VI CURSO DE INTRODUCCION A LA REOLOGÍA Madrid, 8-9 julio 2013€¦ · VI CURSO DE INTRODUCCION A...

VI CURSO DE INTRODUCCION A LA

REOLOGÍA

Madrid, 8-9 julio 2013

Prof. Dr. Críspulo Gallegos

Dpto. Ingeniería Química. Universidad de Huelva &

Institute of Non-Newtonian Fluid Mechanics (UK)

LINEAR VISCOELASTICITY

OUTLINE

1. Introduction

2. The Boltzmann superposition principle

3. Derivative models for the relaxation modulus

4. Relaxation spectrum

5. Small strain material functions

5.1. Stress relaxation

5.2. Creep

5.3. Small amplitude oscillatory shear

6. Some industrial applications

1. INTRODUCTION

Rheology

Solids Emulsions +

Foams Suspensions

Gels

(Polymer) Melts Liquid Crystals Macromolecular

solutions

Food

Detergents

Cosmetics

Paints

Pharmacy

Biological Fluids

Building Materials

Ink

Adhesives

Lubricants

Engineering Materials

Entanglement

in polymers

Structure/network

of colloids

1. INTRODUCTION

1. INTRODUCTION

• Rigid Solid (Euclidean)

• Linear elastic solid (Hookean)

• Non-linear elastic solid

• Viscoelastic

• Non-linear viscous fluid

• Linear viscous fluid (Newtonian)

• Inviscid fluid (Pascalian)

Rheological spectrum of materials

0

tf ,,

0

G

G

1. INTRODUCTION

1. INTRODUCTION

FUNCIONES

VISCOELÁSTICAS

INDEPENDIENTES

DEL ESFUERZO

APLICADO

DESTRUCCIÓN

ESTRUCTURAL

ESFUERZO

APLICADO

MATERIAL

CONSERVACIÓN

DE LA

ESTRUCTURA

> C

< C

VISCOELASTICIDAD

NO LINEAL

VISCOELASTICIDAD

LINEAL

ESTUDIO DE LA

ESTRUCTURA

(DESARROLLO DE

PRODUCTOS)

ESTUDIO DE

OPERACIONES BÁSICAS

(DESARROLLO DE

PROCESOS)

CONTROL DE CALIDAD

DE PRODUCTOS

INDUSTRIALES

1. INTRODUCTION

- Viscoelastic materials possess both viscous and elastic properties

in varying degrees.

- For a viscoelastic material, internal stresses are a function not only

of the instantaneous deformation, but also depend on the whole past

history of deformation. For real materials, the most recent past

history has much more influence (fading memory).

1. INTRODUCTION

- When a viscoelastic material is deformed, thermodynamic forces

immediately begin to operate to restore the minimum-energy state.

Movement from the rest state represents a storage of energy. This

type of energy is the origin of elasticity in many different materials,

i.e. polymer solutions, polymer melts, concentrated suspensions,

concentrated emulsions, asphalts, lubricating greases, etc.

- Linear viscoelasticity is the simplest response of a viscoelastic

material. If a material is submitted to deformations or stresses small

enough so that its rheological functions do not depend on the value

of the deformation or stress, the material response is said to be in the

linear viscoelasticity range.

2. THE BOLTZMANN SUPERPOSITION PRINCIPLE

Let’s consider the function γ(t) as representative of some cause (shear

strain) acting on a given material, and the shear stress, σ(t), the effect

resulting from this cause.

A variation in shear strain, occurring at time t1 will produce a

corresponding effect at some time later, t:

)()()( 11 tttGt

G (t-t1) is the relaxation modulus, which is a property of the material.

It is a function of the time delay between cause and effect.

This influence function is a decreasing function of (t-t1),

representing a “fading memory”, and is independent of the strain

amplitude.


Series of strain increments applied on a given material

(t1)

(t2)

(t3)

(t4) (t5)

TIME

SH

EA

R S

TR

AIN

0

0 t

1

t

4

t

3

t

2

t5


A series of N changes in the shear strain, each occurring at

different time, will all contribute cumulatively to the stress

at some later time:

If the change in strain occurs continuously:

)()()(1

ii

N

itttGt

t

tdttGt )'()'()( ')'()'()(.

dttttGt

t

t’: past time variable running from the infinite past to present time


For a material with a fading memory, there will be a time prior to

which all the strains which have previously occurred will have a

negligible contribution.

Thus, an experiment generally starts at some time (t = 0) when the

material is free of stresses. In this case:

t

tdttGt0

)'()'()(


The Boltzmann superposition principle can be generalized

for any type of deformation that can be applied on the material.

If the shear strain is replaced by the strain tensor, and the

shear stress by the stress tensor:

t

ijij tdttGt )'()'()(

')'()'()(.

dttttGt

t

ijij

3. DERIVATIVE MODELS POR THE RELAXATION MODULUS

The linear response of a given material to any type of deformation

can be predicted if the relaxation modulus has been experimentally

obtained.

It is previously necessary to correlate the evolution of the linear

relaxation modulus with time.

A classical approach to the description of the linear viscoelastic of real

materials is based upon an analogy with the response of combinations of

certain mechanical elements (a spring for elasticity, and a dashpot for

viscosity).

Such models are idealized and purely hypothetical, and are useful for

representing the behaviour of real materials only to the extent that the

observed response of the real material can be approximated by that of

the model.

Viscous flow Elastic deformation

G .

Viscoelasticity

Voigt

model

Maxwell

model

Burgers model

Spring

Dash pot



Maxwell’s model Voigt’s model visel

retG

visel

dt

drel

)/exp()( 0 tGtG

')'(/)'(exp)(.

0 dttttGt

t

ijij

λ: relaxation time (μ/G)

t’: past time variable running from the infinite past to present time


A large number of models can be formulated by combining different

numbers of springs and dashpots together in diverse ways, aiming to

simulate the evolution of the linear relaxation modulus of real

materials.

The Burgers model is a combination of a Maxwell element and a

Voigt element in series:


One of the most useful for viscoelastic fluids is the generalized

Maxwell model, which, in essence, assumes that, instead of a single

relaxation time, the fluid has a response characteristic of a whole

series, or distribution, of relaxation times.

Such a set of values is called a “discrete spectrum” of relaxation

times of the material.


In this case, the shear relaxation modulus is represented by the

following equation:

The linear constitutive equation can be written as:

)/exp()(1

ii

N

itGtG

')'(/)'(exp)(.

dttttGt

t

ijkkij

4. RELAXATION SPECTRUM

Another approach to the quantification of the linear relaxation

modulus is the use of a continuous relaxation spectrum H(λ).

Using it provides a continuous function of relaxation time, λ, rather

than a discrete set.

The relaxation modulus is now defined as:

)(ln/exp()()(

dtHtG

There are several methods to determine the linear relaxation spectrum

of viscoelastic materials from experimental data of different linear

viscoelasticity functions.

5. SMALL STRAIN MATERIAL FUNCTIONS

A number of small strain experiments are used in rheology.

Some of the more common techniques are stress relaxation, creep,

and sinusoidal oscillations.

In the linear viscoelastic region, all small strain experiments must be

related to one another through G(t), or through H(λ).

Different experimental methods are used because they may be more

convenient or better suited for a particular material or because they

provide data over a particular time range.

Furthermore, it is often not easy to transform results from one type

of linear viscoelastic experiment to another, and, consequently,

several functions are often measured.


Method Input Information

Stress relaxation test Const. deformation Material relaxation

Creep test Const. shear stress Deformation, flow

OSC Time test Const. frequency and Monitoring of

const. amplitude chemical reaction

OSC Amplitude sweep Stepwise increasing Network stability,

amplitude linear range

OSC Frequency sweep Stepwise increasing Frequency

frequency

dependence

OSC Temperature sweep const. frequency and Temperature

const. amplitude

dependence (gelation)

5. 1. STRESS RELAXATION

A sample is suddenly deformed

at a given strain, γ0 , and the

resulting stress is measured as

a function of time.

The relaxing stress data can be

used to determine G(t):

c

> c

0

)()(

ttG


GLASS

CROSSLINKING

G0

DILUTE

SOLUTIONS

AND

DISPERSIONS

log G

log t

POLYMERIC

LIQUID

RUBBER

Evolution of the linear relaxation modulus for different materials


Evolution of the linear relaxation modulus for different polymer

Systems. Viscoelastic liquids on left, viscoelastic solids on right


Evolution of the linear and non-linear relaxation modulus for enteral

pudding formulations

10-1

100

101

102

103

100

101

102

103

Strain%

0.8 %

1.0 %

5.0 %

10 %

50 %

100 %

200 %

300 %

400 %

500 %

G(t

)[P

a]

Time [s]

For linear polymers with a broad molecular weight distribution,

highly branched polymers, and many food gels and dispersions,

among others: mcttG )(

5. 2. CREEP

Creep experiments are useful for

certain practical applications

where long times are

involved.

A constant stress, σ0 , is

instantaneously applied on a

sample, and the resulting

strain is recorded versus time:

mainly

elastic

5. 2. CREEP

0.1 1.0 10.0 100.0

Shear Stress ‚ [Pa]

0.001

0.010

0.100

1.000

10.000

100.000

Defo

rma

tio

n

c

> c

ESTIMATION OF THE LINEAR VISCOELASTICITY RANGE

5. 2. CREEP

0 100 200 300 400 500

0

1x10-4

2x10-4

3x10-4

4x10-4

5x10-4

6x10-4

7x10-4

8x10-4

J(t)

(P

a)

time (s)

The strain values obtained

as a function of time can

be used to calculate the

compliance, J(t), as

follows:

0

)()(

ttJ

Typical creep behaviour for viscoelastic fluids

5. 2. CREEP

0 100 200 300 400 500

0

1x10-4

2x10-4

3x10-4

4x10-4

5x10-4

6x10-4

7x10-4

8x10-4

J(t)

(P

a)

time (s)

There is an initial elastic response.

The second region corresponds to

a retarded elastic response.

Finally, a constant shear rate is

reached.

A steady state creep compliance,

Je0, can be defined by

extrapolation of the limiting slope

in this last region to t = 0.

The creep compliance at long

times is given by: 0

0 /)( tJtJ e

5. 2. CREEP

A creep behaviour of a fluid represented by the Maxwell model will

follow the equation:

G

t

GtJ

1)( λ = η/G

At very short time, this behaviour is characterized by an elastic

response, and a purely viscous response at long time (t>> η/G).

5. 2. CREEP

- A creep behaviour represented by the Kelvin-Voigt model will

follow the equation::

τ = η/G (retardation time)

At very short time, this behaviour is characterized by a purely

viscous response.

)/exp(11

)( tG

tJ

5. 2. CREEP

tt

GGtJ )/exp(1

11)(

21

- In the case of the Burgers model:

5. 2. CREEP

- In most cases, a larger number of Kelvin-Voigt elements should be

added in series within the Burgers model, to fit the actual behaviour

of the material. The equation for the creep compliance would result:

tt

Gt

GGtJ .../exp(1

1)/exp(1

11)( 3

3

2

21

0

321

...111

eJGGG

5. 3. SMALL AMPLITUDE OSCILLATORY SHEAR

In this test, the material is submitted to a simple shearing

deformation, by applying a sine-wave-shaped input of stress or strain.

=0

(change of direction)

=0

(change of direction)


phase shift

viscous

response el.

resp.

input

If a sinusoidal strain is applied,

the shear strain as a function of

time would be given by:

)sin()( 0 tt

γ0 is the strain amplitude and ω

is the frequency.

By differentiating, the evolution

of shear rate with time is given

by:

)cos()cos()( 0

.

0

.

wttt is the shear rate amplitude 0

.


phase shift

viscous

response el.

resp.

input The resulting stress is

measured as a function of time

)sin()( 0 tt

σ0 is the stress amplitude and δ

is a phase shift, also known as

loss angle

The amplitude ratio, Gd, defined from the ratio between σ0

and γ0, and the loss angle, δ, are functions of frequency,

although they are independent of strain amplitude in the linear

viscoelasticity range.


The results of an oscillatory shear test can be represented by the

frequency evolution of both the amplitude ratio and the loss angle.

However, they have not direct relationship with the material

functions traditionally used to characterize the linear viscoelasticity

behaviour of a given material.

For this reason, the stress data can also be analyzed by decomposing

the stress wave into two waves of the same frequency, one in phase

with the strain wave (sin ωt), and the other one 90º out of phase with

this wave (cos ωt):

tt cos''sin'''' '

0

''

0tan


This mathematical treatment also suggests two different dynamic

moduli:

0

'

0'

G

0

''

0''

G

elastic, storage, or in-phase modulus

viscous, loss, or out-of-phase modulus

It is obvious that the loss tangent can be rewritten as:

'

''tan

G

G


We can define the complex modulus, G*, as:

00 G

being G’ and G’’ its real and imaginary parts, respectively:

)('')(')( iGGG

From the previous relationships, it is obvious that the amplitude

ratio, Gd, is the magnitude of the complex modulus, G*:

22

0

0 )''()'(* GGGGd

)cos()('')sin()(')( 0 tGtGt


For a purely elastic material, there is no viscous dissipation, no phase

shift, and the loss modulus is zero.

On the contrary, there is no energy storage for a purely viscous

liquid, being the loss storage modulus zero, and the loss angle 90º.


LINEAR VISCOELASTICITY RANGE

The linear viscoelasticity functions,

G’, G” etc., do not depend on shear

stress (or strain)

The linear viscoelasticity range

depends on frequency

↓ω


Another way to interpret the results obtained from small amplitude

oscillatory shear tests is in terms of a sinusoidal strain rate.

Two new material functions are then defined:

)sin()('')cos()(')( 0

.

ttt

''sin'

.

0

0

.

0

''

0 G

'cos''

.

0

0

.

0

'

0 G

The complex viscosity is:

)('')(')( i 22

.

0

0 )''()'(*


The reciprocal of the complex modulus is also defined as an

additional oscillatory material function, the complex compliance, J*:

where the real and imaginary components of the complex

compliance are related to those of the complex modulus by:

)('')(')(*

1)(

iJJ

GJ

)('')('

)(')('

22

GG

GJ

)('')('

)('')(''

22

GG

GJ


If a Maxwell model is used to represent the linear viscoelastic

response of a material submitted to small oscillatory shear, the storage

and loss moduli will be represented by the following equations:

22

22

1)('

GG 221

)(''

GG

At low frequencies, G’’ is much larger than G’, and the material

predominantly shows a viscous response.

At large frequency, the material shows a solid-like behaviour.

The critical crossover frequency, at which G’ = G’’, defines the

characteristic relaxation time of the material.


Typical Maxwellian behaviour in oscillatory

testing, with the crossover point indicated (aqueous

surfactant solutions containing thread-like micelles)


If a generalized Maxwell model is used to represent the linear

viscoelastic response of a material submitted to small oscillatory

shear, the storage and loss moduli will be represented by the

following equations:

N

i i

iiGG

122

22

1)('

N

i i

iiGG

1221

)(''


G‘

G“

||

Frequency sweep tests in oscillatory shear for a biopolymer

solution


Frequency sweep tests in

oscillatory shear for aqueous

xanthan gum solutions, at various

weight percentages

Frequency sweep tests in

oscillatory shear for PMMA (poly

methyl methacrylate), at 185ºC


Oscillatory behaviour of 84

nm PMMA particles

in decalin, at various

effective phase volumes

Oscillatory behaviour of a

nonionic hexagonal

liquid crystalline phase


Oscillatory behaviour of

two toothpastes

Oscillatory behaviour of a

lubricating grease at

room temperature

Unflocculated Emulsions

a) Otsubo-Prud’homme, 1994

b) Tadros, 1993

Weakly Flocculated Emulsions

c) Tadros, 1993

Flocculated Emulsions

d) Partal et al. 1996

e) Franco et al. 1995



0 5 10 15 20 25 30

0

200

400

600

800

1000

1200

Experimental data

Modified Princen Equation

GN

O (

Pa)

d4,3

(m)

)(31

32

0 bd

aG

PRINCEN EQUATION:

G0 = static shear modulus

= interfacial tension

= disperse phase volume fraction

Gc

dN

0

43


The most general small oscillatory shear response of viscoelastic

complex liquid materials is shown below:

G’

G’’

1

2

3

4

5

Frequency, ω

G’

G

’’


Different regions can be noticed:

1) Viscous or terminal region, where G’’ shows larger values than G’.

For frequencies low enough, G’’ varies linearly with frequency.

2) Transition to flow region. The crossover frequency at which G’’ =

G’ is usually denoted as a characteristic relaxation time of the

material.

3) Plateau region, where elastic behaviour predominates. This region

is characterized by higher and almost constant (slight increase

with frequency, actually) values of the storage modulus and a

minimum in the loss modulus at intermediate frequencies.


4) Transition to glassy region. This region is related to high-frequency

relaxation and dissipation mechanisms, in which G’’ rises much

faster than G’. Another crossover frequency is noticed in this

region, which defines another characteristic relaxation time of the

material.

5) Glassy region, which is characterized by larger values of the loss

modulus and a further crossover frequency prior to a final solid-

like material response.


The behaviour of a real system, especially a gel or very high

molecular weight polymer melt, can be simplistically modelled by

dividing it into Maxwell behaviour at low frequencies giving way

to Kelvin-Voigt behaviour at high frequencies:


Temperature dependence of the storage ad loss moduli (at 6.28 rad/s) for: A)

Wheat gluten-based bioplastics, and B) Egg white-based bioplastics.

Temperature sweep tests

COLLOIDAL STRUCTURE FOR BITUMEN

• Bitumen: colloidal dispersion of

asphaltenes micelles peptizided by the

resin molecules.

• At every temperature, the asphaltene

core is insoluble. Peptization is provided

by a shell of resins that surrounds the

asphaltene core.The thickness of this

resin layer is temperature dependent.

•Temperature dependent transition is not

a sol-gel transition (no rubbery-like

plateau in G’). The liquid phase vitrifies

at low temperature (-20ºC).

6. SOME INDUSTRIAL APPLICATIONS


AFM micrographs for

neat bitumen (A), and for

PMBs containing 2 (B),

8 (C) and 10 (D)

wt.% of MDI-PEG,

after a period of curing

of 11 months

A B

C D

Road distresses

Road

Good

Performance

Poor

Performance

Distresses

Loss of

structural

integrity

Destruction

of entire

pavement

Safe and

comfortable

ride over its design life

Problems difficult

to solve

with unmodified bitumen


Road distresses: RUTTING

• Rutting: permanent

deformation due to the repeated

application of traffic loads.

• Rutting is associated with high

temperature and is more

prevalent in hotter

environments.


Road distresses: LOW TEMPERATURE CRACKING

• Low temperature cracking appears

after a single thermal cycle down to

a critical temperature, or after a

series of thermal cycling above the

critical temperature.

• Uniformly spaced cracks initiate at

the pavement surface.


Road distresses: FATIGUE CRACKING

• Fatigue cracking (alligator skin

pattern) is caused by repetitive loading.

• Every cycle of loading develops

accumulated stresses which are not

dissipated completely.

• Starts at the bottom of the road layer

and propagates to the surface.

•Appears in both types of climates.


Tank Asphalt

RTFOT Residue

PAV Residue

Stiffness at 60 s

between 300 and 600 MPa?

m at 60 s

> 0.30?

Rotational Viscosity

135ºC

Viscosity < 3 Pa·s

Dynamic Shear Rheometer

Maximum Pavement Design Temp

G*/sinδ > 1.0 kPa 1/J’’


Maximum Pavement Design Temp

G*/sinδ > 2.2 kPa 1/J’’


Intermed Pavement Design Temp

G*·sinδ < 5.0 kPa G’’

Bending Beam Rheometer

Minimum Design Temperature + 10ºC

Stiffness at 60 s < 300 Mpa

m at 60 s > 0.30

Direct Tension Test

Minimum Design Temperature + 10ºC

strain at failure > 1.0%

pumpability

mixing

rutting

rutting

fatigue cracking

dissipated energy

polymer-modified

nonrecoverable deformation

thermal cracking

YES?

Newtonian Region

Viscoelastic Region

Low-temperature Region

Low-temperature Region

Viscoelastic Region

Viscoelastic Region

Flow Chart for Testing Required to Grade an Asphalt Cement for SUPERPAVETM


Polymer modified bitumen

Bitumen modification by polymers mainly: Reduces its thermal susceptibility

Avoids rutting and cracking

Improves oxidation and ageing resistance

Reduces life cost of pavements


POLYMER MODIFIED BITUMEN:

APPLICATIONS

Physical

modification:

SBS, SBR, EVA,

LDPE, etc.

Reactive (chemical) modification:

• polyethylene-co-methylacrylate

• methyl, ethyl, or butyl acrylate

• isocyanate-terminated polymers

• phenolic resins

Roofing:

Bitumen: 150/200

Polymer: 12-25% w/w

Paving:

Bitumen: 150/200 to 60/70

Polymer: 3-5% w/w


10-6

1x10-4

10-2

100

102

104

106

101

102

103

104

105

106

107

108

109

1x10-4

10-2

100

102

104

10610

-1

100

101

102

103

A Bitumen-Laboratory

Bitumen-pilot plant

CTRMBs

anchor

4-blade

helical

pilot plant

G*

[Pa]

w·aT [rad·s

-1]

B

tan d

w·aT [rad·s

-1]

Master curves of the complex shear modulus (A) and loss tangent (B) for

unmodified bitumen and 9% CTRMBs processed in different devices at 180ºC

(reference temperature: 25ºC; temperature range: -10 – 75ºC)

RECYCLED EVA-BITUMEN

Linear viscoelasticity master curves of unmodified and

modified bitumen at 50ºC, for different EVA concentrations.


10-7

1x10-5

10-3

10-1

101

103

105

10-3

10-2

10-1

100

101

102

103

104

105

106

107

108

109

10-7

1x10-5

10-3

10-1

101

103

105

10-2

10-1

100

101

102

103

104

105

106

107

108

109

Betún 150/200

Tref

=25ºC

Betún fresco

Betún procesado

0.5%

1%

3%

5%

7%

9%

Modelo Maxwell

G'

[Pa]

aT· [rad·s

-1]

Betún 60/70

G' [

Pa]

aT· [rad·s

-1]

260 280 300 320 34010

-5

10-4

10-3

10-2

10-1

100

101

102

103

104

Betún 150/200

Betún procesado

5% Polímero

aT

T [K]

Introduction

Bioplastics

Biopolymers

Plasticizers

Processing

Definition

Definitions of bioplastics according to European Bioplastics (European industrial association):

*Plastics based on renewable resources

*Biodegradable polymers which meet all criteria of

scientifically recognised norms for biodegradability and compostability of plastics and plastic products. In Europe

this is the EN 13432

Bioplastics represent a relatively new class of materials which have much in common with conventional plastics


http://www.european-bioplastics.org/index.php?id=158

Introduction

Bioplastics

Biopolymers


Introduction

Bioplastics

Biopolymers

Plasticizers

Processing

Proteins

Plasticizers

Polysaccharides Lipids

Wheat Gluten

Gliadin and Glutenin

Egg White

Albumen

Dough-like samples – Influence of temperature

Between 100º-135ºC G’ and G’’ increase, fact that may be attributed to protein aggregation

A slight increase in the linear viscoelastic functions above 40ºC

Results: Wheat gluten

Mixing

Thermosetting

Potential

Compression

Moulding

DSC: An exothermic event is found

between 40-60ºC. It would result from

a new rearrangement of protein

molecules (dissociation and unfolding)

20 40 60 80 100 120 14010

4

105

106

G' G''

0.5 G/EW, 10' mixing

G', G

''

(Pa

)

T (º C)

Dough-like samples – Influence of temperature

Up to 50ºC, G’ and G’’ T, reaching similar values G’ and G’’, which indicates an enhancement of the viscous properties of the blend

Above this T, G’ and G’’: this reveals a structural build-up due to protein denaturation and aggregation

Heat-induced gelation of globular proteins:

two-stage process (partial unfolding and

aggregation of unfolded protein molecules)

Results:

Egg white

Mixing

Thermosetting

Potential

Compression

Moulding

Compression Moulding: Influence of protein

nature

Glycerol / Wheat Gluten or Albumen

T (ºC)

Results:

Mechanical

Processing

Thermosetting

Potential

Compression

Moulding

VI CURSO DE INTRODUCCION A LA REOLOGÍA Madrid, 8-9 julio 2013€¦ · VI CURSO DE INTRODUCCION A...

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