======================================================================
= Rubber elasticity =
======================================================================
Introduction
======================================================================
Rubber elasticity, a well-known example of hyperelasticity, describes
the mechanical behavior of many polymers, especially those with
cross-links.
History
======================================================================
Following its introduction to Europe from the New World in the late
15th century, natural rubber (polyisoprene) was regarded mostly as a
fascinating curiosity. Its most useful application was its ability to
erase pencil marks on paper by rubbing, hence its name. One of its
most peculiar properties is a slight (but detectable) increase in
temperature that occurs when a sample of rubber is stretched. If it is
allowed to quickly retract, an equal amount of cooling is observed.
This phenomenon caught the attention of the English physicist John
Gough. In 1805 he published some qualitative observations on this
characteristic as well as how the required stretching force increased
with temperature. By the mid nineteenth century, the theory of
thermodynamics was being developed and within this framework, the
English mathematician and physicist Lord Kelvin showed that the
change in mechanical energy required to stretch a rubber sample should
be proportional to the increase in temperature. Later, this would be
associated with a change in entropy. The connection to thermodynamics
was firmly established in 1859 when the English physicist James Joule
published the first careful measurements of the temperature increase
that occurred as a rubber sample was stretched. This work onfirmed
the theoretical predictions of Lord Kelvin. It was not until 1838 that
the American inventor Charles Goodyear found that its properties could
be immensely improved by adding a few percent sulphur. The short
sulfur chains produced chemical cross-links between adjacent
polyisoprene molecules. Before it is cross-linked, the liquid natural
rubber consists of very long linear chains, containing thousands of
isoprene backbone units, connected head-to-tail. Every chain follows a
random path through the liquid and is in contact with thousands of
other nearby chains. When heated to about 150C, cross-linker molecules
(such as sulfur or dicumyl peroxide) can decompose and the subsequent
chemical reactions produce a chemical bond between adjacent chains.
The result is a three dimensional molecular network. All of the
original polyisoprene chains are connected together at multiple points
by these chemical bonds (network nodes) to form a single giant
molecule. The sections between two cross-links on the same chain are
called network chains and can contain up to several hundred isoprene
units. In natural rubber, each cross-link produces a network node
with four chains emanating from it. The network is the 'sine qua non'
of elastomers. Because of the enormous economic and technological
importance of rubber, predicting how a molecular network responds to
mechanical strains has been of enduring interest to scientists and
engineers. To understand the elastic properties of rubber,
theoretically, it is necessary to know both the physical mechanisms
that occur at the molecular level and how the random-walk nature of
the polymer chain defines the network. The physical mechanisms that
occur within short sections of the polymer chains produce the elastic
forces and the network morphology determines how these forces combine
to produce the macroscopic stress that we observe when a rubber sample
is deformed, e.g. subjected to tensile strain.
Molecular-level models
======================================================================
There are actually several physical mechanisms that produce the
elastic forces within the network chains as a rubber sample is
stretched. Two of these arise from entropy changes and one is
associated with the distortion of the molecular bond angles along the
chain backbone. These three mechanisms are immediately apparent when a
moderately thick rubber sample is stretched manually. Initially, the
rubber feels quite stiff, i.e. the force must be increased at a high
rate with respect to the strain. At intermediate strains, the required
increase in force is much lower to cause the same amount of stretch.
Finally, as the sample approaches the breaking point, its stiffness
increases markedly. What the observer is noticing are the changes in
the modulus of elasticity that are due to the different molecular
mechanisms. These regions can be seen in Fig. 1, a typical stress vs.
strain measurement for natural rubber. The three mechanisms (labelled
Ia, Ib and II) predominantly correspond to the regions shown on the
plot. The concept of entropy comes to us from the area mathematical
physics called statistical mechanics which is concerned with the study
of large thermal systems, e.g. rubber networks at room temperature.
Although the detailed behavior of the constituent chains are random
and far too complex to study individually, we can obtain very useful
information about their 'average' behavior from a statistical
mechanics analysis of a large sample. There are no other examples of
how entropy changes can produce a force in our everyday experience.
One may regard the entropic forces in polymer chains as arising from
the thermal collisions that their constituent atoms experience with
the surrounding material. It is this constant jostling that produces a
resisting (elastic) force in the chains as they are forced to become
straight. While stretching a rubber sample is the most common example
of elasticity, it also occurs when rubber is compressed. Compression
may be thought of as a two dimensional expansion as when a balloon is
inflated. The molecular mechanisms that produce the elastic force are
the same for all types of strain.
When these elastic force models are combined with the complex
morphology of the network, it is not possible to obtain simple
analytic formulae to predict the macroscopic stress. It is only via
numerical simulations on computers that it is possible to capture the
complex interaction between the molecular forces and the network
morphology to predict the stress and ultimate failure of a rubber
sample as it is strained.
===The Molecular Kink Paradigm for rubber elasticity===
The Molecular Kink Paradigm proceeds from the intuitive notion that
molecular chains that make up a natural rubber (polyisoprene) network