Crosslinked Polymers and Rubber Elasticity 11262020 1 Definition

Cross-linked Polymers and Rubber Elasticity 11/26/2020 1

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. 11/26/2020 2

Crosslinking effect 11/26/2020 3

Defects in crosslinks For the purpose of 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. 11/26/2020 4

Force and Elongation Rubber elasticity Stress induced crystallinity Hookian 11/26/2020 5

Rubber Elasticity and Force 11/26/2020 6

The origin of the force At constant V Under isothermal conditions 11/26/2020 Eneregy origin Entropy origin 7

Entropy change or internal energy change is important? Since F is a function of state: 11/26/2020 8

The change in internal energy in effect of l change 11/26/2020 9

Experimental data 11/26/2020 10

Experimental data f is proportional to the temperature and is determined exclusively by the entropy changes taking place during the deformation 11/26/2020 11

Thermodynamic Verification at constant p According to the first and second laws of thermodynamics, the internal energy change (d. E) in a uniaxially stressed system exchanging heat (d. Q) and deformation and pressure volume work (d. W) reversibly is given by: The Gibbs free energy (G) is defined as: 11/26/2020 12

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 11/26/2020 13

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, 11/26/2020 14

Statistical Approach to the Elasticity of a Polymer Chain relates the entropy to the number of conformations of the chain Ω 11/26/2020 15

Entropy of the chain the probability per unit volume, p(x, y, z) <r 2>o represents the mean square end-to-end distance of the chain The entropy decreases as the end-to-end distance increases 11/26/2020 16

The work required for change in length 11/26/2020 It can be concluded that (1) 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 (2) the force is linearly elastic, i. e. , proportional to r. 17

Elasticity of a Netwrok 11/26/2020 18

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. 11/26/2020 19

Model of deformation 11/26/2020 20

And the chain 11/26/2020 21

The entropy change For N chain And 11/26/2020 22

the work done in the deformation process or elastically stored free energy per unit volume of the network. The total work; 11/26/2020 23

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True and Nominal stress 11/26/2020 25

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. 11/26/2020 26

Other quantities: Young Modulus ? 11/26/2020 27

Statistical Approach to the Elasticity a) For a detached single chain 11/26/2020 28

A Spherical Shell and the End of the Chain in it 11/26/2020 29

The probability for finding the chain end in the spherical shell between r and r+ r 11/26/2020 Recall=> 30

Gaussian distribution Recall again => Retractive force for a single chain 11/26/2020 31

b) For a Macroscopic Network 11/26/2020 32

The Stress-Strain Relationship 11/26/2020 33

We have: 11/26/2020 34

11/26/2020 35

And the stress-strain eq. for an elastomer 11/26/2020 36

Equibiaxial tension such as in a spherical rubber balloon, assuming ri 2/r 20 = 1, and the volume changes of the elastomer on biaxial extension are nil. 11/26/2020 37

The Carnot Cycle for an Elastomer 11/26/2020 38

Work and Efficiency 11/26/2020 39

A Typical Rubber Network Vulcanization with sulfur 11/26/2020 40

Radiation Cross-linking 11/26/2020 41

Using Multifunctional Monomers 11/26/2020 42

Comparison between Theory and Experiment 11/26/2020 43

Thermodynamic Verification At small strains, typically less than = L/ L 0 < 1. 1 (L and L 0 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 L 0 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 L 0 was measured at each temperature and if the curves were taken at constant λ (relating to L 0 at the actual temperature). The positive temperature coefficient is typical of entropy-driven elasticity as will be explained in this section. 11/26/2020 44

Stress at constant length as a function of temperature for natural rubber. 11/26/2020 45

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. d. Q=-d. W (when d. E=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. 11/26/2020 46

At constant V and T Wall’s differential mechanical mathematical relationship A Similar Equation 11/26/2020 Thermodynamic eq. of state for rubber elasticity 47

Analysis of Thermodynamic Eq. 11/26/2020 48

Stress-Temperature Experiments 11/26/2020 49

End of Chapter 9 11/26/2020 50