An efficient microfluidic sorter: implementation of double meandering micro striplines for magnetic particles switching

Abstract

The ability to trap, manipulate, and separate magnetic beads has become one of the key requirements in realizing an integrated magnetic lab-on-chip biosensing system. In this article, we present the design and fabrication of an integrated magneto-fluidic device for sorting magnetic particles with a sorting efficiency of up to 95%. The actuation and manipulation of magnetic beads are realized using microfabricated square meandering current-carrying micro striplines. The current is alternated between two neighboring micro striplines to switch the magnetic beads to either one of the two outlets. We performed a series of parametric study to investigate the effect of applied current, flow rate, and switching frequency on the sorting efficiency. Experimental results reveal that the sorting efficiency is proportional to the square of current applied to the stripline, and decreases with increasing buffer flow rate and switching frequency. Such phenomena agree well with our theoretical analysis and simulation result. The fastest switching rate, which is limited by the microchannel geometry and bead velocity, is 2 Hz.

Appendices

Appendix 1: Magnetic forces generated by square meandering micro stripline

In order to have a better understanding of the magnetic field strength and the magnetic forces acting on magnetic beads, we utilized the 3-D magnetostatics toolbox in COMSOL multiphysics (COMSOL Inc, USA) to simulate the magnetic field distribution of the micro stripline. Figure 7 shows the COMSOL multiphysics simulation results for the magnetic flux density (a) B _x , (b) B _y , and (c) B _z with an applied current of 2 A and at a height of 10 µm from the copper stripline surface. The stripline has a width of 50 µm and the height-width ratio is 3. The current is set to flow in the left to right configuration. The highest magnetic flux density for the x-, y-, and z-component is approximately 18 mT. In the simulation, we made the following assumptions: the magnetic field is only generated by the current applied to the stripline, the magnetic field is measured in free-space, the current flow is steady, and the magnetic field is not permeating the air box surface. The boundary conditions of the encapsulating box were set to magnetic insulation and electric insulation while the boundary condition of the stripline was set to continuity.

Fig. 7

Contour plot of the a x-, b y-, and c z-component magnetic flux density at an applied current of 2 A and at a vertical distance of 10 μm from the stripline. The width of the square meandering micro stripline is 50 µm and its height–width ratio is 3. The maximum magnetic flux density generated with 2 A current is approximately 18 mT

Magnetic beads are attracted to the magnetic field generated by the current-carrying micro stripline. The magnetic force acting on a magnetic bead is governed by (Pamme 2005; Shevkoplyas et al. 2007; Beyzavi and Nguyen 2008):

$$ F = {\frac{V \Updelta \chi}{\mu_0}} ({\mathbf{B}} \cdot {\user2{{\nabla}}}) {\mathbf{B}} = F_{x} {{\mathbf{i}}} + F_{y} {{\mathbf{j}}} + F_z {{\mathbf{k}}}, $$

(13)

where

$$ F_x = {\frac{V \Updelta \chi}{\mu_0}} \left( B_x {\frac{\partial{B_x}} {\partial{x}}} + B_y {\frac{\partial{B_x}}{\partial{y}}} + B_z {\frac{\partial{B_x}}{\partial{z}}}\right), $$

(14)

$$ F_y = {\frac{V \Updelta \chi}{\mu_0}} \left( B_x {\frac{\partial{B_y}} {\partial{x}}} + B_y {\frac{\partial{B_y}}{\partial{y}}} + B_z {\frac{\partial{B_y}}{\partial{z}}}\right), $$

(15)

$$ F_z = {\frac{V \Updelta \chi}{\mu_0}} \left( B_x {\frac{\partial{B_z}} {\partial{x}}} + B_y {\frac{\partial{B_z}}{\partial{y}}} + B_z {\frac{\partial{B_z}}{\partial{z}}}\right), $$

(16)

V is the volume of the magnetic bead, $\Updelta \chi$ is the difference in magnetic susceptibilities between the magnetic particle and its surrounding buffer, B is the magnetic flux density, and μ₀ is the magnetic constant $4\pi \times 10^{-7}\, \hbox{T\,mA}^{-1}.$ After obtaining the magnetic flux density from COMSOL, we can compute the force acting on a magnetic bead using Eq. 14–16. The partial derivative of B _x , B _y , and B _z in the x-, y- and z-direction can be computed using the matrix operation as follows:

$$ {\frac{\partial{A}}{\partial{x}}} = MA^T,\; {\frac{\partial{A}} {\partial{y}}} = MA,\; {\frac{\partial{A}}{\partial{z}}} = A_{k+1} - A_k, $$

(17)

where A represents B _x , B _y , and B _z , k is the index representing the kth simulation plane in the z-axis and

$$ M = \left( \begin{array}{lllll} -1 & 1 & & & \\ & -1 & 1 & & \\ & & \ddots & \ddots & \\ & & & -1 & 1 . \end{array}\right) $$

(18)

Figure 8 shows the simulation results for the variation of the maximum y-component of $({\mathbf{B}} \cdot {\varvec{\nabla}}){\mathbf{B}},$ which is proportional to the magnetic bead force in the y-direction, F _y , with the applied current at a vertical distance of 5, 10, and 15 µm from the stripline surface. From the figure, we can deduce that the maximum y-component magnetic force is proportional to the square of the applied stripline current. The magnetic bead forces decreases with height from the stripline surface. The simulation results for the maximum magnetic forces of x- and z-components also show similar variation trend with the applied current (not shown).

Fig. 8

Simulation results for the effect of applied current on the maximum y-component of $({\mathbf{B}} \cdot {\varvec{\nabla}}){\mathbf{B}},$ which is proportional to the magnetic bead force in the y-direction, F _y , at a vertical distance of 5, 10, and 15 µm from the stripline surface. The maximum y-component magnetic force increases with the square of the applied stripline current. Further from the stripline surface, the magnetic forces on the magnetic beads are weaker. The inset is a schematic illustration of the striplines configuration in the microfluidic sorting chamber

Appendix 2: Gravitational settling and wall effect

The magnetic beads tend to sink to the bottom of the channel surface due to the density difference and gravity effect (Li and Daghighi 2010). The sediment velocity of a particle is given by (Huh et al. 2007):

$$ U_{\rm sed} = {\frac{2r^2g \Updelta \rho}{9 \mu}}, $$

(19)

where r is the radius of particles, g is gravitational acceleration, $\Updelta \rho$ is the density difference between particle and carrier liquid, and μ is the viscosity of buffer. With the diameter of 8 µm and density of the magnetic beads is $1.07 \,\hbox{g\;cm}^{-3},$ and carrier liquid density of $1.126 \,\hbox{g\;cm}^{-3}$ and viscosity of 9.01 × 10⁻³ Pa s, the sedimentation velocity is $2.17 \times 10^{-7} {\hbox {ms}}^{-1}.$ Without the presence of the magnetic force, the time taken for a bead to sink to the bottom surface for a 25 µm high channel is about 115 s. Since the lowest flow rate used in the experiment is 200 µl h⁻¹, the slowest average time taken for a particle to travel past the sorting chamber of length 3,500 µm approximately 1.6 s. Hence, the gravitational settling effect is negligible.

In addition, due to the low channel height to bead diameter ratio, there bound to be increased hydrodynamic drag force caused by the wall effect in the vertical direction. However, the wall effect is insignificant in the lateral direction (x − y plane) since the width and length of the sorting chamber are 1,000 and 3,500 µm, respectively, while the bead diameter is only 8 µm. The success rate of sorting the beads to the desired outlet depends on the ability of the device to attract the beads in the y-direction. Therefore, the wall effect on the hydrodynamic force is not considered in this study.