Rheology Simulator

PASTAPASTA(Polymer rheology Analyzer with Slip-

link model of enTAnglement)

Tatsuya ShojiJCII, Doi Project

COGNAC

PASTA

SUSHI

MUFFINGOURMET

0 sec

-3 msec

-6 μsec

-9 nsec

-12 psec

-15 fsec

-15 -12 -9 -6 -3 0fm pm nm μm mm m

IntroductionCatalyst

PolymerMonomerPolymerization

LCBLinear

Molecular weight

Molecule Weight Distribution BrainchingPolymer Structure

Products Mechanical property

Rheologyviscosity,

modulus, etc..

Processing Processability

ObjectivesPrediction of the rheological properties from the knowledge of the structure of polymer

It is not feasible to simulate the rheologicalproperties of polymer by MD simulations.

the Doi-Edwardstheory(tube model)

coarse-grained model for polymer

New stochastic simulation method(PASTA)

Classical tube model

a Obstacle

Tube

Primitive path

Assumption

･The length of primitive path is constant

･Obstacles are permanentReptation is the only mechanism of the relaxation

Important Extensionsof the tube model -1-Contour Length Fluctuation Constraint Release

Convective Constraint Release(CCR) ⇒

Slip links

(1) Each chain obeys the reptation dynamics.(2) Each slip link constrains a pair of chains, and disappears when eitherone of the chain moves away from the slip link.

Important Extensionsof the tube model -2-

Simulation method 1) Each polymer chain is modeled by a primitive path and slip links along it.

Subchain vector 121 ,,, −nrrr L

kr

1r2r 1−nr

1s

2s

∑=k

ktL r

21, ssLength of the tail at each end

Length of the tube

Length of the primitive path 21 ssLL t ++=

Simulation method2) A collection of many chains (ex. 102～104)3) The interaction among the chains is taken into account through the “pairing” of the slip links

Pairing of the slip links(Each slip link has its partner, but only representative

pairs are shown)

Simulation method4 operations in each time step1. Affine deformation of the tubes2. Contour length fluctuation3. Reptation 4. Constraint release / creation

aa Unit of length a

)( eRe MM == ττUnit of time

Me : Entanglement molecular weight

Operation-1Affine deformation of the primitive path

( ) ⇒tr ( )tt ∆+r

Each slip link is moved according the macroscopic flow of the sample.

( ) ( ) ( )tyttxttx ⋅∆+=∆+ γ&

( ) ( )tytty =∆+

Operation-2Contour length fluctuation

L(t+dt)L(t)

( )( ) ( )tgLLtLdtdL

affineeqR

++−−= &τ1

Variation rate of Lby the affine deformation

Gaussian random force

ZaLeq =2ZeR ττ =

Equilibrium length

Rouse relaxation time

eMMZ /≡ Average number of slip links

Operation-3ReptationThe center of mass of each primitive path is randomly moved by Δs along the pathwith diffusion constant Dc

tDs c∆=∆ 22

ZDc

1∝

Operation-4Constraint renewal

If a end of a primitive path passes through the last slip link on the chain, the slip link and its partner are destroyed.

“partner”

Operation-4Constraint renewal

If the length of a tail at the end of the primitive path becomes longer than , a new slip link is created at the end and its partner is created on a randomly selected chain.

a

a

randomly selected chain

Simulation method

( )eq

B

k

kk

LL

aTk

dLLdUF

rF

3−=−=

=rF

( ) ( ) 223

eqeq

B LLaL

TkLU −=

∑−=k

kk rF βααβσ

Tension kF

∑=k k

kkBrrr

LZa

Tk βααβσ 2

3

kr

1r2r 1−nr

Calculation of the Stress

Simulation method4 operations in each time step1. Affine deformation of the tubes2. Contour length fluctuation3. Reptation 4. Constraint release / creation

aa Unit of length a

)( eRe MM == ττUnit of time

Me : Entanglement molecular weight

Star Polymerbranchpoint

Za

Za

Za

3 operations in each time step1. Affine deformation of the tubes2. Contour length fluctuation3. Reptation 3. Constraint release / creation

linear τR=τeZ2

star τR=τe(2Za)2

ZaZaZa=1/2Z

FUNCTION of PASTA

Deformation type・shear・uniaxial elongation・biaxial elongation・planar elongation

・linear polymermonodispersepolydisperse

・star polymermonodispersepolydisperse

・linear/star mixture

Target Sample Flow type

・steady flow・step strain・noflow→thermalization, stress relaxation

Simulation results

Steady shear flow(z=60 monodisperse)

100

101

102

103

104

105

100 101 102η

0

z

3.5

10-4

10-3

10-2

10-1

100

101

10-7 10-6 10-5 10-4 10-3 10-2

σ, N

1, -N

2

shear rate

σ

N1

-N2

Z=60

1/τR

Vertical shift: (15/4) GN0τe

Horizontal shift: τe-1

Me =14400GN

0 =2×106 dyn/cm2

Steady shear viscosityMonodisperse Polystyrene

102

103

104

105

106

107

10-1 100 101 102 103 104

simulation z3.3simulation z12.4simulation z16.8experiment Mw48500experimant Mw179000experiment Mw242000

η (

pois

e)

shear rate (sec-1)

N e e

R. A. Stratton,J. Colloid Interfac. Sci.,22, 517 (1966)

PS 183℃

Self diffusion coefficient

T.P. Lodge, Phys. Rev. Lett. 83, 3218 (1999)

Self diffusion coefficient(monodisperse)

( ) ( )∞→=− ttDt GGG 6)0()( 2RR

5.24.2 −−∝ ZDG )..( 5.30

−∝ Zfc η

10 < Z < 80

10-5

10-4

10-3

10-2

10-1

1 10 100

6DG

Z

Ð2.46

Ð2

Elongational viscosityPolydisperse Polystyrene

0

0.2

0.4

0.6

0.8

1

1.2

104 105 106

dW/d

logM

M

102

103

104

105

106

10-3 10-2 10-1 100 101

experiment

simulation

G',

G''

(Pa)

ω aT (sec-1)

V shifted by 5.3e5 H shifted by 4.9e2

Mw =2.85×105

Mw/Mn =2.0

0

0.2

0.4

0.6

0.8

1

1.2

104 105 106

dW/d

logM

M

Elongational viscosity

Mw =2.85×105

Mw/Mn =2.0

Polydisperse PS

104

105

106

107

108

10-1 100 101 102 103

dε /dt=0.564dε /dt=0.123dε /dt=0.055dε /dt=0.0113η (t)

ηe(t

) (P

as)

t (s)

PS686

experiments

104

105

106

107

108

10-1 100 101 102 103

dε /dt=0.572dε /dt=0.097dε /dt=0.047dε /dt=0.0133η (t)

ηe(t

) (P

as)

t (s)

PS686(98.5)+W320(1.5)

experiments

Mw =2.85×105

Mw/Mn =2.0

+HMW-PS

Mw =3.2×106

1.5 %

Polydisperse PS

Elongational viscosity

Shear viscosity of star polymer

100

101

102

103

104

105

10-7 10-6 10-5 10-4 10-3 10-2 10-1

η

shear rate

Z=5

Z=10

Z=30

Z=20

2Za=30

2Za=20

2Za=10

Z=60

100

101

102

103

104

105

100 101 102ze

ro-s

hear

vis

cosi

ty η

0

Zlinear

or 2Zarm

LinearStar

Branch point

PASTA on GOURMET

Usage

Step1. Creating inputUDF of PASTA

Step2. Running PASTA

Step3. Analysis of outputUDF

Editing InputUDF of PASTA on GOURMET

Sample: Chain type: Linear or ArmZi (=Mi/Me :Average number of slip links)Ni (Number of the i-th Chain)λmax (Max Stretch Ratio)

Z λmax Ni10 4.4 1000

Z λmax Ni1.5 4.4 102.8 4.4 454.9 4.4 1356.8 4.4 2568.9 4.4 168..... ... ......... ... ....

•Monodisperse•Polydisperse

Editing InputUDF of PASTA on GOURMET

Simulation: Flowtype: flow / noflow / stepDeformation type: shear/ uniaxial/ biaxial/ planarStrain: (in the case of step)Strain rate: dt: time step per 1 τeMaxTimeStep: Number of iterationIntervalStep: output every IntarvalStep steps

select function

FORK (a support tools for PASTA)

FORK is a tool to generate inputUDF for PASTA.

properties of polymerMe, M0, GN0, τe...etc

inputUDFfor PASTAFORKMWD

Mw, Mw/Mn, distribution function,

GPC datai Zi Ni1 2.5 32 3.7 103 4.8 474 6.2 285 8.1 9... .... ....

simulation conditiondeformation type

shear, ε=0.01, 10000step

Running PASTA on GOURMET

Engine Run Window

RunMonitor

Analyze output by GOURMET• By “Action”, Python script and the other tools

- shear viscosity, elongational viscosity, relaxation modulus G'(ω), G''(ω), etc....

Example of Action (plot_stress) instant plotby Gnuplot

PASTA:Linking to other layers

COGNAC

PASTA

SUSHI

MUFFIN

Friction constant,Me

Non-equilibriumstructure

StrainStrain rateDeformation type

Linear Non-linear ViscoelasticResponse

SummaryPASTA

• New stochastic simulation methodtube model

+ contour length fluctuation+ constraint renewal

• Successfully calculates most of the linear/nonlinear rheology ofmonodisperse/polydisperse linear/star polymer

• Easy and convenient manipulation on GOURMET

Developer of PASTA

•Theory & Program Prof. Jun-ichi Takimoto(Nagoya Univ.)

•Verification Hiroyasu Tasaki(JCII, Doi Project)

•Connection with GOURMET& FORK Tatsuya Shoji

(JCII, Doi Project)

This work is supported by the national project, which has been entrusted to the Japan Chemical Innovation Institute (JCII) by the New Energy and Industrial Technology Development Organization (NEDO) under MITI's Program for the Scientific Technology Development for Industries that Creates New Industries.

Acknowledgements