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Introduction to Polymer Physics Lecture 1 Boulder Summer School July – August 3, 2012

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Introduction to polymer physics Lecture 1 Boulder Summer School July – August 3, 2012 A.Grosberg New York University Lecture 1: Ideal chains • Course outline: • This lecture outline: •Ideal chains (Grosberg) • Polymers and their •Real chains (Rubinstein) uses • Solutions (Rubinstein) •Scales •Methods (Grosberg) • Architecture • Closely connected: •Polymer size and interactions (Pincus), polyelectrolytes fractality (Rubinstein) , networks • Entropic elasticity (Rabin) , biopolymers •Elasticity at high (Grosberg), semiflexible forces polymers (MacKintosh) • Limits of ideal chain Polymer molecule is a chain: Examples: polyethylene (a), polysterene •Polymericfrom Greek poly- (b), polyvinyl chloride (c)… merēs having many parts; First Known Use: 1866 (Merriam-Webster); • Polymer molecule consists of many elementary units, called monomers; • Monomers – structural units connected by covalent bonds to form polymer; • N number of monomers in a …and DNA polymer, degree of polymerization; •M=N*mmonomer molecular mass. Another view: Scales: •kBT=4.1 pN*nm at • Monomer size b~Å; room temperature • Monomer mass m – (240C) from 14 to ca 1000; • Breaking covalent • Polymerization bond: ~10000 K; degree N~10 to 109; bondsbonds areare NOTNOT inin • Contour length equilibrium.equilibrium. L~10 nm to 1 m. • “Bending” and non- covalent bonds compete with kBT Polymers in materials science (e.g., alkane hydrocarbons –(CH2)-) # C 1-5 6-15 16-25 20-50 1000 or atoms more @ 25oC Gas Low Very Soft solid Tough and 1 viscosity viscous solid atm liquid liquid Uses Gaseous Liquid Oils and Candles Bottles… fuels fuels and greases and solvents coatings Examples Propane Gasoline Motor oil Paraffin Polyethyl wax ene Polymers in living nature DNA RNA Proteins Lipids Polysaccha rides N Up to 1010 10 to 1000 20 to 1000 5 to 100 gigantic Nice Bioinformatics, Secondary Proteomics, Bilayers, ??? Someone physics elastic rod, structure, random/designed liposomes, has to start models charged rod, annealed heteropolymer,HP, membranes helix-coil branched, funnels, ratchets, folding active brushes Uses Molecule Polymer properties depend on… • Chemical composition of a monomer; • Degree of polymerization, N; • Flexibility; •Architecture; •• HomoHomopolymer versus heteroheteropolymer. Architecture Architecture cont’d Homo- vs. Hetero Homopolymer consists of monomers of just one sort: Heteropolymer (copolymer) has two or more monomer species: alternating random grafted Block-copolymers diblock triblock multiblock Flexibility • Sufficiently long polymer is never straight • Different polymers bend differently: Rotation isomers: Polyethylene: bond length l ≅ 1.54A, tetrahedral angle θ ≅ 68o φ H H i H H H H H C H H C H Ci+1 C H C θi y i+1 C H C H H H Ci H H H y θ H H i i-1 C C H φ φ π Ci-1 trans i =0 gauche+ i =2 /3 H yi-1 H H Δε ~ kBT Ci-2 gauche gauche trans ΔE Torsion (measured in angle φ)- Δε main source of polymer flexibility. Elastic flexibility: • Within one rotamer, ΔE~φ2 – Hook’s law. • Many polymers have no freedom to explore rotational isomers, e.g., two strands. • Elastic rod model: How much is polymer bent? • How much is polymer bent? •What is its size R, given N? • Can we measure it? • How/why is it important? • How does it depend on conditions, e.g., temperature? Polymer Size Monomer size b~0.1nm; Number of monomers N~102-1010; Contour length L~10nm – 1m; Depending on how much polymer is bent, its overall size R varies widely and depends on solvent quality Long-range repulsion Good solvent θ-solvent Poor solvent R~1m R~100mm R~10mm R~100nm Astronomical Variations of Polymer Size Increase monomer size by a factor of 108: b ~ 1cm; let N=1010. Poor solvent θ-solvent Good solvent Long-range repulsion Ideal polymer vs. ideal gas •• StrongStrong dependencedependence ofof polymerpolymer sizesize onon environment/solventenvironment/solvent conditionsconditions suggestssuggests aa bigbig rolerole ofof interactions.interactions. •• IdealIdeal polymerpolymer hashas nono interactionsinteractions betweenbetween monomers,monomers, exceptexcept betweenbetween neighborsneighbors alongalong thethe chain.chain. •• JustJust likelike idealideal gasgas maymay havehave allall sortssorts ofof rotationsrotations andand vibrationsvibrations inin thethe molecule,molecule, butbut nono interactionsinteractions betweenbetween molecules.molecules. •• LikeLike idealideal gasgas isis thethe mostmost usefuluseful idealizationidealization inin statisticalstatistical mechanics,mechanics, soso isis thethe idealideal polymer. polymer. no interactions, the •• IdealIdeal chainschains areare goodgood modelsmodels forfor polymerpolymer melts,melts, concentratedconcentrated chain “does not see” solutions,solutions, andand dilutedilute solutionssolutions atat θθ-temperature-temperature itself Ideal chain size Ideal chain: no interactions between monomers if they are not neighbors along the chain, even if they approach one another in space. Conformation of an ideal chain is fully specified by the set of bond vectors {yi} Ideal chain size: mean squared end-to- end distance Ideal chain: no interactions between monomers if they are not neighbors along the chain, even if they approach one another in space. Conformation of an ideal chain is fully specified by the set of bond vectors {yi} Freely-jointed chain or lattice model: Ideal chain size: mean squared end-to- end distance For ideal chains, one can show that there are no long-range correlations between bond directions: For example, freely-rotating chain: SpecificSpecific flexibilityflexibility mechanismmechanism (rotational(rotational isomersisomers etc)etc) is hidden in characteristic ratio C ; is hidden in characteristic ratio C∞∞; seesee P.Flory, P.Flory, Statistical Statistical Mechanics Mechanics of of Chain Chain Mo Moleculeslecules Ideal chain size: worm-like chain (Kratky and Porod model) Key argument: Universal description of ideal polymer Construct equivalent freely jointed chain with the same mean squared end- to-end distance <R2> and the same contour length L as actual polymer. It consists of N effective, or Kuhn, segments, each of length b, such that: For chain of n bonds of Worm-like chain, total length l each: length L, persistence l: Kuhn length and #of Kuhn length and #of effective segments are effective segments are Equivalent freely jointed chain differs from the actual polymer on length scales of Kuhn segment b or smaller, but has the same physical properties on large scales Reverse the question of polymer size: how many monomers are there within radius r? • If r is greater than Kuhn segment, r>b, then sub-coil of g monomers has size R(g) ~ g1/2b; • Therefore, within radius r we expect m(r)~(r/b)2 monomers. r Counting “atoms” in regular objects m~r3 m~r2 m~r1 for 3D object for 2D object for 1D object 3D 2D 1D 0D 3 Koch curve: example of a fractal r2 r1 df m ~ r df – fractal dimension We can either increase “observation field” r, or decrease the size of elementary “atom” Polymeric fractals For ideal chain, m(r) ~ (r/b)2; that means, ideal polymer has fractal dimension df=2. Scaling exponent ν=1/df=1/2 for ideal polymer. We will see later that in a good solvent df=5/3 and ν=3/5 IdealIdeal coilcoil cancan bebe viewedviewed asas N/gN/g blobsblobs ofof gg segmentssegments each.each. BlobBlob sizesize isis x~bgx~bg1/21/2,, thereforetherefore R~bNR~bN1/21/2~x(N/g)~x(N/g)1/21/2 Problem: consider polymer adsorbed on a 2D plane. What are the consequences of the fact that df=D? Radius of gyration End-to-end distance is difficult to measure (and it is ill defined for, e.g., rings or branched polymers). Better quantity is gyration radius: Where position vector of mass center is There is theorem (due to nobody lesser than Lagrange) which says that Exercises: (1) Prove Lagrange theorem; (2) Prove 2 2 that for ideal linear chain Rg =Nb /6; (3) Prove that 2 2 for ideal ring Rg =Nb /12 Entropic elasticity Optical tweezers experiment How much force should we apply to achieve end-to-end distance R? Pincus blob argument Gauss distribution of R follows directly from Central Limit Theorem What if we pull harder? Gaussian theory is satisfactory only up to about 0.1pN S. Smith, L. Finzi, C. Bustamante, \Direct Mechanical Measurements of the Elasticity of Single DNA Molecules by using Magnetic Beads", Science, v. 258, n. 5085, p. 1122, 1992. Non-universal elasticity at higher forces From Langevin to Marko-Siggia • Exact formula for •Very accurate freely-jointed formula for worm- chain: like chain: where Compare: Einstein and Debye theories of heat capacity of a solid Chains Get Softer Under Tension ~K Nonlinear flexible -1 -2 f ~ (1-Rf/Rmax) ) max /R f Linear Nonlinear deformation semi-flexible regime f ~ (1-R ~K-1 bending constant K Linear deformation regime ends at f ~ kBT/b. Problem: Why is there a cross-over from semi-flexible (worm-like) chain to flexible (freely-jointed) chain with increasing tension at fc ~ kBTK/l? Pair Correlations of an Ideal Chain Number of monomers within r range r 2 ⎛ r ⎞ m ≈ ⎜ ⎟ ⎝b ⎠ Probability of finding a monomer m 1 at a distance r from a given one g()≈ r ≈ r3 rb 2 Pair correlation function for a D-dimensional D m~ r m D−3 fractal with g( ) r≈ ~r r3 Scattering provides ensemble average information over wide range of scales Light Scattering Alan Hurd Summary of Ideal Chains • No interactions except along the chain • Equivalent Kuhn chain • Mean square end-to-end:R2= Nb 2 Nb2 R2 = • Mean squared gyration radius: g 6 2/3 r ⎛ 3 ⎞ ⎛ 3R2 ⎞ PNR, = exp⎜− ⎟ 3d ()⎜ 2 ⎟ ⎜ 2 ⎟ • Probability distribution:r ⎝ 2πNb ⎠ ⎝ 2Nb ⎠ 3 R2 F= kT •
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