Transcript Title

Introduction to Viscoelasticity
All viscous liquids deform
continuously under the influence of an
applied stress – They exhibit viscous
behavior.
Solids deform under an applied stress,
but soon reach a position of
equilibrium, in which further
deformation ceases. If the stress is
removed they recover their original
shape – They exhibit elastic behavior.
Viscous fluid
Viscoelastic fluid
Elastic solid
Viscoelastic fluids can exhibit both
viscosity and elasticity, depending on
the conditions.
Polymers display VISCOELASTIC properties
Viscoelasticity
CHEE 390
23.1
Introduction to Viscoelasticity
The response of polymeric
liquids, such as melts and
solutions, to an imposed
stress may resemble the
behavior of a solid or a
liquid, depending on the
situation.
De 
Viscoelasticity
characteristic materialtime C

timescale of thedeformation tS
CHEE 390
23.2
Linear Viscoelasticity
Viscoelasticity is observed in polymer melts and solutions irrespective of
the magnitude and rate of the applied deformation.
– LINEAR viscoelastic behaviour is restricted to mild deformations such that
polymer chains are disturbed from their equilibrium configuration and
entanglements to a negligible extent.
– Under these conditions, stress is LINEARLY proportional to strain, or
s(t) = G(t) go
where s is the stress, G is the modulus and g is the strain.
– Since polymer processing operations very often involve severe
deformations, studies of linear viscoelasticity have little practical utility.
– The value of data acquired under linear viscoelastic conditions lies in its
amenability to fundamental analysis, which leads to useful inferences on
polymer structure (molecular weight distribution, branching) and valuable
insight into the characteristics of polymer flow (relaxation phenomena,
elastic recovery).
Viscoelasticity
CHEE 390
23.3
Dynamic (Oscillatory) Rheometry
Linear viscoelasticity in polymer melts is examined
by dynamic measurements
– Examine the dynamic elasticity as a function of
temperature and/or frequency.
– Impose a small, sinusoidal shear or tensile strain
(linear t-e region) and measure the resulting
stress (or vice versa)
Stress
Strain
Viscoelasticity
CHEE 390
23.4
Dynamic (Oscillatory) Rheometry
A. The ideal elastic solid
A rigid solid incapable of viscous dissipation of energy follows Hooke’s Law,
wherein stress and strain are proportional (s=Ee). Therefore, the imposed
strain function:
e(w)eo sin(wt)
generates the stress response
t(w)Geosin(wt)  to sin(wt)
and the phase angle, d, equals zero.
B. The ideal viscous liquid
A viscous liquid is incapable of storing inputted energy, the result being that
the stress is 90 degrees out of phase with the strain. An input of:
e(w)eo sin(wt)
generates the stress response
t(w)to sin(wt+p/2)
and the phase angle, d, p/2.
Viscoelasticity
CHEE 390
23.5
Dynamic (Oscillatory) Rheometry
Being viscoelastic materials, the dynamic behaviour of polymers is
intermediate between purely elastic and viscous materials.
– We can resolve the response of our material into a component that is inphase with the applied strain, and a component which is 90° out-of-phase
with the applied strain, as shown below:
Viscoelasticity
CHEE 390
23.6
Dynamic (Oscillatory) Rheometry
The dynamic analysis of viscoelastic polymers the static Young’s
modulus is replaced by the complex dynamic modulus:
G* = G’ + i G”
– The storage (in-phase) modulus, G’, reflects the elastic component of
the polymer’s response to the applied strain.
• Reflects the portion of the material’s stress-strain response that is
elastic (stored).
– The loss (out-of-phase) modulus, G”, reflects the viscous component of
the response.
• Reflects the proportion of the material’s stress-strain response that
is viscous (dissipated as heat).
The ratio of the two quantities is the loss tangent, tan d = G”/G’, which
is a function of temperature, frequency and polymer structure.
Viscoelasticity
CHEE 390
23.7
Dynamic (Oscillatory) Rheometry
Zones relevant to
polymer melt
processing
Logarithmic plots of G’ and G” against angular frequency for
uncrosslinked poly(n-octyl methacrylate) at 100°C (above Tg), molecular
weight 3.6x106.
Viscoelasticity
CHEE 390
23.8
Dynamic (Oscillatory) Rheometry: HDPE
Elastic and Loss Moduli (Pa)
10000
1000
1000
100
G'
G"
Dynamic Viscosity (Pa.s)
10000
100000
n*
100
10
0.1
1
10
100
Frequency (rad/s)
Viscoelasticity
CHEE 390
23.9
Simple models of Viscoelasticity
Liquid-like behavior can be described by a Newtonian model, which can
be represented by using a “dashpot” mechanical analog:
t   g
The simplest elastic solid model is the Hookean model, which can be
represented by a “spring” mechanical analog.
tGg
Viscoelasticity
CHEE 390
23.10
Maxwell Model
A simple model of a viscoelastic fluid requires at least two components,
one to describe the elastic component and the other viscous behavior.
One such model is the Maxwell model:
which responds with a stress, t, when deformed by a strain, g:
Viscoelasticity
CHEE 390
23.11
Maxwell Model
The deformation rate of the Maxwell model is equal to the sum of
the individual deformation rates:
γ total  γ dashpot + γ spring
τ τ
γ  +
η G
η
τ + τ  η γ
G
τ + λτ  η γ
  /G (s) is called the
relaxation time
If the mechanical model is suddenly extended to a position and held
.
there (g=const., g=0):
t  t oe - t / 
Viscoelasticity
CHEE 390
Exponential decay
in stress – Stress Relaxation
23.12
Viscoelasticity and Stress Relaxation
Stress relaxation can be measured by shearing the polymer melt in a
viscometer (for example cone-and-plate or parallel plate). If the
.
rotation is suddenly stopped, ie. g=0, the measured stress will not fall
to zero instantaneously, but will decay in an exponential manner.
Relaxation is slower
for Polymer B than for
Polymer A, as a result
of greater elasticity.
These differences may
arise from polymer
microstructure
(molecular weight,
branching).
Viscoelasticity
CHEE 390
23.13
Viscoelasticity and Stress Relaxation
The Maxwell model is
conceptually reasonable, but it
does not fit real data very well.
G(t) 
1
2
Viscoelasticity
3
n
CHEE 390
τ(t)
γo
Instead, we can use the
generalized Maxwell model
23.14
Viscoelasticity and Stress Relaxation
The relaxation of every element is:
ti ( t )  (Gi g o ) e- t / i
The relaxation of the generalized model is:
n
n
t( t )   ti ( t )  g o  Gie-t / i
i 1
i 1
1
2
3
4
t( t )
G( t ) 
  G ie - t / i
go
i 1
N
n
where Gi is a weighting constant or “relaxation
strength” and i the “relaxation time”
Viscoelasticity
CHEE 390
23.15
Viscoelasticity and Dynamic Rheology
The Generalized Maxwell model can also be used to analyze
dynamic oscillatory measurements, by fitting G’ and G” with an
appropriate number of elements, each having a unique relaxation
strength (Gi) and relaxation time (i):
w22i
G( w)   G i
2 2
1
+
w
i
i
w i
G( w)   G i
2 2
1
+
w
i
i
Viscoelasticity
CHEE 390
23.16
Viscoelasticity and Dynamic Rheology
This example illustrates the storage and loss modulus of
uncrosslinked polybutadiene, plotted as a function of oscillation
frequency.
i(s)
Gi(Pa)
8.04x10-3
3.00x105
5.93x10-2
4.83x105
1.46x10-1
2.98x104
7.61x10-1
1.04x102
An adequate representation of G’ and G” as a function of
frequency required four elements, whose Gi and i are tabulated.
Viscoelasticity
CHEE 390
23.17
Dynamic and Stress Relaxation Testing
Recall stress relaxation data from page 23.15 and dynamic rheology from 23.17
G’(w) vs w
G(t) vs t
A is monodisperse with M<Mc; B is monodisperse with M>>Mc and C is
polydisperse
•
•
The information contained in a stress relaxation plot is complementary to that
acquired in a dynamic measurement.
Stress relaxation measurements are used when very low frequencies are
needed to characterize terminal flow behaviour.
Viscoelasticity
CHEE 390
23.18