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
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 -2
-1 ) 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 rg )( ≈≈ rbr 23
Pair correlation function for a D-dimensional D ~ rm m D−3 fractal with rg ≈ ~)( 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: = NbR 22 Nb2 R2 = • Mean squared gyration radius: g 6 2/3 r ⎛ 3 ⎞ ⎛ 3R2 ⎞ (), RNP = exp⎜− ⎟ 3d ⎜ 2 ⎟ ⎜ 2 ⎟ • Probability distribution:r ⎝ 2πNb ⎠ ⎝ 2Nb ⎠ 3 R2 = kTF • Free energy: 2 Nb2 r 3kT r f = R • Entropic “Hook’s Law”: Nb2 13 rg )( = • Pair correlation function: π rb2 • Nonlinear elasticity at high forces Mayer f-function
U(r) Effective interactions potential between two r monomers in a solution of other molecules.
Relative probability of finding two monomers at distance r
2 exp(-U/kT) 1 Mayer f-function 0 1234 rU )( rf exp)( ⎡−= ⎤ −1 1.5 r/b ⎣⎢ kT ⎦⎥ 1 0.5 Excluded volume f 0 1 234 −= ∫ r)( 3rdrfv -0.5 r/b -1 Classification of Solvents −= ∫ r)( 3rdrfv f rb 3 Athermal, high T ≈ bv
r 3 Good solvent, repulsion dominates f 0 < < bv
f r θ-solvent, repulsion compensates v = 0 attraction f r Poor solvent, attraction dominates v < 0
Typically, but not always, repulsion dominates at higher, and attraction -- at lower temperatures. Pair interaction dominance:
• In ideal coil, R~bN1/2. Monomer volume fraction φ~Nb3/R3~N-1/2<<1 • The number of pair contacts Nφ ~N1/2>>1; pair collisions are important. • The number of triple contacts Nφ2~1; they become important only if chain collapses. • Higher order are unimportant unless strong collapse.