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RHEOLOGY1,2
.. is the science that deals with the way materialsdeform or flow when forces (stresses) are applied tothem.
AND IT IS AIMED TO
1
build up mathematical models describing howmaterials respond to any type of solicitation (forcesor deformations).
2
build up mathematical models able to establish alink between materials macroscopic behaviour andmaterials micro-nanoscopic structure.
4.1
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4.2
NORMAL STRESS(N/M2 = Pa)
FA
cross section area
A
F
STRESS F
h
A
cross section area
F
SHEAR STRESS
(N/M2 = Pa)
h
S
AF
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DEFORMATION F
h
A
cross section area
F
SHEAR STRAIN
h
S
h
S
LINEARSTRAIN
L
LL 0
F
L0 L
0
ln
L
L HENCKYSTRAIN
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4.3 RHEOLOGICAL PROPERTIES
A - ELASTICITY
A material is perfectly elastic if it returns to its original shape once the
deforming stress is removed
Normal stress
0 EL
LLE
E = Young modulus (Pa)
Shear stress
G
G = shear modulus (Pa)
HOOKEs Law (small deformations)
Incompressible materialsE = 3G
[SOLID MATERIAL]
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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 = 25C and P = 1 atm
HONEY is a fluid showing high viscosity (~ 19 Pa*s) and low density (~1400Kg/M3)
MERCURY is a fluid showing low viscosity (~ 0.002 Pa*s) and high density (13579Kg/M3)
WATER: viscosity 0.001 Pa*s, density 1000 Kg/m3
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NEWTON Law
td
d
h = viscosity or dynamic viscosity (Pa*s)n = kinematic viscosity = h/density(m2/s)
structureT,,f
Shear rate
LIQUID
MATERIAL
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IF h does not depend on share rate, the fluid is said NEWTONIANWATER is the typical Newtonian fluid.
0.01
0.1
1
10
100
0.1 1 10 100 1000 10000 100000
(s-1
)
(pas)
Legge di potenza
Powell - Eyring
Cross
Carreau
Bingham
Casson
HerschelShangraw
On the contrary it can be SHEAR THINNING
or SHEAR THICKENING (opposite behaviour)
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Usually h reduces with temperature
Why h depends on liquid structure, shear rate and temperature?
friction coefficient
K(T)
K(T)
K(T) K(T)K(T)
K(T)
M
M
MM M
M
M
Idealised polymer chain
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C - VISCOELASTICITY
A material that does not instantaneously react to a solicitation (stress or
deformation) is said viscoelastic
LIQUID VISCOEALSTIC
t
stress
t
deformation
SOLID VISCOEALSTIC
t
stress
t
deformation
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POLYMERIC CHAINS
SOLVENT MOLECULES
STRESS
Material behaviour depends on:
ELASTIC (instantaneous) REACTIONOF MOLECULAR SPRINGS
VISCOUS FRICTION AMONG:- CHAINS-CHAINS- CHAINS-SOLVENT MOLECULES
1
2
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D TIXOTROPY - ANTITIXOTROPY
A material is said TIXOTROPIC when its viscosity decreases with timebeing temperature and shear rate constant.
A material is said ANTITIXOTROPIC when its viscosity increases withtime being temperature and shear rate constant.
The reasons for this behaviour is found in thetemporal modification of system structure
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EXAMPLE: Water-Coal suspensions
t
AT REST: structure
COAL PARTICLE
MOTION structure break up
hIn the case of viscoelastic systems,
no structure break up occurs
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4.4 LINEAR VISCOELASTICITY
THE LINEAR VISCOEALSTIC FIELD OCCURS FOR SMALLDEFORMATIONS / STRESSES
THIS MEANS THAT MATERIAL STRUCTURE IS NOT ALTERED OR DAMAGED
BY THE IMPOSED DEFORMATION / STRESS
.. consequently, linear viscoelasticty enables us to study thecharacteristics of material structure
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MAIN RESULTS
Shear stress
0
tG
Shear modulus Gdoes not depend onthe deformation extension 0
Normal stress 0
tE Tensile modulus Edoes not depend onthe deformation extension 0
tGtE 3 Incompressible materials
G
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G(t) or E(t) estimation
1) MAXWELL ELEMENT1,2
g
h
0
0is instantaneouslyapplied
ggett
0
0
t
getG
E(t) = 3 G(t)
0
20
40
60
80
100
120
0 1 2 3 4 5 6t(s)
G(Pa)[1e
lement]
= 1 s
= 0.1 s
= 10 s
solid
liquid
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2) GENERALISED MAXWELL MODEL1,2
g1
h1
0
0is instantaneoulsyapplied
h2 h3 h4 h5
g2
g3
g4
g5
iii1
i0 i
gegt
N
i
t
E(t) = 3 G(t)
N
i
t
egt
tG
1
i
0
i
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0
20
40
60
80
100
120
0 1 2 3 4 5 6
t(s)
G(Pa)[moreelements] = 1 s
= 0.22 s
= 4.44 s= 88.88 s
= 1600 s
g1 = 90 Pa
g2 = 9 Pag3 = 0.9 Pa
g4 = 0.1 Pa
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SMALL AMPLITUDE OSCILLATORY SHEAR
g1
h1
(t) = 0sin(wt)
h2 h3 h4 h5
g2 g3 g4 g5
w = 2pff = solicitation frequency
-1.5
-1
-0.5
0
0.5
1
1.5
0 1 2 3 4 5 6 7
t(s)
/0
= 1 s-1 = 10 s-1
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On the basis of the Boltzmann1 superposition principle, it can be demonstratedthat the stress required to have a sinusoidal deformation (t) is given by:
(t) = 0sin(wt+d)
(t) = 0*[G(w)*sin(wt) + G(w)*cos(wt)]
d(w) = phase shift
G(w) = Gd*cos(d) = storage modulus
G(w) = Gd*sen(d) = loss modulus
Gd= 0/0=(G2+G2)0.5
tg(d)=G/G
(t) = 0sin( t)
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-1.5
-1
-0.5
0
0.5
1
1.5
0 1 2 3 4 5 6 7
t(s)
0.314
SOLID
G GdG 0
LIQUIDG 0G Gd
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According to the generalised Maxwell Model, G and G can be expressed by:
N
i
gG1
2
i
2
ii'
1
N
i
gG1
2
i
ii"
1 (t) = 0sin(wt)
g1
h1 h2 h3 h4 h5
g2 g3 g4 g5
li = hi/gi
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In the linear viscoelastic field, oscillatory and relaxation tests lead to the sameinformation:
N
i
gG1
2
i
2
ii'
1
N
i
gG1
2
i
ii"
1
N
i
t
egtG1
ii
Oscillatory tests
Relaxation tests
4 5
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4.5 EXPERIMENTAL1
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Rotating plate
Fixed plate
Gel
SHEAR DEFORMATION/STRESS
SHEAR RATE CONTROLLEDSHEAR STRESS CONTROLLED
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STRESS SWEEP TEST: constant frequency (1 Hz)
1000
10000
100000
1 10 100 1000 10000
(pa)
G(Pa)
(elastic or storage modulus)
G(Pa)(loss or viscous modulus)
Linear viscoelastic range
(t) = 0sin(wt)
w = 2pf
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FREQUENCY SWEEP TEST: constant stress or deformation
tt sin 0 0 = constant; 0.01 Hz f 100 Hz
1000
10000
100000
0.01 0.1 1 10 100 1000
(rad/s)
G (Pa)
G (Pa)
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iii
1
12
i
2
i
ie ;)(1
)(' ggGG
n
i
;)(1
''
12
i
ii
n
i
gG
gi
hi
(t)
1000
10000
100000
0.01 0.1 1 10 100 1000
(rad/s)
G (Pa)
G (Pa)
Black lines: model best fitting
Fitting parametersgi, i, n
n
i
gG1
i
l 10* l
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0th Maxwell element (spring) -------> 1 fitting parameter (ge)1st Maxwell element -------> 2 fitting parameters (g1, 1)2nd Maxwell element ------->1 fitting parameters (g
2,
2)
3rd Maxwell element -------> 1 fitting parameters (g3, 3)4th Maxwell element -------> 1 fitting parameters (g4, 4)
li+1 =10* li
0.000001
0.00001
0.0001
0.001
0.01
2 3 4 5 6 7 8
Np*2
Np
Np = generalisedMaxwell model fittingparameters
4 6
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4.6 FLORY THEORY3
Polymer Solvent
Crosslinks
Polymer Solvent
SWELLING EQUILIBRIUM
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SWELLING EQUILIBRIUM
SOLVENT
mgH2O = m
sH2O
D=mgH2O - msH2O = 0
D = DM + DE + DI = 0
Mixing Elastic Ions
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32
p0
p
RT
Gx
rx
= crosslink density in the swollen state
np = polymer volume fraction in the swollen statenp0 = polymer volume fraction in the crosslinking state
T = absolute temperatureR = universal gas constantgi = spring constant of the Maxwell i
th element
DE = -RTrx(np/np0)1/3
n
i
gG1
i
Comments
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Comments
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.
1
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 deformationsare considered, i.e. under linear stress-strain relationship (linear
viscoelastic region)9.
2
G can be determined only inside the linear viscoelastic region.3
4 7
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4.7 EQUIVALENT NETWORK THEORY4
REAL NETWORKTOPOLOGY
SAME CROSS-LINKDENSITY ( x)
EQUIVALENT NETWORKTOPOLOGY
Polymeric chains
Ax
3
1
2
3
4
N
3Ax6 N
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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. CornellUniversity Press, Ithaca (NY), 1953.
4) Schurz J. Progress in Polymer Science, 1991, 16 (1),1991, 1.
REFERENCES
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