1. Polymer chain and rubber elasticity
Lecture 14
Polymer chain and rubber elasticity
Gaussian chain
Entropy
Entropic elasticity

2. Freely jointed chain
Model: N+1 beads (mers) connected by N links (bonds) of length b0.
Vector representing nth link
End to end distance
Since link orientations are random an average over all conformations (denoted by )

3. End to end distance
The average end to end distance is zero but the average distance square is not - it measures the size of the polymer coil .
The last equality comes from the fact that the average dot product of two randomly oriented vectors is zero
Real chains are typically more rigid than model chains

4. Distribution of distances
Probability of a given conformation is a product of probabilities for each link (they are independent .
With a single link probability of given direction given by

5. Distribution of distances - II
Probability that a chain has end to end distance R
Using integral representation of the delta function
We get

6. One integral
Therefore

7. Distribution - final form
For small
For large the above Eq. is also quite OK for large N since both sides are very small. With the above approximation

8. Probability and Entropy
When we calculated probability we counted number of configurations with a given end to end distance. Thus by definition in microcanonical ensemble, the associated entropy is
Where k is the Boltzmann constant, and the Hemholz free energy

9. Entropic spring
When we pull the ends of the polymer chain, an average force we need to exert to stretch the chain end to end distance is
The above formula is like a spring with a spring constant

10. Rubber elasticity Polymer chains connected to cross links
A simple estimate gives entropic contribution to modulus
Where kc is the spring constant, lc is the distance between cross links, Nc is a number of monomers between cross links and nc is the cross link density