Wededucetheeﬀectiveordinarynon-lineardiﬀerentialequationssatisﬁedbythe momentsassociatedtotheend-to-endprobabilitydistributionofapolymerchain inthepresenceofapotentialsimulatingphenomenologicallytheself-avoidanceof thechainwiththe z-axis. Additionally,weshowbyastraightforwardasymptotic analysisofalargenumberoflinksthattheabovecitedchaindistributionmoments eﬀectiveordinarydiﬀerentialequationsforsuch“almost”self-avoidingchain,reaches outfurtherintotheusualthree-dimensionalspacethantheusualgaussianchain. CBPF-NF-045/01 1 Introduction Oneofthemostoutstandingprobleminthe“Polymerphysics”istoexplainthefeature ofthegeometrical(conformational)propertyrelatedtotheexperimentalfactthattheself- avoiding Polymer chain reaches out into the usual space more than the usual gaussian one and described mathematically by a growing of the square moment distribution (in relation to the chain number of links N) of the following form </vectorr2(N)>∼N¯νwith ¯ν∼1+1 6>1 ([1]). WeaiminthisBriefReporttoshowsuchasymptoticgrowingbyasimpleandstraight- forwardanalysisofasetofeﬀectiveordinarydiﬀerentialequationssatisﬁedbythemoment distributionsassociatedtotheaboveproblemandproposedbyus. Inordertoobtainthis resultwefollowF.W.Wiegel([2])bysubstitutingthedelta-functioninteractionbetween the polymer chain monomer ata planar radial distance ρwith thez- a x i sb yapo t e n t i a l oftheformV(ρ)=A ρ2+δ2witha“hard-care”parameter δverysmallinrelationtothe monomerlenght σ. ThisBriefreportisorganizedasfollows: Insection1wededucethesetofeﬀectiveordinarydiﬀerential-integralequationssatis- ﬁedbythemomentdistributionsassociatedtotheend-to-endprobabilitydistributionof apolymer chain in thermal equilibrium with the “solvent-environment” andadded with thepresence ofageneralpotential V(/vectorr). In section 2, we apply the results obtained in section 1 to the problem of “almost” self-avoidancewithalinear([3])andmoddeledbytheabovedescribedphenomenologicalWiegelpotential. 1 The Moments Probability Distribution Equation Letusstartouranalysebyconsideringthegeneralpartialdiﬀerentialdiﬀusionequation governing the end-to-end polymer chain probability distribution ρ(/vectorr;N) in the presence ofapotentialofaforceparmonomerdenotedby V(r) ∂ρ(/vectorr,N) ∂N=σ2 6∆ρ(/vectorr,N)−βV(/vectorr)ρ(/vectorr,N)( 1 ) CBPF-NF-045/01 2 withtheinitialcondition ρ(/vectorr,0)=δ(/vectorr−/vectorr/prime)( 2 ) and prossesing the information where the polymer chain start its replication. Here σis themonomerlenghtformingthepolymerchain. ItisasomewhatcumbersomeproblemofmathematicalmethodsonPhysicsthesearch of exact solutions of eqs. (1)-(2). At this point we alternatively propose to consider directly the associated eﬀective ordinary diﬀerential equations satisﬁed by the moments and the geometrical vector position of the polymer chain averaged with the probability chaindensity. Letus,thus,considerforsimplicityofexpositionthethreebasicobjectsto describe theconformationalgeometryofourpolymeratlarge-orderbehavior for N.W e have,thus,toconsiderthefollowingobjects: thepolymerpartitionfunction Z(N)=/integraldisplay d3/vectorrρ(/vectorr;N)( 3 ) theaveragedchainvectorposition /vectorR(N)=/angbracketleft/vectorr/angbracketright(N)=/parenleftbigg/integraldisplay d3/vectorrρ(/vectorr;N)·/vectorr/parenrightbigg/slashBig Z(N)( 4 ) andthechaingirationradius R2(N)=/angbracketleft(/vectorr·/vectorr)/angbracketright(N)=/parenleftbigg/integraldisplay d3/vectorrρ(/vectorr;N)(/vectorr)2/parenrightbigg/slashBig Z(N)( 5 ) It is straightforward, for instance, to show that /vectorR(N) satisﬁes the following relation afterusingthedeﬁnitioneq. (4)andeq. (1) d dN/vectorR(N)=−β/braceleftbigg/integraldisplay+∞ −∞d3/vectork˜V(/vectork)1 i∇/vectork/angbracketleftei/vectork·/vectorr/angbracketright/bracerightbigg −Z/prime(N) Z(N)/vectorR(N)( 6 ) where˜V(k) denotes the Fourier transform of the environment potential V(/vectorr), and, of course, /angbracketleft/angbracketrightdenotestheend-to-endprobabilitydistributionofourpolymerchaineq. (1). Letusﬁrstlypointoutthatundertheleadingapproximationoftheﬁrstorderofthe cummulantexpansionoftheFouriertransformKernel(orequivalentlyforlowconforma- tionalfrequencyregion |/vectork|2/similarequal0) /angbracketleftei/vectork·/vectorr/angbracketright=ei/vectork·/angbracketleft/vectorr/angbracketright(7) CBPF-NF-045/01 3 the relation eq. (6) turns into the following ﬁrst order ordinary non-linear diﬀerential equationforthechainvectorposition d dN/vectorR(N)=−β/vectorR(N)×V(/vectorR(N))−/parenleftbiggZ/prime(N) Z(N)/vectorR(N)/parenrightbigg (8) By proceeding by the same steps exposed above, we obtain that at ﬁrst order com- mulant approximation, the chain radius giration satisﬁes and exactlysoluble ﬁrst-order ordinarydiﬀerentialequation d dN(R2(N))=σ2−β(/vectorR(N))2V(/vectorR(N))−/parenleftbiggZ/prime(N) Z(N)(R2(N))/parenrightbigg (9) Atthispointweremarktheappearanceofadampingtermwithadampingconstant variablewiththechainlenght Nineq. (8)andeq. (9). The chain partition function Z(N) will be taken to be known in our study from hereafter ([1], [2], [4]). However, its chain lenght evolution equation is easily written down d dNZ(N)=−βV(0)Z(N)−β(/vectorR(N)·∇/vectorv(0))+ ··· (10) where the terms in coma ···have the meaning of involving higher-derivatives of the potentialV(/vectorr)attheoriginandhigherordermoments. Relatedtoproblemoftakinghigherordercorrectionsinthecummulantexpansionof theFourierKernel exp(i/vectork·/vectorr),letusshowhowtoproceedinthecaseofsecondordercor- rections in oureﬀective ordinarydiﬀerential equations. We consider, thus, thecorrected expansion /angbracketleftei/vectork·/vectorr/angbracketright(N)=ei/vectork·/angbracketleft/vectorr/angbracketright(N)e−1 2/braceleftBig |/vectork|2(/angbracketleft(/vectorr)2/angbracketright−/angbracketleft/vectorr/angbracketright2)(N)/bracerightBig (11) the relation eq. (6) takes the integro-diﬀerential form but involving now the giration radius in its functional structure and, thus, becoming a non closed integro-diﬀerential equation,oppositetoeq. (8)intheﬁrstorderapproximation d dN/vectorR(N)=−/parenleftbigg β/vectorR(N)/braceleftbigg/integraldisplay d3yV(/vectory)G(/vectorR,R2,/vectory)/bracerightbigg/parenrightbigg −/parenleftBig β(R2(N)−(/vectorR·/vectorR)(N)) ∇/vectorR(N)/braceleftbigg/integraldisplay d3yV(/vectory)G(/vectorR,R2,/vectory)/bracerightbigg/parenrightbigg −/parenleftbigg/parenleftbiggZ/prime(N) Z(N)/parenrightbigg /vectorR(N)/parenrightbigg (12) CBPF-NF-045/01 4 where G(/vectorR,R2,/vectory)=[R2(N)−(/vectorR·/vectorR)(N)]−3/2exp/braceleftBigg −1 2/bracketleftBigg (/vectorR(N)−/vectory)2 (R2(N)−(/vectorR·/vectorR)(N)/bracketrightBigg/bracerightBigg (13) SimilarcorrectiononthecummulantexpansionfortheFourier-Kernelcanbeconsid- eredforthechaingirationradius. Anotherpointworthremarkistheexplicitsolutionof eq. (9)forthechaingirationradius R2(N)=1 Z(N)/braceleftbigg/integraldisplayN 0dN/primeZ(N/prime)/bracketleftBig (σ2−β(/vectorR(N/prime))2V(/vectorR(N/prime)))+1/bracketrightBig/bracerightbigg (14) Notethatwehavere-introducedthemonomerlenght σineq. (9). We now proceed to show that in these somewhat ﬁrst order approximations an “al- most”self-avoidingchainis“bigger”thantheusualFree(Gaussian)polymerchaininthe nextsectionofourstudy. 2 The “Almost Self-Avoiding Case In this section we apply the previous framework toanalyze the Physico-Chemical im- portancecaseofaPolymerchaininaphenomenologicalinteractionwiththe z-axisthough theWiegelpotentialwitha“hard-core”parameter σintheradialdistanceofthechains monomertothe z-axiswiththefollowingbehaviourofthepolymerchainpartitionfunc- tionalforlargenumberoflinks N Z(N)∼N−γ(15) withthepositiveexponent γequalstoγ=2−εwhereε∼1/5([1]). Ourbasiceﬀectiveequation(8)fortheaveragedpolymerchainsvectorpositionreads as d dNRa(N)=−βA (R2 1+R2 2)(N)+δ2−/parenleftBigγ N/parenrightBig Ra(N) (16.a) d dNR3(N)=−/parenleftBigγ N/parenrightBig R3(N) (16.b) where thesubscript a=1,2indexes thechain vector positioncomponentsonthe( x,y) plane. CBPF-NF-045/01 5 In order to solve eq. (16a) at large N, we ﬁrstly consider the associated (ordinary) diﬀerential totheplanarradius lenght y(N)=(R2 1+R2 2)(N)which iseasily writedown asfollows dy dN(N)=−2βA/parenleftbiggy y+δ2/parenrightbigg (N)=/parenleftBigγ N/parenrightBig y(N) (17) Letussolveeq. (17)bymeansofaasymptoticseriesinthe N-variable y(N)=1 N1+ω/bracketleftbigg a(N)+b(N) N+O/parenleftbigg1 N/parenrightbigg/bracketrightbigg (18) where the exponent ω>0 takes into account the approach to the equilibrium solution g(N)≡0 of eq. (17) at N→∞. Let us note that one should only consider now the situation of zero “hard-core” monomer interaction δ2= 0 and not before, since the diﬀerential equation makes sense only for y(N)/negationslash= 0. Anyway, we always consider in the followingdiscussion thecaseofavery small δbutnonzero. AtlargeN,wecanneglectthepotentialtermineq. (12)andobtain,thus,thebehavior oftheeﬀective gaussiancase( A=0ineq. (16a)) y(N)∼N−γ(19) After substituting eq. (19) into eq. (16a), we ﬁnally get the planar and z-axis large Nbehaviorforthepolymerchainaveragedvectorpositioncomponents Ra(N)∼exp−/parenleftbiggβAN δ2/parenrightbigg N−γ(20) R3(N)∼N−γ(21) Fromeq. (20),wecanseeafastestexponentialrelaxationbehaviortothe“Eﬀective” Gaussiancaseduetothepresence oftheexperimentalterm exp/parenleftbigg −βAN δ2/parenrightbigg . ThelargeN-behaviorofthepolymerchaingirationradiusiseasilyevaluatedbymeans ofeq. (14) R2(N)=N−γ/bracketleftbigg/integraldisplayN (N/prime)γ/parenleftbigg σ2−3β δA(N/prime)−2γ/parenrightbigg/bracketrightbigg ∼σ2 (γ+1)N+/parenleftbigg −3AB δ(1−γ)/parenrightbigg N−2γ+1 (22) which by itsturnsshows thatfor large Nthe self-avoiding polymer chain with δ<<σ2 reachesourfurtherinthethree-dimensionalspacethantheusualGaussiantherelationship CBPF-NF-045/01 6 eﬀective chain,namelyforlarge N,w eha v e R2(N)−/parenleftbiggσ2 γ+1/parenrightbigg N≥N−2γ+1/parenleftbigg3Aβ δ(γ−1)/parenrightbigg (23) an important qualitative result and strongly dependent of the self-ordinary interaction parameters(δ,γ)inanon-analyticalway. Acknowledgments Luiz C.L. Botelho is thankfull to Dr. Welles Morgado (PUC/RJ) for discussions on the topicofthispaperandtoCBPF-RJforwarmhospitality. LuizC.L.Botelhoissupported byCNPqfellowship. CBPF-NF-045/01 7