Time-Temperature Superposition PDF

24. Time-temperature superposition 24.1 Time-temperature superposition It has been suggested earlier that it may be possible to interrelate the time and temperature dependence of the viscoelastic properties of polymers. It is thought that there is a general equivalence between time and temperature....

24. Time-temperature superposition 24.1 Time-temperature superposition It has been suggested earlier that it may be possible to interrelate the time and temperature dependence of the viscoelastic properties of polymers. It is thought that there is a general equivalence between time and temperature. For instance, a polymer that displays rubbery characteristics under a given set of testing conditions can be induced to show glassy behavior by either reducing the temperature or increasing the testing rate or frequency. This type of behavior is shown in Figure 20.11 for the variation of the shear compliance J of a polymer with testing frequency measured at different temperatures in the region of the Tg. The material is rubber-like with a high compliance at high temperatures and low frequencies, and becomes glassy with a low compliance as the temperature is reduced and the frequency increased. It was found empirically that all the curves in Figure 20.11 could be superposed by keeping one fixed and shifting all the others by different amounts, horizontally parallel to the logarithmic frequency axis. If an arbitrary reference temperature Ts is taken to fix one curve then if ωs is the frequency of a point on the curve at Ts with a particular compliance and ω is the frequency of a point with the same compliance on a curve at a different temperature, then the shift required to superpose the two curves is a displacement of (log ωs−log ω) along the log frequency axis. The ‘shift factor’ aT is defined by 1 Much of the early work upon time–temperature superposition was done by Williams, Landel and Ferry and they proposed that aT could be given by an equation of the form (WLF equation) Where C1 and C2 are constants; Ts is the reference temperature. if Ts is taken as T g, the new constants C1g and C2g become ‘universal’ with values of 17.4 and 51.6 K, respectively. Although the WLF equation was developed originally by curve fitting, it is possible to justify it theoretically from considerations of free volume. The fractional free volume fV in a polymer is given as 𝑓𝑉 = 𝑓𝑔 + (𝑇 − 𝑇𝑔 )𝛼𝑓 For the Maxwell model τ0 =η/E where η is the viscosity of the dashpot and E is modulus of the spring. It can be assumed that E is independent of temperature and that only η varies with temperature. If the shifts in the time–temperature superposition are thought of as a process of matching the relaxation times, then the shift factor is given by 𝜏0 (𝑇) 𝜂(𝑇) 𝑎𝑇 = = 𝜏0 (𝑇𝑔 ) 𝜂(𝑇𝑔 ) 2 when the Tg is used as the reference temperature. It is possible to relate the viscosity to the free volume through a semi-empirical equation developed by Doolittle from the study of the viscosities of liquids. He was able to show that for a liquid, η is related to the free volume Vf through an equation of the form 𝐵(𝑉 − 𝑉𝑓 ) ln 𝜂 = ln 𝐴 + 𝑉𝑓 Where V is the total volume and A, B are constants. 1 ln 𝜂(𝑇) = ln 𝐴 + 𝐵( − 1) 𝑓𝑉 1 ln 𝜂(𝑇𝑔 ) = ln 𝐴 + 𝐵( − 1) 𝑓𝑔 𝜂(𝑇) 1 1 ln = 𝐵( − ) 𝜂(𝑇𝑔 ) 𝑓𝑔 + 𝛼𝑓 (𝑇 − 𝑇𝑔 ) 𝑓𝑔 This equation can be rearranged to give 𝜂(𝑇) −(𝐵/(2.303𝑓𝑔 ))(𝑇 − 𝑇𝑔 ) log = log 𝑎 𝑇 = 𝜂(𝑇𝑔 ) 𝑓𝑔 /𝛼𝑓 + (𝑇 − 𝑇𝑔 ) It is found that fg is of the order of 0.025 and αf approximately 4.8 ×10−4 K−1 for most amorphous polymers. 3