9.2 Viscoelasticity and time-temperature superposition

Viscoelastic Behavior of Polymers

Polymers don't behave like simple solids or simple liquids. They do both at once, and the balance between those two responses depends on time and temperature. This dual nature, called viscoelasticity, is central to predicting how polymers perform under real-world loading conditions.

Viscoelasticity in polymer mechanics

Viscoelasticity is the combination of elastic (reversible) and viscous (irreversible) responses in a material under stress or strain.

• Elastic behavior means deformation is instantaneous and fully recoverable. Think of snapping a rubber band: stretch it, let go, and it returns to its original shape.
• Viscous behavior means deformation is time-dependent and permanent. Think of pouring honey: it flows and doesn't spring back.

Polymers sit somewhere between these two extremes, and where they sit changes depending on how fast you load them and what temperature they're at.

Why does this matter in practice?

• It determines the time-dependent stress-strain response in applications like automotive parts and packaging films.
• It affects processing behavior during extrusion and injection molding, where the melt needs to flow but also hold its shape.
• It governs long-term durability: will a part sag over months of use? Will a seal lose its clamping force?

Creep and stress relaxation

These are the two classic experiments that reveal viscoelastic behavior.

Creep is what happens when you apply a constant stress and watch the strain increase over time. The material keeps deforming slowly, even though the load isn't changing.

• Creep compliance $J(t)$ is the ratio of strain to stress as a function of time. A higher $J(t)$ means the material deforms more easily. A practical example: an automotive dashboard slowly sagging under its own weight on a hot day.
• Creep recovery is the partial bounce-back you see after the stress is removed. A memory foam pillow compresses under your head, then gradually returns toward its original shape, but not all the way, because some of that deformation was viscous (permanent).

Stress relaxation is the opposite setup: you impose a constant strain and watch the stress decay over time. The material "relaxes" because polymer chains slide past each other and rearrange.

• Relaxation modulus $E(t)$ is the ratio of stress to strain as a function of time. A dropping $E(t)$ means the material is losing its resistance. A classic example: a bolt tightened against a polymer gasket gradually loses preload as the gasket relaxes.

Dynamic mechanical analysis (DMA) probes viscoelastic response by applying small oscillatory (sinusoidal) deformations and measuring the material's response at different frequencies and temperatures.

• Storage modulus $E'$ captures the elastic, energy-storing part of the response (solid-like behavior).
• Loss modulus $E''$ captures the viscous, energy-dissipating part (liquid-like behavior).
• Tan delta $\tan \delta = E'' / E'$ gives the ratio of energy lost to energy stored per cycle. A high $\tan \delta$ means the material is a good damper (useful for vibration absorption); a low $\tan \delta$ means it behaves more like a stiff solid.

Time-Temperature Superposition

The TTS principle

Here's a powerful idea: for many polymers, raising the temperature has the same effect on viscoelastic behavior as slowing down the loading rate (or equivalently, waiting longer). This is the time-temperature superposition (TTS) principle.

Why is this useful? Because testing a polymer's behavior over decades of real time is impractical. Instead, you can run short experiments at several different temperatures and then stitch the results together into a single master curve that spans a much wider time (or frequency) range than any single experiment could cover.

How it works, step by step:

1. Run viscoelastic measurements (e.g., DMA) at multiple temperatures, each over the same accessible time or frequency window.
2. Choose a reference temperature (often room temperature or $T_g$).
3. Shift each curve horizontally along the log-time (or log-frequency) axis until they overlap and align into one smooth curve.
4. The amount you shift each curve is the shift factor $a_T$ for that temperature.

The shift factors follow well-known equations:

• WLF equation (Williams-Landel-Ferry): applies above $T_g$, where free volume changes dominate molecular mobility.
• Arrhenius equation: applies below $T_g$ in the glassy state, where thermally activated local motions control the response.

TTS only works for thermorheologically simple materials, meaning all relaxation mechanisms shift by the same factor. Some polymers with multiple phases or complex transitions violate this assumption.

Master curves and viscoelastic regions

A master curve gives you the complete viscoelastic spectrum of a polymer at your chosen reference temperature, plotted over many decades of time or frequency.

Three distinct regions typically appear:

1. Glassy region (short times / high frequencies): The modulus is high and damping is low. Chains don't have time to rearrange, so the material is stiff and rigid.
2. Rubbery plateau (intermediate times / frequencies): The modulus levels off at a much lower value. Entanglements between chains act like temporary crosslinks, giving elastomeric behavior. The plateau modulus is directly related to the entanglement density.
3. Terminal (flow) region (long times / low frequencies): The modulus drops rapidly. Chains have enough time to disentangle and flow past each other, so the polymer behaves like a viscous liquid.

The glass transition shows up as the steep drop between the glassy and rubbery regions, and it's where $\tan \delta$ peaks.

Shift factor trends also carry physical meaning:

• Large shift factors at a given temperature mean the viscoelastic response is highly sensitive to temperature changes.
• Materials with more fractional free volume or lower activation energies for chain motion tend to have larger shift factors, reflecting greater molecular mobility at those conditions.