A. Simulation

We start by performing a streamline analysis of an HBM to verify our model against previous work (Fig. 2). We find similar trends in surface interaction as a function of γ to previously published work.11. A. D. Stroock, S. K. Dertinger, A. Ajdari, I. Mezić, H. A. Stone, and G. M. Whitesides, “Chaotic mixer for microchannels,” Science 295, 647–651 (2002). https://doi.org/10.1126/science.1066238 A peak interaction rate of 97% occurs at γ=2.0 for particles with a diameter of 18 μm. When the particle diameter is reduced, the peak becomes much shallower and broader since the particles must get closer to the walls before interacting.We then repeat the simulations using fluid–particle interactions rather than assuming the motion of particles using streamlines (Fig. 2), allowing us to include the effects of gravity. Neutrally buoyant particles closely follow streamlines, as is expected when the forces on the particle are dominated by drag alone; carrying it with the fluid. In the absence of acceleration due to gravity, in a low Reynolds number regime where inertial effects can be ignored, the results show agreement with the curves represented in previous streamline studies.

Particles that are denser than the surrounding fluid experience a net downward gravitational force, altering their positions within the channel over time to stray from that defined by the streamlines. When the particles are 5% denser than the fluid, such as might be expected from polystyrene tracer beads in water, the capture rate increases compared to the neutrally buoyant case. When particles are 5% less dense than the surrounding fluid, the interaction probability is reduced and becomes more constant with varying groove width. This behavior is because the gravitational force biases the particles away from the grooves, toward the top of the channel. It is important to note that at γ=0, a straight channel, the interaction probability is the same when particles are either more or less dense than the fluid, since it is the same system being mirrored.

The change in interaction probability as a function of particle density, ρp, is significant despite retaining a low sedimentation velocity. This behavior is enforced when considering particles of different diameters (Fig. 2). A particle with reduced diameter must be brought closer to the wall in order to interact, and thus, reduces the interaction probability.Testing a particle diameter of 6 μm is significant since it is the approximate size of a spherical equivalent human red blood cell (RBC). If we assume a RBC has a volume of 90 fl,2929. C. E. McLaren, G. M. Brittenham, and V. Hasselblad, “Statistical and graphical evaluation of erythrocyte volume distributions,” Am. J. Physiol. 252, H857–H866 (1987). https://doi.org/10.1152/ajpheart.1987.252.4.H857 then simply approximating its shape to a sphere, gives us a diameter of 5.56 μm. This diameter is inline with other studies which approximate RBC’s as spheres using different methodologies.30–3230. M. Kinnunen, A. Kauppila, A. Karmenyan, and R. Myllylä, “Effect of the size and shape of a red blood cell on elastic light scattering properties at the single-cell level,” Biomed. Opt. Express 2, 1803–1814 (2011). https://doi.org/10.1364/BOE.2.00180331. J. Guck, R. Ananthakrishnan, T. J. Moon, C. Cunningham, and J. Käs, “Optical deformability of soft biological dielectrics,” Phys. Rev. Lett. 84, 5451 (2000). https://doi.org/10.1103/PhysRevLett.84.545132. Y. Park, C. A. Best, K. Badizadegan, R. R. Dasari, M. S. Feld, T. Kuriabova, M. L. Henle, A. J. Levine, and G. Popescu, “Measurement of red blood cell mechanics during morphological changes,” Proc. Natl. Acad. Sci. U.S.A. 107, 6731–6736 (2010). https://doi.org/10.1073/pnas.0909533107 If a RBC has a density of 1.110 g m−33333. N. Norouzi, H. C. Bhakta, and W. H. Grover, “Sorting cells by their density,” PLoS One 12, e0180520 (2017). https://doi.org/10.1371/journal.pone.0180520 and its surrounding blood plasma has a density of 1.080 g cm−3:3434. J. D. Bronzino and D. R. Peterson, Biomedical Engineering Fundamentals (CRC Press, 2014). the RBC is 4.3% more dense than the surrounding plasma. Therefore, our simulation results can be shown to give an estimate response for RBC sedimentation within a HBM (Fig. 2, black stars), with potential interest for applications in blood filtration for on-chip sample preparation in POC devices.The change in interaction rate as a function of particle density can be better understood when considering the interaction surfaces involved (Fig. 3). It can be seen that the majority of interactions in the neutrally buoyant case take place in the vicinity of the interface between the grooves and channel. The interaction surface explains the dependence of interaction probability with groove width, since an impedance matching condition between flow in the grooves and flow in the channel increases the residency time of particles in this region.22. D. Hassell and W. Zimmerman, “Investigation of the convective motion through a staggered herringbone micromixer at low Reynolds number flow,” Chem. Eng. Sci. 61, 2977–2985 (2006). https://doi.org/10.1016/j.ces.2005.10.068 As explained eloquently by Hassel et al., if the resistance of the groove is too high (the groove is too narrow) the groove will insufficiently perturb flow in the channel, and if the resistance of the groove is too low (the groove is too wide), the particles also remain in bulk fluid, only in this case favoring the grooves rather than the channel.

However, we find that the inclusion of gravity alters the favored interaction surfaces. Particles which are denser than the surrounding fluid sediment over time and increase the interaction probability with the channel bottom and groove bottom. Particles which are less dense than the surrounding fluid, experience a force away from the favored surfaces and toward the channel top. Therefore in this instance, the impedance matching condition is no longer a good predictor of peak interaction probability.

These simulations show only the case of perfect binding between particle and wall, but we also consider the case of imperfect binding, modeled by the inclusion of a probability associated with the interaction event, resulting in the particle bouncing instead of binding (Fig. 4). The probability of an interaction event resulting in binding is 0.5%. This probability is not chosen based on any direct comparison to physical systems. Considering this imperfect binding shows a significant reduction in interaction rate for neutrally buoyant particles, as might be expected, although a much less significant change for non-neutrally buoyant particles. This robustness against imperfect binding is because the force of gravity acts as a bias to keep particles close to walls even after a failed capture event. Therefore, the binding affinity becomes less relevant, since the particle will sediment regardless.When replicating this system experimentally, the number of beads captured within a device over time is fitted to the function N=n(1−e−λ(b−c)),(1)where N is the number of particles captured, n is the maximum number of particles which the device is capable of capturing, λ is a decay constant, not relative to the passage of time, but relative to the number of beads which have passed through the device. b is the number of beads which have passed through the device. c is an offset based on time to account for any uncertainty at the start point of a measurement. This fit is obtained for γ in a range between 0.5 and 3.5 and an example fit is given in Fig. 5 inset. We find a peak value of n and a minimum value of λ at γ=2 and 1.5, respectively (Fig. 5). From our fitting function, we can calculate the theoretical capture rate at time, t=0 as with simulation [Fig. 5(d)]. We see a broad peak in interaction probability at γ=2.5.Comparing this figure to what we expect from simulation [Fig. 5(d)], we find some agreement and some disagreement. Both indicate a broad peak capture rate of similar magnitude at intermediate values of γ, while a straight groove has a capture rate of 0. Experimental measurements are suppressed at γ=3.5 due to limited fabrication tolerances resulting in the grooves merging into one large groove, approximating a simple section of straight channel with double the value of h. We also find that the capture rate measured in experiment is significantly reduced compared to simulation. This reduction in capture rate is yet unknown, but could be for a number of likely reasons, such as the elimination of a given interaction surface (Fig. 3), or a finite binding affinity between particles and wall (Fig. 4).The experimental data supporting Fig. 5 are obtained from the median performance across seven devices. Error bars are not calculated due to the limited number of repeats, although the data are available at the University of Exeter Repository.3535. J. L. Binsley, T. O. Myers, S. Pagliara, and F. Y. Ogrin, “Herringbone micromixers for particle filtration,” University of Exeter (2023). https://doi.org/10.24378/exe.4445