Introduction to polymer physics Lecture 4 Boulder Summer School July – August 3, 2012 A.Grosberg New York University Center for Soft Matter Research, New York University Jasna Brujic Paul Chaikin David Grier David Pine Matthieu Wyart Ilya M.Lifshitz Evgenii M.Lifshitz (1917-1982) (1915-1985) Fermi-surfaces, metals, disordered VdW interactions, systems (optimal fluctuations, Lifshitz point… Lifshitz tails), supersolids, polymers… Lecture 4: A bit more mathematics • Course outline: • This lecture outline: •Ideal chains (Grosberg) • Comment on Flory- •Real chains (Rubinstein) Huggins • Solutions (Rubinstein) • Edwards Hamiltonian •Methods (Grosberg) • Polymers and classical • Closely connected: mechanics (Semenov) interactions (Pincus), polyelectrolytes • Polymers and quantum (Rubinstein) , networks mechanics (Lifshitz) (Rabin) , brushes (Zhulina), • Block-copolymers semiflexible polymers (Leibler)… (MacKintosh) Comment on Flory-Huggins (1) • Flory-Huggins interpretation: • Lifshitz (-like) interpretation: • More phenomenological writing: Comment on Flory-Huggins (2) •Van der Waals • Van der Waals equation of state: equation of state: Comment on Flory-Huggins (2) • FH free energy • VdW equation of state Exercises: (1) Re-interpret common tangent construction in terms of equation of state; (2) Develop a competing theory to Flory-Huggins based on VdW equation, discuss its advantages and disadvantages. Ideal chain •• TToo beginbegin with,with, supposesuppose wewe areare interestedinterested inin onlyonly relativelyrelatively fewfew representativerepresentative points:points: yi •• PPositionosition vectorsvectors ofof representative points are {x } representative points are {xii} •• SSupposeuppose piecespieces betbetwweeeenn representativerepresentative pointspoints areare longerlonger thanthan KuhnKuhn segment,segment, thenthen “bond vectors” {y =x -x } are “bond vectors” {yii=xi+1i+1-xii} are (i)(i) independentindependent,, andand (ii)(ii) GaussGauss xk distributeddistributed:: g(g(yy)=(2)=(2ππaa22/3)/3)-3/2-3/2exp[-3exp[-3yy22/2a/2a22].]. xj •H• Heerereaa isis notnotKuhKuhnn segment,segment, itit dependsdepends onon howhow manymany representativesrepresentatives wewe have.have. Ideal chain (cont’d) •• GGiveniven independenceindependence andand 2 -3/2 2 2 g(g(yy)=(2)=(2ππaa2/3)/3)-3/2exp[-3exp[-3yy2/2a/2a2],], yi probabilityprobability distributiondistribution ofof thethe wholewhole conformationconformation onon thisthis levellevel ofof descriptiondescription readsreads G(x ,x ,…,x )=g(y )g(y )…g(y ). G(x11,x22,…,xnn)=g(y11)g(y22)…g(yn-1n-1). xk ThisThis multiplicativemultiplicativestructurestructure isis the signature of ideal chain. the signature of ideal chain. xj Generalization • In general, {x } may include more than just • In general, {xii} may include more than just coordinatescoordinates ofof representativerepresentative points,points, e.g.,e.g., orientation/direction,orientation/direction, internalinternal statestate (helix-(helix- coil),coil), etc.etc. G(x ,x ,…,x )=g(x ,x )g(x ,x )…g(x ,x ). G(x11,x22,…,xnn)=g(x11,x22)g(x22,x33)…g(xn-1n-1,xnn). ThisThis multiplicativemultiplicativestructure structure isis stillstill thethe signaturesignature ofof idealidealchain. chain. ••H Hereere g(x,x’)g(x,x’) isis conditionalconditional probability,probability, itit satisfiessatisfies ∫∫g(x,x’)dx’=1g(x,x’)dx’=1 ••N Noteote ofof care:care: bondsbonds areare NOTNOT inin equilibriumequilibrium Comments: •• AlthoughAlthough g(g(yy)~exp[-3)~exp[-3yy22/2a/2a22]] isis exponential,exponential, itit isis notnot righrightt toto interpinterpretret g(g(yy)) asas equilibriumequilibrium BoltzmannBoltzmann probabilityprobability yi exp(-energy/k T), because it is not exp(-energy/kBBT), because it is not energy,energy, itit isis entropicentropic spring.spring. AsAs aa result,result, therethere isis nono temperature.temperature. •• IIff GG andand gg areare viewedviewed asas TTDD probabilities,probabilities, thenthen normalizationnormalization conditioncondition meansmeans thatthat freefree unrestrictedunrestricted idealideal chainchain isis takentaken asas xk zerozero levellevel ofof freefree energy.energy. •• IImaginemagine wewe wantwant toto switchswitch toto aa lessless detaileddetailed description,description, withwith fewerfewer xj representative points {X }; then representative points {Xii}; then Equation ••G Green’sreen’s functionfunction ••W Wee cancan findfind itit byby integratingintegrating overover allall intermediate {x }. intermediate {xii}. ••I Itt satisfiessatisfies equationequation Operator ••I Iff xx isis justjust coordinatescoordinates andand •T•Thenhen 2 2 ••( (thinkthink ofof k-representation,k-representation, exp[-kexp[-k2aa2/6])/6]) • Under proper conditions G ≈G +∂ G and • Under proper conditions Gn+1n+1≈Gnn+∂nnGnn and exp((aexp((a22/6)/6)ΔΔ)) ≈≈1+(a1+(a22/6)/6)ΔΔthenthen 22 Δ ∂∂nnGGnn=(a=(a /6)/6)ΔGGnn Diffusion equation • Under proper conditions G ≈G +∂ G and • Under proper conditions Gn+1n+1≈Gnn+∂nnGnn and exp((a22/6)Δ) ≈1+(a22/6)Δ then ∂ G =(a22/6)ΔG exp((a /6)Δ) ≈1+(a /6)Δ then ∂nnGnn=(a /6)ΔGnn ••D Diffusioniffusion equationequation returnsreturns thethe wellwell knownknown solutionsolution G (x,x’)~exp[-(3/2na22)(x-x’)22] Gnn(x,x’)~exp[-(3/2na )(x-x’) ] nono newnew information:information: whatwhat wewe putput inin --w wee gotgot outout Challenging exercise: Find operator g for worm-like chain. Hint: in this case, x={x,n} Continuum limit ••G Giveniven g(g(yy)~exp[-3)~exp[-3yy22/2a/2a22],], probabilityprobability ofof conformationconformation G(x ,x ,…,x )=g(y )g(y )…g(y ) G(x11,x22,…,xnn)=g(y11)g(y22)…g(yn-1n-1) ~exp[-3[(x -x )22+(x -x )22+…+(x -x )22]/2a22] ~exp[-3[(x11-x22) +(x22-x33) +…+(xn-1n-1-xnn) ]/2a ] =exp(-=exp(-∫∫[3([3(∂∂xx//∂∂s)s)22/2a/2a22]ds)]ds) ••L Looksooks ververyy muchmuch likelike exp[(i/exp[(i/ħħ))∫∫[[kinetickinetic e energy]dt]nergy]dt] Green’s function in continuum limit • When we integrate over all intermediate {xi} in 2 2 2 2 G(x1,x2,…,xn)~exp[-3[(x1-x2) +(x2-x3) +…+(xn-1-xn) ]/2a ] =exp(-∫[3(∂x/∂s)2/2a2]ds) we arrive at path integral: 2 2 Gn(x,x’)= ∫exp(-∫[3(∂x/∂s) /2a ]ds) Dx(s) • This is the partition sum of our polymer – the sum over all conformations. 2 • In this continuum form, it satisfies ∂nGn=(a /6)ΔGn • Thus, we can do cont. limit via Gn+1≈Gn+∂nGn and exp((a2/6)Δ) ≈1+(a2/6)Δ or via path integral A bit more general: φ(x) • Still assume ideal polymer, but suppose each monomer (really -- representative point) carries energy φ(x) which depends on position (or in general on x). Instead of G(x1,x2,…,xn)=g(x1,x2)g(x2,x3)…g(xn-1,xn). we now have (assuming kBT=1) -φ(x ) -φ(x ) -φ(x ) G(x1,x2,…,xn)=g(x1,x2)e 2 g(x2,x3)e 3 …e n-1 g(xn-1,xn). exp[-φ(x1)] exp[-φ(xn)] Continuum limit with φ(x) •W•Wee hhadad G(x ,x ,…,x )~exp[-3[(x -x )22+(x -x )22+…+(x -x )22]/2a22] G(x11,x22,…,xnn)~exp[-3[(x11-x22) +(x22-x33) +…+(xn-1n-1-xnn) ]/2a ] =exp(-=exp(-∫∫[3([3(∂∂xx//∂∂s)s)22/2a/2a22]ds)]ds) •N•Nowow wwee iinsertnsert φφ(x)(x) inin thethe exponential:exponential: Continuum limit with φ(x) φ(x1) φ(x2) φ(x3) φ(xn) •W•Wee hhadad G(x ,x ,…,x )~exp[-3[(x -x )22+(x -x )22+…+(x -x )22]/2a22] G(x11,x22,…,xnn)~exp[-3[(x11-x22) +(x22-x33) +…+(xn-1n-1-xnn) ]/2a ] =exp(-=exp(-∫∫[3([3(∂∂xx//∂∂s)s)22/2a/2a22]ds)]ds) •N•Nowow wwee iinsertnsert φφ(x)(x) inin everyevery pointpoint inin thethe exponential:exponential: Continuum limit with φ(x) φ(x1) φ(x2) φ(x3) φ(xn) •W•Wee hhadad G(x ,x ,…,x )~exp[-3[(x -x )22+(x -x )22+…+(x -x )22]/2a22] G(x11,x22,…,xnn)~exp[-3[(x11-x22) +(x22-x33) +…+(xn-1n-1-xnn) ]/2a ] =exp(-=exp(-∫∫[3([3(∂∂xx//∂∂s)s)22/2a/2a22]ds)]ds) •N•Nowow wwee iinsertnsert φφ(x)(x) inin everyevery pointpoint inin thethe exponentialexponential andand arrivearrive atat G(x ,x ,…,x )~exp(-∫[3(∂x/∂s)22/2a22+φ(x(s))]ds) G(x11,x22,…,xnn)~exp(-∫[3(∂x/∂s) /2a +φ(x(s))]ds) ••L Looksooks ververyy muchmuch likelike exp[(i/exp[(i/ħħ))∫∫[[kinetickinetic e energynergy ––p potentialotential energyenergy]dt]]dt] exceptexcept forfor disturbingdisturbing signsign difference!difference! Green’s function with φ(x) ••G Green’sreen’s functionfunction isis thethe sumsum overover allall conformationsconformations withwith fixedfixed endsends ––p pathath integral:integral: G (x,x’)= ∫exp(-∫[3(∂x/∂s)22/2a22+φ(x(s))]ds)Dx(s) Gnn(x,x’)= ∫exp(-∫[3(∂x/∂s) /2a +φ(x(s))]ds)Dx(s) ••T Thishis isis partitionpartition sumsum ofof thethe chainchain withwith fixedfixed endsends ••L Looksooks ververyy muchmuch like,like, butbut simplersimpler thanthan ∫∫exp[(i/exp[(i/ħħ))∫∫[[kinetickinetic e energynergy ––p potentialotential energyenergy]dt]]dt] Dx(t)Dx(t) exceptexcept forfor disturbingdisturbing signsign difference!difference! Applicability of cont. limit ••N Nothingothing changeschanges •• BecauseBecause a>>b,a>>b, wewe cancan significantlysignificantly ooverver tthehe approximateapproximate g(y)g(y) asas lengthlength scalescale ofof KuhnKuhn GaussianGaussian segment.segment. •• BecauseBecause λλ>>a,>>a, wewe havehave •L•Letet φφ(x)(x) changechange overover aa22ΔΔ~a~a22//λλ22<<1,<<1, andand wewe lengthlength scalescale λλ>>b.>>b. cancan expandexpand ThenThen wewe cancan choosechoose 2 2 exp((aexp((a2/6)/6)ΔΔ)) ≈≈1+(a1+(a2/6)/6)ΔΔ equivalentequivalent chainchain withwith λλ>>a>>b>>a>>b Edwards Hamiltonian ••G Green’sreen’s functionfunction isis thethe sumsum overover allall conformationsconformations withwith fixedfixed endsends ––p pathath integral:integral: G (x,x’)= ∫exp(-∫[3(∂x/∂s)22/2a22+φ(x(s))]ds)Dx(s) Gnn(x,x’)= ∫exp(-∫[3(∂x/∂s) /2a +φ(x(s))]ds)Dx(s) ••E Externalxternal oror self-consistentself-consistent fieldfield ••T Twowo bodybody interactioninteraction ••VeryVery shortshort rangedranged potentialpotential Example 1 • Imagine a polymer pinned down by one end at the top of a potential bump (e.g., Flagstaff mountain), and hanging down from there. • If the potential is strong and temperature low, the most probable conformation will correspond to the minimum of the expression in the exponent 2 2 G(x1,x2,…,xn)~exp(-∫[3(∂x/∂s) /2a +φ(x(s))]ds) • i.e.