OF GRAVITY ON mm CONFIGURATION AM) CONTACT ANGLE …€¦ · The phenomenon of contact angle hysteresis is not well understood, and is often explained in terms of curvature or line

THE EFFECT OF GRAVITY ON m m CONFIGURATION AM) CONTACT ANGLE HYSTERESIS

Michael R. Sasges

A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy

Graduate Department of Mechanical and Industriai Engineering University of Toronto

O Copyright Michael Reinard Sasges 1997

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ABSTRACT

THE EFFECT OF GRAVITY ON FLUID CONFIGURATION AND CONTACT ANGLE HYSTERESIS

Michael R. Sasges

A thesis submitted for the degree of Doctor of Philosophy, Graduate Department of Mechanicd and Industriai Engineering, University of Toronto.

1997

The phenomenon of contact angle hysteresis is not well understood, and is often

explained in terms of curvature or line tension effects. We apply thermodynamics in an

analysis of this topic. with surprising results. It is predicted that the contact angle of one

fluid region will be affected by a second fluid region in the sarne systern. In particular. it is

possible to predict the contact angle hysteresis between two fluid regions at different ele-

vations in a cylindricai container. This prediction does not require line tension, curvature

effects or non-ideal surfaces for an explanation of hysteresis. These predictions are tested

in an expenment. and the results support the predictions of hysteresis.

A second area to which thermodynamics is applied is that of fluid configuration in the

absence of gravity. There is no pressure-induced hysteresis under these conditions. and it

is shown that there are numerous configurations that satisfy the necessary conditions for

equilibrium in a cylindrical container. By performing an entropy analysis. it is shown that

the equilibrium configuration may be determined. and is dependent on the contact angle

and the arnount of fluid present. These predictions are tested experimentaily, and it is

found that the experimental evidence supports the predictions. and casts doubt on the ear-

lier Bond number theory.

ACKNOWLEDGMENTS

For my mother, who did not live to see me finish my education.

1 would like to thank Laurie Brennan for the years of support, encouragement. joy and

commiseration she has shared with me. 1 could not have done it without her. 1 thank my

parents for their encouragement of higher learning and for supporting me in ai1 my misad-

ventures-1 now have a degree for each of us. 1 also thank rny family. particularly my

father and my sisters Veronica and Margaret for their support, both financial and emo-

tional.

1 am also grateful to my supervisor. Charles Ward. for his academic rigor and for

teaching me about thermodynamics and life from a new perspective. It was also demon-

strated that the use of the passive voice could be learned. 1 also thank Barbara Ward for

keeping order in a lab that would otherwise evolve to a more chaotic configuration.

1 am grateful to al1 of my friends for their diversions and support. In particular. 1 would

like to thank Janet Elliott for many illurninating discussions about thermodynamics. and

many entertaining conversations about life. the universe. and everything. 1 aiso thank her

for helping me cope with adversity and bears, and for teaching me that ibuprofen can be

considered a food group.

1 would also like to thank the third member of the Little Trophy club. Linda Gowman.

She helped me keep my sanity and encouraged me when 1 felt 1 was resembling the trophy.

Thanks to my lab mates and friends: Mark, Gang. Dainis, David. Alberto, and many

others. 1 am also indebted to the faculty and staff in the Mechanical Engineering depart-

ment at the University of Toronto.

This work was supported by scholarships from the Natural Science and Engineering

Research Council, the Ontario Graduate Scholanhip fund, and the University of Toronto.

and by the Canadian Space Agency.

TABLE OF CONTENTS

ABSTRACT .....................~................................................................................................ ii

... .............................................................................................. ACKNOWLEDGMENTS iir

m..

LIST OF FIGURES ...................................................................................................... viii

LIST OF SYMBOLS ..................................................................................................... xii

CHAPTER 1: INTRODUCTION .................................................................................... 1

1.1 Motivation .............................................................................................................. 1

3 1.2 Scope of the Thesis .................................................................................................

CHAPTER 2: METHODS FOR DETERMINNG THE EQUILIBRIUM CONFIGURATION ............................................................................... ...7

............................................................................ ............................. 2.1 Introduction ... -7

....................................................................... ................... 2.2 System Definition ... 8

2.3 Necessary Conditions for Equilibrium ................................................................. 11

........................................................................................... 2.3.1 Pressure Profile 18

2.4 Helmholtz Formulation ........................................................................................ 2 0

33 2.5 Sununary ............................................................................................................... --

CHAPTER 3: EQUILIBRIUM IN THE ABSENCE OF GRAVITY ........e........w...... 24

3.1 Introduction ............................. ,., ........................................................................... 24

3.2 Zero-g limit: Conditions for Equilibnum ............................................................. 27

3.3 Surfaces with Zero Mean Curvature ..................................................................... 29

............................................................... 3.4 Surfaces with non-zero mean curvature -32

...................................................................................... 3.4.1 Delaunay Surfaces 32

3.4.2 Spherical Interfaces ..................................................................................... 33

3.4.2.1 Possible Configurations ................................................................ 33

............. 3 .4.2.2 Determination of the Stable Equilibrium Configuration -36

................................................. 3.5 Expenmental Investigation on a Low-g Aircraft -42

3 . 5. 1 Experimental Procedure ..................... .... ............................................. -43

3.5.2 Results and Discussion ............................................................................... 4t

3.6 Experimental Investigation in a Drop Shaft ......................................................... -47

3.6.1 Experimental Apparatus and Materials .................................................. - 4 3

......................................................... 3.6.2 Experirnental Procedure and Facility S O

....................................................... 3.6.2.1 Gravitation Levels Achieved 1

3.6.3 Results ......................................................................................................... 53

......................... 3 .6.3.1 Configurations after 2.5 seconds of hypogravity -53

.......................... 3 .6.3.2 Configurations after 10 seconds of hypogravity -54

3.6.3.3 Shape of the Interface for Large Contact Angle ........................... 56

..................................................................... 3.7 Chapter Surnmary and Conclusions S6

................................................................................................................... 3.8 Tables -61

3.9 Figures .................................................................................................................. -66

CHAPTER 4: EQUILIBRIUM IN THE PRESENCE OF GRAVITY ................... -82

4.1 Interface S hape of Single-Cornponent Systems ................................................... -82

4.1.1 Caiculated Interface Shapes ....................................................................... -85

........................................... 4.1.2 Criterion for Neglecting Gravitational effects -86

4.2 Effect of a second component .............................................................................. -89

4.2.1 Equilibrium concentration vs . elevation ..................................................... 89

4.2.2 Axi-S ymmetric Liquid-Vapor interfaces ........................ .. ..................... -94

4.2.3 Caiculated Interface Shapes ........................................................................ 98

4.3 Catenoid Interfaces ............................................................................................... 99

.......................................... 4.3.1 Catenoids aligned with Cylindrical Containers 99

.................................................................................... 4.3.2 Catenoid droplets 1 0 0

4.4 Two Interface Equilibrium and Contact Angle Hysteresis ................................. 101

...................................................... 4.4.1 Pressure Profile in a Capillary Syçtem 102

4.4.2 Relation Between Mean Radii of Curvature and Separation Distance ..... 103

...................... 4.4.3 Necessary Difference in Upper and Lower Contact angles 104

............................................................. 4.4.4 Young Equation Considerations 1 0 8

4.4.5 Discussion ................................................................................................. 110

................................................................................. 4.5 Experimentd Investigation I I I

4.5.1 Preliminary Investigation ........................................................................ I l

............................................................................ 4.5.2 Principal Investigation 1 1 2

....................................................... 4.5.3 Preparation of the Aqueous Solvent 1 1 3

4.5.4 Surface Tension of the Aqueous Solution ........................................... 113

............................................ 4.5.5 Preparation of the cylindncal containers 1 14

4.5.6 Bond Number of the Expenmentai Systems ........................................... 116

4.5.7 Results and Discussion ............................................................................ 117

4.5.7.1 System-l ..................................................................................... 117

4.5.7.2 System-2 ........................ ... ..................................................... 119

4.5.7.3 Adhering Droplets ...................................................................... 121

4.5.8 Conclusions ............................................................................................... 122

.......................................................................................... 4.6 Chapter Conclusions 1 1 3

............................................................................................................... 4.7 Figures 1 2 5

CfLAPTER 5: SUMMARY AND CONCLUSIONS ................... ....................... 144

APPENDIX A: SHUlTLE EXPERIMENT ............................................................... 146

.............................................................................................................. REFERENCES 160

vii

LIST OF FIGURES

CHAP'ïER 1: INTRODUCTION .....~.............................................................................. 1

FIGURE 1.1. A: Example of contact angle hysteresis in a sessile liquid droplet on an inclined plate. B: Representation of the effects of Line Tension. .......................................... 6

CHAPTER 2: METHODS FOR DETERMINING THE EQUILIBRIUM CONFIGURATION ..................................................................... **0**.-***a**7

FIGURE 2.1. Geometry of the system. A: General geometry of the container and liquid- vapor interface. B: Vimial displacement of the three phase line. C: Virtual displacement of the interface. D: Differential geometry of the interface. - 2 3

CHAPTER 3: EQUILIBRIUM IN THE ABSENCE OF GRAVIR ................... *0...24 FIGURE 3.1. Scherk's Surface (A and B) and Helicoid ( C ) ........................................ 66

FIGURE 3.2. Catenoid and cylinder with axes of rotation aligned. ............................. 67

FIGURE 3.3. A: Catenoid intersecting a cylinder at 52' contact angle. B: Fhid configuration observed in a drop shaft experiment for a system with 49" contact angle. ......................................................................................... 68

FIGURE 3.4. Nodary intersecting a cylinder with 90° f 5 O contact angle. .................. 69

............. FIGURE 3.5. Some of the possible configurations with sphencal interfaces. 70

RGURE 3.6. Possible configurations as a function of contact angle and amount of fluid. .................................................................................... ........................ 7 1

FIGURE 3.7. A: F1 minus F2 vs. contact angle B: FI minus Fbridge vs. contact angle. ........................................................ 72

FIGURE 3.8. A: Helmholtz potential of the bubble configuration as a function of contact angle, showing the effect of cylinder height and fil1 ratio B: Potential of the droplet configuration as a function of contact angle, showing the effect of cylinder height and fil1 ratio ...................................... 73

FIGURE 3.9. A: Helmholtz potential of the sessile bubble configuration as a function of contact angle, showing the effect of cylinder height and fil1 ratio B: Potential of the sessile droplet configuration as a hinction of contact angle. showing the effect of cylinder height and fil1 ratio. Only the portion of the curve where there is a change in the predicted equilibnum configuration has been shown. ................................................................................................ ..74

FIGURE 3.10. A: Schematic of the experimental apparatus used on the KC 135 aircraft. B: Photograph of the apparatus. ................... .... .................................... 75

FIGURE 3.1 1. Fluid configuration and gravitation levels for an experiment in a 26.5 mm radius on the KC 135 aircraft. .......................... .., ........................................ ..76

FIGURE 3.12. Experimental apparatus used in the &op shaft. .................................... 77

FIGURE 3.13. A: Schematic of the Drop Shaft capsules and the thmsters used to overcome air drag.B: Separation between the inner and outer Capsules during the drop ....................................................................................................... .7 8

FIGURE 3.14. Accelerometer readings with offset removed. ...................................... 79

FIGURE 3.15. Configurations adopted by propanol in horizontal cylinders after 2.5 seconds of hypogravity. Al1 cylinders have zero contact angle except as noted. ........................................................................................................... -80

FIGURE 3.16. Final fluid configurations for propanol in horizontal cylinders after 10 seconds of hypogravity. Al1 cylinders have zero contact angle except as noted. ......................................................................................................... 8 1

...................... CHAPTER 4: EQUILIBRIUM IN THE PRESENCE OF GRAVITY 82

FIGURE 4.1. Geometry used for calculation of interface shape in the presence of ................... .....................................................................*........ gravity. ... 125

FIGURE 4.2. Sample interfaces caiculated frorn numerical integration of the differential equations. showing the effect of the Bond number on the interface shape of lower interfaces (A) and upper interfaces (B). The gravitational acceleration is directly proportional to the Bond number. These calculations are for systems

.............................................................................. with only one interface. 136

FIGURE 4.3. The ratio of the numericai solution of the exact interface equations to the zero-g solution vs. tuming angle for two values of Bond number. Contact

................................................................................................. angle 8 = 0. 127

....... FIGURE 4.4. System used to demonstrate the definition of a saturated system. 128

FIGURE 4.5. Geometry for development of the differential equations for the modified ...................................................................................................... catenoid. 129

FIGURE 4.6. A: Isometric view of a modified catenoid calculated for zero curvature and 52' contact angle at the lowest point, with mole fraction of water in the gas phase of 0.044. Contact angle varies from 49" to 54O. B: Comparison of the observed droplet shape to a modified catenoid. In the left image, the calculated liquid-vapor interface profile has been overlaid on a photograph of the droplet. The nght image is a cross section of the calculated

................................................................... ............................. interface. ,. 130

FIGURE 4.7. Pressure profile in a capillary system in the two-interface configuration partially filled with water. .......................................................................... 13 1

FIGURE 4.8. Contact angle difference versus interface separation: A. for a 1.2 mm diarneter cylinder and B. for a 2.0 mm diarneter cylinder. each partially filled with water at 25°C. The dashed portions of the curves indicate the conditions under which the pressure in the vapor phase becomes greater than the saturation vapor pressure. Curves have been plotted for upper contact angles

........................... of 0, 10, 20, 30, 40, and 60 degrees. ................... .... 132

FIGURE 4.9. A: Equilibnum pressure in the vapor phase vs. upper contact angle. B: Equilibrium pressure in the liquid phase vs. upper contact angle. These calculations have been canied out for water in a 1 . 2 m diarneter cylinder at 25 OC. For contact angles less than 90°. the pressure is less than the saturation value, while for larger contact angles, the pressure is greater than the saturation value. .................................................................................... 133

FIGURE 4.1 O. Liquid configurations in prelirninary experiments. Eac h left and right image pair comprise perpendicular views of the sarne liquid region. Two separate liquid regions are shown. The cylinder wall is barely visible as a vertical line, while the liquid forms a ring dong the cylinder wall. The three phase lines are roughly horizontal. A calculated catenoid has also been shown.

............................................................................................... for reference. 134

FIGURE 4.1 1. Comparison of the liquid shape with the shape of a modified catenoid. The modified catenoid has a contact angle of 36O and has a ratio of Rx to Rs of

FIGURE 4.12. Measured values of the surface tension of the solution as a function of time. and an extrapolation-The error bars represent the standard deviation of the measurements. Inset: Measured values of the solution surface tension vs. time. ............................................................................................................ 136

FIGURE 4.13. Schematic diagram of the quartz container after having been filled and ...................................................... sealed, and of the experimental layout. 137

FIGURE 4.14. Evolution of System-1 containing water at 35°C in a 1.2 mm diameter .......................................................................................... quartz capillary. 138

FIGURE 4.15. Contact angles of System-1 . Contact angles were determined from measurernents of interface height. q. in a 1.2 mm diameter container at 35°C .

............. B: Contact angles of the sarne system after having been inverted 139

............................ FIGURE 4.16. Evolution of S ystem- 1 after having been inverted 140

FIGURE 4.17. Evolution of System-2 containing water at 35" C in a 1.2 mm diameter quartz capiIlary ........................................................................................... 141

FIGURE 4.18. A: Measured contact angles of System-2 . Contact angles were determined from measurements of interface height, >I . in a 1.2 mm diameter container at 35°C with a mole fraction of water of 0.044 in the gas phase .

............. B: Contact angles of the same system after having been inverted 142

............................ FIGURE 4.19. Evolution of System-2 after having k e n inverted 143

APPENDIX A: SHUTIZE EXPERIMENT ............................................................... 146

FIGIJRE A . 1 . Fluid behaviour of a single cylindrical system. partially filled with ......................... Hexadecane. during the ten second period of low gravity 155

FIGURE A.2. Final liquid configuration for Propanol in a cylinder with 49" contact angle. and in a cylinder with zero contact angle ........................................ 156

FIGURE A.3. Photographs of the apparatus ........................................................... 157

............................................. FIGURE A.4. Electrical schematic of the filter circuits 158

FIGURE A.5 . Measured frequency response of the three filter circuits . after construction ......................................................................................... 1 5 9

LIST OF SYMBOLS

A - - - - - - - - - area.

B .. - - - - - - - -the Bond number.

c - - - - - - - - -Concentration, or an arbitrary constant.

C, Ch. Ci - - -Constants defined in tems of properties at a reference.

E - - - - - - - - -total energy.

E; - - - - - - - - -total Helmholtz potential.

H - - - - - - - - -in geometry, the total height of a cylindrical container.

in liquid solutions, Henry's law constant.

L - - - - - - - - -radius of a cylindricai container. A Iength scale.

N - - - - - - - - -number of moles.

N' - - - - - - - -the maximum number of moles of Iiquid that a container could hold.

ni, Ib, - - - - -unit normal vectors of an interface or cylinder, respectively.

p - - * - - - - - - pressure.

R - - - - - - - - -radius.

R, - - - - - - - -radius of a catenoid that originates from the "outside" of the interface.

R, - - - - - - - -radius of a catenoid that originates from the axis of symmetry.

- R - - - - - - - - -ideal gas constant.

s - - - - - - .. - - entropy.

s* - - - - - - - -a modified entropy function that includes ternis to account for constant

energy and number of moles.

T-------- - temperature.

u-- - - - - - - - intemal energy.

V - - - - - - - - -volume.

w - - - - - - - - molecular weight.

X - - - - - - - - -a radial coordinate.

- - - - - - - - - eievation, or axial coordinate.

a - - - - - - - - - a constant.

b - - - - - - - - -the arnount of cylinder wall exposed to vapor in the one-interface configu-

ration.

c - - - - - - - - -a constant.

f - - - - - - - - -intensive Helmholtz potential (per unit volume or unit area).

u -----,-- a -gravitational acceleration.

go- - - - - - - - -standard earth gravitation. 9.8 1 m/s2.

h - - - - - - - - -height of a cylinder wall exposed to vapor in the bridge or two-interface

configuration.

n - - - - - - - - -number of moles per unit volume or area.

9 - * - - - - - - -a function that accounts for the effects of gravity in the differential equa-

tion for an interface.

r - - - - - - - - -the number of chernical species or, in cylindncal coordinates. a radial coor-

dinate. Also, a radius of a sphericai interface.

s - - - - - . . - - -intensive entropy. Seconds.

u - - - - - - - - - intensive intemal energy.

v - . . - - - - - - -specific volume

x - - - - - - - - -a non-dimensional radial coordinate.

xo(@) - - - - - -the radial coordinate in zero gravity.

. . . X l I l

zo(@) - - - - - -the vertical coordinate in zero gravity.

X C - - - - - - - - -an offset of the axis of an interface with respect to that of a cylinder.

z - - - - - - - - -elevation in a gravitational field or dong the axis of a container.

= O - - - - - - - - - the "zero" elevation point.

a - - - - - - - - -a constant.

E - - - - - - - - -a srnall multiplier.

0 - - - - - - - - -the tuming angle measured from the mis of a syrnmetric interface.

also used as a dummy variable.

QZ - - - - - - - - -energy arising from a field.

Y - - - - - - - - -surface tension.

K - - - - - - - - - isothermal compressibility.

A - - .. - - - - - -a Lagrange multiplier (a constant).

- - - - - - -a constant.

p - - - - - - - - - chemical potentiai.

t l - - - - - - - - -in calculus of variations. an arbitrary comparison function.

in themodynamics. a function of liquid pressure.

v - - - - - - - - - specific volume.

0 - - - - - . . - - -contact angle.

P - - - - - - - - - liquid density.

TJ - - - - - - - - -the intensive modified entropy function that includes terms to account for

constant energy and number of moles-the intensive form of s*.

~ ( 9 ) - - - - - - -an exponential function of elevation that affects gas pressure.

X - - - - - - - - -mole fraction in the vapor phase.

5 - - - - - - - - -a variable by which a surface may be parametrized.

xiv

Y - - - - - - - - - intensive potential energy.

v(T, P) - - - - -an unknown function of temperature and pressure.

& - - - - - - - - -a function of elevation that affects the pressure of a gas.

Subscripts and Superscripts - - - - - - - - - - saturation.

1,2 - - - - - - - -one of a pair. e.g. the two orthogonal radii of curvature; two arbitrary eleva-

tions; the one-interface and two-interface configurations: two chernical

components.

2b - - - - - - - - the bridge configuration.

1 - - - - - - - - - interface.

L - - - - - - - - - liquid.

s - - - - - - - - - solid.

v - - - - - - - - - vapor.

LV - - - - - - - - liquid-vapor interface.

SL - - - - - - - - solid-liquid interface.

SV - - - - - - - - solid-vapor interface.

a - - - - - - - - -the axis of the cylinder, on the Iiquid-vapor interface.

b - - - - - - - - -the three phase boundary, or a bubbie.

C - - - - - - - - -the cup-shaped region in a curved interface.

d - - - - - - - - - a droplet.

k - - - - - - - - - chernical species.

1 - - - - - - - - - -the Iower of two interfaces.

O - - - - - - - .. -pure substance.

r,ref- - - - - - - a reference condition.

S - - - - - - - - -saturation.

~ d - - - - - - - - -a sessile droplet.

- - - - - - - -a sessile bubble.

U - - - - - - - - -the upper of two interfaces.

x, y or t - - - -differentiation with respect to a cartesian coordinate.

xvi

CHAPTER 1: INTRODUCTION

1.1 Motivation The configuration that would be adopted by a fluid in the absence of gravity was stud-

ied in the 1960's~*~**~', rnotivated by humans' fint ventures into space. While there was

some disagreement as to the predicted configuration of unconstrained fluids under these

conditions, the engineering solution was to insiail baffles. rneshes and bladders that would

ensure that some fluid would rernain in a known location in zero g a ~ i t ~ ~ ~ .

With the advent of the space shuttle and orbiting experimental platforms. there has

been a renewed interest in the configuration of Buids in low gravity. for such applications

as biological sciences8, semiconductor crystal growthlO, and rnetai purification and crys-

tallization in spaceg. It is often impractical to constrain the fluids in these applications.

since the desired behavior is dependent on free motion of the fluids in the absence of prav-

ity. In light of this renewed interest. we apply thermodynamics to this problem to predict

the configurations that will be adopted by a fluid in the absence of gravitationid forces. and

we test these predictions with expenments.

We aiso extend this anaiysis. to determine the effects of gravity on contact mgle and

contact angle hysteresis. The phenornena of contact angle and surface wetting have impor-

tant applications ranging from paints and water repellants. to contact lenses. An example

of contact angle hysteresis in a sessile droplet on an inclined plate may be seen in Fig. 1.1 -

A. In this configuration. the contact angle at the upper edge of the droplet. O[,. is typically

less than that at the lower edge, This difference has been the subject of debate. Some

predict that the angle of contact between a solid, a liquid and a gas phase is unaffected by

gravity13J8. If the contact angle is a property dependent only on the solid and fluids

present, then observed hysteresis can only be explained as a non-equilibrium phenome-

non. Other investigators do noi considcr contact angle to be a property, and predict that it

may be affected by interface curvature or pressure'7. From this viewpoint. hysteresis has

been explained by non-classicai effects such as line tension, curvature or surface inhomo-

geneity.

Line tension is the one-dimensional (1-D) analog of pressure (3-D) or surface tension

(2-D). It has been suggested that line tension forces act dong the three phase line. tending

to shorten or lengthen the three phase line. For a sessile droplet. line tension would cause

the three phase line to change in length. thus affecting the contact angle. This effect is

illustrated in Fig. 1.1 -B.

We examine the issue of contact angle hysteresis by detennining the conditions for

equiiibnum in a gravitational field of arbitrary intensity. Our analysis follows classical

thermodynamic methods. assuming that the surface is ideai. and neglecting curvature and

line tension effects. By applying the conditions for equilibrium to systems in which grav-

ity and the resulting pressure gradients play a role, we predict contact angle hysteresis in

simple systems. We also conduct an experimental investigation to test these predictions of

hysteresis.

1.2 Scope of the Thesis In this study, the necessary conditions for equilibrium are denved from first pnnciples.

for a system that is partially filled with liquid, and is in the presence of a gravitational field

of arbitrary intensity. The resulting conditions include a condition that Gibbs derived for

finite

where p is the chemical potential: a superscript S V , SL or LV refers a property to the

solid-vapor, solid-liquid or liquid-vapor interface, a superscript V or L refers a property

to the vapor or liquid phases; W is the molecular weight and a subscript on property refers

it to one of the r chernical species in the system: g is the gravitationai intensity and : is

the vertical position and ktt is a constant. The gradient in chemical potential descnbed by

Eq. ( 1.1) has been neglected by some investigators. and it is shown that the consideration

of this equation is key in predicting the affect of gravity on contact angle.

A second condition for equilibrium in the presence of gravity is shown to be the Young

equation,

where y is surface tension, 6 is the contact angle, and the subscript b refers a propeny to

the line of contact between the solid. liquid and gaseous phases. A third condition for

equilibrium is shown to be the Laplace equation, which relates the difference in the pres-

sure, P, across an interface. I , to the radii of curvature of the interface. R I and R2 and the

liquid-vapor surface tension:

The Laplace and Young equations were derived by ~ i b b s " for systerns in the absence of

gravity, and it is shown that these are also necessary conditions for equilibrium in a gravi-

tational field.

Using equations of state for the bulk phases, we are able to apply these conditions to

make predictions about the fluid configuration. The necessary conditions are applied under

two conditions: 1) negligible gravity and 2) finite gravity.

In the absence of gravity, the Bond number criterion is often used to predict the equi-

librium configuration of fluid in a closed container. The Bond number, B. is a ratio of the

gravitational forces to the surface tension forces:

where p is the fluid density, and L is a length scale. The Bond number criterion predicts

that unless the Bond number exceeds a cntical value, which is dependent on the contact

angle, liquid in a cylindrical container will remain at one end of the container.

The predictions in the current analysis contradict the Bond number theory. When the

necessary conditions for equilibrium are used to predict the configurations that could be

adopted by a fluid, it is found that several fluid configurations c m satisfy the necessary

conditions. Subsequently, an entropy analysis is used to predict which of the possible con-

figurations will be adopted by a fluid system in the absence of gravity, and it is found that

the equilibrium configuration is often different from that predicted by the Bond number

criterion. These predictions of equilibrium in zero-gravity are then tested in two experi-

mental investigations. one using a parabolic fiight aircraft, the other using a ten-second

drop shaft. (An experirnent that was designed and constructed, and will be Bown on the

space shuttle, is also described in an appendix.)

The second environment under which the necessary conditions for equilibrium are

investigated is that of normal earth gravity. The necessary conditions for equilibrium are

used to examine how gravity affects fluid configuration and contact angle hysteresis.

These conditions are applied to a multi-component system. and used to derive the expres-

sions for the concentration gradients and liquid-vapor interface shape in a gravitational

field. These expressions are then used dong with the necessary conditions to predict the

effect of gnvity on contact angle hysteresis in one of the fluid configurations that was

snidied in zero-gravity. It is predicted that, through Eq. (1.1). contact angle must be

affected by gravity in this configuration. and that this effect is large enough to be observed

in experiments. These predictions are then tested in an experimental investigation. in nor-

mal earth gravity, in which line tension forces cannot affect contact angle.

Zero Line Tension Finite Line Tension

ree Phase

/ Thtee Phase \ Line, Showing Effects of Line Tension

FIGURE 1 . 1 . A: Exarnple of contact angle hysteresis in a sessile liquid droplet on an inclined plate. B: Representation of the effects of Line Tension.

CHAPTER 2: METHODS FOR DETERMINING THE EQUILIBRIUM CONFIGURATION^

As a first step in perfoxming a thermodynamic analysis of fluid configurations. the nec-

essary conditions for equilibrium must be determined. Subsequently, these necessary con-

ditions rnay be used to predict possible equilibnum configurations. Finally. an entropy

analysis of these configurations may be conducted to determine which configuration max-

irnizes the entropy of the system and its surroundings.

2.1 Introduction Gibbs determined the necessary conditions for equilibrium for a system that contains a

line of contact between solid. liquid and vapour phases and is subjected to His

"dividing surface approximation" replaced the interphase by a mathematical surface and

neglected any dependence of the interfacial energy on the surface curvature. The position

of the dividing surface was chosen such that one component, in this case the liquid. was

not present in the interface. For a smooth. homogeneous, rigid and non-volatile (Le..

"ideal") solid surface, he obtained three necessary conditions for equilibnurn: 1) the

Young equation; 2) the Laplace equation; 3) and a relation between the chemical poten-

tials of the fluid components and the gravitational intensity:

where p is the chemical potential; a superscript S V , SL or LV refers a property to the

solid-vapor, solid-liquid or liquid-vapor interface, a superscript V or L refers a property

to the vapor or liquid phases; W is the molecular weight and a subscript on property refers

-

t Much of the material from this chapter has been submitted for publication in Ref. 36.

7

it to one of the r chernical species in the system; g is the gravitational intensity and z is

the vertical position and hkt is a constant.

It should be emphasized that (as Gibbs pointed out16) these conditions for equilibrium

form a coupled system of equations. Unless al1 three conditions are satisfied, there is no

thermodynamic reason to think any of them would be valid. Since Gibbs' original deriva-

tion, there have been a number of challenges to the validity of the Young

equation. ** 17* l 3 These have usually been raised in reference to the effect of gravity on

the contact angle, to contact angle hysteresis or to the effect of drop size on contact angle.

However, none of the previous investigations have taken into account the full coupling of

the Young equation to the other conditions for equilibrium.

We use the original description of a surface phase that was given by Gibbs. and con-

sider a simple capillary systern in which the solid surface is "ideal"32 and is exposed to

gravity. When al1 three conditions for equilibrium are taken into account. contact angle

hysteresis is predicted to exist in the system: however, if the gravitational intensity is

reduced to zero, contact angle hysteresis is predicted to vanish.

2.2 System Definition Suppose a liquid and a vapor phase are held in a cylindrical container that is closed to

mass and energy transport, and that the container is present in a gravitational field aligned

with the cylinder axis (see Fig. 2.1). The interface between the liquid and vapor phases

will be assumed to be axi-symmetric, and the intemal energy of the liquid-vapor interface

not to depend explicitly on the interface curvature. The dividing surface approximation is

also adopted to describe the solid-liquid and solid-vapor interfaces. For these interfaces.

the position of the dividing surface is taken to be such that there is no adsorption of the

solid component. Hence, intensive (per unit area) intemal energy of any of the three sur-

faces phases is assumed to depend upon the intensive entropy, s~ ,of the phase and on the

number (per unit area) of moles of each component adsorbed. n/ . Thus, for either a sur-

face or a bulk phase, we may write

For the surface phases, the properties appearing in Eq. (2.2) are understood to be

expressed per unit interfacial area, and for the bulk phases are understood to be expressed

per unit volume. From the definition of the intensive properties

From the Euler relation for a surface phase. one finds that the surface tension. y, rnay be

expressed:

j = LV, S V , or SL

and from the Euler relation for a bulk phase. one finds that the pressure may be written:

Since the system is in a gravitational field, the system has an energy that arises from

the field. The potential energy per unit mass would be gz, where g is the magnitude of the

gravitational acceleration and z is the elevation of the mass in the field. The potential

energy per unit volume for the bulk phase and per unit area for the surface phase is given

j = L, V, L V, SV, orSL

where W is the molecuiar weight of the substance. The total energy for the system may be

written

and the total number of moles of one component

The total entropy may be expressed sirnilarly. This integral will be wntten in more detail

for the system corresponding to Fig. 2.1 :

where the subscript I or b on a quantity indicates that it is to be evaluated at the liquid-

vapor interface or the three phase boundary.

In view of Eqs. (2.2) and (2.3), the total entropy, S. may be seen to depend on

u< ni, ni, .. .n! , but in view of Eqs (2.7) and (2.8), not al1 of these variables are indepen-

dent.

2.3 Necessary Conditions for Equilibrium The thermodynarnic description of the surface phase given in Eq. (2.2) is the same as

that used by ~ ibbs . " Using this description. ~ohnson~' presented a derivation of the

Young equation. but did not obtain explicitly either the Laplace eqwtion or Eq. (2.1): thus.

Johnson's derivation could give the impression that the Young equation could be valid. but

that the Laplace equation and the Eq. (2.1) were not necessarily simultaneously valid. As

will be seen al1 three conditions must necessarily be satisfied before one can Say that the

system is in equilibriurn.

Boruvka and ~eurnann~ presented a denvation of the Young and Laplace equations:

however, they did not obtain Eq. (2.1) because they implicitly assurned the chernical

potential hinction to be of the samefimtional f o m for al1 phases. This assumption would

prevent them from obtaining a closed system of equations to define the equilibriurn state

for a three phase system. We discuss this point further below.

One can only speculate as to why confusion has aisen regarding the conditions for

equilibrium in a three phase system. In his classic work15. Gibbs began his determination

of the condition for equilibrium on page 3 14. He then obtained in that section the Young

and Laplace equations, but he did not obtain Eq. (2.1) in that section. For its derivation. he

referred to an earlier section (page 276) in which only fluids were present, but he dis-

cussed the importance of Eq. (2.1) in the system of equations for a three phase system (in

that section on page 3 19); thus, he was clearly aware of the necessity of using Eq. (2.1 ) to

define the equilibrium configuration of a three phase system. However, he did not present

an explicit expression for the chernical potentials. and the expressions for these functions

will be seen to play a critical role in our considerations.

To show that the Young, Laplace, and Eq. (2.1) define the necessary conditions for

equilibrium, we apply the entropy postulate. Thus, we require that the total entropy to be

an extremum subject to constraints of constant total energy. E, total volume and total num-

ber of moles of each component. To satisfy the constraints, we introduce the Lagrange

multipliers, io, hi, i = 1, 2, ... r . and define the function

To determine under what conditions

(2.10)

S* has an extremum, we follow the standard calculus

of variations procedure37 and introduce the set of cornparison functions:

where the subscript e on a quantity indicates it is the equilibrium value of the variable and

qo, q i are arbitrary functions.

If the function d is defined by

then S* may be written (see Fig 2.1)

We note that zo, Q,, and Qb each depend on the M J , nj. Since the number of moles in the

respective phases are allowed to vary, the limiü of the integrals listed in Eq. (2.14) depend

@n and E i q i . After insening Eq. (2.1 1) and (2.12) into Eq. (2.14). taking the partial

derivative of the result with respect to E~ , one finds

"O as* (aF;) E; = O

O

Since S* is an extremum when ei vanishes. we require

and since q i is an arbitras, function, the only way that Eq. (2.16) c m be satisfied for ail

values of this function is if

and since the limits of the integrals also depend on these arbitrary functions. we must have

In addition. since

~ ~ ( 0 ~ ) = L

Eq. (2.15) reduces to

By taking the partial derivative of S* with respect to and making sirnilar are uuments

to those applied to Eq. (2.15), one arrives at the following requirements:

1 ho = lj j = L, V, LV, SV, SL

When Eq. (2.21) is used in Eq. (2.17) and the result is used to simplify the expression for

TJ , one finds from Eq. (2.13) that

( "iN ZJ = S I - - il; + - i = i TJ j = L, V , LV, SL, SV

and when this result is compared with Eq. (2.4) and (2.5), one finds

If Eq. (2.22) is multiplied by deo and Eq. (2.18) by de , , the latter summed over i and

added to the former, one finds

Since we have considered variations about the equilibrium configuration. there is a result-

ing displacement of the interface. From Fig 2.1 C, the following relations are found for this

displacement

and

Aiso, from Fig 2.1A it may be seen that

When Eqs. (2.27), (2.28) 2nd (2.29) are used in Eq. (2.26), one finds

The fact that this integral must vanish for al1 value of the litnits indicates that this condi-

tion cm only be satisfied if the argument of the integral vanishes. In view of the relations

given in Eq. (2.25), one finds from Eq. (2.30) that

which is the Laplace equation.

If Eq. (2.23) is multiplied by deo and Eq. (2.20) by d e i , the latter summed over i and

added to the former, one finds

T ~ ~ R , ( ~ ~ ) d @ ~ - rfvdéb + rbLdz, = O (2.32

It should be noted that the quantities appearing in Eq. (2.32) are al1 evaluated at the three

phase line. Since Eq. (2.27) also applies at the three phase line

and

After using Eq. (2.33) and (2.34) in Eq. (2.32) and the relations given in Eq. (2.25). one

finds

yfV - yf = yk V ~ o ~ 0

which is the Young equation.

From Eq. (2.2 1) and Eq. (2.17) one finds a result equivalent to Eq. (2.1):

For the isolated system. Eqs. (2.2 l), (2.3 1). (2.35) and (2.36) constitute the necessary con-

ditions for equilibrium. It should be noted that unless d l these conditions are satisfied. the

entropy would not necessarily be an extremum.

We now relax one of constraints and suppose the system is exchanging energy with a

surrounding reservoir of known temperature. Then the system and reservoir constitute an

isolated system, and when equilibrium is reached they would have the same temperature.

The necessary conditions for equilibrium for the isothermal system then would be Eqs.

(2.3 l), (2.35) and (2.36).

2.3.1. Pressure Profile

If the differential of Eq. (2.5) is taken and the result combined with Eq. (2.3). then one

obtains the Gibbs-Duhem relation for either bulk phase

12~dp; = - s J d ~ + dP j = L o r V (2.37)

If the liquid is assurned to have a constant isothermal compressibility, K . and a pressure

range is considered for which

where P, is the saturation pressure corresponding to the temperature, then the Gibbs-

Duhem relation may be integrated to obtain

where vL is the specific volume of the Iiquid phase at saturation conditions.

If the vapor phase is approxirnated as an ideal gas. then integrating the Gibbs-Duhem

relation gives

where R is the ideal gas constant. It should be noted that the functional forms of the

chernical potentials of the liquid and vapor phases are different. This difference plays a

critical role in what follows.

For the liquid phases, Eq. (2.1 ) gives

Since ht is a constant, this equation may be evaluated a! two different vertical positions.

z , and Z? , and then subtracted to give

Following the same procedure in the vapor phase yields

If Eq. (2.1) is applied at any position on either side of the liquid-vapor interface. then

one finds

where the subscript I on a property indicates that the property is to be evaluated at the

interface but in the phase indicated by the superscript. Since the chernical potential func-

tions have different dependences on the pressure (Le. functional forms) one finds from

Eqs. (2.39) and (2.40) that a certain relation must exist between the pressures in the

respective phases at the interface:

From the Laplace Eq. the pressure difference across the interface may be expressed in

terms of the mean radius of curvature. R LV. and if Eq. (2.45) is then used to elirninate P Y . one finds

Note that if the value of the mean radius of curvature is known. Eq. (2.46) may be

solved (iteratively) for P : . With this value of pf , one may use Eq. (2.45) to obtain the

corresponding value of P Y . Thus. for a given temperature P: and P: may be viewed as

fùnctions of R ~ ~ .

2.4 Helmholtz Formulation The formalism of thermodynamic potentials affords great simplification in determin-

ing equilibrium configurations. When the potential is minirnized. the entropy of the system

and its surroundings is maximized. The potential is determined by the constraints on the

system, and the Helmholtz potential, F, is the thermodynarnic potential for a system of

constant volume that is closed to m a s transport and is rnaintained isothemal by a reser-

voir.

In classical thermodynamics in the absence of gravity. the intensive Helmholtz poten-

tial (per unit volume or per unit area) is expressed as

In the presence of a field. the potentiai associated with the field must be included in an

entropy/energy analysis. A method of accounting for this effect. which h a . had sorne

success", is to include a potential energy term in addition to the interna1 energy. thus

defining the Helmholtz potential in a field:

where @ is the energy arising from the field. For a gravitational field. this energy is given

Since the value of the intensive Helmholtz potential will Vary with position. the total

potential may be wntten as an integral

The total Helmholtz potential for a configuration that meets the necessary conditions

for equilibrium rnay be found by integrating Eq. (2.50). This value may then be compared

to that of a different configuration, and that with the lowest potentiai rnay be said to be

more stable. Any configuration that satisfies the necessary conditions for equilibrium is at

an extremum in the potential and total entropy, but this rnay be only a local extremum. By

comparing the Helmholtz potential of the possible configurations, the stable configuration

may be deterrnined.

2.5 Summary The conditions necessary for equilibrium in a closed container filled with a liquid and

its vapor in a gravitationai field at constant temperature have been derived from first prin-

ciples. These necessary conditions have k e n shown to be Eq. (2.1), the Young Equation

(Eq. (2.35)) and the Laplace equation (Eq. (2.3 1 )).

Explicit foms for the chernical potential of the buik phases have been introduced. and

expressions for the pressure gradients in the vapor and liquid phases have been derived

(Eq. (2.42) and Eq. (2.43)). In addition. it has been shown that the equilibrium bulk-phase

pressures at an interface are detemiined by the curvature of the interface through

Eq. (2.46).

Finally, the thermodynamic potential fonnaiism was introduced. To determine the

equilibnum configuration of a closed isothermal system. the Helmholtz potential of con-

figurations that satisfy the necessary conditions for equilibnum may be compared.

In the subsequent chapters, these equations will be applied under conditions of zero

and finite gravity to predict the equilibnum configuration. and to predict the effect of grav-

ity on contact angle hysteresis.

Liquid

FIGURE 2.1. Geometry of the system. A: General geometry of the container and liquid-vapor interface. B: Virtual displacement of the three phase line. C: Vimial displacement of the interface. D: Differential geornetry of the interface.

CHAPTER 3: EQUILIBRIUM IN THE ABSENCE OF GRAVITY~

In the previous chapter, we derived the necessary conditions for equilibnum in a

sealed. constant volume, isothemai container partially filled with liquid. The current

chapter will present a derivation of the consequences of these conditions. and describe

some system configurations that satisfy the conditions for equilibrium. In addition. predic-

tions will be made about the stability of these configurations in the absence of gravity. The

predictions arising from this analysis will be found to contradict those of an earlier and

established theory. These predictions will be tested in two experimental investigations. one

on a parabolic Right aircraft. the other in a drop shaft. It will be seen that the experimental

results support the present theoretical predictions, while casting doubt on the previous

anaiysis.

3.1 Introduction The most cornmon approach for predicting the equilibrium fluid configuration in the

absence of gravity is based on continuum r n e c h a n i c ~ ~ ~ ~ * ~ . This method assumes the equi-

librium configuration of the fluid is the configuration corresponding to a minimum in the

sum of the potential and surface energies and does not consider evaporation-condensation

phenornena. Although ~i~~ adopted this approach, he limited his considerations to only the

"one-interface" or "bubble" configurations that are indicated in Fig. 3.5. The experirnental

investigation that we present in Section 3.6 on page 47 strongly indicates that other config-

urations must be considered.

t Some material in this chapter reprinted with permission from Ref. 29. Copyright 1996 Arnencan Institute of Physics.

The continuum mechanics approach was also adopted by concus4 and by Concus and

~ i n n ~ ; however in the approximation that they adopted, only the liquid-vapor interface was

included in the surface energy tenn (Le., the solid-liquid and solid-vapor energies were

neglected). This approximation leads to the prediction that unless the gravity vector is

directed from the liquid to the vapor (i.e. an "inverted system) and is of sufficient rnagni-

tude. the equilibrium configuration for the system is the "one-interface" of Fig. 3.5.

This prediction has been previously investigated using a &op tower that provided

approxirnately 2.5 seconds of reduced, non-negative gravitational intensity2? In these

experiments, the system was never seen to make a transition from the one-interface config-

uration. Thus the observations appeared to support the predictions of ~oncus" and of Con-

cus and ~inn'. However, as will be seen in a later section. this lirnited period of reduced

gravitational intensity was not sufficient to allow the system to reach the equilibrium con-

figuration.

A different approach was adopted by Neu and ~ o o d ~ ' , one they called the "curvature

approach". They lirnit themselves to conditions of zero-g and to liquid-solid combinations

having a zero contact angle. They base their analysis on considerations of surface enegies

and require that the radius of curvature of the liquid-vapor interface be constant. Their

approach leads to the prediction that such a system would. depending on the radius of cur-

vature of the liquid-vapor interface. adopt the "two-interface" configuration or the "bub-

ble" configuration indicated in Fig. 3.5. The curvature approach has lirnited applicability

because no means of predicting the curvature of the liquid-vapor interface was proposed.

Sen and wilcox3' adopted an approach similar to that of Neu and Good. The former

limited their considerations to non-wetting liquids (i.e., liquids with contact angles greater

than ninety degrees). However, they did not require that under conditions of zero-g, the

radius of curvature of the liquid-vapor interface be constant. The latter assumption

amounts to not requiring equiiibrium between evaporation and condensation.

Herein we present the results of a thenodynamic analysis of the problem of Ruid con-

figuration in the absence of gravity. The necessary conditions for equilibrium are used to

find possible fluid configurations. Further analysis is then used to identify the equilibrium

configuration comsponding to a maximum in the total entropy of the system and its sur-

rounding reservoir. This approach allows the radius of the liquid-vapor interface to be pre-

dicted in ternis of the experimentally controllable variables.

The conclusions reached from this approach are at variance with those of previous

investigaton. For exarnple, if gravitational forces are negligible. thcn the arnount of Ruid

present and the contact angle are the factors that determine the equilibrium configuration.

For a lirnited arnount of fluid, three possibilities arise: 1) if the contact angle is less than

approximately 36", then the equilibrium configuration is the "two-interface" configuration

indicated in Fig. 3.5; 2) if the contact angle is between approximately 36" and 144". then

the equilibrium configuration is the "one-interface" configuration; and 3) if the contact

angle is greater than approxirnately 1440T the equilibrium configuration is the "bridge"

configuration (see Fig. 3.5). When the mass of fluid present exceeds a certain value, then

for al1 contact angles, the system is predicted to adopt the "bubble configuration".

2.2 Zero-g limit: Conditions for Equilibrium The necessary conditions for equilibrium in a gravitational field of arbitrary intensity

have been developed in Chapter 2. Consider now the limiting case of negligible or zero

gavitationf. In the absence of gravity, the condition given in Eq. (2.1) becomes

The constant chemical potential results in some significant simplifications. The chemi-

c d potentials of the bulk phases are functions of pressure and temperature:

Since the temperature is rnaintained constant by a reservoir and since the chemical poten-

tials are constant. there m u t be no pressure gradients within the bulk phases.

pL = constant (3.41

pV = constant (3.5)

Surface tension is a function of pressure. Since the pressures in the bulk phases are con-

stant. the surface tensions within each phase must also be constant. Thus the Young equa-

tion requires that the contact angle be constant:

ysv - ySL cos0 = = constant

yLV

f- A criterion for neglecting gravity will be established in Section 4.1.2 on page 86

Finally, in the absence of pressure gradients, the Laplace equation requires that the mean

interface curvature be constant. Thus Eq. (2.3 1) c m be written

I l - + - = constant 4 R2

Any configuration in which the interface meets these necessary conditions of constant

mean curvature and constant contact angle is a possible equilibrium configuration. We

now seek general surfaces that satisQ these conditions for equilibrium. In order to sim-

plify the task somewhat, we will limit Our consideration to cylindrical containers.

The condition of constant mean curvature (Eq. (3.7)) cm be written as a partial differ-

ential equation in Cartesian coordinates where z = z(x.y) and a subscript denotes differen-

tiation with respect to that variable:

The condition of constant contact angle may be satisfied by requiring that the dot prod-

uct of the unit normal vector to the interface Ri and the cylinder P,be constant dong

their intersection:

where, denoting the equation for the cylindrical container as t = c(.r.y). the nomal vectors

may be written:

In order to satisQ the necessary conditions for equilibrium Eq. (3.8) and Eq. (3.9)

must be solved simultaneously. Once this has been accomplished, the Helmholtz potential

of the configuration corresponding to this solution must be compared with that of other

possible configurations in order to determine which is the stable equilibrium configura-

tion.

3.3 Surfaces with Zero Mean Cumature The simplest and most complete solutions to Eq. (3.8) have been found for the case

where the constant C4 is equal to zero. These are known mathematically as minimal sur-

faces. This nomenclature derives from the property that, of al1 surfaces that may span a

closed perimeter. these surfaces have the minimum surface area. Soap films are minimal

surfaces. These surfaces may not necessarily be those that rninirnize the total eneqy for

the entire system, and so rnay not be the equilibrium configuration.

Snidied fint by Lagrange in 1762. the minimal surface problem was not cornpletely

solved until 1939 by J. ~ o u ~ l a s ~ ! There are only a srnall number of known solutions for

this equation in which 2lx.y) c m be written explicitly.

We will consider ail the known classical minimal surfaces. The two surfaces known as

Scherk's surfaces satisfy Eq. (3.8):

However, they do not satisfy Eq. (3.9), and so are not possible equilibrium surfaces.

The Helicoid satisfies both the conditions for equilibnum, but does not resemble any

fluid configurations observed in known micro-gravity experiments. For this reason it was

not considered further, but its equation is shown here for completeness.

Examples of a Helicoid and a Scherk's surface are shown in Fig. 3.1.

The last classical minimal surface investigated is the catenoid. This single-parameter

family of surfaces can be made to satisS, both Eq. (3.8) and Eq. (3.9). and is given by

There are two ways in which the catenoid can intenect the cylinder with constant con-

tact angle. The most obvious way is by placing the cylinder and catenoid with a comrnon

axis of rotation. This results in a Ruid region that is "toroidal" in the sense that it foms a

complete ring of fluid, though the surface is not that of a circle rotated about an offset mis.

as it would be for a true toms. An example of the symmetric catenoid and cylinder may be

seen in Fig. 3.2.

The second way in which the catenoid c m intersect a cylinder with constant contact

angle requires that the cylinder mis be parallel to that of the carenoid but offset by an

amount x, Using the cylinder radius as the length scale for the problem. Eq. (3.9)

becomes:

Requiring that this equation be independent of the x-coordinate results in the follow-

hg:

The equation for the offset catenoid intesecting a cylinder at a constant contact angle

of 52" has been plotted and the result shown in Fig. 3.3A. The cylinder's axis is horizontal

and the image is a cross section of the cylinder and interface. The interface surface has

been "extended" through the cylinder. A photograph of a Buid configuration observed

under conditions of srnall gravity (IO" g) is shown in Fig. 3.3B. This fluid configuration

resembles the calculated interface shown in Fig. 3.3A. This result will be discussed further

in Section 3.6, but this photograph is presented here to give motivation for this analysis.

Note that through appropriate selection of the parameter c in the catenoid equation.

any contact angle may be achieved. This solution resembles the interface shape observed

in experiments, with appropriate curvatures. It also satisfies al1 conditions for equiiibrium.

This interface shape was not investigated further because the catenoid has one critical fail-

ing as a general surface that satisfies the conditions for equilibrium. Since there is only one

parameter in the equation. it is not possible to vary the contact angle and the volume con-

tained between the interface and the cylinder wall independently. That is, once the number

of moles of fluid in a given container is chosen, the contact angle is uniquely detennined.

Since the contact angle is a function of surface tensions (Eq. (3.6)). which are not in gen-

eral an extensive property, this result is not physically reasonable. However, it will be

s h o w that fluid configurations closely resembling this shape have been seen in experi-

ments.

3.4 Surfaces with non-zero mean curvature Although Eq. (3.8) has never been solved in general. there are a few families of sur-

faces which have been found to satisfy this equation when the curvature is not equal to

zero. Three families of surfaces that have constant curvature are cylinders. the Delaunay

surfaces7, and spheres. The cylinder has constant non-zero mean curvature. but it can eas-

ily be shown that a cylindrical interface cannot intersect a cylindrical container at a con-

stant contact angle unless their axes are parallel. In this case, the contact angle at the end

of the cylinder must be 90". and since the contact angle must be constant. the cylindrical

interface would not allow solutions having other contact angles.

3.4.1. Delaunay Surfaces

Delaunay surfaces are represented in terms of Elliptic functions. Two categories of

these surfaces are the nodary and the undulary. The equations for the nodary rnay be writ-

ten in cylindncal coordinates as1'

W W

z ( y ) = bsiny + bseca do - 1 J1 - cos2asin2@d@ Ji-cos'ixsin'~

where b and a are arbitrary constants.

The undulary may be represented in a sirnilar form as

W

z ( ~ ) = i bsiny r bcosa do J I - sec2a sin2$

but here a must be chosen such that the resulting expression in the square root is positive.

It was found that the nodary very nearly meets the conditions of constant contact

angle. For contact angles near 90°, the nodary can be made to intersect the cylinder with a

contact angle that varies by less than fi0. An example of this approximate solution may be

seen in Fig. 3.4. While this solution is valid for contact angles near 90". it cannot satisQ

the conditions for equilibrium for other contact angles.

3.4.2. Spherical Interfaces

The surface that has yielded the most varied solution to the problern of achieving both

constant mean curvature and arbitrary constant contact angle for an arbitrary amount of

Buid is the sphere. A portion of a sphere aligned with any axi-symmetric container c m be

made to enclose an arbitrary volume and have an arbitrary contact angle. Additional con-

figurations are possible by considenng multiple interfaces in the same container. The

Helmholtz potential rnay be used to determine which of these configurations is stable.

3.4.2.1 Possible Configurations

A number of the possible configurations with spherical interfaces may be seen in

Fig. 3.5. Only certain of these configurations will actually be possible. depending on the

container geometry and the number of moles of fluid in the container. We now develop a

method that may be used to determine which of the configurations is possible for given

values of experirnentally controllable variables.

For the bubble configuration, the radius of curvature of the liquid-vapor interface can

be determined as a function of the number of moles of fluid present in the cylinder through

an iterative procedure. An initial value of the bubble radius Rbo is selected and the corre-

sponding equilibrium pressure in the liquid phase is found through Eq. (2.46). The equi-

librium volume of vapor c m be calculated by using the ided gas equation, conservation of

volume and mass, Eq. (2.45) and the nearly incompressible nature of the liquid to yield

where the function q has been defined as:

The volume in Eq. (3.23) must be equal to the volume of the spherical vapor region. i.e.

Equating these we find that the equilibrium radius must be

If this radius is different from the iùitially assumed value. then the procedure is repeated

using this new value until the two radii are equal to within desired accuracy.

In the case of a droplet. m expression for Rd, can be derived using a simiiar procedure.

After choosing an initial value for the droplet radius and calculating the corresponding liq-

uid pressure. the equilibrium radius must be

The process is repeated until the radius has converged to a final value.

For the sessile droplet and sessile bubble configurations, the radius is calculated using

the sarne iterative technique. For these configurations. the initial value of the radius and

the corresponding pressures must agree with the value of radius calculated from geometry:

for the sessile bubble and sessile droplet. respectively.

For the two-interface and the bridge configurations. the total height of the cylinder

wal1 that is exposed to vapor, h, (see Fig. 3.5) may be expressed as

where V, is the volume of vapor contained within the curved interface region. which may

be written

In the one-interface configuration. the height b of the wall exposed to vapor, (see

Fig. 3.5) can be expressed

For a given set of constraints.

identified using Eq. (3.26) to

the configurations that the system can occupy may now

Eq. (3.29). For example, the bubble radius or droplet

radius, Rb or Rd, must be less than the cylinder radius, L.

For the cases that we consider, v i is very small cornpared to v: . Using this approxi-

mation, the configurations that are possible as a function of the number of moles present

and the contact angle have been calculated for a cylinder where HL is 2.09. The results

are presented in Fig. 3.6-A and Fig. 3.6-B. In Fig. 3.6-A, the one-interface, two-interface

and bridge configurations are considered. and in Fig. 3.6-B, the other four possible config-

uration are considered. For a given set of experimental conditions. both of these figures

must be used to determine the possible configurations.

It rnay be seen from Fig. 3.6 that under certain experimental conditions several config-

urations may satisfy the necessary conditions for equilibrium. In that case only one will be

stable, the others metastable. In order to predict which configuration will be adopted. the

thermodynarnic potentials of the possible configurations must be considered.

3.4.2.2 Determination of the Stable Equilibrium Configuration

In the absence of gravity, the expression for the Helmholtz potenrial (Eq. (2.50)) may

be simplified to yield

Once the geometry of the interfaces has been calculated using Eq. (3.26) - (3.32). the areas

in Eq. (3.33) cm be calculated from the geometry of spherical and cylindrical surfaces.

Pressures in the liquid and vapor c m be found using Eq. (2.46) and Eq. (2.45).

As indicated in Fig. 3.6, under some conditions, both the one- and the two-interface

configurations are possible. Under these conditions, explicit relations may be derived for

the Helmholtz potential difference between these two configurations. The interface radius

of curvature may be related to the contact angle by geometry:

Therefore. at equilibnum Eq. (2.46) can be rewntten as

From this relation it rnay be seen that the pressure in the liquid is a function of 7; L and 0 .

We shall assume that the difference in adsorption between the two configurations is

negligible, with the consequence that the surface tensions must be the same in the two

configurations. From the Young equation. the contact angles must therefore be the same in

the two configurations. Since L and 8 have the same values in both the one- and the

two-interface configuration, the pressures in the liquid phase of each configuration must

be equal. Then since the chemical potential must be unifonn throuehout each of these two

equilibnum configurations. and since the chernicd potential in each phase of a single corn-

ponent system is only a function of the temperature and pressure in that phase. one rnay

write that the chemical potential in the one- and two-interface configurations are equal:

From the Laplace equation and Eq. (3.36). it follows that the pressure in the vapor phases

of the two configurations must be equal. Finally. as may be seen from Eq. (3.23). i t then

also follows that the volume of the vapor phase in the two configurations must have the

sarne value. Applying Eq. (3.36) and taking advantage of the constant volume of the sys-

tem results in the following expression for the difference in the potential between the two

configurations:

F I - F , = ( P $ - P ~ ) v + ( P ~ - P ~ ) v Y + ( P ~ - P ~ ) v ~ +

y L V ( ~ f V - ~f V , + ( A ; ~ - A ~ ~ ) ~ ~ ~ c o s € )

According to Eq. (3.34). in both the configurations under consideration each liquid-

vapor interface has the same radius of curvature. Thus, from the Laplace equation. the

sarne difference in pressure between the vapor and liquid phases must exist in each of the

two configurations. Using this fact, and since the volume of the vapor phase in each of the

two configurations has been found to have the same value, one maj wnte

( p i - P[)v+ ( P ~ - P ~ ) v ~ + ( P $ ' - P $ ) ~ ~ - - = O (3.38

After making use of the expression for Vc given in Eq. (3.3 1). one finds from geometry

that the difference in the solid-vapor interfacial areas is given by

Since there is twice as much liquid-vapor interface in the

the one-interface configuration, one finds

(3.39)

two-interface configuration as in

After taking advantage of the fact that the pressures in the Liquid phases are equal in

the two configurations and of Eqs. (3.3 1). (3.32) and (3.33, one finds that Eq. (3.37) c m

be written

2(1 + s ine+ sin") . ~'p' 1 3(1+sin0) J

This difference in potential is illustrated in Fig. 3.7-A. As may be seen there. for contact

angles less than 35.83". the two-interface configuration is more stable. For contact angles

greater than this value. the one-interface configuration is more stable.

An expression that describes the difference in Helmholtz potential between the one-

interface and the bridge configurations can be derived in a similar manner. After some

manipulation, the resulting expression is

where the subscript 26 indicates the bridge configuration, which has two liquid-vapor

interfaces (see Fig. 3.5). This difference in Helmholtz potential is shown in Fig. 3.7-B.

and as may be seen there, for contact angles greater than 144.17". the bridge configuration

is more stable than the one-interface configuration. Since it has already been shown that in

this range of contact angle the rwo-interface configuration is less stable than the one-inter-

=e con- face configuration, this indicates that for contact angles greater than 144' the brid,

figuration is more stable than either of these configurations.

If the number of moles and volume of the container are in a certain range, then as indi-

cated in Fig. 3.6. the bubble and droplet configurations are possible. In these cases, the

radius of the liquid-vapor interface is not the same as in the one-interface configuration:

thus the b u k phase pressures are not the same. A general expression for the difference in

potential between these configurations and the one-interface may also be found by the pro-

cedure oudined above and is presented here for completeness:

where the subscript j indicates the configuration whose potential is to be compared with

that of the one-interface confi,mation.

Equation (3.43) is exact, but presents difficulties in calculation. In order to plot this

function, approximate relations were derived. The approximations necessary are that 1)

the number of moles in the system is equal to the number of moles in the liquid phase.

which requires that the liquid volumes of the two configurations under consideration be

the same; and 2) the pressure in the vapor phase is the saturation vapor pressure. With

these two approximations Eq. (3.43) can be written as

for the bubble configuration, where H is the total height of the cylinder. NN is the frac-

tion of the cylinder volume occupied by the liquid, and the subscript b indicates the bubble

configuration. A plot of this relation as a function of contact angle is shown in Fig. 3.8-A,

showing the effect of fil1 ratio and cylinder height. For positive values of this potential. the

one-interface configuration is more stable than the bubble-configuration. It may be seen

that the contact angle at which one configuration becomes less stable than the other is

affected by the cyiinder height and the amount of fIuid present. This is unlike the one- and

two-interface configurations, where the potential was not a hnction of these quantities.

The corresponding expression for the droplet configuration, is

where the subscnpt d indicates the droplet configuration. A plot of Eq. (3.45) as a function

of contact angle may be seen in Fig. 3.8-B, showing the dependence on fil1 ratio and cylin-

der height. For positive values of this function, the one-interface is more stable than the

droplet configuration. For the experimental conditions considered, the approximate

expressions of Eq. (3.44) and Eq. (3.45) were found to differ from the numencal solution

of Eq. (3.43) by only 0.1%. Therefore, these expressions were used to determine the sta-

bility of bubble and droplet configurations for al1 experimental predictions in Section 3.6.

Making the same approximations as for the bubble and droplet configurations. the cor-

responding potential equations for the sessile droplet and sessile bubble configurations are

Plots of Eq. (3.46) and Eq. (3.47) as a function of contact angle are shown in Fig. 3.9. The

figure also indicates the dependence on fil1 ratio and cylinder height. For negative values

of these functions, the one-interface configuration is more stable.

Equations (3.41)-(3.47) enable us to determine the equilibnurn configuration from

among the bubble, the droplet, the bridge, the one- and the two-interface, and the sessile

droplet and sessile bubble configurations in negligible gravitation in a cylindrical con-

tainer.

3.5 Experimental Investigation on a Low-g ~ircraft? An expenrnental investigation of the predictions of the current chapter was carried out

on a KC135 parabolic flight aircraft. The apparatus used in the expenmental study is

f A more thorough presentation of the materia1 in Section 3.5 has been published in Ref. 34.

shown schematicdy in Fig. 3.10-A. A test fluid was placed in each of four cylinders and

al1 four were held in a rack (Fig. 3.10-B). To subject the apparatus to reduced gravitational

intensities, the apparatus was placed on an aircraft that flew Keplarian parabolas. During

each parabola, the behavior of the four systems was recorded by a camera that was oper-

ated at a framing rate of 17.71 frarnes per second. The component of the gravitational

intensity aligned with the cyiinder axis was also recorded. An example of the gravitational

intensity recorded d u h g the low-g portion of a Keplarian parabola is shown in

Fig. 3.1 LB.

The fluids studied were ethylene glycol. n-hexadecane, and FC-75. A sarnple of each

fluid was placed in each of four differently sized cylinders. The dimensions of the cylin-

ders are listed in Table 3-1. The inside surfaces of two additional sets of cylinders were

coated with a polymer to increase the contact angle. Ethylene glycol was placed in one of

these sets and n-hexadecane in the other. FC-75 was found to degrade the surface coating

and was only examined in the untreated cylinders.

The contact angle of sessile droplets of each liquid on both a cleaned and on an FC-

722 coated flint glass slide was measured in a separate apparatus. Images of the droplets

were analyzed to determine the advancing and receding contact angles. The results are

summarized in Table 3-2.

3.5.1. Experimental Procedure

Each fluid-cylinder combination was exarnined in ten different Keplarian parabolas.

The configuration that FC-75 adopted was only examined in the uncoated cylinders. but it

was exarnined in a total of forty different Keplarian parabolas.

During the high-g portion of a Keplarian parabola, the gravitational intensity was

approximately 1.8 go, where go is 9.8 d s 2 . At this intensity the fluid always adopted the

one-interface configuration. Thus, on entenng the low-g portion of a parabola, the config-

uration of each Buid-cylinder combination was the one-interface configuration. By analyz-

ing the film afier the flight had been cornpleted, we could measure the number of

transitions that took place from the one-interface configuration to either the two-interface

or the bubble-configuration. A typical observation is shown in Fig. 3.10-8. It may be seen

there that the fluid in the two larger cylinders had made a transition to the two-interface

configuration, while the same fluid in the two smaller cylinders remained in the one-inter-

face configuration.

In the fifth column of Table 3- 1, the number of transitions that were observed to take

place from the one-interface-configuration to either of the other two configurations is

recorded. This number has been expressed as a percentage of the number of parabolas to

which a fluid-cylinder combination was exposed.

3.5.2. Results and Discussion

One of the irnmediate observations that can be made from the results Iisted in Table 3-

1 is that for those cylinders with which the liquid formed a contact angle of less than 36".

the larger the radius (or volume) of the cylinder, the more likely the fluid was to make a

transition from the one-interface configuration to one of the other configurations. The size

of the container plays a major role in determining whether a transition from the one-inter-

face configuration takes place.

The contact angle is a second important factor in determining whether a transition will

take place from the one-interface configuration during the low-g portion of a parabola. For

systems in which the contact angle was greater than 3 6 O , no transitions from the one-inter-

face configuration were observed (see Table 3- 1 ).

A partial explanation for both of these factors is available from the Helmholtz poten-

tial analysis presented in Section 3.4.2.2. In Table 3-1. a surnrnary of both the predicted

equilibrium configuration and the predicted value of the potential driving the system from

the one-interface configuration to the equilibrium configuration, i.e. (Feq - FI ) . is listed.

The information required to make this calculation is listed in Table 3- 1 and Table 3-2.

The magnitude of the driving potentiai increases with cylinder radius. and. as may be

seen from the results shown in Table 3- 1, the percentage of the transitions was observed to

increase as cylinder size increased and (Fp4 - F I ) became further negative. (A negative

potential indicates that the one-interface configuration is not stable.)

If the contact angle was greater than 36", as may be seen in Table 3- 1. the one-inter-

face configuration is predicted to be in equilibrium. Thus. (Fe4 - F I ) was zero in each of

these cases. No transitions were observed in these cases.

Although these results appear to support the theoreticai conclusions of Section 3.4.2.2.

they can not be taken as conclusively indicating that the Bond number criterion is invalid.

This critenon claims that a transition from the one-interface configuration would only take

place if the gravitational intensity were negative4 and there were times during many of the

parabolas when the gravitational intensity was negative.

To determine if a negative gravitational intensity is required to bring about a transition

from the one-interface configuration, the recording of the gravitational intensity and the

films were examined to determine if there were any transitions that took place without the

possibility of a negative gravitational intensity being responsible. One of the results found

is sumrnarized in Fig. 3.1 1. The photographs in this figure show the fluid configurations in

the cylinder with a radius of 26.5 mm at each of five different gravitational intensities. The

numbers on the gravitational curve indicate the gravitational intensity at the time that each

photograph was taken. As may be seen there, the transition took place in the Iast three pho-

tographs, with the gravitational intensity positive.The potentid di fference. (Fe4 - F 1 ). driv-

ing the system from the one-interface configuration had a value of -24.3 pl for this

cy linder.

Since the transition seen in Fig. 3.1 1 took place at positive gravitational intensities.

this observation appears to be contrary to the Bond number cntenon. According to this cri-

terion. the one-interface configuration is stable for al1 cylinder sizes and al1 contact angles.

provided the gravitational intensity is positive4.

It is important to note that the sarne fiuid in the smaller sized cylinders also formed a

contact angle of O". However the n-hexadecane in these cylinders did not make a transition

from one-interface configuration on this parabola. The value of (Feq - F I ) in each of these

cases was - 1 1.6, -3.85, and - 1.86 pJ. Thus if the magnitude of the gravitational intensity is

negligible, and (Fq - Fi) 5 -24.3 pY, i t appears that a transition from the one-interface

configuration can be expected. However if (Feq - FI) 2 -1 1.6 @ , no transition from this

configuration c m be expected.

Thus it appears that the magnitude of (Feq - F I ) must be greater than a certain mini-

mum value in order to cause the system to decay from the metastable state. The value of

this difference would of course depend on the time for which the system was in the meta-

stable state. For the case shown in Fig. 3.11, this time was on the order of five seconds.

This conclusion can be examined funher by examining the results that have been

obtained from drop tower studies. Petrash et a1.28 have reported results from drop tower

studies in which a glass cylinder with a radius of 20 mm was partidly filled with tetrabro-

moethane. mercury, or ethyl alcohol, and then each was exposed to lo4 go for 2.5 sec-

onds. Both tetrabromoethane and mercury have contact angles of greater than 36" on

glass; thus according to the andysis of Section 3.4.2.2. under this circumstance the one-

interface configuration is the stable configuration. Petrash et al. reported that these sys-

tems did not make a transition from the one-interface configuration. Thus for these two

Ruids the predictions are in accord with the observations.

The more interesting case examined by Petrash et al. was ethyl dcohol in a glass cylin-

der. The contact angle is 0'. In one of the cases they studied, the two-interface configura-

tion is predicted to be the stable configuration, but the observation was that the system

remained in the one-interface configuration when the system was exposed to 10-5 go [ X I .

However we would note that the value of (Feq - Fi) calculated from Eq. (3.4 1) is -9.25 CLJ.

Since the magnitude of this potentiai is less than the value found necessary to cause a tran-

sition in the case shown in Fig. 3.1 1 and the time in the metastable state is shorter, a tran-

sition would not be expected.

Thus, although the one-interface configuration becomes metastable at the negligible g-

levels, for the small cylinder studied by Petrash et al., the potential difference driving the

system out of the one-interface configuration was not sufficient to cause the system to

make the transition.

3.6 Experimental Investigation in a Drop shaftt While the expenmental results from the parabolic-flight aircraft lend support to the

predictions from thermodynarnics, the

gravitation levels in the aircraft had an

experiments are not

oscillatory character

completely convincing. The

and often induded accelera-

t Section 3.6 has been published in Ref. 29.

tions in directions not aligned with the cylinder. In addition, the vertical gravitation levels

often became negative. At sufficiently negative values of g, the continuum mechanics

approach leads to the conclusion that the one-interface configuration would become unsta-

ble; thus both theories could offer an explmation for the observations in the aircraft. The

Japanese drop ~ h a f t ~ ~ provides 10 second penods during which g has minimal fluctuations

and is less than about 104 go in magnitude in al1 directions. To distinguish the predictions

of the various theoretical approaches. experiments have been conducted using this facility

and the results are reported herein.

The expenmental apparatus used to test the prediction consisted of cylindrical glass

containers of three different sizes, which were each partially filled with one of four liquids.

The fluid-filled containers were exposed to reduced gravitation Ievels in the drop shaft.

and the liquid behavior was recorded by video cameras. The gravitation levels were mea-

sured by three accelerometers, one for each a i s .

3.6.1. Experimental Appara tus and Materials

The expenmental apparatus is shown schematically in Fig. 3.12. The apparatus con-

sisted of a clear polycarbonate rack that holds fifteen Pyrex glass cylinders of three differ-

ent sizes. The glass cylinders are partially filled with a single liquid. Three additionai racks

were assembled, each holding cylinders filled with a different liquid. The cylinders were

held in place using a structural silicone glazing cornpound, GE Silicones SilglazeB N.

grade SCS 2500. Four 8 mm video camcorders were used, one to record the behavior of

each rack. The racks were illuminated from behind by a diffuse lighting source.

The fluids studied were 1-Butanol, (HPLC grade, Fisher Scientific); 1-Propanol.

(Fisher Scientific); Hexadecane, (Aldrich Chernical); and distilled de-ionized water. The

surface tension of each liquid was measured using the capillary rise method and the results

of this measurement are shown in Table 3-3. The tolerance in these values arises from the

uncertainties in the measurement of the capillary rise. The values agree closely with the

literature values for al1 liquids except butanol, which is about 13% less than the published

value. The same value was measured for butanol obtained from two different suppliers.

Contact angles for the liquids were measured with stationary sessile liquid droplets on

horizontal g l a s slides. A video camera with a close-up lens was used to record images of

the liquid droplets. The images were analyzed using Image v. 1.45, a public domain image

processing package from the National Institutes of Health. The measurements were

repeated severai times with different frarnes of video. The measured values of contact

angle are shown in Table 3-4. The uncertainty indicates the standard deviation of a series

of six measurements. Some of the glass surfaces were treated with 3M brand fluorochemi-

cal coating FC 722 in order to raise the contact angle to a value greater than 36". This solu-

tion contains 2% Ruoroaliphatic copolymer. This treatment is applied by rinsing the glass

surface with the solution and then allowing the solvent to evaporate. leaving a thin copoly-

mer film on the surface. Small flakes of this coating were found suspended in the water-

filled systems. and so experimental data for these systems was not used. The coating in

contact with the other fluids remained uniforrn and intact.

The dimensions of the g las containers are listed in Table 3-5. Each cylinder was

blown from standard weight Pyrex tubing to form a right circular cylinder with Bat, closed

ends. A thin pyrex mbe was then attached to a hole in the side of the cylinder near one end

to ailow a path through which liquid could be introduced into the cylinder. The g l a s cylin-

ders were cleaned by rinsing with acetone to remove any oil left from the fabrication pro-

cess.

In order to prevent any possibility of leakage or of contamination by sealing materials.

the following sealing procedure was adopted. The cylinders were filled with the selected

volume of liquid using a s y ~ g e and a length of Teflonm tubing to inject the fluid through

the narrow filling tube. After filling with liquid, a vacuum pump was connected to the fill-

ing tube and the fluid allowed to boil at room temperature for about ten seconds to expel

the air. It was found that the short period of boiling had a negligible effect on the liquid

volume. At this time the filling tube was heated with a propane torch until the g las soft-

ened and was sealed closed by the atmospheric pressure. The filling tube was then

removed frorn the cylinder, leaving the cylinder permanently sealed with a small protrud-

ing neck where the filling tube had been.

3.6.2. Experimental Procedure and Facility

Some expenments were conducted with the cylinders lying "on their side" with the

axes perpendicular to the acceleration vector. This caused the fluid to be initially in a con-

figuration much different from any of the axisymmetric equilibrium configurations. Other

experiments were conducted with the cylinder axes digned with the acceleration vector.

This caused the fluid to be in an axisymmetric configuration sirnilar to the one-interface

configuration.

The container to be dropped consisted of an inner and an outer capsule, and the exper-

imental apparatus was placed inside the inner capsule (see Fig. 3.13-A). To overcome the

effects of air drag, the ths ters mounted at the top of the outer capsule are used. During

the period of free fall, the two capsules are not connected, and the space between them is

at a pressure of 800 Pa. The acceleration levels of the inner capsule were recorded using

three inertial accelerometers, one for each axis. The output of these instruments was

amplified, sampied forty tirnes per second, converted to a digital value, and stored elec-

tronicalIy.

Ln order to relate the recorded fluid motion dunng the drop to the recorded g-levels. an

LED was mounted on the apparatus, in view of the video cameras, which flashed once per

second. The voltage signal for this LED was also recorded by the same data acquisition

system that stored the acceleration levels. The video frarning rate of 30 frarnes per second

allows synchronization within 1/30 second.

3.6.2.1 Gravitation Levels Achieved

niere was a large apparent DC component to the readings. indicating a large non-zero

component of acceleration. This component is probably due to the bias (DC offset) of the

accelerorneters. An upper bound on the mean level of the horizontal acceleration can be

determined from the geometry of the system. The inner and outer capsule are separated by

a horizontal gap of approxirnately 5 cm. If the mean horizontal acceleration of the inner

capsule were greater than 104 go for 10 seconds, the inner and outer capsules would corne

into contact. This contact would transmit the vibration caused by the thrusters into the

inner capsule. Since there is no abrupt change in the acceleration readings. it may be stated

that the mean horizontal acceleration averapd over the drop period did not exceed loJ go.

A rcference for the vertical acceleration levels can also be obtained. When the relative

motion of the two capsules is zero, the inner capsule is then expenencing near-perfect

free-fall. The accelerometer reading under these conditions would represent the bias of the

instrumentation. In Fig. 3.13-B. a typical record of the relative positions of the two cap-

sules is shown. The relative position traces for al1 the drops were exarnined and the mean

acceleration levels during the periods of small relative motion were averaged to give the

zero-g offset of the accelerometer system. The mean value was found to be 0.001 174 go

with a standard deviation of 6 x 1od go. The consistency of these resuits indicates that this

represents a true bias level and that the measurement system did not drift during the time

over which the expenments were conducted. By subuacting this level from the uncor-

rected readings. the m e acceleration levels expenenced by our apparatus c m be deter-

rnined.

The accelerometer readings with the zero-offset removed are shown in Fig. 3.14. Here

it can be seen that the acceleration levels have a low level fluctuation with a maximum

magnitude of about f 1 x lo4 go in the horizontal directions and 3 2 x lo4 go in the verti-

cal. Also apparent from this figure is the high level oscillation that occurs when the cap-

sule is first released. Al1 the drops had very similar acceleration readings with virtually

identicai means and oscillation magnitudes.

The srna11 fluctuations in the gravitation levels are caused by resonances in the expen-

mental frarne. The fluctuations shown in the accelerometer readings have two main fre-

quencies. A spectral analysis of the low level fluctuations indicates a predominant

frequency of 3 Hz with another smaller peak in energy at 13 Hz. In a separate test with a

supplementary set of accelerometers. it was found that these frequencies correspond to

resonance frequencies of the apparatus frarne. An examination of the video records. which

have a framing rate of 30 Hz, did not show similar vibration or oscillation in the fluid

interfaces. This indicates that these low-level vibrations do not have any g r o s effect on the

liquid interfaces.

3.6.3. Results

The gravitation levels measured in the drop shah will be considered to be negligible.

The basis for this approximation will be developed in Section 4.1.2 on page 86. In the

experiments it was found that the fluid interfaces often adopted the spherical shape. The

possible spherical equilibnum configuration can be determined from Fig. 3.6 and the sta-

ble equilibrium configuration c m then be predicted from Equations (3 -4 1 )-(3 -47). The

resulting predictions of the stable equilibrium configuration are indicated in the last col-

umn of Tables 3-6 and 3-7.

The Bond number is a measure of the relative magnitudes of the gravitational and sur-

face tension forces. The predictions of the prevailing continuum mechanics approachJo5

depend on whether the Bond number exceeds a critical value. For the contact angles con-

sidered in the current study, this critical Bond number4 ranges from 0.7 18 to 2.2. Since the

maximum Bond number in this experiment (0.077) is roughly an order of magnitude

smaller than the critical value. the continuum mechanics approach would predict that for

al1 contact angles the fluids would adopt the one-interface configuration at equilibrium. As

may be seen from Tables 3-6 and 3-7, the predictions from Equations (3.41)-(3.47) are

very different from those of the continuum mechanics approach. and the contact angle

plays a pivotal role.

3.6.3.1 Configurations after 2.5 seconds of hypogravity

In contrast to the current study, studies of fluid configuration conducted by other

a ~ t h o r s ~ ' ~ ~ only had access to about 2.5 seconds of hypogravity. In order to compare the

current study with those works, we have examined the configurations adopted after 2.5

seconds of hypogravity and also after the full ten seconds had elapsed.

In Fig. 3.15 an image of fifteen horizontal cylinders containing propanol. arranged in

four colurnns can be seen. The image has had its video noise levels reduced using the algo-

rithm provided with the Image program. The straight vertical lines are the frame members

that hold the cylinders in place. The dark horizontal line in the middle of the figure is the

division between the two illumination panels behind the cylinders. The liquid-vapor inter-

faces appear as broad dark cuwes. Three of the systems shown in Fig. 3.15 had contact

angles of 49". and the others had OC. It may be seen that at this time al1 systems have a sin-

gle Liquid-vapor interface that is concave upward, but clearly not axisymmetric. The pro-

panol in the cylinden was still moving ar this time, and thus had not achieved equilibrium.

The other fluids in horizontally oriented cylinders were observed tu have a sirnilar behav-

ior at this time. Thus, at 2.5 s none of the predictions of the equilibrium configurations can

be critically examined for the horizontally oriented cylinders.

For the verticaliy oriented cylinders, it was observed that after 2.5 s of reduced gravity

and for al1 contact angles 1 12 of the 1 16 fluid systems exarnined in this orientation were in

the axisymmetric, one-interface configuration. These liquid-vapor interfaces in these sys-

tems were al1 oscillating slightly. The fluid in the remaining four systems was in motion.

undergoing a transition to a bubble configuration. Thus they were not in equilibrium.

3.6.3.2 Configurations after 10 seconds of hypogravity

For horizontally oriented cylinders containing propanol an image of the configurations

adopted after ten seconds of negligible gravity is shown in Fig. 3.16. The image has had its

video noise levels reduced. In the three cylinders that have contact angles of 49", the fluid

configuration has not changed significantly in the 7.5 s since the image shown in Fig. 3.15.

was taken. This non-axisymmetric configuration appears to be metastable. The remaining

12 propanol systems had zero contact angle, and 1 1 of thern had reached a steady state

configuration. Of those 11 that had reached steady state. 9 are seen in Fig. 3.16 to have

adopted the two-interface configuration. while the remaining two are in a bubble configu-

ration. These observations of the two-interface configuration are consistent with the pre-

dictions made from Equations (3.41)-(3.47) and listed in Tables 3-6 and 3-7. They are

contrary to those made from the continuum mechanics approach.

For the 1 16 vertically oriented cylinders. the observed fluid configurations at the end

of the 10 s of reduced gravity are indicated in Table 3-7. As seen there. al1 22 of the Ruid

systems that had contact angles greater than 36" were found to remain in the one-interface

configuration. This observation is consistent with the predictions of both theoretical

approac hes.

The remaining 94 vertically oriented cylinders had contact angles near zero. Although

these cylinders were initially in the one-interface configuration at the onset of reduced

gravity, only 88 remained in the one-interface configuration. The other six fluid system

were observed to spontaneously move out of the one-interface configuration. This obser-

vation directly contradicts the continuum mechanics approach, which predicts that none

would have lek the one-interface configuration. Since some of these transitions were

observed to take place oa!y after several seconds in hypogravity (see Fig. A.1 on page

155). it is suggested that with a longer duration of low-g, more systems rnight have made

the transition to the two-interface configuration.

Aithough there was a large initial perturbation in the g-field experienced by the appa-

ratus, this cannot be the cause of al1 Ruid motion. The initial fluid motion damped out

quickly, and the remaining motion was a slow, steady progression toward a final configura-

tion. Al1 of the fluid motion from 2.5 seconds until the end of the drop occurred during

excellent low-gravity conditions with oscillations of 2 x 104 go or less. Much of the fiuid

motion was onto surfaces that had not been wetted by the initial fluid motion. The later

motion did not appear to be a direct result of the minor fluctuations in gravitation.

3.6.3.3 Shape of the Interface for Large Contact Angle

In Section 3.3, equations for the catenoid surface were developed. This surface satis-

fies al1 the necessary conditions for equilibrium but fails as a general interface shape. since

the contact angle and fluid volume are not independent. However, the interface shape that

was observed in some of the systems with large contact angle closely resembles this

shape. In Fig. 3.3, a catenoid has been shown dong with a liquid-vapor interface from the

drop shaft experiments. The calculated catenoid has a contact angle of 52", while the mea-

sured contact angle for a sessile droplet of this liquid is 49'. While it cannot be proven that

this shape is a catenoid. the resemblance suggests it as a possibility. This would also

explain why the system did not evolve from this configuration: since the catenoid meets

the necessary conditions for equilibrium, it is already at an extremum. possibly a mini-

mum, in the Helmholtz potential. If it is a local minimum, then moving from this configu-

ration would require an increase in energy.

3.7 Chapter Summary and Conclusions Thermodynamics has been used to predict the behavior of two-phase fluid systems in

negligible gravity. A nurnber of fluid configurations were investigated, and it was shown

that only certain of these could satisfi dl of the necessary conditions for equilibrium for

an arbitrary amount of fluid and an arbitrary contact angle. A Helmholtz potential analysis

of these possible configurations was conducted. and theoreticai expressions were derived

that may be used to predict the equilibrium configuration. For configurations that satisfy

al1 of the necessary conditions, it was shown that the predicted equilibrium configuration

is dependent on the amount of Ruid present, the container geornetry and dimensions, and

the contact angle.

Two experimentai investigations were aiso undertaken to test these predictions. The

possible sphencal axisymmetnc configurations in a cylindrical container c m be deter-

mined from Eq. (3.26) to Eq. (3.29). and for one of the systems studied in the current

expenmentai investigations, the possible configurations are represented in Fie. 3.6. The

equilibrium configuration can be determined from among these configurations by using

Equations (3.41)4.3.47). We exarnined experimentally only systems that were predicted to

adopt the one- or two-interface or bubble configuration. It was found that both the contact

angle and the initial configuration of the fluid played important roles in determining the

final configuration.

For the experimental systems with contact angles less than 3 6 O , the predicted axisym-

metric equilibrium configuration was the two-interface or the bubble configuration.

depending on the amount of Buid present.

The experiments conducted on the KC 135 aircraft indicated that the current theory of

stability may be correct. Al1 systems were tested in the same orientation. so that the sys-

tems were always in the one-interface configuration at the start of the low-gravity periods.

Systems in which the one-interface configuration was the predicted equilibrium state were

never observed to leave this configuration during the low gravity periods. Systems in

which the two-interface configuration was the predicted equilibrium were observed to

make transitions to this configuration dunng the low-g penods. There aiso appeared to be

a correlation between the magnitude of the driving potential difference (between the start-

ing and equilibnum configuration) and the likelihood of a transition occurring during the

low-g period.

The drop shaft experiments examined the predictions further in a much better quality

low-g environment. In the drop shaft expairnents, the predictions of equilibriurn were

examined using two different initial conditions by using two different orientations of the

cylinders. The first initial condition (cylinders horizontal) placed the Buid in a non-axi-

syrnmetric configuration pior to the low-g period. From this starting condition, rnost of

the zero contact angle systems were observed to move to the predicted equilibrium config-

uration. In those systerns that did not adopt the predicted configuration, the Buid either

was still moving or had adopted a non-axisymmeuic configuration. In no case did the Ruid

adopt the one-interface configuration. The second initial condition (cylinders vertical)

placed the Buid in an axisyrnmetxic configuration similar to the one-interface configura-

tion. From this starting condition. the systerns were generally observed ro adopt the one-

interface configuration. regardless of contact angle. However, there were six cases in

which the zero contact angle systems moved from this axisymmetric configuration toward

another configuration. These cases are contrary to the predictions of continuum mechan-

ics, and suggest that the one-interface configuration may not be stable. Given sufficient

time in negligible gravity, more of the systerns rnay have made a transition away from the

metastable one-interface configuration.

For systems with contact angles between 36" and 144", the predicted axisymmetric

equilibrium configuration for the experimental systems was the one-interface configura-

tion. This prediction was also tested using two different starting conditions. In Our experi-

ments, systems with contact angles in this range that began in a non-axisymmetric

configuration were observed to remain in a non-symmetric configuration, but with a differ-

ent curvature than they initiaily had. This observed configuration was not predicted by any

of the theones. Al1 the theories have only considered axisymmetric configurations. Sys-

terns with contact angles in this range that began in an axisymrnetric one-interface config-

uration remained in a one-interface configuration. In contrast to the low contact angle

systems, there were no exceptions to this behavior. This supports the conclusion that the

one-interface configuration is the equilibnum configuration for systems with contact

angles in this range.

For systems with contact angles greater than 144' and with an appropriate amount of

fluid, the equilibrium configuration predicted by the thermodynarnic theory is the bridge

configuration. This prediction was not tested in the experiments.

Initial fluid configuration was observed to play an important role in determining the

short tenn behavior of fluids in low gravity. Systems in which the fluids were initially in a

configuration much different from the predicted equilibnum configuration were seen more

often to adopt the configuration predicted by thermodynamics. Systems that were initially

in a cûnfiguration sirnilar to the metastable one-interface configuration were more often

observed to adopt this configuration. This behavior suggests that. in the short tem. the

systems tend to evolve toward a local maximum in entropy (a metastable state), not neces-

s a d y the absolute maximum (the equilibrium state).

The duration of hypogravity had a large impact on the configuration adopted by the

fluid. In al1 systems that bqan in the horizontal orientation, the fluids were still moving

after 2.5 s of low gravity. The fluid had not yet reached a stable configuration. By contrast.

after 10 s of reduced gravitation, most of these systems had adopted a stationary configu-

ration. In some cases the fluid was still moving slowly after 10 s of reduced gravitation. (A

longer duration of hypogravity will be available on the shuttle experiments descx-ibed in

the Appendix.)

In system that were oriented vertically, the fluid was observed in two cases to make a

transition out of the one-interface configuration only after more than 2.5 s of low gravity

had passed. This motion occurred at a time when the residual gravity field was small

(-10" go) and stable. This suggests that the observations made by other investigaton

using vertical cylindrical systerns might have been influenced by the short duration of

hypogravity available to them.

The critical Bond number criterion does not lead to a prediction of the equilibnum

configuration of a fluid system. Although the critical Bond number has been used by other

investigators as a criterion to determine whether or not a single interface was stable? a

complete thermodynamic stability analysis was not conducted. A force balance is not a

sufficient condition for equilibnum, since even a metastable configuration meets this con-

dition. In order that stability may be predicted, thermodynamics must be used in an analy-

sis of total system entropy.

3.8 Tables

TABLE 3- 1 . Observed and Predicted Configuration in the KC 135 Experiment. 1 - one- interface; 2 - two-interface; b - bubbIe

Container Driving Poten- % Transitions Vol ./ml;

Radius/mm N/Nt Equilibnum tiai from one-

Configuration (Feq-Fl )'& interface

N-Hexadecane - Untreated G l a s (Advancing Contact Angle: 0')

N-Hexadecane - FC 722 Coated Glass (Advancing Contact Angle: 65.0 c 1 . Io )

Ethylene Glycol - Untreated Glass (Advancing Contact Angle: 25.1 + 1.8')

TABLE 3- 1. Observed and Predicted Configuration in the KC 135 Experiment. 1 - one- interface: 2 - two-interface; b - bubble

- - - . - - - - - . - --

Container Driving Poten- % Transitions

Vol ./ml; N/Nt Equilibrium ti al from one- (Feq-F 1 )/ml interface

Radiuslmrn Configuration

Ethylene Glycol - FC 722 Coated Glass (Advancing Contact Angle: 96.7 + 1 .go)

FC-75 - Untreated Glass: O"

TABLE 3-2. Surface Tensions in air at 25 OC and Contact Angle Measurements

Liquid Surface Ten- Solid Surface Contact Angle, degrees sioii mN/m

Advancing Receding

Ethylene Gly- 47.99 Flint Giass 25.1 + 1.8 5.6 k 0.4 col

Ethylene Gly- 47.99 FC-722 on 96.7 a 1 .9 66.1 k 0.6 col Flint Glass

n-Hexadecane 27.76 Flint Glass 65 .0 I 1.1 56.5 + 2.5

TABLE 3-2. Surface Tensions in air at 25 OC and Contact Angle Measurements

Liquid Surface Ten- Solid Surface Contact Angle, degrees sion mN/m

Advancing Receding

n-Hexadecane 27.76 FC-722 on Flint Glass

FC-75 15.0 Flint Glass

Spreads

Spreads

TABLE 3-3. Measured Surface Tensions at 23°C

Liquid Surface Tension (Wm)

Water 0.07 17 M.00 18

Hexadecane 0.027 1 M.0007

Propanol 0.0229 M.0006

TABLE 3-4. Liquid Contact Angles

Liquid-solid system Contact Angle

degrees

TABLE 3-5. Cylinder Dimensions in the Drop Shaft Experiments

Inside Diarneter 2L hside Length H WL Volume

TABLE 3-6. Configurations adopted after 10 seconds of low gravity by systems in a horizontal orientation. 1 - one-interface; 2 - two-interface; b - bubble: n - non axisymmetric; . . . - not tested

Contact Radius Hexa- Predic ted angle L N/N' Water decane Propanol Butanol Equilibrium

n n - n

2

2

2 2

moving b

b b

2

2

2

2

n n -

n

2 2

2 3 - 2

b

b

b

n

2

moving rnovine

U -

aThe surface treatment for this systern failed.

TABLE 3-7. Configurations adopted after 10 seconds of low gravity by systems in a vertical orientation. 1 - one-interface; 2 - two-interface; b - bubble: n - non axisymrnetric; . . . - not tested

Contact Radius Hexa- Predicted angle L, NIN' Water decane Propanol Butanol Equilibrium

aThe surface treatment for this system failed.

3.9 Figures

FIGURE 3.1. Scherk's Surface (A and B) and Helicoid (C)

FIGURE 3.2. Catenoid and cylinder with axes of rotation aligned.

FIGURE 3.3. A: Catenoid intersecting a cylinder at 52' contact angle. B: Fluid configuration observed in a drop shah experiment for a system with 49" contact angle.

FIGURE 3.4. Nodary intenecting a cylinder with 90' t 5 O contact angle.

One-Interface Configuration

Bridge Configuration Bubble Configuration

Vapor

Sessile Droplet Configuration

Two-Interface Configuration

Droplet configuration

Liquid

Sessile Bubble Configuration

FIGURE 3.5. Some of the possible configuratians with spherical interfaces.

0.5 1 1.5 2 2.5 3

Contact Angle, Radians

FIGURE 3.6. Possible configurations as a function of contact angle and amount of fluid.

Configu- -1.5 1 ration

Contact Angle, degrees

Configuration \

Contact Angle, degrees

150

Bridge Configuration

FIGURE 3.7. A: F1 minus F2 vs. contact angle B: F1 minus Fbridge vs. contact angle.

Contact angle, Radians

Fd - F1 vs Contact angle

Increasing h/L, NMt

.

Contact angle, Radians

FIGURE 3.8. A: Helmholtz potential of the bubble configuration as a hinction of contact angle. showing the effect of cylinder height and fill ratio B: Potential of the droplet configuration as a hinction of contact angle, showing the effect of cylinder height and fill ratio

F1 - Fsb VS. contact angle

F1 - Fsd VS. contact angle

Increasing NL, N/h'

Contact angle, 7 Radians

FIGURE 3.9. A: Helmholtz potential of the sessile bubble configuration as a function of contact angle, showing the effect of cylinder height and fil1 ratio B: Potential of the sessile droplet configuration as a function of contact angle, showing the effect of cylinder height and fil1 ratio. Only the portion of the curve where there is a change in the predicted equilibrium configuration has been shown.

FIGURE 3.10. A: Schematic of the experimental apparatus used on the KC135 airc raft. B: Photograph of the apparatus.

Time, seconds

FIGURE 3.1 1 . Fluid configuration and gravitation levels for an experiment in a 26.5 mm radius on the KC 135 aircra..

FIGURE 3.12. Experimental apparatus used in the drop shaft.

Outer Capsule

Low-Pressure Air Gap

Inner Capsule /

Apparatus

Separation *Y-

Period of Zero-g $ Time (s)

contact -

FIGURE 3.13. A: Schematic of the Drop Shafi capsules and the thnisters used to overcome air drag.B: Separation between the inner and outer Capsules during the drop

-1.. .;. . . . . . . . . . . . . . . . . . . . . . . . . . . . . ; . . . . . . . . . . . - . - . - . . . . . . . . . . . . .

pper bound on mean kvel

- - - - ---- . - "\+p,f"ï ,, . A?.

i 1 1 l

-1 .;'. ' . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. Lower bound on rnean level

Time, seconds

FIGURE 3.14. Accelerometer readings with offset rernoved.

49" Contact Angle

FIGURE 3.15. Configurations adopted by propanol in horizontal cylinders after 2.5 seconds of hypogravity. A11 cylinders have zero contact angle except as noted.

FIGURE 3.16. Final fluid configurations for propanol in horizontal cylinders after 10 seconds of hypogravity. All cylinders have zero contact angle except as noted.

CHAPTER 4: EQUILIBRIUM IN THE PRESENCE OF GRAVITY

In the preceding chapter, the necessary conditions for equilibrium were used to deter-

mine the interface shapes that are stable in the absence of gravitational effects, and these

predictions were tested experimentally. The current chapter will extend this analysis to

systems in which the gravitational effects are not negligible. The effect of gravity on the

interface shapes will be quantified, and it will be predicted that gravity gives rise to contact

angle hysteresis in the two-interface confi,wation. These predictions will be tested in an

expriment, and it will be found that the experimental results agree with the predictions.

4.1 Interface Shape of Single-Component Systems We consider the equilibrium shape of a liquid-vapour interface in a cylindrical con-

tainer with a gravitational acceleration of magnitude g acting dong the axis of the cylin-

der. Expressions may be derived for the isothermal fluid-phase pressures as a hnction of

elevation, relative to the pressures at a reference point. The point at the cylinder axis. on

the liquid-vapour interface will be chosen as the reference point and will be indicated by a

subscript a. Denoting the elevation at the reference as Z,, the expression for the liquid

pressure, Eq. (2.42), can be wntten as

PL = p;-pLg(z-z,)

where is the liquid density. Similarly Eq. (2.43) cm be expressed as

where Ü is the gas constant.

Using Eq. (4.1) and Eq. (4.2) and following the rnethod of Bashforth and ~ d a m s ' .

equations for the interface shape c m be found. This method parametrizes an axisyrnmetric

interface by the turning angle, $ and by the radius of curvature. The interface canot be

parameterized in this way for contact angles of 90°. because the radius becomes infinite.

The radial coordinate, X, is a function of @, and of the radius of the interface, R2. that ong-

inates at the cylinder axis (See Fig. 4.1).

From the differential geornetry of the interface (see Fig. 4. l), the following relations

c m be found:

dZ = R , ($) sin @d@

where R I is the interface radius in the plane of the figure.

The pressure difTerence across an interface at a given elevation must be the same

whether it is written through Eq. (2.31) or from the difference between Eq. (4.1) and

Eq. (4.2). Equating these pressure differences. and using the radius of the cylindncal con-

tainer, L, as the length scale for the problem. equations (4.4) and (4.5) can be written:

and

dz = sin sin @

4(@) - - X

where x and z are the non-dimensionalized radial and vertical coordinates of the interface

and the following definitions have k e n introduced:

P L g ~ 2 Bs- yLV

where B is the Bond number, which is the ratio of gravitational forces to liquid-vapor sur-

face tension forces. For most fluids and practical container sizes. the value of r(@) is very

near unity. For exarnple, for water at room temperature in a 5 cm diarneter cylinder under

normal gravity, r($) - 1 r 1 0 - ~

The procedure for finding the value of the interface radius at the a i s . Ra, and the

shape of the liquid-vapor interface may now be stated. An initial value for Ra is assumed.

and from Eq. (2.46). the liquid pressure corresponding to this value can be calculated.

Using this caiculated value of the pressure, Eq. (4.6) and Eq. (4.7) can be integrated

numerically to obtain x(#) and z(#). If the correct value of Ra has been assumed, the maxi-

mum radial dimension of the interface will be equal to unity, the radius of the cylinder:

where the maximum value of turning angle is related to the contact angle by

If the equality indicated in Eq. (4.11) is not met, a new value of Ra is chosen and the pro-

cess repeated until equality exists.

When the interface shape is determined in this manner, we wouid emphasize that both

the Laplace equation and the condition that the chernical potentials must meet, Eq. (2.1).

are çatisfied. The continuum mechanics approach used by some other inve~ti~ators~- '~

fails to satisQ this latter condition. In contrast to the curvature method of Neu and ~ o o d ~ ' .

the method outlined above ailows the radius of the liquid-gas interface to be predicted and

is not limited to zero contact angle.

4.1.1 Calculated Interface Shapes.

A number of interfaces have been calculated for one-component systems from the dif-

ferential equations expressed in Eq. (4.6) and Eq. (4.7). The results are shown in Fig. 1.2.

The numerical integrations were performed using Mathematicam v.2.2 with an accuracy

goal of Infinity and a precision goal of 12. Some cases were checked using a fourth-order

Runge-Kutta technique executed in Microsoft Excel@ v.4.0. The results from Runge-Kutta

technique with 180, 450 and 900 steps were identical to nine significant figures. and also

agreed with the Mathematica solution to nine significant figures.

The effect of the Bond number on the shapes of the interfaces is shown in Fig. 4.2. The

Bond number is defined as positive for gravitational acceleration directed from the vapor

to the liquid region. "Upper" interfaces correspond to negative Bond numbers. It may be

seen that the shape of the upper interface is more strongly affected by the magnitude of the

Bond number than is the shape of the lower interface. When the Bond number is zero (zero

gravity). the interfaces become spherical.

Note that the interfaces shown in Fig. 4.2 have been calcuiated for systems with only

one interface. If more than one interface were present in the system. the interaction of

these interfaces would have an effect on the curvatures and contact angles of the inter-

faces. Note also that the effect of gravity on upper interfaces is opposite to the effect on

lower interfaces: the upper interface becomes more curved, while the lower interface

becomes flatter with increasing gravity. In Section 4.4. this difference will be exarnined

for systems in the two-interface configuration, and it will be seen that this efTect contrib-

utes to contact angle hysteresis in this configuration.

Upper interfaces are only possible for Bond numbers less than a certain value. which is

dependent on contact angle. For example, for zero contact angle the lirniting value of

Bond number is approximately 0.8, and for larger values, the mean radius of curvature

becomes infinite and the equations can no longer be integrated. ~oncus ' used the lirniting

value of Bond number as a criterion for equilibrium, calling this the Cntical Bond number.

It was shown in the previous chapter that this criterion is not sufficient to predict the equi-

librium configuration of a fluid system.

4.1.2 Criterion for Neglecting Gravi tational effects

The shape of the equilibrium liquid-vapor interface in a gravitational field of arbitrary

intensity can be determined using the numerical procedure defined in the previous section.

No analytic solution to the goveming equations has been found except in the Iirnit of g

approaching zero. In this section the numerical solution of the interface equations will be

compared with the zero-g solution in order to determine the conditions under which the

effects of gravitation on the interface shape can be neglected.

As the gravitational intensity approaches zero, the function q defined in Eq. (4.8)

approaches the limit

L qo = 2- (4- 13)

R a

Thus the effects of gravity that account for the difference between fie differential equation

for the zero-g and the non zero-gravi5 case may be written

For the fluids. container sizes and gravitation Ievels considered in the experimental pro-

gram described in chapter 3, the Bond number is four orders of magnitude larger than the

other term in Eq. (4.14), and thus the Bond number is the dominant influence of gravity on

the interface shape.

The solution to the interface equations in the lirnit of zero gravity may be written

sin @ -q,(@) = - cos e

Consider the ratios of the numerical solution. Eq. (4.6) and Eq. (4.7). and the zero-p

solution, Le.

As the gravitational effects become negligible, these ratios approach unity.

Caiculations were made to investigate the effects of small gravitational accelerations

for a range of fluid properties and contact angles. Fluid properties for water. propanol.

butanol, and n-hexadecane were examined, the fluids used in the drop shaft experirnents.

An effective gravitational magnitude of g = 2 x 104 go was chosen. and contact angles

from zero to 60° were investigated. Cylinder sizes ranging from 40 to 65 mm in diameter

were considered. The maximum Bond number corresponding to these experimentd

parameten is 0.077. The numericai integrations of Eq. (4.6) and Eq. (4.7) were performed

using MathematicaO v.2.2 and were confirmed using a fourth-order Runge-Kutta calcula-

tion.

For two values of Bond number and a contact angle of zero, the vdues of the ratios

given by Eq. (4.17) and Eq. (4.18) have been calculated and the results plotted in Fig. 4.3.

Values of these ratios were calculated for other contact angles and Bond numbers. and al1

had ratios closer to unity than those shown in the figure. The maximum value of Bond

number for which the ratios were plotted is 0.077, while the maximum value of the other

non-dimensional group in Eq. (4.14) is 7 x 10? Frorn Fig. 4.3 it may be seen that the

maximum vaIue of each of the ratios is about 1.013 when the Bond number is 0.077. The

ratio of the solutions decreases with decreasing Bond number.

It may be seen from Fig. 4.3 that for the fluids, containers and gravitational accelera-

tions considered, the effect of gravity is to cause a deviation of the txue interface shape

from the zero-g shape of less than 1.5%. Thus, this deviation was neglected in the analysis

of the drop-shaft results of chapter 3.

4.2 Effect of a second component The preceding equations were derived for a single-component system. In any practicd

systern there will be other components. particularly dissolved gases. The effects of a dis-

solved gas on the interface shape will now be determined, and it will be shown that over a

large range of concentrations, the gas has a negligible effect on the interface shapes.

4.2.1 Equilibrium concentration vs. elevation.

Consider a two component system in which the cornponents are present as a dilute liq-

uid solution and as a perfect gas mixture. The chernical potential p for the solvent in the

Iiquid and the vapor phases at temperature T and pressure P c m be wntten as

where the subscripts I and 2 indicate the solvent and solute components. respectively. and

a superscnpt o denotes a pure substance. The symbol n indicates the number of moles per

unit volume in the liquid, and x I denotes the mole fraction of the solvent component in

the gas phase.

The condition that chernical potential must satisb in a gravitational field, Eq. (2.1).

can be written

where )cl is a constant. Define a reference point, denoted by the subscnpt r. on a liquid-

vapour interface. Neglecting the interface thickness, the liquid, vapour and interface ele-

vation are al1 the same at this point. Evaluating Eq. (4.2 1) at the reference point and at an

arbitrary point in the liquid and approximating the number of moles of solvent per unit

volume as constant yields

In order to calculate the concentration as a function of elevation. an expression for the

chemical potential of solute component in the solution is needed. The chernical potential

for component 2 in a solution of concentration C. can be written

where v(T, P) is an unspecified function of temperature and pressure. In order to deter-

mine the form of this function, consider a systern as shown in Fig. 4.4. This system con-

sists of two chambers separated by a semi-permeable membrane that allows component 3

to pass freely, but does not allow the liquid component 1 to pas . One chamber is filied

with pure gaseous component 2 (the solute), and the other chamber is filled with a solution

of 1 and 2. Both sides of the system are maintained at temperature T and pressure P by a

reservoir. The solution is. by definition. saturated with component 2. The equality of the

chemical potentials of component 2 in the two charnbers can be written

where CJP) is the saturation concentration of component 2 at the pressure P. Therefore.

the unspecified function of temperature and pressure is

Substituting this expression into Eq. (4.23), and rewriting in terms of the nurnber of

moles per unit volume, n. one finds

where ni represents the number of moles of solute per unit volume of a saturated solution

at pressure P.

Consider the first term on the nght hand side of this expression. the chemicai potential

for pure component 2 at the given temperature and pressure. Since we consider a compo-

nent that is gaseous at these conditions, this potential may be wntten in terms of a refer-

ence pressure Pr,! as

Note that this reference pressure is not related to the reference point defined earlier. Using

this expression, Eq. (4.26) may be wntten

Ln order to determine the saturation concentration as a f'unction of pressure. consider

Henry's law for dilute solutions of gases in liquid. Henry's law c m be wntten"

where H is the Henry's law constant for the solution and temperature considered. Using

this relation, the chemical potential of the solute in the liquid, Eq. (4.28). can be written

Note that the explicit dependence on the pressure has been replaced by the concentration

of the solute, which is pressure dependent.

One of the necessary conditions for equilibrium can be wrïtten

where A, is an undetermined constant. The value of this constant can be found by substi-

tuting Eq. (4.30) into Eq. (4.3 1) and evaluating this expression at the reference elevation

and concentration on a liquid-vapour interface. Equating this to the pneral forni of

Eq. (4.3 1) yields

The concentration at the reference point cm be found by equating the gas phase and

the liquid phase potentials of the solute component across the interface at the reference

elevation. The gas phase potential at the reference point is given by

Equating Eq. (4.33) and Eq. (4.30) evaluated at the reference point and solving for the ref-

erence concentration yields

Note that the quanti5 Pref has been eliminated from these equations, while the pressure

and concentration at the reference point remain. Substituting Eq. (4.34) into Eq. (4.22).

the pressure in the liquid phase can be written

The pressure in the gas phase as a function of elevation c m be found by considering

the chemical potentials of both components in the gravitationai field. For an ideai gas mix-

ture. the chemical potential of component j must satisQ the following equaiity:

pg(T, P,) + R ~ l n [ X ' ~ ~ Z ) ] + WjgZ = hj

where X, is the mole fraction of component j in the gas phase. Evaluating Eq. (4.36) at the

reference elevation and at a general elevation and equating, one finds

This expression, Eq. (4.37), represents one equation for each gaseous component.

Since we are dealing with only two components,

Using Eq. (4.38) and solving the two equations represented by Eq. (4.37). one finds

4.2.2 AxiSymmetric Liquid-Vapor interfaces

Having found the equations for the pressure in the bulk phases as a function of eleva-

tion, we now find the equations for the shape of the liquid-vapor interface of a two cornpo-

nent system in the presence of an arbitrary gravitational field. Across a liquid-vapor

interface of arbitrary curvature, there is a pressure difference proportional to the curvature.

At a given elevation, 2. this pressure difference c m be wntten using Eq. (4.35) and

Eq. (4.39) as

Allowing the interface to be of negligible thickness. the elevation of both phases is the

same at the interface. In order to simplify these expressions, make the following defini-

tions:

Consider the axi-symmetric interface shown in Fig. 4.1. The difference in pressure

across the interface is related to the orthogonal radii of curvature by the Laplace equation.

Eq. (2.3 1).

For the reference position, choose the axis of symrnetry, where the two radii becorne

equal and will be referred to as R, At this point. the Laplace equation c m be written

Using Eq's (4.42). (4.43) and (4.44), the expression for the pressure difference across an

interface becomes

Using Eq. (4.3) and equating Eq. (4.44) and Eq. (4.45). the following expression can

be written:

where the following definitions have been used:

Now assign a symbol to the function on the right hand side of Eq. (4.46)

q ( z ) - Cl + Bz + Cv(C,l( 1 - X r ) + zrr l ( z ) + ( 1 - Ch)( 1 - (4.53)

Using this definition, one cm express the relationship between the unknown radius of

curvature and the elevation in t e m of reference quantities:

Consider the relationship between the tuming angle, @, and the radial and vertical

coordinates as shown in Fig. 4.1. From this relationship, and recognizing that the elevation

c m also be expressed in terms of @, Eq. (4.54) can be parametrized in tems of the tuming

angle:

dz = sin $d@ sin @

4(0) - - x(@ )

Note that at this point, the value of the pressures at the reference point have not been

detennined. This can be done by considering the equilibrium of the chernical potentials.

Equating the potential expressions for the solvent (Eq. (4.19) and Eq. (4.20)) across the

interface at the reference point, one finds

A sirnilar relation. Eq. (4.34). was found by considering the equality of the potentials

for the solute. Using Eq. (4.57) and Eq. (4.34), an equation for the gas phase pressure at

the reference point can be found:

The reference liquid pressure can be expressed in terrns of the gas phase pressure and

the radius of the interface at the reference point by using the Laplace equation. Eq. (4.44)

so that the pressure in the gas phase c m be written

If the mole fraction of solvent in the gas phase at the reference point. X , . is known.

then this equation represents a one-to-one relationship between the radius of curvature. R,

and the pressure at this point. In other words, the gas and liquid pressures are uniquely

detennined from the radius of curvanire and the mole fraction of the vapor.

One can select the mole fraction and the pressure in the gas phase and then determine

the shape of an interface by integrating Eq. (4.55) and Eq. (4.56). The integration is

stopped when the h n g angle reaches its maximum value as expressed in Eq. (4.12). At

this point, the radius, x, is examined to determine whether it is equal to unity. which indi-

cates the interface fills the chosen container. If not, a new value of pressure and mole frac-

tion are chosen and the process repeated. Through appropriate choice of pressure, mole

fraction and position of the interface. the constraints on the number of moles present can

be met.

4.2.3 Calculated Interface Shapes

Interface shapes have been calculated for two-component systems of water and nitro-

gen by using Mathematica to integrate Eq. (4.55) and Eq. (4.56) for mole fractions of

water in the gas phase between 0.01 and 1 at 35 OC. corresponding to a total pressure

between 5 kPa and 500 kPa. The results are indistinguishable. ruid are the sarne as those

shown in Fig. 4.2. This result is not surprising, since the curvature of an interface is depen-

dent on the pressure difference across the interface, not on the magnitude of the pressures.

Since the densities of the liquid and vapour phases are changed only slightly by the pres-

ence of the gaseous component, the variation of pressure with elevation and thus the varia-

tion of the interface curvature will be largely unaffected by the presence of the gaseous

component. Saturating a water system with one atmosphere of nitrogen at 25 OC, as will be

done in the experiments described in Section 4.5. would produce a mole fraction of 0.04

which results in interfaces that are indistinguishable from the one-component calculation.

4.3 Catenoid Interfaces It was shown in chapter 3 that the catenoid satisfies the necessary conditions for equi-

librium in the absence of gravity. It was also shown that interfaces resembling the catenoid

have been observed in experimental systems. We will now develop the expressions for a

catenoid shape that has been modified by the effects of gravity. As will be seen later. this

shape corresponds weli with the shape of liquid-vapor interfaces observed in experiments.

4.3.1 Catenoids aligned with Cylindrical Containers

Any axi-syrnrnetric interface aligned with a cylindrical container will intersect the cyl-

inder with constant contact angle. The catenoid is an axi-symrnetric surface of zero mean

curvature and c m be expressed in cylindricd coordinates in terms of a single constant. c:

In the presence of gravity, the catenoid shape would be modified as a result of the g n -

dients of the chernical potentiais of the bulk phases. If the gravity vector is aligned with

the cylinder, the interface will still be axi-symmetric. The modified surface may be param-

etrized by a nirning angle, @, which originates from the liquid side of the interface (see

Fig. 4.5). The radial coordinate, x, measured from the axis of the rnodified catenoid. is a

function of the tuming angle and of the normal radius of the interface that originates from

its axis of symmetry, R,

Considenng the differential geometry of the interface as shown in Fig. 4.5, substituting

Eq. (4.35) and Eq. (4.39) into the Laplace equation, and non-dimensionalizing by the

radius of an intersecting cylinder, the following may be derived:

sin 4 d o ) - - -ml

- sin @

9 w - - ~ ( 0 )

where the definitions given by Eqs. (4.47), (4.48) and (4.53) have been used.

It will be shown in Section 4.5.1 that this modified catenoid shape closely resernbles a

Auid configuration observed in experiments.

4.3.2 Catenoid droplets

Ordinarily. one wouid use the Laplace equation, the Young equation and Eq. (2.1 ) to

predict the shape of a liquid droplet. However, since the variation of surface tensions with

pressure are not known, the Young equation cannot be used to predict the contact angles of

the droplet. Since the droplets are usually of lirnited height, it is expected that the effects

of gravitation on the droplet shape will be small. Therefore, we will consider a droplet

shape that satisfies al1 the necessary conditions for equilibrium in zero gravity, and then

use the Laplace equation and Eq. (2.1 ) to modify the droplet shape.

As was shown in Section 3.3, the catenoid may be intersected with a cylinder of unit

radius offset in the x-direction by a distance x,. The values x, and of the constant c in

Eq. (4.60) that result in a constant contact angle were found to be

and

With these choices of c and x,, the catenoid satisfied dl the necessary conditions for equi-

librium in zero gravity.

To calculate a modified catenoid droplet, the initial conditions of contact angle. mean

curvanire and the gas phase composition at one point may be specified. and Eq. (4.62)

integrated until the boundary condition given by

is satisfied. For a particular choice of the initial conditions. one obtains the results shown

in Fig. 4.6. The modified catenoid sausfies the Laplace equation and Eq. (2.1). and has a

nearly constant contact angle. This interface will be discussed further in Section 4.5.

4.4 Two Interface Equilibnum and Contact Angle Hysteresis Numerical and approximate methods of calculating the shape of a single liquid-gas

interface in a two-component system have been established. We will now apply these

methods to the two-interface configuration, in which two separate liquid regions exist in a

cylindrical container, separated by a gas phase. This configuration was predicted in

chapter 3 to be the equilibrium configuration in zero gravity if the contact angle is less

than approximately 3 6 O , and this prediction was supported by experimental work. If the

gravitational intensity is no longer zero. we investigate the relation that would have to

exist between the contact angles at the upper and lower interfaces in order for equilibnum

to exist.

4.4.1 Pressure Profile in a Capiilary System

The relations descnbing the pressure gradients in a Iiquid and a gaseous region have

been denved and were presented in Section 2.3.1. In addition, it was also shown that the

pressure in the liquid and vapor phases are functions of the equilibrium curvature of the

liquid-vapor interface. We now apply these relations to a two-interface configuration.

Let a property associated with the upper liquid-vapor interface be indicated by a sub-

script u. and one with the lower interface by a subscript i. On the upper interface, the rela-

tion between the liquid and vapor phase pressures can be determined by rewriting

Eq. (2.45)

and similarly on the lower interface

In the vapor phase, the relation between the pressure at a point on the upper interface and

that at a point on the lower interface, a distance h below, is seen from Eq. (7.43) to be

given by

After Eq. (4.68) is substituted into Eq. (4.66) one finds

After making use of Eq. (4.47). Eq. (4.69) rnay be written

For a given value of the pressure in the liquid phase at the upper interface. this equa-

tion indicates that the pressure in the liquid phase ai the lower interface would be the same

as that which would have existed at the lower interface if the intervening region had been

filled with liquid.

4.4.2 Relation Between Mean Radii of Curvature and Separation

Distance

We may now develop a method to determine the curvature at one interface in terms of

that at the other and the difference in interface elevation. If the Laplace equation

(Eq. (2.3 1)) is applied at the lower interface and then. using Eq. (4.68) and Eq. (4.69). the

pressures in the liquid and vapor phases at the lower interface are written in terms of those

at the upper interface, one finds

Wgh Wgk 2 y L V ~ X P [=] - (p: + z) =

After applying the Laplace equation at the upper interface, solving for PL and substituting

the result into Eq. (4.7 1). one cm solve for ~ f - ~ :

Recall that for given values of ~b~ and T , the value of P: can be determined frorn

Eqs. (2.45) and (2.46). Thus, Eq. (4.72) represents an expression for Z?fV in terms of four

variables, the upper interface radius, R ; ~ , the vertical separafion of the interfaces, h. the

temperature and g .

We may now calculate the equilibrium, axial pressure profile for an isothennal capil-

lary system that is in the two-interface configuration. Suppose the axial rnean radius of

curvature at the upper interface is known and the lower interface is at a known distance h

away. Knowledge of ~b~ allows and P: to be determined from Eqs. (2.45) and

(2.46). Then. from knowledge of h . the pressure in the liquid phase at the lower interface

may be determined from Eq. (4.70) and the mean radius of curvature, ~f' from

Eq. (4.72). Since the pressures in the liquid phase at the interface would then be known.

Eq. (2.42) and Eq. (2.43) could then be used to calculate the pressure in each phase away

from the interface. Using this method, a pressure profile has been calculated for a prtrticu-

lar case and the result is shown in Fig. 4.7. Although the vapor phase pressure appears

constant in the figure, it does follow the relation given by Eq. (2.43).

4.4.3 Necessary Difference in Upper and Lower Contact angles

As may be seen from Eq. (4.72), the mean radii of curvature of the upper and lower

interfaces are not in general equd. If the values of T, L, H, N , g, 8, are given. we

now deterrnine the effect of the difference in the mean radii of curvature on the contact

angles at the two three phase lines.

In order to deterrnine the magnitude of the contact angle difference. it is necessary to

find the shape of the liquid-vapor interface under the influence of gravity. Differentiai

equations for the interface shape have been derived. and have been presented for the sin-

gle-component case in Eq. (4.6), Eq. (4.7) and for the two component case in Eq. (4.55)

and Eq. (4.56). These equations may be integrated, subject to the boundary condition on

the contact angle, in order to determine the shape of an interface of known contact angle.

An interface calculated in this way

that are applicable at that interface.

LOS

satisfies ail the necessary conditions for equilibrium

The other interface rnay be determined from an iterative procedure. Let the elevation

of the first interface be z , and make the hypothesis that a second interface exists at a sepa-

ration distance h away. By using Eq. (4.72), the mean curvature at the second interface

rnay be detemiined and Eq. (4.6) and Eq. (4.7) integrated numerically. The pressures in

the bulk phases throughout the systern rnay be determined from Eq. (2.42) and Eq. (2.43).

The number of moles in this hypothesized configuration rnay be determined. The vol-

umes of the adsorbed phases are neglected according to the Gibbs dividing surface

approximation. and the number of moles in the adsorbed phases rnay be neglected corn-

pared to those in the iiquid and vapor phases. The density of the vapor va.ries with pressure

according to the ideal gas law, and it is assumed that over the range of pressures consid-

ered the liquid specific volume is equd to v i . Referring to Fig. 4.7. the number of moles

in this hypothetical configuration rnay be expressed as

'bu 'o 1 (col + 11) ' b u

After carrying out the integrals of Eq. (4.73) numencally, one c m compare the calcu-

lated number of moles with the number known to exist in the system. If these quantities

are not the same, then a new separation distance is hypothesized and the process repeated

until the equilibrium separation that satisfies the constraint is found. Since the liquid den-

sity has k e n assumed constant, the choice of zo does not effect the separation distance.

The cylindrical volume being considered is characterized by its height and radius. H.

and L. Using the calculation technique outlined, and given values of T, L, H. N, g and

Bu, contact angles at the lower interface have been calculated. The position of the lower

interface, zo, has little effect on the contact angle difference. The important factor is the

separation distance between the two interfaces. For a given volume, the separation dis-

tance depends strongly on the cylinder diameter and the number of moles in the system.

but is almost independent of how much of the liquid phase is above or below the vapor

phase. Thus, in Fig. 4.8 the difference in contact angles of the upper and lower interfaces

has been plotted as a function of the ratio of the separation distance to the cylinder radius.

For the case considered, the value of the contact angle difference is seen to be always pos-

itive, indicating that at equilibrium the lower contact angle is predicted to be larger than

the upper. The dashed portions of these figures indicate conditions where the vapor phase

pressure is greater than the saturation pressure corresponding to the temperature.

The iterative procedure used to calculate the liquid-vapor interface corresponding to a

given contact angle also enables the dependence of pressure on contact angle to be deter-

mined. Recall that once the interface that corresponds to a given contact angle has been

calculated, the pressures on the interface at the a i s of the cylinder are also known. These

pressures have been tabulated for water in a 1.2 mm diameter cylinder at 25 OC for both

upper and lower interfaces, and the results may be seen in Fig. 4.9. For contact angles Iess

than 90°, the pressure in the vapor is less than the saturation pressure. while for contact

angles greater than this value, the pressure in the vapor phase is greater than the saturation

value.

To determine the relation between the contact angles of the upper and lower interface

in general, one rnay again consider Eq. (4.72). Since for al1 systems of physicai interest.

the exponential appearing in this equation may be expanded and only the first term

retained. Then, if the vapor is approximated as an ideal gas. Eq. (4.72) rnay be written

and since

one may conclude from Eq. (4.75) that

Since the mean radius of curvature at the lower interface is larger at each point of the

interface than the mean radius of curvature on the upper interface directly above. it fol-

lows that the contact angle on the lower interface will be larger than that of the upper

interface.

The effect of gravity on the predicted contact angle hysteresis rnay be determined from

Eq. (4.72). As may be seen there. when g vanishes the mean radius of curvature ai the

lower interface become equal to mean radius of curvature directly above on the upper

interface. Since the mean radii are equal at ail corresponding points on the two interfaces,

the contact angles would also be the sarne.

4.4.4 Young Equation Considerations

In the previous section, it was shown that if the contact angle at one of the interfaces

was given dong with the experimentally controllable variables T, L, H , N, g . then the

contact angle at the other interface could be predicted. And it has been seen that Eq. (2.1)

and the Laplace equation Iead to the prediction of the contact angle at the lower interface

being larger than that at the upper. To determine if this predicted contact angle hysteresis is

consistent with the Young equation, we first consider the Gibbs adsorption equation. After

differentiating Eq. (2.4) and combining with Eq. (2.3), one finds

dyj = - , J ~ T - nidpi j = SL, Sv, SL

When Eq. (2.1) is applied at the three phase line

= p L = = pL = pV

After making use of Eqs. (4.79) and (2.37). Eq. (4.78) may be wntten

For an isothermd change of state, Eq. (4.80) may be applied at the upper three phase line

and then integrated to obtain

and following the same procedure at the upper interface gives

After subtracting Eq. (4.82) from Eq. (4.81), one finds that the difference in solid-vapor

surface tension at the upper and lower interfaces may be written in terms of the isotherm

relation for the solid-vapor interface. n S V ( ~ , p V ) :

If the same procedure is followed at the solid-liquid interface. then the result may be writ-

ten in ternis of the isotherm relation at the solid-liquid interface. n S L ( ~ , pL) :

and after subtracting Eq. (4.84) from Eq. (4.83) one finds

By combining the Young equation with Eq. (4.85), one obtains

Since it has been predicted that the contact angle at the lower interface is greater than that

at the upper, the left hand side of Eq. (4.86) must be less than zero. Thus,

Equation (4.87) represents a restriction on the isotherm relations at the solid-liquid and

solid-vapor interface that must be saiisfied if the Young equation is to be consistent with

the predicted contact angle hysteresis. Further anaiysis would require the avaiiability of

theoretical isotherm relations at the solid-liquid and solid vapor interface. However. at

present such relations do not exist. One might consider applying either the BET or the

FHH relation at the solid-vapor interface; however, as seen in Fig. 4.9, the pressure in the

vapor phase deviates from the saturation vapor pressure by only approxirnately lo4%

Since the BET and the FHH equations each predict a nonphysical (i.e., infinite) value of

nSV when the vapor pressure is equal to the saturation pressure, these equations are of

doubtful validity when the vapor pressure is so close to the saturation vapor pressure.

4.4.5 Discussion

As was shown in chapter 2, if the two-interface configuration indicated in Fig. 4.7 is in

equilibnum while in a gravitational field, it must satisfy 1) the Young Eq.. 2) the Laplace

Eq. and 3) Eq. (2.1). From the last of these conditions, the difference in the pressure in the

liquid phase at the upper and at the lower interface is the same as that which would have

existed at these two points if the system had been filled with liquid (see Eq. (4.70) ). By

contrast, the pressure in the vapor phase is almost independent of position. Thus. the dif-

ference in pressure across the upper interface is larger than that across the lower interface.

When these differences in pressure are used in the Laplace Eq. one finds that the contact

angle at the upper interface is necessarily smaller than that at the lower interface.

When the Young equation is used to examine the mechanism by which this contact

angle hysteresis c m exist in an equilibrium system. it is found that the adsorption iso-

thems at solid-liquid and solid-vapor interface must satisfy Eq. (4.87). Thus. the contact

angle hysteresis does not necessarily violate the Young equation, but it does establish cer-

tain conditions that the adsorption isotherms must satisfy.

4.5 Experimental Investigationt When the Gibbs formulation is used and al1 three conditions for equilibnum are

included in the analysis, it has been shown that they Iead to the prediction of contact angle

hysteresis. Further, if the contact angle of the upper interface were measured, then frorn

the procedure of Section 4.4, the contact angle of the lower interface could be predicted.

The magnitude of the predicted difference in the contact angles is tens of degrees. even in

simple thermodynamic systems with a solid surface that is ideally smooth, homogeneous.

rigid and non-dissolving. We now describe an experimental program that was undertaken

to examine this prediction

4.5.1 Preliminary Investigation

A preliminary investigation was undertaken to establish the viability of the expenmen-

ta1 procedure. This preliminary program used pyrex capillary tubes as a cylindrical con-

tainer, and water as the fluid. Aside from establishing expenmentd techniques for the

principal expenments, the preliminary investigation gave evidence of a new fi uid configu-

ration that has not previously been reponed, but which we had hypothesized could exist in

a gravitational field.

After two cylindrical containers of 1 mm inner diameter were partidly filled, water

was observed to fom a complete ring-like liquid region dong the clean cylinder wall.

Photographs of two of these liquid regions may be seen in Fig. 4.10. The capillary tube is

vertical in these photographs, and the liquid-vapor interface is visible as a dark region. The

upper two photographs are two perpendicular views of the sarne fluid region, while the

lower two photographs are two views of another fluid region in a second capillary tube.

t Much of the materid presented in this section is contained in Ref. 30.

The three-phase lines, which are horizontal, are visible in the figure. The photographs are

somewhat distorted by the lens effect of the capillary.

These fluid regions remained in the capillary tubes for a period of approximately ten

days, at which time they were removed to prepare for other experiments. The tubes were

not sealed. but there were liquid "piugs" blocking the capillary between the fluid rings and

the room air.

These ffuid rings are similar to the symmetric modified catenoids that were calculated

in Section 4.3 (see Fig. 4.5). Modified catenoids were calculated and compared with the

systems seen in Fig. 4.10. Good agreement between the calculated and observed shapes

was found for a contact angle at the lower interface of 36". and a ratio of R, to R, of 1:-

(see Section 4.3). A cornparison of this calculated interface to a photographed interface is

presented in Fig. 4.1 1. The lighting was changed and the brightness inverted from that

used in the previous figure, to produce a clearer image of the liquid-vapor interface. No

attempt was made to compensate for the optical distortion in the photograph. The vertical

lines in this figure are the walls of the capillary, while the curved lines are the obsenied

and calculated liquid-vapor interface shapes. The calculated shape deviates from the

observed shape primarily at the thickest point of the liquid region. The close resemblance

of the calculated shape to the observed shape suggests that the liquid-vapor interface may

be a modified catenoid, with two radii of curvature of opposite sign.

4.5.2 Principal Investigation

The objective of the experimental procedure was to create a system of constant corn-

position that would remain in the two-interface configuration for a time that was sufficient

to allow the systern to reach equilibrium in a gravitational field. This requires that the sur-

face tension forces be large compared to the gravitation forces. Ideally, a singlexompo-

nent Ruid would have k e n used for this purpose, however, due the difficulty of creating

and maintaking such a system for a penod of time, an aqueous solution was chosen as the

practical alternative.

4.5.3 Preparation of the Aqueous Solvent

Al1 water used in the cleaning process and that to be used as the solvent was purified

by first being passed through a (BarnsteadTM high capacity) de-ionizing filter and then dis-

tilled in an ail-glas still. The water was then further purified in a (BmsteadM Nanopure)

de-ionizing filter and finally passed through a 0.2 micron filter. Using glass capillary tubes

cleaned as descnbed below. the capillary rise of the water at 25 OC was measured using a

cathetometer. The value of surface tension cdculated from this rneasurernent was 7 1.2

mwm. This value is within 1.3% of the literature value. The rneasured resistivity of the

water was 18 Mn-cm.

4.5.4 Surface Tension of the Aqueous Solution

As discussed in the following section, the quartz containers to be used in the experi-

ment were sealed using (Varianm Torr Seal) epoxy. The surface tension of the solution in

the test systems could not be measured directly because of the small quantity of liquid

available. Therefore. expenments were performed in which the surface tension of the sol-

vent in contact with the cunng epoxy was measured. A quantity of uncured vacuum epoxy

and a quantity of purified water were placed into four clean glass containers (sec below)

which were sealed with screw-on caps. The ratio of the epoxy to water was 1 to 4 by vol-

urne. This duplicates the ratio to be used in one of the experimental systems. Zn the other.

the ratio was 2 to 4 (see below).

The surface tension of the solution in al1 systems was measured using the capillary rise

method. Five capillaries were used for the measurements. and they were cleaned. as

descnbed in the following section, between each set of measurements. Surface tension

measurements were made shonly after filling the systems. and periodically during the fol-

lowing eleven days. The values of the surface tension (mean f SD) of the solution may be

seen in Fig. 4.12. The aqueous solution and the epoxy were allowed to equilibrate until

consecutive measurements were the same, to within the experimental error. The value of

surface tension of the test systerns from day-7 to day-1 1 was 40.4 21.4 rnNlm when mea-

sured at 25°C.

In order to quantifi the effect of temperature on the surface tension. the systems were

then imrnersed in a thermal bath maintained at 35 OC and kept at this condition for a period

of 24 hours. The surface tensions were then measured, and the value was indistinguishable

from that at 25°C.

Two systems that were to serve as controls were prepared. They contained only water

and the values of their surface tensions did not change measurably over the eleven day

period.

4.5.5 Preparation of the cylindrical containers

The containers that were to be used in the experiment were made of quartz tubes. with

an inner diarneter of 1.2 mm. As may be seen in Fig. 4.13, the containers consisted of a

central section, sealed at each end, and two branches attached to the main tubing section.

The branches allowed liquids to be injected into and removed from the maiii tubing sec-

tion. The total volume of each container was approximately 200 mm-'.

Before the solvent was injected. the containers were cleaned carefully using a proce-

dure similar to that of Ref. 19. The tubes were first cleaned with acetone (Aldrich HPLC

grade), with detergent (Alconox biodegradable) and finally with a mixture of sulfuric and

chrornic acid ( FisherTM Chromerge).

One branch of the quartz tube used in System-1 was then sealed by fusing the quartz

in a hydrogen-oxygen fiarne. The quartz tube of System-2 was prepared in the same way.

except that one end of this tube was sealed with vacuum epoxy. and allowed to cure in a

nitrogen environment. This difference in sealing procedure allowed different amount of

epoxy to be used with the two containers.

In preparation for filling the quartz tubes. punfied water was saturated with nitrogen

by bubbling nitrogen through water in a clean glass column. The surface tension of this

nitrogen-saturated water was measured by the capillary nse method, and was found to be

indistinguishable from that of the punfied water. Once the water had been saturated with

nitrogen, contact with air was minirnized. The quartz tubes were filled by injecting the

water with a syringe, leaving a gas region between 3 mm and 8 mm in length. Although

the small gas region was initially made up of atmospheric air, we will approximate it as

consisting initially of nitrogen and water vapor at one atmosphere total pressure.

The remaining end of System-1 was seaied by plugging with a Teflon plug that was

held in place with vacuum epoxy, while the remaining end of System-2 was sealed with

vacuum epoxy. System-1 and -2 were then placed in a bath maintained at a constant tem-

perature of 35°C with a (MGW-LaudaTM mode1 S-1) thermostatic heater, and allowed to

evolve toward equilibrium, see Fig. 4.13. It should be noted that the area of contact

between the solvent and the epoxy and the arnount of epoxy was larger in the first set of

containers. This difference will be discussed further below.

The constant temperature bath was filled with a mixture of coconut and sunflower oils

that was prepared so as to have the same index of refraction as the quartz containers. The

surrounding container was rectangular and made from clear Plexiglasm so that the evolu-

tion of the systems could be observed.

Photographs of the systems were taken periodically using a (Cohuw) CCD camera fit-

ted with a close-up lens. The images were recorded with a cornputer (Macintoshnf Quadra

840 AV) and were measured using Image v.1.45 software from the National Institutes of

Health. The scale for the photographs was determined by photographing a rnicrometer cal-

iper set to several different sizes. The resulting images had a resolution of approxirnately

120 pixels per millimeter.

4.5.6 Bond Number of the Experimental Systems

In order for the systems to rernain in the two-interface configuration for an extended

time period, the gravitational forces must be small relative to the surface tension forces.

This requires that the Bond number (a ratio of these forces) be small compared to unity. In

the expenment reported herein, a capillary tube of 0.6 mm radius was used as the cylindri-

cal vesse1 and an aqueous solution with surface tension of approxirnately 40.4 mN/m was

used. This results in a Bond number of 0.09.

If the Bond number is less than 0.08, it has been previously shown in Section 4.1.2 that

the curvanire of the liquid-vapor interface deviates from the spherical by less than 1.5%

(see Section 4.1.2) Thus. if the predictions of Section 4.4 are correct, the upper and lower

interfaces of the system would be approximately spherical, but they would each corre-

spond to different spheres. In the analysis reported below, the deviation from s p h e ~ c d will

be taken into account.

4.5.7 Resdts and Discussion

Photographs of Systems 1 and 2, as a function of time, are presented in Figs. 4.14.

4.16,4.17 and 4.19. The enclosed volume in each image is the vapor region. while the liq-

uid regions extend above and below the vapor, see Fig. 4.13. The interfaces are concave

toward the vapor.

The value of the contact angle corresponding to a measured interface height. from the

three-phase line dong the center line, was determined by calculating numerically the

interface shape. The contact angle of an interface detemiined from such a rneasurement

will be referred as the "measured" contact angle. The numerical method used was

described in Section 4.4.

We report the resuits of repeating an experiment four times with the only difference

between experiments being the method of sealing the two quartz containers. The surfaces

of these system were al1 found to be uniform (see below). Other systems were prepared

and observed. but when they were examined their surfaces were found to be non-uniform.

4.5.7.1 System-1

A senes of photographs depicting the evolution of System-1 may be seen in Fig. 4.14.

and a plot of the measured contact angles of this system are shown in Fig. 4.15-A. The

error bars in these measurements arise from two factors: resolution of the digitized photo-

graphs, and different values of contact angle at different locations on an interface. It rnay

be seen from Fig. 4.15-A that the system evoived to a configuration in which the contact

angles no longer changed. Since no hirther changes in the system were observed. we

assume that equilibnum had been reached. The contact angle at the upper interface was

observed to be less than that at the lower. This relation between the contact angles is con-

sistent with that expected from the theory developed in Section 4.4.3 The observed contact

angle hysteresis between lower and upper interfaces was measured to be 34" I 4". see

Table 4-1.

TABLE 4- 1. Measured and predicted quantities

System- 1 System-2 System- 1 after inversion System-2 after inversion

Measured 4.9 1 B.04 4-91 a.04 3.19 '0.04 3.15 M.04 separation, mm

Measured 0.22 N.03 0.20 M.02 0.23 fl .02 0.24 s . 0 2 height. q, mm

Predicted 0.21 M.02 O. 19 M.02 0.29 M.02 0.29 a . 0 2 height, q mm

Predicted 8, " ' + 30 54 f: 3" 38 k 3" 39 f 3"

Measured 34 39" 29 +4" 45 +4" 45 +4" hysteresis

Predicted 36 +rO 31 fi0 35 fi0 36 fi0 hysteresis

To investigate the possibility that the difference in contact angles between the upper

and lower interface was caused by non-uniformity in the adsorption of the solutes. the

quartz container was inverted and the systern allowed to evolve to equilibrium once again.

The evoiution of this system may be seen in the photographs in Fig. 4.16, and in the plot of

the contact angles of this system in Fig. 4.15-B. As seen in these figures. the contact angle

of the upper interface decreases with time, while that of the lower interface increases. Val-

ues of the contact angles over the last eleven days are the same to within experimental

error which suggest that the system had reached equilibrium in the new orientation.

Values of the measured contact angles and interface heights may be seen in Table 4- 1.

The observed contact angle hysteresis between lower and upper interfaces 98 days after

the inversion was measured to be 29' f 4". nius, there does not seem to be a significant

difference between the contact angle hysteresis in the two orientations. However, there

was a small difference in the value of the contact angles of the upper interface, of at least

4 O , in the two orientations.

To examine these results quantitatively, we will make use of the analytical procedures

described in Section 4.4.3 As seen there, the contact angle of the lower interface rnay he

predicted from measurements of the height of the upper interface and the separation of the

two interfaces. The error bars in this prediction are based on the sensitivity of the predic-

tion to errors in the measurement of the upper interface height and the separation of the

interfaces. The predicted values of these quantities, based on a surface tension of 40.4 mN/

m, may be compared with the values rneasured under equilibnurn conditions and listed in

Table 4- 1. As seen there, there is no significant difference between the measured and pre-

dicted hysteresis in either orientation. The value of surface tension used in these prediction

is same as the value measured after allowing the same relative quantities of water and

epoxy to equilibrate (see Fig. 4.12).

4.5.7.2 System-2

The agreement found between the predicted and measured values of the contact angle

hysteresis in System-1 resulted from using the value of the surface tension measured in

the water exposed to the epoxy. To examine this funher, in system-2 a larger relative

quantity of epoxy was used. Thus, if the epoxy is the source of the surfactant that changes

the surface tension of the water, then one would expect the surface tension of the solution

used in System-2 to be smaller than that in System-1.

The evolution of System-2 may be seen in Fig. 4.17, and the contact angles of the

interfaces are presented in Fig. 4.18-A. It may be seen from Fig. 4.18-A that the system

evolved to a configuration in which the contact angles no longer changed. Since no further

changes in the system were observed, we assume that equilibrium had been reached.

These values of the contact angles are listed in Table 4-1, on page 1 18. Note that the con-

tact angle at the upper interface is less han that at the lower, consistent with the theory

developed in Section 4.4.3.

After these values of the contact angle had been recorded. System-2 was inverted and

allowed to evolve to equilibrium again. The evolution of the system after having been

inverted is depicted in Fig. 4.19. Measured values of contact angles may be seen in

Fig. 4.18-B. After roughiy fifty days, the inverted system appears to have reached equilib-

rium. since the contact angles were no longer changing. The rneasured values of the con-

tact angles and interface heights 72 days after the inversion of System-2 rnay be seen in

Table 4- 1.

Note that the measured contact angles in the two orientations are indistinguishable.

and that the measured separation distance between the interfaces is the same in both orien-

tations. This supports the assumption that equilibrium existed in each orientation when the

final values of the contact angles were measured. Thus. the systern configuration appears

to be independent of the orientation of the container, and there is no indication of surface

non-uniformity playing a role.

Following the procedure descnbed in Section 4.4.3, the measured values of the contact

angle of the upper interface and separation of the interfaces were used to predict both the

height and the contact angle of the lower interface in each orientation. These values may

be seen in Table 4-1. These predictions were based on the surface tension rneasured 264

hours after the exposure of the solvent to the epoxy, see Fig. 4.12. As seen in Table 4- 1. for

both otientations, the predicted hysteresis is approximately 10" smaller than the measured

values.

The measured values of the heights of the upper and lower interfaces and of the inter-

face separation may be used to determine the apparent surface tension of the liquid-vapor

interface. For a hypothesized value of surface tension. the shape of the upper interface

may be calculated from the interface height as descibed earlier. The measured value of the

separation between the interfaces is then used to calculate the predicted value of the height

of the lower interface. If this value does not correspond to the measured height. a new

value of surface tension is hypothesized and the calculation repeated until the predicted

and measured heights are equal.

Using this procedure. it was found that the iiquid-vapor surface tension corresponding

to the measured configuration in System-2 was 26 mN/m . This prediction of the surface

tension is lower than the value of 40.4 mN/m measured for the ratio of water to epoxy cor-

responding to System-1. A lower value of the surface tension is consistent with the larger

quantity of epoxy used in System - 2. This is also consistent with the hypothesis that the

epoxy provided a surfactant in the systerns.

To determine whether the large droplet seen in Fig. 4.14 is in equilibrium with the

other phases. its shape was compared with that of an interface that satisfies the necessary

conditions for equilibrium. One of the shapes that satisfies the necessary condition for

equilibrium is a modified catenoid. see Section 4.3.2 on page LOO. This shape may be

specified by assigning the values of the radii of curvature at a point on its surface. The val-

ues of the radii determine the contact angles at al1 points on the boundary and also deter-

mine the droplet thickness.

A photograph of the largest droplet (in side elevation) may be seen in Fig. 4.6-B. dong

with the same view of a modified catenoid calculated using the techniques descnbed in

Section 4.3.2.where the radii at the starting point for the calculation were chosen such that

the thickness of the catenoid was the same as that measured for the droplet. The vertical

lines in this figure are the cylinder walls, while the lower curved line is a "lower" interface.

with liquid beiow the curve. In addition to the large catenoid-shaped liquid droplet, several

small droplets are also visible in the vapor region.

This calculated catenoid has a contact angle of 52" f 3 O , which agrees closely with the

measured contact angle for the lower interface, 49" rt2" (see Table 4-1). Since this surface

satisfies Eq. (2.1), the Laplace equation, and is isothermal. there is no chernical potential

difference to cause rnass transport between the droplet and its surroundings. Thus this

droplet appears to be near an equilibrium shape.

4.5.8 Conclusions

The pnmary prediction of the analysis given in Section 4.4.3 is that under equilibriurn

conditions in the two interface configuration. the contact angle of the lower interface is

greater than that of the upper. This prediction is supported by the experimental observa-

tions reported herein. Under equilibnum conditions, the measured contact angles of the

lower interfaces were always larger than those of the upper interfaces. Further, since the

observed contact angle hysteresis was essentially the same in both system orientations.

this contact angle hysteresis does not appear to result frorn surface inhomogeneity. surface

roughness or non-uniform adsorption. The difference in contact angle between the upper

and Iower interfaces cannot be explained by line tension effects, since the constant length

of the contact line in this configuration does not allow line tension to affect the contact

angle of the experimental system.

In the exarnination of the surface tension of water that was exposed to epoxy, it was

found that after equilibrium had been reached, the surface tension was reduced to 40.4

rnN/m at the temperature of the expenment. When this value of surface tension was used to

predict contact angle hysteresis in System-1, it was found that the predictions were in

agreement with the observations. In System-2, the water was exposed to a larger quantity

of epoxy and it was found that if a lower value of the surface tension was used in the pre-

dictions then good agreement was found with the measurements.

Contact angle hysteresis in a two-interface system results from including Eq. (2.1) in

the prediction of the equilibrium configuration when the system is in a gravitation field.

There may be other sources of contact angle hysteresis, but the pressure field predicted

from Eq. (2.1 ) is sufficient to produce contact angle hysteresis.

4.6 Chapter Conclusions In this chapter, the necessary conditions for equilibrium have been applied to single-

component and two-component Ruid systerns in a gravitational field. and, not surprisingly.

it was shown that pressure gradients are a necessary consequence of the field. iMore inter-

estingly, it was found that when multiple liquid regions are in equilibnum at different ele-

vations in a closed system. the difference between the pressures of the liquid regions is the

sarne as that which would have existed if the intervening region were filled with liquid.

The conditions for equilibrium were used to derive expressions for the shape of an axi-

symmetric liquid-vapor interface in a gravitational field. Expressions for the modified

catenoid interface were also derived for configurations that were axi-syrnmetnc and for

other configurations that were not. These interface shapes were found to resemble closely

some interface shapes observed in experimental investigations.

The relation between the pressures of separate liquid regions was used to predict the

difference in the curvature of the two liquid-vapor interfaces in the two-interface configu-

ration, and this curvature difference was found to give rise to contact angle hysteresis. This

hysteresis was found to be a direct result of applying the classical thermodynamic treat-

ment of a system, and did not involve line tension, curvature effects, or surface inhomoge-

nei ty.

The analysis was used to predict the difference between the contact angles of two sep-

arate interfaces. for given measurements of one interface height and of the separation

between the interfaces. The predictions of contact angle hysteresis were tested in an exper-

imental program, and it was found that at equilibrium the contact angle of the lower inter-

face was always larger than that of the upper, and the measured values of the hysteresis

could be explained by the theoretical analysis.

4.7 Figures

FIGURE 4.1. Geometry used for calculation of interface shape in the presence of gravity.

Lower 1 hterfaces

FIGURE 4.2. Sample interfaces calculated from numerical integration of the differential equations, showing the effect of the Bond nurnber on the interface shape of lower interfaces (A) and upper interfaces (B). The gravitational acceleration is directly proportional to the Bond number. These calculations are for systems with only one interface.

0.25 0.5 0.75 1 1.25

Tuming Angle, Q, radians

Turning Angle, @, radians

FIGURE 4.3. The ratio of the numericai solution of the exact interface equations to the zero-g solution vs. tuming angle for two values of Bond number. Contact angle 6 = 0.

Membrane, / Permeable to 2, hpervious to 1

I L 1 Gaseous Solute. 2

r\ Moveable

Liquid Solvent, 1 and Dissolved Solute, 2 v

FIGURE 4.4. System used io demonstrate the definition of a saturated system.

FIGURE 4.5. Geometry for development of the differential equations for the modified catenoid.

Liquid

Liquid-vapor boundary

FIGURE 4.6. A: Isometnc view of a rnodified catenoid calculated for zero curvature and 5 2 O contact angle at the lowest point, with mole fraction of water in the gas phase of 0.044. Contact angle varies from 49' to 54O. B: Cornparison of the observed droplet shape to a modified catenoid. In the left image. the calculated liquid-vapor interface profile has been overlaid on a photograph of the droplet. The right image is a cross section of the calculated interface.

f : P=P, - pgz

\

Pressure d Cy Linder Axis

Vapor

Liquid Izd

FIGURE 4.7. Pressure profile in a capillary system in the two-interface configuration partially filled with water.

0 /

/ 0 120 -

. 100 - Increasing 0,

L

Increasing 0,

FIGURE 4.8. Contact angle difference versus interface separation: A. for a 1.2 mm diameter cylinder and B. for a 2.0 mm diameter cylinder, each partially filled with water at 25T. The dashed portions of the curves indicate the conditions under which the pressure in the vapor phase becomes greater than the saturation vapor pressure. Curves have been plotted for upper contact angles of 0. 10, 20.30.40, and 60 degrees.

FIGURE 4.9. A: Equilibrium pressure in the vapor phase vs. upper contact angle. B: Equilibrium pressure in the Iiquid phase vs. upper contact angle. These calculations have been carried out for water in a 1.2rnrn diarneter cylinder at 25 OC. For contact angles less than 90°, the pressure is less than the saturation value, while for larger contact angles, the pressure is greater than the saturation value.

Liquid

Interface

Vapor

Three-phase Line

, Cylinder Wall

FIGURE 4.10. Liquid configurations in preliminary experiments. Each left and nght image pair comprise perpendicular views of the sarne liquid region. Two separate liquid regions are shown. The cylinder wall is barely visible as a vertical line, while the liquid forms a ring dong the cylinder wall. The three phase lines are roughly horizontal. A calculated catenoid has also ken shown, for reference.

Observed Interface

l Cdculated

FIGURE 4.1 1 . Cornparison of the liquid shape with the shape of a rnodified catenoid. The modified catenoid has a contact angle of 36" and has a ratio of RI to R, of 1 :2.

100 150 200 250 Time, hours

FIGURE 4.12. Measured values of the surface tension of the solution as a function of time. and an extrapolation.The error bars represent the standard deviation of the measurements. Inset: Measured values of the solution surface tension vs. time.

Camera

Sealed Quartz Containers

Liquid

Vapor

Liquid

E P ~ ~ Y

Containers in Oil Bath

Therrnostatic Bath

FIGURE 4.13. Schematic diagram of the quartz container after having been filled and sealed, and of the experimental layout.

day O

day 56

day 6

day 66

day 13

day 75

day 3 1

day 112

day 43

day 129

FIGURE 4.14. Evolution of System-1 containing water at 35OC in a 1.2 mm diameter quartz capillary.

+-+-$- --a---- +-- +

/ Lower Interface /

/

0

Tirne, days

Lower Interface

U pper In terface

20 40 60 80 1 O0

Days since Iniersion

FIGURE 4.15. Contact angles of System-1. Contact angles were determined from measurements of interface height, q, in a 1.2 mm diameter container at 35°C. B: Contact angles of the sarne system after having been invened.

Invert day fO day 22 day 28 day 56

FIGU

day 70

'RE 4.16. Evol

day 79

ition of Syster

day 87

n-1 after havi

day 98

3g been invert

day 50 day 60 day 70 day 80 day 90

FIGURE 4.17. Evolution of System-2 containing water at 35" C in a 1.2 mm diameter quartz capillary.

Tirne, Days

10 20 30 40 50 60 70

Days Since Inwsion

FIGURE 4.18. A: Measured contact angles of System-2. Contact angles were detedned from measurements of interface height, q, in a 1.2 mm diarneter container at 35°C with a mole fraction of water of 0.044 in the gas phase. B: Contact angles of the same system after having been inverted.

day -1 Inversion day 2 day 4 day 23

day 30 day 40 day 50 day 65 day 72

FIGURE 4.19. Evolution of System-2 after having been inverted.

CHAPTER 5: SUMMARY AND CONCLUSIONS We have presented a new derivation of the conditions for equilibrium for a closed fluid

system in a gravitational field. It was found that the Young equation is a necessary condi-

tion for equilibrium, and that it is applicable in a gravitational field. When the necessary

conditions were applied to a fluid system in the two-interface configuration in a cylindrical

container, it was found that the contact angle of the upper liquid-vapor interface must be

less than that of the lower. This prediction is contrary to that of other investigators. This

prediction does not require the inclusion of curvature effects. nor does it require the con-

sideration of line tension or surface inhomogeneity to produce this contact angle hystere-

sis.

The predicted hysteresis was obsewed in an experimentd prograrn. It was found that

the equilibrium contact angle of the lower interface was always larger than that of the

upper interface. consistent with the theoreticai prediction. This evidence indicates that a

fluid system must be considered in total, since separate liquid regions can exchange mass

through evaporation and condensation. It is the interaction of the separate liquid regions

through the chernical potential equilibnum that gives rise to the prediction of hysteresis.

This equilibnum in the mass transport across an interface has been considered negligible

by many investigators, and may be responsible for the difference between their predictions

and those of the curent work.

It was found that in the absence of gravity. severd different fluid configurations can

satisQ the necessary conditions for equilibrium, depending on the amount of Buid in the

container. A method was presented that allows the possible configurations to be deter-

mined as a function of the arnount of Buid present in a cylindncal container and of the

contact angle. An entropy analysis was presented that enables the equilibnum configura-

144

tion to be selected from among the possible confipations. It was shown that the equilib-

rium configuration depends both on the contact angle of the systern and on the amount of

Ruid present. These predictions were tested for three configurations, and it was found that

the experimental evidence supported the predictions.

APPENDM A: SHUTTLE EXPERMENT

A. 1 Introduction An experiment has been designed that will be flown on the space shuttle. in order to

answer questions raised by the drop shaft experiment. The most difficult question arising

from that work concems the effect of negative-g episodes. Recall that the Bond Number

theory would explain transitions in terms of negative gravitational magnitudes causing an

instability in the liquid-vapour interface. In Fig. 3.14. a typical accelerometer output from

the drop shaft experiments is shown. In this figure there is a large initial oscillation in the

reading which damps out after about 0.5 second. This suggests that the acceleration levels

becarne negative for a short time, and that this may have been responsible for the transi-

tions observed.

However, some transitions were observed to take place after this initial aisturbance

had finished. In Fig. A.1 the behaviour of a single cylinder partially filled with Hexade-

cane may be seen. This systern remains in the one-interface configuration until about 5

seconds of low gravity have passed. M e r this tirne, the lower interface c m be seen to be

moving downward, indicating that fluid is draining from the lower liquid region. At the

end of the period of low gravity, the liquid is seen to be in two regions, one at either end of

the cylinder with a liquid bridge between them. This behaviour indicates that a significant

t h e may be required before the Ruid will redistribute itself in the absence of vibration or

other disturbances.

Further argument for a longer-duration experiment may be found by considering the

final configurations observed in some of the drop shaft experiments. Consider the fluid

configurations shown in Fig. A.2. In this figure the final liquid configuration may be seen

for Propanol in a cylinder with zero contact angle and in a cylinder with 49' contact angle,

which is larger dian the 36' angle found by thermodynamics to detemine the Ruid behav-

iour. n ie liquid volumes and cylinder dimensions are identical. In the system with 49"

contact angle the liquid has not reached either the one- or the two-interface configuration

after ten seconds but is in some non-equilibrium configuration. This is apparent since in

the absence of a gravitational field the liquid and vapour pressures are uniform, which

requires a constant interface curvature. The interface shown in the system with 49' contact

angle has regions near the cylinder ends that are tightly curved, in contrast to the majority

of the interface, which has a larger radius of curvature.

A much longer penod of low gravity is required to determine the equilibrium configu-

ration. Although an infinite arnount of time may theoretically be necessary to find the

equilibrium state, a penod of low gravity greater than 10 seconds would provide more

convincing evidence of the equilibrium configuration. The shuttle is one of only two facil-

ities available which c m provide extended periods of low gravity. The current experimen-

ta1 program is intended to take advantage of the space shuttle to determine the equilibrium

configuration for a closed cylindncal system in the absence of significant gravitational

fields. Any transitions which might be observed on the space shuttle would not be flawed

by an initial negative-g perturbation.

A.2 Description of Experiment The experimental program is being undertaken with the cooperation of El Paso corn-

munity college, which is coordinating a group of experiments occupying a single "get-

away speciai" GAS canister. The experirnental apparatus incorporates closed, cylindrical

transparent containers, partially filled with liquid. These cylinders are held in two orienta-

tions at right angles to each other in order to avoid bias caused by any residual gravitation

vector which might otherwise be aligned with the cylinders' axis. The behaviour of the flu-

ids in the apparatus will be observed by a compact video recorder and illuminated by fluo-

rescent lights.

The cylinders are mounted inside a GAS canister and will be flown on the space shut-

tle. After a stable orbit has k e n achieved. an astronaut will turn on the apparatus. At this

tirne, an on-board micro-controller will in tum power up three accelerometers and their

associated amplifier and fiiter circuits for a three-minute warm-up and stabilization period.

After this time, the controller will sample the acceleration readings to determine the qual-

ity of the micro-gravity conditions. If the g-Ievels are below 1 û 3 go for an additional three

minutes, the experiment will begin. At this point the fluorescent lights and the video carn-

era will be started and the configuration and behaviour of the fluids recorded. The gravita-

tion levels will also be sampled durhg the recording period, and the values written to the

video tape. Even if the gravitation readings do not meet the above specification after an

extended period, the recording will commence. This will ensure that data is taken even if

there is a failure in an accelerometer circuit.

Several more complicated experiments were considered, including filling the cylinders

in space. but these experirnents were al1 rejected as too costly and complex with no com-

mensurate improvement in the quality of data obtained.

A.2.1. Cylinder Design and Preparation

Cylinder dimensions were chosen to be similar to that used in experiments on the

KC 1 3 ~ ~ ~ . Unlike these experiments, however, the ratio of cylinder diarneter to height has

been kept constant. This allows a non-dimensional analysis where diarneter is the only

parameter necessary to descnbe the change in cylinder size. The cylinder dimensions are

shown in Table A-1. The cylinders used for ths experiment were designed based on expe-

rience gained in the KC135 and drop shaft experiments. In order to avoid problems of

leakage and contamination from sealing materials, an dl-glas cylinder design was

selected. Closed g las cylinders were blow-molded by closing the ends of sections of stan-

dard Pyrex tubing.

A section of narrow tubing approximately 20 cm long and 6 mm in diarneter was

attached to the side of each cylinder. This thin tube allows fluid to be introduced into the

cylinder. After cleaning and filling was complete, the thin tube was sealed by attaching a

vacuum pump to the thin tube and heating from the outside untii the tube collapsed. The

tube was then removed, leaving only a small projection remaining where it was originally

attached.

The procedure for cleaning the cylinders was based on that of Ref. 19. The cylinders

were first rinsed with acetone, which was allowed to rernain in the cylinders for at least 24

hours. The cylinders were then drained and evacuated before introducing a solution of

detergent (Alconox biodegradable) and de-ionized distilled water. This detergent was

allowed to remain in the cylinders for at least 24 hours before being drained out. The cyl-

inders were then rinsed at least five times with distilied de-ionized water to remove the

detergent residues. Finally the cylinders were filled with a solution of Chromic and Sul-

phuric acid (Chromerge) which was allowed to remain in the cylinders for 24 hours before

being drained. The cylinders were then rinsed with distilled de-ionized water at least five

times. The cylinders were finally evacuated to remove the remaining water. At this time

the cylinders appeared clean, since the water completely wetted the cylinder surface with

no area wetting or sheeting more than any other area.

A.2.2. Apparatus Design

The space allocated for this experimental apparatus greatly increased the difficulty of

designing the apparatus. The allowed space is an arc of an annula tube with inside radius

of 6.625" and an outside radius of 9.625". The overall height allowed was 15.75" and the

arc angle is 48 degrees.

In addition to the space constraints, the entire apparatus had to withstand the vibrations

caused by the shuttle launch. The design specifications required that the apparatus with-

stand 12 g of acceleration in any direction, and have resonance frequencies greater than

35 Hz. A detailed analysis of al1 components of Our apparatus and of much of the El Paso

Cornmunity College equipment was performed to ensure that thesr specifications are met.

The final design is a curved aluminum rack which holds 9 glass jars in two orienta-

tions. Three single-axis accelerometers are also mounted and an electrical connection

panel is incorporated near the accelerometers for power input and signal output. A view of

the apparatus is shown in Fig. A.3. This view is similar to that which will be seen by the

video canera, which will be mounted on the canister axis.

The glass jars are held in place by twû rnechanisms. First. a structural silicone glazing

compound bonds the jars to the aluminum rack. A commercial product. GE Silicones

SilglazeO N, grade SCS 2500 was chosen for its excellent adhesion on both g las and ah-

minum. Second, pockets have been milled into the surfaces of the aiurninum rack. These

pockets are slightly larger than the glass jars that they hold, and once the rack has been

assembled. the glas jars cannot be removed from their places without breaking the jar.

The silicone compound holds the jars firmiy in place so that there is no glass to metal con-

tact, and any flexing of the frarne is taken up by the resiliency of the silicone.

A.23. Video monitor

The behaviour of the fluid1fiLled cylinders will be recorded on video tape. The record-

ing equipment and control electronics are being provided through the laboratory of El

Paso Community College. A small camera will be mounted at the axis of the experimental

canister, pointed toward Our apparatus. Using a mock-up of the apparanis, it has been

found that with a wide-angle attachrnent, the video camera will be able to view the entire

apparatus without panning. The signals from this camera will be recorded on a compact S-

VHS recorder.

Lighting for the expenment was chosen to provide adequate illumination and also

withstand the vibration levels of launch. Incandescent bulbs were rejected because the

strong shadows that they cast tended to obscure the interface shapes. In addition. it was

feared that the vibration of launch rnight cause failure of the filament. A fluorescent light

source was chosen for illuminating the apparatus. It was found that a six inch fluorescent

tube powered by a six volt battery provides sufficient illumination and does not cast a

strong shadow. A rigid aluminum holder for the tube ensures that it is not broken by vibra-

tion during shuttle launch, and a shielded housing for the DC to AC converter was also

designed. Two units will be used to provide excess lighting capacity and a measure of

redundancy in case of failure.

A.2.4. Acceleration IeveIs

Significant effort was expended to ensure that the gravitation levels will be measured

accurately. This is necessary since the predictions of equilibrium configuration are made

with the assumption that the gravity levels are negligible. In addition, the Bond number

theory predicts that certain negative gravitatiûn levels are required to cause a transition. In

order to measure the success of Our expenment, the accelerations must be measured accu-

rately and the zero-g level be properly defined.

With the cooperation of the manufacturer, Allied Signal, three suitable accelerorneters.

which are also temperature modeled to quanti@ the effects of changing temperature. were

acquired. The accelerometen each incorporate a temperature sensor which produces a cur-

rent proportional to the temperature. Custom accelerometer mounts were designed to

attach the three accelerometers to the experimental rack in three orthogonal directions.

The output from these accelerometers will be amplified and shifted to remove the zero-

g bias of the accelerorneters. The bias from these accelerometers is specified to drift by

less than IO" go over a penod of one year while measunng vibration levels of 25 go in a

variable temperature environment. The Iaunch of the shuttles will cause a much srnaller

drift in bias, and after consultation with Allied Signal. we have confidence that the bias

levels will closely match the values recorded dunng qualification testing.

The acceleration and temperature signals will aiso be low-pass filtered before being

sent to an 8-bit analog to digital converter. Anti-aliasing filters were designed and con-

stmcted to ensure that the frequency components of the acceleration measurements are not

corrupted by higher frequencies beyond the frequency resolution imposed by the sampling

rate. Sixth-order active Chebyshev filters were chosen for their steep attenuation dope out-

side the pass band. Component values were hand selected to optirnize filter and amplifier

performance. In Fig. A.4 a schematic of the circuit with the theoretical component values

may be seen. The first op-amp is used to convert the output current of the accelerometer to

a voltage level appropriate to the 5 volt span of the A/D converter. It also introduces a volt-

age offset to account for the bias of the accelerometer and to shift the zero point to 2.5 V

which is the midpoint of the converter's span. The following three op-amps are used to

produce a sixth-order Chebyshev filter, using Sallen-Key second order filter circuits. The

measured frequency response of the circuits may be seen in Fig. AS, which shows the

expected smdl ripple in the pass band, dong with a very sharp cutoff.

After the electronics assembly reached the laboratory in Texas, it was not ionger func-

tioning, although it had k e n operational before shippinp from Toronto. The unit was

shipped back to Toronto, where modifications were made, and then retumed to Texas.

Again, it was found that the unit was no longer functioning. Finally, a new electronics

package was designed and built by Miro Kalovsky, at the University of Toronto. that incor-

porated significant anti-static protection circuitry. This unit was sent to Texas. and is oper-

ating successfully.

An eight bit data acquisition system will be used. A higher resolution data acquisition

system was investigated since more bits of resolution would allow a larger range of g-lev-

els to be measured (both larger and smaller). The high cost of such a system was prohibi-

tive. however. Using a measurement range of f2x 1 o - ~ go, the eight bit converter will give a

measurernent resolution of 3 x 1 0 ~ go. As calculated in chapter 4 this value will have negli-

gible effect on the interface and is essentially zero. This makes the 8-bit resolution suffi-

cient for this experiment. The sampling rate of the data acquisition system will be 60 Hz.

allowing frequencies from DC to about 25 Hz to be measured. The sampling rate is limited

by the data storage system, which has a frequency range of up to 20 kHz.

The data conversion will be accomplished by a Motorola MC68HC 1 1 microprocessor.

which incorporates an 8-bit analog to digital converter. The senal digital data strearn from

the microprocessor will be encoded either directly or through a modem chip ont0 the Hi-Fi

audio channels of the video recorder (at the discretion of El Paso college) at the sarne time

that the fluid configuration is recorded on the video tracks. This ensures that the accelera-

tion data is synchronized with the video signal. In order to differentiate between the four

voltage signals, a data header wili be recorded ont0 the tape at regular intervais, establish-

ing the beginning of a group of x-, y-.z-acceleration and temperature readings. This will

overcome potentiai ambiguity problems caused by any dropouts in the recording.

After the data has been transferred to a cornputer for anaiysis, it will be necessary to

synchronize the data and the video record of the Buid behaviour. For this purpose. an LED

mounted on the apparatus in view of the camera will be tumed on bnefly at the sarne time

that the data header is recorded. This will serve as a marker on the videotape which corre-

sponds to the header in the data.

A.3 Schedule The scheduling of this project has proven problematic. owing to repeated delays in the

shuttle launch date. It was originally intended that this experiment would f o m an integral

part of my thesis. The original flight date of Winter 1993 has been postponed numerous

times. The current estimate is that this experiment will be launched in September 1997.

TABLE A- 1. Dimensions of cylinders

Outside Diameter Outside Length Volume

A.4 Figures 1

Axis of Cylinder , Liquid-Vapor Interface

FIGURE A.1. Ruid behaviour of a single cylindricai system, partially filled with Hexadecane, during the ten second period of low gravity.

Treated Cylinder 0 = 49"

Cylinder Outline

Filling Port

Untreated Cylinder e = o

FIGURE A.2. Final liquid configuration for Propanol in a cylinder with 49' contact angle, and in a cylinder with zero contact angle.

FIGURE A.3. Photographs of the apparatus.

accelerometer # 445: R1 = 20K, R.2 = 210K. R3 = 549K accelerometer # 446: R 1 = 13K, R2 = 649K, fi3 = 649K accelerometer # 449: RI = 16.2K, R2 = 2 IOK, R3 = 2 10K Op Amps: LT 1 O MU

- - - I - I I

input from accelerometer

FIGURE A.4. Eleccricai schematic of the filter circuits

- Circuit 1 Circuit 2 - Circuit 3

Frequency (Hz)

FIGURE AS. Measured frequency response of the three filter circuits. after construction.

Bashforth, F. and S. C. Adams An atternpt to test the theories of capillaq action. Cam-

bridge, Cambridge University Press. ( 1883).

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tional Congress of Surface Activity, 3, p. 125-1 30, Buttenvorths Scientific. (1957).

L. Boruvka and A. W. Neumann, "Generalization of the Classical Theory of Capillarity".

J. Chem. Phys. 46,54645476 ( 1977)

P. Concus, "Capillary Stability in an Inverted Rectangular Tank," in Symposiimz on Plvs-

ical and Biological Phenomena in a Weightless State, edited by E.T. Benedikt and R.W.

Halliburton (American Astronautical Society, Los Angeles CA, 1963) 14.2 1-37.

P. Concus and R. Finn, "Exotic Containers for Capillary Surface," J. Fluid Mech. 224.

383-395 (1991).

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processed in rnicrogravity", Proceedings, Spacebound '94, Canadian Space Agency.

l0 T. Duffar, 1. Paret-Harter and P. Dusserre, "Crucible De-wetting during Bridgman

growth of semiconductors in microgravity", J. Crystal Growth 100 (1990)

1. A. W. Elliott, C. A. Ward and D. Yee, "Bubble shapes in rotating two-phase fluid sys-

tems: a thennodynamic approach", J. Fluid Mech. 319, (1996).

l 2 A. R. Forsyth, Calculus of Variations, Cambridge University Press. Cambridge (1927)

l 3 H. Fujii and H. Nakae, "Effect of Gravity on Contact Angle". Philosophical Mag. A, 72.

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l4 W. Gerrard, Solubiliq of Gases and Liqitids, Plenum Press. NY, ( 1976)

l5 J. Williard Gibbs, Trans. Conn. Acad. III, 108-248, 1876; Recently republished in: nie

Scientifc Papers of J. WilZard Gibbs, Ox Bow Press. Woodbridge CN ( 1993). p. 2 19.

l 6 Ibid. p. 3 19.

" W. I. Herzberg and J. E. Marian, "Relationship Between Contact Angle and Drop Size".

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' * J. N. Israelachvili, Intemolecular and Surface Forces, Academic Press, San Diego.

1992

l9 G. B. Jackson Applied Water and Spentwater Chernistry: Laboratory Manrtal, Van NOS-

trand Reinhold, New York, ( 1993) p. 67.

20 R. E. Johnson, "Conflicts Between Gibbsian Thermodynarnics and Recent Treatments

of Interfacial Energies in Solid-Liquid-Vapor Systems", J. Phy. Chem., 63 1655- 1658

( 1959).

21 E. Kreyzig, "On the theory of Minimal Surfaces" in m e Problem of Plateau. ed. T. Ras-

sias, World Scientific Publishing Co., Singapore (1992)

22 D. Langbein, W. Heide and R. Grossbach, "Parabolic Flight Experiments on Wetting.

Stability and Coagulation," in Proceedings VII Eltropean Symposium on Materials and

Flitid Sciences in Microgravity, Oxford, UK, Sept. 1989, (ESA SP-295 Jan. 1990).

23 T. Li, "Hydrostatics in Various Gravitational Fields." J. Chem. Phys. 36.9.2369-2375

( 1962).

24 T. Mon, K. Goto, R. Ohashi, and A. B. Sawaoka, "Capabilities and Recent Activities of

Japan Microgravity Center (JAMIC)," Micrograv. Sci. Tech. 4 238-242 ( 1993).

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26 NASA web page: http://shuttle .nasa.gov/reference/shuuef rbiter/oms/storage . htrnl

( 1997)

27 J. T. Neu and R. J. Good, "Equilibrium Behavior of Fluids in Containers at Zero Grav-

ity," AIAA Journal, 1, (4), 8 14-8 19 (1963).

*' D. A. Peuash, R. C. Nussle and E. W. Otto, "Effect of Contact Angle and Tank Geome-

try on the Configuration of the Liquid-Vapor Interface," National Aeronautics and Space

Administration, Technical Note D-2075 ( 1963).

29 M.R. Sasges, C.A. Ward, H. Azurna and S. Yoshihara "Equilibrium Ruid Configura-

tions in Low Gravity" J. Appl. Phys. 79, 11 June 1996 pp. 8770-8782

30 M. R. Sasges and C. A. Ward, "Effect of Gravity on Contact Angle Hysteresis. An

Experimental Investigation", submitted to J. Chem. Phys., 7/97.

3' R. Sen and W. R. Wilcox, "Behavior of a non-wetting rnelt in free fall: Theoretical:'

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nent Liquid-Vapor-Ideal Solid System", J. Colloid Interf. Sci.. 49.286-290 ( 1974).

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geneous bubble nucleation and conditions for growth in a liquid-gas systern of constant

rnass and volume", J. Appl. Phys. 54 (4), 1833- 1843 (1983)

34 CC. A. Ward, D. Yee, M. R. Sasges, L. Pataki and D. Stanga. "Configurational stability of

fluid systems in near-weightlessness" in proceedings Fliïid Mechanics Plienomena in

Micrograviry, Anaheim 1992 ASME. AMD Vol. 154: FED-Vol. 142

35 C. A. Ward, D. Yee, J. A. W. Elliott and M. R. Sasges, Final report for Canadian Space

Agency contract Stabi l i~ of Multiphased jluid systems siïbjected to Variable Gravita-

tional forces, Phase i( ( 1993)

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containers during weightlessness" Proceedings Second Annual Conference of the CFD

Sociefy of Canada, ed. J. J. Gottlieb, C. R. Ethier (1 994)

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OF GRAVITY ON mm CONFIGURATION AM) CONTACT ANGLE …€¦ · The phenomenon of contact angle hysteresis is not well understood, and is often explained in terms of curvature or line