rheology 1,2
DESCRIPTION
4.1. AND IT IS AIMED TO. … build up mathematical models describing how materials respond to any type of solicitation (forces or deformations). 1. … build up mathematical models able to establish a link between materials macroscopic behaviour and materials micro-nanoscopic structure. 2. - PowerPoint PPT PresentationTRANSCRIPT
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.
4.1
4.2
NORMAL STRESS(N/M2 = Pa)
FA
cross section area
A
Fσ
STRESS F
h
Across section area
F
SHEAR STRESS(N/M2 = Pa)
h
S
A
Fτ
DEFORMATION F
h
Across section area
F
SHEAR STRAIN
h
S
h
Sγ
LINEAR STRAIN
L
LL 0ε
F
L0 L
0
lnεL
L HENCKY 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”
Normal stress
εσ 0 EL
LLE
E = Young modulus (Pa)
Shear stress
γτ G
G = shear modulus (Pa)
HOOKE’s Law (small deformations)
Incompressible materialsE = 3G
[SOLID MATERIAL]
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
td
dηγητ
= viscosity or dynamic viscosity (Pa*s) = kinematic viscosity = /density(m2/s)
structureT ,,γfη
Shear rate
γτη LIQUID MATERIAL
IF 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)
(p
a s)
Legge di potenzaPowell - EyringCrossCarreauBinghamCassonHerschelShangraw
On the contrary it can be “SHEAR THINNING”
… or “SHEAR THICKENING” (opposite behaviour)
Usually reduces with temperature
Why 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
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
POLYMERIC CHAINS
SOLVENT MOLECULES
STRESS
Material behaviour depends on:
ELASTIC (instantaneous) REACTION OF MOLECULAR SPRINGS
VISCOUS FRICTION AMONG:- CHAINS-CHAINS- CHAINS-SOLVENT MOLECULES
1
2
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
t
AT REST: structure
COAL PARTICLE
MOTION structure break up
In the case of viscoelastic systems,
no structure break up occurs
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
0γ
τ tG Shear modulus G does not depend on
the deformation extension 0
Normal stress
0ε
σ tE Tensile modulus E does not depend on
the deformation extension 0
tGtE 3 Incompressible materials
G(t) or E(t) estimation
1) MAXWELL ELEMENT1,2
g
0
0 is instantaneously applied
ggett
ηλγ λ0
λ
0γ
τ t
getG
E(t) = 3 G(t)
0
20
40
60
80
100
120
0 1 2 3 4 5 6
t (s)
G(P
a)
[1 e
lem
en
t]
l = 1 s
l = 0.1 s
l = 10 s
solid
liquid
2) GENERALISED MAXWELL MODEL1,2
g1
1
0
0 is instantaneoulsy applied
2 3 4 5
g2 g3 g4 g5
iii1
λi0 ηλγ i gegt
N
i
t
E(t) = 3 G(t)
N
i
t
egt
tG1
λi
0
i
γ
0
20
40
60
80
100
120
0 1 2 3 4 5 6
t (s)
G(P
a)
[mo
re e
lem
en
ts] l= 1 s
l1= 0.22 sl2= 4.44 sl3= 88.88 sl4= 1600 s
g1 = 90 Pa
g2 = 9 Pa
g3 = 0.9 Pa
g4 = 0.1 Pa
SMALL AMPLITUDE OSCILLATORY SHEAR
g1
1
(t) = 0sin(t)
2 3 4 5
g2 g3 g4 g5
= 2ff = 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
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(t+)
(t) = 0*[G’()*sin(t) + G’’()*cos(t)]
() = phase shift
G’() = Gd*cos() = storage modulus
G’’() = Gd*sen() = loss modulus
Gd= 0/0=(G’2+G”2)0.5
tg()=G”/G’
(t) = 0sin(t)
-1.5
-1
-0.5
0
0.5
1
1.5
0 1 2 3 4 5 6 7
t (s)
/0
/0
0.314
3.14
SOLIDG’≈ Gd
G”≈ 0
LIQUIDG’≈ 0G”≈ Gd
According to the generalised Maxwell Model, G’ and G” can be expressed by:
N
i
gG
12
i
2ii'
ωλ1
ωλ
N
i
gG
12
i
ii"
ωλ1
ωλ (t) = 0sin(t)
g1
1 2 3 4 5
g2 g3 g4 g5
li = i/gi
In the linear viscoelastic field, oscillatory and relaxation tests lead to the same information:
N
i
gG
12
i
2ii'
ωλ1
ωλ
N
i
gG
12
i
ii"
ωλ1
ωλ
N
i
t
egtG1
λi
i
Oscillatory tests
Relaxation tests
4.5 EXPERIMENTAL1
Rotating plate
Fixed plate
Gel
SHEAR DEFORMATION/STRESS
SHEAR RATE CONTROLLEDSHEAR STRESS CONTROLLED
STRESS SWEEP TEST: constant frequency (1 Hz)
1000
10000
100000
1 10 100 1000 10000
0(pa)
G’(Pa) (elastic or storage modulus)
G’’(Pa) (loss or viscous modulus)
Linear viscoelastic range
(t) = 0sin(t)
= 2f
FREQUENCY SWEEP TEST: constant stress or deformation
tt ωsinττ 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)
iii
1
12
i
2i
ie ηλ;)ωλ(1
)ωλ(' ggGG
n
i
;)ωλ(1
λω''
12
i
ii
n
i
gG
gi
i
(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
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)
li+1 =10* li
0.000001
0.00001
0.0001
0.001
0.01
2 3 4 5 6 7 8
N p*c2
Np
Np = generalised Maxwell model fitting parameters
4.6 FLORY THEORY3
Polymer Solvent
Crosslinks
≈
Polymer Solvent
SWELLING EQUILIBRIUM
SOLVENT
gH2O = s
H2O
=gH2O - s
H2O = 0
= M + E + I = 0Mixing Elastic Ions
32
p0
p
ν
νρ
RT
Gx
x = crosslink density in the swollen state
p = polymer volume fraction in the swollen statep0 = polymer volume fraction in the crosslinking stateT = absolute temperatureR = universal gas constantgi = spring constant of the Maxwell ith element
E = -RTx(p/p0)1/3
n
i
gG1
i
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 deformations
are 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 EQUIVALENT NETWORK THEORY4
REAL NETWORK TOPOLOGY
SAME CROSS-LINK DENSITY (x)
EQUIVALENT NETWORK TOPOLOGY
Polymeric chains
Ax
3
ρ
1
2
ξπ
3
4
N
3Axπρ6ξ N
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.
REFERENCES