## 3 Rapid Electrokinetic Patterning
### Opto-Electrokinetic Physics of REP  
The REP-based manipulation of target particles is enabled by the three physical phenomena mentioned previously, i.e., interfacial polarization of particles, long-range EHD flow and ET fluid motion. The interfacial polarization of particles occurring by the interactions of the particle surface  
Figure 8: Hybrid features of REP. (**a**) Particle aggregation by the simultaneous application of a uniform AC electric field and a laser illumination. (**b**) Irregular distribution of particles on an electrode surface observed when turning off the applied laser only. (**c**) Convection of particles caused by turning off the applied electric field only. Reprinted/adapted by permission from (the RightsLink Permissions Springer Customer Service Centre GmbH): (Springer Nature) (Rapid Electrokinetic Patterning) by ([PERSON], [PERSON], [PERSON]) (COPYRIGHT) (2012). charges and ions in a fluid is typically characterized by the Maxwell-Wagner polarization frequency (M-W frequency) [58; 62; 72]. The M-W frequency defines the behavior characteristic of ions in an electric double layer (EDL) surrounding a particle surface [72; 117]. At AC frequencies below the M-W frequency, cations and anions in the EDLs of particles are completely separated, forming dipoles. The dipoles give rise to constant electrostatic repulsion and attraction when the polarized particles are located on the electrode. The electrostatic repulsion and attraction are the forces acting between the particles themselves and between the particles and the electrode surface, respectively. At AC frequencies above the M-W frequency, the ions do not have enough time to move and accumulate in response to the applied AC electric field. Accordingly, no dipoles form, and interfacial polarization does not happen. Recently, [PERSON] et al. used Voronoi diagrams and Delaunay triangulations to confirm that another polarization mechanism exists in the frequency region below the M-W frequency [58; 118]. According to their experimental observation and topological analysis, the repulsive force between the polarized particles decreases with AC frequency even in the frequency band below the M-W frequency. This phenomenon is attributed to the non-equilibrium polarization of ions in the EDLs of the particles. It is closely associated with the critical frequency of REP, which will be introduced later [58]. EHD flow, a second mechanism, is one form of fluid flow that is induced by the mutual attraction which an applied electric field exerts on ions in a fluid. According to studies by [PERSON] et al. and [PERSON] et al., the flow affects the motion of particles on an electrode surface in an AC frequency region above 500 Hz [119; 120; 121; 122; 123; 124; 125; 126]. It gathers the particles laterally, lifting them up slightly from the electrode surface. ET flow, a third mechanism, is driven by the coupling of the temperature-dependent gradients of electrical conductivity and permittivity with an applied electric field (Figure 9) [114; 115; 116]. The axisymmetric non-uniform temperature distribution in a fluid is produced by the highly focused laser beam, which is located on either a top or bottom electrode surface. The temperature gradient in turn induces spatial variations in the physical and electrical properties of the fluid such as density, permittivity and conductivity. The density gradient causes natural convection. Gradients in the electrical conductivity and permittivity produce an ET flow in an applied electric field. The natural convection is generally ignored in microfluidic systems where electric forces are present. Therefore, the ET flow is responsible for particle transport in REP process. The physical description of the flow is given by the time-averaged ET body force in Equation (15) [72; 117].