MSc Sara Kaffashian
email: sara.kaffashian@inserm.fr

MSc Aline Dugravot

PhD Hermann Nabi

PhD G David Batty

PhD Eric Brunner

PhD Mika Kivimäki

PhD Archana Singh-Manoux

Predictive Utility of the Framingham General Cardiovascular Disease Risk Profile for Cognitive Function: Evidence from the Whitehall II Study

Introduction

The importance of vascular risk factors and disease for cognitive impairment and dementia in older adults is widely recognized [START_REF] Bursi | Heart disease and dementia: a population-based study[END_REF][START_REF] Fillit | Cardiovascular risk factors and dementia[END_REF][START_REF] Viswanathan | Vascular risk factors and dementia: how to move forward?[END_REF]. There is growing evidence to suggest that these risk factors are also associated with deficits in cognitive function in mid-life, prior to the onset of overt clinical symptoms of dementia [START_REF] Elias | Untreated blood pressure level is inversely related to cognitive functioning: the Framingham Study[END_REF][START_REF] Knopman | Cardiovascular risk factors and cognitive decline in middle-aged adults[END_REF][START_REF] Knopman | Cardiovascular risk factors and cerebral atrophy in a middle-aged cohort[END_REF][START_REF] Singh-Manoux | High blood pressure was associated with cognitive function in middle-age in the Whitehall II study[END_REF][START_REF] Singh-Manoux | Low HDL cholesterol is a risk factor for deficit and decline in memory in midlife: the Whitehall II study[END_REF][START_REF] Sparks | Increased incidence of neurofibrillary tangles (NFT) in non-demented individuals with hypertension[END_REF]. Several risk algorithms have been developed to predict the risk of stroke and cardiovascular events [START_REF] Conroy | Estimation of ten-year risk of fatal cardiovascular disease in Europe: the SCORE project[END_REF][START_REF] Agostino | General cardiovascular risk profile for use in primary care: the Framingham Heart Study[END_REF][START_REF] Wolf | Probability of stroke: a risk profile from the Framingham Study[END_REF]. Such scores improve the efficiency of risk prediction and provide a more realistic assessment of the collective importance of risk factors as well as easier interpretation of the risk of disease. They may equally help to identify persons at increased risk of disease resulting from risk below the clinical threshold on individual risk factors.

The association between multiple vascular risk factors and cognition has been examined by a number of studies using the Framingham Stroke Risk Profile (FSRP) [START_REF] Brady | Stroke risk predicts verbal fluency decline in healthy older men: evidence from the normative aging study[END_REF][START_REF] Elias | Framingham stroke risk profile and lowered cognitive performance[END_REF][START_REF] Llewellyn | Framingham Stroke Risk Profile and poor cognitive function: a population-based study[END_REF][START_REF] Seshadri | Stroke risk profile, brain volume, and cognitive function: the Framingham Offspring Study[END_REF]. These studies have reported an inverse association between the 10-year risk for stroke and performance on multiple cognitive tests. The majority of these studies have used a cross-sectional design, which provides little information about risk prediction. [START_REF] Elias | Framingham stroke risk profile and lowered cognitive performance[END_REF][START_REF] Llewellyn | Framingham Stroke Risk Profile and poor cognitive function: a population-based study[END_REF][START_REF] Seshadri | Stroke risk profile, brain volume, and cognitive function: the Framingham Offspring Study[END_REF]. In addition, the FSRP is designed for prediction of stroke and therefore does not cover the full range of potentially relevant cardiovascular diseases, such as myocardial infarction, coronary insufficiency, angina, and peripheral artery disease. We used the recently developed Framingham General Cardiovascular Disease Risk Profile to examine associations with cognitive performance and then decline over a ten year period in a large sample of middle aged individuals.

Methods

Data were drawn from the Whitehall II study, established in 1985 to examine the socioeconomic gradient in health and disease among 10,308 civil servants (6,895 men and 3,413 women). Details of the cohort have been described previously [START_REF] Marmot | Cohort Profile: the Whitehall II study[END_REF]. Briefly, all London based office staff aged 35-55 working in 20 civil service departments were invited to participate, of which 73% agreed. Baseline examination took place during 1985-1988 and consisted of a clinical examination and a self administered questionnaire that included sections on demographic characteristics, medical history and health behaviors. Clinical examination included measures of blood pressure, anthropometry, biochemical variables, subclinical makers of cardiovascular disease, and neuroendocrine function. A battery of cognitive tests was introduced to the study at Phase 5 (1997Phase 5 ( -1999)), and repeated at phases 7 (2002-2004), and 9 (2007-2009). Informed consent was obtained from all participants and the University College London ethics committee approved the study.

Assessment of risk factors for the CVD Risk Profile

The Framingham general CVD risk score is designed for use in primary care to identify individuals at high risk for CVD events that include coronary, cerebrovascular and peripheral arterial disease, and heart failure [START_REF] Agostino | General cardiovascular risk profile for use in primary care: the Framingham Heart Study[END_REF]. Its development was based on the prediction of 1,174 CVD events over a 12-years follow-up period of 8,491 participants in the Framingham Heart study.

The risk score, calculated using information on age, HDL cholesterol, total cholesterol, systolic blood pressure, cigarette smoking, and diabetes provides an estimate of the risk of CVD over a 10-year period.

The risk score components in our study were drawn from questionnaire and clinical examination data at phase 5. HDL and total cholesterol (mg/dL) were measured from blood samples collected after either an 8-hour fast for participants presenting in the morning, or at least 4 hours after a light fat-free breakfast for those presenting in the afternoon. Cholesterol was measured using a Cobas Fara centrifugal analyzer (Roche Diagnostics System). HDL cholesterol was measured by precipitating non-HDL cholesterol with dextran sulfate-magnesium chloride with the use of a centrifuge and measuring cholesterol in the supernatant fluid. Systolic blood pressure (mm Hg) was taken as the average of two measurements in the sitting position after a 5 minute rest with the Hawksley random-zero sphygmomanometer. Treated hypertension was determined according to antihypertensive medication use. This included diuretics, beta blockers, ACE inhibitors and calcium channel blockers. Participants were categorized with respect to their cigarette smoking status as current smokers or past/non smokers. Diabetes was defined by a fasting glucose ≥7.0 mmol/L or a 2-hour postload glucose ≥11.1 mmol/L or reported doctor diagnosed diabetes, or use of diabetes medication [START_REF]Report of the expert committee on the diagnosis and classification of diabetes mellitus[END_REF]. Raw scores were calculated and then converted to 10-year risk or predicted probability of incident CVD expressed as a percentage [START_REF] Agostino | General cardiovascular risk profile for use in primary care: the Framingham Heart Study[END_REF]. Missing data for any risk score component were replaced by data from phase 4 (1995-1996), N=27, and in the case of biological measures (HDL cholesterol, total cholesterol, and systolic blood pressure), by data from phase 3 (1991-1993), N=624. Individuals (N=319) with a history of stroke or coronary heart disease (CHD) at phase 5 were excluded. CHD status at phase 5 was defined as nonfatal myocardial infarction (MI) and "definite" angina. MI diagnosis, based on clinical examinations at Phases 1, 3 or 5 and records obtained from general practitioners and hospitals, was assessed using MONICA criteria [START_REF] Tunstall-Pedoe | Myocardial infarction and coronary deaths in the World Health Organization MONICA Project. Registration procedures, event rates, and case-fatality rates in 38 populations from 21 countries in four continents[END_REF].

Angina was assessed based on participant"s reports of symptoms with corroboration in medical records or abnormalities on a resting electrocardiogram, an exercise electrocardiogram, or a coronary angiogram. Stroke diagnosis was self-reported and included history of stroke or a transient ischemic attack (TIA).

Cognitive function

The cognitive test battery, administered at the clinical examinations at Phases 5, 7 and 9, described below, consists of five standard tasks chosen to provide a comprehensive assessment of cognitive function.

The Alice Heim 4-I (AH4-I) is composed of a series of 65 verbal and mathematical reasoning items of increasing difficulty [START_REF] Heim | AH4: Group Test of General Intelligence[END_REF]. It tests inductive reasoning, measuring the ability to identify patterns and infer principles and rules. The time allowed for this test was 10 minutes.

Short-term verbal memory was assessed with a 20-word free recall test. Participants were presented a list of 20 one or two syllable words at two second intervals and were then asked to recall in writing as many of the words in any order and had two minutes to do so.

We used two measures of verbal fluency: phonemic and semantic. Phonemic fluency was assessed via "S" words and semantic fluency via "animal" words [START_REF] Borkowski | Word fluency and brain damage[END_REF]. Subjects were asked to recall in writing as many words beginning with "S" and as many animal names as they could.

One minute was allowed for each test.

Vocabulary was assessed using the Mill Hill Vocabulary test, used in its multiple-choice format, consisting of a list of 33 stimulus words ordered by increasing difficulty and six response choices. ( 22)

Covariates

The following covariates were included; age, marital status, ethnicity, and education.

Although age is a component of the Framingham General CVD risk score, we included it as a covariate because of its established association with cognitive function [START_REF]Alzheimer's Disease International[END_REF]. Ethnicity consisted of two groups; white and non-white. Marital status included two categories; married/cohabiting and single/divorced/widowed. Education was measured as the highest level of education achieved. Categories included (1) elementary or lower secondary, (2) higher secondary (A" levels), and (3) first university degree or higher. We also examined the effect of occupational position at baseline in lieu of education. This variable consisted of three categories: (1) high (administrative), ( 2) intermediate (professional or executive), and (3) low (clerical or support).

Statistical methods

Descriptive analyses were carried out to examine the distribution of the CVD risk score components, all covariates, as well as cognitive function and decline in our study population. In order to carry out cross sectional and longitudinal analyses on the same population, we started with those who had data at Phase 5 and at least one repeat measure so that cognitive decline could be calculated (implying participation in phase 7 or 9 of the study). Approximately 86% of the 4,837 participants included in the study had cognitive data at all three phases of this study. As follow-up time varied between individuals (mean=10.5, SD =0.5) we first estimated the rate of change, standardizing it to represent 10-year change for each individual. The interaction term between the risk score and sex (p<0.001 for all cognitive tests) led us to stratify all analyses by sex.

We first explored correlations of the 10-year CVD risk, assessed at phase 5, with cognitive function at phases 5 and 9 and cognitive decline over the 10-year follow-up.

Subsequently, regression analysis was used to model the impact of a 10 % increment in CVD risk on cognitive function at Phase 5 and 10-year cognitive decline. In these analyses, we first calculated an overall test of association using multivariate analysis of variance (MANOVA) in order to account for the correlation between the cognitive tests and control Type 1 error inflation due to multiple tests. Then, linear regression was used to determine the cross sectional association between the CVD risk, modeled to show the impact of a 10 percentage point increment in risk, and each cognitive test separately. We first examined unadjusted models, followed by models adjusted for age only, and finally the fully adjusted models including all four covariates.

The longitudinal analyses assessed the association between 10 year CVD risk at phase 5 and 10-year cognitive decline, calculated using data from phases 5, 7 and 9 as described earlier.

Linear regression was used to model the association between a 10% increment in CVD risk at baseline and cognitive decline. The adjustment for covariates was performed in three steps, as in the cross-sectional analysis. MANOVA analyses were also carried out to examine the association between CVD risk and overall cognitive decline. In supplementary analyses, occupational position replaced adjustment for education in order to assess the effect of a later life measure of socioeconomic circumstances. Tests of statistical significance were two sided and results were statistically significant at p<0.05. All analyses were conducted using SAS software (version 9; SAS Institute, Cary, NC).

Results

Of 10,308 participants at baseline of the Whitehall II study (Phase 1, 1985-1988), 7,830 (75.9%) individuals at phase 5 (1997-1999) responded to the questionnaire or came to the clinical examination. Of these, 5,146 (65.7%) had complete data on cognitive function and all covariates.

After excluding 319 participants with a history of CHD or stroke at phase 5, our final study sample consisted of 4,827 individuals (3,486 men and 1,341 women). Compared to the sample used in this analysis, participants at phase 5 excluded from this study had a higher mean 10-year CVD risk (12.1% vs. 9.8%, p <0.001). Missing data were also influenced by age, sex, and education as individuals excluded were more likely to be women, older, and have a lower level of education (all p-values <0.001).

The characteristics of the study population are shown in Table 1. Men had a considerably higher mean 10-year CVD risk than women; 1,711 (49.1%) and 514 (14.7%) of men and 71 (5.3%) and 3 (0.2%) of women had a 10-year CVD risk higher than 10% and 20% respectively.

The correlation analysis (supplementary TableS1) suggested robust cross-section and prospective associations between CVD risk and cognition except the association with vocabulary at phase 5 in men. These associations were largely similar in men and women, except for the tests of reasoning and vocabulary.

Regression analysis to model the cross sectional associations between 10% increment in

Framingham CVD risk and cognitive function are presented in Table 2. The MANOVA analyses show significant associations between the CVD risk and overall cognitive function in the fully adjusted model in men (p=0.05) and women (p<0.03). The unadjusted regression estimates show a 10% higher CVD risk to be associated with 1.66 lower score on the test of reasoning (AH4-I) for men (95% confidence interval (CI) =-2.10, -1.22). In the unadjusted models CVD risk was inversely associated with all individual cognitive domains except the vocabulary test in men (p=0.30). These associations were robust to adjustment for age (all p-values <0.01). In the fully adjusted models all tests except reasoning in men (p=0.17) and the verbal fluency tests in women remained associated with CVD risk. Adjustment for occupational position yielded similar results to analyses adjusted for education (supplementary TableS2).

Table 3 shows the results of linear regression used to model the relation between a 10% increment in CVD risk at baseline and cognitive decline over 10 years. The unadjusted MANOVA (p<0.001) suggests an association between CVD risk and overall cognitive decline only in men. In unadjusted models a 10% increment in CVD risk was associated with 1.30 points (95% CI = -1.58, -1.02) greater decline in reasoning. In unadjusted models in men, these affects were evident for all cognitive domains except memory; in fully adjusted models the association was robust only with reasoning (p=0.009). Replacing education with occupational position did not lead to significant changes in the results (supplementary TableS3).

The results on cognitive decline in women prompted us to further explore this association by categorizing the risk score differently in men and women (see supplementary TableS4). These results, adjusted for all covariates, suggest that all cognitive domains except vocabulary decline in all CVD risk groups in men and women.

We carried out several sensitivity analyses to test the robustness of our findings. First, we examined whether use of antihypertensive medication, a component of the Framingham CVD risk algorithm, over the follow-up period, from phase 5 to phase 9, affected the association between CVD risk and cognitive decline. An increasing proportion of participants in the study reported to be on antihypertensive medication, 9.0 % at phase 5, 20.3% at phase 7, 23.3% at phase 8 and 31.3% at phase 9. As expected, adjustment for use of antihypertensive medication over study follow-up slightly attenuated the association between CVD risk and cognitive decline in both men and women. We obtained similar results when we adjusted for use of other classes of CVD medications (nitrates, antiplatelets, and lipid lowering drugs), results not shown but available upon request.

Second, we repeated the analyses of the association between CVD risk and cognitive decline, excluding participants who had a validated CHD event over the follow-up (n=160).

These results were essentially the same as those reported in the main analyses.

Third, since we had imputed the Framingham CVD risk profile for participants who were missing data for one or more components of the risk score, we repeated all analyses with the sample of participants who had complete data at phase 5 (n=4221). Again, we observed similar results to those in the analyses with the imputed data.

Discussion

In this large prospective cohort study of a middle-aged population, an adverse Framingham general CVD risk profile, a validated predictor of future CVD, was associated with poor cognitive function in middle-aged men and women. When these associations were modeled using 10% increment in CVD risk, as has been previously done for stroke risk, (14, 15) the effects were much larger for women than for men. This may be due to the differences in risk distribution in men and women; in our study, and perhaps in others, the mean CVD risk in women was lower, at 4.1% compared with 12 % in men. In our study, cross-sectional correlation coefficients (supplementary TableS1) between CVD risk and cognitive function pointed to comparable associations in men and women. Thus, a 10% increase in CVD risk is a considerably larger increase in risk in women compared to men. In regression analyses, cross-sectional associations were robust and largely persisted after adjustment for demographic variables and education. With respect to 10-year cognitive decline, there was evidence of cognitive decline in all domains except vocabulary at all levels of CVD risk. However, higher CVD risk was associated with greater decline only in reasoning in men.

Comparison with other studies

Findings from this study support results from studies that have examined the importance of multiple vascular and cardiovascular risk factors by examining the collective effect of individual risk factors in relation to cognition [START_REF] Kivipelto | Risk score for the prediction of dementia risk in 20 years among middle aged people: a longitudinal, population-based study[END_REF][START_REF] Kivipelto | Midlife vascular risk factors and late-life mild cognitive impairment: A population-based study[END_REF][START_REF] Kivipelto | Midlife vascular risk factors and Alzheimer's disease in later life: longitudinal, population based study[END_REF][START_REF] Kivipelto | Obesity and vascular risk factors at midlife and the risk of dementia and Alzheimer disease[END_REF][START_REF] Whitmer | Midlife cardiovascular risk factors and risk of dementia in late life[END_REF]. For example, Whitmer and colleagues reported that presence of multiple cardiovascular risk factors at midlife independent of age, race, sex and education substantially increased risk of dementia in old age. Those having simultaneously high cholesterol, hypertension, diabetes, and being smokers had more than a two fold greater risk of dementia than those with no such risk factors [START_REF] Whitmer | Midlife cardiovascular risk factors and risk of dementia in late life[END_REF]. The dementia risk score developed by Kivipelto and colleagues also highlights the role of multiple cardiovascular risk factors in middle age and the future risk of dementia [START_REF] Kivipelto | Risk score for the prediction of dementia risk in 20 years among middle aged people: a longitudinal, population-based study[END_REF].

Given the importance of multiple vascular risk factors in relation to cognitive function, stroke, and CHD and the more global CVD risk scores present an important opportunity to study these associations. Although most of the studies in this domain have focused on stroke risk scores, especially the Framingham Stroke Risk Profile (FSRP), [START_REF] Brady | Stroke risk predicts verbal fluency decline in healthy older men: evidence from the normative aging study[END_REF][START_REF] Elias | Framingham stroke risk profile and lowered cognitive performance[END_REF][START_REF] Llewellyn | Framingham Stroke Risk Profile and poor cognitive function: a population-based study[END_REF][START_REF] Seshadri | Stroke risk profile, brain volume, and cognitive function: the Framingham Offspring Study[END_REF] we can draw some comparisons with these investigations. The cross sectional associations between CVD risk and cognitive function, observed in our study are largely consistent with results obtained in these studies. However, comparison with their findings is limited because of differences in study populations and neuropsychological tests used. In addition, whereas we found sex differences in the associations and stratified our analyses accordingly, none of these studies reported sex differences in the association between stroke risk and cognitive function. We found that after adjusting for age, sex and education, 10 year CVD risk was associated with poorer performance in the test of memory in both men and women. However, while one study found an association with stroke risk and memory,(15) the other two did not find a similar association [START_REF] Elias | Framingham stroke risk profile and lowered cognitive performance[END_REF][START_REF] Seshadri | Stroke risk profile, brain volume, and cognitive function: the Framingham Offspring Study[END_REF].

Our results concerning the association between CVD risk and 10 year cognitive decline suggest a similar rate of decline at all levels of risk in women. In men, there is an indication of a global effect that in individual tests show greater decline in inductive reasoning in those with higher CVD risk at baseline. A previous study on older men showed the Framingham stroke risk score to predict decline in verbal fluency but not memory and visuospatial performance [START_REF] Brady | Stroke risk predicts verbal fluency decline in healthy older men: evidence from the normative aging study[END_REF].

Knopman and colleagues reported a steeper 6 year decline in processing speed and phonemic fluency in diabetics and only in processing speed for individuals with hypertension [START_REF] Knopman | Cardiovascular risk factors and cognitive decline in middle-aged adults[END_REF]. Another study in an older cohort found an association between hypertension and cognitive decline over a 4-year period [START_REF] Tzourio | Cognitive decline in individuals with high blood pressure: a longitudinal study in the elderly. EVA Study Group. Epidemiology of Vascular Aging[END_REF]. Our finding for no greater decline in memory in those with higher CVD risk is consistent with a body of literature suggesting that frontally mediated cognitive functions such as verbal fluency, may be more vulnerable to the pathophysiological processes linked to cardiovascular risk factors than other cognitive abilities such as memory [START_REF] Knopman | Cardiovascular risk factors and cognitive decline in middle-aged adults[END_REF][START_REF] Looi | Vascular dementia as a frontal subcortical system dysfunction[END_REF][START_REF] Elias | NIDDM and blood pressure as risk factors for poor cognitive performance. The Framingham Study[END_REF].

Strengths and limitations

There are a number of limitations to our study. First, the participants of the Whitehall II study are office-based civil servants and thus are not fully representative of the British population which may limit the generalizability of our findings. Second, individuals who were included in our analysis had a more favorable demographic and CVD risk profile, suggesting that our results may be an underestimation of the relationship between CVD risk and cognitive function. In addition, since participants were tested three times over 10 years, there is a possibility of practice effects [START_REF] Rabbitt | Practice and drop-out effects during a 17-year longitudinal study of cognitive aging[END_REF]. As a result, the observed decline in cognitive function may again be an underestimation of the true extent of longitudinal cognitive decline. Finally, the relatively low 10year CVD risk for women in our study population did not allow adequate examination of the relation between CVD risk and 10-year cognitive decline in women.

Conclusions and implications

In summary, our study is the first to examine the relationship between CVD risk as determined by the Framingham general CVD risk profile, and cognitive function and 10-year decline in a large middle-aged cohort. Our results are important as they suggest that not only adverse CVD risk is robustly related to poorer cognitive function in late midlife, it is also associated with decline in at least one cognitive domain in men. To make a difference in outcomes, current thinking about cognitive aging must shift from focusing on thresholds to a continuum of cognitive impairment [START_REF] Hachinski | Shifts in thinking about dementia[END_REF]. Moreover, the current emphasis on risk factors especially treatable ones such as vascular risk factors must shift from late to early stages; subtle cognitive changes have been shown to be present as early as 22 years before diagnosis of Alzheimer"s disease [START_REF] Elias | The preclinical phase of alzheimer disease: A 22-year prospective study of the Framingham Cohort[END_REF]. Our own analyses concerning the role of treatment with antihypertensive medications in attenuating the association between CVD risk and cognitive decline, suggest that early preventive measures and treatment of CVD risk factors may indeed have a positive impact on cognitive outcomes. The Framingham CVD risk score presents a convenient way to identify individuals at an increased risk of cognitive deficits later in life. Given the aging of populations worldwide and the link between impaired cognitive function in midlife and dementia, early targeting and treatment of cardiovascular risk factors, already important in their own right, should gain urgency for prevention of cognitive impairment in late-life. † P value for mean difference between men and women. ± Decline calculated using 3 repeat measures (phases 5 (1997-1999), 7 (2002-2004) and 9 (2007-2009) and standardized to represent 10-year decline in order to take into account variations in the follow-up. (1997-1999), 7 (2002-2004) and 9 (2007-2009) and standardized to represent 10-year decline in order to take into account variations in the follow-up. ‡ β represents the regression coefficient showing the impact of a 10% increase in CVD risk.

Table 1 .

 1 Characteristics of the Study Population

	Variables

Table 2 .

 2 Cross-sectional association between a 10 % increment in the Framingham 10 year CVD risk and cognitive function

			Unadjusted			Adjusted for age			Multiple adjusted *
	Cognitive domain	β †	(95% CI)	p	β ‡	(95% CI)	p	β ‡	(95% CI)	p
	MEN
	MANOVA			<0.001		<0.001		0.05
	Reasoning (AH 4-I)	-1.66 (-2.10, -1.22) <0.001	-0.93 (-1.49, -0.37) 0.001	-0.34 (-0.82, 0.14) 0.17
	Memory	-0.56 (-0.66, -0.45) <0.001	-0.20 (-0.33, -0.06) 0.003	-0.14 (-0.27, -0.01) 0.04
	Semantic fluency	-0.85 (-1.03, -0.67) <0.001	-0.40 (-0.63, -0.17) <0.001	-0.24 (-0.45, -0.02) 0.03
	Phonemic fluency	-0.88 (-1.08, -0.69) <0.001	-0.40 (-0.64, -0.15) 0.001	-0.25 (-0.48, -0.01) 0.04
	Vocabulary (Mill Hill) -0.09 (-0.26, 0.08)	0.30	-0.46 (-0.66, -0.24) <0.001	-0.23 (-0.42, -0.05) 0.01
	WOMEN
	MANOVA			<0.001		<0.001		0.03
	Reasoning (AH4-I)	-8.74 (-10.91, -6.58) <0.001	-5.60 (-7.78, -3.43) <0.001	-2.65 (-4.42, -0.87) 0.003
	Memory	-1.41 (-1.92, -0.91) <0.001	-0.92 (-1.44, 0.39) <0.001	-0.58 (-1.08, -0.07) 0.03
	Semantic fluency	-2.57 (-3.43, -1.72) <0.001	-1.33 (-2.19, -0.47) 0.002	-0.33 (-1.08, 0.42) 0.38
	Phonemic fluency	-2.24 (-3.12, -1.36) <0.001	-1.25 (-2.15, -0.35) 0.006	-0.62 (-1.48, 0.23) 0.15
	Vocabulary (Mill Hill) -2.96 (-3.94, -1.98) <0.001	-2.28 (-3.29, -1.27) <0.001	-0.81 (-1.60, -0.02) 0.05
	*Adjusted for age, ethnicity, marital status, education.
	† β represents the regression coefficient showing the impact of a 10% increase in CVD risk.

Table 3 .

 3 The association between a 10 % increment in the Framingham 10 year CVD risk and cognitive decline † year decline in order to take into account variations in the follow-up . ‡ β represents the regression coefficient showing the impact of a 10% increase in CVD risk.

			Unadjusted			Adjusted for age			Multiple adjusted *
	Cognitive domain	β ‡	(95% CI)	p	β ‡	(95% CI)	p	β ‡	(95% CI)	p
	MEN
	MANOVA			<0.001			0.04		0.04
	Reasoning (AH 4-I)	-1.30 (-1.58, -1.02) <0.001	-0.46 (-0.81, -0.10) 0.01	-0.47 (-0.82,-0.11) 0.009
	Memory	-0.05 (-0.16, 0.07)	0.45	0.08 (-0.06, 0.23) 0.28	0.06 (-0.09, 0.21) 0.43
	Semantic fluency	-0.27 (-0.43, -0.11) <0.001	-0.13 (-0.33, 0.07) 0.20	-0.15 (-0.35, 0.04) 0.14
	Phonemic fluency	-0.17 (-0.34, 0.00)	0.05	-0.14 (-0.36, 0.07) 0.19	-0.16 (-0.38, 0.05) 0.14
	Vocabulary (Mill Hill) -0.25 (-0.35, -0.15) <0.001	-0.09 (-0.22, 0.03) 0.14	-0.08 (-0.21, 0.04) 0.17
	WOMEN
	MANOVA			0.13			0.14		0.04
	Reasoning (AH4-I)	-0.06 (-1.28, 1.16)	0.92	1.09 (-0.16, 2.34) 0.08	1.17 (-0.08, 2.44) 0.07
	Memory	-0.33 (-0.94, 0.28)	0.29	-0.19 (-0.82, 0.44) 0.55	-0.27 (-0.91, 0.36) 0.39
	Semantic fluency	-0.56 (-1.22, 0.10)	0.09	-0.48 (-1.16, 0.21) 0.17	-0.67 (-1.36, 0.02) 0.06
	Phonemic fluency	-0.29 (-1.06, 0.48)	0.46	-0.15 (0.94, 0.65)	0.72	-0.08 (-0.89, 0.72) 0.83
	Vocabulary (Mill Hill) -0.51 (-0.95,-0.06)	0.03	-0.39 (-0.85, 0.08) 0.10	-0.42 (-0.89, 0.04) 0.07
	* Adjusted for age, ethnicity, marital status, education
	† Decline calculated using 3 repeat measures (phases 5 (1997-1999), 7 (2002-2004) and 9 (2007-2009) and standardized to
	represent 10-

Table S2 .

 S2 Cross-sectional association between a 10 % increment in the Framingham 10 year CVD risk and cognitive function Adjusted for age, ethnicity, marital status, occupational position † β represents the regression coefficient showing the impact of a 10% increase in CVD risk.

			Unadjusted			Adjusted for age			Multiple adjusted *
	Cognitive domain	β †	(95% CI)	p	β †	(95% CI)	p	β †	(95% CI)	p
	MEN
	MANOVA			<0.001		<0.001		0.08
	Reasoning (AH 4-I)	-1.66 (-2.10, -1.22) <0.001	-0.93 (-1.49, -0.37) 0.001	-0.22 (-0.69, 0.24) 0.34
	Memory	-0.56 (-0.66, -0.45) <0.001	-0.20 (-0.33, -0.06) 0.003	-0.13 (-0.26, -0.01) 0.04
	Semantic fluency	-0.85 (-1.03, -0.67) <0.001	-0.40 (-0.63, -0.17) <0.001	-0.21 (-0.43, 0.00) 0.05
	Phonemic fluency	-0.88 (-1.08, -0.69) <0.001	-0.40 (-0.64, -0.15) 0.001	-0.21 (-0.45, 0.01) 0.06
	Vocabulary (Mill Hill) -0.09 (-0.26, 0.08)	0.30	-0.46 (-0.66, -0.24) <0.001	-0.21 (-0.39, -0.03) 0.02
	WOMEN
	MANOVA			<0.001		<0.001		0.09
	Reasoning (AH4-I)	-8.74 (-10.91, -6.58) <0.001	-5.60 (-7.78, -3.43) <0.001	-2.08 (-3.77, -0.39) 0.01
	Memory	-1.41 (-1.92, -0.91) <0.001	-0.92 (-1.44, 0.39) <0.001	-0.54 (-1.05, -0.04) 0.03
	Semantic fluency	-2.57 (-3.43, -1.72) <0.001	-1.33 (-2.19, -0.47) 0.002	-0.18 (-0.93, 0.56) 0.62
	Phonemic fluency	-2.24 (-3.12, -1.36) <0.001	-1.25 (-2.15, -0.35) 0.006	-0.44 (-1.27, 0.39) 0.30
	Vocabulary (Mill Hill) -2.96 (-3.94, -1.98) <0.001	-2.28 (-3.29, -1.27) <0.001	-0.53 (-1.28, 0.21) 0.16
	*

Table S3 .

 S3 The association between a 10 % increment in the Framingham 10 year CVD risk and cognitive decline † Adjusted for age, ethnicity, marital status, occupational position † Decline calculated using 3 repeat measures (phases 5

			Unadjusted			Adjusted for age			Multiple adjusted *
	Cognitive domain	β ‡	(95% CI)	p	β ‡	(95% CI)	p	β ‡	(95% CI)	p
	MEN
	MANOVA			<0.001			0.04		0.04
	Reasoning (AH 4-I)	-1.30 (-1.58, -1.02) <0.001	-0.46 (-0.81, -0.10) 0.01	-0.46 (-0.82, -0.11) 0.009
	Memory	-0.05 (-0.16, 0.07)	0.45	0.08 (-0.06, 0.23) 0.28	0.06 (-0.09, 0.21) 0.43
	Semantic fluency	-0.27 (-0.43, -0.11) <0.001	-0.13 (-0.33, 0.07) 0.20	-0.15 (-0.35, 0.04) 0.12
	Phonemic fluency	-0.17 (-0.34, 0.00)	0.05	-0.14 (-0.36, 0.07) 0.19	-0.16 (-0.39, 0.05) 0.13
	Vocabulary (Mill Hill) -0.25 (-0.35, -0.15) <0.001	-0.09 (-0.22, 0.03) 0.14	-0.09 (-0.21, 0.03) 0.16
	WOMEN
	MANOVA			0.13			0.14		0.03
	Reasoning (AH4-I)	-0.06 (-1.28, 1.16)	0.92	1.09 (-0.16, 2.34) 0.08	1.25 (-0.01, 2.52) 0.05
	Memory	-0.33 (-0.94, 0.28)	0.29	-0.19 (-0.82, 0.44) 0.55	-0.27 (-0.91, 0.36) 0.39
	Semantic fluency	-0.56 (-1.22, 0.10)	0.09	-0.48 (-1.16, 0.21) 0.17	-0.67 (-1.36, 0.02) 0.05
	Phonemic fluency	-0.29 (-1.06, 0.48)	0.46	-0.15 (0.94, 0.65)	0.72	-0.09 (-0.91, 0.71) 0.81
	Vocabulary (Mill Hill) -0.51 (-0.95,-0.06)	0.03	-0.39 (-0.85, 0.08) 0.10	-0.42 (-0.89, 0.04) 0.07
	*

ACKNOWLEDGEMENTS

Funding/Support: ASM is supported by a "European Young Investigator Award" from the European Science Foundation and the National Institute on Aging, NIH, USA (R01AG013196; R01AG034454). MK is supported by the Academy of Finland, the BUPA Foundation, and the National Institutes of Health, USA (R01HL036310; R01AG034454). David Batty is funded by a Welcome Trust Fellowship.

We thank all of the participating civil service departments and their welfare, personnel, and establishment officers; the British Occupational Health and Safety Agency; the British Council of Civil Service Unions; all participating civil servants in the Whitehall II study; and all members of the Whitehall II study team. The Whitehall II Study team comprises research scientists, statisticians, study coordinators, nurses, data managers, administrative assistants and data entry staff, who make the study possible.

Role of Sponsors:

The funding organization had no role in the design and conduct of the study, collection, management, analysis, and interpretation of data; and preparation, review, or approval of the manuscript.

Disclosures: None.

Supplementary tables

Table S1. Correlation coefficients between 10-year CVD risk at Phase 5 and cognitive function at phase 5, phase 9, and 10 year decline Phase 5 (1997Phase 5 ( -1999) ) Phase 9 (2007-2009) [START_REF] Conroy | Estimation of ten-year risk of fatal cardiovascular disease in Europe: the SCORE project[END_REF] ± Decline calculated using 3 repeat measures (phases 5 (1997-1999), 7 (2002-2004) and 9 (2007-2009) and standardized to represent 10-year decline in order to take into account variations in the follow-up.


Title: Letter By Hermann Nabi regarding article "Attained Educational Level and Incident Atherothrombotic Events in Low-and Middle-Income Compared With High-Income Countries"

Hermann Nabi, PhD in men and women. The results revealed a clear inverse association between higher education and incidence of cardiovascular events in men from HICs, a modest inverse association in women from HICs, a modest positive association in men from LMICs, and no association in women from LMICs. The authors should be applauded for conducting this research that compares different regions of the world with regard to educational gradient in cardiovascular events.

However, I believe that the authors may have over interpreted their findings. The inference that "In contrast to HICs, higher AEL may not be protective against cardiovascular events in LMICs, particularly in women" is not supported by the data. For instance, there is absolutely no evidence that suggests an educational gradient (in either direction) in cardiovascular events in women from LMICs, even in the model adjusted only for age (hazard ratios in all education strata in table 3 were around 1).

Similarly, the statement that "these data suggest that promoting higher AEL among women in LMICs may be insufficient to protect them from cardiovascular disease, and interventions should be developed to ensure that women of all groups have adequate access to preventive measures and health care" is also not supported by the data. I was very surprised by this recommendation since it has been shown that educated individuals make better use of health-related information than those who are less educated, even in LMICs 2

There are at least two reasons that call for better interpretation of the findings by Goyal et colleagues. First, the LMICs entity includes countries that are at different stage of urbanisation and industrialisation. Furthermore, recent evidence suggests that the health gradient between the well educated and less educated changes as a function of the economic developpment and the increase in the overall level of income of a country 3 . Thus, it is not impossible that grouping all these countries together might have obscured the real association between AEL in LMICs and cardiovascular events in this study.

Second, recent evidence suggests that cardiovascular risk factors and events in LMICs are not more common in affluent and educated individuals. The INTERHEART study [START_REF] Rosengren | Education and risk for acute myocardial infarction in 52 high, middle and low-income countries: INTERHEART case-control study[END_REF] , one of the references used by the authors shows that low education is consistently associated with increased risk for acute myocardial infarction globally. What is more, a recent review [START_REF] Monteiro | Socioeconomic status and obesity in adult populations of developing countries: a review[END_REF] on the association between socioeconomic status (SES), of which education is a marker, and obesity in adult populations in developing countries has observed that the burden of obesity in developing countries tends to shift towards the groups with lower SES as the country's economic development increases.

To conclude, although cautious interpretation of the findings is preferable, this work remains an important contribution to an under-researched topic.

Disclosures: none


Luc Véchot
email: luc.vechot@qatar.tamu.edu

Wilfried Saturnin Minko
email: wilfried.minko@irsn.fr

Jean-Pierre Bigot

Marc Kazmierczak

Vent sizing: Analysis of the blowdown of a hybrid non tempered system

Keywords: Safety vent sizing, Run-away, Blowdown, Non tempered system, Cumene hydroperoxide

  L'archive ouverte pluridisciplinaire

LUC VÉCHOT a (1) , WILFRIED MINKO b (1) , JEAN-PIERRE BIGOT (1) , MARC KAZMIERCZAK (2) , PATRICIA VICOT (2) I. Introduction DIERS c research work introduced a classification for runaway reactions according to which chemical systems are either tempered or non tempered (with vapour and gassy systems as limiting cases). According to Huff or Leung "the so-called tempering condition is achieved if the evaporative heat removal becomes equal to the reaction heat release" [START_REF] Huff | Emergency venting requirements[END_REF] and [START_REF] Leung | Venting of Runaway Reactions with Gas Generation[END_REF]. As a consequence, "the ERS d device is able to control T (dT/dt = 0) and P (dP/dt = 0) at the set pressure". On the other hand, "the ERS device is not able to control T (dT/dt >> 0) and P (dP/dt >> 0) at the set pressure" for hybrid non tempered reactions [START_REF] Fisher | Emergency Relief System Design Using DIERS Technology -The DIERS Project Manual[END_REF]. A "tempering effect" is however sometimes mentioned when vaporization phenomena reduce the reacting mixture temperature and the severity of runaway reaction, without controlling the temperature [START_REF] Leung | Simplified Vent Sizing Methods Incorporating Two-phase Flow[END_REF]. DIERS vent sizing methodology was largely developed and used for tempered systems [START_REF] Fisher | Emergency Relief System Design Using DIERS Technology -The DIERS Project Manual[END_REF]. A method was also proposed for purely gassy systems [START_REF] Leung | Runaway System Characterisation and Vent Sizing Based on DIERS Methodology[END_REF], but very few investigations were published for hybrid non tempered systems. No vent sizing criterion that would take vapour influence into account is available although it was demonstrated that it greatly modifies the necessary vent size [START_REF] Véchot | Sécurité des procédés ; emballement de réaction ; dimensionnement des évents de sécurité pour systèmes « gassy » ou hybrides non tempérés[END_REF], [START_REF] Véchot | Non-tempered systems[END_REF], [START_REF] Minko | Looking for a purely gassy system[END_REF], [START_REF] Minko | Vent sizing for un-tempered systems: Scale-up issues, DIERS vs similarity methods comparison, Possible way of study[END_REF] and [START_REF] Minko | Emballements thermiques de réactions : Etude des méthodes de dimensionnement des évents applicables aux systèmes hybrides non tempérés[END_REF].

Few experiments involving the runaway and blowdown of non tempered systems were published [START_REF] Hare | Evaluation of the UN vent sizing method for organic peroxide portable tanks[END_REF], [START_REF] Etchells | Relief System Sizing for Exothermic Runaway: The UK HSE Strategy[END_REF], [START_REF]the Transport of Dangerous Goods and on the Globally Harmonized System of Classification and Labelling of Chemicals[END_REF], [START_REF] Poteet | An Example of a Test Method for Vent Sizing -OPPSD/SPI Methodology[END_REF] and [START_REF] De Groot | Safety Aspects of Organic Peroxides in Bulk Tanks[END_REF]. These tests were performed with the objective of developing the UN 10 l method based on the similarity principle.

The present work was aimed to gain a better understanding of the blowdown of non tempered hybrid systems, and more precisely to observe the sensitivity of the temperature, pressure and vented mass to the vent area to vessel volume ratio (A/V). The experimental study uses a recently developed 0.1 l scale tool [START_REF] Véchot | Sécurité des procédés ; emballement de réaction ; dimensionnement des évents de sécurité pour systèmes « gassy » ou hybrides non tempérés[END_REF] and [START_REF] Véchot | Runaway reaction of nontempered chemical systems: Development of a similarity vent-sizing tool at laboratory scale[END_REF]. Adiabatic calorimetry tests with closed and open cells according to the DIERS methodology were also carried out prior to the blowdown tests. They allowed the characterization of the thermodynamic and the kinetics of the runaway reaction of concern. ([6] and [START_REF] Véchot | Runaway reaction of nontempered chemical systems: Development of a similarity vent-sizing tool at laboratory scale[END_REF], Figure 1) was used to run blowdown experiments at laboratory scale. This tool is an extension of the VPS2 adiabatic calorimeter in its blowdown configuration which consists in the addition of a main safety relief line, a feed bleed line and a real time vented mass measurement system. The main safety relief line (1/8" diameter, 0.6 m long) includes an actuated ball valve, which simulates the opening of the safety relief system, followed by a metering valve where most of the pressure drop occurs. An equivalent A/V ratio can be defined for different settings of the metering valve. Stainless steel test cells were used for all the tests. This experimental setup allows the measurement of the liquid temperature, the cell pressure and the vented mass during the blowdown. The small scale of the experimental setup allows the study of the sensitivity of the blowdown to the A/V ratio. The measurement of the vented mass is subsequently used to assess the vented mass at turnaround (second pressure peak).

III. Chemical system and experimental conditions

Cumene hydroperoxide (CHP; Tbp = 116 °C at 20 mbar abs pressure) and 2,2,4-trimethyl-1,3pentanediol diisobutyrate (Tbp = 280 °C at atmospheric pressure) were chosen respectively as the substance subject to thermal decomposition and the solvent. CHP is widely used as an initiator for polymerization reactions. It decomposes to mainly produce methane (0.3-0.5 mol/mol), dimethylbenzyl alcohol (0.6-0.9 mol/mol), acetophenone (0.2-0.4 mol/mol), and phenol and acetone when cumene is present (≈0.06 mol/mol each) [START_REF] Kharasch | The chemistry of hydroperoxides. VI. The thermal decomposirion of a-cumyl hydroperoxide[END_REF]. The decomposition of 30% (w/w) CHP under external heat input conditions of dT/dt = 0.5 °C min -1 (0.0083 °C s -1 , fire simulation) was studied. CHP was obtained as a 80% (w/w) solution in cumene. The composition of the investigated chemical mixture is therefore 30% (w/w) CHP, 7.5% cumene and 62.5% solvent. It has to be noted that cumene (T bp = 153 °C) is the most volatile chemical in this mixture. Assuming no decomposition, this mixture would lead to a vapour pressure of around 0.8 bar at 250 °C. For both open and closed cell experiments, the chemical mixture is first quickly heated up (3 °C min -1 ) to 85 °C (at this temperature the decomposition reaction is very slow and can not be detected by the VSP2 in adiabatic mode). External fire exposure is then simulated through constant power input (0.5 °C min -1 ) until the end of the decomposition.

IV. Results of the experimental investigation

I IV V. .
IV.1.2. Results of the closed cell experiments [START_REF] Véchot | Sécurité des procédés ; emballement de réaction ; dimensionnement des évents de sécurité pour systèmes « gassy » ou hybrides non tempérés[END_REF] and [START_REF] Véchot | Runaway reaction of nontempered chemical systems: Development of a similarity vent-sizing tool at laboratory scale[END_REF] We had to lower CHP concentration to 15% (w/w) for the closed cell DIERS calorimetric tests in order to avoid the bursting of the cell. The closed cell experiment however provided some qualitative information about vapour and non-condensable gas production during the runaway reaction. Table 1 and Figure 2a give the test conditions and main results. As expected, the pressure at the end of the closed cell experiment, after cooling P end = 9.4 bar after correction for the pad gas clearly indicated that the decomposition of CHP leads to the formation of non-condensable gases. Two rough assumptions were made in order to estimate possible presence of vapour during the post decomposition period (cooling down to ambient temperature):

 The vapour contents are completely condensed at T end .

 The non-condensable gases are not soluble in the liquid phase, independently of the temperature.

After correcting for the pad gas, it is possible to assess the respective contributions of the non condensable gases and the vapour to the total pressure during the cooling period. Figure 2b shows that a linear Antoine type plot can be obtained for the assessed vapour pressure between 83 °C and T max (242 °C). Vapour would thus contribute to P max as high as 10.4 bar, which would be significant compared to 16 bar for non condensable gases.

The vapour pressure would be around 4 bar at 150 °C after decomposition whereas the pressure (non condensable + vapour) is almost zero at the same temperature before decomposition starts (Figure 2a). This means that the vapour pressure is mainly due to the decomposition products. When dealing with an unknown reaction, it is therefore necessary to investigate the presence of vapour not only before decomposition occurs, but also after its completion.

The presence of vapour is consistent with mechanisms proposed by both Levin et al. [START_REF] Levin | Kinetics of acid-catalyzed cleavage of cumene hydroperoxide[END_REF] and Kharasch et al. [START_REF] Kharasch | The chemistry of hydroperoxides. VI. The thermal decomposirion of a-cumyl hydroperoxide[END_REF]. However the generation of non condensable gases clearly occurs during our experiments (9.4 bar observed after cooling, Figure 2a) [START_REF] Leung | Chemical Process Relief Design Seminar[END_REF]. The decomposition we observed seems thus more consistent with the mechanism proposed by Kharasch et al.

IV.1.3. Results of the open cell experiments

Open cell tests were performed to observe the influence of a pad pressure (P 0 ) on reaction temperature and thus on the reaction kinetics. four tests were run using four different initial pad pressures and two initial filling ratios (Table 2; [Figure 3] and [Figure 4]) with a solution of 30% CHP. Table 2 shows that T max decreases when decreasing P 0 , which is a result of some energy being consumed by more vaporization. Low pad pressures are indeed more favourable for the existence of vaporisation phenomena e . The vaporization would consume an amount of energy such that the liquid temperature cannot exceed 250 °C during a blowdown experiment if the pressure is always below 2 bar ([Figure 3] and [ Figure 4]). The DIERS approach assumes thus that vents can then be sized using the maximum reaction rate under maximum allowed pad pressure at that temperature [START_REF] Creed | An Easy, Inexpensive Approach to the DIERS Procedure[END_REF]. Two facts have to be noted about these experiments:

 The volume of the mixture remaining in the cell at the end of tests at P 0 = 1 bar and P 0 = 4.5 bar is so low that the cell thermocouple is no longer inside the liquid, which could lead to an erroneous temperature measurement and a larger φ adiabaticity factor.

 Two-phase choking flow occurs within test cell pipe during test with P 0 = 4.5 bar when reaction kinetics reach their maximum (Figure 3b).

Open tests under a large initial pad pressure (P 0 = 16 bar and P 0 = 26 bar; Figure a andb) do not show any measurable temperature or time lag between (dP/dt) max and (dT/dt) max . This could be due to vaporization being weak at those pressures. The test at P 0 = 4.5 bar exhibits a 10 °C temperature lag and a 0.26 s time lag between (dT/dt) max and (dP/dt) max but it could be due to choked two-phase flow occurring inside the cell tube or thermocouple time constant. Temperature lag for the test at P 0 = 1 bar is not easy to read (may be 5 °C). Therefore, it seems that using the temperature or the time lag between (dP/dt) max and (dT/dt) max as a criterion to detect the presence of vapour [START_REF] Leung | Chemical Process Relief Design Seminar[END_REF]

Description of a blowdown test

Similarly to the DIERS calorimetry tests described above, the blowdown experiments involved the rapid heating of the chemical mixture (3 °C min -1 ) to 85 °C. External fire exposure is then simulated through constant power input (0.5 °C min -1 ) until the end of the experiment. The chosen set pressure (vent opening pressure) for the blowdown experiments is 4.5 bar. The feed-bleed line is permanently open during experiment, except at the very beginning where it could lead to water from the mass measurement system flowing back into the cell.

The size of the feed-bleed line (A/V < 10 -4 m -1 ) which corresponds to a 1 mm nozzle for a 10 l reactor [START_REF]the Transport of Dangerous Goods and on the Globally Harmonized System of Classification and Labelling of Chemicals[END_REF] and the safety vent line (10 -3 m -1 < A/V < 2.4 × 10 -3 m -1 ) are first established. Figure 5a shows the typical temperature and pressure curves obtained. A test lasts approximately 3 h but the blowdown itself lasts approximately 1 min only (Figure 5b). Liquid is still vented for one more minute.

IV.2.2. Results of the blowdown experiments

Experiments were run for different values of the safety vent A/V ratio and the initial filling ratio (Table 3). Note that test cell swelled during tests A and K inducing uncertainty about A/V ratio. For a given initial filling ratio, all the experiments were intended to be as similar as possible until vent opening. Figure 6 shows that the temperature of the liquid at vent opening (T s ) varies between 209 °C and 224 °C depending on the unavoidable slight changes in feed-bleed line A/V ratio. Accordingly, the higher the temperature at vent opening (T s ), the shorter the time between vent opening and turnaround (Δt, Figure 7). All tests showed a quick pressure drop at vent opening without any temperature decrease until a second pressure peak occurred (when reaction rate is at its maximum, Figure 5b). This means that temperature is not (directly) controlled by pressure, which is typical of a non tempered system [START_REF] Fisher | Emergency Relief System Design Using DIERS Technology -The DIERS Project Manual[END_REF]. However, Figure 8 shows that temperature rise rate (dT/dt) presents a discontinuity at safety vent opening (around 210 °C). The larger the A/V ratio (and thus the larger the pressure drop), the larger the discontinuity. This demonstrates that some vapour is already present at this stage. Increasing A/V ratio (thus lowering average pressure during blowdown) results in decreasing (dT/dt) max and T max . This is very similar to the effect of decreasing nitrogen pad pressure (P 0 ) during a DIERS open cell test (Figure 4). Figure 9 shows the comparison between two tests performed with the same initial filling ratio but with different values of A/V (tests B and C). It can be seen that the vented mass for these two tests is low and almost the same up to 30 s after vent opening (close to turnaround for test B). The observed differences in temperature rise rate (dT/dt) after vent opening cannot be explained by a difference of φ factor resulting from the decrease of reactant mass in the test cell. This effect has thus to be associated with some tempering due to volatile decomposition products. Figure 10 shows that both P max and (dP/dt) max decrease when A/V increases with no detectable influence of initial filling ratio. Figure 11 shows that an Antoine type correlation exists between pressure and temperature at turnaround (P max and T max ), independently of the filling ratio (between 46% and 80%). This pressure is much lower than the vapour pressure observed after the closed cell tests with 15% (w/w) CHP concentration (Figure 2). This could mean that this pressure does not result from a thermodynamic equilibrium. The independence of that pressure to initial filling ratio could mean that this correlation is (at least partially) due to the vaporization kinetics. It is a further evidence of the influence of vaporization on the blowdown. Figure 12a andb show the results obtained from blowdown experiments performed with the same A/V for both the feed-bleed line and the safety vent line, while only varying the initial filling ratio τ 0 . It can be observed that an increase of the initial filling ratio leads to:  A small (<12 °C) decrease of the temperature at vent opening (T s ). Less volume being available for the gas with a higher initial filling ratio, the pressure build-up is quicker and therefore the vent opening pressure is reached at an earlier stage of the reaction;

 An increase of the time between the two pressure peaks as a consequence of the decrease of the temperature at vent opening.

These effects are more obvious for experiments C and G (Figure 12a) than for experiments H and J (Figure 12b). In both cases, an increase of P max , T max and mass vented at turnaround (Δm/m 0 at turnaround) of less than 20% was observed. These tests showed that the sensitivity of other parameters (including (dT/dt) max and (dP/dt) max ) to the initial filling ratio is low (within the measurement error, see Table 3).

V. Discussion

V V. .1 1. . T Te em mp pe er ri in ng g o of f a a n no on n t te em mp pe er re ed d s sy ys st te em ms s There is enough vaporization during both open cell and blowdown experiments to change Tmax (Figure 4 and [Figure 8). The vaporization phenomena are even enough for Pmax to be correlated with Tmax during blowdown experiments. This means that the 30% CHP system is subject to some tempering [START_REF] Leung | Simplified Vent Sizing Methods Incorporating Two-phase Flow[END_REF]. On the other hand vaporization during blowdown experiments does not prevent the reactive mixture temperature increase when the pressure drops at vent opening, resulting in a typical two pressure peak behaviour (Figure 5a andb). CHP is therefore a hybrid non tempered system f [1], [START_REF] Leung | Venting of Runaway Reactions with Gas Generation[END_REF] and [START_REF] Fisher | Emergency Relief System Design Using DIERS Technology -The DIERS Project Manual[END_REF]. V V. .2 2. . M Ma as ss s l lo os ss s h hi is st to or ri ie es s The real time measurement of the vented mass downstream the safety vent allows the investigation of the phenomena occurring during the blowdown. Three types of behaviour have been observed (Figure 13a-c). The attention is first focused on the case of a blowdown experiment with a small A/V ratio (type I, A/V < 0.96 × 10 -3 m -1 ; Figure 13a). No vented mass is recorded during the blowdown until the pressure decrease of the second pressure peak (i.e. after turnaround). This means that flow is only gas during this period. This results from the pressure between the two peaks being never less than 2.1 bar and pressure at turnaround being large (P max = 25.8 bar). This pressure is believed to be sufficient to limit level swell or boiling effects, even when the temperature and reaction kinetics (and thus volatile species production) reach their maximum. Mass loss (around 20 × 10 3 kg m -2 s -1 ) starting 20 s after the safety vent opening (see small pressure accident) is consistent with two-phase flow. The decomposition reaction is finished at that time and no more non condensable gas is produced. Two-phase flow can thus only result from boiling and release of dissolved gases induced by pressure decrease at large temperature. This process results in the venting of up to 75% of initial mass. For medium sized safety vents (type II, 0.96 × 10 -3 < A/V < 1.56 × 10 -3 , Figure 13b) the pressure drops from 4.5 bar down to less than 2 bar in a few seconds after safety vent opening. This lower pressure induces level swell and two-phase flow, even at that early stage of the blowdown where both temperature and reaction kinetics are quite far from their maximum (just before turnaround). 10% of the initial cell contents are already vented between the two pressure peaks. Mass flow through the vent is therefore probably due to level swell induced by gas production. The exponential temperature increase just prior the turnaround leads to a significant increase of the gas generation rate and thus to a pressure increase. This latter prevents boiling. It compresses both gas and vapour, which tends to promote gas disengagement despite an increase in gas and vapour production rate. The above is an attempt to explain the observed mass flow rate decrease at the beginning of the second pressure peak. Two-phase flow due to boiling after the turnaround (around 40% of initial mass) can again be observed. For larger safety vents (type III, A/V > 1.56 × 10 -3 m -1 ; Figure 13c), two-phase venting occurs during the whole blowdown because of the pressure being lower, which promotes level swell. f We moreover observed no change in temperature rise rate during open cell tests at low pressure. This could be consistent with vapor species being products of the reaction directly produced as vapor phase when initial pressure is low. The same species being produced as liquid when pressure is higher.

These three types of behaviour show that the vapour contents have more effect on vent flow during and after the second pressure peak than during runaway reaction itself. Indeed, this is when the temperature is the highest and decomposition reaction has already generated volatile species. Figure 14 shows that mass vented at turnaround increases when A/V ratio increases. This is probably due to the fact that lower pressure induces more level swell. However, Figure 15 shows that the total vented mass (including mass vented after the second pressure peak) remains almost unchanged (between 45% and 65%) for all A/V ratios. This could be related to the total vapour production ratio remaining unchanged.

VI. Conclusions

Our investigation is focused on the behaviour of a hybrid non tempered runaway reaction under blowdown conditions. The DIERS approach (closed and open cell tests) shows that CHP decomposition produces both vapour and non condensable gases. The analysis of the post-decomposition data shows that vapour species are mainly products of the reaction. They are thus most easily detected after decomposition ends. The decomposition seems consistent with the mechanism proposed by Kharasch et al. [START_REF] Kharasch | The chemistry of hydroperoxides. VI. The thermal decomposirion of a-cumyl hydroperoxide[END_REF]. Vaporisation has a significant effect on T max and (dP/dt) max , especially at low initial pad gas pressure. 30% CHP in butyrate solvent is thus a hybrid system. The detection of the hybrid characteristic of the system using temperature or time lag between (dP/dt) max and (dT/dt) max is much less obvious. All the blowdown experiments present a profile with two pressure peaks typical of non tempered gas producing systems, independently of the A/V ratio. The time between the two pressure peaks (Δt) mainly depends on the temperature at vent opening. These experiments are quite reproducible (209 °C < T s < 224 °C). The pressure drop at vent opening leads to a decrease of temperature rise rate. This effect can be related to the vaporisation of decomposition products favoured by the low level of pressure following the vent opening. The maximum temperature and maximum temperature rise rate are also sensitive to the vent size. The kinetics of the studied decomposition is thus sensitive to the vaporisation of decomposition products. This vaporisation is sufficient to result in an Antoine type correlation between P max and T max . This correlation does however not correspond to equilibrium. Increasing the initial filling ratio mainly results in an earlier vent opening (quicker pressure build-up to the vent opening pressure because of a smaller gas space). The temperature at vent opening (T s ) is consequently slightly lower, which results in a longer delay between the two peaks. The sensitivity to initial filling ratio is however visible neither on maximum pressure nor on maximum pressure rise rate, nor on vented mass ratio (at turnaround or total). The relative vented mass (Δm/m 0 ) at turnaround increases with the A/V ratio (from 0 to 45%), probably because pressure being lower. Two-phase flow through the vent is always observed after the second pressure peak despite the reaction is complete. The relative amount of final vented mass is relatively independent (between 45 and 62%) of both the initial filling ratio and the A/V ratio. It could be due to the boiling of the hot reaction products at low pressure. Three types of blowdown behaviours have been observed. For these three behaviours, the pressure profiles show a typical two pressure peak shape and significant mass venting is observed after turnaround, when the reaction is complete. The observed post-turnaround mass flow is probably due to the boiling (at high temperature) of the vessel contents when the pressure decreases. It could alternatively be due to foaming character or to gas desorption. The vented flow at turnaround could be either gas (low A/V ratio, type I) or two-phase (large A/V ratio, type III). It seems gas venting could occur at turnaround even when it is two-phase just before and just after that moment (medium A/V ratio, type II). There is thus a trend towards gas disengagement and gas one-phase flow (revealed by less or no vent mass flow rate) when pressure increases towards the 2nd pressure peak.

Many complements to this study could be of an interest in order to approach a better understanding of the blowdown of non tempered systems:  addition of a real time measurement of vented volume (non condensable gases);  study of other chemical systems in order to identify the specificities of CHP in the present study. Experiments with a chemical system with less vapour production would be of particular interest;

 experiments with the same systems at a larger scale for scale-up validation;

 discussion of vent sizing calculation from these experiments.

Some of these studies will be presented in a next article.

Figures 9 FF

 9 Figures

  Closed and open cell tests according to the DIERS approach were carried out using a Vent Sizing Package 2 (VSP2) adiabatic calorimeter (Fauske & Associates, LLC). Closed cell experiments consist in performing the runaway reaction under adiabatic conditions in a 110 ml closed cell. The gases/vapour generated by the runaway reaction pressurise the test cell. Open cell experiments consist in performing the runaway reaction in a 110 ml test cell open to a 4 l containment vessel. In this case, the generated gases/vapour pressurise a larger volume, thus avoiding the bursting of the test cell. For both configurations the liquid temperature and the generated pressure (either in the test cell or in the containment vessel) are recorded. All the open and closed tests were performed with stainless steel test cells.

		Nomenclature		Index
	A/V	Vent area to reactor (or test cell) volume ratio (m -1 )	ini or 0	at the beginning of test, before reaction begins
	CHP	Cumene Hydro peroxide	max	when pressure is maximum (i.e. at turnaround)
	Tbp	Boiling temperature at ambient pressure (°C)
	τ	Filling ratio (%)
	t	Time between safety vent opening and turnaround (s)
	II. Experimental set up
	I II I. .2 2. . S Se et tu up p f fo or r b bl lo ow wd do ow wn n e ex xp pe er ri im me en nt ts s a at t l la ab bo or ra at to or ry y s sc ca al le e
	A recently developed 0.1 l scale tool

m Mass of reactor (or test cell) contents (kg) end at the end of blowdown, after cooling down P Absolute pressure in reactor (or test cell) (bar) s at vent opening (i.e. at set pressure) T Temperature in reactor (or test cell) (°C) I II I. .1 1. . D DI IE ER RS S a ad di ia ab ba at ti ic c c ca al lo or ri im me et tr ry y

  is much less obvious than comparing T max for different P 0 .

	I IV V. .2 2. . B Bl lo ow wd do ow wn n e ex xp pe er ri im me en nt ts s ( (0 0. .1 1 l l s sc ca al le e) ) w wi it th h P P, , T T a an nd d m m( (t t) ) m me ea as su ur re em me en nt ts s
	IV.2.

  Bl lo ow wd do ow wn n t te es st ts s cco on nd di it ti io on ns s a an nd d m ma ai in n r re es su ul lt ts s. .

	Tables T Ta ab bl le e 3 3: : B
	T Ta ab bl le e 1 1: : C Cl lo os se ed d c ce el ll l t te es st t c co on nd di it ti io on ns s a an nd d m ma ai in n r re es su ul lt ts s. . 0 V Vent A/V Pmax Tests CHP Concentration (% w/w) (%) m 3 m -1 bar	Tmax °C	Ts °C	TPmax 15 °C	t s	(dT/dt) °C/s	bar/s	(%)	Pmax (%)
	A	52.8 110x10 -6		Filling ratio (%) 1.03 x 10 -3 21.20	309.0	52.4 221.6 305.5 21.0		36.8	11.90	48.6	2.8
				Initial mass (g)			52.6
	B	46.4 125x10 -6		Pend after cooling (bar) 1.10 x 10 -3 14.00	306.2	10.4 223.9 303.8 27.4		12.0	2.95	48.9	11.5
	C	46.4 125x10 -6		(dT/dt)max (°C/s) 2.01 x 10 -3 3.83	268.7	5.36 221.7 267.1 59.4		2.1	0.21	48.9	28.8
	D	65.0 125x10 -6		(dP/dt)max (bar/s) 9.05 x 10 -4 17.60	309.0	1.95 217.0 304.2 44.5		12.3	3.51	51.3	14.4
				Tmax °C			250
	T Ta ab bl le e 2 2: : O Op pe en n c ce el ll l t te es st ts s --e ex xp pe er ri im me en nt ta al l c co on nd di it ti io on ns s a an nd d m ma ai in n r re es su ul lt ts s E1 65.0 125x10 -6 1.36 x 10 -3 11.30 296.5 212.8 293.2 69.0		7.1	1.59	49.2	22.6
	E2	Solution 65.0 125x10 -6		1.36 x 10 -3	30% CHP in 2,2,4-trimethyl-1,3-pentanediol diisobutyrate 10.22 294.3 209.0 289.8 95.0 6.4 1.40	49.3	20.5
		P0 (N2) (bar)			1		4.5			16		26
	F	65.0 125x10 -6		1.85 x 10 -3	5.24	280.3	212.4 277.0 88.8		2.8	0.33	45.1	24.6
		Filling ratio (%)			52.4		70		52.4		70
	G	Initial reacting mass (g) 65.0 125x10 -6 2.01 x 10 -3	52.6 4.33	274.5	75 209.6 271.3 97.8 52.6	2.1	75 0.18	51.3	33.4
	H	After test reacting mass (g) 65.0 135x10 -6 2.43 x 10 -3	12.3 5.10	278.1	16.6 217.8 275.7 60.6 36.6	3.9	48 0.47	53.2	36.1
		Mass loss (%)			75.8		77.9		30.4		36.0
	I	(dT/dt)max (°C/s) 67.7 135x10 -6		2.64 x 10 -3	2.3 3.89	273.0	41.8 214.8 271.2 63.6	39	2.2	42 0.18	62.0	45.6
	J	(dP/dt)max (bar/s) 800 135x10 -6	2.43 x 10 -3	0.063 6.08	dP1/dt = 7.9 dP2/dt = 0.83 287.5 213.6 283.5 69.0 2,3	4.0	3.25 0.49	61.7	41.7
	K	Tmax (°C) 52.7 110x10 -6	< 0.96 x 10 -3	260 25.80	315.9	297 219.9 310.9 15.9 317	52.8	328 17.90	77.6	0.00
		Pmax (P1) (bar)			1.95		12.7		20.5		32.5
		Pend (bar)			1.33		5		N/A		N/A

max (m/m0) total (m/m0)

e The decrease of T max at lower pad pressure (P 0 ) could alternatively be attributed to an increase of the ϕ factor following the decrease of reactant mass in the test cell (up to 40.3/52.6 for P 0 = P atm , Table2). The following discussion of blowdown experiments will show that this is probably not the main effect.


D Elvira

X Hachair

V B Verma

R Braive

G Beaudoin

I Robert-Philip

I Sagnes

B Baek

S W Nam

E A Dauler

I Abram

M J Stevens

A Beveratos
email: alexios.beveratos@lpn.cnrs.fr

Higher-order photon correlations in pulsed photonic crystal nanolasers

Keywords: numbers: 42.55.Tv, 42.50.Lc, 42.50.Ar, 42.55.Ah

We report on the higher-order photon correlations of a high-β nanolaser under pulsed excitation at room temperature. Using a multiplexed four-element superconducting single photon detector we measured g (n) ( 0) with n=2,3,4. All orders of correlation display partially chaotic statistics, even at four times the threshold excitation power. We show that this departure from coherence and Poisson statistics is due to the quantum fluctuations associated with the small number of dipoles and photons involved in the lasing process.

The physics of nanoscale lasers is expected to be radically different from that of conventional lasers, due to their very high spontaneous emission ratio (β) into the lasing mode, approaching β → 1. Because of the high β, the lasing threshold is low, the system is capable of lasing with a small number of emitting dipoles and a small number of photons in the cavity mode, and the lasing turn-on time is expected to be very short. However, the quantum fluctuations associated with the small numbers of dipoles and photons cause the output intensity of the laser to fluctuate strongly with chaotic statistics which can be present well above the stimulated emission threshold [1,2] and may even suppress continuous-wave lasing [3]. Understanding these quantum fluctuations, which requires studies of the higher-order photon statistics of the laser output, is thus crucial for the operation of nanolasers, particularly if nanoscale lasers are to be used in demanding applications, such as inter-chip optical interconnects or integrated clocks.

Several studies aimed at understanding the physics behind the dymics and the noise characteristics of nanolasers have been reported in recent years. Measurements of the second order correlation function g (2) (τ ), which is directly linked to the intensity noise properties, were reported some years ago [4,5] and more recently the evolution of g (2) (t, τ ) was followed with picosecond resolution in semiconductor micropillar nanolasers operated at 4K in the weak [6,7] or the strong [8] coupling regimes. Lately, the noise properties of nanolasers operating at room temperature were also reported [9]. In most of these studies, the cavity volume was of the order of a cubic wavelength and the gain material consisted of a small number of semiconductor quantum dots. In some recent work on plasmonic structures, lasers with even smaller volumes have been reported [10][11][12], reducing further the number of emitting dipoles and photons in the cavity. Clearly, in such small lasers, the traditional laser theories based on the "thermodynamic limit" of large-ensemble averaging and continuum approxima-tions for the gain medium and the electromagnetic field are no longer valid, as the discrete nature of the number of dipoles and photons must be taken into account explicitly [13].

In this Letter we report an experimental study of the higher-order photon autocorrelation functions of a singlemode photonic crystal nanocavity laser, with quantum dots as its gain material, operating at room temperature, in the telecommunications wavelength range, and under pulsed excitation. In the second-, third-and fourthorders of correlation, the output intensity fluctuations display a chaotic component and deviate from standard Poisson statistics, even at pump powers of over 4 times the threshold. These noise properties can be accounted for by the quantum fluctuations arising from the small number of photons and dipoles, through a simple model that provides new insight into the physics of nanolasers. The nanolaser cavity is formed by a photonic crystal double heterostructure [14] etched on a 320 nm-thick suspended InP membrane [9] and incorporating a single layer of self-assembled InAsP quantum dots [15] at its vertical center plane. The whole structure is grown by metalo-organic chemical vapor deposition. The quantum dot density is of the order of 1.5×10 9 cm -2 and their spontaneous emission is centered around 1560 nm at 300 K with an inhomogeneous linewidth of about 150 nm. The cavity is fabricated using electron beam lithography, inductively coupled plasma etching and wet etching [16]. The structure consists of a W1 photonic crystal waveguide composed of one missing row of holes in the Γ-K direction of a hexagonal lattice structure with lattice constant a m = 410 nm, and air-hole radius of r = 0.293 a m . The lattice constant is modified over two periods at the center of the photonic crystal waveguide along the Γ-K direction alone, to the value of a l = 440 nm, thus forming a nanocavity with an effective volume of 1.3 (λ/n) 3 , enclosed by two surrounding "mirror" waveguides with smaller lattice constant. The cold cavity quality factor is measured to be Q = 45000 [17], corresponding to a cavity lifetime of 1/Γ c = 35 ps.

The nanocavity laser is studied at room temperature. Optical excitation is provided by a pulsed Ti:Sapphire laser with a center wavelength of 805 nm and a repetition rate of 82 MHz. The pump pulse is broadened to ∼ 50 ps after passage through a 25 m length of singlemode optical fiber and is then focused to a 5µm spot on the sample with a microscope objective (NA = 0.4). When the quantum dots are highly excited and contain several electron-hole pairs, they emit into a broad spectrum with a lifetime of 1/γ = 400 ps [9,18] which feeds the cavity mode [19]. Emission from the nanocavity laser is collected with the same objective, directed through a dichroic beamsplitter that blocks the pump light and sent to a 0.75 m grating monochromator. One output port of the monochromator holds an image intensified near-infrared camera, which records the emitted spectrum and power. Alternatively, an internal mirror can be flipped, directing light to the other output port, where it is filtered with a slit and subsequently collected into a single-mode optical fiber and sent to the detector. The detector is a four-element superconducting nanowire single-photon detector (SNSPD) [20] in which four independent, single-photon-sensitive elements are interleaved over a single spatial mode of the optical beam [21]. Each element consists of a currentbiased superconducting nanowire that is driven into a resistive state, thus delivering an output voltage pulse upon absorption of one photon. The active area of the four-element SNSPD is matched to the mode diameter of the fiber, allowing all four interleaved nanowire elements to equally sample a single spatial mode [21,22]. Fast four-channel electronics record photon arrival times on each element. These time-tag data are post-processed to obtain multi-start, multi-stop correlation histograms between two, three and four SNSPD elements for a continuous range of all associated time delays. The SNSPD is held at a temperature of ≈3 K in a closed-cycle helium cryocooler. Each element has a system detection efficiency of ≈ 1% at 1550 nm and a dark count rate of ∼ 100 Hz. Details on the data processing, including the elimination of crosstalk among the interleaved detectors, are given in the annex on supplementary information [23].

FIG. 2: Third order autocorrelation function g (3) (τ1, τ2) at a pump power P=2 P th . The tallest bar is located at τ1 = τ2 = 0 and represents g (3) (0, 0). The three ridges of elevated bars that fall along τ1 = 0, τ2 = 0 and τ1 = τ2 have values approximately equal to g (2) ( 0). The remainder of the bars have an average value of ∼1.

Figure 2 presents a typical set of normalized experimental three-photon coincidence data, extracted from the time-tagged arrivals of photons on the four multiplexed SSPD detectors. It corresponds to the third order autocorrelation function, g (3) (τ 1 , τ 2 ), in histogram form. The blocks at 0 = τ 1 = τ 2 = 0 are normalized to an average value of one as the three photons detected belong to different excitation cycles; the blocks at τ 1 = 0, at τ 2 = 0, or at τ 1 = τ 2 correspond to two photons being detected in the same excitation cycle (and the third one in another cycle) and thus give directly the value of g (2) ( 0); finally, the block at τ 1 = τ 2 = 0, where all three photons are detected in the same excitation cycle, gives the value of g (3) (0, 0) [22]. For fully coherent light g (n) ( 0) = 1, whereas for chaotic (thermal) light g (n) ( 0) = n!, for all orders n. For partially coherent light, we may thus define

h (n) = g (n) (0) -1 n! -1 (1)
which is a normalized ratio, indicating if the statistics are near coherent or chaotic light : Below threshold the output light is chaotic, so that h (n) = 1, whereas above threshold, one expects a coherent laser output and h (n) → 0 for all n.

On Figure 3 we present the experimental data and the theoretical curves (obtained below) for the Lightin Light-out curve (right hand axis) and h (n) for n = {2, 3, 4} (left hand axis) as a function of the pump power of the nanoscale laser under pulsed excitation. The lasing threshold is clearly identified at the "knee" of the Light-In Light-Out curve (which corresponds to the inflection point of the log-log curve), at P th =100 µW, while the β factor is given by the ratio of the two slopes of the Light-In Light-Out curve (corresponding to the "jump" of that curve in log-log scale) of β = 0.008. The h (n) , which below threshold are expected to be equal to 1, steadily decrease to reach h (2) = 0.05 ± 0.04, h (3) = 0.036 ± 0.02 and h (4) = 0.029 ± 0.015 at P = 4P th = 400µW, all significantly higher than the value of 0 expected for coherent emission. The corresponding g (n) ( 0) are g (2) (0) = 1.05±0.04, g (3) (0) = 1.18±0.10, g (4) (0) = 1.67±0.34. In other words, the intensity fluctuations still have a chaotic component at four times the threshold power, in sharp contrast with the behavior of conventional lasers (with β ≈ 10 -5 ), for which all h (n) rapidly vanish above threshold.

FIG. 3: (Right axis) Experimental (full dots) and calculated

Light-In Light-Out curve. (Left axis) Experimental values of h (n) (0) as function of pump power. Open circles, squares and triangles are for n=2,3,4 respectively. Full lines represent the expected h (n) (0) without any free parameters.

The persistence of the chaotic component can be simply understood following Refs. [1][2][3]: Because of the relatively small numbers of the emitting dipoles and photons in the cavity, a small fluctuation in these numbers introduces a non-negligible change in the stimulated emission rate, inducing chaotic (thermal) statistics even above threshold.

More quantitatively, we may model our system in terms of the traditional laser rate equations [2] under the constraint that the number of excited dipoles, N , and the number of photons in the cavity, s, be integers.

dN (t) dt = P f (t) -γ N (t) -βγ N (t)s(t) (2) ds(t) dt = -Γ c s(t) + βγ N (t) +βγ N (t)s(t) (3)
where f (t) is the temporal profile of the pump pulse intensity, of integral equal to one. For each incident power P , the rate equations are solved numerically by iterating over a time-step dt which is small compared with the characteristic times of the system 1/γ and 1/Γ c , and using a quantum jump approach [24] to ensure that N and s are integers: At each iteration step, the right hand side of Equations ( 2) and ( 3) is evaluated, taking into account the discrete evolution of N and s through a sequence of three binomial processes. First, the number of dipoles having decayed between t and t + dt is drawn from a binomial distribution with probability p 1 = γ (1 + βs(t))dt for each one of them to decay. Second, the decaying dipoles each produce a photon that has a probability p 2 = (1+s(t))β 1+βs(t) of entering the cavity. And third, the photons that have accumulated in the cavity can escape with probability p 3 = Γ c dt. Note that for consistency, we also take into account Poissonian statistics for the pump, although it does not alter the overall conclusions. In the end, for each incident power P , the g (n) ( 0) are calculated, according to

g (n) ( 0) = < t k=n-1 k=0 (s i (t) -k) > (< t s i (t) >) n (4)
where s i (t) is the i -th realization of the number of photons in the cavity, arising from the rate equations, and < x > denotes the average of x over all realizations. Averages over typically 1000 realizations are performed. As can be seen on Fig. 3, the experimental data are very well described by our model for the Light-In Light-Out curve as well the second and third order correlation functions. They are reasonably well described for the fourth order correlation function, considering the very large estimated experimental error associated with the measurements of g (4) (0). Details of the calculations can be found elsewere [25]. Here, it is sufficient to point out the main differences with other methods for calculating the g (n) ( 0). For traditional lasers with a large number of dipoles and photons in the cavity, quantum noise is introduced in the classical rate equations by means of a Langevin driving force with Gaussian statistics [1,2]. This approach, however, does not take into account the discrete nature of the particles and is thus not suited for lasers with a small number of dipoles and photons. In addition, the usual Langevin force statistics have only a second cumulant. This means that the higher-order averages can only be evaluated under specific assumptions, for example, if they correspond to sums of products of lower-order averages [26]. For small lasers, the discretization of the numbers of dipoles and photons is usually treated through a set of master equations for the probabilities of the states with a given number of excited dipoles N and photons s [3,13]. While this approach is ideally suited for very small systems, it requires calculating N 2 0 interdependent probabilities, while our method requires the computation of trajectories (realizations) with at most N 0 jumps. Secondly this approach allows for a more intuitive approach to understanding the underlying physical phenomena. To gain a better insight into the effects of fluctuations, we present in Figure 4 the evolution of the number of photons s(t) for four realizations of the rate equations (dashed lines) at an excitation power P = 1.8P th as well as the mean of s(t) over 1000 realizations (full line). We observe that even at pump power two times that of threshold, there are strong intensity and timing (jitter) fluctuations, which are correlated: The longer it takes for the pulse to be formed, the smaller its intensity, because in the meantime the number of excited dipoles has decreased. The inset of Figure 4 represents the statistical distribution of the integrated pulse area (in number of photons), fitted by a Poisson distribution of the same mean value. Clearly the number of photons in a pulse deviates from a Poisson distribution, and this will cause the autocorrelation functions g (n) ( 0) to deviate from the value of one characteristic of coherent emission. As can be seen in the inset, the deviation from a Poisson distribution consists of a lower probability for high-intensity events and a higher probability for low-intensity events. The correlation between the intensity fluctuations and the jitter implies that late starts are more probable than early starts, even though the gain medium can provide more photons at earlier times. The physical origin of this effect may be similar to the suppression of continuous-wave lasing due to fluctuations in small lasers, discussed by Roy-Choudhoury et al. [3]. Fluctuations may suppress a nascent laser pulse, leading to "false starts" and delaying the time when the successful laser pulse takes off. This asymmetry in the statistics of pulse intensity can be characterized by the higher (third and fourth) moments of the distribution and thus impacts the higherorder autocorrelation functions.

In conclusion we measured up the the fourth order autocorrelation function for a photonic crystal nanolaser operating at room temperature under pulsed excitation. We observe that all values of the autocorrelation function are significantly above unity even at 4 times threshold, indicating the presence of chaotic fluctuations in spite of the predominance of stimulated emission. A simple model taking into account the discrete nature of the number of photons and dipoles succesfully describes the experimental data, providing a novel understanding of the operation of the nanolasers. Measurement of the higherorder photon correlation functions thus opens the way to the study of very small lasers, whose operation deviates strongly from the "thermodynamic limit" that governs conventional lasers.

FIG. 1 :

 1 FIG. 1: Double heterostructure photonic crystal cavity. White dashed lines represent the mirror position. The red arrow shows the laser cavity

FIG. 4 :

 4 FIG. 4: Dashed lines: 4 different realizations of the stochastic rate equations at P = 1.8P th . Full line: The mean pulse averaged over 1000 realizations. It can be clearly seen that the longer it takes for the pulse to emerge, the smaller its amplitude. (Inset) Dots: Statistical distribution of the total number of photons per pulse at P = 1.8P th . Line: Expected distribution for Poisson statistics.

The authors acknowledge financial support from the Triangle de la Physique under the BIRD project and from the French National Research Agency (ANR) through the Nanoscience and Nanotechnology Program (project NATIF ANR-09-NANO-P103-36).