while the higher bound corresponds to a finger-like morphology where the carbon/binder domains are more protruding into the porosity. In this previous work the NMC-based electrode porosities ranged between 32 and 52 % and the CB/PVdF contents were 5+5 wt%. The electrodes studied here have lower porosity, lower CB/PVdF contents (sum is less than 5 wt%) and more complex microstructure due to NMC cluster fragmentation. We can see that the tortuosity factors determined here are mostly averaging the upper and lower bounds from Usseglio-Viretta et al except for NA7b which matches the upper bound. The NMC cluster fragmentation produces a porosity obstruction comparable to the finger-like carbon-binder domain morphology taken into account by Usseglio-Viretta et al [START_REF] Usseglio-Viretta | Li-ion battery electrodes through micro-macro modeling and experiment[END_REF], ex-plaining the very good agreement between our results and theirs despite the difference in CB/PVdF content.

Thorat et al. [START_REF] Thorat | Quantifying tortuosity in porous li-ion battery materials[END_REF] and Zacharias et al. [START_REF] Zacharias | Direct measurements of effective ionic transport in porous Li-ion electrodes[END_REF] [START_REF] Pouraghajan | Quantifying tortuosity of porous li-ion battery electrodes: Comparing polarization-interrupt and blocking-electrolyte methods[END_REF] found a reasonable agreement between this method and other methods that uses impedance spectroscopy to measure the effective ionic [START_REF] Ogihara | Impedance spectroscopy characterization of porous electrodes under different electrode thickness using a symmetric cell for highperformance lithium-ion batteries[END_REF][START_REF] Landesfeind | Tortuosity determination of battery electrodes and separators by impedance spectroscopy[END_REF]. Figure 5b shows a good agreement between our simulations and relationships of Thorat and Zacharias [START_REF] Thorat | Quantifying tortuosity in porous li-ion battery materials[END_REF][START_REF] Zacharias | Direct measurements of effective ionic transport in porous Li-ion electrodes[END_REF], except for NA7b likely for the same reasons that are mentioned above. We must note that we extrapolated their results in the porosity-range that corresponds to our electrodes. In addition, the electrode microstructures studied in these works is different due to a difference in the geometry of the active material particles. Also, the good agreement between our results and those of Zacharias et al. [START_REF] Zacharias | Direct measurements of effective ionic transport in porous Li-ion electrodes[END_REF] and Thorat et al. [START_REF] Thorat | Quantifying tortuosity in porous li-ion battery materials[END_REF] is to be put into perspective. Nevertheless, altogether this suggests the Fourier based method with the simple ohm's law can give a fair prediction of the microstructure influence on the effective ionic conductivity and tortuosity.

Furthermore, in our electrodes we could identify several porosity types [START_REF] Cadiou | Multiscale characterisation of composite electrode microstructures for high density lithium-ion batteries guided by the specificities of their electronic and ionic transport mechanisms[END_REF]: (i) one made of large micrometric cavities within the NMC cluster pack; (ii) one occluded in non-fragmented hollow NMC clusters; (iii) one made of nanometric pores defined by the PVdF/CB mixture morphology and (iv) one due the cracked grain boundaries within fragmented NMC clusters. Quantitatively, the micrometric porosity was defined as pore spaces wider than 60 nm. As a matter of illustration, these different porosities are identified in a 2D slice taken in the NA2b volume in Figure 6a.

Figure 6b provides a current density map within the porosity phase at the same location. The potential difference is applied perpendicularly to this 2D slice. It can be noted that the current density is zero in the occluded porosities, see at the bottom left for example (around coordinates (150, 350)), which are thus useless. Some micrometric porosity areas are crossed by a low current density, less than 0.2 Am -2

Examples are the inside of fragmented NMC clusters. These regions can be dead ends or only accessible through nanopores as those formed by the NMC cluster fragmentation as in middle right (around coordinates (450, 200)). Figure 6b also shows that some other parts of micrometric porosity areas can be crossed by a low current density, typically less than 0.02 Am -2 , such as in the middle left (coordinates (150, 200)).

Figure S2 shows a set of parallel slices to the one shown in Figure 6b, each spaced by 1 µm. It can be seen there that upstream from the middle left region (coordinates (150, 200)) there is an NMC cluster (Figure S2a). The current density is therefore very low in the central part of the porosity just after this cluster because the current lines must bypass it [START_REF] Besnard | Etude des propriétés de transport des charges aux différentes échelles d'une électrode de batterie lithium-ion et de leurs influences sur les performances en puissance pour l'application véhicule électrique[END_REF]. Such micrometric porosity areas thus poorly contribute to carry the ion flux throughout the electrode. On the contrary, other micrometric porosity zones, which are rather small, are crossed by a very high current density, greater than 1 Am -2 . Such zones are localised at interstices at the vicinity of contact areas between NMC clusters. These zones, such as the one in the upper right (coordinates (400, 50)), connect to others of larger dimension situated upstream or on either side and thus act as a linking bottleneck. It can be seen on Figure S2a that on the section situated upstream of the zone in the upper right (coordinates (400, 50)) there is a zone of larger porosity which then converges into a zone of lower porosity. These observations reveal the complex microstructure influence on ionic transport.

We can add that the simulation carried out here does not take into account interface effects which could be very influential insofar as high current density areas are narrow. The simulation also does not take into account ion accumulation nor depletion, upstream and downstream of the bottlenecks, which can modify the electrolyte local conductivity.

Figure 7a-e provides a comparison between current density maps, within the porosity phase, in 2D slices from the different electrode volumes. The current density distributions are also reported in Figure 7f. A comparison can be drawn between the most and the least porous of these microstructures, namely NA3 and NA7b. Current density distribution is centred towards low values for NA7b, in relation with its low effective ionic conductivity, but extends in the domain of the high current densities. These correspond to areas of strong current constriction on the map. This also shows that a part of the porosity is not crossed by the current. This porosity fraction not crossed by the current is remarkable on its distribution by the high value of the ordinate at the curve origin. By arbitrarily defining these "dead areas" as regions with current density values under 0.05 Am -2 , it comes that they represent 8.8 % (NA3) to 27.6 % (NA7b) of the porosity phase. This larger "dead area" fraction in NA7b is the consequence of the low total porosity in this electrode, the NMC cluster strong fragmentation and the high CB/PVdF content, which significantly reduce the micrometric porosity fraction and create numerous bottlenecks and dead ends that form many barriers and obstacles to ion transport. Consequently, the effective porosity, which participates in the carrying of the ion current is lower than the total porosity. Comparatively, the current density, in the porosity phase, is more homogeneous and centred towards high values for NA3 which is more open and thus more fully utilised. Some "dead areas" can also be observed in the other volumes, particularly NA7 and NA2b (Figure 7c andd), near closed pores or very fragmented NMC clusters (representing respectively 15.7 %, 10.5 % and 12.5 % of the total porosity phase for NA2, NA7, NA2b). These "dead areas" illustrate the existence of a non-used micrometric porosity because isolated from the main network by confined porosities or imposed downstream of the NMC clusters due to the current lines having to bypass them. Figure S3 shows other slices with "dead areas" in volume NA7b. Figure 7g gives the percolated microporosity and the porosity crossed by a current density higher than 0.05 Am -2 as a function of the total porosity of the FIB/SEM volumes. As mentioned previously [START_REF] Cadiou | Multiscale characterisation of composite electrode microstructures for high density lithium-ion batteries guided by the specificities of their electronic and ionic transport mechanisms[END_REF], the percolated micrometric porosity varies linearly with the total porosity. Moreover, it can be seen that the presence of "dead areas" clearly increases when there is less percolated micrometric porosity. Indeed, the amount of porosity that is crossed by a current density higher than 0.05 Am -2 matches the amount of percolated micrometric porosity. This indicates that ionic transport is dominated by low tortuosity micrometric channels through the electrode.

This conclusion is in agreement with the modelling work of Lagadec et al. on battery separators which shows that dead ends in a pore network are ineffective for the homogenisation of a large salt concentration of lithium through the separator [START_REF] Lagadec | Topological and network analysis of lithium ion battery components: the importance of pore space connectivity for cell operation[END_REF]. It is also in agreement with the experimental observations by Roberts et al. on LiFePO 4 -based electrodes that lithiation follows preferential pathways along the percolation of large porosities [START_REF] Robert | Multiscale phase mapping of LiFePO 4 -based electrodes by transmission electron microscopy and electron forward scattering diffraction[END_REF]. However, Nguyen et al. simulations on virtual microstructures suggest that dead ends in a pore network might play a crucial role in the performance of real electrodes [START_REF] Nguyen | The electrode tortuosity factor: why the conventional tortuosity factor is not well suited for quantifying transport in porous li-ion battery electrodes and what to use instead[END_REF].

RVE size estimations

The same methodology as in [START_REF] Cadiou | Numerical prediction of multiscale electronic conductivity of lithium-ion battery positive electrodes[END_REF] is here used to assess the FIB/SEM volume representativeness regarding electronic conductivity. The calculations were made for a non percolated microstructure (NA3) and a percolated one (NA2b). With a relative error of 10 % we obtained a Representative Volume Element (RVE) size around 200×200×200 µm 3 for the non percolated state and 30×30×30 µm 3 for the percolated state.

In a non percolated state, RVE is coherent with the RVE size recently determined for NMC electrode [START_REF] Sangrós Giménez | Numerical simulation of the behavior of lithium-ion battery electrodes during the calendaring process via the discrete element method[END_REF]. This dimension is far superior to our electrode thicknesses which means that to capture correctly the electrode behaviour we should acquire, the whole electrode thickness in a 200×200 µm 2 area with enough resolution to discriminate the CB/PVdF phase. This is clearly not feasible with a galium FIB nowadays.

However, our results show a good concordance with experimental ones. This could indicate that, despite the RVE size, we are able to catch a CB/PVdF distribution and morphology that is representative enough while the overall microstructure is not.

FIB/SEM volume dimensions.

We can the conclude that the CB/PVdF morphology greatly influence the RVE size for effective electronic conductivity. In non percolated electrode the NMC skeleton has more influence than the CB/PVdF one and the RVE size grows to match the one in [START_REF] Sangrós Giménez | Numerical simulation of the behavior of lithium-ion battery electrodes during the calendaring process via the discrete element method[END_REF] which is established for the electrode mechanical properties that are naturally more dependent on the NMC phase than on the CB/PVdF one. As for the percolated electrodes, electrons will stream through the CB/PVdF mixture. The RVE size drop should then indicate that it has a characteristic morphology acquirable in smaller volumes and that the NMC skeleton will not play a major role in electron conduction.

The RVE size related to ionic conductivity was also estimated using the same volumes; i.e. NA3 with a broad porosity network and NA2b which has a more tortuous one due to its increased content in CB/PVdF.

The RVE size was found to be around 15 µm wide which is close to our FIB/SEM volumes and thus suggests that the micrometric porosity network is captured well enough at that scale to be representative of the overall ionic diffusion in the electrode. In electrodes containing a more constrained porosity such as NA2b, the RVE size was also found around 15 µm. It can be interpreted as that the more constrained nature of the porous network does not impact much the micrometric porosity representative length scale due to its still large intraconnectivity.

Final verification of such assertions should however be made through the acquisition of microstructural volumes at these dimensions, same resolution and discriminating all the phases.

Conclusion

Effective electronic and ionic conductivity estimations for EV designed electrodes based on NMC 532 composite positive electrodes were conducted using a Fourier based method and actual microstructure volumes acquired by FIB/SEM nanotomography.

For electronic conductivity, simulations were first performed on NMC only 3D microstructures to infer this phase conductivity using macroscopic experimental results. This led to an apparent phase conductivity for the NMC stacking that gathers the NMC clusters bulk conductivity together with their resistive contact interfaces. The NMC phase is indeed seen as an homogeneous one in the segmented FIB/SEM volumes. By coupling this value to the CB/PVdF phase conductivity, effective electronic conductivities were computed in the FIB/SEM volumes and successfully compared to experimental results. The current density fields show that the NMC phase is only solicited when the CB/PVdF is lacking, which is more likely to happen in a non-percolated microstructure. However, even in that apparently more detrimental case, there is still a benefice from the presence of CB/PVdF aggregates that allow the current to bypass the constrictive junctions between the NMC clusters. The segmented microstructures were also numerically modified to mimic composition changes in the CB/PVdF amount. A percolation threshold was thus numerically evidenced for a CB/PVdF volume fraction around 6 to 7 %.

Predictions were also conducted regarding the effective ionic conductivity. A good agreement was found with one experimental measurement. Moreover, the tortuosity factor values deduced form the simulation results are also in good agreement with previous experimental and simulation works for other positive electrodes. Overall, the ionic conductivity appears to mostly rely on the percolated micrometric porosity. The current density fields show the presence of "dead areas" in that micrometric porosity due to accessibility issues caused by more confined porosities.

An RVE size estimation was conducted regarding both electronic and ionic conductivities. The electronic RVE size drops from a 200 µm side to a 30 µm side when the CB/PVdF is percolated. On the contrary RVE size related to ionic conductivity was found around a 15 µm side in both electrode types indicating that the micrometric porous network is captured well enough at that length scale despite differences in pore constrictions. Comparison with experimental results are nonetheless satisfying showing that we captured representatively enough microstructures, especially when it comes to the CB/PVdF phase.

Volumes should however be acquired at these dimensions and resolution to definitely confirm this hypothesis. Table 3: Effective electronic conductivity estimates using Fourier based simulation on FIB/SEM volumes. Phase conductivities for NMC and CB/PVdF phases are respectively 0.0057 Sm -1 et 350 Sm -1 . CB/PVdF percolation state in the simulation volumes is also mentioned as well as its intraconnectivity and the experimental effective conductivity associated to each sample. The electrode experimental effective conductivities were measured by BDS and the percolation state of the CB/PVdF mixture was identified from the activation energy of the experimental effective conductivity [START_REF]Multiscale architectural characterisation of composite Li-ion electrodes using Broadband Dielectric Spectroscopy[END_REF].

Name

Figure 1

 1 Figure 1 also shows 2D maps of the current density flux for slices in volume NA2-0 that are perpendicularor parallel to the imposed potential field. A color map is used to distinguish areas in which the potential difference application causes an enhanced electric current (in yellow-white) from areas where it is moderate (in blue) or inexistent (in black). It can be seen that narrow contacts between two clusters or grains within fragmented clusters are constrictive regions leading to more localised and denser current fluxes, hence the critical role of contact areas on the overall microstructure conductivity.

  established modified Bruggeman-type functions for LiFePO 4 -and LiCO 2 -based electrodes for the porosity dependence of the tortuosity factor, which are reproduced in Figure 5b. The LiFePO 4 based electrodes contained 4 % carbon and 4 % PVdF by weight, and their porosity range was 30-55 %. The LiCoO 2 based electrodes contained 7.8 % carbon and 7.8 % PVdF by weight, and their porosity range was 32-50 %. These results were obtained by measuring electrode-film tortuosity in terms of effective diffusivity. A more recent work by Pouraghajan et al.

Figure 1 :

 1 Figure 1: (a) Example of NA0 like volume, NMC is in white and porosity is left transparent (from NA2-0 data after CB/PVdF phase removal). (b) 2D maps of the current density field extracted from the 3D Fourier based simulation results.

Figure 2 :

 2 Figure 2: Example of CB/PVdF volume fraction modification in FIB/SEM volumes. White is the NMC phase, black is porosity and grey CB/PVdF. (a) CB/PVdF phase erosion, contacts with NMC remain, (b) initial microstructure, (c) CB/PVdF phase dilatation. These subplots are focusing on how interfaces between CB/PVdF and NMC are affected by the procedure. Images are from NA2b volume: (a) after erosion (2.7 % of CB/PVdF) ; (b) initial volume (8.2 % of CB/PVdF) and (c) after dilatation (red11.2 % of CB/PVdF).

Figure 3 :

 3 Figure 3: Effective conductivity evolution from FIB/SEM volume modification on the CB/PVdF phase. Each first point on a dilation plot is for the raw (unmodified) FIB/SEM volume, except for NA2b and NA7b where it is notified by a plain dot.

Figure 4 :

 4 Figure 4: Illustration of the differences in current densities between a non percolated electrode (NA3) and a percolated electrode (NA7b). (a) and (b) are extracted from current density fields in NMC and CB/PVdF phases at the same slice in NA3. (c) and (d) are similar to (a) and (b) for NA7b respectively for NMC and CB/PVdF. (e) Current density histogram comparison within the NMC phase for NA3 and NA7b. Histograms are zoomed on the population main variation window.

Figure 5 :

 5 Figure 5: (a) Ionic effective conductivity evolution from Fourier based simulations and computed from Archie's law (cf. equations 4 and 3) as a function of porosity, the experimental value corresponds to a NMC 532 /PVdF 95:5 (%m) pellet. (b) Tortuosity factor comparison between values from Fourier based simulations and bounds determined in[START_REF] Usseglio-Viretta | Li-ion battery electrodes through micro-macro modeling and experiment[END_REF]

Figure 6 :

 6 Figure 6: Comparison between (a) the microstructure with different porosity types (white is NMC, grey is CB/PVdF, green is (i), black (ii), blue (iii) and red (iv)) and (b) the corresponding current density map (simulation) in identical 2D slices from NA2b FIB/SEM volume.

Figure 7 :

 7 Figure 7: Illustration of the differences in current density (2D slices) between FIB/SEM volumes from simulations and current density histogram comparison over the whole volumes. (a) NA2, (b) NA3, (c) NA7, (d) NA2b, (e) NA7b, (f) histogram. White is NMC and CB/PVdF phases in the 2D slices. (g) compares the percolated micrometric porosity to the one crossed by a current density higher than 0.05 Am -2 (red line is a linear interpolation) as a function of the total porosity.

Table 1 :

 1 Electrode comparison in volumes, thickness and density from manufacturer data

	Name	Volume fraction (%) NMC PVdF/CB Porosity NMC PVdF CB Mass fraction (%)	Thickness (µm) Density (g/cm 3 )
	NA2	64.8	7.2	28	96	1.8	2.2	81.4	3.2
	NA3	64.8	7.2	28	96	1.8	2.2	48.8	3.2
	NA7	70.9	7.9	21.2	96	1.8	2.2	74.4	3.5
	NA2b	63.6	10.4	26.0	94.2	2.6	3.2	81.4	3.2
	NA7b	69.6	11.3	19.1	94.2	2.6	3.2	74.4	3.5

Table 2 :

 2 Comparison between phase volume fractions measured from FIB/SEM and from manufacturer data.

				Volume fraction (%)
	Name	NMC Relative	CB/PVdF Relative	Porosity Relative	Density (gcm -3 )
		FIB/SEM	variation *	FIB/SEM	variation *	FIB/SEM	variation *
			(%)		(%)		(%)
	NA2	70.4 ±0.1	9	4.6 ±0.2	-35 ±2	24.9 ±0.3	-10 ±2	3.4
	NA3	61 ±0.1	-6	4.1 ±0.6	-43 ±7	35 ±0.6	25 ±2	3.0
	NA7	68.8 ±0.3	-3	4.6 ±0.3	-41 ±5	26.5 ±0.7	25 ±3	3.4
	NA2b 68.4 ±0.2	8	8 ±0.4	-24 ±4	23.7 ±0.5	-9 ±2	3.4
	NA7b 72.4 ±0.1	4.0	7.6 ±0.3	-32 ±2	20.1 ±0.4	5 ±2	3.6

* Relative to manufacturer data

Table 4 :

 4 Effective ionic conductivity values from Fourier based simulations carried out considering the total FIB/SEM volumes porosities and tortuosity factor calculated with equation 4 for σ 0 = 1 Sm -1 and σ ef f from Fourier based simulations. The intraconnected micrometric porosity was determined in[START_REF] Cadiou | Multiscale characterisation of composite electrode microstructures for high density lithium-ion batteries guided by the specificities of their electronic and ionic transport mechanisms[END_REF].

	Name	Total porosity (%)	Intraconnected micrometric porosity (%)	Effective ionic conductivity (Sm -1 )	Tortuosity factor
	NA2	24.7	22.3	0,061	3.1
	NA3	34.4	32.8	0,110	3.4
	NA7	25.9	20.5	0,076	4.0
	NA2b	23.2	18.8	0,056	4.1
	NA7b	19.8	14.2	0,032	6.2

Acknowledgement

We are grateful to the Agence Nationale de la Recherche to the funding of the Pepite project (ANR-15-CE05-0001) and for Zeon's support during the project.

Additional information

The software used to perform the FFT numerical simulations is called Morphhom and is available upon reasonable request to François Willlot (francois.willot@mines-paristech.fr)