[paper] Analytic Modeling of Passive Microfluidic Mixers

Alexi Bonament¹, Alexis Prel¹, Jean-Michel Sallese², Christophe Lallement¹, 

and Morgan Madec¹

Analytic modelling of passive microfluidic mixers

Mathematical Biosciences and Engineering (2022)

Vol. 19, No. 4: 3892-3908

DOI: 10.3934/mbe.2022179

   
1. ICube, UMR 7357, Universite de Strasbourg/CRNS (F)
2. STI-IEL-Electronics Laboratory, EPFL (CH)

Abstract: This paper deals with a new analytical model for microfluidic passive mixers. Two common approaches already exist for such a purpose. On the one hand, the resolution of the advection-diffusion-reaction equation (ADRE) is the first one and the closest to physics. However, ADRE is a partial differential equation that requires finite element simulations. On the other hand, analytical models based on the analogy between microfluidics and electronics have already been established. However, they rely on the assumption of homogeneous fluids, which means that the mixer is supposed to be long enough to obtain a perfect mixture at the output. In this paper, we derive an analytical model from the ADRE under several assumptions. Then we integrate these equations within the electronic-equivalent models. The resulting models computed the relationship between pressure and flow rate in the microfluidic circuit, but also takes the concentration gradients that can appear in the direction perpendicular to the channel into account. The model is compared with the finite element simulation performed with COMSOL Multiphysics in several study cases. We estimate that the global error introduced by our model compared to the finite element simulation is less than 5% in every use case. In counterparts, the cost in terms of computational resources is drastically reduced. The analytical model can be implemented in a large range of modelling and simulation languages, including SPICE and hardware description language such as Verilog-AMS. This feature is very interesting in the context of the in silicon prototyping of large-scale microfluidic devices or multi-physics devices involving microfluidic circuits, e.g. lab-on-chips.

Fig:  Schematic of the Y-shaped passive mixer. The device is composed of two inlets (here, one is the water and the other is a dye) and one outlet. As we can see on this cartoon (which is purely illustrative and not a simulation result), the mixing is established along the channel and, for a short channel, the dye concentration is not homogeneous in the x direction.

Acknowledgments: This research was supported by the European Regional Development Fund (ERDF) and the Interreg V Upper Rhine Offensive Sciences Program (Project 3.14 – Water Pollution Sensor).