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Small spatial scales in wetting on non-structured and structured surfaces: thin films of liquid crystals & small-scales wetting properties of ionic liquids

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HAL Id: tel-00662524

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Submitted on 24 Jan 2012

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Small spatial scales in wetting on non-structured and

structured surfaces: thin films of liquid crystals &

small-scales wetting properties of ionic liquids

Céline J. E. Richard

To cite this version:

Céline J. E. Richard. Small spatial scales in wetting on non-structured and structured surfaces: thin

films of liquid crystals & small-scales wetting properties of ionic liquids. Soft Condensed Matter

[cond-mat.soft]. Université Pierre et Marie Curie - Paris VI, 2011. English. �tel-00662524�

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❏❡ .❡♠❡.❝✐❡ 0❣❛❧❡♠❡♥# ❧❡& ♣❡.&♦♥♥❡& ❛✈❡❝ 6✉✐ ❥✬❛✐ ♣✉ #.❛✈❛✐❧❧❡. &✉. ❧❡ &✉❥❡# ✏❝.✐&#❛✉① ❧✐6✉✐❞❡&✑✳ ▼❡.❝✐ ? ❯❧②&&❡ ❉❡❧❛❜.❡ ♣♦✉. ❧❡ ♣❛&&❛❣❡ ❞❡ .❡❧❛✐&✳ ❈❡❧❛ ❛ 0#0 #.%& ❛❣.0❛❜❧❡ ❞❡ #.❛✈❛✐❧❧❡. ❛✈❡❝ 6✉❡❧6✉✬✉♥ ❞✬❛✉&&✐ ♦.❣❛♥✐&0 ❡# .✐❣♦✉.❡✉①✳ ❏❡ ♠❡ .❛♣♣❡❧❧❡.❛✐ 0❣❛❧❡♠❡♥# ❞❡& ♥♦♠❜.❡✉&❡& ❞✐&❝✉&&✐♦♥& &✉. ❧❛ ♣❤②&✐6✉❡ ❡# &✉. ❧❛ #.❛♥&♠✐&&✐♦♥ ❛✉① ♣❧✉& ❥❡✉♥❡& ❛✉#♦✉. ❞✬✉♥ ❜♦♥ ❝❛❢0 ✭♣❛.❢♦✐& ✉♥ ♣❡✉ ❞0❧❛②0 ✦✮✳ ▼❡.❝✐ ? ❨❛♥♥ ❨✐♣ ❈❤❡✉♥❣ ❙❛♥❣ ♣♦✉. ❧❛ ❝♦❧❧❛❜♦.❛#✐♦♥ ❡# ❧❡& ❞✐&❝✉&&✐♦♥& &✉. ❧❡& ❡①♣0.✐❡♥❝❡& ❛✐♥&✐ 6✉❡ ♣♦✉. ❛✈♦✐. ♣❛.#❛❣0 &♦♥ &❡♥& ❞❡ ❧✬❛&#✉❝❡ ♣♦✉. ❧✬❛♠0❧✐♦.❛#✐♦♥ ❞❡& ❞✐&♣♦&✐#✐❢& ❡①♣0.✐♠❡♥#❛✉① ✦ ▼❡.❝✐ ? ❖❦&❛♥❛ ▼❛♥②✉❤✐♥❛ ❞✬❛✈♦✐. ♣❛.#❛❣0 &♦♥ &❛✈♦✐. &✉. ❧❛ ♣❛.#✐❡ #❤0♦.✐6✉❡ ❞✉ &✉❥❡# ✭❛✉ ❧❛❜♦.❛#♦✐.❡ ❡# &♦✉& ❧❡ &♦❧❡✐❧ ❞❡ ❈❛.❣%&❡✮ ❡# &❛ ♣❛&&✐♦♥ ♣♦✉. ❧❛ ♣❤②&✐6✉❡✳

❏❡ .❡♠❡.❝✐❡ ❙0❜❛&#✐❡♥ ▼♦✉❧✐♥❡# ♣♦✉. &❛ ❞✐&♣♦♥✐❜✐❧✐#0✱ ♣♦✉. P#.❡ #♦✉❥♦✉.& ♣.P# ? ❛✐❞❡. ♣❛. &❡& #.%& ❜♦♥& ❝♦♥&❡✐❧& #❡❝❤♥✐6✉❡& ❡# ♣♦✉. ❛✈♦✐. ❛✐❞0 6✉❡❧6✉❡& ❣0♥0.❛#✐♦♥& ❞❡ ❞♦❝#♦.❛♥#& ? ❞0❝.✐✛.❡. ❧❡ ❧❛❜②.✐♥#❤❡ 6✉✬❡&# ❧❛ ❙✶✶ ✦ ❏❡ .❡♠❡.❝✐❡ 0❣❛❧❡♠❡♥# ❧❡& ❛♥❝✐❡♥& ♠❡♠❜.❡& ❞✉ ❣.♦✉♣❡ ♣♦✉. ♠✬❛✈♦✐. ❛❝❝✉❡✐❧❧✐ #.%& ❝❤❛❧❡✉.❡✉&❡♠❡♥#✿ ●❡♦✛.♦② ●✉0♥❛✱ V❡❞❡. ▼♦❧❧❡.✱ ❈❤.✐&#♦♣❤❡ ❈❤❡✈❛❧❧✐❡. ❡# ❉✐❞✐ ❉❡.❦& ✭❡&♣❡❝✐❛❧❧② ❢♦. #❤❡ ♥✐❝❡ ❝❤♦❝♦❧❛#❡✲❝♦✛❡❡ ❜.❡❛❦& ✐♥ #❤❡ ❣❛.❞❡♥✮✳

❏❡ .❡♠❡.❝✐❡ 0❣❛❧❡♠❡♥# #♦✉& ❝❡✉① 6✉✐ ♦♥# ❝♦♥#.✐❜✉0 ? .❡♥❞.❡ ❧❛ ✈✐❡ ❛✉ ❧❛❜♦.❛#♦✐.❡ &②♠♣❛#❤✐6✉❡✿ ❑.✐&#✐♥❛ ❉❛✈✐## ✭❥✬❡&♣%.❡ 6✉❡ ❧✬♦♥ ❝♦♥#✐♥✉❡.❛ ? ♣❛.#❛❣❡. ♥♦#.❡ ♣❛&&✐♦♥ ❞❡& ❣>#❡❛✉①✱ ♠P♠❡ ? ❞✐&#❛♥❝❡ ✦✮✱ ❙❛#♦&❤✐ ❙❛&❛❦✐✱ ❈❧❛✉❞✐❛ V❛♥#❛❧❡✐ ✭♠❡.❝✐ ♣♦✉. ❧✬✐❞0❡ ❞❡& &0❛♥❝❡& ❞❡ .♦❧❧❡. ✦✮✱ ❊❧&❛ ❇❛②❛.# ✭♠❡.❝✐ ♣♦✉. ❧❛ ❝♦✲♦.❣❛♥✐&❛#✐♦♥ ❞✉ &0♠✐♥❛✐.❡ #❤0&❛.❞&✳✳✳ ❡# ? 6✉❛♥❞ ❧❛ ♣.♦❝❤❛✐♥❡ .❛♥❞♦♥♥0❡

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❄✦✮✱ ❳❛✈✐❡) ❘♦❥❛- ❡. ❆)♥❛✉❞ ❆)✈❡♥❣❛- ✭♠❡)❝✐ 7 .♦✉- ❧❡- ❞❡✉① ♣♦✉) ❧❡- ♣❛✉-❡- ✴ ❞✐-❝✉--✐♦♥- ✴ ❝❛❢=✮✳ ❏❡ )❡♠❡)❝✐❡ =❣❛❧❡♠❡♥. ❝❡✉① -❛♥- @✉✐ ❧❡- .❤B-❡- ❞✉ ❧❛❜♦)❛.♦✐)❡ ♥✬✐)❛✐❡♥. ♣❛- ❜✐❡♥ ❧♦✐♥ ✿ ♠❡)❝✐ 7 ❏♦-= ❞❛ ❙✐❧✈❛ ◗✉✐♥.❛- ❡. 7 .♦✉. ❧✬❛.❡❧✐❡) ♠=❝❛♥✐@✉❡ ❀ ♠❡)❝✐ 7 ❈❤)✐-.♦♣❤❡ ❍❡)♠❛♥♥ ❡. 7 .♦✉. ❧✬❛.❡❧✐❡) =❧❡❝.)♦♥✐@✉❡ ❀ ♠❡)❝✐ ❛✉ -❡❝)=.❛)✐❛.✿ ◆♦)❛ ❙❛❞❛♦✉✐ ✭♠❡)❝✐ ♣♦✉) ❧❡ .❡♠♣- ♣❛--= 7 ♠✬❛✐❞❡) ♣♦✉) ❧❡- ♥♦♠❜)❡✉-❡- ❝♦♠♠❛♥❞❡- ❡. ♣♦✉) ❧❡- ♥♦♠❜)❡✉① ♣❛♣✐❡)- ♣♦✉) ❧✬❆✉-.)❛❧✐❡✮✱ ▼❛)✐❡ ●❡✤♦. ❡. ❆♥♥✐❡ ❘✐❜❡❛✉❞❡❛✉ ❀ ♠❡)❝✐ 7 ❩❛P)❡ ❉✐--✐ ❡. 7 .♦✉. ❧❡ -❡)✈✐❝❡ ✐♥❢♦)♠❛.✐@✉❡ ❀ ♠❡)❝✐ 7 ▼✐❝❤❛❡❧ ❘♦-.✐❝❤❡) ❡. 7 .♦✉.❡ ❧✬=@✉✐♣❡ ❞✬❡♥.)❡.✐❡♥ ❞❡ ❧❛ -❛❧❧❡ ❜❧❛♥❝❤❡ ❀ ♠❡)❝✐ 7 ❉✐❞✐❡) ❈♦✉).✐❛❞❡ ❡. 7 .♦✉. ❧❡ -❡)✈✐❝❡ ❡♥.)❡.✐❡♥ ❣=♥=)❛❧ ❡. ♠❛❣❛-✐♥- ✭❡♥ ♣❛).✐❝✉❧✐❡) ♣♦✉) ❧❡- ❝❛- ❞✬✉)❣❡♥❝❡- ❛✈❡❝ ❧❡- ❢✉✐.❡- ❞✬❡❛✉ )=❝✉))❡♥.❡- ❛✉ ❞❡✉①✐B♠❡ -♦✉-✲-♦❧✮ ❀ ♠❡)❝✐ ❛✉ -❡)✈✐❝❡ =❧❡❝.)✐@✉❡✳ ❚❤✐- .❤❡-✐- ❝❛))✐❡❞ ♦♥ ✐♥ ❆✉-.)❛❧✐❛✱ -♦ ■ ✇♦✉❧❞ ❧✐❦❡ .♦ .❤❛♥❦ .❤❡ .❡❛♠✱ ■ ✇❛- ❛❜❧❡ .♦ ✇♦)❦ ✇✐.❤ ✐♥ .❤❡ ■❛♥ ❲❛)❦ ❘❡-❡❛)❝❤ ■♥-.✐.✉.❡✳ ■ .❤❛♥❦ ❏♦❤♥ ❘❛❧-.♦♥ ❢♦) ✇❡❧❝♦♠✐♥❣ ♠❡ ✐♥ ❤✐- ❧❛❜♦)❛.♦)② ❛♥❞ ❢♦) ❣✐✈✐♥❣ ♠❡ .❤❡ ❣)❡❛. ♦♣♣♦)✲ .✉♥✐.② .♦ ✇♦)❦ ✐♥ ❤✐- .❡❛♠✳ ■ .❤❛♥❦ ❤✐♠ ❛❧-♦ ❢♦) ♠❛❦✐♥❣ ♠❡ ❞✐-❝♦✈❡) .❤❡ ❡♥✐❣♠❛.✐❝ ❛♥❞ ❛❧✇❛②-❡♥.❡).❛✐♥✐♥❣ ✐♦♥✐❝ ❧✐@✉✐❞-✳ ■ ❛❧-♦ .❤❛♥❦ ❤✐♠ ❢♦) ❜❡✐♥❣ ❛ ♠❡♠❜❡) ♦❢ ♠② ❥✉)② ❛♥❞ ❢♦) ✇)✐.✐♥❣ ❛ )❡♣♦). ♦♥ ♠② ♠❛♥✉-❝)✐♣. ❛- ✐. ✇❛- ✐♠♣♦--✐❜❧❡ ❢♦) ❤✐♠ .♦ ❛..❡♥❞ .♦ .❤❡ ❞❡❢❡♥-❡✳ ■ .❤❛♥❦ ▼❛).❛ ❑)❛-♦✇-❦❛ ❢♦) ❤❡) ❝♦♥-.❛♥. -✉♣♣♦).✱ ❢♦) ❤❡) ♣❛--✐♦♥ ❢♦) ♣❤②-✐❝-✱ ❡-♣❡❝✐❛❧❧② ❆❋▼ ✭✦✦✦✮ ❛♥❞ ❢♦) ❤❡) ❡♥.❤✉-✐❛-♠✳ ■ ❛❧-♦ ❤❛❞ .♦ -❛②✱ .❤❛. ✏■ ❣✐✈❡ ②♦✉ ♠② ♣❡♥✑ ✦ ■ .❤❛♥❦ ▼✐❤❛✐❧ ]♦♣❡-❝✉ ❢♦) ❤✐- ❤❡❧♣ ❛♥❞ ❢♦) ♥✉♠❡)♦✉- ❞✐-❝✉--✐♦♥- ♦♥ .❤❡ ♣)♦❥❡❝.✳ ■ .❤❛♥❦ .❤❡ -❝✐❡♥.✐✜❝ -❡)✈✐❝❡- ❢♦) .❤❡✐) ❤❡❧♣✱ ❡-♣❡❝✐❛❧❧② ❙❝♦.. ❆❜❜♦.. ✭❢♦) ❝❛)❜♦♥ ✈❛♣♦) ❞❡♣♦✲ -✐.✐♦♥✮ ❛♥❞ ❈❤)✐- ❇❛--❡❧❧ ✭❢♦) ❳]❙✮✳ ■ ✇♦✉❧❞ ❛❧-♦ ❧✐❦❡ .♦ .❤❛♥❦ .❤❡ ♠❡♠❜❡)- ♦❢ .❤❡ ❣)♦✉♣ ❛♥❞ ♦❢ ♦.❤❡) ❣)♦✉♣- ✇❤♦ ✇❡)❡ ✇❡❧❝♦♠✐♥❣ ❛♥❞ ♦❢ ❣)❡❛. ❤❡❧♣ .♦ ♠❡✱ ❡-♣❡❝✐❛❧❧② ❛. .❤❡ ❜❡❣✐♥♥✐♥❣✿ ▼=❧❛♥✐❡ ❘❛♠✐❛-❛ ✭♠❡)❝✐ ♣♦✉) .♦✉. ❡. ✉♥ ❣)♦- ❜✐-♦✉- ❛✉ ♣❡.✐. ▲✉❝❛ ✦✮✱ ▼✉✐)❡❛♥♥ ❖✬▲♦✉❣❤❧✐♥ ✭.❤❛♥❦- ❢♦) .❤❡ ❞✐-❝✉--✐♦♥- ✦✮✱ ▲♦)❡♥❛ ❉✐❡❣✉❡③ ✭.❤❛♥❦ .♦ ②♦✉ ❛♥❞ ❙❛♠✱ ❡-♣❡❝✐❛❧❧② ❢♦) ✇❡❧❝♦♠✐♥❣ ♠❡ ❛. ②♦✉) ♣❧❛❝❡ ✐♥ ❏✉❧② ❛♥❞ ❆✉❣✉-.✮✱ ❆✉❞)❡② ❇❡❛✉--❛). ❛♥❞ ❏❛✐)♦ ●❛)♥✐❝❛ ✭♠❡)❝✐ ♣♦✉) .♦✉. ✦✮✱ ❘=❣✐- ▼=❥❛)❞ ✭♠❡)❝✐ ♣♦✉) ❜❡❛✉❝♦✉♣✳✳✳ ❧❡ .=✢♦♥✱ ❧❡- ❞✐-❝✉--✐♦♥- ❡♥ ❞✐)❡❝. ❡. -✉) -❦②♣❡✱ ❧❛ ❝❤❛--❡ ❛✉① ❢❛✉.❡- ❞✬❛♥❣❧❛✐- ❞❛♥- ❧❡ ✏♠❛♥✉-❝)✐♣.✑ ❡. ♣♦✉) ❧❛ -✉✐.❡✱ ❧✐- ❧❡ ❞❡)♥✐❡) ♣❛)❛❣)❛♣❤❡✳✳✳✮✱ ●❛❜♦) ❘♦③✐♣❛❧✱ ❱❛❧❡♥.✐♥ ❱❛♥❝❡❛ ✭.❤❛♥❦- ❢♦) ❡①♣❧❛✐♥✐♥❣ ❡✈❡)②.❤✐♥❣ ❛❜♦✉. .❤❡ ♣✉)✐✜❝❛.✐♦♥ ♦❢ ✐♦♥✐❝ ❧✐@✉✐❞-✮✱ ❇♦❣❛❧❡ ❚❛❞❡--❡✱ ▲✐ ❏✐❛♥❣✱ ❍✉❛ ▲✐✱ ❩❤❛♥.❛♦ ❲❛♥❣✳ ■ ✇♦✉❧❞ ❛❧-♦ ❧✐❦❡ .♦ .❤❛♥❦ .❤❡ ♥♦♥✲♣❤②-✐❝✐-. ❢)✐❡♥❞- ✇❤♦ ❛❧-♦ ❤❡❧♣❡❞ ♠❡ ❢❡❡❧ ❧✐❦❡ ❤♦♠❡ ❛. .❤❡ ♦.❤❡) ❡♥❞ ♦❢ .❤❡ ✇♦)❧❞✿ ◆❛.❤❛❧✐❡ ❛♥❞ ■-❛❜❡❧❧❡ ❀ ▲❡❡-❛ ❀ ❑❛)❡♥ ❀ ❳❛✈✐❡) ❡. ]❡))✐♥❡ ❀ ❙❛♠ ❀ ❏❛)❡❞ ❀ ❚♦♠✳ ❊♥✜♥✱ ❥❡ .✐❡♥- 7 )❡♠❡)❝✐❡) ♠❛ ❢❛♠✐❧❧❡ ❡. ♠❡- ❛♠✐- ♣♦✉) ❧❡✉) -♦✉.✐❡♥ ❝♦♥-.❛♥. ❡. ♣♦✉) ❧❡✉) ♣)=-❡♥❝❡ .♦✉. -✐♠♣❧❡♠❡♥. ✦ ▼❡)❝✐ .♦✉. -♣=❝✐❛❧❡♠❡♥. 7 ♠❡- ♣❛)❡♥.- ❀ 7 ♠❡- ✏✈✐❡✉①✑ ❛♠✐- ❊❧-❛ ❈✳ ✭❜♦♥ -✉)❢ 7 .♦✐ ❡. ❘♦♠❛✐♥ 7 ▲♦- ❆♥❣❡❧❡-✳✳✳ ♦✉ ♣❧✉.f. ❜♦♥ -=❥♦✉) 7 ▲❆ -❛♥- -✉)❢ ✦✮✱ ❑❡②❛♥ ✭❇♦♥♥❡ ❝❤❛♥❝❡ ♣♦✉) ❧❡ ❈❛♥❛❞❛ ✦✮ ❡. ▼❛.❤✐❧❞❡ ✭❜✐❡♥.f. ❞❡- ✈❛❝❛♥❝❡- ❡♥ ❆✉-.)❛❧✐❡ ❄✦✮ ❀ ❡. ♣✉✐- 7 ❘=❣✐-✭♣)♦♠✐-✱ ❥✬❛♣♣♦).❡ ❞❡- ♠❛))♦♥- ❣❧❛❝=- ❡♥ ❢=✈)✐❡) ✦✳✳✳ ♠❡)❝✐ ❞✬❛✈♦✐) =.= ❧7 ✦✮

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(6)

■♥"#♦❞✉❝"✐♦♥

❉!✐③③❧❡ ❢❛❧❧(✳ ❇✉, ✇❤❛, ✐( ,❤❡ (❤❛♣❡ ♦❢ ,❤❡ ❢❛❧❧✐♥❣ ❞!♦♣( ❄ ❆ ,❤✐♥ ,✉❜❡ ♦❢ ❣❧❛(( ✐( ❞✐♣♣❡❞ ✐♥,♦ ✇❛,❡! ❛♥❞ ✇❛,❡! ✐( (✉❝❦❡❞ ✐♥(✐❞❡ ,❤❡ ,✉❜❡✳ ❲❤② ❄ ❚❤❡ ❛♥(✇❡! ,♦ ,❤❡(❡ <✉❡(,✐♦♥( ✇❛( ❣✐✈❡♥ ✐♥ ,❤❡ (,✉❞✐❡( ♦❢ >✐❡!!❡ ❙✐♠♦♥ ❞❡ ▲❛♣❧❛❝❡ ❛❜♦✉, ❝❛♣✐❧❧❛!② ♣❤❡♥♦♠❡♥❛✳ ❆❧!❡❛❞② ✐♥ ,❤❡ ✶✽,❤ ❝❡♥,✉!②✱ ,❤❡ ❜❡❤❛✈✐♦✉! ♦❢ ✏(♠❛❧❧✑ <✉❛♥,✐,✐❡( ♦❢ ❧✐<✉✐❞( ❛, ❛♥ ✐♥,❡!❢❛❝❡ ✭❧✐<✉✐❞ ✴ ❣❛( ❢♦! ,❤❡ ❞!♦♣ ❀ ❧✐<✉✐❞ ✴ (♦❧✐❞ ❛♥❞ ❧✐<✉✐❞ ✴ ❣❛( ❢♦! ,❤❡ ,✉❜❡✮ ✇❛( <✉❡(,✐♦♥❡❞✳ ❙✐♥❝❡ ,❤✐( ♣✐♦♥❡❡!✐♥❣ ✇♦!❦✱ (♠❛❧❧✲(❝❛❧❡ ✇❡,,✐♥❣ ❜❡❤❛✈✐♦✉! (,❛②❡❞ ❛ ✈✐✈✐❞ ❛!❡❛ ♦❢ ♣❤②(✐❝( ❛♥❞ ,❤✐( ♣❛!,❧② ❜❡❝❛✉(❡ ,❤❡ ❞❡✜♥✐,✐♦♥ ♦❢ ✏(♠❛❧❧✑ ❝❤❛♥❣❡❞ ❛❧♦♥❣ ✇✐,❤ ❛❞✈❛♥❝❡( ✐♥ ,❡❝❤♥♦❧♦❣✐❡(✳ ◆❡✇ ♠❡,❤♦❞( ❛❧❧♦✇✐♥❣ ♦♥❡ ,♦ (❝❛♥ ,❤❡ ♣!♦♣❡!,✐❡( ♦❢ ✐♥,❡!❢❛❝❡( ♦♥ (♠❛❧❧❡! ❛♥❞ (♠❛❧❧❡! (❝❛❧❡( ✇❡!❡ ✐♥❞❡❡❞ ❞❡✈❡❧♦♣❡❞✳ ❖♥❡ ❝♦✉❧❞ ❝✐,❡ ❢♦! ❡①❛♠♣❧❡ ❇!❡✇(,❡! ❆♥❣❧❡ ▼✐❝!♦(❝♦♣❡ ✭❇❆▼✮✱ ❆,♦♠✐❝ ❋♦!❝❡ ▼✐❝!♦(❝♦♣②✱ ✳✳✳ ❲❤❛, ✐( ,❤❡♥ ♠❡❛♥, ❜② ,❤❡ ❡①♣!❡((✐♦♥ ✏(♠❛❧❧✲(❝❛❧❡ ✇❡,,✐♥❣ ❜❡❤❛✈✐♦✉!✑ ❄ ■♥ ,❤✐( ✇♦!❦✱ ,❤❡ (♦✲❝❛❧❧❡❞ ✏(♠❛❧❧✲(❝❛❧❡(✑ ✇✐❧❧ !❡❢❡! ,♦ (✉❜✲♠✐❝!♦♥ ♣❤❡♥♦♠❡♥❛✳ ❙✉❜✲♠✐❝!♦♥ ✐♥,❡!❢❛❝✐❛❧ ♣❤❡♥♦♠❡♥❛ ❛!❡ ♦❢ ❦❡② ✐♠♣♦!,❛♥❝❡ ✐♥ ✐♥❞✉(,!✐❛❧ ♣!♦❝❡((❡( ✭❡✳❣✳ ❝❧❡❛♥✐♥❣✮ ❛♥❞ ❛❧(♦ ✐♥ ,❤❡ ❣!♦✇✐♥❣ ✜❡❧❞ ♦❢ ♥❛♥♦,❡❝❤♥♦❧♦❣✐❡( ❛♥❞ ♠✐❝!♦✢✉✐❞✐❝( ✇❤✐❝❤ ✜♥❛❧ ❛✐♠ ✐( ,♦ ❞❡✈❡❧♦♣ ✏❧❛❜( ♦♥ ❛ ❝❤✐♣✑✳ ❆, ❛ ,✐♠❡ ✇❤❡!❡ ♠✐♥✐❛,✉!✐③❛,✐♦♥ ✐( ♦♥❡ ♦❢ ,❤❡ ♠❛✐♥ ❝❤❛❧❧❡♥❣❡( ✐♥ (❝✐❡♥❝❡✱ ,❤❡ ♣❤②(✐❝( ♦❢ ✇❡,✲ ,✐♥❣ ❛, (♠❛❧❧✲(❝❛❧❡( ✐( ❢❛! ❢!♦♠ ❜❡✐♥❣ ❝♦♠♣❧❡,❡❧② ✉♥❞❡!(,♦♦❞✳ ●♦✐♥❣ ,♦✇❛!❞( ♠✐❝!♦(❝♦♣✐❝ (❝❛❧❡(✱ (②(,❡♠(✬ ❜❡❤❛✈✐♦✉! ❝❤❛♥❣❡(✳ ❚❤❡ (,!✉❝,✉!✐♥❣ !♦❧❡ ♦❢ ,❤❡ ✐♥,❡!❢❛❝❡✭(✮✱ ,❤❡ ❤❡,❡!♦❣❡♥❡✐,✐❡( ♦❢ ,❤❡ (✉❜(,!❛,❡ ♦! ,❤❡ ❣!♦✇✐♥❣ ✐♠♣♦!,❛♥❝❡ ♦❢ ❧✐♥❡ ❡✛❡❝,( ♦✈❡! (✉!❢❛❝❡ ❡✛❡❝,( ❜❡❣✐♥ ,♦ ♣❧❛② ❛ ♠❛❥♦! !♦❧❡✳ ❚❤❡ ❛✐♠ ♦❢ ,❤❡ ♣!❡(❡♥, ,❤❡(✐( ✐( ,❤❡♥ ,♦ ❜!✐♥❣ ♥❡✇ ❝❧✉❡( ,♦ ❤❡❧♣ ,❤❡ ❞❡✈❡❧♦♣♠❡♥, ♦❢ ,❤❡♦!❡,✐❝❛❧ ,♦♦❧( ❛♥❞ ♠♦❞❡❧(✳ ❚✇♦ ❡①❛♠♣❧❡( ♦❢ ❝♦♠♣❧❡① (♠❛❧❧✲(❝❛❧❡ (②(,❡♠( ✇✐❧❧ ❜❡ ♣!❡(❡♥,❡❞✿ ,❤✐♥ ✜❧♠( ♦❢ ♥❡♠❛,✐❝ ❧✐<✉✐❞ ❝!②(,❛❧( ♦♥ ❧✐<✉✐❞ (✉❜(,!❛,❡( ❛( ✇❡❧❧ ❛( ❞!♦♣❧❡,( ❛♥❞ ♣!❡❝✉!(♦!( ♦❢ ✐♦♥✐❝ ❧✐<✉✐❞( ♦♥ (♦❧✐❞ (✉❜(,!❛,❡(✳ ▲✐<✉✐❞ ❝!②(,❛❧( ❛!❡ ✇❡❧❧✲❦♥♦✇♥ ❢♦! ,❤❡✐! ❡✈❡!②❞❛② ❧✐❢❡ ❛♣♣❧✐❝❛,✐♦♥(✳ ❚❤❡ ♠♦(, ❝♦♠♠♦♥ ♦♥❡ ✐( ,❤❡ ▲✐<✉✐❞ ❈!②(,❛❧ ❉✐(♣❧❛② ✭▲❈❉✮ ✢❛, (❝!❡❡♥ ✇❤✐❝❤ ❝❛♥ ❜❡ ❢♦✉♥❞ ♥♦✇❛❞❛②( ♦♥ ❛❧♠♦(, ❡✈❡!② ❝♦♠♣✉,❡!✱ ❝❡❧❧✲♣❤♦♥❡ ♦! ,❡❧❡✈✐(✐♦♥✳ ▲✐<✉✐❞ ❝!②(,❛❧( ❛!❡ ❛ ✈❡!② (♣❡❝✐✜❝ ❦✐♥❞ ♦❢ ♠❛,❡!✐❛❧(✳ ❚❤❡② ✐♥❞❡❡❞ ❛!❡ ❛❜❧❡ ,♦ ✢♦✇ ❧✐❦❡ ❧✐<✉✐❞( ❜✉, ,❤❡② ❛❧(♦ ♣♦((❡(( ❛ ❧♦♥❣ !❛♥❣❡ ♦!❞❡! ❧✐❦❡ (♦❧✐❞(✳ ❈♦♠♣❛!❡❞ ,♦ ❛♥ ✉(✉❛❧ ❧✐<✉✐❞✱ ,✇♦ ❛❞❞✐,✐♦♥❛❧ ✐♥❣!❡❞✐❡♥,( ♠✉(, ❜❡ ❛❞❞❡❞ ,♦ ❞❡(❝!✐❜❡ ,❤❡ ✇❡,,✐♥❣ ♣!♦♣❡!,✐❡( ♦❢ ♥❡♠❛,✐❝ ❧✐<✉✐❞ ❝!②(,❛❧(✿ ❡❧❛(,✐❝✐,② ✇❤✐❝❤ ✐( ,❤❡ (♦✉!❝❡ ♦❢ ❧♦♥❣ !❛♥❣❡ ✐♥,❡!❛❝,✐♦♥( ❛♥❞ ❛♥❝❤♦!✐♥❣ ✇❤✐❝❤ ❞❡(❝!✐❜❡( ,❤❡ (♣❡❝✐✜❝ !♦❧❡ ♣❧❛②❡❞ ❜② ,❤❡ ❜♦✉♥❞❛!② ❝♦♥❞✐,✐♦♥( ❢♦! ❛ ♥❡♠❛,✐❝ ❧✐<✉✐❞ ❝!②(,❛❧✳ ❇❡❝❛✉(❡ ♦❢ ,❤❡ (♣❡❝✐✜❝ ♥❛,✉!❡ ♦❢ ❧✐<✉✐❞ ❝!②(,❛❧(✱ ❢♦! ,❤✐❝❦♥❡((❡( ❧❡(( ,❤❛♥ ♦♥❡ ♠✐❝!♦♠❡,❡!✱ ✜❧♠( ✶

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♦❢ ♥❡♠❛&✐❝ ❧✐*✉✐❞ ❝-②/&❛❧ ❞❡♣♦/✐&❡❞ ♦♥ ❛ /✉❜/&-❛&❡ ✐♥ ❤②❜-✐❞ ❝♦♥✜❣✉-❛&✐♦♥ ✭❛♥&❛❣♦♥✐/& ❛♥❝❤♦-✐♥❣ ❝♦♥❞✐&✐♦♥/ ❛& &❤❡ ❜♦✉♥❞❛-✐❡/✮ ❤❛✈❡ ❛♥ ✉♥✉/✉❛❧ ✇❡&&✐♥❣ ❜❡❤❛✈✐♦✉-✿

• ❚❤❡-❡ ✐/ ❛ ❢♦-❜✐❞❞❡♥ &❤✐❝❦♥❡// ③♦♥❡✿ ✜❧♠/ &❤✐❝❦❡- &❤❛♥ 20 − 30 ♥♠ ♦- &❤✐♥♥❡- &❤❛♥ 3.5 ♥♠ ❢❛- ❢-♦♠ &❤❡ ♣❤❛/❡ &-❛♥/✐&✐♦♥✱ ❛-❡ ♦❜/❡-✈❡❞ ❛♥❞ ♠❛② ❡✈❡♥&✉❛❧❧② ❝♦❡①✐/&✳

• ❚❤❡ /&-✉❝&✉-❡ ♦❢ ✜❧♠/ &❤✐❝❦❡- &❤❛♥ 20−30 ♥♠ ✐/ *✉✐&❡ /♣❡❝✐✜❝✱ ❛& ❧❡❛/& ♦♥ ❧✐*✉✐❞ /✉❜/&-❛&❡/✿ &❤❡② /❤♦✇ /&-✐♣❡❞ ✐♥/&❛❜✐❧✐&② ♣❛&&❡-♥/ ✉♥❞❡-/&♦♦❞ ❛/ ❛ ✇❛② &♦ -❡❧❛① ♥❡♠❛&✐❝ ❡❧❛/&✐❝✐&②✳ ■♥ ❛♥ ✐❞❡❛❧ /✐&✉❛&✐♦♥✱ ✐& ✇♦✉❧❞ ❜❡ ♣♦//✐❜❧❡ &♦ ❜✉✐❧❞ ✉♣ ❛ ✉♥✐*✉❡ ♠♦❞❡❧ ♦❢ &❤❡ ♥❡♠❛&✐❝ ❢-❡❡ ❡♥❡-❣② &♦ ❞❡/❝-✐❜❡ &❤❡/❡ ❜❡❤❛✈✐♦✉-/✳ ❯♥❢♦-&✉♥❛&❡❧②✱ /✉❝❤ ❛ ♠♦❞❡❧✱ &❛❦✐♥❣ ✐♥&♦ ❛❝❝♦✉♥& ♣❤❡♥♦♠❡♥❛ ❛& ❞✐✛❡-❡♥& /❝❛❧❡/✱ ❤❛/ ♥♦& ❜❡❡♥ ❞❡✈❡❧♦♣❡❞ ②❡&✳ ❆/ ❛ ✜-/& ❛♣♣-♦❛❝❤✱ ❛♥ ❛&&❡♠♣& ♦❢ ❞❡/❝-✐♣&✐♦♥ ♦❢ &❤❡ ❜❡❤❛✈✐♦✉- ♦❢ /&-✐♣❡❞ ✜❧♠/ ✐/ ❜❛/❡❞ ♦♥ &❤❡ ❝♦♥&✐♥✉✉♠ &❤❡♦-② ♦❢ ♥❡♠❛&✐❝ ❡❧❛/&✐❝✐&②✳ ❍♦✇❡✈❡-✱ ❛ ❧❛-❣❡ ♣❛-& ♦❢ ❡①♣❡-✐♠❡♥&❛❧ ♦❜/❡-✈❛&✐♦♥/ ❡/❝❛♣❡/ &♦ &❤❡ ❛♥❛❧②/✐/ ❛♥❞ &❤❡ ♠♦❞❡❧ ❞♦❡/ ♥♦& ❛❝✲ ❝♦✉♥& ❢♦- &❤❡ ❢♦-❜✐❞❞❡♥ -❛♥❣❡ ♦❢ &❤✐❝❦♥❡//❡/✳ ❍❡♥❝❡✱ &❤❡ ❡①♣❡-✐♠❡♥&❛❧ ✇♦-❦ ✐♥ ♣-♦❣-❡// ✐/ ❛✐♠❡❞ ❛& ♣-♦❞✉❝✐♥❣ ♣-❡❝✐/❡ ❞❛&❛ &♦ ❜-✐♥❣ ♥❡✇ ❝❧✉❡/ ❢♦- ❢✉&✉-❡ &❤❡♦-❡&✐❝❛❧ ❛♥❛❧②/✐/✳ ❚❤❡ /&-✉❝&✉-❡ ♦❢ &❤❡/❡ &❤✐♥ ♥❡♠❛&✐❝ ✜❧♠/ ❛/ ✇❡❧❧ ❛/ &❤❡ ♦♥❡ ♦❢ &❤❡✐- -✐♠ ✇❡-❡ &❤❡♥ /&✉❞✐❡❞ ❜♦&❤ ✐♥ &❤❡ ♥❡♠❛&✐❝ -❛♥❣❡ ♦❢ &❡♠♣❡-❛&✉-❡ ❛/ ✇❡❧❧ ❛/ ❝♦♠✐♥❣ ❝❧♦/❡ ❛♥❞ ❝-♦//✐♥❣ &❤❡ ♥❡♠❛&✐❝ ✴ ✐/♦&-♦♣✐❝ ♣❤❛/❡ &-❛♥/✐&✐♦♥✳ ■♦♥✐❝ ❧✐*✉✐❞/ ❛-❡ ♠♦❧&❡♥ /❛❧&/ ❛& ❛♠❜✐❡♥& &❡♠♣❡-❛&✉-❡✳ ❚❤❡② ❛-❡ ♥♦✇ ✉❜✐*✉✐&♦✉/ ✐♥ ✐♥❞✉/&-②✿ ❣❛/ ❤❛♥❞❧✐♥❣✱ ♥✉❝❧❡❛- ❢✉❡❧ -❡♣-♦❝❡//✐♥❣✱ /♦❧❛- ❝❡❧❧/✱ ❜❛&&❡-✐❡/✱ ❝❤❡♠✐❝❛❧ ✐♥❞✉/&-②✳✳✳✳ ❈♦♥/✐❞❡-✐♥❣ ❝❤❡♠✐❝❛❧ ✐♥❞✉/&-②✱ ✐♦♥✐❝ ❧✐*✉✐❞/ ❤❛✈❡ ❛♥ ✐♠♣♦-&❛♥& -♦❧❡ &♦ ♣❧❛② ✐♥ &❤❡ ✜❡❧❞ ♦❢ ❣-❡❡♥ ❝❤❡♠✐/&-②✳ ❇❡❝❛✉/❡ ♦❢ &❤❡✐- ✈❡-② ❧♦✇ /❛&✉-❛&✐♥❣ ✈❛♣♦- ♣-❡//✉-❡✱ ✉/✐♥❣ ✐♦♥✐❝ ❧✐*✉✐❞/ ❞♦❡/ ✐♥❞❡❡❞ ♥♦& ❝♦♥&❛♠✲ ✐♥❛&❡ &❤❡ ❛&♠♦/♣❤❡-❡✳ ❖♥❡ ♦❢ &❤❡ ♣♦//✐❜❧❡ ✇❛②/ ♦❢ ❝♦♥&❛♠✐♥❛&✐♦♥ ❢♦- &❤❡ ❡♥✈✐-♦♥♠❡♥& ✐/ &❤❡♥ ❡❧✐♠✐♥❛&❡❞✱ &❤❛& ✐/ ✇❤② &❤❡② ❛-❡ ❝♦♥/✐❞❡-❡❞ ❛/ ❛♥ ❛❧&❡-♥❛&✐✈❡ &♦ ♦-❣❛♥✐❝ /♦❧✈❡♥&/✳ ❍♦✇❡✈❡-✱ ✐❢ &❤❡/❡ /♦✲❝❛❧❧❡❞ ✏❣-❡❡♥ /♦❧✈❡♥&/✑ ❞♦ ♥♦& ❡✈❛♣♦-❛&❡ ✐♥ &❤❡ ❛&♠♦/♣❤❡-❡✱ ✐& ❞♦❡/ ♥♦& ♠❡❛♥ &❤❛& &❤❡② ❛-❡ ♥♦& &♦①✐❝ ✦ M♦❧❧✉&✐♦♥ ❝❛✉/❡❞ ❜② ✐♦♥✐❝ ❧✐*✉✐❞/ ✐/ ❥✉/& ♠♦-❡ ❝♦♥&-♦❧❧❛❜❧❡ &❤❛♥ ❢♦- ✉/✉❛❧ ♦-❣❛♥✐❝ /♦❧✈❡♥&/✳ ■♥ ♦-❞❡- &♦ ❜❡ ❛❜❧❡ &♦ ❝♦♠♣❧❡&❡❧② ❝♦♥&-♦❧ &❤❡ ♣♦❧❧✉&✐♦♥ ❝-❡❛&❡❞ ❜② ✐♦♥✐❝ ❧✐*✉✐❞/✱ &❤❡✐-✐♥&❡-❢❛❝✐❛❧ ♣-♦♣❡-&✐❡/ ❛-❡ ❛ ❦❡② ♣-♦❜❧❡♠✳ ❚❤✐/ ✐/ ♦♥❡ ♦❢ &❤❡ -❡❛/♦♥/ &❤❛& ♠❛❦❡/ &❤❡✐- ✇❡&&✐♥❣ ❜❡❤❛✈✐♦✉- ♦♥ /♠❛❧❧✲/❝❛❧❡/ ❛ ❤♦& &♦♣✐❝✳ ❙♠❛❧❧ *✉❛♥&✐&✐❡/ ♦❢ ✐♦♥✐❝ ❧✐*✉✐❞ ✇❡-❡ ❞❡♣♦/✐&❡❞ ♦♥ /❡✈❡-❛❧ /✉❜/&-❛&❡/✱ /♠♦♦&❤ ♦- -♦✉❣❤✳ ❚❤❡ ♠✐❝-♦/❝♦♣✐❝ ❧❛♥❞/❝❛♣❡ ♦❜&❛✐♥❡❞✱ ❞-♦♣❧❡&/ ❛♥❞ &❤✐♥ ✜❧♠/✱ ✇❛/ &❤❡♥ ✐♠❛❣❡❞ ✉/✐♥❣ ❆&♦♠✐❝ ❋♦-❝❡ ▼✐❝-♦/❝♦♣② ✭❆❋▼✮✳ ❚❤❡ ❆❋▼ ✐♠❛❣❡/ ❛❧❧♦✇❡❞ ❢♦- ❛ ✜-/& ❝❤❛-✲ ❛❝&❡-✐③❛&✐♦♥ ❧❡❛❞✐♥❣ &♦ &❤❡ ❡♠❡-❣❡♥❝❡ ♦❢ ❣❡♥❡-❛❧ &-❡♥❞/ ♦♥ &❤❡ ♠♦❞✐✜❝❛&✐♦♥ ♦❢ &❤❡ ❝♦♥&❛❝& ❛♥❣❧❡ ✇❤❡♥ ❣♦✐♥❣ &♦✇❛-❞/ /♠❛❧❧✲/❝❛❧❡/ ♦- ♦♥ &❤❡ ♣-❡/❡♥❝❡ ❛♥❞ ❡✈♦❧✉&✐♦♥ ♦❢ &❤✐♥ ✜❧♠/✳

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❈❤❛♣$❡& ✶

❆♥ ✐♥#$♦❞✉❝#✐♦♥ #♦ ✇❡##✐♥❣

♣❤❡♥♦♠❡♥❛

✶✳✶ "❤❡♥♦♠❡♥♦❧♦❣✐❝❛❧ ❞❡.❝/✐♣1✐♦♥

✶✳✶✳✶ ❈♦♠♠♦♥ ♦❜'❡)✈❛,✐♦♥'✱ '♦♠❡ /✉❡',✐♦♥' ❛♥❞ ❝♦♠♠❡♥,'

❉❡✇ ♦♥ ❧❡❛✈❡( ♦) ✇❛*❡) ♦♥ ❝❧❡❛♥ ❣❧❛((✱ ✐( *❤❡)❡ ❛ ❞✐✛❡)❡♥❝❡ ❄ ❉❡✇ ♦♥ ❧❡❛✈❡( ❧♦♦❦( ❧✐❦❡ (♠❛❧❧ ❞)♦♣( ✇❤❡)❡❛( ✇❛*❡) ♦♥ ❝❧❡❛♥ ❣❧❛(( (♣)❡❛❞( ❛♥❞ ❛ ✇❛*❡) ✜❧♠ ❝♦✈❡)( *❤❡ ✇❤♦❧❡ (✉)❢❛❝❡✳ ❋♦) *❤❡ (❛♠❡ ❧✐;✉✐❞✱ ❝❤❛♥❣✐♥❣ *❤❡ (✉❜(*)❛*❡ *❤✉( ❛❧❧♦✇( ❢♦) *❤❡ ❝)❡❛*✐♦♥ ♦❢ *✇♦ ❞✐✛❡)❡♥* ✇❡**✐♥❣ (✐*✉❛*✐♦♥(✿ ❞)♦♣ ♦) ✜❧♠✳ ❲❤❛* ✇♦✉❧❞ ❤❛♣♣❡♥ *❤❡♥ ✐❢ ❞)♦♣( ♦❢ *✇♦ ❞✐✛❡)❡♥* ❧✐;✉✐❞( ✇❡)❡ ❞❡♣♦(✐*❡❞ ♦♥ *❤❡ (❛♠❡ (✉❜(*)❛*❡ ❄ ❯(✐♥❣ *❤❡ ❡①❛♠♣❧❡ ♦❢ ✇❛*❡)✱ ✇♦✉❧❞ *❤❡)❡ ❜❡ ❛ ❞✐✛❡)❡♥❝❡ ✐♥ ❜❡*✇❡❡♥ ✇❛*❡) ❞❡♣♦(✐*❡❞ ♦♥ ❝❧❡❛♥ ❣❧❛(( ❛♥❞ ♠❡)❝✉)② ❞❡♣♦(✐*❡❞ ♦♥ ❝❧❡❛♥ ❣❧❛(( ❄ ❚❤❡)❡ ✇♦✉❧❞✿ ♠❡)❝✉)② ✉♥❧✐❦❡ ✇❛*❡) ✇♦✉❧❞ ♥♦* (♣)❡❛❞ ♦♥ *❤❡ ❝❧❡❛♥ ❣❧❛(( (✉)❢❛❝❡✱ ✐* ✇♦✉❧❞ ❢♦)♠ ❛ ❞)♦♣✶✳ ❚❤♦(❡ ♦❜(❡)✈❛*✐♦♥( (❤♦✇ *❤❛* ✐* ✐( *❤❡ ❝♦✉♣❧❡ ❧✐;✉✐❞✴(✉❜(*)❛*❡ ✇❤✐❝❤ ♠❛**❡)( ✐♥ ✇❡**✐♥❣ ♣❤❡♥♦♠❡♥❛✳ ❇✉* ✐( *❤❛* ❛❧❧ ❄ ❲❤❛* ✐❢✱ ✐♥(*❡❛❞ ♦❢ ❝❤❛♥❣✐♥❣ *❤❡ ❧✐;✉✐❞ ♦) *❤❡ (✉❜(*)❛*❡✱ *❤❡ ❛♠❜✐❡♥* ❡♥✈✐)♦♥♠❡♥* ✐( ❝❤❛♥❣❡❞ ❄ ▲❡* ✉( ✜)(* ❝♦♥(✐❞❡) ❞♦❞❡❝❛♥❡ ❞❡♣♦(✐*❡❞ ♦♥ ❛ *❡✢♦♥ (✉❜(*)❛*❡ ✐♥ ✶▼❡"❝✉"② ♦♥ ❣❧❛++ ✐+ ❛❝-✉❛❧❧② ♦♥❡ "❛"❡ ❡①❛♠♣❧❡ ♦❢ ❞"②✐♥❣✿ ❛- -❤❡ ❡5✉✐❧✐❜"✐✉♠✱ -❤❡ ❧✐5✉✐❞ ❞"♦♣ ✐+ +❡♣❛"❛-❡❞ ❢"♦♠ -❤❡ +✉❜+-"❛-❡ ❜② ❛ ♠❛❝"♦+❝♦♣✐❝ ✈❛♣♦" ❧❛②❡"✳ ❋✐❣✉)❡ ✶✳✶✿ ❉♦❞❡❝❛♥❡ ♦♥ *❡✢♦♥ (✉❜(*)❛*❡✳ ❖♥ *❤❡ ❧❡❢* (✐❞❡ ✐♥ ❛✐)✳ ❖♥ *❤❡ )✐❣❤* (✐❞❡ ✐♥ ✇❛*❡)✳ ✸

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❛✐"✳ ■% ❢♦"♠) ❛ ❞"♦♣ ❛) )❤♦✇♥ ✐♥ ✜❣✉"❡ ✶✳✶✳ ❚❤❡ )❛♠❡ ❡①♣❡"✐♠❡♥% ✐) ♥♦✇ "❡♣❡❛%❡❞✱ ❜✉% ✐♥ ❛ ❝❡❧❧ ✜❧❧❡❞ ✇✐%❤ ✇❛%❡"✳ ❚❤❡ ❞♦❞❡❝❛♥❡ ❛❧)♦ ❢♦"♠) ❛ ❞"♦♣ ♦♥ %❤❡ %❡✢♦♥ )✉❜)%"❛%❡✱ ♦♥❧② ✐% ✐) ♠✉❝❤ ✢❛%%❡" ✭)❡❡ ✜❣✉"❡ ✶✳✶✮✳ ❈❤❛♥❣✐♥❣ %❤❡ ❡♥✈✐"♦♥♠❡♥% ♠♦❞✐✜❡❞ %❤❡ ✇❛② %❤❡ ❧✐@✉✐❞ )♣"❡❛❞) ♦♥ %❤❡ )✉❜)%"❛%❡✳ ❚❤❡ ❛♠❜✐❡♥% ❡♥✈✐"♦♥♠❡♥% ✐♥%❡"❛❝%) ✐♥❞❡❡❞ ✐♥❞✐✈✐❞✉❛❧❧② ✇✐%❤ %❤❡ )✉❜)%"❛%❡ ❛♥❞ ✇✐%❤ %❤❡ ❧✐@✉✐❞✳ ❲❡%%✐♥❣ ♣❤❡♥♦♠❡♥❛ ✐) %❤❡♥ ❛ ♣"♦❜❧❡♠ ✇✐%❤ %❤"❡❡ ❝♦♠♣♦♥❡♥%) ✐♥%❡"❛❝%✐♥❣ ✇✐%❤ ❡❛❝❤ ♦%❤❡" ✐♥❞✐✈✐❞✉❛❧❧②✿ %❤❡ ❧✐@✉✐❞ ♣❤❛)❡ ❛♥❞ %❤❡ )✉❜)%"❛%❡✱ %❤❡ ❧✐@✉✐❞ ♣❤❛)❡ ❛♥❞ %❤❡ ❛♠❜✐❡♥% ❡♥✈✐"♦♥♠❡♥% ❛♥❞ %❤❡ )✉❜)%"❛%❡ ❛♥❞ %❤❡ ❛♠❜✐❡♥% ❡♥✈✐"♦♥♠❡♥%✳ ❚❤♦)❡ ✐♥%❡"❛❝%✐♦♥) ✇✐❧❧ ❞❡%❡"♠✐♥❡ ✇❤✐❝❤ )✐%✉❛%✐♦♥ ✐) %❤❡ ♠♦)% ❢❛✈♦"❛❜❧❡ ❢"♦♠ ❛♥ ❡♥❡"❣❡%✐❝ ♣♦✐♥% ♦❢ ✈✐❡✇✳ ❍♦✇❡✈❡"✱ )♦♠❡ ❝❛"❡ ♠✉)% ❜❡ %❛❦❡♥ ✇❤❡♥ ❣♦✐♥❣ ❢"♦♠ ♠❡"❡ ♦❜)❡"✈❛%✐♦♥ %♦ %❤❡♦"❡%✐❝❛❧ ❞❡)❝"✐♣✲ %✐♦♥ ❛) %❤"❡❡ ♣❤❛)❡) ❛"❡ ♣"❡)❡♥%✳ ❲❤❡♥ ♠❡♥%✐♦♥✐♥❣ ❞❡✇✱ ✐✳❡✳✱ ❝♦♥❞❡♥)❛%✐♦♥✱ ♦♥❡ ✐♠♣❧✐❝✐%❧② ❛))✉♠❡) %❤❛% ❛✐" ✐) ❤✉♠✐❞✱ %❤❛% ✐)✱ %❤❡"❡ ❛"❡ ✇❛%❡" ♠♦❧❡❝✉❧❡) ✐♥ %❤❡ ❣❛) ♣❤❛)❡✳ ❆♥❞ %❤❡ ✇❛%❡" ❞"♦♣❄ ■❢ %❤❡ ❛%♠♦)♣❤❡"❡ ✐) ♥♦% )❛%✉"❛%❡❞ ✇✐%❤ ✇❛%❡"✱ ✐% ✇✐❧❧ ❡✈❛♣♦"❛%❡ ❛❢%❡" )♦♠❡ %✐♠❡✳ ❚✇♦ ♣"♦❜❧❡♠) ❛"❡ ❤✐❞❞❡♥ ❤❡"❡✿ %❤❡ ♠✉%✉❛❧ )♦❧✉❜✐❧✐%② ♦❢ %❤❡ ♣❤❛)❡)✱ ❡✳❣✳ ❧✐@✉✐❞ ❡✈❛♣♦"❛%✐♦♥ ✐♥ %❤❡ ❣❛) ♣❤❛)❡✱ ❛♥❞ %❤❡ %✐♠❡ )❝❛❧❡✳ ❚❤❡ ❝♦♥❞❡♥)❛%✐♦♥ ♦❢ ✇❛%❡" ♦♥ ❧❡❛✈❡) ❛❧)♦ ✐♠♣❧✐❡) %❤❛% %❤❡② ❛"❡ )❧✐❣❤%❧② ❝♦❧❞❡" %❤❛♥ ❛✐"✳ ❚❤❡"❡❢♦"❡✱ ♦✉" )✐♠♣❧❡ ♦❜)❡"✈❛%✐♦♥) ❛"❡ ♥♦% )♦ )✐♠♣❧❡ %♦ ❛♥❛❧②③❡✳ ▼♦"❡♦✈❡"✱ ❤❛✈✐♥❣ ❛ ❞"♦♣ )✐%%✐♥❣ ♦♥ %❤❡ )✉❜)%"❛%❡ ❞♦❡) ♥♦% ♠❡❛♥ %❤❛% %❤❡ )✉❜)%"❛%❡ ✐) ❞"② ❛"♦✉♥❞✳ ❚❤❡"❡ ✐) ❛❧✇❛②) ❛ ♠✐❝"♦)❝♦♣✐❝ ✜❧♠✱ ♠♦"❡ ♦" ❧❡)) ❞❡♥)❡✱ ✇❤✐❝❤ ✐) ❛❞)♦"❜❡❞✳ ❚❤❡"❡ ✐) %❤❡♥ ❛ %❤✐"❞ ♣"♦❜❧❡♠✿ %❤❡ ❛♥❛❧②)✐) ♠✉)% %❛❦❡ ✐♥%♦ ❛❝❝♦✉♥% ❛❧❧ )♣❛%✐❛❧ )❝❛❧❡)✱ ❢"♦♠ ♠❛❝"♦)❝♦♣✐❝ %♦ ♠✐❝"♦)❝♦♣✐❝ ♦♥❡)✳ ❚❤✐) ✐♥%"♦❞✉❝%✐♦♥ %♦ ✇❡%%✐♥❣ ♣❤❡♥♦♠❡♥❛ ✇✐❧❧ ♥♦% ❜❡ ❡①❤❛✉)%✐✈❡✳ ❲❡ )❤❛❧❧ ❢♦❝✉) ♦♥ ✇❡❧❧✲❞❡✜♥❡❞ )✐%✉❛%✐♦♥)✱ ✇❤❡"❡ ❡@✉❛%✐♦♥) ❝❛♥ ❜❡ ❝♦♥✈❡♥✐❡♥%❧② ✇"✐%%❡♥✳ ▲❡% ✉) ❣✐✈❡ ❢✉"%❤❡" ❝♦♠♠❡♥%)✳ ❚❤❡"♠♦❞②♥❛♠✐❝ ❡@✉✐❧✐❜"✐✉♠ "❡@✉✐"❡) ❛❧❧ ♣❤❛)❡) %♦ ❜❡ ♠✉%✉❛❧❧② )❛%✉"❛%❡❞✳ ❍♦✇❡✈❡"✱ ❛) ♠❡✲ ❝❤❛♥✐❝❛❧ ❛♥❞ %❤❡"♠❛❧ ❡@✉✐❧✐❜"✐✉♠ ❛"❡ ✉)✉❛❧❧② ♠✉❝❤ ❢❛)%❡" %❤❛♥ ❝❤❡♠✐❝❛❧ ❡@✉✐❧✐❜"✐✉♠✱ ✐% ✐) ♦❢%❡♥ ♣♦))✐❜❧❡ %♦ ♥❡❣❧❡❝% %❤❡ ♠✉%✉❛❧ )♦❧✉❜✐❧✐%② ♦❢ %❤❡ ♣❤❛)❡) ❛❧❧ ♦✈❡" %❤❡ ❡①♣❡"✐♠❡♥%✿ %❤✐) ✐) "❡❢❡""❡❞ %♦ ❛) ✧✐♥✐%✐❛❧✧ ❝♦♥❞✐%✐♦♥)✳ ❚❤❡ )♣❛%✐❛❧ )❝❛❧❡) ❛"❡ ❞❡✜♥❡❞ ✐♥ "❡❢❡"❡♥❝❡ %♦ %❤❡ "❛♥❣❡ ♦❢ ✐♥%❡"❛❝%✐♦♥)✳ ■♥ ✐♥%❡"❢❛❝❡ )❝✐❡♥❝❡✱ ❛ "❡❧❡✈❛♥% )❝❛❧❡ ✐) %❤❡ "❛♥❣❡ l ♦❢ %❤❡ ✐♥%❡"❛❝%✐♦♥ ❜❡%✇❡❡♥ ✐♥✜♥✐%❡ ❤❛❧❢✲)♣❛❝❡)✷ )❡♣❛"❛%❡❞ ❜② ❛ %❤✐"❞ ♣❤❛)❡✱ %❤❛% ✐)✱ ✐♥%❡"❛❝%✐♦♥ ❛❝"♦)) ❛ ✢❛% ✜❧♠✸✳ ❚❤❡ ♠❛❝"♦)❝♦♣✐❝ )❝❛❧❡ ✐) ♠✉❝❤ ❧❛"❣❡" %❤❛♥ l✳ ❆% %❤❡ ♠❡)♦)❝♦♣✐❝ )❝❛❧❡ ≈ l✱ %❤❡ ♠❡❞✐✉♠ ✐) ❝♦♥%✐♥✉♦✉)✳ ❚❤❡ ♠♦❧❡❝✉❧❛" )❝❛❧❡ ❞♦❡) ♥♦% "❡@✉✐"❡ )♣❡❝✐✜❝ ❞❡✜♥✐%✐♦♥✳

✶✳✶✳✷ ❚♦%❛❧ ✇❡%%✐♥❣✱ ♣❛/%✐❛❧ ✇❡%%✐♥❣✱ ❈♦♥%❛❝% ❛♥❣❧❡

❆) ❞❡)❝"✐❜❡❞ ❜❡❢♦"❡✱ ✇❤❡♥ ❧✐@✉✐❞ ✐) ❞❡♣♦)✐%❡❞ ♦♥ ❛ )✉❜)%"❛%❡ ✐% ❝❛♥ ❡✐%❤❡" )♣"❡❛❞ ♦" ❢♦"♠ ❛ ❞"♦♣✳ ❲❤❡♥ %❤❡ ❧✐@✉✐❞ )♣"❡❛❞)✱ %❤❡ )✐%✉❛%✐♦♥ ✐) ❝❛❧❧❡❞ %♦%❛❧ ✇❡%%✐♥❣✳ ❲❤❡♥ %❤❡ ❧✐@✉✐❞ ❢♦"♠) ❛ ❞"♦♣✱ %❤❡ )✐%✉❛%✐♦♥ ✐) ❝❛❧❧❡❞ ♣❛"%✐❛❧ ✇❡%%✐♥❣✳ ■♥ %❤❡ ❧❛%%❡" ❝❛)❡✱ ❧♦♦❦✐♥❣ ❛% ❛ ♠❛❝"♦)❝♦♣✐❝ ❞"♦♣ ❢"♦♠ %❤❡ )✐❞❡ ❡♥❛❜❧❡) ♦♥❡ %♦ ♦❜)❡"✈❡ %❤❛% %❤❡ ❧✐@✉✐❞ ♠❛❦❡) ❛♥ ❛♥❣❧❡ ✇✐%❤ %❤❡ )✉❜)%"❛%❡✳ ■% ✐) ❝❛❧❧❡❞ %❤❡ ❝♦♥%❛❝% ❛♥❣❧❡ ✭)❡❡ ✜❣✉"❡ ✶✳✷✮✳ ❚❤✐) ❛♥❣❧❡ )%❛②) %❤❡ )❛♠❡ ❢♦" ❡❛❝❤ %❤"❡❡✲❝♦♠♣♦♥❡♥% )②)%❡♠✱ ❧✐@✉✐❞ ✴ )✉❜)%"❛%❡ ✴ ❛♠❜✐❡♥% ❡♥✈✐"♦♥♠❡♥%✱ ✷l✐! ♦❢ $❤❡ ♦'❞❡' ♦❢ ❤✉♥❞'❡❞ ♦❢ ♥❛♥♦♠❡$❡'! ❢♦' ❱❛♥ ❞❡' ❲❛❛❧! ✐♥$❡'❛❝$✐♦♥✳■$ ✐! ✇♦'$❤ ♥♦$✐♥❣ $❤❛$ ❞'♦♣❧❡$!✱ ♥❛♥♦♠❡$'✐❝ ❜♦$❤ ✐♥ '❛❞✐✉! ❛♥❞ $❤✐❝❦♥❡!!✱ ❝❛♥♥♦$ ❜❡ $'❡❛$❡❞ ❛! $❤❡ ❥✉①$❛♣♦✲ !✐$✐♦♥ ♦❢ !❧✐❝❡! ♦❢ ✢❛$ ✜❧♠!✳ ❚❤❡ ❝❤❛'❛❝$❡'✐!$✐❝ ❧❡♥❣$❤ ♦❢ $❤❡ '❛♥❣❡ ♦❢ ✐♥$❡'❛❝$✐♦♥ ✐! ♠✉❝❤ !❤♦'$❡' $❤❛♥ l✳

(10)

♥♦ ♠❛$$❡& ✇❤❛$ $❤❡ ✈♦❧✉♠❡ ♦❢ ❞❡♣♦/✐$❡❞ ❧✐1✉✐❞ ✐/ ❛♥❞ ♣&♦✈✐❞❡❞ $❤❡ ❞&♦♣ ✐/ ♠❛❝&♦/❝♦♣✐❝✹✳ ❚❤❡

❝♦♥$❛❝$ ❛♥❣❧❡ ❝❛♥ $❛❦❡ ❛♥② ✈❛❧✉❡ ✐♥ $❤❡ &❛♥❣❡ 0◦ $♦ 180◦ ❛♥❞ $❤❡&❡❢♦&❡ ❛❧❧♦✇/ $♦ ❞❡/❝&✐❜❡ ❛♥②

♣♦//✐❜❧❡ ♣❛&$✐❛❧✲✇❡$$✐♥❣ /✐$✉❛$✐♦♥✳

❆/ ✇❛$❡& ✐/ $❤❡ ♠♦/$ ❝♦♠♠♦♥ ❧✐1✉✐❞✱ $❤❡&❡ ✐/ ❛ /♣❡❝✐✜❝ $❡&♠✐♥♦❧♦❣② ❢♦& ✇❛$❡& ❞&♦♣❧❡$/ ❞❡✲ ♣♦/✐$❡❞ ♦♥ ❛ /✉❜/$&❛$❡ ✐♥ ❛✐&✳ ■❢ $❤❡ ❝♦♥$❛❝$ ❛♥❣❧❡ ✐/ ❧❡// $❤❛♥ 90◦✱ $❤❡ /✉❜/$&❛$❡ ✐/ ❝❛❧❧❡❞

❤②❞&♦♣❤✐❧✐❝ ❛♥❞ ✐❢ $❤❡ ❝♦♥$❛❝$ ❛♥❣❧❡ ✐/ ♠♦&❡ $❤❛♥ 90◦✱ ✐$ ✐/ ❝❛❧❧❡❞ ❤②❞&♦♣❤♦❜✐❝✳ ❲❤❡♥ $❤❡ ❝♦♥$❛❝$

❛♥❣❧❡ ❝♦♠❡/ ❝❧♦/❡ $♦ 180◦✱ $❤❡ /✉❜/$&❛$❡ ✐/ ❝❛❧❧❡❞ /✉♣❡&❤②❞&♦♣❤♦❜✐❝✳ ❙♦♠❡ /✉♣❡&❤②❞&♦♣❤♦❜✐❝

/✉❜/$&❛$❡/ ❛&❡ 1✉✐$❡ /♣❡❝✐✜❝ ❛/ $❤❡② ❛&❡ ❝♦✈❡&❡❞ ✇✐$❤ ♣✐❧❧❛&/✳ ❚❤❡ ♣✐❧❧❛&✬/ /❤❛♣❡ ❛♥❞ $❤❡ ❢&❛❝$✐♦♥ ♦❢ $❤❡ /✉&❢❛❝❡ ♦❢ $❤❡ /✉❜/$&❛$❡ ❝♦✈❡&❡❞ ❜② $❤❡ ♣✐❧❧❛&/ ❝♦♥$&♦❧/ $❤❡ /✉♣❡&❤②❞&♦♣❤♦❜✐❝ ❜❡❤❛✈✐♦&✳

❘❡$✉&♥✐♥❣ $♦ $❤❡ ❝♦♠♠❡♥$/ ✐♥ ✶✳✶✳✶✱ ✐$ ❝♦♠❡/ $❤❛$✿

• ❈♦♥$❛❝$ ❛♥❣❧❡/ ❝❛♥ ❜❡ ❞❡✜♥❡❞ ✉♥❛♠❜✐❣✉♦✉/❧② ♦♥❧② ❛$ $❤❡&♠♦❞②♥❛♠✐❝ ❡1✉✐❧✐❜&✐✉♠✱ ♦& ✐♥ $❤❡ ✧✐♥✐$✐❛❧✧ /✐$✉❛$✐♦♥ ✐❢ ✐$ ✐/ ♣&♦♣❡&❧② ❝♦♥$&♦❧❧❡❞✱

• ❚❤❡② ❛&❡ ❞❡✜♥❡❞ ❛$ $❤❡ ♠❛❝&♦/❝♦♣✐❝ /❝❛❧❡✳

✶✳✶✳✸ ■♥%❡'❢❛❝✐❛❧ %❡♥-✐♦♥

■♥$❡&❢❛❝✐❛❧ $❡♥/✐♦♥ ✐/ ❛ ♠❛❝&♦/❝♦♣✐❝ ❡①❝❡// ❢&❡❡ ❡♥❡&❣② ♣❡& ✉♥✐$ ❛&❡❛ ❞❡✜♥❡❞ ❜❡$✇❡❡♥ $✇♦ ✐♥✜✲ ♥✐$❡ ♣❤❛/❡/ ❛$ $❤❡&♠♦❞②♥❛♠✐❝ ❡1✉✐❧✐❜&✐✉♠✳ ❑❡❡♣✐♥❣ ✐♥ ♠✐♥❞ $❤❡ ♣&❡✈✐♦✉/ ❝♦♠♠❡♥$/✱ ✇❡ /❤❛❧❧ ♣&❡/❡♥$ $❤❡ ❜❛/✐❝ ❡1✉❛$✐♦♥/ ✐♥ ❛♥ ✐❞❡❛❧ /✐$✉❛$✐♦♥ ✇❤❡&❡ ♣❤❛/❡/ ❛&❡ /$&✐❝$❧② ✐♠♠✐/❝✐❜❧❡✱ ❛✈♦✐❞✐♥❣ $❤❡ ❛♠❜✐❣✉✐$② ❜❡$✇❡❡♥ ✐♥✐$✐❛❧ ❛♥❞ ❡1✉✐❧✐❜&✐✉♠ 1✉❛♥$✐$✐❡/✳ ❚❤❡ ♣♦✐♥$ ✇✐❧❧ ❜❡ ❞✐/❝✉//❡❞ ❧❛$❡& ✇❤❡♥ &❡1✉✐&❡❞✳

▲❡$ ✉/ ❝♦♥/✐❞❡& $✇♦ ✐♠♠✐/❝✐❜❧❡ ♣❤❛/❡/ ❛$ ❡1✉✐❧✐❜&✐✉♠✳ ❍❛✈✐♥❣ ❛♥ ✐♥$❡&❢❛❝❡ ♠❡❛♥/ $❤❛$ $❤❡ ❝♦♥$❛❝$ ✐♥ ❜❡$✇❡❡♥ $❤❡ ♠♦❧❡❝✉❧❡/ ♦❢ $❤❡ $✇♦ ❞✐✛❡&❡♥$ ♣❤❛/❡/ ✐/ ♥♦$ ❢❛✈♦&❛❜❧❡✳ ❇&✐♥❣✐♥❣ ♠♦❧❡❝✉❧❡/ ✹❚❤❡ ♠❛❝&♦(❝♦♣✐❝ ❞&♦♣ ✐( ❛ (♣❤❡&✐❝❛❧ (❤❛♣❡ ✐❢ ❣&❛✈✐0② ✐( ♥❡❣❧✐❣✐❜❧❡✳ ❚❤✐( ♠❡❛♥( 0❤❛0 0❤❡ ❧❡♥❣0❤ (❝❛❧❡ ♦❢ 0❤❡

❞&♦♣✱ ✐✳❡✳ ✐0✬( &❛❞✐✉(✱ ❤❛( 0♦ ❜❡ (♠❛❧❧❡& 0❤❛♥ 0❤❡ ❝❛♣✐❧❧❛&② ❧❡♥❣0❤✱ κ−1

✳ ■0 ❝♦&&❡(♣♦♥❞( 0♦ 0❤❡ ❧❡♥❣0❤ (❝❛❧❡ ✇❤❡&❡ 0❤❡ ❤②❞&♦(0❛0✐❝ ♣&❡((✉&❡ ❝♦♠♣❡♥(❛0❡( ❡①❛❝0❧② 0❤❡ ❝❛♣✐❧❧❛&② ♣&❡((✉&❡✿ γ

κ−1 = ρgκ

−1

✇❤❡&❡ γ ✐( 0❤❡ (✉&❢❛❝❡ 0❡♥(✐♦♥ ❀ ρ ✐( 0❤❡ ❞❡♥(✐0② ♦❢ 0❤❡ ❧✐=✉✐❞ ❛♥❞ g 0❤❡ ❛❝❝❡❧❡&❛0✐♦♥ ♦❢ ❣&❛✈✐0②✳ ❚❤❡ ❝❛♣✐❧❧❛&② ❧❡♥❣0❤ ❝❛♥ 0❤❡&❡❢♦&❡ ❜❡ ✇&✐00❡♥ ❛( κ−1

=qρgγ ❀ κ−1

≈ 3 ♠♠ ❢♦& ✇❛0❡&✳

T

❋✐❣✉&❡ ✶✳✷✿ ❚❤❡ ✐❞❡❛❧ ❝❛/❡ ♦❢ $❤❡ /♣❤❡&✐❝❛❧ /❤❛♣❡❞ ❞&♦♣❧❡$ ✭♣❛&$✐❛❧ ✇❡$$✐♥❣✱ $❤❡ ❞&♦♣❧❡$ ✐/ /♠❛❧❧ ❡♥♦✉❣❤ $♦ ❛✈♦✐❞ ❣&❛✈✐$② ❡✛❡❝$/ ❛♥❞ $❤❡ /✉❜/$&❛$❡ ✐/ /♠♦♦$❤✮✳ ❖♥ $❤❡ ❧❡❢$✱ 3D✳ ❖♥ $❤❡ &✐❣❤$✿ $❤❡ ❞&♦♣❧❡$ ✐/ /❡❡♥ ❢&♦♠ $❤❡ /✐❞❡✱ $❤❡ $❛♥❣❡♥$ $♦ $❤❡ /♣❤❡&✐❝❛❧ /❤❛♣❡ ❛$ $❤❡ ❥✉♥❝$✐♦♥ ✇✐$❤ $❤❡ /✉❜/$&❛$❡ ❞❡✜♥❡/ $❤❡ ❝♦♥$❛❝$ ❛♥❣❧❡✱ θ

(11)

❛! !❤❡ ✐♥!❡&❢❛❝❡ !❤✉* ❝♦*!* ❛ ❝❡&!❛✐♥ ❛♠♦✉♥! ♦❢ ❡♥❡&❣②✳ ❚❤✐* ❝♦*! ✐* &❡♣&❡*❡♥!❡❞ ❜② !❤❡ ✐♥!❡&❢❛❝✐❛❧ !❡♥*✐♦♥✱ γ✳ ■! ✐* ✐♥❞❡❡❞ !❤❡ ❡♥❡&❣② ♥❡❡❞❡❞ !♦ ✐♥❝&❡❛*❡ ✐♥ ❛ &❡✈❡&*✐❜❧❡ ✇❛② !❤❡ ❛&❡❛ ♦❢ ❛♥ ✐♥!❡&❢❛❝❡ ❜② ♦♥❡ ✉♥✐! ❛! ❣✐✈❡♥ !❡♠♣❡&❛!✉&❡ ❛♥❞ ♣❤❛*❡ ✈♦❧✉♠❡*✳ ❚♦ ❞❡*❝&✐❜❡ !❤✐* ♣&♦❝❡**✱ !❤❡ ❢&❡❡ *✉&❢❛❝❡ ❡♥❡&❣②✱ F ✱ ✐* ✐♥!&♦❞✉❝❡❞✿

F = γA ✭✶✳✶✮

✇❤❡&❡ A !❤❡ ❛&❡❛ ♦❢ !❤❡ ✢❛! ✐♥!❡&❢❛❝❡ ❛♥❞ γ ✐* !❤❡ *✉&❢❛❝❡ !❡♥*✐♦♥ ✭✐♥ J.m−2 ♦& N.m−1✮✳ ❚❤❡

✐♥!❡&❢❛❝✐❛❧ !❡♥*✐♦♥ ✐* !❤❡ ❡①❝❡** ❢&❡❡ ❡♥❡&❣② ♣❡& ✉♥✐! ❛&❡❛ ❛**♦❝✐❛!❡❞ !♦ !❤❡ ✐♥!❡&❢❛❝✐❛❧ ③♦♥❡✱ ✇❤✐❝❤ ♣&♦♣❡&!✐❡* ❞✐✛❡& ❢&♦♠ !❤❡ ♦♥❡* ♦❢ !❤❡ ❜✉❧❦ ♣❤❛*❡* ✭❬✶✾❪✱ ❬✸✷❪✮✳ ■! ❝❛♥ ❜❡ ❝❛❧❝✉❧❛!❡❞ ❡①♣❧✐❝✐!❧② ✐❢ !❤❡ ✐♥!❡&❛❝!✐♦♥ ♣♦!❡♥!✐❛❧* ❜❡!✇❡❡♥ ♠♦❧❡❝✉❧❡* ❛&❡ ❦♥♦✇♥✳

❋♦& ❧✐H✉✐❞ ❞❡♣♦*✐!❡❞ ♦♥ ❛ *✉❜*!&❛!❡✱ ❛* ♠❡♥!✐♦♥❡❞ ❜❡❢♦&❡✱ !❤❡&❡ ❛&❡ !❤&❡❡ ♣❤❛*❡* !♦ !❛❦❡ ✐♥!♦ ❛❝❝♦✉♥!✿ !❤❡ ❧✐H✉✐❞✱ !❤❡ *✉❜*!&❛!❡ ❛♥❞ !❤❡ ❛♠❜✐❡♥! ❡♥✈✐&♦♥♠❡♥!✳ ❖♥❡ ♠✉*! !❤❡♥ ❝♦♥*✐❞❡& !❤&❡❡ ✐♥!❡&❢❛❝✐❛❧ !❡♥*✐♦♥*✿

• γsg✱ !❤❡ *✉&❢❛❝❡ !❡♥*✐♦♥ ❛**♦❝✐❛!❡❞ !♦ !❤❡ ✐♥!❡&❢❛❝❡ *✉❜*!&❛!❡ ✴ ❛♠❜✐❡♥! ❡♥✈✐&♦♥♠❡♥!✳ ❚❤❡

❛♠❜✐❡♥! ❡♥✈✐&♦♥♠❡♥! ✇✐❧❧ ❜❡ *②♠❜♦❧✐③❡❞ ❜② ❛ ✏❣✑✱ ❛* ✐♥ !❤❡ ♣&❡*❡♥! ❝❛*❡ ✐! ✐* ❛ ❣❛* ♣❤❛*❡✳ • γlg✱ !❤❡ ✐♥!❡&❢❛❝✐❛❧ !❡♥*✐♦♥ ❛**♦❝✐❛!❡❞ !♦ !❤❡ ✐♥!❡&❢❛❝❡ ❧✐H✉✐❞ ♣❤❛*❡ ✴ ❛♠❜✐❡♥! ❡♥✈✐&♦♥♠❡♥!✺✳

• γls✱ !❤❡ ✐♥!❡&❢❛❝✐❛❧ !❡♥*✐♦♥ ❛**♦❝✐❛!❡❞ !♦ !❤❡ ✐♥!❡&❢❛❝❡ ❧✐H✉✐❞ ♣❤❛*❡ ✴ *✉❜*!&❛!❡✳

❊❛❝❤ ✐♥!❡&❢❛❝✐❛❧ !❡♥*✐♦♥ ✐* ❞❡✜♥❡❞ ❢♦& ❛ ❝♦✉♣❧❡ ♦❢ !✇♦ ♣❤❛*❡* ✐♥ !❤❡ ❛❜*❡♥❝❡ ♦❢ !❤❡ !❤✐&❞ ♦♥❡ ✭✇❤✐❝❤ ❝❛✉*❡* ♥♦ ❛♠❜✐❣✉✐!② ✐♥ !❤❡ ✐❞❡❛❧ ❝❛*❡ ❝♦♥*✐❞❡&❡❞ ❤❡&❡ ♦❢ *!&✐❝!❧② ✐♠♠✐*❝✐❜❧❡ ♣❤❛*❡*✮✳ ❚❤❡ ✐♥!❡&❢❛❝✐❛❧ !❡♥*✐♦♥ ❜❡!✇❡❡♥ ❛ ❞❡♥*❡ ♣❤❛*❡✱ ❧✐H✉✐❞ ♦& *♦❧✐❞✱ ❛♥❞ ❛ ❣❛* ♣❤❛*❡ ✐* ❝♦♠♠♦♥❧② ❝❛❧❧❡❞ *✉&❢❛❝❡ !❡♥*✐♦♥✳ ❆* ♠❡♥!✐♦♥❡❞ ❛❜♦✈❡✱ *✉&❢❛❝❡ !❡♥*✐♦♥ ❝♦♠❡* ❢&♦♠ !❤❡ ❜❛❧❛♥❝❡ ♦❢ ✐♥!❡&❛❝!✐♦♥* ✐♥ ❜❡!✇❡❡♥ !❤❡ ♠♦❧❡❝✉❧❡* ✐♥ !❤❡ ♣❤❛*❡*✳ ❆♠♦♥❣ !❤❡*❡ ✐♥!❡&❛❝!✐♦♥*✱ ♦♥❡ ♠❛② ♠❡♥!✐♦♥ ❱❛♥ ❞❡& ❲❛❛❧* ❢♦&❝❡*✱ ❤②❞&♦❣❡♥ ❜♦♥❞* ✭✐♥ !❤❡ ❝❛*❡ ♦❢ ✇❛!❡& ❢♦& ❡①❛♠♣❧❡✮✱ ✐♦♥✐❝ ❜♦♥❞* ♦& ♠❡!❛❧❧✐❝ ❜♦♥❞* ✭✐♥ !❤❡ ❝❛*❡ ♦❢ ❧✐H✉✐❞ ♠❡!❛❧* ❧✐❦❡ ♠❡&❝✉&② ❢♦& ❡①❛♠♣❧❡✮✳

✶✳✶✳✹ ❨♦✉♥❣✬) ❧❛✇

f(

J)

sg

f(

J)

ls

f(

J)

lg

T

x

y

❋✐❣✉&❡ ✶✳✸✿ ❙✐❞❡ ✈✐❡✇ ♦❢ ❛ ♠❛❝&♦*❝♦♣✐❝ ❞&♦♣✳ ❚❤❡ ❢♦&❝❡* ✐♥❞✉❝❡❞ ❜② ✐♥!❡&❢❛❝✐❛❧ !❡♥*✐♦♥* ♦♥ !❤❡ !❤&❡❡✲♣❤❛*❡ ❝♦♥!❛❝! ❧✐♥❡ ❛&❡ ❞&❛✇♥ ❛* ❛&&♦✇*✳ γsg✱γlg❛♥❞ γls*!❛♥❞ &❡*♣❡❝!✐✈❡❧② ❢♦& !❤❡ ✐♥!❡&❢❛❝✐❛❧

!❡♥*✐♦♥ ♦❢ !❤❡ ❣❛* ♣❤❛*❡ ✈✳*✳ !❤❡ *✉❜*!&❛!❡✱ ♦❢ !❤❡ ❧✐H✉✐❞ ♣❤❛*❡ ✈✳*✳ !❤❡ ❣❛* ♣❤❛*❡ ❛♥❞ ♦❢ !❤❡ ❧✐H✉✐❞ ♣❤❛*❡ ✈✳*✳ !❤❡ *✉❜*!&❛!❡

❆! !❤❡ ❛♠❜✐❡♥! !❡♠♣❡*❛!✉*❡✱ ❛♥ ♦*❞❡* ♦❢ ♠❛❣♥✐!✉❞❡ ♦❢ γlg ✐1✿ ≈ 20 ♠◆✴♠ ❢♦* ♠♦1! ♦✐❧1✱ ≈ 72 ♠◆✴♠ ❢♦*

✇❛!❡* ❜❡❝❛✉1❡ ♦❢ ❤②❞*♦❣❡♥ ❜♦♥❞1✱ ≈ 500 ♠◆✴♠ ❢♦* ♠❡*❝✉*② ❜❡❝❛✉1❡ !❤❡ ❝♦❤❡1✐♦♥ ✇✐!❤✐♥ !❤✐1 ❧✐9✉✐❞ ♠❡!❛❧ ✐1 ✈❡*② 1!*♦♥❣✳

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■♥ "❤❡ ✐❞❡❛❧ ❝❛*❡ ♦❢ *"-✐❝"❧② ♥♦♥ ♠✐*❝✐❜❧❡ ♣❤❛*❡*✱ "❤❡ -❡❧❛"✐♦♥*❤✐♣ ❜❡"✇❡❡♥ "❤❡ ✐♥"❡-❢❛❝✐❛❧ "❡♥*✐♦♥* ❛♥❞ "❤❡ ❝♦♥"❛❝" ❛♥❣❧❡ ❝❛♥ ❜❡ ✇-✐""❡♥ ✉♥❛♠❜✐❣✉♦✉*❧②✳ ❖♥ *♦❧✐❞✱ ♥♦♥ ❞❡❢♦-♠❛❜❧❡ *✉❜*"-❛"❡*✱ "❤✐* -❡❧❛"✐♦♥*❤✐♣ ✐* ❝❛❧❧❡❞ ❨♦✉♥❣✬* ❧❛✇✳ ❚✇♦ ❞✐✛❡-❡♥" ❛♣♣-♦❛❝❤❡* ❝❛♥ ❜❡ ✉*❡❞ "♦ ✐♥"-♦❞✉❝❡ ✐"✳ ❚❤❡ ✜-*" ♦♥❡ ✐* ❜❛*❡❞ ♦♥ ❢♦-❝❡* ❛♥❞ "❤❡ *❡❝♦♥❞ ♦♥❡ ♦♥ ❡♥❡-❣②✳ ❚❤❡② ✇✐❧❧ ❜❡ ♣-❡*❡♥"❡❞ ✐♥ "❤❡ ✷❉ ❝❛*❡ ♦❢ ❛ ❧✐?✉✐❞ ✇❡❞❣❡ ♦♥ "❤❡ *♦❧✐❞ *✉❜*"-❛"❡✳ ■♥ "❤❡ ✜-*" ❛♣♣-♦❛❝❤✱ "❤❡ ❢♦-❝❡* ❛❝"✐♥❣ ♣❡- ❧❡♥❣"❤ ✉♥✐" ♦♥ "❤❡ ❝♦♥"❛❝" ❧✐♥❡ ❛-❡ ❝♦✉♥"❡❞ ✭*❡❡ ✜❣✉-❡ ✶✳✸✮✳ ❖♥ ❛ ✢❛"✱ ♥♦♥ ❞❡❢♦-♠❛❜❧❡ *✉❜*"-❛"❡✱ ❛* "❤❡ ❞-♦♣ ✐* ❛" "❤❡ ❡?✉✐❧✐❜-✐✉♠✱ "❤❡ ❝❛♥❝❡❧❧❛"✐♦♥ ♦❢ "❤❡ "❤-❡❡ ❢♦-❝❡* ❝♦♠♣♦♥❡♥"* ♣❡- ❧❡♥❣"❤ ✉♥✐" ❛❧♦♥❣ "❤❡ *✉❜*"-❛"❡ ❧❡❛❞* "♦✿ −fsg+ flg∗ cosθ + fls = 0 ✭✶✳✷✮ ❨♦✉♥❣✬* ❧❛✇ ✐* ♦❜"❛✐♥❡❞✿ −γsg+ γlg∗ cosθ + γls= 0 or cosθ = γsg− γls γlg ✭✶✳✸✮ ❚❤❡ *❡❝♦♥❞ ❛♣♣-♦❛❝❤ -❡❧✐❡* ♦♥ "❤❡ ❝❛❧❝✉❧❛"✐♦♥ ♦❢ "❤❡ -❡✈❡-*✐❜❧❡ ✈✐-"✉❛❧ ✇♦-❦✱ dW ❞♦♥❡ ❜② ❛ ♠♦✈✐♥❣ ❧✐?✉✐❞ ✇❡❞❣❡ ❛❤❡❛❞ ♦❢ "❤❡ ❝♦♥"❛❝" ❧✐♥❡ ❜② ❛ *♠❛❧❧ ❞✐*"❛♥❝❡✱ dx✱ ❝♦--❡*♣♦♥❞✐♥❣ "♦ ❛ *♠❛❧❧ ❝❤❛♥❣❡ dA ♦❢ "❤❡ ❛-❡❛ ❝♦✈❡-❡❞ ♦♥ "❤❡ *✉❜*"-❛"❡✳ ❚❤❡ -❡✈❡-*✐❜❧❡ ✇♦-❦ ❞♦♥❡ ✐* "❤❡ ❝❤❛♥❣❡ ✐♥ ❢-❡❡ ❡♥❡-❣② ❛♥❞ ❝❛♥ ❜❡ ❡①♣-❡**❡❞ ❛*✿ dW = −γsg∗ dA + γlg∗ cosθ ∗ dA + γls∗ dA ✭✶✳✹✮ ❆" "❤❡ ❡?✉✐❧✐❜-✐✉♠✱ dW dA = 0 0 = −γsg+ γlg∗ cosθ + γls ✭✶✳✺✮ ❲❤✐❝❤ ✐* ❛❣❛✐♥ "❤❡ ❡①♣-❡**✐♦♥ ♦❢ ❨♦✉♥❣✬* ❧❛✇✳ ❨♦✉♥❣✬* ❧❛✇ ✐* ❛ ♣♦✇❡-❢✉❧ "♦♦❧ ❛* ✐" ❧✐♥❦* "❤❡ ❡?✉✐❧✐❜-✐✉♠ ❝♦♥"❛❝" ❛♥❣❧❡ ♦❢ ❛ ❞-♦♣ "♦ "❤❡ "❤-❡❡ ✐♥"❡-❢❛❝✐❛❧ "❡♥*✐♦♥* ✐♥✈♦❧✈❡❞✳ ❍♦✇❡✈❡-✱ ♦♥❡ ♠✉*" ❦❡❡♣ ✐♥ ♠✐♥❞ "❤❛" -❡✈❡-*✐❜✐❧✐"② ✐* ❛**✉♠❡❞ ✐♥ "❤❡ ❞❡-✐✈❛"✐♦♥ ✭❡✈❡♥ ✐❢ ✐" ✐* ✐♠♣❧✐❝✐" ✐♥ "❤❡ ❢♦-❝❡ ❛♣♣-♦❛❝❤✮✳ ▼♦-❡♦✈❡-✱ "❤❡ ✉♥-❡❛❧✐*"✐❝ ❛**✉♠♣"✐♦♥ ♦❢ ❝♦♠♣❧❡"❡ ✐♠♠✐*❝✐❜✐❧✐"② ♠✉*" ♥♦✇ ❜❡ -❡❝♦♥*✐❞❡-❡❞✿ "✇♦ ♣❤②*✐❝❛❧❧② ✇❡❧❧✲❞❡✜♥❡❞ *✐"✉❛"✐♦♥* ❛-❡✱ (i)✧✐♥✐"✐❛❧ ❝♦♥❞✐"✐♦♥*✧ ✐♠♣❧②✐♥❣ "❤❡ ❛❝❤✐❡✈❡♠❡♥" ♦❢ ♠❡❝❤❛♥✐❝❛❧ ❛♥❞ "❤❡-♠❛❧ ❡?✉✐❧✐❜-✐✉♠✱ ❜✉" ❛ ♥❡❣❧✐❣✐❜❧❡ "-❛♥*❢❡- ♦❢ ♠♦❧❡❝✉❧❡* ❜❡"✇❡❡♥ ♣❤❛*❡*✱ ❛♥❞ (ii) "❤❡ "❤❡-♠♦❞②♥❛♠✐❝ ❡?✉✐❧✐❜-✐✉♠✳

✶✳✶✳✺ ■♥✐&✐❛❧ )♣+❡❛❞✐♥❣ ♣❛+❛♠❡&❡+✱ &❤❡+♠♦❞②♥❛♠✐❝ ❡5✉✐❧✐❜+✐✉♠

✏■♥✐$✐❛❧ ❝♦♥❞✐$✐♦♥*✑ ✐♠♣❧②✐♥❣ $❤❡ ❛❝❤✐❡✈❡♠❡♥$ ♦❢ ♠❡❝❤❛♥✐❝❛❧ ❛♥❞ $❤❡4♠❛❧ ❡5✉✐❧✐❜4✐✉♠✱ ❜✉$ ❛ ♥❡❣❧✐❣✐❜❧❡ $4❛♥*❢❡4 ♦❢ ♠♦❧❡❝✉❧❡* ❜❡$✇❡❡♥ ♣❤❛*❡*

❚❤❡ ✐♥✐"✐❛❧ *♣-❡❛❞✐♥❣ ♣❛-❛♠❡"❡-✱ S0✱ ❝♦♠♣❛-❡* "❤❡ ❢-❡❡ ❡♥❡-❣② ♣❡- ❛-❡❛ ✉♥✐" ♦❢ ❛ *✉❜*"-❛"❡ ❝♦✈❡-❡❞

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♣❤❛#❡✳ ❚❤✐# ✐# ❛ ♠❛❝*♦#❝♦♣✐❝ ,✉❛♥/✐/②✳ S0=∆F A = γsg− (γlg+ γls) ✭✶✳✻✮ ❚❤❡ #✐❣♥ ♦❢ /❤❡ ✐♥✐/✐❛❧ #♣*❡❛❞✐♥❣ ♣❛*❛♠❡/❡* ❛❧❧♦✇# ♦♥❡ /♦ ♣*❡❞✐❝/ ✐❢ ❛ ♠❛❝*♦#❝♦♣✐❝ ❧✐,✉✐❞ ❞*♦♣ ❥✉#/ ❞❡♣♦#✐/❡❞ ♦♥ /❤❡ #✉❜#/*❛/❡ ✇✐❧❧ #♣*❡❛❞ ❝♦♠♣❧❡/❡❧② ♦* ♥♦/✳ ■❢ S0 ✐# ♣♦#✐/✐✈❡✱ γsg > γlg+ γls✱ ❧✐,✉✐❞ ❝♦✈❡*✐♥❣ /❤❡ #✉❜#/*❛/❡ ✐# ♠♦*❡ ❢❛✈♦*❛❜❧❡ /❤❛♥ ❤❛✈✐♥❣ /❤❡ #✉❜#/*❛/❡ ✐♥ ❞✐*❡❝/ ❝♦♥/❛❝/ ✇✐/❤ /❤❡ ❛♠❜✐❡♥/ ❡♥✈✐*♦♥♠❡♥/✳ ❚❤❡ ❞*♦♣ #♣*❡❛❞#✿ ✐/ ✐# /❤❡ ❝❛#❡ ♦❢ /♦/❛❧ ✇❡//✐♥❣✳ ❖♥ /❤❡ ❝♦♥/*❛*②✱ ✐❢ S0 ✐# ♥❡❣❛/✐✈❡✱ γsg < γlg+ γls✱ ❧✐,✉✐❞ ❝♦✈❡*✐♥❣ /❤❡ #✉❜#/*❛/❡ ✐# ❧❡## ❢❛✈♦*❛❜❧❡ /❤❛♥ ❤❛✈✐♥❣ /❤❡ #✉❜#/*❛/❡ ✐♥ ❞✐*❡❝/ ❝♦♥/❛❝/ ✇✐/❤ /❤❡ ❛♠❜✐❡♥/ ❡♥✈✐*♦♥♠❡♥/✳ ❚❤❡ ❞*♦♣ ❞♦❡# ♥♦/ #♣*❡❛❞✳ ❚❤✐# ✐# /❤❡ ❝❛#❡ ♦❢ ♣❛*/✐❛❧ ✇❡//✐♥❣✳ ■/ ✐# ✉#✉❛❧ /♦ ❞❡✜♥❡ /✇♦ ❝❛/❡❣♦*✐❡# ♦❢ #✉❜#/*❛/❡#✿ ❤✐❣❤✲❡♥❡*❣② #✉❜#/*❛/❡# ❛♥❞ ❧♦✇✲❡♥❡*❣② #✉❜#/*❛/❡#✳ ❋♦* ❝❧❡❛♥ ❤✐❣❤✲❡♥❡*❣② #✉❜#/*❛/❡# ✭❡✳❣✳ ❝❧❡❛♥ ❣❧❛##✮✱ /❤❡ #✉*❢❛❝❡ /❡♥#✐♦♥ ✐# ✈❡*② ❤✐❣❤ γsg ≈ 500 /♦ 5000 ♠◆✴♠✳ ❚❤✐# ❦✐♥❞ ♦❢ #✉❜#/*❛/❡# ❝♦♥/❛✐♥# ✐♦♥✐❝✱ ❝♦✈❛❧❡♥/ ♦* ♠❡/❛❧❧✐❝ ❜♦♥❞#✳ ▼♦#/ ❧✐,✉✐❞# #♣*❡❛❞ ♦♥ /❤❡♠ ✭#❡❡ ❡,✉❛/✐♦♥ ✶✳✻✮✳ ❋♦* ❧♦✇✲❡♥❡*❣② #✉❜#/*❛/❡# ✭❡✳❣✳ /❡✢♦♥✮✱ γsg ≈ 10 /♦ 20 ♠◆✴♠✱ /❤❡② ❤❛✈❡ ❛ ♣♦♦* ✇❡//❛❜✐❧✐/②✳ ❚❤✐# ❦✐♥❞ ♦❢ #✉❜#/*❛/❡# ❛*❡ ♠♦#/❧② ♠♦❧❡❝✉❧❛* ❝*②#/❛❧# ♦* ♣❧❛#/✐❝#✳ ❚❤❡#♠♦❞②♥❛♠✐❝ ❡,✉✐❧✐❜#✐✉♠ ❆/ /❤❡ /❤❡*♠♦❞②♥❛♠✐❝ ❡,✉✐❧✐❜*✐✉♠✱ /❤❡ ♣❤❛#❡# ❛*❡ ♠✉/✉❛❧❧② #❛/✉*❛/❡❞✱ ✇❤✐❝❤ ♠❡❛♥# /❤❛/ /❤❡ ❣❛# ♣❤❛#❡ ✐# #❛/✉*❛/❡❞ ✇✐/❤ ❡✈❛♣♦*❛/❡❞ ❧✐,✉✐❞ ♠♦❧❡❝✉❧❡# ✭❛♥❞ /❤❛/ /❤❡ ❧✐,✉✐❞ ✐# #❛/✉*❛/❡❞ ✇✐/❤ ❣❛# ❀ /❤❡ #♦❧✐❞ ✐# ♥♦/ ❝♦♥#✐❞❡*❡❞✮✳ ■♥ ♣❛*/✐❛❧ ✇❡//✐♥❣✱ ❛ ♠✐❝*♦#❝♦♣✐❝ ✜❧♠ ♦❢ ❧✐,✉✐❞ ✐# ❛❞#♦*❜❡❞ ♦♥ /❤❡ #♦❧✐❞ ❛♥❞ ✐# ❛ ♣❛*/ ♦❢ /❤❡ #♦❧✐❞✲❣❛# ✐♥/❡*❢❛❝❡✱ ✇❤✐❝❤ #✉*❢❛❝❡ /❡♥#✐♦♥ ♠❛② #✐❣♥✐✜❝❛♥/❧② ❞✐✛❡* ❢*♦♠ /❤❡ ♦♥❡ ♦❢ /❤❡ ✧❞*②✧ #♦❧✐❞✳ ❚❤❡*❡❢♦*❡✱ /❤❡ ✐♥✐/✐❛❧ #♣*❡❛❞✐♥❣ ♣❛*❛♠❡/❡* S0 ❞✐✛❡*# ❢*♦♠ /❤❡ ❡,✉✐❧✐❜*✐✉♠ ♦♥❡ S✳ ■♥ ❝♦♠♣❧❡/❡ ✇❡//✐♥❣✱ ❛ ♠❛❝*♦#❝♦♣✐❝ ✜❧♠ ♦❢ ❧✐,✉✐❞ ✐# ♣*❡#❡♥/ ♦♥ /❤❡ #♦❧✐❞ ❛♥❞ ✐# ❛ ♣❛*/ ♦❢ /❤❡ #♦❧✐❞✲❣❛# ✐♥/❡*❢❛❝❡✳ ❚❤❡ ❡,✉✐❧✐❜*✐✉♠ #♣*❡❛❞✐♥❣ ♣❛*❛♠❡/❡* ✐# ③❡*♦ ✭S0> 0✱ S = 0✮✳ ❙/*✐❦✐♥❣ ❡①❛♠♣❧❡# ♦❢ ❛♥ ✐♥✐/✐❛❧ #♣*❡❛❞✐♥❣ ✭S0 > 0✮ ❢♦❧❧♦✇❡❞ ❜② ❛ *❡/*❛❝/✐♦♥ ✭S < 0✮✱ ❛# ❜❡♥③❡♥❡ ♦♥ ✇❛/❡*✱ ❛*❡ ✇❛*♥✐♥❣# ❛❣❛✐♥#/ ✉♥❝♦♥/*♦❧❧❡❞ ✉#❡ ♦❢ ❢♦*♠✉❧❛❡ ❛/ ✐♥/❡*❢❛❝❡#✳

✶✳✶✳✻ ❈♦♥&❛❝& ❛♥❣❧❡ ❤②.&❡/❡.✐.

❙♦ ❢❛*✱ /❤❡ #✉❜#/*❛/❡# ❤❛✈❡ ❜❡❡♥ ❝♦♥#✐❞❡*❡❞ /♦ ❜❡ ✐❞❡❛❧❧② #♠♦♦/❤ ❛♥❞ ❤♦♠♦❣❡♥❡♦✉#✱ ❛# ❢♦* /❤❡ ❞❡*✐✈❛/✐♦♥ ♦❢ ❨♦✉♥❣✬# ❧❛✇✱ ❢♦* ✇❤✐❝❤ *❡✈❡*#✐❜✐❧✐/② ✐# ❛##✉♠❡❞✳ ❍♦✇❡✈❡*✱ ♠♦#/ #♦❧✐❞ #✉❜#/*❛/❡# ❛*❡ ♥♦/✳ ❘♦✉❣❤♥❡## ♦* ❝❤❡♠✐❝❛❧ ❤❡/❡*♦❣❡♥❡✐/✐❡# ❝❛♥ ✐♥❞✉❝❡ ✐**❡❣✉❧❛*✐/✐❡# ❛♥❞ ❡✈❡♥ ♣✐♥♥✐♥❣ ♦❢ /❤❡ /❤*❡❡✲♣❤❛#❡ ❝♦♥/❛❝/ ❧✐♥❡ ✭#❡❡ ✜❣✉*❡ ✶✳✹✮✳ ■♥ /❤❡ ♣*❡#❡♥❝❡ ♦❢ ✇❡//❛❜❧❡ ❞❡❢❡❝/#✱ ✇❤❡♥ /❤❡ ♠✉/✉❛❧ ❛✣♥✐/② ❜❡/✇❡❡♥ /❤❡ ❧✐,✉✐❞ ❛♥❞ /❤❡ ❞❡❢❡❝/# ✐# #/*♦♥❣❡* /❤❛♥ /❤❡ ❛✈❡*❛❣❡ ✈❛❧✉❡ ♦♥ /❤❡ #✉❜#/*❛/❡✱ ❛ *❡❝❡❞✐♥❣ ❝♦♥/❛❝/ ❧✐♥❡ /❡♥❞# /♦ #/❛② ❧♦♥❣❡* ♦♥ /❤❡ ❞❡❢❡❝/#✳ ❖♥ /❤❡ ❝♦♥/*❛*②✱ ✇❤❡♥ /❤❡ ❛✣♥✐/② ❜❡/✇❡❡♥ /❤❡ ❧✐,✉✐❞ ❛♥❞ /❤❡ ❞❡❢❡❝/# ✐# ✇❡❛❦❡*✱ ❛♥ ❛❞✈❛♥❝✐♥❣ ❝♦♥/❛❝/ ❧✐♥❡ ❤❛# ❞✐✣❝✉❧/✐❡# /♦ ❝♦✈❡* /❤❡♠✳ ❉❡❢❡❝/# ❝❛♥ ♣❧❛② ❛/ /❤❡ ♠✐❝*♦#❝♦♣✐❝ #❝❛❧❡✱ ✐♥ ✇❤✐❝❤ ❝❛#❡ /❤❡ ♠❛❝*♦#❝♦♣✐❝ ❝♦♥/❛❝/ ❧✐♥❡ ❧♦♦❦#

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ADVANCING

RECEDING

favorable defect

non-favorable defect

❋✐❣✉$❡ ✶✳✹✿ ❙❝❤❡♠❛/✐❝ /♦♣ ✈✐❡✇ ♦❢ /❤❡ ♠✐❝$♦5❝♦♣✐❝ ♣✐♥♥✐♥❣ ♦❢ /❤❡ ❝♦♥/❛❝/ ❧✐♥❡ ♦♥ ❛ ❤❡/❡$♦❣❡♥❡♦✉5 5✉❜5/$❛/❡ ✐♥ ♣$❡5❡♥❝❡ ♦❢ ❢❛✈♦$❛❜❧❡ ❛♥❞ ♥♦♥✲❢❛✈♦$❛❜❧❡ ❞❡❢❡❝/5✳ ❲❤✐/❡✿ 5✉❜5/$❛/❡❀ ❣$❡②✿ ❧✐>✉✐❞✳ ❚❤❡ ❞♦//❡❞ ❧✐♥❡5 $❡♣$❡5❡♥/ /❤❡ ♣♦5✐/✐♦♥ ♦❢ /❤❡ ❝♦♥/❛❝/ ❧✐♥❡ ✐♥ /❤❡ ❛❜5❡♥❝❡ ♦❢ ❞❡❢❡❝/5✳ ❚♦♣✿ ❚❤❡ ❛❞✈❛♥❝✐♥❣ ❝♦♥/❛❝/ ❧✐♥❡ ✐5 /$❛♥5✐/♦$✐❧② ♣✐♥♥❡❞ ♦♥ /❤❡ ♥♦♥✲❢❛✈♦$❛❜❧❡ ③♦♥❡5 ❛♥❞ ♠❡$❡❧② ❝♦✈❡$5 /❤❡ ❢❛✈♦$❛❜❧❡ ♦♥❡5✳ ❇♦//♦♠✿ ❚❤❡ $❡❝❡❞✐♥❣ ❝♦♥/❛❝/ ❧✐♥❡ 5/❛②5 /$❛♥5✐/♦$✐❧② ♣✐♥♥❡❞ ✐♥ /❤❡ ❢❛✈♦$❛❜❧❡ ③♦♥❡5 ❛♥❞ ♠❡$❡❧② ❡5❝❛♣❡5 ❢$♦♠ /❤❡ ♥♦♥✲❢❛✈♦$❛❜❧❡ ♦♥❡5✳ ❚❤❡$❡ ✐5 ❡❧❛5/✐❝✐/② ♦❢ /❤❡ ❝♦♥/❛❝/ ❧✐♥❡ ❛♥❞ ❛5 >✉♦/❡❞ ✐♥ /❤❡ /❡①/✱ /❤✐5 5❝❤❡♠❛/✐❝ ✜❣✉$❡ $❡❢❡$5 /♦ ✇❡❛❦ ♣✐♥♥✐♥❣ ❬✷✹❪✳ 5♠♦♦/❤✳ ❚❤✐5 ✐5 /❤❡ ❝❛5❡ ❝♦♥5✐❞❡$❡❞ ❤❡$❡ ✭❬✷✹❪✮✳ ❚❤❡ ❣❡♥❡$❛❧ ♣$♦❜❧❡♠ ♦❢ ❤②5/❡$❡5✐5 ✐5 ♠✉❝❤ ♠♦$❡ ❝♦♠♣❧❡① ✭❬✶❪✮✳ ❚❤❡5❡ ♠✐❝$♦5❝♦♣✐❝ ♣$♦❝❡55❡5 ❤❛✈❡ ❛ ♠❛❝$♦5❝♦♣✐❝ ❝♦♥5❡>✉❡♥❝❡ ♦♥ /❤❡ ❝♦♥/❛❝/ ❛♥❣❧❡ ♦❢ ❞$♦♣5✿ ♣♦♦$❧② ✇❡//❛❜❧❡ ❞❡❢❡❝/5 ✐♥❝$❡❛5❡ /❤❡ ❝♦♥/❛❝/ ❛♥❣❧❡ ❛/ ❛♥ ❛❞✈❛♥❝✐♥❣ ❝♦♥/❛❝/ ❧✐♥❡✱ ✇❤✐❧❡ ❤✐❣❤❧② ✇❡//❛❜❧❡ ❞❡❢❡❝/5 ❞❡❝$❡❛5❡ /❤❡ ❝♦♥/❛❝/ ❛♥❣❧❡ ❛/ ❛ $❡❝❡❞✐♥❣ ❝♦♥/❛❝/ ❧✐♥❡✳ ▲❡/ ✉5 /❛❦❡ /❤❡ ❡①❛♠♣❧❡ ♦❢ ❛ ❞$♦♣ 5✐//✐♥❣ ♦♥ ❛ 5✉❜5/$❛/❡✳ ■❢ /❤❡ 5✉❜5/$❛/❡ ✐5 5❧✐❣❤/❧② /✐❧/❡❞✱ /❤❡ ❞$♦♣ ❞♦❡5 ♥♦/ $♦❧❧✱ ✐/ ❞❡❢♦$♠5✳ ❚❤✐5 ✐5 ❜❡❝❛✉5❡ /❤❡ /❤$❡❡✲♣❤❛5❡ ❝♦♥/❛❝/ ❧✐♥❡ ✐5 ♣✐♥♥❡❞✳ ❆5 ❧♦♥❣ ❛5 /❤✐5 ♣✐♥♥✐♥❣ ✐5 5/$♦♥❣ ❡♥♦✉❣❤ /♦ ❝♦✉♥/❡$❛❝/ /❤❡ ❣$❛✈✐/② ❢♦$❝❡ ✐♥❞✉❝❡❞ ❜② /✐❧/✐♥❣ /❤❡ 5✉❜5/$❛/❡✱ /❤❡ ❞$♦♣ 5/❛②5 5/✐❧❧✳ ■❢ /❤❡ /✐❧/✐♥❣ ❛♥❣❧❡ ♦❢ /❤❡ 5✉❜5/$❛/❡ ✐5 /❤❡♥ ✐♥❝$❡❛5❡❞ ✉♥/✐❧ /❤❡ ❞$♦♣ ❜❡❣✐♥5 /♦ ♠♦✈❡✱ ✐/ ✐5 ♣♦55✐❜❧❡ /♦ ♦❜5❡$✈❡

T

a

T

r

❋✐❣✉$❡ ✶✳✺✿ ❍②5/❡$❡5✐5 ♦❢ /❤❡ ❝♦♥/❛❝/ ❛♥❣❧❡✿ ❚❤❡ ♣$♦✜❧❡ ♦❢ ❛ ❞$♦♣ $♦❧❧✐♥❣ ✇✐/❤ ♥❡❣❧✐❣✐❜❧❡ ✈❡❧♦❝✐/② ♦♥ ❛ /✐❧/❡❞ 5✉❜5/$❛/❡ 5❤♦✇5 /✇♦ ❞✐✛❡$❡♥/ ❝♦♥/❛❝/ ❛♥❣❧❡5✱ /❤❡ ❛❞✈❛♥❝✐♥❣ ❝♦♥/❛❝/ ❛♥❣❧❡ θa ❛/ /❤❡ ❢$♦♥/ ❛♥❞ /❤❡ $❡❝❡❞✐♥❣ ❝♦♥/❛❝/ ❛♥❣❧❡ θr ❛/ /❤❡ $❡❛$ ❀ θa> θ > θr✳ ❲✐/❤ ✐♥❝$❡❛5✐♥❣ ❞$♦♣ ✈❡❧♦❝✐/② /❤❡ ❛♥❣❧❡ ❛/ /❤❡ ❢$♦♥/ ✐♥❝$❡❛5❡5 ❛♥❞ /❤❡ ❛♥❣❧❡ ❛/ /❤❡ $❡❛$ ❞❡❝$❡❛5❡5✳

Figure

Figure 4. Series of nCBs on a glycerol substrate. χ = 2π(d/λ) is measured at a given distance from the NI transition
Figure 5. Same as before on a water substrate. An experimental difficulty is to avoid convection in the heated water substrate
Figure 8. Adapted from ref 8. The control parameter κ depends on the elastic constants of the nematic film and will be defined in the main text
Figure 11. Check of the process controlling the dynamics of coalescence. For a collection of coalescing pairs with similar thickness, the relaxation time scales like the square of the radius, which proves that viscous dissipation in the bulk liquid substra
+7

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