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Study of optothermal Marangoni effect using laser light — 1/29
Half-year report of scientific stage at Wroclaw University of Science
Study of the optothermal Marangoni effect using
C L E´ MENT Q UINTARD
S UPERVISOR : A NDRZEJ M INIEWICZ
1.1 Experimental set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2 Theory of the Marangoni effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1 Interface bending due to Marangoni effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Bubble trapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Crystallization inside a gas bubble . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4 Droplets trapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1 Numerical simulations of the Marangoni flows along a flat air / liquid interface . . . . . . . . . . . . . . .
3.2 Numerical simulations of the Marangoni flows around cylindrical gas bubbles . . . . . . . . . . . . . . .
3.3 Numerical simulations of the crystallization inside a gas bubble . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4 Numerical simulations of the droplets’ trapping phenomenon . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1 Several heating sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Several gas bubbles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 3D Comsol simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4 System with random geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5 Experimental work on the bending . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.6 Complete description of COMSOL simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.7 Liquid Crystal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Study of optothermal Marangoni effect using laser light — 3/29
I wish to extend my heartfelt thanks to all the members of the lab in which I was very well received. It was for me of
a great interest to work on this subject and I will continue my studies in the same field. Thanks to both the very
pleasant atmosphere at work and the scientific content, it seems to me that a very short time has elapsed very quickly.
I would like to thank particularly Prof. Andrzej Miniewicz who proposed me a fascinating topic on which I
enjoyed to work. He allowed me great freedom, however he was always here to answer my questions, to help me in
many different ways and to guide me very efficiently thanks to his scientific background. I also would like to thank
Prof. Stanislaw Bartkiewicz who helped me a lot on the experimental side.
Wroclaw is a wonderful city made of gorgeous neighborhoods, exquisite food and many other advantages. From
the frozen winter to the suffocating summer, from the beer festival to the local vodka, Wroclaw is a city you can not
A few decades ago, Ashkin et al.
discovered a method of optical trapping of dielectric particles by focused laser
beam, leading to tremendous progress in the field of light-matter interaction 5-15 . The method of optical tweezers
allowed to trap, translate and rotate not only dielectric particles but also nanoparticles of various materials including
semiconductors, metals and molecular crystals. Multiple optical tweezers found application in three dimensional
manipulation including biomicromanipulation of living cells. This method relies on the so called gradient force
attracting particles toward the beam focal point if the particle has the higher refractive index than surrounding
medium. Such a method did not allow the trapping of gas bubbles in liquid media. Later on, Young et al.16 have
introduced the mechanism of trapping and moving gas bubbles caused by the stresses resulting from the thermal
variation of the surface tension at the bubble surface. These discoveries naturally led to the study of moving and
trapping gas bubbles in a liquid environment17-23 using a gradient of temperature induced by absorption of laser
light. This method is complementary to the traditional laser trapping one because due to light absorption and heat
release it provides much larger forces influencing adjacent objects, moreover the target does not need to fit a special
geometry. This method is less invasive than the optical tweezing method where a high laser power is needed for
trapping what can cause a damage to a studied object.24
Our project is well placed in the context of recent research in the fields of microfluidics and optofluidics.
Scientists are exploring methods of trapping and manipulating bubbles or other particles assembling close or inside
the bubbles. The heat produced by laser light absorption is inducing a Marangoni effect at a liquid-gas interface25-34 .
The Marangoni effect is defined as the flux of liquid at the liquid-gas interface due to surface tension gradient. The
liquid flows from lower surface tension regions toward higher surface tension ones. In our case this phenomenon
may also be called opto-thermo-capillary convection because the surface tension gradient is caused by a temperature
gradient. As a heating source we are using absorption of laser light. The Marangoni effect was observed in the three
dimensional systems, however, we limited its observation to the quite unique 2 dimensional system in which the
Study of optothermal Marangoni effect using laser light — 4/29
liquid layer is confined between closely spaced glass plates. Using optical microscopes and small soot particles we
were able to observe and record the Marangoni laser-induced liquid flows close to the liquid-air interface and around
gas bubbles. Later on, we also investigated a system made of a thin layer of liquid on a glass plate but with its upper
surface left free, in the aim of trapping a droplet of concentrated para nitroaniline.
The aims of this internship include experimental demonstrations of some phenomena caused by the Marangoni
effect in these new configurations. Optical manipulation techniques of small objects are of great interest for a
wide variety of potential applications. For the case of optical trapping of gas bubbles in liquid, however, the
experimental configuration of conventional optical tweezers (e.g. gradient force) can not be applied because of the
lower refractive index of the bubble compared with that of the liquid. To overcome this situation, we employed
the laser-heating-induced shear stress at liquid-gas interface (i.e. Marangoni effect) and demonstrated the trapping
of individual gas bubbles and liquid droplets. The detail mechanism underlying the motion of bubbles was at the
moment of undertaking our studies not well interpreted. In the present report, we explain the mechanism of the
bubble trapping on the basis of detailed numerical simulations. The numerical simulations were performed using the
finite-element method commercially implemented in COMSOL Multiphysics. It was concluded that not only the
fluid velocity field around the bubble, but also the pressure difference along the temperature gradient is responsible
for the bubble migration. The findings are relevant for many applications.
Let’s note that some theoretical work was already done by Young, Goldstein and Block35 and then by Hadland
and Subramanian36 via a procedure of matched asymptotic expansions but basing on several assumptions and giving
only partial answers. Indeed, the complexity of the Navier-Stokes equations describing fluid motion and the heat
diffusion equations lead naturally to a numerical simulation : this is the choice we have made.
1.1 Experimental set-up
We built up our experiment using an inverted optical microscope Olympus fed by cw laser diode with sample
positioned horizontally on manually and piezoelectrically driven stage (Figure 1.a)). A typical sample consists of
two glass plates separated with suitable spacers. Between these glass plates we put a solution of a dye able to absorb
the laser light. Such a system can be considered as 2 dimensional provided that the liquid layer thickness is less
than 100 µm and forming a layer extending within the area of about 10 by 10 mm. Thus the gravity does not play
any significant role in our studies (Appendix 4.3). The liquid phase is made of a dye (para nitroaniline (hereafter
pNA) or dicyano nitro pyrazoline (DCNP)) dissolved in an organic solvent (1,4-dioxane) to which we added small
amount of carbon micro-particles (soot) enabling observations of the fluid movements. The use of soot particles
with wide distribution of sizes and masses (from 1µm to 50 µm) allows to trace the fluid trajectories of different
speeds. The free surface where the Marangoni effect is developing is the one being in contact with air. It can also be
at the surface of a gas bubble in cases in which we introduce a gas bubble in our system in order to study its trapping
by laser light (Figure 1.b). Laser of wavelength 471 nm and several mW power is irradiating the liquid at normal
incidence (along the z-axis) via objective x10 or x40. Due to light absorption the liquid is locally heated, allowing
the Marangoni effect to develop along the free surface when the laser spot is close enough to the interface.
We observe the phenomena of fluid motion through a microscope which is equipped with a Retiga 2000 camera
or alternatively with a fast camera (i.e. a camera allowing to record movies with the speed of 1200 frames per
second) for quantitative measurements of the flows velocities.
Study of optothermal Marangoni effect using laser light — 5/29
Figure 1. a) Scheme of the experimental set-up based on inverted optical microscope for measurements of
laser-induced Marangoni flows in light absorbing liquid layers. b) Top view of the investigated samples with the
bubble and with the flat free interface. The Marangoni flows are shown schematically by arrows, c) Schematic view
of an experimental set-up for gas bubble trapping. Focused laser light attracts the cylindrical gas bubble positioned
at a large distance from the beam spot centre as compared with the beam waist.
1.2 Theory of the Marangoni effect
The heat released from the beam center positioned at (xc , yc ) spreads centrally by conduction heat flux q(r) and
forms the distribution of a temperature according to Fourier’s law:
∂ T (r)
When the temperature field T (x, y) arrives at the liquid-air interface situated at some distance along the x-axis
from the point (xc , yc ) a gradient of temperature ∇T (x, y) appears at the interface line. The time for onset of
temperature non-uniformity is Lα = 0.1s (for L = 0.1mm and α = 8.94.10−8 m2 /s being the thermal diffusivity of
The fluid fluxes are governed by the Navier-Stokes equation coupled with the continuity equation (more details
about the equations used are given in Appendix 4.6) :
ρ ∂~u + ρ(~u.~∇)~u = ~∇.(−p~I + µ(∇u + (∇u)T ))
ρ ∇.~u = 0
The temperature distribution at the interface will change the surface tension. For most of the organic liquids, surface
tension is lowering linearly with temperature increase according to the equation :
q(r) = −k
σ (T ) = σ (Ta ) + b(T − Ta )
where σ (Ta ) is the solvent surface tension value at ambient temperature Ta = 295K and b = ∂∂ σT is the temperature
derivative of the surface tension (b < 0).
This variation of the surface tension along the free surface develops a force with a component parallel to the
surface and proportional to the surface tension gradient. We can generalize it by a boundary condition37 :
ρν~n∇u = b∇k T
where ν = is the kinematic viscosity (and µ the dynamic viscosity), ~n is the vector normal to the surface and ∇k is
the tangential derivative operator. This tangential force is responsible for the liquid flux from the low surface tension
regions (that means the hotter regions) to the high surface tension regions (that means to the colder ones).
Study of optothermal Marangoni effect using laser light — 6/29
We experimentally put on evidence the Marangoni flows in our 2 dimensional system employing our fast camera
which is able to record slow-motion movies with the frame equal to 0.83 ms (Figure 2.a)). Knowing frame time and
distances travelled by particles we were able to calculate the flux speed at any point along the free surface caused by
the Marangoni effect. The results are shown in Figure 2.b).
Figure 2. a) An example of visualization of Marangoni surface flow induced by laser light absorbed by liquid
solution containing dissolved dye. The trajectory of soot particle approaching surface and then accelerated by
Marangoni flow along the liquid-air interface line. The figure has been prepared from the movie of the event in such
a way that each frame lasting 0.83 ms has been labelled with different color so the particle velocity at each point
could be evaluated. The average velocity uav at the place of an arrow along the interface amounts to 0.06 m/s. b)
Quantitative data calculated from the movie presented on the left (a).
The particle is arriving from the laser spot position to the interface line. It has the possibility to go to the left or
to the right side equally. The soot particle, of approximate size of 40 µm, moving toward the right is accelerated by
the Marangoni force, reaches its maximum speed umax = 0.09 m/s at around 250 µm from the starting point and then
slows down because of the decrease of the force given by F = ∂∂ σT ∇T and the friction forces. Knowing this speed we
can calculate the Reynolds number Re = ρLu
µ = 40 which qualitatively captures the characteristics of the flow regime.
This low Reynolds number tells us that the viscous forces dominate in the fluid. The particle speed obtained from the
analysis of its movement does not give the maximum speed of the fluid itself, because the particle movement speed
depends on its weight and shape. Smaller particles will move even faster under the same experimental conditions.
2.1 Interface bending due to Marangoni effect
If the layer of liquid is thick enough to effectively absorb the laser light and thin enough to allow the liquid-gas
interface to bend (the resistance force of thinner layer is weaker than for the thicker one : this statement is confirmed
by the experiment (Appendix 4.5)), we observe a bending of the free interface toward the laser beam position. After
launching intense laser illumination placed at a distance of 300 to 500 µm from the interface it begins to bend
slowly and next abruptly jumps toward the beam center and then it is locked at this place (xc , yc ) as shown in the
movie SIM11 and in the Figure 3.
Once the surface reaches the laser spot, the dynamic process of the bending stops and very stable whirls are
forming. The example of a stable whirl in pNA-dioxane solution excited by a Gaussian laser beam was photographed
by us and composed from many colored frames (using a method of images subtraction on I MAGE J) and presented in
this report, you can click on the underlined mention ”movie SIM” to see the suitable film uploaded on YouTube.
Study of optothermal Marangoni effect using laser light — 7/29
Figure 3. 2D Marangoni effect. The composed microscopic dark field photograph of stable laser-induced whirl in
pNA-1,4-dioxane solution in a thin layer. Multicolor lines represent the movement of particles in a liquid flux
captured by 100 consecutive frames, each lasting 40 ms. The lateral size of the image is about 4 mm. Vertical arrow
shows interface bending amplitude due to local surface tension decrease caused by temperature distribution. Curved
arrow shows the direction of particle movement. In this case the position of laser spot is at the maximum of interface
We call the whole described above phenomenon a light-induced two-dimensional Marangoni whirl formation to
underline a difference from the 3D Marangoni effect, e.g. the one resulting in formation of B´enard cells when the
bottom of a container with liquid is uniformly heated.
2.2 Bubble trapping
For relatively thin dioxane-pNA layers (below 100 µm) it is possible to drag the free border by moving the laser spot
in order to elongate it enormously and finally form a gas bubble. The flows surrounding the bubble are responsible
for the effect of bubble trapping by light (Figure 4.a)). The streams are going from the lower surface tension regions
(i.e. higher temperature, close to the laser spot) toward the higher surface tension regions (i.e. lower temperature, far
from the laser spot). We are able to transport the gas bubble from one to another interface through the movement of
the laser spot position, what is illustrated in Figure 4.b) (see also movie SIM2).
Figure 4. a) The photograph of particles’ trajectories around a gas bubble in the close proximity to laser
beam-induced optothermal Marangoni effect. Arrows indicate the individual particle’s path in the loop around the
bubble which is shown here by a white dashed-line circle. b) Scheme showing the nucleation (left side) and
transport (right side) of a cylindrical gas bubble by shifting the laser beam across the layer.
An example of long distance bubble trapping is shown in Figure 5 as a sequence of 5 photographs separated
by 40 ms each. The position of the laser spot is given by the dashed yellow line. The gas bubble is seen as a black
Study of optothermal Marangoni effect using laser light — 8/29
torus due to the light reflection on the gas / liquid interface. After some incubation time necessary to produce a local
temperature increase at the surface of the bubble, the Marangoni fluxes are able to develop in the system. The Figure
5 only shows the last frames but because of the incubation time, from the laser beam opening to the trapping of
the bubble, the whole process takes about 1 second. This incubation time depends on the laser beam intensity, the
thickness of the liquid layer, the absorbance, the effective heat conduction coefficient but also on the bubble size and
its distance from the laser spot. Indeed, if the heat conduction is relatively fast (i.e. maximum several tens of ms), it
is not enough that the heat arrives at the surface. The Marangoni forces have to exceed a certain threshold to actually
move the bubble. Basing from this sequence of photographs, the bubble moves from approximately 440 µm in 80
ms, which gives us an average speed uav = 5.10−3 m/s. To observe this movement more precisely the fast camera is
Figure 5. An example of bubble trapping phenomenon photographed with the frame rate of 40 ms. After launching
the laser beam at time t = 0 initially bubble stays at its initial position. When a heat via conduction arrives from the
laser beam position to the bubble surface then an attractive force appears (represented by an orange arrow) which
accelerates the bubble toward laser spot. Bubble movement in its final stage is so fast that its shape is heavily
Initially small force is growing in time and when it overcomes the bubble inertia the movement begins. The
bubble movement is hampered by the Stokes’ drag force Fd . The drag force in classical approach depends on the
velocity of the liquid flow around the particle but also on its shape. We used for estimation of the expected magnitude
of drag force the formula Fd = 4π µRb u from the literature 44 . Measuring Rb = 50µm and using u = 5.10−3 m/s
and µ = 0.787.10−3 Pa.s (dynamic viscosity of 1,4-dioxane at 323K) we calculated Fd = 3.10−9 N. We have to
keep in mind that this is a crude estimation and that the reality can be described with much more details, including
correction factors due to the dependence on temperature of the different parameters 45 . Nevertheless, the Marangoni
effect induced force, denoted as FM , must be much stronger than the drag force (FM >> Fd ) as evidently the bubble
accelerates when approaching the laser beam position. The Marangoni force FM is large even in the case when the
beam is situated at the bubble liquid / gas interface. Indeed as explained just before, by shifting the laser spot, the
bubble follows it. Thus this force also defines the bubble trapping strength. It is worth of notice that the bubble
speed when it approaches the laser beam position is so fast that a visible deformation of its spherical or cylindrical
shape is clearly observed (cf. fourth frame of Figure 5). For a spherical bubble of radius Rb in a static liquid,
the surface tension σ allows for a jump in pressure across the bubble’s interface, given by the Laplace pressure
pi − po = 2σ
Rb , where pi and po are the pressures inside and outside the bubble. The bubble deformation can be
caused by the action of an important pressure inside the fluid located close to its surface and possibly the temperature
gradient around the bubble reducing the surface tension. These predictions will be verified by the simulations. No
Study of optothermal Marangoni effect using laser light — 9/29
simple physical formula can be given for the observed trapping event as it is a result of few coupled and complex
phenomena governed by the Navier-Stokes equations. Indeed we can find in the literature very simple estimations of
the fluid particles’ velocity resulting from the Marangoni effect such as u = ∂∂ σT ∆T
µ predicting a speed proportional
to the gradient of temperature . Applying a gradient of temperature of ∆T = 20 K as it is the case in some of
our experiments, one should obtain u = 2 m/s which does not fit at all our observations (by more than an order
of magnitude). It is also evident for us that the speed is not proportional to the gradient of temperature applied.
Therefore to explain and model the phenomenon we decided to perform numerical simulations of the Marangoni
effect for the system studied.
2.3 Crystallization inside a gas bubble
We were able to make a crystallization of pNA inside the gas bubble. This is spectacularly shown in our movie SIM3
from which we extracted three frames shown in Figure 6. We used the same system as before, meaning a liquid
layer (pNA dissolved in 1,4-dioxane) placed between two glass plates. The formation of the bubble is due to the
laser illumination, indeed by using a high power we are able to reach the boiling point of dioxane and thus to form a
gas bubble (it is shown at the end of the movie that by switching off and on the laser, the old gas bubble disappears
and a new one is created). The Marangoni flows are developing around the bubble as shown in section 2.2, moving
the fluid’s particles toward the laser spot with a velocity high enough to allow them to penetrate inside the bubble.
This crystal growth was observed between crossed polarizers, this is why the crystal appears so bright comparing
to the contiguous liquid, due to its birefringence. The Marangoni flows are maintaining the bubble, this is why
by switching off the laser, the gas bubble immediately disappears and thus the crystal is dissolved in the liquid
medium. However, we predicted that the Marangoni flows are also responsible for the crystal growth inside the
bubble, allowing some fluid particles to penetrate the bubble. This is of a great interest given that many people are
studying the crystal growth by laser trapping 47 .
Figure 6. The Marangoni flows allow particles of pNA to penetrate the gas bubble walls and thus forming an
oversaturated solution from which the crystal can grow. Here is shown the crystal growth 36s, 44s and 50s after
launching the laser.
2.4 Droplets trapping
Not only we were able to form and trap gas bubbles in a liquid medium but also to form and trap liquid droplets
(high concentrated in pNA) in the same liquid medium. Let’s note that this part of the work is still in progress, and
that we observed this phenomenon in certain conditions of concentration and laser intensity not well known yet.
The system is still made of a layer of 1,4-dioxane and para-nitroaniline (pNA), however this liquid is now placed on
a glass plate but not covered by another one (so the upper surface of the liquid is let free). Irradiating light on this
liquid one can observe that a brown coloration appears around the laser spot (movie SIM4 and Figure 7), meaning a
higher concentration of pNA in this area (the yellow / brown color is very characteristic of pNA). Considering that
this phenomenon is going against entropy, obviously when the laser is shut off we observe the homogenization of
Study of optothermal Marangoni effect using laser light — 10/29
Lastly, we can see in movie that we can trap this droplet and by shifting the position of the laser spot, the droplet of
pNA follows the laser spot.
Figure 7. An example of droplet trapping phenomenon. After launching the laser beam the droplet of pNA is
forming around the laser spot. When we shift the laser beam position, the droplet of pNA follows it.
Because the upper surface is left free being in contact with the air, it is also important to look at the phenomenon
from the side. We can see in movie SIM5 (and in Figure 8, photographs from this movie) the evolution of the air /
liquid interface after launching the laser. After 5 seconds of illumination, the bending of the interface is clear, and
few additional seconds are needed to observe the formation of a droplet with a high concentration of pNA at the
position of the laser beam.
Figure 8. Three frames extracted from a movie showing the air / liquid interface bending after starting laser
illumination. Few seconds after we launch the laser we can see the formation of a droplet of pNA growing at the
laser beam position.
Study of optothermal Marangoni effect using laser light — 11/29
Basing on our experimental results that provide us the precise trajectories of the liquid flows, we modeled our system
employing the simulation software COMSOL M ULTIPHYSICS. Using finite element analysis to solve coupled
differential equations, we can check if the model fits to our observations and we can then explain the physics of
different phenomena looking at the inaccessible by experiment parameters like temperature distribution, pressure
and velocity fields.
The equations used in the Laminar Flow module are the classic equations of Navier-Stokes (equations (2) and (3)
already presented). Let’s note that the study of our system supposing a laminar flow is justified by the low Reynolds
number associated. Indeed, the maximum velocity amplitude obtained in our experiments is about 0.1 m/s, leading
to a Reynolds number of Re = ρuL
µ = 90. The equation governing the Heat transfers in fluids module is :
+ ρCp~u.~∇T = ~∇.(k~∇T ) + Q
The details of all the equations and the reason why these ones are used in COMSOL are given in Appendix 4.6.
We assume that the light absorption does not change with time and temperature and we do not take into account
the possible evaporation as well as we neglect the heat conduction by the surrounding liquid glass plates. In
calculations we simply introduced a heat source as a circular object to which we have assigned an excess temperature
up to ∆T = 40 K at most. This value is estimated from our experiments ; indeed, playing with the intensity of the
laser beam, we can reach the boiling temperature of the liquid or the melting point of the crystal, leading to some
vapor that we can see looking at the sample. Know the value of these boiling and melting point we know more or less
what gradient of temperature we are imposing. All these assumptions together with the approach of 2D modeling of
the system have simplified the complexity of a real system and allowed us to plot graphical representations of all
crucial parameters within the (x, y) plane.
The walls are defined as no-slip walls, a constraint still not well understood but which describes the reality much
better in most of situations 41-42 . However, the free interface has to be defined as a slip wall to allow the Marangoni
fluxes to develop ; indeed the no-slip wall condition enforces the fluid velocity to tend to zero when approaching the
border, which is conflicting with the Marangoni force defined as it is in COMSOL. These Marangoni forces are
introduced as a boundary condition (weak contribution in COMSOL) in the Laminar flow module given by the
equation (5). The equation (5) sets the conditions for liquid flux movements along the interface and transmitting
their translational movement to the adjacent liquid layer : it is a viscous traction. The other boundary conditions are
that the wall far from the heating source is set to the room temperature Troom and that two pressure constraints points
(at the corners far from the heating source) are put at atmospheric pressure in order to have a solvable mathematical
problem43 (cf Appendix 4.6). Let’s note that by making simulations on random geometries we checked that the walls
far from the heat source do not significantly change the physics observed (Appendix 4.4). Also we paid attention to
the fact that ∆T + Troom < Tboiling During all the numerical simulations we paid attention to reproduce the conditions
of the experiment as well as possible such as the distance between the laser spot and the free interface, the gradient
of temperature, the size of the laser spot and so on.
Study of optothermal Marangoni effect using laser light — 12/29
The velocity field obtained by numerical simulation is of the same order of magnitude than what we observed in
reality (up to few mm/s). In Figure 9 we present the flow of fluid induced by a laser spot situated not far from the
free surface. Liquid fluxes are arriving at the center of the system, going to the free surface and being redirected to
one or another side at high velocities, forming whirls very stable in time.
Figure 9. Pictures taken from a COMSOL simulation 3 seconds after starting the laser illumination. The interface
liquid / air is the flat border on the left. The temperature field (left) shows the transfer of heat by convection caused
by the liquid flows. Flux velocity field is also shown (right). The qualitative accuracy of the numerical model is
demonstrated by these images highlighting the Marangoni whirls induced by a gradient of surface tension of the
liquid / air interface.
The highest velocities are located at two areas symmetrical to the median line of the system, as observed
in our experiments. This can be easily understood by the fact that the Marangoni force is proportional to the
temperature gradient. Given that the laser spot has a circular geometry, the heating of the free surface is Gaussian in
its form. Making the derivative of this Gaussian distribution of temperature at the surface leads us to two high peaks,
corresponding to the two areas where the force is maximum.
In order to prove the accuracy of this numerical model, we also present quantitative results in Figure 10. In
Figure 1 of this report we have shown the experimental data of a soot particle velocity along the air / liquid
interface. As stated before the soot particle had a radius R p of 40 µm, which allows us to estimate its mass around
m = ρ 34 πR3p ∼ 10−10 kg. Thanks to the Particle tracing with mass module on COMSOL we were able to reproduce
the conditions in which was conducted the experiment. By isolating one part of the trajectory of one particle injected
in the software, we extracted the data of its velocity at every point along the interface line. We superimposed these
calculated data with the experimental ones (right side of Figure 10). The agreement between observed and simulated
results is quite satisfactory, but one should take into account that the observation has been done for a single soot
particle of a certain mass. Particles with lower mass can move faster.
Study of optothermal Marangoni effect using laser light — 13/29
Figure 10. Calculated by COMSOL trajectories of two particles ”injected” into the liquid at two points
symmetrical according to the median axis (left) and quantitative comparison of fluid velocity near the gas / liquid
interface calculated by COMSOL (from the part circled in black of the trajectorie aside) within the described
simplified model and that measured experimentally (right).
3.1 Numerical simulations of the Marangoni flows along a flat air / liquid interface
The Figure 11 presents numerical simulations calculated for a 2 dimensional system with a flat air / liquid interface.
The liquid properties are the ones of 1,4-dioxane and are given in Appendix 4.6. COMSOL allows us to see not only
the temperature field and the velocity field (already presented in Figure 9) but also the pressure field (added in Figure
11). This is the key point in order to explain the bending that was observed experimentally. Indeed the Figure 11
shows the pressure field after 2 seconds of laser illumination (on the right). It is well seen that the surface will bend
toward the laser spot because of the negative pressure (relatively to the external atmospheric pressure) at the center.
Figure 11. The pressure field obtained by a COMSOL simulation explained the bending experimentally observed :
the external pressure is higher than the pressure in the liquid along the median axis, allowing the surface to bend
toward the laser spot (the blue colors indicate a pressure smaller to the atmospheric pressure whereas the red colors
are situated at the areas where the pressure is the highest).
This finding led us to consider the case of the gas bubble trapping. Given that the problem of bubble trapping is
the core issue of this report, more details are given in the following section.
Study of optothermal Marangoni effect using laser light — 14/29
3.2 Numerical simulations of the Marangoni flows around cylindrical gas bubbles
Simulations were performed in the 2D approximation. We treated the circular border of the gas bubble as the free
interface (i.e. where the Marangoni constraint is applied). The Marangoni number 48-49 is defined as :
| ∂∂ σT |∆T
µ ρCk p
Then with ∆T = 20 K and other parameters (Appendix 4.6) we get Ma = 1322 for a small bubble (radius
Rb = 50µm). The obtained Marangoni number is much larger than the critical Marangoni number Mac = 80. To
obtain the critical Marangoni number it is sufficient to induce a temperature difference at the bubble surface of 1.2
K. The Prandtl number characterizing the heat transport mechanism in 1,4-dioxane Pr = pk is equal to Pr = 13,
meaning that the momentum diffusivity dominates the behavior of the heat transfer. In Figure 12 we present the
formation of fluid whirls around the gas bubble as a result of COMSOL simulation and compare them with real
observation (cf. also the movie movie SIM6).
Figure 12. a) The normalized liquid flux velocity field showing extended liquid movements around the bubble. The
radius of the bubble is Rb = 500µm and the position of laser beam center is at the distance of Rb from the bubble
gas / liquid border (see the white arrow). b) Photograph of liquid flux traces around the bubble in the real
experiment. Photograph has been composed of several frames of a movie with frame coloration technique. For the
sake of better presentation we introduce the symmetry and composed experimental image from two identical parts,
when in real experiment we have observed a small asymmetry.
These symmetrical flows induce a pressure difference on upper and lower sides of the bubble therefore constituting a net force acting on the bubble and directed on the straight line between the laser spot and the bubble center
toward this spot.
The streams of liquid are obviously directed from the hottest to the coldest regions. However, this is contrary to the
observations which show that the bubble is attracted toward the laser spot and not repelled from it. So another force
directed toward the laser beam has to be discovered in order to explain the experimental findings. The numerical
simulations performed in COMSOL allow the monitoring of the relative pressure around the bubble. This is shown
in Figure 14 for two cases :
• when a laser beam almost touches the bubble (a),
• when a laser beam is at a distance equivalent to bubble radius Rb from it (b).
Study of optothermal Marangoni effect using laser light — 15/29
The first case describes the steady state situation when the gas bubble is pinned up to the laser beam and can be easily
moved with the beam. The second case describes the situation when the gas bubble is attracted toward the laser spot.
The blue color seen in Figure 14 stands for the underpressure while the red one stands for the overpressure. From
these pressure maps it is evident that the pressure difference on both sides of the bubble is strongly asymmetric
along the direction defined by the vertical line connecting the center of the laser spot and the center of the bubble
while in the orthogonal direction there is a mirror symmetry.
In our opinion the amount of underpressure is directly responsible for the magnitude of the driving force
attracting the bubble toward the position of the laser beam. However, the distribution of pressure around the bubble
is not uniform and undergoes changes dependent on the distance between the heat source (the laser spot) and the
bubble outer border as well as on its dimension. In order to calculate the total force FTotal (r) acting on the bubble
one must integrate the relative pressure field in the vicinity of the bubble surface (cf. Figure 13).
Figure 13. a) Exemplary plots of the pressure po distribution around the bubble in the vicinity of laser light
inducing Marangoni flows. Pressures were calculated with increasing time from the opening of laser source. Red
areas show overpressure and blue area underpressure with respect to the mean pressure set here at zero level. The
integration of the area under curves over half circumference distance (i.e. from 0 to 0.157 mm) gives approximately
zero value. b) On this polar plot arrows point the directions of extremal pressure areas used by us to determine the
angle θ between the components of the total force FTotal (r) acting on the bubble.
There are three different forces to consider in this situation :
• the force acting along the median axis moving the bubble toward the laser spot F
~2 and F
~3 acting with an angle θ with respect to the median axis and moving the bubble
• the two other forces F
far from the laser spot
Basing on the fact that only the fluid flows can induce pressure differences one can assume that the underpressure
integrated value pu is exactly twice as large as the two branches of integrated overpressure po located on the two
sides of the bubble and equals to |~po | = 12 |~pu |, i.e. pu = 2po . In this sense the magnitude of the total force FTotal (r)
acting on the bubble and directed toward the position of laser spot will depend mainly on the value of and the angle
Study of optothermal Marangoni effect using laser light — 16/29
θ between vertical line and the normal to the tangent to circle at the maximum value of overpressure (cf. Figure 14) :
~ = (~pu − 2~po )A
where A is the area to which the pressure is applied. Here we assume that the cylindrical bubble has a height of
its radius Rb , then A = πR2b . Thus the value of the total force acting on the bubble will be given by :
~ | = |~pu |(1 − cos(θ ))πR2b
This force is strongly dependent on the distance r between the laser spot and bubble position and can be treated
as introduced earlier Marangoni force FM (r) = FTotal (r). Looking at the numerical simulations, we can see that the
longer is the distance between the laser spot and the bubble surface, the smaller is the angle θ . The angle is not
known a priori, it could be known only on the basis of complex solution of Navier-Stokes, Fourier and Marangoni
equations. With the simple model proposed and described by the equation (9), we see that if θ = 90◦ (i.e. the bubble
is at a quite long distance r from the laser spot) then FTotal = F1 and the bubble is able to accelerate toward the laser
spot. At very short r, θ → 0 so FTotal → 0. These limits fit to the reality and this is in our opinion the reason why the
bubble remains attached to the laser spot when it reaches it.
Figure 14. Relative pressure around the cylindrical gas bubble in: a) close contact with laser beam and b) with laser
beam at some distance Rb from the bubble. The irregularities of the pressure field close to the bubble wall are caused
by the triangular mesh. The total force accelerating particle FTotal is a result of vectorial summation of three forces
acting on the bubble surface ; the vertical one due to underpressure and the two forces due to overpressure directed
toward the bubble center. Depending on laser distance from the bubble surface the directions and magnitudes of
contributing forces are changing.
Calculated on the basis of our simulation, the total force in function of a distance r from laser spot has been
established at seven discreet values of r and is shown in Figure 15. However, when the gas bubble is in motion in
the liquid, it undergoes a resistance force (the drag force Fd = 4π µRb u) that must be subtracted at corresponding
points from the Marangoni force. This drag force was calculated from the movie SIM7 shown in Figure 15 where
we estimated the speed of the bubble with respect to its distance from the laser spot. In our case the bubble is
moving against the flow of liquid which itself has a certain velocity, say −u, then the relative bubble velocity is
approximately twice as large.
This is why we assumed that the drag force will be approximately given by the formula : Fd = 8π µRb u. The
resulting driving force |FM (r) − Fd (r)| is the net force that accelerates the bubble (cf. Figure 15). We would like to
Study of optothermal Marangoni effect using laser light — 17/29
stress that according to our simulations this force is not a constant one and increases from long distance r (∼ 10−9
N) to some maximum value (> 10−8 N) for smaller r. Then the driving force evidently drops to smaller value when
the laser spot is situated at the gas / liquid interface ; when this occurs the drag force Fd = 0 N, because bubble is
trapped and does not move. However the trapping process is so fast that we are not able yet to catch a movie from
this last part in order to calculate the drag force and thus put in evidence this drop.
The values calculated above are showing the order of magnitude of the forces acting on gas bubbles in a particular
system. Choosing other solvents, heating source, bubble diameters these forces can change, at least within an order
Figure 15. Plot of Marangoni force FM calculated from COMSOL simulation model (in blue) versus the distance r
between laser spot and gas bubble. The resulting force |FM (r) − Fd (r)| is diminished by the drag force Fd (r)
estimated from the bubble velocity u. The top right corner shows an exemplary image of trapping gas bubble event
(see also SIM4 movie of Supporting Information). It is composed of sequence of several frames each of 40 ms. The
edge enhancement technique has been applied to monitor the position of accelerating and then trapped bubble.
Let’s note that this mechanism of trapping is different from the ones already described in other papers, namely
trapping by radiation forces, trapping by two-dimensional interference pattern, by photothermal effect with the beam
focal point in the bubble center or using thermoplasmonic Marangoni effect.
Study of optothermal Marangoni effect using laser light — 18/29
3.3 Numerical simulations of the crystallization inside a gas bubble
To model the phenomenon of crystallization inside a gas bubble, we chose the 3 dimensional approach as highlighted
in Figure 16 a). The system is a block (5x5x0.5 mm3 ) of liquid to which we applied the properties of 1,4-dioxane. In
its center we introduced a spherical gas bubble (radius Rb = 0.3 mm). The laser is built as a cylinder going through
the whole block. The Marangoni force is acting on all the surface of the bubble.
We had to put in evidence that the crystallization can occur inside the bubble because of the Marangoni flows.
The Figure 16 depicting the fluid velocity around the gas bubble shows that the strongest fluxes are indeed along the
median axis, toward the center of the bubble. It is clear that the highest velocities are along this axis, allowing the
particles of pNA together with dioxane to penetrate the gas bubble walls and thus forming a droplet rich in pNA
particles leading to the saturation of the solution with solute and finally to the crystal growth.
Figure 16. a) View of the 3 dimensional system in COMSOL. Here is shown the temperature repartition at the
surface of the block. b) Velocity field cut along the median plane. The size of the arrows is proportional to the
velocity of the fluid’s particles. c) Zoom in the zone near the bubble.
3.4 Numerical simulations of the droplets’ trapping phenomenon
To explain this counter-intuitive effect, we proposed a numerical simulation carried out on COMSOL but let’s
pay attention this time that this is a cut of a side view (cf Figure 17). The layer of liquid surrounding the droplet
highly concentrated in pNA is very thin (5 µm). The Marangoni force is acting on the whole upper line. During
our experiments, not only we observed the layer of liquid from the top to demonstrate the higher concentration but
we also put a camera on the side and we observed a deformation of the liquid surface (movie SIM5), leading to the
formation of a droplet highly concentrated in pNA. This deformation was introduced artificially in COMSOL. The
laser is put at the center of the system, meaning at the center of the growing droplet.
Our goal was to understand how the fluxes of liquid were propagating to be able to understand the growth of pNA concentration in the vicinity of the laser spot. As shown in Figure 17 a) we proved that the Marangoni fluxes are able to
develop : the movements of the fluxes are going both ways, thus particles of pNA can be dragged toward the laser spot.
We also have to take into consideration that the dioxane evaporates at much lower temperature than pNA (Tboiling (dioxane) =
374 K, Tboiling (pNA) = 605 K). Thus the dioxane evaporates near the laser spot (if the power is large enough) whereas
the pNA stays, leading to a higher concentration of pNA near the laser spot.
Last but not least, the pressure field (Figure 17.b)) shows a strong underpressure near the laser beam position. This
explains the bending we can see at the beginning of the movie.
Study of optothermal Marangoni effect using laser light — 19/29
Figure 17. a) Liquid fluxes inside the system (here just half of the system is displayed for a better visualization),
showing that the Marangoni fluxes are able to develop even if the layer of liquid is very thin. b) The pressure field
shows an underpressure zone around the laser beam. This is why the system is able to bend at the beginning, then
leading to the formation of the droplet.
Conclusion and perspectives
Having such optical trapping mechanism, we are able to move freely the gas bubble or the liquid droplet pinned-up
to the laser beam through the layer until the other side of the liquid is reached. The process can end with a release of
a bubble or droplet at the other side of liquid. In this fashion, a transport of a pico-liter amount of gas through liquid
membranes becomes feasible. The process can be repeated several times both ways. Having two different gases or
liquids on both sides of liquid membrane and using two laser beams guided separately by systems of mirrors one can
perform reactions of two gases under microscope that could be beneficial in many nanotechnological processes and
may help in controlled synthesis of various single nanoparticles.38-40
In order to understand the physics of these phenomena we developed a numerical model on COMSOL
M ULTIPHYSICS all year long which can be adapted in many situations. We demonstrated that the accuracy of the
model allows us to explain many of our observations. However, it must be strengthened to perform better. Indeed in
this presented work, we did not take into account the liquid evaporation, which can play a key role in such laser
heated systems. We can not introduce a dependence on the temperature of the different parameters we used (except
for the surface tension), neither the deformation of the gas / liquid interface which we build artificially (it is possible
to introduce it with COMSOL but it need complementary modules that we do not have here).
I am very delighted that our work carried all year long resulted in the publication of one article. 50
Study of optothermal Marangoni effect using laser light — 20/29
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Study of optothermal Marangoni effect using laser light — 23/29
4. Appendix A
4.1 Several heating sources
A configuration where we put two lasers heating spots was successfully modeled and we observed two areas where
the Marangoni effect is acting (Figure 10). This situation is very similar to the one where we would have a square
laser beam profile that we could obtain thanks to a spatial light modulator (Figure 11).
Figure 18. Two lasers heating spots can produce whirls at two different places on the free border. On the left side is
shown the temperature field after 1 second of illumination, and on the right side the velocity field at the same time.
Figure 19. A laser beam with the shape of a square can lead to two Marangoni fluxes along the surface. On the left :
temperature field after 4 seconds. On the right : fluid velocity along the interface during the first 4 seconds (each
colored curve corresponds to the velocity along the line at a given time).
Study of optothermal Marangoni effect using laser light — 24/29
4.2 Several gas bubbles
We also investigated the situation where several gas bubbles are positioned at the same distance with respect to the
laser heating spot in a liquid environment. We found out that the system made of three gas bubbles would show the
Marangoni flows which are very unstable in time as shown in Figure 12.
Figure 20. The Marangoni effect is jumping from one bubble to another one because the situation is not stable. It is
first acting on the left upper gas bubble but after 0.5s it is acting on the right upper one, and 0.5s after on the third
4.3 3D Comsol simulations
The 3 dimensional system where a gas bubble is surrounded by a large domain of liquid was also analyzed. It is
not a surprise that we found out that the Marangoni whirls are developing in the 3 directions of space. However,
if we approach a 2 dimensional system by shrinking the thickness along the z-direction, whirls do not appear
perpendicularly to the layer. This is another way to justify our 2 dimensional analysis (Figure 13).
Figure 21. Illustration of 3D whirls surrounding a gas bubble.
We also looked at the case of a sessile droplet lying on a substrate and modeled the flows inside it (Figure 14)
Study of optothermal Marangoni effect using laser light — 25/29
Figure 22. Side view of a sessile droplet heated by a laser beam along its axis of revolution. The upper surface is in
contact with air, allowing the Marangoni flows (represented by the red arrows) to appear.
4.4 System with random geometry
A COMSOL simulation on a system with a random geometry was done to show that the formation of the 2
dimensional whirls do not need a particular symmetry.
Figure 23. The Marangoni flows are present in a system with a random geometry as well.
4.5 Experimental work on the bending
In order to explain better why the liquid layer should be thin enough, we carried on an experiment observing the
bending amplitude according to the thickness of the liquid layer (with the same laser intensity). The results are
shown in Figure 16.
Study of optothermal Marangoni effect using laser light — 26/29
Figure 24. Laser-induced interface distortion. a) Amplitude of liquid-air interface bending with respect to the laser
beam intensity (layer thickness d = 70 µm, λ = 405 nm). Inset : dependence of amplitude of liquid-air interface
bending on thickness of liquid layer confined between two glass plates. Line shows the elasticity limit of the
interface above which it abruptly returns to the initial position. b) Schematic side-view of liquid layer wetting glass
surfaces with dependent on thickness strength of capillary forces.
4.6 Complete description of COMSOL simulations
Table 1. Parameters
Troom is the reference temperature for material properties, this is the initial temperature of the liquid at room
temperature. ∆T is the excess temperature on the laser spot. σT is the temperature derivative of the surface tension. ρ
is the fluid density, µ its dynamic viscosity, k its thermal conductivity, Cp its heat capacity and α its heat expansion.
We chose a free triangular mesh to compute the simulations.
Study of optothermal Marangoni effect using laser light — 27/29
• Boundary conditions
The different boundary conditions are given in Figure 17.
Figure 25. Boundary conditions put in COMSOL.
Let us first consider the Navier-Stokes equation in its most general form :
∂~u ~ ~
+ ∇(ρ (u)~u) = −~∇p + ~∇Σ + ρ~g
where Σ is the viscous stress tensor and where ρ~g = ~0 because the gravity does not play any role in our system.
Let’s note that :
+ ∇(ρ~u~u) = ~u
+~u(~u.~∇ρ) + ρ(~u.~∇)~u + ρ~u(~∇.~u)
= ρ( + (~u.~∇)~u) +~u(
+~u.~∇ρ + ρ~∇.~u)
= ρ( + (~u.~∇)~u) +~u(
Because of the continuity equation, the last term is equal to zero so at the end we obtain :
+ ∇(ρ~u~u) = ρ( + (~u.~∇)~u) = −~∇p + ~∇Σ
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For a newtonian fluid the expression of the viscous stress tensor is the following one :
Σ = µ(∇~u + (∇~u)T ) + µ (2) (~∇.~u)I
where µ (2) is the second viscosity, however this introduction of the second viscosity can be avoided by the
Stokes hypothesis µ + 23 µ (2) = 0 leading to the equation :
+ ρ(~u.~∇)~u = ~∇.[−p~I + µ(∇u + (∇u)T ) − µ(~∇.~u)~I]
An important dimensionless number in fluid dynamics is the Mach number, Mach , defined by Mach = |~u|
a where a
is the speed of sound. In our case Mach ∼ 10 << 0.3 so that we can consider that the fluid is incompressible. This
hypothesis of incompressible fluid reduces the continuity equation to ρ~∇.~u = 0 and equation (16) becomes :
+ ρ(~u.~∇)~u = ~∇.[−p~I + µ(∇u + (∇u)T )]
which is the equation we are using in COMSOL.
There are also some equations describing the behavior of the fluid next to the walls which are:
~u.~n = 0
~ − (K.~
~ n)~n = 0
~ = µ(∇u + (∇u)T ))~n
The equation governing the Heat Transfer in Fluids module is the heat equation:
+ ρCp~u.~∇T = ~∇.(k~∇T ) + Q
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4.7 Liquid Crystal
Recently we started to study the case of liquid crystals. In a system made of a thin layer of liquid crystal with the
upper surface let free, we could apply an electric field which is able to change the orientation of the molecules inside
the liquid crystal. Thus the viscosity should significantly change. Given that it is easier for the fluid’s particles
to move in a liquid less viscous, the size of the Marangoni ring observed from the top should be dependent on
the presence of the electric field or not. The numerical simulations agree with that assumption and show that the
higher is the viscosity, the smaller is the size of the ring due to the Marangoni effect. This numerical approach has
the advantage of being quantitative. Linked to the experiment, it could for instance provide a measurement of the
viscosity of a liquid crystal. However, for now we reach a lot of difficulties generating such Marangoni whirls in
Figure 26. Size of the Marangoni ring in a liquid crystal for three different viscosities.