## 2 Principle of Microdroplet Jetting
### Microdroplet Formation Criterion  
Microdroplets are formed at the mouth of the nozzle due to the periodic vibrations of the piezoelectric actuator [23]. To form a microdroplet, the volume of the droplet formed by this process must exceed a standard droplet volume and satisfy Equations (1) and (2) [24]:  
\[W_{e}=\frac{\rho v_{0}(t)r_{0}}{\sigma}>4, \tag{1}\]  
\[V=\pi{r_{0}}^{2}\int_{0}^{t_{1}}v_{0}(t)dt\geq\frac{4\pi{r_{0}}^{3}}{3} \tag{2}\]  
where \(W_{e}\) is the ratio of the inertia of the fluid to its surface tension, \(\rho\) is the density of the droplet fluid/material, \(\sigma\) is the surface tension coefficient of the droplet fluid/material, \(v_{0}(t)\) is the axial velocity of the fluid, and \(r_{0}\) is the radius of the nozzle channel. Construction of an Electromechanically Coupled Model to Describe Microdroplet Formation and Jetting  
It is speculated that microdroplet production and jetting include electromechanical coupling through piezoelectric actuation and complex fluid changes inside the working chamber. In this part, a mechanical model of microdroplet production and jet creation inside a structural displacement field was built using the principles of conservation of mass and momentum. Integration was then used to determine the link between the structural displacement field and the electric field. Finally, an electromechanically connected model of  
Figure 1: Operating principles of a piezoelectric printhead. microdroplet production and jetting was constructed by substituting electric field terms for displacement field terms in the mechanical model. Figure 2 depicts a simplified diagram of droplet production and jet development that illustrates the electromechanical connections of the model. The model comprises a fluid supply system, a fluid line connecting the supply to the nozzle, a fluid chamber, a nozzle, and a piezoelectric actuator. Considering that a metallic diaphragm with surface area \(A\) vibrates at a rate of \(dz/dt\) and the inlet flow rate is \(v_{i}(t)\), the nozzle flow rate is \(v_{0}(t)\), the inlet area is \(A_{i}\), and the nozzle area is \(A_{0}\), we obtain the following equations using the law of mass conservation:  
\[\frac{d}{dt}(\rho V)+\rho v_{0}A_{0}+\rho v_{i}A_{i}=0, \tag{3}\]  
\[\frac{d}{dt}(\rho V)=V\frac{d\rho}{dt}+\rho\frac{dV}{dt}, \tag{4}\]  
\[\frac{dV}{dt}=-K_{\rm o}A\frac{d}{dt}, \tag{5}\]  
\[\frac{d\rho}{dt}=\rho K_{\rm s}\frac{dP}{dt}, \tag{6}\]  
where \(\rho\) is the fluid density, and \(K_{\rm o}\) is the velocity variation coefficient of the fluid inside the working chamber. When an electrical signal is applied, the piezoelectric actuator generates a displacement perpendicular to the diaphragm. The mathematical description of piezoelectricity is provided by the following equations:  
\[S_{1}=s_{11}^{E}T_{1}+d_{31}E_{3}, \tag{7}\]  
\[D_{3}=d_{31}T_{1}+\epsilon_{33}^{T}E_{3}, \tag{8}\]  
where \(S_{1}\) and \(T_{1}\) are the strain and stress in the \(x\)-direction, respectively; \(E_{3}\) and \(D_{3}\) are the electric field and the electric flux density in the \(z\)-direction, respectively; \(\epsilon_{33}^{T}\) is the dielectric constant when the external stress is equal to zero or a constant; \(d_{31}\) is the piezoelectric constant; and \(s_{11}^{E}\) is the elastic compliance coefficient when the electric field is equal to zero or a constant. If the electric field of the piezoelectric actuator is assumed to be homogeneous, then \(E_{3}\) is a linear function of the voltage \(U\) applied between the ends of the piezoelectric ceramic. If the maximum displacement of the piezoelectric actuator in the vertical direction is \(z\), the force-displacement equation of the piezoelectric actuator is as follows:  
\[z=\alpha U-\beta F, \tag{9}\]  
where \(\alpha\) and \(\beta\) are constants, z is the maximum displacement, \(U\) is the driving voltage, and \(F\) is the external force applied to the piezoelectric ceramic chip. The equation specifies the relationship between the physical quantities \(z\), \(U\), and \(F\).