Simulation of Polymer Processing

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Simulation ofPolymer Processing

• David O. Kazmer, P.E., Ph.D.
• March 26, 2005

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Progress in Polymer Process Simulation!

• General Electric 1988
• Vax 8800 cluster
• ES 3D vector graphics

• UML 2005
• PC

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Simulation of Polymer ProcessingAgenda

• Modeling Overview
• Governing Equations
• Constitutive Models
• Numerical Solution
• Capabilities
• Challenges

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MotivationUnderstand 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

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MotivationVirtual Development

• Model and understand the process
• Perform virtual development
• What-if analyses
• System-level optimization

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MotivationPost-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
impossibleat negligible cost.
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Agenda

• Motivation
• Governing Equations
• Constitutive Models
• Numerical Solution
• Capabilities
• Challenges

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Governing EquationsNavier Stokes Equations

• 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

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

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Governing EquationsMass Equation

• Conservation of mass
• What goes in must come out
• Or stay in there
• Change in density with non-steady velocity

IN
OUT
?r
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Governing EquationsMomentum Equation

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

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Governing EquationsHeat Equation

• Conservation of energy
• Change in temperature balances heat convection,
  heat conduction, and shear heating (and others)

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Agenda

• Motivation
• Governing Equations
• Constitutive Models
• Numerical Solution
• Capabilities
• Challenges

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Constitutive ModelsOverview

• 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

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Constitutive ModelsViscosity

• Most polymers are shear thinning
• Cross model
• WLF temperaturedependence

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Constitutive ModelsViscoelasticity

• Polymers exhibit melt elasticity
• Memory effect
• 5 orders of magnitude!
• Extremely data andCPU intensive
• Need to store andcompute on current andall past
  process states!

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Constitutive ModelsSpecific Volume

• Polymers exhibit thermal expansion and
  compressibility
• Double domainTait Equation

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Constitutive ModelsSpecific Heat

• Specific heat Cp

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Constitutive ModelsThermal Conductivity

• Thermal conductivity k

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Agenda

• Motivation
• Governing Equations
• Constitutive Models
• Numerical Solution
• Capabilities
• Challenges

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Numerical MethodsGeometric 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

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Numerical MethodsSolution

• Iterative solution method
• Flow field
• Temperature field

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Numerical MethodsFinite Element Solution of Flow
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k35
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Numerical MethodsFinite Difference Solution of
Heat
HeatConvection
ViscousHeating
Changein Temp
HeatConduction
AdiabaticCompression
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Agenda

• Motivation
• Governing Equations
• Constitutive Models
• Numerical Methods
• Capabilities
• Challenges

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CapabilitiesOptical Media Molding

• Optical media CDs DVDs
• Injection-compression molding (coining)

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Numerical AlgorithmCoining Process

• Coining Process
• Partly open mold
• Inject polymer
• Profile clamp force
• Simulation

Force
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Coining Process ValidationDisplacement Profiles
Effect of melt temperature experiment vs.
simulation
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Birefringence Models

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

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Birefringence Models (Cont.)

• Shear stress

• First normal stress difference

• Integral stress-optical rule
• (birefringence constitutive model)

• Path difference (retardation)

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Numerical Algorithm

• Incremental formulation for the integral
  equations

• Solved by FDM in time domain

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In-plane Birefringence Validation
Validation experiment vs. simulation
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Vertical Birefringence Prediction
Effect of mold temperature (low-high) simulation
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Simulation of Internal Stressand Post-Molding
Deformation

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

• Out-of-mold FEA (plate bending)

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Finite Element Discretization

• Kirchhoff thin-plate elements

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Finite Element Formulation

• Strain-displacement relationship

• Stress-strain relationship

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

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Relaxation ModelingTruncated WLF Equation

• WLF Fit by data at 150-280oC
• Truncated at at 140, 135, 130, 125oC

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Effect of the Truncation

• Warpage at different truncation temperatures
• Could fudge any desired result!

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Proposed Function for Relaxation Model, aT

• For TltTref
• For TgtTref

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Results for ImplementedRelaxation Function, aT

• Model fit performance in simulation

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Optical Molding SimulationResults Summary

• Optical media simulation used for
• Process development and optimization
• Development of new polymeric materials
• Higher data density lower costs

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Agenda

• Motivation
• Governing Equations
• Constitutive Models
• Numerical Solution
• Capabilities
• Challenges

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ChallengesProcess 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?

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ChallengesConstitutive 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?

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ChallengesNumerical Methods

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

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Final ThoughtsModeling Principles

• Pritskers 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