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