russellstacy-msthesis

7/29/2019 RussellStacy-MSThesis

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Contact Angle Measurement Technique for Rough Surfaces

By

Russell Stacy

A THESIS

Submitted in partial fulfillment of the requirements

for the degree of

Master of Science in Mechanical Engineering

MICHIGAN TECHNOLOGICAL UNIVERSITY

2009

Copyright Russell Stacy 2009


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This thesis, Contact Angle Measurement Technique for Rough Surfaces is herebyapproved in partial fulfillment of the requirements of the Degree of Master of Sciencein Mechanical Engineering.

Department of Mechanical Engineering - Engineering Mechanics

Advisor:

Dr. Jeffrey S. Allen

Committee Member:

Dr. Chang Kyoung Choi

Committee Member:

Dr. Jaroslaw W. Drelich

Department Chair:

Professor William W. Predebon

Date:


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ABSTRACT

Increased transportation fuel costs and environmental awareness has led to an ac-celeration of research into alternative energy production such as low-temperaturefuel cells for automotive use. One of the largest remaining problems facing low-temperature fuel cells is water management. Improvement of water management inlow-temperature fuel cells has led to the investigation of the wettability characteris-

tics of fuel cell components, namely gas diffusion layers, as quantified by the contactangle. Current measurement techniques are not suitable for making accurate contactangle measurements on rough surfaces due to poor optical resolution at the contactline. A technique for accurately measuring contact angles on rough surfaces has beendeveloped using a sessile drop method. The technique requires a drop profile imagewhich is processed to extract the solid surface and drop interface data. The Laplace-Young equation is numerically integrated to generate a Laplacian curve which matchesthe upper portion of the drop profile. The contact angle is then extracted from theLaplacian curve. This technique is advantageous because it removes the dependenceof the contact angle measurement from imaging of the contact line.


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ACKNOWLEDGMENTS

I would like to thank my wife for her continued support and understanding, my friendsand family for believing in me, and my advisor for his wisdom and patience.


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CONTENTS

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

List of Figures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2. Techniques for Measuring Contact Angles . . . . . . . . . . . . . . . . . . 32.1 Wilhelmy Plate Method . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Goniometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.3 Development of the Laplace-Young Model for Sessile Drops . . . . . . 5

2.4 Predicting Contact Angles on Rough Surfaces . . . . . . . . . . . . . 72.5 Axisymmetric Drop Model . . . . . . . . . . . . . . . . . . . . . . . . 8

3. Contact Angle Measurements on GDLs . . . . . . . . . . . . . . . . . . . . 9

4. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.1 Code Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.2 Contact Angle Measurements on FC-721 . . . . . . . . . . . . . . . . 154.3 Contact Angle Measurements on GDL Surfaces . . . . . . . . . . . . 164.4 Contact Angle vs Drop Size . . . . . . . . . . . . . . . . . . . . . . . 17

5. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

Appendices 24

A. Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25


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Contents vi

B. Uncertainty Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27B.1 Uncertainty in c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27B.2 Uncertainty in the scale factor, SF . . . . . . . . . . . . . . . . . . . 28B.3 Uncertainty in the characteristic length, l . . . . . . . . . . . . . . . . 29B.4 Uncertainty in Bo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30B.5 Uncertainty in contact angle, . . . . . . . . . . . . . . . . . . . . . 30

C. Test Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

D. Experimental Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36D.1 Enclosure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36D.2 Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36D.3 Workstation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36D.4 Kohler Illumination . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37D.5 Stage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

E. Contact Angle Measurement Programs . . . . . . . . . . . . . . . . . . . . 39E.1 Contact Angle Program for varying c and b simultaneously . . . . . . 39E.2 Contact Angle Program for varying b . . . . . . . . . . . . . . . . . . 47E.3 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

E.3.1 apex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54E.3.2 c data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55E.3.3 circle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55E.3.4 contact angle sphere cap . . . . . . . . . . . . . . . . . . . . . 56

E.3.5 dat xls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57E.3.6 edge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57E.3.7 fluid prop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58E.3.8 height interface . . . . . . . . . . . . . . . . . . . . . . . . . . 59E.3.9 image analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 60E.3.10 image input . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61E.3.11 laplace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62E.3.12 plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62E.3.13 sphere cap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64E.3.14 trans scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

F. Program which calculates the scale factor for images . . . . . . . . . . . . 66

G. Program used to plot a range of Laplacian curves by varying b . . . . . . . 68

H. Program used to plot a range of Laplacian curves by varying c . . . . . . . 71


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List of Figures vii

LIST OF FIGURES

1.1 Cross sectional view of a basic PEM fuel cell . . . . . . . . . . . . . . 11.2 Youngs model showing the relationship between the three interfacial

tensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2.1 Measuring contact angle via Wilhelmy plate tensiometry . . . . . . . 32.2 Illustration of variance in contact angle . . . . . . . . . . . . . . . . . 42.3 Coordinate system used to solve the Laplace-Young equation . . . . . 6

3.1 Diagram of experimental setup . . . . . . . . . . . . . . . . . . . . . 93.2 Process to find drop interface data . . . . . . . . . . . . . . . . . . . 113.3 Laplacian Curves generated by varying b for c = 14 . . . . . . . . . . 123.4 Laplacian Curves generated by varying c for b = 1 . . . . . . . . . . . 123.5 Water on GDL 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

4.1 Image of water on FC-721 coated glass slide . . . . . . . . . . . . . . 144.2 SEM images of the GDLs . . . . . . . . . . . . . . . . . . . . . . . . 174.3 Contact angle of water on FC-721 . . . . . . . . . . . . . . . . . . . . 18

4.4 Contact angle of water on GDL 4 . . . . . . . . . . . . . . . . . . . . 194.5 Contact angle of water on GDL 6 . . . . . . . . . . . . . . . . . . . . 194.6 Contact angle of water on GDL 8 . . . . . . . . . . . . . . . . . . . . 204.7 Contact angle of water on GDL 11 . . . . . . . . . . . . . . . . . . . 204.8 Contact angle of water on all tested substrates . . . . . . . . . . . . . 21

D.1 Kohler Illumination setup . . . . . . . . . . . . . . . . . . . . . . . . 37


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List of Tables viii

LIST OF TABLES

4.1 Convergence test data for water on FC-721 . . . . . . . . . . . . . . . 154.2 GDL samples tested. . . . . . . . . . . . . . . . . . . . . . . . . . . . 16


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1. INTRODUCTION

High fuel costs and environmental concerns have led to an acceleration of researchinto alternative energies for automotive applications, namely fuel cells. Polymer elec-trolyte membrane (PEM) fuel cells are a leading fuel cell technology for automotiveapplications due to their low operating temperature and their ability to resist corro-sion and contamination [1]. One of the largest remaining problems facing PEM fuel

cells is poor water management.In the case of a basic PEM fuel cell (Figure 1.1), hydrogen and oxygen react toform water, electricity, and heat. The fuel cell membrane must remain fully hydratedand excess water must be removed for proper fuel cell operation. Water is formed nearthe membrane, typically on the cathode side, and excess water travels through thegas diffusion layer (GDL) to form a drop on the GDL surface. The drops that form onthe GDL surface are removed from the cathode channels by the reactant flow. Excesswater can form plugs if not properly removed from the fuel cell channels. These waterplugs can significantly reduce both fuel cell performance, by reducing the number ofreacting sites available in the fuel cell, as well as lifetime of the fuel cell, by increasingthermal gradients and therefore thermal stresses in fuel cell components [1, 2].

An understanding of GDL wettability corresponding to surface roughness, poresize, pore distribution, and surface coating will allow for GDLs to be designed foroptimal water management. GDL wettability will be characterized by contact anglemeasurements, unfortunately current contact angle measurement techniques are notwell defined for rough, porous materials. Therefore a technique for measuring contactangles on rough, porous materials has been developed.

Figure 1.1. Cross sectional view of a basic PEM fuel cell


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1. Introduction 2

A definition of contact angle is needed before discussing contact angle measure-ment techniques. The contact angle can be defined in several ways. Qualitatively,a contact angle is the macroscopic representation of microscopic phenomena. Mi-croscopic characteristics such as surface roughness, surface energies of the materialsinvolved, and surface coatings play a role in the wettability of a material for a givenfluid. Quantitatively, a contact angle is the interior angle formed by the substratebeing used and the tangent to the drop interface at the apparent intersection of allthree interfaces. This intersection is called the contact line. Figure (1.2) illustratesthe tangent line and contact angle of a liquid drop on a surface. Historically a staticcontact angle on a flat surface is defined by the Young Equation (1.1) [3] using inter-facial surface tensions between solid and liquid, SL , solid and vapor, SV , and liquidand vapor, LV. Youngs equation is essentially a force balance in the horizontaldirection. The contact angle may also be directly measured to calculate the ratio of

interfacial surface tensions if the interfacial surface tensions are unknown.

LV cos = SV SL (1.1)

Figure 1.2. Youngs model showing the relationship between the three interfacial ten-sions (solid and liquid, SL , solid and vapor, SV , and liquid and vapor,LV) and the contact angle,


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2. TECHNIQUES FOR MEASURING CONTACT ANGLES

There are several methods of determining contact angles. On a thin, flat substratesuch as a GDL, the two principle methods available are Wilhelmy plate method andgoniometry.

2.1 Wilhelmy Plate Method

Tensiometry was originally developed to find surface tension of liquids. A specificform of tensiometry, the Whilhelmy plate method, can be adapted to calculate contactangles of liquids with known surface tensions using a force balance [3]. The Whilhelmyplate method uses a bulk fluid which is raised towards the plate until the plate issubmerged in the fluid. The method is illustrated in Figure (2.1). Using the generalforce balance Equation (2.1) for tensiometry, the contact angle, , can be calculatedusing the recorded change in weight, W, the wetted object perimeter, p, and surfacetension, .

cos =W

p(2.1)

The roughness and pores of GDLs make it difficult to find a perimeter and also maycause wicking of the fluid into the material and give inaccurate weight measurementsand therefore inaccurate contact angle results.

Figure 2.1. Measuring contact angle via Wilhelmy plate tensiometry.


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2. Techniques for Measuring Contact Angles 4

2.2 Goniometry

Goniometry uses a profile image of a drop to find contact angles. Contact angles

can be directly measured using direct inspection. Other goniometer methods modelthe drop interface in order to find the contact angle using one of three approaches;approximations, curve fitting, or interface modeling.

Direct inspection is the easiest goniometer technique to perform. An image of adrop profile is printed and the substrate surface and tangent line to the drop interfaceat the contact line are drawn using a straight edge. The contact angle is then directlymeasured using a protractor. An alternate approach would be to use a digital imageand a drawing software to draw the surface and tangent lines and measure the contactangle. This technique, although easy to perform, is prone to large inaccuracies incontact angle measurements. User interpretation of the tangent line and surface as

well as improper imaging and poor images lead to large variance in contact anglemeasurements, as illustrated in Figure (2.2).Two approximation methods are the spherical cap approximation and the small

slope approach. As the name implies, the spherical cap approximation models adrop interface as a spherical cap. This approximation may only be used accuratelywhen the drop characteristics meet a specific criteria; the drop must be small enoughthat gravitational effects are minimal. This criteria may be quantified using the Bondnumber, Bo= gl2/, which is a ratio of capillary effects to gravitational effects, where is density of the liquid, g is the acceleration due to gravity, l is the characteristiclength scale, and is the surface tension of the liquid. The capillary length, Lc =

/g, otherwise known as the Laplace constant, may also be used to quantifythe relative effects of gravity verses capillarity. A Bo = 1 implies that gravitationaleffects and capillary effects are equal, with Bo > 1 meaning gravitation effects start to

Figure 2.2. Illustration of variance in contact angle using the direct inspectionmethod. This image is of water on GDL 6 and has a drop height of0.155 centimeters.


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dominate and Bo < 1 meaning capillary effects start to dominate. The characteristiclength scale is the radius at the drop equator for non-wetting drops (contact anglesgreater than 90) and the wetted radius for wetting drops (contact angles less than90).

The small slope approach uses a simplified Laplace-Young equation to model adrop interface. This approach does not have a size constraint like the spherical capapproximation. However, a constraint on contact angle, which must be less than30 [4], is imposed.

Curve fitting techniques typically model the drop interface without constraints onsize or contact angle. Most curve fitting techniques require the drop interface to bedefined using individual data points so a curve fitting routine or algorithm may beused. One approach is to use a polynomial fitting technique to fit the drop interface.The contact angle is determined using the tangent to the drop interface at the contact

line, which is defined as the slope of the polynomial expression. Inaccuracies in contactangle measurements using a polynomial fitting technique come from inaccuratelydefining the tangent to the drop interface at the contact line.

A more accurate approach is to model the drop interface using the Laplace-Youngequation (2.2) which relates the total change in pressure across a curved liquid surface,P, to the two principle radii of curvature, 1/R1 and 1/R2, and surface tension, .

1

R1+

1

R2

= P (2.2)

Bashforth and Adams [5] were the first to develop a numerical solution to the

Laplace-Young equation and published solutions to the equation in the form of ta-bles. Hartland and Hartley [6] later modified the Laplace-Young equation to studyaxisymmetric fluid-liquid interfaces and also published solutions to the Laplace-Youngequation in the form of tables. Li et al. [7] numerically integrate the Laplace-Youngequation using a Runga-Kutta method coded in Fortran, called Axisymmetric DropShape Analysis (ADSA), to find contact angles. Another approach is to use a finiteelement method (FEM) to numerically integrate the Laplace-Young equation [8].

2.3 Development of the Laplace-Young Model for Sessile Drops

The general form of the Laplace-Young equation is shown below with R1 and R2 asthe primary radii of curvature and as the surface tension. The pressure term isdivided into two separate components.

1

R1+

1

R2

= Pg + P (2.3)

where Pg is the change in hydrostatic pressure and P is the change in pressuredue to the curvature of the drop.


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Figure 2.3. Coordinate system used to solve the Laplace-Young equation showing therelationship between X, Z, S, and

Since sessile drops do not always have constant curvature over the entire interface,the pressure and curvature terms need to be expressed in a local coordinate system.

Cartesian coordinates are convenient when the contact angle is less than 90. [4] Forthe gas diffusion media in PEM fuel cells, the contact angle is always greater than90. Therefore, a spherical coordinate system is used to describe local curvatures andpressures. Figure (2.3) illustrates the coordinate system with origin located at theapex of the drop.

The hydrostatic pressure drop from the apex of the drop is expressed as

Pg = gz (2.4)

At the apex of the drop with boundary conditions Pg = 0 and for an axisymmetricdrop, R1 = R2 = b. Thus, the Laplace-Young equation at z = 0 reduces to

2

b= P (2.5)

Therefore at any z, the pressure balance can be expressed as1

R1+

1

R2

=

2

b+ gz (2.6)

The local curvature, 1/R1 + 1/R2, can be expressed in spherical coordinates by rec-ognizing that [3]

1

R1 =

d

ds (2.7)

1

R2=

sin

x(2.8)

Equation (2.6) is transformed to

d

ds=

2

b+

gz

sin

x(2.9)


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where d/ds is the radius of curvature, R1, in spherical coordinates, 2/b is the apexcurvature term, gz/ relates gravitational effects to capillary effects (notice g/ isLc

2), sin /x is the radius of curvature, R2, in spherical coordinates. The final stepis to non-dimensionalize equation (2.9) using the ratio of physical properties of thefluid; defined as c, which has dimensions of 1/length2.

c =g

(2.10)

It should be noted that c is related to the capillary length, Lc, in the form of Lc =c1/2. The dimensionless coordinates and apex curvature are defined as

X = xc1

2 (2.11)

Z = zc1

2 (2.12)

S = sc

1

2

(2.13)B = bc

1

2 (2.14)

Applying this non-dimensionalization to Equation (2.9) and adding two geometric re-lationships results in a set of first-order, ordinary differential equations whose solutionfully defines a drop profile,

d

dS=

2

B+ Z sin

X(2.15)

dX

dS= cos (2.16)

dZ

dS = sin (2.17)

2.4 Predicting Contact Angles on Rough Surfaces

Wenzel [9] developed Equation (2.18) to calculate the theoretical contact angle, W,on rough surfaces. The theoretical contact angle is calculated using the contact angleon a smooth surface of the same material, , and the roughness factor, , which isthe ratio of true surface area to projected surface area.

cos W = cos (2.18)

Later, Cassie [10] developed two equations to calculate theoretical contact angles onheterogeneous surfaces and porous materials. Equation (2.19) calculates the theoret-ical contact angle, C, on heterogeneous surfaces with two different contact angles, 1and 2, with fractional areas of each surface under a drop, f1 and f2. Equation (2.20)calculates the theoretical contact angle, C, on porous surfaces where f1 is the frac-tional area solid-liquid interfaces and f2 is the fractional area of liquid-air interfacesor pores when a drop is placed on the surface.

cos C = f1 cos 1 + f2 cos 2 (2.19)


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cos C = f1 cos 1 f2 (2.20)Wenzels approach is difficult to apply to a non-patterned, rough, porous material

because of the need to know the true surface area of the substrate. Cassies ap-proach can be used for porous materials, however it becomes difficult to apply thisapproach when using a material that is porous with a non-patterned roughness. Bothapproaches require knowledge of the contact angle on a smooth surface of the samematerial, which is impossible to obtain when dealing with typical GDLs which arecoated with PTFE.

2.5 Axisymmetric Drop Model

The models and techniques for measuring contact angles described in this chapter are

not applicable to the measurement of contact angles on rough porous materials suchas GDLs, therefore a technique based on the sessile drop method using the Laplace-Young equation has been developed. The general steps for measuring contact anglesusing the developed technique include capturing a profile image of a sessile dropon the desired substrate, processing the drop image to obtain the drop interfacedata and substrate surface data, and finding a solution to the set of first-order, non-linear differential equations which produces a Laplacian curve which matches the dropinterface. The solution is found by minimizing the distance between the Laplaciancurve and the drop interface data at a point above the contact line. These steps arediscussed in greater detail in the following chapter.


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3. CONTACT ANGLE MEASUREMENTS ON GDLS

This chapter explains in detail the process of making contact angle measurementson GDLs. An illustration of the experimental apparatus used to take profile imagesof drops is shown in Figure (3.1). Components of the setup include the backlight(1), enclosure and drop stage (2), long working distance microscope (3), CCD camera(4), and workstation (5). Details on the experimental apparatus are provided in

Appendix (D).The process of making contact angle measurements begins with capturing a profileimage of a sessile drop and an image of an optical scale using the experimental appa-ratus. The images are processed to extract the drop interface and surface data usinga code developed in MATLAB. A solution that produces a Laplacian curve whichmatches the drop interface is determined by numerically integrating the Laplace-Young equation using MATLAB. The solution to the Laplace-Young equation fullydefines a drop interface from the apex, where = 0, to the maximum theoretical dropheight for that given solution, where = 180.

Figure 3.1. Diagram of experimental setup. 1: Kohler illumination, 2: Enclosure andDrop stage, 3: Long working distance microscope, 4: CCD Camera, 5:Workstation


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3. Contact Angle Measurements on GDLs 10

There are several steps to properly capture a profile image of a sessile drop. Thetemperature of the enclosure must first be set and allowed to stabilize and the enclo-sure must be completely humidified by filling the containers inside of the enclosurewith water and allowing the vapor pressure to equilibrate. Kohler illumination is usedas the backlight which provides an equal intensity light beam and improves contrastbetween the drop interface and the background of the drop image. Before the sub-strate is placed on the drop stage, the stage is leveled front to rear and side to side.The substrate is then placed on the drop stage.

In the case of the FC-721 coated glass, the coated glass slide is simply placedon the drop stage. GDL samples are cut from a larger sheet and then taped to aglass slide to eliminate problems with the GDL moving on the drop stage as well asproblems with the GDL edges rolling up and interfering with imaging.

Once the substrate is placed on the drop stage, drops of water are manually placed

on the substrate using a syringe and needle. The drops are then brought into thefield of view with the substrate surface visible. The entire drop must fit into theimage with at least some of the substrate visible on each side of the drop. The longworking distance microscope is adjusted to focus the image. An image of the dropis then captured using the software that controls the camera (EPIX XCAP). Oncethe first image is captured, the x-y translation stage is used to focus subsequent dropimages on the same substrate. If there is more than one drop on the substrate, onedrop is brought into view at a time and drop images are captured individually. Whenall images are captured, the substrate is removed from the drop stage and an opticalmicrometer (Klarmann Rulings KR-812) is placed on the stage. The micrometer isaligned in the field of view and an image of the micrometer is captured. This image

is then used to calculate a scale factor relating pixels to centimeters.The optical scale image is opened in MATLAB and using a built-in function,

ginput, a user selects two scale marks which correlates to the pixel position on theimage, and enters the number of millimeters between the two marks. MATLAB

uses the pixel information as well as the distance between the two marks to calculatea scale factor in the form of pixels/centimeters. Now an image of a drop profile isloaded into MATLAB, which is converted to black and white using the built-infunction, im2bw . A built-in function, bwtraceboundary, is performed on the blackand white image to extract the substrate surface data and drop interface data. Theentire image conversion process and data extraction is shown in Figure ( 3.2). The

bwtraceboundary function requires two inputs, a starting point and direction to start,which is defined as origin of the image or the upper left corner of the image and startsin the downward direction. This function looks for and stores the points where pixelschange from white to black or black to white, finding the substrate surface locationand the drop interface data.

The data that is found is in units of pixels and using the calculated scale factor thedata is converted to centimeters. The coordinate system is transformed so the apexcoincides with the origin and then non-dimensionalized in the manner of Hartland


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(a) Original Image (b) Black and WhiteImage

(c) Drop Boundary

Figure 3.2. Process to find drop interface data starting with the original image, con-verted to a black and white image, using a tracing function to find thedrop interface. Image shows water on GDL 6 with a drop height of 0.358

centimeters.

and Hartley [6] using c = g/, where is density, g is gravitational acceleration,and is the surface tension of the liquid. Alternatively, the capillary length, Lc =

/g may be used to non-dimensionalize the data. The non-dimensionalization andtransformation of the coordinate system are necessary for physical data to be readilycomparable to the theoretical data. The surface of the substrate is calculated to bean average of the data found on each side of the drop. If the surface location from thedata on the left side of the drop is significantly different, roughly 25 microns, thanthe surface location on the right, the substrate is considered not level and that image

is not used. The camera may also be tilted which may make it difficult to obtainaccurate substrate surface data. Leveling of the camera takes place every time a newset of data is captured to ensure accurate substrate surface data is obtained.

The theoretical data is the solution to the Laplace-Young equation which matchesor defines the physical data. A solution to the Laplace-Young equation for a givenb and c value is found using the built in ordinary differential equation solver inMATLAB, ode45, which is based on a fourth order Runga-Kutta method. Thesolution can be varied by changing the boundary conditions, b and c.

Figure (3.3) illustrates how Laplacian curves change while varying b with a fixed cvalue of 14, roughly the c value of water. Drop profiles produced using small b valuestend to appear small and spherical in nature, whereas drop profiles produced using

large b values tend to appear larger and more like a puddle. This is attributed tothe fact that b is the radius of curvature at the apex, and therefore a small b valueproduces a small drop and a large b value produces a drop with an increasingly flattop. Figure (3.4) illustrates a change in Laplacian curves for a fixed b value of 1and varying c values. The Laplacian curve with a c value of 14 approximates a dropinterface shape of water for b = 1, assuming a density of 1000 kg/m3, gravitationalacceleration of 9.8 m/s2, and a surface tension of 70 dynes/cm. Assuming a fixeddensity and gravitational acceleration, a c value of 6 would provide a surface tension


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of approximately 163 dynes/cm and a c value of 22 would provide a surface tension ofapproximately 45 dynes/cm. The shape and size of the drop profiles in Figure (3.4)do not appear to change significantly for the range of c values shown.

6 4 2 0 2 4 6

1

0

1

2

3

X (dimensionless)

Z(dimensionless)

0.1

0.51.0

5.0 10.0

Figure 3.3. Laplacian Curves generated by varying b for c = 14

3 2 1 0 1 2 3

1

0.5

0

0.5

1

1.5

2

2.5

3

X (dimensionless)

Z(dimensionless)

6.014.0

22.0

Figure 3.4. Laplacian Curves generated by varying c for b = 1


21/83


A fitting routine is used to find the solution to the Laplace-Young equation whichmatches the physical data. This is done by reducing the horizontal distance betweenthe physical data and the theoretical data at a distance above the substrate surface.The purpose of not fitting the theoretical data at the contact line is the poor res-olution of the images at the contact line, which can be seen in Figure (3.5). Forlarge Bo drops, b and c can be solved for simultaneously to find the surface tension,, and contact angle, . However, c should be fixed for small Bo drops due to therelative insignificance of c for small Bo drops as previously mentioned. When fittinga Laplacian curve to the physical data, b is varied to minimize the horizontal distancebetween the physical data and Laplacian curve at a point roughly 5% of the dropheight above the substrate surface. For large Bo drops, b and c may be varied simul-taneously. In this case, b is varied to minimize the horizontal distance between thephysical data and Laplacian curve at a point roughly 5% of the drop height above

the substrate surface and c is varied to minimize the horizontal distance between thephysical data and Laplacian curve at a point roughly half way between the drop apexand the substrate surface. The contact angle measurement program, the scale factorprogram, variation in b plot program, and variation in c plot program are listed inAppendices (E) through (H), respectively.

Figure 3.5. Water on GDL 11 with inset of the poor resolution at the contact line.Drop height is 0.201 centimeters.


22/83

4. DISCUSSION

4.1 Code Verification

The first step in verifying the code was to test the convergence of the code for asingle drop image. A large drop (Figure 4.1) on an FC-721 coated glass slide wascaptured and processed while varying the initial surface tension value, which in effect

varies the initial c value. Table 4.1 shows the results from this drop image that wasprocessed with varying the starting value of between 50 dynes/cm to 80 dynes/cm.The results show a convergence of the contact angle to be 115.45 0.15 and surfacetension to be 69.34 dynes/cm 0.34 dynes/cm. This test demonstrates the programcan resolve contact angle and surface tension simultaneously regardless of the starting, or c, value.

This measurement technique can be used to measure a contact angle and surfacetension simultaneously. This would only be possible for large Bo drops, though. SmallBo drops have little dependence on gravity, therefore the interface shape of a small

Figure 4.1. Image of water on FC-721 coated glass slide used for convergence study.Drop height is 0.324 centimeters.


23/83

4. Discussion 15

Table 4.1. Convergence test data for water on FC-721 (Figure 4.1). The initial valuefor surface tension was varied for each test run. This, in turn, resulted in aunique initial value ofc. Both b and c were allowed to vary while matching

the Laplace curve to the drop profile. Final values after successful matchingare shown for the Bond number, c [1/cm2], contact angle () [deg.], andsurface tension () [dynes/cm]. The final value of the surface tension isderived from c assuming a fixed value for the density and gravitationalacceleration.

test initial value final valuesrun c Bo

1 55 69.00 14.191 1.457 115.62 60 69.40 14.109 1.448 115.53 65 69.67 14.054 1.443 115.4

4 70 69.55 14.079 1.445 115.35 75 69.22 14.147 1.452 115.56 80 69.29 14.131 1.451 115.4

drop becomes invariant to c.

4.2 Contact Angle Measurements on FC-721

After the code was verified for convergence on a single drop, several drops of varyingsize were then placed on the FC-721 coated glass slide and imaged. The results,

Figure (4.3), show a variance in contact angle as a function of the drop size presentedas the non-dimensionless ratio of equator radius, r, over capillary length, Lc. Thisratio is

Bo. The results show contact angle changes with drop size, which has been

reported by Drelich et al. [11], Drelich [12], and Vafaei and Podowski [13]. Figure (4.3)shows that contact angles range within 10 for a given size and shows a general linearincrease in contact angle when plotted as vs log(r/Lc).

Possible causes of this trend could be related to drop deposition [11], line ten-sion [14], or contact line pinning. In the case of small drops, the drop is formed onthe end of the needle and placed on the substrate surface and the needle is removed.Some of the drop volume may be removed on the tip of the needle during this processafter the drop has pinned to the surface, resulting in a smaller contact angle than an-

ticipated. Large drops can not be formed at the end of the needle without detachingprematurely. Therefore the drop is deposited on the surface and volume is added toobtain a large drop. The contact line of the drop could be pinning as fluid is addedresulting in a larger contact angle than anticipated.


24/83

4. Discussion 16

4.3 Contact Angle Measurements on GDL Surfaces

The GDL manufacturer and relative composition can be seen in Table 4.2 with SEM

images of the GDLs shown in Figure (4.2). GDL 4 and GDL 6 are from the samemanufacturer with the only difference being that GDL 6 has a microporous layer(MPL) coating on one side. The PTFE coating can be seen as a webbing betweenthe individual carbon fibers in GDL 4 and GDL 6. The PTFE coating on GDL 8is not as visible as the other GDLs and the carbon fibers appear to be longer andare intertwined in what is called a non-woven fashion. GDL 11 has a similar PTFEcoating, carbon fiber size, and orientation as GDL 4 and GDL 6 but appears to havea larger average pore size. Similar contact angle results were expected for GDL 4 andGDL 6, but the contact angle appears to have a steeper slope for GDL 6 than GDL4 when plotted against log(r/Lc). GDL 4 and GDL 8 have a similar trend in contact

angle plotted against drop size when plotted as vs log(r/Lc). GDL 11 is the onlysample to show decreasing for increasing drop size for a small range of drop sizes.

Table 4.2. GDL samples tested.

GDL 4 Mitshubishi MRC 105 9% PTFE (weight)GDL 6 Mitshubishi MRC 105 9% PTFE (weight) with MPLGDL 8 Freudenberg Plain H2315 nonwoven

GDL 11 Toray TFP-H-060 (7% PTFE)

Contact angles of water on several GDLs were measured and results are shown in

Figure (4.4) through (4.7). The contact angle measurements on the GDLs are higher,in the range of 130 to 170, than the contact angles on FC-721, in the range of 100

to 120. Although data is limited, a similar trend of linearly increasing when plottedas vs log(r/Lc) is seen for GDL 4, GDL 6, and GDL 8. The trend appears to changefor GDL 11 where appears to decrease slightly when plotted as vs log(r/Lc) for asmall range of drop sizes.


25/83

4. Discussion 17

(a) GDL 4 (b) GDL 6

(c) GDL 8 (d) GDL 11

Figure 4.2. SEM images of the GDLs used for contact angle measurements [15]

4.4 Contact Angle vs Drop Size

The trend of contact angles changing with drop size on the GDLs could be explainedby drop deposition and contact line pinning similar to the data obtained for FC-721,but the contact angle appears to increase with increasing drop size more rapidly on

GDL 4, GDL 6, and GDL 8. GDL 11 does not show the same trend of increasingcontact angle with increasing drop size as with GDL 4, GDL 6, and GDL 8. Thismay be explained by a number differences between the GDLs, including differencesin pore size, pore size distribution, coating quality and composition, fiber size andorientation, handling protocol during production, or having a small range of dropsizes. The surface roughness and porosity of the GDLs could increase the effect ofcontact line pinning when compared to the contact angle data on FC-721.

Automating the drop deposition method may not eliminate the problems associ-


26/83

4. Discussion 18

ated with manual drop deposition, but would significantly reduce the contact angledependence associated with moving the contact line of a drop from shaking the needlewhile depositing the drop.

A few ideas for eliminating or significantly reducing the problem of pinning areto vibrate the drop to relax the contact line to its equilibrium state or add a termto the Laplace-Young equation to account for this extra energy. Drop vibrationcould help relax the contact line to an equilibrium state but may be difficult to findthe right frequency and time to vibrate the drop to fully relax the contact line. Theidea of adding an extra energy term to the models used to find contact angle is notnew and has been previously discussed by Drelich et al. [ 11].

101

100

10190

105

120

135

150

165

180

r/Lc

Figure 4.3. Contact angle of water on FC-721 as a function of the ratio of equatorradius, r, to Laplace constant, Lc


27/83

4. Discussion 19

101

100

101

90

105

120

135

150

165

180

r/Lc

Figure 4.4. Contact angle of water on GDL 4 as a function of the ratio of equatorradius, r, to Laplace constant, Lc

101

100

10190

105

120

135

150

165

180

r/Lc



28/83

4. Discussion 20

101

100

101

90

105

120

135

150

165

180

r/Lc


101

100

10190

105

120

135

150

165

180

r/Lc



29/83

4. Discussion 21

101

100

101

90

105

120

135

150

165

180

r/Lc

Figure 4.8. Contact angle of water on all tested substrates as a function of the ratioof equator radius, r, to Laplace constant, Lc. FC-721 (+) GDL 4 ()GDL 6 () GDL 8 () GDL 11 ()


30/83

5. CONCLUSION

A contact angle measurement apparatus has been designed, fabricated, and tested.The apparatus includes a backlight, enclosure to control temperature and humidity, atemperature controllable drop stage, imaging hardware including a long working dis-tance microscope and CCD camera, and processing hardware including a framegrab-ber and workstation. The contact angles are determined using a MATLAB code

developed for use with rough surface. The code is based on the Laplace-Young equa-tion. The contact angles are found after matching the correct Laplacian curve to thedrop profile. This contact angle measurement apparatus, including the MATLAB

program, is suitable for making contact angle measurements of water on PEM fuelcell GDLs at various temperatures and for various drop sizes.

A limited set of data on FC-721 coated glass and four different GDLs has beencollected. The results show a contact angle dependence on drop size for water on FC-721 coated glass and the four GDLs. Results are displayed showing contact angle, ,as a function of drop size using the (r/Lc), or

Bo. In the case of the FC-721 on glass

and GDL 4, GDL 6, and GDL 8 the contact angle tends to decrease as Bo decreasesand increase as Bo increases. This trend is difficult to see in the GDL 11 results due

to a large variance in the contact angle for low Bond numbers and the limited rangeof drop sizes.

The wettability of fuel cell components, namely GDLs, play a crucial role in properwater management for increased fuel cell performance and lifetime. Contact anglesof drops must be measured for a range of drop sizes that are desired to obtain mean-ingful contact angles. The radial and height dimensions of the current set of dataexceed the current dimensions of fuel cell channels, approximately 500-700 micronssquare. Therefore, data for drops with equatorial radii and drop heights in the rangeof the dimensions of current fuel cell channels is needed. The current data set pro-vides information on the trends of contact angles compared to drop sizes but are notmeaningful considering contact angles of drop sizes which may fit in current fuel cellchannels without plugging the channel. The reporting of contact angle data shouldalways include key information such as drop width, either equatorial radius or wettedradius, and drop height.

The continuation of the wettability study on GDLs will require the development ofan automatic drop deposition method to significantly reduce user error from manualdrop deposition. More data is needed for a larger range of Bo numbers at roomtemperature as well as data for temperatures between 0 and 80 Celsius to look at


31/83

5. Conclusion 23

the effect of temperature on contact angles.


32/83

APPENDICES


33/83

A. NOMENCLATURE

ADSA Axisymmetric Drop Shape Analysisb radius of curvature at apexB non-dimensionalized radius of curvature at apexBo Bond numberc ratio of physical properties of fluid

f1 fractional area of surface with contact angle, 1f2 fractional area of surface with contact angle, 2, or fractional area of

pores in materialFEM Finite Element Method

g gravitational accelerationGDL Gas Diffusion Layer

H non-dimensionalized drop heightl characteristic length in centimeters

lpx characteristic length in pixelslnd non-dimensionalized characteristic length scale

Lc capillary lengthMP L Microporous Layerp perimeter

P change in pressurePg hydrostatic pressure dropP change in pressure across curved interfacePEM Polymer Electrolyte Membrane or Proton Exchange Membrane

PTFE polytetrafluoroethylener equator radiusR difference between wetted radius and radius at 0.05 H above the sub-

strate surface

R1 first primary radius of curvatureR2 second primary radius of curvatures arc length of drop interfaceS non-dimensionalized arc length of drop interface

Sc scale length in centimetersScpx scale length in pixelsSF scale factor

T EC Thermoelectric cooler


34/83

A. Nomenclature 26

W change in weightwBo Uncertainty in Bowc Uncertainty in cwg Uncertainty in gravitational accelerationwH Uncertainty in the non-dimensionalized drop heightwl Uncertainty in the characteristic length

wlpx Uncertainty in the characteristic length in pixelsw Uncertainty in density

wSc Uncertainty in the scalewScpx Uncertainty in the pixels in the scalewSF Uncertainty in scale factorw Uncertainty in surface tensionx horizontal distance

X non-dimensionalized horizontal distancez vertical distanceZ non-dimensionalized vertical distance

greek symbols

b non-dimensionalized horizontal minimization coefficient density surface tension contact angle roughness factor

superscripts and subscripts

c Cassies contact anglelv liquid-vaporsl solid-liquidsv solid-vaporw Wenzels contact angle


35/83

B. UNCERTAINTY ANALYSIS

An uncertainty analysis is performed for the Bond number and . The Bond numberanalysis is derived through independent analysis on c, l, and the scale factor, SF.

Bo =gl2

(B.1)

and by substituting in c, Bo = cl2 (B.2)

The uncertainty analysis on is performed with the assumption that tan() (0.05H)/(R), where H is the non-dimensionalized drop height and R is the differ-ence between the non-dimensionalized wetted radius and the non-dimensionalizedradius at 0.05 H above the substrate surface.

B.1 Uncertainty in c

Uncertainty in c is a combination of uncertainties in the fluid properties with units

of [g/cm3

] and with units of [mN/m] as well as g with units of [cm/s2

].

c =g

(B.3)

wc =

c

w

2+

c

gwg

2+

c

w

21/2(B.4)

c

=

g

(B.5)

c

g =

(B.6)

c

=g

2(B.7)

Substituting (B.5 - B.7) into (B.4),

wc =

g

w2

+

wg2

+

g2

w

21/2(B.8)


36/83

B. Uncertainty Analysis 28

Factoring out a c from (B.8) gives,

wcc =

w2

+

wgg2

+

w21/2

(B.9)

w = 0.002 [g/cm3] (B.10)

wg = 2 [cm/s2] (B.11)

w = 5 [mN/m] (B.12)

= 0.998 [g/cm3] (B.13)

g = 981.2 [cm/s2] (B.14)

= 69 [mN/m] (B.15)

Substituting in the values for , g, , w, wg, and w,

wcc

= 0.0725 (B.16)

B.2 Uncertainty in the scale factor, SFIn the analysis of the uncertainty of the scale factor, SF, Scpx is the distance betweenthe two points picked by the user with units of [pixels] and Sc is the distance betweenthe two points picked by the user with units of [cm].

SF =Scpx

Sc(B.17)

wSF =

SF

S cpxwScpx

2+

SF

S cwSc

21/2(B.18)

S F

pSF=

1

Sc(B.19)

S F

S c=Scpx

Sc2(B.20)

wSF =

1

ScwScpx

2+

ScpxSc2

wSc

21/2(B.21)


37/83


wSF

SF

= wScpxScpx

2

+ wScSc

2

1/2

(B.22)

wpxSc and wSc are fixed while Scpx and wSc vary between each data point.

wScpx = 2 [pixels] (B.23)

wSc = 0.00006 [cm] (B.24)

B.3 Uncertainty in the characteristic length, l

In the analysis of the uncertainty of the characteristic length scale, l, lpx

is the num-ber of pixels of the characteristic length and SF is the scale factor with units of[pixels/centimeters].

l =lpxSF

(B.25)

wl =

l

lpxwlpx

2+

l

SFwSF

21/2(B.26)

l

lpx

=1

SF

(B.27)

l

SF=lpxSF2

(B.28)

wl =

1

SFwlpx

2+

lpxSF2

wSF

21/2(B.29)

wll

=

wlpxlpx

2+wSF

SF

21/2(B.30)

wlpx is fixed while lpx varies for each data point. wSF/SF has already been dis-cussed and varies for each data point.

wlpx = 2 [pixels] (B.31)


38/83


B.4 Uncertainty in Bo

The analysis of the Bo is a combination of c and l. The analysis of c is essentially

constant for every data point because it is based on fluid properties. The analysis ofl changes for every drop based on the characteristic length scale or equator radius inthis case.

wBo =

Bo

cwc

2+

Bo

lwl

21/2(B.32)

Bo

c= l2 (B.33)

Bo

l = 2cl (B.34)

wBo =

l2wc2

+ (2clwl)21/2

(B.35)

wBoBo

=

wcc

2+

2wll

21/2(B.36)

B.5 Uncertainty in contact angle,

The analysis of is performed assuming tan()

(0.05H)/(R). Looking at themaximum in this data set, 165, R can be approximated as 0.19 H.

tan() =0.05H

R(B.37)

= tan1

0.05H

R

(B.38)

Using the expansion for tan1, the uncertainty in is calculated as;

w

= wHH

2

+ wRR2

1/2

(B.39)

Substituting in for R and wR, where is approximately 2 b;

w

=

wHH

2+

2b

.19H

21/2(B.40)

b is the non-dimensionalized horizontal minimization coefficient. The drop heightis known within 5 pixels, therefore wH in non-dimensionalized form is;


39/83


wH =5

SF c1/2(B.41)

The uncertainty in is presented in the test matrix.


40/83

C. TEST MATRIX

The test matrix presents the data collected for this presentation. Data is presented inorder of test date and substrate. The contact angle, , is found from the solution to theLaplace-Young equation which produces a Laplacian curve which matches the dropinterface and is reported in degrees (). Drop height and equator radius are foundfrom the original drop image and are reported in units of centimeters. The Bond

number, Bo, is calculated using the final c value and the characteristic length scale,in this case the equator radius. The values ofc and b are final values after running theprogram and have units of [cm2] and [cm], respectively. The c values represent finalc values when c was iterated on while solving the Laplace-Young equation. Remainingc values were not iterated on while solving the Laplace-Young equation and are basedon a surface tension of 70 [mN/m], density of water 0.998 [g/cm3], density of air of0.0012 [g/cm3], and acceleration due to gravity of 981.7 [cm/s2].


41/83

C. Test Matrix 33

Drop

Equator

Date

Su

bstrate

He

ight

Ra

dius

Bo

r/L

c

c

b

Comments

02/04

GDL8

129.9

1.3

0.1

59

0.1

02

0.1

1

0.3

3

10

.27

0.1

04

Fiberprotru

ding

from

thesurfa

ceon

therig

htan

dpoor

imag

ingo

fthesu

b-

stratesurface.

02/04

GDL8

139.9

0.8

0.2

86

0.2

04

0.5

6

0.7

5

13

.57

0.2

23

02/04

GDL8

132.7

1.2

0.1

78

0.1

17

0.2

1

0.4

6

15

.17

0.1

21

Some

fibersprotru

ding

from

thesur-

facetotherig

hto

fthe

drop.

02/04

GDL8

132.9

1.3

0.1

67

0.1

09

0.1

9

0.4

4

16

.17

0.1

13

Dropnotcentere

dhorizontally

inim-

age,

notmuc

hsu

bstratesurface

infor-

mationonthe

lefto

fthe

drop

02/04

GDL11

152.7

1.2

0.2

10

0.1

28

0.2

5

0.5

0

15

.47

0.1

34

02/04

GDL11

150.5

1.1

0.2

18

0.1

34

0.2

6

0.5

1

14

.27

0.1

40

02/04

GDL11

156.1

0.9

0.2

91

0.1

94

0.5

2

0.7

2

13

.67

0.2

10

02/04

GDL11

154.4

1.1

0.2

16

0.1

30

0.2

4

0.4

9

14

.27

0.1

36

02/10

GDL6

148.1

1.2

0.1

99

0.1

23

0.2

5

0.5

0

16

.57

0.1

28

02/10

GDL6

139.8

1.1

0.2

01

0.1

32

0.3

1

0.5

5

17

.57

0.1

38

Su

bstrateorcameramaynotbe

com-

pletelyleve

l.

02/10

GDL6

143.6

0.9

0.2

46

0.1

64

0.4

0

0.6

3

14

.77

0.1

75

Su

bstrateorcameramaynotbe

com-

pletelyleve

l.

02/10

GDL6

140.8

1.0

0.2

17

0.1

41

0.3

0

0.5

4

14

.97

0.1

49

02/15

GDL4

140.2

1.4

0.2

71

0.1

90

0.5

1

0.7

1

14

.07

0.2

07

02/15

GDL4

142.7

1.5

0.2

54

0.1

73

0.4

6

0.6

8

15

.27

0.1

87

Sma

llfiberprotru

ding

from

surfa

ceon

therig

hto

fthe

drop.

02/15

GDL4

131.8

2.0

0.1

82

0.1

21

0.2

3

0.4

8

15

.97

0.1

26

02/15

GDL4

147.5

1.1

0.3

58

0.2

85

1.1

2

1.0

6

13

.87

0.3

36

02/15

GDL4

78.4

1.1

0.1

92

0.1

32

0.2

4

0.4

9

13

.97

0.3

22

Dropp

lace

dtooc

losetoe

dgew

here

fibersan

d

coating

disturbe

dby

cut-

ting

process,

poss

ibly

destroye

d

the

non-wetting

propert

ies.

Data

not

use

d.


42/83

C. Test Matrix 34

Drop

Equator

Date

Su

bstrate

He

ight

Ra

dius

Bo

r/L

c

c

b

Comments

02/15

GDL6

160.5

1.3

0.3

21

0.2

22

0.6

8

0.8

3

13

.77

0.2

48

02/15

GDL6

153.5

1.7

0.2

40

0.1

49

0.3

1

0.5

5

13

.77

0.1

57

02/15

GDL6

149.7

1.4

0.2

81

0.1

88

0.4

7

0.6

9

13

.37

0.2

03

02/15

GDL6

150.2

2.0

0.2

07

0.1

26

0.2

4

0.4

9

15

.27

0.1

32

02/15

GDL6

154.4

1.1

0.3

68

0.2

85

1.1

2

1.0

6

13

.77

0.3

41

02/15

GDL8

141.6

1.1

0.3

50

0.2

76

0.9

9

1.0

0

12

.97

0.3

20

02/15

GDL8

144.5

1.0

0.4

04

0.3

56

1.7

1

1.3

1

13

.47

0.4

61

Large

drop,

notmuc

hsurfaceon

both

sidestofin

danaveragesurface

loca-

tion.

Loo

ks

likesma

llfiberon

the

rig

htsi

deo

fthe

drop

bythe

bas

e.

02/15

GDL11

156.6

1.4

0.3

00

0.2

01

0.5

4

0.7

3

13

.27

0.2

19

02/15

GDL11

164.6

2.2

0.2

01

0.1

17

0.2

1

0.4

6

15

.47

0.1

21

02/15

GDL11

159.1

1.5

0.2

85

0.1

87

0.4

9

0.7

0

13

.97

0.2

02

02/15

GDL11

154.7

1.1

0.3

67

0.2

84

1.0

9

1.0

5

13

.57

0.3

35

Su

bstratesurface

doesnotlook

leve

l.

02/15

GDL11

161.4

1.9

0.2

30

0.1

40

0.3

0

0.5

5

15

.37

0.1

48

Su

bstratesurface

doesnotlook

leve

l.

02/15

GDL11

147.9

1.4

0.2

77

0.1

87

0.4

9

0.7

0

14

.07

0.2

03

Su

bstratesurface

doesnotlook

leve

l.

03/02

FC-7

21

108.1

1.2

0.1

95

0.1

68

0.4

0

0.6

3

14

.27

0.1

79

03/02

FC-7

21

111.4

1.8

0.1

37

0.1

07

0.1

6

0.4

1

14

.27

0.1

10

03/02

FC-7

21

113.1

1.2

0.2

02

0.1

65

0.1

8

0.4

2

14

.27

0.1

75

03/04

FC-7

21

114.2

2.7

0.0

92

0.0

67

0.0

7

0.2

5

14

.27

0.0

68

Contactlinereg

ion

isdistorte

d,poss

i-

blyfrom

lig

htingsource.

03/04

FC-7

21

115.8

1.8

0.1

43

0.1

05

0.1

6

0.4

0

14

.27

0.1

09

Contactlinereg

ion

isdistorte

d,poss

i-

blyfrom

lig

htingsource.

03/04

FC-7

21

112.7

2.3

0.1

07

0.0

80

0.0

9

0.3

0

14

.27

0.0

81

Somere

flectiononthe

leftand

rig

ht

ofthe

drop

image,

substratema

ynot

be

leve

l.

03/04

FC-7

21

112.1

1.0

0.2

53

0.2

27

0.7

3

0.8

6

14

.27

0.2

53

Somere

flectiononthe

lefto

fthe

drop

image,

substratemaynotbe

leve

l.


43/83

C. Test Matrix 35

Drop

Equator

Date

Su

bstrate

He

ight

Ra

dius

Bo

r/L

c

c

b

Comments

03/04

GDL6

144.8

2.2

0.1

43

0.0

84

0.1

3

0.3

6

18

.17

0.0

87

Spotson

image,

lensappears

to

be

dirty.

03/04

GDL6

143.6

2.4

0.1

33

0.0

79

0.1

2

0.3

5

19

.67

0.0

81

Spotson

image,

lensappears

to

be

dirty.

03/04

GDL6

137.2

1.9

0.1

55

0.0

96

0.1

3

0.3

6

14

.17

0.0

98

Spotson

image,

lensappears

to

be

dirty.

03/04

GDL6

147.9

0.9

0.3

58

0.2

81

1.0

8

1.0

4

13

.67

0.3

30

Spotson

image,

lensappears

to

be

dirty.

Large

drop,

notmuc

hsurface

on

bothsi

destofin

danaveragesur-

face

location.

03/09

FC-7

21

105.0

1.9

0.0

89

0.0

72

0.0

7

0.2

7

14

.27

0.0

73

03/09

FC-7

21

114.6

1.0

0.1

89

0.1

49

0.3

2

0.5

6

14

.27

0.1

56

03/09

FC-7

21

108.3

2.3

0.0

77

0.0

59

0.0

5

0.2

2

14

.27

0.0

60

03/09

FC-7

21

106.3

2.2

0.0

79

0.0

63

0.0

6

0.2

4

14

.27

0.0

64

03/09

FC-7

21

110.6

1.7

0.1

05

0.0

81

0.0

9

0.3

0

14

.27

0.0

82

03/09

FC-7

21

109.6

1.9

0.0

92

0.0

70

0.0

7

0.2

6

14

.27

0.0

71

03/09

FC-7

21

112.0

1.0

0.1

87

0.1

51

0.3

3

0.5

7

14

.27

0.1

60

03/12

FC-7

21

120.0

1.0

0.3

38

0.3

26

0.4

1

0.6

4

14

.27

0.4

08

03/12

FC-7

21

118.1

0.9

0.3

39

0.3

38

0.4

3

0.6

5

14

.27

0.4

27

03/12

FC-7

21

116.8

0.9

0.3

52

0.3

69

1.9

4

1.3

9

14

.27

0.4

86

Large

drop,

notmuc

hsurfaceon

both

sidestofin

danaveragesurface

loca-

tion.


44/83

D. EXPERIMENTAL APPARATUS

D.1 Enclosure

In the beginning of this research, evaporation of the drops placed on the substratesproved to be a problem. An enclosure built for this research improves control overhumidity and temperature in the immediate vicinity of the drop and also houses the

drop stage (appendix D.5). Humidity within the enclosure is passively controlled byallowing water filled containers to evaporate over a period of a few hours to overnightdepending on the relative humidity in the room. The walls of the enclosure are madeof polycarbonate which allows imaging and backlighting through the enclosure. Theentire enclosure is mounted on an x-y translation stage (Velmex AXY2509W1). Theenclosure measures 11 inches high 11 inches wide 11 inches deep.

D.2 Imaging

A 10-bit, monochrome, progressive scan, 2/3 inch CCD camera (Pulnix TM-1325CL)with 1392

1040 resolution which can capture up to 30fps is used to capture images

and video. This cameara is mounted to a long working distance microscope (InfinityK2/S) which has a field of view ranging from 7.5mm 5.6mm at a working distanceof 214mm to 1.9mm 1.4 mm at a working distance of 140mm. The camera isconnected to a framegrabber (EPIX EL1DB) through a cameralink connection andthe framegrabber is mounted in the workstation (appendix D.3).

D.3 Workstation

An IBM Intellistation Z Pro (6223-7BU) workstation is used to control the camerafor capturing images using a framegrabber (EPIX EL1DB) through a cameralink

connection. The contact angle measurement program is written in MATLAB, whichis loaded on the workstation. The computer has a 3.8GHz processor with 2MB L2Cache, ability to be upgraded to a dual processor, 2 1024MB PC2-3200 DDR2RAM upgraded to 2 2048MB PC2-4200 DDR2 RAM, 160GB HDD, DVD/CD-RWdrive, gigabit ethernet, and Matrox Millenium P690 PCI. A portion of the RAM isdedicated to image and video capturing. The video card was upgraded to open thePCI-Express slot which was needed for the EPIX framegrabber and allows two DVI


45/83

D. Experimental Apparatus 37

monitors to be used. The workstation originally used Windows XP 64 bit, however alack of 32 bit software compatibility required a change to Windows XP 32 bit.

D.4 Kohler Illumination

Kohler Illumination is a backlight source that provides equal intensity, collimatedlight. This backlight source is used because it provides a crisp, clear image. KohlerIllumination is typically used in chemical or biological fields in a vertical configuration.For this research, Kohler Illumination is created using a series of lenses and aperturesin a horizontal configuration. A projector is used as the light source ( D.1).

Figure D.1. Kohler Illumination setup

The projector uses a 300 watt light bulb. Lens (1) is a diffuser which scattersthe light to eliminate the problem of focusing the filament of the bulb . Lens (2) is

a biconvex lens. Positions (3) and (4) are apertures with variable openings of size1.5 millimeters to 25 millimeters and 1 millimeter to 11 millimeters, respectively.The apertures block the light that comes through edges of the lens where sphericalaberrations distort the light. Lenses (5) and (6) have focal lengths of 38.1 millimetersand 75.6 millimeters, respectively. The combination of the lenses and aperaturesprovide collimated, equal intensity light and can be adjusted to change the lightdiameter and intensity.

D.5 Stage

The drop stage consists of several components. The top of the stage is a copper blockwhich allows for even heating when doing temperature dependent studies. Thereare four thermal electric coolers (Marlow Industries DT12-6-01L), TECs, mountedbetween the copper block and the aluminum fin arrangement. The TECs can beheated or cooled by applying a voltage across them. The aluminum fin arrangementacts as a heat sink to add heat to the TECs when heating the copper block and aremove heat from the TECs when cooling the copper block. The entire drop stage is


46/83

D. Experimental Apparatus 38

mounted to a lab jack (THORLABS L200) which allows the height of the drop stageto be adjusted.


47/83

E. CONTACT ANGLE MEASUREMENT PROGRAMS

E.1 Contact Angle Program for varying c and b simultaneously

% Program written by Russ Stacy

%

% 1 1 1 2 0 8%

5 % Based off of contact angle program written by Derek Fultz and

% Russ Stacy

% c a 4.m

%

% Fourth version of the Contact Angle program

10 %

% CA 4 finds contact angle, drop height, drop diameter at wetted

% area, drop diameter at equator, bond number, and curvature at

% apex while allowing c to vary and find theta and sigma

% simultaneously. This should only be used for relatively large

15 % drops.

%% Must run external scale 1.m file to obtain a scale factor for

% this program.

%

20 % Type help functionname for help with indidual functions used in

% this program.

%% clear and close all opened material

25 clear

clc

close

%% Scale30

% Run scale 1.m to create sc.mat containing the scale factor for

% the image to be processed

load sc.mat;

35

%% Preload Variables


48/83

E. Contact Angle Measurement Programs 40

S s pan=(0:.001:8); % S span is the step variable for ode45 solver

40 eps b=0.000005; % error for b values

eps c=0.000005; % error for c values

ii=1; % counter for error b loop to passthrough if eps b limit is not

met

jj=1; % counter for error c loop to passthrough if eps c limit is not

met

45

% Predefine error c and d cs as row vector for minimazation loop

error c=zeros(1,10);

d cs=zeros(1,10);

50 %% User Inputs (added 01 23 2009 restacy)

[time,substrate,fluid,temperature,excel]=user input;

%% Fluid Properties

55

[rho liquid,rho air,surface tension,grav,c,laplace constant]=

fluid prop;

%% Image import

60 [I,IC,folder,file]=image input;

%% Edge detection with original image

[E,BW,J]=image analysis(IC);

65

%% Boundary Tracing

[boundary]=edge(BW,J);

70 %% Drop Apex

[apex x,apex z]=apex(boundary);

%% Data transform and scaling

75

[x dat,z dat,x dat cm,z dat cm,x dat nd,z dat nd]=trans scale(apex x,apex z,boundary,scale,c);

%%% VARIABLES CHANGED ON 12/9/08

% x p lo t px to x da t

80 % x p lo t cm to x d at c m

% x nd to x da t n d

% y p lo t px to z da t

% y p lo t cm to z d at c m

% y nd to z da t n d


49/83


85 % y values changed to z to reflect zplane notation

%% Z

Plane

[z plane,z plane cm,z plane nd,ind l,ind r]=plane(boundary,z dat,

apex z,scale,c);

90

%% Drop Height and Drop Interface(minus zplane data)

[drop height,drop height cm,drop height nd,x dat cut,z dat cut,

x dc cm,z dc cm,x dc nd,z dc nd,equator radius,bond,x equator left

,x equator right,z equator left,z equator right]=height interface(

z plane,scale,c,z dat,x dat,ind l,ind r);

95 %% Spherical Cap approximation data

[x center,z center,sphere radius]=sphere cap(x equator left,

z equator left,x equator right,z equator right);

%% Circle function

100

[X circle,Z circle]=circle([x center z center],sphere radius,1000,'r.

');

%% Contact Angle with Spherical Cap Approximation

105 [theta sc]=contact angle sphere cap(x center,z center,X circle,

Z circle,drop height);

%% C Cutoff Data

[cut c]=c data(drop height nd,z dat nd);

110

%% LaplaceYoung Equation

% Derek Fultz and Russ Stacy

% 8 1 0 0 7

115

% This program solves 3 simultaneous differential equations. One

% is the LaplaceYoung equation, the other two are geometric% relationships.

120 jjj=1;

LYsoln=0;

while (LYsoln==0);

b1=5;

125 b2=.01;

bb=[b1 b2]; % range of b values to use


50/83


% This is a large loop that solves the LaplaceYoung equation for130 % several values of b to find the solution.

for j=1:2;

b=bb(j); % set b as individual values of bb

135

[S,Y] = ode45(@laplace,S span,[0 1e100 0],[],b,c); % may need toincrease second input if curve does not loop

Z=Y(:,1); % the first column of Y is the Z data. Needs to be defined

to find cutoff point below.

140 % This loop stores the value, i, where the LaplaceYoung data% starts to 'loop' out of control.

i=1;

while Z(i)


51/83


175

[d2,z2]=min(abs( z d c n d (end)z ly(:,2))); % find indice of laplacedata that closely matches raw data

error b2=x dc nd(end)x ly(z2,2); % finds relative horizontalerror of end of trim data

if error b1*error b2>0 && error b10 && error b1>0;

fprintf('choose larger b1 value');

return

end;

185

error b=min(abs([error b1 error b2]));

%% Minimization

190 c change=.1;

b=(b1+b2)/2;

b3=b;

while (abs(error b)>eps b);

195 clear x ly z ly S ly Phi ly Z

% fprintf('ode call\n') % debugging tag[S,Y] = ode45(@laplace,S span ,[0 1e100 0],[],b3,c); %

may need to increase span of S if curve does not loop

% fprintf('out of ode call\n') % debugging tag

200 Z=Y(:,1); % the first column of Y is the Z data. Needs to

be defined to find cutoff point below.

% This loop stores the value, i, where the

% LaplaceYoung data starts to 'loop' out of control.

205 i=1;

while Z(i)


52/83


S ly 1(1:i)=S(1:i,1); % Arc length

dimensionless

220 Phi ly 1(1:i)=Y(1:i,3)*(180/pi); % Contact angle in

degrees

[d3,z3]=min(abs( z d c n d (end)z ly 1)); % find indice oflaplace data that closely matches raw data

error b3=x dc nd(end)x ly 1(z3); % finds relativehorizontal error of end of trim data

225 if error b3*error b1>0;

b1=b3;

error b1=error b3;

error b=error b1;

else

230 b2=b3;

error b2=error b3;

error b=error b2;

end;

b=b3;

235 b3=(b1+b2)/2;

% If eps b limit is never met, passthrough when b=b3

% for 10 iterations. r b is b rounded to 15 decimal

% places, r b3 is b3 rounded to 15 decimal places.

240 % d rs is the difference between r b and r b3. When

% d rs=0, r b=r b3 and when this happens 10 times, a

% minimum has been reached even if it isn't within

% e ps b .

245 r b=round2(b,15);

r b3=round2(b3,15);

d rs=r br b3;

250 if d rs==0;

ii=ii+1;

end;

if ii==10;

255 eps b=abs(error b);

ii=1;end;

fprintf('b=%15.15f error b=%15.15f d r s=%f ii=%f eps b=%f

\n',b,error b,d rs,ii,eps b);260

end;

% Reset eps b back to original value

eps b=.000005;


53/83


265

% cut c is predefined in function (c data). z ly comes from the

% solution to the LY equations using b3. error c is the

% horizontal distance between the actual drop interface and the% LY curve at the point point cut c.

270

[dc,zc]=min(abs(z dat nd(cut c)z ly 1(:,1))); % find indice oflaplace data that closely matches raw data

error c=x dat nd(cut c)x ly 1(zc,1); % finds relativehorizontal error of end of trim data

if abs(error c)>eps c;

275 sign=error c/abs(error c);

if sign==1;

c=c+(c change);

else

c=c(c change);280 end;

[drop height,drop height cm,drop height nd,x dat cut,

z dat cut,x dc cm,z dc cm,x dc nd,z dc nd,equator radius,

bond,x equator left,x equator right,z equator left,

z equator right]=height interface(z plane,scale,c,z dat,

x dat,ind l,ind r);

[x dat nd,z dat nd,x dat cm,z dat cm]=rend(x dat,z dat,scale,

c);

285 fprintf('c=%f\nerror c=%f\n',c,error c)

[S,Y] = ode45(@laplace,S span ,[0 1e100 0],[],b,c); % mayneed to increase span of S if curve does not loop

Z=Y(:,1); % the first column of Y is the Z data. Needs to be

defined to find cutoff point below.

290

% This loop stores the value, i, where the LaplaceYoung% data starts to starts to 'loop' out of control.

i=1;

295 while Z(i)


54/83


x ly 1(1:i)=Y(1:i,2); % Drop height dimensionless

z ly 1(1:i)=Y(1:i,1); % Drop width dimensionless

S ly 1(1:i)=S(1:i,1); % Arc length dimensionless

Phi ly 1(1:i)=Y(1:i,3)*(180/pi); % Contact angle in degrees310

[d3,z3]=min(abs( z d c n d (end)z ly 1)); % find indice oflaplace data that closely matches raw data

error b=x dc nd(end)x ly 1(z3); % finds relativehorizontal error of end of trim data

else

315 LYsoln=1;

end;

end;

b=b3;

320

%% Figures

% Define lines to be used in the plots

z data c cutoff=[z dat nd(cut c);z dat nd(cut c)];

325 z data cutoff=[z dc nd(end);z dc nd(end)];

x data cutoff=[4;4];z plane 1=[drop height nd;drop height nd];

X circle nd=(X circle/scale)*c.5;

Z circle nd=(Z circle/scale)*c.5;

330

% Figure with LaplaceYoung data, cminimization plane,% b

minimization plane, Spherical approximation, Calculated

% z plane, Drop interface data from image. Comment/Uncomment

% any data set you do not want to plot and take out corresponding

335 % legend value.

figure;

% LaplaceYoung dataplot(x ly 1(:,1),z ly 1(:,1),'linewidth',2);

hold all

340 % c minimization plane

plot(x data cutoff,z data c cutoff)

% b minimization plane

plot(x data cutoff,z data cutoff)

% Spherical approximation

345

plot(X circle nd,Z circle nd,'linewidth',2)% Calculated z plane

plot(x data cutoff,z plane 1,'linewidth',2)

% Drop interface data from image

plot(x dat nd,z dat nd,'linewidth',2)

350 % Change axis limits and pbaspect to fit specific drop

axis([5 5 .1 4.9]);set(gca,'YDir','reverse');

pbaspect([2 1 1]);

xlabel('XDirection');


55/83


355 ylabel('ZDirection');legend('LaplaceYoung data','c minimization plane','b minimization

plane','Spherical approximation','calculated z

plane','Raw Data')

title('Nondimensionalized data comparison')hold off

360 % Following figures used for check if problems arise

%%% Figure of raw data reset to apex at origin

% figure

365 % plot(x dat,y dat,'.')

% set(gca,'ydir','reverse')

% axis([800 800 0 800])% pbaspect([2 1 1])

370 %%% Figure of trimmed data with apex at origin

% plot(x dat cut,y dat cut,'.')

% set(gca,'ydir','reverse')

% axis([800 800 0 800])375 % pbaspect([2 1 1])

%% Find Contact Angle

dif=z ly 1drop height nd;380 adif=abs(dif);

low=min(adif);

index=find(adif==low);

contact angle=Phi ly 1(index);

385 %% Data Save

[s]=dat xls(bond,b,error b,c,error c,contact angle,theta sc,

equator radius,drop height cm,time,substrate,fluid,temperature,

excel,scale,file,folder,eps c);

% eof

E.2 Contact Angle Program for varying b

% Program written by Russ Stacy

%

% 1 1 1 2 0 8

%

5 % Based off of contact angle program written by Derek Fultz and

% Russ Stacy ca fixed.m

%

% Fixed c value version of the Contact Angle program


56/83


%

10 % CA FIXED finds contact angle, drop height, drop diameter at

% wetted area, drop diameter at equator, bond number, and

% curvature at apex using a fixed c value.%

% Must run external scale 1.m file to obtain a scale factor for

15 % this program.

%

% Type help functionname for help with indidual functions used in

% this program.

20 %% clear and close all opened material

clear

clc

close

25

%% Scale

% Run scale 1.m to create sc.mat containing the scale factor for

30 % the image to be processed

load sc.mat;

%% Preload Variables

35

S s pan=(0:.001:8); % S span is the step variable for ode45 solver

eps b=0.000005; % error for b values

eps c=0.000005; % error for c values

40

ii=1; % counter for error b loop to passthrough if eps b limit is not

met

jj=1; % counter for error c loop to passthrough if eps c limit is not

met

% Predefine error c and d cs as row vector for minimazation loop

45 error c=zeros(1,10);

d cs=zeros(1,10);

%% User Inputs (added 01 23 2009 restacy)

50 [time,substrate,fluid,temperature,excel]=user input;

%% Fluid Properties

c=14.11914462;

55 laplace constant=1/c.5;

%% Image import


57/83


[I,IC,folder,file]=image input;

60

%% Edge detection with original image

[E,BW,J]=image analysis(IC);

65 %% Boundary Tracing

[boundary]=edge(BW,J);

%% Drop Apex

70

[apex x,apex z]=apex(boundary);

%% Data transform and scaling

75 [x dat,z dat,x dat cm,z dat cm,x dat nd,z dat nd]=trans scale(apex x,

apex z,boundary,scale,c);

%%% VARIABLES CHANGED ON 12/9/08

% x p lo t px to x da t

% x p lo t cm to x d at c m

80 % x nd to x da t n d

% y p lo t px to z da t

% y p lo t cm to z d at c m

% y nd to z da t n d

% y values changed to z to reflect zplane notation85

%% ZPlane

[z plane,z plane cm,z plane nd,ind l,ind r]=plane(boundary,z dat,

apex z,scale,c);

90 %% Drop Height and Drop Interface(minus zplane data)

[drop height,drop height cm,drop height nd,x dat cut,z dat cut,

x dc cm,z dc cm,x dc nd,z dc nd,equator radius,bond,x equator left

,x equator right,z equator left,z equator right]=height interface(

z plane,scale,c,z dat,x dat,ind l,ind r);

%% Spherical Cap approximation data95

[x center,z center,sphere radius]=sphere cap(x equator left,

z equator left,x equator right,z equator right);

%% Circle function

100 [X circle,Z circle]=circle([x center z center],sphere radius,1000,'r.

');


58/83


%% Contact Angle with Spherical Cap Approximation

[theta sc]=contact angle sphere cap(x center,z center,X circle,

Z circle,drop height);105

%% LaplaceYoung Equation

% Derek Fultz and Russ Stacy

% 8 1 0 07

110

% This program solves 3 simultaneous differential equations. One

% is the LaplaceYoung equation, the other two are geometric% relationships.

115 jjj=1;

b1=5;

b2=.01;

120 bb=[b1 b2]; % range of b values to use

% This is a large loop that solves the LaplaceYoung equation for% several values of b to find the solution.

125 for j=1:2;

b=bb(j); % set b as individual values of bb

[S,Y] = ode45(@laplace,S span,[0 1e

100 0],[],b,c); % may need to

increase second input if curve does not loop

130

Z=Y(:,1); % the first column of Y is the Z data. Needs to be defined

to find cutoff point below.

% This loop stores the value, i, where the LaplaceYoung data% starts to 'loop' out of control.

135

i=1;

while Z(i)


59/83


150 % Defines laplace curves for both sides of apex.

%

% x plot lap full=[x plot lap;x plot lap];% y plot lap full=[y plot lap;y plot lap];

end;

155

%% Error

% If the data at the apex fits and the data at the end of the

% trim data fits and all of the fluid properties are correct,

160 % solution should be found. This allows error to only be based on

% the last point of the trim data.

% ERROR FOR B

165 [d1,z1]=min(abs( z d c n d (end)z ly(:,1))); % find indice of laplacedata that closely matches raw data


[d2,z2]=min(abs( z d c n d (end)z ly(:,2))); % find indice of laplacedata that closely matches raw data


170

if error b1*error b2>0 && error b10 && error b1>0;

fprintf('choose larger b1 value');

175 return

end;

error b=min(abs([error b1 error b2]));

180 %% Minimization

% c change=.1; % not used in fixed c program

b=(b1+b2)/2;

185 b3=b;

while (abs(error b)>

eps b);clear x ly z ly S ly Phi ly Z

% fprintf('ode call\n') % debugging tag[S,Y]

russellstacy-msthesis

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