**1. Introduction**

Pumps are devices that provide momentum to transfer fluids and can be integrated in Micro Electromechanical Systems (MEMS) to develop so-called micro-fluidic systems [1]. As an example, micropumps play an important role in biomedical and drug delivery systems: The micro-dosing feature in such devices has improved the effectiveness of the treatment because the concentration of the drug in the patient's body has been kept constant as well as because toxicity has been prevented [2].

Several actuation systems have been investigated in the literature [3], with particular attention to electrostatic forces and piezoelectric materials. The former case, e.g., studied in Reference [4], is connected to some drawbacks related to the high actuation voltage and the possible occurrence of pull-in instability [5]; on the other hand, piezoelectric actuation has been widely utilized based on advantages such as the small size, low power consumptions, no electromagnetic interference, and an insensitivity of fluid viscosity [6] even though the maximum power density of the piezoelectric materials is dependent on the working frequency [7]. Among different materials with a piezoelectric effect, lead zirconate titanate (PZT) ceramics demonstrated optimal performances in view of the large deflection that can be induced in the pump diaphragm, as recently shown by Cazorla et al. [8]. While Nisar et al. have reported different types of micropumps fabricated for biomedical applications [9], there are some recent studies about the design and modelling of piezoelectric micropumps. Revathi and Padmanabhan designed a valveless micropump with a composite piezoelectric actuator which showed the maximum outflow at an aspect ratio of 15 for a nozzle/diffuser [10]. Sateesh et al. [11] designed and modelled a piezoelectrically actuated micropump with PZT-5h and Polydimethylsiloxane (PDMS) with an outflow rate of 0.029 μL s<sup>−</sup>1.

In any case, it has been clearly established that, in addition to the actuator, the presence of valves in the device can affect the performance. Despite the fact that the valveless micropumps have the benefit of no risk of wear and fatigue of the valves, it is hard to control the flow to the desired direction at inlet and outlet, causing energy loss and liquid reflux. Consequently, in order to improve the control of the fluid flow, mechanical valves (either active or passive) must be introduced in the system. The main novelty of the present paper is the comprehensive study of a complete pumping system, composed of a piezoelectric actuator and passive valves, using three-dimensional modelling and a simulation of the device performing a complete cycle of pumping with a consideration for the two-way fluid–structure interaction. An innovative layout is considered, starting from a patent [12] that was originally proposed for electrostatic actuation. The purpose of the present paper is to assess the behavior of the new device and to propose some slight modifications in order to improve the performances.

The paper is organized as follows. After the introduction, the layout of the device is shortly discussed. The subsequent section contains a thorough description of the computational model which then leads to some preliminary results. Then, we consider some modifications of the geometrical features of the pump, with the purpose of optimizing its performances.

#### **2. Description of the Proposed Layout**

In most cases, a pumping device, composed of a micropump and valves, is of complex realization in view of the difficult integration of the different components of the actuating system on a limited number of wafers. Moreover, the integrated valves may present specific issues related to a lack of tightness, which is the cause of a leakage and backflow.

In order to overcome the aforementioned limits, an innovative device based on the use of two wafers has been recently conceived and patented [12]. In spite of the fact that the invention refers to an electrostatic actuation, the main advantage of using just two wafers for an integrated design can be exploited also for the piezoelectric case. According to the invention, the pumping device comprises (i) a pumping chamber, realized between two silicon wafers bonded to each other; (ii) an inlet valve, with a shutter element in correspondence to the connection with the external reservoir; and (iii) an outlet valve, with a shutter element on the external microfluidic circuit. As shown in Figure 1, when the inlet valve is in the open configuration, the shutter is housed by a recess that is fluidly coupled with the pumping chamber by means of an inlet channel. On the other hand, the outlet shutter is located in a recess that is fluidly decoupled with respect to the pumping chamber. The described configuration of the inlet and outlet valves allows the direction of the processed flow to be controlled in a completely passive way. More precisely, the inlet and outlet valves do not require dedicated actuators, and so, the structure is generally simplified for the benefit of both the overall dimensions and the manufacturing costs. For instance, the micropump, as above defined, may be made from just two semiconductor wafers joined together. Moreover, the micropump control is simplified because it does not have to take into account the synchronization of the valves. Dedicated actuators for the valves, in particular for the output valve, may optionally be provided if specific circumstances make this advisable. However, the micropump is still fully operative even with purely passive valves. The inlet valve and the outlet valve are of the orthoplanar type. The sealing is guaranteed by the presence of an initial stress state that can be easily achieved during the bonding phase by introducing a couple of thick elements on the shutters: If the thickness of such elements is larger than the bonding layer, the orthoplanar valves are forced to close the holes. It is worth noting that the layout of the device allows for the introduction of two opposite pumping diaphragms, with a possible increase of the stroke volume. In the present paper, for the sake of simplicity, just one actuation diaphragm is considered.

**Figure 1.** A schematic view of the micropump proposed in Reference [12] and adopted in the present paper with the addition of a piezoelectric actuation.

#### **3. The Three-Dimensional Model**

Multiphysics modelling encompasses two main parts: the "deformable" solid, containing the actuation portion and the valves for controlling the fluid flux, and the "fluid domain", which interacts with the solid domain through the interface at the boundary of the solid and fluid domains. Due to the sake of symmetry, half of the micropump is modelled.

In the initial design, the micropump consists of a circular pumping chamber, delimited by lower and lateral fixed surfaces and by an upper silicon diaphragm. As shown in Figures 2 and 3, the fluid domain is completed by two cylindrical spaces from one side connected to the inlet and outlet, which are connected to the pumping chamber by means of two prismatic channels with a rectangular cross section from the other side. It is worth noting that the inlet and outlet are closed by the silicon valves. The radius of the valve's membrane is larger than the hole, so that the fluid flow is prevented when the valve is in the rest configuration.

**Figure 2.** A 3-D scheme of the proposed micropump.

The geometrical specifications of the fluid domain are presented in Table 1. The considered fluid is water, modeled as an incompressible, viscid fluid with the mechanical properties summarized in Table 2.

The model for the solid parts is depicted in Figure 4. The actuation diaphragm is represented by a suspended disc, with a laminate cross section. The actuation is achieved by means of a piezoelectric layer deposited on a silicon plate. Among the various possibilities, lead zirconate titanate (PZT) is

chosen as the active material, in the form of a thin film [13]. This is fully compatible with the MEMS production process, through the adoption of one of the available techniques, e.g., sputtering [14], pulsed laser deposition [15], and sol-gel process [16]. The piezoelectric layer has a circular shape, coaxial with the silicon disc. The radius of PZT, see Table 3, is selected according to the results presented in a previous study [6], which showed that the highest stroke volume was obtained when the ratio between the radius of the active layer and the radius of the silicon diaphragm was equal to 0.73. The passive valves consist of silicon discs, that are attached to the rigid frame by means of four rectangular beams. The elastic deformation of the beams allows for the vertical movement of the disc that alternatively opens and closes the inlet hole (the same applies to the outlet). The sealing of the valves in the closed rest configuration is ensured by a prestress in the elastic ligaments, given by an imposed transverse displacement on the edge of the disc.

(**b**) The fluid domain mesh in a cross section.

**Figure 3.** The micropump fluid domain.



(**a**) The solid domain of the model.

(**b**) The solid domain mesh.


**Table 2.** The material properties.



**Table 3.** The geometrical specifications of the solid parts.

The mechanical properties of polycrystalline silicon and PZT are given in Table 2.

The investigation of the interaction between the moving solid parts and the fluid is a key point, especially in mechanical micropumps. The motion of the solid, i.e., the oscillating displacement field, induces the motion in the fluid. Also, the fluctuating pressure in the fluid acts as a surface load on the fluid/solid interface. By coupling the governing equations of these two domains, the mutual interaction can be thoroughly studied.

For the proposed micropump, a two-way fluid–Structure Interaction (FSI) is modeled with the commercial code ANSYS® 18.1 and 19.2, with the mechanical and CFX solvers. At each staggered loop, the mechanical solver sends the time derivative of the displacement of the interface nodes to the target nodes in the fluid domain. On the other hand, the CFX solver sends back the stress to the solid nodes based on the traction equilibrium at the interface [17].

The valves are modelled by means of quadrilateral 8-node solid elements (SHELL182 in ANSYS nomenclature), which are suitable for modelling thin to moderately thick shell structures. The aim of choosing this kind of element is to reduce the computational cost, taking into account the specific geometrical features of the problem at hand. The 20-node brick element is chosen for the oscillating membrane (SOLID186), and the coupled-field element is chosen for the piezoelectric actuator (SOLID226). In the fluid domain, the tetrahedral elements with 10 nodes (FLUID221) are used. The number of elements for the solid and fluid domains are 33,968 and 1,425,377, respectively. Some views of the adopted mesh are reported in Figures 3 and 4. The piezoelectric actuator is polarized in the out-of-plane direction. The oscillating membrane is clamped at the edge, as well as the valves at the end of connected bars. The applied voltages to the actuator are 20, 40, and 60 V.

For the boundary conditions on the fluid domain, both the inlet and outlet have a zero relative pressure with respect to the outside of the domain. The faces attached to the silicon membrane and valves are set to the fluid–structure interaction sites as well. The other faces of the fluid domain are set to stationary walls, i.e., the velocity of the fluid is equal to zero.

## **4. Preliminary Results**

The proposed micropump is investigated for different values of the input voltage frequency. The most important figure of merit is the outflow of the micropump that is reported in Figure 5 with the actuation voltage equal to 60 V for the range of frequency between 0.01 to 100 Hz. Ideally, the displaced quantity of fluid can be estimated on the basis of the so-called *stroke volume vs*, i.e., the variation in the volume of the pumping chamber due to the movement of the actuating diaphragm during one cycle. For a given pumping frequency *fp*, the nominal outflow would be

$$q = f\_{\mathcal{V}} v\_{\mathcal{S}} \tag{1}$$

As it is shown by the chart in Figure 5, the computed outflow is strongly different with respect to the nominal quantity. More specifically, the non-monotonic behavior is connected to the inertial force of the fluid so that the maximum outflow is reached at a certain point and a decay is observed after that frequency. The optimal outflow for the designed micropump is achieved at 10 Hz: The numerical results are now thoroughly examined for that specific actuation frequency.

**Figure 5.** The dependency of the outflow on the pumping frequency.

At the first stage of the micropump working cycle, the silicon membrane is deformed in the upward direction due to the PZT deformation. The pressure gradient causes an inlet valve opening and the fluid enters in the chamber. The computed maximum deflection of the diaphragm for different actuation voltages is demonstrated in Figure 6.

(**a**) The actuation voltage = 20 V

(**b**) The actuation voltage = 40 V

(**c**) The actuation voltage = 60 V

**Figure 6.** The deformation of the pumping diaphragm: contour plots of the transverse displacement in μm.

Figure 7 represents the outflow of the designed micropump for different actuation voltages at the working frequency of 10 Hz.

**Figure 7.** The outflow dependency on the actuation voltage.

Figure 8 presents the variation of the chamber volume during a complete pumping cycle. After the silicon membrane reaches the maximum stroke, as mentioned above, the piezoelectric input is instantaneously dropped and the diaphragm tends to recover its initial configuration. Due to the membrane displacement, the fluid pressure increases, and as a consequence, the outlet valve is pushed in its recess and the hole is opened. At the same time, the elastic recovery enables the closing of the inlet channel. The results of Figure 8 allow one to compute the quantity of fluid that is pushed out from the chamber in a cycle connected to the above defined stroke volume.

The analysis shows that the maximum fluid velocity occurs in the channels during the first and the second half cycles (see Figure 9). The maximum backpressure is equal to 6.8 kPa, and the optimal outflow is 1.62 μL min−1.

The apparent power consumption of the actuator can be approximately computed by assuming harmonic variation [18]:

$$P\_A = I\_\mathfrak{e} V\_\mathfrak{e} \tag{2}$$

where *Ie* and *Ve* are the root mean square of the current and voltage, respectively. The magnitude of the current generated by the piezoelectric is

$$I = \omega\_p \int \int D\_3 dx dy \tag{3}$$

where *<sup>ω</sup>p* is the working angular frequency and *D*3 is the electric displacement in the piezoelectric layer. The power consumption for the optimal frequency (*fp* = 10 Hz) is about 0.29, 0.58, and 0.88 mW for the actuation voltages 20, 40, and 60 V, respectively.

**Figure 8.** The variation of the chamber volume during a pumping cycle.

(**a**) The velocity countour in the first half (t = 0.01 s)

(**b**) The velocity countour in the second half cycle (t = 0.06 s)

**Figure 9.** The velocity profiles in the midplane of the micropump.

cycle
