Microfluidic mixing by magnetic particles: Progress and prospects

Abstract

Microfluidic systems have enormous potential for enabling point-of-care diagnostics due to a number of advantages, such as low sample volumes, small footprint, low energy requirements, uncomplicated setup, high surface-to-volume ratios, cost-effectiveness, etc. However, fluid mixing operations are constrained by molecular diffusion since the flow is usually in the laminar regime. The slow nature of molecular diffusion is a technological barrier to implementing fluid transformations in a reasonable time. In this context, magnetically actuated micro-mixers of different sizes, shapes, materials, and actuation techniques provide a way to enhance fluid mixing in microfluidic devices. In this paper, we review the currently existing micro-mixing technologies. From a fundamental perspective, the different magnetization models for permanent and induced dipoles are discussed. The single-particle dynamics in steady and oscillating magnetic fields is studied in order to determine the flow generated and the torque exerted on the fluid due to the magnetic particles. The effect of particle interactions, both magnetic and hydrodynamic, is examined.

FIG. 1.

Magnetization curve for non-hysteretic models: the brown dashed line—permanent dipole, black—signum, red—linear, and blue—Langevin models.

FIG. 2.

The equilibrium moment components (a) parallel and (b) perpendicular to the field for the orientations that make the following angles with the field $0$ (red), $\pi /12$ (black), $\pi /3$ (blue), $5\pi /12$ (brown) and $\pi /2$ (cyan).

FIG. 3.

Types of spheroid: (a) Thin disk $B=-1$. (b) Oblate spheroid $(-1<B<0)$. (c) Sphere $(B=0)$. (d) Prolate spheroid $(0<B<1)$. (e) Thin rod $(B=1)$.

FIG. 4.

Spheroid subjected to shear and constant field. $\hat{\mathit{z}}$ is the orientation vector of the spheroid. (a) $\hat{\mathit{z}}$ makes angle $\theta$ with the $Z$ axis and its projection with $-Y$ subtends angle $\varphi$. Shear is applied in the $X$– $Y$ plane with velocity and gradient in $X$ and $Y$ directions, respectively. (b) $\xi$ and $\eta$ are the meridional and azimuthal angles of the field, H. Reproduced with permission from V. Kumaran, Phys. Rev. Fluids 6, 063701 (2021). Copyright 2021 American Physical Society.

FIG. 5.

The evolution of the phase portrait in the $\theta -\varphi$ plane with $\mathrm{\Sigma }$ for a prolate spheroid with $B=0.6$ in the presence of shear and a streamwise external field $(\xi =\pi /2,\eta =\pi /2)$ : $\mathrm{\Sigma }=$ (a) $0.6$, (b) $\sqrt{0.48}$, (c) $0.75$, and (d) $0.8$. The blue solid circles are stable nodes, the red dashed circles are unstable nodes, the black solid circles are centers, and the brown + are saddle points. Reproduced with permission from V. Kumaran, Phys. Rev. Fluids 5, 033701 (2020). Copyright 2020 American Physical Society.

FIG. 6.

Bifurcation diagram in the $\theta$– $\varphi$– $\mathrm{\Sigma }$ phase space for $B=0.6$ for $\eta =$: (a) $9\pi /20$, (b) $7\pi /15$, and (c) $29\pi /60$. The solid and dashed lines are for equilibrium points in and out of the shear plane, respectively. Blue is for stable and red is for unstable, while brown is for saddle points. The points $B1$, $B2$, $B3$, and $B4$ are bifurcation points. Reproduced with permission from V. Kumaran, Phys. Rev. Fluids 5, 033701 (2020). Copyright 2020 American Physical Society.

FIG. 7.

$\theta -\varphi$ phase portrait of the thin rod $(B=1)$ particle dynamics for $\mathrm{\Sigma }=0.8$ with field parallel to the shear plane $(\xi =\pi /2)$ at $\eta =$ (a) $2\pi /5$, (b) $9\pi /20$, (c) $\pi /2$, and (d) $11\pi /20$. The blue and red solid circles are stable and unstable nodes. The brown $+$ represents the saddle points. The brown dashed lines are the separatrix of the complex system, while the blue solid lines and red dashed represent the loci of stable and unstable points, respectively. These figures are reproduced with permission from V. Kumaran, Phys. Rev. Fluids 5, 033701 (2020). Copyright 2020 American Physical Society.

FIG. 8.

Bifurcation diagram: (a) in the ${\mathrm{\Sigma }}_s-\eta$ plane for a prolate spheroid with $B=0.8$ with the signum model magnetization in the presence of shear and a parallel magnetic field and (b) between a rotating limit cycle (under the black line) and one stable node (above the black line but outside the red island) and two stable nodes (inside the red island). Reproduced with permission from V. Kumaran, Phys. Rev. Fluids 6, 043702 (2021). Copyright 2021 American Physical Society.

FIG. 9.

Phase portrait in the $\theta -\varphi$ plane for $B=0.8$ for the signum model in a parallel magnetic field and shear at different locations in Fig. 9: (a) $\eta =\pi /6$ and ${\mathrm{\Sigma }}_s=0.5$ at $P1$, (b) $\eta =\pi /3$ and ${\mathrm{\Sigma }}_s=0.5$ at P2, (c) $\eta =9\pi /20$ and ${\mathrm{\Sigma }}_s=0.9$ at $P3$, (d) $\eta =19\pi /40$ and ${\mathrm{\Sigma }}_s=0.8$ at $P4$, (e) $\eta =2\pi /3$ and ${\mathrm{\Sigma }}_s=0.5$ at $P5$, (f) $\eta =21\pi /40$ and ${\mathrm{\Sigma }}_s=0.6$ at $P6$, and (g) $\eta =11\pi /20$ and ${\mathrm{\Sigma }}_s=0.2$ at $P7$. The stable and unstable steady points are denoted by the blue and red circles, respectively. The blue lines are the stable limit cycle and the brown $+$ are saddle points with separatrices marked by the brown dashed lines. Reproduced with permission from V. Kumaran, Phys. Rev. Fluids 6, 043702 (2021). Copyright 2021 American Physical Society.

FIG. 10.

The bifurcation diagram in the ${\mathrm{\Sigma }}_s-\eta$ (red lines) and $\mathrm{\Sigma }-\eta$ (blue lines) planes for the transition from rotating to steady states for the Langevin magnetization model in the presence of shear and a magnetic field in the shear plane for (a) $B=0.2$, (b) $B=0.5$, (c) $B=0.8$, and (d) $B=0.95$. Below the line is the rotating state, while above it is the stable fixed point. The blue and red solid lines are for linear and signum models, respectively. The dashed lines are for the Langevin model with $\mathrm{\Sigma }/{\mathrm{\Sigma }}_s$ = $0.3(°)$, $1(\mathrm{△})$, $3(▽)$, $10(◃)$, $30(▹)$, and $100(\diamond )$. Reproduced with permission from V. Kumaran, Phys. Rev. Fluids 6, 043702 (2021). Copyright 2021 American Physical Society.