Simulation ofPolymer Processing

David O. Kazmer, P.E., Ph.D.

March 26, 2005

Progress in Polymer Process Simulation!

General Electric 1988

Vax 8800 cluster E&S 3D vector

graphics

UML 2005 PC

Simulation of Polymer Processing:Agenda

Modeling Overview Governing Equations Constitutive Models Numerical Solution Capabilities Challenges

Motivation:Understand Process

Polymer processing is a nasty black box

Dynamic process Multivariate process Spatially distributed process Complex 3D geometry Thermoviscoelastic materials Multiple quality requirements Expensive mold tooling changes

Polymer Processing

x(t) y(t)

Motivation:Virtual Development

Model and understand the process Perform virtual development

What-if analyses System-level optimization

7

7.1

7.2

7.3

7.4

7.5

7.6

7.7

7.8

7.9

8

Par

t W

eigh

t (g

)

Quality Level

Cost

Defect Costs

Compliance Costs

Total Quality Costs

Motivation:Post-Mortem Analysis

Modeling of existing processes Inspection of internal polymer states

Pressure, temperature, flow rate, shear stress, shear rate, …

Development of corrective strategies Change process conditions Assess material changes Recommend mold tooling changes

Simulation provides the means for trying the impossible

at negligible cost.

Agenda

Motivation Governing Equations Constitutive Models Numerical Solution Capabilities Challenges

For laminar (or time-averaged turbulent) flow:

Net pressure force is the gradient of the pressure

Net viscous force is the Laplacian of the velocity

Governing Equations:Navier Stokes Equations

N-S assumes that all macroscopic length and time scales are considerably larger than the largest molecular length and time scales.

vpDt

vD

v

2

0div

Polymer Processing Simulation:Typical Assumptions

Viscous flow Negligible inertia Negligible viscoelasticity

Known boundary conditions No slip at mold wall Constant inlet resin temperature

Flow travels in a plane No out of plane flow

“2D” simplification

Governing Equations:Mass Equation

0

vxt

Conservation of mass

What goes in must come out Or stay in there… Change in density with non-steady

velocity

IN OUT

Governing Equations:Momentum Equation

x

P

z

v

z

1P

Conservation of momentum

Change in pressure in the flow direction is due to shear stress of flowing viscous melt

L

PP

x

P 12

z

v

2Pv

Governing Equations:Heat Equation

22

2

z

Tk

x

Tv

t

TC p

Conservation of energy

Change in temperature balances heat convection, heat conduction, and shear heating (and others)

1TL

TT

x

T 12

2TT,v

MWTThQ

2 Q

Agenda

Motivation Governing Equations Constitutive Models Numerical Solution Capabilities Challenges

Constitutive Models:Overview

Constitutive model: describes the behavior of the material as a function of polymer state

Viscosity, density, … Trade-offs between:

Model form and complexity Number of model parameters Data redundancy in model fitting Computational efficiency & stability

“Everything should be made as simple as possible -but no simpler!” - Einstein

Constitutive Models:Viscosity

Most polymers are shear thinning Cross model

WLF temperaturedependence

n

PTPT

1*

0

0

)(1

),(),,(

tTT

TTA

TTADpT

)

)(

)(exp(),(

*2

*1

10 10

100

1000

1 10 100 1000 10000

Shear Rate (1/sec)

Vis

cosi

ty (

Pa

Sec

)

n

0

*

-8

-6

-4

-2

0

2

4

6

8

10

20 70 120 170 220 270 320

Temperature (oC)

Lo

g(a

T)

Exp.

Fitted

WLF

Constitutive Models:Viscoelasticity

Polymers exhibit melt elasticity

Memory effect

5 orders of magnitude! Extremely data and

CPU intensive Need to store and

compute on current andall past process states!

dIIhtMTpt

t )(),()()(),( 121

CIσ

m

i

t

iT

i iea

g

dt

tdGtM

1

)()(

)()()()(

10-3 10-2 10-1 100 101 102 10310-1

100

101

102

103

104

105

106

107

108

109

10-1

100

101

102

103

104

105

106

107

108

109

Freq [rad/s]

G' (

)

[Pa]

G" (

) [P

a]

Sto

rag

e M

od

ulu

s Loss M

od

ulu

s

1.E+00

1.E+01

1.E+02

1.E+03

1.E+04

1.E+05

1.E+06

1.E+07

1.E+08

1.E+09

1.E-06 1.E-03 1.E+00 1.E+03 1.E+06 1.E+09

Frequency (rad/s)

G',

G"

(Pa) G'-Exp.

G"-Exp.G'-FittedG"-Fitted

Constitutive Models:Specific Volume

Polymers exhibit thermal expansion and compressibility

Double domainTait Equation

0.80

0.82

0.84

0.86

0.88

0.90

0.92

0.94

0.96

0.98

1.00

0 50 100 150 200 250 300 350

Temperature (oC)

Sp

ec

ific

Vo

lum

e (

10

-3m

3/k

g)

0 MPa Exp. 0 MPa Fitted20 MPa Exp. 20 MPa Fitted40 MPa Exp. 40 MPa Fitted60 MPa Exp. 60 MPa Fitted80 MPa Exp. 80 MPa Fitted100 MPa Exp. 100 MPa Fitted120 MPa Exp. 120 MPa Fitted140 MPa Exp. 140 MPa Fitted160 MPa Exp. 160 MPa Fitted180 MPa Exp. 180 MPa Fitted200 MPa Exp. 200 MPa Fitted

TB

PTvPTv 1ln0894.01, 0

5,2,10 bTbbv ll

5,4,3 exp bTbbTB ll

Constitutive Models:Specific Heat

Specific heat Cp

543521 tanh CTccCTccTC p

1.10

1.20

1.30

1.40

1.50

1.60

1.70

1.80

1.90

2.00

2.10

2.20

2.30

50 100 150 200 250 300 350

Temperature (oC)

Spe

cific

Hea

t (10

3 J/kg

oC

)

Exp.Fitted

Constitutive Models:Thermal Conductivity

Thermal conductivity k

0.20

0.22

0.24

0.26

0.28

0.30

0.32

20 70 120 170 220 270 320

Temperature (oC)

The

rmal

Con

duct

ivity

(W

/moC

) Exp.Fitted

543521 tanh TTTk

Agenda

Motivation Governing Equations Constitutive Models Numerical Solution Capabilities Challenges

Numerical Methods:Geometric Modeling

Polymer domain decomposed into elements

2D elements across flow domain Plastics parts are often thin so nice

assumption Each element has defined thickness

3D elements for entire domain Need many, many elements of higher order

shape functions

Numerical Methods:Solution

Iterative solution method Flow field Temperature field

Read Input

Done?

Advance Time

Write Output

Solve Flow

Solve HeatUpdate BC’s

Numerical Methods:Finite Element Solution of Flow

QPK

0

0

100

00

00

00

00

00

0

0

0

4

3

2

564626564626

565635253525

464624142414

353523122313

26252423262524231212

141312141312

6

5

1

P

P

P

kkkkkk

kkkkkk

kkkkkk

kkkkkk

kkkkkkkkkk

kkkkkk

Q

Q

Q

1 2

3

4 6

5

0

0

0

0

0

0 1

24142414

23132313

242324231212

1413124!1312

3

2

1

P

kkkk

kkkk

kkkkkk

kkkkkk

Q

Q

Q

2

3

4

1

k35

length

jdzjviscjdzjzjrhoW

k

x

dzzdz

x

PW

k

layersn

j

h h

z

_

035

0

35

][][/][][][2

~~

2

VPC

t

x

TvtT

T

T

T

FoFo

FoFoFo

FoFo

VPC

t

x

TvtTFoTTFoFoT

PdVz

Tk

x

Tv

t

TC

t

p

t

ttn

tt

tt

t

ip

t

i

ti

tti

tti

tti

t

tp

12

1

2

1

211

2

1

22

2

1000

210

021

00221

21

5Changein Temp

HeatConvection

HeatConduction

ViscousHeating

AdiabaticCompression

v

Q

Numerical Methods:Finite Difference Solution of Heat

Agenda

Motivation Governing Equations Constitutive Models Numerical Methods Capabilities Challenges

Capabilities:Optical Media Molding

Optical media: CDs & DVDs Injection-compression molding

(coining)

Numerical Algorithm:Coining Process

Coining Process Partly open mold Inject polymer Profile clamp force

SimulationAdjust Thickness

Change Element PropertiesRestore Old Profiles

Y

Calculate Pressure

Calculate Cavity Force

Cavity Force=Clamp Force?

Calculate Temperature

Move on to Next Time Step

NForce

Coining Process Validation:Displacement Profiles

Effect of melt temperature: experiment vs. simulation

1.10

1.15

1.20

1.25

1.30

1.35

1.40

1.45

0 1 2 3 4Time (s)

Mo

ld D

isp

lace

me

nt

(mm

)

300oC310oC

320oC

1.10

1.15

1.20

1.25

1.30

1.35

1.40

1.45

0 1 2 3 4Time (s)

Mo

ld D

isp

lace

me

nt

(mm

)

300oC

310oC

320oC

Birefringence Models

Constitutive model for flow induced stress (Wagner, M. H. et al)

dIIhtMTPt

t )(),()()(),( 121

CIσ

m

i

t

iT

i iea

g

dt

tdGtM

1

)()()()(

)()(

)3exp()1()3exp(),( 2*

1*

21 InmInmIIh

100

01)(

0)()(1

)(

22

1

t

tt

Ct

Birefringence Models (Cont.)

2/

2/)(

d

d zrz dznn • Path difference (retardation):

tt t

rz ddtdthTtM11

2

])[}{3(),( ''

tt t

zzrr ddtdthTtMN11

2 2''1 ])[}{3(),(

00

01

if

if

dtCn

t

)()(

• Shear stress:

• First normal stress difference:

• Integral stress-optical rule

(birefringence constitutive model):rzrz nN cebirefirnen alfor vertic4 22

1

rnN ncebirefringe plane-infor 2

Numerical Algorithm

• Incremental formulation for the integral equations:

m

i jn

nj

t

iT

in deme

a

G njinn

1

2

11

))((/))((1,13 ))((11

1

m

i jn

nj

t

iT

in deme

a

GN njin

n

1

2

1

21

))((/))((1,1 ))((11

1

nnn

jiij

nnij

nnn

jiij

nnjinnjinnjin emeGeeemeG

/

,13/

1,131/

1,131111

nnij

nnnn

jiij

nnij

nnn

jiij

nnjinnjinnjin emeGNeeNemeGN

21,131

2/,1

/1,1

21

/1,1 21111

nnn 11 nnn 11

• Solved by FDM in time domain:

demea

Gn

nj

t

tiT

iijn

njinn

n

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

1,1311

1

demea

GN n

nj

t

tiT

iijn

njinn

n

21

))((/))((1,1 ))((11

1

deme

a

G njinn n

j

t

iT

in

))((/))((

0

mm 1 mm 12

In-plane Birefringence Validation

Validation: experiment vs. simulation

-20

-10

0

10

20

30

40

50

60

70

80

23 28 33 38 43 48 53 58

Radius (mm)

Pa

th D

iffe

ren

ce (

nm

)

Exp.Sim.--TotalSim.--FlowSim.--Cooling

-20

-10

0

10

20

30

40

50

60

70

80

23 28 33 38 43 48 53 58

Radius (mm)

Pat

h D

iffer

ence

(nm

)

Exp.

Sim.--Total

Sim.--Flow

Sim.--Cooling

z

r

t

z

r

t

Vertical Birefringence Prediction

Effect of mold temperature (low-high): simulation

-5

0

5

10

15

20

25

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5

z/d

n

rz (

×10

-4)

Total

Flow Induced

Thermally Induced

-5

0

5

10

15

20

25

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5

z/d

n

rz (

×10

-4)

Total

Flow Induced

Thermally Induced

z

r

t

z

r

t

Simulation of Internal Stressand Post-Molding Deformation

Thermal stress/warpage In-mold: FDM (Baaijens, F. P. T. et al)

dhp σIσ

th dtrTtrp

0)(

1)(

3

1

εσ

deg dtm

i

t

id iεσ /)()(

10

2

Twu D)()(),( rzruzru drrwdr /)()(

)()( rwrw

– Out-of-mold: FEA (plate bending)

Finite Element Discretization

Kirchhoff thin-plate elements

Elements (Divided into m layers)

Inner Edge

1 2

3

n

r1

r2 rn

Finite Element Formulation

2

2

2

1

1

1

2222

22

)32()(6

)341()(61

)62()21(61

)64()21(61

w

u

w

u

r

z

rs

z

rr

z

rs

z

r

s

z

s

z

ss

z

s

z

srr

• Strain-displacement relationship

• Stress-strain relationship

2

1iii uBε

hεHσ

2/

2/

2

1

2d

d

r

r

T

V

Te rdrdzdV HBBHBBk

dzrdrdVd

d

r

r

TT

V

TTe

2/

2/

2

1

)(2)( hBfNhBfNR

rrrrrr

ab

ba

• Element stiffness matrix and element right-hand-side vector

RDK

1E-8

1E-6

1E-4

1E-2

1E+0

1E+2

1E+4

1E+6

1E+8

80 100 120 140 160 180 200 220 240 260 280 300

Temperature (oC)

aT

Exp.

Fitted

Tg

Relaxation Modeling:Truncated WLF Equation

WLF Fit by data at 150-280oC Truncated at at 140, 135, 130,

125oC

Effect of the Truncation

Warpage at different truncation temperatures

Could fudge any desired result!

z

r

t

z

r

t

-300

-250

-200

-150

-100

-50

0

50

100

150

23 28 33 38 43 48 53 58

Radius (mm)

War

pag

e (m

icro

met

er)

Ttrunc=140Ttrunc=135Ttrunc=130Ttrunc=125Exp. Data

T

Radial Direction

Proposed Function for Relaxation Model, aT

For T<Tref

For T>Tref

cref

cref

TTb

TTb

Ted

eada

))((

))(( )1()log(

cref

cref

TTb

TTb

Ted

eada

))((

))(( )1()log(

Results for ImplementedRelaxation Function, aT

Model fit & performance in simulation

1E-8

1E-6

1E-4

1E-2

1E+0

1E+2

1E+4

1E+6

1E+8

70 100 130 160 190 220 250 280 310

Temperature (oC)

aT

Exp.

Fitted

-115

-95

-75

-55

-35

-15

5

23 28 33 38 43 48 53 58

Radius (mm)

Ver

tica

l Dis

pla

cem

ent

(mic

ro m

eter

) Exp.

Sim.

0

20

40

60

80

100

120

0 10 20 30 40 50

Packing Pressure (kgf/cm2)

Ver

tica

l D

isp

lace

men

t (m

icro

met

er)

Exp.

Sim.

Optical Molding Simulation:Results Summary

Optical media simulation used for Process development and optimization Development of new polymeric materials

Higher data density & lower costs

0

20

40

60

80

100

120

95 100 105 110 115 120

Mold Temperature (oC)

Ver

tica

l D

isp

lace

men

t (m

icro

met

er)

Exp.

Sim.

0

20

40

60

80

100

120

295 300 305 310 315 320 325

Melt Temperature (oC)

Ver

tica

l Dis

pla

cem

en

t (m

icro

met

er)

Exp.

Sim.

Agenda

Motivation Governing Equations Constitutive Models Numerical Solution Capabilities Challenges

Challenges:Process Controllability

What are the boundary conditions foranalysis?

Is melt temperature constant? What is the mold wall heat transfer? Is a no-slip condition at mold wall

valid?

0 1 2 3Time

Wall

Center

Wall

Challenges:Constitutive Models

N-S assumes a continuum Is a continuum approach valid on the

nano-level? If not: What are the governing equations? What are the constitutive models? How to apply thermodynamics &

statistics?

Challenges:Numerical Methods

Modeling on the atomic scale? Sandia Labs Atomic weapons Crystal-level modeling of metals Protein folding

Final Thoughts:Modeling Principles

Pritsker’s Modeling Principles, from Handbook of Simulation, edited by Jerry Banks for Wiley Interscience, 1998

Model development requires system knowledge, engineering judgment, and model-building tools.

The modeling process is evolutionary because the act of modeling reveals important information piecemeal.

The secret to being a good modeler is the ability to remodel. A model should be evaluated according to its

usefulness. From an absolute perspective, a model is neither good or

bad, nor is it neutral. All truths are easy to understand once they are

discovered; the point is to discover them. Galileo