Chris Macosko

Department of Chemical Engineering and Materials Science

NSF- MRSEC(National Science Foundation sponsored Materials Research Science and Engineering Center)

IPRIME

(Industrial Partnership for Research in Interfacial and Materials Engineering)

IMA Annual Program Year TutorialAn Introduction to Funny (Complex) Fluids: Rheology, Modeling and Theorems

September 12-13, 2009

Understanding silly putty, snail slime and other funny fluids

What is rheology? (Greek) =

=

rheology =

honey and mayonnaise

rate of deformation

stress= f/area

to flow

every thing flows

study of flow?, i.e. fluid mechanics?

honey and mayo

rate of deformation

viscosity = stress/rate

What is rheology? (Greek) =

=

rheology =

rubber band and silly putty

time of deformation

modulus = f/area

to flow

every thing flows

study of flow?, i.e. fluid mechanics?

honey and mayo

rate of deformation

viscosity

4 key rheological phenomena

fluid mechanician: simple fluids complex flows

rheologist: complex fluids simple flows

materials chemist: complex fluids complex flows

rheology = study of deformation of complex materials

rheologist fits data to constitutive equations which

- can be solved by fluid mechanician for complex flows - have a microstructural basis

from: Rheology: Principles, Measurement and Applications, VCH/Wiley (1994).

ad majorem Dei gloriam

Goal: Understand Principles of Rheology: (stress, strain, constitutive equations)

stress = f (deformation, time)

G dt

d

Simplest constitutive relations:Newton’s Law: Hooke’s Law:

• shear thinning (thickening) • time dependent modulus G(t)

• normal stresses in shear N1

• extensional > shear stress u>

Key Rheological Phenomena

1-8

ELASTIC SOLID1The power of any spring

is in the same proportion

with the tension thereof.

Robert Hooke (1678)

f L

L´

k1 k2

L´

k1

k2

ff kL

modulus

stress

Young (1805)

strain

G

1-9

Uniaxial Extension

Natural rubberG=3.9x105Pa

11 a

fT

L

L

extension

or strain 1L - L

L

a = area

natural rubber G = 400 kPa

1-10

Silicone rubber G = 160 kPaGoal: explain different results in extension and shear

obtain from Hooke’s Law in 3D If use stress and deformation tensors

= 0

= -0.4

= 0.4

Shear gives different stress response

1-11

ˆ ˆ ˆ ˆ ˆ ˆ

ˆ ˆ ˆ ˆ ˆ ˆ

ˆ ˆ ˆ ˆ ˆ ˆ

ˆ ˆ ˆ

xx xy xz

yx yy yz

zx zy zz

x y z

T T T

T T T

T T T

T xx xy xz

yx yy yz

zx zy zz

T xt yt zt

3 3

1 1

ˆ ˆi j ij iji j

T x x T T

Stress Tensor - Notation

zzzyzx

yzyyyx

xzxyxx

TTT

TTT

TTT

T

ˆ ˆ xyTxydirection of stress on plane

plane stress acts on

Other notation besides Tij: ijorij

1 2 3ˆ ˆ ˆ ˆ ˆ ˆx y z x x x, , , ,

333231

232221

131211

TTT

TTT

TTT

ijT

dyad

1-12

1. Uniaxial Extension

Rheologists use very simple T

000

000

0011T

T T22 = T33 = 0

or

33

22

00

00

000

T

TT

T22 = T33

T22

T33

T11 -T22 causes deformation

T11 = 0

1-13

33

22

11

00

00

00

T

T

T

ijT

Consider only normal stress components

0 0 1 0 0

0 0 0 1 0

0 0 0 0 1

1 0 0

0 1 0

0 0 1

ij

p

p p

p

p

T

T I I

T I T

Hydrostatic Pressure T11 = T22 = T33 = -p

If a liquid is incompressible

G ≠ f(p) ≠ f(p)

Then only the extra or viscous stresses cause deformation

T = -pI +

and only the normal stress differences cause deformation

T11 - T22 = 11 - 22 ≡ N1 (shear)

1-14

2. Simple Shear

But to balance angular momentum

000

000

00

000

00

000

12

21

T

TTT ijij

( )T Ti j ij i j jii j i j

T T T x x x x

Stress tensor for simple shear Only 3 components:

T12

T11 – T22 = 11 – 22 ≡ N1

T22 – T33 = 22 – 33 ≡ N2

ˆ ˆ

T T

n T T n

T

Tnt

in general

11 12

12 22

33

0

0

0 0ij

T T

T T T

T

2x̂

3x̂

1̂x

T21

T21

2x̂

3x̂

1̂x

T12

T12

21 12T T

Rheologists use very simple T

Hooke → Young → Cauchy → Gibbs Einstein

(1678) (~1801) (1830’s) (1880,~1905)

Stress Tensor Summary

T11T11

n

2. in general T = f( time or rate, strain)3. simple T for rheologically complex materials:

- extension and shear

4. T = pressure + extra stress = -pI + .

5. τ causes deformation

6. normal stress differences cause deformation, 11-22 = T11-T22

7. symmetric T = TT i.e. T12=T21

1. stress at point on any plane

1-16

s'

P

Q

Motions

P

x3

x2

x1

Rest or past state at t' Deformed or present state at t

Q

ˆ

ˆ

ˆ

Deformation Gradient Tensor

( ) ( )

2

w x w x y s

xx y s Os

x y

= or iij

j

xF

x

F

s w - y F s

x

x

a new tensor !

x(y ) F s

y F s

s = w – yw = y + s

w

y

s

P

Q

s′ = w′ - y′w’ = y’ + s’

w′

y′

s′

P

Q

x = displacement functiondescribes how material points move

s’ is a vector connecting two very close points in the material, P and Q

1-17

Apply F to Uniaxial Extension

Displacement functions describe how coordinates of P in undeformed state, xi‘have been displaced to coordinates of P in deformed state, xi.

1

ij 2

3

0 0

F 0 0

0 0

1 1 1 1 2 1 3

2 1 2 2 2 2 3

3 1 3 2 3 3 3

0 0

0 0

0 0

ij i jF x x

x x x x x x

x x x x x x

x x x x x x

1-18

1

2

3

0 0

0 0

0 0

ij i jF x x

11 2

11 2

1

0 0

0 0

0 0ijF

Can we write Hooke’s Law as ? τ F - IG

Assume: 1) constant volume V′ = V

2) symmetric about the x1 axis

1-19

Can we write Hooke’s Law as ?

Solid Body Rotation – expect no stresses

For solid body rotation, expect F = I

= 0But

F ≠ IF ≠ FT

Need to get rid of rotation create a new tensor!

τ F - IG

1-20Bij gives relative local change in area within the sample.

Finger Tensor

2V

V·I·V

·VV·R·R

V·R·V·RF·F

T

TT

TT

Solid Body Rotation

cos sin 0 cos sin 0 1 0 0

sin cos 0 sin cos 0 0 1 0

0 0 1 0 0 1 0 0 1ijB

stretch

rotation

F V R

V

R

or jT iij ik jk

k k

xxB F F

x x

B F F

1-21

1. Uniaxial Extension

since T22 = 0

Neo-Hookean Solid BITI-Bτ GpG or

1-22

1 1 2 1 22

2 2

3 3

'

sx x x x x

x

x x

x x

2. Simple Shear

21

211 22

G

G

G

τ B - I

agrees with experiment

Silicone rubber G = 160 kPa

1 0

0 1 0

0 0 1ijF

21 0 1 0 0 1 0

0 1 0 1 0 1 0

0 0 1 0 0 1 0 0 1ijB

1. area change around a point on any plane

2. symmetric

3. eliminates rotation

4. gives Hooke’s Law in 3D

fits rubber data fairly well

predicts N1, shear normal stresses

Finger Deformation Tensor Summary

TB F F

Course Goal: Understand Principles of Rheology: (constitutive equations)

stress = f (deformation, time)

G dt

d

Simplest constitutive relations:Newton’s Law: Hooke’s Law:

• shear thinning (thickening) • time dependent modulus G(t)

• normal stresses in shear N1

• extensional > shear stress u>

Key Rheological Phenomena

Gτ B - I

VISCOUS LIQUID2The resistance which arises

From the lack of slipperiness

Originating in a fluid, other

Things being equal, is

Proportional to the velocity

by which the parts of the

fluids are being separated

from each other.

Isaac S. Newton (1687)

yx d x

dy

measured in shear1856 capillary (Poiseuille)1880’s concentric cylinders

(Perry, Mallock, Couette, Schwedoff)

dv

dy x

yx

dv

dy

Newton, 1687 Stokes-Navier, 1845

Bernoulli

Familiar materials have a wide range in viscosity

Adapted from Barnes et al. (1989).

measured in shear1856 capillary (Poiseuille)1880’s concentric cylinders

(Perry, Mallock, Couette, Schwedoff)

measured in extension1906 Trouton

u = 3

dv

dy x

yx

dv

dy

Newton, 1687 Stokes-Navier, 1845

Bernoulli

To hold his viscous pitch samples, Trouton forced a thickened end into a small metal box. A hook was attached to the box from which weights were hung.

“A variety of pitch which gave by the traction method = 4.3 x 1010 (poise) was found by the torsion method to have a viscosity = 1.4 x 1010 (poise).” F.T. Trouton (1906)

polystyrene 160°CMünstedt (1980)

Goal1.Put Newton’s Law in 3 dimensions

• rate of strain tensor 2D• show u = 3

Separation and displacement of point Q from P

( , t)

w y F s

s = w - y = F s

x

s = w - ys′ = w′ - y′

w′

y′

s′

P

Q

recall Deformation Gradient Tensor, F

s'

P

Q

Motions

P

x3

x2

x1

Rest or past state at t' Deformed or present state at t

Q

ˆ

ˆ

ˆ

w

y

s

P

Q

rate of separation

t

s F s

s F ss F

t t

Alternate notation:v

ˆ ˆ jTi j

i j ix

v L x x

dt

d

d d d

sv = F s F x

v F Lx x

is the velocity gradient tensor.

or

lim lim x x x xort t

d d d d

d d d

L

vv v L

v F L

F L F

F I F L

x xx

x x

Velocity Gradient Tensor

Viscosity is “proportional to the velocity by which the parts of the fluids are being separated from each other.” —Newton

θ

r z

v Ωr

v v 0

0 Ω 0

Ω 0 0

0 0 0

Lij

lim lim ( )

lim +

x x x x

x x

t t

• • •

• • •

F V R

F = V R + V R

V R I

F L V R

( )

is anti-symmetric

T T • • • •

•

L V R V R

R

Can we write Newton’s Law for viscosity as = L?

solid body rotation

2 ( ) + 2•

DV L L v vT T Rate of Deformation Tensor D

≠

2 •

D

Other notation:Vorticity Tensor W 2 2

+ ( )

•

R W L L

L D W v

T

T

t t lim 2

BShow D

d

dt

Example 2.2.4 Rate of Deformation Tensor is a Time Derivative of B.

t t

t t

lim +

recall that lim

lim + 2

Thus

B F F

F L

B L L D

T

x x

T

0 0 0 0 0 0

2 0 0 0 0 0 2 - 2 0 0

0 0 0 0 0 0 0 0 0

T TD L L W L L

Show that 2D = 0 for solid body rotation

t t t t

lim lim

BB F F F F F F

F F I

T T T

T

d

dt

Here planes of fluid slide over each other like cards in a deck.

Steady simple shear

Newtonian Liquid

1 1 2 2 2 3 3x x γx x = x x = x

= 2D or T = -pI + 2D

Time derivatives of the displacement functions for simple, shear

2 2

11 2

x

12 3

2

lim v 0 v

vv and v v 0 (2.2.10)

22

x

1 2 2

dx dγ dxx

dt dt dt

dx x

dx

12 12 21 T

0 0 0 0 0 0 0

0 0 0 0 0 2 0 0

0 0 0 0 0 0 0 0 0

1 0 0 0 0

0 1 0 0 0

0 0 1 0 0 0

ij ji ij

ij

L L D

T p

1 1

1 1 11 1 1 1

2 2 2 3 3 3

x x

α or

t

1 1 1

dx dα d x α x x v x x

d dt dt

v x v x

time derivatives of the displacement functionsat

Similarly

1

ij 2

3

0 0

L 0 0

0 0

1

1

1

0 0 0 0

L 0 2 0 0 2 0

0 0 2 0 0 2

2

2 (L L )

ij

ij ij ji

0 0

D 0 0

0 0

1 2 3

2 3

12 3 2 3

(1.7.9)

0

0

υ υ

2

incompressible fluid

or

symmetric, and thus

v

Steady Uniaxial Extension

Newtonian Liquid

Apply to Uniaxial Extension = 2D

11

22 33

2

From definition of extensional viscosity

11 22 3u

2

2 ij

0 0

D 0 0

0 0

Newton’s Law in 3 Dimensions•predicts 0 low shear rate•predicts u0 = 30

but many materials show large deviation

Newtonian Liquid

melt ePolystyren

T11T11

n1. stress at point on plane

Summary of Fundamentals

simple T - extension and shear

T = pressure + extra stress = -pI + .

symmetric T = TT i.e. T12=T21

2. area change around a point on planeTB F Fsymmetric, eliminates rotation

gives Hooke’s Law in 3D, E=3G

3. rate of separation of particles 2 ( ) + T T D L L v v

symmetric, eliminates rotation

gives Newton’s Law in 3D, 3u

Course Goal: Understand Principles of Rheology:

stress = f (deformation, time) NeoHookean: Newtonian:

• shear thinning (thickening) • time dependent modulus G(t)

• normal stresses in shear N1

• extensional > shear stress u>

Key Rheological Phenomena

Gτ B - I = 2D