Enhanced pinch flow fractionation using inertial streamline crossing

Abstract

In this work, we study the influence of inertia on the dynamics of neutrally buoyant spherical microbeads of varying diameter in a pinch flow fractionation device. To that aim, we monitor their trajectory over an unprecedented wide range of flow rates and flow rate ratios. Our experimental results are supplemented by a depth-averaged 2D-model where the flow is described using the Navier-Stokes equation coupled with the shallow channel approximation and where particles trajectories are computed from Newton’s second law of motion with a particle tracing model. Above a certain flow rate, we show that particles inertia enables them to cross streamlines in response to an abrupt change of direction. These streamline crossing events combined with the increasing effect of the inertial lift forces drive particles to deviate from the inertialess trajectory. The amplitude of the resulting inertial deviation increases both with the particles diameter and the total flow rate before reaching a plateau. Consequently, based on our numerical and experimental results, we determine the optimal flow conditions to shift the particles distribution in order to significantly enhance their size-based separation.

Appendices

Appendix A Detailed procedure of the chip conception

We use soft lithography techniques to fabricate our system in polydimethylsiloxane (PDMS). A pre-mold is designed with OpenSCAD and printed using the Two-Photon polymerization (Serbin et al. 2003; Farsari and Chichkov 2009) of the Nanoscribe Photonic Professional GT. Given the dimensions of the system, 25X objective and IP-L resin are chosen to optimize printing time and resolution. After printing, the polymerized resin is developed during 20 min into a SU-8 developer solution (MicroChem) before being washed in ultra pure isopropanol ($\ge$99,8%) (Sigma-Aldrich) for 10 min. Next, the pre-mold is quickly UV-exposed with the BlueWave LED PRIME UVA (Dymax) to reinforce its adhesion to the substrate (Purtov et al. 2018). The system is then placed in a Petri Dish and coated by evaporation of hexamethyldisilazane (Sigma-Aldrich) in a vacuum chamber during 10 minutes (Ting et al. 2011). Afterwards, a mixture of the PDMS (Sylgard 184) and the cross-linking agent at a ratio of 10:1 (w/w) (Mavom NV) is poured onto the pre-mold (Qin et al. 1998). The polymer is degassed in the vacuum chamber for 20 min before being cured for 2 h at 70$^\circ$C in an oven (Termaks AS). Cured-PDMS is peeled off and used as a stamp for the final master. Norland Optical Adhesive 81 (NOA 81) (Norland Products) is poured into the PDMS-moulded microchannel (Bartolo et al. 2008). Both are firmly hand pressed against a glass-slide of 76x52 mm (Marienfeld/VWR) before a 3 minutes-long isolation at full power in the UV-KUB3 (Kloe). The highly resistant and inert final master is obtained after peeling off the PDMS. This procedure enables us to use the same mold for every chip fabrication, which strongly limits the variability induced by dimensional disparities between the chips. Dimensions of the mold are controlled with a VK-X200 3D Laser Scanning Microscope (Keyence). The final device is obtained by moulding degassed PDMS 10:1 (w/w) with the silanized NOA master at 50 $^\circ$C overnight. After the peeling off, inlets and outlet are made using a 1 mm Uni-core Punch (Qiagen). The system is closed with a glass slide by glass-PDMS plasma bonding (Xiong et al. 2014) in a CUTE-1MPR plasma cleaner (Femto Science Inc.).

Appendix B Values of the terms $G_1(s)$ and $G_2(s)$

Table 5 Values of $G_1$ and $G_2$ as function of s numerically estimated by Ho and Leal (1974)

Appendix C Influence of the channel deformation on the flow conditions

The pressure inside the device increases with $Re_\pi$, which may induce a deformation of the flexible PDMS microchannel and eventually modify the flow conditions. According to Gervais et al. (2006), a small variation of the channel height as function of the pressure inside the device can be described as follows when $w \gg h$ (Gervais et al. 2006):

$$\begin{aligned} h(x) =h \left( 1+\alpha \frac{p(x)w}{Eh} \right) , \end{aligned}$$

(C1)

where h(x) is the mean height of the channel cross-section at a certain x-coordinate after the pressure-driven deformation of the system, $\alpha$ is a proportionality constant defined by the channel geometry and the mechanical properties of the PDMS, p(x) is the pressure at the x-coordinate and E is the Young Modulus of the PDMS. Approximations of the values of E and $\alpha$ are obtained respectively from the work of Wang et al. (2014) and from the work of Gervais et al. (2006). We set $E = 2.61 MPa$, which corresponds to the Young Modulus of PDMS with a base/curing agent mass ratio of 1:10 that is baked at 65$^\circ$C for one hour. And $\alpha \approx 1$ given the thickness of the PDMS layer is such that it can be considered as a semi-infinite medium when compared to the channel depth. Those estimations are rough, yet they represent good indicators to assess the order of magnitude of the pressure driven-deformation of the system. The pressure is inferred from our computations on COMSOL with x positioned at the center of the pinched segment, where we decided to assess the deformation of the channel. Table 6 shows how $Re_\pi$ impacts the channel height using Eq. C1. The parameters $p^*$, $h^*$ and $\Delta h$ are respectively the pressure, the mean channel height after deformation and the mean relative alteration of the channel height in x. $Re_\pi ^*$ is the corrected value of the Reynolds in the pinch after the pressure-induced deformation of the channel.

Table 6 Influence of the flow rate on channel deformation

Our simulations indicate a linear increase of the pressure $p^*$ with the flow rate which may eventually lead to a non-negligible modification of the channel height. Yet the relative impact of $h^*$ on the Reynolds number remains quite low even at $Re_\pi = 128$. Therefore this phenomenon cannot explain by itself the discrepancy between our experimental and numerical results at large $Re_\pi$.

Appendix D Numerical decomposition of particle trajectory to separate the individual contribution of each migration step

Fig. 14

Illustration of the numerical procedure performed to isolate the individual contribution of the pinch inlet (a) and pinch outlet (b) streamline crossings to particles inertial deviation. Solid lines and dotted lines depict the trajectories of particles computed with and without inertia, respectively

Particles inertia gives rise to three migration steps along the PFF device, namely the streamline crossing at the pinch inlet, the streamline crossing at the pinch outlet as well as the transversal deviation induced by the lift forces. The trajectories of 10 $\upmu$m particles for $Q_1/Q_2 = 1/6$ are partitioned in different computational steps to extract the influence of $Re_\pi$ on the inertial deviation induced by each of these migration steps. Figure 14a illustrates the different numerical steps performed to isolate the contribution of the pinch inlet streamline crossing. Particles trajectory is modelled from the device inlet without computing the lift forces in the pinch (2). At the pinch inlet they deflect from their initial streamline (1) until they align with the flow (3), meaning that the first migration process is completed. The trajectory of an inertialess particle (4) is then computed starting at this point and the distance between its position at the system outlet and the initial streamline $\Delta _{in}$ corresponds to the inertial deviation induced by the inlet streamline crossing. The contribution of the streamline crossing at the pinch outlet is obtained by comparing the trajectory of a 10 $\upmu$m particle (7) and an inertialess particle (6) whose starting point is set 20 $\upmu$m upstream the pinch outlet. Y-coordinate of this starting position (5) is derived from the full-trajectory simulations (see Fig. 7c) and lift forces are also removed from the computations. The inertial deviation induced by the streamline crossing at the pinch outlet $\Delta _{out}$ is the transverse distance between the 10 $\upmu$m and the massless particle after reaching the equilibrium in the expansion. The contribution of the lift-induced migration is deduced by subtracting the inertial deviations generated by the two streamline crossing processes ($\Delta _{in} + \Delta _{out}$) to the total deviation observed in the full-trajectory simulations, which corresponds to the transverse distance between the curves (1) and (7) at the outlet of the device.