Prediction of Birefringence in Plastic Moldings
Alexander Bakharev*, Chris Friedl, Franco S. Costa and Peter K. Kennedy
Moldflow, 259-261 Colchester Rd Kilsyth, Victoria 3137 Australia
ABSTRACT
This article deals with prediction of birefringence and changes in refractive index in polymer optical parts due to the
injection molding process using finite element analysis. The simulation includes analysis of mold filling, packing and
cooling as well as post-molding deformations (warpage). The geometry of a part and runner systems is represented by a
3D tetrahedral mesh. The material model includes non-linear viscoelasticity in the liquid (molten) and solid (frozen)
domains. The simulation predicts the distribution of birefringence and refractive index over the part as well as integral
retardance for light coming from an arbitrary direction. Verification experiments for PMMA and COP moldings show
reasonable agreement with the numerical results.
Keywords: Birefringence, injection molding, simulation, plastic, viscoelasticity, residual stress, prediction, finite
element
1. INTRODUCTION
Many important optical parts are made of injection molded plastic instead of traditional polished glass. The differences
in manufacturing process provide plastic optics some unique advantages as follows 1-3:
 High-volume production capability and low manufacturing cost: The injection molding process allows very
high volume production, and the unit cost can be very low.
 Design sophistication: The grinding and polishing process makes it difficult and very uneconomical to produce
surface shapes in glass materials that are not spherical or flat. On the other hand, the injection-molding process
makes it feasible and economical to produce more sophisticated optical shapes such as asphere and diffractive
surfaces in plastic provided the mold is properly made. From the design point of view, the more sophisticated
surface shapes may provide much better performance for many applications.
 Unique designs possible: Many useful designs that cannot be realized with glass optics can be achieved with
plastic optics such as lens arrays and Fresnel lenses.
 Lightweight and shatter-resistant: The plastic materials are lighter weight and are more shatter-resistant. This
feature is very important for head-worn optics such as head-mounted displays or prescription glasses.
 Integral mounting: For most optical applications, the individual optical components must be mounted in a
system structure. With glass optics, this is done with separate mechanical mounting hardware. However, with
plastic optics, it is possible to include the mounting features with the optical component. This not only reduces
the overall system cost, but also improves the reproducibility of the assembly.
 Consistent Quality: Plastic optics can be made with very consistent quality since all the lenses are derived from
the same mold cavity.
The major drawbacks of plastic optics are mostly material related. The material flow pattern and shrinkage during
molding limit the surface accuracy that is achievable with plastic optics. Due to the residual stresses and material
orientation during the injection molding process the refractive index distribution within a molded component becomes
inhomogeneous and varying with polarization (birefringence).
To make the correct choice between glass and plastic for crucial parts and to choose the correct grade of plastic, a
designer needs software that correctly predicts the surface accuracy and optical properties of the moldings.
The task of predicting the shrinkages and the warped shape of the injection molding parts is one of the main goals of the
mold filling analysis 4-6. This work was devoted to extending the mold filling simulation to include the prediction of
optical properties.
2. THEORY
2.1 Injection molding basics
We predict melt pressure through the cavity and runners using numerical solution of the low-Reynolds number
momentum-equation:
  (τ  pI)  0
together with a rheological equation:
/1/
τ   (, T )γ
/2/
energy equation:
τ : γ T Dp
T
 v  T   2 T 

t
C C Dt
and mass-conservation equation:
( v) 
 (T , p)
t
/3/
/4/
where:
τ  shear stress; p – pressure, γ    v  (  v)T  shear rate, T – temperature, t – time, v – melt velocity,   thermal
diffusivity,    (T , p) - polymer density.
There are four main phases of the injection molding process (see Fig. 1):
 Filling, when the polymer fills the empty molding cavity;
 Packing, when the contraction of polymer due to cooling are compensated by the flow of additional polymer
melt;
 Cooling inside the mold, when the gates to the injection mold become frozen and polymer cools in the mold
cavity down without additional melt going in;
 De-molding and cooling outside the mold, when the molding is ejected from the mold and cools down while no
longer constrained to the mold shape.
Fig. 1 Phase of the injection molding process; a - Filling, b - packing, c - cooling, d - de-molding.
We solve equations 1-4 using finite element methods on a true 3D mesh of four-node tetrahedral elements 7 (see Fig. 2
below). The different stages of the injection molding process give different boundary conditions for the governing
equations. Filling, packing and cooling inside the mold are simulated by the “Flow” program, while de-molding and
cooling outside the mold are simulated by the “Warp” program 9.
Fig. 2 Simulation uses true 3-D tetrahedral mesh
2.2 Prediction of the optical properties
Following K. Kwon et al 8 we distinguish three causes of changing polymer optical properties in the injection molding
process:
 Elastic deformations caused by residual stresses
 Viscoelastic deformations caused by flow orientation of polymer
 Effect of flow conditions on anisotropic growth of polymer crystallites.
The last cause is not important for optical parts which are usually made out of amorphous polymers.
For the changes in the refractive index caused by the elastic deformation we use a stress-optical law. We adopted the
two-constant formulation of the law that allows taking into account both deviators and the isotropic parts of the stress
tensor10:
nii  C1 ii  C2 ( 11   22   33 )
nij  C1 ij : i  j
/5/
/6/
where nii is the change in the refractive index, σ is stress, and C1 and C2 are stress-optical coefficients. The
coefficient C1 describes the anisotropic effect of the deviator of the stress tensor that causes birefringence, while the
coefficient C2 describes the effect of the isotropic part of the stress tensor. The final distribution of the stresses is
determined at the de-molding stage of the injection molding process 5 so running both Flow and Warp are required to get
accurate results.
For the changes in the refractive index caused by the flow orientation we need a viscoelastic model for the polymer
deformation. We have chosen a multi-mode Kelvin-Voight model similar to the one used by Fan at al 11. The model is
represented by dashpots and spring by the diagram shown on Fig. 3.
Fig. 3 Generalized Kelvin-Voight model
The molecular deformation can be predicted then as:
 (t ) 
 (t )
E0
n
  Ei
i 1
1

Ei
 (1  exp( 
0
i
t
t ))d (t   AT (T )dt )
/7/
0
where the temperature shift function AT (T ) shows how the retardation time  i / Ei changes with changing melt
temperature (T). We have chosen the Williams-Landell-Ferry4 form of the temperature shift function:
AT  exp
A1 (T  Tref )
/8/
A2  (T  Tref )
In order to predict the viscoelastic contribution to the changing optical properties we integrate equation /7/ over time for
all the phases of the injection molding process.
The model often predicts changes in the refractive index of a molding to change dramatically over the thickness of a
molding with high changes in the refractive index in two narrow layers on each side of the molding and little changes
outside these layers (see Fig. 4). The software automatically calculates the changes in retardation as the integral of
changes in the refractive index along rays coming from an arbitrary direction:
R   Δn( x, y, z )dS
S
/9/
We can calculate the maximal phase shift between light waves with different polarization as the difference between the
eigenvalues of the tensor R .
Fig. 4 Typical thickness distribution for the changes in the refractive index
A typical result for the phase shift of a lens is shown on Fig. 5. There is very strong birefringence near the gate, high
birefringence on the edge of the lens, moderate birefringence in a large area on the gate side and low birefringence on the
side opposite the gate. This distribution is typical for real polymer lenses.
Fig. 5 Typical results for the phase shift of a lens scaled to 0..270°.
3. VALIDATION OF THE MODEL
Validation of the model was performed on a flat part cavity fed by runner and fan gate with cavity thickness of 2 & 3
mm (see Fig. 6). We used PEPS 80 tonne injection molding machine to mold parts of poly-methylmethacrylate
Mitsubishi Rayon Acrypet VH001 (PMMA) and a cyclo-olefin polymer Zeonex E48R (COP). For birefringence
measurements a Leica DM LP Polarized Light Microscope with rotating stage was used. The measurements were
performed using the Berek compensation method 12.
Fig. 6 Scheme of the molding used for validation of the model predictions. The picture shows the molding, the runner
system and the nozzle of the injection molding machine.
Moldings were made to examine a range of injection times, melt temperatures and cavity thicknesses. Typically six
moldings were made at each test condition after process stabilization to ensure stability and to enable an average
calculation for simulation. Screw position, melt and mold temperature, and pressure were recorded at 5 millisecond
intervals during each shot. For each of the validation moldings, a grid of 45 birefringence validation values has been
measured. The 45 points were sampled on a 9 x 5 array, 9 points along the cavity length and 5 across the cavity width.
The grid was centred on the flat plate section of the cavity and was offset a small distance inwards from the cavity edges
and gate to avoid high gradient regions or edge effects.
The typical results of the experiments are shown on Fig. 7 and Fig. 8. The software predicts the pattern and magnitude of
the maximal phase shift (retardance) between rays with different polarizations. These simulations gave a good estimation
of the magnitude and pattern of birefringence for the PMMA samples. The retardance for the COP moldings was underpredicted but still showed the correct pattern and order of magnitude.
It is interesting to note that while COP is a much more expensive material, specifically designed for the optical
applications, while PMMA is much cheaper and has absolute values of the stress optical coefficient twice as high as COP
(-4.6 Brewster vs. 2.0 Brewster for COP), the birefringence for PMMA is one third of the COP. The difference is
probably caused by better molding properties of PMMA. This effect was captured in the simulation results also.
a
b
c
Fig. 7 Typical validation result for the maximal phase shift (retardance) between rays with different polarizations, PMMA
molding : a – experiment, b – predictions, c – comparison between the experiment and predictions for the centerline of
the specimen
a
b
c
Fig. 8 Typical validation result for the maximal phase shift (retardance) between rays with different polarizations, COP
molding : a – experiment, b – predictions, c – comparison between the experiment and predictions for the centerline of
the specimen
The simulation model also predicts the principal directions of the changes of the material refractive index. During the
optical measurements we have established the general orientation directions of the maximum refractive index. The
results were:
 PMMA - transverse to flow
 COP
- aligned with flow.
Predictions for the first principal direction of the changes in the refractive index are shown on Fig. 9. The effect of
different orientation of the refractive index is predicted by the simulation.
a
b
Fig. 9 Dart plot of 1st principal component of the retardance tensor: a) PMMA – measured direction of maximum
birefringence is transverse to flow; b) – measured direction of maximum birefringence is aligned with the flow
4. CONCLUSIONS
 Mold filling simulation can be used to predict variations in the optical properties of plastic moldings caused by
the manufacturing process
 The simulation predictions of retardance show similar pattern and magnitude as measurements of actual flat
molding samples. The principal directions of retardance was also correctly predicted.
 Prediction of the optical properties of injection moldings can be used by the designers of optical systems to
choose the best material for the optical parts, by part designers to optimize the design of the parts for better
optical performance, by mold makers to choose the mold design that provides the best optical properties and by
manufacturers to optimize the injection molding processing conditions.
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