Statistical Mechanics of Polymers

Statistical Mechanics of Polymers

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.

STATISTICAL MECHANIC

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 .

REAL POLYMERS

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

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

FREELY ROTATING CHAIN MODEL

bond angles are fixed = θ0

bond lengths are fixed = l0

bond-rotation angles are evenly distributed over 0 ≤ φ≤ 2π

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

RANDOM WALK MODEL

N

1iiaR

ji2 aaR

1N

1i

N

1ijij

222 cosa2NaR

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>

For a freely jointed chain the average of the cross terms above is zero and we recover a classical random walk:

<R2>=Na2

(A matrix of dot-products where the diagonal represents i=j and off axis elements i≠j)

Random walk model

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

2

22/32

Na2

R3exp3

Na2RP

2

22/322

Na2

R3exp3

Na2R4Rg

FOR A THREE DIMENSIONAL POLYMER THE LARGE N RESULT IS SIMILARLY GAUSSIAN.

The Gaussian chain:

THE GAUSSIAN CHAIN:

We can now create the conformational distribution function of the entire chain, by multiplying each bond distribution

2

22/32

Na2

R3exp3

Na2RP

THE KNOT PROBLEM IN STATISTICAL MECHANICS OF POLYMER CHAINS