Paper No 7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
Dynamic Interaction and Coalescence of MicroBubbles and Drops
Derek Y. C. Chan1, Ivan U. Vakarelski2, Grant B. Webber1, Rogerio Manica3,
Raymond R. Dagastine1, Steven L. Carnie1, Franz Grieser1 and Geoffrey W. Stevens1
1University of Melbourne, Particulate Fluids Processing Center, Parkville, VIC 3010, Australia
2Institute of Chemical and Engineering Sciences, 1 Pesek Road, Jurong Island, 627833, Singapore
3Institute of High Performance Computing, 1 Fusionopolis Way, 138632 Singapore
D. Chan@unimelb.edu. au
Keywords: bubble/drop coalescence, direct force measurements, dynamic interactions
Abstract
The time dependent force versus separation characteristics of controlled collisions in water between deformable bubbles,
deformable oil drops and solid silica particles in the 100 tm size range have been measured directly using the atomic force
microscope. Characteristic velocities of these collisions spanned the typical range of Brownian velocities. Viscosity effects
have also been investigated using sucrose solutions of varying concentrations. Detailed theoretical modeling of such systems
gave excellent agreement with experimental results and quantified the importance of the coupling between hydrodynamic
interactions, interfacial deformations and surface forces such as electrical double layer repulsion and van der WaalsLifshitz
attraction in determining whether collisions are stable or result in coalescence. The counterintuitive result of coalescence on
separation has been demonstrated with bubblebubble collisions.
Introduction
The interaction between bubbles, drops and particles in
multiphase systems plays a fundamental role in determining
the flow characteristics of the whole system. However, the
large range of characteristic length scales in such systems
poses significance challenges in interpreting experimental
results and formulating theoretical models. For instance
for bubbles, drops and particles in the 100 tm size range,
their collisions, which may lead to coalescence, will be
affected in part by surface forces (eg van der WaalsLifshitz,
electrical double layer and steric forces) that operate on the
scale of nanometers. Consequently it is important to be able
to describe accurately flow properties of the system that
drive such collisions with the same level of spatial
resolution. In addition, surface deformations that are of
the order of nanometers in bubbles and drops are also
coupled to the flow conditions and to the magnitude of the
surface forces. In addition, factors such as the nature of the
hydrodynamic boundary condition at the surface of
deformable bubbles and drops also contribute to their
interaction. Finally, it is important to appreciate the
dynamic nature of such interactions as both the time and
separation dependence of the forces between colliding
bubbles and drops ultimately determine whether collisions
are stable or lead to coalescence.
In this paper, we report results of direct measurements and
theoretical modeling of controlled dynamic collisions
involving deformable bubbles and drops and also solid
particles. The atomic force microscope is used to control the
collision protocol and measure the time dependent force.
Detailed modeling of the measured forces allowed us to
infer quantitative information about space and time
evolution of surface deformations during collisions and the
manner that coalescence can take place. Results are
obtained between two 100 tm size bubbles or oil drops in
water. Particleoil drop collisions in water and in sucrose
solutions are also studied to elucidate the effects of
viscosity of the continuous phase.
Nomenclature
h separation between drops or bubbles (m)
ho initial distance of closest approach (m)
p pressure (N m2)
r radial coordinate (m)
rmx outer limit of solution domain (m)
t time (s)
z axial coordinate (m)
F dynamic force (N)
K cantilever spring constant (N m1)
R bubble or drop radius (m)
S cantilever deflection (m)
V piezo actuator scan rate (m s 1)
X piezo displacement (m)
Greek letters
u dynamic viscosity (Pa s)
Paper No
a interfacial tension (N m ')
0 contact angle (radian)
H disjoining pressure (N m 1)
Subscripts
d drop
p particle
Experimental Method
Dynamic force measurements were made on an atomic force
microscope (AFM, Fig 1). A silica colloidal sphere or an oil
drop (decane or tetradecane) or a bubble of typical diameter
 100 tm, was placed on the tip of the cantilever whose
spring constant, K was measured by the HutterBechhoefer
(1993) method. Both Vshaped and rectangular cantilevers
have been used. Epoxy glue was used to attach the colloidal
particle while the cantilever could also be treated chemically
to make the tip hydrophobic so that a bubble or an oil drop
could be picked up and anchored to the tip. The attached
particle, drop or bubble interacts with either a drop or
bubble of comparable dimensions located on the substrate.
The interaction takes place in an aqueous solution with
added salt or added surfactant (sodium dodecyl sulfate,
SDS) to impart a negative charge to the air/water or
oil/water interface. Sucrose may also be added to increase
the viscosity of the aqueous solution.
+1   
smF1
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
velocity, dX(t)/dt was used in all data analysis and
modeling.
Time variations of the cantilever deflection, S(t) recorded
by monitoring the reflection of a laser beam focused on the
upper side of the cantilever tip can be converted to the
interaction dynamic force, F(t) = K S(t), using Hooke's Law.
However, at high scan rates and high viscosities, the
cantilever also deflected as a result of its motion. Such
deflections would be independent of any interaction
between the interacting bodies and so need to be subtracted
from the measured total cantilever deflection in order to
derive the dynamic interaction force.
The initial distance of closest approach, ho between the
interacting bodies can be set within the range 1 6 im, but
the design of the AFM did not allow ho to be set to a high
precision. However, detailed analysis based on our model
allowed ho to be determined to within 0.01 im.
The disposition of the particle, drops or bubbles was
checked by microscopic observation and relevant contact
angles can be measured. Other properties of the system such
as interfacial tension and viscosity can be measured
independently or taken from the literature.
Theoretical model
As the particle, bubble and drop dimensions are of order
100 im the interaction between say a particle and an oil
drop is only significant when their separation is of order
tens of nanometer and so the hydrodynamic problem can be
treated in the lubrication approximation. The experimental
set up has axial symmetry and can be described by
coordinates (r,z) so the time evolution of the separation,
h(r,t) between the interacting bodies can be described by the
StokesReynolds equation (Manica, Connor, Dagastine et al
2008)
dh 12r r 3
It 12yr dr ( 9 r
Figure 1: Schematic figure of the atomic force microscope
with a particle on the top of a cantilever with spring constant,
K and a drop on the substrate. Time variations of the
position, X(t) of the other end of the cantilever is controller
by a piezo actuator.
A collision run comprised of an approach branch as the
other end of the cantilever (Fig 1) was moved towards the
substrate by a piezoelectric actuator followed by a retract
branch when the cantilever was moved away from the
substrate. Careful alignment ensured 'headon' collisions
with axial symmetry. The cantilever position, X(t) (see Fig
1) of the cantilever was monitored by a linear variable
differential transformer. The nominal speed or scan rate of
the piezoelectric actuator was varied from less than 1 iim/s
up to 20 tim/s. This range spans the range of Brownian
velocities of particles and drops of this dimension. At a set
scan rate, the actuator velocity could vary during the
approach and retract runs, so values of the instantaneous
where p(r,t) is the hydrodynamic pressure in the continuous
phase (assumed to be Newtonian with viscosity p) between
the oil drop and the particle. This equation assumes the
noslip boundary condition holds at all interfaces.
The deformation of the oil drop during the course of the
dynamic interaction is described by the YoungLaplace
equation. For deformations that are small on the scale of the
drop or bubble size, the separation, h(r,t) between the
interacting particle and deformable drop (or bubble) has the
form
ad rh =(1 1 + )
 r = 2\, + (p +n)
r dr dr Rd R,
where (2o/Rd) is the Laplace pressure of the drop (or bubble).
On the other hand for interactions between two deformable
bubbles or drops, we have
Paper No
a i d dh\ 2a
 r (p+)
2r dr drr R
where a1 = (o1 +021)/2 and R1 = (RI1 + R21)/2 are defined
in terms of the Laplace pressures (2rl/R1) and (2a2/R2) of the
two drops or bubbles. Implicit in (2) is the assumption that
deformations take place under quasiequilibrium conditions
under a dynamic pressure (p + H).
The instantaneous interaction force, F(t) has contributions from
hydrodynamics (p) and disjoining (H) pressures
F(t) = 2x o r [p(r,t) + H(r,t)] dr. (3)
Axial symmetry considerations require the spatial derivative
of the pressure and the separation to be zero at r = 0: 8p/lr =
0 = ch/cr. In the lubrication limit, the initial separation
between the particle and the undeformed drop (or bubble)
has the parabolic form
h(r,t = 0) = h, + r2/Ro (4a)
where Ro1 = (Rp1 + Rd1), and the initial separation
between two deformable drops (or bubbles) is
h(r,t = 0) = ha + r/R (4b)
where R1 = (RI +R2 1)/2.
Far from the central axis, where r * o, the pressure
vanishes as p 1/r4 and this condition is implemented as:
r(8p/cr) + 4p = 0. The three phase contact line of the drop
(or bubble) on the substrate and on the cantilever is assumed
to be pinned and changes in contact angle during interaction
were accounted for using a constant volume constraint for
the drop (or bubble). This gives the following boundary
condition at the outer limit of the solution domain r = rmax
for the particledrop (or bubble) problem
Oh dX l(dF 1 dF\ 5a)
+   B(O) + log (5a)X
dt dt K dt) 21o dt 2Ra
where
F 1 (1+cos8 o
2 \1cosO)]
and 0 is the equilibrium contact angle of the drop (or
bubble) on the substrate. For the case of two drops (or
bubbles), the boundary condition at r = rmax is
h_ dX 1 (dF\
II
9t dt K\ dt
\ dF r \ b
 1 dF) B() + B(O,) + 210og rmax (5b)
2jrcx dt ] 2 R1R2]
and 01 and 02 are the equilibrium contact angles of the drops
(or bubbles).
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
We now have all the necessary initial and boundary
conditions to calculate the force as a function of time as the
collision is controlled by the displacement function X(t) of
the piezo actuator. Equations (1) (5) can be solved by the
method of lines as a set of algebraic differential equations
(Carnie et al 2005). This solution will also give the
separation h(r,t) from which we can see how the
deformation of the drop and bubble evolve as interaction
progresses.
Results and Discussion
Dropdrop collisions
Measurements of collision experiments between two decane
drops were carried out on the AFM in an aqueous solution
with 1 mM NaNO3. The solution also contained 3 mM of
the surfactant sodium dodecyl sulfate (SDS). The added
SDS lowered the oilwater interfacial tension to 17 mN/m
and imparted a negative surface charge to the oil drops that
would prevent coalescence. The surface potential on the oil
drops was 100 mV.
The collisions were driven by programming the piezo
actuator to move the end of the cantilever (Fig 1) with the
following schedule:
X(t) =X Vt,
0 < t < /2 total
Xo + V (t total), 2 total < t < total
where the drops were driven together ("approach") and then
separated ("retract") at a constant scan rate, V(see Fig 1).
A comparison of the measured dynamic force between two
decane drops (radius 50 im) with predictions of the model
outlined in eq (1) (5) is given in Fig 2a for scan rates of V
= 2, 9.3 and 28 im/s (Dagastine et al 2006). For clarity
only about 1% of the acquired experimental data points are
shown. Experimental uncertainties of the measurements are
about the size of the data points.
0 0.2 0.4 0.6 0.8 1
t/ttotal
Figure 2a: Comparison of experimental and predicted
dynamic force, F(t) during collision between two decane
drops, radius = 50 pm in water with 1 mM NaNO3 and 3
mM SDS. The scan rates are 2, 9.3 and 28 tm/s.
Paper No
The repulsion between the drops on approach and the depth
of the attractive minimum on retract both increase with
increasing scan rate. This is consistent with the fact that
these effects are of hydrodynamic origin. Since the decane
drops can also deform, the increase in the magnitude of the
forces with scan rate is not accompanied by corresponding
changes in the separation between the drops during the
collision.
In Fig 2b, we show the predicted values of the distance
between the surfaces of the colliding drops at the axis of
symmetry, h(0,t) corresponding to the different scan rates in
Fig 2a. For all scan rates, the two decane drops did not
approach to closer than about 35 nm, the separation at which
the electrostatic repulsive pressure between the charged
drop became equal to the Laplace pressure of the drops. At
this points the drops will flatten instead of approaching
closer to each other. At higher scan rates, the hydrodynamic
repulsion on approach is stronger so the drops remain
further apart during the approach phase corresponding to t <
1/2 total.
150
100
E 28 pm/s
2 pm/s
01
0.4 0.45 0.5 0.55 0.6 0.65
tttotal
Figure 2b: Predicted values of the distance between the
surfaces of the colliding drops at the axis of symmetry,
h(0,t) corresponding to the results in Fig 2a at scan rates of 2,
9.3 and 28 tm/s.
Particledrop collisions
Using a sucrose solution as the continuous phase, its
viscosity can be changed by up to 50fold while it still
behaves as a Newtonian liquid. We study the dynamic
interaction between a silica particle and a tetradecane drop
in sucrose solutions of up to 40% with 5 mM SDS added to
ensure the particle and drop do not coalesce (Dagastine,
Webber at al 2010).
Variations of the dynamic force, F at a low scan rate of 1
tim/s with relative piezo displacement, AX at different
sucrose concentrations are shown in Fig 3. With increasing
sucrose concentration up to 40% where the viscosity is 6
times that of pure water, the effect on the force is evident in
the increase in the size of hysteresis between the approach
and retract branches of the force curve (Fig 3).
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
At high forces, above 2 nN, where the force for the
approach and retract branches overlap, the relation between
F and AX is given by the approximate formula
SF FRo 1+cosO
AX log + log + 1 .
4Tx [ 8aR4) 1cos 8
Note that the relative piezo displacement AX is arbitrary up
to an additive constant.
At a higher scan rate of 20 tim/s, the attractive force
minimum on retraction is significantly deeper for the 40%
sucrose solution with a viscosity 6 times that of pure water,
than the 20% solution with a viscosity twice that of pure
water (Fig 4).
8 "', 1 pmls
Silica Particle
20%0 +
6 i Tetradecane Drop
0o~s^ N0% Sucrose
" 
0 1  
0.4 0.2 0 0.2
AX (plm)
Figure 3: Comparison of experimental and predicted
dynamic force, F as a function of piezo displacement X
during collision between a silica particle (radius 25 tm) and
a tetradecane drop (radius 107 tm) at a scan rate of 1 tm/s
in different concentration sucrose solutions. Results for
different sucrose concentrations have been displaced
vertically for clarity. The dashed lines are the high force
formula given by eq (7).
w Oi
,0
U
0.04 0.08 0.12
Time (s)
0.16 0.2
Figure 4: Comparison of experimental and predicted
dynamic force, F(t) during collision between a silica particle
(radius 25 itm) and a tetradecane drop (radius 107 tm) at a
scan rate of 20 tm/s in different concentration sucrose
solutions.
Sucrose
20 2m/s /o 40%
S20%
\ 0%
Silica Particle
+ Dr
Tetradecane Drop
Paper No
Bubblebubble collisions
Microbubbles in the 100 im size range were generated by
sonication (Vakarelski et al 2008) and picked up on the tip
of the cantilever. As bubbles carried a negative surface
charge, we worked at a high salt concentration (0.5 M
NaNO3) so that electric double layer repulsion between the
bubbles was screened out. Only attractive van der
WaalsLifshitz forces operated between the bubbles. By
driving the piezo actuator at a constant scan size of about 2
im at 50 im/s, the bubbles were brought together and then
separated (Vakarelski et al 2010).
At an initial distance of closest approach, ho of 2.45 im, the
bubbles came together with a repulsive interaction during
approach and reached a force maximum at the start of the
retraction phase. This collision was stable as the bubbles
separated after the attractive minimum (curve ABCD in Fig
5a). The separation, h(r,t) between bubbles during the
retraction phase is shown in Fig 5b. At the attraction
minimum (point D in Fig 5a), the bubbles were over 200 nm
apart (curve D in Fig 5c). This separation is well beyond the
range of the attractive van der WaalsLifshitz force so
hydrodynamic interaction was responsible for this attraction
on separation.
With the same bubble pair and identical collision conditions
but starting at a smaller initial distance of closest approach,
ho = 2.05 im the repulsive force maximum was higher as
the bubbles were driven closed together and deformed more
(Fig 5a, point E). Upon retraction, the force became
attractive (curve EFGH in Fig 5a) and the bubble coalesced
at point H (Fig 5a). The separation between bubbles during
the retraction phase is shown in Fig 5c. Here we see that the
hydrodynamic attraction on separating the bubbles caused
the bubbles to deform into a dimple shape (curve H, Fig 5c)
and brought the surfaces of the bubbles to within 5 nm
when the attractive van der WaalsLifshitz force was
sufficiently strong to cause the bubbles to coalesce at the
rim of the hydrodynamic dimples.
Thus the attractive hydrodynamic interaction and bubble
deformation combined to bring the bubble close enough
together for the attractive van der WaalsLifshitz force to
cause coalescence. A similar coalescence on separation
phenomenon has also been observed, though not to the same
level of details, between proximal water drops in
hexadecane in microfluidic cells (Bremond et al 2008).
Bubble coalescence can also occur when the bubbles are
continually pushed together or when they are push together
for a time period and then stopped. A comparison of the
measured and predicted coalescence times for these
different modes of coalescence obtained from 30
experiments are shown in Fig 6. Note the close agreement
between experimental measurements and theoretical
predictions. The exact magnitude of the coalescence time
depends of course on the initial separation and the scan size.
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
20
E
15 Two Bubbles
Retract
(Radius 74 pm) Retract
I F
05
S Approach d n ho = 2.05 pm
U
0 B G
ho = 2.45 pm
5 C H
D
0 0.02 0.04 0.06
Time (s)
Figure 5a: Comparison of experimental and predicted
dynamic force, F(t) during approach and then retract
between two 74 im radius bubbles at a scan rate of 50 im/s.
At initial separation h. = 2.45 pm, the collision is stable
(ABCD), but at h = 2.05 im coalescence occurred when the
bubbles are separating (EFGH) and coalesced at point H
(down arrow).
250
\ Two Bubbles
_200
E
150 C
B
100
2 1 0 r (m) 1 2
Figure 5b: Calculated separation, h(r,t) between two
bubbles during the stable collision with curves A, B, C, D
corresponding to time points in the force curve in part (a).
c 30
" A
r(gm)
Figure 5c: Calculated separation h(r,t) between two bubbles
during the coalescing collision with curves E, F, G, H
corresponding to time points in the force curve in part (a).
Paper No
120
100
E 80
on
2on
20 40 60 80 100
Experiment t (ms)
Figure 6: Comparison of experimental and theoretical
coalescence times for two colliding bubbles under different
mode of collision: coalescenceonseparation (A triangles),
coalescenceonapproach (* diamonds) and coalescence on
stopping of the piezo actuator after approach (* circles).
Conclusions
We have presented results of experimental measurements
and theoretical modelling of controlled dynamic collision
studies using the atomic force microscope. Using a
combination of silica particles, oil drops and bubbles in the
size range 100lim we have demonstrated how
hydrodynamic forces and surface deformations combine to
produce the characteristic forms of the dynamic interaction.
In all drops and bubbles (Manor et al 2008) system we
considered, the results are consistent with the noslip or
immobile hydrodynamic boundary condition.
In cases where the collisions result in coalescence, our
model is able to predict the coalescence time accurately.
Both experiment and theory demonstrate that because of
deformations, there are a number of ways in which
coalescence can initiated: by continually pushing the drops
together, by pushing the drops together and then stop and
wait for the drops to coalesce or by separating the drops
after an initial approach (Chan et al 2009).
The present theoretical model has been applied with
quantitative success to other experiments such as the
drainage of thin films in nonaqueous systems (Klaseboer et
al 2000, Manica, Klaseboer et al 2008) and the response of
aqueous thin films between a mica plate and a deformable
mercury drop to electrical and mechanical perturbations
(Connor & Horn 2003, Clasholm et al 2005), Manica et al
2007, Manica, Connor, Clasholm et al 2008).
Acknowledgements
Financial support from the Australian Research Council is
gratefully acknowledged.
Two Bubbles *
S4 *
A
A
A
A Coalescence Times
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
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