9.2 Viscoelasticity and time-temperature superposition

Viscoelastic behavior in polymers combines elastic and viscous responses, influencing their mechanical properties and performance. This unique characteristic affects how polymers react to stress and strain over time, impacting their use in various applications and manufacturing processes.

Creep and stress relaxation are key concepts in viscoelasticity. Creep describes how polymers deform under constant stress, while stress relaxation shows how stress decreases under constant strain. These phenomena are crucial for understanding long-term polymer behavior and durability.

Viscoelastic Behavior of Polymers

Viscoelasticity in polymer mechanics

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• Viscoelasticity combines elastic (reversible) and viscous (irreversible) behavior in polymers under applied stress or strain
  • Elastic behavior exhibits instantaneous and fully recoverable deformation (rubber band)
  • Viscous behavior displays time-dependent and permanent deformation (honey)
• Viscoelasticity critically influences polymer mechanical properties and performance
  • Determines time-dependent stress-strain response in various applications (automotive parts, packaging materials)
  • Affects processing and manufacturing of polymer products (extrusion, injection molding)
  • Impacts long-term durability and stability of polymeric materials (creep resistance, stress relaxation)

Creep and stress relaxation concepts

• Creep characterizes time-dependent deformation under constant stress
  • Creep compliance $J(t)$ measures strain-to-stress ratio over time (automotive dashboard sagging)
  • Creep recovery refers to partial deformation reversal after stress removal (memory foam pillow)
• Stress relaxation describes time-dependent stress reduction under constant strain
  • Relaxation modulus $E(t)$ quantifies stress-to-strain ratio over time (bolt preload loss)
  • Reflects stress dissipation due to molecular rearrangements (polymer chain sliding)
• Dynamic mechanical analysis (DMA) probes viscoelastic response under oscillatory loading
  • Storage modulus $E'$ represents elastic, energy-storing component (solid-like behavior)
  • Loss modulus $E''$ indicates viscous, energy-dissipating component (liquid-like behavior)
  • Tan delta $\tan \delta$ gives the ratio of $E''$ to $E'$, signifying damping capacity (vibration absorption)

Time-Temperature Superposition

Time-temperature superposition principle

• Time-temperature superposition (TTS) principle relates viscoelastic behavior at different temperatures by a time scale shift
  • Allows prediction of long-term properties from short-term experiments at various temperatures
  • Constructs master curves by shifting individual viscoelastic curves ($E'$, $E''$, $\tan \delta$) along time or frequency axis
  • Aligns curves to form a single master curve at a reference temperature (room temperature)
• Shift factors $a_T$ quantify the time or frequency scale change at different temperatures relative to the reference
  • Williams-Landel-Ferry (WLF) equation describes shift factors above glass transition temperature $T_g$
  • Arrhenius equation captures shift factors below $T_g$ (glassy state)

Master curves for viscoelastic response

• Master curves provide a complete viscoelastic spectrum over an extended time or frequency range
  • Enable prediction of polymer behavior beyond experimentally accessible time scales (long-term performance)
  • Reveal temperature-dependent viscoelastic response and transitions (glassy to rubbery)
• Master curve interpretation identifies distinct viscoelastic regions
  1. Glassy region at short times or high frequencies with high modulus and low damping (rigid behavior)
  2. Rubbery plateau region with relatively constant modulus, indicating entanglement network (elastomeric behavior)
  3. Terminal region at long times or low frequencies with rapid modulus decrease, signifying viscous flow (liquid-like behavior)
• Shift factors reflect temperature sensitivity of viscoelastic behavior
  • Higher shift factors imply greater temperature dependence (larger time-temperature superposition)
  • Materials with lower activation energy or higher fractional free volume exhibit larger shift factors (increased molecular mobility)