M Baqué Patrick

Universite Nice

Sophia Antipolis

Faculte De Medecine

Mme Askenazy-Gittard

Florence Pédopsychiatrie

M Barranger

Emmanuel Gynécologie

) M Bérard

Étienne Pédiatrie

) M Bongain

André Gynécologie-Obstétrique

Véronique Rhumatologie

) M Castillo

) M Chevallier

Patrick Radiologie

M Ferrero

Thierry Piche

Gastro

Dominique Hématologie

) M Rosenthal Éric

Mme Alunni Véronique

M Anty

Rodolphe Gastro

) M Bahadoran

Philippe Cytologie

Stéphanie Ophtalmologie

Sylvie Génétique

M Favre

Guillaume Néphrologie

) M Fournier

Jean-Paul Thérapeutique

Mme Giordanengo

Valérie Bactériologie

M Ambrosetti

Damien Cytologie

Mme Bernard-Pomier

Ghislaine Immunologie

M Fosse

Thierry Bactériologie

M Garraffo

Rodolphe Pharmacologie Fondamentale

Humbert Olivier Biophysique

Mme Lamy

Brigitte Bactérilogie

Mme Long-Mira

Elodie Cytologie

Marie -Noëlle

Mme Seitz-Polski Barbara

M Squara

Fabien Cardiologie

M Testa

Jean Épidémiologie

M Darmon

David Médecine

Mme Gros

Auriane Orthophonie

Mme Landi

Rebecca Anglais

Praticien Hospitalier

Universitaire M Durand

Matthieu Urologie

M Amiel

Jean M Gastaud

Pierre M Albertini

Marc M Gérard

Jean-Pierre M Balas

Daniel M Gillet

Jean-Yves M Batt

Michel M Grellier

Patrick M Blaive

Bruno M Grimaud

Dominique M Boquet

Patrice M Hofliger

Philippe M Bourgeon

André M Jourdan

Jacques M Boutté

Patrick M Lambert

Jean-Claude M Bruneton

Jean-Noël M Lazdunski

Michel Mme

Bussiere Françoise

M Lefebvre

Jean-Claude M Camous

Jean-Pierre M Le Fichoux

Yves M Canivet

Bertrand Mme Lebreton

Elisabeth M Cassuto

Jill-Patrice M Mariani

Roger M Chatel

Marcel M Masseyeff

René M Coussement

Alain M Mattei Mathieu

Mme Crenesse

Dominique M Mouiel

Jean M Darcourt

MYQUEL Guy Mme

Martine M Dellamonica

Pierre M Ortonne

Jean-Paul M Delmont

Jean M Pringuey

Dominique M Demard

François M Santini

Joseph M Desnuelle

Claude M Sautron

Jean Baptiste

M Dolisi

Claude M Schneider

Maurice Mme

Euller-Ziegler Liana

M Toubol

Jacques M Fenichel

Patrick M Tran

Dinh Khiem

M Franco

Alain M Van Obberghen

Emmanuel M Freychet

Pierre M Ziegler

Gérard M C U Honoraires

M Arnold

Jacques M Giudicelli

Jean M Basteris

Bernard M Magné

Jacques M Benoliel

José Mme Memran

Nadine Mlle Chichmanian

Rose-Marie M Mengual

Raymond Mme Donzeau

Michèle M Philip

Patrick M Emiliozzi

Roméo M Poirée

Jean-Claude M Franken

Philippe Mme

Roure Marie-Claire

M Gastaud

MAITRES DE CONFÉRENCES DES UNIVERSITÉS -PRATICIENS HOSPITALIERS

Keywords:

Objectives: To characterize cutaneous manifestations of neurofibromatosis type 2 (NF2) in a paediatric population, which are early manifestations of the disease, as ophthalmological features, which can precede neurosensory complications by several years.

Methods

: This observational, prospective and multicentric study was conducted from April 2019 to April 2020 in eight academic French hospitals. Patients ≤ 18 included fulfilled the Manchester diagnostic criteria or had a pathogenic mutation identified in the NF2 gene. All patients had a dermatological examination, guided by a standardized questionnaire, and a collection of anamnestic data.

Results: Twenty-one children were included (median age 13 years [range 2-18]; sex-ratio 11 boys/10 girls), including 18 severe forms and 16 sporadic cases. Except one patient completely asymptomatic, all of them presented at least one skin tumour (mean number of tumours 5.9

; median age at first signs 7 years [1-13]). In the other seventeen cases, diagnosis of NF2 resulted from neurosensory complications (n=10), family screening (n=4) or ocular signs (n=3). Before the NF2 diagnosis, 70% of children have at least one "undiagnosed" cutaneous tumour that did not lead to a specific management. Patients' dermatological examination also found café au lait macules (CALMs) (n=15; 71%) previously described in literature, hypopigmented macules (HPMs) (n=12; 57%), and purple reticulated macule (n=4; 19%).

Conclusion:

Dermatological and ophthalmic lesions are frequent and precocious in children with sporadic severe NF2, but rarely lead to the diagnosis because of lack of clear guidelines.

CALMs and HPMs also seem suggestive, and the sensitivity of reticulated capillary malformation-like lesions remains to be assessed by further studies.

PROFESSEURS ASSOCIÉS

M.

GARDON Un grand merci à vous tous.

A toutes les équipes soignantes rencontrées au cours de mon internat

A l'équipe de dermatologie pédiatrique : Christine, un grand merci pour votre enseignement, vos conseils, et votre bienveillance ! Céline, la meilleure infirmière que je connaisse, avec un coeur en or, merci de ton soutien, ton aide et de m'avoir fait partager ton savoir-faire. Je te souhaite le meilleur pour ta nouvelle vie. In this paediatric population, a precocious diagnosis is important to prevent the onset of neurological complications but is difficult, particularly in absence of a familial history.

Cutaneous or ophthalmological manifestations, that may precede VS or CNS tumours by several years, may contribute to an early diagnosis [START_REF] Anand | Diagnosis of sporadic neurofibromatosis type 2 in the paediatric population[END_REF][START_REF] Ruggieri | Earliest Clinical Manifestations and Natural History of Neurofibromatosis Type 2 (NF2) in Childhood: A Study of 24 Patients[END_REF][START_REF] Gugel | Presenting symptoms in children with neurofibromatosis type 2[END_REF]. However, apart from cutaneous schwannomas, the literature contains few descriptions of dermatological manifestations [START_REF] Mautner | Skin abnormalities in neurofibromatosis 2[END_REF][START_REF] Anand | Diagnosis of sporadic neurofibromatosis type 2 in the paediatric population[END_REF][START_REF] Gugel | Presenting symptoms in children with neurofibromatosis type 2[END_REF][START_REF] Casado-Verrier | Cutaneous and ophthalmic signs as a clue to early diagnosis of severe neurofibromatosis type 2: report of a novel mutation that predicts this poor prognosis[END_REF][START_REF] Plana-Pla | Diagnostic and Prognostic Relevance of the Cutaneous Manifestations of Neurofibromatosis Type 2[END_REF][START_REF] Ruggieri | Natural history of neurofibromatosis type 2 with onset before the age of 1 year[END_REF][START_REF] Rustad | Neurofibromatosis type 2: Multiple intra-dermal tumors in a toddler[END_REF].

The aim of this study was to characterize the dermatological manifestations of NF2 in a paediatric population.

MATERIALS AND METHODS

This observational, prospective and multicentric study was conducted from April 2019 to April 2020 in eight academic French hospitals. Patients of both sexes, ≤ 18 years old, fulfilling the Manchester diagnostic criteria (Table 1) or with a pathogenic mutation identified in the NF2

A Bilateral Vestibular Schwannoma

B First degree family history + unilateral Vestibular Schwannoma or 2 among :

Meningioma, schwannoma, glioma, neurofibroma, posterior subcapsular cataract.

C Unilateral Vestibular Schwannoma + 2 among :

Meningioma, schwannoma, glioma, neurofibroma, posterior subcapsular cataract.

D At least 2 meningiomas + 2 among :

Schwannoma, glioma, neurofibroma, posterior subcapsular cataract.

gene, were included after oral consent. Recruitment was by dermatologists from the Research Group of the French Society of Paediatric Dermatology, the French reference centre for NF2 and the French association of patients with neurofibromatosis.

All patients had a dermatological examination, with Wood's lamp examination, guided by a standardized questionnaire with an atlas of the various types of schwannomas according to the Mautner et al. [START_REF] Mautner | Skin abnormalities in neurofibromatosis 2[END_REF] (Supplementary data 2) and a collection of anamnestic data. According to the literature, the clinical phenotype was considered severe for children with non-VS neurological-related symptoms and mild for the others (10-12). In the same way, NF2 mutations were stratified by using the previously published and validated UK genetic severity score [START_REF] Halliday | Trends in phenotype in the English paediatric neurofibromatosis type 2 cohort stratified by genetic severity[END_REF]. Briefly, group 1A/1B represents no NF2 pathogenic variant detected in blood; groups 2A and 2B represent a mild or moderate, respectively, constitutional or mosaic NF2 pathogenic variant detected in blood; and group 3 represents a constitutional truncating pathogenic variant within exons 2-13.

The study was considered as non-interventional by our ethics committee and was registered a ClinicalTrials.gov (NCT03893643). All patients and/or their caregivers gave their written consent to use their pictures in publications.

RESULTS

Patients

We

Dermatological features

Twenty children had at least one skin tumour, which appeared before age 10 in 14 (66%) children and were congenital in 4 (19%). The mean number of tumours was 5.9 [range 1 -15] per patient, and 5 (23%) children had more than 10 skin tumours. Children with the moderate form had 0 to 3 tumours. Type A or AH tumours were the predominant subtype and were found in 15 (71%) children (Figure 1a Fifteen (71%) children had café-au-lait macules (CALMs). The mean number of CALMs was 2.4 [START_REF] Evans | Schwannomatosis: a genetic and epidemiological study[END_REF][START_REF] Evans | A genetic study of type 2 neurofibromatosis in the United Kingdom. II. Guidelines for genetic counselling[END_REF][START_REF] Evans | Neurofibromatosis type 2 (NF2): A clinical and molecular review[END_REF][START_REF] Evans | Neurofibromatosis Type 2[END_REF][START_REF] Ardern-Holmes | Neurofibromatosis Type 2: Presentation, Major Complications, and Management, With a Focus on the Pediatric Age Group[END_REF] per patient. They were congenital in 3 cases. The size was >15mm in 50%, borders were mainly ragged, and the colour was homogeneous within the same spot but heterogeneous between the different lesions (Figure 1c). CALMs were mainly localized on the trunk (50%) and limbs (41%) and sometimes on the face (9%). Twelve (57%) children showed hypopigmented macules (HPMs), which were congenital (n=1), appeared at the age of 5 years (n=1) or unknown age of onset (n=10), and were mainly large (88% > 15mm), with ragged borders. The mean number was 2.9 [1-7] per patient, with at least 4 spots in 4 children. The spots could be discrete, only highlighted with a Wood's lamp (Figure 2a).

Finally, 4 (19%) children had a heterogeneous, reticulated, purple macule localized on the trunk or lower limbs, which was congenital in one case (Figure 2b). These lesions were discrete and No other skin or mucosal lesion was identified.

Neurosensory features

Seventeen (81%) children had CNS tumours. Tumours were localized in the brain (82%, n=14), spine (82%, n=14) or cranial nerves (41%, n=7). Eight (47%) children had multiple cauda equine schwannomas. Seven (41%) had 1 to 8 tumours and the other 10 (59%) had more than 10 tumours. The different histological types were schwannomas (82%, n=14), meningiomas (70%, n=12) and ependymomas (29%, n=5 did not report such lesions, which suggests that analysis with a Wood's lamp by a dermatologist is more sensitive. Isolated HPMs are a common feature in a paediatric population, but multiple (> 3) HPMs could be an inconstant but specific and precocious sign of NF2.

Finally, we found 4 (19%) children with flat purple lesions of a singular aspect, not previously described, whose sensitivity in NF2 remains to be determined. These lesions were large (>5cm), asymptomatic and discrete. Unfortunately, no Doppler ultrasonography was performed to confirm the vascular origin of these lesions. There is no other case reported in the literature.

Despite their precocity and high prevalence, dermatological lesions, that is, cutaneous tumours, led to the NF2 diagnosis in only 5 (24%) cases after excision and discovery of a schwannoma (n=4) or neurofibroma (n=1). These data indicate that before the NF2 diagnosis, 70% of children have at least one "undiagnosed" cutaneous tumour (but sometimes > 5) that did not lead to a specific management. We found similar results with ophthalmologic symptoms. In accordance with the largest study on this topic (20), we found an ophthalmological involvement in 81% of our children (72% of patients in the MacCollin and Mautner study), mainly retinal hamartomas, epiretinal membranes, strabismus or amblyopia, and posterior cataracts. These lesions were often congenital or diagnosed early in childhood, but they allowed for the diagnosis of NF2 in only 3 (14%) cases.

Indeed, in our series, NF2 was diagnosed in 10 (47%) children because of neurological complications (including hearing loss). Before this diagnosis, 7 of these children had a combination of evocative skin and ophthalmic lesions. These results are similar to those found by Anand et al. (56%) [START_REF] Anand | Diagnosis of sporadic neurofibromatosis type 2 in the paediatric population[END_REF] and Gugel el al. (53%) [START_REF] Gugel | Presenting symptoms in children with neurofibromatosis type 2[END_REF]. However, in the absence of a familial history of NF2, none met the Manchester diagnostic criteria. Furthermore, most children had congenital retinal lesions, which is not part of Manchester criteria.

These data led to several considerations: 1) childhood onset of NF2 is a severity factor, and early diagnosis is mandatory to improve the outcome of patients; 2) the Manchester diagnostic criteria are not adapted to paediatric populations; 3) paediatricians and dermatologists must be aware to not leave skin lesions undiagnosed in children, especially if they are multiple, if their number increases over time and/or if they are associated with congenital ophthalmic lesions; and 4) clear guidelines for NF2 screening in case of suggestive neurological or non-neurological symptoms are mandatory for this population. This would require compiling a list of ophthalmological and dermatological signs suggestive of the diagnosis of NF2 in childhood as proposed by Anand et al. [START_REF] Anand | Diagnosis of sporadic neurofibromatosis type 2 in the paediatric population[END_REF]. In this way, further studies to assess the sensibility and specificity of HPMs and purples macules in larger cohorts of NF2 patients would be of interest.

Finally, regarding the molecular analysis of patients, an NF2 pathogenic variant was found in 11/13 (85%) children, which is consistent with the literature. Recently a genetic severity score was proposed by Halliday et al. [START_REF] Halliday | Trends in phenotype in the English paediatric neurofibromatosis type 2 cohort stratified by genetic severity[END_REF],with 3 groups: 1A/1B, no NF2 pathogenic variant in blood;

2A/2B, mild/moderate NF2 constitutional or mosaic pathogenic variant in blood; and 3, constitutional truncating exon 2-13 pathogenic variant. Patients in group 3 may show disease earlier with greater tumour load and have poorer visual outcomes and more interventions [START_REF] Halliday | Trends in phenotype in the English paediatric neurofibromatosis type 2 cohort stratified by genetic severity[END_REF][START_REF] Hexter | Clinical and molecular predictors of mortality in neurofibromatosis 2: a UK national analysis of 1192 patients[END_REF]. In our series, a severe phenotype was detected in 2 children with no pathogenic variant detected in blood and without molecular analysis of tumours, 1 patient with an in-frame deletion of a nucleotide in mosaic state, and 1 patient with a missense mutation (score 1A to 2A), which suggests that the phenotype-genotype correlation is difficult to clearly establish in the paediatric population.

CONCLUSIONS

Dermatological and ophthalmic lesions are frequent and precocious in children with sporadic severe NF2 but rarely lead to the diagnosis because of lack of clear guidelines. Cutaneous tumours, and in particular schwannomas, are the most frequent dermatological lesions but are often underdiagnosed, and CALMs are frequent but are atypical and mostly in small numbers.

Multiple HPM lesions seem suggestive although inconsistent. The sensitivity of reticulated capillary malformation-like lesions remains to be assessed by further studies.

  ), type B tumours in 11 (52%) children, type C tumours in 10 (47%) children (Figure 1b), and type D in only two children. Histology concluded schwannomas (n=9) or neurofibromas (n=2). Five children specifically consulted for a skin tumour, with a clinical diagnosis of lipomas for 2 children, neurofibromas in 1 child and suspicious lesion in the last 2 children. Four of these children underwent resection with cytopathological examination leading to the diagnosis of schwannoma (n=3) or neurofibroma (n=1) and subsequently an NF2 diagnosis.

Figure 1 :

 1 Figure 1: Classical dermatological manifestations in patients with NF2. a) Type A/AH tumours: pigmented (often pinkish) plaques, rough to the touch, often > 2 cm, with or without hypertrichosis. b) Type C tumours: pink or purplish nodular epicutaneous tumours. c) Atypical café-au-lait macules with small size, irregular margins and ragged borders, and different colours in the same individual.

  vitropression, suggesting a vascular origin, but no Doppler ultrasonography was performed.

Figure 2 :

 2 Figure 2: New dermatological manifestations in patients with NF2. a) Multiple large hypopigmented macules, with ragged borders, highlighted with Woods' lamp. b)Heterogeneous, reticulated, congenital purple macule on the trunk of a 9-year-old girl.

REMERCIEMENTS Au président du Jury de Thèse Monsieur le Professeur Kalamarides, je

  vous remercie de m'avoir fait ce grand honneur, de présider mon jury de thèse. Un immense merci pour votre intérêt et votre soutien lors de cette étude, ainsi qu'à votre accueil au Centre National de Référence de la neurofibromatose de type 2 à l'hôpital de la Pitié Salpêtrière. Je souhaite tout d'abord exprimer ma profonde gratitude à l'ensemble des enfants et leurs parents pour leur participation à l'étude. Je remercie également l'association de patients Neurofibromatoses et le groupe de recherche de la Société Française de Dermatologie Pédiatrique pour l'aide apportée. Un immense merci à vous, Pr Didier Bessis, Dr Fanny Picard, Dr Stéphanie Mallet, Dr Juliette Mazereeuw, Dr Alice Phan, Dr Delphine Dupin Deguine et Dr Thomas Hubiche. Sans votre implication, ce travail n'aurait pu être réalisé.

		Gilles	Médecine Générale (53.03)
	Mme	MONNIER Brigitte	Médecine Générale (53.03)
	MAITRES DE CONFÉRENCES ASSOCIÉS
	Mme	CASTA Céline	Médecine Générale (53.03)
	M.	GASPERINI Fabrice	Médecine Générale (53.03)
	M.	HOGU Nicolas	Médecine Générale (53.03)
	màj27/09/19

Aux membres du jury

Monsieur le Professeur Lacour, merci de me faire l'honneur de participer à mon jury de thèse. Quelle chance d'avoir pu apprendre à vos côtés, lors du semestre passé dans votre service. Merci d'avoir partagé votre expertise avec une infinie bienveillance.

Monsieur le

Professeur Bérard, je vous remercie d'avoir participé à l'évaluation de ce travail, ainsi que pour votre enseignement durant ces années d'internat. Madame le Docteur Soler, un grand merci d'avoir accepté de siéger dans ce jury, et de me faire bénéficier de votre expertise. J'admire votre dévouement et la combativité avec laquelle vous accompagnez vos patients. A ma directrice de thèse Madame le Docteur Chiaverini, Christine, je ne saurais assez vous remercier de m'avoir tant aidée et guidée. Merci pour votre confiance, votre patience, et vos conseils tout au long de ce travail. Merci pour vos enseignements de dermatologie pédiatrique, votre humanité et votre dévouement auprès de vos petits patients. Vous êtes admirablement impressionnante, à jongler entre vie professionnelle plus que remplie et vie personnelle épanouie. Avec mon profond respect et mon immense gratitude.

A mes co-internes de stage : Cloé, ce

	semestre en réanimation néonatale, à partager d'excellents moments, mais aussi à se soutenir
	lors des moments difficiles. A Lorène et Asmao, avec vous les mots entraides, soutien et
	solidarité ont pris tout leur sens. A
	semestre en HDJ a été un réel plaisir. Merci pour ces
	bons moments, ces pauses café ensoleillées, et ces fous-rires. Ivana, Jade, merci pour ce

A notre responsable DES : Lisa, merci pour votre accompagnement et votre soutien tout au long de mon internat. Vous nous poussez à toujours nous dépasser. Votre réussite professionnelle et personnelle est un modèle pour nombre d'entre nous… A l'équipe de pédiatrie libérale Mentonnaise : Henri, merci de m'avoir accueillie dans votre cabinet, et de m'avoir fait partager votre amour de la pédiatrie ! Mais surtout… merci pour cette magnifique ballade à moto, un moment inoubliable ! Jean-Louis, je vous remercie pour votre accueil et votre enseignement. Merci pour ces consultations ponctuées d'humour, que de bons souvenirs ! Vanessa, merci pour ton accueil. J'admire ta capacité à concilier parfaitement vie professionnelle et vie personnelle ! Clélia et Fanny, vous êtes des secrétaires incroyables, mais surtout des femmes en or, bienveillantes et tellement drôles ! Que de bons souvenirs de journées de travail, entrecoupées de déjeuner agréables, le tout entre deux fous rires ! A l'équipe d'hospitalisation programmée de Lenval : Fabienne, Marie-Amélie, et toutes les infirmières que j'apprécie tant, merci pour ce semestre à vos côtés ! Vous m'avez beaucoup appris, dans la joie et la bonne humeur, vous qui avez vu grossir mon ventre jour après jour… A l'équipe de réanimation néonatale : Florence, Stéphanie, Julie, Anne-Marie, Gwenn, Charlotte, Bérengère, Sergio, et toute l'équipe paramédicale. J'ai passé un merveilleux semestre parmi vous. Pour tout votre enseignement, un grand merci ! A l'équipe de pédiatrie de Grasse : Joël, Catherine, Blandine, Olivier, Georges, Jonathan et toute l'équipe paramédicale. Merci à vous tous, j'ai passé un magnifique semestre à vos côtés. J'ai adoré travailler à vos côtés, avec votre soutien et votre bienveillance, le tout dans la joie et la bonne-humeur ! A l'équipe des urgences pédiatriques de l'hôpital Nord de Marseille : Philippe, Rémy, Sophie, merci de m'avoir accueillie dans votre magnifique Provence lors de mon premier semestre d'internat. Vous avez accompagné mes premiers pas d'interne… mais également mes premiers pas de future maman. A l'équipe de réanimation néonatale de Strasbourg : Claire, Anne-Louise, Jennifer, Pierre, Benoît, merci pour votre accueil dans votre belle Alsace et votre enseignement.

A mes co-internes présents et passés

Sidonie, plus qu'une co-interne tu es une véritable amie ! L'amour de la pédiatrie nous a rapprochées, mais également nos maternités. Quelle joie de te voir t'accomplir tant sur le plan professionnel que personnel. Tu es une femme, une mère et une amie incroyable, au grand coeur ! Une véritable Wonder woman !

Faustine, Charlotte, Florence, Bruno, Julie, Carole,

	Nathalie, avec qui j'ai partagé mes premiers pas d'interne. Un semestre inoubliable !
	Corisandre, Olivier, Chloé, merci pour votre accueil chaleureux dans votre belle région
	strasbourgeoise ! Merci pour les bons moments passés ensemble, et ceux à venir !!
	A tous mes co-internes de pédiatrie : Mathilde, Sarah, Anne-Lise, Aymene, Jade, Chloé,
	Robin, Ivana, Jean Marc, Emma, Pierre, Lorenzo, Julie, Jean-Baptiste, …
	A ma famille
	Maman, Papa, si j'en suis là aujourd'hui, c'est grâce à votre soutien sans faille, et à votre
	confiance en moi qui m'ont permis d'accomplir mes rêves. Merci à vous !

A mes frères, Sébastien, Florent, Olivier, vous

  qui avez toujours été là pour m'encourager.

	J'ai de la chance de vous avoir à mes côtés, tout près dans mon coeur, malgré la distance.
	Vous êtes les meilleurs frères et tontons du monde ! Je vous aime !

A mes belles-soeurs, Aude et Susannah, comme

	j'ai de la chance d'avoir des belles soeurs si
	gentilles, bienveillantes et drôles ! Que de bons moments passés et à passer ensemble !

A mes neveux, Théo, Lukas, Kylian, et Jake, qui ont déjà bien grandi… vous êtes de vrais

	rayons de soleil, merci d'illuminer nos vies !
	A Jonathan, mon amour, mon ami… merci de m'avoir toujours soutenue, rassurée,
	encouragée et accompagnée dans mes projets. Merci de partager ma vie, et de la rendre plus
	belle et plus douce. Merci pour nos magnifiques enfants… je t'aime…
	A Roxane et Yann, mes amours, mes chéris… il n'existe pas de plus grand bonheur que de
	vous câliner et vous serrer dans mes bras jour après jour…je vous aime à la folie !

TABLE DES MATIÈRES ABRÉVIATIONS NEUROFIBROMATOSE DE TYPE 2 RÉSUMÉ INTRODUCTION MATÉRIEL ET MÉTHODE RÉSULTATS

  INTRODUCTIONNeurofibromatosis type 2 (NF2), a rare genodermatosis with an incidence of 1/25-33.000 births and a prevalence of 1/60.000[START_REF] Evans | Schwannomatosis: a genetic and epidemiological study[END_REF][START_REF] Evans | A genetic study of type 2 neurofibromatosis in the United Kingdom. II. Guidelines for genetic counselling[END_REF][START_REF] Evans | Neurofibromatosis type 2 (NF2): A clinical and molecular review[END_REF], is linked to an autosomal dominant mutation of the NF2 gene, located on chromosome 22. The average age at diagnosis, based on Manchester criteria or the identification of a pathogenic mutation in NF2, is 17 to 22 years old[START_REF] Evans | Neurofibromatosis Type 2[END_REF]. In up to 20% of cases, especially sporadic ones, no pathogenic mutation in NF2 is identified[START_REF] Ardern-Holmes | Neurofibromatosis Type 2: Presentation, Major Complications, and Management, With a Focus on the Pediatric Age Group[END_REF]. Clinically, NF2 is characterized by the occurrence of bilateral or unilateral vestibular schwannomas (VS), associated with benign central nervous system (CNS) tumours localized in brain, spinal or cranial nerves. The main other presenting features are ophthalmic and dermatological[START_REF] Evans | A genetic study of type 2 neurofibromatosis in the United Kingdom. II. Guidelines for genetic counselling[END_REF][START_REF] Kanter | Central neurofibromatosis with bilateral acoustic neuroma: Genetic, clinical and biochemical distinctions from peripheral neurofibromatosis[END_REF][START_REF] Parry | Neurofibromatosis 2 (NF2): Clinical characteristics of 63 affected individuals and clinical evidence for heterogeneity[END_REF][START_REF] Mautner | Skin abnormalities in neurofibromatosis 2[END_REF][START_REF] Mautner | The Neuroimaging and Clinical Spectrum of Neurofibromatosis 2[END_REF].

	NEUROFIBROMATOSE DE TYPE 2 RÉSUMÉ (nombre moyen de tumeurs 5,9 [1-15]), qui a conduit au diagnostic dans 4 cas. Dans les 17
	autres cas, le diagnostic de NF2 résultait de complications neurosensorielles (n=10), d'un
	NF2 : Neurofibromatose de type 2 La Neurofibromatose de type 2 (NF2) est une maladie génétique à transmission autosomique Introduction: La neurofibromatose de type 2 (NF2) résulte de la mutation du gène NF2 situé dépistage familial (n=4) ou de signes oculaires (n=3). Douze enfants (70%) présentaient avant
	dominante, résultant de l'altération du gène NF2 situé sur le chromosome 22. L'incidence est sur le chromosome 22, à transmission dominante. L'âge moyen au diagnostic est de 17 à 22 le diagnostic de NF2 au moins une tumeur cutanée non diagnostiquée, et non prise en charge
	SV : Schwannome Vestibulaire estimée à 1 / 33.000 et la prévalence à 1 / 60.000. Le gène NF2 code pour une protéine ans, et repose sur les critères de Manchester et/ou l'identification d'une mutation pathogène spécifiquement. L'examen dermatologique des patients retrouvait également des taches café-
	SNC : Système Nerveux Central appelée schwannomin (ou merlin), dont le rôle est d'inhiber la croissance de certaines du gène NF2. Le tableau clinique associe des schwannomes vestibulaires (SV), à d'autres au-lait (TCL) (n=15; 71%) précédemment décrites dans la littérature, mais aussi des macules
	tumeurs. tumeurs bénignes du système nerveux central (SNC), ainsi que des atteintes dermatologiques hypopigmentées (MHP) (n=12; 57%) et des macules réticulées violines (n=4; 19%).
	TCL : Tache Café au Lait et ophtalmologiques. On distingue un phénotype modéré, avec des SV bilatéraux Dix-sept patients (81%) présentaient une atteinte ophtalmologique spécifique et dix-sept
	MHP : Macules Hypopigmentées
	Description clinique de la population
	Atteintes dermatologiques
	Atteintes ophtalmologiques
	Atteintes neurosensorielles
	Données génétiques
	DISCUSSION
	CONCLUSION
	BIBLIOGRAPHIE
	ANNEXES
	SERMENT D'HIPPOCRATE

Génétique

La maladie est de transmission autosomique dominante avec une pénétrance complète. On distingue des formes familiales, et des cas sporadiques avec mutation de novo. On retrouve également des formes mosaïques d'expression clinique variables. Les anomalies génétiques conduisant à une altération du gène peuvent être diverses : mutations ponctuelles, délétion de grande taille, décalage du cadre de lecture, translocation impliquant le chromosome 22, chromosome 22 en anneau, …

Présentation clinique

La NF2 est une phacomatose, caractérisée par le développement de tumeurs bénignes du système nerveux central, associée à une atteinte cutanée et oculaire. L'âge moyen de révélation de la maladie est de 17-22 ans. On distingue un phénotype modéré et un phénotype sévère.

Atteinte neurologique: Les tumeurs peuvent être localisées au niveau cérébral, médullaire ou des nerfs crâniens. Les différents types histologiques retrouvés sont les schwannomes, méningiomes, épendymomes ou plus rarement astrocytomes. La présence de schwannomes vestibulaires (également nommés neurinomes de l'acoustique), souvent bilatéraux, est caractéristique.

Atteinte ophtalmologique: On retrouve des lésions rétiniennes (hamartome, membranes épirétiniennes), mais également des cataractes juvéniles.

Atteinte dermatologique: Les patients peuvent présenter des tumeurs cutanées d'aspect variable. Les manifestations dermatologiques de la NF2 sont peu décrites dans la littérature.

Critères diagnostiques

Le diagnostic repose sur l'identification d'une mutation pathogène du gène NF2 (seulement 3/4 des cas) et/ou remplissant les critères diagnostiques de Manchester (Annexe 1).

Evolution

Le pronostic est variable. L'évolution vers la surdité bilatérale, associée à un syndrome vestibulaire, est quasi-constante en l'absence de traitement. Les complications neurosensorielles (déficit auditif, visuel, paralysies faciales, troubles moteurs, …) conditionnent le pronostic, avec une morbi-mortalité importante.

Prise en charge thérapeutique

Les patients présentant un diagnostic confirmé de NF2 bénéficient d'une surveillance rapprochée avec un suivi neurologique, ORL et ophtalmologique réguliers, associés à l'imagerie cérébro-médullaire. Les symptômes et la croissance tumorale guident la prise en charge. Il n'existe pas de traitement spécifique. Les traitements reposent sur l'ablation chirurgicale des tumeurs, la chimiothérapie et la radiothérapie, en association à des soins de support (kinésithérapie, orthophonie, …). La mise en place d'implants du tronc cérébral permet le maintien d'une audition chez certains patients. symptomatiques chez l'adulte jeune, et un phénotype sévère, avec de multiples tumeurs du SNC, ainsi que des symptômes cutanés et ophtalmologiques, débutant dans l'enfance. Le diagnostic précoce chez l'enfant est difficile, car ceux-ci présentent principalement des signes cutanés ou ophtalmologiques, pouvant précéder de plusieurs années les SV ou les tumeurs du SNC, mais peu décrites dans la littérature. L'objectif de cette étude est de caractériser les manifestations dermatologiques de la NF2 dans une population pédiatrique. Matériel et méthode: Cette étude observationnelle, prospective et multicentrique a été menée d'avril 2019 à avril 2020 dans huit hôpitaux universitaires français. Les patients ≤ 18 inclus remplissaient les critères de diagnostic de Manchester ou présentaient une mutation pathogène identifiée dans le gène NF2. Tous les patients bénéficiaient d'un examen dermatologique, guidé par un questionnaire standardisé. Résultats et discussion: 21 enfants étaient inclus (âge médian de 13 ans [extrêmes 2-18]; sex-ratio 11 garçons / 10 filles), dont 18 formes sévères et 16 cas sporadiques. A l'exception d'un patient complètement asymptomatique, tous présentaient au moins une tumeur cutanée (81%) souffraient de tumeurs du système nerveux central. Les complications neurologiques, incluant les déficits auditifs, conduisaient au diagnostic chez 10 patients. Or, 7 d'entre eux présentaient auparavant des signes cutanés associés à des signes oculaires évocateurs de la maladie. Conclusion: Les lésions dermatologiques (en particulier les schwannomes cutanés) et oculaires sont fréquentes et précoces chez les enfants atteints de NF2, qui présentent principalement des formes sévères et sporadiques, mais ne conduisent que rarement au diagnostic. Ces résultats suggèrent que l'adaptation des critères diagnostiques à la population pédiatrique mériterait d'être discutée. Les TCL et les MHP semblent également évocatrices, et la sensibilité des lésions à type de malformation capillaire reste à évaluer par d'autres études. Two major phenotypes have been described: mild (Gardner type) with bilateral VS becoming symptomatic in young adults, and severe (Wishart type), with multiple CNS tumours, cutaneous and ophthalmological involvement, starting from childhood (10-12).

Table 1 : Manchester diagnosis criteria for neurofibromatosis type 2 Evans

 1

et al, 1992.

  included 21 children (median age 13 years [range 2-18]; sex-ratio 11 boys/10 girls) diagnosed with NF2 between April 2019 and April 2020 (Table 2). NF2 was sporadic in 16

	cases and familial in 5. The phenotype was severe in 18 children and moderate in 2; one child
	was asymptomatic.
	NF2 was diagnosed with neurological complications due to tumour (n=10; median age 9 years
	[6-15]) in patients without a family history, with family screening (n=4; median age 8 years [4-
	11]), with early cutaneous signs (n=4; median age at diagnosis 12 years [3-15]; median age at
	first signs 7 years [1-13]) or with ocular signs (n=3; median age at diagnosis 4 years [2-14];
	median age at first signs 1 year [0-4]).

  ). Complications related to these tumours led to an Wood's lamp by a dermatologist. However, none of our children had ≥ 6 CALMs, and only 4 had > 3 CALMs, with a median of 2.4. Furthermore, analysis of semiological characteristics of CALMs showed that they were atypical[START_REF] Shah | The Diagnostic and Clinical Significance of Café-au-lait Macules[END_REF], with small size, irregular margins and ragged borders, and different colours in the same individual. Because of the high prevalence of CALMs in the general population, 18.9% to 36%[START_REF] Shah | The Diagnostic and Clinical Significance of Café-au-lait Macules[END_REF], the specificity and sensitivity of this clinical sign in NF2 patients seems low, with CALMs not a cardinal feature of NF2.We showed a high prevalence (57%) of HPMs in children with NF2. They were mainly > 15 mm, with irregular margins and ragged borders, and could be unnoticed without careful examination with a Wood's lamp. Although the age of onset was difficult to assess, because they could easily go unnoticed, HPMs were congenital in one case and noticed at age 5 years

	Our study 21 (C) 63 (A+C) examination with a Gugel et al, 2020 Anand et al, 2018 Ruggieri et al, 2005 Mautner et al, 1997 Parry et al, 1994 cohorte 70 (C) 32 (C) 22 (C) 88 (A+C)	Evans et al, 1992
	NF2 diagnosis in 7 patients, with headache (n=2; mean age 6.5 years [6-7]), cervicalgia (n=2;
	mean age 11.5 years [9-14]), brutal motor deficit (n=1; age 9 years), intracranial hypertension
	(n=1) or genito-sphincter disorders (n=1; age 14 years). Epilepsy (n=3; mean age 11 years [10-
	14]) and attention deficit disorder with hyperactivity (n=1; age 8 years) also noticed. Brain
	tumours also caused proptosis (n = 2; mean age 8.5 years [6-11]), oculomotor palsy (n = 1; age
	8 years), XII nerve deficit (n = 1; age 10 years), reduced visual field and retinal detachment (n
	= 1; age 14 years).
	Four (19%) children had optic meningioma (mean age 9.5 years [6-15]) leading to binocular
	blindness and proptosis (n = 1), reduced visual field (n = 1) or amblyopia (n = 1).
	Finally, most children (19/21) had bilateral (84%) or unilateral (16%) VS (mean age at Patient 11 12 13 14 15 Sex M M F M F Patient 16 17 18 19 20 21 Sex F M F F M M DISCUSSION
	Age (yo) at diagnosis / at time of study Phenotype Age (yo) at diagnosis / at time of study Phenotype This study is the first to systematically and prospectively search for skin anomalies by clinical 10 / 13 11 / 13 11 / 13 ND / 14 11 / 14 severe mild severe severe 12 / 17 14 / 15 14 / 17 14 / 17 15 / 17 15 / 18 severe severe severe severe severe severe mild Family history No Yes No Yes Family history No No No No No No Yes Manchester Criteria fulfilled Yes Yes Yes Yes No Mutation Nucleotide change / protein change Mutation in Intron 4 c.447+2T > C / undefined complete deletion of the NF2 gene c.114G>C / p.(Glu38Asp) ND c.1123-3C>G / p.(Met375fs) + MEN2 Manchester Criteria fulfilled Yes Yes Yes Yes Yes Yes Mutation Nucleotide change / protein change ND ND Mutation in Exon 3 c.357_359delCTT / p.Phe119del Mosaicism 25% undefined c.586C>T / p.Arg196ter ongoing c.1137_1138delGA / Mutation in Exon 6 Mutation in Exon 12 examination performed by a dermatologist in a paediatric cohort with NF2. Six other studies
	Genetic severity score Halliday et al. (18) symptoms leading to diagnosis / Age of onset Dermatological features Genetic severity score Halliday et al. (18) symptoms leading to diagnosis / Age of onset complications (hearing loss) 2A/B Cutaneous sign (excision of a skin lesion for aesthetic purposes) / < 6 yo Yes ND Neurosensory / 12 yo Cutaneous sign (sexcision 2A Family screening / 11 yo Yes ND of skin lesions for aesthetic purposes at 8 and 14 yo) / <8 yo complications (cervicalgy + Neurosensory 2A complications (strabismus + proptosis) / 11 yo Yes 2A 3 genito-sphincter disorders) congenital / 14 yo (retinal hamartoma Family screening / ND ND Yes ongoing secondarily complicated with retinal detachment) / + tinnitus) / 15 yo complications (hearing loss Family screening / 11 yo 3 No 31 2A tumours) / 13 yo Cutaneous signs (skin Neurosensory Neurosensory Ophtalmological sign 32
	Skin tumours / Number Dermatological features	Yes	Yes / 1	Yes	Yes / 1	Yes	Yes / 7	Yes	Yes / 15	Yes	No Yes

diagnosis 9.9 years

[START_REF] Evans | Neurofibromatosis type 2 (NF2): A clinical and molecular review[END_REF][START_REF] Evans | Neurofibromatosis Type 2[END_REF][START_REF] Ardern-Holmes | Neurofibromatosis Type 2: Presentation, Major Complications, and Management, With a Focus on the Pediatric Age Group[END_REF][START_REF] Kanter | Central neurofibromatosis with bilateral acoustic neuroma: Genetic, clinical and biochemical distinctions from peripheral neurofibromatosis[END_REF][START_REF] Parry | Neurofibromatosis 2 (NF2): Clinical characteristics of 63 affected individuals and clinical evidence for heterogeneity[END_REF][START_REF] Mautner | Skin abnormalities in neurofibromatosis 2[END_REF][START_REF] Mautner | The Neuroimaging and Clinical Spectrum of Neurofibromatosis 2[END_REF][START_REF] Goutagny | Neurofibromatosis type 2[END_REF][START_REF] Anand | Diagnosis of sporadic neurofibromatosis type 2 in the paediatric population[END_REF][START_REF] Ruggieri | Earliest Clinical Manifestations and Natural History of Neurofibromatosis Type 2 (NF2) in Childhood: A Study of 24 Patients[END_REF][START_REF] Gugel | Presenting symptoms in children with neurofibromatosis type 2[END_REF][START_REF] Casado-Verrier | Cutaneous and ophthalmic signs as a clue to early diagnosis of severe neurofibromatosis type 2: report of a novel mutation that predicts this poor prognosis[END_REF][START_REF] Plana-Pla | Diagnostic and Prognostic Relevance of the Cutaneous Manifestations of Neurofibromatosis Type 2[END_REF]

); 12 were symptomatic at diagnosis (mean age 11.7 years

[START_REF] Goutagny | Neurofibromatosis type 2[END_REF][START_REF] Anand | Diagnosis of sporadic neurofibromatosis type 2 in the paediatric population[END_REF][START_REF] Ruggieri | Earliest Clinical Manifestations and Natural History of Neurofibromatosis Type 2 (NF2) in Childhood: A Study of 24 Patients[END_REF][START_REF] Gugel | Presenting symptoms in children with neurofibromatosis type 2[END_REF][START_REF] Casado-Verrier | Cutaneous and ophthalmic signs as a clue to early diagnosis of severe neurofibromatosis type 2: report of a novel mutation that predicts this poor prognosis[END_REF][START_REF] Plana-Pla | Diagnostic and Prognostic Relevance of the Cutaneous Manifestations of Neurofibromatosis Type 2[END_REF][START_REF] Ruggieri | Natural history of neurofibromatosis type 2 with onset before the age of 1 year[END_REF][START_REF] Rustad | Neurofibromatosis type 2: Multiple intra-dermal tumors in a toddler[END_REF][START_REF] Halliday | Trends in phenotype in the English paediatric neurofibromatosis type 2 cohort stratified by genetic severity[END_REF]

), with hearing loss (n=7; mean age 11years

[START_REF] Kanter | Central neurofibromatosis with bilateral acoustic neuroma: Genetic, clinical and biochemical distinctions from peripheral neurofibromatosis[END_REF][START_REF] Parry | Neurofibromatosis 2 (NF2): Clinical characteristics of 63 affected individuals and clinical evidence for heterogeneity[END_REF][START_REF] Mautner | Skin abnormalities in neurofibromatosis 2[END_REF][START_REF] Mautner | The Neuroimaging and Clinical Spectrum of Neurofibromatosis 2[END_REF][START_REF] Goutagny | Neurofibromatosis type 2[END_REF][START_REF] Anand | Diagnosis of sporadic neurofibromatosis type 2 in the paediatric population[END_REF][START_REF] Ruggieri | Earliest Clinical Manifestations and Natural History of Neurofibromatosis Type 2 (NF2) in Childhood: A Study of 24 Patients[END_REF][START_REF] Gugel | Presenting symptoms in children with neurofibromatosis type 2[END_REF][START_REF] Casado-Verrier | Cutaneous and ophthalmic signs as a clue to early diagnosis of severe neurofibromatosis type 2: report of a novel mutation that predicts this poor prognosis[END_REF][START_REF] Plana-Pla | Diagnostic and Prognostic Relevance of the Cutaneous Manifestations of Neurofibromatosis Type 2[END_REF]

), which led to an NF2 diagnosis in 3 cases, in second case. Four children had > 3 HPMs (range 4 to 7), and 8 had no lesion. We found only 2 articles in literature concerning these lesions. Anand et al.

[START_REF] Anand | Diagnosis of sporadic neurofibromatosis type 2 in the paediatric population[END_REF]

reported a prevalence of 9% of "hypopigmented lesions" in a retrospective study including 32 children with NF2, but no clinical description of lesions was available. Casado-Verrier

[START_REF] Casado-Verrier | Cutaneous and ophthalmic signs as a clue to early diagnosis of severe neurofibromatosis type 2: report of a novel mutation that predicts this poor prognosis[END_REF]

reported multiple HPMs in a case report of a 7-year-old child. The 5 other studies that analysed skin lesions in NF2 patients

SERMENT D'HIPPOCRATE

Au moment d'être admise à exercer la médecine, je promets et je jure d'être fidèle aux lois de l'honneur et de la probité.

Mon premier souci sera de rétablir, de préserver et de promouvoir la santé dans tous ses éléments, physiques et mentaux, individuels et sociaux.

Je respecterai toutes les personnes, leur autonomie et leur volonté, sans aucune discrimination selon leur état ou leurs convictions. J'interviendrai pour les protéger si elles sont affaiblies, vulnérables ou menacées dans leur intégrité ou leur dignité. Même sous la contrainte, je ne ferai pas usage de mes connaissances contre les lois de l'humanité. J'informerai les patients des décisions envisagées, de leurs raisons et de leurs conséquences.

Je ne tromperai jamais leur confiance et n'exploiterai pas le pouvoir hérité des circonstances pour forcer les consciences. Je donnerai mes soins à l'indigent et à quiconque me les demandera. Je ne me laisserai pas influencer par la soif du gain ou la recherche de la gloire.

Admise dans l'intimité des personnes, je tairai les secrets qui me seront confiés. Reçue à l'intérieur des maisons, je respecterai les secrets des foyers et ma conduite ne servira pas à corrompre les moeurs. Je ferai tout pour soulager les souffrances. Je ne prolongerai pas abusivement les agonies. Je ne provoquerai jamais la mort délibérément.

Je préserverai l'indépendance nécessaire à l'accomplissement de ma mission. Je n'entreprendrai rien qui dépasse mes compétences. Je les entretiendrai et les perfectionnerai pour assurer au mieux les services qui me seront demandés. J'apporterai mon aide à mes confrères ainsi qu'à leurs familles dans l'adversité.

Que les hommes et mes confrères m'accordent leur estime si je suis fidèle à mes promesses ; que je sois déshonorée et méprisée si j'y manque.

Cutaneous manifestations of neurofibromatosis type 2 in a paediatric population: a cohort study

Sophie Legoupil 1,2,11 , Didier Bessis 3,11 , Fanny Picard 4,11 , Stéphanie Mallet 5,11 , Juliette Mazereeuw 6,11 , Alice Phan 7,11 , Delphine Dupin Deguine 8 , Michel Kalamarides 9, 10 [START_REF] Kanter | Central neurofibromatosis with bilateral acoustic neuroma: Genetic, clinical and biochemical distinctions from peripheral neurofibromatosis[END_REF][START_REF] Parry | Neurofibromatosis 2 (NF2): Clinical characteristics of 63 affected individuals and clinical evidence for heterogeneity[END_REF][START_REF] Mautner | Skin abnormalities in neurofibromatosis 2[END_REF][START_REF] Mautner | The Neuroimaging and Clinical Spectrum of Neurofibromatosis 2[END_REF][START_REF] Goutagny | Neurofibromatosis type 2[END_REF][START_REF] Anand | Diagnosis of sporadic neurofibromatosis type 2 in the paediatric population[END_REF][START_REF] Ruggieri | Earliest Clinical Manifestations and Natural History of Neurofibromatosis Type 2 (NF2) in Childhood: A Study of 24 Patients[END_REF][START_REF] Gugel | Presenting symptoms in children with neurofibromatosis type 2[END_REF][START_REF] Casado-Verrier | Cutaneous and ophthalmic signs as a clue to early diagnosis of severe neurofibromatosis type 2: report of a novel mutation that predicts this poor prognosis[END_REF][START_REF] Plana-Pla | Diagnostic and Prognostic Relevance of the Cutaneous Manifestations of Neurofibromatosis Type 2[END_REF]). Only a 14-year-old patient with no symptoms of NF2 and a 2-year-old did not have VS.

Genetic data

Mutation analysis of the NF2 gene could be performed for 16 children. The prevalence of skin tumours in our cohort was high (95%) and comparable to that found by Rugierri et al. (95%) [START_REF] Ruggieri | Earliest Clinical Manifestations and Natural History of Neurofibromatosis Type 2 (NF2) in Childhood: A Study of 24 Patients[END_REF], in the only prospective study of children with severe NF2, but higher than the 5 other studies (14-68%). These differences could be explained by these studies also including adults and/or their retrospective design and/or including patients with milder phenotype. We found, as others did, that type A or AH tumours were the most represented, before types B, C and D.

Multiple CALMs, a major clinical diagnostic criterion of neurofibromatosis type 1, have also been reported in NF2, with which it was initially confused. We found CALMs in 71% of our children, a higher prevalence than in other studies (1.4-46%), probably because of the skin

ANNEXES


domain: phys spi

text:
Pierre Guyot

Andrew M Kraynik

Douglas Reinelt

Sylvie Cohen-Addad
email: sylvie.cohen-addad@insp.upmc.fr

Elastic behavior of confined soap froth

The elastic response of ordered 3D soap froth, in which N layers of cells are confined between two rigid walls, is analyzed. Surface Evolver simulations are used to compute the equilibrium structure, which consists of a layer of Fejes-Toth cells at each wall and N -2 core layers of Kelvin cells. The reference state corresponds to the plate spacing h o that achieves isotropic stress; and the foam confinement is varied by changing h. The foam is sheared in two orthogonal directions to determine elastic behavior up to the elastic limit or yield strain, which corresponds to the onset of topological transitions. The shear moduli in both directions decrease as N increases and slowly converge to the values of bulk Kelvin foam; the dependence on N is well described by a threelayer model consisting of Fejes-Toth cells and a core of Kelvin cells that deform like bulk foam. The influence of foam confinement on the elastic limit is studied. The topological transitions are compared to those in bulk Kelvin foam.

Introduction

Aqueous foams are dense packings of gas bubbles in a surfactant solution 1,[START_REF] Weaire | The Physics of Foams[END_REF] . Despite being solely constituted of fluids, they exhibit remarkable elastic properties arising from the surface energy stored in the liquid-gas interfaces under mechanical loading. Under small strain, they behave as a linear elastic solid. Large strains and the corresponding large bubble distortions result in unstable configurations that violate equilibrium and provoke topological transitions. During these transitions bubbles lose contact and separate from neighbors; this is the fundamental mechanism for yielding and is essential for flow.

Due to their rheological properties, aqueous foams are sought for in a variety of applications [START_REF] Stevenson | Foam Engineering: Fundamentals and Applications[END_REF] . Foams have been extensively used at large scale as displacing fluids in geological reservoirs in EOR processes [START_REF] Farajzadeh | [END_REF] , as fracturing fluid in drilling operations [START_REF] Stevenson | Foam Engineering: Fundamentals and Applications[END_REF] , or as chemical contaminant carriers for aquifer remediation 5 . Foam flow in porous media is mimicked in laboratory by model porous microfluidic systems [6][7][8][9] . Flow in confined geometries is used to manipulate bubbles in foams or droplets in concentrated emulsions for the purpose of building miniaturized labs on a chip [10][11][12][13][14] . In these processes where foam deformation is influenced by the confined geometry, their structure must accommodate the shape of a narrow channel or a pore of characteristic size of the order of a few bubble diameters. This raises the question how confinement impinges on the foam rheological behavior.

The rheological properties of 3D bulk foams have been extensively studied experimentally, numerically and theoretically 15,16 . Besides surface tension and bubble size, their static elastic properties such as shear modulus or yield stress depend on the volume fraction of the liquid phase and on the structural order. Rheology of foams under extreme confinement in pores of size equal to that of a single soap film has been investigated by numerical and experimental studies 17 . The impact of confinement specifically on the elastic behavior of a 3D foam has not yet been studied. Here, we consider ideal confined foams in the sense that: i) their liquid content is vanishingly small (soap froth or so-called dry foams); ii) they involve monodisperse ordered packings ; iii) they are confined between two parallel rigid plates separated by a gap distance of the order of a few bubble diameters.

In this paper, we study numerically the elastic behavior of dry confined multilayer foams in response to an applied shear. We address the following questions: How different is the shear modulus of a confined foam compared to that of a bulk 3D foam? What is the impact of the confinement, i.e. the plate spacing, on the shear modulus and how does it depend on the number N of bubble layers? Then we turn to large deformations and ask how the shear induced topological transitions, known as T1s, and their onset compare to those in a bulk foam. Using the Surface Evolver, we determine the shear moduli in the linear elastic regime and the stress-strain relationship up to the onset of T1s. We study the effect of confinement on yield stress and yield strain as a function of the layer structure.

2 Multilayer dry foam structure

Soap froth multilayer between two rigid walls

The equilibrium structure of an ideal bulk 3D foam (soap froth) satisfies Plateau's laws 1 , where each thin film separating adjacent bubbles has uniform curvature, the films meet 3 by 3 at equal angles of 120 • , and the edges at the junctions between films, the so-called Plateau borders, meet 4 by 4 at vertices located at the center of regular tetrahedra. The Kelvin foam is the quintessential perfectly ordered monodisperse soap froth and is encountered in many real situations [START_REF] Weaire | The Physics of Foams[END_REF][18][19][20] . However, it cannot accommodate the geometrical constraint of a rigid wall. At mechanical equilibrium with a rigid plane wall, the films that do not lie along the wall must form 90 • angles with the wall. As first pointed out by Fejes-Toth 21 and later by Weaire 2 , this surface accommodation can be accomplished by a layer of Fejes-Toth (FT) cells. In this study, we consider a thin slab of a multilayer dry 3D foam confined between two parallel rigid plates. The foam consists of N = 2, 3, ... , 11 ordered layers of monodisperse bubbles with FT cells at each wall and N -2 core layers of Kelvin cells in between, as illustrated in Fig. 1.

Kelvin structure

Let us recall the structure and the elastic properties of a Kelvin foam 1 . The Kelvin cell is a tetrakaidecahedron with 6 flat quadrilateral faces and 8 curved regular hexagonal faces (with zero mean curvature), as shown in fig. 2a. All the edges have the same length. Its equilibrium structure can be determined by minimizing surface energy density using the Surface Evolver, a standard software for discrete surface energy minimizations [START_REF] Brakke | The Surface Evolver is freely available at[END_REF] . The cell exhibits cubic symmetry, the symmetry axes (x o , y o , z o ) pointing from the bubble center to the center of each square face. The Kelvin foam is a dense packing of Kelvin cells of identical volume, shape and orientation, where the bubble centers occupy the sites of a body-centered cubic lattice (bcc).

The linear elastic behavior of a Kelvin foam has been extensively studied 22 . The stress σ o and strain ε o tensors of a structure with cubic symmetry can be expressed relative to the crystallographic axes (x o , y o , z o ) using three independent elastic constants c 11 , c 12 and c 44 (cf. definition in the appendix 7). We define the two independent shear moduli: Figures 2a to c illustrate the scheme followed to build a FT cell. It is obtained by rotating the Kelvin cell 45 • about one crystallographic axis (x o -axis for instance in fig. 2a), cutting the cell in two halves along the mid-plane (x-y) perpendicular to the square face, and then stretching it in the z direction such that the new cell has the same volume V as the initial Kelvin cell. The x, y, z axes define the reference frame in the following. The resulting Fejes-Toth cell has tetragonal symmetry, with 11 faces: a (non regular) hexagon in the x-y plane, 2 regular hexagons, 2 quadrilateral faces, 4 irregular pentagons and 2 rectangular faces as illustrated in fig. 2cf). Its equilibrium structure is determined by minimizing surface energy density using the Surface Evolver. This mechanical relaxation produces curved faces required to satisfy Plateaus' laws.

The equilibrium boundary condition at the wall is matched by placing the FT cell with its hexagonal base parallel to the x-y plane so that the films that do not lie along the wall form 90 • angles with the wall. A set of two FT cells packed upside down along the z direction, in the same orientation, allows space filling without overlapping or voids when it is translated in the x and y directions as illustrated in fig. 1a. Similarly a set of two external FT cells with a core of N -2 Kelvin cells in the same orientation fill space when translated along the x-y plane as shown for N = 3 in fig. 1b.

3 Computation of multilayer structure and stress response

Stress derivation

We consider an ordered multilayer foam, confined between two plates perpendicular to the z direction, separated by a gap distance h, and infinitely extended in the (x-y) plane (cf. fig. 1). The structure is described by the three lattice vectors p 1 , p 2 , p 3 which set the unit cell:

p 1 = p x , p y , 0 p 2 = -p x , p y , 0 p 3 = (0, 0, h) (2)
The vectors p 1 and p 2 join the centers of two neighboring (hexagonal) faces at the lower plate. Their length is expressed in units of V 1/3 . An homogeneous deformation, defined by the deformation gradient tensor F, is applied to the foam slab, by displacing the initial lattice vectors to get the new lattice vectors p i such that :

p i = F . p i (3)
The average macroscopic stress of the foam multilayer is then evaluated using Batchelor's general expression of the stress induced by a quasistatic strain in a fluid consisting of two immiscible phases [START_REF] Batchelor | [END_REF] . Omitting the isotropic term due to the gas pressure in each bubble, the contribution of the interfacial tension forces exerted on each individual film is given by 25 :

σ i j = T V f S int δ i j -n i n j ds (4)
where S int is the area of all liquid-gas interfaces in the volume of foam V f , δ i j is the Kronecker delta, n i is a local unit vector normal to the interface, and ds is the differential area element. Since the considered foam structures are periodic, the integral Eq. 4 needs only to be evaluated over the unit cell. In the following, we will consider the situation where each plate surface is wetted by a thin liquid film, assuming total wetting of the wall by the liquid phase. This case is indeed frequently encountered in experimental or practical situations. As a consequence, Eq. 4 can be written as:

σ i j = T V c   2    F ∑ k=1 k / ∈wall S k δ i j -n i n j ds    + 2 S w δ i j -n i n j ds    (5)
Here V c is the volume of the unit cell containing F films, S k is the area of a film k shared by two bubbles in the unit cell. The factor of 2 in the first term of the r.h.s. of Eq. 5 accounts for the two liquid-gas interfaces of each film in the interior of the slab. S w denotes the area of one hexagonal film wetting the wall, such a film having only one liquid-gas interface. By symmetry, the bottom wall film (z = 0) and the top film (z = h) have the same area. Thus the factor of 2 in the second term of the l.h.s. of Eq. 5 stands for the 2 wall films per unit cell.

Surface Evolver simulations

Applying affine deformations using the lattice transformation Eq. 3 generates foam structures that violate Plateau's laws. Equilibrium relaxed structures are then obtained using the Surface Evolver. The structure is initially discretized such that each nsided face is subdivided into n flat (linear mode) triangular facets. This initial mesh can be refined, so that the Ri discretized structures has 4 i times many more facets than the initial R one. Additionally, the facet can be quadratically interpolated (quadratic mode) to gain accuracy at the cost of time calculation. In this study, the surface energy density is minimized with the constraint of fixed bubble volume. This minimization is performed by successive iteration steps using a conjugate gradient algorithm repeated until the convergence is reached. A given strain (shear or stretch) is applied to the structure by applying the deformation gradient tensor F to each lattice vector p i of the unit cell. This defines the new lattice vectors and determines the new position of the vertices in the unit cell. Then the surface is recalculated and its energy minimized. At each minimization step, the stress is evaluated using Eq. 5. The criterion for convergence is based on the stress evaluation as in 22 . For a precision of n digits on the stress, we insure that the n + 1 digit does not vary upon further steps. Lengths and stresses are expressed in V 1/3 and T /V 1/3 units respectively.

Determination of the relaxed lattice parameters

Since the FT structure has a tetragonal symmetry, the plate spacing h and the lattice vector parameters p x , p y must be independently adjusted to achieve isotropic stress. For a given number N of layers, the values of these three parameters are unique. They are determined, for each N value, using repeated tensioncompression annealing cycles in the 3 space directions. For instance, uniaxial extension is applied in the z direction. Then the vertices and edges in contact with the wall are let free to move as the structure is relaxed using the Evolver (in quadratic R2 refinement mode). The extension is repeated with small strain increments until the stress difference σ xxσ yy = 0 with 7 significant digits. A similar elastic recoil is performed in the x and y directions until σ zzσ yy = 0 and σ xxσ xx = 0. This procedure determines with 6 significant digits the relaxed lattice parameters, denoted p xo , p yo , h o , in the reference undeformed state for each N value, as listed in Table 1. Moreover, we deduce from the data for N = 2, the thickness of one relaxed FT layer: δ = h o /2 = 0.97765. Figure 3 shows that the total thickness of the relaxed N-multilayer foam varies linearly with N. In the following, the degree of confinement of a N-multilayer in a gap thickness h is characterized by the confinement ratio defined as:

r = h/h o (6)
Table 1 Lattice parameters (expressed in V 1/3 units) of the N-multilayers as defined in eq. 2. They are determined by requiring the stress of the "undeformed" foam to be isotropic. Since the multilayer structure is anisotropic, we study its elastic response as it is subjected to simple shear in the x-y plane, along either the x or the y direction (cf. Fig. 1). For a given confinement ratio r, we determine the reference relaxed state where the stress in the x-y plane is isotropic. By mechanical annealing with the contact line at the wall being free to move, we perform successive elastic recoils until σ xxσ yy = 0. After this step, the stress tensor components σ xy , σ yz and σ xz are equal to zero with 7 significant 1. The line is a linear regression to the data:

h o = 1.9553+(0.8940± 0.0003)(N -2).
digits. Then we impose that the edges and vertices in contact with each wall are fixed and, we apply a simple shear as given by Eq. 3. This corresponds to the situation where the instantaneous elastic response is probed without foam slip at the wall. For a real foam, pinning at the wall is indeed achieved at the instant where the stress is applied, before any viscous relaxation process in the wetting liquid films occurs. The structure is further relaxed with R2 refinement (in quadratic mode) and the stress components are evaluated using Eq. 5. Stresses and shear moduli are scaled by T /V 1/3 .

For simple shear in the x direction, the deformation gradient tensor F and the infinitesimal strain tensor ε are:

F i j =    1 0 γ 0 1 0 0 0 1    ε i j =    0 0 γ/2 0 0 0 γ/2 0 0    (7)
We define the shear modulus G xz in that direction by:

σ xz = G xz γ (8)
and the shear induced normal stress differences:

N 1 = σ xx -σ zz N 2 = σ zz -σ yy (9)
We deduce G xz from the variations of σ xz with γ as the strain is increased by small increments in the range where the response is linear. We do similarly for simple shear in the y direction with

F i j =    1 0 0 0 1 γ 0 0 1    ε i j =    0 0 0 0 0 γ/2 0 γ/2 0    (10)
and define the shear modulus G yz as:

σ yz = G yz γ (11)
and the shear induced normal stress differences:

N 1 = σ yy -σ zz N 2 = σ zz -σ xx (12)
In the next paragraphs, we study the impact of the confinement ratio r (eq. 6) on both moduli G xz and G yz for N-multilayer as N is varied between 2 and 11.

Bilayer shear modulus.

First we study the linear elastic response of a bilayer. Fig. 4 shows the structure of the periodic cell as the bilayer is squeezed (r < 1) or stretched (r > 1). We recall that r = 1 corresponds to the isotropic state (cf. eq. 6). In this range of r, there are no topological changes of structure. For the non confined bilayer (r = 1), we represent in figure 5 the variations of the shear stress and both normal stress differences as a function of the applied strain γ increased by steps of 0.002 up to 0.02. From the shear stress data, we deduce: G xz,o = 1.37010 ± 2.10 -5 and G yz,o = 0.77688 ± 3.10 -5 . This shows that the elastic behavior is anisotropic. The bilayer is stiffer when sheared along the x direction compared to the y direction. The shear stress results from the traction that the films in contact with the wall and perpendicular to it exert on the contact lines. Since the surface tension acts in the plane tangential to the film at the contact line, a film contributes to the shear stress as long as it is not aligned with the shear direction. Thus for the x-z strain, all 6 films contribute whereas for the y-z strain, only 4 films contribute (cf. fig. 6). Neglecting the exact cell geometry, this simple argument suggests that G xz should be about 1.5 times larger than G yz which is indeed the case. By fitting the normal stress difference data to a quadratic polynomial, we get for x-z shear: N 1 = (1.797 ± 0.001)γ 2 and -N 2 = (1.158 ± 0.001)γ 2 , and for those induced by a y-z shear: N 1 = (1.982 ± 0.008)γ 2 and -N 2 = (1.053 ± 0.001)γ 2 . We observe that the normal stresses induced by shear in the bilayer are anisotropic. For isotropic materials with shear modulus G, the general Poynting relationship holds 26 : N 1 = σ 12 γ, which reduces to N 1 = Gγ 2 for small γ. This relation is predicted to hold for Kelvin foams if the stress is averaged over all possible orientations 27 . For bilayers, the numerical simulations show that N 1 /(Gγ) 2 is 1.31 or 2.55, which confirms that the elastic behavior exhibits significant anisotropy.

Then we vary the degree of confinement with respect to the reference isotropic state. Figure 7a shows the variations of both shear moduli as a function of the confinement ratio r = h/h o . We observe that the moduli strongly increase with r. When the bilayer is compressed from the reference state, the area S w of the hexagonal basis on the wall increases, thus the number of films per unit surface pulling on the wall decreases. At the same time, the tension exerted on the films increases since the perimeter L of the base increases. The results show that the effect of the area dominates, leading to a global decrease of the modulus as the compression increases. We plot in Fig. 7b the variations of the moduli normalized by their value in the reference state (r = 1), denoted G xz,o and G yz,o respectively. The numerical results show that, to second order in r, the relative variations of the moduli normalized by their value in the reference state (r = 1), denoted G xz,o and G yz,o respectively, are well described by the polynomial functions:

G xz G xz,o = 1 + 0.909(r -1) -0.27(r -1) 2 (13)
G yz G yz,o = 1 + 1.107(r -1) -0.11(r -1) 2 (14)

Trilayer shear modulus.

We simulate the shear elastic linear response of a trilayer similarly to that of a bilayer. We deduce its shear moduli G xz and G yz from the variations of the stress response to applied strain up to 0.02. In the isotropic state (r = 1), we find : G xz,oT = 1.22500 ± 2.10 -5 and G yz,oT = 0.67537 ± 2.10 -5 . The results in fig. 7a show the variations of G xz and G yz with the confinement ratio r. The large difference observed between G xz and G yz reflects the anisotropy of the trilayer structure. On both shear directions the trilayer foam becomes stiffer as the ratio r increases. The numerical results show that at second order in r, the variations of the moduli normalized by their value in the reference state are well described by the polynomial functions:

G xz G xz,oT = 1 + 1.008(r -1) -0.25(r -1) 2 (15)
G yz G yz,oT = 1 + 0.99(r -1) + 0.34(r -1) 2 (16)

In the relaxed isotropic reference state (r = 1), our simulations show that the FT cell has an interfacial energy density, scaled by T /V 1/3 , equal to E o,FT = 5.525. This value is larger than that of an undeformed Kelvin cell 22 : E o,K = 5.306. The larger value of E o,FT compared to E o,K reflects the supplementary cost of energy caused by the confining wall. The Kelvin cell in the core of the trilayer lowers the relative contribution of the FT cells to the total interfacial energy. Thus we expect the moduli of a trilayer to be lower than that of a bilayer, which is indeed the case. In the next section, we study the impact of the number of layers on the shear moduli.

N-layer shear modulus.

In this section, we consider multilayers in their isotropic undeformed reference state corresponding to a confinement ratio r = 1 (cf. Fig. 8) and, we analyze their response to an applied shear strain in the linear domain. As for the previous bi-and trilayers, we apply increasing shear strains ranging between 0.002 to 0.02, in either x or y directions and we deduce the corresponding shear moduli G xz and G yz from linear fits to the stress-strain data.

In Figure 10, we plot both shear moduli as a function of the number of layers N. As N increases from 2 to 11, we see that G xz and G yz both decrease monotonously. To evaluate the impact of confinement on the multilayer stiffness, we compare their shear moduli to those of a bulk Kelvin foam, evaluated in directions that correspond to similar orientations in the multilayer and in the Kelvin foam. For instance, let us consider a shear deformation ε xz applied to the N-multilayer along direction x (as defined in Fig. 1 or 2d-f). In a Kelvin foam, such a deformation would be equivalent to that obtained after rotating the cell crystallographic axis 45 • about the x o axis (cf. Fig. 2a), and then shearing the cell along the x direction. The calculation given in the appendix 7 shows that:

σ xz = G 1 γ (17)
where G 1 is defined by Eq. 1. Thus the multilayer shear modulus G xz can be compared to the shear modulus G 1 of a bulk Kelvin foam. Similarly a simple shear deformation ε yz applied to a Kelvin cell in the reference frame of Fig. 2d-f is equivalent to that obtained after rotating the cell crystallographic axis 45 • about the x o axis (cf. Fig. 2a), and then shearing the cell along the y direction. The calculation given in the appendix 7 shows that:

σ yz = G 2 γ (18)
with γ = 2ε yz . Hence the multilayer shear modulus G yz can be compared to the shear modulus G 2 of a bulk Kelvin foam. The data in Fig. 10 show that the shear moduli indeed asymptotically converge toward both values G 1 and G 2 expected for a bulk Kelvin foam.

To model the multilayer elasticity, we consider the structure to consist of three independent stacked layers (cf. Fig. 9): A core layer made of a Kelvin foam, with shear modulus G K , sandwiched between two FT surface layers with modulus G FT . A uniform shear stress σ applied to such a composite structure induces a local strain in the core layer:

γ K = σ /G K , (19)
and in each FT layer:

γ FT = σ /G FT . (20)
The local displacements in each layer add to build up the total displacement:

∆u = 2δ γ FT + (h o -2δ )γ K , (21)
where h o is the total thickness of the foam and δ is the thickness of one FT layer. We define the average strain:

γ av ≡ ∆u h o , (22)
and the effective shear modulus of the composite layer:

G e f f ≡ σ γ av . (23)
Combining Eq. 19 to 23 yields the expression:

h o G e f f = h o -2δ G K + 2δ G FT . (24)
As shown by the data of Table 1 and by Fig. 3, the total thickness varies linearly with N according to: h o = 2δ + p(N -2) with δ = 0.97765 and p = dh o /dN = 0.8940. Thus the variations of the effective modulus with the number of layers can be written as:

G e f f = 2δ + p(N -2) 2δ G FT + p(N-2) G K (25)
We expect this relation to hold for both shear directions with the corresponding FT moduli G xz,o and G yz,o found for the the bilayer (cf. section 4.2) and with the Kelvin moduli G 1 and G 2 (cf. section 2). As can be seen in Fig. 10, this prediction captures remarkably well the softening of the multilayer foam, for both shear moduli, without any free parameter. The small difference between the data and the prediction can be attributed to the fact that the 3 layers are not completely independent and are not strictly separated at the distance δ from each plate. δ is the distance between a plate and the top of the pentagonal faces of the FT cells. A part from it, there is a thin zone where the FT cells and the Kelvin cells overlap. The local non-affine distortion of the structure in the overlapping zone is not taken into account in our model. Furthermore, we deduce from these results the minimum number of bubble layers an ordered dry foam must have for its shear moduli to be representative of those of a bulk foam sample. We see that the relative difference between G xz and G 1 is less than 5% for N ≥ 14 and that the relative difference between G yz and G 2 is less than 5% for N ≥ 12. This is important for practical applications where mechanical properties of a foam confined in channels or pores must be assessed. As an extension of this study, it would be interesting to determine the Young modulus of a N-multilayer in response to a uniaxial stretching or compression along the z-direction.

Large deformations

Topological transitions.

As in section 4.1, we apply quasistatic deformations to multilayer foams with confinement in the range 0.7 < r < 1.5 and increase strain by increments of 0.02. Using the Surface Evolver, the structure is relaxed with R2 refinement(quadratic mode) after each strain step, and the stress is evaluated using eq. ( 5). We observe the evolution of the structure as faces and edges deform up to the strain where Plateau's laws are violated, which provokes irreversible topological transitions.

Let us recall the topological transitions that occur in a Kelvin foam under shear; these T1s cause cells to switch neighbors but they remain Kelvin. Three types of transitions can be triggered 28 . 1) A standard T1 occurs when opposite edges of a shrinking quadrilateral face go to zero length. This leads to a pair of 5- fold vertices that violate Plateau's laws and is analogous to the T1 in 2D soap froth 29 . 2) A point T1 occurs when a quadrilateral face shrinks to a point. 3) A triple T1 happens when opposite long curved edges of a shrinking hexagonal face touch at their center. Figure 11 shows the evolution of a bilayer and a trilayer, initially in the isotropic reference state (r = 1) as they are sheared in the x direction. The areas of a quadrilateral face and a hexagonal face diminish as the strain γ increases. Two opposite long edges of the hexagonal face eventually touch, which produces an unstable junction and triggers a triple T1. This transition is consistent with triple T1 induced in bulk Kelvin foam. We recall that the shear stress σ xz corresponds to the orientation 1.2 defined in ref. 28 . Triple T1s occur for all confinements: 0.7 < r < 1.5 for bilayers and 0.8 < r < 1.4 for trilayers.

Figure 12 shows the evolution of a bilayer and a trilayer, sheared in the y direction. As the strain γ increases, a square face shrinks to a point, triggering a point T1. Again, this transition is consistent with point T1 induced in bulk Kelvin foam since the shear stress σ yz corresponds to the orientation 1.3 in ref. 28 . Point T1s occur for all investigated confinements.

Yield stress and yield strain of multilayer foams.

Figure 13 shows the variations of the shear stress with the strain, applied either in the x or y direction, for bilayers and trilayers with different confinements. The largest strain γ y for each curve corresponds to the onset of topological transitions, the elastic limit, and the yield stress σ y . Note that the shear stress σ yz for the bilayers has a local maximum; consequently the yield stress associated with the yield strain is lower than the maximum stress. This is an artifact of the ordered structure. For both shear directions, these variations are consistent with those observed in a Kelvin foam 28 .

Finally, we show in figure 14, the variations of the yield strain of the multilayers as a function of r. Again, the mechanical response is anisotropic. γ y is larger in the x-z direction than in the y-z direction. Moreover we observe that the yield strain, thus the onset of T1s, appears earlier since the multilayer foam is stretched. Upon stretching in the z direction, i.e. for increasing ratio r, the lateral faces become narrow. Thus they vanish for a smaller applied strain. As a consequence, the yield strain γ y decreases as r increases. Now we turn to the effect of the number of layers N. In the reference state with r = 1 where the multilayer can be compared to a bulk Kelvin foam, we see that the multilayer yields for smaller strains than the Kelvin foam. For the bilayer, where the foam is more restricted, γ y is decreased by a factor of 0.7 and 0.4 in the x-z and y-z directions respectively. Figures 13 and14 clearly show that the non-linear elastic response of ordered confined foams is highly orientation dependent. This anisotropic elastic behavior is also a characteristic of the Kelvin foam.

Summary

We have shown that the elastic response of ordered soap froth confined between two rigid walls is stiffer than bulk foam. The Surface Evolver was used to calculate the equilibrium structure and nonlinear elastic behavior of N layers of confined cells. The increased stiffness is well described by a simple micro-mechanical model with no free parameter, by assuming that the foam consists of three independent layers: a layer of Fejes-Toth cells at each wall and a core of Kelvin cells that deform like a bulk foam. The nonlinear elastic response was analyzed up to the elastic limit, which coincides with the onset of topological transitions and determines the yield stress and yield strain. This response is highly anisotropic, similar to the Kelvin foam; the shape of the stressstrain curve, yield stress and yield strain are all highly dependent on foam orientation. The point and triple topological transitions that occur in the present study are a consequence of the highly symmetric deformations considered. The standard T1 was not found, but is by far the most prevalent during shearing flow of a Kelvin foam. Predicting the elastic behavior of confined soap froth is relevant to foam rheology in porous media and in the narrow channel geometries encountered in miniaturized lab-ona-chip applications for instance. Further developments should address the determination of the tensorial constitutive law in the domain of large strains, and the investigation of the elastic behavior of wet confined foams.

Appendix

For a crystal with cubic symmetry, the stress and strain tensor components expressed in the coordinate axes corresponding to the crystallographic axes (x o y o z o ) (cf. fig. 2a) write 22,[START_REF] Nye | Physical properties of crystals[END_REF] :

σ
In the reference frame (xyz) (cf. fig 2d,e,f), a shear strain γ applied along the x direction in the plane normal to the z direction is described by the strain tensor:

ε i j =    0 0 γ/2 0 0 0 γ/2 0 0    (27)
The corresponding strain tensor ε o in the crystallographic frame (x o y o z o ) is such that:

ε o = Q x ε Q -1 x , (28)
where Q x is the rotation matrix that performs counterclockwise rotation of 45 deg around the x axis. The stress tensor σ o in the crystallographic frame is then given by Eq. 26. Finally the stress tensor σ in the reference frame is deduced from:

σ = Q -1 x σ o Q x , (29)
which reads:

σ xx = σ yy = σ zz = 0 , σ xy = σ yz = 0 , σ xz = c 44 γ , (30)
Similarly, a shear strain γ applied along the y direction in the plane normal to the z direction is described by the strain tensor:

ε i j =    0 0 0 0 0 γ/2 0 γ/2 0    (31)
Calculating the strain and stress tensors in the crystallographic frame using eq. 28 and 29 gives:

σ xx = σ yy = σ zz = 0 , σ xy = σ xz = 0 , σ yz = 1 2 (c 11 -c 12 ) γ , (32)

Conflicts of interest

There are no conflicts to declare.

Fig. 1

 1 Fig. 1 Structure of multilayer foams confined between two parallel walls (represented by the rectangular frames) normal to the z direction. The cell faces in contact with the wall are shown in yellow. a) A bilayer (N = 2) consisting of 2 Fejes-Toth cell layers; b) A trilayer (N = 3) consisting of two Fejes-Toth cell layer in contact with each wall and a core layer of Kelvin cells. The multilayers have relaxed isotropic structure with lattice parameters given in table 1 (See text for definition.) corresponding to gap separation h o .

Fig. 2

 2 Fig. 2 Structure of Kelvin and Fejes-Toth (FT) bubbles in a dry multilayer. a) A Kelvin cell with its crystallographic axes x o , y o , z o . b) A Kelvin cell rotated by 45 • around the x o axis. c) A FT cell obtained by cutting the Kelvin cell in b) along a plane perpendicular to the square face, and stretched along the z direction such that it has the same volume as the original Kelvin cell. The axes x, y, z refer to the fixed reference frame. d), e), f) give three orthogonal views of the FT cell. The FT structure matches the boundary condition of 90 • angles between the films and the x-y plane.

Fig. 3

 3 Fig.3 Thickness of the relaxed isotropic N-multilayer as a function of the total number N of layers. The symbols correspond to the data of Table1. The line is a linear regression to the data: h o = 1.9553+(0.8940± 0.0003)(N -2).

Fig. 4 Fig. 5

 45 Fig. 4 Visualization of the relaxed periodic cell of a bilayer for different confining ratios r = h/h o , varied between 0.7 and 1.5 as indicated, viewed along an orthogonal projection: a, b) on the x-z plane ; c, d) on the y-z plane. The cell is confined between two plates (not represented) normal to the z direction separated by a distance h. The value r = 1 corresponds to the isotropic reference state. See section 3.1. a) and c) show the undeformed bilayers. b) (resp. d) show the relaxed structure as the bilayer is sheared in the x (resp. y) direction. A large strain γ = 0.1 is applied for the visualization, but much smaller strains are used to calculate the shear moduli.

Fig. 6 a

 6 Fig. 6 a) Orthogonal projection of the undeformed FT bilayer showing the hexagonal basis in contact with the wall. Under a x-z shear, all 6 perpendicular films contribute to the shear stress σ xz . Under a y-z shear, films labeled 2 and 5 remain parallel to the direction of motion. Thus the surface tension remains perpendicular to the plate and does not contribute to the shear stress σ yz . b) Schematic drawing of surface tension acting on film 2 undergoing a x-z strain (top) or a y-z strain (down).

Fig. 7

 7 Fig. 7 Bilayer (N = 2) and trilayer (N = 3) shear moduli. a) Variation of G xz and G yz as a function of the confinement ratio r = h/h o when the foam is sheared in either x (filled symbols) or y (empty symbols) direction. The moduli are scaled by T /V 1/3 . b) Moduli normalized by their value for r = 1 plotted as a function of r -1. The continuous lines are quadratic fits to the data (Eq. 13 and 14). Same symbols as in a).

Fig. 8

 8 Fig. 8 Basic cell of the N-multilayer foam structure. It consists of one Fejes-Toth cell at the bottom, one at the top, and N -2 Kelvin internal cells. The basic cell is infinitely repeated in the x and y directions. The parallel confining plates (not shown) lie in the x-y planes. The represented structures correspond to the lattice parameter h o (or r = 1) in the z direction such that the multilayer is in its reference isotropic state.

Fig. 9

 9 Fig.9 Schema of the multilayer foam consisting of 3 independent layers. An applied shear stress σ induces local displacements in each layer that add up to build the total displacement ∆u.

FTFig. 10

 10 Fig. 10 Variation of the shear modulus of the multilayer foam as a function of the number of layers N. The symbols correspond to the data as labeled. The equivalent shear moduli for a bulk Kelvin foam are indicated as G 1 and G 2 (dotted lines, see text). The continuous full lines represent the prediction by the layer model (eq. 25) with the bilayer and Kelvin moduli: G FT = G xz,o = 1.370 and G K = G 1 = 0.965 for the x-z shear and, G FT = G yz,o = 0.777 and G K = G 2 = 0.571 for the y-z shear. The moduli are scaled by T /V 1/3 .

Fig. 11

 11 Fig. 11 Large shear deformations applied in the x direction, for different strain amplitudes γ = 2ε xz as indicated to: a) a bilayer with r = 1; b) a trilayer with r = 1. The area of the square (in red) and hexagonal (in blue) faces decrease. The two opposite long edges of the hexagonal faces will touch and trigger a triple T1, as in bulk Kelvin foam; c) a Kelvin cell sheared in the same orientation 28 . A triple T1 is triggered at the strain: a) γ y = 1.04, b) γ y = 1.16, c) γ y,K = 1.47.

Fig. 12

 12 Fig. 12 Large shear deformations applied in the y direction, for different strain amplitudes γ = 2ε yz as indicated to: a) a bilayer with r = 0.7; b) a trilayer with r = 1. The square face (in red) contracts to a point and triggers a point T1, as in bulk Kelvin foam; c) a Kelvin cell sheared in the same orientation 28 . A point T1 is triggered at the strain a) γ = 0.40, b) γ = 0.32, c) γ y,K = 0.548.

  xx = c 11 ε xx + c 12 (ε yy + ε zz ), σ yy = c 11 ε yy + c 12 (ε xx + ε zz ), σ zz = c 11 ε zz + c 12 (ε xx + ε yy ), σ xy = 2c 44 ε xy , σ yz = 2c 44 ε yz , σ xz = 2c 44 ε xz ,

Fig. 13 Fig. 14

 1314 Fig.13 Stress-strain curves simulated for y-z shear (left axis) and x-z shear (right axis): a) bilayers and, b) trilayers, for different confinement ratios r as labeled. The stress is expressed in T /V 1/3 units.

1-12

| 1-12

1-12 J o u r n a l N a me , [ y e a r ] , [ v o l . ] ,

Acknowledgements

We thank R. Höhler and T. Baumberger for fruitful discussions. We acknowledge financial support from Centre national d'Etudes Spatiales (research agreement CNES/CNRS number 180113), and the European Space Agency (Soft Matter Dynamics MAP grant AO 09-943).

Notes and references

1 I. Cantat, S. Cohen-Addad, F. Elias, F. Graner, R. Höhler, O. Pitois, F. Rouyer and A. Saint-Jalmes, Foams: Structure and