4010Exclusively/pdf/Ch9
Download
Report
Transcript 4010Exclusively/pdf/Ch9
Dynamic-Mechanical Analysis of
Materials (Polymers)
Big Assist: Ioan I. Negulescu
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:
1 d
EM dt M
d
dt
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:
E*=E’ + iE”
I E* I = Peak Stress / Peak Strain
o
STRESS
STRAIN
o
0
/
2 /
STORAGE ( Elastic) MODULUS
I E' I = I E* I cos
t
LOSS MODULUS
I E" I = I E* I sin
Parallel-plate geometry for shearing of viscous
materials (DSR instrument).
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.
Definition of elastic and viscous materials under shear.
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.
DMA of recyclable HDPE. Dependence of tan on . The
transition is at 62C, the transition at -117C.
Dependence of G", G' and tan on for
HDPE at 180C. More elastic at high !
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.
1.0
ure
t
a
r
pe
m
Te
0.8
E'
80
E"
tan
0.6
40
tan
0.4
0.0
15.5oC
tan
-40
E'
E"
Temperature
-80
0.2
0.0
-10
0
10
20
30
40
50
60
Time, min
70
80
90
100
110
Temperature, oC
Continental Carbon
Sample A-97058
DMA of low cryst. poly(lactic acid): Dependence of tan
upon T and for 1st heating run
Tg
0.8
o
62 C
0.6
o
69 C
C rystallization
tan
o
75 C
C rystallization
0.4
1.0 H z
50 H z
Tg
o
66 C
0.2
0.0
PLALC
30
40
50
60
70
80
o
Tem perature, C
90
100
DMA of Low Cryst. Poly(lactic acid). Dependence of E’ on
thermal history. Bottom line – high info. content, little work.
2G
Storage Modulus (Pa)
st
E' @ 10 Hz (1 h)
nd
E' @ 10 Hz (2 h)
st
1 heating
Crystallization
(Stiffening)
1G
Glass
Transition
Tg
T CR
2
nd
heating
10 Hz
PLALC
0
40
50
60
70
80
o
Tem perature, C
90
100