Interpreting DMA Curves, Part 2

Introduction 

Part 1 of this series of articles (UserCom15) covers non-isothermal DMA measurements and the dependence of the mechanical modulus on temperature.

This second article deals with the frequency dependence of the mechanical properties and quantities of stable samples. Because of the enormous scope of this field, only the basic principles and general rules that explain the behavior of materials are discussed.

In practice, materials are subjected to stresses at many different frequencies. It is therefore extremely important to have a detailed understanding of the effect of frequency on mechanical properties. In addition, it means that materials need to have different properties under different conditions. For example, an adhesive should behave elastically without breaking when it suffers a blow (high frequencies), but should at the same time be able to “accommodate” stress arising from temperature fluctuations (low frequencies) like a liquid.

Complex Modulus and Compliance

An ideal elastic solid

In shear mode, DMA measures the shear modulus, G*, and the shear compliance, J*. An ideal elastic material stores the entire mechanical energy responsible for the deformation. When the shear stress is removed, this energy is liberated. The modulus is independent of frequency; stress and deformation (strain) are in phase. In this situation, G* = G′, whereby G′ is known as the storage modulus. An example that illustrates this behavior is that of a spring (Fig. 1a).

An ideal viscous liquid 

In an ideal liquid, the applied stress and the strain are phase-shifted by 90°. Because the molecules are free to move, no mechanical energy is stored in the material - the energy is completely converted to heat. The corresponding model is the damping device shown (Fig. 1b). Typically, a liquid is described with the frequency-independent viscosity η0. In the case of the shear modulus, G*(ω) = iG″(ω). The imaginary number, i = √-1, is a mathematical expression that represents the fact that the mechanical energy is dissipated, i.e. converted to heat. G″ is the loss modulus, where G″ = ωη0. In a liquid, the loss modulus therefore increases linearly with the frequency, f, where f = ω/2π.

Viscoelastic materials

In real materials, the reaction to an external stress is accompanied by molecular rearrangements that take place over a wide frequency range. Examples of this are lattice vibrations in solids at about 1014 Hz and cooperative rearrangements at the glass transition at about 10-2 Hz. Molecular rearrangements are the reason for the different relaxation processes. At higher temperatures, the frequency of the molecular rearrangements increases. The resulting temperature dependence of the relaxation process is discussed in another article in this publication (UserCom16, page 10). The properties of real materials lie between those of an ideal liquid and an ideal solid. Due to the fact that they have both elastic and viscous properties, the materials are said to be viscoelastic. Their behavior is described mathematically by the complex, frequency-dependent modulus G*(w)= G'(w) +iG" (w) (1) where G′ is the elastic part of the modulus and G″ the energy dissipation part (viscous component). Technical models that illustrate viscoelastic behavior are combinations of springs and damping devices (Fig. 1c).

The complex compliance 

The G* modulus describes the relaxation of the mechanical stress for a given strain. In everyday language, a material with a larger storage modulus is said to be “harder”.

Conclusions 

Knowledge of the frequency dependence of mechanical behavior is of great value for the practical application of materials and for material optimization. Information is obtained for material optimization because the frequencies at which the different processes occur correlate with the characteristic volumes of the corresponding molecular regions. At higher frequencies, smaller molecular regions are observed.

In comparison to temperature-dependent measurements, frequency-dependent measurements provide additional information about material properties in general and on molecular processes in particular

_{Interpreting DMA curves, Part 2 | Thermal Analysis Application No. UC 161 | Application published in METTLER TOLEDO Thermal Analysis UserCom 16}