UNIVERSITE LIBRE DE BRUXELLES
Ecole Polytechnique de Bruxelles
Experimental Study of the Evaporation of Sessile Droplets
of Perfectly-Wetting Pure Liquids
Thèse présentée par
Ioannis Tsoumpas
en vue de l’obtention du grade de
Docteur en Sciences de l'ingénieur et technologie
Promoteur de thèse:
Pierre Colinet
Composition du jury :
Jean-Marie Buchlin (Président du jury)
Pierre Colinet
Joël De Coninck
Pierre Lambert
Detlef Lohse
Benoit Scheid
(Secrétaire du jury)
Uwe Thiele
Acknowledgments
During my doctoral studies, and with Brussels as my personal headquarters, I was given the opportu-nity to explore various countries, meet different people and consequently discover different sides of myself. I do sense that all the knowledge and experience that I have gained in these years have not only expanded my professional skills but they have also broaden my cultural and personal horizons; time will tell whether I were indeed able to absorb completely the fruits of this journey.
“Who lives sees, but who travels sees more” Independently of the outcome, however, I feel the need to thank the people that have influenced me in one way or another during these years. First and foremost, I would like to express my deepest gratitude to Professor Pierre Colinet for accepting me as his PhD student and offering me the chance to be a member of his group as well as of the Marie Curie Actions’ MULTIFLOW initial training network. His constructive and far-sighted comments about the scientific work presented in this dis-sertation were more than useful, especially at times when the next step seemed unclear. Working under the supervision of Professor Colinet I discovered a more fundamental approach to science that complemented my engineering points of view. It is mainly thanks to him that I now realize that an experiment without an underlying theory is like a theory without an accompanying experiment. I am also beholden to Dr. Sam Dehaeck and Dr. Alexey Rednikov for their constant support, encour-agement and mentoring; they were always there for me. I have been very lucky to collaborate with them and to me they are something more than just colleagues. I would also like to take the opportunity and thank all my colleagues (past and present) in the Transfers, Interfaces and Processes department for the warm and friendly atmosphere they have created over these years.
Professor Uwe Thiele and Dr. Mariano Galvagno definitely deserve my deep appreciations for the warmth of their welcome and the hospitality they showed me during my stay at the University of Loughborough. I am truly thankful for this fruitful and enlightening collaboration; I indeed enjoyed every single second of it.
I also feel extremely grateful to all the members of MULTIFLOW. Participating in a network that consisted mainly of young scientists relieved the feeling of loneliness that most PhD students must have felt at some point during their research.
I am also much obliged to Olivier Berten and his technical team as well as to Professor Heidi Ottevaere for kindly offering me their help, Dr. Michalis Vlachogiannis for his inspirational advice and of course Dr. Maria Rosaria Vetrano for being the stepping stone to this long journey that is now coming to an end.
all the invaluable emotional support that you have offered me in order to finalize my research project and above all to experience a joyful life, both social and academic. This dissertation is also partially dedicated to the memory of Dimitris Kazantzis; the more I grow up, the more I feel the need to discuss with him.
P.S.: The pictures shown at the beginning of each chapter are taken from different parts of the world during (I mean after) various conferences that I have attended. At this point, I would like to gratefully thank my friend Mariano who initiated me in the philosophy of photography. I hope the results will not let him down. In particular, one sees the Passage de Lorette in Marseilles (Chapter 1), a pleasant walk around the Porta Nuova railway station in Turin (Chapter 2), a female street artist at the middle of Grafton Street in Dublin (Chapter 3), one of the numerous mystic alleys of Jerusalem (Chapter 4), the scenic coastline of Traigh Mhor beach on the Isle of Lewis (Chapter 5), the crowded streets of Manhattan in New York City (Chapter 6), the bridge located at the outstanding Musée des Civilisations de l’Europe et de la Méditerranée in Marseilles (Chapter 7) and a normal morning around the Porto Antico in Genoa (Appendix A). The first word coming to my mind when looking at the these pictures is together...
Really, after f irst approach everything looked assorted
Contents
Acknowledgments v
List of Figures xiii
1 Introduction 1
2 Basic Concepts of Capillarity 7
2.1 Surface Tension . . . 7
2.2 Laplace Pressure . . . 8
2.3 Contact Between Three Phases . . . 9
2.3.1 Equilibrium Contact Angle: Young’s Equation . . . 9
2.3.2 Non-Equilibrium Contact Angle . . . 11
3 Experimental Methods 17 3.1 Liquids . . . 17
3.2 Interferometric Methods . . . 18
3.2.1 Comparison Between Different Methods . . . 21
3.3 Post Processing . . . 33
3.3.1 Profile Extraction . . . 34
3.3.2 Contact Radius and Contact Line Velocity . . . 35
3.3.3 Contact Angle . . . 38
3.3.4 Volume and Evaporation Rate . . . 38
3.4 Substrate . . . 38
4 Freely Receding Volatile Drops of Wetting Liquids 41 4.1 Evaporating Droplets . . . 42
4.1.1 Local and Global Evaporation Rate . . . 43
4.1.2 Evolution of the Droplet Radius During Evaporation . . . 45
CONTENTS 4.2 Results . . . 50 4.2.1 HFE - 7100 . . . 50 4.2.2 HFE - 7200 . . . 54 4.2.3 HFE - 7500 . . . 57 4.3 Discussion . . . 61
4.3.1 Global Evaporation Rate . . . 61
4.3.2 Apparent and Evaporation-Induced Contact Angles . . . 63
4.3.3 Radius and Apparent Contact Angle Evolution in Time . . . 66
4.3.4 Travelling Waves and Fluctuations of the Contact Line . . . 68
5 Effect of Evaporation on Droplet Shape 71 5.1 Marangoni Effect . . . 71
5.2 Experimental and Classical-Static Shapes for Freely Receding Drops . . . 76
5.2.1 Classical Static Shape of a Sessile Droplet . . . 76
5.2.2 Comparison with the Experiments . . . 78
5.2.2.1 HFE-7100 . . . 78
5.2.2.2 HFE-7200 . . . 80
5.2.2.3 HFE-7500 . . . 82
5.2.2.4 Discussion . . . 83
5.3 Experimental and Classical-Static Shapes for Pinned Drops . . . 93
5.3.1 Results . . . 95
5.3.2 Discussion . . . 101
5.4 General Discussion . . . 101
5.4.1 Notes on Existing Literature . . . 101
5.4.2 Concluding Remarks . . . 103
6 Effect of Evaporation on Contact Line Depinning 105 6.1 Gibbs’ Criterion . . . 105
6.2 Experimental Methods . . . 106
6.2.1 Groove Fabrication . . . 106
6.2.2 Experimental Apparatus and Procedure . . . 107
6.3 Results . . . 110
6.4 Discussion . . . 111
6.4.1 Comparison with the Classical-Static Shape Theory . . . 111
CONTENTS
6.4.2.1 Theoretical Approach . . . 115
6.4.3 Contact-Line-Speed-Influenced Gibbs’ Criterion . . . 116
6.4.4 Square Groove . . . 117
6.4.5 Concluding Remarks . . . 122
7 Conclusions 123
Bibliography 125
A Validation Check A.1
List of Figures
1.1 (a) A water droplet on a lotus leaf, attaining a spherical shape; (b,c) electron mi-croscopy images of an Indian Cress leaf demonstrating its rough surface structure. From Otten and Herminghaus [2004]. (d) A strider on the surface of water; (e) its ability to repel water lies in the microstructures that are found on its leg, and (f) espe-cially in the nanogrooves that exist on these structures. From Feng et al. [2007]. . . 2 1.2 (a) A Wilson’s phalarope, found mostly in the prairies of North America, during its
meal. The prey is inside the captured water droplet; (b) a mechanical wedge that mod-els the opening and closing cycle of the beak, illustrating how the droplet eventually reaches the bird’s mouth. The underlying principle, referred to as contact angle hys-teresis, is the same that allows water droplets stick on vertical windows after a rainy Sunday morning. From Prakash et al. [2008]. . . 3 1.3 (a) A scheme of a flip-flop device. A flip-flop device can store a binary bit by
remem-bering in which of the two possible states the device was last set up. Here, the memory bit is represented by the droplet sitting on one of the two positions that lie between the input and the outputs; a time sequence demonstrating the flip-flop memory operation is also shown (droplet radius' 1.5mm, track: width' 1mm, depth' 0.4mm). From Mertaniemi et al. [2012]. (b) In this flow-focusing microfluidic device an initially stable flow of an aqueous solution, coming from the left, breaks up into small drops as it is squeezed by the outer continuous phase (oil). The size of these drops can be fine-tuned through the relative flow rate of the two immiscible liquids allowing in this way well-controlled experiments in a very small scale (lab-on-a-chip). From Arratia et al. [2008]. . . 3 1.4 (a) A Leidenfrost water drop on a flat metallic plate heated up to 300◦C. From Quéré
[2013]; (b) a self-propelled Leidenfrost drop of R134a (radius' 1.5mm) on a saw-tooth shaped brass surface (moving direction is to the right); (c) the ratchet effect can be explained in terms of pressure differential in the vapor layer. Namely, the high pressure region at A induces both a forward and a backward flow, with the net viscous forces due to the later being negligible. The levitated drop eventually moves to the right dragged by the forward flow of the vapor layer. From Linke et al. [2006]. . . . 5 1.5 (a) A sketch of an evaporating droplet on a flat solid surface that illustrates the
LIST OF FIGURES
1.6 (a) A dried drop of blood. The bloodstain pattern analysis could assist in the diag-nosis of blood diseases, e.g. anaemia or hyperlipidaemia. From Brutin et al. [2011]. (b) Scanning electron microscopy images of silver (left) and ZnO (right) particles produced by chemical decomposition of a Leidenfrost drop. As the drop slides on a tilted surface, dense wires of nanoclusters are left as footprints. From Elbahri et al. [2007]. (c) Pattern of an onion tear. From the personal web page of Rose-Lynn Fisher. Interestingly, this flower-like shape can also be seen in cases of dewetting nanofluids (Pauliac-Vaujour et al. [2008]). . . 5
2.1 (a) A soap film will tend to resist any attempt of surface area increase owing to surface tension; (b) molecules at the surface of a liquid bath are subjected to a net force directed towards the bulk, whereas the attractive interactions that the molecules inside the liquid feel balance each other. This is why an excess energy is associated with the liquid-air interface; (c) the air pressure is always higher in the interior of an inflated balloon. When the balloon is untied the pressure difference drives the air out. One could say that the surface energy of the balloon is converted to the kinetic energy of the air jet, which makes the balloon hover around as a result of the produced thrust. . 8 2.2 (a) Schematic representation of a sessile drop, which partially wets a substrate, and
of the acting interfacial tensions at the contact line; (b) Derivation of Young’s equa-tion based on free energy consideraequa-tions; (c) schematic representaequa-tion of a complete wetting situation where the equilibrium contact angle is zero. . . 9 2.3 Scheme of the effect of gravity on the shape of a droplet for various radii. . . 10 2.4 Liquid addition to a droplet: (a) in the absence of hysteresis contact line advances
immediately whereas (b) for non-ideal surfaces it moves for angles larger than the Young’s angle. . . 11 2.5 A drawing of a spreading droplet the macroscopic shape of which is described by an
apparent contact angle. Nevertheless, near the contact line a dynamic angle which varies logarithmically with the distance from the contact line can be defined. The region surrounded by the dashed circle refers to a region in the vicinity of the contact line, shown in Fig. 2.6. . . 12 2.6 Microscopic parameters need to be included in the hydrodynamic analysis of a
mov-ing droplet to estimate the flow near the edge. In one of the models a precursor film is expected to be present ahead of the apparent contact line where the interactions between the liquid and solid molecules are not to be neglected. . . 13 2.7 (a) Thomas Young (1773-1829) and (b) Pierre Simon de Laplace (1749-1827) are
both known for their contributions not only to the subject of capillarity but also to numerous other scientific fields, such as linguistics (Parkinson [1999]) and political sciences (Ball [2002]), respectively. . . 15 3.1 Measurements of the refractive index of various liquid HFEs at different temperatures. 18 3.2 A schematic representation of the configuration used to obtain an expanded and
noise-free laser beam. From left to right we have the laser light source, the microscope lens, the pinhole and the double convex lens. . . 19
LIST OF FIGURES
3.3 Schematic representation of a side view of (a) a Double - Beam interferometer and (b) a Michelson interferometer. Initially the collimated beam is displayed as rays and at the end as plane wave propagation in terms of red parallel lines. . . 20 3.4 Fringe pattern obtained when (a) no drop is present and (b) after depositing a drop
with the initial fringes, however, “tuned off” (zero fringe mode). . . 21 3.5 Scheme of a Mach-Zehnder interferometer (side view). . . 22 3.6 On a small fragment of the drop we present the optical path of the interfering beams
in a Double - Beam (left) and a Michelson interferometer (right). . . 25 3.7 The optical path difference introduced by the presence of a liquid droplet in the path
of the measurement beam in a Mach - Zehnder interferometer. . . 26 3.8 Numerical aperture limitation: for a given objective, light deflected by the drop can be
(a) either fully captured by the lens or (b) partially escape from it, depending on the contact angle of the drop; (c) the latter case can be avoided if we eventually decrease the distance between the objective and the drop; (d) in general, our target here is to determine for different interferometries and distances d, the contact angle up to which the objective gathers all the light emitted by the drop . . . 29 3.9 A sketch that demonstrates the path of the incident beam inside the drop in various
configurations. On the left we assume a Michelson or a Double Beam interferometer whereas on the right a Mach - Zehnder. The liquid-air interface is approximated by a straight line as we are considering only the points of the drop that are located in the vicinity of the contact line. . . 30 3.10 Critical contact angles, regarding various distances and interferometers, for which
light is deflected towards the pupil of our objective ( f/#= 2.8, F = 10.5cm). . . 31 3.11 (a) Maximum measurable contact angle that can be safely sampled by our sensor
(5.5µm true pixel size) for various magnifications and interferometers; (b) compar-ison between the Numerical Aperture limitation (dictated by the lens) and the Sam-pling limitation (dictated by the camera’s pixel size ) for the Mach-Zehnder interfer-ometer. . . 33 3.12 (a) Two examples of a Morlet wavelet used in the CWT algorithm; (b) The
interfero-gram of Fig. 3.4b presented now in polar coordinates. . . 34 3.13 (a) Intensity versus pixel position of row 180 on which we perform the 1D CWT; (b)
modulus and (c) phase of the complex array obtained after applying the 1D CWT. . . 35 3.14 (a) Wrapped and (b) unwrapped phase of row 180; (c) Calibrated 3D representation
of the processed droplet (maximum height 0.04mm) along with a part of the substrate. 36 3.15 Amplitude of the fitting wavelets versus pixel position for (a) a single row or (b) for
all rows (in grayscale) of Fig. 3.12b based on which we can extract the contact line position shown as blue points in (c). (d) CWT-processed image of the interference pattern transformed back to cartesian coordinates; contours correspond to the local slope of the drop indicated in terms of angles. . . 37 3.16 We apply the trapezoidal rule to calculate first (a) the “area” below a profile of the
LIST OF FIGURES
4.1 Three schematic representations of liquid evaporation taking place in a cell. When the cell is sealed and no vapor is present in the gas phase (a) liquid molecules evaporate filling up the gas phase until (b) we are at an equilibrium where both evaporation and condensation occur at the same rate; (c) As soon as the seal is removed, evaporation continues until all the liquid is vanished (for a practically low ambient humidity). . . 42 4.2 (a) Contour plot of the vapor concentration distribution above a droplet of 1mm radius
and 0.364mm height during the diffusion-controlled evaporation process. From Hu and Larson [2002]; (b) part of the top view of an evaporating water droplet. The induced flow inside the drop drags suspended particles towards the pinned contact line. From Yunker et al. [2011]. . . 44 4.3 Distribution of the evaporation flux along the surface of a sessile water droplet. A
divergence at the contact line is apparent. From Hu and Larson [2002]. . . 44 4.4 Sketch of the microstructure near the contact line where the evaporation-induced
an-gles are located. . . 48 4.5 Apparent contact angle versus the elapsed time for (a) a single experiment and (b) ten
different experiments (HFE-7100). . . 51 4.6 Apparent contact angle versus (a) the time remaining for the drop to evaporate
com-pletely and (b) the drop radius after averaging the results of different experiments as shown in the inset (HFE-7100). . . 51 4.7 (a) Apparent contact angle and (b) radius versus the time remaining for the drop to
evaporate completely after averaging the results of different experiments (logarithmic scale) (HFE-7100). . . 52 4.8 Contact line velocity versus (a) the time remaining for the drop to evaporate
com-pletely and (b) the drop radius after averaging the results of different experiments (HFE-7100). . . 53 4.9 (a) Ratio of the angle increment induced by the speed of the contact line to the
ex-perimentally determined contact angle; (b) Measured apparent contact angle and the evaporation-induced contact angle after extracting from the former the influence of the contact line speed (HFE-7100). . . 53 4.10 (a) Volume versus radius after averaging the results of different experiments. The inset
shows the volume versus the time remaining for the drop to evaporate completely for all the experimental runs; (b) global evaporation rate versus the radius again after averaging the results of different experiments (HFE-7100). . . 54 4.11 Apparent contact angle versus (a) the time remaining for the drop to evaporate
com-pletely and (b) the drop radius after averaging the results of different experiments (HFE-7200). . . 55 4.12 Contact line speed versus (a) the time remaining for the drop to evaporate completely
and (b) the drop radius after averaging the results of different experiments (HFE-7200). . . 55 4.13 (a) Ratio of the angle increment induced by the speed of the contact line to the
ex-perimentally determined contact angle; (b) Measured apparent contact angle and the evaporation-induced contact angle after extracting from the former one the influence of the contact line speed (HFE-7200). . . 55
LIST OF FIGURES
4.14 (a) Apparent contact angle and (b) radius for HFE-7200 versus the time remaining for the drop to evaporate completely after averaging the results of different experiments (logarithmic scale) (HFE-7200). . . 56 4.15 (a) Volume versus the radius after averaging the results of different experiments. The
inset shows the volume versus the time remaining for the drop to evaporate completely for all the experimental runs; (b) global evaporation rate versus the radius again after averaging the results of different experiments (HFE-7200). . . 56 4.16 Apparent contact angle versus (a) the elapsed time and (b) the radius for a single
experiment (HFE-7500). . . 58 4.17 Apparent contact angle versus (a) the time remaining for the drop to evaporate
com-pletely and (b) the drop radius after averaging the results of different experiments (HFE-7500). . . 58 4.18 (a) Apparent contact angle and (b) radius versus the time remaining for the drop to
evaporate completely after averaging the results of different experiments (logarithmic scale) (HFE-7500). . . 59 4.19 Contact line speed versus (a) the time remaining for the drop to evaporate completely
and (b) the drop radius after averaging the results of different experiments (HFE-7500). . . 60 4.20 (a) Ratio of the angle induced by the speed of the contact line to the experimentally
determined contact angle; (b) Measured apparent contact angle and the induced con-tact angle by evaporation after extracting from the former one the influence of the contact line speed (HFE-7500). . . 60 4.21 (a) Volume versus radius after averaging the results of different experiments. The inset
shows the volume versus the time remaining for the drop to evaporate completely for all the experimental runs; (b) global evaporation rate versus the radius again after averaging the results of different experiments (HFE-7500). . . 60 4.22 Comparing the experimental results of different liquids; (a) global evaporation rate
with respect to a linear fit of the evaporation rate for small radii (up to 0.5mm) and (b) percentage difference between these two; (c) global evaporation rate with respect to the pure diffusion case and (d) percentage difference between these two. All quantities are plotted versus the radius of the drop. . . 61 4.23 The dimensionless evaporation rate versus (a) the droplet radius and (b) the Grashof
number. . . 63 4.24 Comparing the experimental results for different liquids; (a) apparent contact angle,
(b) contact line speed and (c) evaporation-induced contact angle in the normal as well as in (d) logarithmic scale versus the drop radius. . . 64 4.25 Scaling of the (a) evaporation-induced and (b) apparent contact angle with the
evap-oration rate for the three HFEs at normal and heated conditions, as far as a sessile configuration is concerned. Pendant drops of HFE-7100 and 7200 as well as hexane and heptane sessile drops in normal ambient conditions are also shown. . . 66 4.26 Comparing the experimental results of different liquids; (a) apparent contact angle
LIST OF FIGURES
4.27 (a) A receding drop of HFE-7500 which (b) suddenly spreads south-west. (c) After the local spreading has reached the maximum (d) axisymmetry is quickly achieved; however, another part of the drop is now pinned. (e) Contact line speed and apparent contact angle versus the vanishing time of drop. The arrows follow the fluctuation introduced by the travelling wave but not by the pinning. . . 69 5.1 Schematic representation of the hexagonal pattern observed in Marangoni–Bénard
convection. From Colinet et al. [2001]. . . 72 5.2 A liquid layer of HFE 7000 evaporating freely from a container exposed to open air.
As the thickness of the layer decreases the Marangoni effect decreases as well, while the characteristic convection cells become smaller in size. From Chauvet et al. [2012]. 73 5.3 (a) Experimental image and (b) model predictions of the flow field inside an
evapo-rating octane droplet of a contact radius of 2mm. From Hu and Larson [2006]. . . 73 5.4 (a) Critical relative thermal conductivity according to Eq. (5.2) (inset sketches indicate
the direction of the flow in each region) (Ristenpart et al. [2007]) and (b) critical Rn
values versus the contact angle according to Eq. (5.3) (inset sketches indicating the direction of the temperature field and of the respective flow in each region), for which the Marangoni flow reverses direction (Xu et al. [2009]). . . 74 5.5 (a) Critical relative thermal conductivity versus contact angle for a constant hR= 1.25,
according to (5.3). Inset sketches indicate the direction of the flow in each region; (b) schematic representation of the non-monotonic temperature gradient along the liquid-air interface at critical values of the contact angle as described in Zhang et al. [2014]. 75 5.6 Sketch of a sessile droplet on a flat solid surface. . . 76 5.7 Droplet profiles calculated according to Eq. (5.10) (solid line) assuming a contact
an-gle of 10◦and a contact radius equal to (a) three times and (c) half the capillary length. In (b) and (d) the respective local angles are shown. The dashed line corresponds to a parabolic fit. . . 78 5.8 (a) Experimental profile of a cross-section of an evaporating droplet of HFE-7100
(θt= 8.86◦, R = 1.63mm) fitted by a parabola and compared to the predictions of the
classical static shape assuming the same contact radius and true contact angle; (b) corresponding local slope of each profile. . . 79 5.9 (a) Experimental profile of a cross-section of an evaporating droplet of HFE-7100
(θt= 8.30◦, R = 0.65mm) fitted by a parabola and compared to the predictions of the
classical static shape assuming the same contact radius and true contact angle; (b) corresponding local slope of each profile. . . 80 5.10 True contact angles for evaporating droplets of HFE-7100 as compared to (a) the ones
predicted by the classical static shape theory, assuming the same volume and contact radius, and (b) the parabolic ones. . . 81 5.11 Experimental profiles of an evaporating droplet of HFE-7200 at different moments,
(a) θt = 6.00◦, R = 1.63mm and (c) θt= 5.80◦, R = 0.65mm, fitted by a parabola and
compared to the predictions of the classical static shape assuming the same contact radius and true contact angle; (b,d) corresponding local slope of each profile. . . 81
LIST OF FIGURES
5.12 True contact angles for evaporating droplets of HFE-7200 as compared to (a) the ones predicted by the classical static shape theory, assuming the same volume and contact radius, and (b) the parabolic ones. . . 82 5.13 Experimental profiles of an evaporating droplet of HFE-7500 at different moments,
(a) θt = 3.48◦, R = 1.63mm and (c) θt= 3.60◦, R = 0.65mm, fitted by a parabola and
compared to the predictions of the classical static shape assuming the same contact radius and true contact angle; (b, d) corresponding local slope of each profile. . . 83 5.14 True contact angles for evaporating droplets of HFE-7500 as compared to (a) the ones
predicted by the classical static shape theory, assuming the same volume and contact radius, and (b) the parabolic ones. . . 84 5.15 (a) Profile of a receding PDMS drop at various instants, fitted by parabolas; (b)
schematic illustration that indicates the point of inflection of the liquid-air interface in the vicinity of the contact line, when the droplet starts to recede. From Guéna et al. [2007c]. . . 84 5.16 Exaggerated sketch that illustrates the effect of thermocapillary stresses on the shape
of an evaporating sessile droplet, in the case of a radially inward flow along the liquid-air interface. . . 85 5.17 Scheme of a flattened evaporating liquid droplet resting on a flat solid surface. . . . 86 5.18 Theoretical profiles of an evaporating droplet of HFE-7100 by varying the Marangoni
effect strength relative to its actual value; (a) θ = 8◦, R = 1.6mm and (b) correspond-ing local angles; (c) θ = 8.3◦, R = 0.87mm and (d) corresponding local slopes. . . . 89 5.19 (a, c) The experimental profiles for HFE-7100 and (b, d) the corresponding local
slopes, shown before (Figs. 5.8 and 5.9), are compared now to the predictions of the model. . . 90 5.20 Same as in Fig. 5.19 with the model now modified in order to include the deviations
from the diffusion-limited theory. . . 91 5.21 (a, c) The experimental profiles for HFE-7200 and (b, d) the corresponding local
slopes, shown before (Fig. 5.11), are now compared to the predictions of the (modi-fied) model. . . 92 5.22 (a, c) The experimental profiles for HFE-7500 and (b, d) the corresponding local
slopes, shown before (Fig. 5.13), are now compared to the predictions of the (modi-fied) model. . . 93 5.23 Measured contact angles as compared to the ones predicted by the classical static
shape theory and by the model for the same droplet volume and contact radius in the case of (a) 7100 (original model), (b) 7100 (modified model), (c) HFE-7200 (modified model) and (d) HFE-7500 (modified model). . . 94 5.24 Indicative images of the 0.86mm groove, obtained with a 3D confocal microscope. . 95 5.25 Measured contact angles as compared to the ones predicted by our model for the same
LIST OF FIGURES
5.26 Snapshot of the experimental profile and the corresponding local slope of an evapo-rating droplet of (a, b) HFE-7100, (c, d) HFE-7200 and (e, f) HFE-7500 pinned on the 2.37mm groove. The results are also compared to the predictions of our model as well as to the classical static theory based on the measured contact angle. . . 98 5.27 Snapshot of the experimental profile and the corresponding local slope of an
evapo-rating droplet of (a, b) HFE-7100, (c, d) HFE-7200 and (e, f) HFE-7500 pinned on the 1.62mm groove. The results are also compared to the predictions of our model as well as to the classical static theory based on the measured contact angle. . . 99 5.28 Snapshot of the experimental profile and the corresponding local slope of an
evapo-rating droplet of (a, b) HFE-7100, (c, d) HFE-7200 and (e, f) HFE-7500 pinned on the 0.86mm groove. The results are also compared to the predictions of our model as well as to the classical static theory based on the measured contact angle. . . 100 5.29 Comparison of the experimental results concerning both the freely receding and the
pinned droplets with the theory of Xu et al. [2009] described shortly in section 5.1. . 102 5.30 (a) A theoretical profile of an evaporating droplet of HFE-7500 according to the
model, assuming a very small contact angle (θ = 1.7◦, R = 2.5mm); (b) the cor-responding local slope, where an inflection point near the contact line is present. . . 103
6.1 Illustration of the Gibbs’ criterion for a droplet on a substrate with a sharp edge. The shaded zone indicates the range of possible equilibrium values (between θ and θ + α) that the apparent contact angle can adopt when the contact line is pinned at the edge. 106 6.2 The excimer laser has the ability to break the chemical bonds in a polymer, while a
mask allows us to control the area of the material that burns out. From Crafer and Oakley [1992]. . . 107 6.3 (a) Several positions of the mask with respect to its circular path; (b) when the
tra-jectory of the mask is aligned with its diagonal a V-shaped cross section is generated whereas (c) a U-shaped profile is observed when the trajectory is aligned to the sides of the mask; (d,e) 3D images of the parts of the groove corresponding to triangular and rectangular profiles, respectively, taken with a laser confocal microscope. . . 108 6.4 FM40 EasyDrop Tensiometer and its working principle. . . 109 6.5 Experimentally-obtained side views showing a part of a quasi-steadily evaporating
droplet at different stages together with schematic representations (insets): (a) free, (b,c) pinned at the groove edge, and (d) depinned contact line. . . 109 6.6 (a) Apparent contact angle and (b) apex of the drop at the moment of depinning as well
as (c) total experimental time in linear and (d) logarithmic scale for various HFEs; (e) total experimental time only for HFE-7500; (f) calculated total injected volume for various HFEs. All values are plotted versus the injection rate. . . 112 6.7 Assuming the same apparent contact angle we plot together the results of the static
model and the experiments for the apex and volume: (a,b) HFE-7100, (c,d) HFE-7200 and (e,f) HFE-7500. . . 113
LIST OF FIGURES
6.8 (a) Typical height profiles (solid lines) of steady evaporating drops of a perfectly wet-ting liquid on a substrate with a downward bend and the corresponding evaporation flux density profiles (broken lines) calculated with auto07p for various influxes; (b) apparent contact angle with respect to the horizontal versus the position of the contact line x0 (relative to the bend position c), for various cases as indicated in the legend.
All quantities, including the angles, are in rescaled units. . . 115 6.9 Cubic difference between the advancing apparent contact angle (determined
implic-itly from the experiment) and the apparent contact angle induced by evaporation ver-sus the injection rate for HFE-7500. . . 117 6.10 The profile of a pinned droplet when this is seen from two different perspectives: (a)
along one of the sides and (b) perpendicular to the diagonal. . . 118 6.11 Experiments performed for different flow rates while we are focused on the middle
point of an edge are compared to experiments during which we are looking at a single corner. The comparison takes place in terms of (a) the apex of the drop, (b) the total experimental time, (c) the calculated total injected volume and (d) the apparent contact angle at the moment of depinning. The results, which concern only HFE-7100, are also compared to the predictions of the classical static model under the premise of the same apparent contact angle at the middle point of the edge. . . 119 6.12 A three-dimensional plot of the surface of a pinned square-base drop. The apparent
contact angle in the middle point of the edge is chosen similar to the experimental one, i.e., 45◦. The bottom right plot shows the distribution of the apparent contact angle along a side of the drop. . . 120 6.13 (a) Middle and diagonal cross-sections of the droplet as well as (b) the respective
zooms near the contact line. . . 120 6.14 (a) Variation of the local contact angle with the angular coordinate along the perimeter
of the groove, for a given maximum contact angle around the perimeter and different corner-rounding ratios R; (b) minimum contact angle as a function of the ratio R, with the maximum contact angle θAvarying from 10◦to 90◦. From Soltman et al. [2013]. 121
6.15 Josiah Willard Gibbs (1839-1903); his significant contribution to physical science and mathematics is undisputed. A vast number of his papers can be found in the book "The Scientific Papers of J. Willard Gibbs". . . 122 A.1 (a) A computer generated interference pattern where the amplitude of the signal is
five times larger that that of the noise (radius=2mm, height=0.03mm). (b) Theoretical and calculated profiles extracted from two different interference patterns. . . A.2 A.2 Histograms that show the distribution of the height difference between the true and
the calculated profiles that are shown in Fig. A.1b. The left plot corresponds to the smaller drop and the right plot to the larger one. . . A.2 A.3 (a) Interferogram of a concave mirror as examined under a Michelson interferometer;
(b) same fringe pattern in polar coordinates; (c) modulus obtained after applying the 1D CWT on a single row. . . A.4 A.4 (a) Extracted calibrated profile of the mirror and (b) the evolution of its slope with the
LIST OF FIGURES
A.5 (a) Modulus that corresponds to a single row of an interferogram that experiences distortion due to violation of the sampling limitation and (b) the related unwrapped phase; (c) after the necessary calibrations we calculate the evolution of the angle with the radial distance. . . A.5
To my brother, who once told me
“you will remember your dissertation mostly
1
Introduction
The study described in this dissertation is related to the charming world of capillarity and wetting phenomena and more specifically to droplets. The adjective “charming” may indeed sound quite exaggerated to those who are unfamiliar with this subject area; however, this should change as soon as one considers this particular branch of science as the cross-section of physics, surface chemistry and engineering (Bonn et al. [2009]). To make the argument even stronger, some daily phenomena in addition to several industrial applications that are closely related to this field are mentioned below. One of the most commonly encountered examples is perhaps the so-called lotus effect. When water droplets are falling on plant leaves they will, most of the times, adopt a nearly spherical shape. The observed behavior of water is actually attributed to a rough structure of the leaves at a microscale, although at first sight these seem rather smooth (Fig. 1.1a-c). Eventually, droplets are likely to roll off the leaves carrying away dirt particles. Not only does this phenomenon favor the self-cleaning of the plants, but it also keeps them dry preventing moisture-friendly pathogens to reproduce and spread (Neinhuis and Barthlott [1997], Samaha and Gad-el Hak [2014]). The ability of water striders to walk on water (Fig. 1.1d-f), as well as the super-hydrophobic butterfly wings, can also be explained in the same context (Feng et al. [2007], Zheng et al. [2007]).
Additionally, capillary interactions between the beak of certain types of shorebirds (Fig. 1.2a) and water droplets can explain how these birds eventually manage to feed themselves (Fig. 1.2b), defying the laws of gravity (Prakash et al. [2008]).
As far as the practical applications are concerned, the list is rather inexhaustible. For example, the lotus effect principle has enabled scientists to create self-cleaning surfaces for the needs of automotive (e.g. water-repellent door windows and windshields; see Aegerter et al. [2008]), robotic (e.g. aquatic miniature robots that could ideally mimic the water strider abilities; as mentioned by Feng et al. [2007]) and construction industries (e.g. dust-free facades resistant to biodeterioration; nicely de-scribed by Solga et al. [2007]). In agriculture, however, the opposite result to the lotus effect is often desired in order to improve the deposition of pesticides on the leaf surface. This can be achieved, for instance, if one enhances the applied pesticide with certain types of additives (surfactants), promoting this way the spreading of the pesticide on the leaf (Bergeron et al. [2000]).
phenom-CHAPTER 1. INTRODUCTION
(a) (b) (c)
(d) (e) (f)
Figure 1.1: (a) A water droplet on a lotus leaf, attaining a spherical shape; (b,c) electron microscopy images of an Indian Cress leaf demonstrating its rough surface structure. From Otten and Herminghaus [2004]. (d) A strider on the surface of water; (e) its ability to repel water lies in the microstructures that are found on its leg, and (f) especially in the nanogrooves that exist on these structures. From Fenget al. [2007].
ena. On this topic, a recent study published by Mertaniemi et al. [2012] demonstrates how through guided droplet collisions on superhydrophobic surfaces one can fabricate memory devices as well as devices that execute elementary Boolean operations, where droplets act as bits of digital information (Fig. 1.3a). This simple yet remarkable work could potentially open the way for autonomous logic devices that are electricity-free. In a different field, that of optics, droplets can be used as adjustable lenses; that is to say, varying the shape of a single droplet results in changes to its focal length and consequently to different optical behaviors (Ren et al. [2010]). To better control the shape of a liquid lens a combination of electronics and optics is usually involved. In particular, by applying an external voltage field, a method called electrowetting (Mugele and Baret [2005]), one can alter the liquid-solid interactions and hence manipulate the droplet shape. A non-trivial application rising from the cou-pling of optics and electronics is a variable-focus liquid lens mounted on a miniature camera, which for example can be exploited by portable device manufacturers (Kuiper and Hendriks [2004]). The list extends even more with applications regarding chemical, biological, pharmaceutical and med-ical sciences. There, microfluidic devices (Beebe et al. [2002], Whitesides [2006]) are often employed to perform well-controlled reactions at the scale of a chip (Kockmann et al. [2006]) or to improve tar-geted drug delivery to diseased cells of the body (Khan et al. [2013]). A typical microfluidic setup is shown in Fig. 1.3b.
If we now consider evaporating droplets, which is the principal subject of the present work, things become interesting in their own way. In the case of pure liquids, for instance, evaporating droplets can prove to be useful in the semiconductor and electronics industry for cooling overheated chips and even to specifically target localized hotspots where nanodroplets are transported. This adaptive cooling is realized through electrowetting and microfluidic techniques and it can dynamically cool
(a) (b)
Figure 1.2: (a) A Wilson’s phalarope, found mostly in the prairies of North America, during its meal. The prey is inside the captured water droplet; (b) a mechanical wedge that models the opening and closing cycle of the beak, illustrating how the droplet eventually reaches the bird’s mouth. The underlying principle, referred to as contact angle hysteresis, is the same that allows water droplets stick on vertical windows after a rainy Sunday morning. From Prakashet al. [2008].
(a)
(b)
CHAPTER 1. INTRODUCTION
different hot areas of the integrated circuit (Chakrabarty et al. [2007]).
Let us now think of a liquid drop that is lying on a solid substrate, which is often referred to as sessile. If one starts increasing gradually the temperature of the surface, the droplet would evaporate faster and faster and finally boil. If, however, the temperature of the surface would be further increased exceeding considerably the boiling point of the liquid, then a vapor layer would form between the droplet and the solid making eventually the drop levitate on its own vapor (Fig. 1.4a); this is known as the Leidenfrost effect (Quéré [2013]). Owing to this insulating layer, boiling is hindered and several remarkable effects emerge, e.g. the self-propelled drop of Fig. 1.4b,c. The ratchet effect, described by Linke et al. [2006], can lead to the development of pumps powered by waste heat, which for example can be found on an integrated circuit. The advantage of this method over electrowetting is that no external power is necessary to transport the drop. On the other hand, however, the insulating vapor layer decreases the heat exchange between the hot surface and the cooling liquid. Independently of any objections, the Leidenfrost effect can effectively be used to transport liquid droplets thanks to a low-friction motion.
The evaporation process of a complex liquid yields even more fascinating results, as it happens in the case of a droplet with suspended particles in it. There, in fact, lies the reason why a stain is found on a kitchen table after a typical coffee drop has dried out. More specifically, the evaporation along the surface of a sessile evaporating water droplet, as the one of Fig. 1.5a, is not uniform, but more intense at the perimeter of the drop∗. This eventually generates a hydrodynamic flow inside the droplet that pumps liquid and inevitably the coffee particles too, from the center to its perimeter (Fig. 1.5a). As soon as water has completely dried, the accumulated coffee particles form a ring-like pattern, as the one of Fig. 1.5b (Marín et al. [2011]). In general, depending on the mechanisms that lead to the final pattern, other shapes can be obtained as well.
From a practical perspective, the so called coffee-stain effect can be highly advantageous to biological sciences by forcing, for example, DNA molecules to stretch on substrates for further study (Jing et al. [1998]). In other cases, however, like coating or inkjet printing applications, the coffee-stain effect is considered a “persona non grata” and should be suppressed (Eral et al. [2011]).
Other studies, such as the one by Brutin et al. [2011] have examined the pattern formation of more complex liquids; human blood in this case (Fig. 1.6a). Variations in the structure of the pattern could be potentially linked to blood diseases. Another interesting application is the one suggested by Elbahri et al. [2007]. There, the deposition of nanoparticles on an inclined flat substrate is realized by means of a Leidenfrost drop of a chemical solution in order to produce nanowires. The innovation here lies in the fact that the drop is not loaded with particles prior to its deposition on the hot surface. Instead, these are formed from the chemical decomposition of the initial solution due to the overheated vapor layer between the solid surface and the drop. In other words, the layer beneath the droplet acts as a chemical reactor that rapidly generates nanoparticles which form the trail of the sliding drop (Fig. 1.6b).
Clearly, pattern formation in evaporating drops could not leave artists unmoved. In Fig. 1.6c we present a picture shot by photographer Rose-Lynn Fisher with the help of a standard light microscope. In her album†, entitled The Topography of Tears, one can find patterns of various evaporated tears, induced either by emotional or physical reasons.
Along with the on-ground experimental studies, space and micro-gravity experiments (Carle et al. [2012]) are also dedicated to the further exploration of the aforementioned phenomena. Needless to say that numerical and theoretical studies abound, as well. The topics that were not covered in this
∗This holds for contact angles less than 90◦(Deegan et al. [1997]) †http://rose-lynnfisher.com/tears.html
(a) (b) (c)
Figure 1.4: (a) A Leidenfrost water drop on a flat metallic plate heated up to 300◦C. From Quéré [2013]; (b) a self-propelled Leidenfrost drop of R134a (radius' 1.5mm) on a saw-tooth shaped brass surface (moving direction is to the right); (c) the ratchet effect can be explained in terms of pressure differential in the vapor layer. Namely, the high pressure region at A induces both a forward and a backward flow, with the net viscous forces due to the later being negligible. The levitated drop eventually moves to the right dragged by the forward flow of the vapor layer. From Linkeet al. [2006].
(a) (b) (c)
Figure 1.5: (a) A sketch of an evaporating droplet on a flat solid surface that illustrates the behavior of the suspended particles due to the non-uniform local evaporation fluxes; (b) the particles are dragged to the perimeter of the drop where they are deposited creating a ring-shape pattern; (c) the self-organization can be better seen when the so-called “coffee stain” is examined under higher magnifications. From Marínet al. [2011].
(a) (b) (c)
CHAPTER 1. INTRODUCTION
introduction are still numerous of course. Indicatively, we mention a few of them, such as acoustically levitated droplets, magnetic droplets and droplets on fibers as well as on elastic substrates. We hope, however, that the presented facts are already sufficient to erase any doubt that could possibly question the importance of droplets in everyday life.
Yet, in order to realize new and optimize the already existing application systems that are linked to wetting and capillarity phenomena, we should first be able to understand in depth the underlying physics that governs these processes. The fundamentally oriented work presented in the following chapters aims in that direction. Particularly, we examine, under normal ambient conditions, the be-havior of droplets (pinned and freely receding) of various HFE liquids (instead of the widely used water), which are considered so far as environmentally friendly and are often used as heat-transfer fluids in thermal management applications. They are pure perfectly-wetting and volatile liquids with low thermal conductivity and high vapor density. Actually, these properties affect in their own way many aspects concerning droplet evaporation such as the evaporation-induced contact angles, evapo-ration rate of a droplet, contact line pinning and Marangoni flow, all of which are treated in the present dissertation.
In general, the thesis is structured as follows. In Chapter 2 we provide a general overview of capillarity-related concepts. Then, in Chapter 3 we present the interferometric setup, along with the liquids and the substrate, that is used in the experiments, and also explain the reasons why this particular method is chosen. In Chapter 4 we address, among others, the issue of evaporation-induced contact angles under complete wetting conditions. The behavior of the global evaporation rate is also examined here, whereas in Chapter 5 we discuss the influence of thermocapillary stresses on the shape of strongly evaporating droplets. Finally, before concluding in Chapter 7, we address in Chap-ter 6 the still open question of the influence of non-equilibrium effects, such as evaporation, on the contact-line pinning at a sharp edge, a phenomenon usually described in the framework of equilibrium thermodynamics.
The experimental results obtained are also compared with the predictions of existing theoretical mod-els giving rise to interesting conclusions and promising perspectives for future research.
