= ON K,,._ = ON K lltONC

• ••

Aequisitiortsand ~.1fI BibliOgraphiC:SeMon MrvcesbibIiOgmphiquM

3lI5~SIr_ 395... W- o -

= ON K,,._ = ON K lltONC

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().612-63948-7

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Monte C arlo and M ean Field Studies of Polymers in Solution

hy

©xturcPepin

:'.I.Sr.(Q ueen'sLniversuy.Kings ton.Canada)199 1 B.Se.H.(Q U I"'U'SLnivr-rsir y,Kingston.Canuda]1990

Athes issubrult ted totill' Sch oolofGrad ua teStud ies in partialFulfilmentofthe n-qnircmcursforthe degreeof

Doct or uf Philosophy

Dopar tmeur of PhysicsandPhysical Oc ean ography :>.Iclllo ria l l"n iwrsityof Xowfou udl and

{September.19!J9)

St..101m's Xc wfo uudla ud

Ab stract

This thes isPll"Sf'lits a("()lIlp«,II(,IISi\~st udyof111(, etrtn-tureandphysit:alprop.'niC'S orpolvrnce-sinsolunon.TIl<' focusis on~loTltrCarlo()ole )sim ula t ions,Thef1'Sllhs nrccompa redwithmenufield theoreticalpredk-rjons andusedro studytheIimi t atiolls oftill'meanfieldthco nes.

FOUldist inctS~·Sh'IIISareInvo- ngarcd Thefirstonel'ousistsofA,-I~Ddihl'Kk copolymer"cre w-cut "min·llrsin.-1solvent.Ther...latiwlylongBblockislncoru- paublowiththe solventand formstllf'coreofrho mil-dll's.andtilerclanvclvsho rt .-lblockformsathinr-omna.ResultsIromsimulati onsforthesizeof theI:Uf('aliiI

functio nofthemolecularwd~ htortill'Bhlockarecompared with simpk-mcunfield throril'S illtheliterntu rr-alldextensionsinthis thl'Sis_Tlwysup porttheargll llll' lI- tthat theweak...rdl'lK'udl'lI'-'" ollSl'rn'tlill reecnt.'XI}f'ri nw lltsisa nou-equilib muu ('lfoct.

\\·lIcIIasma ll amountofBhomopolymer ili add,,,,ttotill'1>10<'1.. copofymeesolution.

it cau Iw sollll.ilizl't! withinth('mil-...II(' l"OH'S andswr-llthemicelles.orseparate into amacro phasewiththe'"OIHJIYlUer.;attit!'11IJIIIOIHJr~-lUl'r-sohl" lltinterface au.l/orill micelles.ResultsIroru)Olonl!'Carlo simulationsshowitthrr-sholr l\-0 1111111"Iracnouof hom opo lymerbe lowwhichthohomopolymer issolubilizedwithinthe micellecores and abovewhichmarrophasl'separa tionoccu rs, TILl'S!' resul ts arcinquulltat fve agreement withprevious experimr-ursandaslmple mounfieldtheory.

In111(' thirdxys t eru,0111'endofeachpolymer isr-nd-terhcredtoasurface andtlJ('

iii

remaining secnon orthe polYIUC'rstretches Int o~OO(Isolventrormi ngalleud-te rhored I,,~"(,r.Thefourthsystemindlldl'S rr('(' polymerillsolut ion.TIlt' rO<'II,;orthisworkis 011,;:,:st('1IITl'g illlt'SwhichcorrespondtotheN:'st udiedillfIIO>'Iexperi ments.Inhoth svsn-ms.Ih('results orth...:l.ICsimulutionsagree\\"('11withthose ortill'nuuu-rirnl soH-co nsl stom field(XSCF')r-air-ulutionsIorsnrra('(' rourontrut ionsabovea th roshohl.

Asl"il linJ!; analysisorthr-t.lyl'rIhi('knt"SSshows thatII...sySlt' IJISarcnot inthr-limit or highmolecular wl'ilthtandhi~h l ~'strNdl('l!chains.Furthermore.insystems wuh rda l h"f'lyhilthmolecula r wl.'ig!ltIrec po lyrnors.Ih" dog ecc

or

penerr a non ortill'free polymer int o till'end - tether ed lay f'r is great er than pred ict ed by a.'iYIllI'to lil'SCF (Iwori,.,;.althoug h stilllt'S.'ithan ulea-rvedin recentexperi ments.

Tomy gra nd mothe r Flor ette andto myparents.

Cont ents

Abstract Listof Tab les Listof Figu r es Ackn owle d geme n t s 1 Introduction

1.1 Polyme rsillSolution-Gcn r-ru l Remarks 1.2 Thcorc ucutXlodcls.

1.3 Crew-c ut~l in'll('s.

1...1 5wo\11'11\[k dlf's

1.;j Encl-tr-tlu-rcd La Y<'fS

l.a

ThillFilius 2 Mon teCarloSimulat io ns

2.1 lntroductor vRem a rks

'J'J ~IolLtf'Cnrl~Simu la tio nsof Pclvmc rs. 2.2 .1 Fr N.'Pul YlUf'r s. . 2.2.2 Eud-tethen-dPolvrur-rs

3 MeanField The or y

.1.1 Intro d uc t oryRemarks 3.1 :\Iin, lll's

.'3.2.1 Gcm-ralCase .1.1.2 StrongSegregation limit 3.3 Swolle n:\licel hos.

3.... XumertcalSelf-consls re nt FieldTheo n "

.'3.-1.1 \[(' a llFieldApprc ximat.io u. -

3.-1.1 Summary-Self.eo ns ts rour:\II"\ILField Theory

viii

Ix xii 1 1 G 13

"

19

'J ' )

26 :!6

,.

31 38 41 H

.,

.1 .8

51 58 GO

'"

:l.-L} Pro hlorus inOllt'Dimension .5

.. Cre w-cue ;\Iic elles 79

-1.1 Inr rodu c rion and Review. 79

-1.1 ~Il'a nFieldStllcl~... S6

-'.3 ~Ionl('CarloSI\I(I\' 91

-1.3.1 Monte CarteSimulations, 91

-1,3.1 :\IC Sim ulationsofitTypicalS:,~U'UI 9-1

-I.3.:} Sysll'lIlatk~Iont('CarloSt ud y 10.1

-1.-1 Snmmarv 108

5 Swo ll enMtc etle s 111

5.1 lutrodm-tionandRovn-w. II I

'i.:.!MonteCitr inSilll u\;uions. II:}

3.:1 Result saud Discussion, 11'i

".3.1 Swollen~li('dl l'Svstr-m. IV)

;;.3.:.! \lin u phaSl'\"S.~ia('foplli\S('Scpurnrion. 1:.!6

5.-1Suuuna rv 1.1-1

I) En d - t erheee d Polyme r s 137

6.1 lnt roducfionamiRr-vu-w• 13.

6.2 ~lollt.,CarloSimu lations. 1-1-1

6.3 XlUll('rkal

s-rr.

r-OIl S is h 'lllFif'ldTh{'(Iry 1·16

6.-1 R<'Suh s andDiSl:"llS:..iou. 1-17

6.-1.1 Free PolvruersinGOOtI 501\"('1\1 1-17

6.-1.2 Entl-t.·t1;.'rrd Potvmers.. 1,;0

G.-I.:} O<'llsit~·Pro filt"S: 1.; 1

6.-1.-1 Lay('rThir-kness, 15·;

G,:; Sununarv IGI

7 Thin Films 165

•.1 lurr oduc-tion awl Rr-view . !G.;

•.1 ~IontrCarlo Sil!lu lat iollS, 173

r,3 Xumcrh-alSd f-collsis(l:"n1Fid d Theory 17-1

•.-1 Results and DiS<'us...ion. I7G

•.-'.1

xrc

Shnul arlcnsandXSCF Theorv Irr

,.-1.1 Com pa risonwithExporimeuta. . 18-1

7.-1.3 Sysll' lIIalkStudy;Volume Fract ionProfiles ,Overlap and Layer

Height 186

7.-1.-1 5yst('lllilt j(-St udy;Scalingofthe Laver H..ighr . 191

r.3 Su mm a ry 201

8 Conclusions 8. 1 IutroducrorvRemarks S.:? Sunuuary ofR,'SIl!tS S.3 FinalRemarks. Bibliography A MonteCa rl o Algorithm

A. 1Generating\"('wStates A.1.1 Choosing\"ew Configu rations A.I.:?Illllslfa t iwCases oftheCh a nge inEncrgj A.1.:1 Autoco r relation Functions A.1A AveragedQuantitil's B InterfacialTension

RI Spr-rialCasl'

vii

20.

lOG :WS 11:3

215 227 :?:!i 2:!S 2:!!) :!J J :B. j 23.

J·U

vif

List of Tables

1.1 Scalingla ws furthecore ra d iusofmicellesfrumva r io usrhooru-sami

expcrirue urs. 82

1.:1 Ra d ius ofcore.thirkncss ofcoro n aandnumber ofcopolymers per

micellefrom experi ment(1]and theo ry. 8i

-1..1 Res ultsfortheradius of aggr egate co res(Re~,,).thonumberofcop olv- mors peraAAft'gate(X u)and the reducedintera ction('Ilt'rg )' (telfor r;=8.000andT<=12.0lMJfrom~ICsimula t ions atdifferr-nrmolecular

weights. 103

.5.1 Resultsfro m rhofit of theaurocor rolanonfunctionsto a slimofrwo oxpoueunalfunct io ns . l

=

0.31. Zc..\= -12.ZCB

=

30.Z/f

=

-I.

oc=O.025 an doll lJ=0.OI2 5 Ii i

G.I Gen eral spocifications of svstr-ms modelr-dilltheXSCFr-ak-ulanons

and\!ont('Carlosim ula t io ns. 111

i.l Powerlaw dependencesfor thelaver he ightIIfromsca lingandSCF rhr-onoxfordiffore ur regimes andcquationsforboundartos be t ween

regmu-s lUi

i.2 Descript ion of \ICsimularion sandXSCF calcularlonsrorthinfilms. 171 i.3 PO\\"C'f lawdepo ndoucosforth elayr-rheight II fromseatingandSCF

thour-ics.andr-alculatorlpo werlaw sfro m theXSC Fcalculnt.io ns ofhr ",.

for differ e ntregimes.There gimesaredctenumcdhy thescalingand

anaiytica ltheories, 19.';

:\.1 Profi leofthe

cpe

usa geforatypica l~ICsouulauon. 2;33

lx

List of Figure s

1.1 Cilt('~()rilat i(J 1Iof"ariOllSthco roticalmClhod s . 6

1.2 Sd lt'lIlillie picturo of blockcopoly-mer-mlcelh-s. I~

1.:3 Sc he rnaticpk-tur e ofmicellesswollenhyhomopolymerand mncrophuso

separatedhomopolvmr-r. 18

IA Polvtuerchains eud-tcthereclbyO/ICendtoasurface :a)o'

«

Iand

il)0' :»I. 21

1.1 vartouselement ary :'>IC1!I0 \ ·C'S . :}.1

3.1 Struc tureofthe("[("W-c.'11tmin -lies. . ~3

-I_I Coreradii....ofmirr-lle(l u )a...afunc tio n oftill'.ll'gr('(' ofpolyru ertzarion

ofthe[Jblock(Zrn). 88

1_1 Corerudlu s ;lndthicknessofco ro naitSfu nctio nsofthed(,,~R""ofpoly- rucriaarionofthe.-landDbloc ks.has

=

I.:.'G). 90

~.:3 (a)Semi-Jogplot IIftherele vant autocorrela rloutillll"Sa...fuucuous of thr-iut e rucr ion('II('rg.'".(II)Fr actionofpolymersiiimicelles an dsma ll

aAAr<'gatt'!<\~r, 95

~A Semi-logplot oftILl'weighted and Ilon-w('ighlrdchainexrruc rionand

exchangeautccorretanonlil1l('S\-Sc, 9.

~.5 Xurrualjzed distrih ill io llof aggregatesasaInnerionofthe numberof copolj-mers intbeaggregates .

« ( =

0.1~8.Zr.~

=

10.Ze'II

=

~) 99

~.6 Volumefnu-li oll profilesof.-land[Jblocksandnorma lized distrib u- tionsof.-\audDends,and.-l~[Jjo intsasIuucuons ofthe dtsranee fromtill' ('('II!('rofmass (C.:'>!.)oflilt'lIIic<,I\.'S.[e=0.2-\8 .ZC.-I=10.

Zcn=50) 101

·1.T Product ofrho("("1111('("1\internet ionl'nergylint! thl'chainle ng th.ofthe Dblock(il) and radi usofgy rat ionoftill'voreof themice lles(L)as

functionsof tilt>dlainlr-ngthoftheBblock . lOG

.

j .1 Exrruction.I'XI'h,U1gf' ant!eud-to-endvee-tor....uroccrrelaricnrUlIct iuns versustimefo rcopolymerandhomopolymer.f =0.31.2c.~=-11.

Zcn=30.21t=-I.Oc=O.01.'jand0/10=0.0I 1.j. 116

·j.1 St-Illi-!ogplotoftbcrelevant autocorrr -lauontimes>I...afuuctto nofrhc im('ral·tiollf'1If'rK,.\·forhoth.copo lymerandhomopolyme r. 119

·'j.3 Xonualtecddistribut ion oragg rega tesa."arlllU·t ionofthenumberor copolvrner(a)orhomo pol vmr-r(hIin agg rt"gatl'S. 111 .

'j...J Xu rnbcrofsmallaggregatesillrhosyst e m>I...a Iuucri cuofthe imerur- tionenergj-,f.TileIuset sho wsthenumberormicellesinthr-s~'SIt'1lI

asa functt onoff. 111

.

j.::; Fracrfcuorpolymerilla.Q:r<>ga u."Sas....fuucticn ortilt'uucracnon~for

copolymeriII...1homopolymer. 11.1

·

j .6 Yolunn- Ira r-rlouprofilesof.-1.andBblocks ofcopolymer(a)andno r- ma lizeddist r ibu tionsor.-I.andBcOPOI.\"IIII'rendsund.·I-Dcopo ly me r joint s (h)usfunctin ns ofthedis tancefromthecentr-rofmass (C.?>.!.) ortheuiirr-llos.TheIllSI'!in(b)showsrho volume fractio nprofilesof

.-I..andD(,Hdsuud.-1.·0juints. Ll-1

'i.. Sn....p-shotorthepolymer con figu ration:a)01/1flo~'f1=0.1.2" u=6.

b)01fBlo~'lJ=0.5.ZIIU=6. 11.

5.8 St-lIli-lo~plotorthe rr! t'\'antuutoco rr ola r iontillll-'S

'L "

Innerionsor tht' imcracuonenergy forcopolymerandhoruopolcmer:a)o~/"IO~'H""

0.1.211 fl=G andh)O~' Bloh,=0.;';.2"" =6 119

::;.9 Volumerr••ctionprofill'S'IS[u nct.ionsofth<' dist illll"l'Irom rhr-l"l'U1l'r of lila ...>;(C.?>.!.) ofthemk e I11"S:a)O~'Hlo':.-lI

=

0.1.2" H

=

6and h)

o~llllo':-H

=

0.::;.2,,11

=

6. _, , . . . ... 131

.j.1O~orlllali 7.('(1distribution of aggregates;L>;IIIuncr icu ofthcnumberof 11tJlIlo poly rnl'tl>pillSccpolynn- rs ill rheaMrl'g all'S;....)01'Hlo~-H=O. L

2 UR=6.h)o~Hllo~R= 0.5.2u n= 6. 131

;j.1IPh>l.o;('dia gra mindicat ingthe lnacro/microph iL>;t'tpgionsIL'iInucrions

oftilt'ratioso~/IIlo~1Iand21/(,12c lI. 13.1

6.1 log-logplotoftill-'il\l'ragt'radius orgyranousquared.

R ;,

and1' 1111- to-enddist a lll-'Csquared.

n

²•as functionsof2andZ-I.rt'Sp('(·tiwly.

forIrecPUI~'1ll1'11'inathe rmalsolution("a k u la t l,,1 illthevlo meCarlo

simulatio ns . US

6.1 .Autoccr rr-larto n fUllct ion s for end-t e t her edpolymerlayC'rs as functions

of rimeferasvst omwitho:

=

1.5 and2.1

=

80. 1::;1

6..3 PolYlIll' rvolumehanion profilesIrom?>.ICsimulat ionand:\"SC fcal-

culationsrot2,.\

=

1Il0 ando:

=

I to 10. 151

6.-1 Pol ym e r\'OhIlIlCfra cti on profiles from ?>.ICsimu lations and~SCfcal-

culat io nsfor~.

=

3.5ami21

=

SOto100. n3

xi 6..) Therm»laverthic knosses.II"".,.as functtonsofthereducedsurfurr-

ccn c-urre noua".fro m:\ICsimu lat io ns and :'\SCFcalr-ulatious. 1::;6

G.6 The"1fl.~laver thicknessesdividedb~'the unperturbedradiu s ofAyra-

lion .h",.x/Rg ,a.sfunc tlo usofIII{'redueodsurfacecouo- ntraticua".

Outvtil!':\fCresultsan'shown l.j D

6.7 :\101'111'Ca rloand:'\SCf layer thicknessesplottedas Iuurrionsof the

finedpowerlaw. lUO

7.1 Sketchof thebound ari esfordiffere ntregimespredic-tedhys('al ill~tho-

cr yfo rZF ;:::ZA. 166

7.:2Voluu,c frac tionprofiles ofeud- rerhorcdand freepolymers fo r:\lCsim- ul.utcnsand:,\SCF calculat ions:ZF=:2::;.Z ..\=100 .Of'=O,05~and

a'

=

12(a)anda"

=

1(b). {7S

7..J nll.~thic knessof the end-tet he redlayer(a) an drolauve degrer-ofover- lap(h)as functionsofthereduced su r faceccuccnrrunon for:'lIesim- ulatio nsand:\SCF calculations.Theinsetsho ws a log-logplotofthr- area ofoverla pversusthoreducedsur facecow' rage.TheplotsarcdOI1(, forZ,. « Z.·I.ZF

=

Z.,Iand2f·>Z..t- 180 7..1 Volumefruction profiles ofend-tethered and fH'I'polym ers;'Idirect

com par iso n between :\SCf theory andexpertmcntsof LeeandKent{21.

a)2 F=·H 3 .2,.1=1.6:2·).OF:::0.06an da"

=

12.h)ZF

=

.J.8·W.

Z..•= 1. 6 25. ,,/.,:::0.06andc '=12. 185

7..) \'oluIII('frnctionprofilesofoud-terherodandfreepolym ersfrom:\SCF calculanous:a)21'=-100.Z..,= -I.uno.OF= O.OGanda" =:2.

II)Zf'=~Oo.Z.I=-1.000 .OF

=

0.06ando'=1:2.c)2F=-1.000 Z,I=-100.0/.'=0.06anda'=2.d)ZF=-1.000.Z..•=-100,OF=0.00

ando:

=

12. 187

7.6 Log-logplots(Iftherelativedegree of over la p.oo- (a) andovcrl.ap.fll.

(h)H' rS\lStheredurr-dsu rfa ceconcenr rarion fo rZF-c2....ZF

=

Z,.l.

2f·:»Z.,\and0.005::::OF::::0.06. 190

7.7 r11l.~llci!l;h tofrlu-end-u-r horedlaye rversustilt'reducedsu rfa c erou- ccntrarionforZF-e

z..

21'

=

Z,I'2 F

»

Z..•and0.005:SOf'

s

^{0.06 . 192}

7.8 Lo/-!;-Iol;plot ofthenTl.~heightoftheend- tet ho redlayer (a).ma ximum volumefr ar-tion ofthr-polymerinthecud-tethe redla.n'r(II)and re t- ativedegreeofoverlap (c)asfunctionsof thevolumefract io noffn'l' PolYIIII'r0Ffor 2f

·«

2....21'»Z...anda' =:2. 1:2. 198

A. I Variouselr-momary:\lC1II0H' S . 231

0.1 \'OhIllICfrac t ionprofiles ofthe.-l.andBhomopolymer. 2.39

xii Acknowledge ments

I wouldlike totakr- th isnp portunitytotha n k all thos ewhoha n'madecontrihu- tio nstothisdisscr t ur lou.

Iwish to tha nk mythe sissupervisor.Dr.vlurkD.Whtuno rc.for sh a r ing his knowledgeandexpertiso. Hisconsmur su pport hasmad e th is wor kpossiblo.Ialso tluurkhimIor hisInterestand effortsillIllyca reeras a teacherand physic-ist.

tthank Dr..101m \\'h itdl('adandDr..lo hndeBruvn fo r acting asmembers oflily Supe r visoryCcmmtu ec and fortln-irguidancethroughouttheprogram.

Iamvervgra t efu lto Dr.Gary Slater for sha r ing histimennd r-xpert.iseandto Dr. :>.lkhadS.Kentforinsi ghtfuldiscussions.

I~ratcfllllyackn o wledge th e financial«sststuuccbythcScho o lof Graduate Stud- tcsandDe par t mentofPhy sicsPhysicalOceanographyandintill' form of grad uate fellowships.

Ialsowish to acknowledg e researchassistancefromCa na d a 's C3,("aprogram .III particular,I wishto th an kthe following members ofCa.cafor theirgeucrosit vill su p p lying eoruputingresources andr-xpcrtise: Xlcruoria l Cni\"t'rsit yofXowfoundland.

Cuiversi tv of Alb erta, Luivcrsit v ofCalgar~'andL"n i\"ersitcdevlontrcal . Iwould liketothank allmyFriendsandcoll ea guesin51..Iohn's. :>'Iyfriends havemade this speci a l timeinSt.John'sa wonderfu lexperience.Itha nkDr. Allan B.xla c ls aucfo r hishelp in gcnt ng st a rt ed011mv research. Speci a l rhanks to Dr, RomanBaranowsk i for hisconstanthelp duringIllystud iesand forallow in glilt'to

xiii

usehis codefo rstllll~ofIllyca lcu la t io ns .Ialso t1mnk :l.l'lTtinKenwardforrunn i ng simulatio ns and 11l'lpingwiththe£"O(1f'.

I wouldlik...totha n kUi~·df'ilr loveRauya furah,·ays rrlllilldill F;meth atun-rc is flllid.more(0life-thanaPh.D.thesis.

fillall~·.spec ia lth a nks to1lI~'paren tsandfamily.who hevoalwa ysIK,(,11the refur mo.TheirIon' and<'II,'OuraJ;C" lIIt'lIt art.'greatlyapproclated.

Chapter 1

Introduction

1.1 Polymersin So l u t io n- GeneralRemarks

Inrecen t}'('i lrSpolymersha n 'gainedremar ka ble popularityilldiversefiddsof STudy.TIll' Interestill polymersisutrributedtothe wideran ge ufpossib lf'morpholo- gil's forme r!hypoljmers whichcall1Jl'tailo red forspecificap p licatio ns.Tilenuiety and compk-xitvof till'variouspolyr nr-rsyst e mshuvo goneratr-dextensiveexperiruental andrheorcticalst udies

Om'aspect ofpolymerswhichconrr iburcstoallextonstvc numberuf ap plica tions isthe\'i1rk ty ofmorpho logiesandprope r t ies arisiugfrompolyme rblendsand poly . mcrsill>lOln'IIL for example.acornbmarlcnofpolyme rsco ns ist ingof a har dhu t hrjttlepolymerandaso ft hut mulloablopolym er('<Il lresultinaprodu ctwhk-his hardbut not britt leandloanbe used in numerousCUIIIIIlE'rcialapplica t ions"Inge net- 111.differentspeeios ofpuly-mers are imtniscihlc.and forexample.dlblock copolymers mighthI'useda.'! compatibllizorstoim provethemiscibilltv he-tweentwo polvnu-r-

sand ehaugethemorp holo/l:Yand properties of tlu-system. \ 'anous morphologies pro,lu t'l·c\ fromcom b in atio ns of pulymer-,vanincludesphe-res .cvlmdcrsand lavers.

Those nauo-structurr-scanbeuseful illapplicnnonssuc-h as lithogra phy

P I,

Polymers art' al,.;,) foundincompositematerials.adhesivesandcoariugs.Rr-cr-nt imere.rshave beeuintheIISI' ofpolymers as coating sintI\I' manufactur e ofmrxlical dedl'l'ssuch ashioimplantswheretill'polymersllr f'l l'los form st ro ng repulsivetayl'rswhichl~a ll

actiL";lubr h-autswhr-nillconta ctwithutherbiomarrcr. Thetililorin l-!; oftill'mor-

pholo14il'Sandtill'ahilitytoformprotcrrive coatings canalsoheusefulillllt' \"l'lo pi ul;

efficient drugdelivery svstctus. Iiiord e rInunder sta ndth.:- mechanismsresponsible furtln-tllllrp hu lol-!;."and propertiesofpolvmcrs,,"ste llls.anundorxtandingof their therruocivnennicproper tics is required.

Po lym ersurr-urok-cnk-scomposedof repoatiug chemical uuitx.known asIIW lI OI " " r - s,The1'1'1)(>a t unit s arebonded together toformvariou sarclnu-ct ur os.TIll' simple s!

architecturc-isthelinear homopolymer.in whkllidenticalmonomersan'joinedill asillg:lecha in,TIll' number of monomersthat makelipaPolYIIH'r isreferr ed toas thcdl' gn ,,'ofpolvun-rjzution.Z, andiiigeneralisrelarivr-lv hil-!; h(10~-10'),Otlll'r struvru resiucludr- brunchedpolvmors .which canhe used tofo r mst a r and comb..like st r uct ures. Copolymerscontain twoor moretvp csof 1l101101!lI'rs .IIIparticular.di- bloekcrJJlol.IITII.'r,~co ns is t ofablockof 1Il0n OIlI<'rsA(c-a. polystvrr-nc} ofdegrcr-of polymcrizationZ('.landa blockof 1Il0110fllN'SB[c.g. polvbutadlr-no ]of degree of polvmcrizatiouZ('/I.Thov arcchemically bondedtOI-\('(I]('rtoformamoleculewit h

Zr''" Zr ,t+ZCB"

\\"ht' llpol:nllc'r.;areillllll('rsr'l"l illsolventthe behaviorislargd )"dt' f>(,llOil'nl011the inlerat'liolLs"t'l w('('1I11t('sol\l' n1andpo lyurer,The rill/tlity

or

rhoso ln'llIsisr!('S("ril...d hy,1Ir,'I.'("'lIl'1~orir.,,:good.('oorand

e

solvout..-\polymer ofinfinitemolecularweight wittdissolvr- illagoodsol\"('1lIandnor ina po orsolven t.Tilt'8solventropreson tsthe cross-ever(,<'I W('('IIgoodandI.oorSOln'ILLThe radiusofgyrationR,ofa polvmer alsor1l'IH'lUlsonthet(Hall!)' of lilt'solvent andS(""lr-sapproximatelya...;

! ^zv:

^{forgoodsolvent}

R,)( Zln fm

e

solventorhulk polymer ZI/1 forpoorsuh"t'llt[.fl"

(1.1)

Theexpone nt (I/:?)fo r

e

solveru orhul k polymercorresponds10a random walk of uupcrrurbed"ideal' chains.Ingoodsolvent.theS(.ln 'lllthereforeswd lsthr-ehaius.

\\"IWIIslmlllamouuo,ofA-b-Bdiblockcopolvmer,typica lly27<volumeIrucriou.

,HI'inuur-rsedillaselecnvcsolvent whichispoorforthcDblockuuclgood furIht'.-l hlor"I.:,min (}(loln ain s withsi7.t'SoforderW-::.0 11I11r-an form.Th("S("nucrod omains an'referred 10<I.";micelles.TheDblocksfor m tIL,.. core ofthelIlicdlt'Sandthe.-l hlor-k.~form aroeeua. Ofinterest isthl'size ofthecoreand ('(.rOllaa.~funetions ofthr-elegrt'('ofpolyuw rizatirlllandsolve nt(Illillil)"withrf'S(l('(;(toboth blocksof the(~OP()IY III ('I,Generally,copolynu-rswuhrf'lill i\"l'1y shortcoreblocks formmicelles withsmalln)[l'Sandrdath"f'lythickcorona..hutrec entinteresthas1>("('11inthest udy ofso-r-allr-d-erew-cut"micelles with la rge ("orcsauclthincoronaswhich consistof

l'OIJOIYIll('rwit hrl'lati\"dyIOIi~core-for mj ng blocks.Thb...svsterub<I!ll'tirs roffour

",y",lf'ms investigatedinrhis thesis.

\\'lIt'liasmall;1111011111of lowmolecula r w('ight0homopclyun-ri...addedtoasys- 11'11\of,-t.JJ..8dibkx-kl'OIJOI~'merIlIkdlt'S. thl' homopolymercallf'itllf'rbc solubihzed withinIhf'rmccltccoreswSII[tingillauucropha-e.orph,...S('(Mra ft'"'itllthe {"opot.,·.

un-rsatthehomopolymer-solventinterfaceand/or illmicelles.111themit-wphaM-. thehOlllop olyulf'rs\\"(·llsthr-mtcctlcs.Thet,lfl'(:t oftilt'copolymer atthr-ill!l'r f;K(' between thr-homopolymerl\lI dsolve nt istoreducethointerfacial ten sion.Ofinter- estisthr-inter playhorwoentheuucroaudrnllcrupllilSI'S.whichisdependent011thc molecular weightsuud volumeIr ac ricns ofthe0hcruo pcl yuu-rund

n

copotynu-r,111111 tilt'solventlilla lit~·andSl,!t'l.·liI·ity.Thisconsetme cstllf'secondsys temill,·t'Sti~a t f'l"1 inthis rhcsis.

III1lI,lI1ycast'S.polymersatsurfacesand illt('r fal"C.-s can 1)('11....·.1(0produc-e c!t's irl'tl cffl'Cts ill polymersystl' IIIS.fo r('xamplt'.in a polym('r/""h- ('1I1 "'ySIt'IIl, when011('end ofeachpolj-meris (,lI(l·t NIil'n orl to11sllrfat"e. I!I(' pro l'(' rt iC'Saud the strucrureufthe polymerlayerwill dt'I)('1II1 011IIll'polymerusodandtheslirfa l"edt'lls it y of till'I)(}I~·III('J.

Asthesu rfacedl'lIsityiIltTt'IISO:'S.rho polymers h('('OIII('sironglystretc hedaW1\.'·from the sllrfa("('and 1\po lYIllt'r-brush' isformed.Thethid alt'ssofrho1'1Id-I('(I I('r('(II 1\~'('r isd('pt' ltdl'ntOil thosur face dt'llsityand lilt'degr eeofpolymerizu t lon ofthe pol ymer.

III111('studyofthis thirdsystctu.flit'fo cu sis 011thest ructure and behavior oftho

l'lLd-t('( h('r('(l la~...ringood solvent.

Theeud- retberedlayerscanbe USl'l1 to alter sur fac eproporriosandprovidepro t ec- rive coat ings hutthf'Ycanalsohe used ill othe rapphcu tions.InSOUlf'('a>'-I'S.whena rhiufihuof liquid(_ 300mil) is app lil'll lln ifor mly toaslIrf.u 't'.till'filmoftl ·udew...ts the sur face.Thepn'sence ofanend-nxherrdh'yl'rillr-onjum-eionwith smallamouurs offreehOUlOpolYIlINillsolut iontendstostabihzethr-filmsfor extended I)('r iods uf uruo. A thres holdauiuuutofrreepolym er isr('(luir ('(ltostabilizetill'films[5.G}, End-t ct hen-dor frN'polyull'r alonerloosnotsrahllizethefilms anditishdi"\"f'(1!5.6!

that the coupling hetwei-ulilt'1'lId-tNIWH'dandIre«polymersalong withtln-nvcrlnp of thefreepol~·tI1t'n;stro nglyalf ..-ctthe-st a hilil Y of thefilm.:\st ud y of thinfihus<t1Ll1 till'imr-racnonsl,('t\\"I'\'11rhoend-tethered layer and freepolymermake-upthe fnurth syst l'I11sru d it'tlillthisdis.Wftation.

:\11fou rsyMl'mS art"dir ectlyrclatr-d to eXlx' riuwtllsand ha ve im porta n lapptl- ca rious.The IIIkfOpha.-.<'behaviorfoundin min 'lI t'andswulle umicelh-syst('lIIsis

\"{Ory usefulill 1II0011'rial sscience.oilfl'l'O\1.'f.'·and dru l!:dchverv.End-n-theredlavers can1)(' useful illtheslahi lizationof mlloids[•.81.adbcsion(91andlub r jca tton.and hilll,h.'"sic.:s. App lica l io nsfo r uniformrlnufilmsan'Ionudill S("w'ra ltl'c.:lmoloKkal fj"I.ls.asillIithogf-aphyinthe uucrocketronic sindusrry.Athl'Ort't ic-a1stud.'"ofeh«

propc rtiesofIhl'S('S.'·SII'IllS whichcan1>1' compa rrxlwith oxisriugl'XllCrime llt aldat a is ther efo re \"Ctyimportaut.Inad dition.we comparedilfC'n'lIltbco renc almod olsat stud y ingIIws C'syMelllsuudquautlfytherungeof appllca hilryof {'ad Lap proach USI'!!

illthis st ud y undhOI\"IhC'y co mparetoex pe r imo uts. Dttrer cnrmod elsart'discussed

illthe folluwing secnon.

1.2 Theoretic all\lodel s

Th issectionpro\·i,I.'Sa"riddcscnpuon ofthe rhcorcuc...l modelsalong withthe al!\'antagC'Sanddlsadvnnmgosofeach approach.Of'lailsandreferences an 'given in latersect ions.

TIl('modelsca ll1..0('rnregor tzr-diL~shownin Fi/-:llr(' 1.1.IIIgeneral.tln-analvt- ITHEORYi

---- ----

tANAL

mc ! i

NUMERICALI

/\ .r ^{i >:}

ISCAlI NG:

~ I

NSCF; IMONTEC\ RLO IMOLECULA

DYNAMICS Figur('l.1:Breakdownofvario ustheorerlcalmcrhods.

ieillmode lsprovideglobalinfor mat ionilltermsufsntl illg lawswithrespecttoIII<"

physk alcharactor jsricsoftho systems anti mayprovid ederailedinformat ionabout tln-propertiesofthe'syst emsilllimitiu!; caseson ly.011till'otherhand"till'1I11111(' r·

ienl techniques callprovide moredt,tailNi iufor rua tiun abou tthe phvsiculproperties

ind ud in g scaling lawsatthe t'xprUSl'of ilion'exte nsivecompnranuualdemands.hut they are 1I0trcstrtct odtu limilinp;l·;L....·S .

The anal:\"tiralmodelsca nI)('subdividedinto twol·alt~ori.~:Scaling!I Ojand IIIl'allfieldtheories.Tilt'~'alingtheoriesarci..la...-d on~"<llillgargumentsandprovide a~oodsta rtingpointforthestud yof l'nd -t t'lht'rt'<llx,lyml'rsIlLI:?Jandrlunfilm- s(i:1J,TllI'yart'luuued inIht'ifrangeofalJplkahililydue10 assllmp tionsabout tilt' srruc turooftbr-sysn-rusandingr-nor.aldolintprovide tlN<lilsabo uttill' structure ofthesys tcrrrs. "·\na IYl inllllll'i1nfieldIh("tJril~havehN'1IsillTt'Ssfu linthesi udyof oud-tctbercd layers(I.L 151and thinfilms[16.liJ.Till'tlu.'Oric'SweredC'\"l'lopl'llill termsofananalyt k alSl'lf-t'Ollsist t'ulfield(SCF)tlll'llry.Tilt'ruodelisb'L<;('(1 0111111 analogydrawnb.'"St'IlIt'1I0\'(181"The analogystemsfromthet'orrrspo nc!l'l\n'!wtwt'l'1l till'configura tio n ufa \\"('aklyst retchedorunst rct ehed chainallc(III('pos.~iblctrajec- tortesofa quaJU IIlII-nu't:hankalparticle"011thcothe r hand.lILt' ro nfignra no u uftill' ("OlIIp ll' tl'I.'" strt"u-hC'lirhaln isreminiscenrofIIIl'rrajcctorvofada....sical partidt'.In till'CiL"l'ofend-t r-thorr-d1<1.'"('(1;withchai nsofdl'gH,(, ofIK,I.'"IlIl'ri7.'\IionZ.when the dl'usit.'·..fchains ishighenoughsotha t theyerel"Ollsid l'rablystretchedtilt'la~"('r uuckuossIIscall'SasII-.. Z.In this regimetilt'chains art'highly stretc hed ("OIIII" u ro to till'end-to-cuddist illl('l'Rof all unperturbede-haln.Fortheunperturbedchains.

R...ZI/1and tlll'YthereforertuctuareaboutSOIUl'path,Intheasymptoticlimitof infinite-molecula r\\"('iKht(Z-jocc]and for highlyst rNchNI chains.theanalyt ical SCFIIllxl('1usesthc mostprobable path.Thisapproximationis stricrl yvalidintllC'

limit ofinfinitemoleculnrweigbrwheretlll'chains an'high lystrotc-ln-d. Th isimpost's alim itonthr-runge011sur facedens ities of ond-rcrhorcd polymertowhichthis SCF theorvapp lies.Xone-thelcss,thisanalyticalformalismprovidedsealing lawsandmore dr-tailerllnform anon un thestructur a! prop e-rt iesufthesvs toms

\ritl!till' adventof hig hporfonnuucecomputi ng"numerical rochn iquoshavehe- eorur-ilLt' l"l'as illg l~"im port a ntht'c,Ul~'ofthe irca p,dlili tytosimu latesyS(('IlLSl'Olllp f iSl ' d ofa largenumber ofintr-rar-tin gmacromoleculesandsolventmolecules .Theselargl' systemsare therefor e sui tablefordescriptionillthr-la nguageof statisticalphysics.

Ingenera l. till'descri p t ionofthe syst em proceedsillt1Lr1,(,ste ps:

I.~[o d ('li llgand dctcr unnarlonoftheenergyofthe variousmicros copiceo nfigu rn- nonsof thesyst etu. The modeledconfig u r a rio nslII11s thecomparlbk- with rill' lIIan Osc op ll"st a t e.

.) Dctcnuination ofthepartiti unFuuction Z•

3.Evaluatio nof the HelmholtzIrceenergy

(1.2)

~lostret...vautpllys ica lquantitiescall bedorerminodfromthefree cnerg vexpression illEquadon(1.2). Theapproach toevalunr ingtill' frr-e I'lll'rgy isnot unique ami till' waytill' particularapp roac h uddrosscspoints(1)and(2)ranbe lIs{'dtoclassify the mode!'TIl"numerica lmodelsof int ere s tintilisthesi scanbr-dividedinto th ree rutegorirs: mean field th-orv.~lollteCarlo(~ I C)simu la t ions and moleculardy n a m ics

(:\.ID).ThepoIYIll("O'eunI" · modeledhydiscreteunitsuriLit'olitill IlOIL~Sl.lil(· (" (·lIn"t"S.

TIll'meanfieldIhl'Orif"S.inthecont ex tuft hisstudy.cons r n u teitcc mbina tionof1II1'all fil'l<lehoom-sand numerica 1*lfo('OIISis((>lItfieldP;SCFj

useones. .-\

simpl(" nu- anfield apPfua.dlisusedforIllkdlf'S119- 31)andswollen mi("('II<'S(;32J.Th('S('moddsUSII<lII:,' ("ollsistof('a k llia lilll;'"fl"f"l'f'1If'rgyfro mva ri o usccnmbu ncuswhjchdepl'nd011 II!(' structureofII",poivun-rsysH·Ili.TII('ronrriburionsaredet crrnim-dInnnFlury typo;' uu-anfield('x p rf'Ssiolls[1:IJ.TIll' frr-ef'!lI'rKYisthenmlnimizodwith respect10thr- vuriuhleswhich characteriz ethesy Stl'lIl.Th('S('theo ries onceagaillofferaglo bal ll'·.scriptionofthepll~'siealpropcrties ofS}·St('IHS.induclillgpolyme rswithf('alisti!..· dl'gn'I'Sofpolyuu-riaarion. hutre-quire-a prioria....~ltl npti(lllsabouttill'srructurcof thl' S}·SIf'IlI.

:\singlechain meanfieldthf'Or:r(SC :\IF) was .1f'\·f'lup{'(1 hyCarignanuandS·

zlt' ifrr[:J-IJand app!if'llto end- tethered la.n·rs(3-1-381and tlnnfilu L-;{39].TII('thco rv involvesevaluat ingalll'xprf'SSiullfur theHl'lmhu lu:[reef'1lf't~,,\'illnth'iugtll("proha- hililY disrrib u rio n funcnon ofchaillco uforma tl on s.To ob tainthisIunenon.thefrN"

('Il('r~yisminim iz...1and111('resultkngex pressionis a[u!lc·tiunof till'chainconfor ma- liuus onlv.The fini l{'setofsill~I('rhaincouformar ionsis generated using atarncr- modelorrot atiunalisotuotir-stall'(R ISI1II0d['1(olf. lattit't> model].

TheXSCFIhl'Ori('Sha vehecnwrysuccessfulinthesl udyofend-tetheredpo l,,\'- mers[-IO-·.I6].thi n films[17.-I7J.mlcolk-s[.IS-.ll]andswcllenmicelh-s[:;2-:;-11 wit hrhc

<\ch'illltagpofsu pplyingII1lI0Tedl't a ik'd structureoft11Cphysic'illsys t l'IIIS.IIlII they

!O

act'abolim ned in'he irrangeofappli(·ahilit~·.Thel"ilklllatiollS become1lI0H'('01111>11.

tationallyintensivetha n Ih slmpl ...llI....anfieldtheo rieshut uour-the-lr-e-S)"SlrIllSwith [ralist ie dl'grf'f"Sofpolj'tnerjationarcmodeled

Inthese1I1001.-ls .ftucmarious about the1"o:,u ilibriml1conformarionof the chains are included .Therefo re.forend-rerhe rodSY"SU' IllS .rho n-smcnon10st rungly "lH'ld"..l.

highmolecular wr-ightpulY!Il(,rs does1I0[applv.011thcotln-rhan d.hoth tIL('XSCf and the analyt icalSCf1ll'pruadll'saSSII III("lat er allyho mo geneoussvsrems aiulcO llI , I I'1 imcnu-rio ns hr-twer-nrotuponcurs.Hen ce.t.hr-df'lIsit rIlur-tnat iunxparallel tolilt'r-url- tl'lIu"rillgsurfac ewhichocrur atlowcuncenr nuicns,HI'lgno rod.

Inthe:'\SCFmodels . tll('polym erCOlifigu rationsaremodeledbytr ajectoriesill continuous space.asinthisthesis .orbywalks01111lattin'_SdlC'lltjC'IISand FIl'f'r appliPd alar tit-c11I1)(1r-1 to sys t emsofeud-t etheredIInlllollOl)'IlIl' rs[-.11--.1 -.1). ("OIl'CII.\"lIlf'r albo rbt ...1 at all interface-(5.)-5,] andIhin films[II.-.li].Thepolymor chai ns arcd~

scri bodaswa lk....onalatuce.Each('ffl."l:tin·1lI0 IlUIll"[ l'xpt'r it"nt"t'Sanin ho mogl'l\Rl\Is tid.1 whichisanan·ra A...1 quantity 010 <1('1£'11fromnearestueigh bo rpairintc raerions andsllrfa("(....monomerinteractions.Till'sur faceili('r"c tiollsart'limitedto latt iC(' sitrs dirl'(:tlyill ronrar-twiththesurface. Tilt" meanfieldtherefor edependsOiltill'local coucentratio usof polvmorwhich.ill ru ru.arcderernnned (romtheIIlI'''"field.Till' problem issolvedS('lf-collsistl'lltly.The :'\SCFapproachisdff'Cti, ·efortreatiugfinite molecularwei gh tcorre ctio nshiltisresr rlcr cd hytill'lin-anfich lap pr oximu tlou.

\!ol('cularDyna mics P,ID)methodshave11('('1\ usedext ensivelytost udypolymers.

II

IIIprinciple.the~IDmethod!m p!i,·ssolvingtheclassicalxewtc ntau equutiouof mononanddoe-,notrequire1I11'approx ima tionof atuoaufield. \\"!lell (fI'at ing marromolrculcs with afully atomisticmodel. problems ar ise'illtln-le ng t h",:ales involved.Forasim p!!'flexiblerhaln the-re b structu reonscalesfrom tha tofitsingle dll'llJiea lbond(::::::0.1nm)tothe coilradius(:=::::10 mil}. .Aclditioual l" ug th Sl'al""

occur due tostruct ureintit!'systemsuch asmicelleformarlon. .-\saresul t. large syslt'!IlSwithmany pol ym er sarcn-quirr-d for ruodellngreal svst ems.Theprobloru withthef('ii.~ib ilit rofthosimulatio nsisevengreate r whenthetitucscalesinvolved are consider ed.IIIpolymers . motionsoccu r011 1lI.lIlYdiffer crutime scales. Forexample.

vibrutions on thelengthscale oftheC~CbondmayO(TUf011timescalesoforder of10-¹¹S Oflos swhereas rrausit io us fro mtransto9(lIIch~statesOfvice-versaoccur over 10-¹¹s.Inmicelle svsrcmsthc timc scale invol ved in the exchange of polyme rs hr-twr-enmicellesca llhI' ashighas 10²s ormore . Th isefret'tcausesserio us pro blem s inatotuisticrnodels.

Allaltr-r nntivr-10~IDistheXlo nreCarloappro ac h"illwhich the ruarromolcculcs art'I!lm!t'll'da.~roarse-graluedpoly me rs onalatti ce .Thismethod offer s sovoralad-

\"<ultagf:':iilltermsof com puta t io naldfid('l\(:Yand fem,ihili ty.Inthistypeofsim ula t io n eadl "l'fr,'<:t i\"t~bond-corresponds10severalchemicalbondsalong thecha inofthe polymer .TIll'lengthsr-ale of the-otlcct tvcbond"is tha tof thr- persist encelen gth rat he rtha n the cbennca l boud.Byelim inar iugthestructureat shor t le ngthscales.

the slllalltimesealemotions<IfI'dfl'uh"{'!y elim inat ed "Thela tt ice st ructu re alsoal-

lows forf'ffidl'ntalgorithms.Thismode lcanbell~('{lto study till' svstemsofinter est inIIIistlwsisbUIthe uuruborofthe crfccuve 1Il01l01l1!"f Spercha in is still restricted tolOU orfewer.Anothcr drawbacktothis mode-l istill'lossofthestructuroat sho rt len gth SI'all''';whichresultsina lossof informationilln'rtus uftill'dy namles involved atalltimcandleng t h ,.;{"alPs.

Other{"uarSl'-p;railll'dmodelssuchasthc bond-Huetuanonmodel have alsoIn-en successfu l in till'st ud~rorla rge syste ms.Intilebond-Hucruurionmodelthedlt'mil-al str uct ure can1)('kepthyindirr-ct.lychoosing ap p rop r ia t edistributionsofthele ngthof thc(·ffeet in ' bondsand/orthedistribution of the anglehctweousubseque nteffec ti \'e bonds.These modelshave b(,,(,11success ful in thest udyof end-tetherr-dpotvmr-rs(.':i8]

butth is approachis st ill restrictedtosim u la t io ns of rd at i\"elyfew,low molecular weightpol ymers.

The rehan.'IWl'lIrec entadvances inXfC simu la t iontechniques. Doruk er and :\Iatt ice{59]haw perfor-uux l sim u lati o nsof polyothylr-no {P El thinfilmsin an urtotnpt tohrid w'thegirph{'IW{''-' lItill'comparutivol y sm a llsystr-msinfu llyatomisricmodr-ls aut!tilt' :\IC sim ula tions Forcoarsr--gr uine dst ructurelcs spolvun-rs .:\IC shuulutio ns weredone011a!li ~hcoordination la t ticewhichroftoctsthe struct ure of PE.The:'lIe snuulutionsw...roperformedaft er tilr-PE dl<lin sweremapped ont o th ...coarse-gr uiucd latt ice.Thro ughoutasi mul at ionit is fous tblc toreversemap thesna psho ts hack10 the atomisticmodelallo win g:the st ud y oflargebulk sys temswhilekeepingatomis tic featnrr-s ofthopolymerchains.

Ea chIIlct hod descri bedal)O\'r ha....itsadva nt a ges audsho rtcomings.Tho nll'all field tbccnes ran treatsystl' IIISwith real isr jc t!('grf't"'Sof polyme riza tion atrelativel v low con rp utariuna l("OSl.hut theyur.... res tri("t('fl ill tlwi rrange ofal'plk ahil itydue to the 1ll'!-II('("tofHuctuat iont·ff(,(·tsand/orrequirementofaprioriknowledgr-ubuntthe st r uctu reofthesyst(,lIl. Till' \ICapproadlincorporut esfluct.ua t lon (·Ifl'(·ts,hiltis rcst ricteri10f('lat in'I~;lowmolecularweight polymers.III Ihisthcsis.thet:"llIph a s is isonthC'list'ofxlo ru cCarlostmulanonsas a oom ple mcutto01 ...\11fieldtheories.

~Iolll(,C"r1o simulations aTC'usedto studythelim itations oftheIIU'''U Ifj('lrl theorir-s and theIll('(-han ism sinvolv ed wir hpolvmorsillsolur io u. Titefllllowill l!;su!Jsl'(:ti u IiS diseu-e..howthese 1II0011'Isha ve1>f'('11used to st w l:,'eachofthefOIlTsyste ms. Each subsectionalso discusseshow \ICsim ula t ions. illco uj unc tjo nwitht'xis tilll!;1II0d el s OTother mo(h·1sdl'\"l'lop"dillth isthesis,wil lbeIIs('(1 toguin afurt her uudersrundiug ofthe mechanismsrcspousiblofOTthc structureandPTOpl'Tt i<'Sof rhoso svstetns.

1.3 Crew-cu t I\licelles

Blockcopolyme rs han'I}('(' II 11.,,*'(1inawidr-\<\r il't:" of ap p li<:at ions,"lIldthr-irl'ff('("(

onthe be-h a vjorofsolutions<111£1blends isinrportanr.Thes:,-slt'lIlsofinte res tha n ' low coucenr ra rionsofA-b-Oblockropolym or. The illllllisdbilit yb('( \l1'1"11theBblock orthcl:opolyllll' randtill'ma trix [solve nt orhomopo jytuer]inrr mjunc tionwitha TC'I l\t i\"l'ly lull' cnr ropic !ll'lIa lt y illlocalizing thepolymers results illth t' for mation of nucelk-s.Till'IJblocksfo rm thl' ('orpscfthcllIi(·C'lIt'S.uudtheA "locksIcrm coro nas.IS

14

showninFigure1.2.1£the overallcopolymerconcentrationislow eno ugh,themicelles

o_~_:

^°

O~oBI~kOo ^:

⁰⁰

~ .. :.;:.f;.

::~~_ ^{. .} : ^.. _~~:~~ ^:.

. . ..

o

.;>

0_ 0 0

o °

Core

00

1.e!J

° 0

COPOL YMER SOLVENT OR HOMOPOLYMER

Figure 1.2:Sche matic pictu re of block copoly mer micelles .

are randomlydistributed insolution.Theo reticalandexperimental interes thas been focusedonthest udyof micellesfor which theBblock ofthecopoly meris short compar edto theAblock.Hence,the micelleshave arelarivel y smallcore compared to thesizeofthe corona. Recently,therehasbeeninterestinthe experimenta l study[1,601 ofso called"crew-cut"micelles,fot"whichtheBblockislongerthan the .t-block.Thecrew-cut micellesthe reforehaverelativelyla rge corescomparedtothe thicknessofthe corona.

Of parti cularinterestinallthese systemsis thecriticalmicelle concentration

(C~IC )and the sizl'ofthc micelle-s 'IS a function of til!' degn'E"Sof polvmnrjza tio n of.-1 andBblor-ks.ZC4anrlZen ,rt'SpI'(:ti \'dy.Touhraintill'radius oftln-corr- audthethick nessof theeorc na asim p l.. mean fieldtheory.which isouutucdill Chupt er.3,has1l('I 'llused. IIIthis thr-or v.it is,ISS U llI f, drhut thecorr- andcorona regionshave uniforml!1'lls ityprofiles .and thr- totalfroeeller gy is ca k ul a tN IFromi\

to t alofsixcontrtburionx.Ina stro ng ly !«'I;re gatef! syS\('1!l,till'sizr-ufthe IIIkdll'sis mai nl ye-ontrollcdby the interplay bet wee ntheinterfacialtensio n attilt'core-coron a intl'rfa('1'and theolasuc r-ncrgy associat ed withst rc tchiug oftil!'core-fo r m ing block oftln-copolymer.Theinterfac-ialtens io nn-ndstoincrea se thesizl'oftill'mice-lle hy reducingtheto/tilint erfa cialart'awhichresultsin fewerbutla rgr-rnucclk-s.011 the othorhand.asthesil l' iue-roasos .theBblockofthe copolym ermustsr rr-tchtothe centerofthe micelk-.Till'IK' lla hyin Irce energy assoc iat edwithstr e tc hillgtlu-B blockrendstok('('!'themir-clhe,slllal l.Sim p lemea nfieldthoorv !'[{'(lietsthattho radiu s ofther-oreof till'micelle.(/I,scalesas

(U )

Theanalytical l'x pr t:'s.-;iu nfor //1inEquatio n(1..1)dol'Snot show a dopr-ndo ncoUIL tln-It'ugt hoft.hccoron a blockZ(",Ihilt a numortcal analysiswithinthr-sa me mod el sho ws a weak. inversedopondr-ucoUlltheAblock.OtherflLI',HIfirldtheories prodlct simi larsea ling lawsand inallcases0::::0,6 .

TIll'!<I.'al illg lawsobr.uuodarcinagree mentwjthIlLauyoxperinu-ntsOlL mirr-llos withrelu tivc lv sho rtcore-for ming blo cks ,HOWI'\'('r.recent oxpo rinn.nts

III

Oilcrew-

16

curmicellesresultedin aIIHf't' U'1l1 scaling relat ion wit h1#:x~';' ZC~-l-"i.IIIrlliscase.

thesca ling ("XIJOI1('11!oftill'ZCBte rm wasmud .10\\"('[ lha ll thatpredic tedby silllpll' mean fieldtheory.Till;"au t hors 1I01l'dth at thr-scsyste m...W(' T€'nOIat('(Iuilihrhu lland IhAIthe discTt'pallcyillthe!'oCaliull:('XPOIICIlISmightl>ea non-equilib riu meffect.III thcee exporuue n rs. theIMllytlle l"S weredisso lvedillaSUh~1I 1wh id.higood fo rbosh l'OIlLIJ(JrI<'n!Sofcopo lrull'rs.TIl(' solutton wasIIIl' Il,lia lyz('(!llll:ailist watr-r,whid .is apoorsolvent forotn-l)ftheblocks.foraper iodofrime. Astill'solveutqualityfur thr-cureblockdecreased.micellesformed. \\'1](,11till'solventqualityW<lSdt'('U';lSl'l:!

furthor,thecopolvmcrs\\"1'rI'tmpill'dillthemtcclleswhichresultsin non-oquilihrium struct ur es.The measuredsi~ ,'SIIfthecoresmaythereforercflcr-tcquihbrmmeondi- nons at the pointwheretlu-copolymersI)('g ilnto1)(' Inlpp''11in the micell,'S,

Sililple"" ' IUI HeldIhl'Of~'[:HJsllAAl~ISthu t II,e sol\"e nlC1U<llit~·at which themi- ("('lIesbeco m euon-equjlibmuusrmcru rcss(foIlKI~'dl"pellds011thedCF;f('1'ofIM'!)-"

mcrtzanonof thf'cerehlock.III 11.1.' studvof crew-cutmicellesPfl'Sl'lIh'1 lhere.the 1l0IH "lui lib rillllleffecr saresimul aledill our:\olesiruulatjonsby i<!euti fyi llp;the ccndl- tiollSwhere theCOP'llI)-'IIII' rsIwgintoI,... tra pped intill'mi('"('III"S,The:\oICsillllliat iu lls lifethon1IS1't110illHostiga tt'S('alinRrelationsandCOIUJ><II"t'wit hmeanfid elresul t s. FurthprtllOrl'. the"ICsimularlo us areusod10probo II\('struc t u reofthennccllesand iuvcsngar otheassttrnptionsofthe meanfield app rmwh.

1.4 SwollenMicelles

\"nrlna lly_abinurybleud of_-tandBhomopotvmc rsIIl'Kruphil."iol'separatesIK't:a u""

oftlu-r('pubi\"l'l'[('('"tin'tutora ct tous a.>; discussedilltheprl'\"iu lissccnou. Till.'pha.'iC' sepa ratedhl{'1II1consists of domai n">;of theminor it y("OIII!KJIll'llt dis!K'rsl'd illtill'ma- joriryone.TIl('sil t'S oftilt.'domains de pen d011theproccs...:iul!;hbtor.\-_hut Iypit"all.\"

an'Oilthr- orderof mkruns orlar gt·r.\\-hl'''..:rna llallLo un tsof_-t-/~ nblock(,ol'olyllwr art' addNI10the.-tIn hleud.tln-ropolyun-rsrnu OI("tm'surfact ant to dislK'rwtill' tuinnrit j-coru poneutintosma llerdourains.

Thenext syste mofinterestillthisthes isisthe.-t- /~ol.-t/nterna ryblend,with onlj-a fewpcreenrofbOlhcopolvme-r and

n

homopolj-mcr.illa hostof..-thomopoly- IlIt'r orSOI\"l'lIt,IIIthissyst em.0111't-anillla g illf'"1I111111...rofpos."ihilit it'S" ind lldi llg 11!'oinglt,mucrophascIJOplllal t'1lhytuicelh-sS\\"OIII'IIhy homopolynu-randwluN'silt 'S arcoftlu-orde rof lllolf'Cula rdinn-usi.. ns:or llIanollha ,.,{' ..rpar'llio n(S('("F'Igu re1.3)_

Of ,,';!'H it"lllarimt'fl'Stbthe inter playbetweenmicrophaseauclmacrophascseparauon alltl l!L... mil't'lIl' sizt'Sandnumber ,If'u!'oit.\"_

:\ pre vio usmeanfieldand ('xperiult.'lltalstud .\-h.\"\\'hit m or c and Smith(3"1)ha."

1}("('11 dunetostu.lyblockcopolyme rmicelless\\"OIIf'1IlIy addedhomopolymer.The theorvprr-dict x that. for r('la t h -clylowmolecular wr igh thomopolyruer.

uu-re

cxbt- sathresholdvolume fractionofhom opolym erbelowwhich III('lmmop nlvuu-r will bl'solubitlzed withinthenncc ltocuresandabove which ituuu-rophase separat es.

Till'rhcorotu-nl rosul rswr-reinqualitntive agrCt'Ulrntwithcxpo r uncntswhetePEO

~~~ ? 'J ). ^{. . ..} . '1&U. ^~~(.:.

18

COPOLYMER

SOLVENT OR

HOMOPOLYMER HOM OPO LY ME R

Figure1.3: Schematic pictureof micelles swollen b.... homopolymer and macrophase separatedho mopolymer.

homopolymerwas added toamixt ure cons istingof polystyrene-blo ck.poly(ethylene oxide ) (PS.b-P EO) blockcopolymersin PSpolymerhosts.Transmissio nelect ro n mi- erogr aphs(T E~ t)showed micelle cores ranging in diameterfrom 20to40urn witho ut addedPEO homopolymerand from 20 to60om witharelativel y sm allvolume Free- tion ofPEOhomopolymer.Wit ha volumefraction ofhomopolymerwellabovethe predi ctedthreshold,domainsrangingupto 400 urn wereobserved .

Alt ho ughthemeanfieldapproachwassuccessful inpredictingthe solubi litylim its insuch systems some question sariseabout some of the mean field approximations.

Thesolubi lizationlimits areca lculated on thebasisofasimpleexpressionfor the free

L9

('UI' Tg ,y. IIItheea...' ofswollen mit:l'lIt'Sin tin' stro ng*gn-,;atioll limi t.the<It'"sil .'' I'Tnti!.'Softl,('copo lymerandhcmopolvmerwithin('a dllIJic.-lIl.'areassuuu..1to 1M' unifo r m.withitshan)core-co ronaint erface;lIId all lIti('('lI("Sl\a \"jugtheS;llllt'sil"'.

Thf'S('a...ulllpliuIIS abou t IIII'strur-turr-of tilt" nucelles("jill1)('iUH'St i,;.u('dwit htilt' use of),ICsimula t io ns.

IIIthr-st udy ofswollenmicelles.WI'pcrfor ru:\IC!'ilUu laliu lI"ofhkx-k('\>pHI:n llt'Ts ill,;uln 'lltwithadd''flllOmup ol.Hllerwhichis t:ompa t ih lf'withrho("orl'- formill,ll;block ofthemicelle. IIIth<'M'simulationsthe llIi('dl<"SM'lf-IIs--;l'llIh [('an dno!Ipriori .\....

sump riousabouttln-it-str ucture orstao dlsrrtb urio n an' required.\\....luve-rtg.uothe struct uroof the lIlil'l,1l1'S ,;wo11(,11hythe homopolymer alldthr-sol ubiliz atio n limi ts pft'<.lk t('(1 hy moantieldtheorv.\r('rl'dsitthe phll.s<'hl'hadufreport ediiitht'1II1' ,ill field('ak 1l111. t iolls hyi"' ..'St i~al i ngthethresholdvolume Iracno usofhomo polymerand il Sd...pendem-e-Oiltherr-l....tiv...dl'g H'CSof polymcriza rion ofthecopolymer <lIHIIIII' al\III'I1homopo lyuu-r.

1.5 End-t etheredLa y ers

Thorois l"Illls i<!('rahll'Interes tinthcthooreri cal .11111experituent alstudy of"('11I1- terbr-rcd"polymr-rr-hainsatsurfaces.Tvplcalsysrr-mswilhone end of each polymer eud-tcthorcdtoasu r fa!"(' andthe polymerchalusfo rm ing'Ilave r illsolventurerro- 'IllI'llllyrefe rredtoaspol yme rbrushes._.\,; discussod in Sccnon1.:2.seve-rn]tlu-orct - teal[12-1~.~ O.~6.;j8.6 1-6·')1st udies han?beendone toitl\·{'Stigatl'th .. prcpcrt tesof

:!O

thr-end- t ethered laye r:;._·\s w{'11.numerousexpe ri me nts

{5' .66 ·'11

wr-ro performe d tosill ilyend- t ethe re dlay ers.

Asyst eruisdlanu:l l'r izl'dhy themo lecula r wf'iS!,ht JIand the[)()I'\·IIIl'r sll rf,u-c

(kl1~ i l y.Tbr-an'ral1;l'areape Tmolecule is deno tedhy~andthea\·l'li lj1;l'numberof chainsP"Tunitareah~·rr""l/~.Thesurfa ce d"llsitycun alsoI){, c"ha ra.·\('rizNI h.\"

thereducedsurfan'("O)\"('TaS!,t'whithisdefinedas

(1.-1) whereR,is therrecpolvnu-r ra d ius ofgy ra t ion,The-behavioroftheS.\·str' 1Hdepends

011r!LI'reducedsllrfan'CO\·f'r agl', Toillus rrarethis\H'considerthe IWOlimits as

shownillFigu re1.-1_Fo r(1'<:1111('average distancedborwecnpolvmr-rchains is greatertha nR,audthepol.\'lIwr s areisola tedIrom radlother.Till'lay,' r islate rall.\"

illl101ll0l;l'lll'OllSanditsthickuoss.It.is indopendout ofthr-sllr ral"(~r-ouceutrariouand

~i\"('llhy

(U i) whoreZisthcdrgH'f'"fpol"'·II\.'r iza ti ollof tbo("\II~lillgblockandv :::::3j.'jill~oocl solven t,Thisiscallodthemushroomregi me.Inlhf'lim ito·

»

1.the avr-rn g...dts tanc e bet weenpolymer cnd-tr-rhermg po intsbmuchsma llertha nR,.IIIthislimittho da ngling ('!lainsarestn-t dll'llint othesolvent "md thelaYl'ris lat('rallyhomo ge neous exceptlittile surfuco andthc tip.Thisisolt eureferred to as till'bnt.•/tregime.

:\I1 <1I~rtical thoo rleshave beendevelopedforhighl ysrretchr-dbrushes.Ale xander auddr- Ct.'IIIl1'S(AdG)[l l.13J11M'llthr-eoncepr of-blobs"alongwithscalingargu-

11

I')

lb)

Figure-l.~:Polymerrhalusr-nd-tr-theredby one cud10ilslIrf'il"t':il)o'<:Iandh) o'>1.Thela~"{"[thil:ku('S,.o;isII.Till" ;lwra gl" distant"('betweenpolymer chainsis(I.

meurs1I>obtain eh..la~"f'rheight,~Iilller.\\"ill(,11andCart'S(~I\\"C)[l~].1('\"('101"".1 allaualyticsd f..rOllsblr'lIrfield(Se n thooryofthe end..ll"l h('rl'(lla~"("r.Both tllt-'SCf andS(·a li ll~the...,rit'SI'rl'(li(·(1'(1(ha t.ill (ill"a.~Ylllpto ticlm lshlimitoflilrgl"Zandlit

(1.6)

ThescillillgandiUlaly lkal

scr

tbecncshawhN.'Usuccessful indescri bing high ly stretchedend..tc t hc n...1 polymers.Onthe utlu-rhund.I\:('ntd rd. {69]remarkthat..

inallcast'sexc-eptsltldicshy.-\lIro~·f'tIll.[6 6 .6 71wheretill'la~l' rsarc for me d from M'tlli..diluteand('ollc('TLt ra t(-'(1solutions.thereducedslIrf,u·('coverageislilllit(-'(110

o·::::.15..-\!<aresult.the-l':qH'riULl'UlSIIlaynotf..11ill IIII'asvmpeor iclimit. R('Su!ls from IIII' experinu-nrs wr-rerorfohu ratt'11wit h IIllllll'ri l'alSC'lf-t.·Olis ist l'llllit,l,1(:'\SCFI studies of Bar a nows ki and\\'hit llLorp[-.161.The I'xlwrilllt'lLl a landtll['Orl'tinlln~llh s imply th a tlIu-,;('S~'st ('lllS<10not c....rn~I)OlI<I10IlIl' a.";yllll)lOt iclimit forIhisusual rangeofo·::::.IS.

IIIthe presentstudy.till'prima ry int erestsare in dl'taiiPd1'01l1lmriStlllS.H IltJlig aualyric

scr

thcorics .:\SCF e-ak-ulatjons.:\IC sillLulatio llsamI experimentalresults.

The r-xpe-rimeutsimp lythutthcpolYllll'rs arc not ashighlystrl'le!a('t!asassumed ill till'usytuptotictheoriesand that:\SCFcak-ulationsmayhemoreappropriate.TIll' :\SCFcalcu la uous do1101incorporutodensitvHuct ua rions orlate ra l ilLhollLog ('lll' it i('S whichIllIll'toccu r.11till'lipof till'layr r forall0·.andthroug ho ut till' layrrforlow 0·.:\ICsim ula tio nsarr-thr-reforeperformedinconj uuction with:'\SCFcOl k ula liolls tocIlI<llilify " fallgl' ofred ucedsurraceco nce ntrario ns werethe :'\SCFal.pr<);Icllis valid

1.6 Thin Films

,-\s discussodillthepreviousSccnon. the reiscO!ls id cra h l('interestillIII('theo- reticaland exporiuremalstudyuf POIYIllNSatslIrfa,('('S. Polvmr-rs atsurfacesca u altr-r thesurfacepropt,rt i('shy challging tin-illt f'r fad a l tension. andsomcumcsIIII' syste ms ca llI){' tailored 10 obtuln doslrodresults.IIItill'ea Sl' of uniformthinfilmsof liquid(....30011111thir-k]till' filmswillrendtodr-we-tover a perio doftitm-.Will'lItile

13

lilln i,1isdcminarcd~.'"UId('r \\-aa ls turera cnonsuu-roarl'twoillll>ort. m tpa rarn ....

te rs: (1) theHam ake rcon stant..-t.whichd,~;(:riI K'S10111; rangeiuter.u-rionsandtoa firstapprc xinumonisconrrolk-dh~·the polari1.ahililyufthr-threeI'ha,,",'S.and(1)the sprC'i1, l ill~r-oeffi cient.S.whhhis ameasure ofsho rlran ge su r faceinteractions.The fr('.> ('l\I'rll;Y.C.per unit areaofI!.liq uid fiitnof thickness 110due 10the van dor\\·i1,11s iutoracrionsis(73j

(1.7)

\\·IIt'1l.-1ispcsinv o .the'"UIdcr\\'aa lsintcfilct i<lIIs(('lUItothin Ihcfilm,TIII's p rl'" ,I- illg l'odfid eJll isl\t'fiuC'tIa...

(1.8)

where~.•,..A,o/uud~'I,"arr-thesolid-vapo r.solid-liquid and Iiquid-n lJlorsurface H'lI- sioHs .rospr-ctivr-lv,If5isnegutivetill'liq uiddlll'snotWI'I Ih('sur face.TI\('('asl' where.-tispusitin' nud5isneg u uvois classlfiodasI'tlrti fl l llw tti llY.Inthis (·aSt.·the filmsalf'unxt ablr-. Hole-shf'Kill10 fo rmIhlt'ttlther maltllll·"Mtionsalldtilt' films c\"t'lIll1all y"n'a k upintodropl l't s[51_

Inthinfilm:-applicatio nsitisl-:C'lJc ra ll ~-prt'f.'rr« l l h;atthe films1>1' stah l"agains t d"\l~ltillgill(>f(I.'r(0kN' p th(' filmsataunifunurlnck noss.Tilt' I'r('S('II("('ofend- tethr-rrxlorfreepulytuer ca nstro ll~lyalterthe prOIIl' r t i""of til"syS(C'IlI.Ithas 1)('('11 remarkedinr-xperiruom alst udies{5}.tha tan eud-rc rhcredlayr-rorFree IKJIYIII['ralone cannot beusedto stabilizethe filmsand acombinn rionof bothisrequiredtomake tI\I'film...s(.II,I('forextendedpNiodsof time.Furthe r mo re.therr- is 11minimumhulk

\UIIlIlL'-'fract ionofrrccpclvnn-rrequired. The th resholdccnce nrrun cnisilppro xi- lIl.ltrl~·five tilllt 'Sthr-overlapconccur r a u onoflilt'frt'{'polynu-r which.illrur u.is dependent0(1thedf'Rrl'rofpolymeriza t ionofthefree pol YIIII'r. Th,~'l'xlH'rillw llts Sll~l'Stth a ttheIw y('jl'llIrllts inund ersta ndin g till'sti\ h ilizOltiollofthese systl'lllSart';

(I)thecou p linghN\\W IIthe freeandend-t etheredpolymers asitfunctionofrhode- I;rt'(' of(Jo ly u lI'riziit io noftile IlOlyno....rs.thegra ftillgdl'llsi tyandth... volume frartill ll of fn'f' polynn-r.and(1)till' rtfff tof the owrlapccuccmrnt tou.

Sl"a lillp;[nlalldanalyt ical SCF[16.1•.

' -IJ

th("()ril'Shan'beeudovolopodfortill' stlldyof[hillfilms.Tbeillllilysrswr-reros rrtctcd to 2f?;SZ,.,whereZf·and2.~an- till'dl'J;rl~ofpolynu-rizutlonoftill'frr-c andl'lIl!-tl'tllt'n'drhains.rcs per-tivr-ly. III thl' caS{'whr-rl'Z,.>Z.I.thepartialpenr-trntiuuoflongrreodliliuswasaSMI!IIN!

1I1'glili:ihll'(191.ForZ".;SZ~ .tlu ('('ma jorrl.'gillLl'S Wl'n'idl'l lti fil"C,LIIItill'firstrf'J1;il lil'.

till'!'ou rfall:'coverage is high.till'ond-rerhcredIM)IYllll' tsare high ly Strt't d lNI. the ponotra rion offree ("ha iIlS isIlr-gligibh ' andthe layr-r hoight isnot atfC't:t...-1 hythe frw I'ol yllu-r. IIIthesecondregime.theO\'l'rallvo lunrr- frac ti onof frl't· (·ha ins islar~l' l'fI' llIgh so rharrhr- ('II<1·ll' tl lr rf'<1 lay er isaff'''C,·tl'llI,~·the pr('S('II['(-'ofthr-frl't'(·haills hilttill' sllr fill'"c"'·t·ri\~1'ishil;h enoughtopr"\ "(,llt!M·IIr-tratiouof rreo("hainsillthr- ('IIt1-1NI!Crl'(!layer.III the- thirrlregime.thesu rfacerevr-ragr-islow enough toallow till'free chai nsto(·olllpll'tc-lyr-nvelopthoend- rothr-redlnvcr.

Pcncrratiouof rl'la ti\"('[yhighmolecularweight fr('f'homopol ymerinthe end- tethe redlaYf'rlIasbeen obscrwxlillrece nt expcrhncuts[2Jat conuuouly ohSl'n·I'(1

end- tetherin gdensities(0".

:S

15 ).Thisobscrvancusuggests thatcor rectio ns tothe asvmptoticbrushlim it1U;-.~·beimpo rtantandthat thepenet rati onofrlu'freer-hains sho uldbeinn'sliga te dforr-nd-t r-thc rtng denstrioswhicharemore r-losclyn-lutodtu experiment.A

xscr

approach is approprratr- todealwithfinitemokx-ular woiuht correctio ns expect edfor thisrangeofred uced surfaceconcoueruncusexceptthat.

again,lateralinhomogenc-itiesillthe planespar-alh-Itothc eud- re t heriugsur faceat loweonccntr .u tous impose arest ncr ton011the appliruhilitvoftill'.\"SCFapproach.In this thesis.a~I(JllteCarlostud yisdonoillordertoobtaina rangeof reduced surfno- couc-eut rntions overwhichthe.\"SCFappro ac hisvalid.Till'volumefractiou profiles.

laye r heightsandinrerpr-netrat ionof chains atthe intr-rfneean"invest igarccl.uuda comparisonwithXSC F theoryforvarious mole cularweightsand surfacecoverages is donetoquantifvtherang!'ofapplit~abilityof rho

.\"scr

thcnrj-,The

.\"scr

approach isalso IIsp{1toma kedirectcomp.msonswit htheexpcriuicur.al work ufKentet: al ..and wit h power lawspredic t ed h.\" sOi ling and

scr

theoriesforthed,>g r("("of pcnct rat ion of tilt'cnd-tr-ther edandfree chains.III particulur, th eheightufthe cnd-te-theredla.wr andthe' O\'l'rlap lwt\\ '(,l'l!tilt' fn·t' and end-rr-rhe rodpolymersarc ill\"l'sti gat edfor the threemajur regi mespn-dh-tcd byscalingandSCF rhoortos.

Chapter 2

Monte Carlo Simulacio ns

2.1 Introd uctoryRemarks

TIll' generalformal ism for~lll ll! ('CarloSilllUI,u iullS isrevie wed inthis sr-r-tion.and thesim ula tions spN;ifkInIKJI~-IIl. ·r/soIH·1Its:,-stl'lIISwillhediscus..;o....1illIII','following,

Inslat is tit-al pll:,"sks.IIprohlf'llIofinte res t islilt" ('lIk uhuiu lI ofthermala\l'rall;<'S inIIU'cano n icalcnserulsle.TIIt':'-all'gin 'nh:,"

<..t>r=

~ J

^{d{.r }.}.t({.r})f'Xp[- J1i({.r })j CUI

\\"I...re{r}isavector,1('SCrihillltitstateillpha~'Slla('C'.A({r})isa localopr-ra tcr.

?oH{r })istheHamilt onian.j

=

l{k nTwherekllisBoltzmann'srousran t.andZis givenby

(2.2) IfOIll' ormor eofthe .E"sill{r.isdiscrete.rho in regrariouill Equat ion(2.1)isreplaced

hytheappropriat e SIIIII.

III pracncc.itisnotalways1KJS...;ih!e[0evaluatethermalaveragesh)' the!IlCIUIS sho wnin Equat ion(2.1).An alrer na t ivr- i.., to('\-lII1Iaff'them uumerjcallvhy mf'alLS of ),Io lllf' Ca rlo sim ulal iolls. III the ),ICsimu lalio llsafillitf'su l)Sl'tof phaSl' "llalV issam pled, andthr-approp r iatt'iln'rag<'San'tlu-ur-vn luun-d.Therearr-IWOmajor tur-thodsofsumpling.Thefirstmethodiscalle-d~silll plrsam pling"inwhichI·;IS(".

.\1st ares aresample'llwjthequulpro ha h ilit y(P""1/.\1).Thean'ragpsarr then cuk-ula redwith IIIC'appropria testa tis tical wf'ig hlPrq{J}where

(2.3)

Thisis calledthenorma lized BohamanuIa ctor,Thismethodcan he\"cryilll'ffidl'lItif mostofIII('st all'S donoteontributesignifk-anrlytotheuu cgra ndill Equation(2.1).

"t at<'Sarechosen withast atis tjcalweightproportionaltoIhf'normalized Boltzmann [actorsotha tIllUS'ofthest ates thataresalllpll-ti fromth<"finitephasespacearcl!LoSf' whidl contr-ibu tesigllifi",IIIII)'toIhI'ill[('grand illulualioll(2.1) .Thermal a\1?mgNi iUCtheuapproximatodhy

1 ·^1t

<A>T:::::Ti ~ A({r}^{..) (2.-1)}

where.\1isthe1I11fl11wrofst ntessampled.Theaq:llT<I('Yoftheresultclepr-ncls onthe numberofpoint s saruplr-d and also011the finiteSill'oftltesvsrem. Theaccu racyrun b"iUC"Tl'asf',jby saml' lillll;IIIOT I'pointsand iuc:rl'a singthe Sill' ofthe syst e m.Limited

(2.6) 18

ccmputiugresources maynotallow calculnnous OilsyS\('llIS which arelargl't'ntlll glt10 f'lilll in1lh'till' finite sizr-effects:illtltisr-asr-the resultsrunbf' extrapol at edby1Ilt'ilnS of finin- si7.(·Sl:a lingnx-hnlqucs[,.'),,61,

:\kt nJpo liset'It.(••Jadvanced IIII' co ncep t ofim po rt ance saltlplingandsuggested rhata setofsta tf'S{J'}"ca nIwgcno ra n-d hychoosingsta tes bymeans of11trajec-tory in phaS(' sJlm'erathertha n ehuu sin gS{i1I f'Siudr-pench-ntly ofeach othr-r.Tilt' result is

iI:\Ia r ko\'proe-r-sswhere astu te {.r}v+-Iiscoustructedfrom theprevious st a u-{r }"

viaa suit.ahlr-rra us tnonprobablhr yJI'({ r }v- t(r}"..tl, ,-\res tricrion011thctra us i- tion prlltmhilit.\"II' must1)('imposedsuch thatilltho limit,\I- tccthoprohahilit~

disrrihution funcriouofthe stairsgenera ted h.\'theprne:t'Ssn-ndstowardstil(' oqui- hhnum disrrlbuuonfuncnoudotiuodillEqua tion(1.:1),To achicvr-th iscondrtinntilt' priuctph- or detailedhal<lln:1'is imposedwith

whereallYst a t f'{or}",is obtainedIroui allYorhe rSt ilt I'{z-],byafinitcsoriosuf s!t,ps ill phnsospart'_ Alth o ug htheprinci p le ofdctaih-dhalnnc-edocsnutuniquelyspt'dfy lr({./"}v-+{j"},~ )11suitublocoudirio n whichSillisfit's Equation(2.5) is

{

I if.:::.1i>0

lI"({.,' }"~{ .,}".)~ -

I'Xp(-.iSH) if SH<0

whereS1/. ""1i({r},,..)-1i({.r)v).Having dofinod atransirionprohablliry,agene ral algorithmforMoureCar lo simulations r-an heconstructed as follo ws :

.)Calculateth.·changeill cu..rg..'".~H.iii thetra nsit ion[romIIII' sta tl'{..r ~..10

=1 Gel wr at (' arandom uurubc rr suchthatr ([fl.I).

-I.Ifr:5Il·{{r}..-+(r....)thenthenewstate is ac"t-'t'pl ('<l:othr-r wisr- itisrt'j.'t·tN I andthr-old sta tris("()lIutN !anew,Thisst ('11isknownasthe"Ietrop..li.~mb-.

:J.Rl'I)(',H stl'PS1to.tmany times.

The method usedtog"lll' nl[('newst a t es dl'!)('lld sonthemodl'1hdn j!;USN!.lindfo r till' wor k herelilt' nn-thod isdescribedforjlol~'lI\l' r/sol\"t'lltsystemsill thefol lowiuA sectinns.\\"l'alsotI.'tim'rimein:l.ICsimulationsin1I11itsor thenumberofattr-mpta atgCIl('ratill~1Il' \\'sta l\'s.

st oursean1>("eorrelatr-d.IIIgl'llr ral{.r}..and{r.... ,ar.,hi l!.hl~·cor relat edMid":l.1C ste psarerequiredIol,fon'thetwost a res{r}..and{rt ....arenolonger correlat ed.

IIIun ll'rtogene rate.\1uncorrclar cdslat es . "tuue,iJlisrequired.Inpral·t ke.the initia lst all'ofaSYSICIillUayhefenfrom("<Iuilihri llltl"1lI1anumber,flO.of:\ICstl'!JS an'requiredI)('for('asyst em("('acht'SaILequ ilihriulIISIalI' .Therefo re."0 :l.ICsteps arerequiredbeforegeucrut odsta tes canbeused illstatis tit'a lan·ra g ing .III prjm-iplr- allsubsequentstatrs canhe used for st ntistica lan'raging.Howeve rill prac ticr-for computar ioual,·ffil'iI'IIt:Y.there arc6t:l.ICstr-ps illbetweenst at eswhich nreused ill tlu-statistical averages.Forcom p uta tiona lefficiencyallappropri atechokeor"g."

'10 and61 isrequiredIJlIliiigl'Tlf'ta l."0-fIandtitart' notknown aprIOri.This"rnlll"1111.0;

wsol\"{'(!bymonitorin g variousrelaxar iou[1I1\('l ioll ,;which dorermiuetil"aPI' Topr iatr tiuu-sca h-s.\\"r,Id i nl'til('norma lizodrela xationfuucriouofa quamitv.-1as

<..t(O)..1{t )>_<..I>2 o,dtl

=

<._p>_<.-t>l . Then-iuxaric nlimea~ud a t .'(twithtlu-quantiryA isdefinedas

(:2.;)

(2.8)

Seve ralre-la xati ontilll<'S('au1M,'('alcuhu rdcor respondingtova rio us proporrjcs.rmd liI(' lILaxilllUIUrelaxatjonti l1l(,(T...dZ )det ermi nestheminimumnumberIIf:\ICsu'ps

)oleItorntionsforequilibriuru srarisrics .TIl('ca k u la t iolloftill'relaxat iontitur-sabo prO\'id ,~alILt ' 1!.1lSofcilltu lat illl1; rite sta tistka lerrorilltheuvcrugcdquantitir-s.TIl<'

TILr correcrtouterm2,.,/,)1a("('OII1l ISforcorrelauous1,..·tw''('11states. Iftitrs-

T _ ,.

Ihl'llthocorrec tio ntermisIlf'gl ig ihlf'andtllf'standardformulaillthc squareof lhf'st atis ticalerror foruur-crrulnt odst all'Sisrecovered. If! l!Il' op posi teC.L">l'where tit¢:T__I.wefind