University of Groningen

On the phase behavior of polydisperse copolymersKok, Christiaan

IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.

Document VersionPublisher's PDF, also known as Version of record

Publication date:2001

Link to publication in University of Groningen/UMCG research database

Citation for published version (APA):Kok, C. (2001). On the phase behavior of polydisperse copolymers: (tapering, oscillations and destruction).s.n.

CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).

The publication may also be distributed here under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license.More information can be found on the University of Groningen website: https://www.rug.nl/library/open-access/self-archiving-pure/taverne-amendment.

Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.

Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.

Download date: 13-06-2022

Section4 75

4 Destruction of diblock copolymers

Themainresultsof this sectionarepublishedas:“Molecular inhomegeneityandamplitudeof scatteringof theproductsof diblock copolymerdegradation”,C.Kok, S.I. Kuchanov andG. tenBrinke,Macromolecules, 33 (19),2000,pp. 7195-7206[79].

4.1 Intr oduction

The investigationof thedegradationprocessesof polymersis of utmostsignificancefor macromolecularchemistry[80, 81]. Thechemicaltransformationof polymersin-ducedby their fragmentationusuallyleadsto anunwanteddeteriorationof theservicepropertiesof polymericmaterials.Thechangeof someimportantstatisticalcharacter-isticsof a polymersample(e.g.,its molecularweightdistribution, MWD, describingthe molecularinhomogeneityof homopolymers)dueto the degradationof polymerchainsis obviously responsiblefor this deterioration.On the otherhand,for someapplications,suchasbiomedical,degradationmayevenbeoftena desirableproperty[82, 83]. However, alsoin thatcaseknowledgeof theevolution of theperformancepropertiesis essential.In thecaseof binarydiblockcopolymers,themacromoleculesdiffer in numbersl1 and l2 of monomericunits M1 andM2 which unambiguouslycharacterizetheirsizel ¥ l1 î l2 andchemicalcompositionζ1 ¥ l1 Ã l © ζ2 ¥ l2 Ã l . Thejoint distribution of moleculesfor sizeandcomposition,(SCD),describesthemolec-ular inhomogeneity. To theauthors’knowledgeno attemptsto considertheoreticallytheevolutionof suchaSCDin thecourseof theprocessof diblockcopolymerdegra-dationhave beenundertakensofar. For homopolymeranalogues,thoughmoresim-ple, the problemof the alterationof the MWD of the productsof degradationwastackled[84, 85, 86].

A specificfeatureof diblockcopolymersis thepossibilityof phaseseparationduringdegradation.This phenomenonresultsfrom anappreciableincreasein compositioninhomogeneitydueto the contribution of homopolymers,whosefractionsprogres-sively increasewith the growth of the degradationdepth. Whenthis factorprevailsover thedecreaseof theaveragesizeof themacromolecules,which favorsthestabil-ity of thehomogeneousstate,thesystemmayexhibit phaseseparation.Thecapacityto predictsucha destructioninducedphasetransition,which may leadto a lossoftransparency of the polymersystemanda deteriorationof someof its performanceproperties,is of indisputablepracticalinterestbecausepolymericmaterialsareknownto undergo degradationduring their processingandaging. Theoppositesituationisalsoquite conceivablewhenin the courseof degradationthe interplayof the above

76 Section4

two factorswill causetheannihilationof spatiallyperiodicstructuresformedin meltsof diblock copolymers.Revealingthe conditionsof suchannihilationby meansofmathematicalmodelingconstitutesa challengingtaskwhenpredictingthepropertiesof advancedpolymermaterialsbasedonblock copolymers.

Themoststraightforwardway to measurethethermodynamicmiscibility of polymerliquidsis viascatteringtechniques,namely, light, X-ray orneutronscattering[87, 88].Measuringthe angulardependenceof the static structurefactor provides valuableinformationon thethermodynamicstateof a polymerliquid. Thecalculationof thisdependencewithin the framework of the randomphaseapproximation(RPA) wasreportedfor block copolymersof differentarchitecture[89, 90, 91]. However, withrespectto the productsof their degradationsucha calculationhasnot beendonesofar. Its inherentpeculiarityconsistsin the fact that to have this problemsolved it isnecessaryto first find thedependenceon time of theSCDof macromoleculesbeingformedduringthedegradationprocessof theinitial block copolymersystem.

The solutionof this problemof statisticalchemistryof polymerswill be discussedin the next sectionof this paper. In the third sectionthe equationswill be derivedwhich describewithin theframework of RPA theevolution of thestructurefactorofthe productsof degradationof diblock copolymerswith an arbitrarydistribution ofblock lengths.Furthermore,equationsdescribingthespinodalandtheLifshitz pointsin themelt of suchcopolymerswill bepresented.Thelastsectionwill bedevotedtotheillustrationof theapplicabilityof thegeneraltheorydevelopedto thedegradationof diblock copolymerswhoseblock lengthdistributionsareeithermonodisperseorexponential.

4.2 Statistical Chemistry of DegradedDiblock Copolymers

The theoreticalinvestigationof diblock copolymerdegradationrequiresas a firststepto write down a kinetic schemefor thechemicaltransformationsof themacro-molecules.Below wewill considerrandomchainscissionratherthanadepolymeriza-tion mechanismof degradation[80], assumingthat the Flory principle works. Thismeansthat any intramolecularbondenteringinto the α-th type block ç α ¥ 1© 2è isbroken within an infinitely small interval of time dt with infinitesimal probabilitykαdt. Consequently, the modelcomprisesonly two kinetic parameters,k1 andk2 ,beingthechemicalbondcleavagerateconstantsof bondsof thefirst andthesecondtype,respectively. This simplestmodelof degradation,basedon theapplicabilityofthefundamentalFlory principle,is mostwidespreadin polymerchemistry. However,violationof thisprinciplecanoccurin somesystemsdueto, for instance,shearactingon macromolecules.

If a diblock copolymermolecule,whoseblock lengthsarel1 andl2, is designatedas¨ l1 © l2 ª , thekinetic schemeof themolecularreactionswill be

Section4 77

¨ l1 © l2 ª k1l1À l ú1 © l2 î l ú ú1 © 0 © l ú1 î l ú ú1 ¥ l1 © l2 û 0 (163)

¨ l1 © l2 ª k2l2À l1 © l ú2 î 0© l ú ú2 © l ú2 î l ú ú2 ¥ l2 © l1 û 0

Let C1 ç l1 è , C2 ç l2 è andC12 ç l1 © l2 è denotethedimensionlessconcentrationsof theho-mopolymers l1 © 0ª and ¨ 0© l2 ª , anddiblockcopolymer l1 © l2 ª (reducedto theoverallconcentrationof monomericunits) which arepresentin the reactionsystemat timet. Thekineticequationsfor theseconcentrationscorrespondingto scheme(Eq(163))read

dC12 ç l1 © l2 èdt ¥ k1

∞

l1

C12 ç ξ1 © l2 è dξ1 î k2

∞

l2

C12 ç l1 © ξ2 è dξ2 ÎKç k1l1 î k2l2 è C12 ç l1 © l2 è(164)

dC1 ç l1 èdt ¥ k1

∞

0

dl2

∞

l1

C12 ç ξ1 © l2 è dξ1 î 2k1

∞

l1

C1 ç ξ1 è dξ1 Î k1l1C1 ç l1 è (165)

dC2 ç l2 èdt ¥ k2

∞

0

dl1

∞

l2

C12 ç l1 © ξ2 è dξ2 î 2k2

∞

l2

C2 ç ξ2 è dξ2 Î k2l2C2 ç l2 è (166)

Theseequationswill besolvedbelow underthefollowing initial conditions

C012 ç l1 © l2 è ¥ Y0 f 0

1 ç l1 è f 02 ç l2 èO© C0

1 ç l1 è ¥ C02 ç l2 è ¥ 0 (167)

where f 0α ç lα è representsthe MWD of the α-th type blocks ç α ¥ 1© 2è in the initial

copolymerwhile Y0 ¥ Π0 Ã M standsfor the ratio of the concentrationsof all itsmoleculesΠ0 to the concentrationM of monomericunits involved in them. Evi-dently, Y0 is nothingbut thereciprocalnumberaveragedegreeof polymerizationl

0

of theinitial copolymersystem.As for thefunctionC012 ç l1 © l2 è , its assumedfactoriza-

tion (Eq (167))suggeststhe independenceof thelengthdistributionsof thedifferenttypesof blocks. This condition is likely to hold for mostwaysof diblock copoly-mer synthesis.In caseof the presenceof somefraction of homopolymers(typicalfor commercialblock copolymers)in a specimenundergoingdegradation,theinitialcondition(Eq (167))admitstheextensionallowing for thiscircumstance.

Thesolutionof theCauchyproblem(Eqs(164)- (167)),givenin theAppendix,maybepresentedasfollows

78 Section4

C12 ç l1 © l2 è ¥ C12 f1 ç l1 è f2 ç l2 èO© where C12 ¥ C012 ¥ Y0 (168)

Cα ç lα è ¥ Cα f hα ç lα è ç α ¥ 1© 2èü© where Cα ¥ kαXαl

0t ¥ Zατ (169)

Here f1 ç l1 è and f2 ç l2 è denotetheMWD of blocksof monomericunitsM1 andM2 attime t

fα ç lα è ¥ exp ç�Î ταlα è f 0α ç lα è î τα

∞

lα

f 0α ç ξ è dξ ç α ¥ 1© 2è (170)

whereasf h1 ç l1 è and f h

2 ç l2 è aretheMWDs of thefirst andsecondtypehomopolymers

f hα ç lα è ¥ exp ç�Î ταlα è

l0α

∞

lα

ý1 î τα ç ξ Î lα è=þ f 0

α ç ξ è dξ ç α ¥ 1© 2è (171)

In expressions168- 171useis madeof thefollowing designations

τα ¥ kαt © τ ¥ k1l01 î k2l

02 t © l

0α ¥ Xαl

0 © Zα ¥ kαXαk1X1 î k2X2

(172)

Here l0α standsfor the numberaveragelength of the α-th type block in the initial

blockcopolymer, wherethemolarfractionsof unitsM1 andM2 areX1 andX2. Thesequantities,Xα, representingnumberaveragevaluesof thedegradationproducts’com-position,remainunchangedin thecourseof degradation.Thesamepropertyis alsoinherentto theoverall concentrationC12 (Eq (168))of block copolymermolecules.

It is possibleto separatehomopolymersformedasaresultof thedegradationprocessfrom block copolymersby meansof a chromatographictechnique. Consequently,by plotting the experimentaldependencieson time of the concentrationof the twohomopolymersformedduringthedegradationof theinitial block copolymeronecanfind thevaluesof bothkineticparameters,k1 andk2, characterizingthemodelathand.

A comparisonof theMWDs of thesehomopolymerscalculatedby Eq (171)with theexperimentalMWDs allows a verificationof theadequacy of thedegradationmodelchosen.Suchacomparisonis easierto realizewith respectto thestatisticalmomentsof the MWD thanwith respectto the very distribution itself. Explicit formulasforthemcanbe derived (seeAppendix)provided analyticalexpressionsareknown forthe generatingfunctionsg0

α ç pè of the MWD f 0α ç lα è of blocksin the initial diblock

copolymer. Thesimplestcriterionfor theapplicabilityof themodelof randomdegra-dationis theproportionalityof theoverall concentrationof homopolymermolecules,Cα, to thedegradationtime t. Proceedingfrom Eq (169), it is easyto determinethe

Section4 79

constantof degradationkα from theslopeof thestraightline of Cα versust. Recourseto thecross-fractionationmethod[92, 93] providesthepossibility to separateblockcopolymermacromoleculesby their sizel ¥ l1 î l2, irrespective of composition.Theequationallowing to find thedistribution f s

12 ç l è of theblockcopolymermoleculesforl canbederivedanalytically(seeAppendix).

4.3 Statistical Physicsof DegradedDiblock Copolymers

In orderto calculatewithin theframework of theRPA thescatteringintensityI ç qè ofa melt of anarbitraryincompressiblemixtureof macromoleculeswith two blocksofdifferenttypesof monomericunitsM1 andM2, it is possibleto resortto well-knownrelations[89, 90,91,94].

I ç qè ¥ M ç a1 Î a2 è 2D ç qè © D ç qè ¥ H ç qèGÎ 2χ (173)

H ç qè ¥ X11 î X22 î 2X12

X11X22 Î X212

(174)

whereq representsthemodulusof thescatteringwavevector, a1 anda2 arethescatter-ing lengthsof unitsM1 andM2, while χ is theFlory-Hugginsinteractionparameter.As for the elementsκαβ ¥ MXαβ of the structurematrix, their dependenceon q isdefinedby theexpressions

Xαα ç qè ¥ 2

∞

0

Cbαdlα

lα

0

dηη

0

dξexpý Î Qξþ ç α ¥ 1© 2è (175)

X12 ç qè ¥∞

0

∞

0

C12 ç l1 © l2 è dl1dl2

l1

0

dη1

l1 ÿ l2

l1

dη2expý Î Q ç η2 Î η1 è=þ (176)

wherethefollowing designationsareemployed

Q ¥ a2q2

6Cb

α ç lα è ¥ Cα ç lα è î Y0 fα ç lα è (177)

for the dimensionlesssquareof the wave vector q with monomericunit size a asa scaleand dimensionlessconcentrationCb

α ç lα è of the α-th type blocks with lαmonomericunits. Knowing theLaplacetransformof this function,Cb ç pè , it is easyto find thediagonalelementsof thestructurematrix

80 Section4

Xαα ç Qè ¥ 2Q2 Cb

α ç QèGÎ Cbα ç 0è î XαQ ç α ¥ 1© 2è (178)

In conformity with Eq (177),Cbα ç pè representsa linear combinationof generating

functionsgα ç pè andghα ç pè of distributions fα ç lα è (Eq (170))and f h

α ç lα è (Eq (171)),the expressionsfor which arederived in the Appendix. Making useof expressions293and294aswell asof therelations177and178wegetthefinal expressionfor thematrixelementsX11 ç Qè andX22 ç Qè

Xαα ç Qè ¥ 2Y0

Qα2 g0

α Qα Î 1 î l0αQα © Qα ¥ Q î τα (179)

Justin thesamewaybymeansof theexpressions(176),(168),and(293),theequationfor thenondiagonalelementsof thestructurematrix

X12 ç Qè ¥ X21 ç Qè ¥ Y0

Q1Q21 Î g0

1 Q1 1 Î g02 Q2 (180)

canbederived. To switchto naturalscalesof thevaluesof thevariables,it is conve-nient to usevariableslα Ã l 0

α ratherthan lα. In termsof theserescaledvariables,Eqs(179)and(180)assumetheform

Xαα ç yè ¥ 2l0X2

α

ç yα è 2 g0α ç yα èÎ 1 î yα ç α ¥ 1© 2è (181)

X12 ç yè ¥ X21 ç yè ¥ l0X1X2

y1y21 Î g0

1 ç y1 è 1 Î g02 ç y2 è (182)

Hereadditionaldesignationsareintroduced

yα ¥ l0αQα ¥ Xαy î Zατ © y ¥ Ql

0 ¥ q2R2G © R2

G ¥ a2l0

6(183)

wherethedimensionlesstime τ andtheparameterZα have, consequently, themean-ing of theaveragenumberof cleavagespermacromoleculeandthe fractionof thesecleavagesoccurringin α-th typeblocks.In theexpressions181and182g0

α ç pè stands

for thegeneratingfunctionof thestochasticvariablelα Ã l 0α distributionin moleculesof

theinitial blockcopolymer. Substitutingexpressions181and182for theelementsofthestructurematrix into Eq(174)enablesoneto find theexpressionfor thescatteringintensity(Eq (173))of theproductsof a diblock copolymerdegradationat arbitrarydepthof this reaction. Examplesof the employmentof this expression,derived for

Section4 81

arbitraryMWDs of blocksfor lengths,will beprovidedin thenext Sectionwherewespecializeto systemswith specifiedMWDs.

An importantelementof a phasediagramis thespinodal,i.e., a hypersurfacewithinthespaceof externalparametersof a systemwherethelatter looseslocal stability ofthespatially-homogeneousstate.Themathematicalconditionof thespinodal,wheretheamplitudeof scattering(Eq (173)),becomesinfinite is asfollows

Hm ¥ 2χ © whereHm � minq

H ç qè ¥ H ç q¹ è (184)

The function H ç qè (Eq (174)) canreachits minimum valueHm eitherat zerowavevector, q¹ ¥ 0, or at q¹ �¥ 0. In the first casewe dealwith the trivial branchof thespinodal,whereasin the secondcasewe dealwith its nontrivial branch. Equationsfor thesehypersurfaceswithin theparametricspaceread,respectively,

1è 2χ ¥ H ç 0è 2è 2χ ¥ H ç q¹ èO© H ú ç q¹ è ¥ 0 H ú ú ç q¹ è ñ 0 (185)

whereH ú ç qè andH ú ú ç qè denotethefirst andthesecondorderderivativesof thefunc-tion H ç qè . The trivial andnontrivial branchesof the spinodalareseparatedby theLifshitz point hyperlinewhichcanbefoundusingtheequations

2χ ¥ H ç 0èü© H ú ú ç 0è ¥ 0© H ú ú ú ú ç 0è ñ 0 (186)

For the systemunderconsideration,the function H (Eq (174)) dependson q onlythroughthe variabley ¥ q2R2

G so that for theoreticalconsiderationsit is convenient

to usethe function � ç yè � � q2R2G ¥ H ç qè insteadof H ç qèü¦ Thederivativesof

thesefunctionsareconnectedby simplerelations

H ú ç qè ¥ 2qR2G � ú ç yèO© H ú ú ç qè ¥ 2R2

G 2y� ú ú ç yè î � ú ç yè (187)

whichallows to rewrite Eq (185)for thespinodalin termsof thefunction � ç yè1è 2χ ¥ � ç 0è 2è 2χ ¥ � ç y¹ èO©�� ú ç y¹ è ¥ 0©�� ú ú ç y¹ è ñ 0 (188)

andEq(186)for theLifshitz point

2χ ¥ � ç 0èO© � ú ç 0è ¥ 0©�� ú ú ç 0è ñ 0 (189)

Thefunction � dependingonthevariabley ¥ q2R2G (Eq(183))andthedimensionless

timeτ (Eq(172))is alsocontrolledby asetof externalparametersbothchemicalandphysicalin nature.To thechemicalparameters,thereactivity ratio r ¥ k1 Ã k2 belongs,characterizingthedifferencein stability of thechemicalbondsinvolved in blocksof

82 Section4

different type. More appropiatefor numericalcalculationsis the kinetic parameterU1 ¥ 1 Î U2 ¥ k1 ÃOç k1 î k2 è , sinceits value lies within the interval [0,1]. The setcomprisingthe compositionof the initial block copolymerX andthe parametersoftheMWD of its blocksbelongsto thephysicalparameters.

Thetheorydevelopedaboveallowsusto answerat leasttwo questionsof practicalin-terest.Thefirst question,whichis relevantto thethermodynamicsof reacting”li ving”systems,is whetherthe lossof thermodynamicstability of the initial homogeneousstatewill happenin thecourseof thedegradationprocess.And, if so,will it occuratzeroor non-zerowave vectorq¹ ? Thesecondquestionis what thespinodalwill beof the”dead” systemcomprisingtheproductsof thedegradationof the initial blockcopolymerformedduringa specifiedtime of degradation.This systembeingcooledwith a rateperceptiblyexceedingthatof degradationwill reachthespinodalat a cer-taintemperatureTs. Theanswerto thesecondquestionfor aknown dependenceof theparameterχ on temperatureenablesusto reveala region of thosevaluesof externalparametersinsidewhich thepolymerspecimencanbecooledto a fixedtemperatureT with no lossin local stability of its spatiallyhomogeneousstate.Otherwisestated,it enablesusto find theareaof theparametricspacewhereTs Æ T.

It is naturalto starttheexaminationof theevolution of thespinodalin thecourseofdegradationwith theanalysisof its appearancein themeltof theinitial blockcopoly-mer. To this endexpressionsmay be usedfor the function � ç yè andits derivative� ú atpoint y ¥ 0

� ç 0è ¥ K1X21 î K2X2

2 î 2X1X2

l0 ç X1X2 è 2 ç K1K2 Î 1è ©�� ú ç 0è ¥ ∆ ç X è

3 ç X1X2 è 2 ç K1K2 Î 1è 2 (190)

∆ ç X è ¥ K ú1K21X1 ç X1 î K2X2 è 2 î K ú2K2

2X2 ç K1X1 î X2 è 2 (191)

Î 3 ç K1X1 î K2X2 èOç X1 î K2X2 èOç K1X1 î X2 èThe coefficients of the cubic polynomial (Eq (191)) are controlledexclusively bypolydispersitycoefficientsKα andK úα of theMWD of α-th typeblocks ç α ¥ 1© 2è intheinitial copolymerwhicharerelatedto numberaverage,Pnα, weightaverage,Pwα ©andz-average,Pzα © degreesof polymerizationof theseblocksby simplerelationships

Kα � PwαPnα

¥ l2α

lα lαK úα � Pzα

Pwα¥ l3

α lα

l2α l2

α(192)

Here for simplicity superscript”0” is omitted from the designationof the n-orderstatisticalmomentsln

α ç n ¥ 1© 2© 3è of theMWD f 0α ç lα è .

The sign of thequantity∆ is of centralimportancefor thespinodalanalysis.Thus,for instance,in caseof monodisperseblock copolymersthespinodalis known[10] to

Section4 83

have no trivial branchat all. Will polydispersecopolymercontainsucha branch?Anecessaryconditionfor this is thepositivenessof ∆. If thefunction � ç yè hasnomorethanoneminimum,thentheimplementationof theinequality∆ ñ 0 is notonly anec-essarybut alsoa sufficient conditionfor a spatially-homogeneousstateof a systemto looseits stability just at zerowave vectorvalueduring cooling. For systemsex-hibiting sucha typicalbehavior of thefunction � ç yè , zeroesof thepolynomial∆ ç xè(Eq(191))insidetheunit interval ç 0© 1è correspondto theLifshitz pointsdividing thisunit segmentinto intervals within eachof which the quantity∆ doesnot changeitssign.Thoseintervalswhereit is positiveor negativecorrespondto trivial or nontrivialspinodalbranches,respectively. To find themrecourseshouldbemadeto Eqs(188),(190),and(191)while it is possibleto calculatetheLifshitz pointsusingEq(189).

All initial diblock copolymers(proceedingfrom the shapeof their spinodalχ ¥FS ç X è ) canbesubdivided into 4 typesaccordingto thenumberof rootsi ¥ 0© 1© 2© 3of polynomial∆ ç xè (Eq (191)) insidetheunit interval. This numbermaybeevenorodddependingon whetherthequantityδ ¥ ç K ú1 Î 3èOç K ú2 Î 3è is positive or negative.Consequently, theconditionδ Æ 0 is sufficientfor theexistenceof at leastoneLifshitzpoint and,thus,for theexistenceof a trivial branchof thespinodalχ ¥ FS ç X è .In order to reveal how many roots polynomial ∆ ç xè (Eq (191)) hasinside the unitsegmentit is necessaryto calculatethesignof theexpression

� ¥ B2C2 Î 4AC3 Î 4B3D Î 27A2D2 î 18ABCD (193)

wherethefollowing designationsareemployed

A ¥ K ú1 Î 3 K21 © B ¥ K1 2K ú1K1K2 î K ú2K1K2

2 Î 3 ç K1K2 î K2 î 1è (194)

C ¥ K2 K ú1K21K2 î 2K ú2K1K2 Î 3 ç K1K2 î K1 î 1è © D ¥ K ú2 Î 3 K2

2

If� Æ 0 thenpolynomial∆ ç X è hasonly oneroot having a physicalmeaning.Con-

versely, when� ñ 0 the numberof suchrootscoincideswith the numberof sign

reversalsin thesequenceA,B,C,D definedin (Eq(194)).Hence,thequestionof clas-sificationof conceivablespinodalformsmaybeconsideredassettled.

Concludingthis Sectionit is pertinentto point out unambiguouslythe correlationbetweenthe shapeof curves H ç qè and I ç qè . So, every minimum of the first onecorrespondsto a maximumof thesecondone. This meansthatwhenmeasuringtheangulardependenceof the amplitudeof scatteringit is possibleto predictat whichparticularspinodalbranch,trivial or nontrivial, thesystemwill loosethestability ofthespatially-homogeneousstateundercooling.Essentially, it is possibleto carryoutthe scatteringexperimentsat a temperaturewhich is higherthanthat correspondingto the cloud point curve. The advantagesof performingtheseexperimentsin theregion of absolutethermodynamicstability of the homogeneousstateinsteadof inthemetastableregionarebeyondany doubt.

84 Section4

4.4 Applications

Thedistribution of theblocksfor lengthsin an initial copolymeris evidently prede-terminedby theconditionsof its synthesis.If theblock formationfollows themech-anismof ”li ving” anionicpolymerization,theirdistribution is closeto monodisperse,i.e. it is describedby theDirac delta-function

f 0α ç lα è ¥ δ ç Nα Î lα èü© g0

α ç yα è ¥ exp çëÎ Nαyα èO© g0α ç yα è ¥ exp ç�Î yα è (195)

whereNα is the numberof units involved in an α-th type block. Whenblocksarepreparedby the methodsof free-radicalpolymerizationor polycondensation,theirdistribution is oftenanearlyexponentialone

f 0α ç lα è ¥ εα exp ç�Î εαlα èO© g0

α ç yα è ¥ 1

1 î l0αyα

© g0α ç yα è ¥ 1

1 î yα(196)

whereεα ¥ 1Ã l 0α is the reciprocalaveragenumberof units in an α-th type block.

Polydispersitycoefficients(Eq (192))of themonodisperse(Eq (195))andtheFlorydistribution (Eq (196))are,respectively

1è Kα ¥ K �α ¥ 1 2è Kα ¥ 2© K �α ¥ 32

(197)

Proceedingfrom Eqs(170)or (293)it is possibleto derive theexpressions

fα ç lα è ¥ exp ç�Î ταlα è ý δ ç Nα Î lα è î ταηs ç Nα Î lα è=þ (198)

fα ç lα è ¥ εα exp ç�Î εαlα èO© where εα ¥ εα î τα (199)

characterizing,respectively, the evolution of initial distributions (Eq (195) and Eq(196))of theα-th typeblocksfor lengthslα in thecourseof thedegradation.Hence-forward ηs standsfor the Heaviside step function. The MWD of homopolymersformedduringthedegradationprocessof monodisperseandexponentiallydistributedblocksaredescribedby thefollowing formulas

f hα ç lα è ¥ exp ç�Î ταlα è

Nα

ý1 î τα ç Nα Î lα è=þ ηs ç Nα Î lα è (200)

f hα ç lα è ¥ εα exp ç�Î εαlα è (201)

whichmaybereadilyderivedusingtheexpression171or 294with allowancefor Eq(195)andEq (196),respectively. Noteworthy, thedistributionsEqs(199)and(201),

Section4 85

unlikeEqs(198)and(200),coincidewith oneanotherandwith theinitial distribution(Eq (196)) upon the replacementin the latter of the parameterεα by εα. Generalformulas295and296in caseof monodisperseblocksyield thefollowing expressionsfor thestatisticalcharacteristicsof thedegradationproducts

lα ¥ Nαθα

1 Î e¾ θα © σ2α ¥ N2

αθ2

α1 Î 2θαe¾ θα Î e¾ 2θα © where θα ¥ Nατα

(202)

Phnα ¥ Nα

θ2α

θα Î 1 î e¾ θα © Khα ¥ 2θα θα 1 î e¾ θα Î 2 1 Î e¾ θα

θα Î 1 î e¾ θα 2 (203)

Analogousformulasfor exponentiallydistributedblocksread

lα ¥ Phnα ¥ ε ¾ 1

α © σ2α ¥ ε ¾ 2

α © Khα ¥ 2 (204)

Below wewill considerthreetypesof initial diblockcopolymers

I è MD î MD II è MD î F I I I è F î F (205)

composedof monodisperseblocks(MD) andthosecharacterizedby theFlory expo-nentialdistribution (F). TheSCDof theproductsof degradationof thesecopolymersis describedby theexpression

f12 ç l © ζ è ¥ l f1 ç lζ1 è f2 ç lζ2 è (206)

wherethepair of functions f1 ç l1 è and f2 ç l2 è arepickedout of {Eq (198),Eq(199)},dependingon the type of initial copolymers(Eq (205)). So, for instance,to getan idea of the evolution of the distribution (Eq (206)) for type III it is possi-ble to turn to Fig 38. An interestingpeculiarity of this evolution is the fact thatunder the condition ç X2 Î X1 èOç U2 Î U1 è ñ 0 a unique instant of time doesexist,τ ¹ ¥ ç X2 Î X1 èOç U1X1 î U2X2 èOÃéç X1X2 ç U2 Î U1 è�è , whentheSCD(Eq (206))doesnotdependon compositionζ (Fig 38.2), thereforedegeneratinginto a one-dimensionaldistribution

f12 ç l © ζ è ¥ ε 2l exp ç�Î εl èO© where ε ¥ ε1 ¥ ε2 ¥ ç U2X2 Î U1X1 èl0 ç U2 Î U1 è X1X2

(207)

Substitutinginto formula (293) expressions(195) and (196) for g0α resultsin the

Laplacetransformsof thedistributionsfor theblock lengthlα in degradedcopolymer

86 Section4

0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.2

0.4

0.6

0.8

1

ζ

1

0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.2

0.4

0.6

0.8

1

ζ

2

0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.2

0.4

0.6

0.8

1

ζ

3

Figure38: Contourplots(equalheightlines)of theSize(l )-Composition(ζ ¥ ζ1) Dis-tribution (SCD)(Eq (206))of degradeddiblockcopolymersof typeIII (doubleFlorydistribution) at valuesof parameterU ¥ U1 ¥ k1 ÃOç k1 î k2 è ¥ 1Ã 10 andinitial com-positionX ¥ X1 ¥ 1Ã 3 at rescaledtimes(Eq (172))τ=0(1),1.19(2),2.38(3).Darkerregionscorrespondto largerprobabilities.

Section4 87

molecules.Further, makinguseof formula(297) it is possibleto derive expressionsfor the distribution of thesemoleculesf s

12 ç l è for their size l for the threetypes(Eq(205))of initial copolymers

I è f s12 ç l è ¥ exp ç -τ1N1-τ2N2 è δ ç l -N è (208)

î τ1exp ç -τ1l èτ2 Î τ1

¨ τ2ηs ç N1-l è Î τ1expý ç τ1-τ2 è N2þ ýηs ç l -N2 èGÎ ηs ç l -N è=þ ª

î τ2exp ç -τ2l èτ1 Î τ2

¨ τ1ηs ç N2-l è Î τ2expý ç τ2-τ1 è N1þ ýηs ç l -N1 èGÎ ηs ç l -N è=þ ª

I I è f s12 ç l è ¥ ε2

ε2 Î τ1¨ τ1exp ç -τ1l è ηs ç N1-l è

î exp ç�Î ε2l è ý ε2expý ç ε2 Î τ1 è N1þ ηs ç l -N1 èÎ τ1þ ª (209)

I I I è f s12 ç l è ¥ ε1ε2

ε2 Î ε1

ýexp ç -ε1l èGÎ exp ç -ε2l è=þ (210)

Therandomvariablel ¥ l1 î l2, beingthesumof two independentrandomvariablesl1 and l2, hasthe centerl1 î l2 andthe dispersionσ2

1 î σ22, where lα andσ2

α weredeterminedabovefor monodisperse(Eq(202))andexponentiallydistributed(Eq204)blocks.

Theintegrationof expression(206)over thevariablel resultsin theone-dimensionaldistribution f c

12 ç ζ è of thedegradationproductswhich for the initial diblock copoly-merof typeIII takestheform

f c12 ç ζ è ¥ ε1ε2

ç ε1ζ1 î ε2ζ2 è 2 (211)

Owingto itsmonotonicitythisfunctionreachesitsmaximumvalueatoneof theedgesof theunit segmentζ1 î ζ2 ¥ 1. This edgecorrespondsto theα-th homopolymerforwhich thevalueof theparameterεα is least.Thecompositiondistribution (Eq (211))is controlledbyonly oneparameterwhichis theratiobetweenthequantitiesε1 andε2.Themorethey differ thenarrower this distribution is. Thelargestcompositioninho-mogeneityoccursfor ε1 ¥ ε2, whenany valueof block copolymercompositionturnsouttobeequiprobable.In orderfor thiscaseto berealizedduringthedegradationpro-cessit is necessaryandsufficient for the initial copolymerto containin excessthoseunitswhoseblocksaremoreproneto thecleavage. Themathematicalconditionforthis is the inequality ç X2 Î X1 èOç k2 Î k1 è ñ 0. If this conditionis met, thentheblockcopolymercompositioninhomogeneityfirst risesin thebeginningof thedegradationupto themomentτ ¹ ¥ ç X2 Î X1 èOç U1X1 î U2X2 èOÃ X1X2 ç U2 Î U1 è , whenthequantitiesε1 andε2 becomeequalto oneanother, andthenstartsto decrease.Conversely, if theinequality ç X2 Î X1 èüç k2 Î k1 è Æ 0 holdsthenthecompositiondistribution (Eq (211))getsmonotonicallynarrower duringthewholedegradationprocess.

88 Section4

As for the distribution (Eq (210)) of the sizeof the degradationproductsof macro-moleculesof type III, theevolution of its width in caseç X2 Î X1 èOç k2 Î k1 è ñ 0 pro-ceedsin oppositedirectionascomparedto the alterationof the width of the com-position distribution (Eq (211)). This conclusionensuesfrom the analysisof theexpression

K ¥ 2 1 Î ε1 Ã ε2 î ε2 Ã ε1

¾ 2

(212)

for the polydispersitycoefficient of distribution (Eq (210)). It decreasesfrom thevalueK0 ¥ 1 î X2

1 î X22 atτ ¥ 0 upto thevalueK ¹ ¥ 3Ã 2 atτ ¥ τ ¹ andthenrisestend-

ing to thequantityK∞ ¥ 1 î U21 î U2

2 . Conversely, in caseç X2 Î X1 èOç k2 Î k1 è Æ 0 thecoefficient (Eq(212))changesmonotonicallyfrom K0 to K∞. Examples,demonstrat-ing theabovementionedregularitiesof theevolutionof one-dimensionaldistributionsfor sizel or for compositionζ, areprovidedin Fig 39.

Turningto theamplitudeof scatteringof thedegradationproducts,it is necessarytopresenttheexpressionsfor theelementsof thestructurematrix. To this endrecourseshouldbemadeto thegeneralexpressionsEqs(181)and(182)in whichthefunctionsEq (195)and/orEq (196)shouldbesubstituteddependingon thetype(Eq (205))ofthe initial block copolymer. In caseof type I, the formulaswill bearresemblanceto thosederivedearlier[10] concerningthephasebehavior of monodispersediblockcopolymers

Xαα ç yè ¥ NX2αg ç yα è ç α ¥ 1© 2èü© g ç xè ¥ 2

x2 x Î 1 î e¾ x (213)

X12 ç yè ¥ NX1X2h ç y1 è h ç y2 èO© h ç xè ¥ 1x

1 Î e¾ x (214)

wherethedependenceof the variableyα on thewave vectormodulusandtime wasdeterminedabove(Eq(183)).For theinitial copolymerspertainingto typeII wehave

X11 ç yè ¥ l0X2

1g ç y1 èO© X22 ç yè ¥ 2l0X2

2 ç 1 î y2 è ¾ 1 (215)

X12 ç yè ¥ l0X1X2h ç y1 èOç 1 î y2 è ¾ 1 (216)

while for thoseof typeIII

Xαα ç yè ¥ 2l0X2

α ç 1 î yα è ¾ 1 ç α ¥ 1© 2è (217)

X12 ç yè ¥ l0X1X2 ç 1 î y1 è ¾ 1 ç 1 î y2 è ¾ 1 (218)

Section4 89

0.2 0.4 0.6 0.8 1

0.5

0.75

1

1.25

1.5

1.75

2

f12(ζ)c

ζ

2

3

1

(a)

0.5 1 1.5 2

0.2

0.4

0.6

0.8

1

1.2

1.4f12

� � �s

� � 0

(b)

2

3

1

Figure 39: One-dimensionalsectionsof Size(l )-Composition(ζ ¥ ζ1) Distribution(SCD)(Eq (206)),describingthedistribution for composition(a) andsize(b) of de-gradeddiblock copolymerof type III (doubleFlory). Valuesof the parametersUandX andrescaledtimesτ arethesameasthosepresentedin Fig 38, i.e,U ¥ 1Ã 10,X ¥ 1Ã 3, andτ=0(1),1.19(2),2.38(3).

90 Section4

0 0.2 0.4 0.6 0.8 1X

0

0.2

0.4

0.6

0.8

1

U

I

0 0.2 0.4 0.6 0.8 1X

0

0.2

0.4

0.6

0.8

1

U

II

0 0.2 0.4 0.6 0.8 1X

0

0.2

0.4

0.6

0.8

1

U

III

Figure40: Classificationdiagramcharacterizingqualitative peculiaritiesof the de-pendenceof thereducedtemperatureT ¥ 100Tmax

l0Θ

on thereduceddegradationtime τfor diblockcopolymersof typeI, II, andIII (Eq(205)).Correspondingexplanationisprovidedin thetext andfurtherillustratedin Figs41-43.

Thekey problemto beinvestigatedwhenconsidering”li ving” systemsis thedepen-denceon time of the quantityH ¾ 1

m , which is the reciprocalof Hm (Eq (184)). Thisquantity is equalto the ratio TmaxÃ Θ of the maximumtemperatureTmax for whichthe lossof thermodynamicstability of the homogeneousstatehappensto the FlorytemperatureΘ. ThetemperatureΘ is definedasthetemperatureat which theFlory-Huggingsparameterχ, describingthe monomer-monomerinteractions,reachesavalueof a half. If during the degradationthe valueTmax in its growth exceedsthetemperatureof the experimentT, the reactionsystembecomesabsolutelyunstablewith respectto smallcompositionfluctuations(spinodaldecomposition).

Section4 91

2 4 6 8 10

3.5

4.0

4.5

Tb

a

c

τFigure41: ReducedtemperatureT versusreduceddegradationtime τ (Eq (172)) indifferentregionsof theclassificationdiagram(Fig 40): white region (a),grayregion(b), andblackregion (c). Thepointwherethecurve T ç τ è reachesits maximumvalueis denotedby an opensquare,whereasthe Lifshitz point is representedby a filledcircle. The curvespresentedin this figure arecalculatedfor diblock copolymersoftypeI at thefollowing valuesof theparameters:(a)X=0.4;U=0.8;(b) X=0.4;U=0.1;(c) X=0.3; U=0.4

Let usconsiderfirst thedegradationof typeI (Eq (205))copolymer. In this case,theunit squareof possiblevaluesof parametersX1 ¥ X andU1 ¥ U maybeseparatedintothreekindsof regions(Fig 40.I) concerningtheshapeof thecurvescharacterizingthedependenceof Tmax onthedegradationtime(seeFig 41). Whereasin thewhiteregionof Fig 40.I this dependenceis monotonic,in the othertwo regions,gray andblack,thefunctionTmax ç τ è hasa maximum.Thedistinctionbetweenthelattertwo regionsconsistsin thefactthatin thegrayareatheLifshitz point is situatedto theright of themaximum,while in the black areait is locatedto the left of it. On the borderlines,dividing the regionsof differentcolors,bifurcationsof threetypesdepictedin Fig42 occur. Among them there is the nontrivial bifurcation occuringon the borderbetweenthewhite andtheblack regionsat the intersectionof which the locationofthemaximumof thecurveTmax ç τ è changesdiscontinuouslyfrom τm ¥ 0 (in thewhiteregion) to τm

�¥ 0 (in theblackregion). Whenmoving alongtheborderlineX ¥ 1à 2from U ¥ 0 to U ¥ 1à 2 the time τL of reachingthe Lifshitz point monotonically

92 Section4

decreasesfrom 5¦ 778 to 3¦ 785. An analogousdecreaseof τL takesplacealongtheborderlinebetweenthegrayandtheblackregionswhenwithin the interval 0¦ 327 ðX ð 1Ã 2 the quantityτL changesfrom 4¦ 772 to 3¦ 785. Along the third borderline,U ¥ 1Ã 2, thecharacterof thechangeof τL is analogousto thatmentionedabove, i.e..the valueof τL decreasesfrom ∞ to 3¦ 785 asX changesfrom 0 to 1Ã 2. The abovementionedregularitiesof the behavior of τL at the borderlinesof diagramI of Fig40 areillustratedin Fig 43. In thepoint of intersectionof all threeborderlinecurvesç X ¥ 1Ã 2© U ¥ 1Ã 2è , the dependenceTmax ç τ è hasthe appearanceshown in Fig 42,whereT ç 0è ¥ 100Tmax¼ 0¿

Θl0 ¥ 4¦ 76.

2 4 6 8 10

4

4.5

5

T 3

τ

1

2

Figure42: Thedependenceof thereducedtemperatureT versusthereduceddegrada-tion timeτ ontheboundariesbetweenthedifferentregionsof classificationsdiagram40.I. (1) white-blackboundary(X=0.35; U=0.5) (2) white-grayboundary(X=0.5;U=0.1) (3) gray-blackboundary(X=0.35; U=0.1) The dashedcurve correspondstothepoint(X=0.5;U=0.5)belongingto all threeboundaries.ThepointwherethecurveT ç τ è reachesits maximumvalueis denotedby anopensquare,whereastheLifshitzpoint is representedby a filled circle.

The diagrampresentedin Fig 40 allows us to predict the possiblephasebehaviorof the productsof the degradationof the type I block copolymersin the courseof degradationat fixed temperature. If the valuesof the external parametersX

Section4 93

andU are locatedwithin the white region of this diagramthe homogeneousstateof the systemremainslocally stablefor the whole period of degradation. Whentheseparametersfall into the gray region of the diagramthereis a small intervalTmax ç 0è Æ T Æ Tmax ç τmè of valuesof temperaturewherethe systemloosesthe sta-bility of its homogeneousstateat nonzerovector. Within theblackregion theaboveinterval getsnoticeablywider. Along with it thereis, however, anothertemperatureinterval, Tmax ç τL è Æ T Æ Tmax ç τmè , wherethesystemloosesits stabilityatzerowavevector. In otherwordsit canapproachwithin theblackregion eitherthenontrivial orthetrivial branchof thespinodal.

τL

0.2 0.4 0.6 0.8 1

4.0

4.5

5.0 2

X, U

3

1

Figure43: Thevaluesof thetime of reachingtheLifshitz point τL alongthebound-ariesbetweenthe regionsof diagram3.I. The numberingof the curvesherecorre-spondto thoseusedfor thedesignationsof theseboundariesin Fig 42.

The dependenceof the quantitiesτm andτL on X is depictedin Fig 4.4. The valueof τL is a smoothfunction of X, which divergesfor X À 0 or 1. The valueof τm,however, hasadiscontinuousderivative at thegray/blackboundary, which is denotedby X ¹ , whichis alsothepointwhereτm ¥ τL. For X ñ 0¦ 5 thevalueof τL is identicallyzero.Thevalueof X ¹ ¥ 0¦ 419.

94 Section4

0.2 0.4 0.6 0.8

1

2

3

4

5

6

7

* X

ττm

L

X

Figure44: Dependenceof the reducedtime of reachingthe maximumof the curveT τ � , τm (solid line), andtheLifshitz point, τL (dashedline) on thethecomposition(X) for diblock copolymersof typeI. Thecurveswerecalculatedwith theparameterU equalto 0.3.

In Fig 45, therescaledtemperaturesof phaseseparationat thetimesτm, τL (depictedin Fig 4.4), andτ � 0 arepresented.The solid curve (T τm � ) will touch the longdashedcurve (T τL � ) in onepoint (X ) only andwill beat higherT elsewhere.Thelong dashedcurve lies very closeto the solid curve for X � X , becausefrom Fig4.4 it is clear that τL and τm do not differ too muchandFig 41 demonstratesthatthe dependenceof T on τ is ratherweakaroundτ � τm. At rescaledtemperaturesabove the solid curve phaseseparationcannotbe inducedby destruction.For X �X , i.e. theblack area,macrophaseseparationcanbe inducedby destructionin theminutewindow of temperatureseparatingthesolidandthelongdashedcurveandfortemperaturesbetweenthe long dashedcurve andtheshortdashedcurve (T τ � 0� )microphaseseparationcanbeinduced.For X �� X � 0� 5, i.e., within thegrayarea,only microphaseseparationcanbeinducedif therescaledtemperatureis betweenthesolid andthe shortdashedcurve. If X � 0� 5, i.e., within the white area,no phase

Section4 95

separationcanbe induced,becausethe shortdashedcurve and the solid curve aremerged.

0.3 0.4 0.5 0.6

3.5

4.0

4.5

5.0

X*

T

X

Figure45: Dependenceof T τ � at thepointsτ � τm (solid line), τ � τL (long dashedline), andτ � 0 (shortdashedline) on thecompositionof diblockcopolymersof typeI for parameterU � 0� 3.

Underthedegradationof block copolymersof typeII theclassificationdiagram(Fig40.II) remainsbasicallythe sameasfor type I (Fig 40.I). The only qualitative dis-tinction is theabsenceof symmetrywith respectto thecenterof thesquare.On theborderlineX � 0� 408betweenthewhite andgrayregionsthevalueτL U � decreasesfrom τL 0��� 1� 956 to τL 1� 2��� 1� 630 andthenincreasesto τL 1��� 2� 621 . Thesamedependencetakesplacemoving alongtheborderlineseparatingthegrayandthe

96 Section4

black regionswhereτL 0��� 2� 000, τminL � τL 0� 622��� 1� 607 andτL 1��� 1� 891.

On the borderlineU � 0� 500, the quantity τL X � changingfrom τL 0��� ∞ toτL 1��� ∞ alsogoesthrougha minimum τmin

L � τL 0� 575��� 1� 502. At the point X � 0� 408� U � 1� 2� wherethethreeboundarycurvesintersectτL � 1� 629.

Themostsimplediagramis shown in Fig 40.III correspondingto block copolymersof type III. It is symmetricalwith respectto the centerof the squareanddoesnotcontainthegrayregion. On theverticalborderlineX � 1� 2 thevalueof τL is not in-fluencedbyU andequalszero,while on thehorizontalborderlineU � 1� 2 thequan-tity τL X � monotonicallydecreaseswithin theinterval 0 � X � 1� 2 from τL 0��� ∞to τL 1� 2��� 0. As the analysisshows, underthe degradationof diblock copoly-merwhoseblocksaredistributedexponentially, thewidth of thetemperatureintervalwherethesystemreachesthetrivial branchof thespinodalduringthedegradationcanbemarkedly higherthanin caseof monodisperseblocks.

Up to this point, we werediscussingthermodynamicstability of thespatiallyhomo-geneousstateof a ”li ving ” systemwherethe changeof the size andcompositionof macromoleculesoccursasa resultof their degradation.Now let us considertheproblemof thestabilityof a ”dead” systemrepresentingthepolymermelt composedof theproductsof thedegradationof theblock copolymersformedat a certaintimeof thedegradationreaction.Apart from the initial parametersX andU , theSCDoftheseproductsis entirelycharacterizedby thedimensionlesstimeτ of degradationoftheinitial block copolymers.At any fixedvalueτ it is possibleto constructa surfacedescribingthedependenceof thereducedvalueT on X andU (SeeFig 46). As canbeseen,every spinodalsurfacepresentedin this figurecomprisestwo regionscom-posedof thepointsof trivial (top) andnontrivial (bottom)spinodalbranches.Theseregionsareseparatedby aline of theLifshitz pointswhichappearatspinodalsurfacesI, II andIII uponelapseddegradationtime τI � 3� 78, τI I � 1� 50 andτI I I � 0. Forblock copolymersof the typesI andII it happensat pointsXI � 1� 2 , UI � 1� 2 andXI I � 0� 56,UI I � 0� 58atvaluesof Tmax equal,respectively, to 4.76and6.96.For thecaseof typeIII blockcopolymertheLifshitz pointsfill thesegmentof thestraightlineXI I I � 1� 2, 0 � UI I I � 1, T � 12� 5 alreadybeforethebeginningof thedegradation,sothatthetrivial branchof thespinodalsurfacewill appearimmediately.

Section4 97

0.2 0.4 0.6 0.8

0.2

0.4

0.6

0.8

1234

0.2 0.4 0.6 0.8

0.2

0.4

0.6

0.8

T

U

X

I

0.2 0.4 0.6 0.8

0.2

0.4

0.6

0.8

2

4

6

0.2 0.4 0.6 0.8

0.2

0.4

0.6

0.8

T

U

X

II

0.2 0.4 0.6 0.8

0.2

0.4

0.6

0.8

258

11

0.2 0.4 0.6 0.8

0.2

0.4

0.6

0.8

T

U

III

X

Figure46: Examplesof surfacesdepictingthedependenceof T onstoichiometric,X,andkinetic, U, parametersfor theproductsof degradationof diblock copolymersoftypesI, II, andIII obtainedat valuesof thereduceddegradationtime τ � 4� 0(I), τ �1� 6(II), andτ � 0� 18(III). The trivial andnontrivial spinodalbranchesareindicatedby thin andthick lines,respectively.

98 Section4

02

46

8

0.2

0.4

0.6

0.8

0

2

4

02

46

8

y*

τX

Figure47: Surfaceportrayingthedependenceon compositionandreduceddegrada-tion time of the reducedquantityy (proportionalto the squareof the wave vector(Eq (183)))at which thespinodalis reachedwhenthedegradationof type I diblockcopolymersis stoppedat time τ. Thevalueof thekinetic parameterU=0.1.

Having thespinodalsurfaceconstructed,it is easyto indicatethe temperatureup towhich the melt canbe cooledwithout lossof thermodynamicstability of its homo-geneousstateaswell asto answerthequestionwhetherthis lossoccursat zerowavevectorq � 0 or at one,distinct from zeroq � q ��� 0. In the latter case,which cor-respondsto the nontrivial branchof the spinodal,of specialinterestis the valueq determiningtheperiodof thespatialstructuresof smallamplitudeformedundermi-crophaseseparation[10]. Hererecoursecanbemadeto thetheoreticalresultsof thepresentwork, suchasthosepresentedin Fig 47. By intersectingthesurfacedepictedon thisfigureby planesτ � τi, thecurvesof thedependencey onthecompositionofcopolymerX will beobtained.Oneof them,correspondingto τi � 0, waspresentedalreadyby Leibler [10]. Up to the momentτ � 4� 47, the curvesy X � aresituatedentirelyabove theplane X � τ � . Within theinterval τ � 4� 47,thevaluesy aredistinctfrom zeroonly on a portionof eachof thesecurves. Evidently, theline alongwhichthe surfacey on Fig 47 intersectsthe plane X � τ � is nothingbut the curve of thereducedtime of attainingtheLifshitz pointsτL X � (SeeFig 43). Theintersectionsoftheabovesurfaceby theplanesX � Xi showsthecharacterof theevolutionof y withtimeof thedegradationτ of theblockcopolymeratgivenvaluesof parametersX and

Section4 99

U . Thecharacterof thedependenceof thetemperatureat theLifshitz point TL on theparametersX andU is depictedFig 48.

0.2 0.4 0.6 0.8

0.20.40.60.8

1

2

3

4

0.2 0.4 0.6 0.8

0.20.40.60.8

T

X

U

L

0.2 0.3 0.4 0.5 0.6 0.7 0.80

0.2

0.4

0.6

0.8

1

U

X

Figure48: Thesurfaceof thereducedLifshitz pointsTL � T τL � (a) andits contourplot (b) above the planeof parameters(X,U) for the productsof the degradationofdiblock copolymersof typeI.

The theory elaboratedhereallows us to find the amplitudeof scatteringI q� (Eq(173))of theproductsof thedegradationof block copolymerbeingformedat differ-enttimes.Fig 49illustratestheevolutionof thiscurvein thecourseof thedegradationreaction.Thepositionof themaximumshiftsgraduallyto theregionof smallscatter-

100 Section4

ing angleswhereasits heightsteadilyincreases.This happensup to a certaininstantwhenthe maximumof the curve I q� occursat q � 0, to remaintherefor any latertimesof degradation.

4.5 Conclusion

An importantconclusionfrom this theoreticalstudyis theexistenceof thepossibilityof phaseseparationin themelt of block copolymerssubjectto degradation.Depend-ing onthevaluesof kineticandstoichiometricparameters,suchasystemcanundergoeithermicrophaseor macrophaseseparation.Thereasonfor this phenomenonis thegrowth (inducedby chemicaltransformations)of thepolydispersityfor bothsizeandchemicalcompositionof themacromoleculesinvolved. This factorshouldbe takeninto accountwhen dealingwith the processesof aging of block copolymerbasedmaterials.

Section4 101

00.01

0.020.03

0.040.050

2

4

6

0

5

10

15

00.01

0.020.03

0.040.05

J

τy

Figure 49: The evolution of the reducedscatteringintensity (Eq (173)), J �I ��� M a1 � a2 � 2 , of the productsof degradationof diblock copolymersof type I,calculatedat valuesof theparametersX=0.3 andU=0.1.

102 Section4