Rubber elasticity
Polymers can be modeled as freely jointed chains with one fixed end and one free end (FJC model):
where b is the length of a rigid segment, n is the number of segments of length b, r is the distance between the fixed and free ends, and Lc is the "contour length" or nb. Above the glass transition temperature, the polymer chain oscillates and r changes over time. The probability of finding the chain ends a distance r apart is given by the following Gaussian distribution:
Above the glass transition temperature, the polymer chain oscillates and r changes over time. The probability of finding the chain ends a distance r apart is given by the following Gaussian distribution:
Note that the movement could be backwards or forwards, so the net time average <r> will be zero. However, we usually use the root mean square as a useful measure of that distance.
Ideally, the polymer chain's movement is purely entropic (no enthalpy involved). By using the following basic equations for entropy and Hemholtz free energy, we can model the driving force of entropy "pulling" the polymer into an unstretched conformation. Note that the force equation resembles that of a spring: F=kx.
The worm-like chain (WLC) takes the energy required to bend a molecule into account. The variables are the same except that Lp, the persistence length, replaces b. When graphed, the force equation resembles a sideways S curve beginning at the origin and going to infinity as r approaches Lc.