PhD Maëlenn Guerchet

MD Alain Maxime Mouanga

Pascal M'belesso

MD André Tabo

MD ; Pascal M'belesso

MD ; Bébène Bandzouzi

PhD Moussiliou N Paraïso

MD, PhD Dismand S Houinato

MD, PhD Pascale Cowppli-Bony

MD, PhD Philippe Nubukpo

MD, PhD Victor Aboyans

MD, PhD Jean-Pierre Clément

MD, PhD Jean-François Dartigues

MD, PhD Pierre-Marie Preux
email: preux@unilim.fr

Pierre-Marie Pr

Preux

Factors associated with dementia among elderly people living in two

Keywords: dementia, Alzheimer's Disease, risk factors, Africa, epidemiology

come

Introduction

A few risk factors have been established for dementia. Ageing is the main one, with the prevalence of dementia roughly doubling every five years over the age of 65 [START_REF] Prince | World Alzheimer Report. Alzheimer's Disease International[END_REF].

Genetic causes have been found for Alzheimer's Disease (AD). Dementia has also been shown to be influenced by the environment, life events, and socio-economic and modifiable risk factors such as cardiovascular risk factors [START_REF] Qiu | The epidemiology of the dementias: an update[END_REF]. However, these epidemiologic data were provided by studies conducted in high-income countries. In contrast, in low-income countries, the epidemiology and risk factors of dementia are poorly studied. Regarding sub-Saharan Africa, studies carried out in Nigeria reported the significant association of older age and female sex with dementia and AD [START_REF] Ochayi | Risk factors for dementia in central Nigeria[END_REF][START_REF] Hall | Risk factors and Alzheimer's disease: a comparative study of two communities[END_REF][START_REF] Ogunniyi | Epidemiology of dementia in Nigeria: results from the Indianapolis-Ibadan study[END_REF][START_REF] Ogunniyi | Risk factors for incident Alzheimer's disease in African Americans and Yoruba[END_REF]. A BMI<18.5 [START_REF] Ogunniyi | Risk factors for incident Alzheimer's disease in African Americans and Yoruba[END_REF], living with others [START_REF] Ogunniyi | Epidemiology of dementia in Nigeria: results from the Indianapolis-Ibadan study[END_REF] and lifetime history of alcohol use [START_REF] Gureje | The profile and impact of probable dementia in a sub-Saharan African community: Results from the Ibadan Study of Aging[END_REF] were also reported to be associated with AD, whereas a self-reported medical history of hypertension was found to be protective against incident AD [START_REF] Ogunniyi | Risk factors for incident Alzheimer's disease in African Americans and Yoruba[END_REF]. A lack of association between AD or dementia and the APOE ε4 allele [START_REF] Osuntokun | Lack of an association between apolipoprotein E epsilon 4 and Alzheimer's disease in elderly Nigerians[END_REF][START_REF] Gureje | APOE epsilon4 is not associated with Alzheimer's disease in elderly Nigerians[END_REF] was reported in elderly Nigerians, and the allele was not linked with cognitive impairment in elderly Beninese people [START_REF] Guerchet | Cognitive impairment and dementia in elderly people living in rural Benin, West Africa[END_REF]. In the same populations, education was not significantly associated with dementia in multivariate analysis [START_REF] Ochayi | Risk factors for dementia in central Nigeria[END_REF][START_REF] Hall | Risk factors and Alzheimer's disease: a comparative study of two communities[END_REF][START_REF] Ogunniyi | Epidemiology of dementia in Nigeria: results from the Indianapolis-Ibadan study[END_REF][START_REF] Ogunniyi | Risk factors for incident Alzheimer's disease in African Americans and Yoruba[END_REF][START_REF] Guerchet | Cognitive impairment and dementia in elderly people living in rural Benin, West Africa[END_REF]. However, few subjects had an education beyond primary school in these studies.

The present study aimed to investigate factors associated with dementia in a crosssectional sample of elderly people living in two cities in Central Africa: Bangui (Central African Republic) and Brazzaville (Republic of Congo). We hypothesized that, given the particularities of living conditions in these countries, the factors associated with dementia in sub-Saharan African people may differ from those reported in high-income countries or other low-income countries.

Methods

Study design

The EDAC (Epidemiology of Dementia in Central Africa) survey is a multicenter study carried out in Bangui and Brazzaville between September 2008 and March 2009.

Bangui's population is estimated to be 622 800, including 1.4% aged 65 years [START_REF]Recensement Général de la Population et de l'Habitat[END_REF].

Brazzaville's population is about 1 373 382, including 2.3% ≥ 65 years old [START_REF]Recensement Général de la Population et de l'Habitat[END_REF]. Life expectancy at birth is estimated to be 48 years (49 for men and 48 for women) in the Central African Republic (CAR) whereas it is 55 (53 for men and 57 for women) in the Republic of Congo [13]. The main industries of Congo are linked to oil and wood, other industries are poorly developed and of only local importance. Forestry and mining are the only developed industries in CAR; there are essentially no others. The Human Development Index is 179/182 for CAR and 136/182 for Congo [START_REF]Overcoming barriers: Human mobility and development[END_REF].

Cross-sectional surveys among the general population were conducted in the 3 rd district of Bangui and the 4 th district of Brazzaville, using a door-to-door approach.

The estimated populations of elderly people in the 3 rd and 4th districts of Bangui are 2603 (45% men) and 3911 (44% men), respectively [START_REF]Recensement Général de la Population et de l'Habitat[END_REF][START_REF]Recensement Général de la Population et de l'Habitat[END_REF]. Each house visited was marked by the investigators (name of the study, number of the investigator and subject) with chalk, just above the door, ensuring exhaustive coverage, even in the absence of addresses. Town halls were informed of the study and district chiefs were visited prior to starting the screening in order to increase awareness of the study.

Local associations were contacted and information broadcast on the radio.

Participants were not remunerated, but a few basic drugs (such as analgesics and vermifuge) were distributed after physical examination if necessary. Subjects were required to be aged 65 years or older and to be currently resident in the area under study. Exclusion criteria were refusal to participate, and the presence of severe comorbidity precluding cognitive testing. Consent was obtained from the subject and/or family after explanation of the research goals, and before the interview.

Dementia diagnosis

All subjects aged 65 years or more and living in the study areas were contacted by the investigators and asked to participate in the research. If necessary, age was ascertained by official documents, from an informant or through a local event calendar. Age was estimated using two historical landmarks in each country (Congo: independence and Marien Ngouabi assassination; CAR: independence and Bokassa priesthood), using a validated method [START_REF] Ogunniyi | Determination of ages of elderly Nigerians through historical events: validation of Ajayi-Igun 1963 listings[END_REF][START_REF] Paraïso | Validation of the use of historical events to estimate the age of subjects aged 65 years and over in Cotonou (Benin)[END_REF] .

Cognitive testing was performed using the Community Screening Interview for Dementia (CSI-D) [START_REF] Hall | The development of a dementia screening interview in two distinct languages[END_REF] and the 5 Words Test (FWT) [START_REF] Dubois | The 5 words": a simple and sensitive test for the diagnosis of Alzheimer's disease[END_REF], adapted and pretested in the local languages (Sango, Lingala and Kituba). A relative of each elderly person included was interviewed at the same time using the CSI-D informant section to assess daily activities and any personality change. Every subject with a poor performance in the CSI-D cognitive tests (<25.5/30) or the FWT (less than 10/10) was considered cognitively impaired and invited for further clinical assessment to confirm or rule out dementia. This confirmation stage occurred during the weeks following the screening, at the hospital. Further psychometric tests were conducted, including the Free and Cued Selective reminding test [START_REF] Grober | Screening for dementia by memory testing[END_REF], the Zazzo cancellation task [START_REF] Zazzo | Test des deux barrages[END_REF] and Isaac's Set Test of verbal fluency [START_REF] Isaacs | The Set test as an aid to the detection of dementia in old people[END_REF]. Neurological examination was performed by neurologists, and stroke history and depressive disorders were sought. Orientation skills and daily activities were investigated in order to evaluate their dependency.

The diagnosis of dementia was made according to the DSM-IV criteria [START_REF]American Psychiatric Association: Diagnostic and statistical manual of mental disorders[END_REF] and according to the clinical criteria proposed by the NINCDS-ADRDA (National Institute of Neurological and Communicative Disorders and Stroke and the Alzheimer's Disease and Related Disorders Association) for AD [START_REF] Mckhann | Clinical diagnosis of Alzheimer's disease: report of the NINCDS-ADRDA Work Group under the auspices of the Department of Health and Human Services Task Force on Alzheimer's disease[END_REF]. Diagnosis of other types of dementia (especially vascular dementia) was based on Hachinski Ischemic Scale [START_REF] Hachinski | Cerebral blood flow in dementia[END_REF], history of stroke and other clinical assessments. Medical records and test performances were reviewed by a group of experienced neurologists in uncertain cases and a consensus on the diagnosis was reached.

Associated factors

Sociodemographic data (i.e. marital status, schooling, occupation before retirement) and medical history (cardiovascular disease (CVD), head trauma, treatments if available, alcohol and tobacco consumption) were collected for each subject. Height (cm) and weight (kg) were measured in order to determine the Body Mass Index (BMI = weight divided by squared height). Hypertension was defined in terms of a reported history of hypertension and/or when systolic blood pressure at rest was 140 mmHg and/or diastolic blood pressure was 90 mmHg. Diabetes was defined according to self-reported medical history or in cases of elevated capillary blood glucose level (Accu-Chek® Performa, Roche), above 140 mg/dL if the fasting period was >2 hours or above 200 mg/dL in non-fasting participants. Capillary blood cholesterol level was also determined (considered elevated if >240 mg/dL) (Accutrend® Plus, Roche). Goldberg's anxiety and depression scale was used [START_REF] Goldberg | Detecting anxiety and depression in general medical settings[END_REF].

The presence of 18 psychosocial factors was investigated for three different life periods (under 16 years old, between 16 and 64 years, 65 years and over) according to Persson and Skoog [START_REF] Persson | A prospective population study of psychosocial risk factors for late onset dementia[END_REF].

Statistical analysis

As the main goal of this survey was to estimate the prevalence of dementia [START_REF] Guerchet | Prevalence of dementia in elderly living in two cities of Central Africa: the EDAC survey[END_REF], we aimed to include a minimum of 456 participants in each site in order to detect a prevalence of dementia of 5% with a precision of 2% (EpiInfo 6.04, Epiconcept).

The mean and standard deviation (SD) were used as summary statistics for quantitative variables (e.g. age), and compared using Student's t-test. Percentage counts were used for all qualitative variables of interest, and Fisher exact tests were used for comparisons. Level of significance for these comparisons was 0.05. Two separate multivariate regression analyses were conducted, using dementia as the dependent variable (reference group: subjects without dementia). Variables that had a p-value ≤ 0.25 in the univariate analysis were included in multivariate logistic regression models. Backward stepwise procedures were realized, leading to final multivariate models with a level of significance of 0.05. Confounding effects and interactions between independent variables in final models were examined. In the presence of interactions, odds ratios and their confidence intervals were determined for each group resulting from the interaction of the variables.

The first model was fitted using sociodemographic, medical and psychosocial data from before the onset of dementia as independent variables, and the second model included variables concomitant to the assessment of dementia.

Statistical analyses were carried out using SAS® software (version 9.2, SAS Institute, Cary, NC).

Results

Study population

During this study, 509 people aged 65 or more were contacted in Bangui and 546 in Brazzaville, of whom 496 and 520, respectively, agreed to participate (N=1016; Figure 1). Fifty-nine percent of those studied were females (ratio M/F: 0.69), the mean age was 73.6 ± 6.5 years (range: 65 to 96 years) and 47.3% had primary school and higher education. Complete descriptions of the studied populations in each city are given in Table 1.

After the screening phase, 188 (18.5%) elderly subjects in Bangui and 148 (14.6%) in Brazzaville were identified as cognitively impaired; 251 (74.7%) were women. The mean age of cognitively impaired participants (75.7 ± 7.2 years) was significantly higher than that of subjects free of cognitive impairment (72.5 ± 5.9) (p<0.0001).

Among the cognitively impaired subjects, 75 (7.4%) were diagnosed with dementia after neurological examination; 57 (5.6%) had probable or possible AD and 18 (1.8%) had vascular dementia. Sixty participants with dementia were women (80.0%). Mean age was 77.4 ± 7.3 and 18.7% of them were aged ≥85 years. They were mostly widowed (66.7%) and only 24.0% had been to primary school. Forty-eight (64.0%) had hypertension, 10 (13.3%) had diabetes and 68 (90.7%) showed depressive symptoms at the time of the study.

Subjects with suspected cognitive impairment but not dementia were categorized separately. Thirty-eight participants (3.7%) were lost between the two stages of the survey and could not be diagnosed (14 refused, 9 had moved away, 3 had died, 6 were too ill to participate and 6 were not retrieved); only one subject remained unclassifiable at the end of the study. These subjects were excluded from the analysis.

Rates of missing data were high for diabetes (15.7%) and cholesterol level (24.6%).

Importantly, the subjects with missing data for those variables did not differ from the others regarding age, sex, and education (data not shown) but the proportions with dementia were significantly lower (98.0% vs. 2.0%, p=0.004 and 96.6% vs. 3.4%, p=0.005, respectively).

Factors associated with dementia

Subjects with and without dementia were compared by univariate analyses as presented in Table 2. Age, female gender, marital status (divorced or widowed) and no schooling were significantly associated with dementia with a p-value<0.05.

Regarding medical variables, a low BMI, the presence of anxiety disorders and depressive symptoms were also associated with dementia. The following life events were associated with dementia in the univariate analysis: death of a parent or having been raised by one parent before 16 years of age, the death of a spouse between the ages of 16 and 64, and a recent house move or deterioration in financial status.

In the multivariate analysis, the first model included variables occurring prior to the study (Table 3). Independent factors associated with prevalent dementia were, adjusted for study site, age and the death of a parent before the age of 16. In this model, schooling was no longer significant, with OR remaining greater than one, but with a wide confidence interval, whereas it was a highly significant variable associated with dementia in univariate analysis. Thus an interaction between gender and schooling was suspected and found in a further model: the risk of having dementia was significantly higher for women who had no education compared to those who had at least primary education, while no significant effect of education was found for men. Similarly, among subjects with no education, women were at far higher risk of dementia than men, while the risk was equivalent among subjects with at least primary education (Table 3).

The following model included factors which were found concomitant to the diagnosis of dementia during the study (Table 4). Hypertension and a BMI <18.5 kg.m² were significantly associated with dementia, as were depressive symptoms. The only psychosocial factor associated with dementia in this model was having moved residence (OR=1.8, 95% CI: 1.1-3.0, p=0.028).

No interaction was found in this model.

Discussion

To our knowledge, this is the first multicenter population-based study of dementia in central Africa. We showed that beyond traditional risk factors for dementia already found in other epidemiological studies conducted elsewhere, several life events which may be more frequent in these populations are associated with dementia.

The two countries surveyed were chosen for several reasons: dementia in the older general population had never been studied there; moreover, they recently experienced political disturbances and civil wars, leading to interesting research perspectives for the association of dementia and life events. At the time of the study, it was not safe to carry out surveys other than in the capital cities. Door-to-door sampling could not ensure representative sampling [START_REF] Prince | Methodological issues for population-based research into dementia in developing countries. A position paper from the 10/66 Dementia Research Group[END_REF]. However, the Bangui participants were not significantly different in age and sex from elderly people living in the 3 rd district (p>0.05 for each age group, and p=0.94 for sex) [START_REF]Recensement Général de la Population et de l'Habitat[END_REF] and the Brazzaville participants from the 4 th district did not differ in sex ratio from the elderly people living in Brazzaville as a whole (p>0.05) [START_REF]Recensement Général de la Population et de l'Habitat[END_REF]. Ethnic characteristics of our samples were faithful to the cities' overall characteristics. Some ethnic groups might have been over-represented, such as Bacongos in Brazzaville, resulting from the geographical localisation of the city in the country.

The CSI-D cognitive test was chosen for its relevance and adaptability in populations with different socioeconomic and linguistic backgrounds [START_REF] Hall | Community screening interview for dementia (CSI-D), Performance in five disparate study sites[END_REF]. It is commonly used in studies carried out in developing countries with low rates of literacy, especially in Nigeria [START_REF] Osuntokun | Lack of an association between apolipoprotein E epsilon 4 and Alzheimer's disease in elderly Nigerians[END_REF] and Benin [START_REF] Gureje | APOE epsilon4 is not associated with Alzheimer's disease in elderly Nigerians[END_REF] in Africa, and by the 10/66 Dementia Research Group. We used the same cut-off for the CSI-D cognitive part as successfully used in a previous study in Benin [START_REF] Guerchet | Cognitive impairment and dementia in elderly people living in rural Benin, West Africa[END_REF]. We also included the FWT, allowing rapid screening of patients for AD. The threshold of 10 was validated in the general population, with a maximal sensitivity of 63%, specificity of 91.1% and a minimum of false negative and false positive subjects [START_REF] Cowppli-Bony | Validity of the five-word screening test for Alzheimer's disease in a population based study[END_REF]. This version of the verbal memory test based on oral language produces accurate results in illiterate or poorly educated subjects. The combination of the two tests for screening elderly people has been validated in Benin; neuropsychological testing in 50 elderly people negative to screening found no additional cases of dementia [START_REF] Guerchet | Cognitive impairment and dementia in elderly people living in rural Benin, West Africa[END_REF].

We acknowledge that the number of subjects was guided by the estimation of dementia prevalence, which was the first aim of the study. The design does not allow us to determine risk factors for dementia, but does identify several factors which might be associated with an increased or decreased risk of dementia in these populations. These are the first such findings in for this region of sub-Saharan Africa.

The choice of factors to investigate was guided by the feasibility of data collection in the specific environment of low-income African countries. Worldwide, age is the most consistent risk factor for dementia [START_REF] Kalaria | Alzheimer's disease and vascular dementia in developing countries: prevalence, management, and risk factors[END_REF], and the prevalence and incidence generally increase as populations grow older [START_REF] Launer | Rates and risk factors for dementia and Alzheimer's disease: results from EURODEM pooled analyses. EURODEM Incidence Research Group and Work Groups. European Studies of Dementia[END_REF][START_REF] Plassman | Prevalence of dementia in the United States: the aging, demographics, and memory study[END_REF][START_REF] Rodriguez | 10/66 Dementia Research Group. Prevalence of dementia in Latin America, India, and China: a population-based cross-sectional survey[END_REF]. In this cohort, the prevalence of dementia increased dramatically with age [START_REF] Persson | A prospective population study of psychosocial risk factors for late onset dementia[END_REF]. Age determination is critical in epidemiological studies, but is not always easy when working with low-income populations. In this study, a reliable age for each participant was obtained from official documents and informant reporting, followed by estimation based on historical events [START_REF] Ogunniyi | Determination of ages of elderly Nigerians through historical events: validation of Ajayi-Igun 1963 listings[END_REF][START_REF] Paraïso | Validation of the use of historical events to estimate the age of subjects aged 65 years and over in Cotonou (Benin)[END_REF] in discordant cases.

We found a significant interaction between gender and schooling, with a poor education being significantly associated with dementia among women, but not men.

Both female gender [START_REF] Ochayi | Risk factors for dementia in central Nigeria[END_REF][START_REF] Hall | Risk factors and Alzheimer's disease: a comparative study of two communities[END_REF][START_REF] Ogunniyi | Epidemiology of dementia in Nigeria: results from the Indianapolis-Ibadan study[END_REF][START_REF] Ogunniyi | Risk factors for incident Alzheimer's disease in African Americans and Yoruba[END_REF]] and low education level [START_REF] Kalaria | Alzheimer's disease and vascular dementia in developing countries: prevalence, management, and risk factors[END_REF] have been reported to be risk factors for dementia in African populations. In our sample, primary education was far less common among women (26.2%) than men (77.8%), and the gap was even greater when only subjects with dementia were considered (10.0% vs. 80.0%, respectively). Men are probably less affected by a lack of formal education because their occupational activities allowed them to compensate for it as they grew older, which was not the case for women. The occupational and leisure activities of poorly educated women are certainly not sufficiently stimulating, unlike those of men in a similar position. A social bias is also possible, as women with access to education are generally from wealthy families. The role of education needs to be thoroughly investigated in countries where schooling was not available for large portions of the population in the past few decades, and may not be a good indicator of cognitive ability.

The death of a parent before the age of 16 was associated with dementia in the first model. Similar results were observed in a Swedish population, with a relative risk of 6.6 of developing severe dementia between the ages of 70 and 79 years [START_REF] Persson | A prospective population study of psychosocial risk factors for late onset dementia[END_REF]. We are aware that self-reported biographical events are subject to memorisation bias, especially when they happen early in life. However, the events collected were chosen for their ability to leave their mark on people's minds and there was no other way to obtain this information as no national database is available.

Our finding that malnutrition, indicated by a BMI<18.5 kg/m², was associated with dementia is in accord with a previous study performed in Nigeria [START_REF] Ochayi | Risk factors for dementia in central Nigeria[END_REF] reporting an association between a BMI<18.5 kg/m² and AD. Relationships between nutritional status and dementia have been more widely investigated in high-income countries.

Declining BMI was associated with an increasing risk of incident AD and a faster rate of disease progression in elderly Americans [START_REF] Buchman | Change in body mass index and risk of incident Alzheimer disease[END_REF]. Some authors also suggested that low BMI, or a faster decline in BMI in later life, may be a preclinical marker of underlying dementia or AD [START_REF] Hughes | Association between late-life body mass index and dementia: The Kame Project[END_REF][START_REF] Chu | Late-life body mass index and waist circumference in amnestic mild cognitive impairment and Alzheimer's disease[END_REF]. The cross-sectional design of our study does not allow us to draw conclusions about the causal relationship between a low BMI and dementia, and we cannot exclude the possibility that malnutrition was a consequence of dementia.

Regarding cardiovascular risk factors, hypertension was associated with dementia.

Several studies in high-income countries have reported an association between hypertension and low cognitive performance [START_REF] André-Petersson | A comparison of cognitive ability in normotensive and hypertensive 68-year-old men: results from population study 'men born in 1914' in Malmö, Sweden[END_REF][START_REF] Budge | Total plasma homocysteine, age, systolic blood pressure, and cognitive performance in older people[END_REF][START_REF] Kuo | Effect of blood pressure on cognitive functions in elderly persons[END_REF]. However, we have to interpret this result with caution, as blood pressure was measured only once. The analysis did not highlight other CVD risk factors associated with dementia (i.e. diabetes or obesity). This effect could be due to a selective survival of people not affected by these conditions in these specific populations with low levels of healthcare. Interaction of nutritional factors, CVD risk factors and dementia may also be presumed.

In this study, the presence of depressive symptoms at the time of the survey was significantly associated with dementia. This factor has already been associated with cognitive impairment in another elderly African population [START_REF] Gureje | APOE epsilon4 is not associated with Alzheimer's disease in elderly Nigerians[END_REF]. The relationship between depression and dementia is discussed in the literature [START_REF] Tsuno | What is the association between depression and Alzheimer's disease?[END_REF][START_REF] Panza | Late-life depression, mild cognitive impairment, and dementia: possible continuum?[END_REF][START_REF] Jorm | Is depression a risk factor for dementia or cognitive decline? A review[END_REF], but it remains controversial whether depressive symptoms represent a risk factor for dementia, or if they are an early symptom of neurodegeneration or a response to first cognitive deficit symptoms. Due to the cross-sectional design of this study, we are unable to assess the temporal association between depression and dementia.

Change of residence was significantly associated with dementia in our model. This life event requires adaptations to daily living. It can be traumatic and lead to increased stress for elderly people [START_REF] Persson | A prospective population study of psychosocial risk factors for late onset dementia[END_REF]. The association between moving and dementia is in accordance with the hypothesis that such events may be involved in the dementia process and could trigger cognitive decline [START_REF] Letenneur | Are sex and educational level independent predictors of dementia and Alzheimer's disease? Incidence data from the PAQUID project[END_REF][START_REF] Ngandu | Education and dementia: what lies behind the association?[END_REF]. No stressful events have been previously reported to be associated with dementia in African studies. However, we cannot rule out the possibility that the change of residence of the elderly was consequent to the occurrence of dementia.

With regard to statistical analysis, we chose to combine AD and VaD as the diagnosis of dementia was mainly clinical, without using cerebral imaging or laboratory investigation to confirm the subtype of dementia. Moreover, given the low number of subjects with dementia, distinguishing between the conditions could have led to low statistical power. Studying the severity of cognitive impairment was not relevant as validated scales measuring cognitive decline are not available in these populations.

We are aware of the high number of independent variables studied in this study, but colinearity was not an issue in the analyses.

Our study has several limitations. We cannot exclude a lack of power regarding two measures: diabetes and high blood cholesterol. Indeed, they both have a high rate of missing data in Bangui due to problems inherent to specific technical issues encountered in Africa. Because it is a cross-sectional study, the temporality of the associations found cannot be assessed. We also acknowledge that the absence of ApoE genotyping is a limit to our study. High frequencies of the ApoE ε4 allele have been reported in Pygmies in the Central African Republic (40.7%) and Bantu (29%) populations [46].

Conclusion

This study identified seven variables independently associated with dementia in older Central Africans residing in two large cities. Most are consistent with the worldwide literature (age, female gender, low level of education, hypertension, low BMI and depressive symptoms) but the effects of stressful events which may differ in these specific global regions are the most original findings. Factors associated with dementia in African countries seem different from established factors in high-income countries and need further longitudinal study.

  Multiethnic considerations (to avoid cluster effects) and feasibility guided the identification of participating districts. Approvals were obtained from ethical committees of the Central African Republic and Republic of Congo, supervised by the Ministries of Public Health or Scientific Research.

[ 46 ]

 46 Singh PP, Singh M, Mastana SS. APOE distribution in world distribution with new data from India and the UK. Ann Hum Biol 2006;33:279-308.

Figure 1 :

 1 Figure 1: Flow diagram of EDAC study in Bangui and Brazzaville, 2008-2009.

Table 1 :

 1 Characteristics of the population studied according to dementia and city, EDAC survey, Bangui & Brazzaville, 2008-2009

	Both		Bangui		Brazzaville	Missing data
	n	%	n	%	n	%	n

Table 2 :

 2 Univariate analysis between dementia and sociodemographic, medical and psychosocial variables, EDAC study, 2008-2009. CI = 95 % confidence interval * the univariate logistic model is not converging because of the quasi-complete separation of data points 29

	OR	95% CI	p-value

Table 3 :

 3 Multivariate analysis of risk factors present before the diagnosis of dementia. The EDAC study, 2008-2009.

			Multivariate analysis
		Initial model			Final model
	OR	95% CI	p-value	OR	95% CI	p-value

Table 4 :

 4 Multivariate analysis of factors concomitant to the diagnosis of dementia. The EDAC study. 2008-2009.

			Multivariate analysis
		Initial model			Final model
	OR	95% CI	p-value	OR	95% CI	p-value

Acknowledgements

The authors thank the health ministries of the Central African Republic and the Republic of Congo, and Bangui and Marien Ngouabi (Brazzaville) Universities for their moral support. We also thank all the participants who made these surveys possible: the investigators, the staffs of Bangui and Brazzaville hospitals for their assistance, and the elderly people and their families who devoted time to the study.

We particularly acknowledge Miss Anaïs Labrunie for her help with statistical analysis.

The project was funded by the French Ministry of Foreign Affairs as part of the "Coopération pour la Recherche Universitaire et Scientifique" programme (CORUS 6024-4). These investigations were also supported by the Fondation pour la Recherche Médicale and the Doctoral School of Limoges University.

Disclosure

The authors report no conflict of interest.


Abdelhamid Hassairi
email: abdelhamid.hassairi@fss.rnu.tn

Mahdi Louati

Mixture of the Riesz distribution with respect to the multivariate Poisson

Keywords: Mixed distribution, Riesz distribution, Bessel function, natural exponential family, variance function

Mixture of the Riesz distribution with respect to the multivariate Poisson

Introduction

Let µ λ be a probability distribution on a finite dimensional linear space E depending on a parameter λ which belongs to a subset Λ of IR r . Suppose that

µ λ = f (x, λ)σ(dx),
where σ is some reference measure, and that for each x in E, the map λ → f (x, λ) defined on Λ is measurable. For a probability distribution ν(dλ) on the set Λ, define

h(x) = Λ f (x, λ)ν(dλ).
Then the probability measure µ ν (dx) = h(x)σ(dx)

is called the mixture of the distribution µ λ with respect to ν. (See Feller [START_REF] Feller | An Introduction to Probability Theory and its Applications[END_REF], Vol. II, page 53 or Johnson et al. [START_REF] Johnson | Univariate discrete distributions[END_REF], page 360). Usually, µ λ is said the mixed distribution and ν the mixing distribution (see Karlis and Meligkotsidou [9]).

A special case of interest is when µ is not concentrated on an affine hyperplane of E, µ λ is the λ-power of convolution of µ, and Λ is the so called Yørgensen set of µ. Specifically, let L µ (θ) = E exp( θ, x )µ(dx) (1.1) denote the Laplace transform of µ, where , is the duality crocket, and suppose that the set Θ(µ) = interior{θ ∈ E * ; L µ (θ) < +∞} (1.2) is nonempty. Then the set Λ = {λ > 0; ∃ µ λ such that L µ λ (θ) = (L µ (θ)) λ for all θ ∈ Θ(µ)} (1.3) is called the Yørgensen set of µ and the measure µ λ is its λ-power of convolution. This parameter λ appears in the most common models, it is in particular, the intensity in a Poisson model, the variance in a Gaussian model, and the shape parameter in a gamma model. For λ and λ ′ in Λ, we have that

µ λ * µ λ ′ = µ λ+λ ′ .
The set Λ contains always the set IN * of positive integers and it is equal to ]0, +∞[ if and only if µ is infinitely divisible (see Seshadri [START_REF] Seshadri | The Inverse Gaussian Distribution. A case study in exponential families[END_REF], page 155).

In fact, if X 1 , . . . , X N are iid random variables with distribution µ, then the distribution of X 1 + . . . + X N is µ N = µ ⋆N , the N -power of convolution of µ. Accordingly, for any distribution µ and any positive integer N , one may consider the distribution µ N as defined in (1.3). When µ is discrete, i.e., with countable support, then the mixture of µ with respect of a distribution ν on the parameter N is known as a compound distribution. The most famous compound distribution is the one corresponding to the case where ν is Poisson (see Feller [START_REF] Feller | An Introduction to Probability Theory and its Applications[END_REF], Vol. I, page 286 or Vol. II, page 451 or Aalen [START_REF] Aalen | Modelling heterogeneity in survival analysis by the compound Poisson distribution[END_REF], or Yørgensen [START_REF] Yørgensen | The Theory of Dispersion Models[END_REF], page 140). In fact, the real Poisson distribution appears in numerous works either as a mixed distribution (see Bhattacharya and Holla [START_REF] Bhattacharya | On a discrete distribution with special reference to the theory of accident proneness[END_REF] or Johnson et al. [START_REF] Johnson | Univariate discrete distributions[END_REF], page 366) or as a mixing distribution (see Perline [START_REF] Perline | Mixed Poisson distributions tail equivalent to their mixing distributions[END_REF]). In the present work, we will be interested in a very special case in which the mixed distribution is defined on the cone of (r, r) positive definite symmetric matrices Ω with a parameter which belongs to a subset of IR r . More precisely, the mixed distribution will be the absolutely continuous Riesz model introduced in Hassairi and Lajmi [START_REF] Hassairi | Riesz exponential families on symmetric cones[END_REF] R(s, σ), s ∈

r i=1 i -1 2 , +∞ .
This model contains the Wishart model, since R(s, σ) reduces to a Wishart distribution when s 1 = s 2 = . . . = s r , and it has a convolution property which is analogous to the one satisfied by the ordinary powers of convolution. In fact, if s and s ′ are in

r i=1 i -1 2 , +∞ , then R(s, σ) * R(s ′ , σ) = R(s + s ′ , σ).
The mixing distribution will be the multivariate Poisson distribution on IN r . For simplicity, we will be interested in the case where σ is equal to the identity matrix of size r denoted I r . We first show that the mixture distribution is expressed in terms of the modified Bessel function. We then determine the domain of the means of the generated natural exponential family, and we calculate its variance function. This provides a rich class of natural exponential families.

The Riesz exponential dispersion model

A dispersion model is a class of natural exponential families where each family is a power of convolution of the other. In this section, we will first recall some general facts concerning the exponential dispersion models in an Euclidean space. Then we introduce the Riesz dispersion model on symmetric matrices.

Exponential dispersion model

Let E be an Euclidean space with finite dimension n, and let , denote the scalar product in E. If µ is a positive measure on E, we denote by M(E) the set of measures µ such that Θ(µ) given in (1.2) is not empty and µ is not concentrated on an affine hyperplane of E.

If µ is in M(E), we denote k µ (θ) = log L µ (θ), for all θ in Θ(µ)
the cumulant function of µ, where L µ is the Laplace transform of µ defined in (1.1).

To each µ in M(E) and θ in Θ(µ), we associate the probability distribution on

E P (θ, µ)(dx) = exp ( θ, x -k µ (θ)) µ(dx). The set F = F (µ) = {P (θ, µ); θ ∈ Θ(µ)}
is called the natural exponential family (NEF) generated by µ. We also say that µ is a basis of F .

The function k µ is strictly convex and real analytic. Its first derivative k ′ µ defines a diffeomorphism between Θ(µ) and its image

M F . Since k ′ µ (θ) = E xP (θ, µ)(dx), M F is
called the domain of the means of F . The inverse function of k ′ µ is denoted by ψ µ and setting P (m, F ) = P (ψ µ (m), µ) the probability of F with mean m, we have

F = {P (m, F ); m ∈ M F } ,
which is the parametrization of F by the mean. Now the covariance operator of P (m, F ) is denoted by V F (m) and the map

M F -→ L s (E); m -→ V F (m) = k ′′ µ (ψ µ (m))
is called the variance function of the NEF F . It is easy proved that for all m ∈ M F ,

V F (m) = (ψ ′ µ (m)) -1 ,
and an important feature of V F is that it characterizes F in the following sense: If F and F ′ are two NEFs such that V F (m) and V F ′ (m) coincide on a nonempty open set of

M F ∩M F ′ , then F = F ′ .
In particular, knowledge of the variance function gives knowledge of the natural exponential family.

Let µ be an element of M(E) and let Λ be its Yørgensen set defined by (1.3). Then the set

{P (θ, λ) = exp (< θ, x > -k µ λ (θ)) µ; θ ∈ θ(µ), λ ∈ Λ}
is called the dispersion model generated by µ. For more details, we refer to Letac [START_REF] Letac | Lectures on natural exponential families and their variance functions[END_REF].

Riesz natural exponential families

Let E be the Euclidean space of (r, r) real symmetric matrices equipped with the scalar product x, y = tr(xy), and the inner product x.y = 1 2 xy +yx , where xy is the ordinary product of two matrices. We denote by e 1 , e 2 , . . . , e r the canonical basis of IR r ; e i = (0, . . . , 0, 1, 0 . . . 0), (1 in the i th place), and we set c i = diag(e i ) for all 1 ≤ i ≤ r. For x ∈ E, we consider the endomorphism L(x) of E defined by L(x) : y -→ x.y and we set

P (x) = 2(L(x)) 2 -L(x 2 ).
We denote by Ω the cone of (r, r) real symmetric positive definite matrices. For x = (x ij ) 1≤i,j≤r in E and 1 ≤ k ≤ r, we define the sub-matrices

P k (x) = (x ij ) 1≤i,j≤k and P * k (x) = (x ij ) r-k+1≤i,j≤r .
For convenience, P k (x) and P * k (x) are also considered as elements of the space E, where the other entries are equal to zero and we set P * 0 (x) = 0. Let ∆ k (x) and ∆ * k (x) denote the determinant of P k (x) and the determinant of P * k (x) respectively. Then the generalized power of x in Ω is defined, for s

= (s 1 , s 2 , . . . , s r ) ∈ IR r , by ∆ s (x) = ∆ 1 (x) s 1 -s 2 ∆ 2 (x) s 2 -s 3 . . . . . . ∆ r (x) sr . (2.4)
Note that if for all i ∈ {1, . . . , r}, s i = p ∈ IR, then ∆ s (x) = (det x) p . We also define

∆ * s (x) = (∆ * 1 (x)) s 1 -s 2 (∆ * 2 (x)) s 2 -s 3 . . . (∆ * r-1 (x)) s r-1 -sr (∆ * r (x)) sr . (2.5)
It is shown (see Hassairi and Lajmi [START_REF] Hassairi | Riesz exponential families on symmetric cones[END_REF]) that for all x ∈ Ω and all s ∈ IR r , we have

∆ s (x -1 ) = ∆ * -s * (x), (2.6)
where s * = (s r , s r-1 , . . . , s 1 ). .

We denote by T + l the set of lower triangular matrices with positive diagonal elements, and for u ∈ T + l , we define on E the automorphism

u(y) = uyu * ,
where u * denotes the transpose matrix of u.

It is well known that for all x ∈ Ω, there exists a unique u ∈ T + l such that

x = u(I r ),
where I r is the identity matrix of order r, it is the Cholesky decomposition of x.

We also have (see Hassairi and Lajmi [6]) that for all 1 ≤ i ≤ r,

P * i (u(I r )) -1 -1 = u   r k=r-i+1 c k   , (2.7)
and for all s = (s

1 , s 2 , . . . , s r ) ∈ IR r , ∆ s (u(I r )) = ∆ * -s * (u * -1 (I r )). (2.8)
Recall also that for x ∈ Ω and u ∈ T + l , we have

∆ i (u(x)) = ∆ i (u(I r ))∆ i (x) = u 2 1 . . . u 2 i ∆ i (x), (2.9)
where for all i ∈ {1, . . . , r},

u i = u, c i . (2.10)
(See Faraut and Koranyi [START_REF] Faraut | Analysis on Symmetric Cones[END_REF], page 114). Now let Ξ be the set of elements s = (s 1 , s 2 , . . . , s r ) ∈ IR r defined as follows:

Consider the function ξ defined from IR + into {0; 1}, by

ξ : a -→ 0 if a = 0, 1 if a > 0.
For all (u 1 , u 2 , . . . , u r ) ∈ IR r + , we define

       s 1 = u 1 s k = u k + ξ(u 1 ) + ... + ξ(u k-1 ) 2 , ∀ k ∈ {2, . . . , r}.
A result due to Gindikin [START_REF] Gindikin | Analysis on homogeneous domains[END_REF] and proved in Faraut and Korányi [START_REF] Faraut | Analysis on Symmetric Cones[END_REF], page 124, says that there exists a positive measure R s such that for all θ ∈ -Ω,

L Rs (θ) = E e θ,x R s (dx) = ∆ s (-θ -1 )
if and only if s is in Ξ. This measure is called the Riesz measure with parameter s.

When s = (s 1 , s 2 , . . . , s r ) ∈ Ξ \ r i=1 i -1 2
, +∞ , the measure R s is concentrated on the boundary ∂Ω of Ω and when s = (s 1 , s 2 , . . . , s r ) is such that for all i, s i > i -1 2 , the measure R s is absolutely continuous with respect to the Lebesgue measure and is given by

R s (dx) = ∆ s-n r (x) Γ Ω (s) 1 Ω (x)(dx) with n = r(r + 1) 2 the dimension of E and Γ Ω (s) = (2π) r(r-1) 4 r i=1 Γ s i - i -1 2 . (2.11)
It is shown in Hassairi and Lajmi [START_REF] Hassairi | Riesz exponential families on symmetric cones[END_REF] that the measure R s generates a natural exponential family if and only if s 1 = 0. In this case

F = F (R s ) = R(s, σ) = e -<σ,x> ∆ s (σ -1 ) R s , σ ∈ Ω, s ∈ Ξ, s 1 = 0 .
The distribution R(s, σ) is called the Riesz distribution with parameters s and σ. It is shown in Hassairi and Lajmi [START_REF] Hassairi | Riesz exponential families on symmetric cones[END_REF] that the Laplace transform of R(s, σ) is defined for θ in σ -Ω, by

L R(s,σ) (θ) = ∆ s ((σ -θ) -1 ) ∆ s (σ -1 ) . (2.12)
This implies that if σ is an element of Ω and if s and s ′ are in Ξ, then we have

R(s, σ) * R(s ′ , σ) = R(s + s ′ , σ).
Let σ be an element of Ω. If s satisfies the conditions

s i > i -1 2 , for 1 ≤ i ≤ r, then the Riesz distribution is given by R(s, σ)(dx) = e -<σ,x> ∆ s-n r (x) Γ Ω (s)∆ s (σ -1 )
1 Ω (x)(dx).

When s 1 = s 2 = . . . = s r = p > 0, R(s, σ) reduces to the Wishart distribution with parameters p > r -1 2 and σ ∈ Ω,

W (p, σ)(dx) = 1 Γ Ω (p) det(σ -p ) e -<σ,x> det(x) p-n r 1 Ω (x)(dx),
with Laplace transform equal for all θ ∈ σ -Ω, to

L W (p,σ) (θ) = det I r -σ -1 θ -p .
3 The mixture of the Riesz distribution with respect to the multivariate Poisson

Consider the Poisson distribution on IN r with parameter λ = (λ 1 , λ 2 , . . . , λ r ) ∈ (IR + ) r ν(dx) = e -r i=1 λ i q∈IN r λ q q! δ q (dx), (3.13)
where q! = q 1 ! q 2 ! . . . q r ! and

λ q = λ q 1 1 λ q 2 2 . . . λ qr r . Then for all θ ∈ IR r , L ν (θ) = r i=1 e λ i (e θ i -1) . (3.14)
The variance function of the natural exponential family generated by ν evaluated for m ∈]0, +∞[ r is equal to

V (m) = diag(m).
Note that this variance function is homogeneous of order 1 and consequently the natural exponential family F (ν) belongs to the Tweedie scale of IR r (see Hassairi and Louati [START_REF] Hassairi | Multivariate stable exponential families and Tweedie scale[END_REF]).

Next, we will show that the mixture of the distribution R(s, I r ) with respect to ν is related to the modified Bessel function of the first kind and of order 1.

Consider the second-order linear differential equation

z 2 y ′′ (z) + zy ′ (z) + (z 2 -α 2 )y(z) = 0, (3.15)
where z is a complex variable and α is a real parameter. One of the solution of (3.15) is the function J(α, z) known as the Bessel function of the first kind of order α, and defined for z in I C by the series

J(α, z) = +∞ k=0 (-1) k z 2 2k+α k!Γ(k + α + 1) . (3.16)
In many applications, one frequently encounters two functions I(α, z) and K(α, z) called respectively the modified Bessel functions of the first and second kinds of order α, defined on the complex plane cut along the negative real axis by It is an immediate fact that I(α, z) and K(α, z) are linearly independent solutions of the differential equation:

I(α, z) = i -α J(α, iz), (3.17
z 2 y ′′ (z) + zy ′ (z) -(z 2 + α 2 )y(z) = 0. (3.18)
For more details about the Bessel functions, we refer to Lebedev [START_REF] Lebedev | Special functions and their applications[END_REF], page 108 and to Feller [START_REF] Feller | An Introduction to Probability Theory and its Applications[END_REF], Vol. II, page 58.

We also need to establish the following result related to these functions.

Proposition 3.1 Let x > 0 and let b > 0, then k∈IN x k k! Γ(k + b) = x 1-b 2 I(b -1, 2 √ x).

Proof

Consider the function

g b (x) = k∈IN x k k! Γ(k + b) . (3.19) Deriving g b gives g ′ b (x) = k∈IN x k k! Γ(k + 1 + b) = k∈IN x k k!(k + b) Γ(k + b) .
This implies that

x b g ′ b (x) ′ = k∈IN x k+b-1 k! Γ(k + b) = x b-1 g b (x).
Consequently, g b (x) is a solution of the differential equation

x b y ′′ (x) + bx b-1 y ′ (x) -x b-1 y(x) = 0.
Or equivalently, g b (x) is solution of the equation

xy ′′ (x) + by ′ (x) = y(x). (3.20)
Using now the fact that I(α, z) and K(α, z) are two linearly independent solutions of (3.18), we deduce that x

1-b 2 I(b -1, 2 √ x) and x 1-b 2 K(b -1, 2 √
x) are two linearly independent solutions of (3.20). It follows that the general solution of (3.20) is of the form

c 1 x 1-b 2 I(b -1, 2 √ x) + c 2 x 1-b 2 K(b -1, 2 √ x),
where c 1 and c 2 are two constants.

According to (3.19), (3.16) and (3.17), we deduce that

g b (1) = k∈IN 1 k! Γ(k + b) = I(b -1, 2).
Therefore c 1 = 1 and c 2 = 0.

Consequently, for all x > 0, (2π)

g b (x) = k∈IN x k k! Γ(k + b)! = x 1-b 2 I(b -1, 2 √ x). ( 3
r(r-1) 4 det(x) r i=1 √ λ i e -λ i ∆ i-1 (x) I 1, 2 λ i ∆ e i (x) 1 Ω (x)(dx),
where ∆ 0 (x) = 1.

Proof

Let a = (a 1 , a 2 , . . . , a r ) be an element of ]0, +∞[ r and let

R k,a = R(k + ρ + a, I r ).
Denote µ a the mixture of R k,a with respect to ν. Then

µ a (dx) = h a (x)1 Ω (x)(dx),
where

h a (x) = e -r i=1 λ i q∈ IN r λ q e -tr(x) ∆ q+ρ+a-n r (x) q! Γ Ω (q + ρ + a) , (3.22)
and lim

a-→0 µ a (dx) = µ(dx) = e -r i=1 λ i q∈ IN r
λ q e -tr(x) ∆ q+ρ-n r (x) q! Γ Ω (q + ρ)

1 Ω (x)(dx).

(3.23)

According to (2.4) and (2.11), (3.22) may be written as

h a (x) = e -tr(x) (2Π) r(r-1) 4 det(x) r i=1   ∆ i (x) -1 2 e -λ i q i ∈IN λ q i i q i ! Γ(q i + a i ) ∆ i (x) ∆ i-1 (x) q i +a i   . = e -tr(x) (2Π) r(r-1) 4 det(x) r i=1   ∆ i (x) -1 2 e -λ i q i ∈IN λ q i i q i ! Γ(q i + a i ) ∆ e i (x) q i +a i   . = e -tr(x) (2Π) r(r-1) 4 det(x) r i=1   ∆ i (x) -1 2 e -λ i ∆ e i (x) a i q i ∈IN (λ i ∆ e i (x)) q i q i ! Γ(q i + a i )   .
Therefore

h a (x) = e -tr(x) (2Π) r(r-1) 4 det(x) r i=1 e -λ i ∆ i (x) ∆ e i (x) a i g a i (λ i ∆ e i (x)) , (3.24)
where g a i is defined in (3.19). Inserting now (3.21) in (3.24), we obtain

h a (x) = e -tr(x) (2Π) r(r-1) 4 det(x) r i=1   λ 1-a i 2 i e -λ i ∆ i (x) ∆ e i (x) 1+a i 2 I a i -1, 2 λ i ∆ e i (x)   .
According to (3.23), we deduce that

µ(dx) = e -tr(x) (2Π) r(r-1) 4 det(x) r i=1 √ λ i e -λ i ∆ i-1 (x) I -1, 2 λ i ∆ e i (x) 1 Ω (x)(dx).
Invoking the fact that for all x > 0, we have

I(1, x) = I(-1, x),
(see Lebedev [START_REF] Lebedev | Special functions and their applications[END_REF], page 110), the proof of the theorem is complete. 2

The variance function of F (µ)

In this subsection, we study the natural exponential family F generated by µ. We first give the Laplace transform of µ, then we determine the domain of the means and the variance function of the family F .

In what follows, for r ≥ 1, we denote κ r = r j=1 j 2 e j and we set κ 0 = 0.

Theorem 3.3 For all θ ∈ I r -Ω, we have

L µ (θ) = ∆ * κ r-1 (I r -θ) exp r i=1 λ i ∆ * -e r-i+1 (I r -θ) -1 . (3.25)

Proof

Let X k be a random variable such that X k ∼ R(k + ρ, I r ). Then, according to (2.12), we have that for all θ ∈ I r -Ω,

L µ (θ) = E e θ,X k = E E e θ,X k | k = E ∆ k+ρ ((I r -θ) -1 ) .
Using (2.5) and (2.6), we can write

L µ (θ) = E ∆ * -(k+ρ) * (I r -θ) . = E (∆ * 1 (I r -θ)) k r-1 -kr-1 2 . . . ∆ * r-1 (I r -θ) k 1 -k 2 -1 2 (∆ * r (I r -θ)) -k 1 . = r-1 i=1 1 ∆ * i (I r -θ) 1 2 E   r i=1 ∆ * i-1 (I r -θ) ∆ * i (I r -θ) k r-i+1   . = r-1 i=1 1 ∆ * i (I r -θ) 1 2 E r i=1 ∆ * -e i (I r -θ) k r-i+1 .
It follows that for all θ ∈ I r -Ω,

L µ (θ) = r-1 i=1 1 ∆ * i (I r -θ) 1 2 E r i=1 e k r-i+1 log(∆ * -e i (Ir-θ)) . (3.26) Setting α(θ) = log(∆ * -er (I r -θ)), log(∆ * -e r-1 (I r -θ)), . . . , log(∆ * -e 1 (I r -θ)) , then (3.26) becomes L µ (θ) = r-1 i=1 1 ∆ * i (I r -θ) 1 2
E e α(θ),k .

As k has the multivariate Poisson distribution ν, then

L µ (θ) = r-1 i=1 1 ∆ * i (I r -θ)
According to (3.14), we can write for all θ ∈ I r -Ω,

L µ (θ) = r-1 i=1 1 ∆ * i (I r -θ) 1 2 r i=1 e λ i   e log ∆ * -e r-i+1 (Ir -θ) -1   . Therefore L µ (θ) = r-1 i=1 1 ∆ * i (I r -θ) 1 2 r i=1 e λ i ∆ * -e r-i+1
(Ir-θ) -1

.

(3.27)

On the other hand, using (2.5), we have

∆ * κ r-1 (I r -θ) = r-1 i=1 1 ∆ * i (I r -θ) 1 2
.

Inserting this in (3.27), we get (3.25). 2

Theorem 3.4 The domain of the means of the natural exponential family F = F (µ) generated by the mixture µ is Ω.

Proof

From (3.25), we deduce that for all θ ∈ Θ(µ) = I r -Ω,

k µ (θ) = r i=1 λ i ∆ * -e r-i+1 (I r -θ) -1 + log(∆ * κ r-1 (I r -θ)). (3.28)
As for all i ∈ {1, . . . , r}, the map

ϕ i : x -→ log ∆ * i (x) is differentiable on Ω and ϕ ′ i (x) = (P * i (x)) -1 , (3.29)
then, for all i ∈ {1, . . . , r}, we have

∆ * -e i (x) ′ = ∆ * i-1 (x) ∆ * i (x) ′ = ∆ * -e i (x) (P * i-1 (x)) -1 -(P * i (x)) -1 , (3.30)
and for r ≥ 2, we have

log(∆ * κ r-1 (x)) ′ = - 1 2 r-1 i=1 (P * i (x)) -1 (3.31)
Differentiating (3.28) and taking into account (3.30) and (3.31), we get

k ′ µ (θ) = r i=1 λ r-i+1 ∆ * -e i (I r -θ) -λ r-i ∆ * -e i+1 (I r -θ) + 1 2 (P * i (I r -θ)) -1 - 1 2 (P * r (I r -θ)) -1 , (3.32)
where λ 0 = 0. Let θ ∈ I r -Ω, and let u be the unique element of T + l such that I rθ = u * -1 (I r ). Then, for all i ∈ {1, . . . , r}, we have

(P * i (I r -θ)) -1 = P * i (u * -1 (I r )) -1 = P * i (u(I r )) -1 -1 .
According to (2.7), this implies that for all i ∈ {1, . . . , r},

(P * i (I r -θ)) -1 = u   r j=r-i+1 c j   . (3.33)
On the other hand, using (2.8), we can write for all i ∈ {1, . . . , r},

∆ * -e i (I r -θ) = ∆ * -e i u * -1 (I r ) = ∆ e * i (u(I r )) = ∆ e r-i+1 (u(I r )) = ∆ r-i+1 (u(I r )) ∆ r-i (u(I r )) .
This with (2.9) imply that for all i ∈ {1, . . . , r},

∆ * -e i = u 2 r-i+1 , (3.34)
where for all i ∈ {1, . . . , r}, u i are defined in (2.10).

Using (3.33) and (3.34), we deduce from (3.32) that

k ′ µ (θ) = r i=1 λ r-i+1 u 2 r-i+1 -λ r-i u 2 r-i + 1 2 u   r j=r-i+1 c j   - 1 2 u r i=1 c i .
This after a standard calculation, gives

k ′ µ (θ) = r i=1 λ i u 2 i + i -1 2 u(c i ) = u r i=1 a i (θ)c i , (3.35)
where

a i (θ) = i -1 2 + λ i u 2 i . (3.36)
As the a i are strictly positive, we deduce that

k ′ µ (Θ(µ)) = k ′ µ (I r -Ω) ⊆ Ω. (3.37)
Conversely, consider y ∈ Ω, then using the Cholesky decomposition, there exists a unique w ∈ T + l such that

y = w(I r ) = w r i=1 c i = w P r i=1 1 a i (θ) c i r i=1 a i (θ)c i . Let θ such that I r -θ = u * -1 (I r ) ∈ Ω, where u = w r i=1 1 a i (θ) c i . Then y = u r i=1 a i (θ)c i .
This, using (3.35), gives

y = k ′ µ (θ) ∈ k ′ µ (I r -Ω). Therefore Ω ⊆ k ′ µ (I r -Ω) = k ′ µ (Θ(µ)
). This with (3.37) imply that the domain of the means of the NEF F = F (µ) is

M F = k ′ µ (Θ(µ)) = Ω. 2
The following theorem gives the variance function of the natural exponential family F = F (µ).

Theorem 3.5 For all m ∈ Ω,

V F (m) = - 1 2 P r i=1 1 b i (m) P * r-i+1 (m -1 ) -1 -P * r-i (m -1 ) -1 + r i=1 λ r-i+1 ∆ e r-i+1 (m) b r-i+1 (m) - λ r-i ∆ e r-i (m) b r-i (m) + 1 2 ×   P   r j=r-i+1 1 b j (m) P * r-j+1 (m -1 ) -1 -P * r-j (m -1 ) -1     + r i=1 λ r-i+1 ∆ e r-i+1 (m) (b r-i+1 (m)) 3 P * i (m -1 ) -1 -P * i-1 (m -1 ) -1 ⊗ P * i (m -1 ) -1 -P * i-1 (m -1 ) -1 , (3.38)
where for all i ∈ {1, . . . , r}, b i (m

) = i -1 4 + i -1 4 2 + λ i ∆ e i (m) .
Usually, for the calculation of the variance function, we set m = k ′ µ (θ) and we determine its reciprocal θ = ψ µ (m). This is difficult to do in the present situation, however, we are able to determine (P * i (I rψ µ (m))) -1 , ∆ * -e i (I rψ µ (m)), and a i (ψ µ (m)) where a i is defined in (3.36). It follows that

Theorem 3.6 For all i ∈ {1, . . . , r}, i) a i (ψ µ (m)) = i -1 4 + i -1 4 2 + λ i ∆ e i (m). ii) (P * i (I r -ψ µ (m))) -1 = r j=r-i+1 1 a j (m) P * r-j+1 (m -1 ) -1 -P * r-j (m -1 ) -1 . iii) ∆ * -e i (I r -ψ µ (m)) = ∆ e r-i+1 (m) a r-i+1 (m) . ( 3
v = u r i=1 a i (ψ µ (m)) c i .
Or equivalently,

u = v r i=1 1
a i (ψ µ (m)) c i . (3.40)
On the other hand, using (2.9), (3.36) becomes

a i (ψ µ (m)) = i -1 2 + λ i ∆ e i (u(I r )).
Then using (3.40), we can write

a i (ψ µ (m)) = i -1 2 + λ i ∆ e i     v r j=1 1 a j (ψ µ (m)) c j     r j=1 1 a j (ψ µ (m)) c j v *     . = i -1 2 + λ i ∆ e i   v r j=1 1 a j (ψ µ (m)) c j v *   . = i -1 2 + λ i ∆ e i   v   r j=1 1 a j (ψ µ (m)) c j     .
Therefore a i (ψ µ (m)) satisfies the equation

a i (ψ µ (m)) = i -1 2 + λ i v 2 i a i (ψ µ (m)) , (3.41)
where v i is defined in (2.10).

As a i (ψ µ (m)) > 0, we deduce that

a i (ψ µ (m)) = i -1 4 + i -1 4 2 + λ i v 2 i .
On the other hand, since m = v(I r ), then using (2.9), we have that

v 2 i = ∆ e i (m).
Consequently, for all i ∈ {1, . . . , r},

a i (ψ µ (m)) = i -1 4 + i -1 4 2 + λ i ∆ e i (m).
ii) With the notations used above, we can write for all i ∈ {1, . . . , r},

(P * i (I r -ψ µ (m))) -1 = P * i (u * -1 (I r )) -1 . = u   r j=r-i+1 c j   . =   v r i=1 1 a i (ψ µ (m)) c i     r j=r-i+1 c j     r i=1 1 a i (ψ µ (m)) c i v *   . = v r j=r-i+1 1 a j (ψ µ (m)) c j v * . = v   r j=r-i+1 1 a j (ψ µ (m)) c j   Thus (P * i (I r -ψ µ (m))) -1 = r j=r-i+1 1 a j (ψ µ (m)) v(c j ). ( 3

.42)

As m = v(I r ), then for all j ∈ {1, . . . , r}, we have

v(c j ) = v   r i=j c i - r i=j+1 c i   = P * r-j+1 (m -1 ) -1 -P * r-j (m -1 ) -1
.

Inserting this in (3.42), we deduce that

(P * i (I r -ψ µ (m))) -1 = r j=r-i+1 1 a j (ψ µ (m)) P * r-j+1 (m -1 ) -1 -P * r-j (m -1 ) -1 . (3.43) Consequently (P * i (I r -ψ µ (m))) -1 -P * i-1 (I r -ψ µ (m)) -1 = 1 a r-i+1 (ψ µ (m)) P * i (m -1 ) -1 -P * i-1 (m -1 ) -1 .
(3.44) iii) According to (2.5) we have

∆ * i (I r -ψ µ (m)) = ∆ * i (u * -1 (I r )) = ∆ * ϑ i (u * -1 (I r )),
where ϑ i = i j=1 e j . Using (2.8) and (3.40), we deduce that = ∆ e r-i+1 (m) a r-i+1 (ψ µ (m))

∆ * i (I r -ψ µ (m)) = ∆ -ϑ * i (u(I r )). = ∆ -ϑ * i (uu * ). = ∆ -ϑ * i     v r j=1 1
. 2

We are now in position to give the variance function of the natural exponential family F stated in Theorem 3.5.

Proof of Theorem 3.5

We have that for all m ∈ M F = Ω,

V F (m) = k ′′ µ (ψ µ (m)).
Differentiating (3.32) and using (3.30) and the fact that for all i ∈ {1, . . . , r} and x ∈ Ω, (P * i (x)) -1 ′ = -P (P * i (x)) -1 , we get for all θ ∈ I r -Ω, k ′′ µ (θ) = -

  α, x) -I(α, x) sin(αΠ) , α = 0, ±1, ±2, . . . .

.21) 2 3. 1 2 ,

 212 The probability density of the mixture of the Riesz distribution with respect to the multivariate PoissonLet ρ = 0, 1 2 , . . . , r-1 k = (k 1 , k 2 , . . . , k r ) be in IN r and let R k = R(k + ρ, I r ).Suppose that k = (k 1 , k 2 , . . . , k r ) has the multivariate Poisson distribution ν defined in(3.13). For simplicity, in what follows, we will denote by µ the mixture of R k by ν. The following theorem gives the expression of µ in terms of the modified Bessel function of the first kind.

  As from Theorem 3.4, m = k ′ µ (θ) is in Ω, there exists a unique v ∈ T + l such that m = v(I r ). According to (3.35), we have v(I r ) = m = u r i=1 a i (ψ µ (m))c i = u P r i=1 a i (ψ µ (m)) c i r i=1 c i . Therefore v(I r ) = u P r i=1 a i (ψ µ (m)) c i (I r ) .

1 a. 9 )

 19 a j (ψ µ (m)) j (ψ µ (m))c j v * rψ µ (m)) = ∆ -ϑ * It follows that for all i ∈ {1, . . . , r}, ∆ * -e i (I rψ µ (m)) = ∆ * i-1 (I rψ µ (m)) ∆ * i (I rψ µ (m,we deduce that for all i ∈ {1, . . . , r},∆ * -e i (I rψ µ (m)) = v 2 r-i+1 a r-i+1 (ψ µ (m)) . Therefore ∆ * -e i (I rψ µ (m)) = ∆ r-i+1 (m) a r-i+1 (ψ µ (m))∆ r-i (m).

2 P 1 . 2 P 2 P 1 .

 21221 i+1 ∆ * -e i (I rθ)λ r-i ∆ * -e i+1 (I rθ) +1 (P * i (I rθ)) -1 + r i=1 λ r-i+1 ∆ * -e i (I rθ) (P * i (I rθ)) -1 -P * i-1 (I rθ) -1 ⊗ (P * i (I rθ)) -1 -r i=1 λ r-i ∆ * -e i+1 (I rθ) P * i+1 (I rθ) -1 -(P * i (I rθ)) -1 ⊗ (P * i (I rθ)) -It follows that for all θ ∈ I r -Ω,k ′′ µ (θ) = -1 (P * r (I rθ)) -1 + r i=1 λ r-i+1 ∆ * -e i (I rθ)λ r-i ∆ * -e i+1 (I rθ) + 1 (P * i (I rθ)) -1 + r i=1 λ r-i+1 ∆ * -e i (I rθ) × (P * i (I rθ)) -1 -P * i-1 (I rθ) -1 ⊗ (P * i (I rθ)) -1 -P * i-1 (I rθ)-We need only to replace θ by ψ µ (m), then insert (3.39), (3.43) and (3.44) to get the expression of the variance function of F = F (µ) given in (3.38). 2

L ν (α(θ)) .


Patrick Oswald
email: patrick.oswald@ens-lyon.fr

Alain Dequidt

Andrzej Zywocinski

Andrzej Żywociński

Sliding planar anchoring and viscous surface torque in a cholesteric liquid crystal

Keywords: numbers: 61, 30, Hn, 42, 70, Df

come

Sliding planar anchoring and viscous surface torque in a cholesteric liquid crystal

I. INTRODUCTION

In a nematic liquid crystal, the rod-like molecules tend to have the same direction, their centers of mass being distributed randomly. Order is purely orientational and characterized by a unit vector n parallel to the average orientation of the molecules (with n ⇔ -n). If one adds a chiral impurity to a nematic phase, one usually obtains a cholesteric phase in which the director turns around a space direction called the helical axis [START_REF] Oswald | Nematic and Cholesteric Liquid Crystals: Concepts and Physical Properties Illustrated by Experiments[END_REF]. The distance over which the director rotates by 2π is called the cholesteric pitch. As we shall see later, the pitch can sometimes strongly depend on temperature. One important property of these materials is their ability to be oriented by the surfaces in contact with them. In particular, considerable effort has been expended in the past to find surface treatments allowing for strong planar or homeotropic anchoring of the molecules. In the first case, the molecules orient parallel to the surface, whereas in the second case, the molecules are perpendicular to it. These two anchorings are characterized by a single direction named "surface easy axis" n s ."Strong" means that the molecules shift very little from n s when the director field is deformed in the bulk under the action of an external mechanical, electric or magnetic torque. In this "geometric" limit, the surfaces play a passive role as they only fix the orientation of the molecules in contact with them. These boundary conditions were used in most experiments till recently and continue to play a major role in display applications. In particular, the strong planar anchoring is widely used in making the twisted or super-twisted nematic display.

Other types of anchoring exist [START_REF] Jérôme | [END_REF]. For instance, oblique monostable or bistable anchorings have been observed on evaporated SiO layers [3] or on cleaved surfaces of crystals such as mica [4] with the possibility of passing from one to another by changing a physical parameter such as the temperature or the humidity (for a review about anchoring transitions, see chapter V of Ref. [START_REF] Oswald | Nematic and Cholesteric Liquid Crystals: Concepts and Physical Properties Illustrated by Experiments[END_REF]). More recently, planar and oblique azimuthally degenerate anchorings have also been reported in the literature. Different techniques have been proposed to reach such boundary conditions. One of them has consisted of adding to the liquid crystal a small amount of oligomeric molecules [5]. This method was shown to lead to very small zenithal (out of substrate plane) and azimuthal (in plane) anchoring energies with the liquid crystal 5CB. Another technique consists of depositing a "thick" layer (more than 100 Å, typically) of a surfactant belonging to the family of the organofunctional silanes. One of them, the (3-glycidoxypropyl) trimethoxysilane (also called 3-GPS or GLYMO) gives a planar degenerated anchoring with the liquid crystal 5CB [6]. One can also graft on the surface highly mobile polymer chains. For instance, grafted polystyrene was shown to lead to a conicallydegenerated anchoring of the nematic 5CB with a pretty small zenithal anchoring energy [7]. Thin layers of the photopolymer NOA60 was also used recently to produce a planar anchoring with an ultraweak azimuthal anchoring [8]. These new anchorings are interesting for two reasons:

-first, they open the possibility of making new and more performing displays in which the surface now plays an active role, the anchoring conditions changing when switching between two states of the display [9][10][11];

-second, they allow to perform new experiments in the field of fundamental physics.

A good example was the first direct observation of the smectic blocks in a TGB A phase. In this experiment, the GLYMO surface treatment was used [12]. Another example was the experimental evidence of backflow effects during the collapse of two disclination lines [13]. In this case, the surfaces were treated with the UV glue NOA60. It must be noted that in this experiment, an electric field was used to move the lines, which shows that the anchoring was not completely sliding.

In the present paper, we describe a new surface treatment we recently used to evidence the thermomechanical Lehmann effect in cholesteric liquid crystal [14,[START_REF] Dequidt | Lehmann effect in a compensated cholesteric liquid crystal: experimental evidence with fixed and gliding boundary conditions[END_REF]. This surface treatment turned out to give planar and sliding anchoring, which was essential for observing the Lehmann effect. In our case, it consisted of spin coating a thin layer of the polymercaptan hardener of an epoxy resin (Structuralit 7).

The goal of the paper is to establish the unique properties of this new surface treatment as it gives a planar alignment of the molecules on the surface with a strong zenithal anchoring energy, while allowing the molecules to rotate viscously on the surface (which implies a zero or extremely small azimuthal anchoring energy). To show these results we conducted microscopic observations of defect annihilations and capacitance measurements. We also developed a new thermo-optical method to determine the liquid crystal surface viscosity γ S , a quantity difficult to measure and very little documented in the literature in comparison with the anchoring energy.

The plan of the article is the following. In section II, we describe the sample preparation and we show from microscopic observation of the optical texture and from capacitance measurements that the anchoring of the molecules on the polymercaptan is planar and sliding. In section III, we measure the bulk rotational viscosity γ 1 and show that the presence of the polymeric layer does not change its value. In section IV, we show how to measure the surface viscosity γ S of the liquid crystal by measuring the variations of the optical transmittance of samples of different thicknesses during temperature ramps. The measured value of γ S is discussed within a simple model in section V, while in section VI, we discuss the problem of the annihilation of two ±1/2 disclination lines. Finally, conclusions are drawn in section VII.

II. SAMPLE PREPARATION AND CAPACITANCE MEASUREMENTS

The liquid crystal chosen was a mixture of 4-n-octyloxy-4'-cyanobiphenyl (8OCB from Synthon Chemicals GmbH &Co) and of cholesteryl chloride (CC from Aldrich) in proportion 1:1 in weight with a compensation point at T c = 59 • C and a clearing point at 66.5 • C. We recall that at T c , the equilibrium twist vanishes and changes sign. The phase has thus a nematic-like structure at this particular temperature. The 8OCB was purified by one of us (AZ) and the CC was used without further purification. The mixture was filtered in the cholesteric phase through 0.2 µm nylon filter to remove dust particles.

The samples were prepared between two glass plates treated either for strong planar unidirectional anchoring or for planar sliding anchoring.

The first anchoring is classical and was obtained by covering the surface with a rubbed polyimide layer baked at 300 • C during 2 hours (ZLI 2650 from Merck). To realize the second anchoring, we treated the surface with the polymercaptan hardener of an epoxy glue (Structalit 7 from Eleco). This polymer was first dissolved in a ketone, the 2-butanone (5% in mass of hardener). The obtained solution was then filtered through 0.2 µm PTFE membrane to eliminate dust particles and spread by spin-coating on the plate at 500 rpm for 1 mn. In this way, the polymercaptan harderner formed a homogeneous thin layer wetting perfectly the surface (up to 100 • C, at least). Its thickness was measured by Michelson interferometry and was found to be of the order of 0.2 µm. In addition, the layer was held at 60 • C for one hour under vacuum in order to completely evaporate the ketone. This precaution is important to not pollute the liquid crystal [16].

Finally, nickel and tungsten wires of calibrated diameters were used as spacers to fix the sample thickness.

In order to prove that this new surface treatment gives a sliding planar anchoring, we performed the two following experiments.

In the first one, we made a 10 µm-thick sample with the two glass plates treated with the polymercaptan hardener.

The liquid crystal was first introduced by capillarity in the isotropic phase and then cooled down into the cholesteric phase. As a result we observed the formation of a "Schlieren" texture containing a large quantity of ±1/2 disclination lines perpendicular to the surfaces (Fig. 1a). This observation clearly shows that the anchoring is planar (and not oblique) and azimuthally degenerate since it is the only orientation allowing such defects. Note that ±1 defects are also present, but they are much less numerous that the previous because they cost more elastic energy [START_REF] Oswald | Nematic and Cholesteric Liquid Crystals: Concepts and Physical Properties Illustrated by Experiments[END_REF]. In addition, we observed that isolated pairs of opposite defects were able to spontaneously annihilate (Fig. 1b). This is a clear indication that the anchoring is sliding, as otherwise the defects could not move. This is compatible with a zero (or extremely small) azimuthal anchoring energy and the absence of memory effects, at least over the time scale of our experiments, i.e. 2-3 days. In this respect, our anchoring seems to be more sliding than that reported in Ref. [5] or in Ref. [13] (in this work, in particular, where the UV glue NOA60 was used, it was necessary to apply a small horizontal electric field to observe the annihilation of the defects). We also observed that defects of positive strength moved faster than defects of negative strength, in agreement with previous experiments [13,17] and theoretical works [18][19][20]. Finally, we emphasize that the annihilation is a slow process in spite of the sliding anchoring as long as the distance between the two defects is larger than a few tens of a micrometer. This point will be discussed in detail in Section VI.

To confirm that the anchoring was planar, we performed capacitance measurements. More precisely, we made two 10 µm-thick samples, the first one treated for strong planar anchoring on one plate and for sliding anchoring on the opposite one (called sample of type I in the following), whereas the second sample was treated for strong planar (and parallel) anchoring on both sides (called sample of type II in the following). In both cases, the glass plates were coated with ITO layers to impose the electric field. In Fig. 2, we show the capacitances of the two cells measured at the compensation temperature as a function of the applied voltage at frequency f = 10 kHz. In the figure, the capacitances C are normalized to their values C ⊥ measured below the threshold of Fredericksz instability V c (of the order of 0.92 Vrms in both samples, which proves, by the way, that the presence of polymercaptan hardener does not change the value of the splay constant K 1 as well). Within experimental errors, the two curves superimpose. In addition, measurements of the capacitances of the cells before and after filling with the liquid crystal led to similar values (within 1%) of ε ⊥ (4.52 and 4.50, respectively), in very good agreement with the value given previously (4.5±0.2 [14,[START_REF] Dequidt | Lehmann effect in a compensated cholesteric liquid crystal: experimental evidence with fixed and gliding boundary conditions[END_REF]).

From this experiment, we concluded that the anchoring of the liquid crystal on the polymercaptan hardener is planar and rather strong zenithally since it behaves like the polyimide layer up to an electric field of 2 V/µm. More precisely, we estimated a lower bound for the zenithal anchoring energy W θ by noting that at electric field E max = 2 V/µm, the liquid crystal was not fully realigned by the field close to plate treated with the polymercaptan hardener. As this phenomenon occurs for electric fields larger than the saturation field

E s = W θ / √ ε 0 ε a K [21]
(with ǫ a the dielectric anisotropy and K the average of the splay and bend elastic constants), we deduced that

W θ > E max √ ε 0 ε a K since E max

III. ROTATIONAL VISCOSITY γ1

In practice, the melting temperature of the cholesteric phase is lower by typically 1 • C in samples treated for sliding anchoring. In contrast, the compensation temperature does not change within ±0.2 • C. This suggests that a small amount of the polymercaptan hardener dissolves in the liquid crystal. In the following section, we shall need the value of the bulk rotational viscosity γ 1 . For this reason and to investigate the influence of the dissolved polymer, we measured γ 1 in planar samples of the two types (as defined in the previous section) at the compensation temperature. The method used was classical and consisted of first destabilizing the sample with a large electric field (20 Vrms, f = 10 kHz) and then, of observing the relaxation of the distorted director field after the electric field was removed. In practice, the sample is placed between crossed polarizers at 45 • of the anchoring direction and its optical transmittance is measured by a photodiode connected to a memory oscilloscope. Curves obtained with samples of types I and II are shown in Fig. 3. To fit these curves, we took advantage that we previously measured the elastic constants, the dielectric constants and the optical indices of the liquid crystal at the compensation point: 55 and n e = 1.64 [14,[START_REF] Dequidt | Lehmann effect in a compensated cholesteric liquid crystal: experimental evidence with fixed and gliding boundary conditions[END_REF]. Using these data, it was possible to solve numerically the governing equations for the director field (in anisotropic elasticity) and to then calculate the optical transmittance. We took as fit parameters the sample thickness d and the classical relaxation time τ = γ1d 2 π 2 K1 . Two fits are shown respectively in Fig. 3(a) for a type I sample and in Fig. 3(b) for a type II sample. In the first case, we found d = 29.2 µm and τ = 1.91 s, while d = 30.5 µm and τ = 2.07 s in the second sample. In these two examples, the fitted thickness was very close to the nominal one (30 µm). We thus calculated γ 1 by taking the value of the thickness given by the fit and K 1 = 3.4 × 10 -12 N [14,[START_REF] Dequidt | Lehmann effect in a compensated cholesteric liquid crystal: experimental evidence with fixed and gliding boundary conditions[END_REF]. This procedure gave the same value in type I and type II samples: γ 1 = 0.075 ± 0.009Pa.s. We performed similar measurements in 20 µm-thick commercial type II cells (from Instec, Inc) and found again a similar value within experimental errors. In conclusion, the rotational viscosity γ 1 does not chan-ge in a measurable way in the presence of the polymercaptan hardener. This result agrees with previous conclusions of Jǎkli et al [22] about the role in the value of γ 1 of a polymer dissolved in the liquid crystal. In the following, we shall take γ 1 = 0.075 ± 0.009 Pa.s.

K 1 = 3.4 × 10 -12 N, K 3 = 5.9 × 10 -12 N, ε = 9.4, ε ⊥ = 4.5, n o = 1.

IV. SURFACE VISCOSITY γS

In practice, the director does not rotate freely at the surface of the liquid polymercaptan layer, but experiences a viscous surface torque which can be written in the usual form [START_REF] De Gennes | The Physics of Liquid Crystals[END_REF]

Γ S = -γ S n × ∂ n ∂t (1)
The surface viscosity γ S has the dimension of a bulk viscosity times a length. This length l S characterizes the distance over which the liquid crystal interacts with the polymer layer. It will be discussed in Section 5. In the next three subsections, we successively describe the principle of the experiment, the technique of measurements and the experimental results.

A. Principle of the experiment

The experiment consisted of measuring the optical transmittance in monochromatic light (λ = 546 nm) of a type I sample as a function of the temperature when it was changed at a constant rate. The sample was placed between crossed polarizer and analyzer with the polarizer parallel to the anchoring direction imposed on the lower glass plate.

Due to the sliding anchoring on the upper plate, the transmittance changed with the temperature and vanished (by assuming an adiabatic rotation of the plane of polarization of the light) when the director on this surface was perpendicular to the analyzer. We checked numerically that this optical condition was fulfilled at a temperature T 0 (close to T c ) being a function of the imposed temperature ramp r = dT /dt (with T 0 → T c when r → 0). The thus consisted of measuring T 0 (r) in samples of different thicknesses d. In the following, we show that the derivative dT 0 /dr is a well-defined function of d, γ 1 and γ S .

In order to calculate this quantity, we first recall the governing equations of the problem. Let ϕ be the angle between the director and the anchoring direction on the bottom plate. Because the director remains parallel to the glass plates, it is only function of z and t (with the z axis perpendicular to the plates). If, in addition, the temperature and thus, the equilibrium twist q, can be supposed independent of z, an assumption we shall justify in the Appendix, then the bulk torque equation simply reads

γ 1 ∂ϕ ∂t = K 2 ∂ 2 ϕ ∂z 2 (2)
At the bottom plate, the director orientation is fixed, so that

ϕ = 0 at z = 0 (3)
while at the top plate, the anchoring is sliding, imposing

γ S ∂ϕ ∂t = -K 2 ∂ϕ ∂z -q[T (t)] at z = d (4)
where q[T (t)] is the equilibrium twist of the cholesteric.

we impose a constant temperature ramp r = dT /dt, so that

T (t) = T c + rt (5)
addition, we know from previous measurements that q is, to a very good approximation, proportional to T -T c in vicinity of the compensation point [14,[START_REF] Dequidt | Lehmann effect in a compensated cholesteric liquid crystal: experimental evidence with fixed and gliding boundary conditions[END_REF]. This allows us to write that qd = ωt with ω = rd dq dT

Let us now define the angle δ(z, t) = ϕ(z, t) -qz characterizing the shift to the perfect helix. According to Eqs. 1 and 6, this angle must satisfy the following equation

γ 1 ∂δ ∂t = K 2 ∂ 2 δ ∂z 2 -γ 1 ω z d (7)
In the stationary regime (δ independent of t), this equation has the general solution:

δ = ω 3ω b z d 3 +az + b (8)
where ω b = 2K2 γ1d 2 is a bulk relaxation rate. Constants b and a are respectively given by the boundary conditions 3 and 4: b = 0 and a = -ω d

1 ω b + 1 ω S (9)
where ω S = K2 γ S d is a surface relaxation rate. Coming back to angle ϕ, the general reads in the stationary regime:

ϕ = ω 3ω b z d 3 +ω t - 1 ω b + 1 ω S (10)
Close to T c , the optical transmittance vanishes when ϕ(d, t) = 0. This condition is fulfilled at time

t 0 = 1 ω S + 2 3ω b
corresponding to temperature T 0 = T c + r 1 ω S + 2 3ω b according to Eq. 6. Finally we obtain the researched quantity:

dT 0 dr = 1 ω S + 2 3ω b = γ S K 2 d + γ 1 3K 2 d 2 (11)
In the following section, we show how to measure this quantity.

B. Experiment

We used a Mettler oven to impose temperature ramps to the samples. The oven was placed on the stage of a polarizing Leica microscope and the optical transmittance was measured with a photodiode. To measure the temperature of the sample we used a local sensor consisting of a 10 µm in diameter tungsten wire placed between two glass plates a few mm apart from the zone of interest. Its resistance was measured via a four-wire method with a Keihtley 2001 multimeter interfaced with a PC and was found to linearly depend on the temperature. The temperature stability of this home-made sensor was excellent as no visible drift was observed over many days. An example of a measured temperature profile is shown in Fig. 4. For slow ramps (inferior to 3 • C/min in absolute value), we found very good agreement between nominal and measured values. For faster ramps, small differences were observed. For this reason, we measured systematically the temperature ramp to minimize errors.

An attentive reader could object that the temperature measured in this way could be different from the temperature in the middle of the sample where optical measurements were performed. For this reason, we realized a dummy sample with one wire on the side (as in usual samples) and another in the middle. We then recorded as a function of time the temperatures given by the two wires with a Keihtley 2001 multimeter equipped with a scanner card. In Fig. 5 we plotted the temperature of the side sensor as a function of the temperature of the middle sensor. ramps (±7 • C/min). This test proved that our sensor did give the temperature of the sample at the place where measurements were performed with a negligible error for slow ramps and an error that does not exceed ±0.1 • C for the fastest ramps. The next step was to determine for each sample its compensation temperature. The used method consisted of recording its optical transmittance as a function of temperature under a very slow temperature ramp (typically 0.2 • C/min).

Note we used two Keihtley multimeters interfaced with a PC to record the photodiode voltage (proportional to the optical transmittance) and the temperature given by the tungsten wire. At this speed, the cholesteric phase may be considered at equilibrium during the ramp and its optical transmittance is given by the following formula:

I I 0 = 1 2 -cos (2qd) κ 2 + cos 2qd √ 1 + κ 2 2(1 + κ 2 ) -sin (2qd) sin 2qd √ 1 + κ 2 2 √ 1 + κ 2 (12)
with κ = π∆n qλ (λ = 546 nm). A typical experimental curve is shown in Fig 6 . To fit it to the previous law, we chose as free parameters the compensation temperature T c (knowing that the equilibrium twist is given by q(µm -1 ) = 0.1365(T -T c )+0.00284(T -T c ) 2 , with the temperatures in • C [14,[START_REF] Dequidt | Lehmann effect in a compensated cholesteric liquid crystal: experimental evidence with fixed and gliding boundary conditions[END_REF]), the sample thickness d, the intensity I 0 and the birefringence (which we took in the form ∆n = ab(T -T c )). This led to T c = 58.9 • C, I 0 = 16.9mV, d = 23.8µm, and ∆n = 0.090 -0.0029(T -T c ). Note that these values of the thickness and of the birefringence at the compensation point are very close to the expected ones: 25µm for the thickness and 0.09 for the birefringence [14,[START_REF] Dequidt | Lehmann effect in a compensated cholesteric liquid crystal: experimental evidence with fixed and gliding boundary conditions[END_REF]. In the following, we took for the thickness the value given by the fit.

Once the compensation temperature and the thickness of a sample were measured, we systematically recorded its transmittance curves for different temperature ramps. Two curves measured for two successive ramps of opposite signs are shown in Fig. 7. As expected the curve obtained by cooling is shifted to the left (towards lower temperatures), while the other obtained by heating is shifted to the right (towards higher temperatures). From these curves, we measured temperatures T 0 (r) by fitting the corresponding minima of the transmittance curve with parabolas (see in Points are experimental and the solid line is the best fit to Eq. 12 .

Fig. 7). Curve T 0 (r) is shown in Fig. 8. It is well fitted by a linear law the slope of which gives the researched quantity dT 0 /dr defined in Eq. 11. The experiment was then repeated for samples of different thicknesses. Our results are given in the next subsection.

C. Results

Fig. 9 collects all our experimental results obtained with samples of thicknesses ranging between 7 and 33 µm.

In practice, it was not really possible to perform the experiment with thicker samples because of a spontaneous destabilization of the helical structure into cholesteric fingers when the temperature was changed (even as slowly as

±1 • C/min.
In order to find the value of the surface viscosity, we fitted this experimental curve to the theoretical law Eq. 11 by taking for the value of the bulk rotational viscosity γ 1 that measured previously: 0.075 ± 0.009 Pa.s and for the twist 14,[START_REF] Dequidt | Lehmann effect in a compensated cholesteric liquid crystal: experimental evidence with fixed and gliding boundary conditions[END_REF]. In this way, we found from the fit of the experimental data and by taking into account the uncertainties in the values of γ 1 and K 2 :

constant K 2 = 2.8 ± 0.2 × 10 -12 N [
γ S = 3.2 ± 0.8 × 10 -7 Pa.s.m
This value is discussed in the next section.

V. SIMPLE MODEL FOR THE SURFACE VISCOSITY

This value of the surface viscosity is very large. Indeed, let us consider the viscosity η of the polymercaptan hardener. Measurements in a rotating rheometer gave η ∼ 1 Pa.s at the compensation temperature (59 • C) for shear rates ranging between 0.1 and 10 s -1 . From these two quantities, we can form a length

l S = γ S η (13)
which turns out to be very large, of the order of 0.3 µm. This value is of the same order of magnitude as the thickness l P of the polymer layer we measured by Michelson interferometry.

This result thus suggests that the liquid crystal diffuses into the polymer layer over a typical distance l D . This distance must be smaller than l P in order that the liquid crystal molecules do not adsorb on the glass (no memory effects were observed over many days on condition that the polymercaptan layer be thick enough [START_REF]Samples prepared by using a concentration of 2% in mass of hardener in the ketone showed clear memory effects after one day at the compensation temperature[END_REF]).

To estimate l D , let us assume that each molecule dissolved in the polymer layer experiences a viscous torque Γ mol proportional to its rotational velocity φ. For a rod-like molecule of length L and diameter Φ, the viscous torque reads [START_REF] Larson | The Structure and Rheology of Complex Fluids[END_REF] Γ mol = πηL 3 φ 3(ln(L/Φ) -0.8) (14) Let n S be the number of molecules dissolved in the polymer layer per unit surface area. In a crude model, the surface torque Γ S introduced phenomenologically in Eq. 1 reads:

Γ S ∼ n S Γ mol (15)
which gives the surface viscosity

γ S ∼ n S πηL 3 3(ln(L/Φ) -0.8) (16)
Finally, writing that l D ∼ n S LΦ 2 , we obtain from the preceding equation

l D ∼ γ S η Φ L 2 ln L Φ -0.8 (17)
In practice L ∼ 30 Å, Φ ∼ 5 Åwhich gives from the measured value of γ S and η given before: l D ∼ 0.01 µm = 100 Å.

As expected, we find that l D is smaller than the thickness of the polymer layer, although this calculation obviously underestimates its value.

To summarize, the large value of l S defined in Eq. 13 is certainly a clear indication that the nematic liquid crystal dissolves partially in the polymer surface layer. On the other hand, l D must be smaller than the thickness of the polymer layer in order to prevent that the liquid crystal molecules adsorb on the glass. These conclusions are very similar to those drawn before by Vilfan et al [START_REF] Vilfan | [END_REF] who also measured by dynamic light scattering very large values of l S for the liquid crystal 5CB in contact with a photoaligning poly-(vinyl-cinnamate) layer.

VI. ANNIHILATION TIME OF TWO DISCLINATION LINES OF OPPOSITE SIGNS

Before concluding the article, let us return to the problem of the annihilation of the two ±1/2 disclination lines shown in Fig. 1b. Indeed, their annihilation may seem too long as it takes about 11.5 min while the initial distance between the two defects is of "only" 110 µm. Thus, one may wonder whether this observation is compatible with a sliding anchoring.

show it is indeed the case, let us estimate this time. A straightforward calculation neglecting the backflow effects and the material elastic anisotropy gives:

t annihil = γ ⋆ 1 4K r 2 0 ln r 0 r c (18)
where r 0 is the initial distance between the two defects, r c a core radius of molecular size, K = K1+K3 Pa.s.m, we calculate t annihil ≈ 16 min by taking r c ≈ 50 Å. This time is longer than that observed (11.5 min). This difference is nevertheless not surprising as we know from numerical simulations that backflow effects tend to reduce by a typical factor of 1.5 the annihilation time between the two defects [18,20]. This interpretation is reinforced by the important experimental fact that the +1/2 defect moves faster than the -1/2 defect, another phenomenon predicted by the theory [18][19][20]. We thus conclude that our observations on defect annihilation are compatible with a sliding anchoring of the molecules on the polymer layer.

VII. CONCLUSION

We have described a new surface treatment which allows us to obtain a sliding anchoring of the molecules of a compensated cholesteric mixture. We checked experimentally that this anchoring was efficient for other liquid crystals such as 5CB and 8CB, but we did not performed systematic measurements with these materials. By using the large variation of the cholesteric pitch around the compensation point, we were able to measure the surface viscosity of the liquid crystal. We emphasize that this experiment was made possible owing to the development of temperature sensors able to measure in-situ the sample temperature. We have found that the surface viscosity γ S was large, which we interpreted by assuming that the liquid crystal diffuses within the polymer layer over a characteristic length l D much larger than a molecular length. This contrasts with usual surface treatments (as, for instance, bare glass or covered with a SiO or a polymerized PVA or PI layer) where l D or more precisely l S (defined in this case as the ratio γ S /γ 1 ) is of the order of a molecular length [27]. In the latter case, the origin of the surface viscosity is certainly very different and was explained as due a backflow effect close to the surface [28]. Nevertheless this explanation cannot apply here as we are dealing with azimuthal (and not zenithal) molecular rotations. As a consequence, we only have twist deformations which do not generate any backflow. Finally, note that all our experiments were performed in the stationary regime (i.e., at times t ≫ 1/ω b and 1/ω S ), which allowed us to escape the complications caused by the incompatibility (at the initial time, when the ramp is started) between the bulk equation ( 2) and the surface equation (4) [29].

One of our objective in the future will be to improve the model of the surface viscosity. This could be done by using a tensorial description [30] of the Ericksen-Leslie theory of the nematohydrodynamics coupled with a Cahn-Hilliard description [31] of the nematic-polymer interface [32]. It would be also interesting for applications to better determine the zenithal anchoring energy, for instance, by measuring the saturation voltage above which the director is fully realigned by the field. Another crucial point for applications would be to study aging properties of this new anchoring over longer period of time than two or three days in order to check whether memory effects develop at long time. Such studies are planed in the future, in particular as a function of the thickness of the polymercaptan layer.

APPENDIX

In our experiments, the average sample temperature changes quickly during the steepest ramps. As a consequence the temperature can no longer be considered as constant along the z axis. In these conditions Eq. 2 is only approximate because it neglects the spatial variations of K 2 and q. As in practice K 2 changes much less rapidly than q, we shall only consider the q variation assuming K 2 = constant. In these conditions, a new term proportional to ∂q/∂z occurs in the bulk torque equation ( 2) which becomes

γ 1 ∂ϕ ∂t = K 2 ∂ 2 ϕ ∂z 2 -K 2 ∂q ∂z (A.1)
In order to determine whether the new term -K 2 dq dz is pertinent, let us calculate its order of magnitude with respect to the two others by using the simplified solution given by Eq. 10. By denoting by ∆T the typical temperature variation across the sample thickness d, we find: where D o = K2 γ1 is the orientational diffusivity. In our sample, D o ≈ 3.7 × 10 -11 m 2 s -1 . As for its thermal diffusivity, it must be of the same order of magnitude as in usual nematics: D T ≈ 10 -7 m 2 s -1 [33,34]. From these two values we can estimate that This fully justifies neglecting the term in dq dz in the bulk torque equation.

  FIG. 1: a) Texture observed at the compensation temperature after a quench from the isotropic liquid. b)Sequence of photographs showing the spontaneous collapse at of two ±1/2 disclination lines. Note the +1/2 defect moves faster than the -1/2 defect. Time is given in min at the bottom left hand side of the photographs. d = 10 µm. Crossed polarizers.

FIG. 2 :

 2 FIG. 2: Capacitance as a function of the applied voltage (f = 10 kHz) measured at the compensation temperature Tc in a standard planar sample and in a sample treated on one plate for sliding anchoring. The curves superpose within experimental errors.

FIG. 3 :

 3 FIG. 3: Optical transmittance measured at T = Tc between crossed polarizers at 45 • of the anchoring direction as a function of time after switching off the voltage. Points are experimental and correspond to several curves measured at different places in the sample. The solid line is the best fit to the numerical model. a) Type I sample; b) Type II sample.

FIG. 4 :

 4 FIG.4: Temperature profile measured in situ with the tungsten wire. The measured value of the temperature ramp is here slightly larger than its nominal value given by the Mettler oven. This difference increases when the ramp is faster, but never exceeds 5%.

FIG. 5 :FIG. 6 :

 56 FIG. 5: Temperature of the side wire as a function of the temperature of the middle wire. The curves measured at ±2 • C/min are superimposed (within the experimental noise) whereas those measured at ±7 • C/min are systematically shifted by about 0.1 • C. Note that these curves are noisier than in Fig. 4 because of the use of the scanner card.

FIG. 7 :

 7 FIG.7: Curves of optical transmittance measured between crossed polarizers as a function of temperature for two ramps of equal amplitude, but opposite signs. Temperatures T0 are obtained by fitting with parabolas (solid lines) the local minima.

FIG. 8 :

 8 FIG.8: Temperature T0 as a function of the temperature ramp r for a sample of a given thickness and its fit to a linear law.

FIG. 9 :

 9 FIG. 9: Derivative dT0/dr as a function of the sample thickness d. The solid line is the best fit to Eq. 11 by taking γ1 = 0.075 Pa.s.

2

  the average of the splay and bend constants and γ ⋆ 1 an effective viscosity taking into account the surface viscosity: γ ⋆ 1 = γ 1 + 2γ S /d. With the experimental values d = 10 µm, r 0 = 110 µm, K = 4.6 × 10 -12 N [14, 15], γ 1 = 0.075 Pa.s and γ S = 3.2 × 10 -7

2 (A. 5 )

 25 D T is the thermal diffusivity of the liquid crystal. From this equation, we find that in order of magnitude∂T ∂t = D T ∆T dThis allows us to calculate the ratio between the two terms:

Acknowledgments

This work has been supported by the Polonium Program N • 11622QC.