ICMF 2010, Tampa, FL, May 30 June 4, 2010
Film drainage between a bubble and a fluid interface
H. Ko6drkovf* F. Pigeonneau* F. Rouyert
Surface du Verre et Interfaces, UMR 125 CNRS/SaintGobain, Aubervilliers, France
t Laboratoire de Physique des Mat6riaux Divis6s et des Interfaces, FRE 3300,
University ParisEst, Marne la Vall6e, France
helena.kocarkova@saintgobain.com
Keywords: bubble, glass, high viscous fluid, foam, film drainage
Abstract
In order to understand the stability of foam of high viscous fluids, experiments have been performed with silicon
oil and molten glass, where a bubble moves toward a liquid interface. Temporal evolution of thickness between a
single bubble and the fluid interface is investigated via interference method under various physical conditions, range
of viscosities and bubble sizes.
The first part of the work concerns a drainage of a bubble in a silicon oil at a room temperature, which serves as a liquid
model for molten glass. Second part of this work uses similar experimental setup to investigate temporal evolution
of a thickness, but for a bubble in molten glass at high temperature. Influence of various temperature on drainage of
a single molten glass film is studied. The comparison with results of the silicon oil experiment shows the influence of
chemistry on nature of film drainage.
Evolution of thickness for silicon oil and molten glass is exponential. Speed of drainage is decreasing as a function of
Bond number for molten glass, but remains constant for silicon oil. In this later case, we assume that the long chains
of silicon oil influence the drainage for low thickness (lesser than few pm).
1 Introduction
Foam consists of bubbles entrapped in a liquid occur
ring in daily life of everybody as well as in many in
dustrial processes. In most of cases, the foam stability
is a searched property. Today, many investigations try
to form foams more and more stable. Nevertheless, in
glass melting process in particular, foam can be a nui
sance. Most of glass furnaces are heated by a combus
tion chamber above the glass bath (Trier 1984). Conse
quently, if a large part of the bath surface is covered with
foam, heat transfer, mainly radiative', decreases due to
the insulator property of glass foam.
Glass melting is a chemical process where glass is
made in most of cases with silica, soda ash, and lime.
The raw materials are generally carbonaceous elements
leading to the carbon dioxide release. The low solubil
ity of CO2 (Beerkens 2003) leads to a creation of large
quantity of bubbles entrapped in a molten glass. To re
move these gaseous inclusions, sulfate compounds are
'The typical value of temperature is the combustion chamber is
greater than 2000 K.
added to raw materials, see Shelby (1997). Gases re
leased by sulfate decomposition lead to a raising of bub
ble size. Due to buoyancy forces, bubbles can escape
from the bath surface.
The onset of foaming has been studied by Kim and
Hrma (1992) from the knowledge of chemical reactions
produced by sulfate species. Pilon (2002) developed a
model to study the foam formation by bubbling and es
tablished a relationship between foam layer and physical
properties of liquid using a dimensional analysis.
The threshold between frost and foam can be seen as
a balance between the gas flux coming from the bath
and the bursting of bubbles at the free surface, various
studies have been focused on the stability of one bub
ble on a free surface. This kind of work has been in
vestigated by Princen (1963) where the shape of a bub
ble was determined from the pressure balance. Chi and
Leal (1989) gave a numerical study using a boundary
integral method applied for a drop rising toward a fluid
interface. They pointed out the importance of the ra
tio between the viscosities of liquid and of the drop.
Howell (1999) proposed a theoretical model for a bub
ble drainage using the lubrication theory. He gave a
ICMF 2010, Tampa, FL, May 30 June 4, 2010
scaling for the film drainage but this analysis was per
formed in bidimensional framework. In glass science
literature, Kappel et al. (1987) achieved an experimen
tal study of the vertical film drainage on molten glasses
where they found that the film thickness decreases ex
ponentially with time. They studied also the bubble
drainage but analyzed only the lifetime of the bubble.
Laimbock (1998) did a similar experiments on vertical
film where the thickness had been determined by elec
tric resistivity. He found that glass film can reach a sta
bilized thickness. Since molten glass is a high viscous
fluid, Debr6geas et al. (1998) studied the drainage of a
bubble in polydimethylsiloxane (PDMS) liquid seen as
a fluid model. They proposed a simple theory model
to explain the exponential decrease of the film thick
ness. Nevertheless, the slope obtained experimentally
was slower than the theory result. This point was not
discussed by the authors. More recently, van der Schaaf
and Beerkens (2006) proposed a model to describe the
drainage of a large bubble where an unknown parameter
was introduced to chance the interface mobility.
Despite these works, the drainage of a bubble at room
temperature for a fluid model as PDMS and for a bub
ble in molten glass is not completely understood. The
influence of the bubble size compared to the capillary
length is lacked. In this article, the drainage of a bubble
is investigated experimentally at room temperature with
PDMS liquid and at high temperature with molten glass.
In the following of this article, the physics of bubble
drainage is briefly described in section 2. The details
of experimental setup are presented in 3. The results
are given in section 4 and discussed in 5. Section 6 is
devoted to the conclusion.
2 Physics of bubble drainage
Before the presentation of the experimental setup, it is
important to point out the relevant parameters driving
the physics of a bubble close to a fluid interface. The
shape of the bubble is mainly controlled by the balance
of the buoyancy and surface tension forces:
smaller than the capillary length defined by
1cap 
Apsg'
the interface above the bubble is slightly deformed and
the bubble keeps the spherical shape. Conversely, at
large Bond number observed for a bubble larger than
the capillary length, the interface should be strongly de
formed and the shape of bubble as well. For more detail,
the shape of a bubble close to an interface has been very
well studied by Princen (1963).
The drainage of a bubble or a drop has been studied
since many years. The rate of drainage depends strongly
on the behaviour of the interface between the two fluids.
For a vertical film, Schwartz and Roy (1999) showed
that when the interface behaves as a solid surface, the
film thinning is an algebraic function of time. Con
versely, if the interface is completely mobile, the film
thinning is an exponential function of time. In this last
situation, the film thickness is not a relevant parameter to
scale the characteristic time of drainage. Howell (1996)
established the lubrication equations for vertical film.
He pointed out that the characteristic length scale is the
height of the film. For a bubble shape, Debr6geas et al.
(1998) pointed out that the characteristic time from the
drainage of a bubble depends on the bubble size. This
point can be easily understood if the balance of viscous
and gravity force is done. By assuming the fully mobile
interface, the flow is a pure extensional flow, where the
velocity profile in the film is a plug flow (Howell 1996).
The following equation
pD pgD,
gives the balance of gravity and viscous forces where p
is the dynamical viscosity, p the density of the liquid,
and U the velocity in the film. From this last equation, it
is possible to define a velocity scale:
pgD2
D3Apg ~ Da,
where g is the gravitational constant, D the character
istic length of a bubble, Ap the different of density be
tween the two phases and ca the surface tension.
The ratio of these two forces is directly the Bond num
ber defined as:
ApgD2
At small Bond number corresponding to a bubble
and a time scale:
pgD
In the following of this article, the time will be nor
malised as follows
t
t (7)
where t is the dimensionless time and t the physical
time.
ICMF 2010, Tampa, FL, May 30 June 4, 2010
3 Experimental setup
Our work with a single bubble moving through a liquid
interface is decomposed in two experiments. The first
one is done with a PDMS liquid and the second one,
with a similar experimental setup, is done with molten
glass.
The experiment done with a PDMS liquid is very sim
ilar to the work of Debr6geas et al. (1998). However, in
our work, a different ranges of viscosities and bubble
sizes are investigated. Sketch of the experimental setup
is given in Figure 1.
Figure 1: Experiment setup used for PDMS liquid.
The inflation of bubbles in a high viscous liquid is a
difficult task since the viscous force prevents the rising
of the bubble when it is yet kept at the needle (Snabre
et al. 1998). So, the size of the bubble must be suffi
ciently large in order that the buoyancy force would be
larger than the viscous force. Moreover, in our exper
iment, the bubble must be created on demand. Conse
quently, it is not possible to work with a constant flow
rate. So, a special device is used to inflate a bubble.
In order to create a bubble with a controlled size, a
solenoidvalve is used where the pressure can change be
tween a low and high values. Graph of the pressure pulse
is shown in Figure 2 where the pressure is low when the
solenoidvalve is off and high when the solenoidvalve
is on. In order to avoid the introduction of liquid in
the needle, its diameter is small (100 pm). By chang
ing the pressure level and the duration of the opening of
the solenoidvalve, the bubble size can be controlled.
When the observed bubble rises toward the free sur
face a photograph of it is taken by the camera placed be
side the pool. The image is compared to a photograph of
a scale inside the pool and the bubble size is determined
high
pressure
max 2bar
low
pressure
time
Figure 2: Typical signal of the pressure pulse.
precisely. As soon as a bubble reaches the free surface, a
liquid film between the free surface and a bubble is cre
ated. The temporal evolution of its thickness is measured
using an interference method thanks to a laser source at
532 nm of wavelength where a photodiode is used to
record the light power reflecting by the two interfaces
of the liquid film (see Figure 3). From the reflecting in
tensity of light, the film thickness can be determined as
a function of time. Figure 3 gives an example of the
signal recorded thanks to the photodiode. The intensity
of reflecting light is translated in electric voltage. From
the classical theory of light interference (Isenberg 1992),
the maximum and minimum of intensity can be used to
determine the thickness thanks to equations recalled be
low:
A
A. (2k
4n k
2n
where h is the film thickness, n the refractive index, A
the wavelength of the laser and k is the order of inter
ference. The subscript Imax is used when the maximum
of reflecting intensity is reaches and Imin is used for the
minimum of interference.
Five viscosities in the range from 10 to 100 Pas and
six different sizes of bubbles in the range from 2.5 to
7.5 mm in diameter have been investigated in this ex
periment. From these bubble sizes, the Bond number
defined by eq. (2) varies from 2 to 26.
The possible errors are related mainly to the determi
nation of the thickness. Indeed, the last point of the inter
ference pattern is assumed to be obtained when the order
of interference, k, is equal to one. In the experiment of
Sende et al. (1999), which used the same method for the
thickness determination, they achieved the experiment
using two wavelengths of light source to determine the
point where the first order of interference occurs. We use
,,,,,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
t0r00 350 400 450 500
teratlon (1t =0 2s)
550 600
Figure 3: Tension of the photodiode as a function of
time corresponding to the light intensity due
to the interference on the liquid film.
only one wavelength and we determined the last point of
the interference pattern assuming that after this point the
regularity of the interference pattern is disrupted. The
error can appear if we establish k = 1 at a wrong posi
tion, where k can in reality equal to a larger value. As the
thickness is determined using eqs. (8), and (9), the ab
solute value may be incorrect. The error in determining
the thickness is then A/(2n). This error may lead to an
inexact value of the thickness but the relative behaviour
of the thickness important to determine the thinning rate
stays the same even if this error occurs. We are not in
terested in the determination of a critical thickness, but
in the drainage in this work, so we can accept this error.
Another error is related to a determination of a bubble
size. The comparison of a photograph of a bubble
with a picture of a scale can vary in a few pixels. The
difference of min and max pixels gives an error of
0.25%. The error for Bo is computed using eq. (2) and
is equal to 0.5%. The error for a characteristic time T
can be determined using eq. (6) and is equal to 0.25%.
In a second part of this work, the experimental evolu
tion of the thickness is investigated for a molten glass.
The experiment setup is shown in Figure 4 and is sim
ilar to the previous one apart from the source of light,
which in this case is the radiation of the refractory of the
furnace and the radiation of molten glass. The interfer
ence pattern is observed via videocamera and recorded
in a computer. In front of the videocamera is placed a
very narrow green filtre.
The thickness is computed as it is mentioned above.
Glass used in the experiment is an ordinary sodalime
silica glass with a low iron content (0.01 weight % of
Fe20O). Experiment is achieved for three various tem
peratures corresponding to dynamical viscosities from
62 to 943 Pas. Bubble sizes varies from 5.5 to 14 mm
Figure 4: Experiment setup for molten glass.
in diameter corresponding to Bo from approximately 2
to 15.
Errors in this part of the experiment can be due to the
thickness determination as it has already been mentioned
above. Moreover, the bubble size can not be determined
using the same method presented for the PDMS experi
ment, because the design of the furnace does not allow a
side view but only a top view. This complicates the de
termination of a bubble size and increases errors. Video
camera is focused on the level of molten glass, so that in
terference pattern is visible. The bubble size can not be
determined when it reaches the level, because its shape
is no more spherical. We have to determine the size of
a bubble when it is close to the bottom of the crucible
and far from the free surface of glass, so the shape of it
is still spherical. From one frame of the movie obtained
with the videocamera, a size of the bubble, while it is
still close to the bottom, is compared to a size of a bot
tom and to a top of the crucible, because their diameters
are known. We compare the bubble size to the bottom
and to the top of the crucible as well, because we do not
know the exact position of the bubble. The bottom of the
crucible is smaller than the top from the top view, which
causes an error of 8% on the diameter of a bubble. The
properties of glass are computed using data published by
(Volf 1984) giving an error of 2% for viscosity, refrac
tive index and density and 1% on surface tension. This
leads to an error of 19 and 12% for Bond number and
characteristic time 7 respectively.
glass
ICMF 2010, Tampa, FL, May 30 June 4, 2010
4 Results
As it has been mentioned in section 2, the thickness evo
lution is expected to be exponential with time for a fully
mobile interface. The experimental result can be fitted
using the relationship
h hoeat
where ho and a are fitting parameters. The dimension
of ho is in metre and a is a dimensionless number that
correspond to the normalized thinning rate. This last pa
rameter is called "thinning rate".
The film thickness evolutions as a function of time
for PDMS liquid are shown in Figures 5, 6, 7, 8 and 9
for dynamical viscosities 10, 23, 51, 74 and 102 Pas
respectively.
S Bo=24, a=010
S Bo4 9, a0 11
Bo=8 8, a=0 092
Bo=13 2, a=0 13
104 c .. n
S 10
So ,,%;.
0 0 20 300 40 50
tau
60 70 80 90
Figure 7: Temporal evolution of film thickness for
PDMS with a dynamical viscosity equal to 51
Pas, and various Bond number.
o B=2 8,a=012
SBo=5 6,a= 09
u Bo=21 0, a=0 098
* Bo=246, a=0095
10:
0 10 20 30 40 50 60 70 80 90
ttau
A
Figure 5: Temporal evolution of film thickness for
PDMS with a dynamical viscosity equal to 10
Pas, and various Bond number.
10
10'
lO
t au
SBo=36 a= 01
SBo59 a 009
SBo=84 a=0089
SBo=12 7 a= 084
u Bo=197,a=0 1
SBo=235,a=0 095
60 70 90 90
Figure 6: Temporal evolution of film thickness for
PDMS with a dynamical viscosity equal to 23
Pas, and various Bond number.
The exponential behaviour is very well reproduced for
each viscosity. The smallest film thickness is 95 nm
computed from eq. (8) if the following parameters are
10
D 10 20 30 40 50 60 70 80 90
ttau
Figure 8: Temporal evolution of film thickness for
PDMS with a dynamical viscosity equal to 74
Pas, and various Bond number.
used: A=532 nm, n=1.4 and k=l. We do not measure
the critical thickness of the rupture, but it is lower than
this value.
Figure 10 shows the film thickness evolution for a
glass bubble at three measured viscosities and for vari
ous Bond number. The last determined thickness is now
smaller than in the PDMS experiment, because the re
fractive index of glass is computed from its composition
as n=1.52 (Volf 1984). The smallest determined thick
ness is then approximately 87.5 nm.
The first result very important to point out is that the
liquid film of molten glass behaves as a fully mobile
interfaces since the film thickness decreases exponen
tially with time. The second point is that the drainage
is faster when the Bond number is small. The experi
ment done with a dynamical viscosity equal to 208 Pas
and a small bubble giving a Bond number equal to 1.7
gives a drainage rate approximatively four times smaller
than this obtained with a Bond number equal to 14.67
observed with a dynamical viscosity equal to 943 Pas.
The remarkable point is that the influence of the Bond
SBo=33, a=0 12
Bo=5 9, a=O 87
* Bo=9 6, a= 0096
Bo12 7, aO 092
nBo=191, a= 085
SBo=248, a=0087
10
'10
I o
ICMF 2010, Tampa, FL, May 30 June 4, 2010
o B=28, a=0 1
*Bo=56, a=0088
l i ,. ..
a=016
a=O 10
* PDMS glass PDMS at a higher thickness
0,7
0,6
0,5
0.3
0,2
10 10 20 30 40 50 60 70 80 90
tau
Figure 9: Temporal evolution of film thickness for
PDMS with a dynamical viscosity equal to
102 Pas, and various Bond number.
o10
10 A
(n. .+ *
Svisk=943Pas, Bo=8 5, a=12
Svisk943Pa s, Bo2 3, a=0 38
visk=208Pa s, Bo6 6 a=0 21
o visk=208Pas, Bo=36, a=073
Svisk=208Pa s, Bo=1 7, a=0 57
Svisk=208Pas, Bo=86, a=0 13
+ visk=62Pa s, Bo=4 1, a=0 42
+ visk=62Pas, Bo=4 4 a=0 31
x visk=62Pa s. Bo=4 5 a=045
0 5 10 15 20 25 30 35 40 45 50
ttau
Figure 10: Temporal evolution of film thickness for
silicasodalime glass at three dynamical vis
cosity, and various Bond number.
number is not observed for the PDMS liquid. The depen
dence of the coefficient a on Bond number for PDMS
and molten glass is reported in Figure 11. Whereas a
is approximately constant and equal to 0.1 +/0.01 for
PDMS liquid, a decreases strongly with Bo and reaches
a constant value for Bo larger than 10 for molten glass.
5 Discussion
The experimental work of Kumar et al. (2002) shows,
that for thicker films, larger than 100 pm, the drainage is
slower for a curved film. The situation is contrary, when
the thickness of a film is smaller than 100 pm. They
explain the slow down of the drainage for curved films
from the Reynolds theory for immobile interface based
on the reduction of the driving force in case of curved
film. Moreover they explain the different drainage be
haviour for thin films by a thermodynamical effects that
start to play a role at a small thickness when effect of
micelles occurs.
10 15
Bo number
20 25 30
Figure 11: Dependence of thinning rate on Bo for
PDMS, glass and PDMS with thicker film.
As showed in Figure 11, the thinning rate is higher for
a smaller Bond number for a molten glass. As it is men
tioned in the second part of the article of Princen (1963),
the surface of the liquid film is more deformed when
the Bond number is large. Since the drainage occurs on
a large area, the drainage must be slower. Conversely,
when the Bond number is very small meaning that the
fluid interface stays approximatively flat and the bubble
spherical, the drainage occurs on a very tiny area leading
a fast drainage. It is important to remark also that for non
deformed interfaces, the hydrodynamic force for a bub
ble rising toward an interface can be determined exactly
in the Stokes's regime. Bart (1968) determined the exact
solution of the hydrodynamic interactions between an
inclusion and a flat plane. It is possible to show that the
hydrodynamic force diverges very slowly with the dis
tance between the bubble and the fluid interface. In fact,
as it is shown by Kim and Karrila (2005), the divergence
is only logarithmic with the thickness between the bub
ble and the fluid interface. Consequently, the integration
of the bubble motion gives a contact between the two in
terfaces in a finite time. Therefore, results obtained for
molten glass agree with the theory. Nevertheless, results
for a PDMS liquid behave otherwise. Thinning rate re
mains independent of the Bond number. This behaviour
could be explained by the length of chains of PDMS that
start to influence the drainage at a small thickness sim
ilarly to the one observed by Kumar et al. (2002), who
observed an influence of micelles at a thickness lower
than 100 pm. Figure 9 shows, that for a small Bo num
ber corresponding to a small curved surface, the thinning
rate for higher thickness is equal to 0.16 and 0.10 for
smaller thickness for a dynamical viscosity equal to 102
Pas. This phenomena is observed for a small Bo for
every viscosity tested in our experiment, but this effect
is more remarkable for the largest viscosity. The thick
ness at which the speed of drainage changes is approx
imately 4 or 5 pm depending on the viscosity. Remark
that a similar behaviour can be observed in the work of
Debr6geas et al. (1998). If the new thinning rates 0.15
and 0.16 obtained for dynamical viscosities 74 and 102
Pas respectively are reported in Figure 11, we obtain
two new points, that are plotted slightly above the aver
age trend of PDMS.
6 Conclusion
The drainage of a single liquid film obtained between a
bubble and a free surface was investigated for PDMS liq
uid and a molten glass. The film thickness is determined
thanks to an interference method.
The drainage observed for the two liquid is character
istic of a fully mobile interface leading to an exponen
tial decrease of the liquid film with a time. Thinning
rate of the film with molten glass decreases as a function
of Bond number. For small values of Bond number, the
drainage is faster due to a minor curvature of the surface.
The thinning rate remains independent of Bond num
ber for PDMS liquid, which can be explained by the in
fluence of long chains at small thickness. However it
respects the trend with Bond number at larger thickness.
Future work plans to create more experiments with
various glass compositions under various conditions and
compare experimental results with a mathematical sim
ulation.
Acknowledgments
We would like thank M. VignesAdler for the helpful
advices on the bubble experiment. We are also indebted
to P Baranger, L. Canova and PH. Guering working at
SaintGobain Recherche (Aubervilliers, France) for the
design of the optical setup.
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