1. Polymer network consists of long polymer chains which are crosslinked with each other and form a continuous molecular framework. All polymer networks (which are not in the glassy or partially crystalline states) exhibit the property of high elasticity, i.e the ability to undergo large reversible deformations at relatively small applied stress. High Elasticity of Polymer Networks

2. High elasticity is the most specific property of polymer materials; it is connected with the most fundamental features of ideal chains considered above. In everyday life, highly elastic polymer materials are called rubbers. Molecular picture of high-elastic deformations Elasticity of the rubber is composed from the elastic responses of the chains crosslinked in the network sample.

3. Characteristic values for deformation are much larger for rubber. Characteristic values for strain are much larger for steel. Characteristic values for Young moduli are enormously larger for steel ( ) than for rubber ( ). For steel linearity and reversibility are lost practically simultaneously, while for rubbers there is a very wide region of nonlinear rever- sible deformations. For steel there is a wide region of plastic defor- mations (between points B and C) which is practically absent for rubbers. Typical stress-strain curves For rubberFor steel A - upper limit for stress-strain linearity B - upper limit for reversibility of deformations C - fracture point 0.01

4. Elasticity of a Single Ideal Chain For crystalline solids the elastic response ap- pears, because external stress changes the equilibrium inter-atomic distances and increases the internal energy of the crystal (energetic elasticity). Since the energy of ideal polymer chain is equal to zero, the elastic response appears by purely entropic reasons (entropic elasticity). Due to the stretching the chain adopts the less probable conformation its entropy decreases.

5. According to Boltzmann, the entopy Where k is the Boltzmann constant and is the number of chain conformations compa- tible with the end-to-end distance. The free energy F : But, Thus,

6. The chain is elongated in the direction of and (kind of a Hooke law). “Elastic modulus” 1) is proportional to, i.e very small for large values of L. Long polymer chains are very susceptible to external actions. 2) is proportional to kT which is the indication to entropic nature of elasticity. Limitations: should be Gaussian which is the case for not too strongly elongated chains.

7. Let us consider densely packed system of crosslinked chains (freely jointed chains of contour length L and Kuhn segment length l ). Flory theorem: the statistical properties of a polymer chain in the dense system are equivalent to those for ideal chains. Let the deformation of a the sample along the axes x, y, z be, i.e the sample dimensions along the axes are Affinity assumption: the crosslink points are deformed affinely together with the network sample. I.e if in the initial state the end-to-end vector has the coordinates in the deformed state its coordinates are Elasticity of a Polymer Network (Rubber)

8. Thus, the change of the free energy of the chain between two crosslink points upon extension is For the whole sample where is the number of chains per unit volume and V is the volume of the sample. But,

9. It is interesting that the answer does not depend on the parameters L and l that describe an individual subchain. This indicates that the theory is universal. It works whatever is the particular structure of the subchains (regardless of whether they are freely jointed or wormlike), for whatever contour length and Kuhn lengths, and so on. If we glance again at our calculations, we can see that basically all we needed to draw the main conclusion was just to regard the subchains as ideal.

10. Since we have from the uncompressibility condition (at characteristic values of stress applied to rubbers the intermolecular distances practically do not change: 1% change at 10 7 Pa ). Let us apply the general formula (see above) for the case of uniaxial extension or compression along the axis x.

11. Modulus of elasticity is For loosely crosslinked networks it is small (from the incompressibility condition where is the volume of a monomer unit, thus ). This is just the origin of the high elasticity of rubbers. The final formula predicts not only modulus, but also nonlinear elasticity. Analogous formula can be obtained for other kinds of deformation (shear, twist etc). The final formula is universal, i.e independent of specific chain model. Reason: entropic elasticity is caused by large - scale properties of polymer coils.

12. Main assumption in the above derivation: i) the chains are Gaussian; ii) the chain entanglements are neglected. If and T increases, the value of should decrease, i.e the rubber shrinks upon heating (contrary to gases) and vice versa. Also: at adiabatic extension the rubber is heated ( contrary to gases). This is the consequence of entropic character of elasticity. Correlations with experiment: Reason: chain entanglements - theory significantly underestimates stress, at a given strain. Reason: finite extensibility of the chains. experiment theory - very good agreement - theory slightly overestimates stress at a given strain.