simulation of polymer processing david o. kazmer, p.e., ph.d. march 26, 2005
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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
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