Analytic modelling of passive microfluidic mixers

Alexi Bonament; Alexis Prel; Jean-Michel Sallese; Christophe Lallement; Morgan Madec; Alexi Bonament; Alexis Prel; Jean-Michel Sallese; Christophe Lallement; Morgan Madec

doi:10.3934/mbe.2022179

Mathematical Biosciences and Engineering

2022, Volume 19, Issue 4: 3892-3908. doi: 10.3934/mbe.2022179

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Analytic modelling of passive microfluidic mixers

1.
Laboratory of Engineer Sciences, Computer Science and Imagine (ICube), UMR 7357 (UniversitȦ de Strasbourg/Centre National de Recherche Scientifique), Strasbourg, France
2.
STI-IEL-Electronics Laboratory, Ecole Polytechnique FȦdȦrale de Lausanne (EPFL), Lausanne, Switzerland

Received: 14 October 2021 Revised: 04 January 2022 Accepted: 24 January 2022 Published: 11 February 2022

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 silico prototyping of large-scale microfluidic devices or multi-physics devices involving microfluidic circuits, e.g. lab-on-chips.
- microfluidics,
- passive mixer,
- modelling,
- analytic models,
- hardware description languages,
- COMSOL simulations,
- electronic-equivalent microfluidic circuits,
- Verilog-AMS
Citation: Alexi Bonament, Alexis Prel, Jean-Michel Sallese, Christophe Lallement, Morgan Madec. Analytic modelling of passive microfluidic mixers[J]. Mathematical Biosciences and Engineering, 2022, 19(4): 3892-3908. doi: 10.3934/mbe.2022179

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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 silico prototyping of large-scale microfluidic devices or multi-physics devices involving microfluidic circuits, e.g. lab-on-chips.

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