Actuation Modeling of a Microfluidically Reconfigurable Radiofrequency Device Free

The concept of microfluidics was introduced more than 70 years ago, with the advent of the inkjet printing nozzle by IBM [1]. Microfluidic devices offer fast reaction times due to their reduced diffusion distances, precise fluid control, and small reaction volumes. These devices enable rapid and efficient chemical or biochemical reactions, making them valuable for applications such as chemical synthesis, diagnostics, and drug discovery. Their ability to integrate sensors, parallelize reactions, and control temperature also contributes to their effectiveness in achieving faster reaction rates [2]. Later Puneeth et al. [3] reported on micro-electromechanical pump technologies and possible actuation concepts. Afterward, many new fabrication micropump techniques have been developed that can be grouped into three primary concepts: mechanical, electrokinetic, and magnetokinetic micropumps [4–7].

In mechanical micropumps, mechanical motion produces the required pressure. Mechanical micropumps can be categorized into moving diaphragm micropumps, rotary micropumps, and peristaltic micropumps [8]. Electrokinetic and magnetokinetic micropumps work based on the conversion of electrical or magnetic energy to fluid movement. They generate gradients in either electrical or conductivity permittivity of the working fluid. A recent micromachined electrohydrodynamic system was proposed by Russel et al. [9]. They surveyed different methods for enhancing system performance, such as the number of interdigitated electrode pairs, electrode surface topology, the effect of doping ferrocene in the working fluid, and the effect of external flow on the discharge characteristics.

A literature survey of recent statistics regarding wireless networks worldwide reveals that more than five billion devices demand wireless connections to run voice, data, and other applications. One active component is a radio frequency (RF) switch, a device that routes high-frequency signals through transmission paths. An RF switch is commonly used in microwave systems for signal routing between instruments and devices. Nowadays, RF configuration can be achieved whether by micro-electromechanical systems switches, micro-electromechanical systems capacitors, material loading method, varactors, positive- intrinsic-negative diodes, or ferro-electric varactors. Although these methods yield good performance in terms of speed, cost, and size, they are limited in power handling capability, radiation efficiency, and tuning range [10].

Alternatively, by integrating micro-electromechanical pumping technologies with a RF device, RF switches and other active devices can be actuated based on microfluidics. Microfluidics devices can be utilized to reconfigure the RF signals by pumping conductive fluids such as liquid metals [7]. Dey et al. presented a microfluidically reconfigured frequency-tunable liquid-metal monopole antenna [11]. The system relied on continuous movement of the liquid metal over the capacitively coupled microstrip line. Current microfluidic-based devices have been mostly limited to frequencies well below 10 GHz due to challenges with RF modeling, utilization of liquid metals, and their inherent limitations in terms of lower conductivity and reliability [10,12–14]. In addition, most recent papers focus on electronic aspects of designing microfluidically reconfigurable RF devices that are actuated by manual syringe pumps. For instance, in 2018, Kataria et al. designed a reconfigurable microfluidic filter based on ring resonators [15]. Their device employed eutectic gallium indium liquid metal mixed with sodium hydroxide for improvement in the liquid metal fluidity. However, their device was actuated by ten separate manual syringe pumps [15].

Recently Mumcu and coworkers have introduced the concept of selectively metalized plates (SMP) that move within microfluidic channels to realize reliable devices with superior efficiency and power handling capabilities [16,17]. Their works have used commercial micropumps (Bartels mp-6 piezopumps) [17,18]. Another recent study by Gonzalez-Carvajal and Mumcu [18] presented a mm-wave tunable microfluidic reconfiguration bandpass filter. In this work, the researchers used FC-40 in a microfluidic channel to move the SMP using a piezo-electric disk to pump fluid between two reservoirs. In their work, the channel and SMP sizes were not optimized. An improved understanding of the channel, fluid, and SMP parameters can improve the reconfiguration speed of such a pump [18]. Dey et al. designed a microfluidic broadband tunable metallic liquid monopole antenna [19], but again, the influence of the channel, liquid, and SMP parameters on the response was not explored.

A literature survey of the recent advances in the microfluidic reconfigurable RF area reveals a lack of an accurate fluid mechanics model of the actuation dynamics. Such a model would enable faster development of systems with higher performance and is the focus of this paper. In particular, this paper explores the dynamic response of an SMP in a fluidic channel and how it is influenced by SMP and fluidic channel dimensions, upstream and downstream reservior sizes, fluid viscosity, and actuation pressure. The model allows rapid estimatioun of critical design parameters so that RF designers can optimize a microfluidic-based reconfigurable RF devices. As an input, the designer specifies the desired output RF-critical variables, including desired SMP size, SMP response time, SMP speed, etc. The designer can then specify input variables that include microfluidic channel size, fluid properties, driving pressure actuation, etc. Then the designer can solve the analytical model presented here to calculate the response time, SMP velocities, etc., and compare these results with the desired design values.

The remainder of this paper is organized as follows. In Sec. 2, a dynamical model of SMP actuation response is presented as a function of all geometric and fluid parameters. Experiments that validate the model were also performed and Sec. 3 presents the experimental setup and approach and4 provides a comparison between the analytical model and experimental results. Also, comparisons with literature data from some device scale RF devices are also presented in this section. Finally, conclusions of the work are given in Sec. 4.

In this section, an analytic model of the SMP response in a microfluidic channel will be presented. Figure 1 shows a side-view schematic of the microfluidic channel of height 2h, width into the page of b, and length L. The SMP of length l is surrounded by the working fluid (of density ρ and viscosity μ) and is able to move horizontally between the two “channel stoppers” shown in the figure. When a positive pressure difference (denoted here as an initial equivalent height difference between the two reservoirs, H₀) is applied between the upstream and downstream reservoirs, fluid will move from the upstream to downstream reservoir, causing SMP motion as the height difference between the two reservoirs moves toward equilibrium. Leakage of fluid between the SMP and the channel walls exists, which depends strongly on the gap width δ. The fluid velocity in the channel varies across the channel width and is a maximum at the center of the channel and is zero at the top and bottom walls due to the classic no-slip fluid condition. The gap width δ affects both the amount of fluid leakage around the SMP and the shear stress that resists SMP motion, with decreasing δ resulting in higher shear stress and smaller leakage. All of the variables utilized in the model are defined in Table 1.

When a pressure difference between the two reservoirs is applied, acceleration of all of the fluid in the channel and the SMP will occur. Simultaneously shear forces exerted by the channel walls on the fluid exist as the fluid moves and causes flow resistance. Shear forces exist on all walls, but the local shear stress is inversely related to the spacing between solid walls. Since the gap width δ is much smaller than the channel spacing 2h, the fluid shear forces along the length of the SMP dominate all of the shear stresses and thus we consider only the shear that exists in the gaps between the SMP and the channel walls. To account for the shear stress that prevails on the channel walls along the length of the SMP, an expression for the transient fluid velocity distribution in the gap is needed. Thus, determining the transient fluid velocity distribution in the gap region will be considered first. Subsequently, control volume formulations of the conservation of mass and linear momentum principles will be developed.

where (⁠$dp/dx)$ is a constant pressure gradient that is suddenly applied along the channel of length $L$and is equal to −Δp/L. $ugap(y,t)$ is the velocity of fluid in the gap and it is initially zero, but then increases once the pressure gradient is applied and is a function of time and the wall-normal coordinate y.

The first term on the left side of the equation represents the driving pressure force due to the pressure differential that exists between the front and back of the SMP. The second term is the resisting shear force that exists on the channel walls along the length of the SMP. The terms on the right side of the equation represent the fluid and SMP masses multiplied by their respective accelerations. In Eq. (5), γH is the driving pressure that is imposed due to an external source like an actuator, $A$ is the cross section area of the channel, $m’f$ is the total mass of fluid in the channel, $m’f_res$ is the total mass of fluid in the upstream reservoir, $m’f_outlet$ is the mass of fluid in the outlet reservoir, $msmp$ is the mass of the SMP, $u¯f$ is the average velocity of the fluid in the channel, $h$ is the height of the channel, and $b$ is the width of the channel.

The analytical model described above was exercised over the range of physical parameters that are shown in Table 3 and some representative results of this process will be presented and discussed in this section. Figure 3(a) shows the dimensionless SMP velocity, $Usmp*$ as a function of the dimensionless time, τ. For the results shown in Fig. 3(a), the dimensionless SMP mass, SMP length, and dimensionless fluid mass were held constant at the values shown in Table 4 and the dimensionless gap size, δ*, was allowed to vary from 0.015 to 0.12. This variation corresponds to a physical gap that varied between 3% and 24% of the channel height. The results of Fig. 3(a) reveal that all of the $Usmp*$ data lie on a single curve and thus for these conditions the $Usmp*$ versus τ curve are not dependent on the value of δ^*. Further, the data show that the steady-state (maximum) SMP velocity is attained at a nondimensional time of τ ≈ 1.4 and that the steady-state value of U^* is approximately 0.206.

The influence of varying the dimensionless SMP mass and the dimensionless SMP length is illustrated in Figs. 3(b) and 3(c). Shown in Fig. 3(b) is the dimensionless SMP velocity as a function of τ for several values of $Msmp*$ and at constant values of M^*_f, L^*, and δ^*, as shown in Table 4. Here the data show, as expected, the steady-state value of $Usmp*$ increases as the mass of the SMP decreases, due to the smaller inertia of the SMP relative to the driving pressure force. For example, a 50% decrease in $Msmp*$ results in a 124% increase in the steady-state value of $Usmp*$⁠. Shown in Fig. 3(c) is $Usmp*$ as a function of τ over a range of L^* values, with the other independent parameters fixed as specified in Table 4. As L^* increases the steady-state speed of the SMP decreases. This is caused by the increased shear force that prevails due to increased SMP surface area.

Interestingly, while the data of Figs. 3(b) and 3(c) show that the steady-state value of $Usmp*$ depends strongly on L^*and $Msmp*$⁠, the value of τ required for the steady-state speed to be reached is independent of both of these values. In fact, the value of τ when the SMP velocity is within 5% of its steady-state value is τ = 1.45 and this value is independent of the value of any of the other independent dimensionless variables listed in Table 4. The importance of this is that the dimensional time required for the SMP to attain its maximum velocity is easily predicted as t = 1.45ρδ²/μ. Thus, as the gap between the SMP and channel walls decreases, the response time decreases as well and increasing the viscosity of the liquid in the fluidic channel will yield an increase in the response time. It should be noted that for the results presented here the smallest gap width considered was 3% of the channel height and this value was chosen based on the limitations prevalent in the manufacture of real devices.

The analytical model was exercised over the entire range of parameters shown in Table 3 and the resulting independent dimensionless variables. The purpose of this was to understand how the steady-state (maximum) SMP velocity varies as a function of all variables. An example of these simulations is shown in Figs. 4(a) and 4(b). Figure 4(a) shows the steady-state value of the dimensionless SMP velocity, U^*_smp-ss as a function of the dimensionless SMP mass, M^*_smp. for values of M^*_f varying as shown in the figure legend. The results shows that U^*_smp-ss increases following an inverse power law relation as M^*_smp decreases. Decreasing M^*_smp can be accomplished by decreasing the SMP mass, increasing the driving height difference between the two reservoirs (driving pressure), and increasing the height or width of the fluidic channel. Figure 4(b) shows U^*_smp-ss as a function of the dimensionless SMP length and the results clearly show that the steady-state velocity of SMP increases as L^* decreased, which can be accomplished either by decreasing the SMP length or increasing the driving pressure.

where C₁ and C₂ are constant coefficients that can be calculated from results from the model.

In practice, the designer can follow the process illustrated in the process diagram shown in Fig. 6 to select geometric sizes for the specific application. The process begins with specifiying the initial values for the geometric and fluid variables such as channel size, gap size, working fluid, driving pressure, etc. The designer can then select a commercial micropump and use its performance curve to calculate the differential pressure that the selected pump can provide. For instance, Fang and Lee reported a performance curve for a micropump for both water and oil as a working fluid [21]. After selecting the input pressure source, the designer can calculate the dimensionless parameters shown in Table 2, and check the actual time constant. Then the steady-state speed of the SMP response can be calculated by Eq. (11). The time constant and steady-state speed can then be compared to the design requirments and then the process can be repeated to yield SMP speed and time response that fits the specific design. In this manner, the design knowledge accumulated from the analytical model and regression results are highly valuable in the practical design process so that the designer can quickly identify variable values for a SMP design and actuation to reach a required SMP velocity.

In microscale fabrication, subtractive techniques such as etching, and photolithography can be utilized to fabricate microfluidic channels in silicon or glass [22]. Recently, a wide range of interest has been focused additive manufacturing (i.e., 3D printing techniques). In this study, we have deployed additive manufacturing techniques in addition to traditional computer numerical control machining to make a millimeter scale model of the microfluidic channel and SMP. Figure 7 shows the schematic figure of the proposed channel designed in addition to the actuation method and measuring techniques. The SMP can move in the channel which is connected to a pressure source by a valve. In order to decrease the model parameter uncertainty, a simple actuation method is selected with a nearly constant gravitational pressure from a large diameter reservior. A high-speed camera is utilized to the measure the velocity of the SMP and the volumetric flowrate.

Figure 8 shows the prototype. In this study, the experimental data have explored the effects of varying flow viscosity, and potential force in addition to the effects of channel height, gap size, SMP size, and SMP mass.

The SMP is fabricated by stereolithography resin with an embedded magnet to help to reset the position of SMP after each test and increase the plate density. Figure 9 shows the SMP position inside of channel both in the rest position and under acceleration. As illustrated in Fig. 9, the SMP moving through the channel is subject to a locking mechanism based on friction called “wedging and jamming” [23]. Asymmetry in the drag and/or acceleration forces will rotate the SMP until it contacts the edges and lock the system due to friction. In order to overcome these drawbacks, four small feet were added to the SMP to align the plate to the channel with minimal friction.

The case-study tactics and design knowledge accumulated from the analytical model are highly valuable if they can verify the experimental data. In order to verify the analytical model, the system of ordinary differential equation initial value problems (Eqs. (8) and (9)) is solved numerically using the parameters from Table 2.

The test begins by filling the large cylindrical reservoir with fluid to a specific the height required to obtain a desired actuation pressure. The moving plate is positioned in its starting position. To start the test, a trigger synchronizes the start of the camera footage and opens the valve. The high-speed camera measures both position of the fluid in the outlet pipe and the position of the SMP over time. It should be noted that no flow instabilities were observed in any of the experiments that were performed. Table 5 shows the experimental parameters that were varied.

An experiment was conducted to study the effect of the potential driving force on SMP velocity. As can be seen in Fig. 10, the experimental measurements closely match the theoretical models. Figure 10(a) shows the SMP velocity for different pressure heads. Decreasing the driving force by 50% only decreased the steady-state velocity 34% due to the nonlinearity of this system. In the next set of experiments, the effect of fluid viscosity on SMP velocity was explored. Figure 10(b) shows the SMP velocity for 20.2% glycerol and 36.3% glycerol as well as water. As can be seen in this figure, the SMP steady-state velocity increases when the viscosity increases since less fluid passes through the gap between the SMP and channel wall; however, that only happened because of current combination of gap size (⁠$h/δ≈8.3$⁠). The simulations show that with smaller gap $(h/δ>10.7$⁠) the steady-state velocity will always decrease when viscosity increases. In this figure, we illustrate the influence of viscosity on fluid behavior. During the tests conducted with a 36.3% glycerol solution, the presence of small air bubbles was observed. These bubbles will initially compress, before expanding as time progresses. We believe it is possibly the existence of air bubbles in the highly viscous glycerol solution that leads to the different shape of the velocity versus time curves for the experimental and analytical curves. However, once the valve is fully open, the system eventually reaches steady-state stage, and the delay caused by the air bubbles does not affect the steady-state results. It should be noted that in the context of RF fluidic reconfigurable devices, high-viscosity fluids are typically not utilized.

Figure 11 shows the comparison between the experimental and theoretical approaches of SMP velocity as a function of time for various SMP masses and lengths. For instance, decreasing the length of SMP by 20% increases the steady-state velocity by 17.2% and decreasing the mass of SMP by 54% increases the steady-state SMP velocity by 69.9%. Table 6 shows the experimental parameters used for these tests. In addition, Fig. 12 compares nondimensional experimental and analytical models based on Table 6. The experimental results demonstrate that the prototype can achieve excellent performance in steady-state velocity and time response; however, in the transient region, the value predicted by the experiment may be a little different from the theoretical model due to some minor losses such as reservoir valve opening time, and surface roughness which are unavoidable for large scale experiment test.

To examine the model accuracy at the device scale, the results reported in recent papers by Mumcu and his group on SMP controlled reconfigurable microfluidics have been studied. In fact, many studies in the area of microfluidically reconfigurable RF device, do not report important actuation parameters such as flowrate and pressure, and mainly, they study the frequency and response time rather than modeling the fluid system. However, Palomo and Mumcu [17] reported all required variables for the proposed model.

In this work, Bartels mp-6 piezopumps were utilized to move SMP 11.57 mm in ∼1 s, the flowrate, and pressure are 8000 $μl/min$⁠, 500 $mbar$ accordingly. Their channel was made from SU8 with 275 $μm$ thickness and SMP thickness is 250 $μm$⁠. Table 7 shows the schematic figure of their setup and reported parameters while the comparison between the proposed model and the reference model is shown in Table 8.

Another highly valuable piece of work was done by González and Mumcu in 2019 [24]. In this study, the SMP located inside the microfluidic channel is repositioned with the fluid flow and actuated by a piezo-electric actuation method, Fig. 13 shows the schematic model of the actuation method. However, the authors did not measure and report the working pressure. In this case, the finite element method methodology can offer a highly precise result, so the comsolmultiphysics software will be utilized to estimate the operating pressure range for the T216-A4NO-05-piezo-electric bending disk. Figure 14 shows the pressure as a function of voltage for different membrane thickness. In the reference, the author reported the reconfiguration time 1.12 ms while 52 V applied. As can be seen in this figure, the estimated pressure should be between 23 cpa and 142 cpa for membrane thickness 100–250 $μm$⁠. The comparison between the proposed model and the reference model is shown in Table 8.

Since we were able to validate our proposed model successfully, now the proposed model can be utilized to improve the performance of the RF devices that are designed in the literature. In other words, the design knowledge accumulated from the analytical model and dimensionless parameters are highly valuable in the practical design process so that the designer can work out a specific SMP design to reach a required SMP velocity. For instance, González and Mumcu in 2019 [24] reported the response time as 1.12 ms. To increase the speed of response by 50% to 0.56 ms, the mass of SMP should be decreased by 42.1%, or the length of SMP should decrease by 55.6%. Moreover, our study shows that by increasing the diameter of reservoir by 20.3%, the speed of response can be increased by 50% to 0.56 ms or instead of increasing the diameter of reservoir which may not be feasible due to the space limitation, the designer can increase the pressure by 37.5% to reach to that result. This can be done by increasing the working voltage or changing the micropump type. Since the ratio of channel height to the gap size is greater than 10.7 $(h/δ≈41)$⁠, increasing the viscosity of the working fluid decreases the actuation speed. For instance, by decreasing the viscosity by 50%, the actuation speed can be increased by 19%, so we propose to use Novec-649, instead of FC-40 to increase the actuation speed or instead of changing the working fluid, the same result can be achieved by increasing the working temperature.

In this paper, a design methodology for a microfluidically reconfigurable RF plate was presented. A simple actuation method was utilized in this research; however, based on the required flowrate and time response, the designer can select a commercial micropump and use its performance curve to calculate the head as the differential pressure that the selected pump has to overcome in order to move the fluid. One limitation of the present model is that the applied driving pressure is a sudden jump in pressure to a constant value. When operated in a system with a micropump, the applied pressure may ramp up much more slowly. Future work should incorporate a time-dependent pressure function into the model. To explore the parameters effect in the proposed model, the nondimensional parameters were defined. The experimental results demonstrated that the model matched the measured performance very well in the steady-state region. By decreasing the mass of SMP by 42.1%, the steady-state velocity increases by 50%. On the other hand, when the length of SMP increases by 55.6%, the steady-state velocity decreases by 50%. Moreover, when the pressure increases 100%, the steady-state velocity increases 129%.

In addition, the linear regression analysis can quantify the strength of the relationship between the normalized steady-state velocity of SMP response and the nondimensional parameters. The case study tactics and design knowledge accumulated from the analytical model and dimensionless parameters are highly valuable in the practical design process so that the designer can work out a specific SMP design to reach a required SMP velocity. In other words, this model facilitates understanding of the actuation response of a given RF tuning system independent of the actuator. In addition, the proposed model was verified against published work on RF devices that utilize SMP actuation with good agreement.

In future work, this analytical model can be a valuable tool if applied to the conversion of a single-input/single-output transfer function within a controller unit. Specifically, it can be employed with pressure as the input and the velocity of SMP as the output. Integration and calibration of this model with an RF device offer the potential for users to tune the RF device without requiring in-depth knowledge of the underlying design details.

Presently, microfluidic-based devices predominantly rely on mechanical micropumps, including diaphragm micropumps, rotary micropumps, and peristaltic micropumps. These pumps necessitate expensive clean-room fabrication methods and are constrained in terms of their size. In light of these limitations, exploring alternative solutions such as the utilization of an electrowetting on dielectric pump holds promise. Such an approach has the potential to address some of the aforementioned issues by enhancing performance and reducing costs in microfluidic systems.

The authors wish to thank Dr. Gokhan Mumcu for providing additional information on the experimental setup at the Department of Electrical Engineering at the University of South Florida.

• National Science Foundation Division of Electrical, Communications and Cyber Systems Grant No. 1920953; Funder ID: 10.13039/100000148.

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.