dynamic-mechanical analysis of materials (polymers) big assist: ioan i. negulescu
TRANSCRIPT
Viscoelasticity
According to rheology (the science of flow), viscous flow and elasticity are only two extreme forms of rheology. Other cases: entropic-elastic (or rubber-elastic), viscoelastic; crystalline plastic.
SINGLE MAXWELL ELEMENT(viscoelastic = “visco.”)
All real polymeric materials have viscoelasticity, viscosity and elasticity in varying amounts. When visco. is measured dynamically, there is a phase shift () between the force applied (stress) and the deformation (strain) in response.
The tensile stress and the deformation (strain) for a Maxwellian material:
dt
d
dt
d
E MM
1
Generally, measurements for visco. materials are represented as a complex modulus E* to capture both viscous and elastic behavior:
E* = E’ + iE”
* = 0 exp(i (t + )) ; * = 0 exp(it)
E*2 = E’2 + E”2
It’s solved in complex domain, but only the real parts are used.
In dynamic mechanical analysis (DMA, aka oscillatory shear or viscometry), a sinusoidal or applied.
For visco. materials, lags behind . E.G., solution for a single Maxwell element:
0 = EM 0 / [1 + 22]
E’ = EM 2 2 / [1 + 22] = 0 cos/0
E” = EM / [1 + 22] = 0 sin/0
= M/EM = Maxwellian relax. t
Schematic of stress as a function of t with dynamic (sinusoidal) loading (strain).
COMPLEX MODULUS:
I E' I = I E* I cos I E" I = I E* I sin
LOSS MODULUSSTORAGE ( Elastic) MODULUS
I E* I = Peak Stress / Peak Strain
E*=E’ + iE”
t
0
o
o
2 /
/
STRESS STRAIN
The “E”s (Young’s moduli) can all be replaced with “G”s (rigidity or shear moduli), when appropriate. Therefore:
G* = G’ + iG"where the shearing stress is and the deformation (strain) is . Theory SAME.
In analyzing polymeric materials:
G* = (0)/(0), ~ total stiffness.
In-phase component of IG*I = shear storage modulus G‘ ~ elastic portion of input energy
= G*cos
The out-of-phase component, G" represents the viscous component of G*, the loss of useful mechanical energy as heat
= G*sin = loss modulus
The complex dynamic shear viscosity * is G*/, while the dynamic viscosity is
= G"/ or = G"/2f
For purely elastic materials, the phase angle = 0, for purely viscous materials, 90.
The tan() is an important parameter for describing the viscoelastic properties; it is the ratio of the loss to storage moduli:
tan = G"/ G',
A transition T is detected by a spike in G” or tan(). The trans. T shifts as changes. This phenomenon is based on the time-temperature superposition principle, as in the WLF eq. (aT).
The trans. T as (characteristic t ↓)
E.G., for single Maxwell element:
tan = ( )-1 and W for a full period (2/) is:
W = 02 E” = work
Dynamic mechanical analysis of a viscous polymer solution (Lyocell). Dependence of tan on - due to complex formation.
• DMA very sensitive to T.
• Secondary transitions, observed with difficulty by DSC or DTA, are clear in DMA.
• Any thermal transition in polymers will generate a peak for tan, E“, G“
• But the peak maxima for G" (or E") and tan do not occur at the same T, and the simple Maxwellian formulas seldom followed.
Data obtained at 2C/min showing Tg ~ -40C (max. tan) and a false transition at 15.5C due to the nonlinear
increase of T vs. t.
-10 0 10 20 30 40 50 60 70 80 90 100 110
0.0
0.2
0.4
0.6
0.8
1.0
Continental CarbonSample A-97058
15.5oC
tan
E"E'
Te
mp
era
ture
, o C
80
40
0.0
-40
-80
tan E' E" Temperature
tan
Time, min
DMA of low cryst. poly(lactic acid): Dependence of tan upon T and for 1st heating run
30 40 50 60 70 80 90 100
0.0
0.2
0.4
0.6
0.8
Crystallization
Tg
Crystallization
Tg
PLALC
75oC
69oC
66oC
62oC
tan
Temperature, oC
1.0 Hz 50 Hz
DMA of Low Cryst. Poly(lactic acid). Dependence of E’ on thermal history. Bottom line – high info. content, little work.
40 50 60 70 80 90 1000
1G
2G
(Stiffening)Crystallization
10 HzPLALC
2nd heating
1st heating
TCR
GlassTransitionT
g
E' @ 10 Hz (1st h)
E' @ 10 Hz (2nd h)
Sto
rag
e M
od
ulu
s (
Pa
)
Temperature, oC