Statistical Mechanics of Polymers

1. Statistical Mechanics of Polymers

2. Polymer • Polymers are giant molecules usually with carbons building the backbone exceptions exist (poly dimethylsiloxane). • Linear chains, branched chains, ladders, networks. • Homopolymers, copolymers, random copolymers, micro phase separated polymers.

3. One way to study of polymers long molecule is the statistical mechanics. It has become particularly exciting recently because biopolymers, such as DNA, allow the investigation of individual polymers. In turn, the statistical mechanics of such polymers is important in the biological function. Statistical mechanic

4. Many physical chemists have labored to relate the properties of polymer systems to the chemical structure of constituent chain molecules. • properties, such as • molecular dimensions • hindered internal rotations • localized or nonlocalized coupling • bond lengths • bond angles Real polymers

5. Rotational isomers Newman projections of the C-C bond in the middle of butane. Rotation about σ bonds is neither completely rigid nor completely free. A polymer molecule with 10000 carbons have 39997 conformations

6. Freely rotating chain Model • bond angles are fixed = θ0 • bond lengths are fixed = l0 • bond-rotation angles are evenly distributed over 0 ≤ φ≤ 2π

7. Freely jointed chain Model • bond angles are evenly distributed over 0 ≤ θ ≤ 2π • bond lengths are fixed = l0 • bond-rotation angles are evenly distributed over 0 ≤ φ ≤ 2π

8. Random walk model

9. Random walk model R is made up of N jump vectors ai . The average of all conformational states of the polymer is <R>=0 The simplest non-zero average is the mean-square end to end distance <R2> (A matrix of dot-products where the diagonal represents i=j and off axis elements i≠j) For a freely jointed chain the average of the cross terms above is zero and we recover a classical random walk: <R2>=Na2

10. Gaussian distribution • The distribution of end-to-end distances (R) in an ensemble of random polymer coils is Gaussian. The probability function is: • The probability decreases monotonically with increasing R (one end is attached at origo). The radial distribution g(R) is obtained by multiplying with 4πR2

11. The Gaussian chain: For a three dimensional polymer the large N result is similarly Gaussian.

12. We can now create the conformational distribution function of the entire chain, by multiplying each bond distribution The Gaussian chain:

14. The knot problem in statistical mechanics of polymer chains