9.5 KiB
| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Dynamic mechanical analysis | 2/3 | https://en.wikipedia.org/wiki/Dynamic_mechanical_analysis | reference | science, encyclopedia | 2026-05-05T10:04:19.735755+00:00 | kb-cron |
σ
(
t
)
γ
0
=
[
ω
∫
o
∞
G
(
s
)
sin
(
ω
s
)
d
s
]
⏟
shear storage modulus
G
′
sin
(
ω
t
)
+
[
ω
∫
o
∞
G
(
s
)
cos
(
ω
s
)
d
s
]
⏟
shear loss modulus
G
″
cos
(
ω
t
)
.
{\displaystyle {\frac {\sigma (t)}{\gamma _{0}}}=\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).\,}
with converging integrals, if
G
(
s
)
→
0
{\displaystyle G(s)\rightarrow 0}
for
s
→
∞
{\displaystyle s\rightarrow \infty }
, which depend on frequency but not of time. Extension of
σ
(
t
)
=
σ
0
⋅
sin
(
ω
⋅
t
+
Δ
φ
)
{\displaystyle \sigma (t)=\sigma _{0}\cdot \sin(\omega \cdot t+\Delta \varphi )}
with trigonometric identity
sin
(
x
±
y
)
=
sin
(
x
)
⋅
cos
(
y
)
±
cos
(
x
)
⋅
sin
(
y
)
{\displaystyle \sin(x\pm y)=\sin(x)\cdot \cos(y)\pm \cos(x)\cdot \sin(y)}
lead to
σ
(
t
)
γ
0
=
σ
0
γ
0
⋅
cos
(
Δ
φ
)
⏟
G
′
⋅
sin
(
ω
⋅
t
)
+
σ
0
γ
0
⋅
sin
(
Δ
φ
)
⏟
G
″
⋅
cos
(
ω
⋅
t
)
{\displaystyle {\frac {\sigma (t)}{\gamma _{0}}}=\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)\,}
. Comparison of the two
σ
(
t
)
γ
0
{\displaystyle {\frac {\sigma (t)}{\gamma _{0}}}}
equations lead to the definition of
G
′
{\displaystyle G'}
and
G
″
{\displaystyle G''}
.
== 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. Increasing the amount of SBR in the blend decreased the storage modulus due to intermolecular and 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. 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. DMA can also be used to effectively evaluate the miscibility of polymers. The E40S blend had a much broader transition with a shoulder instead of a steep drop-off in a storage modulus plot of varying blend ratios, indicating that there are areas that are not homogeneous.
== Instrumentation ==
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.
=== 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.