M3 Linear Viscoelasticity: Relaxation Spectrum and WLF

Topic

The generalized Maxwell model—a sum of many individual Maxwell elements, each with its own relaxation time and characteristic modulus—describes the polymer's relaxation spectrum (i.e., how its mechanical response depends on time or, equivalently, on the applied frequency). This same relaxation spectrum is utilized by another model in the set—employing a single, simplified Maxwell element—to describe the polymer's stress relaxation under constant strain; it is also this same spectrum that, when decomposed into its elastic energy-storing and energy-dissipating components, defines the loss factor used by yet another model in the set to describe the polymer's hysteresis heating under cyclic loading. The fact that the polymer's relaxation spectrum includes longer relaxation times distributed across a greater number of individual Maxwell elements means that its mechanical response is consequently more strongly dependent on time or the applied frequency.

Measuring the polymer's full relaxation spectrum separately at every temperature of interest is not feasible; this is where the WLF equation comes into play—a principle already anticipated by another model in the set via the time-temperature superposition principle. The WLF equation allows for the construction of a master curve for the polymer—referenced to a single temperature—based on relaxation spectrum measurements taken at various temperatures; this is the same master curve constructed by yet another model in the set using moduli measured via oscillatory rheometry. The ability of the WLF equation to superimpose a wider range of measured temperatures onto this master curve means that the curve consequently encompasses a correspondingly wider range of effective times or frequencies. Consequently, the relaxation spectrum and the master curve used to initiate this model allow that same master curve—once constructed—to predict the polymer's viscoelastic behavior at any temperature other than the reference temperature, without the need to repeat the full relaxation spectrum measurement at each of those temperatures. This same prediction is utilized by another model in the set to estimate the polymer's creep compliance and stress relaxation over decades of practical service, and it also enables the use of the Cox-Merz rule to relate the oscillatory viscosity (described by this relaxation spectrum) to the steady-shear viscosity (described by a different model in the set). The fact that this relaxation spectrum and master curve share the same fundamental physical basis—the generalized Maxwell model used at the outset—ensures that all the other models utilizing them (stress relaxation, loss factor, creep, and the Cox-Merz scaling) describe, each from its own perspective, the same underlying linear viscoelasticity of the polymer.

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