Plastron Regeneration on Submerged Superhydrophobic Surfaces

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Surfaces, Interfaces, and Applications

Plastron Regeneration on Submerged Superhydrophobic Surfaces using in-situ Gas Generation by Chemical Reaction Divya Panchanathan, Anoop Rajappan, Kripa K Varanasi, and Gareth H. McKinley ACS Appl. Mater. Interfaces, Just Accepted Manuscript • DOI: 10.1021/acsami.8b12471 • Publication Date (Web): 05 Sep 2018 Downloaded from http://pubs.acs.org on September 6, 2018

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ACS Applied Materials & Interfaces

P❧❛str♦♥ ❘❡❣❡♥❡r❛t✐♦♥ ♦♥ ❙✉❜♠❡r❣❡❞ ❙✉♣❡r❤②❞r♦♣❤♦❜✐❝ ❙✉r❢❛❝❡s ✉s✐♥❣ ✐♥✲s✐t✉ ●❛s ●❡♥❡r❛t✐♦♥ ❜② ❈❤❡♠✐❝❛❧ ❘❡❛❝t✐♦♥ ❉✐✈②❛ P❛♥❝❤❛♥❛t❤❛♥✱ ❆♥♦♦♣ ❘❛❥❛♣♣❛♥✱ ❑r✐♣❛ ❑✳ ❱❛r❛♥❛s✐✱ ❛♥❞ ●❛r❡t❤ ❍✳

∗

▼❝❑✐♥❧❡②

❉❡♣❛rt♠❡♥t ♦❢ ▼❡❝❤❛♥✐❝❛❧ ❊♥❣✐♥❡❡r✐♥❣✱ ▼❛ss❛❝❤✉s❡tts ■♥st✐t✉t❡ ♦❢ ❚❡❝❤♥♦❧♦❣②✱ ❈❛♠❜r✐❞❣❡ ❊✲♠❛✐❧✿ ❣❛r❡t❤❅♠✐t✳❡❞✉

❑❡②✇♦r❞s✿ ♣❧❛str♦♥✱ ❞r❛❣ r❡❞✉❝t✐♦♥✱ ❤②❞r♦❣❡♥ ♣❡r♦①✐❞❡✱ ❝❛t❛❧②s✐s✱ s✉♣❡r❤②❞r♦♣❤♦❜✐❝✱ r❡✲ ❣❡♥❡r❛t✐♦♥✱ s❡❧❢✲r❡❝♦✈❡r②

❆❜str❛❝t ❙✉♣❡r❤②❞r♦♣❤♦❜✐❝ s✉r❢❛❝❡s s✉❜♠❡r❣❡❞ ✉♥❞❡r ✇❛t❡r ❛♣♣❡❛r s❤✐♥② ❞✉❡ t♦ t♦t❛❧ ✐♥t❡r✲ ♥❛❧ r❡✢❡❝t✐♦♥ ♦❢ ❧✐❣❤t ❢r♦♠ ❛ t❤✐♥ ❧❛②❡r ♦❢ ❛✐r ✭♣❧❛str♦♥✮ tr❛♣♣❡❞ ✐♥ t❤❡✐r s✉r❢❛❝❡ t❡①t✉r❡✳ ❚❤✐s ❡♥tr❛♣♣❡❞ ❛✐r ✐s ❛❞✈❛♥t❛❣❡♦✉s ❢♦r ❢r✐❝t✐♦♥❛❧ ❞r❛❣ r❡❞✉❝t✐♦♥ ✐♥ ✈❛r✐♦✉s ❛♣♣❧✐❝❛t✐♦♥s r❛♥❣✐♥❣ ❢r♦♠ ♠✐❝r♦✢✉✐❞✐❝ ❝❤❛♥♥❡❧s t♦ ♠❛r✐♥❡ ✈❡ss❡❧s✳ ❍♦✇❡✈❡r✱ t❤❡s❡ ❛❡r♦♣❤✐❧✐❝ t❡①t✉r❡s ❛r❡ ♣r♦♥❡ t♦ ✐♠♣r❡❣♥❛t✐♦♥ ❜② ✇❛t❡r ❞✉❡ t♦ t✉r❜✉❧❡♥t ♣r❡ss✉r❡ ✢✉❝t✉❛t✐♦♥s ❢r♦♠ ❡①t❡r✲ ♥❛❧ ✢♦✇s ❛♥❞ ❞✐ss♦❧✉t✐♦♥ ♦❢ t❤❡ tr❛♣♣❡❞ ❣❛s ✐♥t♦ t❤❡ ✇❛t❡r✳ ❲❡ ❞❡♠♦♥str❛t❡ ❛ ♥♦✈❡❧ ❝❤❡♠✐❝❛❧ ♠❡t❤♦❞ t♦ r❡♣❧❡♥✐s❤ t❤❡ ♣❧❛str♦♥ ✐♥✲s✐t✉ ❜② ✉s✐♥❣ t❤❡ ❞❡❝♦♠♣♦s✐t✐♦♥ r❡❛❝t✐♦♥ ♦❢ ❤②❞r♦❣❡♥ ♣❡r♦①✐❞❡ ♦♥ s✉♣❡r❤②❞r♦♣❤♦❜✐❝ s✉r❢❛❝❡s ♣r❡♣❛r❡❞ ✇✐t❤ ❛ ❝❛t❛❧②t✐❝ ❝♦❛t✐♥❣✳ ❲❡ ❛❧s♦ ♣r♦✈✐❞❡ ❛ t❤❡r♠♦❞②♥❛♠✐❝ ❢r❛♠❡✇♦r❦ ❢♦r ❞❡s✐❣♥✐♥❣ s✉♣❡r❤②❞r♦♣❤♦❜✐❝ s✉r❢❛❝❡s ✇✐t❤ ♦♣t✐♠❛❧ t❡①t✉r❡ ❛♥❞ ❝❤❡♠✐str② ❢♦r ✉♥❞❡r✇❛t❡r ♣❧❛str♦♥ r❡❣❡♥❡r❛t✐♦♥✳ ❲❡ ✜♥❛❧❧② ❞❡♠♦♥str❛t❡ t❤❡ ♣r❛❝t✐❝❛❧ ✉t✐❧✐t② ♦❢ t❤✐s ♠❡t❤♦❞ ❜② ❢❛❜r✐❝❛t✐♥❣ ♣❡r✐♦❞✐❝ ♠✐❝r♦t❡①t✉r❡s ✶

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ACS Applied Materials & Interfaces 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

♦♥ ❛❧✉♠✐♥✉♠ s✉r❢❛❝❡s t❤❛t ✐♥❝♦r♣♦r❛t❡ ❛ ❝❤❡❛♣ ❝❛t❛❧②st✱ ♠❛♥❣❛♥❡s❡ ❞✐♦①✐❞❡✳ ❲❡ ♣❡r✲ ❢♦r♠ ❞r❛❣ r❡❞✉❝t✐♦♥ ❡①♣❡r✐♠❡♥ts ✉♥❞❡r t✉r❜✉❧❡♥t ✢♦✇ ❝♦♥❞✐t✐♦♥s ✐♥ ❛ ❚❛②❧♦r✲❈♦✉❡tt❡ ❝❡❧❧ ✭❚❈ ❝❡❧❧✮ ✇❤✐❝❤ s❤♦✇ t❤❛t ♠♦r❡ t❤❛♥ ❤❛❧❢ ♦❢ t❤❡ ❞r❛❣ ✐♥❝r❡❛s❡ ❡♥s✉✐♥❣ ❢r♦♠ ♣❧❛str♦♥ ❝♦❧❧❛♣s❡ ❝❛♥ ❜❡ r❡❝♦✈❡r❡❞ s♣♦♥t❛♥❡♦✉s❧② ❜② ✐♥❥❡❝t✐♦♥ ♦❢ ❞✐❧✉t❡ ❍2 ❖2 ✐♥t♦ t❤❡ ❚❈ ❝❡❧❧✳ ❚❤✉s✱ ✇❡ ♣r❡s❡♥t ❛ ❧♦✇✲❝♦st s❝❛❧❛❜❧❡ ♠❡t❤♦❞ t♦ ❡♥❛❜❧❡ ✐♥✲s✐t✉ ♣❧❛str♦♥ r❡❣❡♥❡r❛t✐♦♥ ♦♥ ❧❛r❣❡ s✉r❢❛❝❡s ❢♦r ♠❛r✐♥❡ ❛♣♣❧✐❝❛t✐♦♥s✳

■♥tr♦❞✉❝t✐♦♥ ❙✉♣❡r❤②❞r♦♣❤♦❜✐❝ s✉r❢❛❝❡s ❛r❡ ✇❡❧❧ ❦♥♦✇♥ ❢♦r t❤❡✐r ♣r♦♣❡rt✐❡s ♦❢ s❡❧❢✲❝❧❡❛♥✐♥❣✱ ✶ ❢♦❣ ❝♦❧❧❡❝✲ t✐♦♥✱ ✷ ❛♥t✐✲✐❝✐♥❣✱ ✸ ❛♥❞ ❛♥t✐✲❜✐♦❢♦✉❧✐♥❣✳ ✹ ❚❤❡ ❞❡s✐❣♥ ♦❢ s②♥t❤❡t✐❝ s✉♣❡r❤②❞r♦♣❤♦❜✐❝ s✉r❢❛❝❡s ✐s ♦❢t❡♥ ✐♥s♣✐r❡❞ ❜② ♥❛t✉r❛❧❧② ♦❝❝✉rr✐♥❣ ✇❛t❡r r❡♣❡❧❧❡♥t s✉r❢❛❝❡s s✉❝❤ ❛s ❧♦t✉s ❧❡❛✈❡s ❛♥❞ ✇❛✲ t❡rstr✐❞❡r ❢❡❡t✳ ✺ ❚❤❡s❡ ♥❛t✉r❛❧❧② ♦❝❝✉rr✐♥❣ s✉r❢❛❝❡s ❤❛✈❡ ❛ ❤✐❡r❛r❝❤✐❝❛❧ t❡①t✉r❡ ✇✐t❤ ♠✐❝r♦✲ ❛♥❞ ♥❛♥♦s❝❛❧❡ ❢❡❛t✉r❡s✱ ❝♦✉♣❧❡❞ ✇✐t❤ ❛ ❤②❞r♦♣❤♦❜✐❝ ❝❤❡♠✐❝❛❧ ❝♦❛t✐♥❣ ✭❨♦✉♥❣✬s ❝♦♥t❛❝t ❛♥❣❧❡

θw > 90◦ ✮ ✇❤✐❝❤ ❝♦❧❧❡❝t✐✈❡❧② ✐♠♣❛rt t❤❡ ♣r♦♣❡rt② ♦❢ s✉♣❡r✲r❡♣❡❧❧❡♥❝② t♦✇❛r❞s ✇❛t❡r ✭❤❡r❡ s✉♣❡r✲r❡♣❡❧❧❡♥❝② ✐s t❛❦❡♥ t♦ ♠❡❛♥ ❛♣♣❛r❡♥t ❝♦♥t❛❝t ❛♥❣❧❡ θw∗ > 150◦ ✱ ❛♥❞ ❛ r♦❧❧✲♦✛ ❛♥❣❧❡

< 5 − 10◦ ✮✳ ✶✱✻✱✼ ❲❤❡♥ ❛ s✉♣❡r❤②❞r♦♣❤♦❜✐❝ s✉r❢❛❝❡ ✐s s✉❜♠❡r❣❡❞ ✉♥❞❡r✇❛t❡r✱ ❛ t❤✐♥ ❧❛②❡r ♦❢ ❛✐r ✐s tr❛♣♣❡❞ ✇✐t❤✐♥ ✐ts ♠✐❝r♦t❡①t✉r❡ ❝♦rr❡s♣♦♥❞✐♥❣ t♦ t❤❡ t❤r❡❡ ♣❤❛s❡ ❝♦♥✜❣✉r❛t✐♦♥ ❦♥♦✇♥ ❛s t❤❡ ❈❛ss✐❡ st❛t❡✳ ✽ ❍❡♥❝❡✱ t❤❡s❡ t❡①t✉r❡s ❛r❡ ❛❧s♦ ❦♥♦✇♥ ❛s ✉♥❞❡r✇❛t❡r s✉♣❡r❛❡r♦♣❤✐❧✐❝ s✉r❢❛❝❡s✳ ✾✱✶✵ ❚❤❡ ❛✐r ❡♥tr❛♣♣❡❞ ✇✐t❤✐♥ t❤❡ t❡①t✉r❡ ✐s t❡r♠❡❞ ❛ ♣❧❛str♦♥ ✐♥ t❤❡ ❜✐♦❧♦❣✐❝❛❧ ❝♦♠✲ ♠✉♥✐t② ❛♥❞ t❤✐s ♣❤❡♥♦♠❡♥❛ ✐s ♦❜s❡r✈❡❞ ✐♥ ♥❛t✉r❡ ❛♠♦♥❣ ♠❛♥② ✐♥s❡❝ts ❛♥❞ ♣❧❛♥ts ✇❤✐❝❤ ❤❛✈❡ ❡✈♦❧✈❡❞ t❤❡ ❛❜✐❧✐t② t♦ ❜r❡❛t❤❡ ✉♥❞❡r✇❛t❡r✳ ✶✶✕✶✸ ■♥ ❛❞❞✐t✐♦♥ t♦ ♣r❡✈❡♥t✐♥❣ ✇❛t❡r ❢r♦♠ ✇❡tt✐♥❣ t❤❡ ✉♥❞❡r❧②✐♥❣ s✉❜str❛t❡✱ t❤❡ ♣❧❛str♦♥ ❛❝ts ❛s ❛ ❜❛rr✐❡r t♦✇❛r❞s ❝♦rr♦s✐♦♥ ♦♥ ♠❡t❛❧ s✉r❢❛❝❡s ✶✹ ❛♥❞ t❤❡ s✉r❢❛❝❡ ❛✐r ❧❛②❡r r❡❞✉❝❡s t❤❡ s❦✐♥ ❢r✐❝t✐♦♥ ❞r❛❣ ❡①♣❡r✐❡♥❝❡❞ ❜② s✉♣❡r❤②✲ ❞r♦♣❤♦❜✐❝ s✉r❢❛❝❡s ✉♥❞❡r✇❛t❡r✳ ✶✺✱✶✻ ❈♦♥t✐♥✉❡❞ st❛❜✐❧✐t② ♦❢ t❤❡ ❈❛ss✐❡ st❛t❡ ✐s ❡ss❡♥t✐❛❧ ❢♦r ❡①♣❧♦✐t✐♥❣ t❤❡ ❞❡s✐r❛❜❧❡ ♣r♦♣❡rt✐❡s ♦❢ ✉♥❞❡r✇❛t❡r s✉♣❡r❛❡r♦♣❤✐❧✐❝ s✉r❢❛❝❡s✳ ❇✉t t❤❡ ❛✐r ✐♥ t❤✐s t❤r❡❡✲♣❤❛s❡ st❛t❡ ✐s ❡❛s✐❧② ❞✐s♣❧❛❝❡❞ ❜② ✇❛t❡r ✉♥❞❡r ♣r❡ss✉r❡ ✢✉❝t✉❛t✐♦♥s✱ s❤❡❛r ✢♦✇✱ ❛♥❞ ❧♦✇ ❞✐ss♦❧✈❡❞ ❣❛s ❝♦♥❝❡♥tr❛t✐♦♥s ✶✼✕✷✶ ✷


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❝❛✉s✐♥❣ t❤❡♠ t♦ tr❛♥s✐t✐♦♥ ✐♥t♦ t❤❡ ❲❡♥③❡❧ ♦r ❢✉❧❧②✲✇❡tt❡❞ st❛t❡✳ ✷✷ ■♥ ♦r❞❡r ❢♦r s✉❝❤ ❛ s✉r✲ ❢❛❝❡ t♦ r❡❝♦✈❡r ✐ts ❞r❛❣✲r❡❞✉❝✐♥❣ ❛♥❞ ❛♥t✐✲❝♦rr♦s✐♦♥ ♣r♦♣❡rt✐❡s✱ ✐t ✐s ♥❡❝❡ss❛r② t♦ ✐♥❞✉❝❡ ❛ ❲❡♥③❡❧✲t♦✲❈❛ss✐❡ st❛t❡ tr❛♥s✐t✐♦♥ ♦♥ t❤❡ t❡①t✉r❡❞ s✉r❢❛❝❡ ✇❤✐❧❡ ✐t r❡♠❛✐♥s s✉❜♠❡r❣❡❞ ✐♥ t❤❡ ❧✐q✉✐❞ ♣❤❛s❡✳ ❲❡♥③❡❧✲t♦✲❈❛ss✐❡ st❛t❡ tr❛♥s✐t✐♦♥s ♦♥ t❡①t✉r❡❞ s✉r❢❛❝❡s ❤❛✈❡ ❜❡❡♥ ❛❝❤✐❡✈❡❞ ❢♦r s❡ss✐❧❡ ✇❛t❡r ❞r♦♣❧❡ts ✐♥ ❛✐r ✉s✐♥❣ ❡❧❡❝tr♦✇❡tt✐♥❣✱ ✷✸✱✷✹ ❜♦✐❧✐♥❣✱ ✷✺ ♠♦❞✉❧❛t✐♦♥ ♦❢ t❤❡ ▲❛♣❧❛❝❡ ♣r❡ss✉r❡✱ ✷✻ ✈✐❜r❛t✐♦♥✱ ✷✼ ❛♥❞ ✈♦❧❛t✐❧❡ ❧♦✇ s✉r❢❛❝❡ t❡♥s✐♦♥ ❧✐q✉✐❞s✳ ✷✽ ❍♦✇❡✈❡r✱ t❤❡s❡ ♠❡t❤♦❞s ✭❡①❝❡♣t ❜♦✐❧✐♥❣✮ ❝❛♥♥♦t ❡❛s✐❧② ❜❡ ❛♣♣❧✐❡❞ ✇❤❡♥ t❤❡ s✉r❢❛❝❡ ✐s s✉❜♠❡r❣❡❞ ✉♥❞❡r✇❛t❡r ❜❡❝❛✉s❡ t❤❡r❡ ✐s ♥♦ s♦✉r❝❡ ♦❢ ❛✐r ❛✈❛✐❧❛❜❧❡ ✐♥ t❤❡ s✉rr♦✉♥❞✐♥❣s t♦ ❞✐s♣❧❛❝❡ t❤❡ ✇❛t❡r ✐♥ t❤❡ t❡①t✉r❡✳ ❚❤❡r❡❢♦r❡ ✐♥ ♦r❞❡r t♦ ❛❝t✐✈❡❧② r❡st♦r❡ ❛ ♣❧❛str♦♥ ❜② r❡♣❧❡♥✐s❤✐♥❣ t❤❡ ❈❛ss✐❡ st❛t❡✱ t❤❡r❡ ✐s ❛ ❝❧❡❛r ♥❡❡❞ t♦ ✐♠♣❧❡♠❡♥t ❛ ❣❛s ♦r ✈❛♣♦r s♦✉r❝❡ ✉♥❞❡r✇❛t❡r✳ ▼❡t❤♦❞s t❤❛t ❤❛✈❡ ❜❡❡♥ ❞❡✈❡❧✲ ♦♣❡❞ t♦ ❛❞❞r❡ss t❤✐s ♥❡❡❞ ❝❛♥ ❜❡ ❜r♦❛❞❧② ❝❧❛ss✐✜❡❞ ✐♥t♦ t❤r❡❡ ❝❛t❡❣♦r✐❡s✿ ✭✐✮ ✐♥✲s✐t✉ ❣❛s✴✈❛♣♦r ❣❡♥❡r❛t✐♦♥✱ ✭✐✐✮ ❣❛s ✐♥❥❡❝t✐♦♥✱ ❛♥❞ ✭✐✐✐✮ ✐♥❝r❡❛s✐♥❣ t❤❡ ❝♦♥❝❡♥tr❛t✐♦♥ ♦❢ ❞✐ss♦❧✈❡❞ ❣❛s ✐♥ t❤❡ ❝♦♥t❛❝t✐♥❣ ❧✐q✉✐❞✳ ❚❤❡ ✜rst ❛♣♣r♦❛❝❤ ✉s❡s ✐♥✲s✐t✉ ❣❡♥❡r❛t✐♦♥ ♦❢ ❣❛s✴✈❛♣♦r t♦ r❡♣❧❡♥✐s❤ t❤❡ ❣❛s❡♦✉s✴✈❛♣♦r ♣❤❛s❡ t❤❛t ✐♥✐t✐❛❧❧② ❢♦r♠❡❞ t❤❡ ❧♦st ♣❧❛str♦♥✳ ❚❤❡s❡ r❡❣❡♥❡r❛t✐✈❡ t❡❝❤♥✐q✉❡s ✐♥❝❧✉❞❡ ❡❧❡❝tr♦❧②✲ s✐s✱ ✷✾✱✸✵ ♣❤♦t♦❡❧❡❝tr♦❝❛t❛❧②t✐❝ ✇❛t❡r s♣❧✐tt✐♥❣✱ ✸✶ ❛♥❞ ✜❧♠ ❜♦✐❧✐♥❣✳ ✸✷✱✸✸ ▲❡❡ ❡t ❛❧✳ ❞❡♠♦♥str❛t❡❞

✐♥✲s✐t✉ ♣❧❛str♦♥ r❡❝♦✈❡r② ✉s✐♥❣ ❡❧❡❝tr♦❧②s✐s ♦♥ Pt✴❆✉ ❝♦❛t❡❞ s✐❧✐❝♦♥ ♣✐❧❧❛rs✳ ✷✾ ❆❧t❤♦✉❣❤ t❤✐s ♠❡t❤♦❞ ❤❛s t❤❡ ❛❞✈❛♥t❛❣❡ ♦❢ ❜❡✐♥❣ s❡❧❢✲❧✐♠✐t✐♥❣✱ ✐t ✐s ❡①♣❡♥s✐✈❡ t♦ ♣❡r❢♦r♠ ✇❛t❡r✲s♣❧✐tt✐♥❣ ✉s✐♥❣ ♥♦❜❧❡ ♠❡t❛❧s✳ ❖♥ t❤❡ ♦t❤❡r ❤❛♥❞✱ ✜❧♠ ❜♦✐❧✐♥❣ ❤❛s ♥♦ ♠❛t❡r✐❛❧ ❧✐♠✐t❛t✐♦♥s ❜✉t ✐s ❡♥❡r❣② ✐♥t❡♥s✐✈❡ t♦ ♣❡r❢♦r♠ ♦♥ ❛ ❧❛r❣❡ s❝❛❧❡✳ ✸✹ ❚❤❡ s❡❝♦♥❞ ♠❡t❤♦❞✱ ❣❛s ✐♥❥❡❝t✐♦♥✱ r❡❧✐❡s ♦♥ ♣r♦✈✐❞✲ ✐♥❣ ❛ ❝♦♥t✐♥✉♦✉s s✉♣♣❧② ♦❢ ❣❛s✴❛✐r t♦ t❤❡ ♣❧❛str♦♥✳ ❚❤✐s ❝❛♥ ❜❡ ❛❝❤✐❡✈❡❞ ✉s✐♥❣ ❛ ❝♦♥♥❡❝t❡❞

♣❧❛str♦♥ ✭♣❧❛str♦♥ ❝♦♥♥❡❝t❡❞ t♦ ❛♥ ❛✐r r❡s❡r✈♦✐r t❤r♦✉❣❤ ❛ ♣♦r♦✉s ♠✐❝r♦str✉❝t✉r❡✱ ✸✺ ❛ ❧❛r❣❡ ❜✉❜❜❧❡✱ ✸✻ ♦r ❛ r❡❣✐♦♥ ♦❢ ❝♦♠♣r❡ss❡❞ ❣❛s ✸✼ ✮✱ ♦r ❜② ✐♥❥❡❝t✐♥❣ ❣❛s ❜✉❜❜❧❡s ❞✐r❡❝t❧② ✐♥t♦ t❤❡ ❜♦✉♥❞❛r② ❧❛②❡r ♦❢ t❤❡ ✢♦✇✳ ✸✽ ❈♦♥♥❡❝t✐♥❣ ❛ ♣❧❛str♦♥ t♦ ❛♥ ❡①t❡r♥❛❧ r❡s❡r✈♦✐r ❞♦❡s ♥♦t r❡q✉✐r❡ ❛♥② ❡①♣❡♥s✐✈❡ ♠❛t❡r✐❛❧s ♦r ❡♥❡r❣② ✐♥♣✉t✱ ❜✉t ♥❡❡❞s ♣r❡❝✐s❡ ❝♦♥tr♦❧ ♦❢ ♣r❡ss✉r❡ ✐♥ t❤❡ r❡s❡r✈♦✐r t♦ ❢❛✈♦r ♣❧❛str♦♥ s♣r❡❛❞✐♥❣✳ ❆❧s♦✱ ❜✉❜❜❧❡s ✐♥❥❡❝t❡❞ ✐♥t♦ t❤❡ ❜♦✉♥❞❛r② ❧❛②❡r ❞♦ ♥♦t ❛❞❤❡r❡ t♦ t❤❡ s✉r❢❛❝❡ ✐♠♠❡❞✐❛t❡❧② ❞✉❡ t♦ t❤❡ ❧✉❜r✐❝❛t✐♦♥ ♣r❡ss✉r❡ ❡①❡rt❡❞ ♦♥ t❤❡ ❜✉❜❜❧❡ ❜② t❤❡ ✸



Page 4 of 35

❞r❛✐♥✐♥❣ ❧✐q✉✐❞ ✜❧♠✳ ✸✾ ❚❤❡s❡ ♠❡t❤♦❞s ❛r❡ t②♣✐❝❛❧❧② ♥♦t s❡❧❢✲❧✐♠✐t✐♥❣ ❛♥❞ ❛r❡ s✉s❝❡♣t✐❜❧❡ t♦ t❤❡ ❢♦r♠❛t✐♦♥ ♦❢ ❧❛r❣❡ ❜✉❜❜❧❡s t❤❛t r❡s✐❞❡ ♦♥ t❤❡ t❡①t✉r❡ ✐♥st❡❛❞ ♦❢ ❢♦r♠✐♥❣ ❛ t❤✐♥ ♣❧❛str♦♥ ❧❛②❡r ✇✐t❤✐♥ t❤❡ t❡①t✉r❡✳ ✸✽ ❚❤❡s❡ ♠❛❝r♦s❝♦♣✐❝ ❜✉❜❜❧❡s ✐♥❝r❡❛s❡ t❤❡ ❢♦r♠ ❞r❛❣ ❜② ❞✐sr✉♣t✐♥❣ t❤❡ ❜♦✉♥❞❛r② ❧❛②❡r✱ ❡✈❡♥ t❤♦✉❣❤ t❤❡ s❦✐♥ ❢r✐❝t✐♦♥ ❞r❛❣ ❝♦♥tr✐❜✉t✐♦♥ ✐s ❧♦✇❡r✳ ✹✵ ❚❤❡ ✜♥❛❧ ❛♣♣r♦❛❝❤ ✐s t♦ s❛t✉r❛t❡ t❤❡ ✇❛t❡r ✇✐t❤ ❞✐ss♦❧✈❡❞ ❛✐r✴❣❛s t♦ ❛❝❤✐❡✈❡ ♣❧❛str♦♥ r❡❝♦✈❡r② ❜② ❞✐❢✲ ❢✉s✐♦♥ t❤r♦✉❣❤ t❤❡ ❧✐q✉✐❞ ❛♥❞ ♠❛ss tr❛♥s❢❡r ❛t t❤❡ ✐♥t❡r❢❛❝❡✳ ✹✶✱✹✷ ❚❤✐s t❡❝❤♥✐q✉❡ r❡q✉✐r❡s t❤❡ ❣❛s ❝♦♥❝❡♥tr❛t✐♦♥ t♦ ❜❡ ❝♦♥tr♦❧❧❡❞ ❧♦❝❛❧❧② ♥❡❛r t❤❡ s✉r❢❛❝❡ ❜② ✐♥❝r❡❛s✐♥❣ t❤❡ t❡♠♣❡r❛t✉r❡ ♦r ❞❡❝r❡❛s✐♥❣ t❤❡ ♣r❡ss✉r❡ ♦❢ t❤❡ ✇❛t❡r ✐♥ s✐t✉✳ ■♥ t❤✐s ✇♦r❦✱ ✇❡ ❞❡♠♦♥str❛t❡ ❛ ♥♦✈❡❧ ❝❤❡♠✐❝❛❧ ♠❡t❤♦❞ t♦ r❡♣❧❡♥✐s❤ ❛✐r ❧♦ss ❢r♦♠ ❛♥ ❡st❛❜✲ ❧✐s❤❡❞ ♣❧❛str♦♥ ❜② ✉s✐♥❣ ❝❛t❛❧②t✐❝ ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ ❤②❞r♦❣❡♥ ♣❡r♦①✐❞❡✳ ❍②❞r♦❣❡♥ ♣❡r♦①✐❞❡ ✐s ❛♥ ✉♥st❛❜❧❡ ❝♦♠♣♦✉♥❞ ✇❤✐❝❤ s❧♦✇❧② ❞❡❝♦♠♣♦s❡s ✐♥t♦ ✇❛t❡r ❛♥❞ ♦①②❣❡♥ ✭❊q✉❛t✐♦♥ ✶✮ ❡✈❡♥ ❛t r♦♦♠ t❡♠♣❡r❛t✉r❡ ❛♥❞ ♣r❡ss✉r❡✳ ❝❛t❛❧②st✴❤❡❛t

✷❍2 ❖2 −−−−−−−→ ✷❍2 ❖ + ❖2

✭✶✮

❚❤✐s r❡❛❝t✐♦♥ ✐s s❧♦✇ ✉♥❞❡r r♦♦♠ t❡♠♣❡r❛t✉r❡ ❝♦♥❞✐t✐♦♥s ✭✇✐t❤ ❛ ❞❡❝♦♠♣♦s✐t✐♦♥ ❤❛❧❢✲ ❧✐❢❡ ♦❢ t1/2 ∼ ✸ ②❡❛rs ✹✸ ✮ ❜✉t ❝❛♥ ❜❡ ❛❝❝❡❧❡r❛t❡❞ ❜② ✐♥❝r❡❛s✐♥❣ t❤❡ t❡♠♣❡r❛t✉r❡ ♦r ✉s✐♥❣ ❛ s✉✐t❛❜❧❡ ❝❛t❛❧②st✳ ❚❤❡r❡ ❛r❡ ❛ r❛♥❣❡ ♦❢ ♣♦ss✐❜❧❡ ❝❛t❛❧②sts ✐♥❝❧✉❞✐♥❣ t❤❡ ❡♥③②♠❡ ❝❛t❛❧❛s❡✱ ✹✹ ♣❧❛t✐♥✉♠ ❛♥❞ ✐ts ❛❧❧♦②s✱ ✹✺ ✐♦❞✐❞❡ ✹✻ ❛♥❞ ♠❡t❛❧ ♦①✐❞❡s✳ ✹✼ ❚❤❡ ✐♥t❡r♠❡❞✐❛t❡s ♦❢ t❤✐s r❡❛❝t✐♦♥ ❞❡♣❡♥❞ ♦♥ t❤❡ ❦✐♥❞ ♦❢ ❝❛t❛❧②st ✉s❡❞ ❢♦r t❤❡ ❞❡❝♦♠♣♦s✐t✐♦♥ r❡❛❝t✐♦♥✳ ■♥ t❤✐s ✇♦r❦✱ ✇❡ ✉s❡ Pt ❛♥❞ ▼♥❖2 ❛s ❝❛t❛❧②sts✳ ❆ r❡❝❡♥t st✉❞② ❤❛s s❤♦✇♥ t❤❛t t❤❡ r❡❛❝t✐♦♥ ♦❢ ♣❧❛t✐♥✉♠ ✇✐t❤ ❍2 ❖2 t♦ ❣✐✈❡ Pt✭❖✮ ✐s t❤❡ r❛t❡ ❧✐♠✐t✐♥❣ st❡♣ ✐♥ t❤❡ r❡❛❝t✐♦♥✳ ✹✽ ❖♥ t❤❡ ♦t❤❡r ❤❛♥❞✱ ❝❛t❛❧②s✐s ❜② ▼♥❖2 ♣r♦❝❡❡❞s ✈✐❛ ❤②❞r♦♣❡r♦①✐❞❡✴s✉♣❡r♦①✐❞❡ ❛♥✐♦♥ ✐♥t❡r♠❡❞✐❛t❡s✳ ✹✾ ❘❡❣❛r❞❧❡ss ♦❢ t❤❡ ✐♥t❡r♠❡❞✐❛t❡s✱ t❤❡ ✜♥❛❧ r❡❛❝t✐♦♥ ♣r♦❞✉❝ts ❛r❡ ✇❛t❡r ❛♥❞ ♦①②❣❡♥ ❣❛s✳ ❚❤✐s r❡❛❝t✐♦♥ ❤❛s ❜❡❡♥ ♣r❡✈✐♦✉s❧② ❡①♣❧♦✐t❡❞ ✐♥ st✉❞✐❡s ♦❢ ❛❝t✐✈❡ ♠❛tt❡r ✇❤✐❝❤ ✉s❡ ❣❛s ❣❡♥❡r❛t✐♦♥ t♦ ♣r♦♣❡❧ ♥❛♥♦r♦❞s ❛♥❞ ♠✐❝r♦t✉❜❡s ✉♥❞❡r✇❛t❡r✳ ✺✵✱✺✶ ■♥ t❤✐s ✇♦r❦✱ ✇❡ ✉s❡ t❤❡ ♦✈❡r♣r❡ss✉r❡ ♦❢ t❤❡ ♦①②❣❡♥ ❣❛s ❣❡♥❡r❛t❡❞ ❜② t❤❡ ❞❡❝♦♠♣♦s✐t✐♦♥ r❡❛❝t✐♦♥ t♦ ❞❡✇❡t t❤❡ ♠✐❝r♦t❡①t✉r❡ ❛♥❞ ❞r✐✈❡ ❛ ✇❡tt✐♥❣ st❛t❡ tr❛♥s✐t✐♦♥ ❢r♦♠ t❤❡ ❲❡♥③❡❧ st❛t❡ ❜❛❝❦ t♦ t❤❡ ❈❛ss✐❡ st❛t❡✳ ❋✐❣✉r❡ ✶ ❞❡♣✐❝ts t❤❡ st❛❣❡s ✹


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♦❜t❛✐♥❡❞ ❛♥♦t❤❡r ❝♦♥❞✐t✐♦♥ ♦♥ t❤❡ ♣r❡♠✐s❡ t❤❛t t❤❡ ❧❛t❡r❛❧ ❢r♦♥t ♦❢ t❤❡ ♣❧❛str♦♥ ♠✉st t♦✉❝❤ t❤❡ s✉❝❝❡ss✐✈❡ ♣✐❧❧❛r ❜❡❢♦r❡ t❤❡ ✈❡rt✐❝❛❧ ❢r♦♥t r❡❛❝❤❡s t❤❡ t♦♣ ♦❢ t❤❡ ♠✐❝r♦t❡①t✉r❡✳ ▲❡❡ ❡t ❛❧✳ ✷✾ ❛♥❞ ❱❡r❤♦ ❡t ❛❧✳ ✸✻ ❢♦✉♥❞ t❤❛t ❛ s✉♣❡r✐♠♣♦s❡❞ ♥❛♥♦t❡①t✉r❡ ✐s ❛ ♥❡❝❡ss❛r② ❝♦♥❞✐t✐♦♥ ❢♦r ♣❧❛str♦♥ r❡❝♦✈❡r②❀ ❜② ❝♦♥tr❛st✱ ❋♦rs❜❡r❣ ❡t✳ ❛❧✳ ♦❜t❛✐♥❡❞ ♣❧❛str♦♥ r❡❝♦✈❡r② ♦♥ ♣✐❧❧❛r❡❞ ♠✐❝r♦t❡①t✉r❡s ✇✐t❤ s♠♦♦t❤ ✇❛❧❧s✳ ❖♥❡ ♦❢ t❤❡ r❡❛s♦♥s ❢♦r t❤❡ ✐♥❝♦♥s✐st❡♥❝② ✐♥ t❤❡s❡ ♦❜s❡r✈❛✲ t✐♦♥s ✐s t❤❛t t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ t❤❡♦r✐❡s ✇❡r❡ ❞❡r✐✈❡❞ ❜❛s❡❞ ♦♥ ❣❡♦♠❡tr✐❝ ♣r✐♥❝✐♣❧❡s ❛♥❞ t❤❡ ✐♥❤❡r❡♥t t❤❡r♠♦❞②♥❛♠✐❝ st❛❜✐❧✐t② ♦❢ t❤❡ ♣❧❛str♦♥ ✐♥ t❤❡ t❡①t✉r❡ ✇❛s ✐❣♥♦r❡❞✳ ❏♦♥❡s ❡t ❛❧✳ ✺✷ ❛♥❞ ▼❛r♠✉r ❡t ❛❧✳ ✺✸ s❤♦✇ t❤❛t t❤❡ st❛❜✐❧✐t② ♦❢ t❤❡ ❈❛ss✐❡ st❛t❡ ✐s t❤❡ ♣r✐♠❛r② ❝♦♥❞✐t✐♦♥ ❢♦r ✉♥❞❡r✇❛t❡r st❛❜✐❧✐t② ♦❢ t❤❡ ♣❧❛str♦♥✳ ❲❡ ❡①♣❧♦r❡ t❤❡ ✐♠♣♦rt❛♥❝❡ ♦❢ t❤✐s ❝♦♥❞✐t✐♦♥ ❢♦r t❤❡ ❧❛t❡r❛❧ s♣r❡❛❞✐♥❣ ♦❢ t❤❡ ♣❧❛str♦♥ ✐♥ ❛♥ ✉♥❞❡r✇❛t❡r s✉♣❡r❤②❞r♦♣❤♦❜✐❝ t❡①t✉r❡✳ ❲❡ ❢❛❜r✐❝❛t❡ ♠✐❝r♦♣✐❧❧❛rs ✐♥t❡rs♣❡rs❡❞ ✇✐t❤ ♣❧❛t✐♥✉♠ ❝❛t❛❧②st ♦♥ s✐❧✐❝♦♥ ✇❛❢❡rs✱ ❤②❞r♦♣❤♦✲ ❜✐③❡ t❤❡♠ ✇✐t❤ ❛ ❚❡✢♦♥ ❧❛②❡r✱ ❛♥❞ ❞❡♠♦♥str❛t❡ ♣❧❛str♦♥ r❡❝♦✈❡r② ✉♥❞❡r✇❛t❡r ✇❤❡♥ ❞✐❧✉t❡ ❝♦♥❝❡♥tr❛t✐♦♥s ♦❢ ❤②❞r♦❣❡♥ ♣❡r♦①✐❞❡ ❛r❡ ✐♥tr♦❞✉❝❡❞✳ ❲❡ ♣r♦♣♦s❡ ❛ t❤❡r♠♦❞②♥❛♠✐❝ ♠♦❞❡❧ ❜❛s❡❞ ♦♥ ❡♥❡r❣❡t✐❝s ♦❢ ❣❛s ❤❡♠✐✇✐❝❦✐♥❣ ✉♥❞❡r✇❛t❡r✱ ❛♥❞ ❞❡r✐✈❡ ❛ ❝r✐t❡r✐♦♥ ❢♦r ♣❧❛str♦♥ r❡✲ ❝♦✈❡r②✳ ❲❡ ❡①♣❧♦r❡ t❤✐s ❝r✐t❡r✐♦♥ ✉s✐♥❣ ♣❧❛t✐♥✉♠✲❝♦❛t❡❞ ♠✐❝r♦♣✐❧❧❛r ❛rr❛②s ♦❢ ✈❛r②✐♥❣ ❤❡✐❣❤t ❛♥❞ ♣✐t❝❤✳ ❋✐♥❛❧❧②✱ ✇❡ ❛❧s♦ ❞❡♠♦♥str❛t❡ ♣r❛❝t✐❝❛❧ ♣❧❛str♦♥ r❡❝♦✈❡r② ❞✉r✐♥❣ ♠❡❛s✉r❡♠❡♥ts ♦❢ ❢r✐❝t✐♦♥❛❧ ❞r❛❣ ♦♥ ❛♥ ✐♥❡①♣❡♥s✐✈❡ ❛♥❞ s❝❛❧❛❜❧❡ ❛❧✉♠✐♥✉♠ s✉r❢❛❝❡ ✇✐t❤ ♠❛♥❣❛♥❡s❡ ❞✐♦①✐❞❡ ❝❛t❛❧②st ✉s✐♥❣ ❛ ❚❛②❧♦r✲❈♦✉❡tt❡ ❝❡❧❧✳ ✺✹

❚❤❡r♠♦❞②♥❛♠✐❝ ❙t❛t❡s ♦❢ ●❛s ❇✉❜❜❧❡s ♦♥ ❆❡r♦♣❤✐❧✐❝ ❚❡①✲ t✉r❡s ❇✉❜❜❧❡s ✐♥t❡r❛❝t ✇✐t❤ t❡①t✉r❡❞ s♦❧✐❞ s✉r❢❛❝❡s ✐♥ ❞✐✛❡r❡♥t ✇❛②s ❞❡♣❡♥❞✐♥❣ ♦♥ t❤❡ r♦✉❣❤♥❡ss ❛♥❞ ❝❤❡♠✐str② ♦❢ t❤❡ s✉r❢❛❝❡✳ ✺✺ ❲❡ ❝❧❛ss✐❢② t❤❡ st❛t❡s ♦❢ ✇❡tt✐♥❣ ♦❢ ❛ ❣❛s ❜✉❜❜❧❡ ❣❡♥❡r❛t❡❞ ♦♥ ❛ t❡①t✉r❡❞ s♦❧✐❞ s✉r❢❛❝❡ ✐♠♠❡rs❡❞ ✉♥❞❡r✇❛t❡r ❜❛s❡❞ ♦♥ t❤❡✐r t❤❡r♠♦❞②♥❛♠✐❝ st❛❜✐❧✐t②✳ ❈♦♠♣❛r✐s♦♥ ✇✐t❤ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ s②st❡♠ ✐♥ ✇❤✐❝❤ ❛ ❞r♦♣ ♦❢ ❧✐q✉✐❞ ✇❡ts ❛ t❡①t✉r❡❞ s♦❧✐❞ s✉r❢❛❝❡ ✐♥ ❛✐r ❡♥❛❜❧❡s t❤❡ ✐❞❡♥t✐✜❝❛t✐♦♥ ♦❢ ❢♦✉r ❞✐st✐♥❝t r❡❣✐♠❡s✱ ❛♥❞ t❤❡ st❛❜✐❧✐t② ♦❢ ❡❛❝❤ ♦❢ ✻


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t❤❡s❡ ❣❛s ❜✉❜❜❧❡✲✐♥✲✇❛t❡r st❛t❡s ❞❡♣❡♥❞s ♦♥ ❜♦t❤ t❤❡ s✉r❢❛❝❡ ❝❤❡♠✐str② ❛♥❞ r♦✉❣❤♥❡ss ♦❢ t❤❡ t❡①t✉r❡✳ ❋✐❣✉r❡ ✷❛ s❤♦✇s ❛ s❝❤❡♠❛t✐❝ ✐❧❧✉str❛t✐♥❣ t❤❡ ❢♦✉r r❡❣✐♠❡s ♦♥ ❛ ♠✐❝r♦t❡①t✉r❡❞ s♦❧✐❞ s✉r❢❛❝❡ ✲ ❜✉❜❜❧❡✱ ❜❧✐st❡r✱ ❤❡♠✐✇✐❝❦✐♥❣ ❜❧✐st❡r✱ ❛♥❞ ♣❧❛str♦♥✳ ❇✉❜❜❧❡ ✐❞❡♥t✐✜❡s t❤❡ st❛t❡ ✐♥ ✇❤✐❝❤ t❤❡ ❣❛s ❞♦❡s ♥♦t ✇❛♥t t♦ ✇❡t t❤❡ s♦❧✐❞ s✉r❢❛❝❡ ❛♥❞ ✐s ♣✐♥♥❡❞ t♦ t❤❡ s✉r❢❛❝❡ ✇✐t❤ ♠✐♥✐♠❛❧ s♦❧✐❞✲❛✐r ❝♦♥t❛❝t ❛r❡❛✳ ❇❧✐st❡rs ❛r❡ ❜✉❜❜❧❡s t❤❛t ♣❛rt✐❛❧❧② s♣r❡❛❞ ♦♥ t❤❡ s♦❧✐❞ s✉r❢❛❝❡ ✉♥t✐❧ t❤❡② ❡st❛❜❧✐s❤ ❛♥ ❡q✉✐❧✐❜r✐✉♠ ♣r❡ss✉r❡ ❛♥❞ ❛♣♣❛r❡♥t ❝♦♥t❛❝t ❛♥❣❧❡✳ ❚❤❡② ❞♦ ♥♦t ✐♠❜✐❜❡ s♣♦♥t❛♥❡♦✉s❧② ✐♥t♦ t❤❡ t❡①t✉r❡✳ ❚❤❡ P❧❛str♦♥ ✐s t❤❡ st❛t❡ ✐♥ ✇❤✐❝❤ t❤❡ ❣❛s ✐s ❝♦♠♣❧❡t❡❧② ✐♠❜✐❜❡❞ ✐♥ t❤❡ t❡①t✉r❡ ❛♥❞ ♦♥❧② t❤❡ t♦♣ ♦❢ t❤❡ s✉r❢❛❝❡ t❡①t✉r❡ ❝♦♥t❛❝ts t❤❡ ❧✐q✉✐❞ ♣❤❛s❡✳ ❋✐♥❛❧❧②✱ t❤❡ ✐♥t❡r♠❡❞✐❛t❡ st❛t❡ ✐❞❡♥t✐✜❡❞ ❛s ❍❡♠✐✇✐❝❦✐♥❣ ❇❧✐st❡rs ❛r❡ ❜❧✐st❡rs t❤❛t s♣r❡❛❞ ♦♥ t♦♣ ♦❢ ❛ ♠✐❝r♦t❡①t✉r❡ ✐♠❜✐❜❡❞ ✇✐t❤ ❛✐r ✐✳❡✳ ♣❧❛str♦♥✳ ❚❤❡ ❞✐st✐♥❝t✐♦♥ ❜❡t✇❡❡♥ ❛ ❜❧✐st❡r ❛♥❞ ❛ ❤❡♠✐✇✐❝❦✐♥❣ ❜❧✐st❡r ✐s t❤❛t t❤❡ ✢✉✐❞ ♣❤❛s❡ ♣r❡❢❡r❡♥t✐❛❧❧② ✐♠❜✐❜❡❞ ✐♥ t❤❡ t❡①t✉r❡ s✉rr♦✉♥❞✐♥❣ t❤❡ ❣❛s ❜❧✐st❡r ✐s ❞✐✛❡r❡♥t✳ ❲❡ ❞❡✜♥❡ t❤✐s st❛t❡ ✉s✐♥❣ ❛ ♥♦♠❡♥❝❧❛t✉r❡ ❞❡r✐✈❡❞ ❢r♦♠ t❤❡ ✇♦r❦ ♦❢ ❇✐❝♦ ❡t ❛❧✳ ✇❤♦ ❝♦♥s✐❞❡r❡❞ ❝♦rr❡s♣♦♥❞✐♥❣ st❛t❡s ❢♦r ❧✐q✉✐❞ ❞r♦♣❧❡ts ✇❡tt✐♥❣ ♠✐❝r♦t❡①t✉r❡❞ ❞r② s✉❜str❛t❡s ✐♥ ❛✐r✳ ✺✻ ■t ❝❛♥ ❜❡ s❡❡♥ ❢r♦♠ ❞❡t❛✐❧❡❞ ❡①❛♠✐♥❛t✐♦♥ ♦❢ ❋✐❣✉r❡ ✷❛ ✭✐✮✲✭✐✈✮ t❤❛t t❤❡ ♣✐❧❧❛rs ❛r❡ ✈✐s✐❜❧❡ ❢♦r t❤❡ ❜✉❜❜❧❡ ❛♥❞ ❜❧✐st❡r st❛t❡s ✇❤✐❧❡ t❤❡ ♣✐❧❧❛rs ❛r❡ ❤✐❞❞❡♥ ✐♥s✐❞❡ t❤❡ ❣❛s ❧❛②❡r ❢♦r t❤❡ ❤❡♠✐✇✐❝❦✐♥❣ ❜❧✐st❡r ❛♥❞ ♣❧❛str♦♥ st❛t❡s✳ ❲❤❡♥ ❣❛s ✐s ♣r♦❞✉❝❡❞ ✐♥ t❤❡ t❡①t✉r❡ t❤r♦✉❣❤ t❤❡ ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ ❤②❞r♦❣❡♥ ♣❡r♦①✐❞❡✱ ✐t ♠❛② ❧❡❛✈❡ t❤❡ s✉r❢❛❝❡ t❤r♦✉❣❤ ❜✉♦②❛♥❝② ❡✛❡❝ts ❛s ❛ ♠❛❝r♦s❝♦♣✐❝ ❜✉❜❜❧❡✱ ♦r s♣r❡❛❞ t❤r♦✉❣❤ t❤❡ ♠✐❝r♦s❝♦♣✐❝ t❡①t✉r❡ ✐♥ ♦♥❡ ♦❢ t❤❡ ♦t❤❡r st❛t❡s ❞❡♣❡♥❞✐♥❣ ♦♥ t❤❡ t❡①t✉r❡ ❝❤❡♠✐str② ❛♥❞ r♦✉❣❤♥❡ss✱ ❛♥❞ t❤❡ r❛t❡ ♦❢ ❣❡♥❡r❛t✐♦♥ ❝♦♠♣❛r❡❞ t♦ t❤❡ r❛t❡ ♦❢ s♣r❡❛❞✐♥❣✳ ■♥ t❤✐s ✇♦r❦✱ ✇❡ ❢♦❧❧♦✇ ◗✉éré✬s ❤❡♠✐✇✐❝❦✐♥❣ ♠♦❞❡❧ ❢♦r ❤②❞r♦♣❤✐❧✐❝ t❡①t✉r❡s ✺✻ t♦ ♦❜t❛✐♥ ❛♥ ❡q✉✐✈❛❧❡♥t ❤❡♠✐✇✐❝❦✐♥❣ ♠♦❞❡❧ ❢♦r s✉♣❡r❤②❞r♦♣❤♦❜✐❝ t❡①t✉r❡s ✉♥❞❡r✇❛t❡r✳ ■♥ ✐❞❡❛❧ ♣❧❛str♦♥ r❡❝♦✈❡r②✱ t❤❡ ❣❛s ✇♦✉❧❞ s♣r❡❛❞ ❛❧♦♥❣ t❤❡ s✉r❢❛❝❡ ♠✐❝r♦t❡①t✉r❡ ✇✐t❤♦✉t ❞✐s♣❧❛❝✐♥❣ t❤❡ ✇❛t❡r ♦♥ t♦♣ ♦❢ t❤❡ t❡①t✉r❡ ❛s ❞❡♣✐❝t❡❞ ✐♥ ❋✐❣✉r❡ ✶✳ ❚❤✐s ♣r♦❝❡ss ✐♥✈♦❧✈❡s t❤❡ ❝r❡❛t✐♦♥ ♦❢ ♥❡✇ s♦❧✐❞✲ ❛✐r ❛♥❞ ✇❛t❡r✲❛✐r ✐♥t❡r❢❛❝❡s ✇❤✐❧❡ r❡♣❧❛❝✐♥❣ t❤❡ ✇❛t❡r✲s♦❧✐❞ ✐♥t❡r❢❛❝❡s ❞✉r✐♥❣ ❞❡✲✇❡tt✐♥❣✳ ❚❤❡ ❝❤❛♥❣❡ ✐♥ ❡♥❡r❣② ♦❢ t❤❡ s②st❡♠ ✇❤❡♥ t❤❡ ❣❛s ❢r♦♥t ❞✐s♣❧❛❝❡s ✇❛t❡r ❢r♦♠ ❛ s♠❛❧❧ ❛r❡❛ ldx ✐s ❣✐✈❡♥ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣ ❡q✉❛t✐♦♥✳ ✼




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dE = γw,g (1 − φs )ldx + γs,g (r − φs )ldx − γs,w (r − φs )ldx

✭✷✮

✇❤❡r❡ γw,g ✱ γs,g ❛♥❞ γs,w r❡♣r❡s❡♥t t❤❡ r❡s♣❡❝t✐✈❡ ✐♥t❡r❢❛❝✐❛❧ ❡♥❡r❣✐❡s ❜❡t✇❡❡♥ t❤❡ t❤r❡❡ ♣❤❛s❡s✱

r ❛♥❞ φs ❞❡♥♦t❡ t❤❡ ❲❡♥③❡❧ r♦✉❣❤♥❡ss ❛♥❞ t❤❡ ♣r♦❥❡❝t❡❞ s♦❧✐❞ ❢r❛❝t✐♦♥ r❡s♣❡❝t✐✈❡❧②✱ ❛♥❞ l ✐s t❤❡ ❧❡♥❣t❤ ✐♥t♦ t❤❡ ♣❧❛♥❡ ♦❢ ❋✐❣✉r❡ ✶✳ ❚❤❡ ❲❡♥③❡❧ r♦✉❣❤♥❡ss r ✐s ❞❡✜♥❡❞ ❛s t❤❡ t♦t❛❧ s✉r❢❛❝❡ ❛r❡❛ ♣❡r ✉♥✐t ♣r♦❥❡❝t❡❞ ❛r❡❛ ♦❢ s♦❧✐❞ s✉r❢❛❝❡✱ ❛♥❞ s♦❧✐❞ ❢r❛❝t✐♦♥ φs ✐s ❞❡✜♥❡❞ ❛s t❤❡ ❛r❡❛ ✇❡tt❡❞ ❜② t❤❡ ❧✐q✉✐❞ ✐♥ t❤❡ ❈❛ss✐❡ st❛t❡ ♣❡r ✉♥✐t ♣r♦❥❡❝t❡❞ s✉r❢❛❝❡ ❛r❡❛ ♦❢ s♦❧✐❞✳ ❋♦r ❡①❛♠♣❧❡ ♦♥ ❛ ♠✐❝r♦t❡①t✉r❡ ✇✐t❤ ❝✐r❝✉❧❛r ♣✐❧❧❛rs✱ r ❛♥❞ φs ❛r❡ ❝❛❧❝✉❧❛t❡❞ ❜② ✉s✐♥❣ t❤❡ ❢♦❧❧♦✇✐♥❣ ❡q✉❛t✐♦♥s✳

r =1+

2πRH L2

πR2 φs = 2 L

✭✸✮

✭✹✮

❚❤❡ ❝❤❛♥❣❡ ✐♥ ❡♥❡r❣② dE ♦❢ t❤❡ s②st❡♠ ♠✉st ❜❡ ♥❡❣❛t✐✈❡ ✐♥ ♦r❞❡r ❢♦r t❤❡ ❣❛s t♦ ♣r❡❢❡r❡♥✲ t✐❛❧❧② ✇✐❝❦ ✐♥t♦ t❤❡ t❡①t✉r❡✳ ❍❡♥❝❡ ❜② s❡tt✐♥❣ dE < 0 ❛♥❞ s✐♠♣❧✐❢②✐♥❣ t❤❡ ❡①♣r❡s✐♦♥ ✉s✐♥❣ t❤❡ ❡①♣r❡ss✐♦♥ ❢♦r ❨♦✉♥❣✬s t❤r❡❡ ♣❤❛s❡ ❝♦♥t❛❝t ❛♥❣❧❡ ❡q✉❛t✐♦♥✱ ✺✼ ✇❡ ♦❜t❛✐♥ t❤❡ t❤❡r♠♦❞②♥❛♠✐❝ ❝♦♥❞✐t✐♦♥ ❢♦r s♣♦♥t❛♥❡♦✉s ♣❧❛str♦♥ r❡❝♦✈❡r②✳

cos θw < −

1 − φs r − φs

✭✺✮

❊q✉❛t✐♦♥ ✺ ✐♠♣❧✐❡s t❤❛t ♣❧❛str♦♥ r❡❝♦✈❡r② ✐s ♦♥❧② ♣♦ss✐❜❧❡ ❢♦r t❡①t✉r❡s ✇✐t❤ ❤②❞r♦♣❤♦❜✐❝ ❝❤❡♠✐str✐❡s ✭θw > 90◦ ✮ s✐♥❝❡ t❤❡ ❡①♣r❡ss✐♦♥ ♦♥ t❤❡ r✐❣❤t ✐s ❛❧✇❛②s ♥❡❣❛t✐✈❡✳ ❚❤❡ ❢❛✈♦r❛❜❧❡ r❡❣✐♠❡s ♦❢ ✉♥❞❡r✇❛t❡r ❣❛s ✇❡tt✐♥❣ ❝❛♥ ❜❡ r❡♣r❡s❡♥t❡❞ ♦♥ t❤❡ ♣❧♦t ✐❧❧✉str❛t✐♥❣ t❤❡ r❡❧❛t✐♦♥s❤✐♣ ❜❡t✇❡❡♥ t❤❡ ❝♦s✐♥❡ ♦❢ t❤❡ ❛♣♣❛r❡♥t ❝♦♥t❛❝t ❛♥❣❧❡ ♦❢ ❛ ✇❛t❡r ❞r♦♣❧❡t ♦♥ ❛ t❡①t✉r❡❞ s♦❧✐❞ ✐♥ ❛✐r✱ cos θw∗ ❛♥❞ t❤❡ ❝♦s✐♥❡ ♦❢ t❤❡ ❨♦✉♥❣✬s ❝♦♥t❛❝t ❛♥❣❧❡ ♦❢ ❛ ✇❛t❡r ❞r♦♣❧❡t ♦♥ ❛ ✢❛t s✉r❢❛❝❡ ✐♥ ❛✐r✱ cos θw ❛s s❤♦✇♥ ✐♥ ❋✐❣✉r❡ ✷❜✳ ❍❡r❡ ✐t ❝❛♥ ❜❡ s❡❡♥ t❤❛t ♣❧❛str♦♥ ❛♥❞ ❤❡♠✐✲✇✐❝❦✐♥❣ ❜❧✐st❡rs ❛r❡ ❢❛✈♦r❛❜❧❡ ✐♥ t❤❡ t❡①t✉r❡s ✇❤✐❝❤ ❛r❡ ❈❛ss✐❡✲st❛❜❧❡ t♦✇❛r❞s s❡ss✐❧❡ ✇❛t❡r ❞r♦♣❧❡ts

✾



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✐♥ ❛✐r✱ ❜❧✐st❡rs ❛r❡ ❢❛✈♦r❛❜❧❡ ✐♥ t❤❡ t❡①t✉r❡s ✇❤✐❝❤ ❛r❡ ❲❡♥③❡❧✲st❛❜❧❡ t♦✇❛r❞s s❡ss✐❧❡ ✇❛t❡r ❞r♦♣❧❡ts ✐♥ ❛✐r✱ ❛♥❞ ❜✉❜❜❧❡s ❛r❡ ❢❛✈♦r❡❞ ✐♥ t❤❡ t❡①t✉r❡s ✇❤✐❝❤ ❛r❡ ❤❡♠✐✇✐❝❦✐♥❣ t♦✇❛r❞s s❡ss✐❧❡ ✇❛t❡r ❞r♦♣❧❡ts ✐♥ ❛✐r✳ ❙✐♥❝❡ ❣❛s ❞✐s♣❧❛❝❡s ✇❛t❡r ✐♥ t❤❡ t❡①t✉r❡ ❞✉r✐♥❣ ♣❧❛str♦♥ s♣r❡❛❞✐♥❣✱ ✐t ✐s ♠♦st ❛♣♣r♦♣r✐❛t❡ ✐♥ ❊q✉❛t✐♦♥ ✺ t♦ ✉s❡ t❤❡ r❡❝❡❞✐♥❣ ❝♦♥t❛❝t ❛♥❣❧❡ ♦❢ ❛ s❡ss✐❧❡ ✇❛t❡r ❞r♦♣❧❡t ♦♥ ❛ s♠♦♦t❤ s✉r❢❛❝❡ ✇✐t❤ t❤❡ s❛♠❡ ❝❤❡♠✐str②✳ ❘❡❛rr❛♥❣✐♥❣ ❊q✉❛t✐♦♥ ✺ ✉s✐♥❣ t❤❡ r❡❝❡❞✐♥❣ ✇❛t❡r ❝♦♥t❛❝t ❛♥❣❧❡ ❣✐✈❡s ✉s t❤❡ ❝♦♥❞✐t✐♦♥ t❤❛t ♥❡❡❞s t♦ ❜❡ s❛t✐s✜❡❞ ❢♦r t❡①t✉r❛❧ ❞❡s✐❣♥ ♣❛r❛♠❡t❡rs s✉❝❤ ❛s t❤❡ s♦❧✐❞ ❛r❡❛ ❢r❛❝t✐♦♥ φs ❛♥❞ t❤❡ s♦❧✐❞ r♦✉❣❤♥❡ss r✳ ❚❤✐s ❛❧s♦ ♠❛t❝❤❡s ✇✐t❤ t❤❡ ♠♦♥♦st❛❜❧❡ ❈❛ss✐❡ st❛t❡ ❝♦♥❞✐t✐♦♥ ❞❡r✐✈❡❞ ❜② ▲✐ ❡t ❛❧✳ ❢♦r t❤❡ ❲❡♥③❡❧✲t♦✲ ❈❛ss✐❡ tr❛♥s✐t✐♦♥ ♦❢ ❧✐q✉✐❞ ❞r♦♣❧❡ts ♦♥ ♠✐❝r♦t❡①t✉r❡s✳ ✷✻ ❋♦r ♣❧❛str♦♥ r❡❝♦✈❡r② t♦ ❜❡ ❢❛✈♦r❛❜❧❡ ♦✈❡r ❜❧✐st❡r ❢♦r♠❛t✐♦♥✱ ✇❡ t❤❡r❡❢♦r❡ r❡q✉✐r❡ ♠✐❝r♦t❡①t✉r❡s ✇✐t❤ r♦✉❣❤♥❡ss ❡①❝❡❡❞✐♥❣ ❛ ❝r✐t✐❝❛❧ ✈❛❧✉❡✿

r >1−

1 + 1 (1 − φs ) cos θwrec

∀ θwrec > 90◦

✭✻✮

✇❤❡r❡ t❤❡ ❝♦s✐♥❡ ♦❢ t❤❡ r❡❝❡❞✐♥❣ ❛♥❣❧❡ ✐s ♥❡❣❛t✐✈❡✱ −1 < cos θwrec < 0✱ ❛♥❞ ❤❡♥❝❡ t❤❡ r✐❣❤t ❤❛♥❞ s✐❞❡ ❡①♣r❡ss✐♦♥ ✐s ❛❧✇❛②s ♣♦s✐t✐✈❡✳

❘❡s✉❧ts ❛♥❞ ❉✐s❝✉ss✐♦♥ ■♥ t❤✐s ✇♦r❦✱ ✇❡ ✉s❡ t✇♦ t②♣❡s ♦❢ t❡①t✉r❡❞ s✉r❢❛❝❡s ❢♦r t❡st✐♥❣ ♣❧❛str♦♥ r❡❝♦✈❡r② ✲ ❧✐t❤♦✲ ❣r❛♣❤✐❝❛❧❧② ♣❛tt❡r♥❡❞ s✐❧✐❝♦♥ ❛♥❞ ❧❛s❡r✲t❡①t✉r❡❞ ❛❧✉♠✐♥✉♠ ✭❙❡❡ ❋✐❣✉r❡ ✸✮✳ ■♥ ♦r❞❡r t♦ s②s✲ t❡♠❛t✐❝❛❧❧② st✉❞② t❤❡ ❡✛❡❝t ♦❢ s✉r❢❛❝❡ r♦✉❣❤♥❡ss ♦♥ ♣❧❛str♦♥ r❡❝♦✈❡r②✱ ♠✐❝r♦✲♣✐❧❧❛r ❛rr❛②s ✇✐t❤ ✈❛r②✐♥❣ ♣✐❧❧❛r ❤❡✐❣❤ts ❛♥❞ s♣❛❝✐♥❣ ✇❡r❡ ❢❛❜r✐❝❛t❡❞ ♦♥ s✐❧✐❝♦♥ ✇❛❢❡rs ✉s✐♥❣ ♣❤♦t♦❧✐t❤♦❣✲ r❛♣❤②✳ P❧❛t✐♥✉♠✱ ❛ ✇❡❧❧✲❦♥♦✇♥ ❝❛t❛❧②st ❢♦r t❤❡ ❤②❞r♦❣❡♥ ♣❡r♦①✐❞❡ ❞❡❝♦♠♣♦s✐t✐♦♥ r❡❛❝t✐♦♥✱ ✇❛s s❡❧❡❝t✐✈❡❧② ❞❡♣♦s✐t❡❞ ♦♥ t❤❡ ❜♦tt♦♠ ❛♥❞ t❤❡ s✐❞❡s ♦❢ t❤❡ ♠✐❝r♦✲♣✐❧❧❛rs ✉s✐♥❣ ❧✐❢t✲♦✛ ♣❤♦✲ t♦❧✐t❤♦❣r❛♣❤② ✭s❡❡ t❤❡ ❡①♣❡r✐♠❡♥t❛❧ ♠❡t❤♦❞s s❡❝t✐♦♥ ❢♦r ♠♦r❡ ❞❡t❛✐❧s✮✱ t♦ ❡♥s✉r❡ t❤❛t t❤❡ ♣❧❛str♦♥ r❡✲✐♥✢❛t✐♦♥ ♣r♦❝❡ss ✐s s❡❧❢✲❧✐♠✐t✐♥❣✳ ❚❤❡ t❡①t✉r❡s ✇❡r❡ t❤❡♥ s♣✐♥✲❝♦❛t❡❞ ✇✐t❤ ❛ ✶ ✇t✳✪ ❚❡✢♦♥ s♦❧✉t✐♦♥ ❛♥❞ ❝✉r❡❞ t♦ r❡♥❞❡r t❤❡♠ ❤②❞r♦♣❤♦❜✐❝✳ ✶✵


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❙✐✲Pt✲❚❡✢♦♥ s✉r❢❛❝❡ tr❛♥s✐t✐♦♥✐♥❣ ❢r♦♠ ❛ ♣❛rt✐❛❧❧②✲✇❡tt❡❞ st❛t❡ t♦ ❛ ❢✉❧❧②✲r❡❝♦✈❡r❡❞ ♣❧❛str♦♥ st❛t❡✳ ❙✉♣♣❧❡♠❡♥t❛r② ❱✐❞❡♦ ❙✶ s❤♦✇s t❤❡ ♣r♦❝❡ss ♦❢ ♣❧❛str♦♥ ♥✉❝❧❡❛t✐♦♥ ❛♥❞ ❧❛t❡r❛❧ s♣r❡❛❞✐♥❣ t❤r♦✉❣❤ t❤❡ t❡①t✉r❡✳ ❊①♣❡r✐♠❡♥ts ✇❡r❡ ❛❧s♦ ♣❡r❢♦r♠❡❞ ♦♥ s✐❧✐❝♦♥ ♠✐❝r♦♣✐❧❧❛rs ✇❤♦s❡ t♦♣s ✇❡r❡ ♥♦t str✐♣♣❡❞ ♦❢ ♣❧❛t✐♥✉♠ ❝❛t❛❧②st ✭❙❡❡ ❙✉♣♣❧❡♠❡♥t❛r② ❋✐❣✉r❡ ❙✷✮ ❛♥❞ ✐t ✇❛s ❢♦✉♥❞ t❤❛t ♣❧❛str♦♥ r❡❝♦✈❡r② ✐s ♣♦ss✐❜❧❡ ✐♥ t❤❡s❡ ❝❛s❡s t♦♦✳ ❲❡ ❜❡❧✐❡✈❡ t❤❛t t❤✐s ✐s ♣♦ss✐❜❧❡ ❜❡❝❛✉s❡ t❤❡ ♥✉❝❧❡❛t✐♦♥ ♦❢ ❣❛s ✐s ❢❛✈♦r❡❞ ❛t t❤❡ s❤❛r♣ ❝♦r♥❡rs ❛t t❤❡ ❜♦tt♦♠ ♦❢ t❤❡ t❡①t✉r❡✱ ❡✈❡♥ t❤♦✉❣❤ t❤❡r❡ ✐s ❝❛t❛❧②st ❛t t❤❡ t♦♣✳ ❲❡ ❛❧s♦ ♥♦t❡ t❤❛t ❙✐ ♠✐❝r♦♣✐❧❧❛rs ❢❛❜r✐❝❛t❡❞ ✉s✐♥❣ t❤❡ ❇♦s❝❤ ❘■❊ ♣r♦❝❡ss ❛❧s♦ ❤❛✈❡ ❛ ♥❛♥♦s❝❛❧❡ t❡①t✉r❡ ✇❤✐❝❤ ❝❛♥ ♣r♦♠♦t❡ ♣❧❛str♦♥ r❡❝♦✈❡r②✳ ❍❡♥❝❡✱ ❢✉rt❤❡r t❡sts ✇❡r❡ ❝♦♥❞✉❝t❡❞ ✉s✐♥❣ s♠♦♦t❤✲✇❛❧❧❡❞ ♠✐❝r♦♣✐❧❧❛rs ♠❛❞❡ ♦❢ ❙❯✽ ♣❤♦t♦r❡s✐st ✭❙❡❡ ❙✉♣♣❧❡♠❡♥t❛r② ❋✐❣✉r❡ ❙✹✮✳ ■t ✇❛s ♦❜s❡r✈❡❞ t❤❛t s♠♦♦t❤✲✇❛❧❧❡❞ ♠✐❝r♦♣✐❧❧❛r t❡①t✉r❡s ❛❧s♦ s❤♦✇ ♣❧❛str♦♥ r❡❝♦✈❡r② s✐♠✐❧❛r t♦ t❤❡ ❞r②✲❡t❝❤❡❞ ❙✐ ♠✐❝r♦♣✐❧❧❛rs✱ ❢✉rt❤❡r s✉♣♣♦rt✐♥❣ ♦✉r ♦❜s❡r✈❛t✐♦♥ t❤❛t ♥❛♥♦s❝❛❧❡ r♦✉❣❤♥❡ss ✐s ♥♦t ♥❡❝❡ss❛r② ❢♦r ♣❧❛str♦♥ r❡❝♦✈❡r②✳ ❍♦✇❡✈❡r ❛s ♣r❡❞✐❝t❡❞ ❜② t❤❡ t❤❡r♠♦❞②♥❛♠✐❝ ❛♥❛❧②s✐s✱ ✐t ✇❛s ♦❜s❡r✈❡❞ t❤❛t s♦♠❡ t❡①t✉r❡s ❞♦ ♥♦t ♣r♦♠♦t❡ ♣❧❛str♦♥ r❡❝♦✈❡r② ❡✈❡♥ ✐❢ t❤❡② ❛r❡ ❤②❞r♦♣❤♦❜✐❝✳ ❋✐❣✉r❡ ❙✺ s❤♦✇s ❛♥ ✐♠❛❣❡ ♦❢ ❛ s✉❜♠❡r❣❡❞ ❙✐✲Pt✲❚❡✢♦♥ t❡①t✉r❡ ✭✷✺ µ♠ ❞❡♣t❤✱ ✺✵ µ♠ ♣✐❧❧❛r ✇✐❞t❤✱ ✶✺✵ µ♠ ♣✐t❝❤ ❝♦rr❡s♣♦♥❞✐♥❣ t♦ φs ❂ ✵✳✵✽✼✱ r ❂ ✶✳✶✼✺✮ ♦♥ ✇❤✐❝❤ ❜❧✐st❡rs ❛r❡ ❢♦r♠❡❞ ✉♣♦♥ t❤❡ ❛❞❞✐t✐♦♥ ♦❢ ❤②❞r♦❣❡♥ ♣❡r♦①✐❞❡ ✉♥❞❡r s✐♠✐❧❛r ❝♦♥❞✐t✐♦♥s✳ ❋✐❣✉r❡ ✹ s❤♦✇s ❛ r❡❣✐♠❡ ♠❛♣ ❝❧❛ss✐❢②✐♥❣ t❤❡ s✐❧✐❝♦♥ ♠✐❝r♦✲♣✐❧❧❛r ❡①♣❡r✐♠❡♥ts ✐♥t♦ t✇♦ ❝❛t❡❣♦r✐❡s ❜❛s❡❞ ♦♥ ✇❤❡t❤❡r ♣❧❛str♦♥ s♣r❡❛❞✐♥❣ ♦r ❜❧✐st❡r ❢♦r♠❛t✐♦♥ ✇❛s ❢❛✈♦r❡❞ ❜❛s❡❞ ♦♥ t❤❡ t❡①t✉r❡ ♣❛r❛♠❡t❡rs ❝❤❛r❛❝t❡r✐③✐♥❣ ❡❛❝❤ s✉r❢❛❝❡✳ ❚❤❡ r❡❝❡❞✐♥❣ ❝♦♥t❛❝t ❛♥❣❧❡ ♦❢ ❛ ✇❛t❡r ❞r♦♣❧❡t ♦♥ ❛ ❚❡✢♦♥ ❝♦❛t❡❞ s♠♦♦t❤ s✐❧✐❝♦♥ ✇❛❢❡r ✇❛s ♠❡❛s✉r❡❞ t♦ ❜❡ θwrec = 111◦ ± 2◦ ❛♥❞ t❤✐s ✈❛❧✉❡ ❤❛s ❜❡❡♥ ✉s❡❞ t♦ ♣❧♦t t❤❡ r❡❝♦✈❡r② ❝♦♥❞✐t✐♦♥ ❣✐✈❡♥ ❜② ❊q✉❛t✐♦♥ ✻✳ ❚❤❡ s❤❛❞❡❞ r❡❣✐♦♥ ❛r♦✉♥❞ t❤❡ t❤❡♦r❡t✐❝❛❧ ♣r❡❞✐❝t✐♦♥ ❧✐♥❡ ✐♥❞✐❝❛t❡s t❤❡ s❡♥s✐t✐✈✐t② ♦❢ ❊q✉❛t✐♦♥ ✻ t♦ ❛ r❡❝❡❞✐♥❣ ❝♦♥t❛❝t ❛♥❣❧❡ ✈❛r✐❛t✐♦♥ ♦❢ ±2◦ ✳ ■t ❝❛♥ ❜❡ s❡❡♥ ❢r♦♠ ❜♦t❤ t❤❡ t❤❡♦r② ❛♥❞ t❤❡ ❡①♣❡r✐♠❡♥ts t❤❛t t❡①t✉r❡s ✇✐t❤ ❧❛r❣❡r r♦✉❣❤♥❡ss ❛♥❞ ❤✐❣❤❡r s♦❧✐❞ ❢r❛❝t✐♦♥ ❢❛✈♦r ♣❧❛str♦♥ r❡❝♦✈❡r②✱ ✇❤✐❧❡ t❤♦s❡ ✇✐t❤ ❧♦✇❡r r♦✉❣❤♥❡ss ❛♥❞ s♠❛❧❧❡r s♦❧✐❞ ❢r❛❝t✐♦♥ ♣r♦♠♦t❡ ❜❧✐st❡r ❢♦r♠❛t✐♦♥✳ ❲❡ ❛❧s♦ s❡❡ t❤❛t t❤❡ t❡①t✉r❡s ✇❤✐❝❤ ❤❛✈❡ ❝❛t❛❧②st ♦♥ t♦♣ ♦❢ t❤❡ ♣✐❧❧❛rs ❢❛✈♦r ♣❧❛str♦♥ r❡❝♦✈❡r② ❛s ✇❡❧❧✳ ✶✷


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❆❧t❤♦✉❣❤ ✐t ❤❛s ❜❡❡♥ r❡♣♦rt❡❞ ✐♥ ♣r❡✈✐♦✉s st✉❞✐❡s t❤❛t t❤❡ ♣r❡s❡♥❝❡ ♦❢ ❛ ♥❛♥♦✲❈❛ss✐❡ st❛t❡ ✐s ♥❡❝❡ss❛r② ❢♦r ♣❧❛str♦♥ r❡❝♦✈❡r②✱ ✷✾✱✸✻ ✇❡ ♦❜s❡r✈❡ t❤❛t s❛t✐s❢②✐♥❣ t❤❡ t❤❡r♠♦❞②♥❛♠✐❝ ❝♦♥❞✐t✐♦♥ ❢♦r ♣❧❛str♦♥ ❣❛s s♣r❡❛❞✐♥❣ ❛❧♦♥❡ ✐s s✉✣❝✐❡♥t t♦ ❛❧❧♦✇ ❢♦r ♣❧❛str♦♥ r❡❝♦✈❡r② ♦♥ ♠✐❝r♦t❡①t✉r❡s ❡✈❡♥ ✇✐t❤♦✉t ❛ ♥❛♥♦s❝❛❧❡ t❡①t✉r❡ t♦ ❤♦❧❞ ❛ st❛❜❧❡ ♣❧❛str♦♥✳ ❚❤✐s ✐s ❛❧s♦ ✐♥ ❛❣r❡❡♠❡♥t ✇✐t❤ t❤❡ ♦❜s❡r✈❛t✐♦♥ ❜② ❋♦rs❜❡r❣ ❡t✳ ❛❧✳✱ ✸✼ ✇❤♦ r❡♣♦rt❡❞ ♣❧❛str♦♥ r❡❝♦✈❡r② ♦♥ s♠♦♦t❤✲✇❛❧❧❡❞ ♣♦❧②♠❡r ♣✐❧❧❛rs✳ ■t ❝❛♥ ❜❡ ♦❜s❡r✈❡❞ t❤❛t ❛ ❢❡✇ t❡①t✉r❡s s❤♦✇ r❡❝♦✈❡r② ❡✈❡♥ t❤♦✉❣❤ t❤❡② ❧✐❡ ❜❡❧♦✇ t❤❡ t❤❡♦r❡t✐❝❛❧❧② ❝♦♥❞✐t✐♦♥ ❢♦r ♣❧❛str♦♥ r❡❝♦✈❡r②✳ ❚❤✐s ❝❛♥ ❜❡ ❡①♣❧❛✐♥❡❞ ❜② t❤❡ ❢❛❝t t❤❛t t❤❡s❡ t❡①t✉r❡s ❝❛♥ tr❛♥s✐t✐♦♥ ❢r♦♠ ❛ ♠❡t❛st❛❜❧❡ ❲❡♥③❡❧ st❛t❡ t♦ ❛ st❛❜❧❡ ❈❛ss✐❡ st❛t❡ ✐❢ t❤❡② s❛t✐s❢② ❊q✉❛t✐♦♥ ✻ ✇✐t❤ θwrec = θwy ✱ t❤❡ ❨♦✉♥❣ ❝♦♥t❛❝t ❛♥❣❧❡ ❢♦r ❚❡✢♦♥✳ ❚❤✐s ❝✉r✈❡ ✇♦✉❧❞ ❧✐❡ ✐♥ ❜❡t✇❡❡♥ t❤❡ t❤❡♦r❡t✐❝❛❧ ❝✉r✈❡s ❢♦r θwadv ❛♥❞ θwrec s❤♦✇♥ ✐♥ ❋✐❣✉r❡ ✹✱ ❛♥❞ ❞❡✜♥❡ t❤❡ ❜✐✲st❛❜❧❡ r❡❣✐♦♥ ✇❤❡r❡ t❤❡ ❲❡♥③❡❧ st❛t❡ ✐s ♠❡t❛st❛❜❧❡✳ ✷✻ ■♥ ✉♥❞❡r✇❛t❡r ❲❡♥③❡❧✲t♦✲❈❛ss✐❡ tr❛♥s✐t✐♦♥s✱ t❤❡ ❡♥❡r❣② ❜❛rr✐❡r ❜❡t✇❡❡♥ t❤❡ ❈❛ss✐❡ ❛♥❞ ❲❡♥③❡❧ st❛t❡s ✐♥ t❤❡ ❜✐st❛❜❧❡ r❡❣✐♦♥ ❝❛♥ ❜❡ ♦✈❡r❝♦♠❡ ❜② t❤❡ ❣❛s ♦✈❡r✲♣r❡ss✉r❡ ✐♥ t❤❡ ♣❧❛str♦♥ ❞✉❡ t♦ ❝♦♥t✐♥✉♦✉s ❣❡♥❡r❛t✐♦♥ ♦❢ ❣❛s✳ ❲❡ ❜❡❧✐❡✈❡ t❤❛t t❤✐s t❤❡r♠♦❞②♥❛♠✐❝ ❝♦♥❞✐t✐♦♥ ❞❡r✐✈❡❞ ❛ss✉♠✐♥❣ ❛ s✐♠♣❧❡ ♣❛tt❡r♥❡❞ t❡①t✉r❡ ✐s ❛ ♥❡❝❡ss❛r② ❜✉t ♥♦t ❛ s✉✣❝✐❡♥t ❝♦♥❞✐t✐♦♥ ❢♦r ♣❧❛str♦♥ r❡❝♦✈❡r②✳ ❲❤✐❧❡ t❤✐s ❝♦♥❞✐t✐♦♥ ✐s ❛♥ ✐♠♣♦rt❛♥t ❣❧♦❜❛❧ ❢❛❝t♦r✱ t❤❡r❡ ❛r❡ ✈❛r✐♦✉s ♦t❤❡r ❧♦❝❛❧ ❢❛❝t♦rs ✇❤✐❝❤ ❝❛♥ ❛❧s♦ ✐♥✢✉❡♥❝❡ t❤❡ ❞②♥❛♠✐❝s ♦❢ ♣❧❛str♦♥ s♣r❡❛❞✐♥❣ s✉❝❤ ❛s t❤❡ ♣✐❧❧❛r ❜r❡❛❦t❤r♦✉❣❤ ♣r❡ss✉r❡✱ ✷✾ ❝♦♥t❛❝t ♦❢ ❣❛s ♠❡♥✐s❝✉s ✇✐t❤ ♣✐❧❧❛rs ✐♥ t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ s♣r❡❛❞✐♥❣✱ ✸✼ ❝♦♥t❛❝t ❧✐♥❡ ♣✐♥♥✐♥❣✱ ❛♥❞ t❡①t✉r❡ r❡✲❡♥tr❛♥❝❡✳ ✺✽ ■♥ ♦r❞❡r t♦ ♠❛❦❡ t❤✐s ♠❡t❤♦❞ s❝❛❧❛❜❧❡ ❢♦r ❧❛r❣❡ s❝❛❧❡ ❞r❛❣ r❡❞✉❝t✐♦♥ ❛♣♣❧✐❝❛t✐♦♥s✱ ❝❛t❛❧②t✲ ✐❝❛❧❧② ❛❝t✐✈❡ t❡①t✉r❡s ✇❡r❡ ❢❛❜r✐❝❛t❡❞ ♦♥ t❡①t✉r❡❞ ❛❧✉♠✐♥✉♠ ✉s✐♥❣ ❛♥ ✐♥❡①♣❡♥s✐✈❡ ❝❛t❛❧②st✱ ♠❛♥❣❛♥❡s❡ ❞✐♦①✐❞❡✳ ❋❧❛t ♣♦❧✐s❤❡❞ ❛❧✉♠✐♥✉♠ ♣❧❛t❡s ✇❡r❡ ❧❛s❡r t❡①t✉r❡❞✱ ❡❧❡❝tr♦♣❧❛t❡❞ ✇✐t❤ ♠❛♥❣❛♥❡s❡ ❞✐♦①✐❞❡✱ ❛♥❞ ❝♦♥❢♦r♠❛❧❧② ❝♦❛t❡❞ ✇✐t❤ ❛ ✢✉♦r✐♥❛t❡❞ s✐❧❛♥❡ t♦ ♦❜t❛✐♥ ❝❛t❛❧②t✐❝❛❧❧② ❛❝t✐✈❡ s✉♣❡r❤②❞r♦♣❤♦❜✐❝ t❡①t✉r❡s ✭❉❡t❛✐❧s ♦❢ t❤❡ ♣r♦❝❡❞✉r❡ ❛r❡ ❡①♣❧❛✐♥❡❞ ✐♥ t❤❡ ❡①♣❡r✐♠❡♥t❛❧ s❡❝t✐♦♥✮✳ ❋✐❣✉r❡ ✸❝ s❤♦✇s ❛ ♠✐❝r♦❣r❛♣❤ ♦❢ ❛ s♦✲❝❛❧❧❡❞ ❡❣❣✲❝r❛t❡ ✺✾ ❛❧✉♠✐♥✉♠ ♠✐❝r♦t❡①t✉r❡ ❝♦❛t❡❞ ✇✐t❤ ▼♥❖2 ❛♥❞ s✐❧❛♥❡✳ ❚❤❡ ❲❡♥③❡❧ r♦✉❣❤♥❡ss ♦❢ t❤✐s t❡①t✉r❡ ✇❛s ❝❛❧❝✉❧❛t❡❞ ❢r♦♠ t❤❡ s✉r❢❛❝❡ ❛r❡❛ ♠❡❛s✉r❡❞ ✉s✐♥❣ ❛♥ ♦♣t✐❝❛❧ ♣r♦✜❧♦♠❡t❡r ❛♥❞ ❝♦rr❡s♣♦♥❞s t♦ r ❂ ✺✳✶✳ ❚❤❡ ✶✹


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❡❧❡❝tr♦♣❧❛t❡❞ ▼♥❖2 ❞❡♣♦s✐t ❤❛s ❜♦t❤ ♠✐❝r♦s❝❛❧❡ ❛♥❞ ♥❛♥♦s❝❛❧❡ ❢❡❛t✉r❡s ❛s ❝❛♥ ❜❡ s❡❡♥ ✐♥ ❙✉♣♣❧❡♠❡♥t❛r② ❋✐❣✉r❡ ❙✸✳ ❙✐♥❝❡ t❤❡ t❡①t✉r❡ ❤❛s ❛ ❞✉❛❧ s❝❛❧❡ r♦✉❣❤♥❡ss✱ ✐t ✐s ❡①♣❡❝t❡❞ t❤❛t t❤❡ ♥❛♥♦✲❈❛ss✐❡ st❛t❡ ❝❛♥ ❜❡ s✉♣♣♦rt❡❞ ✐♥ t❤❡ ♥❛♥♦❢❡❛t✉r❡s ♦❢ t❤❡ t❡①t✉r❡ ❡✈❡♥ ❛❢t❡r ✐♠♣r❡❣✲ ♥❛t✐♦♥ ❛t t❤❡ ♠✐❝r♦✲s❝❛❧❡✳ ✸✻ ❙✉♣♣❧❡♠❡♥t❛r② ❱✐❞❡♦ ❙✷ s❤♦✇s t❤❡ ✐♥✢❛t✐♦♥ ♦❢ ♣❧❛str♦♥ ✇✐t❤✐♥ t❤❡ ❡❣❣✲❝r❛t❡ ❛❧✉♠✐♥✉♠ ♠✐❝r♦t❡①t✉r❡✳ ■♥ ♦r❞❡r t♦ ❡♥s✉r❡ ♠❛①✐♠✉♠ ✇❛t❡r ✐♠♣r❡❣♥❛t✐♦♥ ✐♥ t❤❡ t❡①t✉r❡✱ t❤❡ s❛♠♣❧❡s ✇❡r❡ ✐♠♠❡rs❡❞ ✐♥t♦ ❜♦✐❧❡❞ ❞❡✐♦♥✐s❡❞ ✇❛t❡r ✉♥❞❡r ✈❛❝✉✉♠ ❢♦r ✶✷ ❤♦✉rs ❜❡❢♦r❡ t❤❡ ✐♥tr♦❞✉❝t✐♦♥ ♦❢ ❤②❞r♦❣❡♥ ♣❡r♦①✐❞❡✳ ❋✐❣✉r❡ ✸❞ s❤♦✇s t❤❡ ♣r♦❝❡ss ♦❢ ♣❧❛str♦♥ r❡❝♦✈❡r② ♦♥ s✉❝❤ ❛ t❡①t✉r❡ ❛t ❛ ❤②❞r♦❣❡♥ ♣❡r♦①✐❞❡ ❝♦♥❝❡♥tr❛t✐♦♥ ♦❢ ✶✳✺ ✇t✳✪✳ ◆♦t❡ t❤❛t t❤❡ ▼♥❖2 ❝♦❛t✐♥❣ ✇❛s ♥♦t r❡♠♦✈❡❞ ❢r♦♠ t❤❡ t♦♣s ♦❢ t❤❡ ❛❧✉♠✐♥✉♠ ♣②r❛♠✐❞s ✐♥ t❤✐s ❡①♣❡r✐♠❡♥t❀ ❤♦✇❡✈❡r✱ ♣❧❛str♦♥ r❡❝♦✈❡r② ✇❛s st✐❧❧ ❢❛✈♦r❡❞ ♦✈❡r ❜❧✐st❡r ❢♦r♠❛t✐♦♥✳ ❚❤✐s ✐s ✐♥ ❛❣r❡❡♠❡♥t ✇✐t❤ ♦✉r ♣r❡✈✐♦✉s ♦❜s❡r✈❛t✐♦♥ ♦♥ s✐❧✐❝♦♥ ♣✐❧❧❛rs ✇✐t❤ ♣❧❛t✐♥✉♠✲❝♦❛t❡❞ t♦♣s✳ ■♥ ❛❧❧ ♦❢ t❤❡ ❡①♣❡r✐♠❡♥ts ❢♦r ✇❤✐❝❤ ♣❧❛str♦♥ r❡❝♦✈❡r② ✇❛s ♦❜s❡r✈❡❞✱ t❤❡ ❣❛s s♣r❡❛❞s ❧❛t❡r❛❧❧② ❛❧♦♥❣ t❤❡ t❡①t✉r❡ ✉♥t✐❧ t❤❡ ❝♦♥t❛❝t ❧✐♥❡ ❜❡❝♦♠❡s s✉✣❝✐❡♥t❧② ♣✐♥♥❡❞ t♦ ♣r❡✈❡♥t ❢✉rt❤❡r ❛❞✈❛♥❝❡♠❡♥t✳ ❲❤❡♥ t❤❡ ❝♦♥t❛❝t ❧✐♥❡ ✐s ✉♥❛❜❧❡ t♦ ❞❡✲♣✐♥ ❢✉rt❤❡r✱ t❤❡ ❣❛s ❜r❡❛❦s t❤r♦✉❣❤ t❤❡ t♦♣ ♦❢ t❤❡ ♣✐❧❧❛rs ❛♥❞ ❢♦r♠s ❛ ❤❡♠✐✇✐❝❦✐♥❣ ❜❧✐st❡r✳ ❆t t❤✐s ♣♦✐♥t t❤❡ ❣❛s ❣❡♥❡r❛t❡❞ ✇✐t❤✐♥ t❤❡ t❡①t✉r❡ ✐s ♣r❡❢❡r❡♥t✐❛❧❧② ❝❤❛♥♥❡❧❡❞ ✐♥t♦ t❤❡ ❤❡♠✐✇✐❝❦✐♥❣ ❜❧✐st❡r ✇❤✐❝❤ ❤❛s ❛ ❧♦✇ ▲❛♣❧❛❝❡ ♣r❡ss✉r❡ ❞✉❡ t♦ t❤❡ ❧❛r❣❡ r❛❞✐✉s ♦❢ ❝✉r✈❛t✉r❡ ♦❢ t❤❡ ❜❧✐st❡r✳ ❚❤✐s st♦♣s t❤❡ ❝♦♥♥❡❝t❡❞ ♣❧❛str♦♥ ❢r♦♠ s♣r❡❛❞✐♥❣ ❢✉rt❤❡r ✐♥t♦ ♦t❤❡r ✇❡tt❡❞ ♣❛rts ♦❢ t❤❡ t❡①t✉r❡✳ ❍❡♥❝❡✱ ✐t ✐s ✐♠♣♦rt❛♥t t♦ ❢❛❜r✐❝❛t❡ s✉r❢❛❝❡ t❡①t✉r❡s ✇✐t❤ ❤✐❣❤ ♣❧❛str♦♥ ❝♦♥♥❡❝t✐✈✐t② ❛♥❞ ❧♦✇ ♣✐♥♥✐♥❣ ❞❡❢❡❝ts ❢♦r ❝♦♠♣❧❡t❡ ♣❧❛str♦♥ r❡❝♦✈❡r②✳ ❚♦ ♠❡❛s✉r❡ t❤❡ ❞r❛❣ r❡❞✉❝t✐♦♥ ♣❡r❢♦r♠❛♥❝❡ ❛♥❞ s❧✐♣ r❡❝♦✈❡r② ❝❛♣❛❜✐❧✐t✐❡s ♦❢ t❤❡s❡ ♣❧❛str♦♥ ♣r♦♠♦t✐♥❣ t❡①t✉r❡s✱ ❛❧✉♠✐♥✉♠ r♦t♦rs ✇✐t❤ ❝❛t❛❧②t✐❝❛❧❧② ❛❝t✐✈❡ t❡①t✉r❡s ✇❡r❡ t❡st❡❞ ✐♥ ❛ ❚❛②❧♦r✲❈♦✉❡tt❡ ❝❡❧❧ ✉♥❞❡r t✉r❜✉❧❡♥t ✢♦✇ ❝♦♥❞✐t✐♦♥s✳ ✶✻ ❈②❧✐♥❞r✐❝❛❧ r♦t♦rs ✭Rrotor ❂ ✶✹ ♠♠✮ ✇❡r❡ t❡①t✉r❡❞ ❜② ♠❛❝❤✐♥✐♥❣ ♠✐❝r♦❣r♦♦✈❡s ♦♥t♦ t❤❡ s✉r❢❛❝❡ ✐♥ ❜♦t❤ t❤❡ ❝✐r❝✉♠❢❡r❡♥t✐❛❧ ❛♥❞ ❧♦♥❣✐t✉❞✐♥❛❧ ❞✐r❡❝t✐♦♥s t♦ ♦❜t❛✐♥ ❛ r❡❣✉❧❛r ❛rr❛② ♦❢ sq✉❛r❡ ♠✐❝r♦♣✐❧❧❛rs ♦❢ ✇✐❞t❤ ✶✵✵ µ♠✱ ❤❡✐❣❤t ✷✵✵ µ♠✱ ❛♥❞ s♣❛❝✐♥❣ ✶✵✵ µ♠ ✭r ❂ ✷✱ φs ❂ ✵✳✷✺✮✳ ❚❤❡ r♦t♦r ✇❛s t❤❡♥ ❡❧❡❝tr♦♣❧❛t❡❞ ✇✐t❤ ▼♥❖2 ❛♥❞ ❤②❞r♦♣❤♦❜✐③❡❞ ✉s✐♥❣ ❛ ✢✉♦r✐♥❛t❡❞ s✐❧❛♥❡ ✐♥ ❛ s✐♠✐❧❛r ♠❛♥♥❡r t♦ t❤❡ ✢❛t ✶✺



❧❛s❡r✲t❡①t✉r❡❞ ❛❧✉♠✐♥✉♠ s❛♠♣❧❡s✳ ❋✐❣✉r❡ ✺❛ s❤♦✇s ❛ ♠✐❝r♦❣r❛♣❤ ♦❢ t❤✐s t❡①t✉r❡❀ ✇❤❡r❡ t❤❡ ♠✐❝r♦✲❞❡♣♦s✐ts ♦♥ t❤❡ ♣✐❧❧❛rs ❝♦rr❡s♣♦♥❞ t♦ ▼♥❖2 ❡❧❡❝tr♦♣❧❛t❡❞ ♦♥ t❤❡ t❡①t✉r❡✱ ✇❤✐❧❡ ❋✐❣✉r❡ ✺❜ s❤♦✇s ❛ ❞r♦♣❧❡t ♦❢ ✇❛t❡r s✐tt✐♥❣ ♦♥ t❤❡ s✉♣❡r❤②❞r♦♣❤♦❜✐❝ s✉r❢❛❝❡ ♦❢ t❤❡ t❡①t✉r❡❞ ❛❧✉♠✐♥✉♠ r♦t♦r✳ ❆❧❧ ❞r❛❣ r❡❞✉❝t✐♦♥ ❡①♣❡r✐♠❡♥ts ✇❡r❡ ♣❡r❢♦r♠❡❞ ✉♥❞❡r ❢✉❧❧② t✉r❜✉❧❡♥t ✢♦✇ ❝♦♥❞✐t✐♦♥s ✐♥ ❛ ❝✉st♦♠✲❢❛❜r✐❝❛t❡❞ ❚❛②❧♦r✲❈♦✉❡tt❡ ❈❡❧❧ ✭s❤♦✇♥ ✐♥ ❋✐❣✉r❡ ✺❝✮ ❛t ❘❡ = ✹✼✱✵✵✵✱ ✇❤❡r❡ t❤❡ ❘❡②♥♦❧❞s ♥✉♠❜❡r ✐s ❝❛❧❝✉❧❛t❡❞ ❛s ❘❡ = Ri Ω∆R/ν ✳ ❍❡r❡ Ri ✐s t❤❡ r♦t♦r r❛❞✐✉s✱

∆R ✐s t❤❡ ❝❡❧❧ ❣❛♣✱ Ω ✐s t❤❡ ❛♥❣✉❧❛r ✈❡❧♦❝✐t② ♦❢ t❤❡ r♦t♦r ✭Rrotor ✮✱ ❛♥❞ ν ✐s t❤❡ ❦✐♥❡♠❛t✐❝ ✈✐s❝♦s✐t② ♦❢ t❤❡ ✇♦r❦✐♥❣ ✢✉✐❞✳ ❚❤❡ ♠❡❛♥ ✈❛❧✉❡ ♦❢ t❤❡ ❢r✐❝t✐♦♥❛❧ ✇❛❧❧ s❤❡❛r str❡ss ✭τw ✮ ♦♥ t❤❡ r♦t♦r ✐s ❞❡✜♥❡❞ ❛s τw = T /2πRi2 L ✇❤❡r❡ T ❛♥❞ L ❛r❡ t❤❡ ❛✈❡r❛❣❡❞ t♦rq✉❡ ❡①♣❡r✐❡♥❝❡❞ ❜② t❤❡ r♦t♦r ❛♥❞ t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ r♦t♦r r❡s♣❡❝t✐✈❡❧②✳ ❚❤❡ ♣❡r❝❡♥t❛❣❡ ❞r❛❣ r❡❞✉❝t✐♦♥ ✐s ❞❡✜♥❡❞ ❛s DR = 100 × (τ❙♠♦♦t❤ − τ❈❛ss✐❡ )/τ❙♠♦♦t❤ ✱ ✇❤❡r❡ τ❙♠♦♦t❤ ✐s t❤❡ s❤❡❛r str❡ss ♦♥ ❛♥ ✉♥t❡①t✉r❡❞ ❝②❧✐♥❞r✐❝❛❧ r♦t♦r ❛♥❞ τ❈❛ss✐❡ ✐s t❤❡ s❤❡❛r str❡ss ♦♥ t❤❡ ♣❛tt❡r♥❡❞ r♦t♦r ✇✐t❤ ❛ ❢✉❧❧② ✐♥✢❛t❡❞ ♣❧❛str♦♥✳ ◆♦t❡ t❤❛t t❤❡ s♦❧✐❞ ❢r❛❝t✐♦♥ φs ♦❢ t❤❡ ❣r♦♦✈❡❞ r♦t♦r ✐s ❤✐❣❤ ✭φs = 0.25✮ t♦ ♣r♦♠♦t❡ ♣❧❛str♦♥ r❡❣❡♥❡r❛t✐♦♥ ✭❙❡❡ ❋✐❣✉r❡ ✹✮✱ ❜✉t t❤❡ ❡①♣❡r✐♠❡♥t❛❧❧② ♠❡❛s✉r❡❞ t✉r❜✉❧❡♥t ❞r❛❣ r❡❞✉❝t✐♦♥ ✐s ♦♥❧② ❛❜♦✉t ✸✪ ❝♦♠♣❛r❡❞ t♦ ❛ s♠♦♦t❤ ✉♥t❡①t✉r❡❞ ❤②❞r♦♣❤✐❧✐❝ r♦t♦r ❛t t❤❡ s❛♠❡ ❘❡②♥♦❧❞s ♥✉♠❜❡r ♦❢ ❘❡ ❂ ✹✼✱✵✵✵✳ ❍♦✇❡✈❡r✱ t❤✐s r♦t♦r ❝❛♥ st✐❧❧ ❜❡ ✉s❡❞ t♦ ❡✈❛❧✉❛t❡ t❤❡ ❡✛❡❝t✐✈❡♥❡ss ♦❢ t❤❡ ♣❧❛str♦♥ r❡❝♦✈❡r② ♠❡t❤♦❞ ❜② ❝♦♠♣❛r✐♥❣ t❤❡ t♦rq✉❡s ♦♥ t❤❡ r♦t♦r ✐♥ ✐ts ♦r✐❣✐♥❛❧ ♣❧❛str♦♥ ✐♥✢❛t❡❞ st❛t❡ ✭❈❛ss✐❡✮✱ ✐ts ✇❡tt❡❞ st❛t❡ ✭❲❡♥③❡❧✮ ❛♥❞ ✐ts ✜♥❛❧ r❡❝♦✈❡r❡❞ st❛t❡✳ ❚❤❡ ♣❧❛str♦♥ r❡❝♦✈❡r② t❡sts ✇❡r❡ ♣❡r❢♦r♠❡❞ ✐♥ t❤r❡❡ st❛❣❡s ❛s s❤♦✇♥ ✐♥ ❋✐❣✉r❡ ✺❡✳ ❚❤❡ ♣❧❛str♦♥ ✐s ✐♥✐t✐❛❧❧② t❡st❡❞ ❢♦r ✐ts st❛❜✐❧✐t② ❜② r✉♥♥✐♥❣ ❛ ❧♦♥❣ t❡st ✭✹ ❤♦✉rs✮ ✉♥❞❡r ❛✐r✲s❛t✉r❛t❡❞ ✇❛t❡r ❝♦♥❞✐t✐♦♥s✳ ❚❤❡r❡ ✐s ♠✐♥✐♠❛❧ ❝❤❛♥❣❡ ✐♥ t❤❡ s❤❡❛r str❡ss✱ τ❈❛ss✐❡ ✱ ❛s t❤❡ ♣❧❛str♦♥ ✐s st❛❜❧❡ ❞✉r✐♥❣ t❤✐s t✐♠❡ ♣❡r✐♦❞✳ ❆ ✇❡tt✐♥❣ tr❛♥s✐t✐♦♥ ✇❛s t❤❡♥ ❛rt✐✜❝✐❛❧❧② ✐♥❞✉❝❡❞ ❜② ✢✉s❤✐♥❣ ❞❡✲ ❛❡r❛t❡❞ ✇❛t❡r t❤r♦✉❣❤ t❤❡ s②st❡♠ ✉♥t✐❧ t❤❡ s❤❡❛r str❡ss r✐s❡s ❛♥❞ r❡❛❝❤❡s ❛ ♥❡✇ st❡❛❞② st❛t❡ ✈❛❧✉❡ ❝♦rr❡s♣♦♥❞✐♥❣ t♦ ❛ ❢✉❧❧② ✇❡tt❡❞ ❲❡♥③❡❧ st❛t❡✱ τ❲❡♥③❡❧ ✳ ❱✐s✉❛❧ ✐♥s♣❡❝t✐♦♥ ♦❢ t❤❡ t❡①t✉r❡❞ r♦t♦r ✐♥ t❤❡ ❝❡❧❧ ❛❧s♦ s❤♦✇❡❞ ❛❧♠♦st ❝♦♠♣❧❡t❡ ❧♦ss ♦❢ t❤❡ s✐❧✈❡r② ♣❧❛str♦♥ ✜❧♠✳ ❚❤❡ r♦t♦r ✇❛s ♠❛✐♥t❛✐♥❡❞ ✐♥ t❤✐s st❛t❡ ❢♦r ❛ ❢❡✇ ♠✐♥✉t❡s ❜❡❢♦r❡ ✐♥tr♦❞✉❝✐♥❣ ❤②❞r♦❣❡♥ ♣❡r♦①✐❞❡ ✐♥t♦ ✶✻


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t❤❡ s②st❡♠✳ ✶ ♠❧ ♦❢ ✼✳✺ ✇t✪ ❤②❞r♦❣❡♥ ♣❡r♦①✐❞❡ ✇❛s ✐♥tr♦❞✉❝❡❞ ✐♥t♦ t❤❡ ❝❡❧❧ ♦✈❡r ❛ ♣❡r✐♦❞ ♦❢ ✺ ♠✐♥s t♦ ✐♥❝r❡❛s❡ t❤❡ ❍2 ❖2 ❜✉❧❦ ❝♦♥❝❡♥tr❛t✐♦♥ ✉♣ t♦ 1.3 × 10−5 ♠♦❧ ♠❧−1 ✳ ❚❤❡ ❛✈❡r❛❣❡ s❤❡❛r str❡ss ❡①❡rt❡❞ ♦♥ t❤❡ ✐♥♥❡r r♦t♦r ✐s ♦❜s❡r✈❡❞ t♦ ❞❡❝r❡❛s❡ s❧♦✇❧② ❛♥❞ r❡❛❝❤❡s ❛ st❡❛❞② st❛t❡ ✈❛❧✉❡✱ τ❘❡❝♦✈❡r❡❞ ✱ ♦✈❡r ❛ ♣❡r✐♦❞ ♦❢ ✷✵ ♠✐♥ ❛❢t❡r ✇❤✐❝❤ ❛♥② ❛❞❞✐t✐♦♥❛❧ ♦①②❣❡♥ ♣r♦❞✉❝❡❞ ♦♥❧② ❝♦♥tr✐❜✉t❡❞ t♦ ❜❧✐st❡r ❢♦r♠❛t✐♦♥ ♦♥ t❤❡ r♦t♦r✳ ❚❤❡ ♣❧❛str♦♥ r❡❝♦✈❡r② ✐s ♦❜s❡r✈❡❞ t♦ ❜❡ ♣❛t❝❤② ✐♥ ♠♦st ♣❛rts ❞✉❡ t♦ ♣✐♥♥✐♥❣ ❞❡❢❡❝ts ❛s ❝❛♥ ❜❡ s❡❡♥ ✐♥ ❙✉♣♣❧❡♠❡♥t❛r② ❋✐❣✉r❡ ❙✻✳ ❋✐❣✉r❡ ✺❞ s❤♦✇s ✧✉♥✇r❛♣♣❡❞✧ ✈✐❡✇s ♦❢ t❤❡ r♦t♦r ✐♥ ✐ts ❈❛ss✐❡ st❛t❡✱ ❲❡♥③❡❧ st❛t❡ ❛♥❞ ❛❢t❡r ♣❛rt✐❛❧ r❡❝♦✈❡r② ✭r❡❝♦♥str✉❝t❡❞ ❢r♦♠ ❛ s❡q✉❡♥❝❡ ♦❢ ✐♠❛❣❡s✮✳ ❆❧t❤♦✉❣❤ t❤❡ ♣❧❛str♦♥ r❡❝♦✈❡r② ✐s ✐♠♣❡r❢❡❝t✱ t❤❡ ♣r❡s❡♥❝❡ ♦❢ t❤❡ ❛✐r ❧❛②❡r ❝♦♥tr✐❜✉t❡s t♦ ✐♠♣r♦✈✐♥❣ t❤❡ ❛✈❡r❛❣❡ ❧❡✈❡❧ ♦❢ ✇❛❧❧ s❧✐♣ ✐♥❞✉❝❡❞ ♦♥ t❤❡ r♦t♦r s✉r❢❛❝❡✳ ❚❤❡ ♣❡r❝❡♥t❛❣❡ s❧✐♣ r❡❝♦✈❡r② ✐s ❝❛❧❝✉❧❛t❡❞ ❛s SR =

100 × (τ❲❡♥③❡❧ − τ❘❡❝♦✈❡r❡❞ )/(τ❲❡♥③❡❧ − τ❈❛ss✐❡ )✱ ✇❤❡r❡ τ❲❡♥③❡❧ ✐s t❤❡ t♦rq✉❡ ♦♥ t❤❡ ❝♦♠♣❧❡t❡❧② ✇❡tt❡❞ r♦t♦r ❛♥❞ τ❘❡❝♦✈❡r❡❞ ✐s t❤❡ t♦rq✉❡ ♦♥ t❤❡ r♦t♦r ❛❢t❡r ♣❧❛str♦♥ r❡❝♦✈❡r②✳ ❚❤❡ ❛✈❡r❛❣❡ s❧✐♣ r❡❝♦✈❡r② ♣❡r❝❡♥t❛❣❡ ♦✈❡r ❢♦✉r r❡♣❡❛t❡❞ ❡①♣❡r✐♠❡♥ts ✇❛s ❢♦✉♥❞ t♦ ❜❡ ✺✶ ± ✶✹ ✪✳ ■t ❝❛♥ ❜❡ ❡st✐♠❛t❡❞ ❢r♦♠ t❤❡ r♦t♦r t❡①t✉r❡ ♣❛r❛♠❡t❡rs t❤❛t 6.1 × 10−7 ♠♦❧ ♦❢ ♦①②❣❡♥ ✐s r❡q✉✐r❡❞ ❢♦r ❡✈❡r② sq✉❛r❡ ❝❡♥t✐♠❡t❡r ♦❢ r♦t♦r s✉r❢❛❝❡ ❢♦r ♣❧❛str♦♥ r❡❝♦✈❡r② t♦ ♦❝❝✉r ♦✈❡r t❤❡ ❝♦♠♣❧❡t❡ s✉r❢❛❝❡ ❛r❡❛ ♦❢ t❤❡ r♦t♦r ❛ss✉♠✐♥❣ ♥❡❛r ❛t♠♦s♣❤❡r✐❝ ♣r❡ss✉r❡ ❝♦♥❞✐t✐♦♥s ❛♥❞ ❛ t❡①t✉r❡ ❞❡♣t❤ ♦❢ ❍ ❂ ✷✵✵ µ♠✳ ❍♦✇❡✈❡r✱ s✐♥❝❡ ❛❧❧ t❤❡ ❤②❞r♦❣❡♥ ♣❡r♦①✐❞❡ ✐♥❥❡❝t❡❞ ❞♦❡s ♥♦t ❝♦♠❡ ✐♥t♦ ❝♦♥t❛❝t ✇✐t❤ t❤❡ ❝❛t❛❧②st s✉r❢❛❝❡ ❛♥❞ t❤✉s ❞♦❡s ♥♦t ❝♦♥tr✐❜✉t❡ t♦ ♣❧❛str♦♥ r❡❝♦✈❡r②✱ ✐t ✐s ♥❡❝❡ss❛r② t♦ ✐♥❥❡❝t ❛♥ ❛♠♦✉♥t ♦❢ ❤②❞r♦❣❡♥ ♣❡r♦①✐❞❡✱ n✐♥❥❡❝t❡❞ ❍2 ❖2 ✱ ❤✐❣❤❡r t❤❛♥ t❤❡ r❡q✉✐r❡❞ st♦✐❝❤✐♦♠❡tr✐❝ q✉❛♥t✐t②✱ n✉t✐❧✐③❡❞ ❍2 ❖2 ✱ ✐♥ ♦r❞❡r t♦ ❛❝❤✐❡✈❡ ❛ s✐❣♥✐✜❝❛♥t r❛t❡ ♦❢ ♣❧❛str♦♥ r❡❝♦✈❡r②✳ ❚❤❡ ♠❛①✐♠✉♠ ❡✣❝✐❡♥❝② ♦❢ ❝❤❡♠✐❝❛❧ r❡❣❡♥❡r❛t✐♦♥ ♦❢ ♣❧❛str♦♥ ❝❛♥ ❜❡ ❝❛❧❝✉❧❛t❡❞ ❛s ✐♥❥❡❝t❡❞ ηmax = n✉t✐❧✐③❡❞ ❍2 ❖2 /n❍2 ❖2 ✱ ❛♥❞ ✐t ✇❛s ❢♦✉♥❞ t♦ ❜❡ ✶✳✶✪ ✐♥ t❤❡ ❝✉rr❡♥t ❝♦♥✜❣✉r❛t✐♦♥✳ ❙✐♥❝❡ t❤❡

♣❧❛str♦♥ r❡❝♦✈❡r② ✐s ♣❛rt✐❛❧✱ t❤✐s ❡✣❝✐❡♥❝② ♠✉st ❛❧s♦ ❜❡ ♠✉❧t✐♣❧✐❡❞ ❜② t❤❡ ♣❡r❝❡♥t❛❣❡ ❛r❡❛ r❡❝♦✈❡r❡❞ t♦ ❣❡t t❤❡ ❡✛❡❝t✐✈❡ ❡✣❝✐❡♥❝②✳ ❋♦r ❡①❛♠♣❧❡✱ ✐♥ ❋✐❣✉r❡ ✺❞✱ t❤❡ ♣❡r❝❡♥t❛❣❡ ♦❢ t❤❡ ✇❡tt❡❞ ❛r❡❛ ♦✈❡r ✇❤✐❝❤ ♣❧❛str♦♥ r❡❝♦✈❡r② ✐s ♦❜s❡r✈❡❞ ✐s ♦♥❧② ❡st✐♠❛t❡❞ t♦ ❜❡ ✺✽✪✱ ❛♥❞ ❤❡♥❝❡ t❤❡ ❡✣❝✐❡♥❝② η ❂ ✵✳✻✪✳ ❚❤❡ ♣❧❛str♦♥ r❡❝♦✈❡r② ❝♦✈❡r❛❣❡ ❝♦✉❧❞ ❜❡ ✐♥❝r❡❛s❡❞ ❜② ✐♥❝♦r♣♦r❛t✐♥❣ ❛ ❞✉❛❧✲s❝❛❧❡❞ t❡①t✉r❡ ❧✐❦❡ t❤❛t ✐♥ t❤❡ ❧❛s❡r t❡①t✉r❡❞ ❛❧✉♠✐♥✉♠ s✉❜str❛t❡s✱ ✇❤✐❝❤ ❛❧❧♦✇s ❢♦r t❤❡ ✶✽


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❣❛s t♦ s♣r❡❛❞ ♠♦r❡ ❡❛s✐❧② t❤r♦✉❣❤ t❤❡ ♥❛♥♦t❡①t✉r❡ ✇✐t❤♦✉t ♣✐♥♥✐♥❣✱ t❤✉s r❡❞✉❝✐♥❣ ❝♦♥t❛❝t ❛♥❣❧❡ ❤②st❡r❡s✐s✳ ❚❤❡ ♠❛ss tr❛♥s❢❡r ❡✣❝✐❡♥❝② ❝♦✉❧❞ ❜❡ ❛❧s♦ ♣♦t❡♥t✐❛❧❧② ✐♠♣r♦✈❡❞ ❜② ✐♥❥❡❝t✐♥❣ t❤❡ ❤②❞r♦❣❡♥ ♣❡r♦①✐❞❡ ❞✐r❡❝t❧② ✐♥t♦ t❤❡ ❜♦✉♥❞❛r② ❧❛②❡r ❢r♦♠ ✇✐t❤✐♥ ❛ ♣♦r♦✉s s✉r❢❛❝❡✳

❈♦♥❝❧✉s✐♦♥ ❲❡ ❤❛✈❡ ❞❡♠♦♥str❛t❡❞ ❛ ♥♦✈❡❧ ❝❤❡♠✐❝❛❧ ♠❡t❤♦❞ t♦ r❡♣❧❡♥✐s❤ ❛ s✉r❢❛❝❡ ❛✐r ❧❛②❡r ♦r ♣❧❛str♦♥ ✐♥✲s✐t✉ ♦♥ s✉♣❡r❤②❞r♦♣❤♦❜✐❝ t❡①t✉r❡s t❤❛t ❛r❡ ❢✉❧❧② ✐♠♠❡rs❡❞ ✉♥❞❡r✇❛t❡r ✉s✐♥❣ ❝❛t❛❧②t✐❝ ❞❡✲ ❝♦♠♣♦s✐t✐♦♥ ♦❢ ❤②❞r♦❣❡♥ ♣❡r♦①✐❞❡✳ ❲❡ ❛❧s♦ s❤♦✇❡❞ t❤❛t ♣❧❛str♦♥ s♣r❡❛❞✐♥❣ ✐s ♣♦ss✐❜❧❡ ♦♥ ♠✐❝r♦t❡①t✉r❡s ✇✐t❤♦✉t ❛ ♥❛♥♦✲❈❛ss✐❡ st❛t❡ ❜② ♣❡r❢♦r♠✐♥❣ ❡①♣❡r✐♠❡♥ts ♦♥ s✐❧✐❝♦♥ ♠✐❝r♦t❡①✲ t✉r❡s ✇✐t❤ ♣❧❛t✐♥✉♠ ❝❛t❛❧②st ✇✐t❤♦✉t ❛ s❡❝♦♥❞ ❧❡✈❡❧ ♦❢ ♥❛♥♦✲t❡①t✉r❡✳ ❖✉r t❤❡r♠♦❞②♥❛♠✐❝ ❢r❛♠❡✇♦r❦ ❢♦r ♣r❡❞✐❝t✐♥❣ r❡❝♦✈❡r② ♦♥ ❤②❞r♦♣❤♦❜✐❝ t❡①t✉r❡s ❝❛♥ ❜❡ ✉s❡❞ t♦ ❞❡s✐❣♥ t❡①t✉r❡s ✇✐t❤ ♦♣t✐♠❛❧ r♦✉❣❤♥❡ss ❛♥❞ s✉r❢❛❝❡ ❝❤❡♠✐str② ❢✉♥❝t✐♦♥❛❧✐③❛t✐♦♥ ❢♦r ♣❧❛str♦♥ r❡❝♦✈❡r②✳ ❲❡ ❛❧s♦ ❞❡♠♦♥str❛t❡ t❤❡ s❝❛❧❛❜✐❧✐t② ♦❢ t❤✐s ♠❡t❤♦❞ ❜② ❢❛❜r✐❝❛t✐♥❣ ❛ s✉✐t❛❜❧❡ ✬❡❣❣ ❝r❛t❡✬ str✉❝✲ t✉r❡ ♦♥ ❧❛s❡r✲t❡①t✉r❡❞ ❛❧✉♠✐♥✉♠ s✉r❢❛❝❡s ✇✐t❤ ❛♥ ✐♥❡①♣❡♥s✐✈❡ ♠❛♥❣❛♥❡s❡ ❞✐♦①✐❞❡ ❝❛t❛❧②st ❛♥❞ ❞❡♠♦♥str❛t✐♥❣ ♣❧❛str♦♥ r❡❝♦✈❡r② ♦♥ t❤✐s s✉r❢❛❝❡✳ ❋✐♥❛❧❧② ✇❡ ❤❛✈❡ ♣❡r❢♦r♠❡❞ ✢✉✐❞ ❢r✐❝✲ t✐♦♥❛❧ ❞r❛❣ ♠❡❛s✉r❡♠❡♥ts ♦♥ ❛ ♠✐❝r♦t❡①t✉r❡❞ ❛❧✉♠✐♥✉♠ r♦t♦r ❧♦❛❞❡❞ ✇✐t❤ ▼♥❖2 ❝❛t❛❧②st✳ ❙❦✐♥ ❢r✐❝t✐♦♥ ♠❡❛s✉r❡♠❡♥ts ✐♥ ❛ t✉r❜✉❧❡♥t ❚❛②❧♦r✲❈♦✉❡tt❡ ✢♦✇ s❤♦✇❡❞ t❤❛t ✐t ✐s ♣♦ss✐❜❧❡ t♦ ♦❜t❛✐♥ ❛♥ ❛✈❡r❛❣❡ s❧✐♣ r❡❝♦✈❡r② ♦❢ ✺✶ ± ✶✹✪ ❢r♦♠ t❤❡ t❡①t✉r❡✬s ❢✉❧❧② ✇❡tt❡❞ st❛t❡ ✇❤❡♥ ❤②❞r♦❣❡♥ ♣❡r♦①✐❞❡ ✇❛s ❛❞❞❡❞ t♦ t❤❡ s②st❡♠✳ ❲❡ ❛♥t✐❝✐♣❛t❡ t❤❛t t❤❡s❡ ❝❛t❛❧②t✐❝❛❧❧②✲❛❝t✐✈❡ ❞r❛❣✲r❡❞✉❝✐♥❣ s✉r❢❛❝❡ ♠✐❝r♦t❡①t✉r❡s ❝❛♥ ❜❡ s❝❛❧❡❞✲✉♣ ❢✉rt❤❡r ❛♥❞ ✐♠♣❧❡♠❡♥t❡❞ ✐♥ t❤❡ ♠❛r✐♥❡ ❡♥✈✐r♦♥❡♠♥t t♦ ❡①t❡♥❞ t❤❡ st❛❜❧❡ ❧✐❢❡t✐♠❡s ♦❢ ❢r✐❝t✐♦♥❛❧ ❞r❛❣ r❡❞✉❝t✐♦♥ ❛♥❞ ❛♥t✐✲❝♦rr♦s✐♦♥✳

✶✾



❊①♣❡r✐♠❡♥t❛❧ ▼❡t❤♦❞s ❋❛❜r✐❝❛t✐♦♥ ♦❢ ❙✐❧✐❝♦♥ ▼✐❝r♦♣✐❧❧❛rs ✇✐t❤ ❈❛t❛❧②st✲❢r❡❡ ❚♦♣s

❲❡ ❢❛❜r✐❝❛t❡❞ ♠✐❝r♦t❡①t✉r❡s ✇✐t❤ s❡❧❡❝t✐✈❡ ❝❛t❛❧②st ❝♦✈❡r❛❣❡ ♦♥ s✐❧✐❝♦♥ ✇❛❢❡rs ✉s✐♥❣ ✐♠❛❣❡ r❡✈❡rs❛❧ ♣❤♦t♦❧✐t❤♦❣r❛♣❤②✱ r❡❛❝t✐✈❡ ✐♦♥ ❡t❝❤✐♥❣ ✭❘■❊✮✱ ❡✲❜❡❛♠ ❡✈❛♣♦r❛t✐♦♥ ♦❢ ❝❛t❛❧②st ♦♥t♦ t❤❡ ♣✐❧❧❛rs✱ ❛♥❞ s✉❜s❡q✉❡♥t ❧✐❢t✲♦✛ ♦❢ t❤❡ ❝❛t❛❧②st ❢r♦♠ t❤❡ t♦♣s ♦❢ t❤❡ ♠✐❝r♦♣✐❧❧❛rs✳ ✻✧ ❙✐❧✐❝♦♥ ✇❛❢❡rs ✭❯♥✐✈❡rs✐t② ✇❛❢❡rs✮ ✇❡r❡ ✜rst ❝♦❛t❡❞ ✇✐t❤ t❤❡ ✐♠❛❣❡ r❡✈❡rs❛❧ ♣❤♦t♦r❡s✐st✱ ❆❩✺✷✶✹✱ ❛♥❞ ❛♥ ❛rr❛② ♦❢ ❝✐r❝✉❧❛r ♣✐❧❧❛rs ♦❢ ♣❤♦t♦r❡s✐st ✇❛s ❝r❡❛t❡❞ ♦♥ t❤❡ ✇❛❢❡r ❜② ✉s✐♥❣ ✐♠❛❣❡ r❡✈❡rs❛❧ ♣❤♦t♦❧✐t❤♦❣r❛♣❤②✳ ❘■❊ ✭❙❚❙ P❡❣❛s✉s✮ ✇❛s t❤❡♥ ✉s❡❞ t♦ ❡t❝❤ t❤❡ ❡①♣♦s❡❞ ❙✐ ✐♥ t❤❡ ♣❤♦t♦r❡s✐st ♠❛s❦ t♦ ❝r❡❛t❡ ♠✐❝r♦♣✐❧❧❛rs ✐♥ t❤❡ s✐❧✐❝♦♥✳ ❚❤❡ ❞❡♣t❤ ♦❢ t❤❡ ♣✐❧❧❛rs ✇❛s ❝♦♥tr♦❧❧❡❞ ❜② ❛❞❥✉st✐♥❣ t❤❡ ❘■❊ ♣r♦❝❡ss t✐♠❡✳ P❧❛t✐♥✉♠ ✇❛s ✉♥✐❢♦r♠❧② ❞❡♣♦s✐t❡❞ ♦♥t♦ t❤❡ ♣✐❧❧❛rs ✉s✐♥❣ ❡✲❜❡❛♠ ❡✈❛♣♦r❛t✐♦♥ ✭t❤✐❝❦♥❡ss ❂ ✷✵ ♥♠✮✳ ❚❤❡ ❝❛t❛❧②st ♦♥ t❤❡ t♦♣s ♦❢ t❤❡ ♠✐❝r♦♣✐❧❧❛rs ✇❛s s✉❜s❡q✉❡♥t❧② r❡♠♦✈❡❞ ✉s✐♥❣ ❛ ❧✐❢t✲♦✛ ♣r♦❝❡❞✉r❡✳ ❚❤❡s❡ ♣✐❧❧❛rs ✇❡r❡ t❤❡♥ s♣✐♥✲❝♦❛t❡❞ ✇✐t❤ ❚❡✢♦♥ ❆❋ s♦❧✉t✐♦♥ ✭✶ ✇t✳✪✱ ❈❤❡♠♦✉rs✮ ❛t ✸✵✵✵ r♣♠ t♦ r❡♥❞❡r t❤❡ s✉r❢❛❝❡ ❝❤❡♠✐str② ♦❢ t❤❡ ♠✐❝r♦t❡①t✉r❡ ❛rr❛② ❤②❞r♦♣❤♦❜✐❝✳ ❋✐❣✉r❡ ✸❛ s❤♦✇s ❛ ♠✐❝r♦❣r❛♣❤ ♦❢ ❛ ❙✐✲Pt✲❚❡✢♦♥ t❡①t✉r❡ ✇✐t❤ ✶✺✵ µ♠ ❞❡♣t❤✱ ❛♥❞ Pt✲❧❡ss t♦♣s✳ ❚❤❡ ❧✐❢t✲♦✛ st❡♣ ✇❛s ♦♠✐tt❡❞ ❢♦r t❡st✐♥❣ t❤❡ s❛♠♣❧❡s ✐♥ ✇❤✐❝❤ t❤❡ ❝❛t❛❧②st ✇❛s r❡t❛✐♥❡❞ ♦♥ t❤❡ t♦♣s ♦❢ t❤❡ ♣✐❧❧❛rs ❛♥❞ t❤❡s❡ ❛r❡ s❤♦✇♥ ✐♥ ❋✐❣✉r❡ ❙✷✳ ❚❤❡ ❞✐♠❡♥s✐♦♥s ♦❢ t❤❡s❡ ♣✐❧❧❛rs ❛r❡ ❣✐✈❡♥ ✐♥ ❚❛❜❧❡ ❙✶ ✐♥ t❤❡ s✉♣♣❧❡♠❡♥t❛r② ♠❛t❡r✐❛❧✳ ❙♠♦♦t❤✲ ✇❛❧❧❡❞ ♠✐❝r♦♣✐❧❧❛r ❛rr❛②s ✇❡r❡ ❢❛❜r✐❝❛t❡❞ ✉s✐♥❣ ♣❤♦t♦❧✐t❤♦❣r❛♣❤② ✇✐t❤ ❙❯✽ ♣❤♦t♦r❡s✐st ♦♥ Pt✲❝♦❛t❡❞ ❙✐ ✇❛❢❡rs✳

❋❛❜r✐❝❛t✐♦♥ ♦❢ ❛❧✉♠✐♥✉♠ ▼✐❝r♦♣✐❧❧❛rs ✇✐t❤ ❈❛t❛❧②st ❈♦❛t✐♥❣

❲❡ ❢❛❜r✐❝❛t❡❞ ❝❛t❛❧②st✲✐♠♣r❡❣♥❛t❡❞ s✉♣❡r❤②❞r♦♣❤♦❜✐❝ ❛❧✉♠✐♥✉♠ ♠✐❝r♦t❡①t✉r❡s ❜② ✉s✐♥❣ ❧❛s❡r t❡①t✉r✐♥❣✱ ❡❧❡❝tr♦♣❧❛t✐♥❣ ❛♥❞ s✐❧❛♥✐③❛t✐♦♥✳ ❆♥ ❡❣❣✲❝r❛t❡ ♠✐❝r♦t❡①t✉r❡ ✭r ❂ ✺✳✶✮ ✇❛s ♣❛tt❡r♥❡❞ ♦♥t♦ ✢❛t s❤❡❡ts ♦❢ ❛❧✉♠✐♥✉♠ ✭❆❧ ✻✵✻✶✲❚✻✮ ❜② s❝r✐❜✐♥❣ ❡q✉❛❧❧② s♣❛❝❡❞ ❧✐♥❡s ✭♣✐t❝❤ ❂ ✼✺ µ♠✮ ✐♥ t❤❡ ❤♦r✐③♦♥t❛❧ ❛♥❞ ✈❡rt✐❝❛❧ ❞✐r❡❝t✐♦♥s ✉s✐♥❣ ❛ ❧❛s❡r s❝r✐❜❡r ✭❊❧❡❝tr♦① ❙❝r✐❜❛ ❊✲❇♦①✱ ✷✵✵ ♠♠✴s✱ ✷✵ r❡♣❡t✐t✐♦♥s✮✳ ❚❤✐s s✉r❢❛❝❡ ✇❛s t❤❡♥ ❞✐♣♣❡❞ ✐♥t♦ ✺ ✇t✪ s♦❞✐✉♠ ❤②✲ ✷✵


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❞r♦①✐❞❡ ❢♦r ✹✺ s t♦ r❡♠♦✈❡ ❛♥② ♦r❣❛♥✐❝ ❝♦♥t❛♠✐♥❛♥ts✳ ❚❤❡ s✉❜str❛t❡ ✇❛s t❤❡♥ ❡❧❡❝tr♦♣❧❛t❡❞ ✇✐t❤ ♠❛♥❣❛♥❡s❡ ❞✐♦①✐❞❡ ❜② ✉s✐♥❣ ❛♥♦❞✐❝ ❡❧❡❝tr♦❧②t✐❝ ❞❡♣♦s✐t✐♦♥✳ ✻✵ ❚❤❡ s✉❜str❛t❡ ❛♥❞ ❝♦✉♥t❡r ❡❧❡❝tr♦❞❡ ✭❛ ✢❛t ❆❧ ♣❧❛t❡ ❝❧❡❛♥❡❞ ✉s✐♥❣ t❤❡ s♦❞✐✉♠ ❤②❞r♦①✐❞❡ ♣r♦t♦❝♦❧ ❞❡s❝r✐❜❡❞ ❡❛r❧✐❡r✮ ✇❡r❡ ❞✐♣♣❡❞ ✐♥t♦ ✶✵✵ ♠❧ ♦❢ ✵✳✵✺ ▼ ♠❛♥❣❛♥❡s❡ ❛❝❡t❛t❡ s♦❧✉t✐♦♥ ✇✐t❤ ❛ ❣❛♣ ♦❢ ✷✵ ♠♠ ❜❡t✇❡❡♥ t❤❡♠✳ ❚❤❡ s♦❧✉t✐♦♥ ✇❛s ♠❛✐♥t❛✐♥❡❞ ❛t ✻✵ ◦ ❈ ❛♥❞ st✐rr❡❞ ✉s✐♥❣ ❛ st✐r ❜❛r ❛t ✻✵ r♣♠✳ ❊❧❡❝tr♦❧②t✐❝ ❞❡♣♦s✐t✐♦♥ ✇❛s ♣❡r❢♦r♠❡❞ ❛t ✺ ♠❆✴❝♠2 ❢♦r ✺ ♠✐♥✱ ❛♥❞ t❤❡♥ t❤❡ ❡❧❡❝tr♦❞❡s ✇❡r❡ ✐♠♠❡❞✐❛t❡❧② r❡♠♦✈❡❞ ❢r♦♠ t❤❡ s♦❧✉t✐♦♥ ❛♥❞ ❞r✐❡❞ ❛t ✶✵✵ ◦ ❈ ✐♥ ❛✐r✳ ❋r❡s❤ s♦❧✉t✐♦♥s ♦❢ ▼♥✭❈❍3 ❈❖❖✮2 ✇❡r❡ ✉s❡❞ ❢♦r ❡✈❡r② ♥❡✇ ❞❡♣♦s✐t✐♦♥✳ ❋✐❣✉r❡ ❙✸ s❤♦✇s ▼♥❖2 ❡❧❡❝tr♦❞❡♣♦s✐t❡❞ ♦♥ ❛ ✢❛t ❆❧ s✉❜str❛t❡ ✉♥❞❡r s✐♠✐❧❛r ❞❡♣♦s✐t✐♦♥ ❝♦♥❞✐t✐♦♥s ✭❝❛t❛❧②st ❧♦❛❞✐♥❣ ≃ ✺ ♠❣ ❝♠−2 ✮✳ ❚❤❡ ❡❧❡❝✲ tr♦♣❧❛t❡❞ s✉❜str❛t❡s ✇❡r❡ t❤❡♥ ❝❧❡❛♥❡❞ ✉s✐♥❣ ❛ ♣❧❛s♠❛ ❝❧❡❛♥❡r ✭❍❛rr✐❝❦✮ ❢♦r ✺ ♠✐♥s ❛♥❞ ♣❧❛❝❡❞ ✐♥t♦ ❛ ❣❧❛ss ❜❡❛❦❡r ❢♦r s✐❧❛♥❡ ❝❤❡♠✐❝❛❧ ✈❛♣♦r ❞❡♣♦s✐t✐♦♥✳ ❆♥ ♦♣❡♥ ✈✐❛❧ ✇✐t❤ ✸ ♠❧ ♦❢ ❛ ✢✉♦r✐♥❛t❡❞ s✐❧❛♥❡ ✭❚r✐❝❤❧♦r♦✭✶❍✱✶❍✱✷❍✱✷❍✲♣❡r✢✉♦r♦♦❝t②❧✮s✐❧❛♥❡✮ ✇❛s ♣❧❛❝❡❞ ✐♥t♦ t❤❡ ❣❧❛ss ❜❡❛❦❡r ❛❧♦♥❣ ✇✐t❤ t❤❡ ♦t❤❡r s✉❜str❛t❡s✳ ❚❤❡ ❜❡❛❦❡r ✇❛s t❤❡♥ ❝♦✈❡r❡❞ ✇✐t❤ ❛♥ ❛❧✉♠✐♥✉♠ ❢♦✐❧ ❛♥❞ ♣❧❛❝❡❞ ♦♥ ❛ ❤♦t♣❧❛t❡ ❛t ✶✷✵ ◦ ❈ ❢♦r ✸ ❤♦✉rs✳ ❚❤❡ s✉❜str❛t❡s ✇❡r❡ t❤❡♥ r✐♥s❡❞ ✇✐t❤ ✐s♦♣r♦♣❛♥♦❧ t♦ r❡♠♦✈❡ ❡①❝❡ss s✐❧❛♥❡ ❛♥❞ ❞r✐❡❞ ✐♥ ❛✐r✱ t♦ ♦❜t❛✐♥ ❛ s✉♣❡r❤②❞r♦♣❤♦❜✐❝ t❡①t✉r❡ ✇✐t❤ ✭θadv = 162◦ ± 3◦ , θrec = 148◦ ± 4◦ ✮

■♠❛❣✐♥❣ ♦❢ ♣❧❛str♦♥ r❡❝♦✈❡r②

❚❤❡ s✉❜str❛t❡s ✇❡r❡ ✐♠♠❡rs❡❞ ✐♥t♦ ❛ q✉❛rt③ ❝❡❧❧ ✭✷✧▲ × ✶✧❲ × ✷✧❍✱ r❛♠é✲❤❛rt✮ ❝♦♥t❛✐♥✐♥❣ ✸✵ ♠❧ ♦❢ ❞❡✲✐♦♥✐③❡❞ ✭❉■✮ ✇❛t❡r ❛♥❞ s✉♣♣♦rt❡❞ ❛t ❛♣♣r♦①✐♠❛t❡❧② ✹✺◦ t♦ t❤❡ ❤♦r✐③♦♥t❛❧ ♣❧❛♥❡✳ ❆ ♠✐❝r♦s❝♦♣❡ ✭❩❊■❙❙ ❆①✐♦ ❩♦♦♠✳❱✶✻✮ ✇❛s ✉s❡❞ t♦ ♦❜s❡r✈❡ t❤❡ ♣❧❛str♦♥ ❛t ❤✐❣❤ ♠❛❣♥✐✜❝❛t✐♦♥✳ ❆ s♣♦t ❧✐❣❤t s♦✉r❝❡ ✭❉♦❧❛♥✲❏❡♥♥❡r ▼■✲✶✺✵✮ ✇❛s ✉s❡❞ t♦ ✐❧❧✉♠✐♥❛t❡ t❤❡ ♣❧❛str♦♥ ❛t ❛♥ ❛♥❣❧❡ ❣r❡❛t❡r t❤❛♥ t❤❡ ❝r✐t✐❝❛❧ ❛♥❣❧❡ ❢♦r t♦t❛❧ ✐♥t❡r♥❛❧ r❡✢❡❝t✐♦♥ ✭θc = 48.75◦ ✮ t♦ ✈✐s✉❛❧✐③❡ t❤❡ ❛✐r✲ ✇❛t❡r ✐♥t❡r❢❛❝❡ ❛s s❤♦✇♥ ✐♥ ❋✐❣✉r❡ ❙✶✳ ❚❤❡ ♣❧❛str♦♥ t❤❛t ✐s ✐♥✐t✐❛❧❧② ♣r❡s❡♥t ♦♥ t❤❡ s✉❜str❛t❡ ❛❢t❡r ✐♠♠❡rs✐♦♥ ✐♥t♦ ❉■ ✇❛t❡r ✇❛s r❡♠♦✈❡❞ ❢r♦♠ t❤❡ s✉❜str❛t❡s ❜② ❛♣♣❧②✐♥❣ s✉❝t✐♦♥✴♣r❡ss✉r❡ ❛t t❤❡ s✉r❢❛❝❡ ✉s✐♥❣ ❛ ♣✐♣❡tt❡✳ ❚♦ tr✐❣❣❡r ♣❧❛str♦♥ r❡❝♦✈❡r②✱ ✶✲✸ ♠❧ ♦❢ ✸✵ ✇t✳✪ ❤②❞r♦❣❡♥ ♣❡r♦①✐❞❡ ✭✐♥❤✐❜✐t❡❞ ❤②❞r♦❣❡♥ ♣❡r♦①✐❞❡✱ ❙✐❣♠❛ ❆❧❞r✐❝❤✮ ✇❛s ❣❡♥t❧② ❛❞❞❡❞ t♦ t❤❡ ✇❛t❡r ❛♥❞ ✷✶



❛❧❧♦✇❡❞ t♦ ❞✐✛✉s❡ t♦ t❤❡ s✉r❢❛❝❡✳ ❱✐❞❡♦s ✇❡r❡ t❤❡♥ r❡❝♦r❞❡❞ ✉s✐♥❣ ❛ ❈❈❉ ❝❛♠❡r❛ ❛t ✷✹ ❢♣s✳

❉r❛❣ ❘❡❞✉❝t✐♦♥ ❊①♣❡r✐♠❡♥ts

❉r❛❣ r❡❞✉❝t✐♦♥ ❡①♣❡r✐♠❡♥ts ✇❡r❡ ♣❡r❢♦r♠❡❞ ♦♥ ❛ ❝♦♥tr♦❧❧❡❞ str❡ss r❤❡♦♠❡t❡r ✭❚❆ ■♥str✉✲ ♠❡♥ts ❆❘✲●✷✮ ✉s✐♥❣ ❛ ❜❡s♣♦❦❡ ❚❛②❧♦r✲❈♦✉tt❡ ❝❡❧❧ ✇✐t❤ r♦t♦r r❛❞✐✉s ✶✹ ♠♠✱ ❝❡❧❧ ❣❛♣ ✷✵ ♠♠ ❛♥❞ r♦t♦r ❧❡♥❣t❤ ✹✷ ♠♠✱ ✶✻ ❛♥❞ ❛ ✢✉✐❞ ✐♥❧❡t ♣♦rt t♦ ✐♥❥❡❝t ❤②❞r♦❣❡♥ ♣❡r♦①✐❞❡ ✐♥t♦ t❤❡ ❝❡❧❧ ❞✉r✐♥❣ t❤❡ ❡①♣❡r✐♠❡♥t ❛s s❤♦✇♥ ✐♥ ❋✐❣✉r❡ ✺❝✳ ■♥ t❤✐s ❚❛②❧♦r✲❈♦✉tt❡ ❝❡❧❧✱ tr❛♥s✐t✐♦♥ t♦ ❢✉❧❧② t✉r❜✉❧❡♥t ✢♦✇ ♦❝❝✉rs ❛t ❘❡ > ✶✵✱✵✵✵✳ ▼✐❝r♦❣r♦♦✈❡s ✇❡r❡ ♠❛❝❤✐♥❡❞ ♦♥t♦ t❤❡ r♦t♦r ✭❆❧ ✻✵✻✶✲❚✻✮ ✉s✐♥❣ ❛ ❈◆❈ ❧❛t❤❡ ✭❇✉r❧✐♥❣t♦♥ ▼❛❝❤✐♥❡✱ ❲✐❧♠✐♥❣t♦♥✱ ▼❆✮✳ ❚❤❡ r♦t♦r ✇❛s t❤❡♥ ❡❧❡❝tr♦♣❧❛t❡❞ ✐♥ ♠❛♥❣❛♥❡s❡ ❛❝❡t❛t❡ s♦❧✉t✐♦♥ ✭✵✳✵✺ ▼✱ ✻✵ ◦ ❈✱ st✐rr✐♥❣ ❛t ✻✵ r♣♠✮ ✇✐t❤ ❛ ❝②❧✐♥❞r✐❝❛❧ ❝♦✉♥t❡r ❡❧❡❝tr♦❞❡ ❛t ❛ ❝✉rr❡♥t ❞❡♥s✐t② ♦❢ ✺ ♠❆✴❝♠2 ❢♦r ✺ ♠✐♥✳ ■t ✇❛s t❤❡♥ ❞r✐❡❞ ❛♥❞ ♣❧❛s♠❛ ❝❧❡❛♥❡❞ ✐♥ ❛✐r ❢♦r ✺ ♠✐♥ ❜❡❢♦r❡ ❞❡♣♦s✐t✐♥❣ s✐❧❛♥❡ ✉s✐♥❣ t❤❡ s❛♠❡ ♣r♦❝❡❞✉r❡ ❛s t❤❡ ✢❛t ❛❧✉♠✐♥✉♠ ♣❧❛t❡s✳

❆❝❦♥♦✇❧❡❞❣❡♠❡♥t ❚❤❡ ❛✉t❤♦rs t❤❛♥❦ ❑❋❯P▼ ✭❑✐♥❣ ❋❛❤❞ ❯♥✐✈❡rs✐t② ♦❢ P❡tr♦❧❡✉♠ ❛♥❞ ▼✐♥❡r❛❧s✮ ❛♥❞ t❤❡ ❖✣❝❡ ♦❢ ◆❛✈❛❧ ❘❡s❡❛r❝❤ ✭❖◆❘✮ ❢♦r t❤❡✐r ✜♥❛♥❝✐❛❧ s✉♣♣♦rt✳ ❚❤❡ ❛✉t❤♦rs ❛❧s♦ t❤❛♥❦ t❤❡ st❛✛ ❛t ▼❚▲ ✭▼✐❝r♦s②st❡♠s ❚❡❝❤♥♦❧♦❣② ▲❛❜♦r❛t♦r✐❡s✱ ▼■❚✮ ❢♦r t❤❡✐r ✐♥♣✉t ✐♥ s✐❧✐❝♦♥ ♣✐❧❧❛r ❢❛❜r✐❝❛t✐♦♥✱ t❤❡ ❵P❤♦t♦✈♦❧t❛✐❝s ❘❡s❡❛r❝❤ ▲❛❜♦r❛t♦r②✬ ✭Pr♦❢✳ ❚♦♥✐♦ ❇♦✉♥❛ss✐s✐✮ ❢♦r t❤❡ ❧❛s❡r s❝r✐❜✐♥❣ ❢❛❝✐❧✐t②✱ ❛♥❞ ❉r✳ ❙✐❞❞❛rt❤ ❙r✐♥✐✈❛s❛♥ ❢♦r t❤❡ ✐♥s✐❣❤t❢✉❧ ❞✐s❝✉ss✐♦♥s✳

❙✉♣♣♦rt✐♥❣ ■♥❢♦r♠❛t✐♦♥ ❆✈❛✐❧❛❜❧❡ • ❙✉♣♣❧❡♠❡♥t❛r② ■♥❢♦r♠❛t✐♦♥ ✿ ❋✐❧❡ ❝♦♥t❛✐♥s ✜❣✉r❡s ♦♥ ♣❧❛str♦♥ ✐♠❛❣✐♥❣✱ ❡①tr❛ ♠✐❝r♦✲ ❣r❛♣❤s ♦❢ s✉r❢❛❝❡s ❛♥❞ ❛ t❛❜❧❡ ♦❢ ❞✐♠❡♥s✐♦♥s ♦❢ ❛❧❧ t❤❡ s✐❧✐❝♦♥ t❡①t✉r❡s ✉s❡❞✳

• ❱✐❞❡♦ ❙✶✿ ❱✐❞❡♦ ♦❢ ♣❧❛str♦♥ r❡❝♦✈❡r② ♦♥ ❛ ❙✐❧✐❝♦♥ ♣✐❧❧❛r ❛rr❛②✳ ✷✷


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• ❱✐❞❡♦ ❙✷✿ ❱✐❞❡♦ ♦❢ ♣❧❛str♦♥ r❡❝♦✈❡r② ♦♥ ❛ ❛❧✉♠✐♥✉♠ ❧❛s❡r t❡①t✉r❡❞ s✉r❢❛❝❡✳ • ❱✐❞❡♦ ❙✸✿ ❱✐❞❡♦ ♦❢ ❛ ❞r❛❣ r❡❞✉❝t✐♦♥ ❡①♣❡r✐♠❡♥t ✐♥ t❤❡ ❚❛②❧♦r✲❈♦✉tt❡ ❝❡❧❧✱ ✇✐t❤ ❛ ❝❧✐♣ ♦❢ t❤❡ ♣❛rt✐❛❧❧② r❡❝♦✈❡r❡❞ ♣❧❛str♦♥ ✈✐s✐❜❧❡ ❛❢t❡r t❤❡ ❡①♣❡r✐♠❡♥t✳

❘❡❢❡r❡♥❝❡s ✭✶✮ ❉r❡❧✐❝❤✱ ❏✳❀ ▼❛r♠✉r✱ ❆✳ P❤②s✐❝s ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥s ♦❢ ❙✉♣❡r❤②❞r♦♣❤♦❜✐❝ ❛♥❞ ❙✉♣❡r❤②✲ ❞r♦♣❤✐❧✐❝ ❙✉r❢❛❝❡s ❛♥❞ ❈♦❛t✐♥❣s✳ ❙✉r❢❛❝❡ ■♥♥♦✈❛t✐♦♥s ✷✵✶✹✱ ✷✱ ✷✶✶✕✷✷✼✳ ✭✷✮ P❛r❦✱ ❑✳ ❈✳❀ ❈❤❤❛tr❡✱ ❙✳ ❙✳❀ ❙r✐♥✐✈❛s❛♥✱ ❙✳❀ ❈♦❤❡♥✱ ❘✳ ❊✳❀ ▼❝❑✐♥❧❡②✱ ●✳ ❍✳ ❖♣t✐♠❛❧ ❉❡s✐❣♥ ♦❢ P❡r♠❡❛❜❧❡ ❋✐❜❡r ◆❡t✇♦r❦ ❙tr✉❝t✉r❡s ❢♦r ❋♦❣ ❍❛r✈❡st✐♥❣✳ ▲❛♥❣♠✉✐r ✷✵✶✸✱ ✷✾✱ ✶✸✷✻✾✕✶✸✷✼✼✳ ✭✸✮ ❇❡♥❣❛❧✉r✉ ❙✉❜r❛♠❛♥②❛♠✱ ❙✳❀ ❑♦♥❞r❛s❤♦✈✱ ❱✳❀ ❘ü❤❡✱ ❏✳❀ ❱❛r❛♥❛s✐✱ ❑✳ ❑✳ ▲♦✇ ■❝❡ ❆❞❤❡✲ s✐♦♥ ♦♥ ◆❛♥♦✲❚❡①t✉r❡❞ ❙✉♣❡r❤②❞r♦♣❤♦❜✐❝ ❙✉r❢❛❝❡s ✉♥❞❡r ❙✉♣❡rs❛t✉r❛t❡❞ ❈♦♥❞✐t✐♦♥s✳

❆❈❙ ❆♣♣❧✐❡❞ ▼❛t❡r✐❛❧s ❛♥❞ ■♥t❡r❢❛❝❡s ✷✵✶✻✱ ✽✱ ✶✷✺✽✸✕✶✷✺✽✼✳ ✭✹✮ ❆r♥♦tt✱ ❏✳❀ ❲✉✱ ❆✳ ❍✳ ❋✳❀ ❱✉❝❦♦✱ ▼✳ ❏✳❀ ▲❛♠❜✱ ❘✳ ◆✳ ▼❛r✐♥❡ ❆♥t✐❢♦✉❧✐♥❣ ❢r♦♠ ❚❤✐♥ ❆✐r✳ ❇✐♦❢♦✉❧✐♥❣ ✷✵✶✹✱ ✸✵✱ ✶✵✹✺✕✶✵✺✹✳ ✭✺✮ ❋❡♥❣✱ ❳✳ ❏✳❀ ❏✐❛♥❣✱ ▲✳ ❉❡s✐❣♥ ❛♥❞ ❈r❡❛t✐♦♥ ♦❢ ❙✉♣❡r✇❡tt✐♥❣✴❆♥t✐✇❡tt✐♥❣ ❙✉r❢❛❝❡s✳ ❆❞✲

✈❛♥❝❡❞ ▼❛t❡r✐❛❧s ✷✵✵✻✱ ✶✽✱ ✸✵✻✸✕✸✵✼✽✳ ✭✻✮ ❇✉tt✱ ❍✳ ❏✳❀ ❙❡♠♣r❡❜♦♥✱ ❈✳❀ P❛♣❛❞♦♣♦✉❧♦s✱ P✳❀ ❱♦❧❧♠❡r✱ ❉✳❀ ❇r✐♥❦♠❛♥♥✱ ▼✳❀ ❈✐❝✲ ❝♦tt✐✱ ▼✳ ❉❡s✐❣♥ Pr✐♥❝✐♣❧❡s ❢♦r ❙✉♣❡r❛♠♣❤✐♣❤♦❜✐❝ ❙✉r❢❛❝❡s✳ ❙♦❢t ▼❛tt❡r ✷✵✶✸✱ ✾✱ ✹✶✽✕ ✹✷✽✳ ✭✼✮ ❲♦♥❣✱ ❚✳ ❙✳❀ ❙✉♥✱ ❚✳❀ ❋❡♥❣✱ ▲✳❀ ❆✐③❡♥❜❡r❣✱ ❏✳ ■♥t❡r❢❛❝✐❛❧ ▼❛t❡r✐❛❧s ✇✐t❤ ❙♣❡❝✐❛❧ ❲❡tt❛✲ ❜✐❧✐t②✳ ▼❘❙ ❇✉❧❧❡t✐♥ ✷✵✶✸✱ ✸✽✱ ✸✻✻✕✸✼✶✳ ✭✽✮ ❈❛ss✐❡✱ ❆✳ ❇✳ ❉✳❀ ❇❛①t❡r✱ ❙✳ ❲❡tt❛❜✐❧✐t② ♦❢ P♦r♦✉s ❙✉r❢❛❝❡s✳ ❚r❛♥s❛❝t✐♦♥s ♦❢ t❤❡ ❋❛r❛❞❛②

❙♦❝✐❡t② ✶✾✹✹✱ ✹✵✱ ✺✹✻✕✺✺✶✳ ✷✸



✭✾✮ ❚✐❛♥✱ ❨✳❀ ❙✉✱ ❇✳❀ ❏✐❛♥❣✱ ▲✳ ■♥t❡r❢❛❝✐❛❧ ▼❛t❡r✐❛❧ ❙②st❡♠ ❊①❤✐❜✐t✐♥❣ ❙✉♣❡r✇❡tt❛❜✐❧✐t②✳

❆❞✈❛♥❝❡❞ ▼❛t❡r✐❛❧s ✷✵✶✹✱ ✷✻✱ ✻✽✼✷✕✻✽✾✼✳ ✭✶✵✮ ●❡♦r❣❡✱ ❏✳ ❊✳❀ ❈❤✐❞❛♥❣✐❧✱ ❙✳❀ ●❡♦r❣❡✱ ❙✳ ❉✳ ❘❡❝❡♥t Pr♦❣r❡ss ✐♥ ❋❛❜r✐❝❛t✐♥❣ ❙✉♣❡r❛❡r♦✲ ♣❤♦❜✐❝ ❛♥❞ ❙✉♣❡r❛❡r♦♣❤✐❧✐❝ ❙✉r❢❛❝❡s✳ ❆❞✈❛♥❝❡❞ ▼❛t❡r✐❛❧s ■♥t❡r❢❛❝❡s ✷✵✶✼✱ ✹✱ ✶✻✵✶✵✽✽✳ ✭✶✶✮ ❚❤♦r♣❡✱ ❲✳ ❍✳ P❧❛str♦♥ ❘❡s♣✐r❛t✐♦♥ ✐♥ ❆q✉❛t✐❝ ■♥s❡❝ts✳ ❇✐♦❧♦❣✐❝❛❧ ❘❡✈✐❡✇s ✶✾✺✵✱ ✷✺✱ ✸✹✹✕✸✾✵✳ ✭✶✷✮ ❋❧②♥♥✱ ▼✳ ❘✳❀ ❇✉s❤✱ ❏✳ ❲✳ ▼✳ ❯♥❞❡r✇❛t❡r ❇r❡❛t❤✐♥❣✿ t❤❡ ▼❡❝❤❛♥✐❝s ♦❢ P❧❛str♦♥ ❘❡s✲ ♣✐r❛t✐♦♥✳ ❏♦✉r♥❛❧ ♦❢ ❋❧✉✐❞ ▼❡❝❤❛♥✐❝s ✷✵✵✽✱ ✻✵✽✱ ✷✼✺✕✷✾✻✳ ✭✶✸✮ ❇❛rt❤❧♦tt✱ ❲✳❀ ❙❝❤✐♠♠❡❧✱ ❚✳❀ ❲✐❡rs❝❤✱ ❙✳❀ ❑♦❝❤✱ ❑✳❀ ❇r❡❞❡✱ ▼✳❀ ❇❛r❝③❡✇s❦✐✱ ▼✳❀ ❲❛❧✲ ❤❡✐♠✱ ❙✳❀ ❲❡✐s✱ ❆✳❀ ❑❛❧t❡♥♠❛✐❡r✱ ❆✳❀ ▲❡❞❡r✱ ❆✳❀ ❇♦❤♥✱ ❍✳ ❋✳ ❚❤❡ ❙❛❧✈✐♥✐❛ P❛r❛❞♦①✿ ❙✉♣❡r❤②❞r♦♣❤♦❜✐❝ ❙✉r❢❛❝❡s ✇✐t❤ ❍②❞r♦♣❤✐❧✐❝ P✐♥s ❢♦r ❆✐r ❘❡t❡♥t✐♦♥ ✉♥❞❡r ❲❛t❡r✳ ❆❞✲

✈❛♥❝❡❞ ▼❛t❡r✐❛❧s ✷✵✶✵✱ ✷✷✱ ✷✸✷✺✕✷✸✷✽✳ ✭✶✹✮ ❲❛♥❣✱ P✳❀ ◗✐✉✱ ❘✳❀ ❩❤❛♥❣✱ ❉✳❀ ▲✐♥✱ ❩✳❀ ❍♦✉✱ ❇✳ ❋❛❜r✐❝❛t❡❞ ❙✉♣❡r✲❍②❞r♦♣❤♦❜✐❝ ❋✐❧♠ ✇✐t❤ P♦t❡♥t✐♦st❛t✐❝ ❊❧❡❝tr♦❧②s✐s ▼❡t❤♦❞ ♦♥ ❈♦♣♣❡r ❢♦r ❈♦rr♦s✐♦♥ Pr♦t❡❝t✐♦♥✳ ❊❧❡❝tr♦❝❤✐♠✐❝❛

❆❝t❛ ✷✵✶✵✱ ✺✻✱ ✺✶✼✕✺✷✷✳ ✭✶✺✮ ❘♦t❤st❡✐♥✱ ❏✳ P✳ ❙❧✐♣ ♦♥ ❙✉♣❡r❤②❞r♦♣❤♦❜✐❝ ❙✉r❢❛❝❡s✳ ❆♥♥✉❛❧ ❘❡✈✐❡✇ ♦❢ ❋❧✉✐❞ ▼❡❝❤❛♥✐❝s

✷✵✶✵✱ ✹✷✱ ✽✾✕✶✵✾✳ ✭✶✻✮ ❙r✐♥✐✈❛s❛♥✱ ❙✳❀ ❑❧❡✐♥❣❛rt♥❡r✱ ❏✳ ❆✳❀ ●✐❧❜❡rt✱ ❏✳ ❇✳❀ ❈♦❤❡♥✱ ❘✳ ❊✳❀ ▼✐❧♥❡✱ ❆✳ ❏✳ ❇✳❀ ▼❝❑✐♥✲ ❧❡②✱ ●✳ ❍✳ ❙✉st❛✐♥❛❜❧❡ ❉r❛❣ ❘❡❞✉❝t✐♦♥ ✐♥ ❚✉r❜✉❧❡♥t ❚❛②❧♦r✲❈♦✉❡tt❡ ❋❧♦✇s ❜② ❉❡♣♦s✐t✐♥❣ ❙♣r❛②❛❜❧❡ ❙✉♣❡r❤②❞r♦♣❤♦❜✐❝ ❙✉r❢❛❝❡s✳ P❤②s✐❝❛❧ ❘❡✈✐❡✇ ▲❡tt❡rs ✷✵✶✺✱ ✶✶✹✱ ✵✶✹✺✵✶✳ ✭✶✼✮ ❇♦❜❥✐✱ ▼✳ ❙✳❀ ❑✉♠❛r✱ ❙✳ ❱✳❀ ❆st❤❛♥❛✱ ❆✳❀ ●♦✈❛r❞❤❛♥✱ ❘✳ ◆✳ ❯♥❞❡r✇❛t❡r ❙✉st❛✐♥❛❜✐❧✐t② ♦❢ t❤❡ ✧❈❛ss✐❡✧ ❙t❛t❡ ♦❢ ❲❡tt✐♥❣✳ ▲❛♥❣♠✉✐r ✷✵✵✾✱ ✷✺✱ ✶✷✶✷✵✕✶✷✶✷✻✳ ✭✶✽✮ ❙❛♠❛❤❛✱ ▼✳ ❆✳❀ ❚❛❢r❡s❤✐✱ ❍✳ ❱✳❀ ●❛❞✲❡❧ ❍❛❦✱ ▼✳ ■♥✢✉❡♥❝❡ ♦❢ ❋❧♦✇ ♦♥ ▲♦♥❣❡✈✐t② ♦❢ ❙✉♣❡r❤②❞r♦♣❤♦❜✐❝ ❈♦❛t✐♥❣s✳ ▲❛♥❣♠✉✐r ✷✵✶✷✱ ✷✽✱ ✾✼✺✾✕✾✼✻✻✳ ✷✹


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✭✶✾✮ ❇♦❝q✉❡t✱ ▲✳❀ ▲❛✉❣❛✱ ❊✳ ❆ ❙♠♦♦t❤ ❋✉t✉r❡❄ ◆❛t✉r❡ ▼❛t❡r✐❛❧s ✷✵✶✶✱ ✶✵✱ ✸✸✹✕✸✸✼✳ ✭✷✵✮ ●♦✈❛r❞❤❛♥✱ ❘✳ ◆✳❀ ❙r✐♥✐✈❛s✱ ●✳ ❙✳❀ ❆st❤❛♥❛✱ ❆✳❀ ❇♦❜❥✐✱ ▼✳ ❙✳ ❚✐♠❡ ❉❡♣❡♥❞❡♥❝❡ ♦❢ ❊✛❡❝t✐✈❡ ❙❧✐♣ ♦♥ ❚❡①t✉r❡❞ ❍②❞r♦♣❤♦❜✐❝ ❙✉r❢❛❝❡s✳ P❤②s✐❝s ♦❢ ❋❧✉✐❞s ✷✵✵✾✱ ✷✶✱ ✵✺✷✵✵✶✳ ✭✷✶✮ ▲②✉✱ ❙✳❀ ◆❣✉②❡♥✱ ❉✳ ❈✳❀ ❑✐♠✱ ❉✳❀ ❍✇❛♥❣✱ ❲✳❀ ❨♦♦♥✱ ❇✳ ❊①♣❡r✐♠❡♥t❛❧ ❉r❛❣ ❘❡❞✉❝✲ t✐♦♥ ❙t✉❞② ♦❢ ❙✉♣❡r✲❍②❞r♦♣❤♦❜✐❝ ❙✉r❢❛❝❡ ✇✐t❤ ❉✉❛❧✲❙❝❛❧❡ ❙tr✉❝t✉r❡s✳ ❆♣♣❧✐❡❞ ❙✉r❢❛❝❡

❙❝✐❡♥❝❡ ✷✵✶✸✱ ✷✽✻✱ ✷✵✻✕✷✶✶✳ ✭✷✷✮ ❲❡♥③❡❧✱ ❘✳ ◆✳ ❘❡s✐st❛♥❝❡ ♦❢ ❙♦❧✐❞ ❙✉r❢❛❝❡s t♦ ❲❡tt✐♥❣ ❜② ❲❛t❡r✳ ■♥❞✉str✐❛❧ ✫ ❊♥❣✐✲

♥❡❡r✐♥❣ ❈❤❡♠✐str② ✶✾✸✻✱ ✷✽✱ ✾✽✽✕✾✾✹✳ ✭✷✸✮ ❱r❛♥❝❦❡♥✱ ❘✳ ❏✳❀ ❑✉s✉♠❛❛t♠❛❥❛✱ ❍✳❀ ❍❡r♠❛♥s✱ ❑✳❀ Pr❡♥❡♥✱ ❆✳ ▼✳❀ P✐❡rr❡✲▲♦✉✐s✱ ❖✳❀ ❇❛st✐❛❛♥s❡♥✱ ❈✳ ❲✳ ▼✳❀ ❇r♦❡r✱ ❉✳ ❏✳ ❋✉❧❧② ❘❡✈❡rs✐❜❧❡ ❚r❛♥s✐t✐♦♥ ❢r♦♠ ❲❡♥③❡❧ t♦ ❈❛ss✐❡✲ ❇❛①t❡r ❙t❛t❡s ♦♥ ❈♦rr✉❣❛t❡❞ ❙✉♣❡r❤②❞r♦♣❤♦❜✐❝ ❙✉r❢❛❝❡s✳ ▲❛♥❣♠✉✐r ✷✵✶✵✱ ✷✻✱ ✸✸✸✺✕ ✸✸✹✶✳ ✭✷✹✮ ▼❛♥✉❦②❛♥✱ ●✳❀ ❖❤✱ ❏✳ ▼✳❀ ❱❛♥ ❉❡♥ ❊♥❞❡✱ ❉✳❀ ▲❛♠♠❡rt✐♥❦✱ ❘✳ ●✳ ❍✳❀ ▼✉❣❡❧❡✱ ❋✳ ❊❧❡❝tr✐❝❛❧ ❙✇✐t❝❤✐♥❣ ♦❢ ❲❡tt✐♥❣ ❙t❛t❡s ♦♥ ❙✉♣❡r❤②❞r♦♣❤♦❜✐❝ ❙✉r❢❛❝❡s✿ ❆ ❘♦✉t❡ ❚♦✇❛r❞s ❘❡✈❡rs✐❜❧❡ ❈❛ss✐❡✲t♦✲❲❡♥③❡❧ ❚r❛♥s✐t✐♦♥s✳ P❤②s✐❝❛❧ ❘❡✈✐❡✇ ▲❡tt❡rs ✷✵✶✶✱ ✶✵✻✱ ✵✶✹✺✵✶✳ ✭✷✺✮ ❑r✉♣❡♥❦✐♥✱ ❚✳ ◆✳❀ ❚❛②❧♦r✱ ❏✳ ❆✳❀ ❲❛♥❣✱ ❊✳ ◆✳❀ ❑♦❧♦❞♥❡r✱ P✳❀ ❍♦❞❡s✱ ▼✳❀ ❙❛❧❛♠♦♥✱ ❚✳ ❘✳ ❘❡✈❡rs✐❜❧❡ ❲❡tt✐♥❣✲❉❡✇❡tt✐♥❣ ❚r❛♥s✐t✐♦♥s ♦♥ ❊❧❡❝tr✐❝❛❧❧② ❚✉♥❛❜❧❡ ❙✉♣❡r❤②❞r♦♣❤♦❜✐❝ ◆❛♥♦str✉❝t✉r❡❞ ❙✉r❢❛❝❡s✳ ▲❛♥❣♠✉✐r ✷✵✵✼✱ ✷✸✱ ✾✶✷✽✕✾✶✸✸✳ ✭✷✻✮ ▲✐✱ ❨✳❀ ◗✉éré✱ ❉✳❀ ▲✈✱ ❈✳❀ ❩❤❡♥❣✱ ◗✳ ▼♦♥♦st❛❜❧❡ ❙✉♣❡rr❡♣❡❧❧❡♥t ▼❛t❡r✐❛❧s✳ Pr♦❝❡❡❞✐♥❣s

♦❢ t❤❡ ◆❛t✐♦♥❛❧ ❆❝❛❞❡♠② ♦❢ ❙❝✐❡♥❝❡s ✷✵✶✼✱ ✶✶✹✱ ✸✸✽✼✕✸✸✾✷✳ ✭✷✼✮ ❇♦r❡②❦♦✱ ❏✳ ❇✳❀ ❈❤❡♥✱ ❈✳ ❍✳ ❘❡st♦r✐♥❣ ❙✉♣❡r❤②❞r♦♣❤♦❜✐❝✐t② ♦❢ ▲♦t✉s ▲❡❛✈❡s ✇✐t❤ ❱✐❜r❛t✐♦♥✲■♥❞✉❝❡❞ ❉❡✇❡tt✐♥❣✳ P❤②s✐❝❛❧ ❘❡✈✐❡✇ ▲❡tt❡rs ✷✵✵✾✱ ✶✵✸✱ ✶✼✹✺✵✷✳ ✭✷✽✮ ❙t❛♠❛t♦♣♦✉❧♦s✱ ❈✳❀ ❙❝❤✉t③✐✉s✱ ❚✳ ▼✳❀ ❑ö♣♣❧✱ ❈✳ ❏✳❀ ❊❧ ❍❛②❡❦✱ ◆✳❀ ▼❛✐tr❛✱ ❚✳❀ ❍❡♠r❧❡✱ ❏✳❀

✷✺



P♦✉❧✐❦❛❦♦s✱ ❉✳ ❖♥ t❤❡ ❙❤❡❞❞✐♥❣ ♦❢ ■♠♣❛❧❡❞ ❉r♦♣❧❡ts✿ ❚❤❡ ❘♦❧❡ ♦❢ ❚r❛♥s✐❡♥t ■♥t❡r✈❡♥✐♥❣ ▲❛②❡rs✳ ❙❝✐❡♥t✐✜❝ ❘❡♣♦rts ✷✵✶✻✱ ✻✱ ✶✽✽✼✺✳ ✭✷✾✮ ▲❡❡✱ ❈✳❀ ❑✐♠✱ ❈✳✲❏✳ ❯♥❞❡r✇❛t❡r ❘❡st♦r❛t✐♦♥ ❛♥❞ ❘❡t❡♥t✐♦♥ ♦❢ ●❛s❡s ♦♥ ❙✉♣❡r❤②❞r♦♣❤♦✲ ❜✐❝ ❙✉r❢❛❝❡s ❢♦r ❉r❛❣ ❘❡❞✉❝t✐♦♥✳ P❤②s✐❝❛❧ ❘❡✈✐❡✇ ▲❡tt❡rs ✷✵✶✶✱ ✶✵✻✱ ✵✶✹✺✵✷✳ ✭✸✵✮ ▲❧♦②❞✱ ❇✳ P✳❀ ❇❛rt❧❡tt✱ P✳ ◆✳❀ ❲♦♦❞✱ ❘✳ ❏✳ ❑✳ ❆❝t✐✈❡ ●❛s ❘❡♣❧❡♥✐s❤♠❡♥t ❛♥❞ ❙❡♥s✐♥❣ ♦❢ t❤❡ ❲❡tt✐♥❣ ❙t❛t❡ ✐♥ ❛ ❙✉❜♠❡r❣❡❞ ❙✉♣❡r❤②❞r♦♣❤♦❜✐❝ ❙✉r❢❛❝❡✳ ❙♦❢t ▼❛tt❡r ✷✵✶✼✱ ✶✸✱ ✶✹✶✸✕✶✹✶✾✳ ✭✸✶✮ ▲❡❡✱ ❏✳❀ ❨♦♥❣✱ ❑✳ ❈♦♠❜✐♥✐♥❣ t❤❡ ▲♦t✉s ▲❡❛❢ ❊✛❡❝t ✇✐t❤ ❆rt✐✜❝✐❛❧ P❤♦t♦s②♥t❤❡s✐s✿ ❘❡✲ ❣❡♥❡r❛t✐♦♥ ♦❢ ❯♥❞❡r✇❛t❡r ❙✉♣❡r❤②❞r♦♣❤♦❜✐❝✐t② ♦❢ ❍✐❡r❛r❝❤✐❝❛❧ ❩♥❖✴❙✐ ❙✉r❢❛❝❡s ❜② ❙♦✲ ❧❛r ❲❛t❡r ❙♣❧✐tt✐♥❣✳ ◆P● ❆s✐❛ ▼❛t❡r✐❛❧s ✷✵✶✺✱ ✼✱ ❡✷✵✶✳ ✭✸✷✮ ❱❛❦❛r❡❧s❦✐✱ ■✳ ❯✳❀ P❛t❛♥❦❛r✱ ◆✳ ❆✳❀ ▼❛rst♦♥✱ ❏✳ ❖✳❀ ❈❤❛♥✱ ❉✳ ❨✳❀ ❚❤♦r♦❞❞s❡♥✱ ❙✳ ❚✳ ❙t❛✲ ❜✐❧✐③❛t✐♦♥ ♦❢ ▲❡✐❞❡♥❢r♦st ❱❛♣♦✉r ▲❛②❡r ❜② ❚❡①t✉r❡❞ ❙✉♣❡r❤②❞r♦♣❤♦❜✐❝ ❙✉r❢❛❝❡s✳ ◆❛t✉r❡

✷✵✶✷✱ ✹✽✾✱ ✷✼✹✕✷✼✼✳ ✭✸✸✮ ❙❛r❛♥❛❞❤✐✱ ❉✳❀ ❈❤❡♥✱ ❉✳❀ ❑❧❡✐♥❣❛rt♥❡r✱ ❏✳ ❆✳❀ ❙r✐♥✐✈❛s❛♥✱ ❙✳❀ ❈♦❤❡♥✱ ❘✳ ❊✳❀ ▼❝❑✐♥✲ ❧❡②✱ ●✳ ❍✳ ❙✉st❛✐♥❡❞ ❉r❛❣ ❘❡❞✉❝t✐♦♥ ✐♥ ❛ ❚✉r❜✉❧❡♥t ❋❧♦✇ ✉s✐♥❣ ❛ ▲♦✇✲❚❡♠♣❡r❛t✉r❡ ▲❡✐❞❡♥❢r♦st ❙✉r❢❛❝❡✳ ❙❝✐❡♥❝❡ ❆❞✈❛♥❝❡s ✷✵✶✻✱ ✷✱ ❡✶✻✵✵✻✽✻✳ ✭✸✹✮ ❍♦❢✱ ❇✳ ❋❧✉✐❞ ❉②♥❛♠✐❝s✿ ❲❛t❡r ❋❧♦✇s ❖✉t ♦❢ ❚♦✉❝❤✳ ◆❛t✉r❡ ✷✵✶✼✱ ✺✹✶✱ ✶✻✶✕✶✻✷✳ ✭✸✺✮ ▲✐♥❣✱ ❍✳❀ ❙r✐♥✐✈❛s❛♥✱ ❙✳❀ ●♦❧♦✈✐♥✱ ❑✳❀ ▼❝❑✐♥❧❡②✱ ●✳ ❍✳❀ ❚✉t❡❥❛✱ ❆✳❀ ❑❛t③✱ ❏✳ ❍✐❣❤✲ ❘❡s♦❧✉t✐♦♥ ❱❡❧♦❝✐t② ▼❡❛s✉r❡♠❡♥t ✐♥ t❤❡ ■♥♥❡r P❛rt ♦❢ ❚✉r❜✉❧❡♥t ❇♦✉♥❞❛r② ▲❛②❡rs ♦✈❡r ❙✉♣❡r✲❍②❞r♦♣❤♦❜✐❝ ❙✉r❢❛❝❡s✳ ❏♦✉r♥❛❧ ♦❢ ❋❧✉✐❞ ▼❡❝❤❛♥✐❝s ✷✵✶✻✱ ✽✵✶✱ ✻✼✵✕✼✵✸✳ ✭✸✻✮ ❱❡r❤♦✱ ❚✳❀ ❑♦r❤♦♥❡♥✱ ❏✳ ❚✳❀ ❙❛✐♥✐❡♠✐✱ ▲✳❀ ❏♦❦✐♥❡♥✱ ❱✳❀ ❇♦✇❡r✱ ❈✳❀ ❋r❛♥③❡✱ ❑✳❀ ❋r❛♥ss✐❧❛✱ ❙✳❀ ❆♥❞r❡✇✱ P✳❀ ■❦❦❛❧❛✱ ❖✳❀ ❘❛s✱ ❘✳ ❍✳ ❆✳ ❘❡✈❡rs✐❜❧❡ ❙✇✐t❝❤✐♥❣ ❇❡t✇❡❡♥ ❙✉♣❡r❤②❞r♦♣❤♦❜✐❝ ❙t❛t❡s ♦♥ ❛ ❍✐❡r❛r❝❤✐❝❛❧❧② ❙tr✉❝t✉r❡❞ ❙✉r❢❛❝❡✳ Pr♦❝❡❡❞✐♥❣s ♦❢ t❤❡

◆❛t✐♦♥❛❧ ❆❝❛❞❡♠② ♦❢ ❙❝✐❡♥❝❡s ✷✵✶✷✱ ✶✵✾✱ ✶✵✷✶✵✕✶✵✷✶✸✳ ✷✻


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✭✸✼✮ ❋♦rs❜❡r❣✱ P✳❀ ◆✐❦♦❧❛❥❡✛✱ ❋✳❀ ❑❛r❧ss♦♥✱ ▼✳ ❈❛ss✐❡✲❲❡♥③❡❧ ❛♥❞ ❲❡♥③❡❧✲❈❛ss✐❡ ❚r❛♥s✐✲ t✐♦♥s ♦♥ ■♠♠❡rs❡❞ ❙✉♣❡r❤②❞r♦♣❤♦❜✐❝ ❙✉r❢❛❝❡s ✉♥❞❡r ❍②❞r♦st❛t✐❝ Pr❡ss✉r❡✳ ❙♦❢t ▼❛tt❡r

✷✵✶✶✱ ✼✱ ✶✵✹✕✶✵✾✳ ✭✸✽✮ ❉✉✱ P✳❀ ❲❡♥✱ ❏✳❀ ❩❤❛♥❣✱ ❩✳❀ ❙♦♥❣✱ ❉✳❀ ❖✉❛❤s✐♥❡✱ ❆✳❀ ❍✉✱ ❍✳ ▼❛✐♥t❡♥❛♥❝❡ ♦❢ ❆✐r ▲❛②❡r ❛♥❞ ❉r❛❣ ❘❡❞✉❝t✐♦♥ ♦♥ ❙✉♣❡r❤②❞r♦♣❤♦❜✐❝ ❙✉r❢❛❝❡✳ ❖❝❡❛♥ ❊♥❣✐♥❡❡r✐♥❣ ✷✵✶✼✱ ✶✸✵✱ ✸✷✽✕ ✸✸✺✳ ✭✸✾✮ ❞❡ ▼❛❧❡♣r❛❞❡✱ ❍✳❀ ❈❧❛♥❡t✱ ❈✳❀ ◗✉éré✱ ❉✳ ❙♣r❡❛❞✐♥❣ ♦❢ ❇✉❜❜❧❡s ❛❢t❡r ❈♦♥t❛❝t✐♥❣ t❤❡ ▲♦✇❡r ❙✐❞❡ ♦❢ ❛♥ ❆❡r♦♣❤✐❧✐❝ ❙❧✐❞❡ ■♠♠❡rs❡❞ ✐♥ ❲❛t❡r✳ P❤②s✐❝❛❧ ❘❡✈✐❡✇ ▲❡tt❡rs ✷✵✶✻✱

✶✶✼✱ ✵✾✹✺✵✶✳ ✭✹✵✮ ❉❛✈✐s✱ ❆✳ ▼✳❀ ▲❛✉❣❛✱ ❊✳ ●❡♦♠❡tr✐❝ ❚r❛♥s✐t✐♦♥ ✐♥ ❋r✐❝t✐♦♥ ❢♦r ❋❧♦✇ ♦✈❡r ❛ ❇✉❜❜❧❡ ▼❛t✲ tr❡ss✳ P❤②s✐❝s ♦❢ ❋❧✉✐❞s ✷✵✵✾✱ ✷✶✱ ✵✶✶✼✵✶✳ ✭✹✶✮ ▲✐♥❣✱ ❍✳❀ ❑❛t③✱ ❏✳❀ ❋✉✱ ▼✳❀ ❍✉❧t♠❛r❦✱ ▼✳ ❊✛❡❝t ♦❢ ❘❡②♥♦❧❞s ◆✉♠❜❡r ❛♥❞ ❙❛t✉r❛t✐♦♥ ▲❡✈❡❧ ♦♥ ●❛s ❉✐✛✉s✐♦♥ ■♥ ❛♥❞ ❖✉t ♦❢ ❛ ❙✉♣❡r❤②❞r♦♣❤♦❜✐❝ ❙✉r❢❛❝❡✳ P❤②s✐❝❛❧ ❘❡✈✐❡✇

❋❧✉✐❞s ✷✵✶✼✱ ✷✱ ✶✷✹✵✵✺✳ ✭✹✷✮ ❱❛❦❛r❡❧s❦✐✱ ■✳ ❯✳❀ ❈❤❛♥✱ ❉✳ ❨✳ ❈✳❀ ▼❛rst♦♥✱ ❏✳ ❖✳❀ ❚❤♦r♦❞❞s❡♥✱ ❙✳ ❚✳ ❉②♥❛♠✐❝ ❆✐r ▲❛②❡r ♦♥ ❚❡①t✉r❡❞ ❙✉♣❡r❤②❞r♦♣❤♦❜✐❝ ❙✉r❢❛❝❡s✳ ▲❛♥❣♠✉✐r ✷✵✶✸✱ ✷✾✱ ✶✶✵✼✹✕✶✶✵✽✶✳ ✭✹✸✮ ▲✐♥✱ ❈✳ ❈✳❀ ❙♠✐t❤✱ ❋✳ ❘✳❀ ■❝❤✐❦❛✇❛✱ ◆✳❀ ❇❛❜❛✱ ❚✳❀ ■t♦✇✱ ▼✳ ❉❡❝♦♠♣♦s✐t✐♦♥ ♦❢ ❍②❞r♦✲ ❣❡♥ P❡r♦①✐❞❡ ✐♥ ❆q✉❡♦✉s ❙♦❧✉t✐♦♥s ❛t ❊❧❡✈❛t❡❞ ❚❡♠♣❡r❛t✉r❡s✳ ■♥t❡r♥❛t✐♦♥❛❧ ❏♦✉r♥❛❧ ♦❢

❈❤❡♠✐❝❛❧ ❑✐♥❡t✐❝s ✶✾✾✶✱ ✷✸✱ ✾✼✶✕✾✽✼✳ ✭✹✹✮ ▲♦❡✇✱ ❖✳ ❆ ◆❡✇ ❊♥③②♠❡ ♦❢ ●❡♥❡r❛❧ ❖❝❝✉r❡♥❝❡ ✐♥ ❖r❣❛♥✐s♠s✳ ❙❝✐❡♥❝❡ ✶✾✵✵✱ ✶✶✱ ✼✵✶✕ ✼✵✷✳ ✭✹✺✮ ▼❝❑❡❡✱ ❉✳ ❲✳ ❈❛t❛❧②t✐❝ ❉❡❝♦♠♣♦s✐t✐♦♥ ♦❢ ❍②❞r♦❣❡♥ P❡r♦①✐❞❡ ❜② ▼❡t❛❧s ❛♥❞ ❆❧❧♦②s ♦❢ t❤❡ P❧❛t✐♥✉♠ ●r♦✉♣✳ ❏♦✉r♥❛❧ ♦❢ ❈❛t❛❧②s✐s ✶✾✻✾✱ ✶✹✱ ✸✺✺✕✸✻✹✳

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✭✹✻✮ ▲✐❡❜❤❛❢s❦②✱ ❍✳ ❆✳ ❚❤❡ ❈❛t❛❧②t✐❝ ❉❡❝♦♠♣♦s✐t✐♦♥ ♦❢ ❍②❞r♦❣❡♥ P❡r♦①✐❞❡ ❇② t❤❡ ■♦❞✐♥❡✲ ■♦❞✐❞❡ ❈♦✉♣❧❡ ❆t ✷✺◦ ✳ ❏♦✉r♥❛❧ ♦❢ t❤❡ ❆♠❡r✐❝❛♥ ❈❤❡♠✐❝❛❧ ❙♦❝✐❡t② ✶✾✸✷✱ ✺✹✱ ✶✼✾✷✕✶✽✵✻✳ ✭✹✼✮ ❘♦②✱ ❈✳ ❇✳ ❈❛t❛❧②t✐❝ ❉❡❝♦♠♣♦s✐t✐♦♥ ♦❢ ❍②❞r♦❣❡♥ P❡r♦①✐❞❡ ♦♥ s♦♠❡ ❖①✐❞❡ ❈❛t❛❧②sts✳

❏♦✉r♥❛❧ ♦❢ ❈❛t❛❧②s✐s ✶✾✻✽✱ ✶✷✱ ✶✷✾✕✶✸✸✳ ✭✹✽✮ ❙❡rr❛✲▼❛✐❛✱ ❘✳❀ ❇❡❧❧✐❡r✱ ▼✳❀ ❈❤❛st❦❛✱ ❙✳❀ ❚r❛♥❤✉✉✱ ❑✳❀ ❙✉❜♦✇♦✱ ❆✳❀ ❘✐♠st✐❞t✱ ❏✳ ❉✳❀ ❯s♦✈✱ P✳ ▼✳❀ ▼♦rr✐s✱ ❆✳ ❏✳❀ ▼✐❝❤❡❧✱ ❋✳ ▼✳ ▼❡❝❤❛♥✐s♠ ❛♥❞ ❑✐♥❡t✐❝s ♦❢ ❍②❞r♦❣❡♥ P❡r♦①✲ ✐❞❡ ❉❡❝♦♠♣♦s✐t✐♦♥ ♦♥ P❧❛t✐♥✉♠ ◆❛♥♦❝❛t❛❧②sts✳ ❆❈❙ ❆♣♣❧✐❡❞ ▼❛t❡r✐❛❧s ❛♥❞ ■♥t❡r❢❛❝❡s

✷✵✶✽✱ ✶✵✱ ✷✶✷✷✹✕✷✶✷✸✹✳ ✭✹✾✮ ❉♦✱ ❙✳✲❍✳❀ ❇❛t❝❤❡❧♦r✱ ❇✳❀ ▲❡❡✱ ❍✳✲❦✳❀ ❑♦♥❣✱ ❙✳✲❤✳ ❍②❞r♦❣❡♥ P❡r♦①✐❞❡ ❉❡❝♦♠♣♦s✐t✐♦♥ ♦♥ ▼❛♥❣❛♥❡s❡ ❖①✐❞❡ ✭ P②r♦❧✉s✐t❡ ✮✿ ❑✐♥❡t✐❝s ✱ ■♥t❡r♠❡❞✐❛t❡s ✱ ❛♥❞ ▼❡❝❤❛♥✐s♠✳ ❈❤❡♠♦✲

s♣❤❡r❡ ✷✵✵✾✱ ✼✺✱ ✽✕✶✷✳ ✭✺✵✮ ❙❛♥❝❤❡③✱ ❙✳❀ ❙♦❧♦✈❡✈✱ ❆✳ ❆✳❀ ▼❡✐✱ ❨✳ ❋✳❀ ❙❝❤♠✐❞t✱ ❖✳ ●✳ ❉②♥❛♠✐❝s ♦❢ ❇✐♦❝❛t❛❧②t✐❝ ▼✐❝r♦❡♥❣✐♥❡s ▼❡❞✐❛t❡❞ ❜② ❱❛r✐❛❜❧❡ ❋r✐❝t✐♦♥ ❈♦♥tr♦❧✳ ❏♦✉r♥❛❧ ♦❢ ❆♠❡r✐❝❛♥ ❈❤❡♠✐❝❛❧

❙♦❝✐❡t② ✷✵✶✵✱ ✶✸✷✱ ✶✸✶✹✹✕✶✸✶✹✺✳ ✭✺✶✮ ❲❛♥❣✱ ❨✳❀ ❍❡r♥❛♥❞❡③✱ ❘✳ ▼✳❀ ❇❛rt❧❡tt✱ ❉✳ ❏✳❀ ❇✐♥❣❤❛♠✱ ❏✳ ▼✳❀ ❑❧✐♥❡✱ ❚✳ ❘✳❀ ❙❡♥✱ ❆✳❀ ▼❛❧❧♦✉❦✱ ❚✳ ❊✳ ❇✐♣♦❧❛r ❊❧❡❝tr♦❝❤❡♠✐❝❛❧ ▼❡❝❤❛♥✐s♠ ❢♦r t❤❡ Pr♦♣✉❧s✐♦♥ ♦❢ ❈❛t❛❧②t✐❝ ◆❛♥♦♠♦t♦rs ✐♥ ❍②❞r♦❣❡♥ P❡r♦①✐❞❡ ❙♦❧✉t✐♦♥s✳ ▲❛♥❣♠✉✐r ✷✵✵✻✱ ✷✷✱ ✶✵✹✺✶✕✶✵✹✺✻✳ ✭✺✷✮ ❏♦♥❡s✱ P✳ ❘✳❀ ❍❛♦✱ ❳✳❀ ❈r✉③✲❈❤✉✱ ❊✳ ❘✳❀ ❘②❦❛❝③❡✇s❦✐✱ ❑✳❀ ◆❛♥❞②✱ ❑✳❀ ❙❝❤✉t③✐✉s✱ ❚✳ ▼✳❀ ❱❛r❛♥❛s✐✱ ❑✳ ❑✳❀ ▼❡❣❛r✐❞✐s✱ ❈✳ ▼✳❀ ❲❛❧t❤❡r✱ ❏✳ ❍✳❀ ❑♦✉♠♦✉ts❛❦♦s✱ P✳❀ ❊s♣✐♥♦s❛✱ ❍✳ ❉✳❀ P❛t❛♥❦❛r✱ ◆✳ ❆✳ ❙✉st❛✐♥✐♥❣ ❉r② ❙✉r❢❛❝❡s ✉♥❞❡r ❲❛t❡r✳ ❙❝✐❡♥t✐✜❝ ❘❡♣♦rts ✷✵✶✺✱ ✺✱ ✶✷✸✶✶✳ ✭✺✸✮ ▼❛r♠✉r✱ ❆✳ ❯♥❞❡r✇❛t❡r ❙✉♣❡r❤②❞r♦♣❤♦❜✐❝✐t②✿ ❚❤❡♦r❡t✐❝❛❧ ❋❡❛s✐❜✐❧✐t②✳ ▲❛♥❣♠✉✐r ✷✵✵✻✱

✷✷✱ ✶✹✵✵✕✶✹✵✷✳

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✭✺✹✮ ●r♦ss♠❛♥♥✱ ❙✳❀ ▲♦❤s❡✱ ❉✳❀ ❙✉♥✱ ❈✳ ❍✐❣❤✲❘❡②♥♦❧❞s ◆✉♠❜❡r ❚❛②❧♦r✲❈♦✉❡tt❡ ❚✉r❜✉❧❡♥❝❡✳

❆♥♥✉❛❧ ❘❡✈✐❡✇ ♦❢ ❋❧✉✐❞ ▼❡❝❤❛♥✐❝s ✷✵✶✻✱ ✹✽✱ ✺✸✕✽✵✳ ✭✺✺✮ ❙❤✐✱ ❈✳❀ ❈✉✐✱ ❳✳❀ ❩❤❛♥❣✱ ❳✳❀ ❚❝❤♦✉❦♦✈✱ P✳❀ ▲✐✉✱ ◗✳❀ ❊♥❝✐♥❛s✱ ◆✳❀ P❛✈❡♥✱ ▼✳❀ ●❡②❡r✱ ❋✳❀ ❱♦❧❧♠❡r✱ ❉✳❀ ❳✉✱ ❩✳❀ ❇✉tt✱ ❍✳✲❏✳❀ ❩❡♥❣✱ ❍✳ ■♥t❡r❛❝t✐♦♥ ❜❡t✇❡❡♥ ❆✐r ❇✉❜❜❧❡s ❛♥❞ ❙✉♣❡r✲ ❤②❞r♦♣❤♦❜✐❝ ❙✉r❢❛❝❡s ✐♥ ❆q✉❡♦✉s ❙♦❧✉t✐♦♥s✳ ▲❛♥❣♠✉✐r ✷✵✶✺✱ ✸✶✱ ✼✸✶✼✕✼✸✷✼✳ ✭✺✻✮ ❇✐❝♦✱ ❏✳❀ ❚❤✐❡❧❡✱ ❯✳❀ ◗✉éré✱ ❉✳ ❲❡tt✐♥❣ ♦❢ ❚❡①t✉r❡❞ ❙✉r❢❛❝❡s✳ ❈♦❧❧♦✐❞s ❛♥❞ ❙✉r❢❛❝❡s ❆✿

P❤②s✐❝♦❝❤❡♠✐❝❛❧ ❛♥❞ ❊♥❣✐♥❡❡r✐♥❣ ❆s♣❡❝ts ✷✵✵✷✱ ✷✵✻✱ ✹✶✕✹✻✳ ✭✺✼✮ ❨♦✉♥❣✱ ❚✳ ■■■✳ ❆♥ ❊ss❛② ♦♥ t❤❡ ❈♦❤❡s✐♦♥ ♦❢ ❋❧✉✐❞s✳ P❤✐❧♦s♦♣❤✐❝❛❧ ❚r❛♥s❛❝t✐♦♥s ♦❢ t❤❡

❘♦②❛❧ ❙♦❝✐❡t② ♦❢ ▲♦♥❞♦♥ ✶✽✵✺✱ ✾✺✱ ✻✺✕✽✼✳ ✭✺✽✮ ❚✉t❡❥❛✱ ❆✳❀ ❈❤♦✐✱ ❲✳❀ ▼❛❜r②✱ ❏✳ ▼✳❀ ▼❝❑✐♥❧❡②✱ ●✳ ❍✳❀ ❈♦❤❡♥✱ ❘✳ ❊✳ ❘♦❜✉st ❖♠♥✐♣❤♦❜✐❝ ❙✉r❢❛❝❡s✳ Pr♦❝❡❡❞✐♥❣s ♦❢ t❤❡ ◆❛t✐♦♥❛❧ ❆❝❛❞❡♠② ♦❢ ❙❝✐❡♥❝❡s ✷✵✵✽✱ ✶✵✺✱ ✶✽✷✵✵✕✶✽✷✵✺✳ ✭✺✾✮ ❑✐♠✱ ❏✳ ●✳❀ ❈❤♦✐✱ ❍✳ ❏✳❀ P❛r❦✱ ❑✳ ❈✳❀ ❈♦❤❡♥✱ ❘✳ ❊✳❀ ▼❝❑✐♥❧❡②✱ ●✳ ❍✳❀ ❇❛r❜❛st❛t❤✐s✱ ●✳ ▼✉❧t✐❢✉♥❝t✐♦♥❛❧ ■♥✈❡rt❡❞ ◆❛♥♦❝♦♥❡ ❆rr❛②s ❢♦r ◆♦♥✲✇❡tt✐♥❣✱ ❙❡❧❢✲❈❧❡❛♥✐♥❣ ❚r❛♥s♣❛r❡♥t ❙✉r❢❛❝❡ ✇✐t❤ ❍✐❣❤ ▼❡❝❤❛♥✐❝❛❧ ❘♦❜✉st♥❡ss✳ ❙♠❛❧❧ ✷✵✶✹✱ ✶✵✱ ✷✹✽✼✕✷✹✾✹✳ ✭✻✵✮ ❇❛❜❛❦❤❛♥✐✱ ❇✳❀ ■✈❡②✱ ❉✳ ●✳ ❆♥♦❞✐❝ ❉❡♣♦s✐t✐♦♥ ♦❢ ▼❛♥❣❛♥❡s❡ ❖①✐❞❡ ❊❧❡❝tr♦❞❡s ✇✐t❤ ❘♦❞✲❧✐❦❡ ❙tr✉❝t✉r❡s ❢♦r ❆♣♣❧✐❝❛t✐♦♥ ❛s ❊❧❡❝tr♦❝❤❡♠✐❝❛❧ ❈❛♣❛❝✐t♦rs✳ ❏♦✉r♥❛❧ ♦❢ P♦✇❡r

❙♦✉r❝❡s ✷✵✶✵✱ ✶✾✺✱ ✷✶✶✵✕✷✶✶✼✳

✷✾




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Schematic of an active superhydrophobic microtexture which generates gas on-demand upon addition of the reactant, hydrogen peroxide. 224x151mm (144 x 144 DPI)



(a) Schematic of the thermodynamic states of gas bubbles spreading on textured surfaces. The stability of the states is dictated by both the microtexture and surface chemistry. The scale bar represents 0.5 mm; (b) Plot of the apparent contact angle of water (θ*w) on a textured surface (Wenzel roughness r = 2, solid fraction Φs = 0.4) versus Young's contact angle of water on a flat surface (θw). The theoretically predicted regime of stable underwater gas wetting or aerophilicity is also indicated by the shading in the plot. Plastron recovery is favored only in the region shaded in green. 249x146mm (144 x 144 DPI)


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(a) Micrograph of silicon texture with platinum coating on the sides and bottom of the texture, and Teflon coating on the entire texture (Si-Pt-Teflon); (b) Plastron recovery on Si-Pt-Teflon texture underwater at 4.5 wt.% H2O2; (c) Micrograph of laser textured aluminum pillars coated with MnO2 and Fluorosilane (Al-MnO2Fluorosilane); (d) Plastron recovery underwater on Al-MnO2-Fluorosilane texture at 1.5 wt.% H2O2. 259x120mm (144 x 144 DPI)



Regime map for plastron reinflation experiments on Si-Pt-Teflon surfaces with varying texture parameters. The colors indicate whether plastron recovery was observed: plastron recovery (green) and blisters (red); The shape of the markers indicate the shape of the posts: cylindrical pillars (circles), square cylindrical pillars (square). The marker filling indicates whether the catalyst was deposited on the pillar tops: no catalyst on top (filled), catalyst on top (hollow). 175x134mm (144 x 144 DPI)


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(a) Micrograph of square pillars (width 100 µm, height 200 µm, and spacing 100 µm) machined on an aluminum rotor of radius Ri = 14 mm and coated with MnO2 and a fluorinated silane; (b) Drop of water (V = 12 µL)sitting on the hydrophobized machined rotor used for plastron recovery (θadv = 159±2; θrec = 109±5); (c) Schematic of the Taylor-Couette cell used for drag reduction measurements with fluid port for H2O2 injection; (d) Unwrapped views of the cylindrical surface of the rotor before plastron failure, after plastron failure, and after partial plastron recovery. This image was produced by imaging the rotating rotor at constant speed, and stitching together vertical sections of the rotor captured across one entire rotation. (e) Plot of the average wall shear stress, ζw, experienced by the rotor in the Taylor-Couette cell at Re = 47,000 during the initial period of drag reduction (Cassie state), plastron failure (Wenzel state) and plastron recovery (partial Cassie) stages. 249x239mm (144 x 144 DPI)


Plastron Regeneration on Submerged Superhydrophobic Surfaces

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