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{{Short description|Technique used to study & characterize materials}}
{{Infobox chemical analysis
| name = Dynamic mechanical analysis
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| hyphenated =
}}
==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
===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.<ref name="Meyers1999">{{cite book|last=Meyers|first=M.A.|author2=Chawla K.K.|title=Mechanical Behavior of Materials|publisher=Prentice-Hall|year=1999}}</ref> Viscoelastic polymers have the characteristics in between where some [[phase lag]] will occur during DMA tests.<ref name=Meyers1999/> When the strain is applied and the stress lags behind, the following equations hold:<ref name="Meyers1999"/>
*Stress: <math> \sigma = \sigma_0 \sin(t\omega + \delta) \,</math>
*Strain: <math> \varepsilon = \varepsilon_0 \sin(t\omega)</math>
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.
Consider the purely elastic case, where stress is proportional to strain given by [[Young's modulus]] <math>E</math> . We have <br>
<math>
\sigma(t) = E \epsilon(t)
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*Phase angle: <math> \delta = \arctan\frac {E''}{E'} </math>
Similarly, in the shearing instead of tension case, we also define [[Shear modulus|shear storage]] and loss moduli, <math>G'</math> and <math>G''</math>.
Complex variables can be used to express the moduli <math>E^*</math> and <math>G^*</math> as follows:
:<math>E^* = E' + iE'' = \frac {\sigma_0} {\varepsilon_0} e^{i \delta} \,</math>
:<math>G^* = G' + iG'' \,</math>
where
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====Derivation of dynamic moduli====
Shear stress <math>\sigma(t)=\int_{-\infty}^t G(t-t') \dot{\gamma}(t')dt'</math> of a finite element in one direction can be expressed with relaxation modulus <math>G(t-t')</math> and strain rate, integrated over all past times <math>t'</math> up to the current time <math>t</math>. With strain rate <math> \dot{\gamma(t)}=\omega \cdot \gamma_0 \cdot \cos(\omega t)</math>and
:<math>
\frac{\sigma(t)}{\gamma(t)}=\underbrace{[\omega\int_o^{\infty}G(s)\sin(\omega s) ds]}_{\text{shear storage modulus }G'} \sin(\omega t)+\underbrace{[\omega\int_o^{\infty}G(s)\cos(\omega s) ds]}_{\text{shear loss modulus }G''} \cos(\omega t).
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===Measuring glass transition temperature===
One important application of DMA is measurement of the [[Glass transition#Transition temperature Tg|glass transition temperature]] of polymers. Amorphous polymers have different glass transition temperatures, above which the material will have [[rubber]]y 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.[[File:2019-10-17 20 23 45-DMA Reference Measurements Linear Drive - Anton Paar RheoCompass™.png|alt=|thumb|325x325px|Figure 2. Typical DMA thermogram of an amorphous thermoplastic (polycarbonate). Storage Modulus (E’) and Loss Modulus (E’’) and Loss Factor
===Polymer composition===
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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
===Types of analyzers===
There are two main types of DMA analyzers used currently: forced resonance analyzers and free resonance analyzers. Free resonance analyzers measure the free oscillations of damping of the sample being tested by suspending and swinging the sample. A restriction to free resonance analyzers is that it is limited to rod or rectangular shaped samples, but samples that can be woven/braided are also applicable. Forced resonance analyzers are the more common type of analyzers available in instrumentation today. These types of analyzers force the sample to oscillate at a certain frequency and are reliable for performing a temperature sweep.
[[Image:
Analyzers are made for both stress (force) and strain (displacement) control. In strain control, the probe is displaced and the resulting stress of the sample is measured by implementing a force balance transducer, which utilizes different shafts. The advantages of strain control include a better short time response for materials of low viscosity and experiments of stress relaxation are done with relative ease. In stress control, a set force is applied to the sample and several other experimental conditions (temperature, frequency, or time) can be varied. Stress control is typically less expensive than strain control because only one shaft is needed, but this also makes it harder to use. Some advantages of stress control include the fact that the structure of the sample is less likely to be destroyed and longer relaxation times/ longer creep studies can be done with much more ease. Characterizing low viscous materials come at a disadvantage of short time responses that are limited by [[inertia]]. Stress and strain control analyzers give about the same results as long as characterization is within the linear region of the polymer in question. However, stress control lends a more realistic response because polymers have a tendency to resist a load.<ref name="book">{{cite book|last=Menard|first=Kevin P.|title=Dynamic Mechanical Analysis: A Practical Introduction|publisher=CRC Press|year=1999|chapter= 4 |isbn=0-8493-8688-8}}</ref>
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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.
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.
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==External links==
* [https://polymerdatabase.com/polymer%20physics/DMA.html Dynamical Mechanical Analysis] Retrieved May 21, 2019.
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