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open science

# Interfaces in motion in elementary liquid foams

Théo Lenavetier

► To cite this version:

Théo Lenavetier. Interfaces in motion in elementary liquid foams. Physics [physics]. Université de Rennes, 2023. English. NNT: . hal-03013537

HAL Id: hal-03013537

https://hal.science/hal-03013537

Submitted on 29 Jan 2024

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
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abroad, or from public or private research centers.

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destinée au dépôt et à la diffusion de documents
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émanant des établissements d'enseignement et de
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Public Domain
<figure><img src="image_1.png" /></figure>

HAL
open science

# Interfaces in motion in elementary liquid foams

Théo Lenavetier

► To cite this version:

Théo Lenavetier. Interfaces in motion in elementary liquid foams. Physics [physics]. Université de Rennes, 2023. English. NNT: . hal-03013537

HAL Id: hal-03013537

https://hal.science/hal-03013537v1

Submitted on 29 Jan 2024

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

Public Domain
THESE DE DOCTORAT DE

L'UNIVERSITE DE RENNES

ECOLE DOCTORALE N° 638

Science de la Matière, des Molécules et Matériaux

Spécialité : Physique

Par

Théo LENAVETIER

Mouvements aux interfaces de mousses liquides élémentaires

Traduit de l'anglais : Interfaces in motion in elementary liquid foams

Thèse présentée et soutenue à Rennes, le 13 décembre 2023

Unité de recherche : Institut de Physique de Rennes – UMR 6251

Rapporteurs avant soutenance :

Christophe YBERT

Directeur de Recherche, CNRS, ILM

Jacco SNOEIJER

Professeur, Université de Twente

Composition du Jury :

Président : Elisabeth CHARLAIX

Professeur des universités, Université Grenoble Alpes, LiPhy

Examinateurs : Elise LORENCEAU

Thomas SALEZ

Christophe YBERT

Jacco SNOEIJER

Directrice de Recherche, CNRS, LiPhy

Chargé de Recherche, CNRS, LOMA

Directeur de Recherche, CNRS, ILM

Professeur, Université de Twente

Dir. de thèse : Isabelle CANTAT

Professeur des universités, Université de Rennes, IPR
THESE DE DOCTORAT DE

L'UNIVERSITE DE RENNES

ECOLE DOCTORALE N° 638

Science de la Matière, des Molécules et Matériaux

Spécialité : Physique

Par

Théo LENAVETIER

Mouvements aux interfaces de mousses liquides élémentaires

Traduit de l'anglais : Interfaces in motion in elementary liquid foams

Thèse présentée et soutenue à Rennes, le 13 décembre 2023

Unité de recherche : Institut de Physique de Rennes – UMR 6251

Rapporteurs avant soutenance :

Christophe YBERT

Directeur de Recherche, CNRS, ILM

Jacco SNOEIJER

Professeur, Université de Twente

Composition du Jury :

Président : Elisabeth CHARLAIX

Professeur des universités, Université Grenoble Alpes, LiPhy

Examinateurs : Elise LORENCEAU

Thomas SALEZ

Christophe YBERT

Jacco SNOEIJER

Directrice de Recherche, CNRS, LiPhy

Chargé de Recherche, CNRS, LOMA

Directeur de Recherche, CNRS, ILM

Professeur, Université de Twente

Dir. de thèse : Isabelle CANTAT

Professeur des universités, Université de Rennes, IPR
<figure><img src="image_2.png" /></figure>
<figure><img src="image_2.png" /></figure>
Ah! Mais ch'est èn Anguiais enco en pu c't'affaire?

(Normand du centre Manche)
Marie-Thérèse Gouesmel, ma grand-mère, en
découvrant la langue de ce manuscrit.
Ah! Mais ch'est èn Anguiais enco en pu c't'affaire?

(Normand du centre Manche)
Marie-Thérèse Gouesmel, ma grand-mère, en
découvrant la langue de ce manuscrit.
Remerciements

J'ai tricoté tous ces paragraphes de "Mercis" petit à petit, le long de l'écriture de ce manuscrit et jusqu'à quelques temps après ma soutenance. Cette période intense aura été ponctuée de moments de gratitude dont ces (longs) remerciements sont le fruit. Ma thèse représente pour moi la fin d'un parcours scolaire puis universitaire le long duquel j'ai pu faire de nombreuses rencontres, sans lesquelles il aurait été bien différent. Pour cette raison et parce que je m'y suis plu, je voudrais remercier pour cela les personnes qui m'ont permis de le réaliser.

Je remercie tout d'abord le jury de cette thèse, pour leur temps, leur intérêt pour ces travaux, et leurs lectures de mon manuscrit. Merci à Christophe Ybert et Jacco Snoeijer, les rapporteurs, pour leurs remarques qui ont mené à des discussions très riches, en amont, pendant la soutenance, et à continuer après. Mon travail et ma réflexion sur les systèmes physiques de ma thèse en sortent grandis. Merci à Thomas Salez et Elise Lorenceau, examinateurs, et Elisabeth Charlaix, présidente du jury, pour la séance de questions qui allaient de la physico-chimie utilisée/à essayer aux ingrédients physiques profonds derrière les équations des écoulements présentés dans cette thèse. Je n'ai pas vu le temps de question passer, c'était juste passionnant.

Je remercie Isabelle Cantat, ma directrice de thèse. J'ai passé trois années scientifiques et humaines incroyables à l'IPR, et je te dois beaucoup pour cela. J'ai eu les encouragements, les ressources, la présence et l'espace qu'il faut pour m'approprier les notions et les codes nécessaires à la thèse. Avec cela, j'ai eu la chance de m'engager dans la vie du laboratoire, enseigner, partir en conférence souvent, et bien sûr une superbe expérience d'avoir pu partir à Cambridge en stage. Tu m'as fait découvrir ton métier et communiqué ta passion. Tu as été une mentor et une collègue bienveillante, toujours présente et à l'écoute. J'ai énormément d'admiration et de gratitude pour tout cela, et je suis extrêmement fier d'avoir été à ton école.

Je remercie les employés de l'IPR et sa direction, pour leur accueil et leur aide: Bénédicte Faure et Valérie Ferry, pour leur assistance et présence auprès des non-permanents, Nathalie Mabic, pour sa gentillesse et sa disponibilité, Jean-Charles Potier, pour sa jovialité et les pièces de méca toujours nickel, Patrick Chasles, qui m'a bien aidé sur l'électronique de ma manipe, Jean-Christophe Sangleboeuf, pour son accueil et nos discussions, Florian Scholkopf, pour les pauses café, et Jérémy Gardais, pour nos discussions occasionnelles mais toujours très agréables.

Merci à Emmanuel Schaub, ingénieur de recherche au département matière-molle. Cela a été un vrai plaisir de travailler avec toi sur mon montage expérimental. Tu as toujours pris le temps de répondre à toutes mes questions sur l'optique, l'électronique et l'interfaçage, et ça a été très formateur pour moi.

Merci aux enseignants-chercheurs avec qui j'ai eu la chance de travailler et/ou partager mon environnement de travail: Axelle Amon, pour les discussions scientifiques et rôlistiques, Laurent Courbin, notre adoré et révéré chef, Sylvie Beaufils, avec qui c'est un plaisir d'enseigner, et Jérôme Crassous, généreux en miel et en explications apicoles.

Merci aux non-permanents de l'IPR qui ont vu mon arrivée au labo et m'y ont accueilli: Donatien Mottin, mon camarade rôliste et de théâtre de l'ESPCI, Nolwenn Delouche, qui m'a fait don des séminaires non-permanents, Raphaël Poryles, qui m'a lancé sur les manipes, Jean-Baptiste Besnard, et sa grande culture musicale, Jeanne Kergomard, pour nos importantes discussions,
Remerciements

J'ai tricoté tous ces paragraphes de "Mercis" petit à petit, le long de l'écriture de ce manuscrit et jusqu'à quelques temps après ma soutenance. Cette période intense aura été ponctuée de moments de gratitude dont ces (longs) remerciements sont le fruit. Ma thèse représente pour moi la fin d'un parcours scolaire puis universitaire le long duquel j'ai pu faire de nombreuses rencontres, sans lesquelles il aurait été bien différent. Pour cette raison et parce que je m'y suis plu, je voudrais remercier pour cela les personnes qui m'ont permis de le réaliser.

Je remercie tout d'abord le jury de cette thèse, pour leur temps, leur intérêt pour ces travaux, et leurs lectures de mon manuscrit. Merci à Christophe Ybert et Jacco Snoeijer, les rapporteurs, pour leurs remarques qui ont mené à des discussions très riches, en amont, pendant la soutenance, et à continuer après. Mon travail et ma réflexion sur les systèmes physiques de ma thèse en sortent grandis. Merci à Thomas Salez et Elise Lorenceau, examinateurs, et Elisabeth Charlaix, présidente du jury, pour la séance de questions qui allaient de la physico-chimie utilisée/à essayer aux ingrédients physiques profonds derrière les équations des écoulements présentés dans cette thèse. Je n'ai pas vu le temps de question passer, c'était juste passionnant.

Je remercie Isabelle Cantat, ma directrice de thèse. J'ai passé trois années scientifiques et humaines incroyables à l'IPR, et je te dois beaucoup pour cela. J'ai eu les encouragements, les ressources, la présence et l'espace qu'il faut pour m'approprier les notions et les codes nécessaires à la thèse. Avec cela, j'ai eu la chance de m'engager dans la vie du laboratoire, enseigner, partir en conférence souvent, et bien sûr une superbe expérience d'avoir pu partir à Cambridge en stage. Tu m'as fait découvrir ton métier et communiqué ta passion. Tu as été une mentor et une collègue bienveillante, toujours présente et à l'écoute. J'ai énormément d'admiration et de gratitude pour tout cela, et je suis extrêmement fier d'avoir été à ton école.

Je remercie les employés de l'IPR et sa direction, pour leur accueil et leur aide: Bénédicte Faure et Valérie Ferry, pour leur assistance et présence auprès des non-permanents, Nathalie Mabic, pour sa gentillesse et sa disponibilité, Jean-Charles Potier, pour sa jovialité et les pièces de méca toujours nickel, Patrick Chasles, qui m'a bien aidé sur l'électronique de ma manipe, Jean-Christophe Sangleboeuf, pour son accueil et nos discussions, Florian Scholkopf, pour les pauses café, et Jérémy Gardais, pour nos discussions occasionnelles mais toujours très agréables.

Merci à Emmanuel Schaub, ingénieur de recherche au département matière-molle. Cela a été un vrai plaisir de travailler avec toi sur mon montage expérimental. Tu as toujours pris le temps de répondre à toutes mes questions sur l'optique, l'électronique et l'interfaçage, et ça a été très formateur pour moi.

Merci aux enseignants-chercheurs avec qui j'ai eu la chance de travailler et/ou partager mon environnement de travail: Axelle Amon, pour les discussions scientifiques et rôlistiques, Laurent Courbin, notre adoré et révéré chef, Sylvie Beaufils, avec qui c'est un plaisir d'enseigner, et Jérôme Crassous, généreux en miel et en explications apicoles.

Merci aux non-permanents de l'IPR qui ont vu mon arrivée au labo et m'y ont accueilli: Donatien Mottin, mon camarade rôliste et de théâtre de l'ESPCI, Nolwenn Delouche, qui m'a fait don des séminaires non-permanents, Raphaël Poryles, qui m'a lancé sur les manipes, Jean-Baptiste Besnard, et sa grande culture musicale, Jeanne Kergomard, pour nos importantes discussions,
Maxime Vassaux, pour avoir mis l'ambiance et les discussions scientifiques toujours passionnantes, Yoann Raffray, Theany To et Alexis Duval, avec qui je n'ai que des bons moments et n'ai pas fait assez de soirées, Gaël Privault, avec qui j'ai commencé la thèse et vécu les épreuves de la fin en simultané, et Elliot Speirs, qui m'a fait découvrir les whiskies écossais.

Merci aux non-permanents que j'ai vus arriver et dont je ne pourrai maintenant suivre le parcours qu'à distance en leur souhaitant bon vent: Amélie Godard-Palluet, la super représentante des non-permanents, Alberto Macario-Farto, pour ta sympathie et d'avoir repris les séminaires non-permanents, Marion Baudoin et Laurianne Chonchon, avec qui on se croise ponctuellement pour discuter.

Merci à Thomas Mabit, doctorant arrivé 1 an après moi, pour ta présence douce et tes explications de méthodes numériques toujours riches, claires et passionnantes. Bravo pour ton poste !

Merci à Alexy Brunel, doctorant arrivé 2 ans après moi, pour le support technique (et moral!) le jour de la soutenance. Ca m'a beaucoup aidé. Ta culture générale scientifique m'impressionne (de la physique à la bio/médecine), je suppose qu'il faut au moins ça pour rendre de l'eau comestible ;)

Merci à Noémie Ourvois-Maloisel, doctorante arrivée 1 an après moi en thèse, mais 1 an avant moi à l'IPR en stage. La vraie doyenne des non-permanents. Ta jovialité et les moments de détente que l'on a sont très rafraîchissants. C'était bien sympa de me faire découvrir l'analyse senso et de servir de cobaye pour tes formulations de faumage.

Merci à Youna Louyer et Nathan Chapelle, qui arrivent reprendre le flambeau en tant que doctorants et qui dynamisent déjà l'équipe. Venir en thèse en quête de défi intellectuel après une ou plusieurs années d'enseignement est une démarche pour laquelle j'ai un immense respect. Nul doute que ces trois années vous sourient.

Merci à Marion Berry, François Giavassis, Milène Aubry, Gaëlle Audéoud et Gabriel de Oliveira Rodrigues, stagiaires dans notre groupe tout au long de ma thèse. Partager ma salle de manipe avec chacun d'entre vous à été un plaisir, et encadrer Gaëlle et Gabriel ont été d'excellentes expériences dont je sors grandi.

Un gros merci en particulier à Thomas, Alexy, Noémie, Nathan, Lucas, Florian, Ambroise, Axelle, Isabelle, Milène et Youna pour l'aide apportée le jour de la soutenance et le magnifique pot que vous m'avez réalisé. Merci beaucoup à Marion Berry, Alice Etienne-Simonetti, Corentin Trégouët, Adrien Bussonnière, Louis Ginabat et Pierre Glidic, qui m'ont fait la magnifique surprise de venir à Rennes assister à ma soutenance: vous êtes des amis et consoeurs/confrères adorables ! Cette journée s'est passée exactement comme je l'ai rêvée, et c'est notamment grâce à votre aide et votre présence à toutes et tous :)

Merci à Corentin Trégouët, qui est un peu devenu mon grand frère du milieu académique, et pour tout ce que tu as fait pour moi depuis 3 ans. De la visite d'appartement pour mon compte lors de mon arrivée à Rennes en temps de covid, aux week-ends passés chez Chloé et toi pour relâcher un peu la pression pendant la rédaction. Partager avec toi tous ces moments en conférence, en GDR, ou simplement autour d'un tableau à craie sont des épisodes de vie scientifique et humaine que je chérie. Ton soutien, ton expérience et ton amitié m'ont rendu la vie plus douce.
Maxime Vassaux, pour avoir mis l'ambiance et les discussions scientifiques toujours passionnantes, Yoann Raffray, Theany To et Alexis Duval, avec qui je n'ai que des bons moments et n'ai pas fait assez de soirées, Gaël Privault, avec qui j'ai commencé la thèse et vécu les épreuves de la fin en simultané, et Elliot Speirs, qui m'a fait découvrir les whiskies écossais.

Merci aux non-permanents que j'ai vus arriver et dont je ne pourrai maintenant suivre le parcours qu'à distance en leur souhaitant bon vent: Amélie Godard-Palluet, la super représentante des non-permanents, Alberto Macario-Farto, pour ta sympathie et d'avoir repris les séminaires non-permanents, Marion Baudoin et Laurianne Chonchon, avec qui on se croise ponctuellement pour discuter.

Merci à Thomas Mabit, doctorant arrivé 1 an après moi, pour ta présence douce et tes explications de méthodes numériques toujours riches, claires et passionnantes. Bravo pour ton poste !

Merci à Alexy Brunel, doctorant arrivé 2 ans après moi, pour le support technique (et moral!) le jour de la soutenance. Ca m'a beaucoup aidé. Ta culture générale scientifique m'impressionne (de la physique à la bio/médecine), je suppose qu'il faut au moins ça pour rendre de l'eau comestible ;)

Merci à Noémie Ourvois-Maloisel, doctorante arrivée 1 an après moi en thèse, mais 1 an avant moi à l'IPR en stage. La vraie doyenne des non-permanents. Ta jovialité et les moments de détente que l'on a sont très rafraîchissants. C'était bien sympa de me faire découvrir l'analyse senso et de servir de cobaye pour tes formulations de faumage.

Merci à Youna Louyer et Nathan Chapelle, qui arrivent reprendre le flambeau en tant que doctorants et qui dynamisent déjà l'équipe. Venir en thèse en quête de défi intellectuel après une ou plusieurs années d'enseignement est une démarche pour laquelle j'ai un immense respect. Nul doute que ces trois années vous sourient.

Merci à Marion Berry, François Giavassis, Milène Aubry, Gaëlle Audéoud et Gabriel de Oliveira Rodrigues, stagiaires dans notre groupe tout au long de ma thèse. Partager ma salle de manipe avec chacun d'entre vous à été un plaisir, et encadrer Gaëlle et Gabriel ont été d'excellentes expériences dont je sors grandi.

Un gros merci en particulier à Thomas, Alexy, Noémie, Nathan, Lucas, Florian, Ambroise, Axelle, Isabelle, Milène et Youna pour l'aide apportée le jour de la soutenance et le magnifique pot que vous m'avez réalisé. Merci beaucoup à Marion Berry, Alice Etienne-Simonetti, Corentin Trégouët, Adrien Bussonnière, Louis Ginabat et Pierre Glidic, qui m'ont fait la magnifique surprise de venir à Rennes assister à ma soutenance: vous êtes des amis et consoeurs/confrères adorables ! Cette journée s'est passée exactement comme je l'ai rêvée, et c'est notamment grâce à votre aide et votre présence à toutes et tous :)

Merci à Corentin Trégouët, qui est un peu devenu mon grand frère du milieu académique, et pour tout ce que tu as fait pour moi depuis 3 ans. De la visite d'appartement pour mon compte lors de mon arrivée à Rennes en temps de covid, aux week-ends passés chez Chloé et toi pour relâcher un peu la pression pendant la rédaction. Partager avec toi tous ces moments en conférence, en GDR, ou simplement autour d'un tableau à craie sont des épisodes de vie scientifique et humaine que je chérie. Ton soutien, ton expérience et ton amitié m'ont rendu la vie plus douce.
Merci à Adrien Bussonnière, mon prédécesseur sur mon montage expérimental. Tu m'as refourgué un sacré bestiau ! Et j'ai eu 3 ans pour apprécier tout le travail, la minutie et la pugnacité que tu as mis dedans. Merci pour tes conseils toujours judicieux, tes offres de postdocs, tes matrices élevées à la puissance 22... et tous les coups que tu m'as payés en conf/GDR! (j'ai arrêté de compter)

Merci à Jacques-Teïva Baué, qui m'a précédé à la présidence du bureau 007 en son temps. Mon camarade, mon partner, pour tous les coups géniaux à foireux sur ces deux ans passés ensemble. Du mariage en tant que chauffeurs ensemble jusqu'au fin fond du Grand Canyon, en passant par un road trip andalou, tout est gravé et précieusement gardé. Tu as une énergie positive, un sens de la répartie et une détermination qui font de toi une rencontre qui m'a marqué, et un ami formidable. Les fous rire que tu m'as offerts pendant cette thèse résonneront longtemps dans ma mémoire !

Merci à Adrien Gans, le post-doc prodigue en café et en bons conseils, maintenant maître pokémon de conférence à Nancy. Tu es une force tranquille de l'ESR, une source intarissable d'enthousiasme pour les sciences... et de mèmes de qualité. J'ai beaucoup apprécié nos discussions craie-tableau, tes cours d'économie politiquement très neutres, et bien sûr partager ces instants en dehors du labo et à ton mariage. Cette année passée ensemble m'a beaucoup apporté.

Merci à Ambroise Mathey, mon ami arrivé 1 an après moi en tant que doctorant. "Little knowledge of lab earthquakes, large knowledge of Rennes' pubs", il se présente comme ça ! Tu as une grande humilité et un si bel éclectisme face aux sciences, discuter avec toi est toujours très riche. J'y ai trouvé des questions qui m'ont fait avancer sur ma compréhension de nos sujets respectifs, découvert des problèmes numériques que je ne m'étais jamais posés. Ta culture et ta capacité d'apprentissage sur ces sujets est impressionnante. Nos nombreuses soirées avec tes anecdotes incroyables à partir de 2 pintes vont beaucoup me manquer.

Merci à Lucas Jannin, mon ami arrivé 2 ans après moi en tant que doctorant et actuel président du bureau 007 (puisque Ambroise a préféré le rôle d'éminence grise). Derrière tout le sel que tu balances gentiment, il y a un coeur un or. J'ai beaucoup apprécié ces moments à l'escalade, autour d'un verre ou du tableau pour décider comment on va détecter tes fichues ellipses. Merci pour tout ça!

Merci à Maryam Sadeghiyan Dehaghani, doctorante arrivée peu de temps avant moi. My dear friend ! Échanger avec toi sur nos parcours a été très enrichissant. Merci de m'avoir fait découvrir l'incroyable cuisine iranienne et des bribes de persan à travers de belles musiques.

Merci à Pierre Glidic, mon binôme de TP à l'ESPCI et ami depuis les premiers jours où j'y ai mis les pieds. Ta curiosité, ta culture et ton humour sont précieux. Construire ma pensée scientifique, découvrir la physique avec un camarade comme toi a été une grande chance. Tu fais un fier docteur depuis le 24 novembre dernier, et je n'ai eu que trop hâte de te rejoindre en tant que confrère, mon ami.

Merci à Anaïs Gauthier, ma co-bureau pendant l'écriture de ce manuscrit. Tu es une force positive et une marraine des sciences qui m'a appris énormément pour ce qui est de naviguer dans ce vaste milieu académique. Discuter de la vie, trouver des idées de manipes, et s'enthousiasmer avec toi sur des résultats a été très riche et rafraîchissant. Je ne t'aurai pas convaincue avec ce que tu appelles mes "thés de princesse", mais ils auront été bus en si bonne compagnie.
Merci à Adrien Bussonnière, mon prédécesseur sur mon montage expérimental. Tu m'as refourgué un sacré bestiau ! Et j'ai eu 3 ans pour apprécier tout le travail, la minutie et la pugnacité que tu as mis dedans. Merci pour tes conseils toujours judicieux, tes offres de postdocs, tes matrices élevées à la puissance 22... et tous les coups que tu m'as payés en conf/GDR! (j'ai arrêté de compter)

Merci à Jacques-Teïva Baué, qui m'a précédé à la présidence du bureau 007 en son temps. Mon camarade, mon partner, pour tous les coups géniaux à foireux sur ces deux ans passés ensemble. Du mariage en tant que chauffeurs ensemble jusqu'au fin fond du Grand Canyon, en passant par un road trip andalou, tout est gravé et précieusement gardé. Tu as une énergie positive, un sens de la répartie et une détermination qui font de toi une rencontre qui m'a marqué, et un ami formidable. Les fous rire que tu m'as offerts pendant cette thèse résonneront longtemps dans ma mémoire !

Merci à Adrien Gans, le post-doc prodigue en café et en bons conseils, maintenant maître pokémon de conférence à Nancy. Tu es une force tranquille de l'ESR, une source intarissable d'enthousiasme pour les sciences... et de mèmes de qualité. J'ai beaucoup apprécié nos discussions craie-tableau, tes cours d'économie politiquement très neutres, et bien sûr partager ces instants en dehors du labo et à ton mariage. Cette année passée ensemble m'a beaucoup apporté.

Merci à Ambroise Mathey, mon ami arrivé 1 an après moi en tant que doctorant. "Little knowledge of lab earthquakes, large knowledge of Rennes' pubs", il se présente comme ça ! Tu as une grande humilité et un si bel éclectisme face aux sciences, discuter avec toi est toujours très riche. J'y ai trouvé des questions qui m'ont fait avancer sur ma compréhension de nos sujets respectifs, découvert des problèmes numériques que je ne m'étais jamais posés. Ta culture et ta capacité d'apprentissage sur ces sujets est impressionnante. Nos nombreuses soirées avec tes anecdotes incroyables à partir de 2 pintes vont beaucoup me manquer.

Merci à Lucas Jannin, mon ami arrivé 2 ans après moi en tant que doctorant et actuel président du bureau 007 (puisque Ambroise a préféré le rôle d'éminence grise). Derrière tout le sel que tu balances gentiment, il y a un coeur un or. J'ai beaucoup apprécié ces moments à l'escalade, autour d'un verre ou du tableau pour décider comment on va détecter tes fichues ellipses. Merci pour tout ça!

Merci à Maryam Sadeghiyan Dehaghani, doctorante arrivée peu de temps avant moi. My dear friend ! Échanger avec toi sur nos parcours a été très enrichissant. Merci de m'avoir fait découvrir l'incroyable cuisine iranienne et des bribes de persan à travers de belles musiques.

Merci à Pierre Glidic, mon binôme de TP à l'ESPCI et ami depuis les premiers jours où j'y ai mis les pieds. Ta curiosité, ta culture et ton humour sont précieux. Construire ma pensée scientifique, découvrir la physique avec un camarade comme toi a été une grande chance. Tu fais un fier docteur depuis le 24 novembre dernier, et je n'ai eu que trop hâte de te rejoindre en tant que confrère, mon ami.

Merci à Anaïs Gauthier, ma co-bureau pendant l'écriture de ce manuscrit. Tu es une force positive et une marraine des sciences qui m'a appris énormément pour ce qui est de naviguer dans ce vaste milieu académique. Discuter de la vie, trouver des idées de manipes, et s'enthousiasmer avec toi sur des résultats a été très riche et rafraîchissant. Je ne t'aurai pas convaincue avec ce que tu appelles mes "thés de princesse", mais ils auront été bus en si bonne compagnie.
Merci à Sylvain Deville et Félix Ginot, qui m'ont encadré durant mon premier stage de recherche en 2018 au LSFC. C'était mon premier contact professionnel avec le monde académique, et ma première expérience de recherche tout court. En plus de l'aspect scientifique qui m'a beaucoup passionné, j'ai eu avec vous 6 mois très importants qui ont très largement contribué à me faire commencer cette thèse et vouloir continuer en recherche.

Merci à Franck Lesieur, mon prof de maths en spé. Votre passion pour les maths m'a marqué, et j'en garde clairement un enthousiasme teinté de respect pour cette discipline. Votre dévotion pour vos élèves m'a beaucoup aidé dans un moment difficile d'orientation scolaire. Ah, et votre "boîte à outil" m'aura servi exactement 3 fois dans cette thèse !

Merci à Aline Barazer, ma prof de chimie en sup. Vous avez cru en mes capacités, et sans vos mots au bon moment je ne me serais probablement pas donné les moyens d'intégrer l'ESPCI.

Merci à Nicolas Guillouet, mon prof de physique en sup. Votre sens de la pédagogie et votre enthousiasme pour transmettre votre savoir ont beaucoup contribué à m'orienter vers la physique. J'essaye d'arriver devant mes élèves avec le même élan que vous.

Merci à Emmanuel Brouard, mon prof de maths en seconde et terminale. L'entrain et l'énergie que vous avez dans votre métier m'ont beaucoup marqué. Je m'en inspire le plus possible pour enseigner, et je m'y découvre une passion. Vous m'avez écouté, encouragé et orienté pour mon envol vers le supérieur. Cela a été très important pour moi. Surveillant le jour de mon bac de maths, un sourire aux lèvres: "Je me mets à côté de Théo, il triche tout le temps!". Passer cette épreuve sous votre égide a été un temps fort de mon lycée.

Merci à Fabrice Vimond, mon prof de physique-chimie de collège. Je me souviens de vos passionnantes petites expériences en fin de cours, et votre passion pour transmettre aux jeunes de la belle science. Mention spéciale pour celle où on relie un petit et un grand ballon de baudruches et on se demande lequel va se vider dans l'autre. C'était mon premier contact avec la pression de Laplace, qui, 15 ans plus tard, est un des phénomènes centraux de ce manuscrit. Merci pour la leçon d'avance!

Enfin, Merci à ma famille pour son soutien de chaque instant, à chaque étape de mon parcours: Annie, ma mère, Paul et Zoé, mon frère et ma soeur, Jean-Marie, Dominique, Colette et Françoise, mes oncles et tantes, Benjamin et Margot, mes cousins, Jean-Claude et Marie-Thérèse, mes grands-parents, et Swanny, ma conjointe. Votre amour, vos encouragements et votre présence sont un moteur qui m'a permis d'identifier tôt et vivre pleinement ma passion pour les sciences, et me donne les moyens pour que je puisse en faire mon métier. Sans tout ça, l'apprenti scientifique qui arrivait le matin au labo n'aurait pas eu le même épanouissement, la même envie, la même passion. En cela, cette thèse vous doit beaucoup.
Merci à Sylvain Deville et Félix Ginot, qui m'ont encadré durant mon premier stage de recherche en 2018 au LSFC. C'était mon premier contact professionnel avec le monde académique, et ma première expérience de recherche tout court. En plus de l'aspect scientifique qui m'a beaucoup passionné, j'ai eu avec vous 6 mois très importants qui ont très largement contribué à me faire commencer cette thèse et vouloir continuer en recherche.

Merci à Franck Lesieur, mon prof de maths en spé. Votre passion pour les maths m'a marqué, et j'en garde clairement un enthousiasme teinté de respect pour cette discipline. Votre dévotion pour vos élèves m'a beaucoup aidé dans un moment difficile d'orientation scolaire. Ah, et votre "boîte à outil" m'aura servi exactement 3 fois dans cette thèse !

Merci à Aline Barazer, ma prof de chimie en sup. Vous avez cru en mes capacités, et sans vos mots au bon moment je ne me serais probablement pas donné les moyens d'intégrer l'ESPCI.

Merci à Nicolas Guillouet, mon prof de physique en sup. Votre sens de la pédagogie et votre enthousiasme pour transmettre votre savoir ont beaucoup contribué à m'orienter vers la physique. J'essaye d'arriver devant mes élèves avec le même élan que vous.

Merci à Emmanuel Brouard, mon prof de maths en seconde et terminale. L'entrain et l'énergie que vous avez dans votre métier m'ont beaucoup marqué. Je m'en inspire le plus possible pour enseigner, et je m'y découvre une passion. Vous m'avez écouté, encouragé et orienté pour mon envol vers le supérieur. Cela a été très important pour moi. Surveillant le jour de mon bac de maths, un sourire aux lèvres: "Je me mets à côté de Théo, il triche tout le temps!". Passer cette épreuve sous votre égide a été un temps fort de mon lycée.

Merci à Fabrice Vimond, mon prof de physique-chimie de collège. Je me souviens de vos passionnantes petites expériences en fin de cours, et votre passion pour transmettre aux jeunes de la belle science. Mention spéciale pour celle où on relie un petit et un grand ballon de baudruches et on se demande lequel va se vider dans l'autre. C'était mon premier contact avec la pression de Laplace, qui, 15 ans plus tard, est un des phénomènes centraux de ce manuscrit. Merci pour la leçon d'avance!

Enfin, Merci à ma famille pour son soutien de chaque instant, à chaque étape de mon parcours: Annie, ma mère, Paul et Zoé, mon frère et ma soeur, Jean-Marie, Dominique, Colette et Françoise, mes oncles et tantes, Benjamin et Margot, mes cousins, Jean-Claude et Marie-Thérèse, mes grands-parents, et Swanny, ma conjointe. Votre amour, vos encouragements et votre présence sont un moteur qui m'a permis d'identifier tôt et vivre pleinement ma passion pour les sciences, et me donne les moyens pour que je puisse en faire mon métier. Sans tout ça, l'apprenti scientifique qui arrivait le matin au labo n'aurait pas eu le même épanouissement, la même envie, la même passion. En cela, cette thèse vous doit beaucoup.
Résumé en Français

Les travaux de cette thèse portent sur les écoulements dans le plan de films de savon. C'est un point de vue d'hydrodynamicien qui vise à aider à mieux comprendre la rhéologie et le vieillissement des mousses liquides en s'attaquant à des échelles locales: le film de savon unique et la mousse liquide élémentaire, c'est à dire trois films reliés entre eux à 120° par un ménisque.

Dans l'introduction, je présente l'état de l'art sur la physique des films de savon et des mousses liquides élémentaires. Le lecteur y trouvera en premier lieu des définitions de quantités thermodynamiques importantes pour coupler la rhéologie d'une interface gas-liquide aux écoulements dans la phase liquide, puis une mise en équation pour les mouvements dans le plan des films de savon, qui représentent un cas extrême de confinement entre deux de ces interfaces. Cela fait poser et calculer quelques ordres de grandeurs nécessaires pour la suite en discutant notamment les échelles spatiales et temporelles pertinentes, ainsi que les propriétés physico-chimiques des solutions savonneuses (choix et propriétés des surfactants, viscosités etc...).

Ensuite, il est question de regarder plus spécifiquement la zone de jonction entre ces films plats et un ménisque courbé. Des régimes stationnaires et instationnaires d'évolution du profil d'épaisseur dans le film sont discutés selon l'écoulement imposé côté film.

Enfin, j'expose un état de l'art sur les mousses liquides élémentaires sous déformation imposée. Deux expériences précédentes (par Durand & Stone [1] et Besson & Debrégeas [2]) et des simulations numériques récentes (Titta [3]) ont montré la pertinence de cette échelle locale des mousses, que ce soit pour mieux comprendre la mécanique globale de ces matériaux, ou pour sonder la rhéologie de surface des interfaces qui les composent. Dans tous les cas, la réponse mécanique de la mousse élémentaire est montrée comme étant notamment déterminée par les quantités de surfactants échangées entre films, et entre les films et le ménisque. Ces échanges font systématiquement l'objet d'hypothèses fortes dans ces modèles, et tenter de les estimer par le calcul confronte la communauté à des variables physico-chimiques pas ou mal connues. Les connaître permettrait de résoudre le champ de vitesse dans la mousse élémentaire, pour lequel un modèle existe (Bussonnière & Cantat [4]), et de donner des lois rhéologiques locales physiques pour la rhéologie des mousses liquides.

La principale problématique de cette thèse est ainsi posée: peut-on quantifier ces échanges de surfactants entre films et entre films et le ménisque?

Le plan du manuscrit est construit avec un Chapitre 1 portant sur l'élasticité individuelle d'un film de savon, un Chapitre 2 sur une description théorique et une mesure d'une tension de ligne d'origine purement capillaire dans les films, et la modélisation des écoulements qu'elle génère dans le plan du film, et un Chapitre 3 portant sur la quantification des échanges de surfactants au sein d'une mousse liquide élémentaire entre films et entre les films et le ménisque.

Le Chapitre 1 commence par présenter un modèle d'élasticité de film de savon (Prins et al. [5]) qui résout complètement le couplage entre étirement et échanges volume-interface de surfactants pour des régimes sous-micellaires. Ce modèle est validé par des expériences faites à l'époque, et également présentées dans le manuscrit.

Mon apport est ici d'étendre ce modèle au régime micellaire grâce à des résolutions numériques, et de prédire les courbes de charge $\Delta\sigma = f(\epsilon)$, les variation de tension de surface $\Delta\sigma$ en fonction
Résumé en Français

Les travaux de cette thèse portent sur les écoulements dans le plan de films de savon. C'est un point de vue d'hydrodynamicien qui vise à aider à mieux comprendre la rhéologie et le vieillissement des mousses liquides en s'attaquant à des échelles locales: le film de savon unique et la mousse liquide élémentaire, c'est à dire trois films reliés entre eux à 120° par un ménisque.

Dans l'introduction, je présente l'état de l'art sur la physique des films de savon et des mousses liquides élémentaires. Le lecteur y trouvera en premier lieu des définitions de quantités thermodynamiques importantes pour coupler la rhéologie d'une interface gas-liquide aux écoulements dans la phase liquide, puis une mise en équation pour les mouvements dans le plan des films de savon, qui représentent un cas extrême de confinement entre deux de ces interfaces. Cela fait poser et calculer quelques ordres de grandeurs nécessaires pour la suite en discutant notamment les échelles spatiales et temporelles pertinentes, ainsi que les propriétés physico-chimiques des solutions savonneuses (choix et propriétés des surfactants, viscosités etc...).

Ensuite, il est question de regarder plus spécifiquement la zone de jonction entre ces films plats et un ménisque courbé. Des régimes stationnaires et instationnaires d'évolution du profil d'épaisseur dans le film sont discutés selon l'écoulement imposé côté film.

Enfin, j'expose un état de l'art sur les mousses liquides élémentaires sous déformation imposée. Deux expériences précédentes (par Durand & Stone [1] et Besson & Debrégeas [2]) et des simulations numériques récentes (Titta [3]) ont montré la pertinence de cette échelle locale des mousses, que ce soit pour mieux comprendre la mécanique globale de ces matériaux, ou pour sonder la rhéologie de surface des interfaces qui les composent. Dans tous les cas, la réponse mécanique de la mousse élémentaire est montrée comme étant notamment déterminée par les quantités de surfactants échangées entre films, et entre les films et le ménisque. Ces échanges font systématiquement l'objet d'hypothèses fortes dans ces modèles, et tenter de les estimer par le calcul confronte la communauté à des variables physico-chimiques pas ou mal connues. Les connaître permettrait de résoudre le champ de vitesse dans la mousse élémentaire, pour lequel un modèle existe (Bussonnière & Cantat [4]), et de donner des lois rhéologiques locales physiques pour la rhéologie des mousses liquides.

La principale problématique de cette thèse est ainsi posée: peut-on quantifier ces échanges de surfactants entre films et entre films et le ménisque?

Le plan du manuscrit est construit avec un Chapitre 1 portant sur l'élasticité individuelle d'un film de savon, un Chapitre 2 sur une description théorique et une mesure d'une tension de ligne d'origine purement capillaire dans les films, et la modélisation des écoulements qu'elle génère dans le plan du film, et un Chapitre 3 portant sur la quantification des échanges de surfactants au sein d'une mousse liquide élémentaire entre films et entre les films et le ménisque.

Le Chapitre 1 commence par présenter un modèle d'élasticité de film de savon (Prins et al. [5]) qui résout complètement le couplage entre étirement et échanges volume-interface de surfactants pour des régimes sous-micellaires. Ce modèle est validé par des expériences faites à l'époque, et également présentées dans le manuscrit.

Mon apport est ici d'étendre ce modèle au régime micellaire grâce à des résolutions numériques, et de prédire les courbes de charge $\Delta\sigma = f(\epsilon)$, les variation de tension de surface $\Delta\sigma$ en fonction
de la déformation $\epsilon$ selon les axes du plan du film. L'élasticité d'un film est quantifié en prenant la pente à l'origine de ces courbes. Notamment, il apparaît ici qu'étant données les hypothèses du modèle, les surfactants en régime micellaire ne contribuent pas à cette élasticité. On attend donc à être dominé principalement par les contributions soit des impuretés, soit d'un co-surfactant peu soluble de la solution.

<figure><img src="image_3.png" /><figcaption>Figure 1: Courbes de charge: (A) pour différentes vitesses de déformation à chimie donnée [$DOH$] = 35 mg/L ($U$ = 5 et 10 mm/s resp. en rouge et noir); (B) Pour différentes chimies à vitesse $U$ = 10 mm/s (de rouge à noir, [$DOH$] = 0, 15, 35, 50 mg/L)</figcaption></figure>

Enfin, je compare ce modèle étendu à des expériences menées avant le début de ma thèse avec des solutions micellaires de SDS auxquelles du dodécanol est ajouté comme co-surfactant. Ces dernières ont permis de tracer expérimentalement les courbes des charges et d'effectuer la comparaison avec le modèle. Un bon accord quantitatif est trouvé si on ajuste sur la quantité de dodécanol ajoutée plutôt que de rentrer la valeur nominale des solutions. Cela laisse supposer que les impuretés ont un rôle non-négligeable, et que les différentes variables physico-chimiques les caractérisant sont mal connues. Le modèle et ses résultats ont fait l'objet d'une publication dans *Soft Matter* [6] en 2022.

Le Chapitre 2 porte sur une tension de ligne dans les films de savon comportant des hétérogénéités d'épaisseurs. J'y établis en premier lieu l'expression d'un tenseur des contraintes capillaire 2D pour un élément de film comprenant un gradient d'épaisseur.

Grâce à ce dernier, il est possible d'effectuer un bilan des forces dans une zone séparant deux régions du film de savon qui ont des épaisseurs différentes. De par l'anisotropie des forces en jeu, on montre qu'un excès de contrainte est stocké dans cette zone de transition d'une épaisseur à l'autre. En prenant la limite d'un largeur de transition nulle, on construit ainsi un contour contenu dans le plan du film de savon le long duquel une tension de ligne est présente. Cette dernière pourrait être impliquée dans les écoulements dans le plan des films de savon dès lors que ceux-ci sont d'épaisseur non-uniforme, et il devient donc intéressant de la quantifier.

La suite du chapitre porte sur un montage expérimental où cette tension de ligne est mise en évidence. Il s'agit d'un film de savon à l'horizontale que l'on crée fin (~ 200 nm) sur un cadre rectangulaire déformable au rapport d'aspect initial très allongé. En étendant le rectangle brusquement en largeur, on extrait du film épais (~ 10 µm) des ménisques du cadre vers le plan du film pré-existent. On crée ainsi une situation où un film de savon possède deux régions
de la déformation $\epsilon$ selon les axes du plan du film. L'élasticité d'un film est quantifié en prenant la pente à l'origine de ces courbes. Notamment, il apparaît ici qu'étant données les hypothèses du modèle, les surfactants en régime micellaire ne contribuent pas à cette élasticité. On attend donc à être dominé principalement par les contributions soit des impuretés, soit d'un co-surfactant peu soluble de la solution.

<figure><img src="image_3.png" /><figcaption>Figure 1: Courbes de charge: (A) pour différentes vitesses de déformation à chimie donnée [$DOH$] = 35 mg/L ($U$ = 5 et 10 mm/s resp. en rouge et noir); (B) Pour différentes chimies à vitesse $U$ = 10 mm/s (de rouge à noir, [$DOH$] = 0, 15, 35, 50 mg/L)</figcaption></figure>

Enfin, je compare ce modèle étendu à des expériences menées avant le début de ma thèse avec des solutions micellaires de SDS auxquelles du dodécanol est ajouté comme co-surfactant. Ces dernières ont permis de tracer expérimentalement les courbes des charges et d'effectuer la comparaison avec le modèle. Un bon accord quantitatif est trouvé si on ajuste sur la quantité de dodécanol ajoutée plutôt que de rentrer la valeur nominale des solutions. Cela laisse supposer que les impuretés ont un rôle non-négligeable, et que les différentes variables physico-chimiques les caractérisant sont mal connues. Le modèle et ses résultats ont fait l'objet d'une publication dans *Soft Matter* [6] en 2022.

Le Chapitre 2 porte sur une tension de ligne dans les films de savon comportant des hétérogénéités d'épaisseurs. J'y établis en premier lieu l'expression d'un tenseur des contraintes capillaire 2D pour un élément de film comprenant un gradient d'épaisseur.

Grâce à ce dernier, il est possible d'effectuer un bilan des forces dans une zone séparant deux régions du film de savon qui ont des épaisseurs différentes. De par l'anisotropie des forces en jeu, on montre qu'un excès de contrainte est stocké dans cette zone de transition d'une épaisseur à l'autre. En prenant la limite d'un largeur de transition nulle, on construit ainsi un contour contenu dans le plan du film de savon le long duquel une tension de ligne est présente. Cette dernière pourrait être impliquée dans les écoulements dans le plan des films de savon dès lors que ceux-ci sont d'épaisseur non-uniforme, et il devient donc intéressant de la quantifier.

La suite du chapitre porte sur un montage expérimental où cette tension de ligne est mise en évidence. Il s'agit d'un film de savon à l'horizontale que l'on crée fin (~ 200 nm) sur un cadre rectangulaire déformable au rapport d'aspect initial très allongé. En étendant le rectangle brusquement en largeur, on extrait du film épais (~ 10 µm) des ménisques du cadre vers le plan du film pré-existent. On crée ainsi une situation où un film de savon possède deux régions
d'épaisseurs différentes, et l'arène de film fin initialement allongée relaxe vers une forme de cercle sous l'effet de la tension de ligne à sa frontière.

<figure><img src="image_4.png" /></figure>

Figure 2: Timelapse de l'expérience de relaxation d'un patch de film fin (coloré) au sein d'un film plus épais (gris). Une tension de ligne *T* est présente le long du contour séparant les deux régions.

La tension de ligne est mesurée grâce à une caméra spectrale qui donne accès au profil d'épaisseur le long d'un segment perpendiculaire à la ligne de tension. En intégrant les gradients des profils d'épaisseurs, on obtient une mesure à 30% de la tension de ligne qui est de l'ordre de $T \sim 10^{10} \text{N}$.

Afin de valider cette mesure, nous avons modélisé les écoulements, dont la tension de ligne est la force motrice. Connaissant la force de friction opposée à cette dernière, la friction sur l'air au-dessus et au-dessous du film, nous regardons la vitesse à laquelle l'arène relaxe vers un cercle. Un accord quantitatif entre mobilité prédite et mesurée valident notre mesure de la force motrice, et donc de la valeur de tension de ligne trouvée.

Le Chapitre 3 détaille la réponse expérimentale à la problématique principale de ma thèse. J'y rappelle d'abord la situation physique: nous nous intéressons à un système comportant trois films de savon reliés entre eux par un ménisque libre, où on impose des écoulements bouchons loin de ce dernier.

Un modèle décrivant le champ de vitesse dans les films, près du ménisque, a été développé avant le début de ma thèse par Bussonnière & Cantat [4]. Ses principaux éléments sont rappelés: des écoulements bouchon (sans cisaillement) sont imposés loin du ménisque, mais la présence de points d'arrêts aux interfaces dus à la symétrie de déformation impliquent l'existence de zones cisaillées au voisinage du ménisque. Dans ces zones, un cisaillement simple est attendu, dont la pente et la longueur sur laquelle il se développe dépendent directement de variables physico-chimiques et de la capacité du ménisque à échanger des surfactants avec les films. En effet, les points d'arrêt imposant des destructions/créations d'interface, connaître ces flux d'échange détermine la contrainte de Marangoni le long des films, et donc les cisaillements qu'ils subissent.

Le challenge expérimental qui se pose alors est de pouvoir mesurer simultanément ces quantités d'interface $L_{in}$ et $L_{out}$. Cela est rendu possible par un montage commencé avant le début
d'épaisseurs différentes, et l'arène de film fin initialement allongée relaxe vers une forme de cercle sous l'effet de la tension de ligne à sa frontière.

<figure><img src="image_4.png" /></figure>

Figure 2: Timelapse de l'expérience de relaxation d'un patch de film fin (coloré) au sein d'un film plus épais (gris). Une tension de ligne *T* est présente le long du contour séparant les deux régions.

La tension de ligne est mesurée grâce à une caméra spectrale qui donne accès au profil d'épaisseur le long d'un segment perpendiculaire à la ligne de tension. En intégrant les gradients des profils d'épaisseurs, on obtient une mesure à 30% de la tension de ligne qui est de l'ordre de $T \sim 10^{10} \text{N}$.

Afin de valider cette mesure, nous avons modélisé les écoulements, dont la tension de ligne est la force motrice. Connaissant la force de friction opposée à cette dernière, la friction sur l'air au-dessus et au-dessous du film, nous regardons la vitesse à laquelle l'arène relaxe vers un cercle. Un accord quantitatif entre mobilité prédite et mesurée valident notre mesure de la force motrice, et donc de la valeur de tension de ligne trouvée.

Le Chapitre 3 détaille la réponse expérimentale à la problématique principale de ma thèse. J'y rappelle d'abord la situation physique: nous nous intéressons à un système comportant trois films de savon reliés entre eux par un ménisque libre, où on impose des écoulements bouchons loin de ce dernier.

Un modèle décrivant le champ de vitesse dans les films, près du ménisque, a été développé avant le début de ma thèse par Bussonnière & Cantat [4]. Ses principaux éléments sont rappelés: des écoulements bouchon (sans cisaillement) sont imposés loin du ménisque, mais la présence de points d'arrêts aux interfaces dus à la symétrie de déformation impliquent l'existence de zones cisaillées au voisinage du ménisque. Dans ces zones, un cisaillement simple est attendu, dont la pente et la longueur sur laquelle il se développe dépendent directement de variables physico-chimiques et de la capacité du ménisque à échanger des surfactants avec les films. En effet, les points d'arrêt imposant des destructions/créations d'interface, connaître ces flux d'échange détermine la contrainte de Marangoni le long des films, et donc les cisaillements qu'ils subissent.

Le challenge expérimental qui se pose alors est de pouvoir mesurer simultanément ces quantités d'interface $L_{in}$ et $L_{out}$. Cela est rendu possible par un montage commencé avant le début
de ma thèse et complété pour répondre à cette exigence. Je détaille donc dans une section l'ensemble de caméras utilisées pour permettre cette mesure: Trois caméras en fluorescence, une regardant le film comprimé, une sur un film étiré, une sur le côté du ménisque, et trois caméras spectrales regardant les profils d'épaisseur de chaque film. A cela s'ajoute un montage laser pour le photoblanchiement de l'agent fluorescent placé dans nos solutions pour pouvoir créer des traceurs passifs dans les films. Je détaille ensuite le protocole expérimental (vitesses des moteurs, temps de préparation, de déformation des films etc...). Une autre section décrit les traitements numériques relatifs à chaque caméra ainsi que les tracking de taille et position du ménisque, des films extraits lors des déformations et des points photoblanchis.

Avec ces informations, une nouvelle section décrit comment nous remontons aux quantités d'interface échangées et à la différence de tension entre les films. Cette dernière est trouvée principalement grâce à la position du ménisque, qui est une information qui reste délicate à trouver étant donnée la présence de glissements aux temps initiaux des déformations imposées. Ensuite, pour estimer les quantités d'interface $L_{in}$ et $L_{out}$, l'information la plus délicate à trouver est l'état de compression des films. En effet, les longueurs recherchées sont proportionnelles aux quantités de surfactant qu'elles portent, et il faut donc corriger les longueurs directement mesurées par des facteurs de déformation $\epsilon$ estimés pour chaque film (étiré ou comprimé). Cela se fait en estimant pour chaque film la différence de longueur par rapport au cas initial, avec quelques subtilités quant à localiser précisément un système fermé pour chacun d'entre eux.

<figure><img src="image_5.png" /><figcaption>Figure 3: Quantités d'interface échangées mesurées simultanément. Le facteur 1/2 correspond au cas de l'absence d'échanges entre le ménisque et les films.</figcaption></figure>

Une fois que ces déformations sont connues, $L_{in}$ et $L_{out}$ le sont également. Il apparaît alors clairement que pour toute les gammes de vitesses de déformations et de concentration en dodécanol utilisées ($V = 10 - 50$ mm/s, $[DOH] = 15 - 50$ mg/L), le cas $L_{out} = \frac{1}{2}L_{in}$ correspondant à l'absence d'échanges ménisque-film est atteint. Cela suggère donc que pour des gammes de taux de déformation autour de $\dot{\gamma} = 1 - 10$ Hz et de chimies similaires (un ordre
de ma thèse et complété pour répondre à cette exigence. Je détaille donc dans une section l'ensemble de caméras utilisées pour permettre cette mesure: Trois caméras en fluorescence, une regardant le film comprimé, une sur un film étiré, une sur le côté du ménisque, et trois caméras spectrales regardant les profils d'épaisseur de chaque film. A cela s'ajoute un montage laser pour le photoblanchiement de l'agent fluorescent placé dans nos solutions pour pouvoir créer des traceurs passifs dans les films. Je détaille ensuite le protocole expérimental (vitesses des moteurs, temps de préparation, de déformation des films etc...). Une autre section décrit les traitements numériques relatifs à chaque caméra ainsi que les tracking de taille et position du ménisque, des films extraits lors des déformations et des points photoblanchis.

Avec ces informations, une nouvelle section décrit comment nous remontons aux quantités d'interface échangées et à la différence de tension entre les films. Cette dernière est trouvée principalement grâce à la position du ménisque, qui est une information qui reste délicate à trouver étant donnée la présence de glissements aux temps initiaux des déformations imposées. Ensuite, pour estimer les quantités d'interface $L_{in}$ et $L_{out}$, l'information la plus délicate à trouver est l'état de compression des films. En effet, les longueurs recherchées sont proportionnelles aux quantités de surfactant qu'elles portent, et il faut donc corriger les longueurs directement mesurées par des facteurs de déformation $\epsilon$ estimés pour chaque film (étiré ou comprimé). Cela se fait en estimant pour chaque film la différence de longueur par rapport au cas initial, avec quelques subtilités quant à localiser précisément un système fermé pour chacun d'entre eux.

<figure><img src="image_5.png" /><figcaption>Figure 3: Quantités d'interface échangées mesurées simultanément. Le facteur 1/2 correspond au cas de l'absence d'échanges entre le ménisque et les films.</figcaption></figure>

Une fois que ces déformations sont connues, $L_{in}$ et $L_{out}$ le sont également. Il apparaît alors clairement que pour toute les gammes de vitesses de déformations et de concentration en dodécanol utilisées ($V = 10 - 50$ mm/s, $[DOH] = 15 - 50$ mg/L), le cas $L_{out} = \frac{1}{2}L_{in}$ correspondant à l'absence d'échanges ménisque-film est atteint. Cela suggère donc que pour des gammes de taux de déformation autour de $\dot{\gamma} = 1 - 10$ Hz et de chimies similaires (un ordre
de grandeur d'élasticité de film balayé avec le dodécanol), l'hypothèse de nullité pour ce flux peut être faite, ouvrant la voie pour clore le modèle théorique qui prédit la viscosité effective des mousses liquides élémentaires.

Enfin et avant de conclure, j'expose également dans ma thèse comment cette expérience de déformation de mousses liquides élémentaires permet de retrouver les élasticités de film définies et mesurées dans le Chapitre 1, et montre l'accord entre les deux mesures. D'autres déformations sont également montrées, et servent à discuter les différence attendues pour d'autres régimes et symétries, ou la place des recirculations gravitaires dans les mécanismes d'échange.

En conclusion, je récapitule ces différents chapitres et discute les développements expérimentaux et théoriques attendues après ces travaux.
de grandeur d'élasticité de film balayé avec le dodécanol), l'hypothèse de nullité pour ce flux peut être faite, ouvrant la voie pour clore le modèle théorique qui prédit la viscosité effective des mousses liquides élémentaires.

Enfin et avant de conclure, j'expose également dans ma thèse comment cette expérience de déformation de mousses liquides élémentaires permet de retrouver les élasticités de film définies et mesurées dans le Chapitre 1, et montre l'accord entre les deux mesures. D'autres déformations sont également montrées, et servent à discuter les différence attendues pour d'autres régimes et symétries, ou la place des recirculations gravitaires dans les mécanismes d'échange.

En conclusion, je récapitule ces différents chapitres et discute les développements expérimentaux et théoriques attendues après ces travaux.
# Contents

<table><tr><td>0 Introduction</td><td>1</td></tr><tr><td>0.1 The fluid dynamics around a surfactant-laden interface</td><td>3</td></tr><tr><td>0.1.1 Surfactants</td><td>4</td></tr><tr><td>0.1.2 Definitions and generalities about interfaces and surface tension γ</td><td>5</td></tr><tr><td>0.1.3 Coupling the bulk to the interface</td><td>9</td></tr><tr><td>0.1.4 Constitutive equation and 2D stress tensor σ&#773;<sub>s</sub></td><td>13</td></tr><tr><td>0.2 2D hydrodynamics and rheology of a foam film</td><td>15</td></tr><tr><td>0.2.1 Foam films only have plug flows far from the meniscus</td><td>15</td></tr><tr><td>0.2.2 Thin bulk means fast chemical equilibrium</td><td>16</td></tr><tr><td>0.2.3 First model for film elasticity and its physical origin</td><td>17</td></tr><tr><td>0.2.4 Far from the meniscus, film tension is uniform</td><td>19</td></tr><tr><td>0.2.5 The incompressible interface limit</td><td>21</td></tr><tr><td>0.3 Flows in foam films at a junction with a meniscus</td><td>21</td></tr><tr><td>0.3.1 Lubrication equations in a foam film</td><td>21</td></tr><tr><td>0.3.2 Steady state film extraction: the Frankel problem</td><td>23</td></tr><tr><td>0.3.3 Capillary suction near the meniscus: Aradian's marginal pinching</td><td>25</td></tr><tr><td>0.3.4 Marginal regeneration</td><td>26</td></tr><tr><td>0.3.5 Rayleigh-Taylor-like instability with thickness heterogeneities and gravity</td><td>29</td></tr><tr><td>0.3.6 Evaporation</td><td>30</td></tr><tr><td>0.4 Elementary liquid foams</td><td>30</td></tr><tr><td>0.4.1 Plateau's laws: the local equilibrium state for the shape of liquid foams</td><td>31</td></tr><tr><td>0.4.2 T1 events and the Durand-Stone experiment</td><td>31</td></tr><tr><td>0.4.3 Axisymmetric elementary liquid foam under small deformation: the Besson-Debrégeas experiment</td><td>34</td></tr><tr><td>0.4.4 Recent numerical advances</td><td>39</td></tr><tr><td>0.4.5 Rheology of an elementary liquid foam: state of the art in Isabelle Cantat's group at the beginning of my Ph.D</td><td>41</td></tr><tr><td>0.5 Spontaneous evolution of foams and films</td><td>44</td></tr><tr><td>0.5.1 Normal motion of the films</td><td>44</td></tr><tr><td>0.5.2 Gravity drainage</td><td>44</td></tr><tr><td>0.5.3 Coarsening of liquid foams</td><td>45</td></tr><tr><td>0.5.4 Link to liquid foams in the industry and daily life</td><td>46</td></tr></table>
# Contents

<table><tr><td>0 Introduction</td><td>1</td></tr><tr><td>0.1 The fluid dynamics around a surfactant-laden interface</td><td>3</td></tr><tr><td>0.1.1 Surfactants</td><td>4</td></tr><tr><td>0.1.2 Definitions and generalities about interfaces and surface tension γ</td><td>5</td></tr><tr><td>0.1.3 Coupling the bulk to the interface</td><td>9</td></tr><tr><td>0.1.4 Constitutive equation and 2D stress tensor σ&#773;<sub>s</sub></td><td>13</td></tr><tr><td>0.2 2D hydrodynamics and rheology of a foam film</td><td>15</td></tr><tr><td>0.2.1 Foam films only have plug flows far from the meniscus</td><td>15</td></tr><tr><td>0.2.2 Thin bulk means fast chemical equilibrium</td><td>16</td></tr><tr><td>0.2.3 First model for film elasticity and its physical origin</td><td>17</td></tr><tr><td>0.2.4 Far from the meniscus, film tension is uniform</td><td>19</td></tr><tr><td>0.2.5 The incompressible interface limit</td><td>21</td></tr><tr><td>0.3 Flows in foam films at a junction with a meniscus</td><td>21</td></tr><tr><td>0.3.1 Lubrication equations in a foam film</td><td>21</td></tr><tr><td>0.3.2 Steady state film extraction: the Frankel problem</td><td>23</td></tr><tr><td>0.3.3 Capillary suction near the meniscus: Aradian's marginal pinching</td><td>25</td></tr><tr><td>0.3.4 Marginal regeneration</td><td>26</td></tr><tr><td>0.3.5 Rayleigh-Taylor-like instability with thickness heterogeneities and gravity</td><td>29</td></tr><tr><td>0.3.6 Evaporation</td><td>30</td></tr><tr><td>0.4 Elementary liquid foams</td><td>30</td></tr><tr><td>0.4.1 Plateau's laws: the local equilibrium state for the shape of liquid foams</td><td>31</td></tr><tr><td>0.4.2 T1 events and the Durand-Stone experiment</td><td>31</td></tr><tr><td>0.4.3 Axisymmetric elementary liquid foam under small deformation: the Besson-Debrégeas experiment</td><td>34</td></tr><tr><td>0.4.4 Recent numerical advances</td><td>39</td></tr><tr><td>0.4.5 Rheology of an elementary liquid foam: state of the art in Isabelle Cantat's group at the beginning of my Ph.D</td><td>41</td></tr><tr><td>0.5 Spontaneous evolution of foams and films</td><td>44</td></tr><tr><td>0.5.1 Normal motion of the films</td><td>44</td></tr><tr><td>0.5.2 Gravity drainage</td><td>44</td></tr><tr><td>0.5.3 Coarsening of liquid foams</td><td>45</td></tr><tr><td>0.5.4 Link to liquid foams in the industry and daily life</td><td>46</td></tr></table>
# 1 Elasticity of a single foam film and how to measure it 51

1.1 Surface and film elasticity measurements in the literature 52

1.1.1 SDS/DOH mixtures: a few more comments 52

1.1.2 Film elasticity and model in the submicellar regime: the Prins experiment 52

1.2 Thermochemical models for foam film elasticities 54

1.2.1 Our model for SDS/DOH mixtures 55

1.2.2 Numerical results 59

1.3 A foam film rheometer 61

1.3.1 Experimental setup 61

1.3.2 Experimental results and comparison with the predictions 63

1.3.3 Theoretical and experimental film elasticities 65

1.4 Conclusions of Chapter 1 68

# 2 In-plane motions in a foam film 69

2.1 Capillary force and line tension in foam films 70

2.1.1 Building the capillary stress tensor for an elementary piece of film with a thickness gradient 70

2.1.2 Gradients of thickness localised in a very thin region: building a line tension T 72

2.1.3 Important remarks 75

2.2 Creating line tension in a foam film: experimental setup 78

2.2.1 Principle of the experiment, technical limitations and notations 78

2.2.2 Setup 80

2.3 Image processing 81

2.3.1 Detection of the thin region contour and kinematics of the relaxation 81

2.3.2 Comment on the chemistry 83

2.3.3 Extracting the thickness profiles 84

2.3.4 Computing the resulting line tension 87

2.3.5 Results 88

2.4 Modelling the relaxation of the thin film 90

2.4.1 Equation of motion for a foam film with line tension 90

2.4.2 Closing the problem: incompressibility of the interfaces 91

2.4.3 Damping forces: the Boussinesq number $Bq$ 92

2.4.4 Mapping on the Saffman and Hughes model 94

2.4.5 Quantitative agreement in the air friction-dominated regime 96

2.4.6 Important remarks and some calculations 98

2.5 Conclusions of Chapter 2 101

# 3 Dissipative phenomena in an elementary foam 103

3.1 Rheology of an elementary liquid foam 104

3.1.1 An elementary liquid foam with three films 104

3.1.2 Position of the problem 104

3.1.3 First hint 105

3.2 Theoretical state of the art 107

3.2.1 Theoretical description of the flat film, the sheared zone, the dynamic and static menisci 107
# 1 Elasticity of a single foam film and how to measure it 51

1.1 Surface and film elasticity measurements in the literature 52

1.1.1 SDS/DOH mixtures: a few more comments 52

1.1.2 Film elasticity and model in the submicellar regime: the Prins experiment 52

1.2 Thermochemical models for foam film elasticities 54

1.2.1 Our model for SDS/DOH mixtures 55

1.2.2 Numerical results 59

1.3 A foam film rheometer 61

1.3.1 Experimental setup 61

1.3.2 Experimental results and comparison with the predictions 63

1.3.3 Theoretical and experimental film elasticities 65

1.4 Conclusions of Chapter 1 68

# 2 In-plane motions in a foam film 69

2.1 Capillary force and line tension in foam films 70

2.1.1 Building the capillary stress tensor for an elementary piece of film with a thickness gradient 70

2.1.2 Gradients of thickness localised in a very thin region: building a line tension T 72

2.1.3 Important remarks 75

2.2 Creating line tension in a foam film: experimental setup 78

2.2.1 Principle of the experiment, technical limitations and notations 78

2.2.2 Setup 80

2.3 Image processing 81

2.3.1 Detection of the thin region contour and kinematics of the relaxation 81

2.3.2 Comment on the chemistry 83

2.3.3 Extracting the thickness profiles 84

2.3.4 Computing the resulting line tension 87

2.3.5 Results 88

2.4 Modelling the relaxation of the thin film 90

2.4.1 Equation of motion for a foam film with line tension 90

2.4.2 Closing the problem: incompressibility of the interfaces 91

2.4.3 Damping forces: the Boussinesq number $Bq$ 92

2.4.4 Mapping on the Saffman and Hughes model 94

2.4.5 Quantitative agreement in the air friction-dominated regime 96

2.4.6 Important remarks and some calculations 98

2.5 Conclusions of Chapter 2 101

# 3 Dissipative phenomena in an elementary foam 103

3.1 Rheology of an elementary liquid foam 104

3.1.1 An elementary liquid foam with three films 104

3.1.2 Position of the problem 104

3.1.3 First hint 105

3.2 Theoretical state of the art 107

3.2.1 Theoretical description of the flat film, the sheared zone, the dynamic and static menisci 107
<table><tr><td>3.2.2</td><td>Set of equations</td><td>109</td></tr><tr><td>3.2.3</td><td>Boundary condition on the meniscus side s → s<sub>m</sub>: the surfactant flux j<sub>m</sub> as an unknown parameter</td><td>111</td></tr><tr><td>3.2.4</td><td>Numerical resolution of the system</td><td>112</td></tr><tr><td>3.3</td><td>Experimental challenge: measuring j<sub>m</sub></td><td>114</td></tr><tr><td>3.4</td><td>Experimental setup and notations</td><td>117</td></tr><tr><td>3.4.1</td><td>Chemistries</td><td>117</td></tr><tr><td>3.4.2</td><td>Notations for the 3-films elementary foam</td><td>118</td></tr><tr><td>3.4.3</td><td>Motors, optics and photobleaching setup</td><td>119</td></tr><tr><td>3.4.4</td><td>Interfacing and synchronising the experiment</td><td>123</td></tr><tr><td>3.5</td><td>Detailed protocol for the “Push 3” experiment</td><td>124</td></tr><tr><td>3.6</td><td>Data and image processing</td><td>126</td></tr><tr><td>3.6.1</td><td>Tracking of the free meniscus in the horizontal plane</td><td>126</td></tr><tr><td>3.6.2</td><td>Tracking the vertical position of the free meniscus and its radius by ray tracing</td><td>128</td></tr><tr><td>3.6.3</td><td>Tracking the edges of Frankel films in the horizontal film 3</td><td>130</td></tr><tr><td>3.6.4</td><td>Tracking the Frankel film in the top left film 1</td><td>132</td></tr><tr><td>3.6.5</td><td>Thickness profiles with hyperspectral cameras</td><td>133</td></tr><tr><td>3.6.6</td><td>Tracking of the photobleached dots</td><td>135</td></tr><tr><td>3.7</td><td>Monitoring the invariance by translation along z</td><td>136</td></tr><tr><td>3.8</td><td>Computing tensions, deformations and elasticites of the films</td><td>139</td></tr><tr><td>3.8.1</td><td>Relative tensions between the films</td><td>139</td></tr><tr><td>3.8.2</td><td>Deformation of film 3 ε<sub>3</sub> and film 1 ε<sub>1</sub></td><td>143</td></tr><tr><td>3.8.3</td><td>Determination of the film elasticity E<sub>f</sub></td><td>146</td></tr><tr><td>3.8.4</td><td>A second estimation of the deformation ε<sub>1,b</sub></td><td>148</td></tr><tr><td>3.9</td><td>Results: Surfactant balance and the fate of j<sub>m</sub></td><td>149</td></tr><tr><td>3.9.1</td><td>Computing L<sub>in</sub> and L<sub>out</sub></td><td>149</td></tr><tr><td>3.9.2</td><td>Vanishing flux j<sub>m</sub></td><td>151</td></tr><tr><td>3.10</td><td>Other deformations</td><td>153</td></tr><tr><td>3.10.1</td><td>“Push 1”: Potential influence of gravity</td><td>153</td></tr><tr><td>3.10.2</td><td>“Pull 1”: Same symmetry, opposite directions for the imposed velocities</td><td>155</td></tr><tr><td>3.11</td><td>Conclusions of Chapter 3</td><td>158</td></tr></table>
<table><tr><td>3.2.2</td><td>Set of equations</td><td>109</td></tr><tr><td>3.2.3</td><td>Boundary condition on the meniscus side s → s<sub>m</sub>: the surfactant flux j<sub>m</sub> as an unknown parameter</td><td>111</td></tr><tr><td>3.2.4</td><td>Numerical resolution of the system</td><td>112</td></tr><tr><td>3.3</td><td>Experimental challenge: measuring j<sub>m</sub></td><td>114</td></tr><tr><td>3.4</td><td>Experimental setup and notations</td><td>117</td></tr><tr><td>3.4.1</td><td>Chemistries</td><td>117</td></tr><tr><td>3.4.2</td><td>Notations for the 3-films elementary foam</td><td>118</td></tr><tr><td>3.4.3</td><td>Motors, optics and photobleaching setup</td><td>119</td></tr><tr><td>3.4.4</td><td>Interfacing and synchronising the experiment</td><td>123</td></tr><tr><td>3.5</td><td>Detailed protocol for the “Push 3” experiment</td><td>124</td></tr><tr><td>3.6</td><td>Data and image processing</td><td>126</td></tr><tr><td>3.6.1</td><td>Tracking of the free meniscus in the horizontal plane</td><td>126</td></tr><tr><td>3.6.2</td><td>Tracking the vertical position of the free meniscus and its radius by ray tracing</td><td>128</td></tr><tr><td>3.6.3</td><td>Tracking the edges of Frankel films in the horizontal film 3</td><td>130</td></tr><tr><td>3.6.4</td><td>Tracking the Frankel film in the top left film 1</td><td>132</td></tr><tr><td>3.6.5</td><td>Thickness profiles with hyperspectral cameras</td><td>133</td></tr><tr><td>3.6.6</td><td>Tracking of the photobleached dots</td><td>135</td></tr><tr><td>3.7</td><td>Monitoring the invariance by translation along z</td><td>136</td></tr><tr><td>3.8</td><td>Computing tensions, deformations and elasticites of the films</td><td>139</td></tr><tr><td>3.8.1</td><td>Relative tensions between the films</td><td>139</td></tr><tr><td>3.8.2</td><td>Deformation of film 3 ε<sub>3</sub> and film 1 ε<sub>1</sub></td><td>143</td></tr><tr><td>3.8.3</td><td>Determination of the film elasticity E<sub>f</sub></td><td>146</td></tr><tr><td>3.8.4</td><td>A second estimation of the deformation ε<sub>1,b</sub></td><td>148</td></tr><tr><td>3.9</td><td>Results: Surfactant balance and the fate of j<sub>m</sub></td><td>149</td></tr><tr><td>3.9.1</td><td>Computing L<sub>in</sub> and L<sub>out</sub></td><td>149</td></tr><tr><td>3.9.2</td><td>Vanishing flux j<sub>m</sub></td><td>151</td></tr><tr><td>3.10</td><td>Other deformations</td><td>153</td></tr><tr><td>3.10.1</td><td>“Push 1”: Potential influence of gravity</td><td>153</td></tr><tr><td>3.10.2</td><td>“Pull 1”: Same symmetry, opposite directions for the imposed velocities</td><td>155</td></tr><tr><td>3.11</td><td>Conclusions of Chapter 3</td><td>158</td></tr></table>
0
Introduction

Here is a liquid foam

<figure><img src="image_6.png" /><figcaption>Figure 1: A dry liquid foam. Credits to S. Cohen-Addad, R.M. Guillermic & A. Saint-Jalmes [7].</figcaption></figure>

What we see is a network of flat liquid films separating faceted bubbles. This is essentially a close look at what is in a sink while doing the dishes, or what we hold in our hands while shampooing! And we all know the recipe: a **liquid**, water most of the time, in which we add a
O
Introduction

Here is a liquid foam

<figure><img src="image_6.png" /><figcaption>Figure 1: A dry liquid foam. Credits to S. Cohen-Addad, R.M. Guillermic & A. Saint-Jalmes [7].</figcaption></figure>

What we see is a network of flat liquid films separating faceted bubbles. This is essentially a close look at what is in a sink while doing the dishes, or what we hold in our hands while shampooing! And we all know the recipe: a **liquid**, water most of the time, in which we add a
**surfactant**, soap most of the time, and some **mechanical work** to incorporate some **air** into the liquid. This creates a suspension of gas in a liquid, as we are accustomed to in everyday life, whether it is for washing, cooking, or simply enjoying the view of bubbles.

For an engineer, predicting the overall mechanical properties of these structures is a complex problem — whether it involves guiding their flow from point A to B... or preventing their formation in the first place.

Let us first have an even closer look, zooming on one single film with Fig. 2A.

<figure><img src="image_7.png" /></figure>

Figure 2: **A:** single flat foam film on a supporting frame. The interference colours correspond each to a thickness of film in the range ~200–800nm. Thicker pieces appear grey, thinner appear black. Credits to Eric Sorensen<fnref n="2" />. **B:** At the typical time scale of a foam film lifetime, each tiny coloured patch mentally cut out of it can be seen as a closed system which is advected in the plane of the film. Describing the in-plane motions of a foam film is a sliding puzzle game!

This is a flat film of liquid supported by a solid rectangular frame. The thickness of liquid varies from a few hundreds of nanometers to a few microns, around one hundred times thinner than a human hair!

Also, what remains hidden in the picture is the meniscus, which is a curved liquid beading at the edges present either on the supporting solid frame for our single film, or at the junctions between the films in the liquid foam. What we can see on the other hand are the heterogeneities of thickness in the film itself, evidenced by the (delightfully) colourful patterns at the surface. All of these factors contribute to the inherent complexity of this system even at this scale.

A first important piece of knowledge for instance, is that if we track by eye a tiny piece of film in the coloured patterns, it is a closed system which does not exchange any matter with its neighbours. This is true on the typical time scales ~1–10s at which we witness these colours moving, even if the said patch of film runs through the whole film!

As a child, I saw bright and wiggling colours in my bath. As an undergrad hydrodynamicist, I thought it was about the inner flows of a fluid sandwich made of two liquid-gas interfaces. As

<footnote n="2">²A beautiful zoology of foam films by Eric Sorensen at http://esorensen.com/soap-film-photography/. Many thanks to him.</footnote>
**surfactant**, soap most of the time, and some **mechanical work** to incorporate some **air** into the liquid. This creates a suspension of gas in a liquid, as we are accustomed to in everyday life, whether it is for washing, cooking, or simply enjoying the view of bubbles.

For an engineer, predicting the overall mechanical properties of these structures is a complex problem — whether it involves guiding their flow from point A to B... or preventing their formation in the first place.

Let us first have an even closer look, zooming on one single film with Fig. 2A.

<figure><img src="image_7.png" /></figure>

Figure 2: **A:** single flat foam film on a supporting frame. The interference colours correspond each to a thickness of film in the range ~200–800nm. Thicker pieces appear grey, thinner appear black. Credits to Eric Sorensen<fnref n="2" />. **B:** At the typical time scale of a foam film lifetime, each tiny coloured patch mentally cut out of it can be seen as a closed system which is advected in the plane of the film. Describing the in-plane motions of a foam film is a sliding puzzle game!

This is a flat film of liquid supported by a solid rectangular frame. The thickness of liquid varies from a few hundreds of nanometers to a few microns, around one hundred times thinner than a human hair!

Also, what remains hidden in the picture is the meniscus, which is a curved liquid beading at the edges present either on the supporting solid frame for our single film, or at the junctions between the films in the liquid foam. What we can see on the other hand are the heterogeneities of thickness in the film itself, evidenced by the (delightfully) colourful patterns at the surface. All of these factors contribute to the inherent complexity of this system even at this scale.

A first important piece of knowledge for instance, is that if we track by eye a tiny piece of film in the coloured patterns, it is a closed system which does not exchange any matter with its neighbours. This is true on the typical time scales ~1–10s at which we witness these colours moving, even if the said patch of film runs through the whole film!

As a child, I saw bright and wiggling colours in my bath. As an undergrad hydrodynamicist, I thought it was about the inner flows of a fluid sandwich made of two liquid-gas interfaces. As

<footnote n="2">²A beautiful zoology of foam films by Eric Sorensen at http://esorensen.com/soap-film-photography/. Many thanks to him.</footnote>
a Ph.D student, I see now that it is all about a sliding puzzle game with patches of foam films (Fig.2B). A puzzle game with many mysteries yet to be solved: What forces set the pieces of film in motion within their own plane? How does a patch of film behave close to the meniscus? Can neighbouring films exchange patches if we deform them? With whom do they exchange? With the meniscus or with other films?

Part of these questions have had their answers in the literature, but an even bigger part is
still open!

This thesis is about the adventure of such a 2D patch of foam film, a piece of a sliding puzzle: We are going to apply forces on it, advect it, stretch it and even try to shear it in its own plane. In the end, it will be allowed to deliver the surfactants it carries from one film to another in a foam, thus representing a transport mechanism relevant for foam rheology (Fig.3).

<figure><img src="image_8.png" /><figcaption>Figure 3: A local scale of a liquid foam such as shown above. Deforming the films (red) sets in motion patches of film. Some of them will even leave their plane (dark blue), thus representing a surfactant transport mechanism. These exchanges are of the utmost importance for the sake of describing the flows in the system.</figcaption></figure>

0.1 The fluid dynamics around a surfactant-laden in-
terface

As stated earlier in the very basic description of a bubble or a foam film, we are dealing with a very
thin gas-liquid-gas sandwich. The major ingredient to add in order to stabilise this structure is the
presence of surfactant molecules at the interface. But how do they organise for one interface first?
What tools do we need to properly describe the features of an interface containing surfactants?
a Ph.D student, I see now that it is all about a sliding puzzle game with patches of foam films (Fig.2B). A puzzle game with many mysteries yet to be solved: What forces set the pieces of film in motion within their own plane? How does a patch of film behave close to the meniscus? Can neighbouring films exchange patches if we deform them? With whom do they exchange? With the meniscus or with other films?

Part of these questions have had their answers in the literature, but an even bigger part is
still open!

This thesis is about the adventure of such a 2D patch of foam film, a piece of a sliding puzzle: We are going to apply forces on it, advect it, stretch it and even try to shear it in its own plane. In the end, it will be allowed to deliver the surfactants it carries from one film to another in a foam, thus representing a transport mechanism relevant for foam rheology (Fig.3).

<figure><img src="image_8.png" /><figcaption>Figure 3: A local scale of a liquid foam such as shown above. Deforming the films (red) sets in motion patches of film. Some of them will even leave their plane (dark blue), thus representing a surfactant transport mechanism. These exchanges are of the utmost importance for the sake of describing the flows in the system.</figcaption></figure>

**0.1 The fluid dynamics around a surfactant-laden interface**

As stated earlier in the very basic description of a bubble or a foam film, we are dealing with a very
thin gas-liquid-gas sandwich. The major ingredient to add in order to stabilise this structure is the
presence of surfactant molecules at the interface. But how do they organise for one interface first?
What tools do we need to properly describe the features of an interface containing surfactants?
### 0.1.1 Surfactants

First, let us define what a surfactant is. A very usual example given at school is soap, often with the simplified chemical representation of a polar round head with an elongated non-polar carbon tail. Heads like water which is polar, tails prefer air which is non-polar. They ease the formation of bubbles and liquid foams by going at the liquid-air interfaces and lowering **surface tension**, denoted $\gamma$, which is usually presented as “the energy per unit area” it takes to create some interface.

Usual species of surfactants encountered in daily life can have many different origins. A lot of proteins are efficient surfactants: dairy and egg proteins are rather good examples, as both milk and beaten eggs foam well. Surfactants in cleaning agents also have many different origins, from the traditional savon de Marseille made with oleic acid from olive oil to “sulfates” (or SAS for secondary alkanesulfonates) used in shampoos and dishwashing liquids, which are petroleum derivatives.

In research, quite a wide range of surfactants is also utilised: TERGITOL's, Triton X-100, the $C_n$TAB family (for TrimethylAmmonium Bromide, exist with carbon chains from $n = 10$ to 16) and the sodium sulfate families. In the frame of this thesis work, we will focus on a surfactant from this last family, maybe the most widely used in soft-matter: **SDS for Sodium Dodecyl Sulfate**. Its structure is represented in Fig.4A. It is an anionic surfactant synthesised from dodecanol (hereafter DOH) shown in Fig.4B.

<figure><img src="image_9.png" /><figcaption>Figure 4: Structure of the two surfactants used in the frame of our work: **A**: SDS (for Sodium Dodecyl Sulfate) is a very soluble anionic surfactant (forms micelles in water **C**) and **B**: DOH for Dodecanol (poorly soluble, is the electrolysis product of SDS, hides in SDS micelles **D**)</figcaption></figure>

SDS is a very soluble surfactant in water thanks to its very polar head. When its concentration reaches a critical value called Critical Micellar Concentration $CMC = 8.1$ mmol/L, SDS molecules start forming spherical structures called micelles, of which a schematic view is shown in Fig.4C and which represent an actual new phase in the system. Thus, above the $CMC$, the monomeric concentration of SDS in water remains capped at $CMC = 8.1$ mmol/L and the
### 0.1.1 Surfactants

First, let us define what a surfactant is. A very usual example given at school is soap, often with the simplified chemical representation of a polar round head with an elongated non-polar carbon tail. Heads like water which is polar, tails prefer air which is non-polar. They ease the formation of bubbles and liquid foams by going at the liquid-air interfaces and lowering **surface tension, denoted** $\gamma$, which is usually presented as “the energy per unit area” it takes to create some interface.

Usual species of surfactants encountered in daily life can have many different origins. A lot of proteins are efficient surfactants: dairy and egg proteins are rather good examples, as both milk and beaten eggs foam well. Surfactants in cleaning agents also have many different origins, from the traditional savon de Marseille made with oleic acid from olive oil to “sulfates” (or SAS for secondary alkanesulfonates) used in shampoos and dishwashing liquids, which are petroleum derivatives.

In research, quite a wide range of surfactants is also utilised: TERGITOL's, Triton X-100, the $C_n$TAB family (for TrimethylAmmonium Bromide, exist with carbon chains from $n = 10$ to 16) and the sodium sulfate families. In the frame of this thesis work, we will focus on a surfactant from this last family, maybe the most widely used in soft-matter: **SDS for Sodium Dodecyl Sulfate**. Its structure is represented in Fig.4A. It is an anionic surfactant synthesised from **dodecanol (hereafter DOH)** shown in Fig.4B.

<figure><img src="image_9.png" /><figcaption>Figure 4: Structure of the two surfactants used in the frame of our work: **A**: SDS (for Sodium Dodecyl Sulfate) is a very soluble anionic surfactant (forms micelles in water **C**) and **B**: DOH for Dodecanol (poorly soluble, is the electrolysis product of SDS, hides in SDS micelles **D**)</figcaption></figure>

SDS is a very soluble surfactant in water thanks to its very polar head. When its concentration reaches a critical value called Critical Micellar Concentration $CMC = 8.1$ mmol/L, SDS molecules start forming spherical structures called micelles, of which a schematic view is shown in Fig.4C and which represent an actual new phase in the system. Thus, above the $CMC$, the monomeric concentration of SDS in water remains capped at $CMC = 8.1$ mmol/L and the
molecules in excess form micelles. On the other hand, DOH is a poorly soluble surfactant, having a solubility in pure water of $s = 2.2 \times 10^{-5}mol/L$. When solubilised in a micellar SDS solution however, DOH can take refuge in the micelles, as shown in Fig.4D, and its overall solubility can go up to $s \sim 10^{-3}mol/L$.

As for this choice of surfactant, it is very important to note that a tiny amount of DOH is still present in commercial SDS powders. The manufacturer only gives the global purity which is > 99%. Later estimations made in the frame of our work (cf. chapter 1 subsection 1.2.1) tend to show that DOH represents approximately ~ 0.1 – 0.2% of the SDS-DOH population. It is a crucial factor as these tiny amounts of DOH drastically change the properties of the foam films and thus represent a control parameter in the following when we will be adding DOH ourselves to the solutions.

## 0.1.2 Definitions and generalities about interfaces and surface tension $\gamma$

God made the bulk; surfaces were invented by the devil

Wolfgang Pauli

### Surface tension $\gamma$ defined as an energy

Now that we have defined what our surfactants are and how they organise in the bulk liquid, which is always water in the frame of this work, let us see what happens when our solutions are put in contact with a gas interface, which is always air here. Thus, a first naive/schematic view of such an interface is given by Fig.5A, which can be imagined with or without the surfactants covering it: a discrete jump from one phase to the other, a 2D object which can possibly be populated by surfactants.

Most importantly, this description is associated with the definition of a central physical quantity that is the surface tension $\gamma$, first introduced by Gibbs [8, 9] and classically defined as the energy per unit area required to create some interface:

$$ \gamma = \left. \frac{\partial F}{\partial A} \right|_{T,V,N} \qquad (1) $$

Where $F$ is the free energy, to be switched with any other thermodynamic potential depending on the variables of the problem (e.g. $T, V, N$ the entropy, volume and quantity of matter), and $A$ the interface area. The canonical molecular interpretation of this energy cost is to imagine a single molecule near the discrete interface and how the attractive interactions acting on it vanish in the direction of the interface, as air is very dilute (or similarly how they diminish in this direction for liquid-liquid interfaces). This results in an energy frustration of any molecule at the surface, meaning there is an energy cost per unit area to create the interface.
molecules in excess form micelles. On the other hand, DOH is a poorly soluble surfactant, having a solubility in pure water of $s = 2.2 \times 10^{-5}mol/L$. When solubilised in a micellar SDS solution however, DOH can take refuge in the micelles, as shown in Fig.4D, and its overall solubility can go up to $s \sim 10^{-3}mol/L$.

As for this choice of surfactant, it is very important to note that a tiny amount of DOH is still present in commercial SDS powders. The manufacturer only gives the global purity which is > 99%. Later estimations made in the frame of our work (cf. chapter 1 subsection 1.2.1) tend to show that DOH represents approximately ~ 0.1 – 0.2% of the SDS-DOH population. It is a crucial factor as these tiny amounts of DOH drastically change the properties of the foam films and thus represent a control parameter in the following when we will be adding DOH ourselves to the solutions.

## 0.1.2 Definitions and generalities about interfaces and surface tension $\gamma$

God made the bulk; surfaces were invented by the devil

Wolfgang Pauli

### Surface tension $\gamma$ defined as an energy

Now that we have defined what our surfactants are and how they organise in the bulk liquid, which is always water in the frame of this work, let us see what happens when our solutions are put in contact with a gas interface, which is always air here. Thus, a first naive/schematic view of such an interface is given by Fig.5A, which can be imagined with or without the surfactants covering it: a discrete jump from one phase to the other, a 2D object which can possibly be populated by surfactants.

Most importantly, this description is associated with the definition of a central physical quantity that is the surface tension $\gamma$, first introduced by Gibbs [8, 9] and classically defined as the energy per unit area required to create some interface:

$$ \gamma = \left. \frac{\partial F}{\partial A} \right|_{T,V,N} \qquad (1) $$

Where $F$ is the free energy, to be switched with any other thermodynamic potential depending on the variables of the problem (e.g. $T,V,N$ the entropy, volume and quantity of matter), and $A$ the interface area. The canonical molecular interpretation of this energy cost is to imagine a single molecule near the discrete interface and how the attractive interactions acting on it vanish in the direction of the interface, as air is very dilute (or similarly how they diminish in this direction for liquid-liquid interfaces). This results in an energy frustration of any molecule at the surface, meaning there is an energy cost per unit area to create the interface.
## What is hidden in an interface?

However, the molecular reality is in fact more complicated than that. How does it make sense to speak of distance from the interface for a single molecule when these molecules define what the interface is? Part of the answer can be found if we dare opening the Pandora box of “what is hidden in an interface in a 3D world?”. Let us do so gently, opening it ajar only, just to be sure at least of the sizes we are dealing with.

<figure><img src="image_10.png" /><figcaption>Figure 5: A liquid-gas interface wide open. **A:** Schematic view as a 2D object. **B:** Schematic view of the real 3D world as we open the interface. The typical thickness over which we jump from one phase to the other is of the order of ~3–5 $r_M$ molecular sizes. $\rho_l$, $\rho_g$ and $\rho_s$ are respectively normalised densities of liquid, gas and surfactants.</figcaption></figure>

Pauli warned us already: these are tricky objects to think about and describe. Let us have a closer look. In Fig. 5B, in the case where there are no surfactants (meaning we temporarily forget about the red curve), the density of water molecules varies from a constant value, which is the bulk value, to zero over a width ~1–10 nm which is the size of several water molecules. This remains much thinner than regular foam films as we recall their typical thickness $h \sim 1$ µm. The interface itself $z=0$ is here defined as the origin of a step function jumping from $\rho=0$ to $\rho_{l,bulk}$ that makes the two grey excess areas have equal surfaces.

What is surface tension in this 3D description? This is a matter of molecular dynamics: we would need to describe the potential of interaction between the molecules, draw the profile of density of energy of interactions across the interface and integrate it along $z$. The excess of energy compared with bulk liquid would be $\gamma$. We won’t risk writing it in math mode though.

Now let us consider surfactants. We add the red curve to Fig. 5B alongside the blue one. In the vicinity of $z=0$, the surfactants are much more concentrated. When we will get back to a 2D description, this local excess of molecules will be called surface excess (meant by comparison to the bulk value $\rho_{s,bulk}$). The existence of a surface excess for a species is the definition of what we have called so far a surfactant.

**NB 1:** This definition of “surfactant” is canonical in physical chemistry. For other communities in chemistry, a species also need to be soluble enough to form micelles in order to be called surfactant, and for example dodecanol would not be called surfactant, but rather “co-surfactant” as it relies on SDS to ensure its solubility. For hydrodynamicists like us, a surfactant would be anything changing our boundary conditions at the interface, which is more in line with the definition of the physical chemists.
## What is hidden in an interface?

However, the molecular reality is in fact more complicated than that. How does it make sense to speak of distance from the interface for a single molecule when these molecules define what the interface is? Part of the answer can be found if we dare opening the Pandora box of “what is hidden in an interface in a 3D world?”. Let us do so gently, opening it ajar only, just to be sure at least of the sizes we are dealing with.

<figure><img src="image_10.png" /><figcaption>Figure 5: A liquid-gas interface wide open. **A:** Schematic view as a 2D object. **B:** Schematic view of the real 3D world as we open the interface. The typical thickness over which we jump from one phase to the other is of the order of ~3–5 $r_M$ molecular sizes. $\rho_l$, $\rho_g$ and $\rho_s$ are respectively normalised densities of liquid, gas and surfactants.</figcaption></figure>

Pauli warned us already: these are tricky objects to think about and describe. Let us have a closer look. In Fig. 5B, in the case where there are no surfactants (meaning we temporarily forget about the red curve), the density of water molecules varies from a constant value, which is the bulk value, to zero over a width ~1–10 nm which is the size of several water molecules. This remains much thinner than regular foam films as we recall their typical thickness $h \sim 1$ µm. The interface itself $z=0$ is here defined as the origin of a step function jumping from $\rho=0$ to $\rho_{l,bulk}$ that makes the two grey excess areas have equal surfaces.

What is surface tension in this 3D description? This is a matter of molecular dynamics: we would need to describe the potential of interaction between the molecules, draw the profile of density of energy of interactions across the interface and integrate it along $z$. The excess of energy compared with bulk liquid would be $\gamma$. We won’t risk writing it in math mode though.

Now let us consider surfactants. We add the red curve to Fig. 5B alongside the blue one. In the vicinity of $z=0$, the surfactants are much more concentrated. When we will get back to a 2D description, this local excess of molecules will be called surface excess (meant by comparison to the bulk value $\rho_{s,bulk}$). The existence of a surface excess for a species is the definition of what we have called so far a surfactant.

**NB 1:** This definition of “surfactant” is canonical in physical chemistry. For other communities in chemistry, a species also need to be soluble enough to form micelles in order to be called surfactant, and for example dodecanol would not be called surfactant, but rather “co-surfactant” as it relies on SDS to ensure its solubility. For hydrodynamicists like us, a surfactant would be anything changing our boundary conditions at the interface, which is more in line with the definition of the physical chemists.
The contribution of the surface excess of surfactant to the aforementioned energy integral of $\gamma$ is negative, and thus adding surfactants results in a lower surface tension.

**NB 2:** This decrease of $\gamma$ is very often proposed as an explanation for the foaming properties of soapy solutions. “Creating water-air interfaces costs less, thus allowing foams to exist”. **This is utterly wrong.** A lot of pure liquids with surface tensions close to soapy solutions such as regular oils do not foam at all! Allowing a foam film to exist is not about how much it costs to create it, but rather giving its surface tension a law of evolution $\gamma(\epsilon, \dot{\epsilon})$ under a certain deformation of the interface $\epsilon$ to resist gravity and capillary forces. Describing these laws is a big part of this thesis work, as we will later see.

Adding a surfactant also raises the question of the evolution of the typical width $l_z$ where we start deviating from the bulk properties, and to see to what extent the interface remains thin enough to be considered 2D. In our specific case of an ionic surfactant such as SDS, the typical width of the interface is the Debye length $\lambda_s$ as predicted and experimentally shown by Bergeron et al. [10, 11] with DLVO theory [12, 13]. We can write it and estimate it in our case using our typical bulk concentration of surfactant (which is lower than the concentration in the interface):

$$l_z \sim \lambda_s = \sqrt{\frac{\epsilon_0 \epsilon_r k_B T}{\rho_s N_A z_s^2 e^2}} \sim 10 \text{ nm} \qquad (2)$$

Where $\epsilon_0 \epsilon_r = 6.9 \times 10^{-10} \text{ F m}^{-1}$ is the electric permittivity of water, $k_B T = 4.1 \times 10^{-21} \text{ J}$ is the thermal energy, $e = 1.6 \times 10^{-19} \text{ C}$ is the elementary charge, $z_s = 1$ is the ionic valence of SDS, $N_A = 6.64 \times 10^{23} \text{ mol}^{-1}$ is the Avogadro constant and $\rho_s > \rho_{s,bulk} = 1 \text{ mol/m}^3$ is an order of magnitude for the concentration of SDS.

This is the order of magnitude of the range at which our interface can interact. Thus, this allows us to distinguish what is called a black film (so thin that it appears black), the world of physical chemists with $h < 100 \text{ nm}$, from regular “thick” films, our world as hydrodynamicists with $h > 100 \text{ nm}$. Below this limit, with black films, an important ingredient to add is disjoining pressure, a key physical quantity to study the stability of these systems which is directly linked to the ageing of liquid foams [14–16]. Above this limit $h > 100 \text{ nm}$, which will always be the case from now on, we are allowed to close the box of the 3D interface and get back to our surface tension $\gamma$ defined properly along a 2D interface.

## Surface tension $\gamma$ as a force per unit length and how to measure it

All these clarifications about interfaces being made, we can have a look at what happens depending on their geometries. In fact, as we implied earlier, our interfaces are going to be curved as the foam films have heterogeneous thicknesses and are connected to menisci at the edges. This is where surface tension comes into play, but rather as a force per unit length this time. The beauty is that although it is defined only in the 2D interface, its contributions are both on the normal and tangential constraints of the surface!

The stress at the interface can be written (neglecting surface viscosities for now):

$$\overline{\sigma}.n = \nabla_s \gamma + \kappa \gamma n \qquad (3)$$
The contribution of the surface excess of surfactant to the aforementioned energy integral of $\gamma$ is negative, and thus adding surfactants results in a lower surface tension.

**NB 2:** This decrease of $\gamma$ is very often proposed as an explanation for the foaming properties of soapy solutions. “Creating water-air interfaces costs less, thus allowing foams to exist”. **This is utterly wrong.** A lot of pure liquids with surface tensions close to soapy solutions such as regular oils do not foam at all! Allowing a foam film to exist is not about how much it costs to create it, but rather giving its surface tension a law of evolution $\gamma(\epsilon, \dot{\epsilon})$ under a certain deformation of the interface $\epsilon$ to resist gravity and capillary forces. Describing these laws is a big part of this thesis work, as we will later see.

Adding a surfactant also raises the question of the evolution of the typical width $l_z$ where we start deviating from the bulk properties, and to see to what extent the interface remains thin enough to be considered 2D. In our specific case of an ionic surfactant such as SDS, the typical width of the interface is the Debye length $\lambda_s$ as predicted and experimentally shown by Bergeron et al. [10, 11] with DLVO theory [12, 13]. We can write it and estimate it in our case using our typical bulk concentration of surfactant (which is lower than the concentration in the interface):

$$l_z \sim \lambda_s = \sqrt{\frac{\epsilon_0 \epsilon_r k_B T}{\rho_s N_A z_s^2 e^2}} \sim 10 \text{ nm} \qquad (2)$$

Where $\epsilon_0 \epsilon_r = 6.9 \times 10^{-10} \text{ F m}^{-1}$ is the electric permittivity of water, $k_B T = 4.1 \times 10^{-21} \text{ J}$ is the thermal energy, $e = 1.6 \times 10^{-19} \text{ C}$ is the elementary charge, $z_s = 1$ is the ionic valence of SDS, $N_A = 6.64 \times 10^{23} \text{ mol}^{-1}$ is the Avogadro constant and $\rho_s > \rho_{s,bulk} = 1 \text{ mol/m}^3$ is an order of magnitude for the concentration of SDS.

This is the order of magnitude of the range at which our interface can interact. Thus, this allows us to distinguish what is called a black film (so thin that it appears black), the world of physical chemists with $h < 100 \text{ nm}$, from regular “thick” films, our world as hydrodynamicists with $h > 100 \text{ nm}$. Below this limit, with black films, an important ingredient to add is disjoining pressure, a key physical quantity to study the stability of these systems which is directly linked to the ageing of liquid foams [14–16]. Above this limit $h > 100 \text{ nm}$, which will always be the case from now on, we are allowed to close the box of the 3D interface and get back to our surface tension $\gamma$ defined properly along a 2D interface.

## Surface tension $\gamma$ as a force per unit length and how to measure it

All these clarifications about interfaces being made, we can have a look at what happens depending on their geometries. In fact, as we implied earlier, our interfaces are going to be curved as the foam films have heterogeneous thicknesses and are connected to menisci at the edges. This is where surface tension comes into play, but rather as a force per unit length this time. The beauty is that although it is defined only in the 2D interface, its contributions are both on the normal and tangential constraints of the surface!

The stress at the interface can be written (neglecting surface viscosities for now):

$$\bar{\sigma}.n = \nabla_s \gamma + \kappa \gamma n \qquad (3)$$
Where $\overline{\sigma}$ is the stress tensor in the liquid, $\kappa = \kappa_x + \kappa_y$ is the local mean curvature, $n$ is the normal vector of the interface (oriented toward the gas) and the $s$ indices account either for the evaluation of quantities at the interface or in the local basis of the interface. The first term on the right hand side is the Marangoni stress, that is the gradient of surface tension in the local basis of the interface. The second term is a normal force to the interface which is related to the curvature of the surface.

<figure><img src="image_11.png" /><figcaption>Figure 6: Jump of pressure $\Delta p = p_i - p_0$ across a curved interface subject to surface tension $\gamma$. $R_x$ and $R_y$ are the local radii of curvature along both axis $x$ and $y$. $d\theta_x$ and $d\theta_y$ are the angular apertures of the interface element along $x$ and $y$.</figcaption></figure>

For an interface at equilibrium, the Marangoni stress disappears on the right hand-side of eq.3 and on the left hand-side the normal stress is only balanced by the difference of pressure between the two fluids:

$$ (\overline{\sigma} \cdot n)_{eq} = \kappa \gamma n = \Delta p n \quad (4) $$

Where $\Delta p = p_{gas} - p_{liq}$ is called the Laplace pressure jump. In Fig.6, we give a more geometrical view of the different forces acting along the edges of an element of interface. $R_x$ and $R_y$ are the radii of curvature along the two directions of the interface and $d\theta_x$, $d\theta_y$ are the angular apertures of the element along the same axis. The norms of the elementary forces can directly be written: $F_{x,1} = F_{x,2} = \gamma R_y d\theta_y$ and $F_{y,1} = F_{y,2} = \gamma R_x d\theta_x$. By projecting all the forces in the normal direction of the interface and writing the balance, the difference of pressure compensating for the curvature is given by the Laplace law:

$$ \Delta p = \gamma \left( \frac{1}{R_x} + \frac{1}{R_y} \right) \quad (5) $$

Note that for a spherical interface, both radii of curvature are equal and we have a factor 2: $\Delta p = \frac{2\gamma}{R}$.

This pressure jump is an important ingredient as many experimental measurement methods for $\gamma$ rely on a Laplace pressure measurement. In Fig.7, we display an overview of six different measurement methods for surface tension [17]:

* **Wilhelmy plate:** Relies on a direct measurement of the vertical component of the capillary force $F_{cap} = 2\gamma l \cos \theta_c$. The main difficulties are to know $\theta_c$ accurately and to wet the plate properly.
Where $\overline{\sigma}$ is the stress tensor in the liquid, $\kappa = \kappa_x + \kappa_y$ is the local mean curvature, $n$ is the normal vector of the interface (oriented toward the gas) and the $s$ indices account either for the evaluation of quantities at the interface or in the local basis of the interface. The first term on the right hand side is the Marangoni stress, that is the gradient of surface tension in the local basis of the interface. The second term is a normal force to the interface which is related to the curvature of the surface.

<figure><img src="image_11.png" /><figcaption>Figure 6: Jump of pressure $\Delta p = p_i - p_0$ across a curved interface subject to surface tension $\gamma$. $R_x$ and $R_y$ are the local radii of curvature along both axis $x$ and $y$. $d\theta_x$ and $d\theta_y$ are the angular apertures of the interface element along $x$ and $y$.</figcaption></figure>

For an interface at equilibrium, the Marangoni stress disappears on the right hand-side of eq.3 and on the left hand-side the normal stress is only balanced by the difference of pressure between the two fluids:

$$ (\overline{\sigma} \cdot n)_{eq} = \kappa \gamma n = \Delta p n \qquad (4) $$

Where $\Delta p = p_{gas} - p_{liq}$ is called the Laplace pressure jump. In Fig.6, we give a more geometrical view of the different forces acting along the edges of an element of interface. $R_x$ and $R_y$ are the radii of curvature along the two directions of the interface and $d\theta_x$, $d\theta_y$ are the angular apertures of the element along the same axis. The norms of the elementary forces can directly be written: $F_{x,1} = F_{x,2} = \gamma R_y d\theta_y$ and $F_{y,1} = F_{y,2} = \gamma R_x d\theta_x$. By projecting all the forces in the normal direction of the interface and writing the balance, the difference of pressure compensating for the curvature is given by the Laplace law:

$$ \Delta p = \gamma \left( \frac{1}{R_x} + \frac{1}{R_y} \right) \qquad (5) $$

Note that for a spherical interface, both radii of curvature are equal and we have a factor 2: $\Delta p = \frac{2\gamma}{R}$.

This pressure jump is an important ingredient as many experimental measurement methods for $\gamma$ rely on a Laplace pressure measurement. In Fig.7, we display an overview of six different measurement methods for surface tension [17]:

* **Wilhelmy plate:** Relies on a direct measurement of the vertical component of the capillary force $F_{cap} = 2\gamma l \cos \theta_c$. The main difficulties are to know $\theta_c$ accurately and to wet the plate properly.
* **Maximum bubble pressure:** The goal is to blow a bubble at the tip of a capillary by imposing an air flow rate and to measure the pressure in the gas plug. At earlier times when less than half a sphere is out, the radius at the end of the gas plug decreases until a hemispherical bubble is created of radius $R$ the radius of the capillary. After this time, as more air comes in, the radius of the bubble increases until the bubble detaches from the capillary. A minimum of radius is attained with the hemispherical bubble, and thus a maximum of pressure, which is measured. The Laplace law gives $\gamma$ as both $R$ and $\Delta p$ are known.

* **Spinning drop:** An elongated plug of air is entrapped in the liquid in a horizontal spinning cylinder. The evolution of the length of the plug is a balance between the centrifugal forces applied on the bubble and its surface tension. Writing both energies, we have the relationship: $\gamma = \frac{\rho_l\omega^2}{4}R^3$ with $\rho_l$ the liquid density and $\omega$ the angular velocity. By monitoring $R$, a precise measurement of $\gamma$ can be done. This method is particularly useful for low surface tensions and is also often used with two immiscible liquids, replacing the fluid density by the difference of densities in the relationship above.

* **Du Noüy ring:** This method is very close to the Wilhelmy plate except the force is exerted on the interior and exterior of a circular ring of radius $R$. The extracted film takes shape of a catenoid, and a precise tedious calculation yields a relationship between the measured force and surface tension, knowing the geometry of the ring.

* **Capillary rise:** By dipping a cylindrical capillary of known interior radius, we see the rise of a liquid column above the surface of the bath which reaches an equilibrium quite rapidly. This is the historical experiment of Jurin, and a very academic example to highlight the presence of surface tension. The vertical liquid column balances the pressure drop due to the curvature of the wetting meniscus in the capillary. Knowing the interior radius of the capillary $r_c$, the height $h$ and density $\rho$ of the liquid column and the wetting angle $\theta$ of the liquid, it is possible to have a measurement of surface tension by writing the balance: $\rho gh = 2\gamma \cos\theta / r_c$. This method is unreliable in this form, but can be adapted for porous media and is well known in soil physics [18]

* **Pendant drop:** This method consists of creating a drop in air (or another liquid for liquid-liquid measurements) attached to a capillary of known geometry. The volume injected in the drop and the density of the liquid are known, and the profile of the drop is measured by a camera on the side. Fitting the profile, it is possible to determine the wetting angle precisely and by making the force balance with the weight it is possible to determine the surface tension. This is a very reliable method for a wide range of liquids. Besides, it enables dynamic measurements by controlling the amount of liquid pumped in/out of the drop, probing the surface rheology of the drop interface.

### 0.1.3 Coupling the bulk to the interface

#### For a single interface

As suggested earlier, the surface tension is a function of the surface excesses of the surfactant species dwelling at the interface. Let us give the notation $\Gamma_i$ for the surface excess of a species
* **Maximum bubble pressure:** The goal is to blow a bubble at the tip of a capillary by imposing an air flow rate and to measure the pressure in the gas plug. At earlier times when less than half a sphere is out, the radius at the end of the gas plug decreases until a hemispherical bubble is created of radius $R$ the radius of the capillary. After this time, as more air comes in, the radius of the bubble increases until the bubble detaches from the capillary. A minimum of radius is attained with the hemispherical bubble, and thus a maximum of pressure, which is measured. The Laplace law gives $\gamma$ as both $R$ and $\Delta p$ are known.

* **Spinning drop:** An elongated plug of air is entrapped in the liquid in a horizontal spinning cylinder. The evolution of the length of the plug is a balance between the centrifugal forces applied on the bubble and its surface tension. Writing both energies, we have the relationship: $\gamma = \frac{\rho_l\omega^2}{4}R^3$ with $\rho_l$ the liquid density and $\omega$ the angular velocity. By monitoring $R$, a precise measurement of $\gamma$ can be done. This method is particularly useful for low surface tensions and is also often used with two immiscible liquids, replacing the fluid density by the difference of densities in the relationship above.

* **Du Noüy ring:** This method is very close to the Wilhelmy plate except the force is exerted on the interior and exterior of a circular ring of radius $R$. The extracted film takes shape of a catenoid, and a precise tedious calculation yields a relationship between the measured force and surface tension, knowing the geometry of the ring.

* **Capillary rise:** By dipping a cylindrical capillary of known interior radius, we see the rise of a liquid column above the surface of the bath which reaches an equilibrium quite rapidly. This is the historical experiment of Jurin, and a very academic example to highlight the presence of surface tension. The vertical liquid column balances the pressure drop due to the curvature of the wetting meniscus in the capillary. Knowing the interior radius of the capillary $r_c$, the height $h$ and density $\rho$ of the liquid column and the wetting angle $\theta$ of the liquid, it is possible to have a measurement of surface tension by writing the balance: $\rho gh = 2\gamma \cos\theta / r_c$. This method is unreliable in this form, but can be adapted for porous media and is well known in soil physics [18]

* **Pendant drop:** This method consists of creating a drop in air (or another liquid for liquid-liquid measurements) attached to a capillary of known geometry. The volume injected in the drop and the density of the liquid are known, and the profile of the drop is measured by a camera on the side. Fitting the profile, it is possible to determine the wetting angle precisely and by making the force balance with the weight it is possible to determine the surface tension. This is a very reliable method for a wide range of liquids. Besides, it enables dynamic measurements by controlling the amount of liquid pumped in/out of the drop, probing the surface rheology of the drop interface.

### 0.1.3 Coupling the bulk to the interface

#### For a single interface

As suggested earlier, the surface tension is a function of the surface excesses of the surfactant species dwelling at the interface. Let us give the notation $\Gamma_i$ for the surface excess of a species
<figure><img src="image_12.png" /><figcaption>Figure 7: Various experimental techniques commonly used to characterise surface tension. Credits to Berry et al. [17].</figcaption></figure>

$i$ (usually expressed in $mol/m^2$). Surface tension can be written as an unambiguous function of $\Gamma_i$ using the thermodynamic identity (later detailed in chapter 1):

$$d\gamma = -\sum_{i} \Gamma_{i}d\mu_{i} \qquad (6)$$

Where $\mu_i$ are the chemical potentials of the different species.

Thus, our problem is now to describe the evolution of these $\Gamma_i$ as the population at the interface is coupled through diffusion to the population $c_i$ in the bulk (the concentration of species $i$ in $mol/m^3$). **Note that for now we speak of an infinite bulk, thus not limiting the depth at which we probe it by diffusion.** This coupling is rather difficult to establish though, as it depends on the evolution of the area of the interface, the time of diffusion, of convection, the competition with other species etc... resulting in a complex function $\Gamma_i = f(c_i, A, \Gamma_j, t, ...)$ which is hard to describe in a general case. Even for a single species on an interface at rest in contact with an infinite bulk, the proper description of the diffusion problem is not trivial, as suggests the work of Ward, Tordai [19].

If we now want to capture the physics of these processes with scaling laws, we need to define several quantities. First the diffusion coefficient $D_i$ ($m^2/s$) giving the time $\tau$ for a species $i$ to diffuse through a width $l$ by: $\tau = l^2/D_i$. Second, what we will call the reservoir length $h_{\Gamma,i} = \Gamma_i/c_i$ which is a length giving an estimate of how deep we need to go in the bulk to gather enough surfactants to populate the interface. For surfactant molecules, this length is actually a good indicator of their solubility: soluble surfactants have a big population in the bulk, and we do not need to go “that far” to populate the interface ($h_{\Gamma} \to 0$), insoluble surfactants are all the interface, and are barely present in the bulk ($h_{\Gamma} \to \infty$).

These two limits, with SDS closer to the soluble limit and DOH closer to the insoluble limit, change drastically the physics of the interface if we want to deform it. This is well illustrated in
<figure><img src="image_12.png" /><figcaption>Figure 7: Various experimental techniques commonly used to characterise surface tension. Credits to Berry et al. [17].</figcaption></figure>

$i$ (usually expressed in mol/m²). Surface tension can be written as an unambiguous function of $\Gamma_i$ using the thermodynamic identity (later detailed in chapter 1):

$$d\gamma = -\sum_i \Gamma_i d\mu_i \qquad (6)$$

Where $\mu_i$ are the chemical potentials of the different species.

Thus, our problem is now to describe the evolution of these $\Gamma_i$ as the population at the interface is coupled through diffusion to the population $c_i$ in the bulk (the concentration of species $i$ in mol/m³). **Note that for now we speak of an infinite bulk, thus not limiting the depth at which we probe it by diffusion.** This coupling is rather difficult to establish though, as it depends on the evolution of the area of the interface, the time of diffusion, of convection, the competition with other species etc... resulting in a complex function $\Gamma_i = f(c_i, A, \Gamma_j, t, ...)$ which is hard to describe in a general case. Even for a single species on an interface at rest in contact with an infinite bulk, the proper description of the diffusion problem is not trivial, as suggests the work of Ward, Tordai [19].

If we now want to capture the physics of these processes with scaling laws, we need to define several quantities. First the diffusion coefficient $D_i$ (m²/s) giving the time $\tau$ for a species $i$ to diffuse through a width $l$ by: $\tau = l^2/D_i$. Second, what we will call the reservoir length $h_{\Gamma,i} = \Gamma_i/c_i$ which is a length giving an estimate of how deep we need to go in the bulk to gather enough surfactants to populate the interface. For surfactant molecules, this length is actually a good indicator of their solubility: soluble surfactants have a big population in the bulk, and we do not need to go “that far” to populate the interface ($h_{\Gamma} \to 0$), insoluble surfactants are all the interface, and are barely present in the bulk ($h_{\Gamma} \to \infty$).

These two limits, with SDS closer to the soluble limit and DOH closer to the insoluble limit, change drastically the physics of the interface if we want to deform it. This is well illustrated in
the Fig.8 from Manikantan et al. [20]. For a pure liquid (case A), the creation of an amount ΔA of interface has an energy cost given by the definition ΔF = γ₀ΔA. For a soluble surfactant (case B), creating the same amount of interface requires to bring surfactants from the bulk to the interface, which is ensured by the vast amounts of surfactants present there, and the cost is ΔF = γₐₐqΔA with γₐₐq a surface tension of equilibrium for a given population c of surfactant in the bulk. For a perfectly insoluble surfactant (case C), the population at the interface is fixed and no exchange with the bulk is authorised, meaning that the surface tension γ varies as the interface is deformed leading to a complex energy contribution ΔF = ∫γ(Γ)dA (NB: We took the example of the creation of an interface, meaning a deformation in extension of it, but the same reasoning goes for a compression, a destruction of interface requiring to evacuate surfactants towards the bulk).

<figure><img src="image_13.png" /><figcaption>Figure 8: Energy cost of the creation of interface in the case of A: Pure liquid B: Soluble surfactants C: Insoluble surfactants. Credits to Manikantan et al. [20].</figcaption></figure>

Any surfactant is between these two limits. Besides, even in the case of an infinite bulk, these limits are time dependent as it depends on the time it takes for the exchanges of surfactant to take place. In theory, any interface deformed fast enough can be considered to be loaded with insoluble surfactants. For regular surfactants such as SDS or DOH, which have approximately the same molecular sizes R<sub>hydro</sub> ~ 1 nm, the diffusion coefficients can be estimated as:

$$D = \frac{k_B T}{6\pi\eta R_{hydro}} \sim 10^{-10} \text{ m}^2/\text{s} \qquad (7)$$

Where k<sub>B</sub>T = 4.1 × 10<sup>-21</sup> J is the thermal energy, η = 10<sup>-3</sup> Pa·s is the dynamic viscosity of water and R<sub>hydro</sub> the aforementioned hydrodynamic radius of the molecule. With such a diffusion coefficient, probing one millimetre takes approximately τ<sub>mm</sub> ~ 10<sup>4</sup> s and one micron τ<sub>μm</sub> ~ 10<sup>-2</sup> s, for examples. Below these times, which depend on the depth of depletion/evacuation, we cannot consider the interface to be at equilibrium, even though the surfactants are very soluble.
the Fig.8 from Manikantan et al. [20]. For a pure liquid (case A), the creation of an amount $\Delta A$ of interface has an energy cost given by the definition $\Delta F = \gamma_0 \Delta A$. For a soluble surfactant (case B), creating the same amount of interface requires to bring surfactants from the bulk to the interface, which is ensured by the vast amounts of surfactants present there, and the cost is $\Delta F = \gamma_{eq} \Delta A$ with $\gamma_{eq}$ a surface tension of equilibrium for a given population c of surfactant in the bulk. For a perfectly insoluble surfactant (case C), the population at the interface is fixed and no exchange with the bulk is authorised, meaning that the surface tension $\gamma$ varies as the interface is deformed leading to a complex energy contribution $\Delta F = \int \gamma(\Gamma)dA$ (NB: We took the example of the creation of an interface, meaning a deformation in extension of it, but the same reasoning goes for a compression, a destruction of interface requiring to evacuate surfactants towards the bulk).

<figure><img src="image_13.png" /><figcaption>Figure 8: Energy cost of the creation of interface in the case of A: Pure liquid B: Soluble surfactants C: Insoluble surfactants. Credits to Manikantan et al. [20].</figcaption></figure>

Any surfactant is between these two limits. Besides, even in the case of an infinite bulk, these limits are time dependent as it depends on the time it takes for the exchanges of surfactant to take place. In theory, any interface deformed fast enough can be considered to be loaded with insoluble surfactants. For regular surfactants such as SDS or DOH, which have approximately the same molecular sizes $R_{hydro} \sim 1$ nm, the diffusion coefficients can be estimated as:

$$D = \frac{k_B T}{6\pi\eta R_{hydro}} \sim 10^{-10} \text{ m}^2/\text{s} \qquad (7)$$

Where $k_B T = 4.1 \times 10^{-21}$ J is the thermal energy, $\eta = 10^{-3}$ Pa·s is the dynamic viscosity of water and $R_{hydro}$ the aforementioned hydrodynamic radius of the molecule. With such a diffusion coefficient, probing one millimetre takes approximately $\tau_{mm} \sim 10^4$ s and one micron $\tau_{\mu m} \sim 10^{-2}$ s, for examples. Below these times, which depend on the depth of depletion/evacuation, we cannot consider the interface to be at equilibrium, even though the surfactants are very soluble.
Furthermore, another ingredient very common with surfactant and present in our case is the presence of micelles. They fix the monomeric concentration of surfactants at the *CMC*, which is an important change in the diffusion field. In our specific case of SDS in micellar regime with DOH, it also represents a new time of diffusion in the system, as bulk-DOH is mainly stored in the micelles.

During the beginning of my Ph.D, a campaign of pendant drop measurements was performed with our foaming solutions in order to precisely know the equilibrium surface tensions for all our SDS/DOH mixtures. The results are shown in Fig.9. To fix ideas, the surface tension of pure water at T = 20°C is $\gamma_0$ = 72.8 mN/m.

<figure><img src="image_14.png" /><figcaption>Figure 9: Surface tension of SDS/DOH mixtures measured by the pendant drop method. **Left:** Micellar regime [$SDS$] = 2.4 CMC for different concentration in DOH, increasing from top to bottom: 0 (red), 15 mg/L (orange), 30 mg/L (green), 50 mg/L (cyan). **Right:** Sub-micellar regime [$SDS$] = 0.75 CMC for different concentration in DOH, increasing from top to bottom: 0 (red), 5 mg/L (orange), 10 mg/L (green), 20 mg/L (cyan).</figcaption></figure>

It has first to be noted that this method was performed at constant volume for the drop, thus injecting continuously liquid to compensate for the loss by evaporation. This may tend to concentrate the bulk in surfactant as time goes by. In the sub-micellar regime (Fig.9 Right), the equilibrium is essentially not attained. Whether or not DOH is present, the interface requires to probe deep in the drop to stabilise its population. For a drop of size $r \sim 1$ mm, we computed that it can go up to waiting for $\tau \sim 10^4$s which is not incompatible. Note that at a high concentration of [$DOH$] = 20 mg/L, we start to see what appears to be a phase transition in the form of an angular point for $\gamma(t)$. This could be related to the formation of highly concentrated regions in dodecanol at the surface (see Lu et al. [21], Fainerman et al. [22]).

In the micellar regime, surface tension stabilises over a few tens of seconds, as SDS is more concentrated overall. The remaining changes in surface tension are due to DOH and impurities, which are expected to be stored in the micelles in the bulk, enhancing their effective solubility. In the following, for most of our solutions, we work at at least twice the CMC in SDS where a $\gamma_{eq}$ is very well defined.

All the parameters at stake here, micelle concentration, fraction of insoluble surfactants in
Furthermore, another ingredient very common with surfactant and present in our case is the presence of micelles. They fix the monomeric concentration of surfactants at the *CMC*, which is an important change in the diffusion field. In our specific case of SDS in micellar regime with DOH, it also represents a new time of diffusion in the system, as bulk-DOH is mainly stored in the micelles.

During the beginning of my Ph.D, a campaign of pendant drop measurements was performed with our foaming solutions in order to precisely know the equilibrium surface tensions for all our SDS/DOH mixtures. The results are shown in Fig. 9. To fix ideas, the surface tension of pure water at T = 20°C is $\gamma_0 = 72.8$ mN/m.

<figure><img src="image_14.png" /><figcaption>Figure 9: Surface tension of SDS/DOH mixtures measured by the pendant drop method. **Left:** Micellar regime [$SDS$] = 2.4 CMC for different concentration in DOH, increasing from top to bottom: 0 (red), 15 mg/L (orange), 30 mg/L (green), 50 mg/L (cyan). **Right:** Sub-micellar regime [$SDS$] = 0.75 CMC for different concentration in DOH, increasing from top to bottom: 0 (red), 5 mg/L (orange), 10 mg/L (green), 20 mg/L (cyan).</figcaption></figure>

It has first to be noted that this method was performed at constant volume for the drop, thus injecting continuously liquid to compensate for the loss by evaporation. This may tend to concentrate the bulk in surfactant as time goes by. In the sub-micellar regime (Fig. 9 Right), the equilibrium is essentially not attained. Whether or not DOH is present, the interface requires to probe deep in the drop to stabilise its population. For a drop of size $r \sim 1$ mm, we computed that it can go up to waiting for $\tau \sim 10^4$s which is not incompatible. Note that at a high concentration of [DOH] = 20 mg/L, we start to see what appears to be a phase transition in the form of an angular point for $\gamma(t)$. This could be related to the formation of highly concentrated regions in dodecanol at the surface (see Lu et al. [21], Fainerman et al. [22]).

In the micellar regime, surface tension stabilises over a few tens of seconds, as SDS is more concentrated overall. The remaining changes in surface tension are due to DOH and impurities, which are expected to be stored in the micelles in the bulk, enhancing their effective solubility. In the following, for most of our solutions, we work at at least twice the CMC in SDS where a $\gamma_{eq}$ is very well defined.

All the parameters at stake here, micelle concentration, fraction of insoluble surfactants in
the micelles, reservoir lengths, diffusion distances, are all interdependent and represent a complex coupling between bulk and interface.

However, we have to bear in mind that all these interdependencies we just highlighted were meant in the case where our interface is in contact with an infinite homogeneous bulk. Thus we anticipate that because of the specific geometry of foam films, some of the exchange mechanisms will be blocked, and we will later see how it plays a role in the problem.

Now that we know more about the story and dependencies of $\gamma$, we need to build a proper constitutive law for the rheology of our interface, which involves other ingredients besides $\gamma$: surface viscosities.

### 0.1.4 Constitutive equation and 2D stress tensor $\overline{\overline{\sigma}}_s$

Let us provide our interface with a constitutive equation for a proper 2D stress tensor. It must take into account surface tension $\gamma$ as well as what is called intrinsic viscosities $\eta_s$ and $\kappa_s$. This description is known as a Boussinesq-Scriven model for the interface [23, 24], which has been thoroughly described by Lopez et al. [25] and Manikantan et al. [20]. In Appendix 1, we also address the construction of this tensor and the momentum conservation associated to it. It can be written:

$$ \overline{\overline{\sigma}}_s = (\gamma + (\kappa_s - \eta_s)\nabla_s \cdot \mathbf{v}_s)\overline{\overline{\mathbf{I}}}_s + \eta_s(\overline{\overline{\nabla}}_s \mathbf{v}_s + t\overline{\overline{\nabla}}_s \mathbf{v}_s) \quad (8) $$

Where $\gamma$ is the surface tension, $\mathbf{v}_s$ the velocity at the interface, $\eta_s$ the intrinsic surface shear viscosity, $\kappa_s$ the intrinsic dilatational viscosity, $\nabla_s$ the nabla operator in the basis of the interface, and $\overline{\overline{\mathbf{I}}}_s$ is the identity tensor in the same basis.

**NB:** we do not add any shear elastic response for the interface. With proteins or gels forming structures at the interface, this is a relevant ingredient though (see for example [26, 27]).

This constitutive equation is a complete 2D analogue of a 3D viscous compressible fluid. We can now write the stress balance for the interface:

$$ 0 = f_{\text{ext}} + \nabla_s \cdot \overline{\overline{\sigma}}_s \quad (9) $$

Where $f_{\text{ext}}$ are the exterior forces applied to the interface (such as friction with the bulk liquid and bulk gas for instance) and will be treated more thoroughly in chapter 1. If we develop the term with the stress tensor, which is done in detail in Appendix 1, the projection in the plane of the interface is [28–30]:

$$ 0 = f_{\text{ext},s} + \nabla_s \gamma + \kappa_s \nabla_s (\nabla_s \cdot \mathbf{v}_s) + \eta_s \Delta_s \mathbf{v}_s \quad (10) $$

Where $f_{\text{ext},s}$ is now the projection of the exterior forces in the plane of the interface. This expression eq. 10 will be made explicit when needed only in the simple case of a quasi-plane interface.

Let us now focus on the second term of the right-hand side $\nabla_s \gamma$, which is the Marangoni stress at the interface. This is actually where a lot of the physical richness lies in this equation, and we need to treat it carefully. All variations of surface tension ultimately have one origin in
the micelles, reservoir lengths, diffusion distances, are all interdependent and represent a complex coupling between bulk and interface.

However, we have to bear in mind that all these interdependencies we just highlighted were meant in the case where our interface is in contact with an infinite homogeneous bulk. Thus we anticipate that because of the specific geometry of foam films, some of the exchange mechanisms will be blocked, and we will later see how it plays a role in the problem.

Now that we know more about the story and dependencies of $\gamma$, we need to build a proper constitutive law for the rheology of our interface, which involves other ingredients besides $\gamma$: surface viscosities.

### 0.1.4 Constitutive equation and 2D stress tensor $\overline{\overline{\sigma}}_s$

Let us provide our interface with a constitutive equation for a proper 2D stress tensor. It must take into account surface tension $\gamma$ as well as what is called intrinsic viscosities $\eta_s$ and $\kappa_s$. This description is known as a Boussinesq-Scriven model for the interface [23, 24], which has been thoroughly described by Lopez et al. [25] and Manikantan et al. [20]. In Appendix 1, we also address the construction of this tensor and the momentum conservation associated to it. It can be written:

$$ \overline{\overline{\sigma}}_s = (\gamma + (\kappa_s - \eta_s)\nabla_s \cdot \mathbf{v}_s)\overline{\overline{\mathbf{I}}}_s + \eta_s(\overline{\overline{\nabla}}_s \mathbf{v}_s + t\overline{\overline{\nabla}}_s \mathbf{v}_s) \quad (8) $$

Where $\gamma$ is the surface tension, $\mathbf{v}_s$ the velocity at the interface, $\eta_s$ the intrinsic surface shear viscosity, $\kappa_s$ the intrinsic dilatational viscosity, $\nabla_s$ the nabla operator in the basis of the interface, and $\overline{\overline{\mathbf{I}}}_s$ is the identity tensor in the same basis.

**NB:** we do not add any shear elastic response for the interface. With proteins or gels forming structures at the interface, this is a relevant ingredient though (see for example [26, 27]).

This constitutive equation is a complete 2D analogue of a 3D viscous compressible fluid. We can now write the stress balance for the interface:

$$ 0 = f_{\text{ext}} + \nabla_s \cdot \overline{\overline{\sigma}}_s \quad (9) $$

Where $f_{\text{ext}}$ are the exterior forces applied to the interface (such as friction with the bulk liquid and bulk gas for instance) and will be treated more thoroughly in chapter 1. If we develop the term with the stress tensor, which is done in detail in Appendix 1, the projection in the plane of the interface is [28–30]:

$$ 0 = f_{\text{ext},s} + \nabla_s \gamma + \kappa_s \nabla_s (\nabla_s \cdot \mathbf{v}_s) + \eta_s \Delta_s \mathbf{v}_s \quad (10) $$

Where $f_{\text{ext},s}$ is now the projection of the exterior forces in the plane of the interface. This expression eq. 10 will be made explicit when needed only in the simple case of a quasi-plane interface.

Let us now focus on the second term of the right-hand side $\nabla_s \gamma$, which is the Marangoni stress at the interface. This is actually where a lot of the physical richness lies in this equation, and we need to treat it carefully. All variations of surface tension ultimately have one origin in
the frame of our problem class: a variation of surface excess of surfactants $\Gamma$. The link between these variations is called the Marangoni modulus $E_M$ [20, 28, 31] defined as:

$$E_M = -\Gamma \frac{d\gamma}{d\Gamma} \qquad (11)$$

This is a thermodynamic quantity which can be computed using eq. 6 and of which a model will be proposed for foam films throughout chapter 1.

<figure><img src="image_15.png" /><figcaption>Figure 10: Ingredients for the interfacial rheology of a surfactant-laden interface: **A:** Let us look at the interface of a piece of film, and see the ingredients of the constitutive law (Credits to Eric Sorensen<fnref n="4" />). **B:** The interface element has a purely viscous response to shear, with a surface shear viscosity $\eta_s$. **C:** Under stretch, the film has both an intrinsic viscous response (with $\kappa_s$ the dilatational surface viscosity) and a response of surface tension $\frac{d\gamma}{dA}$ that includes an elastic part and an extrinsic viscous contribution. **NB:** For the sake of representation, here we take a piece of film big enough to encompass quite a large thickness variation. This has to be avoided in practice and a typical local scale in the following will be chosen below these scales. Also, we need to bear in mind that we use a foam film to illustrate the ingredients of a constitutive law which is valid in the more general case of a single interface in contact with a bulk of infinite size.</figcaption></figure>

When a piece of interface A is deformed, its surface excess of surfactants $\Gamma = f(A)$ may vary. This also leads to a variation of surface tension with the area $\frac{d\gamma}{dA}$ which is taken into account in the $\nabla_s\gamma$ term. Fig. 10 summarises all the different contributions to the rheology of the interface, both under shear or dilation.

Notably, we can look at two limits. With insoluble surfactants, the population N at the interface is constant, leading to a linearly varying $\Gamma = N/A$ with $d\Gamma = -NdA/A^2$. With soluble surfactants, the bulk population compensates for the change in $\Gamma$ during/after the deformation, leading to no change in surface excess and thus no change in surface tension. Thus, this is also a matter of how fast the deformation occurs, as soluble surfactants need time to restore their population through diffusion. This could be summarised by “any surfactant is insoluble for fast enough deformations of the interface”.

<footnote n="4">⁴http://esorensen.com/soap-film-photography/</footnote>
the frame of our problem class: a variation of surface excess of surfactants $\Gamma$. The link between these variations is called the Marangoni modulus $E_M$ [20, 28, 31] defined as:

$$E_M = -\Gamma \frac{d\gamma}{d\Gamma} \qquad (11)$$

This is a thermodynamic quantity which can be computed using eq. 6 and of which a model will be proposed for foam films throughout chapter 1.

<figure><img src="image_15.png" /><figcaption>Figure 10: Ingredients for the interfacial rheology of a surfactant-laden interface: **A:** Let us look at the interface of a piece of film, and see the ingredients of the constitutive law (Credits to Eric Sorensen<fnref n="4" />). **B:** The interface element has a purely viscous response to shear, with a surface shear viscosity $\eta_s$. **C:** Under stretch, the film has both an intrinsic viscous response (with $\kappa_s$ the dilatational surface viscosity) and a response of surface tension $\frac{d\gamma}{dA}$ that includes an elastic part and an extrinsic viscous contribution. **NB:** For the sake of representation, here we take a piece of film big enough to encompass quite a large thickness variation. This has to be avoided in practice and a typical local scale in the following will be chosen below these scales. Also, we need to bear in mind that we use a foam film to illustrate the ingredients of a constitutive law which is valid in the more general case of a single interface in contact with a bulk of infinite size.</figcaption></figure>

When a piece of interface A is deformed, its surface excess of surfactants $\Gamma = f(A)$ may vary. This also leads to a variation of surface tension with the area $\frac{d\gamma}{dA}$ which is taken into account in the $\nabla_s\gamma$ term. Fig. 10 summarises all the different contributions to the rheology of the interface, both under shear or dilation.

Notably, we can look at two limits. With insoluble surfactants, the population N at the interface is constant, leading to a linearly varying $\Gamma = N/A$ with $d\Gamma = -NdA/A^2$. With soluble surfactants, the bulk population compensates for the change in $\Gamma$ during/after the deformation, leading to no change in surface excess and thus no change in surface tension. Thus, this is also a matter of how fast the deformation occurs, as soluble surfactants need time to restore their population through diffusion. This could be summarised by “any surfactant is insoluble for fast enough deformations of the interface”.

<footnote n="4">⁴http://esorensen.com/soap-film-photography/</footnote>
In the purely insoluble regime, the variations of $\gamma$ are a conservative contribution and the deformed interface reacts with an elastic behaviour of **surface elasticity** $E_M = A \frac{d\gamma}{dA} \bigg|_{insoluble}$. In the soluble regime, deforming the interface leads to a variation of surface tension $\frac{d\gamma}{dA} = f(\dot{\epsilon})$ depending on the deformation rate, which is the signature of a net loss of energy, i.e. an **extrinsic dilatational surface viscosity**.

This latter is not to be confused with the intrinsic viscosities $\eta_s$ and $\kappa_s$ which are features of the interface itself. The extrinsic dilatational viscosity originates from the bulk, whose role as a surfactant reservoir is to smooth out the surface variations of surfactant population by adsorption/desorption.

Having an experimental access to all these quantities is also very hard, and subject to a vast and perilous literature [32–36]!

Using a Langmuir-Blodgett apparatus (a quite ancient method [37, 38] which has had a century of technological improvements) can be relevant for specific cases such as the Marangoni modulus of very insoluble surfactants deposited from the gas phase [32], or complex molecules such as proteins or polymers which have very long diffusion dynamics (e.g. Renault *et al.* [39]). However, measuring the surface intrinsic viscosities with this apparatus poses first the problem of an unfortunate coupling between shear and compression because of the 1D stretching, and second a coupling with the extrinsic viscosity. In a classical Langmuir configuration, we end up measuring an effective viscosity which is a combination $\eta_{s,eff} = \eta_s + \kappa_s + E_M\tau$, with $\tau$ a typical time of adsorption/desorption [20].

Last but not least, the surface viscosity of a wide range of surfactants, notably soluble surfactants, are now known to be poorly measured, as bulk friction with the gas or liquid have been overlooked. This is the observation made by Zell *et al.* [36], who set an upper bound for the surface shear viscosity of soluble surfactants $\eta_{s,max} = 4 \times 10^{-9}$ Pa·m·s which is below or of the order of the majority of measurements found in the literature.

We have encountered this problem of bulk friction in the frame of this work as well, and this will be subject to extensive discussions and modelling in chapter 2.

## 0.2 2D hydrodynamics and rheology of a foam film

Now that we have provided a proper rheological description of a single interface, let us look at the particular case this thesis work is all about: foam films. These systems are peculiar because of the extreme aspect ratios between the thickness separating two interfaces ($h \sim 10^{-6}$m) and the typical length scales in the tangential directions to the interfaces ($l \sim 10^{-4} - 10^{-2}$m). This has a lot of consequences on the viscoelastic response of the interfaces under deformation, as the total quantity of surfactants present at the interfaces can easily be of the order of/much greater than the quantities contained in the thin bulk separating them.

### 0.2.1 Foam films only have plug flows far from the meniscus

A first important piece of knowledge about foam films is that their Poiseuille flows due to local pressure gradients are very slow. The direct consequence is the uniformity of the velocity field
In the purely insoluble regime, the variations of $\gamma$ are a conservative contribution and the deformed interface reacts with an elastic behaviour of **surface elasticity** $E_M = A \frac{d\gamma}{dA} \bigg|_{insoluble}$. In the soluble regime, deforming the interface leads to a variation of surface tension $\frac{d\gamma}{dA} = f(\dot{\epsilon})$ depending on the deformation rate, which is the signature of a net loss of energy, i.e. an **extrinsic dilatational surface viscosity**.

This latter is not to be confused with the intrinsic viscosities $\eta_s$ and $\kappa_s$ which are features of the interface itself. The extrinsic dilatational viscosity originates from the bulk, whose role as a surfactant reservoir is to smooth out the surface variations of surfactant population by adsorption/desorption.

Having an experimental access to all these quantities is also very hard, and subject to a vast and perilous literature [32–36]!

Using a Langmuir-Blodgett apparatus (a quite ancient method [37, 38] which has had a century of technological improvements) can be relevant for specific cases such as the Marangoni modulus of very insoluble surfactants deposited from the gas phase [32], or complex molecules such as proteins or polymers which have very long diffusion dynamics (e.g. Renault *et al.* [39]). However, measuring the surface intrinsic viscosities with this apparatus poses first the problem of an unfortunate coupling between shear and compression because of the 1D stretching, and second a coupling with the extrinsic viscosity. In a classical Langmuir configuration, we end up measuring an effective viscosity which is a combination $\eta_{s,eff} = \eta_s + \kappa_s + E_M\tau$, with $\tau$ a typical time of adsorption/desorption [20].

Last but not least, the surface viscosity of a wide range of surfactants, notably soluble surfactants, are now known to be poorly measured, as bulk friction with the gas or liquid have been overlooked. This is the observation made by Zell *et al.* [36], who set an upper bound for the surface shear viscosity of soluble surfactants $\eta_{s,max} = 4 \times 10^{-9} \text{ Pa ms}$ which is below or of the order of the majority of measurements found in the literature.

We have encountered this problem of bulk friction in the frame of this work as well, and this will be subject to extensive discussions and modelling in chapter 2.

## 0.2 2D hydrodynamics and rheology of a foam film

Now that we have provided a proper rheological description of a single interface, let us look at the particular case this thesis work is all about: foam films. These systems are peculiar because of the extreme aspect ratios between the thickness separating two interfaces ($h \sim 10^{-6}\text{m}$) and the typical length scales in the tangential directions to the interfaces ($l \sim 10^{-4} - 10^{-2}\text{m}$). This has a lot of consequences on the viscoelastic response of the interfaces under deformation, as the total quantity of surfactants present at the interfaces can easily be of the order of/much greater than the quantities contained in the thin bulk separating them.

### 0.2.1 Foam films only have plug flows far from the meniscus

A first important piece of knowledge about foam films is that their Poiseuille flows due to local pressure gradients are very slow. The direct consequence is the uniformity of the velocity field
along the thickness of the film: a plug flow. Thus, a moving piece of interface carries its bulk all along its way! This is the sliding puzzle dynamics.

Let us put some numbers on the Poiseuille component of the flow. Using the geometry of Fig.11 for an element of film, we get the typical difference of pressure: $\Delta p \sim \gamma_0 h / \ell^2$. This leads to the typical Poiseuille velocity $v_p$ between the interfaces:

$$v_p \sim \frac{h^2 \Delta p}{\eta \ell} \sim \frac{\gamma_0 h^3}{\eta l^3} \sim 10 \, \mu\text{m/s} \qquad (12)$$

Where $\eta \sim 10^{-3} \text{Pa} \cdot \text{s}$ is the viscosity of water and $\gamma_0 \sim 10^{-2} \text{N/m}$ the equilibrium surface tension of the foamy solution. **This velocity will be small compared to any other velocity seen in our foam films. Only block movements of both interfaces and the bulk they sandwich are enabled. This notably allows the existence of localised gradients of thickness in the film for times much longer than our typical time of experiments $\tau_{def} \sim 1 \text{s.}$**

### 0.2.2 Thin bulk means fast chemical equilibrium

Following up on the constitutive law for one interface eq.8, and the stress balance attached to it, what we need is an equation for $\gamma$ in order to get a proper set of 2D Navier-Stokes equations. However, we just saw in subsection 0.1.4 that the coupling with an infinite bulk makes it very difficult to describe. To further advance, **we need to start not speaking of one single interface anymore, but rather look at our specific case of foam films where the bulk is thin enough for its surfactant reservoir to display finite size effects.**

As mentioned in subsection 0.1.4, the surface tension of a deformed interface is a function of time tending towards an equilibrium value imposed by the chemical equilibrium with the bulk. Let us have a look at the typical times of exchanges in a foam film. The geometry is shown in Fig.11.

Here, the thickness of the foam film $h \sim 1 \, \mu\text{m}$ imposes a confinement in the vertical direction. Also, we anticipate that the dynamics we will be looking at take place at the order of $\tau_{def} \sim 1 \, \text{s}$, and that all typical sizes in the tangential directions of film will be greater than/of the order of $\ell \sim 100 \, \mu\text{m}$.

Given these orders of magnitude, it is now possible to compute the time it takes for the interface to equilibrate with the bulk. In the case of a fresh interface in contact with a bulk very limited in depth, the main adsorption mechanism for times $t < \tau_{def}$ is diffusion [22, 40]. In this case, the time it takes for the chemical equilibrium to be established is $\tau_{d,\perp} = h^2/D \sim 10^{-2} \, \text{s}$ with $D \sim 10^{-10} \, \text{m}^2/\text{s}$ from eq.7. This time is much lower than the typical time of deformation, and **we thus consider the chemical equilibrium between the bulk and the interfaces to be instantaneous.**

In the tangential direction, all typical sizes are greater than $100 \, \mu\text{m}$ meaning that the transport by diffusion is here a matter of $\tau_{d,\parallel} = \ell^2/D \sim 10^2 \, \text{s}$. Also, we need to evaluate the bulk advection transport due to the Poiseuille flows computed in subsection 0.2.1 with its velocity $v_p \sim 10 \, \mu\text{m/s}$ and thus to a typical time of transport over $\ell$:
along the thickness of the film: a plug flow. Thus, a moving piece of interface carries its bulk all along its way! This is the sliding puzzle dynamics.

Let us put some numbers on the Poiseuille component of the flow. Using the geometry of Fig.11 for an element of film, we get the typical difference of pressure: $\Delta p \sim \gamma_0 h / \ell^2$. This leads to the typical Poiseuille velocity $v_p$ between the interfaces:

$$v_p \sim \frac{h^2 \Delta p}{\eta \ell} \sim \frac{\gamma_0 h^3}{\eta l^3} \sim 10 \, \mu\text{m/s} \qquad (12)$$

Where $\eta \sim 10^{-3} \text{Pa} \cdot \text{s}$ is the viscosity of water and $\gamma_0 \sim 10^{-2} \text{N/m}$ the equilibrium surface tension of the foamy solution. **This velocity will be small compared to any other velocity seen in our foam films. Only block movements of both interfaces and the bulk they sandwich are enabled. This notably allows the existence of localised gradients of thickness in the film for times much longer than our typical time of experiments** $\tau_{def} \sim 1 \text{s}$.

### 0.2.2 Thin bulk means fast chemical equilibrium

Following up on the constitutive law for one interface eq.8, and the stress balance attached to it, what we need is an equation for $\gamma$ in order to get a proper set of 2D Navier-Stokes equations. However, we just saw in subsection 0.1.4 that the coupling with an infinite bulk makes it very difficult to describe. To further advance, **we need to start not speaking of one single interface anymore, but rather look at our specific case of foam films where the bulk is thin enough for its surfactant reservoir to display finite size effects.**

As mentioned in subsection 0.1.4, the surface tension of a deformed interface is a function of time tending towards an equilibrium value imposed by the chemical equilibrium with the bulk. Let us have a look at the typical times of exchanges in a foam film. The geometry is shown in Fig.11.

Here, the thickness of the foam film $h \sim 1 \, \mu\text{m}$ imposes a confinement in the vertical direction. Also, we anticipate that the dynamics we will be looking at take place at the order of $\tau_{def} \sim 1 \, \text{s}$, and that all typical sizes in the tangential directions of film will be greater than/of the order of $\ell \sim 100 \, \mu\text{m}$.

Given these orders of magnitude, it is now possible to compute the time it takes for the interface to equilibrate with the bulk. In the case of a fresh interface in contact with a bulk very limited in depth, the main adsorption mechanism for times $t < \tau_{def}$ is diffusion [22, 40]. In this case, the time it takes for the chemical equilibrium to be established is $\tau_{d,\perp} = h^2/D \sim 10^{-2} \, \text{s}$ with $D \sim 10^{-10} \, \text{m}^2/\text{s}$ from eq.7. This time is much lower than the typical time of deformation, and **we thus consider the chemical equilibrium between the bulk and the interfaces to be instantaneous**.

In the tangential direction, all typical sizes are greater than $100 \, \mu\text{m}$ meaning that the transport by diffusion is here a matter of $\tau_{d,\parallel} = \ell^2/D \sim 10^2 \, \text{s}$. Also, we need to evaluate the bulk advection transport due to the Poiseuille flows computed in subsection 0.2.1 with its velocity $v_p \sim 10 \, \mu\text{m/s}$ and thus to a typical time of transport over $\ell$:
<figure><img src="image_16.png" /><figcaption>Figure 11: Side view of an elementary piece of film (a slab of the sliding puzzle game). For typical dimensions $l \sim 100 \mu m$ in the tangential directions of the film, the typical times of transport by diffusion and Poiseuille flow $\tau_{d,\parallel}$ and $\tau_{p,\parallel}$ are long compared to the time of dynamics $\tau_{def} \sim 1 s$, whereas in the normal direction, the thickness $h \sim 1 \mu m$ make diffusion be almost instantaneous with $\tau_{d,\perp} \sim 10^{-2} s$.</figcaption></figure>

$$ \tau_{p,\parallel} \sim \frac{l}{v_p} \sim 10 s \qquad (13) $$

Meaning that we have in both cases $\tau_{p,\parallel}$, $\tau_{d,\parallel} \gg \tau_{def}$ and we can consider the film element to be a closed system with respect to surfactant conservation far enough from the meniscus (where the thickness rapidly increases).

### 0.2.3 First model for film elasticity and its physical origin

Let us consider a piece of film of the same dimensions as described in Fig. 11 with a single surfactant in the micellar regime (we discuss qualitatively the case of micellar SDS/DOH mixtures at the end of this subsection and quantitatively with the full chapter 1 dedicated to it). As demonstrated earlier in subsection 0.2.2, it can be considered as a closed system whose interfaces are at equilibrium with the bulk for deformations in the typical time range $\tau \sim 10^{-2} - 10 s$. This piece of bulk at rest is shown in Fig. 12 Left, where the surface excess of surfactants $\Gamma = \Gamma_{eq}$ and the bulk concentration $c = c_{eq}$ are at equilibrium.

We want to impose a deformation $\epsilon$ in the direction tangential to the film, meaning the lengths in this direction become $L_{\parallel} = L_{\parallel,0}(1+\epsilon)$ and in the direction normal to the film $L_{\perp} = L_{\perp,0}/(1+\epsilon)$. The deformation remains of the order of $\epsilon = 1$ at most. Here we implicitly use a 2D model invariant by translation in the direction normal to the plane of the figure.

When we impose the deformation, the piece of film is stretched, and both areas $A_0$ of the interfaces are extended by the same factor $A = A_0(1+\epsilon)$. Their surface excesses $\Gamma$ remain the same for both interfaces at all times. Volume is conserved $V = Ah = A_0h_0$ imposing $h = h_0/(1+\epsilon)$. This leads to the conservation of the total population of surfactants:

$$ 2\Gamma_{eq}A_0 + c_{eq}A_0h_0 = 2\Gamma A + cAh \qquad (14) $$
<figure><img src="image_16.png" /><figcaption>Figure 11: Side view of an elementary piece of film (a slab of the sliding puzzle game). For typical dimensions $l \sim 100 \mu m$ in the tangential directions of the film, the typical times of transport by diffusion and Poiseuille flow $\tau_{d,\parallel}$ and $\tau_{p,\parallel}$ are long compared to the time of dynamics $\tau_{def} \sim 1 s$, whereas in the normal direction, the thickness $h \sim 1 \mu m$ make diffusion be almost instantaneous with $\tau_{d,\perp} \sim 10^{-2} s$.</figcaption></figure>

$$ \tau_{p,\parallel} \sim \frac{l}{v_p} \sim 10 \, \text{s} \qquad (13) $$

Meaning that we have in both cases $\tau_{p,\parallel}$, $\tau_{d,\parallel} \gg \tau_{def}$ and we can consider the film element to be a closed system with respect to surfactant conservation far enough from the meniscus (where the thickness rapidly increases).

### 0.2.3 First model for film elasticity and its physical origin

Let us consider a piece of film of the same dimensions as described in Fig. 11 with a single surfactant in the micellar regime (we discuss qualitatively the case of micellar SDS/DOH mixtures at the end of this subsection and quantitatively with the full chapter 1 dedicated to it). As demonstrated earlier in subsection 0.2.2, it can be considered as a closed system whose interfaces are at equilibrium with the bulk for deformations in the typical time range $\tau \sim 10^{-2} - 10 \, \text{s}$. This piece of bulk at rest is shown in Fig. 12 Left, where the surface excess of surfactants $\Gamma = \Gamma_{eq}$ and the bulk concentration $c = c_{eq}$ are at equilibrium.

We want to impose a deformation $\epsilon$ in the direction tangential to the film, meaning the lengths in this direction become $L_{\parallel} = L_{\parallel,0}(1+\epsilon)$ and in the direction normal to the film $L_{\perp} = L_{\perp,0}/(1+\epsilon)$. The deformation remains of the order of $\epsilon = 1$ at most. Here we implicitly use a 2D model invariant by translation in the direction normal to the plane of the figure.

When we impose the deformation, the piece of film is stretched, and both areas $A_0$ of the interfaces are extended by the same factor $A = A_0(1 + \epsilon)$. Their surface excesses $\Gamma$ remain the same for both interfaces at all times. Volume is conserved $V = Ah = A_0h_0$ imposing $h = h_0/(1 + \epsilon)$. This leads to the conservation of the total population of surfactants:

$$ 2\Gamma_{eq}A_0 + c_{eq}A_0h_0 = 2\Gamma A + cAh \qquad (14) $$
<figure><img src="image_17.png" /><figcaption>Figure 12: Elastic response of a single surfactant in the sub-micellar regime due to finite size effects of the bulk. Stretching a piece of film by a factor $(1+\epsilon)$ leads an increase of the area, thus diluting the surface excess $\Gamma_{eq}$ of surfactants present at the interface. This dilution is compensated by the population present in the bulk, as the equilibrium is instantaneous given the small thickness of a foam film (cf. subsection 0.2.2). Because of the tiny volume, the bulk population of surfactants $c_{eq}$ undergoes a significant drop, down to a value $c < c_{eq}$, thus balancing a decreased population $\Gamma < \Gamma_{eq}$ at the interface. This decrease of $\Gamma$ leads to a reversible increase of surface tension $\gamma > \gamma_{eq}$. The same reasoning goes for a compression of the film ($\epsilon < 0$).</figcaption></figure>

We can now reintroduce the reservoir length $h_Γ = Γ/c$ which is considered here as a constant of the chemical equilibrium linking the surface and bulk populations of surfactants (this is the simplest model for the chemical equilibrium, and it already captures the physical origin of foam film elasticity, chapter 1 shows a more accurate one based on Langmuir isotherms). This allows us to rewrite the equilibrium as:

$$ \Gamma\left(1 + \epsilon + \frac{h_0}{2h_Γ}\right) = \Gamma_0\left(1 + \frac{h_0}{2h_Γ}\right) \qquad (15) $$

Thus, deforming the piece of foam film results in a change of both the concentration and the surface excess of surfactants if the term $h_0/h_Γ$ remains of the same order of magnitude as $(1+\epsilon)$. That is if the film is thin enough for its bulk to be significantly depleted by a change of surface of the interfaces. With a change of surface excess, the surface tension of the interfaces $\gamma_{top/bot} = \gamma$ also changes at first order through the Marangoni elastic modulus defined by eq. 11. Thus, a simplified expression for $\gamma$ can be written:

$$ \gamma = \gamma_{eq} + E_M \frac{\epsilon}{1 + \epsilon + \frac{h_0}{2h_Γ}} \qquad (16) $$

This is the physical origin of foam film elasticity [8, 9], and it is not a dynamic process due to a sorption time, but a finite size effect of the bulk acting as a reservoir. This completely blocks the extrinsic dissipation originating from the bulk. Also, it has to be noted that this analytical expression relies on two bold approximations: that the surface excess is given by $\Gamma = ch_Γ$ and that $\gamma$ varies linearly with $\Gamma$ at first order.
<figure><img src="image_17.png" /><figcaption>Figure 12: Elastic response of a single surfactant in the sub-micellar regime due to finite size effects of the bulk. Stretching a piece of film by a factor $(1+\epsilon)$ leads an increase of the area, thus diluting the surface excess $\Gamma_{eq}$ of surfactants present at the interface. This dilution is compensated by the population present in the bulk, as the equilibrium is instantaneous given the small thickness of a foam film (cf. subsection 0.2.2). Because of the tiny volume, the bulk population of surfactants $c_{eq}$ undergoes a significant drop, down to a value $c < c_{eq}$, thus balancing a decreased population $\Gamma < \Gamma_{eq}$ at the interface. This decrease of $\Gamma$ leads to a reversible increase of surface tension $\gamma > \gamma_{eq}$. The same reasoning goes for a compression of the film ($\epsilon < 0$).</figcaption></figure>

We can now reintroduce the reservoir length $h_{\Gamma} = \Gamma/c$ which is considered here as a constant of the chemical equilibrium linking the surface and bulk populations of surfactants (this is the simplest model for the chemical equilibrium, and it already captures the physical origin of foam film elasticity, chapter 1 shows a more accurate one based on Langmuir isotherms). This allows us to rewrite the equilibrium as:

$$ \Gamma\left(1 + \epsilon + \frac{h_0}{2h_{\Gamma}}\right) = \Gamma_0\left(1 + \frac{h_0}{2h_{\Gamma}}\right) \qquad (15) $$

Thus, deforming the piece of foam film results in a change of both the concentration and the surface excess of surfactants if the term $h_0/h_{\Gamma}$ remains of the same order of magnitude as $(1+\epsilon)$. That is if the film is thin enough for its bulk to be significantly depleted by a change of surface of the interfaces. With a change of surface excess, the surface tension of the interfaces $\gamma_{top/bot} = \gamma$ also changes at first order through the Marangoni elastic modulus defined by eq. 11. Thus, a simplified expression for $\gamma$ can be written:

$$ \gamma = \gamma_{eq} + E_M \frac{\epsilon}{1 + \epsilon + \frac{h_0}{2h_{\Gamma}}} \qquad (16) $$

This is the physical origin of foam film elasticity [8, 9], and it is not a dynamic process due to a sorption time, but a finite size effect of the bulk acting as a reservoir. This completely blocks the extrinsic dissipation originating from the bulk. Also, it has to be noted that this analytical expression relies on two bold approximations: that the surface excess is given by $\Gamma = ch_{\Gamma}$ and that $\gamma$ varies linearly with $\Gamma$ at first order.
**Chapter 1 provides a more accurate model for film elasticity that I developed, taking into account the non-linear couplings between $\gamma$ and $\Gamma$, and $\Gamma$ and $c$, as well as the presence of micelles.**

This elastic behaviour is thus conditioned by the reservoir length $h_\Gamma$ which needs to be of the order of or greater than the typical thickness of the film $h_0 \sim 1 \mu m$. This is especially true for poorly soluble surfactants such as DOH where we estimate $h_{\Gamma,DOH} > 5 \mu m$ in Bussonnière et al. [41, 42]. Micellar SDS is much more concentrated in the bulk, and a first estimation using $\Gamma_{SDS} \sim 10^{-6} \text{mol/m}^2$ [42], $c_{SDS} \sim 10 \text{mol/m}^3$ yields $h_{\Gamma,SDS} \sim 0.1 \mu m$ meaning the $h_0/2h_{\Gamma,SDS}$ term dominates and the elastic contribution of micellar SDS can be neglected a priori.

Also, this elasticity and the expression eq.16 for the surface tension remain valid as far as the piece of film can be considered to be a closed system at chemical equilibrium. That is at times $\tau \gg 10^{-2} \text{s}$ long enough for the bulk-interface equilibrium to be reached, and $\tau \ll 10^2 \text{s}$ short enough for surfactant exchanges with neighbouring pieces of films to be neglected.

Lastly, the time evolution of $\gamma$ due to the dissipation intrinsic to the interface is discarded here (i.e. the response of $\gamma$ to $\dot{\epsilon}$ through $\eta_s$ and $\kappa_s$). In the range of attainable strain rates $\dot{\epsilon}$, $E_M$ dominates [4].

### 0.2.4 Far from the meniscus, film tension is uniform

Now that we have clarified the physical origin of foam film elasticity, we need to look at the tension balance in the film in order to describe the mechanics of not only a piece of film (size $l \sim 100 \mu m$), but of the foam film as a whole (size $L \sim 1 \text{ cm}$). To do so, let us introduce the film tension $\sigma^f$, which is the total action of a piece of film (black dotted rectangle in Fig.13) on its neighbour. **This $\sigma^f$ includes the capillary stress in the bulk, as well as the surface tension of both interfaces (and the surface elastic and viscous contributions this latter contains).** At rest and for symmetric flat interfaces, we simply have $\sigma_{eq}^f = 2\gamma_{eq}$.

We also define a 2D model for a foam film of length $L$, of local coordinates $(s, \zeta)$ as shown in Fig.13, and invariant by translation along the third axis normal to the figure. The velocity in the liquid bulk is written $v = u e_s + w e_\zeta$ and the thickness of the film $h(s)$ depends on the local abscissa $s$ in the general case.

We start by writing the conservation of momentum projected along the $s$ direction for a piece of film whose boundaries are set between the local abscissa $s$ and $s+ds$ and with ordinates $\zeta$ so that they include both interfaces (black dotted rectangle in Fig.13). It can be expressed as an integral over the liquid bulk $\zeta \in [0, h]$:

$$ \int_0^h \rho \left( \frac{\partial u}{\partial t} + u \frac{\partial u}{\partial s} + w \frac{\partial u}{\partial \zeta} \right) d\zeta = \frac{\partial \sigma^f}{\partial s} + \rho g_s h + f_g \quad (17) $$

Where $g_s$ is gravity projected along $s$, $f_g$ is the friction of the piece of film with the gas phase.

Expressing the conservation of volume with the invariance by translation along the normal to $Os\zeta$, we have $\frac{\partial w}{\partial \zeta} = -\frac{\partial u}{\partial s}$ and after a few integrations in eq.17:
Chapter 1 provides a more accurate model for film elasticity that I developed, taking into account the non-linear couplings between $\gamma$ and $\Gamma$, and $\Gamma$ and $c$, as well as the presence of micelles.

This elastic behaviour is thus conditioned by the reservoir length $h_\Gamma$ which needs to be of the order of or greater than the typical thickness of the film $h_0 \sim 1 \mu m$. This is especially true for poorly soluble surfactants such as DOH where we estimate $h_{\Gamma,DOH} > 5 \mu m$ in Bussonnière et al. [41, 42]. Micellar SDS is much more concentrated in the bulk, and a first estimation using $\Gamma_{SDS} \sim 10^{-6} \text{mol/m}^2$ [42], $c_{SDS} \sim 10 \text{mol/m}^3$ yields $h_{\Gamma,SDS} \sim 0.1 \mu m$ meaning the $h_0/2h_{\Gamma,SDS}$ term dominates and the elastic contribution of micellar SDS can be neglected a priori.

Also, this elasticity and the expression eq.16 for the surface tension remain valid as far as the piece of film can be considered to be a closed system at chemical equilibrium. That is at times $\tau \gg 10^{-2} \text{s}$ long enough for the bulk-interface equilibrium to be reached, and $\tau \ll 10^2 \text{s}$ short enough for surfactant exchanges with neighbouring pieces of films to be neglected.

Lastly, the time evolution of $\gamma$ due to the dissipation intrinsic to the interface is discarded here (i.e. the response of $\gamma$ to $\dot{\epsilon}$ through $\eta_s$ and $\kappa_s$). In the range of attainable strain rates $\dot{\epsilon}$, $E_M$ dominates [4].

### 0.2.4 Far from the meniscus, film tension is uniform

Now that we have clarified the physical origin of foam film elasticity, we need to look at the tension balance in the film in order to describe the mechanics of not only a piece of film (size $l \sim 100 \mu m$), but of the foam film as a whole (size $L \sim 1 \text{ cm}$). To do so, let us introduce the film tension $\sigma^f$, which is the total action of a piece of film (black dotted rectangle in Fig.13) on its neighbour. This $\sigma^f$ includes the capillary stress in the bulk, as well as the surface tension of both interfaces (and the surface elastic and viscous contributions this latter contains). At rest and for symmetric flat interfaces, we simply have $\sigma_{eq}^f = 2\gamma_{eq}$.

We also define a 2D model for a foam film of length $L$, of local coordinates $(s, \zeta)$ as shown in Fig.13, and invariant by translation along the third axis normal to the figure. The velocity in the liquid bulk is written $v = u e_s + w e_\zeta$ and the thickness of the film $h(s)$ depends on the local abscissa $s$ in the general case.

We start by writing the conservation of momentum projected along the $s$ direction for a piece of film whose boundaries are set between the local abscissa $s$ and $s+ds$ and with ordinates $\zeta$ so that they include both interfaces (black dotted rectangle in Fig.13). It can be expressed as an integral over the liquid bulk $\zeta \in [0, h]$:

$$ \int_0^h \rho \left( \frac{\partial u}{\partial t} + u \frac{\partial u}{\partial s} + w \frac{\partial u}{\partial \zeta} \right) d\zeta = \frac{\partial \sigma^f}{\partial s} + \rho g_s h + f_g \quad (17) $$

Where $g_s$ is gravity projected along $s$, $f_g$ is the friction of the piece of film with the gas phase.

Expressing the conservation of volume with the invariance by translation along the normal to $Os\zeta$, we have $\frac{\partial w}{\partial \zeta} = -\frac{\partial u}{\partial s}$ and after a few integrations in eq.17:
<figure><img src="image_18.png" /><figcaption>Figure 13: Force balance on a film element of Fig.11 (a slab of the sliding puzzle, in case the reader had forgotten our beloved analogy). A typical length for a foam film is given $L \sim 10^{-2}$ m (which is among the longest lengths of film encountered with real macroscopic liquid foams).</figcaption></figure>

$$ \rho \left( \frac{\partial(h\langle u \rangle)}{\partial t} + \frac{\partial(h\langle u^2 \rangle)}{\partial s} \right) = \frac{\partial \sigma^f}{\partial s} + \rho g_s h + f_g \quad (18) $$

Where the brackets stand for the mean value over $\zeta \in [0, h]$.

This is where we need to put some orders of magnitude to further advance and estimate each term. To do so, we take the typical scales:

* $\tau_{def} \sim 1$ s the typical time scale

* $L \sim 10^{-2}$ m the typical length scale along $s$, yielding $U \sim \frac{L}{\tau_{def}} \sim 10^{-2}$ m/s

* $h \sim 10^{-6}$ m the typical length scale along $\zeta$

* $\rho = 10^3$ kg/m$^3$ the bulk density (water in most cases, including ours)

* $g_s \sim 10$ m/s$^2$ the projected gravity

Inertial terms scale both as $\frac{\rho h}{\tau_{def}} \sim 10^{-5}$ Pa and weight scales as $\rho g_s h \sim 10^{-2}$ Pa.

The air friction scales as $f_g \sim \eta_g \frac{U}{\delta}$, where $\eta_g \sim 10^{-5}$ Pas is the gas viscosity and $\delta$ is a Blasius length over which momentum diffuses in the neighbouring gas. We estimate $\delta \sim \sqrt{\frac{\eta_g \tau_{def}}{\rho_g}} \sim 10^{-3}$ m with $\rho_g = 1$ kg/m$^3$ the air density and we get $f_g \sim 10^{-4}$ Pa.

The gradient of film tension is estimated by estimating the value of the film elasticity $E_f \simeq 2E_M$, which is of the order of $E_f \sim 10^{-2}$ N m$^{-1}$ [4, 6], yielding the scaling $\frac{\partial \sigma^f}{\partial s} \sim \frac{E_f \epsilon}{L} \sim 1$ Pa with $\epsilon \sim 1$ which is very large compared to any other term. Note that this can be seen as a lower estimate, as the intrinsic viscosities of the interfaces or the capillary stress may also contribute to film tension (we will later see that they do not far from the edges of the foam film).
<figure><img src="image_18.png" /><figcaption>Figure 13: Force balance on a film element of Fig.11 (a slab of the sliding puzzle, in case the reader had forgotten our beloved analogy). A typical length for a foam film is given $L \sim 10^{-2}$ m (which is among the longest lengths of film encountered with real macroscopic liquid foams).</figcaption></figure>

$$ \rho \left( \frac{\partial(h\langle u \rangle)}{\partial t} + \frac{\partial(h\langle u^2 \rangle)}{\partial s} \right) = \frac{\partial \sigma^f}{\partial s} + \rho g_s h + f_g \quad (18) $$

Where the brackets stand for the mean value over $\zeta \in [0, h]$.

This is where we need to put some orders of magnitude to further advance and estimate each term. To do so, we take the typical scales:

* $\tau_{def} \sim 1$ s the typical time scale

* $L \sim 10^{-2}$ m the typical length scale along $s$, yielding $U \sim \frac{L}{\tau_{def}} \sim 10^{-2}$ m/s

* $h \sim 10^{-6}$ m the typical length scale along $\zeta$

* $\rho = 10^3$ kg/m$^3$ the bulk density (water in most cases, including ours)

* $g_s \sim 10$ m/s$^2$ the projected gravity

Inertial terms scale both as $\frac{\rho h}{\tau_{def}} \sim 10^{-5}$ Pa and weight scales as $\rho g_s h \sim 10^{-2}$ Pa.

The air friction scales as $f_g \sim \eta_g \frac{U}{\delta}$, where $\eta_g \sim 10^{-5}$ Pas is the gas viscosity and $\delta$ is a Blasius length over which momentum diffuses in the neighbouring gas. We estimate $\delta \sim \sqrt{\frac{\eta_g \tau_{def}}{\rho_g}} \sim 10^{-3}$ m with $\rho_g = 1$ kg/m$^3$ the air density and we get $f_g \sim 10^{-4}$ Pa.

The gradient of film tension is estimated by estimating the value of the film elasticity $E_f \simeq 2E_M$, which is of the order of $E_f \sim 10^{-2}$ N m$^{-1}$ [4, 6], yielding the scaling $\frac{\partial \sigma^f}{\partial s} \sim \frac{E_f \epsilon}{L} \sim 1$ Pa with $\epsilon \sim 1$ which is very large compared to any other term. Note that this can be seen as a lower estimate, as the intrinsic viscosities of the interfaces or the capillary stress may also contribute to film tension (we will later see that they do not far from the edges of the foam film).
We can thus conclude that for a typical deformation of more than $\epsilon_{min} = 10^{-2}$, no other force can balance the tension gradient, and thus **film tension $\sigma^f$ is uniform in a foam film, far from its edges**. This is the limit of high elasticities, which will essentially be always valid in the frame of our study, as well as for a very wide range of foams.

However, it has to be noted that in other situations, gravity can be important [43], or inertia for film rupture [44–46].

### 0.2.5 The incompressible interface limit

Since Mysels, Frankel and Shinoda [47–49] and the first actual measurements of foam film elasticities $E_f$ by Prins et al. [5, 50], a first key observation is that foam films are hard to stretch. This means that the cost $E_f$ to pay is high.

Most of the time, we assume incompressible interfaces, meaning we close the problem imposing:

$$ \nabla_s \cdot v_s = 0 \qquad (19) $$

This is a robust assumption for a wide range of problems [51–53] which we will later use in chapter 2. This hypothesis only finds its limits when the films are stretched or compressed by imposing a significant change of area as we will later see in chapters 1 and 3. In Appendix 2, we propose a general framework for compressible interfaces in the limit of high elasticities.

## 0.3 Flows in foam films at a junction with a meniscus

Since the beginning of this manuscript, I have been carrying the metaphor of the sliding puzzle for a foam film to establish the idea that pieces of films are advected patches interacting with one another not through bulk Poiseuille flows, but rather with their surface rheology. This is a faithful representation up to a certain extent, and this section proposes to see the situations where it is not anymore. That happens notably for times much longer than our typical $\tau_{def} \sim 1$ s, or at the edges of a foam film where it is put in contact with a meniscus. The typical dimensions for a meniscus (and its radius of curvature) is of the order of $r_m \sim 100$ µm, which is two orders of magnitude above the typical thickness of a film!

All the following subsections correspond each to a phenomenon which will be observed in my experiments. Some of them are used to our advantage (Frankel film extraction), others hinder my measurements (Marginal regeneration, Rayleigh-Taylor instability...). And all need a proper description.

### 0.3.1 Lubrication equations in a foam film

We aim here at establishing a general set of equations for the thickness profile of two symmetrical incompressible interfaces set in motion. In the bulk of a foam film, the liquid flow is governed by the Stokes equations:
We can thus conclude that for a typical deformation of more than $\epsilon_{min} = 10^{-2}$, no other force can balance the tension gradient, and thus **film tension $\sigma^f$ is uniform in a foam film, far from its edges**. This is the limit of high elasticities, which will essentially be always valid in the frame of our study, as well as for a very wide range of foams.

However, it has to be noted that in other situations, gravity can be important [43], or inertia for film rupture [44–46].

### 0.2.5 The incompressible interface limit

Since Mysels, Frankel and Shinoda [47–49] and the first actual measurements of foam film elasticities $E_f$ by Prins et al. [5, 50], a first key observation is that foam films are hard to stretch. This means that the cost $E_f$ to pay is high.

Most of the time, we assume incompressible interfaces, meaning we close the problem imposing:

$$ \nabla_s \cdot v_s = 0 \qquad (19) $$

This is a robust assumption for a wide range of problems [51–53] which we will later use in chapter 2. This hypothesis only finds its limits when the films are stretched or compressed by imposing a significant change of area as we will later see in chapters 1 and 3. In Appendix 2, we propose a general framework for compressible interfaces in the limit of high elasticities.

## 0.3 Flows in foam films at a junction with a meniscus

Since the beginning of this manuscript, I have been carrying the metaphor of the sliding puzzle for a foam film to establish the idea that pieces of films are advected patches interacting with one another not through bulk Poiseuille flows, but rather with their surface rheology. This is a faithful representation up to a certain extent, and this section proposes to see the situations where it is not anymore. That happens notably for times much longer than our typical $\tau_{def} \sim 1$ s, or at the edges of a foam film where it is put in contact with a meniscus. The typical dimensions for a meniscus (and its radius of curvature) is of the order of $r_m \sim 100$ µm, which is two orders of magnitude above the typical thickness of a film!

All the following subsections correspond each to a phenomenon which will be observed in my experiments. Some of them are used to our advantage (Frankel film extraction), others hinder my measurements (Marginal regeneration, Rayleigh-Taylor instability...). And all need a proper description.

### 0.3.1 Lubrication equations in a foam film

We aim here at establishing a general set of equations for the thickness profile of two symmetrical incompressible interfaces set in motion. In the bulk of a foam film, the liquid flow is governed by the Stokes equations:
$$ \left\{ \begin{array}{l} \eta \Delta \mathbf{v} - \nabla p = 0 \\ \nabla \cdot \mathbf{v} = 0 \end{array} \right. \qquad (20) $$

Where $\eta$ is the bulk viscosity and $p$ the pressure in the liquid.

Given the geometry of millimetric/centimetric foam films (the same aspect ratio as if a paper sheet was 10 to 100 times thinner), the bulk flows are described using the lubrication equations. We will use the notations of Fig. 14, with the $Oxz$ plane being the medium plane of the foam film, and $y = h(x)$ the ordinate locating the position of the interface. For now we consider the 2D problem of two symmetrical incompressible interfaces. They are described by any semi-thickness field $h(x)$ varying slowly enough to remain in the frame of the lubrication approximation, and with an imposed velocity $v_x(x, h(x)) = U$ which is uniform along the interface as this latter is incompressible.

The lubrication equations can be written as follows (established in [54, 55]):

$$ \left\{ \begin{array}{l} \eta \partial_{yy} v_x - \partial_x p = 0 \\ \partial_y p = 0 \\ \partial_x v_x = -\partial_y v_y \\ v_x(x, h(x)) = U \end{array} \right. \qquad (21) $$

Where lines 3 and 4 are respectively the conservation of volume and of the area of interface. The pressure field is only depending on $x$, and as the curvature of the interfaces remains small, we can write $p$ as the Laplace pressure jump: $p = \gamma_0 \partial_{xx} h$, where $\gamma_0$ is the surface tension of the interfaces. Given the independence of the pressure gradient with $y$, the boundary condition $v_x(x, h(x)) = U$ and the geometry of the problem, the velocity field can be integrated as:

$$ v_x(x, y) = U + \frac{1}{2} \frac{\gamma_0}{\eta} \partial_{xxx} h (y^2 - h^2) = v_{ad} + v_{cap} \qquad (22) $$

This velocity field can be integrated to get the flow rate:

$$ Q = \int_{-h}^{h} v_x(x, y) dy = 2Uh - \frac{2}{3} \frac{\gamma_0}{\eta} h^3 \partial_{xxx} h \qquad (23) $$

Whose conservation law (line 3 of eq. 21) yields the equation for $h$:

$$ \partial_t h + U \partial_x h - \frac{1}{3} \frac{\gamma_0}{\eta} \partial_x (h^3 \partial_{xxx} h) = 0 \qquad (24) $$

This is the generic equation for the evolution of a profile $h(x, t)$ given the velocity $U$ which can be a function of time, negative or positive, but must remain uniform at the interfaces because of the surface incompressibility condition.

This equation opens up pathways to the solutions of problems where bulk transport in foam films matter. This happens for instance near the edges of a foam film, where the flat film is connected to a meniscus, and we propose to see two examples in the following: the Frankel problem and the Aradian problem.
$$ \left\{ \begin{array}{l} \eta \Delta v - \nabla p = 0 \\ \nabla \cdot v = 0 \end{array} \right. \qquad (20) $$

Where $\eta$ is the bulk viscosity and $p$ the pressure in the liquid.

Given the geometry of millimetric/centimetric foam films (the same aspect ratio as if a paper sheet was 10 to 100 times thinner), the bulk flows are described using the lubrication equations. We will use the notations of Fig. 14, with the $Oxz$ plane being the medium plane of the foam film, and $y = h(x)$ the ordinate locating the position of the interface. For now we consider the 2D problem of two symmetrical incompressible interfaces. They are described by any semi-thickness field $h(x)$ varying slowly enough to remain in the frame of the lubrication approximation, and with an imposed velocity $v_x(x, h(x)) = U$ which is uniform along the interface as this latter is incompressible.

The lubrication equations can be written as follows (established in [54, 55]):

$$ \left\{ \begin{array}{l} \eta \partial_{yy} v_x - \partial_x p = 0 \\ \partial_y p = 0 \\ \partial_x v_x = -\partial_y v_y \\ v_x(x, h(x)) = U \end{array} \right. \qquad (21) $$

Where lines 3 and 4 are respectively the conservation of volume and of the area of interface. The pressure field is only depending on $x$, and as the curvature of the interfaces remains small, we can write $p$ as the Laplace pressure jump: $p = \gamma_0 \partial_{xx} h$, where $\gamma_0$ is the surface tension of the interfaces. Given the independence of the pressure gradient with $y$, the boundary condition $v_x(x, h(x)) = U$ and the geometry of the problem, the velocity field can be integrated as:

$$ v_x(x, y) = U + \frac{1}{2} \frac{\gamma_0}{\eta} \partial_{xxx} h (y^2 - h^2) = v_{ad} + v_{cap} \qquad (22) $$

This velocity field can be integrated to get the flow rate:

$$ Q = \int_{-h}^{h} v_x(x, y) dy = 2Uh - \frac{2}{3} \frac{\gamma_0}{\eta} h^3 \partial_{xxx} h \qquad (23) $$

Whose conservation law (line 3 of eq. 21) yields the equation for $h$:

$$ \partial_t h + U \partial_x h - \frac{1}{3} \frac{\gamma_0}{\eta} \partial_x (h^3 \partial_{xxx} h) = 0 \qquad (24) $$

This is the generic equation for the evolution of a profile $h(x, t)$ given the velocity $U$ which can be a function of time, negative or positive, but must remain uniform at the interfaces because of the surface incompressibility condition.

This equation opens up pathways to the solutions of problems where bulk transport in foam films matter. This happens for instance near the edges of a foam film, where the flat film is connected to a meniscus, and we propose to see two examples in the following: the Frankel problem and the Aradian problem.
### 0.3.2 Steady state film extraction: the Frankel problem

To close the problem of eq.24, we either need to get an initial profile for $h(x, t = 0)$ or to look at its steady solutions, as well as getting more spatial conditions for $h$.

Here, we look at the problem of a steady film extraction from a meniscus as shown in Fig.14. It is called the Frankel’s problem and has been extensively studied, both experimentally and theoretically [47, 51, 52, 56]. We take the steady state of eq.24 where we impose a given $U$ chosen positive along $+x$ for now, and a limit of constant curvature $\partial_{xx}h \to 1/r_m$ when $x \to -\infty$ in the meniscus.

<figure><img src="image_19.png" /><figcaption>Figure 14: Notations for the Frankel problem. A flat film is connected to a meniscus along the $x$ axis. A steady plug flow $U$ is imposed far from the meniscus, and this latter has a constant curvature $\partial_{xx}h = 1/r_m$ in $x \to -\infty$. The steady flow at the junction (called the dynamic meniscus) is the sum of an outward plug flow $v_{ad}$ and an inward capillary flow $v_{cap}$.</figcaption></figure>

The equations of the problem thus write:

$$ \left\{ \begin{array}{l} U\partial_x h - \frac{1}{3}\frac{\gamma_0}{\eta}\partial_x(h^3\partial_{xxx}h) = 0 \\ \partial_{xx}h(x \to -\infty) = 1/r_m \\ v_x(x \to +\infty, y) = U \end{array} \right. \qquad (25) $$

Which can be solved either numerically or with scaling laws. Let us proceed with the latter. As we impose a constant velocity far from the meniscus, mass conservation imposes the existence of the limit $h(x \to +\infty) = h_\infty$, giving us a typical scale for $h$ of which we want to make the scaling explicit.

The meniscus imposes $\partial_{xx}h = 1/r_m$, meaning the typical scale $\ell_x$ over which $h$ varies is given by $\ell_x \sim \sqrt{h_\infty r_m}$. By injecting this in eq.25 (line 1), it comes:

$$ h_{\infty} \sim r_m Ca^{2/3} \qquad (26) $$

Where $Ca = \frac{\eta U}{\gamma_0}$ is a capillary number comparing viscous and interfacial forces. Numerical calculations give the prefactor which is $\simeq 1.34$.
### 0.3.2 Steady state film extraction: the Frankel problem

To close the problem of eq.24, we either need to get an initial profile for $h(x, t = 0)$ or to look at its steady solutions, as well as getting more spatial conditions for $h$.

Here, we look at the problem of a steady film extraction from a meniscus as shown in Fig.14. It is called the Frankel’s problem and has been extensively studied, both experimentally and theoretically [47, 51, 52, 56]. We take the steady state of eq.24 where we impose a given $U$ chosen positive along $+x$ for now, and a limit of constant curvature $\partial_{xx}h \to 1/r_m$ when $x \to -\infty$ in the meniscus.

<figure><img src="image_19.png" /><figcaption>Figure 14: Notations for the Frankel problem. A flat film is connected to a meniscus along the $x$ axis. A steady plug flow $U$ is imposed far from the meniscus, and this latter has a constant curvature $\partial_{xx}h = 1/r_m$ in $x \to -\infty$. The steady flow at the junction (called the dynamic meniscus) is the sum of an outward plug flow $v_{ad}$ and an inward capillary flow $v_{cap}$.</figcaption></figure>

The equations of the problem thus write:

$$ \left\{ \begin{array}{l} U\partial_x h - \frac{1}{3}\frac{\gamma_0}{\eta}\partial_x(h^3\partial_{xxx}h) = 0 \\ \partial_{xx}h(x \to -\infty) = 1/r_m \\ v_x(x \to +\infty, y) = U \end{array} \right. \qquad (25) $$

Which can be solved either numerically or with scaling laws. Let us proceed with the latter. As we impose a constant velocity far from the meniscus, mass conservation imposes the existence of the limit $h(x \to +\infty) = h_\infty$, giving us a typical scale for $h$ of which we want to make the scaling explicit.

The meniscus imposes $\partial_{xx}h = 1/r_m$, meaning the typical scale $\ell_x$ over which $h$ varies is given by $\ell_x \sim \sqrt{h_\infty r_m}$. By injecting this in eq.25 (line 1), it comes:

$$ h_{\infty} \sim r_m Ca^{2/3} \qquad (26) $$

Where $Ca = \frac{\eta U}{\gamma_0}$ is a capillary number comparing viscous and interfacial forces. Numerical calculations give the prefactor which is $\simeq 1.34$.
This law means that the faster a foam film is extracted from a meniscus, the thicker are the newly created pieces of film with a power law 2/3 with the velocity of extraction. In practice when it comes to stretching a foam film at a given velocity, it is qualitatively very useful, as it allows us to extract films quite uniform in thickness in a very repeatable way. **This is what we will call a Frankel film, and we will have an extensive use of this name.**

Also, it has to be noted that although the interface is kept incompressible, it is subject to a difference of surface tension $\Delta\gamma^{out}$ between the meniscus and the flat film. This is no contradiction, but the fact that the incompressible limit is equivalent to taking $E_f \to \infty$ and the deformation $\epsilon \to 0$ with their product being constant and equals to the difference of tension $\Delta\gamma^{out} \sim E_f\epsilon$. It can be numerically computed and is [48]:

$$ \Delta\gamma^{out} = 3.84\gamma_0 Ca^{2/3} \qquad (27) $$

The main use of the Frankel's problem we make in practice is when we stretch a foam film. Initially, the foam film is of thickness $h_{init}$ and is linked to a meniscus. Stretching it means imposing a plug flow in the initial film, because of the arguments developed in subsection 0.2.1. The steady response of the system described by the Frankel's law leads to the situation shown in Fig.15: some film is extracted at the same steady velocity $U$ than the preexisting film, but with a thickness $h_\infty$ set by the law 26. Both films exist in the same plane, with a thickness jump between them. This thickness heterogeneity is stable, and would tend to smooth out at much longer times $\tau \gg 1$ s with a dynamics given in [57]. Thus, the frontier between the two films is a material point whose tracking will be useful many times in this manuscript. Also, in practice, even for small imposed velocities, the Frankel films are generally thicker than the preexisting films.

<figure><img src="image_20.png" /><figcaption>Figure 15: Extraction of a Frankel film at a steady velocity $U$. In the general case, the preexisting flat film set in motion has a thickness $h_{init} \neq h_\infty$. In practice, we often have $h_\infty > h_{init}$.</figcaption></figure>

Now a complete other range of problems: what about taking a negative value for $U$? The problem is actually solved in a completely different way, as $h_\infty$ has to be provided besides $U$,
This law means that the faster a foam film is extracted from a meniscus, the thicker are the newly created pieces of film with a power law 2/3 with the velocity of extraction. In practice when it comes to stretching a foam film at a given velocity, it is qualitatively very useful, as it allows us to extract films quite uniform in thickness in a very repeatable way. **This is what we will call a Frankel film, and we will have an extensive use of this name.**

Also, it has to be noted that although the interface is kept incompressible, it is subject to a difference of surface tension $\Delta\gamma^{out}$ between the meniscus and the flat film. This is no contradiction, but the fact that the incompressible limit is equivalent to taking $E_f \to \infty$ and the deformation $\epsilon \to 0$ with their product being constant and equals to the difference of tension $\Delta\gamma^{out} \sim E_f\epsilon$. It can be numerically computed and is [48]:

$$ \Delta\gamma^{out} = 3.84\gamma_0 Ca^{2/3} \qquad (27) $$

The main use of the Frankel's problem we make in practice is when we stretch a foam film. Initially, the foam film is of thickness $h_{init}$ and is linked to a meniscus. Stretching it means imposing a plug flow in the initial film, because of the arguments developed in subsection 0.2.1. **The steady response of the system described by the Frankel's law leads to the situation shown in Fig. 15: some film is extracted at the same steady velocity $U$ than the preexisting film, but with a thickness $h_\infty$ set by the law 26.** Both films exist in the same plane, with a thickness jump between them. This thickness heterogeneity is stable, and would tend to smooth out at much longer times $\tau \gg 1$ s with a dynamics given in [57]. **Thus, the frontier between the two films is a material point whose tracking will be useful many times in this manuscript.** Also, in practice, even for small imposed velocities, the Frankel films are generally thicker than the preexisting films.

<figure><img src="image_20.png" /><figcaption>Figure 15: Extraction of a Frankel film at a steady velocity $U$. In the general case, the preexisting flat film set in motion has a thickness $h_{init} \neq h_\infty$. In practice, we often have $h_\infty > h_{init}$.</figcaption></figure>

Now a complete other range of problems: what about taking a negative value for $U$? The problem is actually solved in a completely different way, as $h_\infty$ has to be provided besides $U$,
and is not a mere consequence of the balance between viscous and capillary forces. The resulting tension in this case writes [48]:

$$ \Delta\gamma^{in} = \gamma_0 (3Ca)^{2/3} (2.55\alpha^{1/3} - 2.68) \qquad (28) $$

Where $\alpha = (h\infty/r_m) (3Ca)^{-2/3}$.

This is an important result. Tension has a unique value in the film (as seen in subsection 0.2.4), and a unique value in the meniscus. The gap between these two values can be filled either by a $\Delta\gamma^{in}$ or a $\Delta\gamma^{out}$ with respectively a negative or positive $U$ and for different thicknesses $h_\infty$ (the nullity of $U$ is no practical case when a film is being deformed). This leads to an instability which spontaneously breaks the symmetry along the dimension parallel to the meniscus: this is called marginal regeneration, and we will describe it later in subsection 0.3.4.

**Important remark:** The problem of taking into account the visco-elasticity of the interfaces (meaning we have $U(x)$ no longer uniform) is quite a recent theoretical advance, and can be found in Seiwert et al. [51, 52].

### 0.3.3 Capillary suction near the meniscus: Aradian's marginal pinching

Let us look now at the instationary problem of a flat film at rest ($U = 0$) put into contact with the meniscus. We will call this problem the Aradian problem [58]. It has the same configuration as previously, but now we keep the time varying term of eq.24 and discard the convection term:

$$ \left\{ \begin{aligned} \partial_t h - \frac{1}{3} \frac{\gamma_0}{\eta} \partial_x (h^3 \partial_{xxx} h) &= 0 \\ \partial_{xx} h (x \to -\infty) &= 1/r_m \\ v_x (x \to +\infty, y) &= 0 \\ h (x \to +\infty) &= h_\infty \end{aligned} \right. \qquad (29) $$

It describes what happens to a film at rest when put into contact with a meniscus. As the film is flat (at air pressure) and the meniscus is curved with a negative pressure, the meniscus sucks back fluid from the film, which is what has already been called **capillary drainage** in this manuscript. However, the film does not drain uniformly along $x$, and we rather see the formation of a depleted zone near the meniscus whose geometry includes different lengths, as shown in Fig.16.

The main problem leading to such a multiplicity of lengths is that although the draining Poiseuille flow develops over a large length $W$, the majority of the gradient of curvature $\partial_{xxx}h$ linking the meniscus to the flat film, and which is the main driving force, develops over a very localised zone $w$. All that means the problem needs a careful handling, and a mathematical treatment which involves an asymptotic matching between the two scales. If we now want to find the main results using scaling laws, we also need to do it carefully, by considering the following:

* $h_\infty$ and $W$ are the typical scales for large $x$. $h_\infty$ is a given parameter from the initial conditions, $W$ is a variable of which we seek the scaling. They are linked by the conservation of flow rate (line 1 of eq.29), leading to: $W \sim h_\infty (\frac{\gamma}{\eta h_\infty})^{1/4} t^{1/4}$.