# Numerical and Experimental Research on the Laser-Water Jet Scribing of Silicon
## 2 Numerical Simulation and Experimental Setup  
### Theoretical Background  
The schematic of LWJ machining is demonstrated in Figure 1. During the impingement of LWJ, the laser beam within the waterjet interacts with the substrate. Meanwhile, the waterjet scours the machining region continuously. It was obvious that thermal conduction and convection were the main regimes accounting for the substrate ablation and groove construction. Based on the first law of thermodynamics, the governing equation describing the thermal conduction in the interaction between the silicon substrate and LWJ was written in cylindrical coordinates, as shown in Equation (1) [15,16]:  
\[\rho_{\rm S}c_{\rm PS}\frac{\partial T}{\partial t}+\rho_{\rm S}c_{\rm PS}{\bf u }\
abla T-[\frac{\partial}{\partial r}(k_{\rm s}\frac{\partial T}{\partial r}) +\frac{\partial}{\partial z}(k_{\rm s}\frac{\partial T}{\partial z})]=Q \tag{1}\]  
where \(\rho_{\rm S}\) is the silicon density; \(c_{\rm PS}\) the specific heat of silicon; \(k_{\rm s}\) is the thermal conductivity of silicon; \(T\) is temperature; \(t\) is time; \({\bf u}\) is velocity vector; and \(Q\) is time-dependent heat source, \(Q=\frac{\pi d^{2}Q(t)}{4}\), where \(Q(t)\) is the effective laser power at the machining region. The schematic of \(Q\) is shown in Figure 2. In specific, the laser transmits along the waterjet to achieve total reflection. The laser energy distribution on the cross-section of waterjet was assumed to be nearly top-hatted, as demonstrated in Figure 2a, which was experimentally measured in our previous work [7]. The scouring effect of the waterjet brings about the continuous force of thermal convection throughout the LWJ machining process, which could be described as shown by Equation (2) [17]:  
\[H_{\rm ctc}=Nk_{\rm wat}/l_{\rm fs} \tag{2}\]  
where \(l_{\rm fs}\) is the feature size of the analyzed region. \(k_{\rm wat}\) is the thermal conductivity of water and \(N\) is the Nusselt number. The empirical description of \(N\) is expressed as [17]:  
\[Nu= \begin{cases}0.797{R_{e}}^{1/2}{P_{rm}}^{1/3}&P_{run}{>3}\\ 0.715{R_{e}}^{1/2}{P_{run}}^{0.4}&0.15<P_{run}<3\end{cases}  \tag{3}\]  
Figure 1: Schematic of LWJ machining. Figure 2: Schematic description of \(Q\). (**a**) Definition of d according to experimental test; and (**b**) definition of \(Q(t)\). where \(R_{e}\) is the Rayleigh number, \(Re=\frac{\rho_{w}V_{w}l}{\mu_{w}}\), \(\rho_{w}\) is the water density, \(V_{w}\) is the velocity of the water jet, and \(\mu_{w}\) is the dynamic viscosity of water. \(P_{run}\) is the Prandtl number \(P_{run}=\frac{c_{pw}l_{w}}{k_{w}}\), and \(c_{pw}\) is the specific heat of water. Furthermore, the temperature field at the machining region was not uniformly distributed considering the scouring effect of the waterjet. Thermal stress generated correspondingly. Three equation sets were introduced to characterize the thermal stress and the induced thermal strain within the substrate:  
1. Generalized Hooke equations describing the thermal stress [18]: \[ \begin{array}{l}\sigma_{x}=2G\varepsilon_{x}+\lambda e-\beta t\\ \sigma_{y}=2G\varepsilon_{y}+\lambda e-\beta t\\ \sigma_{z}=2G\varepsilon_{x}+\lambda e-\beta t\\ \tau_{xy}=2G\varepsilon_{xy}\\ \tau_{yz}=2G\varepsilon_{yz}\\ \tau_{zx}=2G\varepsilon_{zx}\\ \end{array} \] (4) where \(\sigma_{x}\), \(\sigma_{y}\), and \(\sigma_{z}\) are the normal stresses along X, Y, and Z axes, respectively. \(G\) is the shear modulus, \(G=\frac{E}{2(1+\mu)}\), where E is the elasticity modulus, and \(\mu\) is the Poisson's ratio. \(\varepsilon_{x}\), \(\varepsilon_{y}\), and \(\varepsilon_{x}\) are the normal strain along X, Y, and Z axes, respectively. \(\varepsilon\) is the volumetric strain, \(e=\varepsilon_{x}+\varepsilon_{y}+\varepsilon_{z}\). \(\lambda\) is the Lame constant, \(\lambda=\frac{E_{H}}{(1+\mu)(1-2\mu)}\). \(\beta\) is the thermal stress coefficient, \(\beta=\frac{\alpha_{\rm s}E}{1-2\mu}\), where \(\alpha_{\rm s}\) is the linear expansion coefficient.