PerÐlhyh H moriak arqitektonik kai o sqediasmìc polôplokwn morðwn pou perièqoun foullerènia antiproswpeôei èna pedðo thc upermoriak c epist mhc sto op

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1 DIDAKTORIKH DIATRIBH MORIAKH MONTELOPOIHSH THS UGROKRUSTALLIKHS SUMPERIFORAS UPERMORIAKWN SUSTHMATWN POU PERIEQOUN FOULLERENIA StaÔrou D. PeroukÐdh upoblhjeðsa sto Diatmhmatikì Prìgramma Metaptuqiak n Spoud n sthn Epist mh kai TeqnologÐa twn Polumer n epiblèpwn: D. I. Fwteinìc Panepist mio Patr n Okt brioc 2006

2 PerÐlhyh H moriak arqitektonik kai o sqediasmìc polôplokwn morðwn pou perièqoun foullerènia antiproswpeôei èna pedðo thc upermoriak c epist mhc sto opoðo prosf twc anaptôssetai èntonh ereunhtik drasthriìthta. H kat llhlh qhmik tropopoðhsh tou foullerèniou} èqei odhg sei ston sqhmatismì miac meg lhc poikilðac ugrokrustallik n parag gwn tou. Sthn paroôsa diatrib, me thn eisagwg miac apl c moriak c jewrðac tou sunarthsiakoô thc puknìthtac kai me upologistikèc prosomoi seic, prospajoôme na sundèsoume thn autoorg nwsh pou ekdhl noun aut ta sust mata me thn moriak touc dom kai tic diamoriakèc allhlepidr seic. Sto pr to kef laio k noume mia anadrom sto mìrio foullerènio kai stic qhmikèc domèc autoô, pou parousi zoun endiafèron se sqèsh me ugrokrustallik sumperifor. Parousi zoume en suntomða, me mia fainomenologik perigraf kai taxinìmhsh, tic ugrokrustallikèc f seic pou ja sunant soume sta epìmena kef laia. Tèloc, anaferìmaste stic mejìdouc pou qrhsimopoioôme gia th jewrhtik melèth foullerenik n ugr n krust llwn. Sto deôtero kef laio perigr fetai h statistik mhqanik kai jermodunamik eôkamptwn megalomoriak n susthm twn. Proqwr me sth diatôpwsh thc akriboôc èkfashc thc eleôjerhc enèrgeiac Helmholtz sthn kanonik sullog (NVT) gia èna monosustatikì sôsthma eôkamptwn megalomorðwn. Eis goume thn prosèggish twn allhlometatrepìmenwn moriak n sqhm twn}, omadopoi ntac tic moriakèc diamorf seic se ènan orismèno arijmì sqhm twn}, orðzoume thn sun rthsh katanom c tou enìc morðou kai anaptôssoume thn eleôjerh enèrgeia se seir qrhsimopoi ntac th jewrða twn metabol n. Lamb noume mia proseggistik èkfrash gia thn eleôjerh enèrgeia krat ntac touc ìrouc pou proèrqontai apì du dec tautìqrona allhlepidr ntwn morðwn, agno ntac touc upìloipouc ìrouc. DÐnoume tic antðstoiqec ekfr seic an loga me tic summetrðec thc ugrokrustallik c f shc, gia th melèth moriak n susthm twn twn opoðwn ta mìria mporoôn na lamb noun di fora sq mata. AnaptÔssoume tic jermodunamikèc ekfr seic gia thn perigraf twn metatrop n f shc. Na shmei soume ìti h sun rthsh katanom c tou enìc morðou eðnai ginìmeno dôo parag ntwn: o pr toc ìroc sqetðzetai me thn pijanìthta eôreshc tou apomonwmènou morðou se orismèno sq ma} kai o deôteroc me thn pijanìthta eôreshc tou morðou se orismèno sq ma} kai dieujèthsh sto ulikì. GenikeÔoume thn parap nw mejodologða, sth melèth susthm twn pou apoteloôntai apì dôo perissìtera eðdh eôkamptwn morðwn. Sto trðto kef laio efarmìzoume thn jewrða twn allhlometatrepìmenwn sqhm twn, gia th melèth thc fasik c sumperifor c thgm twn adropoihmènwn foullerenik n ugr n krust llwn. K je mìrio foullerenikoô ugroô krôstallou (UK) upodiaireðtai se ènan arijmì tmhm twn orismènou tôpou. Eis goume treðc tôpouc tmhm twn pou antistoiqoôn se foullerenikèc mon dec, se mesogìnec mon dec kai se mh mesogìnec sundetikèc om dec. H diamoriak allhlepðdrash upologðzetai me jroish ep nw se ìla ta dunat zeôgh tmhm twn metaxô dôo morðwn. H kðnhsh kai h peristrof twn morðwn jewroôme ìti pragmatopoieðtai se èna kubikì plègma, kai h moriak eukamyða eis getai

3 II epiptrèpontac na lamb noun ènan el qisto arijmì antiproswpeutik n moriak n sqhm twn, sto plaðsio thc anapar stas c touc sto kubikì plègma. PragmatopoioÔntai upologismoð gia mia poikilða moriak n arqitektonik n, pou perilamb noun foullerenikoôc UK me dðdumouc mesogìnouc braqðonec, foullerenikoôc UK me dðdumouc dendritikoôc braqðonec, kai kwnikoôc puramidikoôc foullerenikoôc UK me apeujeðac sundedemèna mesogìna sthn epif neia tou foullerèniou. Oi jewrhtikoð upologismoð anapar goun se entupwsiak kal sumfwnða ton ugrokrustallikì polumorfismì; ta sônora eust jeiac twn ugrokrustallik n f sewn, kaj c kai h moriak org nwsh se aut, katanooôntai me ìrouc diamoriak n allhlepidr sewn kai moriak c eukamyðac. Sto tètarto kef laio diereunoôme thn fasik sumperifor migm twn kamptwn swmatidðwn. Ta swmatðdia kinoôntai kai peristrèfontai se èna kubikì plègma; qtðzontai} apì ènan orismèno arijmì tmhm twn ìpou kajèna èqei to mègejoc miac plegmatik c kuyelðdac tou kubikoô plègmatoc. Arqik, exet zoume mðgmata adropoihmènwn foullerèniwn (pou katalamb noun mia plegmatik kuyelðda) me rabdìmorfa kai allhlepidroôn me sklhrèc} allhlepidr seic. DiereunoÔme thn ekd lwsh smhktik c, nhmatik c isìtrophc sumperifor c gia k je sustatikì. BrÐskoume ìti emfanðzoun sônjeth fasik sumperifor pou sunðstatai sthn sunôparxh dôo perissotèrwn f sewn ek twn opoðwn mia toul qiston eðnai ugrokrustallik (nhmatik smhktik ). H smhktik f sh apoteleðtai apì diadoqik str mata r bdwn pou enall ssontai apì upostr mata foullerèniwn. Elègqoume thn axiopistða thc jewrðac, sugkrðnontac ta apotelèsmata pou lamb noume me thn up rqousa bibliografða se jewrhtik apotelèsmata gia parìmoia sust mata. Diapist noume sumfwnða wc proc thn fasik sumperifor kai thn org nwsh mèsa sthn smhktik f sh kai proqwr me se problèyeic. DiereunoÔme ton rìlo thc mh filikìthtac} twn swmatidðwn sthn fasik sumperifor, enisqôontac thn fobikìthta} metaxô diaforetik n swmatidðwn (mèsw tropoi shc tou diaswmatidiakoô dunamikoô allhlepðdrashc). Oi ugrokrustallikèc f seic tìte mporoôn na uposthrðxoun} mikrìterec posìthtec foullerenðwn, en h smhktik f sh (me upostr mata foullerenðwn) kajðstatai eustaj c se mia sten perioq tou diagr mmatoc f shc pðeshc - sugkèntrwshc. Sthn sunèqeia exet same mðgmata diskìmorfwn swmatidðwn kai foullerèniwn kai thn sqetik eust jeia sthn ekd lwsh isìtrophc, nhmatik c, smhktik c kai kionik c f shc. Ta swmatðdia allhlepidroôn me sklhrì} dunamikì en sthn sunèqeia, exet zoume kai thn epðdrash thc fobikìthtac} metaxô diaforetik n swmatidðwn sth fasik sumperifor. H smhktik f sh eðnai apoôsa se ìlec tic peript seic pou exet same; parathroôme jermodunamik eustajeðc kionik kai nhmatik. Kai p li diapist noume ìti h enðsqush thc fobikìthtac} metaxô diaforetik n swmatidðwn èqei san apotèlesma stic ugrokrustallikèc f seic na anamignôontai mikrìterec posìthtec foullerèniwn. Gia lìgouc plhrìthtac se sqèsh me ta parap nw sust mata, exet same mðgmata swmatidðwn apì sustatik pou to kajèna mporeð na ekdhl sei UK sumperifor, dhlad mðgmata diskìmorfwn me rabdìmorfa swmatðdia. Ta swmatðdia allhlepidroôn apokleistik me sklhrì} dunamikì. Lamb noume poiotik apodekt apotelèsmata ìson afor thn topologða tou diagr mmatoc f shc sthn perioq sunôparxhc nhmatik c kalamitik c kai nhmatik c diskotik c f shc. Epiprìsjeta problèpoume thn sunôparxh nhmatik c kalamitik c me kionik f sh. Ta apotelèsmata autoô tou kefalaðou mporoôn na qrhsimeôsoun gia thn melèth

4 III perissìtero polôplokwn susthm twn, pou ja perilamb noun mìria ta opoða mporoôn na l boun perissìterec apì mia diamìrfwsh, kai diaforetik parametrikopoðhsh sta dunamik allhlepðdrashc. Sto pèmpto kef laio exet zoume thn fasik sumperifor adropoihmènwn foullerenik n UK pou perilamb noun dendrimer tôpou Percec. T gmata twn parap nw foullerenik n UK ekdhl noun kionik f sh. Sth montelopoðhsh twn morðwn omadopoioôme tic moriakèc diamorf seic se èna moriakì sq ma. Oi jewrhtikoð upologismoð pragmatopoioôntai se èna kubikì plègma. Ta mìria apoteloôntai apì èna foullerènio (pou katalamb nei èna tm ma -mia plegmatik kuyelðda- ) kai apì ènan orismèno arijmì tmhm twn, gia thn perigraf tou dendritikoô mèrouc tou morðou. Se mia eureða perioq twn upologismènwn, me th moriak jewrða, diagramm twn f shc ekdhl noun kionik sumperifor ; ta mìria sthn kionik f sh autosunarmologoôntai se diskìmorfec peplatusmènec upermoriakèc domèc oi opoðec me thn seir touc topojetoôntai h mia p nw sthn llh sqhmatðzontac kðonec. Ta apotelèsmata aut eðnai se poiotik sumfwnða me peiramatik apotelèsmata an logwn susthm twn. Metab llontac th fobikìthta} foullerèniwn - dendritik n tmhm twn parathroôme thn emf nish dôo smhktik n f sewn pou diafèroun ston bajmì allhlodieðsdushc twn dendritik n mer n mèsa sto smhktikì str ma. Sthn sunèqeia exet same thn epðdrash pou èqei sth fasik sumperifor aut n twn thgm twn foullerenik n UK h prosj kh foullerèniwn. H organwmènh kionik f sh tou mðgmatoc mporeð na uposthrðxei mikrèc posìthtec eleôjerwn} foullerenðwn ta opoða diat ssontai gôrw apì touc xonec twn kiìnwn. Sto èkto kef laio exet zoume thn fasik sumperifor monosustatik n kwnik n ( puramidik n) foullerenik n UK me thn ektèlesh moriak n prosomoi sewn Monte Carlo (NVT) se kubikì plègma. Arqik, ekteloôme moriakèc prosomoi seic ajermikoô sôsthmatoc (me diamoriakèc allhlepidr seic sterik c pwshc) gia mia seir puknot twn kai diapist noume ìti to sôsthma metabaðnei apì mia kionik f sh sthn isìtroph. Sthn sunèqeia jètoume thn parametrikopoðhsh twn diamoriak n allhlepidr sewn Ðdia me ekeðnh twn antðstoiqwn moriak n susthm twn pou melet same me jewrða sto trðto kef laio. Exet zoume thn fasik sumperifor tou sust matoc se orismènh puknìthta, gia mia seir jermokrasi n. ParathroÔme ìti emfanðzetai mia metatrop f shc, isìtrophc se mia apolik tetragwnik kionik f sh tìso kat thn yôxh ìso kai kat thn jèrmansh. Ta upologistik apotelèsmata pou lamb noume den mporoôn na sugkrijoôn me to pl rec jewrhtikì di gramma pou parèqei h moriak jewrða gia aut ta sust mata, epeid stic moriakèc prosomoi seic pou ektelèsame esti same se orismènh puknìthta. Epitugq netai poiotik sumfwnða wc proc thn prìbleyh emf nishc kionik c f shc. Tèloc, sto èbdomo kef laio parousi zoume ta sumper smata thc diatrib c.

5 IV Abstract The covalent linkage of [C60] fullerene to liquid crystalline (LC) chemical compounds has produced a great variety of non-conventional liquid crystals. Herein, we introduce a coarse grained description for the molecular modeling of these systems which takes into account the elements of primary importance for their mesomorphic behavior: the global molecular shape, the molecular flexibility and the submolecular partitioning into chemically distinct regions. A simple molecular theory and computer simulations studies are used to relate the self-organization exhibited by these systems to their molecular structure. The molecules are constructed by a number of submolecular blocks to which specific blockblock interactions are assigned. We have introduced three different types of blocks: fullerene units, mesogenic units and non-mesogenic units (i.e. flexible spacers and linkage groups). The total interaction of the molecular ensemble is then built up as a combination of all block - block interactions within the ensemble. For reasons of simplicity, the molecules are constrained to move and rotate in a cubic lattice. The theoretical calculations are initially applied to a variety of mono dispersed fullerene containing LCs, including: twin mesogenic branch monoadducts of C60, twin dendromesogenic branch monoadducts of C60, simple dendromesogenic branch monoadducts and conical (badminton shuttlecock) multiadducts of C60. The latter system is also studied by computer simulations. In spite of its many simplifications, the coarse-grained molecular model accounts remarkably well for the phase polymorphism of these systems and offers valuable insights into the conformational aspects of the phase transitions. Keeping the simplicity of the above modeling, the phase behavior of binary mixtures of "fullerene" with anisotropic particle (rodlike or disklike) and binary mixtures of rodlike and disklike particles is also studied. We compare the predicted phase behavior with the existing theoretical (hard rodlike and disklike particles) and experimental (colloidal systems) results. Finally attempts are made to study how the molecular organization of mono dispersed fullerene containing LC systems is influenced by the addition of non-bonded fullerenes. The theoretical approach is quite general and can be applied to a variety of systems such as, liquid crystalline dendrimers and mixtures of colloidal particles.

6 I Didaktorik diatrib pou egkrðjhke omìfwna apì thn Eptamel Exetastik Epitrop pou orðsthke apì thn Eidik Diatmhmatik Epitrop tou DiatmhmatikoÔ Progr mmatoc Metaptuqiak n spoud n sthn Epist mh kai TeqnologÐa Polumer n tou PanepisthmÐou Patr n, sthn up r. 5/ sunedrðas thc sômfwna me tic diat xeic tou rjrou 12 par.5b tou N. 2038/92. Ta mèlh thc EptameloÔc Exetastik c Epitrop c san: A. Banak rac, Lèktorac tou Tm matoc Epist mhc twn Ulik n PanepisthmÐou Patr n J. Jeod rou, Kajhght c tou Tm matoc Qhmik n Mhqanik n E. M. P M. Kosm c, Kajhght c tou Tm matoc QhmeÐac PanepisthmÐou IwannÐnwn B. Maurantz c, Anaplhrwt c Kajhght c tou Tm matoc Qhmik n Mhqanik n PanepisthmÐou Patr n A. Terz c, EpÐkouroc Kajhght c tou Tm matoc Fusik c PanepisthmÐou Patr n Q. Topraktsiìglou, Kajhght c tou Tm matoc Fusik c PanepisthmÐou Patr n D. Fwteinìc, Kajhght c tou Tm matoc Epist mhc twn Ulik n PanepisthmÐou Patr n

7 perieqìmena 1 Eisagwg To foullerènio C Sust mata qhmik tropopoihmènwn C Ugrokrustallik sumperifor Ta UK dendrimer Mèjodoi jewrhtik c prosèggishc thc autoorg nwshc Jewrhtikì upìbajro Statistik mhqanik eôkamptwn megalomorðwn H prosèggish allhlometatrepìmenwn moriak n sqhm twn} Isìtroph f sh Nhmatik f sh Smhktik f sh Kionik f sh Metatropèc f sewn monosustatik n susthm twn MÐgmata Isìtroph f sh Nhmatik f sh Smhktik f sh Metatropèc f shc migm twn dôo sustatik n Monosustatik sust mata foullerenik n ugr n krust llwn Eisagwg Moriak montelopoðhsh se kubikì plègma FoullerenikoÐ UK me DÐdumouc Mesogìnouc BraqÐonec (TMBC60) Apotelèsmata thc moriak c jewrðac gia to TMBC FoullerenikoÐ UK me DÐdumouc DendritikoÔc BraqÐonec (TDBC60) Apotelèsmata thc moriak c jewrðac gia to TDBC KwnikoÐ FoullerenikoÐ UK (CSMC60)

8 perieqìmena Apotelèsmata thc moriak c jewrðac gia to CSMC SÔnoyh - Sumper smata Ugrokrustallik sumperifor kai moriak org nwsh se mðgmata kamptwn swmatidðwn diafìrwn sqhm twn Eisagwg Moriak montelopoðhsh duadik n susthm twn MÐgmata foullerèniwn kai rabdìmorfwn swmatidðwn MÐgmata foullerèniwn kai diskìmorfwn swmatidðwn MÐgmata rabdìmorfwn kai diskìmorfwn swmatidðwn SÔnoyh - Sumper smata Autosunarmolìghsh kai autoorg nwsh foullerenik n mesogìnwn se kionikèc mesof seic Eisagwg Moriakì montèlo tou DESC Apotelèsmata Monosustatik sust mata tou DESC MÐgmata foullerèniwn kai morðwn DESC SÔnoyh - Sumper smata Moriakèc prosomoi seic twn kwnik n foullerenik n ugr n krust llwn (CSMC60) Teqnikèc leptomèreiec twn upologistik n prosomoi sewn Par metroi t xhc kai sunart seic susqètishc Apotelèsmata - suz thsh Ajermikì montèlo tou CSMC Montèlo tou CSMC60 me malakèc} diamoriakèc allhlepidr seic SÔnoyh - sumper smata EpÐlogoc Sumper smata BibliografÐa 110

9 Kef laio 1 Eisagwg Sto kef laio pouakoloujeð gðnetai mia sôntomh anaskìphsh tou eurôterou ereunhtikoô pedðou pou ent ssontai ta sust mata pou pragmateôetai h paroôsa diatrib. Eidikìtera anaferìmaste sto mìrio foullerènio, se sut mata qhmik tropopoihmènwn foullerèniwn, kai sthn ugrokrustallik sumperifor sumbatik n kai mh moriak n susthm twn. Parousi zoume tic basikèc jewrhtikèc kai upologistikèc mejìdouc pou qrhsimopoioôme sta epìmena kef laia gia th melèth foullerenik n ugr n krust llwn. 1.1 To foullerènio C60 Ta foullerènia eðnai qhmikèc en seic pou apoteloôntai apì toma njraka dieujethmèna ètsi ste na sqhmatðzoun ènan klwbì (cage-like molecules). To foullerènio C60, eðnai apì touc kuriìterouc ekpros pouc thc foullerenik c oikogèneiac}. AniqneÔthke pr th for se mikrèc posìthtec apì touc 1 Kroto et. al. sta 1985 me ex tmish grafðth prokaloômenh apì laser. Argìtera, sta 1990, par qjhke se proparaskeuastik klðmaka (thc t xhc tou miligkr m) apì touc 2 Kratschmer et. al. Perigr fthke san mia nèa morf stereoô njraka apoteloômenh apì takta diatetagmènec domikèc mon dec pou h k je mia apoteleðtai apì ex nta toma njraka. Se mian kal prosèggish, mporoôme na jewr soume ìti ta ex nta toma njraka topojetoôntai stic korufèc enìc kanonikoô kìlourou eikos edrou 3 (icosahedral Ih). K je tomo njraka sundèetai me qhmikì desmì me lla trða geitonik tou. Apì tic 32 èdrec pou sqhmatðzontai sto kìlouro eikos edro 12 apì autèc eðnai pent gwna kai 20 ex gwna. K je ex gwno sqhmatðzei mia dom Kekule me enallassìmenouc aploôc kai diploôc qhmikoôc desmoôc. H summetrða tou morðou diathreðtai par tic mikrèc apoklðseic stic timèc tou m kouc twn qhmik n desm n, me th mèsh tim thc apìstashc C C na eðnai 4 0,144nm. To foullerènio C60 loipìn, eðnai mia kleist moriak dom pou epalhjeôei to je rhma tou Euler, sômfwna me to opoðo mia kleist epif neia pou apoteleðtai apì pent gwna kai ex gwna èqei akrib c 12 pent gwna kai adi foro} arijmì exag nwn. 5 Jewr ntac ta toma tou njraka wc shmeða, aut h sfairik } nanodom, èqei di metro 0,71nm. 6 Mia apeikìnish thc 4

10 1.1. To foullerènio C60 5 dom c tou C60 dðnetai sto sq ma 1.1a. To antikeðmeno pou perigr foume jumðzei mp la podosfaðrou (sq ma 1.1b). Ta mìria C60 krustall nontai se stere Van der Waals (ta opoða eðnai monwtèc hmiagwgoð lìgw twn sumplhrwmènwn moriak n floi n) se kubik dom me plegmatik stajer 1,417nm, kai mèsh apìstash metaxô kontin n geitìnwn njraka - njraka 6 1,002nm. H diafor 0,292nm (se sqèsh me thn di metro) ofeðletai stic aktðnec Van der Waals. Se jermokrasða dwmatðou, ta mìria peristrèfontai eleôjera (plastikìc krôstalloc -plastic crystal) kai ta gewmetrik kèntra touc diat ssontai se kubik edrokentrwmènh krustallik dom, me èna mìrio se k je stoiqei dh kuyelðda. Se qamhlìterec jermokrasðec (260K) pragmatopoieðtai mia metatrop f shc pr thc t xhc 3 se apl kubik krustallik dom. To shmeðo t xhc tou stereoô eðnai idiaðtera uyhlì, 7 stouc 1453K. To C60 genik parousi zei meg lh ikanìthta sto na dèqetai hlektrìnia. Parìti eðnai hmiagwgìc me meg lh hlektrik antðstash h eisagwg metallik n stoiqeðwn (kajorismènhc stoiqeiometrðac) èqei san apotèlesma thn meðwsh aut c. Brèjhke 8, 9 ìti h embol alkalik n met llwn (hlektroniodìtec) sto C60 me stoiqeiometrða MxC60 (ìpou M=alk lia kai x fusikìc arijmìc) metab llei sunolik tic hlektroniakèc idiìthtec tou ulikoô metatrèpont c to se ag gimo uperag gimo, me sqetik uyhl jermokrasða uperagwgimìthtac (gia par deigma to K 3 C 60 kajðstatai uperagwgìc stouc 19K). Shmantikèc eðnai kai oi optikohlektronikèc tou idiìthtec. H ekmet leush aut n mporeð na breð, gia par deigma, efarmog sthn prostasða fwtoeuaðsjhtwn ulik n prostateôont c ta apì uyhl c èntashc palmoôc 10 laser. H dunatìthta qhmik c tropopoðhshc} tou C60 to kajist autom twc ikanì gia thn paragwg plhj rac nèwn ulik n. 'Enac meg loc arijmìc qhmik n en sewn (upokatast tec) mporoôn na sundejoôn sthn epif neia autoô me qhmikì desmì. 'Etsi, mporoôn na sundejoôn dôo perissìteroi upokatast tec me aplì desmì se dôo perissìtera toma njraka, opìte anaferìmaste se par gwga tou foullerenðou me aploôc desmoôc (singly bonded functionalized derivatives). Up rqoun par gwga tou foullerèniou ìpou h sôndesh upokatast th odhgeð ston sqhmatismì propanikoô daktulðou (cyclopropanated derivatives). M lista eðnai dunatìc kai o sqhmatismìc tetraploô, pentaploô exaploô daktulðou. Gia mia ekten anafor stic fusikoqhmikèc idiìthtec tou C60 all kai se 3, 10, 11 teqnologikèc efarmogèc parapèmpoume sthn bibliografða. Parìlo thn ploôsia hlektrik kai hlektroqhmik sumperifor tou C60 h epexergasða tou èqei anadeiqjeð se kôrio prìblhma gia thn qr sh autoô se praktikèc efarmogèc. 10 To C60 eðnai sqedìn adi luto dôskola dialutì stouc perissìterouc dialôtec kai susswmat netai eôkola pr gma pou to kajist akìma ligìtero dialutì Eidik sto nerì parousi zei uyhl udrofobikìthta kai emfanðzontai meg lec dun meic sunoq c metaxô twn mori n C60. Autì to dôskolo empìdio ègine dunatì na xeperasteð, toul qiston enmèrei, me thn qhmik tropopoðhs tou. H qhmik tropopoðhsh tou

11 1.2. Sust mata qhmik tropopoihmènwn C60 6 a) b) Sq ma 1.1: a) Apeikìnish thc dom c tou foullerèniou C60 b) To sq ma tou jumðzei mp la podosfaðrou. C60 od ghse sthn sônjesh enìc meg lou arijmoô nèwn ulik n se poll apì ta opoða diathroôntai oi arqikèc idiìthtec tou foullerèniou, kai tautìqrona eðnai perissìtero eôkola sthn epexergasða. Aut h dunatìthta pou parousi zei to C60 èdwse jhsh ston kl do thc upermoriak c qhmeðac tou foullerèniou. 1.2 Sust mata qhmik tropopoihmènwn C60 O sqediasmìc, h melèth thc org nwshc kai sumperifor c mh sumbatik n upermoriak n mon dwn pou perièqoun foullerènia antiproswpeôei s mera èna pedðo suneq c auxanìmenou endiafèrontoc thc upermoriak c qhmeðac qwrðc katfl an gkhn na periorðzetai se aut. Amèswc parak tw dðnoume orismèna antiproswpeutik paradeðgmata upermoriak c qhmeðac tou tropopoihmènou foullerenðou. Ta foullerènia eðnai pl rwc adi luta sto nerì, all h qhmik tropopoðhsh aut n kajist aut dialut kai mporeð na odhg sei ston sqhmatismì upermoriak n dom n. Prìsfata èqoun parathrhjeð sfairikèc nanodomèc (vesicles) apoteloômenec apì tropopoihmèno foullerènio C 60 (C 6 H 5 ) 5 K sto nerì. 15 Lìgw thc udrofobikìthtac tou foullerèniou aut autodomoôntai arqik se didi statec diplèc stoib dec. Oi diplèc stoib dec apoteloôn ton floiì} twn sfairik n nanodom n me diast seic thc t xhc 10 1 nm (sq ma 1.2a). Ta polik udrìfila mèrh twn tropopoihmènwn foullerèniwn koitoôn proc ta èxw} ekatèrwjen tou floioô (sq ma 1.2a). M lista, to mègejoc twn sfairik n nanodom n exart tai apì to eðdoc twn upokatastat n pou perib lloun thn polik om da (K). 'Etsi sfairikèc nanodomèc apì tropopoihmèna C 60 (CH 3 ) 5 K èqoun mikrìterh aktðna 16. 'Opwc ja doôme h dipl stoib da (ed o floiìc tou upermorðou) den eðnai k ti asun jisto sta upermoriak sust mata pou perièqoun foullerènia. O mhqanismìc pou perigr yame sto prohgoômeno par deigma moi zei me touc mhqanismoôc autoorg nwshc sta sumbatik amfðfila mìria ìpwc ta lipðdia ta Apì ed kai sto ex c ìpou qrhsimopoioôme th lèxh foullerènio sto keðmeno, ja anaferìmaste sto foullerènio C60. "Beyond molecular chemistry based on the covalent bond, supramolecular chemistry aims at developing highly complex chemical systems from components interacting through noncovalent intermolecular forces" (J-M. Lehn)

12 1.2. Sust mata qhmik tropopoihmènwn C60 7 opoða sqhmatðzoun thn kuttarik membr nh pou eðnai ugrokrustallik c fôshc (sq ma 1.2a). Oi om dec fainulðou sto C 60 (C 6 H 5 ) 5 K eðnai dunatìn na tropopoihjoôn me megalôterec kamptec om dec (mesogìna) (sq ma 1.2b), ste na sqhmatðzoun kwnik mìria ikan na filoxen soun mèsa sthn kwnik touc koilìthta llo tropopoihmèno mìrio foullerèniou 17. H kinhtikìthta se moriakì epðpedo gia autèc tic domèc epitugq netai lìgw twn anjrakik n alusðdwn sta kra twn mesogìnwn /kai lìgw thc parousðac dialôth 18. Genikìtera, ta mìria aut èqoun kwnikì puramidikì sq ma kai ìpwc ja doôme leptomerèstera se eidikì kef laio, autodomoôntai sqhmatðzontac monodi statouc kðonec kai parousi zoun ugrokrustallik sumperifor (sq ma 1.2b). Kionik nhmatik } kai exagwnik kionik f sh parathroôntai ìtan dwdek nio qrhsimopoieðtai wc dialôthc. Stouc kðonec dhmiourgoôntai mikrofasikèc perioqèc me ta foullerènia na diat ssontai se st lec (eswterik perioq ) kai touc upokatast tec na apoteloôn to perðblhm touc (exwterik perioq ). H katanìhsh twn mhqanism n org nwshc tètoiou eðdouc susthm twn, h susqètis touc me tic jermodunamikèc metablhtèc thc jermokrasðac kai sugkèntrwshc, eðnai idiaðtera qr simh gia thn an ptuxh upermoriak n dom n, me epijumhtèc, prosqediasmènec idiìthtec. Apì thn melèth aut n twn susthm twn anadeiknôontai trìpoi gia to xepèrasma twn duskoli n pou parousi zei h epexergasða mh tropopoihmènou foullerèniou. Sto xepèrasma aut n twn duskoli n ent ssetai kai o sqhmatismìc diafìrwn nanodom n me th mèjodo thc autodìmhshc sto nerì se dialôtec miac poikilðac tropopoihmènwn foullerenðwn. Se dialômata kationik n amfðfilwn en sewn tou foullerenðou, dimejulomonoxeðdiou tou jeðou kai benzðnhc parathr jhke 19 o sqhmatismìc rabdìmorfwn nanodom n me di metro nm kai m koc perissìtero twn 70μμ. Sthn Ðdia perioq èreunac, oi Sano et. al. me qhmik tropoðhsh tou foullerèniou parat rhsan 20 ton sqhmatismì sfairik n nanodom n me mègejoc 20-50nm. Ex llou, shmantik suneisfor se aut thn kateôjunsh èqei gðnei kai apì touc Prato et al 21, 22 ìpou èqei epiteuqjeð o sqhmatismìc miac poikilðac nanodom n (sfairik n, rabdìmorfwn ktl) me thn autodìmhsh qhmik tropopoihmènwn foullerenðwn se di forouc dialôtec. Apì tìte pou efeurèjhkan ta foullerènia idiaðtero b roc d jhke ston sqhmatismì film (umenðwn) elegqìmenou p qouc. Oi pr tec prosp jeiec 25 gia sqhmatismì omoiìmorfwn monostrwmatik n umenðwn tou foullerèniou den tan ikanopoihtikèc. Oi Nakamura et. al. br kan 26 ìti to tropopoihmèno foullerènio me karboxulikì oxô pou sunèjesan, mporeð na sqhmatðzei umènia Langmuir- Blodgett (LB) sthn diepif neia neroô-aèra. M lista h diereônhsh thc dom c me atomik mikroskopða (AFM) èdeixe ìti h epif neia perièqei polô ligìterec atèleiec apì ta LB umènia apì mh tropopoihmèno foullerènio k tw apì tic Ðdiec sunj kec. Argìtera 27, akoloôjhse o sqhmatismìc LB umenðwn apì upermìria dendritik c arqitektonik c pou perièqoun sto èna hmisfaðrio} touc foullerènia dec sthn parak tw enìthta

13 1.2. Sust mata qhmik tropopoihmènwn C60 8 a) b) g) d) Sq ma 1.2: a) Apì arister : eikìna 15 apì atomikì mikroskìpio tou C 60 (C 6 H 5 ) 5 K kai montèlo tou distoibadikoô floioô} autoô. Sto kèntro: moriakì montèlo tou amfðfilou foullerèniou C60 (C 6H 5 ) 5. Me pr sino qr ma apeikonðzontai ta oudètera toma njraka, me mplè to fortismèno kuklopent nio, kai me kðtrino oi upokatast tec. Sta dexi : antiproswpeutikì biologikì lipidðo thc lekijðnhc. b) Foullerenik par gwga me kwnikì sq ma g) Ierarqikèc domèc 23 sumpolumer n d) Ierarqikèc domèc foullerenik n parag gwn 24 (gia perissìterec leptomèreiec blèpe sto keðmeno).

14 1.2. Sust mata qhmik tropopoihmènwn C60 9 (udrìfobo mèroc) kai sto llo dendritik topologða. To endiafèron gia thn kataskeu (sônjesh) foullerenodendritik n umenðwn exakoloujeð na paramènei energì 16. Xeqwristì endiafèron parousi zei h org nwsh se diaforetikèc klðmakec megèjouc (apì to moriakì epðpedo se autì tou nanomètrou kai mikromètrou). EÐnai gnwstì, ìti ta kat sust dec sumpolumer eðnai mia kathgorða morðwn pou mporoôn na ekdhl soun auto-org nwsh se diaforetikèc klðmakec megèjouc. 'Etsi eðnai dunatìc o sqhmatismìc ierarqik n susthm twn (domèc mèsa se domèc) me mesofasik morfologða. Tètoiou eðdouc ierarqik sust mata 23, 28, 29 èqoun anaferjeð (sq ma 1.2g). Ex llou èqoun anaptuqjeð 30, 31 kai UK dendrimer ta opoða mèsa apì thn diadikasða thc autodìmhshc telik droôn wc upermoriak mesogìna. Me thn ekmet leush twn mèqri s mera qhmik n tropopoi sewn tou foullerèniou oi Nakanishi et. al. 24 pètuqan ton sqhmatismì ierarqik n foullerenik n uperdom n me elegqìmenec diast seic se di forouc dialôtec pèran tou neroô. H autodìmhsh twn tropopoihmènwn foullerèniwn me anjrakikèc alusðdec eðqe san apotèlesma thn dhmiourgða diafìrwn dom n ìpwc sfairikèc, diskoeideðc, kwnikèc (10 2 nm) me thn epif neia touc ( floiìc} ) na sunðstatai apì distoibadik str mata tropopoihmèno foullerèniou (sq ma 1.2d). Autèc oi kwnikèc upermoriakèc domèc eðnai kat dôo t xeic megèjouc megalôterec twn apl n kwnik n morðwn pou anafèrame prohgoômena. H dunatìthta peraitèrw sqhmatismoô kiìnwn me domik mon da autèc eðnai èna zhtoômeno proc diereônhsh. Basik proupìjesh eðnai h eust jeia (kaj c kai h diaspor twn megej n sto di luma) twn kwnik n upermorðwn. Shmantikì rìlo sthn eust jeia aut n twn dom n paðzoun oi asjeneðc} allhlepidr seic pou emfanðzontai an mesa sta udrìfila kai udrìfoba tm mata mèsa stouc dialôtec. Sto st dio èreunac 32 brðskontai kai llec upermoriakèc domèc pou perièqoun foullerènia. AxÐzei na anaferjeð ìti h autoorg nwsh sumpolumer n pou apoteloôntai apì èna kampto kai èna eôkampto tm ma (parousða dialut n) se gig ntiec sfairikèc domèc èqei epitrèyei 33, 34 thn pagðdeush kai to susswm twma foullerenðwn sto eswterikì thc sfairik c koilìthtac se arijmoôc thc t xhc Polumer proskollhmèna se èna foullerènio (ìpwc to polusturènio topoluaijulenoxeðdio ka) se dialôtec èqei apodeiqjeð 35 peiramatik ìti mporoôn na filoxen soun foullerènia sqhmatðzontac mikkôlia. Mia llh meg lh kathgorða upermoriak n susthm twn eðnai ta ugrokrustallik foullerenodendrimer ta opoða sundu zoun tic hlektroqhmikèc kai fusikoqhmikèc idiìthtec tou foullerèniou me tic idiìthtec twn dendrimer n all kai thn org nwsh se mikroskopikì epðpedo pou sunantiètai stouc ugroôc krust llouc. Ed anafèroume gia par deigma ìti me kat llhlh sqedðash} twn uperdom n eðnai dunatìn na eleqjeð h taktopoðhsh tou foullerèniou se orismènec perioqèc tou ulikoô se di krish me llec, ìpou brðskontai diaforetik c qhmik c suggèneiac tm mata tou dendrimeroôc. Me dec sthn parak tw enìthta

15 1.3. Ugrokrustallik sumperifor 10 autì ton trìpo,epitugq netai o mikrofasikìc diaqwrismìc (se str mata) gia par deigma 36 hlektroniak n dekt n (foullerènio) kai hlektroniak n dot n (ferosônh). Na shmei soume ìti kai sthn perioq twn ugrokrustallik n dendrimer n parathroôntai ierarqikèc domèc. Autodomhmènec om dec dendritik n morðwn droôn san upermoriak mesogìna. O V. Percec et. al. 37 èqoun sunjèsei ènan meg lo arijmì ugrokrustallik n dendrimer n. DiapÐstwsan ìti kwnikoô puramidikoô sq matoc dendritik mìria sqhmatðzoun sfairikèc upermoriakèc domèc pou organ nontai se kubikèc f seic, en kionikèc f seic ekdhl noun dendritik mìria hmidiskoeidoôc sq matoc. Oi R. Deschenaux et. al. 38 aôxhsan thn poluplokìthta aut n twn morðwn sundèontac me qhmikì desmì èna foullerènio me aut. Me autì ton trìpo eis gagan mia nèa sqediastik arq gia thn epðteuxh kionik n f sewn apì foullerenik dendrimer. Sthn epìmenh enìthta anaferìmaste sunoptik sthn ugrokrustallik sumperifor sumbatik n kai mh sumbatik n moriak n susthm twn, dðnontac èmfash sta ugrokrustallik dendrimer. Skopìc mac eðnai na eis goume basikoôc orismoôc kai ènnoiec pou qrhsimopoioôme sthn diatrib. 1.3 Ugrokrustallik sumperifor O ìroc ugrokrustallik kat stash apodðdei eôstoqa, an kai apoteleð antðfash apì touc ìrouc, mia eureða perioq thc fusik c thc sumpuknwmènhc Ôlhc. Sugkekrimèna, perigr fei thn ekd lwsh moriak c t xhc prosanatolismoô, /kai jèshc, se sunduasmì me moriak kinhtikìthta pou se makroskopikì epðpedo ekdhl netai wc reustìthta. Arketèc forèc antð gia ugrokrustallikèc katast seic f seic mil me gia mesomorfikèc f seic katast seic. Mesomorfikèc f seic katast seic eðnai ekeðnec pou qarakthrðzontai apì endi mesh} moriak t xh metaxô pl rouc t xhc (moriak n jèsewn kai prosanatolism n) pou parousi zontai sta krustallik stere kai pl rouc ataxðac pou parousi zoun ta isìtropa ugr morfa stere. To sônolo twn ugrokrustallik n f sewn sunjètei autì pou onom zetai 39 ugrokrustallik kat stash thc Ôlhc. Oi ugroð krôstalloi (UK) eðnai loipìn reust ta opoða parousi zoun anisotropða, oi fusikoqhmikèc idiìthtèc touc exet zontai apì thn Epist mh Ulik n kai tautìqrona apoteloôn 40 sustatik mèrh} se meg lo arijmì teqnologik n efarmog n aiqm c. To moriakì sq ma, omikrofasikìc diaqwrismìc (microphase segragetion) (metaxô diaforetik c qhmik c suggèneiac mer n), kai h auto-dìmhsh (self-assembly) (auto-sunarmolìghsh domik n mon dwn se upermoriakèc domèc) eðnai treðc shmantikoð par gontec pou odhgoôn ston sqhmatismì diafìrwn morf n autoorg nwshc. H diamìrfwsh kai sunduasmìc aut n twn parag ntwn lìg wtwn diamoriak n allhlepidr sewn upì thn parousða orismènwn jermodunamik n metablht n epifèrei plhj ra mesof sewn kai dom n se mia eureða klðmaka megej n org nwshc thc Ôlhc

16 1.3. Ugrokrustallik sumperifor 11 b) Sq ma 1.3: a) ParadeÐgmata orismènwn tôpwn moriak n dom n pou sqhmatðzoun ugrokrustallikèc f seic. b) Mesof seic sqhmatizìmenec apì rabdìmorfa, diskìmorfa, tôpou mpan nac, kwnik kai amfðfila mìria. Suntm seic: SmA = Smhktik A f sh, SmC =Smhktik C f sh (oi strwmatweideðc f seic apeikonðzontai lamb nontac thn tom enìc tm matoc twn strwm twn ta opoða ekteðnontai sthn trðth di stash tou q rou), Col =kionik mesof sh, kai Cub =kubik mesof sh apoteloômenh apì mikkôlia.

17 1.3. Ugrokrustallik sumperifor 12 Apì thn asummetrða twn domik n mon dwn pou touc apartðzoun, oi UK diakrðnontai se kalamitikoôc (rabdoeideðc), diskotikoôc, tôpou mpan nac, kwnikoôc puramidikoôc ktl (sq ma 1.3a). An log ame ton trìpo pou dieujetoôntai oi domikèc mon dec (t xh wc proc thn jèsh) kai thn t xh wc proc ton prosanatolismì touc, oi UK diakrðnontai se nhmatikoôc (N), smhktikoôc (Sm), kionikoôc (Col) (exagwnikoð, orjogwnikoð, me qwrðc t xh stouc kðonec ktl) kubikoôc (Cub) k.a (sq ma 1.3b). An loga me thn jermodunamik par metro (jermokrasða sugkèntrwsh) pou metab lletai gia na pragmatopoihjeð h metatrop sthn ugrokrustallik kat stash, diakrðnoume tic ex c kathgorðec mesof sewn: a) Jermotropikèc mesof seic (thermotropic mesophases) ìpou h emf nish diafìrwn mesof sewn exart tai apì thn jermokrasða. 'Etsi, poikilða UK f sewn emfanðzetai sun jwc kat thn met bash apì uyhl organwmèna krustallik stere (shmeðo zèsewc) se anisìtropa ugr (jermokrasða kajarismoô) (clearing temperature). Fusik up rqei kai h antðstrofh diergasða ìpou me yôxh (anisìtropou) ugroô parathreðtai met bash se ugrokrustallikèc f seic. Na shmei soume ìti sthn perðptwsh polumerik n ulik n antð gia shmeðo zèsewc mil me gia jermokrasða ual douc met bashc. b) Luotropikèc f seic (lyotropic phases) ìpou emfanðzontai se polusustatik sust mata (qwrðc katan gkhn k je sustatikì na eðnai UK apì mìno tou); h ekd lwsh diafìrwn UK f sewn exart tai apì thn sqetik sugkèntrwsh twn sustatik n. Tèloc, oi UK pou parousi zoun tìso luotropik ìso kai jermotropik sumperifor onom zontai amfitropikoð UK (amphotropic). H epist mh thc qhmeðac eðnai se jèsh na sunjètei qhmikèc domèc pou ekdhl noun ugrokrustallik sumperifor all kai tautìqrona na efeurðskei nèec domèc thc ugrokrustallik c kat stashc thc Ôlhc. Suneq c prostðjetai ènac meg loc arijmìc morðwn pou dðnoun ugrokrustallikèc f seic. Mia klassik prosèggish perilamb nei anisìtropa mesogìna mìria qamhloô moriakoô b rouc. Ta teleutaða apoteloôntai apì èna rabdìmorfo diskìmorfo kampto tm ma, pou sundèetai me eôkamptec anjrakikèc alusðdec sta kra sthn perifèreia tou antðstoiqa. To kampto tm ma eðnai upeôjuno gia thn ekd lwsh meg lhc embèleiac moriak c t xhc prosanatolismoô. Aut h sqediastik } arq odhgeð stic gnwstèc nhmatikèc, smhktikèc kai kionikèc f seic kai ta sust mata aut parousi zoun tic perissìterec forèc jermotropik sumperifor. An kai oi mesof seic rabdìmorfwn diskìmorfwn UK eðnai aplèc} kai oi mhqanismoð org nwshc gia aut (moriakì sq ma, kat lhyh q rou, dipolikèc allhlepidr seic) èqoun sqetik kal c katanohjeð, den paôoun na apoteloôn shmantikì kekthmèno gia thn katanìhsh thc ugrokrustallik c sumperifor c perissìtero polôplokwn ugrokrustallik n susthm twn. 'Etsi, oi par gontec pou eis goume se aut thn enìthta, kaj c kai oi klassikoð mhqanismoð} pou mìlic perigr yame, qrhsimopoioôntai gia thn katanìhsh thc org nwshc twn UK dendrimer n Ta UK dendrimer

18 1.3. Ugrokrustallik sumperifor 13 a) b) Sq ma 1.4: a) Dendritik dom b) Foullerenikì dendrimerèc pou èqoun sunjèseioi Dardel et. al. 45 K je braqðonac fèrei dendrimerèc 3hc geni c. Ta dendrimer an koun sthn kathgorða uperdiakladwmènwn makromorðwn me elegqìmena kajorismènh topologða, diast seic, moriakì b roc kai drastikìthta. Suntèjhkan pr th for to 1979 apì ton D. A. Tomalia sta ergast ria thc poluejnik c Dow Chemical Company. Oi par metroi pou kajorðzoun thn trisdi stath moriak arqitektonik, oi opoðec kai epidèqontai elegqìmenh rôjmish kat thn qhmik sônjesh, eðnai: hpollaplìthta twn diaklad sewn (arijmìc twn kl dwn an kìmbo), to m koc twn kl dwn kai h dendritik gene (sq ma 1.4). Ta teleutaða qrìnia mia kathgorða twn dendrimer n (pou ja mac apasqol sei kai sthn diatrib ) eðnai ta ugrokrustallik dendrimer, ta opoða proselkôoun èntono ereunhtikì endiafèron. H susthmatik melèth UK dendrimer n diaforetik c moriak c arqitektonik c èqei odhg sei sthn anagn rish orismènwn mhqanism n pou eujônontai gia thn upermoriak auto-org nwsh. Ed, anaptôssoume mia taxinìmhsh 46 h opoða perilamb nei touc ex c mhqanismoôc: a) Ton mikro-fasikì diaqwrismì, pou ofeðletai sthn Ôparxh diakrit n diaforetik c qhmik c suggèneiac tmhm twn tou dendrimeroôc. 47 H spoudaiìthta autoô tou mhqanismoô ìpwc èqoume anafèrei den periorðzetai mìnon sta dendrimer. b) Thn amoibaða eujugr mmish mesogìnwn mon dwn kai sunep c thn emf nish proexèqontoc prosanatolismoô. Oi mesogìnec mon dec mporeð na eðnai sundedemènec sthn perifèreia thn dendritik c arqitektonik c na eðnai mèroc thc eswterik c dendritik c arqitektonik c. H t xh prosanatolismoô ofeðletai tìso sthn anisotropða twn allhlepidr sewn metaxô twn mesogìnwn pou an koun sto Ðdio dendrimerèc (intra-dendritic) se diaforetik dendrimer (inter-dendritic). Oi endodendritikèc allhlepidr seic odhgoôn se allag sthn sunolik morfologða tou dendrimeroôc eunno ntac thn t xh wc proc ton prosanatolismì. Apì thn llh merða, sta dendritik sust mata me eôkampth dom kai me mesogìnec mon dec sundedemènec sthn perifèreia touc, o mikrofasikìc diaqwrismìc, pou proèrqetai apì th diaforetik qhmik suggèneia thc perifèreiac kai tou eswterikoô tou dendimeroôc, kajðstatai prwteôwn se sqèsh me tic allhlepidr seic twn mesogìnwn eunno ntac thn emf nish t xhc wc proc tic jèseic h opoða odhgeð ston sqhmatismì strwmatoeid n kionik n f sewn Ta

19 1.4. Mèjodoi jewrhtik c prosèggishc thc autoorg nwshc 14 zht mata mikrofasikoô diaqwrismoô kaj c kai h spoudaiìthta twn allhlepidr sewn metaxô twn mesogìnwn ja ta pragmateutoôme ìtan exet soume dendritik upermoriak sust mata pou perièqoun foullerènia se epìmeno kef laio. Sto sq ma 1.4b dðnoume èna par deigma foullerenikoô UK dendrimeroôc. g) Thn autodìmhsh dendritik n mon dwn se megalomoriakèc domèc oi opoðec auto-organ nontai se UK f seic ìpwc anafèrame sthn enìthta 1.2. d) ApeujeÐac auto-org nwsh 51, 52 sqetik kamptwn megalomoriak n dom n, ìpwc polumerikèc kamptoi r bdoi, pou sqhmatðzontai apì dendritikèc mon dec. 1.4 Mèjodoi jewrhtik c prosèggishc thc autoorg nwshc Sthn paroôsa diatrib, h melèth thc fasik c sumperifor c kai genikìtera twn jermodunamik n idiot twn foullerenik n upermoriak n susthm twn pou ekdhl noun ugrokrustallik sumperifor pragmatopoieðtai me dôo jewrhtikèc proseggðseic: a) me moriak jewrða pou basðzetai stic arqèc thc statistik c mhqanik c kai b) me moriakèc prosomoi seic Monte Carlo (MC). Parak tw parousi zoume sunoptik autèc tic proseggðseic. Oi moriakèc jewrðec proupojètoun thn eisagwg enìc diamoriakoô dunamikoô allhlepidr sewn metaxô twn morðwn enìc sust matoc, prokeimènou na problèyoun thn statistik kai jermodunamik sumperifor autoô. O pl rhc kajorismìc ìmwc enìc diamoriakoô dunamikoô den eðnai p nta dunatìc miac pou exart tai apì thn poluplokìthta thc moriak c dom c. To basikì, loipìn, z thma gia thn efarmog miac moriak c jewrðac se polôploka mìria (ìpwc oi foullerenikoð UK) eðnai o bajmìc leptomèreiac sthn perigraf thc dom c, twn moriak n diamorf sewn kai sunep c twn diamoriak n allhlepidr sewn pou antistoiqoôn se autì to bajmì leptomèreiac. Lamb nontac ìmwc upìyh ìti kat kanìna analutikìc upologismìc me statistik mhqanik eðnai adônatoc se moriak sust mata, akìma kai gia apl dunamik, eðmaste anagkasmènoi na katafôgoume se proseggistikèc mejìdouc. Me ton trìpo autì mporoôme na epitôqoume mia poiotik katanìhsh thc sumperifor c pragmatik n moriak n susthm twn. H plhrìthta} twn moriak n jewri n sun jwc sumplhr netai kai elègqetai me thn qr sh moriak n prosomoi sewn. Tic teleutaðec dekaetðec oi moriakèc prosomoi seic (MC) apoteloôn basikì ergaleðo gia th melèth moriak n susthm twn. H spoudaiìtht touc ègkeitai sto gegonìc ìti parèqoun akrib hmipeiramatik } apotelèsmata gia kal c orismèna moriak montèla. 'Etsi, qrhsimopoioôntai ìqi mìno gia thn prìbleyh thc fasik c sumperifor c moriak n susthm twn all kai gia ton èlegqo diafìrwn moriak n jewri n jewr ntac thn Ðdia parametrikopoðhsh tou diamoriakoô dunamikoô allhlepðdrashc. Thn teleutaða eikosaetða, h qr sh aut n èqei epektajeð kai sthn perioq twn ugr n krust llwn, 53, 54 arqik me apl diamoriak dunamik sterik c ap shc (hard core interactions) gia kampta

20 1.4. Mèjodoi jewrhtik c prosèggishc thc autoorg nwshc 15 morða (sfairokôlindroi, dðskoi k.a). Oi upologistikèc prosomoi seic èdeixan ìti akìma kai mia adropoihmènh moriak dom mporeð na qrhsimopoihjeð san sôsthma anafor c gia thn katanìhsh kai ermhneða tìso thc ekd lwshc miac poikilðac UK f sewn ìso kai twn metaxô touc metatrop n f shc. M lista, h upost rixh twn apotelesm twn miac moriak c jewrðac apì prosomoi seic epitrèpei sthn sunèqeia thn diereônhsh me jewrða tou moriakoô sust matoc se megalôterh leptomèreia, qwrðc na eðnai anagkaða h ektèlesh moriak n prosomoi sewn pou genik apaitoôn auxhmènh upologistik isqôc. H sunèqeia thc diatrib c èqei wc ex c: Sto deôtero kef laio eis goume to jewrhtikì upìbajro gia th melèth twn moriak n susthm twn pou melet me sfl aut th diatrib. Sto trðto kef laio melet me t gmata adropoihmènwn foullerenik n UK me th moriak jewrða. Sto tètarto kef laio epekteðnoume thn jewrhtik mac melèth se mðgmata anisìtropwn kamptwn nanoskopik n swmatidðwn. Sto pèmpto kef laio diereun me thn epðdrash pou èqei sthn fasik sumperifor t gmatoc adropoihmènou foullerenikoô UK, h prosj kh eleôjerwn} foullerenðwn. Sto èkto kef laio me thn ektèlesh moriak n prosomoi sewn MC elègqoume ta apotelèsmata thc moriak c jewrðac gia epilegmèno adropoihmèno foullerenikì UK. Tèloc, sto èbdomo kef laio parousi zoume ta sumper smata apì thn paroôsa diatrib.

21 Kef laio 2 Jewrhtikì upìbajro Se autì to kef laio anaptôssoume tic basikèc arqèc gia thn klassik statistik mhqanik kai jermodunamik perigraf susthm twn, pou apoteloôntai apì eôkampta mìria enìc perissotèrwn eid n, kai brðskontai se isorropða. Diatup noume èna sônolo apì exis seic, pou sthrðzontai stic arqèc thc statistik c mhqanik c, gia th melèth thc ugrokrustallik c sumperifor c aut n twn susthm twn. 2.1 Statistik mhqanik eôkamptwn megalomorðwn JewroÔme èna poluatomikì mìrio to opoðo apoteleðtai apì M domikèc mon dec (moriak tm mata), diaforetikoô tôpou sthn genik perðptwsh, me thn k je domik mon da na qarakthrðzetai apì m za m i kai kôriec ropèc adr neiac I (i) α, α = x, y, z. OrÐzoume to moriakì sôsthma suntetagmènwn me kèntro R to kèntro m zac enìc apì ta moriak tm mata kai prosanatolismì Ω ton prosanatolismì tou antðstoiqou moriakoô tm matoc. Se aut thn anapar stash to sônolo twn jèsewn kai twn prosanatolism n twn upoloðpwn domik n mon dwn wc proc to moriakì sôsthma anafor c orðzoun mia diamìrfwsh tou morðou ν ({r}, {ω}). Upojètoume epð plèon ìti gnwrðzoume to dunamikì allhlepðdrashc, u (r i,j,ω i,j ), metaxô twn diaforetik n tmhm twn kai ìti to sunolikì dunamikì tou morðou proseggðzetai ikanopoihtik krat ntac mìno suneisforèc zeug n allhlepidr sewn (prosjetikìthta kat zeôgh). Se aut thn perðptwsh h eswterik dunamik enèrgeia thc diamìrfwshc ν gr fetai wc ex c, M E (ν) = u (r i,j,ω i,j ), (2.1) i=1 j>i kai h qamhltonian tou sust matoc, sthn klassik prosèggish eðnai H = K r + K t + E (ν), (2.2) ìpou K t h metaforik kai K r h peristrofik kinhtik enèrgeia tou morðou oi opoðec dðnontai apì M K t = p 2 i /2m i (2.3) i=1 16

22 2.1. Statistik mhqanik eôkamptwn megalomorðwn 17 kai K r = M Jia 2 /2I(i) a (2.4) i=1 me p i sumbolðzetai h orm tou tm matoc i kai J ia eðnai h sunist sa thc stroform c pou anafèretai ston kôrio xona a tou moriakoô tm matoc i. H moriak sun rthsh epimerismoô, se jermokrasða T a eðnai q cl = ( M i=1 h ci P ) 1 dνd {p} d {J} exp [ βh (ν, {p}, {J})] (2.5) ìpou h P eðnaihstajer tou Planck, β =1/k B T me k B hstajer tou Boltzmann kai c i to pl joc twn klassik n bajm n eleujerðac tou moriakoô tm matoc i (kai eðnai treðc gia shmeiak m za, pènte gia moriak tm mata kulindrik c summetrðac kai èxi sthn genik perðptwsh). SÔmfwna me touc Gray kai Gubbins 55 h sun rthsh epimerismoô mporeð na paragontopoihjeð wc q cl = q t q r q c (2.6) ìpou q t = i h 3 dp i exp [ βp 2 i /2m i ] = i Λ 3 t;i (2.7) me Λ t;i = ( βh 2 2πm i ) 1 2 to jermikì m koc tou tm matoc i, kai q r = i 1 h ci 3 [ dj i exp β a J 2 ia/2i (i) a ] = i Λ 1 r;i (2.8) me Λ r;i = π 1 2 ( βh 2 8π 2 I (i) x ) 1 ( 2 βh 2 8π 2 I (i) y ) 1 2 ( βh 2 8π 2 I (i) z ) 1 2, kai thn sun rthsh epimerismoô twn moriak n diamorf sewn na eðnai q c = dν exp [ βe (ν)]. (2.9) JewroÔme èna sôsthma N ìmoiwn morðwn periorismènwn se ìgko V kai se dexamen jermìthtac se jermokrasða T. Kajèna apì aut perigr fetai apì touc deðktec I, J... =1, 2,...N. Sto q ro twn jèsewn (configurational space) to I mìrio prosdiorðzetai apì tic suntetagmènec jèshc R I, prosanatolismoô Ω I kai diamìrfwshc autoô ν I. H sunolik sun rthsh epimerismoô ja eðnai Q (N,V,T) =Q t Q r Z N (2.10)

23 2.1. Statistik mhqanik eôkamptwn megalomorðwn 18 me Q t = q N t,q r = q N r. Sunektik gr foume gia autèc tic suneisforèc u T = q t q r. Me Z N sumbolðzoume to olokl rwma twn diamorf sewn (configurational integral) Z N = 1 N! [ N d {I} exp β E (ν I ) β ] U I,J. (2.11) I=1 I<J 'Opou gia èna sôsthma N morðwn, h olokl rwsh gðnetai sthn sullog {I} = {R, Ω,ν} twn jèsewn {R} =(R 1, R 2,...,R N ) twn prosanatolism n {Ω} =(Ω 1, Ω 2,...,Ω N ) kai twn diamorf sewn {ν} =(ν 1,ν 2,...,ν N ) touc. H allhlepðdrash metaxô dôo morðwn (I,J), perigr fetai apì èna dunamikì kat zeôgh U I,J = U (R I,J ;Ω I,J ; ν I,ν J ), me R I,J, Ω I,J na perigr foun thn jèsh kai ton prosanatolismì tou morðou J se sqèsh me to I. U I,J U (R I,J, Ω I,J ; ν I,ν J )= u (i I,j J ). (2.12) i I,j J Ta moriak tm mata ta perigr foume me touc deðktec i I,j I... =1, 2,...M. Sto i I emperièqetai h jèsh kai o prosanatolismìc tou tm matoc i (wc proc to ergasthriakì sôsthma axìnwn) ìtan to mìrio I eðnai sth jèsh R I me prosanatolismì Ω I sth diamìrfwsh ν I. H sôndesh statistik c mhqanik c me thn jermodunamik pragmatopoieðtai mèsw tou jermodunamikoô dunamikoô thc kanonik c sullog c: thn eleôjerh enèrgeia Helmholtz, opìte èqoume βf (N,V,T)= ln Q (N,V,T) =N ln u T ln Z N. (2.13) Gia th melèth metatrop n f shc apaiteðtai mìnon o ìroc thc eleôjerhc enèrgeiac pou exart tai me to olokl rwma diamorf sewn. O ìroc pou sqetðzetai me tic suneisforèc apì tic kinhtikèc enèrgeiec, dhlad o ln u T, paraleðpetai apì touc upologismoôc thc eleôjerhc enèrgeiac miac pou, ètsi ki alli c, prostafaireðtai ìtan efarmìzoume tic sunj kec sunôparxhc dôo f sewn. H èkfrash gia to olokl rwma diamorf sewn mporeð na diatupwjeð se mia enallaktik morf wc ginìmeno dôo parag ntwn Z N = q N c Z C, (2.14) ìpou to q c dðnetai apì thn ex.(2.9), en Z C eðnai h mèsh tim tou oloklhr matoc diamorf sewn twn morðwn wc proc ta statistik b rh twn diamorf sewn twn morðwn kai dðnetai apì Z C = 1 N! N d {I} P o (ν I ) exp [ βu I,J ] (2.15) I=1 I<J ìpou P o (ν I )=exp[ βe (ν I )] /q (2.16)

24 2.2. H prosèggish allhlometatrepìmenwn moriak n sqhm twn} 19 orðzetai wc h apì kataskeu c} pijanìthta 46, 56, 57 èna apomonwmèno mìrio (apousða allhlepidr sewn me lla mìria) na brejeð sthn diamìrfwsh ν I. H èkfrash gia to olokl rwma diamorf sewn ex.(2.14) mac epitrèpei ton diaqwrismì thc eleôjerhc enèrgeiac Helmholtz tou sust matoc pou melet me, βf = ln Z N (2.17) se èna tm ma pou sqetðzetai me thn suneisfor lìg w twn diamorf sewn tou enìc apomonwmènou} morðou βf D = ln q kai se èna diamorikì tm ma βf C = ln Z C, (2.18) opìte lamb noume F = NF D + F C. (2.19) O ìroc F D, sômfwna me thn ex.(2.9), eðnai profan c anex rthtoc tou ìgkou tou sust matoc kai eðnai sun rthsh mìno thc jermokrasðac. Stouc upologismoôc pou upeisèrqontai stic metatropèc f shc, ìpou mìnon diaforèc sthn eleôjerh enèrgeia mac endiafèrei, o ìroc autìc prostafaireðtai kai epomènwc mìnon of C thc eleôjerhc enèrgeiac eðnai qr simoc. Me ton ìro F C ja asqolhjoôme analutikìtera sthn epìmenh enìthta. H ex.(2.15) kai oi sqetikèc ekfr seic stic ex.(2.16)-(2.19) apoteloôn èna sônolo exis sewn gia thn perigraf tou sust matoc mac me ìrouc statistik c mhqanik c, me tic allhlepidr seic na eðnai prosjetikèc kat zeôgh morðwn. 2.2 H prosèggish allhlometatrepìmenwn moriak n sqhm twn} An jewr soume ìti oi diamorf seic tou morðou eðnai diakritèc, ìti h metablht twn diamorf sewn ν I lamb nei diakritèc timèc, tìte to olokl rwma thc ex.(2.15) tou sust matoc ja metasqhmatisteð se èna jroisma se ìlec tic diamorf seic, {ν}, kai se èna olol rwma, d { ω} se ìlec tic jèseic kai prosanatolismoôc tic opoðec ja sumbolðzoume wc ω =(R, Ω). Sthn sunèqeia omadopoioôme tic diamorf seic, ètsi ste h allhlepðdrash U I,J na èqei thn Ðdia tim gia dôo mìria I,J pou an koun se orismènec om dec diamìrfwshc (adi foro an eðnai Ðdiec) gia thn Ðdia sqetik jèsh kai prosanatolismì twn morðwn. 'Etsi, gia eukolða ja anaferìmaste stic di forec om dec diamorf sewn me ton ìro sq mata}, an kai aut h omadopoðhsh eðnai ikan kai gia malak } dunamik allhlepðdrashc. Ta diaforetik sq mata enìc morðou I ta sumbolðzoume me s I kai tic diakritèc diamorf seic pou omadopoioôntai sto sq ma s I tic perigr foume wc ν (s I ). To Autìc o ìroc qrhsimopoieðtai kurðwc gia allhlepidr nta mìria me sklhr } dunamik.

25 2.2. H prosèggish allhlometatrepìmenwn moriak n sqhm twn} 20 jroisma ep nw se ìlec tic diamorf seic pou upeisèrqetai sthn ex.(2.15) mporeð na pragmatopoihjeð pr ta ep nw stic diamorf seic pou antistoiqoôn se èna sq ma kai èpeita ep nw se ìla ta dunat sq mata, wc ex c P o (ν I ) exp [ βu I,J ]= si s I ν(s I) P o (s I )exp[ βu I,J ] (2.20) ìpou h apì kataskeu c} pijanìthta P o (s I ) tou sq matoc s I dðnetai apì to jroisma twn ek kataskeu c pijanot twn twn diamorf sewn pou antistoiqoôn se autì to sq ma wc P o (s I )= P o (ν I ) (2.21) ν(s I ) alli c P o (s I )= exp [ βe (s I)] s I exp [ βe (s I )] (2.22) [ ] me βe (s) = ln ν(s) exp [ βe (ν)], ìpou h E (s) katanoeðtai wc h eleôjerh enèrgeia tou sq matoc s pou ofeðletai sthn dunatìthta tou morðou na brejeð stic diamorf seic ν (s). Tèloc to G (I,J)=exp[ βu I,J ], perigr fei tic allhlepidr seic metaxô morðwn opoiasd pote diamìrfwshc sq matoc s I me opoiad pote llh diamìrfwsh sq matoc s J gia orismèno ω I,J pou perigr fei tic sqetikèc jèseic kai prosanatolismoôc twn morðwn I,J. To olokl rwma twn diamorf sewn Z C ex.(2.15) se aut th nèa perigraf {I} = {R, Ω, s} gðnetai Z C = 1 N! {s I} N d { ω} P o (s I ) exp [ βu I,J ]. (2.23) I=1 I<J Se k je moriakì sôsthma pou exet zoume, tìso h apì kataskeu c pijanìthta P o (s) ìso kai to diamoriakì dunamikì U I,J eis gontai wc dedomèna sthn ex.(2.23). Gia mh tetrimmèna dunamik allhlepðdrashc, gia ton upologismì tou Z C efarmìzoume mia proseggistik mèjodo sthn opoða anaferìmaste parak tw. Sto shmeðo autì eis goume thn dokimastik sun rthsh φ (I), gr fontac thn enèrgeia allhlepðdrashc dôo morðwn βu I,J me ton ex c trìpo βu I,J = H o (I,J)+H (I,J) (2.24) O pr toc ìroc tou ajroðsmatoc H o (I,J)=φ (I)+φ (J) perièqei mìnon tic dokimastikèc sunart seic kai o deôteroc H (I,J)=βU I,J φ (I) φ (J) epiprìsjeta tic diamoriakèc allhlepidr seic metaxô Pio swst h posìthta aut den eðnai h enèrgeia allhlepðdrashc dôo morðwn all deðqnei pìso aut diafèrei apì thn k B T.

26 2.2. H prosèggish allhlometatrepìmenwn moriak n sqhm twn} 21 dôo morðwn. Antikajist ntac thn ex.(2.24) sthn ex.(2.23) èqoume Z C = 1 N! {s I} N d { ω} exp [ (N 1) φ ( ω; s I )] P o (s I ) exp [ H (I,J)] (2.25) I=1 I<J H èkfrash aut mporeð na grafteð wc ginìmeno Z C = Z o exp [ H (I,J)] I<J = Z o Z H (2.26) ìpou Z o = 1 N! zn o (2.27) o me z o = s I d ω exp [ (N 1) φ ( ω; si )] P o (s I ). O deôteroc ìroc tou dexioô mèlouc thc ex.(2.26) ekfr zei thn mèsh tim thc posìthtac mèsa stic agkôlec kai se genik morf gr fetai... o = {s I } N d { ω} (...) P ( ω I ; s I ). (2.28) I=1 H P ( ω; s) P (I) eðnai h katanom pijanìthtac eôreshc enìc morðou me suntetagmènec (I) (R, Ω; s) kai gr fetai wc P ( ω; s) = H eleôjerh enèrgeia tou sust matoc F C gr fetai wc exp [ (N 1) φ ( ω; s)] z o P o (s) =f ( ω; s) P o (s). (2.29) F C = F o + F H (2.30) ìpou lìgw thc ex.(2.27) èqoume βf o = ln Z o = ln 1 N! zn o =lnn! N ln z o (2.31) kai βf H = ln exp [ H (I,J)] I<J o (2.32) Sthn sunèqeia gia ton upologismì tou dexioô mèlouc thc ex.(2.32) efarmìzoume thn prosèggish: To deôtero mèloc thc ex.(2.32) mporeð na anaptuqjeð se seir sômfwna me th mèjodo 58, 59 variational cluster expansion (VCE) (jewrða metabol n sto an ptugma kat sm nh (om dec) thc sun rthshc epimerismoô) wc ex c ln exp [ H (I,J)] I<J o = N! 2! (N 2)! ln exp [ H (I,J)] o +

27 2.2. H prosèggish allhlometatrepìmenwn moriak n sqhm twn} 22 N! 3! (N 3)! [ 3ln exp [ H (I,J)] o +ln exp [ H (I,J)] H (I,K) H (J, K) o ]+... (2.33) O pr toc ìroc tou anaptôgmatoc thc ex.(2.33) sqetðzetai me thn suneisfor sthn eleôjerh enèrgeia apì ìlec tic du dec swmatidðwn, o deôteroc apì ìlec tic tri dec ktl. Stouc upologismoôc pou ja pragmatopoi soume h eleôjerh enèrgeia helmholtz lamb netai sthn prosèggish twn allhlepidr sewn du dwn swmatidðwn kai apì tic ex.(2.30)-(2.33) ja eðnai βf (2) C =lnn! N ln z N (N 1) o ln exp [ H (I,J)] 2 o. (2.34) Se autì to shmeðo me b sh ton formalismì sqhm twn morðwn èqoume exp [ H (I,J)] o = d ω I d ω J P ( ω I ; s I ) P ( ω J ; s J )exp[ H ( ω I, ω J ; s I,s J )]. (2.35) s I,s J H eleôjerh enèrgeia eðnai sunarthsiakì thc dokimastik c sun rthshc φ (I). Y qnoume na broôme thn katanom pijanìthtac P (I) pou elaqistopoieð aut ìtan to sôsthma (NV T) eðnai se kat stash jermodunamik c isorropðac. Autì epitugq netai me thn apaðthsh gia mhdenismì thc sunarthsiak c parag gou thc eleôjerhc enèrgeiac wc proc thn φ (I). Sthn perðptwsh pou melet me efarmìzoume thn (2) δf C (2) δφ(i), sthn proseggistik èkfrash gia thn eleôjerh enèrgeia F C ( ex.(2.34) ) h opoða dðnei φ (I) =ln exp [ H (I,J)] o exp [ βu I,J ]exp[φ (J)] J (2.36) ìpou exp [ βu I,J ]exp[φ (J)] J = s J d ω J P ( ω J ; s J )exp[ βu I,J ]exp[φ ( ω J ; s J )]. (2.37) H sun rthsh φ ( ω; s) upologðzetai me autosumbibastì trìpo (emfanðzetai kai sta dôo mèlh thc ex.(2.36) ). Me antikat stash sthn ex.(2.29) eðmaste se jèsh na upologðsoume thn katanom pijanìthtac eôreshc enìc morðou me suntetagmènec (I). H morf thc ex rthshc thc puknìthtac pijanìthtac tou enìc morðou P ( ω; s) orismènou sq matoc s, apì tic suntetagmènec jèshc kai prosanatolismoô ω =(R, Ω), antanakl thn summetrða thc f shc pou exet zoume. Sthn paroôsa diatrib melet me thn ugrokrustallik sumperifor moriak n susthm twn pou ekdhl noun tic epìmenec f seic: a) isìtropec, stic opoðec h f ( ω; s) (thc ex.(2.29) ) eðnai anex rthth twn jèsewn kai twn prosanatolism n, me f ( ω; s) f (s).

28 2.2. H prosèggish allhlometatrepìmenwn moriak n sqhm twn} 23 b) nhmatikèc, gia tic opoðechf ( ω; s) eðnai anex rthth thc jèshc, f ( ω; s) f (Ω; s). g) smhktikèc, gia tic opoðec h f ( ω; s) eðnai anex rthth twn suntetagmènwn jèshc X, Y sto smhktikì str ma, f ( ω; s) f (Z, Ω; s), ìpou Z eðnai h suntetagmènh jèshc kata m koc thc kajètou sto smhktikì str ma. d) kionikèc, gia tic opoðec h f ( ω; s) eðnai anex rthth thc suntetagmènhc jèshc Z kat m koc tou xona tou kðona, f ( ω; s) f (X, Y, Ω; s), ìpou oi suntetagmènec X, Y eðnai se epðpedo k jeto ston xona thc kionik c f shc. Oi shmantikèc om dec diamorf sewn, sq mata, den eðnai kat an gkh ekeðna pou emfanðzoun tic megalôterec apì kataskeu c pijanìthtec P o (s) gia èna apomonwmèno mìrio, all ekeðnec pou pou èqoun axioshmeðwth pijanìthta na sumboôn sthn f sh pou ekdhl nei to sôsthma twn allhlepidr ntwn morðwn. Autì mporeð na to diapist sei kaneðc parathr ntac thn ex.(2.29), ìpou h katanom pijanìthtac eôreshc tou enìc morðou eðnai ginìmeno dôo parag ntwn. O par gontac f ( ω; s) emfanðzetai lìgw thc Ôparxhc diamoriak n allhlepidr sewn. 'Etsi, sq mata ta opoða èqoun meg lh pijanìthta na emfanistoôn ìtan ta mìria eðnai apomonwmèna mporeð na kajðstatai ligìtero ( akìma perissìtero) pijan lìgw thc parousðac diamoriak n allhlepidr sewn Isìtroph f sh Akìmh kai sthn isìtroph f sh ìpou h puknìthta pijanìthtac den exart tai oôte apì ton moriakì prosanatolismì oôte apì th jèsh tou morðou, aut den eðnai eujèwc an logh me thn apì kataskeu c pijanìthta P o (s) lìgw thc Ôparxhc allhlepidr sewn metaxô twn morðwn. 'Etsi h puknìthta pijanìthtac gr fetai P (Iso) (s) = 1 V Ω f (Iso) (s) P o (s), (2.38) ìpou Ω=8π 2. Mìnon sthn perðptwsh polô arai n ugr n me ρ 1 èqoume P (Iso) (s) 1 V ΩS P o (s), me S to pl joc twn sqhm twn pou mporeð na emfanisteð èna mìrio. H dokimastik sun rthsh φ (I) ja exart tai apì to moriakì sq ma. Apì thn ex.(2.36) èqoume φ (s) = exp [ βu I,J]exp[φ (I)] exp [φ (J)] o exp [ βu I,J ]exp[φ (J)] J. (2.39) Sthn sunèqeia proqwr me se diereônhsh tou dexioô mèlouc thc parap nw exðswshc. Eis goume thn sun rthsh Mayer tou diamoriakoô dunamikoô dôo metablht n (twn moriak n sqhm twn s I kai s J ) h opoða gr fetai wc ex c g (Iso) (s I,s J )= dω I dω J dr I,J [1 exp [ βu (R I,J, Ω I, Ω J ; s I,s J )]], (2.40)

29 2.2. H prosèggish allhlometatrepìmenwn moriak n sqhm twn} 24 kai metr ei ton ìgko pou apokleðei èna mìrio (sq matoc s I ) lìgw thc parousðac enìc deôterou (sq matoc s J )oloklhr nontac ep nw se ìlouc touc sqetikoôc prosanatolismoôc kai tic jèseic twn dôo morðwn. H posìthta eðnai thc t xhc tou moriakoô ìgkou}. Me thn eisagwg thc g (Iso) (s I,s J ) sthn ex.(2.39) mporoôme na anaptôxoume to dexiì mèloc se dun meic tou 1/V. Krat ntac touc grammikoôc ìrouc brðskoume gia thn dokimastik sun rthsh φ (s I )= φ 1 V Ω 2 g (Iso) (s I,s J ) + 1 o V Ω 2 g (Iso) (s I,s J ). (2.41) J H teleutaða exðswsh mac epitrèpei na broôme thn analutik èkfrash gia thn katanom pijanìthtac tou enìc morðou sthn isìtroph f sh (h ex.(2.29) mèsw thc ex.(2.41) gðnetai) [ P (Iso) (s I )= 1 exp ( ρ/ω 2) g (Iso) (s I ) V Ω ζ ] P o (s I )= 1 V Ω f (Iso) (s I ) P o (s I ) (2.42) ìpou ζ Iso = s I [ exp ( ρ/ω 2) ] g (Iso) (s I ) P o (s I ). (2.43) H sun rthsh g (Iso) (s I )= g (Iso) (s I,s J ) J, eðnai h mèsh tim thc g(iso) (s I,s J ) upologismènh p nw se ìla ta moriak sq mata s J tou morðou J : g (Iso) (s I )= s J g (Iso) (s I,s J ) f (Iso) (s J ) P o (s J ). (2.44) [ Apì thn exðswsh f (Iso) (s I )=exp ( ρ/ω 2) ] g (Iso) (s I ) /ζ Iso mazð me tic ex.(2.43)-(2.44), me autosumbibastì trìpo upologðzetai h sun rthsh f (Iso) (s I ). EÐmaste se jèsh na gr youme thn èkfrash gia thn eleôjerh enèrgeia βf (2) C N =lnρ ln Ω 1 ln ζ Iso ρ 2Ω 2 g (Iso) (s I,s J ) o (2.45) me g (Iso) (s I,s J ) o = s J f (Iso) (s J ) P o (s J ) g (Iso) (s I ). ParathroÔme ìti gia ton upologismì thc eleôjerhc enèrgeiac eðnai aparaðthth mìnon hf (Iso) (s I ) kai P o (s I ). H pðesh kai to qhmikì dunamikì dðnontai apì tic ex.(2.79) kai (2.80) antðstoiqa, kai eðnai βp = ρ + ρ2 2 g (Iso) (s I,s J ) o (2.46) kai βμ =lnρ ln Ω ln ζ Iso. (2.47)

30 2.2. H prosèggish allhlometatrepìmenwn moriak n sqhm twn} Nhmatik f sh Sth nhmatik f sh ta kèntra m zac twn morðwn eðnai omoiìmorfa katanemhmèna sto q ro pou aut katalamb noun (omoiogen sust mata). Se autèc tic f seic h epikr thsh moriak n diamorf sewn sqhm twn pou qarakthrðzontai apì anisometrða sq matoc, sunodeôetai me taktopoðhsh twn moriak n prosanatolism n se mia perissìterec proexèqousec dieujônseic an loga an h f sh eðnai monaxonik diaxonik nhmatik. H puknìthta pijanìthtac eðnai anex rthth twn jèsewn twn morðwn, exart tai mìnon apì ton prosanatolismì kai to moriakì sq ma P (Ω; s) = 1 V f (Nem) (Ω,s) P o (s). (2.48) H parap nw sqèsh proupojètei ìti h dokimastik sun rthsh φ (I) ja exart tai mìno apì ton prosanatolismì kai apì to sq ma twn morðwn. 'Opwc gnwrðzoume h majhmatik thc èkfrash (k nontac qr sh kai thc ex.(2.36) ) eðnai φ (I) =ln exp [ βu I,J]exp[φ (I)] exp [φ (J)] o exp [ βu I,J ]exp[φ (J)] J. (2.49) To dexð mèloc thc ex.(2.49) perilamb nei mèsec timèc posot twn stic opoðec upeisèrqetai to dunamikì allhlepðdrashc U I,J twn morðwn I kai J. MporoÔme na aplopoi soume tic ekfr seic autèc eis gontac thn sun rthsh Mayer tou diamoriakoô dunamikoô h opoða se aut thn perðptwsh ja exart tai apì touc prosanatolismoôc kai ta moriak sq mata s I kai s J wc ex c g (Nem) (Ω I, Ω J ; s I,s J )= dr I,J [1 exp [ βu (R I,J, Ω I, Ω J ; s I,s J )]] (2.50) 1 1 V g(nem) (Ω I, Ω J ; s I,s J )= 1 V dr I,J exp [ βu (R I,J, Ω I, Ω J ; s I,s J )]. (2.51) H gewmetrik shmasða tou oloklhr matoc thc ex.(2.50), ìtan to dunamikì antistoiqeð se sklhr } mìria, eðnai o apokleiìmenoc ìgkoc tou enìc morðou lìgw thc parousðac deôterou, ìtan aut èqoun prosanatolismoôc ω I kai ω J se diamorf seic pou omadopoioôntai sta sq mata s I kai s J. Gia dunamik mikr c embèleiac, to mègejoc tou apokleiìmenou ìgkou eðnai genik thc t xhc twn merik n moriak n ìgkwn}. An antikatastðsoume thn ex.(2.50) sthn ex.(2.49) kai anaptôxoume se dun meic tou 1/V to dexiì mèloc krat ntac touc grammikoôc ìrouc, ja èqoume gia thn φ (I) : φ (Ω I,s I )= φ 1 V g (Nem) (Ω I, Ω J ; s I,s J ) + 1 o V g (Nem) (Ω I, Ω J ; s I,s J ) J. (2.52)

31 2.2. H prosèggish allhlometatrepìmenwn moriak n sqhm twn} 26 H katanom pijanìthtac ex.(2.29) apì thn ex.(2.52) ja gðnei P (Nem) (Ω I ; s I )= 1 V [ ] exp ρg (Nem) (Ω I ; s I ) ζ Nem P o s = 1 V f (Nem) (Ω I,s I ) P o (s I ). (2.53) ìpou ζ Nem = s I [ ] dω I exp ρg (Nem) (Ω I ; s I ) P o (s I ). (2.54) H g (Nem) (Ω I ; s I )= g (Nem) (Ω I, Ω J ; s I,s J ) J eðnai h mèsh tim thc g(nem) (Ω I, Ω J ; s I,s J ) upologismènh p nw se ìlouc touc prosanatolismoôc kai moriak sq mata s J tou morðou J : g (Nem) (Ω I ; s I )= s J dω J g (Nem) (Ω I, Ω J ; s I,s J ) f (Nem) (Ω J,s J ) P o (s J ). (2.55) 'Opwc sthn isìtroph f sh h sun rthsh f (Nem) (Ω,s) upologðzetai me autosumbibastì trìpo. H èkfrash gia thn proseggistik eleôjerh enèrgeia tou sust matoc ex.(2.34), an anaptôxoume ìpou qrei zetai se dun meic tou 1/V, krat ntac touc grammikoôc ìrouc kai qrhsimopoi ntac ton tôpo tou Stirling gr fetai: βf (2) C N =lnρ 1 ln ζ Nem ρ g (Nem) (Ω I, Ω J ; s I,s J ) 2, (2.56) o ìpou g (Nem) (Ω I, Ω J ; s I,s J ) o = s J dωj f (Nem) (Ω J,s J ) P o (s J ) g (Nem) (Ω J ; s J ). H pðesh kai to qhmikì dunamikì upologðzontai apì tic ex.(2.79) kai (2.80), kai eðnai βp = ρ + ρ2 2 g (Nem) (Ω I, Ω J ; s I,s J ) o (2.57) kai βμ =lnρ ln ζ Nem. (2.58) Smhktik f sh An logh mejodologða me aut pou exet same thn isìtroph kai th nhmatik f sh qrhsimopoioôme gia thn melèth anisìtropwn omoiogen n didi statwn ugr n. Tupik perðptwsh tètoiwn susthm twn eðnai h smhktik A ìpou ta mìria diat ssontai se str mata apokt ntac proexèqousa dieôjunsh prosanatolismoô k jeth se aut, me thn epanalambanìmenh dom (se k je str ma) na èqei p qoc d. H puknìthta pijanìthtac eðnai anex rthth twn suntetagmènwn twn morðwn se k je str ma (didi stato ugrì), all perimènoume na eðnai sun rthsh thc jèshc tou morðou z kat m koc thc

32 2.2. H prosèggish allhlometatrepìmenwn moriak n sqhm twn} 27 kajètou sto smhktikì str ma, tou moriakoô prosanatolismoô kai tou sq matoc s P (z,ω; s) = d V f (Sm) (z,ω,s) P o (s) (2.59) ìpou d V 1 3, genik eðnai thc t xhc twn moriak n diast sewn. H dokimastik sun rthsh φ (z,ω; s) exart tai kai aut apì tic parap nw metablhtèc kai gr fetai φ (z,ω; s) =ln exp [ βu I,J]exp[φ (I)] exp [φ (J)] o exp [ βu I,J ]exp[φ (J)] J. (2.60) 'Opwc kai stic prohgoômenec peript seic eis goume thn sun rthsh Mayer g (Sm) (z IJ, Ω I, Ω J ; s I,s J ) g (Sm) (z IJ, Ω I, Ω J ; s I,s J )= dx IJ dy IJ [1 exp [ βu (R I,J, Ω I, Ω J ; s I,s J )]], (2.61) h opoða èqei diast seic embadoô (tetragwnikèc mon dec) kai gewmetrik ekfr zei thn epif neia pou apokleðetai apì èna mìrio lìgw thc parousðac enìc deôterou oloklhr nontac sto epðpedo gia orismèno sqetikì prosanatolismì, ìtan oi jèseic twn morðwn wc proc thn proexèqousa dieôjunsh diafèroun kat z IJ = z J z I. Me antikat stash thc ex.(2.61) sthn ex.(2.60) kai afoô anaptôxoume se seir dun mewn tou 1/V krat ntac touc grammikoôc ìrouc, ja èqoume gia thn φ (z,ω; s) φ (z,ω; s) = φ d V g (Sm) (z IJ, Ω I, Ω J ; s I,s J ) + d o V g (Sm) (z IJ, Ω I, Ω J ; s I,s J ) J (2.62) GnwrÐzontac thn φ (z, Ω; s), apì thn ex.(2.59) mporoôme na upologðsoume thn katanom pijanìthtac P (Sme) (z,ω; s) = 1 V [ ] exp ρdg (Sm) (z,ω; s) ζ Sm P o s = 1 V f (Sm) (z,ω; s) P o (s) (2.63) ìpou ζ Sm = s I d 0 [ ] dω I dz I exp ρdg (Sm) (z I, Ω I ; s I ) P o (s I ). (2.64) H sun rthsh g (Sm) (z I, Ω I ; s I ) eðnai h mèsh tim thc g (Sm) (z IJ, Ω I, Ω J ; s I,s J ) p nw se ìlouc touc prosanatolismoôc ta sq mata kai kat m koc thc proexèqousac dieôjunshc g (Sm) (z I, Ω I ; s I )= s J dz J dω J g (Sm) (z IJ, Ω I, Ω J ; s I,s J ) f (Sm) (z J, Ω J,s J ) P o (s J ). (2.65) H èkfrash gia thn eleôjerh enèrgeia sthn smhktik A f sh gr fetai βf (2) C N =lnρ 1 ln ζ Sm d ρ 2 d g (Sm) (z IJ, Ω I, Ω J ; s I,s J ) o (2.66)

33 2.2. H prosèggish allhlometatrepìmenwn moriak n sqhm twn} 28 ìpou g (Sm) (z IJ, Ω I, Ω J ; s I,s J ) o = s J d 0 dz JdΩ J g (Sm) (z J, Ω J ; s J ) f (Sm) (z J, Ω J,s J ) P o (s J ). Na shmei soume ìti h eleôjerh enèrgeia ( ex.(2.66) ) elaqistopoieðtai kai wc proc thn par metro d, dhlad to p qoc tou smhktikoô str matoc. DÐnoume tic ekfr seic gia thn pðesh kai to qhmikì dunamikì (ex.(2.79) kai (2.80) ), βp = ρ + ρ2 2 d g (Sm) (z IJ, Ω I, Ω J ; s I,s J ) o (2.67) kai βμ =lnρ ln ζ Sm d. (2.68) Kionik f sh Stic kionikèc f seic ta mìria organ nontai se kðonec se mia kuyelðda thc kionik c f shc me m koc pleur n d x kai d y. H katanom pijanìthtac sthn kionik f sh exart tai apì thn jèsh tou morðou sthn epif neia thc kuyelðdac (suntetagmènec x, y ), ton prosanatolismì tou morðou kai apì to sq ma tou wc ex c P (Col) (x, y, Ω; s) = 1 V [ ] exp ρg (Col) (x, y, Ω; s) P o (s) = d xd y V f (Col) (x, y, Ω; s) P o (s) (2.69) ζ Col me ζ Col = s I dx dy 0 0 [ ] dω I dx I dy I exp ρd x d y g (Col) (x I,y I, Ω I ; s I ) P o (s I ). (2.70) ìpou kai h g (Col) (x, y, Ω; s) eðnai mia sun rthsh pou prokôptei wc mèsh tim thc g (Col) (x IJ,y IJ, Ω I, Ω J ; s I,s J ) oloklhr nontac se ìlouc touc sqetikoôc prosanatolismoôc, ta sq mata kai tic jèseic twn morðwn g (Col) (x I,y I, Ω I ; s I )= s J dx J dy J dω J g (Col) (x IJ,y IJ, Ω I, Ω J ; s I,s J ) f (Col) (z J, Ω J,s J ) P o (s J ), (2.71) me thn Mayer sun rthsh g (Col) (x IJ,y IJ, Ω I, Ω J ; s I,s J )= dz IJ [1 exp [ βu (R IJ, Ω I, Ω J ; s I,s J )]]. H eleôjerh enèrgeia gia thn kionik f sh eðnai βf (2) C N =lnρ 1 ln ζ Col ρd xd y g (Col) (x IJ,y IJ, Ω I, Ω J ; s I,s J ) d x d y 2 o (2.72)

34 2.3. Metatropèc f sewn monosustatik n susthm twn 29 ìpou g (Col) (x IJ,y IJ, Ω I, Ω J ; s I,s J ) = o s I dx dy 0 0 dx I dy I dω I g (Col) (x I,y I, Ω I ; s I ) f (Col) (z I, Ω I,s I ) P o (s I ). (2.73) H eleôjerh enèrgeia ( ex.(2.72) ) elaqistopoieðtai kai wc proc tic paramètrouc d x kai d y, pou ekfr zoun tic diast seic thc kuyelðdac thc kionik c f shc. Gia thn pðesh kai to qhmikì dunamikì ( ex.(2.79) kai (2.80) )èqoume, βp = ρ + d xd y ρ 2 2 g (Col) (x IJ,y IJ, Ω I, Ω J ; s I,s J ) o (2.74) kai βμ =lnρ ln ζ Col d x d y. (2.75) 2.3 Metatropèc f sewn monosustatik n susthm twn Gia na melet soume thn jermodunamik eust jeia monosustatik n susthm twn orismènou arijmoô swmatidðwn N euriskìmena se ìgko V kai jermokrasða T, qrhsimopoioôme thn arq elaqistopoðhshc thc eleôjerhc enèrgeiac HelmholtzF (N, V, T). 'Estw ìti to parap nw sôsthma brðsketai se epaf me mia dexamen jermìthtac jermokrasðac T o, tìte gia thn eleôjerh enèrgeia èqoume h (ρ, T o )= (1/N ) F (N,V,T o ). M lista, parìlo pou to sôsthma brðsketai se jermik isorropða mporeð na mhn eðnai omoiogenèc all na up rqei h dunatìthta na diaqwristeð se dôo omoiogen mèrh ta opoða na diathroôntai se epaf metaxô touc (kai kajèna na brðsketai se diaforetik f sh). Genik loipìn, h h (ρ, T o ) mporeð na parousi zei akrìtata (sq ma 2.1a). 'Etsi, eðnai dunatìc o diaqwrismìc se dôo uposust mata (diaqwrismìc f shc), ta opoða ja suneqðsoun na brðskontai sthn Ðdia jermokrasða kai mporoôn na antall soun m za kai jermìthta. Aut prosdiorðzontai apì ta megèjh N 1,V 1 kai N 2,V 2 (arijmìc swmatidðwn kai ìgkoc antðstoiqa gia kajèna apì aut ) me arijmhtikèc puknìthtec ρ 1 kai ρ 2. Me autì ton trìpo epitugq netai h elaqistopoðhsh thc eleôjerhc enèrgeiac sto di sthma ( 1 ρ 1, 1 ρ 2 )pou gewmetrik perigr fetai me thn kataskeu thc koin c efaptomènhc gia ta akraða shmeða autoô tou diast matoc. Hnèah 12 (ρ, T o ) se autì to di sthma eðnai h 12 (ρ, T o )= h (ρ ( 2,T o ) h (ρ 1,T o ) ρ + h (ρ 2,T o ) h (ρ ) 2,T o ) h (ρ 1,T o ) ρ 2. (2.76) ρ 2 ρ 1 ρ 2 ρ 1 H ektatikìthta thc eleôjerhc enèrgeiac isqôei sto jermodunamikì ìrio, ìtan V kai N.

35 2.3. Metatropèc f sewn monosustatik n susthm twn 30 a) b) g) Sq ma 2.1: Grafik anapar stash thc eleôjerhc enèrgeiac kai kataskeu thc koin c efaptomènhc sunôparxhc se monosustatik moriak sust mata: a) perðptwsh dôo f sewn Ðdiac summetrðac, b) perðptwsh dôo f sewn diaforetik c summetrðac kai g) suneq c metatrop f shc dôo f sewn diaforetik c summetrðac. ( ) 1 'Opou h h 12 (ρ, T o ) <h(ρ, T o ) sto di sthma ρ 1, 1 ρ 2. HeÔreshtwn shmeðwn ρ 1,ρ 2 gia thn kataskeu thc koin c efaptomènhc (kataskeu Maxwell)g Ðnetai me thn epðlush twn exis sewn h(ρ, T o ) ρ = h(ρ, T o) ρ=ρ1 ρ = h (ρ 2,T o ) h (ρ 1,T o ) (2.77) ρ=ρ2 ρ 2 ρ 1 O trìpoc pou perigr yame parap nw eðnai isodônamoc me thn efarmog twn sunjhk n mhqanik c kai qhmik c isorropðac metaxô twn dôo uposusthm twn (upì orismènh jermokrasða), dhlad P (ρ 1,T o )=P (ρ 2,T o ) και μ (ρ 1,T o )=μ (ρ 2,T o ), (2.78) ìpou h pðeseic kai ta qhmik dunamik twn uposusthm twn èqoun thn Ðdia tim. H pðesh kai to qhmikì dunamikì prokôptoun apì thn eleôjerh enèrgeia b sei twn exis sewn kai P (ρ, T o )= F V = ρ 2 h(ρ, T o) N,T ρ (2.79) To μ (ρ, T o )= F N = h (ρ, T o )+ρ h(ρ, T o) V,T ρ. (2.80) To Oi puknìthtec ρ 1 kai ρ 2 onom zontai puknìthtec sunôparxhc (binodal). H sunj kh eust jeiac, ( ) 1 gia èna sôsthma pou sumpièzetai ekton netai xekin ntac èxw apì thn perioq ρ 1, 1 ρ 2, eðnai 2 h(ρ,t o) ρ 2 To > 0. Oi puknìthtec ìpou isqôei MetaxÔ aut n isqôei sumbaðnei diaqwrismìc f shc. 2 h(ρ,t o) ρ 2 To = 0 kajorðzoun ta ìria eust jeiac. 2 h(ρ,t o) ρ 2 To < 0 kai sunep c akìma kai h topik eust jeia q netai opìte Na shmei soume ìti h sumperifor pou perigr fetai sto sq ma 2.1a eðnai qarakthristik perðptwsh dôo f sewn pou èqoun thn Ðdia summetrða (ìpwc gia par deigma h sunôparxh ugroô-aerðou enìc reustoô V anderw aals ). Sto sq ma 2.1b mporeð na kataskeuasteð h koin efaptomènh gia thn

36 2.4. MÐgmata 31 sunôparxh dôo f sewn diaforetik c summetrðac enallaktik efarmìzoume tic sunj kec sunôparxhc ex.(2.78). Oi sunart seic thceleôjerhc enèrgeiac gia autèc tic f seic genik eðnai diaforetikèc. Tèloc, sto sq ma 2.1gparousi zoume mia eidik perðptwsh tou 2.1b ìpou sumbaðnei suneq c metatrop f shc sto shmeðo ρ = ρ 1 = ρ MÐgmata GenikeÔontac th mèjodo pou anaptôxame stic prohgoômenec enìthtec ja melet soume ed sust mata apoteloômena apì dôo perissìtera eðdh morðwn ta opoða mporoôn na lamb noun di forec diamorf seic. Me b sh thn je rhsh thc enìthtac 2.1 melet me èna sôsthma to opoðo apoteleðtai apì K diaforetik eðdh morðwn N 1,N 2,...,N K me sunolikì arijmì N = K k=1 N k, pou katalamb noun ìgko V se jermokrasða T. Kajèna apì aut perigr fetai apì touc deðktec I 1,J 1,...I k,j k,... = 1,...,N. K je mìrio I k (eðdouc k) prosdiorðzetai apì tic suntetagmènec jèshc R Ik, prosanatolismoô Ω Ik kai diamorf sewn ν Ik. H eswterik enèrgeia E k (ν Ik ) tou morðou I k (eðdouc k) proèrqetai apì tic allhlepidr seic twn tmhm twn tou kai upologðzetai apì thn ex.(2.1). H allhlepðdrash metaxô dôo morðwn I k,i t (eðdouc k kai t) perigr fetai apì to U Ik,I t = U (R Ik,I t Ω Ik,I t,ν Ik,I t ) me R Ik,I t kai Ω Ik,I t na perigr foun thn jèsh kai ton prosanatolismì tou morðou I k wc proc to I t. To diamoriakì dunamikì eðnai upèrjesh metaxô zeug n tmhm twn twn morðwn I k,i t kai dðnetai apì thn ex.(2.12). H olik enèrgeia twn N morðwn U N tou sust matoc, ta opoða prosdiorðzontai apì thn sullog { } {I} = {R, Ω,ν} twn jèsewn {R} = R (1 ) 1,R(1 ) 2,...,R(k) 1, R(k) 2,... twn prosanatolism n {Ω} = { } { } Ω (1) 1, Ω(1) 2,...,Ω(k) 1, Ω(k) 2,... kai twn diamorf sewn {ν} = ν (1) 1,ν(1) 2,...,ν(k) 1,ν(k) 2,..., eðnai U N = K K k=1 t=1 I k I t [ U Ik,I t + E ] k (ν Ik )+E t (ν It ) = N 1 ìpou ìtan k = t, I t = I t +1,...,N t kai ìtan k t, I t =1,...,N t. K K k=1 t=1 I k I t Ũ Ik,I t, (2.81) To olokl rwma diamorf sewn Z N thc ex.(2.11) prosarmosmèno sto sôsthma pou melet me eðnai Z N = 1 K k=1 N k! d {I} exp k,t I k,i t [ βũi k,i t ]. (2.82) An anakatat xoume touc ìrouc sta oloklhr mata, teleutaða èkfrash mporeð na grafteð wc ginìmeno dôo parag ntwn Z N = K k=1 q N k k Z C (2.83)

37 2.4. MÐgmata 32 ìpou to q k eðnai to olokl rwma diamorf sewn tou enìc morðou eðdouc k kai eðnai q k = dν Ik exp [ βe k (ν Ik )]. (2.84) Me Z C eðnai h suneisfor sto olokl rwma diamorf sewn twn diamoriak n allhlepidr sewn wc proc tic katanomèc thc apì kataskeu c pijanìthtac emf nishc twn diamorf sewn twn morðwn kai gr fetai wc Z C = 1 K k=1 N k! d {I} k Pk o (ν I k ) exp [ βu Ik,I t ] (2.85) I k I k,i t k,t me P o k (ν I k )= exp [ βe k (ν Ik )] q k, (2.86) orðzetai wc h apì kataskeu c pijanìthta gia èna apomonwmèno mìrio (eðdouc k) na brejeð sthn diamìrfwsh ν Ik. 'Otan to N k /V eðnai polô mikrì gia ìla ta k (araiì mðgma), tìte to U N ( ex.(2.81) )jaeðnai apl to jroisma twn eswterik n} energei n tou k je morðou gia ìla ta eðdh. Se aut thn perðptwsh to Z N eðnai Z N = N k k=1 1 N k! qn k k V N k. (2.87) To epìmeno b ma sthn prosèggish mac eðnai na jewr soume, ìpwc kai sta monosustatik sust mata, ìti k je mìrio (eðdouc k) mporeð na l bei èna sônolo diakrit n diamorf sewn, opìte to olokl rwma thc ex.(2.85) metasqhmatðzetai se èna jroisma {ν Ik }, kai se èna olokl rwma d {ϖ} se ìlec tic jèseic kai touc prosanatolismoôc twn morðwn, ìpou ϖ (R, Ω). Sthn sunèqeia omadopoioôme tic diamorf seic, ètsi ste h allhlepðdrash U Ik,J t metaxô twn morðwn I k,j t na èqei thn Ðdia tim ìtan aut brðskontai se orismènec om dec diamìrfwshc (qwrðc naeðnai aparaðthta Ðdiec) gia tic Ðdiec sqetikèc jèseic kai prosanatolismoôc. Akolouj ntac thn suz thsh monosustatik n susthm twn, h omadopoðhsh eðnai tètoia ste ta mèlh miac om dac ja èqoun Ðdia moriak sq mata. Genik, an kai diaforetik eðdh morðwn mporeð na èqoun to Ðdio sq ma den èqoun katan gkhn kai tic Ðdiec diast seic. Ta diaforetik sq mata tou morðou I k ta sumbolðzoume me s Ik kai tic diakritèc diamorf seic tic sundèoume me to sq ma mèsw thc ν (s Ik ). Opìte gr foume Pk o (ν I k )exp[ βu Ik,I t ]= Pk o (ν (s I k )) exp [ βu Ik,I t ]= ν Ik s Ik ν(s Ik ) = s Ik P o k (s I k )exp[ βu Ik,I t ] (2.88) ìpou h apì kataskeu c pijanìthta P o s Ik tou sq matoc s Ik tou morðou (eðdouc k) dðnetai apì to jroisma twn diamorf sewn pou antistoiqoôn se autì to sq ma, wc ν(s Ik ) P o k (ν (s I k )). Aut

38 2.4. MÐgmata 33 orðzetai apì P o k (s I k )= exp [ βe k (s Ik )] s Ik exp [ βe k (s Ik )], (2.89) [ ] me βe k (s Ik )= ln ν(s Ik ) exp [ βe (v I k )], kai to jroisma trèqei} p nw se ìla ta sq mata tou morðou I k. Tìte to olokl rwma diamorf sewn sth nèa perigraf I k =(ϖ Ik ; s Ik ) eðnai Z C = 1 K k=1 N k! {s Ik } d {ϖ Ik } k P o (s Ik ) exp [ βu Ik,I t ]. (2.90) I k k,t I k,i t Eis goume tic dokimastikèc sunart seic φ k (I k ) kai gr foume to βu Ik,I t me ton ex c trìpo βu Ik,I t = H o k,t (I k,i t )+H k,t (I k,i t ) (2.91) ìpou o pr toc ìroc tou ajroðsmatoc H o k,t (I k,i t )=φ k (I k )+φ t (I t ) perilamb nei tic dokimastikèc sunart seic kai o H k,t (I k,i t )=βu Ik,I t φ k (I k ) φ t (I t ) perilamb nei epiprìsjeta tic diamoriakèc allhlepidr seic. Tìte to olokl rwma twn diamorf sewn Z C mporeð na grafteð wc ginìmeno dôo parag ntwn Z C = Z o exp [ H k,t (I k,i t )] k,t I k,i t o = Z o Z H (2.92) ìpou Z o = 1 K k=1 N k! {s Ik } d {ϖ} [ ] K exp H o 1 k,t = N k! zn k k, (2.93) k,t I k,i t k=1 me z k = s Ik dϖik exp [ (N 1) φ k (I k )] P o k (s I k ). OrÐzoume thn katanom pijanìthtac tou enìc morðou wc P k (I k )= 1 z k exp [ (N 1) φ k (I k )] P o k (s Ik ). (2.94) O deôteroc ìroc tou ginomènou Z H, ex.(2.92), eðnai h mèsh tim thc posìthtac stic agkôlec wc proc tic katanomèc pijanot twn twn morðwn, pou sthn genik perðptwsh eðnai... o = {s Ik } d {ϖ Ik } (...) k I k P k (I k ). (2.95) H eleôjerh enèrgeia tou sust matoc gr fetai βf C = F o F H (2.96) ìpou F o =lnz o = k (N k ln z k ln N k!) (2.97)

39 2.4. MÐgmata 34 kai sthn prosèggish suneisfor n apì zeôgh morðwn h F H eðnai F H =ln exp [ H k,t ] I k,i t k,t o k t I k I t ln exp [ H k,t ] o = = k t 1 2 N k (N k δ k,t )ln exp [ H k,t ] o (2.98) Apì tic ex.(2.91)-(2.93) kai ex.(2.96)-(2.97) h èkfrash gia thn eleôjerh enèrgeia eðnai βf (2) C = k { } N k ln z k ln N k!+ 1 2 N k (N t δ k,t )ln exp [ H k,t ] o t (2.99) Me b sh ton formalismì pou èqoume exp [ H k,t ] o = s Ik,s It dϖ k dϖ t P k (ϖ k ; s Ik ) P t (ϖ t ; s It )exp[ H k,t ] (2.100) Ekmetaleuìmenoi thn arq elaqistopoðhshc thc eleôjerhc enèrgeiac, apaitoôme lamb noume φ k (I k )= ln t δf (2) C δφ k = 0 kai x t exp [ φ t (I t )] exp [ βu Ik,I t ] t exp [ H k,t ] o (2.101) ìpou h posìthta x t = N t /N ekfr zei thn sugkèntrwsh morðwn eðdouc t sto meðgma. H φ k (I k ) emfanðzetai kai sta dôo mèlh thc ex.(2.101) kai upologðzetai me autosumbibastì trìpo Isìtroph f sh Sthn isìtroph f sh mðgmatoc morðwn h katanom pijanìthtac tou morðou I k eðdouc k exart tai apì thn apì kataskeu c pijanìthta P o k (s I k ) kai epiplèon apì mia sun rthsh f (Iso) (s Ik ) tou moriakoô sq matoc s Ik. Sunep c gr fetai ìpou Ω=8π 2. P k (s Ik )= 1 V 1 Ω f (Iso) (s Ik ) P o (s Ik ), (2.102) Oi dokimastikèc sunart seic φ k (I k ), ja exart ntai mìnon apì apì to sq ma tou morðou I k. 'Opwc kai sthn perðptwsh monosustatik n susthm twn gia thn diereônhsh twn dokimastik n sunart sewn mac dieukolônei h eisagwg thc sun rthshc Mayer q kt (s Ik,s It ), pou exart tai apì dôo metablhtèc

40 2.4. MÐgmata 35 (tou sq matoc s Ik kai s It twn morðwn eðdouc k kai t) kai orðzetai wc ex c g (Iso) kt (s Ik,s It )= dr Ik,I t dϖ k dϖ t (1 exp [ βu Ik,I t ]) (2.103) ìpou to gewmetrikì nìhma eðnai an logo me autì thc ex.(2.40). An t ra k noume tic pr xeic sthn ex.(2.103), kat llhla antikatastðsoume thn ex.(2.103) sthn ex.(2.101), kai anaptôxoume to dexiì mèloc se dun meic tou 1/V krat ntac touc grammikoôc ìrouc ja èqoume φ k (I k )= t x t φ t + 1 V t x t 1 Ω 2 g (Iso) kt t 1 V t x t 1 Ω 2 g (Iso) kt o (2.104) 'Eqontac upologðsei thn dokimastik sun rthsh eôkola mporoôme na broôme thn f (Iso) (s Ik ) me antikat stash thc ex.(2.104) sthn katanom pijanìthtac ex.(2.102). 'Etsi lamb noume P k (s Ik )= 1 V 1 Ω sik [ exp ρ ] t x t 1 Ω g (Iso) 2 kt t Pk o (s Ik ) (2.105) ζ k me ζ (Iso) k = s Ik [ Pk o (s I k )exp ρ t x t 1 Ω 2 g (Iso) kt ]. (2.106) t H sun rthsh g (Iso) kt t eðnai g (Iso) kt = g (Iso) kt t = s It f (Iso) (s It ) P o t (s I t ) g (Iso) kt (s Ik,s It ). (2.107) Telik h eleôjerh enèrgeia gr fetai βf (2) C N =lnρ 1+ k x k ln x k k x k ln ζ (Iso) k k x k ln Ω sik ρ x t x k 2 Ω 2 k t g (Iso) kt o H pðesh P kai to qhmikì dunamikì μ t tou sustatikoô t, (ex.( ) ) eðnai βp = ρ 1+ ρ 1 2 Ω 2 x k x t k,t g (Iso) kt o. (2.108) (2.109) βμ t =lnρ +lnx t ln ζ (Iso) t ln Ω. (2.110)

41 2.4. MÐgmata Nhmatik f sh H ekd lwsh nhmatik c f shc apì mðgma K diaforetik n morðwn proupojètei thn emf nish moriak n diamorf sewn sqhm twn pou parousi zoun anisometrða gia toul qiston èna apì ta eðdh morðwn tou meðgmatoc. Hkatanom pijanìthtac P k (I k ) tou enìc morðou eðnai anex rthth thc jèshc tou kai gr fetai P k ( Ω sik ; s Ik ) = 1 V f (Nem) ( Ω sik ; s Ik ) P o (s Ik ) (2.111) wc ginìmeno thc apì kataskeu c pijanìthtac kai miac sun rthshc f (Nem) ( Ω sik ; s Ik ) tou moriakoô prosanatolismoô Ω sik kai sq matoc s Ik. H diereônhsh thc ex.(2.101) pou dðnei tic dokimastikèc sunart seic φ k (I k ) gðnetai me thn eisagwg ( ) thc sun rthshc Mayer q kt Ω sik, Ω sit ; s Ik,s It pou orðzetai se aut thn perðptwsh wc ) g (Nem) kt (Ω sik, Ω sit ; s Ik,s It = dr k,t (1 exp [ βu k,t ]) (2.112) kai ekfr zei ton apokleiìmeno ìgko enìc morðou eðdouc k apì èna llo eðdouc t me kajèna na èqei ) orismèno prosanatolismì (Ω sik, Ω sit kai sq ma (s Ik,s It ). Akolouj ntac thn Ðdia mèjodo me thn isìtroph f sh brðskoume thn ex c èkfrash gia thn dokimastik sun rthsh φ k ( Ω sik ; s Ik ) = t x t φ t + 1 V x t t g (Nem) kt t 1 V x t t g (Nem) kt o (2.113) Apì tic ex.(2.111) kai ex.(2.113) h katanom pijanìthtac gðnetai P k ( Ω sik ; s Ik ) = 1 V [ exp ρ t x t g (Nem) kt ] t ζ k P o k (s I k ) (2.114) me ζ (Nem) k = s Ik [ dω sik Pk o (s I k )exp ρ t x t g (Nem) kt ]. (2.115) t H sun rthsh g (Nem) kt = g (Nem) kt g (Nem) kt t eðnai t = s It dω sit f (Nem) ( ) ) Ω sit ; s It P o t (s It ) g (Nem) kt (Ω sik, Ω sit ; s Ik,s It (2.116) H eleôjerh enèrgeia dðnetai apì thn exðswsh:

42 2.4. MÐgmata 37 βf (2) C N =lnρ 1+ k x k ln x k k x k ln ζ (Nem) k ρ 2 x k x t k t g (Nem) kt o. (2.117) H pðesh P kai to qhmikì dunamikì μ t tou sustatikoô t ( ex.( ) ) eðnai βp = ρ 1+ ρ x k x t 2 k,t g (Nem) kt (2.118) o βμ t =lnρ +lnx t ln ζ (Nem) k (2.119) Smhktik f sh H katanom pijanìthtac morðou I k eðnai P k ( z,ω sik ; s Ik ) = d V f (Sme) ( z,ω sik ; s Ik ) P o k (s I k ) (2.120) ìpou h f (Sme) ( z,ω sik ; s Ik ) eðnai sun rthsh tou moriakoô prosanatolismoô, tou sq matoc all kai thc jèshc tou morðou z wc proc ton xona proexèqousac dieôjunshc. ) Me thn eisagwg thc sun rthshc g (Sm) kt (z kt, Ω sik, Ω sit ; s Ik,s It pou orðzetai wc ) g (Sm) kt (z kt, Ω sik, Ω sit ; s Ik,s It = dx kt dy kt (1 exp [ βu k,t ]) (2.121) kai ekfr zei thn epif neia pou apokleðei èna mìrio eðdouc k apì èna llo eðdouc t, mporoôme na ft soume (me antikat stash thc ex.(2.121) sthn ex.(2.101) ) sthn φ k ( z k, Ω sik ; s Ik ) = t x t φ t + d V x t t g (Sm) kt t d V x t t g (Sm) kt o. (2.122) Tìte apì tic ex.(2.122) kai ex.(2.120) h katanom pijanìthtac, se èna smhktikì str ma p qouc d, gðnetai P k ( z k, Ω sik ; s Ik ) = d V [ exp ρ t x t g (Sm) kt ] t P o (s Ik )= d ( ) ζ k V f (Sme) z k, Ω sik ; s Ik P o k (s I k ) (2.123) me ζ (Sm) k = s Ik d 0 dz k dω sik P o k (s Ik )exp [ ρ t x t g (Sm) kt ]. (2.124) t

43 2.5. Metatropèc f shc migm twn dôo sustatik n 38 a) b) Sq ma 2.2: Grafik anapar stash thc eleôjerhc enèrgeiac Gibbs ana swmatðdio gia di forec pièseic se duadik mðgmata san sun rthsh thc sôstashc x. Kataskeu thc koin c efaptomènhc gia thn eôresh twn sust sewn sunôparxhc: a) Gia pðesh P 2 ta sustatik eðnai pl rwc anamðxima, en gia thn pðesh P 1 sumbaðnei sunup rqoun dôo f seic Ðdiac summetrðac gia sust seic x I <x<x II, b) se aut thn perðptwsh èqoume sunôparxh dôo f sewn diaforetik c summetrðac se pðesh P 3. H sun rthsh g (Sm) kt g (Sm) kt t = s It t eðnai dz t dω sit f (Sme) ( [ ) z t, Ω sit ; s It P o t (s It )exp ρd t x t g (Sm) kt ] t (2.125) H eleôjerh enèrgeia dðnetai βf (2) C N =lnρ 1+ k x k ln x k k x k ln ζ(sm) k d ρd 2 x k x t k t g (Sm) kt o, (2.126) kai elaqistopoieðtai kai wc proc to p qoc tou smhktikoô str matoc d. H pðesh P kai to qhmikì dunamikì μ t tou sustatikoô t (ex.( ) eðnai βp = ρ 1+ ρd 2 x k x t k,t g (Sm) kt (2.127) o βμ t =lnρ +lnx t ln ζ(sm) k. (2.128) d 2.5 Metatropèc f shc migm twn dôo sustatik n Se aut thn par grafo suzht me thn jermodunamik susthm twn pou perièqoun dôo diaforetik eðdh swmatidðwn (duadik mðgmata). Me apl genðkeush apì ta monosustatik sust mata, h eleôjerh enèrgeia Helmholtz ja eðnai sun rthsh tou arijmoô twn swmatidðwn tou k je sustatikoô N 1,N 2 se ìgko V kai jermokrasða T. Lìgw thc ektatikìthtac thc eleôjerhc enèrgeiac se orismènh jermokrasða èqoume g (x, ρ) =F (N 1,N 2,V,T)/N. 'Opou N = N 1 + N 2 eðnai o sunolikìc arijmìc swmatidðwn, kai x = N 2 /N h sqetik sôstash tou 2 (gia to 1 èqoume N 1 /N =1 x). 'Olec oi jermodunamikèc idiìthtec aut n twn susthm twn mporoôn na prosdioristoôn, gia par deigma, san

44 2.5. Metatropèc f shc migm twn dôo sustatik n 39 sun rthsh thc jermokrasðac T, thc pðeshc P, kai tou arijmoô twn swmatidðwn diaforetikoô eðdouc N 1 kai N 2. 'Opwc eðdame sthn enìthta 2.3 h sunj kh jermodunamik c isorropðac metaxô dôo f sewn (se orismènh jermokrasða) proupojètei thn mhqanik kai qhmik isorropða twn f sewn. Sto genikìtero prìblhma me dôo eðdh swmatidðwn h sunj kh isorropðac dôo f sewn eðnai isìthta thc jermokrasðac thc pðeshc kai twn qhmik n dunamik n tou k je sustatikoô se autèc. Me latinikoôc qarakt rec sumbolðzoume tic di forec f seic kai me arabikoôc deðktec to eðdouc tou sustatikoô. Tìte gia thn perðptwsh sunôparxhc dôo f sewn h sunj kh jermodunamik c isorropðac 61 eðnai P I (x I,ρ I )=P II (x II,ρ II ) (2.129) μ I 1 (x I,ρ I )=μ II 1 (x II,ρ II ) (2.130) μ I 2 (x I,ρ I )=μ II 2 (x II,ρ II ) (2.131) Stic parap nw exis seic, dôo f seic I kai II, me arijmhtikèc puknìthtec ρ I kai ρ II kai sqetikèc sust seic x I kai x II brðskontai se isorropða. Sthn pragmatikìthta èqoume na epilôsoume èna sôsthma tri n exis sewn me tèsseric agn stouc, oi opoðec mporoôn na ikanopoioôntai gia èna meg lo pl joc puknot twn kai sust sewn. Na shmei soume, sômfwna me ton kanìna tou Gibbs gia èna sôsthma apoteloômeno apì dôo eðdh swmatidðwn, ìqi perissìterec apì tèsseric f seic mporeð na brðskontai se isorropða. H èkfrash gia thn pðesh tou sust matoc eðnai P (x, ρ) = F V = ρ 2 h(ρ, T o) N,T ρ (2.132) x,t kai gia ta qhmik dunamik kai μ 1 = h (x, ρ, T o )+ 1 ρ P (x, ρ) x h(x, ρ, T o) x μ 2 = h (x, ρ, T o )+ 1 ρ P (x, ρ)+(1 x) h(x, ρ, T o) x (2.133) ρ,t. (2.134) ρ,t H melèth thc jermodunamik c sumperifor c tou mðgmatoc mporeð na perigrafeð me thn eisagwg thc eleôjerhc enèrgeiac Gibbs (NPT) ìpou h ektatik metablht tou ìgkou V antikajðstatai apì thn entatik metablht thc pðeshc P. Lìgw thc ektatikìthtac thc eleôjerhc enèrgeiac Gibbs, h Gibbs

45 2.5. Metatropèc f shc migm twn dôo sustatik n 40 an swmatðdio gr fetai g (x, P, T )=G (N 1,N 2,P,T) /N kai apì ton orismì aut c èqoume g (x, P, T )= i μ i x i = f (x, ρ, T )+ 1 P. (2.135) ρ Gia duadik mðgmata, h sunj kh jermodunamik c eust jeiac metaxô dôo f sewn Ðdiac summetrðac eðnai h mhqanik kai qhmik isorropða metaxô twn sustatik n stic f seic I kai II. Aut h sunj kh eðnai isodônamh me thn kataskeu thc koin c efaptomènhc (sq ma 2.2) ìpou mporoôme na upologðsoume tic sust seic sunôparxhc x I kai x II gia orismènh pðesh P kai jermokrasða T. Dhlad èqoume g x x=xi = g x = g (x II,P,T o ) g (x I,P,T o ). (2.136) x=xii x II x I Oi sust seic x I kai x II eðnai oi sust seic sunôparxhc. Ta ìria eust jeiac kajorðzontai apì thn sunj kh 2 g x 2 )P,T =0. H kampôlh thc eleôjerhc enèrgeiac Gibbs gia thn pðesh P 1 eðnai qarakthristik sunôparxhc dôo f sewn thc Ðdiac summetrðac (sq ma 2.2a). H kampôlh gia pðesh P 2 den parousi zei akrìtata kai h mia f sh pou ekdhl netai eðnai eustaj c se ìlo ta di sthma twn sust sewn 0 x 1. Tèloc sthn sq ma 2.2b parousi zoume tic kampôlec thc eleôjerhc enèrgeiac Gibbs gia dôo f seic diaforetik c summetrðac sthn Ðdia pðesh P 3.

46 Kef laio 3 Monosustatik sust mata foullerenik n ugr n krust llwn H jewrða twn metatrepìmenwn moriak n sqhm twn efarmìzetai gia th melèth monosustatik n susthm twn adropoihmènwn foullerenik n UK. Parousi zontai ta upologismèna diagr mmata f shc gia mia poikilða moriak n arqitektonik n pou perilamb noun foullerenikoôc UK me dðdumouc mesogìnouc braqðonec, foullerenikoôc UK me dendritikoôc braqðonec pou perièqoun mesogìna, kai kwnikoôc foullerenikoôc UK. H fasik sumperifor kai h moriak auto org nwsh stic f seic, katanoeðtai me ìrouc diamoriak c allhlepðdrashc kai eukamyðac twn adropoihmènwn morðwn. 3.1 Eisagwg Xekin ntac to 1996 me thn ergasða twn 63 Chuard kai Deschenaux, h qhmik sôndesh tou foullerèniou me qhmikèc om dec pou parousi zoun ugrokrustallik (UK) sumperifor od ghse sthn sônjesh nèwn qhmik n en sewn (foullerenikìc UK) pou emfanðzoun UK moriak t xh. Apì tìte èqoun gðnei pollèc prosp jeiec se aut thn kateôjunsh apodðdontac ènan meg lo arijmì foullerenik n UK. Mèqri prìsfata gnwrðzoume ìti oi sundeìmenec sto foullerènio moriakèc domèc mporeð na eðnai eðte apl mesogìna mìria, eðte pl joc mesogìnwn sundedemènwn se dendritik topologða, eðte dendrimer tôpou Percec. An loga me thn dom twn om dwn pou sundèontai sto foullerènio, oi foullerenikoð UK omadopoioôntai stic parak tw kathgorðec (me tic antðstoiqec arqitektonikèc na dðnontai sto sq ma 3.1): a) FoullerenikoÐ UK me dðdumouc mesogìnouc braqðonec oi opoðoi eðnai sundedemènoi mèsw enìc triploô daktôliou. K je braqðonac xekin ei me mia eôkampth anjrakik alusðda kai termatðzetai me mia mesogìno mon da. Oi foullerenikoð UK autoô tou tôpou 36, 45, ekdhl noun apokleistik smhktik A f sh (sq ma 3.1a). b) FoullerenikoÐ UK me dðdumouc braqðonec, ìpwc kai sthn (a) perðptwsh, mìnon pou ed k je braqðonac èqei dendritik topologða kai apoteleðtai apì pl joc mesogìnwn sta kra tou (sq ma 41

47 3.1. Eisagwg 42 45, b). Oi foullerenikoð UK autoô tou tôpou emfanðzoun qamhl c dendritik c geni c dendrimer emfanðzoun nhmatik f sh. smhktik A f sh en epiprìsjeta g) FoullerenikoÐ UK me aploôc dendritikoôc braqðonec (sq ma 3.1g). Aut eðnai tropoihmènec en seic thc perðptwshc (b) ìpou ènac apì touc braqðonec èqei dendritik topologða en o lloc mh dendritik. 'Eqei anaferjeð 45, 66, ìti autèc oi en seic ekdhl noun apokleistik smhktik A f sh. d) FoullerenikoÐ UK me dðdumouc dendritikoôc braqðonec, ìpwc sthn perðptwsh (b), all sundedemènouc se duo diaforetik shmeða sthn epif neia tou foullerèniou (sq ma 3.1d). H mesof sh pou ekdhl noun aut ta mìria den èqei prosdioristeð 80 oristik. e) FoullerenikoÐ UK me braqðonec sundemènouc se èxi diaforetik shmeða thc epif neiac autoô (sq ma 3.1e). 'Exi zeug ria apì dðdumouc mesogìnouc braqðonec pou kajènac termatðzetai apì mia mesogìno mon da sundèontai mèsw enìc triploô daktulðou me to foullerènio. Autèc oi en seic 81, 82 ekdhl noun smhktik A f sh. st) FoullerenikoÐ UK me kwnik 83, 84 puramidik dom 85 (sq ma 3.1st). H kwnik puramidik koilìthta sqhmatðzetai me apeujeðac ( mèsw miac dimejulopuritik c om dac) sôndesh pènte mesogìnwn mon dwn gôrw apì èna pent gwno tou foullerenðou. Gia ta kwnik mìria èqei anaferjeð o sqhmatismìc jermotropik n exagwnik n kionik n f sewn kai luotropik n nhmatik n kionik n en gia ta puramidik jermotropik n exagwnik n kionik n f sewn. z) FoullerenikoÐ UK sundemènoi, mèsw anjrakik c alusðdac, me dendrimer tôpou 38 Percec. (sq ma 3.1z). Autèc oi en seic ekdhl noun tetragwnik kionik f sh. H poikilða twn foullerenik n UK pou èqei melethjeð peiramatik eðnai arket eureða ste na epitrèpei ton èlegqo kai epal jeush jewrhtik n problèyewn. Skopìc mac eðnai me thn eisagwg miac apl c moriak c jewrðac na sundèsoume thn autoorg nwsh pou ekdhl netai apì aut ta sust mata me thn moriak dom. Den skopeôoume sthn posotik sumfwnða sthn diereônhsh tou rìlou epimèrouc moriak n qarakthristik n twn en sewn aut n twn susthm twn. Sthn enìthta 3.2 perigr foume ta basik shmeða thc jewrhtik c prosèggishc, anaferìmaste sthn moriak adropoðhsh pou pragmatopoioôme kai sth genik morf tou dunamikoô allhlepðdrashc metaxô twn adropoihmènwn morðwn. Stic epìmenec enìthtec exet zoume xeqwrist thn moriak adropoðhsh gia di forec arqitektonikèc kai parousi zoume ta apotelèsmata thc moriak c jewrðac. Tèloc, sunoyðzoume ta sumper smata sthn enìthta 3.6.

48 3.2. Moriak montelopoðhsh se kubikì plègma 43 a) b) g) d) e) st) z) Sq ma 3.1: Di forec moriakèc arqitektonikèc foullerenik n UK. 3.2 Moriak montelopoðhsh se kubikì plègma Se aut thn enìthta efarmìzoume thn prosèggish twn metatrepìmenwn moriak n sqhm twn (enìthta 2.2) gia mia poikilða arqitektonik n dom n foullerenik n UK. Ta aparaðthta sustatik gia thn efarmog thc jewrhtik c prosèggishc eðnai ta parak tw: a) H omadopoðhsh tou ter stiou arijmoô twn dunat n diamorf sewn tou enìc morðou se ènan arijmì diakrit n katast sewn sqhm twn} pou antistoiqoôn stic basikèc moriakèc katast seic. b) Se k je sq ma s} apodðdoume mia apì kataskeu c pijanìthta} P o (s), pou dðnei thn pijanìthta na broôme èna apomonwmèno mìrio sto sq ma s}. IsodÔnama, mia apì kataskeu c eleôjerh enèrgeia E(s) apodðdetai se k je sq ma. 'Opwc èqoume anafèrei (suz thsh sthn ex.(2.22) [ ] ) h E (s) gr fetai wc βe (s) = ln ν(s) E (ν (s)), ìpou to jroisma sthn agkôlh eðnai h sun rthsh epimerismoô twn diamorf sewn tou enìc morðou pou omadopoioôntai sto sq ma s}. g) Ta mìria upodiairoôntai se tm mata me kajorismènec allhlepidr seic pou exart ntai apì to eðdoc thc qhmik c om dac pou anaparistoôn. H diamoriak allhlepðdrash upologðzetai wc upèrjesh twn allhlepidr sewn metaxô ìlwn twn dunat n zeugari n tmhm twn ìpwc perigr fetai sthn ex.(2.12). 'Opwc èqoume anafèrei sthn paroôsa prosèggish stoqeôoume se mia poiotik orj perigraf thc fasik c sumperifor c twn moriak n susthm twn. Gia autì ton lìgo, eis goume mìnon èna mikrì

49 3.2. Moriak montelopoðhsh se kubikì plègma 44 arijmì antiproswpeutik n moriak n sqhm twn, antistoiqoôme mia seir apì kataskeu c eleôjerec enèrgeiec se aut, kai mazð mia adropoihmènh upodiaðresh tou morðou se tm mata sunodeuìmenh apì mia apl parametrikopoðhsh twn allhlepidr sewn metaxô aut n. 'Etsi, h moriak upodiaðresh (diamerismatopoðhsh), kai o susqetizìmenoc me aut mikrofasikìc diaqwrismìc, antanakl tic diaforetikèc allhlepidr seic metaxô moriak n tmhm twn. Sta sust mata pou melet me to foullerènio kajorðzei thc diast seic enìc tm matoc, en oi diast seic twn upìloipwn qhmik n mon dwn kajorðzontai me b sh me to mègejoc autoô tou tm matoc. Oi upologismoð thc eleôjerhc enèrgeiac twn moriak n sust matwn kai oi sunart seic autosunèpeiac, gia thn aplìthta twn upologism n, pragmatopoioôntai se èna kubikì plègma. Ta mìria periorðzontai na kinoôntai se autì to plègma me monadiaða plegmatik kuyelðda Ðsh me th di metro tou foullerèniou. Ta adropoihmèna mìria qtðzontai} qrhsimopoi ntac domik tm mata Ðsou megèjouc me thn monadiaða kuyelðda tou plègmatoc. Stic peript seic pou pragmateuìmaste, oi moriakèc domèc apoteloôntai apì èna tm ma pou antistoiqeð sto foullerènio kai apì èna epiplèon pl joc tmhm twn gia to qtðsimo} twn sundedemènwn sto foullerènio qhmik n mon dwn. AfoÔ kataskeu soume ta adropoihmèna mìria, to diamoriakì dunamikì allhlepðdrashc U I,J metaxô twn morðwn I,J upologðzetai apì thn ex.(2.12), pou prosarmosmèno se kubikì plègma gr fetai wc U I,J = ) (u (0) (i I,i J ) δ (r ii,i J )+u (1) (i I,i J ) δ ( r ii,i J 1), (3.1) i I,i J ìpou δ eðnai h sun rthsh Dirac, r ii,i J eðnai h sqetik apìstash twn tmhm twn i I kai i J, kai u (0) (i I,i J ) kai u (1) (i I,i J ) eðnai to mètro thc allhlepðdrashc gia èna zeug ri tmhm twn pou katalamb noun thn Ðdia plegmatik kuyelðda kontinoô geðtona, antðstoiqa. Sta sust mata pou melet same o deðkthc i mporeð na eðnai f m l, sumbolðzontac, antðstoiqa, foullerènia, mesogìnec mon dec kai mh mesogìnec sundetikèc mon dec. Na shmei soume ìti o deðkthc I J perièqei thn plhroforða se poiì sq ma an kei to mìrio autì. Eis goume thn parak tw morf gia to diatmhmatikì dunamikì allhlepðdrashc u (n) (i I,i J ),n=0, 1: u (n) (i I,i J )=q (n) (i, j)+w (n) (i, j) P 2 (z i z j ), (3.2) me q (n) (i, j) kai w (n) (i, j) eðnai par metroi allhlepðdrashc metaxô twn i kai j tmhm twn ìtan katalamb noun thn Ðdia, n =0, geitonik, n =1, kuyelðda. O ìroc q (n) (i, j) eðnai to mètro thc mh kateujuntik c diatmhmatik c allhlepðdrashc (isìtroph allhlepðdrash). Ja prèpei ìmwc na shmei soume ìti autìc o ìroc eðnai arketìc gia na emfanisteð metaxô twn morðwn kateujuntik allhlepðdrash ìtan ta moriak tm mata sundèontai proc mia dieôjunsh (ìpwc eðnai stic peript seic pou exet zoume parak tw: sq ma 3.3 kai 3.9). O ìroc w (n) (i, j) epitrèpei thn Ôparxh endogenoôc kateujuntikìthtac sth diatmhmatik allhlepðdrash kai eis getai lìgw thc parousðac mesogìnwn mon dwn. K ti tètoio epitugq netai me to polu numo Legendre

50 3.3. FoullerenikoÐ UK me DÐdumouc Mesogìnouc BraqÐonec (TMBC60) 45 a) b) Sq ma 3.2: a) QhmikoÐ tôpoi foullerenik n 45 UK TMBC60, b) jermotropik fasik sumperifor 45 aut n. Mwb: ual dhc perioq, portokalð: ugrokrustallik perioq (smhktik A) kai bioletð: isìtroph f sh. H jermokrasða se bajmoôc KelsÐou. deôterhc t xhc, P 2 (z i z j ), ìpou z i kai z j eðnai anôsmata prosanatolismoô twn braqiìnwn tou kôriou moriakoô xona. Polikèc allhlepidr seic metaxô twn tmhm twn paraleðpontai all ja mporoôsan na lhfjoôn upìyh se perissìtero sônjetouc upologismoôc sumperilamb nontac ton ìro P 1 (z i z j ) sthn ex.(3.2). H par leiy touc apì touc upologismoôc gðnetai gia lìgouc aplìthtac qwrðc katan gkh na jewroôme ìti o rìloc touc eðnai amelhtèoc. EÐnai gegonìc, ìti oi perissìterec moriakèc domèc pou pragmateuìmaste se aut thn diatrib perièqoun qhmikèc mon dec me isqur dipolik rop. GnwrÐzoume ìti tètoiou eðdouc allhlepidr seic mporoôn na ephre soun shmantik thn eust jeia twn ugrokrustallik n f sewn orismènwn sumbatik n mesogìnwn. 3.3 FoullerenikoÐ UK me DÐdumouc Mesogìnouc BraqÐonec (TMBC60) Oi foullerenikoð UK me dðdumouc mesogìnouc braqðonec TMBC60, apoteloôntai apì dôo braqðonec ìmoiac qhmik c fôshc ìpou o kajènac fèrei mia mesogìno mon da sta kra tou (sq ma 3.2a); ekdhl noun jermotropik mesofasik sumperifor (sq ma 3.2b). Profan c, to pl joc twn dunat n diamorf sewn gia tètoia sust mata eðnai idiaðtera meg lo exaitðac thc eukamyðac twn anjrakik n alusðdwn pou sundèoun to C60 me tic mesogìnec mon dec. UpologismoÐ moriak c mhqanik c me eurèwc apodekt empeirik dunamik 86 (force fields) deðqnoun ìti diamorf seic pou omadopoioôntai se diaforetik sq mata den parousi zoun shmantikèc energeiakèc diaforèc. Epiprìsjeta, moriakèc katast seic metic mesogìnec mon dec par llhlec metaxô touc kai sthn Ðdia kateôjunsh wc proc to C60, parìlo to mikrì energeiakì kèrdoc lìg wtou van der Waals elktikoô dunamikoô (se sqèsh me tic katast seic ìpou oi mesogìnec om dec eðnai apomakrusmènec), pragmatopoioôntai gia ènan periorismèno arijmì diamorf sewn twn eôkamptwn alusðdwn kai gia autì den eunooôntai entropik.

51 3.3. FoullerenikoÐ UK me DÐdumouc Mesogìnouc BraqÐonec (TMBC60) 46 a) b) g) Sq ma 3.3: Atomistik eikìna kai anapar stash tou adropoihmènou TMBC60, a) sto sq ma EA} b) sto sq ma FP} kai g) sto sq ma L}. H perissìtero èntonh skðash sta tm mata tou (b) (pl n tou foullerèniou) eðnai gia na deðxei thn filoxenða dôo moriak n tmhm twn se mia plegmatik kuyelðda. Basizìmenoi sta parap nw, melet me to adropoihmèno TMBC60 omadopoi ntac tic moriakèc diamorf seic se trða xeqwrist moriak sq mata (sq ma 3.3). K je sq ma apoteleðtai apì mia foullerenik mon da kai dôo braqðonec. K je tm ma filoxeneð èna foullerènio ètsi ste to m koc tou k je braqðona na eðnai treðc forèc h di metroc tou foullerenðou. Sunep c, o braqðonac apoteleðtai apì trða tm mata (building blocks) me to eggôtato sto foullerènio nakatalamb netai merik c apì anjrakikèc alusðdec kai ta epìmena dôo apì th mesogìno mon da. Sto pr to apì ta sq mata, pou sto ex c sumbolðzetaime EA} (sq ma 3.3a), oi braqðonec ekteðnontai se antðjetec kateujônseic prosdðdontac kulindrik summetrða se autì (akribèstera, tetragwnik afoô anaferìmaste se kubikì plègma) kai apolikìthta kat thn epim kh dieôjunsh. Sto deôtero sq ma, pou to sumbolðzoume me L} (sq ma 3.3g), oi braqðonec eðnai k jetoi metaxô touc. Sto trðto sq ma, pou to sumbolðzoume me FP} (sq ma 3.3b), oi dôo braqðonec eðnai proc thn Ðdia kateôjunsh, kai parousi zei tìso kulindrik summetrða ìso kai polikìthta kat thn dieôjunsh epim kunshc tou morðou. Epeid k je braqðonac prokôptei apì mia grammik di taxh twn tmhm twn, ta trða sq mata pou anafèrame kalôptoun ìlec tic peript seic se èna kubikì plègma. 'Etsi, oi diamorf seic ìpou emfanðzoun endi mesh klðsh metaxô twn braqiìnwn (V-shaped), kai oi opoðec eðnai h pleioyhfða twn

52 3.3. FoullerenikoÐ UK me DÐdumouc Mesogìnouc BraqÐonec (TMBC60) 47 diamorf sewn tou TMBC60, katanooôntai wc upopðptousec se aut ta sqhm ta, an log a me to mègejoc thc gwnðac klðshc. Gia ta sust mata pou melet me se aut thn enìthta, oi braqðonec den eðnai pukn kateilhmmènoi apì tic antðstoiqec qhmikèc en seic. 'Etsi, jewroôme ìti dôo moriak tm mata (diaforetik n morðwn) allhlepidroôn mìnon ìtan katalamb noun to Ðdio plegmatikì shmeðo kai epomènwc to u (1) (i, j) thc ex.(3.1) lamb netai Ðso me mhdèn. Gia thn parametrikopoðhsh tou u (0) (i, j) jewroôme ìti plegmatik shmeða ta opoða katalamb nontai apì foullerènia, den eðnai diaperat oôte apì foullerènia oôte apì opoiad pote llo tm ma twn braqiìnwn. Epiprìsjeta, jewroôme ìti ta tm mata tou braqðona (diaforetik n morðwn) allhlepidroôn metaxô touc me elafr c apwstikì dunamikì qwrðc ex rthsh prosanatolismoô. Sthn parametrikopoðhsh thc ex.(3.2), me aut thn je rhsh èqoume w (0) (i, j) =0 gia opoiod pote diamoriakì zeôgoc i, j, q (0) (f,j) = gia j = f m l, kai q (0) (l, l) =q (0) (m, m) =q (0) (m, l) =u>0, me to u na ekfr zei to mètro thc apwstikìthtac twn allhlepidr sewn aut n twn tmhm twn. Epiprìsjeta, exet zontai oi epipt seic sthn moriak org nwsh tou TMBC60, me thn eisagwg kateujuntik n allhlepidr sewn metaxô tmhm twn twn braqiìnwn, epitrèpontac w (0) (i, j) 0, gia i, j = m. H apì kataskeu c eleôjerec enèrgeiec twn sqhm twn EA} L} jewroôme ìti eðnai Ðdiec, E (EA) = E (L) =0. Gia thn apì kataskeu c eleôjerh enèrgeia E (FP) jewroôme tic dunatèc peript seic me E (FP) > 0,E (FP)=0kai E (FP) < 0 pou antistoiqoôn se autì to sq ma na eðnai mikrìterec, Ðsec, perissìtero pijanèc apì ta upìloipa dôo sq mata Apotelèsmata thc moriak c jewrðac gia to TMBC60 Exet zoume wc proc thn sqetik touc jermodunamik eust jeia thn isìtroph, th nhmatik kai thn smhktik f sh (upoenìthtec ), metab llontac thn tim thc paramètrou allhlepðdrashc u alli c thc antðstrofhc jermokrasðac. Sthn pragmatikìthta mac endiafèrei to ginìmeno βu, kai twn apì kataskeu c pijanot twn twn moriak n sqhm twn (par metroc E (FP)). Gia ìlouc touc sunduasmoôc twn βu kai E (FP) pou melet me, oi mesof seic pou parathroôntai eðnai nhmatik kai orjog nia smhktik. H teleutaða parousi zei poikilða endostrwmatik c dom c. ExaitÐac thc dunatìthtac metatrop c tou moriakoô sq matoc pou èqoun ta mìria, me aisjht diaforetik moriak m kh gia k je sq ma, o polumorfismìc twn smhktik n f sewn pou parathreðtai diafèrei shmantik apì ton polumorfismì pou emfanðzoun sthn smhktik f sh sumbatik ugrokrustallik mìria sta opoða oi moriakèc diast seiceðnai sqetik kal c kajorismènec. EÐnai gnwstì ìti polik rabdoeid mìria emfanðzoun polumorfismì sthn endostrwmatik org nwsh twn mesof se n touc. 'Etsi, h kathgoriopoðhsh twn smhktik n A f sewn gðnetai se SmA 1,SmA d kai SmA 2 f seic, Ta opoða mporoôn na katanohjoôn me thn prosèggish tou kamptou morðou}.

53 3.3. FoullerenikoÐ UK me DÐdumouc Mesogìnouc BraqÐonec (TMBC60) 48 a) b) g) d) Sq ma 3.4: Upologismèna diagr mmata f shc (anhgmènh pðesh san sun rthsh thc antðstrofhc jermokrasðac (P βu)) gia to TMBC60. Ta sq mata EA} kai L} jewroôme ìti èqoun Ðdia apì kataskeu c pijanìthta Ðdia eleôjerh enèrgeia (E(EA) =E(L) =0), en exet zoume tèsseric diaforetikèc peript seic gia thn apì kataskeu c pijanìthta FP}a)E(FP)/u = 3, b) E (FP)/u = 1,g) E (FP)/u =1kai d) E (FP)/u =3. H par metroc allhlepðdrashc u èqei thn tim 0,1 gia ìlec tic peript seic.

54 3.3. FoullerenikoÐ UK me DÐdumouc Mesogìnouc BraqÐonec (TMBC60) 49 a) b) g) d) Sq ma 3.5: Anapar stash twn smhktik n f sewn pou emfanðzoun foullerenikoð UK. Me pr sino qrwmatðzoume ta foullerènia, me bussinð touc braqðonec kai me gal ziotamesogìna (ìtan eis goume kateujuntikèc allhlepidr seic): a) Smhktik f sh apl c stoib dac foullerenðwn (SmA d1 ), b) Smhktik f sh dipl c stoib dac foullerenðwn (SmA d2 ), g) Mh enarmonismènh smhktik f sh mon c kai dipl c stoib dac (SmA di ) kai d) Smhktik dipl c stoib dac me kateujuntikèc allhlepidr seic. ìpou oi deðktec 1,d kai 2 deðqnoun ìti to m koc enarmìnishc thc periodik c puknìthtac d} eðnai, antðstoiqa, èna (d = c), (1 <d/c<2) dôo forèc (d =2c), to moriakì m koc c}. Htaxinìmhsh twn smhktik n A f sewn pou anafèrame proupojètei thn Ôparxh enìc kai monadikoô moriakoô sq matoc. 'Opwc èqoume anafèrei k ti tètoio den isqôei gia to TMBC60 ìpou apì to sq ma 3.3 mporeð kaneðc na diapist sei ta teleðwc diaforetik gewmetrik qarakthristik twn moriak n sqhm twn. MolataÔta, gia tic an gkec thc paroôsac melèthc perigr foume tic smhktikèc f seic me ton eurèwc apodektì sumbolismì twn sumbatik n smhktik n f sewn, orðzontac san moriakì m koc} to jroisma tou m kouc twn tmhm twn twn braqiìnwn kai tou foullerenðou (gia to sq ma FP}, me kulindrik summetrða). SÔmfwna me aut thn epilog, sumbolðzoume me SmA d1 èntona allhloepikaluptìmenec (interdigitated) smhktikèc f seic ìpou mèsa sto str ma sqhmatðzetai apì apl stoib da foullerenðwn (sq ma 3.5a). Smhktikèc f seic parìmoiec me thn SmA d1 all me dipl stoib da foullerenðwn sumbolðzontai me SmA d2 (sq ma 3.5b). Smhktikèc f seic memhenarmonismènh periodikìthta antistoiqoôn se upèrjesh twn SmA d1 kai SmA d2 kai sumbolðzontai me SmA di (sq ma 3.5g). Tèloc, me thn eisagwg kateujuntik n allhlepidr sewn pragmatopoieðtai epiprìsjetoc mikrofasikìc diaqwrismìc se trða upostr mata (sq ma 3.5d). H fasik sumperifor twn susthm twn tou TMBC60 akolouj ntac thn sômbash pou perigr yame parap nw, sunoyðzetai sto sq ma 3.4 ìpou anaparðstatai h pðesh wc sun rthsh thc antðstrofhc

55 3.3. FoullerenikoÐ UK me DÐdumouc Mesogìnouc BraqÐonec (TMBC60) 50 Sq ma 3.6: Upologismènh pijanìthta emf nishc sthn sumpuknwmènh f sh tou moriakoô sq matoc EA} (kôkloc), tou L} (trðgwno) kai tou FP} (tetr gwna), san sun rthsh thc pðeshc (P ) gia to sôsthma tou opoðou to di gramma f shc dðnetai sto sq ma 3.4ggia orismènh tim thc antðstrofhc jermokrasðac βu =0, 20. Sto di gramma dðnontai epðshc h apì kataskeu pijanìthta twn sqhm twn EA}, L} (orizìntia suneq c gramm ) kai FP} (orizìntia diakekommènh gramm ). jermokrasðac gia di forec timèc tou lìgou E (FP)/u, qwrðc thn eisagwg kateujuntik n allhlepidr sewn. Se ìlec tic peript seic, ta sust mata emfanðzoun mia isìtroph f sh (I), di forec smhktikèc f seic (Sm), kai sthn perioq twn qamhl n jermokrasi n mia monoaxonik nhmatik f sh (N) h opoða, ìmwc, eðnai eustaj c se mia sten perioq tou diagr mmatoc f shc (en sto sq ma 3.4a den emfanðzetai kajìlou). Oi orjog niec smhktikèc f seic diafèroun ston trìpo pou oi foullerenikèc mon dec diat ssontai mèsa sta str mata. Eidikìtera, gia tic perioqèc twn uyhl n jermokrasi n kai uyhl c pðeshc smhktik n f sewn, ta foullerènia diat ssontai se apl stoib da (sq ma 3.5a) en se qamhlìterec jermokrasðec diat ssontai se dipl stoib da mèsa sta str mata (sq ma 3.5b). 'Olec oi metatropèc f shc eðnai pr thc t xhc. Oi SmA d1 SmA d2 kai isìtrophnhmatik eðnai ligìtero èntonec se sqèsh me thn metatrop isìtrophc - smhktik c kai nhmatik c - smhktik c. H nhmatik f sh, emfanðzetai mìnon se sqetik qamhlèc jermokrasðec ìpou ta moriak sq mata kajðstantai ìlo kai perissìtero adiapèrasta kaihautoorg nwsh kajorðzetai apì to sunolikì sq ma (allhlepidr seic sterik c} pwshc). Gia E (FP)/u = 3, dhlad, ìtan ta FP} sq mata eðnai apì kataskeu c perissìtero pijan na emfanistoôn apì ta upìloipa dôo ( EA} kai L} ),h moriak org nwsh sta str mata antistoiqeð sthn SmA d2 f sh ektìc apì mia sten lwrðda uyhl n pièsewn kai uyhl n jermokrasi n ìpou h SmA d1 f sh eðnai perissìtero eustaj c. Aux nontac thn eleôjerh enèrgeia E (FP) all diathr ntac thn arnhtik, E (FP)/u = 1, hsunolik eikìna tou diagr mmatoc paramènei h Ðdia all me mia megalôterh perioq gia thn SmA d1 f sh. Se aut thn perðptwsh h nhmatik f sh emfanðzetai se qamhlèc jermokrasðec. H t sh akìma megalôterhc eust jeiac thc SmA d1 enisqôetai ìtan E (FP)/u =1. Tìte, me ta sq mata FP} na èqoun uyhlìterh eleôjerh enèrgeia (qamhlìterh apì kataskeu c pijanìthta) apì ìti to sq ma EA}, h SmA d1 perioq ekteðnetai se uyhlìterec

56 3.3. FoullerenikoÐ UK me DÐdumouc Mesogìnouc BraqÐonec (TMBC60) 51 jermokrasðec. Gia tic peript seic pou anafèroume parap nw ìson afor thn fasik sumperifor twn susthm twn, mei nontac thn jermokrasða k tw apì stajer pðesh parathroôme: (a) sthn perioq uyhl n pièsewn I SmA d1 SmA d2, (b) sthn endi mesh perioq I SmA d2, kai (g) sthn qamhl perioq I N SmA d2. HtopologÐa tou diagr mmatoc f shc all zei idiaðtera ìtan E (FP)/u =3. Se aut thn perðptwsh h perioq eust jeiac thc SmA d1 enisqôetai kalôptontac, gia endi mesec pièseic, ìlo to jermokrasi ko eôroc. 'Etsi, h met bash I SmA d1 kai I N SmA d1, pou den parathroôntan prðn, kajðstatai dunatèc ìtan sta moriak sq mata me touc braqðonec ektetamènouc ( EA} kai ligìtero L} ) dðnontai uyhlèc apì kataskeu c pijanìthtec. Sth met bash smhktik c - smhktik c f shc aux netai to p qoc twn strwm twn. Autì den ofeðletai sthn mh allhlepik luyh moriak n tmhm twn twn braqiìnwn all ston sqhmatismì jermodunamik eustajoôc smhktik c f shc, h opoða sumbolðzetai wc SmA di, ìpou mèsa sta str mata emfanðzontai kai oi dôo tôpoi smhktik c f shc, dhlad, tìso SmA d1 ìso kai SmA d2. Aut h f sh eðnai eustaj c se mia sten perioq, se mia suneq met bash metaxô twn SmA d1 kai SmA d2. H moriak org nwsh stic smhktikèc f seic all kai o plhjusmìc} h pijanìthta emf nishc diafìrwn moriak n sqhm twn metab lletai shmantik apì thn mia f sh sthn llh. Autì mporeð kaneðc na to diapist sei apì to sq ma 3.6 ìpou apeikonðzetai h pijanìthta emf nishc twn moriak n sqhm twn sto ulikì san sun rthsh thc pðeshc se orismènh jermokrasða βu =0, 20 gia èna sôsthma me E (FP)/u =1. Sto Ðdio Sq ma, oi orizìntiec grammèc dðnoun thn apì kataskeu c pijanìthta gia èna EA} kai gia èna L} sq ma (orizìntia suneq c gramm ) kai gia èna FP}sq ma (orizìntia diakekommènh gramm ). EÐnai fanerì ìti to moriakì sq ma L} eðnai to ligìtero pijanì stic organwmènec f seic en èqei thn Ðdia pijanìthta meto EA} sthnisìtroph f sh. Sto sq ma 3.6 h mia f sh xeqwrðzei apì thn llh, me estigmènec grammèc k jetec ston xona twn pièsewn. Sthn SmA d2 f sh kuriarqoôn oi FP} katast seic en eðnai axioshmeðwto ìti sthn SmA d1 emfanðzontai kai EA}katast seic mepijanìthta uyhlìterh thc antðstoiqhc se apomonwmèno mìriou tou Ðdiou sq matoc. AxÐzei na poôme ìti oi pijanìthtec emf nishc diafìrwn moriak n sqhm twn sto ulikì mporoôn na katanohjoôn wc sust seic} enìc mðgmatoc me trða eðdh morðwn ìpou h bèltisth dieujèthsh kajist tic f seic jermodunamik eustajeðc. Me b sh aut thn optik, h prosèggish twn metatrepìmenwn sqhm twn odhgeð ston upologismì twn bèltistwn sust sewn diaforetik n eðdwn (moriak n sqhm twn) san sun rthsh twn jermodunamik n metablht n tou sust matoc. Epanal bame touc upologismoôc krat ntac mìnon ta dôo grammik moriak sq mata ( ta FP} kai EA} ) paraleðpontac to L} sq ma. Se aut thn perðptwsh, pèra apì mia elafri stajeropoðhsh twn organwmènwn f sewn wc proc thn isìtroph, ta diagr mmata f shc eðnai poiotik parìmoia me aut pou lamb noume kai gia ta trða sq mata. Autì deðqnei ìti moriak sq mata twn opoðwn oi

57 3.3. FoullerenikoÐ UK me DÐdumouc Mesogìnouc BraqÐonec (TMBC60) 52 a) b) g) Sq ma 3.7: H parametrikopoðhsh eðnai Ðdia me aut tou sust matoc tou sq matoc 3.4b all epiprìsjeta up rqoun kateujuntikèc allhlepidr seic w (0) (m, m) (sta dôo akraða tm mata twn braqiìnwn) twn opoðwn to mètro se sqèsh me th u eðnai a) w (0) (m, m) = 0, 5 b) w (0) (m, m) = 1 kai g) w (0) (m, m) = 2.

58 3.4. FoullerenikoÐ UK me DÐdumouc DendritikoÔc BraqÐonec (TDBC60) 53 summetrðec apoklðnoun apì thn summetrða thc f shc èqoun mikr epðdrash sthn moriak org nwsh aut n, afoô leðpoun apì autèc, akìma kai an oi apì kataskeu c pijanìthtec aut n eðnai uyhlèc. Tèloc, melet same thn allag sthn fasik sumperifor TMBC60 susthm twn me thn eisagwg kateujuntikìthtac stic diatmhmatikèc allhlepidr seic gia ta dôo akraða tm mata tou k je braqðona, ta opoða antistoiqoôn stic mesogìnec mon dec. Jewr same allhlepidr seic lìgo prosanatolismoô me sqetikì mètro w (0) (m, m) w (0) (m, m) /u, wc proc thn isìtroph allhlepðdrash, na metab lletai apì w (0) (m, m) = 0.5 mèqri 2. Ta upologismèna diagr mmata f shc dðnontai sto sq ma 3.7. EÐnai profanèc apì ta diagr mmata ìti h topologða touc parousi zei orismènec xek jarec diaforèc se sqèsh me ta sust mata pou den èqoun kateujuntikèc allhlepidr seic. 'Etsi, ìtan to mètro twn kateujuntik n allhlepidr sewn lìgo prosanatolismoô eðnai uyhlì, h nhmatik f sh exafanðzetai pl rwc apì to di gramma en eunoeðtai h smhktik. Autì deðqnei ìti oi kateujuntikèc allhlepidr seicsta akraða tm matatwn braqiìnwn enisqôoun thn t sh gia mikrofasikì diaqwrismì. Mia axioshmeðwth sunèpeia aut n twn allhlepidr sewn sthn strwmatik org nwsh twn morðwn eðnai h meðwsh tou bajmoô allhlepik luyhc. Autì sumbaðnei epeid ta mìria diadoqik n strwm twn allhlepikalôptontai mèqri sto shmeðo pou ta mesogìna kra touc topojetoôntai to èna dðpla sto llo. To apotèlesma eðnai h aôxhsh tou p qouc tou str matoc kat mia plegmatik kuyelðda. Aut h eðkona thc moriak c org nwshc eðnai se sumfwnða me thn enðsqush thc SmA d2 se sqèsh me thn SmA d1 epeid ètsi up rqei h dunatìthta perissìtera mesogìna na brðskontai to èna dðpla sto llo (sq ma 3.5d). 'Opwc èqoume anafèrei parap nw, h moriak org nwsh mèsa sta str mata thcsma d2 f shc perilamb nei mia dipl stoib da foullerenðwn en to p qoc autoô eðnai pènte èxi moriak tm mata(ìtan eis goume kateujuntik allhlepðdrash). Lamb nontac upìyh ìti to m koc akm c k je tm matoc eðnai perðpou 1nm (h foullerenik di metroc), to upologizìmeno p qoc tou str matoc eðnai 5, 4nm. Peiramatik apotelèsmata 45 skèdashc aktðnwn Q gia aut ta mìria dðnoun p qoc strwm twn perðpou 5, 7nm. 'Etsi loipìn h moriak dieujèthsh dipl c stoib dac ermhneôei ta peiramatik apotelèsmata. K je str ma upodiaireðtai se trða diaforetik c suggèneiac upostr mata pou kajèna apì aut apoteleðtai apì foullerènia, sundetikèc om dec kai mesogìnec om dec (sq ma 3.5d). 3.4 FoullerenikoÐ UK me DÐdumouc DendritikoÔc BraqÐonec (TDBC60) Oi FoullerenikoÐ UK me dðdumouc dendritikoôc braqðonec TDBC60 èqoun thn moriak arqitektonik pou dðnetai sto sq ma 3.8. Sto sq ma 3.8a dðnetai mia qarakthristik qhmik ènwsh TDBC60 deôterhc dendritik c geni c mazð me peiramatik apotelèsmata pou aforoôn thn jermotropik UK sumperifor c autoô. Parajètoume sto sq ma 3.9a thn qhmik dom tou TDBC60 se mia diamìrfwsh ìpou oi dendritikoð braqðonec eðnai ektetamènoi. Epiprìsjeta, diamorf seic me touc dendritikoôc

59 3.4. FoullerenikoÐ UK me DÐdumouc DendritikoÔc BraqÐonec (TDBC60) 54 a) b) Sq ma 3.8: a) QhmikoÐ tôpoi foullerenik n 45 UK TDBC60 b) Jermotropik fasik sumperifor 45 autoô. PortokalÐ: Smhktik f sh (SmA), kìkkino: Nhmatik (N) kai bioletð: isìtroph f sh. H jermokrasða se bajmoôc KelsÐou. a) b) g) Sq ma 3.9: Atomistik eikìna kai anapar stash adropoihmènou TDBC60. H perissìtero èntonh skðash sta tm mata tou (g) (pl n tou foullerenðou) eðnai gia na deðxei thn filoxenða dôo moriak n tmhm twn se mia plegmatik kuyelðda. braqðonec sthn Ðdia pleur eðnai dunatèc. JewroÔme loipìn ìti oi moriakèc diamorf seic mporoôn na omadopoihjoôn se dôo sq mata ìpou oi dendritikoð braqðonec ekteðnontai sthn Ðdia pleur ( FP} sq ma) se antðjetec pleurèc ( EA} sq ma) kai qtðzoume} ta moriak sq mata aut ìpwc deðqnetai sto sq mata 3.9b kai 3.9g. Se autì to epðpedo leptomèreiac thc moriak c dom c, h kôria diafor me tic domèc pou dðnontai sto sq ma 3.3 eðnai ìti oi braqðonec pou sundèontai me to foullerènio katalamb noun megalôtero ìgko}. QrhsimopoioÔme diaforetikoôc tìnouc sta qr mata (sq ma 3.9g) gia na diakrðnoume ta tm mata me diaforetikì perieqìmeno. Me perissìtero èntonh skðash gia ta tm mata tou FP}deÐqnoume ìti kai oi dôo dendritikoð braqðonec moir zontai ton Ðdio q ro kai sunep c h puknìthta touc eðnai dôo forèc aut c gia to kajèna dendritikì braqðona tou EA} sq matoc. Arqik pragmatopoi same upologismoôc perilamb nontac kai èna trðto, to moriakì sq ma L}. Oi upologismoð èdeixan ìti h eisagwg autoô den ephre zei idiaðtera thn fasik sumperifor tou sust matoc kai gia autì paraleðpetai.

60 3.4. FoullerenikoÐ UK me DÐdumouc DendritikoÔc BraqÐonec (TDBC60) 55 PÐnakac 3.1: PÐnakac allhlepidr sewn gia to TDBC60 ìtan ta tm mata diafìrwn sqhm twn katalamb noun thn Ðdia geitonik kuyelðda (se parènjesh). Gia to C60 oi allhlepidr seic metr ntai anex rthta apì to moriakì sq ma. C60 FP} EA} C60 (0) (u/2) (u/4) FP} (u/2) u (u/4) u/4(0) EA} (u/4) u/4(0) u/4(u/8) Sthn parametrikopoðhsh twn allhlepidr sewn tou TDBC60 jewroôme ìti to foullerènio eðnai adiapèrasto kai katalamb nei èna moriakì tm ma. 'Otan k poio apì ta upìloipa tm mata tou sq matoc FP} brejeð se geitonik me to foullerènio kuyelðda tìte emfanðzetai mia elafr c apwstik allhlepðdrash u/2 me u>0, en an brejeð k poio apì ta upìloipa tm mata tou EA} sq matoc se geitonik kuyelðda me to foullerènio emfanðzetai mia u/4 allhlepðdrash. Amèswc parak tw perigr foume tic allhlepidr seic metaxô ìlwn twn tmhm twn pl n tou foullerèniou: Ta moriak tm mata tou FP} sq matoc ìtan brejoôn sthn Ðdia kuyelðda allhlepidroôn me u, en ìtan brejoôn se geitonikèc me u/4. Lìgo thc mikrìterhc kat lhyhc ìgkou ta moriak tm mata tou EA} sq matoc ìtan brejoôn sthn Ðdia kuyelðda allhlepidroôn me u/4, en se geitonik me u/8. Tèloc deqìmaste ìti ta moriak tm mata tou sq matoc EA} kai FP} allhlepidroôn mìnon ìtan brejoôn se Ðdia plegmatik kuyelðda me u/4. Ston pðnaka 3.1 sunoyðzoume thn parap nw parametrikopoðhsh. Na shmei soume ed ìti h parap nw anapar stash thc moriak c dom c den eðnai kat llhlh gia na perigr yei dendritik mìria geni c megalôterhc thc trðthc. Se aut thn perðptwsh, to mègejoc tou foullerenðou eðnai polô mikrì se sqèsh me touc dendritikoôc braqðonec oi opoðoi kalôptoun pl rwc ta foullerènia, kai gia autì apotrèpoun apeujeðac allhlepðdrash metaxô twn foullerenðwn Apotelèsmata thc moriak c jewrðac gia to TDBC60 To di gramma f shc sto sq ma 3.10a upologðsthke gia βu =0, 025, kai antistoiqeð se piec apwstikèc allhlepidr seic. To sôsthma emfanðzei dôo smhktikèc f seic apì tic opoðec aut thc uyhl c pðeshc eðnai mia SmA d1 (apl c stoib dac foullerenðwn) kai aut qamhl c pðeshc eðnai SmA d2 (dipl c

61 3.4. FoullerenikoÐ UK me DÐdumouc DendritikoÔc BraqÐonec (TDBC60) 56 a) b) g) d) Sq ma 3.10: Upologismèna diagr mmata f shc pðeshc san sun rthsh thc apì kataskeu c pijanìthtac tou EA} sq matoc gia ta TDBC60 me diaforetikèc timèc thc paramètrou allhlepðdrashc βu: a) 0,025, b) 0,05 g) 0,1 kai d) 0,2.

62 3.4. FoullerenikoÐ UK me DÐdumouc DendritikoÔc BraqÐonec (TDBC60) 57 stoib dac foullerenðwn). H eust jeia twn f sewn eðnai apotèlesma tou mikrofasikoô diaqwrismoô pou ofeðletai sthn diamerismatopoðhsh tou adropoihmènou morðou. 'Opwc faðnetai apì to sq ma 3.10b aux nontac thn par metro allhlepðdrashc βu =0, 05 lamb noume èna di gramma to opoðo diafèrei apì autì tou sq matoc 3.10a sto ìti h SmA d2 f sh eðnai eustaj c gia mia stenìterh perioq pièsewn kai mia mikr perioq nhmatik c f shc N emfanðzetai an mesa sthn isìtroph kai thn SmA d2 gia uyhlèc apì kataskeu c pijanìthtec thn ektetamènhc antipar llhla EA} kat stashc. Gia βu = 0, 1 parathroôme shmantikèc allagèc sthn fasik sumperifor (sq ma 3.10g) wc proc ta sust mata me mikrìterou megèjouc apwstikèc allhlepidr seic. Pr ton, h dom thc smhktik c f shc qamhl n pièsewn den parousi zei allhlepik luyh: To m koc tou smhktikoô str matoc eðnai perðpou Ðso me to m koc tou adropoihmènou morðou sto moriakì sq ma EA}. DeÔteron, h allhloepikaluptìmenh SmA d1 emfanðzetai sthn perioq twn meg lwn pièsewn kai eðnai astaj c gia meg lec timèc tou moriakoô sq matoc EA}. Tèloc, h perioq thc nhmatik c f shc dieurônetai. Autì deðqnei ìti ìso perissìtero enisqôetai to mètro apwstikìthtac sthn allhlepðdrash an mesa sta dendritik moriak tm mata orìloc tou sunolikoô sq matoc kajðstatai kajoristikìc par gontac gia thn autorg nwsh en exasjeneð h epirro thc moriak c diamerismatopoðhshc, kai sunep c h suneisfor tou mikrofasikoô mhqanismoô sthn moriak org nwsh. Ta parap nw gðnontai perissìtero xek jara apì thn fasik sumperifor enìc sust matoc gia βu =0, 2. 'Opwc eðnai fanerì apì to sq ma 3.10d, to sôsthma den emfanðzei smhktik f sh me allhlepik luyh twn dendritik n tmhm twn. Oi dunatèc metatropèc f shc eðnai I SmA, se qamhlèc apì kataskeu c pijanìthtec antipar llhlwn katast sewn, I N SmA, se uyhlìterec pijanìthtec. H fasik sumperifor eðnai parìmoia me aut rabdìmorfwn morðwn 91 pou allhlepidroôn me sterikèc} ap seic. Genik, sto ìrio mikr c apì kataskeu c pijanìthtac thc antipar llhlhc kat stashc EA}, ìtan P o (EA) 1 gia to TDBC60 montèlo ìtan E (FP) E (EA) gia to TMBC60 ta mìria eðnai praktik kampta, me tic mesogìnec mon dec na ekteðnontai sthn Ðdia kateôjunsh. Se autì to moriakì sq ma omadopoioôntai oi kôriec moriakèc diamorf seic poudðnoun oi foullerenikoð UK sto sq ma 3.1g. Oi upologismoð gia aut ta sust mata den emfanðzoun nhmatik f sh, se sumfwnða me peiramatik apotelèsmata, deðqnontac ìti h endogeneðc polikìthta pou up rqei se aut ta mìria lìgo diamerismatopoðhshc den eunoeð thn nhmatik f sh. H fasik sumperifor gðnetai plousiìterh ìtan lamb netai h suneisfor apì ìla ta dunat sq mata, kurðwc exaitðac tou sqhmatismoô diafìrwn smhtik n f sewn. O polumorfismìc twn f sewn pou parathroôme proèrqetai apì thn diamerismatopoðhsh twn adropoihmènwn morðwn kaj c kai apì thn eukamyða pou prosdðdetai se aut mèso twn diaforetik n dunat n sqhm twn. 'Etsi, o bajmìc allhlepik luyhc metaxô moriak n tmhm twn pou brðskontai se diadoqik str mata prosdiorðzetai apì ton sunduasmì moriak c eukamyðac kai tou mikrofasikoô diaqwrismoô. 'Opwc deðqoun oi upologismoð, oi metab seic f shc metaxô diafìrwn smhktik n f sewn sunodeôontai apì idiaðtera shmantikèc allagèc sthn moriak diamìrfwsh all qwrðc aparaðthta shmantikèc allagèc sto m koc

63 3.5. KwnikoÐ FoullerenikoÐ UK (CSMC60) 58 tou str matoc. 'Oso afor ta sust mata TMBC60 kai TDBC60 h SmA d2 eunoeðtai kurðwc apì thn uyhl pijanìthta twn moriak n sqhm twn FP} en h SmA d1 eðnai perissìtero eustajeðc ìtan ta moriak sq mata EA} èqoun uyhl apì kataskeu c pijanìthta. Tìso hmiaìsokaih llh f sh emfanðzoun ektetamènh allhlepik luyh twn mer n touc kai to p qoc tou str matoc diafèrei kat mia foullerenik di metro. 3.5 KwnikoÐ FoullerenikoÐ UK (CSMC60) Parousi zontai sta sq mata 3.11 kai 3.12 mazð me ta antðstoiqa moriak montèla. Ed jewroôme ìti ta mìria qtðzontai apì to foullerènio (f) kai ta sundedemèna tm mata sto foullerènio (m). Se aut ta sust mata to adropoihmèno foullerenikì mìrio eðnai kampto kai perigr fetai apì èna moriakì sq ma. Aut h anapar stash eðnai se sumfwnða me th qhmik fôsh twn en sewn pou eðnai sundedemènec sto foullerènio. Pènte mesogìna eðnai apeujeðac mèsw tou dimejulopuritðou sundedemènec gôrw apì èna pent gwno tou foullerenðou sqhmatðzontac antðstoiqa èna kwnikì èna puramidikì mìrio (sq mata 3.11b kai 3.11d). H Ôparxh anjrakik n alusðdwn sta kra twn mesogìnwn, en eis gei th moriak eukamyða sthn f sh, den epifèrei apoklðseic apì to kurðarqo sq ma. Ousiastik lìgw thc Ôparxhc thc sundetik c om dac tou dimejulopuritðou, epekteðnetai h koilìthta tou morðou, kai autì to perigr foume sto epðpedo akrðbeiac pou èqoume deqteð me thn eisagwg tou sq matoc thc puramðdac. O moriakìc xonac tou CSMC60 eðnai o xonac summetrðac twn dom n sta sq mata 3.11b kai 3.11d Apotelèsmata thc moriak c jewrðac gia to CSMC60 H anapar stash twn allhlepidr sewn metaxô tmhm twn diaforetik n morðwn pou qrhsimopoioôme sta CSMC60 eðnai parìmoia me aut gia ta TMBC60, me thn parametrikopoðhsh u (0) (f,f) = u (0) (f,m) =, u (0) (m, m) =u>0, u (1) (i, j) =0 en den up rqoun l tm mata. 'Opwc kai prðn to u eðnai to mètro thc apwstikìthtac metaxô diafìrwn tmhm twn. Epiplèon, eis goume tic allhlepidr seic pou perigr fontai sto sq ma Sto sq ma 3.14a kai 3.15a dðnoume to upologismèna di grammata pðeshc san sun rthsh thc antðstrofhc jermokrasðac. Ta diagr mmata eðnai ìmoia wc proc thn akoloujða twn f sewn. EÐnai profanèc ìti, an loga me thn pðesh, ta sôsthmata akoloujoôn dôo diaforetikèc fasikèc sumperiforèc. Se qamhlèc pièseic kai qamhlèc jermokrasðec, metabaðnoun apì thn isìtroph f sh sthn kionik (Col) mèsw miac metatrop c f shc pr thc t xhc. Se uyhlìterec pièseic mia smhktik (SmA), apl c stoib dac, me allhlodieisdôonta malak } mèrh emfanðzetai an mesa sthn isìtroph kai thn kionik (sq mata 3.14b kai 3.14g). H org nwsh twn CSMC60 wc proc thn jèsh se epðpedo k jeto stouc kðonec eðnai anagkastik tetragwnikoô plègmatoc lìgw akrib c twn periorism n pou epib llontai apì thn kðnhsh aut n sto kubikì plègma. O xonac par llhloc stouc kðonec mporeð na eðnai C4 C2 summetrðac kai giautì h didi stath org nwsh wc proc tic jèseic eðnai tetragwnik c summetrðac. Sunep c, eðnai adônaton na

64 3.5. KwnikoÐ FoullerenikoÐ UK (CSMC60) 59 a) b) g) d) Sq ma 3.11: Atomistik eikìna kwnikoô foullerenikoô UK CSMC60 a) me apeujeðac kai b) mèsw thc om dac tou dimejulopuritðou sundedemèna se autì mesogìna. H anapar stash tou antðstoiqou adropoihmènou parag gou tou (a) gðnetai apì mia kwnik dom kai tou (g) apì mia puramidik. Ta di fana tm mata qrhsimopoioôntai gia thn kalôterh optikopoðhsh tou moriakoô sq matoc.

65 3.5. KwnikoÐ FoullerenikoÐ UK (CSMC60) 60 a) b) g) d) Sq ma 3.12: a) QhmikoÐ tôpoi kwnik n foullerenik n UK CSMC60 83, 85 me a) apeujeðac b) mèsw thc om dac tou dimejulopuritðou sundedemèna se autì mesogìna. Jermotropik fasik sumperifor aut n 83, 85 g) kai d) antðstoiqa. Mwb: ual dhc perioq, portokalð: ugrokrustallik perioq (kionik f sh) kai bioletð: isìtroph f sh. H jermokrasða se bajmoôc KelsÐou. a) b) Sq ma 3.13: AnaparÐstantai ta tm mata thc moriak c dom c tou CSMC60 lamb nontac mia tom (epðpedo pou perilamb nei ton moriakì xona) se orismènec sqetikèc dieujet seic. Ta mìria qtðzontai} se kubikì plègma. Gia kalôterh optikopoðhsh ta malak } tm mata diaforetik n morðwn èqoun qr ma gal zio portokalð (an loga me ton prosanatolismì) kai ta foullerènia pr sino. a) 'Otan to èna mìrio brðsketai sthn koilìthta tou llou, me to èna foullerènio sthn eggôtath kuyelðda tou llou foullerèniou, tìte ta mìria allhlepidroôn me sklhrì} dunamikì gia ìlouc touc dunatoôc sqetikoôc moriakoôc prosanatolismoôc ektìc an oi moriakèc touc dieujônseic eðnai sthn Ðdia kateôjunsh kai b) ta mìria pou brðskontai se aut th dieujèthsh allhlepidroôn me apwstikì dunamikì au,ìpou a eðnai o arijmìc twn tmhm twn thc b shc tou morðou.

66 3.5. KwnikoÐ FoullerenikoÐ UK (CSMC60) 61 a) b) g) Sq ma 3.14: a) Di gramma f shc pðesh wc sun rthsh thc antðstrofhc jermokrasðac gia to kwnikì mìrio (sq ma 3.11b). Anapar stash b) thc smhktik c kai g) thc kionik c f shc pou emfanðzoun oi foullerenikoð UK (eikìna 3.11). Me pr sino qrwmatðzoume ta foullerènia, me gal zio ta mesogìna kai tic aleifatikèc alusðdec. xeqwrðsoume an mesa se mia exagwnik kai mia tetragwnik kionik f sh sto plegmatikì montèlo pou qrhsimopoioôme. MolataÔta, ta sônora an mesa sthn kionik kai thn isìtroph f sh kaj c kai an mesa sthn kionik kai sthn smhktik, den perimènoume na ephre zontai shmantik epeid h diafor sthn eleôjerh enèrgeia metaxô miac exagwnik c kionik c kai miac tetragwnik c kionik c eðnai polô mikrìterh se sqèsh me mia kionik (exagwnik tetragwnik ) kai se mia nhmatik smhktik. H endokionik org nwsh eðnai tètoia pou to èna CSMC60 mìrio topojeteðtai sthn koilìthta tou llou kajist ntac sunolik ton kðona mh polikì. Na shmei soume ìti san apotèlesma thc mh Ôparxhc polik n allhlepidr sewn an mesa sta tm mata, h enèrgeia sqetik c olðsjhshc dôo geitonik n kiìnwn den exart tai apì thn sqetik touc polikìthta. Opìte, h sunolik polikìthta twn kionik n f sewn kajorðzetai pl rwc apì entropikoôc par gontec me tic mhpolikèc kionikèc f seic naeðnia perissìtero eustajeðc apìticpolikèc. Sto sq ma 3.16 dðnontai di foroi trìpoi dieujèthshc se mia stoiqei dh kuyelðda thc kionik c f shc me Ðdia eleôjerh enèrgeia (gia mìria sq ma 3.11d). Ektìc apì to di gramma pðeshc - antðstrofhc jermokrasðac gia to puramidikì mìrio upologðsame to di gramma arijmhtik c puknìthtac - antðstrofhc jermokrasðac (sq ma 3.15b). 'Opwc parathroôme sthn perioq twn uyhl n jermokrasi n kai pièsewn sto sôsthma sunup rqoun mia isìtroph me mia

67 3.5. KwnikoÐ FoullerenikoÐ UK (CSMC60) 62 a) b) Sq ma 3.15: a) Di gramma f shc pðeshc wc sun rthsh thc antðstrofhc jermokrasðac tou puramidikoô morðou (sq ma 3.11d) kai b) di gramma f shc thc arijmhtik c puknìthtac wc sun rthsh thc antðstrofhc jermokrasðac. a) b) Sq ma 3.16: Duo dunatèc dieujet seic se epðpedo k jeto stouc xonec twn kiìnwn me thn Ðdia eleôjerh enèrgeia.

68 3.6. SÔnoyh - Sumper smata 63 smhktik f sh kai mia smhktik me mia kionik f sh. Se qamhlìterec puknìthtec faðnetai ìti h isìtroph f sh sunup rqei me thn kionik. 3.6 SÔnoyh - Sumper smata Melet same thn fasik sumperifor kai thn moriak org nwsh gia mia poikilða foullerenik n UK me thn qr sh apl c moriak c jewrðac. Parìlo thn adropoðhsh pou eis goume gia thn anapar stash twn moriak n dom n me diamerismatopoðhsh aut n se èna mikrì arijmì tmhm twn, periorismèna na kinoôntai se kubikì plègma kai na allhlepidroôn me apl diatmhmatik dunamik, h jewrða apodðdei orj (toul qiston poiotik ) basikèc peiramatikèc parathr seic gia ìlec tic kathgorðec twn susthm twn pou melet same. Anapar getai h nhmatik, h smhktik kai h kionik sumperifor kai prosdiorðzontai ta moriak qarakthristik pou eunooôn aut thn sumperifor. O polumorfismìc pou parathreðtai metaxô twn smhktik n f sewn (stic arqitektonikèc domèc me dðdumouc braqðonec) katanoeðtai se sqèsh me to moriakì sq ma kai tic allhlepidr seic. O mhqanismìc tou mikrofasikoô diaqwrismoô eis getai sthn moriak jewrða pou efarmìzoume, mèso sugkekrimènwn diatmhmatik n allhlepidr sewn metaxô twn morðwn. H diaforopoðhsh orismènwn diatmhmatik n allhlepidr sewn, pq me thn eisagwg en gènei kateujuntik n allhlepidr sewn, eunnoeð wc proc thn jermodunamik eust jeia smhktikèc f seic ìpou up rqei diplì upìstrwma foullerenðwn. Shmantikì qarakthristikì twn diafìrwn smhktik n f sewn eðnai h Ôparxh stoib dwn apì foullerènia diaforetikoô p qouc. To teleutaðo kajorðzetai apì tic jermodunamikèc metablhtèc thc pðeshc kai thc jermokrasðac. Genik, eðnai gnwstì ìti parathreðtai o sqhmatismìc stoib dwn apì foullerènia kai se lla 15 upermoriak susthm ta pou perièqoun qhmik tropopoihmèna foullerènia. K je smhktikì str ma upodiaireðtai se stoib dec ìpou taktopoioôntai moriak mèrh Ðdiac qhmik c suggèneiac} (ìmoiac diatmhmatik c allhlepðdrashc) fainìmeno pou sunantiètai kai se UK qamhloô moriakoô b rouc. H moriak org nwsh se smhktikèc f seic sunodeôetai apì allagèc ston sqetikì plhjusmì twn moriak n sqhm twn. 'Etsi, h pijanìthta emf nishc enìc moriakoô sq matoc metab lletai mèsa sthn f sh se sqèsh me thn antðstoiqh tim pou èqei gia to apomonwmèno mìrio kai mporeð na eðnai eðte megalôterh eðte mikrìterh. Orismèna sq mata (kai kat sunèpeia oi moriakèc diamorf seic pou omadopoioôntai se aut ) kajðstantai ligìtero pijan se ìlec tic organwmènec f seic. Eidikìtera, autì isqôei gia moriak sq ma twn opoðwn oi summetrðec diafèroun shmantik apì tic summetrðec twn f sewn. H dieujèthsh twn kwnik n morðwn kajorðzetai apì to sq ma touc. AutodomoÔntai se polikoôc kðonec oi opoðoi autoorgan nontai par llhla metaxô touc sqhmatðzontac kionikoôc UK. Mèsa se

69 3.6. SÔnoyh - Sumper smata 64 k je kðona ta foullerènia diat ssontai se seir (grammik ) kai perib llontai apì ta malak } tm mata tou morðou. Ta apotelèsmata pou lamb noume me thn jewrða, sthn paroôsa apl thc morf, mporeð na apodeiqjoôn qr sima gia thn moriak sqedðash foullerenik n UK.

70 Kef laio 4 Ugrokrustallik sumperifor kai moriak org nwsh se mðgmata kamptwn swmatidðwn diafìrwn sqhm twn Melet me thn fasik sumperifor migm twn apì kampta swmatðdia montelopoihmèna se kubikì plègma. Parousi - zoume ta upologismèna me th jewrða diagr mmata f shc, gia mðgmata rabdìmorfwn swmatidðwn -foullerèniwn, kai diskì-morfwn swmatidðwn -foullerèniwn me sklhrèc} diaswmatidiakèc allhlepidr seic. Sthn sunèqeia, diereunoôme thn epðdrash thc mh filikìthtac metaxô diaforetik n swmatidðwn sthn fasik sumperifor. Tèloc, parousi zoume ta diagr mmata f shc migm twn rabdìmorfwn -diskìmorfwn swmatidðwn. Elègqoume tic problèyeic thc jewrðac gia th melèth aut n twn susthm twn, sugkrðnontac ta apotelèsmata mac me peiramatik kai jewrhtik se parìmoia sust mata. Ta sust mata pou melet me mporoôn na jewrhjoôn wc sust mata anafor c, gia th melèth perissìtero sônjetwn moriak n susthm twn, ìpwc mðgmata adropoihmènwn foullerenik n UK me eleôjera} foullerènia. 4.1 Eisagwg Sto trðto kef laio methn efarmog thc jewrðac twn metatrepìmenwn sqhm twn melet same thn fasik sumperifor monosustatik n susthm twn adropoihmènwn foullerenik n UK me dðdumouc mesogìnouc braqðonec (TMB) kai me dðdumouc dendritikoôc braqðonec (TDB). Parathr same ìti ìtan h apì kataskeu c pijanìthta sq matoc EA} mep o (EA) 1, ìtan h eleôjerh enèrgeia E (FP) E (EA) ( thc E (L)) gia ta TMB, tìte mporoôme na jewr soume ìti kuriarqeð èna sq ma me touc braqðonec na koitoôn sthn Ðdia kateôjunsh. Se aut n thn prosèggish omadopoioôntai kai oi kôriec diamorf seic foullerenik n UK pou perièqoun ènan braqðona (qwrðc katan gkhn autìc na èqei dendritik topologða uyhl c geni c). Ex llou, gia arket meg lec timèc tou βu thc t xhc 65

71 4.1. Eisagwg (gia ta jewrhtik apotelèsmata) to mìrio den eðnai apl kampto} (enìc sq matoc) all kai ta tm mat tou teðnoun na gðnoun adiapèrasta afoô oi diamoriakèc (diatmhmatikèc) allhlepidr seic eðnai thc morf c twn sterik n} ap sewn. To ìrio gia to opoðo mil same parap nw ( kampta mìria pou allhlepidroôn me sterikèc ap seic) mporeð na qrhsimopoihjeð wc shmeðo anafor c gia thn exètash thc fasik c sumperifor c duadik n susthm twn foullerenðwn me foullerenikoôc UK. 'Etsi loipìn, se mia pr th pragm teush, diereunoôme thn epðptwsh pou èqei sth fasik sumperifor h an mixh mh tropopoihmènwn ( eleôjerwn} ) foullerenðwn se mia seir apì kamptec moriakèc domèc. Apì di forec ereunhtikèc om dec dðnetai idiaðtero b roc sthn melèth thc di taxhc foullerenðwn se stoib dec orismènou p qouc. Autì to z thma proseggðzetai apì di forec pleurèc. Gia par deigma 92 èqei exetasteð h autoorg nwsh mikkulðwn foullerenðwn se biomoriak ulik kai sthn sunèqeia apì thn epexergasða aut n o sqhmatismìc tetrapl c stoib dac foullerenðwn. Apì thn llh meri gðnontai prosp jeiec 26, 27 gia ton sqhmatismì lept n umenðwn apì tropopoihmèna foullerènia. Ed proteðnoume ènan trìpo pou epitrèpei na kateujônoume thn autoorg nwsh foullerenðwn (wc sustatik ) se duadik sust mata. Oi upologismoð pragmatopoioôntai se èna kubikì plègma jewr ntac ìti ta foullerènia anaparðstantai me mia kubik dom (sfairik ston suneq q ro) pou katalamb nei mia plegmatik kuyelðda. Pio sugkekrimèna, arqik diereunoôme thn jermodunamik isorropða UK f sewn se mðgmata foullerèniwn rabdìmorfwn swmatidðwn kai touc par gontec pou uposthrðzoun thn parousða foullerèniwn se autèc kai ephre zoun thn eust jeia aut n (enìthta 4.3); m lista, exet zoume kai ta gewmetrik qarakthristik dom n pou sqhmatðzontai apì thn autoorg nwsh foullerenðwn (pq stoib dec foullerenðwn se full deic f seic). Autì pou mac endiafèrei eðnai o susqetismìc thc ekd lwshc thc UK sumperifor c se sqèsh me thn amfifilikìthta twn foullerenðwn, thn sqetik touc sôstash sto mðgma, thn pðesh genikìtera me tic jermodunamikèc metablhtèc tou sust matoc, qwrðc na stoqeôoume se mia diexodik melèth se kajautì sust mata kubik n rabdìmorfwn swmatidðwn. H fasik sumperifor rabdìmorfwn all kai sfairik n morðwn èqei melethjeð leptomer c peiramatik (gia qamhloô all kai uyhloô moriakoô b rouc en seic) me jewrða kai me moriakèc prosomoi seic H perioq twn kolloeid n prosfèrei arket paradeðgmata 93, duadik n susthm twn rabdìmorfwn kai sfairik n kolloeid n (Ioi kai pq polusturènio antðstoiqa) ta opoða mporoôn na droôn wc sklhr } swmatðdia k tw apì orismènec sunj kec. 'Opwc èqoume dei, nèec upermoriakèc domèc me kajorismènh topologða twn foullerèniwn eðnai dunatìn na epiteuqjoôn se monosustatik sust mata apì mh sumbatik UK mìria ìpwc ta kwnik kai ta pu- 83, 85 ramidik. Lìg wthcautoorg nwshc aut n se kðonec ta foullerènia diat ssontai se grammikèc alusðdec} se kajorismènh apìstash metaxô touc ìpwc prokôptei apì thn topologða thc kionik c f shc. 'Omwc kionik mesomorfik sumperifor èqei parathrhjeð 109 kai apì thn autoorg nwsh se kðonec me thn an mixh dôo mh mesomorfik n ulik n: dhlad mðgma enìc foullerenikoô parag gou

72 4.2. Moriak montelopoðhsh duadik n susthm twn 67 Sq ma 4.1: Anapar stash diskìmorfou kai rabdìmorfou swmatidðou sto kubikì plègma. (dôo diskìmorfa mìria eðnai sundedemèna sto foullerènio) me diskìmorfo mìrio. UposthrÐzetai ìti ta foullerènia diat ssontai se didi stato exagwnikì plègma an mesa stouc kðonec pou sqhmatðzontai apì ta diskìmorfa mìria. To pedðo èreunac gia thn exètash uyhl organwmènwn f sewn (ìpwc h kionik ) eðnai anoiqtì; h jewrhtik prìbleyh thc fasik c sumperifor c duadik n susthm twn foullerenðwn kai diskìmorfwn swmatidðwn endeðknutai san pr to b ma gia thn melèth perissìtero sônjetwn susthm twn (enìthta 4.4). Gia par deigma, san pr to b ma exet zetai h epðptwsh pou èqei h prosj kh mikr n posot twn foullerenðwn sthn autoorg nwsh t gmatoc diskìmorfwn swmatidðwn. Ex llou, ìpwc kai sthn perðptwsh twn rabdìmorfwn swmatidðwn, èqei gðnei emperistatwmènh melèth susthm twn diskìmorfwn swmatidðwn peiramatik 110, 111 me jewrða 112 kai me moriakèc 113 prosomoi seic. Skopìc mac ed eðnai na katano soume touc lìgouc pou enisqôoun diatar ssoun thn jermodunamik eust jeia organwmènwn f sewn se mðgmata foullerenðwn - diskìmorfwn swmatidðwn sto epðpedo akrðbeiac thc jewrðac se kubikì plègma. Tèloc, sthn enìthta 4.5 melet me mðgmatadiskìmorfwn kai rabdìmorfwn swmatidðwn, me gewmetrik qarakthristik koin n jermotropik n UK kolloeid n pou ekdhl noun UK sumperifor. K ti tètoio gðnetai gia lìgouc plhrìthtac se sqèsh me ta mðgmata pou anaferj kame stic parap nw par grafouc qwrðc naeiserqìmaste sthn diadikasða diereônhshc diafìrwn parag ntwn pou mporeð pq na eujônontai gia thn emf nish ìqi diaxonik c nhmatik c f shc. To moriakì montèlo gia duadik sust mata pou anaferj kame parap nw perigr fetai sthn enìthta 4.2. Stic epìmenec enìthtec suzht me ta apotelèsmata gia mðgmata rabdìmorfwn swmatidðwn - foullerenðwn, diskìmorfwn swmatidðwn - foullerenðwn kai rabdìmorfwn - diskìmorfwn swmatidðwn. To kef laio oloklhr netai sunoyðzontac ta apotelèsmata kai me thn par jesh sumperasm twn. 4.2 Moriak montelopoðhsh duadik n susthm twn Se aut thn enìthta esti zoume sthn melèth susthm twn apì dôo eðdh swmatidðwn, apì ta opoða to èna kai ta dôo parousi zoun anisotropða sq matoc. K je sustatikì tou mðgmatoc upodiaireðtai se tm mata me kajorismènec diatmhmatikèc allhlepidr seic. 'Opwc èqoume anafèrei to foullerènio sqhmatðzetai apì èna tm ma (enìthta 3.2). H diamoriak allhlepðdrash upologðzetai wc upèrjesh twn allhlepidr sewn metaxô ìlwn twn dunat n zeugari n tmhm twn ìlwn twn eid n twn swmatidðwn.

73 4.2. Moriak montelopoðhsh duadik n susthm twn 68 Gia thn aploôsteush twn upologism n thc eleôjerhc enèrgeiac kai twn sunart sewn autosunèpeiac jewroôme ìti ta mìria kinoôntai se èna kubikì plègma ìpou to foullerènio katalamb nei mia monadiaða plegmatik kuyelðda. Sunep c, ta anisìtropa adropoihmèna mìria qtðzontai} qrhsimopoi ntac domik tm mata megèjouc Ðsou me thn monadiaða kuyelðda tou plègmatoc. Sta sust mata pou melet me eis goume dôo eðdh anisìtropwn swmatidðwn (sq ma 4.1) ta rabdìmorfa kai ta diskìmorfa (tetragwnik ) swmatðdia. To mètro thc anisotropðac twn rabdìmorfwn swmatidðwn eðnai Ðso me ton arijmì twn tmhm twn pou apoteloôntai, kai sumbolðzetai me L, en gia ta diskìmorfa swmatðdia eðnai Ðso me ton arijmì twn tmhm twn pou apoteleðtai mia pleur tou tetrag nou, kai sumbolðzetai me w. AfoÔ kataskeu soume ta adropoihmèna mìria, to diamoriakì dunamikì allhlepðdrashc twn morðwn I,J eðdouc k, t antðstoiqa orðzetai sto kubikì plègma wc ex c U Ik,J t = { ) ( ) u (0) (i Ik,i Jt ) δ (r iik,i Jt + u (1) (i Ik,i Jt ) δ r iik,i Jt 1 (δ k,t 1) i Ik,i Jt ( +u (1) (i Ik,i Jt ) δ r iik,i Jt ) 2 (δ k,t 1) ( +u (1) (i Ik,i Jt ) δ r iik,i Jt ) } 3 (δ k,t 1) (4.1) ìpou r iik,i Jt eðnai h sqetik apìstash twn tmhm twn i Ik,i It,u (0) (i Ik,i It ) eðnai to mètro thc apwstik c allhlepðdrashc gia èna zeug ri tmhm twn pou katalamb noun thn Ðdia kuyelðda (h dèlta ) sun rthsh δ (r iik,i Jt qrhsimopoieðtai gia na upodhl sei autì). O ìroc u (1) (i Ik,i It ) metr thn allhlepðdrash ìtan ta tm mata diaforetik n morðwn brðskontai se plegmatik jèsh pr tou deôterou trðtou geðtona. Sta sust mata pou melet me ta swmatðdia apoteloôntai apì tm mata Ðdiac morf c allhlepðdrashc. Gia ìlec tic peript seic pou exet zoume o ìroc u (0) = en o u (1) = u>0. Ousiastik me thn morf tou dunamikoô thc prohgoômenhc exðswshc enisqôoume thn filikìthta metaxô ìmoiwn morðwn; alli c èmmesa eis goume mia elktik dunamik enèrgeia} allhlepðdrashc metaxô ìmoiwn morðwn ìtan aut brðskontai sto mðgma afoô lìgw thc Ôparxhc tou ìrou δ kt 1 mìno ta anìmoia mìria dèqontai pwsh ìtan tm mat touc brejoôn se jèseic kontin n geitìnwn. Gia ton upologismì thc fasik c sumperifor c qrhsimopoioôme thn eleôjerh enèrgeia pou dðnetai apì thn sqèsh ex.(2.99). Eidikìtera, an loga me thn summetrða twn f sewn pou melet me qrhsimopoioôme tic ekfr seic gia thn eleôjerh enèrgeia pou dðnontai stic upoenìthtec mèqri Gia ton upologismì twn sunìrwn sunôparxhc metaxô f sewn diaforetik c sugkèntrwshc epilôoume tic sunj kec mhqanik c kai qhmik c isorropðac ex.( ).

74 4.3. MÐgmata foullerèniwn kai rabdìmorfwn swmatidðwn MÐgmata foullerèniwn kai rabdìmorfwn swmatidðwn Sto sq ma 4.2a parousi zoume to di gramma f shc (pðesh wc sun rthsh thc sqetik c sugkèntrwshc twn r bdwn) gia èna duadikì mðgma foullerenðwn me r bdouc anisotropðac L =5. H parametrikopoðhsh twn diatmhmatik n allhlepidr sewn gðnetai wc ex c: jewroôme ìti ta tm mata apì ta opoða sqhmatðzontai ta swmatðdia eðnai adiapèrasta kai jètoume u (0) =, en gia tic allhlepidr seic metaxô kontin n geitìnwn u (1) =0. Sunep c ta swmatðdia, anex rthta apìto eðdoc touc, allhlepidroôn me sklhrì} apwstikì dunamikì. Sthn perioq twn qamhl n pièsewn βp < 0.5 to sôsthma anamignôetai pl rwc gia ìlec tic sust seic. Gia endi mesec pièseic 0.5 <βp <1.5 kai gia mikr sugkèntrwsh r bdwn (x R ) to sôsthma brðsketai sthn isìtroph f sh (I) en gia meg lec x R se mia monoaxonik nhmatik (N). Sthn Ðdia perioq pièsewn kai gia endi mesh sugkèntrwsh twn r bdwn parathroôme sunôparxh isìtrophc - nhmatik c f shc, en se sqetik uyhl sugkèntwsh x R to sôsthma metabaðnei apì thn isìtroph se mia ploôsia se r bdouc monoaxonik nhmatik mèsw miac metatrop c f shc pr thc t xhc (oi kampôlec sunôparxhc ekfulðzontai se mia gramm ) (sq ma 4.2a). Sqetik jewrhtik apotelèsmata (wc proc thn topologða twn diagramm twn) gia mðgma kôbwn kai r bdwn (x R > 0.5), èqoun eðdh lhfjeð apì ton Alben 114 apì to Se akìma uyhlìterec pièseic kai gia x R > 0.7 to sôsthma metabaðnei apì thn nhmatik se mia smhktik f sh (Sm). H parousða tou triploô shmeðou me suntetagmènec (βp,x R )=(2.30; 0.32) upodhl nei thn sunôparxh diafìrwn f sewn: thc isìtrophc me th nhmatik, thc isìtrophc me thn smhktik kai thc nhmatik c me thn smhktik. Gia pièseic megalôterec tou triploô shmeðou ekdhl netai diaqwrismìc f shc metaxô isìtrophc - smhktik c f shc. H smhktik f sh apoteleðtai apì diadoqik str mata r bdwn pou enall ssontai me upìstrwma foullerèniwn orismènou p qouc. Stic jermodunamik eustajeðc smhktikèc f seic oi r bdoi, kat mèso ìro, eðnai dieujethmènec par llhla metaxô touc (sqhmatðzontac èna didi stato str ma isìtropou ugroô) kai k jeta sta upostr mata twn foullerèniwn. AxÐzei na shmei soume ìti parathr same metastajeðc smhktikèc f seic me ton kateujunt twn r bdwn par llhlo stic stoib dec twn foullerenðwn. 'Opwc mporeð na diapist sei kaneðc apì to sq ma 4.2a h kampôlh pou afor thn smhktik f sh sunodeôetai apì asuneqeðc metabolèc. To p qoc tou str matoc thc smhktik c f shc all zei tim, mei netai asuneq c, kaj c aux netai h pðesh kai h sôstash twn r bdwn. H kampôlh den perigr fetai apì mia omal gramm epeid h par metroc tou p qouc tou str matoc pou upeisèrqetai stouc upologismoôc autosunepoôc pedðou ek twn pragm twn lamb nei diakritèc timèc (pollapl sia twn diast sewn twn foullerenik n tmhm twn). H meðwsh tou p qouc thc smhktik c f shc sunodeôetai me meðwsh tou p qouc tou upostr matoc twn foullerenðwn. Dunatìthta dieôrunshc tou p qouc twn strwm twn epishmaðnetai kai apì touc Koda et. al. 115 exet zontac èna mðgma tèleia prosanatolismènwn sfairokulðndrwn me sfaðrec. KatalabaÐnoume loipìn, ìti eðnai dunatìc o èlegqoc tou p qouc tou upostr matoc twn foullerenðwn auxomei nontac thn pðesh kai thn sôstash tou enìc sustatikoô se sqèsh me to llo.

75 4.3. MÐgmata foullerèniwn kai rabdìmorfwn swmatidðwn 70 a) b) g) d) Sq ma 4.2: Upologismèna diagr mmata f shc pðeshc BP wc sun rthsh thc sugkèntrwshc twn r bdwn x R se mðgmata foullerenðwn rabdìmorfwn swmatidðwn, gia di fora m kh r bdou kai paramètrou allhlepðdrashc metaxô kontin n geitìnwn anìmoiwn morðwn: (a) gia m koc r bdou L =5kai βu (1) =0, (b) gia L =5kai βu (1) =0, 02, (g) gia L =9kai βu (1) =0kai (d) gia L =9kai βu (1) =0, 02. Ta sômbola I, N, Sm, kai DF} upodhl noun isìtroph, monoaxonik nhmatik, smhktik f sh kai difasik perioq, antðstoiqa. Se ìlec tic peript seic deqìmaste u (0) =.

76 4.3. MÐgmata foullerèniwn kai rabdìmorfwn swmatidðwn 71 H nhmatik f sh enisqôetai wc proc thn smhktik me thn aôxhsh thc sôstashc twn foullerèniwn mèqri to triplì shmeðo; faðnetai ìti h eisagwg foullerenðwn sumb llei sthn diat raxh thc jermodunamik c eust jeiac thc smhktik c f shc. Aut h t sh anafèretai kai apì touc Cinnachi et. al. 116 ìpou melet ne mðgmata sfairokulðndrwn me sklhrèc} sfaðrec. Se melètec 117 pou èqoun prohghjeð se mðgmata sfair n kai par llhla prosanatolismènwn sfairokulðndrwn (me anhgmèno m koc toul qiston 10 forèc megalôtero thc sfairik c diamètrou) sthn prosèggish tou deôterou suntelest virial, me b sh epiqeirhmatologða susqètishc thc metatrop c f shc me thn sunolik puknìthtac kat lhyhc tou sust matoc, uposthrðzetai ìti prosjètontac polô mikrèc sfaðrec enisqôetai h eust jeia thc smhktik c f shc, se sqèsh me thn nhmatik. Aux nontac thn anisotropða twn r bdwn L =9 (jètontac thn parametrikopoðhsh twn diatmhmatik n allhlepidr sewn Ðdia me aut pou perigr yame sto prohgoômeno sôsthma) parathroôme orismènec allagèc sto di gramma f shc sthn perioq sunôparxhc isìtrophc - nhmatik c f shc. H klðsh tou kl dou sunôparxhc thc nhmatik c f shc (me thn isìtroph) all zei prìshmo. Sumpièzontac to mðgma apì thn isìtroph f sh (gia x R 0, 18) autì emfanðzei arqik mia sunôparxh isìtrophc me nhmatik, akoloujeð mia met bash se nhmatik f sh kai tèloc to sôsthma eisèrqetai ek nèou se mia nèa sunôparxh thc isìtrophc me th nhmatik k tw apì thn pðesh tou triploô shmeðou (βp,x R )=(1.15; 0.26). To teleutaðo metatopðzetai se akìma qamhlìterec sugkentr seic r bdwn. Gia pièseic βp < 1 h nhmatik eðnai eustajeðc akìma kai se idiaðtera qamhlèc sugkentr seic x R. H smhktik f sh kajðstatai jermodunamik astajeðc se elafr c uyhlìterec pièseic me thn aôxhsh thc sqetik c sôstashc twn foullerèniwn, fainìmeno pou to parathroôme kai sto mðgma ìpou L =5. Genik h topologða twn dôo diagramm twn eðnai parìmoia (sq ma 4.2a kai 4.2g). Parak tw exet zoume thn epðdrash malak n apwstik n allhlepidr sewn metaxô swmatidðwn diaforetikoô eðdouc ìpwc perigr fthke sthn enìthta thc montelopoðhshc. H parametrikopoðhsh twn diatmhmatik n allhlepidr sewn gðnetai wc ex c: u (0) =, kai gia tic allhlepidr seic kontin n geitìnwn metaxô anìmoiwn swmatidðwn u (1) = u>0. 'Etsi, sta mðgmata pou melet me, h filikìthta metaxô ìmoiwn swmatidðwn enisqôetai kai èmmesa eis getai mia elktik allhlepðdrash} foullerèniou - foullerèniou kai r bdou - r bdou. Autì epèfere tìso poiotikèc ìso kai posotikèc allagèc sta diagr mmata f shc. 'Opwc mporoôme na diapist soume apì ta sq mata 4.2b kai 4.2d (gia mðgmata foullerèniou - r bdou me anisotropða L =5kai L =9antÐstoiqa) dieurônetai h perioq diaqwrismoô f shc. Organwmènec f seic (smhktik kai nhmatik ) gðnontai jermodunamik eustajeðc se uyhlìterec sust seic r bdwn se sqèsh me ta antðstoiqa sust mata pou melet same prohgoumènwc se ìlo to eôroc twn antðstoiqwn pièsewn (sugkrðnoume ta diagr mmata sta sq mata 4.2a kai 4.2gme ta antðstoiqa sta sq mata 4.2b kai 4.2d). EÐnai endiafèron ìti h eisagwg epiprìsjetwn apwstik n allhlepidr sewn metaxô kontin n geitìnwn an mesa se anìmoia swmatðdia eunnoeð ton diaqwrismì

77 4.4. MÐgmata foullerèniwn kai diskìmorfwn swmatidðwn 72 se mia kajar isìtroph f sh kai mia smhktik pou periorðzetai se sugkentr seic x R > 0, 75 se pièseic βp > 0, 90 (sq ma 4.2d). Hisìtroph f sh sunup rqei me mia nhmatik ploôsia se r bdouc (sq ma 4.2b kai 4.2d). H dieôrunsh tou q smatoc an mesa stic kampôlec sunôparxhc thc nhmatik c kai thc isìtrophc f shc mporeð na katanohjeð jewr ntac ton sunagwnismì an mesa sthn eleôjerh enèrgeia Helmholtz an mixhc kai tou mèrouc pou perilamb nei diaswmatidiakèc allhlepidr seic. To mèroc thc eleôjerhc enèrgeiac diaswmatidiak n allhlepidr sewn aux netai exaitðac tou epiprìsjetou apokleiìmenou ìgkou lìgw twn apwstik n allhlepidr sewn foullerenðwn kai r bdwn. Kaj c aux netai to mètro twn mal kwn apwstik n allhlepidr sewn twn anìmoiwn morðwn to mèroc thc eleôjerhc enèrgeiac Helmholtz twn diaswmatidik n allhlepidr sewn megal nei ìtan ta sust mata eðnai anamigmèna (profan c perissìtero apì ìti sthn perðptwsh q ric thn epiplèon apwstikìthta) kai kajðstatai akìma megalôtero se sqèsh me autì thc eleôjerhc enèrgeiac diaqwrismoô. Telik to teleutaðo epikrateð lìgw thc meðwshc pou prokaleðtai sthn eleôjerh enèrgeia an mixhc. To sôsthma (sq ma 4.2b) perièqei èna triplì shmeðo (βp,x R )=(1.67; 0.73) se uyhlìterec sust seic twn r bdwn se sqèsh me ta mðgmata (sq ma 4.2a) ìpou den up rqei allhlepðdrash metaxô kontin n geitìnwn. Se pièseic p nw apì to triplì shmeðo kai sugkentr seic x R > 0, 73 sumbaðnei sunôparxh f shc miac isìtrophc me mia smhktik. 'Opwc kai stic prohgoômenec peript seic sthn smhktik f sh emfanðzetai foullerenikì upìstrwma. Sthn smhktik f sh emfanðzetai sqhmatismìc upostrwm twn foullerèniwn se k je str ma. Ja prèpei na shmei soume ed, ìti to mègisto upologizìmeno p qoc twn strwm twn twn jermodunamik eustaj n smhktik n f sewn, all kai to mègisto p qoc tou upostr matoc twn foullerèniwn, mei netai me thn eisagwg twn apwstik n allhlepidr sewn metaxô anìmoiwn morðwn, se sqèsh me autì pou parathreðtai sto antðstoiqo sôsthma ìpou up rqei apokleistik sklhrì} apwstikì diaswmatidiakì dunamikì allhlepðdrashc. 4.4 MÐgmata foullerèniwn kai diskìmorfwn swmatidðwn Se aut thn enìthta melet me mðgmata foullerenðwn diskìmorfwn swmatidðwn. Exet zoume thn sqetik jermodunamik eust jeia thc isìtrophc, thc nhmatik c, thc smhktik c kai thc kionik f shc. Sto sq ma 4.3a parousi zoume to di gramma f shc gia èna mðgma foullerenðwn kai diskìmorfwn swmatidðwn me anisotropða w =5. H diatmhmatik allhlepðdrash twn swmatidðwn eðnai thc morf c sterik c} pwshc, dhlad u (0) =, kai u (1) =0. ParathroÔme treðc jermodunamik eustajeðc f seic: mia isìtroph, mia diskotik nhmatik (N) kai mia tetragwnik kionik (Col). Se qamhlèc pièseic βp < 0, 1 ta swmatðdia anamignôontai pl rwc se ìlec tic sqetikèc sugkentr seic twn diskìmorfwn swmatidðwn x P. Prosjètontac foullerènia sthn perioq pièsewn 0, 1 <βp <0, 2 h diskotik nhmatik f sh paramènei eustajeðc mèqri thn perioq qamhl n sugkentr sewn x P

78 4.4. MÐgmata foullerèniwn kai diskìmorfwn swmatidðwn 73 a) b) g) Sq ma 4.3: Upologismèna diagr mmata f shc foullerenðwn - diskìmorfwn swmatidðwn pðeshc βp san sun rthsh thc sqetik c sugkèntrwshc twn diskìmorfwn swmatidðwn x P, gia di fora w kai paramètrou allhlepðdrashc metaxô kontin n geitìnwn anìmoiwn morðwn: (a) gia w =5kai βu (1) =0, (b) gia w =5kai βu (1) =0, 01, (g) gia w =7kai βu (1) =0. Ta sômbola I, N, Col kai DF} upodhl noun isìtroph, diskotik nhmatik, tetragwnik kionik f sh kai difasik perioq, antðstoiqa. Se ìlec tic peript seic deqìmaste u (0) =. 'Enjeta: antiproswpeutik upologismèna diagr mmata thc puknìthtac pijanìthtac eôreshc twn tmhm twn twn diskìmorfwn swmatidðwn (gal zio) kai twn foullerenðwn (bioletð) se tèsseric g eitonikèc kuyelðdec thc kionik c f shc (se epðpedo k jeto ston xona thc kionik c f shc). ìpou sumbaðnei diaqwrismìc f shc an mesa se aut kai sthn isìtroph. ParathroÔme thn Ðdia topologða ìpwc sta mðgmata foullerenðou - r bdwn ìson afor thn perioq qamhl n pièsewn ìpou emfanðzetai mia fardi kai se meg lo eôroc sugkentr sewn x P lwrðda nhmatik c f shc. BrÐskoume ìti kajar sust mata diskìmorfwn swmatidðwn akoloujoôn diadoqik mia metatrop f shc apì thn isìtroph sth nhmatik kai sthn sunèqeia apì th nhmatik sthn kionik, akoloujða f sewn pou èqei parathrhjeð kai me thn ektèlesh moriak n prosomoi sewn gia anisìtropec kommènec} sfaðrec apì ton Veerman et. al. 102 ston suneq q ro. M lista, peiramatik apotelèsmata 110 diskìmorfwn kolloeid n (an kai me sqetik diaspor sta megèjh touc) ekdhl noun aut thn UK sumperifor. H efarmog thc jewrðac mac sto kubikì plègma uperektim thn eust jeia thc nhmatik c f shc gia mikrèc timèc tou w.

79 4.4. MÐgmata foullerèniwn kai diskìmorfwn swmatidðwn 74 Se sugkèntrwsh diskìmorfwn morðwn x P =0, 17 kai pðesh βp =0, 37 up rqei èna triplì shmeðo. BrÐsketai ekeð pou h kampôlh sunôparxhc thc kionik c f shc me thn nhmatik tèmnetai me thn gramm sunôparxhc thc nhmatik c me thn isìtroph. Sunep c, sto triplì shmeðo miaisìtroph f sh sunup rqei me mia diskotik nhmatik kai me mia kionik f sh. Se uyhlèc pièseic (p nw apì to triplì shmeðo) sumbaðnei diaqwrismìc an mesa se mia isìtroph kai mia tetragwnik kionik. H kampôlh diaqwrismoô f shc isìtrophc nhmatik c ekfulðzetai se mia gramm met bashc f shc se sqetik meg lec sust seic x P. Sto di gramma f shc (sq ma 4.2a) up rqoun wc ènjeta antiproswpeutik upologismèna diagr mmata thc puknìthtac pijanìthtac eôreshc twn tmhm twn twn diskìmorfwn swmatidðwn kai twn foullerenðwn se tèsseric geitonikèc kuyelðdec thc kionik c f shc (se epðpedo k jeto ston xona thc kionik c f shc). Apì aut mporoôme na ex goume plhroforðec gia thn autoorg nwsh twn swmatidðwn sto mðgma. Sthn kionik f sh oi dðskoi organ nontai o ènac p nw ston llo se kðonec oi opoðoi me thn seir touc stoiqðzontai par llhla se èna didi stato tetragwnikì plègma af nontac an mes touc stenoôc diadrìmouc; autoð katalamb nontai apì mikrèc posìthtec foullerenðwn en kurðwc ta foullerènia organ nontai se st lec, pou topojetoôntai sta shmeða diastaôrwshc twn kiìnwn, kai me prosanatolismì k jeto sto didi stato tetragwnikì plègma thc kionik c f shc. AxÐzei na shmei soume ìti xekin ntac apì pðesh gia par deigma βp 0, 5 kai aux nontac thn sugkèntrwsh x P oi diast seic (d x,d y ) thc plegmatik c kuyelðdac pou antistoiqeð sthn eustajèsterh kionik f sh surrik nontai elafr c; ètsi, gia x P 0, 4 eðnai (d x,d y ) = (7, 7), gia x P 0, 6 eðnai (d x,d y )=(13, 13), kai gia x P > 0, 80 eðnai (d x,d y )=(6, 6). Apì to di gramma (sq ma 4.2a), parathroôme ìti èstw kai mikrèc posìthtec foullerenðwn enisqôoun thn jermodunamik eust jeia thc nhmatik c f shc wc proc thn kionik ; èqontac upìyh kai an logh t sh pou sumbaðnei se mðgmata foullerenðwn - r bdwn sumperaðnoume ìti h prosj kh foullerenðwn diatar ssei thn eust jeia uyhl organwmènwn UK f sewn. Sthn sq ma 4.3gparousi zoume to di gramma gia enìc mðgmatoc foullerenðwn kai dðskwn anisotropðac w =7, me parametrikopoðhsh twn allhlepidr sewn ìpwc sto amèswc prohgoômeno sôsthma. H topologða twn diagram twn eðnai parìmoia. EntoÔtoic, se aut thn perðptwsh h perioq diaqwrismoô f shc ( DF} ) isìtrophc kai nhmatik c ftwq c se diskìmorfa swmatðdia eðnai perissìtero sten. Sto sq ma 4.3b parousi zoume to di gramma f shc me anisotropða w =5kai foullerèniwn. H parametrikopoðhsh twn diatmhmatik n allhlepidr sewn èqei wc ex c: u (0) = kai u (1) = u>0. Apì thn sôgkrish twn diagramm twn sta sq mata 4.3a kai 4.3b diapist noume ìti eðnai parìmoia. 'Omwc, dieurônetai h perioq diaqwrismoô f shc thc kionik c me thn isìtroph f sh. Na shmei soume ìti h klðsh thc kampôlhc thc nhmatik c all zei prìshmo, fainìmeno pou to parathroôme kai sta mðgmata r bdwn kai foullerèniwn. H surrðknwsh twn diast sewn (d x,d y ) thc eustajèsterhc kionik c f shc

80 4.5. MÐgmata rabdìmorfwn kai diskìmorfwn swmatidðwn 75 a) b) Sq ma 4.4: Upologismèna diagr mmata f shc rabdìmorfwn diskìmorfwn swmatidðwn pðeshc βp wc sun rthsh thc sugkèntrwshc twn diskìmorfwn swmatidðwn x P, gia di fora L kai w : (a) gia L =7 kai w =5, kai (b) gia L =11kai w =5. Ta sômbola I, N P,N R,Colkai DF} upodhl noun isìtroph, diskotik nhmatik, kalamitik nhmatik, tetragwnik kionik f sh kai difasik perioq, antðstoiqa. Oi diatmhmatikèc allhlepidr seiceðnai thc morf c sterik c} pwshc. 'Enjeta: antiproswpeutik upologismèna diagr mmata thc puknìthtac pijanìthtac eôreshc twn tmhm twn twn diskìmorfwn swmatidðwn (gal zio) kai twn r bdwn (bioletð) se tèsseric geitonikèc kuyelðdec thc kionik c f shc (se epðpedo k jeto ston xona thc kionik c f shc). aux nontac thn sugkèntrwsh x P autì to sôsthma. se orismènh pðesh gia par deigma βp 0, 4 parathreðtai kai se 4.5 MÐgmata rabdìmorfwn kai diskìmorfwn swmatidðwn Sta sust mata pou èqoume melet sei mèqri t ra, to foullerènio paðzei polô shmantikì rìlo ston sqhmatismì organwmènwn f sewn kai sthn jermodunamik isorropða metaxô aut n. Se aut thn enìthta epekteðnoume th melèth mac exet zontac duadik mðgmata pou apoteloôntai apì swmatðdia ta opoða se kajarèc f seic emfanðzoun UK sumperifor. Tètoia mðgmata, idiaðtera diskìmorfwn swmatidðwn èqoun melethjeð 118, 119 prìsfata se sqèsh me ton sqhmatismì diaxonik c nhmatik c f shc. Ed, den skopeôoume se mia ektetamènh melèth tètoiwn susthm twn, all ston èlegqo thc jewrðac mac wc proc thn probleyimìthta thc fasik c touc sumperifor c. Sto sq ma 4.4 parousi zoume to jewrhtik upologismèno di gramma f shc enìc mðgmatoc diskìmorfwn kai rabdìmorfwn swmatidðwn me anisotropða w =5kai L =7antÐstoiqa. Oi diatmhmatikèc allhlepidr seic jewroôme ìti eðnai thc morf c sterik c} pwshc. Exet zoume wc proc thn jermodunamik eust jeia thn ekd lwsh isìtrophc, nhmatik c, smhktik c kai kionik c f shc. Gia thn isìtroph f sh parathroôme ìti eðnai dunat h met bash mìnon stic ligìtero organwmènec UK f seic (sq ma 4.4a). 'Etsi, stic qamhlèc pièseic mia isìtroph - nhmatik met bash f shc epitugq netai gia ìlo to eôroc twn sust sewn. Onom zoume thn nhmatik, kalamitik ìtan eðnai

81 4.6. SÔnoyh - Sumper smata 76 ploôsia se rabdìmorfa mìria (N R ) kai diskotik ìtan eðnai ploôsia se diskìmorfa (N P ). Se pièseic βp < 0, 1 ta sustatik tou mðgmatoc pl rwc anamðxima. Sthn perðoqh twn pièsewn 0, 1 <βp <0, 25 kai se qamhlèc sugkentr seic x P tou diagr mmatoc f shc (sq ma 4.4a) parathreðtai diaqwrismìc nhmatik c - nhmatik c sqetik ploôsiwn se rabdìmorfa swmatðdia. Autì eðnai k ti pou perimènoume epeid oi apokleiìmenoi ìgkoi twn zeugari n omoðwn swmatidðwn den èqoun thn Ðdia tim. Ja prèpei na shmei soume ìti metab seic an mesa se nhmatik monoaxonik (kalamitik diskotik ) kai diaxonik nhmatik eðnai metastajeðc. Se aut thn ergasða, den èqoume skopì na ereun soume thn eust jeia thn diaxonik c nhmatik c wc proc ton diaqwrismì f shc metaxô monoaxonik n nhmatik n. Gia perissìterec leptomèreiec parapèmpoume sthn bibliografða To di gramma f shc (sq ma 4.4a) perièqei triplì shmeðo (βp,x P )=(0.28; 0.43). Mia kalamitik nhmatik sunup rqei me mia diskotik kai mia tetragwnik kionik f sh; epiprìsjeta miadiskotik nhmatik me mia kionik. Sthn tetragwnik kionik f sh ta diskìmorfa swmatðdia stoib zontai to èna p nw sto llo sqhmatðzontac kðonec. Sto sq ma 4.4a ta ènjeta di grammata anaparistoôn antiproswpeutikèc upologismènec puknìthtec pijanìthtac eôreshc tmhm twn diskìmorfwn kai rabdìmorfwn swmatidðwn se tèsseric geitonikèc kuyelðdec thc kionik c f shc (se epðpedo k jeto ston xona thc kionik c f shc). Oi kðonec organ nontai se tetragwnikì plègma en oi r bdoi sqhmatðzoun diastauroômenouc diadrìmouc ston kenì q ro an mesa sta tetragwnik mìria. Se k je perðptwsh o moriakìc xonac twn r bdwn eðnai kat mèso ìro k jetoc stouc kðonec. Se pièseic βp > 0.30 h kalamitik nhmatik sunup rqei me thn tetragwnik kionik (Col). Aux nontac thn anisotropða twn r bdwn h kalamitik nhmatik gðnetai perissìtero eustajeðc ìpwc mporeð na dei kaneðc sto sq ma 4.4b. 4.6 SÔnoyh - Sumper smata Se autì to kef laio melet same mðgmata kamptwn swmatidðwn. Ta sustatik twn migm twn eðnai sunduasmìc dôo apì tic akìloujec treðc domèc: pl rwc summetrikìc kôboc (foullerènio), r bdoc tetr gwno. Sugkekrimèna, exet same duadik sust mata: foullerenðwn - rabdìmorfwn swmatidðwn, foullerenðwn - diskìmorfwn swmatidðwn kai rabdìmorfwn - diskìmorfwn swmatidðwn. H diatmhmatik allhlepðdrash metaxô aut n eðnai thc morf c twn sterik n} ap sewn. Epiplèon, ìtan èna apì ta sustatik eðnai to foullerènio se orismènec peript seic eis goume kai malak } apwstik allhlepðdrash gia tm mata tou deôterou sustatikoô pou brðskontai se kontinoôc geðtonec tou foullerèniou. H fasik sumperifor migm twn foullerenðwn - r bdwn eðnai ploôsia, parìlo pou diereunoôme wc proc thn eust jeia mìnon dôo UK f seic, thn nhmatik kai thn smhktik kai ìqi thn kionik.

82 4.6. SÔnoyh - Sumper smata 77 ParathroÔme diaqwrismì f shc an mesa se mia isìtroph kai mia kalamitik nhmatik ploôsia se foullerènia, all kai se uyhlìterec pièseic an mesa se mia isìtroph kai mia smhktik ìpou sumbaðnei mikrofasikìc diaqwrismìc metaxô r bdwn kai foullerenðwn. Ta teleutaða sqhmatðzoun upostr mata orismènou p qouc an log ameto mètro thc pðeshc kai thc sqetik c sugkèntrwshc. Autì to fainìmeno èqei parathrhjeð jewrhtik 116 kai peiramatik 105 en problèfjhke gia pr th for apì touc 115 Koda et. al. Diathr ntac thn pðesh stajer (gia timèc p nw apì to triplì shmeðo) kai aux nontac thn sqetik sugkèntrwsh twn r bdwn sto mðgma mei netai tìso to p qoc thc smhktik c ìso kai tou upostr matoc twn foullerenðwn. H aôxhsh thc energoô diamètrou} twn foullerenðwn alli c h aôxhsh thc fobikìtht c touc wc proc tic r bdouc eunnoeð akìma perissìtero ton diaqwrismì se mia smhktik f sh ploôsia se r bdouc kai se mia isìtroph ploôsia se foullerènia. H rôjmish twn jermodunamik n metablht n (pðesh - sqetik sugkèntrwsh) all kai h tropopoðhsh thc epif neiac tou foullerenðou eðnai kajoristikoð par gontec gia ton èlegqo kai sqhmatismì dom n epijumht c topologðac. Oi jermodunamik eustajeðc UK f seic pou ekdhl nontai se sust mata foullerèniwn - diskìmorfwn swmatidðwn eðnai h nhmatik kai h kionik. To fainìmeno tou mikrofasikoô diaqwrismoô parathreðtai kai sta mðgmata foullerenðwn - diskìmorfwn swmatidðwn. Se aut thn perðptwsh ta diskìmorfa swmatðdia sqhmatðzoun kðonec kai ta foullerènia topojetoôntai an mesa se autoôc. 'Etsi sqhmatðzontai swl nec} (apì diskìmorfa swmatðdia) pou perib llontai apì foullerènia; ta teleutaða, èqoun perissìterh pijanìthta emf nishc sta shmeða diastaôrwshc tou plègmatoc thc tetragwnik c kionik c f shc. H parousða foullerenðwn ephre zei tic diast seic twn plegmatik n stajer n (d x,d y ) thc kionik c f shc. Gia stajer pðesh (se timèc p nw apì to triplì shmeðo) aux nontac thn sqetik sugkèntrwsh twn diskìmorfwn swmatidðwn mei nontai oi plegmatikèc stajerèc kai oi sqetikèc apost seic twn axìnwn twn kiìnwn. H aôxhsh thc fobikìthtac twn foullerenðwn wc proc ta diskìmorfa èqei san apotèlesma na dieurônetai h difasik perioq isìtrophc me thn kionik f sh. Ta apotelèsmata pou lamb noume eðnai qr sima gia th melèth thc fasik c sumperifor c migm twn kolloeid n, gia par deigma rabdìmorfwn kai sfairik n swmatidðwn (ìpwc ioð kai polumer ) pou k tw apì orismènec sunj kec allhlepidroôn san sklhr } swmatðdia 105. Tèloc, se mðgmata rabdìmorfwn me diskìmorfa swmatðdia exet same wc proc thn eust jeia th nhmatik, thn smhktik kai thn kionik f sh. Lamb noume poiotik apodekt apotelèsmata gia thn topologða twn diagramm twn se qamhlèc pièseic, en problèpoume kai thn sunôparxh nhmatik c - kionik c f shc se uyhlìterec pièseic.

83 Kef laio 5 Autosunarmolìghsh kai autoorg nwsh foullerenik n mesogìnwn se kionikèc mesof seic Me th moriak jewrða, exet zoume thn fasik sumperifor thgm twn adropoihmènwn foullerenik n UKpouperi- lamb noun dendrimer tôpou Percec. Suzht me thn moriak autoorg nwsh me ìrouc polufilikìthtac. Diereun me thn epðdrash pou èqei sth fasik sumperfor thgm twn foullerenik n UK, h prosj kh mikr n posot twn mh tropopoihmènwn} foullerenðwn. 5.1 Eisagwg H qhmik tropopoðhsh tou foullerèniou kai h sôndes tou me mesogìnec mon dec (se dendritik mh arqitektonik ) èqei d sei mia ploôsia sullog apì upermoriakoôc UK oi opoðoi ekdhl noun smhktik A, nhmatik kai qeirìmorfh nhmatik f sh. 'Eqoume anaferjeð ekten c kai melet sei mia meg lh poikilða foullerenik n UK (dec sto trðto kef laio). IdiaÐtero ereunhtikì endiafèron parousi zoun kai oi kionikoð foullerenikoð UK pou apoteloôn en dun mei ulik gia hlektronikèc efarmogèc 38. Arqik, anaptôqjhke h sqediastik sôndeshc apeujeðac mesogìnwn gôrw apì èna pent gwno sthn epif neia tou foullerèniou, kai o sqhmatismìc kwnik n puramidik n morðwn (me koruf èna foullerènio). Ta kwnik mìria stoib zontai to èna p nw sto llo sqhmatðzontac kðonec. 'Etsi, epitugq netai grammik di taxh twn foullerenðwn se kionik f sh. Argìtera, me an mixh dôo mh mesomorfik n sustatik n, enìc parag gou tou foullerèniou (foullerènio sundemèno mèsw anjrakik n alusðdwn me diskìmorfa mìria) kai ènoc diskìmorfou morðou parathr jhke autoorg nwsh se kionik f sh, me ta foullerènia na diat ssontai an mesa stouc kðonec thc UK f shc 121. Oi kðonec thc f shc sqhmatðzontai me thn topojèthsh enìc diskìmorfou morðou ep nw sto llo. 78

84 5.1. Eisagwg 79 a) b) g) Sq ma 5.1: a) QhmikoÐ tôpoi DESC60 morðwn 38 kai b) allhlouqða f sewn sômfwna me peiramatik apotelèsmata 38. Mwb: perioq ual douc met bashc, portokalð: ugrokrustallik perioq (tetragwnik kionik ) kai bioletð: isìtroph f sh. H jermokrasða se bajmoôc KelsÐou. g) ProtajeÐsa di taxh twn DESC60 morðwn, se epðpedo k jeto ston kôrio xona thc kionik c f shc 38. Prìsfata oi Deschenaux et. al. 38 melèthsan thn fasik sumperifor enìc nèou tôpou qhmik tropopoihmènou foullerèniou. SÔndesan sto foullerènio dendrimer pou èqoun sunjèsei o Percec et. al. ta opoða autodomoôntai se kulðndrouc sfairikèc uperdomèc kai autoorgan nontai se kionik kubik UK f sh antðstoiqa. H sqediastik arq pou eis gagan o Deschenaux et. al. eðnai perissìtero genik, se sqèsh me tic prosp jeiec pou anafèrame sthn prohgoômenh par grafo, exaitðac tou meg lou arijmoô twn dendrimer n pou mporoôn na qrhsimopoihjoôn gia autìn to skopì. Oi foullerenikoð UK pou sônjesan, oi DESC60 (sq ma 5.1a), exet sthkan me optik mikroskopða (POM), me diaforik jermidometrða s rwshc (DSC) kai me skèdash aktðnwn Q (X-ray). Apì thn epexergasða twn apotelesm twn diapist jhke ìti emfanðzoun kionik UK sumperifor. H fasik sumperifor sunoyðzetai sthn (sq ma 5.1b). M lista, prìteinan kai ènan trìpo autosunarmolìghshc twn morðwn se romboeid elleiyoeid uperdom ( nhsðdec} ), pou apoteloôntai apì foullerènia pou perib llontai apì dendritik mèrh. Oi nhsðdec} topojetoôntai h mia p nw sthn llh kai sqhmatðzoun kðonec pou autoorgan nontai se tetragwnikèc kionikèc f seic (sq ma 5.1g). Me autì to trìpo dhmiourgoôntai kðonec me elleiyoeid diatom, me to eswterikì touc kateilhmmèno apì foullerènia.

85 5.2. Moriakì montèlo tou DESC60 80 Se autì to kef laio meth moriak jewrða prospajoôme katarq n na katano soume thn fasik sumperifor thgm twn DESC60 me ìrouc diatmhmatik n moriak n allhlepidr sewn. Kat deôteron, epekteðnoume thn jewrða sto perissìtero sônjeto probl mata twn epipt sewn sth moriak autoorg nwsh - fasik sumperifor, lìgw prosj khc mh tropoihmènwn ( eleôjerwn} ) foullerèniwn se t gmata DESC60. Xekin ntac me b sh ta apotelèsmata migm twn diskìmorfwn swmatidðwn - foullerenðwn problèpoume tic metabolèc sth moriak org nwsh tou t gmatoc lìgw thc parousðac mikr c posìthtac eleôjerwn} foullerèniwn. Sthn amèswc epìmenh enìthta perigr foume to moriakì montèlo. Sthn sunèqeia parousi zoume ta apotelèsmata thgm twn DESC60 metab llontac tic allhlepidr seic twn foullerèniwn me ta dendritik mèrh. Sthn enìthta 5.4 sunoyðzoume ta apotelèsmata kai anaptôssoume ta sumper smata. 5.2 Moriakì montèlo tou DESC60 Sthn sq ma 5.2 parousi zoume to moriakì montèlo tou DESC60. H moriak ènwsh perigr fetai se mia pr th prosèggish apì èna moriakì sq ma se kubikì plègma. Perimènoume ìti stic kionikèc f seic naepikratoôn moriakèc diamorf seic pou omadopoioôntai se autì to moriakì sq ma} (sq ma 5.2). Eis goume dôo eðdh tmhm twn, ta foullerènia (f) kai aut me ta opoða qtðzetai o dendritikìc pur nac (d) (me tic anjrakikèc alusðdec sta kra tou) apodðdontac ètsi èna trigwnikì sq ma gia to adropoihmèno mìrio. To diatmhmatikì dunamikì allhlepðdrashc dðnetai apì thn ex.(3.1). Oi diatmhmatikèc allhlepidr seic parametrikopoioôntai wc ex c: u (0) (f,f) =u (0) (f,d) =, u (0) (d, d) =u kai u (1) (f,d) =α me α>0 kai me thn par metro u>0 na eðnai mètro thc apwstikìthtac metaxô twn tmhm twn. Stic upìloipec allhlepidr seic apodðdoume thn tim mhdèn. To foullerenikì tm ma jewroôme ìti eðnai adiapèrasto; ìtan katalamb nei thn Ðdia plegmatik kuyelðda me foullerènio llou morðou dendritikì tm ma tìte allhlepidroôn me sklhrì} dunamikì. H allhlepik luyh dôo dendritik n tmhm twn diaforetik n morðwn sunodeôetai apì malak } apwstik allhlepðdrash; me autì ton trìpo epitrèpetai allhlodieðsdush dendritik n tmhm twn. Tèloc, me thn par metro u (1) (f,d) eis goume th mh filikìthta} dendritik n tmhm twn - foullerèniwn ìtan brejoôn se diplanèc plegmatikèc kuyelðdec (pr toi geðtonec). Sthn perðptwsh pou exet zoume mðgmata eleôjerwn} foullerèniwn kai morðwn DESC60, oi diatmhmatikèc allhlepidr seic twn eleôjerwn} foullerèniwn jewroôme ìti eðnai Ðdiec me autèc pou èqoun ta dèsmia. 5.3 Apotelèsmata Monosustatik sust mata tou DESC60 Sto sq ma 5.3a dðnoume to upologismèna diagr mmata f shc thc pðeshc wc sun rthsh thc antðstrofhc jermokrasðac. Exet zoume pr ta thn perðptwsh ìpou u (1) (f,d) =u. Sflaut thn perðptwsh to

86 5.3. Apotelèsmata 81 Sq ma 5.2: Anapar stash tou adropoihmènou DESC60 me to topikì moriakì sôsthma axìnwn. a) b) Sq ma 5.3: Upologismèna diagr mmata f shc, me xonec anhgmènh pðesh - par metroc βu, gia t gma DESC60. a) 'Otan u (1) (f,d) =u kai b) ìtan u (1) (f,d) =2u. 'Enjeta diagr mmata: Apeikìnish thc f (x, y) = Ω f (Col) (x, y, Ω) (blèpe keðmeno) se treðc diast seic.

87 5.3. Apotelèsmata 82 foullerènio dèqetai} apwstikì dunamikì apfl ta dendritik tm mata ìtan eðnai pr toc geðtonac me aut. To sôsthma ekdhl nei isìtroph f sh, smhktik f sh, kai kionik f sh h opoða eðnai eustaj c se èna meg lo eôroc pièsewn kai jermokrasi n. Sthn smhktik f sh emfanðzetai allhlodieðsdush twn dendritik n tmhm twn kai ta foullerènia sqhmatðzoun dipl stoib da. Sthn kionik f sh ta mìria DESC60 autosunarmologoôntai se upermoriakèc domèc. Sugkekrimèna, om dec apì mìria DESC60 dieujetoôntai me tètoio trìpo ste toul qiston dôo foullerènia na sqhmatðzoun foullerenikì pur na} o opoðoc perib lletai apì dendritik tm mata (sq ma 5.4). Autèc oi upermoriakèc domèc topojetoôntai h mia p nw sthn llh sqhmatðzontac kðonec. To eswterikì twn kiìnwn katalamb netai apì foullerènia kai h perifèreia apì dendritik tm mata. Prokeimènou na posotikopoi soume ta gewmetrik qarakthristik twn kiìnwn kai ton trìpo pou organ nontai ta mìria entìc aut n upologðsame: 1) thn puknìthta pijanìthtac twn dèsmiwn foullerèniwn twn morðwn DESC60 anex rthta tou moriakoô prosanatolismoô se epðpedo k jeto stouc kðonec, dhlad f (x, y) = Ω f (Col) (x, y, Ω), 2) thn ekkentrìthta thc diatom c twn kiìnwn, dhlad λ = I xx /I yy, ìpou I xx kai I yy eðnai oi kôriec ropèc adr neiac} miac fètac} tou kðona p qouc ΔZ =1, kai orðzontai wc ex c: I xx = x,y (y cm y) 2 f (x, y) kai Iyy = x,y (x cm x) 2 f (x, y) wc proc xonec ìpou Ixy kai I yx =0. 'Opou x cm = x,y x f (x, y) / x,y f (x, y) kai y cm = x,y y f (x, y) / x,y f (x, y) oi suntetagmènec tou kèntrou m zac} thc kionik c fètac wc proc to ergasthriakì sôsthma axìnwn. 3) upologðsame ton arijmì twn foullerèniwn an fèta} tou kðona p qouc Ðsh me thn akm miac plegmatik c kuyelðdac (ΔZ =1), dhlad N f = ρaδz, wc sun rthsh thc posìthtac ξ = βu/ (βu) tran, gia di forec pièseic. 'Opou A eðnai to embadìn thc epif neiac thc kionik c kuyelðdac thc f shc, (βu) tran eðnai h jermokrasða metatrop c apì thn kionik se mia smhktik isìtroph f sh, se orismènh pðesh. Apì tic timèc twn upologismènwn f (x, y), sumperaðnoume ìti h kionik f sh eðnai tetragwnik c summetrðac (sq ma 5.4). Sto sq ma 5.5a dðnoume to di gramma thc N f wc sun rthsh thc posìthtac ξ. Stic pièseic P/u =16, 78 kai P/u =10, 50 gia ξ<1, 46 kai ξ<1, 26 antðstoiqa, emfanðzontai auxomei seic ston arijmì N f pr gma pou ofeðletai sthn Ôparxh kionik n f sewn me diaforetikì mègejoc kuyelðdac thc kionik c f shc. 'Otan apomakrunìmaste apì thn perioq metatrop c f shc, parathroôme ìti, gia stajer pðesh, aux netai o arijmìc twn foullerèniwn an fèta} kðona. PlhroforÐec gia thn dom tou pur na} thc uperdom c lamb noume me ton upologismì thc ekkentrìthtac λ = I xx /I yy wc sun rthsh thc ξ = βu/ (βu) tran, gia di forec pièseic (sq ma 5.5b). Gia ìlec tic pièseic pou exet same, an exairèsoume tic jermokrasðec kìnta sthn metatrop f shc, h ekkentrìthta aux netai wc sun rthsh thc posìthtac ξ. Aut h t sh mporeð na sundejeð me to ìti

88 5.3. Apotelèsmata 83 a) b) g) d) Sq ma 5.4: Antiproswpeutikì di gramma thc upologismènhc puknìthtac pijanìthtac f (x, y) (blèpe keðmeno) se pðesh P/u =10, 5 kai βu =0, 30: a) apeikìnish se treðc diast seic kai b) apeikìnish twn isouy n gramm n kai thc autodìmhshc twn DESC60 sthn kionik f sh: Ta tm mata twn morðwn qrwmatðzontai wc ex c: Foullerènia me pr sino, dendritik tm mata me portokalð (kai ìtan up rqei allhlodieðsdush dendritik n tmhm twn me skoôro portokalð). (g) kai (d) ìpwc (a) kai (b) all se pðesh P/u =5, 5 kai jermokrasða βu =0, 34. a) b) Sq ma 5.5: a) Arijmìc twn foullerenðwn N f an fèta} tou kðona, wc sun rthsh thc posìthtac ξ = βu/ (βu) tran gia di forec pièseic: P/u = 16, 78(tetr gwno), P/u = 10, 50(trÐgwno) kai P/u =4, 5(kÔkloc). b) Ekkentrìthta twn kiìnwn λ wc sun rthsh thc ξ, gia di forec pièseic: P/u =16, 78(tetr gwno), P/u =10, 50(trÐgwno) kai P/u =4, 5(kÔkloc). Oi upologismènec timèc aforoôn to sôsthma sto sq ma 5.3a.

89 5.3. Apotelèsmata 84 h f (x, y) apokt ìlo kai perissìtero oxeðec korufèc me th mei sh thc jermokrasðac, se stajer pðesh. Sthn perioq twn uyhl n pièsewn kai me meðwsh thc jermokrasðac to sôsthma metabaðnei apì thn isìtroph se mia smhktik f sh (SmA d ). Ta foullerènia sqhmatðzoun dipl stoib da (sq ma 5.6) kai ta dendritik tm mata gemðzoun to q ro an mesa se diadoqik upostr mata foullerèniwn. H dieujèthsh se smhktik f sh autoô tou tôpou epitrèpei thn Ôparxh perissìterwn prosanatolism n an mìrio, se sqèsh me mia orjog nia smhktik f sh, miac pou oi moriakoð prosanatolismoð aux nontai kat dôo gia k je moriak jèsh sthn stoib da twn foullerenðwn. Sthn sunèqeia, me peraitèrw meðwsh thc jermokrasðac to sôsthma metabaðnei sthn kionik f sh. Exet same thn epðdrash tou apwstikoô ìrou u (1) foullerenðwn - dendritik n tmhm twn sthn fasik sumperifor. Gia 0 <u (1) < 2u, h topologða tou diagr mmatoc paramènei parìmoia me aut gia u (1) = u me diaforèc sthn org nwsh thc smhktik c f shc. Ed, parousi zoume thn perðptwsh ìpou u (1) =2u, af nontac to mètro twn upìloipwn allhlepidr sewn amet blhto. Sthn perioq twn qamhl n pièsewn P/u < 2.65 kai me meðwsh thc jermokrasðac to sôsthma metabaðnei apì thn isìtroph sthn kionik f sh ìpwc kai sto prohgoômeno sôsthma (sq ma 5.3b). Shmantik diaforopoðhsh parathroôme stic endi mesec kai uyhlèc perioqèc pièsewn. EkeÐ, me meðwsh thc jermokrasðac to sôsthma dièrqetai apì thn isìtroph se mia smhktik f sh (SmA d ) me allhlodieðsdush dendritik n tmhm twn, ìmoia me aut tou parap nw sust matoc. Kai stic dôo peript seic ta foullerènia sqhmatðzoun upìstrwma dipl c stoib dac. H parapèra meðwsh thc jermokrasðac odhgeð se mia smhktik f sh sthn opoða den parousi zetai allhlodieðsdush dendritik n tmhm twn. H aôxhsh tou energeiakoô kìstouc pou ja proklhjeð an diathrhjeð h allhlo-dieðsdush (me th meðwsh thc jermokrasðac) èqei telik san apotèlesma to p qoc twn strwm twn thc nèac smhktik c f shc na megal nei ste na mhn sumbaðnei allhlodieðsdush. 'Etsi, apofeôgetai h epaf tìso metaxô dendritik n tmhm twn ìso kai metaxô dendritik n tmhm twn kai foullerèniwn. Se qamhlìterec jermokrasðec sumbaðnei met bash sthn kionik f sh me thn qarakthristik autodìmhsh twn foullerèniwn ìpwc perigr fetai sto sq ma 5.4. H shmantikìterh diafor, pèran k poiwn allag n stic perioqèc met bashc f shc (dec sq ma 5.3a kai 5.3b), pou epifèrei h enðsqush thc apwstikìthtac (stic allhlepidr seic kontin n geitìnwn) metaxô foullerenðwn kai dendritik n tmhm twn, antikatoprðzetai sthn poikilða smhktik n f sewn pou ekdhl nontai sta dôo sust mata. Esti zoume sthn perioq twn diagramm twn f shc ìpou ekdhl netai smhktik f sh (sq ma 5.3). Pio sugkekrimèna, sto sôsthma me auxhmènh fobikìthta foullerèniwn - dendritik n tmhm twn u (1) (f,d) = 2u : Se stajer pðesh kai me meðwsh thc jermokrasðac to sôsthma metabaðnei apì thn SmA d (me auxhmènouc kat dôo touc dunatoôc prosanatolismoôc an mìrio) se mia orjog nia smhktik SmA me p qoc dôo forèc to moriakì m koc. Apì thn llh, ìtan u (1) (f,d) =u, h aôxhsh thc entropðac sthn SmA d eðnai ikan na akur sei to energeiakì kìstoc se sqèsh me mia orjog nia (qwrðc allhlodieðsdush dendritik n tmhm twn) smhktik f sh, opìte kai parathroôme thn ekd lwsh enìc eðdouc smhktik c f shc. Kai stic dôo peript seic,

90 5.3. Apotelèsmata 85 a) b) Sq ma 5.6: Antiproswpeutikì upologismèno di gramma thc f (x, y) (tom k jeth sta smhktik str mata) se pðesh P/u =11, 8 kai jermokrasða βu =0, 14: a) apeikìnish se treðc diast seic kai b) apeikìnish twn isouy n gramm n. Ta moriak tm mata qrwmatðzontai wc ex c: ta foullerènia me pr sino, kai ta dendritik tm mata me portokalð. Epiprìsjeta, apeikonðzontai mìria DESC60 se tèsseric qarakthristikoôc prosanatolismoôc kai epishmaðnetai h dunatìthta peristrof c touc gôrw apì ton kôrio moriakì xona (kat ±90 o sto kubikì plègma). Apì touc moriakoôc prosanatolismoôc pou apeikonðzontai up rqoun dôo an jèsh thc stoib dac twn foullerèniwn. h auxomeðwsh thc entropðac kai thc enèrgeiac odhgeð sthn elaqistopoðhsh thc eleôjerhc enèrgeiac Helmholtz me sunèpeia thn allag ston trìpo dieujèthshc twn morðwn sthn f sh. Me lla lìgia, hepðteuxh thc miac thc llhc smhktik c f shc eðnai dunatìn narujmisteð me qhmik tropopoðhsh thc epif neiac tou foullerèniou, alli c me ìrouc allhlepidr sewn metab llontac to mètro apwstikìthtac foullerèniwn - dendritik n tmhm twn, se kontinoôc geðtonec. 'Oson afor tic kionikèc f seic den parathroôme diaforèc sthn autoorg nwsh twn morðwn se autèc MÐgmata foullerèniwn kai morðwn DESC60 H fasik sumperifor all kai h di taxh twn morðwn stic uyhl organwmènec f seic (kionik ), pou problèpei h moriak jewrða gia t gmata adropoihmènwn foullerenik n UK, eðnai se ikanopoihtik poiotik sumfwnða me ta up rqonta peiramatik dedomèna. EkeÐno pou den èqei exetasteð akìma peiramatik, kai apoteleð prìklhsh gia thn diereônhs tou me jewrða, eðnai h epðdrash pou ja èqei sthn moriak org nwsh pou parathreðtai sto t gma h prosj kh mikr n posot twn eleôjerwn} foullerenðwn. Na shmei soume ìti èqoume dh diereun sei (tètarto kef laio) mðgmata diskìmorfwn swmatidðwn - foullerenðwn pou allhlepidroôn eðte me sterikèc ap seic eðte èmmesa me elktikèc} allhlepidr seic giata ìmoia mìria. Exet same to mðgma foullerèniou - morðwn DESC60 se βu =0, 40 ìpou sto t gma (sq ma 5.3a) parathroôme met bash apì thn isìtroph sthn kionik f sh me aôxhsh thc pðeshc. Sto sq ma 5.7 parousi zoume thn kampôlh sunôparxhc f sewn se èna di gramma pðeshc wc sun rthsh thc sqetik c sugkèntrwshc twn DESC60 (x r ) se βu =0, 40. Ja prèpei na tonðsoume, ìti èqei deiqjeð peiramatik ìti ta foullerènia eðnai el qista dialut se mia meg lh poikilða dialut n kai susswmat nontai eôkola lìg wtwn isqur n elktik n allhlepidr sewn pou anaptôssontai metaxô touc 17. Sthn mon-

91 5.3. Apotelèsmata 86 Sq ma 5.7: Di gramma f shc duadikoô mðgmatoc foullerèniwn - DESC60 pðeshc wc proc thn sqetik sugkèntrwsh twn DESC60 x r. Me I1 kai I2 sumbolðzoume thn isìtroph f sh ploôsia kai ligìtero ploôsia se eleôjera} foullerènia antðstoiqa, me DF th difasik perioq, me Col1 kionik ploôsia se foullerènia kai me Col r tetragwnik kionik f sh ploôsia se DESC60. 'Enjeta diagr mmata: anaparistoôn thn f C60 (x, y) twn eleôjerwn} foullerèniwn (mwb) kai thn f (x, y) twn dèsmiwn foullerèniwn twn morðwn DESC60 (gal zio). telopoðhsh pou efarmìzoume autì lamb netai upìyh mìnon èmmesa. Kat sunèpeia, h qrhsimìthta twn apotelesm twn pou parousi zoume parak tw, sqetðzetai kurðwc me th dunatìthta prìbleyhc me th moriak jewrða tou trìpou taktopoðhshc} twn foullerenðwn entìc twn kionik n f sewn, se mikrèc sugkentr seic foullerèniwn. Gia touc lìgouc pou anafèrame, h fasik sumperifor pou parathroôme se meg lec sugkentr seic foullerèniwn den perimènoume na anaparist thn fasik sumperifor pragmatik n susthm twn. H kampôlh sunôparxhc f sewn (sq ma 5.7) emfanðzei olikì el qisto (krðsimo shmeðo) to opoðo qarakthrðzetai apì mia krðsimh pðesh kai jermokrasða (βp,x r )=(0.51; 0.10). SÔmfwna me touc upologismoôc, ta sustatik tou mðgmatoc gia pièseic qamhlìterec thc krðsimhc anamignôontai pl rwc se ìlo to eôroc twn sugkentr sewn se isìtroph f sh. Aux nontac thn pðesh (apì to krðsimo shmeðo) sumbaðnei diaqwrismìc f shc se dôo isìtropec f seic, tic I1 kai I2, apì tic opoðec h mia eðnai ligìtero ploôsia se foullerènia (h I2). Sthn sunèqeia, sthn endi mesh perioq pièsewn pragmatopoieðtai diaqwrismìc f shc an mesa se mia f sh meg lhc periektikìthtac se foullerènia (Col1) (x r < 0, 05) kai se mia isìtroph f sh (I2). Sthn Col1 f sh parathreðtai di krish metaxô diaforetik c suggèneiac moriak n tmhm twn. Ta dendritik mèrh twn DESC60 topojetoôntai se swl nec} oi opoðoi perib llontai apì foullerènia. Autì to diapist noume apì thn trisdi stath apeikìnish thc f (x, y) twn dèsmiwn foullerenðwn sta mìria DESC60 all kai apì thn f C60 (x, y) twn eleôjerwn} foullerenðwn (sq ma 5.7 ènjeto). AxÐzei na poôme ìti h exètash uyhlìtera organwmènwn UK f sewn, ìpwc h kubik, èdwse lôseic parapl siac eleôjerhc enèrgeiac me thn Col1, me ta dendritik mèrh na sqhmatðzoun mikkôlia ta opoða perib llontai apì foullerènia (anestrammènh kubik f sh). Me thn aôxhsh thc pðeshc dieurônetai h difasik perioq Col1 I2, mèqri to triplì shmeðo x r 0, 70 kai βp 1, 05. Sto

92 5.3. Apotelèsmata 87 a) b) g) d) Sq ma 5.8: Antiproswpeutik diagr mmata thc f (x, y) twn dèsmiwn foullerèniwn twn morðwn DESC60 (gal zio), a) se trisdi stath apeikìnish kai b) apeikìnish twn isouy n gr mmwn. Gia thn Ðdia pðesh kai sugkèntrwsh, diagr mmata thc f C60 (x, y) twn eleôjerwn} foullerenðwn (mwb), g) se trisdi stath apeikìnish kai d) apeikìnish twn isouy n gramm n.

93 5.4. SÔnoyh - Sumper smata 88 triplì shmeðo sunup rqoun tautìqrona treðc f seic hcol1, h I2 kai h tetragwnik kionik Col r. M lista, sthn pðesh βp 1 kai gia sugkentr seic x r > 0, 80 pragmatopoieðtai mia metatrop f shc pr thc t xhc apì thn I2 sthn Col r. P nw apì to triplì shmeðo (sthn kionik ) epitrèpetai h di lush akìma mikrìterwn posot twn eleôjerwn} foullerenðwn ìpou lìgw thc amfifilikìthtac topojetoôntai gôrw apì ta foullerènia twn morðwn DESC60 (sq ma 5.8). H jewrða loipìn, problèpei thn Ôparxh eustajoôc kionik c f shc, kai thn dunatìthta elègqou taktopoðhshc twn foullerenðwn sto mðgma. Autì pou diapist noume apì thn epilektik exètash foullerèniou - DESC60 (se orismènh jermokrasða) eðnai h dunatìthta anamixhmìthtac foullerenðwn se UK f sh (kionik ), gia sugkentr seic x r > 0, SÔnoyh - Sumper smata Se autì to kef laio exet same foullerenodendrimer ta opoða ekdhl noun kionik ugrokrustallik sumperifor. H fasik sumperifor twn adropoihmènwn morðwn eðnai se poiotik sumfwnða me ta peiramatik dedomèna. Ta DESC60 autosunarmologoôntai se upermoriakèc domèc, ta foullerènia twn morðwn topojetoôntai sto eswterikì touc kai perib llontai apì dendritik mèrh. Oi upermoriakèc domèc stoib zontai h mia p nw sthn llh sqhmatðzontac kðonec. Epiplèon, apì ta jewrhtik apotelèsmata parathroôme thn Ôparxh smhktik c f shc ìpou ta foullerènia organ nontai se diplèc stoib dec. ParathroÔme poikilða twn smhktik n f sewn all zontac to mètro thc apwstikìthtac foullerèniwn - dendritik n tmhm twn se allhlepidr seic kontin n geitìnwn. Moriakèc domèc (ìpwc ta mìria DESC60 ) pou diafèroun apì ta moriak sq mata sumbatik n UK (ìpwc rabdoeid c diskoeid c) lìgw thc polufilikìthtac mporoôn na ekdhl soun mia poikilða ugrokrustallik n f sewn. Me skopì na epekteðnoume thn melèth sto pio sônjeto prìblhma thc moriak c autoorg nwshc ìtan anamignôontai mikrèc posìthtec eleôjerwn} foullerenðwn se t gmata DESC60, exet same mðgma eleôjerwn} foullerenðwn - morðwn DESC60. Diapist noume ìti gia βu = 0, 40, eðnai dunat periorismènh an mixh aut n se UK f sh; ètsi, parathroôme thn ekd lwsh tetragwnik c kionik c f shc, se arket uyhlèc sqetikèc sugkentr seic morðwn DESC60 x r > 0, 80 kai se uyhlèc pièseic. Ta eleôjera} foullerènia topojetoôntai plhsðon twn dèsmiwn foullerèniwn twn morðwn DESC60. H diereônhsh mikrìterwn sugkentr sewn x r ègine gia thn plhrìthta tou diagr mmatoc f shc, qwrðc na shmaðnei ìti ta apotelèsmata japerigr foun pragmatik sust mata ploôsia se foullerènia. Ex llou, gnwrðzoume apì peiramatik dedomèna ìti to foullerènio eðnai el qista dialutì se mia meg lh poikilða dialut n, sunep c ja perimèname ton sqhmatismì susswmatwm twn foullerenðwn. Se endi mesec kai uyhlèc pièseic, gia x r < 0, 05 emfanðzetai mia kionik f sh me mikrofasikì diaqwrismì diaforetik c qhmik c suggèneiac mer n (dendrimer - foullerènia). Gia epilegmènec

94 5.4. SÔnoyh - Sumper smata 89 timèc pðeshc kai sôstashc, diereunoôme thn sqetik eust jeia aut c thc kionik c me lôseic pou prokôptoun gia kubikèc f seic upologðzontac tic diaforèc sthn metaxô touc eleôjerh enèrgeia Helmholtz. Diapist noume ìti eðnai dunat h Ôparxh anestrammènhc kubik c f shc (me mikkôlia apì dendrimer pou perib llontai apì foullerènia) me mikr sqetik diafor sthn eleôjerh enèrgeia Helmholtz. Me th jewrhtik mac prosèggish eðnai dunat h prìbleyh mikkulðwn llwn upermoriak n dom n.

95 Kef laio 6 Moriakèc prosomoi seic twn kwnik n foullerenik n ugr n krust llwn (CSMC60) Me thn ektèlesh moriak n prosomoi sewn Monte Carlo (NVT) melet me thn fasik sumperifor adropoihmènwn (CSMC60) se kubikì plègma. Arqik exet zoume ajermik sust mata me puramidik arqitektonik gia mia pl rh seir puknot twn. Sthn sunèqeia parametrikopoioôme to diamoriakì dunamikì allhlepðdrashc ìpwc kai sto antðstoiqo montèlo thc moriak c jewrðac, eis gontac malak } dunamik allhlepðdrashc; esti zoume se mia puknìthta ìpou sto ajermikì sôsthma ekdhl netai kionik f sh, kai èkei diereun me thn sumperifor tou sust matoc kat thn yôxh kai thn jèrmansh. Diapist noume thn dunatìthta isìtrophc - kionik c metatrop c f shc (me antistreptì trìpo). Epitugq netai sumfwnða me ta apotelèsmata thc moriak c jewrðac toul qiston wcprocthnprìbleyh ekd lwshc kionik c f shc. 6.1 Teqnikèc leptomèreiec twn upologistik n prosomoi sewn 122, 123 Se autì to kef laio efarmìzoume th mèjodo moriak n prosomoi sewn Metropolis Monte Carlo gia thn exètash twn fusik n idiot twn susthm twn adropoihmènwn kwnik n foullerenik n morðwn CSMC60. Eidikìtera, asqoloômaste me moriakèc prosomoi seic stajeroô arijmoô swmatidðwn N se ìgko V kai apìluth jermokrasða T ( NVT kanonik jermodunamik sullog ). SugkrÐnoume ta apotelèsmata twn moriak n prosomoi sewn me ekeðna thc moriak c jewrðac (pou perigr fontai sto trðto kef laio), gia thn Ðdia parametrikopoðhsh twn moriak n allhlepidr sewn. Amèswc parak tw parousi zoume orismènec teqnikèc leptomèreiec twn moriak n prosomoi sewn gia ta sust mata pou melet same. Sthn sunèqeia parousi zoume ta apotelèsmata g ia èna ajermikì sôsthma kai akoloujeð h exètash susthm twn pou parousi zoun jermokrasiak ex rthsh. 90

96 6.1. Teqnikèc leptomèreiec twn upologistik n prosomoi sewn 91 a) b) g) d) Sq ma 6.1: Apeikìnish tou adropoihmènou puramidikoô morðou tou opoðou ta tm mata na anaparðstantai me sfaðrec: a) ìtan autì eðnai sthn arq twn axìnwn me to moriakì tou xona z na eðnai par llhloc ston ergasthriakì OZ, b) Par deigma moriak c metatìpishc kat ton xona OY, g) par deigma allag c prosanatolismoô tou morðou proc ton xona OY apì ton OZ (sthn kat stash (a) ) kai d) anastrof tou morðou apì thn arqik tou kat stash (a). Ta moriak sust mata pou melet me up rqoun} se ènan trisdi stato plegmatikì q ro. To plègma eðnai tôpou aploô kubikoô. K je plegmatik kuyelðda mporeð na filoxenðsei èna moriakì tm ma; apodðdoume se aut m koc akm c 1nm ìso perðpou kai to mègejoc miac foullerenik c diamètrou. To CSMC60 apoteleðtai apì 25 moriak tm mata kai èqei to sq ma tetragwnik c puramðdac (sq ma 6.1a), ìpwc ta antðstoiqa adropoihmèna mìria pou melet me sto trðto kef laio. Epeid h tetragwnik puramðda den eðnai sumpag c, sqhmatðzetai mia koilìthta pou perib lletai apì ta toiq mata thc puramðdac. Oi moriakèc prosomoi seic ekteloôntai apì ènan arijmì adropoihmènwn morðwn thc t xhc twn merik n qili dwn. O q roc ston opoðo brðskontai N mìria eðnai sq matoc orjogwnðou parallhlepipèdou (koutð prosomoðwshc), me ìgko V = Wu o, perilamb nei W = D x D y D z plegmatikèc kuyelðdec, ìpou D l (l = x, y, z) oi diast seic twn akm n autoô kai u o o ìgkoc miac plegmatik c kuyelðdac. To dunamikì allhlepðdrashc dôo CSMC60 morðwn I kai J, U IJ kajorðzetai wc: U IJ = i I,i J u (o) (i I,i J ) (6.1)

97 6.1. Teqnikèc leptomèreiec twn upologistik n prosomoi sewn 92 ìpou h jroish trèqei p nw se ìla ta tm mata twn dôo morðwn i I kai i J. Ta u (o) (i I,i J ) eðnai to mètro thc diatmhmatik c allhlepðdrashc ìtan aut brejoôn sthn Ðdia plegmatik kuyelðda. Na shmei soume ìti aut h diatmhmatik allhlepðdrash se ìla ta sust mata pou exet zoume eðnai apwstik. Jewr ntac ìti to sôsthma eðnai sto q ro twn jèsewn sto shmeðo {I} h olik tou enèrgeia eðnai: U N ({I}) = I<J U I,J (6.2) 'Estw ìti arqik to sôsthma brðsketai sto jeseografikì shmeðo {I} (o) me U N ({I} (o)) = U (o). Kat thn di rkeia thc prosomoðwshc dialègoume tuqaða èna mìrio apì ta N kai prokaloôme mia tuqaða Monte Carlo (MC) kðnhsh autoô. Efarmìzoume treðc basikèc moriakèc kin seic: a) Epilègetai tuqaða mia dieôjunsh tou sust matoc suntetagmènwn kai pragmatopoieðtaimia metotìpish twn antðstoiqwn suntentagmènwn twn moriak n plegmatik n shmeðwn se mia nèa jèsh new}, r (new) r (o) + NINT [Δ x (ξ 0, 5)] me r (o) h pali jèsh o} kai Δ x eðnai dôo forèc h mègisth dunat metatìpish pou epib loume (sq ma 6.1b), b) epilègetai tuqaða mia apì tic èxi kateujônseic pou orðzontai apì to kartesianì ergasthriakì sôsthma suntetagmènwn kai apodðdetai o antðstoiqoc moriakìc prosanatolismìc (sq ma 6.1g) kai g) anastrof tou morðou (sq ma 6.1d). Mia MC kðnhsh mporeð na eðnai mia apl metatìpish tou morðou (a), mporeð na eðnai ìmwc kai sunduasmìc twn (a) kai (g) twn (a) kai (b). Paragmatopoi ntac thn kðnhsh MC, to sôsthma metabaðnei se mia nèa kat stash {I} (new) kai upologðzetai h U N ({I} (new)) = U (new), kai h sunolik metabol thc enèrgeiac eðnai: ΔU = U (new) U (o). Sthn pragmatikìthta, h ΔU eðnai Ðsh me thn metabol sthn enèrgeia allhlepðdrashc mìnon tou morðou (pou dialèxame tuqaða) me ta upìloipa mìria sthn kat stash new} apì thn kat stash o}. M lista, gia thn exoikonìmhsh upologistikoô qrìnou den eðnai aparaðthto na akolouj soume mia diadikasða exètashc twn allhlepidr sewn tou tuqaðou morðou me to sônolo twn upoloðpwn morðwn pou brðskontai mèsa sto koutð prosomoðwshc giatð ta perissìtera apì aut ja brðskontai se apost seic} mh allhlepðdrashc. AntÐ loipìn aut c thc diadikasðac, eis goume ènan diaforetikì trìpo gia ton upologismì twn allhlepidr sewn: Gia k je plegmatikì shmeðo apojhkeôoume se mia lðsta to pl joc twn moriak n tmhm twn pou katalamb noun autì, krat ntac gia k je èna apì aut thn plhroforða se poiì mìrio an kei kai pio eðnai to eðdoc tou. Opìte, gia ton upologismì twn allhlepidr sewn enìc morðou me ta upìloipa apaiteðtai mìnon h gn sh twn suntetagmènwn twn tmhm twn tou. Apì thn lðsta gnwrðzoume to pl joc kai to eðdoc twn tmhm twn pou katalamb noun tic Ðdiec me autì to mìrio plegmatikèc jèseic kai to dunamikì allhlepðdrashc prokôptei wc upèrjesh aut n twn allhlepidr sewn. Kat thn di rkeia thc prosomoðwshc an log aangðnontai apodektèc ìqi oi MC kin seic oi lðstec enhmer nontai. Sthn eidik perðptwsh ìpou oi allhlepðdraseic twn tmhm twn eðnai tou tôpou sterik c pwshc} se k je plegmatik kuyelðda filoxeneðtai mèqri èna moriakì tm ma. Tìte, mia kðnhsh MC gðnetai apodekt mìnon ìtan den up rqei up rqei allhlepik luyh tmhm twn diaforetik n morðwn.

98 6.1. Teqnikèc leptomèreiec twn upologistik n prosomoi sewn 93 Stic moriakèc prosomoi seic eðnai aparaðthth mia arqik di taxh twn morðwn sto koutð prosomoðwshc. Oi idiìthtec tou sust matoc den prèpei na exart ntai apì thn arqik tou di taxh kai sunep c k je logik di taxh (pou den odhgeð se apeirismì sthn enèrgeia tou sust matoc) eðnai apodekt. O pio aplìc trìpoc gia ton sqhmatismì miac di taxhc, na topojet soume tuqaða ta mìria sto koutð prosomoðwshc, odhgeð (sqedìn sðgoura) se apeirismì thc enèrgeiac tou sust matoc lìgw parousðac sterik n ap sewn metaxô moriak n tmhm twn, kai gifl autì apofeôgetai. Sta sust mata pou melet me xekin me apì mia krustallik plegmatik dom. To m koc miac MC prosomoðwshc metriètai se MC kôklouc (monte carlo cycles). An N eðnai ta mìria tou sust matoc se k je MC kôklo epiqeiroôntai kat mèson ìro N MC kin seic. 'Ena sôsthma eðnai se statistik isorropða ìtan h tim thc puknìthtac pijanìthtac eôreshc miac opoiasd pote mikrokat stashc den exart tai apì ton qrìno. 'Ena kleistì sôsthma p nta brðsketai se isorropða, kalôtera, xodeôei sqedìn ìlo tou ton qrìno se èna mikrì sônolo katast sewn me th megalôterh sqetik puknìthta pijanìthtac. An ìmwc to kleistì sôsthma arqik den eðnai se isorropða ( an gia par deigma teqnht xekin ei apì mia tuqaða di taxh) tìte ja qreiasteð k poio qrìno sthn orologða twn prosomoi sewn arketoôc mc kôklouc gia na exisorrop sei. Sunep c, hdi- adikasða epilog c nèwn katast sewn epanalamb netai mèqri to sôsthma na brejeð se katast seic me meg lh statistik pijanìthta ste sthn sunèqeia na k noume metr seic twn fusik n tou idiot twn. 'Enac aplìc kanìnac gia na diapist soume autì, eðnai na elègqoume kat pìso kumaðnontai oi anamenìmenec timèc diafìrwn posot twn. 'Enac tupikìc èlegqoc eðnai h tim thc sunolik c dunamik c enèrgeiac allhlepðdrashc h opoða ja prèpei na stajeropoihjeð gia ènan sqetik meg lo arijmì mc kôklwn. Opìte, apaiteðtai ènac ikanopoihtikìc arijmìc mc kôklwn ste to sôsthma na exisorrop sei kai afoô sumbeð autì ènac epiprìsjetoc arijmìc kôklwn M pou diasfalðzei ton upologismì mèswn tim n parathr simwn posot twn. Sugkekrimèna h mèsh tim thc parathr simhc posìthtac A ({I}) dðnetai apì thn sqèsh me thn diakômansh σ A aut c A = 1 M σ A = M a=1 ( A {I} (a)) (6.3) A 2 A 2. (6.4) Sthn amèswc epìmenh enìthta eis goume tic aparaðthtec paramètrouc t xhc kai sunart seic susqètishc gia ton poiotikì kai posotikì qarakthrismì twn f sewn pou parathroôme stic moriakèc prosomoi seic. antð tou ìrou metropolis mc prosomoðwsh ja qrhsimopoioôme: MC prosomoðwsh.

99 6.2. Par metroi t xhc kai sunart seic susqètishc Par metroi t xhc kai sunart seic susqètishc Gia na posotikopoi soume to mèso moriakì prosanatolismì sta sust mata pou prosomoi noume eis goume mia posìthta pou onom zetai kôria par metroc t xhc prosanatolismoô S. H teleutaða, deðqnei to mèso bajmì prosanatolismoô tou kôriou moriakoô xona z wc proc thn proexèqousa dieôjunsh thc f shc pou orðzetai apì to nusma tou kateujunt N. O prosdiorismìc thc S gðnetai mèso tou tanust deôterhc t xhc Q zz, me stoiqeða pðnaka 3 Q zz AB = 2 (z I A)(z I B) 1 2 δ AB (6.5) ìpou A, B =(X, Y, Z) eðnai anôsmata sthn kateôjunsh tou ergasthriakoô hmi xona OX OY OZ kai δ AB eðnai to dèlta tou Kronecker. H... eðnai h mèsh tim thc sun rthshc stic agkôlec wc proc ìla ta mìria tou sust matoc pou prosomoi noume. Sthn perðptwsh pou ta mìria up rqoun} se èna kubikì plègma o pðnakac autom twc kajðstatai diag nioc. Apì ton diagwnopoihmèno pðnaka upologðzontai treðc diaforetikèc λ +,λ,λ o idiotimèc. H mègisth idiotim λ + dðnei thn kôria par metro t xhc prosanatolismoô S en to antðstoiqo idiodi nusma aut c prosdiorðzei thn proexèqousa dieôjunsh thc f shc alli c ton kateujunt N. AfoÔ to upì exètash sôsthma èqei exisorrop sei sthn sunèqeia susswreôoume stigmiaðec timèc thc paramètrou t xhc gia thn eôresh thc mèshc tim c kai thc diakômanshc aut c thc posìthtac (apì tic ex.(6.3)-(6.4) ). H par metroc t xhc S deðqnei kat pìso ta mìria eðnai prosanatolismèna se mia proexèqousa dieôjunsh anex rthta an eðnai par llhla antipar llhla se aut n. Sunep c, den mac parèqei plhroforðec gia thn polikìthta thc f shc. Sto plegmatikì moriakì montèlo pou qrhsimopoioôme, gia na exet soume to bajmì polikìthtac miac f shc upologðzoume thn posìthta P A z wc proc k je ergasthriakì hmi xona apì thn exðswsh P z A = N I=1 z I A N A. (6.6) Hpolikìthta PA z, ekfr zei to bajmì prosanatolismoô twn morðwn sthn kateôjunsh tou anôsmatoc A wc proc ton sunolikì arijmì twn morðwn N A me xona z sthn dieôjunsh tou A. Sthn perðptwsh pou den up rqei kanènac moriakìc xonac sthn dieôjunsh tou anôsmatoc A apodðdoume sthn polikìthta thn tim mhdèn. Genik to di sthma tim n thc eðnai sto ( 1, 1) me tim mhdèn gia mia idanik mh polik f sh. Kat thn di rkeia thc prosomoðwshc susswreôoume timèc apì thn parap nw exðswsh kai mèsw twn ex.(6.3)-(6.4) brðskoume thn mèsh tim kai thn diakômansh aut c. H dom enìc moriakoô sust matoc mporeð na qarakthristeð mèsw sunart sewn susqètishc, h aploôsterh twn opoðwn eðnai h an zeôgh aktinik sun rthsh susqètishc g (R). Aut h sun rthsh dðnei thn pijanìthta na broôme èna zeug ri foullerenik n mon dwn (dôo diaforetik n morðwn) se apìstash metaxô R kai R +ΔR se sqèsh me thn pijanìthta pou anamènetai gia èna isìtropo reustì thc Ðdiac

100 6.2. Par metroi t xhc kai sunart seic susqètishc 95 puknìthtac. Stic moriakèc prosomoi seic qrhsimopoioôme thn parak tw èkfrash gia thn exagwg thc g (R) g (R) = δ (R R IJ) IJ (N 1) ΔV 1 ρ ( ìpou ρ h puknìthtatou isìtropou reustoô kai ΔV 1 =4π/3 R 3 (R +ΔR) 3) (apì thn gewmetrik kataskeu enìc sfairikoô floioô). Ed, to... IJ ekfr zei th mèsh tim thc sun rthshc stic agkôlec wc proc ìla ta zeug ria morðwn (I <J) mèqri mia orismènh sqetik apìstash Ðsh me to hmim koc thc mikrìterhc akm c tou koutioô prosomoðwshc. Praktik, h sun rthsh dèlta antikajðstatai apì mia sun rthsh pou den eðnai mhdèn gia mikrèc diamerðseic kai èna istìgramma thc g (R) stoiqeiojeteðtai apì thn suss reush moriak n zeugari n pou upeisèrqontai se k je diamèrish kat thn di rkeia thc prosomoðwshc. (6.7) Gia na exet soume thn t xh wc proc tic jèseic me megalôterh leptomèreia eis goume thn aktinik sun rthsh susqètishc g (R ). Aut perigr fei thn pijanìthta na broôme èna zeug ri foullerenik n mon dwn (diaforetik n morðwn) se apìstash R kai R +ΔR pou eðnai to mètro thc k jethc probol c tou diamoriakoô dianôsmatoc ston kateujunt. Stic prosomoi seic upologðzoume thn posìthta g (R )= δ (R IJ N) δ (R R IJ ) IJ (N 1) ΔV 2 ρ (6.8) ìpou R IJ = RIJ 2 (R IJ N) 2. Ed, gia k je diamèrish kataskeu zetai ènac kulindrikìc floiìc ( ) ìpou to ΔV 2 = π (R +ΔR ) 2 R 2, me to Ôyoc tou floioô na jewreðtai mia grammik mon da. UpologÐsame thn aktinik sun rthsh susqètishc prosanatolismoô-jèshc apì thn exðswsh g θ (R )= δ (R R IJ )cosθ IJ IJ δ (R R IJ ) IJ (6.9) ìpou θ IJ =cos 1 (z I z J ) h gwnða pou sqhmatðzoun oi moriakoð xonec z I kai z J twn morðwn I,J. H sun rthsh aut mac epitrèpei na elègxoume tìso thn polikìthta miac f shc ìso kai thn polikìthta twn kiìnwn aut c (ìtan prìkeitai gia kionikèc f seic). Se mia mh polik f sh perimènoume ìti apì polô mikrèc apost seic h tim aut c thc sun rthshc fjðnei gr gora sto mhdèn. H didi stath plegmatik apeikìnish twn f sewn pou prosomoi noume epitugq netai me thn sun rthsh f (x, y) = δ (x Δx IJ) δ (y Δy IJ ) IJ (N 1) ΔV 3 ρ (6.10) ìpou ΔV 3 =ΔxΔyσ, kai h opoða dðnei thn pijanìthta emf nishc enìc zeugarioô foullerenik n mon dwn diaforetik n morðwn se apìstash x kai x +Δx kai tautìqrona y kai y +Δy (anex rthta

101 6.3. Apotelèsmata - suz thsh 96 Sq ma 6.2: Di gramma upologismènhc paramètrou t xhc prosanatolismoô wc proc thn puknìthta (bajmìc kat lhyhc) ( S n) gia ajermikì sôsthma. apì ton prosanatolismì touc) wc proc thn pijanìthta pou anamènetai gia èna isìtropo reustì thc Ðdiac puknìthtac. Me σ sumbolðzoume to hmim koc thc mikrìterhc akm c tou koutioô prosomoðwshc. 6.3 Apotelèsmata - suz thsh Ajermikì montèlo tou CSMC60 Arqik melet me sust mata ìpou to diamoriakì dunamikì allhlepðdrashc perigr fetai apì tic paramètrouc u (o) (f,f) =, u (o) (f,m) = kai u (o) (m, m) = en eðnai mhdèn ìtan den up rqei allhlepik luyh metaxô tmhm twn diaforetik n morðwn (dec upoenìthta gia ton orismì twn f,m kai gia tic epiplèon diamoriakèc allhlepidr seic). Melet me dhlad sust mata qwrðc jermokrasiak ex rthsh. H puknìthta (bajmìc kat lhyhc (packing fraction) ) orðzetai wc n = Kv o ρ, ìpou ρ eðnai h arijmhtik puknìthta ρ = N/V kai K = 25 o arijmìc twn moriak n tmhm twn. SqhmatÐzoume mia krustallik dom pl rouc kat lhyhc (n =1). H seir twn prosomoi sewn xekin ei apì thn krustallik dom all me atèleiec} (n <1) afoô èqoume afairèsei tuqaða orismèno arijmì morðwn apì aut. Sthn sunèqeia, h exisorrophmènh diamìrfwsh orismènhc puknìthtac, qrhsimopoieðtai san arqik diamìrfwsh thc epìmenhc prosomoðwshc, qamhlìterhc puknìthtac, afoô afairèsoume tuqaða ènan orismèno arijmì morðwn diathr ntac ton ìgko tou sust matoc stajerì. Gia k je prosomoðwsh apaitoôntai kat mèso ìro 10 6 MC kôkloi gia thn exisorrop sh kai oi mèsec timèc idiot twn tou sust matoc upologðzontai me epiprìsjeta 10 6 MC kôklouc. Sta sust mata pou prosomoi noume o arijmìc twn morðwn xekin ei me N = 1668 mìria sthn puknìthta kat lhyhc n =0, 9653, ta opoða mei nontai kat 20 (se k je b ma), ft nontac ta N = 528 sthn puknìthta n =0, Na shmei soume ìti exet same kai mikrìtera sust mata me

102 6.3. Apotelèsmata - suz thsh 97 a) b) Sq ma 6.3: Tupik stigmiìtupa se puknìthtec kat lhyhc a) n =0, 4098 kai b) n =0, 8264 gia ajermik sôsthmata me N = 708 kai N = 1428 adropoihmèna mìria antðstoiqa. K je mìrio anaparðstantaime to sq ma tou kìlourou k nou kai k je mia apì tic èxi dunatèc moriakèc kateujônseic apodðdetai me diaforetikì qr ma. Sta stigmiìtupa, k je o kìlouroc k noc emfanðzetai sumpag c en den apeikonðzetai h koilìthta autoô. t xh megèjouc arijmoô morðwn Ta apotelèsmata pou lamb noume eðnai statistik isodônama me ta megalôtera sust mata. Gia na qarakthrðsoume posotik thn t xh wc proc ton moriakì prosanatolismì upologðsame thn par metro t xhc S. To di gramma thc paramètrou t xhc S - puknìthtac n dðnetai sto sq ma 6.2. Apì autì katalabaðnoume ìti to sôsthma metabaðnei apì mia f sh qwrðc kajìlou moriak t xh prosanatolismoô se mia llh me uyhl t xh prosanatolismoô. To mègejoc thc S all zei xekin ntac apì mia tim kont sto mhdèn (se qamhlèc puknìthtec) kai apì ekeð aux netai san sun rthsh thc puknìthtac. 'Enac aplìc trìpoc gia na diapist soume tic domikèc diaforèc an mesa stic dôo f seic eðnai optikopoi ntac ta apotelèsmata me thn l yh antiproswpeutik n stigmiìtupwn exisorrophmènwn katast sewn pou dðnontai sto sq ma 6.3. Ta mìria anaparðstantai me to sq ma tou kìlourou k nou kai an loga me thn kateôjunsh tou kôriou xon touc apodðdetai se aut diaforetikì qr ma; o kìlouroc k noc emfanðzetai sumpag c sto sq ma 6.3, en den apeikonðzetai h koilìthta autoô. Stic qamhlèc puknìthtec (n <0, 55) lamb noume thn eikìna enìc isìtropou reustoô. Apì ta stigmiìtupa aut c thc perioq c eik zoume ìti to sôsthma den emfanðzei uyhl t xh tìso wc proc tic moriakèc jèseic ìso kai wc proc ton prosanatolismì. 'Oson afor ton prosanatolismì autì epibebai netai apì thn S. Apì thn llh meri gia n>0, 80 to sôsthma brðsketai se mia f sh me uyhl t xh prosanatolismoô all kai jèshc (ìpwc eðnai fanerì apì to sq ma 6.3b). Ta mìria autoorgan nontai topojetoômena to èna mèsa sthn kwnik koilìthta tou llou sqhmatðzontac kðonec, oi opoðoi me thn seir touc diat ssontai par llhla metaxô touc, qarakthristikì thc kionik c f shc. Lìgw thc fôshc tou diamoriakoô dunamikoô oi sqetikèc jèseic kai prosanatolismoð metaxô geitonik n morðwn eðnai arket periorismènec. Autì èqei san apotèlesma h met bash apì thn kionik sthn isìtroph f sh na mhn entopðzetai se mia sqetik sten perioq

103 6.3. Apotelèsmata - suz thsh 98 a) b) Sq ma 6.4: Upologismèna diagr mmata a) thc aktinik c sun rthshc susqètishc diamoriak n apost sewn foullerenik n mon dwn g (R) kai b) thc aktinik c sun rthshc susqètishc diamoriak n probol n twn apost sewn foullerenik n mon dwn k jeta ston kateujunt g (R, ) se di forec puknìthtec gia ajermik sôsthmata. (kìkkinh gramm : isìtroph f sh kai mple gramm : kionik f sh). puknot twn all na parousi zei axioshmeðwto eôroc (0, 6 <n<0, 8), sto opoðo h diakômansh thc S èqei sqetik auxhmènh tim. Hexètash thc t xhc wc proc tic moriakèc jèseic gðnetai me thn eisagwg thc aktinik c sun rthshc susqètishc g (R). To di gramma thc g (R) dðnetai sto sq ma 6.4a gia dôo diaforetikèc puknìthtec. Gia thn qamhl puknìthta (n =0, 4098) h g (R) teðnei sthn mon da qwrðc èntonec diakum nseic all parousi zei epiprìsjeta mia sqetik protðmhsh twn foullerenik n mon dwn na brðskontai se eggôtatec plegmatikèc kuyelðdec gia R =1. Lamb nontac upìyh kai thn suz thsh thc prohgoômenhc paragr fou jewroôme ìti h f sh aut eðnai isìtroph. To di gramma thc g (R) gia uyhlìterh puknìthta (n =0, 8264) mac parèqei mia pr th èndeixh kanonikìthtac sthn di taxh twn morðwn sthn f sh. H meg lhc èntashc korufèc gia R 2, ofeðlontai sthn autoorg nwsh twn morðwn se kðonec. IdiaÐtera aut gia R =1faner nei ìti h kurðarqh diamoriak apìstash mèsa se ènan kðona eðnai tou megèjouc miac foullerenik c mon dac. Oi epìmenec dôo korufèc me sqetik uyhl èntash gia R 5 upodhl noun kanonikìthta sthn dieujèthsh twn kiìnwn sthn f sh. Autèc oi apost seic eðnai qarakthristikèc pr tou kai deôterou geðtona tou tetragwnikoô plègmatoc. Peraitèrw diereônhsh thc teleutaðac prìtashc gðnetai me thn eisagwg thc aktinik c sun rthshc g (R ), pou dðnetai sto sq ma 6.4b. Oi tèssereic uyhl c èntashc korufèc pou parathroôntai ekfr zoun tic antðstoiqec kontinèc apost seic metaxô plegmatik n shmeðwn enìc tetragwnikoô plègmatoc. ApodeiknÔetai ètsi o sqhmatismìc enìc didi statou krustallikoô plègmatoc apì thn taktopoðhsh twn kiìnwn. H diakionik apìstash eðnai 5 kuyelðdec ìso kai h pleur thc b shc tou adropoihmènou morðou. Tèloc, h sqetik mikr c èntashc koruf gia R =1ofeÐletai sthn

104 6.3. Apotelèsmata - suz thsh 99 a) b) Sq ma 6.5: Trisdi stath apeikìnish twn sunart sewn susqètishc a) f (x, y) kaib)hg θ (R ) gia ajermikì sôsthma, sthn kionik f sh. dieujèthsh se seir morðwn me prosanatolismì k jeto ston kateujunt (sto sq ma 6.3b mporeð kaneðc na parathr sei tètoiec dieujet seic metaxô twn morðwn). H apeikìnish thc plegmatik c dom c thc kionik c f shc mporeð na gðnei me thn sun rthsh f (x, y). To an glufo thc trisdi stathc morfologðac aut c perilamb nei uyhl c èntashc korufèc se plegmatikèc jèseic enìc tetragwnikoô plègmatoc (sq ma 6.5a). Ex llou, ta apotelèsmata thcf (x, y) mporoôn na sugkrijoôn me aut thc g (R ) ìpou afoô ta orðsmata sundèontai me thn sqèsh R 2 =(x)2 +(y) 2, mporoôme na diapist soume thn metaxô touc sumfwnða. Strèfoume t ra thn prosoq mac sthn diereônhsh tou trìpou me ton opoðo autoorgan nontai ta mìria mèsa stouc kðonec. Autì gðnetai me thn aktinik sun rthsh g θ (R ). H sun rthsh aut parousi zei mia mh mhdenik koruf ìtan R =0ìpwc faðnetai sto sq ma 6.5b. Se llec sqetikèc apost seic eðte eðnai mhdèn eðte emfanðzei mikrìterhc èntashc korufèc. 'Omwc k ti tètoio sumbaðnei: a) ìtan k je kðonac parousi zei polikìthta me ta diadoqik mìria (apì ta opoða apoteleðtai o kðonac) na stoib zontai to èna mèsa sthn koilìthta tou llou ètsi ste o prosanatolismìc touc na eðnai sthn Ðdia kateôjunsh kai b) ìtan sunolik h f sh eðnai mh polik opìte kai oi suneisforèc (gia timèc di forec tou R diaforetikèc tou mhdenìc) allhloexoudeter nontai ìtan prostðjentai algebrik lìg wtou cos (z I z J ) sthn ex.(6.9). Epeid sta sust mata pou melet me o kateujunt c (ek twn pragm twn) eðnai par llhloc me ènan apì touc treðc ergasthriakoôc xonec (pq. me ton OZ) kai k jetoc stouc upìloipouc dôo, elègqoume kat pìso oi topikoð moriakoð xonec prosanatolðzontai par llhla antipar llhla me touc hmi xonec tou ergasthriakoô sust matoc suntetagmènwn. Gia thn posotikopoðhsh qrhsimopoioôme thn par - metro thc polikìthtac PA z wc proc touc hmi xonec (A = X, Y, Z). Ta upologismèna diagr mmata gia mia seir puknot twn dðnontai sto sq ma 6.6. Apì aut eðnai fanerì ìti h kionik f sh den eðnai polik (sq ma 6.6g). H sunolik polikìthta thc kionik c f shc prosdiorðzetai apì entropikoôc

105 6.3. Apotelèsmata - suz thsh 100 a) b) g) Sq ma 6.6: Upologismèna diagr mmata thc polikìthtac - puknìthtac gia to ajermikì sôsthma a) P z X wc proc ton ergasthriakì xona OX, b) P z Y wc proc ton ergasthriakì xona OY kai g) P z Z wc proc ton ergasthriakì xona OZ, me tic antðstoiqec apoklðseic gia k je tim (error bars).

106 6.3. Apotelèsmata - suz thsh 101 par gontec kajist ntac tic mh polikèc f seic eustajèsterec apì tic polikèc. Gia touc parap nw lìgouc h dieujèthsh twn polik n kiìnwn sto tetragwnikì plègma mporeð na gðnei me di forouc trìpouc Montèlo tou CSMC60 me malakèc} diamoriakèc allhlepidr seic Esti zoume t ra se mia epilegmènh perioq puknot twn kai melet me thn sumperifor sust matoc allhlepidr ntwn CSMC60 wc sun rthsh thc apìluthc jermokrasðac. H jewrhtik diereônhsh thc fasik c sumperifor c CSMC60 (enìthta 3.5), èdeixe ìti aut stoib zontai to èna sthn koilìthta tou llou sqhmatðzontac kðonec. Epiplèon hdieujèthsh eðnai tètoia ste na eunoeðtai h dìmhsh polik n kiìnwn. H parap nw sumperifor, dhlad h parousða ugrokrustallik c f shc me uyhl t xh wc proc ton prosanatolismì kai th jèsh ìpwc eðnai h kionik, parathreðtai gia èna meg lo eôroc jermokrasi n kai pièsewn. Epiprìsjeta ìmwc, sthn perioq uyhl n jermokrasi n kai pièsewn parathreðtai mia lwrðda smhktik c f shc me apl stoib da foullerenðwn. 'Etsi loipìn eðnai dunatèc oi jermotropikèc akoloujðec I SmA Col kai h I Col. IdiaÐtero endiafèron parousi zei to di gramma arijmhtik n puknot twn (twn f sewn se jermodunamik isorropða) wc sun rthsh thc antðstofhc jermokrasðac (blèpe sto trðto kef laio sq ma 3.15b). Apì autì diapist noume th dunatìthta, se mia perioq puknot twn, to sôsthma na metabeð apì thn isìtroph sthn kionik f sh (me meðwsh thc jermokrasðac) afoô pr ta dièljei apì mia perioq ìpou sumbaðnei sunôparxh aut n. Aut ta apotelèsmata mac epitrèpoun thn exètash thc fasik c sumperifor c tou CSMC60 me thn qr sh moriak n prosomoi sewn sthn kanonik sullog, afoô jewrhtik up rqei mia endi mesh perioq puknot twn ìpou to sôsthma metabaðnei apì thn isìtroph sthn kionik en eðnai jermodunamik astajèc se mia sten perioq jermokrasi n. H allhlepðdrash metaxô moriak n tmhm twn diaforetik n morðwn parametrikopoieðtai ìpwc kai sthn perðptwsh pou exet same me thn efarmog thc moriak c jewrðac, dhlad wc ex c: u (o) (f,f) =, u (o) (f,m) = kai u (o) (m, m) = u>0. Ed to u eðnai to mètro thc apwstikìthtac metaxô twn tmhm twn diaforetik n morðwn kai apodðdoume se autì aujaðreta mia jetik tim. Epiplèon eis goume kai tic allhlepidr seic pou perigr foume sthn upoenìthta Sthn pragmatikìthta mac endiafèrei h sumperifor tou sust matoc wc sun rthsh thc posìthtac βu. Se sqetik uyhlèc timèc tou βu (dhlad se sqetik qamhlèc jermokrasðec) to sôsthma akoloujeð thn sumperifor enìc ajermikoô sust matoc (anex rthtou thc jermokrasðac, opìte kai metabaðnoume sthn perðptwsh twn sklhr n} allhlepidr sewn). Ta sust mata pou prosomoi same perièqoun N = 1728 mìria se puknìthta n =0, Ektelèsame dôo seirèc moriak n prosomoi sewn: h pr th antistoiqeð se jèrmansh tou sust matoc xekin ntac apì mia qamhl jermokrasða kai h deôterh se yôxh autoô xekin ntac apì uyhlèc jermokrasðec.

107 6.3. Apotelèsmata - suz thsh 102 Sq ma 6.7: Upologismèno di gramma paramètrou t xhc prosanatolismoô - paramètrou allhlepðdrashc (βu) gia sust mata pou apoteloôntai apì N = 1728 mìria se puknìthta n =0, Ta mwb stðgmata lamb nontai me yôxh kai ta portokalð me jèrmansh (se k je shmeðo up rqoun kai oi antðstoiqec apoklðseic apì thn mèsh tim (error bars) ). a) b) Sq ma 6.8: Tupik stigmiìtupa gia dôo timèc thc paramètrou allhlepðdrashc (jermokrasðac) a) βu = 0, 4 kai b) βu = 1, 6 susthm twn pou apoteloôntai apì N = 1728 mìria se puknìthta n =0, Gia k je mia apì tic èxi dunatèc moriakèc kateujônseic qrhsimopoioôntai diaforetik qr mata. Sta stigmiìtupa, k je o kìlouroc k noc emfanðzetai sumpag c kai den apeikonðzetai h koilìthta autoô.