Statistical Mechanics of Polymers
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Transcript of 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