# Modelling Sessile Droplet Profile Using Asymmetrical Ellipses

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theoretical Background

#### 2.1. Defining the Ellipses

_{0}= y

_{0}= 0. For the bottom ellipse, the semi-major and semi-minor axes are defined as a′ and b′. The bottom ellipse is centred at (a−a′,0) instead of the origin such that both the top and bottom ellipses share the same equatorial point.

#### 2.2. Determining Surface Tension Using Droplet Apex and Equatorial Point

_{1}and R

_{2}are the principal radii of curvature. These radii cannot be directly measured and therefore derived from the geometry of the droplet profile. Using the elliptic model, R

_{1}and R

_{2}can be easily approximated as it is relatively trivial to determine the radii of curvature of an ellipse at its apexes.

_{1}is not equal to the radius of curvature of the fitting ellipse, we introduced a set of constant correction factors C to improve the approximation such that ${R}_{1}={R}_{2}\cong \frac{{C}_{1}{a}^{2}}{b}$ at the apex. At the equator, we set the latitudinal radius to ${R}_{1}=a$, whereas the longitudinal radius was set to ${R}_{2}\cong \frac{{C}_{2}{b}^{2}}{a}$.

_{1}and C

_{2}are determined such that the resultant surface tensions are closest to the nominal values, using the numerical Excel goal seek function.

#### 2.3. Determining Contact Angles

_{3}is also determined using the Excel goal seek function, with the target to minimise the difference between the generated contact angles and nominal contact angles. From Equation (1), we require 4 constraints for the bottom ellipse to be fully defined. For the parameters associated with the bottom ellipse, we assigned the prime symbol such that the semi-major and semi-minor axes are defined as a′ and b′, respectively.

- (i)
- y
_{0}= 0, since the bottom ellipse foci are located on the x-axis; - (ii)
- When x = a, y = b, since the equatorial point is known;
- (iii)
- When x = i, y = j, since the three-phase contact point is known;
- (iv)
- $\frac{{{b}^{\prime}}^{2}}{{a}^{\prime}}=\frac{{C}_{3}{b}^{2}}{a}$ from the radius of curvature.

## 3. Methods

## 4. Results and Discussion

#### 4.1. Surface Tension Determination

_{1}and C

_{2}, resulting in C

_{1}= 0.923579 and C

_{2}= 0.859953. The generated surface tensions with the correction factors implemented agreed well with the nominal value of 72 mN/m, Table 1. The errors were less than 2% for droplets smaller than or equal to 100 µL. The error was slightly larger at 4.5% for a large droplet of 300 µL. We also performed an additional comparison between errors of the generated surface tension with correction factors and without correction factors, as shown in Table S1. Without correction factors, there were significant errors in the calculated surface tension at more than 10%.

#### 4.2. Generation of Elliptic Droplet Profile and Contact Angle

_{3}= 0.963238 was determined similarly to C

_{1}and C

_{2}. Without the correction factor C

_{3}, the errors for volume and surface area calculation were significant as they reached up to 10% (Table S3). We compared parameters derived from droplet profiles such as surface areas and droplet volumes to demonstrate the accuracy of the elliptic method. Table 2 shows that the volume and surface area are in good agreement with the numerical method. Errors tend to increase with the volume and contact angle because the droplet shape starts to deviate from an ellipsoid in these extreme regions.

#### 4.3. Compatibility of Elliptic Model with Actual Sessile Droplet

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Lubarda, V.A.; Talke, K.A. Analysis of the Equilibrium Droplet Shape Based on an Ellipsoidal Droplet Model. Langmuir
**2011**, 27, 10705–10713. [Google Scholar] [CrossRef] - Dixit, S.S.; Pincus, A.; Guo, B.; Faris, G.W. Droplet Shape Analysis and Permeability Studies in Droplet Lipid Bilayers. Langmuir
**2012**, 28, 7442–7451. [Google Scholar] [CrossRef] [Green Version] - Zhou, W.; Loney, D.; Degertekin, F.L.; Rosen, D.W.; Fedorov, A.G. What controls dynamics of droplet shape evolution upon impingement on a solid surface? AIChE J.
**2013**, 59, 3071–3082. [Google Scholar] [CrossRef] - Saad, S.M.I.; Policova, Z.; Neumann, A.W. Design and accuracy of pendant drop methods for surface tension measurement. Colloids Surf. A Physicochem. Eng. Asp.
**2011**, 384, 442–452. [Google Scholar] [CrossRef] - Berry, J.D.; Neeson, M.J.; Dagastine, R.R.; Chan, D.Y.C.; Tabor, R.F. Measurement of surface and interfacial tension using pendant drop tensiometry. J. Colloid Interface Sci.
**2015**, 454, 226–237. [Google Scholar] [CrossRef] [PubMed] - Brandon, S.; Marmur, A. Simulation of Contact Angle Hysteresis on Chemically Heterogeneous Surfaces. J. Colloid Interface Sci.
**1996**, 183, 351–355. [Google Scholar] [CrossRef] [PubMed] - He, B.; Yang, S.; Qin, Z.; Wen, B.; Zhang, C. The roles of wettability and surface tension in droplet formation during inkjet printing. Sci. Rep.
**2017**, 7, 11841. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Jain, V.; Raj, T.P.; Deshmukh, R.; Patrikar, R. Design, fabrication and characterization of low cost printed circuit board based EWOD device for digital microfluidics applications. Microsyst. Technol.
**2017**, 23, 389–397. [Google Scholar] [CrossRef] - Nguyen, C.T.; Kim, B. Stress and surface tension analyses of water on graphene-coated copper surfaces. Int. J. Precis. Eng. Manuf.
**2016**, 17, 503–510. [Google Scholar] [CrossRef] - Gates, C.H.; Perfect, E.; Lokitz, B.S.; Brabazon, J.W.; McKay, L.D.; Tyner, J.S. Transient analysis of advancing contact angle measurements on polished rock surfaces. Adv. Water Resour.
**2018**, 119, 142–149. [Google Scholar] [CrossRef] - Hünnekens, B.; Peters, F.; Avramidis, G.; Krause, A.; Militz, H.; Viöl, W. Plasma treatment of wood–polymer composites: A comparison of three different discharge types and their effect on surface properties. J. Appl. Polym. Sci.
**2016**, 133. [Google Scholar] [CrossRef] - Bashforth, F.; Adams, J. An Attempt to Test the Theory of Capillary Action; Cambridge University: Cambridge, UK, 1892. [Google Scholar]
- Shanahan, M.E.R. An approximate theory describing the profile of a sessile drop. J. Chem. Soc. Faraday Trans. 1 Phys. Chem. Condens. Phases
**1982**, 78, 2701–2710. [Google Scholar] [CrossRef] - Joshi, Y.P. Shape of a liquid surface in contact with a solid. Eur. J. Phys.
**1990**, 11, 125–129. [Google Scholar] [CrossRef] - Yildiz, B.; Bashiry, V. Shape analysis of a sessile drop on a flat solid surface. J. Adhes.
**2019**, 95, 929–942. [Google Scholar] [CrossRef] - Hartland, S.; Hartley, R.W. Axisymmetric Fluid-Liquid Interfaces: Tables Giving the Shape of Sessile and Pendant Drops and External Menisci, with Examples of Their Use; Elsevier Science Limited: Amsterdam, The Netherlands, 1976. [Google Scholar]
- Jennings, J.W.; Pallas, N.R. An efficient method for the determination of interfacial tensions from drop profiles. Langmuir
**1988**, 4, 959–967. [Google Scholar] [CrossRef] - Yu, K.; Yang, J.; Zuo, Y.Y. Automated Droplet Manipulation Using Closed-Loop Axisymmetric Drop Shape Analysis. Langmuir
**2016**, 32, 4820–4826. [Google Scholar] [CrossRef] [Green Version] - Skinner, F.K.; Rotenberg, Y.; Neumann, A.W. Contact angle measurements from the contact diameter of sessile drops by means of a modified axisymmetric drop shape analysis. J. Colloid Interface Sci.
**1989**, 130, 25–34. [Google Scholar] [CrossRef] - Río, O.I.D.; Neumann, A.W. Axisymmetric Drop Shape Analysis: Computational Methods for the Measurement of Interfacial Properties from the Shape and Dimensions of Pendant and Sessile Drops. J. Colloid Interface Sci.
**1997**, 196, 136–147. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Aussillous, P.; Quéré, D. Liquid marbles. Nature
**2001**, 411, 924–927. [Google Scholar] [CrossRef] [PubMed] - Aussillous, P.; Quéré, D. Properties of liquid marbles. Proc. R. Soc. A Math. Phys. Eng. Sci.
**2006**, 462, 973–999. [Google Scholar] [CrossRef] - Singha, P.; Ooi, C.H.; Nguyen, N.-K.; Sreejith, K.R.; Jin, J.; Nguyen, N.-T. Capillarity: Revisiting the fundamentals of liquid marbles. Microfluid. Nanofluidics
**2020**, 24, 81. [Google Scholar] [CrossRef] - Singha, P.; Nguyen, N.-K.; Sreejith, K.R.; An, H.; Nguyen, N.-T.; Ooi, C.H. Effect of Core Liquid Surface Tension on the Liquid Marble Shell. Adv. Mater. Interfaces
**2021**, 8, 2001591. [Google Scholar] [CrossRef] - Ooi, C.H.; Vadivelu, R.; Jin, J.; Sreejith, K.R.; Singha, P.; Nguyen, N.-K.; Nguyen, N.-T. Liquid marble-based digital microfluidics—Fundamentals and applications. Lab Chip
**2021**, 21, 1199–1216. [Google Scholar] [CrossRef] - Ooi, C.H.; Singha, P.; Nguyen, N.-K.; An, H.; Nguyen, V.T.; Nguyen, A.V.; Nguyen, N.-T. Measuring the effective surface tension of a floating liquid marble using X-ray imaging. Soft Matter
**2021**, 17, 4069–4076. [Google Scholar] [CrossRef] - Singha, P.; Nguyen, N.-K.; Zhang, J.; Nguyen, N.-T.; Ooi, C.H. Oscillating sessile liquid marble—A tool to assess effective surface tension. Colloids Surf. A Physicochem. Eng. Asp.
**2021**, 627, 127176. [Google Scholar] [CrossRef] - Ooi, C.H.; Vadivelu, R.K.; St John, J.; Dao, D.V.; Nguyen, N.-T. Deformation of a floating liquid marble. Soft Matter
**2015**, 11, 4576–4583. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Cengiz, U.; Erbil, H.Y. The lifetime of floating liquid marbles: The influence of particle size and effective surface tension. Soft Matter
**2013**, 9, 8980–8991. [Google Scholar] [CrossRef] - Jin, J.; Nguyen, N.-T. Manipulation schemes and applications of liquid marbles for micro total analysis systems. Microelectron. Eng.
**2018**, 197, 87–95. [Google Scholar] [CrossRef] - Nguyen, N.-K.; Singha, P.; Zhang, J.; Phan, H.-P.; Nguyen, N.-T.; Ooi, C.H. Digital Imaging-based Colourimetry for Enzymatic Processes in Transparent Liquid Marbles. ChemPhysChem
**2021**, 22, 99–105. [Google Scholar] [CrossRef] - Nguyen, N.-K.; Ooi, C.H.; Singha, P.; Jin, J.; Sreejith, K.R.; Phan, H.-P.; Nguyen, N.-T. Liquid Marbles as Miniature Reactors for Chemical and Biological Applications. Processes
**2020**, 8, 793. [Google Scholar] [CrossRef] - Nguyen, N.-K.; Singha, P.; An, H.; Phan, H.-P.; Nguyen, N.-T.; Ooi, C.H. Electrostatically excited liquid marble as a micromixer. React. Chem. Eng.
**2021**, 6, 1386–1394. [Google Scholar] [CrossRef] - Ooi, C.H.; Nguyen, N.-T. Manipulation of liquid marbles. Microfluid. Nanofluidics
**2015**, 19, 483–495. [Google Scholar] [CrossRef] - Arbatan, T.; Li, L.; Tian, J.; Shen, W. Liquid Marbles as Micro-bioreactors for Rapid Blood Typing. Adv. Healthc. Mater.
**2012**, 1, 80–83. [Google Scholar] [CrossRef] [PubMed] - Zang, D.; Li, J.; Chen, Z.; Zhai, Z.; Geng, X.; Binks, B.P. Switchable Opening and Closing of a Liquid Marble via Ultrasonic Levitation. Langmuir
**2015**, 31, 11502–11507. [Google Scholar] [CrossRef] [PubMed] - Sato, E.; Yuri, M.; Fujii, S.; Nishiyama, T.; Nakamura, Y.; Horibe, H. Liquid marbles as a micro-reactor for efficient radical alternating copolymerization of diene monomer and oxygen. Chem. Commun.
**2015**, 51, 17241–17244. [Google Scholar] [CrossRef] [PubMed] - Han, X.; Koh, C.S.L.; Lee, H.K.; Chew, W.S.; Ling, X.Y. Microchemical Plant in a Liquid Droplet: Plasmonic Liquid Marble for Sequential Reactions and Attomole Detection of Toxin at Microliter Scale. ACS Appl. Mater. Interfaces
**2017**, 9, 39635–39640. [Google Scholar] [CrossRef] - Tian, J.; Fu, N.; Chen, X.D.; Shen, W. Respirable liquid marble for the cultivation of microorganisms. Colloids Surf. B Biointerfaces
**2013**, 106, 187–190. [Google Scholar] [CrossRef] [PubMed] - Sheng, Y.; Sun, G.; Wu, J.; Ma, G.; Ngai, T. Silica-Based Liquid Marbles as Microreactors for the Silver Mirror Reaction. Angew. Chem. Int. Ed.
**2015**, 54, 7012–7017. [Google Scholar] [CrossRef] [PubMed] - Gu, H.; Ye, B.; Ding, H.; Liu, C.; Zhao, Y.; Gu, Z. Non-iridescent structural color pigments from liquid marbles. J. Mater. Chem. C
**2015**, 3, 6607–6612. [Google Scholar] [CrossRef] - McHale, G.; Elliott, S.J.; Newton, M.I.; Herbertson, D.L.; Esmer, K. Levitation-Free Vibrated Droplets: Resonant Oscillations of Liquid Marbles. Langmuir
**2009**, 25, 529–533. [Google Scholar] [CrossRef] [PubMed] - Farid, M. A new approach to modelling of single droplet drying. Chem. Eng. Sci.
**2003**, 58, 2985–2993. [Google Scholar] [CrossRef] - Hu, H.; Larson, R.G. Evaporation of a Sessile Droplet on a Substrate. J. Phys. Chem. B
**2002**, 106, 1334–1344. [Google Scholar] [CrossRef] - Marinaro, G.; Riekel, C.; Gentile, F. Size-Exclusion Particle Separation Driven by Micro-Flows in a Quasi-Spherical Droplet: Modelling and Experimental Results. Micromachines
**2021**, 12, 185. [Google Scholar] [CrossRef] [PubMed] - Marchese, A.J.; Dryer, F.L.; Nayagam, V. Numerical modeling of isolated n-alkane droplet flames: Initial comparisons with ground and space-based microgravity experiments. Combust. Flame
**1999**, 116, 432–459. [Google Scholar] [CrossRef]

**Figure 1.**Schematic of the coordinate system for the elliptic model. O is the coordinate of the origin, whereas (i, j) is the coordinate of the three-phase contact point.

**Figure 2.**Comparison between droplet profiles with the Young–Laplace theoretical model (continuous lines) and elliptic model (scattered data points) at different nominal contact angles.

**Figure 3.**(

**a**–

**c**) Images of 10 μL droplets with constructed fitting curves using ImageJ’s plugin LBADSA. Scale bar is 1 mm.

Volume (µL) | Input Parameters (cm) | Surface Tension (mN/m) | ||
---|---|---|---|---|

a | b | Generated | Error (%) | |

1 | 0.0636 | 0.0628 | 72.00 | 0.000 |

10 | 0.1388 | 0.1314 | 72.29 | 0.403 |

30 | 0.2087 | 0.1866 | 73.69 | 2.347 |

100 | 0.3280 | 0.2589 | 71.99 | 0.139 |

300 | 0.5135 | 0.3323 | 70.47 | 2.125 |

**Table 2.**Droplet parameters at different volumes and various contact angles: (i) Young–Laplace theoretical droplet; (ii) generated droplet using the elliptic model.

Pre-Defined Volume (µL) | Contact Angle (°) | Volume (µL) | Surface Area (cm^{2}) | ||||||
---|---|---|---|---|---|---|---|---|---|

(i) | (ii) | Error (%) | (i) | (ii) | Error (%) | (i) | (ii) | Error (%) | |

1 | 90 | 90 | 0 | 1.0 | 1.0 | 0 | 0.039 | 0.039 | 0 |

135 | 134.13 | 0.65 | 1.0 | 1.0 | 0 | 0.044 | 0.043 | 0.23 | |

180 | 174.75 | 3 | 1.0 | 1.0 | 0 | 0.050 | 0.049 | 0.41 | |

10 | 90 | 90 | 0 | 9.9 | 10.0 | 1.00 | 0.178 | 0.178 | 0.28 |

135 | 137.12 | 1.55 | 10.4 | 10.4 | 0 | 0.199 | 0.199 | 0.10 | |

180 | 179.21 | 0.44 | 10.0 | 10.0 | 0 | 0.218 | 0.219 | 0.23 | |

30 | 90 | 90 | 0 | 30.8 | 30.8 | 0 | 0.379 | 0.379 | 0.03 |

135 | 138.76 | 2.71 | 31.6 | 31.7 | 0.32 | 0.410 | 0.410 | 0.19 | |

180 | 180.00 | 0 | 30.9 | 31.1 | 0.64 | 0.452 | 0.454 | 0.57 | |

100 | 90 | 90 | 0 | 100.7 | 101.0 | 0.30 | 0.846 | 0.848 | 0.19 |

135 | 141.35 | 4.49 | 101.2 | 101.5 | 0.30 | 0.871 | 0.871 | 0.01 | |

180 | 180.00 | 0.00 | 100.4 | 101.3 | 0.89 | 0.963 | 0.972 | 0.88 | |

300 | 90 | 90 | 0 | 300.7 | 294.4 | 2.14 | 1.821 | 1.801 | 1.10 |

135 | 139.82 | 3.45 | 298.9 | 297.0 | 0.64 | 1.784 | 1.772 | 0.67 | |

180 | 180.00 | 0 | 300.8 | 300.0 | 0.27 | 1.959 | 1.964 | 0.28 |

**Table 3.**Comparison of Bond number between (i) Young–Laplace theoretical droplet and (ii) generated droplet using the elliptic model.

Pre-Defined Volume (µL) | Bond Number | ||
---|---|---|---|

(i) | (ii) | Error (%) | |

1 | 0.0536 | 0.0534 | 0.374 |

10 | 0.2461 | 0.2462 | 0.041 |

30 | 0.5195 | 0.5201 | 0.115 |

100 | 1.1423 | 1.1444 | 0.184 |

300 | 2.3654 | 2.3505 | 0.64 |

**Table 4.**Comparison of droplet parameters between two methods: (i) LBADSA; (ii) axisymmetrical elliptic model.

Run | Volume (µL) | Surface Area (cm^{2}) | Contact Angle (°) | Surface Tension (mN/m) | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

(i) | (ii) | Error (%) | (i) | (ii) | Error (%) | (i) | (ii) | Error (%) | (i) | (ii) | Error (%) | |

1 | 10.7 | 10.7 | 0 | 0.203 | 0.204 | 0.49 | 137 | 138.46 | 1.07 | 81.75 | 78.7 | 3.87 |

2 | 10.5 | 10.6 | 0.95 | 0.201 | 0.202 | 0.5 | 137.72 | 139.46 | 1.26 | 72.67 | 69.6 | 4.41 |

3 | 10.8 | 10.8 | 0 | 0.205 | 0.206 | 0.49 | 138.91 | 140.84 | 1.39 | 72.67 | 70 | 3.81 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Tran, D.T.; Nguyen, N.-K.; Singha, P.; Nguyen, N.-T.; Ooi, C.H.
Modelling Sessile Droplet Profile Using Asymmetrical Ellipses. *Processes* **2021**, *9*, 2081.
https://doi.org/10.3390/pr9112081

**AMA Style**

Tran DT, Nguyen N-K, Singha P, Nguyen N-T, Ooi CH.
Modelling Sessile Droplet Profile Using Asymmetrical Ellipses. *Processes*. 2021; 9(11):2081.
https://doi.org/10.3390/pr9112081

**Chicago/Turabian Style**

Tran, Du Tuan, Nhat-Khuong Nguyen, Pradip Singha, Nam-Trung Nguyen, and Chin Hong Ooi.
2021. "Modelling Sessile Droplet Profile Using Asymmetrical Ellipses" *Processes* 9, no. 11: 2081.
https://doi.org/10.3390/pr9112081