Flow and solidification of semi-crystalline polymer during micro-injection molding

Abstract

This work investigates flow and solidification characteristics of semi-crystalline thermoplastic polymer melt during the micro injection molding process. When the melt flow enters micro-cavities and solidifies, it has been observed that micro-balls are formed on the flow front and coalesced with each other creating many micro-weld lines. Consequently, a characteristic pattern is observed by optical microscopy. This work has examined the mechanism of their creations experimentally as well as numerically. It has been hypothesized based on the experimental results that the onset of this phenomenon is initiated by nucleation in the micro-cavities. For the same micro-cavity, both injection molding and imprinting have been conducted and the shape and the morphologies of the molded parts have been analyzed. Based on observation of the morphologies, this work concludes that nucleation is due to high shear rate while entering the micro-cavity and the balls are formed by the growth of the spherulites. Moreover, the validity of the hypothesis has been verified by the numerical simulation of the ball formation process. The numerical simulation has reproduced the geometric shape found in the experiments. This result supports the proposed hypothesis.

Full text

1
Flow and solidification of semi-crystalline polymer during micro-injection molding
Jiseon Hong, Sun Kyoung Kim, Young-Hak Cho
Mechanical System and Design Engineering, Seoul National University of Science and Technology,
This work investigates flow and solidification characteristics of semi-crystalline thermoplastic
polymer melt during the micro injection molding process. When the melt flow enters micro-cavities and
solidifies, it has been observed that micro-balls are formed on the flow front and coalesced with each other
creating many micro-weld lines. Consequently, a characteristic pattern is observed by optical microscopy.
This work has examined the mechanism of their creations experimentally as well as numerically. It has been
hypothesized based on the experimental results that the onset of this phenomenon is initiated by nucleation
in the micro-cavities. For the same micro-cavity, both injection molding and imprinting have been
conducted and the shape and the morphologies of the molded parts have been analyzed. Based on
observation of the morphologies, this work concludes that nucleation is due to high shear rate while entering
the micro-cavity and the balls are formed by the growth of the spherulites. Moreover, the validity of the
hypothesis has been verified by the numerical simulation of the ball formation process. The numerical
simulation has reproduced the geometric shape found in the experiments. This result supports the proposed
hypothesis.
Keywords: moving boundary; solidification; crystallization; morphology; micro-injection molding
Introduction
The injection molding process of the thermoplastics is the only polymer process that facilitates mass-
production of consumer products with various geometries [1, 2, 3]. As the polymer melt in the mold shrinks
significantly during solidification, extra melt has to be supplied to the mold to compensate the shrinkage.
The injection molding process facilitates this compensation by continuously feeding the melt into the mold
through the injection gate as demanded. Thus, parts with great dimensional stability can be manufactured
 corresponding author: sunkkim@seoultech.ac.kr
Dasan Bldg. Room 234, 232 Gonreung-Ro, Nowon-Gu, Seoul, 01811, Korea

2
at unmatched rates. Modern machine and mold tooling technology facilitate production of parts with higher
precision. These days injection molding studies expand to thin-walled parts, micro-cellular parts, and so on
[4, 5]. Moreover, injection molding technologies such as vibration, induction heating and gas-assisted
methods have been extensively investigated to improve the productivity and part quality [6, 7, 8].
Based on the improved dimensional stability and geometric resolution, injection molding of polymer
parts with micro-scale features have been tried and successful in many applications [9]. Especially, medical,
electronic and optical parts such as microfluidic chips and light guide plates have been commercially
successful. However, several challenges are difficult to overcome and still remained as open problems.
These are further pronounced when the feature size approaches the intrinsic meso-scale dimensions of
morphologies in the solidified polymer. The shrinkages in the conventional injection molding process are
predictable in a range although not fully controlled. The micro-injection molding processes of amorphous
polymers are well established in comparison with that of semi-crystalline polymers. Some additional cares
in tooling, filling and ejection would allow feasible processing windows for amorphous polymers [10].
Meanwhile, micro-injection molding of semicrystalline polymers is rather complicated. Semicrystalline
thermoplastic polymers shrink more than amorphous ones due to crystallizations inside and outside the
mold. It not only shrinks dimensionally larger but also is less predictable. Therefore, more consideration in
mold design and tooling is required. In micro-injection molding processes, the dimensions of the crystalline
morphologies are close to those of the geometric feature sizes. This can incur inhomogeneity in the
solidified part. Furthermore, the filling of the micro-cavity can involve phenomena other than normal
viscous flow.
In a micro molding process, polymer melt experiences an extreme contracting flow. Normally the
macro-cavity is filled first then the melt enters micro-cavities. The macro-cavity can be filled under a
relatively lower pressure whereas micro-cavities demand higher pressure to be filled. Prior to filling of
micro-cavities, the macro-cavity needs to be filled first and then pressurized to a certain level. Thus, the
melt already starts to solidify and crystallize while waiting for the pressure to rise up staying near the
entrance of the micro-cavity. In this stage, it has been found that the flow front inside the micro-cavity is
not round and smooth as is observed in the macro-cavity. It has been observed that small balls are formed
on the flow fronts inside the micro-cavities [11]. The geometry of the micro-cavity there is a rectangular
groove, which is a basic two-dimensional shape in the micro-engineering. It can be applied to micro-fluidic
channels, optical structures and micro-electric conduits. When the packing pressure is high enough, the
balls merge together and eventually the micro-cavity is completely replicated. Although microscopic
inspection gives no visual flaws, micro-welds are formed all over the micro-cavity. They can bring about

3
gas permeable and mechanically weak surfaces.
The balls have been reproduced in repeated experiments. It looked like teeth on the gum or balls on
the rectangular stick. Let us call this phenomenon ball formation (BF). When the BF was first observed for
PP (polypropylene) copolymer, it was simply thought of as one of the many injection molding troubles [11].
That means, as long as we can keep out that happening, analysis of the phenomenon itself can be regarded
marginally important. However, it has been repeated with certain popular resins such as polyamide
especially when the feature size was small [12]. It has been also found that the BF is apparently influenced
by the flow geometry. If the flow path was wider than a certain value, the balls have not been formed. Thus,
there is a critical dimension for its onset [11, 12]. In addition, it should be emphasized that BF is important
because it happens with one of the most popular commodity polymers, PP [13, 14].
Based on the observations, the previous work argued that the created spherulite entered the micro-
cavity ensuing the BF [12]. However, some experimental evidences contrary to this argument has been
found. The details will be explained later in this work while explaining the new hypothesis. To further
understand the ball formation from creation to final solidification, another set of experiments has been
planned. This work reports the results of the experiments and establishes a revised theory for the
phenomenon. A new mold insert that can yield more experimental evidences is fabricated. Besides, an
imprinting process is conducted with the same mold insert and the results will be compared with those by
the injection molding processes. Based on the new experimental observation, a theoretical hypothesis
including a flow model is proposed. The model is numerically implemented by a computer code to verify
the hypothesis. The numerical method in this work simulates the crystalline growth on the flow front, which
is the first attempt in the micro-injection molding simulation to the best of the authors’ knowledge. To do
so, this work employs a Lagrangian flow front tracking instead of the conventional Eulerian approach as in
[15]. The numerically obtained flow fronts are compared with the experimental ones to prove the hypothesis.
Theoretical
Micro-molding method
The filling and solidification in micro-cavities have been still challenging due to rheological,
thermal, structural issues. To overcome these, considerable amount of researches and developments have
been fulfilled yielding some meaningful and productive results [9]. The melt flow in the micro-injection
molding is quite complicated as the melt has to pass through the runner and flow all the way from the gate

4
to the end of the micro-cavity. When this is not viable for some parts, imprinting can be chosen as an
alternative processing method. Taking the disadvantage in cycle time and process control, hot embossing
or imprinting methods are sometimes chosen to shorten the flow path [16]. The imprinting method is usually
chosen when the production volume is small, or the injection molding is impracticable due to the filling
problem. Figure 1 compares these two processes for parts with the micro-features in this study. The mode
of delivering the material differs in each of the two methods. In the imprinting method, the material should
be ready prior to imprinting because the injection gate is not available. In this work, the material for the
imprinting process is molded by the injection molding process with a dummy insert without any micro-
patterns on it. However, it should be mentioned that the purpose of the imprinting experiment in this work
is to reveal the characteristics of micro-injection molding process by comparison.
Fig. 1 Process schematics of injection molding and imprinting.
Filling flow in micro-molding process
The micro-cavity in this work is a thin and long rectangular groove with a finite depth, H, as shown
in Fig. 2. The width, W, is an order of magnitude smaller than the macro-cavity thickness, C. As the pressure
for the filling flow into and inside the micro-cavity can be high, it is now always easy to let the melt flow
there. When the flow front first touches the micro-cavity, the pressure there is not much higher than the
atmospheric pressure. Sometimes, it might not be high enough to push the melt immediately into the micro-
cavity. Thus, the melt stays near the entrance waiting for the cavity pressure to rise up. Apparently, the
fabricated
mold insert
filling
ejection
packing and
solidification
ejection
injection molded plate imprinting and
solidification

5
pressure at the entrance rises up as the macro flow advances during the filling phase.
In the meantime, the total shear force exerted on the wall of the micro-cavity and the surface tension
on the flow front are in balance with the pressure at the entrance. Thus, once the flow occurs at the micro-
cavity during the filling phase of the injection molding process, the flow advances in accordance with the
pressure form the macro-flow. However, sometimes, the pressure at the entrance might not reach the
required pressure to induce the micro-flow. In that case, the flow will not occur until the packing phase.
The filling of the micro-cavity would depend on the pressure at the entrance reached then.
Fig. 2 Micro-injection molding of rectangular micro-cavity (C = 1 mm).
Previous observations
It is unquestionable from the previous study that the BF is caused by the morphological development
[11, 12, 17]. The spherulites have engaged in BF and its emergence depends on the conditions affecting
crystallization. A spherulite is a semicrystalline structure characterizing the mesoscale feature of the
polymer. This consists of crystalline lamellae that are aggregated forming a radiating structure. Once
nucleated, it spheroidally grows to a size of 1 to 1000 m. Figure 3 shows the concept of the spherulite
`
micro-cavity
entrance
to micro-cavity
macro flow
W
flow hesitation
H =50~150m
C
W = 4~ 80m

6
growth in an injection molded specimen. The flow front characterized by the BF has been generated under
two conditions. First, the width of flow channel should be smaller than a certain limit since the balls have
disappeared for channels wider than the limit. Second, the polymer should form a spherulites in a scale
similar to that of the channel. It should be emphasized that the BF does not occur with any polyethylene
tried due to the small spherulites [12].
However, it is still unclear how BF is initiated. More specifically, it can be guessed that the spherulites
are grown inside the micro-channel or in the core region. The previous work argued that the high pressure
during the molding has pushed the grown spherulite into the micro-cavity causing deformation and recovery
[12]. As a result of the experimental observations, the key dimensionless number for a semicrystalline
polymer melt, Km, has been suggested as
Km=
s
W
D
(1)
where W is the channel width and the Ds is the characteristic diameter of the spherulite. In order for BF to
occur, Km needs to be smaller than the critical value. The criterion is specific to each setup including
selection of polymer and the molding conditions. However, there has been a criticism based on some
experimental results that the spherulites are not observed or those near the entrance is significantly smaller
than those engaged in the BF. Since this could not be refuted by simple explanation, a revised hypothesis is
proposed in this work.
Hypothesis on ball formation
In this work, a new hypothesis that the spherulite might have been grown inside the micro-cavity has
been proposed. Here, the spherulite growth is comprised of two phases. In the first phase, the nucleation
should take place. Then, the spherulites should develop to their final forms. The nucleation in the solidifying
polymer depends on several conditions such as flow, temperature and additives. Nucleation is initiated by
locally developing parallel chains resulting from molecular motions [18]. Thus, it should be relevant to all
the conditions that affect the molecular motion in nanoscale. Over last century since 1930s, the effects of
these conditions on nucleation have been intensively investigated for many polymers [19, 20]. In injection
molding of a commercial polymer, which is compounded with nucleation agents, the flow and temperature
become the most critical factors affecting the crystalline morphology. The injection molding process
accompanies dramatic changes of temperature and shear in the polymer being processed while each cycle
is executed. Therefore, the process conditions are crucial to the nucleation. The number of nucleated crystals

7
depends on both the temperature and the shear histories. Since each contributes to the nucleation via
different mechanisms, they do not interfere with nucleation by each other as long as space for nucleation is
still available [21]. Thus, the nucleation by shear can take place even after cooling has substantially
progressed.
Thermoplastic crystals observed in the injection molding process can be either spherulites or shish-kebabs.
While the shish-kebabs appear in the high shear region mostly near the wall, the spherulites are commonly
observed in the slow flow regions. As shown in Fig. 3, the crystalline structure extends in the radial direction
from the nucleation site. The regular crystal unit is chain folded ribbon lamella connected by the tie molecules
that are randomly arranged in the amorphous region [18]. Thus, the density inside the spherulite is higher than
that outside. The growth rate in the radial direction can be described by the Hoffman-Lauritzen model [22].
This model can compute the crystal growth rate, which is a radial velocity. In injection molding, the growth
rate can be simply expressed as a function of the temperature. The crystal growth can be prevented by
neighboring crystals, mold walls and the kinetic conditions. Figure 3 shows the impingement between two
growing spherulites forming linear boundaries.
Fig. 3 Spherulites schematic in injection molded specimen.
Coming back to the BF hypothesis, what should be questioned first is whether nucleation is feasible inside
the micro-cavity. From the described theory, nucleation requires appropriate temperature or shear conditions.
nucleus
boundaries between
impinged spherulities
growth rate, G
(m/s)
injection-molded
part
spherulite

8
Regarding the temperature, the molds in the micro-injection molding processes are usually heated in practice
to guarantee the required fluidity inside the micro-cavity. This allows favorable conditions for nucleation.
Furthermore, the narrow dimension of the flow path in the micro-cavity incurs very high shear rate. As long as
the flow occurs, nucleation by shear deformation would be initiated.
While solidification inside the micro-cavity progresses, the nucleated crystals start to grow.
Assuming the quiescent condition, the crystals would grow to a spherulite. The growth rate at temperature,
T, is modeled by the competition between the kinetic driver by the Gibbs free energy and the molecular
mobility for long range movement, which is expressed as [22]
( ) ( )
*
0
( ) exp exp g
m
K
U
G T G R T T T T T

   
= − −
   
   
−−
   
(2)
where
*
U
is an energy constant, R is the universal gas constant,
m
T
is the melting point,
T
is the
temperature at which no further molecular displacement occurs, G0, and Kg are the constants [23]. Note that
they are material specific constants except R. It should be stressed that he growth rate is greatly dependent
on G0, and Kg. Temperature here is a bulk temperature of the regime occupied by the spherulite. The growth
rate
()GT
cannot be calculated without knowing the temperature,
( )
,Ttx
, where x is the current position.
In actual calculation, the growth rate will be a function of position and time, i.e.,
( ) ( )
( )
,,G t G T t=xx
. The
final dimension of the spherulite is governed by the temperature history and the impingement between the
spherulites. Without the impingement, the spherulite diameter can reach
2 ( ) / ( / )
m
T
sT
D G T dT dt dT

=
(3)
Since the material specific temperatures,
T
and
m
T
, are given, the cooling rate governs the final diameter.
Because of the elevated temperature in the micro-cavity, the cooling rate,
( / )dT dt−
, is very small. The
spherulite inside the micro-cavity can have enough time for growth that Ds can reach W in Fig. 1. From the
previous experimental observations, Ds, is close to W when BF occurs. In summary, nucleation occurs
during the entrance flow, and then it grows to a large spherulite inside the micro-cavity due to the low
cooling rate. This is the hypothesis in this work and will be verified by further experimental and numerical
investigations.

9
Experimental
Mold and mold insert
An injection mold that can contain the mold insert with the micro-patterns has been built. The
mold will be installed onto a vertical injection molding machine. As aforementioned, micro-injection
molding processes usually require heating of the mold. It is desirable to localize heating on the mold surface
since it should be cooled later to eject the molded article. In this work, the mold insert is heated from the
back side of the mold insert by a coiled tube. The tube is heated and cooled by changing the internal air
flow. In the previous studies and manufacturing practices, the infrared, induction, vibration and steam have
been utilized [5, 7, 8, 24]. Further review for the variotherm method for micro-injection molding is found
in [25].
Figure 4(a) shows the conceptual diagram of the heating device and the built coil assembled in the
mold. Heating and cooling by air flow is usually unfavorable because air cannot carry much energy due to
the low volumetric heat capacity. However, it is convenient in that cooling can be also facilitated by the
same tube. Figure 4(b) shows how heating and cooling modes can be switched by controlling the valves.
To enhance heating efficiency, the flow rate and temperature should be increased. The inlet temperature
during heating is 600oC and the mass flow rate is 0.62 g/s. The tube coil is encapsulated by sintered power
metal to enhance the heat transfer and to structurally protect the tube from the molding pressure.
Figure 5 shows the mold design including the delivery system and the mold insert. The macro-cavity
is placed on the bottom side of the parting line while the mold insert with the micro-cavities are placed on
the top side. The mold insert is laterally surrounded by a retainer to be securely fixed on the mold. A side
gate, as illustrated in Fig. 1, is employed to deliver melt to the thin rectangular cavity with periodic micro-
features. Based on the previous studies [11, 12], the cavity in this work, which is characterized by Fig. 2,
does not causes any venting problems. This is presumably because long grooves lower the likelihood of air
trap. Thus, no device and design for air evacuation are employed in this study. Moreover, the air trap can
affect the replication performance when fully filled but not the ball formation.

10
(a)
(b)
Fig. 4 (a) Implementation of the variotherm system by hot/cool air flow by tube coil; (b) Operation
of variotherm by air flow for micro-molding.
Heating
air flow
Mold
insert
Exhaust
flow
Insulation
Compressed
air
Compressed
air
Chilled
compressed
air
Chilled
compressed
air

11
Fig. 5 Design of the assembled mold.
Fig. 6 Nickel mold insert with micro-cavity arrays fabricated by a LIGA process.
Figure 6 shows the photo image and layout of the mold insert with dimensions of each micro-cavity
region. One mold insert with 16 different kinds of micro-cavities with various dimensions on the surface
have been fabricated to scrutinize the micro-mold filling flow in detail. In each numbered region in Fig. 6,
an array of all vertically oriented micro-cavities with the same width are fabricated. The total count of
Runner
Sprue
Macro-
cavity
Retainer
Micro-cavity on mold insert
Mold insert
Tube coil
Gate
Insulation plate
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
60mm
gate
(4)
W=17.6
H=81.1
(3)
12.7
75.7
(2)
7.8
66.0
(1)
4.2
50.3
(8)
W=39.1
H=91.3
(7)
34.2
91.2
(6)
29.4
90.1
(5)
22.4
89.8
(12)
W=55.7
H=98.3
(11)
53.7
93.0
(10)
49.8
91.9
(9)
43.0
91.7
(16)
W=80.1
H=147.3
(15)
72.3
118.9
(14)
68.4
105.4
(13)
62.5
104.1
60mm
(m)

12
micro-cavities in each region depends on the width of micro-cavity there. Since it has to sustain the high
molding pressure, a metallic structure is preferred. Besides, the metallic surface can exchange more heat
than others can. A LIGA process followed by electroplating nickel on it facilitates fabrication of such a
mold insert. Then, the mold insert has to be further machined and trimmed to fit in the retainer. The widths
and depths are affected by the etching and the plating processes. They have been measured and averaged
after all the fabrication was finished. The depth are roughly constant for the micro-cavities (5)-(11) due to
the employed etching method. Refer to [26] for the details of the fabrication method. Note that the square
design of the mold insert allows changing the position relative to the gate by rotating it.
The temperature of the mold insert is monitored on the back side by a K type thermocouple to control
it. In this experiment, the temperature is controlled by adjusting the power input of the air heating furnace,
which is detached from the injection molding machine. The heated or chilled air enter the tube coil through
an insulated line. The injection molding in this work is conducted in a typical vertical machine with a
holding force of 50 ton. Furthermore, the same mold insert is reused for the hot embossing process using a
hydraulic hot press where temperature and pressure are controlled. Here, the polymer is injection-molded
first with a dummy insert to make the bulk thickness same. Then, the plate is hot-embossed with the mold
insert in the hydraulic hot press.
Material
It has been already verified that the BF happens with semicrystalline polymers yielding fairly large
spherulites [12]. This work is focused on the morphological development to investigate the mechanism of
the BF. Thus, this work considers a homo polypropylene (PP) only (Lotte Chemical SJ150), which is a
standard injection molding grade PP composed of more than 95% of isotactic PP with moderate xylene
solubility. As is well-known, PPs are renowned for the pronounced crystalline morphologies [27, 28, 29].
To avoid any possible problems from moisture, the material in a pellet form has been dried for 10
hours under 90oC in a convection oven. Then, it has been vacuum-packed to prevent water uptake until the
molding time. Moreover, the melt flow index (MFI) of multiple samples from the same bag has been
examined to check the consistency. By doing so, possible inconsistency induced by the material variation
can be suppressed in the molded parts. The MFI of 11.8 g/10 min, has been reproduced with an error of
0.74g / 10 min, with a standard device compliant with ASTM D1238 (Tinius Olsen, MP987). It is quite
close to the value reported by the manufacturer, 10 g/ 10 min.

13
Experimental conditions
In micro-injection molding, the pressure during the filling and packing phases drives the melt into
the micro-cavity. The pressure at the entrance of the micro-cavity, pi, is an important factor. When pi is
higher, the shear rate is higher and accordingly the possibility of shear-induced nucleation is higher.
Moreover, the higher packing pressure enables removing the traces of the BF by fully merging the balls
without any visible boundaries between them. To visually inspect the BF in the molded specimen, short
shot in the micro-cavity is necessary and the packing pressures has to be set properly.
Once the nucleation is accomplished at one or more sites, temperature of the melt, T, is the most
critical variable for the BF since the spherulite size, Ds, greatly relies on the cooling rate as expressed in
Eq. (3). The melt temperature history depends on the initial melt temperature, Ti, and the temperature of the
mold insert, Tw, and the mold flow. To simplify the experimental conditions, while the temperature at the
injection nozzle, Tn, is fixed as 200 oC , only Tw is controlled at 60, 80, 100 and 120 oC. Raising Tw over
120oC can make cooling and ejection very difficult. The injection rate is not varied in this work as its effect
has been examined elsewhere [12]. While the flow rate is fixed as 12 cc/s during the filling phase, three
different pressures, 0, 1 and 5 MPa, have been tried to check the effect of packing.
Polarized optical microscopy and XRD
One of the popular methods for scrutinizing the morphologies of semicrystalline polymer is to
visualize the thin crosssection by the polarized optical microscopy (POM). An optical microscope (Nikon,
Optiphot 2) with polarization accessory (POT-1) has been employed to have the specimen scrutinized. The
image has been taken by a CCD camera connected to a PC via USB. Besides, the measurement of
characteristic dimensions have been done in the magnified images. The thickness of the solidified layer
attached to the wall will be also measured. Besides, to compare the crystalline characteristics of the
specimen by the injection molding and the hot embossing, XRD (x-ray diffraction, Bruker D2 Phase) test
will be conducted and the test data will be analyzed. For the XRD specimen, a microtome with glass knife
has been employed. For the POM, small pieces from the molded specimen has been embedded in epoxy
and has been cured and thinned to allow light transmission that helps visualize the spherulites.

14
Numerical Simulation
Overall approach
To simulate the aforementioned BF, a well-defined numerical scheme is required. The flow problem
including the flow front tracking has to be solved. Moreover, to simulate the spherulite growth, the
temperature field is required. With a suitable spherulite growth model, the flow front can be numerically
simulated. Thus, the numerical problem can be stated as a convective heat transfer problem with moving
boundaries. To visually demonstrate the deformed shape, the flow front should be captured along with the
flow. Moreover, the method should realize the underlying phenomena involving the crystalline development
as well as the viscous polymer flow. As the crystalline development is essential to the characteristic shape,
the spherulite growth and the corresponding density change should be a staple of the simulation.
Some recent works simulate the micro-injection molding by the molecular dynamics [30, 31].
However, a continuum method is required to simulate the flow within an affordable computational time.
Even with a continuum method, the numerical simulation of the micro-injection molding is a quite
demanding task. In the earlier studies, an approximate model has been used [32]. Usually, it is advantageous
to adopt an Eulerian method with a fixed mesh to track the flow front while solving the Navier-Stokes
equation [15, 33, 34]. However, as the simulation here has to involve crystalline development, a moving
mesh with a Lagrangian scheme is employed. To focus on the BF, the domain of interest is confined to the
micro-regime, which is treated as a two-dimensional domain for simplicity. If the pressure, pi, has to be
determined accurately, it should be from the simulation in the macro-regime [15]. However, since the
simulation here focuses numerical reproduction of the geometric characteristics in the BFs, this work
assumes pi properly as a boundary condition. It also requires some other considerations regarding the initial
setup. The simulation starts from a situation that flow has considerably progressed in the micro-cavity. In
addition, a single nucleus is seeded in the beginning on the flow front since the experimental observations
show only one spherulite per ball. Further details of the simulation settings are in the coming sections.
Flow and thermal model
Since the crystalline growth in this work is fully coupled with the flow and heat transfer problems.
These are already established in the conventional simulation method of the injection molding process. The
mathematical models for them are well described in the previous literatures [15, 35, 36]. However, they are
briefly repeated here since some modifications are required. The flow inside the mold is expressed in terms
of velocity, u. Given the density,

, and time, t, the velocity should satisfy the continuity,

15
/ ( ) 0t

  +   =u
. Here, the density is given as a function of temperature T and pressure p. The density
can be determined by the p-v-T equation of the states since the density is the reciprocal of the specific
volume, v, of the polymer. In injection molding of thermoplastic polymer, the p-v-T relations are
conventionally modeled by the Tait equation, which is expressed as [35, 37]
( , ) ( ) 1 ln(1 ) ( , )
()
og
p
v T p v T C v T p
BT

= − + +


(4)
where
()
o
vT=
12
b b T+
,
()BT =
34
expb b T


−
,
( )
7 8 9
( , ) exp
g
v T p b b T b p=−
, and
5
T T b=−
. Here,
the constants, bi’s are fitted from the test. In this model, the melt and solid phases are divided at
56t
T b b p=+
. In this work, this equation cannot be used below
t
T
as is. Below
t
T
, solidification and
crystallization take place resulting in coexistence of crystalline and amorphous phases. The model should
be able to accommodate the density differences of those two phases. Although the model for each phase
requires lengthy explanation but it is a part of this study, its detail will be presented in the results section.
The velocity should satisfy the momentum equation,
Dp
Dt

= − + 
uτ
(5)
where the extra stress tensor,
τ
, is described in terms of the velocity. Assuming the GNF (generalized
Newtonian fluid), the relation between the extra stress tensor and the rate of deformation tensor,
γ
, is
described by
( , )T

=τ γ
where
γ
is defined by
t
=  + γuu
and the
( , )T

is the viscosity. Here,
the viscosity is a function of the temperature and the shear rate,

, which is the second invariant of
γ
that represents the magnitude of the deformation rate. It can be simply calculated by
( )
1/2
(1/ 2) ][ij ji
ij

=
. The Cross-WLF model for viscosity is adopted to describe the non-Newtonian as
well as temperature-dependent behaviors of the polymer melt. For the given T and

, the corresponding
viscosity function takes the form [38]
01
00
()
1 ( ( ) / ) n
T
T

  
−
=+
(6)
where the zero shear viscosity is described by
( )
0T

=
1 1 2 2 2
exp[ ( ) / ( )]D A T D A T D− − + −
for
NF
TT
and
0

=
for
NF
TT
. Here, D1, D2, A1, A2 and

are constant values that should be determined by

16
fitting the experimental data for a specific polymer. The no flow temperature,
NF
T
, is introduced to describe
the flow stoppage due to rapid solidification by crystallization [39]. This is a lot higher than the glass
transition temperature, D2, and corrects the error in
( )
0T

caused by fitting with the data of amorphous
melt state [40]. Since the flow by crystallization can occur even below
NF
T
, the velocity filed might be
acquirable. In this case, Eq. (5) becomes
00
lim( / ) / 0p D Dt

→
  =  + =τu
, which gives
20=u
that
can be solved together with the corresponding kinematic condition by the spherulite growth. Of course, p
and
τ
cannot be calculated here. It is also noted that the viscosity is important not only in simulation but
also in monitoring the process [41].
To be more accurate, the viscosity in the micro-injection molding simulation should accommodate the
increase by the size effect [32]. However, the melt viscosity measured in the micro-channels have been
reported lower than that by the capillary rheometer due to the possible slip at the wall [42]. Also
acknowledge that finding slip correlation by the Mooney analysis is not a simple matter. The slip on the
boundary and the surface tension are present in the flow of micro-injection molding but this work does not
consider them in the model to focus on the ball formation. However, the surface tension will be discussed
further regarding the feasibility of the flow initiation inside the micro-cavity.
The heat transfer is an important aspect of the injection molding analyses since the mold temperature
history should replicate the previous cycles for quality control [43]. The temperature field during the
process is obtained by solving the energy equation,
2
DT
c k T
Dt
 
=    +
(7)
Here, the thermal conductivity, k, and the specific heat capacity, c, are required in addition to the density.
Moreover, the volumetric heat dissipation by shear heating,
2

, is calculated from the shear rate that is
obtained from the velocity field. On the mold wall, a Dirichlet condition is inappropriate to impose. Because
of the mold wall roughness and the polymer surface state, there exists significant thermal resistance. The
thermal resistance is described by introducing a heat transfer coefficient, hc, as
/ ) ( )(wwc
T n T Tkh  −−=
where Tw is the mold wall temperature [44]. The wall temperature can vary
over the position and the time.

17
Fig. 7 Spherulite growth model and flow conditions.
Discretization
The finite element formulation with a mixed method have been employed [35]. The numerical
procedure is presented in this section briefly. Multiplying the weighted residuals, q, to the continuity
equation followed by integration over the interior domain,  gives
0d
t




+     =



uq
(8)
Similarly, the momentum equation with GNF constitutive equation can be also written in a weak form
( ) ( )
( )
( )
d d p d d
t
   
   
  +     +   −    =  

   
uw u u w u w w n
(9)
where w is the weighted residual function and  is the boundary domain. The integral forms are discretized
by the Galerkin method as
( )
/
pt−  + + + =M u u Cu Au Qp B
and
0
T=Qu
, where M, C, A, Q, and B
are the mass, the convection, the shear stress, the pressure and the boundary matrixes, respectively. The p
and t denotes the value from the previous time and the time step size, respectively. The velocity vector u
( )
( )
G T t
( ) ( )
( )
/1
a c a
u G T t

=−
extra-outflow due to density difference
i
R
L
W
( )
( )
G T t
1

2

2

1

1

3

4

5

wall
pressure
specified
inlet
continuity
and specified
deformation by G
specified
deformation by G
free
deformation

18
and pressure vector p at the nodes are achieved by
T
     
=
     
     
K Q u F
Q 0 p 0
(10)
at each time step where
/t=  + +K M C A
and
/
pt=  +F Mu B
. The discretization for the
temperature field is not presented here since it is similar to the above momentum equation. Note that the
developed code implementing the above formulation solves the slightly compressible flow in connection
with the p-v-T equation [35].
Geometric model
In the micro-cavities, the flow path is so narrow that multiple spherulites cannot grow on the flow
front. In fact, the POM images shows only one large spherulite on the flow front [12, 17]. Thus, this work
assumes the spherulite growth from single nucleation site on the flat initial flow front. Figure 7 shows the
schematic diagram that illustrates flow involving the spherulite growth in the micro-cavity with such
nucleation. Once nucleation takes place at a point, the spherulite grows simply in an isotropic manner by
Eq. (2). To update the spherulite domain, the geometry has to be updated accordingly by the Lagrangian
scheme. To make this Lagrangian scheme easier, the simulation starts with a spherulite of initial radius, Ri
not with a point nucleus. Fluid in the main body (amorphous region) of the micro-cavity should flow toward
to the spherulite. Consequently, the free surface in the main body should deform in accordance with the
outflow. According to the changed boundary, the computational domain needs to be renewed at every time
step. Simply nodes are moved according to the velocity by u

t. The model also assumes the flow has
already progressed by the distance, L, from the entrance into the cavity as shown in Fig. 7.
It is predicted that the characteristic geometry in the flow front will be formed mainly due to the
spherulite growth together with the density difference between crystalline and amorphous region. In the
crystalline region, the polymer chains are packed efficiently than in the amorphous region. Accordingly, the
density in the crystalline region should be higher. Let us denote the densities in the amorphous and
crystalline regions,
a

and
c

, respectively. Acknowledge that polymer chains should be supplied into
the spherulite to form lamellae in the spherulite. To allow this, flow into the spherulite should happen to
meet the mass conservation. Thus, the whole domain is considered consisted of two connected subdomains
with different densities as shown in Fig. 7. The spherulite growth rate,
( )
GT
, and the flow into the
spherulite by the density difference contribute to the flow front deformation. Considering the required mass

19
flow rate into the spherulite, the normal velocity relative to the spherulite boundary is estimated by
( ) ( )
/1
a c a
u G T

=−
(11)
Apparently, since
c

is higher than
a

,
a
u
will always meet
0
a
u
. Also note that when the growth
stops,
a
u
will vanish.
(a)
(b)
Fig. 8 SEM image of ball formation for case (4) with short shot at Tw=120 oC: (a) SEM image of the
micro-cavity; (b) SEM image of the cross-section with gold plating.
Results and Discussions
Observation of ball formation
Micro-injection molding experiments described in Fig. 1 have been conducted and plastic parts have
been attained. The detailed shape of the balls have already been presented in [12], but to help understand
the phenomenon, similar results are repeated here. Figure 8 shows SEM images of the micro-cavity cut
from a molded sample. In this case, to form and expose the balls, short shot was intentionally tried under a
fairly high wall temperature,
o
120 C
w
T=
. Figure 8(a) and (b) shows the different micro-cavities from the
same array. The three-dimensional outline of the ball formation is visualized in Figure 8(a). Moreover, to
clearly show the cross-section of the balls, the micro-cavity was cut along the approximate center line and
gold-plated for the improved contrast. Figure 8(b) shows the cross-section clearly and also allows
comprehension of inside terrain on the flow front. While cutting, the micro-cavity was slightly deflected as
100m

20
can be seen in the figure. Two things should be emphasized here. First, the balls are formed on the
progressed flow front, which means nucleation by shear might need a distance to flow. In other words, there
is a minimum depth for the BF and it would be shear-rate dependent. As the shear rate is proportional to the
reciprocal of the width, there is a minimum aspect ratio (=H/W) for nucleation. In addition, there should be
enough space for the balls to grow. Second, the surrounding boundaries of the micro-cavity stops the growth
of the balls. When the depth is too short, the micro-cavity will be fully filled before a noticeable BF.
(a) (b)
(c) (d)
Fig. 9 (a) Molded specimen by short shot at Tw = 60 oC for cavities (3-6);(b) molded specimen by
short shot at Tw = 80 oC for cavities (3-6); (c) molded specimen by short shot at Tw = 100 oC for
cavities (3-6); (d) molded specimen at Tw = 120 oC for cavity (5) under different packing pressure.
cavity (3) cavity (4)
cavity (5) cavity (6)
cavity (3) cavity (4)
cavity (5) cavity (6)
cavity (3) cavity (4)
cavity (5) cavity (6)
1 MPa 5 MPa
Short shot 0 MPa

21
To observe the balls, short shots are necessary. Otherwise, balls are very difficult to visualize out of
the specimen. Figure 9(a) shows the short shot results for some micro-cavities at
o
60 C
w
T=
. There have
been ejection problems for the micro-cavities (1) and (2) in Fig. 6 that the results could not be properly
obtained. The results for the (3-6) micro-cavities are presented in the figure and none of them shows BF.
This is continued for the case of
o
80 C
w
T=
and smooth flow fronts with no anomalies are shown in Fig.
9(b). However, BF is very active at the elevated temperature of
o
100 C
w
T=
as shown in Fig. 9(c). The
formed sites are not quite regularly placed but balls are scattered all over the micro-cavities. For the case
for
o
120 C
w
T=
, Fig. 9(d) shows that the BF has become more regular and the entire flow fronts are covered
with the balls. As expressed in Eq. (3), the increased wall temperature reduces the cooling rate and thus
enhances the spherulite growth.
When the packing pressure is increased, the balls become closer to each other and they form weld
lines between them. Those weld lines eventually disappear at a further increased packing pressure as can
be seen in 5MPa case of the Fig. 9(d). The replication could be completely achieved for cases (3-16) for
5MPa without molding troubles such as air trap or specks. Note that the micro-cavities (1-2) did not allow
intact ejection under any molding conditions. Simply by examining the final sample obtained with proper
packing, it would be almost impossible to determine whether BF has occurred or not. This suggests that BF
itself does not necessarily mean processing failure. However, these micro-weld lines can negatively impact
the material properties of the molded part. Apart from the obvious deterioration of the mechanical properties,
barrier properties would be also decreased. Thus, when molded parts are designed for micro-channels, gas
permeation should be checked if necessary.
Spherulites in the micro-cavities
The morphology inside and near the micro-cavity provides plenty of information about what
happened while the filling and solidifying processes. The molded specimen has been embedded in epoxy
followed by curing, cutting and thinning, sequentially. Then, crystalline morphology has been imaged by
the POM. Figure 10 shows such a morphology near the surface with micro-cavities. Where the balls are
formed, clear boundaries can be seen as annotated in the figure. In the core regime roughly more than 100
m away from the entrance to the micro-cavity, fairly large spherulites are created. This is typical in
injection molding and the development of such morphology is quite well understood, but it is worth
repeating here.

22
As explained previously, nucleation is more active where high shear and thermal gradient are
existent. Thus, the core regime is sparsely nucleated while the wall regime is densely nucleated. As dense
nucleation causes premature impingements between the spherulites, they cannot grow large. However, in
the core regime, sparse nucleation allows larger spherulites since impingement does not take place for
longer time. Besides, the slower cooling in the core regime also enhances the growth.
Figure 10 also shows the frozen layer where crystalline morphology is not observed inside. There
is a quite distinct boundary between the core regime and the frozen layer. The frozen layer is formed by
solidification while the flow into the micro-cavity is stagnant at the entrance waiting for the inlet pressure
to build up. This is a sort of hesitation flows where the melt cannot flow to the closer location due to easiness
of the flow to other locations. Conventionally in injection molding, a good mold design should not allow
this since it can cause mechanical and cosmetic flaws. However, in micro-injection molding processes, this
is unavoidable in many cases especially when the characteristic feature is smaller.
Fig. 10 Ball formation and detailed morphologies on the cross-section.
flow
frozen layer
ball
spherulite
nucleus
dislocation
through frozen
layer by flow
W=22.4m

23
Fig 11. Morphologies of different micro-cavities filling at Tw =120oC and packing at 1MPa.
Through this frozen layer, thick lines have penetrated from the core regime to the entrance of the
micro-cavities as indicated in Fig. 10. It is postulated that these lines are traces of plug flows of the
solidifying melt into the micro-cavity from the frozen layer. The plug flow has taken place due to the high
viscosity and the long relaxation time of the polymer there. Because each plug flow to different micro-
cavities might have happened at different time and rate, the flows are separated from each other leaving
traces of dislocation. This dislocation might have happened at locations such as grain boundaries where
shear strength was lower. Similar lines can be also found elsewhere [45]. What is additionally noticeable
here is the lines are bent mildly toward in the direction of the main flow as can be seen in Fig. 10.
Figure 11 shows all the images obtained for the fourteen different micro-cavities from cavity (3) to
(16) in Fig. 6 in order to visualize the characteristics of the morphology including the balls, spherulites and
the frozen layers. Balls are exhibited only for (3)-(5) in this figure. As can be seen in Fig. 11, a few balls
3번4번
(3) (4) (5) (6)
(7) (8) (9) (10)
(11) (12) (13) (14)
(15) (16)

24
are sparsely formed in (6) but in this cross-section none is found. One of the evident facts here is no frozen
layer is found for (13)-(16). The wider channel width should have eliminated the hesitation. Consequently,
the crystalline structure are seamlessly built from the core regime to the deep inside to the micro-cavity.
The spherulitic crystals have smoothly transformed into the cylindrites in the micro-cavities there. Although
such cylindrites are appeared in the micro-cavities of all cases (6)-(16) without balls, there apparently exist
the frozen layers in the cases of (6)-(12). Cases (7) and (8) show an interesting feature. Near the entrance
of the micro-cavity, a thick dark layer is observed in the figure. Below that layer another boundary
interfacing the core regime can be seen. After fully filling the micro-cavity the second frozen layer is
constituted between the initial frozen layer and the core regime. Both the first and second frozen layers
become thinner as the width decreases as can be seen in the figure.
As a whole, it can be seen that the thickness of the frozen layer decreases along with the increase of
channel width. To check the tendency, the thicknesses have been measured at five locations sampled over
the whole imaged domain per each case, and their average is taken as the representative value. The solid
dots in Fig. 12 display the thickness of the frozen layers along with the widths of the micro-cavity. Also in
Fig. 12, the dashed line shows the proposed model based on this observation. The flow hesitation creates
thick frozen layers around the narrow micro-cavities while the reduced hesitation makes the frozen layer
around the wider ones thinner as plotted in Fig. 12.
Based on the above observations, simple models for the morphological development are suggested.
First, for cases (13)-(16), the flow does not hesitate at the entrance since mesoscale feature is smaller than
the characteristic dimension as in conventional injection molding processes. Second, for cases (6)-(12), Fig.
13 (a) illustrates the sequence of morphological development. Although a frozen layer is formed, the frozen
layer can flow into the micro-cavity and can fill it prior to the BF. This means filling does not require long
time and any BF could not occur. Finally, for cases (3)-(5), the flow can be described similarly, but the main
difference lies in that the narrower width requires higher shear rate. This might have induced nucleation
and provided enough time for the spherulite growth. Figure 13 (b) describes the sequence consisted of the
frozen layer formation, nucleation and spherulite growth.
As is investigated in the previous studies and earlier in this work, the position of the micro-cavity
relative to the gate is not the main factor for the BF [11, 12]. However, this can affect the onset of the
hesitation flow. Since the pressure would greatly vary over the domain during the injection phase, whether
or not a specific micro-cavity is filled would depend on its position. In general, the position relative to the
gate affects the replication in micro-injection molding. If the melt enters the micro-cavity without hesitation,
which is feasible in cases (13)-(16), the momentum in the macro-cavity would play a role. In addition, to

25
achieve a complete replication of the micro-cavity as in 5MPa case of Fig. 9(d), the pressure development
should be considered. This is a problem of relating the macro-cavity and micro-cavity flows as studied in
[15, 46]. Since this work is focused on the flow inside the micro-cavity and such effect is already
experimentally studied in [12], it has not been deeply investigated in this work. One observation that should
be reported here is that results similar to Fig. 9(c) have been obtained for Tw =100oC with the mold insert
rotated by 90o and 180o. Thus, the position effects are not dominant in the current experimental setup.
Let us further discuss over the effects of the flow history and the geometric conditions on the BF.
Although the width of the micro-cavity is the critical factor for the hesitation, the flow velocity in the macro-
cavity, viscosity and surface tension can partly influence the hesitation. To put it differently, the position
and orientation relative to the gate, temperatures of the melt and mold, surface material and so on can be
factors. Thus, prediction of hesitation is not a simple matter. However, this work has clarified that the flow
in the macro-cavity is involved in the BF through the flow hesitation.
Once the frozen layer is formed by the hesitation, the flow history in the macro-cavity does not affect
the flow inside the micro-cavities since the frozen layer does not remember the momentum during the flow.
Likewise, the flow inside a micro-cavity would be independent of those inside the adjacent micro-cavities.
After the nucleation in the micro-cavity, the BF requires that the crystalline growth should stands out from
the advancing flow. In other words, the flow should be stagnant and the condition should be favorable to
the crystalline growth. The slow flow is caused by the frozen layer and the crystalline growth is maintained
by the temperature of the mold insert. It is inferred that once nucleation happens the BF might manifest
under the favorable speed and temperature for the micro-cavities of any shape other than those by Fig. 2.

26
Fig 12. Measured thickness of frozen layer along with the width (solid dot) and the model (dashed
line).
(a) (b)
Fig. 13 Flow and ball formation model: (a) frozen layer formed without balls; (b) ball formation
model.
0
20
40
60
80
100
120
020 40 60 80
frozen layer thickness (m)
micro-cavity width (m)
hesitation frozen layer formation
frozen layer pushed up
complete filling and
second frozen layer
formation
12
34
hesitation frozen layer formation
frozen layer pushed up
nucleation by shear
spherulite growth and
frozen layer stagnation
low T
high T
low T
dislocation
12
34

27
(a)
(b)
Fig. 14 Specimen of micro-cavity (5) imprinted at 160oC under approximately 2MPa followed by
ejection at 40oC: (a) molded micro-cavity by optical microscopy; (b) image by polarized optical
microscopy.
Imprinting
In imprinting cases, to observe possible BFs, a variety of cases have been tried changing the
temperature and pressing forces. However, no ball has been found with this process even with longer time
and higher temperature. As shown in Fig. 14(a), BF cannot be seen for the micro-cavity (4) showing smooth

28
flow front. Figure 14(b) shows the morphology of the imprinted specimen. Unlike injection molded
specimen, the spherulites are all obscured and balls are not found anywhere. The slow flow under an
isothermal condition followed by slow cooling has resulted in such a morphology. Recall that the process
here starts from the specimen molded with a dummy insert. As the process begins with the morphology of
the injection molded specimen, the condition at the entrance to the micro-cavity is very similar to the
injection molding case. No BF in the printing process means BF requires condition specific to the injection
molding processes, which are the high shear rate, pressure and temperature gradient at the entrance.
Figure 15 shows the XRD curves of after and before the imprinting process. Overall, the crystalline
peaks are lowered by the imprinting process. What is most noticeable is the  crystalline structure near the
surface has been almost disappeared. The  crystalline structure is known to be formed near the skin by the
shear flow whereas the  crystalline structure is formed in the core. It is concluded that the surface
morphology in the imprinting process is inherently different from that in the injection molding process.
Fig. 15 XRD measurement of injection molded and imprinted specimen.
0
200
400
600
800
1000
510 15 20 25 30
Imprinted
Injection-molded
Counts
2
 (110)
 (300)  (040)
 (130)  (131/141)

29
Table 1 Constants for Eq. (2) [47]
U* (J)
6720
R (J/mol)
8.314472
T
(K)
264
G0 (m/s)
283
Kg (K2)
550000
0
m
T
(
K
)
483
Numerical
Simulation setup and crystal growth
Numerical simulation of the whole phenomena is quite complicated. Although several models are
engaged in the micro-injection molding simulation, the flow front in the simulated results is usually bland
as in [15, 48, 49]. The purpose of simulation here is to visualize the terrain of interest based on the numerical
results. Thus, this work is focused on simulation of the BF. More specifically, simulation in this work aims
at realizing the fourth state in Fig. 13(b) starting from the third state assuming established nucleation on the
flow front. By doing so, the hypothesis described earlier in this work can be sustained.
The initial geometry of the simulation is based on the previous experimental observations. Since
a nucleus is to be seeded on the flow front, a small circle is placed on the flow front. This circle will grow
at the rate of the spherulite growth. Despite the well-defined crystal growth model, its model constants are
not easy to experimentally determine. This work employs the constants for a PP from a literature, which
are presented in Table 1 [47].
Constants for flow equations
As described before the Cross-WLF model is selected to mathematically express the viscosity as
in other previous works [15, 35]. The corresponding model constants are presented in Table 2 [37].
Additionally, the aforementioned no flow temperature, TNF, is set as 111oC. The known densities for
amorphous and crystalline PP at
T=
298K and
p=
101.3 kPa are
,0c

=
9 46 kg/m3 and
,0a

=
855
kg/m3, respectively [14]. The density is modeled by the p-v-T equation as
( , ) 1/ ( , )T p v T p

=
. The

30
specific volume in amorphous regime is modeled by linear relationship between
,
( , )
atm a atm
Tv
and
( , ( , ))
g g atm
T v T p
as in shown Fig. 16. This line is very close the extrapolated line from the data above Tm
as can be seen in the figure but does not exactly matches. The CTE of the amorphous region,
a

, can be
estimated by this slope using
( / ) / (3 )
a a p a
v T v

=  
.
Table 2 Values of constants for the Cross-WLF model Eq. (6) [37].
A1
26.12
A2 (K)
51.6
D1 (Pa s)
4.66

1012
D2 (K)
263.15
n
0.2751

(Pa)
24200
Fig. 16 p-v-T model for the crystallizing PP.
0.0009
0.001
0.0011
0.0012
0.0013
0.0014
300 350 400 450 500
Bulk
Amorphous
Crystalline
specific volume (m3/kg)
temperature (K)

31
The thermal expansion of crystalline region in a PP is not simple. According to the studies, it
shows negative CTE over a certain range of temperature [50, 51]. This has been also encountered in this
work while trying to determine the CTE of the crystalline region,
c

, from the bulk CTE,
( )
1
b c a
cc
  
= + −
, where c is the degree of crystallinity. However, since the CTE of that region is
definitely positive near Tm, the model needs to be built considering all these. The magnitude of
c

is one
order smaller than that of
a

(
5
10
c

−
−
K-1 and
4
2 10
a

−

) and the typical c value of homo PPs
for injection molding is 48.8% [50]. The crystallinity of PP cannot be as high as that of polyethylene families
because of the steric effect and the stereo-irregularity [23]. The p-v-T measured during the solidification
represents that of a crystalline and amorphous mixture.
Thus, it is not straightforward to obtain the crystalline or amorphous part of the density from the
p-v-T equation. Moreover, the thermal expansion behavior near Tm of the crystalline region alone is not
known quantitatively. In other words, the decrease of the specific volume along with the decrease of
temperature is attributed to both the crystallization and the thermal contraction. Therefore, for simplicity,
/
c
vT
is assumed zero over the entire range of temperature below Tm. In addition, compressibility in the
crystalline region is also ignored. Now, these are all passed on to the value of the amorphous state,
a
v
, since
the compressibility and the thermal expansion in the amorphous regions are significant. Consequently, the
specific volumes of each part can be modeled as shown in Fig. 16. The bulk specific volume,
( )
,
b
v T p
,
lies between the amorphous and the crystalline curves as can be seen in the figure.
The compressibility,
( 1/ )( / )
a a a T
v v p

= −  
, of the bulk PP in the solid phase can be estimated
from the mixture of the crystalline and amorphous phases. Since the crystalline phase is set incompressible,
all the compressibility is imposed onto the amorphous phase. To have convergence of the compressibility
at low temperature and the reported crystallinity of 48.8%, the parameters in the p-v-T equation has been
set accordingly [52]. As can be seen in Fig. 17, the compressibility of the amorphous region and the bulk
PP agrees well below 370K. Although the crystallinity curve in the figure does not bear strict quantitative
meaning, a feasible trend of crystallinity change is obtained based on the p-v-T equation for the bulk. The
constants for the bulk are from [37] and those for the separated amorphous and crystalline phases are
obtained by the above procedure. The specific values are presented in Table 3.

32
Fig. 17 Compressibility and crystallinity along with temperature.
Table 3. Constants of the p-v-T equation for PP [37]
C
0.0894
b5 (K)
443.15
b6 (K/Pa)
1.1210-7
b1m (m3/kg)
0.001304
b2m (m3/kgK)
1.03710-6
b3m (Pa)
8.48518107
b4m (1/K)
0.00635
amorphous
bulk
crystalline
b1s (m3/kg)
0.001289
0.001171
0
b2s (m3/kgK)
8.22710-7
3.61310-7
0
b3s (Pa)
2.08383108
-
b4s (1/K)
0.002403
-
b7 (m3/kg)
0
0.0001322
0.001057
b8 (1/K)
-
0.05777
0
b9 (1/Pa)
-
1.14610-8
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0
0.2
0.4
0.6
0.8
1
300 320 340 360 380 400 420 440
compressibility at 20MPa
bulk compressibility
compressibility in amorphous region
bulk crystalinity
compressibility(10-9/Pa)
bulk crystalinity
temperature (K)

33
Flow and spherulite growth
Let us start the simulation from the geometry shown in Fig. 7. Given the two dimensional geometry,
the mesh can be easily created by the Delaunay triangulation. Several open source for it is available and
this work has employed and modified the one in [53]. The details of the method is abbreviated here. As
aforementioned, hesitation incurs formation of the frozen layer. Since it is very difficult to model the frozen
layer, this work introduces a film temperature that is an algebraic average of the initial melt temperature,
Tn, and the controlled insert temperature, Tw. In the third stage in Fig. 14, the frozen layer is pushed up into
the micro-cavity. The simulation starts taking the film temperature as the initial condition (
( ) / 2
i n w
T T T=+
).
The cavity width, the initial height and the initial diameter of the spherulite in the simulation is W = 22.4
m, L = 20 m and Ri = 2 m, respectively (roughly
LW
).
To compute the spherulite growth rate while simulation, the temperature inside the spherulite is
necessary. The thermal conductivity and the specific heat capacity in Eq. (7) are 0.108 W/mK and 3.4
kJ/kgK, respectively. The thermal boundary conditions on the boundaries are three kinds. The inlet
temperature on  in Fig. 7 is fixed as the film temperature. In the meantime, a convection boundary
condition should be imposed on the mold walls to consider the contact resistance [36, 54, 55, 56]. The heat
transfer coefficient in the micro-injection molding has been set around 25000 W/m2K in the previous works,
which is five times the value for the conventional injection molding [57, 58, 44]. This work also takes this
value for on 1 in Fig. 7. Since the flow front is surrounded by stationary air in the micro-cavity, both the
radiation and convection boundary conditions are imposed on  and . The environmental radiation
has been imposed there with the emissivity of 1. The convection heat transfer in a closed cavity is roughly
estimated by Nu = 2 assuming a square region and the conduction limit. By an approximation that
( 3 ) / 4
air s w
T T T+
, the heat transfer between the flow front and the micro-cavity wall is described by
( ) (3 / 4) ( )
air s air air s w
q h T T h T T
 = − = −
. The heat transfer coefficient,
(3/ 4)
w air
hh=
, is approximated to
2625 W/m2K taking the characteristic length as 20 m and the thermal conductivity of air at Tf as 0.035
W/mK [59].
With the average temperature in , the growth rate, G, can be calculated by Eq. (2). The impingement
of the spherulite into the end wall of the cavity is not treated in this work. In actual filling flow, the spherulite
eventually touches the wall and the radiation boundary condition is subject to alteration to the contact
boundary condition as in the side wall. This impingement will rapidly terminate the spherulite growth by
lowering the temperature. The moving boundaries,  and  deforms radially at a velocity of G.

34
A pressure inlet condition is imposed at the micro-cavity entrance, . If the inlet pressure is not
high enough, the flow would be stagnant, and the crystal growth would dominate the flow front. In contrast,
if it is higher, the flow inside the micro-cavity can continue. In this case, both the unidirectional flow and
the spherulite growth contribute to the terrain. The flow rate into the micro-cavity relative to the volumetric
growth rate of the spherulite affects the shape. To predict this, the flow rate according to the pressure
gradient should be calculated.
This can be simply checked prior to the simulation by an algebraic calculation. Such a flow rate can
be approximated by assuming a fully developed flow between parallel plates. Given the pressure gradient
and the Cross viscosity, the mean velocity,
m
u
, is obtained by [60, 61]
( )
( )
( )
( )
23
021
2
233
1 1, ;1 ;
23
1
mm
w
w
mw
m
ww
n
Wm
uF
m m m

 


+   

= − + − + −
   

   
+


(12)
where
21
F
is a hypergeometric function and
( )
/ 2 /
wW p x

= −  
. Moreover, m and

are given by
1mn=−
and
00
/
  
=
, respectively. The wall shear rate,
w

, should be calculated by solving an
algebraic equation,
( )
,
w w f w
T
   
=
. The inlet temperature and the initial temperature are set as
o
160 C
if
TT==
, which gives
012327.2Pas

=
and
0.5094s

=
. Even at a very low pressure such
as 1 kPa, the mean velocity is 0.172 m/s, which is close to the maximum spherulite growth rate. Due to
the low shear rate, the Newtonian assumption gives 0.164 m/s and the error by the assumption is not large.
This implicates that the actual flow is not stopped by the shear stress inside the micro-cavity. This is
why the short shot conditions are required to visualize balls in Fig. 8 and Fig. 12. Therefore, for the inlet
flow effect to be negligible, the pressure should be set almost zero. There can be errors in the viscosity due
to the molecular orientation in micro-scale [32], slip during measurement [42] and the WLF superposition
[39]. The first increases the viscosity while the second and the third does the reverse. Addressing these
might require some change in the inlet pressure, but the change would not significantly affect its magnitude.
In the end, the flow in the micro-cavity is stopped by the condition that T < TNF or material shortage.

35
Fig. 18 Simulated results for micro-cavity (5) with no inlet pressure.
Let us present results for these two cases separately. The no inlet flow case, where pi is 0kPa, is
presented first. Figure 18 shows the flow front change and the spherulite growth along with time. The initial
geometry shown in Fig. 7 is deformed as the spherulite grows. The inlet flow is not fully stopped but it does
not supply enough mass to compensate the out flow due to the crystalline growth. It should be
acknowledged that the shear flow of a generalized Newtonian fluid cannot theoretically be stopped as long
as the viscosity is finite. As the spherulite grows, bays are formed between the spherulite and the main flow
front as can be seen in the results at the later times. This is attributed to the flow from the front into the
spherulite as can be seen from the velocity arrows in the figure. A key result of this simulation is that the
characteristic shape in Fig. 8 is reproduced by this simulation in Fig. 18. This proves validity of the
aforementioned hypothesis illustrated by Fig. 7.
155oC
151oC
144oC
138oC
135oC
130oC
5s 10s 20s
30s 40s 60s

36
Fig. 19 Simulated results for micro-cavity with inlet pressure of 1 kPa.
In the case with finite inlet pressure of
i
p=
1 kPa, the bay in the Fig. 18 is not pronounced as shown
in Fig. 19. Because the material inflow is sufficient to supply the material to the spherulite, the convex
shape is maintained in the main flow front. As a result, the flow fronts of Fig. 18 and Fig. 19 are quite
different from each other as can be seen in the figure. In both cases, inclusion of the surface tension would
not have greatly affect flow front shape since the contact angle of PP melt with solid surface is close to 90o
[62]. Also note that the pressure is quite low that the effect of b3s and b4s in Table. 2 or Eq. (4) are negligible.
Both the patterns in Fig. 18 and Fig. 19 can take place in actual molding flows. The former happens for
short shot,
0
i
p
and T < TNF whereas the latter can happen otherwise. The computation has been
conducted for 717s per each case. The domain was meshed by 3543 triangular mixed elements where the
velocity and the pressure were interpolated by the quadratic and linear shape functions, respectively.
In actual filling of micro-cavity, the flow cannot be initiated simply by overcoming the wall shear
stress of a solidifying melt regarding the melt as a GNF. This implies the initial state of the simulation,
which is L=20 m in Fig. 7, is not achievable with a low inlet pressure. The inlet pressure should exceed
the combined force by the surface tension on the melt front and the shear strength in the frozen layer. The
surface tension, 0.073 N/m, demands additional pressure of approximately 3 kPa for the PP to enter the
cavity, which seems also very small in injection molding [15]. In the frozen layer, the temperature is higher
128oC
132oC
157oC
150oC
143oC
127oC
136oC
30s 40s 60s

37
near the entrance whereas lower at the wall as designated in Fig. 14. Thus, the shear strength adjacent to
the wall can be substantially high and the flow cannot occur easily. There is no straightforward way to
estimate the shear strength but it would not be negligible since the traces of dislocation are remained as can
be seen in Fig. 10. It cannot be simulated currently because it entails complex models and viscoelastic
material characterizations. Thus, it is inferred that a high pressure from outside should be applied for the
pressure driven flow to happen in the presence of the frozen layer.
Conclusions
This work has investigated the mechanism of ball formation in micro-injection molding of
semicrystalline polymer. Both the experimental and numerical methods have been employed to explain the
ball formation. To perform the micro-injection molding process, a heating system with variotherm
capability and a mold insert with micro-cavities of two-dimensional grooves were devised and built. A
vertical injection molding machine were employed and a mold accommodating the mold insert was attached
to it. Then the molding experiments have been conducted with a polypropylene. The optical microscopic
images of the molded specimen have reaffirmed the previous observations. This work have obtained images
of polarized optical microscopy to investigate the morphologies along with the micro-cavity dimensions.
The investigation has shown that the ball formation is related to the frozen layer, and the balls are built
inside the micro-cavity by growing the spherulite. Moreover, the imprinting experiments does not yield
flow fronts with balls. Based on these observations, a hypothesis has been postulated. The hypothesis is
comprised of:
• The spherulites are nucleated passing through the micro-cavity entrance because of the sudden
increase in shear rate and the temperature gradient.
• The spherulites growth on the flow front forms the ball inside the micro-cavity.
• Without the frozen layer, which is formed by the flow hesitation, ball formation does not occur.
To verify the above hypothesis, a numerical model has been developed. A numerical simulation in
the micro-cavity has been designed to start from a progressed flow domain with nucleus on the flow front.
The simulated results have reproduced the flow front shapes in the experiments. Thus, the numerical results
support the proposed hypothesis.

38
Acknowledgement
This work was supported by a NRF grant funded by the Korea Government (NRF-
2014R1A2A1A11054451 and NRF-2018R1A5A1024127)
References
[1]
C. A. Hieber, Fundamentals in Injection and Compression Molding, New York: Marcel Dekker, 1987.
[2]
D. O. Kazmer, Injection Mold Design Engineering, Hanser, 2007.
[3]
T. A. Osswald, L.-S. Turing and P. Gramann, Injection Molding Handbook, Second Edition ed.,
Munich: Hanser, 2008.
[4]
M.-S. Huang and C.-Y. Lin, "A novel clamping force searching method based on sensing tie-bar
elongation for injection molding," International Journal of Heat and Mass Transfer, vol. 109, pp.
223-230, 2017.
[5]
L. Zhang, G. Zhao, G. Wang, G. Dong and H. Wu, "Investigation on bubble morphological evolution
and plastic part surface quality of microcellular injection molding process based on a multiphase-
solid coupled heat transfer model," International Journal of Heat and Mass Transfer, vol. 104, pp.
1246-1258, 2017.
[6]
S.-C. Nian, M.-H. Li and M.-S. Huang, "Warpage control of headlight lampshades fabricated using
external gas-assisted injection molding," International Journal of Heat and Mass Transfer, vol. 86,
pp. 358-368, 2015.
[7]
H.-T. Bui and S.-J. Hwang, "Modeling a working coil coupled with magnetic flux concentrators for
barrel induction heating in an injection molding machine," International Journal of Heat and Mass
Transfer, vol. 86, pp. 16-30, 2015.
[8]
X. Sánchez-Sánchez , M. Hernández-Avil, L. E. Elizalde, O. Martínez, I. Ferrer and A. Elías-Zuñiga,
"Micro injection molding processing of UHMWPE using ultrasonic," Materials and Design, vol. 132,
pp. 1-12, 2017.

39
[9]
J. Giboz, T. Copponnex and P. Mélé, "Microinjection molding of thermoplastic polymers: a review,"
Journal of Micromechanics and Microengineering, vol. 17, pp. R96-R109, 2007.
[10]
J. Greener and R. Wimberger-Friedl, Precision Injection Molding: Process, Materials, and
Applications., New York: Hanser, 2006.
[11]
S. K. Kim, S.-J. Choi, K.-H. Lee, D.-J. Kim and Y.-E. Yoo, "Observation of instabilities in flow front
during micro injection molding process," Polymer Engineering & Science, vol. 50, no. 7, pp. 1377-
1381, 2010.
[12]
S. K. Kim, D.-J. Kim, J. S. Jung and Y. Cho, "Flow instability of semicrystalline polymer melt during
micro-injection molding," Journal of Micromechanics and Microengineering, vol. 24, no. 8, p.
085015, 2014.
[13]
J. Karger-Kocsis, Polypropylene: An A-Z Reference, Kluwer, 1999.
[14]
D. Tripathi, Practical guide to polypropylene, Shrewsbury, UK: RAPRA Technology, 2002.
[15]
S.-J. Choi and S. K. Kim, "Multi-scale filling simulation of micro-injection molding process," Journal
of Mechanical Science and Technology, vol. 25, pp. 117-124, 2011.
[16]
P. Muanchan and H. Ito, "Replication of Micro-/Nanostructures," in Micro and Nano Fabrication
Technology, Springer, 2018, pp. 1-29.
[17]
J. Hong, D.-J. Kim and S. K. Kim, "Polypropylene morphology in micro-injection molding of
rectangular groove," SPE Antec, vol. 63, pp. 1586-1589, 2017.
[18]
C. De Rosa and F. Auriemma, "Crystal Structures of Polymers," in Handbook of Polymer
Crystallization, E. Piorkowska and . G. C. Rutledge, Eds., Wiley, 2013, pp. 31-72.
[19]
V. D. W. Becker, "Kinetische Behandlung der Keimbildung in übersättigten Dämpfen," Annalen der
Physik, vol. 416, no. 8, pp. 719-752, 1935.
[20]
D. Turnbull and J. C. Fisher, "Rate of Nucleation in Condensed Systems," The Journal of Chemical
Physics, vol. 17, no. 1, pp. 71-73, 1949.

40
[21]
H. Zuidema, G. Peters and H. Meijer, "Development and validation of a recoverable strain-based
model for flow-induced crystallization of polymers," Macromolecular Theory and Simulations, vol.
10, pp. 447-460, 2001.
[22]
J. D. Hoffman and R. L. Miller, "Kinetic of crystallization from the melt and chain folding in
polyethylene fractions revisited: theory and experiment," Polymer, vol. 38, no. 13, p. 3151–3212,
1997.
[23]
A. K. Ahmed, M. Atiqullah, D. R. Pradhan and M. A. AlHarth, "Crystallization and melting behavior
of i-PP: a perspective from Flory's thermodynamic equilibrium theory and DSC experiment," RSC
Advances, vol. 7, p. 42491–42504, 2017.
[24]
P.-C. Chang and S.-J. Hwang, "Simulation of infrared rapid surface heating for injection molding,"
International Journal of Heat and Mass Transfer, vol. 49, no. 21-22, pp. 3846-3854, 2006.
[25]
F. D. Santis and R. Pantani, "Development of a rapid surface temperature variation system
andapplication to micro-injection molding," Journal of Materials Processing Technology, vol. 237,
pp. 1-11, 2016.
[26]
D.-K. Lee, H. Ko and Y. Cho, "Single Si submicron wire photodetector fabricated by simple wet
etching process," Materials Letters, vol. 160, p. 562–565, 2015.
[27]
C. W. Hock, "Morphology of polypropylene crystallized from the melt," Journal of Polymer Science
Part A-2: Polymer Physics, vol. 4, no. 2, pp. 227-242, 1966.
[28]
M. Kojima, "Morphology of polypropylene crystals. III. Lamellar crystals of thermally decomposed
polypropylene," Journal of Polymer Science Part A-2: Polymer Physics, vol. 6, no. 7, pp. 1255-1271,
1968.
[29]
X. Liu, K. Dai, X. Hao, G. Zheng, C. Liu, D. W. Schubert and C. Shen, "Crystalline Structure of
Injection Molded β-Isotactic Polypropylene: Analysis of the Oriented Shear Zone," Ind. Eng. Chem.
Res., vol. 52, no. 34, pp. 11996-12002, 2013.
[30]
J. Pina-Estany and A. A. García-Granada, "Molecular dynamics simulation method applied to
nanocavities replication via injection moulding," Int. Commun. Heat Mass Transfer, vol. 87, pp. 1-5,

41
2017.
[31]
J. Yang, C. Weng, J. Lai, T. Ding and H. Wang, "Molecular Dynamics Simulation on the Influences
of Nanostructure Shape, Interfacial Adhesion Energy, and Mold Insert Material on the Demolding
Process of Micro-Injection Molding," Polymers , vol. 11, p. 1573, 2019.
[32]
D. Yao and B. Kim, "Simulation of the filling process in micro channels for polymeric materials,"
Journal of Micromechanics and Microengineering, vol. 12, p. 604–610, 2002.
[33]
Y. -X. Chen, S.-C. Chang and W.-B. You, "Application of lattice Boltzmann method in free surface
flow simulation of micro injection molding," Computers and Mathematics with Applications, vol. 75,
pp. 2374-2386, 2018.
[34]
G. Toselloa and F. S. Costa, "High precision validation of micro injection molding process
simulations," Journal of Manufacturing Processes, vol. 48, pp. 236-248, 2019.
[35]
S. Y. Kang, S. K. Kim and W. I. Lee, "Penalty formulation for postfilling analysis during injection
molding," International Journal for Numerical Methods in Fluids, vol. 57, no. 2, pp. 139-155, 2008.
[36]
Y. Liu and M. Gehde, "Evaluation of heat transfer coefficient between polymer and cavity wall for
improving cooling and crystallinity results in injection molding simulation," Applied Thermal
Engineering, no. 80, pp. 238-246, 2015.
[37]
Autodesk Inc., MoldFlow 2016, 2016.
[38]
T. Osswald and N. Rudolph, Polymer Rheology: Fundamentals and Applications, Munich: Hanser,
2015.
[39]
P. K. Kennedy, "Practical and scientific aspects of injection molding simulation," Technische,
Eindhoven, 2008.
[40]
A. Naranjo, J. F. Campuzano and I. Lópe, "Analysis of heat transfer coefficients and no-flow
temperature in simulation of injection molding," SPE Antec, vol. 63, pp. 1394-1400, 2017.
[41]
J.-Y. Chen, K.-J. Yang and M.-S. Huang, "Online quality monitoring of molten resin in injection
molding," International Journal of Heat and Mass Transfer, vol. 122, pp. 681-693, 2018.

42
[42]
S. C. Chen, R. I. Tsai, R. Chien and T. Lin, "Preliminary study of polymer melt rheological behavior
flowing through micro-channels," International Communications in Heat and Mass Transfer, vol. 32,
p. 501–510, 2005.
[43]
B. Pignon, V. Sobotka, N. Boyard and D. Delaunay , "Improvement of heat transfer analytical models
for thermoplastic injection molding and comparison with experiments," International Journal of Heat
and Mass Transfer, vol. 118, pp. 14-26, 2018.
[44]
S. C. Somé, D. Delaunay, J. Faraj, J. L. Bailleul and S. Quilliet, "Modeling of the thermal contact
resistance time evolution at polymer–mold interface during injection molding: Effect of polymers'
solidification," Applied Thermal Engineering, vol. 84, pp. 150-157, 2015.
[45]
B. A. G. Schrauwen, L. C. A. v. Breemen, A. B. Spoelstra, L. E. Govaert, G. W. M. Peters and H. E.
H. Meijer, "Structure, Deformation, and Failure of Flow-Oriented Semicrystalline Polymers,"
Macromolecules, vol. 37, pp. 8618-8633, 2004.
[46]
Y. -E. Yoo, T.-H. Kim, J. Tae-Jin, C. Doo-Sun, C.-W. Kim and S. K. Kim, "Injection molding of micro
patterned PMMA plate," Trans. Nonferrous Met. Soc. China, vol. 21, pp. s148-s152, 2011.
[47]
R. Zheng and . P. K. Kennedy, "A model for post-flow induced crystallization: General equations and
predictions," J. Rheol., vol. 48, p. 823–842, 2004.
[48]
Y. -K. Shen, C.-Y. Chang, Y.-S. Shen, S.-C. Hsu and M.-W. Wu, "Analysis for microstructure of
microlens arrays on micro-injection molding by numerical simulation," Int. Comm. Heat Mass
Transfer, vol. 35, no. 6, pp. 723-727, 2008.
[49]
L. Yu, L. J. Lee and K. W. Koelling, "Flow and heat transfer simulation of injection molding with
microstructures,," Polymer Engineering and Science, vol. 44, no. 10, p. 1866–1876, 2004.
[50]
F. C. Chen, C. L. Choy and K. Young, "Negative thermal expansion of polymer crystals: Lattice
model," Journal of Polymer Science: Polymer Physics Edition, vol. 18, no. 12, pp. 2313-2322, 1980.
[51]
D. J. Lacks and G. C. Rutledge, "Mechanisms for axial thermal contraction in polymer
crystals:polyethylene vs isotactic polypropylene," Chemical Engineering Science, vol. 49, no. 17, pp.
2881-2888, 1994.

43
[52]
A. Rudin and P. Choi, The Elements of Polymer Science & Engineering, Third Edition ed., Academic
Press, 2012.
[53]
J. Shewchuk, "Mesh generation and Delaunay triangulation," 2019. [Online]. Available:
https://people.sc.fsu.edu/~jburkardt/c_src/triangle/triangle.html. [Accessed 7th 10 2019].
[54]
H. Massé, E. Arquis, D. Delaunay and S. L. Quilliet , "Heat transfer with mechanically driven thermal
contact resistance at the polymer–mold interface in injection molding of polymers," International
Journal of Heat and Mass Transfer, vol. 47, no. 8-9, pp. 2015-2027, 2004.
[55]
S. K. Parihar and N. T. Wright, "Thermal contact resistance at elastomer to metal interfaces,"
International communications in heat and mass transfer,, vol. 24, no. 8, pp. 1083-1092, 1997.
[56]
E. M. Park and S. K. Kim, "Effects of Mold Heat Transfer Coefficient on Numerical Simulation of
Injection Molding," Transactions of the Korean Society of Mechanical Engineers, vol. 43, no. 3, pp.
201-209, 2019.
[57]
T. Nguyen‐Chung, G. Jüttner, C. Löser, T. Pham and M. Gehde, "Determination of the heat transfer
coefficient from short‐shots studies and precise simulation of microinjection molding," Polymer
Engineering & Science,, vol. 50, no. 1, pp. 165-173, 2010.
[58]
S. Hong, J. Kang and K. Yoon, "Correlation between thermal contact resistance and filling behavior
of a polymer melt into multiscale cavities in injection molding," International Journal of Heat and
Mass Transfer, vol. 87, pp. 222-236, 2015.
[59]
Engineering ToolBox, "Air - Thermal Conductivity," 2009. [Online]. Available:
https://www.engineeringtoolbox.com/air-properties-viscosity-conductivity-heat-capacity-
d_1509.html. [Accessed 16 10 2019].
[60]
S. K. Kim, "Flow rate based framework for solving viscoplastic flow with slip," Journal of Non-
Newtonian Fluid Mechanics, vol. 269, pp. 37-46, 2019.
[61]
S. K. Kim, "Forced convection heat transfer for the fully-developed laminar flow of the Cross fluid
between parallel plates," Journal of Non-Newtonian Fluid Mechanic, vol. 276, p. 104226, 2020.

44
[62]
G. Zitzenbacher, H. Dirnberger, M. Längauer and C. Holzer, "Calculation of the Contact Angle of
Polymer Melts on Tool Surfaces from Viscosity Parameters," polymers, vol. 10, no. 1, pp. 38-39,
2018.