Hydrodynamics of droplets under different flow conditions

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Electrohydrodynamic jetting

Electrohydrodynamic (EHD) jetting relies on the competition between electrostatic pressure and surface tension which relative strenght is defined by the dimensionless Bond electric number, Be = 0rE2l/ with 0 the vaccuum permittivity, r the fluid permittivity, E the electric field and l the characteristic dimension of the fluid. Above a critical electric field, electrostatic stresses are too large to maintain the interface stability; the interface deforms and a droplet is ejected. EHD jetting is usually used to print droplets on flat substrates[119, 120, 121] and create complex patterns but microfluidic versions of such devices have been developped[37].
The inspection of Be dependence with device scale shows that the miniaturization of EHD jetting systems demands the imposition of larger voltages and its feasibility needs to be demonstrated. However, this method raises few technical questions regarding the implementation of a miniaturized version of existing device and was one of the first envisioned to produce fL droplets.


Pico-injection is widely used in multi-step droplet-based microfluidics assays as it allows the injection of a precise picolitric volume of reagent inside droplets without synchronization issues. Perpendicularly to the main flow channel, an injection channel is added and pressurized to obtain a stable meniscus of the fluid to be injected. When a droplet flows, upon imposition of an electric field, the meniscus can break and the injected volume depends on the contact time between the droplet and the reagent, and on the injection channel pressure. One of its major drawback is the cross-contamination induced by the contact between the droplet and the injection channel. The following injected droplet may receive part of the material of the preceding droplet. Miniaturization of the pico-injection raises few technical questions. The injected volume varies with the contact time between the droplet and the injection channel and with the pressure of the injector; the minimal injected volume is on the order of the nozzle dimension. The reduction of the injected volume should be possible by decreasing the nozzle dimension and increasing electric fields to maintain a high-enough Be.

Piezoelectric actuation

A piezoelectric material is a material that converts an electric input into a mechanical displacement. Piezo stacks allow microsecond response times and low deflections (1-100 μm). Implementation of such actuators in microfluidic chips is usually made by positionning the piezo stack above a large reservoir, separated by a thin PDMS membrane[29, 30]. In this configuration, the deflection of the stack is directly transfered to the fluidic reservoir and the stroke volume corresponds to the product of the actuator extension times its diameter[30]; corresponding volumes are in the range of 100 pL – 100 nL and could hardly be miniaturized considering the macroscopic size of actuators. This objection might change in the years to come as PZT thin films were recently integrated into microfluidic chip, enabling the creation of micropumps actuated at low voltage[31]. This could open up new miniaturization possibilities.
In the absence of availability of such techniques in our lab, the ultra-fast response time of piezo stacks led us to look for new geometries that minimize the injected volumes with keeping macroscopic stacks.

Other active actuators in microfluidics

Other drop-on-demand methods exist, such as surface acoustic wave generation[35], thermally mediated droplet formation[32], hydrophobic valving[34] or magnetically modified elastomeric valves[36]. Magnetic and thermomechanical actuation were envisioned at first but are discarded from the manuscript for their lack of interesting results. Other methods were not investigated for lack of time.

Mixing inside droplets during the step emulsification process- direct visualization

In this section, we present preliminary experiments that pretended to visualize recirculations inside droplets during their production in step-emulsification.

Experimental protocol

To decrease the level of complexity of the experiment, 10 pL water-in-oil droplets were produced in step-emulsification. First experiments were carried out with 100 nm fluorescent beads and a sensitive Hamamatsu Orca camera. However, the droplet production frequency and the weak fluorescence of beads could not allow us to extract workable data. To be able to work at high acquisition frequencies, carbon black particules were added inside the water phase in a dilute regime. The solution was filtered at 1.2 μm to suppress eventual aggregates and avoid clogging issues. Droplets production was observed with a Photron fast camera.

Results and discussion

We could image some interesting recirculations in pL droplets, such as pictured in Figure 4.3. To make appearent particule movements, each image of Figure 4.3 is the superposition of stacks during 2 milliseconds. Particules act as passive tracers and are advected along the streamlines inside the droplet as it grows (Figure 4.3. 0ms – 12ms). As soon as the liquid bridge breaks, any advectional movement is stopped (Figure 4.3. 14 – 16 ms).
This allows us to assume that advection mixing only happens during the droplet growth and stops as soon as the droplet detaches. Though particule movements could only be imaged from the top view, recirculations are necessarily 3 dimensional, as pictured in Figure 4.4. The droplet can be divided in four advection rolls, further decreasing the distance over which molecular diffusion is expected to work.
Direct visualization of the mixing process in the droplets only brought qualitative understanding of the process. To measure a quantitative mixing time, further experiments were carried out.

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Table of contents :

I Droplet-based microfluidics and experimental section 
1 Droplet-based microfluidics 
1.1 Droplet-based microbiology before microfluidics
1.1.1 Pure-culture
1.1.2 In Vitro Compartmentalization (IVC)
1.1.3 Input of microfluidics
1.2 Droplet-based microfluidics
1.2.1 Physical principles
1.2.2 Basic operations of droplet-based microfluidics
1.3 On the advantages and drawbacks of droplets miniaturization
1.3.1 Throughput of the encapsulation step
1.3.2 Throughput of the incubation step
1.3.3 Throughput of droplet interrogation
1.3.4 Sensitivity
2 Experimental techniques 
2.1 Emulsification process
2.1.1 Step-emulsification
2.1.2 Emulsion stabilization
2.2 Microfabrication techniques
2.2.1 PDMS chips
2.2.2 Systems bonding and obtention of different surface properties
2.2.3 Microfluidic pneumatic valves
2.2.4 Other fabrication techniques
2.3 Biological assays
2.3.1 Polymerase Chain Reaction
2.3.2 Agarose gel electrophoresis
2.3.3 DNA purification
II Miniaturization of droplet-based operations to the femtoliter scale 
3 Femtoliter drop-on-demand
8 Contents
3.1 Drop-on-demand devices
3.1.1 Pneumatic valves
3.1.2 Electrohydrodynamic jetting
3.1.3 Pico-injection
3.1.4 Piezoelectric actuation
3.1.5 Other active actuators in microfluidics
3.2 Experimental part
3.2.1 EHD jetting
3.2.2 Femto-injection
3.2.3 Piezoelectric actuation
3.2.4 Pneumatic actuation
3.3 Summary of all envisioned DoD methods
4 Mixing during the step emulsification process 
4.1 Passive mixing inside droplets
4.1.1 Reminders on the internal recirculations of a circulating droplet
4.1.2 La transformation du boulanger
4.1.3 Diffusion times in femtoliter droplets produced in step-emulsification
4.2 Mixing inside droplets during the step emulsification process – direct visualization
4.2.1 Experimental protocol
4.2.2 Results and discussion
4.3 Mixing inside droplets during the step emulsification process – indirect measurement
4.3.1 Experimental protocol
4.3.2 Results and discussion
4.4 Conclusion on the mixing process during step-emulsification
5 Manipulation of droplets with electric fields 
5.1 Impact of an electric field on water-in-oil droplets
5.1.1 Behavior of an isolated conductive droplet in an electric field
5.1.2 Behavior of a pair of droplets under an electric field
5.2 Dielectrophoretic sorting of femtoliter droplets
5.2.1 Theoretical analysis of the sorting process
5.2.2 Experimental protocol
5.2.3 Results and discussion
5.3 Electrocoalescence of femtoliter droplets
5.3.1 Theoretical reminders
5.3.2 Experimental protocol
5.3.3 Results and discussion
5.4 Conclusion on the manipulation of femtoliter droplets with electric fields
6 Hydrodynamic splitting 
6.1 Reminders on droplet splitting operations
6.1.1 Significance of this operation
6.1.2 Miniaturization considerations
6.2 Splitting of femtoliter droplets
6.2.1 Protocol
6.2.2 Results
6.3 Conclusion
7 Droplet stability and biochemical compatibility 
7.1 Femtoliter droplet stability
7.1.1 Theoretical approach on droplet stability
7.1.2 Stability of femtoliter droplets
7.2 Reinjection of femtoliter droplets
7.3 Polymerase Chain Reaction (PCR)
7.3.1 Experimental protocol
7.3.2 Results
7.4 Conclusion on the stability of femtoliter droplets and their use as biological
III Direct droplet labeling with oligobarcodes fabricated in situ 
8 Introduction to single-cell analysis 
8.1 Single-cell analysis
8.1.1 Screening on single-cells and directed evolution
8.1.2 Single-cell transcriptomics
8.2 Droplet encoding and analysis per sequencing
8.2.1 Droplet encoding
8.2.2 Sequencing
8.3 Hydrodynamics of droplets under different flow conditions
8.3.1 Hydrodynamics of droplets
8.3.2 Coupling of confined and unconfined flows
9 Direct droplet labeling with oligobarcodes fabricated in situ 
9.1 Interest of our method compared to the state of the art
9.1.1 State of the art
9.1.2 Interest of our method
9.2 Injection of multiple femtoliter droplets on demand
9.2.1 Experimental protocol
9.2.2 Results
9.3 Pairing and releasing of multiple droplets into a slug
9.3.1 Design of the chip
9.3.2 Pairing and releasing by imposition of an electric field
9.4 Construction of barcodes in situ on a 8 injectors / 8 valves design
9.4.1 Definition of the elementary DNA units
9.4.2 Experimental protocol
9.4.3 Results
9.5 Perspectives
9.5.1 Multiplexed version of the droplet printer
9.5.2 Going towards higher throughputs
9.6 Conclusion
IV Contribution to other biological projects 
10 Introduction to the study of evolutionary processes 
10.1 About the origin of life
10.2 RNA world evolutionary processes
10.2.1 The RNA world
10.2.2 Problematic
10.3 DNA world evolutionary processes – enzyme engineering
10.3.1 Strategies for the engineering of proteins
10.3.2 Interest of miniaturization
11 Study of evolutionary processes with femtoliter droplets 
11.1 Early RNA world study – the origin of chromosomes
11.1.1 Description of the workflow
11.1.2 Splitting of picoliter droplets into multiple femtoliter droplets
11.1.3 Pairing and coalescence with fresh picoliter droplets
11.2 Genotype-phenotype mapping of the enzyme SGAP
11.2.1 A multistep directed evolution workflow
11.2.2 Production of 200 fL droplets
11.2.3 Pairing and releasing of 200 fL droplets in 2 pL droplets
11.2.4 Perspectives – Towards a more complex genotype-phenotype mappin
A Multiple layer soft lithography
B Article : Droplet-based microfluidics at the femtoliter scale
C Article : Dynamics of a small number of droplets in microfluidic Hele-Shaw cells
D Elementary DNA units sequences


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