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2012, Polymer Engineering & Science

Polymer Engineering & …

Bubble dissolution in molten polymers and its role in rotational molding1999 •

Journal of Materials Research and Technology

Bubble behavior in the slab continuous casting mold: Physical and mathematical model2020 •

Journal of The Korean Society of Manufacturing Technology Engineers

A Study on the Behavior of Bubbles Trapped in the In-Mold Coating Process2012 •

Steel Research International

Bubbly Mold Flow in Continuous Casting: Comparison of Numerical Flow Simulations with Water Model Measurements2020 •

The injection of gas at the stopper rod tip in continuous casting of steel slabs is a common practice to prevent clogging and to control the produced steel quality. The injected gas forms bubbles and is transported by the liquid steel through the submerged entry nozzle in the mould region of the strand (Figure 1). Most of the gas bubbles rise to the mold surface and leave the liquid steel there. In a resting liquid, single gas bubbles start rising due to their buoyancy and reach quickly the terminal rising velocity where the buoyancy force and drag force are in equilibrium. Basically, the terminal rising velocity is higher for larger bubbles. Drag forces and resulting terminal rising velocities for bubbles in liquid have been intensively investigated for air bubbles in water, but also for bubbles in liquid metals, as shown in Figure 2a. The unstable shape of bubbles in contrast to rigid spheres causes a saturation-like effect as the terminal rising velocity is only slightly increasi...

Journal of Coatings Technology and Research

The effect of solidification on the casting window after bubble entrainment2013 •

Numerical Modeling of Bubbly Flow in Continuous Casting Mold using Two Population Balance Approaches

2018 •

To model the spatial evolution of the gas bubbles in continuous casting mold, two population balance approaches of MUlti-SIze-Group (MUSIG) model and Average Bubble Number Density (ABND) model, -have been employed and merged with the Euler-Euler two-fluid model. The bubble induced turbulence model and various interfacial forces including drag, lift, virtual mass, and turbulent dispersion are incorporated in both of models. A 1/4th scaled water model was built to measure and investigate the bubble behavior and size distribution in the mold. Predictions by MUSIG model and ABND model were in reasonable agreement with experimental data of gas void fraction, liquid flow pattern, and local bubble size distribution profiles. The “intermediate peak” and “core peak” behaviors of void fraction inside the submersed entry nozzle (SEN) had been captured very well. From a practical perspective, the ABND model may be considered as a more viable approach for industrial applications of gas-liquid tw...

Journal of Micro and Nano-Manufacturing

Centrifugal Casting of Microfluidic Components With PDMS2013 •

This work describes the centrifugal casting and fast curing of double-sided, polydimethylsiloxane (PDMS)-based components with microfeatures. Centrifugal casting permits simultaneous patterning of multiple sides of a component and allows control of the thickness of the part in an enclosed mold without entrapment of bubbles. Spinning molds filled with PDMS at thousands of revolutions per minute for several minutes causes entrapped bubbles within the PDMS to migrate toward the axis of rotation or dissolve into solution. To cure the parts quickly (<10 min), active elements heat and cool cavities filled with PDMS after the completion of spinning. Microfluidic channels produced from the process have a low coefficient of variation (<2% for the height and width of channels measured in 20 parts). This process is also capable of molding functional channels in opposite sides of a part as demonstrated through a device with a system of valves typical to multilayer soft lithography.

PDA journal of pharmaceutical science and technology / PDA

Syringe siliconization process investigation and optimizationThe interior barrel of the prefilled syringe is often lubricated/siliconized by the syringe supplier or at the syringe filling site. Syringe siliconization is a complex process demanding automation with a high degree of precision; this information is often deemed "know-how" and is rarely published. The purpose of this study is to give a detailed account of developing and optimizing a bench-top siliconization unit with nozzle diving capabilities. This unit comprises a liquid dispense pump unit and a nozzle integrated with a Robo-cylinder linear actuator. The amount of coated silicone was determined by weighing the syringe before and after siliconization, and silicone distribution was visually inspected by glass powder coating or characterized by glide force testing. Nozzle spray range, nozzle retraction speed, silicone-coated amount, and air-to-nozzle pressure were found to be the key parameters affecting silicone distribution uniformity. Distribution uniformity is particul...

Bubble Removal in Centrifugal Casting: Combined
Effects of Buoyancy and Diffusion
Aaron D. Mazzeo, Michelle E. Lustrino, David E. Hardt
Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge,
Massachusetts 02139
This article models bubble removal in centrifugal casting of the thermosetting silicone polydimethylsiloxane
(PDMS). Given spin speeds, mold geometry, and resin
properties for centrifugal casting, it is possible to
predict the amount of spin time required to produce
bubble-free parts. For the particular conditions and
materials considered in this work, required spin times
range from seconds to minutes. The model itself balances two complementary physical behaviors: bubble
dissolution and buoyancy. Diffusion dominates the removal of small bubbles below a critical size, while
buoyancy dominates the removal of bubbles above a
critical size. To simulate the removal of bubbles from a
spinning solution, two ﬁrst-order nonlinear differential
equations are solved numerically. To verify simulated
results, designed sets of experiments related the
number of remaining bubbles in cured parts to spin
times and speeds. Dow Corning Sylgard 184, Sylgard
186, and 3-6121 were the silicones tested in this
work. POLYM. ENG. SCI., 52:80–90, 2012. ª 2011 Society of
Plastics Engineers
INTRODUCTION
This work is motivated by the desire for rapid production of bubble-free silicone components with micro/nano
features (e.g. microﬂuidic devices) [1, 2]. This is achieved
by the use of a centrifugal casting technique. Centrifugal
or spin casting techniques have molded a variety of materials including metals, ceramics, thermoplastics, and
thermosets. With these materials, centrifugal casting has
produced steel tubes, optical telecommunication ﬁbers,
polyester and polyvinyl pipes, functional gradient metalceramic materials, porous ceramic supports for membrane
applications, gears, rubber tires, and hollow rubber dolls
[3–6]. Centrifugal casting has also formed objects from
rubbery polyurethanes [7, 8] and can even mold metals
with a rubber mold [9].
Correspondence to: D.E. Hardt; e-mail: hardt@mit.edu
Contract grant sponsor: Singapore-MIT Alliance.
DOI 10.1002/pen.22049
Published online in Wiley Online Library (wileyonlinelibrary.com).
C 2011 Society of Plastics Engineers
V
Centrifugal casting is a recognized method for eliminating bubbles in polymer solutions. In some cases, buoyant transport of the bubbles toward the center axis of
rotation is the physical explanation for removal [4, 7, 8].
At high enough spin speeds with a polymer with low
enough viscosity, buoyant forces aid in removing bubbles.
In other cases, diffusion is the physical explanation for
bubble removal [8, 10–14]. These are cases in which rotational speeds are not high enough to generate gravitational/buoyant forces sufﬁcient to overcome the high
viscosity of a polymer and signiﬁcantly move bubbles toward the center axis of rotation. In diffusion-based bubble
removal, there is a gradient of dissolved gas between the
bubble wall and the polymer surrounding the bubble. If
the concentration of gas dissolved within the bubble wall
is greater than the concentration of gas in the surrounding
solution, the bubble will lose gas.
The aspects of diffusion described in this work are
similar to those present in devolatilization of polymers.
Nevertheless, there are at least two key differences
between devolatilization and centrifugal casting as
presented in this work. In devolatilization, the volatiles—
gaseous or low-molecular weight undesirables—are
removed from the processed polymer. In centrifugal casting, the gases from the bubbles are either removed
through buoyancy or dissolved into the polymer. Another
difference between these two processes involves the relevant sets of boundary conditions. Machines for devolatilization are designed to bring a portion of the processed
polymer in contact with atmospheric pressure or vacuum
through an outlet. In centrifugal casting as described in
this article, there is only one surface exposed to atmospheric pressure, which is the liquid-air interface near the
center of the spinning mold cavity.
In this work, the centrifugal accelerations are on the
order of 1000g’s (10,000 m/s2) with spinning speeds of
thousands of revolutions per minute and resulting ﬂuid
pressures on the order of 700 kPa (100 psi). At these
pressures, air bubbles in PDMS can readily dissolve into
solution, however, the low dynamic viscosity of the polymer (1 or 10 Pa s) combined with high centrifugal
accelerations also leads to buoyancy-driven transport
of bubbles out of the ﬂuid. Given the manufacturing
POLYMER ENGINEERING AND SCIENCE—-2012
c ¼ kD p;
FIG. 1. Mold cavity for centrifugal casting being spun about an axis of
rotation. The parameters gr and gb represent centrifugal and gravitational
accelerations, respectively. [Color ﬁgure can be viewed in the online
issue, which is available at wileyonlinelibrary.com.]
objectives of quality (bubble-free) and production rate
(time), the key issue is the length of time required to
remove all bubbles by both mechanisms.
Centrifugal Casting and Bubble Dissolution
In centrifugal casting, a reservoir receives a quantity of
PDMS, which is then spun to ﬁll the mold cavity. The
speciﬁc geometry used in this work is shown in Fig. 1.
During spin-up of the mold, the PDMS solution traps
some air from the mold cavity, which causes the PDMS
to be slightly more saturated with air relative to its initial
state. Two things then occur during spinning: (1) the
PDMS attempts to return to a normal saturated state via
diffusion at all liquid–air interfaces and (2) bubbles move
toward the center of the centrifuge. The movement of
bubbles toward the center (larger ones faster than smaller
ones) combined with the dissolution of some bubbles
leaves a distribution of smaller and smaller bubbles in the
PDMS as time progresses. To achieve a bubble-free part,
all bubbles, regardless of size, must be removed either by
buoyancy-driven transport or by dissolution. As will be
discussed below, there is a critical bubble diameter that is
deﬁned by the longest time for removal either by buoyancy-driven transport (above the critical diameter) or by
dissolution (below the critical diameter).
Bubbles shrink or grow depending on the difference
between the air concentration inside the bubble wall and
the air concentration in the PDMS solution surrounding
the bubble. If the concentration inside the bubble wall is
less than the concentration in the surrounding solution,
the bubble will grow as air diffuses into the bubble
through its wall. If the concentration of air in the wall of
the bubble is greater than the concentration of air in the
solution surrounding the bubble, the bubble will tend to
shrink. The concentration of air in the bulk of the PDMS
solution is initially slightly supersaturated from the centrifuge’s startup, and during spinning, the air concentration
remains nearly constant. However, the air concentration c
in the polymer of the bubble’s wall varies linearly with
internal pressure p of the bubble according to Henry’s
Law:
DOI 10.1002/pen
(1)
where kD is Henry’s coefﬁcient.
Higher internal pressures within a bubble will correspond to higher concentrations of air in the wall of a bubble and greater diffusion of the gas into the polymer. This
pressure will increase not only as the pressure of surrounding polymer increases, but also as the diameter of
the bubble decreases because of the effects of surface tension. Thus, the air concentration in the bubble wall will
be larger for smaller bubbles, and if the air concentration
in the bubble wall is larger than the air concentration of
the surrounding PDMS ﬂuid, the bubble will shrink and
dissolve completely given sufﬁcient time.
The goal here is to determine the minimum time
required to remove all bubbles from a mold cavity whether
by diffusion of the air within the bubbles into the polymer
or by buoyancy-driven transport of the bubbles to the free
surface at the center of the centrifuge. For the geometries
considered here, it will be shown that centrifugal casting
will remove a large bubble, (1 mm in diameter) from a
solution spinning at thousands of revolutions per minute by
buoyant forces before the bubble has sufﬁcient time to dissolve completely, whereas smaller bubbles, (10 lm in diameter) will dissolve into solution before buoyant forces
drive them to the liquid–air interface. The basic geometry
for analyzing the bubble dynamics is shown in Fig. 2. For
description of the parameters, please see Nomenclature.
Bubble Diffusion
In 1950, Epstein and Plesset [15] published a seminal
work describing diffusional growth or dissolution of an
air bubble in water. Using their results and including both
concentration gradients and surface tension effects, the
following relationship for the rate of change in bubble radius can be derived [1]:
(
)
dR kðcs1 ðcf 1Þ 2kRD sÞ 1
1
¼
þ
;
(2)
r1 þ 2t=ð3RÞ
dt
R ðpktÞ1=2
FIG. 2. Gas bubble in a mold cavity of ﬂuid being spun in a centrifuge.
[Color ﬁgure can be viewed in the online issue, which is available at
wileyonlinelibrary.com.]
POLYMER ENGINEERING AND SCIENCE—-2012 81
where: s ¼ 2Mr/(BT). For descriptions of the parameters,
please see Nomenclature.
To simulate the full dynamics of a bubble, it is also
necessary to track the change in mass, and this is given
by a modiﬁed form of Eq. 2:
dm
2
¼ 4pR kkD pfluid ðcf
dt
1Þ
2s
R
(
)
1
1
þ
:
R ðpktÞ1=2
(2a)
In Eq. 2, if cf 1, the bubble will always shrink
regardless of its radius. However, if cf [ 1 (supersaturation of the ﬂuid with dissolved air), the following condition must hold for the bubble to shrink:
R<
2s
pfluid ðcf
1Þ
:
(3)
fuge. The buoyant force in the radial direction (driven by
the high value of gr) is
FBr ¼ ðrf
ra ÞVr gr ;
(4)
where rf is the density of the ﬂuid, rf is the density of
the air within the bubble, and Vr is the volume of the
bubble. The drag force can be modeled according to the
Hadamard-Rybczynski relationship [14, 15, 19]:
FDr ¼ 3pmf dur
2mf þ 3mb
;
3mf þ 3mb
(5)
where lb is the dynamic viscosity of the air within the
bubble and lf is the viscosity of the ﬂuid. For air (lb
1025 Pa s) and for PDMS Sylgard 184 (lf 1 Pa s),
the drag force acting on an air bubble in PDMS reduces
to the following:
Depending on the diameter of the bubble, pressure,
and concentration of air dissolved in solution, calculated
times for dissolution of a bubble in stagnant PDMS may
require only a fraction of a second or may take longer
than an hour [1]. Regardless of these times, it is assumed
here that the bubble velocity will not alter the rate of gas
diffusion, and therefore the quiescent ﬂuid condition of
the Epstein-Plesset model. For the small bubbles encountered in this system, this is a reasonable assumption.
By balancing the buoyant and drag forces on a bubble
and assuming negligible inertial contributions, the following relationship holds for the velocity of a bubble vr at a
given location r being spun at a speed x:
Viscous Effects on Dissolution
Simulations for Bubble Transport and Diffusion
Kloek et al. [16], Zana and Leal [17], Kim et al. [18],
and others have derived relationships for growth of bubbles
by diffusion or dissolution including the viscosity of the
ﬂuid. Kontopoulou and Vlachopoulos [12] applied models
of diffusive dissolution of bubbles with viscosity to rotational molding. They modeled bubble dissolution in polyethylene using viscosity values of 500 Pa s, 5000 Pa s, and
10,000 Pa s (all of which are at least a factor of 5 greater
than the viscosity of PDMS used in this study) and determined that for these values ‘‘the effect of viscosity is marginal and is more prominent at longer times, when bubble
diameter becomes small.’’ Gogos [14] also addressed dissolution of bubbles in rotational molding, cited the work of
Kontopoulou and Vlachopoulos [12], and ignored the
effects of viscosity in his calculations. In both works by
Kontopoulou and Gogos, the authors assumed the bubble
was stationary. They also modeled the case of a single bubble in an inﬁnite medium. Given that the effect of
the material’s viscosity has only ‘‘marginal effects’’ on bubble growth/dissolution, the modeling in this work ignores
the effects of viscosity on bubble dissolution.
The removal of bubbles from the polymer involves both
the change in radius and mass of the air bubble described
by Eq. 2a and the transport by buoyancy described by
Eq. 7. Since both sets of equations are nonlinear, they are
best solved numerically, and the following describes a
simple Euler-integration scheme used to obtain simulated
results. This approach also allows for modeling a nonconstant centrifuge speed owing to ﬁnite start-up times.
Initially, the bubble diameter is determined by using
the ideal gas law for the calculated mass of air within a
bubble. The mass of bubble at time interval i þ 1 is
found by integrating dm
dt from Eq. 2a:
Buoyancy Effects
Unless or until the bubble dissolves into the polymer,
it is subjected to buoyant and drag forces in the centri82 POLYMER ENGINEERING AND SCIENCE—-2012
FDr 2pmf dur :
ur
(6)
ðra rf Þ 2 2
ro d :
12mf
miþ1 ¼ mi þ
dm
j Dt;
dt i
(7)
(8)
Once the diameter is known, the radial position of the
bubble at time interval i þ 1 is given by
riþ1 ¼ ri þ ui Dt;
(9)
where vi is the radial velocity given by Eq. 7 at time
interval i.
The numerical solution of the combined model for a
single bubble of speciﬁed initial radius and radial position
has the following steps:
DOI 10.1002/pen
Initialization
The initial position, radial distance, and all ﬂuid properties in Eqs. 2 and 7 are speciﬁed, and the initial concentration of air within the liquid is determined from
ci ¼ cf
init kD pinit ;
(10)
where cf_init is the initial amount of fractional supersaturation (i.e. cf_init ¼ 1.05 would indicate a supersaturation of
5%) caused by pouring, mixing, or the sloshing of ﬂuid at
the start of spinning. The parameter pinit is the initial pressure (atmospheric for this work). Additional parameters
including chamber pressure, temperature, maximum spin
speed, slew rate (centrifuge’s angular acceleration), and
time step for the numerical integration (Dt) are speciﬁed.
It is important to note that the simulation is for a single
bubble, and as will be shown, the initial bubble diameter
do with the mold cavity at rest is a key initial condition.
First Iteration
Number of Moles of Gas. Given the initial value do of
the bubble, the initial number of moles n within the bubble is determined with the following expressions:
Pb ¼
4s
þ Pcham ;
d0
(11)
pPb d03
;
6BT
(12)
n¼
where B is the Universal Gas Constant (287 J/(kg K)).
Fluid Pressure. Using the current spin speed (can
include a startup or ramp proﬁle as well as the steadystate speed), the pressure of the liquid Pﬂuid at the bubble’s location is determined from:
1
Pfluid Pcham þ rf o2 r 2
2
rf2 :
(13)
Bubble Pressure. The internal pressure Pb of the air
within the bubble is found from:
4pPb
Pb ¼ 2s
3nBT
1=3
þPfluid :
(14)
Bubble Diameter. Using Pb and the ideal gas law, a
new diameter d for the bubble is determined.
Repetitive Iterations
Bubble Speed. Given the diameter, the speed vi of the
bubble at the current location and time can be found from
DOI 10.1002/pen
Eq. 7, and using the speed, the new location ri
found by Eq. 9.
þ 1
is
Nondimensional Degree of Saturation cf. Using the
ﬂuid pressure and the Henry’s coefﬁcient, the value for cf
is given by:
cf ¼
ci
:
kD pfluid
(15)
Mass of the Bubble. From the current number of moles
and molecular mass of the gas, the current mass of the
bubble m is determined. Equation 8 is then used in conjunction with Eq. 2a to update the mass for time interval
i þ 1. Using the new mass of air within the bubble and
the molecular weight of the air (28.97 g/mol), the new
number of moles within the bubble is determined.
Update the Internal Bubble Pressure. Using the
updated number of moles of gas within the bubble and
the solution to Eq. 14, along with the ﬂuid pressure at the
current bubble location, the updated internal bubble pressure can be computed. From this resulting internal bubble
pressure, a new diameter of the bubble is found using the
ideal gas law.
The algorithm cycles through the above sequence of
repetitive iterations until the bubble diameter goes to zero
or the bubble position reaches the desired end point.
Simulation Results
Using the above algorithm, it is possible to simulate
bubble travel in a viscous polymer given a set of centrifugal casting parameters. The following three ﬁgures show
a variety of outputs for simulations using the values in
Table 1. The termination point of a line signiﬁes that the
bubble has either reached the desired end position (the
liquid–air interface in this case) or dissolved into solution
(d 0). The selected parameter values are representative
of the designed and tested system.
Figure 3 shows the value of cf plotted against time for
different spin speed proﬁles and initial bubble diameters.
The initial value of cf starts at 1.05 and then decreases
because of the acceleration of the centrifuge. As the centrifuge accelerates at a rate of 300 rpm/s, the ﬂuid pressure builds up causing a decrease in cf as described by
Eq. 15. The centrifuge then reaches steady-state, and the
bubble continues moving toward the liquid–air interface
near the center of the centrifuge. As the bubble moves
closer to the center of the centrifuge, the ﬂuid pressure
surrounding the bubble decreases as described by Eq. 13.
This decrease in ﬂuid pressure leads to an increase in the
cf parameter, and if the bubble survives without dissolving
and arrives at the liquid–air interface, the cf parameter
returns to 1.05.
POLYMER ENGINEERING AND SCIENCE—-2012 83
TABLE 1. Parameter values used in simulating combined bubble speed and diffusion behavior.
Parameter
Value
Description
Bubble_start_pos
Bubble_end_pos
rf_start
T
r
l
qf
Pcham
Spin_speed
Slew_rate
Offset
j
kD
M
dt
cf_init
0.0615 m
0.0343 m
0.0343 m
298 K
0.02 N/m
3.9 Pa s
1030 kg/m3
101.3 kPa
1000–8000 rpm
300 rpm/s
100 rpm
1.47 3 1029 m2/s
1.927 3 1026 kg(gas)/m3(polymer)/Pa
0.02897 kg/mol
0.01 s
1.05
Bubble start distance from center axis of rotation
Bubble ending distance from center axis of rotation
Liquid–air interface distance from center axis of rotation
Temperature during spinning
Liquid–gas surface tension
Liquid viscosity
Liquid density
Chamber pressure
Set spin speed
Spin speed acceleration (slew rate)
Spin speed at start
Diffusion coefﬁcient
Henry’s coefﬁcient
Molecular weight of gas
Time step
Initial supersaturation fraction
Figure 4 shows the bubble diameter as a function of
bubble position. The bubble starts at the right of each
graph (61.5 mm from the axis of rotation) and moves toward the center/left of the graph (34.3 mm from the axis
of rotation). Both plots in Fig. 4 show bubbles with do ¼
100 lm dissolving into solution within 5 mm of the bubble’s starting location. Assuming the original solution is
slightly supersaturated, it is also possible for a bubble to
shrink by diffusion and then reach a region where the
change in the pressure of the ﬂuid causes the gradient in
the concentration of gas between the bubble wall and the
PDMS surrounding the bubble to ﬂip. This change in the
direction of the gradient results in diffusional growth of
the bubble as the bubble gets closer to the center of the
centrifuge. In general, diffusional shrinking of bubbles
competes against the natural growth of bubbles as they
move toward the axis of rotation. Without diffusion, the
volume of the bubbles would increase as they approach
the axis of rotation because of lower pressures near the
axis of rotation.
Figure 5 shows trajectories of bubbles with different
diameters. The bubbles start at the upper horizontal line
and progress toward the lower line. The bubbles either
dissolve into solution or have buoyancy carry them to the
lower horizontal line. Both graphs shown in Fig. 5 have
an initial bubble diameter (do ¼ 132.1 lm for a spin
speed of 1000 rpm and do ¼ 166.2 lm for a spin speed
of 8000 rpm) that results in a maximum amount of time
required for a given spin proﬁle. These values of do
resulting in maximized times for bubble removal were
obtained using an iterative optimization routine. Figure 6
shows the maximum time values for these optimizations
plotted against spin speed.
near the point of maximum removal time (peak of the
curve in Fig. 7 a). For example, near do ¼ 132.1 lm, a
change of 1 3 1029 lm corresponds to a difference in
time of 151 s for bubble removal. It is possible to avoid
spurious bubble removal times with unrealistic sensitivity
in the numerical simulation by instead considering the
Sensitivity to Initial Diameter
FIG. 3. The cf parameter plotted against time for a variety of initial
bubble diameters and set spin speeds. The bubble starts 61.5 mm away
from the center of rotation, and the liquid–air interface is 34.3 mm from
the center of rotation. (a) 1000 rpm, (b) 8000 rpm.
The numerical simulation is very sensitive to changes
or perturbations to initial diameter do when do is chosen
84 POLYMER ENGINEERING AND SCIENCE—-2012
DOI 10.1002/pen
moval time as shown in Fig. 7. That said, the maximum
time for bubble removal is associated with a speciﬁc initial diameter, which is dependent on the proﬁle of the
spinning centrifuge and the bubble’s starting location. To
the left of the peak in bubble removal time, diffusion
dominates bubble removal whereas to the right of the
peak, buoyancy and bubbles reach the liquid–air interface
before dissolving.
To calculate removal times associated with speciﬁed
percentages of trapped bubbles, the simulation considers
the travel of bubbles with initial diameters from 0 lm to
1 mm in increments of 25 nm. Figure 8 shows the maximums for the amount of time required for bubble removal
over the speciﬁed domain of do. These values are depicted
by the jagged curves just inside the curve showing the
maximum exit times for optimized values of do. The
lower three curves in Fig. 8 indicate the amount of spin
time required to remove approximately 90% (line width
FIG. 4. The bubble diameter plotted against position for a variety of
initial bubble diameters and set spin speeds. The bubble starts 61.5 mm
away from the center of rotation, and the liquid–air interface is 34.3 mm
from the center of rotation. (a) 1000 rpm, (b) 8000 rpm.
probability of removing certain fractions of bubbles initially trapped within the solution.
Assuming a uniform probability distribution of values
for do (equal probability of any particular bubble diameter
occurring) between 0 lm and 1 mm, domains of values
for do can be associated with a probability of occurrence.
For example, if a speciﬁed domain has a span of 100lm,
there is a 10% chance of a bubble existing within the
speciﬁed domain over the total domain between 0 lm and
1 mm. Accordingly, if it is desirable to remove 99% of
all the bubbles between 0 lm and 1 mm, a domain length
of 10 lm is permissible.
Given a domain length, it is possible to estimate the
time required to remove the corresponding fraction of
bubbles from solution. Figure 7b and c show the required
amounts of time to remove 90% of bubbles with diameters from 0 lm to 1 mm. The cutoff line with a length of
100 lm in Fig. 7b suggests spinning at 1000 rpm for 65 s
to remove 90% of the bubbles. A similar cutoff line in
Fig. 7c suggests spinning at 8000 rpm for 13 s to remove
90% of the bubbles starting at the outer edge of the mold.
Using the maximum times for bubble removal associated with the optimized value of do alone will likely overpredict the amount of time necessary to remove bubbles,
especially for simulations with sharp peaks in bubble reDOI 10.1002/pen
FIG. 5. The bubble position plotted against time for a variety of initial
bubble diameters and two set spin speeds. The bubble starts 61.5 mm
away from the center of rotation, and the liquid–air interface is 34.3 mm
from the center of rotation. (a) 1000 rpm, (b) 8000 rpm. [Color ﬁgure
can be viewed in the online issue, which is available at wileyonline
library.com.]
POLYMER ENGINEERING AND SCIENCE—-2012 85
less than 20 s for bubbles that start at 4 cm from the center axis of rotation. Because the differences in maximum
times for bubble removal are minimal at low spin speeds
FIG. 6. Top graph shows initial bubble diameter values (do) requiring
the longest time to exit the solution of Sylgard 184. Bottom graph shows
the optimized times for the bubble to exit. For the simulated data shown,
the bubble started 61.5 mm from the center.
of 100 lm), 95% (line width of 50 lm), and 99% (line
width of 10 lm) of the bubbles that started at the outer
edge of the mold.
Effect of Starting Position of the Bubble
Up to this point, all the simulations shown for Sylgard
184 have started with the bubbles sitting at the outer edge
of the mold. In reality, the bubbles requiring the most
time for removal do not necessarily start at the outer edge
of the mold. In a case study of different geometries [1], it
is shown that the slowest bubbles for removal may start
in the middle of the part.
Using the same parameters as the above simulations,
Fig. 9 shows the simulated times required for removal
plotted against multiple spin speeds and starting locations.
The times for bubble removal are the simulated amount
of time required for a bubble at a speciﬁed starting location with a speciﬁed initial diameter to either dissolve into
solution or reach the desired location (edge of the mold
or the liquid-air interface). The peaks (simulated
maximum times) shown in Fig. 9 are dependent on the
discretization of the initial diameters. To estimate actual
removal times for bubbles and eliminate the unrealistic
sensitivity to the discretization of the initial diameters of
the bubbles, speciﬁed line widths representing a percentage of the possible diameters between 0 and 1 mm were
utilized.
As shown in Fig. 10, the maximum time to remove
99% of the bubbles for a set spin speed of 1000 rpm
actually occurs with a bubble starting approximately
4.5 cm from the center axis of rotation. However, at
higher spin speeds, the maximum time occurs with a bubble starting at the outer edge of the mold. For example,
with a set spin speed of 5000 rpm, the bubble removal
time is 20 s for bubbles that start at the outer edge but
86 POLYMER ENGINEERING AND SCIENCE—-2012
FIG. 7. (a) Bubble removal time plotted against initial bubble diameters near do ¼ 132.1 lm for a spin speed of 1000 rpm with Sylgard 184.
(b) Calculated time to exit for a spin speed of 1000 rpm with do varied
in steps of 25 nm. (c) Calculated time to exit for a spin speed of 8000
rpm with do varied in steps of 25 nm. In all these cases, the bubbles
started near 61.5 mm from the central axis of rotation.
DOI 10.1002/pen
FIG. 8. Simulated bubble removal times based on optimization of do
and different fractions of removed bubbles. The bubbles started at the
outer edge of the molding region (61.5 mm from the axis of rotation).
or at multiple steps after ﬁlling goes beyond the scope of
this work. The agreement between the experimental
results and the simulations reported suggests that the
assumptions made about the initial distribution of relevant
bubbles with diameters less than 1 mm are reasonable.
Images of the centrifugally cast parts were acquired
using a Canon CanoScan LiDE 600F scanner at a resolution of 1200 dpi. Figure 11a shows an example image of
parts scanned without any bubbles in the regions of interest (outlined rectangular regions of images). Figure 11b
shows an example image of parts scanned with a number
of bubbles left in them. An algorithm (described in Ref.
1) was used to process the images and count the number
of bubbles in each part. Figure 12 presents the average
number of bubbles found in four parts at each combination of spinning parameters. The numbers next to the
asterisk marks are the average number of bubbles meas-
(1000 rpm) for bubbles starting in the middle of the part
versus the outer edge of the part, it is simpler to use the
outer edge as the starting position associated with the
maximum bubble removal time.
Experimental Validation
To verify the predictive capabilities of the described
numerical simulation, a series of experiments were performed varying spin time and spin speed. The spin speeds
and spin times were chosen to produce approximately half
of the resulting parts with bubbles and half without bubbles. The spin times were 0.5, 1, 2, 3, and 5 min, and the
spin speeds were 1000, 1600, 2500, and 4000 min. With
ﬁve discrete spin times and four discrete spin speeds,
there were 20 possible combinations. The spin time refers
to the total amount of time the centrifuge’s motor was
energized and does not include the time for the centrifuge
to decelerate and stop. The centrifuge accelerated to the
desired spin speed with a slew rate of 300 rpm/s. The rest
of the relevant geometric and ﬂuid properties are included
in Table 1.
Two parts were produced per trial and each set of parameters was run twice. With 20 possible combinations
and each combination run twice, there were 40 trials.
Since each trial yielded two parts, there were a total of 80
parts produced. The number of bubbles in each part was
counted, and the number of bubbles observed for a given
combination of spin time and spin speed was calculated
by averaging the numbers of bubbles in four parts.
Counted numbers of bubbles in the cured parts were compared with the simulated results for percentages of bubbles removed. While a comparison between experimental
results and simulations strictly using either numbers of
bubbles or percentages of removed bubbles would be
more desirable, the careful characterization of the distributions of bubbles in parts immediately after ﬁlling the mold
DOI 10.1002/pen
FIG. 9. Simulated times for bubbles to be removed from PDMS Sylgard 184 plotted against a domain of initial bubble diameters. The
lengths indicated at the various peaks indicate the bubble starting position. (a) 1000 rpm, (b) 4000 rpm. [Color ﬁgure can be viewed in the
online issue, which is available at wileyonlinelibrary.com.]
POLYMER ENGINEERING AND SCIENCE—-2012 87
FIG. 10. Plot of simulated times for 99% removal of bubbles up to 1
mm in diameter from PDMS Sylgard 184 versus bubble starting positions and spin speeds. The dots represent simulated results.
ured in four samples. Figure 12 also shows simulated
times from Fig. 8 for bubble removal with bubbles starting at the outer edge of the mold.
When Fig. 12 is replotted on log-log coordinates as
shown in Fig. 13, the simulated curves become straight
lines over much of the region of experimental interest.
The line with pentagrams (time required to remove 99%
of the bubbles) roughly separates the parts with more than
one bubble from those that averaged less than one bubble.
This dividing line shows agreement between the experimental data and the numerical simulations.
Effect of Material Property Changes
Finally, the simulations and modeling described in this
work also apply to other materials besides Sylgard 184.
Figures 14 and 15 show simulated calculations and experimental results for centrifugally casting Dow Corning 36121 and Dow Corning Sylgard 186. The only parameter
changed in the simulations for these different materials
was the viscosity. Dow Corning 3-6121 has a speciﬁed
dynamic viscosity of 25 Pa s, the Sylgard 186 has a
dynamic viscosity of 65 Pa s, and the Sylgard 184 has a
speciﬁed dynamic viscosity of 3.9 Pa s. The line with diamonds (simulated time required to remove 99% of the
bubbles) in Fig. 14 for Dow Corning 3-6121 roughly separates the parts with more than one bubble from those
averaging less than one bubble just as shown in Figs. 12
and 13 for Sylgard 184.
Figure 15a with Sylgard 186 shows data exhibiting the
same trends as those shown for Sylgard 184 and 3-6121.
However, the level of agreement between the simulations
for removing 99% of the bubbles and the data for spin
speeds less than 2500 rpm is not as pronounced (the
simulated line does not sharply delineate the parts with
bubbles from those without bubbles). This disagreement
88 POLYMER ENGINEERING AND SCIENCE—-2012
FIG. 11. (a) Scanned image of parts produced in a trial (1000 rpm and
5 min) with no counted bubbles in the regions of interest. (b) Scanned
image of parts produced in a trial (1000 rpm and 0.5 min) with 144 bubbles counted within the region of interest in the part on the left and 21
bubbles counted within the region of interest in the part on the right.
FIG. 12. Experimental data plotted against simulated predictions for
Sylgard 184 with a viscosity of 3.9 Pa s.
DOI 10.1002/pen
FIG. 13. Log-log plot showing experimental data and predictions for
Sylgard 184 with a viscosity of 3.9 Pa s.
may partially result from the simulations in Fig. 15a only
starting with the bubble at the outer edge of the mold,
61.5 mm from the axis of rotation. For simulations with
the bubbles starting at the outer edge of the mold, the
required spin time to remove 99% of the trapped bubbles
with a set spin speed of 1000 rpm is 1090 s. Additional
simulations run with a set spin speed of 1000 rpm and
bubbles starting 49.5 mm from the axis of rotation
showed a required spin time of 1190 s to remove 99% of
the trapped bubbles. Thus, a bubble starting 45 mm away
from the center axis of rotation with a set spin speed of
1000 rpm will take longer to exit than a bubble starting at
the outer edge. More likely, the disagreement between the
simulated results and the experimental data in this case
results from unmodeled interactions between bubbles,
ﬂuid, and surfaces of the mold, unmodeled effects of vis-
FIG. 15. (a) Plot showing experimental data and predictions for Sylgard 186 with a viscosity of 65 Pa s. The numbers represent the average
number of bubbles counted in four parts, and the predictions assume
bubbles start 61.5 mm from the axis of rotation. (b) Plot of simulated
times for 99% removal of bubbles up to 1 mm in diameter from Sylgard
186 versus bubble starting positions and spin speeds. The dots represent
simulated results.
cosity on bubble dissolution, or variations in the viscosity
or Henry’s coefﬁcient.
CONCLUSIONS
FIG. 14. Plot showing experimental data and predictions for Dow
Corning 3-6121 with a viscosity of 25 Pa s. The numbers represent the
average number of bubbles counted in four parts. The lines demarcate
the simulated results to remove different percentages of bubbles from the
regions of interest.
DOI 10.1002/pen
The model presented in this work captures the combined effects of buoyancy, drag, and diffusion on bubble
removal in centrifugal casting. The simulated results predict the time required for bubbles to either dissolve into
solution or leave the region of interest with a given spin
proﬁle and starting location. For a given set of conditions
for centrifugal casting, there is a critical initial size for
bubbles. This critical initial size, or diameter, separates
the bubbles removed predominantly by diffusion from
those primarily removed by buoyancy. The required times
for bubble removal are estimated using a numerical
POLYMER ENGINEERING AND SCIENCE—-2012 89
solution of a pair of ﬁrst-order nonlinear differential equations for rate of bubble mass and position change with
time. Experiments on silicones with three different viscosities veriﬁed the theoretical basis for the model and numerical simulations described in this work.
q!
cS!
cf
ACKNOWLEDGMENTS
The authors thank the Singapore-MIT Alliance for
funding this work. MIT students Eehern Wong, A.J.
Schrauth, Matthew Dirckx, Melinda Hale, Hayden Taylor,
David Lee Hennan, Vikas Srivastava, and J.P. Urbanski
also contributed to the development of this work. In addition, Dr. Brian Anthony and Dr. James Bales were very
helpful in getting high-speed video images, which aided
in further understanding the physics behind bubble removal and centrifugal casting.
NOMENCLATURE
The following parameters are pertinent to the geometry
in Fig. 2:
x
gr
vr
FDr
FBr
d
Pb
rf
r
Pcham
T
Angular velocity of the centrifuge
Radial acceleration owing to angular velocity ¼
rx2
Radial velocity of the bubble
Viscous drag on the bubble in the radial direction
Buoyant force on the bubble in the radial direction
Bubble diameter
Bubble pressure
Radial distance from the free surface to the axis
of rotation
Radial distance of the bubble from the axis of
rotation
Ambient pressure in the centrifuge
Polymer temperature (assumed to be uniform and
constant).
For Eq. 2, we have the following parameters:
M
B
j
Molecular weight of the gas (28.97 kg/1000 mol)
Universal gas constant (8.314 J/(mol K))
Coefﬁcient of diffusion
90 POLYMER ENGINEERING AND SCIENCE—-2012
ci
Density of the gas at the pressure and temperature
of the surrounding ﬂuid
kDpﬂuid (the concentration of dissolved gas in the
wall of a bubble if the radius of the bubble were
inﬁnite)
ci
cS1 (the normalized saturation of the solution
surrounding the bubble)
Initial concentration of gas dissolved in the
solution
REFERENCES
1. A.D. Mazzeo, Centrifugal Casting and Fast Curing of Polydimethylsiloxane (PDMS) for the Manufacture of Micro and
Nano Featured Components, PhD Thesis, Massachusetts
Institute of Technology, Cambridge, MA (2009).
2. J.C. McDonald and G.M. Whitesides, Acc. Chem. Res., 35,
491 (2002).
3. P.M. Biesheuvel, A. Nijmeijer, and H. Verweij, AIChE. J.,
44, 1914 (1998).
4. C.D. Spencer, Soc. Plast. Eng. J., 18, 774 (1962).
5. W.M. Larson, U.S. Patent 3,956,448 (1976).
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7. W.M. Haines, Elastomerics, 110, 26 (1978).
8. E.A. Sheard, Elastomerics, 122, 49 (1990).
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10. L. Xu and R.J. Crawford, J. Mater. Sci., 28, 2067 (1993).
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13. R.J. Crawford and J.L. Throne, Rotational Molding Technology, Plastics Design Library, Norwich, NY (2002).
14. G. Gogos. Polym. Eng. Sci., 44, 388 (2004).
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16. W. Kloek, T. van Vliet, and M. Meinders, J. Colloid Interface Sci., 237, 158 (2001).
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DOI 10.1002/pen

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