Rheology A short Introduction Dirk van den Ende What is Rheology ?

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UNIVERSITEIT TWENTE. Physics of Complex Fluids

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Rheology A short Introduction

Dirk van den Ende

Dept. Science and Technology University of Twente

·  What is Rheology ·  A bit of continuum mechanics ·  Rheometry / µRheology ·  Structure Rheology


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What is Rheology ?

Rheology is the study of the flow of matter in response to an applied force. It applies to substances which have a complex microstructure, such as muds, sludges, suspensions, polymers and other glass formers (e.g., silicates), as well as many foods and additives, bodily fluids (e.g., blood) or other materials which belong to the class of soft matter.

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•  Quality control A simple practical test will do mostly.

•  Design and control of processes •  Production of materials •  Transport (e.g. pumping) The process should be better understood, more detailed testing is imperative.

•  Search for new materials and/or new applications To tune the properties of the material, one needs understanding of the underlying microscopic processes.

Rheology comes into play during...


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Interesting fluids (from a Rheological perspective)

plastics dairy products, (low-fat) polymer melts margarine, yoghurt, paint cream, salad dressings, bitumen tomato ketchup, emulsions dough, cosmetics, soap

These materials contain rather tall units, like long polymers or particles of (sub-) micron

size, which can interact with each other

We call them: COMPLEX FLUIDS

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Solid: shape preserved Liquid: adapts it shape

Solid, Liquid and in between…

Time scale: •  short times: solid like •  on the long run: liquid like

Silly putty: Bounces on the table but eventually it adapts its shape.

t > t char

t < t char


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Characteristic time of the material tells you what is short and what is long: Water 10 -12 sec Dough products 1 sec – 100 sec Polymer liquids 1 – 5 min Glacier 10 year Glass 500 year Bronze 2000 year

Spider web:

1 2 3

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Continuum mechanics tells us how to describe stress and strain.

Stress state is described by 9 components, giving the stress tensor:


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x

z

y Tyy

Tyx

Txx

Txz

-Tzy

Tzz

Some components of the stress tensor

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Basic forms of deformation:

Pure strain: Txx

-Txx

Pure shear: Tyx

T-yx

C.W. Macosko: Rheology; 1994


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x

y

v = γ y ex .

Vo

h

τyx = F/A

γ = Vo /h = dvx/dy .

Simple shear flow

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x

y

v = γ y ex .

. V[2] = ½ γ (y ex - x ey)

V[1] = ½ γ (y ex + x ey) .

2

1

v = v[1] + v[2]

1 extension 2 rotation


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Stress tensor in simple shear flow:

Newtonian liquids: η is constant Ψ1 and Ψ2 are zero

Steady state:

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Uniaxial elongation

ez

er

vz = ε z vr = -½ ε r

..

However, it is impossible to create a steady extensional flow.


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stress: τ, σ [Pa] shear: γ [-] shear rate: γ [1/s] strain: ε [-] strain rate: ε [1/s] shear modulus: G [Pa] viscosity: η [Pa s]

. .

About names, symbols and units

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Flow curves of non-Newtonian liquids

shear thinning polymer melts


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shear thinning viscosity curve

polymer solutions

shear rate [1/s]

η [P

a s]

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γ/γ#

η/η o

plastic behavior


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shear rate [1/s]

η [P

a s]

plastic behavior

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0. 001 0. 01 0. 1 1 10 10 0 10 000. 01

0. 1

1

10

10 0

0. 001 0. 01 0. 1 1 10 10 0 10 000. 01

0. 1

1

10

10 0

0. 001 0. 01 0. 1 1 10 10 0 10 000. 01

0. 1

1

10

10 0

Flow behavior

Newtonian

= o '

Shear thinning

= (c' n) ' (power law)

Plastic

= o +o '

green: shear stress red: viscosity

’


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Normal stesses in a PMMA solution

ω

non-Newtonian phenomena

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Normal stresses

Due to the rotating lower disk, a shear flow exists between the disks. In case of visco-elastic fluids, this gives rise to normal stress differences.

Ω


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Dough for bread baking, shows rod climbing during its preparation.

Rod climbing due to normal stresses

Newtonian visco-elastic

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Another visco-elastic effect: the tubeless siphon


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Geometries for nearly simple shear flow

Rheometry

·  Cone-plate Shear rate constant Little sample needed

·  Plate-plate Shear rate not constant Little sample needed

·  Couette geometry Shear rate nearly constant More sample needed Higher sensitivity

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shear rate: γ' (r) = Ω r/h torque: M = 2 r2 τ(r) dr Can be used for normal force measurements

plate-plate geometry

h

torque and normal force sensor

Ω


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cone-plate geometry

torque and normal force sensor

shear rate: γ' (r) = Ω/θ torque: M = 2/3 R3 τ Can be used for normal force measurements

Ω

θ

R

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Couette geometry

torque sensor

Ω

L Ro

Ri shear rate: γ' (r) ≈ ½Ω(Ro+Ri)/(Ro-Ri) torque: M = 2r2 Lτ(r)


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0.90 0.95 1.000.0

5.0

10.0

15.0

20.0

25.0

30.0

n = 1.00

n = 0.50

n = 0.20

n = 0.10

n = 0.05

Shear rate in Couette for a power law fluid with index n

'/

r/Ru

•  n

•  0.00

•  -0.50

•  -0.80

•  -0.90

•  -0.95

Ri/Ru = 0.9

= (c' n) '

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"controled shear rate" : shear rate is applied resulting stress is measured.

"controled stress" : torque (shear stress) is applied resulting shear rate is measured. Useful in case of yield measurements.

0 20 40 60 80 100

schuifspanning [Pa]

0

50

100

150

200

afschuifsnelheid [1/s]

0

1

2

3

4

viscositeit [Pa s]


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Linear Visco-elasticity

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Ideal elastic Hookean behavior F = G (A/h) u τ = Gγ

Elasticity and viscosity

y

x h

u, v

γ = u/h, γ = v/h .

τ =F/A

Ideal viscous Newtonian behavior F = η (A/h) v τ = ηγ .


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Visco-elastic measurements: reveal important time scales.

u,v

γ = u/h, γ = v/h .

If you aply γ = γo cos (ωt) you measure τ = τo cos (ωt+φ)

τ

But how?

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Emulsion droplet

slow

fast

0 1 2 3 4 5

frequentie [mHz]

0.0

0.5

1.0

1.5

2.0

G [kPa] v [Pa s]

G


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Stress is a functional of the shear history. For small shear this functional is linear:

Or equivalently for small stresses: shear is a linear functional of the stress history:

Retardation function

Relaxation function

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G(t): relaxation function

G(t)

[Pa]

t [s]

Viscoelastic solid: G()>0

Viscoelastic liquid: G() = 0

relaxation times


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How to measure the relaxation function G(t)?

1: step response:

t [s]

τ [P

a] γ

(γ is a step)

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2: harmonic driving:

storage modulus

loss modulus


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harmonic shear experiment

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G(t)

[Pa]

t [s]

ω [rad/s]

G’,

G’’

[Pa]

G’

G’’


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How to measure the retardation function J(t)?

step response: (τ is a step)

-1 0 1 2 3 4 5 6 7 8

t [s]

0.00

0.20

0.40

0.60

0.80

1.00

1.20

γ’→0

γ’≠0 solid like

liquid like

τ0

γ(t)

creep measurement

t [s]

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Relation between G(t) and J(t)

Laplace transform

- = G’+jG’’ = J’-jJ’’


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Generalized Stokes Einstein Relation and particle tracking micro-rheology

10|-‐2 10|-‐1 10|0 10|1

10|0

10|1

(t-‐tw)/tc

<Δr2>/<Δr2>0

10 |-‐1 10 |0 10 |1 10 |2

10 |-‐1

10 |0

ωtc

G'/G'∞, G''/G'∞

?

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Stokes Einstein relation

links a transport coefficient (η) to an equilibrium property (D)


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5 m

rn(t)

rn(t+s)

particle tracking µ-rheology

< >: ensemble averaging and/or time averaging

: fluoresent tracer observed by CSLM

rn = (xn,yn)

Bursac et al; Nature materials 2005

Stokes Einstein Relation (Newtonian fluid):

Generalized Stokes Einstein Relation:

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Retardation function of a Newtonian fluid

generalization

prove via Laplace transforms; T.G. Mason, 2000


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concentrated emulsion

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10|0

10|1

(t-‐tw)/tc

<Δr2>/<Δr2>0

Dense suspension of polyNipam microgel particles

3 <Δx2(t)> /a2

(t-tw)/tc


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10 |-‐1

10 |0

ωtc

G'/G'∞, G''/G'∞

the resulting G’and G’’

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Two critical assumptions: - GSER is valid - Complex fluid around probe can be

considered as a continuum

If valid: - You measure from equilibrium properties a non-equilibrium transport property - There exist several approaches to calculate J*() from J(t)


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Microscopic view on the stress tensor

We consider a polymer solution

end to end vector

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polymer as a freely jointed chain

ui = ri-ri-1

r0

rN

N segments with length b Q

O


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spring constant of the entropic spring:

equipartition of energy:

Gaussian probability distribution:

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Consider the vectors Q near imaginairy interface:

x,y z

the number of vectors with value Q punching through the interface: n(Q)Qzd3Q. so, dTzβ= n(Q)QzFβ d3Q


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Polymer contribution to the stress tensor:

Hence, the rheologist should study the probability distribution p(Q,γ’)

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Probability distribution p(Q,γ’)

At rest this probability is fully symmetric, so T contains only diagonal components.

rest small γ’ large γ’


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Under small shear rate the distribution streches along the velocity direction, leading to a linear increase of the shear stress


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Under large shear rate the streched distribution rotates towards the velocity direction, leading to shear thinning and a normal stress difference.


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Rheology A short Introduction Dirk van den Ende What is Rheology ?

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