Analytical model describing the nonlinear behavior of an elastomeric pump membrane in a microfluidic network

Abstract

For several decades, there has been a strong research interest in microfluidic systems and their applications. To bring these systems to market, a high development effort is often necessary for conceptualization and fabrication of such systems as well as the implementation of automated biological analysis processes. In this context, the simulation of microfluidic processes and entire microfluidic networks is becoming increasingly important, as this allows a significant reduction in development efforts, as well as easy adaptation of existing systems to specific requirements. This work presents an analytical model for elastomeric membrane-based micropumps as well as guidelines on how to apply this model to calculate microfluidic networks. The model is derived from the Young–Laplace equation for a non-prestressed elastomeric membrane with a purely nonlinear deflection as a function of applied pressure. The resulting cubic pressure–volume relation is validated by static measurements of the membrane deflection and the transported volume. The model is further used to calculate transient volume flows induced by a membrane micropump in a microfluidic network by adding Hagen Poiseuille’s law. Pressure measurements in the pump chamber confirm the basic assumptions of the model and allow definition of its validity scope. This work lays the foundation for designing elastomeric membrane-based micropumps together with microfluidic networks, estimating maximum flow rates in the system and optimizing pum** frequencies for different microfluidic configurations.

Appendix

Figs. 10, 11, 12.

Fig. 10

Numerical simulation to estimate the error resulting from the assumptions for the analytical model. The simulation was set up in COMSOL, computing the displacement of an elastic membrane securely clamped with the form of the investigated pump chamber and displaced through a surface load. As the pump chamber is symmetrical in x and y direction, two symmetry planes were inserted into the model to reduce computing times. The pressure in the COMSOL experiment is given in arbitrary units, as the elastic modulus taken for the membrane in the simulation was estimated and deviates from the actual value. So the membrane displacement as a function of the applied pressure is shown qualitatively but not quantitatively. First, the pressure is determined when the membrane reaches the limit stop which is at 0.6 mm (height of the pump chamber from the zero line of the membrane). The displacement of the center point of the membrane is shown in the graph as a function of pressure. The positions of the membrane at the pressure steps $p=0$ (zero line of the membrane) and $p=0.14$ (just before the membrane reaches the maximum displacement) are also displayed

Fig. 11

Two different simulation experiments compared (I) the pressure-dependent displacement of the membrane with a limit stop in the dimensions of the pump chamber (upper row and darker curve) with (II) the pressure-dependent displacement of the membrane without a limit stop (lower row and light curve). The displaced volume was extracted for every simulated pressure step and is shown in the displayed graph. The position of the displaced membrane is shown for the pressure steps $p=0$, $p=0.14$, $p=1$ for simulation (I) and for the pressure steps $p=0.14$, $p=0.4$, $p=1$ for simulation (II). The displaced volume for simulation (I) at $p=0.14$ is the same as the displaced volume for simulation (II) at $p=0.14$ which is $V_\text {pumped}={2.785}{\upmu }L$. After $p=0.14$, when the membrane center reaches the maximum displacement, the two curves start to deviate from each other as the increasing contact area between membrane and pump chamber results in an increasing counterforce and limits the maximal displaceable volume to 3.75 $\upmu$L (corresponding to $\frac{1}{8}$ of the pump chamber volume for the simulated part of the pump chamber with two symmetry planes and displacement of the membrane from the zero line)

Fig. 12

Cross-section of the pump chamber and graphical visualization of the residual volume in the pump chamber at different positions of the displaced TPU-membrane: The pressure-driven displacement of the TPU-membrane was simulated with the same model as described in Fig. 10. For a pressure $p=0$ the membrane is in its zero position, meaning that 50 % of the pump chamber volume is already pumped out, if the membrane started from its maximum position, being displaced by negative pressure. The membrane is being displaced through applied overpressure, while its center reaches the half of the maximum displacement position from the zero line at $p=0.04$ and its maximum displacement position at $p=0.14$. From here the rest of the membrane area is displaced further until every point reaches its maximum displacement position, which would be when the membrane is fully clinging to the pump chamber. The pumped fluid volume as well as the fluid volume that is still in the pump chamber is displayed in the table at $p=0$, $p=0.04$, $p=0.14$ and $p=0.3$. As the model contains two symmetry planes and the displaced volume is computed from the zero line of the membrane, the total pumped and remaining volume in a pump chamber is shown in the right columns of the table. The row containing the volumes at the contact point of the membrane center with the pump chamber at $p=0.14$ is highlighted with a box