Particles in microfluidic devices
2.2 Particle separation by a hydrodynamic switch
In order to understand some features of the behaviour of small magnetic beads in microfluidic devices, we will investigate the separation properties of the microfluidic geometry shown on the left side of Figure 2.10(a). Two microfluidic channels of a width of 80 µm run parallel to each other with a distance h and are connected via an additional channel segment cutting both channels under an angle α. For the inflow velocity in both entrances, a parabolic Poiseuille profile according to equation (2.11) is assumed where different maximum values at upper and lower entrance are chosen. Denoting these inflow velocities by uup and udown at the inlets A and B, respectively, a relation factor ξ can be defined by
up down
u
ξ =u . (2.25)
In particular, an equilibrium value ξ0 may be identified where no volume flux from the lower part B of the geometry to the upper part A can be found. The influence of ξ on the flow behaviour is shown in Figure 2.10(b). The proposed geometry acts as a hydrodynamic switch:
the flow direction can be adjusted by specific choices of ξ. In detail, we obtain a flow from B to A or A to B if either ξ ξ< 0or ξ >ξ0 is chosen, respectively. Additionally, the switch is shut for ξ =ξ0.
According to (2.14), a magnetic particle passing the separation area follows the streamline profile of the fluid flow if no further external forces act on the particle. To manipulate magnetic beads along the junction area, an inhomogeneous magnetic field is required, which can be introduced by current leading wire geometries on the microscale. Many designs have been proposed (see e.g. [CLee01]), in our approach, we consider a wire geometry of two opposite rings as indicated. The design of the wire configuration follows a guideline according to [AWed06], Figure 2.11 shows a finite element simulation of the behaviour of the magnetic field between the two rings with a current of 1 mA in each component.
Figure 2.11: Inhomogeneous magnetic field created by ring-shaped electric wires with a current of 1 mA. (a) shows the norm of the field, (b) different components of H in the centre of the
If only electric currents are taken into account, we can easily conclude that a particle will always feel an attractive force pointing towards the nearest wire: as the magnetic moment mpart of the particle aligns with the inhomogeneous magnetic field, it is mpartHext. Therefore, we may write
ext
part part
ext
| |
| |
= H
m m
H
The force acting on a magnetic particle is given by (compare section 4.2)
part=µ0( part∇) ext
F m H . (2.26)
As εijk∂jHext,k =0, it is
2
ext, ext, ext, ext, ext,
1 2
j j i j i j i j
H ∂ H =H ∂H = ∂H
⇒ part 0 part ext 0 part ext2
ext
| |
( )
| |
µ µ
= ∇ = m ∇
F m H H
H
As per definition, the gradient vector ∇Hext2 points into the direction of the highest increase, the direction of the current density itself, which is due to the radially decaying field ~r−1. Therefore, to achieve forces pushing particles from one side of the geometry to the opposite one, it is necessary to uncouple the direction of the magnetic moment from the inhomogeneous wire field. For this purpose, the whole setup is brought into a strong homogeneous magnetic field pointing perpendicular to the plain of the microfluidic structure, compare Figure 2.10(a). Such a field does not exert any force onto the particles but fixes the magnetization direction along the z-axis and we may write mpart =mzmˆ . Thus, formula (2.26) describing the force onto the particle simplifies to
ext part 0( part ) ext 0mz
µ µ ∂ z
= ∇ =
∂
F m H H (2.27)
The force in y-direction is determined by the derivative ∂zHy. The numerical simulations presented in Figure 2.11 show that this component is the only derivative contributing in the centre of the channel geometry; all other components are on the size scale of numerical noise.
Combing equation (2.27) with equation (2.14), the velocity of particles passing the separation area depends on two particle properties:
the size and the magnetization.
Therefore, the microfluidic geometry can be used as a separation device for particles: We assume that all particles enter the geometry through the lower entrance B. For the considerations here, the particle concentrations are chosen sufficiently low so that
Concentration dynamics:
, 0
6 k TB
c c c
t πηR
∂ − ∆ +〈 ∇ 〉=
∂ v with mag
6πηR
= + F
v u on Ω
1 / 0
c= c= inlet concentration on ∂ΩA / ∂ΩB
ˆ,D c 0
〈n ∇ 〉= only convective outflow on ∂Ωex
ˆ, D c c 0
〈n − ∇ +v 〉= insulation elsewhere where the velocity u follows from
η p
∆ = ∇u and ∇ =u 0 on Ω
4(1 ) max
ux = −s s u⋅ , uy =0 Poiseuille profile with on ∂ΩA
4(1 ) max
ux = −s s⋅ξu , uy =0 parametrization s ∈ [0,1] on ∂ΩB
(
( ) ( ( ))T)
ˆ 0η ∇u r + ∇u r n= Neutral-flow-condition on ∂Ωex 0
=
u ‘No slip’-condition elsewhere
which provides the parameters T = 300 K, ρ = 998.2 kg/m³ and η = 1.002 mPa s. If we denote the inlet boundaries by ∂ΩA and ∂ΩB and summarize the exits by ∂Ωex, the complete set of equations is given by
Since the flow profile is stationary, we focus on stationary solutions of the equation system only.
For small particles, all equations are discretized using quadratic Lagrangian elements except for the pressure p which is approximated via linear functions. Analyzing particles of a size exceeding ~ 0.5 µm leads to convection dominated systems; diffusive fluxes play a minor role.
Therefore, for these particle sizes a Petrov-Galerkin approach is employed for the numerical discretization of the Stokes equation to guarantee convergence.
The results of the simulations are shown in Figure 2.12, which presents the relation between particles leaving the geometry through the upper exit and the inflow concentration with respect to the particle size is shown. If all particles leave through the lower exit a value of “1” results, whereas a value of “0” is obtained if all particles leave through the lower exit instead. Therefore, these simulations determine what particles of which sizes can be separated for the given set of parameters.
The bigger the junction area (d → 320 µm, h → 320 µm), the more important the diffusive effects acting on the particles become. In these cases, very small particles can always be found at both exits if a relation factor of ξ = 1.5 is chosen, resulting in a bad separation yield of the device. However, an increase of the velocity inflow ratio ξ can always suppress particle diffusivity (Figure 2.12(b), (d), (e)). Thus, the proposed microchannel geometry is suitable for even particles on the size scale of several 10s of nanometers. Furthermore, in all cases the size interval of particles that can be found in both exits is very narrow for high ξ. Therefore, it is also possible to separate particles with desired accuracy by a proper adjustment of the fluidic and geometric parameters.
Figure 2.12: Relation between mass flow through upper and lower exit. A value of “1” corresponds to all particles reaching the upper part of the geometry, a value of “0” indicates no particle switching from the down to up. The plots show the dependence of the separation properties on different system parameters.
Values not explicitly given coincide with the reference values d = 200 µm, h = 80 µm, ξ = 1.5, udown = 100 µm/s and MS∂zHy = 10000 kA/m2.
The experimental verification of the separation device has been realized by F. Wittbracht in the framework of his master thesis [FWit09]. All experiments were performed with an optical microscope and an attached CCD-camera. The setup enables the simultaneous recording of images and current through the conducting lines at 8 frames per second. For the creation of a homogeneous external magnetic field in z-direction, a cylindrical coil was used. The magnetic gradient field is generated by conducting lines
on the microfluidic chip as shown in Figure 2.10. Samples are prepared using optical laser lithography and magnetron sputtering for gold conducting lines and UV-lithography for microfluidic channels. The epoxy-based resist SU-8 25 is used as channel material due to its mechanical and chemical stability [CLin02]. As the fluid flow is generated by hydrostatic pressure, a fixed velocity ratio ξ is difficult to realize experimentally when using two inlet
two channel branches as shown schematically in Figure 2.10, with an angle α = 26,6° obtained also from finite element calculations. For size separation experiments, a mixture of Dynabeads®
MyOne™ (1.05 µm diameter) and M-280 (2.8 µm diameter) superparamagnetic beads is used.
Both bead species have narrow very size distributions (CV ≈ 1.9 %) and comparable susceptibilities [GFon05]. The homogeneous magnetic field is adjusted to 557 Oe which is the maximum of the described setup. For these field values, both bead types are almost completely saturated. This verifies the assumption of the simulations of saturated magnetic beads and therefore validates (2.27). The current applied through the conducting lines is varied during experiments. Employing the particle parameters for the simulations similar plots can be obtained as shown in Figure 2.13.
The maximum velocity value umax is 400 µm/s. In this case, the geometry parameters are chosen as d = 160 µm and h = 80 µm. For a separation at higher flow rates, the geometry parameters need to be adjusted in respect to Fig. 2.12. Therefore, a separation is theoretically possible at any given flow rate and only limited by experimental constraints. Due to the experimental realization, the bead solution fills both channels and the analysis of experimental results needs to be based on single bead tracking. Manual tracking is achieved using ImageJ [ImagJ] and the MTrackJ plugin [MTrak]. In order to grant comparability of the data, tracks of beads at similar positions in the channel are evaluated (Figure 2.14(d)). M-280 and MyOne™
beads enter and leave the separation region through the lower channel (Figure 2.14(a)) if no current is applied to the (ring-shaped) conducting lines. When a current of 180 mA is applied, it leads to a magnetic force which drags the M-280 beads to the upper channel exit (compare Figure 2.14(b)). At the same current, MyOne™ beads enter and leave through the lower channel exit as shown in Figure 2.14(c). Figure 2.15 is obtained by observing the behaviour of bypassing beads and shows the ratio of beads flowing from B to A. It is clearly demonstrated that at a current of 180 mA the M-280 can be completely separated from the MyOneTM. The small degree of impurity (18 %) can be attributed to a factor ξ too small to completely suppress a volume flow from B to A.
Figure 2.15: Comparison between bead flows from B to A for different electric currents and bead species.
Figure 2.14: Typical bead trajectories through the separation region for different bead species and currents. (a) M-280 beads enter and leave the separation region through the lower channel at 0 mA. (b) These beads can be dragged to the upper channel by applying a current of 180 mA to the conduction lines, while MyOneTM enter and leave through the lower channel. (d) shows a superposition to ease the comparison.
Figure 2.16: Microfluidic lab-on-a-chip geometry consisting of one inlet reservoir (left) and two outlets (right). In the centre, a reaction chamber for e.g. particle supply or chemical reactions is embedded. The streamline course of the product leaving the reaction chamber can be found in the subplots and determines the particle trajectory for particles of radius down to several 100 nm. The two junctions J1 and J3 can be used to manipulate the flow in the separation area on the right side in the channel segments A and B (see below). At junction J2, the focusing of the particle beam takes place. Additionally, J2 is optimized t to suppress particle diffusion, preventing particle flow into the channel segment A.