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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
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