Magnetic Beads inside Droplets for Agitation and Splitting Manipulation by Utilizing a Magnetically Actuated Platform

We successfully developed a platform for the magnetic manipulation of droplets containing magnetic beads and examined the washing behaviors of the droplets, including droplet transportation, magnetic bead agitation inside droplets, and separation from parent droplets. Magnetic field gradients were produced with two layers of 6 × 1 planar coils fabricated by using printed circuit board technology. We performed theoretical analyses to understand the characteristics of the coils and successfully predicted the magnetic field and thermal temperature of a single coil. We then investigated experimentally the agitation and splitting kinetics of the magnetic beads inside droplets and experimentally observed the washing performance in different neck-shaped gaps. The performance of the washing process was evaluated by measuring both the particle loss ratio and the optical density. The findings of this work will be used to design a magnetic-actuated droplet platform, which will separate magnetic beads from their parent droplets and enhance washing performance. We hope that this study will provide digital microfluidics for application in point-of-care testing. The developed microchip will be of great benefit for genetic analysis and infectious disease detection in the future.


Introduction
Nucleic acid-based diagnostics is rapidly expanding and has various applications in infectious disease diagnosis, pharmacogenomics, oncology, and genetic testing. Miniaturized total analysis systems (µTAS) for genetic assays have been developed by using microfluidic technologies, which integrate nucleic acid extraction, amplification, and detection [1]. Microfluidics, i.e., continuous fluid flow or droplet-based actuation, has been introduced to develop µTAS for biomedical applications successfully. Although these two approaches are promising, their dependence on equipment for point-of-care testing (POCT) remains a major challenge in real-world biosample tests. In such applications, droplet-based chips have shown significant potential for storing and processing samples and reagents without the aid of pumping devices. Recently, a considerable amount of attention has been paid to droplet-based microfluidics, also known as digital microfluidics. Droplets not only serve as reaction chambers and transportation units but also as self-contained systems. In addition to their simplicity, portability, and ability to store samples and reagents on chips, droplet-actuated platforms are particularly useful for POCT. Droplet movement is effectively controlled by numerous actuated methods, including electrowetting on dielectric [2], thermocapillary force [3], surface acoustic waves [4], electrophoresis [5], optical force [6], and magnetic force [7][8][9][10][11][12]. Among all actuation devices, magnetic-actuated droplet manipulators [7][8][9][10][11][12] exhibit some advantages due to their flexibility, long distances, high driving forces, and easy use. Using droplets as micro-reactors, reagents can be transported, merged, mixed, and analyzed effectively [13,14].

Theoretical Analysis
Microparticle kinematics obeys Newton's second law, which depends on the balance among z-axial magnetic force (F B,z ), Stokes' force (F St ), gravity force (F g ), and buoyancy force (F b ). It can be expressed as follows: Droplet movement is driven by the x-axis magnetic force (F B,z ), which can be expressed by [26] Here, m b and ρ b are the mass and density of the clustered magnetic beads, respectively; D d is the diameter of the droplet, and B m is the z-axial magnetic field generated by the Helmholtz coil. Moreover, the maximum z-direction magnetic field (B z,Max ) is generated by the coil applied with a given DC current.
The movement of the microparticle in the surrounding medium is governed by Stokes' law. Stokes' force (F St ) is expressed as follows: where η is the medium viscosity, d b is the microparticle diameter, and u b is the microparticle velocity. The difference between the gravity force and buoyancy force of a microparticle is expressed as follows: Here, ∆ρ and V b represent the difference between the densities of a microparticle and medium and the volume of a microparticle, respectively.

Design and Fabrication
The on-chip method for cell lysis and nucleic acid extraction and purification adhered to a standard protocol and used silica-coated magnetic particles. A small amount of surfactant (Span-80, Sigma-Aldrich, Burlington, MA, USA) was introduced into the mineral oil (M5904, Sigma-Aldrich, Burlington, MA, USA) coating on the cartridge to prevent droplet splitting. Sessile reagent droplets were sequentially added to load each compartment. The first compartment of the device received a droplet of biosample mixed with lysis buffer, Tris-EDTA, proteinase K, and magnetic particles. Subsequently, compartments 2 through 6 were loaded in sequence with four individual washing buffers and the PCR reagent mixture [11]. The last compartment was purposefully left empty to serve as a waste collection reservoir. The cartridge is illustrated in Figure 1a. In this study, the magnetically actuated chip consisted of three layers, namely, a cartridge, coil chip, and cooling system, as shown in Figure 1b. The cartridge was designed with seven compartments. A neck-shaped channel interconnected every compartment. During each washing step, the microparticles merged with the wash buffer then advanced to the next reagent droplet. A magnetic field gradient induced by a planar coil was employed to control the magnetic particles on this platform. A cooling system was used to dissipate the high DC current-induced thermal heat.
Here, τ is the periodic time.

Design and Fabrication
The on-chip method for cell lysis and nucleic acid extraction and purification adhered to a standard protocol and used silica-coated magnetic particles. A small amount of surfactant (Span-80, Sigma-Aldrich, Burlington, MA, USA) was introduced into the mineral oil (M5904, Sigma-Aldrich, Burlington, MA, USA) coating on the cartridge to prevent droplet splitting. Sessile reagent droplets were sequentially added to load each compartment. The first compartment of the device received a droplet of biosample mixed with lysis buffer, Tris-EDTA, proteinase K, and magnetic particles. Subsequently, compartments 2 through 6 were loaded in sequence with four individual washing buffers and the PCR reagent mixture [11]. The last compartment was purposefully left empty to serve as a waste collection reservoir. The cartridge is illustrated in Figure 1a. In this study, the magnetically actuated chip consisted of three layers, namely, a cartridge, coil chip, and cooling system, as shown in Figure 1b. The cartridge was designed with seven compartments. A neck-shaped channel interconnected every compartment. During each washing step, the microparticles merged with the wash buffer then advanced to the next reagent droplet. A magnetic field gradient induced by a planar coil was employed to control the magnetic particles on this platform. A cooling system was used to dissipate the high DC current-induced thermal heat. We used the printed circuit board (PCB) technique to print a two-layer 6 × 1 coil array chip to generate the magnetic field gradients for particular applications. The coil chip contained six coils on its top and bottom layers with a distance of 200 µm, as shown in Figure  2a. The square-profile coils were designed with eight windings, and the coils were designed with a width and spacing of 150 µm and partially overlapped between adjacent coils. Master molds were created by using 3D rapid prototyping and acryliconitrile butadiene styrene, then used to cast poly-dimethylsiloxane (PDMS) prepolymer (Sylgard 184, Dow Corning Corporation, Midland, MI, USA) with a base-to-crosslinker ratio of 9:1. After the device had been cured at 80 °C for 30 min, it was attached to a glass coverslip by using a standard O2 plasma treatment process. We dip-coated the device with a solution of 1% w/w Teflon AF 1600 (DuPont Corp., Wilmington, DE, USA) in FC-40 solvent (3M Company, St. Paul City, MN, USA) and baked it overnight at 80 °C to prevent biomolecule surface adsorption and to prevent reagent droplets from spreading on the cartridge surface. The cartridge, which was composed of seven compartments interconnected serially We used the printed circuit board (PCB) technique to print a two-layer 6 × 1 coil array chip to generate the magnetic field gradients for particular applications. The coil chip contained six coils on its top and bottom layers with a distance of 200 µm, as shown in Figure 2a. The square-profile coils were designed with eight windings, and the coils were designed with a width and spacing of 150 µm and partially overlapped between adjacent coils. Master molds were created by using 3D rapid prototyping and acryliconitrile butadiene styrene, then used to cast poly-dimethylsiloxane (PDMS) prepolymer (Sylgard 184, Dow Corning Corporation, Midland, MI, USA) with a base-to-crosslinker ratio of 9:1. After the device had been cured at 80 • C for 30 min, it was attached to a glass coverslip by using a standard O 2 plasma treatment process. We dip-coated the device with a solution of 1% w/w Teflon AF 1600 (DuPont Corp., Wilmington, DE, USA) in FC-40 solvent (3M Company, St. Paul City, MN, USA) and baked it overnight at 80 • C to prevent biomolecule surface adsorption and to prevent reagent droplets from spreading on the cartridge surface. The cartridge, which was composed of seven compartments interconnected serially by six sieve structures, were bound onto the two-layer 6 × 1 coil array chip. Figure 2b,c shows the top-side and lateral side views of the cartridge, respectively.
Micromachines 2023, 12, x FOR PEER REVIEW by six sieve structures, were bound onto the two-layer 6 × 1 coil array chip. Figu shows the top-side and lateral side views of the cartridge, respectively.

Experiments
The experimental setup (Detailed in supplementary document S1), as depi Figure 3, included a Helmholtz coil laid out around the magnetically actuated chip erate a uniform transverse magnetic field of approximately 50.0 mT. The magnetic tuated chip was placed on an experimental platform designed to observe microp kinematics. A diagnosed droplet containing 2.88 µm magnetic particles (MF-DEX MagQu LLC, Surprise City, AZ, USA) was added to the first compartment of the car Six transparent liquid droplets were sequentially added into the specific compartm the cartridge. The magnetic field's magnitude and direction can be controlled by ad the DC currents, whereas the magnetic fields produced by each coil were manage custom-designed analog circuit. A digital camera connected to a charge-coupled (CCD, Cool SNAP HQ2, Photometrics, Huntington Beach, CA, USA) was used to c images of droplet movement and magnetic particle kinetics for further analysis. In tion, the images of the dye were converted into grayscale values by using ImageJ National Institute of Health, Bethesda, MD, USA). Custom software developed i VIEW (National Instruments, Austin, TX, USA) was applied to oversee the mag duration, and sequence of the DC currents. A crucial component of this setup is th moelectric cooler integrated with a K-type thermocouple that functioned as a cooli tem beneath the PCB. We used a PID controller with a thermoelectric module to m a consistent 37 °C, crucial for cell-friendly conditions. This system effectively coun Joule heating, ensuring a steady temperature during the experiment [12].

Experiments
The experimental setup (Detailed in Supplementary Document S1), as depicted in Figure 3, included a Helmholtz coil laid out around the magnetically actuated chip to generate a uniform transverse magnetic field of approximately 50.0 mT. The magnetically actuated chip was placed on an experimental platform designed to observe microparticle kinematics. A diagnosed droplet containing 2.88 µm magnetic particles (MF-DEX-3000, MagQu LLC, Surprise City, AZ, USA) was added to the first compartment of the cartridge. Six transparent liquid droplets were sequentially added into the specific compartment of the cartridge. The magnetic field's magnitude and direction can be controlled by adjusting the DC currents, whereas the magnetic fields produced by each coil were managed by a custom-designed analog circuit. A digital camera connected to a charge-coupled device (CCD, Cool SNAP HQ2, Photometrics, Huntington Beach, CA, USA) was used to capture images of droplet movement and magnetic particle kinetics for further analysis. In addition, the images of the dye were converted into grayscale values by using ImageJ (v1.8.0, National Institute of Health, Bethesda, MD, USA). Custom software developed in LabVIEW (National Instruments, Austin, TX, USA) was applied to oversee the magnitude, duration, and sequence of the DC currents. A crucial component of this setup is the thermoelectric cooler integrated with a K-type thermocouple that functioned as a cooling system beneath the PCB. We used a PID controller with a thermoelectric module to maintain a consistent 37 • C, crucial for cell-friendly conditions. This system effectively counteracts Joule heating, ensuring a steady temperature during the experiment [12].

Magnetic Characterization
The theoretical calculation of the z-axial magnetic field of the eight wound coils is difficult and complex. Thus, a simple model is introduced that uses a single coil for calculation. For different heights, BZ,Max at the center of the coil can be expressed as follows: (Detailed in supplementary document S2). Here, z represents the height at different locations, either A or B. li signifies the distance of the i-th winding coil from the center. The distance can be calculated using the formula li = 225 + (i − 1) × 300 µm. In the present study, zA = 300 µm at location A, and zB = 400 µm at location B. After substituting these values into Equation (10), we can obtain BZ,A = 3.66 × I for location A and BZ,B = 3.10 × I for location B, respectively.
The magnetic field in the z-direction (BZ,Max) was theoretically predicted by Equation (10), and the obtained results were then compared with the measurements. The dynamics of the magnetic field generated by the coil is a crucial issue for the magnetically actuated mechanism of droplets. Consequently, understanding the magnetic field is a priority for understanding the underlying physical phenomenon. A tesla meter (tm-401, Kanetec, Tokyo, Japan) was employed to measure the magnetic fields at the center of the coil. Figure  4 presents the maximum experimental magnetic field (Bz,Max) in relation to various applied currents at different locations. Locations A and B represent the top and bottom layers of the coil, respectively, as indicated in Figure 4. The height difference between the two locations of A and B was set to be approximately 100 µm. In experiments involving planar coils of varying heights (i.e., different locations), the magnetic fields exhibited a quasilinear correlation with the applied DC currents. As anticipated, the applied DC current and magnetic field demonstrated a linear relationship. The magnetic fields also decreased with the increase in coil height, emphasizing the importance of considering the coil's position when designing and optimizing the magnetically actuated mechanism for droplet control. Furthermore, the theoretical analysis was reasonably in agreement with the experimental measurements. The theoretical predictions were within 17.2% and 25.7% of the experimental measurements for locations A and B, respectively.

Magnetic Characterization
The theoretical calculation of the z-axial magnetic field of the eight wound coils is difficult and complex. Thus, a simple model is introduced that uses a single coil for calculation. For different heights, B Z,Max at the center of the coil can be expressed as follows: (Detailed in Supplementary Document S2).
Here, z represents the height at different locations, either A or B. li signifies the distance of the i-th winding coil from the center. The distance can be calculated using the formula l i = 225 + (i − 1) × 300 µm. In the present study, z A = 300 µm at location A, and z B = 400 µm at location B. After substituting these values into Equation (10), we can obtain B Z,A = 3.66 × I for location A and B Z,B = 3.10 × I for location B, respectively.
The magnetic field in the z-direction (B Z,Max ) was theoretically predicted by Equation (10), and the obtained results were then compared with the measurements. The dynamics of the magnetic field generated by the coil is a crucial issue for the magnetically actuated mechanism of droplets. Consequently, understanding the magnetic field is a priority for understanding the underlying physical phenomenon. A tesla meter (tm-401, Kanetec, Tokyo, Japan) was employed to measure the magnetic fields at the center of the coil. Figure 4 presents the maximum experimental magnetic field (Bz, Max ) in relation to various applied currents at different locations. Locations A and B represent the top and bottom layers of the coil, respectively, as indicated in Figure 4. The height difference between the two locations of A and B was set to be approximately 100 µm. In experiments involving planar coils of varying heights (i.e., different locations), the magnetic fields exhibited a quasi-linear correlation with the applied DC currents. As anticipated, the applied DC current and magnetic field demonstrated a linear relationship. The magnetic fields also decreased with the increase in coil height, emphasizing the importance of considering the coil's position when designing and optimizing the magnetically actuated mechanism for droplet control. Furthermore, the theoretical analysis was reasonably in agreement with the experimental measurements. The theoretical predictions were within 17.2% and 25.7% of the experimental measurements for locations A and B, respectively.  Here, zA = 300 µm at location A and zB = 400 µm at location B.
The 1-D, i.e., z-directional, steady-state heat conduction equation can be expressed as The power density (ġ) of the coil is obtained as Here, and AC are the electrical conductivity and cross-sectional area of a winding copper, respectively. AS and t are the surface area and thickness of a coil, respectively. The top and bottom surfaces dissipated by heat convection are expressed as (13) where k and h are the thermal conductivity and heat convention coefficient, respectively.
In this study, the geometric parameters were calculated as LC = 17.4 mm, AS = 36 mm 2 , Ac = 5.25 × 10 −3 mm 2 , σ = 5.996 × 10 7 Sm −1 There are several factors that affect the convection coefficient h, such as flow pattern, fluid properties, surface geometry, bulk velocity, and temperature difference between surfaces and their surroundings [27]. In general, the convection coefficient (h) ranges from 2 to 25 Wm −2 K −1 [28]. So, it is difficult to be determined. Here, h = 10 and 15 Wm −2 K −1 are conducted to investigate the surface temperature.
A higher magnetic field was achieved under higher applied DC currents. However, higher applied DC currents can also generate increased thermal power, leading to elevated temperatures. Therefore, temperature management is a critical aspect of the biosample platform, making temperature control an essential challenge. The 1-D, i.e., z-directional, steady-state heat conduction equation can be expressed as The power density ( . g) of the coil is obtained as Here, σ and A C are the electrical conductivity and cross-sectional area of a winding copper, respectively. A S and t are the surface area and thickness of a coil, respectively. The top and bottom surfaces dissipated by heat convection are expressed as where k and h are the thermal conductivity and heat convention coefficient, respectively. Substituting Equation (12) into Equation (11) and integrating yields the following formula: (Detailed in Supplementary Document S3).
In this study, the geometric parameters were calculated as L C = 17.4 mm, A S = 36 mm 2 , Ac = 5.25 × 10 −3 mm 2 , σ = 5.996 × 10 7 Sm −1 There are several factors that affect the convection coefficient h, such as flow pattern, fluid properties, surface geometry, bulk velocity, and temperature difference between surfaces and their surroundings [27]. In general, the convection coefficient (h) ranges from 2 to 25 Wm −2 K −1 [28]. So, it is difficult to be determined. Here, h = 10 and 15 Wm −2 K −1 are conducted to investigate the surface temperature.
A higher magnetic field was achieved under higher applied DC currents. However, higher applied DC currents can also generate increased thermal power, leading to elevated temperatures. Therefore, temperature management is a critical aspect of the biosample platform, making temperature control an essential challenge. Figure 5 displays the maximum temperature measured in relation to the applied current. K-type thermocouples were affixed to the tops of the planar coils to measure their temperatures. In the experimental results, the maximum temperatures exhibited a quadratic increase with the increase in applied currents, emphasizing the importance of maintaining a balance between magnetic field strength and temperature control. The theoretical predictions based on the experimental data agreed with the experimental data with a variation of 12.8% for the convention h = 15 Wm −2 K −1 .
the maximum temperature measured in relation to the applied current. K-type thermocouples were affixed to the tops of the planar coils to measure their temperatures. In the experimental results, the maximum temperatures exhibited a quadratic increase with the increase in applied currents, emphasizing the importance of maintaining a balance between magnetic field strength and temperature control. The theoretical predictions based on the experimental data agreed with the experimental data with a variation of 12.8% for the convention h = 15 Wm −2 K −1 .

Agitation Characterization
In the present study, we set a standard reference scenario by keeping a steady direct current of 1.5 A, alternating at a frequency of 1.0 Hz, and enclosing a 100 µg bead in a 20-uL droplet at a consistent temperature of 37 °C. Our goal was to examine and understand the fundamental behavior of the washing process under these particular conditions. By agitating magnetic particles under alternating attractive and repulsive forces, the coil platform can actuate magnetic particles efficiently ( Figure 6). Permanent magnets were used to affix the magnetic polarization of the microparticles in a particular direction. Repulsion and attraction were created as a result of changing the direction of the DC current. The droplet was agitated by a single coil that was centered over it. The coil generated magnetic field gradients that were alternatively upward or downward due to the change in the direction of the DC current. The microparticles inside the droplet can be repeatedly agitated by alternating the direction of the current. During this study, 300 µg of microparticles were agitated with an applied DC current of 1.

Agitation Characterization
In the present study, we set a standard reference scenario by keeping a steady direct current of 1.5 A, alternating at a frequency of 1.0 Hz, and enclosing a 100 µg bead in a 20-uL droplet at a consistent temperature of 37 • C. Our goal was to examine and understand the fundamental behavior of the washing process under these particular conditions. By agitating magnetic particles under alternating attractive and repulsive forces, the coil platform can actuate magnetic particles efficiently ( Figure 6). Permanent magnets were used to affix the magnetic polarization of the microparticles in a particular direction. Repulsion and attraction were created as a result of changing the direction of the DC current. The droplet was agitated by a single coil that was centered over it. The coil generated magnetic field gradients that were alternatively upward or downward due to the change in the direction of the DC current. The microparticles inside the droplet can be repeatedly agitated by alternating the direction of the current. During this study, 300 µg of microparticles were agitated with an applied DC current of 1.5 A at a frequency of 1.0 Hz. the maximum temperature measured in relation to the applied current. K-type thermocouples were affixed to the tops of the planar coils to measure their temperatures. In the experimental results, the maximum temperatures exhibited a quadratic increase with the increase in applied currents, emphasizing the importance of maintaining a balance between magnetic field strength and temperature control. The theoretical predictions based on the experimental data agreed with the experimental data with a variation of 12.8% for the convention h = 15 Wm −2 K −1 .

Agitation Characterization
In the present study, we set a standard reference scenario by keeping a steady direct current of 1.5 A, alternating at a frequency of 1.0 Hz, and enclosing a 100 µg bead in a 20-uL droplet at a consistent temperature of 37 °C. Our goal was to examine and understand the fundamental behavior of the washing process under these particular conditions. By agitating magnetic particles under alternating attractive and repulsive forces, the coil platform can actuate magnetic particles efficiently ( Figure 6). Permanent magnets were used to affix the magnetic polarization of the microparticles in a particular direction. Repulsion and attraction were created as a result of changing the direction of the DC current. The droplet was agitated by a single coil that was centered over it. The coil generated magnetic field gradients that were alternatively upward or downward due to the change in the direction of the DC current. The microparticles inside the droplet can be repeatedly agitated by alternating the direction of the current. During this study, 300 µg of microparticles were agitated with an applied DC current of 1. The motion of microparticles within droplets can cause a significant mixing effect. Thus, microparticles can effectively perform capturing and washing functions. Two samples containing blue dye (oil phase) and a buffer solution (sodium borate, 1 mM, pH 8.2) were used to evaluate the mixing effect to verify the mixing efficiency of the proposed magnetically actuated mixer. First, 5.0 µL of blue dye was pipetted into a 15 µL droplet of buffer solution, as shown in Figures 7a(i) and 8a(i). As can be clearly seen in the figures, no microbeads were agitated under the initial conditions. In the droplet-based mixer, the blue dye (oil phase) mainly diffused in the upper part of the droplet (see Figure 7a(ii),a(iii) but cannot diffuse in the lower part because the density of oil was less than that of water (see Figure 7b(ii),b(iii)). Conversely, significant mixing was observed in Figure 8, when a DC current of 1.5 A with a frequency of 1.0 Hz was applied to the coil. Figure 8 shows the images taken successively by agitating magnetic beads to disturb the fluid inside droplets. By moving the magnetic beads back and forth, the fluid inside the droplets was disturbed to distribute the dye uniformly after 4.0 s. Figure 8b(ii),b(iii) illustrate that the homogeneous mixing of the blue dye resulted in high mixing efficiency. A droplet-based mixer was agitated in four back-and-forth cycles to demonstrate uniform mixing quickly by using magnetic beads. The experimental results were consistent with previously reported findings [12]. The motion of microparticles within droplets can cause a significant mixing effect. Thus, microparticles can effectively perform capturing and washing functions. Two samples containing blue dye (oil phase) and a buffer solution (sodium borate, 1 mM, pH 8.2) were used to evaluate the mixing effect to verify the mixing efficiency of the proposed magnetically actuated mixer. First, 5.0 µL of blue dye was pipetted into a 15 µL droplet of buffer solution, as shown in Figures 7a(i) and 8a(i). As can be clearly seen in the figures, no microbeads were agitated under the initial conditions. In the droplet-based mixer, the blue dye (oil phase) mainly diffused in the upper part of the droplet (see Figure 7a(ii),(iii) but cannot diffuse in the lower part because the density of oil was less than that of water (see Figure 7b(ii),(iii)). Conversely, significant mixing was observed in Figure 8, when a DC current of 1.5 A with a frequency of 1.0 Hz was applied to the coil. Figure 8 shows the images taken successively by agitating magnetic beads to disturb the fluid inside droplets. By moving the magnetic beads back and forth, the fluid inside the droplets was disturbed to distribute the dye uniformly after 4.0 s. Figure 8b(ii),(iii) illustrate that the homogeneous mixing of the blue dye resulted in high mixing efficiency. A dropletbased mixer was agitated in four back-and-forth cycles to demonstrate uniform mixing quickly by using magnetic beads. The experimental results were consistent with previously reported findings [12]. The motion of microparticles within droplets can cause a significant mixing Thus, microparticles can effectively perform capturing and washing functions. Two ples containing blue dye (oil phase) and a buffer solution (sodium borate, 1 mM, p were used to evaluate the mixing effect to verify the mixing efficiency of the pro magnetically actuated mixer. First, 5.0 µL of blue dye was pipetted into a 15 µL drop buffer solution, as shown in Figures 7a(i) and 8a(i). As can be clearly seen in the fi no microbeads were agitated under the initial conditions. In the droplet-based mixe blue dye (oil phase) mainly diffused in the upper part of the droplet (see Figure 7a(ii but cannot diffuse in the lower part because the density of oil was less than that of (see Figure 7b(ii),b(iii)). Conversely, significant mixing was observed in Figure 8, w DC current of 1.5 A with a frequency of 1.0 Hz was applied to the coil. Figure 8 show images taken successively by agitating magnetic beads to disturb the fluid inside dro By moving the magnetic beads back and forth, the fluid inside the droplets was dist to distribute the dye uniformly after 4.0 s. Figure 8b(ii),b(iii) illustrate that the homo ous mixing of the blue dye resulted in high mixing efficiency. A droplet-based mixe agitated in four back-and-forth cycles to demonstrate uniform mixing quickly by magnetic beads. The experimental results were consistent with previously reported ings [12].

Splitting Characterization
For the purpose of diagnosis, magnetic beads bound to a specific virus have split from the parent droplet. Therefore, the kinetics of magnetic bead splitting from lets is highly challenging. Figure 9 shows how a droplet containing magnetic particle deformed by neck-shaped channels and split into a smaller plug that contained from its parent droplet. This was achieved by using surface topography to help the d split. When the maximum of the magnetic field was reached at the destination com ment, magnetic particles were gathered into a plug and pulled through an elongated shaped structure until scission occurred [11]. As a result of splitting from its parent let, a magnetic particle joined the following droplet. With an optimal concentration o factant, in this case 0.5% w/w mixed with mineral oil, magnetic particles can be su fully extracted from a parent droplet and transported to the subsequent compartme while preventing premature merging with the preceding droplet [29]. By doing so, d compartments could be packed more compactly onto smaller cartridges. Magnetic cles could be collected in the center of a compartment by creating a field maximum center. Two distinct phenomena can occur when utilizing varying magnetic force applied DC current, to separate magnetic beads from the droplet. In Figure 9, the DC rent of 1.5 A is employed to separate all magnetic beads from the droplet. In con Figure 10 uses the DC current of 1.0 A to segregate only a portion of the magnetic b thereby enabling the continuous separation of satellite droplets. The continuous im show that the satellite droplets successfully separated from their parent droplets. Su mentary Video S1 presents the separation sequence of the droplets.

Splitting Characterization
For the purpose of diagnosis, magnetic beads bound to a specific virus have to be split from the parent droplet. Therefore, the kinetics of magnetic bead splitting from droplets is highly challenging. Figure 9 shows how a droplet containing magnetic particles was deformed by neck-shaped channels and split into a smaller plug that contained water from its parent droplet. This was achieved by using surface topography to help the droplet split. When the maximum of the magnetic field was reached at the destination compartment, magnetic particles were gathered into a plug and pulled through an elongated neck-shaped structure until scission occurred [11]. As a result of splitting from its parent droplet, a magnetic particle joined the following droplet. With an optimal concentration of surfactant, in this case 0.5% w/w mixed with mineral oil, magnetic particles can be successfully extracted from a parent droplet and transported to the subsequent compartment, all while preventing premature merging with the preceding droplet [29]. By doing so, droplet compartments could be packed more compactly onto smaller cartridges. Magnetic particles could be collected in the center of a compartment by creating a field maximum at the center. Two distinct phenomena can occur when utilizing varying magnetic forces, i.e., applied DC current, to separate magnetic beads from the droplet. In Figure 9, the DC current of 1.5 A is employed to separate all magnetic beads from the droplet. In contrast, Figure 10 uses the DC current of 1.0 A to segregate only a portion of the magnetic beads, thereby enabling the continuous separation of satellite droplets. The continuous images show that the satellite droplets successfully separated from their parent droplets. Supplementary Video S4 presents the separation sequence of the droplets.

Washing Characterization
We estimated the color intensity of the blue dye carried over between each a droplet to evaluate the effectiveness of the washing process. The neck-shaped channel had a gap of 500 µm and a length of 1500 µm. The washing process involved the motion of a droplet, agitation of magnetic beads within a droplet, and extraction of magnetic beads from a droplet. The blue dye intensity of a droplet indicates how well the magnetic beads have been washed after each step. Incomplete washing is indicated by higher intensity. By con-

Washing Characterization
We estimated the color intensity of the blue dye carried over between each a droplet to evaluate the effectiveness of the washing process. The neck-shaped channel had a gap of 500 µm and a length of 1500 µm. The washing process involved the motion of a droplet, agitation of magnetic beads within a droplet, and extraction of magnetic beads from a droplet. The blue dye intensity of a droplet indicates how well the magnetic beads have been washed after each step. Incomplete washing is indicated by higher intensity. By con-

Washing Characterization
We estimated the color intensity of the blue dye carried over between each a droplet to evaluate the effectiveness of the washing process. The neck-shaped channel had a gap of 500 µm and a length of 1500 µm. The washing process involved the motion of a droplet, agitation of magnetic beads within a droplet, and extraction of magnetic beads from a droplet. The blue dye intensity of a droplet indicates how well the magnetic beads have been washed after each step. Incomplete washing is indicated by higher intensity. By contrast, lower intensity indicates more complete washing. The effect of washing was evaluated by measuring the mean gray value of the blue dye in compartment 1 against serial dilutions of the buffer. A DC current of 1.5 A was applied to split microparticles from a droplet, and 1.5 A was applied to agitate magnetic beads inside a droplet. The driving frequency was also set to 1.0 Hz. The magnetic particles were washed in order from compartment 2 to compartment 6 after separating them from compartment 1 containing blue dye. In the first chamber, a 15 µL droplet of blue dye was dispensed. Six 15 µL transparent droplets were dispensed in the second to sixth compartments as previously shown in Figure 1a. We use a 20 µL droplet, consisting of 15 µL water and 5 µL dye, due to its compatibility with our planar coil and the visibility it offers for real-time performance analysis. A DC current of 1.5 A with an alternation frequency of 1.0 Hz was applied during eight agitation cycles to distribute the dye in each compartment uniformly. By using a DC current of 1.5 A and a frequency of 1.0 Hz, the magnetic beads inside a droplet were actuated across the neck-shaped channel to split from the previous droplet. The intensity of a blue-colored droplet depends on the magnetic beads that carry liquid dye to the next droplet. The blue dye was also uniformly distributed within the next transparent droplet through the agitation of the magnetic beads. The dye intensity of the droplet should have reduced compared with that of the previous droplet. The dye intensity images of the first droplet were compared with those of the following droplets to quantify the amount of liquid carried by the magnetic beads during the washing process. The intensity of blue dye in different chambers is shown in Figure 11a. Moreover, a graph showing the normalized concentration of blue dye in each chamber can be seen in Figure 11b. These two parameters are related by the Eq. C* = 100e −1.15N . The coefficient of determination was R 2 = 0.99. Here C* represents the normalized concentration, and N represents the washing chamber. The continuous images show that the magnetic beads inside a droplet successfully separated from their previous parent droplet to perform the washing effect. Supplementary Video S5 presents the washing sequence of the magnetic beads inside the droplets. trast, lower intensity indicates more complete washing. The effect of washing was evaluated by measuring the mean gray value of the blue dye in compartment 1 against serial dilutions of the buffer. A DC current of 1.5 A was applied to split microparticles from a droplet, and 1.5 A was applied to agitate magnetic beads inside a droplet. The driving frequency was also set to 1.0 Hz. The magnetic particles were washed in order from compartment 2 to compartment 6 after separating them from compartment 1 containing blue dye. In the first chamber, a 15 µL droplet of blue dye was dispensed. Six 15 µL transparent droplets were dispensed in the second to sixth compartments as previously shown in Figure 1a. We use a 20 µL droplet, consisting of 15 µL water and 5 µL dye, due to its compatibility with our planar coil and the visibility it offers for real-time performance analysis. A DC current of 1.5 A with an alternation frequency of 1.0 Hz was applied during eight agitation cycles to distribute the dye in each compartment uniformly. By using a DC current of 1.5 A and a frequency of 1.0 Hz, the magnetic beads inside a droplet were actuated across the neck-shaped channel to split from the previous droplet. The intensity of a bluecolored droplet depends on the magnetic beads that carry liquid dye to the next droplet.
The blue dye was also uniformly distributed within the next transparent droplet through the agitation of the magnetic beads. The dye intensity of the droplet should have reduced compared with that of the previous droplet. The dye intensity images of the first droplet were compared with those of the following droplets to quantify the amount of liquid carried by the magnetic beads during the washing process. The intensity of blue dye in different chambers is shown in Figure 11a. Moreover, a graph showing the normalized concentration of blue dye in each chamber can be seen in Figure 11b. These two parameters are related by the Eq. C* = 100e −1.15N . The coefficient of determination was R 2 = 0.99. Here C* represents the normalized concentration, and N represents the washing chamber. The continuous images show that the magnetic beads inside a droplet successfully separated from their previous parent droplet to perform the washing effect. Supplementary Video S2 presents the washing sequence of the magnetic beads inside the droplets. A larger neck-shaped gap allowed a droplet to pass through without magnetic bead separation. Conversely, a smaller neck-shape gap prevented the parent droplet from moving and allowed the magnetic beads to be extracted. As discussed in this section, we investigated the effect of the neck-shaped gaps on the magnetic beads' ability to carry over droplets. Figure 12a-c show the droplet dilution diagrams obtained after the magnetic beads were split from the neck-shaped gaps of 500, 750, and 1000 µm. Analysis was performed by separating the magnetic beads from the fourth washing solution and diluting them. ImageJ software was used to determine whether any changes should be made to the color intensity of the photos, and the results were compared with a previously established standard curve. The experiments revealed variations of 1.17%, 1.52%, and 2.06% in blue dye intensities in the neck-shaped gaps of 500, 750, and 1000 µm, respectively, as shown in Figure 12d. Washing performance can be improved by a smaller neck-shape gap of 500 µm. Our analysis will provide valuable insight into the influence of neck-shaped gaps on droplet agitation and splitting in microfluidics. conditions are set as follows: a DC current of 1.5 A, a beads' mass of 100.0 µg in the droplet, and a frequency of 1.0 Hz alternation for the adjacent coil. The red scalar represents 1.0 mm.
A larger neck-shaped gap allowed a droplet to pass through without magnetic bead separation. Conversely, a smaller neck-shape gap prevented the parent droplet from moving and allowed the magnetic beads to be extracted. As discussed in this section, we investigated the effect of the neck-shaped gaps on the magnetic beads' ability to carry over droplets. Figure 12a-c show the droplet dilution diagrams obtained after the magnetic beads were split from the neck-shaped gaps of 500, 750, and 1000 µm. Analysis was performed by separating the magnetic beads from the fourth washing solution and diluting them. ImageJ software was used to determine whether any changes should be made to the color intensity of the photos, and the results were compared with a previously established standard curve. The experiments revealed variations of 1.17%, 1.52%, and 2.06% in blue dye intensities in the neck-shaped gaps of 500, 750, and 1000 µm, respectively, as shown in Figure 12d. Washing performance can be improved by a smaller neck-shape gap of 500 µm. Our analysis will provide valuable insight into the influence of neck-shaped gaps on droplet agitation and splitting in microfluidics. Particle correction and the optical detection were set up in chamber No. 6. Particle loss plays a pivotal role when separating magnetic beads from the parent droplet, especially in microfluidic or lab-on-a-chip setups. We define the particle loss ratio as the ratio of collected particle mass to the initial particle mass. Figure 13a showed a particle-loss ratio of 0.67%, 0.52%, and 0.42% for gap sizes of 500, 750, and 1000 µm, respectively, suggesting larger gaps might enhance bead transport efficiency, reducing bead loss. Moreover, as indicated by the Beer-Lambert law, optical density (OD) or absorbance correlates Particle correction and the optical detection were set up in chamber No. 6. Particle loss plays a pivotal role when separating magnetic beads from the parent droplet, especially in microfluidic or lab-on-a-chip setups. We define the particle loss ratio as the ratio of collected particle mass to the initial particle mass. Figure 13a showed a particle-loss ratio of 0.67%, 0.52%, and 0.42% for gap sizes of 500, 750, and 1000 µm, respectively, suggesting larger gaps might enhance bead transport efficiency, reducing bead loss. Moreover, as indicated by the Beer-Lambert law, optical density (OD) or absorbance correlates directly with the concentration of the absorbent substance within a solution. Figure 13b portrays this relationship, displaying OD values of 73.8%, 80.1%, and 83.9% for gaps of 500, 750, and 1000 µm, respectively. It suggests a lower concentration of blue dye results in lower OD, whereas a higher concentration increases OD.
directly with the concentration of the absorbent substance within a solution. Figure 13b portrays this relationship, displaying OD values of 73.8%, 80.1%, and 83.9% for gaps of 500, 750, and 1000 µm, respectively. It suggests a lower concentration of blue dye results in lower OD, whereas a higher concentration increases OD.
(a) (b) Figure 13. The relationship of (a) particle loss ratio and (b) optical density (OD) with respect to different gap sizes of 500, 750, and 1000 µm. The operational conditions are set as follows: a DC current of 1.5 A, a beads' mass of 100.0 µg in the droplet, and a frequency of 1.0 Hz alternation for the adjacent coil.

Conclusions
We demonstrated a magnetic droplet-based actuation system for agitating microbeads inside droplets and splitting magnetic particles from droplets. We also formulated a straightforward theoretical model for both the magnetic field and thermal temperature. This model aligns reasonably well with experimental measurements, displaying variations of 25.7% for the magnetic field and 12.8% for the thermal temperature. In addition, the use of microbeads within droplets enhanced washing performance based on agitation and separation. In the current study, particle-loss ratios of 0.67%, 0.52%, and 0.42% correspond to gap sizes of 500, 750, and 1000 µm, respectively. Concurrently, OD values

Conclusions
We demonstrated a magnetic droplet-based actuation system for agitating microbeads inside droplets and splitting magnetic particles from droplets. We also formulated a straightforward theoretical model for both the magnetic field and thermal temperature. This model aligns reasonably well with experimental measurements, displaying variations of 25.7% for the magnetic field and 12.8% for the thermal temperature. In addition, the use of microbeads within droplets enhanced washing performance based on agitation and separation. In the current study, particle-loss ratios of 0.67%, 0.52%, and 0.42% correspond to gap sizes of 500, 750, and 1000 µm, respectively. Concurrently, OD values are 73.8%, 80.1%, and 83.9% for the same gaps. These results demonstrate that particle loss increases with larger gaps, whereas OD increases with smaller gaps. Additionally, the proposed platform can be integrated with thermal control and optical detection, adding to its benefits. We hope to deliver the developed platform for POCT in the future.
Supplementary Materials: The following supporting information can be downloaded at: https:// www.mdpi.com/article/10.3390/mi14071349/s1. Document S1: Configuration for the experimental setup of the entity; Document S2: Magnetic-field derivation of a single coil with eight turns; Document S3: Temperature derivation of a single coil with eight turns; Video S4: Satellite droplets continuously splitting from a parent droplet; Video S5: The washing performance evaluated within different chambers of a droplet.

Data Availability Statement:
The data used in this study can be found within the article. For any further data that bolster the conclusions of this study, please direct requests to the author in charge of correspondence.