Polymer Physics

3.2.3 Entropic Elasticity of a Deformed Polymer Coil

From the viewpoint of chemistry, rubbers are formed by the cross-linked polymeric
networks. The cross-links release the entropic elasticity of polymer coils in-
between them. Based on the ideal-chain model, theoretical descriptions about the
entropic elasticity of rubbers have been well developed (Meyer et al. 1932 ; Guth
and Mark 1934 , 1937 ; Kuhn 1939 ; Guth and James 1941 ; Treloar 1943 ; Flory 1944 ,
1961 ). Meyer et al.first assigned the high elasticity of the rubber to the capability of
large deformation of random coil polymers (Meyer et al. 1932 ). Guth and Mark
attempted to make a statistical theory on the spatial arrangement of polymer
conformations for such a high elasticity (Guth and Mark 1934 , 1937 ). This
approach has been then developed by Kuhn (Kuhn 1939 ). The statistical theory
for the high elasticity of the ideal-chain network was eventually delivered under the
great efforts of Guth, James, Treloar, and Flory, et al.(Guth and James 1941 ;
Treloar 1943 ; Flory 1944 , 1961.
In the three-dimensional network of long-chain molecules, the chain ends of
each single coil are separately fixed at (0,0,0) and (x,y,z). All the possible
conformationsOof this coil with fixed end locations should be proportional to
the probabilityW(x,y,z) associated with this end-to-end distance. According to the
Boltzmann’s law,S¼klnO, wherekis the Boltzmann constant, as well as to the
relationship in (3.3), we obtain

S¼Bkb^2 ðx^2 þy^2 þz^2 Þ (3.16)

whereBis a constant. Assuming a deformation makingx¼l 1 x,y¼l 2 y, and
z*¼l 3 z, we have

S¼Bkb^2 ðl^21 x^2 þl^22 y^2 þl^23 z^2 Þ (3.17)

Accordingly, the conformational-entropy change of the polymer coil is

DS¼SS¼kb^2 ½ðl^21  1 Þx^2 þðl^22  1 Þy^2 þðl^23  1 Þz^2 Š (3.18)

3.2.4 Statistical Thermodynamics of a Cross-Linked Polymer

Network

Let’s set up an ideal model for a three-dimensional network of the rubber. The
model is based upon the following assumptions.

1. The cross-links are well distributed in the elastic body;

3.2 Statistical Mechanics of Rubber Elasticity 37