Summary of (past and ongoing) research

Contents:
Brownian Granular Materials
Foams Stability
Plant Gravitropism
Optical Tweezers and Out-of-Equilibrium Statistical Physics

Brownian granular materials

I am currently working on "Brownian Granular Materials", i.e. dense suspensions of colloids that are heavy enough to sediment and form a well-defined pile (like granular materials do), but small enough to be sensitive to Brownian motion. The flowing properties of such materials in the intermediate regime where both thermal forces and contacts between grains play a role remains largely unexplored, but may be relevant both for natural phenomenons (like soil erosion) and industrial processes (like powders and pastes). In particular, these systems offer an interesting opportunity to study granular materials relaxing under small external constraints.


Micro-fluidic drums (PDMS, 100 µm in diameter, 50 µm in depth) are filled with a suspension of 2.12 µm silica particles in deionized water. When the drums are flipped upside-down, a Rayleigh-Taylor like instability appears: the particles falls forming finger-shaped structures.

Unlike classical granular materials that have a repose angle (typically ~20°) below which no avalanche flow is observed, an inclined pile of Brownian granular material creeps even at very small angles. This creeping regime below the repose angle is logarithmic in time, and heavily depends on the size of the particles. This behavior is interesting, as it is reminiscent of the logarithmic relaxations observed in granular materials (both dry and immersed) under various external forcing.

Sketch of a rotating drum experiment with 4.4 µm silica particles, and avalanches curves showing the pile angle as a function of the time after the initial rotation. Schematical description of a rotating drum experiment (PDMS, 100 μm in diameter) filled with a suspension of silica particles (4.4 μm in diameter) in deionized water: the pile is initially at rest, then rapidly inclined, and the pile angle is measured while it relaxes. Temporal evolutions of the pile angle are shown on the right for several particles diameter. Below a critical angle (~ 8°) the pile enters a logarithmic creeping regime that has no equivalent in athermal granular materials.

These results are presented in the following article:
"Brownian Granular Flows Down Heaps" (Phys. Rev. Lett., 2019)

Ongoing projects include investigating more complex geometries (made possible thanks to micro-fluidic devices), such as constrictions where clogging and intermittency can occur.


Micro-fluidic hourglasses (PDMS, heigth: 200 µm; depth: 10 µm, central gap: 10 µm) are filled with a suspension of 3.97 µm silica particles in deionized water. The movie is accelerated 5 times.

Foams Stability

During my second post-doctorate (with Isabelle Cantat), I (briefly) worked on foam stability and soap films.

Liquid foams are made of an assembly of gas bubbles, separated by liquid edges. While being very common in industry and everyday-life (soap, beer, etc.), some of their properties remain hard to predict, such as their viscosity, or their stability. In particular the role of surface tension gradients in the apparent viscosity of liquid foams remains largely unexplained.

We have theoretically studied a very simple toy-model: a foam made of an array of 2D hexagonal bubbles, separated by flat films of constant thickness (which ensures that the only driving forces are the Marangoni forces). When such a foam is submitted to an external shear, the bubbles are deformed (which shears the liquid films) but can also rotate on themselves (which also shear the liquid films). By analytically solving the dynamics of the system we have shown that the continuity of the surface tension around the whole bubble is the relevant condition to determine the bubble rotation rate and the energy dissipation. This result illustrates that thin film dynamics should be solve at the scale of the whole bubble interface (and not only at the scale of a single film) when interface rheology matters.


External shearing of the model foam made of an array of 2D hexagonal bubbles separated by flat liquid films of constant thickness. Due to the shearing, the bubbles deform themselves, and rotate on themselves, which both contribute to shearing the liquid films. The viscous stress created by this shear of the liquid film is linked to the surface tension gradient through the Marangoni law.

I have also been involved in experimental work: when a foam film on a frame is rapidly extended, it will pull on its surrounding menisci, and a new film will be extracted. This new film is generally thicker than the existing film, and under gravity, a Rayleigh-Taylor like instability appears: the thicker film at the top will destabilize and fall forming finger-shaped structures.


Video explaining the Rayleigh-Taylor instability that appears when a thicker foam film is on top of a thinner one. This video has been presetend to the Gallery of Fluid motion at the 71th Annual Meeting of the APS Division of Fluid Dynamics in 2018.

These results are presented in the following articles:
"Marangoni stress induced by rotation frustration in a liquid foam" (Soft Matter, 2019)
"Rayleigh-Taylor-like instability in a foam film" (Phys. Rev. Fluids, 2019)

Plant Gravitropism

During my first post-doctorate (with Yoël Forterre and Olivier Pouliquen), I worked on plant biomechanics, and in particular, plant gravitropism.

Plants are able to sense the direction of gravity, so that the roots grow "down" and the shoots grow "up". This phenomenon, called "plant gravitropism", has been studied for a long time (especially by plant biologist), but is still not fully understood. In particular it is known that specialized cells (called "statocytes") that contain dense amyloplasts (called "statoliths") which sediment to the bottom of the cells, are involved in plant gravisensing, as plants which lack those cells (or which have had those cells ablated by a laser) becomes less sensitive to gravity. It is also known that the reorientation of plants in response to the gravity field is achieved through a differential growth: one side of the shoot/root grows faster than the other, which allows the plant to bend. This differential growth is itself induced by a differential auxin (which is a plant hormone) flux: one side of the shoot/root receives more auxin than the other. However, the exact cascade of events that leads from the statoliths sedimentation in the statocytes to the auxin differential flux remains unknown.


Example of shoot gravitropism: time laps of mustard shoots growing in the dark. The left sample is flipped several time but the shoots always grow up (source: GPhase Youtube Channel).

For a long time, it has been assumed that statocytes behaved as a force sensor, i.e. that the gravity was detected by sensing statoliths’ weight on the cell walls. However, this hypothesis has recently been falsified by experiments revealing that shoot gravitropism is actually insensitive to the intensity of gravity. These results are in favor of the hypothesis that it is the position of the statoliths inside the cell that is important for the gravisensing.

I worked on the motion of statoliths in wheat coleoptiles cuts. In particular, I studied the way a pile of statoliths relaxes when the plant is inclined, as well as the vertical fluctuations of statoliths inside (and oustide of) statocytes.


A pile of statoliths inside a wheat coleoptile cut relaxes after being tilted, showing the fluctuating motion of the statoliths. The movie is observed under microscrope (objective 40x) and is accelerated 80 times (the real duration of the movie is 857 s, the real time between successive images is 3.3 s).

We have shown that piles of statoliths in plant cells relax towards an horizontal surface even for small tilt angles, which is consistent with the fact that plants are sensitive even to very small inclinations. We also have shown that this peculiar behavior is made possible by the cell activity. Plant cells are active medium with complex rheology, and statoliths inside them show fluctuations that are about 10 times higher than thermal fluctuations, which explains their high flowability.

These results are presented in the following articles:
"A new scenario for gravity detection in plants: the position sensor hypothesis" (Physical Biology, 2017)
"Gravisensors in plant cells behave like an active granular liquid" (PNAS, 2018)

Optical Tweezers and Out-of-Equilibrium Statistical Physics

Subcontents:
Minimal energy dissipation for manipulating a memory system
Effective heat fluxes between two particles trapped at different effective temperatures
Anomalous high fluctuations at the sol-gel transition of gelatin

During my PhD (with Sergio Ciliberto and Artem Petrosyan) I worked with custom-built optical tweezers to study out-of-equilibrium statistical physics.

Optical tweezers allow for trapping and manipulating micron-size dielectric particles (typically between 10 nm and 10 µm) thanks to the radiation pressure exerted by light on matter. Using a single highly focused laser beam going through an acousto-optic deflector, it is possible to change rapidly (up to the MHz) the position of the trap, and to manipulate several particles simultaneously in aqueous solutions.

Sketch of the experimental set-up used to trap and manipulated several micron-size particles in aqueous solutions. Sketch of the experimental optical tweezers set-up: AOD is an acousto-optic deflector, λ/2 is an half-wave plate, DM is a dichroïc mirror. The particle(s) trapped by the focused laser beam are tracked using digital video microscopy.

Optical tweezers can be used to study stochastic thermodynamics, as they provide a controllable system at a scale where thermal agitation is predominant. For example, micron-size particles immersed in water are submitted to Brownian motion: they continuously undergo a random motion due to the collisions with the surrounding fluid's particles.


A silica particle (2 µm in diameter) immersed in water undergoes Brownian motion before being trapped by the optical tweezers. At the end of the video the laser in turned off and the particle again becomes free to move randomly.

We have used small variations of the same experimental set-up to study three different out-of-equilibrium problems:

1) Minimal energy dissipation for manipulating a memory system:

By trapping one particle in a double-well potential it is possible to create a memory system able to store one bit of information (for example if the particle is in the left well the information stored is "0" and if the particle is in the right well the information stored is "1").

Using our system, we have measured the energy dissipation that occurs when such a 1-bit memory system undergoes a "logically irreversible procedure", i.e. a procedure for which the initial state (the input) cannot be inferred from the final state (the output) only. Namely, we have studied the amount of heat that is dissipated into the surrounding thermal bath when the 1-bit memory system is "reset to zero" (for this erasure procedure the output is always "0" independently of the input value).

Sketch of the
                experimental erasure procedure used to reset to zero the 1-bit memory system made by one particle trapped in a double-well potential. Sketch of the experimental erasure procedure used to reset to zero the 1-bit memory system made by one particle trapped in a double-well potential. Initially (1) the central energy barrier between the two wells is high, and the state of the memory system is either "0" or "1" (depending on the particle position). Then (2) the energy barrier is lowered, and the double-well potential is progressively tilted to the left (3 to 5). At the end of the procedure (6) the energy barrier is raised again and the particle ends in the "0" state independently of its previous initial state.
We have shown that the amount of (stochastic) heat dissipated by the particle lowers when the procedure duration is longer, and approaches the Landauer Limit of kBT ln(2) in the quasi-static limit (kB being the Boltzmann constant and T the temperature).

Average heat dissipated by the memory erasure procedure, as a function of the procedure duration in seconds. Average heat Q dissipated by the memory erasure procedure, as a function of the procedure duration τ.

These results are presented in the following articles:
"Experimental verification of landauer’s principle linking information and thermodynamics" (Nature, 2012)
"Detailed jarzynski equality applied to a logically irreversible procedure" (EPL, 2013)
"Information and thermodynamics : experimental verification of landauer’s erasure principle" (J. Stat. Mech., 2015)

2) Effective heat fluxes between two particles trapped at different effective temperatures:

By adding a white noise on one trap position, one can mimic a higher temperature than the real temperature of the bath in which the particle is trapped. This technique allows for reaching very high effective temperatures (up to thousands of Kelvin).

Power Spectral Densities of one particle's position when submitted to different effective temperatures. Power Spectral Densities (PSD) of one particle (2 µm in diameter) position when the trap position is "shaken" with different white noise intensities. By fitting the PSD to the theoretical formula, one can measure the effective temperature of the particle.
By switching rapidly (10 kHz in our case) the position of the trap between one fixed position, and one position submitted to a white noise, it is possible to trap nearby two particles held at two different effective temperatures. We have studied the hydrodynamic coupling of such a pair of particles, and measured the energy fluxes between them.

We have shown that, in average, the heat exchanged between the particles is proportional to the effective temperature difference, as in the classical Fourier's law: ‹Q› ∝ ΔT

We have also measured the Probability Density Functions (PDF) of the dissipated heat and shown that they verify an exchange Fluctuation Theorem of the form:

Exchange Fluctuation Theorem for the heat exchanged between two baths at different temperatures. Where Q is the heat exchanged between the two particles at (effective) temperatures T1 and T2 during a given time τ, and kB is the Boltzmann constant. This is verified in the stationary regime (large values of τ) for both particles, and in the transient regime (right after the effective temperature gradient is switched on) only for the "hot" particle.

Probability Density Functions (PDF) of the heat dissipated by the two particles trapped at different effective temperatures. Probability Density Functions (PDF) of the dissipated heat by both particles. Each term Qij can be interpreted as the heat dissipated by particle i due to the coupling with particle j (here particle 1 is the particle at the hot effective temperature, and particle 2 is at rest in the real thermal bath).

These results are presented in the following articles:
"Energy flow between two hydrodynamically coupled particles kept at different effective temperatures" (EPL, 2014)
"Stationary and Transient Fluctuation Theorems for Effective Heat Fluxes between Hydrodynamically Coupled Particles in Optical Traps" (Phys. Rev. Lett., 2016)
"Theoretical description of effective heat transfer between two viscously coupled beads" (Phys. Rev. E, 2016)

3) Anomalous high fluctuations at the sol-gel transition of gelatin:

Phase-transitions are intrinsically out-of-equilibrium situations, where deviations from equilibrium properties can sometimes occur. For example, it had been previously observed that particles trapped in a droplet of liquid gelatin exhibit anomalously high fluctuations when the gelatin undergoes its sol-gel transition.

Using two aligned lasers (one for trapping one particle, and one to locally heat the gelatin sample), we realized rapid quenches of the gelatin below its gel temperature (Tgel), and precisely measured the fluctuations of one trapped particle during the gelation of the sample.

Sketch of the procedure for quenching the gelatin below its gel temperature. Sketch of the quenching procedure for the gelatin droplet. All the system is held at a feed-back temperature Tfb = 27.5°C < Tgel. When the heating laser is on, the temperature of the sample increase by 11°C and becomes above Tgel ensuring the melting of a gelatin droplet around the focal point of the laser. When the heating laser is switched off, the heat immediately diffuses in the sample, and the temperature rapidly drops below Tgel, initiating the sol-gel transition of gelatin. The trapping laser is always on, allowing for tracking the position fluctuations of the trapped particles during the quench.
In such ageing systems, one has to be careful with the definition of the average quantities. Indeed, the classical ergodic hypothesis that allows to consider equally the "ensemble averages" and "time averages" might fail in this case. Here, we define these two different averages as the following:
- the ensemble average is the average done over several realizations of the same experiment, at a given time (here, we average together several particle positions taken at the same time after the quench).
- the time average is the average done over time, for one particular realization of the experiment (here, we average over time one particle trajectory after the quench).

Sketch that shows the definitions of the ensemble and time averages. Sketch of the definitions for ensemble and time averages. Given a set of trajectories having a same time reference, the ensemble average considers all the trajectories at a given time (green box), whereas the time average considers only one trajectory over a period of time δt (fuchsia box).
Contrary to what was previously observed, we found no sign of anomalously high fluctuations during the sol-gel transition of the gelatin droplet. We found that the Probability Density Function (PDF) of position fluctuations remains constant after the quench. Moreover, we have found that the ensemble variance of the position fluctuations verifies the equipartition theorem at any time after the quench: ‹x²› = kBT / k (where k is the trap stiffness, ‹.› stands for the ensemble average, kB is the Boltzmann constant, and T the temperature).

Therefore, we have shown that this aging system is actually a stationary non-ergodic system: the ensemble averaged quantities are time independent, but the time averaged quantities are trajectory dependent.

These results are presented in the following article:
"Fluctuations in an aging system: the absence of an effective temperature in the sol–gel transition of a quenched gelatin sample" (J. Stat. Mech., 2015)