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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