Material Functions Part 3 Introduction to the Rheology of Complex Fluids Dr Aldo Acevedo - ERC...

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Material Functions Part 3 Introduction to the Rheology of Complex Fluids Dr Aldo Acevedo - ERC SOPS 1

Transcript of Material Functions Part 3 Introduction to the Rheology of Complex Fluids Dr Aldo Acevedo - ERC...

Material Functions

Part 3

Introduction to the Rheology of Complex Fluids

Dr Aldo Acevedo - ERC SOPS 1

To find constitutive equations, experiments are performed on materials using standard flows.

Numerous standard flows may be

constructed from the two sets of flows, by varying the functions σ(t) and ε(t) (and b)

stress responses → materials & type of flow

timestrain

strain rate (or other kinematic parameters)chemical nature of the material

functions of the kinematic parameters that characterize the rheological behavior are

material functionsDr Aldo Acevedo - ERC SOPS 2

Material Functions

Definitions of material functions consist of three parts:

1. Choice of flow type

2. Details of the σ(t) and ε(t) (and b) that appear in the definition of the flows.

3. Material function definitions – based on the measured stress quantities

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Material Functions

To predict them: Use the kinematics and const. eq. to predict the stress

components Calculate the material functions

To measure them:1. Impose the kinematics on material in a flow cell2. Measure the stress components

To choose a constitutive equation to describe a material, we need both to measure the material function and to predict it.

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Material Functions for Shear Flow

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Steady Shear

Kinematics for steady shear 0)( t

constant

Produced in a rheometer by:1. Forcing the fluid through a capillary at a constant rate and

the steady pressure required to maintain the flow is measured.

2. Using cone-and-plate and parallel-plate geometry, rotate at constant angular velocity while measuring the torque generated by the fluid.

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Steady-State Shear

0

21)(

20

332220

22

)()(

N

20

221120

11

)()(

N

Viscosity

For S.S., the stress tensor is constant in time, and the three stress quantities are measured.

The three material functions that are defined are:

First normal-stress coefficient

Second normal stress coefficient

Either + or -, depending on the flow direction and the choice of coordinate system.

Zero-shear viscosity 00 )(lim

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Steady-State Shear

Either + or -, depending on the flow direction and the choice of coordinate system.

shear rate strain stress

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Unsteady Shear

Made in the same geometries as steady shear.

Measured pressures and torques are functions of time.

Many types of time-dependent shear flows: Shear-stress growth Shear-stress relaxation/decay Shear creep Step shear strain Small-amplitude oscillatory shear

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Shear-Stress Growth

Before steady-state is reached, there is a start-up portion of the experiment in which the stress grows from its zero at-rest value to the steady-state value.

This start-up experiment is one time-dependent shear flow experiment

0

00)(

0 t

tt

may be positive or negative

Kinematics for shear-stress growth

no flow initially

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Shear-Stress Growth

0

21),(

t

20

332220

22

)(),(

Nt

Viscosity

The three material functions that are defined are:

20

221120

11

)(),(

NtFirst normal-

stress growth coefficient

Second normal-stress growth

coefficient

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Shear-Stress Growth

)(),(lim tt

)(),(lim 22 tt

Viscosity

At steady-state these material functions become steady-state functions:

)(),(lim 11 tt

First normal-stress growth

coefficient

Second normal-stress growth

coefficient

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Shear-Stress Growth

shear rate strain stress

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Shear-Stress Decay

Relaxation properties of non-Newtonian fluids may be obtained by observing how the steady-state stresses in shear flow relax when the flow is stopped.

00

0)( 0

t

tt

may be positive or negative

Kinematics for shear-stress decay

no flow initially

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Shear-Stress Decay

0

21),(

t

20

332220

22

)(),(

Nt

Viscosity

The three material functions are analogous to stress growth and are defined as:

20

221120

11

)(),(

NtFirst normal-

stress decay coefficient

Second normal-stress decay

coefficient

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Shear-Stress Decay

shear rate strain stress

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Shear-Stress Decay

Newtonian fluids relax instantaneously when the flows stops.

For many Non-Newtonian fluids relaxation takes a finite amount of time.

The time that characterizes a material’s stress relaxation after deformation is called the relaxation time, λ.

A dimensionless number that is used to characterize the importance of λ is the Deborah number De

flowtDe

material relaxation time

flow time scale

Deborah number can help predict the response of a system to a particular deformation.

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Deborah Number – An unusual interpretation

The prophetess Deborah said:

“The mountains flowed before the Lord” (Judges 5:5)

The interpretation of Prof. Markus Reiner:

Deborah knew two things:

1. Mountains flow as everything flows.

2. But they flowed before the Lord, not before man

Reiner’s interpretation: “Man in his short lifetime cannot see them flowing, while the time of observation of God is infinite.”

Thus, even some solids “flow” if they are observed long enough.

Reiner, M. “The Deborah Number,” Physics Today, pp 62, January, 1964.

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Shear Creep

An alternative way of producing steady shear flow is to drive the flow at constant stress.

Constant driving pressure in a capillary flow, or by driving the fixtures with a constant-torque motor.

The unsteady response to shear flow when a constant stress is imposed is necessarily different from the response when a constant strain rate is imposed.

In the constant-stress experiment, the time-dependent deformation of the sample is measured during the transient flow.

The unsteady shear experiment where the stress is held constant is called creep.

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Shear Creep

The material function will prescribe the stress:

0constant

00)(

021 t

tt

Prescribed stress

function for creep

In creep deformation of a sample is measured, that is how the sample changes shape over some time interval as a result of the imposition of the stress.

To do that the concept of strain must be defined.

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Shear Creep

To measure deformation we use the shear strain.

Strain is a measure of the change of the shape of a fluid particle, that is, how much stretching or contracting a fluid experiences.

Shear strain is denoted by γ21(tref, t) Refers to the strain at time t with respect to the shape of the fluid

particle at some other time (i.e. tref) may be abbreviated as γ21(t), where tref = 0

For short time intervals:

where u1 is the displacement function in the 1-direction2

121 ),(

x

uttref

Shear Strain

(small deformations)

21

Shear Creep

Then the displacement function is:

1233

2

1

1233

2

1

)(

)(

)(

)(

)(

)(

)(

)(

tx

tx

tx

tr

tx

tx

tx

tr

ref

ref

ref

ref

)()(),( refref trtrttu

u1 is just the 1-direction component

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Shear Creep

Physical Interpretation:

2

1

2

121 ),(

x

u

x

uttref

)()(),( refref trtrttu

is the slope of the side deformed particle

Thus, strain is related to the change in shape of a fluid particle in the vicinity of points P1 and P2.

23

Shear Creep

where r is the initial particle position and the velocity is 120 exv

1233

2

1

1233

2

201

1233

2

1

)(

)(

)(

)(

)(

)(

)()(

)(

)(

)(

tx

tx

tx

tr

tx

tx

xtttx

tx

tx

tx

r

ref

ref

refref

For steady shear flow over short time intervals, the particle position vector is:

For steady simple shear flow over a short time interval from 0 to t then, we calculate the displacement function and strain:

tx

ut

xttttu refref

02

121

201

),0(

)(),(

Strain in steady-shear over short interval 24

Shear Creep

where tp = pΔt and Δt = t/N.

),)1((...),(...),(),0(),0( 121121 ttNtttttt pp

The deformation in the creep experiment occurs over a long time interval, and the previous equation is for small deformations is not sufficient for calculating strain in this flow.

However, we can break a large strain into a sequence of N smaller strains:

The steady shear-flow displacement function is given for short time intervals by:

tx

utt

xtttu

pp

pp

02

11

2011

),(

),(

Therefore, for each small-strain interval,

independent of timeDr Aldo Acevedo - ERC SOPS 25

For creep, an unsteady flow, the relationship between γ21(0,t) and the measured shear rate is a bit more complicated since the shear rate varies with time.

The displacement function is the same, however is replaced by the

measured time-dependent shear rate function

Shear Creep

Same results as for short time intervals and it is valid in steady-shear flows.

)(21 t

00

1

012121 ),(),0( ttNttt

N

ppp

The total strain over the entire interval from 0 to t is given by:

0

Now we will consider the general case of strain between two times t1 and t2.

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Shear Creep

The strain for each interval is:

)(),( 1212

1121

ppp ttx

utt

212111

1

)(),(

1....2,1,0

xttttu

Nptptt

ppp

p

Break the interval into N pieces of duration Δt:

With varies with time. Thus, for unsteady shear flow, a large strain between times t1 and t2 is given by:

1

0121

1

01212121 )(),(),(

N

pp

N

ppp tttttt

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Shear Creep

This expression for strain is valid in unsteady shear flows such as creep.

In the limit Δt goes to zero

Shear strain in the creep experiment may be obtained by measuring the instantaneous shear rate as a function of time and integrating it over the time interval.

2

1

')'()(lim),( 21

1

012102121

t

t

N

ppt dtttttt

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Shear Creep

0

210

),0(),(

ttJ

In creep because the stress is prescribed rather than measured, the material functions relate the measured sample deformation (strain) to the prescribed stress.

The creep compliance is:

The creep compliance curve has many features, and several other material functions related to it can be defined.

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Shear Creep

)(),()(

state-steady00

t

tJJ s

Steady-state compliance is defined as the difference between the compliance function at a particular time at steady-state and t/η, the steady-flow contribution to the compliance function at that time:

Creep recovery – when the driving stress is removed, elastic and viscoelastic materials will spring back in the opposite direction to the initial flow direction, and the amount of strain that is recovered is called the steady-state recoverable shear strain or recoil strain.

Sample is constrained such that no recovery takes place in the 2-direction.

t

tr tdtt2

)()(

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Shear Creep

Recoil Function

000 ),(lim)(

)(lim

tRR

t

t

rt

),(),(

)(),(

00

00

tJtR

ttJ

r

rr

Recoverable Creep Compliance

Recoverable Shear

Ultimate recoil function

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Shear Creep

Nonrecoverable shear strain due to steady shear flow

For small stresses (i.e. linear viscoelastic limit) the strain at all times is just the sum of the strain that is recoverable and the strain that is not recoverable (due to steady viscous flow at infinite time)

the shear rate attained at s.s. in creep experiment

t

Advantages to creep flow:• more rapid approach to steady-state• Creep-recovery gives important insight into elastic memory effects• Sometimes materials are sensitive to applied stress levels rather than

shear-rate levels• It is straightforward to determine critical stresses

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Step Shear Strain

One of the interesting properties of polymers and other viscoelastic materials is that they have partial memory (stresses that do not relax immediately but rather decay over time) .

The decay is a kind of memory time or relaxation time for the fluid.

To investigate relaxation time, one of the most commonly employed experiments is the step-strain experiment in shear flow.

constant

00

0

00

lim)(

0

00

t

t

t

tKinematics of step shear strain

The limit expresses that the shearing should occur as rapidly as possible. The condition __ relates to the magnitude of the shear strain imposed.

33

Step Shear Strain

Taking the time derivative and applying Leibnitz rule:

As previously discussed:

For step-strain experiment:

0021

0 0

0

21

21

212121

)(

'0''0)(

)(),(

')'(),(2

1

t

dtdtdtt

tdt

ttd

dtttt

t

ref

t

t

It is called the step-strain experiment: this flow involves a fixed strain applied rapidly to a test sample at time t=0.

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Step Shear Strain

Where the function multiplying the strain is an asymmetric impulse or delta function

The prescribed shear-rate function in terms of the strain is:

)()(

1)(

and

0

01

00

lim)(

0

01

00

lim)(

0

-

0

00

tt

dtt

t

t

t

t

t

t

t

t

Thus we can write:

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Step Shear Strain

Heaviside step function

The strain can be expressed:

01

00)(

where

)(),(

0

00)(),(

021

0021

t

ttH

tHt

t

ttdtt

t

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Step Shear Strain

Relaxation modulus

The response of a non-Newtonian fluid to the imposition of a step strain is a rapid increase in shear and normal stresses followed by a relaxation of these stresses.

The material functions are based on the idea of modulus rather than viscosity. Modulus is the ratio of stress to strain and is a concept that is quite useful for elastic materials.

20

33220

20

22110

0

0210

)(),(

)(),(

),(),(

2

1

tG

tG

ttG

First normal-stress step shear relaxation modulus

Second normal-stress step shear relaxation modulus

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Step Shear Strain

The second normal stress … modulus is seldom measured since it is small and requires specialized equipment.

For small strain, the moduli are found to be independent of strain, this limit is called the linear viscoelastic regime.

In the linear viscoelastic regime G(t,strain) is written as G(t), and often high strain data are reported relative to G(t) through the use of a material function called the damping function:

)(

),()( 00 tG

tGh

only reported when it is independent of time.

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Small-Amplitude Oscillatory Shear

One of the most common material functions set.

The flow is again shear flow, and the time-dependent shear-rate function used for this flow is periodic (a cosine function).

tt

xt

v

cos)(

0

0

)(

0

123

2

Kinematics for SAOS

Usually done (but not limited to) in parallel plate or cone-and-plate.

frequency (rad/s)constant amplitude of the shear rate function

39

Small-Amplitude Oscillatory Shear

From the strain, the wall motion required to produce SAOS can be calculated.

Small shear strains can be written as

ttt

tdttdtt

h

tbt

x

u

tt

sinsin),0(

cos)(),0(

)(),0(

00

21

0 00 2121

21

2

121

If b(t) is the time-dependent displacement of the upper plate (for example) and h the gap between the plates

And strain can be calculated from the strain rate:

strain amplitudeDr Aldo Acevedo - ERC SOPS 40

Small-Amplitude Oscillatory Shear

Thus the motion of the wall is:

Moving the wall of a shear cell in a sinusoidal manner does not guarantee that the shear-flow velocity profile will be produced, but one can show that a linear velocity profile will be produced for sufficiently low frequencies or high viscosities.

thtb sin)( 0

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Small-Amplitude Oscillatory Shear

ttt

ttt

tt

cos)sin(sin)cos()(

)cossincos(sin)(

)sin()(

0021

021

021

Expanding using trigonometric identities:

When a sample is strained at low strain amplitudes, the shear stress that is produced will be a sine wave of the same frequency as the input strain wave.

The shear stress usually will not be in phase with the input strain.It can be expressed as:

there is a portion of the stress wave that is in phase with the imposed strain (sen) and a portion of the stress wave that is in phase with the strain rate (cos)

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Newtonian fluids: stress is proportional to shear rate.

For elastic materials: •shear stress is proportional to the imposed strain, that is to the deformation

•similar to mechanical springs (which generate stress that is proportional to the change in length)

Small-Amplitude Oscillatory Shear - Significance

2121 GHooke’s law(shear only)

The stress response generated in SAOS has both a Newtonian-like and an elastic part. Thus, SAOS is ideal for probing viscoelastic materials (i.e. materials that show both viscous and elastic properties) 43

Small-Amplitude Oscillatory Shear

SAOS material functions

sin)(

cos)(

cossin

0

0

0

0

0

21

G

G

tGtG

storage modulus

loss modulus

portion of the stress wave that is in phase with the strain wave divided by the amplitude of the strain wave

portion of the stress wave that is out of phase with the strain wave divided by the amplitude of the strain wave

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Small-Amplitude Oscillatory Shear – The Limits

For a Newtonian fluid, the response is completely in phase with the strain rate:

For an elastic solid that follows Hooke’s law (a Hookean solid) , the shear stress response is completely in phase with the strain.

0

0

GGG

GG

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Small-Amplitude Oscillatory Shear

Several other material functions related to G’ and G” are also used by the rheological community, although they contain no information not already present in the two dynamic moduli already defined.

Table 5.1

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Material Functions for Elongational Flow

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Material Functions for Elongational Flow

Based on the velocity field:

Only the stress differences can be measured.

Stress measurements are very challenging to make in elongational geometries.

In many experiments flow birefringence is used. Flow birefringence is an optical property that is proportional to stress.

Measurements of strain are sometimes made by videotaping a marker particle in the flow and analyzing the images using computer software.

123

3

1

1

)(

)1)((2

1

)1)((2

1

xt

xbt

xbt

v

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Steady Elongation

Steady-state elongational flow is produced by choosing the following

kinematics:

For these flows, constant stress differences are measured. The material functions defined are two elongational viscosities based on the measured normal stress-differences.

For both uniaxial and biaxial extension, the elongational viscosity base on 22-11 is zero for all fluids.

0

11330

0

11330

0

)()(

)()(

constant)(

B

t

uniaxial elongational viscosity

biaxial elongational viscosity

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Steady Elongation

Steady-state elongational flow is difficult to achieve because of the rapid

Rate of particle deformation that is required.

Very few reliable data are available for this important flow.

00 ln

)(),(

l

lt

tdtttt

trefref

The strain in elongational flow is defined as:

Integrating:

Hencky strain

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Elongation Stress Growth

Start-up elongational flow has the same problems as steady elongational flow, but some start-up curves have been reported.

The material functions for the startup of steady elongational analogously to shear flow.

Material for stress decay could be defined, but steady-state is seldom reached. Thus it is not very useful.

0

00)(

)(

)1)((2

1

)1)((2

1

0

123

3

2

1

t

tt

xt

xbt

xbt

v

Kinematics of startup of steady uniaxial elongation

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Elongation Stress Growth

Material functions

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Elongational Creep

If instead of a constant elongational rate, a constant stress is applied to drive the flow, the flow is called elongational creep.

May be obtained by hanging a weight in a cylindrical sample. The deformation of the length (expressed as strain) measured quantity.

The material function is:

00

01133

),0(),(

0constant

00

ttD

t

tKinematics of elongational creep

Elongational creep compliance

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Elongational Creep

An experiment that gives some information about relaxation after elongational deformation is the unconstrained or free recoil experiment.

The material is able to relax in all three directions. The amount contraction that occurs can be expressed as an amount of recoil strain and is an indication of the amount elasticity in the material:

)0(

)(lnl

tlrUltimate recoverable

elongational strain

• L(tinf) is the length of the sample at the time at which the sample is cut free after it has had a chance to relax completely

• L(0) is the length of the sample at the time at which the sample is cut free of the driving mechanism

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Step Elongational Strain

The kinematics are:

constant

0

0

00

lim)(

)(

)1)((2

1

)1)((2

1

00

00

123

3

2

1

e

et

et

t

t

xt

xbt

xbt

v

e

Kinematics of step elongational strain

0 is the magnitude of the elongational strain imposed on the fluid.

short time interval

Dr Aldo Acevedo - ERC SOPS 55

Step Elongational Strain

Uniaxial and biaxial each have one non-zero step elongational modulus,

Planar has two moduli.

By convention , the strain measure is not the simple elongational strain e0, but rather the difference between two components of a strain tensor called the Finger strain tensor C-1.

02

001

00

00

21122

111

122

11220

21133

111

133

11330

21133

111

133

11330

21133

111

133

11330

1

)()(),(

)()(),(

)()(),(

)()(),(

eCCtE

eeCCtE

eeCCtE

eeCCtE

P

P

B

Uniaxial step elongational relaxation modulus

Biaxial step elongational relaxation modulus

Planar step elongational relaxation moduli

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Small-Amplitude Oscillatory Elongation

The kinematics are:

tt

xt

xt

xt

v

cos)(

)(

)(2

1

)(2

1

0

123

3

2

1

Kinematics of SAOE

Similar to SAOS, the deformation rate can be calculated as:

tt

ttdttt

sin),0(

sincos),0(

0

0

0

0

Dr Aldo Acevedo - ERC SOPS 57

Small-Amplitude Oscillatory Elongation

The stress tensor is assumed to be of the form:

12311

11

11

200

00

00

For small deformations and small deformation rates, the stresses generated in SAOE flow will be oscillatory functions of time with the same frequency as the input deformation wave.

The stress will be in general out of phase with respect to both deformation and deformation rate.

If we designate d as the phase difference between stress and strain, the 11-component of the stress as:

tt sin)( 011Dr Aldo Acevedo - ERC SOPS 58

Small-Amplitude Oscillatory Elongation

The stress difference on which material functions will be based::

1111111133 32

The material functions for SAOE:

sin3

)(

cos3

)(

cossin

0

0

0

0

0

1133

E

E

tEtESAOE material functions

Elongational storage modulus

Elongational loss modulus

Dr Aldo Acevedo - ERC SOPS 59

Small-Amplitude Oscillatory Elongation

The SAOS is related to the SAOE as:

GE

GE

3

3

33 GEFor a Newtonian fluid:

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