Ferrofluid droplet manipulation using an adjustable alternating magnetic field

Journal Pre-proof Ferrofluid droplet manipulation using an adjustable alternating magnetic field Mohamad Ali Bijarchi, Amirhossein Favakeh, Erfan Sedighi, Mohammad Behshad Shafii

PII:

S0924-4247(19)31280-4

DOI:

https://doi.org/10.1016/j.sna.2019.111753

Reference:

SNA 111753

To appear in:

Sensors and Actuators: A. Physical

Received Date:

20 July 2019

Revised Date:

7 October 2019

Accepted Date:

11 November 2019

Please cite this article as: Bijarchi MA, Favakeh A, Sedighi E, Shafii MB, Ferrofluid droplet manipulation using an adjustable alternating magnetic field, Sensors and Actuators: A. Physical (2019), doi: https://doi.org/10.1016/j.sna.2019.111753

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Ferrofluid droplet manipulation using an adjustable alternating magnetic field

Mohamad Ali Bijarchia, Amirhossein Favakeha, Erfan Sedighia, Mohammad Behshad Shafiia,* a

Department of Mechanical Engineering, Sharif University of Technology, Tehran, Iran. P.O.

Box: 11155-9567

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Corresponding Author: [email protected]

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*

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Graphical abstract

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Highlights:

The alternating magnetic field is utilized for droplet manipulation. A water-based ferrofluid with bio-compatible surfactant is used. An analytical model is presented for droplet manipulation on hydrophobic surfaces. Droplet manipulation and mixing are studied on a hydrophobic surface and in oil. Alternating magnetic field can be considered as an alternative for previous methods.

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    

Abstract

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Magnetically actuated droplet manipulation offers a promising tool for biomedical and engineering applications, such as drug delivery, biochemistry, sample handling in lab-on-chip devices and tissue engineering. In this study, characteristics of an adjustable alternating magnetic field generated by a magnetic coil for droplet manipulation was investigated which enables more control on droplet transport, and it can be considered as a suitable alternative for moving magnets or an array of micro-coils. By adjusting the magnetic flux density, the duty cycle and applied magnetic

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frequency, the manipulation of water-based ferrofluid droplets with a bio-compatible surfactant

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for different volumes was comprehensively examined. Also, the platform was able to manipulate the ferrofluid droplets completely immersed in oil. This solves the problem with droplet

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evaporation which has previously been reported for droplet manipulation on the surface.

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Furthermore, an analytical model is proposed for the movement of the ferrofluid droplet on the hydrophobic surface. The model predictions are in good agreement with experimental results.

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Also, the effects of magnetic flux density, duty cycle, frequency and the distance between the coils on the mixing process were studied. Results showed that the droplet movement on the hydrophobic

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surface was fully synchronized with the generated signal while the droplet moved backward in the oil after the magnetic field was turned off. By decreasing magnetic flux density, droplet volume, duty cycle, as well as increasing applied magnetic frequency, the step lengths become more uniform, the resolution of droplet displacement increases, but the average-velocity decreases.

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Keywords: Ferrofluid; magnetic digital microfluidics; droplet manipulation; alternating magnetic field

1. Introduction

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‘Digital microfluidics’ at first was proposed as a concept for organizing and operating complex microfluidic systems in a manner that is analogous to a digital computer [1]. Despite the usual procedure used on microchannel devices, which is using continuous flow and interconnected streams within microchannels, liquids in digital microfluidics are divided into specified droplets that are independently controlled using simple electrical or mechanical mechanisms. Digital microfluidics (DMF) has shown high flexibility and capability of performing multiplex and

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parallel biochemical operations, and therefore, has been considered as a suitable candidate for

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point of care applications [2-5]. The fluid movement mechanism on DMF systems does not require embedding complicated parts such as pumps or valves [6]. Additionally, High precision and speed

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can be achieved for sample preparation on such microfluidic systems [7]. Digital microfluidic

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devices are categorized into open and closed configurations [8]. Several different techniques such as electrowetting on dielectric (EWOD) [9-14], dielectrophoresis (DEP) [15, 16],

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electrohydrodynamic (EHD) breakup [17], surface acoustic waves [18-21], thermocapillary force [22], optoelectrowetting [23, 24], and magnetic actuation [25-30] have been proposed by

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investigators for controlling the droplets.

Electrowetting is the physical phenomenon where the wettability of a surface is adjusted using an applied electrical field [31]. Electrowetting on Dielectric (EWOD) is a well-known type of digital microfluidic platform in which liquids are processed as individual unit-sized droplets that are

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dispensed from a source, merged together, split apart or transported between different locations [1]. In this method, controlling electrodes along with hydrophobic surfaces are utilized for the droplet manipulation. This technique enables programmable transportation of droplets throughout a network of electrodes by using the electrodes subsequently in a specified procedure [32]. Fan et al. [33] performed DEP and EWOD methods concurrently on a general digital microfluidic

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platform, where dielectric (silicone oil) and conductive (water) droplets were driven. They demonstrated the merging of water and oil droplets, transporting and splitting the merged waterin-oil droplet in their investigation. Luan et al. [34] fabricated an optical sensing system integrated with an EWOD digital microfluidic system to detect changes in the index of refraction by testing glucose solutions in small droplets with different concentrations that were delivered by the

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microfluidics system. Ferrofluids, which are colloidal suspensions of magnetic material in a liquid medium, are liquids

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that respond to an external magnetic field. The coupling of liquid and magnetic behavior results in

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controlling the liquid’s location by an applied magnetic field [35-37]. Ferrofluids have been leveraged in a wide range of applications [38]. In their earliest application, NASA used them as

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rotating shaft seals in satellites [39]. Recently, ferrofluids application in other fields such as thermal management and microfluidics has been investigated extensively [40]. Ghofrani et al. [41]

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experimentally investigated forced convection heat transfer of an aqueous ferrofluid flow passing through a circular copper tube in the presence of an alternating magnetic field. Their results showed

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that applying an alternating magnetic field enhanced the convective heat transfer rate since the particle migration and the disturbance of the boundary layer was intensified. Zhang et al. [42] presented a novel magnetic repulsion-actuated microfluidic technique that is capable of achieving on-demand manipulation, including splitting, dispensing, trapping, release and demulsification of

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nano to pico-liter volumes of water droplets. They implemented those procedures by using ferrofluid, microfluidic chips, and a magnet. Aboutalebi et al. [43] used a T-junction for the asymmetric splitting of ferrofluid droplets using an asymmetric magnetic field. They showed the capability of the magnetic field to produce unequal droplets with a splitting ratio up to 0.75. A

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correlation was derived to predict the splitting ratio based on the capillary number, droplet length, and magnetic bond number. Zhang and Nguyen [44] comprehensively showed a wide range of application of magnetic digital microfluidics in their review paper. They categorized the studies in magnetic digital microfluidics into three types, including magnetic droplets, magnetic liquid marbles [45], and magnetic

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substrates. The application of ferrofluids in digital microfluidics (first category) is also appealing to investigators. Magnetic digital microfluidics utilizes magnetic forces for actuation, droplet

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formation, and splitting, and mixing [46], and proposes particular options for digital microfluidic

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platforms. Magnetic particles used in magnetic digital microfluidics, in addition to serving as actuators, provide a functional solid substrate for molecule binding, which is a desired

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characteristic in molecular diagnostics and immunodiagnostics [44]. Furthermore, magnetic digital microfluidics can be utilized without using any external source of power, which is also a desired

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characteristic in point-of-care diagnostics [44]. Lehman et al. [47], demonstrated a droplet manipulation system in which magnetic microparticles act both as to force mediators and mobile

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substrates for biomolecules. They claimed that the actuating magnetic field in their system does not have any direct interaction with the dielectric properties of the DNA. They used a switchable matrix of simple coils for the actuation of droplets which helped them to reach high operational flexibility. Nguyen et al. [48] experimentally studied the effect of magnetowetting and the sliding

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motion of a sessile ferrofluid droplet. Their results presented that the shape of a sessile ferrofluid droplet and the contact angle at the solid-liquid interface depend on the flux density of the magnetic field. They also investigated the kinematic behavior of the ferrofluid droplets dragged by a permanent magnet at a constant velocity. In fact, they used an external mechanism for moving the permanent magnets with different magnetic strengths at a constant speed which was accompanied

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by the movement of different sized ferrofluid droplets. Afterward, the contact angles were measured and used to estimate the magnitude of the forces involved in the sliding motion. Similarly, Koh et al. [49] analyzed circular, rectangular, triangular and number-eight-shape trajectories of a ferrofluid droplet on the hydrophobic surface. They showed that smooth trajectory could result in better steady movement. Also, they used ferrofluid droplet as the driving engine for other diamagnetic liquid droplets. Also, Chakrabarty et al. [50] numerically studied the magnetic

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manipulation of an immiscible, microliter-scale ferrofluid droplet over a thin aqueous film on a

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solid substrate using a periodically switchable array of square-electromagnetic micro-spirals embedded coils. They declared that at relatively low current, ferrofluid droplets showed zigzag

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motion and passed through the centers of the excited coils.

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As mentioned earlier, manipulation of a ferrofluid droplet on a digital microfluidics system has been used in many applications and has been considered by many researchers in the last decade

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[51]. In the previous studies, usually, a movable permanent magnet has been used for droplet manipulation. In this case, to move the magnet to the desired location requires an actuator which

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increases the complexity of the system. Besides, the disengagement was observed in related works for higher velocity magnitude of the magnet. Also, recently, some researchers have used a large number of micro coils to transfer droplets which might increase the level of complexity of the structure and the cost of the final product. The resolution of droplet manipulation using micro coils

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depends on the length between micro coils used in the device. Although the setup used in the present work might not be as elegant as some of the previous works in the same field, introduces a relatively simple solution to droplet manipulation with very high precision (less than 50µm). The desired displacement pattern could be simply achieved by just tuning the novel adjustable magnetic

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field, while only the pre-specified displacement between the coils could be obtained using micro coils in the previous studies. The current study brings a new idea to meet the request of moving the ferrofluid droplets to the exact location on a hydrophobic surface by using a normal size magnetic coil with ferrite core and producing an alternating magnetic field. By tuning the magnitude and time in which the electrical

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current is active and producing a magnetic field in the coil, the ferrofluid droplet can be transferred with micro-sized steps. In other words, this paper examines the possibility of using one coil with

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an alternating magnetic field to adjust the displacement of ferrofluid droplets instead of using a

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large number of coils. For this purpose, at first, the effects of the magnetic flux density, duty cycle, and frequency of applied magnetic field along with the ferrofluid droplets’ volume on the droplet

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movement were investigated experimentally. Afterward, a mathematical model was proposed for investigating the kinematic behavior of the ferrofluid droplets dragged by the magnetic coil. Also,

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due to the drawback of evaporation of ferrofluid transport on the surface, its motion in the oil was also examined. At last, the capability of the present platform for mixing components inside the

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ferrofluid droplet was studied experimentally.

2. Experimental setup and procedure

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In the previous studies usually, an array of electrodes or coil conductors had been utilized for droplet manipulation. In Fig. 1(a) the array of the coil conductors for ferrofluid droplet transport is shown [50]. The ferrofluid droplet passes through the centers of the active coils. In this study, instead of using an array of coils, only two coils were utilized. By switching the magnetic field on and off the same function of droplet manipulation was achieved with fewer coils but larger size.

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The concept of the current study has been drawn schematically in Fig.1 as well as compared with that of the previous study [50]. It can be seen that when the right magnetic coil is switched on the droplet moves to the right, and when it is switched off the droplet stops moving. The period in which the magnetic field is on determines the step length of the ferrofluid droplet. This platform can manipulate droplets with arbitrary step length. Therefore, the position of ferrofluid droplets on the surface can be adjusted by tuning the magnetic flux density, duty cycle, and frequency of

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each magnetic coil. Furthermore, this platform is able to transfer, split ferrofluid droplets and also

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mix the component inside the droplet. Droplet formation can also be achieved using this platform. In the present research, the transfer and mixing of ferrofluid droplets have been explicitly

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investigated.

Figure 2, Schematically displays the complete experimental set-up for the current study. The images of the

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ferrofluid droplet were captured by a CCD camera (Nikon J4). The camera was placed on the top of the platform and the droplet motion was recorded that point of view. The videos were captured with high quality

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(1920× 1080 pixels). A professional kit of the optical lens was used for adjusting the contrast and image focusing during video capturing. The largest error caused by pixel resolution was 30 μm in experimental data. The ferrofluid droplets’ dimensions and also their displacements were precisely measured using the image processing software proposed by Basu [52]. The program converts the recorded grayscale image into

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a binary image for further processing. Therefore, the pixels within the droplet were marked and the

center of mass of the area covered by the marked pixels was calculated as follows.

𝑥̅ =

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∑𝑁 𝑖=1 𝑥𝑖 𝐴𝑖 , ∑𝑁 𝑖=1 𝐴𝑖

(1)

∑𝑁 𝑖=1 𝑦𝑖 𝐴𝑖 𝑦̅ = 𝑁 ∑𝑖=1 𝐴𝑖

(2)

Where N is the number of pixels inside the droplet. 𝐴𝑖 , 𝑥𝑖 , 𝑦𝑖 are the area, x coordinate and y coordinate of the center of the ith pixel, respectively. A DC current (𝐼0 ) was generated by the DC power supply. The value of 𝐼0 could be adjusted from 0 to

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5.24A. A circuit was designed to convert the DC current into an alternating current. The frequency could be tuned up to 500Hz. Also, the duty cycle of the current which is defined as bellow (Eq. (3)) could be

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adjusted from zero to one (DC current). Where 𝑡𝑜𝑛 and 𝑡𝑜𝑓𝑓 are the periods when the magnetic field is on

𝑡𝑜𝑛 𝑡𝑜𝑛 = 𝑡𝑜𝑛 + 𝑡𝑜𝑓𝑓 𝑇

(3)

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𝐷=

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and off respectively.

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According to the system characteristics discussed above, the magnitude of magnetic flux density, frequency and duty cycle could be easily adjusted.

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For droplet manipulation, only one magnetic coil (I) was placed beneath the glass slide. During the mixing procedure, two magnetic coils (I and II) were used. Utilized magnetic coils consist of copper wire with a length of 125m and 1mm diameter along with a ferrite core with 2.9cm length,

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2.75cm width, and 7.5cm height. The magnetic flux density on the glass surface was measured with a precise gauss meter (MG3002-LUTRON). The magnetic flux density for different input currents (𝐼0 ) is shown in Fig. 3(a). The droplet moves toward the right side and by approaching the magnetic coil, experiences higher magnetic flux density. The legend shows the magnetic flux density (B0) which is measured on the edge of the top surface of the coil. Indeed, during the experiments, the Lab’s temperature was adjusted by the air conditioner. In other words, the 10

ambient temperature fluctuation was about 2°C peek to peek during experiments with an average temperature of 18°C. Also, the effect of heat generated by the magnetic coil on the temperature variation on the place of droplet movement was negligible. The experiments were done under atmospheric pressure.

By using a caliper, the top face of the magnetic coil was positioned at a distance of 2 mm away

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from the hydrophobic plane.

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Ferrofluid droplets were placed on a hydrophobic glass slide. The thickness of the glass slide was

droplets with the surface which is almost 104.4°.

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1mm. The glass slide was coated with SiO2 nanoparticles. Fig. 3(b) shows the contact angle of the

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Water-based ferrofluid was synthesized using the co-precipitation method [53]. A bio-compatible surfactant was utilized to have a stable colloidal solution. This surfactant makes it possible to use

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this droplet to carry the bio component. The average diameter of nanoparticles coated with the surfactant is 66.97nm (Fig. 3(c)). The magnetization curve of ferrofluid is shown in Fig. 3(d)

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(Princeton Applied Research, Vibrating Sample Magnetometer, Model No. 155, Magnet: Varian, V-7300 Series 12” Electromagnet). The water-based ferrofluid with a 1.3% volume fraction was used in all the experiments. The viscosity and the density of the ferrofluid droplets were measured by DV-E viscometer (Brookfield, LVDVE230) and density meter (Anton Paar, DMA 38), which

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are respectively, 4.66±0.01 mPa.s and 1097.0±0.1 kg/m3. The surface tension of the ferrofluid was measured by the tensiometer (KRUSS GmbH Germany, K6) and it is equal to 45.0±0.1 mN/m.

Additionally, another platform was used in this study in which the hydrophobic layer was replaced by a pool of olive oil (Fig. 4) as a carrying substrate for water-based ferrofluid droplets. The pool’s 11

dimensions were 1.5cm×2cm×8cm and a volume of 21 milliliters of oil was poured into the pool for each experimental section. The viscosity and the density of olive oil were measured by the same instruments which are 58.7±0.01 mPa.s and 909.0±0.1 kg/m3, respectively. Also, the surface tension between the ferrofluid and the olive oil was measured by the same tensiometer which is equal to 20.0±0. 1 mN/m. Due to the immiscibility of the water-based ferrofluid and olive oil, they

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were not mixed with each other resembling a superhydrophobic surface for the ferrofluid droplets. In this platform ferrofluid was completely immersed in the oil and droplet evaporation will be

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minimized in the thermal cycle used in bio applications. It should be noted that during the experiments, the temperature variation due to the heat generated by the magnetic coil on the surface

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of the platform was negligible. Therefore, the physical properties of ferrofluid and oil remained

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constant.

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During the experiments, the effect of four parameters including magnetic flux density (B), droplet volume (V), duty cycle (D), and frequency (f) was investigated. Different magnetic flux densities

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(ranging from 10mT to 90mT), different droplet volumes )ranging from 5μL to 150μL(, different duty cycles (varying from 0.07 to 0.93), and different frequencies (changing from 0.5Hz to 10Hz( were used during the droplet manipulation. Also, the droplet movement was studied under two different situations, on a hydrophobic surface and in a pool of olive oil. As it was mentioned earlier,

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despite the procedure used in the previous investigations [48, 54], the magnetic coil position in this study was fixed which resulted in the variation of the magnetic force applied on the droplet during its movement due to the change of its distance from the center of the magnetic coil. Hence, there could not be a balance between the involved forces and therefore, in case of any movement, the relocation is always accompanied by accelerating. Two main advantages of using a fixed

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magnetic coil instead of using a moving magnet are avoiding the pinching moment [48] and preventing the disengagement of the magnet and the droplet. The pinching moment and the disengagement event may cause some difficulties in predicting the exact droplet’s trace line. However, it should be considered that the required magnetic strength for moving the droplets varies with respect to some physical parameters (e.g. the plain’s roughness and the droplet’s physical parameters). Furthermore, since the droplets move with acceleration, their velocity during

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displacement may become high and thereby difficult to control. Considering the latter point, the

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generated current was considered to apply in an intermittent way in order to prevent the droplets from accelerating to high velocities. Furthermore, by using this method, step lengths as small as a

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micron can be achieved. In other words, AC signals with different frequencies instead of DC

3. Results and discussion

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signals were utilized in order to have more control over the movement of the droplets.

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3-1. Synchronization between droplet motion and applied magnetic field At first, it is important to check that the droplet movement was synchronous with the electrical circuit when switching from off to on and vice versa. This was possible by embedding an LED in the circuit and recording its function during the experiments. Also, after the experiments, by

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obtaining the droplet position with respect to time, the synchronization of the circuit and the movement of the droplets was studied. This was investigated in both cases, i.e., using a hydrophobic surface and immersing the droplet in a pool of olive oil. In Figure 5(a) displacement of 20μL ferrofluid droplet based on the mass center of the ferrofluid droplet for B0=50mT, D=0.5, and f=1Hz on the hydrophobic surface and in the oil is compared. 13

Magnetic flux density decreases as the distance from the coil increases for both cases as depicted in Fig. 3(a) and it is only depending on the distance from the coil. The variation of magnetic flux densities for both cases in the corresponding location of the ferrofluid droplet is illustrated in Fig. 5(a). The time periods during which the magnetic field is turned on are displayed with orange color. It can be seen that as the droplet approaches the coil, the amplitude of magnetic flux density increases and the droplet moves by taking larger steps. In the magnified diagram in Fig. 5(a), it

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can be seen that the droplet movement on the hydrophobic surface is completely synchronized

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with the magnetic field, while the droplet in the oil moves backward after the magnetic field is turned off. ∆𝑙 shows the droplet displacement after the magnetic field is turned off. This backward

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movement is due to the capillary restoring force (the difference between advancing and receding

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contact angles) which forces the droplet to retrieve its symmetric shape. Therefore, as it can be inferred from the graphs, the droplet movement is fully congruent with the generated signal on the

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hydrophobic surface, but it is not synchronized with the signal while moving in the oil. For more clarification, the velocity of the mass center of the ferrofluid droplet is also computed

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and is plotted versus time for both cases in Fig. 5(b). Stage A represents the time when the magnetic field is turned on and stage D is the time when the magnetic field is turned off. It should be noted that the velocity is measured based on the displacement of the mass center of the droplet which is

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associated with the movement of the whole body of the droplet (rigid motion) or the shape evolution. It can be seen that when the magnetic field is turned on, the velocity of the droplet in the oil immediately increases (from A to B) and afterward decreases (from B to C). The shape of the ferrofluid droplet changes from sphere to ellipsoid in the presence of the magnetic field [55]. This rapid rise in the velocity from A to B is due to the change of the shape of the ferrofluid droplet from the sphere to ellipsoid. In fact, the center of mass abruptly moves forward due to the droplet

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shape transformation. This rapid increase in the velocity cannot be seen while using the hydrophobic surface (from A’ to B’). This can be interpreted in two ways. First, the value of the static friction force caused by the hydrophobic surface is much higher than the viscous force applied by the olive oil medium. Second, at a constant magnetic field, ferrofluid droplet deformation increases with decreasing surface tension [56-59]. As mentioned in section 2, the surface tension of the droplet on the hydrophobic surface is 45mN/m which is 2.25 times higher

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than surface tension between the droplet and the oil. Hence, droplet deformation inside the oil is

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larger than on the hydrophobic surface. After that, the droplet reaches its equilibrium shape and the changes in velocity are only due to the rigid body motion of the droplet. Since the movement

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of the droplet after shape equilibrium is not as fast of the abrupt motion of the droplet due to the

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shape deformation, the velocity in the oil decreases from point B to C. The reduction in the velocity also can be seen for the droplet on the hydrophobic surface from B’ to C’ but it is smaller than the

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velocity decrease in oil. The velocity increases from point C to D in oil as the droplet gets closer to the coil. The velocity on the hydrophobic surface remains almost constant because the droplet

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is relatively far from the coil at this period with respect to the droplet in oil. When the magnetic field is turned off the velocity of the droplet in the oil becomes negative (from D to E) and eventually, it stops moving (from E to F). Conversely, the velocity of the droplet immediately becomes zero for the hydrophobic surface (with a little bit oscillation from D’ to E’ to F’). During

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the motion, the advancing contact angle becomes larger than the receding angle (it will be discussed in section 3.2) and applied capillary force to the droplet reverses the direction of movement. When the magnetic field is turned off the droplet tends to form its symmetric shape like in point A. Therefore, the capillary force causes the droplet to move backward until the advancing and receding contact angles become equal. As can be seen, droplet deformation in the

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oil (stage D) is much more than it is on the hydrophobic surface (stage D’). Therefore, the capillary force in the oil is larger than on the hydrophobic surface. As a result, the capillary force in the oil can dominate the drag force and the droplet moves backward. While the capillary force on the hydrophobic surface cannot overcome the large static friction force and the droplet remains almost

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in its place.

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3-2. Analytical model for droplet manipulation on the hydrophobic surface

In this section, a simple analytical model is introduced to obtain droplet displacement considering

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the forces involved in the droplet manipulation on the hydrophobic surface. In the present experiments, the droplets’ dynamic behavior was determined by interactions between the magnetic

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force (𝐹𝑚 ) acting on the ferrofluid droplet, the capillary force (𝐹𝑐 ) induced by the droplet

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deformation and the frictional force (𝐹𝑓 ) between the ferrofluid droplet and the plain surface [48]. Considering the forces shown in Fig. 6, the force balance in the direction of motion of the sliding

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ferrofluid droplet is shown in Eq. (4). It is clear that the magnetic force applies in two perpendicular directions, parallel to the surface and perpendicular to it. The perpendicular force was assumed to have a very small amount of variation during the droplet movement. Therefore, only the component contributing to the droplets displacement was considered in the equations.

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𝐹𝑚 − 𝐹𝑐 − 𝐹𝑓 = 𝑚𝑥̈

)4(

The magnetic force could be evaluated from Equation (5). 𝐹𝑚 =

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𝜒𝑉 𝛼𝑉 −2 𝛼𝑉 𝐵∇B = × = (𝑅 − 𝑥)2 (𝑅 − 𝑥)3 (𝑅 − 𝑥)5 𝜇0

)5(

Where, α and V represent the strength of the magnetic field applied by the magnetic coil and the volume of the ferrofluid droplet, respectively. Additionally, 𝑥 represents the distance traveled by the droplet where the initial distance between the magnet and the droplet is considered to be 𝑅. As it can be seen from Eq. (5), the magnetic force is inversely proportional to the square of the distance. As it was discussed earlier, this relation shows that the magnetic force is not constant

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during the droplet’s displacement.

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Further, the friction force could be determined by Equation (6).

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𝐹𝑓 = 𝛽𝑎2 𝑣

)6(

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Where, 𝛽 , 𝑎 , and 𝑣 are the friction factor between the droplet and the surface, the base diameter of the droplet, and the droplet’s velocity, respectively. It is obvious, the base diameter increases as

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the droplet gets closer to the magnet. This alteration during the experiments was not considerable, although the dimensions of droplets were always being monitored.

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Equation (7) presents the capillary force acting against the droplet’s motion. 𝐹𝑐 = 𝛾(cos 𝜃𝑟 − cos 𝜃𝑎 )𝑎

)7(

Where 𝛾 is the surface tension coefficient, 𝜃𝑟 and 𝜃𝑎 are the receding and advancing contact angles

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of the droplet during its movement, respectively. In addition to the variation of the base diameter of the droplet, the advancing and receding contact angles alter as well during its displacement. However, if the distance between the droplet and the coil does not decrease substantially, the variation of (cos 𝜃𝑟 − cos 𝜃𝑎 ) can be neglected. Fig. 7 shows the variation of advancing and receding contact angles of a 20µL droplet during its movement towards the constant magnetic field by a DC current. Four different magnetic fields were applied. Six different positions are presented 17

for contact angle monitoring which are: before movement (1), initial movement (2), at the 1⁄4 of the path (3) , at the 1⁄2 of the path (4), at the 3⁄4 of the path (5), and at the end of the path (6). This figure shows that the stronger magnetic field causes larger advancing contact angles and smaller receding contact angles all over the path. Also, the advancing contact angle increases and the receding contact angle decreases as the droplet approaches to the magnetic coil. Since in the first

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quarter of the path, (cos 𝜃𝑟 − cos 𝜃𝑎 ) does not change considerably for weaker magnetic fields (less than 8 degrees for 70mT and 60mT), this difference could be considered constant in this

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specific region in the experiments. Therefore, based on the experimental measurement in Fig. 7 capillary force could be calculated (𝐹𝑐 = −.00016𝑎𝑁 = 9.1𝜇𝑁). This assumption is used for

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deriving the magnitude of the involved factors in the present study.

written as Eq. (8). 𝛼𝑉 − 𝛽𝑎2 𝑥̇ − 𝐹𝑐 (𝑅 − 𝑥)5

)8(

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𝑚𝑥̈ =

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According to the previously mentioned equations, the kinematic relation for each droplet can be

According to the stepwise movement of the droplets due to the applied alternating magnetic field, Eq. (8) have to be solved for each step distinctively with new boundary conditions.

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Equation (8) does not have an analytical solution for x(t). Therefore, the following objective function was defined to minimize the root mean square of the difference between the proposed model and experimental data.

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𝑁

𝑥𝑖,𝑒𝑥𝑝𝑒𝑟𝑖𝑚𝑒𝑡𝑎𝑙 − 𝑥𝑖,𝑎𝑛𝑎𝑙𝑦𝑡𝑖𝑐𝑎𝑙 1 𝑓(𝛼, 𝛽) = √ ∑ ( × 100) 𝑁 𝑥𝑖,𝑒𝑥𝑝𝑒𝑟𝑖𝑚𝑒𝑡𝑎𝑙

2

(9)

𝑖=1

for finding the appropriate 𝛼 and 𝛽 the following procedure was implemented.

of

1. An initial guess is considered for 𝛼 and 𝛽.

the values obtained from the previous step.

-p

3. The objective function is calculated by Eq. (9).

ro

2. Equation (8) is solved numerically to get the displacement as a function of time considering

re

4. 𝛼 and 𝛽 are specified by means of the pattern search algorithm [60-65]. 5. If the convergence criterion is satisfied, the algorithm stops, otherwise, it goes back to step

lP

2.

Figure 8 illustrates the experimental and analytical model outputs for the displacement of the

ur na

droplet versus time. The droplet’s volume is 20µL, the applied magnetic flux density is 40mT, the duty cycle is 0.33, and the applied magnetic frequency is 0.67Hz. It should be noted that the factors were all assumed to be constant during the indicated displacement range and the error (objective

Jo

function) obtained was less than 0.2%. After applying the optimization procedure, the factors were obtained as 𝛼 = 6.02 × 10−8

𝑁 𝑚

𝑁𝑠

, 𝛽 = 41.82 𝑚3 . This figure approves the hypothesis that the

assumed analytical model, as in Eq. (8), is logical when the droplet moves in the region of 0 ≤ 𝑥 ≤ 2𝑚𝑚. In this interval, the calculated driving magnetic force Fm (from the Eq. (5)) varies from 34.4µN to 46.1 µN. There is a slight mismatch between the analytical model and the experimental

19

data due to the droplet shape deformation in the experiments which was not considered in the model. 3-3. Droplet manipulation on the hydrophobic surface and immersed in oil In this section kinematic characteristics of the ferrofluid droplet for different magnetic flux densities, droplet volumes, duty cycles, and frequencies were studied. The difference between the

of

motion of the ferrofluid droplet on the hydrophobic surface and in the oil was also investigated.

ro

The droplet was located at a fixed specified point that is 35mm away from the magnetic coil edge. Fig. 9(a) depicts the droplet displacement relative to the time for different magnetic flux densities

-p

for constant droplet volume of 20µL, frequency equal to 0.67Hz, and duty cycle of 0.33 on the

re

hydrophobic surface (left side), and in the oil (right side). In Fig. 9(a1), it can be seen that by increasing the magnetic flux density, the droplet travels the distance faster and takes fewer steps

lP

to reach the origin of the magnetic field. In fact, by increasing the magnetic flux density, the friction and capillary forces remain almost constant but the magnetic force increases. Therefore,

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the maximum amount of velocity increases by increasing B.

The step lengths versus the number of steps were calculated and are displayed in Fig. 9(b). It can be seen that as the droplet approaches the coil, the step lengths increase.

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The main purpose of this study is to introduce a relatively simple solution to the droplet manipulation with adjustable step lengths. The aim of the parametric study is to find an appropriate range of parameters in which the steps are:

(1) as small as possible (better resolution), (2) as uniform as possible, 20

(3) and the fastest way possible to move the droplet to its destination.

To this end, all the step lengths are shown in Fig. 9(b1, b2). It is an inherent characteristic of the system that the step lengths become larger when the ferrofluid droplet gets closer to the magnetic coil (magnetic force is proportional to 1/r2). Hence, the step lengths are growing as the ferrofluid moves toward the coil and the rate of growing drastically increases in the final steps. Therefore,

of

for three aforementioned purposes, three criteria are proposed to understand quantitively how the

ro

goals are satisfied.

(1) To introduce an index for understanding how small the steps are, the average of all step lengths

-p

is calculated. Hence, smaller average-step length means that the ferrofluid droplet has traveled

re

the distance with smaller step lengths, which means the droplet can map the surface with higher resolution. The variation of the average-step length (savg) versus magnetic flux density,

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droplet volume, duty cycle, and frequency is demonstrated in the right y-axis of Figs. S2(a1, a2), S2(b1, b2), S2(c1, c2), S2(d1, d2), respectively (in the appendix).

ur na

(2) In order to quantify the uniformity of step lengths, the standard deviation of the step lengths was calculated using Eq. (10). By decreasing STD, the step lengths become more uniform.

𝑁

1 2 𝑆𝑇𝐷 = √ ∑(𝑠𝑖 − 𝑠𝑎𝑣𝑔 ) 𝑁

(10)

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𝑖=1

(3) The number of steps (N) represents the time that the droplet reaches its destination. By decreasing N, the average-velocity of droplet increases. The variation of N versus magnetic flux density, droplet volume, duty cycle, and frequency are demonstrated in the left y-axis of Figs. S2(a1, a2), S2(b1, b2), S2(c1, c2), S2(d1, d2), respectively (in the appendix).

21

It is worth noting that lower values of savg, STD, and N are more desirable in terms of droplet manipulation. It can be seen that by increasing B, the number of steps decreases. In other words, the number of steps represents the time that the droplet reaches the destination. Also, by increasing B, both the average-step length and the standard deviation increase. In fact, as much as the average-step length decreases, the droplet moves with smaller steps and so the droplet can map the surface with higher resolution. Therefore, the average-step length is a criterion for resolution.

of

Also, the standard deviation determines the uniformity of step lengths during the motion. The step

ro

lengths are more uniform as much as the standard deviation decreases. As a result, by increasing B the time of traveling reduces but the uniformity of the step lengths and the resolution of the

-p

droplet movement decreases. Hence, there should be an optimum value for B to satisfy both

re

conditions. Besides, in some applications, fast traveling is desired which needs stronger B to be selected. However, in other applications transferring droplets to a certain position with high

for selecting an appropriate B.

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precision is desired which requires lower values of B. Fig. S2(a1) shows the characteristic curves

ur na

The experiment was done for droplet movement in the oil with the same condition on the hydrophobic surface with V=20µL, D=0.33, and f=0.67Hz which is shown on the right side of Fig. 9. The same trend can be seen for oil.

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In the oil medium, the lower magnetic flux density can trigger the droplet motion, and the number of steps decreases (the average velocity increases), but the average-step length increases (The resolution decreases) relative to the hydrophobic surface. These happened because the droplet starts moving with each pulse and stops at the end of the pulse. At the beginning of each pulse, the static friction force is applied on the droplet on the hydrophobic surface and once the droplet starts moving it converts to kinetic friction [66]. Therefore, when the drop moves with a high number of 22

steps, the friction force is in the order of static friction. Also, the drag force in the oil is proportional to the second power of droplet velocity and it is very low when the droplet starts moving (the velocity is low). As a result, the droplet moves faster in the oil than on the hydrophobic surface. In Fig. 9(a2), it is depicted that the magnetic flux density of 10mT can pull the droplet while in the case of hydrophobic surface the minimum of 40mT is needed to launch the droplet transfer. From this figure, it is also obvious that, after the magnetic field is set to off, the droplet takes a step back.

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This is due to the viscous friction applied by the continuum phase (oil) while trying to re-establish

ro

its initial state.

-p

Figure 10 depicts the effect of droplet volume on droplet manipulation. For all the droplets, the

re

magnetic flux density is 40mT, the duty cycle is 0.33, and the applied magnetic frequency is 0.67Hz. By increasing droplet volume, the magnetic force increases (it is a volumetric force), and

lP

also the friction force increases (it is proportional to the contact surface area of the droplet and the plane). In Fig. 10(a) it can be seen that the effect of the magnetic force is dominant, and by

ur na

increasing volume, the droplet reaches the destination faster with a fewer number of steps and the maximum of the velocity increases. Smaller droplets need more time to travel along the entire path. In other words, smaller droplets approach the coil with smaller steps and velocities. For smaller droplets, the velocity is almost constant for the initial steps at a relatively long distance

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from the magnetic coil. It can also be deduced that when the droplets get close to the magnet, their velocity and displacement would increase considerably (Fig. 10(b1)) which makes their manipulation more difficult.

23

In fact, by increasing droplet volume, the average-step length increases (the resolution decreases) and the number of time step decreases (the droplet reaches the destination faster). The standard deviation increases accordingly (the step lengths are less uniform). The results for the same condition for ferrofluid droplet in the oil is depicted on the right side of Fig. 10. In general, for the same magnetic field, droplets in oil move towards the coil with larger

of

steps and higher velocities, compared to those on hydrophobic surface droplet with small volume of 5µL can be absorbed by magnetic field when the droplet is immersed in oil while the minimum

ro

volume of 20µL can be transferred on the hydrophobic surface (Fig 10(a2)). Besides, by increasing

-p

droplet volume the backward displacement increases after the magnetic field is turned off.

re

The effect of the duty cycle is presented in Fig. 11 for the fixed magnetic flux density of 50mT,

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droplet volume of 20µL, and the frequency of 0.67Hz. Magnetic, friction and capillary forces do not change by duty cycle variation. The total duration of the displacement and the number of steps increase as the duty cycle decreases. For constant frequencies, decreasing the duty cycle means

ur na

reducing the applied magnetic force time in a given time period. Hence, the average-velocity and distance traveled in each step are reduced. By decreasing the duty cycle the average-step length decreases (the displacement resolution increases) and the step lengths are more uniform (the

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standard deviation decreases). The results for the droplet in the oil with the same condition as that on the hydrophobic surface is shown on the right side of Fig. 11.

Figure 12 shows the kinematic characteristics of the ferrofluid droplet for different frequencies at a fixed magnetic flux density of 50mT, the volume of 20µL, and the duty cycle of 0.5. (ton, toff) for magnetic frequencies of 0.5, 0.8, 1, 1.25, 1.6, 5, and 10Hz are (1000ms,1000ms), (625ms,625ms), 24

(500ms,5000ms), (400ms,400ms), (312.5ms,312.5ms), (100ms,100ms), and (50ms,50ms), respectively. In fact, the magnetic field’s operation time decreases as the magnetic frequency increases. Hence, the droplet has less time to get attracted by the coil. For instance, in the time period of 2s, for the magnetic frequency of 0.5Hz, the droplet is pulled by the coil in a single step for 1s, whereas, it takes 20 steps for 0.05s in the magnetic frequency of 10Hz. Although by changing the applied frequency, the total time of attraction in a constant period remains the same

of

(1s=20×0.05s), for lower values of magnetic frequency, the droplet is displaced by fewer pulses,

ro

and with higher velocity magnitude. For the magnetic frequency of 20Hz, the droplet is displaced by 20 pulses, which means it stops multiple times and the droplet cannot reach such high velocities

-p

that are recorded for lower frequencies. Therefore, the total duration of the displacement and the

re

number of steps increases. Whereas, the average-step length decreases (resolution increases) and the step lengths become more uniform as the frequency increases. In the oil medium, the droplet

ur na

2220 number of steps.

lP

can be manipulated with the frequency of 25Hz which results in 15.8µm average-step length and

The correlations for the number of steps and average-step length as a function of the magnetic flux density, droplet volume, duty cycle, and frequency were obtained based on data from Figure S2.

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The following forms of the equations are assumed for the number of steps and average-step length, respectively.

𝑁 = 𝑎1 𝐵 𝑎2 𝑉 𝑎3 𝐷𝑎4 𝑓 𝑎5

(11)

𝑠𝑎𝑣𝑔 = 𝑏1 𝐵 𝑏2 𝑉𝑏3 𝐷𝑏4 𝑓 𝑏5

(12)

25

In Eqs. (11) and (12), magnetic flux density, droplet volume, and frequency are in mT, µL, and Hz, respectively. The coefficients for droplet manipulation on the hydrophobic surface and in oil were calculated based on the experimental results. The coefficients are listed in Table 1.

The mean relative errors between the predicted values and the experimental data for correlations provided for the number of steps are 14.36% and 11.06% for droplet manipulation on the

of

hydrophobic surface and oil, respectively. Besides, the mean relative errors for correlations of average-step lengths are 12.38% and 15.84% for droplet manipulation on the hydrophobic surface

ro

and oil, respectively.

-p

The effect of four parameters including magnetic flux density, droplet volume, duty cycle, and

re

frequency was investigated. Hence, the operation map can be illustrated for any combination of two of these parameters. The operation map is shown in Figure 13(a) for various magnetic flux

lP

densities and droplet volumes at a constant duty cycle of 0.5 and frequency of 10Hz. The manipulation boundary between the two regions is also shown in this figure. Since the magnetic

ur na

force is proportional to droplet volume and magnetic flux density (Eq. (5)), the droplet cannot be manipulated for lower values of magnetic flux density and droplet volume.

By repeating the same procedure as in Figure 13(a) for different frequencies, the effect of

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frequency on the manipulation boundary between the two regions is shown in Fig. 13(b). It can be seen that by increasing the frequency, ton decreases and the possibility of droplet manipulation decreases. Therefore, by considering the constant droplet volume, the required magnetic flux density for manipulation increases with frequency. Besides, the effect of the duty cycle on the manipulation boundary is also illustrated in Fig. 13(c). Considering the constant droplet volume,

26

by increasing the duty cycle, ton in a period increases and AC magnetic field approaches to DC magnetic field, and as a result, the required magnetic flux density for manipulation decreases.

3-4. Mixing In this section, two magnetic coils were activated complementary. Both of them have the duty

of

cycle of 0.5 and when the left coil is on, the right one is off and vice versa. The effects of magnetic flux density (B0), droplet volume (V), coils distance (L) and the applied magnetic frequency (f)

ro

were investigated. Also, the droplet motion on the hydrophobic surface and in the oil was

-p

compared with each other.

Figure 14(a) shows the droplet’s displacement versus time under the influence of two alternating

re

magnetic fields for the droplet volume of V=20µL, the distance between the magnets equals to

lP

L=15mm and the applied frequency of f=1Hz. From Fig. 14(a) it can be inferred that the magnetic flux density of 30mT is not powerful enough to create a complete loop for the 20µL droplet. This shows that there is a threshold for the applied magnetic field in order to create a complete loop for

ur na

a droplet with a fixed volume. Furthermore, as the magnetic field gets stronger, the droplet arrives at the other end in a shorter time range letting it rest more. In other words, choosing magnetic fields with high magnitudes might not be a wise decision since it leads to droplet’s mobility

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reduction. As the droplet rests more, the mixing procedure efficiency might drop drastically. Therefore, there is an optimum for magnetic flux density for each droplet volume, coils distance and applied magnetic frequency. Figure 14(b) shows the velocity versus position. In this figure, it can be seen that by increasing the magnetic flux density, the maximum velocity increases. This figure also shows that the maximum velocity occurs when the droplet is near to the destination coil. Theoretically, all the figures should be symmetrical with respect to the origin of the 27

displacement axis (x), but slight asymmetry can be seen in the figures due to a little bit difference between two magnetic coils structure and generated magnetic field.

Figure 15 shows the displacement versus time for investigating the effect of different droplet volumes. Again, in this figure, it can be seen that by reducing the droplet size, the magnetic force

of

reduces and eventually the droplet could not be able to complete the loop for droplets with such a

ro

small volume (in this case 20µL). Additionally, by further increasing the droplet size to 120µL, the magnetic force gets very large which results in the droplet over the displacement. The over

-p

displacement might also be regarded as a drawback since it causes the droplet to exceed the anticipated (programmed) displacement range. Also, by increasing the droplet volume, it has more

re

time to rest. Considering the points discussed, the 60µL droplet is the best choice for a system with

lP

B0=47mT, L=20mL, and f=0.67Hz. It has the minimum time for resting and also it does not exceed the pre-specified displacement range. In Fig. 15(b), it can be seen that by increasing the droplet

ur na

volume the velocity of the droplet increases.

Figure 16 depicts the effect of the coils’ distance on the droplet displacement and velocity for magnetic flux density of 47mT and applied magnetic frequency of 1Hz. In this figure, the results

Jo

are shown for the droplet volume of 60µL and 80µL on the left and right sides, respectively. For the droplet volume of 60µL (left side), it can be seen that the droplet could complete the loop for L=15mm and 20mm, but by increasing the distance to 25mm it could not reach the second coil. By increasing the droplet volume to 80µL, the droplet could complete the cycle for all distances

28

of 15, 20 and 25mm. Fig. 16(b2) clearly displays that by increasing L, the velocity reaches higher values.

The displacement and velocity of the ferrofluid droplet are shown in Fig. 17 for different applied magnetic frequencies and constant value of B0=47mT, V=60µL, L=20mm. For high frequency

of

(3.33Hz), it can be seen that the droplet could not reach to the second coil. By decreasing

ro

frequency, the droplet could complete the cycle and it had more time to rest on the coil. Therefore, there is an optimum frequency for the selected value of B0=47mT, V=60µL, L=20mm. The

-p

optimum frequency is 2Hz, where the droplet immediately comes back just after it reaches the coil. From Fig. 17(b), it can be inferred that the maximum velocity was not changed for different

re

frequencies. Also, the maximum velocity occurs in the same position for different frequencies.

lP

In the mixing process, the droplet moves the distance between two coils (L=20mm) in one step, while in the previous section (3-3) the droplet travels step by step (in some cases with more than

ur na

500 steps). As earlier discussed, in the case of the AC magnetic field, the droplet in the oil reaches the destination faster compared to the droplet on the hydrophobic surface. At the beginning of each step, the droplet experiences static friction force on the hydrophobic surface and after the droplet starts moving it converts to kinetic friction force [66]. This means that a higher number of steps

Jo

results in a higher contribution of the static friction factor. Additionally, the drag force applied on the droplet by the oil is proportional to the second power of droplet velocity, which is very low when the droplet starts moving. As a result, the droplet moves faster in the oil than on the hydrophobic surface.

29

In this section, the droplet reaches the destination in one step and as can be seen in Fig. 18(b), the magnified image shows, once the magnetic field is on (t=2s) the droplet velocity in oil is higher than one which is located on the hydrophobic surface. This is consistent with the previous results in section 3-3. As the droplet moves faster, the drag force in the oil (which is greater than the drag force in the air) increases with the second power of the droplet velocity while the kinetic friction force on the hydrophobic surface increases linearly with the velocity [48]. As can be seen in Fig.

of

18(b), the droplet velocity in the oil is lower than on the hydrophobic surface for long-distance

ro

(L=20mm). Therefore, unlike the previous section, considering single-step displacement, the

-p

droplet reaches the destination faster on the hydrophobic surface than in the oil.

re

In Fig. 18 the mixing characteristics in two different mediums were compared. By looking at Fig.

lP

18 (c), it can be inferred that the velocity of the droplet decays drastically by using the oil medium for mixing. Although this might be regarded as a drawback for the oil medium, the droplet velocity in the oil despite the droplet velocity on the hydrophobic surface is almost uniform which could

ur na

be regarded as a benefit of mixing in the oil medium.

3-5. Other operations

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It should be noted that this platform by utilizing the alternating magnetic field has the ability to be used for other main operations in digital microfluidics such as droplet formation and splitting. The results are not presented in this paper due to space limitations. 4. Conclusion

30

In this study ferrofluid droplet manipulation on a hydrophobic surface and immersed in a pool of oil was investigated and compared with each other. An analytical model was presented for droplet transport on the hydrophobic surface. The effect of magnetic flux density, droplet volume, duty cycle, and applied magnetic frequency were investigated. Also, the effect of the mentioned parameters on the mixing was studied. Considering the results, here is a summary of core findings:

of

(1) The droplet movement was fully synchronized with the generated signal on the hydrophobic surface. However, the backward movement of droplets in the oil was observed

ro

after the magnetic field was turned off.

-p

(2) An analytical model was presented for droplet transport on the hydrophobic surface. The unknown factors in the nonlinear equation of motion were obtained by following an

re

optimization algorithm (pattern search method) and using the experimental data. The

error of 0.2%.

lP

results from the analytical solution can track the experimental results with the maximum

(3) By increasing magnetic flux density, droplet volume, duty cycle, and decreasing the

ur na

applied magnetic frequency, the number of steps and the total traveling time of droplets decrease but the step lengths become less uniform and the average step length increases. In other words, the resolution of droplet displacement decreases. (4) The present platform has the ability to transport the ferrofluid droplets completely

Jo

immersed in the pool of oil. Transferring the droplet inside the oil has the advantage of avoiding evaporation which enables droplet manipulation with minimum volume loss. The same trend as the hydrophobic surface can be seen, but the number of steps decreases (the average velocity increases), and the average-step length increases (the resolution decreases) relative to the hydrophobic surface for each individual case. Also, the droplet

31

can be transferred in the oil while using lower magnetic flux density or lower droplet volume with respect to the hydrophobic surface. (5) An optimum magnitude of magnetic flux density, droplet volume, magnetic coil distance, and applied magnetic frequency was found in the mixing process. Which avoids incomplete loops for the droplets and minimizes the resting time at each end in order to maximize the

ro

of

mixing efficiency.

Conflicts of interest

-p

There are no conflicts to declare.

re

Appendix

The ferrofluid droplet velocity versus time for various magnetic flux densities, droplet volumes,

lP

duty cycles, and frequencies are illustrated in Figure S1(a), S1(b), S1(c), S1(d), respectively. The droplet is on the hydrophobic surface for the left diagrams and it is immersed in the oil for the

ur na

right diagrams. The results are derived based on the displacement of ferrofluid droplet which are shown in Fig. 9 to Fig. 12. In all cases, the velocities are almost constant when the droplets are placed at a relatively far distance from the magnet and the velocities increase very rapidly as the

Jo

droplets come closer to the magnets. Also, the backward movement after the magnetic field is turned off can also be deduced from the velocity curves shown in Fig. S1. Besides, it can be seen that by increasing the magnetic flux density and droplet volume, the maximum of droplet velocity increases, whereas by increasing duty cycles, the maximum of the droplet velocity remains almost constant.

32

of ro (a2)

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ur na

lP

re

-p

(a1)

33

(b1)

(b2)

of ro (c2)

ur na

lP

re

-p

(c1)

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(d1) (d2) Figure S1. The velocity of the ferrofluid droplet versus time for (a) for different magnetic flux densities and constant V=20µL, D=0.33, f=0.67Hz, (b) for different droplet volumes and constant B0=40mT, D=0.33, f=0.67Hz. (c) for different duty cycles and constant B0=50mT, V=20µL, f=0.67Hz. (d) different frequencies and constant B0=50mT, V=20µL, D=0.5. (1) The droplet is on the hydrophobic surface for the left diagrams and (2) it is immersed in the oil for the right diagrams.

Based on Figs. 9(b), 10(b), 11(b), and 12(b), the number of steps and average-step length versus magnetic flux density, droplet volume, duty cycle, and frequency are plotted in Figs. S2(a), S2(b), 34

S2(c) and S2(d), respectively. In Fig. S2(a), the x-axis denotes magnetic flux density, the left yaxis represents the number of steps (N), and the right y-axis is the average-step length (savg). The droplet is on the hydrophobic surface for the left diagrams and it is immersed in the oil for the right diagrams. By increasing magnetic flux density, droplet volume, duty cycle, and decreasing the applied magnetic frequency, the number of steps and the total traveling time of droplets decrease but the step lengths become less uniform and the average step length increases. Fig. S2

of

shows the characteristic curves for selecting an appropriate B, V, D and f in order to obtain fast

ur na

lP

re

-p

ro

droplet manipulation or smaller step lengths and more uniformity.

(a2)

(b1)

(b2)

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(a1)

35

(c2)

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ro

of

(c1)

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5. References

lP

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(d1) (d2) Figure S2. Variation of the number of steps and average-step length versus (a) magnetic flux density and constant V=20µL, D=0.33, f=0.67Hz, (b) droplet volume and constant B0=40mT, D=0.33, f=0.67Hz. (c) duty cycle and constant B0=50mT, V=20µL, f=0.67Hz. (d) frequency and constant B0=50mT, V=20µL, D=0.5. (1) The droplet is on the hydrophobic surface for the left diagrams and (2) it is immersed in the oil for the right diagrams.

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[1] M.G. Pollack, V.K. Pamula, V. Srinivasan, A.E. Eckhardt, Applications of electrowetting-based digital microfluidics in clinical diagnostics, Expert review of molecular diagnostics, 11(2011) 393-407. [2] E. Samiei, M. Tabrizian, M. Hoorfar, A review of digital microfluidics as portable platforms for lab-on a-chip applications, Lab on a Chip, 16(2016) 2376-96. [3] U. Lehmann, C. Vandevyver, V.K. Parashar, M.A. Gijs, Droplet‐based DNA purification in a magnetic lab‐on‐a‐chip, Angewandte Chemie International Edition, 45(2006) 3062-7. [4] C.-H. Chiou, D.J. Shin, Y. Zhang, T.-H. Wang, Topography-assisted electromagnetic platform for bloodto-PCR in a droplet, Biosensors and Bioelectronics, 50(2013) 91-9. [5] S.L. Freire, Perspectives on digital microfluidics, Sensors and Actuators A: Physical, 250(2016) 15-28. [6] S.K. Cho, H. Moon, C.-J. Kim, Creating, transporting, cutting, and merging liquid droplets by electrowetting-based actuation for digital microfluidic circuits, Journal of microelectromechanical systems, 12(2003) 70-80. [7] J. Gong, All-electronic droplet generation on-chip with real-time feedback control for EWOD digital microfluidics, Lab on a Chip, 8(2008) 898-906. [8] M. Abdelgawad, A.R. Wheeler, The digital revolution: a new paradigm for microfluidics, Advanced Materials, 21(2009) 920-5.

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Mohamad Ali Bijarchi was born in Tehran, Iran, in 1990. He received his B.Sc. and M.Sc. degrees in Mechanical Engineering in 2012 and 2014, from Tehran University, Iran. He is currently pursuing his Ph.D. degree in the Department of Mechanical Engineering at Sharif University of Technology, Tehran, Iran. From October 2017 to October 2018, he was a Visiting Researcher at Harvard University, USA. His graduate studies focus on numerical and experimental study on ferrofluid droplet formation in lab-on-a-chip devices.

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Email: [email protected]

Amirhossein Favakeh was born in Shiraz, Iran, in 1991. He received his B.Sc. degree in Mechanical Engineering in 2013 and his M.Sc. in Mechanical Engineering in 2015, from the Sharif University of Technology International Campus (both in Iran), respectively. Since October 2015 he is working as a research fellow at the SERI lab, under the guidance of Prof. Mohammad

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Behshad Shafii. His research interests include droplet manipulation, magnetic digital microfluidics, and phase change material. Currently he is working on different kind of ferrofluid droplet formation from the nozzle, in the microchannels and on the hydrophobic surface. Email: [email protected]

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He received his B.Sc. and M.Sc. degrees in Mechanical Engineering in 2015 from Shiraz University and in 2018 from Sharif University of Technology (both in Iran), respectively. His research interests include droplet manipulation, desalination, ferrofluid applications, computational fluid dynamics, and optimization techniques.

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Mohammad Behshad Shafii received his Ph.D. degree in Mechanical Engineering from Michigan State University, East Lansing, in 2005. He is currently a Professor in the Mechanical Engineering Department, Sharif University of Technology, Tehran, Iran. His research interests include fluid diagnostic techniques (Molecular Tagging Velocimetry, Particle Image Velocimetry, and LaserInduced Fluorescence), microfluidics, heat transfer, phase change, micropumps, and heat pipes.

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Email: [email protected]

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Figure 1. (a) Array of conductors which has been used previously for ferrofluid droplet manipulation [50], (b) Two magnetic coils with adjustable magnetic field used in the present study

Figure 2. Schematic of the testing platform with a hydrophobic surface

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(d) Figure 3. (a) Variation of the measured magnetic flux density versus distance from the coil edge for 9 different magnetic coil currents, (b) contact angle on the hydrophobic surface, (c) the distribution of diameter of magnetic nanoparticles, and (d) magnetization curve of the utilized ferrofluid

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Figure 4. Schematic illustration of the ferrofluid droplet in olive oil manipulation platform

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Figure 5. (a) Displacement of 20μL ferrofluid droplet versus time for B0=50mT, D=0.5, f=1Hz showing synchronization with the applied magnetic field while using the hydrophobic surface and the droplet’s backward movement in the oil and (b) velocity of the same droplets versus time, and the corresponding droplet shape for different stages during transfer

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𝑥̇

𝐹𝑚

Magnetic coil

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Figure 6. Forces applied on a moving ferrofluid droplet

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Figure 7. Advancing and receding contact angles of a 20µL droplet at six different positions during its movement under applying four different magnetic fields (A, B, C, and D) all in a DC form

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Figure 8. Comparison between experimental data and the proposed analytical model for the droplet with the volume of 30µL, B0=40mT, f=0.67Hz, and the D=0.33

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(b1) (b2) Figure 9. (a) Displacement of the ferrofluid droplet versus time for different magnetic flux densities, (b) variation of step lengths versus the number of steps for different magnetic flux densities. All of the cases are for the constant V=20µL, D=0.33, f=0.67Hz. (1) The droplet is on the hydrophobic surface for the left diagrams and (2) it is immersed in the oil for the right diagrams.

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(b1) (b2) Figure 10. (a) Displacement of the ferrofluid droplet versus time for different droplet volumes, (b) variation of step lengths versus the number of steps for different droplet volumes. All of the cases are for the constant B0=40mT, D=0.33, f=0.67Hz. (1) The droplet is on the hydrophobic surface for the left diagrams and (2) it is immersed in the oil for the right diagrams.

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(b1) (b2) Figure 11. (a) Displacement of the ferrofluid droplet versus time for different duty cycles, (b) variation of step lengths versus the number of steps for different duty cycles. All of the cases are for the constant B0=50mT, V=20µL, f=0.67Hz. (1) The droplet is on the hydrophobic surface and (2) it is immersed in the oil.

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(b1) (b2) Figure 12. (a) Displacement of the ferrofluid droplet versus time for different frequencies, (b) variation of step lengths versus the number of steps for different frequencies. All of the cases are for the constant B0=50mT, V=20µL, D=0.5. (1) The droplet is on the hydrophobic surface for the left diagrams and (2) it is immersed in the oil for the right diagrams.

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Figure 13. (a) Operation map which determines the regions that the ferrofluid droplet can be manipulated or not for different magnetic flux densities and droplet volumes at a constant duty cycle of 0.5 and frequency of 10Hz. The red dash line named the manipulation boundary which separates the two regions. (b) The manipulation boundary for different frequencies and constant duty cycle of 0.5. (c) The manipulation boundary for different duty cycles and constant frequency of 10Hz.

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(b) Figure 14. (a) Displacement of the ferrofluid droplet on the hydrophobic surface versus time for different magnetic flux densities, (b) velocity versus displacement for the constant value of V=20µL, L=15mm, f=1Hz

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(b) Figure 15. (a) Displacement of the ferrofluid droplet on the hydrophobic surface versus time for different droplet volumes, (b) velocity versus displacement for the constant value of B0=47mT, L=20mL, f=0.67Hz

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(b1) (b2) Figure 16. (a) Displacement of the ferrofluid droplet on the hydrophobic surface versus time for different coils’ distances, (b) velocity versus displacement for the constant value of B0=47mT, f=1Hz, and (1) V=60µL, (2) V=80µL

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(b) Figure 17. (a) Displacement of the ferrofluid droplet on the hydrophobic surface versus time for different frequencies, (b) velocity versus displacement for the constant value of B0=47mT, V=60µL, L=20mm

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Figure 18. (a) Displacement, and (b) velocity of the ferrofluid droplet on the hydrophobic surface and immersed in oil versus time, (c) velocity versus displacement for the constant value of B0=47mT, V=80µL, L=20mm, f=0.5Hz

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Table 1. Coefficients for the correlations in Eqs. (11) and (2)

coefficients

oil

coefficients

8.77×104 -1.97 -0.70 -1.11 1.22

𝑏1 𝑏2 𝑏3 𝑏4 𝑏5

savg hydrophobic surface 5.74×10-9 3.88 1.77 1.14 -0.93

oil 1.73×10-4 2.17 0.74 1.17 -1.18

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𝑎1 𝑎2 𝑎3 𝑎4 𝑎5

N hydrophobic surface 7.69×108 -3.48 -1.72 -1.57 1.25

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