MODELING AND SIMULATION OF MORPHOLOGICAL EVOLUTION FOR
POLYMER BLENDS IN INJECTION MOLDING
Yi Zhang1, Fen Liu1, Shubiao Cui1,2, Yun Zhang1, and Huamin Zhou1,2*
1
State Key Lab of Material Processing and Die & Mould Technology,
Huazhong University of Science and Technology, Wuhan, China
2
DG-HUST Manufacturing Engineering Institute, Dongguan, China
Abstract
Numerical prediction of morphology in polymer
blends is of vital importance in mastering of material
structure and property optimization from both theoretical
and industrial views. A systematical study was conducted
on the morphological simulation in injection molding for
polymer blends, including model selection, process
simulation and experimental validation. The principle and
crucial factors in the evolution of dispersions (specifically
the deformation, break and coalescence) were
investigated. Simulation results were compared and
evaluated with the experiment data, which suggest the
effectiveness of predicting microstructure in the process.
Introduction
In injection molding, performance variations of
product are caused by its morphology (crystallization,
orientation and dispersion phases etc.) formed in different
regions, including appearance, permeability and
mechanical property. Thus, the prediction or even
adjustment of microstructures is a critical part in the
molding process of immiscible polymeric blends [1].
Temporarily, the injection molding analysis mainly
focuses on the feasibility of the forming process. The
performance prediction of the final part is not
implemented. Even if the preliminary calculation of
residual stress and wrapage could be achieved [2,3], the
computation precision cannot satisfy the engineering
requirements without considering the influence of the
microstructure. And current theoretical investigation of
morphology in polymer blends focuses on discrete
dispersion without considering the interaction of droplets
[4], which could not be applied in the simulation of
practical process.
In this paper, a coupled model of flow field and
morphological evolution is proposed to shed some light
on the problem mention above. With finite element
method, a 3D numerical simulation is conducted with
morphology simulation, including the deformation,
breakup and coalescence of the dispersion phase droplet
in injection molding process. And corresponding
governing equations and numerical simulation strategies
are applied. Finally, simulation results are compared with
the practical product in validation of the feasibility of
morphological evolution model constructed.
Coupled Modeling
Injection molding process involves microstructured
fluid–fluid dispersions in complex laminar flows [1]. By
microstructured, we mean that the homogeneous regions
of each phase in the dispersion are much smaller than the
macroscopic flow domain.
Modeling these systems is particularly challenging
because relevant physics must be accounted for at two
vastly different length scales. On the macroscale, the flow
in the mixing device must be modeled. On the mesoscale,
the deformation, reorientation, and advection of the
microstructure must be calculated. While each case can be
addressed individually using existing tools, little progress
has been made on the combined problem.
To solve this two scales synthetic problem, a model
with microstructure detail representation, total variation
and strategy of precise modeling complex flow field in
macroscope and microstructure evolution is needed. In
this paper, the modeling objects are immiscible two phase
blend that one dispersing in the other.
The research subject here is dilute solution, which the
volume ratio of the dispersion phase is lower than 30%. It
adopts a general coupled modeling method and only
considers the rheological behaviors of dispersion phase on
the blend without the reaction of dispersion phase
granules on flow field. In this method, the calculation of
the flow field and microstructure is mutually independent,
which means the output of the flow field is the input of
microstructure.
The calculation steps in each time interval are as
below. First calculate the macroscopic variations (velocity,
temperature, pressure etc.) distribution with constitutive
equations at one moment; then import the local flow field
parameters as the conditions of microstructure evolution,
and calculate the microstructure; and this microstructure
_______________________________
*
Author to whom correspondence should be addressed: hmzhou@hust.edu.cn
will affect the flow field by changing the rheological
behavior (viscosity) of the blend. With that, the
macroscopic flow field and the mesoscopic structure are
coupled.
Model selection
Lij  w ijA  f 2eijA 
Because dispersion phase iterative evolution is
dependent on the flow field condition (shearing rate,
temperature, viscosity etc.). A thermodynamic model that
can precisely describe the reality of flow field is
necessary before the analysis of morphology.
Applying the incompressibility for injection molding
process in 3D, the control equations of mass, momentum
and energy can be simplified as
ui , i  0
(1)
- P,i ij  2Dij , j  0
.
(3)
where i  1,2,3 and j  1,2,3 are coordinate components,
signal ‘,’ means derivation; u , T , P are velocity,
temperature and pressure;  ， Cv and K are density,
special heat and thermal conductivity coefficient of
material respectively;  is the heat source in the energy
equation,  is viscosity coefficients, H c is the heat
source of crystallization (based on shearing induced
theory),  ij is Kronecke function, D is the strain rate.
For morphological model, the droplet shape is found
very close to an ellipsoid [5]. This kind of droplet shape
can normally be described by a second-order tensor G
[6]. Variables as the radius of spherical granule R and
the lengths of each axis are used to describe the
transformation pattern of within local areas.
In the filling stage, the affine deformation model is
applied for simulating transient slender deformation
caused by high shearing. The lengths of two shorter axes
are assumed equal during the deformation. Setting L , B
and W semi-axial lengths of the droplet ( L  B ), it
follows [7]

tan   0.5  0.5
B R0  0.5  0.5 4   2
4 2

f1  3G ij

 δij 
2  G kk

(7)
where   R  is a characteristic time,  is the
interfacial tension; eij and w ij are deformation rate
tensor and vorticity tensor of the applied flow field
respectively,
following
,
eij  Lij  L ji 2


w ij  Lij  L ji  2 ; superscript A means deformation
rate imposed by outside;
f1 and f 2 have to be
determined through asymptotic limits.
(2)
Cv (T,t  uiT,t )    KT,ii  H c
L R0  0.5  0.5 4   2
In the postfilling stage, when small deformation and
the relaxation recovery process of the droplet is the
dominant factor, here MM model is put into use here. In
the model, the velocity gradient can be expressed as [8]


0.5
1
(4)
(5)
The breakup and coalescence of droplets are the
principal mechanisms that lead to the size variations of
the granules. Assume these two mechanisms are linearly
addible, the evolution equation of R is as
 dR   dR 
 dR 
 
  
 dt   dt  break  dt  coal
(8)
For the breakup of droplet, the simplified capillary
*
number k may help to select the proper breakup
models[9,10,11]. And experimental model or statistical
probability model can be conveniently applied for the
description of droplet coalescence [12].
In order to assure the accuracy of the simulation
results, viscosity ratio is determined by viscosity model.
Considering the dependency between the viscosity of melt
and shearing rate, temperature and pressure, Cross-WLF
model is adopted to describe the rheological behavior in
injection molding [13].
Numerical Simulation
For morphological evolution, in order to achieve
more accurate and reliable simulation results, the
calculation space should be extended from 2D to 3D. First
the constitutive equations of physical model constructed
above should be discrete in 3D. Based on PSPG/SUPG
method a stable finite element format is then established.
And tetrahedron element mesh control volume method is
used to trace the flow front.
(6)
where  is the accumulated shear history experienced.
According to the characteristics of 3D simulation of
injection molding process, the assumptions of flow field
are given as follows [14]. In filling phase, the melt is
incompressible pure viscous fluid. The viscosity force
cause by gravity and inertial force can be negligible, due
to the large viscosity nature of the melt. Assume the
specific heat capacity and heat conductivity coefficient to
be constant for simplification. In each moment, the flow
field is assumed to be stable. In the postfilling stage, the
polymer material becomes compressible.
 N

 ui N  ,i Tn 1d
 t

CV N    SUPGu j N  , j 

e
e
   KN ,i N  ,iTn1d
e
e
 
e
e
For microstructure[15], deformation, breakup and
coalescence mentioned above are only restricted to grainy
dispersion phase, thus the research object are low
concentration dispersion phase (volume content below
30%) system. The medium is assumed to be continuum.
Suppose that the dispersion phase enter the mold cavity in
uniform spherical shape.
The initial diameter D0 of dispersion phase can be
approximately resulted from the observation of
hydrostomia on injector nozzle, and is ranging from 1-10
 m for appropriate simulation outcomes [16,17]. On the
mold wall, the melt freezes instantly, thus the dispersion
phase remains to be in an initial state, which is spherically
shaped.
The finite element method is adopted to discrete the
control equations for the injection molding process. Due
to the complexity of flow field formats in injection
molding, a 3D 4 nodes tetrahedron element mesh is
applied to construct the spatial area to achieve the
accuracy and reliability in 3D simulation. The finite
element formats of velocity and pressure equations are as
follows, respectively.

e
e
N , j N  , j uia  N  ,i u ja d    N , j N  P  ij d (9)
e
e
   N h d
e
e

e
e
N



  SUPGu j N  , j  0.5 N  ,k ui  N  ,i u k   H c d
   e CV N    SUPGu j N  , j 
e
2
N
t
(11)
Tn d    N  q0 d
e
e
N

u j N , j    ui N  ,i  Rn1d

t


(12)
N n
   N   SUPGu j N , j 
R d    N N  F d
e
e
t
e
e
  N  
e
e
SUPG
where F represents the loading part on the right side of
the radius evolution formula.
In postfilling stage, the convection terms of energy
and microstructure equations could be neglected (flow
field is approximately static), and classical decomposition
format Galerkin would not lead to false oscillation of
results. Thus the discretization of equations doesn’t need
to consider the stable formats above.
Experimental Verification
To validate the accuracy of coupled simulation of
macro flow field and microstructure evolution, polymer
blend PP/POE with mass ratio (75/25) is chosen, PP (T36f)
is from the Sinopec, POE (8150) is provided by the
Dupont-DOW company. And injection machine TY200
(provided by Zhejiang Dayu machinery company Ltd.) is
applied to mold the test samples (80×10×4mm) as shown
in Figure 1. A and B are the near gate and far gate
sampling positions for microstructure observation,
respectively.
nel
N N  ,i ui d     SUPG  1 N ,i N  ,i P d  0 (10)
e
e
Where  represents space area,  e is the element space
area, e  1,2,..., nel and
nel is the total number of the
elements; N  is the interpolation function,   1,2,3,4 is
node number of one tetrahedron element; suppose
h is
the stress on the boundary.
In the filling process, the stable finite element format
for energy equation and dispersion phase granules radius
is as below.
Figure 1. Molding sample for microstructure observation
The experiments are conducted with various process
conditions, such as injection velocity, injection
temperature, dwelling pressure, dwelling time and cooling
time. When one process condition varies, other process
parameters are set to be standard. The standard process
parameters are, injection temperature 220℃ and dwelling
pressure 50Mpa, dwelling time 10s, cooling time 30s. And
in this paper, the injection velocity experiment is collected
as an instance for the morphological observation. The
experiment is conducted for 10 molding samples, and the
last 5 samples (stabilized) are for observation, the
injection velocity level is shown in Table 1.
Table 1. Injection velocity level
Low high
Injection velocity (cm3/s)
6
26
Coupled calculating of macro and micro in molding
process with the numerical method proposed previously in
the paper, simulation results in different injection velocity
are obtained. The comparisons with experiment results are
from Figure 2 to 3.
a)
a)
b)
Figure 3. Microstructure mean deformation comparison
under different injection velocities
a) Low injection velocity(6cm3/s)
b) High injection velocity(26cm3/s)
It can be seen that the simulation results agree well
with the experimental data. And higher injection velocity
brings stronger shear, thus the deformation and breakup of
dispersion phase granules are promoted with smaller mean
radius of the granules and larger mean deformation rate.
And also the size of granules in the shear layer
dramatically decline because of the stronger shear.
Conclusions
b)
Figure 2. Microstructure mean radius comparison
under different injection velocities
a) Low injection velocity(6cm3/s)
b) High injection velocity(26cm3/s)
Numerical
simulation
of
polymer
blends
morphological evolution in injection molding process is
systematically investigated. A flow field and
morphological evolution coupled model was constructed
in macro and micro scales. Based on the tetrahedron
element mesh and PSPG/SUPG method, a stable FEM
was established for iteratively solving the interaction of
flow and microstructure. Experimental results validated
the accuracy of macro flow field simulation. And
comparisons between the simulation and experiment
suggest the effectiveness in predicting microstructure in
the process.
Acknowledgement
The authors would like to acknowledge financial
support from the National Natural Science Foundation
Council of China (Grant No. 51125021, 51105152) and
the Guangdong Province-Ministry of Education IndustryAcademic-Research Project (Grant No. 2011B090400381).
References
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
V. Cristini , R. W. Hooper, et al, Industrial and
Engineering Chemistry Research, 41, 6305(2002).
F. Liu, H. Zhou, D. Li, Journal of Reinforced
Plastics and Composites, 28, 571(2009).
H. Zhou, G. Xi, F. Liu, Journal of Materials
Engineering and Performance, 17, 422(2008).
C. L. Tucker III, P. Moldenaers, Annual Reveiws
of Fluid Mechanics, 34, 177(2002).
S. Guido, M. Villone, Journal of Rheology, 42,
395(1998).
E. D. Wetzel, C. L. Tucker III, Journal of Fluid
Mechanics, 426, 199(2001).
D. V. Khakhar, J. M. Ottino, Journal of Fluid
Mechanics, 166, 265(1986).
P. L. Maffettone, M. Minale, Journal of NonNewtonian Fluid Mechanics, 78, 227(1998).
J. M. H. Janssen , PhD. Thesis, The Netherlands:
Eindhoven University of Technology, 1993.
P. H. M. Elemans, PhD. Thesis, The Netherlands:
Eindhoven University of Technology, 1989.
M. A. Huneault, Z. H. Shi, L. A. Utracki,Polymer
Engineering and Science, 35, 115(1995).
B. J. Bentley, L. G. Leal, Journal of Fluid
Mechanics, 167, 241(1986).
B. Yan, PhD. Thesis, China: Huazhong University
of Science & Technology, 2008.
J. Guo, PhD. Thesis. Newark: New Jersey Institute
of Technology, 2000.
J. K. Lee, C. D. Han, Polymer, 41, 1799(2000).
J. T. Lindt, A. K. Ghosh, Polymer Engineering and
Science, 32, 1802(1992)
U. Sundararaj, Y. Dori, C. W. Macosko, Polymer,
36, 1957(1995).