Transcript sfd - Cain Department of Chemical Engineering

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(it)
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 + 22]
E’ = EM 2 2 / [1 + 22] = 0 cos/0
E” = EM   / [1 + 22] = 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"/2f
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  - peaks 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, tan  vs. . The  transition is
at 62C, the  transition at -117C.
Dependence of G", G' and tan on  for
HDPE at 180C. More elastic at high !
Data obtained at 2C/min showing Tg ~ -40C (max. tan)
and a false transition at 15.5C 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
SAOS – Typical “Master Curve” Results
Approximations to Rubbery Region (plateau)
• Using ideal rubber eq. of state:
F = -T (S/L)P,T , can obtain the entropy elasticity eq.
(Staudinger eq.)
 = G0 ( - 1/2) ;  = L/Lu , so  =  - 1
is at plateau , ~  k T/Ne (Russo) . Note that it predicts
little change in G itself in rubber region. Better approxs.
exist (Guth, Mooney-Rivlin eqs.)
-G0
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). E’ vs. 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
Relation between oscillatory, steady shear
(’) = |*()|’ =  - Cox-Merz Rule
-Works best in the “viscous” region – long
t, low 
-Can also extrapolate / interpolate data to
other T’s or other time scales using WLF
theory (viscosities, moduli, relax. times).
Master Curves of Mechanical Properties
• shift in log(t) [=] shift in T
• so: E(t/aT1 , T1) = E(t/aT2, T2)
• T > Tg, aT < 1, shift curve to right (for t)
• T < Tg, aT > 1, shift curve to left (for t)
• A better approximation to the time-T superpos. eq:
• {E/(T )} (t/aT1 , T1) = {Eref/(Tref ref)} (t/aT2, T2)
However, the T  correction is secondary and usually
neglected over small T-ranges or where the WLF
constants are less precisely known.
Example of Master Curve – shift on  axis
Boltzmann Superposition Principle (for “linear”
viscoelasticity)
•  additive in parallel Maxwell elements, moduli also
• Can interconvert using J* = 1/E* = E*/(mod)2
• Since total  is linear WRT applied i, then (for creep):
n
 (t )    i J (t  ui )
i 1
 (u )
 (t )  
J (t  u ) du

u
let a  t  u ; note J ()  0
t
 (t )  J (0)  0 


0
 (t  a )
J ( a )
da
a
“ui” represents the t where the i was applied