# Investigation of Ferrofluid Sessile Droplet Tensile Deformation in a Uniform Magnetic Field

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Experimental and Numerical Methods

_{3}O

_{4}nanoparticles with a diameter of 10 nm, a volume fraction of 1.8%, and an initial susceptibility of 0.36. The viscosity of ferrofluid is 5 mPa·s at 27 °C. The density of ferrofluid and mineral oil (M5904, Sigma, Alexandria, VA, USA) at 25 °C are 1.1 × 10

^{3}kg/m

^{3}and 0.84 × 10

^{3}kg/m

^{3}, respectively. The ferrofluid droplet is generated by a pipette (Finnpipette, Thermo Scientific, Waltham, MA, USA), which can accurately control the droplet volume within a range of 0.5–10 μL. The magnitude of the magnetic field can be tuned by varying the gap distance and current. In our experiment, the gap is kept at 20.1 mm to ensure the stability, uniformity, and accurate control of the magnetic field. With a fixed gap, the electromagnetic field strength that alters with the current input (0–70 A) is measured by a Gauss meter (Model 410, Lake Shore, Westerville, OH, USA). The calibration of the field is implemented with a comparison with the data given in the manufacturer’s manual sheet. The shape of the droplet is recorded by the CCD camera (Pulnix, progressive scan camera, JAI Inc., Yokohama, Japan) and further analyzed by a customized MATLAB 2019b program.

_{3}O

_{4}nanoparticles. The initial shape of the droplet depends on surface tension, viscous force, and gravitational force which is presumed to be constant. The grid-independent study is accomplished by calculation with grids at 288,060 and 550,523. The height and width of the droplets show a relative error of less than 5%. As a result, the following results are provided for the model with grid number 288,060.

_{d}, and d are the volume fraction, magnetic moment, saturation magnetization, and diameter of magnetic nanoparticles, k is the Boltzmann constant, and T is the temperature which is taken to have the value of the room temperature. In this work, we customized the magnetization curve of the ferrofluid droplet according to the droplet shape obtained in the experiment with the corresponding ${\chi}_{m}$-H curve shown in Figure 3. A certain degree of deviation is observed although the magnetization curve fitted by experimental data has the same trend as the regular Langevin equation. It is evident in our study that the magnetization involves the nonlinear region since the maximum field strength reaches 6.15 × 10

^{5}A/m at the applied potential of 3680 A. For H less than 6666.7 A/m, the magnetic susceptibility of ferrofluid is around 0.36. The magnetic susceptibility drops sharply when the magnetic field strength exceeds the critical value.

## 3. Results Analysis and Discussion

^{3}for potential in the range of 20–3680 A. The magnetization saturation leads to a sharp reduction of the magnetic susceptibility, resulting in a stable magnetic force. The extent of the droplet deformation is almost linearly determined by the magnitude of the magnetic Bond number. The experimental and numerical results agree well with each other, indicating a more comprehensive and systematical study of the partial-wetting sessile ferrofluid droplet deformation in a uniform magnetic field.

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Aksay, I.A.; Hoge, C.E.; Pask, J.A. Wetting under chemical equilibrium and nonequilibrium conditions. J. Phys. Chem.USA
**1974**, 78, 1178–1183. [Google Scholar] [CrossRef] - David, S.; Sefiane, K.; Tadrist, L. Experimental investigation of the effect of thermal properties of the substrate in the wetting and evaporation of sessile drops. Colloids Surf. A
**2007**, 298, 108–114. [Google Scholar] [CrossRef] - Nguyen, N.T.; Ting, T.H.; Yap, Y.F.; Wong, T.N.; Chai, J.C.K.; Ong, W.L.; Zhou, J.; Tan, S.H.; Yobas, L. Thermally mediated droplet formation in microchannels. Appl. Phys. Lett.
**2007**, 91, 084102–084103. [Google Scholar] [CrossRef] - Hayes, R.A.; Feenstra, B. Video-speed electronic paper based on electrowetting. Nature
**2003**, 425, 383–385. [Google Scholar] [CrossRef] [PubMed] - Yi, U.C.; Kim, C.J. Characterization of electrowetting actuation on addressable single-side coplanar electrodes. J. Micromech. Microeng.
**2006**, 16, 2053. [Google Scholar] [CrossRef] - Wang, X.; Zhang, G.; Ren, H. Large-Area Optical Switch Using Surface-Expandable Liquid Droplets. J. Disp. Technol.
**2017**, 12, 1565–1569. [Google Scholar] [CrossRef] - Nguyen, N.T.; Zhu, G.; Chua, Y.C.; Phan, V.N.; Tan, S.H. Magnetowetting and sliding motion of a sessile ferrofluid droplet in the presence of a permanent magnet. Langmuir
**2010**, 26, 12553–12559. [Google Scholar] [CrossRef] - Capobianchi, P.; Lappa, M.; Oliveira, M.S.N. Deformation of a ferrofluid droplet in a simple shear flow under the effect of a constant magnetic field. Comput. Fluids
**2018**, 173, 313–323. [Google Scholar] [CrossRef] - Filali, Y.; Er-Riani, M.; El Jarroudi, M. The deformation of a ferrofluid drop under a uniform magnetic field. Int. J. Non-Linear Mech.
**2018**, 99, 173–181. [Google Scholar] [CrossRef] - Vafeas, P.; Bakalis, P.; Papadopoulos, P.K. Effect of the magnetic field on the ferrofluid flow in a curved cylindrical annular duct. Phys. Fluids
**2019**, 31, 117105. [Google Scholar] [CrossRef] - Volkova, T.I.; Böhm, V.; Naletova, V.A.; Kaufhold, T.; Becker, F.; Zeidis, I.; Zimmermann, K. A ferrofluid based artificial tactile sensor with magnetic field control. J. Magn. Magn. Mater.
**2017**, 431, 277–280. [Google Scholar] [CrossRef] - Sharova, O.A.; Merkulov, D.I.; Pelevina, D.A.; Vinogradova, A.S.; Naletova, V.A. Motion of a spherical magnetizable body along a layer of magnetic fluid in a uniform magnetic field. Phys. Fluids
**2021**, 33, 087107. [Google Scholar] [CrossRef] - Akbar, N.S.; Al-Zubaidi, A.; Saleem, S.; Alsallami, S.A.M. Variable fluid properties analysis for thermally laminated 3-dimensional magnetohydrodynamic non-Newtonian nanofluid over a stretching sheet. Sci. Rep.
**2023**, 13, 3231. [Google Scholar] [CrossRef] - Bacri, J.C.; Salin, D.; Massart, R. Study of the deformation of ferrofluid droplets in a magnetic field. J. Phys. Lett.
**1982**, 43, 179–184. [Google Scholar] [CrossRef] - Nguyen, N.T.; Beyzavi, A.; Ng, K.M.; Huang, X. Kinematics and deformation of ferrofluid droplets under magnetic actuation. Microfluid. Nanofluidics
**2007**, 3, 571–579. [Google Scholar] [CrossRef] - Nguyen, N.T.; Ng, K.M.; Huang, X. Manipulation of ferrofluid droplets using planar coils. Appl. Phys. Lett.
**2006**, 89, 648. [Google Scholar] [CrossRef] - Lehmann, U.; Hadjidj, S.; Parashar, V.K.; Rida, A. Two dimensional magnetic manipulation of microdroplets on a chip. Sens. Actuators B Chem.
**2006**, 117, 457–463. [Google Scholar] [CrossRef] - Saroj, S.K.; Asfer, M.; Sunderka, A.; Panigrahi, P.K. Two-fluid mixing inside a sessile micro droplet using magnetic beads actuation. Sens. Actuators A Phys.
**2016**, 244, 112–120. [Google Scholar] [CrossRef] - Lehmann, U.; Vandevyver, C.; Parashar, V.K.; Gijs, M.A. Droplet-based DNA purification in a magnetic lab-on-a-chip. Angew. Chem.
**2006**, 45, 3062–3067. [Google Scholar] [CrossRef] - Lehmann, U.; De Courten, D.; Vandevyver, C.; Parashar, V.K.; Gijs, M.A.M. On-chip antibody handling and colorimetric detection in a magnetic droplet manipulation system. Microelectron. Eng.
**2007**, 84, 1669–1672. [Google Scholar] [CrossRef] - Bacri, J.; Cebers, A.O.; Perzynski, R. Behavior of a magnetic fluid microdrop in a rotating magnetic field. Phys. Rev. Lett.
**1994**, 72, 2705–2708. [Google Scholar] [CrossRef] [PubMed] - Cēbers, A.; Javaitis, I. Dynamics of a flexible magnetic chain in a rotating magnetic field. Phys. Rev. E
**2004**, 69, 021404. [Google Scholar] [CrossRef] [PubMed] - Shikida, M.; Nagao, N.; Imai, R.; Honda, H.; Okochi, M.; Ito, H.; Sato, K. A palmtop-sized rotary-drive-type biochemical analysis system by magnetic bead handling. J. Micromechanics Microengineering
**2008**, 18, 35034–35041. [Google Scholar] [CrossRef] - Pelevina, D.A.; Turkov, V.A.; Kalmykov, S.A.; Naletova, V.A. Motions of objects with magnetizable materials along a horizontal plane in a rotating magnetic field. J. Magn. Magn. Mater.
**2015**, 390, 20–25. [Google Scholar] [CrossRef] - Shaposhnikov, I.; Shliomis, M. Hydrodynamics of A Magnetizable Medium. 1975. Available online: https://www.researchgate.net/publication/255293888_Hydrodynamics_of_a_magnetizable_medium (accessed on 7 November 2021).
- Rosensweig, R.E. Ferrohydrodynamics; Cambridge University Press: New York, NY, USA, 1985. [Google Scholar]
- Chakrabarty, D.; Dutta, S.; Chakraborty, N.; Ganguly, R. Magnetically actuated transport of ferrofluid droplets over micro-coil array on a digital microfluidic platform. Sens. Actuators B: Chem.
**2016**, 236, 367–377. [Google Scholar] [CrossRef] - Stan, C.; Cristescu, C.P.; Balasoiu, M.; Perov, N.; Fetisov, L. Physics. Investigations of a Fe
_{3}O_{4}-ferrofluid at different temperatures by means of magnetic measurements. UPB Entific Bull. Ser. A: Appl. Math.**2011**, 73, 117–124. [Google Scholar] - Deshpande, K.B.; Zimmerman, W.B. Simulation of interfacial mass transfer by droplet dynamics using the level set method. Chem. Eng. Sci.
**2006**, 61, 6486–6498. [Google Scholar] [CrossRef] - Baldoni, F. A Slip Boundary Condition for the Motion of a Newtonian Droplet on a Solid Surface. Zamm-J. Appl. Math. Mech.
**1999**, 79, 193–203. [Google Scholar] [CrossRef]

**Figure 2.**Computational models and magnetic field parameters: (

**a**) Computational domain and setting; (

**b**) The correspondence between magnetic scale potential and initial magnetic field.

**Figure 4.**The experimental results for deformation of droplets at different volumes with and without magnetic field.

**Figure 5.**The volume fraction of the droplet on the XOZ plane: (

**a**) At t = 400 ms and ${V}_{m}$ = 20–3680 A, the contour of the droplet is compared with the experiment; (

**b**) At ${V}_{m}$ = 1000 A and t = 0–400 ms, the shape of the sessile droplet.

**Figure 6.**Distribution on XOZ plane at ${V}_{m}$ = 1000 A and t = 400 ms: (

**a**) Magnetic flux density, B (T); (

**b**) Magnetic field strength, H (A/m); (

**c**) Magnitude of velocity.

**Figure 7.**Distribution of magnetic force in

**X**direction ${f}_{\mathrm{x}}$, the magnetic force due to field gradient ${f}_{\mathrm{x}1}$, the magnetic force due to susceptibility gradient ${f}_{\mathrm{x}2}$, and interfacial tension in X direction along the droplet width on XOZ plane at ${V}_{m}$ =1000 A and t = 400 ms.

**Figure 9.**Comparison of numerical and experimental results at various magnetic Bond numbers: (

**a**) Height; (

**b**) Width.

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**MDPI and ACS Style**

Zhu, G.-P.; Wu, S.-H.; Zheng, S.-Z.; Li, L.; Nguyen, N.-T.
Investigation of Ferrofluid Sessile Droplet Tensile Deformation in a Uniform Magnetic Field. *Magnetochemistry* **2023**, *9*, 215.
https://doi.org/10.3390/magnetochemistry9100215

**AMA Style**

Zhu G-P, Wu S-H, Zheng S-Z, Li L, Nguyen N-T.
Investigation of Ferrofluid Sessile Droplet Tensile Deformation in a Uniform Magnetic Field. *Magnetochemistry*. 2023; 9(10):215.
https://doi.org/10.3390/magnetochemistry9100215

**Chicago/Turabian Style**

Zhu, Gui-Ping, Shi-Hua Wu, Shu-Ze Zheng, Lai Li, and Nam-Trung Nguyen.
2023. "Investigation of Ferrofluid Sessile Droplet Tensile Deformation in a Uniform Magnetic Field" *Magnetochemistry* 9, no. 10: 215.
https://doi.org/10.3390/magnetochemistry9100215