Dr Guillaume Leurent
email: guillaume.leurent@chu-rennes.fr

Ronan Garlantézec

Vincent Auffret

Jean Philippe Hacot

Isabelle Coudert

Emmanuelle Filippi

Antoine Rialan

Benoît Moquet

Gilles Rouault

Martine Gilard

Ronan Garlantezec

Philippe Castellant

Philippe Druelles

Bertrand Boulanger

Josiane Treuil

Bertrand Avez

Marc Bedossa

Dominique Boulmier

Marielle Le Guellec

Hervé Le Breton

data from 5000 patients included in the ORBI prospective French

Keywords: gender, differences, ST-elevation myocardial infarction, age-gender interaction, outcome, mortality genre, différences, syndrome coronaire aigu avec sus-décalage persistant du segment ST, interaction âge-genre, pronostic

Abstract:

Background: Gender differences in presentation, management and outcome in patients with ST-elevation myocardial infarction (STEMI) are reported.

Aim: To determine whether female gender is associated with higher in-hospital mortality.

Methods: Data from ORBI, a regional STEMI registry of 5 years' standing, were analyzed.

The main data for presentation, management, in-hospital outcome and prescription at discharge were compared between genders. Various adjusted hazard ratios were then calculated for in-hospital mortality (women vs. men).

Results: The analysis included 5,000 patients (mean age = 62.6 ± 13 years), with 1,174 women (23.5%). Women were a mean 8 years older than men, with more frequent comorbidities. Median ischemia time was 215 minutes (26 minutes longer in women, p<0.05).

Reperfusion strategies in women involved less fibrinolysis, coronary angiography, radial access and thrombo-aspiration. Female gender, especially in patients younger than 60 years, was associated with poorer in-hospital prognosis (including higher in-hospital mortality: 9% vs. 4% in men; p<0.0001), and under-utilisation of the recommended treatment at discharge. Moreover, excess female in-hospital mortality was independent of presentation, revascularisation time and reperfusion strategy (hazard ratio for women: 1.33, 95% confidence interval, 1.01 to 1.76; p=0.04).

Conclusions: One in 4 patients admitted for STEMI was female, with significant differences in presentation. Female gender was associated with less optimal treatment, both in the acute phase and at discharge. Effort should be made to reduce those differences, especially as female gender was independently associated with elevated risk of in-hospital mortality.

Abstract word count: 238

Résumé:

Contexte: Des différences liées au sexe sont signalées dans la présentation, la gestion et le pronostic des patients hospitalisés pour infarctus du myocarde avec élévation du segment ST (IDM ST +).

Objectif: déterminer si le sexe féminin est associé à une mortalité intra-hospitalière plus élevée.

Méthode: Nous avons analysé les données d'ORBI, un registre régional de 5 ans concernant les IDM ST+. Les principales données concernant la présentation, la gestion, le devenir intrahospitalier et le traitement à la sortie ont été comparées en fonction du sexe. Ensuite, nous avons analysé la mortalité intra hospitalière (femmes vs hommes), avec différentes variables d'ajustement.

Résultats: L'analyse a inclus 5000 patients (âge moyen = 62,6 ± 13 ans), dont 1174 femmes (23,5%), qui présentaient des co-morbidités plus fréquentes et étaient 8 ans plus âgées que les hommes. Le temps d'ischémie médian était de 215 minutes (26 minutes de plus chez les femmes). Comparativement aux hommes, les stratégies de reperfusion chez les femmes comportaient moins de fibrinolyse, de coronarographie, d'accès radial et de thromboaspiration. Le sexe féminin, en particulier chez les moins de 60 ans, était associée à un mauvais pronostic intra-hospitalier (y compris une plus forte mortalité intra hospitalière: 9% contre 4% chez les hommes, p <0,0001), et une sous-utilisation des traitements recommandés à la sortie. Par ailleurs, la surmortalité intra hospitalière observée chez les femmes était indépendante de la présentation, des délais de revascularisation et des stratégies de reperfusion (hasard ratio=1,33, intervalle de confiance 95% :1,01 à 1.76, p = 0,04).

Conclusions: Une personne sur 4 patients admis pour un IDM ST+ est une femme, avec des différences significatives dans la présentation. Le sexe féminin est associé à un traitement moins optimal, tant à la phase aiguë qu'à la sortie d'hôpital. Un effort particulier devra être effectué afin de réduire ces différences, d'autant plus que le sexe féminin semble constituer dans cette analyse un risque indépendant de surmortalité intra-hospitalière.

Introduction: Several studies have reported increased in-hospital mortality in myocardial infarction (MI) with persistent ST-segment elevation (STEMI) in women [START_REF] Vaccarino | Sex differences in mortality after myocardial infarction: evidence for a sex-age interaction[END_REF][START_REF] Claassen | Gender gap in acute coronary heart disease: Myth or reality?[END_REF][START_REF] Puymirat | Association of changes in clinical characteristics and management with improvement in survival among patients with ST-elevation myocardial infarction[END_REF][START_REF] Merz | Women's Ischemic Syndrome Evaluation: current status and future research directions: report of the National Heart, Lung and Blood Institute workshop: October 2-4, 2002: executive summary[END_REF][START_REF] Kang | Sex differences in management and mortality of patients with ST-elevation myocardial infarction (from the Korean Acute Myocardial Infarction National Registry)[END_REF][START_REF] Milcent | Gender differences in hospital mortality and use of percutaneous coronary intervention in acute myocardial infarction: microsimulation analysis of the 1999 nationwide French hospitals database[END_REF][START_REF] Zhang | Age-specific gender differences in in-hospital mortality by type of acute myocardial infarction[END_REF] . Several hypotheses have been put forward to account for this excess female mortality, including more serious comorbidity, longer time to revascularization, or less "optimal" reperfusion strategies.

It is, however, not yet clearly established whether female gender is in itself a risk factor for inhospital death in case of STEMI. The present study is an update on gender-linked differences in the characteristics, means of treatment, mortality and in-hospital prognosis of patients admitted for STEMI.

Method: We used the data from the Brittany Regional Infarction Observatory (Observatoire Régional Breton sur l'Infarctus: ORBI) [START_REF] Leurent | Door-toballoon delays before primary angioplasty in the Regional Acute Myocardial Infarction Registry of Brittany. An analysis of the Observatoire Regional Breton sur l'Infarctus du myocarde (ORBI)[END_REF] . Brittany is an administrative Region with a population of 3.2 million and 9 interventional cardiology centers (see list in Appendix), for an area of 34,023 km². ORBI prospectively includes all patients admitted within 24 hours of symptom onset to any of the 9 centers for STEMI (final diagnosis). Demographic and electrocardiographic data, treatments, time intervals and in-hospital events are recorded prospectively. Overall ischemia time is defined as the time between symptom onset and initiation of reperfusion: balloon inflation, in case of primary angioplasty, or administration of fibrinolytic treatment. All patients registered by ORBI between July 1 st , 2006 and August 31 st , 2011 were included in the present study.

Statistical analysis:

Qualitative data were expressed as percentages and quantitative data as means ± standard deviations, except for times, which were expressed as medians [range].

Multiple imputation was performed to take into account missing data [START_REF] Ibrahim | Missing Data in Clinical Studies: Issues and Methods[END_REF] As a first step, univariate analysis according to gender, clinical characteristics, time to revascularization, revascularization strategy, in-hospital prognosis and discharge prescription was performed.

Student (or Wilcoxon, as appropriate) and Chi² were used for quantitative and qualitative data respectively. In a second step, in-hospital mortality was analyzed according to gender, using a Cox model. The association was presented in the form of a Hazard Ratio (HR) with 95% confidence intervals (95% CI). Various models were constructed according to adjustment strategy: model 1, non-adjusted; model 2, adjusted for patient characteristics (age, diabetes, arterial hypertension, active smoking, anterior STEMI location, 3-vessels or common left main coronaropathy); model 3, adjusted for the variables of model 2 and for overall ischemia time; and model 4, adjustment for the variables of model 3 and for revascularization parameters (coronarography, primary angioplasty, radial arterial approach, thromboaspiration, anti-Gp2b3a administration, fibrinolysis).

The significance threshold was systematically set at 5%. Statistical analysis used SAS® software, version 9.1 (SAS Institute Inc., Cary, NC, USA).

Results: Over the study period, 5,000 patients (mean age, 62.6 ± 13 years) were included in ORBI, with 1,174 (23.5%) female. The main patient characteristics and coronaropathy and MI data are presented according to gender in Table 1. Emergency myocardial revascularization was performed in 4,344 patients (87%): 3,583 (71%) by primary angioplasty and 761 (15%) by fibrinolysis.

The initial emergency call was on the dedicated emergency telephone number 15 (emergency medicalized ambulance service: SAMU) for 40% of female versus 44% of male patients (p=0.004). ECG when performed before hospital admission showed significant ST-segment elevation in 88% of female versus 95% of male patients (p=0.03). The SAMU provided preadmission treatment in 55% of female versus 61% of male patients (p<0.001). Finally, direct access to the cath lab in case of primary angioplasty was available for 57% of female versus 66% of male patients (p<0.0001).

Median overall ischemia time was 215 minutes [63-1,530 min], and was significantly longer for female patients (235 min, vs. 209 min for male patients, p<0.05), with delayed treatment at all stages: median pain-to-call time, 60 vs. 44 min (p<0.0001); call-to-door, 130 vs. 125 min (p<0.05); and door-to-balloon (or thrombo-aspiration) in primary angioplasty, 45 vs. 40 min (p<0.05). Thus the median time from first medical contact to balloon inflation or thrombo-aspiration was 95 min: 100 min in female vs. 94 min in male patients (p<0.05), and less than 120 min for respectively 65% and 72% (p<0.0001). Table 2 presents revascularization strategies according to gender, showing lower rates of use of the various reperfusion techniques in women.

In-hospital evolution:

As seen in Table 3, in-hospital morbidity and mortality were significantly higher in women, with higher rates of high-grade atrioventricular block and atrial fibrillation and lower left-ventricle ejection fractions. Likewise, total hospital stay was longer. Above all, mortality from all causes was higher: 106 (9%) vs. 167 (4%) (p<0.0001). These deaths were mainly from cardiovascular causes: 96 (90% of deaths) in female and 149 (89%) in male patients. Death from hemorrhage was rare: n=11; 3 male versus 8 female (p<0.001), including 8 intracranial hemorrhages (2 male, 6 female; p=0.002) and 1 retroperitoneal hematoma (in a male patient). As seen in Table 4, which compares in-hospital mortality by age and gender, excess mortality mainly affected female patients under 60 years of age, in univariate analysis. Finally, the rate of hemorrhagic events of whatever severity, systematically screened in 427 patients, was 3% in female versus 2% in male patients (p=0.48).

Discharge treatment:

Table 5 shows the rate of prescription of the various treatments recommended by the European Society of Cardiology 10 at discharge home, according to gender, again showing under-prescription for women in all cases.

Adjusted analysis of in-hospital

Discussion

The present study highlights the specificities of STEMI and its prognosis in female patients: onset associated with higher comorbidity rates, later and less optimal treatment, and above all much poorer in-hospital prognosis, especially in younger women.

The present female prevalence for STEMI (about 1 in 4) is in full agreement with French national data [START_REF] Puymirat | Association of changes in clinical characteristics and management with improvement in survival among patients with ST-elevation myocardial infarction[END_REF][START_REF] De Peretti | Personnes hospitalisées pour infarctus du myocarde en France: tendances 2002-2008[END_REF] . Likewise, the present rate of comorbidity in female patients is in agreement with recent registry reports 2 , notably a higher mean age in females: 65.5 vs. 61.2 years in the American registry [START_REF] Zhang | Age-specific gender differences in in-hospital mortality by type of acute myocardial infarction[END_REF] , 71 vs. 62 years in the German Mitra registry [START_REF] Heer | Gender differences in acute myocardial infarction in the era of reperfusion (the MITRA registry)[END_REF] , and 67 vs. 56 years in the Korean registry [START_REF] Kang | Sex differences in management and mortality of patients with ST-elevation myocardial infarction (from the Korean Acute Myocardial Infarction National Registry)[END_REF] . Time to revascularization : ORBI revealed significantly longer ischemia times in women, due to less use of the SAMU emergency service, lower ECG input and longer delays to all types of treatment, as also found in other registries [START_REF] Jneid | Sex differences in medical care and early death after acute myocardial infarction[END_REF][START_REF] Dreyer | Evaluation of Gender Differences in Door-to-Balloon Time in ST-Elevation Myocardial Infarction[END_REF] . Thus, the median pain-to-door time in the American registry [START_REF] Jneid | Sex differences in medical care and early death after acute myocardial infarction[END_REF] was 195 min for women, versus 150 min for men, and 217 versus 161 min in the Korean registry [START_REF] Kang | Sex differences in management and mortality of patients with ST-elevation myocardial infarction (from the Korean Acute Myocardial Infarction National Registry)[END_REF] . Most notable was the significantly longer (by 16 min) pain-to-call time in the ORBI data, indicating a need for greater awareness of the issue of STEMI on the part of women and for early call for specialized help (SAMU) in case of chest pain, to allow direct admission to the cath lab [START_REF] Leurent | Door-toballoon delays before primary angioplasty in the Regional Acute Myocardial Infarction Registry of Brittany. An analysis of the Observatoire Regional Breton sur l'Infarctus du myocarde (ORBI)[END_REF][START_REF] Steg | ESC Guidelines for the management of acute myocardial infarction in patients presenting with STsegment elevation: The Task Force on the management of ST-segment elevation acute myocardial infarction of the European Society of Cardiology (ESC)[END_REF] .

Management:

The present study also found a very clear gender-related difference in coronary revascularization strategy. Here again, under-implementation of fibrinolysis and angioplasty in female patients has frequently been reported [START_REF] Kang | Sex differences in management and mortality of patients with ST-elevation myocardial infarction (from the Korean Acute Myocardial Infarction National Registry)[END_REF][START_REF] Milcent | Gender differences in hospital mortality and use of percutaneous coronary intervention in acute myocardial infarction: microsimulation analysis of the 1999 nationwide French hospitals database[END_REF][START_REF] Heer | Gender differences in acute myocardial infarction in the era of reperfusion (the MITRA registry)[END_REF][START_REF] Jneid | Sex differences in medical care and early death after acute myocardial infarction[END_REF][START_REF] Ayanian | Differences in the use of procedures between women and men hospitalized for coronary heart disease[END_REF] . Most registries, however, fail to specify revascularization techniques, and data on arterial approach or thrombo-aspiration, for example, are seldom reported. The present study found significant differences in these two aspects of revascularization, yet recommended (European Society of Cardiology's guideline: IIa 10 ): although of proven benefit in terms of mortality [START_REF] Mamas | Influence of access site selection on PCI-related adverse events in patients with STEMI: metaanalysis of randomised controlled trials[END_REF] , the radial approach is less often used in female patients: 40% vs. 51%; p<0.0001. Likewise, despite proof of benefit [START_REF] Vlaar | Cardiac death and reinfarction after 1 year in the Thrombus Aspiration during Percutaneous coronary intervention in Acute myocardial infarction Study (TAPAS): a 1-year follow-up study[END_REF] , thromboaspiration is again less often used in women (46% vs. 55%; p<0.0001).

Prognosis: Poorer in-hospital prognosis for female STEMI patients has been frequently

reported 1, 5-7, 18, 19 . The main contribution of the present study lies in its analysis of this excess female mortality, with a 2-fold greater rate of in-hospital death in women.

Firstly, it emerged that the excess mortality exclusively concerned younger women, being significant only in those under 60 years of age, with a trend in 60-to-70 year-olds; in older female patients, no differential mortality was found. This age-gender interaction was also observed in the American registry [START_REF] Zhang | Age-specific gender differences in in-hospital mortality by type of acute myocardial infarction[END_REF] , with post-STEMI in-hospital mortality 98% greater in 50 year-old women than in men of the same age, but identical in both sexes after 80 years: 12.2% in female and 12.3% in male patients. This is an especially worrying factor as a recent analysis of French STEMI data found a constant increase in the proportion of young women in the STEMI population over the last 15 years: 11.8% of women were aged less than 60 years in 1995, versus 25.5% in 2010 (p<0.001) [START_REF] Puymirat | Association of changes in clinical characteristics and management with improvement in survival among patients with ST-elevation myocardial infarction[END_REF] .

Moreover, it emerged that this excess mortality was not simply due to greater comorbidity (model 2), longer time to revascularization (model 3) or revascularization strategies that were less "aggressive" and less in line with scientific recommendations (model 4). Rather, female gender as such emerged as a factor of elevated risk of in-hospital mortality. Few registries have undertaken such analysis of the female in-hospital mortality they display. However, an analysis of the Korean data, using a methodology similar to the present study, showed that the excess female mortality that persisted after adjustment on age became non-significant when adjusted on age, medical history, hemodynamic status and "clinical performance" [START_REF] Kang | Sex differences in management and mortality of patients with ST-elevation myocardial infarction (from the Korean Acute Myocardial Infarction National Registry)[END_REF] . A previous analysis of data from the State of New York registry also showed significant excess mortality in under-75 year-old women that disappeared after adjustment on age, comorbidity and hemodynamic status [START_REF] Berger | Gender-age interaction in early mortality following primary angioplasty for acute myocardial infarction influence of sex on in-hospital outcomes and long-term survival after contemporary percutaneous coronary intervention[END_REF] . Likewise, a recent report from a French registry also highlighted an impact of the age-gender interaction, but with higher hospital mortality in the subgroup of over-65 year old women [START_REF] Juliard | Comparison of Hospital Mortality During ST-Segment Elevation Myocardial Infarction in the Era of Reperfusion Therapy in Women Versus Men and in Older Versus Younger Patients[END_REF] . This monocentric registry studied 2600 patients (including 199 under-65 year-old women) in a 23 years period. Therefore, the impact of this age-gender interaction is different according to the inclusion period. One quality of our multicentric registry is the inclusion of a high number of patients, in a quite short period, with homogenous STEMI-support.

We have no clear explanation for this excess female mortality. It has been suggested that, in STEMI, pre-hospital mortality is higher in male subjects [START_REF] Sonke | Sex differences in case fatality before and after admission to hospital after acute cardiac events: analysis of community based coronary heart disease register[END_REF] , counterbalancing the excess female in-hospital mortality. It is also established that young women enjoy hormonal protection against atherosclerosis [START_REF] Mendelsohn | The protective effects of estrogen on the cardiovascular system[END_REF] ; it may thus be the case that pre-menopausal women admitted for STEMI are those with especially severe atherosclerosis or in whom risk factors are accumulated. Moreover, as young women show less stenosis in the coronary network than men, they develop a lower ischemic myocardial protection response, entailing greater vulnerability to acute ischemia [START_REF] Vaccarino | Sex differences in mortality after myocardial infarction. Is there evidence for an increased risk for women?[END_REF] . And finally, the mechanism of coronary occlusion may vary according to age, with plaque erosion predominating in younger subjects and rupture in the over-50 year-olds [START_REF] Burke | Effect of risk factors on the mechanism of acute thrombosis and sudden coronary death in women[END_REF] .

Study limitations:

As ORBI is a specifically French registry, the present findings may not be extrapolated elsewhere. Moreover, ORBI is by definition restricted to patients admitted to an interventional cardiology center, and this may induce selection bias: STEMI cases managed medically in a non-interventional setting are not included. It is, however, unlikely that women are referred differently from male patients of the same age group.

Conclusion

The present analysis of ORBI registry data found that about a quarter of patients admitted for STEMI were female, with a significantly greater rate of comorbidity than in men. Moreover, time to revascularization was significantly longer, partly due to women delaying their call for medical help, and treatment strategies were less optimal, both in the acute phase and at discharge. Women, and especially younger women, thus show excess in-hospital mortality that is only partly accounted for by the above factors and persists after adjustment.

It would therefore seem necessary to raise awareness of the issue: in the female population in particular, but also in emergency staff and cardiologists. Only management that is at least as "optimal" for women as for men can bring down this mortality rate.

  mortality: Female STEMI patients showed significantly higher non-adjusted in-hospital mortality than males (model 1): HR = 1.96 [1.53-2.52]; p<0.0001. This excess mortality persisted after adjustment on patient characteristics (model 2): HR = 1.36 [1.04-1.79]; p=0.02. It also persisted after adjustment on time to revascularization (model 3): HR = 1.37 [1.05-1.80]: p=0.02. Finally, "complete" adjustment, on patient characteristics and time to revascularization and revascularization technique (model 4), gave an HR of 1.33 [95% CI, 1.01-1.76] (p=0.04), revealed excess female in-hospital mortality in STEMI, independently of all study parameters (figure 1).

Table 1 : Main patient characteristics according to gender.

 1

		Male (n=3,826)	Female (n=1,174)	p
	Age	60.8 ±12	68.8 ±14	<0.001
	Arterial hypertension	1,377 (36%)	634 (54%)	<0.001
	Dyslipidemia	1,989 (52%)	528 (45%)	<0.001
	Diabetes	420 (11%)	152 (13%)	0.06
	Active smoking	1,568 (41%)	305 (26%)	<0.001
	Body-mass index (kg/m²)	26.8 ±3	25.5 ±5	<0.001
	Known coronaropathy	734 (19%)	208 (18%)	0.2
	History of myocardial infarction	306 (8%)	47 (4%)	<0.001

Table 2 : Univariate analysis of coronary revascularization techniques according to patient gender

 2

	Male (n=3,826)	Female (n=1,174)	p

*percentage of coronarography patients with data available (n=2,618 male, 743 female). ** percentage of primary angioplasty patients.

† percentage of acute-phase coronarography patients (n=2,799 male, 924 female).

Table 4 : Mortality according to age group and gender.

 4

	Deaths	Men	Women	Total	p-value
	Age < 60 yrs 34/1,844 (1.8%) 13/317 (4.1%) 47/2,161 (2.1%)	0.01
	60 ≤ age < 70	32/930 (3.4%)	12/202 (5.9%) 44/1,132 (3.9%)	0.09
	70 ≤ age < 80	54/743 (7.2%)	23/332 (6.9%) 77/1,075 (7.1%)	0.84
	80 ≤ age	47/308 (15.2%) 58/323 (17.9%) 105/631 (16.6%)	0.36

Table 5 : Discharge prescription according to gender, for patients discharged alive from the hospital

 5

		Male (n=3,659)	Female (n=1,068)	p
	Aspirin	3,584 (98%)	1,022 (95%)	<0.0001
	Clopidogrel/Prasugrel	3,506 (96%)	993 (93%)	<0.0001
	Beta-blockers	3,349 (91%)	940 (88%)	0.001
	Angiotensin conversion enzyme inhibitor/Sartan	2,481 (67%)	662 (62%)	<0.0001
	Statin	3,475 (95%)	955 (89%)	<0.0001
	Cardiovascular rehabilitation	1,577 (43%)	267 (25%)	<0.0001

Acknowledgments:

-Thanks to the team of Clinical Research Assistants who ensured data collection: E. Babin and A. Piel (CHU de Rennes), P. Héry (Clinique St Laurent, Rennes), F. Langlais (CH de St Brieuc), P. Dias and C. Kergoulay (CHU de Brest), and P. Rameau and M. Paquin (CH de Quimper);

-and to all the cardiologists and emergency and SAMU staff who collated the data.

APPENDIX

The following medical centres participated in this study


Périne Landois

Mathieu Pinault

Mickaël Huard

Valérie Reita

Stéphan Rouzière

Pascale Launois

Martine Mayne-L'hermite

Nedjma Bendiab
email: nedjma.bendiab@grenoble.cnrs.fr

Structure in Nascent Carbon Nanotubes Revealed by Spatially Resolved Raman Spectroscopy

Keywords: Aligned-MWNT, spatially resolved Raman spectroscopy, mapping, colocalisation, X-Ray microdiffraction

The understanding of carbon nanotubes (CNT) growth is crucial for the control of their production. In particular, the identification of structural changes of carbon possibly occurring near the catalyst particle in the very early stages of their formation is of high interest. In this study, samples of nascent CNT obtained during nucleation step and samples of vertically aligned CNT obtained during growth step are analysed by combined spatially resolved Raman spectroscopy and X-Ray diffraction measurements. Spatially resolved Raman spectroscopy reveals that iron-based phases and carbon phases are co-localised at the same position, and indicates that sp 2 carbon nucleates preferentially on iron-based particles during this nucleation step. Depth scan Raman spectroscopy analysis, performed on nascent CNT, highlights that carbon structural organisation is significantly changing from defective graphene layers surrounding the iron-based particles at their base up to multi-walled nanotube structures in the upper part of iron-based particles.

Introduction:

Aligned multi-walled carbon nanotubes (A-MWNT) are promising nanostructured networks for a large range of applications in various fields such as separation or filtration [START_REF] Holt | Fast Mass Transport Through Sub-2-Nanometer Carbon Nanotubes[END_REF], supercapacitors [START_REF] Su | Symmetric Self-Hybrid Supercapacitor Consisting of Multiwall Carbon Nanotubes and Co-Al Layered Double Hydroxides Batteries and Energy Storage[END_REF], composite materials [START_REF] Cebeci | Multifunctional properties of high volume fraction aligned carbon nanotube polymer composites with controlled morphology[END_REF][START_REF] Peng | Highly aligned carbon nanotube/polymer composites with much improved electrical conductivities[END_REF]. They are mainly prepared by CVD (Chemical Vapour Deposition) processes involving, either a deposition of a metal film before the reaction with a carbonaceous source [START_REF] José-Yacamán | Catalytic growth of carbon microtubules with fullerene structure[END_REF][START_REF] Eres | In situ control of the catalyst efficiency in chemical vapor deposition of vertically aligned carbon nanotubes on predeposited metal catalyst films[END_REF][START_REF] Yasuda | Improved and Large Area Single-Walled Carbon Nanotube Forest Growth by Controlling the Gas Flow Direction[END_REF], or the formation of catalyst particles from a metal precursor which is injected together with the carbon precursor [START_REF] Andrews | Continuous production of aligned carbon nanotubes: a step closer to commercial realization[END_REF][START_REF] Mayne | Pyrolytic production of aligned carbon nanotubes from homogeneously dispersed benzene-based aerosols[END_REF][START_REF] Kamalakaran | Synthesis of thick and crystalline nanotube arrays by spray pyrolysis[END_REF][START_REF] Pinault | Growth of multiwalled carbon nanotubes during the initial stages of aerosol-assisted CCVD[END_REF]. Concerning this later method, liquid precursors are generally injected in the reactor either with syringes or as liquid aerosols and A-MWNT are subsequently grown with, generally, a high growth rate and almost no by-products (amorphous carbon) among nanotubes [START_REF] Andrews | Continuous production of aligned carbon nanotubes: a step closer to commercial realization[END_REF][START_REF] Mayne | Pyrolytic production of aligned carbon nanotubes from homogeneously dispersed benzene-based aerosols[END_REF][START_REF] Kamalakaran | Synthesis of thick and crystalline nanotube arrays by spray pyrolysis[END_REF][START_REF] Pinault | Growth of multiwalled carbon nanotubes during the initial stages of aerosol-assisted CCVD[END_REF][START_REF] Castro | Dynamics of catalyst particle formation and multi-walled carbon nanotube growth in aerosol-assisted catalytic chemical vapor deposition[END_REF]. Such processes offer the advantage to be continuous and to be easily scalable. Their development, in order to extend their potentialities, is strongly related to the precise control of carbon nanotube (CNT) growth which is a direct result of the understanding of their growth mechanisms. This is one of the motivations for the study of nanotube formation which has been widely investigated by researchers all over the world [START_REF] Tessonnier | Recent Progress on the Growth Mechanism of Carbon Nanotubes: A Review[END_REF]. Ex-situ physico-chemical analyses of samples synthesized under precise conditions are focused on the size and shape of catalyst particles as well as on their physical state and their chemical composition and structure [START_REF] Pinault | Growth of multiwalled carbon nanotubes during the initial stages of aerosol-assisted CCVD[END_REF][START_REF] Kim | Observation and formation mechanism of stable face-centered-cubic Fe nanorods in carbon nanotubes[END_REF][START_REF] Heresanu | Nature of the Catalyst Particles in CCVD Synthesis of Multiwalled Carbon Nanotubes Revealed by the Cooling Step Study[END_REF][START_REF] Kumar | Chemical Vapor Deposition of Carbon Nanotubes: A Review on Growth Mechanism and Mass Production[END_REF]. The analysis of carbon in such samples has been also investigated by different techniques such as electron microscopy (SEM or TEM), Raman spectroscopy, X-Ray Diffraction or X-Ray Photoelectron Spectroscopy [START_REF] Pinault | Growth of multiwalled carbon nanotubes during the initial stages of aerosol-assisted CCVD[END_REF][START_REF] Heresanu | Nature of the Catalyst Particles in CCVD Synthesis of Multiwalled Carbon Nanotubes Revealed by the Cooling Step Study[END_REF][START_REF] Pinault | Carbon nanotubes produced by aerosol pyrolysis: growth mechanisms and postannealing effects[END_REF][START_REF] Koos | Effect of the experimental parameters on the structure of nitrogen-doped carbon nanotubes produced by aerosol chemical vapour deposition[END_REF][START_REF] Pichot | Evidence of strong nanotube alignment and for iron preferential growth axis in multiwalled carbon nanotube carpets[END_REF]. Nevertheless, complementary information regarding global quantitative carbon structure changes occurring on nanotubes near the catalyst particles, from the CNT nucleation step up to the CNT growth step, still need to be completed.

Our previous works regarding A-MWNT carpet synthesised by aerosol-assisted CCVD (Combustion Chemical Vapor Deposition) process studied through ex-situ analysis demonstrated that (i) different iron-based phases are occurring in the same sample [START_REF] Heresanu | Nature of the Catalyst Particles in CCVD Synthesis of Multiwalled Carbon Nanotubes Revealed by the Cooling Step Study[END_REF][START_REF] Pichot | Evidence of strong nanotube alignment and for iron preferential growth axis in multiwalled carbon nanotube carpets[END_REF], (ii) iron oxide is formed during the cooling step [START_REF] Heresanu | Nature of the Catalyst Particles in CCVD Synthesis of Multiwalled Carbon Nanotubes Revealed by the Cooling Step Study[END_REF], (iii) nanotubes are obtained through a base-growth mechanism [START_REF] Pinault | Growth of multiwalled carbon nanotubes during the initial stages of aerosol-assisted CCVD[END_REF][START_REF] Pinault | Evidence of Sequential Lift in Growth of Aligned Multiwalled Carbon Nanotube Multilayers[END_REF]. X-Ray diffraction allows one to quantify the alignment degree of nanotubes in macroscopic carpets [START_REF] Pichot | Evidence of strong nanotube alignment and for iron preferential growth axis in multiwalled carbon nanotube carpets[END_REF][START_REF] Pichot | Structural and mechanical properties of single-wall carbon nanotube fibers[END_REF]. Raman spectroscopy demonstrates that such nanotubes exhibit a high structural quality as compared to nanotubes generally grown by CVD [START_REF] Castro | Dynamics of catalyst particle formation and multi-walled carbon nanotube growth in aerosol-assisted catalytic chemical vapor deposition[END_REF], and XPS experiments (X-rays Photoelectron Spectroscopy), performed on samples resulting from the early stages of nanotube growth, show the occurrence of sp 2 carbon in the early stages of CNT growth [START_REF] Pinault | Growth of multiwalled carbon nanotubes during the initial stages of aerosol-assisted CCVD[END_REF]. More recently, in-situ X-ray diffraction experiments on A-MWNT carpet synthesised by aerosol-assisted CCVD process allowed us to show that particles giving rise to CNT growth are cementite nanoparticles [START_REF] Landois | Growth of aligned multi-walled carbon nanotubes: First in situ and time-resolved X-ray diffraction analysis[END_REF].

The present study couples global and local analysis, based on ex-situ X-ray diffraction and on Raman spectroscopy. The recent developments in spatially resolved Raman spectroscopy allowed us to analyse samples obtained during CNT nucleation step (very early stages of CNT synthesis), formed of nascent CNT on the catalyst particles. Thus, it was possible to get Raman mapping of both catalyst nanoparticles and CNT. Our work gives evidence of carbon structural changes between nucleation and growth stages. By analysing the D and G band obtained on CNT samples produced during the nucleation step, we report that carbon structural organisation is significantly changing from defective graphene layers surrounding the particles at their base up to multi-walled nanotube structures in the upper part of particles.

Alignment of CNT, as determined by X-ray diffraction, differs between the CNT nucleation and growth step.

Experimental Section

Aligned carbon nanotube samples are synthesized by aerosol-assisted catalytic chemical vapour deposition (AA-CCVD) from toluene and ferrocene precursors [START_REF] Castro | Dynamics of catalyst particle formation and multi-walled carbon nanotube growth in aerosol-assisted catalytic chemical vapor deposition[END_REF][START_REF] Pinault | Evidence of Sequential Lift in Growth of Aligned Multiwalled Carbon Nanotube Multilayers[END_REF]. The synthesis set-up and standard procedure have been described previously [START_REF] Mayne | Pyrolytic production of aligned carbon nanotubes from homogeneously dispersed benzene-based aerosols[END_REF][START_REF] Pinault | Carbon nanotubes produced by aerosol pyrolysis: growth mechanisms and postannealing effects[END_REF]. Briefly, the precursor mixture is made of ferrocene (2.5 wt. %) dissolved in toluene, the resulting solution is injected through an aerosol and carried by a helium flow (purity 99.999; oxygen impurities 3 ppm-mol H 2 O and 2 ppm-mol O 2 ) in the reactor heated at 850 °C and placed at atmospheric pressure. The cooling step is also performed under helium (same quality). This flexible technique allowed us to control the CNT length thanks to the duration of the continuous injection of precursors and thus to obtain different samples at different growth steps of the CNT. It is important to note that even if the metal catalyst source is injected continuously, the resulting A-MWNT carpets exhibit a high purity with almost no catalyst particles deposited in between the MWNT.

For this study two types of samples are obtained corresponding to two key steps of the growth chronology with different AA-CCVD durations. The first one (sample 1) is obtained after 15 seconds of synthesis and the second one after 15 minutes corresponding respectively to MWNT nucleation and growth steps. These two samples are then cooled down [START_REF] Heresanu | Nature of the Catalyst Particles in CCVD Synthesis of Multiwalled Carbon Nanotubes Revealed by the Cooling Step Study[END_REF].

The general morphology and arrangement of the resulting CNT samples were examined in a field emission scanning electron microscope (FEG-SEM; Carl Zeiss Ultra 55). Raman spectroscopy and X-Ray diffraction are also carried out ex-situ at room temperature on the resulting samples. Sample corresponding to nucleation step (catalyst nanoparticles and few nucleating CNT) was analysed by Raman spectroscopy first on its top surface and second in the depth of nanoparticles initiating CNT growth (one spectrum every 230 nm) in order to evaluate the carbon changes along nanoparticle agglomerates (few µm). Sample corresponding to the growth step (CNT carpet) was analysed by Raman spectroscopy along the carpet height (one spectrum every 40-50 µm) in order to evaluate the carbon structure changes along the carpet (400 µm thick). Therefore, Raman analysis scale along nanoparticles (nucleation step) is more local as compared to the one performed in the case of carpet (growth step).

Raman spectroscopy was performed with a commercial Witec Alpha 500 spectrometer in a back-scattering configuration. We used excitation wavelengths at 532 nm from a solid state laser diode. The power of the laser beam was kept below 600 µW to prevent any damage on the A-MWNT carpet. All Raman spectra were recorded with a Nikon x50 objective (NA=0.75) focusing the light with a 350 nm diameter spot. A dual axis XY piezo stage coupled to a confocal microscope allows mapping Raman intensity with subwavelength accuracy (ca. 120 nm) and in the Z direction with accuracy of about 350 nm. Raman spectra are recorded on each pixel and the integrated intensity of a chosen mode is displayed on a false color scale.

Furthermore, spectral resolutions of our spectrometer (UHTS300) are (i) < 3 cm -1 (ie 0.009 nm) with a 600 grooves/mm grating and (ii) < 0.9 cm -1 (ie 0.025 nm) with a 1800 grooves/mm grating. In this paper, maps are performed with a 600 grooves/mm grating and spectra with a 1800 grooves/mm grating. X-Ray diffraction experiments were performed on two different setups:

(i) microdiffraction measurements are performed with a monochromatic Cu-Kα (λ=1.542 Å) X-ray microbeam of 20 µm diameter size obtained with a multilayer optics coupled with a laboratory rotating-anode generator [START_REF] Rouzière | A laboratory X-ray microbeam for combined X-ray diffraction and fluorescence measurements[END_REF]. The 2D direct-illumination CCD detector allows to measure weak scattering signals such as the (002) CNT peak from the carpet. Measurement along the height of the carpet with a 10 µm step is performed thanks to a motorized translation stage.

(ii) standard laboratory diffraction experiments with a millimeter-size and monochromatic Mo-Kα (λ=0.711 Å) beam are performed in order to measure iron phases [START_REF] Heresanu | Nature of the Catalyst Particles in CCVD Synthesis of Multiwalled Carbon Nanotubes Revealed by the Cooling Step Study[END_REF]. The intense Xray beam allows detecting the weak scattered signals from the small amounts of iron. The short Mo-Kα wavelength gives access to a large Q wavevector domain (1.5 -4.5 Å -1 ). It also avoids the X-ray fluorescence background that hinders the weak signals to be observed with Cu-Kα radiation.

Results and discussion:

Figure 1 shows SEM images of two different samples formed respectively during CNT nucleation step and CNT growth step on silicon substrates. The first one (sample 1) is obtained during the early growth stages of CNT formation (15 s) corresponding to CNT nucleation step (figure 1 (a)). The second one (sample 2) is obtained after a longer CCVD synthesis duration (15 min) corresponding to CNT growth step (figure 1(b)).

In figure 1 (a) (sample 1), we observe a layer of iron-based catalyst nanoparticles covering the whole silicon wafer surface, and constituting the main part of the sample. We also observe a significant amount of catalyst particle agglomerates with various dimensions from 500 nm to 2 µm [START_REF] Pinault | Growth of multiwalled carbon nanotubes during the initial stages of aerosol-assisted CCVD[END_REF]. These two area (A: agglomerated particles and B: film of catalyst nanoparticles) are designed designated on figure 1(a) by two white crosses and are shown with higher magnification on figure 1(c) and (d) respectively. Some nanotubes are already occurring even for this early stage and are localized mainly on catalyst particles which are agglomerated. In figure 1 (b), large scale SEM image of sample 2 shows a typical well-aligned MWNT carpet exhibiting a thickness of 400 µm. These MWNT are partially filled with iron based phases as described by Pinault et al. [START_REF] Pinault | Growth of multiwalled carbon nanotubes during the initial stages of aerosol-assisted CCVD[END_REF][START_REF] Bouchet -Fabre | NEXAFS and X-ray scattering study of structure changes after post-annealing treatments of aligned MWNTs[END_REF] .

In this work, we emphasize first on the study of the CNT obtained during the nucleation step by performing both Raman spectroscopy and X-Ray Diffraction analysis (XRD) on sample 1.

These two techniques are complementary: Raman spectroscopy gives spatially localized information (laser spot ~350nm) whereas XRD allows one to perform statistical average (XRD beam ~1 mm for this experiment). For this sample grown during the early stages and thus exhibiting a low amount of carbon, Raman spectroscopy is sensitive to both iron-based and carbon phases whereas XRD is mainly sensitive to iron-based phases.

Figure 2 shows Raman spectroscopy performed on the top surface and XRD measurements on sample 1. White crosses designed in figure 1 (a) correspond to the two areas (A and B) studied by Raman spectroscopy. The first spectrum is achieved on the top surface of agglomerated catalyst particles (figure 2 (a,c)) and the second one on the film of catalyst nanoparticles (figure 2 (b,d)). On particle agglomerates (figure 2), we can observe several peaks between 100 and 600 cm -1 (figure 2 (a)) and one at 1316 cm -1 (figure 2 (c)).We can assign most of them to iron oxides phases. Peaks at 226, 245, 294, 410, 610, 1316 cm -1 are due to α-Fe 2 O 3 whereas the one at 663 cm -1 is assigned to magnetite Fe 3 O 4 [START_REF] De Faria | Raman microspectroscopy of some iron oxides and oxyhydroxides[END_REF][START_REF] Oh | Characterization of Iron Oxides Commonly Formed as Corrosion Products on Steel[END_REF][START_REF] Nasibulin | Simple and rapid synthesis of α-Fe 2 O 3 nanowires under ambient conditions[END_REF] (table 1).

On the film of nanoparticle, the peak positions assigned to α-Fe 2 O 3 are very difficult to identify whereas the peak related to Fe 3 O 4 at 663 cm -1 is clearly occurring. This assignment obtained through spatially localised analysis is in agreement with XRD [START_REF] Pinault | Growth of multiwalled carbon nanotubes during the initial stages of aerosol-assisted CCVD[END_REF][START_REF] Heresanu | Nature of the Catalyst Particles in CCVD Synthesis of Multiwalled Carbon Nanotubes Revealed by the Cooling Step Study[END_REF]. This is the first XRD identification of α-Fe 2 O 3 in such samples, due probably to a too low content of such phase in previous studies [START_REF] Pinault | Growth of multiwalled carbon nanotubes during the initial stages of aerosol-assisted CCVD[END_REF][START_REF] Heresanu | Nature of the Catalyst Particles in CCVD Synthesis of Multiwalled Carbon Nanotubes Revealed by the Cooling Step Study[END_REF][START_REF] Pinault | Carbon nanotubes produced by aerosol pyrolysis: growth mechanisms and postannealing effects[END_REF][START_REF] Pichot | Evidence of strong nanotube alignment and for iron preferential growth axis in multiwalled carbon nanotube carpets[END_REF].

The presence of iron oxides has also been reported in the case of CNT growth by CCVD [START_REF] Fan | Self-Oriented Regular Arrays of Carbon Nanotubes and Their Field Emission Properties[END_REF][START_REF] Mauron | Synthesis of oriented nanotube films by chemical vapor deposition[END_REF][START_REF] Schnitzler | Utilization of iron oxide film obtained by CVD process as catalyst to carbon nanotubes growth[END_REF][START_REF] Dijon | How to switch from a tip to base growth mechanism in carbon nanotube growth by catalytic chemical vapour deposition[END_REF]. However, such oxide phases are not the catalytic phases in the process we use, where nanoparticles giving rise to CNT growth were shown to be cementite nanoparticles [START_REF] Landois | Growth of aligned multi-walled carbon nanotubes: First in situ and time-resolved X-ray diffraction analysis[END_REF]. We have shown that the occurrence of oxides is due to iron oxidation during the cooling step [START_REF] Heresanu | Nature of the Catalyst Particles in CCVD Synthesis of Multiwalled Carbon Nanotubes Revealed by the Cooling Step Study[END_REF], and the present results indicate that the oxidation is not complete for all iron-based particles since Fe 3 O 4 is detected. In addition, in our study no oxygen-containing promoters such as traces of water have been added to the reactive gas phase. Catalyst particles are formed in-situ in the gas phase from ferrocene decomposition initiating quasi instantaneously the decomposition of toluene and liberating hydrogen in the gas-phase, thus preventing any oxidation of the different phases during CNT growth even with oxygen contamination, as shown through our in-situ study [START_REF] Landois | Growth of aligned multi-walled carbon nanotubes: First in situ and time-resolved X-ray diffraction analysis[END_REF]. This is, therefore, not comparable to studies involving catalyst films predeposited on substrates which could be oxidized during annealing through oxygen contamination [START_REF] Bayer | Support-Catalyst-Gas Interactions during Carbon Nanotube Growth on Metallic Ta Films[END_REF][START_REF] Oliver | Statistical Analysis of Variation in Laboratory Growth of Carbon Nanotube Forests and Recommendations for Improved Consistency[END_REF].

It is therefore important to localise and analyse the carbon structure in the same areas as the ones of iron-based particles. It is first noteworthy that in both areas (agglomerated particles or films of nanoparticles), carbon could be always detected even if it corresponds to the early stages of CNT formation and if oxidation is occurring during the cooling step, indicating that only a partial oxidation is involved which is corroborated by the occurrence of Fe 3 O 4 .

Concerning the carbon phase, figure 2 (c) shows the Raman response measured on agglomerated catalyst particles of sample 1. Three peaks at 1576, 1605 and 2663 cm -1 (not shown) assigned to sp 2 carbon modes can be distinguished. The first one is due to the in plane bond stretching of all pairs of sp 2 atoms (E 2g zone-centred) usually called the G band which is common to both rings and chains. The second one at ~1605 cm -1 called D', is assigned to noncentred phonons [START_REF] Dresselhaus | Light scattering in graphite intercalation compounds[END_REF][START_REF] Eklund | Vibrational modes of carbon nanotubes[END_REF][START_REF] Thomsen | Raman spectroscopy on single-and multi-walled nanotubes under high pressure[END_REF]. The inset of figure 3 (d) shows the 2D band which is the overtone of the D band (around 1353 cm -1 included on the large band at 1316 cm -1 which is assigned to α-Fe 2 O 3 and called M band in this work (see Table 1)). The D band is due to the breathing mode of sp 2 rings [START_REF] Ferrari | Interpretation of Raman spectra of disordered and amorphous carbon[END_REF]. Some other bands around 1100 cm -1 (figure 3 (d) and figure 4 (d-e)) and 1500 cm -1 (mode at 1412 cm -1 on figure 2 (d)) are detected which are usually observed in amorphous carbon and disappear with increasing order in sp 2 carbon materials [START_REF] Ferrari | Interpretation of Raman spectra of disordered and amorphous carbon[END_REF][START_REF] Tan | Polarization properties, high-order Raman spectra, and frequency asymmetry between Stokes and anti-Stokes scattering of Raman modes in a graphite whisker[END_REF]. and the D' band is around 1614 cm -1 . In this spectrum, we clearly identify the D band at 1350 cm -1 and its overtone at 2656 cm -1 (not shown). The difference in G band frequency between these two areas of the sample is due to two different carbon structural organisations.

The one at 1576 cm -1 occurring on the agglomerated catalyst particles, exhibiting the higher oxidation degree (presence of α-Fe 2 O 3 ), corresponds to the one already reported for aligned MWNT carpet [START_REF] Castro | Dynamics of catalyst particle formation and multi-walled carbon nanotube growth in aerosol-assisted catalytic chemical vapor deposition[END_REF], whereas the G band at 1580 cm -1 , detected on the film of nanoparticles exhibiting a lower oxidation degree (presence of mainly Fe 3 O 4 ), corresponds to the one reported for graphitic systems exhibiting a significant amount of defects [START_REF] Obraztsova | Raman identification of onion-like carbon[END_REF]. These results obtained on the top surface of the two areas are corroborated by SEM images (figure 1 (a) and(c)) where we can clearly observe the occurrence of CNT on particle agglomerates while only very few CNT-like morphology are observed on the catalyst nanoparticle film.

Therefore, Raman spectroscopy reveals that on samples obtained during the nucleation step, both iron-based phases and carbon phases are together detected, even if oxidation occurs during the cooling step. In addition, the carbon structure is less defective in the areas in which α-Fe 2 O 3 is predominant (higher oxidation degree) while it is more defective (see D and G band intensities and frequencies in addition with the presence of a large background centered around 1400 cm -1 due to amourphous carbon) in the areas in which Fe 3 O 4 is predominant (lower oxidation degree). Assuming that oxidation of carbon-encapsulated iron-based nanoparticles involves the formation of vacancies in carbon structure through the formation of CO or CO 2 vapours and then the oxidation of iron-based nanoparticles [START_REF] Feng | In Situ TEM observation of the gasification and growth of carbon nanotubes using iron catalysts[END_REF][START_REF] Fedoseeva | Effect of oxidation and heat treatment on the morphology and electronic structure of carbon-encapsulated iron carbide nanoparticles[END_REF], the trends obtained in our study cannot be explained only by an oxidation phenomenon. Therefore, the identification by Raman spectroscopy of both iron-based and carbon phase in the same areas combined to SEM and XPS measurements [START_REF] Pinault | Growth of multiwalled carbon nanotubes during the initial stages of aerosol-assisted CCVD[END_REF] indicates that, as soon as iron-based catalytic nanoparticles are deposited, carbon structures organised into more or less defective sp 2 carbon forming graphene sheets (Raman and XPS) and arranged concentrically (presence of nanotubes shown by SEM) are formed [START_REF] Pinault | Growth of multiwalled carbon nanotubes during the initial stages of aerosol-assisted CCVD[END_REF].

In order to go further in the analysis of the CNT nucleation step, a spatially resolved analysis of the carbonaceous structure around the catalyst particles is of high importance. This is possible thanks to the recent developments in spatially resolved Raman spectroscopy which exhibit potentialities to get Raman mapping of both the iron-based and carbon phase. Thus, we studied the spatial distribution of C and Fe elements on sample 1 by XY-spatially resolved Raman imaging. Raman intensity maps for iron-based phases are reported on figure 3 (a) and (b) by using the most intense peak at 226 cm -1 for α-Fe 2 O 3 and 663 cm -1 for Fe 3 O 4 . Typical spatially resolved Raman intensity maps of carbon using the G band are reported on figure 3 (c). In addition, the acquisition of a Raman spectrum at a determined position on the agglomerated catalyst particles (figure 3 (d)), indicates that all iron oxides and carbon phases are occurring all together.

These intensity maps (figures 3 a, b, c) demonstrate clearly that carbon and iron are localized at the same position. Indeed the intensities related to iron oxides are proportional to the one related to carbon and the maximum of their intensity corresponds to each other and to the area on which agglomerated particles are observed. Therefore, these results confirm that sp 2 carbon nucleates preferentially on iron-based particles and that a high amount is more specifically located on areas in which particles are agglomerated, at least during this very early CNT growth stages. Therefore, during this early stage of CNT formation for which a steady state has, still, not been reached this result suggests that a higher amount of carbon is generated through catalytic decomposition of the carbon source in areas where the quantity of nanoparticles is predominant.

In order to go into details with these results, a depth scan in (z, x) directions (figure 4) has been performed on agglomerated particles. Figure 4 (a) to (c) indicate that carbon (G band) and iron-based phases (α-Fe 2 O 3 at 226 and 1316 cm -1 ; same results for Fe 3 O 4 not shown) are localized at the same z position and slightly shifted in x direction. This shift is in agreement with SEM observations showing that several bended CNT are growing from agglomerated particles and they are not precisely in a vertical direction (figure 1 (c)). We can directly observe the intensity variation of these two phases as a function of z position, that is to say according to the thickness of catalyst particles and CNT deposited on the surface.

In figure 4 (d) and (e), we present the Raman spectra of carbon recorded along the blue line (designed in figure 4 (a-c), z is a relative coordinate). These Raman spectra were performed at the agglomerated nanoparticle scale (very local scale as compared to the analysis along the carpet, see below). All the presented spectra along this z line are representative of depth scans realized on several different agglomerates, from their base to their top, of this sample. The [START_REF] Ferrari | Interpretation of Raman spectra of disordered and amorphous carbon[END_REF][START_REF] Obraztsova | Raman identification of onion-like carbon[END_REF]. In addition up to around 1.5 µm high, the decrease of the G band frequency down to 1568 cm -1 is attributed to the progressive curvature of graphene planes in carbon onion like structure [START_REF] Dresselhaus | Light scattering in graphite intercalation compounds[END_REF][START_REF] Obraztsova | Raman identification of onion-like carbon[END_REF]. Then, from around 1.5 µm up to 3 µm, the G band intensity saturates while its frequency increases from 1568 cm -1 to 1574 cm -1 . Finally, the Raman spectra on the top of the particle, corresponds to the MWNT synthesis by the process of this study [START_REF] Castro | Dynamics of catalyst particle formation and multi-walled carbon nanotube growth in aerosol-assisted catalytic chemical vapor deposition[END_REF].

Regarding the D band, the intensity is quite constant in the upper part of the agglomerate even if between 1.25 to around 1.5 µm a decrease of the intensity occurs. This decrease of the D band intensity is also visible in the I D /I G ratio indicating an improvement of the structural carbon organisation, due probably to a decrease of sp 3 /sp 2 ratio [START_REF] Ferrari | Interpretation of Raman spectra of disordered and amorphous carbon[END_REF] when CNT are growing.

The ratio between the intensity of α-Fe 2 O 3 band (at 226 cm -1 ) and of the G band change decreases along the agglomerated catalyst particles from their base to their top. This indicates that the significant increase of the G band occurs while the intensity of α-Fe 2 O 3 (and Fe 3 O 4 not shown) decreases only slightly. Thus, the base of the agglomerate is composed of low amount of sp² carbon versus iron oxide. This amount increases progressively up to around 1 µm high and then remains stable in the upper part of the particles (Figure 4 (g)).

Figure 4 (i) reports the I D /I G ratio which is related to structural defects in nanotubes: the smaller this ratio is the higher the degree of crystallinity is [START_REF] Fedoseeva | Effect of oxidation and heat treatment on the morphology and electronic structure of carbon-encapsulated iron carbide nanoparticles[END_REF][START_REF] Tunistra | Raman Spectrum of Graphite[END_REF]. Figure 4 (i) indicates that this ratio remains almost constant (around 1) up to 1.5 µm high which is the signature of the progressive increase of D and G band intensities. Then from 1.5 µm up to the top of the particles, the I D /I G ratio reaches a value of 0.5 which corresponds to a better organisation on sp² carbon as a CNT structure.

All these depth scan analysis highlights that the structural organization of carbon changes significantly from the base of particles to their top. In the first µm at their base, Raman results indicate onion like structures with a significant amount of defects whereas in the upper parts of the particles, carbon becomes more organised. It is also important to note that the amount of sp² carbon increases in the basal region of the particles. Defects in sp 2 carbon structure occurring at the base of the particle could be explained by the metal catalysed oxidation of carbon [START_REF] Feng | In Situ TEM observation of the gasification and growth of carbon nanotubes using iron catalysts[END_REF][START_REF] Fedoseeva | Effect of oxidation and heat treatment on the morphology and electronic structure of carbon-encapsulated iron carbide nanoparticles[END_REF][START_REF] Zhou | Diffusion of Metal in a Confined Nanospace of Carbon Nanotubes Induced by Air Oxidation[END_REF][START_REF] Booth | Discrete Dynamics of Nanoparticle Channelling in Suspended Graphene[END_REF]. This phenomenon could occur during the cooling step in helium gas containing oxygen traces since iron-based particles are converted into iron oxide particles.

However, such a phenomenon should be very limited since the partial pressure of oxygen in helium is very low (3 ppm-mol H 2 O + 2 ppm-mol de O 2 ) contrary to the oxidation studies reported in the literature for which the oxygen partial pressure is high [START_REF] Feng | In Situ TEM observation of the gasification and growth of carbon nanotubes using iron catalysts[END_REF][START_REF] Fedoseeva | Effect of oxidation and heat treatment on the morphology and electronic structure of carbon-encapsulated iron carbide nanoparticles[END_REF][START_REF] Zhou | Diffusion of Metal in a Confined Nanospace of Carbon Nanotubes Induced by Air Oxidation[END_REF]. According to the studies reported in the literature, air oxidation involves that a part of carbon atoms bindings with oxygen atoms leaves the sample in form of carbon monoxide or dioxide, creating vacancies in the carbon shells which serves as channels for outward diffusion of metallic core and penetration of oxygen in the interior, thus creating large size hollow Fe 2 O 3 /C ox nanoparticles [START_REF] Feng | In Situ TEM observation of the gasification and growth of carbon nanotubes using iron catalysts[END_REF][START_REF] Fedoseeva | Effect of oxidation and heat treatment on the morphology and electronic structure of carbon-encapsulated iron carbide nanoparticles[END_REF][START_REF] Zhou | Diffusion of Metal in a Confined Nanospace of Carbon Nanotubes Induced by Air Oxidation[END_REF][START_REF] Booth | Discrete Dynamics of Nanoparticle Channelling in Suspended Graphene[END_REF]. However, in our study we could not identify neither hollow particle by electron microscopy nor carbon-oxygen bonding by Raman spectroscopy.

In addition, according to the oxidation phenomenon reported in the literature, carbon should be uniformly defective all along the iron-based particle, which is not the case in our study.

Therefore, the carbon structural difference occurring along the iron-based particles from their base to their top cannot be explained only by oxidation, especially because the cooling process is performed with only a very low partial pressure of oxygen as compared to literature. We believe that such changes in carbon structure along the particles indicate that the very early nucleation steps of carbon involves the formation of defective graphene layers surrounding the particles and forming cap-like morphologies. This suggests that the structure of nucleating carbon depends most probably on its location with respect to catalyst particles and that the carbon structural organisation is progressively improving from the particles up to the upper parts of CNT. Such a behavior regarding the carbon structure around the catalyst particle analyzed at this Raman scale is difficult to compare with the literature. Papers studying locally structural changes around the catalyst particles either through in-situ TEM or XRD [START_REF] Yoshida | Atomic-Scale In-situ Observation of Carbon Nanotube Growth from Solid State Iron Carbide Nanoparticles[END_REF][START_REF] Wirth | The Phase of Iron Catalyst Nanoparticles during Carbon Nanotube Growth[END_REF][START_REF] Sharma | Site-Specific Fabrication of Fe Particles for Carbon Nanotube Growth[END_REF] are focused on the structural changes occurring on the catalyst particles and show the forming graphitic structure, but no quantitative analysis of the carbon structure comparable to Raman spectroscopy has been performed. In addition, literature does not report any quantitative and spatially resolved analysis of carbon structure around catalyst nanoparticles for nanotube growth.

To go further in the understanding of carbon organisation, we also perform Raman analysis in conjunction with X-Ray microdiffraction analysis on aligned MWNT carpet (sample 2) grown for longer durations (15 min).

A Raman statistical analysis was achieved along the carpet from the base to the top (black arrow in figure 1 (b)). The scale analysis of the carpet to follow the carbon structure along the carpet is not the same that for agglomerate. The number of spectra concerning the agglomerate (size of few µm) is close to this performed along the carpet (size of 400 µm) to have changes of carbon structure along agglomerate and along the carpet. Figure 5 (a) shows representative Raman signature of the top, the middle and the base of the carpet indicated by white rectangles on the SEM image of the carpet cross-section (figure 1 (b)). We identified on these spectra, the D, G and 2D bands whatever the position along the carpet. The background of the spectra and the width of these bands are increasing near the top of the carpet. The I D /I G ratio varies from 0.5 to 1.35 from the base to the top of the carpet (inset figure 5 (a)). The I D /I G value for our raw sample is similar to the one reported for aligned MWNT raw samples synthesized by CVD [START_REF] Amara | Interaction of carbon clusters with Ni(1 0 0): Application to the nucleation of carbon nanotubes[END_REF] and is significantly lower than the ones generally reported for MWNT produced by CVD [START_REF] Singh | Towards the production of large-scale aligned carbon nanotubes[END_REF]. The values for both I D /I G ratio and width bands are related to structural defects in nanotubes: the smaller these parameters are, the higher the degree of crystallinity is [START_REF] Tunistra | Raman Spectrum of Graphite[END_REF][START_REF] Tan | Probing the phonon dispersion relations of graphite from the double-resonance process of Stokes and anti-Stokes Raman scatterings in multiwalled carbon nanotubes[END_REF]. The Raman spectrum of the top shows an important background due to iron based-phases luminescence and to carbon induced-defects indicating also a higher carbon structural disorder at the top of the carpet. This could be assigned to MWNT which are quite bended and decorated with residual catalyst encapsulated by carbon (inset figure 5 (e)).

X-Ray micro-diffraction as well as GISAXS measurements performed by different groups show the possibility to extract the degree of MWNT alignment on such a carpet [START_REF] Pichot | Evidence of strong nanotube alignment and for iron preferential growth axis in multiwalled carbon nanotube carpets[END_REF][START_REF] Meshot | High-Speed in Situ X-ray Scattering of Carbon Nanotube Film Nucleation and Self-Organization[END_REF]. In our case XRD analyses were performed in transmission from the base to the top of the carpet as indicated in figure 1 (b) by the black arrow. As reported by Pichot et al. [START_REF] Pichot | Evidence of strong nanotube alignment and for iron preferential growth axis in multiwalled carbon nanotube carpets[END_REF], the intensity of the (002) peak at Q=1.83 Å -1 , corresponding to the scattering from the MWNT walls, is angularly modulated. By fitting this modulation, it is possible to determine the alignment degree of the CNT according to the normal of the substrate surface [START_REF] Pichot | Structural and mechanical properties of single-wall carbon nanotube fibers[END_REF][START_REF] Nakamura | Disorder-induced line broadening in firstorder Raman scattering from graphite[END_REF]. The two Lorentzian peaks of the (002) measured in the reciprocal space correspond to the orientation distribution function in the direct space which is a Lorentzian 3/2 [START_REF] Pichot | Thesis : Etudes structurales par diffusion des rayons X d'une assemblée de nanotubes de carbone alignés. relation avec les mécanismes de croissance et les propriétés Physiques[END_REF]. Therefore, more defective carbon (high I D /I G ratio) is located by Raman spectroscopy at the top of the carpet (inset figure 5 (a)) and corresponds to bended (SEM observations) and less aligned (XRD analysis) nanotubes as well as carbon encapsulated residual catalyst particles.

An improvement of the carbon structural quality (progressive decrease of I D /I G ratio) is occurring in the core and at the base of the carpet and corresponds to quite straight, well aligned and pure nanotubes. At the base of the carpet corresponding to the place where catalyst particles are localized, i.e. in the areas in which nanotubes and catalyst particles are together present, the I D /I G ratio reaches the lowest value of 0.5 which is similar to the value obtained from depth scan analysis at the top of agglomerated particles, indicating that the organisation of carbon located in the catalyst particle areas is quite high and remains similar whatever the formation stage of nanotube (nucleation or growth step). Taking into account that these aligned nanotubes are growing through a base growth mechanism [START_REF] Pinault | Evidence of Sequential Lift in Growth of Aligned Multiwalled Carbon Nanotube Multilayers[END_REF], this result strongly suggests that the structural organisation of carbon formed around the active catalyst particles at the base of the carpet remains quite similar all along CNT nucleation and growth.

The base growth mechanism of our CNT synthesized by AA-CCVD was previously described [START_REF] Tessonnier | Recent Progress on the Growth Mechanism of Carbon Nanotubes: A Review[END_REF][START_REF] Pinault | Evidence of Sequential Lift in Growth of Aligned Multiwalled Carbon Nanotube Multilayers[END_REF], briefly the carbon species are integrated by the catalyst particles and then CNT are growing on the top of these particles, the catalyst particles remain attached at the CNT base all along the growth and are staying at the base of the carpet. Considering such a base growth mechanism, the top of the carpet is composed of nanotubes firstly formed. Therefore, we can

Conclusions

Spatially resolved Raman spectroscopy analyses on samples prepared during CNT nucleation stage show that it is possible to co-localise and analyse simultaneously the structure of ironbased catalyst particles and carbonaceous materials resulting from the nucleation step of aligned MWNT. The structural organisation of carbon changes significantly from the base of agglomerated particles to their top. At their base, Raman reveals onion like structures with a significant amount of defects suggesting that the very early nucleation steps of carbon involves the formation of defective graphene layers. Then in the upper parts of these particles, carbon becomes organised suggesting that the structure of nucleating carbon depends on its location with respect to the catalyst nanoparticle. In addition, the amount of carbon is found to be lower at the vicinity of the catalyst nanoparticle (nanotube base) and increases in the upper part of nanotube. These results are of main interest for the understanding of growth mechanism of aligned carbon nanotube, which is crucial for applications in different fields of nanotechnology. Table 1. Raman wavenumbers of iron oxide phases (most intense peaks are underlined).

Tables

Iron oxides

Wavenumbers from literature (cm -1 ) Our experiments (cm -

analysis (Figure 2

 2 (e)) involving statically statistically averaged information where all diffraction peaks are attributed to α-Fe 2 O 3 and Fe 3 O 4 (calculation give us a α-Fe 2 O 3 / Fe 3 O 4 mass ratio = 4/3, figure 2 (e))

Figure 2 (

 2 Figure 2 (d) presents Raman spectrum regarding carbon on catalyst nanoparticles film of

1

 1

  carbon and iron structures are changing according to their location along the height of the iron oxide agglomerates. (i) From the base of the agglomerate up to around 1.5 µm high, the G band intensity increases and simultaneously the M band present gradually a shoulder corresponding to the D band (dotted arrow, figure 4 (d)). (ii) Then in the upper 1.25 µm, the G band intensity saturates whereas the D and M bands both decreases (α-Fe 2 O 3 peak at 226 cm -1 decreases also).

Figure 4 (

 4 Figure 4 (f) and (h) show changes of intensity and frequency of respectively the D and the G

  Figure 5 (b) reports the orientation of MWNT according to the normal of the substrate surface by giving the Full Width at Half Maximum of this function and shows that the alignment degree is constant (+/-10° according to the normal of the substrate surface) from the base of the carpet up to approximately 60 µm before the top surface. This is consistent with SEM observations showing that MWNT alignment is better at the base (figure 5 (c)) and in the core (figure 5 (d)) than at the top (figure 5 (e)) of the carpet, and that, at the top, MWNT are quite bended and decorated with particles which are most probably residual catalyst encapsulated by carbon (inset figure 5 (e)).

  compare their structural organisation (I D /I G around 1.3) to the one of CNT formed on the top of catalyst particle agglomerates during the early stages (I D /I G around 0.5). By combining these results to SEM observations, the high I D /I G ratio at the top of the carpet can be explained by the presence of residual catalyst particles coated with carbon and located at the CNT surface (inset figure 5 (e)) as well as amorphous carbon responsible of the important background of the Raman spectrum (black spectrum figure 5 (a)) comparatively to the flat one on the top of the particles in the sample 1 (figure 4 (e)). These contaminations (residual catalyst particles and amorphous carbon) probably appear during the cooling step of the sample; they are predominant at the top of the carpet and decrease progressively, which explain the higher I D /I G ratio at the top of the carpet than the one at the top of the particles.

Figure captions Figure 1 -

 captions1 Figure captions

Figure 2 -

 2 Figure 2 -Raman spectra on sample 1 localized on catalyst particle agglomerates

Figure 3 -

 3 Figure 3-Bi-dimensional (x y) Raman mapping of sample 1 for α-Fe 2 O 3 (a), Fe 3 O 4 (b) and G

Figure 4 -

 4 Figure 4 -Bi-dimensional (x z) Raman mapping of sample 1 for α-Fe 2 O 3 at 226 cm -1 (a), α-Fe 2 O 3 at 1316 cm -1 (denoted M band) (b) and G band (c) with a 600 grooves /mm grating -

Figure 5 -

 5 Figure 5 -(a) Raman spectra (1800 grooves/mm grating) -(b) Micro-diffraction analysis:

Acknowledgement

The authors thank C'Nano Ile de France (microdiffraction detector) and C'Nano Rhône Alpes (Ti-saphire Laser), CPER Rhône-Alpes, Région Ile de France (FEG-SEM) and the french agency ANR (Alucinan project in the P-Nano program) as well as RTRA Triangle de La Physique for financial support.


Denys Dutykh
email: denys.dutykh@univ-savoie.fr

Elena Tobisch
email: elena.tobisch@jku.aturl

OBSERVATION OF THE INVERSE ENERGY CASCADE IN THE MODIFIED KORTEWEG-DE VRIES EQUATION

Keywords: Modulational Instability, energy cascade, modified KdV equation, inverse cascade PACS: 47.35.Bb (Gravity waves), 47.20.Cq (Inviscid instability), 47.27.er (Spectral methods)

In this Letter we demonstrate for the first time the formation of the inverse energy cascade in the focusing modified Korteweg-de Vries (mKdV) equation. We study numerically the properties of this cascade such as the dependence of the spectrum shape on the initial excitation parameter (amplitude), perturbation magnitude and the size of the spectral domain. Most importantly we found that the inverse cascade is always accompanied by the direct one and they both form a very stable quasi-stationary structure in the Fourier space in the spirit of the FPU-like reoccurrence phenomenon. The formation of this structure is intrinsically related to the development of the nonlinear stage of the Modulational Instability (MI). These results can be used in several fields such as the internal gravity water waves, ion-acoustic waves in plasmas and others.

Introduction

Currently there exist three main models which describe the energy cascade in nonlinear wave systems. Each model covers its own range of the nonlinearity parameter. Namely, for strong nonlinearities the Kolmogorov cascade with the widely known exponent k -5 3 (k being the vortex scale) introduced for the hydrodynamic turbulence theory [START_REF] Kolmogorov | The Local Structure of Turbulence in Incompressible Viscous Fluid for Very Large Reynolds Numbers[END_REF]. In the opposite limit of weak nonlinearities, the kinetic weak turbulence spectra k -α (now k is the wavenumber) were introduced, [START_REF] Zakharov | Kolmogorov Spectra of Turbulence I[END_REF]; in this case the spectrum exponent α is not universal anymore and the nonlinearity parameter ak ≲ 10 -2 , where a is the typical wave amplitude. For the intermediate values of the nonlinearity ak = O(10 -1 ) the Dynamic energy cascade (D-cascade) model was proposed for the equations which feature the Modulational Instability (MI) property [START_REF] Kartashova | Energy spectra of 2D gravity and capillary waves with narrow frequency band excitation[END_REF]. This construction was originally described and studied in details for the NLS-type equations [START_REF] Kartashova | Energy transport in weakly nonlinear wave systems with narrow frequency band excitation[END_REF][START_REF] Kartashova | Time scales and structures of wave interaction exemplified with water waves[END_REF].

In particular it was shown [START_REF] Kartashova | Energy spectra of 2D gravity and capillary waves with narrow frequency band excitation[END_REF] that in the focusing (i.e. possessing the MI) NLS equation only one cascading mode is forming due to the intrinsic narrow band spectrum approximation. This prediction was checked numerically and the development of the MI in the NLS is shown on Figure 1. On this bottom panel one can see only one spike in the Fourier spectrum along with the broadening wings.

On the other hand, the modified Korteweg-de Vries (mKdV) equation does not have any restrictions on the wave spectrum. In this study we consider the following focusing mKdV equation:

mKdV(u) ≐ u t + u xxx + 6u 2 u x = 0 . (1.1)
The formation of the direct D-cascade in this equation has been recently observed and investigated numerically in [START_REF] Dutykh | Direct dynamical energy cascade in the modified KdV equation[END_REF]. The direct cascade forms starting from any base wavenumber k 0 > 0. The main properties of the direct cascade will be briefly described in the next Section. Concerning the inverse cascade, it was not observed yet to our knowledge. In this Lettre we aim at presenting solid numerical evidences towards the existence of the inverse D-cascade and studying some of its properties comparing to the direct one.

Direct cascade in the mKdV equation

For our numerical simulations we adopt the set-up used also earlier in [START_REF] Grimshaw | Wave group dynamics in weakly nonlinear long-wave models[END_REF][START_REF] Grimshaw | Short-Lived Large-Amplitude Pulses in the Nonlinear Long-Wave Model Described by the Modified Korteweg-De Vries Equation[END_REF] for the study of the modulation instability in the physical space, with the following initial condition posed on a periodic domain [-ℓ, ℓ] ≐ -π k 0 , π k 0 :

u(x, 0) ≡ u 0 (x) = a 1 + δ sin(K 0 x) sin(k 0 x), (2.1)
where a is the base wave amplitude, δ is the perturbation magnitude and the wavenumbers k 0 , K 0 are chosen such that their ratio k 0 K 0 ∈ Z. It is important to realize that the amplitude a is the measure of the nonlinearity in the mKdV equation, in the contrast to the NLS equation where the nonlinearity is related to the wave steepness aK 0 defined in terms of the modulational wavenumber K 0 .

The main characteristics of the direct D-cascade observed in [START_REF] Dutykh | Direct dynamical energy cascade in the modified KdV equation[END_REF] are as follows:

• (a) the main structure of the cascade is already observable in Fourier spectra from the first instances of the dynamical evolution; • (b) the development of the Modulation Instability (MI) in the physical space occurs at the time scale t ∼ ε -2 and corresponds to the spectral broadening of the cascading modes; the frequencies and energies of the cascading modes are quasi-stationary; • (c) the energy spectrum of the cascade has exponential form, E k ∝ exp(-α k)

which does not depend on the base amplitude a for fixed values of other parameters; • (d ) the increase of the spectral domain yields the increase of the cascade length but does not changes the exponent α;

• (e) the increase of the perturbation magnitude m from 5% to 50% effects the time scale on which the MI occurs in the physical space: higher perturbation accelerates the process; • (f ) the increase of the base wavenumber k 0 to 2k 0 , 3k 0 , . . ., 30k 0 ; changes the exponent α and yields at some point (beginning with ≈ 35k 0 ) the occurrence of the inverse cascade;

The appearance of the direct D-cascade has been clearly observed (characteristic cascades are shown in Figure 2), for the wide range of the initial parameters described in Table 1. For the sake of illustration, on Figure 2 we show the formation of the direct Dcascade for the initial condition (2.1) with the following parameters: k 0 = 1.884, m = 5×10 -2 , K 0 = 0.00785 and a = 8 × 10 -2 on the top panel (a) and a = 1.6 × 10 -1 on the panel (b). These values of parameters were also used in [START_REF] Grimshaw | Short-Lived Large-Amplitude Pulses in the Nonlinear Long-Wave Model Described by the Modified Korteweg-De Vries Equation[END_REF]. However, the numerical resolution was significantly lower in their study and the authors did not report the evolution of the system in the Fourier space. From our simulations it can be easily seen that an increase in the base wave amplitude does not change the shape of the energy spectrum along with the spacing between cascading modes. Moreover, the time instances t = 50 and 90, where these snapshots were taken, indicate that the MI development goes faster with higher amplitudes. The dashed black lines shown on bottom panels show the same exponential fit for both panels.

In the next Section we present the results of our studies of the inverse cascade, using the same parameters as above but with base wavenumbers k 0 shifted toward short waves.

Double cascade in the mKdV equation

In order to solve numerically the mKdV equation on a periodic domain we used the classical Fourier-type pseudo-spectral method along with the 2 3-rule for dealising of nonlinear products. The discretization in time was done with the embedded adaptive 5 th order Cash-Karp Runge-Kutta scheme with the adaptive PI step size control. In most numerical simulations presented below we will use N = 32 768 Fourier harmonics, unless a special remark is made.

On Figure 3 we show the simultaneous formation of the direct and inverse cascades (which will be referred below as the double cascade) at different times. In order to achieve this result the base wavenumber k 0 was shifted approximatively into the middle of the spectral domain (k 0 = 35 × 1.884). One can see that starting from t ≈ 20 (see Figure 3(d )) the MI is already fully developed in the physical space, which corresponds to the appearance of a quasi-stationary structure -double cascade in the Fourier space. For times t ⩾ 20 the evolution of the system follows the FPU-like pattern, without any significant change in the Fourier domain (see Figure 3(e,f )).

As in the case of the isolated direct cascade, the double cascade equally has the exponential decay in the Fourier space. However, the two exponents are slightly different in a way that the direct cascade decays generally faster than its inverse counterpart.

Effect of the amplitude

We performed the same simulation as shown on Figure 3, but with twice the initial wave amplitude a = 2 × 0.08. The results are demonstrated on Figure 4. One can see that the main characteristics of the double cascade remain unaffected, i.e. the number and position of cascading modes along with the exponential shape of the Fourier spectrum. This observation can be made even more precise as we plot the same fitted exponents on bottom panels. One can see on Figure 4(c) a very good agreement between the observed spikes with the observed above exponential decay rate.

In contrast to the previous case with a = 0.08, all the processes are accelerated. Most importantly, the development of the MI in the physical space happens earlier with respect to the former computations shown on Figure 3. Consequently, the process enters in the FPU-like recurrence regime earlier showing that the cascade formation is intrinsically linked to the MI development.

Effect of the perturbation

We studied also the effect of the perturbation magnitude m. We do not report the numerical results here due to the limitations in the Lettre length. However, we report that for m = 0.05 ÷ 0.5 (i.e. 5% to 50% of the base wave amplitude) the same processes described above still take place. Basically, the higher values of the parameter m yield the larger initial amount of the energy in the system which lead to the faster development of the MI.

Effect of the spectral domain

The last series of numerical experiments reported in this Lettre aim at the studying the effect of the spectral domain on the energy cascade shape. With this purpose in mind we keep the same initial conditions as above, however we double the size of the spectral domain (consequently, the number of Fourier modes becomes N = 65 536 correspondingly). These results are shown on Figure 5. Our initial intuition, based on the experience with an isolated direct cascade, indicated that the increase of the spectral domain will lead to the change of the spectrum exponents by preserving the global structure of the double cascade.

Surprisingly, our simulations show that the cascade observed on Figure 3 preserves its shape and location in the Fourier space (0 ⩽ k < 120), while in the additional space (120 ⩽ k ⩽ 240) another (apparently slightly smaller) replica of the initial double cascade appears, shifted farther in the Fourier domain.

Each replica is an equally stable structure in the Fourier space as the double cascade described above (i.e. with evolution the number and location of cascading modes remain the same along with the spectrum exponents). However, they are different from the isolated double cascade observed for precisely the same initial conditions. The main difference consists, for example, in the number of cascading modes which obviously affects the exponent of the energy spectrum shape.

By comparing the development of the MI in the physical space on Figures 3(d,e) and 5(b,c) we can see that in the case of the smaller spectral domain the MI is more developed comparing to the latter. This phenomenon can be explained by the presence of two recurrent coherent structures (instead of one) each of them having a stabilizing effect on the MI.

Summary

Integrability of a specific evolutionary nonlinear PDE is a mathematical issue related to the spectrum of the corresponding Lax operator [START_REF] Ablowitz | Solitons and the Inverse Scattering Transform[END_REF]. It gives a priori no information about the possible scenarios of the energy transport over the scales (in direct, inverse or both directions) as well as about the shape of the energy spectra. A good illustration is the nonlinear Schrödinger (NLS) equation which is integrable but this fact does not prevent numerous researchers from studying the corresponding energy spectra numerically, experimentally and analytically even nowadays [START_REF] Kuksin | Derivation of the Kolmogorov-Zakharov equation from the resonant-averaged stochastic NLS equation[END_REF]. Let us recall that the complete integrability of the NLS equation was proved by Zakharov & Shabat in 1972 [START_REF] Zakharov | Exact Theory of Two-dimensional Self-focusing and Onedimensional Self-modulation of Waves in Nonlinear Media[END_REF], while the observation of the inverse cascade in the NLS was done only 20 years later [START_REF] Dyachenko | Optical turbulence: weak turbulence, condensates and collapsing filaments in the nonlinear Schrödinger equation[END_REF].

The energy transport in the mKdV, also known to be integrable by the inverse scattering transform, is even less studied than the NLS, and the existence of the inverse energy cascade in the mKdV is a novel phenomenon which we believe should be studied further on, both numerically and experimentally. In this Lettre we presented some preliminary but very promising results on the formation of the inverse energy cascade in the mKdV equation. Briefly our findings can be summarized as:

• Contrary to the direct D-cascade which can be observed for any k 0 > 0, the inverse cascade starts to appear only for large values of k 0 . • It is not possible to observe an isolated inverse D-cascade, since the direct cascade forms for any choice of k 0 . As a result, our observations show the so-called double cascade which is composed of at least one instance of the direct and inverse cascades. • The exponents of the inverse and direct cascades are slightly different. In general, the direct cascade tends to decay faster. Moreover, the values of the exponents were shown not to depend on the initial wave amplitude, but only on the initial wavenumber k 0 . • The double cascade appears to be a very robust quasi-stationary structure in the Fourier space. In other words, the system at this stage enters into the FPU-like recurrent regime. • The formation of the D-cascade is accompanied by the nonlinear development of the MI. • The increase of the spectral domain yields the formation of two (and possibly even more) replica of the double cascade, each containing roughly one half of the initial energy. Our conjecture is that the further increase in the spectral domain will lead to the appearance of more and more replica of this double cascade structure. Or in other words, we will see the appearance of more and more recurrent structures sharing the total system energy. This process may be regarded as a mechanism of the MI stabilization.

The problem of the MI stabilization was considered by several research groups [START_REF] Segur | Stabilizing the Benjamin-Feir instability[END_REF] and the main stabilization mechanism which was put forward has been related to some sort of dissipation. In this study we propose another mechanism which can work in fully conservative systems.

The present investigation is restricted to the case of moderate nonlinearities (a ∼ O(10 -1 )), since in this case it is straightforward to find an unstable mode k 0 . When we go to the field of the finite amplitude waves (a ∼ O(1)), the stability study becomes a complicated mathematical problem [START_REF] Johnson | On the modulation equations and stability of periodic generalized Korteweg-de Vries waves via Bloch decompositions[END_REF]. The investigation of the D-cascade formation in strongly nonlinear regimes, occuring for example in plasma physics [START_REF] Ruderman | Dynamics of modulationally unstable ionacoustic wavepackets in plasmas with negative ions[END_REF], is among our next priorities.

Figure 1 .

 1 Figure 1. Modulational instability in the NLS equation.

Parameter Range Amplitude a = 10 - 2 ÷ 2 . 4 × 10 - 1

 10224101 Perturbation magnitude m = 5 × 10 -2 ÷ 5 × 10 -1 Base wavenumber k 0 = 1.8 ÷ 60.0 Simulation time horizon T = 40 ÷ 2000 Fourier harmonics N = 1024 ÷ 131 072

a = 1 . 6 × 10 - 1 , t = 281 Figure 2 .Figure 3 .

 1610128123 Figure 2. Direct cascade in the mKdV equation. The energy spectrum exponential fit shown on bottom panels is the same for both amplitudes and equal to e -1.45k .

  Numerical solution of the mKdV equation at t = 10.000 t = 30

Figure 4 .

 4 Figure 4. Development of the double cascade for the initial wave amplitude a = 2 × 0.08.

t = 25 Figure 5 .

 255 Figure 5. Formation of two copies of the double inverse and direct cascades in a two times larger spectral domain. The exponents shown with dashed lines were measured in the simulations from Figure3. The observed discrepancy shows the effect of the spectral domain on the energy spectrum shape.

Table 1 .

 1 Parameters range used in numerical simulations.

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Acknowledgments

This research has been supported by the Austrian Science Foundation (FWF) under projects P22943-N18 and P24671.