Geometrical optimization of microstripe arrays for microbead magnetophoresis

Abstract

Manipulation of magnetic beads plays an increasingly important role in molecular diagnostics. Magnetophoresis is a promising technique for selective transportation of magnetic beads in lab-on-a-chip systems. We investigate periodic arrays of exchange-biased permalloy microstripes fabricated using a single lithography step. Magnetic beads can be continuously moved across such arrays by combining the spatially periodic magnetic field from microstripes with a rotating external magnetic field. By measuring and modeling the magnetophoresis properties of thirteen different stripe designs, we study the effect of the stripe geometry on the magnetophoretic transport properties of the magnetic microbeads between the stripes. We show that a symmetric geometry with equal width of and spacing between the microstripes facilitates faster transportation and that the optimal period of the periodic stripe array is approximately three times the height of the bead center over the microstripes.

FIG. 1.

(a) Illustration of the periodic stripe system, the used coordinate system, and the magnetic field, B_s, from the array of magnetic stripes. Red colors indicate a high field magnitude. (b) x-component and (c) z-component of the magnetic field from the stripes for different height-to-period ratios. The black bars indicate the location of the stripes. The calculations were carried out for a ferromagnetic layer of thickness t_FM/λ = 0.003 and a volume magnetization M.

FIG. 2.

Illustration of concept for magnetic bead magnetophoresis across the array of magnetized stripes. The stripes are magnetized in the x-direction and extend to infinity in the y-direction. The superposition of the external magnetic field that rotates in the xz-plane and the static field from the magnetized stripes creates an energy minimum that travels across the stripe array along the positive x-direction and drags the beads (labeled as 1 and 2) along. The potential energy landscape for a bead for each field condition is illustrated with the dashed curve.

FIG. 3.

Illustration/picture of the experimental setup.

FIG. 4.

Example of a frequency sweep for a stripe geometry with w = s = 6 μm. The line is a fit of the binomial distribution with probability given by Eq. (8) to the data with the parameters given in Table I for chip 2. The inset shows an example of an analyzed frame with stationary beads (red circles), phase-locked beads (green circles), and beads that could not be categorized (blue circles).

FIG. 5.

Maximum magnetophoretic velocity as function of the stripe width w for stripe geometries with (a) equal width and spacing, (b) constant spacing, and (c) constant period. The blue circles are measured velocities, and the red areas are simulated using ξ_M–270 = 54 μm²/Pa s with no fitting parameters. The error bars correspond to ± f_σ · λ.

FIG. 6.

Contour plot of simulations of the maximum magnetophoretic bead velocity, f_{V =1} · λ, as function of w and s. The black lines and circles indicate the simulated data slices and parameters for the experimental data points in Fig. 5. A corresponding figure including all experimental values is given as supplementary Figure S2. The simulations were done for M-270 beads with ξ_M–270 = 54 μm²/Pa s, a rotating external magnetic field with B₀ = 5 mT and stripes with thickness t_FM = 15 nm.

FIG. 7.

Contour plot of the maximum bead velocity as function of stripe period, λ, and bead height z. The dashed lines correspond to z = 0.13λ and z = 0.32λ. The data were calculated for B₀ = 5 mT, t_FM = 1 nm, w = s = λ/2, and ξ_M–270 = 54 μm²/Pa s. The black lines and circles indicate the simulated data slices and parameters for the experimental data points in Fig. 5(a).

FIG. 8.

Maximum magnetophoretic bead velocity, f_V = 1 · λ, and the optimal period as function of the height z of the bead center over the stripe array, λ_optimal = 3.1z. The simulations were done with t_FM = 15 nm for a rotating external magnetic field with B₀ = 5 mT. The bead magnetophoretic mobility was assumed equal to ξ = ξ_M–270 r²/1.4 μm².