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Alexander L.
Dubov
*^{a},
Taras Y.
Molotilin
^{a} and
Olga I.
Vinogradova
*^{abc}
^{a}A.N.Frumkin Institute of Physical Chemistry and Electrochemistry, Russian Academy of Sciences, 31 Leninsky Prospect, 119071 Moscow, Russia. E-mail: alexander.dubov@gmail.com; oivinograd@yahoo.com
^{b}Department of Physics, M.V.Lomonosov Moscow State University, 119991 Moscow, Russia
^{c}DWI – Leibniz Institute for Interactive Materials, RWTH Aachen, Forckenbeckstr. 50, 52056 Aachen, Germany

Received
17th May 2017
, Accepted 5th September 2017

First published on 5th September 2017

We propose a novel microfluidic fractionation concept suitable for neutrally buoyant micron-sized particles. This approach takes advantage of the ability of grooved channel walls oriented at an angle to the direction of an external electric field to generate a transverse electroosmotic flow. Using computer simulations, we first demonstrate that the velocity of this secondary transverse flow depends on the distance from the wall, so neutrally buoyant particles, depending on their size and initial location, will experience different lateral displacements. We then optimize the geometry and orientation of the surface texture of the channel walls to maximize the efficiency of particle fractionation. Our method is illustrated in a full scale computer experiment where we mimic the typical microchannel with a bottom grooved wall and a source of polydisperse particles that are carried along the channel by the forward electroosmotic flow. Our simulations show that the particle dispersion can be efficiently separated by size even in a channel that is only a few texture periods long. These results can guide the design of novel microfluidic devices for efficient sorting of microparticles.

Complex velocity profiles, which potentially can be suitable for particle separation, can be generated using a special channel geometry.^{22,23} The use of patterned surfaces can also induce novel flow properties that the microfluidic channels do not have when their walls are flat.^{10,24–26} In particular, highly anisotropic periodic grooves in the Cassie state,^{24,27} in which the texture is filled with gas, or the Wenzel state,^{28,29} when liquid follows the topological variations of the surface, generally generate secondary flows transverse to the direction of the applied force.

These textured walls have been already studied in combination with an electroosmotic flow when the grooves were perpendicular to the direction of an external field, and it has been shown that such a system demonstrates a rich set of properties that are determined by its geometry,^{29} or the charge of the texture regions.^{30,31} In addition to that, one can expect that inclination of the grooves at some angle relative to the direction of an external field ought to create a secondary transverse flow suitable for particle fractionation like in the case of pressure-driven flows.^{32}

In this paper, we propose a novel concept of particle fractionation based on the electroosmotic flow in a channel with grooved walls. We use the emerging anisotropy of the flow to divert particles of different sizes towards different streamlines, thus achieving a lateral segregation of a uniformly mixed colloidal suspension. To study this system, we employ a hybrid lattice Boltzmann–molecular dynamics simulation method suitable for both charged^{33,34} and uncharged systems.^{35,36} To avoid the explicit simulations of charged electrolyte species,^{11} we propose to apply a set of appropriate boundary conditions at the channel walls instead. This strategy critically reduces the numerical cost of our simulations which in turn makes it possible to illustrate our concept via a full-scale computer experiment involving a continuous fractionation of a particle suspension in a grooved microchannel.

Fig. 1 Sketch of the front (a) and top (b) views of a charged microchannel. See the text for notations and details. |

The channel is filled with a 1:1 electrolyte solution of concentration c_{0} and solvent permittivity ε, so its inverse Debye screening length is λ_{D}^{−1} = (8πl_{B}c_{0})^{1/2} with being the Bjerrum length, where e_{0} and k_{B}T denote the elementary charge and thermal energy. An electric field is tangent to the walls and applied at an angle α relative to the grooves. We now neglect the convection (Pe ≪ 1) so that the distribution of the electrostatic potential is independent of the fluid flow. We also assume weak external fields (E ≤ 50 V mm^{−1}) and weakly charged surfaces with a low ζ-potential (of the order of −50 mV or smaller) to provide a linear response of electroosmotic flow with respect to the applied electric fields. Finally, we assume that the diffuse layer near the charged walls is thin, λ_{D}/min{L,H,e} ≪ 1, which is typical for microchannels. The application of a horizontal electric field E will generate an outer (outside of the EDL) flow with the velocity u_{EO}, which can be expressed by the Smoluchowski equation:^{37}

(1) |

Neutrally buoyant microparticles entering such a channel near the bottom wall will be translated along the streamlines of the flow, and particles of different radii R will follow different streamlines: the smaller particles will dive deeper into the grooves and will be strongly deflected by the transverse flow, but the larger particles will stay higher and move closer to the direction of an external field. To analyze the lateral particle deflection, we now use coordinates which correspond to longitudinal ξ and transverse η alignments with the applied electric field (see Fig. 1b):

(2) |

Finally, we would like to stress that our approach assumes that particles always stay close to the grooved wall and that their surface potential is much lower than ζ to neglect their own electrophoretic motion.

To simulate the system, we use a hybrid lattice Boltzmann (LB)–molecular dynamics (MD) method.^{35,36,40} The hydrodynamic flow induced by an apparent electrokinetic slip at the walls is simulated via the D3Q19 LB implementation. The unit length is set by the lattice step a = 1.0 and the timestep is defined by the dynamic viscosity μ = 3.0 and mass density ρ = 1.0 as . In simulations, we reproduce the actual geometry described in the previous section: in the z-direction, the channel is confined between a plane upper wall (at z = H = 30a) and a bottom wall (at z = 0) modified with a periodical array of rectangular grooves of period L = 48a, ridge width w = 6a and groove depth e = 7a. Note that the ridge was placed in the middle of the periodic cell, so that the cell was symmetrical in y-direction. In the x- and y-directions, the simulation domain is limited by periodical boundaries. On the walls, we impose a constant electroosmotic velocity boundary condition. Namely, on the upper wall and horizontal sides of the bottom texture we apply u = (u_{x},u_{y},0), and on the side walls of the grooves we set u = (u_{x},0,0). We thus simulate a horizontal electric field with its normal component, with respect to the plane of the channel, being strictly zero. In our simulations, we vary velocities u_{x,y} in the range from to , which gives the channel Reynolds number Re_{ch} = 0.5–5. Note that for a real microfluidic system with typical channel height of the order of 200 μm and above and surface zeta-potential of the order of −50 mV, the employed simulation parameters would correspond to a velocity of 1 mm s^{−1} at an applied electric field of 50 V mm^{−1}. We also stress that convective transport of ions in such systems can indeed be safely neglected, as the upper boundary for Pe does not exceed uL/D ∼ 10^{−3} for typical values of ionic diffusion coefficient in water D ≃ 10^{−6} cm^{2} s^{−1}.

The colloidal particles in our simulation model consist of MD beads, uniformly filling the sphere of radius R according to the “raspberry” approach introduced earlier^{33,35} and further developed in subsequent publications.^{36,41,42} In this method, the hydrodynamic coupling to the LB subsystem^{40} is introduced as a sum of frictional and random forces F = −γ(u_{MD} − u_{LB}) + F_{R} acting on each MD bead, where γ is a tunable coupling parameter and u_{MD,LB} are the bead and fluid velocities, respectively, with the latter being extrapolated from 8 closest grid points to the MD bead position. The random term is a zero-mean white noise whose amplitude satisfies the fluctuation–dissipation theorem through 〈F_{α}(t)F_{β}(t′)〉 = 2δ(t − t′)2δ_{αβ}k_{B}Tγ. The key concept of this coupling mechanism is that both frictional and random forces are strictly pairwise for the MD beads and LB fluid, thus the total momentum is always preserved; and this coupling also works as a thermostat. We used a very small value of k_{B}T, which was equal to 10^{−5} (in simulation units). This implies that the smallest Peclet number of particles in our simulations was uL/D_{p} ≃ 10^{5}, where D_{p} is the diffusion coefficient of the smallest particles. Therefore, the diffusion of particles is fully suppressed.

To simulate properly the no-slip boundary condition at the surface of a spherical particle implemented through the filled raspberry model,^{36} we have used the friction coefficient γ = 20 μa. Further, in the study, we use particles whose radii lie in the range 2a–5a. Their hydrodynamic radii have been validated by Stokes drag measurements, as well as translational and rotational diffusion measurements.^{36} We note that regardless of the particle radii, the characteristic size of flow non-uniformities, generated by the wall texture, is negligible compared to the channel height H, so we can safely neglect the lift forces that might push the particles away from the bottom wall and thus affect the fractionation.

The simulations of motion of single particles have been performed in a periodic system consisting of two grooves, with the size of the simulation box being N_{x} = 32a, N_{y} = 96a, N_{z} = 30a. The locations of the centers of mass (x_{0}, y_{0}, z_{0}) of injected particles have been controlled as follows. We have randomly varied the value of x_{0} from 12a to 16a, and that of y_{0} from 0 to 2a. However, z_{0} has been fixed and always equal to R. In other words, particles have always been injected at the bottom of the groove. For each particle of a given radius, we have conducted at least ten simulation runs with different injection positions. The direction of the field was always at an angle α = π/4.

In order to simulate the fractionation of polydisperse suspensions of particles, we have used a simulation box of the size: N_{x} = 32a, N_{y} = 576a, N_{z} = 30a. The simulation system includes two buffer areas at the beginning and at the end of the channel (y = 0 − 48a and y = 480a − 528a), where both top and bottom walls are flat. The middle section of the channel (y = 48a–480a) includes 9 periodic cells of bottom grooves. We have continuously injected particles at the same x_{0} and z_{0} as in the computer experiment with single particles, but now y_{0} varies from 48a to 50a. The time interval between the subsequent injections of particles has been randomly varied with a mean periodicity 3 × 10^{−3}τ^{−1}. In order to prevent the interpenetration of two neighboring particles, and also to simulate the mechanical interaction between the MD beads and LB walls we have implemented the Weeks–Chandler–Anderson^{43} potential in the following form:

(3) |

Near a grooved texture, the liquid deflects from the direction of the electric field E (Fig. 2(b)). To quantify this deflection, we introduce an angle β, which defines the ratio of the velocity components, averaged over a period, so that tanβ = u_{η}/u_{ξ} for each streamline with a fixed starting point z above the center of the ridge. In Fig. 3(a), we plot the measured values of tanβ as a function of the vertical coordinate z. We conclude that it decays nearly linearly, and vanishes completely at the upper flat wall. This suggests that a transverse shear flow has been generated in the channel. One can easily show that the ratio of transverse and forward flow rates is given by , and the value of Q_{η}/Q_{ξ} can be deduced from the simulation data. Our aim is to optimize the angle α between the directions of the grooves and the external electric field, so that Q_{η}/Q_{ξ} is maximal, providing the strongest transverse flow. The simulation results shown in Fig. 3(b) indicate that the optimal angle is α ≃ π/4, and this value of α has been used in all the subsequent simulations throughout the paper.

Fig. 3 (a) tanβ as a function of the streamline height above the center of a ridge calculated for α = 0, π/8, π/4, and 3π/4; (b) Q_{η}/Q_{ζ}vs. α. Symbols show simulation data, and the dashed curve plots theoretical predictions, eqn (4). |

We should also note that the value of Q_{η}/Q_{ξ} at a given α can be predicted theoretically. A precise discussion of the liquid flow rate in our microchannel (of a variable thickness) requires a complicated analysis. Here, we use an approach that is based on an approximate solution. To do so, we consider a plug forward flow with superimposed uniform transverse shear in a microchannel of a constant thickness. Such a simplification immediately leads to

(4) |

In Fig. 5(a), we plot the yz-projections of the trajectories of the particle centers, which are periodical in the y-direction. One can see that the distance between the trajectories and the texture increase with the particle radius. We remark that particle trajectories (solid curves) slightly deviate from the streamlines (dotted curves), which is likely due to the finite size effects and non-uniformity of the flow near a grooved wall. Similar deviations in the lateral displacement of particles to the lateral displacement of streamlines have been found (Fig. 5(b)). We see a decay of lateral displacement of particles and streamlines with z, but tanβ for particles is smaller. However, these differences are quite small, so we conclude that the fractionation of particles should be controlled by Q_{η}/Q_{ξ}.

In our computer experiment, we have also detected the rotation of particles near the edges of the grooved wall (Fig. 6). Namely, it has been found that particles located close to the corners of the grooves, where streamlines of the flow warp strongly, experience a significant torque. As a result, clockwise rotation near the upper corners (convex streamlines) and counter-clockwise rotation close to the lower corners (concave streamlines) of our rectangular grooves have been observed (Fig. 6(a)). Note that the absolute value of the angular velocity does not exceed 0.04τ^{−1} (Fig. 6(b)). This implies that the rotation angles are π/2 and smaller. We can also conclude that smaller particles rotate faster since they move closer to the walls, where the curvature of streamlines is larger. The detected rotation does not strongly influence the translation of spherical particles. However, one can speculate that such a rotational motion could become important for the fractionation of non-spherical objects.

Generally, the sensitivity of our system, i.e. the minimal size difference for particles to be separated, depends strongly on the length of the channel. Therefore, it is possible to separate particles of any, however small, size difference provided the channel is long enough. Note that surprisingly, the electric field amplitude does not affect the efficiency of separation. This is likely because it does not increase the shear rate of the secondary transverse flow. Therefore, by increasing the electric field we could only accelerate the separation, but not its selectivity.

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## Footnote |

† Electronic supplementary information (ESI) available: A video file representing full-scale simulations of a polydisperse suspension. See DOI: 10.1039/c7sm00986k |

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