Dynamic mechanical analysis: Difference between revisions

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'''Dynamic mechanical analysis''' (abbreviated '''DMA''') is a technique used to study and characterize materials. It is most useful for studying the [[viscoelastic]] behavior of [[polymer|polymers]]. A sinusoidal stress is applied and the strain in the material is measured, allowing one to determine the [[Dynamic modulus|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<ref>{{cite web |url=https://coventivecomposites.com/explainers/dynamic-mechanical-analysis-dma/|accessdate=2018-10-01|title=What is Dynamic Mechanical Analysis (DMA)?|date=22 April 2018 }}</ref> of the material, as well as to identify transitions corresponding to other molecular motions.

==Theory==

===Viscoelastic properties of materials===

[[Image:Dynamic+Tests+Setup+Chem+538.jpg|thumb|325px|Figure 1. A typical DMA tester with grips to hold the sample and an environmental chamber to provide different temperature conditions. A sample is mounted on the grips and the environmental chamber can slide over to enclose the sample.]]

Polymers composed of long molecular chains have unique viscoelastic properties, which combine the characteristics of [[Elasticity (physics)|elastic solid]]s and [[Newtonian fluid]]s. The classical theory of elasticity describes the mechanical properties of elastic solidsolids where stress is proportional to strain in small deformations. Such response ofto stress is independent of [[strain rate]]. The classical theory of hydrodynamics describes the properties of viscous fluid, for which thestress response of stress is dependentdepends on strain rate.<ref name="Ferry1980">{{cite book|last=Ferry|first=J.D.|title=Viscoelastic properties of polymers|publisher=Wiley|year=1980|edition=3}}</ref> This solidlike and liquidlike behaviorbehaviour of polymers can be modeledmodelled mechanically with combinations of springs and dashpots, making for both elastic and viscous behaviour of viscoelastic materials such as bitumen.<ref name="Ferry1991">{{cite journal|last=Ferry|first=J.D|year=1991|title=Some reflections on the early development of polymer dynamics: Viscoelasticity, dielectric dispersion and self-diffusion|doi=10.1021/ma00019a001|journal=Macromolecules|volume=24|issue=19|pages=5237–5245|bibcode = 1991MaMol..24.5237F }}</ref>

===Dynamic moduli of polymers===

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where

:<math> \omega </math> is the frequency of strain oscillation,

:<math>t</math> is time,

:<math> \delta </math> is phase lag between stress and strain.

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*Storage modulus: <math> E' = \frac {\sigma_0} {\varepsilon_0} \cos \delta </math>

*Loss modulus: <math> E'' = -\frac {\sigma_0} {\varepsilon_0} \sin \delta </math>

*Phase angle: <math> \delta = \arctan\frac {E''}{E'} </math>

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[[Image:Schematic of DMA.png|thumb|Figure 3. General schematic of a DMA instrument.]]

The instrumentation of a DMA consists of a [[displacement sensor]] such as a [[linear variable differential transformer]], which measures a change in voltage as a result of the instrument probe moving through a magnetic core, a temperature control system or furnace, a drive motor (a linear motor for probe loading which provides load for the applied force), a drive shaft support and guidance system to act as a guide for the force from the motor to the sample, and sample clamps in order to hold the sample being tested. Depending on what is being measured, samples will be prepared and handled differently. A general schematic of the primary components of a DMA instrument is shown in figure 3.<ref>{{cite web|url=http://www.mse.iastate.edu/research/research-groups/gom/laboratory-facilities/charaterization-lab/dma.html|title=DMA|accessdate=2010-02-02|url-status=dead|archiveurl=https://web.archive.org/web/20100610052549/http://www.mse.iastate.edu/research/research-groups/gom/laboratory-facilities/charaterization-lab/dma.html|archivedate=2010-06-10}}</ref>

===Types of analyzers===

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[[Image:Freq Sweep Chem538.jpg|thumb|325px|Figure 5. A frequency sweep test on Polycarbonate under room temperature (25 °C). Storage Modulus (E’) and Loss Modulus (E’’) were plotted against frequency. The increase of frequency “freezes” the chain movements and a stiffer behavior was observed.]]

A sample can be held to a fixed temperature and can be tested at varying frequency. Peaks in <math>\tan(\delta)</math> and in E’’ with respect to frequency can be associated with the glass transition, which corresponds to the ability of chains to move past each other. Note that thisThis implies that the glass transition is dependent on strain rate in addition to temperature. Secondary transitions may be observed as well.

The [[Maxwell material|Maxwell model]] provides a convenient, if not strictly accurate, description of viscoelastic materials. Applying a sinusoidal stress to a Maxwell model gives: <math> E'' = \frac{E \tau_0 \omega}{\tau_0^2 \omega^2 + 1} ,</math> where <math>\tau_0 = \eta/E</math> is the Maxwell relaxation time. Thus, a peak in E’’ is observed at the frequency <math>1/\tau_0</math>.<ref name="Young" /> A real polymer may have several different relaxation times associated with different molecular motions.