## 2 Microfluidic Manipulation Techniques
### Acoustophoresis  
Acoustic waves with varying pressure profiles induce the movement of suspended particles relative to a stationary fluid, exerting an acoustic force on the particles. Such particle migration is defined as acoustophoresis (Figure 5) [39,41]. The acoustic force (\(F_{a}\)) on particles is given by [41]  
\[F_{a}=-\left(\frac{\pi P_{0}^{2}V_{p}\beta_{m}}{2\lambda}\right)u(\beta,\rho)sin \left(\frac{2\pi x}{\lambda}\right), \tag{13}\]  
\[u(\beta,\rho)=\frac{5\rho_{p}-2\rho_{m}}{2\rho_{p}+\rho_{m}}-\frac{\beta_{p}} {\beta_{m}}, \tag{14}\]  
where \(V_{p}\) is the volume of a particle, \(x\) is the distance to a nearest pressure node, \(P_{0}\) and \(\lambda\) are the pressure amplitude and wavelength of an acoustic wave, and \(\beta\) and \(\rho\) are liquid compressibility and density, respectively. The subscripts \(p\) and \(m\) indicate a particle and suspending medium, respectively. As seen in the equation, the acoustic force depends on various factors including the magnitude and frequency of the acoustic waves, the size and elasticity of the particle, and the surrounding fluid.  
Particles in the influence of an acoustic wave are pushed to either the pressure nodes or the pressure anti-nodes (Figure 5a) [39,40]. The pressure nodes are the locations where pressure and density do not vary in the wave field. At the nodes, the acoustic wave forms a zero or minimum pressure. The pressure anti-nodes represent the locations where pressure and density show the largest variations. The acoustic wave produces the maximum pressure at the nodes. Suspended particles show acoustic wave-based motion depending on the pressure distribution. Then, the direction of particle motion can be explained based on the sign of the \(u\) factor defined in Equaion (14) (Figure 5b). If \(u\) is positive, the particles move toward the pressure nodes. On the contrary, if \(u<0\), they show migration to the pressure anti-nodes.  
Acoustophoresis provides the ability for size-, density-, and compressibility-based separation of particles [39,49,107]. Compressibility-based separation maximizes the strengths of acoustophoresis. Even though two particles have similar sizes and densities, if one particle is more compressible than a suspending fluid and the other particle is less compressible, their separation is achievable based on the compressibility difference. As such, a mixture of particles can be separated by an acoustic wave in a microchannel, and the separated particle groups can be collected at different outlets in a flowing fluid. A drawback of acoustophoretic manipulation is that it needs to integrate an acoustic wave resonator into a microfluidic structure for miniaturization and portability of the manipulation system [27,107]. In addition, controlling nano-sized particles using acoustophoresis is inefficient because the magnitude of the acoustic force acting on the particles is very small [24,27].  
Figure 4: Magnetophoretic motion of a magnetic particle in a non-uniform magnetic field. (**a**) Positive magnetophoresis in which a magnetic particle migrates toward a higher magnetic field region. (**b**) Negative magnetophoresis in which a magnetic particle moves to a lower magnetic field region. Reproduced from [40], with the permission of AIP Publishing (2012).