# Integrated Hybrid Tweezer for Particle Trapping with Combined Optical and Acoustic Forces
## 2 Principle and Model
The optical force \(\mathbf{F_{o}}\) can be calculated by integrating the time-independent Maxwell stress tensor over the surface of particles [42]:  
\[\mathbf{F_{o}}=\oint\limits_{S}(\langle\overrightarrow{T_{M}}\rangle\mathbf{\bullet n })dS, \tag{1}\]  
\[\langle\overrightarrow{T_{M}}\rangle=\mathbf{D}\mathbf{E}^{*}+\mathbf{H}\mathbf{B}^{*}-\frac{ 1}{2}(\mathbf{D}\mathbf{\cdot}\mathbf{E}^{*}+\mathbf{H}\mathbf{\cdot}\mathbf{B}^{*})\mathbf{I}, \tag{2}\]  
where \(\overrightarrow{T_{M}}\) is the Maxwell stress tensor, \(\mathbf{n}\) is the normal vector, \(S\) is the surface of the particle, \(\mathbf{E}\) is the electric field, \(\mathbf{D}\) is the electric displacement, \(\mathbf{H}\) is the magnetic field, and \(\mathbf{B}\) is the magnetic flux field, and \(\mathbf{I}\) is an isotropy tensor. The total acoustic force \(\mathbf{F_{a}}\) exerted on particles can be divided into two parts. One is the acoustic radiation force, and the other is the Stokes drag force caused by the acoustic streaming. Due to the relatively small viscosity of water, the viscous and thermal boundary layer thicknesses are smaller than 10% of the size of the particle, which is within the validity range of the ideal fluid assumption [43]. Therefore, the acoustic radiation force \(\mathbf{F_{ar}}\) can be calculated by integrating the Brillouin stress tensor on the surface of the particles [44]:  
\[\mathbf{F_{ar}}=-\oint\limits_{S}(\overrightarrow{T_{B}}\mathbf{\bullet n})dS, \tag{3}\]  
\[T_{Bij}=\rho_{0}\
u_{1i}\
u_{1j}+\frac{1}{2}\Bigg{[}\frac{1}{\rho_{0}}\bigg{(} \frac{p_{1}}{c_{0}}\bigg{)}^{2}-\rho_{0}\
u_{1k}\
u_{1k}\Bigg{]}\delta_{ij}, \tag{4}\]  
Figure 1: (**a**) The 3D view of the on-chip integrated hybrid tweezer. (**b**) The side view in the \(x\)-\(y\) plane. where \(\hat{\mathbf{T}_{B}}\) is the Brillouin stress tensor, \(\rho_{0}\) is the density of water, \(v_{1i}\) is the \(i\)-th direction component of the first-order velocity, \(p_{1}\) is the first-order pressure, \(c_{0}\) is the speed of sound in water, and \(\delta_{ij}\) is the Kronecker delta. The Stokes drag force \(\mathbf{F_{as}}\) exerted on particles can be calculated by the time-averaged second-order velocity. In the case of a particle near the surface, the Stokes drag force can be expressed as [45; 46]:  
\[\mathbf{F_{as}}=\frac{6\pi\mu R\cdot\left(\mathbf{v_{2}}-\mathbf{v_{p}}\right)}{\left[1- \frac{9}{16}\left(\frac{R}{h}\right)+\frac{1}{8}\left(\frac{R}{h}\right)^{3}- \frac{45}{256}\left(\frac{R}{h}\right)^{4}-\frac{1}{16}\left(\frac{R}{h} \right)^{5}\right]}, \tag{5}\]  
where \(\mu\) is the viscosity of water, \(R\) is the radius of particle, \(\mathbf{v_{2}}\) is the second-order velocity of water, \(\mathbf{v_{p}}\) is the velocity of the particle, and \(h\) is the distance between the center of the particle and the surface. From Equation (5), it can be seen that if the velocity of the particle is different from the time-averaged second-order velocity of water, the particle will be subjected to the Stokes drag force. After calculating optical and acoustic forces, the trajectory of particles in water can be expressed through the motion equation of particles. \[ma=\mathbf{F_{g}}+\mathbf{F_{b}}+\mathbf{F_{o}}+\mathbf{F_{ar}}+\mathbf{F_{as}}+\mathbf{F_{s}}, \tag{6}\]  
where \(m\) is the mass of the particle, \(\mathbf{a}\) is the acceleration of the particle, \(\mathbf{F_{g}}\) is the gravity of the particle, \(\mathbf{F_{b}}\) is the buoyancy of the particle in water, and \(\mathbf{F_{s}}\) is the supporting force. In Equation (6), \(\mathbf{F_{g}}\), \(\mathbf{F_{b}}\), and \(\mathbf{F_{s}}\) are in x-direction. \(\mathbf{F_{o}}\), \(\mathbf{F_{ar}}\), and \(\mathbf{F_{as}}\) have components in y-direction. To trap particles, the y-component of acoustic radiation force \(F_{ary}\), Stokes drag force \(F_{asy}\), and optical force \(F_{oy}\) are required to be balanced.