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

Outline

1. Definitions

2. Stress relaxation

3. Maxwell element

4. Dynamic moduli

5. Compliance

6. Polymer solutions, gels, and melts

Definitions

t log t

log G

Stress: Strain: Strain rate:

(shear) modulus: G = / viscosity: = /

stress relaxation modulus: G(t,) = (t,) /

linear response: small enough so that G is independent of

Limiting cases

time

time

Hookean

time

Newtonian

time

viscoelastic solid

viscoelastic liquid

Maxwell Element

o el vis

Ý o 0 Ý el Ý vis

0 1

GoÝ

1

o

(t) (0) exp( t /)

(0) Go o

o / Go

Go o

G(t)

t

Dynamic shear modulus

o sint Ý o cost

o sin(t )

o ' sint o"cost

G' o '

o, G"

o"

o, tan

G"

G'

Elastic “storage” modulus, viscous “loss” modulus, loss tangent

'

"

time

.

Maxwell element

G' Go22

122 ; G" Go

122

Limiting slopes:

low , G’ ~ 2 , G” ~

high , G’ ~ 0 , G” ~

10-4

10-3

10-2

10-1

100

101

10-3 10-2 10-1 100 101 102

G'/Go

G"/Go

Complex notation

G*() *

* G' i G"

*() *

Ý * ' i "

G* i*; G' "; G" '

Dynamic viscosity:

* 2 2 1/2( ' " ) /G G

Maxwell element

10-4

10-3

10-2

10-1

100

101

10-3 10-2 10-1 100 101 102

G'/Go

G"/Go

10-4

10-3

10-2

10-1

100

101

10-3 10-2 10-1 100 101 102

'/ Go

''/ Go

Creep compliance

J(t) (t)

o

(t) el vis o

Go

1

odt

J(t) 1

Go1 t

Maxwell element:

J(t) J eo

t

General LVE:

time

o

time

J

1/

Jeo

Jeo

Outline

1. Definitions

2. Stress relaxation

3. Maxwell element

4. Dynamic moduli

5. Compliance

6. Polymer solutions, gels, and melts

Meet the suspects6 typical materials

Surfactant Solution

Dilute Polymer Solution

Entangled Polymer (M/S)

EmulsionSuspension

Gel

G’, G” for a single flexible chain in a solvent

P. E. Rouse, Jr. J. Chem. Phys. 21, 1872 (1953)

B. H. Zimm, J. Chem. Phys. 24, 269 (1956)

www.joogroup.com/graphics/single_poly_cg.jpg

Random coil Bead-spring model

G’, G” for a high M chain in an oligomer

10-1100101102103104105106100101102103104105106G', G" (Pa) aTPS-650,0003.1% in PS-3,000Tref = 180 ºC

10-1

100

101

102

103

104

105

106

100 101 102 103 104 105 106

G',

G"

(Pa

)

aT

PS-650,000

3.1% in PS-3,000

Tref

= 180 ºC

= G”/

“Internal modes”

Longest relaxation time“Terminal” regime

G” always > G’

D. Tan, unpublished results

G’, G” for a single chain in a theta solvent

Sahouani and Lodge, Macromolecules, 25, 5632 (1992)

10-9

10-8

10-7

10-6

10-5

10-4

10-2 10-1 100 101 102 103

[G'] R

, [G

"]R

1

Polybutadiene in DOP

T = 18.0 oC ()

Zimm theory(Dynamic scaling, = 0.50

Polyisoprene: an entangled melt

101

102

103

104

105

106

107

10-2 100 102 104 106

G' a

nd G

" ,

Pa

aT

Polyisoprene

Mw

= 80,000

Tr = 20

oC

G'

G"

J. C. Haley, Ph. D. Thesis, Univ. Minn., (2005)

New solid-like regime

QuickTime™ and a decompressor

are needed to see this picture.

The “Gel” Samples Can be Interpreted Simply

Worm-like micelles

Bernheim-Groswasser, A., Zana, R., and Talmon, Y., J. Phys. Chem. B 104, 4005 (2000).

4 nm

The “Gel” Samples Can be Interpreted Simply

Maxwellian response

100 mmol cetyl pyridinium chloride60 mmol sodium salicylate 100 mmol sodium chloride

QuickTime™ and a decompressor

are needed to see this picture.

Candau et al., J. Phys. IV, 3, 197 (1993).

Gelation of ABA triblock copolymers

G’, G”

Liquid

Physical gel

Gel point

C > C*

Triblock copolymers

C << C*

SOS gel point in an ionic liquid

0.001

0.01

0.1

1

10

100

1000

G',

G"

(Pa

)

0.01 0.1 1 10 100

(rad/s)

G' G''10 wt% 4 wt% 1 wt%

G' ~ 2

G'' ~

G'=G'' ~

G' ~

A

O

Y. He, P. G. Boswell, P. Bühlmann, T. P. Lodge, J. Phys. Chem. B, 111, 4645, (2007)

The “Gel” Samples Can be Interpreted Simply

Newtonian droplets in a Newtonian fluid

I. Vinckier et al., J. Rheol. 40, 613 (1996)

10% low molar mass PIB in low molar mass PDMS

Polyisoprene: an entangled melt

101

102

103

104

105

106

107

10-2 100 102 104 106

G' a

nd G

" ,

Pa

aT

Polyisoprene

Mw

= 80,000

Tr = 20

oC

G'

G"

J. C. Haley, Ph. D. Thesis, Univ. Minn., (2005)

Meet the suspects6 typical materials

Surfactant Solution

Dilute Polymer Solution

Entangled Polymer (M/S)

EmulsionSuspension

Gel

NIST standard

11 wt% high MW PIB (MW~106, Aldrich) in Pristane

LVE properties

Entangled polymers

Data from Snijkers et al, J. Rheology, 53, pp. 459-480 (2009)

L. Raynaud et al, J. Coll. Int. Sci, 81, 11 (1996))

Colloidal Suspension

(rad/s)

10-2 10-1 100 101 102

G' (P

a)

10-2

10-1

100

101

102

103

104

0.4210.4260.4370.4520.4550.4710.4820.5020.5150.534

φc

Polystyrene-butylacrylate latices