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Cross-linked Polymers and Rubber Elasticity. Definition. An elastomer is defined as a cross-linked amorphous polymer above its glass transition temperature. 1. Capability for instantaneous and extremely high extensibility
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Definition • An elastomer is defined as a cross-linked amorphous polymer above its glass transition temperature. 1. Capability for instantaneous and extremely high extensibility 2. Elastic reversibility, i.e., the capability to recover the initial length under low mechanical stresses.when the deforming force is removed.
Defects in crosslinks For the purpose of the theoretical treatments presented here, the elastomer network is assumed to be structurally ideal, i.e., all network chains start and end at a cross-link of the network.
Force and Elongation Rubber elasticity Stress induced crystallinity Hookian
The origin of the forceAt constant V Under isothermal conditions Eneregy origin Entropy origin
Entropy change or internal energy change is important? Since F is a function of state:
Experimental data f is proportional to the temperature and is determined exclusively by the entropy changes taking place during the deformation
Thermodynamic Verificationat constant p According to the first and second laws of thermodynamics, the internal energy change (dE) in a uniaxially stressed system exchanging heat (dQ) and deformation and pressure volume work (dW) reversibly is given by: The Gibbs free energy (G) is defined as:
The partial derivatives of G with respect to L and T are: The partial derivative of G with respect to L at constant p and constant T
The derivative of H with respect to L at constant p and constant T Experiments show that the volume is approximately constant during deformation, (V /L)p,T= 0 . Hence,
Statistical Approach to the Elasticity Elasticity of a Polymer Chain relates the entropy to the number of conformations of the chain Ω
Entropy of the chain the probability per unit volume, p(x, y, z) <r2>o represents the mean square end-to-end distance of the chain The entropy decreases as the end-to-end distance increases
The work required for change in length • It can be concluded that • is proportional to the temperature, so that as T increases the force needed to keep the chain with a certain value of r increases, and • the force is linearly elastic, i.e., proportional to r.
Assumptions l. The network is made up of N chains per unit volume. 2. The network has no defects, that is, all the chains are joined by both ends to different cross-links. 3. The network is considered to be made up of freely jointed chains, which obey Gaussian statistics. 4. In the deformed and undeformed states, each cross-link is located at a fixed mean position. 5. The components of the end-to-end distance vector of each chain change in the same ratio as the corresponding dimensions of the bulk network. This means that the network undergoes an affine deformation.
The entropy change For N chain And
the work done in the deformation process or elastically stored free energy per unit volume of the network. The total work;
The Phantom Model When the elastomer is deformed, the fluctuation occurs in an asymmetrical manner. The fluctuations of a chain of the network are independent of the presence of neighbor in chains.
Statistical Approach to the Elasticity a) For a detached single chain
The probability for finding the chain end in the spherical shell between r and r+r Recall=>
Gaussian distribution Recall again => Retractive force for a single chain
Equibiaxial tension such as in a spherical rubber balloon, assuming ri2/r 20 = 1, and the volume changes of the elastomer on biaxial extension are nil.
A Typical Rubber Network Vulcanization with sulfur
Thermodynamic Verification At small strains, typically less than = L/ L0 < 1.1 (L and L0 are the lengths of the stressed and unstressed specimen, respectively), the stress at constant strain decreases with increasing temperature, whereas at λ values greater than 1.1, the stress increases with increasing temperature. This change from a negative to a positive temperature coefficient is referred to as thermoelastic inversion. Joule observed this effect much earlier (1859). The reason for the negative coefficient at small strains is the positive thermal expansion and that the curves are obtained at constant length. An increase in temperature causes thermal expansion (increase in L0 and also a corresponding length extension in the perpendicular directions) and consequently a decrease in the true λ at constant L. The effect would not appear if L0 was measured at each temperature and if the curves were taken at constant λ (relating to L0 at the actual temperature). The positive temperature coefficient is typical of entropy-driven elasticity as will be explained in this section.
Stress at constant length as a function of temperature for natural rubber.
Thermodynamic Verification The reversible temperature increase that occurs when a rubber band is deformed can be sensed with your lips, for instance. It is simply due to the fact that the internal energy remains relatively unchanged on deformation, i.e. dQ=-dW (when dE=0). If work is performed on the system, then heat is produced leading to an increase in temperature. The temperature increase under adiabatic conditions can be substantial. Natural rubber stretched to λ=5 reaches a temperature, which is 2-5 K higher than that prior to deformation. When the external force is removed and the specimen returns to its original, unstrained state, an equivalent temperature decrease occurs.
At constant V and T Wall’s differential mechanical mathematical relationship Thermodynamic eq. of state for rubber elasticity A Similar Equation
