analytical rheology - people

Analytical RheologyLinear Viscoelasticity of Model and Commercial

Long-Chain-Branched Polymer Melts

SachinSachin ShanbhagShanbhag, , SeungSeung JoonJoon Park, Park,QiangQiang Zhou and Ronald G. Larson Zhou and Ronald G. Larson

Chemical Engineering, University of Michigan

03/06

Motivation

Polyethylene and polypropylene > 50% of thetotal synthetic polymer produced world-wide.

linear

comb

Long Chain Branching

LCB > 100 C-atoms

In LDPE ~ 10 LCB/1000backbone carbon atoms

Motivation

comb


Rheological properties → processingproperties

LDPE: strain hardening, shear thinning

Wood-Adams and Dealy, Macromolecules 2000

mPE (150°C) same MW

Analytical Analytical RheologyRheology of Long-Chain of Long-ChainBranched Branched PolyethylenesPolyethylenes

• spectroscopy, chromatography

• rheology is sensitive

• need accurate rheological models of branched

structures

metallocene catalysts

Polymer solutions/melts are viscoelastic

Viscosity = “fluids”Elasticity = “solids”

store energy, memory

relax or dissipateenergy

Elastic Recoil

http://burghdt3.chem-eng.northwestern.edu

EntangledEntangled Polymer Melts Polymer Melts

3.4

1

log M

log η

0

http://zeus.plmsc.psu.edu/~manias

UNENTANGLED

high interpenetration

ENTANGLED

Linear Linear ViscoelasticityViscoelasticity

σ(t)

γ0

t

G(t) = σ(t)/γ0

Relaxation Modulus

dtùt)cos(G(t)ù)(ùG

dtùt)sin(G(t)ù)(ùG

0

0

!

!"

"

=##

=#

Dynamic Moduli

Storage Modulus

Loss Modulus

Branched PolymersBranched Polymers

linear

star

H-polymer

comb


The Tube ModelThe Tube Model

courtesy: Richard Graham

matrix chain

test chaintube

http://zeus.plmsc.psu.edu/~manias

The Tube ModelThe Tube Model

Stress RelaxationStress Relaxation

Arm Retraction

τlinear ~ M3.4τarm ~ exp(M)

Reptation

σ(t)

γ0

t

G(t) = σ(t)/γ0

Relaxation Modulus

ReptationReptation

courtesy: IRC Leeds

Arm RetractionArm Retraction

courtesy: IRC Leeds

Third Mode: Constraint ReleaseThird Mode: Constraint Release

A = test chainB, C, D = matrix chains

Equations for Equations for MonodisperseMonodisperse Stars Stars

443

16

9)( !"#!" aeearly S=

eaaMMS /=

2/12)1/(2

22

2/15

2/3

1

1

3

1)1(

)](exp[

6)(

!!

"

#

$$

%

&'(

)*+

,

+-''

(

)**+

, ++.

''(

)**+

,=

.+

/

/00

01202

//

/

a

eff

aelate

S

US

)2)(1(

])1(1[)1(12)(

1

!!

"!"#"

!

++

++$$=

+

aeff SU

2/3=!

)(/)](exp[)(1

)](exp[)()(

!"!!"

!!"!"

lateeffearly

effearly

aU

U

+=

Ball and McLeish, Macromolecules, 1989

Milner and McLeish, Macromolecules, 1997

Milner and McLeish, Macromolecules, 1998

!"

#$%

&

+'+= ( )(1

)()1()1(/)(*

1

0

0

)*+

)*+)),* ,

i

idGG

N

"exp " onentdilution=!

2)( !"!aSU =Early-time fluctuations

Late-time retraction

Crossover equation

Complex modulus

Retraction Potential

Dynamic Dilution

!

Me

=4

5

"RT

GN

0

!

"e

=#(M

e/M

0)2b2

3$2kBT

!

" = 0

!

" =1

!"" /)(ee

MM =

• Star polymer– Milner and McLeish, Macromolecules (1997)

• Linear polymer– Milner and McLeish, Phys Rev Lett (1998)

• Star/linear blend– Milner et al., Macromolecules (1998)

• H polymer– McLeish et al., Macromolecules (1999)

• Comb polymer– Daniels et al., Macromolecules (2001)

• Mixture of general branched polymers– Larson, Macromolecules (2001): hierarchical model

Theories for BranchedTheories for Branched Polymer Polymerss

Park, Shanbhag, Larson, Rheol. Acta., 2005Larson, Macromolecules, 2001

+

+

+

Successes of the Tube Theory: PreviewSuccesses of the Tube Theory: PreviewLaun/Schmidt Benchmark ExperimentAnalytical Rheology of Commercial Linear Polymers

MWD from GPC (BASF)

0

20

40

60

80

100

1E+3 1E+4 1E+5 1E+6 1E+7

molar mass (Dalton)

we

igh

t fr

ac

tio

n w

(lg

M)

(%)

PS1 linear narrow

PS2 linear broad

PS3 linear trimodal

PS5 linear broad

Polystyrene samples from Dr. Christian Schade, BASF

Successes of the Tube Theory: PreviewSuccesses of the Tube Theory: Preview

PS2 @ 170°C

1.E-3

1.E-1

1.E+1

1.E+3

1.E+5

1.E+7

1.E+9

1.E-5 1.E-3 1.E-1 1.E+1 1.E+3 1.E+5 1.E+7 1.E+9

angular frequency ? (rad/s)

mo

du

li G', G''

(P

a)

BASF G'

BASF G''

Carrot G'

Carrot G''

Larson G'

Larson G''

Leonardi G'

Leonardi G''

Pattamaprom G'

Pattamaprom G''

Schulze G'

Schulze G''

Likhtman G'

Likhtman G''

Ruymbeke G'

Ruymbeke G''

Overview on moduli predictions for PS2

1

10

100

1000

104

105

106

107

10-5

10-3

10-1

101

103

M1

M2

G', G"

(Pa)

! (sec-1

)

PS, linear

G'

G"

Successes of the Tube TheorySuccesses of the Tube Theory

Polydisperse polystyrene, Graessley and coworkers(lines theory of Pattamaprom, et al.)

1% high molecularweight tail added!

Determination of Determination of ModelModelParametersParameters (1,4- (1,4-PBdPBd))

molecular weight

104 105 106

zero

-shear v

iscosity

(Pa.s

)

101

102

103

104

105

106

107

108

Struglinski and Graessley (1985)

Colby et al. (1987)

Roovers (1987)

Rubinstein and Colby (1988)

Baumgaertel et al. (1992)

Milner and McLeish (1998)

arm molecular weight

104 105

zero

-she

ar visco

sity (Pa

.s)

102

103

104

105

106

107

108

Raju et al. (1981)

Roovers (1985)

Roovers (1987)

Struglinski et al. (1988)


GN=1.15E+6 (Pa), Me=1650, τe=3.7E-7 (sec)

zero-shear viscosity of linear1,4-polybutadiene at T=25 °C

zero-shear viscosity of star 1,4-polybutadiene at T=25 °C

h0

a = 4/3Park, Shanbhag, Larson, Rheol. Acta., 2005

molecular weight

104 105 106

zero

-shear v

iscosity

(Pa.s

)

102

103

104

105

106

107

Gotro and Graessley (1984)


GN=0.44E+6 (Pa), Me=4054, τe=1.0E-5 (sec)

zero-shear viscosity of linear1,4-polyisoprene at T=25 °C

zero-shear viscosity of star 1,4-polyisoprene at T=25 °C


104 105

zero

-she

ar visco

sity (Pa

.s)

102

103

104

105

106

107

108

3-arm Fetters et al. (1993)

4-arm Fetters et al. (1993)


Determination of Determination of ModelModelParametersParameters (PI) (PI)

a = 4/3Park, Shanbhag, Larson, Rheol. Acta., 2005

Hierarchical MHierarchical Model Predictionodel Prediction

PBd linear (M1=37K)/linear(M2=168K) blend at T=25°C

Struglinski et al., Macromolecules, 1985


PBd 3-arm star (M=127K)/linear(M=100K) blend at T=25°C

Frequency (1/s)

10-2 10-1 100 101 102 103

G' (P

a)

102

103

104

105

106

107

!2=0.0

!2=0.1

!2=0.3

!2=0.5

!2=0.7

!2=1.0

model

Frequency (1/s)

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

G" (P

a)

102

103

104

105

106

!s=1.0

!s=0.75

!s=0.5

!s=0.2

!s=0.0

model

G’ G’’

Gr = M2Me2/M1

3 = 0.01 < Grc = 0.064

Constraint release is by CR-Rouse motion

Park, Shanbhag, Larson, Rheol. Acta., 2005

1.00E+02

1.00E+03

1.00E+04

1.00E+05

1.00E+06

1.00E+07

0 0.2 0.4 0.6 0.8 1

volume fraction of star

zero

sh

ear

vis

co

sit

y (

Pa.s

)PBd 3-arm star (127K) - linear blends at T=25 0C

L: 168K

L: 100K

L: 37K

Star/Linear BlendsStar/Linear Blends

h0


“H”

“POM-POM”

“COMB”

span

backbone

backbone

span

backbone

arm

arm or

tooth

Multiple Side BranchesMultiple Side Branches

Relaxation of backbone requires motion of branch points


McLeish et al., Macromolecules, 1999

Frequency (1/s)

10-5 10-4 10-3 10-2 10-1 100 101 102 103

G' &

G" (P

a) 103

104

105

106

G'

G"

model (poly)

model (mono)

Frequency (1/s)

10-5 10-4 10-3 10-2 10-1 100 101 102 103

G' &

G" (P

a) 103

104

105

106

G'

G"

model (poly)

model (mono)

PI H polymer at T=25 °CMa Mb

We take Dbr = p2a2/2qta, with p2 = 1/12

Ma=20000, Mb=110000PIa=1.01, PIb=1.13

Ma=40000, Mb=164000PIa=1.05, PIb=1.30


Start

Generate randomnumber R: U(0,1)

R>ppPropagation Termination

Save molecule

Add monomer Add macromonomer

Generate randomnumber R: U(0,1)

R>lp

N Y

N Y

monomer addition

addition of unsaturated chain

generation of dead saturated chain

β-hydride elimination

Reaction kinetics of LCB PEusing single-site catalyst

Algorithm for Monte Carlosimulation of LCB PE using

single-site catalyst

Monte Carlo probabilities

Costeux et al., Macromolecules, 2002propagation probability monomer selection probability

Commercial Single-Site Commercial Single-Site MetallocenesMetallocenes

Lightly-branched Lightly-branched metallocenemetallocene--catalyzed catalyzed HDPEsHDPEs

0.9999160.9991920.1160.0424320086000HDB3

0.9999260.9991790.0990.0374259082000HDB2

0.9999480.9991720.0670.0263890077000HDB1

10.999398004470093000HDL1

lppp βλ (=LCB/1000C)MnMwresin

β: average number of branches per molecule

λ: average branch point density per 1000C

lppppp

lpppMn

!+"

"=

!#

21

)1(

1014 3

$%

)1(2

10)1)(2(1014 33

lpppMw

!=+"

=##

$

Data from Wood-Adams et al., Macromolecules, 2002

Frequency (1/s)

0.01 0.1 1 10 100

G' (P

a)

100

101

102

103

104

105

106

HDL1

HDB1

HDB2

HDB3

G’

LCB mHDPE is a mixture of linear, star, H-polymers, combs, and hyper-branched structures

Frequency (1/s)

0.01 0.1 1 10 100 1000

G' (P

a)

100

101

102

103

104

105

106

HDL1

HDB1

HDB2

HDB3

Hierarchical model

Wood-Adams et al., Macromolecules, 2002

Park and Larson, J. Rheol., 2005

Metallocene-catalyzed polyethylene

GN=2.0E+6 (Pa), Me=1150, τe=4.0E-9 (s) at T=150 0C

slope=2

Frequency (1/s)

0.01 0.1 1 10 100 1000

G" (P

a)

101

102

103

104

105

106

HDL1

HDB1

HDB2

HDB3

Hierarchical model

slope=1

G’ G’’


10,000 chains in the ensemble

Frequency (1/s)

10-2 10-1 100 101 102

Lo

ss a

ng

le (d

eg

ree

)

30

40

50

60

70

80

90

100

HDL1

HDB1

HDB2

HDB3

Hierarchical model

Loss

tang

ent

Metallocene-catalyzed polyethylene

GN=2.0E+6 (Pa), Me=1150, τe=4.0E-9 (s) at T=150 0C


Wood-Adams et al., Macromolecules, 2002

Park and Larson, J. Rheol., 2005

Frequency (1/s)

10-4 10-3 10-2 10-1 100 101 102 103 104

G' &

G" (P

a) 103

104

105

106

107

G' (exp)

G" (exp)

model (mono)

model (poly)

Daniels et al., Macromolecules, 2001

Park, Shanbhag, Larson, Rheol. Acta, 2005

PBd comb polymer 25 °C

Ma=17.7K, Mb=124.6K, q=5,PIa=1.01, PIb=1.06

Ma

Mb

q arms

Frequency (1/s)

10-4 10-3 10-2 10-1 100 101 102 103 104

G' &

G" (P

a) 103

104

105

106

107

G' (exp)

G" (exp)

model (mono)

model (poly)

Ma=5.8K, Mb=62.7K, q=8,PIa=1.03, PIb=1.03

For calculations with polydispersity, to obtain anensemble of chains, polydisperse arms are randomlyattached to polydisperse backbones via a Poissonprocess.


Test of Dynamic DilutionTest of Dynamic DilutionLinear-Linear Binary Blend with Large Molecular Weight Ratio

G’

Blend of monodisperse 1,4 PBd: 20k/550kGr = M2Me

2/M13 = 0.16 > Grc

= 0.064

Frequency (rad/s)

10-3 10-2 10-1 100 101 102 103

G' (P

a)

101

102

103

104

105

106

107

a = 4/3

Solid lines: reptation in fat tube

Frequency (rad/s)

10-3 10-2 10-1 100 101 102 103

G' (P

a)

101

102

103

104

105

106

107

a = 1

G’

Limitations of the Tube TheoryLimitations of the Tube Theory

Dynamic Tube Dilution Branch Point Motion Fluctuation Potential Nature of entanglement network,

dilution exponent

Issues for Tube Theory with LCB

Need insights from more detailedmodels

Slip-Link ModelEntanglement

Bond-Fluctutation Model“Monomer”

Shanbhag et al., PRL, 2001Shanbhag and Larson, Macromolecules, 2004Shanbhag and Larson, PRL, 2005Shanbhag and Larson, Macromolecules, 2006

Primitive PathPrimitive Path

E1 E2

Primitive Path: the shortest pathconnecting the two ends of the polymerchain subject to the topologicalconstraints imposed by the obstacles

Primitive PathPrimitive Path

E1 E2

Length of the primitive path is shorter from the length of the polymer chain

Primitive Path in a MeltPrimitive Path in a Melt

Polymer chains themselves form obstacles for other chains

Obstacles are not “fixed”

Outstanding problem in polymer physics, until recently (?)

Everaers et al., Science, 2004

Primitive Path in a MeltPrimitive Path in a Melt

Shanbhag and Larson, PRL, 2005Shaffer, J. Chem Phys., 1994

Bond Fluctuation Model

Efficient equilibration of chains

Primitive Path


Extract the primitive paths

Locating Entanglement PointsLocating Entanglement Points

How do I spatially locate entanglements on a primitive path?

3

1

5 7

6

4

2

Shanbhag and Larson, Macromolecules, 2006

Z = 2 entanglements

Nature of CouplingNature of Coupling

“if an entanglement is released, how many additionalentanglements are lost as a consequence?”

“binary coupling”*

related to dilution exponent α = <Zlost/Zdel> α=1 for binary coupling*assumed in slip link models


“1 entanglement releases 3 entanglements”

completely unravels



N=300, Np~300Lbox=65, Φ=0.5


Statistically run through all the chains

dilution exponent α = <Zlost/Zdel>=1.03

supports the notion that entanglements are binary-contacts justifies the assumption made in slip link models implies that α = 4/3, which works best with the tube model, and the “true”value of α may be different

Shanbhag and Larson, Macromolecules, 2006Shanbhag et al., PRL, 2001

SummarySummary

Advances in catalyst technology (metallocene) allow us muchgreater control over branching details.

There is a need to understand the rheology of branched polymersbecause rheology is the most sensitive probe of moleculararchitecture.

The analytical tube model needs significant revision to addressbranched polymers, and needs help from detailed simulations.

•• Prof. R. G. LarsonProf. R. G. Larson

•• Dr. S. J. Park, Q. Zhou, Dr. Dr. S. J. Park, Q. Zhou, Dr.PattampromPattamprom

•• ChihChih-Chen and The Larson Group-Chen and The Larson Group

•• NSF, NSF, ChEChE

AcknowledgementsAcknowledgements

Comparison: Milner-McLeish model andComparison: Milner-McLeish model andthe hierarchical modelthe hierarchical model


104 105

zero

-shear v

iscosity

(Pa.s

)

102

103

104

105

106

107

108

Raju et al. (1981)

Roovers (1985)

Roovers (1987)

Struglinski et al. (1988)


Hierarchical model

zero-shear viscosity of star 1,4-polybutadiene at T=25 °C


Comparison with Standard MethodComparison with Standard Method

Shanbhag and Larson, Macromolecules, 2006

Standard Method

Z = <Lpp2>/<R2>

(ensemble average)

N=# beads on chain(1 bead ~ 5 C atoms)

analytical rheology - people

Documents