Migration Behavior of Low-Density Particles in Lab-On-A-Disc Devices: Effect of Walls

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Article Migration Behavior of Low-Density Particles in Lab-on-a-Disc Devices: Effect of Walls

Vyacheslav R. Misko 1 , Agata Kryj 1, Aude-Muriel Tamandjo Ngansop 1, Sogol Yazdani 1, Matthieu Briet 1, Namanya Basinda 1,2, Humphrey D. Mazigo 2 and Wim De Malsche 1,*

1 µFlow Group, Department of Bioengineering Sciences, Vrije Universiteit Brussel, 1050 Brussels, Belgium; [email protected] (V.R.M.); [email protected] (A.K.); [email protected] (A.-M.T.N.); [email protected] (S.Y.); [email protected] (M.B.); [email protected] (N.B.) 2 Department of Medical Parasitology and Entomology, School of Medicine, Catholic University of Health and Allied Sciences, Mwanza 33000, Tanzania; [email protected] * Correspondence: [email protected]; Tel.: +32-2-6293781

Abstract: The effect of the lateral walls of a Lab-On-a-Disc device on the dynamics of a model system of particles with a density lower than that of the solvent (modelling parasites eggs) is analyzed theoretically and experimentally. In the absence of lateral walls, a particle always moves in the direction of the centrifugal force, while its trajectory is deflected in the tangential direction by the inertial Coriolis and Euler forces. Lateral walls, depending on the angle forming with the radial direction, can guide the particle either in the same or in the opposite direction to the centrifugal force, thus resulting in unusual particle trajectories including zig-zag or backwards particle motion. The effect is pronounced in the case of short operation times when the acceleration Citation: Misko, V.R.; Kryj, A.; of the angular rotation, and thus the Euler force, is considerable. The predicted unusual motion is Ngansop, A.-M.T.; Yazdani, S.; Briet, demonstrated by numerically solving the equation of motion in the presence of lateral walls and M.; Basinda, N.; Mazigo, H.D.; De verified in the experiment with particles of density lower than that of the solvent. Our analysis is Malsche, W. Migration Behavior of useful for design and operational considerations of Lab-On-a-Disc devices aiming for or involving Low-Density Particles in (bio)particle handling. Lab-on-a-Disc Devices: Effect of Walls. Micromachines 2021, 12, 1032. https://doi.org/10.3390/mi12091032 Keywords: particle separation; parasite egg identification and quantification; diagnostic microfluidic device; extreme point-of-care Academic Editor: Nam-Trung Nguyen

Received: 22 July 2021 1. Introduction Accepted: 25 August 2021 A Lab-On-a-Disc (LOD) device is a centrifugal microfluidic device that can be referred Published: 28 August 2021 to as a subclass of integrated Lab-on-a-Chip (LOC) platforms [1–4]. In addition to the advantages of a LOC platform such as portability, the use of small amounts of materials Publisher’s Note: MDPI stays neutral and reagents, faster reaction times, and programmability [2], the LOD platform employs with regard to jurisdictional claims in pseudo-forces generated during the rotation of the device: centrifugal force, the Coriolis published maps and institutional affil- force, and the angular acceleration-generated Euler force [3]. iations. A wide range of applications including clinical chemistry, immunoassay, cell analysis [5,6], and nucleic acid tests [7–9], could be demonstrated on a spinning disc [10]. Typical examples of applications of these LOD platforms are sample-to-answer systems for biomedical point-of-care and global diagnostics [11], liquid handling automation for Copyright: © 2021 by the authors. the life sciences (e.g., concentration/purification and amplification of DNA/RNA from Licensee MDPI, Basel, Switzerland. a range of bio-samples), process analytical techniques and cell line development for bio- This article is an open access article pharma as well as monitoring the environment, infrastructure, industrial processes and distributed under the terms and agrifood [12,13]. conditions of the Creative Commons The LOD platform has been applied for detection and molecular analysis of Attribution (CC BY) license (https:// pathogens [14] such as, e.g., Salmonella, a major food-borne pathogen [15,16]. Thus, a creativecommons.org/licenses/by/ centrifugal microfluidic device was developed [15], which integrated the three main steps 4.0/).

Micromachines 2021, 12, 1032. https://doi.org/10.3390/mi12091032 https://www.mdpi.com/journal/micromachines Micromachines 2021, 12, 1032 2 of 13

of pathogen detection, DNA extraction, and isothermal recombinase polymerase amplification (RPA) and detection, onto a single disc. Very recently, possibilities of employing LOD platforms for the detection of Covid-19 have been discussed in the literature [17]. Recently, a LOD device was proposed [18] for the detection of soil-transmitted helminths (STH) which are intestinal worms that infect humans and are spread through contaminated soil [19,20]. The device represents a centrifugal microfluidic platform based on centrifugation and flotation which isolates and collects eggs within an imaging zone using saturated sodium chloride as flotation solution. The main advantage of this device (for other diagnostics methods, see: [21–25]) is that it provides quick diagnostics where needed (point-of-care testing) and requires small amounts of sample and materials. This device provided fast and efficient operation and an image of a packed monolayer of eggs collected within a single imaging zone. To enable the separation and to improve the purification efficiency, secondary flotation and size-based separation mechanisms were implemented in the platform. To provide a complete analysis platform, a bench-top imaging setup was coupled to the centrifugation unit, designed and constructed. This platform was equipped with a high-resolution imaging unit and light source and was ready for wireless data transfer. The distinct feature of this system is that a parasite egg monolayer can be formed by restricting the chamber height of the imaging zone to the size of a single egg (as low as 60 µm). The consumer camera that we have used can in future be replaced by a smartphone. This would allow for a further reduction of the instrument cost and can allow for a more widespread implementation of the technique. The platform was successfully tested in Ethiopia, Tanzania, Uganda and Kenya on infected human and animal samples for evaluation of the developed technology (unpublished results). Several studies on particle centrifugation are focused on the optimization of the efficiency of the devices intended for performing specific tasks, while the mechanism of centrifugation is assumed to be well-established. Indeed, it is the well-understood inertial (pseudo-)force, called centrifugal force, that causes the radial motion of particles in a centrifuge and in this way, e.g., separates various types of particles depending on their density. Thus, at first sight, the mechanism of centrifugation is very simple. However, a more detailed look at the dynamics of particles in a chamber of a centrifuge device reveals a rather complex motion pattern due to the presence of other forces, in addition to the centrifugal force, exerted on a particle in a centrifuge device. Here we analyze—experimentally and theoretically—the effects related to these additional forces, including the inertial Coriolis and Euler forces, and the effect of lateral walls of the chamber on centrifugation of particles. First, we consider centrifugation of heavy particles (density particles higher than of the surrounding liquid, ρp > ρf) where the above effects are intuitively more transparent and can thus be easier understood, and then we analyze these effects for particles with the density lower than that of the surrounding fluid, representative of parasite eggs immersed in a solution, to understand complex motion patterns and their impact on the efficiency of centrifugation, with the ultimate goal of optimization of the LOD design.

2. Effect of Lateral Walls 2.1. Governing Forces The motion of a particle of mass m in a centrifuge device obeys Newton’s equation of motion: → → m a = ∑ F i, (1) i → → where a is the acceleration, and ∑ F i is the sum of all the forces exerted on the particle. i Rotation with angular velocity ω results in effective inertial forces, also called pseudo- forces, acting on the particle even in the absence of external forces:

→ → dω → → → → → → → m a = −m × r − 2mω × v − mω × ω × r + ∑ F i (2) dt i Micromachines 2021, 12, 1032 3 of 13

→ dω → Here, −m dt × r is the Euler force, which is due to the angular acceleration, i.e., essential during the initial phase of rotation, before the centrifuge reaches the maximum rotation speed ω, and also during slowing down at the end of rotation. Note that the Euler → force is directed normal to the radius-vector of the particle position, r , in the plane of the → → rotation. The term −2mω × v represents the Coriolis force that depends on the velocity of the particle and not on the particle position. Its direction is determined by the direction of the motion. For example, if a particle moves outwards from the center of the disc, the Coriolis force deﬂects the motion in the direction opposite to the rotation. Finally, the term → → → −mω × ω × r expresses the centrifugal force that only depends on the particle position and the angular velocity ω, and is directed along the radius. In most cases, the centrifugal force is considered as the main governing force leading to the centrifugation effect [26]. Indeed, only the centrifugal force is directed along the radius and thus ultimately causes the motion of particles from the center to the periphery or in the opposite direction. However, even the Euler force and the Coriolis force that are normal to the radius of the rotation and to the direction of the motion, respectively, can lead to striking effects such as zig-zag trajectories and even backwards movements of particles, as demonstrated below. These unusual effects result from the interaction of the moving particles with the lateral walls of the centrifuge chamber. The above inertial forces in Equation (2), which are mentioned in the literature on centrifugal devices (see, e.g., [3]), come purely from the rotation. They result from the formal transformation of the equation of motion, Equation (1), to a frame rotating with angular velocity ω. Next, a particle of density ρp immersed in a ﬂuid of density ρf experiences the Archimedes buoyancy force: Fb, A = Vpg ρp − ρ f (3)

where g and Vp are the gravitational acceleration and the particle volume, respectively. In a rotating frame, in addition to the Archimedes buoyancy, valid for an inertial motion in the gravity field, one should take into account also the rotating buoyancy [10] related to the accelerations resulted from the rotation. Since all the inertial forces in Equation (2) are due to the accelerations induced by the rotation, all of them should be modified for particles immersed in a fluid. Thus, the modified centrifugal force taking into account the rotating buoyancy becomes:

→ → → Fc,br = −Vp ρp − ρ f ω × ω × r (4)

The modiﬁed Euler and Coriolis forces are, correspondingly:

→ dω → F = −V ρ − ρ × r (5) E,br p p f dt and: → → FC,br = −Vp ρp − ρ f ω × v (6) In addition to the Archimedes buoyancy (3) and the inertial forces (4) to (6), the Stokes drag force: Fd = −6ηπrpv (7)

is exerted on a particle of radius rp moving with velocity v in the ﬂuid, as well as the forces due to the friction with the chamber walls. Micromachines 2021, 12, x FOR PEER REVIEW 4 of 12

Micromachines 2021, 12, 1032 4 of 13

2.2. Effect of Lateral Walls 2.2. Effect of Lateral Walls To distinguish effects resulting from particle–wall interactions from other possible To distinguish effects resulting from particle–wall interactions from other possible effects,effects, let let us ﬁrstus first consider consider a heavy a particle. heavy particle. InIn the the absence absence of lateral of walls, lateral the particle walls, trajectory the particle is represented trajectory by the curveis represented shown by the curve inshown Figure1 in. Figure 1.

FigureFigure 1. Trajectory1. Trajectory of a particle of a particle moving frommoving the center from of the a rotating center disc of towardsa rotating its edge, disc in towards a disc its edge, in a disc rotatingrotating with with angular angular velocity velocitω. They direction ω. The of direction the velocity, ofv, isthe deﬂected velocity, from v the, is radial deflected direction from the radial direc- oftion the centrifugalof the centrifugal force by the force Coriolis byforce the FCorioliscor. force Fcor.

If we now imagine a rectangular chamber placed on the disc then, depending on the widthIf we of now the disc, imagine the direction a rectangular of the particle chamber motion placed and of theon Coriolisthe disc force then, at depending on the thewidth wall of will the differ disc, (as the shown direction in Figures of S1a,bthe par in theticle Supplementary motion and Materials). of the Coriolis For a force at the wall narrowwill differ chamber, (as the shown direction in of Figures motion will S1 bea,b close in to the parallel Supp tolementary the wall (Figure Materials S1a), ). For a narrow while for a wider channel the particle would reach the chamber wall at a larger angle γ (Figurechamber, S1b). the This direction means that of the motion projection will of the be particle close velocityto parallel on the to chamber the wall wall, (Figure S1a), while vforw = va· cos(widerγ), will channel be small. the The particle direction would of motion reach of the the particle chamber along wall the wall at ina larger this angle γ (Figure caseS1b). is determined This means by the that balance the between projection the projections of the particle of the centrifugal velocity force on (towards the chamber wall, vw = thev∙cos( edge)γ), and will the be Coriolis small. force The (towards direction the center)of motion on the of direction the particle of the wall,along during the wall in this case is the collision with the wall (Figure S1c). Note that the Coriolis force results in this case in adetermined transient effect, by related the balance to the inertial between motion the of the projections particle after of the the collision, centrifugal which is force (towards the rapidlyedge) dampedand the by Coriolis the Stokes force drag (towards and the friction the center) with the on wall. the However, direction the Eulerof the wall, during the forcecollision due to with eventual the radial wall acceleration (Figure S of1 thec). discNote is notthat evanescent the Coriolis and can force compensate results in this case in a and even overcome the projection of the centrifugation force. In the case that it does not overcometransient the effect, radial migration,related to the the motion inertial still persists motion in theof directionthe particle towards after the edgethe collision, which is ofrapidly the disc: damped the particle by drifts the alongStokes the discdrag with and a smallerthe friction velocity with than itthe would wall. do However, in the Euler theforce absence due of to the eventual wall. Note radial that this accele drift velocityration couldof the be disc very is small, not dueevanescent to additional and can compensate effectsand even of velocity overcome slowing downthe projection near the wall of [27 the]. Wall centrifugation imperfections canforce. lead In to stickingthe case that it does not of the particle at the wall. overcomeIf the trajectory the radial of a moving migration, particle the deviates motion more still strongly persists towards in thethe wall direction (due to towards the edge aof faster the rotationdisc: the or p angulararticle acceleration), drifts along then the the disc particle with arrives a smaller at the velocity wall at a largerthan it would do in the angle,absence close of to normal.the wall. In this Note case that the projection this drift of thevelocity Coriolis could and Euler be forces very are small, larger due to additional thaneffects the projectionof velocity of theslowing centrifugal down force near on thethe direction wall [2 of7]. the Wall wall imperfections (Figure S2a), and can lead to sticking the resulting velocity of the particle after the collision with the wall is directed inwards, i.e.,of the opposite particle to the at centrifugal the wall. force. In other words, the interaction with a chamber wall can leadIf tothe a changetrajectory in the of direction a moving of the particle centrifugation deviates velocity more to thestrongly opposite. towards After the wall (due to thea faster collision, rotation the particle or driftsangular along acceleration), the wall, not to the then direction the particle of the centrifugation arrives at but the wall at a larger towards the center of the disc. angle,However, close suchto normal. a drift towards In this the case center, the contrarily projection to the of drift the in Coriolis the direction and of Euler the forces are larger centrifugationthan the projection (Figure S1d) of isthe not centrifugal stable. When force moving on towards the direction the center, of the the Coriolis wall (Figure S2a), and the resulting velocity of the particle after the collision with the wall is directed inwards, i.e., opposite to the centrifugal force. In other words, the interaction with a chamber wall can lead to a change in the direction of the centrifugation velocity to the opposite. After the collision, the particle drifts along the wall, not to the direction of the centrifugation but towards the center of the disc. However, such a drift towards the center, contrarily to the drift in the direction of the centrifugation (Figure S1d) is not stable. When moving towards the center, the Coriolis force exerted on the particle also changes the direction to the opposite, and the particle moves away from the wall (Figure S2b). In this way, the particle becomes “free” again and then it follows a typical trajectory for a free particle until it hits the wall again.

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Micromachines 2021, 12, 1032 5 of 13 It is clear that the above scenario can lead to unusual motion patterns such as zig- zag-type trajectories, when a particle moves towards the wall and repeatedly collides with force exerted on the particle also changes the direction to the opposite, and the particle movesthe wall away but from still the walldrifts (Figure towards S2b). In the this way,edge the of particle the disc. becomes Under “free” other again andcircumstances, if the thenrotation it follows is f aast typical enough trajectory or when for a free the particle disc experiences until it hits the sufficient wall again. angular acceleration, a par- ticleIt can is clear even that display the above backwards scenario can motion, lead to unusual which motion, however patterns, is suchnot asstable zig- and can result in zag-typea rather trajectories, complex whentrajectory a particle consisting moves towards of repeated the wall forwards and repeatedly and collides backwards movements with the wall but still drifts towards the edge of the disc. Under other circumstances, if theand rotation eventually is fast enough zig-zag or whenmovements the disc experiences between sufﬁcientthe opposite angular lateral acceleration, walls a of the chamber. particleTo can support even display this backwards qualitative motion, analysis, which, we however, solved is not the stable equation and can of result motion, Equation (2), innumerically, a rather complex for trajectory model consistingparameters, of repeated in a rotating forwards andframe backwards in a rectangular movements chamber. Typical andtrajectories eventually of zig-zag particles movements are shown between in the Figure opposite 2. lateralA trajectory walls of theof a chamber. free particle, in the absence To support this qualitative analysis, we solved the equation of motion, Equation (2), numerically,of lateral walls, for model is parameters,presented in by a rotating a curve frame (orange in a rectangular dashed chamber.line in TypicalFigure 2) describing the trajectoriesmotion when of particles the aredistance shown infrom Figure the2. Acenter trajectory of ofrotation a free particle, monotonically in the absence increases in time. In ofthe lateral presence walls, isof presented a lateral by wall, a curve a particle (orange dashed under line the in same Figure rotation2) describing velocity the reaches the wall motion when the distance from the center of rotation monotonically increases in time. In theand presence then drifts of a lateral along wall, the a particlewall towards under the the same perip rotationhery, velocity being reaches driven the by wall the centrifugal force and(red then line). drifts However, along the wall when towards a nonzero the periphery, angular being acceleration driven by the centrifugal is present force (as during the initial (redphase line). of However, the rotation) when a and nonzero the angular additional acceleration Euler is force present is (asexerted during on the initialthe particle opposite to phasethe direction of the rotation) of the and rotatio the additionaln, the trajectory Euler force ischanges, exerted on and the particleafter reaching opposite tothe wall the particle the direction of the rotation, the trajectory changes, and after reaching the wall the particle startsstarts moving moving towards towards the center the of center rotation, of i.e., rotation, opposite toi.e., the opposite direction of to the the centrifugal direction of the centrifu- force.gal force. This supports This supports the above the analysis above predicting analysis the predicting possibility of the backwards possibility motion of of backwards motion particlesof particles in a centrifuge in a centrifuge device (cp. device Figure S2b).(cp. Figure S2b).

FigureFigure 2. Calculated2. Calculated trajectories trajectories of a particle of (dap particle= 123 µm, (ρdpp= = 1.05123 g/mL, μm, ρρf p= = 1 g/mL)1.05 g in/mL, a rotating ρf = 1 g/mL) in a rotating disc.disc. The The orange orange dashed dashed line shows line a shows trajectory a oftrajectory a free particle of a in free a disc particle rotating in around a disc the rotating point around the point (0,0)(0,0) in in the the counterclockwise counterclockwise direction direction with angular with velocity angularω = 800 velocity rpm. A trajectoryω = 800 ofrpm. the sameA trajectory of the same particleparticle in ain rectangular a rectangular chamber chamber of 2 mm widthof 2 mm (empty width box) is(empty modified: box) after is the modified: particle reaches after the particle reaches the lateral wall of the chamber, it drifts under the action of the centrifugal force (red line). When the lateral wall of the chamber, it drifts under the action of the centrifugal force (red line). When an an additional Euler force due to the angular acceleration (assuming linear increase of ω from 0 to 800additional rpm in 0.5 Euler s) is exerted force on due the particle,to the itangular moves backwards acceleration after reaching(assuming the wall linear (blue increase line). of ω from 0 to 800 rpm in 0.5 s) is exerted on the particle, it moves backwards after reaching the wall (blue line). 2.3. Particles with Density Lower Than That of the Fluid 2.3.The Particles above analysiswith Density revealed Lower unusual than behavior That of of the particles Fluid in a centrifuge chamber of rectangular shape, for the case of particles with density larger than that of the fluid, ρp > ρf,The where above the predicted analysis effects revealed (such asunusual a zig-zag behavior and backwards of particles movement) in cana centrifuge be chamber of easilyrecta understood.ngular shape, The situation for the with case “light” of particles particles, i.e.,with those density with density larger smaller than than that of the fluid, ρp > that of the fluid, ρp < ρf, is less intuitively clear. Indeed, in this case the particles can be ρf, where the predicted effects (such as a zig-zag and backwards movement) can be easily understood. The situation with “light” particles, i.e., those with density smaller than that of the fluid, ρp < ρf, is less intuitively clear. Indeed, in this case the particles can be considered as “bubbles” in a fluid. Contrarily to the situation described above, this is now the heavier fluid that experiences stronger inertial forces than the particles themselves. This means that now, e.g., during the initial phase of rotation characterized by the angular acceleration, the fluid will be affected by the Euler force stronger than the particles and, as a result, the fluid will move in the opposite direction to the angular acceleration, and the particles will move in the direction of the acceleration. The same is applied to other inertial

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consideredforces: they as “bubbles” act on the in a higher fluid. Contrarily density to fluid, the situation and the described lower above, density this (and is now thus lighter) parti- the heavier fluid that experiences stronger inertial forces than the particles themselves. Thiscles means are excluded that now, e.g.,by the during fluid the via initial the phase pressure of rotation exerted characterized on the byparticles the angular from the fluid. Thus, acceleration,the mechanism the fluid of will motion be affected of particles by the Euler lighter force than stronger the than fluid the ( particlesρp < ρf) and,is qualitatively differ- as a result, the fluid will move in the opposite direction to the angular acceleration, and ent from that of “heavy” particles with ρp > ρf—their motion is mediated by the surround- the particles will move in the direction of the acceleration. The same is applied to other inertialing fluid. forces: they act on the higher density fluid, and the lower density (and thus lighter) particlesHowever, are excluded the by theoretical the fluid via the formalism, pressure exerted as formulated on the particles above from the in fluid. Equations (4)–(7), re- Thus,mains the unchanged mechanism of and motion applicable of particles also lighter for than particles the fluid with (ρp < ρρf)p is < qualitatively ρf, same as the Archimedes different from that of “heavy” particles with ρp > ρf—their motion is mediated by the buoyancy (3). Because of the condition ρp < ρf, the direction of the forces in Equations (3)– surrounding fluid. (7) changesHowever, theto theoreticalthe opposite. formalism, as formulated above in Equations (4)–(7), remains unchangedThe trajectory and applicable of alsoa particle for particles that withmovesρp < towardsρf, same as the the center Archimedes under buoy- the action of the cen- ancytrifuge (3). Becauseforce is of deviated the condition oppositeρp < ρf, to the the direction direction of the of forces rotation in Equations (Figure (3)–(7) 3a), under the action changes to the opposite. of theThe Coriolis trajectory force of a particle (that would that moves normally towards lead the center to the under deviation the action in the of the direction of rotation centrifugefor a particle force iswith deviated ρp > oppositeρf). Then to, therepeating direction the of rotation above (Figurediscussion,3a), under valid the for the case of ρp > actionρf, for of particlesthe Coriolis with force (thatρp < wouldρf, we normally arrivelead at theto the situation deviation in shown the direction in Figure of 3b: for a large rotation for a particle with ρ > ρ ). Then, repeating the above discussion, valid for the enough angle between pthe fradial direction and the lateral wall of the chamber, α > αc, the case of ρp > ρf, for particles with ρp < ρf, we arrive at the situation shown in Figure3b: forparticle a large executes enough angle a backward between the motion radial direction along andthe thewall, lateral opposite wall of the to chamber,the action of the centrifu- αgal> α cforce,, the particle and executesthen it adetaches backward motionfrom the along wall, the wall, due opposite to the toCoriolis the action force, of the and moves in the centrifugalinner region force, of and the then chamber. it detaches After from the that, wall, it duebecomes to the Coriolis free and force, repeats and moves the motion in a way inas the shown inner region in Figure of the 3 chamber.a, resulting After that,in a itzig becomes-zag trajectory free and repeats or in the a motiontrajectory in a containing back- way as shown in Figure3a, resulting in a zig-zag trajectory or in a trajectory containing backwardswards movements. movements.

FigureFigure 3. The3. The motion motion of a particle of a particle with density with smaller density than smaller that of the than liquid, thatρp < ofρf, the in a centrifugeliquid, ρp < ρf, in a centrifuge chamberchamber rotating rotating counterclockwise: counterclockwise: (a) the trajectory (a) the of trajectory a free particle of movinga free particle towards the moving center towards the center underunder the the action action of the of centrifugal the centrifugal force deviates force opposite deviates to the directionopposite of rotation;to the direction (b) the backward of rotation; (b) the back- α α > α motionward ofmotion the particle of the for particle an angle forexceeding an angle a critical α exceeding value: ac (cp.critical Figure value: S2b). α > αc (cp. Figure S2b). The dynamics of light particles, i.e., particles with the density smaller than that of the ﬂuid, hasThe been dynamics further analyzed of light in particles, numerical i.e., calculations. particles The with results the are density presented smaller in than that of the Figuresfluid, 4has and 5been. further analyzed in numerical calculations. The results are presented in Figures 4 and 5.

(a) (b)

Figure 4. Calculated trajectories of a particle lighter than the fluid (dp = 60 μm, ρp = 1.05 g/mL, ρf = 1.075 g/mL) in a rotating disc. (a) The orange dashed line shows a trajectory of a free particle in a

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forces: they act on the higher density fluid, and the lower density (and thus lighter) particles are excluded by the fluid via the pressure exerted on the particles from the fluid. Thus, the mechanism of motion of particles lighter than the fluid (ρp < ρf) is qualitatively different from that of “heavy” particles with ρp > ρf—their motion is mediated by the surrounding fluid. However, the theoretical formalism, as formulated above in Equations (4)–(7), remains unchanged and applicable also for particles with ρp < ρf, same as the Archimedes buoyancy (3). Because of the condition ρp < ρf, the direction of the forces in Equations (3)– (7) changes to the opposite. The trajectory of a particle that moves towards the center under the action of the centrifuge force is deviated opposite to the direction of rotation (Figure 3a), under the action of the Coriolis force (that would normally lead to the deviation in the direction of rotation for a particle with ρp > ρf). Then, repeating the above discussion, valid for the case of ρp > ρf, for particles with ρp < ρf, we arrive at the situation shown in Figure 3b: for a large enough angle between the radial direction and the lateral wall of the chamber, α > αc, the particle executes a backward motion along the wall, opposite to the action of the centrifugal force, and then it detaches from the wall, due to the Coriolis force, and moves in the inner region of the chamber. After that, it becomes free and repeats the motion in a way as shown in Figure 3a, resulting in a zig-zag trajectory or in a trajectory containing backwards movements.

Figure 3. The motion of a particle with density smaller than that of the liquid, ρp < ρf, in a centrifuge chamber rotating counterclockwise: (a) the trajectory of a free particle moving towards the center under the action of the centrifugal force deviates opposite to the direction of rotation; (b) the backward motion of the particle for an angle α exceeding a critical value: α > αc (cp. Figure S2b).

Micromachines 2021, 12, 1032 The dynamics of light particles, i.e., particles with the density smaller than that of7 ofthe 13 fluid, has been further analyzed in numerical calculations. The results are presented in Figures 4 and 5.

(a) (b)

Micromachines 2021, 12, x FOR PEER REVIEWFigure 4. Calculated trajectories of a particle lighter than the fluid (dp = 60 μm, ρp = 1.05 g/mL,7 of ρ12f = Figure 4. Calculated trajectories of a particle lighter than the ﬂuid (dp = 60 µm, ρp = 1.05 g/mL, 1.075 g/mL) in a rotating disc. (a) The orange dashed line shows a trajectory of a free particle in a ρf = 1.075 g/mL) in a rotating disc. (a) The orange dashed line shows a trajectory of a free particle in a disc rotating around the point (0,0) in the counterclockwise direction with angular velocity discω = rotating 800 rpm. around The red the line point shows (0,0) the in trajectorythe counterclockwise of a particle direction in a rectangular with angular chamber velocity of 4 mm ω = width 800 rpm.(the The grey red empty line box). shows After the reachingtrajectory the of lateral a particle wall in of a the rectangular chamber, thechamber particle of keeps 4 mm moving width in(the the greydirection empty of box). the centrifugalAfter reaching force. the (b lateral) The redwall line of showsthe chamber, the trajectory the particle of a particlekeeps moving in the absencein the direction of the centrifugal force. (b) The red line shows the trajectory of a particle in the absence of of the Euler force. The particle starts moving from the initial position (x = 0, y = 5 mm) and reaches the Euler force. The particle starts moving from the initial position (x = 0, y = 5 mm) and reaches the the bottom of the chamber. When the Euler force (when ω linearly increases from 0 to 800 rpm in bottom of the chamber. When the Euler force (when ω linearly increases from 0 to 800 rpm in 0.5 s) is0.5 exerted s) is exerted on the onparticle, the particle, it achieves it achieves the wall the and wall then and moves then moves along alongthe wall the in wall the in direction the direction of the of centrifugalthe centrifugal force force(the blue (the line). blue A line). stronger A stronger Euler force Euler (ω force increases (ω increases from 0 fromto 800 0 rpm to 800 in rpm0.33 s) in leads 0.33 s) toleads the backward to the backward motion motion (the light (the lightblue blueline). line). When When the trajectory the trajectory of the of theparticle particle starts starts at ata acloser closer positionposition to to the the center center of of rotation rotation (x (x == 0, 0, yy == 2 2mm), mm), a aweaker weaker Euler Euler force force (ω (ω = =0 0to to 800 800 rpm rpm in in 0.5 0.5 s) s) is is sufficientsufﬁcient for for the the backward backward motion motion (the (the green green line). line).

(a) (b) (c)

FigureFigure 5. 5. CalculatedCalculated trajectories trajectories of of light light particles particles (using (using the the same same parameters parameters as as in in Figure Figure 4)4 )in in a a rotatingrotating disc, disc, in in chambers chambers of of varying varying width. width. (a ()a )Chamber Chamber width width ww == 5 5mm. mm. The The red red line line shows shows the the trajectory of a particle moving in a disc rotating with a constant angular velocity ω = 800 rpm. A trajectory of a particle moving in a disc rotating with a constant angular velocity ω = 800 rpm. A weak angular acceleration (see the legend) is sufficient to cause a backward motion. (b) The same weak angular acceleration (see the legend) is sufﬁcient to cause a backward motion. (b) The same angular acceleration (the blue line) appears to be insufficient for a backward motion in a narrower chamber,angular accelerationw = 2.5 mm. (theThe blueeffect line) is achieved appears for to bea stronger insufﬁcient Euler for force a backward (the light motion blue line). in a narrower (c) In a verychamber, narroww chamber,= 2.5 mm. w The = 1 effectmm, even is achieved in the presence for a stronger of rather Euler strong force angular (the light accelerations blue line). (cthe) In particlea very drifts narrow towards chamber, thew center= 1 mm, of rotation. even in theTo turn presence the motion of rather back, strong a very angular strong accelerations Euler force theis requiredparticle (see drifts the towards legend). the center of rotation. To turn the motion back, a very strong Euler force is required (see the legend). As predicted and illustrated in Figure 3a, a light particle moves towards the center of rotationAs under predicted the action and illustrated of the centrifugal in Figure 3force,a, a light and particle its trajectory moves is towards deviated the in center the di- of rectionrotation opposite under theto the action angular of the rotation centrifugal (Figure force, 4a). and To its cause trajectory a backward is deviated motion, in the an direc- an- gulartion oppositeacceleration to the (resulted angular in rotation the Euler (Figure force)4a). sho Tould cause be applied a backward as illustrated motion, an in angularFigure 4b.acceleration The Euler force (resulted needed in the to Eulerflip the force) direction should of bethe applied particle as near illustrated the wall in ofFigure the chamber4b . The depends on the distance of the particle from the center of rotation. Thus, to cause a backward motion of a particle at the distance y = 5 mm from the center of rotation, a rather strong angular acceleration is required, ω = 0 to 800 rpm in 0.33 s. For a particle positioned closer to the center of rotation, y = 2 mm, a smaller angular acceleration, ω = 0 to 800 rpm in 0.5 s, is sufficient to cause a backward motion. Therefore, the effect of eventual backward movements of centrifugated particles in chambers manifests itself preferably in the vicinity to the center of rotation. Next, we analyze the effect of chamber width on the backward movements near the chamber boundary. Figure 5 presents results for three various chamber widths: w = 5 mm, 2.5 mm and 1 mm, for a particle initially positioned at (x = 0, y = 2 mm). In the case of a wide chamber, w = 5 mm (Figure 5a), a relatively small angular acceleration, ω = 0 to 800 rpm in 0.83 s, causes a backward motion of the particle after the collision with the boundary. Note that in this case the particle detaches from the wall and moves in the interior of the chamber as predicted and illustrated in Figure 3b. This motion is caused by the Cori- olis force, which in this case appears to be strong enough to overcome the effect of the relatively weak Euler force. For narrower chambers, as shown in Figure 5b,c, stronger

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Euler force needed to flip the direction of the particle near the wall of the chamber depends on the distance of the particle from the center of rotation. Thus, to cause a backward motion of a particle at the distance y = 5 mm from the center of rotation, a rather strong angular acceleration is required, ω = 0 to 800 rpm in 0.33 s. For a particle positioned closer to the center of rotation, y = 2 mm, a smaller angular acceleration, ω = 0 to 800 rpm in 0.5 s, is sufficient to cause a backward motion. Therefore, the effect of eventual backward movements of centrifugated particles in chambers manifests itself preferably in the vicinity to the center of rotation. Next, we analyze the effect of chamber width on the backward movements near the chamber boundary. Figure5 presents results for three various chamber widths: w = 5 mm, 2.5 mm and 1 mm, for a particle initially positioned at (x = 0, y = 2 mm). In the case of a wide chamber, w = 5 mm (Figure5a), a relatively small angular acceleration, ω = 0 to 800 rpm in 0.83 s, causes a backward motion of the particle after the collision with the boundary. Note that in this case the particle detaches from the wall and moves in the Micromachines 2021, 12, x FOR PEER REVIEW 8 of 12 interior of the chamber as predicted and illustrated in Figure3b. This motion is caused by the Coriolis force, which in this case appears to be strong enough to overcome the effect of the relatively weak Euler force. For narrower chambers, as shown in Figure5b,c, angularstronger acceleration angular acceleration is required is requiredto turn the to motion turn the of motion the particle of the to particle the direction to the directionopposite toopposite the centrifugal to the centrifugal force. These force. observations These observations confirm the confirm above the prediction above prediction that the possibil- that the itypossibility of a backward of a backward motion,motion, i.e., opposite i.e., opposite to the centrifugation to the centrifugation force, of force, a particle of a particle in a centri- in a fugecentrifuge chamber chamber depends depends on the on angle the angle between between the radial the radial direction direction and andthe direction the direction of the of wall.the wall. Our Our analysis analysis suggests suggests that that the the effect effect can can be be easier easier avoided avoided for for smaller smaller angles angles that can be achieved ( i) for particles situated further away from the center of rot rotationation or (ii) in narrow chambers.

3. Experiment To analyzeanalyze thethe effecteffect of of lateral lateral walls walls on on the the dynamics dynamics of of particles particles in ain LOD-type a LOD-type device, device,we employed we employed a LOD a platformLOD platform with rectangular with rectangular chambers chambers (Figure6 (Figurea). The injection6a). The chamberinjection chamberwas placed was further placed awayfurther from away the from ﬂotation the flotation chamber chamber to allow to allow for the for observation the observation and andvalidation validation of egg of presenceegg presence in the in separationthe separation chamber chamber entry entry using using the imagingthe imaging setup. setup. The Thedevice device was createdwas created using using computer computer numerical numerical method method milling milling in polymethyl in polymethyl methacrylate meth- (PMMA) with a milling robot (Daltron Neo). The channel was 4 mm wide and 5 mm deep acrylate (PMMA) with a milling robot (Daltron Neo). The channel was 4 mm wide and 5 and sample was inserted at the peripheral inlet using a syringe, after which the inlet an mm deep and sample was inserted at the peripheral inlet using a syringe, after which the outlet channels were closed with an end ﬁtting (red item in Figure6a). inlet an outlet channels were closed with an end fitting (red item in Figure 6a).

(a) (b) (C)

Figure 6. ((aa)) LOD LOD platform platform with with four four rectangular rectangular chambers. The channel width is 4 mm, the channel length is 35 mm, and the pitch of the scalescale barsbars isis 11 mm.mm. (b) The LOD platform in the centrifugecentrifuge (Eppendorf® Centrifuge MiniSpin G). (c) The photo imaging system: a camera with macro lens. (Eppendorf® Centrifuge MiniSpin G). (c) The photo imaging system: a camera with macro lens. The observation of the consequent locations of a 123 μm polystyrene particle (micro- Particles GmbH) with density 1.05 g/mL, which is close to that of the fluid, 1.07 g/mL, recorded after each 5 s in a disc rotating at 800 rpm (Eppendorf® Centrifuge MiniSpin G), revealed an unusual zig-zag motion pattern and backward motion (Figure 7).

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angular acceleration is required to turn the motion of the particle to the direction opposite to the centrifugal force. These observations confirm the above prediction that the possibility of a backward motion, i.e., opposite to the centrifugation force, of a particle in a centrifuge chamber depends on the angle between the radial direction and the direction of the wall. Our analysis suggests that the effect can be easier avoided for smaller angles that can be achieved (i) for particles situated further away from the center of rotation or (ii) in narrow chambers.

3. Experiment To analyze the effect of lateral walls on the dynamics of particles in a LOD-type device, we employed a LOD platform with rectangular chambers (Figure 6a). The injection chamber was placed further away from the flotation chamber to allow for the observation and validation of egg presence in the separation chamber entry using the imaging setup. The device was created using computer numerical method milling in polymethyl methacrylate (PMMA) with a milling robot (Daltron Neo). The channel was 4 mm wide and 5 mm deep and sample was inserted at the peripheral inlet using a syringe, after which the inlet an outlet channels were closed with an end fitting (red item in Figure 6a).

(a) (b) (C)

Micromachines 2021, 12, 1032 Figure 6. (a) LOD platform with four rectangular chambers. The channel width is 4 mm, the channel9 of 13 length is 35 mm, and the pitch of the scale bars is 1 mm. (b) The LOD platform in the centrifuge (Eppendorf® Centrifuge MiniSpin G). (c) The photo imaging system: a camera with macro lens.

The observation of of the the consequent consequent locations locations of of a 123 a 123 μmµ mpolystyrene polystyrene particle particle (micro- (mi- croParticlesParticles GmbH) GmbH) with with density density 1.05 1.05 g/mL, g/mL, which which is isclose close to to that that of of the ﬂuid,fluid, 1.071.07 g/mL,g/mL, recorded afterafter eacheach 55 ss inin aa discdisc rotatingrotating atat 800800 rpmrpm (Eppendorf(Eppendorf®® Centrifuge MiniSpin G), revealed anan unusualunusual zig-zagzig-zag motionmotion patternpattern andand backwardbackward motionmotion (Figure(Figure7 7).).

Figure 7. Left: Reconstruction of consequent locations of a particle in chamber recorded each after 5 s in a disc rotating at 800 rpm. Right: Snapshots of consequent particle locations showing a backwards

movement (positions 4 and 5).

The revealed motion pattern is consistent with the predicted behavior for particles with ρp < ρf. Indeed, starting from position 1 near the inlet in Figure7, the particle moves towards the center, and its trajectory deviates to the right, according to Figure5a, until it reaches the lateral wall (position 2 in Figure7). At that position, the angle α between the radial direction and the lateral wall is smaller than needed for backwards motion, and the particle, after it reaches the wall, drifts along the wall in the direction of the outlet. However, at some point the angle α exceeds the critical angle, α > αc, thus providing the condition for eventual backwards motion and detachment from the wall, and the particle can eventually arrive at the opposite side of the chamber (position 3 in Figure7). After that, the particle repeats the motion pattern similar to that between positions 1 and 2, and after that it becomes “trapped” at position 4 (Figure7), where the condition α > αc holds and, therefore, the particle experiences a strong backwards net force being close to the center of rotation which hits it back to position 5. Finally, after some iterations of back and forth movements, the particle reaches the outlet position. Similar motion patterns have been revealed in the experiments with 60 µm polystyrene particles, immersed in a solution with density 1.075 g/mL, in a disc rotating at 1000 rpm, when recording particle positions after each 15 s. Figure8 shows an example of such behavior. Finally, we performed experiments with 123 µm particles in discs with rectangular chambers of various width rotating at 800 rpm and found similar motion patterns (Figure9). We found that the effect of walls manifests itself more strongly in wide channels, resulting in striking backwards movements. These ﬁndings are in qualitative agreement with our theoretical predictions that lateral walls’ misalignment with respect to the radial direction leads to zig-zag and backwards motion patterns. While the above considerations allow for explaining the occurrence of backwards movement, the migration (velocity) is also inﬂuenced by physical (mostly reversible but occasionally even irreversible) interactions with the walls during the course of the experiment, which are less straightforward to predict. This will be a subject of a future study. Micromachines 2021, 12, x FOR PEER REVIEW 9 of 12

Figure 8. Left:: Reconstruction of consequent locationslocations ofof aa particleparticle inin chamber.chamber. RightRight:: SnapshotsSnapshots ofof Micromachines 2021, 12, x FOR PEER REVIEWconsequent particle locations (1–4) recorded after 15 s in a disc rotating at 1000 rpm. The pitch10 of the12 consequent particle locations (1–4) recorded after 15 s in a disc rotating at 1000 rpm. The pitch of the scale bars is 1 mm. scale bars is 1 mm. Finally, we performed experiments with 123 μm particles in discs with rectangular chambers of various width rotating at 800 rpm and found similar motion patterns (Figure 9). We found that the effect of walls manifests itself more strongly in wide channels, resulting in striking backwards movements. These findings are in qualitative agreement with our theoretical predictions that lateral walls’ misalignment with respect to the radial direction leads to zig-zag and backwards motion patterns. While the above considerations allow for explaining the occurrence of backwards movement, the migration (velocity) is also influenced by physical (mostly reversible but occasionally even irreversible) interactions with the walls during the course of the experiment, which are less straightforward to predict. This will be a subject of a future study.

(a) (b)

FigureFigure 9. 9. ReconstructionReconstruction of ofconsequent consequent locations locations of a of particle a particle in chambers in chambers with with various various widths: widths: w = 4w mm= 4 mm(a) and (a) andw = 12.5w = 12.5mm mm(b). ( ab)). In (a the) In narrow the narrow channel, channel, the theparticle particle moves moves towards towards the thecenter center of the rotation following the centrifugal force, with just two backward movements (at y = 20 mm and of the rotation following the centrifugal force, with just two backward movements (at y = 20 mm and y = 4 mm). (b) In the wide channel, backwards movements are observed in each step: the particle y = 4 mm). (b) In the wide channel, backwards movements are observed in each step: the particle moves back and forth as predicted in Section 2 (Figures 4 and 5). moves back and forth as predicted in Section2 (Figures4 and5).

4.4. Conclusions Conclusions TheThe effect effect of of the the lateral lateral walls walls of of a a centrifuge chamber chamber on on the the trajectories of of moving particlesparticles has has been been analyzed. analyzed. First, First, for for clarity, clarity, we we considered massive particles, particles, i.e., with thethe density density higher higher than than that that of of the the solution. solution. In In the the absence absence of of lateral lateral walls, walls, such such a a particle particle movesmoves along along a curved trajectorytrajectory resultingresulting from from the the interplay interplay of of the the centrifugal centrifugal and and Coriolis Cori- olisforces, forces, and and the distancethe distance from from the centerthe center of rotation of rotation to the to particle the particle in this in casethis iscase a monotonic is a mon- otonicincreasing increasing function function of time. of time. When When placed placed in a chamberin a chamber of, e.g., of, e.g., rectangular rectangular shape, shape, the thetrajectories trajectories of theof the particles particles are are impacted impacted by by the the lateral lateral walls. walls. DependingDepending onon the angle angle betweenbetween the the wall wall and and the the radial radial direction direction as well as well as on as the on rotation the rotation speed speed and on and the on even- the tualeventual angular angular acceleration acceleration (that (thatinduced induced additional additional Euler Euler force force exerted exerted on the on particle) the particle) the particlethe particle can move can move either either away away or towards or towards the center the center of rotation. of rotation. In the In former the former case, case, the particle,the particle, when when reaching reaching the wall, the wall, drifts drifts along along the wall the wallin the in same the same direction direction as a free as a par- free ticle,particle, i.e., i.e.,along along the thecentrifugal centrifugal force. force. In Inthe the latter latter case, case, when when the the relative relative angle angle between between thethe wall wall and and the the radial radial direction direction is is large large enough enough and and if if there there is is an an additional additional acceleration acceleration inin t thehe direction direction of of rotation, rotation, the the particle particle can can move move backwards, backwards, i.e., i.e., opposite opposite to to the the action action of of thethe centrifugal centrifugal force. force. This This regime regime can can be be easily easily achieved achieved during during the the initial initial acceleration acceleration of of thethe centrifuge centrifuge and and can can essentially essentially contribute contribute to to the the motion motion for for short short ce centrifugationntrifugation times. times. However, while in the case of the forward motion along the wall, the particle can drift all the way towards the outlet, the backwards motion is not as stable. When the direction of motion changes to the opposite, the direction of the inertial Coriolis force also changes to the opposite, and then it acts in the direction of rotation and thus it facilitates detachment of the particle from the wall and moving it in the interior of the chamber. After that, the particle becomes free again and repeats the motion pattern typical for a free particle until it reaches the wall again. This can lead to a rather unusual zig-zag motion pattern with backwards movement parts. In the case of particles lighter than the solution, i.e., when the particle density is lower than that of the solution, the centrifugation effect forces the particles to move from the periphery of the centrifuge device towards its center. Correspondingly, all the inertial forces change the sign, and the above motion patterns remain but become “inverted” as compared to the above case of particles heavier than the fluid. They look less intuitively clear, but the above analysis allows us to better understand the resulting patterns which were revealed in the experiments with synthetic particles with density slightly lower than the density of the fluid.

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However, while in the case of the forward motion along the wall, the particle can drift all the way towards the outlet, the backwards motion is not as stable. When the direction of motion changes to the opposite, the direction of the inertial Coriolis force also changes to the opposite, and then it acts in the direction of rotation and thus it facilitates detachment of the particle from the wall and moving it in the interior of the chamber. After that, the particle becomes free again and repeats the motion pattern typical for a free particle until it reaches the wall again. This can lead to a rather unusual zig-zag motion pattern with backwards movement parts. In the case of particles lighter than the solution, i.e., when the particle density is lower than that of the solution, the centrifugation effect forces the particles to move from the periphery of the centrifuge device towards its center. Correspondingly, all the inertial forces change the sign, and the above motion patterns remain but become “inverted” as compared to the above case of particles heavier than the fluid. They look less intuitively clear, but the above analysis allows us to better understand the resulting patterns which were revealed in the experiments with synthetic particles with density slightly lower than the density of the fluid. The observed effect is in particular important for particles lighter than the solvent, such as eggs. Light particles move towards the center of rotation, where the revealed effect is much stronger than far from the center of rotation. Due to the revealed effect, they can accidentally accumulate near the outlet and thus deteriorate the efficiency of the device. To reduce this undesired effect, the chamber should be properly designed. The obtained results provide a deeper understanding of the centrifugation mechanisms and can be useful for the design optimization of LOD devices. In terms of importance and potential applications of the presented results: (i) Our work provides an understanding of the effects which were observed and did not have proper explanation (e.g., were attributed to the device imperfections such as surface roughness or eventual leakages, etc.). Our work explicitly revealed the physical mechanisms behind the unusual observed behavior, such as back and forth movements (in particular, when close to the center of rotation). Knowing these mechanisms allows one to better control the motion of particles in LOD devices. (ii) Our findings provide instructions as to how to avoid the undesired effects, e.g., by narrowing the chambers, or by other design improvements. The strength of our approach is that it allows predicting trajectories of particles in any geometry, by a proper description of the boundary conditions, and in this way improve device design to eliminate or to minimize undesired effects. (iii) More challenging, the results can be considered for potential selectivity of particles based on, e.g., their position with respect to the center of rotation, or their individual properties. In this way one can think of collecting various particles separately by a proper device design.

Supplementary Materials: The following are available online at https://www.mdpi.com/article/ 10.3390/mi12091032/s1, Figure S1: Trajectory of a particle projected on a narrow (a) and wide (b) rectangular chamber (no parti-cle-wall interaction); particle-wall interaction: the centrifugal, Fcen, Euler, FEuler, and Coriolis, Fcor, forces and their projections on the lateral wall during the collision of the particle with the wall (c); the resulting velocity in wall contact mode, v1, and the Coriolis force, Fcor, after the collision with the wall (d): the particle moves in the direction towards the edge, and the Coriolis force is directed towards the wall. The green arrow shows the reaction force exerted on the particle from the wall. Figure S2: Forces and their projections on the lateral wall at the moment of collision with the wall (as in Figure S1 (c)) but for a larger angle between the radius (white dashed line) and the wall (a); the resulting velocity, v1, and the Coriolis force, Fcor, after the collision with the wall (b): the particle moves in the direction towards the center, and the Coriolis force is directed towards the inner part of the chamber. Micromachines 2021, 12, 1032 12 of 13

Author Contributions: Conceptualization, W.D.M. and V.R.M.; methodology, W.D.M. and V.R.M.; formal analysis, V.R.M., A.K., A.-M.T.N., S.Y., N.B.; investigation: experiment, A.K., A.-M.T.N., S.Y., N.B.; theory and simulations, V.R.M.; resources, W.D.M.; writing—original draft preparation, V.R.M. and W.D.M.; writing—review and editing, V.R.M. and W.D.M.; visualization, A.K., A.-M.T.N., S.Y., M.B., N.B.; supervision, W.D.M. and H.D.M.; project administration, W.D.M.; funding acquisition, W.D.M. and H.D.M. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by the Innovation counsel at VUB, grant number IOFPOC35 and by Vlir-Uos, grant number SI-2020-01-86 (‘SIMPAQ in Tanzania’). Conﬂicts of Interest: The authors declare no conﬂict of interest.

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