Polymer Physics

The mean end-to-end distance is thus

¼

ð^1

0

RWðRÞdR¼

2

ffiffiffi

p

p

b

(3.8)

Similarly, one can derive the mean-square end-to-end distance<R^2 >¼3/2/b^2.
In the following, we will apply the Gaussian chain above to interpret the entropic
origin of high elasticity of rubbers.

3.2 Statistical Mechanics of Rubber Elasticity

3.2.1 Mechanics of Elasticity

Rubbers are featured with their high elasticity. Let us look at an elastic cylindrical
body with a lengthland a cross-sectional areaA, as demonstrated in Fig.3.4.A
stretching forcefis applied at both ends of the cylinder with the initial sizes ofA 0
andl 0. Under the engineering stresss¼f/A 0 , we obtain the engineering strain

e¼

ll 0

l 0

¼

l

l 0

 1 ¼l 1 (3.9)

For an ideal elastic deformation, there exists the well-known Hooke’s law,

s¼Ee (3.10)

whereEis the elastic modulus.

3.2.2 Thermodynamics of Elasticity

According to the first law of thermodynamics, a small change in the internal energy
of the systemdUcontains two major contributions, i.e. the heat exchangedQand
the external workdW.

Fig. 3.3 Illustration of the radial distribution of the end-to-end distances of a polymer coil.R*is
the most probable end-to-end distance

3.2 Statistical Mechanics of Rubber Elasticity 35