## 2 Microfluidic Manipulation Techniques
### Magnetophoresis  
Magnetophoresis is the motion of a magnetic particle relative to a stationary fluid, driven by a non-uniform magnetic field (Figure 4) [35; 36; 37; 38]. Magnetophoretically induced motion depends on the magnetic properties of the particle [38; 40]: diamagnetism, paramagnetism, and ferromagnetism. Diamagnetism represents the property of magnetic materials that are magnetized in the opposite direction to an external magnetic field and are therefore repelled by an approaching magnet [102]. Most materials including water, mercury, copper, lead, gold, and silver are diamagnetic; superconductors have very high diamagnetism. The magnetic permeability of the materials is less than the permeability of free space (\(\mu_{0}=4\pi\times 10^{-7}\) Hm\({}^{-1}\)). Paramagnetism indicates the property of magnetic materials magnetized in a direction generally similar to an external magnetic field; accordingly, they are weakly attracted to an approaching magnet [103]. The magnetized status is not retained in the absence of the applied magnetic field because electron spins are randomized by thermal fluctuations. Representative paramagnetic materials include aluminum, oxygen, titanium, and iron oxide. Their relative magnetic permeability is slightly greater than 1. Ferromagnetism refers to the property of materials strongly magnetized in the same direction as an external magnetic field [38; 103]. The magnetization is retained for a long time after removal of the applied magnetic field. Thus, ferromagnetic materials behave like a permanent magnet. Iron, cobalt, and nickel are included in the category, and their magnetic permeability is very high compared to diamagnetic and paramagnetic materials.  
When the three kinds of magnetic particles are subject to an external magnetic field, they experience a magnetic force (\(F_{mag}\)) given by [104; 105]  
\[F_{mag}=\frac{\left(\chi_{p}-\chi_{m}\right)V_{p}}{\mu_{0}}(B\cdot\
abla)B, \tag{12}\]  
where \(V_{p}\) is the volume of a magnetic particle, \(\mu_{0}\) is the permeability of free space, and \(\chi_{p}\) and \(\chi_{m}\) are the magnetic susceptibilities of a magnetic particle and suspending fluid, respectively. Magnetic susceptibility defines the degree of magnetization of a material by an applied magnetic field [38]. If the magnetic susceptibility of a particle is higher than that of a surrounding fluid, the particle migrates toward a higher magnetic field region. This phenomenon is termed positive magnetophoresis (Figure 4a) [38; 40]. On the contrary, negative magnetophoresis occurs when the magnetic susceptibility of a particle is lower than that of the surrounding fluid (Figure 4b) [38; 40], causing to particle to a lower magnetic field. The definitions of positive and negative magnetophoretic motion of a particle are analogous to the DEP introduced earlier.  
Magnetophoretic manipulation has been often utilized for colloidal separation, sorting, and transport [36; 37; 38; 104; 105; 106]. In applications, magnetophoresis offers advantages for angular positioning and rotation of magnetic particles. Also, it can minimize the risk of particle damage during the manipulation process. However, magnetophoresis involves the magnetic hysteresis of micro-sized particles [55]. It causes the particles to agglomerate together into clusters even when the applied magnetic field is removed.