# Use of Heating Configuration to Control Marangoni Circulation during Droplet Evaporation

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

## Abstract

**:**

## 1. Introduction

_{s}/λ

_{l}), reversing the direction at a critical contact angle over the range (1.45 < λ

_{s}/λ

_{l}< 2). In the same analytical context, the authors of [34] showed that the direction of the Marangoni flow reversed at a critical contact angle, depending not only on (λ

_{s}/λ

_{l}) but also on the ratio of the thickness of the substrate to the contact line radius. Marangoni convection in a low-volatile sessile droplet was numerically studied in [35] by varying the Marangoni number and the contact angles. They showed that Bénard–Marangoni convection and thermocapillary convection can coexist in a droplet due to its curved surface. The second possibility is the change in the surface tension with the composition of a droplet, such as multicomponent droplets [36] or droplets with surfactants [37].

#### Synthesis and Objectives of the Work

## 2. Mathematical Model

#### 2.1. Physical Domain

#### 2.2. ALE Formulation

- move with the materials (at the interface liquid–gas) to accurately reproduce the moving boundaries and interfaces of multi-domain systems;
- be fixed in space inside the material domain;
- be fixed in one direction and move with the material in other directions.

_{m}) [25,31,56]. Therefore, the velocities are defined as U(X, t) in the area and U(X

_{m}, t) in the mesh, as follows:

_{c}, as follows:

_{m},t) is the flow velocity. In the advection term, the velocity is denoted as U

_{c}. In the rest of the paper, for simple notations, the velocity U(X

_{m},t) is denoted as U. Consequently, the material derivative of each physical quantity φ is written as follows:

#### 2.3. Governing Equations System

- Conduction equation in the solid substrate:

- Continuity, Navier–Stokes, and energy equations in the liquid droplet and gas domain:

- Advection-diffusion equation in the gas domain, with air surrounding the droplet:

- At the liquid–gas interface.

_{v,sat}involving the saturation temperature T

_{sat}is given by the following expression [25]:

_{sat}is calculated based on the saturation pressure P

_{v,sat}, such that:

_{w}is the molar mass of liquid (water) and D is the diffusion coefficient.

_{ev}. The thermal and dynamic conditions at the interface are as follows [25,31,52]:

#### 2.4. Initial and Boundary Conditions

_{0}and humidity H

_{0}. Initially, the substrate is also at temperature T

_{0}.

#### 2.4.1. Boundary Conditions

- At r = 0,

- At r = 0, z = h,

- At z = −e,

- At z = 0, 0 < r < R,

- At z = 0, R < r < L,

- At z = ∞,

#### 2.4.2. Heating Configurations

_{h}of 50 °C. On the other hand, when dealing with the TH configuration, we imposed at the top of the droplet (on a line of 50 μm) a heating flux Φ of 40 mW. The objective of choosing these boundary conditions was to qualitatively validate our numerical results with the experimental results of [46].

_{h}= 50 °C) and we decreased the value of the flux (Φ = 3 mW) in order to highlight the two Marangoni circulations.

## 3. Numerical Simulation

#### 3.1. Mesh Velocity and Balanced Stresses

_{ev}is the local evaporated flux defined by Equation (16), n presents the normal vector of the interface, $\mathsf{\tau}$ is the total stress tensor, and f

_{st}represents the force per unit area due to the surface tension σ.

_{st}is the surface gradient operator.

_{st}is described by two components, normal and tangential, and we write the balance of the forces as follows:

#### 3.2. Computer Code and Used Grid

_{ev},V) and i represents the number of the grid, i = 1,2.

## 4. Results and Discussion

#### 4.1. Model Validation

^{3}and an initial contact angle equal to 57.2°, deposited on a glass substrate. The environment surrounding the droplet (i.e., temperature and humidity) was controlled, and the droplet evaporated under ambient conditions (T = 25 °C and H = 40%). The numerical work in [31] and the present work have reproduced the same experimental conditions presented in [57].

#### 4.2. Marangoni Circulation in the Case of One Single Heat Source (TH and BH Configurations)

#### 4.3. TBH Configuration and Nature Substrate Effect

#### 4.3.1. Marangoni Circulation

#### 4.3.2. Substrate Type Effect

#### 4.3.3. Effect of the Substrate Thickness

_{SZ}to the triple point as the thickness of the substrate increased.

_{SZ}for different thicknesses of PTFE substrates. At the start of evaporation (t = 1 s), this zone approached the top of the droplet for small substrate thicknesses, which favored the upward Marangoni flow, and the opposite case was observed when the thickness was large. After t = 10 s, and when the heat propagated towards the liquid–solid interface, the position of the stagnation zone X

_{ZS}was almost similar to all the thicknesses considered.

## 5. Conclusions

- Using one single source of heat, the direction of Marangoni circulation is monotonic and can be chosen; upward flow occurs when the substrate is heated (BH) and a downward flow occurs when heat is supplied at the top of the droplet (TH);
- The combination of the two types of heating (TBH) triggered a Marangoni flow with two vortices separated by a stagnation point;
- The balance between the magnitude of the two heat sources was changed by the nature and the thickness of the substrate. The results show that the respective importance of the two vortices and the position of the stagnation point can be controlled.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

Symbols | Abbreviation |

ALE | Arbitrary Lagrangian–Eulerian (-) |

BH | Bottom Heating (-) |

Bo | Bond number (-) |

C | concentration (mol·L^{−1} or g·L^{−1}) |

Cp | heat capacity (J·kg^{−1}·K^{−1}) |

D | diffusion coefficient (m^{2}·s^{−1}) |

e | substrate thickness (m) |

f_{st} | force per unit area (N·m^{−2}) |

g | gravity acceleration (m·s^{−2}) |

h | droplet height (m) |

H | humidity (%) |

L | substrate length (m) |

Lc | capillary length (m) |

L_{v} | latent heat (J·kg^{−1}) |

m_{ev} | local evaporation rate (kg·m^{−2}·s^{−1}) |

Mw | molar Mass (kg·mol^{−1}) |

n | normal direction (-) |

p | pressure (Pa) |

r_{c} | curvature radius (m) |

R | contact radius (m) |

Rm | universal gas constant (J·mol^{−1}·K^{−1}) |

(r,z) | cylindrical coordinates (m) |

SZ | Stagnation zone (-) |

t | time (s) |

$\U0001d4c9$ | tangential direction (-) |

T | temperature (K or °C) |

TH | Top Heating (-) |

TBH | Top and Bottom Heating (-) |

(u,w) | velocity components (m·s^{−1}) |

U | norm of velocity (m·s^{−1}) |

V | droplet volume (mm^{3} or µL) |

x | space coordinates (m) |

X | domain coordinates (m) |

X_{m} | mesh coordinates (m) |

Greek symbols | |

ε | relative difference (%) |

θ | contact angle (°) |

λ | thermal conductivity (W·m^{−1}·K^{−1}) |

μ | dynamic viscosity (Pa·s) |

ρ | density (kg·m^{−3}) |

σ | surface tension (N·m^{−1}) |

τ | total stress tensor (-) |

Φ | heat flux (W) |

Subscripts | |

c | convective |

g | gas (air) |

h | hot |

l | liquid (water) |

∝ | infinite |

0 | ambient, reference |

s | solid (substrate) |

sat | saturation |

## References

- Leenaars, A.F.M.; Huethorst, J.A.M.; Van Oekel, J.J. Marangoni drying: A new extremely clean drying process. Langmuir
**1990**, 6, 1701–1703. [Google Scholar] [CrossRef] - Kim, J. Spray cooling heat transfer: The state of the art. Int. J. Heat Fluid Flow
**2007**, 28, 753–767. [Google Scholar] [CrossRef] - Park, J.; Moon, J. Control of Colloidal Particle Deposit Patterns within Picoliter Droplets Ejected by Ink-Jet Printing. Langmuir
**2006**, 22, 3506–3513. [Google Scholar] [CrossRef] [PubMed] - Wu, L.; Dong, Z.; Kuang, M.; Li, Y.; Li, F.; Jiang, L.; Song, Y. Printing Patterned Fine 3D Structures by Manipulating the Three Phase Contact Line. Adv. Funct. Mater.
**2015**, 25, 2237–2242. [Google Scholar] [CrossRef] - MacBeath, G.; Schreiber, S.L. Printing Proteins as Microarrays for High-Throughput Function Determination. Science
**2000**, 289, 1760–1763. [Google Scholar] [CrossRef] - Dugas, V.; Broutin, A.J.; Souteyrand, E. Droplet Evaporation Study Applied to DNA Chip Manufacturing. Langmuir
**2005**, 21, 9130–9136. [Google Scholar] [CrossRef] - Sempels, W.; De Dier, R.; Mizuno, H.; Hofkens, J.; Vermant, J. Auto-production of biosurfactants reverses the coffee ring effect in a bacterial system. Nat. Commun.
**2013**, 4, 1757. [Google Scholar] [CrossRef] [Green Version] - Liu, H. Science and Engineering of Droplets—Fundamentals and Applications; Noyes Publications: Norwich, CT, USA, 2000. [Google Scholar]
- de Gennes, P.-G.; Brochard-Wyart, F.; Quere, D. Capillarity and Wetting Phenomena: Drops, Bubbles, Pearls, Waves; 2004th ed.; Springer: New York, NY, USA, 2013. [Google Scholar]
- Sirignano, W.A. Fluid Dynamics and Transport of Droplets and Sprays, 2nd ed; Cambridge University Press: Cambridge, UK, 2014. [Google Scholar]
- Brutin, D. (Ed.) Droplet Wetting and Evaporation: From Pure to Complex Fluids; Academic Press: San Diego, CA, USA, 2015. [Google Scholar]
- Murisic, N.; Kondic, L. On evaporation of sessile drops with moving contact lines. J. Fluid Mech.
**2011**, 679, 219–246. [Google Scholar] [CrossRef] - Picknett, R.; Bexon, R. The evaporation of sessile or pendant drops in still air. J. Colloid Interface Sci.
**1977**, 61, 336–350. [Google Scholar] [CrossRef] - Rowan, S.M.; Newton, M.I.; McHale, G. Evaporation of Microdroplets and the Wetting of Solid Surfaces. J. Phys. Chem.
**1995**, 99, 13268–13271. [Google Scholar] [CrossRef] - Deegan, R.; Bakajin, O.; Dupont, T.F.; Huber, G.; Nagel, S.R.; Witten, T.A. Contact line deposits in an evaporating drop. Phys. Rev. E
**2000**, 62, 756–765. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Hu, H.; Larson, R.G. Evaporation of a Sessile Droplet on a Substrate. J. Phys. Chem. B
**2002**, 106, 1334–1344. [Google Scholar] [CrossRef] - Erbil, H.Y. Evaporation of pure liquid sessile and spherical suspended drops: A review. Adv. Colloid Interface Sci.
**2012**, 170, 67–86. [Google Scholar] [CrossRef] - Gelderblom, H.; Bloemen, O.; Snoeijer, J.H. Stokes flow near the contact line of an evaporating drop. J. Fluid Mech.
**2012**, 709, 69–84. [Google Scholar] [CrossRef] [Green Version] - 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 Physicochem. Eng. Asp.
**2007**, 298, 108–114. [Google Scholar] [CrossRef] - Lu, G.; Duan, Y.-Y.; Wang, X.-D.; Lee, D.-J. Internal flow in evaporating droplet on heated solid surface. Int. J. Heat Mass Transf.
**2011**, 54, 4437–4447. [Google Scholar] [CrossRef] - Lopes, M.C.; Bonaccurso, E.; Gambaryan-Roisman, T.; Stephan, P. Influence of the substrate thermal properties on sessile droplet evaporation: Effect of transient heat transport. Colloids Surfaces A Physicochem. Eng. Asp.
**2013**, 432, 64–70. [Google Scholar] [CrossRef] - Maatar, A.; Chikh, S.; Saada, M.A.; Tadrist, L. Transient effects on sessile droplet evaporation of volatile liquids. Int. J. Heat Mass Transf.
**2015**, 86, 212–220. [Google Scholar] [CrossRef] - Khilifi, D.; Foudhil, W.; Harmand, S.; Ben Jabrallah, S. Evaporation of a sessile oil drop in the Wenzel-like regime. Int. J. Therm. Sci.
**2020**, 151, 106236. [Google Scholar] [CrossRef] - Saada, M.A.; Chikh, S.; Tadrist, L. Numerical investigation of heat and mass transfer of an evaporating sessile drop on a horizontal surface. Phys. Fluids
**2010**, 22, 112115. [Google Scholar] [CrossRef] - Yang, K.; Hong, F.; Cheng, P. A fully coupled numerical simulation of sessile droplet evaporation using Arbitrary Lagrangian–Eulerian formulation. Int. J. Heat Mass Transf.
**2013**, 70, 409–420. [Google Scholar] [CrossRef] - Askounis, A.; Sefiane, K.; Koutsos, V.; Shanahan, M.E. Effect of particle geometry on triple line motion of nano-fluid drops and deposit nano-structuring. Adv. Colloid Interface Sci.
**2015**, 222, 44–57. [Google Scholar] [CrossRef] [PubMed] - Girard, F.; Antoni, M.; Faure, S.; Steinchen, A. Influence of heating temperature and relative humidity in the evaporation of pinned droplets. Colloids Surfaces A Physicochem. Eng. Asp.
**2008**, 323, 36–49. [Google Scholar] [CrossRef] - Diddens, C. Detailed finite element method modeling of evaporating multi-component droplets. J. Comput. Phys.
**2017**, 340, 670–687. [Google Scholar] [CrossRef] - Diddens, C.; Tan, H.; Lv, P.; Versluis, M.; Kuerten, J.G.M.; Zhang, X.; Lohse, D. Evaporating pure, binary and ternary droplets: Thermal effects and axial symmetry breaking. J. Fluid Mech.
**2017**, 823, 470–497. [Google Scholar] [CrossRef] [Green Version] - Diddens, C.; Kuerten, J.; van der Geld, C.; Wijshoff, H. Modeling the evaporation of sessile multi-component droplets. J. Colloid Interface Sci.
**2017**, 487, 426–436. [Google Scholar] [CrossRef] - Foudhil, W.; Chen, P.; Fahem, K.; Harmand, S.; Ben Jabrallah, S. Study of the evaporation kinetics of pure and binary droplets: Volatility effect. Heat Mass Transf.
**2021**, 57, 1773–1790. [Google Scholar] [CrossRef] - Hu, H.; Larson, R.G. Analysis of the Microfluid Flow in an Evaporating Sessile Droplet. Langmuir
**2005**, 21, 3963–3971. [Google Scholar] [CrossRef] - Ristenpart, W.D.; Kim, P.G.; Domingues, C.; Wan, J.; Stone, H.A. Influence of Substrate Conductivity on Circulation Reversal in Evaporating Drops. Phys. Rev. Lett.
**2007**, 99, 234502. [Google Scholar] [CrossRef] [Green Version] - Xu, X.; Luo, J.; Guo, D. Criterion for Reversal of Thermal Marangoni Flow in Drying Drops. Langmuir
**2009**, 26, 1918–1922. [Google Scholar] [CrossRef] - Shi, W.-Y.; Tang, K.-Y.; Ma, J.-N.; Jia, Y.-W.; Li, H.-M.; Feng, L. Marangoni convection instability in a sessile droplet with low volatility on heated substrate. Int. J. Therm. Sci.
**2017**, 117, 274–286. [Google Scholar] [CrossRef] - Tan, H.; Diddens, C.; Lv, P.; Kuerten, J.G.M.; Zhang, X.; Lohse, D. Evaporation-triggered microdroplet nucleation and the four life phases of an evaporating Ouzo drop. Proc. Natl. Acad. Sci. USA
**2016**, 113, 8642–8647. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Karapetsas, G.; Sahu, K.C.; Matar, O.K. Evaporation of Sessile Droplets Laden with Particles and Insoluble Surfactants. Langmuir
**2016**, 32, 6871–6881. [Google Scholar] [CrossRef] [PubMed] - Deegan, R.D.; Bakajin, O.; Dupont, T.F.; Huber, G.; Nagel, S.R.; Witten, T.A. Capillary flow as the cause of ring stains from dried liquid drops. Nature
**1997**, 389, 827–829. [Google Scholar] [CrossRef] - Mampallil, D.; Eral, H.B. A review on suppression and utilization of the coffee-ring effect. Adv. Colloid Interface Sci.
**2018**, 252, 38–54. [Google Scholar] [CrossRef] [PubMed] - Hu, A.H.; Larson, R.G. Marangoni Effect Reverses Coffee-Ring Depositions. J. Phys. Chem. B
**2006**, 110, 7090–7094. [Google Scholar] [CrossRef] - Zhang, Z.; Zhang, X.; Xin, Z.; Deng, M.; Wen, Y.; Song, Y. Controlled Inkjetting of a Conductive Pattern of Silver Nanoparticles Based on the Coffee-Ring Effect. Adv. Mater.
**2013**, 25, 6714–6718. [Google Scholar] [CrossRef] - Lee, K.-H.; Kim, S.-M.; Jeong, H.; Jung, G.-Y. Spontaneous nanoscale polymer solution patterning using solvent evaporation driven double-dewetting edge lithography. Soft Matter
**2011**, 8, 465–471. [Google Scholar] [CrossRef] - Cui, L.; Zhang, J.; Zhang, X.; Li, Y.; Wang, Z.; Gao, H.; Wang, T.; Zhu, S.; Yu, H.; Yang, B. Avoiding coffee ring structure based on hydrophobic silicon pillar arrays during single-drop evaporation. Soft Matter
**2012**, 8, 10448–10456. [Google Scholar] [CrossRef] - Huang, Y.; Zhou, J.; Su, B.; Shi, L.; Wang, J.; Chen, S.; Wang, L.; Zi, J.; Song, Y.; Jiang, L. Colloidal Photonic Crystals with Narrow Stopbands Assembled from Low-Adhesive Superhydrophobic Substrates. J. Am. Chem. Soc.
**2012**, 134, 17053–17058. [Google Scholar] [CrossRef] - Crivoi, A.; Duan, F. Three-dimensional Monte Carlo model of the coffee-ring effect in evaporating colloidal droplets. Sci. Rep.
**2014**, 4, 4310. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Yen, T.M.; Fu, X.; Wei, T.; Nayak, R.U.; Shi, Y.; Lo, Y.-H. Reversing Coffee-Ring Effect by Laser-Induced Differential Evaporation. Sci. Rep.
**2018**, 8, 1–11. [Google Scholar] [CrossRef] [PubMed] [Green Version] - van Gaalen, R.; Diddens, C.; Wijshoff, H.; Kuerten, J. Marangoni circulation in evaporating droplets in the presence of soluble surfactants. J. Colloid Interface Sci.
**2020**, 584, 622–633. [Google Scholar] [CrossRef] [PubMed] - Legros, J.; Limbourg-Fontaine, M.; Petre, G. Influence of a surface tension minimum as a function of temperature on the marangoni convection. Acta Astronaut.
**1984**, 11, 143–147. [Google Scholar] [CrossRef] - Limbourg-Fontaine, M.C.; Pétré, G.; Legros, J.C. Thermocapillary movements under microgravity at a minimum of surface tension. Sci. Nat.
**1986**, 73, 360–362. [Google Scholar] [CrossRef] - Savino, R.; Cecere, A.; Di Paola, R. Surface tension-driven flow in wickless heat pipes with self-rewetting fluids. Int. J. Heat Fluid Flow
**2009**, 30, 380–388. [Google Scholar] [CrossRef] - Ouenzerfi, S.; Harmand, S. Experimental Droplet Study of Inverted Marangoni Effect of a Binary Liquid Mixture on a Nonuniform Heated Substrate. Langmuir
**2016**, 32, 2378–2388. [Google Scholar] [CrossRef] - Ruiz, O.E.; Black, W.Z. Evaporation of Water Droplets Placed on a Heated Horizontal Surface. J. Heat Transf.
**2002**, 124, 854–863. [Google Scholar] [CrossRef] - Mollaret, R.; Sefiane, K.; Christy, J.R.; Veyret, D. Experimental and Numerical Investigation of the Evaporation into Air of a Drop on a Heated Surface. Chem. Eng. Res. Des.
**2004**, 82, 471–480. [Google Scholar] [CrossRef] - Ganesan, S.; Tobiska, L.U.T.Z. Finite element simulation of a droplet impinging a horizontal surface. Proc. Algoritm.
**2005**, 2005, 1–11. [Google Scholar] - Bozorgmehr, B.; Murray, B.T. Numerical Simulation of Evaporation of Ethanol–Water Mixture Droplets on Isothermal and Heated Substrates. ACS Omega
**2021**, 6, 12577–12590. [Google Scholar] [CrossRef] [PubMed] - Comsol Multiphysics. Micro-Fluidics Module, User’s Guide, Theory for the Two Phase Flow, Moving Mesh User Interface; Comsol Multiphysics: Kingston, Australia, 2012; pp. 113–121. [Google Scholar]
- Song, H.; Lee, Y.; Jin, S.; Kim, H.-Y.; Yoo, J.Y. Prediction of sessile drop evaporation considering surface wettability. Microelectron. Eng.
**2011**, 88, 3249–3255. [Google Scholar] [CrossRef]

**Figure 6.**Time evolution of the apex temperature: comparison with the numerical study of [22]. (V

_{0}= 3.64 mm

^{3}, θ = 57.2°, T

_{0}= 25 °C, H = 40%).

**Figure 7.**Temperature field for two configurations. (t = 1 s, θ = 90°, V

_{0}= 1 mm

^{3}) (

**a**): BH (T

_{h}= 50 °C) (

**b**): TH (Φ = 40 mW).

**Figure 8.**Temperature evolution along the interface for two configurations. (t = 1 s, θ = 90°, V

_{0}= 1 mm

^{3}, T

_{h}= 50 °C, Φ = 40 mW).

**Figure 9.**Average flux rate evolution vs time for two configurations. (t = 1 s, θ = 90°, V

_{0}= 1 mm

^{3}

_{,}T

_{h}= 50 °C, Φ = 40 mW).

**Figure 10.**Streamlines and flow direction inside water droplet for two configurations. (t = 1 s, V

_{0}= 1 mm

^{3}, θ = 90°) (

**a**): BH (T = 50 °C) (

**b**): TH (Φ = 40 mW).

**Figure 11.**Droplet internal flow: reversing coffee ring effect. (

**a**): Experimental results: [46] (laser apex heating), (

**b**): present simulation (TH configuration Φ = 0.04 W).

**Figure 12.**TBH configuration on glass substrate (t = 1 s, θ = 90°, V

_{0}= 1 mm

^{3}, T

_{h}= 50 °C, Φ = 3 mW). (

**a**) Temperature field, (

**b**) temperature along the interface.

**Figure 13.**TBH configuration: streamlines and flow direction. Influence of the substrate nature (t = 1 s, θ = 90°, V

_{0}= 1 mm

^{3}, T

_{h}= 50 °C, Φ = 3 mW). (

**a**): PTFE, (

**b**) glass.

**Figure 14.**TBH configuration: velocity evolution along the interface. Influence of the substrate nature (t = 1 s, θ = 90°, V

_{0}= 1 mm

^{3}, T

_{h}= 50 °C, Φ = 3 mW).

**Figure 15.**TBH configuration: streamlines and flow direction on PTFE substrate. Influence of the substrate thickness (t = 1 s, θ = 90°, V

_{0}= 1 mm

^{3}, T

_{h}= 50 °C, Φ = 3 mW).

**Figure 16.**TBH configuration: velocity evolution along the interface. Influence of the PTFE substrate thickness (t = 1 s, θ = 90°, V

_{0}= 1 mm

^{3}, T

_{h}= 50 °C, Φ = 3 mW).

**Figure 17.**Evolution of stagnation zone with PTFE substrate thickness. (t = 1 s, θ = 90°, V

_{0}= 1 mm

^{3}, T

_{h}= 50 °C, Φ = 3 mW).

**Figure 18.**Temporal evolution of stagnation zone for PTFE substrate thickness. (T

_{h}= 50 °C, Φ = 3 mW, t = 1 s, θ = 90°).

Grid 1 29,261 Elements | Grid 2 35,557 Elements | Grid 3 41,034 Elements | |
---|---|---|---|

TH (Φ = 40 mW) | |||

V (mm^{3}) | 0.7239 | 0.7219 | 0.7235 |

T (°C) | 33.061 | 33.161 | 33.080 |

m_{ev} (g·m^{−2}·s^{−1}) | 3.9119 | 3.9045 | 3.8955 |

BH (T_{h} = 50 °C) | |||

V (mm^{3}) | 0.3211 | 0.3282 | 0.3271 |

T (°C) | 45.532 | 45.738 | 45.732 |

m_{ev} (g·m^{−2}·s^{−1}) | 2.3767 | 2.2721 | 2.2824 |

TBH (T_{h} = 50 °C, Φ = 3 mW) | |||

V (mm^{3}) | 0.3193 | 0.3151 | 0.3184 |

T (°C) | 47.683 | 47.708 | 47.692 |

m_{ev} (g·m^{−2}·s^{−1}) | 3.2700 | 3.2499 | 3.2213 |

λ (W·m^{−1}·K^{−1}) | ρ (kg·m^{−3}) | C_{p} (J·kg^{−1}·K^{−1}) | |
---|---|---|---|

Glass | 1.38 | 2203 | 703 |

PTFE | 0.25 | 2200 | 1010 |

**Table 3.**Evolution of stagnation zone with substrate thickness (t = 1 s, θ = 90°, V

_{0}= 1 mm

^{3}, T

_{h}= 50 °C, Φ = 3 mW).

e (mm) | 0.4 | 0.6 | 0.8 | 1 |

X_{SZ} (mm) | 0.837 | 0.762 | 0.576 | 0.240 |

(X_{i+1} − X_{i})/X_{i+1} | - | 8.96% | 24.41% | 58.33% |

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

© 2022 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**

Foudhil, W.; Aricò, C.; Perré, P.; Ben Jabrallah, S.
Use of Heating Configuration to Control Marangoni Circulation during Droplet Evaporation. *Water* **2022**, *14*, 1653.
https://doi.org/10.3390/w14101653

**AMA Style**

Foudhil W, Aricò C, Perré P, Ben Jabrallah S.
Use of Heating Configuration to Control Marangoni Circulation during Droplet Evaporation. *Water*. 2022; 14(10):1653.
https://doi.org/10.3390/w14101653

**Chicago/Turabian Style**

Foudhil, Walid, Costanza Aricò, Patrick Perré, and Sadok Ben Jabrallah.
2022. "Use of Heating Configuration to Control Marangoni Circulation during Droplet Evaporation" *Water* 14, no. 10: 1653.
https://doi.org/10.3390/w14101653