Condensed and Melting Droplet Behavior on Superhydrophobic Surfaces [1st ed.] 9789811584923, 9789811584930

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Springer Theses Recognizing Outstanding Ph.D. Research

Fuqiang Chu

Condensed and Melting Droplet Behavior on Superhydrophobic Surfaces

Springer Theses Recognizing Outstanding Ph.D. Research

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Fuqiang Chu

Condensed and Melting Droplet Behavior on Superhydrophobic Surfaces Doctoral Thesis accepted by Tsinghua University, Beijing, China

123

Author Dr. Fuqiang Chu School of Energy and Environmental Engineering University of Science and Technology Beijing Beijing, China

Supervisor Prof. Xiaomin Wu Department of Energy and Power Engineering Tsinghua University Beijing, China

ISSN 2190-5053 ISSN 2190-5061 (electronic) Springer Theses ISBN 978-981-15-8492-3 ISBN 978-981-15-8493-0 (eBook) https://doi.org/10.1007/978-981-15-8493-0 Jointly published with Tsinghua University Press The print edition is not for sale in China. Customers from China please order the print book from: Tsinghua University Press. © Tsinghua University Press 2020 This work is subject to copyright. All rights are reserved by the Publishers, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publishers, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publishers nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publishers remain neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Supervisor’s Foreword

With the development and maturity of nanotechnology and material science, the research and application of superhydrophobic surfaces have received numerous attentions, especially in recent years. Due to the great application potential in the fields of condensation heat transfer enhancement, ice and frost suppression, power device and chip cooling, dehumidification, mist water collection, self-cleaning, seawater desalination, droplet transport, etc., the self-propelled droplet movements on superhydrophobic surfaces have become a frontier and hot issue in international scientific research. For example, the condensed droplet jumping phenomenon induced by coalescence on superhydrophobic surfaces further enhances the condensation heat transfer on the shoulders of the hydrophobic surfaces (compared to the film condensation on hydrophilic surfaces, the heat transfer coefficient of droplet condensation on hydrophobic surfaces has already been improved by one to two orders of magnitude due to the efficient gravity-driven droplet removal mechanism), as the droplet jumping is able to self-clean microscale droplets, exposes large area of clean surface and achieves continuous condensation. The self-propelled droplet jumping phenomenon on superhydrophobic surfaces also has huge potential in suppressing the condensation frosting. Generally, the condensation frosting undergoes stages including condensation nucleation, droplet growth, droplet coalescence, ice nucleation, droplet freezing and freezing propagation. While the droplet jumping phenomenon makes it possible that the supercooled droplets depart from the surface before ice nucleation, it also increases the distance between adjacent droplets, resulting in a reduced freezing propagation velocity. In addition to the condensed droplets, the melting droplets on superhydrophobic surfaces present various self-propelled movements similarly, such as the droplet rotating, the droplet jumping and the droplet sliding. All these dynamic movements facilitate the removal of meltwater, which improves the comprehensive anti-frosting performance of the superhydrophobic surface. Under the background of the application of superhydrophobic surfaces in the fields of condensation enhancement, anti-frosting, hot-spot cooling in electronic chips, dehumidification, water harvesting, self-cleaning, seawater desalination, etc., the present thesis focuses on the general fundamental problem: the dynamic droplet v

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Supervisor’s Foreword

behaviors on the superhydrophobic surface and their physical mechanisms. The key point of this problem is actually the correlation between the droplet behavior and the property of the superhydrophobic surface, mainly the micro-nanostructure and the wettability, i.e., the contact angle and its hysteresis. It is this issue which limits the further development and practical application of superhydrophobic surfaces. The present thesis devotes to solve this problem, in other words, characterizes the dynamic droplet behaviors on superhydrophobic surfaces and reveals their physical mechanisms, and clarifies the relationship between the droplet behavior and the property of the superhydrophobic surface. The present thesis involves multidisciplinary theory, such as fluid mechanics, thermodynamics, transformation kinetics, surface and interface science, heat and mass transfer rule, nanotechnology and material science. From the point of science, the research results in the present thesis are expected to expand the scientific frontier and promote the formation and development of new academic growth points among interdisciplinary subjects. From the aspect of engineering application, the research results in the present thesis could promote the application of superhydrophobic surfaces in multi-industrial fields, provide guidance for the development of new technologies such as condensation enhancement, anti-frosting, defrosting, anti-corrosion, anti-microbial and flow drag reduction, and finally contribute to the sustainable development of the national economy and society. Beijing, China September 2020

Prof. Xiaomin Wu

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Research Background and Proposal of Topics . . . 1.1.1 Condensation . . . . . . . . . . . . . . . . . . . . . . 1.1.2 Frosting and Icing . . . . . . . . . . . . . . . . . . 1.1.3 Proposal of Topics . . . . . . . . . . . . . . . . . . 1.2 Research Status . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Fabrication of Superhydrophobic Surfaces . 1.2.2 Condensation and Droplet Behaviors on Superhydrophobic Surfaces . . . . . . . . . . . 1.2.3 Frost/Ice Melting and Droplet Behaviors on Superhydrophobic Surfaces . . . . . . . . . 1.2.4 Summary of Research Status . . . . . . . . . . 1.3 Research Contents of Present Work . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2 Experimental System and Superhydrophobic Surfaces . . . . . . . . . 2.1 Experimental System and Data Processing . . . . . . . . . . . . . . . . 2.1.1 Overview of Experimental System . . . . . . . . . . . . . . . . 2.1.2 Data Processing Methods . . . . . . . . . . . . . . . . . . . . . . . 2.2 Fabrication and Characterization of Superhydrophobic Surfaces . 2.2.1 Fabrication Methods of Superhydrophobic Surfaces . . . . 2.2.2 Al-Based Superhydrophobic Surfaces . . . . . . . . . . . . . . 2.2.3 Cu-Based Superhydrophobic Surfaces . . . . . . . . . . . . . . 2.3 Selection of Superhydrophobic Surfaces for Experiments . . . . . 2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3 Behaviors of Condensed Droplets on Superhydrophobic Surfaces . 3.1 Experimental Surfaces and Conditions . . . . . . . . . . . . . . . . . . . 3.2 Condensed Droplet Behaviors on Superhydrophobic Surfaces . . 3.2.1 Immobile Droplet Coalescence . . . . . . . . . . . . . . . . . . .

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3.2.2 Self-propelled Droplet Jumping . . . . . . . . . . . . . . . . . . 3.2.3 Self-propelled Droplet Sweeping . . . . . . . . . . . . . . . . . . 3.3 Statistics of Condensed Droplet Behaviors on Superhydrophobic Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Critical Conditions for Self-propelled Droplet Behaviors . . . . . . 3.4.1 Theoretical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Minimum Critical Droplet Radius . . . . . . . . . . . . . . . . . 3.4.3 Critical Ratio of Droplet Radius . . . . . . . . . . . . . . . . . . 3.4.4 Critical Static Contact Angle . . . . . . . . . . . . . . . . . . . . 3.5 Effect of Self-propelled Droplet Behaviors on Droplet Growth . 3.5.1 Droplet Diameter Distribution . . . . . . . . . . . . . . . . . . . . 3.5.2 Average Droplet Diameter . . . . . . . . . . . . . . . . . . . . . . 3.5.3 Surface Coverage Fractions . . . . . . . . . . . . . . . . . . . . . 3.5.4 Effects of Working Conditions . . . . . . . . . . . . . . . . . . . 3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Numerical Simulations of Multi-droplet Coalescence-Induced Jumping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Simulation Objects and Conditions . . . . . . . . . . . . . . . . . . . . . 4.2 Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Control Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Computational Domain, Boundary Conditions, and Grids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3 Energy Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Model Validation—Two-Droplet Coalescence-Induced Jumping . 4.4 Multi-droplet Coalescence-Induced Droplet Jumping . . . . . . . . . 4.4.1 Effect of Coalesced Droplet Number . . . . . . . . . . . . . . . 4.4.2 Effect of Droplet Position Distribution . . . . . . . . . . . . . 4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Dynamic Melting of Freezing Droplets on Superhydrophobic Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Experimental Surfaces and Conditions . . . . . . . . . . . . . . . . . . . 5.2 Freezing of Condensed Droplets on Superhydrophobic Surfaces . 5.3 Self-propelled Behaviors During Melting Process of Freezing Droplets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Melting Droplet Rotating . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Melting Droplet Jumping . . . . . . . . . . . . . . . . . . . . . . . 5.3.3 Melting Droplet Sliding . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Effects of Self-propelled Melting Droplet Behaviors on Surface Coverage Fraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Summary of This Chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Contents

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6 Meltwater Evolution During Defrosting on Superhydrophobic Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Experimental Surfaces and Conditions . . . . . . . . . . . . . . . . . 6.2 Meltwater Evolution on Superhydrophobic Surfaces . . . . . . . 6.3 Edge Curling Phenomenon of Meltwater Films . . . . . . . . . . 6.4 Non-breaking Phenomenon of Chained Droplets . . . . . . . . . . 6.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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7 Relation Between Surface Wettability and Droplet Behaviors, and Hysteresis Number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Morphologies and Behaviors of Condensed Droplets and Melted Droplets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.1 Morphologies and Behaviors of Condensed Droplets . . . 7.1.2 Morphologies and Behaviors of Melted Droplets . . . . . . 7.2 Relation Between Surface Wettability and Droplet Behaviors . . 7.3 Hysteresis Number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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8 Conclusions and Outlooks . . . . . . . . . . . . . 8.1 Main Conclusions in the Present Work 8.2 Innovations in the Present Work . . . . . 8.3 Outlooks for Future Research . . . . . . .

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Nomenclature

A Bo CA CAH D d E F F f I k L M m n N Oh p r rw RA RH S T t U u

Area, m2 Bond number Static contact angle, o Contact angle hysteresis, o Diameter of droplet, m Diameter of liquid column, m Energy, J Force, N Source term of surface tension Fraction of contact area Moment of inertia, kg m2 Correction factor Droplet distance, m Moment, N m Mass, kg Number of coalesced droplets Normal vector Ohnesorge number Pressure, pa Radius of droplet curvature, m Surface roughness factor Rolling angle, o Relative humidity, % Moving distance, m Temperature, °C or K Time, s or min Velocity of droplet jumping, m/s Velocity along the x-axis, m/s

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Nomenclature

Volume, m3 Velocity along the y-axis, m/s Velocity vector Width of droplet base, m Velocity along the z-axis, m/s Diameter ratio of small droplet to large droplet

V v v W w x

Greek Letters a b c h j l p q x / D

Volume fraction Index of power function Surface tension, N/m Contact angle, o Interface curvature Dynamic viscosity, Pa s PI value Density, kg/m3 Rotation speed, r/s Dissipation function Amount of change

Subscripts 0 a r C W Y gs lg sl ave bri cah cri gra ini jump

Static contact angle Advancing contact angle Receding contact angle Droplet under Cassie Equation Droplet under Wenzel Equation Droplet under Young’s Equation Gas–solid interface Liquid–gas interface Solid–liquid interface Average value Liquid bridge between droplets Contact angle hysteresis Critical value Gravity Initial value Jumping

Nomenclature

kin min neg sur tpl vis

Kinetic energy Minimum value Negative Surface Triple-phase line Viscosity

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Chapter 1

Introduction

1.1 Research Background and Proposal of Topics 1.1.1 Condensation Back to Pre-Qin Period in China (more than 2000 years ago), people have summarized the yearly routine of season, climate, and phenology changes via observing solar movements, and set the twenty-four solar terms on the calendar. After ‘Li-Qiu (the beginning of fall)’, one of the twenty-four solar terms, the weather turns cold, followed by ‘Bai-Lu’ and ‘Han-Lu’—the other two terms. The meaning of ‘BaiLu’ and ‘Han-Lu’ is essentially about the condensation: on cold surfaces (ground, surfaces of plants) whose temperatures are lower than the dew point, supersaturated vapor will condense and appear as liquid droplets [1–4]. Apart from nature, the phenomenon of condensation widely exists in engineering field such as energy and electrical power system, refrigeration, water recycling, petrochemical, etc. [5–10], and has significant impact on corresponding industry processes and the equipment involved. For instance, in thermal power plant, the exhaust gas after the gas turbine can be condensed into liquid water in the condenser and therefore recycled to the system (Fig. 1.1a), and the condensation within this system has a great impact on the safe and economic operation of the plant [5]. For air-conditioning system, the working gas condenses and takes away the heat via phase change, and the condensation process greatly affects the refrigeration coefficient as well as the cooling inside the room [6]. In the water collection tower, vapor from the air condenses on the cooled surface (Fig. 1.1b), and the condensed water can be recycled to ameliorate the water shortage in arid areas, where the amount of water collected is directly determined by the process of condensation [8]. In the distillation tower, the rising steam in the top of the tower goes into the condenser, and the cooled liquid flows back to the top of the tower, or flows out as distilled liquid. Similarly, the process of condensation has a direct impact on the efficiency of component separation [10].

© Tsinghua University Press 2020 F. Chu, Condensed and Melting Droplet Behavior on Superhydrophobic Surfaces, Springer Theses, https://doi.org/10.1007/978-981-15-8493-0_1

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1 Introduction

Fig. 1.1 Examples of condensation and the equipment involved in engineering: a condenser and pipes for condensation; b water collection tower and condensation rope network

1.1.2 Frosting and Icing In the twenty-four solar terms mentioned in the last section, ‘Han-Lu’ is followed by ‘Shuang-Jiang’; also, there is a famous verse in ‘The Books and Songs’—the earliest poetry collection in ancient China: ‘Green reeds are thick and dense, white dews become frost thence’. The examples above both refer to the phenomenon of frosting and icing in nature. When the temperature of the cooled surface is lower than the freezing point of water (0 °C), the condensed droplets freeze and become ice droplets; frost crystals grow on those droplets, and eventually form the frost [11– 14]. In addition, under the condition of extremely low surface temperature (at least lower than −30 °C for common aluminum surfaces) and high relative humidity in air, desublimation occurs, which means the vapor directly phase change into the ice [15, 16]. This is also a widespread phenomenon in engineering fields such as energy and electrical power, refrigeration, aeronautics, etc., however, frosting and icing have an adverse effect on most of the engineering processes, accompanied by severe consequences [9, 17–21]. For instance, in energy and electrical power industry, blade surfaces of wind turbine are prone to be covered by frost/ice (Fig. 1.2a), which can alter the aerodynamic characteristics of blades and finally suppress the efficiency of electricity generation and irritate the normal operation [17]. Surfaces of power lines are also prone to have frost/ice deposition (Fig. 1.2b), which leads to the tower damaging, tilting of insulating strings, wire sagging, etc., and the electricity blackout and grounding accidents may occur if massive frost or ice accumulates [20]. In refrigeration industry, especially in cold and humid zones, the vaporizer in heat pump system is subject to frost/ice (Fig. 1.2c), which increases thermal resistance and blocks the flow channel, and thus interrupts the operation of vaporizer, decreases the heating coefficient of heat pump system, and hinders the heat generation of the system. In aeronautics, frosting and icing are common phenomenon that brings huge hazards to the industry [18, 19]. Aircrafts accumulate frost/ice on crucial parts (such as wings, horizontal tails, propellers) when passing through cold and humid area

1.1 Research Background and Proposal of Topics

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Fig. 1.2 Frosting and icing phenomenon in industry: a blades of wind turbine; b power lines; c vaporizer; d wings of aircraft

or encountering rainy and snowy days (Fig. 1.2d). This alters the characteristics of the aircraft, increases the weight, deteriorates the flight performance, restrains the aircraft operation, and finally leads to instrument errors or even accidents such as aircraft crashes [9, 21].

1.1.3 Proposal of Topics As an efficient way of phase-change heat transfer that can help to recycle water, condensation has been applied to multiple manufacturing processes in industry. Therefore, people hope to exploit the technology of enhancing condensation with high efficiency. Unlike the condensation, frosting/icing pose threats to most of the industrial processes; however, they cannot be suppressed totally. In view of that, we need to melt or remove the frost/ice accumulated on the equipment; thus, there is a demand of increasing the efficiency of frost/ice melting and removal. Since the processes of condensation, frosting, and icing are greatly determined by surface characteristics (microstructures and wettability), controlling droplet behaviors by designing surface structures and wettability without imposing extra energy is an efficient and promising way to achieve enhanced condensation as well as frost/ice melting. For the condensation, the dominant source of thermal resistance comes from the condensate itself [1, 2]. Compared with filmwise condensation on hydrophilic surfaces, condensed droplets cannot entirely cover hydrophobic surfaces by dropwise condensation, and those condensates can be removed via the gravity. Thus, the

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1 Introduction

Fig. 1.3 Superhydrophobic surfaces in nature: a lotus leaf; b legs of water strider; c cicada wings

heat transfer coefficient of surfaces with dropwise condensation is higher than that with filmwise condensation (1–2 magnitudes) [1, 2, 22]. In other words, choosing the appropriate surface can help to achieve dropwise condensation and improve the mobility of condensates, whose fast removal is conducive to enhancing condensation. As for frost/ice melting, the liquid generated by phase change also needs to have a high mobility that can be quickly removed, instead of sticking on the surface, being hard to remove, and leading to the second freezing that is far more hazardous. That is to say, a proper surface can help to improve the mobility of liquid from melted frost/ice, decrease the adhesion of droplets, and enhance the efficiency of frost/ice melting. To summarize, a proper surface creates a high mobility of droplets on it, which is highly beneficial for the efficiency of both condensation and frost/ice melting. Naturally, this ‘proper surface’ refers to the superhydrophobic surface. Superhydrophobic surfaces are the surfaces with extremely high static contact angle (larger than 150°) and low contact angle hysteresis (less than 10°), and droplets on them can exhibit high mobility [23–27]. There are various superhydrophobic surfaces in nature, such as lotus leaf (Fig. 1.3a), water strider legs (Fig. 1.3b), cicada wings (Fig. 1.3c), etc., and the surface of lotus leaf is an exemplar. Water droplets cannot wet that surface, but they can roll on it and carry away the dust. Hence, the lotus leaf has self-cleaning effect that is called Lotus Effect [28–30]. Given that superhydrophobic surfaces have a huge potential in enhancing the condensation and improving the efficiency of frost/ice melting, the present work will firstly focus on the following topics: (1) How to fabricate superhydrophobic surfaces with excellent performance which can be put into application? (2) What are the condensation characteristics of superhydrophobic surfaces, and is there any special behavior of condensates that can further enhance the condensation? (3) What are the frost/ice melting characteristics of superhydrophobic surfaces, and is there any special behavior that helps to further improve the efficiency of melting?

1.2 Research Status Under the background of enhancing the condensation and improving the efficiency of frost/ice melting, for the three topics proposed in Sect. 1.1.3, this section will

1.2 Research Status

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make literature review in three aspects: fabrication of superhydrophobic surfaces, condensation characteristics and droplet behaviors on superhydrophobic surfaces, and frost/ice melting characteristics and droplet behaviors on superhydrophobic surfaces. The status of relevant research will be understood, with a summary of drawbacks of the current research, which helps to introduce the specific research contents of the present work.

1.2.1 Fabrication of Superhydrophobic Surfaces Wettability is an important characteristic of solid surface, and the word ‘superhydrophobic’ is used to describe it. Apart from ‘superhydrophobic’, there are other descriptive words, such as hydrophobic, hydrophilic, and superhydrophilic, and the difference between them is quantified by the value of contact angle [31, 32]. Superhydrophobic surfaces are the ones that possess a static contact angle larger than 150° and a contact angle hysteresis lower than 10° (or rolling angle lower than 5°) [23–27]. As early as the year 1805, Thomas Young started to study the wetting phenomenon when water droplets and ideal smooth surface are in contact, and deduced the relation between the contact angle of ideal smooth surface and surface tension, which is the well-known Young’s equation [33]: cos θY =

γgs − γsl γlg

(1.1)

In the equation above, θ Y is the static contact angle of ideal smooth surface, which is also called static contact angle or Young’s contact angle; γ represents surface tension, and subscripts gs, sl, and lg refer to gas–solid, solid–liquid, and gas–liquid interfaces, respectively. We can see from Young’s equation that the smaller the surface tension γ gs , the lower the surface energy of solid, which creates larger static contact angle. Nonetheless, even if we utilize single molecule of long-chain fluorosilane (contains –CF3 group)—the known substance with lowest surface energy—to modify smooth surfaces, the static contact angle on the surface cannot exceed 120° [34], which is much less than the minimum contact angle required for superhydrophobic surfaces. Since that Young’s equation can only apply to ideal smooth surface, actual surfaces are always rough, Wenzel studied the effect of surface roughness on wetting in 1936. Surface roughness factor (r w ) was introduced to improve Young’s equation, and the modified equation became Wenzel’s equation [35]: cos θW = rw

γgs − γsl = rw cos θY γlg

(1.2)

In the equation above, θ W is the static contact angle of actual rough surface; surface roughness factor is defined to be the ratio of the actual solid surface area to

6

1 Introduction

the projected area, whose value is constantly larger than 1. We can see from Wenzel’s equation that for surfaces whose intrinsic contact angle exceeds 90°, the larger the surface roughness factor, the larger the static contact angle on rough surfaces. That is to say, Wenzel’s equation provides us a thought of fabricating superhydrophobic surfaces: to design structures as rough as possible on hydrophobic surfaces, until the value of static contact angle meets the requirement for superhydrophobic surfaces. Based on this thought, researchers have fabricated superhydrophobic surfaces as early as the year 1988, when Morra et al. [36] treated PTFE surface with oxygen plasma to obtain rough surface structures, and the surface became superhydrophobic after 5-min treatment. In 1996, Shibuichi et al. [37, 38] fabricated superhydrophobic surfaces by creating rough fractal structures with molten alkyl ketene dimer. In 2003, Erbil et al. [39] dissolved isotactic polypropylene in a mixed solvent of p-xylene and methyl ethyl ketone, and fabricated the isotactic polypropylene surface with porous structures via the evaporation of the solvent. The static contact angle of this surface was measured to be larger than 160°. In addition to the static contact angle, contact angle hysteresis (or rolling angle) is also an important indicator of surface superhydrophobicity, which needs to be considered for a comprehensive judgment of superhydrophobic surfaces. However, in the description of Wenzel’s equation, droplets have a complete contact with the solid surface, and the rough structures are totally wetted by the liquid, which means the surface has a relatively strong adhesion to droplets. Hence, superhydrophobic surfaces fabricated by the thought from Wenzel’s equation may not be satisfying in terms of contact angle hysteresis. Also, considering the dependence on the intrinsic contact angle of the material, from the view of application (most of the metals used in engineering are hydrophilic materials), that thought is rather confined. In 1944, Cassie and Baxter modified Young’s equation in a way different from Wenzel. They took the effect of diverse chemical compositions on the surface into account and proposed Cassie–Baxter equation [40]: cos θC = f 1 cos θY1 + f 2 cos θY2

(1.3)

In the equation above, θ C is the static contact angle of the surface; θ Y1 and θ Y2 are the intrinsic contact angle of composition 1 and 2 on the surface; f 1 and f 2 are the area fraction of composition 1 and 2, with a sum equaling to 1. Due to the unchanged existence of air in space between surface structures, we can regard the solid surface material as composition 1, and air as composition 2. In that way, Cassie simplified Cassie–Baxter equation for rough surfaces that have microstructures filled with air (since the intrinsic contact angle of air is 180°), which is also called Cassie’s equation [41]: cos θC = f cos θY + f − 1

(1.4)

In the equation above, f is the area fraction and θ Y is the intrinsic contact angle of the solid material. We can see that the static contact angle increases with the decrease of area fraction of solid material on the surface. Thus, from Cassie’s equation, the

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area fraction of solid should be decreased to a value as low as possible for superhydrophobic surfaces. In Cassie’s equation, liquid is unable to wet microstructures inside the surface, but instead makes as much contact as possible with air, which leads to an extremely low surface adhesion to the liquid and a low contact angle hysteresis that is convenient for droplets to roll on. In other words, a real superhydrophobic surface should maintain the state as described in Cassie’s equation, and the fabrication of superhydrophobic surfaces ought to start from the perspective of Cassie’s equation. That being the case, what structures can help to maintain droplets in the state described in Cassie’s equation? Nature is the best teacher for us. With the technology development of scanning electron microscope (SEM), in 1997, Barthlott and Neinhuis [42, 43] captured SEM photos of superhydrophobic surface structures on plant leaves of Nelumbo nucifera and Colocasia esculenta in nature for the first time. The leaves are not only distributed with micron-sized mastoid structures, but also densely covered with nanosized fluff on the mastoid structures [42, 43]. Besides the two-tier micro-nanostructures, these leaves are covered with waxy layer of low surface energy. The hierarchical structures collaborate with the waxy layer, and they together contribute to the superior superhydrophobicity of the surface (extremely large static contact angle and low contact angle hysteresis) [23]. Therefore, based on Cassie’s equation and the insight gained from the nature, the second thought of fabricating superhydrophobic surfaces appears: Create micro- and nanohierarchical structures of two scales or more on the surface first, and then modify the surface via low-surface-energy substance. Following the second thought of fabricating superhydrophobic surfaces, researchers have exploited various methods, including sol–gel [44], galvanic erosion [45], solution immersion [46], chemical deposition [47], chemical vapor deposition [48], chemical etching [49], anodization [50], laser etching [51], electrospinning [52], template [53], etc., and have successfully fabricated numerous superhydrophobic surfaces with excellent performance. In 2000, Tadanaga et al. [44] utilized sol–gel method to cover alumina on the glass substrate, created flower-like rough structures, and finally modified the surface by fluorosilane to achieve a static contact angle as large as 165°. Wu et al. [45] fabricated copper phosphate nanosheets on the copper surface by galvanic erosion method and further designed micro-nanohierarchical structures; the static contact angle and the rolling angle are 155° and 3° after the modification by dodecyl mercaptan that has a low surface energy. Jiang et al. [46] immersed the copper plate into myristic acid solution; chemical reaction occurred in the solution, which generated multi-tier grass-like micro-nanostructures on the copper surface that had a static contact angle of 162°, and the contact angle hysteresis was only 2°. Larmour et al. [47] used tetrachloroauric acid and silver nitrate solution to react with zinc respectively for the deposition of micro-nanostructures on the zinc surface; after being modified by fluorosilane, both of them exhibited superhydrophobicity, reached a static contact angle even as large as 173°, with the rolling angle less than 1°. Liu et al. [48] utilized gold-catalyzed chemical vapor deposition method to deposit zinc oxide film with two-tier micro-nanostructures on the sapphire substrate, and finally obtained the superhydrophobic surface with a static contact

8

1 Introduction

angle of 164°. Qian and Shen [49] used chemical etching method on aluminum, copper and zinc surfaces to create multi-tier micro-nanostructures, and static contact angles exceeded 150°, with rolling angles less than 10° for all types of metallic surfaces. Wang et al. [50] treated the aluminum surface with anodization and plasma orderly and fabricated porous and rough micro-nanostructures on the surface; the static contact angle reached 158° after fluorosilane modification. Song et al. [54] also treated the aluminum surface with anodization to generate labyrinthine micronanostructures on the surface, and finally fabricated the superhydrophobic surface with a static contact angle of 152° by octadecenoic acid that has a low surface energy. The second thought of fabricating superhydrophobic surfaces is not restricted to certain types of materials, so that it can be applied to various metallic substrates for the fabrication of superhydrophobic surfaces. In addition to the examples mentioned above, numerous representative surfaces with metallic substrates have been fabricated, and Table 1.1 lists out a part of them.

1.2.2 Condensation and Droplet Behaviors on Superhydrophobic Surfaces Before the fabrication of superhydrophobic surfaces, researchers have discovered that droplets condensed on hydrophobic surfaces cannot entirely cover the surface, and they can achieve self-removal via gravity, which leads to an increase of 1– 2 magnitudes in heat transfer coefficient compared to filmwise condensation on hydrophilic surfaces [1, 2, 63]. The mobility of condensed droplets on superhydrophobic surfaces should be even better than the ones on hydrophobic surfaces, which helps to further enhance the condensation. Therefore, after the massive fabrication of superhydrophobic surfaces, their condensation characteristics have naturally become the hot spot of research [64–67]. One of the representatives was the research by Chuan-Hua Chen in Duke University. In 2007, Chen et al. [68] fabricated micron-sized square pillars on the silicon surface via mechanical methods, and then deposited carbon nanotubes on the pillar surface to create nanostructures, finally achieved superhydrophobicity after hexadecane modification. They conducted condensation experiments on the fabricated horizontal surface and discovered that droplets can maintain in Cassie state and is easy to roll off, as well as leaving a large area of clean surface. Thus, they considered superhydrophobic surfaces successful in continuous dropwise condensation, which is conducive to the enhanced condensation heat transfer. Subsequently, they continued to dig into the mechanism of droplet removal on superhydrophobic surfaces. In 2009, Boreyko and Chen [69] observed the phenomenon from experiments that the droplet removal on the superhydrophobic surface is not triggered by external force, but the natural coalescence of droplets themselves. With the help of the extremely large static contact angle on superhydrophobic surfaces, the area of gas–liquid interface shrinks when droplets coalesce, and the released surface energy can be transformed into kinetic energy, which drives

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Table 1.1 Examples of metal-substrate superhydrophobic surfaces in recent years Year

Authors

Substrate material

Method

CA/RA

Fabrication time

Remarks

2014

Yuan et al. [55]

Copper

Template-etching

160°/3°

>24 h

Complex process, long fabrication time

2016

Guo et al. [56]

Copper

Chemical etching

157°/4°

>1 h

Demanding conditions of fabrication

2015

Liu et al. Aluminum [57]

Anodization

171°/6°

>3 h

Demanding conditions of fabrication

2015

Wang et al. [58]

Aluminum

Chemical etching

160°/3°

>2 h

Hazardous

2014

Qi et al. [59]

Zinc

Chemical etching

161°/3°

>3 h

Complex process, high cost

2015

Li et al. [60]

Zinc

Chemical etching-hydrothermal

152°/10°

>10 d

Complex process, long fabrication time

2015

Liu et al. Magnesium [61]

Electrochemical deposition

159°/2°

>1 h

Demanding conditions, high cost

2016

Chen et al. [62]

Laser etching

157°/1°

>3 h

High cost

Stainless steel

the coalesced droplet to jump out from the surface. The observed critical diameter of droplets in droplet jumping behavior was approximately 10 µm, and the size of droplets should reach this value before jumping. They also measured the velocity of droplet jumping and claimed it to be comparable to 0.2 times the inertia-capillary scaling speed from energy conservation calculation and dimensional analysis [69]. Boreyko and Chen reported this phenomenon for the first time and named it as ‘selfpropelled droplet jumping’ (the droplet jumping behavior they captured is shown in Fig. 1.4) [69]. Droplet jumping can succeed in the self-cleaning of droplets on the surface, which is believed to be promising in application in the fields of enhanced condensation heat transfer, self-cleaning, anti-frosting/icing, etc. Because of that, since the report of droplet jumping phenomenon, it has attracted attention from numerous researchers and initiated a series of follow-up studies.

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1 Introduction

Fig. 1.4 Droplet jumping phenomenon of condensates on the superhydrophobic surface [69]. Reproduced with permission. Copyright 2009, The American Physical Society

(a) Experimental research Some researchers have conducted more detailed research of droplet jumping by adopting more advanced equipment and methods to further illustrate the mechanism of this phenomenon and understand the behavioral characteristics of droplet jumping. Lv et al. [70] conducted experiments directly on the treated lotus leaf and witnessed another triggering mechanism of droplet jumping: The coalesced droplet jumped and fell back onto the surface, contacted with another droplet on the surface, and jumped again as a larger coalesced droplet. This mechanism was thought to be helpful in achieving continuous droplet jumping. Yanagisawa et al. [71] used two high-speed camera to capture top-view and side-view images of droplet jumping simultaneously. Experimental observations displayed that droplet jumping was not only triggered by the coalescence of two droplets, but also by multiple droplets; detailed information about the diameter, velocity, and height of droplet jumping was also recorded in their research. Kim et al. [72] also recorded more than 1000 incidents of droplet jumping from the top and the side, and summarized three triggering mechanisms of droplet jumping: two-droplet coalescence, multi-droplet coalescence, and coalescence from the droplet falling back onto the surface, which confirmed the conclusion drawn by Lv and Yanagisawa. Lv et al. [73] continued to research multi-droplet-induced droplet jumping, and the statistics exhibited the relation between droplet jumping probability and the number of droplets coalescing, as well as the relation between critical droplet diameter and the number of coalesced droplets. Results of the research indicated three and four droplets coalescing and jumping to be the most probable incidents, and multi-droplet-induced jumping could break through the critical droplet diameter of 10 µm. Chen et al. [74] also calculated the relation between droplet jumping probability and the number of coalesced droplets, and reached a consistent conclusion with Lv et al.’s [73]. Cha et al. [75] exploited the mobile-focus-plane imaging technology, which can trace the trajectory of the jumped droplet in one camera. They observed droplet jumping phenomenon on the surface structure within a range of 10 nm–1 µm and droplet diameter within a range of 6–320 µm, and collected massive data of droplet jumping velocity. The research of Cha et al. [75] indicates that the critical diameter of droplet jumping can be influenced by multiple factors such as size of surface structures, interactions on non-ideal surfaces, droplet

1.2 Research Status

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evolution morphology, etc. They captured a diameter of 500 nm for coalesced droplet jumping on the superhydrophobic surface covered with carbon nanotubes [76], which further reduced the scale of droplets needed for jumping. There have been researchers attempting to control or enhance droplet jumping by various methods to achieve applications that are more promising. Chen et al. [77] designed pyramid micro-nano two-tier structures on superhydrophobic surfaces to enhance droplet jumping, and experiments shown that the volume of removed droplets via jumping was 450% more than the volume on normal superhydrophobic surfaces. Similarly, Wang et al. [78] utilized triangle pillar structures to enhance droplet jumping. Tian et al. [79] fabricated superhydrophobic surface covered with densely stacked zinc oxide nanoneedles that was able to enhance droplet jumping and effectively reduce droplet diameter; statistics shown that over 80% of the droplets on the surface were the ones whose diameters were less than 10 µm. He et al. [80] designed superhydrophobic surfaces with micro-nanoporous structures, which helped to achieve the control of droplet coalescence and the enhancement of droplet jumping. Miljkovic et al. [81] imposed electric field to the condensation area where the jumping droplets captured positive charges, so that the droplet jumping phenomenon could be manipulated and enhanced via the control of electric field. Liu et al. [82] designed superhydrophobic surface with anisotropic surface structures and regulated the direction of droplet jumping by gradient surface energy. In addition, there are some other researchers showing direct interests of whether droplet jumping phenomenon can help to enhance condensation heat transfer. Dietz et al. [83] used environmental scanning electron microscope to observe the condensation on silicon-substrate superhydrophobic surface with copper hydroxide structure, and calculated the number of condensed droplets as well as diameter distribution within the field of view. Their research showed that coalesce-induced droplet jumping could accelerate the removal of droplets from the surface, and decrease the droplet diameter on the surface, which was highly conducive to improving condensation heat transfer coefficient. Miljkovic et al. [84] treated the copper tube surface with leafshaped copper oxide microstructures to perform condensation experiments outside the tube. The frequent occurrence of droplet jumping was observed and compared to the condensation of ordinary hydrophobic copper tube surface. Under the identical working conditions, the heat flux outside the superhydrophobic copper tubes was measured to be 25% higher than hydrophobic copper tubes, and the condensation heat transfer coefficient was approximately 30% higher. Miljkovic et al. [81, 85] imposed electric field on the copper tube surface to further enhance droplet jumping, and the corresponding results were measured. In comparison with copper tubes without electric field, when the external field was 100 V/cm, the condensation heat transfer coefficient could be further increased by 50%. Apart from the enhancement of condensation heat transfer, researchers also explored potential applications of droplet jumping in engineering fields, such as surface self-cleaning and anti-frost/icing. Wisdom et al. [86] conducted condensation experiments on the superhydrophobic surface of cicada wings and observed droplet jumping phenomenon, laying emphasis on the self-cleaning effect brought by droplet jumping. Their research indicated that droplet jumping could not only clear

12

1 Introduction

free pollutant particles on the surface in the process of jumping, but also separate pollutant particles that adhere on the surface. In addition, scattered pollutant particles can be combined during droplet coalescence, and then brought away by jumped droplets. Watson et al. [87] studied the droplet jumping phenomenon of condensates on the lotus leaf surface, and claimed that due to the existence of droplet jumping, even without the brush of rain, leaves can still remove pollutant particles, bacteria and fungi by self-cleaning mechanism. Boreyko and Collier [88] investigated the value of droplet jumping on superhydrophobic surfaces in the field of anti-frosting/icing, and the results shown that benefitting from droplet jumping on the surface, subcooled droplets could coalesce and jump away from the surface before freezing took place, which contributed to an effective delay of frosting/icing. The research from Zhang et al. [89] revealed that for a surface temperature of −15 °C, droplet jumping could delay the freezing of droplets for as long as 1 h. (b) Theoretical and numerical simulation research Since the report of droplet jumping phenomenon, scholars also employed methods of theoretical analysis to quantitatively investigate its mechanism. Wang et al. [90] established an energy model for droplet jumping at an early time, which took the effect of viscous dissipation that Boreyko and Chen did not consider initially into account. They suggested that the released surface energy from droplet coalescence was firstly dissipated by viscosity, and the remaining part was transformed into kinetic energy to drive the jumping of coalesced droplet. According to the model from Wang et al., when the diameter of the coalesced droplet is too small, viscous dissipation will be dominant, which deprives the jumping of coalesced droplet. The velocity of droplet jumping predicted by Wang et al. was consistent with the experimental value measured by Boreyko and Chen in tendency, but much larger in the magnitude of value [90]. Liu et al. [91] further set up a mathematical model that considered tripleline blocking force of motion during droplet coalescence and discussed the effect of droplet wetting state on droplet jumping. When droplets maintain in Wenzel state of high adhesion, jumping cannot be triggered; instead, droplets in Cassie state are subject to droplet jumping phenomenon. Lv et al. [70] also established an energy conservation model that contained triple-phase line blocking work induced by kinetic energy, surface energy, viscous dissipation, contact angle hysteresis, etc., and the calculation results of droplet jumping velocity fitted well to the experimental value from Boreyko and Chen. Compared with theoretical research, numerical simulation methods can reproduce the process of droplet coalescence and jumping, and obtain data that are hard to collect in experiments and theories. Therefore, scholars adopt multiple simulation methods to simulate droplet jumping phenomenon, and the research contents include energy conversion analysis, fluid dynamics analysis, influencing factors and optimization, and the possibility of nanoscale droplet jumping. For details, please refer to Table 1.2. Nam et al. [92] simulated droplet coalescence of two 30-µm-sized droplets on a superhydrophobic surface whose static contact angle was 161°; after energy conversion analysis, it was concluded that 40–60% of the surface energy was

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Table 1.2 Numerical simulation research of droplet jumping Author

Simulation method

Simulation object

Core content

Nam et al. [92, 102]

Level contour reconstruction

Flat superhydrophobic surface, coalesced jumping by two identical droplets

Energy conversion analysis

Peng et al. [93]

Lattice Boltzmann Method

Flat superhydrophobic surface, coalesced jumping by two identical droplets

Energy conversion analysis

Enright et al. [94]

Computational fluid dynamics

Flat superhydrophobic surface, coalesced jumping by two identical droplets

Fluid dynamics analysis, energy conversion analysis

Liu et al. [95]

Phase fluid

Flat superhydrophobic surface, coalesced jumping by two identical droplets

Fluid dynamics analysis

Farokhirad et al. [96]

Lattice Boltzmann method

Flat superhydrophobic surface, coalesced jumping by two identical droplets

Fluid dynamics analysis

Liu et al. [100]

Lattice Boltzmann method

Superhydrophobic surface with microstructures, coalesced jumping by two identical droplets

Fluid dynamics analysis, parameter optimization of microstructures

Cheng et al. [98]

Diffuse-interface

Flat superhydrophobic surface, coalesced jumping by two identical droplets

Influential factor analysis

Khatir et al. [103]

Volume of fluid

Flat superhydrophobic surface, coalesced jumping by two identical droplets

Influential factor analysis

Attarzadeh et al. [101]

Volume of fluid

Superhydrophobic surface with microstructures, coalesced jumping by two identical droplets

Fluid dynamics analysis, parameter optimization of microstructures

Liang et al. [104]

Molecular dynamics

Flat superhydrophobic surface, coalesced jumping by two identical droplets

Possibility of nanosized droplet jumping (continued)

14

1 Introduction

Table 1.2 (continued) Author

Simulation method

Simulation object

Core content

Gao et al. [105]

Molecular dynamics

Superhydrophobic surface with microstructures, coalesced jumping by two identical droplets

Possibility of nanosized droplet jumping

transformed into kinetic energy before the droplet jumped away from the surface. The research of Peng et al. [93] revealed that for the kinetic energy gained by the droplet, 25% of them were effective energy in the vertical direction. Enright et al. [94] suggested that only less than 6% of the surface energy released during droplet coalescence could be transformed into vertical kinetic energy that was effective. Liu et al. [95] conducted a detailed study of the evolution of droplet morphology during the process of droplet jumping via simulation and explained the mechanism of droplet jumping from the view of mechanics: the liquid bridge extended in the process of droplet coalescence, which strikes the surface, so that the reaction force was generated to bounce the droplet up. They also extracted the velocity variation of the droplet in the vertical direction and divided droplet jumping into four stages according to the morphology evolution and velocity variation of the droplet. Farokhirad et al. [96] mainly discussed inertia and viscosity effect during droplet jumping, and the results indicated that when Ohnesorge number (Oh number) [97]—the parameter characterizing the ratio of viscosity to inertia-capillary force—was larger than 0.3, viscous dissipation would be dominant and droplet jumping could not occur. Cheng et al. [98] paid attention to the influential factors of droplet jumping, and the discussion included static contact angle, contact angle hysteresis, droplet diameter, viscosity, gravity, etc. Liu et al. [99, 100] performed two-dimensional and three-dimensional lattice Boltzmann simulation of droplet jumping phenomenon on the superhydrophobic surface with microstructures, whose results of droplet jumping velocity were in good agreement with the experimental values from Boreyko and Chen. Attarzadeh et al. [101] also simulated droplet jumping phenomenon on the similar surface. The research indicated that when the scale of droplets was comparable with the scale of microstructures, relatively large impact was posed by microstructures to droplet jumping, and there was a group of optimal parameters to maximize the velocity of droplet jumping. Besides, some scholars adopted Molecular Dynamics method to simulate droplet jumping phenomenon triggered by the coalescence of nanosized droplets. For instance, Liang et al. [104] and Gao et al. [105] both simulated the coalescence of two droplets on superhydrophobic surfaces by Molecular Dynamic method, whose significance is more about exploring the possibility of nanosized-droplet-induced jumping triggered by their coalescence. However, there is still no reached conclusion whether the coalescence of nanosized droplets can trigger droplet jumping phenomenon, which needs to be proved by further experiments.

1.2 Research Status

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1.2.3 Frost/Ice Melting and Droplet Behaviors on Superhydrophobic Surfaces Since the fabrication of superhydrophobic surfaces, massive research has been conducted regarding the frosting/icing characteristics on superhydrophobic surfaces. Due to the extra-large static contact angle, superhydrophobic surfaces have extremely high energy barrier of heterogeneous nucleation, which delays the nucleation of ice [106–111]. That large static contact angle also leads to a small contact area between the droplet and the surface, and accounts for the suppression of latent heat transfer [108, 112, 113]. Meanwhile, superhydrophobic surfaces possess extra-low adhesion force that is conducive to the removal of frost/ice [114–116]. In addition, as mentioned in Sect. 1.2.2, the droplet jumping phenomenon occurred on superhydrophobic surfaces can trigger the coalescence-induced jumping of subcooled droplets prior to freezing, which further delays the freezing of droplets and suppresses the growth of frost/ice [88, 89, 117–120]. Hence, multiple researches indicated that frosting/icing could be delayed on superhydrophobic surfaces to some extent [121– 124]. However, that is only the delay, not a total inhibition. When the surface is subject to extremely low temperature and high humidity with a long time, frosting/icing still happens on the surface [121–124]. Therefore, it is important to investigate the frost/ice melting characteristics on superhydrophobic surfaces. As early as the year 2002, scholars have studied the frost/ice melting characteristics on hydrophobic surfaces [125, 126], while some other researchers paid attention to the frost/ice melting characteristics of microgroove surfaces [127–131]. However, it was not until 2013 that researchers started to study the frost/ice melting characteristics on superhydrophobic surfaces. This was a late start compared with the study of frosting/icing characteristics, and the relevant research was less. In 2013, Boreyko et al. [132] fabricated the nanostructured superhydrophobic surface and explored its frost/ice melting characteristics. The research shown that droplets generated after the melting of frost maintained in Cassie state and would roll off the surface when slightly tilting the surface (tilting angle less than 15°). Boreyko et al. [132] divided this melting process on horizontal superhydrophobic surfaces into three stages: partial melting, spontaneous dewetting, and rolling off the surface, which is displayed in Fig. 1.5. They also investigated the effect of frost layer height on frost melting process and revealed that when the layer had a large height (>2 mm), the melted frost would shrink and dehumidify into a large droplet; otherwise ( 0, self-propelled droplet behaviors can be triggered; the critical condition for self-propelled droplet behaviors occurring when this energy equals to zero, that is, E cri = 0. The amount of the surface energy released from the initial condition to the critical condition is deduced to be as follows:    E sur = πr 2 γlg 1+nx 2 − x 2 2(1 − cos θ0 ) − sin2 θ0 cos θ0    2  − 2 1 + nx 3 − x 3 (1 − cos θ0 )2 (2 + cos θ0 ) 3

(3.2)

r is the radius of the larger droplet; n is the number of droplets in the initial coalescence; x is the ratio of radius of the smaller droplet to the larger droplet; θ 0 is the static contact angle of the surface; γ lg is the surface tension of gas–liquid interface. It is worth mentioning that for ideal smooth surfaces, the adhesion work (external work imposed to separate the liquid and the surface) is actually calculated from the change of surface energy. Therefore, we only need to calculate the change of surface energy, without considering the adhesion work repeatedly. That is to say, expression (3.2) has already considered the adhesion work. Since fluids (water) has viscosity, and the shear motion inside the fluid can generate viscous dissipation, referring to the literature [12], the viscous dissipation of the deformation process during droplet coalescence is estimated as below: E vis

 21

 γlg 3 3 3 = 36πμ r 2 1 + nx 2 − x 2 ρ

(3.3)

Change of gravitational potential energy mainly stems from the uplift of the gravity center after droplet coalescence and is deduced to be as follows: 1 E gra = ρgπr 4 (1 − cos θ0 )2 (2 + cos θ0 )· 3

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3 Behaviors of Condensed Droplets on Superhydrophobic Surfaces

⎧   13 ⎨ 2 4 3 3 3 (1 − cos θ0 ) (2 + cos θ0 ) 1 + nx − x ⎩ 4   (3 + cos θ0 )(1 − cos θ0 )  − 1 + nx 4 − x 4 4(2 + cos θ0 )

(3.4)

In expressions (3.3) and (3.4), μ is the dynamic viscosity of the fluid; ρ is the density of fluid; g is the gravity acceleration. Utilizing expressions (3.2)–(3.4), we can discuss the impact of the influential factor mentioned above on energy variations in the process of self-propelled droplet behaviors. Further combined with the critical condition of E cri = 0, the critical values of those influential factors can be therefore calculated.

3.4.2 Minimum Critical Droplet Radius In self-propelled droplet behaviors, the driving energy is the released surface energy via droplet coalescence, and from expression (3.2), we can see that it is proportional to the square of droplet radius; from expressions (3.3) and (3.4), relations also exist between droplet radius and other types of obstructing energy, such as viscous dissipation and gravitational potential energy (E vis ∝ r 1.5 , E gra ∝ r 4 ). This means there is a minimum radius for droplets, once the droplet radius is smaller than that value, the surface energy released cannot overcome viscous dissipation, and self-propelled droplet behaviors cannot occur. On the other hand, there is also a maximum radius for droplets, once the droplet radius exceeds that value, the surface energy released cannot overcome gravitational potential energy, disabling self-propelled droplet behaviors again. For self-propelled behaviors of condensed droplets on superhydrophobic surfaces, the minimum critical radius is more significant in reality, such as in engineering practice of enhanced condensation heat transfer. The minimum critical droplet radius determines the minimum radius that droplets can be removed from the surface, which further determines the extent of enhancement for condensation heat transfer. Hence, the present work only focuses on the minimum critical droplet radius of self-propelled droplet behaviors and does not calculate droplets with a radius exceeding 200 μm. Correlating expressions (3.1)–(3.4), and providing the static contact angle of the surface and the ratio of droplet radius, the variation curve of minimum critical droplet radius of self-propelled behaviors with respect to the number of coalesced droplets can be drawn. Figure 3.10 is the curve of minimum critical droplet radius of droplet jumping with respect to the number of coalesced droplets, in which the droplets are identical (x = 1), and the static contact angle of the surface is 160°. It can be seen that as the number increases, the minimum critical droplet radius decreases. For example, the critical radius is 12 μm for three-droplet coalescence, while this value decreases to 5 μm for seven-droplet coalescence. When the total volume of droplets is identical,

3.4 Critical Conditions for Self-propelled Droplet Behaviors

55

Fig. 3.10 Variations of minimum critical droplet radius regarding number of coalesced droplets

the more the number of droplets coalescing together, the larger surface area they possess, releasing more surface energy to overcome viscous dissipation. Therefore, multi-droplet coalescence is conducive to the decrease of minimum critical droplet radius. Statistical fitting results indicate that as the number of coalesced droplets rises, the critical radius falls in a power function (r min = 72.88 × n−1.5 ). Providing the number of coalesced droplets and the radius ratio of droplets, the relation between minimum critical droplet radius and the static contact angle can also be calculated. As shown in Fig. 3.11, as the static contact angle of the surface increases, the minimum critical droplet radius decreases. From expression (3.2), we can see that the larger the static contact angle, the more surface energy can be released after the coalescence to overcome dissipation, thus reducing the minimum critical droplet radius of self-propelled droplet behaviors. However, the decreasing trend of critical radius slows down as the static contact angle increases. When the static contact angle exceeds 150°, that is, the surface becomes superhydrophobic, the Fig. 3.11 Variations of minimum critical droplet radius regarding static contact angle of surface

56

3 Behaviors of Condensed Droplets on Superhydrophobic Surfaces

decrease in critical diameter is not obvious. This is because at this time, the static contact angle is large enough, and further increasing the static contact angle will not have an apparent effect in releasing surface energy, as calculated by expression (3.2).

3.4.3 Critical Ratio of Droplet Radius Self-propelled droplet behaviors are triggered by droplet coalescence; therefore, we can infer that the ratio of droplet radius can also affect self-propelled droplet behaviors, and there exists a minimum ratio. When the ratio of droplet radius does not meet that threshold, there will be a lack of surface energy released by coalescence, which disables self-propelled droplet behaviors. According to expression (3.1)–(3.4), we can derive the variation of critical ratio of droplet radius with respect to droplet radius, the number of coalesced droplets, and the static contact angle. In Fig. 3.12a, within the range of 30–200 μm for droplet radius, as the radius increases, the critical ratio of droplet radius decreases. This can be explained by the weakened effect of viscous dissipation as droplet radius increases, which allows a smaller ratio of droplet radius. Besides, in Fig. 3.12a, two droplets coalesce on a superhydrophobic surface with a 160° static contact angle; when the droplet radius is 50 μm, the corresponding critical ratio of radius is 0.6, that is, when the radius of the larger droplet is 50 μm, the radius of another droplet should exceed 30 μm to trigger self-propelled droplet behaviors. In fact, this is a rather strict condition; therefore, due to the limitation of ratio of droplet radius, numerous droplet coalescence cannot trigger self-propelled droplet behaviors even on superhydrophobic surfaces. As shown in Fig. 3.12a, critical ratio of droplet radius also decreases with the increase of the number of coalesced droplets. That is because more surface energy is released from multi-droplet coalescence, which makes up for the reduction of released surface energy due to the decrease of droplet radius ratio.

Fig. 3.12 Variations of critical ratio of droplet radius with a number of coalesced droplets and b static contact angle under the variation of droplet radius

3.4 Critical Conditions for Self-propelled Droplet Behaviors

57

Providing the number of coalesced droplets and the radius of coalesced droplets, Fig. 3.12b exhibits the relation between critical ratio of droplet radius and the static contact angle of the surface. With the rise of static contact angle, the surface energy released from droplet coalescence increases, so that the reduced energy due to the decrease of droplet radius ratio is offset, which lowers the critical ratio of droplet radius. However, when the static contact angle of the surface is larger than 150°, the increment of released surface energy is no longer obvious; thus, the decreasing trend of critical ratio of droplet radius also becomes subtle.

3.4.4 Critical Static Contact Angle Figure 3.13 provides the relation between critical static contact angle and the number of coalesced droplets during the occurrence of self-propelled droplet behaviors. We can see that as droplet radius increases, the critical static contact angle decreases; with the increase of the number of droplets, the critical static contact angle also decreases. It should be noted that attributed to the assumption of ideal smooth surface (no contact angle hysteresis) in the theoretical model, the critical value of static contact angle decreases to the range of hydrophobic in Fig. 3.13 (Figs. 3.11 and 3.12b also involve ordinary hydrophobic surfaces). However, for hydrophobic surfaces in reality, extralarge contact angle hysteresis and blocking force of droplet movements are imposed on the surface, so that self-propelled droplet behaviors cannot occur (in reference to experiments in Sect. 3.5). In fact, factors such as droplet radius, ratio of droplet radius, number of coalesced droplets, and static contact angle are coupled together to affect energy variations Fig. 3.13 Variations of critical static contact angle regarding droplet radius and number of coalesced droplets

58

3 Behaviors of Condensed Droplets on Superhydrophobic Surfaces

such as surface energy released by droplet coalescence, as well as viscous dissipation. When one factor variates, the critical values of other factors are changed automatically to satisfy the critical condition of self-propelled droplet behaviors (E cri = 0).

3.5 Effect of Self-propelled Droplet Behaviors on Droplet Growth The motional characteristics, statistics, and the critical occurring conditions of selfpropelled droplet behaviors have been introduced in Sects. 3.2–3.4, and this section pays attention to the effect of self-propelled droplet behaviors on the growth of droplet clusters on superhydrophobic surfaces. Besides the superhydrophobic surface (static contact angle: 160°, contact angle hysteresis: 6.0°), this section also selects the bare aluminum surface and the hydrophobic surface as control surfaces. The bare surface is a polished aluminum alloy surface without any chemical treatment, and its values of static contact angle and contact angle hysteresis are 85° and 42° respectively. The hydrophobic surface is fabricated by chemical etching method, with a static contact angle of 115° and a contact angle hysteresis of 38°. Figure 3.14 exhibits the images of condensed droplets on bare aluminum, hydrophobic, and superhydrophobic surfaces, under the condition of T sur = 2.0 ± 0.5 °C, T air = 20.0 ± 1.0 °C, RH = 50.0 ± 5.0%. As shown in Fig. 3.14a, a majority of droplets on the bare aluminum surface appear to have elliptic or irregular shapes; as droplets grow, immobile droplet coalescence takes place among droplets, and the merged droplets are in similar shapes. Figure 3.14b is the image of condensed droplets on the hydrophobic surface, and most of the droplets are spherical, among which the coalescence behaviors are all immobile coalescence. Figure 3.14c is the image of condensed droplets on the superhydrophobic surface, since that droplet jumping and droplet sweeping occur frequently, condensed droplets can succeed in the fast self-removal, which exposes large areas of clean surface.

3.5.1 Droplet Diameter Distribution As mentioned above, on the bare aluminum and the hydrophobic surfaces, only immobile droplet coalescence occurs, while self-propelled droplet jumping and droplet sweeping can also occur on the superhydrophobic surface. Statistical methods will be adopted below to obtain characteristic parameters of droplet growth on these three types of surfaces, such as droplet diameter distribution, average diameter of droplets, and surface coverage fraction of droplets. Comparisons are drawn among different surfaces to analyze the effect of self-propelled droplet jumping and droplet sweeping on the growth characteristics of condensed droplets.

3.5 Effect of Self-propelled Droplet Behaviors on Droplet Growth

59

Fig. 3.14 Images of condensed droplets on surface (T sur = 2.0 °C, T air = 20.0 °C, RH = 50.0%): a bare aluminum surface; b hydrophobic surface; c superhydrophobic surface

Counting droplets within certain area on the surface, and labeling those droplets in an order of droplet diameter from the large to the small, the droplet diameter distribution can be obtained. Figure 3.15a exhibits the distribution after 20 and 40 min of condensation on the bare aluminum surface, and it can be seen that there is a high linearity of droplets from the large to the small. This result indicates that droplets on the bare aluminum surface maintain at a relatively consistent growth rate. In Fig. 3.15b, condensed droplets on the hydrophobic surface also exhibit a quasi-linear distribution of droplet diameter from the large to the small. However, the situation is different on the superhydrophobic surface; as shown in Fig. 3.15c, the distribution appears to be curve-shaped. This can be interpreted as: a majority of droplets on the superhydrophobic surface have small sizes, but a few large droplets also exist on the surface. For a more direct display of the droplet distribution trend, an auxiliary line is added to Fig. 3.15c as the distribution line on the bare aluminum and the hydrophobic surfaces. Divided by the auxiliary line, points in the upper part represent a few large droplets, and points in the lower part represent a majority of droplets that have small sizes.

60

3 Behaviors of Condensed Droplets on Superhydrophobic Surfaces

Fig. 3.15 Condensed droplet diameter distribution on surfaces (T sur = 2.0 °C, T air = 20.0 °C, RH = 50.0%): a bare aluminum surface; b hydrophobic surface; c superhydrophobic surface

In fact, it is the frequently occurred self-propelled droplet behaviors on the superhydrophobic surface that lead to the unique distribution of droplet diameter. On one hand, as small droplets grow and contact with each other on the superhydrophobic surface, their coalescence is prone to trigger droplet jumping or droplet sweeping, and the condensation starts again at the site of clean surface where droplet behaviors take place. These repeated cycles of condensation and droplet behavior suppress the growth of small droplets, and small droplets are therefore dominant on the surface. On the other hand, droplets bouncing and falling back, or sweeping droplets promote the coalescence of multiple droplets simultaneously, forming large droplets. The growth of droplet on the bare aluminum and hydrophobic surfaces relies primarily on the direct condensation and coalescence of droplets. The growth via direct condensation has a relatively slow speed, and all of the droplet coalescence is immobile with a small number of droplets involved. Thus, the droplet growth on the bare aluminum and the hydrophobic surfaces appears to be slow and stable.

3.5 Effect of Self-propelled Droplet Behaviors on Droplet Growth

61

3.5.2 Average Droplet Diameter The average droplet diameter is obtained by averaging the diameter of counted droplets. Figure 3.16 shows the change of average droplet diameter with respect to time on the bare aluminum, the hydrophobic, and the superhydrophobic surfaces. It can be seen that the average diameter of condensed droplets decreases with the increase of static contact angle, and the growth rate slows down. This is mainly attributed to the larger nucleation energy as the static contact angle increases, and the smaller the contact area between the droplet and the surface, the less helpful it is to heat transfer. In addition, as time goes by, the increase of average droplet diameter on the bare aluminum and hydrophobic surfaces follow the power function (Dave ∝ t β ), which is consistent to the result in the literature. The index can be calculated via data fitting, and the values for the bare aluminum and the hydrophobic surfaces are 0.94 and 0.76 respectively. On the superhydrophobic surface, the variation of average droplet diameter regarding time fluctuates. This is due to the frequently occurred selfpropelled droplet behaviors that promote the coalescence of droplets on the surface and clear out droplets on the surface quickly, which leads to the deviation from power function for average droplet diameter on the superhydrophobic surface. Figure 3.17 exhibits the relative standard deviation (RSD) of droplet diameter on the bare aluminum, the hydrophobic, and the superhydrophobic surfaces at different times. The relative standard deviation of droplet diameter is defined as the ratio of the standard deviation of droplet diameter to the average droplet diameter, which is adopted to reflect the relative difference of droplet diameter within the condensed droplets. As the figure shows, the relative standard deviation on the superhydrophobic surface is larger than the ones on bare aluminum and hydrophobic surfaces, which can also be discovered directly from Fig. 3.14 (images of droplet growth) and Fig. 3.15 (droplet diameter distribution). This larger value of relative standard deviation of droplet diameter can also be attributed to the frequent occurrence of self-propelled droplet behaviors.

0.6 Bare Al Hydrophobic Superhydrophobic

0.5

Dave (mm)

Fig. 3.16 Variations of average droplet diameter of coalesced droplets regarding condensation time on bare aluminum, hydrophobic, and superhydrophobic surfaces (T sur = 2.0 °C, T air = 20.0 °C, RH = 50.0%)

0.4 0.3 0.2 0.1 0.0

5

10

15

20

25 30 t (min)

35

40

45

Fig. 3.17 Relative standard deviation of droplet diameter on bare aluminum, hydrophobic, and superhydrophobic surfaces (T sur = 2.0 °C, T air = 20.0 °C, RH = 50.0%)

3 Behaviors of Condensed Droplets on Superhydrophobic Surfaces

1.0 Bare Al Hydrophobic Superhydrophobic

0.8

RSD

62

0.6 0.4 0.2 0.0

20

25

30 t (min)

35

40

3.5.3 Surface Coverage Fractions Surface coverage fraction of droplets is another important parameter of measuring surface condensation characteristics. Statistics give the variation of surface coverage fractions with respect to time on the bare aluminum, the hydrophobic, and the superhydrophobic surfaces, which is displayed in Fig. 3.18. As the static contact angle increases, the surface coverage fractions decrease. In addition, the surface coverage fractions increases as time goes by on the bare and the hydrophobic surfaces, while the parameter fluctuates around a low value on the superhydrophobic surface. The extremely low and fluctuating surface coverage fractions can also be attributed to the frequently occurred self-propelled droplet behaviors, which can effectively remove droplets on the surface, and expose large area of surface for condensation. 100

Surface coverage fraction (%)

Fig. 3.18 Surface coverage fractions on bare aluminum, hydrophobic, and superhydrophobic surfaces (T sur = 2.0 °C, T air = 20.0 °C, RH = 50.0%)

Bare Al Hydrophobic Superhydrophobic

80 60 40 20 0

5

10

15

20 25 30 35 40 45 t (min)

3.5 Effect of Self-propelled Droplet Behaviors on Droplet Growth

63

Fig. 3.19 Effect of subcooled surface temperature on condensed droplet growth (T air = 20.0 °C, RH = 50.0%): a bare aluminum surface; b hydrophobic surface; c superhydrophobic surface

3.5.4 Effects of Working Conditions This section primarily discusses the effect of working conditions on the droplet growth characteristics on the superhydrophobic surface. The bare aluminum and the hydrophobic surfaces are still control surfaces. Figure 3.19 shows the effect of subcooled surface temperature on the condensed droplet growth on these three types of surfaces. On the bare aluminum and the hydrophobic surfaces, the average droplet diameter increases as the surface temperature decreases, followed by a faster growth of droplets. However, the dependence of condensed droplet diameter on the surface temperature is not obvious on the superhydrophobic surface. It can be forecasted that on the superhydrophobic surface, within certain range of working conditions, the effect of other working conditions, for instance, wet air humidity, on the growth of condensed droplets is also subtle. The reason for this phenomenon is that the frequently occurred self-propelled droplet behaviors on the superhydrophobic surface irritate the growth of droplets, resulting in a different growth trend, compared with the trend of droplet growth on ordinary surfaces.

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3 Behaviors of Condensed Droplets on Superhydrophobic Surfaces

3.6 Summary Using the Al-based superhydrophobic surface as the experimental surface, this chapter primarily investigates behaviors of condensed droplets on superhydrophobic surfaces. Through microscale observations, droplet behaviors on superhydrophobic surfaces are classified; through statistical calculations, the occurring laws of droplet behaviors are determined; through theoretical modeling, critical conditions of triggering self-propelled droplet behaviors are analyzed; further investigation is made about the effect of self-propelled droplet behaviors on the characteristics of droplet growth on superhydrophobic surfaces. The main conclusions in this chapter are drawn as follows: (a) There are mainly three types of behaviors occurring during the coalescence of condensed droplets on superhydrophobic surfaces: immobile droplet coalescence, droplet jumping, and droplet sweeping. The latter two behaviors are selfpropelled behaviors that can occur without the external driving forces such as gravity or wind force. Besides, the self-propelled droplet sweeping phenomenon is independently observed in the present work, which is mainly triggered by the coalescence of droplets with uneven distribution of size and position, or by the droplet falling back after droplet jumping. Compared with droplet jumping, droplet sweeping can achieve the self-removal of more droplets, thus possessing a wider application prospect. (b) On the Al-based superhydrophobic surface in the present work, immobile droplet coalescence is still dominant, and the occurring probability of selfpropelled droplet jumping and droplet sweeping is less than half. Immobile droplet coalescence mainly takes place between 2 and 3 droplets, and the occurring probability decreases with the increase of coalesced droplet number. Multidroplet coalescence is conducive to triggering droplet jumping; however, owing to the probability variation of occurring multi-droplet coalescence, the occurring probability of droplet jumping first rises and then falls down as the number of coalesced droplets increases. The number of droplets involved in droplet sweeping far surpasses the number in immobile droplet coalescence and droplet jumping with an average of 12, and this number can even reach the value of tens or hundreds. (c) Self-propelled behaviors of condensed droplets on superhydrophobic surfaces are driven by the released surface energy from droplet coalescence; however, it needs to overcome some types of energy, such as viscous dissipation and gravitational potential energy. Therefore, the occurrence of self-propelled droplet behaviors lies on the foundation of certain critical condition. This critical condition is affected by factors such as droplet radius, ratio of droplet radius, static contact angle of the surface, the number of coalesced droplets, etc. These influential factors couple with each other, and affect the variation of diverse types of energy, such as surface energy released by droplet coalescence and viscous dissipation. When one of the factors changes, the critical values of other factors will vary too.

3.6 Summary

65

(d) The frequently occurred self-propelled droplet jumping and droplet sweeping can promote the coalescence of numerous droplets on the surface, and achieve a rapid self-removal of droplets, bringing a fluctuated curve of droplet diameter variation and an extremely low surface coverage fraction of droplets. What is more, due to the existence of self-propelled droplet behaviors, the effect of working condition changes on the condensed droplet growth on superhydrophobic surfaces is weaker than the effect on other surfaces.

References 1. Boreyko J, Chen C-H (2009) Self-propelled dropwise condensate on superhydrophobic surfaces. Phys Rev Lett 103(18):184501 2. Lv FY, Zhang P (2014) Fabrication and characterization of superhydrophobic surfaces on aluminum alloy substrates. Appl Surf Sci 321:166–172 3. Kim S, Kim KJ (2011) Dropwise condensation modeling suitable for superhydrophobic surfaces. J Heat Transfer 133(8):081502 4. Miljkovic N, Enright R, Wang EN (2013) Modeling and optimization of superhydrophobic condensation. J Heat Transfer 135(11):111004 5. Lv C, Hao P, Yao Z, Song Y, Zhang X, He F (2013) Condensation and jumping relay of droplets on lotus leaf. Appl Phys Lett 103(2):021601 6. Yanagisawa K, Sakai M, Isobe T, Matsushita S, Nakajima A (2014) Investigation of droplet jumping on superhydrophobic coatings during dew condensation by the observation from two directions. Appl Surf Sci 315:212–221 7. Kim MK, Cha H, Birbarah P, Chavan S, Zhong C, Xu Y, Miljkovic N (2015) Enhanced jumpingdroplet departure. Langmuir 31(49):13452–13466 8. Lv C, Hao P, Yao Z, Niu F (2015) Departure of condensation droplets on superhydrophobic surfaces. Langmuir 31(8):2414–2420 9. Chen X, Patel RS, Weibel JA, Garimella SV (2016) Coalescence-induced jumping of multiple condensate droplets on hierarchical superhydrophobic surfaces. Scientific Reports. 6:18649 10. Cha H, Chun JM, Sotelo J, Miljkovic N (2016) Focal plane shift imaging for the analysis of dynamic wetting processes. ACS Nano 10(9):8223–8232 11. Cha H, Xu CY, Sotelo J, Chun JM, Yokoyama Y, Enright R, Miljkovic N (2016) Coalescenceinduced nanodroplet jumping. Physical Review Fluids 1(6):064102 12. Wang F-C, Yang F, Zhao Y-P (2011) Size effect on the coalescence-induced self-propelled droplet. Appl Phys Lett 98(5):053112

Chapter 4

Numerical Simulations of Multi-droplet Coalescence-Induced Jumping

Self-propelled droplet jumping is one of the typical droplet behaviors on superhydrophobic surfaces. Since that it can achieve self-removal of droplets on the surface; thus, it has an excellent application prospect in engineering fields of enhanced condensation heat transfer, anti-frosting/icing, self-cleaning, water collection, etc. The last chapter has observed and investigated the motional characteristics of droplet jumping at microscales, and statistical laws have been obtained through calculations, with the critical conditions of occurrence analyzed through theoretical modeling. However, there are still multiple questions for droplet jumping, especially for the jumping triggered by multi-droplet coalescence. The essence of droplet jumping is the motional behaviors driven by the surface energy released from droplet coalescence, and multiple types of energy involve in the process of droplet jumping. Then, what are the proportions of each type of energy, and what is the percentage of conversion from surface energy to effective kinetic energy for jumping? For multi-droplet coalescence-induced droplet jumping that has a larger probability of occurrence, how does the number of coalesced droplets affect the energy conversion? Initial droplet distribution of position is tightly related to multi-droplet coalescence, and how does the distribution affect droplet jumping? These questions remain unsolved until now. What is more, due to the randomness of droplet position distribution, as well as the extremely tiny scale and short time of occurrence, it is still difficult to solve questions above via experimental methods; due to the complex fluid deformation involved in droplet jumping, relevant question cannot be solved simply by theoretical analysis. Therefore, numerical simulation becomes the optimal choice for solving these questions. This chapter conducts numerical simulations of the multi-droplet coalescenceinduced jumping process, analyzes the energy variation and energy conversion during droplet jumping, discusses the effect of coalesced droplet number and droplet position distribution on droplet jumping, and thereby solves questions mentioned above. This provides further insight for the optimization of droplet jumping process, as well as promoting practical engineering applications of droplet jumping.

© Tsinghua University Press 2020 F. Chu, Condensed and Melting Droplet Behavior on Superhydrophobic Surfaces, Springer Theses, https://doi.org/10.1007/978-981-15-8493-0_4

67

68

4 Numerical Simulations of Multi-droplet Coalescence …

4.1 Simulation Objects and Conditions Since the primary goal in this chapter is to analyze energy variation and energy conversion in multi-droplet coalescence-induced droplet jumping and discuss the effect of the coalesced droplet number and droplet position distribution on droplet jumping, five simulation cases are designed as in Fig. 4.1. The specific parameters of these five cases are listed in Table 4.1. Case 1 is the coalescence and jumping of two identical droplets, which is mainly used for model validation and comparison with other cases. Case 2 is the coalescence and jumping of three identical droplets, with the three droplets arranged as an equilateral triangle; that is, the maximum angle between the lines connecting centers of droplets is 60°. Case 3 is the coalescence and jumping of four identical droplets, with the location of centers of four droplets being a square. Case 4 is the coalescence and jumping of three identical droplets that locate in a straight line; that is, the maximum angle between the lines connecting centers of droplets is 180°. Case 5 is also the coalescence and jumping of three identical droplets, with the maximum angle between the lines connecting centers of droplets being 120°. This chapter mainly discusses the effect of coalesced droplet number and droplet position distribution on droplet jumping; therefore, droplets in the simulation objects in the present work are all identical in sizes (100 µm radii), and the coalescence of droplets with different sizes will not be discussed here. Fig. 4.1 Top-view of five simulation cases for droplet jumping

Table. 4.1 Simulation cases and parameters for droplet jumping Simulation cases

Droplet radius, r (µm)

Number of droplets

Angle of lines connecting droplet centers (o )

1

100

2

–

2

100

3

60

3

100

4

–

4

100

3

180

5

100

3

120

4.1 Simulation Objects and Conditions Table. 4.2 Physical properties of fluids at 20 °C (g = 9.8 m/s2 )

69

Fluids

Surface tension, γ lg (N/m)

Density, ρ (kg/m3 )

Dynamic viscosity, μ (Pa s)

Water

0.072

998.2

1.0 × 10–3

1.2

1.8 × 10–5

Air

The droplet jumping phenomenon simulated in this chapter is the coalescence of droplets condensed from vapor in the air on the superhydrophobic surface; hence, the simulated droplets are condensed water droplet, and the ambient gas is the air. The physical properties of fluids are extracted at 20 °C under atmospheric pressure, and the gravity acceleration is 9.8 m/s2 , consistent with the literature [1]. The main physical properties of fluids used in the simulation include surface tension, density, and dynamic viscosity. Specific values of those parameters are listed in Table 4.2.

4.2 Mathematical Model 4.2.1 Control Equation Considering the dramatic deformation during the process of droplet coalescence, which involves complex variations of interfaces, the volume of fluids (VOF) multiphase fluid model [2] is utilized in this chapter to achieve the designed simulation cases. The model consists of two phases: the air phase and the liquid water phase. Primary control equations are continuity equation and momentum equation, which are: Continuity equation: ∇ ·v =0

(4.1)

Momentum equation:  ρ

    ∂v + v · ∇v = −∇ p+∇ μ ∇ · v + ∇ · vT + ρg + F ∂t

(4.2)

where v is the velocity vector, t is the time, and p is the pressure; F is the source term of surface tension; ρ and μ are density and viscosity, both of which are weighted by the water phase volume fraction (α) as: ρ = αρwater + (1 − α)ρair

(4.3)

μ = αμwater + (1 − α)μair

(4.4)

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4 Numerical Simulations of Multi-droplet Coalescence …

In addition, the phase interface changes with the motion of fluids. Hence, one more equation is needed for the water phase volume fraction: ∂α + v · ∇α = 0 ∂t

(4.5)

The continuum surface force (CSF) model proposed by Brackbill et al. [3] is adopted for the source term of surface tension F in the momentum equation. CSF model is based on the Laplace pressure at two sides of the phase interface to obtain surface tension coefficient, calculate volume fraction gradient in normal direction of the phase interface for the result of phase interface curvature, then express the surface tension in the form of body force according to the divergence theorem, and finally add it in into the momentum equation. The source term F is: F = γlg κ∇α

(4.6)

where κ is the curvature of the phase interface, calculated by: κ = −∇ · n

(4.7)

n is the unit normal vector on the phase interface, which is expressed as: n=

∇α |∇α|

(4.8)

There exist three phases on the contact line of the droplet, as shown in Fig. 4.2. At this time, n is calculated by: n = nw cos θ0 + tw sin θ0

(4.9)

where θ 0 is the static contact angle of the surface, nw and tw are the unit normal vector and the unit tangent vector of the surface. Numerical simulations in this chapter are performed using interFoam solver in OpenFOAM, an open-source CFD simulation software. The simulation is based on the finite volume method, and solutions of pressure and velocity are coupled by Semi-Implicit Method for Pressure-Linked Equation (SIMPLE) algorithm. Fig. 4.2 Schematic of normal vectors at triple-phase line on droplet

4.2 Mathematical Model

71

4.2.2 Computational Domain, Boundary Conditions, and Grids The 3D numerical simulation of droplet jumping is performed on the superhydrophobic surface; the computational domain is shown in Fig. 4.3, that is, 1 mm in length, 1 mm in width, and 2 mm in height. Compared to the size of droplets (100 µm radii), the domain is large enough to avoid the effect of edges on droplet jumping. In the computation domain, the superhydrophobic surface at the bottom is considered as no-slip ideal surface, with a static contact angle of 160°. Given that droplet jumping generally occurs in an open space, so that the other boundaries in the computational domain are set to be pressure outlet boundaries; that is, there is no pressure or velocity gradient on these boundaries. Structured hexahedral meshes are adopted in the computational domain. To ensure the independence of simulation results from mesh sizes, grids of 10, 5, and 2.5 µm are used to perform numerical simulations of droplet jumping of two identical droplets (Case 1). As displayed in Fig. 4.4, the calculated droplet jumping velocities have a deviation less than 5% when using 5 and 2.5 µm mesh sizes, while the result of using 10 µm mesh size has a relatively large deviation. Therefore, to ensure the calculation accuracy as well as saving computational time, 5-µm mesh size is used in the present work for numerical simulations of other cases. Fig. 4.3 Computation domain of numerical simulations and boundary conditions

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4 Numerical Simulations of Multi-droplet Coalescence …

Fig. 4.4 Validation of mesh independence

4.2.3 Energy Analysis Droplet jumping is a self-propelled behavior driven by surface energy, and complex energy variations and conversions are involved in this process. Therefore, to better comprehend the droplet jumping mechanism, calculations and analyses on the energy variation and energy conversion, including surface energy, kinetic energy, viscous dissipation, and gravitational potential energy, during droplet jumping, are needed. On superhydrophobic surfaces, the gas–solid contact area between the liquid and the surface is much smaller than the surface area of the gas–liquid interface; hence, when calculating surface energy, the surface energy on gas–solid and solid–liquid interfaces can be neglected to simplify the calculation, and the surface energy on the gas–liquid interface is the only one to be considered, and it can be expressed as: E sur = γlg Alg

(4.10)

where γ lg is the liquid–gas surface tension for water and air, that is, the surface tension of liquid water (in reference to Table 4.1), and Alg is the surface area of the liquid–gas interface. The kinetic energy of the droplet can be calculated by the following equation, which is the sum of the kinetic energy in every mesh element in the droplet:  E kin =

 1  2 ρ u + v2 + w 2 dV 2

(4.11)

V

where u, v, and w are velocities in x-axis, y-axis, and z-axis in a mesh element; ρ is the weighted density of fluids in every mesh element; dV is the volume of mesh element, and V is the volume of the droplet.

4.2 Mathematical Model

73

Dramatic deformation occurs during the coalescence-induced droplet jumping. Since water is viscous fluid, therefore, viscous dissipation exists in the process of droplet jumping, which is calculated by [4]: t E vis = 

0

⎞ ⎛  ⎝ φdV ⎠ dt V

    ∂u ∂v 2 ∂v ∂w 2 1 + φ = 2μ + + + ∂y ∂z 2 ∂y ∂x 2  2   ∂w ∂v ∂u ∂w + + + + ∂y ∂z ∂z ∂x ∂u ∂x

2



(4.12)

2

(4.13)

where φ is the dissipation function; μ is the weighted viscosity of fluids in every mesh element. The gravitational potential energy also changes during the process of droplet jumping, but for the simulation cases in the present work, the droplet radius is only 100 µm, much less than the capillary length of water (2.7 mm) [5]. Therefore, compared with surface energy, the effect of gravity can be ignored, and this chapter will skip the discussion about changes of gravitational potential energy. Besides the energies mentioned above, droplet jumping velocity is also an important parameter for droplet jumping. The droplet jumping velocity is calculated as the mass-weighted velocity of the coalesced droplets in the vertical direction (z-axis), which is:  ρwdV (4.14) U = V V ρdV The effective kinetic energy for droplet jumping is considered as the corresponding kinetic energy of droplet jumping velocity, that is: E jump =

1 mU 2 2

(4.15)

4.3 Model Validation—Two-Droplet Coalescence-Induced Jumping The numerical simulation of droplet jumping triggered by the coalescence of two identical droplets (Case 1) is conducted first to validate the accuracy of the model in the present work. Figure 4.5 exhibits the droplet morphology evolution during

74

4 Numerical Simulations of Multi-droplet Coalescence …

Fig. 4.5 Droplet morphology evolution during two-droplet coalescence-induced jumping: a sideview image; b top-view image

the droplet jumping process from the top and the side. As shown in this figure, at initial time (0 ms), two identical droplets with radii of 100 µm are placed on the superhydrophobic surface, with their interfaces just overlapped. Then, when droplets start to coalesce, the liquid bridge between two droplets extends, until the size of the liquid bridge is comparable with the size of droplets, and touches the droplet (0.1 ms). Subsequently, due to the counteraction with the surface, the reaction force is generated, and the coalesced droplet begins to move upwards. Finally, at some time, the coalesced droplet jumps out of the surface (the approximate time is 0.45 ms) and keeps moving in the vertical direction; however, the droplet then oscillates in the air and gradually decelerates. Based on the droplet morphology evolution during droplet jumping, this process can be divided into four stages [6], as shown in Fig. 4.6. Stage I is the formation and extension of liquid bridge between coalesced droplets. Stage II is the bouncing and acceleration of droplet; in this stage, due to the effect of the reaction force from the surface, the coalesced droplet bounces and accelerates upwards. As this motion continues, the effect of the reaction force eventually dies away. When the reaction force disappears, Stage III starts, which is the departure of the coalesced droplet from the surface. When the droplet jumps out of the surface completely, Stage IV occurs, that is, the oscillation and deceleration of the departed droplet in the air. For the simulation of Case 1 in this chapter, the dividing times of four stages are 0.2, 0.4, and 0.5 ms. Comparing the morphology evolution (Fig. 4.5) and stage division (Fig. 4.6) of droplet jumping via simulations in the present work with the

4.3 Model Validation—Two-Droplet Coalescence-Induced Jumping

75

Fig. 4.6 Jumping velocity curve and stage division of droplet jumping

results of experimental data and numerical simulations in the literature [6–10], a good agreement is reached, verifying the reliability of simulation in this chapter in a preliminary and qualitative way. For further validation of the accuracy of simulation results in this chapter, the jumping velocity variation in the process of two-droplet coalescence-induced jumping is extracted according to formula (4.14), and it is non-dimensionalized (divided by surface tension–inertia dimension speed [7], (γ lg /ρr)0.5 ) and compared with results in the literature [6, 10]. We can see from Fig. 4.7 that the variation trend of non-dimensional jumping velocity simulated in this chapter is consistent with the results in the literature; however, since that the static contact angle of the superhydrophobic surface used in this model is smaller than the value in the literature, the simulation result of non-dimensional absolute speed is slower, with a slightly later departure of the droplet from the surface. For the scenario of droplet jumping by the coalescence of two identical droplets with 100 µm radii on superhydrophobic surface with a static contact angle of 160°, Boreyko and Chen [7] measured the jumping velocity as 0.15 m/s; the theoretically calculated result from Lv et al. [11] was 0.173 m/s; Liu et al. [12] gave a simulation result of 0.170 m/s; Khatir et al. [13] obtained the jumping velocity from their simulation as 0.159 m/s. The simulation result in the present work is 0.156 m/s, which 0.5 Present model

∗ Departure

Liu et al. [6] Cheng et al. [10]

0.4 Dimensionless U

Fig. 4.7 Comparison of non-dimensional jumping velocity variation trend between the results of simulation and literature

0.3

∗

CA=180o

0.2

∗

CA=180o

∗

CA=160o

0.1 0.0

Case 1 0

1

2 3 4 Dimensionless t

5

6

76

4 Numerical Simulations of Multi-droplet Coalescence …

is in good agreement with the literature, and the relative deviation is only 4%. To conclude, the model in the present work is verified to be reliable.

4.4 Multi-droplet Coalescence-Induced Droplet Jumping It can be discovered from the statistics in Sect. 3.3 that during the condensation process on superhydrophobic surfaces in reality, approximately 80% of the droplet jumping is triggered by the coalescence of multiple droplets; therefore, investigating multi-droplet coalescence-induced droplet jumping would be more practical. During the process of multi-droplet coalescence, the number of coalesced droplets and the droplet position distribution are two crucial parameters. Hence, the section is going to discuss the effect of coalesced droplets number and droplet position distribution on droplet jumping in order.

4.4.1 Effect of Coalesced Droplet Number First, we investigate the effect of coalesced droplet number on droplet jumping, and the selected simulation cases are two-droplet coalescence-induced droplet jumping (Case 1), three-droplet coalescence-induced droplet jumping (Case 2), and fourdroplet coalescence-induced droplet jumping (Case 3). The sizes of droplets in these three cases are identical, and the positions of droplets are ideal distributions (two adjacent droplets just contact with each other, three droplets are distributed as an equilateral triangle, and four droplets are distributed as a square), and the only difference among these cases is the number of coalesced droplets. Figures 4.8 and 4.9 are the droplet morphology evolution of Case 2 and Case 3 (the corresponding morphology evolution of Case 1 is exhibited in Fig. 4.5). For the jumping mechanism, there is no difference among the droplet jumping triggered by two, three, and four droplets, and all of their process of jumping consist of the four stages stated in Sect. 4.3. However, differences still exist in the morphology evolution during jumping. Compared with the droplet jumping in Case 1, the droplet deformation during jumping is greater in Case 2 and Case 3 in the early stage. As shown in Fig. 4.8, in the early stage of threedroplet coalescence, surface dimple occurs (0.2 ms); in Fig. 4.9, ripples appear on the droplet surface in the early stage (0.2 ms) for four-droplet coalescence, and there is a high extent of irregularity for droplet morphology (0.3–0.4 ms). In the later stage of droplet jumping (0.6 ms), the droplet morphology variation of Case 2 and Case 3 is instead milder than Case 1, which resembles more to spherical compared with the morphology variation after two-droplet coalescence, with a smaller oscillation amplitude and shorter oscillation time. The variation of unit mass surface energy of the droplet for Cases 1, 2, and 3 is derived from formula (4.10), and the trend with respect to time is displayed in Fig. 4.10, which corresponds to the variation of droplet morphology and motion

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Fig. 4.8 Droplet morphology evolution during three-droplet coalescence-induced jumping process: a side-view diagram; b top-view diagram

stage. In the early stage of droplet coalescence (Stage I), the surface energy decreases dramatically due to the rapid coalescence of droplets; then, as the droplet moves upwards in Stage II and exposes a part of the surface again, the surface energy rises to a certain extent; in Stage III, the droplet departs from the surface, and the surface energy continues raising to a local maximum; subsequently, in stage IV, the droplet oscillates in the air, but eventually becomes spherical, and the surface energy undergoes a fluctuating decline. Although the variation trend of surface energy in the droplet jumping process is consistent among three simulation cases, the absolute value of surface energies is different among them. The surface energy of the droplet decreases as the number of coalesced droplets increases; in other words, the released surface energy increases with the rising of coalesced droplet number. In addition, from the variation diagram of the surface energy, we can see that the amplitude and time of oscillation are reduced as the number of coalesced droplets increases in the later stage of droplet jumping. These variations are synchronized with the droplet morphology variations depicted in Figs. 4.5, 4.8, and 4.9. The variation of kinetic energy of droplets in Cases 1, 2, and 3 is derived from formula (4.11), and the trend is displayed in Fig. 4.11. This kinetic energy refers

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Fig. 4.9 Droplet morphology evolution during four-droplet coalescence-induced jumping process: a side-view diagram; b top-view diagram

Fig. 4.10 Variation of unit mass surface energy of droplets during jumping regarding time

to the total kinetic energy, including the translational kinetic energy and the vibration kinetic energy. The kinetic energy of the droplet first increases rapidly and then declines in a fluctuating style. As the number of coalesced droplets increases, the obtained kinetic energy for unit mass droplet increases, and its fluctuation weakens during the declining period. In fact, the peak of the kinetic energy fluctuation corresponds to the valley of the surface energy fluctuation. This is because the kinetic

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Fig. 4.11 Variation of unit mass kinetic energy of droplets during jumping regarding time

Fig. 4.12 Variation of unit mass viscous dissipation of droplets during jumping regarding time

energy of the droplet is transformed from the released surface energy, and these two types of energy are convertible to each other. The viscous dissipation during the droplet jumping process for Cases 1, 2, and 3 is derived from formula (4.12), with the trend displayed in Fig. 4.12, and it gradually increases as time goes by. From expression (4.13), we can see that the increasing rate of the viscous dissipation is correlated with the velocity gradient of fluid, and this gradient can further be reflected by the droplet deformation and droplet oscillation. The more dramatic the deformation and oscillation for the droplet, the larger velocity gradient exists inside the fluid, and the larger increasing rate of the viscous dissipation. In the early stage of droplet coalescence, with the increasing number of coalesced droplets, the extent of droplet deformation increases (as shown in Figs. 4.8 and 4.9, there are even dimples and ripples on the droplet surface), and the increasing rate of viscous dissipation is therefore elevated; in the later stage, however, as the number of coalesced droplets increases, the extent of droplet oscillation becomes weaker, resulting in the reduction of increasing rate of viscous dissipation. The jumping velocity (in the vertical direction) during the droplet jumping process for Cases 1, 2, and 3 with respect to time is derived from formula (4.14), and the effective jumping kinetic energy of droplet is further calculated by formula (4.15), which is shown in Fig. 4.13. The trend of velocity variation is the same among the coalescence

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Fig. 4.13 Variation of a jumping velocity and b effective jumping kinetic energy of droplets during jumping regarding time

jumping by two, three, and four droplets, but the velocity increases as the number of coalesced droplets increases. It should be noted that the velocity when the droplet just departs from the surface is not the maximum velocity on the curve, but instead, it is slightly later than the maximum velocity point. This is due to the disappearance of reaction force between the droplet and the surface before the droplet departs from the surface, so that the droplet is actually decelerating when it is departing from the surface. For the coalescence-induced droplet jumping process by two, three, and four droplets, the variation trends of effective jumping kinetic energy are in accordance with the trends of corresponding variations of velocity. Figure 4.13 indicates that the increase of the number of coalesced droplets is conducive to accelerating the droplet jumping velocity, as well as increasing the effective jumping kinetic energy. For further understanding of the effect of coalesced droplet number on droplet jumping, the conversion rate from the surface energy to the total kinetic energy and effective jumping kinetic energy, as well as the proportion of effective jumping kinetic energy in the total kinetic energy, is calculated, which is exhibited in Fig. 4.14. For two-droplet coalescence-induced jumping, only less than 20% of the released surface energy is transformed into kinetic energy, and this proportion improves as the number of coalesced droplets increases. For four-droplet coalescence-induced Fig. 4.14 Energy conversion rates in droplet jumping Conversion rate (%)

50 2 droplets 3 droplets 4 droplets

40 30 20 10 0

Ekin/ΔEsur

Ejump/Ekin

Ejump/ΔEsur

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jumping, approximately 36% of the released surface energy is transformed into kinetic energy. The proportion of effective jumping kinetic energy increases as the number of coalesced droplets increases, which also reflects the fact that as the number rises, the proportion of vibration kinetic energy in kinetic energy decreases. This is consistent with the morphology evolution diagram of droplets and the energy variation mentioned above. The trend of conversion rate from surface energy to effective jumping kinetic energy is the same, which is positively correlated to the change of coalesced droplet number. Nonetheless, this conversion rate is not high enough, even for four-droplet coalescence-induced jumping, the conversion rate from surface energy to effective jumping kinetic energy is no more than 10%, and for the twodroplet coalescence-induced jumping, the rate is less the 5%. The conversion rates calculated in the present work are consistent with the results in the literature [6, 10,14].

4.4.2 Effect of Droplet Position Distribution Droplet position distribution is also an important factor in the process of multidroplet coalescence-induced jumping. Simulation Cases 2, 4, and 5 are selected in this section, since they have the same number of coalesced droplet, while they are different from each other in terms of the droplet position distribution. Case 2 is a three-droplet coalescence with a 60-degree distribution; Case 4 is a three-droplet coalescence with a 180-degree distribution; Case 5 is a three-droplet coalescence with a 120-degree distribution. Before further discussions, for coalescence with the same number of droplets, two parameters are defined to characterize the droplet position distribution: the degree of concentration and the degree of symmetry. The degree of concentration characterizes the extent of droplets concentrated in position; for coalesced droplets with the same number and radii, the smaller the external circle (Fig. 4.15), the more concentrated for droplets, and the degree of concentration is higher. From the definition above, droplets of 60-degree distribution (Case 2) have Fig. 4.15 Schematic of multi-droplet coalescence and the external circle

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the highest degree of concentration, droplets of 120-degree distribution (Case 5) rank second, and droplets of 180-degree distribution (Case 4) have the lowest degree of concentration. The degree of symmetry characterizes the extent of centrosymmetric for droplet distribution. For coalesced droplets with the identical number and radii, rotate them around the central axis vertical to the surface, until reaching the initial distribution. The smaller the rotation angle needed, the higher the degree of symmetry. As shown in Fig. 4.16, only 120-degree rotation is needed for 60-degree droplet distribution, 180-degree rotation is needed for 180-degree droplet distribution, and 360-degree rotation for 120-degree distribution. Therefore, 60-degree droplet distribution has the highest degree of symmetry, 180-degree droplet distribution ranks second, and 120-degree distribution is the lowest. Figures 4.17 and 4.18 are the droplet morphology evolutions of coalescenceinduced jumping for 180-degree droplet distribution and 120-degree droplet distribution (the diagram of droplet morphology evolution for 60-degree droplet distribution is shown in Fig. 4.8). Comparisons among them reveal that in the early stage of droplet coalescence, 60-degree droplet distribution has the most dramatic morphology deformation in the early stage, with dents on the surface; it has the least oscillating amplitude and oscillating time in the later stage. In addition, since that 120-degree droplet distribution has the lowest degree of symmetry, it is the least harmonious with irregular droplet morphology evolutions. We can draw preliminary conclusions that the morphology evolution of coalesced droplets is related to the initial position distribution of droplet the more concentrated the droplet distribution, the smaller the oscillation of droplets; the more symmetric the droplet distribution, the more symmetric and harmonious the morphology evolution of droplets. Figure 4.19 consists of variation curves of surface energy during the coalescenceinduced droplet jumping of three types of droplet position distributions, which are derived from formula (4.10). During the droplet jumping process, the values of the released surface energy after coalescence are almost identical among three types of droplet distributions; that is to say, position distribution does not affect the release of surface energy. However, the surface energy variation of 60-degree droplet distribution occurs earlier than other distributions, and this is due to the higher extent of

Fig. 4.16 60-degree droplet distribution can reach the initial distribution after 120-degree rotation around the central axis

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Fig. 4.17 Droplet morphology evolution during coalescence-induced jumping process with 180degree droplet distribution: a side-view diagram; b top-view diagram

concentration, where droplets have shorter movements during the coalescence and spend less time, resulting in an earlier surface energy variation. Looking from the fluctuation of surface energy curve, 60-degree distribution has a weaker fluctuation after the coalescence, with a shorter fluctuation time, which is consistent with the corresponding diagram of droplet morphology evolution. Since the kinetic energy is transformed from the surface energy, and they can succeed in mutual conversions, thus, 60-degree distribution also has an earlier variation of kinetic energy after the coalescence, with a weaker energy fluctuation, which is displayed in Fig. 4.20. Figure 4.21 consists of variation curves of viscous dissipation during the coalescence-induced droplet jumping with three types of droplet position distributions, which are derived from formula (4.12). As mentioned above, the increasing rate of viscous dissipation is positively correlated to the velocity gradient within the fluid, which can be reflected from the deformation and oscillation of the droplet. From Figs. 4.8, 4.17, and 4.18, we can see that in the early stage of droplet coalescence, 60-degree droplet distribution has the most dramatic deformation, thus having the largest increasing rate of viscous dissipation. What is more, due to the continuous oscillation in the later stage of droplet jumping, the viscous dissipation of 180-degree and 120-degree droplet distribution during the process of jumping gradually exceeds

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Fig. 4.18 Droplet morphology evolution during coalescence-induced jumping process with 120degree droplet distribution: a side-view diagram; b top-view diagram

Fig. 4.19 Variation of unit mass surface energy of droplets during jumping regarding time

the dissipation of 60-degree droplet distribution. In addition, compared with 180degree droplet distribution, 120-degree droplet distribution has the less symmetric and harmonious oscillation morphology after the coalescence, leading to a larger increasing rate of viscous dissipation in the later stage, which eventually surpasses the dissipation of 180-degree droplet distribution.

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Fig. 4.20 Variation of unit mass kinetic energy of droplets during jumping regarding time

Fig. 4.21 Variation of unit mass viscous dissipation of droplets during jumping regarding time

Fig. 4.22 Variation of a jumping velocity and b effective jumping kinetic energy of droplets during jumping regarding time

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Fig. 4.23 Energy conversion rates in droplet jumping

Conversion rate (%)

40

180° 120° 60°

30 20 10 0

Ekin/ΔEsur

Ejump/Ekin

Ejump/ΔEsur

Figure 4.22 consists of the jumping velocities of droplets (in the vertical direction) and the variations of effective jumping kinetic energy with three types of droplet position distributions, which are calculated by formula (4.14) and (4.15). 60-degree droplet distribution has the largest jumping velocity after the coalescence, with the largest effective jumping kinetic energy of the droplet; 180-degree droplet distribution ranks the second, and 120-degree droplet distribution has the lowest jumping velocity as well as the effective jumping kinetic energy. Besides, 60-degree droplet distribution is the earliest in terms of departing from the surface after the coalescence-induced droplet jumping. For the coalescence-induced droplet jumping of three types of droplet position distribution, the conversion rate from surface energy to total kinetic energy and effective jumping kinetic energy, as well as the ratio of effective jumping kinetic energy to total energy, are calculated, which are shown in Fig. 4.23. The results indicate that the conversion rates from surface energy to total kinetic energy after coalescence are almost identical among three types of droplet position distribution; that is, this conversion rate is independent from the droplet position distribution. However, there is a significant difference in the conversion rate from surface energy to effective jumping kinetic energy among three cases. The coalescence-induced jumping with 60-degree droplet distribution has the highest conversion rate from surface energy to effective jumping kinetic energy; 180-degree droplet distribution ranks second, and 120-degree droplet distribution has the lowest conversion rate. Similar result appears in the ratio of effective jumping kinetic energy to total energy. The ratio after droplet coalescence with a 60-degree droplet distribution is the highest; that is to say, vibration kinetic energy makes up the least proportion in this case; the ratio after droplet coalescence with a 120-degree droplet distribution is the lowest, where vibration kinetic energy makes up the largest proportion. This result reveals that a more concentrated and symmetric droplet position distribution is conducive to improving the conversion rate from surface energy to effective jumping kinetic energy during the process of droplet jumping, which helps to accelerate the jumping velocity of the droplet.

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4.5 Summary In this chapter, VOF model is adopted to simulate the self-propelled droplet jumping on the superhydrophobic surface, and the focus of this chapter is the multi-droplet coalescence-induced droplet jumping. The energy variation and energy conversion during jumping process are analyzed, and the effects of the coalesced droplet number and droplet position distribution on droplet jumping are discussed. The research reveals that: (a) Compared with the two-droplet coalescence-induced droplet jumping phenomenon captured in experiments or obtained via simulations in the literature, the simulation model in the present work is able to simulate droplet jumping phenomenon properly, and the results of droplet morphology evolution and stage division are in good agreement with previous work from literature. The simulated result of droplet jumping velocity in the present work has an only 4% deviation from the experimental and computational results in the literature. (b) For multi-droplet coalescence-induced droplet jumping, as the number of coalesced droplets increases, more surface energy is released from the coalescence, and the conversion rate from surface energy to kinetic energy increases. As the number of coalesced droplets increases, the proportion of effective kinetic energy to the total energy rises, as well as the jumping velocity. Only little proportion of surface energy can be transformed into effective jumping kinetic energy for the droplet even for four-droplet coalescence-induced jumping, the conversion rate is less than 10%; for two-droplet coalescence-induced jumping, the rate is less than 5%. (c) When the number and the radii of coalesced droplets are the same, the droplet position distribution does not affect the release of surface energy and has almost no effect on the conversion rate from released surface energy to total kinetic energy. Nonetheless, droplet position distribution affects the conversion rate from released surface energy to effective jumping kinetic energy. The more concentrated and symmetric the coalesced droplets are, the higher the conversion rate and droplet jumping velocity. Droplet with position distribution of low degree of concentration or symmetry undergoes a large oscillation amplitude, a long oscillation time and an unharmonious oscillation morphology, thereby producing a larger amount of viscous dissipation.

References 1. Attarzadeh R, Dolatabadi A (2017) Coalescence-induced jumping of micro-droplets on heterogeneous superhydrophobic surfaces. Phys Fluids 29(1):012104 2. Hirt CW, Nichols BD (1981) Volume of Fluid (VoF) method for the dynamics of free boundaries. J Comput Phys 39(1):201–225 3. Brackbill JU, Kothe DB, Zemach C (1992) A continuum method for modeling surface-tension. J Comput Phys 100(2):335–354

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4. Nam Y, Kim H, Shin S (2013) Energy and hydrodynamic analyses of coalescence-induced jumping droplets. Appl Phys Lett 103(16):161601 5. Hager WH (2012) Wilfrid Noel Bond and the Bond number. J Hydraul Res 50(1):3–9 6. Liu F, Ghigliotti G, Feng JJ, Chen C-H (2014) Numerical simulations of self-propelled jumping upon drop coalescence on non-wetting surfaces. J Fluid Mech 752:39–65 7. Boreyko J, Chen C-H (2009) Self-propelled dropwise condensate on superhydrophobic surfaces. Phys Rev Lett 103(18):184501 8. Wang K, Liang Q, Jiang R, Zheng Y, Lan Z, Ma X (2016) Self-enhancement of droplet jumping velocity: the interaction of liquid bridge and surface texture. RSC Adv 6(101):99314–99321 9. Peng B, Wang S, Lan Z, Xu W, Wen R, Ma X (2013) Analysis of droplet jumping phenomenon with lattice Boltzmann simulation of droplet coalescence. Appl Phys Lett 102(15):151601 10. Cheng Y, Xu J, Sui Y (2016) Numerical investigation of coalescence-induced droplet jumping on superhydrophobic surfaces for efficient dropwise condensation heat transfer. Int J Heat Mass Transf 95:506–516 11. Lv C, Hao P, Yao Z, Niu F (2015) Departure of condensation droplets on superhydrophobic surfaces. Langmuir 31(8):2414–2420 12. Liu X, Cheng P, Quan X (2014) Lattice Boltzmann simulations for self-propelled jumping of droplets after coalescence on a superhydrophobic surface. Int J Heat Mass Transf 73:195–200 13. Khatir Z, Kubiak KJ, Jimack PK, Mathia TG (2016) Dropwise condensation heat transfer process optimisation on superhydrophobic surfaces using a multi-disciplinary approach. Appl Therm Eng 106:1337–1344 14. Enright R, Miljkovic N, Sprittles J, Nolan K, Mitchell R, Wang EN (2014) How coalescing droplets jump. ACS Nano 8(10):10352–10362

Chapter 5

Dynamic Melting of Freezing Droplets on Superhydrophobic Surfaces

Even for condensed droplets on superhydrophobic surfaces, it is not likely to remove all the droplets via self-propelled behaviors. When the surface temperature further decreases, the remaining droplets on the surface freeze and the frost crystals grow from the tips of freezing droplets as time goes by. If the freezing time is long enough, the frost crystals will propagate until they cover the entire surface. The freezing droplets or frost crystals on the surface may greatly affect the performance of various engineering surfaces, such as the fin surface of heat exchangers and the front surface of aircraft wings, which can even lead to safety accidents. Therefore, the melting and removal of frosts are needed for the surface. In the process of melting and removing frosts, the melted droplet is the fundamental object for research, whose behavioral characteristics determine the effect of frost melting and removal. However, there has been no scholar investigating motional behaviors of droplets melted from the freezing droplets on superhydrophobic surfaces yet. Hence, this chapter aims to conduct continuous experiments on droplet condensation, condensed droplet freezing, and melting of freezing droplets on superhydrophobic surfaces, and lay emphasis on the motional behaviors of the melting of freezing droplets, which helps to broaden the understanding of freezing droplet melting on superhydrophobic surfaces, and provide theoretical instructions to the improvement of melting on frosted surfaces, as well as the efficiency of frost removal.

5.1 Experimental Surfaces and Conditions The Al-based superhydrophobic surface with better performance is still selected as the experimental surface for experiments in this chapter. The surface is evenly covered with flower-like micro-nano structures, which is similar to the structures in Figs. 2.9 and 2.10. The static contact angle of the experimental surface in this chapter

© Tsinghua University Press 2020 F. Chu, Condensed and Melting Droplet Behavior on Superhydrophobic Surfaces, Springer Theses, https://doi.org/10.1007/978-981-15-8493-0_5

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is measured to be 160–165°, with a contact angle hysteresis of 4.0–6.0°. The bare aluminum surface is used as the control surface, which is the polished aluminum alloy surface. The superhydrophobic surface is connected to the cold terminal of semiconductor cooler via thermal grease in experiments. To eliminate the effect of gravity on the behaviors of melted droplets, the superhydrophobic surface is placed horizontally, and the level is calibrated by a bubble level. The microscope and the CCD high-speed camera are utilized to trace and capture the droplets, and the behavioral characteristics are investigated via observing and analyzing the captured images. Experiments are conducted in an enclosed laboratory. To achieve the continuous process of droplet condensation, condensed droplet freezing, and melting of freezing droplets, experiments are divided into three stages, and the working conditions for each stage are different from each other. The main working conditions for experiments include cold surface temperature (T sur ), wet air temperature (T air ), and relative humidity (RH), and the specific values in each stage are listed in Table 5.1. In Stage 1 of the experiments, the temperature of the superhydrophobic surface is −5.0 °C. Although the surface temperature is below 0 °C, due to the fact that superhydrophobic surfaces can effectively delay droplet freezing when the surface temperature is not too low (a minimum of −10 °C), therefore, only drop condensation takes place in Stage 1. In Stage 2, the surface temperature further decreases to −12 °C, which is cold enough, and the process of droplet freezing occurs on the superhydrophobic surface. At the end of Stage 2, condensed droplets on the surface have been completely frozen. Stage 3 is initiated by disconnecting the power supply from the semiconductor cooler, and consequently, the surface temperature rises rapidly. When the surface temperature is above 0 °C, freezing droplets start to melt naturally. In the entire process of experiments, three stages are continuous. The variation of working conditions in these continuous stages is shown in Fig. 5.1. Table 5.1 Working conditions and duration in each stage of experiments

Stage of experiments Working conditions

Duration (min)

Stage 1

T sur = −5.0 ± 0.1 °C T air = 18.8 ± 0.5 °C RH = 85.0 ± 5.0%

Stage 2

T sur = −12.0 ± 0.1 °C 15 T air = 18.8 ± 0.5 °C RH = 85.0 ± 5.0%

Stage 3

T sur = −12.0–18.8 °C