30

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.

The cone is forced into oscillatory shear (angular

frequency ω) or rotation.

Rotating or osc. cone

Fixed plate

φ

θSample fluid

31

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).

32

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.

33

Loss tangent: tan ' ''

( )δ =GG

measure of damping

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)

m(t) =S t t-

t n (t')dt'( ')−∞∫ γ (3)

m = The shear stress

γ(t′) = the rate of deformation of the sample

34

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 frequency-

independent value of tan δ versus time or versus

temperature for a thermoreversible gel.

35

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).

36

300 350 400 4500.0

0.2

0.4

0.6

0.8

1.0c

c(PVA) = 4%; c(GA) = 22 mM

tg = 359min

n', n

"

700 800 9000.0

0.2

0.4

0.6

0.8

1.0a

c(PVA) = 4%; c(GA) = 9 mM

tg = 739min

n' n"

n', n

"

350 400 450

b

c(PVA) = 4%; c(GA) = 13 mM

tg = 404min

600 700 800

d

c(PVA) = 5%; c(GA) = 9 mM

tg = 644min

The time of gelation decreases with increasing

polymer concentration and cross-linker

concentration.

37

3 4 5 6 7 8 9 10 11 12 13100

200

300

400

500

600

700

800

c(GA) = 9 mM c(GA) = 13 mM c(GA) = 22 mM

Gel

tim

e (m

in)

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:

38

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′′.

39

10-1 100102

103

tg =126min

n' =0.43 ± 0.01n" =0.43 ± 0.01

c(PVA) = 10%c(GA) = 22 mM

Frequency (s-1)

10-2 10-1 100100

101

102

tg =304min

n' =0.53 ± 0.01n" =0.54 ± 0.01

c(PVA) = 6%c(GA) = 22 mM

10-2 10-1 100

102

tg =361min

n' =0.45 ± 0.01n" =0.46 ± 0.01

c(PVA) = 10%c(GA) = 9 mM

10-1 100100

101 tg = 644min

n' = 0.62 ± 0.01n" =0.62 ± 0.01

c(PVA) = 5%c(GA) = 9 mM

G'G"

G',

G" (

Pa)

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.

40

The gel strength parameter increases with

polymer concentration and it rises with cross-

linker density at low GA-concentrations.

3 100.40

0.45

0.50

0.55

0.60

0.65

0.70a)

c(GA) = 9 mM c(GA) = 13 mM c(GA) = 22 mM

n

3 101

10

100

1000

S ~ c4.7

b) c(GA) = 9 mM c(GA) = 13 mM c(GA) = 22 mM

S (P

a sn )

Concentration of PVA (% w/w)

41

Temperature-induced gelation of an EHEC/SDS

sample:

32 34 36 380.0

0.5

1.0

1.5

30 35 40 450.0

0.5

1.0

10-1 100101

102

103

G´~ωn´

G´´~ωn´´

gel point

4 % EHEC8 mmolal SDS

ω = 0.09 s-1

ω = 0.1 s-1

ω = 0.3 s-1

ω = 0.6 s-1

ω = 0.7 s-1

ω = 0.8 s-1

ω = 1.0 s-1

tan δ

Temperature (oC)

gel point n' n"

n', n

"

Temperature (oC)

T = 36 oC

n' = 0.40 ± 0.01n" = 0.40 ± 0.01

G' G"G

', G

" (P

a)

Frequency (s-1)

Macromolecules 31, 1852 (1998)