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4.1
RHEOLOGY1,2
.. is the science that deals with the way materials
deform or flow when forces (stresses) are applied to
them.
AND IT IS AIMED TO
1
… build up mathematical models describing how
materials respond to any type of solicitation (forces
or deformations).
2
… build up mathematical models able to establish a
link between materials macroscopic behaviour and
materials micro-nanoscopic structure.
4.2
A
cross section area
STRESS
F
h
F
A
cross section area
F
σ
A
NORMAL STRESS
(N/M2 = Pa)
F
S
h
F
τ
A
SHEAR STRESS
(N/M2 = Pa)
A
cross section area
DEFORMATION
F
h
L0
L
S
h
F
L  L0
ε
L
L
ε  ln  
 L0 
F
LINEAR
STRAIN
HENCKY
STRAIN
S
γ
h
SHEAR STRAIN
4.3
RHEOLOGICAL PROPERTIES
A - ELASTICITY
“ A material is perfectly elastic if it returns to its original shape once the
deforming stress is removed”
HOOKE’s Law (small deformations)
Normal stress
L  L0
σE
 Eε
L
E = Young modulus (Pa)
Shear stress
Incompressible materials
E = 3G
τ  Gγ
[SOLID MATERIAL]
G = shear modulus (Pa)
B - VISCOSITY
“ This property expresses the flowing (continuous deformation) resistance
of a material (liquid) ”
Very often VISCOSITY and DENSITY are used as synonyms but this is WRONG!
EXAMPLE: at T = 25°C and P = 1 atm
HONEY is a fluid showing high viscosity (~ 19 Pa*s) and low density (~1400
Kg/M3)
MERCURY is a fluid showing low viscosity (~ 0.002 Pa*s) and high density (13579
Kg/M3)
WATER: viscosity 0.001 Pa*s, density 1000 Kg/m3
NEWTON Law
d
τ  ηγ  η
dt


LIQUID
MATERIAL
ητ γ
Shear rate
h = viscosity or dynamic viscosity (Pa*s)
n = kinematic viscosity = h/density(m2/s)


η  f  γ, T , structure 


IF h does not depend on share rate, the fluid is said NEWTONIAN
WATER is the typical Newtonian fluid.
On the contrary it can be “SHEAR THINNING”
100
h(pa s)
10
1
Legge di potenza
Powell - Eyring
Cross
Carreau
Bingham
Casson
Herschel
Shangraw
0.1
0.01
0.1
1
10
100
°(s )
-1
1000
10000
100000
… or “SHEAR THICKENING” (opposite behaviour)
Usually h reduces with temperature
Why h depends on liquid structure, shear rate and temperature?
M
M
M
K(T)
M
M
M
K(T)
K(T)
K(T)
K(T)
M
K(T)
Idealised polymer chain
z friction coefficient
C - VISCOELASTICITY
“ A material that does not instantaneously react to a solicitation (stress or
deformation) is said viscoelastic”
LIQUID VISCOEALSTIC
SOLID VISCOEALSTIC
deformation
t
stress
t
deformation
t
stress
t
STRESS
Material behaviour depends on:
SOLVENT MOLECULES
POLYMERIC CHAINS
1
ELASTIC (instantaneous) REACTION
OF MOLECULAR SPRINGS
2
VISCOUS FRICTION AMONG:
- CHAINS-CHAINS
- CHAINS-SOLVENT MOLECULES
D – TIXOTROPY - ANTITIXOTROPY
A material is said TIXOTROPIC when its viscosity decreases with time
being temperature and shear rate constant.
A material is said ANTITIXOTROPIC when its viscosity increases with
time being temperature and shear rate constant.
The reasons for this behaviour is found in the
temporal modification of system structure
EXAMPLE: Water-Coal suspensions
h
In the case of viscoelastic systems,
no structure break up occurs
AT REST: structure
MOTION structure break up
COAL PARTICLE
t
4.4
LINEAR VISCOELASTICITY
THE LINEAR VISCOEALSTIC FIELD OCCURS FOR SMALL
DEFORMATIONS / STRESSES
THIS MEANS THAT MATERIAL STRUCTURE IS NOT ALTERED OR DAMAGED
BY THE IMPOSED DEFORMATION / STRESS
.. consequently, linear viscoelasticty enables us to study the
characteristics of material structure
MAIN RESULTS
Shear stress
Normal stress
τt 
G
γ0
σt 
E
ε0
Et   3Gt 
Shear modulus G does not depend on
the deformation extension 0
Tensile modulus E does not depend on
the deformation extension e0
Incompressible materials
G(t) or E(t) estimation
1) MAXWELL ELEMENT1,2
g
h
 t   γ 0 ge
t

λ
λη g
t

τ
Gt  
 ge λ
γ0
E(t) = 3 G(t)
120
solid
0
l = 10 s
G (Pa) [1 element]
0 is instantaneously
applied
100
80
l =1s
60
40
liquid
20
l = 0.1 s
0
0
1
2
3
t (s)
4
5
6
2) GENERALISED MAXWELL MODEL1,2
g1
g2
h1
g3
h2
g4
h3
g5
h4
h5
0
0 is instantaneoulsy
applied
N
 t   γ 0  g i e

t
λi
λ i  ηi g i
i 1
Gt  
 t 
γ0
N
  gi e
i 1

t
λi
E(t) = 3 G(t)
120
l 1 = 0.22 s g1 = 90 Pa
l 2 = 4.44 s g2 = 9 Pa
l 3 = 88.88 s g3 = 0.9 Pa
G (Pa) [more elements]
100
l =1s
l 4 = 1600 s
80
g4 = 0.1 Pa
60
40
20
0
0
1
2
3
t (s)
4
5
6
SMALL AMPLITUDE OSCILLATORY SHEAR
1.5
w = 1 s-1
g1
g2
g3
g4
w = 10 s-1
1
g5
h1
h2
h3
h4
h5
 / 0
0.5
0
0
1
2
3
4
-0.5
-1
(t) = 0sin(wt)
-1.5
w = 2pf
f = solicitation frequency
t (s)
5
6
7
On the basis of the Boltzmann1 superposition principle, it can be demonstrated
that the stress required to have a sinusoidal deformation (t) is given by:
(t) = 0sin(wt+d)
d(w) = phase shift
(t) = 0*[G’(w)*sin(wt) + G’’(w)*cos(wt)]
(t) = 0sin(wt)
Gd= 0/0=(G’2+G”2)0.5
G’(w) = Gd*cos(d) = storage modulus
G’’(w) = Gd*sen(d) = loss modulus
tg(d)=G”/G’
1.5
/ 0
d3.14
LIQUID
G’≈ 0
G”≈ Gd
1
0.5
d 0.314
0
0
-0.5
-1
1
2
3
4
/ 0
SOLID
G’≈ Gd
G”≈ 0
-1.5
t (s)
5
6
7
According to the generalised Maxwell Model, G’ and G” can be expressed by:
g i λ i ω
G 
2
i 1 1  λ i ω
2
N
'
li = hi/gi
gi λiω
G 
2
i 1 1  λ i ω 
N
"
g1
g2
h1
g3
h2
h3
g4
h4
(t) = 0sin(wt)
g5
h5
In the linear viscoelastic field, oscillatory and relaxation tests lead to the same
information:
g i λ i ω
G 
2
i 1 1  λ i ω
2
N
'
Oscillatory tests
gi λiω
G 
2
i 1 1  λ i ω 
N
"
N
Relaxation tests
G t    g i e
i 1

t
λi
4.5
EXPERIMENTAL1
SHEAR DEFORMATION/STRESS
Rotating plate
Gel
Fixed plate
SHEAR RATE CONTROLLED
SHEAR STRESS CONTROLLED
STRESS SWEEP TEST: constant frequency (1 Hz)
100000
G’(Pa)
(elastic or storage modulus)
Linear viscoelastic range
(t) = 0sin(wt)
w = 2pf
10000
G’’(Pa)
(loss or viscous modulus)
1000
1
10
100
 0 (pa)
1000
10000
FREQUENCY SWEEP TEST: constant stress or deformation
τt   τ 0 sin ωt 
0 = constant;
0.01 Hz ≤ f ≤ 100 Hz
100000
G’ (Pa)
10000
G’’ (Pa)
1000
0.01
0.1
1
10
w (rad/s)
100
1000
n 1
(λ i ω) 2
G'  Ge   g i
;
2
1  (λ i ω)
i 1
λ i  ηi g i
ω λi
G' '   gi
;
2
1  (λ i ω)
i 1
gi
n
hi
(t)
100000
G’ (Pa)
10000
Fitting parameters
gi, hi, n
Black lines: model best fitting
G’’ (Pa)
1000
0.01
0.1
n
1
10
w (rad/s)
100
1000
G   gi
i 1
li+1 =10* li
0th Maxwell element (spring) -------> 1 fitting parameter (ge)
1st Maxwell element
-------> 2 fitting parameters (g1, l1)
2nd Maxwell element
------->1 fitting parameters (g2, l2)
3rd Maxwell element
-------> 1 fitting parameters (g3, l3)
4th Maxwell element
-------> 1 fitting parameters (g4, l4)
0.01
Np
0.001
Np = generalised
Maxwell model fitting
parameters
0.0001
0.00001
0.000001
2
3
4
5
N p*c
6
2
7
8
4.6
FLORY THEORY3
Polymer
Solvent
Polymer
Solvent
≈
Crosslinks
SWELLING EQUILIBRIUM
SOLVENT
mgH2O = msH2O
D=mgH2O - msH2O = 0
D = DM + DE + DI = 0
Mixing Elastic Ions
DE = -RTrx(np/np0)1/3
G
ρx 
RT
 νp 


ν 
 p0 
23
n
G   gi
i 1
rx = crosslink density in the swollen state
np = polymer volume fraction in the swollen state
np0 = polymer volume fraction in the crosslinking state
T = absolute temperature
R = universal gas constant
gi = spring constant of the Maxwell ith element
Comments
1
The use of Flory theory for biopolymer gels, whose
macromolecular characteristics, such as flexibility, are far from
those exhibited by rubbers, has been repeatedly questioned.
2
However, recent results have shown that very stiff biopolymers
might give rise to networks which are suitably described by a
purely entropic approach. This holds when small deformations
are considered, i.e. under linear stress-strain relationship (linear
viscoelastic region)9.
3
G can be determined only inside the linear viscoelastic region.
4.7
EQUIVALENT NETWORK THEORY4
x
x
x
Polymeric chains
SAME CROSS-LINK
DENSITY (rx)
3
4 ξ
1
π  
3  2  ρx NA
REAL NETWORK
TOPOLOGY
ξ  3 6 πρ x N A
EQUIVALENT NETWORK
TOPOLOGY
REFERENCES
1) Lapasin R., Pricl S. Rheology of Industrial
polysaccharides, Theory and Applications. Champan &
Hall, London, 1995.
2) Grassi M., Grassi G. Lapasin R., Colombo I.
Understanding drug release and absorption mechanisms:
a physical and mathematical approach. CRC (Taylor &
Francis Group), Boca Raton, 2007.
3) Flory P.J. Principles of polymer chemistry. Cornell
University Press, Ithaca (NY), 1953.
4) Schurz J. Progress in Polymer Science, 1991, 16 (1),
1991, 1.

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