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Introduction to Lecture 4 Boulder Summer School July – August 3, 2012

A.Grosberg New York University Center for 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, … 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.

•• 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. x(s) corresponds to min ∫[3(∂x/∂s)2/2a2+φ(x(s))]ds A.N. Semenov, Sov. Phys. JETP, 61, p. 733, 1985 Example 1 (cont’d)

• i.e. x(s) corresponds to min ∫[3(∂x/∂s)2/2a2+φ(x(s))]ds • This looks exactly like: if applies, then particle chooses trajectory x(t) such as to minimize action min ∫[m(∂x/∂t)2/2-U(x(t))]dt • Dictionary: s->t, 3/a2 -> m, ∂x/∂s ->v, φ(x)->-U(x). It is surprising, but has physical meaning that φ(x) is MINUS potential. Example 1 (cont’d further)

• Optimal trajectory in mechanics satisfies F=ma or mv2/2+U=const • Optimal conformation of a polymer satisfies

• Per little chain piece δs: i.e., chain stretching energy is large at large potential and small where potential is small, thus φ(x) is analogous to –U(x).

• Free energy F=-kBTlnG reads First serious application

•• CConsideronsider aa brush,brush, andand looklook atat oneone chainchain there.there. ItIt cannotcannot wigglewiggle sidewayssideways becausebecause ofof peerpeer pressure.pressure. ThereThere isis somesome sortsort ofof aa (self-(self- consistent)consistent) fieldfield φφ(x),(x), wherewhere xx isis heightheight aboveabove thethe plane,plane, whichwhich favorsfavors chainchain rushingrushing up,up, toto betterbetter life.life. WhatWhat isis thisthis field?field? •• EEveryvery chainchain isis aa “particle”“particle”thatthat φ startsstarts atat x=0x=0 andand reachesreaches somesome heightheight x=h,x=h, differentdifferent forfor differentdifferent chains,chains, inin oneone andand thethe x samesame “time”“time”t=N…t=N… Isochronous pendulum

•• RReverse:everse: thethe timetime (period)(period) doesdoes notnot dependdepend onon heightheight (amplitude).(amplitude). IsochronousIsochronous pendulum!!!pendulum!!! •• FForor aa harmonicharmonic oscillator,oscillator, ωω=(k/m)=(k/m)1/1/22independentlyindependently ofof amplitude;amplitude; thisthis isis truetrue forfor harmonicharmonic oscillatoroscillator ONLY.ONLY. •• FForor realreal pendulum,pendulum, periodperiod becomesbecomes longerlonger atat veryvery largelarge amplitudes.amplitudes. As φ candelabra swings more, does it also x swing faster? Potential in the brush

Thus,Thus, brushbrush isis likelike aa harmonicharmonic oscillator,oscillator, U(x)~xU(x)~x22 or or φφ(x)~(x)~-x-x22 FamousFamous “parabolic“parabolic profile”profile” ItIt isis veryvery cleverclever thatthat φφ(x)->-U(x),(x)->-U(x), becausebecause φφ(x)(x) inin thethe brushbrush mustmust bebe aa bumpbump

WeWe cancan eveneven establishestablish coefficientcoefficient inin φφ(x)=-kx(x)=-kx22/2,/2, because quarter-period is N: k=(3π22/4)(k T/N22a22) because quarter-period is N: k=(3π /4)(kBBT/N a )

In this form, the agument is due to S.T.Milner, T.A.Witten, and M.E.Cates Europhys. Lett., 5 (5), pp. 413-418 (1988) “A Parabolic Density Profile for Grafted Polymers”.

E.Zhulina in her lecture is likely to present another independently developed view on the same topic Potential and density

GivenGiven thethe harmonicharmonic oscillatoroscillator potentialpotential

thethe “particle“particle energyenergy conservation”conservation”reads reads

“Total“Total energy”energy”i iss establisheestablishedd fromfrom boundaryboundary conditioncondition thatthat “particle”“particle”arrives arrives toto thethe amplitudeamplitude positionposition atat zerozero velocity,velocity, oror chainchain isis notnot stretchedstretched atat thethe end:end:

Density:Density:#monomers#monomers δδsspperer heightheight intervalinterval δδx:x: δδs/s/δδx,x, i.e.,i.e., invertedinverted velocity:velocity: Density

InvertedInverted velocityvelocity isis densitydensity ofof oneone chainchain whosewhose endend isis locatedlocated atat givengiven heightheight h:h:

DifferentDifferent chainschains endend atat didifferentfferent heightsheights h.h. GivenGiven σσ chainschains perper unitunit area,area, totaltotal densitydensity isis thethe sum:sum:

wherewhere HH isis totaltotal brushbrush heightheight (maximal(maximal h),h), andand ψψ(h)(h) isis thethe fractionfraction ofof chainschains endingending atat hh ––b bothoth toto bebe foundfound fromfrom thethe self-consistencyself-consistency conditioncondition whichwhich relatesrelates fieldfield φφ(x)(x) toto the total density ρ(x): φ(x)-φ(H) = k Tv ρ(x) the total density ρ(x): φ(x)-φ(H) = kBBTv ρ(x)

Exercise: Show that for a chain with free end at the height h, monomer number s is located on average at the height x=h sin(πs/2N) Self-consistency

Self-consistencySelf-consistency conditioncondition relatrelateses fieldfield φφ(x)(x) toto thethe totaltotal density ρ(x): φ(x)-φ(H) = k Tv ρ(x): density ρ(x): φ(x)-φ(H) = kBBTv ρ(x):

Since we know both φ(x) and ρ (x,h), we get equation: Since we know both φ(x) and ρ11(x,h), we get equation:

TheThe restrest isis aa fewfew lineslines ofof algebra.algebra. IntegrateIntegrate bothboth sidessides overover hh fromfrom 00 toto HH toto getget

ThisThis completescompletes thethe meanmean fieldfield solution.solution. Results for brush:

•Figure from Milner- Witten-Cates, EPL 1988 ρ(x)/ρ(0)

σa2=0.04 σa2=0.08

ψ(h)

x/H,h/H Applicability of mean field

• Mean field applies if thermal blob gT (the distance needed to develop self- avoidance correlations) is larger than the distance g’ to the neighboring chains. • The former is estimated from vg1/2/a3~1, the latter from a2g~1/σ • Thus the conditions: v2/a6<<σa2<<1 Green’s function with φ(x) (repeated)

••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! ••I Inn QM,QM, itit satisfiessatisfies SchroedingerSchroedingerequation. equation. ByBy analogy,analogy, ∂G /∂n= (a22/6)ΔG -φ(x)G ∂Gnn/∂n= (a/6)ΔGnn-φ(x)Gnn Bilinear expansion

••S Sinceince itit lookslooks likelike SchroedingerSchroedingerequation, equation, letlet usus proceedproceed asas wewe areare taughttaught inin QM:QM: expandexpand x-dependencex-dependence overover eigenmodeseigenmodes

•T•Thenhen Ground state dominance

••I Iff therethere isis discretediscrete “energy“energy level”,level”, thenthen largestlargest ll (smallest(smallest energy)energy) dominatesdominates andand

•w•wherehere λλanandd ψψareare “grou“groundnd statestate energy”energy”a andnd “wave“wave function”function”of of

••R Remindereminder fromfrom QM:QM: wavewave functionfunction ofof groundground ststateate cancan bebe alwaysalways chosenchosen positivepositive (has(has nono zeros).zeros). •• Examples:Examples:chainchain inin aa box,box, inin aa tube,tube, inin aa slit,slit, etcetc Phase transition

• Remember that φ(x) -> φ(x)/k T. Presence or absence of • Remember that φ(x) -> φ(x)/kBBT. Presence or absence of discretediscrete levellevel isis determineddetermined byby temperature:temperature:

••C Challenginghallenging exercises:exercises: (1)(1) SupposeSuppose fieldfield f(x)f(x) isis somesome potentialpotential wellwell inin 3D,3D, butbut φφ(x)=0(x)=0 outsideoutside aa certaincertain finitefinite volumevolume (or(or rapidlrapidlyy goesgoes toto zero).zero). WhatWhat isis thethe orderorder ofof phasephase transition?transition? (2)(2) SuppSupposeose fieldfield φφ(x)(x) dependsdepends onlyonly onon oneone coordinate,coordinate, saysay z,z, andand isis equalequal toto zerozero outsideoutside finitefinite iintervalnterval ofof z;z; isis therethere phasephase transition?transition?(3) (3) SupSupposepose fieldfield φφ(x)(x) dependsdepends onlyonly onon oneone coordinacoordinate,te, saysay z,z, andand isis equalequal toto infinityinfinity atat z<0,z<0, zerozero atat z>z>A,A, andand somesome negativenegative valuevalue betweenbetween z=0z=0 andand z=A;z=A; isis therethere phasephase transition?transition? Free energy, density

••G Giveniven thatthat freefree energyenergy isisF=TN F=TNλλ

••T Too findfind density,density, considerconsider

•T•Thus,hus, nn(x)=(x)=ψψ22(x)(x) (assuming(assuming ψψisis properlyproperly normalized)normalized) Entropy

••W Wee knowknow freefree energyenergy isis F=TNF=TNλλandand densitydensityn(x)= n(x)=ψψ22(x)(x) •E•Entropy:ntropy:

•N• Notes:otes: (1)(1) ψψisis anan ordorderer parameter;parameter;(2) (2) S~NaS~Na22/R/R22isis recovered.recovered. Self-consistency

••P Particlearticle whichwhich digsdigs wellwell forfor itself.itself. ••F Freeree energy:energy:

••Minimize Minimize thisthis withwith respectrespect toto ψψ,, withwith LagrangeLagrange multipliermultiplier λλensuringensuring thatthatψψ isis properlyproperly normalized:normalized:

••E Equilibriumquilibrium freefree energy:energy: Virial expansion

••V Virialirial e expansionxpansion forfor μμ** Solution for large globule

••I Inn thethe interior,interior, ψψ=const=const:: ••E Equilibriumquilibrium freefree energy:energy: (1/2)Bn(1/2)Bn22+(1/3)+(1/3)CnCn33,, hashas minimumminimum atat B<0B<0 atat n~-B/C:n~-B/C: exactlyexactly thethe samesame answeranswer asas before.before. ••I Inn thethe surfacesurface layer,layer, fullfull non-linearnon-linear PDEPDE hashas toto bebe addressed.addressed. Challenging problem

••C Consideronsider polymerpolymer withwith excludedexcluded volume,volume, butbut nono attractionsattractions (B>0,C>0)(B>0,C>0) collapsecollapsedd inin aa potentialpotential boxbox withwith infiniteinfinite walls.walls. WhatWhat isis thethe profileprofile ofof thethe self-consistself-consistentent potentialpotential μμ*?*? Summary

• Chain connectivity is one thing and interactions is the other: this is a possible and fruitful point of view. • “Quasi-classical” limit for stretched chains. • “Ultra-quantum” limit for collapse. • One has to watch for physics not only doing magic estimates, but also performing routine calculations.