# Dry Test Methods for Micropumps
## Materials and Methods
### Methods
#### 2.2.6 Pumping Mechanism Stiffness  
Two main functional characteristics of the piezo actuator are the free deflection and the blocking force, which is defined as the maximum force the actuator can deliver at zero deflection. Quality control is performed during piezo manufacturing by sampling. The actuators are clamped and activated to measure the deflection of the piezo tip in both directions. The blocking force is evaluated during piezo activation with a fixed-force probe placed at the end of the bender. After assembly, these two characteristics can no longer be directly evaluated. Due to the complexity of the mechanical loop, an assessment of the piezo actuator performances after assembly is, however, crucial to set up the manufacturing process and to anticipate any drift in production. Once coupled to the pumping membrane, the essential performances of the piezo actuator can be formulated a little differently: the actuator will be able to maintain a force after a given displacement, typically a half stroke. This force, which allows the desired volume to be delivered whatever the specified pressure conditions, is indirectly estimated through the measurement of the maximum pressure that can generate the micropump at the outlet. This test, which is described later, provides a good indication of the blocking force. Since the full description of the pumping membrane movement upon actuation is not required here (see [29]), the system can be significantly simplified to model the blocking force. The equivalent stiffness \(k_{eq}\) of the piezo actuator coupled to the pumping membrane will be calculated first. It is assumed that the stiffnesses of the pumping membrane and the piezo actuator are constant and noted as \(k_{m}\) and \(k_{p}\), respectively. The flexible blade is assumed to fully transmit the piezo force without storing elastic energy in a first approximation. For a given position \(z\) of the pumping membrane (see Figure 7), the restoring forces of the pumping membrane \(F_{m}\) and the piezo actuator \(F_{p}\) onto the flexible blade are:  
\[F_{m}=k_{m}\bigg{(}\frac{\delta_{1}}{2}-z\bigg{)} \tag{19}\]  
And  
\[F_{p}=k_{p}(\delta_{2}-\delta_{0}-z) \tag{20}\]  
In the presence of a gradient of pressure \(\Delta P\) between the pumping cavity and the external environment of the micropump, the sum of the forces onto the blade at equilibrium is equal to zero:  
\[-\Delta PS_{m}+F_{m}+F_{p}=0 \tag{21}\]  
where \(S_{m}\) is the surface of the pumping membrane. This equation can be rewritten as:  
\[\Delta PS_{m}=-z\ k_{eq}-C \tag{22}\]  
where the equivalent stiffness \(k_{eq}\) of the system is:  
\[k_{eq}=k_{m}+k_{p} \tag{23}\]  
And \(C\) is a constant equal to:  
\[C=-k_{q}(\delta_{2}-\delta_{0})-k_{m}\frac{\delta_{1}}{2} \tag{24}\]  
Applying a voltage \(V\) to the piezo actuator leads to an additional force \(F(V)\) that satisfies, at equilibrium, the following equation:  
\[F(V)=\Delta PS_{m}+z\ k_{eq}+C \tag{25}\]  
After assembly, using \(V=0\), \(\Delta P=0\), and \(z=\delta_{2}\), Equation (25) leads to:  
\[k_{m}\bigg{(}\frac{\delta_{1}}{2}-\delta_{2}\bigg{)}=k_{p}\delta_{0} \tag{26}\]  
Based on this simple model, compression tests #3 and #4 can be built to derive the equivalent stiffness \(k_{eq}\) experimentally. Compression test #3:  
* Activate the piezo to pull down the pumping membrane up to \(z=0\) (PULL position);
* Apply a vacuum to the outlet valve port to vent the pumping chamber;
* Apply a large pressure (>+ 1 barg) to the outlet valve port to force the valve closed;Set the piezo voltage to \(V=0\) to release the actuator and compress the air in the pumping chamber; the pumping membrane moves from position \(z=0\) to \(z=\delta_{3}\).