A Hamiltonian-Based Algorithm for Equilibrium Molecular Dynamics Simulation at Constant Chemical Potential

Transcription

1 nvey of enneee Knoxvlle ae: enneee Reeah an Ceave Exhange ae hee Gauae Shool A Halonan-ae Algoh fo Equlbu oleula Dyna Sulaon a Conan Cheal Poenal Johanna ae Reye Reoene Caon Reye Johanna ae "A Halonan-ae Algoh fo Equlbu oleula Dyna Sulaon a Conan Cheal Poenal. " ae' he nvey of enneee 006. h://ae.enneee.eu/u_gahe/784 h he bough o you fo fee an oen ae by he Gauae Shool a ae: enneee Reeah an Ceave Exhange. I ha been aee fo nluon n ae hee by an auhoze anao of ae: enneee Reeah an Ceave Exhange. Fo oe nfoaon leae ona ae@u.eu.

2 o he Gauae Counl: I a ubng heewh a he wen by Johanna ae Reye enle "A Halonan-ae Algoh fo Equlbu oleula Dyna Sulaon a Conan Cheal Poenal." I have exane he fnal eleon oy of h he fo fo an onen an eoen ha be aee n aal fulfllen of he equeen fo he egee of ae of Sene wh a ao n Cheal Engneeng. We have ea h he an eoen aeane: Wlla V. Seele Ognal gnaue ae on fle wh offal uen eo. Dav J. Keffe an J. Ewa ao Pofeo Aee fo he Counl: Dxe L. hoon Ve Povo an Dean of he Gauae Shool

3 o he Gauae Counl: I a ubng heewh a he wen by Johanna ae Reye Sanago enle A Halonan-ae Algoh fo Equlbu oleula Dyna Sulaon a Conan Cheal Poenal. I have exane he fnal eleon oy of h he fo fo an onen an eoen ha be aee n aal fulfllen of he equeen fo he egee of ae of Sene wh a ao n Cheal Engneeng. Dav J. Keffe Co-ao Pofeo an J. Ewa Co-ao Pofeo We have ea h he an eoen aeane: Wlla V. Seele Aeane fo he Counl: Anne ayhew Ve Chanello an Dean of Gauae Sue Ognal gnaue ae on fle wh offal uen eo

4 A HAILOIA-ASED ALGORIH FOR EILIRI OLECLAR DYAICS SILAIO A COSA CHEICAL POEIAL A he Peene fo he ae of Sene Degee he nvey of enneee Knoxvlle Johanna ae Reye Sanago Augu 006

5 DEDICAIO o y ea faly he Sanago he Reyee he We an he Lhfel an o y fen Ee Jun Gg Joy Gale Ronnel an Luy

6 ACKOWLEDGEES I woul le o han y aae avo D. Dav Keffe an D. an Ewa fo he guane an aene. I alo han D. Wlla Seele fo ng n y oee. I woul le o exe y aeaon o y lab ae a he Couaonal aeal Reeah Gou fo he enouageen an uo.

7 ASRAC he obeve of he eeah ebe n h he o eene whehe lang a an oue an equlbu oleula yna algoh fo goou onan heal oenal ulaon. he hyohe ee by evelong an equlbu oleula yna algoh fo he gan eneble onan heal oenal volue an enegy eneble o VE eneble followng a ehoal oeue eveloe by Keffe e al. [] an unnng ulaon on obly a VE eneble. A novel one fo a heoa onolle ebe. An equaon fo he nananeou heal oenal no avalable hu a oey alle he nananeou aal ef Halonan ha elae o he heal oenal wa efne. he Halonan fo he VE eneble wa foulae an fo h he equaon of oon wee eve. he evaon of he algoh fo he negaon hee ngle e ale eveble efeene ye oagao RESPA eene. We wee able o ulae uefully a able algoh.e. he heoa onolle funon oely vng he ye o he e on ou of he aal ef Halonan an a an how an equvalene of he hange n a an he hange n nube of ale wh ee o he hange n oenal enegy. he ehoal oeue fo algoh eveloen ha gea oenal fo exenng he VE eneble algoh o goou gan anonal eneble ED ulaon. v

8 ALE OF COES Inouon.... agoun: ovaon fo Foulang Conan Cheal Poenal oleula Dyna Algoh.... Syno of Chae... agoun Calulaon an Devaon.... Foulang he Halonan fo Gan Eneble Equlbu oleula Dyna..... he Cheoa Conolle..... Inananeou Paal Sef Halonan VE Halonan Devaon of he Equaon of oon Reveble Refeene Sye Poagao Algoh RESPA Inegaon Shee..... he Louvlle Oeao..... he e Evoluon Equaon RESPA Algoh... 8 VE Eneble Sulaon.... VE Ieal Ga..... Sulaon whee e Equal o o Sulaon whee e < o Dlue Ga Sulaon Fo of he Poenal Enegy he Sef Poenal Enegy Poenal Enegy a a Funon of an ρ VE Dlue Ga Sulaon ng he Sef Poenal Enegy....4 VE Dlue Ga Sulaon ng Poenal Enegy a a Funon of an ρ 4.4. he Pa Coelaon a a Funon of an ρ Sulaon Reul an Duon Conluon an Fuue Wo REFERECES APPEDICES... 5 VIA v

9 LIS OF ALES able.- Sulaon Inu Paaee fo ehane... 4 able.4- Reul fo Sulaon a Dffeen Value of e... 8 able.4- Reul fo Sulaon a Dffeen Dene... 8 v

10 LIS OF FIGRES Fgue.- a v. ulaon e fo an eal ga ye wh e o Fgue.- v. ulaon e fo an eal ga ye wh e o Fgue.- v. ulaon e fo an eal ga ye wh e Fgue.-4 v. ulaon e fo an eal ga ye wh e Fgue.-5 a v. ulaon e fo an eal ga ye wh e Fgue.- v. ulaon e fo a lue ga ye ef oenal enegy ae wh e o Fgue.4- v. ulaon e fo he ulaon of a lue ga ye ρ ae wh e o Fgue.4- v. ulaon e fo he ulaon of a lue ga ye ρ ae wh e o Fgue.4- a hoga fo he ulaon of a lue ga ye ρ ae wh e o Fgue.4-4 Pa oelaon funon fo he ulaon of a lue ga ye ρ ae wh e o Fgue.4-5 he oenal enegy fo VE VE an OZPY v. eny Fgue.4-6 he hange n oenal enegy wh ee o a hange n ye eny v. eny v

11 Inouon. agoun: ovaon fo Foulang Conan Cheal Poenal oleula Dyna Algoh hee ha been exenve eeah on evelong exene-ye oleula yna D algoh [ ]. Aong exng algoh hee ae oe ha have oven o be goou.e. hee geneae aeoe n he oeonng aal ehanal eneble. hee ae noable feaue n hee ueful algoh. Dlang e wa eenal fo he oé-hoove heoa whh allowe fo he goou ulaon n he anonal V eneble une a le e of onan [ 4] an wa eenly genealze fo equlbu oleula yna ED une he oon of an abay exenal foe [5]. Dlang ae wa eenal fo he oé baoa whh allowe fo he goou ulaon n he oba-oheal an he oba-enhal H eneble wh a le e of onan [6]. h ha been eenly genealze fo ED [7]. he obeve of he eeah ebe n h he o eene whehe lang a an oue goou D algoh fo onan heal oenal eneble. he hyohe ee by evelong an ED algoh an unnng ulaon on a gan eneble VE; by gan eneble we ean onan heal oenal volue an enegy ye. hee ae exng onan heal oenal D hee n he leaue [8-0]. hee hee ue ffeen ehnque uh a a obnaon of one Calo an D an allowng fo faonal ale o vay he nube of ale n he ulaon. he ay ffeene an avanage of ou ooe algoh ha we woul o away wh ale neon an eleon whh ha been he oon aoah o onan heal oenal ulaon. In h eeah he oeonene beween lang a an hangng he nube of ale n a D ulaon of a VE eneble wh a onan heal

12 oenal evaluae. he eul of he onan heal oenal algoh wll be efee o a evoluon equaon fo he heoa.. Syno of Chae Chae agoun Calulaon an Devaon ebe he eho ue n evelong he ED VE algoh. A novel one fo a heoa onolle ebe. An equaon fo he nananeou heal oenal no avalable hu we foulae a oey alle he nananeou aal ef Halonan ha loely elae o he heal oenal. he VE eneble Halonan hen foulae fo whh he equaon of oon ae eve. In he la eon he negaon hee a ngle e ale eveble efeene ye oagao RESPA fo he eulng equaon of oon eene. Chae VE Eneble Sulaon onan he eul of he VE D ulaon. he algoh eve n Chae ale o an eal ga ye an a lue ga ye. Fo lue ga ulaon wo ffeen equaon fo he oenal enegy ae ue. In he f equaon oenal enegy exee a he ou of he a of a ale an he ef oenal enegy. he eon equaon exee he oenal enegy a a funon of boh ale ane an ye eny. he eon equaon eule n a able ulaon fo he lue ga ye an wa ue o un oe ulaon o e he algoh. Chae 4 onan onluon an eoenaon fo fuue wo.

13 agoun Calulaon an Devaon. Foulang he Halonan fo Gan Eneble Equlbu oleula Dyna.. he Cheoa Conolle he onolle fo he onan heal oenal algoh eveloe analogouly o he evou exene-ye ED algoh uh a he oé-hoove heoa an he oé baoa a hown below. In he V ED algoh a evouly eve evoluon equaon fo he heoa oenu of he fo ν e - whee he nananeou eeaue e he e on eeaue an ν a heoa onolle fequeny. Fo Equaon - we ee ha hee no hange n he heoa oenu when he ye a he e eeaue. Lewe n he H ED algoh a evouly eve evoluon equaon fo he baoa oenu P of he fo P P ν P f V e P - whee an nananeou agonal eleen of he eue eno e he e on eue V he nananeou volue f he nube of egee of feeo olzann onan ν P a baoa onolle fequeny an P he ae laon vaable. Agan we ee ha hee no hange n baoa oenu when he ye a he e eue. heefoe when we leen he heoa

14 oble ha we wll eque an evoluon equaon fo he heoa oenu of he fo f ν f e - whee he nananeou heal oenal of oonen e he e on heal oenal of oonen ν a heoa onolle fequeny fo oonen he a laon vaable fo oonen an f an f ae oe funon o be eene. Howeve a hown n Aenx A we fn ha h fo fo he heoa onolle equaon - wll no funon oely. A heoa onolle of he fo f ν f e -4 whee he lae ale a of he ale an he unlae ale a wll be he fou of nee n h he. In Equaon - - an -4 nananeou heoyna funon fo eeaue eue an heal oenal wee noue an u be uably efne. he efnon of he nananeou eeaue an he nananeou eue oe fo he genealze equaon heoe []. A uh he eeaue funon efne a f -5 whee he oenu of ale n he enon he a of a ale he oal nube of ale an f he nube of egee of feeo. he nananeou agonal eleen of he eue eno efne a 4

15 whee F V an F ae he oon an foe of ale n he enon -6 eevely. I obvou ha we nee an exeon fo he nananeou heal oenal n e of he oleula level oee nlung he oon oena ae an foe la o Equaon -5 an -6. Suh an exeon oe no uenly ex... Inananeou Paal Sef Halonan In h eon we eve an exeon fo he nananeou aal ef Halonan a oey ha no he ae a he aonal heal oenal bu loely elae o. heefoe f we have an algoh fo ulaon wh onan ef aal Halonan we houl have eve an algoh fo ulaon n he VE eneble. We now eve an exeon fo he nananeou ef aal Halonan of oonen whh wll be enoe by he ybol. We a wh laal heoyna n whh he ef heal oenal of oonen n a uloonen xue of C oonen efne a G ι A V ι H S ι S V ι -7 whee G he Gbb fee enegy A he Helholz fee enegy H he enhaly nguhe fo a Halonan by la of ub he nenal enegy an S he enoy all on an exenve ba. Alo he oal a of oonen whh an be exee a -8 5

16 whee he nube of oleule no ao of ye an he a of a oleule no ao of ye. We nguh beween oleule an ao hee beaue laal heoyna efne he ef heal oenal on a oleula ba wheea ulaon yally ea ale ha oeon o ao ahe han oleule. In he wo of Keffe e al. [] on evng goou ED algoh hey obeve ha he Halonan of a ye ha a e elaonh o n VE A n V G n an H n H ulaon. heefoe we nee only ffeenae he Halonan wh ee o n oe o oban an exeon ha elae o he ef heal oenal. We wll efe o h aal evave a he aal ef Halonan nea of heal oenal beaue whle elae he wo ae no equvalenl. Hoally eole naually eae a ffeenaon wh ee o a equvalen o a ffeenaon wh ee o whee he a of a oleule wa aue o be onan.e.. h vey eaonable hng o o n uaon whee he a of a oleule onan. In ulaon we have oe avanage ove he laboaoy. We an agne an leen leve aheaal anfoaon ha ae no uenly oble n he exeenal laboaoy. Fo exale oé lae e an ae o evelo h heoa an baoa. Hee we nen o lae a whou hangng he nube of ale n he ye. A uh we one he ffeenaon of a equvalen o a ffeenaon wh ee o whee he nube of oleule aue o be onan.e. wll have H ι. heefoe when we ubue h no equaon -7 we H V V ι H H S ι H VE S V ι -9 In a ngle-oonen onao ye n he oanonal eneble he Halonan an be wen a 6

17 H VE v -0 whee he nube of ale he oenal enegy fo ale. an v he veloy of he h ale n he enon. We ubue Equaon -0 no Equaon -9 an evaluae he evave o ge he exeon fo he aal ef Halonan fo a ngle oonen: v A h on we have eve - an nananeou heoyna oey whh ealy evaluae fo a ulaon an neee fo a onan heal oenal algoh... VE Halonan Havng eve an nananeou ef aal Halonan we u now foulae he Halonan of an exene ye. nle he f e of he oeue n Keffe e al. [] whee he Halonan f ha o be exee n e of he eula an ene of a CO oonae n he aheaal oenally ahyal fae of efeene he VE Halonan an be efne n laboaoy oonae n he aheaal fae of efeene. Exeng he Halonan n e of eula an CO oonae ha o be one fo he heoa an baoa algoh ne he e an ae enon wee only lae fo he eula vaable; one hen able o ee nuvely whee o ne oely he e an ae laon vaable. We we he VE Halonan n e of ale veloe ahe han oena o wll be lea whee o ne he a laon vaable whn he Halonan. he a laon vaable fo oonen a ulle fo he unlae a 7

18 8 vaable.e.. hu we we he Halonan of a uloonen olyao ye n he VE eneble a ln e V v H E - whee he oenu of he a laon vaable fo oonen he neal a of he a laon vaable fo oonen an C he oal nube of oonen. We have e he vaable o nae ha hey ae efne n a fae of efeene befoe we aly a non-anonal anfoaon. he enulae uaon on he RHS of he Halonan he ne enegy of he a laon vaable. he la uaon on he gh han e RHS of he Halonan he oenal enegy of he a laon vaable. he ne enegy of he a laon vaable ha a funonal fo ha enely analogou o ha of he e an ae laon vaable efne by oé [ ]. he oenal enegy of he a laon vaable ha a funonal fo ha ly a Legene anfoaon of he Halonan oleely analogou wh he oeue ue wh he ae laon vaable by oé. he Legene anfoaon efne ou fee enegy E a e C E - whee we eognze ha e he e on ef aal Halonan of oonen. In he la e of Equaon - whh oeon o he oenal enegy of he a laon vaable we exe he a laon vaable a ln nea of. h eul n a Halonan ha yel a heoa onolle of he fo

19 9 e -4 We nex we he Halonan n he e fae of efeene n e of he oon an oena ln e V H E -5 whee he oenu n he e fae of efeene. h neeay beaue n hee vaable ha he Halonan an he equaon of oon ae anonal o yle.. Devaon of he Equaon of oon Fo he Halonan of -5 we eve he equaon of oon n e of he laboaoy oonae n he e fae of efeene. Hee we ely on he yle elaonh beween he Halonan an he equaon of oon: V H E -6 H V E -7 H V E -8

20 0 e V H E -9 hee equaon of oon woul geneae goou aeoe heelve bu hey ae nonvenen ne hey ae n he ahyal e fae of efeene. o anfo he o he hyally eanngful une fae of efeene we efne a nonanonal anfoaon. he f e n efong a non-anonal anfoaon on he equaon of oon o ae he evave of he anfoaon equaon. he anfoaon equaon ae gven by -0 hu -4 an he evave of hee equaon ae equaon -5 hu -9:

21 he eon e of he non-anonal anfoaon o ubue he e equaon of oon equaon -6 o -9 no equaon -5 o -9:

22 e - Fo he h an fnal e of he non-anonal anfoaon we anfo he eanng e vaable n equaon -0 o - o une hyally eanngful vaable ung he anfoaon equaon -0 o -4: ν e e f -7 In he nex eon we eve he negaon hee fo he equaon of oon neeay o efo ulaon o e h algoh.

23 . Reveble Refeene Sye Poagao Algoh RESPA Inegaon Shee We wll e he VE ED algoh on a le onao flu of ale ung he ngle e ale eveble efeene ye oagao algoh RESPA [] o negae nueally he equaon of oon. In h eon we how how we eve he RESPA algoh fo VE ED le flu ulaon... he Louvlle Oeao We begn by lfyng he equaon of oon equaon -4 o -7 an exe hee fo a le onao flu: ν f e e ex we foulae a Louvlle oeao fo he le onao flu n a VE ye. he Louvlle oeao L fo he VE ye

24 4 L -4 h L oeao l no wo a L an L L L L. -4 hee wo a of he L oeao ae aange a L L L L. -44 We bue he e n equaon -4 beween L an L ung. L -45 L -46 hen we ubue he lfe equaon of oon equaon -8 o -4 no equaon -45 an -46: L -47 ν e e f L -48 Fnally we ave a he fnal fo of ou oeao:

25 5 ν ν e e e e f f L -49 We exan equaon -49 lng he ffeen e of he oeao: ν ν e e e e f f L -50 In he nex eon we how how h oeao ue o oban he e evoluon equaon... he e Evoluon Equaon We ue he oeao n equaon -50 on he neenen vaable an o eve he e evoluon equaon of hee quane. I oan ha we aly he e of he oeao n he oe hown n equaon -50 fo o o boo lef o gh. In he equaon ha we eve below neenen vaable wll have

26 6 ue nube n aenhee. hee nube eeen he ouaonal oe a eah e e. Fo exale ean he value of he heoa oenu ue n he f e of he algoh whle he value of he heoa oenu ha wll be ue n he eon e of he algoh. We now oee o eve he RESPA algoh.he f an eon e n he oeao -50 hange he heoa oenu aong o ex Δ Δ -5 Δ Δ e e e e f f ex ν ν -5 he h fouh an ffh e of he oeao -50 hange he oenu aong o ex 4 Δ Δ -5 Δ Δ ex ex -54

27 7 ex Δ Δ -55 he xh e of he oeao -50 hange he a laon vaable ung Δ Δ 7 ex ex -56 he evenh e of he oeao -50 hange he oon : ex Δ Δ -57 he eghh e of he oeao -50 hange he a laon vaable : Δ Δ ex ex -58 he nnh enh an elevenh e of he oeao -50 wll hange he oenu : ex Δ Δ -59

28 8 Δ Δ 0 0 ex ex -60 ex 9 Δ Δ -6 he welfh an heenh e of he oeao -50 wll hange he heoa oenu aong o Δ Δ e e e e f f ex ν ν -6 4 ex Δ Δ -6 We gve a uay of he RESPA algoh below. h algoh wa ue n ulaon eene n he nex hae... RESPA Algoh efoe we begn he negaon hee we oue he foe beween ale bae on he uen oon of he ale. Followng he ouaon of foe he negaon hee begn wh he heoa oenu:

29 Δ -64 v Δ e ν f e We ubue oe e n equaon -66 wh he nananeou aal ef Halonan of -65. Fo a onao le flu onan an an be faoe ou an -66 hen lfe o -67 Δ ν f ex we olve fo he oena : e Δ ex Δ Δ e When equaon -68 hough -70 ae obne we ge he equaon

30 6 Δ Δ ex he oenu equaon an alo be exee n e of he foe F a F Δ -7 Subung -7 no -7 we ave a he followng exeon fo oena : -7 6 F Δ Δ ex ex we alulae he new value of he a laon vaable aong o 7 Δ ex hen we alulae he new oon of he ale a F Δ Δ 6-75 Sne we have new oon fo he ale we alulae he new foe beween he ale. We oee o alulang he new a laon vaable a 9 7 Δ ex hen we oue fo he new oena of he ale F 9 Δ Δ ex F 9 Δ -77 0

31 Fnally we alulae he heoa oenu aong o 4 Δ ν f Δ e 9 e We ue he negaon algoh uaze above n ou VE ulaon

32 VE Eneble Sulaon We an ulaon ung he algoh eve n Chae fo a onao le flu. F we ee whehe he heoa funone oely.e. houl ve he ye owa he e on ou of he aal ef Halonan an a. We hen evaluae he ulaon eul o eene whehe he ulaon wa uly n he VE eneble. he algoh wa f ee fo he le ye an eal ga whee oenal enegy neglgble. hen he algoh wa ee on a lue ga a ye ha ha oenal enegy. wo ye of ulaon wee un wh he lue ga ye. In he f ye of ulaon he oenal enegy wa exee a a ou of he a an he ef oenal enegy ˆ. In he eon ye of ulaon he oenal enegy wa exee a a funon of boh he ane beween ale an he nube eny ρ.. VE Ieal Ga Fo an eal ga he Halonan n equaon -5 eue o equaon -: H e VE ln - he equaon of oon ae eve anonally fo he Halonan n he e fae of efeene. h followe by a nonanonal anfoaon of he equaon of oon fo he e fae of efeene o he hyal o une fae of efeene. he evaon la o he oeue lluae n Chae. elow ae he equaon of oon eve fo he Halonan -:

33 e he aal ef Halonan fo h eal ga ye only ha he ne enegy e ne oenal enegy neglgble. I a funon of he ale veloe. o v -6 If we eve he evoluon equaon fo he veloy we ee ha he veloy oe no hange n e a hown n he evaon v v v v v v v v 0 Sne he veloy wll neve hange wll no hange. ng he equaon of oon - o -5 we ulae a onao le flu. he ulaon ye oee ae uaze n able.. -7

34 able.- Sulaon Inu Paaee fo ehane Value Reue Value a.6568 ga/oleule.0 Sga.780 Ango.0 Elon.6 Joule.0 eeaue 5 Kelvn.75 Deny.677E-5 ale/ango.446e- Δ feoeon.7e- Lengh of ulaon o llon e o 4 o 6 nanoeon ube of ale Sulaon whee e Equal o o We f un a ulaon whee he e on aal ef Halonan equal e o he nal value of he aal ef Halonan. In h ae o ne we ae ulang an eal ga he aal ef Halonan wll ean a a onan value hough ou he ulaon. he aal ef Halonan wll be equal o he e on houghou he ulaon an he heoa onolle no exee o ea. he nal value fo he aal ef Halonan o.68 whh alo ou e on aal ef Halonan e. o oban he aal ef Halonan we un he ulaon unl he ye eahe equlbu an we ea off he value fo he aal ef Halonan. A ee he heoa onolle no ea ne hee wa neve a e e when.. In Fgue.- we how ha he a eane onan an n o Fgue.- ha he ou no hange; eane a he e on e value of.68. h bae ae hu oue eaonable eul a ee. 4

35 . a ulaon e Fgue.- a v. ulaon e fo an eal ga ye wh e o ulaon e Fgue.- v. ulaon e fo an eal ga ye wh e o. 5

36 .. Sulaon whee e < o In he ae whee e le han o wll ll ean onan ne h ll an eal ga ulaon bu he heoa onolle exee o ea an ve owa e. he ou e wa e equal o.6 lghly le han.68. Fgue.- how ha onan a.68 houghou he ulaon. We an ee ha he onolle ve owa he e on n Fgue.-4. he ou fluuae abou he e on; ha an aveage an ana evaon value of.6000 ± he a aue value Fgue.-5 fo.0 o he ulaon aveage a of ± hu he heoa onolle funonng oely fo he eal ga ye bae on he nal e of he VE algoh. he followng eon ebe lue ga ulaon ulaon e Fgue.- v. ulaon e fo an eal ga ye wh e.6. 6

37 .6 Inananeou Aveage ulaon e Fgue.-4 v. ulaon e fo an eal ga ye wh e Inananeou Aveage a ulaon e Fgue.-5 a v. ulaon e fo an eal ga ye wh e.6. 7

38 . Dlue Ga Sulaon Fo of he Poenal Enegy In he ulaon of le flu he oenal enegy noally exee a a funon of he ane beween ale uh a he awe neaon oenal nown a he Lenna-Jone oenal enegy. In a ye whee he heal oenal o n ou ae he aal ef Halonan hel onan he oenal enegy of he ye houl hange a he nube of ale whn he ye hange. Fo ou ulaon whee a lae nea of hangng he nube of ale n he ye he oenal enegy ha o au aongly. hu an exeon fo he oenal enegy a a funon of a neee. We lafy oe e befoe we oee o u he wo fo fo he oenal enegy ue n ou lue ga ulaon. In a one oonen ye he oal a equal o whee he oal nube of ale n he ye. We ve he oal a wh he unlae a o ge he elaonh beween he nube of ale n he hyal fae of efeene an he e fae of efeene whee he onan nube of ale n he une fae of efeene an he effeve nube of ale. he effeve nube eny ρ V.. he Sef Poenal Enegy he oenal enegy an be exee n e of he a of a ale ulle by he ef oenal enegy whee ˆ ˆ he oenal enegy fo ale on a ef e un a ba. We efne he nananeou ef oenal enegy a 8

39 9 6 4 ˆ σ σ ε - hee an aonal / e n equaon - o eove any a eenene. Sne ˆ neenen of a houl no be affee by hange n he a laon vaable. In ou evaon n Chae we ue a geneal exeon fo he oenal enegy. We een below he equaon ha have he oenal enegy e exee a he ou of he a an he ef oenal enegy ˆ. F we have he equaon fo he VE Halonan fo equaon -0 an he aal ef Halonan fo equaon -: VE v H ˆ - H v VE ˆ - ex he VE Halonan fo equaon -5 exee n he e fae of efeene ln ˆ e V H E -4 La we have he equaon of oon fo he oenu equaon -7 an -5 ae anfoe o equaon -5 an -6 eevely an he heoa oenu equaon -9 an -7 ae anfoe o equaon -7 an -8 eevely. We now een he equaon of oon n he e fae of efeene n lfe fo fo a onao le flu n he une fae of efeene:

40 0 H V ˆ ' E -5 ˆ ' -6 ˆ e V H E -7 ˆ ν e e f -8.. Poenal Enegy a a Funon of an ρ he oenal enegy an alo be exee a a funon of he ane beween ale an he nube eny of he ye. A enone evouly he oenal enegy an be gven n e of a a-we neaon oenal n a le ye uh a he Lenna-Jone oenal enegy. u -9 We alo now ha he enegy of he ye an be exee hough he a oelaon funon g a g u V 0 4 π -0 whee he aal ehanal ean oenal enegy. Sne all ale ae enal h an alo be wen a

41 V 0 u g 4π o ave a an exeon fo he nananeou oenal enegy we begn wh efnng an nananeou a oelaon funon. A any nan n e we an efne an nananeou a oelaon funon abou ale gven by g ρ4π δ - whee we ae now exlly nang he e eenene of he nananeou a oelaon funon g. We an ee ha h exeon fo g val beaue afe he efnon of he a oelaon funon ρ 0 g 4π - h a oelaon funon alo wen a a funon of he nube eny. o noue a eny eenene no he a oelaon funon we we a g ρ g ρ g ρ g ρ effeve he f e on he RHS of equaon -4 he a oelaon funon a he ale eny of he ye; h eny onan. he e ρ effeve he effeve ale eny ρ. he eon e he ao of hegh n he a oelaon funon a ffeen ene. Effevely wegh he neaon aong o a ffeen eny. We hen elae only he f a oelaon funon wh he nananeou a oelaon funon g ρ ρ4π δ g ρ effeve g ρ hu he nananeou oenal enegy a a funon of he ane beween ale an he nube eny

42 effeve g g u ρ ρ ρ -6 he effeve aveage oenal enegy a he aveage effeve ye eny gven by: ρ ρ ρ ρ ρ g g u effeve effeve -7 In ou evaon n Chae we ue a geneal exeon fo he oenal enegy. We how below he equaon ha have he oenal enegy e exee a ρ. F we have he equaon fo he VE Halonan fo equaon -0 an he aal ef Halonan fo equaon -: VE v H ρ -8 ρ ρ v v -9 ex he VE Halonan fo equaon -5 exee n he e fae of efeene ρ ln e V H E -0 La we have he equaon of oon fo he oenu equaon -7 an -5 ae anfoe o equaon - an - eevely an he heoa oenu equaon -9 an -7 ae anfoe o equaon - an -4. We een he

43 equaon of oon n he e fae of efeene an lfe fo fo a onao le flu n he une fae of efeene: H V ρ E - ρ - ρ e V H E - ρ ν e e f -4. VE Dlue Ga Sulaon ng he Sef Poenal Enegy ng he ae ye n able. we f an ulaon whee we efe e o be equal o. he ulaon eule n a aly neang value of a hown n Fgue.. he heoa onolle wa no able o ea oon enough; he eul wa an unable ulaon. nle he eal ga ulaon when hee wa no oenal enegy n a lue ga ulaon no onan. he ne enegy an oenal enegy boh onbue o v ˆ -5

44 ulaon e Fgue.- v. ulaon e fo a lue ga ye ef oenal enegy ae wh e o. he oenal enegy onbuon o neenen of a. h ay aue o be unable. We now ha hee a hange n oenal enegy when he nube of ale hange n a ye; hee houl alo be a hange n oenal enegy when we hange he a of ale. he la e n he aal ef Halonan n equaon 5 aually he hange n oenal enegy wh ee o he hange n a; h lealy houl have a a eenene n. In he nex eon we e he eon foulaon of he oenal enegy a a funon of an ρ..4 VE Dlue Ga Sulaon ng Poenal Enegy a a Funon of an ρ.4. he Pa Coelaon a a Funon of an ρ he oenal enegy exee a a funon of an ρ ρ g ρ effeve u g ρ -6 4

45 In evey ulaon e ale a hange a well a he effeve nube eny. Afe evaluang he effeve nube eny we fn he a oelaon funon a h effeve nube eny an a he efe ulaon eny. hee ae a nube of eho o fn he a oelaon funon fo a gven eny of a laal onao o le flu ye. We an aque aa fo ulaon o we an nueally olve an negal equaon fo he a oelaon. One uh exeon fo he obnaon of wo equaon he Onen Zene OZ Equaon an Peu-Yev PY Aoxaon. he OZ equaon an equaon ha efne he e oelaon funon. he e oelaon funon he a of g ha nvolve only he oelaon of a enal ale an eae neghbo. We efne an aonal funon h g -7 uh ha he OZ equaon h 0 ρ h aally he OZ equaon exa beaue anfe all of ou la of nowlege of g no anohe vaable. In ohe wo we have one equaon he OZ equaon bu we have wo unnown g an. he OZ equaon eque a eon equaon uually an aoxae equaon o be olve. One of he le aoxaon ue o loe he OZ equaon he PY Aoxaon. efoe we we he PY equaon we f noue oe new vaable. F we efne a new vaable he oenal of ean foe w a g ex[ βw ] h exa beaue ly a efnon of w. he oenal of ean foe he effeve oenal eque o geneae a gven a oelaon funon. I anno be hough of a a a-we oenal. In he l ha he eny goe o zeo he oenal of ean foe aoahe he a oenal. he PY equaon aoxae he e oelaon funon a he ffeene beween a oal a oelaon funon g an

46 an ne a oelaon funon y. h ne a oelaon funon efne a ex{ β[ w u ]} y -40 he PY aoxaon hu g y f y -4 whee f he foe ue o he oenal of ean foe efne a g f w ln[ g ] β βg -4 he PY equaon equaon 4 noue an aoxaon fo o ha we now have only one equaon he OZ equaon an only one unnown g.[4] Solvng fo gwe leen an exng oe ha ue Gllan Inegaon eho o olve he OZ-PY [5] ha ue he Lenna-Jone neaon oenal. We eae a aabae of g a ffeen ene. A we olve fo an effeve eny n he ulaon we eahe an neolae whn ou aabae of g ρ. effeve.4. Sulaon Reul an Duon Cheoa Conolle F we wan o eene f we an un a able ulaon when we have oenal enegy a a funon of ane beween ale an eny. We uaze he obeve behavo of fo evou ulaon: n he eal ga ulaon wa a onan Seon.; an n he lue ga ef oenal enegy ulaon wa unable Seon.. In h lue ga ρ ulaon we e f ulae. In Fgue.4 we how able fluuang o abou a nealy onan value. 6

47 ulaon e Fgue.4- v. ulaon e fo he ulaon of a lue ga ye ρ ae wh e o. Wh a able we exe ha he heoa onolle an ve he ye owa e. In Fgue.4 fluuae abou.69 loe o he e on of.68. We vae he e on e n ffeen ulaon whle eeng he ye eny onan. he aveage fo ffeen ulaon ae eoe n able.4. he ulaon aveage value fo ae nea e nang ha he heoa onolle funonng oely. ow we woul le o now f he onolle wll oue eaonable eul f we e un ulaon a hghe ene. Whle we vae he eny we e a.6. In able.4 we how ha we ll ge aveage value fo loe o e ; he heoa onolle funon oely a ffeen ene. 7

48 ulaon e Fgue.4- v. ulaon e fo he ulaon of a lue ga ye ρ ae wh e o. able.4- Reul fo Sulaon a Dffeen Value of e e Aveage <> Sana evaon <> able.4- Reul fo Sulaon a Dffeen Dene eny Aveage Sana Devaon <> <>.446E E E E E E E

49 We have a ha lang a houl oeon o hangng he nube of ale n he ye. A oee ae ale n a D ulaon he o obable value fo a oey oeon o he alulae aveage value of h oey. We exe a o behave a a oey onolle n a D ulaon u le he eeaue n a heoa D ulaon. We how n Fgue.4 ha he a buon fo one of ou ulaon eeble a Gauan buon an ha he a ha oue he o nube of e n he oue of he ulaon oeon o he aveage a alulae n he ulaon. We lo he a oelaon funon fo one of ou ulaon n Fgue.4 4 an we how ha he ulaon ha oue a eaonable ale buon fo a lue ga ye even a a hange n ou ulaon fequeny eue a Fgue.4- a hoga fo he ulaon of a lue ga ye ρ ae wh e o. 9

50 g Fgue.4-4 Pa oelaon funon fo he ulaon of a lue ga ye ρ ae wh e o. he e fo a VE Sye A h on we nee o analyze ulaon eul o eene whehe we ae ulang n he VE eneble. Sefally we wane o oae he aal ef Halonan an he heal oenal. he oe way o oae hee wo oee hough he hange n oenal enegy he eon e n boh equaon hown below. Equaon 4 he equaon fo he aal ef Halonan: ρ v -4 he eon e on he RHS of equaon -4 he hange n oenal enegy wh ee o a hange n a. he aal ef Halonan evaluae a evey e e an we eo aveage value of ung a VE ulaon. Equaon -44 a aal ehanal exeon fo he heal oenal: 40

51 V / Λ ln < Δ > ex β ln -44 he eon e on he RHS of equaon 44 eenally he aveage value fo he hange n oenal enegy wh ee o a hange n nube of ale n he ye. Equaon -44 he exeon ue n Wo ale neon eho o alulae fo he heal oenal of a ye. We efo Wo ale neon eho a we un a VE ulaon o eae he heal oenal of he ye. Pevouly we ae ha a hange n a houl have he ae effe a a hange n he nube of ale. If we have a box of fxe volue wh a nube of ale we now ha ang anohe ale wll lghly neae he oenal enegy ne he box. We elae h neae n nube of ale o an neae n a by oneng he equaon ' ' -45 Equaon 45 ou anfoaon equaon fo he a of he ale. If he a laon vaable neae he a n he hyal une fae neae. h neae n a n he hyal fae oeon o an neae n he nube of ale n he ahyal e fae of efeene a hown n equaon 46 So fo ou box of fxe volue ang anohe ale equvalen o an neae n a. An neae n a houl alo bng an neae n oenal enegy. We oae ulaon aa fo boh he hange n oenal enegy wh ee o a hange n a an he hange n oenal enegy wh ee o a hange n he nube of ale by onveng hee o a hange n oenal enegy wh ee o a hange n ye eny: ρ wo V

52 ρ a V ' V ' ' V ' We nguh fo an fo -48. he f one a hange n he nube of ale fo aually hangng he nube of ale n he ye an no a laon. he eon he hange n he nube of ale n he hyal fae of efeene an h zeo ne a onan. he h he hange n nube of ale n he e o ahyal fae of efeene; equvalen o he exeon. We an alo oae ρ wo an ρ a wh abulae value of he aveage oenal enegy fo VE an VE ulaon. In he VE ulaon whee e wa vae whle he ye eny wa e a a onan he a of a ale wa allowe o lae o ve owa e. Fo hee ulaon we oban aveage value fo he lae a an wh equaon 45 an 46 he oeonng nube of ale n he e o ahyal fae of efeene ae alulae. Alo wh equaon -9 we alulae he effeve eny ρ effeve ha oeon o he aveage lae a. Fo hee VE ulaon we olle E v. ρ aa. We an alo eae V effeve ρ fo VE ung enee fne ffeene on ou VE ρ effeve aa. We an VE ulaon a he ffeen effeve eny value ρ effeve obane fo he VE ulaon. Coae o he VE ulaon eah VE ulaon wa a a ffeen e eny. hee VE ulaon gave aveage oenal enegy v. eny aa fo whh we an alo eae ρ fo VE ung enee fne ffeene on ou ρ aa. We alo olve fo he oenal enegy VE effeve 4

53 ung he exeon fo he oenal enegy a a funon of he a oelaon funon g V 0 u g 4π -49 he OZPY aoxaon wa ue o olve g. hen we ue enee fne ffeene o eae ρ foozpy. In Fgue.4 5 we lo he oenal enegy fo VE ulaon he fo VE ulaon an he fo negang he OZPY v. eny. In h v. eny lo we how ha he aveage value fo VE ulaon ae oaable o he aveage obane a VE ulaon an OZPY alulaon. he oenal enegy value fo VE VE an OZPY n Fgue.4 5 follow he ae en wh ee o a hange n eny; heefoe he hange n oenal enegy ue o a laon - Poenal Enegy fo OZPY fo gan eneble ulaon fo VE ulaon E-0.44E-0.46E-0.48E-0.50E-0.5E-0.54E-0.56E-0 eny Fgue.4-5 he oenal enegy fo VE VE an OZPY v. eny. 4

54 ρ fo VE equvalen o he hange n oenal enegy fo hange n nube of ale n he ye ρ fo VE o. ρ foozpy In Fgue.4 6 we lo fo Wo ale neon he ρ ρ wo hange n oenal enegy wh ee o he hange n a enegy fo ulaon aa ρ fo VE an ρ fo VE ρ a aveage oenal an alulae oenal enegy ρ foozpy. he aveage value ρ wo -577 ρ fo VE -874 an ρ foozpy -69 all eeen he hange n oenal enegy ue o hange n he nube of ale n he ye. he ρ fo VE eeen he hange n oenal enegy ue o a hange n ale a. hee fou value ae negave an eany beween an an ρ ρ ρ ρ wo fo VE fo VE fo OZPY obly ue o he ffeene n he eho of aveagng he oenal enegy fo Wo ale neon. he aveage of a vey all ove value ρ an a lnealy neang en. h value wa obane fo an eeen he evaon of a he ye goe fuhe fo eal onon ye effeve eny neae. a 44

55 /ρ fo a E-0.44E-0.46E-0.48E-0.50E-0.5E-0.54E-0.56E-0 eny /ρ fo a fo wo fo uve E fo VE fo OZPY E-0.44E-0.46E-0.48E-0.50E-0.5E-0.54E-0.56E-0 eny Fgue.4-6 he hange n oenal enegy wh ee o a hange n ye eny v. eny. 45

56 We have hown n ou ulaon aa an alulaon ha he hange n a an he hange n nube of ale gve he ae effe on he hange n oenal enegy. A he a neae he effeve eny neae an he oenal enegy neae n agnue. Fo h eul we onlue we oban eaonable ulaon eul fo he VE algoh. 46

57 4 Conluon an Fuue Wo he ay goal of h wo wa o eae a Halonan-bae equlbu oleula yna algoh fo onan heal oenal ulaon ha oe away wh ale neon an eleon. h he nvegae he obly of lang a nea of hangng he nube of ale n a VE ED ulaon. Followng he oeue by Keffe e al. [] algoh eveloen began wh foulang a VE Halonan n he e ahyal efeene fae whee a a laon vaable wa noue. he equaon of oon wee eve fo h Halonan. One of he equaon of oon ha a heoa onolle whh onol he aal ef Halonan. Cuenly hee no nown eho fo evaluang he heal oenal n e of oleula level oee. hu he aal ef Halonan whh an be alulae fo he oleula level oee uh a veloy an oenal enegy wa foulae. he aal ef Halonan a oey la bu no equal o he heal oenal. he VE ED algoh wa ee on wo ye: an eal ga an a lue ga. We e he algoh whehe able o onol he aal ef Halonan an whehe a gan eneble ulae. In boh eal ga an lue ga ulaon he heoa onolle funone oely; ove he ou owa he e on e. In he ae of he lue ga ulaon he oenal enegy exee a a funon of eny an he ane beween ale eule n able ulaon ha allowe u o e he algoh fuhe. Sefally we wane o eene whehe we ae efong a VE ulaon. We ough o eene whehe ou ooe algoh of neang a oeon o an neae n he nube of ale. We now ha an neae n ale n a fxe volue houl eul n an neae n oenal enegy. We foun ha a hange n a n ulaon equvalen o a hange n nube of ale n e of he effe on he oenal enegy. 47

58 he nex e fo fuue wo o ue Keffe e al.' [] ehoal oeue fo algoh eveloen an exen he VE algoh o a gan anonal eneble algoh. 48

59 REFERECES 49

60 . Keffe D.J. C. ag.j. Ewa an P. Ahangale. A Halonan-ae Algoh fo Rgoou oleula Dyna Sulaon n he VE V an H Eneble. n AIChE Annual eeng Cnnna OH.. Hunenbege P. heoa algoh fo oleula yna ulaon n Avane Coue Sulaon Aoahe Fo Sof ae Sene I oé S. A oleula yna eho fo ulaon n he anonal eneble. ol. Phy : Hoove W.G. Canonal Dyna - Equlbu Phae-Sae Dbuon. Phyal Revew A 985. : Keffe D.J. C. ag P. Ahangale an.j. Ewa A Genealze Halonan-ae Algoh fo Rgoou Equlbu oleula Dyna Sulaon n he Canonal Eneble. Coue eho n Ale ehan an Engneeng 005. ube. 6. oe S. A nfe Foulaon Of he Conan eeaue oleula- Dyna eho. Jounal Of Cheal Phy : Keffe D.J. C. ag P. Ahangale an.j. Ewa A Genealze Halonan-ae Algoh fo Rgoou Equlbu oleula Dyna Sulaon n he Ioba-Ioheal Eneble. ol. S aee. 8. Heffelfnge G.S. an F. Vanwol Dffuon In Lenna-Jone Flu ng Dual Conol-Volue Gan-Canonal oleula-dyna Sulaon Dv- G. Jounal Of Cheal Phy : Cagn. an.. Pe oleula-dyna Wh A Vaable ube Of oleule. oleula Phy 99. 7: Paaooulou A. E.D. ee. Luow an F. Vanwol oleula- Dyna An one-calo Sulaon In he Gan Canonal Eneble - Loal Veu Global Conol. Jounal Of Cheal Phy : üne A. Saal heoyna. Vol ew Yo: Aae Pe. 50

61 . oé S. A nfe Foulaon of he Conan eeaue oleula- Dyna eho. J. Che. Phy : uean..j. ene an G.J. ayna Reveble ulle e Sale oleula-dyna. J. Che. Phy : uae D.A. Saal ehan. Hae' Chey See e. S.A. Re. 976 ew Yo: Hae & Row. 5. Lee L.L. oleula heoyna of noneal flu. uewoh ee n heal engneeng. 988 oon: uewoh. x eha. an D.A. Kofe oleula Sulaon In A Peuo Gan-Canonal Eneble. oleula Phy :

62 APPEDICES 5

63 5 APPEDIX A F Veon fo he Cheoa Conolle We ha begun wh a heoa onolle of he fo ν f f e A- he Halonan ha woul gve h fo fo he heoa onolle e V H E A- Fo a onao le flu eal ga equaon A- lfe o e V H E A- We an alo we h Halonan n he hyal fae of efeene a e V H E A- 4 he equaon of oon ae eve anonally fo he Halonan n he e fae of efeene equaon A-. h followe by a nonanonal anfoaon of he equaon of oon fo he e fae of efeene o he hyal o une fae of efeene. he evaon la o he oeue lluae n Chae. elow ae he equaon of oon eve fo equaon A-: A- 5 A- 6

64 e A- 8 he aal ef Halonan fo h eal ga ye only ha he ne enegy e ne oenal enegy neglgble. I a funon of he ale veloe. v o A- 9 If we eve he evoluon equaon fo he veloy we ee ha he veloy oe no hange n e: v v v v v v v v 0 A- 0 In h ae a VE eal ga ye wll no be ven owa e ne he veloy wll neve hange; onequenly wll no hange. ae on he above wa aaen ha he heoa onolle A- flawe. he Halonan wa eve uh ha he heoa onolle nea of beng ooonal o he ffeene beween an e wa ooonal o he ffeene beween an ' e. he a e wa lae wh o ha he lae a woul ve owa e. A- 7 54

65 55 APPEDIX Poeue fo Gan Canonal Equlbu oleula Dyna Halonan fo Gan Canonal ED We begn wh he VE Halonan n e oonae n Chae an a heoa e u le n he oé-hoove V Halonan. he Halonan of he uloonen olyao ye n he gan anonal eneble ln e e V f f v H - whee we have e he vaable o nae ha hey ae efne n a fae of efeene befoe we aly a nonanonal anfoaon. Ae fo he a laon vaable we now have he e laon vaable n he Halonan. he Halonan n e of oena ln e e V f f H - whee f eeen a funon of. A h on we ae oneng wo oble Halonan. he f fo of he Halonan when f ln e e V f H -

66 56 Anohe fo of he Halonan when ln f an when we ae he lae a an he ognal unlae a ln ln e e V f H - 4 We le hee wo fo beaue when alyng oé Halonan oof boh eue o he laal fo of he V eneble. he f veon wll gve a heoa onolle of he fo equaon - whle he eon veon wll gve a heoa onolle of he fo -4. We have no been able o ge goo ulaon eul fo hee wo fo. he ue ha he heoa an heoa onolle ounea eah ohe. elow ae he evoluon equaon fo he heoa onolle an he heoa onolle whh we wll eve. he heoa oenu ν e e f f - 5 h equaon lfe o ν e e f f - 6 Fo a onao le flu wh ale he equaon fo he heoa oenu

67 57 ν e e f f - 7 When we ubue equaon -8 he aal ef Halonan no equaon-7 he equaon lfe o equaon -9: - 8 ν e e f f - 9 he heoa oenu fo a onao le flu wh ale e e f f ν - 0 he nananeou eeaue equaon an be ubue no equaon -0: f - oe ha f nally e > hen he heoa oenu wll have a ove value an he a wll neae. When he a hange he nananeou eeaue wll be affee an wll hange alo. h wll aue he heoa onolle o ea o au he veloy. When veloy aue he nananeou aal ef Halonan wll be affee an wll aue he heoa onolle o a one agan hangng he a. h aue he nably of he algoh. We leave h fo fuue wo. evehele we gve he e-by-e oeue of he evaon of he equaon of oon fo he geneal fo of he Halonan equaon - n he nex

68 58 eon. In he fnal eon we eonae how oé aal ehanal oof followe o ove ha he V Halonan - an -4 eue o he laal fo of he V aon funon. Devaon of he equaon of oon A we have one n Chae we ely on he yle elaonh beween he Halonan an he equaon of oon o eve he followng evoluon equaon n he e fae of efeene: V H - H V - H V - 4 e V f H - 5 V H - 6 f H e V - 7

69 59 We efne a nonanonal anfoaon o a hee equaon of oon fo he e o ahyal fae o he hyally eanngful une fae. he f e o anfo o by he elaon gven by oé : e - - e f - he eon e o ae he evave of he anfoaon equaon equaon -4 o

70 he evave ae

71 6 he h e o ubue equaon -8 o - no equaon - o -7: f e - 4 e f Fnally ung he anfoaon equaon on he eanng e vaable he equaon of oon n he une fae ae

72 ν e e f f - 47 e e f f ν Saal ehanal Poof Halonan Equaon - We begn wh he Halonan exee n laboaoy oonae n he aheaal fae of efeene equaon - ln e e V f H ex we evelo he V aon funon. elow we ae ue of he efnon

73 6-50 K - 5 K - 5 ] [! E H Z V V δ K K - 5 We anfo he vaable fo he aheaal fae of efeene o he hyal fae of efeene ung he anfoaon equaon fo he evou eon. he anfoaon fo he oena boen no wo e fo lay n ueeng e. F we aly he e laon an hen he a laon: he anfoaon ale o he Halonan an o he ffeenal: e e V f H ln ˆ - 56

74 64 D - 57 D - 58 he anfoe aon funon hu ] [! D V D V E H Z δ K K - 59 ex we negae he aon funon ove. We nex ue a δ-funon oey δ[g] Σ δ- / G/ whee ae all he oo afyng G0 leng E H G V G - 6 G ha only one oo E f e e e - 6 o ha f G e - 6 an [ ] e f G G δ δ δ - 64

75 65 We ubue he la exeon no he aon funon o oban! D D e V f Z δ K K - 65 We ue anohe δ-funon oey f f δ o efo he negaon ove : D D f f - 66 he aon funon now beoe! D D e V f Z K K - 67 E f D D e V e e e f Z! K K - 68 ex we negae ove. Se f D. We aly he oluon a x e ax π fo negaon able o ave a e e e π - 69 Afe negang ove he aon funon beoe! e e D e V e f C Z K K - 70

76 66 whee C e E e e C π - 7 hen we negae ove. ng a x e ax π agan e e e π - 7 Afe negang ove he aon funon beoe e e D V e C Z K - 7 whee C e e f C C! π - 74 We hen negae ove : D K onan C he aon funon hen beoe e e V e C C Z - 75

77 67 o e e e V e e C C Z - 76 whh ooonal o he laal aon funon fo he V eneble e e e whee e e Halonan Equaon - 4 We begn wh he Halonan exee n laboaoy oonae n he aheaal fae of efeene equaon -4 ln ln e e V f H hen we evelo he V aon funon. Le - 77 K - 78 K - 79 ] [! E H Z V V δ K K - 80

78 68 ex we anfo he vaable fo he aheaal fae of efeene o he hyal fae of efeene ung he anfoaon equaon fo he evou eon. he anfoaon fo he oena boen no wo e fo lay n ueeng e: he anfoaon ale o he Halonan an he eque ffeenal hough he elaonh e e V f H ln ln - 8 D - 84 D - 85 he anfoe aon funon

79 69 ] [! D V D V E H Z δ K K - 86 ow we negae he aon funon ove. We f ue a δ-funon oey δ[g] Σ δ- / G/ whee ae all he oo afyng G0 le E H G V G - 88 G ha only one oo E f e e e ln - 89 o ha f G e - 90 an [ ] e f G G δ δ δ - 9 We ubue he la exeon no he aon funon! D D e V f Z δ K K - 9 We ue anohe δ-funon oey f f δ o efo he negaon ove

80 70 D D f f - 9 he aon funon now beoe! D D e V f Z K K - 94 E f D D e V e e e f Z ln! K K - 95 ex we negae ove. Se f D. We aly he oluon a x e ax π fo negaon able o ave a e e e π - 96 Afe negang ove he aon funon beoe ln! e e D e V e f C Z K K - 97 whee C e E e e C π - 98 ow we negae ove. ng a x e ax π agan

81 7 e e e π - 99 Afe negang ove he aon funon beoe ln e e D V e C Z K - 00 whee C e e f C C! π - 0 Laly we negae ove. h he only a whee Halonan - an -4 ffe. he n ln wa neeay n oe fo he e e o uvve h negaon e: ln e e D e K onan C he aon funon hu e e V e C C Z - 0

82 7 he aon funon an alo be exee a equaon -0 whee he ooonaly o he laal aon funon of he V eneble lea: e e e V e e C C Z - 0 e e e whee e e h oof nae ha he algoh eve fo hee Halonan oul oue goou V ulaon.

83 APPEDIX C Sufaan an Ele Fel Sengh Effe on Sufae enon a Lqu/Lqu/Sol Inefae h aba fo a ae by he ae nae ublhe n he ounal Langu n 006 by Johanna. Sanago Dav J. Keffe an Robe. Coune: Sanago J.. Keffe D.J. Coune R.. Sufaan an Ele Fel Sengh Effe on Sufae enon a Lqu/Lqu/Sol Inefae Langu Reoue n a wh eon fo Langu Coygh 006 Aean Cheal Soey. Aba We efoe a ee of exeen egne o eluae he effe of he eene of ou oeyl ulfae SDS ufaan an ale eleal fel on he weng behavo n a ye onanng a ele ole of henylehyl olyloxane PPS ol on a olhe anle eel ufae ubee n aqueou oluon. he volage ffeene ange fo -V o V whh a lea hee oe of agnue alle han oaable een wo. We eo he eaue equlbu ona angle of he ole a a funon of ufaan onenaon an fel engh. We hen oele he ye. We olve Lalae equaon o oban he hee-enonal fel whn ou ye. We exane he hee ufae enon ol ole-aqueou oluon oa ol ole-eal ufae o an aqueou oluon-eal ufae a n aylo ee wh ee o ufaan onenaon an loal fel engh. We ue hee hee ufae enon n Young equaon o oban he heoeal ona angle of he ogan ole. We eonae ha he lage hange n ona angle ue o he ulaneou eene of all onenaon of ufaan an all volage ffeene an be aoune fo by hange n he oa an a ufae enon. ey wo: ona angle ole eahen ufae oenal eleoweng ufaan SDS PPS 7

84 VIA Johanna Sanago wa bon an ae n uezon Cy Phlne. She gauae fo he nvey of he Phlne Dlan wh a ahelo of Sene egee n Cheal Engneeng n 00. She hen woe a F Ga Powe Cooaon n he Phlne fo wo yea befoe aenng he Cheal Engneeng.S. oga a he nvey of enneee Knoxvlle n Augu 004. She wll eeve he.s. egee fo he nvey of enneee Knoxvlle n Augu