ROLE OF SURFACE TOPOLOGY ON WETTING OF COMPLEX SURFACES
by
JAVIER SANCHEZ SANTIAGO
A THESIS
Submitted in partial fulfillment of the requirements for the degree of Master of Science in The Department of Chemical and Material Engineering to The School of Graduate Studies of The University of Alabama in Huntsville
HUNTSVILLE, ALABAMA
2013
ACKNOWLEDGMENTS
First, I would like to thank my advisor, Dr. Ramon Cerro, for his guidance, his patience and all the knowledge he has given to me. Also, I would like to thank my advisor in Spain, Dr. Elena Diaz, for helping me to come to the United States to pursue a graduate degree. Without her none of this could have been possible.
I would like to thank the entire Chemical Engineering department, especially to
Dr. C.P. Chen and Mrs. Jennifer Perkins, for all their help with everything I needed
through my journey at UAHuntsville.
I have to say thank you to all my family. To my parents, Maria Teresa Santiago
Martin and Antonio Sanchez Berrocal, for all their support and encouragement in
providing the means to an outstanding education. I do not forget my grandparents, my
aunt Maribel, my uncle Jose, and my cousin Sofia. Thank you very much to all of you.
Finally, I have to thank my girlfriend, Mar Piernavieja, for helping me with
everything and always supporting me.
v
TABLE OF CONTENTS
Page
LIST OF FIGURES ...... ix
LIST OF TABLES ...... xiii
LIST OF SYMBOLS ...... xiv
INTRODUCTION ...... 1
1.1 Taylor and Hauksbee experiments...... 1
1.2 Concus and Finn results ...... 3
1.3 Pomeau geometrical relationships...... 4
1.4 Contact angle hysteresis...... 6
1.5 Rough and heterogeneous surfaces...... 8
1.6 Super-hydrophobic surfaces...... 11
1.7 The lotus and petal effects...... 13
1.8 Applications to flow in porous media...... 14
1.9 Wicking...... 16
CONTACT ANGLES AND CAPILLARY PHENOMENA ...... 17
2.1 Young’s relationship ...... 17
2.2 The Stress-free Interface Analysis...... 19
2.3 Contact Angles and Boundary Conditions for the Young-Laplace Equation. ... 22
2.4 The Young-Laplace equation...... 24
vi
2.5 Solutions to the Young-Laplace equation...... 27
CAPILLARY RISE IN A CORNER. EXPERIMENTAL ...... 32
3.1 Experimental setup ...... 32
3.2 The curve locus of the static contact line...... 37
3.3 Geometrical relationships...... 39
3.4 The small elevation region...... 42
3.5 The incipient capillary rise region...... 50
3.6 Results for large values of elevation...... 54
3.7 Experimental results for different corner angles...... 55
3.8 Contact angle measurement...... 60
3.9 Experimental errors...... 62
CAPILLARY RISE IN A CORNER. THEORETICAL ...... 66
4.1 Introduction...... 66
4.2 Solution for large values of elevation...... 68
4.3 Solution for small values of elevation...... 69
4.4 Intermediate region...... 71
AN EXPLANATION TO HAUKSBEE EXPERIMENTS ...... 76
5.1 Introduction to Hauksbee experiments...... 76
5.2 Corner wedge in the lower position...... 76
5.3 Corner wedge in the lower position, figure number 4...... 84
vii
5.4 Corner wedge in the upper position...... 84
5.5 Plates previously fully immersed. Hauksbee case 7...... 88
CONCLUSIONS AND RECOMMENDATIONS ...... 94
6.1 Conclusions ...... 94
6.2 Recommendations for future work ...... 95
APPENDIX A – EXPERIMENTAL RESULTS ...... 97
A.1 Measurements of l versus z for different corner angles ...... 97
A.2 Pictures of the different experiments...... 100
A.3 Pictures of the drops used to measure the contact angle ...... 112
A.4 Pictures showing experimental errors ...... 115
APPENDIX B – THEORETICAL RESULTS ...... 122
B.1 Solution for large values of elevation...... 122
B.2 Solution for the intermediate region...... 124
APPENDIX C – HAUKSBEE EXPERIMENTS ...... 125
C.1 Numerical results for Case 1 ...... 125
C.2 Numerical results for Case 2 ...... 127
C.3 Numerical results for Case 3 ...... 128
C.4 Numerical results for Case 7 ...... 129
REFERENCES ...... 130
viii
LIST OF FIGURES
Figure Page
Figure 1.1 Diagram performed by Francis Hauksbee in his first experiment ...... 1
Figure 1.2 Different inclination angles performed by Hauksbee in his third experiment .. 2
Figure 1.3 Figure utilized by Concuss and Finn in their mathematical study ...... 4
Figure 1.4 Figure utilized by Pomeau in his geometrical study ...... 5
Figure 1.5 Contact angle in a drop over a flat surface ...... 6
Figure 1.6 Liquid column suspended in a vertical capillary ...... 7
Figure 1.7 Influence of solid surface roughness on contact angle hysteresis...... 13
Figure 2.1 Forces affecting a point near the contact line...... 20
Figure 3.1 Experimental setup ...... 32
Figure 3.2 Experimental setup sketch ...... 33
Figure 3.3 Pieces of glass used ...... 34
Figure 3.4 Camera Nikon D70 and lens used ...... 34
Figure 3.5 Motor electro-craft E-552-S ...... 35
Figure 3.6 Source of additional spotting light used ...... 35
Figure 3.7 Equilibrium contact line ...... 37
Figure 3.8 Equilibrium contact line ...... 38
Figure 3.9 Top view of solid wedge in contact with a liquid pool...... 40
Figure 3.10 Capillary rise at the opposite side of the corner, angle of 3.06 degrees ...... 45
ix
Figure 3.11 Side view of the glasses with an angle of 3.06 degrees...... 46
Figure 3.12 Capillary rise at the opposite side of the corner, angle of 1.18 degrees ...... 47
Figure 3.13 Side view of the glasses with an angle of 1.18 degrees...... 48
Figure 3.14 View of the opposite side to the corner, angle of 0.66 degrees ...... 49
Figure 3.15 Side view of the glasses with an angle of 0.66 degrees...... 50
Figure 3.16 Side view of the intermediate region with an angle of 3.06 degrees ...... 51
Figure 3.17 Side view of the intermediate region with an angle of 1.18 degrees ...... 52
Figure 3.18 Side view of the intermediate region with an angle of 0.66 degrees ...... 53
Figure 3.19 Pictures of the curve locus for large values of elevation ...... 55
Figure 3.20 Example of the grid method used ...... 56
Figure 3.21 Results for a corner angle of 3.07 degrees...... 57
Figure 3.22 Results for a corner angle of 1.18 degrees...... 58
Figure 3.23 Results for a corner angle of 0.66 degrees...... 59
Figure 3.24 Comparison of the results of different corner angles...... 60
Figure 3.25 Picture of drop 1 used to measure the contact angle ...... 61
Figure 4.1 Geometry of the problem...... 67
Figure 4.1 Non dimensional forms of hyperbola and computational solutions ...... 73
Figure 4.2 Computational versus experimental results ...... 74
Figure 5.1 Figure 1 of Hauksbee (1713b) ...... 77
Figure 5.2 Sketch of the geometrical variables...... 78
x
Figure 5.3 Capillary rise described by equation 5.7 – Case 1...... 81
Figure 5.4 Capillary rise described by equation 4.7 – Case 2...... 82
Figure 5.5 Capillary rise described by equation 5.7 – Case 3...... 83
Figure 5.6 Figures 5 and 6 of Hauksbee ...... 84
Figure 5.7 Sketch of Case 5 showing the geometric variables ...... 85
Figure 5.8: Sketch of the cases of infinite plates where the touching edge is submerged at angles different than 90o ...... 86
Figure 5.9: Geometrical sketch used to determine the wedge angle projected on the horizontal direction...... 87
Figure 5.10: Case 7 of Hauksbee (1713) ...... 88
Figure 5.11. A: sketch showing contact angle and shape of interface in lower liquid rise.
B: sketch showing contact angle and shape of interface for higher liquid rise...... 90
For the higher liquid rise, the Young-Laplace equation can be written as ...... 90
Figure 5.12: Values of lower and higher capillary rise for a water/air/glass system...... 93
Figure A.1 Experiment 1 angle 3.06 degrees ...... 100
Figure A.2 Experiment 2 angle 3.06 degrees ...... 101
Figure A.3 Experiment 3 angle 3.06 degrees ...... 102
Figure A.4 Experiment 4 angle 3.06 degrees ...... 103
Figure A.5 Experiment 1 angle 1.18 degrees ...... 104
Figure A.6 Experiment 2 angle 1.18 degrees ...... 105
xi
Figure A.7 Experiment 3 angle 1.18 degrees ...... 106
Figure A.8 Experiment 4 angle 1.18 degrees ...... 107
Figure A.9 Experiment 1 angle 0.66 degrees ...... 108
Figure A.10 Experiment 2 angle 0.66 degrees ...... 109
Figure A.11 Experiment 3 angle 0.66 degrees ...... 110
Figure A.12 Experiment 4 angle 0.66 degrees ...... 111
Figure A.13 Picture of drop 2 used to measure the contact angle ...... 112
Figure A.14 Picture of drop 3 used to measure the contact angle ...... 113
Figure A.15 Picture of drop 4 used to measure the contact angle ...... 114
Figure A.16 Non uniform interface shape ...... 115
Figure A.17 Air trapped ...... 116
Figure A.18 Air trapped ...... 117
Figure A.19 Non complete capillary rise ...... 118
Figure A.20 Non complete capillary rise ...... 119
Figure A.21 Clamp pressure deforming the interface ...... 120
Figure A.22 Close view of the glasses edges ...... 121
xii
LIST OF TABLES
Table Page
Table 3.1 Separation and angles between the glasses ...... 36
Table 3.2 Physical properties of water at 22˚C...... 43
Table 3.3 Contact angle measurements and average value...... 62
Table A.1 Measurements of l(z) versus z for an angle of 3.06 degrees...... 97
Table A.2 Measurements of l(z) versus z for an angle of 1.18 degrees...... 98
Table A.3 Measurements of l(z) versus z for an angle of 0.66 degrees...... 99
Table B.1 Results of the hyperbola solution ...... 123
Table B.2 Sample of the results for the computational solution ...... 124
Table C.1 Initial parameters for Case 1 ...... 125
Table C.2 Numerical results for Case 1 ...... 126
Table C.3 Initial parameters for Case 2 ...... 127
Table C.4 Numerical results for Case 2 ...... 127
Table C.5 Initial parameters for Case 3 ...... 128
Table C.6 Numerical results for Case 3 ...... 128
Table C.7 Initial parameters for Case 7 ...... 129
Table C.8 Numerical results for Case 7 ...... 129
xiii
LIST OF SYMBOLS
α angle between the pieces of glass
β contact angle
an energy parameter between phases and
푖푗 훽 surface tension between liquid and vapor푖 phases푗
휎 surface tension between i and j phases
푖푗 휎 small deviation of the contact angle
훿( ) distance from the centerline to the wedge
훿 푧 characteristic curvature
휅 contact angle
휃 equilibrium contact angle
휃푒 advancing contact angle
휃퐴 receding contact angle
휃ε 푅 Gibbs-Thomson equation parameter
π pi constant
ρ density surface angle
characteristic size of solid surface roughness Ω diameter of the pore 푑
푝 function based on angle 푑( ) height 훼 퐺, 훼 droplet size/distance from vertex to interface ℎ 퐻 latent heat per unit mass 푙 parameter in Johnson and Dettre experiments 퐿
푟 xiv
tangential force
퐹푇 force number i
퐹푖 molecular density in phase
푛푖 radius 푖
푅 horizontal radius
푅퐻 vertical radius
푅푆 Gibbs-Thomson radius
푅s 퐺푇 arc length parameter, s direction u u direction
temperature
푇 equilibrium temperature
푇푒 distance to the contact line
푥푇 capillary length
퐿2퐶 curvature
2퐻 non dimensional curvature
� 퐻 non dimensional height
휉l(z) distance from the wedge
Hamaker constant to i j interface
푖푗 pressure phase i 퐴 integration constant 푝C푖 Cartesian direction x Cartesian direction y Cartesian direction z
xv
CHAPTER ONE
INTRODUCTION
1.1 Taylor and Hauksbee experiments.
Over 300 years ago, the English scientist Francis Hauksbee (1687–1763), reported
several experiments related to capillary rise in corners, performed by Brook Taylor. In
his first experiment, he held two glass planes by one of their ends, with the other one
opening a certain angle, and he put them into a trough of water. He realized that the
water rose in the closed side of the glasses, and then performed two different
measurements using 20 and 40 minutes angles. Hauksbee (1710) published the following
table and diagram of his experiment.
Figure 1.1 Diagram performed by Francis Hauksbee in his first experiment
1
In his second experiment, he prepared 32 pieces of “Brats lamine”, in order to separate two parallel glass planes, and then put them into a trough of spirit of wine
(Hauksbee, 1713a). The water rose between the two planes and Hauksbee discovered that by decreasing the separation between the glasses the height of the water would increase.
The third experiment was a continuation of what he did in the first experiment, but now he changed the angle of inclination between the glasses when immersed into water (Hauksbee, 1710b). As a result, he discovered that the curve formed by the water holds in all directions of the planes, being the asymptotes of one of the curves the surface of the water and the other one the line drawn along the touching sides. The following figure shows an explanation of the third experiment (Hauksbee, 1710b).
Figure 1.2 Different inclination angles performed by Hauksbee in his third experiment
2
Finally, in his fourth experiment, he repeated his first experiment with spirit of wine and created a table of distances from the touching ends versus the angle of elevation with two different aperture angles, 18 minutes and 10 minutes (Hauksbee, 1710c).
With these four experiments, Mr. Francis Hauksbee set up the basis of capillary rise in corners.
Meanwhile, Mr. Brook Taylor was performing a similar experiment, with two glass planes, forming an angle of two and a half degrees, and he realized that the shape of the upper part of the water approached very closely the shape of the common Hyperbola
(Taylor, 1712).
1.2 Concus and Finn results
In 1969, Paul Concus and Robert Finn performed an extensive analysis of the behavior of the capillary surface in a wedge. They studied the height of the equilibrium free surface of a liquid, which partially fills a cylindrical container whose cross section contained a corner with an interior angle 2 α. They discovered that the behavior of the water rise was a function of the angle α and the contact angle β. In their mathematical study, based on the geometry of the Figure 1.3 (Concus and Finn, 1969), they postulated that for a contact angle β on the side walls of the wedge, the behavior of the capillary rise depends on the interior wedge angle. Diaz et al (2010) defined the contact angle as the angle defined by the normal to the solid surface and the normal to the interface air/liquid.
The behavior changes if the contact angle is smaller or greater than . Concus and 휋 2 − 훽
3
Finn discovered that in order to have capillary rise with a hyperbola shape, the geometry of the wedge angle with the contact angle needs to satisfy equation 1.1
+ < (1.1) 2 휋 훼 훽
Figure 1.3 Figure utilized by Concuss and Finn in their mathematical study
Also, they performed an experiment with two microscope slides forming a wedge with an interior angle of about ten degrees. They dipped the bottom of the wedge into the surface of a reservoir of liquid (either water or California olive oil). In both cases the hyperbola shape was obtained in the liquid and + < was achieved. 휋 훼 훽 2
1.3 Pomeau geometrical relationships.
In 1986, Yves Pomeau studied the phenomena of wetting in corners. Using the
Gibbs-Thomson condition, the radius of a liquid droplet in metastable equilibrium with its vapor is fixed. The radius follows the Equation 1.2
4
2 = (1.2) 휎푇푒 푅퐺푇 휀휌퐿 Where is the surface tension, is the equilibrium temperature, = , is
푒 푒 the liquid density휎 and is the latent heat푇 per unit mass. For a negative 휀 the푇 result− 푇 휌is a negative radius of curvature.퐿 An extensive study in this conditions, leaded휀 Pomeau to the condition that for a given negative radius of curvature the geometric construction of
퐺푇 the droplet in the corner requires the inequality shown in푅 Equation 1.3
+ < (1.3) 2 2 훼 휋 훽 Figure 1.4 (Pomeau, 1986) shows that the angle α used by Pomeau is the double of the angle that Concus and Finn used, so basically equations 1.1 and 1.3 are the same.
All authors reached to the same conclusion using different approaches.
Figure 1.4 Figure utilized by Pomeau in his geometrical study
Pomeau also showed that by measuring the droplet size from the vertex to the
liquid/vapor interface, it is possible to obtain the relationship shown푙 in Equation 1.4
5
2 = (1.4) 훼 푐표푠훽 − 푠푖푛 퐺푇 2 푙 푅 훼 푠푖푛 1.4 Contact angle hysteresis.
When a liquid drop is placed on a clean, flat and solid surface, it is possible to
observe a contact angle, , which is the same used in Young’s formula (see Chapter
푒 2.1). 휃
θ
Figure 1.5 Contact angle in a drop over a flat surface
If the drop grows by adding more liquid, the contact angle can be greater than
without moving the contact line. The drop can grow until the contact휃 angle reaches a
푒 휃final value, called , and when this occurs the contact line moves. is described as
퐴 퐴 advancing contact angle.휃 휃
Otherwise if the drop is deflated, can be decreased to a limiting value . The
푅 line of contact will shift in the case where휃 = . is defined as the receding휃 contact
푅 푅 angle. 휃 휃 휃
6
The difference between the limiting angles and , is defined as the contact
퐴 푅 angle hysteresis (Quere et al, 2003). 휃 휃
It is possible to capture a liquid column suspended in a vertical capillary thanks to the hysteresis (Quere et al, 2003).
θ1
θ2
Figure 1.6 Liquid column suspended in a vertical capillary
The upper force can be obtained as:
= 2 (1.5)
퐹1 훾푐표푠휃1 휋푅 The lower force can be obtained as:
= 2 (1.6)
퐹2 훾푐표푠휃2 휋푅 The weight of the water column is
7
(1.7) 2 휌푔휋푅 퐻
The upper and lower forces have to balance the weight of the column
2 ( ) = (1.8) 훾 푐표푠휃1 − 푐표푠휃2 휌푔퐻 푅 In order to have equilibrium, the upper line needs to resist a receding movement and the lower line an advancing movement. This implies that the contact angles must satisfy:
> < (1.9)
휃1 휃푅 푐표푠휃1 푐표푠휃푅 < > (1.10)
휃2 휃퐴 푐표푠휃2 푐표푠휃퐴 Equations 1.8, 1.9, 1.10 imply that in order to have equilibrium:
2 ( ) = (1.11) 훾 푐표푠휃푅 − 푐표푠휃퐴 휌푔퐻 푅 Equation 1.11 shows that without contact angle hysteresis ( = ) it is not
푅 퐴 possible to capture liquid inside a capillary. 휃 휃
1.5 Rough and heterogeneous surfaces.
The influence of roughness in surfaces is crucial when the contact angle is being studied. An ideal solid surface would be flat, rigid, perfectly smooth and chemically
8
homogeneous, and theoretically it would have no contact angle hysteresis, which means
that the advancing and receding contact angles would be identical.
In 1980, Schwartz defined the “intrinsic contact angle hysteresis”, as hysteresis
that cannot be ascribed to roughness, heterogeneity, or penetrability of the solid surface
(Schwartz, 1980). Obviously, the contact angle is affected by the roughness and heterogeneity of the surface, but there are some surfaces that show no hysteresis with certain liquid-fluid pairs, and at the same time they show considerable hysteresis with
other liquid-fluid pairs. In order to compare the hysteresis of two different liquid-fluid
pairs, molecular volume has to be taken into consideration. Pairs have to have similar
molecular volume, because the thickness of the liquid layer adjacent to the solid surface
is affected by its molecular volume. This is the layer responsible for the surface pressure.
As it will be shown in Chapter 2, the contact angle is function of the surface tension
between the three phases, solid, liquid and vapor (Young’s relationship). In a system at
equilibrium and at constant temperature, the surface tensions between liquid/vapor and
solid/vapor phases are constant. This requires that in order to have intrinsic contact angle
hysteresis, the surface tension between the solid/liquid phases should be a function of the
contact angle, which contradicts Young’s relationship.
Schwartz final statement is that, if hysteresis is caused by roughness or
heterogeneities, its value should remain more or less constant over a wide range of
temperatures, if the temperature does not affect the rugosity or heterogeneity of the
surface.
9
In a paper in 2002, and later in a book in 2003, David Quere et al. summarized some of the models that have been developed, in order to explain the role of the surface roughness. Some of these models are:
- The basic model based on the experiment of Johnson and Dettre (1964): the
authors defined the parameter as the ratio of the real surface area to the apparent
surface area. This parameter was푟 not measured rigorously, but it was discovered
that the advancing angle increases with while the receding angle decreases
(which means that the hysteresis increases).푟 It was also found that after one point,
both angles increase and the hysteresis drops to a smaller value.
- The Wenzel’s model (1936): this model is used to predict wetting properties for
rough surfaces, but chemically homogeneous. Wenzel’s conclusion was that
surface roughness always magnifies the underlying wetting properties.
- The Cassie-Baxter model (1944): this model is used to predict wetting properties
for chemically heterogeneous surfaces, but planar surfaces. Cassie and Baxter
realized that when dealing with heterogeneous surfaces, Wenzel’s model was not
sufficient. They developed a more complex model to measure how the apparent
contact angle changes when various materials are involved. The results of this
model are still being discussed
In 2010, Diaz et al. published “Hysteresis during contact angles measurements”
which provides a molecular interpretation of intrinsic hysteresis. This publication
explained that there is a difference between the contact angle measurements depending
on if surface has been previously in contact with a vapor phase saturated with the
10 molecules of the liquid phase or not. The advancing contact angle is larger than the static contact angle, if the solid surface is free of adsorbed liquid molecules. If a drop is left on the surface as it evaporates, the vapor/liquid interface contracts and the apparent contact line moves towards the center of the drop. The film left behind by the drop is thicker than the adsorbed film. The resultant receding contact angle will be smaller than the static contact angle due to the molecular attraction. This phenomenon will be explained in Chapter 2.
1.6 Super-hydrophobic surfaces.
Following the Wenzel’s model (Wenzel, 1936) it is possible to classify the surfaces into hydrophobic and hydrophilic. In hydrophobic surfaces, air can remain trapped under the drop which leads to a contact angle > 90 degrees. On the other
푒 hand, hydrophilic surfaces lead to a contact angle of <휃90 degrees.
휃푒 Some surfaces are super-hydrophobic, and can lead to contact angles close to 180 degrees. David Quere et al. (2003) refer to drops capable of achieving those angles as liquid pearls. One method for creating this liquid pearls is using surfaces made of hydrophobic spherules (derived from Teflon). The combination of hydrophobicity, roughness and trapping, leads to advancing contact angles of up to 175 degrees. Another method uses a material with low roughness and with pinning of lines of contact, which will lead into super-hydrophobicity. These surfaces have hydrophobic pillars that imply the wetting of the surface is practically zero, with an advancing angle of up to 170
11
degrees. If we assume that the drops rest on the top of the pillars only, this phenomenon
can be called “fakir effect” (Quere et al, 2003).
Another way to make liquid pearls is based on texturing the surface of the liquid.
A hydrophobic powder is attached to the surface of the drop, forming a shell which protects the liquid. No matter what surface is used to deposit the drop, it will display zero wetting. Quere et al. (2003) call these objects liquid marbles. The final way to create this kind of drops is by placing a water drop on a very hot plate. A vapor film around the drop will be created and the drop will retain its spherical shape. But after a short period of time it will evaporate, these are the so-called Leidenfrost drops.
Diaz et al. (2012) established the two surface parameters that determine the
configuration of a small drop (less than 1mm in diameter) when it is in contact with a
rough surface. These parameters are the distance between the contact points on the
surface, and the angle determined by the relationship:
( ) = (1.12) 2 휋 Ω 푑 − 훼 − 휃푒 These parameters are not affected by the orientation with respect to gravity. With
the analysis that Diaz et al. performed, it is possible to justify the effect of the
characteristic size of solid surface roughness on contact angle hysteresis. Diaz et al.
utilized Figure 1.7 to illustrate this effect. The푑 parameter is defined as the surface
angle. Ω
12
Figure 1.7 Influence of solid surface roughness on contact angle hysteresis.
1.7 The lotus and petal effects.
It is possible to find in nature super-hydrophobic surfaces, usually in plants and insects, which can achieve contact angles larger than 150 degrees. In particular, two surfaces have very interesting properties, lotus leaves and petals.
Lotus leaves are being studied because in their super-hydrophobic surfaces, the contact angle with water drops is 161 degrees (Uchida et al, 2010). The water drop can move and roll over the surface. This is called the lotus effect. It is used in nature as a self-cleaning effect because, since the drops can move, they can remove dust particles on the surface (Feng et al, 2008).
Rose petals also have super-hydrophobic surfaces, with contact angles with water of around 152.4 degrees. But in this case, the droplets are pinned to the surface and
13
cannot roll off, even if the petal is turned upside down. This phenomenon is called the
pinned effect or petal effect. It is produced by hierarchical micro- and nanostructures that
provide sufficient roughness for super-hydrophobicity, but have high adhesive force with
water (Feng et al, 2008).
Diaz et al. (2010) by using Equation 1.12, studied the surface angle in the lotus
and petal effect. Using two different scales and , being > , the surface angle
1 2 1 2 can be positive or negative depending on the 푑values of푑 and 푑. 푑
훼 휃푒 If the large scale angle ( ) and lower scale angle ( ) are negative,
1 2 penetration of the liquid into the surfaceΩ 푑 structure is not allowed, beingΩ 푑 this the case of
the lotus effect. In the other hand, if the large scale angle is negative but the lower scale
angle is positive, the pinned contact lines will determine the shape of the droplet,
apparently creating a large macroscopic contact angle. Meanwhile, the surface under the
drop is totally wet. This type of wetting creates the typical adhesion of the petal effect.
A final case is the one in which, both angles are positive creating an apparent contact
angle smaller than the actual contact angle.
1.8 Applications to flow in porous media.
If a paper tissue is introduced into a cup of water, the water will ascend through
the tissue. The rise of the liquid will stop at a certain height. The capillary forces pull
the liquid up meanwhile gravity pulls the liquid down. It is possible to estimate the height
of the liquid using Equation 1.13 (Quere et al, 2010).
ℎ
14
−2 (1.13) 퐿푐 ℎ ≈ 푐표푠휃푒 푑푝 Where is the diameter of the pore, is the capillary length and is the
푝 푐 푒 equilibrium contact푑 angle. 퐿 휃
The equilibrium contact angle can only be defined in equilibrium situations, because if a drop is placed on a dry porous surface it will penetrate the medium. Also, the contact angle will depend on the preparation of the media. This means that the contact angle will be different if the drop is placed in a wet, horizontal porous surface or if it is achieved by capillary rise in a porous media. Diaz et al. (2010) performed an analysis on the differences between contact angles with surfaces with previously adsorbed liquid layers.
Also, hysteresis can occur locally within the pores and air bubbles can be trapped during capillary rise. Porous media always has some random surface, which will imply changes in the contact angle.
In order to study the flow in porous media, suction experiments on drops and films have been performed. The results of these experiments are discussed in Quere et al. book (2003).
In 2000, Das developed a mathematical model for the imbibition characteristics of colloidal suspension inside the porous ceramic network (Das, 2000). This process has some similarities with filtration, since there is filtrate flow and a cake forming around the ceramic (a filter cake is formed by the accumulation of substances retained on a filter).
The model has two different steps: in the first one, the flow is due to capillary rise inside
15 the monolith channel. At the same time, the filtrate is percolating the porous media.
Capillary rise stops when the driving force due to surface tension is equal to gravitational force. In the slurry-flow it was studied that, at the same time as the filtrate is entering the porous media cake is being formed on the surface. In the second step, the liquid stops rising inside the capillary. Instead, the flow continues to form cake and imbibing the porous network. Due to this secondary process, the liquid level continues to rise, so at the end all the surface of the monolith gets wet.
1.9 Wicking.
There are several ways to define wicking. In 1993 Chibowski et al defined wicking as “phenomenon of liquid penetration into a solid porous layer”. In 1995 Yeh et al explained wicking as “liquid penetrating the fine and less permeable material in a liner system that tends to spread out laterally in the fine material after the wetting front advances to the interface between the fine and coarse material”. In 2002 Staples et al defined wicking as “a factor affecting the flow of fluid in porous media”. In 2003
Tavisto et al defined it as “the spontaneous flow of a liquid in a porous substrate, driven by capillary forces”. This phenomenon depends on the geometrical and physical of the liquid and the porous solid (Fries et al. 2008).
Wicking has plenty of uses, from designing sports clothing (skin moisture and sweat go from the skin to the clothes surface and then evaporates), industrial drying processes, wetting, etcetera.
16
CHAPTER TWO
CONTACT ANGLES AND CAPILLARY PHENOMENA
2.1 Young’s relationship
A contact line is the ideal line where three phases exist at rest. For example, when a drop is deposited on a horizontal, flat, solid surface the three phases are the solid, the liquid contained in the drop and a gas or vapor phase surrounding the drop. Contact angles are typically measured by amplifying pictures of a sessile drop and extrapolating the vapor/liquid interface until it contacts the solid surface. In rigor, as it has been demonstrated by Diaz et al. (2010b) contact angles must be measured by fitting a solution to the Young-Laplace equation to the drop profile and determining the contact angle as the angle formed by the scalar product between the unit vector normal to a solid surface and the unit vector normal to the vapor/liquid interface at the ideal triple point of contact.
The earliest attempts to characterize contact angles are due to Thomas Young
(1805). Young, a scientist of extraordinary broad knowledge for his times, conceived the contact line as an equilibrium of forces involving interfacial tensions, between the three phases. In Young’s words: “We may therefore inquire into the conditions of equilibrium of the three forces acting on the angular particles, one on the direction of the surface of
17 the fluid only, a second in that of the common surface of the solid and fluid, and the third in that of the exposed surface of the solid. Now supposing the angle of the fluid to be obtuse, the whole superficial cohesion of the fluid being represented by the radius, the part of which acts in the direction of the surface of the solid will be proportional to the cosine of the inclination; and its force added to the force of the solid, will be equal to the force of the common surface of the solid and fluid, or to the difference of their forces; consequently, the cosine added to twice the force of the fluid; will be equal to ....”. Young never actually wrote down the equation but his words can be translated to an equation, to get the relationship shown in Eq. 2.1, (Young, 1805):
= (2.1)
휎푐표푠휃푒 휎푆푉 − 휎푆퐿 Where , and are the surface tensions between the liquid and vapor, solid
푆푉 푆퐿 and liquid, and휎 solid휎 and vapor휎 phases respectively and is the equilibrium contact
푒 angle. From a physical point of view, the contact line is not휃 a line (one-dimensional) but a small, molecular-size region. Thus, the untenable concept of an equilibrium of forces eventually evolved into the thermodynamic interpretation of interfacial tension as specific free energies per unit area. The fact that interfaces are not surfaces, (i.e. two-dimensional surfaces in a three-dimensional space) was recognized by Rayleigh and van der Waals
(for a historical review as well as a clear explanation of these concepts see Rowlinson and Widow, 1982) who developed the concept of “diffuse” interfaces, i.e. continuous variations of density as opposed to the concept of a mathematical two-dimensional interface. However, scientists have been slow to recognize the fact that contact lines are also “diffuse” regions where molecules of the three phases interact to balance molecular attractions defining a stress-free region where the contact angles are shaped.
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Despite large amounts of research efforts devoted to rationalize and justify the
equilibrium of forces proposed by Young (1805) (See for example Girifalco and Good
(1957) and van Oss et al. (1987) and references therein), there are no proven ways to
define, let alone measure interfacial tensions between a solid and a liquid or a solid and a
vapor. There have been also many attempts to “develop” Young’s relationship either
from thermodynamic (See for example Schwartz 1980 and Roura and Fort, 2004) or
mechanical concepts (There is an remarkable perturbation analysis performed by
Merchant and Keller, 1991) most of these derivations fall short because they do not
include molecular interactions between molecules in the solid and molecules in the
liquid.
2.2 The Stress-free Interface Analysis.
The simplest and yet very effective analysis done by Miller and Ruckenstein
(1974) is based on the fact that at equilibrium, there cannot be any non-compensated
stresses in the liquid phase since, by definition, a fluid will deform even in the presence
of very small shear stress. Their analysis is based on the simple sketch of molecular
forces acting on a fluid interface near a solid surface shown Figure 2.1:
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Figure 2.1 Forces affecting a point near the contact line.
A molecule of liquid identified by Q in Figure 2.1 is subject to attractions exerted by all the other liquid molecules and all the solid molecules as well. The forces due to attraction from the solid molecules, FS, and attraction from other liquid molecules, FL, combine to give a resultant attraction force with tangential and normal stresses. Normal stresses are compensated by pressure but shear stresses must vanish for the interface to be in static equilibrium. Molecular forces due to the gas or vapor molecules can be safely neglected due to the large differences in density between the solid, liquid and gas phases.
Assuming the vapor/liquid interface is a straight line, individual molecular attractions can be integrated taking into account all molecules over the surrounding phases. Miller and
Ruckenstein (1976) obtained Equation 2.2 as an expression for the residual tangential force acting on Q:
= ( ) ( ) + ( ) (2.2) 4 4 −휋 2 휋 푇 4 퐿 퐿퐿 퐿 푆 푆퐿 4 퐿 푆 푆퐿 퐹 푇 푛 훽 − 푛 푛 훽 퐺 훼 푇 푛 푛 훽 퐺 휋 − 훼 where FFT= LT − F ST is 푥the resulting tangential force, 푥 is the molecular density in phase
푖 , the distance to the contact line, binary interaction푛 parameters between phases
푇 푖푗 and푖 푥 and ( ) results from the space 훽integration and is a function of the contact angle .푖
푗 퐺 훼 훼
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Since the mechanical condition for equilibrium requires a stress-free interface, it is assumed that the force acting on molecule Q has no tangential component, FT = 0,
Equation 2.2 reduces to 2.3.
Ga(π − ) n2 β +=1 L LL G(αβ) nn L S SL (2.3) 3 G (ααα) =++csc33( ) cot( ) cot ( α) 2
Apparently unaware of Miller and Ruckenstein (1974) paper, Jameson and Del
Cerro (1976) published a very similar analysis where they extended the concept of
interfacial equilibrium to develop a defining equation for the equilibrium contact angle of
a non-polar liquid over a solid surface. Further, Jameson and Del Cerro (1976) argued
that Young’s equation (Equation 1) was not correct, and submitted Equation 2.4 as the
proper definition of the equilibrium contact angle:
ASL 13 1 3 =+−cosθθee cos (2.4) ALL 24 4
2 Where ASL= nn L Sβ SL and AnLL= LLβ LL are the Hamaker constants related to the
solid-liquid and liquid-liquid interactions, and is the contact angle. Notice that
푒 Equation 2.4 can be derived from Equation 2.3. 휃
Interestingly, Equation 2.4 is not the expression for the equilibrium contact angle,
but for the advancing contact angle with no adsorbed film present as later demonstrated
by Diaz et al (2010).
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2.3 Contact Angles and Boundary Conditions for the Young-Laplace Equation.
Although it can be argued on whether a sessile drop is in actual equilibrium with its surroundings (Shanahan, 2002) it is generally accepted that there is an equilibrium contact angle since the processes of evaporation and condensation of liquid from and to a drop are much slower than the dynamic of molecular forces interacting near the contact line. The argument about the physical processes determining equilibrium contact angles was brought into a new level by the analysis of Diaz et al (2010 b) on the definition and precise measurement of contact angles on a sessile drop of large diameter, i.e. a puddle.
The approach used by Diaz et al (2010b) consists on recognizing three regions based on the profile of a two-dimensional drop. The largest region is the macroscopic region where only capillary and gravity forces are important and it is described by the Young-
Laplace equation. There is an intermediate transition region where capillary and gravity forces are of the same order of magnitude than molecular forces, and it is best described by the so-called Augmented Young-Laplace equation, AYL (Mohanty, 1981). Finally, there is an interior molecular region where only molecular forces are important and is described by the Fully Augmented Young-Laplace Equation, FAYL (Diaz et al, 2010).
Molecular forces to be included in the analysis of equilibrium contact angles are
(Churaev and Sobolev, 1995) non-polar molecular forces, ionic-electrostatic forces and structural forces.
There are three important concepts to highlight from Diaz et al. (2010b) paper.
The first concept is that the equilibrium static contact angle is totally determined by binary force interactions between molecules of the solid and liquid phases. Thus, even when the macroscopic shape of the vapor/liquid interface is determined by the Young-
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Laplace (YL) equation, the equilibrium contact angle is independent of the specific
geometry of the system. The second concept is that in the intermediate and molecular
regions the shape of the vapor/liquid interface departs from the solutions to the YL
equation and is better described by the Fully Augmented Young-Laplace equation
(FAYL). Because of this departure from the solution of the YL equation, near the contact
line (transition and molecular region) the angle of inclination of the interface varies with position along the interface voiding the practice of improving contact angle measurements by increasing the resolution of pictures taken of capillary shapes. Since
the angle of inclination is continuously changing down to the solid surface, there is no
physically sound position along the streamline where the contact angle can be defined.
The third and last concept is the definition of the equilibrium contact angle as the angle
where a solution to the YL equation modeling the interface, intersects with the solid
surface. The proper way to determine the contact angle experimentally is to reproduce by
computation the shape of the interface and to compute the contact angle as the arc-cosine
of the scalar product of the unit normal to the interface and the unit normal to the solid
surface, at the extrapolated point where the interface and the solid surface meet. This
definition results in a precise and unambiguous characterization of the contact angle but it
does not guarantee its physical existence. In fact, for a puddle, there is no place along the
interface where the angle of inclination is equal to the equilibrium contact angle
computed as described above. However, the difference between the angle of inclination
of the interface at the transition region and the equilibrium contact angle is minute.
Although Jameson and Del Cerro (1976) argument for a new relationship for contact angles, is in principle correct, it does not take into account the fact that the solid
23 surface adjacent to the three-phase region is not a “clean” surface but is “contaminated” by the presence of molecules of the liquid phase adsorbed on the solid surface. Diaz et al. (2010a) developed an expression for disjoining pressure in the presence of a molecular-size adsorbed film on the solid surface and integrated the fully augmented
Young-Laplace equation to get a functional dependence of static contact angles, molecular forces and the thickness of the adsorbed liquid film. Thus, in the presence of an equilibrium adsorbed film, the contact angle is given by the following expression:
[ ] [ ] [ ] 1 3 1 = 1 + ln 1 × + 12 (6 6 ) 12 6 2 4 4 퐴퐿퐿 − 퐴푆퐿 퐴퐿퐿 3 푐표푠휃푒 � − 2 − 2 �− 푐표푠휃푒 − 푐표푠 휃푒�� 휋휎 퐷푎푑푠 − 퐷푚 휋휎퐷푚 (2.5)
is the thickness of the horizontal adsorbed layer, and is the total film
푎푑푠 푚 thickness퐷 (adsorbed layer plus cutoff layer). 퐷
2.4 The Young-Laplace equation.
The shape of liquid-vapor interfaces is described by an equation presented in words by Thomas Young (1805) and first written in equation form by Laplace (1807).
The propriety of using the names of Young and Laplace to describe this equation was discussed in a scholarly paper by Pujado and Scriven (1972). Although there was some early controversy about whether interfacial tension was really a force acting per unit area, instead of the more general concept of specific free energy, within the macroscopic domain of continuous mechanics it is generally accepted that the Young-Laplace
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equation is a local force balance derived from conservation of momentum (Scriven,
1982):
2Hσ = ppAB − (2.6)
where ppABand are pressure in phase A and B, 2Hr= 1/12 + 1/ r is the mean curvature,
rr12and are the principal curvature radii and σ is the vapor/liquid interfacial tension. The
sign of the mean curvature is defined by convention assuming that 20H > if ppAB> .
If the difference in pressure between phases A and B is due to gravitational forces, i.e.
p= p − ρ gh and p= p − ρ gh, curvature is a function of elevation and a A refA A B refB B
drop cannot be represented as a sphere of constant radius without introducing an error,
albeit small. The error is a function of the difference in densities between the two fluid
phases and the size of the drop. The concept of a two-dimensional interface is an idealization and in rigor one should describe a diffuse interface as it was discussed before. However, the thickness of diffuse interfaces is several orders of magnitude smaller than the characteristic size of macroscopic capillary shapes and Equation 2.6 can
be safely used as a continuum model of vapor/liquid interfaces. Also, it should become
evident from Equation 2.6 that surface curvature and pressure are directly related. Since
physical states with lower energy are more stable, a water drop will adapt the shape of a
sphere at low gravity or when it is very small in order to reduce its surface energy.
Otherwise, the shape of a drop is either a pendant drop or a sessile drop. The curvature of
a surface is defined as the inverse of the radius of a circle tangent to the surface. There
are two independent radius of curvature and their directions are normal to each other.
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The sum of the two characteristic curvatures is equal to the mean curvature of the surface, defined as 2H.
Description of interfaces using the Young-Laplace equation is valid on interfaces away from other phases and from external force fields. Near solid surfaces molecular forces are introduced to account for an additional pressure term, generically described as
“disjoining pressure” although it can be either attractive or repulsive effects. The basic assumption, that disjoining pressure can be simply added to hydrostatic pressure was first introduced by Derjaguin and coworkers (For a historical introduction and review see the treatise by Derjaguin, Churaev and Muller, 1982) and has been used by many researchers to derive the augmented Young-Laplace equation (Mohanty, 1981):
2Hσθ=( ppAB −) +Π( h,) (2.7) where the disjoining pressure, Π (h,θ ) is assumed to be a function of the distance to the solid surface, h, and of the angle of inclination of the vapor/liquid interface with respect to the solid surface. Disjoining pressure can be computed based on the interaction of
London-van der Waals forces, electrostatic forces and structural forces (Derjaguin and
Churaev, 1974). More recently, Diaz et al (2010b) recognizing that, within the molecular region, surface tension is also affected by the presence of a solid phase, introduced the fully augmented Young-Laplace equation
∞ θ= − +Π θ θσ= + Π θ 2Hgh( ,) ( pAB p) ( h,;) gh( ,) ∫ (h ,) dh (2.8) h where gh( ,θ ) is a variable surface tension (Jameson and del Cerro, 1976) depending on disjoining pressure.
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Because liquid/liquid and solid/liquid binary forces determine the range of variation,
interfacial tension reaches its bulk value, gh( ,θσ) → , within a few nanometers of the
solid surface.
2.5 Solutions to the Young-Laplace equation.
In its most general form, the Young-Laplace equation is a nonlinear, second order, ordinary differential equation. For a Cartesian coordinate system where the mathematical surface representing the interface can be expressed as z= f( xy, ) , and
potential/gravitational energy is a function of elevation, z, the Young Laplace equation
has the following mathematical form, known also as Monge’s representation (Scriven,
1982):
2 2 dz d22 z dz dz d z dz d 2 z +1 − 21+ + 22 dy d x dx dy dx dy dx d y ρ g 2Hz= 3/2 = (2.9) 2 2 σ dz dz 1++ dx dy
There are no complete analytical solutions to Equation 2.9. Simpler forms of the
YL equation can be expected for surfaces with cylindrical, i.e. translational symmetry where the Cartesian representation is given by y = y(x):
d2 y dy 1 dx2 d dx 2H = = = (2.10) R x, y 223/2 dx 1/2 ( ) dy dy 11++ dx dx
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Similarly, when the surface is axisymmetric, i.e. it has rotational symmetry and
can be represented by a function of the type R(z):
dR2 1 dz2 ρ g 2Hz= −= (2.11) 221/2 3/2 σ dR dR R11++ dz dz
Even simpler representations can be found for a surface with translational
symmetry, such as the two-dimensional sessile drop (puddle), where the interface can be
represented by the inclination angle as a function of elevation over the solid surface,
sinθ = fh( ) :
dgcosθρ −=h (2.12) dh σ
Equation 2.12 was integrated by Diaz et al. (2010b) to get an expression for the
angle of inclination of the interface of a puddle as a function of elevation, h, and also to
find an expression for the equilibrium contact angle assuming is given by the intersection
of the interface with the solid surface:
2 2 dgcosθρ ρghC h =(hCC −⇒ h) cosθ =− 1 −hh + (2.13) dh σσ22
where hC is height of the top of the puddle. Applying the boundary condition at the contact line we get an expression for the equilibrium contact angle:
2 ρ g hC cosθe = 1 − (2.14) σ 2
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Capillary rise on a flat wall is also a translationally symmetric surface. Diaz et al.
(2013) developed a solution for capillary rise on a flat, vertical wall where elevation is
measured by the coordinate x:
ρ gx2 sinθ = 1 − (2.15) σ 2
At the maximum capillary rise, xm, that is when the liquid touches the wall, the
equilibrium contact angle is given by a simple function:
2 ρ g xm sinθe = 1 − (2.16) σ 2
Equation 2.16 was first developed by McNutt and Andes (1959) using a complex
analysis based on Legendre transforms and it is the simplest and more reliable method to
measure experimental equilibrium contact angles.
Air/liquid interfaces develop capillary surfaces where mean curvature is a linear
function of elevation, as shown in Equations 2.9, 2.11 and 2.13. This simple
functionality was exploited by Ramos, Redner and Cerro (1993) to develop a precise technique for the measurement of surface tension on the basis of the interfacial shape of pendant drops. The technique consists on fitting the cross sectional shape of a pendant
drop by second order spline functions and plotting curvature as a function of elevation.
Since curvature is a linear function of elevation, the slope of the line 2H = m z + c, is in
turn a function of interfacial tension, mg= ρσ/ .
One of the most comprehensive analysis of solutions of the Young-Laplace equation is due to Padday and co-workers. In the 1970’s and early 1980’s the late
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professor L. E. Scriven and his graduate students performed extensive analysis of
capillary shapes associated to various geometries and boundary conditions as well as the
mathematical solutions of the Young-Laplace equation. Many of the results, published or
unpublished were later included as part of class noted for an introductory graduate course
on Interfacial phenomena (Scriven, 1982). One of the most interesting and a practical
result is an analysis of interfacial shapes of translational symmetry under gravity where
the governing equation is:
dz d dx −=κo az (2.17) dx 2 1/2 dz 1+ dx
where κo= 2H ref is a characteristic curvature, i.e. at a given point of the surface where
hydrostatic pressure is known, for example an inflection point in the profile described by
Equation 2.17 where the curvature is cero and −=κo az. In addition to being
translationally symmetric, Equation 2.17 is invariant to reflections on the x-coordinate, i.e. ± x . If we choose to write this equation as a function of the inclination angle and the
arc length along the interface, s, Equation 2.17 simplifies to:
d θ =κ az + (2.18) ds o
Differentiation of Equation 2.18 renders an equation that can be used to describe a variety of elastic shapes, including cylindrical interfaces, bending of thin rods and the oscillation of circular pendulums (Scriven, 1982).
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d 2θ = sinθ (2.19) ds2
Remarkably, Eq. Equation 2.19 has a solution in terms of Jacobian elliptic sine
functions (Abramowitz and Stegun, 1964). All solutions of the Young-Laplace equation
in a force field described as the gradient of a scalar potential, share this property.
For a given liquid fluid, i.e. for constant values of density and interfacial tension,
the mathematical function satisfying the Young-Laplace equation and describing all
possible capillary shapes is unique, although the particular form of the equation depends
on symmetry and boundary conditions. The Young-Laplace equation includes a single
parameter, the capillary length, LgC = σρ/ , thus for a given capillary length, all solutions of the Young-Laplace equation should have the same functional form. For an interesting analysis of the progression of solutions of Equation 2.17 without reference to
boundary conditions the reader is referred to the class notes of the late professor Scriven
(1982). As a corollary to this analysis, Scriven (1982) stated: If for given boundary conditions there is any interface that satisfies the (Young-Laplace)equation, it must have a meridional section that is a segment of one of the profiles in the solution space. A simpler description consists in the observation that in a gravitational field all solutions of the Young-Laplace equation in the curvature-elevation space are straight lines with
2 slopes given by mg=ρσ/ = 1/ LC .
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CHAPTER THREE
CAPILLARY RISE IN A CORNER. EXPERIMENTAL
3.1 Experimental setup
The experimental setup used to perform the experiments is shown in Figure 3.1
Figure 3.1 Experimental setup
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1. Pieces of glass and glass recipient 3 2. Digital camera 3. Moving system with clamps 4. Source of additional spotting light 4 2 1
Figure 3.2 Experimental setup sketch
The experimental setup consists of the following elements:
- Two pieces of glass, which dimensions are 15.3x27.9 cm, and a glass recipient of
8x18x18cm.
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Figure 3.3 Pieces of glass used
- A digital camera Nikon D70, with a Nikon AF micro Nikkor 60 mm lens and
approximation rings used when needed.
Figure 3.4 Camera Nikon D70 and lens used
- A moving system with two clamps, controlled by a motor electro-craft E-552-S.
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Figure 3.5 Motor electro-craft E-552-S
- A source of additional spotting light model Cuda I-150. This is a light source with
two fiber optic cables used to bring an intense amount of light on the prototype.
Figure 3.6 Source of additional spotting light used
- Rubber bands and different metal pieces to form different angles between the
glass pieces.
The experimental procedure consists of:
1) Cleaning the glass surfaces, using distillate water, Fisherbrand Versa-Clean liquid
concentrate and Kimtech delicate task wipers.
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2) Filling the glass recipient with distillate water.
3) Select two metal pieces and put them between the pieces of glass to form the
angle, holding the glasses with rubber bands.
4) Immerse the pieces of glass in the water tank, making sure that the position is
completely vertical, using the help of the clamps to maintain the position, but
without holding them or making any pressure on them.
5) Wait until the capillary rise is complete and in equilibrium, and stick a known
length mark (usually 1 cm mark) on the glass surfaces in order to have a reference
for the measurements.
6) Take pictures with the camera.
The angle formed between the planes was calculated by trigonometry, using a caliber measurement of the distance between the glasses at the opposite side of the corner and knowing the width of the glasses. Table 3.1 shows the separations between the glasses and the calculated angles:
Table 3.1 Separation and angles between the glasses
Separation (inches) Separation (mm) Angle (degrees)
0.069 1.75 0.66
0.124 3.15 1.18
0.321 8.15 3.06
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3.2 The curve locus of the static contact line.
Once the equilibrium has been achieved, the static contact line of the water raised between the two glass planes is shown in Figure 3.7.
Figure 3.7 Equilibrium contact line
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The shape of the contact line is the similar in all experiments performed.
Increasing the angle between the glass planes decreases the distance of the contact line
“F” (Figure 3.8) to the touching edge of the glasses that forms the angle “AC” (Figure
3.8) and vice versa. Also, the capillary rise of the water in the horizontal plane next to the water pool “EE” changes, increasing the height of the contact line “F” when the angle is decreased.
D C
F
E E
B A
Figure 3.8 Equilibrium contact line
The curve locus of the static contact line obtained in the experiments is also very similar to experimental and theoretical results of authors, like Hauksbee (1713), Concus and Finn (1969), Bico and Quere (2002), Higuera et al (2008).
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3.3 Geometrical relationships.
The Young-Laplace equation for capillary rise in a corner includes two principal
radius of curvature. One of the radius, , lays on a surface with tangent parallel to the
푉 axis of the wedge, while the other radius푅 lays on a horizontal surface, . The Young-
퐻 Laplace equation a capillary rise related to the hydrostatic pressure 푅term, as a linear
function of elevation:
11 ρ g +=z;0 z = at liquid pool (3.1) RRVH σ
The horizontal radius of curvature, RH, can be related using a simple trigonometric construction to the wedge, α , and contact θ angles. The distance from
the point of contact to the centerline, d, can be related to the wedge angle using a simple
geometrical relationship (See Figure 3.9):
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l -RH
α θ d
δ
Figure 3.9 Top view of solid wedge in contact with a liquid pool.
The contact angle is the projection of the static contact angle, θo, on the horizontal coordinate and the half angle of the solid wedge is α.
dz( ) R= ;d= lz( ) sinα (3.2) H π sin −−αθhor 2
Where, l is the distance from the corner to the interface, measured along the side of the solid surface. A simple geometric construction was used by Pomeau (1986) to
demonstrate that capillary rise could only take place if π/2>+ αθo . Substitution of the expression for the distance “d” into Equation 3.2 gives an expression for the horizontal radius:
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sinα (aR) H = δ ( z) cosθα− sin sinα = (b) RH lz( ) (3.3) cos(αθ+ ) cosθα− sin (c) δ ( x) = lx( ) cosα cos θ− sin αθ sin
Where δ(z) is the distance from the touching edge, “AC”, to the centerline of the
air/liquid interface “F” and l(z) is the distance from “AC” to the contact line “F”
measured along the glass surface. Since the horizontal is not affected by gravity forces the horizontal cross section of the interface is a surface with translational symmetry, and the horizontal shape of the interface is a segment of a circle. The two-dimensional shape of the interface is generated by the vertical translation of the segment of a circle of varying radius. The pressure within the liquid column is hydrostatic and determined by gravitational forces. Within the liquid column all horizontal surfaces have the same pressure. Thus, at the center of symmetry, the Young-Laplace equation is satisfied by a mean curvature that it is only function of the elevation parameter, z.
The angle of inclination of the centerline with respect to the horizontal is given by:
dRH ( z) 1 = = cotφH (3.4) dz tanφH
The mean curvature of the interface, for a surface with cylindrical or translational
symmetry is given by (Scriven, 1982):
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1 sinφ = 2 1/2 dR 1+ H dz sinφφHHd sin 2H = + d sinφ dR2 (3.5) R dR H HHd sinφ dz 2 = = − dz dR 2 3/2 dRH H dR dz 1+ H dz
Finally, substitution of Equation 3.5 into the Young-Laplace equation gives:
2 dRH 1 2 z 2H = −=dz (3.6) 221/2 3/2 2 LC dRHH dR RzH ( ) 11++ dz dz
3.4 The small elevation region.
The results for the small elevation region, z~0, for the two smaller wedge angles
(0.63 and 1.18 degrees) are similar to each other but different for the case of a larger angle and larger separation between the solid surfaces. The explanation for these differences can be found on the “footprint” of capillary rise, as shown schematically in
Figure 3.9. The relationship between capillary rise and the angle of inclination of the interface is given by (McNutt and Andes, 1959):
zz22ξ θξ=−=− = sin 12 1 ; (3.7) 22LLCC
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The tangent of the angle of inclination of the interface is given by the derivative of the film thickness at the centerline, δ(z), with respect to elevation
ξ 2 1− 2 ddδδ 2 ξ− 2 = =−=−tanθ = (3.8) dz dξ L 2 222 C ξ 42−−( ξ ) 11−− 2
Integration of Eq. (3.8) gives us an expression for the thickness of a single film near the wall where the liquid raises due to capillary rise.
ξ = 0 2 ξ 2 − 2 =−= ξ = ∫∫dR0 Ro d 1.48 (3.9) 2 2 Ro 0.37 42−−( ξ )
The integration limits were chosen to account the fact that the film thickness is zero at the maximum elevation, where ξ = 2 and the lower limit for an elevation of about 0.0006, i.e. barely detectable by the human eye. Otherwise, the film thickness goes asymptotically to infinity as the elevation approaches zero. This distance is equal to half the distance for the large wedge angle of separation used in our experiments.
All experiments were performed at room temperature, typically 22˚C. At this temperature, the physical properties for water are shown in table 3.2 (Perry, 1999)
Table 3.2 Physical properties of water at 22˚C.
Density (kg/m3) Surface tension (N/m)
977.770 0.0725
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With these values, the capillary length for water at 22 ˚C is
= = 0.0027 휎 퐿퐶 � 푚 휌 ∙ 푔 The larger separations between the glass surfaces at the edge opposite to the touching edge, “BD”, are 0.0018 m and 0.0032 m. Thus, half of the separation between
plates, d, is smaller than dL<=(1.48C ) 0.004 m and the shape of the interface is a cylinder. However for the larger wedge angle (3.06 degrees) the separation between the glass plates is 0.0082 m and the half separation, d, is equal to a computed film thickness with a maximum elevation of 0.0006 m. This is the value used in the computations in
Equation 3.9, d=≈=0.0041 mL( 1.48C ) 0.004 m. As a consequence, for this separation between plates the interface has a flat horizontal section surrounded by two capillary rise surfaces.
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Figure 3.10 Capillary rise at the opposite side of the corner, angle of 3.06 degrees
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Figure 3.11 Side view of the glasses with an angle of 3.06 degrees
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Figure 3.12 Capillary rise at the opposite side of the corner, angle of 1.18 degrees
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Figure 3.13 Side view of the glasses with an angle of 1.18 degrees
When the angle between the planes is 0.66 degrees, the separation between them at the opposite side of the corner is only 0.0018 m, smaller than the footprint found for capillary rise on a flat surface, 0.004 m. For small angles, capillary rise starts at the edge of the glass wedge opposite to the touching edge. There is an end effect in which water forms a curved surface between the glasses as shown in Figure 3.14
48
Figure 3.14 View of the opposite side to the corner, angle of 0.66 degrees
Because of the light reflexing between the pieces of glass and the water at that zone, it is difficult to capture the phenomenon in a picture, but through the lens of the camera it is possible to see how the capillary rise is much larger inside of the glasses than outside just at the edge of them.
49
Figure 3.15 Side view of the glasses with an angle of 0.66 degrees
More pictures are provided in Appendix A section 2.
3.5 The incipient capillary rise region.
The incipient capillary rise region is the most interesting region and the region that needs a deeper study. In this region, the shape of the curve changes from horizontal to vertical. The following pictures illustrate the different behavior depending on the corner angle.
50
Figure 3.16 Side view of the intermediate region with an angle of 3.06 degrees
51
Figure 3.17 Side view of the intermediate region with an angle of 1.18 degrees
52
Figure 3.18 Side view of the intermediate region with an angle of 0.66 degrees
The distance from the touching wedge, “AC”, to the interface denoted by δ(z),
increases as the angle of separation between the glass plates is decreased (See Equation
3.3). When the angle of separation between the glass plates decreases, the distance
between the glass plates, 2d, also decreases in agreement with Equation 3.3. Smaller plate separations in turn result in decreasing values of the radius of curvature (increasing curvature) and consequently, larger values of δ(z).
53
The locus of the intermediate region is different for different angles, being a
larger and more rounded transition region when the angle is smaller. For larger angles,
the intermediate region shows a sharp transition between the lower region (small values of elevation) and the upper region (large values of elevation).
More pictures are provided in Appendix A section 2.
3.6 Results for large values of elevation.
The results for large values of elevation are similar in shape regardless of the
corner angle. The contact line adopts the form of a hyperbola, and for tall glass surfaces,
the angle of the contact line with respect to the horizontal is close to 90 degrees. In some
cases, the experiments have an end effect just at the end of the glasses that makes the
distance l(z) go close to zero.
54
Figure 3.19 Pictures of the curve locus for large values of elevation
Figure 3.19 shows the curve locus for large values of elevation. On the left, the end effect for an angle of 3.06 degrees is shown; while on the right the effect is shown for a corner angle of 0.66 degrees.
More pictures are provided in Appendix A section 2.
3.7 Experimental results for different corner angles.
The results of four different experiments with three different corner angles are presented. In order to measure the length versus the height of the capillary rise, a grid method was used. This method consists of placing a known distance over one of the
55
glasses (in this case a piece of paper with a line of 1 cm long drawn on it). Once the
picture is taken, a grid is placed over the picture. Amplifying the picture with the grid
using a computer, it is possible to take precise measurements of the shape of the water-air
interface between the glasses.
Figure 3.20 Example of the grid method used
The tables of height versus length are presented in Appendix A section 1. Figures
3.21 to 3.23 show experimental data of δ(z) vs. z, for three different corner angles. To assure repeatability, four different experiments were performed for each angle of
separation between plates. Except for the experiment #4 shown in Figure 3.23 all
experimental data fall within 10 % separation for values of δ(z). Film thickness is a very
sensitive function of plate separation as well as of contact angle between the liquid and
the glass surface.
56
Corner angle 3.07 degrees 100.0 90.0 80.0 70.0
60.0 Experiment 1 50.0 Experiment 2 z [mm] z 40.0 Experiment 3 30.0 20.0 Experiment 4 10.0 0.0 0.0 50.0 100.0 150.0 200.0 l (z) [mm]
Figure 3.21 Results for a corner angle of 3.07 degrees.
57
Corner angle 1.18 degrees 120.0
100.0
80.0
Experiment 1 60.0 Experiment 2 z [mm] z
40.0 Experiment 3 Experiment 4 20.0
0.0 0.0 50.0 100.0 150.0 200.0 l (z) [mm]
Figure 3.22 Results for a corner angle of 1.18 degrees.
58
Corner angle 0.66 degrees 120.0
100.0
80.0
Experiment 1 60.0 Experiment 2 z [mm] z 40.0 Experiment 3 Experiment 4 20.0
0.0 0.0 20.0 40.0 60.0 80.0 100.0 120.0 140.0 160.0 l (z) [mm]
Figure 3.23 Results for a corner angle of 0.66 degrees.
In order to compare the difference between corner angles, the results of experiment 1 of each corner angle were plotted.
59
Comparison between corner angles 120.0
100.0
80.0
60.0 Experiment 1 3.06 degrees z [mm] z Experiment 1 1.18 degrees 40.0 Experiment 1 0.66 degrees
20.0
0.0 0.0 50.0 100.0 150.0 200.0 l (z) [mm]
Figure 3.24 Comparison of the results of different corner angles.
Figure 3.24 shows the different behavior of the capillary rise depending on the corner angle. In all experiments, no matter what corner angle was used, the capillary rise reached the top of the glasses. The most important difference between them is that the incipient capillary rise region makes a sharp transition with lower angles values, while the transition is smooth and more rounded with smaller corner angles.
3.8 Contact angle measurement.
Being an important parameter, the equilibrium static contact angle was measured between the glass and the water. The experiment consisted on taking a picture of the drop over the glass surface and measuring the angle by projecting a tangent to the air/liquid interface near the solid surface. The measurements were performed by printing
60 each picture and measuring the angle between the surface and the tangent line. The proper way to measure contact angles consists of matching a solution of the Young-
Laplace equation to the picture of the drop and then projecting the solution to the glass surface. However, the values of contact angles measured by these simple experiments are within the expected range of experimental data.
Four different pictures of drops over the glass surface were used. The final contact angle value is an average of the four measurements. Figure 3.25 shows one of the drops used, the other three can be found in Appendix A section 3.
Figure 3.25 Picture of drop 1 used to measure the contact angle
Once the four contact angles were measured, an average value of the four was used as the contact angle. Table 3.3 contains the four values and their average.
61
Table 3.3 Contact angle measurements and average value.
Drop 1 Drop 2 Drop 3 Drop 4 Average Contact angle 34.9 42.5 42.5 37.9 39.5
The differences in the measurements of the contact angle can be explained as the combination of: small differences in each drop volume (a transfer pipet was used to deposit the drop over the glass surface); the distance from the drop to the camera lens for each measurement is not exactly the same, because it is difficult to place a small drop just in the same spot where the before drop was placed (it is necessary to clean the glass between each measurement); finally there is some random roughness of the glasses surfaces, as it will be explained in section 3.9.
3.9 Experimental errors.
During the development of the different experiments, several errors were encountered. Some of them were solved improving the experimental setup, but others are still present in the final results.
- Cleaning the glasses.
One of the mayor issues in the experiments was how to clean the glasses. In order to have good results, the glasses must be perfectly clean, in order to have a perfect contact line between the water and the glass surface. To clean the glasses, distillate water, soap and delicate tasks wipes were used. In some of the experiments it is possible to see how the wipes left some residual pieces over the surfaces, making the contact angle
62
vary. In other cases, it is possible to see how the water cannot rise because something is
blocking its path. Also, sometimes the glass seemed to be clean, but when the pictures
were taken it is possible to see some soap remaining over the surface. This made the
experiments really hard to prepare. Also, it was not possible to clean the glasses and
leave them to dry, because dust particles will deposit over the surface.
- Air bubbles trapped inside the capillary region.
In some of the experiments it was possible to see how small air bubbles were
trapped inside the capillary region. This phenomenon can affect the development of the
interface because the pressure in that small region is different than the one outside. This
is a random error that sometimes occurred, but easy to fix repeating the experiment.
- Roughness of the surfaces of the glasses.
It is assumed that the surfaces of the glasses are perfectly flat and homogeneous.
In some of the experiments, the pictures showed that the surfaces were clean but still
some irregularities on the interface appeared. Probably this is due to some roughness at
the molecular level of the glass surfaces. In addition, it is possible to see in some pictures
that the edges of the glasses got scratches with time, which leads to a non-perfect contact between the edges forming the angle. This can lead to different values of the contact angle at those points.
- Pressure applied by the clamps.
At the beginning, all experiments were performed holding the two pieces of glass with the clamps, without touching the bottom of the glass container. It was realized that
63
the pressure of the clamps affected the distance between the glasses. The distance was
reduced leading to accumulation of water in those regions that affected the whole locus of
the interface. While the glasses are rigid, they have a low flexibility in order to get
curved with pressure. This is a very important error because even using pieces of
stainless steel, with enough pressure over them, the distance between the pieces can be
affected.
- Orientation of the camera taking the pictures.
A small difference in the orientation of the camera when the pictures are being taken can lead to errors in the measurements of the distance. Since the order of magnitude of the measurements are in terms of millimeters, not taking the pictures in a perfect perpendicular way to the glasses can lead to significant differences in the measurements.
- Ruler used to perform the measurements.
Some of the pictures were expanded up to 1.6 – 1.7 times the original size.
Assuming that the precision of the ruler used to make the measurements is between 0.5 –
1 mm, this precision can lead to a significant difference between the real value and the measurement.
64
- Grid used to perform the measurements.
One way to improve the accuracy of the measurements is to decrease the grid size placed over each of the pictures. The problem of using a coarse grid is that it can interfere in the observation of the shape of the interface, making the measurements hard to execute.
Appendix A section 4 contains pictures showing the effects explained above.
65
CHAPTER FOUR
CAPILLARY RISE IN A CORNER. THEORETICAL
4.1 Introduction.
In a cylindrical coordinate system, the expression for the curvature in the Young-
Laplace equation can be expressed in terms of the radial coordinate variable. In our system the radial direction coincides with the horizontal radius of curvature of the interface. Thus, the complete expression for the Young-Laplace equation can be written in the following form:
2 dRH 1 2 z 2H = −=dz (4.1) 221/2 3/2 2 LC dRHH dR RzH ( ) 11++ dz dz
66
l
-RH α θ d
δ
Figure 4.1 Geometry of the problem
In figure 4.1, α is half of the wedge angle, θ is the contact angle, l is the distance from the wedge to the interface air-glass-liquid, δ is the distance from the wedge to half the interface air-liquid, d is half of the distance between the ends of the air-glass-liquid interface and RH is the horizontal radius of curvature.
In turn, the horizontal radius of curvature (See Figure 4.1) can be related to the distance of the center of the interface to the touching edge, δ(z), and to the locus of the line of contact between the glass and the liquid, l(z):
sinα (aR) H = δ ( z) cosθα− sin sinα = (b) RH lz( ) (4.2) cos(αθ+ ) cosθα− sin (c) δ ( z) = lz( ) cosα cos θ− sin αθ sin
67
Introducing a characteristic length equal to the Capillary length, LgC = σρ/ , to define the dimensionless variables:
−1 ξξ=z/; LCH R( ) = R H( zL) C ⇒=2 H LC 2 H =ξ, it is possible to obtain an expression for the Young-Laplace equation in a dimensionless form:
2 dRH 2 1 dξ 2H = −=ξ (4.3) 221/2 3/2 dRHHdR RH (ξ ) 11++ ddξξ
Solving the dimensionless Equation 4.3 and then applying the definitions in
Equation 4.2, will lead to the relation between z and l(z) for any wedge angle. Since
Equation 4.3 is a nonlinear equation, some approximations are needed in order to solve it.
4.2 Solution for large values of elevation.
The first approximation that can be done to solve equation 4.3 establishes that, for large values of elevation the relationship between the non-dimensional elevation ξ and
R(ξ) is a hyperbola.
2 1/2 2 −14d R 12d R dR 11/2 R(ξξ) = ⇒=−; = ;1 + =ξ +1 (4.4) ξ ξ23 ξξ ξ ξ2 dd d
68
Using equation 4.4, the two principal radius of curvature in Equation 4.3 become:
1 ξ 3 (a) = 241/2 1/2 dR ξ +1 R(ξ ) 1+ ξ d dR2 2 dξξ232ξ 3 (b) − =−=− (4.5) 243/2 1 3/2 3/2 ξ 4 + ξ +1 dR 6 1 1+ ξ ξ d
33 ξξ2 (cH) 2 =1/2 −=3/2 ξ 44 ξξ++11
For values of ξ ≥ 4, the relationship between ξ and R(ξ) follows Equation 4.5c.
Numerical results for equation 4.5c can be found in Appendix B.
4.3 Solution for small values of elevation.
Equation 4.5c does not hold for values of ξ ≤ 4, because the contact angle is no
longer horizontal. Thus an approximation for small values of elevation is needed must be
introduced. It is assumed that for small values of the elevation variable, the shape of the meniscus is similar to capillary rise on a flat surface, where the dominant curvature in the
Young-Laplace equation is the vertical curvature
dR2 d sinφ dξ 2 2H == −=ξ (4.6) dR 2 3/2 dR 1+ ς d
69
Writing the above expression for the mean curvature for a curve where ξ (R)
(Scriven, 1982)
2ξ d dR2 2H = = ξ (4.7) 2 3/2 dξ 1+ dR
Equation (4.7) can be integrated once to get (McNutt and Andes, 1959):
−2 ξ 2 = + C (4.8) 2 1/2 dξ 1+ dR
Where C is an integration constant. The solution given by McNutt and Andes is
based on a Legendre transformation but it has a simple form for the range of interest:
2 1 24+−ξ 2 R(ξ)=ln −4 −ξξ2 ; 0 ≤≤ 2 (4.9) 2 ξ
Computing the first and second derivatives of Eq. 4.9 and using them to build the expression for the mean curvature in Eq. (4.8):
dR(ξ ) ξ 2 − 2 = dξ ξξ4 − 2 dR2 (ξ ) 8 = (4.10) dξ 2 223/2 ξξ4 − 8 223/2 ξξ4 − 2H = 3/2 2 ξ 2 − 2 1− ξξ− 2 4
70
These expressions give results similar to the ones found for the vertical curvature in Eq. 4.5.
4.4 Intermediate region.
Equation 4.3 has no parameters and could be integrated either as a boundary value problem or as a backward computation problem, where an initial value approach is used.
The boundary values are:
( ) 0 and ( ) 0
휉 → ∞ 푅� 휉 → 휉 → ∞ 푅� 휉 → The initial value problem is defined as
1 1 ( ) =
휉 ≫ 푅� 휉 휉 1 1 = 푑푅� 휉 ≫ − 2 푑휉 휉 In order to solve the Young-Laplace equation as an initial value problem, equation 4.2 needs to be decomposed as a system of three linear ordinary differential equations, all of them as function of the arc length s.
d ξ =sinφ ds dR = cosφ (4.11) ds dφφ sin =2H − ds R
71
Solving the system of equations provides a solution for the values of ( ), and
. 푅� 휉 휉
휙 The system was solved using a Runge-Kutta method with fixed step. The initial conditions used were = 40, ( ) = 1/40 and = /2, and the method used consisted of 1000 steps with 휉= 0.004푅�. The휉 software used휙 was휋 MathCad.
푑푠 Once the solution for ( ) is achieved, those values need to be transformed to dimensional values using the푅� 휉equations:푎푛푑휉
( + ) ( ) = ( ) × ( ) 푐표푠 훼 휃 푙 휉 푅 휉 푠푖푛 훼 = × (4.12)
푧 휉 퐿푐 ( ) = ( ) ×
푙 푧 푙 휉 퐿푐
72
hyperbola vs computational solution 40
35
30
25
20 ξ =2H hyperbola solution
15 computational solution
10
5
0 0 10 20 30 40 50 60 l (ξ)
Figure 4.1 Non dimensional forms of hyperbola and computational solutions
Figure 4.1 shows that for large values of elevation, the hyperbola solution and the computational solution overlap each other, but for elevation values below = 4 they differ. This is a reasonable result since the hyperbola solution is only valid for휉 values of
> 4.
휉
73
Comparison of the results 0.12
0.1
0.08
0.06 z [m]
0.04
0.02
0 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 l (z) [m] Experiment 2 Experiment 4 ComputationalComputational solutionsolucion solution with with correction correction
Figure 4.2 Computational versus experimental results
The computational values obtained are for a system with = 0.59 degrees,
= 39.5 degrees and = 0.0027 m, corresponding to the second 훼set of experiments
푐 performed휃 in chapter 3.퐿 Note that for the computational results, is half of the wedge angle (1.18 degrees in experiments). 훼
The computational results can be found in Appendix B.
For the smaller values of elevation a correction needs to be implemented, since the Young-Laplace equation does not take into account the capillary rise phenomena
74 between the plates for elevation values close to zero. The equation only obtains the capillary rise as a function of the horizontal radius.
One suggestion to correct the values is the following equation, developed by
McNutt and Ande (1959):
= 2 × (1 sin ( )) (4.13)
푧푐표푟푟푒푐푡푖표푛 퐿퐶� − 휃 Equation 4.13 is only valid for values of < 2. This means that this correction should only be added to the lower values of elevation.휉 The value obtained for water and a corner angle of 1.18 degrees is z=0.0023 m.
This addition creates a discontinuity in the results, between the values of < 2 and > 2, since for = 2 the value of z is=0.0054 m. In order to solve that 휉 disconti휉 nuity, an intermediate휉 region is created for 4 > > 2. For this particular region, the correction becomes: 휉
= (2 /2) × 2 × (1 sin ( )) (4.14)
푧푐표푟푟푒푐푡푖표푛 − 휉 퐿퐶� − 휃 With this intermediate region, the discontinuity vanishes, and for = 4 the values of elevation correspond to the solution of the Young-Laplace equation. 휉
75
CHAPTER FIVE
AN EXPLANATION TO HAUKSBEE EXPERIMENTS
5.1 Introduction to Hauksbee experiments.
Hauksbee (1713b) performed several experiments of capillary rise in corners
using different angles of inclination, different immersions of the plates and different
positions. In this chapter, a theoretical explanation of these experiments is going to be
developed.
5.2 Corner wedge in the lower position.
This section corresponds to Figures 1, 2 and 3 in Hauksbee (1713b) experiments.
These first three cases are almost identical, except that the angle of inclination of the plates and the immersion is different for every case.
76
The first case, Figure 1 of Hauksbee, shows an small angle of inclination with
respect to the horizontal, and also a small amount of area of the glass plates immerse in
the liquid Assuming that the figures are precise enough, the angle of inclination appears
to be 9.2o, while in case 2 and 3 are 40o and 43o respectively. It seems that the difference
between case 2 and 3 is mostly than the immersion of the lower corner is larger in case 2.
Figure 5.1 Figure 1 of Hauksbee (1713b)
In Figure 5.1, the line “ac” is the corner where the class plates touch each other, the line “ab” is the water/air interface outside the glass plates and the line “dd” is the water/air interface inside the glass plates.
In order to proceed with the description of the interface shapes as a solution of the
Young-Laplace equation, the geometrical variables are going to be described as Figure
5.2 shows:
77
z s d d u β γ β c x b a d
c
Figure 5.2 Sketch of the geometrical variables.
The sketch is a mirror image of Figure 1 in order to align the figure in the positive x-z directions. The meaning of lines “ab”, “ac”, and “dd” has been preserved. The angle
β is the angle of inclination of the “ac” line of the plates with respect to the horizontal and the angle γ is the angle of inclination of the interface inside the glass plates with respect to the horizontal line, measured at the edge of the plate. The distance d is the depth of immersion of corner “c” into the liquid. On the glass plane we assume there is local coordinate system, (u, s) inclined an angle β with respect to the laboratory coordinate system, (x, y, z).
The glass wedge formed by the two glass plates touching at the line “ac” form an angle α and the contact angle between the glass plates and the liquid is denoted by θ .
The line formed by the ends of the glass plates and the “ac” line intersect at a 90o angle.
On a (u, s) coordinate system located on the glass plates, the direction along the “ac” line will be “s” and the direction on the normal line will be “u”. The separation distance between the plates growths along the normal line, following the u-direction:
78
uz=cosβ = zo cos ββ − x sin δ=usin α ⇒=δzxo sin α cos β − sin αβ sin sz=sinββ = x cos
(5.1)
The value of zo is arbitrary and depends on the depth of immersion of the glass plates in the liquid, d. Horizontally, along the x-direction, the separation between the glass plates decreases because of the inclination of the plates with respect to the laboratory coordinate system. When the separation between plates decreases, the radius of the liquid meniscus in contact with the glass plates also decreases:
δ zxo sinα cos β− sin αβ sin Rx( ) = = (5.2) cos(αθ++) cos(αθ)
The shape of the liquid/air interface is given by a solution of the Young-Laplace equation where the elevation zx( ) of the interface at any given point is given by
σ cos(αθ+ ) ρσgh( x) = = (5.3) Rx( ) zo sinα cos β− x sin αβ sin
Equation 5.3 can be reordered to find the elevation h(x) as a function of distance along the plate, x:
σσcos(αθ+ ) hx( ) = ; LC = ρgzo sin α cos β− x sin αβ sin ρg x X = (5.4) cos(αθ+ ) LC λ ( X ) = ZXsinα cos β− sin αβ sin z o Z = LC
79
At the right hand side end of the plate in Figure 5.1, there is a small elevation
corresponding to the largest separation between plates and, consequently, the smaller
elevation, λo ( x = 0):
cos(αθ+ ) λo (0) = (5.5) Zo sinαβ cos
Substitution of Equation 5.5 into Equation 5.4 we get the final expression for
capillary rise along the plate:
cos(αθ++) cos(αθ) λλ( X ) −=o (0) − (5.6) ZXoosinαβ cos− sin αβ sin Z sin αβ cos
Equation 5.6 can be reduced to the following quasi-linear form:
aX λλ( XX) −=( ) (5.7) o b− cX
where the coefficients are defined as follows:
a =cos(αθ + ) sin α sin β 2 bZ= ( o sinαβ cos ) (5.8)
cZ= sinαβ sin( o sin α cos β)
Although Equation 5.7 is not a straight line, for small inclination angle, β , and
relatively small contact and wedge angles, αθ+ , it appears to be a line with very small curvature.
80
elevation λ(X) vs X 14.00
12.00
10.00
8.00
X) λ( 6.00 elevation λ(X)
4.00
2.00
0.00 0 1 2 3 4 5 X
Figure 5.3 Capillary rise described by equation 5.7 – Case 1
Figure 5.3 shows the results for an experiment similar to case 1. The angle of separation of the glass plates is α =1o , the water/air contact angle is θ =10o , the angle of inclination of the line “ac” with respect to the liquid surface is β =10o , while the liquid
rise at X = 0 is Zo = 5 . The numerical results can be found in Appendix C.
Case 2 is similar to case 1, with the difference that the angle of immersion of the glass plates is larger, β = 40o and the depth of immersion of the lower corner of the plates is also larger.
81
elevation λ(X) vs X 12.00
10.00
8.00
X) 6.00 λ( elevation λ(X) 4.00
2.00
0.00 0 1 2 3 4 5 X
Figure 5.4 Capillary rise described by equation 4.7 – Case 2
In Figure 5.4, the angle of separation of the glass plates is α =1o , the water/air
contact angle is θ =10o , the angle of inclination of the line “ac” with respect to the liquid
o surface is β = 40 , while the liquid rise at X = 0 is Zo =10 . There is a hint of the
hyperbolic shape in Case 2, but it still mostly resembles a line with very little curvature.
The numerical results can be found in Appendix C.
Case 3 is almost identical to case 2. The difference among them is the depth of immersion of the plate, larger in case 3 than in case 2.
82
elevation λ(X) vs X 60.00
50.00
40.00
X) 30.00 λ( elevation λ(X) 20.00
10.00
0.00 0 1 2 3 4 5 X
Figure 5.5 Capillary rise described by equation 5.7 – Case 3
In Figure 5.5 the angle of separation of the glass plates is α =1o , the water/air
contact angle is θ =10o , the angle of inclination of the line “ac” with respect to the liquid
o surface is β = 40 , while the liquid rise at X = 0 is Zo = 5 . The numerical results can be
found in Appendix C.
The shape of the interface is definitely now closer to a hyperbolic shape. As the liquid rise on the corner that is above the surface, it runs into a region of small separation between the plates.
83
5.3 Corner wedge in the lower position, figure number 4.
It is believed that there is an error in the paper regarding this Case (Figure 4 of
Hauksbee, 1713). The line “ac” should be on top of the figure and this case should be the first of a sequence that includes Figures 4, 5, 6 and 7. Instead, in the text, the author identifies as “and so of all the rest of the curves, as in the Figures 2, 3, 4, 5, 6. as a result of the several angles, made by the touching sides of the planes. Now when the touching sides are placed upwards… ”. In the interpretation used in this chapter, figures 1, 2, and
3 are cases where the edge of contact of the planes is immersed, at different angles and
different depths of immersion. On the other hand, in Figures 4, 5, 6, and 7, the edge of
contact is above the water surface.
5.4 Corner wedge in the upper position.
Figure 5.6 Figures 5 and 6 of Hauksbee
84
It is important to notice that the touching edge of the plates (line “ac”) is now above the liquid level. There is no explanation on the paper but the difference between
Figures 5 and 6 seems to be the angle of inclination β.
Case 5 is analogous to cases, 6 and 7, except for the angle of inclination with respect to the liquid interface (horizontal) and the depth of penetration of the plates in the liquid.
c
d
β d a a b
d
Figure 5.7 Sketch of Case 5 showing the geometric variables
Figure 5.7 is a sketch of cases 5, 6, and 7 showing the main geometric variables.
Notice that the figure has been flipped to conform with the definition of angles in Figure
5.2.
This case is similar to the typical capillary rise in a corner phenomenon, with two differences, (a) the touching edge is not vertical, (b) the plates are finite and both corners
85 of the touching edge are outside of the liquid. To analyze this phenomenon, first it is necessary to focus on the related case where two infinite plates are immersed at an angle different from 90o with respect to the liquid/air interface. There are two basic cases shown in Figure 8, one where the angle of inclination is smaller than 90o, βπ< /2, and another case where βπ> / 2.
β>π/2 β<π/2
Figure 5.8: Sketch of the cases of infinite plates where the touching edge is submerged at angles different than 90o
86
Figure 5.9: Geometrical sketch used to determine the wedge angle projected on the
horizontal direction.
One important consideration in these cases is that the angle between the plates, α,
must be measured on a horizontal direction. If the two wedges have the same value of the horizontal projection of the wedge angle, α’, the solution of the Young-Laplace equation for large elevation values, z →∞, should be identical for both cases. Notice
that the limiting case of βπ> / 2. is when βπ= . This will be similar to Figure 7 of
Hauksbee (1713). However, unlike the case of infinite plates, in this case the depth of immersion is important. Similarly, the limiting case for βπ< /2 is when β = 0 . This
will be similar to Case 1, and will describe capillary rise inside a wedge where the
touching edge is immersed in the fluid and parallel to the liquid/air interface. From the
geometrical sketch shown in Figure 5.9, it is possible to compute the value of the wedge
angle projected on the horizontal direction, α ' , as a function of the angle of inclination of
the touching edge with respect to the liquid surface, β :
87
π π β=⇒=sin α ' sin ααα ; ' = sinα '=⋅− sin αβ cos 2 (5.9) 2 βπ=⇒==sin'0 α ; α '0
For any inclination angle, β, the projected wedge angle, αα'≤ .
5.5 Plates previously fully immersed. Hauksbee case 7.
When βπ= , Case 7 of Hauksbee (1713) is recovered. Notice that in this case,
the plates where first fully submerged into the liquid and then raised up to a certain
elevation, L, measured from the line where the plates touch, line “ac”, to the surface of
the liquid. The half distance between plates, d, as a function of elevation is a function of
the wedge angle and elevation:
d=0, when z = L d=( Lz − )sinα (5.10) d= Lsinα , when z = 0
Figure 5.10 shows the original picture of Case 7 of Hauksbee (1713), where the definition of the elevation L has been included.
Figure 5.10: Case 7 of Hauksbee (1713)
88
Figure 5.10 results of keeping the touching edge outside the liquid and at an
inclination angle of π, with respect to the liquid interface. The touching edge is at a
distance L measured from the liquid interface.
Notice that for small values of L or very small wedge angles, we may have only one liquid region between the flat plates without the formation of an air region between the lower and upper liquid regions. The original Figure from Hauksbee (1713b) indicates
that the air region has a vertical curvature in the form of “lobes” indicating that the
pressure in the liquid at a3 is lower than atmospheric pressure. Because of these lower
pressure, after a certain amount of time, the bridge between the two horizontal lobes
breaks down, and the shapes of the upper and lower liquid regions is shown as a dotted
line. When the lobes touch, there is a clear air region between the top and bottom
entrained liquid regions. In any case, the column of liquid rising from the liquid interface
to the first liquid/air interface, shown as a1 in Figure 5.10, satisfies the Young-Laplace
equation for a liquid rising between two plates inclined at an angle α. Notice that the
air/liquid interface at a1 has a different construction from the interface shown in Figure
5.9. In this case, the liquid phase is on the “outside” of the wedge as shown in Figure
5.11 A.
89
Figure 5.11. A: sketch showing contact angle and shape of interface in lower liquid rise.
B: sketch showing contact angle and shape of interface for higher liquid rise.
For the higher liquid rise, the Young-Laplace equation can be written as
σcos( θαo − ) ρ gzo = (5.11) (Lz− o ) sinα
where zo is the elevation of the interface. Solving for zo, it is possible to obtain:
cos(θα− ) σ z22−+ Lz L o =0;L = o oC sinαρC g (5.12) 2 LL 2 cos(θαo − ) zLoC=±− 2 2 sinα
The main constraint to this solution is that the term inside the square root sign, must be positive.
cos(θα− ) LL≥ 2 o (5.13) C sinα
90
o For water, LC =σρ/ gm ≅ 0.0027 . If the contact angle θ ≅=20 0.35rad and
α =20' ≈ 0.00582rad , then the minimum height Lm≅ 0.07 . In other words, for a
system as it was just described, the wedge will be full of liquid as long as L is less than 7
cm. Past that point, there will be two regions of liquid and the height of the lower region
will be given by one of the roots of Equation 5.12. For the system just described when the total elevation is for example 5 in = 8.89 cm, the roots of the lower column elevation
= = = are zoo120.1773 mz ; 0.007 m . Indeed, the second root, zmo 2 0.007 , is the one
that has physical meaning.
Capillary rise of the air liquid interface at a2 is also described by the Young-
Laplace equation but in this case the liquid phase is inside of the wedge, as shown in
Figure 5.11 B. Under these conditions, capillary rise is described by:
σcos( θαo + ) ρ gL( −= z1 ) (5.14) (Lz− 1 ) sinα
and the value of z1 is
cos(θα+ ) z2−20 Lz +− L 22 L o = 11 C sinα (5.15) cos(θα+ ) zL= ± L2 o 1 C sinα
The condition for the existence of a system with two liquid regions has already
been determined, Equation 5.13. There are now two restrictions to the results shown in
Equation 5.15. The first restriction is that the first root has not physical meaning since
≤ the root cannot be larger than the elevation of the plates, zL1 1 . The second restriction
is that the second root should be larger than zero;
91
cos(θα++) cos( θα) LL≥=2 oo L (5.16) CCsinααsin
However, this condition is superseded by Equation 5.13. For a water system as
= described before, the value of the second root will be zm1 1 0.0544 . Clearly, the solution with two liquid regions, the values of the roots of Equations 5.12 and 5.15 describing the elevation of the liquid regions are: z= 0.028 mz ;= 0.054 m. o 1 1 1
Figure 5.12 shows the computed values of the position of the air/water interface for the lower and higher liquid regions for Case 7. At lower elevations of the contact edge above the water level, the two regions converge into one. As the elevation of the contact edge is increased, the lower region become smaller since the separation of the plates increases. However, the upper region increases linearly with elevation indicating, as it should be expected. that the amount of liquid trapped by capillarity near the contact edge, is constant.
92
Elevation versus distance 0.25
0.2
0.15
Z (m) lower region 0.1 higher region
0.05
0 0.06 0.11 0.16 0.21 0.26 0.31 L (m)
Figure 5.12: Values of lower and higher capillary rise for a water/air/glass system.
In Figure 5.12, the macroscopic contact angle is 20o and the wedge angle of 20’.
The numerical results can be found in Appendix C.
93
CHAPTER 6
CONCLUSIONS AND RECOMMENDATIONS
6.1 Conclusions
The experiments performed for three different corner angles show that capillary rise in corner angles have interesting and important characteristics. Experiments, theoretical analysis and numerical simulations show if the wedge, α, and contact angles, θ, are sufficiently small, water raises at the corner wedge to the top of the glass surfaces, regardless of the height of the surfaces. The parameters affecting capillary rise in corners are the contact angle between the liquid and the solid surface, the liquid characteristics
(density, surface tension) and the wedge angle.
Three different numerical methods were used to solve the Young-Laplace equation. One was a Runge-Kutta method with fixed step, a second method was a Runge-Kutta with adaptive step and the third and Adams-Bashforth method. All numerical computations gave similar results. Also, a perturbation of the initial condition was introduced to evaluate the effect of boundary conditions on results but within a certain range of perturbations the results were in agreement.. This gives confidence that the numerical results obtained are in agreement with the real solution of the equation. Differences
94 between the data obtained and the experimental data were observed, which could be due to all the experimental errors explained in Chapter 3. In the experiments described in this thesis the only external body forces is gravity. A further study in these phenomena could lead to an implementation of the capillary rise in corners to some industrial processes such as drying, wetting, surfaces impingement. This could be accomplished with low energy cost, since there is no pump or external device needed to produce a flow through the corners. Also, capillary rise has applications in spacecraft tank design (Weislogel,
2004).
6.2 Recommendations for future work
In order to improve the results presented in this thesis, these are the recommendations for future work:
- Perform the experiments with glass surfaces optical grade to avoid even minor
changes in thickness and more homogeneous wetting properties.
- Improve the experimental setup in such a way that the pieces of glass can be hold
exactly in the same position for all the experiments. This will reduce the random
error between the measurements since each time a new experiment was
performed, the pieces of glass were manually adjusted to have a vertical
orientation. This could be accomplished by using a rigid experimental setup on
the optical table used.
95
- Improve the measurement system by introducing qualitative and precise image
capture and calibration. Nowadays some digital cameras are able to measure
distances with proper software.
- Repeat the experiments with an experimental setup that can change the liquid
conditions, such as temperature, in order to obtain different capillary lengths with
the same substance and compare the results.
- Possibly improve the numerical simulations using a new method more suited to
the solution of highly stiff ordinary differential equations.
- Scale up the experimental setup for large systems, with significant values of
elevation (order of one meter) and validate capillary rise under evaporation.
- Develop an experimental setup capable of measuring the horizontal and vertical
radius along the air-liquid interface.
96
APPENDIX A – EXPERIMENTAL RESULTS
A.1 Measurements of l versus z for different corner angles
Table A.1 Measurements of l(z) versus z for an angle of 3.06 degrees.
Experiment 1 Experiment 2 Experiment 3 Experiment 4 z (mm) l(z) (mm) z (mm) l(z) (mm) z (mm) l(z) (mm) z (mm) l(z) (mm) 0.0 150.5 0.0 148.4 0.0 148.5 0.0 148.2 7.2 35.4 4.3 45.1 6.8 33.0 4.7 34.2 12.1 17.7 8.7 26.5 11.3 22.0 9.3 20.0 16.9 11.8 13.0 18.6 15.8 16.5 14.0 14.3 21.7 8.9 17.3 15.9 20.3 13.8 18.7 11.4 26.6 8.3 21.7 11.7 24.8 12.1 23.3 9.1 31.4 5.9 26.0 9.5 29.3 11.0 28.0 8.6 36.2 4.7 30.4 8.0 33.8 8.3 32.6 8.0 41.0 3.0 34.7 6.4 38.3 6.6 37.3 6.3 45.9 2.4 39.0 5.8 42.8 5.5 42.0 5.7 50.7 1.8 43.4 5.3 47.3 4.4 46.6 4.6 55.5 1.8 47.7 5.3 51.8 3.9 51.3 4.0 60.3 1.8 52.0 4.2 56.3 3.3 56.0 3.4 65.2 1.8 56.4 3.7 60.8 3.3 60.6 2.9 70.0 1.8 60.7 3.2 65.3 3.3 65.3 2.9 74.8 1.8 65.0 2.7 69.8 3.3 70.0 2.9 79.7 1.8 69.4 2.7 74.3 3.3 74.6 2.9 84.5 1.8 73.7 2.7 78.8 3.3 79.3 2.9 89.3 1.8 78.1 2.7 83.3 3.3 83.9 2.9 82.4 2.7 85.5 3.3 86.7 2.7
97
Table A.2 Measurements of l(z) versus z for an angle of 1.18 degrees.
Experiment 1 Experiment 2 Experiment 3 Experiment 4 z (mm) l(z) (mm) z (mm) l(z) (mm) z (mm) l(z) (mm) z (mm) l(z) (mm) 0.0 148.4 0.0 148.5 0.0 148.2 0.0 148.8 4.6 143.1 4.5 140.3 4.7 136.8 5.1 142.6 9.2 42.4 9.0 60.5 9.3 39.9 10.1 49.6 13.8 23.9 13.5 41.3 14.0 28.5 15.2 31.0 18.4 18.6 18.0 33.0 18.7 25.7 20.3 24.8 23.0 15.9 22.5 24.8 23.3 17.1 25.4 18.6 27.6 13.3 27.0 16.5 28.0 14.3 30.4 14.9 32.2 12.2 31.5 13.8 32.6 11.4 35.5 12.4 36.8 10.6 36.0 11.0 37.3 10.8 40.6 12.4 41.4 9.5 40.5 11.0 42.0 10.3 45.7 11.2 46.0 8.0 45.0 8.3 46.6 10.3 50.7 10.5 50.6 6.4 49.5 7.7 51.3 8.6 55.8 9.3 55.2 5.3 54.0 7.2 56.0 6.8 60.9 8.7 59.8 5.3 58.5 6.6 60.6 6.3 65.9 8.1 64.4 5.3 63.0 6.6 65.3 6.3 71.0 7.4 69.0 5.3 67.5 6.6 70.0 6.3 76.1 6.2 73.6 5.3 72.0 6.6 74.6 6.3 81.2 6.2 78.2 5.3 76.5 6.6 79.3 6.3 86.2 6.2 82.8 5.3 81.0 6.1 83.9 6.3 91.3 6.2 83.1 5.3 87.4 6.0 88.6 6.3 83.1 6.2 86.5 5.3 91.2 6.0 86.5 6.8 86.5 6.2 90.0 5.3 95.0 6.0 90.0 6.8 90.0 6.2 93.5 5.3 98.8 6.0 93.5 6.8 93.5 6.2 96.9 5.3 102.6 6.0 96.9 6.8 96.9 6.2 100.4 5.3 106.4 6.0 100.4 6.8 100.4 6.2 103.8 5.3 110.2 6.0 103.8 6.8 103.8 6.2
98
Table A.3 Measurements of l(z) versus z for an angle of 0.66 degrees.
Experiment 1 Experiment 2 Experiment 3 Experiment 4 z (mm) l(z) (mm) z (mm) l(z) (mm) z (mm) l(z) (mm) z (mm) l(z) (mm) 0.0 148.5 0.0 150.5 0.0 148.1 0.0 148.4 4.5 143.0 7.2 141.6 5.2 138.6 5.6 144.9 9.0 82.5 12.1 59.0 10.5 81.9 11.3 96.6 13.5 60.5 16.9 44.3 15.7 50.4 16.9 65.6 18.0 46.8 21.7 35.4 20.9 37.8 22.6 51.8 22.5 38.5 26.6 30.1 26.2 31.5 28.2 41.4 27.0 33.0 31.4 25.4 31.4 25.2 33.9 33.8 31.5 27.5 36.2 21.2 36.7 20.8 39.5 27.6 36.0 24.8 41.1 18.9 41.9 18.9 45.2 24.2 40.5 23.1 45.9 17.7 47.1 16.4 50.8 20.7 45.0 21.5 50.7 14.8 52.4 15.8 56.5 17.3 49.5 19.8 55.5 13.0 57.6 13.9 62.1 15.2 54.0 18.7 60.4 11.8 62.8 13.2 67.7 13.8 58.5 17.1 65.2 9.4 68.1 12.6 73.4 12.4 63.0 14.9 70.0 8.9 73.3 12.0 79.0 11.0 67.5 13.8 74.9 8.3 78.5 10.1 84.7 9.7 72.0 12.1 79.7 7.7 83.8 9.5 90.3 9.0 76.5 11.0 84.5 7.1 89.0 9.5 96.0 9.0 81.0 9.9 89.4 6.5 90.0 9.4 101.6 9.0 83.1 9.9 89.2 6.8 93.5 9.4 107.3 9.0 86.5 9.9 92.9 6.8 96.9 9.4 86.5 8.9 90.0 9.9 96.6 6.8 100.4 9.4 90.0 8.9 93.5 9.9 100.3 6.8 103.8 9.4 93.5 8.9 96.9 9.9 104.0 6.8 96.9 8.9 100.4 9.9 107.7 6.8 100.4 8.9 103.8 9.9 111.5 6.8 103.8 8.9
99
A.2 Pictures of the different experiments
The following pictures are an illustration of the experiments performed. In order to obtain the measurements shown in Appendix A section 1, more pictures were used.
Figure A.1 Experiment 1 angle 3.06 degrees
100
Figure A.2 Experiment 2 angle 3.06 degrees
101
Figure A.3 Experiment 3 angle 3.06 degrees
102
Figure A.4 Experiment 4 angle 3.06 degrees
103
Figure A.5 Experiment 1 angle 1.18 degrees
104
Figure A.6 Experiment 2 angle 1.18 degrees
105
Figure A.7 Experiment 3 angle 1.18 degrees
106
Figure A.8 Experiment 4 angle 1.18 degrees
107
Figure A.9 Experiment 1 angle 0.66 degrees
108
Figure A.10 Experiment 2 angle 0.66 degrees
109
Figure A.11 Experiment 3 angle 0.66 degrees
110
Figure A.12 Experiment 4 angle 0.66 degrees
111
A.3 Pictures of the drops used to measure the contact angle
Figure A.13 Picture of drop 2 used to measure the contact angle
112
Figure A.14 Picture of drop 3 used to measure the contact angle
113
Figure A.15 Picture of drop 4 used to measure the contact angle
114
A.4 Pictures showing experimental errors
Figure A.16 Non uniform interface shape
115
Figure A.17 Air trapped
116
Figure A.18 Air trapped
117
Figure A.19 Non complete capillary rise
118
Figure A.20 Non complete capillary rise
119
Figure A.21 Clamp pressure deforming the interface
120
Figure A.22 Close view of the glasses edges
121
APPENDIX B – THEORETICAL RESULTS
B.1 Solution for large values of elevation.
The hyperbola solution is computed as follows:
= (B.1) 푧 1 휉 퐿푐 휉 ( ) = (B.2) 1 푅� 휉 휉 2 = (B.3) [ 3] / [ 3] / 휉 2휉 1 4 1 2 4 3 2 퐻� � 휉 +1 − 휉 +1 � 휉
122
Table B.1 Results of the hyperbola solution
ξ R(ξ) 2H=ξ error % 10.00 0.10 10.00 0.00 9.00 0.11 9.00 0.01 8.00 0.13 8.00 0.01 7.00 0.14 7.00 0.02 6.00 0.17 6.00 0.04 5.00 0.20 5.00 0.08 4.50 0.22 4.49 0.12 4.00 0.25 3.99 0.20 3.50 0.29 3.49 0.33 3.40 0.29 3.39 0.37 3.30 0.30 3.29 0.42 3.20 0.31 3.18 0.48 3.10 0.32 3.08 0.54 3.00 0.33 2.98 0.62 2.90 0.34 2.88 0.71 2.80 0.36 2.78 0.82 2.70 0.37 2.67 0.95 2.60 0.38 2.57 1.10 2.50 0.40 2.47 1.29 2.40 0.42 2.36 1.53 2.30 0.43 2.26 1.83 2.20 0.45 2.15 2.20 2.10 0.48 2.04 2.68 2.00 0.50 1.93 3.31
123
B.2 Solution for the intermediate region.
The following table contains a sample of the calculations used to solve the
Young-Laplace equation.
Table B.2 Sample of the results for the computational solution
ξ R(ξ) z R(z) l(z) z corrected l(ξ) 40 0.018 0.108 4.86E-05 0.003611 0.108 1.337308 39.96 0.026 0.107892 7.02E-05 0.005216 0.107892 1.931667 39.92 0.031 0.107784 8.37E-05 0.006218 0.107784 2.303141 … … … … … … … 4.051 0.245 0.010938 0.000662 0.049146 0.0109377 18.20225 4.01 0.247 0.010827 0.000667 0.049547 0.010827 18.35084 3.969 0.25 0.010716 0.000675 0.050149 0.010752004 18.57372 3.928 0.252 0.010606 0.00068 0.05055 0.010688525 18.72231 3.887 0.255 0.010495 0.000689 0.051152 0.010625046 18.9452 … … … … … … … 0.273 1.14 0.000737 0.003078 0.22868 0.003040571 84.69617 0.241 1.166 0.000651 0.003148 0.233895 0.002954171 86.62784 0.209 1.192 0.000564 0.003218 0.239111 0.002867771 88.5595 0.178 1.218 0.000481 0.003289 0.244326 0.002784071 90.49117 0.147 1.245 0.000397 0.003362 0.249742 0.002700371 92.49713 0.117 1.273 0.000316 0.003437 0.255359 0.002619371 94.57739 0.087 1.301 0.000235 0.003513 0.260976 0.002538371 96.65765 0.058 1.33 0.000157 0.003591 0.266793 0.002460071 98.8122 0.029 1.359 7.83E-05 0.003669 0.27261 0.002381771 100.9668
124
APPENDIX C – HAUKSBEE EXPERIMENTS
C.1 Numerical results for Case 1
Table C.1 Initial parameters for Case 1
degrees radians alpha 1 0.017 Zo 5 beta 10 0.174 theta 10 0.174
125
Table C.2 Numerical results for Case 1
X λ(X) X λ(X) 0 11.42 2.3 12.43 0.1 11.46 2.4 12.48 0.2 11.50 2.5 12.53 0.3 11.54 2.6 12.58 0.4 11.59 2.7 12.62 0.5 11.63 2.8 12.67 0.6 11.67 2.9 12.72 0.7 11.71 3 12.77 0.8 11.75 3.1 12.82 0.9 11.80 3.2 12.88 1 11.84 3.3 12.93 1.1 11.88 3.4 12.98 1.2 11.93 3.5 13.03 1.3 11.97 3.6 13.08 1.4 12.02 3.7 13.14 1.5 12.06 3.8 13.19 1.6 12.11 3.9 13.24 1.7 12.15 4 13.30 1.8 12.20 4.1 13.35 1.9 12.24 4.2 13.41 2 12.29 4.3 13.46 2.1 12.34 4.4 13.52 2.2 12.38 4.5 13.58
126
C.2 Numerical results for Case 2
Table C.3 Initial parameters for Case 2
Degrees radians alpha 1 0.017 Zo 10 beta 40 0.698 theta 10 0.174
Table C.4 Numerical results for Case 2
X λ(X) X λ(X) 0 7.34 2.3 9.10 0.1 7.40 2.4 9.19 0.2 7.47 2.5 9.29 0.3 7.53 2.6 9.39 0.4 7.60 2.7 9.49 0.5 7.66 2.8 9.60 0.6 7.73 2.9 9.70 0.7 7.80 3 9.81 0.8 7.87 3.1 9.92 0.9 7.94 3.2 10.04 1 8.01 3.3 10.15 1.1 8.09 3.4 10.27 1.2 8.16 3.5 10.40 1.3 8.24 3.6 10.52 1.4 8.32 3.7 10.65 1.5 8.40 3.8 10.78 1.6 8.48 3.9 10.91 1.7 8.56 4 11.05 1.8 8.65 4.1 11.19 1.9 8.73 4.2 11.34 2 8.82 4.3 11.49 2.1 8.91 4.4 11.64 2.2 9.00 4.5 11.80
127
C.3 Numerical results for Case 3
Table C.5 Initial parameters for Case 3
degrees radians Zo alpha 1 0.017 10 beta 40 0.698 theta 10 0.174
Table C.6 Numerical results for Case 3
X λ(X) X λ(X) 0 14.68 2.3 23.92 0.1 14.94 2.4 24.59 0.2 15.19 2.5 25.30 0.3 15.46 2.6 26.05 0.4 15.74 2.7 26.85 0.5 16.03 2.8 27.70 0.6 16.33 2.9 28.61 0.7 16.64 3 29.57 0.8 16.96 3.1 30.61 0.9 17.30 3.2 31.72 1 17.65 3.3 32.91 1.1 18.01 3.4 34.20 1.2 18.39 3.5 35.59 1.3 18.78 3.6 37.10 1.4 19.19 3.7 38.74 1.5 19.62 3.8 40.53 1.6 20.08 3.9 42.50 1.7 20.55 4 44.67 1.8 21.04 4.1 47.08 1.9 21.56 4.2 49.75 2 22.10 4.3 52.75 2.1 22.68 4.4 56.14 2.2 23.28 4.5 59.98
128
C.4 Numerical results for Case 7
Table C.7 Initial parameters for Case 7
degrees radians Lc (m) alpha 0.33 0.006 0.0027 theta 20 0.349
Table C.8 Numerical results for Case 7
L z0 z1 0.07 0.02829 0.035722 0.08 0.019505 0.045722 0.09 0.015931 0.055722 0.1 0.013668 0.065722 0.11 0.012046 0.075722 0.12 0.010806 0.085722 0.13 0.009818 0.095722 0.14 0.009008 0.105722 0.15 0.008329 0.115722 0.16 0.00775 0.125722 0.17 0.00725 0.135722 0.18 0.006813 0.145722 0.19 0.006428 0.155722 0.2 0.006085 0.165722 0.21 0.005778 0.175722 0.22 0.005501 0.185722 0.23 0.00525 0.195722 0.24 0.005022 0.205722 0.25 0.004813 0.215722 0.26 0.00462 0.225722 0.27 0.004443 0.235722
129
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