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Dynamics of superparamagnetic and ferromagnetic nano-objects in
continuous-flow microfluidic devices
Clime, Liviu; Le Drogoff, Boris; Veres, Teodor
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https://nrc-publications.canada.ca/eng/view/object/?id=9f4bb239-fdf1-4cec-9996-9ebb4a4441dd https://publications-cnrc.canada.ca/fra/voir/objet/?id=9f4bb239-fdf1-4cec-9996-9ebb4a4441ddIEEE TRANSACTIONS ON MAGNETICS, VOL. 43, NO. 6, JUNE 2007 2929
Dynamics of Superparamagnetic and Ferromagnetic Nano-Objects
in Continuous-Flow Microfluidic Devices
Liviu Clime, Boris Le Drogoff, and Teodor Veres
NRC, Industrial Material Institute, Boucherville, QC J4B 6Y4, CanadaWe present a numerical study on the dynamics of magnetic micro- and nano-objects subjected to magnetic field gradients in con-ventional continuous-flow microfluidic devices. By a mixed finite-element/discrete-element approach we solve the equations of the field driven motion for magnetic nano-objects floating in liquids at very low Reynolds numbers and compare the magnetic trapping effi-ciency of commercially available superparamagnetic microbeads to that of ferromagnetic nanowires. The drag force and the remanent magnetization of ferromagnetic nanowires are found to be responsible for the huge increase of their magnetic trappability whereas the slip-length associated with the Navier boundary condition at the transition to the nanoscale regime is found to be a much less important parameter.
Index Terms—Ferromagnetic nanowires, magnetic separation, microfluidics, superparamagnetic microbeads.
I. INTRODUCTION
M
AGNETOPHORESIS has been shown to be a promising technique for separation of magnetically “tagged” DNA, proteins, or cells. An emerging class of applications that make use of magnetic nano-objects is the identification of biological species in Micro Total Analysis Systems ( TAS) [1] or tion devices [2], [3]. A typical application of magnetic separa-tion in a lab-on-a-chip system consists of a microfluidic channel provided with one or more magnetic elements at the bottom wall [4]–[6]. Magnetic micro- or nano-objects coated with specific biomolecules are then flowed through the micro-channel and confined in the regions of high field-gradient generated by the magnetic elements. The efficiency of this process is dependent not only on the degree of selectivity with which the chemical or biological targets can be attached to the magnetic carriers but also by the efficiency of the magnetic manipulation and capture [5], [6]. While the selectivity and the specificity to the targets is related to the surface functionalization and can be solved by the right choice of the chemical or linker attached to the carrier sur-face, the efficiency of the magnetic manipulation relies upon the design of both confinement device and magnetic carriers. Ma-nipulation and capture of magnetic carriers in flow conditions require a careful design of the magnetic field gradients and ap-propriate flow conditions for the magnetic carriers. Numerical simulations aimed to find accurate solutions for the differential equations of motion of the magnetic carriers subjected to both magnetic field gradient and flow conditions can provide an in-creased understanding and consequently a better design of these devices [7].Most of the experimental and theoretical work related to the manipulation of magnetic carriers in continuous-flow magnetic devices was done for superparamagnetic microbeads. Relatively few works are related to magnetic nano-objects (nanobeads or nanowires). The present study is aimed to give an insight into
Digital Object Identifier 10.1109/TMAG.2007.892322
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the main aspects of the transition toward the nanoscale regime by a theoretical comparison between the capture probabilities of classical superparamagnetic microbeads and ferromagnetic nanowires in the same microfluidic device.
In the following sections, the Poisson equations related to the magnetostatic problem for the magnetic field and the Stokes equation for the liquid flow in a conventional microfluidic de-vice are numerically solved by finite elements [8]. The equa-tions of motion for the particles consist of a system of coupled second-order ordinary differential equations and are solved by a finite difference algorithm [9]. The full trajectories of particles are modeled and the resulting trappabilities of superparamag-netic (SP) microbeads and ferromagsuperparamag-netic nanowires compared at different flow velocities.
II. THEORETICALCONSIDERATIONS
We consider a laminar liquid flow between two parallel plates (see Fig. 1) along the axis. At below the bottom plate out-side the microfluidic channel (C), there is a micro-electromagnet consisting of a ferromagnetic Ni micropost P of diameter and a current carrying solenoid S made of a conducting nonmagnetic metal.
In order to find the magnetic field inside the microchannel, one can take advantage from the axial symmetry of the electro-magnet and find the electro-magnetic field by solving the 2-D problem in the plane , where is the circumferential component of the magnetic vector potential
(so that ), the current density carrying the
solenoid and the absolute magnetic permeability of the post. In the absence of any inertial contribution to the motion of micro- or nano-objects in viscous liquids, the law of motion for these particles can be obtained from the static equilibrium condition between the magnetic force and the drag force [2], [5], [10]. For spherical SP beads, this condition may be expressed as
(1) where is the liquid viscosity, the particle radius, and the liquid velocity. in the expression above represents the magnetic force acting upon the particle and is expressed as
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Fig. 1. Longitudinal section of the electromagnetic separation device used in this study. The micropost (P) symmetry axis coincides withOz axis and it is magnetized by the magnetic field generated by a metallic (and nonmagnetic) solenoid S. Qualitative graphical representations of the magnetic field gradient r ~H inside the microfluidic channel and the micropost magnetization vector are shown in the scalar and vector cut sections, respectively.
(2) where is the magnetic susceptibility of the bead.
When ferromagnetic nanowires are employed as magnetic carriers, the drag force will depend on the nanowire orientation relative to the liquid flow and the equations of motion become more complicated than (1) as the drag force and the rel-ative velocity of the liquid flow are not collinear any-more. If we denote with and the unit vectors of the local magnetic field and particle relative velocity with respect to the liquid flow respectively, the drag force acting upon a fer-romagnetic nanocylinder oriented parallel to the magnetic field direction can be written as
(3) Analytical expressions for and may be obtained by approximating the cylinder with prolate spheroids or elongated rods [10], [11]. As the magnetic objects reduce theirs dimen-sions, Navier boundary conditions must be considered at the solid-liquid interface and (3) has to be corrected by a factor that takes into account the finite value of the slip length at the tran-sition toward the nanoscale regime [12].
The magnetic force acting upon ferromagnetic nanowires de-pends on its total magnetic moment and on the magnetic field
gradients where . When nanowires are
parallel to the magnetic field direction, the magnetic force may be expressed as
(4) The value of the total remanent magnetic moment for each fer-romagnetic object is obtained by full numerical micfer-romagnetic simulations [13] and due to the relatively large values of the
coercive fields, this value hardly change under the influence of the magnetic field generated by conventional micro-electromag-nets . As for the liquid velocity in (1), when the mi-crochannel is large enough on the direction, we can use a second degree polynomial function (Poiseuille flow) whose coefficients are computed from the no-slip boundary con-ditions at the surface of the walls and at the middle distance between the walls [10].
A number of magnetic objects (microbeads or nanowires) are launched from different upstream points of the mi-crochannel and the equations of motion solved iteratively for
(microchannel floor) and , with the maximum time allowed for the simulation. If
while then the particles are considered as trapped. If while then the particles are considered as escaped.
The probability of capture (or trapping ratio) is given by
(5) where represents the number of trapped particles. This quantity is evaluated for uniformly increased flow velocity values and the dependence , that is the dependence of capture probability on the liquid flow velocity, evaluated for superparamagnetic microbeads and ferromagnetic nanowires. In order to compare different types of magnetic carriers, we define a critical velocity as the maximal velocity of the liquid at which only 50% of the total launched carriers fall down on the channel floor.
III. RESULTS ANDDISCUSSIONS
We are considering three types of magnetic objects that have been proposed as potential carriers for magnetic microma-nipulation: i) Dynabeads MyOne microbeads [14] of 1 m in diameter and magnetic susceptibility ; ii) cylindrical Permalloy nanowires of 1 m length and radius of 40 nm (PermL1000); and iii) cylindrical codebar nanowires with the total length of 1 m containing a cylindrical ferromagnetic in-clusion of 100 nm length in the middle (NiL100). The geometry of the microfluidic configuration is defined by the following parameters (see Fig. 1 for their meaning): m
m m m m. The density of the
electrical current carried by the solenoid is A/m . The particles are launched from points situated in the plane m, far enough from the electromagnet in order to consider that at these points the particles are driven by drag forces only. Launch points are uniformly distributed in
this plane from m to m inside
the microfluidic channel C whose thickness on direction is considered much larger than 200 m. The dependence of the magnetic trappability on the liquid maximal velocity for the superparamagnetic microbeads and ferromagnetic nanowires described above is shown in Fig. 2.
As we can see in this figure, even at very slow flows m/s only 75% of the launched SP beads (Dynabeads MyOne, dashed line) are captured. At this velocity, all the metallic nanowires are captured in a proportion of 100%, their
CLIME et al.: DYNAMICS OF SUPERPARAMAGNETIC AND FERROMAGNETIC NANO-OBJECTS IN CONTINUOUS-FLOW MICROFLUIDIC DEVICES 2931
Fig. 2. Dependence of the magnetic trappabilityT on the liquid maximal ve-locityV for Permalloy nanowires of length L = 1 m embedded in prolate nonmagnetic spheroids of various radii (PermL1000), nonmagnetic nanowires of 1m length containing a single magnetic segment of 100 nm Ni in the middle (NiL100) and commercial Dynabeads MyOne superparamagnetic microbeads [14] of 1m diameter.
dependencies being shifted toward larger velocities with at least two orders of magnitude. The critical velocity is 7.5 m/s for the Dynabeads, 25.7 m/s and 4.4 mm/s for NiL100 and PermL1000 nanowires, respectively. Obviously, the large difference between SP microbeads and ferromagnetic nanowires originates mainly in the permanent-magnet moment of ferromagnetic materials. In order to evaluate the contribution of the drag force on the particle trappabilities, we consider the m Permalloy nanowire embedded in nonmagnetic prolate spheroids (see Fig. 2) with uniformly increased semi-minor axes from 40 to 500 nm. As expected, larger spheroids undergo larger drag forces and the trappability is decreased accordingly: the critical velocity diminishes from 4.4 mm/s at nm to 1.2 mm/s at nm that is the critical velocity increases about 4 times if the drag force becomes about 12 times smaller. According to [12], the correction factor for the drag force due to the transition to the nanoscale regime is in the interval [0.66, 1] and hardly changes the trappability of the
nanocarriers. The increase of the trappability for ferromagnetic contiguous or codebar nanowires originates mainly in two factors: their permanent-magnet moment and the important decrease of drag forces at the transition from micro- to the nanoscale regime.
ACKNOWLEDGMENT
This work was supported by the Natural Sciences and Engi-neering Research Council and by the National Research Council of Canada.
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Manuscript received October 30, 2006 (e-mail: liviu.clime@imi.cnrc-nrc. gc.ca)
