Dynamic mechanical analysis (abbreviated DMA) is a technique used to study and characterize materials. It is most useful for studying the viscoelastic behavior of polymers. A sinusoidal stress is applied and the strain in the material is measured, allowing one to determine the complex modulus. The temperature of the sample or the frequency of the stress are often varied, leading to variations in the complex modulus; this approach can be used to locate the glass transition temperature of the material, as well as to identify transitions corresponding to other molecular motions.
Theory
Viscoelastic properties of materials
Polymers composed of long molecular chains have unique viscoelastic properties, which combine the characteristics of elastic solids and Newtonian fluids. The classical theory of elasticity describes the mechanical properties of elastic solid where stress is proportional to strain in small deformations. Such response of stress is independent of strain rate. The classical theory of hydrodynamics describes the properties of viscous fluid, for which the response of stress is dependent on strain rate. This solidlike and liquidlike behavior of polymers can be modeled mechanically with combinations of springs and dashpots.
Dynamic moduli of polymers
The viscoelastic property of a polymer is studied by dynamic mechanical analysis where a sinusoidal force (stress σ) is applied to a material and the resulting displacement (strain) is measured. For a perfectly elastic solid, the resulting strain and the stress will be perfectly in phase. For a purely viscous fluid, there will be a 90 degree phase lag of strain with respect to stress. Viscoelastic polymers have the characteristics in between where some phase lag will occur during DMA tests. When the strain is applied and the stress lags behind, the following equations hold:
Stress: Strain:
where
is frequency of strain oscillation, is time, is phase lag between stress and strain.
Consider the purely elastic case, where stress is proportional to strain given by Young's modulus . We have
Now for the purely viscous case, where stress is proportional to strain rate.
The storage modulus measures the stored energy, representing the elastic portion, and the loss modulus measures the energy dissipated as heat, representing the viscous portion. The tensile storage and loss moduli are defined as follows:
Storage modulus: Loss modulus: Phase angle:
Similarly, in the shearing instead of tension case, we also define shear storage and loss moduli, and .
Complex variables can be used to express the moduli and as follows:
where
Derivation of dynamic moduli
Shear stress of a finite element in one direction can be expressed with relaxation modulus and strain rate, integrated over all past times up to the current time . With strain rate and substitution one obtains . Application of the trigonometric addition theorem lead to the expression
with converging integrals, if for , which depend on frequency but not of time. Extension of with trigonometric identity lead to . Comparison of the two equations lead to the definition of and .
Applications
Measuring glass transition temperature
One important application of DMA is measurement of the glass transition temperature of polymers. Amorphous polymers have different glass transition temperatures, above which the material will have rubbery properties instead of glassy behavior and the stiffness of the material will drop dramatically along with a reduction in its viscosity. At the glass transition, the storage modulus decreases dramatically and the loss modulus reaches a maximum. Temperature-sweeping DMA is often used to characterize the glass transition temperature of a material.
Polymer composition
Varying the composition of monomers and cross-linking can add or change the functionality of a polymer that can alter the results obtained from DMA. An example of such changes can be seen by blending ethylene propylene diene monomer (EPDM) with styrene-butadiene rubber (SBR) and different cross-linking or curing systems. Nair et al. abbreviate blends as E0S, E20S, etc., where E0S equals the weight percent of EPDM in the blend and S denotes sulfur as the curing agent.