Volume89A, number9 PHYSICSLETTERS 14 June1982

CONFORMATION SPACERENORMALIZATION THEORY OF SEMIDILUTE POLYMER SOLUTIONS

T. OHTA’DepartmentofPhysics,Kyushu University,Fukuoka812,JapanandDepartmentofPhysicsandAstronomy,UniversityofPittsburgh,Pittsburgh,PA 15260, USA

and

Y. 00N02DepartmentofPhysics,KyushuUniversity,Fukuoka812,JapanandDepartmentofPhysicsandMaterialResearchLaboratory, UniversityofIllinois at Urbana-Champaign,Urbana, IL 61801,USA

Received11 March 1982Revisedmanuscriptreceived-3 April 1982

A transparentconformationspacerenormalizationgroupformalismis givenwhichcan incorporatearbitrarymolecularweight distributionsto thetheoryof semi-dilutepolymersolutions.Theclosedformsfor theosmoticpressureir andthemeansquareend-to-enddistanceof a singletestchain aregivento ordere= 4 — d, dbeingthespatialdimensionality,asfunctionalsof themolecularweightdistributionfunction.Thescalingform of ii is comparedto therecentexperimentalresultby Nodaetal.A distributionsensitiveuniversalplot is alsoproposed.

Thepurposeof thepresentletteris to give acanonicalensemblerenormalizationgroup theoreticformalismfor semidilutepolymersolutionswhich isnot dependentonthepolymer—magnetanalogy[1,2] andis capableofdealingwitharbitrarypolydispersity.As examplesof theapplicationof this formalism,the osmoticpressureirandthemean-squareend-to-enddistance[R2] of a testchainin thesolution are given in closedformsto order� = 4 — d, d beingthe spatialdimensionality.The obtainedscalingform of ir is comparedwith the recentexperi-mentby Nodaet a!. [31.Furthermore,convenientapproximantsto ir andER2] are proposedwhich dependon= M~IMn, whereM~is the weight averagemolecularweight andMn the numberaveragemolecularweight.A polydispersitysensitiveuniversalplot which is experimentallyaccessibleis proposedat the endof theletter.

It isgenerallyvery difficult to preparea strictly monodispersepolymersolution,sothatit isimportantto assesstheeffect of thepolydispersityin theanalysisof experimentalresults.Sofar, however,therehavebeenfew reliablecalculationsof the polydispersityeffectson physicalpropertiesof semidilutesolutionswith goodsolvents.Quiterecently,Knoll et a!. [4] havecalculatedthe scalingform of the osmoticpressurefor arbitrarypolydispersitywith the aid of thepolymer—magnetanalogy.The analogyis, however,notfreefrom controversy[5,6], especially,in thecaseof calculatingcorrelationfunctions [5]. AlthoughWheelerandPfeuty [7] haveshownthat theviola-tionof stability in then -÷0 limit of then-vectormodel doesnot appearin the polymersystems,their resultdoesnotcoverthesemidiluteregime.Furthermore,the difficulty of thefield theorydue to theGoldstonemodebelowthe critical temperatureis nota genuinedifficulty butanartificial onefor polymerproblems.In thesecircumstan-cesit is desirableto developa theoryof semidilutesolutionsfree from the magneticanalogy.Indeed,des

Permanentaddress:Departmentof Physics,KyushuUniversity, Fukuoka812,Japan.2 Permanentaddress:DepartmentofPhysics,Universityof Illinois atUrbana-Champaign,Urbana,IL 61801,USA.

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Cloizeauxdevisedsucha theory [8] andcalculatedphysicalquantitiesup to onelooporder [9]. However,theresultantperturbationseriesarenot renormalizedto covergoodsolventsituationsasin Moore’spapers[10,11].Moreover,thefollowing canonicalensembleformalismis simplerthendesCloizeaux’sgrandcanonicalensembleformalismin the actualcalculations.The osmoticpressureobtainedby our methodiscomparedwith thatby thefield theoreticapproach.

Ourmodel of thegenerallypolydispersesemidilutesolutionin a box Vconsistsof continuouspolymerchainswith differentlength interactingwitheachotherandalsowith themselvesby the localized~-function-likerepul-sive potentialv~6(r), wherev0 is a positivebare interactionparameter(excluded-volumeparameter).More speci-fic ally, if wedecoupleall the chainsby theintroductionof thenormal stochasticfield [12], thepartitionfunctionof our modelis givenby [13]

z = fD[~] H ~ ~)exp(_ ~L f~2 ddr), (1)

whereD impliesthe uniform functionalmeasureoverreal scalarfunctions ,I’, N~is the baredegreeof polymeriza-tion of the ~th polymer,andn isthe totalnumberof polymersin the box.G isgivenby

N~

G(N~,i~1i)=f D[c] exp[_~.f (dcr)2 dr—if IP(c(r)dr], (2)

whereD impliesthe uniform measureon theset of all (continuous)conformationsof the athpolymerchaininthebox.Conformationsare designatedby c(r) withr beingthe contourvariable.It is understoodthat thereis animplicit cut-off contourlengtha to avoidself-interactions.

If we introducethedensitydistributionfunctionP0(N0)of thedegreeof polymerizationN0, then(1) canbe

rewrittenas

z= fD[~i]exp(n f P0(N0)ln G(N0, ~)dN0 — ~ f~p2ddr), (3)

which is themostconvenientstartingpoint to studytheeffect of polydispersity.To evaluatethepartitionfunction(3), we follow Edwards[12]. Ignoringthe effect of thewalls of thecontain-

er, we get the bare free energyFb to the lowest non-trivial orderin v0 aftera straightforwardcalculationas

r (N0)2n2V

0 ~ v I 2nv0Fb=kT[nlnV— 2V 2(~’1df ddkln~1+~ (So(k~No)))j~ (4)

‘ /

where( ) denotestheaverageoverF0, andS0(k,N0) = 2N0/k2— 4 [1 — exp(—N

0k2/2)] 1k4. Thebareosmotic

pressureIrb is given by —aFb/aV.The baremean-squareend-to-enddistance[R2]b canbeobtainedanalogously.The renormalizedmean-square end-to-end distance [R2I andtherenormalizedosmoticpressureir mustbewell

definedin thea -~ 0 limit. Thisexpectationwasverified in the samefashionasis explainedin our previouspapers[14,15]. Thuswe areleft with the following closedforms.

= 1 + — fj f dz z3 {ln[l + 2Xg~(z2)]— 2Xg~(z2)/[l+ 2Xg~(z2)]— 21z4(l — e_z)}, (5)

and

[R2]/[R2]~o exp(_~f dx(1 —x)x2 f dzz5eXz2 2Xg~(z2)/[l+2Xgp(z2)]), (6)

where c is the polymer number density, [R2]~,o is the mean-square end-to-end distance in the c -~ 0 limit. Xisgiven by X = c(NYhlIT2e/2with dv= 2 — �14+ O(e2), where N is the renormalizeddegree of polymerization, and( ) is the average with respect to the observed distribution function P of the degree of polymerization. gp is given by

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g~(z2)z2 —[1 —(exp(—Nz2/(N)))]z-4. (7a)

Eqs. (5) and(6) with (7a) are thegeneralscalingformsup to O(�).It is found that the following approximant(7b) to(7a)gives accurateresultsandenablesusanalyticevaluationof(S) and(6),

g~(z2)= (z2 + 2p)~, (7b)

wherep = (N)2/(A12) = Mn/MW. It mustbestressedthat this approximantwith p = 1/2 is exactwhenP is givenby theexponentialdistribution.Forgenera!functionofF theapproximant(7b) reproducesthecorrectasymptoticbehaviorin thebothlimit z-* 0 andz -~ 00• In themonodispersecasefor which p = 1, the numericalresultsdueto(5) and(6) with (7b) lie within about 3% errorof thosewith (7a).

Introducing(7b) into (5) and(6) yieldsthefollowing analyticclosedforms

ir/ckT= 1 + ~Xexp~ �[p/X+ (p2/X2) ln p + (1 — p2/X2) ln(X+ p)]} (8)

and

[R2]/[R2]~,~o=exp~�X~3/(X+p)—[3/2(Xi-p)2-t- l/(X+p)][1n2(X+p)+~]

+e2(~~I2)Ei(_2(X+p))[1—2/(X-i-p)+3/2(X+p)2]}, (9)

wherej’ ~ 0.5772is Euler’sconstant,X = �ir2c(N)th’[l + e(1J2+ ln 2)/4]/2 withdv = 2 — �14+ O(�2)andEi(—x)= —f°dtet/t. In theseexpressionstheorder e correctionsare exponentiated.This is necessaryespeciallyfor [R2].Otherwise,it becomesnegativefor largerX (or, we couldperturbativelycalculateln IT andln [R2] as wasdoneinref. [15]).

The graphof ir/ckTversusXis shownin fig. 1. The recentexperimentalresultsby Nodaet a!. [3] arealso

fl / ckT

• a—104 *

o n—12100 ~ a—103

A a—hO *

10 4 a—112

1~1 ~0.01 0.1 1.0 10 c/ce

Fig. 1. Therelation rr/ckT versusc/c” to ordere (e is setequalto 1). Thefull line showsthecaseM = 1 (monodisperse)andthebrokenline p = 0.01 (verybroaddistribution).Theexperimentalresults[3] for poly (c~-methylstryrenes)in tolueneat25°Carealsoplotted.a-104etc.specify thesampleswith differentmolecularweights.Seeref. [3] for details.Thepolydispersitymakesthecrossoverconcentrationfromthediluteto thesemidiluteregimesomewhatsmaller,but it is clearthat the polydispersityeffect isvery small.

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plotted.Wehaveadjustedtheunknownproportional constant between X and c/c”‘ withc” thecritical concentra-tion of chainoverlapping.Sincetheexperimenthasbeendonefor the sampleshavingsharpmolecularweight dis-tributions,the datashouldbe comparedto the monodispersescalingfunction(thefull line). Theagreementis quitesatisfactoryexceptfor theregimeof the largevaluesof X. Themain reasonis dueto thefirst order approximationusedhere.In fact eq.(8) gives IT X1+e14 for the largevaluesof X while the experimentis morecloseto Irx l/(dv 1) with v = 0.585.The secondordercalculationwould improvetheagreement.Theobtainedformula(8) shouldbecomparedwith theresultof the field theoreticcalculation.By usingthe free

energyformula for the n-vector0-modelgivenby Brésin et al. [16] anddesCloizeaux’scorrespondencebetweenpolymersandfields [2], ir/ckT canreadilybecalculatedandturnsoutto beidenticalto eq. (2) with p = 1/2(whichis exactto order �)andX beingreplacedwith X’ =gcN2 — e/4[1 + �(3/2+ ln 2)] /3, whereg is thecouplingconstantin the Ø4-model.Since thedifferencebetweenthe distributionof the degreeof polymerizationfor thefield theo-retic calculationandthe exponentialdistributionis of order� [17], this agreementis asexpected.Thesameresultcanalso be obtainedfrom theimplicit formulas(2.36), (2.39ii) and(2.40) of Knoll et a!. [4]. However, it shouldbe stressedthat thepresentmethodprovidesnot only thermodynamicquantitiesbutalso quantitiesrelatedto thecorrelationfunctionwithoutbeingaffectedby therecentcriticismsof field theoreticmethods[5].

In the largeX limit, IT behavesasX1~/4,which agreeswith the scalingargument[18]. In this limit [R2] be-havesasM(N>~/8X~i8andis alsoin agreementwith thescalingargument[19], whereMis the renormalizedde-gree of polymerizationof thetestchain. [R2] B containstermswhichdependonaX/M

0. In thecalculationabove,thea/M0 -+0 limit wastakenwith fixed X. In thehigh densitylimit, however,aX/M0 canbelarge(especiallywhen(N)/M~ 1 evenif themonomerdensityis finite). Therefore,theresult (6)or (9) cannotbeusedif the con-dition aX/M0 ~ 1 is notsatisfied.

Fig. 1 showsthat, to order �, theeffect of polydispersityis negligible.Since theresultsto order�are knownto

beat leastsemiquantitativelyreliable(e.g., refs. [20] and [211),the qualitativeconclusionis expectedto remainintact in thehigherordercalculations.However,thereis an observableuniversalfunctionwhich is fairly sensitivetotheratiop = Mn/MW.This is h (x) = NAØr/ckT— 1)/M~A2c,whereNA is Avogadro’sconstantandA2 is theosmoticsecondvirial coefficient.SinceA2 behavesaspEI

4 ascanbe seenfrom (8), thedependenceof h (x) on pis observable.

A p-sensitiveuniversalplot [R2]/ [R2]c0 versush(x) is given in fig. 2. Sinceit is well knownthat theratio of[R2] andtheradiusof gyrationis fairly insensitiveto theexcluded-volumeeffect [17,15], [R2]f[R2]c

0 may bereplacedby thecorrespondingratio of theradii of gyration.

In summary,we havecalculatedthepolydispersityeffectson semidiulatepolymersolutionsrenormalization-group-theoreticallywith theaid of our newformalism,theconformationspacecanonicalensembleformalism.If thepolydispersityis givenby theexponentialdistribution,our resultagreeswith thatby thefield theoreticalmethodasis expected.It hasbeenshownthat thereis a polydispersity-sensitiveuniversalplot which is experimentallyaccessible.

More detailedcalculationsandresultsaboutotherquantitiesaswell aspoor to goodsolventcrossoverbehaviorbegiven elsewhere.

~o.8. ~

0.7 - euitiuniv~rlp LLR;0h:~.

o c I aregivenin thefigure.p = 1 is monodisperse,p = 0.5 corre-I 2 3 spondsto theexponentialdistribution(exactto ordere). ph = 0.1 showstheeffect of higherpolydispersity.

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T.O. Wishesto thankProfessor Kyozi KawasakiandProfessor David Jasnowforvaluablediscussions.A travelsupportto T. 0. by Yosida ScienceFoundationis also acknowledged.Y. 0. isgrateful to ProfessorH. Mori,ProfessorK. KawasakiandProfessor1. Ohtafor their hospitalityat Kyushu University.Particularthanksare dueto ProfessorM.E. McDonnell for sendingT. 0. his preprintwith Dr. J.Amirzadeh,throughwhich we were awareof ref. [3]. Thiswork is supported,in part,by the University of Illinois MRL GrantNSF-DMR-80-20250.

References

[11 P.G.deGennes,Phys.Lett. 38A (1972)339.[2] J. desCloizeaux,J. dePhys.(Paris)36(1975)281.[3] I. Noda, N. Kato, T. Kitano andM. Nagasawa,Macromolecules14 (1981)668.[4] A. Knoll, L. SchaferandT.A. Witten, J. dePhys.(Paris)42 (1981)767.[5] P.D. Gujrati, preprint (1981)andJ. Phys.A14 (1981)L345.[6] M.A. Moore andC.A. Wilson, J. Phys.A13 (1981)3501.[7] J.C.WheelerandP. Pfeuty,Phys.Rev.A23 (1981)1531.[8] J. desCloizeaux,J. dePhys.(Paris)41(1980)749.[9] 1. desCloizeaux,J. dePhys.(Paris) 41(1980)761.

[10] M.A. Moore, J. dePhys.(Paris) 38 (1977) 265.[11] M.A. Moore andG.F. Al-Noaimi, J. dePhys.(Paris)39 (1978)1015.[12] M. Kac, in: Applied probability, ad.L.A. MacCoil (McGraw-Hill, NewYork, 1957).[13] S.F.Edwards,Proc.Phys.Soc.88(1966)265.[14] Y. Oono,T. OhtaandK.F. Freed,J. Chem.Phys.74 (1981) 6458.[15] T. Ohta,Y. OonoandK.F. Freed,Phys.Rev. A, to bepublished.[16] E.Brezin, J.C.LeGuilleu and J. Zinn-Justin,in: Phasetransitionsandcritical phenomena,Vol.6, eds.C. Domb andM.S.

Green(AcademicPress,New York, 1976).[17] T.A. Witten andL. Schafer,J. Phys.All (1978) 1843.[18] P.G.deGennes,Scalingconceptsin polymerphysics(CornellUP, Ithaca,1979).[19] J.F.Joanny,P. Grant,L.A. TurkevichandP. Pincus,J. dePhys.(Paris)42 (1981)1045.[20] Y. Oono,T. Ohtaand K.F. Freed,Macromolecules14 (1981)880.[21] Y. OonoandT. Ohta,Phys.Lett. 85A (1981)480.

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