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{{Short description|Technique used to study & characterize materials}}
{{Infobox chemical analysis
| name = Dynamic mechanical analysis
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| hyphenated =
}}
'''Dynamic mechanical analysis''' (abbreviated '''DMA
==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
*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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<math>
\sigma(t) = K \frac{d\epsilon}{dt}
\implies \sigma_0 \sin{(\omega t + \delta)} =
\implies \delta = \frac{\pi}{2}
</math>
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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.<ref name=Meyers1999/> The tensile storage and loss moduli are defined as follows:
*Storage
*Loss
*Phase
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
:<math>{i}^2 = -1 \,</math>
====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 substitution <math>\xi(t')=t-t'=s </math> one obtains <math>\sigma(t)=\int_{\xi(-\infty)=t-(-\infty)}^{\xi(t)=t-t} G(s) \omega \gamma_0 \cdot \cos(\omega(t-s))(-ds)=\gamma_0\int_0^{\infty} \omega G(s)\cos(\omega(t-s))ds</math>. Application of the trigonometric addition theorem <math>\cos(x \pm y)=\cos(x)\cos(y) \mp \sin(x)\sin(y)</math> lead to the expression
:<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).
\,</math>
with converging integrals, if <math>G(s) \rightarrow 0</math> for <math>s \rightarrow \infty </math>, which depend on frequency but not of time. Extension of <math>\sigma(t)=\sigma_0 \cdot \sin (\omega \cdot t + \Delta \varphi) </math> with trigonometric identity <math> \sin(x \pm y)=\sin(x)\cdot \cos(y) \pm \cos(x)\cdot \sin(y)</math> lead to
:<math> \frac{\sigma(t)}{\gamma(t)}=\underbrace{\frac{\sigma_0}{\gamma_0} \cdot \cos(\Delta \varphi)}_{G'}\cdot \sin (\omega \cdot t)+ \underbrace{\frac{\sigma_0}{\gamma_0} \cdot \sin(\Delta \varphi)}_{G''} \cdot \cos (\omega \cdot t)
\,</math>.
Comparison of the two <math>\frac{\sigma(t)}{\gamma(t)}</math> equations lead to the definition of <math>G'</math> and <math>G''</math>.<ref name="Ferry">{{cite book|last=Ferry|first=J.D.|author2 = Myers, Henry S |year=1961|title=Viscoelastic properties of polymers|publisher=The Electrochemical Society |volume=108 }}</ref>
==Applications==
===Measuring glass transition temperature===
===Polymer composition===
Varying the composition of monomers and [[cross-link]]ing 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
Increasing the amount of SBR in the blend decreased the storage modulus due to [[intermolecular]] and [[Intramolecular force|intramolecular]] interactions that can alter the physical state of the polymer. Within the glassy region, EPDM shows the highest storage modulus due to stronger intermolecular interactions (SBR has more [[steric]] hindrance that makes it less crystalline). In the rubbery region, SBR shows the highest storage modulus resulting from its ability to resist intermolecular slippage.<ref name="Nair" />
When compared to sulfur, the higher storage modulus occurred for blends cured with dicumyl peroxide (DCP) because of the relative strengths of C-C and C-S bonds.
Incorporation of reinforcing fillers into the polymer blends also increases the storage modulus at an expense of limiting the loss tangent peak height.
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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
Stress and strain can be applied via torsional or axial analyzers. Torsional analyzers are mainly used for liquids or melts but can also be implemented for some solid samples since the force is applied in a twisting motion. The instrument can do creep-recovery,
Changing sample geometry and fixtures can make stress and strain analyzers virtually indifferent of one another except at the extreme ends of sample phases, i.e. really fluid or rigid materials. Common geometries and fixtures for axial analyzers include three-point and four-point bending, dual and single cantilever, parallel plate and variants, bulk, extension/tensile, and shear plates and sandwiches. Geometries and fixtures for torsional analyzers consist of parallel plates, cone-and-plate, couette, and torsional beam and braid. In order to utilize DMA to characterize materials, the fact that small dimensional changes can also lead to large inaccuracies in certain tests needs to be addressed. Inertia and shear heating can affect the results of either forced or free resonance analyzers, especially in fluid samples.<ref name="book" />
===Test modes===
Two major kinds of test modes can be used to probe the viscoelastic properties of polymers: temperature sweep and frequency sweep tests. A third, less commonly studied test mode is dynamic
====Temperature sweep====
A common test method involves measuring the complex modulus at low constant frequency while varying the sample temperature. A prominent peak in <math>\tan(\delta)</math> appears at the glass transition temperature of the polymer. Secondary transitions can also be observed, which can be attributed to the temperature-dependent activation of a wide variety of chain motions.<ref name = "Young">{{cite book|last=Young|first=R.J.|author2=P.A. Lovell|title=Introduction to Polymers|publisher=Nelson Thornes|year=1991|edition=2}}</ref> In [[semi-crystalline polymer]]s, separate transitions can be observed for the crystalline and amorphous sections. Similarly, multiple transitions are often found in polymer blends.
For instance, blends of [[polycarbonate]] and poly([[acrylonitrile-butadiene-styrene]]) were studied with the intention of developing a polycarbonate-based material without
====Frequency sweep====
[[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.
====Dynamic
By gradually increasing the amplitude of oscillations, one can perform a dynamic
====Combined sweep====
Because glass transitions and secondary transitions are seen in both frequency studies and temperature studies, there is interest in multidimensional studies, where temperature sweeps are conducted at a variety of frequencies or frequency sweeps are conducted at a variety of temperatures. This sort of study provides a rich characterization of the material, and can lend information about the nature of the molecular motion responsible for the transition.
For instance, studies of [[polystyrene]] (T<sub>g</sub>
==See also==
* [[Maxwell material]]
* [[Standard
* [[Thermomechanical analysis]]
* [[Dielectric
* [[
* [[Electroactive polymers]]
==References==
{{reflist}}
==External links==
* [https://polymerdatabase.com/polymer%20physics/DMA.html Dynamical Mechanical Analysis] Retrieved May 21, 2019.
{{DEFAULTSORT:Dynamic Mechanical Analysis}}
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