Rheology Simulator

PASTA(Polymer rheology Analyzer with Sliplink model of enTAnglement)

Tatsuya Shoji
JCII, Doi Project0 sec-3 msec

GOURMET-6 μsec

MUFFIN
SUSHI-9 nsec

PASTA-12 psec

COGNAC-15 fsec
-15fm

-12pm

-9nm

-6μm

-3mm

0m

Introduction
Monomer

Catalyst
Polymerization

Polymer

Polymer Structure
Molecule Weight Distribution

BrainchingLinearLCB

Molecular weight

Processing

Processability

Products Mechanical property

Rheology
viscosity,
modulus, etc..

Objectives

Prediction of the rheological properties fromthe knowledge of the structure of polymer

It is not feasible to simulate the rheologicalproperties of polymer by MD simulations.
the Doi-Edwards
theory(tube model)coarse-grainedmodel for polymer

New stochastic simulation method(PASTA)

Classical tube model
ObstacleaTube
Primitive path
Assumption･The length of primitive path is constant･Obstacles are permanent
Reptation is the only
mechanism of the relaxation

Important Extensionsof the tube model -1Contour Length Fluctuation

Convective Constraint Release(CCR)

Constraint Release⇒Important Extensionsof the tube model -2Slip links(1) Each chain obeys the reptation dynamics.
(2) Each slip link constrains a pair of chains, and disappears when either
one of the chain moves away from the slip link.

Simulation method
1) Each polymer chain is modeled by a primitive path andslip links along it.
rkr1

r2

s1

r n −1s2

Subchain vectorr 1 , r 2 , L , r n −1

Length of the tube

Lt = ∑ rkk

Length of the tail at each end

s1, s 2

Length of the primitive path

L = Lt + s1+ s 2

Simulation method
2) 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 representativepairs are shown)Simulation method
4 operations in each time step
1. Affine deformation of the tubes
2. Contour length fluctuation
3. Reptation
4. Constraint release / creationaa

Unit of length a
Unit of time τ e = τ R ( M = M e )Me : Entanglement molecular weightOperation-1
Affine deformation of the primitive path
Each slip link is moved according the macroscopic
flow of the sample.

r (t )⇒r (t + ∆t )x(t + ∆t ) = x(t ) + γ&∆t ⋅ y (t )y (t + ∆t ) = y (t )L(t)Operation-2L(t+dt)Contour length fluctuation()

1dL=−L(t ) − Leq + L& affine + g (t )τRdt
Variation rate of Lby the affine deformation

Gaussian random force

Leq = Za

Equilibrium lengthτ R = τ eZ 2

Rouse relaxation timeZ ≡ M / Me

Average number of slip links

Operation-3
Reptation
The center of mass of each primitive path israndomly moved by Δs along the pathwith diffusion constant Dc∆s 2 = 2 Dc ∆t1
Dc ∝ZOperation-4
Constraint renewal
If a end of a primitive path passes through the last slip linkon the chain, the slip link and its partner are destroyed.
“partner”

Operation-4
Constraint renewal
If the length of a tail at the end of the primitive path becomes
longer than a , a new slip link is created at the end and its
partner is created on a randomly selected chain.

randomly selected chaina

Simulation method
Calculation of the Stressrk

σ αβ = −∑ Fkα rkβr1

r2r n −1kr
Fk = F krkdU ( L )3k BT LF =−=−dLa Leq
3k BT(U (L ) =L − Leq ) 22aLeq

Tension Fkrkα rkβ
3k BTLσ αβ =2 ∑rkZa k

Simulation method
4 operations in each time step
1. Affine deformation of the tubes
2. Contour length fluctuation
3. Reptation
4. Constraint release / creationaa

Unit of length a
Unit of time τ e = τ R ( M = M e )Me : Entanglement molecular weightStar Polymerbranchpoint
ZaZaZa1. Affine deformation of the tubes
2. Contour length fluctuation
Za=1/2Z 3. ReptationZa
3. Constraint release / creation

linear τR=τeZ2Za

3 operations in each time stepstar τR=τe(2Za)2

FUNCTION of PASTA
Target Sample

Flow type・linear polymer
monodisperse
polydisperse
・star polymer
monodisperse
polydisperse・linear/star mixture

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

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

Simulation results

Steady shear flow

(z=60 monodisperse)
10N0110σ-143η0210-N1σ , N , -N

1051

Z=601010101010210-2

3.51/τ-310R

10-4

102-710-610-510-4

shear rate

10-310-2

1010010z1102

Steady shear viscosity

Monodisperse Polystyrene
10N e

7PS 183℃

Vertical shift: (15/4) GN0τe
Horizontal shift: τe-1

10e

sim ulation z3.3
simulation z12.4

6

simulation z16.8
experiment M w48500
experimant Mw179000

η (poise)

Me =14400
GN0 =2×106 dyn/cm2

101010R. A. Stratton,J. Colloid Interfac. Sci.,
22, 517 (1966)

experiment Mw242000

543210-11010010110210-1

shear rate (sec )3104

Self diffusion coefficient

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

Self diffusion coefficient(monodisperse)(R G (t ) − R G (0))2 = 6 DG t (t → ∞ )10-1

10-2Ð2.466DG

10 < Z < 80

10DG ∝ Z-3−2.4 − 2.5(c. f . η 0 ∝ Z −3.5 )Ð21010-4-5110Z100

Elongational viscosity
Polydisperse Polystyrene
Mw =2.85×105
Mw/Mn =2.0106V shifted by 5.3e5 H shifted by 4.9e2

1.2

10G', G'' (Pa)1dW /dlogM

0.8

0.6

100.4

10543experiment

0.2

simulation
20104

10510M

610-3
1010-2

10-110-1

ω a (sec )
T

010

1

Elongational viscosity
Polydisperse PS
Mw =2.85×105
Mw/Mn =2.0

1076experimentse1

8PS68610

η (t) (Pas)

1.2

10

dW /dlogM0.8

d ε /dt=0.564

100.6

d ε /dt=0.123

5d ε /dt=0.055
d ε /dt=0.011

0.4

3 η (t)0.210010410510M6410-1100101t (s)102103

Elongational viscosity
108PS686(98.5)+W 320(1.5)

Polydisperse PSHMW-PS
Mw =3.2×106
1.5 %1076experiments

e+10η (t) (Pas)

Mw =2.85×105
Mw/Mn =2.01010

d ε /dt=0.572
d ε /dt=0.097
d ε /dt=0.047
d ε /dt=0.013
3 η (t)5410-1100101t (s)102103

Shear viscosity of starpolymer
105Z=60

104Branch point

2Z =30aZ=301010302Z =20 Z=20a22Z =10a1Z=10Z=510010-710-610-510-410shear rate

-310-210-1zero-shear viscosity ηη

1010510410310210110010Star010Z

linearorLinear

1102Zarm

2

PASTA on GOURMETUsage
Step1. Creating inputUDF of PASTA
Step2. Running PASTA
Step3. Analysis of outputUDF

Editing InputUDF of PASTA
on GOURMET
Sample:Chain type: Linear or Arm
Zi (=Mi/Me :Average number of slip links)
Ni (Number of the i-th Chain)
λmax (Max Stretch Ratio)Z λmaxNi10 4.4 1000
•Monodisperse•Polydisperse

Z λmax
1.5 4.4
2.8 4.4
4.9 4.4
6.8 4.4
8.9 4.4..... ...
..... ...

Ni
1045135
256
168
....
....

Editing InputUDF of PASTA
on GOURMET
select function

Simulation:
Flowtype: flow / noflow / step
Deformation type: shear/ uniaxial/ biaxial/ planar
Strain: (in the case of step)
Strain rate:
dt:
time step per 1 τe
MaxTimeStep: Number of iteration
IntervalStep:output every IntarvalStep steps

FORK (a support tools for PASTA)FORK is a tool to generate inputUDF for PASTA.
properties of polymerMe, M0, GN0, τe...etc
MWDMw, Mw/Mn,distribution function,
GPC data
simulation condition
deformation typeshear, ε=0.01, 10000step

FORK

inputUDFfor PASTA
i
1
2
3
4
5...

Zi
2.5
3.7
4.8
6.2
8.1
....

Ni
3
1047
28
9....

Running PASTA on GOURMETMonitor
Engine Run Window

Run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 layersLinearNon-linear
Viscoelastic
Response

MUFFINSUSHI

Friction constant,MePASTA
Non-equilibrium
structure

COGNAC

StrainStrain rate
Deformation type

SummaryPASTA• New stochastic simulation method
tube model+ contour length fluctuation+ constraint renewal• Successfully calculates most of thelinear/nonlinear rheology ofmonodisperse/polydisperse linear/starpolymer• Easy and convenient manipulation on GOURMETDeveloper of PASTA•Theory & Program

Prof. Jun-ichi Takimoto(Nagoya Univ.)•Verification•Connection with GOURMET& FORK

Hiroyasu Tasaki(JCII, Doi Project)

Tatsuya Shoji(JCII, Doi Project)

Acknowledgements
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.