MATS3004 · Polymer Science and Engineering 1
Elastomers and Rubber Elasticity
Week 5 explains what makes a material an elastomer: long, flexible chains used above their Tg, lightly crosslinked so they recover after large strains, with an entropic origin for the restoring force. It covers crosslinking and vulcanization (sulfur bridges) and where elastomers sit on the stress-strain map relative to plastics and fibres. The examinable calculation is the statistical rubber-elasticity relation σ = G(λ − 1/λ²) with modulus G = ρRT/Mc.
What this chapter covers
- 01Elastomer definition: lightly crosslinked amorphous polymer used above Tg, with large recoverable extension
- 02Entropic (entropy-spring) origin of rubber elasticity: stretching lowers conformational entropy, retraction recovers it
- 03Retractive force rises with temperature (unlike energetic solids) — the entropic signature
- 04Statistical rubber elasticity: engineering stress σ = G(λ − 1/λ²) versus stretch ratio λ
- 05Network modulus G = n·k_B·T = ρRT/Mc, where Mc is the molar mass between crosslinks
- 06Mooney-Rivlin correction for real rubbers: σ/(λ − 1/λ²) = 2C₁ + 2C₂/λ
- 07Requirements for a good elastomer: amorphous, flexible backbone, Tg well below use temperature, light crosslinking
- 08Crosslinking and vulcanization (sulfur crosslinks); swelling (Flory-Rehner) as a route to Mc
Modulus and stress of a crosslinked rubber from network theory
- +1Network modulus is G = ρRT/Mc. Keep ρ in kg·m⁻³ and Mc in kg·mol⁻¹ so the units give pascals: (kg·m⁻³)(J·mol⁻¹)/(kg·mol⁻¹) = J·m⁻³ = Pa.
- +1(a) G = (910 × 8.314 × 300) / 5.0 = 2 269 722 / 5.0 = 4.54 × 10⁵ Pa ≈ 0.45 MPa.
- +1Statistical rubber elasticity gives the engineering stress on uniaxial extension as σ = G(λ − 1/λ²). At λ = 2: λ − 1/λ² = 2 − 1/4 = 1.75.
- +1(b) σ = G × 1.75 = 4.54 × 10⁵ × 1.75 = 7.9 × 10⁵ Pa ≈ 0.79 MPa.
Key terms
- Elastomer
- A lightly crosslinked amorphous polymer used above its glass transition temperature, capable of large (up to several hundred percent) recoverable extension. Rubber is the archetype.
- Rubber elasticity (entropy spring)
- The restoring force in a stretched elastomer arises from entropy, not bond stretching: extending the network uncoils and aligns the chains, lowering their conformational entropy, and retraction is driven by the recovery of that entropy. Uniquely, the retractive force increases with temperature.
- Stretch ratio (λ)
- The deformed length divided by the original length (λ = L/L₀). It replaces small-strain measures for the large deformations elastomers undergo, and it appears in the stress law σ = G(λ − 1/λ²).
- Network modulus (G)
- The small-strain shear modulus of a rubber network, G = n·k_B·T = ρRT/Mc, where n is the network-chain density and Mc the average molar mass between crosslinks. It rises with temperature and with crosslink density (smaller Mc).
- Crosslink density / Mc
- A measure of how tightly the network is tied together, expressed as the molar mass Mc between crosslinks. Fewer, more widely spaced crosslinks (large Mc) give a softer, more extensible rubber; it can be estimated from equilibrium swelling (Flory-Rehner).
- Vulcanization
- Crosslinking natural rubber with sulfur to form bridges between chains, converting a tacky, flowable polymer into a recoverable elastic network with improved strength and set resistance. It is the classic route to a usable elastomer.
Elastomers and Rubber Elasticity FAQ
Why does a stretched rubber band pull back — and why is that unusual?
The restoring force is entropic. In the relaxed state the network chains are coiled in many possible conformations (high entropy). Stretching straightens them out into far fewer conformations (low entropy), which is thermodynamically unfavourable, so the network pulls back to recover its entropy. This is quite different from a metal spring, where the restoring force comes from stretching bonds (energy). A striking consequence is that a rubber's retractive force increases as you heat it, because the entropic term scales with temperature (G ∝ T), whereas an ordinary solid softens on heating.
What makes a polymer a good elastomer?
Four things together: the polymer must be amorphous (no crystallites to lock the chains), have a flexible backbone so segments can move, have a glass transition temperature well below the use temperature (so it is rubbery, not glassy, in service), and be lightly crosslinked. The crosslinks are essential — without them the chains would simply flow past each other and the material would not recover — but they must be sparse, so local chain motion and large recoverable extension are still possible. Vulcanization (sparse sulfur crosslinks) is the classic way to achieve this balance.
How does crosslink density change the stiffness of a rubber?
Through the modulus G = ρRT/Mc. More crosslinks means a smaller molar mass Mc between them, which raises G and makes the rubber stiffer and less extensible; fewer crosslinks means a larger Mc, a lower modulus and a softer, stretchier rubber. This is why the amount of sulfur in vulcanization is a design lever: too little and the material flows, too much and it becomes hard (in the extreme, ebonite). Equilibrium swelling in a solvent falls as crosslink density rises, which gives an experimental route to Mc.
How is rubber elasticity examined in MATS3004?
As a calculation using σ = G(λ − 1/λ²) and G = ρRT/Mc — finding the modulus from network parameters and the stress at a given stretch ratio — and as conceptual questions on the entropic origin of the restoring force and the requirements for a good elastomer. Watch units (keep ρ and Mc mass-consistent) and remember the temperature dependence G ∝ T. Confirm any provided constants and the assessed scope on the UNSW course outline / Moodle.
Exam move
Fix the two rubber-elasticity relations firmly: the modulus G = ρRT/Mc and the non-linear stress law σ = G(λ − 1/λ²), and practise chaining them (parameters → G → σ at a given λ) with a units check that lands you in pascals. Understand the physics well enough to explain the entropy-spring picture in two sentences and to state the counter-intuitive temperature dependence (force rises with T, G ∝ T). Keep the good-elastomer checklist (amorphous, flexible backbone, Tg below use temperature, light crosslinking) and know vulcanization as the sulfur-crosslinking route. Be able to place an elastomer against a plastic and a fibre on a stress-strain diagram (low modulus, very high recoverable strain). Confirm the examinable formulas and constants on the UNSW course outline / Moodle.
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