Analytical Modeling of Cuboid Microfluidic Multilayered Glass Structure for Microwave Permittivity Sensors

A microfluidic structure plays a crucial role in supporting liquid sensors. However, modeling a multilayered microfluidic structure faces challenges for permittivity sensor applications, particularly concerning the non monotonic behavior of multilayered dielectrics. These challenges arise due to the varying characteristics of the electric field (E) direction in dielectrics with low and high permittivity. The existing models exhibit significant deviations from measurement results across a wide permittivity range. To address the issue, this article proposes a quasi-static conformal approach with exponentially tapered capacitance to minimize deviations caused by nonlinear behavior. This study uses the exponential tapered permittivity ratio to adjust and modify the capacitance value. The proposed model was examined across samples with a wide range of permittivity, spanning from air of 1.0 to water of 80.0. To verify the proposed model, finite element method (FEM) simulations and experimental measurements were conducted. A three-layer configuration was prepared, i.e., glass (<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\varepsilon _{{r}{2}} =7.3$ </tex-math></inline-formula>)/liquid sample (<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\varepsilon _{{r}{3}} =1.0$ </tex-math></inline-formula>–80.0)/glass (<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\varepsilon _{{r}{4}} =7.3$ </tex-math></inline-formula>). The sample was positioned at the middle layer by using a microfluidic channel with a cuboid shape. As a result, the comparison of the quasi-static conformal approach without and with the exponentially tapered capacitance model reveals deviations in the effective permittivity (<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\varepsilon _{{r}\text {-eff}}$ </tex-math></inline-formula>) of 27.7% and 1.3%, in the characteristic impedance (<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${Z} _{{0}}$ </tex-math></inline-formula>) of 11.1% and 0.8%, and in the total capacitance (<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${C} _{\text {T}}$ </tex-math></inline-formula>) of 28.5% and 1.4%, respectively. Subsequently, the proposed sensor structure was fabricated and measured for permittivity sensor application using the resonant frequency shift approach. The measurement results, ranging from the air (<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\varepsilon _{{r}{3}} =1.0$ </tex-math></inline-formula>) to the water sample (<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\varepsilon _{{r}{3}} =80.0$ </tex-math></inline-formula>), showed a frequency shift of 425.50 MHz and an average normalized sensitivity (NS) of 0.64%. This study presents a robust and accurate model, offering a practical solution for permittivity sensing. The proposed approach meets the high accuracy and sensitivity demands in diverse industrial and environmental applications. Additionally, the model is recommended for various sectors, including the biomedical industry, medicine, and material quality control.