Tripti Rastogi

Frederick K Ho

Patrick Rossignol

Thomas Merkling

Javed Butler

Andrew Clark

Timothy Collier

Christian Delles

Wouter Jukema

Stephane Heymans

Roberto Latini

Alexandre Mebazaa

Pierpaolo Pellicori

Peter Sever

Jan A Staessen

Lutgarde Thijs

John G Cleland

Naveed Sattar

Faiez Zannad

Nicolas Girerd
email: n.girerd@chru-nancy.fr

Affiliations

Comparing and contrasting risk factors for Heart failure in patients with and without history of myocardial infarction: Data from HOMAGE and the UK Biobank

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Heart failure (HF) is one of the leading causes of morbidity and mortality throughout the world.

With ≥ 37 million people already affected worldwide, the number will continue to rise for the next 20 years [START_REF] Ziaeian | Epidemiology and aetiology of heart failure[END_REF]2 . Increasing life expectancy and prevalence of risk factors such as hypertension, diabetes, obesity and coronary artery disease, especially in younger populations, will further add to the steadily increasing number of HF patients 3 . Despite advances in treatment, mortality rates remain particularly high with poor 5-year survival [START_REF] Shah | Heart Failure With Preserved, Borderline, and Reduced Ejection Fraction: 5-Year Outcomes[END_REF] . Hence, early identification and management of risk factors is critical for preventing or delaying the onset of HF.

The implementation of preventive measures relies on the correct identification of individuals at higher risk. Several scores have been developed with the aim of identifying populations at higher risk taking into account variables such as age, sex, body mass index (BMI), lipid levels, kidney function and other comorbidities including diabetes and hypertension [START_REF] Jacobs | Risk for Incident Heart Failure: A Subject-Level Meta-Analysis From the Heart "OMics" in AGEing (HOMAGE) Study[END_REF][START_REF] Khan | 10-Year Risk Equations for Incident Heart Failure in the General Population[END_REF] . These scores assess HF risk irrespectively of the presence of prior myocardial infarction (MI) i.e., these risk score assess the HF risk based on a fixed value assigned to each risk factor including MI. However, the strength of a relation between a risk factor and outcome might depend upon whether a patient has previously had an MI.

Different aetiologies, risk factors and pathophysiological mechanisms, in combination with concomitant diseases, renders HF a complex and heterogeneous syndrome with several phenotypes [START_REF] Shah | Phenotype-Specific Treatment of Heart Failure With Preserved Ejection Fraction: A Multiorgan Roadmap[END_REF][START_REF] Ho | Predicting Heart Failure With Preserved and Reduced Ejection Fraction: The International Collaboration on Heart Failure Subtypes[END_REF] . However, HF with history of prior MI appears distinct from HF without prior MI in many aspects including: ischemic injury, inflammation and neurohormonal pathways involved in cardiac remodelling [START_REF] Bhatt | Adverse Remodeling and Reverse Remodeling After Myocardial Infarction[END_REF] . Importantly, ischemic/non-ischemic HF is often the only characterization reported in large registries regarding HF aetiology, thus emphasizing the cardinal importance of MI history. Given the different pathophysiological background of HF with and without prior MI, classical risk factors (such as age, blood pressure, diabetes, dyslipidemia, kidney disease, etc.) may contribute differentially to HF onset in patients with and without MI. History of MI is known to be associated with worse outcomes in patients with overt HF and could be related to MI history rather than the intrinsic prognosis of ischemic HF [START_REF] Kapoor | Precipitating Clinical Factors, Heart Failure Characterization, and Outcomes in Patients Hospitalized With Heart Failure With Reduced, Borderline, and Preserved Ejection Fraction[END_REF][START_REF] Fonarow | Factors identified as precipitating hospital admissions for heart failure and clinical outcomes: findings from OPTIMIZE-HF[END_REF] .

We compared the clinical characteristics associated with the progression to HF in patients with or without a history of MI in patients at higher risk of HF in the HOMAGE database and validated our results in the UK Biobank population-based cohort.

Methods

Derivation cohort: Patients included in the Heart 'OMics' in AGEing (HOMAGE) merged database were studied. Briefly, the HOMAGE database included 52 631 study participants from 21 studies from eight European countries which enrolled subjects with overt cardiovascular (CV) disease or at risk of CV disease or healthy individuals. A detailed description of the database is provided elsewhere [START_REF] Ferreira | Renal function estimation and Cockroft-Gault formulas for predicting cardiovascular mortality in population-based, cardiovascular risk, heart failure and postmyocardial infarction cohorts: The Heart 'OMics' in AGEing (HOMAGE) and the high-risk myocardial infarction database initiatives[END_REF][START_REF] Jacobs | Heart 'omics' in AGEing (HOMAGE): design, research objectives and characteristics of the common database[END_REF] . Among this large database, patients identified to be at a higher risk for HF (i.e. without HF but with significant risk for HF based on their comorbidities/clinical history) were included in the analysis reported here [START_REF] Ferreira | Renal function estimation and Cockroft-Gault formulas for predicting cardiovascular mortality in population-based, cardiovascular risk, heart failure and postmyocardial infarction cohorts: The Heart 'OMics' in AGEing (HOMAGE) and the high-risk myocardial infarction database initiatives[END_REF][START_REF] Jacobs | Heart 'omics' in AGEing (HOMAGE): design, research objectives and characteristics of the common database[END_REF] . Patients included were originally from four separate cohorts (ASCOT, DYDA, HVC, PROSPER) in which included patients had higher risk of HF and provided sufficient follow-up to assess the incidence of HF onset (Supplementary figure 1). Patients were included if the baseline data on age, sex, body weight and height was available. Patients who had HF at baseline or for whom follow-up data on outcome was unavailable were excluded.

Outcomes:

The objective of the study was to identify whether the risk factors for HF onset differed depending on the presence (MI+) or absence of prior MI (MI-). For this objective, the selected outcome was time to HF onset as defined by hospitalization for HF. The endpoints for each study were adjudicated in the respective cohort and trials and the committee within the HOMAGE consortium retrospectively assessed the quality of endpoint adjudication for each study (supplementary table 1) [START_REF] Jacobs | Risk for Incident Heart Failure: A Subject-Level Meta-Analysis From the Heart "OMics" in AGEing (HOMAGE) Study[END_REF][START_REF] Jacobs | Heart 'omics' in AGEing (HOMAGE): design, research objectives and characteristics of the common database[END_REF] .

Validation cohort:

The results of our analysis were replicated in UK Biobank, a large population-based, prospective observational study with 502,493 middle-aged and elderly participants from the United Kingdom [START_REF] Fry | Biobank Biomarker Project: Companion Document to Accompany Serum Biomarker Data[END_REF][START_REF] Ho | Child maltreatment and cardiovascular disease: quantifying mediation pathways using UK Biobank[END_REF] . The health outcomes were ascertained through data linkage to hospitalization records and incident HF was defined according to the International Classification of Diseases-10 codes. Subjects with prevalent heart failure at baseline were excluded from this analysis.

Brief study design and baseline characteristics of the study participants included from the UK Biobank are presented in the supplementary material and supplementary table 2.

All of the studies were conducted in accordance with Good Clinical Practice guidelines and applicable national regulations and all study participants provided written informed consent.

Statistical analysis:

For the descriptive analyses, continuous variables are expressed as mean ± standard deviation (SD) for normally distributed data or as median (Q1-Q3) for skewed data.

Categorical variables are expressed as proportions (%).

Univariable Cox regression models were used to calculate the hazard ratio (HR) for HF onset associated with each risk factor. Due to differences in duration of follow-up between each cohort, the cohorts were added as strata in the Cox regression models for the HOMAGE database. The stratified Cox-proportional hazard models were used because the baseline risk and hazards in each cohort constituting HOMAGE cohort were different and unrelated [START_REF] Rossello | Survival analyses in cardiovascular research, part II: statistical methods in challenging situations[END_REF] . Age, systolic blood pressure (SBP) and estimated glomerular filtration rate (eGFR) were categorized due to non-linearity. A backward selection procedure was applied to determine the variables to be included in the multivariable models.

Total cholesterol, body weight and height were not included in the backward selection due to their high collinearity with other variables (only low density lipoprotein-cholesterol (LDL-c) and BMI were considered). The variables found to be significantly associated with either the patients with prior MI or the patients with cardiovascular risk factors but no prior MI were then included in the final multivariable model with same variables for each group. Interactions between each risk factor and history of MI were examined in both backward selection model (interaction for unadjusted model) and fully adjusted model (additive and multiplicative interaction for adjusted model), adding multiplicative interaction terms to survival models. The discriminative value of multivariable models for HF onset was assessed using Harrell's c-index. We also ran sensitivity analyses with ACEi/ARBs and Betablockers use, forced in the Cox regression models. The competing risk analysis with non-HF death as competing event was conducted using Fine and Gray modification of Cox-proportional hazard models.

For validation analysis, a dedicated adjustment was performed to calculate hazard ratios of risk factor associated with HF onset in UK Biobank.

Population attributable risk percentage (PAR) was calculated using the formula PAR= Pd*(HR-1)/HR where Pd is the proportion of cases (i.e., HF onset) exposed to a risk factor [START_REF] Natarajan | A simple method of determining confidence intervals for population attributable risk from complex surveys[END_REF] . Using Bonferroni inequality, 95% confidence intervals (CI) for PAR percentage were calculated from the general PAR formula using the lower and upper limits of the 97.5% CI for Pd and HR (Supplementary Figure 2). This metrics could be calculated for categorical variables.

Statistical analyses were performed using R 3.6.1 and 3.5.3 (URL https://www.R-project.org/).

A two-sided p value of less than 0.05 was considered statistically significant.

Results

HOMAGE cohort results

A total of 26 478 subjects was included for analysis, out of whom 7 241 (27%) had a history of MI. During a median follow-up time of 5.2 (3.5-5.9) years, 2.44% (N=177/7 241) of participants with a history of MI and 1.92% (N=370/19 237) without a history of MI developed HF.

Baseline characteristics (Table 1)

Participants with HF onset were older than those without, irrespective of prior MI status.

Participants without prior MI who developed HF had higher SBP than participants without HF (167.5 (23.0) for HF versus 161.3 (19.9) for non-HF) whereas the opposite was observed in patients with MI history (154.0 (23.8) for HF vs. 159.2 (19.3) for non-HF) (Table 1).

Predictors of HF with or without history of MI (Supplementary Table 3)

In multivariable analysis, male sex, older age and higher heart rate were significantly associated with increased risk of HF onset both in participants with, and those without, prior MI. However, older age had a stronger association with the risk of HF in participants with prior MI, as compared to those without prior MI.

Active smoking status was significantly associated with HF onset (HR=2.00 (1.36 -2.94) p<0.001) in participants with prior MI, while the association was neutral in univariable analysis in those with no prior MI. Similarly, diabetes was a predictor of HF onset only in those with prior MI. Also, worsening renal function was associated with increased risk of HF only in participants with prior MI (HR = 2.0 (0.96-4.14) for 60-75ml/min/1.73m²; HR= 3.13 (1.54-6.37) for <60ml/min/1.73m²; overall p=0.002). In participants with no prior MI the association was non-linear (HR = 0.81 (0.55-1.18) for 60-75 ml/min/1.73m²; HR= 1.3 (0.9-1.87) for <60ml/min/1.73m²; overall p=0.006). In contrast, higher BMI, higher SBP and higher blood glucose levels were significantly associated with HF onset and retained in multivariable models in participants without prior MI but not in participants with prior MI.

Head-to-head comparison of HF risk factors in multivariable models

When predictors of HF were simultaneously included in multivariable models to allow a direct comparison of the magnitude of associations in participants with and without prior MI (Figure 1 and graphical abstract), a similar pattern of results was observed: SBP, blood glucose and BMI were more strongly associated with incident HF in participants without prior MI (p value for interaction respectively 0.003, 0.006 and 0.027) whereas, smoking status (p value for interaction 0.021) and worse renal function (p value for interaction 0.002) were associated with incident HF in patients with prior MI.

The c-index of the model improved when the interaction terms of variables were added to the model (delta C-index = 0.02 (from 0.70 to 0.72); p-value = 0.001). Similarly, a modest, but still significant, improvement of C-index was observed in the validation cohort (delta C-index = 0.004 (from 0.797 to 0.802); p-value = 0.01).

Forcing for ACEi/ARBs and beta-blockers use into the model did not affect the results (Supplementary Table 4).

In the competing risk analysis model, the pattern of association of variable and HF-onset was similar to multivariable model without competing risk. However, sex was neutrally associated with HFonset in patients without prior MI in the competing risk model (Supplementary Table 6).

The attributable risk of high SBP was markedly higher in patients without prior MI (Supplementary Figure 2).

Validation analysis in the UK Biobank study

From the UK Biobank study, 500 001 subjects were included in the validation analysis, of whom 4 555 (0.91%) had a history of MI at baseline. The validation cohort artici ants ere youn er ( 9 years) and fewer participants had hypertension and diabetes compared to participants in the HOMAGE database. Among people with prior MI, 14.9% (680/4555) developed incident HF while 2.7% (13536/495446) participants without prior MI developed HF over a median (Q1 -Q3) follow-up of 11.8 (11.1-12.5) years. Increasing age and smoking were significantly associated with increased risk of HF in people with and without prior MI. A higher BMI, HbA1c, diabetes and hypertension had a stronger association with HF onset in participants without prior MI compared to participants with prior MI (p value of interaction: 0.0005, 0.03, 0.03 and 0.0005 respectively -Table 2 andsupplementary Table 5). In addition, male sex was significantly associated with a higher risk of HF onset only in participants with prior MI but not in participants without MI (p for interaction <0.0001) and lower eGFR was more strongly associated with higher risk of HF onset in patients with prior MI (HR: 0.73 (0.67-0.81) vs 0.67 (0.66-0.69), p value of interaction <0.0001).

Discussion

In this pooled cohort study of patients at higher risk of developing HF, we found that the pattern of association of risk factors with HF was dependent on the presence of prior MI. Increasing age, male sex and higher heart rate were associated with HF onset irrespective of MI history, and should consequently be perceived as stable and ubiquitous risk factors for HF onset. Poorer renal function predisposed to HF onset more inpatients with MI. Blood pressure/hypertension, BMI and blood glucose/HbA1c were more strongly associated with HF onset in participants with cardiovascular risk but no prior MI both in the HOMAGE cohort and UK Biobank cohort. Our results are in line with previous reports suggesting that metabolic factors and hypertension are more associated with heart failure with preserved ejection fraction (HFpEF) than heart failure due to reduced ejection fraction (HFrEF) [START_REF] Elgendy | Medical Therapy for Heart Failure Caused by Ischemic Heart Disease[END_REF][START_REF] Pfeffer | Heart Failure With Preserved Ejection Fraction In Perspective[END_REF] . Importantly, despite the differences in baseline characteristics and risk factors in the HOMAGE (participants at high risk of HF) and UK Biobank study (population-based cohort), we saw a strikingly similar pattern of association between metabolic-and hypertension-related factors and new onset HF in patients without MI.

These results consequently suggest that, as in patients with overt HF, prior MI should be a variable systematically emphasized in epidemiological studies focusing on the prediction of HF onset.

Importantly, metabolic-and hypertension-related factors have a greater impact on HF onset in the absence of prior MI, and their prevention/treatment should consequently be particularly optimized in people who have not had an MI.

Risk factors more associated with HF onset in the participants with prior MI

Diabetes, renal dysfunction and smoking status were strong predictors of HF onset in the prior MI group in the HOMAGE cohort. Our results are in keeping with previous studies where diabetes and renal dysfunction were found to be an important determinant of HF onset (2-fold increase in risk of HF) in post-MI patients [START_REF] Patel | Impact of diabetes on heart failure incidence in adults with ischemic heart disease[END_REF][START_REF] Wellings | Risk Factors and Trends in Incidence of Heart Failure Following Acute Myocardial Infarction[END_REF] . This differential impact is not currently captured in the risk scores calculated with multivariable risk models where diabetes is associated with a fixed risk estimate irrespectively of MI history. However, it is noteworthy to mention that diabetes was more strongly associated with participants with no prior MI in the validation cohort. The difference of association of diabetes in derivation and validation may be due to differences in the population requiring insulin for management of diabetes. Previous studies have shown that cardiovascular mortality is substantially higher in patients with prior-MI and insulin-treated diabetes compared to patients with prior MI and diabetes treated with oral medications due to more frequent presence atherosclerotic disease in patients with diabetes requiring insulin [START_REF] Rossello | Editor's Choice-Impact of insulin-treated diabetes on cardiovascular outcomes following high-risk myocardial infarction[END_REF] . Yet, we did not had access to diabetes treatment in the HOMAGE cohort, and could consequently not verify this hypothesis.

Smoking is significantly associated with new onset HF following MI, possibly due to increased myocardial injury and myocardial haemorrhage, even though the smokers seemingly had a better risk profile than non-smokers [START_REF] Haig | Current Smoking and Prognosis After Acute ST-Segment Elevation Myocardial Infarction: New Pathophysiological Insights[END_REF][START_REF] Gerber | Smoking status and long-term survival after first acute myocardial infarction a population-based cohort study[END_REF] . However, smoking status was similarly associated with incident HF regardless of prior MI in the validation cohort and the pattern of association seen for smoking in HOMAGE (p-interaction=0.021) was not replicated in the UK Biobank (p-interaction=0.59). This difference was perhaps due to differences in the baseline smoking behaviour of the participants in the two cohorts.

The pattern of association between male sex and HF onset was dissimilar between the two cohorts. In the HOMAGE cohort, male sex was similarly associated with higher risk in both the groups whereas, in the UK Biobank, male sex was associated significantly with participants with CV risk but no prior MI. The difference in the statistical association could be due to an overall higher proportion of men in the HOMAGE compared to UK Biobank (60% vs 45%).

Renal impairment/CKD is frequently associated with patients with prior MI. Although the exact mechanism linking HF and CKD is not known, it has been speculated that neurohormonal activation, renal hypoperfusion, increased inflammatory markers involved in myocardial repair, anaemia and volume overload contribute towards progressive HF, especially post-MI. Similar findings were reported in an observational study where patients with HFrEF and CKD had higher frequency of a prior MI compared to patients with HFpEF [START_REF] Löfman | Associations with and prognostic impact of chronic kidney disease in heart failure with preserved, mid-range, and reduced ejection fraction[END_REF] . Further, CKD was more strongly associated with HFrEF in comparison of HFpEF. Besides pathophysiological mechanisms, CKD restricts the use of optimal dosage of renin-angiotensin inhibitors post-MI, thereby increasing the risk of HF These findings imply that reno-protection is particularly important in patients with prior MI.

Risk factors more strongly associated with HF onset in participants with cardiovascular risk but no prior MI

Our study found hypertension as a strong risk factor for HF onset in patients with cardiovascular risk but no prior MI. Similarly, an analysis of the Framingham Heart study cohort previously demonstrated that hypertension is the most important risk factor associated with HF onset [START_REF] Levy | The progression from hypertension to congestive heart failure[END_REF] .

Our results highlight that the magnitude of the association between blood pressure and incident HF is far greater in patients without prior MI than in patients with prior MI. There was no relation between SBP and incident HF in patients with prior MI, perhaps because low SBP was due to reduced left ventricular function and hence, greater risk of HF onset [START_REF] Regnault | Opposite predictive value of pulse pressure and aortic pulse wave velocity on heart failure with reduced left ventricular ejection fraction: insights from an Eplerenone Post-Acute Myocardial Infarction Heart Failure Efficacy and Survival Study (EPHESUS) substudy[END_REF] . Importantly, in the present analysis, higher SBP (in HOMAGE) and history of hypertension (in the UK Biobank study) was not associated with an increased risk of incident HF in patients with a prior MI. Previous studies have shown that lower BP could be related to greater LV dilation following MI, thus increasing the risk of subsequent HF [START_REF] Regnault | Opposite predictive value of pulse pressure and aortic pulse wave velocity on heart failure with reduced left ventricular ejection fraction: insights from an Eplerenone Post-Acute Myocardial Infarction Heart Failure Efficacy and Survival Study (EPHESUS) substudy[END_REF][START_REF] Dobre | Heart rate, pulse pressure and mortality in patients with myocardial infarction complicated by heart failure[END_REF] .

There is a BP aradox in atients ith HFrEF: hi her BP a ears to be ' rotective' [START_REF] Ambrosy | Clinical profile and prognostic value of low systolic blood pressure in patients hospitalized for heart failure with reduced ejection fraction: insights from the Efficacy of Vasopressin Antagonism in Heart Failure: Outcome Study with Tolvaptan (EVEREST) trial[END_REF][START_REF] Ather | Association of systolic blood pressure with mortality in patients with heart failure with reduced ejection fraction: a complex relationship[END_REF] . However, the paradox has been described only after HF onset, whereas we focus here on the pre-HF period.

Metabolic factors such as BMI and blood glucose were stronger predictors of HF onset in patients with cardiovascular risk but no prior MI in our study. Similarly, the validation analysis found that hypertension, diabetes and higher BMI were more strongly related to HF onset without prior MI. Similar findings were observed in a study combining data from middle-aged men and women from four different American cardiovascular disease cohorts [START_REF] Ahmad | Hypertension, Obesity, Diabetes, and Heart Failure-Free Survival: The Cardiovascular Disease Lifetime Risk Pooling Project[END_REF] , in which participants who did not have any of the three risk factors (hypertension, obesity and diabetes) had a substantially lesser risk of HF, thus placing greater emphasis on metabolic factors in the genesis of HF in patients without MI. In the presence of a prior MI, a lower prevalence of metabolic risk factors is perhaps sufficient to trigger remodelling and tip the patient in to HF. In addition, previous studies have reported an obesity paradox where lower to normal BMI is associated with poorer CV outcome post-MI compared to overweight patients perhaps due to a younger age of disease presentation and other confounding variables such as smoking and cardiorespiratory fitness [START_REF] Lavie | Management of cardiovascular diseases in patients with obesity[END_REF][START_REF] Stienen | Mean BMI, visit-to-visit BMI variability and BMI changes during follow-up in patients with acute myocardial infarction with systolic dysfunction and/or heart failure: insights from the High-Risk Myocardial Infarction Initiative[END_REF] .

Diabetes, hypertension and obesity are associated with impaired myocardial energy consumption, cardiac hypertrophy and oxidative damage. These mechanisms play a role in impaired relaxation leading to diastolic dysfunction [START_REF] Von Bibra | Cardiometabolic Syndrome and Increased Risk of Heart Failure[END_REF] . Further, a recent study found that diabetes/elevated blood sugar is reported to be associated with increased levels high sensitive cardiac troponin-T suggesting sub-myocardial damage [START_REF] Selvin | Diabetes mellitus, prediabetes, and incidence of subclinical myocardial damage[END_REF] . It seems that even in absence of direct myocardial damage (prior MI), considerable injury to heart is caused by cardiometabolic risk factors.

Risk scores and risk prediction models

A number of scores have been developed for predicting the risk of HF. The Framingham Study CHF (1999) risk score based on key clinical features was the first to provide estimates of the 4-year probability of heart failure [START_REF] Kannel | Profile for estimating risk of heart failure[END_REF] . Subsequently, the ARIC heart failure risk calculator, the Health ABC risk score, the PCP-HF calculator and the HOMAGE score were developed based on clinical and laboratory features [START_REF] Jacobs | Risk for Incident Heart Failure: A Subject-Level Meta-Analysis From the Heart "OMics" in AGEing (HOMAGE) Study[END_REF][START_REF] Khan | 10-Year Risk Equations for Incident Heart Failure in the General Population[END_REF][START_REF] Butler | Incident heart failure prediction in the elderly: the health ABC heart failure score[END_REF][START_REF] Agarwal | Prediction of incident heart failure in general practice: the Atherosclerosis Risk in Communities (ARIC) Study[END_REF] . Importantly, however, none of these scores differentiates HF prediction according to the history of MI. Having access to a large dataset enabled us to assess interactions with MI history efficiently. Interaction p values were significant (for both additive and multiplicative interaction) for many factors and addition of interaction terms led to a modest but significant improvement of the cindex in both derivation and validation cohort. Further, we found similar pattern of association and significant interaction-p values in competing risk model with non-HF death as competing risk factor.

Fine and Gray competing risk model may have predicted an upward bias in estimating HRs for each risk factor however, directionality of the association and relative magnitudes of the risk factors can be inferred [START_REF] Austin | Practical recommendations for reporting Fine-Gray model analyses for competing risk data[END_REF] . Overall, these findings suggest substantial differences between the association of each risk factor in patients with or without prior MIsomething that has not been previously emphasized.

Limitations

There are several limitations to our study. First, the results are subject to biases inherent to all observational cohort studies. Second, the included cohorts featured certain differences in the inclusion criteria and length of the follow up-period. For example, the DYDA study considered MI as an exclusion criterion while the ASCOT cohort had a longer follow-up time than the other 3 cohorts.

However, all models were adjusted for clinical characteristics. Third, the diagnosis of HF was not based on the Framingham criteria but rather on hospitalization of HF. Hence, non-hospitalized HF patients were not labelled as HF in the current analysis. Moreover, differences in HF hospitalization due to different hospital admissions policy, treating physicians, adherence to guidelines for therapy occur in different cohorts. Further, we do not have data regarding how long ago MI happened before the recruitment in the cohort and if there were recurrent MIs during follow-up. Fourth, it is possible that the some of the patients enrolled had had clinically silent MIs and were consequently included in the cardiovascular risk but no prior MI group, thus diluting the difference between the two groups. In addition, the overall SBP in the HOMAGE database was high due to the phenotype of patients included in the original cohorts aggregated within the HOMAGE consortium. Therefore, we used a cut-off of 160 mm Hg in the HOMAGE cohort given the structure of the data. However, the results were similar in the UK Biobank using the classic definition for hypertension. Finally, LVEF at the time of HF onset was not available; we cannot consequently provide HFrEF-or HFpEF-specific associations. Whether the pattern of association we observed is primarily driven by a subtype of HF (i.e. HFrEF or HFpEF) should be further studied in future.

Research and clinical implications

Our results suggest that "classical" risk factors for HF carry a different weighting depending on the clinical setting. This fact has not been sufficiently emphasized previously. In light of our results, a history of prior MI should be particularly highlighted in epidemiological studies focusing on predictors of HF onset, as is done in studies performed in patients with overt HF. In addition, the modifying impact of MI on HF predictors should be systematically assessed in future reports. A personalized HF risk stratification may help in designing preventive strategies depending on the clinical setting. In patients without MI, hypertension treatment and the control of metabolic features appear to be of much greater importance. These results could help in the prioritization of healthcare interventions in the prevention of HF.

Conclusions

The importance of clinical risk factors and the increase in subsequent mortality risk following HF onset is dependent on the presence or absence of a history of prior MI. These results suggest that patients should be differentiated in terms of risk assessment based on their history of prior MI and may ultimately benefit from different targeted interventions to prevent HF.

Table 1: Baseline characteristics in relation to history of MI and the occurrence of HF during follow-up in the HOMAGE cohort

Figure 1 :

 1 Figure 1: Forest plot of the head-to-head comparison of risk factors for HF onset in multivariable

Table 2 : Multivariable Cox regression models for HF event with or without history of MI in UK biobank study Patients with history of MI Patients without history of MI P-value of interaction

 2

		HR (95% CI)	P	HR (95% CI)	P
	Age in year	1.04 (1.03-1.05)	< 0.0001	1.12 (1.11-1.12)	< 0.0001	< 0.0001
	Male	1.00 (0.83-1.21)	0.97	2.06 (1.99-2.13)	< 0.0001	< 0.0001
	eGFR	0.73 (0.67-0.81)	< 0.0001	0.67 (0.66-0.69)	< 0.0001	< 0.0001
	HbA1c	1.13 (1.07-1.20)	< 0.0001	1.19 (1.17-1.20)	< 0.0001	0.03
	Hypertension	1.14 (1.08-1.22)	0.29	1.61 (1.54-1.69)	< 0.0001	0.0005
	BMI	1.22 (1.12-1.33)	< 0.0001	1.49 (1.46-1.51)	< 0.0001	0.0005
	Smoker	1.88 (1.48-2.38)	< 0.0001	1.97 (1.87-2.08)	< 0.0001	0.59
	Ex-smoker	1.36 (1.13-1.63)	0.0009	1.32 (1.27-1.37)	< 0.0001	0.74
	Type2 diabetes	1.50 (1.22-1.84)	0.0001	1.81 (1.72-1.91)	< 0.0001	0.03
	Continuous variables were standardized to per-SD increase
	Detailed adjustment model is presented in supplementary table 4

Acknowledgements

Dr Girerd, Rossignol, and Zannad are supported by the French National Research Agency Fighting Heart Failure (ANR-15-RHU-0004), by the French PIA roject Lorraine Université d'Excellence GEENAGE (ANR-15-IDEX-04-LUE) programs, and the Contrat de Plan Etat Région Lorraine and FEDER IT2MP. Dr Girerd reports honoraria from Novartis, AstraZeneka, Boehringer and Vifor. Dr Rossignol received personal fees (consulting) for Idorsia and G3P, honoraria from Astra-Zeneca, Bayer, CVRx, Fresenius, Grunenthal, Novartis, NovoNordisk, Servier, StealthPeptides, Ablative Solutions, Corvidia, Relypsa and Vifor Fresenius Medical Care Renal Pharma, outside the submitted work, and is the cofounder of CardioRenal. Dr Zannad reports steering committee personal fees from Applied Therapeutics, Bayer, Boehringer, Boston Scientific, Novartis, Janssen and CVRx, advisory board personal fees from, Astra-Zeneca, Vifor Fresenius, Cardior, Cereno pharmaceutical and Merck, stock options at G3Pharmaceutical, and being the founder of CardioRenal and CVCT. Dr Sattar and Dr Cleland are supported by the British Heart Foundation Research Excellence Award (RE/18/6/34217). Dr Staessen reports funding from APPREMED (URL: www.appremed.org) and a non-binding grant from OMRON Healthcare Co., Ltd., Kyoto, Japan. Dr. Butler is a consultant to Abbott, Adrenomed, Amgen, Array, AstraZeneca, Bayer, Berlin Cures, Boehringer Ingelheim, Bristol-Myers Squib, CVRx, G3 Pharmaceutical, Innolife, Janssen, LivaNova, Luitpold, Medtronic, Merck, Novartis, Novo Nordisk, Relypsa, Roche, and Vifor.

Sources of Funding

The research leadin to these results has received fundin from the Euro ean Union Commission's Seventh Framework Programme under grant agreement No. 305507 (HOMAGE [Heart Omics in Ageing consortium]). UK Biobank was established by the Wellcome Trust medical charity, Medical Research Council, Department of Health, Scottish Government and the Northwest Regional Development Agency. It has also had funding from the Welsh Assembly Government and the British Heart Foundation.


Yvan Goyhénèche

Aspa

Georges Duffa
email: gduffa@club-internet.fr

Gérard L Vignoles

Jean-Marc Goyhénèche

Yvan Aspa
email: yvanaspa@yahoo.fr

Ablation of carbon-based materials : investigation of roughness set-up from heterogeneous reactions

Ablation of carbon-based materials : investigation of roughness set-up from heterogeneous reactions

Subscripts and superscripts

• g = relative to a gas-phase species,

• s = relative to a solid phase ,

• 0 = reference quantity,

• + = relative to the right-hand side of a singular point Carbon/carbon (C/C) composites are the almost unique class of thermostructural materials that are used as thermal protection of atmospheric reentry bodies. Indeed, the local conditions are dramatically severe : temperatures up to 4500 K and pressures ranging between 0.1 and 100 bars, and a heat flux received by the protection ranging from 0.1 to 500 M W.m -2 . In such conditions, a non-negligible part of the heat flux is consumed by interfacial mass transfer, which has two principal forms : oxidation (and other chemical reactions) and sublimation. These phenomena are grouped under the generic name of ablation [START_REF] Duffa | Ablation, Monograph[END_REF]. Their particularity is to display a strong coupling with many other physical phenomena, namely fluid flow and heat transfer.

The surface roughness related to the ablation phenomena of the atmospheric reentry body thermal protections has considerable consequences. It is able to promote a laminarto-turbulent regime transition [START_REF] Jackson | Roughness induced transition on blunt axisymmetric bodies[END_REF][START_REF] Reda | Correlation of nosetip boundary-layer transition data measured in ballisticrange experiments[END_REF], a fact which induces a strong enhancement of heat and mass exchanges between the protection wall and the surrounding environment. It has been experimentally observed that the heat flux may be multiplied by a factor up to three in turbulent regime [START_REF] Wool | Summary of experimental and analytical results[END_REF]. The consequences on the surface temperature and heat-affected zone depth are obvious, as well as on the overall ablation velocity.

However, little is known of the creation mechanisms of such a surface state, which results from a strong coupling between the material and the surrounding flow. The aim of this work is to set up and use simple models of interaction, involving the least possible phenomena at first, that is, merely the heterogeneous reaction and mass transport from the flowing fluid.

The first question that one can ask about material/fluid interaction is : what is the relation between the surface morphology and the underlying physico-chemical phenomena ? This question has been dealt with in numerous works on CVD (Chemical Vapor Deposition) [START_REF] Hwang | Three-dimensional morphological instabilities in chemical vapor deposition films[END_REF][START_REF] Palmer | Local equilibrium model of morphological instabilities in chemical vapor deposition[END_REF][START_REF] Palmer | Kinetic model of morphological instabilities in chemical vapor deposition[END_REF][START_REF] Srolovitz | Analytical and numerical modeling of columnar evolution in thin films[END_REF][START_REF] Thiart | Chemical vapor deposition and morphology problems[END_REF], which is closely related to ablation, the difference being the sign of the wall velocity.

Palmer and Gordon [START_REF] Palmer | Local equilibrium model of morphological instabilities in chemical vapor deposition[END_REF][START_REF] Palmer | Kinetic model of morphological instabilities in chemical vapor deposition[END_REF] has found out by linear perturbation theory and by numerical simulation that at flat surface may be destabilized by gas-phase diffusion tangent to the surface, while surface diffusion had the opposite effect. Another analytical approach [START_REF] Srolovitz | Analytical and numerical modeling of columnar evolution in thin films[END_REF] focused on the modification of the local growth rate due to curvature, and a competition of such an effect with surface diffusion : the result is that wavy surface may develop. Thiart, Hlavaček, and Viljoen [START_REF] Thiart | Chemical vapor deposition and morphology problems[END_REF] have developed efficient numerical tools for the simulation of CVD growth in various conditions and have shown that "fingering" appears in the case of diffusion-limited growth, a fact that has been confirmed by linear stability analysis [START_REF] Hwang | Three-dimensional morphological instabilities in chemical vapor deposition films[END_REF][START_REF] Thiart | Chemical vapor deposition and morphology problems[END_REF].

In the case of graphite ablation, experimental determinations of the roughness have been performed by profilometry , using well-defined flow conditions and heat transfer rates [START_REF] Dourthe | Etude des propriétés d'ablation des graphites à l'aide d'une torche à plasma[END_REF].

In some experimental conditions for which the flow is laminar (Fig. 1a), the surface of a polycrystalline graphite looks smooth up to a few µm resolution. The apparent roughness seems to rely only on the material heterogeneity, which is known to contain small domains of better cristallinity than the surrounding matter. Roughness here may only be the result of the material heterogeneity (variations of the density and local heterogeneous kinetic constant) : this issue will be addressed in the following parts, mostly in the case of carbon/carbon (C/C) composites, which are made of different types of carbon. An example is given at fig. 2: in a composite the matrix of which is made of a very lowdensity (and high-reactivity) carbon, the fibers, which are more resistant, are salient with respect to the matrix, and acquire an ogival shape. In a composite, the space scales are related to the individual fiber diameter (e.g. ≈ 7µm for carbon fibers), and to the apparent diameter of tows, warps, needlings, etc. . . (e.g. ≈ 300 -500µm for many composites).

Nonetheless, in turbulent flow conditions, the surface of the same material displays a "cellular" roughness (fig. 1b) with length scales (here, roughly 1 mm) that cannot be related to the material heterogeneity nor to turbulent length scales. This kind of "scalloped" surface is strikingly similar to glacier cave walls. In addition, a closer look at the fibers in the ablated C/C composite shown at fig . 2 reveals other scalloped features on single composite fibers, this time with a length scale around 10 µm, that are clearly seen in fig.

3. Again, it cannot be related to the material heterogeneity ; so the explanation of this has to arise from an extrinsic phenomenon.

It will be tried to show that natural length scales arising from the coupling of transport and heterogeneous reaction are of comparable orders of magnitude, since other solutions than the trivial flat surface can be produced from a simple model. Most of the development presented will be devoted to mass transfer by diffusion, and heat transfer will be treated by analogy.

It is worthwhile to note that although CVD growth morphologies are sometimes similar to these ablation morphologies (though with opposite concavities), they can not be accounted for with the same kind of model.

In the following part, a diffusion-reaction model will be recalled ; then an evolution equa-tion will be produced for the surface points. Stationary solutions will be presented in the case of a growing boundary layer (initial transient state) and in the case of a constant boundary layer (steady-state) and some discussions on stability will be made. Singular points will be next considered, and a condition for their stability will be worked out.

Numerical simulations will then be presented, which confirm the steady-state analysis and give an insight into the transient regimes. Then, an application of the model will be presented and discussed with respect to the available experimental facts. Finally, a brief presentation of a heat-transfer counterpart of the model is given, and the results are discussed in terms of shape and scale.

2 Model set-up

Model frame and assumptions

Let us consider a parallelepipedic domain (x, y, z) containing a part of the fluid, and a part of the solid (fig. 4). The interface (or surface) is globally perpendicular to the z direction and may be described by a set of points (x, y, z) such that S(x, y, z, t) = 0 [START_REF] Katardjiev | A kinematic model of surface evolution during growth and erosion : Numerical analysis[END_REF].

Indeed, a particular case of this general formulation will be used frequently, when S may be explicited under the form S = h(x, y, t)z. The S = 0 condition coincides then with the definition of a unique height h at which the interface lies. In this case, S has the dimension of a length ; however, other choices are possible for such a "potential" function : for example, the local proportion of solid could be an acceptable candidate for S, and the formulation would be a VOF (Volume of Fluid) formulation [START_REF] Hirt | Volume of fluid (VOF) method for the dynamics of free boundaries[END_REF]. The fluid lies in the upper part of the domain, with z pointing downwards. The height of the diffusion path h is assumed to have an order of magnitude at least comparable to the roughness characteristic length. The (x, y) extension of the domain may be much larger than the z extension. If needed, periodic boundary conditions may be defined in the x and y directions. See fig.

4 for a 2D scheme. For most purposes of this work, only two dimensions of space will be needed.

The fluid transfer will be limited to pure diffusion. The chemical reactants will be considered as having a fixed, prescribed concentration either at the top of the domain (z = 0)

or at a given height z = η(t), and diffusing downwards, while the species created by the ablation process diffuse upwards and their concentration at the upper boundary will be held at zero value. This represents indeed the fact that convection is non-negligible outside of the studied domain, and that h is also the size of the diffusive boundary layer D/v conv .

Two physical cases in which this is a valid assumption may be quoted :

(1) In real reentry conditions, the height of this boundary layer is very small since the convective velocity is extremely high ; however, a stagnation point has to exist at the top of the reentering object, with moderate convection in its vicinity. In such a region, the size of the diffusive layer may be larger.

(2) In laboratory experiments, the convective velocity is moderate -and even low -at some parts of the ablated sample, especially in "stop-point" configurations.

It is not yet possible to perform full calculations featuring the whole physics of atmospheric reentry (i.e. including turbulence with Kolmogorov scale around 10 µm) and covering a typical macroscopic cell (size ≈ 1 cm). These two length scales are such that any DNS simulation is not feasible ; and since no data is available on the precise effect of surface roughness on turbulence, it is not possible either to perform LES simulations. Nonetheless, experimental evidence shows that the surface morphology is not very different below the stagnation point (for which it is acceptable to neglect convection) and at the rear part of the reentering body (with Mach numbers superior to one outside the viscous boundary layer). This suggests that a first approach not featuring convection may have some interest.

Stefan flux, that is, the normal mass flux entering the gas phase at the wall, will be neglected for two reasons :

(1) In many practical cases, it may be considered that the gaseous molecules involved in reaction and/or sublimation are diluted in a carrier gas.

(2) An experimental study of the modification of heat transfer under the effect of gas blowing from a porous substrate [START_REF] Lin | An aerothermal model for ablation heat shields[END_REF] shows that the transfer rate is altered by at most 10 % when realistic values for the blowing/input fluxes are chosen. If Lewis and Prandtl numbers are close to unity, it is straightforward to extend this result to mass transfer, that is : Stefan flux alters by less than 10 % the total flux amount in any case of interest. We may thus neglect it safely in a first approach.

The fluid domain will be considered as isobaric and isothermal. This is not a strong assumption in the case of a small domain height ; the converse case will be studied in another paper. As a consequence, the transport and reaction properties will be considered constant throughout the domain.

Curvature and surface diffusion effects will be neglected at first in this approach, although they are known to have important effects in CVD. However, it will be seen that they may only be seen as second-order corrections to the principal results of the present study.

Surface evolution

The surface having a local normal velocity v, its time and space evolution obeys the following equation at all points where S is differentiable [START_REF] Katardjiev | A kinematic model of surface evolution during growth and erosion : Numerical analysis[END_REF]:

∂S ∂t + v • ∇S = 0 (1)
This equation only means that in a local coordinate frame moving with velocity v, the implicit equation defining the surface is preserved. The expression of the velocity relies on the ablation phenomenon ; thus, it depends on the ablation molar rate R. Its expression follows from a simple mass balance equation :

v = υ s Rn (2)
where υ s is the solid molar volume and the normal vector is defined as :

n = ∇S ∇S (3)
By convention, it will be pointing towards the fluid ; as a consequence, the quantity S will be larger in the fluid than in the solid. The ablation molar rate may be related to the impinging mole flux J :

R = ν s ν g J • n (4)
where the term between parentheses represents the stoichiometric ratio of ablated solid with respect to the gaseous species. For a chemical reaction (or a phase change) noted i ν i A i = 0, the stoichiometric coefficients ν i will be counted negative for the reactants and positive for the products. They are detailed at table 1. Two cases are distinguished :

i) a first-order reaction with an etching species, like O 2 for example, and ii) spontaneous sublimation. The case of CVD is given as a comparison.

Collecting eqs. ( 1), (2), and (4) together, one has the following equation for S(x, y, z, t):

∂S ∂t = -υ s ν s ν g (J • ∇S) (5)
Under the considered approximations, the gas mole flux is given by Fick's first law :

0 = J + D∇C (6)
and the mass conservation in the gas phase reads :

∂C ∂t = -∇ • J (7)
Let us now define a reference height h 0 . This may be for instance the size of a diffusion boundary layer, resulting from the interaction of ablation and of an outer tangent gas stream. Let also C 0 be a reference gas-phase concentration. The variables may be scaled in the following form :

x = h 0 x C = C 0 C S = h 0 S ∇ = h -1 0 ∇ t = h 2 0 DC 0 υ s t J = DC 0 h 0 J (8)
Then, the equations to solve are :

                           (υ s C 0 ) ∂ C ∂ t = -∇ • J 0 = J + ∇ C ∂ S ∂ t = - ν s ν g J • ∇ S (9)
Since the dimensionless group υ s C 0 (the condensation ratio) is a small quantity, the first term in the first equation may be considered as zero -that is, gas diffusion being rapid with respect to surface evolution, it can be considered in steady state. Dropping out all tildes for sake of simplicity, one now has the following set of equations :

               ∇ 2 C = 0 ∂S ∂t = + ν s ν g ∇C • ∇S (10)
subject to boundary conditions sets which are linked to the precise nature of the reaction or change of state.

Boundary conditions and mass sources

The boundary conditions are listed at Table 1. Two limiting cases are considered. Indeed, it is well known that carbon loss in atmospheric reentry arises principally from two phenomena : i) oxidation or similar reaction (with nitrogen for example), and ii) carbon sublimation. The first case may have complicated kinetic laws, the apparent order of which ranges usually between 0 and 1 [START_REF] Nagle | Oxidation of carbon between 1000-2000 • C[END_REF]. The limiting case of first-order wall reaction will be dealt with first ; in a second time, a local equilibrium model, applicable either to sublimation or to a weak chemical reaction, will be presented.

For a first-order irreversible reaction, it is considered that the concentration is held fixed at a constant value C 0 -which is used as a reference for dimensionless quantities -at the boundary layer top, which lies at some height η(t), and is considered as flat.

The boundary conditions are :

               C(z = η) = 1 -∇C • ∇S = ν g Da ∇S C at S = 0 (11)
where the Damköhler number has been introduced : Da = kh 0 D , with k being a first-order heterogeneous reaction constant in m.s -1 . The stoichiometric coefficients ν s and ν g are both negative in the case of oxidation. For example, for the reaction

C(s) + CO 2 (g) -→ 2CO(g) , we have ν g = ν(CO 2 ) = ν s = -1. The CVD reaction CH 4 (g) -→ C(s) + 2H 2 (g) yields ν s = +1 and ν g = -1.
In the case of sublimation, the species partial pressure is supposed to be zero at the upper limit of the domain. The net sublimation mole flux (in a direction normal to the wall) is given by the Knudsen-Langmuir relationship [START_REF] Schrage | A Theoretical Study of Interphase Mass Transfer[END_REF] in dimensional form :

J • n = γ(p -p) ( 12
)
where p is the equilibrium partial pressure and :

γ = α 4 1 RT 8RT πM (13)
Here, α is a sticking coefficient1 . The term in square root is the root mean square molecular velocity c = 8RT πM .

By comparison with the preceding case, and replacing the partial pressure p by CRT from the perfect gas law, the sublimation flux is rewritten :

J • n = k( C -C) (14)
where it has been possible to define an "equivalent kinetic constant" :

k = α 4 c (15)
Summarizing in dimensionless form, one has :

               C(z = η) = 0 -∇C • ∇S = ν g Da ∇S (1 -C) at S = 0 (16)
In such a case, the stoichiometric coefficients are such that ν g is positive while ν s is negative. For example,

3C(s) -→ C 3 (g)
gives ν s = -3 and ν g = +1. In eqs. ( 16) it is assumed that the reference concentration C 0 is the equilibrium gas concentration C obtained from the Clausius-Clapeyron relation.

Surface height as a function of the abscissa

Now if the surface is such that z can be solved unequivocally out of S(x, y, z, t) = 0, that is , if :

S(x, y, z, t) = 0 = h(x, y, t) -z (17)
then the evolution equation ( 5) for S may be rewritten for the height h of the surface :

∂h ∂t + ν s ν g J x ∂h ∂x + J y ∂h ∂y -J z = 0 (18)
The normal vector is :

n = 1 1 + ∂h ∂x 2 + ∂h ∂y 2               ∂h/∂x ∂h/∂y -1               (19)
and points towards the fluid. The normal flux is then :

J.n = 1 1 + ∂h ∂x 2 + ∂h ∂y 2 J x ∂h ∂x + J y ∂h ∂y -J z (20)
The boundary conditions ( 11) and ( 16) rewrite then :

               C(z = η) = 1 J x ∂h ∂x + J y ∂h ∂y -J z = ν g Da 1 + ∂h ∂x 2 + ∂h ∂y 2 C at S = 0 (21)
and :

               C(z = η) = 0 J x ∂h ∂x + J y ∂h ∂y -J z = ν g Da 1 + ∂h ∂x 2 + ∂h ∂y 2 (1 -C) at S = 0 (22)

Vertical flux approximation

Let us now consider that the gaseous mole flux is vertical, that is,

|J x | << |J z | and |J y | << |J z |.
In such a case, it is possible to reduce the system to only one equation, provided that the surface is not too steep , that is :

∂h ∂i << J z J i , i = x, y (23)
For instance, if one chooses a slope 100 times inferior to a Jz J i ratio, and a value of 100 for such a ratio, one obtains a critical surface inclination of 45 • . From Fick's laws, it appears that J z is a constant :

J z = - C(z) -C(η) z -η = constant ∀x, y, z (24)
and that C(x, y, z) = J z z + constant. The boundary conditions (21) rewrite :

               C(η) = 1 C(h) -1 = ν g (h -η).Da.C(h) 1 + ∂h ∂x 2 + ∂h ∂y 2 (25)
The concentration at the interface is then :

C(h) = 1 1 -(h -η)ν g Da 1 + ∂h ∂x 2 + ∂h ∂y 2 (26)
Substituting back into eqs. [START_REF] Baker | The high temperature thermochemical properties of carbon[END_REF][START_REF] Hirschfelder | Molecular theory of gases and liquids[END_REF][START_REF] Biolsi | Transport properties for a mixture of the ablation products C, C 2 and C 3[END_REF](21), one arrives at the following equation for h at the interface (z = h):

∂h ∂t = ν s ν g 1 h -η - 1 νgDa 1+( ∂h ∂x ) 2 +( ∂h ∂y ) 2 (27)
Or, letting h = hη and ν g = ν s = -1 :

∂h ∂t = 1 h + 1 Da 1+( ∂h ∂x ) 2 +( ∂h ∂y ) 2 - dη dt (28)
However, dη/dt may be viewed as a mean ablation velocity, so it is interesting to extract from eq. ( 28) the local information. Choosing a reference frame with origin at z = η and moving with velocity dη/dt (i.e. x = xtdη/dt; t = t), one has :

∂h ∂t =   h +   Da 1 + ∂h ∂x 2 + ∂h ∂y 2    -1    -1 (29)
This is a Hamilton-Jacobi equation for h . In the case of sublimation, things are similar.

The boundary conditions (22) may be rewritten :

               C(η) = 0 C(h) -1 = ν g (h -η).Da. (1 -C(h)) 1 + ∂h ∂x 2 + ∂h ∂y 2 (30)
Making the same algebra as in the preceding case leads to the same equation as eq. ( 29) but with a reversed sign ; however, the stoichiometric ratio is now negative, so the result is :

∂h ∂t = ν s ν g   h +   |ν g |Da 1 + ∂h ∂x 2 + ∂h ∂y 2    -1    -1 (31)
and is formally equal to the preceding one.

The obtained equation differs from previously proposed equations [START_REF] Srolovitz | Analytical and numerical modeling of columnar evolution in thin films[END_REF][START_REF] Thiart | Chemical vapor deposition and morphology problems[END_REF] for several reasons :

-The vertical flux in eq. ( 9) is not treated as an externally provided constant, but as dependent on the boundary layer height h and on the interface concentration C(h), which has indeed to be done in the case where neither diffusion nor reaction overcomes on each other ;

-The variation of the kinetic constant k with the local curvature has been neglected ;

-The effect of surface diffusion, also sensitive to curvature, has been neglected.

Stationary profiles and velocities

Equation ( 29) is indeed a Hamilton-Jacobi equation, for which "stationary solutions" means solutions with a fixed velocity in global frame, or with null velocity in local frame with respect to dη/dt.

Two distinct methods will be presented for the derivation of such solutions. First, a global criterion, based on the existence of a "prime integral" is used. Second, a local criterion based on the time stability of the slope (or the equality of the velocity for all points) will be used.

Global (geometrical) method

Let us introduce the angle ϕ such that J•n = J n cos ϕ. Since we are in the hypothesis that J is strictly vertical, then :

cos ϕ = -n -1 = -    1 + ∂h ∂x 2 + ∂h ∂y 2    -1 (32)
In two dimensions, that is, dropping out the y coordinate, from fig. 4 it is easy to see that θ = πϕ = arctan ∂h ∂x is the angle between the slope of the interface and the horizontal axis. Introducing this angle into eq. ( 28) yields in fixed coordinates :

∂h ∂t = h + cos θ Da -1 - ∂η ∂t ( 33
)
where cos θ is positive, i.e. θ ∈ (-π/2; π/2).

Then, stationarity of the interface ( ∂h ∂t = 0) implies conservation of the quantity :

h + Da -1 cos θ = 1/V (34)
where V = ∂η ∂t is the global velocity.

A trivial solution is θ = 0 everywhere, that is, a flat interface. A natural choice for the reference length h 0 is then such that h = 1 when cos θ = 1, i.e. at the surface. The velocity is then V = Da 1+Da , that is, turning back to dimensional quantities :

dη dt = 1 Da + 1 . ν s ν g . (υ s C 0 ) .k (35)
This choice for characteristic dimensions will be retained also in the following.

On the other hand, since ∂h ∂x = tan θ, using a chain derivation rule, one obtains that ∂x ∂θ = -Da -1 cos θ. Then, by integration, x = x 0 + Da -1 sin θ, and the solution is thus a circle arc with center at (x 0 , 1 -1/Da) and radius 1/Da :

               h = 1 + Da -1 (1 -cos θ) x = x 0 + Da -1 sin θ (36)
Since cos θ is positive, only the circle arc with a center lying in the fluid is correct. When θ = 0, that is, for a point with null slope, the velocity is equal to the flat-surface value, but is also equal to the velocity of the other points ; thus, the circle-arc solution has the same velocity as the flat one.

The analytic form of the solution is, in the moving coordinate frame :

h (x) = -1 -(x -x 0 ) 2 + h 0 + 1 (37)
It is obvious that an interface with indefinite x extension cannot be obtained from continuous solutions of eq. 28. This issue is readily dealt with, considering that a solid surface does not have to be smooth everywhere : it just has to be continuous. We will address further this point at next section ; however, a glance at fig. 6 helps to understand the shape of a particular, non-trivial solution.

Local criterion

Another way of working out stationary solutions is to consider directly eq. 28 (neglecting the y coordinate for simplicity), derive it with respect to x, and equal the result to zero, which means that the slope does not vary with time and that the time evolution for all points is the same :

∂ 2 h ∂t∂x = 0 = - ∂h ∂t 2 ∂h ∂x 1 + Da -1 κ ( 38
)
where κ is the curvature :

κ = - ∂ 2 h/∂x 2 (1 + (∂h/∂x) 2 ) 3/2 (39)
Stationarity of the profile is thus equivalent to having a constant curvature equal to -Da,

i.e, a circle arc with a center lying in the fluid and radius 1/Da. This is exactly the same result as before.

Singular points

As it stands clear from fig. 1b), a description of the surface involving singular points seems to make enough sense. Moreover, the result of the preceding section is that a stationary continuous surface may be a circle arc ; thus, an interface with a large lateral extension may be described as a collection of circle arcs, at the expense of the existence of a certain number of singular points. Let us assume that there exists an enumerable quantity of such points (in practice it will be a finite quantity), then, since the surface is continuous, it is possible to define for every point of it a left-hand slope and a right-hand slope :

tan θ + (x) = ∂h ∂ξ ξ→x ξ≥x (40) tan θ -(x) = ∂h ∂ξ ξ→x ξ≤x (41)
The solid mass balance (eq. 33) gives now two equalities, one for each "half-slope" :

∂h ∂t = h + cos θ + Da -1 = h + cos θ - Da -1 (42)
From these relations, it is readily seen that two kinds of points may exist :

• θ + = θ -, that is, the surface is continuous at x;

• θ + = πθ -, that is, there exists a singular point, and the curve is locally symmetrical with respect to a vertical axis at x.

It is also straightforward to see from the conservation relation (34) that all circle arcs have to display the same curvature if the material is homogeneous. Fig. 5 is a sketch of a typical possible solution, for which i) the linear density of singular points is O(Da),

and ii) most of the circle arcs contain a point with θ = 0. An exception to statement ii)

is shown as a grey-filled circle : it is a reentrant singularity. It will be shown in a later section that such a singularity is not likely to be realized in practice.

Indeed, the experimentally observed patterns are such that the properties mentioned above hold. Why is it so ? Some qualitative arguments may be exposed. First, the linear density of singular points has to be superior or equal to (2Da) -1 , since the individual circle arc definition range is at most 2Da. Second, if the linear density is much superior to Da -1 , then the curve becomes apparently smoother (i.e. the peak-to-valley parameter [START_REF] Jackson | Roughness induced transition on blunt axisymmetric bodies[END_REF][START_REF] Wool | Summary of experimental and analytical results[END_REF] diminishes strongly).

Phenomena that have not been taken into account, like surface diffusion, curvatureinduced reactivity excess, or -more probably -thermal effects and deviation from the verticality of the diffusion fluxes, etc. . . are believed to smooth locally the surface and transform a singular point into a region with finite but strong curvature and width λ << Da.

The presented model is thus not applicable when the density of singular points becomes comparable or superior to λ -1 .

Profile stability discussion

This discussion will be split in two distinct parts : first, the smooth parts of the solution curves are studied through a standard perturbation calculus. Then, the stability of singular points will be treated using Huyghens' principle [START_REF] Katardjiev | A kinematic model of surface evolution during growth and erosion : Numerical analysis[END_REF][START_REF] Katardjiev | The application of the Huyghens principle to surface evolution in inhomogeneous, anisotropic and time-dependent systems[END_REF].

Continuous parts of the surface

A standard stability analysis may be performed on the 2D stationary profile solutions z = h(x, t), by introducing perturbations of C and h :

∆C = C -C = δ C exp (ω C t + k C,x x + k C,z z) ∆h = h -h = δ h exp (ω h t + k h,x x) (43)
The time and space pulsations ω i and k i will be related by dispersion relations, after having checked their interdependency. Introducing the perturbation ∆C into Fick's second law yields :

k 2 C,x + k 2 C,z = (υ s C 0 )ω C ≈ 0 (44)
A correct choice for a perturbation should be with a decreasing amplitude for decreasing hz, so that k C,z should be real and positive. As a consequence, k C,x = ±jk C,z is purely imaginary, i.e. the perturbation is periodic in x direction. Using now eq. ( 18) yields :

ω h ∆h = hx k C,x -k C,z ∆C (45)
where hx = ∂ h/∂x is a short notation for the local slope. This result, valid for all x, implies that ∆C and ∆h do have equal space periods :

k h,x = k C,x = k x k C,z = k z (46)
On the other hand, making use of eqs. ( 21) or ( 22) gives :

-

Cz   1 -k x hx Da 1 + h2 x 3/2    ∆h =   1 + k z -k x hx Da 1 + h2 x   ∆C (47)
Combining together eqs. (44-47) and setting hx = cos θ yields finally the following dispersion relation :

ω h = - -Cz k z 1 + ( kz Da ) 2 + 2( kz Da ) cos θ ... ( 48
) ... 1 + ( k z Da ) cos θ cos 2 θ + tan 2 θ ... ( 49
) ... ±j tan θ + ( k z Da ) cos θ sin θ cos θ + ( k z Da )
Remembering that cos θ, -Cz , and k z are positive, one finds that e(ω h ) is always negative, which means that any perturbation decreases, no matter what the local slope is.

Both solutions, the flat and the curved one, are stable.

graph, for θ = ± π 2 , and the other ones are for θ 0 verifying :

cos 3 θ 0 -2hDa + 3hDa cos 2 θ 0 = 0 (60)
As shown at figure 8, this angle belongs to the interval ]θ lim ; π/2], where θ lim = arccos 2/3 ≈ 35 • 44 , a value inferior to the angle domain corresponding to the vertical flux approximation.

Application of Huygens' principle to the evolution of singular points

The principle of the approach has been described in details in [START_REF] Katardjiev | A kinematic model of surface evolution during growth and erosion : Numerical analysis[END_REF][START_REF] Katardjiev | The application of the Huyghens principle to surface evolution in inhomogeneous, anisotropic and time-dependent systems[END_REF]. By analogy with the propagation of electromagnetic waves, the surface is assimilated to a wavefront, and it is stated that each point of the wavefront is a source of a secondary wave called Huygens wavelet. The evolving wavefront is defined as the common envelope of the wavelets. In the present case, the wavelets have no frequency (or an infinite one), so there is no phenomenon of interference, as in the electromagnetic application case. Now, Huygens wavelets have to be defined. Continuing the analogy, one says that the wavelet is an elementary wavefront which is the locus of the points reachable by a source after a short time increment δt. This definition is equivalent to the approach described in section 4.2.

So, the Huygens wavelet is the locus of the points defined by the vectorial equation (56).

The evolved surface is then obtained as the common tangent envelope of the wavelets created by every point. An example of construction is given at fig. 9.

For a local study, it will be considered that h and Da are fixed values, so that the Huyghens wavelet is of fixed size and shape in the vicinity of the studied point. Thus, applying the construction rules allows to discuss the stability of edges in the (θ -, θ + ) plane, as illustrated at fig. 10. It is possible to classify the singular points into four types, depending on the behavior :

• Type I is a "sharp" reentrant edge, i.e. with two sides such as |θ i | > θ 0 . Its evolution leads to a smooth curve and two singular points noted S on the scheme. Note that those evolved points are not of type I, so that type I edges are not stable.

• Type II is associated with reentrant edges with |θ i | < θ 0 on one side only. A piece of smooth curve appears, and an evolved singular point S is present. They may be considered as stable because the singular point is conserved.

• Type III edges are edges with small slopes : they evolve into smooth curves.

• Type IV edges are stable edges, marked with an S on the figure : among them, one finds all salient edges.

The only edges which are present in a stationary curve are symmetrical edges, represented by the second diagonal of the (θ -, θ + ) plane. By inspection of fig. 10 it is easy to see that :

• Symmetrical type IV (salient) edges are stable ;

• Symmetrical type III edges are not stable and transform themselves into pieces of smooth curves ;

• Symmetrical type I edges are not stable : they split into two unsymmetrical type II edges which later on are prone to transform themselves into something else.

Accordingly, this confirms the fact that a stationary surface does not contain in practice any reentrant edge.

Asymptotic limits

It is easy to identify from the Da → 0 and Da → ∞ limits, the two asymptotic regimes that may be reached from the presented model. In the case of diffusion-limited ablation (Da → ∞), eq. ( 28) becomes :

∂h ∂t = 1 h - dη dt (61)
from which the classical results on diffusion-limited etching or growth are recovered : the interface is flat, and its velocity is not steady, unless h becomes large. Considering dη dt = 0, there is a parabolic law h 2 = h (0) 2 + t.

The converse case makes the chosen reference time unusable, and one has to set t * 0 = Da t 0 = h 0 k(C 0 υs) . Note that h 0 is not any more a diffusion layer thickness. Indeed, the surface evolution does not depend any more on the distance to the chemical source.

Eq. ( 28) then becomes :

∂h ∂t * = 1 + ∂h ∂x 2 + ∂h ∂y 2 - dη dt * (62)
Considering again dη dt = 0, one faces now a classical Hamilton-Jacobi equation, the solutions of which have been the object of a broad literature.

Transient simulations

The evolution equation for S (equation 10-11) has been studied by a direct numerical integration. However, such an equation was obtained at first for a regular surface, and it has been seen that this was not necessarily the case. Moreover, in a discretized surface, every point can be singular, even if the continuous expression is differentiable. An explicit numerical scheme has been designed, using a discretized surface and applying eq. ( 56) to find the position of the evolved points and construct the new surface.

Once the surface is discretized, all points become singular. Their evolution, in the limit of locally constant h and Da parameters, is summarized at fig. 10; rules for the numerical scheme may be deduced from it.

The algorithm of surface construction is built as follows :

Step 1 : Discretization of the initial surface. Let δx be a regular space increment and (i, j) the discretized variables corresponding to (x, t). The discretized surface is a set of points M j i = t (x j i , z j i ). To each segment (Σ j i ) = [M j i-1 , M j i ] is associated θ j i , the angle between the normal and the vertical axis.

Step 2 : Evolution of adjacent segments At the time increment j, the point M j i is surrounded by two segments (Σ j i ) and (Σ j i+1 ). As seen before, the evolved segments will obey to the following equations :

(Σ j+1 i ) : sin(θ j i )(x -x j i ) + cos(θ j i )(z -z j i ) = W n (θ j i ) • nδt (63) (Σ j+1 i+1 ) : sin(θ j i+1 )x + cos(θ j i+1 )z -c 2 = W n (θ j i+1 ).δt (64)
where , c 1 = sin(θ j i )x j i + cos(θ j i )z j i , and c 2 = sin(θ j i+1 )x j i + cos(θ j i+1 )z j i .

The evolution of a type IV edge is given by the simple intersection of the evolved segments.

For a type I,II or III edge, the evolution is given by the intersection of the evolved segments and the locus of points defined by W as seen in figures 10 b).

Step 3 : Reconstruction Once the evolved surface has been built, there remains to choose new points O(i, j + 1) to represent it. In order to avoid point merging, we take

x j i = x i constant ∀j.
With such a choice, we have :

• For a type II edge :

• if θ j i > θ j i+1 , z j+1
i is given by the equation (64) with x = x i ;

• if θ j i < θ j i+1 , z j+1 i is given by the equation (63) with x = x i ;

• For a type I or III edge : z j+1 i = W z (θ = 0);

• For a type IV edge :

• if θ j i > θ j i+1 , z j+1
i is given by the equation (63) with x = x i ;

• if θ j i < θ j i+1 , z j+1 i is given by the equation (64) with x = x i ;

One of the most interesting points is to find out in what kind of situation a stationary solution made of circle arcs appears. An example is illustrated at fig. 11 : an initial condition has been taken with an "inverse" profile, that is, with symmetrical singular points pointing towards the solid instead of the fluid, also featuring a curvature superior to the opposite of the stationary curvature. It is seen that relaxation towards the stationary "circle-arcs" solution occurs within a few dimensionless time units, with two phenomena : i) the smooth arcs increase curvature, and a singular point appears, which then relaxes towards the stationary situation, and ii) the angle at the initial singular point (type I) diminishes rapidly and a smooth circle arc appears eventually.

7 Application case : onset of cellular roughness from sublimation A numerical evaluation of the curvature radius Da -1 h 0 = D/k can be made, taking typical values of plasma jet conditions, such that the surface temperature is about 4300 K and the pressure approximately 5 M P a. In such conditions, the major carbon-containing species is C 3 [START_REF] Duffa | Ablation, Monograph[END_REF]. The value of the sticking coefficient is known experimentally [START_REF] Palmer | Vaporization of carbon[END_REF][START_REF] Baker | The high temperature thermochemical properties of carbon[END_REF] : α 0.028.

It is considered that, even if the flow is turbulent, there exists a thin layer close to the wall where laminar flow occurs ; this layer has to be traversed by the gaseous species, so the diffusion coefficient in laminar flow is the critical transport property. Its value can be estimated by the classical theory of Chapman-Enskog [START_REF] Hirschfelder | Molecular theory of gases and liquids[END_REF], provided the parameters of interaction. Here the data of [START_REF] Biolsi | Transport properties for a mixture of the ablation products C, C 2 and C 3[END_REF] has been chosen : a Lennard-Jones potential (σ = 0.445 nm and k B = 128 K). These values lead to a diffusion coefficient D = 0.026 cm 2 • s -1 . Then the resulting value for the curvature radius is about 3 µm. This value scales reasonably with the features of fig. 3, provided the large uncertainty on the sticking coefficient. In fact, measuring this scale can be a way to have an indirect assessment of α.

A remark has to be done about such a scale. If the heat flux is strongly varied, the fact that sublimation is the main ablation cause renders it "robust", because flux variations will primarily lead to changes in sublimation flux, but not in surface temperature ; since D/k is mainly a function of T , it will not vary strongly when the heat flux varies.

It is also straightforward to evaluate the possible importance of curvature effects, as depicted in [START_REF] Srolovitz | Analytical and numerical modeling of columnar evolution in thin films[END_REF][START_REF] Thiart | Chemical vapor deposition and morphology problems[END_REF]. The correction that has to be added to the source term is :

R = R(1 + Γκ) ( 65
)
where κ is the curvature and Γ = γ f s υs RT is the length scale for the correction. Such a Γ factor ranges between 10 -10 and 10 -8 m depending on the chosen value for the surface tension of carbon. It is thus clear that the curvature correction is negligible in the presented case. Now a question remains : if the identified scale is micrometric, how to explain the millimeter-scale features of fig. 1b) ? An answer, at least partial, may arise from the consideration of heat transfer by diffusion.

Thermal conduction scalloping

Let us now consider that the transported quantity that brings a rate limitation is temperature instead of gas species mass. We are working now at a length scale that is superior to the mass-transfer boundary layer, and may consider that there is an effective rate constant

k = R/C 0 = k -1 + h 0 D -1 -1
that incorporates the effect of gas diffusion through this layer. On the other hand, it will be assumed that k depends on T through, for example, an Arrhenius law. Summarizing, the reaction rate is written under the form :

R = k C 0 = A exp - E a RT C 0 (66)
By analogy with the mass transfer case, it will be considered that the temperature is held constant at a certain height :

T (z = η) = T 0
Applying again the hypothesis of vertical flux approximation, the heat flux is given by :

q z = - λ h (T (z = h) -T 0 ) (67)
In addition to this, let us assume that there is a constant, known, vertical heat flux at the top of the thermal boundary layer q z (z = η) = q 0 , and that the thermal conductivity is not a function of temperature (indeed, it depends on temperature much less than the reaction rate). Then, the temperature will be a simple function of h :

T (h ) = T 0 - q 0 λ h (68)
On the other hand, recalling eqs. (1 -3) and ( 17) as well as the definition of θ, one has :

V = ∂h ∂t = R (T (h (x))) cos θ (69)
The stationarity of the profile implies, as before, that the velocity is independent of the x coordinate. This yields the following result :

κ = 1 1 + ( ∂h ∂x ) 2 ∂h ∂x -1 ∂ ln R ∂x (70)
Making use of eqs. ( 68) and (66) finally gives :

κ = - E a RT (h(x)) • q 0 λT (h(x)) cos θ (71)
Choosing a reference length h 0 = λT 0 q 0 gives the following relation :

κ = -γ a h -1 0 (1 -h ) -2 cos θ (72)
where γ a is the scaled activation energy E a /RT 0 . In this solution, it is found that the curvature radius is not any more a constant, but a function of the depth h . Fig. 12 is a plot of a numerical evaluation of h (x), where it is seen that the shape differs neatly from the semi-circle in the region of high slopes. In practice, the dimensionless depth is no more than 0.1, so that the surface temperature is 90% of the top fluid temperature.

This suggests that the maximal order of magnitude for the scallop curvature radius is h 0 0.2γ a . Numerical application in the case of fig. 1b) is the following :

q 0 ≈ 10 M W.m -2 , T 0 ≈ 4800 K, λ 0 ≈ 10 W.m -1 .K -1 , γ a ≈ 12.
Then r ≈ h 0 /2.4 ≈ 16 mm. This matches approximately the observed scale.

Fortunately, the fact that this length scale is much higher than the preceding one helps to validate the isothermal hypothesis that has been given in the case of mass diffusional limitations.

Conclusion

The question of roughness set-up due to ablation in atmospheric reentry has been addressed. Apart from the trivial cause due to the material inhomogeneities, another cause seems to act even in the case of homogeneous materials. It is basically a dynamical effect based on the concurrence between transfer and reaction : when the surface is higher, the consumption rate is higher, but when it is more inclined with respect to the principal transfer direction, the rate is lowered. Thus, points with different altitudes may well have the same recession velocity, leading to a rough surface profile.

A simplified, isothermal model featuring the interaction of 1D (normal) diffusion with heterogeneous reaction or sublimation has been built, and it shows that in the intermediate regime (that is , neither diffusion-limited nor reaction-limited), cellular roughness may appear, under the form of a stable surface made of circle arcs in 2D (and by extension of sphere caps in 3D). Such a morphology has been indeed found experimentally in various cases. One of the successes of the presented model is that it helps understand why a characteristic length scale, different from the turbulence-related scales, appears in ablation, even without any heterogeneity of the substrate. Also, since this roughness is capable of being formed in laminar flow conditions, it is a new candidate to the explanation of laminar-to-turbulent transition in the flow regime around the reentry body.

Two length scales have been found for the curvature radii of the roughness features : the first one, arising from mass transfer, is much smaller than the second one, arising from heat transfer. Indeed, both of them are known experimentally.

Consequences of such a behavior have to be carefully considered with respect to other experimental results, and the solidity of the model with respect to addition of new physical features, such as the curvature-enhanced chemical reactivity or an explicit coupling between mass, heat and momentum transfers, has to be tested in future work. Also, extensions of the simple model to anisotropic and composite surfaces will be treated in other papers. and the reference length is given by heat conduction h 0 = λ 0 T 0 q -1 0 .

Tables

1 D 1 H 1 k = wave vector of a perturbation, m - 1 M 2 R 1 W

 111121 phase concentration , mol • m -3 c = Root-mean-square molecular velocity, m.s -= diffusion coefficient, m 2 • s -1 Da = Damköhler number E a = activation energy, J.mol -= Hamiltonian operator, m • s -1 h = height from upper domain limit, m h 0 = reference height, m m(x) = imaginary part of a complex number J = mole flux, mol • m -2 • s -1 k = heterogeneous reaction coefficient, m • s -= position of a surface point, m M = molar mass, kg • mol -1 List of Symbols (cont'd) n = normal vector of the interface p = pressure, P a q = negative normal velocity amplitude, m • s -1 q 0 = heat flux, W.m -= mole ablation rate per unit surface, mol • m -2 • s -1 e(x) = real part of a complex number S = function allowing to define the surface as an isosurface, m T = temperature, K t = time, s V = global ablation velocity, m • s -= velocity of a point of the surface , m • s -1 x = lateral space coordinate, m y = lateral space coordinate, m z = vertical space coordinate, m List of Symbols (cont'd) Greek α = sticking coefficient Γ W = graph of W γ = Knudsen-Langmuir parameter γ a = scaled activation energy γ f s = fluid-solid interface energy, J.m -2 δ = perturbation η = position of the diffusive boundary layer top , of the surface at current point, rad θ 0 = critical tilt angle for Γ W , rad τ = time scale factor, s υ = molar volume, m 3 • mol -1 List of Symbols (cont'd) φ = angle between the normal to the surface and the flux, rad ω = time pulsation of a perturbation, s -1

•

  -= relative to the left-hand side of a singular point • = dimensionless variable • = relative to a moving coordinate frame • = relative to equilibrium or stationarity Constants k B = 1.3806581 • 10 -23 J • K -1 : Boltzmann's constant N = 6.02213674 • 10 23 mole -1 : Avogadro's number R = 8.3145107 J • mole -1 • K -1 : perfect gas constant

  ) + O 2 (g) → 2CO(g) -1 -1 C(z = η) = 1 J • n = ν g kC 2 Sublimation nC(s) C n (g) -n +1 C(z = η) = 0 J • n = ν g (α c /4) C -C 3 Deposition C n H m (g) → nC(s) + m 2 H 2 (g) +n -1 C(z = η) = 1 J • n = ν g kC

Fig. 1 .

 1 Figures

Fig. 2 .Fig. 3 .

 23 Fig. 2. SEM micrograph of an ablated C/C sample (ex-phenolic resin/carbon composite)

Fig. 6 .Fig. 8 .$Fig. 9 .Fig. 10 . 5 Fig. 11 .Fig. 12 .

 6891051112 Fig. 6. Representation of the concentration field for a steady, quasi-1D, nontrivial solution of the ablation profile

Table 1

 1 Balance equations, stoichiometric coefficients, and boundary conditions for two causes of ablation. Comparison with CVD

this coefficient is not a sticking probability but is very close to it, when its value is small compared to 1[START_REF] Schrage | A Theoretical Study of Interphase Mass Transfer[END_REF].

Acknowledgements

The authors wish to thank D. Rousselle (CEA) for transmitting profilometry data, which has been obtained at Laboratoire de Microanalyse des Surfaces (ENSMM Besançon), as well as J. Lachaud (LCTS) for providing micrograph at figs. 2 and 3. Snecma Propulsion Solide is acknowledged for Masters and PhD grants to Y. A.

Such an analysis would have to be modified in the presence of curvature effects, since they have been proved to destabilize CVD growth processes. To simplify notations, we will drop the primes corresponding to the moving coordinate frame. The local normal velocity v writes :

The vector v is plotted at figure 7 for all possible values of θ. Even though v is the local normal velocity of the surface, we will see that, for a point M of the surface, W = dM dt = v.

Since v and n are collinear, il will be practical to define the quantity :

and rewrite eq. ( 5) under a Hamilton-Jacobi form :

If we suppose v to be differentiable, then the associated characteristic system is :

with x i = x, y, z. After some algebra, eq. ( 53) may be rewritten in vector form :

where ∇ n = ∂ ∂n is the (formal) derivative with respect to the local normal components. To illustrate the calculus, we consider the 2D case. In our application, with n = t (sin θ; cos θ), one obtains :

which gives, using eq. ( 51) :

The graph of W, noted Γ W , is also presented at figure 7. This figure indicates that this graph may be constructed as the envelope of the set of lines defined by :

One notes that Γ W shows three singular points. The first one lies at the bottom of the

Table Captions

• Table 1. Balance equations, stoichiometric coefficients, and boundary conditions for two causes of ablation.

Figure Captions

•