Rheology.
Oscillatory Shear measurements:
By subjecting a specimen to an oscillatory stress
(σ) and determining the response, both the elastic
and viscous or damping characteristics can be
obtained.
φ
Rotating or osc. cone
Sample fluid
θ
Fixed plate
The cone is forced into oscillatory shear (angular
frequency ω) or rotation.
30
The sample is placed between the plate and the
cone.
Linear Viscoelasticity.
When oscillatory shear measurements are
performed in the linear viscoelastic regime, the
storage modulus G' (elastic response) and loss
modulus G'' (viscous behavior) are independent of
the strain amplitude.
Viscosity experiments are carried out in the
zero-shear-rate Newtonian plateau (low shear rate).
31
Oscillatory Shear Experiments.
In an oscillatory shear experiment a sample,
which is exposed to a sinusoidal strain (γ) at an
angular frequency of ω will respond with a
gradual approach to a steady sinusoidal stress (σ)
γ = γo sin ωt
(1)
σ = γo ( G′(ω) sin ωt + G′′(ω) cos (ωt))
(2)
From this type of experiment the storage
modulus G′, the loss modulus G′′ and the dynamic
viscosity η′ = G′′/ω can be determined.
32
G''
( measure of damping )
Loss tangent: tan δ =
G'
Theoretical Models for Linear Viscoelasticity
In the linear viscoelastic regime (small strain
values) the viscoelastic properties of the incipient
gel can be described by the gel equation (Winter
and Chambon; J. Rheol. 1986, 30, 367)
(-∞ < t' < t)
t
m(t) =S ∫ (t − t') nγ (t')dt'
-∞
(3)
m = The shear stress
γ(t′) = the rate of deformation of the sample
33
S = the gel strength parameter (depends on the
crosslinking density and the molecular chain
flexibility)
n = the relaxation exponent
For incipient gels G′ and G′′ are expected to obey
power laws in frequency
G′ ~ G′′ ~ ωn
(4)
The gel point of a chemical gel can be
determined by observation of a frequencyindependent value of tan δ versus time or versus
temperature for a thermoreversible gel.
34
An alternative method is to plot against
temperature the “apparent” viscoelastic exponents
n’
and
n’’
obtained
from
the
frequency
dependence of G’ and G’’ at each temperature of
measurement and observing a crossover where
n’=n’’=n.
Rheological features of a chemically cross-linked
system: Aqueous Poly(vinyl alcohol) (PVA) in the
presence of Glutaraldehyde (GA).
35
1.0
n', n"
a
n'
n"
0.8
0.6
0.4
tg = 739min
0.2 c(PVA) = 4%; c(GA) = 9 mM
0.0
700
800
900
1.0
tg = 359min
0.8
n', n"
b
tg = 404min
c(PVA) = 4%; c(GA) = 13 mM
350
400
450
d
c
0.6
0.4
0.2
0.0
tg = 644min
c(PVA) = 4%; c(GA) = 22 mM
300
350
400
c(PVA) = 5%; c(GA) = 9 mM
450
600
700
800
The time of gelation decreases with increasing
polymer
concentration
concentration.
36
and
cross-linker
800
c(GA) = 9 mM
c(GA) = 13 mM
c(GA) = 22 mM
Gel time (min)
700
600
500
400
300
200
100
3
4
5
6
7
8
9
10
11
12
13
Concentration of PVA (% w/w)
The behavior at the gel-point:
At the gel point, the straight lines representing the
frequency dependencies of G′ and G′′ are parallel
(log-log plot) and we may distinguish between
three situations:
37
a) For chemical gels (“strong gels”) one usually
have G′<G′′.
b) For stoichiometrically balanced chemical gels
the dynamic moduli are usually congruent
(G′=G′′).
c) For physical gels we usually have (“weak” gels)
G′>G′′.
38
c(PVA) = 5% t = 644min
g
c(GA) = 9 mM
101
G' n' = 0.62 ± 0.01
G" n" =0.62 ± 0.01
100
10-1
102
c(PVA) = 6%
c(GA) = 22 mM
1
100
tg =304min
G', G" (Pa)
10
n' =0.53 ± 0.01
n" =0.54 ± 0.01
100
10-2
102
103
102
10-1
100
c(PVA) = 10% t =361min
g
c(GA) = 9 mM
n' =0.45 ± 0.01
n" =0.46 ± 0.01
10-2
10-1
100
c(PVA) = 10% t =126min
g
c(GA) = 22 mM
n' =0.43 ± 0.01
n" =0.43 ± 0.01
10-1
100
Frequency (s-1)
The value of the viscoelastic exponent decreases
with increasing polymer concentration and at low
cross-linker concentration it also decreases as the
cross-linker density increases. This trend is
probably due to enhanced entanglement effects.
39
The gel strength parameter increases with
polymer concentration and it rises with crosslinker density at low GA-concentrations.
0.70
a)
0.65
n
0.60
0.55
0.50
c(GA) = 9 mM
c(GA) = 13 mM
c(GA) = 22 mM
0.45
0.40
1000
3
c(GA) = 9 mM
c(GA) = 13 mM
c(GA) = 22 mM
100
S (Pa sn)
10
b)
S ~ c4.7
10
1
3
10
Concentration of PVA (% w/w)
40
Temperature-induced gelation of an EHEC/SDS
sample:
1.0
n´
G´~ω
n´´
G´´~ω
gel point
n', n"
1.5
4 % EHEC
8 mmolal SDS
0.5
n'
n"
30
35
40
45
o
Temperature ( C)
1.0
3
0.5
G', G" (Pa)
tan δ
0.0
10
n' = 0.40 ± 0.01
n" = 0.40 ± 0.01
ω = 0.09 s
2
-1
10
ω = 0.1 s
-1
10
ω = 0.3 s
G'
G"
o
1
T = 36 C
-1
gel point
32
-1
-1
ω = 0.8 s
0
-1
ω = 1.0 s
34
36
o
Temperature ( C)
Macromolecules 31, 1852 (1998)
41
-1
ω = 0.6 s
ω = 0.7 s
10
10
-1
Frequency (s )
0.0
-1
38