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A contact angle study at solid-liquid-air interface by laser goniometry and ADSA-CD computer simulation

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A contact angle study at solid-liquid-air interface by laser goniometry and ADSA-CD computer simulation

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Content A CONTACT ANGLE STUDY AT SOLID-LIQUID-AIR INTERFACE BY LASER GONIOMETRY AND ADSA-CD COMPUTER SIMULATION by Chun-Chi Lo A Thesis Presented to the FACULTY OF THE SCHOOL OF ENGINEERING UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree MASTER OF SCIENCE IN CHEMICAL ENGINEERING j December 1991 I i I 1 t I I Copyright 1991 Chun-Chi Lo UMI Number: EP41832 All rights reserved INFORMATION TO ALL USERS The quality of this reproduction is dependent upon the quality of the copy submitted. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if material had to be removed, a note will indicate the deletion. U M T Dissertation Publishing UMI EP41832 Published by ProQ uest LLC (2014). Copyright in the Dissertation held by the Author. Microform Edition © ProQ uest LLC. All rights reserved. This work is protected against unauthorized copying under Title 17, United States Code ProQ uest LLC. 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, Ml 48106 - 1346 L7?S UsrIJtfl * This thesis, written by under the guidance of h ± sFaculty Committee and approved by all its members, has been presented to and accepted by the School of Engineering in partial fulfillment of the re quirements for the degree of JdMiM ‘' f s e i z e s ......... C tim zC A L z - M G iffzenurq Date. f o / % y / g i A c k n o w le d g m e n t I am grateful to my advisor, Professor Wenji V ictor Chang, for his guidance, j criticism and insightful suggestions th a t have contributed to this thesis and, more j im portant, to my own understanding of surface chemistry. I thank him for training | me from ignorant to become skillful in research and for having spared the tim e ! i for th e numerous discussions w ith me over the past two years. I have had the pleasure of having known a gifted set of colleagues, past and | I present, from whom I have learnt a lot and would like to thank - Dr. Fute Chen, Dr. Chang (a visiting professor from Korea), Shuhui Peng, Sho-Chang i Sun, Xuzhi Qin, Jianfen Tsai, H shChun Wang, Carmel Sanjeevi, Janice W est, and j Chris Bailey. And I would like to thank Professor Ronald Salovey and Professor I K atherine Shing, who kindly agreed to serve on my thesis com m ittee. Finally, on the personal side, I am grateful to my friend, Ju n Chen, for his support during the course of my research. J C o n te n ts L ist O f T ables v L ist O f F igu res vii 1 G en eral In tro d u ctio n 1 1.1 Introduction ................................................................................................... 1 1.2 Work Presented ......................................................................... 3 2 T h erm o d y n a m ic A p p roach es o f C on tact A n g le S tu d ies 4 2.1 Introduction ................................................................................................... 4 2.2 The Young E q u a tio n ..................................................................................... 5 2.3 Hysteresis of Contact A n g le ....................................................................... 7 2.3.1 Nonuniform S urfaces...................................................................... 7 2.4 O ther Effects on Contact A n g le ................................................................ 9 2.5 C o n clu sio n s.................. 10 3 S tu d y o f C on tact A n g le M ea su rem en t on P M M A In traocu lar C o n ta ct L enses and A sp h a lts by Laser C on tact A n g le G o n io m etry 12 3.1 M easurem ent of Contact A n g le 12 | 3.2 Experim ental Set U p 13 ! 3.3 S u b stra te s 15 | 3.3.1 PM M A Intraocular Contact L enses 15 j 3.3.2 A sphalts 15 j 3.4 W etting Liquids 15 I 3.5 E x p e rim e n t 16 ^ 3.6 Observations and S u g g estio n s................................................................... 16 I 3.7 Results and D iscussion 20 j 3.7.1 A sphalt M a te r ia ls ................... ...................................................... 20 i i 3.7.2 PM M A Contact L e n s e s 21 ‘ 3.8 Conclusion and Future W o r k ................................................................... 23 ! 4 C o n ta ct A n g le D eterm in a tio n s from th e C o n ta ct D ia m ete r of S essile D rop s by M ean s o f a M od ified A x isy m m e tr ic D rop S h ap e i A n a ly sis 46 4.1 Introduction 46 • 4.2 Theory . 47 ' 4.2.1 Laplace Equation of Capillarity and Its Derivations . . . . 47 ! 4.2.2 Numerical Procedure in th e Com puter P ro g ra m ................... 50 m 4.2.3 Theory Application to the Curved Solid Surface ................ 51 4.3 ADSA-CD Com puter Sim ulation Program and Its M odifications . 52 4.3.1 For F lat Substrates ....................................................................... 52 4.3.2 For Curved S u b stra te s.................................................................... 53 4.4 Results and D iscussion................................................................................. 53 4.4.1 For F lat Substrates ....................................................................... 53 4.4.2 Results for Curved Substrates .................................................. 54 4.5 Conclusion and Future w o rk ...................................................................... 55 A p p en d ix A 73 ! i IV L ist O f T a b le s 3.1 A sphalts - Distilled W ater .......................................................... 25 3.2 A sphalt - F o rm am id e................................................................................... 25 3.3 A sphalt - G ly c e r o l...................................................................................... 25 3.4 First B atch of Unmodified Lenses Received From IO PT E X . . . 33 3.5 Num bers of M easurem ents on the F irst B atch of Unmodified Lenses Received From IO P T E X ............................................................ 33 3.6 First Batch of Surface Modified Lenses Received From IO PTE X 34 3.7 Num bers of M easurem ents on the F irst Batch of the Surface Modified lenses Received From I O P T E X .......................................... 34 3.8 Second B atch of Unmodified Lenses Received From IO PT E X . 35 3.9 Num bers of M easurem ents on the Second B atch of Unmodified lenses Received From I O P T E X ............................................................ 35 3.10 Second B atch of Surface Modified Lenses Received From IO PTE X 36 3.11 Numbers of M easurem ents on the Second B atch of Surface M od ified lenses Received from IO P T E X ..................................................... 36 3.12 Third B atch of Controlled Lenses Received From IO PTE X . . . 37 3.13 Num bers of M easurem ents on the T hird B atch of Controlled lenses Received From I O P T E X ............................................................ 37 3.14 Third B atch of Surface Modified Lenses Received From IO PT E X 38 3.15 Num bers of M easurem ents on the T hird B atch of Surface Modi fied Lenses Received From I O P T E X ................................................. 38 3.16 Unmodified PM M A Lenses (First Batch)- 2 fxl S o lv en ts............... 39 3.17 Unmodified PM M A Lenses (First Batch)- 4 fx 1 S o lv en ts............... 39 3.18 Unmodified PM M A Lenses (First Batch)- 6 fx\ S o lv en ts............... 39 3.19 Modified PM M A Lenses (First Batch)- 2 fx 1 S o lv e n ts ................... 40 3.20 Modified PM M A Lenses (First Batch)- 4 \x 1 S o lv e n ts ................... 40 3.21 Modified PM M A Lenses (First Batch)- 6 fx 1 S o lv e n ts .................. 40 3.22 M easurem ents on first batch of unmodified contact lenses by op erator # 1 41 3.23 M easurem ent on second batch of unmodified contact lenses by operator ^ 2 ........................................................... 41 3.24 M easurem ent on third batch of controlled contact lenses by op erator # 3 ................................................................................ 42 4.1 A Comparison of Experim ental Values of C ontact Angle Found by Various Experim ents and the calculated values obtained by A uthor ......................................................................................................... 56 v : ( I 4.2 The contact radius of the sessile drop w ith a replaced volume, V+ V ’ , and a fixed contact angle, 70.0° is calculated when the curvature of the solid surface is in the range from 0.6 cm to 4.0 cm 57 L ist O f F ig u r e s 2.1 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 Contact angle equilibrium on a smooth, homogeneous, planar, and rigid s u r fa c e ....................................................................................... 11 Experimental set up of Laser Beam Goniometry............................. 14 Contact angle vs. drop volume for four different asphalt materials in an asphalt-water system 26 j Contact angle vs. drop volume for four different asphalt materials in an asphalt-formamide sy stem ............................................................ 27 Contact angle vs. drop volume for four different asphalt materials in an asphalt-glycerol s y s t e m ........................ 28 Time dependence in contact angle m ea su rem en ts......................... 29 Time dependence in contact angle m ea su rem en ts......................... 30 i Tem perature dependence in the contact angle m easurem ents . . 31 Volume-dependency in contact angle m easurem ent: asphalts vs. distilled w ater................................................................... 32 The distribution of contact angles m easured on the fly cut sample l,an d 2 w ith volume 4 /d distilled w a t e r .......................................... 43 The distribution of contact angles m easured on th e fly cut sample 3, and 4 w ith volume 4 fil distilled w a t e r .......................................... 44 The effect of the beam diam eter on the reading of the contact a n g le ................................................................................................................ 45 Definition of the coordinate system of a sessile d r o p ..................... 58 Schem atic of volume, V, versus radius of curvature, b, for sessile j drops w ith contact radius, xc 59 j Flow chart of the com puter p r o g r a m 60 1 Definition of the coordinate system of a sessile drop sitting on a j curved solid su rface 61 I Calculated droplet curves in the m ercury - glass contact angle ; s y s te m 62 ! Calculated droplet curves in the w ater - carbon steel contact angle s y s t e m 63 j The normalized droplet curves for three different droplet sizes of i m e r c u ry 64 ' T he normalized droplet curves for two different droplet sizes of J distilled w a te r 65 j The correlation between the contact diam eter and contact angle when the volume of m ercury droplet is f ix e d ................................... 6 6 4.10 T he com parison of the droplet profiles of three different volumes of droplets on curved solid surface ..................................................... 67 4.11 T he com parison of the droplet profiles of three different volumes of droplets on curved solid surface ..................................................... 6 8 4.12 The com parison of th e droplet profiles w ith different contact an gle on curved solid s u rfa c e ...................................................................... 69 4.13 T he configuration of the sessile drop which is located off the center of the contact lens ...................................................................... 70 I ! i i | i i l i i i ! i viii I I . : ! i Chapter 1 ! General Introduction 1.1 Introduction J ! j The m easurem ent of contact angle which is formed between the fluid interface and j the restricting wall are of prim e im portance in the science of surface chem istry and j related subjects. The concept of the contact angle and its equilibrium is valuable ; since it gives a definition of the wettability. In particular, the distinct advantage , of contact angle m easurem ent is th at this m ethod is sensitive to only the surface; i molecular layer. This is not true of other, more sophisticated, surface characteriza-’ j tion techniques. Conventional goniom etry m ethods for contact angle m easurem ent' have a num ber of lim itations; typically they depend on a hum an observer’s ab ility ; I to estim ate a tangent on the sessile drop profile at the point of contact with the solid surface. Thus, angle determ ination is highly subjective.. The accuracy of conven tional techniques is ± 2 ° or more and this is adequate if only a rough estim ate of the contact angle is required. However, if accuracy is an im portant factor, alternative m ethods m ust be considered. One such m ethod was proposed by Israel [1] and was term ed Laser C ontact Angle G oniom etry (LCAG). This m ethod utilizes the beam of an inexpensive helium-neon ! laser, projected through air, to produce two scattering lines upon striking th e solid- l liquid-air interface. These two lines form an angle, the contact angle, which can be j m easured conveniently and reproductively [1] [2 ]. In particular, LCAG is less subjec- ! tive than conventional goniom etry because it is not based on hand-placed tangents ! jto curved surfaces. However, LCAG exclusively rely on reading contact angles at the j solid-liquid-air contact point. This will reduce the accuracy of contact angles since 1 the contact point m ust be precisely determ ined. One other approach of determ ining contact angles is based on the shape of a sessile drop profile. A.W . N eum ann [3] proposed two techniques: (1 ) axisym m etric drop shape analysis - profile (ADSA-P) [3] and (2) axisym m etric drop shape analysis - contact diam eter (ADSA-CD) [4][5]. ADSA-P was developed to determ ine contact angles from th e shape of axisym m et ric sessile drops. A part from local gravity and densities of liquid and fluid phases th e other input inform ation required is several arb itrary accurate coordinate points selected from the drop profile. The strategy employed is to num erically minimize an objective function which expresses the deviation of physically observed curve from a theoretical Laplacian curve, i.e., a curve satisfying the Laplace equation of capillarity (see full descriptions of Laplace equation of capillarity in chapter 4). j However, for sessile drops which possess very low contact angles, say below 30°, the ; precision of ADSA-P decreases since it becomes m ore difficult to acquire accurate I coordinate points along th e edge of the drop profile. ADSA-CD was developed for i i this situation. ADSA-CD is not restricted to angles of any particular size because it does not require coordinates along the drop profile; in stead, the drop volume and contact diam eter are required along w ith the liquid surface tension as input data. O ur objective is to develop technologies and methodologies in order to obtain the consistency of contact angles accurately and reproducibly. For this purpose, Laser Contact Angle Goniom etry was selected to be the experim ental work in directly I m easuring contact angles because it is more convenient and less subjective than i conventional goniom etry to conduct the contact angle m easurem ent. In p a rtic u la r,! it is useful for conducting the m easurem ent on curved solid substrates such as i intraocular contact lenses which are our m ajor solid substrates to be exam ined [2 ]. i On the other hand, ADSA-CD, which is less sophisticated th an ADSA-P, was taken I to develop a sim ilar com puter program as N eum ann’s to be our theoretical work. ; This work will enable us to study the num erical relationships between the drop size, \ drop profile, curved solid surface and contact angles. Later, we will discuss the two ! m ethods in detail in chapter 3 and chapter 4. 1.2 Work Presented The work presented, here is divided into four chapters. The title of the second chapter is “Therm odynam ic Approaches of Contact Angle Study.” In this p art, | we will review the basic therm odynam ic fundam entals of surface tension, c o n ta c t! angle, and the reversible work of adhesion. j The title of the third chapter of this work is “The Contact Angle M easurem ent; on A sphalts and C ontact Lenses by Laser Contact Angle G oniom etry (LCA G)’V In this p art, we will describe in detail the m ethod which we have utilized to co n -! duct the contact angle m easurem ent - Laser C ontact Angle Goniometry. This part will present the full descriptions of the experim ental setup of LCAG, the procedure !of th e contact angle m easurem ent, the experim ental observations, and th e results I and discussions. We will carefully evaluate the procedure of the experim ent, make suggestions to improve our techniques and also analyze the d ata obtained by differ ent operators in contact angle m easurem ents on asphalts and different batches of contact lenses. The title of the forth chapter is “Contact Angle D eterm ination From th e Contact D iam eter of Sessile Drops by Means of an A xisym m etric Drop Shape Analysis - Contact D iam eter”. The purpose of this part is to determ ine the contact angle o f 1 a sessile drop by num erically solving the Laplace equation of capillarity. In this | p art, we will present the theory of ADSA-CD, the m athem atical derivations of j the Laplace equation of capillarity, and the num erical procedure of the co m p u ter! program . Additionally, the modification of ADSA-CD com puter program has been 1 developed which will enable us to sim ulate the drop profiles on curved solid surfaces. : Finally, we will analyze the d ata of contact angles obtained by com puter sim ulation j and also discuss the relationships between the contact angle, droplet profile, contact diam eter, droplet volume, and gravitational force on both flat and curved so lid . surfaces. ' 3 Chapter 2 Thermodynamic Approaches of Contact Angle i Studies • t i i 2.1 Introduction A liquid in contact with, a solid will form a contact angle (Figure 2.1). If the system is at rest, a static contact angle is obtained. If th e system is in m otion, a dynam ic | contact angle is obtained. Here, static contact angles are discussed. A system at rest ‘ m ay be in stable equilibrium (the lowest energy state), or in m etastatic equilibrium ; (an energy separated from neighboring states by energy barriers). Stable equilibrium ; will be obtained if the solid surface is ideally sm ooth, homogeneous, planar, and j nondeform able; the angle formed is the equilibrium contact angle, 6e. On the other hand, if the solid surface is rough or com positionally heterogeneous, th e system m ay reside in one of m any m etastatic states; the angle formed is a m etastatic contact angle. M etastatic contact angles vary w ith drop volume, external m echanical energy (such as vibration), and the way the angle is formed (w hether by advancing or receding the liquid front on the solid). The angle formed by advancing the liquid front on the solid is term ed the ad- ' vancing contact angle, 9a. The angle formed by receding the liquid front on the] solid is term ed the receding contact angle, 0r. Advancing contact angles are usually i greater than receding contact angle when the system is in a m etastatic state. How- j (ever, the advancing and th e receding angles are identical when equilibrium contact ■ jangles are formed. M any real surfaces are rough or heterogeneous. Thus, variable I \ k contact angles are often observed. : 2.2 The Young Equation If a drop of liquid is in contact w ith a solid surface, one out of two phenom ena will occur. The liquid will either spread or stay as a drop on th e surface (Figure 2.1). In the form er case, we say th a t w etting occurs and th e contact angle of the liquid on th a t solid is 0°, while in the latter case, we have a contact angle system. T he contact i angle m easurem ents is based on the equilibrium at a three phase boundary. The three phases can be solid, liquid and vapor or solid and two im m iscible liquids. The j angle formed between the liquid and the solid surface is dependent on th e surface tension of th e liquid, 7 l v , the surface tension of the solid - vapor interface, 7 sv> and the solid - liquid interfacial surface tension, 7 s l - Young [6 ] and D upre [7] derived the following equation to describe the contact angle in term s of th e surface tensions, j I I l v co s 9 = 75V 75L (2 .1 ) j I I In Equation (2.1), 9 is the equilibrium contact angle for a drop of the given liquid j on the given solid surface. The above equation can be obtained either by a force | balance or a therm odynam ic treatm ent [8 ]. This equation has been used extensively in treating contact angle data as well as in the theoretical developm ent of contact angle. However, it is very difficult to observe the phenom ena which Young’s equation predicts. In the definition of 7 s l and -fsv , neither of which we can conveniently and reliably m easure because solid surfaces are rarely smooth and homogeneous. Tensile stresses in the solid which have penetrated from below the surface layer commonly exist and they cause the system not to be in equilibrium . Moreover the w etting liquid m ust be a neutral probe, so th a t neither physical nor chemical interactions w ith the solid can occur. A nother approach avoids specifying the field of interm olecular force between solid and liquid and instead resorts to therm odynam ics. J. W illard Gibbs > introduced the reversible work of adhesion of liquid and solid, Wa , and its relation i I to 7 5 V and 7 Sl ■ \ Wa = 7 sv + 7 lv ~ I sl (2 -2 ) j t This equation simply indicates the fact th a t the reversible work of separating I 1 the liquid and solid phases m ust be equal to the change in the free energy of the j system. In a liquid-liquid system all the surface tension term s in Eq. (2.2) can j 5 be independently m easured; hence, Wa is determ ined. However, in a solid-liquid system , neither the surface energy of the solid nor the interfacial tension can be directly m easured. The Young equation can also be derived therm odynam ically for the ideal plane solid surface of Fig. 2.1, provided th at the system is treated as one in a state of therm al and mechanical equilibrium , and the quantities Jsli 7 sv, ancl 7 lv are defined as follows: ~ ( £ L 7 tv -= ( s T , w (2 ' 5)l where F is the Helmholtz free energy of the system, A sv is the area of the solid- j vapor interface, etc., T is the tem perature, and jj,t is th e chemical potential of each '■ i com ponent in the phases present. Im plicit in this treatm en t, and also in Young’s ( I derivation, is the assum ption th a t the contact angle is nindependent of the volume i of the drop and depends only on the tem perature and the nature of th e liquid, solid, j and vapor phases in contact. j Later Bangham and Razouk [9, 10] called attention to the im portance of n o t! neglecting the adsorption of vapor on the surface of the solid phase in deriving the \ equilibrium relations concerning the contact angle. Boyd and Livingston [10] gave: I a m ore precise system to distinguish between the solid-vacuum and solid-liquid; interface. Thus, js° is the surface tension of the substrate in equilibrium at the, solid-vacuum interface, 7 sy> the corresponding term for th e interface of the solid ; w ith th e saturated vapor, and Jl v° th a t for the interface of the liquid w ith t h e ; saturated vapor. ; 7sv° — J s l — J l v « COS $ (2-6) j W a = 7 5° + 7lv° — Jsl (2-7), and hence, W a = (7 s» - 7 s v ) + 7 l v «(1 + cos#) (2 .8 ) 6 1 W hen a liquid forms a finite contact angle on a solid, the reversible work ofj adhesion of th e liquid to the solid can be determ ined by ! I W A = 72,1 / 0(1 4 - cos<9) + t v (2.9) | I I 7T = 7 so - 7 S V ° ( 2 .1 0 ) J ! where 7r is the spreading pressure of the liquid vapor on th e surface of the solid and j it is the decrease of surface tension due to vapor adsorption. It is generally accepted j th at if the contact angle is larger than 1 0 ° th e spreading pressure can be neglected, j t i 2.3 Hysteresis of Contact Angle j 1 As m entioned earlier in this section, it is very difficult to observe th e behavior th at ’ Young’s equation predicts. The reason for this lies in the fact th a t E quation (2.1) j is correct only if the solid surface is “ideal” . The physical constraints to be m et j to define a surface as “ideal” are th a t it m ust be sm ooth and characterized by j a well defined value of surface tension. On the other hand, Young’s equation is j a therm odynam ic description of the three phases boundary equilibrium , therefore; cannot account for phenom ena related to non-equilibrium situations. It has been observed th a t the contact angle th a t is obtained when th e liquid drop is advancing on a clean solid surface is larger th an the receding angle obtained when . the liquid is being w ithdraw n from the surface of the solid. The difference betw een! advancing and receding angles is called the hysteresis of the contact angle. Current theories to explain hysteresis of contact angle are prim arily based on th e concepts of surface roughness, surface heterogeneity, friction, and adsorption phenom ena. The last factor includes contam ination and non-equilibrium effects. Especially, hetero-1 geneous surfaces and rough surfaces are well known to exhibit hysteresis of contact angle and wetting. 2.3.1 N on u n ifo rm Surfaces T he Derivation given for the contact angle equation can be adapted in an empirical ! m anner in the case of a nonuniform solid surface. First, the surface m ay be rough 7 and the effect of roughness of the substrate on the equilibrium contact angle was first modeled by Wenzel [11]. Its basic idea was th a t “w ithin a m easured unit area of (rough) surface, there is actually more surface, and in th a t sense therefore a greater intensity of surface energy resulted, th an in the sam e m easured unit area on a sm ooth surface”. This fact does not involve m odifications of specific surface quantities but do modify the relative m agnitude of the vectors composing the Young equation. He proposed the following equation: K ts - 7 s l ) = 7 l v cos 0W (2 -1 1 ) where r is the so called roughness factor: , r = actualsurf ace/ geometricsurf ace (2 .1 2 ) | The subscript “W ” indicates th a t the angle appearing in Eq. (2.11) is different i from th a t of th e Young equation (2.1). The roughness factor r is always greater I than one excepts on an ideally sm ooth surface when it is equal to one. The W enzel’s j angle and the Young’s angle are related as follows: I i I C O S 6yy = C C O S Oy (2.13) From this relation, it follows th a t if a contact angle (Young’s angle) on a smooth surface is less th an 90°, roughening the surface would decrease th e observed contact angle. If the angle on a sm ooth surface is more th an 90°, roughening the surface should increase th e observed contact angle. A lternately, the surface may be composite, th a t is, consist of small patches of various kinds. The equation describing the equilibrium contact angle of a liquid drop on a two-phase surface has been proposed by Cassie [12]: cos 9c = Qi cos + Q2 cos 62 (2-14) where subscript 1 and 2 refer to the two components of the surface, Q ’s are the fractional coverages and $’s are the Young contact angles. The angle on the le ft! 1 J I 1 side is the Cassie’s angle, th a t is the equilibrium contact angle of the given liquid on th e heterogeneous surface. If the Young angle is large th an 90° and the surface is sufficiently rough, the liquid m ay trap air so as to give a composite surface effect, as illustrated in Fig. 2.3. In this case the w etting liquid does not penetrate the crevices and interacts with a surface composed both by the solid and air atm osphere. If the intrinsic contact angle of th e unw etted region is taken as 180°, then Eq. (2.14) becomes j cos 9 apparent = r Q x COS 9 X - Q 2 (2.15) \ I ] where 1 represents the w etted region and 2 the unw etted region. A possible m echa-' nism for such air trapping is suggested in the figure. If it is assumed th a t the lo cal! or true angle 9t rem ains invariant as the liquid advances over a roughness asperity, i then if 9t is large, the liquid surface can be so reentrant th a t it intercepts the next i asperity and traps air between the two. It should be emphasized th a t the equations of this subsection are quite em pirical \ and m odelistic. It is by no means clear, for exam ple, w hether for a composite j surface it is cos 9 th a t is averaged as in Eq. (2.14), or 9 , or some other function ofj 9. Roughness is not adequately defined by r, b u t is also a m atter of topology; thej same roughness in the form of parallel grooves gives an entirely different b ehavior! than one in the form of pits. j 2.4 Other Effects on Contact Angle 1 . Effect of tem perature on contact angle: C ontact angle m ay either decrease or increase w ith tem perature, depending on the relative m agnitudes of the surface entropies of the two phases. However, at tem peratures near the boiling point, the liquid surface tension decreases rapidly w ith tem perature, and the contact j i angle will rapidly approach zero. 2. Effect of solute adsorption on contact angle: W hen th e w etting liquid is a i solution, the solute m ay adsorb at the liquid-vapor, solid-liquid, and solid- j i vapor interfaces and thus affect the contact angle. Two behaviors deserve our ! attention. First, if there is no specific interaction between, the solute and the i 9 i solid, the adsorptions at liquid-vapor and solid-liquid interfaces will be equal, j since in this case the adsorption arise from repulsion between the solute and j th e solvent molecules. Second, the solute m ay also adsorb onto the solid-vapor ' interface from the vapor phase or by diffusion from the liquid front. ' 3. D istortion of liquid surfaces - M arangoni effect: Surface tension gradient (due to com position or tem perature variation) can cause local distortion of the ■ liquid surface, known as the Marangoni effect [13]. Local variations of surface tension will produce unbalanced tension, causing the liquid to flow from a 1 j lower surface tension region to a higher surface tension region. A depression ; is thus formed on the liquid surfaces [14]. ! I I I 2.5 Conclusions j It is very im portant for us to understand the therm odynam ic fundam entals and I correlations of Young’s equation, contact angle, adhesive work, and the hysteresis! of contact angle since we want to study the contact angle m easurem ents from the | experim ental point of view. First of all, Young’s equation can only be applied to : an ’ ’ideal” solid surface when the solid-liquid-air system is in equilibrium . However, it is very difficult to define an ’ ’ideal” solid surface which is perfectly sm ooth and homogeneous as well as the therm odynam ic equilibrium of contact angle system, j Especially, the solid substrates, such as asphalts and contact lenses, which were utilized to do the contact angle m easurem ents in our lab definitely do not have ’ ’ideal” surfaces. Thus, in order to obtain consistent d ata of contact angles, it is essential th a t im portance is given factors which m ay affect the contact angle values such as surface roughness, surface contam ination, heterogeneous surface, surface interactions w ith solvent, solute adsorption, tem perature, and drop distortion. An indepth knowledge and understanding about these factors will assist us to interpret the experim ental d ata and to develop the experim ental techniques.The inform ations j accrued from the experim ents of contact angle m easurem ents will throw m ore lig h t1 in the related area of surface chemistry. ! Yl v VAPOR LIQUID Figure 2.1: C ontact angle equilibrium on a sm ooth, homogeneous, planar, and rigid surface i I ■ 1 11 Chapter 3 j Study of Contact Angle Measurement on PM M A Intraocular Contact Lenses and \ Asphalts by Laser Contact Angle Goniometry i 3.1 Measurement of Contact Angle i C ontact angle m easurem ent is perhaps th e m ost widely used technique for investi- J gating the surface interactions between two phases such as liquid-liquid and solid- i liquid interfaces. The contact angle (Zc) of a sessile liquid drop on a horizontal! surface can be obtained directly by m easurem ents of th e slope of a tangent to th e j profile at a point where the liquid and solid m eet. However, the inevitably s u b -; jective quality of such m easurem ents can lead to significant errors, particularly f o r ; small contact angles. The various techniques for m easuring contact angle have been J * reviewed in detail by N eum ann and Good [11]. The m ost commonly used m ethod J is th a t m easuring 9 directly for a drop of liquid resting on a flat surface of the solid. | Zisman and co-workers [12] simply viewed a sessile drop through a com parator mi- | croscope fitted w ith a goniom eter scale, thus m easuring th e angle directly. O ttew ill | [13] m ade use of a captive bubble m ethod wherein a bubble formed by m anipula- | I tion of a m icrom eter syringe is m ade to contact th e solid surface. The contact angle m ay be m easured from photographs of the bubble profile, or directly, by m eans ' of a goniom eter telem icroscope [14]. However, these surface characterization tech- j I niques need th a t th e operator m ust place th e two tangents by hand and, thus, angle j determ ination is highly subjective. Agreem ent among different operators and dif- i ferent laboratories is not always good. A typical standard deviation of m ore th an j I _______________________________________________, ____________________1 2 J 2 degrees is expected. In 1982 Israel [1] first observed th at when a laser beam was projected at a solid-liquid-air or solid-liquid-liquid interface two beam s resulted. It can be shown th a t these two lines of scattering are norm al to the profile of the fluid and solid surface at the point of contact. From their observations a new m ethod of contact angle m easurem ent term ed Laser Contact Angle G oniom etry (LCAG) has been developed. This new m ethod does not require hand-placed tangents to curved surfaces and hence it is less subjective. It allows for the m easurem ent of contact angles on all types of surfaces utilizing very small liquid drops on samples of m ini m al surface area. T he technique is particularly useful for curved surfaces since the quality of the d ata obtained for curved surfaces is usually b etter th an th a t obtained for fiat surfaces. This is because these scattering lines are thinner if th e laser beam , i touches a small portion of the test surface. Bush Sz Huff [2] later com pared LCAG 1 w ith conventional goniom etry in doing th e contact angle m easurem ent on curved PM M A contact lenses of 6.5 to 8.9 m m radius. They found th a t for each m ate rial studied, th e angle : (1 ) was independent of the front surface radius, (2 ) was j independent of droplet volume at 2 and 10 /d, and (3) decreased w ith tim e after: drop was placed. They also suggested th a t the laser-assisted m ethod is superior to 1 goniom etry because it is less subjective and is based on scattering lines g en erated ! at the point of contact, rather th an on hand-placed tangents to curved surfaces. ! 3.2 Experimental Set Up 1 T he experim ental set up is illustrated in Fig. 3.1. A 1 m W helium -neon laser i (Newport product model U-1321p) is m ounted to a three axis tran slato r on ani optical bench to serve as the source of a beam of coherent light w ith a substantially ! parallel beam w ith a divergence less th an 1.0 m illiradian. The laser beam is passed l through a filter to absorb heat, and through three optical plano-convex lenses in : order to focus th e beam and to reduce its diam eter. The sam ple was placed at t h e ! focal point of th e optical lenses which enabled the narrow beam of light em erging! from these lenses to strike exactly at the point of contact of solid-air-liquid interface, j T he num ber of optical lenses was not relatively im portant to reduce the diam eter of i laser beam . However, a clear image of two scattering lines would be obtained if the i diam eter of laser beam was reduced as much as possible. An environm ental cham ber Environmental Cham be: He-Neon Polar Graph Paper X-Y-Z Translator Sample Stage Screen Laser Optical Lense Figure 3.1: Experim ental set up of Laser Beam Goniometry. 3.3 Substrates 3.3.1 P M M A In tra o cu la r C on ta ct L en ses j The m aterials exam ined were unmodified and modified PM M A contact lenses which were received from IO PTE X . Before the experim ents, th e lenses were cleaned by the following procedures: first, we used distilled w ater and m ethanol to clean the lens surfaces, then we rinsed th e lenses w ith distilled w ater and dried the lenses w ith i Kimwipe tissues and finally p u t them into an oven w ith the range of tem perature j varying from 30 °C to 40 °C w ith air circulation for 10 m inutes in order to dry the) solid surfaces. j i | I 3 .3 .2 A sp h a lts j | T he asphalts exam ined were: AAK, AAG, AAM, and AAD. T he asphalt substrates I were prepared by the following procedures: first, a sm all am ount of asphalt w as' placed on a clean microslide and then the slide was kept in an oven w ith tem p er-; atures in the range of 70 °C to 80 °C for five hours. It should be noted th a t at j th e tem peratures higher than 80 °C, asphalt starts to liquidize and forms a lot of i ]small air bubbles in the bulk of asphalt. Consequently, it is believed th a t this phe- nom ena will cause a change in the m olecular structure of th e asphalts. Hence i t ' should be m ade sure th a t th e tem perature does not exceed m ore than 80 °C. D uring; i i (this period of heating, asphalt m aterial was m elted and spread along the surface j of microslide. Finally, it was allowed to rest at room tem perature until th e surface' was even, and smooth. I 3.4 W etting Liquids ! T he following w etting liquids were used in our m easurem ents: distilled w ater, fo r-! i m am ide, glycerol and diidom ethane. For each liquid type, three’different volumes: 2 4 fx 1, and 6 {A were used on contact lenses and 2 /xl, 5 /x\, and 10 /xl were used on asphalts to investigate the volume-dependence property. In order to avoid the vaporization of the liquids during the m easurem ents, the sam ple (substrate and | liquid) was placed in a pre-saturated environm ental cham ber. M easurem ents were 15 recorded for six m inutes to observe the tim e-course phenom ena. In addition, tricre- syl phosphate was used to do the m easurem ent on asphalts in order to observe the phenom ena of liquid interaction w ith solid surfaces. 3.5 Experiment In order to m easure the contact angle of a liquid drop on a solid surface, th e solid substrate was first placed on the platform inside the environm ental chamber. Then the cham ber was filled w ith the liquid of study and rested, for at least ten m inutes, I for it to get satu rated w ith its vapor. During this period, the solid substrate was J able to reach th e energy equilibrium w ith the liquid vapor. Next the cham ber i was placed on a X-Y-Z translational stage w ith m icrom eter adjustm ents. Then th e ' liquid drop was placed on the solid substrate. The stage was first adjusted vertically 1 until th e laser beam im pinged on the sam ple and then adjusted horizontally u n til, the beam im pinged on the solid-liquid-air interface. W hen the laser beam was accurately aim ed at the solid-liquid-air interface .of the liquid on a solid sample, it ■ produced two lines on the screen which are theoretically norm al to the profile of| the liquid and the solid surfaces at the point of contact. T he stage was further j adjusted to maxim ize the angle formed between the two lines on the screen, due to ■the scattering of the laser beam . A fter placing of a drop on th e substrate surface, j the contact angle was m easured at 1, 2, 3, 4, 5, and 6 m in. A sheet of polar graph j j paper w ith circles of 360° w ith 1° gradations was pinned to the screen in order to j i read th e d ata directly. In all cases, m easurem ents were m ade on two opposite sides i of the liquid drop and were averaged in order to balance any differences in contact angle due to positioning of the sample or surface imperfections. I ! t i 3.6 Observations and Suggestions | t i W hile th e contact angle m easurem ents were conducted on both PM M A contact lenses and asphalts, a num ber of phenom ena which were very im portant to improve th e techniques and to. avoid system errors was observed. The observations and suggestions are listed below. 1. Common observations: (a) The exact procedure to m easure th e contact angle is described in the following. I i. The direction of the laser beam was adjusted to point at the central I point of the polar graph paper. Then the solid substrate was placed in the environm ental cham ber which was later placed on the X-Y-Z translator. j ii. T he X-Y-Z translator was m anipulated w ith the help of the microm- j eter adjustm ents, both vertically and horizontally, until the beam ; ju st grazed on the top portion of th e solid substrate. Perfection was j adjusted by the vertical line which is norm al to the tangent of thej solid surface stretched exactly on the center of th e polar graph pa- ' per. If the solid surface on which the laser beam im pinged is f la t1 and sm ooth, this line should lie on th e zero degree line of the p o la r, | graph paper. j | iii. T he drop was discharged onto th e top portion of the solid substrate, | i.e. the center of the contact lens. Then the sample stage was ad-'! justed both horizontally and vertically until the laser beam im pinged | the point of interface which gave rise to a second line which is nor- j m al to the tangent of the droplet profile. The angle between these two lines which is respectively norm al to the tangent to the solid surface and to the droplet profile at th e point of contact is equal to the contact angle. (b) It was noted th at if large portion of the beam touched either the droplet or th e solid surfaces, broad, wedge-shape lines appeared due to the w idth of the beam touching the surface. Narrow and straight lines could be produced and m easured reliably only if th e beam grazed exactly th e point of contact. In addition, if th e solid substrate was a thick transparent j m aterial, such as IO PT E X fly cut samples, then a lot portion of the laser I beam would transm it through the solid which m eant only small am ount j of laser beam touch the contact point and therefore a dim p attern of j im age resulted. . j 2. PM M A Intraocular contact lenses: (a) We suggested the use of small volumes of liquid to run th e experim ents on PM M A contact lenses because large volumes, such as 10 /A, showed a significant tendency to slide tow ard to one side of the lens which would lead to a large difference of contact angle betw een the two sides of the j drop and therefore have a large deviation from th e actual value of the | | contact angle. For instance, it was found th a t the contact angle observed \ would deviate approxim ately 4-9 degrees between two opposite sides of j th e 10 jA distilled w ater droplet on an unm odified contact lens. 1 (b) It is pertinent th a t the surface of the contact lens was kept clean and d ry .! Observations were m ade th a t if the lens was not clean and m easurem ents were taken directly; the reading was sm aller than w hat was m easured in ' a well cleaned lens. To highlight it, for a 4 fxl distilled w ater on modified i lenses, the average contact angle for uncleaned lenses was 46.5°, w h ile 1 the average contact angle for well-cleaned lenses was 68.75°. Therefore,, in order to avoid the effect of surface contam inations of contact lenses i I and to obtain consistent data, we suggest cleaning the lenses well before i l taking m easurem ents. In addition, we suspect th a t it is not enough to \ use only hydrophilic solvents such as w ater and m ethanol to clean the lenses. ! (c) Before the syringe was m ounted to a stand, due to th e lack of proper 1 ! m ounting accessories, our operators had to deliver the droplet su b jec-. tively onto the center of the contact lens. If the drop was not on the j center of contact lens, th e gravitational force would tend to pull the : drop onto one side of contact lens and therefore cause an unsym m etrical droplet shape. This would cause a large deviation in th e contact angle I which was m easured. It was very difficult for us to deliver th e droplet ! and place it exactly on the center of the contact lenses because, first, : the drop came out of the side of the needle of the syringe which we used and, second, our hands are not stable enough to place the drop exactly , at the center of the lens so the shape of the droplet was usually im perfect ■ and thus the inconsistent d ata resulted. We suggest the use of a vertical adjustable holder to overcome the above difficulty. (d) In order to deliver the drop onto the center of the contact lens, a practical m ethod has been developed. It was described below. i. T he vertical height of the syringe was adjusted until the relative dis tance between th e contact lens and the syringe was adequate enough for the placem ent of the drop. ii. th e drop was discharged. However, the drop is still suspended on the needle of the syringe. Then we carefully adjusted the sam ple stage horizontally and observed th e relative position between the drop and contact lens until a final position was determ ined. iii. Very fine vertical translation was m ade to the sample stage until th e drop touched the solid surface. This enabled a sym m etric round drop to be placed on the center of contact lens. We found this m ethod was advantageous because the ratio of failure was only tw enty percent. Sufficient practice is needed by the operator to produce consistent and reproducible data. 3. A sphalt m aterials: (a) It is very im portant, not to use oily liquids to conduct m easurem ents on asphalt substrates because they have th e tendency to interact w ith the asphalt surfaces and therefore change th e chemical and physical prop-1 l erties of asphalts. For example, tricresyl phosphate, an oily liquid, had [ I been used to m easured the contact angle on asphalts which are hydropho- » bic. It was found th a t the angle decreased very fast w ithin a few m inutes j and the color changed, indicating th a t some com ponents were e x tra c te d , from asphalt surfaces to the w etting liquid. Therefore, the hydrophilic' liquids are preferred in the study of asphalt surfaces. | I (b) It was found th a t after the m easurem ents, if the drop was removed, a circle was left by the drop on the surface of the asphalt and it was found to be proportional to the size of the drop. This m ay be due to th e gravity of the droplet or the softness of the asphalt m aterials. In case of AAD, 19 th e softest of the asphalt substrates, it was observed th a t if 15 fi 1 distilled w ater was placed on it for 5 m inutes, the drop trickled into the substrate which attrib u ted to the inconsistency of data. (c) It was found th a t if the angle was taken corresponding to th e different positions on th e droplet, the contact angle m ight be different. It is be lieved th a t this could be due to the im proper delivery of drop, and surface im perfections of asphalt substrates. 3.7 Results and Discussion j i 3.7.1 A sp h a lt M a teria ls i i 1. The value of the contact angles are relatively insensitive to the size of the drop ' for all types of asphalts (Figs. 3.2, 3.3, 3.4). T h at is,- there is no significant \ relations between the droplet volume and th e contact angle. In our figures,: bars show th e zero-tim e contact angle which were obtained by th e extrapola- > tion of th e finite tim e d ata points using th e linear regression m ethod, and the - brackets represent the standard errors of each m easurem ent. ' 2. It is found th a t for distilled w ater the order of the m agnitude of contact angle ' is: AAD > AAK > AAM > AAG; for glycerol: AAK > AAD > AAM > A AG; ’ for formamide: AAD > AAK > AAM > AAG (Tables 1.1 -1.3). The d ata; suggested th a t AAD and AAK contain fewer polar com ponents th an AAM j and AAG. The other possible factor in the contact angle m easurem ent is the softness of asphalt. It was observed th at AAD was the softest one in all types of asphalts and the STD of its contact angles was th e largest. j 3. Tim e-course observations have shown th a t th e contact angle do not change sig- j nificantly over tim e if the drops stay stable w ith asphalt and the m easurem ents were taken in a environm ental chamber. For exam ple, in the m easurem ents of distilled w ater on AAK, the contact angle changes approxim ately one degree over six m inutes .(Fig. 3.5) while in the m easurem ents of tricresyl phosphate on AAK, the angle decreases rapidly in first three m inutes and continues de creasing thereafter (Fig. 3.6). Fig. 3.7 represent the environm ental effect on 20 contact angle m easurem ents, it compares the contact angles observed in and not in an environm ental chamber. 4. For tem perature dependency of contact angles on asphalts, it has not been found th e significant relations between tem perature and contact angles due to lack of tem perature control in our laboratory (Fig. 3.7). However, it is j believed, if the asphalt m aterials were m elted when the tem perature exceeded j t 80 °C , its physical properties would be changed. It is believed th a t ,in this : case, th e value of contact angle will be different from the value when not i heated. j I T he results of contact angle m easurem ents on asphalt surfaces show th a t LCAG | I is able to distinguish different type of asphalts well when utilizing distilled w ater, | I form am ide, and glycerol as solvents which do not interact with asphalt surfaces when conducting the experim ents (Table 3.1, 3.2, and 3.3). 3 .7 .2 P M M A C o n ta ct L en ses In the following section, we will discuss the results of different batches of contact lenses obtained by different operators. In future discussions, operator # 1 , # 2 , and # 3 will represent Janice, Chris, and Chunchi respectively. Tables 3.4 - 3.15 represents the descriptions about different batches of contact lenses received from IO P T E X company and contact angles m easured by each operator are tabulated. 3.7.2.1 First Stage of E xperim ent by O perator # 1 F irst, it was found th a t for distilled w ater, glycerol and diidom ethane the order of the m agnitude of the contact angle is : unm odified PM M A < modified PM M A ; but for form am ide : unm odified PM M A > modified PM M A. In addition, for form am ide at the second m easurem ents, the order of the m agnitude of th e contact angle is modified PM M A > unm odified PM M A which is extrem ely reverse to the results obtained at the first tim e. Second, according to our m easurem ents, no significant relationship betw een contact angles w ith respect to different volume were noticed. B ut we believe th a t the force of gravity on the drop could be a contributing factor in th e contact angle m easured on the contact lenses. Theoretically speaking, more 21 accurate and consistent d ata could be obtained by using smaller droplets to run the experim ents. As in our previous discussions, the technique used to place the drop at the center of contact lens should be im proved to discover the volum e-dependent property. It was noticed th a t at different room tem peratures, it did not adversely affect the results, as the m easurem ent were taken in an pre-saturated environm ental cham ber. If the d ata is not taken under satu rated conditions, the angle decreases quickly in th e first three m inutes and continues decreasing thereafter.- On th e other hand, p u ttin g the sample in a pre-saturated cham ber is helpful for finding th e zero tim e contact angle according to the tim e dependence observations. Finally, we will present th e d ata taken on the droplet of three liquids on modified and unmodified PM M A lenses in Tables 3.16 - 3.21. 3.7.2.2 Second Stage of E xperim ent by O perator # 1 , # 2 , and ^ 3 In this section, we will discuss the results obtained by three operators, using 4 (A distilled w ater to do the experim ent on three different batches of contact lenses. O perator # 1 , # 2 , and # 3 carried out the m easurem ents on th e first, second batch of unmodified lenses, and th ird batch of controlled lenses. In order to exam ine the consistency of results, each operator m easured th e contact angle of the lens twice for two days in succession. This was done w ith a prim e idea of com paring the reading of th e different operators at different operating condition. T he results were shown in Tables 3.22 -3.42. 3.7.2.3 Third Stage of E xp erim en t by O perator 2 In this section, we will discuss the results of contact angles m easured by operator | # 2 on the fourth batch of contact lenses. Theses lenses received were m ostly fly cut I samples, i.e. the flat PM M A round disks which would be processed to be the curved j lenses in course of tim e. Due to the inconsistent data which were obtained from i I th e previous experim ents, It was of interest to us to find out if th e d ata obtained by contact angle m easurem ent on fly cut samples would be more consistent than j i on curved contact lenses. Unlike curved contact lenses, in flat fly cut samples, the j necessity of placing the droplet on the center does not occur as the surface is flat i i I and effects due to gravitational force is insignificant. Therefore, the reading taken for fly cut samples are expected to be consistent. However, the results did not agree w ith our predictions. For example, for fly cut sample 2 and 3, the range of the contact angle is varying from 65.5° to 79° (Fig. 3.12) and 64° to 80° (Fig.3.13).j B ut, from th e vast survey of a lot quantity of data, it was found th a t the most! probable contact angle of Figures 3.12-3.13 were located on 71° (sam ple 1), 70° (sam ple 2), 69.5° (sam ple 3), and 69.5° (sam ple 4). T he results of contact angle m easurem ents on intraocular lenses are not consis ten t w ith different batches and different operators. This indicates th a t it is necessary for us to find out the underlying causes for the inconsistency of data. First, it is noticed th a t the proper technique involving the discharge of droplet should be de veloped. Care should be taken th at the droplet is not allowed to fall freely from the syringe on to th e solid surface. This will cause the spreading of the droplet and will violate the equilibrium of the solid-liquid-air system . In addition, the positioning! of the droplet play a vital role. It would give rise to the undesirable errors if the position of the drop was off the center. Second, im proper alignm ent of the laser j beam would contribute to a beam which would scan a large portion of the contact! point. This would result in an unclear image on the screen. As we m entioned before, a clear image of two lines of scattering would be obtained if the diam eter of laser beam was reduced as much as possible. Last, other factors th at m ay contribute to the error of our results are the surface roughness and the nonhomogeneous surface of solid substrate. It is believed th a t the m ain reason for the inconsistent d ata obtained on fly cut samples is because th a t the surfaces of fly cut samples are not com pletely smooth. 3.8 Conclusions and Future Work LCAG is developed to conduct the contact angle m easurem ents on four types ofj asphalt m aterials and different batches of intraocular lenses. It is found th a t it is! useful for distinguishing different types of asphalt m aterials if there is no m olecular! interactions occurred at the solid-liquid interface. The d ata also suggested th at! AAD and AAK contain fewer polar com ponents th an AAM and AAG and thej standard deviation for asphalts is from 1.0° to 2.9° (Tables 3.1, 3.2, and 3.3). On the! other hand, for th e contact angle m easurem ents on introacular lenses, th e standard! I 2 3 1 deviation for contact lenses is from 0.5° to 2.2° and the inconsistent d ata shows th at it is necessary for us to improve th e existing LCAG techniques. The possible causes for the inconsistent d ata of m easurem ents on contact lenses had been discussed in th e previous subsection and the proposal for im proving the existing technique of th e contact angle m easurem ent is present in the following paragraph. F irst of all, reducing the diam eter of the laser beam would im prove the reading ! arising from the image of two lines of scattering. This could be achieved by changing th e optical lenses w ith short focal length or introducing a m icrobjective to obtain j a m inim al diam eter of th e laser beam in order to obtain a bright and sharp image! I (Fig 3.11). However, th e focal length of the m icrobjective is too short (only several m icrom eters) to enable us to conduct th e experim ent in an environm ental chamber, j C urrent w idth of the scattering line is approxim ately 2 ° on the projection screen.! Hence, it will im prove the reading much b etter if the w idth of the scattering linej can be reduced to be 1° or less by selecting the proper optical lenses. A nd this will enable us to reduce the standard deviation for contact lenses to be less th an 1°. Second, in order to understand the effect of delivery and discharging the droplet onto th e solid surface, it is recom m ended to record the movement of th e droplet from th e syringe by a video camera. And the technique of digital image processing can be utilized to exam ine the droplet shape from both top and side view. M onitoring j of this droplet fall will give us a clear idea of th e spread of the liquid on th e solid surface. T hird, F T IR (Fourier Transform Infrared Spectrom eter) and microscopy could be utilized to analyze a small portion of th e surface of contact lens (such as 100 m ircon) to study if the solid surface is homogeneous or not. Reading would i | be taken at different point on the same lens, and also for different contact lenses, ! to give us a thoughrogh understanding of the com position and the structure of the contact lenses. I i 1 Table 3.1: A sphalts - D istilled W ater Volume AAK AAK AAG AAG AAM AAM AAD AAD (A-l) Zc STD Zc STD Zc STD Zc STD (°) (°) (°) (°) (°) (°) (°) (°) 2 96.2 1.3 8 6 .1 1.5 88.7 1.4 99.2 2.9 5 95.2 1.7 85.4 1.9 90.9 1.4 99.1 2 .6 10 97.9 2 .6 86.7 1.4 90.7 1 .6 99.7 2.9 Table 3.2: A sphalt - Formam ide Volume AAK AAK AAG AAG AAM AAM AAD AAD w STD Zc STD Zc STD Zc STD (°) (°) (°) (°) (°) (°) (°) (°) 2 88.3 2.3 76.0 1.5 81.4 1.5 90.3 2.7 5 89.1 2 .6 77.6 1 .0 81.2 1 .6 89.2 2.9 10 86.4 2.4 76.4 1 .2 80.1 1.5 89.5 2.3 Table 3.3: A sphalt - Glycerol Volume w AAK Zc (°) AAK STD (°) AAG Zc (°) AAG STD (°) AAM Zc (°) AAM STD (°) AAD Zc (°) AAD STD (°) 2 93.6 1.6 81.4 1 .6 85.0 1 .0 8 8 .8 2.4 5 95.0 1.1 83.1 1.4 85.2 1 .0 92.5 2 .6 10 94.1 1.5 82.0 1.4 86.7 1 .2 93.5 2.5 C o m p a riso n o f C o n ta c t A ngle o f AAK. AAG. AAM. a n d AAD o n F o rm a m id e AAD AAK 1 02 AAM c r > 94 AAG 86 o 78 70 Figure 3.2: Contact angle vs. drop volume for four different asphalt m aterials in an 1 asphalt-w ater system. Above figure represents 5 observations for volum e 2, 5, and | l L O /d, error bars represent SD ’s. 1 C o m p o ria o n o f C o n to c t A ngle o f AAK. AAG. AAM. a n d AAD o n F o r m a m id e 1 0 AAD AAK 98 AAM 86 AAG 74 o 62 50 9 10 11 12 13 0 2 3 4 5 6 7 8 Figure 3.3: C ontact angle vs. drop volume for four different asphalt m aterials in an asphalt-form am ide system. Above figure represents 5 observations for volume 2, 5, and 10 fA, error bars represent SD’s. C o m p a r is o n o f C o n ta c t A n g le o f AAK, AAG, AAM, e n d AAO o n G ly cero l 1 0 AAD AAK 01 AAM c 92 < AAG 74 65 0 1 2 3 4 5 6 7 8 9 10 11 12 13 Figure 3.4: Contact angle vs. drop volume for four different asphalt m aterials in an asphalt-glycerol system. Above figure represents 5 observations for volume 2, 5, and 10 /d, error bars represent SD’s. I 1 0 5 r 1 01 97 92 c o o 88 84 80 0 A s p h a l t t y p e — A A K K D i s t i l l e d W a t e r □ F o r m a m i d e 3 4 T i m e ( m i n ) Figure 3.5: Tim e dependence in contact angle m easurem ents where th e liquids arej stable w ith the asphalt surfaces. I i i A s p h a l t T y p e — A A K 70 D i i d o m e t h a n e T r i c r e s y l P h o s p h a t e 60 JE 50 c : < o 40 o c o O 30 20 0 2 3 4 T i m e ( m i n ) 5 6 7 (Figure 3.6: Tim e dependence in the contact angle m easurem ents where th e liquids react w ith th e asphalt surfaces. 30 C o n t a c t a n g l e v s t e m p e r a t u r e 1 00 A A K A A M W a t e r W a t e r 98 96 c 88 o C J 86 84 82 80 22 24 26 27 29 31 33 34 36 T e m p e r a t u r e ( C ) Figure 3.7: Tem perature dependence in the contact angle m easurem ents: asphalts; vs. distilled water. J I 31' Contact Angle Measurement Distilled Water on AAK S .100 T i m e 2ul a Sul ^ 10u I 1 0 0 9 8 9 6 0> 9 4 i c n c 9 ? < 9 0 c 8 H < 3 8 6 8 4 8 2 8 0 Contact Angie Measurement Distilled w ater on AAM 3 Time 2ul a 5ul «*» 10ul Contact Angle Measurement Distilled Water on AAG ■ £ . 9 0 o 8 5 { ) Sul o 5ul A 10uI | Figure 3.8: Volume-dependency in contact angle m easurem ent: asphalts vs. dis tilled water. 32 Table 3.4: F irst B atch of Unmodified Lenses Received From IO PT E X Dp Grp Lot/Ser u lla UV 302-03 19.00 0410-4153 u l 2 a UV 302-03 18.00 0397-8257 u!3a UV 302-03 18.00 0397-8248 ul4a UV 302-03 18.50 0410-4293 u!5a UV 302-03 18.00 0397-8259 u l 6 a UV 302-03 19.00 0410-4169 ul7a UV 302-03 18.00 560 0382 u l 8 a UV 302-03 18.00 235 0328 ul9a UV 302-03 19.50 473 0379 u2 0 a UV 302-03 19.50 0397-8252 l f I Table 3.5: Num bers of M easurem ents on th e F irst B atch of Unmodified Lenses Received From IO PT E X Wetting Liquids Operators Distilled Water Formamide Glycerol Diidomethane # 1 # 2 #3 u lla 3 0 3 6 0 12 0 u l2 a 7 0 3 6 0 16 0 ul3a 8 0 3 3 0 14 0 ul4a 5 0 0 0 0 5 0 ul5a 14 6 6 0 18 8 0 u l6 a 24 6 6 0 18 18 0 ul7a 7 4 0 0 0 11 0 u l 8 a 4 3 0 0 0 7 0 ul9a 4 3 0 0 0 7 0 u2 0 a 4 3 0 0 0 7 0 Operator # 1 : Chunchi ; # 2 : Janice ; # 3 : Chris Table 3.6: F irst B atch of Surface Modified Lenses Received From IO P T E X Dp Grp Lot/Ser m31c UV 302-03 16.50 0391-8036 m32c UV 302-03 16.50 0391-8037 m33c UV 302-03 19.50 0384-7871 m34c UV 302-03 16.50 0391-8031 m35c UV 302-03 18.50 0410-4285 m36c UV 302-03 16.50 0391-8032 m37c UV 302-03 16.50 0391-8035 m38c UV 302-03 16.50 0391-8026 m39c UV 302-03 19.50 560 0382 m40c UV 302-03 19.50 560 0382 m41c UV 302-03 16.50 0391-8034 m42c UV 302-03 16.50 0391-8033 Table 3.7: N um bers of M easurem ents on the F irst B atch of the Surface Modified lenses Received From IO PT E X Wetting Liquids Operators Distilled Water Formamide Glycerol Diidomethane # 1 # 2 # 3 m31c 16 6 6 6 12 16 6 m32c 12 0 0 6 0 18 0 m33c 11 0 0 6 0 17 0 m34c 8 9 6 0 18 5 0 m35c 2 3 0 0 0 5 0 m36c 2 3 0 0 0 5 0 m37c 6 3 0 0 0 9 0 m38c 4 0 0 0 0 4 0 m39c 4 0 0 0 0 4 0 m40c 4 0 0 0 0 4 0 m41c practice lens m42c practice lens O perator # 1 : Chunchi ; # 2 : Janice ; #3 : Chris 34 Table 3.8: Second B atch of Unmodified Lenses Received From IO P T E X i Dp ACL(mm) Grp Lot/Ser u2 1 b UV 405-25 23.00 12.5 367 0328 u2 2 b UV 405-25 23.50 12.5 367 0328 u23b UV 405-25a 23.00 12.5 297 0328 u24b UV 405-25b 23.50 12.5 367 0328 u25b UV 405-25c 23.50 12.5 367 0328 u26b UV 405-25d 23.00 12.5 297 0328 u27b UV 405-25e 23.50 12.5 367 0328 u28b UV 405-25f 23.00 12.5 297 0328 u29b UV 405-25g 23.50 12.5 367 0328 Table 3.9: N um bers of M easurem ents on the Second B atch of Unmodified lenses Received From IO PT E X Wetting Liquids Operators Distilled Water Formamide Glycerol Diidomethane # 1 # 2 # 3 u2 1 b practice lens u.2 2 b practice lens u23b 4 0 0 0 0 0 4 u24b 10 6 6 0 18 0 4 u25b 10 6 6 0 18 0 4 u26b 4 0 0 0 0 0 4 u27b 4 0 0 0 0 0 4 u28b 4 0 0 0 0 0 4 u29b 4 0 0 0 0 0 4 Operator #1 : Chunchi ; # 2 : Janice ; # 3 : Chris l 35' _ _ j Table 3.10: Second B atch of Surface Modified Lenses Received From IO P T E X j Dp ACL(mm) Grp Lot/Ser m41d UV 405-25 23.50 12.5 367 0328 m42d UV 405-25 23.00 12.5 297 0328 m43d UV 405-25 24.00 12.5 296 0328 m44d UV 405-25 23.00 12.5 297 0328 m45d UV 405-25 24.00 12.5 296 0328 xn46d UV 405-25 23.50 12.5 367 0328 m47d UV 405-25 24.00 12.5 296 0328 m48d UV 405-25 23.00 12.5 297 0328 m49d UV 405-25 23.50 12.5 367 0328 1 Table 3.11: Num bers of M easurem ents on th e Second B atch of Surface M odified' lenses Received from IO PT E X Wetting Liquids Operators Distilled Water Formamide Glycerol Diidomethane # 1 # 2 # 3 m41d 2 0 0 0 0 2 0 m42d 10 0 0 0 0 10 0 m43d 0 0 6 0 0 6 0 m44d 12 6 9 0 18 3 6 m45d 0 0 3 0 0 3 0 m46d 0 0 0 0 0 0 0 m47d 6 6 6 0 18 0 0 m48d 0 0 0 0 0 0 0 m49d 0 0 0 0 0 0 0 Operator # 1 : Chunchi ; # 2 : Janice ; # 3 : Chris : I i ! < i i i Table 3.12: T hird B atch of Controlled Lenses Received From IO P T E X j < i i i Table 3.13: Num bers of M easurem ents on th e Third B atch of Controlled lenses' Received From IO PT E X Wetting Liquids Operators Distilled Water Formamide Glycerol Diidomethane # 1 # 2 # 3 culOl practice lens c u l02 20 0 0 0 4 16 0 cul03 15 0 0 0 0 15 0 cul04 19 0 0 0 4 15 0 cul05 20 6 6 0 8 8 16 cul06 19 0 0 0 4 15 0 cul07 17 0 0 0 4 0 13 cul08 7 0 0 0 4 0 3 cul09 10 6 6 0 22 0 0 cullO 4 0 0 0 4 0 0 c u lll 4 0 0 0 4 0 0 c u ll2 4 0 0 0 4 0 0 cull3 4 0 0 0 4 0 0 cu ll4 4 0 0 0 4 0 0 — Dp — Grp — Lot/Ser culOl c u l02 cul03 cul04 The list represents controlled lenses cul05 without specific descriptions. cul06 cul07 cul08 cul09 cullO c u lll c u ll2 cu ll3 cu ll4 Operator # 1 : Chunchi ; # 2 : Janice ; # 3 : Chris Table 3.14: T hird B atch of Surface Modified Lenses Received From IO PT E X — Dp — Grp — Lot/Ser m51e m52e m53e m54e The list represents surface modified lenses m55e without specific descriptions. m56e m57e m58e m59e m60e xn61e m62e m63e I Table 3.15: Num bers of M easurem ents on the T hird B atch of Surface Modified; Lenses Received From IO PT E X Wetting Liquids Operators Distilled Water Formamide Glycerol Diidomethane # 1 # 2 # 3 m51e 0 0 0 0 0 0 0 m52e 0 0 0 0 0 0 0 m53e 0 0 0 0 0 0 0 m54e 0 0 0 0 0 0 0 m55e 0 0 0 0 0 0 0 m56e 0 0 0 0 0 0 0 m57e 0 0 0 0 0 0 0 m58e 0 0 0 0 0 0 0 m59e 0 0 0 0 0 0 0 m60e 0 0 0 0 0 0 0 m61e 6 6 6 0 18 0 0 m62e 6 6 6 0 18 0 0 m63e 6 6 6 0 18 0 0 Operator # 1 : Chunchi ; # 2 : Janice ; # 3 : Chris ! I J I i i 38 Table 3.16: Unmodified PM M A Lenses (First B atch)- 2 /zl Solvents Tim e Distilled W ater Glycerol Formam ide D iidom ethane (min) Zc(°) Zc(°) Zc(°) Zc(°) 0 66.70 60.50 50.80 36.20 1 66.63 60.42 50.80 36.20 2 66.63 60.42 50.80 36.20 3 66.63 60.42 50.80 36.20’ 4 66.63 60.42 50.80 36.20 5 66.50 60.25 50.80 36.20 6 66.44 60.25 50.80 36.20 STD 0.77 0.82 0.70 1.89 Table 3.17: Unmodified PM M A Lenses (First Batch)- 4 /d Solvents Tim e D istilled W ater Glycerol Form am ide D iidom ethane (min) Z c (°) Z c (°) Z c (°) Zc(°) 0 66.95 61.73 51.63 34.50 1 66.80 61.67 51.63 34.50 2 66.80 61.67 51.63 34.50 3 66.80 61.67 51.63 34.50 4 66.80 61.67 51.63 34.50 5 66.70 61.67 51.63 34.50 6 66.70 61.50 51.63 34.50 STD 1 .2 0 1 .1 2 0.50 1.83 Table 3.18: Unmodified PM M A Lenses (First B atch)- 6 /d Solvents Tim e Distilled W ater Glycerol Formam ide D iidom ethane (min) Zc(°) Z c (°) Z c (°) Z c (°) 0 66.70 59.75 51.94 34.30 1 66.69 59.75 51.94 34.30 2 66.69 59.75 51.94 34.30 3 66.63 59.75 51.94 34.30 4 66.63 59.75 51.94 34.30 ■ 5 66.63 59.75 51.94 34.30 6 66.63 59.75 51.94 34.30 STD 0.81 1.08 0.62 1.63 Table 3.19: Modified PM M A Lenses (First B atch)- 2 fA Solvents Tim e Distilled W ater Glycerol Form am ide D iidom ethane (min) /c(°) Z c(°) L c(°) Zc(°) 0 70.84 63.14 49.31 37.33 1 70.63 63.06 49.31 37.33 2 70.63 63.06 49.31 37.33 3 70.63 62.88 49.31 37.33' 4 70.38 62.75 49.31 37.33 5 70.13 62.75 49.31 37.33 6 70.13 62.75 49.31 37.33 STD 0.72 0.56 1.41 1.79 Table 3.20: Modified PM M A Lenses (First B atch)- 4 /A Solvents Tim e Distilled W ater Glycerol Form am ide D iidom ethane (min) L c (°) Zc(°) Zc(°) Z c (°) 0 69.74 63.25 50.25 37.00 1 69.69 63.13 50.25 37.00 2 69.69 63.13 50.25 37.00 3 69.69 63.13 50.25 37.00 4 69.63 62.94 50.25 37.00 5 69.56 62.94 50.25 37.00 6 69.56 62.81 50.25 37.00 STD 0.89 0.53 1.75 1.54 Table 3.21: Modified PM M A Lenses (First B atch)- 6 (A Solvents Tim e D istilled W ater Glycerol Form am ide D iidom ethane (min) Z c (°) Z c(°) Z c (°) Z c(°) 0 70.96 62.98 51.13 36.67 1 70.88 62.94 51.13 36.67 2 70.88 62.94 51.13 36.67 3 70.82 62.94 51.13 36.67 4 70.82 62.88 51.13 36.67 5 70.75 62.81 51.13 36.67 6 70.63 62.81 51.13 36.67 STD 1.62 0 .6 6 0.60 1.49 Table 3.22: M easurem ents on first batch of unm odified contact lenses by operator # 1 Lens No. 12/19 18.5 °C Zc (°) 12/20 15 °C Zc(°) STD (°) ul 2a 60.5 61.5 64.5 6 6 .0 2 . 2 2 u l3 a 64.0 62.0 67.5 66.5 2.15 u !4 a 70.5 6 8 .0 6 8 .0 6 6 .0 1.60 u l5 a 68.5 67.5 69.5 6 8 .0 0.74 u l 6 a 67.0 69.5 69.0 67.5 1.03 u l7 a 69.0 66.5 63.0 - u l 8 a 68.5 69.5 69.0 68.5 0.41 u l9 a 63.0 64.5 67.5 64.5 1.63 u 2 0 a 58.0 69.0 68.5 65.0 1.56 STD 3.11 2.92 2.08 i Table 3.23: M easurem ent on second batch of unm odified contact lenses by operator ■ # 2 ; _______________________________________________________ j Lens No. 12/14 17 °C Zc (°) 12/20 17.5 °C Zc (°) STD (°) c u ll3 69.0 68.5 72.5 71.0 1.60 c u ll4 6 8 .0 66.5 68.5 68.5 0.82 uv25a 69.5 73.5 70.5 6 8 .0 2 .0 1 uv25b 70.5 70.0 70.5 66.5 1.67 uv25c 69.0 69.5 70.5 67.0 1.27 uv25d 69.5 - 6 8 .0 70.5 uv25e 69.5 67.5 65.5 69.5 1.65 uv25f 71.0 68.5 65.5 68.5 1.94 uv25g - 66.5 69.0 69.5 STD 2.23 1.42 Table 3.24: M easurem ent on th ird batch of controlled contact lenses by operator Lens No. 12/14 o O 1 2 / 2 0 17 °C STD Zc (°) Zc (°) (°) c u l0 2 71.5 70.5 68.5 68.5 1.36 cu!04 74.5 74.0 73.25 75.0 0.65 cul05 70.0 69.0 69.75 70.0 0.41 cul06 70.5 70.5 70.0 69.5 0.51 cul07 70.5 70.5 6 8 .0 68.75 1 .1 0 culOS - 71.25 67.5 68.75 cul09 68.5 70.5 68.5 68.5 0.76 cullO 105.0 102.5 94.75 92.0 culll 70.5 71.25 70.0 70.0 0.52 cul 12 70.25 71.75 69.0 69.0 1.13 S T D (+ cul 10) 10.34 9.96 8.09 7.19 STD(- cu l 1 0 ) 2.61 1.30 1.60 1.93 42 Flycut sa m p le 1 1 0 9 8 7 6 5 4 3 2 1 0 —'—1 —1 —' —1 — J 1 1 “ H 1 11 H ^ 6 0 6 2 6 4 6 6 6 8 7 0 7 2 7 4 7 6 7 8 8 0 C o n t a c t A n g l e ( d e g r e e ) I F l y c u t s a m p l e 2 10 9 8 7 6 5 4 3 2 1 6 0 6 2 6 4 6 6 6 8 7 0 7 2 7 4 7 6 7 8 8 0 0 C o n t a c t A n g l e ( d e g r e e ) Figure 3.9: T he distribution of contact angles m easured on th e fly cut sam ple l,an d • 2 w ith volume 4 [il distilled w ater. Reading was based on 35 experim ental d ata obtained from each sample. Flycut sa m p le 3 10 9 8 7 3 2 1 0 6 0 6 2 6 4 6 6 6 8 7 0 7 2 7 6 7 8 8 0 C o n t a c t A n g l e ( d e g r e e ) I l F l y c u t s a m p l e 4 10 9 8 7 6 5 4 3 2 Q | , , , t u o j 1 i B U 11 a I— t L - 6 0 6 2 6 4 6 6 6 8 7 0 7 2 7 4 7 6 7 8 8 0 C o n t a c t A n g l e ( d e g r e e ) Figure 3.10: T he distribution of contact angles m easured on th e fly cut sam ple 3, and 4 w ith volume 4 (A distilled water. Reading was based on 35 experim ental d ata obtained from each sample. Figure 3.11: T he effect of the beam diam eter on the reading of the contact angle. From the above figure, it shows th a t th e scattering lines are sharper if the diam eter j ,of th e laser beam is smaller. Circle 1 and 2 represent two different diam eters of | 'laser beam which touches th e contact point. W idths 1 and 2 represent the widths j of the scattering lines which appear on th e projection screen. j Chapter 4 Contact Angle Determinations from the Contact Diameter of Sessile Drops by Means of an Axisymmetric Drop Shape Analysis — Contact Diameter 4.1 Introduction Based on the laser-assisted m ethod, m any variables which cause th e inconsistency: of contact angle d ata have been discovered and discussed in chapter 3. In the! presence of any such variables, it will raise the standard deviation of d ata obtained j from the experim ents. In order to improve the accuracy of LCAG, it is essential toj i quantify the effect of each variable by com puter sim ulation analysis. Furtherm ore, th e analysis enables us to analyze the effect of the gravitational force and th e droplet size, to sim ulate th e droplet profile, and to study the effect of the substrate curve in contact angle m easurem ent. According to the theory of the sessile drop configuration, F. K. Skinner and A. W . N eum ann [5] proposed an approach to m easure contact angles by the A x -' isym m etric Drop Shape Analysis - C ontact D iam eter (ADSA-CD). ADSA-CD was developed to determ ine contact angles from th e shape of axisym m etric drops. A part i from local gravity and densities of liquid the only input inform ation required is the contact diam eter at the solid-liquid interface. Additionally, by th e com parison w ith : I th e other models [15] [16] in sim ulating the contact angle, there is an am ount ofj advantages in ADSA-CD: (A) This m odel has physical significance because it takes I 46 into account the liquid surface tension, the gravitational acceleration and the con tact diam eter which enables us to consider the surface interaction between the liquid and solid substrate. This will give us a com plete equilibrium scheme of the sessile drop at the solid-liquid-air interface. (B) It is able to determ ine low contact angles j which are below 30°. As the contact angle decreases, th e sessile drop becomes very I flat, m aking it very difficult to acquire the coordinates along the drop profiles or * Jto align a tangent at the point of contact w ith the substrate. ADSA-CD does not require coordinates along the drop profile; instead, the drop volume and contact diam eter are required along w ith th e liquid surface tension as input data. (C) It is not necessary to identify the apex of the drop profile, th e drop w idth, or drop height. (D) It only needs to deal w ith three first-order differential equations in com puting the value of contact angle. A com puter program w ritten in C-code language was developed to im plem ent the m ethod. The program assumes a coordinate system as shown in Fig. 4.1 w ith thej x-axis perpendicular to the axis of sym m etry of th e interface; z is m easured positive I in the direction away from th e x-axis and into th e concave side of th e interface.; This program will enable us to determ ine contact angles and to study th e effects of t gravitational force and droplet size in the contact angle m easurem ents. Additionally', it is modified to sim ulate drop profiles and to determ ine contact diam eters of sessile; drops on both flat and curved solid substrates. j f 4.2 Theory 4.2.1 L ap lace E q u a tio n o f C a p illa rity an d Its D er iv a tio n s j M echanical equilibrium between two homogeneous fluids separated by an interface can be described by the Laplace equation of capillarity. It relates the pressure difference across a curved interface to the surface tension and th e curvature of the interface. Next paragraph will carry out the derivation of the Laplace of capillarity from the fundam entals of therm odynam ics. F irst, let us consider a soap bubble in the absence of fields, such as gravitational, jit is spherical. If the radius were to decrease by dy, then the change in surface free | energy would be 8-irr'ydr. Since shrinking decrease the surface energy, the tendency | to do so m ust be balanced by a pressure difference across th e film A P, such th a t th e ' work against this pressure difference A P 47rr 2dr is equal to th e decrease in surface free energy. Thus, A P4:7rR2dr = 8irR'ydR (4-1) j or ; AP = ii (4- 2)j Equation ( 4.2) can be considered as th a t the capillary is circular in cross section and ! not too large in radius. However, in general cases, th e gravitational force can not | jbe neglected so th a t the droplet profile is not spherical. Therefore, it is necessary! jto have two principal radii of curvature to represent th e configuration of a sessile' i drop, ■ "(i + i;) = A P (4-3 ) ' where 7 is th e interfacial tension, Rj and R s represent the' two principal radii of; curvature, and A P is the pressure difference across the interface. For an axisym- m etrical fluid interface in the absence of external forces, other th an gravity, the pressure difference is a linear function of the elevation, , j ! A P = A P 0 + (A p)gz (4.4) where A Po is the pressure difference at a selected datum plane, A p is the difference; in the densities of th e two bulk phases, g is the gravitational acceleration, and z isj th e vertical height m easured from the datum plane. 1 Referring to Fig. 4.1, the x axis is tangent to th e curved interface and norm al' to th e axis of sym m etry and the origin is placed at the apex, then from Eqs. ( 4.3) and ( 4.4), | + h = t+ (4-5)| i where R\ turns in the plane of the paper and i ?2 = x f sin < f> (refer to Fig. 4.1) rotates | in a plane perpendicular to the plane of the paper and about th e axis of sym m etry;; b is th e radius of curvature at th e origin of the x-z coordinate system (i?i — R 2 — b . j at th e origin), and < j> is the turning angle m easured between th e tangent to t h e ' 48; interface at th e point (x ,z) and th e datum plane. Thus, from Eq. ( 4.5), at z = 0, A P — 2 7 / 6 , and any other value of z, the change in A P is given by A pgz. The Laplace equation can be w ritten (refer to Fig. 4.1 for the definition of geom etrical variables, x , z, s, and 0 ) in the differential form dx . — — — cos < p (4-6) as dz . A — = sm 0 (4-7) as By definition i - 2 »«>; is the rate of change of the turning angle 0 w ith respect to th e arc-length p a ra m eter, s. Hence, combining Eq. ( 4.8) w ith Eq. ( 4.5) and rearranging yields 1 I dj, = 2 ( A ^ _ s in ^ | ds b 7 x The proper boundary conditions for the above differential equations are * ( 0 ) = z(0 ) = 0(0) = 0 (4.10) For a configuration such as a sessile drop, th e angle 0 at th e point of interaction of the solid surface w ith th e fluid interface defines the contact angle. For the system considered here, for which the value of the surface tension is known, the coordinates of th e system m ay be expressed as J I x = x(s, b),z — z(s, b), 0 = 0 (5 , b) (4-11) | I I where s is th e arc length variable and b, the radius of curvature, is a geom etrical j param eter of th e system . Thus, Eqs. ( 4.6), ( 4.7), ( 4.9) and ( 4.10) form a set I of first-order differential equations for x, z, and 0 as functions of the argum ent s. \ If th e value of b is assum ed to be known, then the set of differential equations can I | be integrated to determ ine the complete shape of th e curve as well as the c o n ta c t! j angle. To solve for the proper value of param eter b, we m ake use of the volume of the drop, which is calculated as I V " = j 7rx 2dz (4-12) In this situation it is assum ed th a t the value of th e surface tension is known and th e volum e of the drop, V O L , as well as the contact diam eter, 2 X C , of the drop are m easurable by experim ental means. We express th e objective function (this is th e function to be m inim ized) as rsc(b) E(b) = / 7 tx dz — V O L (4.13) Jo i i ' [where S e is a boundary point which depends on th e value of 6 , and is equal to; |th e value of the arc length at th e contact point, i.e., th e value of s for which; '£ (s, b) — xc — 0 . The initial value of b is obtained by the assum ption th a t there is no gravitational force effected on the droplet. T h at is, according to the input param eters, it first determ ines the radius of curvature, the initial value of b, for a hem ispherical droplet | and then sta rt th e iteration of Secant m ethod until th e m inim ization of the objective function is satisfied. T he num erical procedures of ADSA-CD com puter program are . present in th e following subsection. I 4 .2 .2 N u m e r ic a l P ro c ed u r e in th e C o m p u te r P ro g ra m i The com plete flow chart of this program is illustrated in Figure 4.4. This p ro g ram ! was coded by C and it contains three m ajor functions: j i 1. Function M ain: M ain is the m ajor structure of this program . It first invokes j a function in p u t_ p a ra m eters to ask th e user to input the necessary param - I eters such as th e contact diam eter (cm), the droplet volume (m/), the gravity | constant (c m /s2), the density difference (5 /c m 3), and th e liquid surface ten sion (ergs/cm 2). T hen it invokes another function o p tim a and transfers the input param eters to o p tim a for further usage. 2. Function O ptim a: The purpose of function o p tim a is to minim ize the ob- ! jective function E(b), Eq. (4.14), and to num erically solve its root by Secant j I 5 0 i m ethod. W hile utilizing the Secant m ethod, two initial values bo and & i are needed to iterate Eq. (4.14) until the root, b, is found. Hence, th e function b _ estim a te is invoked in order to obtain two reasonable initial values of b. In b _ estim a te, an assum ption th a t there is no gravitational force affected on th e droplet is m ade. This enables us to obtain the radius of curvature from a hem ispherical sessile drop w ith the same type of liquid, droplet vol um e, contact diam eter as the applied contact angle system . Next, the initial values calculated in b _ estim a te are transferred to th e function o p tim a , then function vol_calc is invoked. 3. Function V oL calc: T he purpose of vol_calc is to calculate th e value of ar, j z, the angle < f > and the volume, vol, w ith respect to th e input param eters j and th e value of b transferred from o p tim a . T he calculated values of x a n d ; vol obtained by solving Eqs. (4.7 - 4.8) will be com pared w ith the in p u t 1 param eters, i.e., contact diam eter, X c and the droplet volume, V O L . O p tim a ' will be kept iterating the nth value of b until th e calculated and input valu es; are close enough, say, less then 10~5. A fter th e root of E{b) is found, the! contact angle can also be determ ined by solving Eq. (4.10). ‘ i The ADSA-CD com puter program is presented in appendix A. ’ 4 .2 .3 T h e o ry A p p lic a tio n to th e C u rved S o lid S urface I The theory of th e Laplace equation of capillarity is not lim ited to the curvature of! the solid surface. In fact, w ith a given radius of the solid curvature, it is able to I I determ ine th e contact angle and contact diam eter at the solid-liquid-air Interface of I a curved solid surface. F irst, let us consider a liquid drop sitting on the center of the | i curved solid surface (Fig. 4.4). The contact angle is defined as th e angle between | the two tangents to the solid and to the droplet surface at the contact point. In this situation th e replacem ent of the droplet volume V to V + V ’ is necessary for solving for th e objective function E(b). From figure 4.4, it is obvious th a t the | replacem ent of V + V ’ is able to give us a contact angle system of a sessile drop j on a flat solid surface. Then if it is applied to the com puter sim ulation program as I described in section 4.2.1 and 4.2.2, the contact angle or contact diam eter will be determ ined. 4.3 ADSA-CD Computer Simulation Programi and Its Modifications According to th e theory of Laplace equation of capillarity, three com puter program s were developed in order to obtain more inform ation from th e com puter sim ulation analysis. T he functions and num erical procedures of these program s are similar, i.e., all of them possess the same m ajor functions as discussed in subsection 4.2.2. ’ However, th e input param eters and the outp u t of the com putation are different f o r ! each program and th e applicable system s are divided into two situations, th a t is, j the sim ulation on flat and curved solid substrates. 4.3.1 For F la t S u b str a tes 1. A xisym m etric Drop Shape Analysis - C ontact D iam eter (ACSA-CD): This program is sim ilar to N eum ann’s [5]. The input param eters include drop , volume, contact diam eter, gravitational acceleration, liquid surface tension, | and density difference betw een liquid and air. T he ou tp u t of the com putation : is the contact angle. It is not only able to determ ine contact angles by solving! Laplace equation of capillarity but also to provide the com plete shape of a j sessile drop. I 2. A xisym m etric Drop Shape Analysis - C ontact Angle (ADSA-CA) This pro gram is utilized to determ ine the contact diam eter when the contact angle and volume of the sessile drop are fixed. The input param eters include droplet vol- I um e, contact angle, gravitational acceleration, liquid surface tension, and t h e ! density difference betw een liquid and air. T he o u tp u t of com putation is the 1 contact diam eter. Additionally, in order to sim ulate the relationship between j ! contact diam eters and contact angles when the droplet size is fixed, a loop is | added into the program . A fter the contact diam eter is determ ined, this loop calculates the corresponding contact angle of the contact diam eter ± a very i I small region, say 0.1 cm. It will enable us to study th e sensitivity of contact i angles in a region which is very close to th e solid-liquid-air contact point. j I ! 4 .3 .2 For C u rved S u b str a tes j i T he sim ulation program developed for the curved solid substrates is term ed as | I “A xisym m etric Drop Shape Analysis - C ontact D iam eter - Curved Surface (ADSA- I CD -CS)” . As we discussed in section 4.2.3, in order to apply ADSA-CD to a curved substrate, a void volume, V ’ is added to th e droplet volume, V. The input param - ; i eters include droplet volume, contact angle, th e radius of curvature of the solid | surface, gravitational acceleration, liquid surface tension, and th e density difference between liquid and air. T he output is the contact diam eter ( see Fig. 4.4) and it also can provide th e com plete droplet profile of th e sessile drop according to the 1 input param eter. ; I •) 4.4 Results and Discussion ; In the following, we will discuss the results obtained from ADSA-CD, ADSA-CA,j and ADSA-CD-CS. Due to the lack of the experim ental facilities and skills to ac curately m easure the value of the contact diam eter of a sessile drop, we took the ; experim ental d a ta of droplet volume and contact diam eter from Denis J. Ryley [18] j and C. Dahlgren [19]. j I i 4.4.1 For F la t S u b str a tes ! 1. ADSA-CD: j (a) T he results of the calculated contact angle and th e calculated droplet j I volum e is shown in the Table 4.1. T he values of calculated volumes and 1 calculated contact angles were obtained from au th o r’s com puter simu- j lation and th e values of experim ental contact diam eters, volumes, and ! contact angles were obtained from references [16] and [17]. In Table 4.1, ; it shows th a t th e result of th e ADSA-CD com puter analysis agrees w ith | th e d ata obtained from [16] [17]. 53 (b) Figures 4.5 and 4.6 represent th e configurations of droplet profiles for three different volumes of m ercury on glass surface and two different volumes of w ater on carbon steel surfaces. T he definition of bond num ber is: Bond num ber= A p * g * Zmax2 7 where Z max2 is th e m axim um height of the sessile drop. According to Figures 4.5 and 4.6, the droplet profiles of different droplet sizes were norm alized to analyze the effect of th e gravitational force on droplets (Fig. 4.7 and Fig. 4.8). A nd it is found th a t th e gravitational force does not significantly effect the droplet profiles of different droplet sizes (Fig. 4.7 and Fig. 4.8). j 2. ADSA-CA: For the droplet w ith a given volume, th e contact angle decreases j as the contact diam eter increases. Fig. 4.9 shows the correlations between | contact angles and contact diam eters when the droplet volume is fixed. j j j 4 .4 .2 R e su lts for C u rved S u b str a te s i \ d 1. Table 4.2 represents th e prediction of th e contact radius (contact diam eter /2) | I of a sessile drop w ith a fixed volume, 0.00675 ml, and a fixed contact angle, j 70.0°, on the curved surface whose curvature is from 0.6 cm to 4.0 cm. It shows < th a t th e contact diam eter increases as th e radius of solid surface increases. 2. Figures 4.10 -4.11 present the effect of the droplet size on th e droplet profile j in the contact angle m easurem ent. Figure 4.9 shows th e droplet profiles of I i three different volumes w ith 70.0° contact angle on th e solid substrate whose \ radius of curvature is 0.8 cm. Figure 4.10 shows th e droplet profiles of four different volumes w ith 2 0 .0 ° contact angle on the solid substrate whose radius ( of curvature is 1.6 cm. From Figures 4.10-4.11, it is found th a t th e shapes * i of different droplet sizes are slightly different and, at the contact point, th e . tangents to the droplet surfaces are the same. This indicates th a t, for the* same type of liquid, the size of droplet does not significantly effect its contact j angle when the solid surface has a regular curvature. ' (4.14) I 3. Fig. 4.12 presents the configuration of two droplets w ith different contact j angles and w ith the same droplet size when th e solid surface curvature is 0 .8 | i cm. i i I I To sum up, th e functions of th e com puter sim ulation program has been demon- } strated well. It can not only predict the contact angle and contact diam eter when j the solid surface is flat b u t also sim ulate the droplet profiles either on a flat or curved J solid surface. This will enable us to obtain m ore inform ation from the com puter J ! sim ulation. i I i j 4.5 Conclusions and Future work i ADSA-CD is useful for determ ining the contact angle by solving the Laplace equa tion of capillarity numerically. According to the theory of ADSA-CD, a C-code com puter sim ulation program was developed to analyze th e relationships between ; the contact angle, th e contact diam eter, the droplet profile, and th e effect of the gravitational force. ; ADSA-CD-CS com puter sim ulation is useful for determ ining the contact angle ' and contact diam eter of a sessile drop on the curved solid surface. T he results for j th a t the droplet sitting on th e center of th e curved solid surface has been discussed | i in the previous section. T he m ore com plicated situation is th a t the droplet is | located off the center of th e solid surface as illustrated in Figure 4.13. In this situation, it should be carefully considered th a t the gravitational force would cause a different droplet shape (not axisym m etric) and different volum e distribution (not regular) from th e case th a t the drop is on the center. T he future work will be m ostly concentrated on this situation. Additionally, it is recom m ended to utilize a video cam era to record the droplet profile both top and side view and utilize the technology of im age processing to obtain a clear image of the droplet shape. This will enable us to obtain accurate contact diam eters to conduct sim ulations and to com pare th e observed curve w ith the com putational curve in order to m inim ize the i deviation of th e contact angles. j Table 4.1: A Com parison of Experim ental Values of Contact Angle Found by Vari ous Experim ents and the calculated values obtained by A uthor Liquid No. Experi. Contact Diameter X c(cm) Experi. Volume (mi) Contact Angle Z c(°) Calcul. Volume (ml) Calcul. Angle L c(°) Mercury a 1 0.1444 0.00251 130.3 0.002508 130.69 2 0.1768 0.00483 133.8 0.004829 135.13 3 0.2382 0.01307 133.9 0.010376 135.04 4 0.361 0.02723 131.7 0.027240 135.8 Water 1 0.3496 0.00675 72.0 0.006755 71.68 2 6 0.4480 0.01350 71.3 0.013501 70.34 3 c 0.730 0 .0 1 15 0 .0 1 1 15.916 4 d , 0.650 0 .0 1 2 1 0 .0 1 0 0 0 1 22.023 ! j*Experimental data was obtained from [18] and [19], the calculated data was obtained j |by author’s computer simulation. j | a. On Glass j j b. On Carbon Steel ( ] c. On Gel Bond Film i d. On Glass Slide 1 Temp = 25 °C Water Mercury Density Difference A p 0.9882 13.534 Surface Tension 7 72.785 484 Table 4.2: T he contact radius of the sessile drop w ith a replaced volume, V + V \ and a fixed contact angle, 70.0° is calculated when th e curvature of the solid surface J is in the range from 0.6 cm to 4.0 cm. radius of solid curvature contact radius contact angle r (cm) X C (cm) zc n 0 .6 0.164066 70.0 1 .0 0.168945 70.0 1.4 0.170822 70.0 2 .0 0.173359 70.0 2.4 0.173157 70.0 2 .8 0.174273 70.0 3.2 0.175019 70.0 3.6 0.174516 70.0 4.0 0.176004 70.0 I ___ 57 ! ___ i 0 z Figure 4.1: Definition of the coordinate system s of a sessile drop w ith contact radius, ' X c, and contact point, S c. At a point (X i,Z i), the turning angle is < f > . S is the arc j length m easured along the drop. R i and R 2 are th e two principal radii of curvature; j i?i tu rn s in th e plane of th e paper, and R 2 rotates in th e plane perpendicular to i the plane of th e paper (R 2 — x j sin (f> ). j 58 I V(b0) < E > < 90 b Figure 4.2: Schem atic of volume, V , versus radius of curvature, b, for sessile drops w ith contact radius, x c. A drop w ith contact radius equal to 90° has a radius of curvature bo and a volume V(bo). For drops w ith a contact angle greater th an 90°, the radius of curvature b > bo and the drop volume is greater th an F(&o)- For drops w ith a contact angle less th an 90°, th e radius of curvature b > b0 and th e drop volum e is less th an V(b0). 59j C STAR]) MAINO INVOKE FUNCTION b_ESTIMATE() INVOKE FUNCTION: INPUT PARAMETERS INVOKE FUNCTION: CONTACT ANGLE INVOKE FUNCTION PRINT THE VALUE OF CONTACT ANGLE VOL-CalcO to determine droplet profile. FUNCTION: b.ESTIMATEO To estimate the initial values, bO, b l, by assuming that the drop is spherical. FUNCTION: VOL_CALC() To calculate the value of x, z, < 3 > , and the volume by solving three differential equations. Function: INPUTJPARAMETERSO the Contact Diameter: Xc (cm) the Gravity Constant: g (cm / s**2) the Density Difference: Dr (g / cm**3) the Liquid Surface Tension: g (erg / cm* *2) FUNCTION OPTIMAO: To use the Secant method to determine the root, the radius of the curvature, b, of die objective function E(b). INVOKE FUNCTION OPTIMAO to optimize the objective function E(b) in order to obtain the root, b. INVOKE FUNCTION VOL_CALC() To determine the values of x, z, < £ > according to the radius of curvature, b, the contact diameter, XC, and the droplet volume, VOL. Figure 4.3: Flow chart of the com puter program. 60 Contact Point XC Contact Lens Figure 4.4: Definition of the coordinate system of a sessile drop sitting on a curved j solid surface w ith droplet volume, V, and contact angle, < j > . T he replacem ent of volume, V + V ’ enables to give a system whose contact angle equals to ( < f> + 9). 0.2 (1) XC=0.1444 cm; VOL=0.00251 ml; angle=130.693 degree - (2) XC=0.1768 cm; VOL=0.00483 ml; angle=135.18 degree (3) XC=0.2382 cm; VOL=0.01037 ml; angle=137.10 degree 0.15 0.1 0.05 -0.05 - 0.1 -0.15 - 0.2 - 0.2 -0.15 - 0.1 -0.05 0.05 0.1 0.15 0.2 Figure 4.5: Calculated droplet curves in the m ercury - glass contact angle system . 1 Bond num ber: ( 1) 0.52281; (2 ) 0.82088; (3) 1.31184. In the above figure, XC and VOL represent th e contact diam eter and droplet volume obtained from [19], and angle was calculated by au th o r’s com puter program . ! 0.2 (1) XC=0.3496 cm; VOL=0.00675 ml; angle=71.59 degree 0.15 (2) XC=0.4480 cm; VOL=0.01350 ml; angle=70.51 degree 0.1 0.05 S' C O < N -0.05 - 0.1 -0.15 - 0.2 0.3 -0.3 - 0.2 - 0.1 0.1 0.2 X Axis (cm) i 1 | i Figure 4.6: C alculated droplet curves in th e w ater - carbon steel contact angle 1 system . Bond num ber: ( 1 ) 0.194348; (2) 0.295057. In the above figure, XC and[ VOL represent th e contact diam eter and droplet volume obtained from [19], a n d 1 angle was calculated by au th o r’s com puter program . i i I 63j . : XC = 0.1444 cm VOL = 0.00251 ml angle = 130.693 degree o": XC = 0.1768 cm VOL =0.00483 ml angle = 135.180 degree + XC = 0.2382 cm VOL = 0.01037 ml angle = 137.10 degree Figure 4.7: T he norm alized droplet curves for three different droplet sizes of m er-| cury. In the above figure, XC and VOL represent the contact diam eter and droplet J volume obtained from [19], and angle was calculated by au th o r’s com puter program . 64 - 0,1 - 0.2 -0.3 XC = 0.3496 cm VOL = 0.00675 ml angle = 71.59 degree -0.4 -0.5 - 0.6 "o": XC = 0.4480 VOL = 0.01350 ml angle = 70.51 degree -0.7 -0. : -0.9 -0.9 -0.8 1 -0.7 - 0.6 -0.5 -0.4 -0.3 - 0.2 0.1 0 I I I I Figure 4.8: T he norm alized droplet curves for two different droplet sizes of distilled w ater. In th e above figure, XC and VOL represent th e contact diam eter and droplet volum e obtained from [19], and angle was calculated by au th o r’s com puter program . . i t i i C o n t a c t A n g l e ( d e g r e e ) \ M ercury 190- 180- 170- 160- 150- 140- 130- 120- 110- 100- 90- 80- 70- 60- (1) VOL = 0.01037 m l (2) VOL = 0.00483 m l (3) VOL = 0.00251 m l 0.450 0.000 0.150 0.300 C o n t a c t D i a m e t e r ( c m ) Figure 4.9: The correlation betw een th e contact diam eter and contact angle when the volume of m ercury droplet is fixed. 0.16 0.14 0.12 0.1 o o.o8 < > * 0.06 0.04 0.02 0 Curvature of solid surface: 0.8 cm + + + ' . + + ■ .+ + + + + + + + + + + + 4 4 4 4 4 4 4 + + + 4 4 + Contact angle: 70.0 degree V : 0.0038 ml "o": 0.004 ml 0.008 ml 4 4 4 4 4 -0.2 -0.15 -0.1 -0.05 0 0.05 X Axis (cm) 0.1 0.15 0.2 Figure 4.10: The com parison of the droplet profiles of three different volumes of| droplets. T he curvature of th e solid surface is 0.8 cm and th e contact angle is 70.0°. Curvature of the solid stirfac^Jii^ttrti+HH*,^ ,0.05 0.04 ■ & 0.03 0.02 Contact angle: 20.0 degree 0.0038 ml "o": 0.004 ml 0.0042 ml 0.008 ml 0.01 0 - 0.2 -0.3 - 0.1 0 0.1 0.2 0.3 Figure 4.11: The com parison of th e droplet profiles of three different volumes of droplets. T he curvature of th e solid surface is 1.6 cm and th e contact angle is 20.0°. 0.3 Curvature of the solid surface: 0.8 cm 0.25 Droplet volume: 0.008 ml (1) Contact angle : 70.0 degree 0.2 (2) Contact arigle : 20.0 degree 0.1 0.05 - 0.2 0.1 0.2 0.3 -0.3 - 0.1 X Axis (cm) Figure 4.12: The com parison of th e droplet profiles w ith different contact angle. T he curvature of th e solid surface is 0.8 cm and the droplet volum e is 0.008 ml. 69 2 Contact Point 2 ContactPoint XC Contact Lens Figure 4.13: T he configuration of th e sessile which is off th e center of the contact; lens. and fa represent two different contact angles of the drop at two opposite solid-liquid-air contact points. j Reference List [1] S. C. Israel, W . C. Yang, C. H. Chae, “C haracterization of Polym ers Surface I By Laser C ontact Angle G oniom etry” , Dallas TX M eeting of ACS, Division of! Polym er Chem istry, Polymer Reprints, 30(1), 328 (1989). j [2] John F. Bush, Joseph W . Huff, “Laser-Assisted C ontact Angle M easurem ents” , j American Journal of Optometry & Physiological Optics, 65, No. 9, 722 (1989). 1 [3] F. K. Skinner, Y. Rotenberg, and A. W. N eum ann, “C ontact Angle M ea surem ent from the C ontact D iam eter of Sessile Drops by M eans of a Modified i A xisym m etric Drop Shape Analysis” , Journal of Colloid and Interface Science, i j 130, N O .l, 25 (1989). * [4] F. K. Skinner, Y. Rotenberg, and A. W. N eum ann, “C ontact Angle Mea- J surem ent from the C ontact D iam eter of Sessile Drops by M eans of a Modified I A xisym m etric Drop Shape Analysis” , Journal of Colloid and Interface Science, 130, N O .l, 25 (1989). [5] F. K. Skinner, Y. Rotenberg, and A. W . N eum ann, “C ontact Angle Mea- : surem ent from the C ontact D iam eter of Sessile Drops by M eans of a Modified ! A xisym m etric Drop Shape Analysis”, Journal of Colloid and Interface Science, 130, N O .l, 25 (1989). [6 ] T. Young, Miscellaneous Works, Vol. I, G. Peacock, Ed., M urray, London, pp. 418, 1855. [7] A. D upre, Theorie Mecanique de la Chaleur, Paris, pp. 368, 1869. [8 ] J. W . Gibbs, The Collected Works of J. W. Gibbs, Longm ans, Green, N ew 1 York, Vol. 1 , pp. 63, 1931. [9] D. H. Bangham and R, I. Razouk, Trans. Faraday Soc., 33, 1459 (1937). J i [10] G. E. Boyd and H. K. Livingston, J. Am. Chem. Soc., 64,2383 (1942). I [11] R. N. W enzel, Ind. Eng. Che,., 28, 988 (1936); J. Phys. Colloid Chem., 53, 1466 (1949). [12] S. B axter and A. B. D. Cassie, J. Text. Inst., 36, T67 (1945); A. B. D. Cassie and S. B axter, Trans. Faraday Soc., 40, 546 (1944). [13] M. K. B ernett and W. A. Zisman, Adv. Chem., Ser., 43, 332 (1964). ; i [14] A. W . N eum ann and R. J. Good, Surface and Colloid Science, Vol. 2, R .J. Good and R. R. Strom berg, Eds., Plenum , New York, 1979. i [15] J. D. M alcolm and H. M. Paynter, “Sim ultaneous D eterm ination of C ontact Angle and Interfacial Tension from Sessile Drop M easurem ents” , Journal of] Colloid and Interface Science, 82, No. 2, 269 (1981). ! [16] Denis J. Ryley, “A New M ethod of D eterm ining th e C ontact Angle M ade by a Sessile Drop upon a Horizontal Surface”, Journal of Colloid and Interface Science, 59, No.2 , 243 (1977). I [17] C. D ahlgren, H. Elwing, K. E. Hagnusson, “Com parison of C ontact Angles j C alculated From The D iam eter of Sessile Drops And Subm erged Air Bubbles 1 in C ontact W ith A Solid Surface” , Colloid and Surfaces, 17, 295 (1986). ! * * * Appendix A: Computer Simulation Program * * (C - code) * * * #include <stdio.h> #include <math.h> #define PI 3.141596 #define delta_b le-5 s ( c s ? f c j J js f c j J c s ) e % J j < s f c f * 3 V l3 .H l * * * * * * I K * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * main() { double radii_vol, radii_cont, beta, vol_in, vol90, contact_angle; double D1FF_1, DTFF_2, delta_arc_length; double optimaQ; ******** invoke function input_parameters ******** input_parameters(&radii_vol, &radii_cont, &beta, &vol_in, &DIFF_1, &DIFFJ2, &delta_arcjength); %******4 c jnyQifg function optima **************** contact_angle = optima(radii_vol, radii_cont, beta, vol_in, DIFF_1, DIFF_2, delta_arc_length); printf("ViVi ***The contact angle is %lf degrees***\n\n", contact_angle); } s************ Function: Input Parameter *********************** This function is to ask user to input the necessary parameters in order to determine the contact angle by numerically calculating the Laplace equation o f capillarity. The input parameters include: droplet volume, contact diameter, liquid surface tension, density between air and liquid, gravitational acceleration, decisions, and the increment o f ds. ********************************************************* input_parameters(radii_vol, radii_cont, beta, vol_in, D IFF_li, DIFF_2i, delta_arc_lengthi) double *radii_vol, *radii_cont, *beta, *vol_in; double *DEFF_li, *DIFF_2i, *delta_arc_lengthi; { double dia_in, g, vol, delta_density, sur_tension; double d l, d2, d3; printf ("Vi\n Input the Contact Diameter (cm): "); scanf ("% lf\ &dia_in); printf (" Input the Drop Volume (ml): "); scanf ("% lf\ &vol); printf (" Input the Gravity Constant (cm/sA 2): ”); scanf ("% lf\& g); printf (" Input the Density difference (g/cmA 3): ''); scanf ("%lf \ &delta_density); printf (" Input the Surface Tension Constant (): "); scanf ("%lf”, &sur_tension); printf(" Input the minimum difference between x and XC (DIFF_1): "); scanf("%lf", &dl); printf(” Input the minimum difference between v and VOL (DIFF_2): ”); scanf(”% lf\ &d2); printf(" Input the increment of the arc length (delta_arc_length): ”); scanf(”% lf’, &d3); *vol_in = vol; *beta = delta_density*g/sur_tension; *radii_vol = pow(3.0*vol/2.0/PI, 1.0/3.0); *radii_cont = dia_in/2.0; *DIFF_li = dl; *DIFF_2i = dl; *delta_arc_lengthi = d3; } Function* Optimn This function is to optimize the objective equation, E(b). Secant method is utilized to iterate the value o f nth b. Then b is transferred to function voI_calc to calculate x, z, and volume by the Laplace equation of capillarity. double optima(radii_vol, radii_cont, beta, vol_in, DIFF_1, DIFF_2, delta_s) double radii_vol, radii_cont, beta, vol_in, DIFF_1, DIFF_2, delta_s; { double v0=0.0, v l= 0 .0 , b=0.0,tmp = 0.0, phi=0.0; double vol_calc(), dabs(), b_estimate(); 74 void write_file(); double bO, b l, vol90; int i=0, s, sign(); s jc s fc s fs j ( e J fc % s f e J f c f l i n C t l O f i f o C S t l D l ^ t C s |s J |c s t s J f c s f e J js s |s ! |e j j c « Jc J j c ; ( { j | j s f t b_estimate(radii_cont, vol_in, &bO, & bl); vol90 = 4.0/3.0*PI*pow(radii_cont, 3.0); s=sign(vol90-vol_in); * is s fc s i® jjtjf* function vol oslc vO =vol_calc(&bO, beta, radii_cont, s, &phi, DIFF_1, delta_s); vl=vol_calc(& bl, beta, radii_cont, s, &phi, DIFF_1, delta_s); ********* Secant Method *********************** while (dabs(vl-vol_in)>=DIFF_2) { b = ((vO-vol_in)*bl -(vl-vol_in)*bO)/(vO-vl); bO = b l; b l = b; vO = v l; v l = vol_calc (& bl, beta, radii_cont, s, &phi, DIFF_1, delta_s); ++i; } write_file(s, & bl, radii_cont, beta, phi, DIFF_1, delta_s); return (phi* 180/PI); } ***** Yoj C ^alc ************************************* This function is to utilize the value o f b which was transferred from the function optima in order to calculate the droplet volume according to the Laplace equation o f capillarity. Then the calculated volume is compared to the input volume. If the difference between these two data is not satisfied by the decision of DIFF_in, then ask function optima to iterate another b. * $ $ 4 c s f c * H t $ * J fc j | t * $ s |c $ s (e Hi : f c 3 fc * s fe : f c Hi : f c * j J s * * $ s fc $ * : f c j f c * : f c * : f c : f c j f c $ j f c % j f c i f : i f : j f c : f t double vol_calc (bn, betan, radii_contn, sn, phin, DIFF_ln, delta_arc_leng thn) double *bn, betan, radii_contn, *phin, DIFF_ln, delta_arc_lengthn; int sn; { double xO, x, zO, z; double phiO, vol90, vol_calc; double pow(), dabs(); CHUNCHILO: x0=le-6; x= z0= z=0.0; phiO= vol_calc=0.0; *phin=phiO; •isjfssfss 4 «sfssfcSecant method while ( (dabs(x-radii_contn) >= DIFF_ln) I I (sn*(phiO* 180/PI - 90.0) >= 0)) { x = xO+cos(phiO)*delta_arc_lengthn; z = zO+sin(phiO)*delta_arc_lengthn; *phin = phi0+(2.0/(*bn) + betan*zO - sin(phiO)/xO)*delta_arc_lengthn; vol_calc += PI*(x0+x)*(x0+x)/4.0*(z-z0); xO = x; zO = z; phiO = *phin; if(((phiO* 180/PI) > 90.0) && (x < radii_contn)) { *bn = *bn + delta_b; goto CHUNCHILO; } } return (vol_calc); } double dabs(a) double a; { if ( a<0) retum(-a); else retum(a); } int sign(y) double y; { if(yX » { printf("\n\n **The contact angle is less than 90 degrees**\n\n”); retum(l); } else if(y<0){ printf("\n\n **The contact angle is larger than 90 degrees**\n\n"); return(-l); } else printf("\n\n **The contact angle is equal to 90 degrees**\n\n"); exit(l); } ************ **pyjj£jjQji write *************** ********** This function is to open two files: x.dat and z.dat and to put the calculated point (x, z) o f the droplet profile into these two files. The data of (x,z) was Obtained by the calculation from the Laplace equation of capillarity when the root ,b, of the objective function is determined. * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * void write_file(sl, bw, radii_contw, betaw, phiw, DIFF_lw, delta_sw) double *bw, radii_contw, betaw, phiw, D IFF_lw, delta_sw; int si; { FILE *ifpx, *ifpz, *open(); double x=0.0, x0= le-6, z=0.0, z0=0.0, phi0=0.0, phin; double volw=0.0; if((ifpx = fopen("x.dat”, "w")) == NULL){ printf("\n Cannot open x.dat."); exit(l); } if((ifpz = fopen(”z.dat”, "w")) = NULL){ printf("\n Cannot open z.dat."); exit(l); } while((dabs(x-radii_contw) >= DIFF_lw) I I (sl*(phi0* 180/PI -90.0) >= 0)) { fprintf(ifpx, ” \n %.61f", x); fprintf(ifpz, "\n %.61f ", -z); fprintf(ifpx, "\n %.61f", -x); fprintf(ifpz, "\n %.61f ", -z); x = xO+cos(phiO)*delta_sw; z = zO+sin(phiO)*delta_sw; phin = phiO +(2.0/(*bw) + betaw* zO-sin(phiO)/xO)*delta_sw; volw += PI*(x0+x)*(x0+x)/4.0*(z-z0); xO =x; zO =z; phiO=phin; if(phin >= phiw) break; } printf("\n X=% lf', x*2.0); printf("\n vol=% lf, volw); fprintf(ifpx, "\n %.61f ", x ); fprintf(ifpz, "\n %.61f ”, -z); f]printf(ifpx, "\n %.61f x); fprintf(ifipz, "\n %.61f", -z); fclose(ifpx); fclose(ifpz); } ********* Function- B Estimate **************************** This function is to determine the initial valude of b according to the input parameters and the estimate value of contact angle. The initial value o f b is calculated based on the assumption that there is no gravitational force on the droplet. double b_estimate(radii_contm, vol_inm, bOm, blm ) double radii_contm, vol_inm, *bOm, *blm ; { double theta, h, hO, h i, temp, rO, r l, r3, eO, el; double pow(); printf("\n Please input the estimate value of contact angle: "); scanf("%lf", &theta); rO = radii_contm/sin(theta*PI/180); hO = rO* (1 -cos(theta*PI/180)); temp = 3.0*voLinm/PI/pow(l-cos(theta*PI/180), 2.0)/(2.0+cos(theta*PI/180)); rl = pow( temp, 1.0/3.0); h i = r 1 *( 1 -cos(theta*PI/l 80)); eO = vol_inm - 1.0/6.0*PI*h0*(3.0*pow(radii_contm, 2.0)+ hO*hO); e l = vol_inm - 1.0/6.0*PI*hl*(3.0*pow(radii_contm, 2.0)+ h l* h l); h=hl; while( dabs(eO) >= le-5) { h = (eO* h 1 -e 1 * hO)/(eO-e 1); h i = h; eO = el; e l = vol_inm - 1.0/6.0*PI*hl*(3.0*pow(radii_contm, 2.0)+ h l* h l); } r3 = h/2.0+ pow(radii_contm, 2.0)/(2.0*h); *bOm = rO ; *blm = r3;

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University of Southern California Dissertations and Theses

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Asset Metadata

Creator Lo, Chun-Chi (author)

Core Title A contact angle study at solid-liquid-air interface by laser goniometry and ADSA-CD computer simulation

Degree Master of Science

Degree Program Chemical Engineering

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag engineering, chemical,OAI-PMH Harvest

Language English

Contributor Digitized by ProQuest (provenance)

Advisor Chang, Wenji Victor (committee chair), Salovey, Ronald (committee member), Shing, Katherine S. (committee member)

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c20-316496

Unique identifier UC11259340

Identifier EP41832.pdf (filename),usctheses-c20-316496 (legacy record id)

Legacy Identifier EP41832.pdf

Dmrecord 316496

Document Type Thesis

Rights Lo, Chun-Chi

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the au...

Repository Name University of Southern California Digital Library

Repository Location USC Digital Library, University of Southern California, University Park Campus, Los Angeles, California 90089, USA