Polymer models

Talkgivenfor: Hauptseminar instatistical physics 8/52006 PeterBjødstrup Jensen Overview

Polymers
Polymerbasics anddefinitions
The idealchain
Freely jointed chain (FJC)
Freely rotating chain (FRC)
Kuhnlength andpersistence length
Endtoend vector distributionfunction
Gaussian chain
Forceextension relation
Wormlikechain
Realchains
Conclusion Polymers polymerization lsisiesle…Proteins,polysaccharides, Biopolymers: monophosphate Deoxyadenosine Plastics,fibers,glues… polymers: Synthetic bonded. sm covalently relatively simplemolecules units(monomers)of weigh repeated molecular of (high consisting molecules long usually Polymersare Definition: h rcs fcvln onn ftemnmrskonas monomersisknown the of joining covalent of process The tyeemnmrPolyethylene monomer Ethylene Actin,DNA… alland t) Polymers te tutrlfactors structural Other random….     ifrn yefmonomerrepeats:homopolymers,alt typesof Different toca –Rand–Hgroups of Stereoisomerism:Orientation (doublebonds) isomerisms Structural architect branched kindsof havedifferent Polymerscan  idrdrotationshindered ures ernating, bnplanerbon The ideal chain

No correlation between polymermonomersseperated bylong

Polymers distancesalong the polymer.

 Shortrangecorrelations between neighboring monomersare not excluded  Idealchain modelsdonottake interactions caused by conformations inspace into account  Idealchains allow the polymertocross itself Modelling a polymer

Imagining ablown up picture of asection of the polymerpolyethylene ina

Polymers certain conformation,could looklike this: r r Conformations: 5  Torsionangleφ r r4  Bondangleθ r r r R5 3 Bond vectors: r r2 Starting fromone endwe use vectors r to i r represent the bonds r1 End-to-end vector:

The sumof allbond vectors r n r = The ensamble average of =0duetoisotropy Rn ∑ ri n i=1 Mean square end-to-end distance : r r n n r r Simplestnonzero average 2 = ⋅ = ⋅ R Rn Rn ∑∑ ri rj i=1j = 1 Freely jointed chain

No correlation between the directions of different bond vectors.Θ and

Polymers φ are free torotate.Allbond vectors havelength l

r r r n n r r R2 = R ⋅R = r ⋅ r n n ∑∑ i j i=1 j =1 r r r n n r ⋅ r = llcosθ ⇒ R2 = l2 cosθ i j ij ∑∑ ij i=1 j =1 No correlation between different bond vectors,i≠j r r r r ⋅ = ⋅ = ri rj ri rj 0

R2 = nl 2 R ∝ n Freely rotating chain

Bondangleθ isfixed.Torsionangleφ stillfree torotate. Polymers r r n n r r r r r 2 = ⋅ → ⋅ = 3 R ∑∑ ri rj ri rj ? i=1j = 1 r r2

Ex:what isthe correlation between vector r3 andr 0? l ⋅cos θ Duetothe free rotationaround the torque angle,

only the perpendicular component of r3 ispassed down. r r r r1 r ⋅r = l(cos θ )2 ⋅l cos θ = l 2 (cos θ )3 3 0 l ⋅(cos θ )2 The generelexpression becomes: r r0 r r ⋅ = 2 ( θ )i− j ri rj l cos r r ⋅ = 2 ( θ )i− j ri rj l cos

Inserting this expression inour equation for Polymers n n r r n n 2 = ⋅ = 2 θ i− j R ∑∑ ri rj l ∑∑(cos ) i=1j = 1 i=1j = 1

This issolved bymanipulating sums,andbywriting the rapidly decaying cosine termsasaninfinite series.

Forcalculation see Rubinsteinp.56

The endresult is: 1+ cos θ R2 = nl 2 1− cos θ Polymers rprinlt.W aejsde osat>1 hasnotchan aconstant correlation havejustadded of proportionality.We introduction the see,that We oagiie neatosti ilawy etecas the be always will this interactions Forrangelimited C oaigcanw have: mo we polymerinagivenidealchain chain the rotating of stiffness the of C j lim R − ∞ ∞ i scle lr’ hrceitcrtona ese a seen be ratio,andcan characteristic Flory’s iscalled 2 → = ∞ = 1 1 cos + − l 2 cos cos ∑∑ ∑∑

i = = = θ n

1 1 1

ij

θ θ j n = 0 cos

⇒ θ ∑ ij j n = 1 = cos l 2

i θ n ij C = i ' = C nl i ' 2 C ∞ del.Forthe e e h n the ged sameasure ½ Kuhn length: idealchains can be rescaled into afreely jointed chain,aslong asthe chain islong compared tothe scale of short rangeinteractions Polymers

Newsegmentlength bischoosen so long,that neighbooring segmentsare noncorrelated  Newchain isafreely jointed chain

R2 = Nb 2 R ∝ N

biscalled the Kuhnlength,andobviously holdsinformationon short scale interactions andstiffness.

= 2 Rmax Nb R nl 2C b = = ∞ 2 2 2 R = nl C∞ = Nb Rmax Rmax Persistence length

The vector correlation termfromthe freely rotating chain,decays quickly andcan be written intermsof anexponential function Polymers r r ⋅ = 2 ( θ )i− j ri rj l cos

−  j − i  ()cos θ i j = exp []j − i ln(cos θ ) = exp −   s p  1 s = l = l ⋅ s p ln(cos θ ) p p This isaconsequence of the rangelimited interactions,andwill always be the case

Sp isthe number of bonds inapersistence segment

The persistence length lp isthe length of the persistence segment

The persistence length lp isthe length scale with which the decay occurs End-to-end vector distribution:

We use the CentralLimitTheorem which states: CLT:Givenaseriesof random variables;X1,X2,…Xnsampled fromthe Polymers samepoolof probability with adefined mean andvariance σσσ2,the distributionof the sumS=X1+X2+…Xnwill converge toagaussian distibution.

Mean andvariance of the endtoend vector isalready known: Mean ==0 Variance σσσ2= 2==Nb 2

we get the probability distributionfunction in3D: r r  3  2/3  3R  =   −  P3D (N, R) 2 exp  2   2πNb   2Nb  The Gaussian chain:

The gaussian chain isachain madeup of kuhn bonds that are assumed gaussianly distributed Polymers 3 r 2 r   2   = 3 − 3r  P(r)  2  exp  2   2πb   2b 

We can now create the conformational distributionfunction of the entire chain,bymultiplying each bond distribution

3 r 2 N  3  2  − 3r  Ψ(){}= ∏  n  rn  2  exp  2  n=12πb   2b  3N r 2  3  2  N 3r  = exp − n  π 2   ∑ 2  2 b   n=1 2b  The beadspring modelisamechanical representation of the Gaussian chain Each springrepresents aGaussianly distributed Kuhnsegment Polymers If the springpotentialbetween two beads is defined as: r 3 r U (r ) = k rT 2 0 n 2b2 B n The bond distributionfunction forasingle segmentcan be found. r r  U   3r 2  ∝ − 0  =  n  P(rn ) exp   exp  2   kBT   2b 

Normalizing we get the known Gaussian distribution. One will also findthe samemean quare endtoend distance. Force extension relations:

What happens when we apply aforceFtostretch the polymer Polymers Looking atfirst atone segment r r = − ⋅ Esegment ri F

Orientations are boltzmann weigted

r r r r  ⋅   ⋅  =  r F  =  r F  θ θ ϕ Zsegment ∑ exp   ∫ exp  sin d d states =  kBT   kBT  orientatio ns Energy forthe entire chain n r r r r = ⋅ = ⋅ Echain ∑ ri F R F i=1 Polymers Partition function forthe entire chain

 r N r  N =  1 ⋅  θ θ ϕ Z(N, F) ∫ exp  F ∑ ri ∏sin id id i  kBT i=1  i=1

This can be factorized andsolved togive:

N   r r   = π θ  1 ⋅  θ Z(N, F) ∫ 2 sin i exp  F ri d i    kBT   ⇓ ⇓

N   fb  4π sinh    kT  Z(N, F) =    fb   kT  The free energy isfound inthe standardway fromthe partition function,andthe average endtoend distanceforagivenforce can finally be found bydifferentiating the free energy Polymers

     = − = − π  Fb  −  Fb  G(F, N) kBT ln Z(N, F) kBTN ln( 4 sinh   ln     kBT   kBT    ∂G   Fb  1   Fb  R = − = bN coth   −  = bNL   ∂   Fb   F   kBT    kBT     kBT 

3 1 x x 5 L(x) = coth( x) − = − + Ο(x) Langevin function x 3 45 R   = Fb L  Rmax  kBT  One important limittothe forceextension expression isthat of asmallforce.Taking only the first order of the Langevin functions givesus: Polymers Fl R = Nl 3kBT 3k T 3k T ⇒ F = B R = B R Nl 2 R2

We obtain springlike behavior with the springconstant 3k T k = B spring R2

Springconstant isproportionaltotemperature.Higher temperatures  greater forcesnecessary tostretch Entropic effect  entropic spring The worm like chain:

Continous development of the freely rotating chain forsmallbond

Polymers anglesθ, used forpolymerswith high stiffness

The meaningfull limitstotake inthis development are:

l  0,θ  0,butcontour length nl andpersistence length lp remain the same We calculate the mean square endtoend distance: We change the sumoversegmentsinto anintegralovercontour n n n n n n r r −  j − i  2 = 2 ⋅ = 2 ()θ i j = 2 − R l ∑∑ ri rj l ∑∑ cos l ∑∑exp  l i=1j = 1 i=1j = 1 i=1j = 1  l p 

n R n R l → max ds and l → max ds ' ∑ ∫0 ∑ ∫0 i=1 j=1 − − RRmax max  s s'   2 =   R ∫ ∫ exp  ds ds ' 0 0  l   Polymers ⇒ For For For For R hti,i sflyextended isfully is,it That h oielimit: rodlike The limit: ideelchain The endtoend maximum forthe limitsare interesting two The h nerlcnb ovdtieteresult: togivethe solved be integralcan The R R max 2 2 R R b max >>l 2 ≅ = =

≅ 2 2 << 2 p R l l l p p andR p max 2 l R R p the

max max →

max − freely for exp 2 <> exp     ≅ l p    

-1 − chain R R l max p l max p +     1 2 is     recovered     R l max p     2

distance Polymers nesfterdufcrauesurdwihih cu isthe which squared squared curvature radiusof the inverseof h nrypruiegho eddba isproportion beam abended of perunitlength energy The U u ⇒ r uvtr r then: are u(s)andthecurvature tangentvector s.The the curve pointon space anarbitrary of radiusvector r(s)isthe ) ( bending / h omlk hi sasaecurve: a as space chain like worm The s unitlength U contour = ∂ ∂ r s r =

= 1 2

1 2

ε and ε ( radius ∫ 0 s

   ∂ ∂ c r u s r ( of s    2 ) ds

= curvature) ∂ ∂ u s r =     ∂ ∂ 2- s 2 r r 2 =    

1 2 ε    ∂ ∂ u s r    2 rvature altothe Real chains:

 Long rangeinteractions between monomersare taken into account

Polymers  Interactions between solute andpolymerandbetween different polymers  Excluded volume andself avoiding random walks  Dynamics    Polymers ecbsteextension the descibes  function Langevin bythe    expression:=C ∞ nl 2  oogcan egenerally we chains forlong 2  >=Nb isGaussianr ecln nofreely into rescaling swl discribed iswell n pringconstant 2

Polymers

www.its.caltech.edu/~bois/pdfs/poly.pdf 6rdi anrilm thesis Wagner.Diploma [6]Fredrik [5]ThomasFranosch. Bois[4]Justin http://www.molsci.polym.kyotou.ac.jp/archives/redbo Yamakawa. [3]Hiromi Doi. [2]Masai [1]MichaelRubinsteinandRalphH.Colby.

References:

Polymerphysics

oenTer fPolymerSolutions of Theory Modern topolymerphysicsIntroduction oyesiofndgeometryPolymersinconfined Polymers polymerphysicsRudimentsof

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