Chapter 3
3D Hydrofocusing by sheath ow
warping
Contents3.1 Introduction . . . 41
3.2 Numerical Flow Model . . . 42
3.2.1 Laminar model . . . 42 3.2.2 Turbulent model . . . 43 3.3 Preliminary experiments . . . 44 3.3.1 2D hydrofocusing . . . 44 3.3.2 3D hydrofocusing . . . 45 3.4 3D hydrofocusing . . . 46 3.4.1 Experimental Setup . . . 47 3.4.2 Results . . . 49
3.4.3 Comparison between numerical and experimental data . . . . 50
3.4.4 Spreading beam correlation . . . 53
3.5 Conclusion . . . 55
3.1 Introduction
In Chapter 1, several techniques for particle manipulation have been discussed and their main properties presented. In this chapter, we will focus on sheath ow con- gurations for obtaining 3D focusing in a micro-particle uid dispersion. Several devices have been investigated and developed to perform a ow warping of the sam-ple. The commercially available ibidi3in1 device has been rst used to perform 2D
3.2 Numerical Flow Model
Numerical modeling was an important step for understanding and foreseeing the be-havior of experimental device and for helping interpreting the results in the post pro-cessing phase. In the frame of this work we used the commercial software FluentT M
together with the mesh generator GambitT M. FluentT M belongs to the class of
full Navier-Stokes (NS) equations solvers. It includes many standard discretization schemes and physical models, but is allows for implementing applications-speci c terms in both NS equations and boundary conditions, strongly enhancing its nu-merical capability. Moreover, it allows for dealing with very complex geometry. Simulations are performed on a IntelT M quad-core 64bits, 16Gb of RAM. Based on
this hardware con guration, the time required to achieve a meaningful simulation ranges from hours-2D unsteady laminar case- to days for unsteady 3D turbulent cases for particle tracking.
3.2.1 Laminar model
Sheath ow modeling is part of the standard, built-in, physical models of FluentT M.
Basically, it is necessary to model two or multiple streams identical for uids prop-erties, co-current, owing from an injection point to a given pressure outlet. However, because the streams have dierent physical meaning and purposes (sheath ow /sample) a special treatment to numerically distinguish them is needed. The basic modeling hypothesis is that the two miscible streams could be modeled as two dierent species with the same physical properties. This expedient enables for a clear visualization of the diusion/dispersive eects at the mixing line between the sheath and sample ows [143] [144]. All simulations are computed using water-based uids (water and seawater).
For incompressible and isothermal streams mixing in a channel, the set of equations to be solved read as follows:
Continuity:
∇.(⃗u) = 0 (3.1)
Where ∇ operator contains the three partial space derivative of the vector velocity u. Momentum: d⃗u dt +∇.(⃗u.⃗u) = ν∆⃗u − 1 ρ ⃗ ∇p (3.2)
Where ∆ is the laplacian operator that corresponds to the divergence of the gradient of velocity eld u, ρ is the average density, ν is the dynamic kinematic viscosity and p the pressure.
3.2. Numerical Flow Model 43 Where the index i = 1, 2 identi es the two uids, respectively to the averaged properties of the global uid. The mass diusion contribution has been computed by considering the diusion coe cient as constant. The associated diusive ux is evaluated as:
⃗
Ji=−ρ D⃗∇(
ρi
ρ) (3.4)
Here, D is the binary auto-diusion coe cient for water, estimated at 2.2 10−9m2.s−1.
Discrete phase:
Spherical particles have been numerically modeled as a discrete phase of a low vol-ume fraction (< 10 percent) with a diameter normal distribution centered around 30µm and their trajectories have been individually calculated at speci ed time dur-ing the uid phase computation. The force balance on each particle is:
d⃗u
dt = FD(⃗u− ⃗uP) + ⃗g ρP − ρ
ρP (3.5)
Where ⃗u is the uid phase velocity, ⃗uP is the particle velocity, ρP the particle
density, ⃗g is the gravitational constant (g = 9.81m.s−2 in the vertical direction and
g = 0 in the other directions) and FD(⃗u− ⃗uP) the drag force.
Boundary conditions:
At the inlets, a volumetric ow rate of the streams is assigned in the range 10 to 500µl.min−1. Dierent combinations of injection rates have been tested both
numerically and experimentally. At the outlet section of the channel, a constant, atmospheric pressure is applied. This condition is also consistent with the real experimental con gurations tested. At the wall the no-slip boundary conditions is applied as well as the condition of no-diusion ux.
⃗u = 0 (3.6)
⃗ ∇(ρi
ρ) = 0 (3.7)
3.2.2 Turbulent model
As pointed out by [145], the smallness of ReDH =
Vav·DH
ν - DH being the hydraulic
diameter - is not always a su cient condition in micro uidics for the ow to be fully laminar, because of the perturbations induced by surface roughness and local disturbances as, in our case, the mixing of two streams having dierent momentum. To check for the existence, in uence and persistence of recirculating ows in the computational domain, simulations have been performed by taking advantage of the k − ε (k for the turbulent kinetic energy and ε for its dissipation rate) model as implemented in FluentT M [146]. This model is eective for small Reynolds ow
the transport of eddy viscosity is added to the standard set of NS equations. The transport of turbulent kinetic energy and its dissipation rate are modeled as:
∂ ∂ t(ρk) + ∂ ∂ xi (ρkui) = ∂ ∂ xj [ (µ +µt σk ) ∂k ∂ xj ] + Gk+ Gb− ρε − YM + Sk (3.8) ∂ ∂ t(ρε)+ ∂ ∂ xi (ρεui) = ∂ ∂ xj [ (µ + µt σε ) ∂ε ∂ xj ] +C1ε ε k(Gk+C3εGb)−C2ερ ε2 k+Sε (3.9) In this equation, the production of turbulent kinetic energy due to mean velocity gradients and buoyancy are taken into account by Gkand Gb respectively while YM
represents the contribution of the uctuating dilatation in compressible turbulence to the overall dissipation rate. C1ε, C2ε and C3ε are constants, σk and σε are the
turbulent Prandtl numbers for k and ε, while Sεand Skare additional custom source
term. A thorough and detailed description of the model can be found in [147].
3.3 Preliminary experiments
3.3.1 2D hydrofocusing
To check for the ability of generating an hydrofocusing eect in a uid sample in 2D con guration, we used a commercially available ow through cell, the ibidi3in1 (see
Fig.1.5). At rst, we used two water streams without dispersed phase. For the sake of visualization, the sample stream was blue-colored with the help of a negligible quantity of methyl-blue. The main result was we manage to contain the central ow stream (the sample that in the nal experiment should contain the particles) in the middle of the channel and, by setting dierent ow rates at the entrances, we were able to precisely regulate the width of the central streamline.
3.3. Preliminary experiments 45
Fig. 3.2: Experimental tests: control of central streamline width using dierential ow rate ratios at the inlet
On picture a) of Fig.3.2the ow rate of the central streamline is 10 times higher than the ow rate of the side inlets. As a result, the width of the central streamline is almost the width of the channel (measured 2.8mm). On picture b), the ratio between the injected ow rate is 1 : 1. As a result, the central streamline width is one third of the channel width (measured 1mm). On picture c) the ow rate of the side inlets is 10 times higher than the ow rate of the central streamline. Consequently the central streamline width is very thin. By continuously changing the ow ratios between the sheath and sample ow we achieved a minimal sample width of 50µm that is 20 times thinner that its initial width.
3.3.2 3D hydrofocusing
This 2D hydrofocusing device however does not prevent the streamline from sedi-mentation since the vertical position is not controlled by additional ow. To tackle this issue, we developed a device where the injection of the central streamline is surrounded with a carrier ow in order to make a fully 3D hydrofocusing device. This type of ow through channel is not commercially available. As a result, only the sample inlet and the main channel in glass are bought, and the connections between them has been designed and manufactured at MRC (see Fig.3.3).
In this device, a 2mm×2mm glass channel is connected to a component that con-tains two inlets: one on the top and the other on the bottom where the carrier ow (water or seawater) is injected. This component is also crossed by a smaller square channel (0.4mm×0.4mm) that acts as a sample injection unit. The outlet of this channel is several centimeters further than the support ow injection to ensure the sample is injected in a fully developed laminar ow (Fig. 3.3).
An extensive numerical study was performed with Fluent Software with the aim to calculate the uid velocity pro le and the mixing ratio between carrier and sample uid at the end of the channel. The main result is that the con guration studied allowed having the needed laminar feature of the carrier uid at the nozzle inlet (Fig.3.4) even for a uidic cell whose dimensions are relatively larger than the ones usually used in similar applications.
Fig. 3.3: Scheme of the 3D hydrofocusing device realized by MRC
Fig. 3.4: Fluid computation of 3D Hydrofocusing device with FluentT M
(Fig.3.5). A dye was rst added to the sample ow to show the boundaries between the sample and the carrier ow (Fig a) and Fig c)). Then tests with dierent ow rates have been performed and visualized with the help of a DHM. Picture b) shows small organisms exiting the nozzle. A magni cation ×5 has been used for a), b) and c) pictures. On picture d) however, a magni cation ×2 was used to observe the boundaries of a 3µm ux of synthetic particles. It is clear on this picture that the particles are not touching the channel walls, at least in the width direction. The DHM reconstruction allowed to con rm focusing eect also in the orthogonal direction.
3.4 3D hydrofocusing
3.4. 3D hydrofocusing 47
Fig. 3.5: Experimental tests of the 3D Hydrofocusing device: a) and b) show the inlet of the nozzle; c) and d) in the middle of the main channel; eld of view: 720µm for a), b) and c) and 1.4mm for d)
3.4.1 Experimental Setup
The experimental setup consists of a uidic circuit and a digital holographic micro-scope for the 3D visualization of the micro-objects trajectories.
The uidic system for 3D focusing is composed by two squared, coaxial borosil-icate glass channels (VitroCom, USA) having an internal diameter of 2mm and 400µm respectively (Fig. 3.6). The inner microchannel has an external diameter of 800µm. The 2mm minichannel carries the sheath ow devoted to wrap the sample, injected through the 400µm microchannel (the capillary nozzle) see Fig.3.6(a). The 2channels are separately glued (standard bicomponent epoxy glue 3M, USA) into a custom-designed device that allows for an independent ow feeding. Commercially available Luer Ibidi⃝R connectors complete the uidic cell. The sheath ow is
injected orthogonally to the capillary 42mm prior to the nozzle injection in order to enable the sheath ow to completely wrap the sample ow after the injection and to ensure it would be fully developed before injecting the sample inlet. They join in a main glass channel with a square section of 4mm2 of a total length of 50mm
-Fig.3.6(b).
(a) Flow cell with the nozzle INLET Sheath Flow INLET Sample x z y Cross Section B Main Channel Observation Nozzle x=2mm x=30mm OUTLET mix Cross Section A
(b) Geometrical con guration of the device: Sections A and B where the anal-ysis is carried out
Fig. 3.6: Flow cell presentation
pumping systems(KD Scienti c, USA, KDS250P and KDS270P ). At the sheath ow inlet, distilled and degassed water is injected while at the sample inlet a ux of water seeded with 27 − 45µm neutrally buoyant (1g.cm−3) synthetic particles,
opaque blue polyethylene microspheres (Cospheric, USA), mimicking biological objects is imposed. The full setup is shown in Fig. 2.2.
3.4. 3D hydrofocusing 49 pumping systems at a ow rate of 5µl.min−1. Then distilled water is ushed at
a ow rate of 50µl/min during 10min. Dry air is nally injected into the system to remove completely the liquid from the channel test section. At the beginning of each new test, the channel is previously emptied so that the following lling procedure could be observed. The sample solution is injected into the capillary with a ow rate of 5µl.min−1. Once the capillary is completely lled, the sheath pump is
activated with a low ow rate of 5µl.min−1. These complementary ow rates have
been selected so that air bubbles can not be trapped at the exit of the capillary tube. As long as the capillary and the main channel are both lled, the relevant test parameters are set to each pumping system, see Table3.1. When the hydrodynamic balance has been reached, based on the observation of the particle ow through the observation window, the frame acquisition is started up to 24Hz during 25s so that 600holograms will be recorded. Each set of experiment has been repeated 4 times.
Table 3.1: Flow rate and velocity ratio corresponding to the investigated cases
Case number Sheath Flow rate (µl/min) Sample Flow rate (µl/min) V sheath V sample ReDH Rep Case1 83.3 10 0.4 0.9 2.78E-04 Case2 333 10 1.6 3.2 1.03E-03 Case3 1670 10 7.9 15.5 5.02E-03 Case4 2400 10 11.4 22.3 7.21E-03 Case5 1330 1 63.5 12.3 3.99E-03
Observation of particles has been lead using a DHM. The recorded focus plane for the experiments is set manually to the middle of the channel depth, at the nozzle injection so that the major particles would be optically focused in this area. The magni cation used during these experiments is ×10, which leads to a eld of view of (720 × 720µm2) and a capability to calculate with a 1µm accuracy, the Z-location
of particle in a range of -400µm to +400µm from the recorded plane. 3.4.2 Results
the glass nozzle walls contributes to create locally re-circulating ows that signi -cantly alter the trajectories of the particles, both at higher and close-to unity ow rate ratios. Results are presented as beam spreading ratios taking as a reference, alternatively, the cross-section of the sample ow micro-channel or section A so to evaluate the focusing capability of the device against the more realistic situation where re-circulating ows are present. Table 3.2 summarizes the total number of particles for which the detection process was successfully achieved for each case and section. It is worthy to note that the observation and the consequent detection of particles at Section A and B is not simultaneous due the limitation of the eld of view of the microscope. As a result, the cumulative number of particles detected at section A and B is not strictly correlated, quantitatively. Nonetheless, the raw data presented in Table 3.2 have been used to check for the statistical relevance of the spreading area of particles. Also the substantial drop in the number of parti-cles detected at Section B as in Case 5, could be interpreted as a warning for the incipient formation of recirculating ows. As a matter of fact in this latter case, particles are trapped close to the nozzle and prevented from reaching Section B. As a result, a statistical analysis of this turbulent-like case 5 would require further speci c experiments involving a quanti cation of particles trapped in the nozzle, which is beyond the subject of the present study.
Table 3.2: Total number of particles detected at the two sections.
Case number Number of particles Section A Number of particles Section B
Case 1 648 174
Case 2 100 166
Case 3 173 30
Case 4 134 27
Case 5 112 7
3.4.3 Comparison between numerical and experimental data First, the validity of the numerical model has been investigated close to the nozzle injection, at section A. On Fig. 3.7, the typical reconstruction process is shown where numerical trajectories (line) are compared with the tracking of experimental positions of particles (dot) and plotted in the depth of the channel cross-section. X-direction refers to channel longitudinal axis. The successive particle positions are separated by a distance corresponding to the product of the average velocity of the ow, times the recording period. Numerical trajectories in Case 3 - Table 3.1
3.4. 3D hydrofocusing 51 transported along the ow streamlines (i.e. no cross-migration observed). Table3.1
shows their respective particle Reynolds number - Rep = 2Vav·a 2
3ν·DH , (where p stands
for particle, DH the hydraulic diameter of the channel and 2a the diameter of the
particle) that are much lower than unity.
(a) Case3 (b) Case5
Fig. 3.7: Comparison between experimental and numerical particles trajectories from the nozzle injection
(a) Case1 (b) Case2
(c) Case3 (d) Case4
(e) Case5
3.4. 3D hydrofocusing 53 3.4.4 Spreading beam correlation
The focusing e ciency could be de ned as the ratio between the sample beam cross-section at the investigation point along the channel length - section B in our paper - and the one at a reference section - in our case, the sample inlet or the section A alternatively. From this de nition, it follows that smaller values of the focusing e ciency correspond to very narrow sample beams at the investigation point - Sinvestigation
Sref erence ≪ 1. In general, the sample beam cross-section is de ned as the
area of the smallest poly-line enclosing all the observed particles at a given section. In our work, for the sake of simplicity and without any signi cant loss of generality, the sample beam cross-section is calculated from the extremal lateral and vertical positions of the particles in the cross-section, i.e. as the area of the circumscribed rectangle de ned by the actual particle positions.
A visual presentation of the focusing capabilities is shown in Fig. 3.9, where the spreading of particles from experimental data is simultaneously displayed at section A (blue ring) and Section B (red disc) for the dierent cases of Table 3.1. Particles position at section A has been set to center of the channel cross section in order to be used as a reference. As a result, the misalignment of particles beam at Section B could be interpreted as a default of channel set-up parallelism. The focusing e ciency for the dierent cases of Table 3.1 is given in Fig. 3.10. The dashed line corresponds to a non-focused beam and is given as a reference. For what concerns section A, it is possible to spot that for Case 1 and Case 2 the focusing e ciency is greater than 1. This could be interpreted as follows: when the
Vsheath
Vsample ≤1 - see Table 3.1, the pressure recovery of the sample stream at the exit
of the nozzle is smoothed out very slowly. As a consequence, the sample stream is rstly aected by a spreading followed by a subsequent focusing. As long as the
Vsheath
Vsample ≈ 7.9 - Case 3 and up to around 20, the focusing e ciency at section A
becomes less than unity and decreases monotonically as the velocity ratio increases. However, when the Vsheath
Vsample ≥ 20 as in Case 5 of Table3.1, local recirculation at the
exit of the nozzle induced by the strong shear between the two streams avoid any further focusing eects. Indeed, In case 5, the great ratio between the sheath and the sample ow velocity creates a turbulent-like behavior of the ow at the exit of the nozzle, due to its wall thickness (200µm). Small recirculating vortices trap particles and release them only when reaching the external border of the sample nozzle wall. The same trend could be spotted at the investigation point - section B. In Case 1, the net result is that the sample beam results completely non-focused - Sinvestigation
Sref erence ≈ 1, and in Case 2, only partially focused
-Sinvestigation
Sref erence ≈ 0.5. With
increasing ratio of Vsheath
Vsample ≈10 and up to 20, the sample beam is consequently more
and more focused.
The maximal focusing capability was found in the Case 4, where it was possible to achieve a focusing e ciency of 4.4% with a relative low ow velocity ratio of 11.4. For higher Vsheath
Vsample the sample beam is again spreading up to a ratio of ≈ 10%. In
(a) Case1 (b) Case2
(c) Case3 (d) Case4
(e) Case5
3.5. Conclusion 55
Fig. 3.10: Particle spreading plotted as a rectangular area at Section A and B for the several cases
respect to the one at section A. In this case, the smallest ratio is found for Case 3 with a Sinvestigation
SsectionA ≈ 9%. This result con rms that the optimal focusing e ciency
could be found for 5 ≤ Vsheath
Vsample ≤ 20.
The results of numerical simulations are plotted against the experimental ev-idences in Fig.3.12. As stated in the numerical section, simulations were performed in both laminar and turbulent regimes. The laminar model is in good agreement with the results of experiments for the Case 2−4, errors in the order of few percents, while overestimates the spreading of the beam at the investigation region (higher pressure recovery at the outlet of the sample nozzle) in Case 1 and underestimates the spreading of the beam in Case 5 (local recirculating ows rapidly smoothed out by the numerical damping of the energy transport). In this latter case, the simulations carried out in turbulent regime are able to catch the main feature of the experimental evidences, i.e. a surging in the spreading beam ratio.
3.5 Conclusion
Flow visualization and especially biological ow visualization require an appropri-ate match between the ow cell and the optical system. In order to enhance its lifespan, and to reduce the experiments duration, the ow cell geometry has been investigated to take advantage of an hydrodynamic focusing eect.
Fig. 3.11: Spreading of particles
Fig. 3.12: Comparison between experimental and modeled (laminar and turbulent) spreading of particles at section B
system has been intensively investigated through the injection of sample and sheath ow rate in terms of shape of the focused streamlines in order to validate the pro-totype design.
Numerical models using FluentT M software have been developed to describe the ow
patterns. They also provided a tool able to foresee the position of the suspended particles in the sample ow.
