Carnegie Mellon1. 2 3 4 5 6 7 8 9 10 Carnegie Mellon11.

Carnegie Mellon 1

Rheological Behavior and Polymer Properties

G. C. Berry

Department of Chemistry

Carnegie Mellon University

Colloids, Polymers and Surfaces

e-mail: [email protected]

web site: http://www.chem.cmu.edu/berry

Carnegie Mellon 2

• Introduction 3 (12)

• Rheological methods 16 (19)

• Linear elastic parameters 26 (5)

• Linear viscoelastic functions 33 (12)

• Several viscoelastic experiments 44 (16)

• Relations among linear viscoelastic functions 62 (10)

• Examples of linear viscoelastic functions 73 (9)

• Time-temperature equivalence 83 (9)

• The glass transition temperature 93 (13)

• The viscosity 107 (26)

• Effects of polydispersity 134 (4)

• Network formation 139 (13)

• Isochronal Behavior 153 (6)

• Examples from the literature 160 (45)

Branched and linear metallocene polyolefins 161 (10)

Colloidal dispersions 172 (9)

Wormlike Micelles 182 (4)

Deformation of rigid materials 187 (4)

Nonlinear shear behavior 192 (16)

Linear and nonlinear bulk properties 209 (6)

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• Introduction

• Rheological methods

• Linear elastic parameters

• Linear viscoelastic functions

• Several viscoelastic experiments

• Relations among linear viscoelastic functions

• Examples of linear viscoelastic functions

• Time-temperature equivalence (Thermo-rheological simplicity)

• The glass transition temperature

• The viscosity

• Effects of polydispersity

• Network formation

• Isochronal Behavior

• Examples from the literature

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PROTEINS

POLYNUCLEOTIDES

POLYSACCHARIDES

GUMS

RESINS

ELASTOMERS

THERMOPLASTIC THERMOSETTING

NATURAL SYNTHETIC

POLYMERS

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Some Common Elastomers, Plastics and Fibers

ELASTOMERS PLASTICS FIBERS

Polyisoprene polyethylene

polyisobutylene polytetrafluoroethylene

poybutadiene polystyrene

poly(methyl methacrylate)

Phenol-formaldehyde

Urea-formaldehyde

Melamine-formaldehyde

←⏐⏐⏐⏐⏐⏐⏐⏐ Poly(viny l chlorid )e ⏐⏐⏐⏐⏐⏐⏐⏐⏐⏐⏐→

←⏐⏐⏐⏐⏐⏐⏐⏐ Polyurethanes⏐⏐⏐⏐⏐⏐⏐⏐⏐⏐⏐⏐⏐⏐⏐→

←⏐⏐⏐⏐⏐⏐⏐⏐ Polysiloxanes ⏐⏐⏐⏐⏐⏐⏐⏐⏐⏐⏐⏐⏐⏐⏐→

←⏐⏐⏐⏐⏐⏐⏐⏐⏐⏐ Polyamide⏐⏐⏐⏐ ⏐⏐→

←⏐⏐⏐⏐⏐⏐⏐⏐⏐⏐ Polyester ⏐⏐⏐⏐⏐ ⏐⏐→

←⏐⏐⏐⏐⏐⏐⏐⏐⏐⏐ Polypropylen e ⏐⏐⏐ ⏐→

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Molecular Weight M

Mn

Mw

Mz

Fraction of Molecules With

Molecular Weight M

A Schematic Illustration of a Typical Distribution

of Molecular Weights, showing Mn, Mw, and Mz

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A generalized Average of molecular weights:

wµ is the weight fraction of polymer with molecular weight Mµ:

M(α) = ⎝⎜⎛

⎠⎟⎞Σ

µ wµMµ

α 1/α

Specia l Case :s

Numb er average:

Mn = M(0)/M−1

(−1) = 1/⎝⎜⎛

⎠⎟⎞Σ

µ wµM µ

−1

Weight avera ge:

Mw = M(1)/M(0) = Σµ

wµMµ

z-averag e:

Mz = M 2(2)/M(1) = ⎝⎜

⎛⎠⎟⎞Σ

µ wµMµ

2 /Σµ

wµMµ

G. C. Berry "Molecular Weight Distribution" Encyclopedia of Materials

Science and Engineering, ed. M. B. Bever, Pergamon Press, Oxford, 3759-68 (1986)

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Temperature

Specific Volume

Tm

A schematic v-T diagram for a typical nonpolymeric material.

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Temperature

Specific Volume

TmTg

A schematic v-T diagram for a typicalsemi-crystalline polymeric material.

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Temperature

Specific Volume

Tg

A schematic v-T diagram for a typical

noncrystalline polymeric material.

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Rigid Plastic

Flexible Plastic

Elastomer

Strain

Stress

Typical Stress-Strain Behavior for Plastics and Elastomers

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F. W. Billmeyer Jr. (1976):J. Polym. Sci.: Symp. (1976) 55: 1-10

"…characterization of polymers is inherently more

difficult than that of other materials. Polymers are

roughly equivalent in complexity to, if not more complex

than, other materials, at every physical level of

organization from microscopic to macroscopic…"

"We would wish, ideally, to characterize all aspects of a

polymer structure in enough detail to predict its

performance from first principles. I seriously doubt that

this will ever be possible, and I am sure that even if it

were, it would never be economically feasible."

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Microscopic Characterization Needed at Many Resolutions:

A

A

B

A

BB

A A B A BB

A

A

B

A

B B

C C C

C C

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2-D projection of a random arrangement of a chain

with 1000 non-overlapping bonds, each step

otherwise randomly selected

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Mean chain dimensions:

For a linear chain with contour length L

(without excluded volume effects):

Mean square-end-to-end dimension:

R2L = 2âL

â is the persistence length (2â is the Kuhn length)

for a flexible chain, â << L.

Mean square-radius of gyration:

R2G = R2

L/6 = âL/3

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• Introduction









• The viscosity





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Computer System

for

Data Acquisition

and

Instrument Control

Rheometer

Torque

Transducer

Position

Transducer

Temperature

Transducer

Force

Transducer

Shape

Transducer

Output

Interfaces

Shear Stress

vs

Time (Frequency)

Shear Strain

vs

Time (Frequency)

Normal Force

vs

Time (Frequency)

Temperature

vs

Time

Schematic of Rheometer System

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CONTROLLED STRESS

IN TENSION

Sample

"Frictionless"

Bearing

Removable

Weight

Position

Transducer

Tare

Device Input Output

Removable Weight Controlled weight Controlled force

Position Transducer Measure of shaft position Voltage (current)

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CONTROLLED DEFORMATION

IN TENSION

Sample

Position

Transducer

Crosshead

Drive Screws

Device Input Output

Crosshead Drive Controlled Drive Controlled force

Position Transducer Measure of shaft position Voltage (current)

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CONTROLLED STRESS RHEOMETER

Sample

Shaft

"Frictionless mount"

Controlled

Torque

Drive

Fixed Shaft

(Alternate: controlled rotation)

Fixtures

Angle

Position

Transducer

Device Input Output

Controlled Torque Drive Controlled voltage Controlled torque

Angle Position Transducer Measure of shaft angle Voltage (current)

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CONTROLLED DEFORMATION RHEOMETER

Sample

Shaft

"Frictionless mount"

Controlled

Rotation

Drive

Fixtures

Angle

Position

Transducer

Torque Transducer

(Force Transducer)

Device Input Output

Controlled Deformation Drive Controlled voltage Controlled shaft rotation

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Electromagnetic Coils

I : A-F

II : a-f

,

AB

a

b

C

D

c

d

EF

e

f

G

H

g

h

Aluminum Cylinder

Attached to Rotor

Iron Core

• Phasing of the currents in Coils I and II can produce a time-

dependent torque:

◊ Constant torque amplitude

◊ Sinusoidal torque amplitude

• Torque amplitude may readily be varied over a factor of 106.

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Sample

2R

Height h

Ω

Fixtures

Parallel Plates

Sample

Fixtures

Angle α

2R

Ω

& Cone Plate

Sample

Fixtures

Δ R

2R

h

Ω Concentric Cylinders

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Geometric Factors in Rheometry

Geometry Measured Calculateda

Translational geometriesParallel Plate Force: F Stress: σ = F /wb

widt ,hw; breadt h b; separati on h Displacemen :t D Strain: γ= D/h

Concentric Cylinders Force: F Stress: σ = F /2πRh

inner radius R; gap Δ; heigh th Displacemen :t D Strain: γ= D/Rln(1 + Δ/R)

Rotationa l geometries

Parallel Plate Torque: M Stress: σ = (2 /r R)M /πR3

outer radius R; separation h Rotation: Ω Strain: γ(r) = ( /r h) Ω

Cone & Plate Torque: M Stress: σ = (3/2)M /πR3

outer radius R; cone angle π - α Rotation: Ω Strain: γ= (1/α) Ω

Concentric Cylinders Torque: M Stress: σ = (R/2h)M /πR3

inner radius R; gap Δ; heigh th Rotation: Ω Strain: γ(r) = (R/ΔR) Ωf(R,r)

f(R,r) = ( /R r)21 + Δ/R1 + Δ/2R

a σ and γ are the shea r stress and strai ,n respectively

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Functions and Parameters Used

Function/Parameter Symbol Units

Time t T

Frequency ω T-1

Stra in Component εij---

Elongationa l strain ε ---Shear strain γ ---Rat eof shear γ, ε T-1

Stre ss Component SijML-1T-2

Shear stress σ ML-1T-2

Modulus G, K, E ML-1T-2

Compliance J , B, D M-1LT2

Viscosity η ML-1T-1

Carnegie Mellon 26

• Introduction









• The viscosity





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Linear elastic phenomenology

Shear stress σShea r strain γ

γ = Jσ = (1/G)σ

Elongati onal stress σΤ

Elongati onal strain ε

ε = DσΤ = (1/E)σΤ

Pressure ∆PVolume cha nge ∆V

∆V/V = B∆ P = (1/K)∆P

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Linear Elastic Functions

Shear Compliance

Shear Modulus

J

G

Bulk Compliance

Bulk Modulus

B

K

Tensile Compliance

Tensile Modulus

D = J/3 + B/9

1/E = 1/3G + 1/9K

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εij = 12 ⎝

⎜⎛

⎠⎟⎞∂ui

∂xj +

∂uj∂xi

; u is the displacement

vector

2εij = J [Sij – 13 δij Sαα] + (2/9)δij B Sαα

Sij = 2G [εij – 13 δij εαα ] + δij K εαα

δij = 1 if i = j, and δij = 1 if i ≠ j

In this notation,

Shear stress σ = S12

Shear strain γ = 2ε12

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Relations Among Linear Elastic Constants

K, G E, G K, E K, ν E, ν G, ν

K K EG3[3G – E]

K K E3[1 – 2ν]

2G[1 + ν]3[1 – 2ν]

E 9KG3K + G

E E 3K(1 – 2ν) E 2G(1 + ν)

G G G 3KE9K – E

3K[1 – 2ν]2[1 + ν]

E2[1 + ν]

G

ν 3K – 2G6K + 2G

E2G – 1

3K – E6K

ν ν ν

J = 1/G, = 1/B K, =D 1/E

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Carnegie Mellon 32

• Introduction









• The viscosity





Carnegie Mellon 33

Linear Elastic Functions

Shear Compliance

Shear Modulus

J

G

Bulk Compliance

Bulk Modulus

B

K

Tensile Compliance

Tensile Modulus

D = J/3 + B/9

1/E = 1/3G + 1/9K

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Linear Viscoelastic Functions

Shear Compliance

Shear Modulus

J(t)

G(t)

Bulk Compliance

Bulk Modulus

B(t)

K(t)

Tensile Compliance

Tensile Modulusa

D(t) = J(t)/3 + B(t)/9

1/E(s) = 1/3G(s) + 1/9K(s)

a. The superscript " " denotes a Laplace transform.

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Linear viscoelastic phenomenology—

Stress Controlled

γ(t) = ∑i=1

Ν

J(t – ti) Δσi = ∫0

σ(t)

d[σ(u)] J(t – u)

γ(t) = ∫-∞ tduJ(t – u)

∂σ(u)∂u

γ(t) = Joσ(t) + ∫0 ∞

du σ(t – u) ∂J(u)

∂u

γ (t)

Δσ 1

t

t 1 t 2

γ (t)

Δσ 2

σ (t)

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Linear viscoelastic phenomenology—

Strain Controlled

σ(t) = ∑i=1

Ν

G(t – ti) Δγi = ∫0

γ(t)

d[γ(u)]G(t – u)

σ(t) = ∫-∞ tduG(t – u)

∂γ(u)∂u

σ(t) = Goγ(t) + ∫0 ∞

du γ(t – u) ∂G(u)

∂u

γ (t)

Δγ 1

t

t 1 t 2

σ (t)

Δγ 2

σ (t)

Carnegie Mellon 37


εij = 12 ⎝

⎜⎛

⎠⎟⎞∂ui

∂xj +

∂uj∂xi

; u is the displacement

vector

2εij = J [Sij – 13 δij Sαα] + (2/9)δij B Sαα

Sij = 2G [εij – 13 δij εαα ] + δij K εαα

δij = 1 if i = j, and δij = 1 if i ≠ j

In this notation,

Shear stress σ = S12

Shear strain γ = 2ε12

Carnegie Mellon 38

Linear viscoelastic phenomenology

εij = 12 ⎝

⎜⎛

⎠⎟⎞∂ui

∂xj +

∂uj∂xi

; u is the displacement vector

2εij(t) = ∫-∞ tds{J(t – s)[

∂Sij(s)∂s –

13 δij

∂Sαα (s)∂s ]

+ (2/9)δij B(t – s) ∂Sαα (s)

∂s }

Sij(t) = ∫-∞ tds{2G(t – s)[

∂εij(s)∂s –

13 δij

∂εαα (s)∂s ]

+ δij K(t – s) ∂εαα (s)

∂s } In this notation,

Shear stress σ(t) = S12 (t)

Shear strain γ(t) = 2ε12(t)

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Relation between G(t) and J(t)

1t ∫

0 tdu G(t – u) J(u) = 1

with Laplace transform:

s2G(s)J(s) = 1

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Shear Compliance J(t) andRecoverable Shear Compliance R(t)

R(t) = J(t) – t/η = J∞ – [J∞ – Jo]α( ) tα(t): Retardation Function

Shear Modulu s ( )G t

(G )t =Ge + [Go – Ge]ϕ( )t ϕ(t): Relaxation Function

η the (linear) viscosity, with 1/η = 0 for a solid,

Ge the equilibrium modulus, with Ge = 0 for a fluid,

Go the "instantaneous" modulus, with JoGo = 1, and

J∞ the limit of R(t) for large t:

Solid: J∞ = Je = 1/Ge; equilibrium compliance

Fluid: J∞ = Js; steady-state recoverable compliance

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Creep Shear Compliance J(t)

R(t) = J(t) – t/η = J∞ – [J∞ – Jo]α(t)

Shea r Modulus G( )t

(G t) = Ge + [Go – Ge]ϕ(t)

• Linear elastic solid: 1/η = 0, J∞ = Je = 1/Ge, α(t) = ϕ(t) = 0

• Linear viscous fluid: 1/η > 0, Go = 0, α(t) = ϕ(t) = δ(t)

• Linear viscoelastic solid: 1/η = 0, J∞ = Je = 1/Ge, 0 < α(t) < ϕ(t) ≤ 1

• Linear viscoelastic fluid: 1/η > 0, J∞ = Js (= Joe), 0 < α(t) < ϕ(t) ≤ 1

Bulk Compliance B(t) = Be – [Be – Bo]β(t)

Bulk Modulus K(t) = Ke + [Ko – Ke]κ(t)

Carnegie Mellon 42

Retardation function α(t)

R(t) = J( )t – /tη = J∞ – [J∞ – Jo]α(t)

Relaxation function ϕ(t)

(G t) = Ge + [Go – Ge]ϕ(t)

ϕ (t)

α (t)

Time

Viscoelastic Fluid or Solid

0

1

Linear Viscous Fluid

Linear Elastic Solid

Carnegie Mellon 43

Simple example of the relation between G(t) and J(t)

Maxwell fluid:

G(t) = Goexp(- t/τ); τ = η/Go

( )J t = J s + /t η; J s = J o = 1/Go

( )R t = J s

Note: ϕ(t) = exp(-t/τ) and α (t) = 0 for this model.

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Often used relations for ϕ(t) and α(t)

A weight se t of exponentials wit h N relaxation times:

α(t) = ∑m

N-1

α i exp(–t/λ i) = 1

J∞ – Jo ∫

-∞ ∞

d(ln λ) L(λ)exp(–t/λ)

ϕ(t) = ∑1

N

ϕ i exp(–t/τi) = 1

Go – Ge ∫

-∞ ∞

d(ln τ) H(τ)exp(–t/τ)

Notes: Σα i = Σϕ i = 1, and

m is equal to 0 or 1 for a solid and fluid, resp.

(1 − m)λ0 > τ1 > λ1 > … > λi > τi > λi+1 > … > λN-1 > τN

(The contribution λ0 is absent for a fluid)

Carnegie Mellon 45

• Introduction









• The viscosity





Carnegie Mellon 46

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Creep and recovery with a step shear stress

Stress history: σ(t) = 0 t < 0

σ(t) = σo 0 ≤ t≤ Teσ(t) = 0 t > Te

q = t - T e

Time

0

0

t

t = T e

γ (t)

γ R ( θ ) = γ (t = T e ) - γ (t)

Stress

Strain

γ (t) = a + bt

σ (t) = σo

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The strain in creep for t ≤ Te:

γ(t) = σo∫0 tdu (J t – u) δ( u - 0)

= σo (J t) = σo[R(t) + t/η]

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γ(t) = σo∫0 tdu (J t – u) δ( u - 0)

= σo (J t) = σo[R(t) + t/η]

The strain fo r ϑ = t– Te > 0 in recovery:

γ(t) = σo∫0 Tedu (J t – u) δ( u - 0) – σo∫Te

tdu (J t – u) δ( u - Te)

γ(ϑ) = σo[ (J ϑ + Te) – (J ϑ)] = σo[R(ϑ + Te) – R(ϑ) + Te/η]

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γ(t) = σo∫0 tdu (J t – u) δ( u - 0)

= σo (J t) = σo[R(t) + t/η]

The strain fo r ϑ = t– Te > 0 in recovery:

γ(t) = σo∫0 Tedu (J t – u) δ( u - 0) – σo∫Te

tdu (J t – u) δ( u - Te)

γ(ϑ) = σo[ (J ϑ + Te) – (J ϑ)] = σo[R(ϑ + Te) – R(ϑ) + Te/η]

The recoverable strain γR(ϑ) = γ(Te) – γ(t) for ϑ > 0:

γR(ϑ) = σo{ J(Te) – [ (J ϑ + Te) – (J ϑ)]}

= σo{R(ϑ) + R(Te) – R(ϑ + Te)}

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The stress response for t > 0:

σ(t) = γo∫0 t

duG( t– )u δ(u - 0) = γoG(t)

= γo{Ge + (Go – Ge)ϕ(t)}

σ(∞) = γoGe

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The stress response for t ≤ Te :

σ(t) = γ ∫0 t du (G –t u) = γ [Get + (Go – Ge)∫0

tds ϕ( )s ]

For a fluid in steady-state deformat , ion σ = ηγ, or

η = σ(∞)/γ = Go ∫0 ∞ ds ϕ( )s

The strain fo r t > Te:

σ(t) = 0 = γ ∫0 Te du (G –t u) + ∫Te

tdu G(t – u)

∂γ(u)∂u

For large Te and t, (full recoil after steady flow) it can be shown

that fo r a fluid thi s gives:

τc = ηJs = ∫0 ∞ds sϕ( )s /∫0

∞ds ϕ( )s

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The strain response for t > 0:

γ(t) = ωσo∫0 t

duJ( – t u)c (osωu)

In the st -eady st ate limit with l arge t:

γ(t) = σo{ J '(ω)sin(ωt) – J ''(ω)c (osωt)}

I -n pha se(or rea l or storage) dynam iccomplianc :e

J'(ω) = J∞ – ω[J∞ – Jo]∫0 ∞ds α( )s sin(ωs)

Out-of-phas e( orimaginar y orl ) oss dynami ccompliance

J"(ω) = (1/ωη) + ω[J∞ – Jo]∫0 ∞ds α( )s c (osωs)

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Alternatively

γ(t) = σo |J* (ω)|sin [ωt – δ(ω)]

"Dynamic compliance":

|J* (ω)|2 = [ (J' ω)]2 + [ "(J ω)]2

Phas ea ngleδ(ω):tan δ(ω) = "(J ω)/ (J' ω)

For sma ll ω:

J (' ω) ≈ J∞, J"(ω) ≈ 1/ωη, and J"(ω) – 1/ωη ≈ ω

Carnegie Mellon 58

Oscillation with a sinusoid shear strain

Strain history: γ(t) = 0 < 0t

γ(t) = γosin(ωt) t≥ 0

T hestre ssrespons ef or t> 0 is give n by

σ(t) = ωγo ∫0 t

duG( t– )uc (osωu)

I n thestea -dystat elim itw ithlar ,ge t

σ(t) = γo{G'(ω)sin(ωt) + G''(ω)c (osωt)}

I -n pha se(or rea l or storage) dynam iccomplianc :e

G'(ω) = Ge + ω[Go – Ge]∫0 ∞ds ϕ(s)sin(ωs)

Out-of-phas e( orimaginar y orl ) oss dynami ccompliance

G''(ω) = ω[Go – Ge]∫0 ∞ds ϕ(s)cos(ωs)

Carnegie Mellon 59

Alternatively

σ(t) = γo |G* (ω)|sin [ωt + δ(ω)]

"Dynamic compliance":

|G*(ω)|2 = [G (' ω)]2 + [G"(ω)]2

Phas ea ngleδ(ω):tan δ(ω) = "(G ω)/ (G'ω)

For sma ll ω:

(G'ω) ≈ Ge + ω2[Go – Ge]∫0

∞ds sϕ(s) fluid⇒ (ωη)2Js

G''(ω) = ω[Go – Ge]∫0 ∞ds ϕ( )s fluid⇒ ωη

Carnegie Mellon 60

Exact relations among the dynamic moduli and compliances:

|G*(ω)||J* (ω)| = 1

(J' ω) = (G'ω)/| *G (ω)|2

"(J ω) = "(G ω)/| *G (ω)|2

(G'ω) = (J' ω)/|*J (ω)|2

"(G ω) = "(J ω)/|*J (ω)|2

tan δ(ω) = "(J ω)/ (J' ω) = "(G ω)/ (G'ω)

Carnegie Mellon 61

The dynamic viscosity:

In-phase with the strain rate:

η (' ω) = "(G ω)/ω

Out-o -f phase wi th the strain rate:η"(ω) = (G'ω)/ω

For sma ll ω:

η (' ω) = [Go – Ge]∫0 ∞ds ϕ( )s fluid⇒ η

η (''ω) ≈ Ge/ω + ω[Go – Ge]∫0 ∞ds sϕ(s) fluid⇒ ωη2Js

Carnegie Mellon 62

• Introduction









• The viscosity





Carnegie Mellon 63

Linear Viscoelastic Functions

Shear Compliance

Shear Modulus

J(t)

G(t)

Bulk Compliance

Bulk Modulus

B(t)

K(t)

Tensile Compliance

Tensile Modulusa

D(t) = J(t)/3 + B(t)/9

1/E(s) = 1/3G(s) + 1/9K(s)

a. The superscript " " denotes a Laplace transform.

Carnegie Mellon 64

Relation between G(t) and J(t)

1t ∫

0 tdu G(t – u) J(u) = 1

with Laplace transform:

s2G(s)J(s) = 1

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An often used relation between G(t) and J(t)

A weight set of exponentials with N relaxation times:

α(t) = ∑m

N-1

α i exp(–t/λi) = 1

J∞ – Jo ∫

-∞ ∞

d(ln λ) L(λ)exp(–t/λ)

ϕ(t) = ∑1

N

ϕ i exp(–t/τi) = 1

Go – Ge ∫

-∞ ∞

d(ln τ) H(τ)exp(–t/τ)

Notes: Σα i = Σϕ i = 1, and

m is equal to 0 or 1 for a solid and fluid

λ0 > τ1 > λ1 > … > λ i > τi > λ i+1 > … > λN-1 > τN

(λ0 absent for a fluid)

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Determination of L(λ) (or the αi-λi )set from ( )J t

(Simila r considerations apply to the determination of

(H τ) (or the ϕ i-τi )set from G( )t )

Derivative methods for L(λ):

1st Approx.: L(λ) ≈ ( )M m [∂ ( )R t /∂ln ]t t = λ

( )M m = ∂lnL(λ)/∂ln λ (interative)

2nd Approx.: L(λ) ≈ [∂ ( )J t /∂ln t – ∂2 ( )J t /∂(ln )t 2] t = 2λ

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Determination of L(λ) (or the αi-λi )set from ( )J t

(Simila r considerations apply to the determination of

(H τ) (or the ϕ i-τi )set from G( )t )

Invers e transform methods for αi-λi:

Th e inver se transform is "ill-posed", and a stable

solution s require s constraints (e. .g , αi ≥ 0)

In a n ofte n use d strategy, a set of logarithmically spaced

λ i ar e chosen suc h that the spa n in 1/λI does no t exceed

the tim e span in the experimental data. A constrained

nonlinea r le ast squares analysis is then use d to

determin e the αi. Commercia l package s are available for

thi s transform.

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• The viscosity





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Low Molecular Weight Glass Formerslope = 1/3H(τ)(L λ) = 1slope( )G t( )J tη tan δ(ω)t/ηGoJolog λ or log τ log t log ωt/ηη (' ω) = (G''ω)

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Low Molecular Weight Glass Formerslope = 1/3H(τ)(L λ) = 1slope( )G t( )J tη tan δ(ω)t/ηGoJolog λ or log τ log t log ωt/ηη (' ω) = (G''ω) < Polymeric Fluid with M Me = 1/3slope(H τ)(L λ) = 1slope = -1/2slope( )G t( )J tη tan δ(ω)GoJolog λ or log τ log t log ωt/ηη (' ω) = (G''ω)

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Polymeric Fluid with M >> Meslope = 1/3H(τ)(L λ)log λ or log τ = -1/2slope = 1slopeJ NGN( )G t( )J t log tη (' ω) = (G''ω) tan δ(ω) log ωGoJoηt/η Low Molecular Weight Glass Former = 1/3slope(H τ)(L λ) = 1slope( )G t( )J tη tan δ(ω)t/ηGoJolog λ or log τ log t log ωt/ηη (' ω) = (G''ω) < Polymeric Fluid with M Me = 1/3slope(H τ)(L λ) = 1slope = -1/2slope( )G t( )J tη tan δ(ω)GoJolog λ or log τ log t log ωt/ηη (' ω) = (G''ω)

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Peak I with L(λ) linear in λ1/3 before the peak decreases sharply tozer .o

• The behavior ascribed to peak I, first reported by Andrade, isseen in a variety of materials, such as metals, ceramics,crystall ine and glassy polymers and small organic molecules;the decrease of L(λ) to zero being evident in examples of the

latter.

• The area under peak I provides the contribution JA – Jo to the

total recoverable compliance Js.

• It seems likely that the mechanism giving rise to peak I may be

distinctly different from the largely entropic origins of peaks IIand III described in the following.

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Peak II that increases in peak area with increasing M until reaching

a certain level, beyond which the peak is invariant with increasing

M, both in area and position in λ

• Peak II is ascribed to Rouse-like modes of motion, either fluid-

like for low molecular weight in the range for which the area

increases with M, or pseudo-solid like (on the relevant time

scale) in the range of M after peak II I develops.

• For low molecular weight, the Rouse model gives the area of

peak II as

Js – (JA + Jo) = (2M/5ϕRT).

• For the pseudo-solid like behavior, obtaining when peak III has

developed, reflecting the effects of intermolecular

entanglement, the area of peak II becomes invariant with M and

given by

JN – (JA + Jo) = (Me/ϕRT).

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Peak III that develops as peak II area ceases to increase with

increasing ϕM, wi th peakIII deve loping an area invariant with

ϕM, anda maximum a tλMAX that move s to larger λ asλMAX ∝

(ϕM/Mc)3.4 fo r ϕM > Mc

• The area under peak III, also invariant with M, ascribed to theeffects of chain entanglements is given by

Js – (JN + JA + Jo) = (kMe/ϕ2+sRT),

where k is in the range 6-8 in most cases, and

s ≈ 2(ε – 1)/(3ε – 2) ≈ 0 to 1/4 with ε = ∂ln R2G/∂ln M

• Overall,

Js – (JA + Jo) = (2M/5ϕRT)[1 + (ϕ1+sM/kMc)ε]−1/ε

1.0 1.5 2.0 � 2.5 3.0

Log (X)

~

3.5

1.0

1.2

1.4

1.6

1.8

2.0

2.2

Log (J ϕ ) + .Cst

S

2

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• The viscosity





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Consider the following reduced expressions:

[J(t/τc) – Jo]/Js = [R( /t τc) – Jo]/ Js + t/η Js

[ (J t/τc) – Jo]/ Js = [R( /t τc) – Jo]/ Js + t/τc

τc = Jsη'(0) (= Jsη)

T he"tim –e temperatur eequivalence" approximation:

[ (J t/τc) – Jo]/Js is asingl -e value d func tion o f t/τc ove r ara nge of

temperature.

A lthoughrare , ly i f eve , r t rulyaccurat efo r a ll temperatur , e itisnever-the-les s a usefu l andwide ly use d approxima tionf or us ewithmateria lsexhibiting no phas etransition ove r th etemperaturerang e ofinterest.

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Since τc may not be know n over the ra nge o f temperature of

interest, it i s often useful to "reduce" data to a common reference

temperature TREF. Formally, this may be accomplished with

[ (J t) – Jo]/bTJs(TREF) = [R(t) – Jo]/bTJs(TREF) + /tηbTJs(TREF)

[ (J t) – Jo]/bTJs(TREF) = [R(t) – Jo]/bTJs(TREF) + t/hTbTτc(TREF)

[ (J t/aT) – Jo]/bT = [R(t/aT) – Jo]/bT + /thTbTη(TREF)

[ (J t/aT) – Jo]/bT = [R(t/aT) – Jo]/bT + /taTη(TREF)

bT = b(T, TREF) =Js(T)/Js(TREF)

hT = h(T, TREF) =η (' 0)[T]/η'(0)[TREF] {=η(T)/η(TREF)}

aT = bT hT

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Temperature

Specific Volume

Tg

A schematic v-T diagram for a typical

noncrystalline polymeric material.

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A Free Volume Model:

(vf)i = (v – vo)

i at a certain position ri,

v = (specific) volume

vf = free volume

vo

= occupied volume

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The glass transition temperature Tg

Tg depends on both intramolecular conformation and

intermolecular interactions.

Various Models/Treatments:

• Iso Free Volume: f(Tg) = constant

• Iso Viscous: η(Tg) = constant

• Iso Entropic: ∆S(Tg) = constant

None of these are fully satisfactory are free of arbitraryassumptions, and all contain parameters that can not beindependently evaluated.

The free volume and entropic models provide similar expectations

re the dependence of Tg on chain length and diluent.

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40

60

80

100

120

0 0.2 0.4 0.6 0.8 1

Syndiotactic fraction

T (°C)g

PMMA

40

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Estimation of Tg and Tm via Group Contributions

Tg ≈ M-1ΣYg,i

Tm ≈ M-1ΣYm,i

• The Yx,i represent molar group contributions to the relevant property

• Higher order approximations are available for both cases

D. W. van Krevelen, Properties of polymers : their correlation with chemical structure, theirnumerical estimation and prediction from additive group contributions, 3rd Ed., Elsevier;Amsterdam ; New York, 1990.

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Group Yg,i Group Yg,i Group Yg,i

1. Polyisobutylene only

Group contributions: gY (K·g/mol)

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Group Ym,i Group Group

Group contributions:Y (K·g/mol) m

Y m,iYm,i

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1.2

1.4

1.6

1.8

2.0

2.2

2.4

200 300 400 500 600 700

boyer

krevelen avg

krevelen calc

T /Km

T /Tg

m

D.W. Van Krevelen, op citR. F. Boyer, Rubber Reviews 36:1303-421

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Both free volume and entropic models give results that may be cast

in the forms:

Tg(M) ≈ Tg (∞){1 + kM/Mn}

1Tg(w, Μ) =

⎝⎜⎛

⎠⎟⎞w

Tg(Mn) + R

1 - wTg;DIL

⎝⎜⎜⎛

⎠⎟⎟⎞

1w + R (1 - )w

Both KM and R are model specific parameters, bes tevaluated

experimental .ly

For example, in thefree volume model, KM and R aris e from theextr afre e vol ume provided by chain ends and diluent, respectively:typical , ly R i s in therange 0.5 to 1.5.

Note, t hat i f Tg;DIL > Tg(Mn), then Tg(w, Mn) is increased bythe

diluent.

[ .G C. Berry J. Phys. Che .m 70:1194-8 (1966)]

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60

80

100

120

T (°C)g

0 1 2 3 4 5

10 /Mn4

p(Syndio) ~ 0.50

Free Radicalp(Syndio) ~ 0.76

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The temperature dependence of the viscosity:

η(T) ≈ ηLOC(T) F(large scale structure, T)

≈ ηLOC(T) F(large scale structure)

"Arrhenius" form:

ηLOC(T) ∝ exp[W/T] if T > (1.5-2)Tg

For melts of crystalline polymers, Tm > (1.5-2)Tg, permitting use

of this simple form.

"Vogel-Fulcher" form:

For amorphous polymers with 0 ≤ (T – Tg)/K < ≈ 200:

ηLOC(T) ∝ exp[C/(T – To)] if T < (1.5-2)Tg

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The temperature dependence of the viscosity:

η(T) ≈ ηLOC(T) F(large scale structure, T)

≈ ηLOC(T) F(large scale structure)

For amorphous polymers with 0 ≤ (T – Tg)/K < ≈ 200:

ηLOC(T) ∝ exp[C/(T – To)] if T < (1.5-2)Tg

"WLF form":

ηLOC(T)/ηLOC(TREF) = exp[C/(T – To) – C/(TREF – To)]

= exp⎝⎜⎜⎛

⎠⎟⎟⎞

– C(T – TREF)

ΔREF(T – TREF + ΔREF)

with C and To being constants, and ∆REF = TREF – To.

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If TREF = Tg then

ηLOC(T)/ηLOC(Tg) = exp⎝⎜⎜⎛

⎠⎟⎟⎞

– K (T – Tg)T – Tg + Δ

where Δ = Tg – To and K = C/Δ.

For many polymers:

K = 2300 K and Δ = 57.5 K

These parameters may be interpreted in terms of the "free-volume"model

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Viscosity of Polymers and Their Solutions

η( , M c, T) ≈ ηLOC(T) F(M, c, T)

Dilute solutions

ηLOC(T) ≈ ηSolvent(T)

F(M, c, T) ≈ 1 + [η]c + …

[η] = πNAKR2GRH/M

G. C. Berry J. Rheology 40:1129-54 (1996)

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F(M, c, T) ≈ 1 + [η] + …c

[η] = πNAKR2GRH/M

Spherica l Particles

R = RH = (5/3)1/2RG; K = 50/9

[η]c = (5/2)ϕ

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F(M, c, T) ≈ 1 + [η]c + …

[η] = πNAKR2GRH/M

Flexi ble Chain Linea r Polymers

R2G = (âL/3)α2; α the chain expansion factor

â the persistence length

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F(M, c, T) ≈ 1 + [η] + …c

[η] = πNAKR2GRH/M

Flexibl eC hainLinea r Polymers

R2G = (âL/3)α2; α th echa in expa nsionfactor

â th epersistenc e length

H :igh M

3RH/2 ≈ RG ∝ L1/2; K ≈ 10/3

ML[η] = πNA(20/9)(â/3)3/2α3L1/2 = Φ'(â/3)3/2α3L1/2

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F(M, c, T) ≈ 1 + [η] + …c

[η] = πNAKR2GRH/M

Flexibl eC hainLinea r Polymers

R2G = (âL/3)α2; α th echa in expa nsionfactor

â th epersistenc e length

H :igh M

3RH/2 ≈ RG ∝ L1/2; K ≈ 10/3

ML[η] = πNA(20/9)(â/3)3/2α3L1/2 = Φ'(â/3)3/2α3L1/2

Low :M

RH ≈ L; K ≈ 1

ML[η] = πNA(â/3)L (Debye)

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Flexible Chain Branched Polymers

ML[η] = πNAKR2GRH/L

g = R2G/(R

2G)LIN; calculated α = 1

High M:h = RH/(RH)LIN; K ≈ KLIN (f g, shap )eh ≈ g1/2

[η] = (f g, shap )e g3/2[η]LIN

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g = R2G/(R

2G)LIN; calculated α = 1

High M:h = RH/(RH)LIN; K ≈ KLIN (f g, shap )eh ≈ g1/2

[η] = (f g, shap )e g3/2[η]LIN

Sta :r [η] = g1/2[η]LIN

Comb: [η] = g3/2[η]LIN

Random: [η] = g[η]LIN

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g = R2G/(R

2G)LIN; calcula tedα = 1

H :igh M

h = RH/(RH)LIN; K ≈ KLIN (f ,g shap )eh ≈ g1/2

[η] = (f ,g shap )e g3/2[η]LIN

Star: [η] = g1/2[η]LIN

Com :b [η] = g3/2[η]LIN

Ra :ndom [η] = g[η]LIN

Low :M

[η] = πNAKR2GRH/LML

RH ≈ L; K ≈ 1

[η] = g[η]LIN

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η(M, ,c T) ≈ ηLOC(T) F(M, c, T)

Concentrated solutions and undiluted linear flexiblechain polymers

ηLOC(T) ≈ ηLOC(Tg)exp{ –K(T – Tg)/(T – Tg +∆)}

F(M, c, T) ≈ 1 + [η](c)c

Low M (Rouse behavior; α = 1):

F(M, c, T) ≈ 1 + ~X ≈

~X

~X = [η](c)c; a modified Fox parameter

ML[η](c) = πNA(â/3)L; ([η](c) independent of c in thisrange)

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High M (Entanglement regime)

F(M, c, T) ≈ 1 + ~XE(~X/~Xc ) ≈ ~XE(~X/~Xc )

E(~X/~Xc ) = {1 + (~X/~Xc )4.8}1/2

~Xc = πNA(â/3)ρMc ≈ 100 f or ma ny polymers

Mc = ~Xc/πNA(â/3)ρ ≈ 100/πNA(â/3)ρ

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The dependence of Tg on the diluent concentration must be

considered for polymer solutions:

ηLOC(T)/ηLOC(Tg) = exp⎝⎜⎜⎛

⎠⎟⎟⎞

– K (T – Tg)T – Tg + Δ

where Δ = Tg – To and K = C/∆.

For many polymers:

K = 2300 K and Δ = 57.5 K

∆ is approximately independent of the polymer concentration

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0 0.2 0.4 0.6 0.8 1

Volume Fraction Polymer

To

Tg

T – Tog

0

100

200

300

400

Temperature/K

Polystyrene/Dibenzyl ether

G. C. Berry and T. G Fox Adv. Polym. Sci. 5:261-357 (1968)

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3 4 5 6

log( ϕ )Mw

0

2

4

-2

(log η / )Pa·s

1

3

5

-1

1.0

0.75

0.50

0.25

0.125

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η(M, c, T) ≈ ηLOC(T) F(M, c, T)

Branched C hain Po lymer s (Concen tra ted or und iluted)

ηLOC(T) ≈ [ηLOC(T)]LIN; Rare exceptions to this known

F(M, c, T) ≈ 1 + [η](c)c

ML[η](c) = πNA(â/3) gL

F(M, c, T) ≈ 1 + ~XE(

~X/

~Xc );

~X = [η](c)c

E(~X/

~Xc ) = {1 + B(g, MBR/Mc)(

~X/

~Xc )

4.8}1/2

B(g, MBR/Mc) ≈ 1 unless the branch molecular MBR > Mc

~Xc = πNA(â/3)ρMc ≈ 100 for many polymers

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Moderately Concentrated Solutions

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η(M, c, T) ≈ ηLOC(T) F(M, c, T)

Modera te ly Concentra ted Solutions

ηLOC(T) ≈ [ηLOC(T)]1 - µc = 0 [ηLOC(T)]µ

c = ρ; µ ≈ ϕ = c/ρ

F(M, c, T) ≈ 1 + [η](c)c

ML[η](c) = πNA(â/3)α(c)2(RH(c)/L)L

F(M, c, T) ≈ 1 + H(c)~XE(~X/

~Xc );

~X = [η](c)c

E(~X/

~Xc ) = {1 + (

~X/

~Xc )

4.8}1/2

~Xc = πNA(â/3)ρMc ≈ 100 for many polymers

[G. C. Berry J. Rheology 40:1129-54 (1996)]

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• Introduction









• The viscosity





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Molecular Weight Polydispersity

• ηLOC(T) scales with Mn through Tg

• η/ηLOC(T) scales with Mw, except perhaps for unusual

distributions

• Peak I in L(λ) is essentially unaffected by molecular weight

dispersion

• Peak II in L(λ) may comprise two pieces:

i) an area proportional to ϕLMzMz+1/Mw, with the averages

calculated for chains with M < Me at volume fraction ϕL, and

ii) an area proportional to (1 – ϕL)Me for chains with M > Mc at

volume fraction 1 – ϕL

• Peak III in L(λ) has an area proportional to

(1 – ϕL)-2(Mz/Mw)2.5

• The maxima for peaks II and III separate in λ as (1 – ϕL)Mw

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Theoretical treatments are usually cast in terms of G(t), often in the

form:

G(t) = {Σi wiGi(t)

ν}1/ν

Gi(t) = shea r modulu s for chains with Mi

at weight fraction wi

For example:ν = 1 in the "repta tion modelν = 1/2 in the "doubl -e reptation" model

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Theoretical treatments are usually cast in terms of G(t), often in the

form:

G(t) = {Σi wiGi(t)

ν}1/ν

Gi(t) = shea r modulu s for chains with Mi

at weight fraction wi

For example:ν = 1 in the "repta tion modelν = 1/2 in the "doubl-e reptation" model

The effects o f increased dispersity of molecula r specie s is usuallymost promi nent in Peak III in L(λ), followed by effect s in Pe akIIin L(λ). This i sseen in (L λ) fo r a polymer undergoing

crosslinking to form a branched polymer, le ading to anetworkpolymer

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• Introduction









• The viscosity





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Power-law behavior

G(t) = [Go – Ge]ϕ(t) + Ge

J ( )t = Jo + ψ(t) + t/η

ψ(t) = (Js – Jo)[1 – α(t)]

Suppose that for all t (note, this involves permissible, but peculiarbehavior for large t):

ψ(t) = (t/λ)μ

With this expression, and 1/η = 0:

[J'(ω) – Jo]/Jo =µΓ(µ)cos(µπ/2) (ωλ)-µ

J"(ω)/Jo =µΓ(µ)sin(µπ/2) (ωλ)-µ

Use of the convolution integral relating J(t) and G(t) gives

ϕ(t) =Eµ(-kµ(t/λ)µ)

with Ge = 0 and 1/η = 0, where kµ = µΓ(µ) and

Eµ(x) =∑n=0

∞

xn

Γ(nµ + 1) : The Mittag-Leffler function

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For small µ,G(t) ≈ Go{1 + ( /tλ)µ}−1

For a nyµ, for large t/λ

G(t) ≈Gos (in µπ)/µπ( /tλ)µ

G(t)J( )t ≈s (in µπ)/µπ < 1

(G'(ω) – Ge)/(Go – Ge) ≈Γ(1-μ) si [(n 1-μ)π/2] (ωλ)μ

"G (ω)/(Go – Ge) ≈Γ(1-μ) c [os(1-μ)π/2] (ωλ)μ

[J '(ω) – Jo]/Jo =µΓ(µ)cos(µπ/2) (ωλ)-µ

J"(ω)/Jo =µΓ(µ)s (in µπ/2) (ωλ)-µ

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Bounded power-law behavior for ϕ(t) might be obtained in th eform

ϕ(t) =1; for t ≤ τΝ=(τΝ/t)μ; for τΝ < t ≤ τ1, with 0 < µ < 1

=(τq/t)m; for > t τ1, with m> 1

where τq = τ1(τΝ/τ1)μ/m.The ,n

G'(ω) – Ge ∝ ω2 andG"(ω) ∝ ω for ω << 1/τ1;

G'(ω) = Go andG"(ω) = 0 forω >> 1/τΝ;

(G'(ω) – Ge)/(Go – Ge) ≈Γ(1-μ) si [(n 1-μ)π/2] (ωτΝ)μ

"G (ω)/(Go – Ge) ≈Γ(1-μ) c [os(1-μ)π/2] (ωτΝ)μ

for the interva 1/l τ1 < ω < 1/τΝ.

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An alternative relation that also exhibits partial power-law behavior is givenby:

ϕ(t) =⎝⎜⎜⎛

⎠⎟⎟⎞∑

i = 1

Ν

(τ1/τi) n/m e (– /xp tτi) /⎝⎜⎜⎛

⎠⎟⎟⎞∑

i = 1

Ν

(τ1/τi) n/m

where τi = τ1/im; m = 2 a nd n = 0 in the Rouse mode .l

For the intermediat e interva 1l /τ1 < ω < 1/τΝ,

(G'(ω) – Ge)/(Go – Ge) ≈{π/2m sin[(1-μ)π/2]} (ωτ1)μ

"G (ω)/(Go – Ge) ≈{π/2m sin[(2-μ)π/2]} (ωτ1)μ

where µ = (1 + n)/m ( =µ 1/2, for the Rouse mode )l .

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• The viscosity





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ISOCHRONAL BEHAVIOR

• In some cases, the temperature is scanned while the dynamic

properties are determined at fixed frequency; such experiments

might typically be reported as G'(ω;T) an d tan δ(ω; T) or η'(ω;T)

versus T, depending on the application.

• Insofar a s G'( ωτc(T)) a nd G "(ωτc(T)) a s func tions of ωτc(T) a re

inde pen dent of T, the isoc hronal plots a re s e en t o be mapp ings

in which ωτc(T) increases with dec reas ing tem pera ture with:

τc(T) ∝ exp⎝⎜⎜⎛

⎠⎟⎟⎞

KT - (Tg - Δ)

• For a reference temperature equal to the glass temperature Tg,

so that aΤ = τc(T)/τc(Tg):

ln aΤω = ln ω – KΔ2

⎝⎜⎜⎛

⎠⎟⎟⎞

T - Tg1 + (T - Tg)Δ ≈ k1 + k2(T - Tg) +

…

with the linear approximation valid for (T - Tg) << Δ; k1 = ln ω and

k2 = K /Δ2.

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1

0

-1

-2

-3

420-2-4

− ( log a ω )

-10 0 10 20

- T T g

- T T g = 0

ω = 1 s

-1

1

0

-1

-2

-3

/Log G' G o and Log tan δ

Log tan δ

/Log G' Go

/Log G' Go

Log tan δ

T

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• Introduction









• The viscosity





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Examples from the literature

• Branched and linear metallocene polyolefins

• Colloidal dispersions

• Wormlike Micelles

• Deformation of rigid materials

• Nonlinear shear behavior

• Linear and nonlinear bulk properties

Carnegie Mellon 162

log ω-1012012345log G'(ω)log G''(ω)Unmodified Linear

Metallocene polyethylenesClaus Gabriel and Helmut Münstedt Rheol. Acta 38: 393-403 (1999)

Carnegie Mellon 163

log ω-1012012345log G'(ω)log G''(ω)Unmodified Linear2345log G'(ω)log G''(ω)Modified Branched 1


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log ω-1012 (log G'ω) (log G''ω)2345M 2odified Branched2345 (log G'ω) (log G''ω)M 1odified Branched


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-2-1012-4-5log ω (log J' ω) log η (' ω) Unmodified Linear 2Modified Branched34 1Modified Branched


Carnegie Mellon 166

23456-4-5log J'(ω) log η (' ω)/η (' 0) Unmodified Linear 1Modified Branched 2Modified Branched-10 log ω η(' 0)


Carnegie Mellon 167

log ω η(' 0) b23456-4-5 (log J' ω)/b log η (' ω)/η (' 0) Unmodified Linear 1Modified Branched 2Modified Branched-10 U 1M 2M

log η (' 0) 3.28 3.68 4.00

log b -0.7 0 0

Carnegie Mellon 168

Carnegie Mellon 169

From creep/recovery

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Carnegie Mellon 173

Colloidal dispersions: Linear and nonlinear

viscoelastic behavior.

Dilute dispersion of spheres interacting via a hard-core

potential:

η = ηLOC{1 + (5/2)ϕ + k'(5/2)2ϕ 2 + …}

ϕ = volum efracti on = c/ρ(5/2)ϕ = [η]c

ηLOC ≈ ηsolv.

k' ≈ 1.0

Carnegie Mellon 174

Concentrated dispersion of hard-core spheres:Empirical relations:

η ≈ ηLOC{1 – ϕ/n1}–5n1/2

η ≈ ηLOC{1 – (5/2)ϕ[1 – ϕ/n2]–5n2k /' 2}

desi gnedto force agreemen twith t he viria l expansion at least to orde r ϕ and ϕ2,

respectivel ,y

n1 = 5/8 to gi vek' ≈ 1.0

n1 = ϕmax ≈ 0.64

Theoretica l relations:

η = ηLOC{1 + (5/2)ϕ + k'[ψ1(ϕ) + ψ2(ϕ)](5/2)2ϕ2}

ψ1(ϕ): hydrodynamics

ψ2(ϕ): thermodynamics

ψ1(0) + ψ2(0) =1

U: ψ1(ϕ) ≈ (4/5)(1 – ϕ/ϕmax)

ψ2(ϕ) ≈ (1/5)(1 – ϕ/ϕmax)2

(se -miempircial)

Carnegie Mellon 175

Carnegie Mellon 176

Concentrated dispersion of hard-core spheres:

Linear Viscoelastic Response:

• η'(ω) = η'(0) for small ω, as expected, but also show a plateau η'(ω) ≈ η'(ωL) for a

regime at an intermediate range of ω ≈ ωL, before decreasing to zero with increasing

ω.

• η'(ωL) is estimated with ψ2(ϕ) = 0, reflecting the suppression of thermodynamic

interactions at high ω

• G'(ωL) ≈ G1; G1R3/kTϕ2 ≈ ψ0(ϕ) for spheres of radius R

• ψ0(ϕ) ≈ 0.78(η'(ωL)/ηsolv)g(2, ϕ)

g(2, ϕ) is the radial distribution at the contact condition r/R = 2

Theory:

g(2, ϕ) = (1 – ϕ/2)2/(1 – ϕ)3 for ϕ < 0.5 and

g(2, ϕ) = (6/5)(1 – ϕ/ϕmax) for ϕ ≥ 0.5

Carnegie Mellon 177

Carnegie Mellon 178

Concentrated dispersion of hard-core spheres:

Linear Viscoelastic Response:

Theory:• η'(0) = ηLOC{ 1 + (5/2)ϕ + k'[ψ1(ϕ) + ψ2(ϕ)](5/2)2ϕ2}

• η'(ωL) = ηLOC{ 1 + (5/2)ϕ + k'[ψ1(ϕ)](5/2)2ϕ2}

• J'EFF(ω) ∝ ω-1/2 for a range of ω < ωL

• J'EFF(ωL) ≈ 1/G'(ωL) ≈ 1/G1 ≈ R3/kTϕ2ψ0(ϕ)

Carnegie Mellon 179

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Concentrated dispersion of interacting spheres:

• Van der Waals interactions

• Electrostatic interactions among charged spheres

• Interactions among spheres and a dissolved polymer

• True or apparent yield behavior may obtain

Carnegie Mellon 181170 nm beads (0.05 to 0.2 volume fraction), in 15% polystyrene solutionD. Meitz, L. Yen, G. C. Berry and H. Markovitz J. Rheol. 32:309-51 (1988)

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Carnegie Mellon 183

Wormlike micelles

Certain amphillic molecules organize to form curvilinear cylinders, or wormlike

micelles. For example, in an aqueous medium, the amphiphile might organize

with its hydrophobic parts aggregated in the interior of the cylinder, and its

hydrophopic pieces arranged on the "surface" of the cylinder

The micelle structure will exhibit a lifetime τrupture for rupture of its components

If τruptur e is les s th an a longes t rheologica l tim e constan t τrheol the intac t wormlike

micelle would exhib ,it then the rupture dynamics may dominate the observed

rheological behavior,

The chain ma y respond to a deformation b y micellar dynamics similar t o those

for a structu re without rupture, abetted by the rupture process.

With one mode ,l this approximate s Maxwell behavio r wit h a tim e constant

τeffectiv e≈ (τruptureτrupture)1/2

Carnegie Mellon 184

Cetyl triethylammonium tosylate

+hydrophobic

hydrophilic

+++

+++

++ + + + +

+ +++++++

+

-

-

micelles grow10 nm

micellar network

CTA+ -T

Schematic courtesy Dr. Lynn M. Walker

–

Carnegie Mellon 185

In an extreme case, the system might approximate behavior for the Maxwell model, with a single

relaxation time τeffectiveso that

J (t) = J s + t/η; wit h J sη = τeffective

(G t) = (1/J s)ex (p -t/τeffective)

With this simple model,

J '(ω) = J s

η'(ω) = (1/J s)/[1 + (ωτeffective)2]

Carnegie Mellon 186J. F. A. Soltero and J. E. Puig Langmuir 12: 141-8 (1996)

20%

10%These data reveal several deviations from simple Maxwell behavior, including:

◊ The rate of decrease of ⏐ '(⏐ ) with increasing ⏐ for larger ⏐ , to the extentof an increase in ⏐ '(⏐ ) with increasing⏐ for the data on the less concentratedSample

◊ The increase of J'(⏐ ) above the imputed Js for smaller ⏐ for the data on the more

concentrated sample

◊ It may be likely that these samples exhibitsolid-like behavior with a Je at smaller ⏐

than the experimental range, and that Jp

is truly Js

◊ The relatively constant J'(⏐ ) is expectedwith the Maxwell model, but this may be fortuitous

-3 -2 -1 0 1 2

-3

-2

-1

0

1

2

-3

-2

-1

0

30 35 40

10%

20%3.5

4.0

4.5

-3

-2

Temperature (°C)

log η/η

solvent

/log J Pas

log ω η Jpp

log η '(ω) / η (or log J' ω )/J

p

p

30

35

40

(° )T Cη ' (ω) / η (J' ω )/Jpp

3

-4

Calculated

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Carnegie Mellon 188

Deformation of Rigid Materials

Creep and Recovery in Tension

Creep for 0 ≤ t ≤ Te

ε(t) = σo (D t) = σo[DR(t) + DNR(t)]

Recover y fo r ϑ = t– Te > 0

ε(ϑ, Te) = σo[DR(ϑ + Te) – DR(ϑ) + DNR(Te)]

εR(ϑ, Te) = ε(Te) – ε(ϑ, Te)

= σo{DR(Te) – DR(ϑ + Te) + DR(ϑ)}

Carnegie Mellon 189

G. C. Berry J. Polym. Sci.: Polym. Phys. Ed. 14:451-78 (1976)

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(t/sec)1/3 (ϑ/sec) or [ϑ + T )/sec] – (ϑ/sec) 1/31/3 1/3

0 5 10 15 20 250 5 10 15 200

1

2

3

D(t)/MPa

-1

(a) (b)

Andrade Creep (with DNR(t) ≠ 0)

A nonrecoverable logarithmic creep is frequently observed under

larger stress:

DNR(t) ≈ DL ln(1 + µt/DL) µt/DL <<1

⇒ µt

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Carnegie Mellon 193

An "Incompressible" Isotropic Elastic Material

Suppose K >> G, then for infinitisimal strains

Sij = 2 G {εij – δij3 εαα} – δij P

Mor e generall ,y for finit e strain :s

Sij = W1 Bij + W2Bij-1 – δij P

Wi = Wi (I B;1, IB;2) – ∂W ∂IB;i

For simple extension:

f/ A ≈ 2(λ2 – λ-1)(W1 + W2/ λ)

For simple shea :r

S12 = 2(W1 + W2) γ = G γ

S11 – S33 = 2W1 γ2 ; S22 – S33 = – 2W2 γ2

Carnegie Mellon 194

An expansion of the strain energy function gives theMooney–Rivlin Equation for small deformations:

W ≈ C1 (IB;1 – 3) + C2 (IB;2 – 3)

W1 = C1 and W2 = C2

For the original Kinetic Theory of Rubber Elasticity thecontributions to C1 are entropic in origin, and.:

2C1 = νEkT = ρR /T MXL

2C2 = 0

νE = Numb erof chains understress

MXL = Molecula r weight ofchains

betw eencrosslinks

The precedin g estimat esfor C1 an d C2 a renotaccura ,te an d hav ebeen modified in m oremoderntheori ,es e. .,g th esegive C2 > 0.

Carnegie Mellon 195

An "Incompressible" Viscoelastic Material

Suppose K(t) >> G(t), then for infinitisimal strains

Sij(t) = 2⌡⎮⎮⌠

-∞

t

G(t – s)⎝⎜⎜⎛

⎠⎟⎟⎞∂εij( )s

∂s – δij3 ∂εαα( )s

∂s ds – δijP

Several relations are proposed for finit e strains,including that du e to Bernste ,in Kearsley an dZapas::

Sij(t) = 2⌡⎮⎮⌠

-∞

t

⎝⎜⎜⎛

⎠⎟⎟⎞∂U

∂IB;1 B(t)ij( )s –

∂U ∂IB;2

B(t)ij( )s-1 ds – δijP

Carnegie Mellon 196

An "Incompressible" Viscoelastic Material

Suppose K(t) >> G(t), then for infinitisimal strains

Sij(t) = 2⌡⎮⎮⌠

-∞

t

G(t – s)⎝⎜⎜⎛

⎠⎟⎟⎞∂εij( )s

∂s – δij3 ∂εαα( )s

∂s ds – δijP

Several relations are proposed for finit e strains,including that du e to Bernste ,in Kearsley an dZapas::

Sij(t) = 2⌡⎮⎮⌠

-∞

t

⎝⎜⎜⎛

⎠⎟⎟⎞∂U

∂IB;1 B(t)ij( )s –

∂U ∂IB;2

B(t)ij( )s-1 ds – δijP

Carnegie Mellon 197

Nonlinear Response in Simple Shear for aFluid

(In the approximation with t >> τR)

Sh ear Stress σ(t) = S12(t):

σ(t) = – ⌡⎮⌠

0

∞

[Δγ(t,u)] F1[Δγ(t,u)] ∂G(u)

∂u du

σ(t) = ⌡⎮⌠

-∞

t

G(t – u) ∂γ(u) ∂u M1[Δγ(t,u)] du

Δγ(t,u) = γ(t) – γ(u)

M1[Δγ(t,u)] = ∂γ F1(γ)

∂γ = F1(γ)⎩⎨⎧

⎭⎬⎫

1 + ∂ l n F1(γ)

∂ l n γ

Carnegie Mellon 198

Nonlinear Response in Simple Shear for aFluid

(In the approximation with t >> τR)

First–Normal Stress Difference ν (1)(t) = σ11(t) – σ22(t) :

ν(1)(t) = – ⌡⎮⌠

0

∞

[Δγ(t,u)]2 F1[Δγ(t,u)] ∂G(u)

∂u du

ν(1)(t) = ⌡⎮⌠

-∞

t

G(t – u) ∂γ(u) ∂u M2[Δγ(t,u)] du

M2[Δγ(t,u)] = ∂ γ2 F1(γ)

∂γ = γF1(γ)⎩⎨⎧

⎭⎬⎫

2 + ∂ l n F1(γ)

∂ l n γ

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Response to a Ramp Deformation

γ(t) = ·γ t t > 0

Stress Growth:

σ(t) = ·γ ∫t

0 G(s) ×

F1(·γ s)

⎩⎪⎨⎪⎧

⎭⎪⎬⎪⎫

1 + ∂ l n F1(

·γ s)

∂ l n ·γ s ds

ν(1)(t) = ·γ2 ∫t0 sG(s) ×

F1(·γ s)

⎩⎪⎨⎪⎧

⎭⎪⎬⎪⎫

2 + ∂ l n F1(

·γ s)

∂ l n ·γ s ds

Carnegie Mellon 202

Steady-State Flow

Viscosity

limt >> τc

σ( )t =σSS(·γ)

η(·γ) = σSS(·γ)/·γ

limγ=0

η(·γ) = η(0) = η

η(·γ) = η(0) H(τc·γ /γ'')

H(τc·γ /γ'') =

⌡⎮⎮⌠

0

∞

(G u)M1[·γ ]u du

⌡⌠

0

∞

(G u)du

Carnegie Mellon 203

Steady-State Flow

First-Normal Stress Difference

limt >> τc

ν(1)(t) = ν(1)SS(·γ)

N(1)(·γ) = ν(1)SS(·γ) /2{σ

SS(·γ)}2

limγ=0

N(1)(·γ) = Js

N(1)(·γ) = Js SN(τc

·γ/γ'')

SN(τc

·γ/γ'') =

⌡⎮⎮⌠

0

∞

uG(u)M2[·γ u]du

⌡⌠

0

∞

uG(u)du

⎝⎜⎜⎜⎜⎛

⎠⎟⎟⎟⎟⎞

⌡⎮⎮⌠

0

∞

G(u)M1[·γ u]du

⌡⌠

0

∞

G(u)du

-2

Carnegie Mellon 204

Steady-State Flow

Steady-State Recoverable Compliance

limt;θ >>τc

γR( ,tθ) = γ

R(·γ)

RSS(·γ) = γ

R(·γ)/σ

SS(·γ)

limγ=0

RSS(·γ) = Js

RSS(·γ) = Js SR(τc

·γ/γ'')

SR(τc·γ/γ'') = Result of a n iterativ e calculation

involving (G t) andF1(γ)

Carnegie Mellon 205

Suppose

G(t) = Go∑ϕ iexp(–t/τi); ∑ϕ i = 1

The , n with t he approxim ateF1(γ) given above

η(γ · ) =Go∑ϕ i τi (H γ ·τi/γ )''

(H γ ·τi/γ )'' ≈1

[1 + (βγ·τi/γ ' )' ε]2/ε ; ε ≈ 6/5, β ≈ 1

By comparison,

η (' ω) =Go∑ϕ i τi 1

[1 + (ωτi)2]

I n both cases, the factors ϕ i τi in the term s in t he summation are

weighted by functions tha t decreas eter – –m by term with increasing γ · orω.

Consequent , ly thes eexpression sexhibit the Cox-Merz approximation:

η(γ ·) ≈η'(ω = γ ·)

Carnegie Mellon 206

Carnegie Mellon 207

-3 -2 -1 0 1 2 3

0

-1

0

-1

log( τ γ )

c

·

[log η(γ )/ η(0 )]·

[ log S (γ )/J ]

·

(1)

s

[log J (γ )/J ]

·

s

s

0

-1

PolyethyleneK. Nakamura, C.-P. Wong and G. C. Berry J. Polym. Sci: Polym. Phys. Ed. 22:1119-48 (1984)

Carnegie Mellon 208

-2 -1 0 1 2 3

0

-1

log( τ γ )

c

·

[log η (' ω )/ η(0 )]

[ (log J' ω )/J ]

s

[ log S (γ )/J ]

·

(1)

s

0

-1

[log η ( γ )/ η(0 )]·

-1

0

-2

-1

0

-2

Linear and nonlinear behavior for a polymer with a relatively narrow MWD

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Carnegie Mellon 210

Carnegie Mellon 211

D. J. Plazek and G. C. Berry , in Glass: Science and Technology, Vol. 3 Viscosity and Relaxation, ed. D. R. Uhlmann and N.J. Kreidl, Academic Press (1986).

Carnegie Mellon 212

An Inherent Nonlinearity in Response

B(t) = B(0) + ΔB β(t)

β(t) = β(t/ τκ)

But τκ = τκ(V,T)

An attempt to accoun t for this effec t m akes use of anmaterial t imeconstant averaged over t he timeinterval o f interes :t

⟨τκ−1(t2 ,t1)⟩ =1

(t2 - t1) ∫

t2t1 τκ−1(u) du

(V t) – (V 0)(V 0) = ∫t

-∞ [(B t – s) ⟨τκ−1(t ,s)⟩]∂P(s) ∂s ds

Frequently,

B(t) = BA{1 + (t/τA)1/3}; t < το

Carnegie Mellon 213

D. J. Plazek and G. C. Berry , in Glass: Science and Technology, Vol. 3 Viscosity and Relaxation, ed. D. R. Uhlmann and N.J. Kreidl, Academic Press (1986).

Carnegie Mellon 214D. J. Plazek and G. C. Berry , in Glass: Science and Technology, Vol. 3 Viscosity and Relaxation, ed. D. R. Uhlmann and N.J. Kreidl, Academic Press (1986).

Carnegie Mellon 215

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