1

SIMULATION OF CELL GROWTH

IN HIGH-PRESSURE FOAM INJECTION MOLDING

Chongda Wang1, Vahid Shaayegan

1, Sejin Han

2, Franco Costa

3, Chul B. Park

1

1 - Microcellular Plastics Manufacturing Laboratory, Department of Mechanical and Industrial

Engineering, University of Toronto, 5 King’s College Road, Toronto, Ontario, Canada M5S 3G8

2 - Autodesk, Inc. 2353 North Triphammer Road, Ithaca, NY 14850, USA

3 - Autodesk, Inc. 259-261 Colchester Rd., Kilsyth, VIC. 3137, Australia

Abstract

In this work, the prediction of final cell size of high-

pressure foam injection molded parts has been attempted.

An in-situ visualization technique was used to capture

real-time cell growth data from high-pressure foam

injection molding experiments conducted with PS and

CO2. The simulated cell growth profile was compared

with experimental measurements. For the PS/CO2 system,

quantitative agreement (over 80%) between predicted and

measured growth profile were achieved. With the

validated simulation, the effect of cooling history on final

cell size and cell size distribution was investigated. It is

shown that in high-pressure foam injection during which

all the gate nucleated cells are dissolved, final

morphology is characterized by having large cells in the

center and smaller cells near the skin.

Introduction

Parts produced by foam injection molding (FIM) are

utilized in many industries such as automotive,

construction, aerospace, sports and etc. Light-weighting,

geometry accuracy, short production cycles and other

countless advantages bring FIM a substantial amount of

attention from industry and academia. In spite of its

flourish, FIM has been recognized as a highly complex

unsteady process due to its cyclically interrupted

plasticization and gas mixing, and the number of variables

involved yet have to be controlled. As a result, when it

comes to the designing of a suitable foam morphology

tailored a specific application, engineers have to go

through a large number of painstaking experiments and

optimization to reach the desired morphology. Such

iterative process is not only time, labor, and material

consuming, but also inevitably increases production costs

and more importantly protract time to market (TTM). One

of the solutions is computer simulation and prediction.

Bubble growth modeling has been studied by a

number of researchers [1-9]. Amon and Denson

introduced the Cell Model which describes the growth of

closely spaced bubbles with limited gas concentration [1].

They assumed that each cell is surrounded by a thin shell

of Newtonian fluid with limited gas concentration only

from which the bubble can absorb gas molecules.

Afremanesh and Advani adopted the cell model and more

importantly took the viscoelasticity of the shell into

account [2]. The viscoelastic fluid was represented by the

quasi-linear, upper-convected Maxwell model. Moreover,

they compared the growth results obtained by finding the

concentration gradient via explicitly solving advection-

diffusion partial derivative equation (PDE) against finding

it using a fitted polynomial concentration profile.

Consequently, they concluded it is necessary to solve the

diffusion PDE with proper boundary conditions. Ramesh

et al. applied the cell model in the simulation of batch

foaming [3]. They compared the difference in predicting

growth profile using between Power-law based model

(Newtonian fluid) and Maxwell based model (viscoelastic

fluid). Without an in-situ visualization batch foaming

chamber, they quenched their sample in liquid nitrogen at

different timings of bubble growth and performed SEM

on the fractured surface. Their results showed that growth

equations derived with viscoelastic model offer more

accuracy in predicting final bubble size when compared

with a pure Newtonian model. Leung and et al. also

simulated the bubble growth in batch foaming process [4].

Using the Maxwell viscoelastic solution and solving the

diffusion equation, more importantly, incorporating

Sanchez-Lacombe equation of state (SL-EOS), they were

able to not only accurately predict the growth profile but

also that of bubbles nucleated at different timings.

Amon and Denson also attempted to simulate the

bubble growth in low-pressure FIM (LP-FIM) with the

help of a visualized cavity [5]. They obtained good

qualitative agreement between experimental and model-

predicted results, but quantitative discrepancies were

shown between them. They believed that the negligence

of melt elasticity and bubble coalescence in low-pressure

FIM (LP-FIM) were the main contributions to the

discrepancy. Moreover, due to hardware limitations, they

were unable to obtain real-time growth data. Afremanesh

et al. also simulated bubble growth in a highly viscous

solution in LP-FIM, furthermore, they considered the

pressure variations at the different thickness of the mold

cavity [6]. They reported growth profile and foam density

in normalized dimensionless form. They demonstrated

larger bubble size in the melt front and smaller bubbles

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near the gate which is consistent with experimental

findings in other literature. Han et al. employed the cell

model and described the viscosity of the solution with

modified Cross-WLF equation [7]. Even though they

assumed second-order polynomial concentration profile,

they were able to predict the cell size distribution along

the thickness of the cavity.

All these works have significantly contributed to the

understanding of bubble growth phenomena in different

foaming process. Nonetheless, most of the FIM bubble

growth simulations either were not validated with

experimental data or neglected the melt elasticity. In this

work, a bubble growth simulation in HP-FIM was

performed based on the Cell Model using transient

pressure and temperature data, material properties that

vary with temperature, pressure, and gas concentration.

More importantly, the simulation of cell growth is

validated by conducting visualized HP-FIM experiments.

Theory

The cell model suggested by Amon and Denson

describes the growth of closely spaced bubbles with

limited gas concentration [1]. It assumes polymer solution

is divided into a large number of spherical unit cells

within which the bubble will only nucleate and grow, as

shown in Figure 1, in which C is the concentration of

blowing agent in the shell, Rbub and Rshell represent the

bubble and shell radius respectively. The polymer/gas

shell supplies a limited amount of gas molecules to the

bubble. This model accurately describes the boundary

condition when the diffusion process occurs between

bubbles that are in close proximity to each other.

Figure 1. Cell model by Amon and Denson [1]

To further simplify the phenomenon, the following

assumptions are made:

1. The bubble remains spherically symmetric;

2. The polymer/gas solution is single-phase,

homogeneous, and incompressible;

3. No mass flux at the shell boundary (i.e.

|

);

4. Transient cooling for the unit cell is accounted,

however, the temperature is uniform in the unit cell;

5. There is no interactions between neighboring bubbles

during growth;

6. The pressure at the boundary of the shell is equal to

system pressure;

7. The gas inside the bubble is ideal gas and obeys ideal

gas law;

8. The gas molecules distribute uniformly in the bubble,

and concentration obeys Henry’s law (i.e. ( ) );

9. The initial stress in the polymer/gas shell is zero;

10. The inertial forces and the effect of gravity on bubble

growth are negligible;

The continuity and momentum equations are as

represented as (1) and (2), respectively. The stress field

within the shell is described by Maxwell model (i.e. Eq. 3

and 4) was used to describe the viscoelastic behavior of

the melt [4].

(1)

( )

∫

(2)

(

)

(3)

(

)

(4)

(5) describes the mass balance across the bubble-shell

interface during the diffusion process. The concentration

gradient at the bubble-shell interface was solved by the

advection-diffusion equation (6) and its initial and

boundary conditions (7-9) were used.

(

)

|

(5)

(

) (6)

( ) (7)

( ) ( ) (8)

( )

(9)

r

C

Rbub Rshell

Polymer/gas

mixture shell

Gas bubble

𝜕𝐶

𝜕𝑟 Polymer/gas

shell

Gas bubble

SPE ANTEC® Anaheim 2017 / 2526

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The models of material properties were adopted from

[4, 13-15] for surface tension, diffusion coefficient,

relaxation time, and zero-shear viscosity, respectively.

(1-9) constitute the mathematical description of the

bubble growth based on cell model. The system of the

governing equations is nonlinear and coupled [8, 9].

Therefore, the equations were solved by explicit finite

difference and 4th

order Runge-Kutta method in

accordance with [4].

Experimental

A 50-ton Arburg ALLROUNDER 270/320C

injection molding machine, equipped with Trexel Mucell

technology, was used to carry out HP-FIM experiments.

Moreover, the fix mold platen is a visualization mold that

consists of a prism, three fused silica windows. Imaging

equipment includes a CV M10 camera from JAI,

equipped with a Navitar magnifying lens. A detailed

description of the visualization setup can be found in [10].

Figure 3 shows the visualization mold and geometry of

injected part.

Figure 3. (a) Top cross-section view of visualization mold

[11] and (b) Mold cavity geometry with sprue

The polymer and blowing agent being injected were

Polystyrene (PS 675) from Americas Styrenics, with an

MFI of 7.0 gr/10min and a density of 1.04 g/cm3 and CO2

from Linde Gas Canada.

The processing conditions of the HP-FIM trial used

for simulation validation is summarized in the table

below.

Table 1. Processing conditions

Processing parameters HP-FIM

Gas content 3 wt%

Melt temperature 220 oC

Mold temperature 30 oC

Injection speed 80 cm3/s

Packing pressure 20 MPa

Packing time 8 s

Cooling time 30 s

Result and discussion

Visualization of bubble growth dynamic in HP-FIM

The bubble nucleation and growth mechanisms in

HP-FIM process have been studied by Shaayegan et al.

[10-12]. He reported that the application of packing

pressure reduces the number of cells nucleated during

mold filling (i.e. Gate Nucleated Cells, GNCs). When the

applied packing pressure is sufficiently high, total

dissolution of GNCs is achievable. Nucleation is primarily

induced by shrinkage. The resulting morphology is

significantly more uniform compared with the ones with

GNCs left in the system.

Figure 4 shows the snapshots of bubble growth at

Location B of the cavity during the HP-FIM trial. As

shown in Figure 4b), all the bubble that nucleated while

passing the gate were dissolved back into the system,

hence nucleation is primarily induced by shrinkage.

Figure 4 d), e), and f) clearly demonstrate that the bubble

grows at a faster rate at the beginning stage of growth.

Whereas Figure 4 g), h), and i) indicate that the reduced

growth rate and stabilization of the bubble size.

Simulation of bubble growth in HP-FIM

To simulate the experimental conditions of the HP-

FIM trial, temperature and pressure profiles along with

initial gas concentration were used as program inputs.

Figure 5a) and 5b) show the transient temperature and

pressure inputs for the bubble growth solver, respectively.

Initial blowing agent concentration was set to be 3% wt.

The temperature profile is simulated with Moldflow,

Autodesk under the same process setting as experiment

trial. The pressure profile is obtained by a pressure

transducer in the mold cavity at Location B.

Figure 6 plots the simulated 40 s growth profile

against the measured growth data. As can be seen, a

quantitative agreement was achieved. At the beginning

stage of growth, system temperature is relatively high, the

bubble starts to grow with a slower rate, despite the high

diffusivity, and low melt strength. Leung et al. also

reported similar trend and attributed this to the

contribution of surface tension to the bubble [4]. The 3rd

term on left-hand side of (2) represent the pressure

difference across the bubble-shell interface. When the

bubble is small, the pressure difference is greater and

imposes larger impedance on the growth. However, its

effect diminished as the bubble grows.

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Then the bubble grows rapidly due to the reduced

retarding effect of surface tension and large concentration

gradient at the bubble-shell interface caused by the high

diffusivity of gas molecules at high temperature. Bubble

growth at this stage is dominated by the diffusion rate and

concentration flux at the bubble surface.

As the temperature decreases, the melt strength

(viscoelasticity) increases which acts as another retarding

force for bubble growth leading to the growth to slow

down. At the end, the growth rate decreased to zero and

the structure stabilizes.

Cell size distribution in HP-FIM

At a different thickness of the part, the polymer/gas

experiences different cooling history. The polymer/gas

mixture cools faster at locations closer to skin due to the

large temperature gradient, whereas it cools slower at the

core region.

Referencing the core as d=0mm, Figure 7 shows the

simulated temperature history by Moldflow, Autodesk at

varying distance from the core using the same processing

conditions. Figure 8 demonstrates the effect of

temperature history on cell growth. By comparing d=0 mm and d=0.53 mm, it can be noted that the growth is not

sensitive to slight changes temperature while other

parameters are maintained. However, when the initial

temperature decrease is large, the bubble experiences a

slower growth rate and stops growing earlier due to

solidification of the structure. Furthermore, at the

beginning stage of the growth, both the cell size and the

discrepancy of cell sizes at different locations is small.

PackingInjection Cooling t=0 s

t=27.315 s

a b c

d e f

g h i

t=2 s

t=8 s

t=4 s t=6 s

t=10 s t=12 s

1mm

1mm

1mm

Figure 4 Visualization snapshots of HP-FIM at Location B

SimulatedMeasured

Bubble growth profile at location B

Figure 6 Simulated and measured

bubble growth profile at Location B

Figure 5 (a) Temperature and

(b) Pressure profile at Location B

Tem

per

ature

(K

) P

ress

ure

(M

Pa)

B

ub

ble

rad

ius

(mm

)

time (s)

time (s) time (s)

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Plotting the final cell size against distance from the

core, Figure 9 shows the final cell size distribution. As

can be seen, near the core region, when the temperature

was relatively more uniform, the final cell size was almost

the same. However, near the skin where the temperature

non-uniformity is large, the cell size varies largely along

the thickness direction. Such structure is an indication of

HP-FIM where all GNCs were dissolved. On the contrary,

when the packing pressure is not high enough, GNCs

remains in the system. As GNCs experiences cavity filling

phase, they are deformed in the sheared melt flow. Thus,

the deformed GNCs have a larger surface area. Gas

diffusion is promoted as the result. Which in turn causes

the bubble to grow faster. Moreover, during the filling

phase, due to the fountain effect, these GNCs are pushed

to locations near the wall. This explains the non-uniform

morphology in traditional foam injection molded parts

whose bubbles are larger and have a sheared and/or

elongated shaped when compared with the shrinkage

induced cells in the foam core [11].

Based on above observation, instructive advice can be

made that a uniform cell structure can be obtained by

keeping a uniform temperature history throughout the

mold cavity. One the other hand, a rapid temperature drop

can be used to ‘freeze’ the cell structure and obtain parts

with much smaller cell size. Such ‘freezing’ can be

achieved by rapid mold temperature control technologies.

Conclusion

In this work, numerical simulation and experimental

trials of bubble growth in HP-FIM were conducted to

study the bubble growth behavior. The HP-FIM

confirmed that the when packing is sufficiently applied,

removal of GNCs is achievable. Both qualitative and

quantitative agreement were obtained between simulated

and experimentally measured growth profiles.

Furthermore, with the validated simulation, prediction of

cell size distribution along the thickness direction of the

FIM part was attempted. Results showed that cells near

the skin that experienced a faster cooling rate grew to a

smaller size compared to the cells near the core region.

This is the exact opposite observation in tradition foam

injection molding, where cells neat the skin grows faster

due to increased surface area and promoted molecular

transfer blowing agents. This simulation work confirmed

the importance of application of proper packing phase in

achieving uniform final cell morphology.

0

0.05

0.1

0.15

0.2

0.25

0.3

0 0.2 0.4 0.6 0.8 1 1.2

0

0.05

0.1

0.15

0.2

0.25

0.3

0 5 10 15 20 25 30

Cel

l ra

diu

s (m

m)

time (s)

Effect of temperature history

d=0mm

d=0.53mm

d=1.07mm

d=0mm

d=0.53mm

d=1.07mm

Tem

per

atu

re (

)

time (s)time (s)

Bub

ble

rad

ius

(mm

)

time (s)

Tem

per

ature

(℃

)

Figure 7 Cooling history at different thickness Figure 8 Bubble growth under different cooling history

Distance from the core (mm)

Bub

ble

rad

ius

(mm

)

Figure 9 Final cell size distribution

d=0mm

d=0.53mm

d=1.07mm

d=0mm

d=0.53mm

d=1.07mm

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Nomenclature

Radial distance in spherical coordinates

Polar angle in spherical coordinates

time

Radius

Velocity

Pressure

Concentration

Shear stress

Surface tension

Zero-shear viscosity

Relaxation time

Density

Diffusivity

Henry’s law constant

Subscript:

Bubble

Shell

In radial direction

In tangent direction

Initial condition ( )

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