Polymer Viscoelasticity Chapter 10 (Sperling)
• Stress Relaxation and Creep (sections 1 and 2) – – – –
Chemical versus Physical Processes Analysis with Springs and Dashpots Relaxation and Retardation Times Physical Aging of Glassy Polymers
• Time-Temperature Superposition Principle (sections 3 and 4) – Method of Superposition – WLF Equation and Application
Stress Relaxation and Creep Reasons for Stress Relaxation and Creep Chain Scission: Oxidation, Hydrolysis.
Bond Interchange: Polyester, Polysiloxane Polyamides, Polyethers.
Stress Relaxation and Creep Reasons for Stress Relaxation and Creep Segmental Relaxations: (especially around Tg, but also to a smaller extent, secondary relaxations)
Thirion Relaxation
Viscous Flow (slippage of chains past one another, fast reptation)
Remember, all physical reasons for stress relaxation and creep are related to molecular motion (strain) induced by stress, therefore, must be related to conformational changes.
Stress Relaxation and Creep • Analytical Models for Stress Relaxation and Creep – Using (a) Springs (purely elastic) and (b) Dashpots (purely viscous) For a spring σ = E ε For a dashpot σ = η dε/dt
(Hooke’s Law) (Newton’s Law)
Springs store energy and respond instantaneously. Dashpots dissipate energy in the form of heat, characterizes retarded nature of response. Spring
Dashpot
Creep Experiments • Maxwell Model – Developed to explain the behavior of pitch and tar. – Maxwell assumed that such materials can undergo viscous flow and also respond elastically. Combine Hooke’s and Newton’s laws and assume that the strains are additive (both elements feel the same stress).
Serial Combination
ε = εelast + εvisc Assume the stress is constant. ε(t) = σ/G + σt/η This is a creep experiment. Note, that the first term is the instantaneous elastic response, while the second term is the viscous retarded response.
Creep Experiments • Kelvin-Voigt Model Material can undergo viscous flow but can also respond elastically. Combine Hooke’s and Newton’s laws, assuming that the same strain is felt by both elements and that the stresses are additive. σ= σelast + σvisc σ = ε Ε + η dε/dt Assume overall stress is constant ε(t) = σ0 J [1 - exp(-t /Jη)] This is a Creep Experiment. The strain reaches a limiting value at very long times. Retarded Elastic behavior. τ2 = 1 / ηJ is the Retardation Time
Parallel Combination
Stress Relaxation Experiments Consider a Maxwell Element Apply an instantaneous deformation, ε0 and keep the strain constant, while you record the decaying stress: ε = εelast + εvisc dε/dt = (1/G) dσ/dt + σ/η Since the shear strain is kept constant: dσ/dt = -Gσ/η σ(t) = σ0 exp(-Gt/η)
Stress relaxation modulus G(t) = σ(t) / ε0
This is a Stress Relaxation Experiment. The quantity τ1 = η/G has units of time and is called the Relaxation Time.
Stress Relaxation and Creep Stress-Strain Behaviors Predicted by (a) the Maxwell and (b) the Kelvin-Voigt Models
Stress Relaxation and Creep The Four Element Model The combination of Maxwell and Kelvin elements in series is often used to describe in most simple terms the viscoelastic deformation of polymers. OA: AB: BC: CD: DE:
Instantaneous extension (Maxwell element, E1) Creep Kelvin element Creep Maxwell element Instantaneous recovery (Maxwell element, E1) Delayed recovery (Kelvin element)
Note that the recovery is not complete. One says that there is a permanent set, or a partial recovery.
O
A
C
D
E
Stress Relaxation and Creep Physical Aging Amorphous materials below Tg (glasses) are not equilibrium structures. They physically age !!! (Their properties depend on time.) Aging can be studied with a number of techniques (dilatometry, calorimetry and through creep experiments). Aging Rate ? Superposition ?
Time-Temperature Superposition The Stress relaxation modulus depends on temperature and time. First: Choose an arbitrary reference temperature T0. Second: Shift, along the time axis, the stress relaxation modulus curve recorded at T just above or below T0, so that these two curves superpose partially. Third: Repeat the procedure until all curves have been shifted to be partially superposed with the previous ones. Four: Keep track of aT, the amount a curve recorded at T is shifted. No shift for curve recorded at T0.
Time-Temperature Superposition The shift factor, aT, is defined mathematically by: log10 [aT] = log10 [t] – log10 [t0] = log10 [t/t0]
The temperature dependence of the shift factor, aT, is given by the WLF equation. This relates to the fact that there is a specific relationship between the times (t and t0) and the temperatures (T and T0), for which E(T, t) = E(T0, t0). If the reference temperature, T0, is chosen to be Tg, then we can write: log10 [aT] = -[C1 (T-Tg)]/[C2+T-Tg]
Time-Temperature Superposition Similar master curves can be obtained with dielectric data. When superposition does not work, it implies that more than one relaxation mechanism is at work in the temperature range investigated. The system is said to be rheologically complex (as opposed to simple).
Time-Temperature Superposition Applications: Prediction of viscosity at different temperatures Prediction of mechanical properties: E(T,t), G(T,t), D(T,t), J(T,t) Accelerated Aging Issues: The WLF parameters , C1g, C2g depend slightly on the measurement technique. WHY ? WLF equation works better for intrapolation than for extrapolation.