Verification of Contact Angle Hysteresis in Capillary Effect and the Young-Laplace Equation

Larry Zhang, Bhagya Athukorallage, Ram Iyer

Department of Mathematics and Statistics/Clark Scholars Program, Texas Tech University, Lubbock, Texas, United States

ARTICLE INFO ABSTRACTFinal Form:8/7/2013

In this work, we verify (a) contact angle hysteresis in capillary effect, (b) the numerical value of surface tension of water, and (c) the Young-Laplace equation. Contact angle hysteresis is studied via addition and removal of methylene-blue dyed water to a heptane-washed glass slide. Recording is done simultaneously using a top-view digital 30-fps RTV camera and a side-view DSLR camera. Results show significant and clear hysteresis between advancing and receding angles. A novel method for obtaining the numerical value for the surface tension of water is then implemented, using images of a pipette tip containing a droplet of water. A value for the surface tension is calculated using Newton’s Laws and hydrostatics equations, along with measurements of contact angles between menisci and the pipette. A correction for the viewing angle with respect to the horizontal is implemented which leads to a mean value for the surface tension within 0.2% of previously reported values of 72.8 mN/m. A statistical two-sided t-test failed to reject the null hypothesis that the surface tension of water is 72.8 mN/m. The Young-Laplace equation is verified by comparing the meniscus shape predicted by the Young-Laplace equation and the observed meniscus shape.

Keywords:Surface TensionWettingContact AngleHysteresisCapillarityMeniscusYoung-LaplaceRunge-Kutta

1. Introduction

Description of concepts and formulas required in this experiment.

1.1 Surface TensionLiquids conform to any container, yet they

form quite stable and characteristic shapes. Their surfaces are characterized by an interface-specific force called surface tension that opposes any deformation. For a liquid and a rarer bounding fluid medium, the boundary’s shape may be explained as follows. Within the liquid, intermolecular forces are balanced on any molecule due to the surrounding molecules. At the surface, each molecule has only half its interactions and thus has half the energy of molecules inside the liquid. Any distortion to the liquid, then, must either bring new molecules to the surface or spread the molecules out (typically a combination thereof), and thus requires a work to be done proportional to the change in surface area. The ratio of the amount of work done and the change in the surface area is defined as the surface tension of the liquid-fluid boundary, and is thus defined as energy per unit area or force per unit length for an isotropic surface.

Note that surface tension values depend on not only the liquid itself but also the surrounding fluid-medium.

1.1.1 Surface Tension Measurement

Measurements of surface tension utilize a variety of methods, but in all cases any contamination must be avoided as liquid surfaces must be ideal, smooth, and chemically homogenous; any surfactants present on the surface of the liquid will affect any surface tension calculations made. Methods and values are in sections 2.2.1-2.2.3 and 3.2 respectively.

1.1.2 Laplace PressureDue to the curvature of the boundary, a

pressure difference is present between the liquid and surrounding fluid, and is described by the following equation describing work done due to a change in the surface2:

(1) δ W =−pa d V a−pl d V w+γdA

(1)

where γ is the interfacial tension between air and

liquid,d V a the change in the volume of the

surrounding medium, d V w the change in the volume of

the liquid, pa the pressure of the medium, pl the

pressure of the liquid, dA the change in surface area, and δW the work required. For example, we take a sphere,

(2)

d V a=4 π R2dR=−dV w

dA=8 πRdR(2)

1

For equilibrium,δW=0, and simplification gives us the following:

(3)

Δp=2 γR

(3)

1.1.3 The Young-Laplace EquationThe Young-Laplace equation is a combination

of equation (1) with the concept of virtual work for surfaces in equilibrium, and is as follows:

(4)

Δp=pa−pl=γ (2 H )=γ ( 1R

+ 1R ' ) (4

)

where H is the mean curvature and R and R ' are the principal radii of curvature.

1.2 WettingDue to interfacial molecular interactions,

liquid droplets on a solid surface behave in certain ways based on the chemical properties of the liquid, substrate, and medium surrounding the liquid and the substrate’s smoothness and uniformity. The combination of all these factors provides the basis for the definition of the property of a liquid-surface interface known as wetting, which describes the degree to which a liquid spreads on a substrate. Wetting is an integral physical phenomenon to life and is applied in the cooling of electronic circuits, the construction of self-cleaning hydrophobic surfaces and even the absorption of liquids into porous materials.

1.2.1 Total and Partial WettingWetting can occur in two ways, total or partial,

and is characterized by the spreading parameter S, the difference between the interfacial energy of the substrate when dry and wet. More useful is the equivalent equation relating interfacial tensions:

(1) S=γ sa−(γ sl+γ )

(5)

where γ sa is the interfacial tension between the

substrate and air, γ sl between substrate and liquid, and

γ between air and liquid. If S is positive, the surface has more energy

when dry and the liquid experiences total wetting, spreading completely in order to minimize the total surface energy. The result is a film of miniscule thickness resulting from a balance between solid-liquid molecular interactions and surface tension. If S is negative, the surface has more energy when wet and the liquid experiences partial wetting, attempting to take up as little surface area as physically allowed by the three

interfacial tensions. Most liquids are mostly wetting (i.e. the contact angle formed on is under 90o), and will naturally enter any capillary. Note that whether a liquid exhibits total or partial wetting depends not only on the liquid but also on the substrate.

1.3 The Contact AngleAny partially wetting liquid ideally forms a

characteristic contact angle with its substrate. This contact angle can easily be derived given the three interfacial surface tensions and Newton’s Laws, resulting in the Law of Young-Dupré3:

(2)

γ cos (θ )=γ sa−γ sl(6)

Note that we can use a perturbation equation:

(3)

∂ W∂ x

=γ cos (θ )−(γ ¿¿ sa−γ sl)¿(7)

Fig. 1: Relationship between interfacial surface tensions and the contact angle.

If we have equilibrium, then the work done per

perturbation ∂ W∂ x

=0 and we obtain equation (6).

Note that the contact angle only exists when the spreading parameter S is less than 0, easily shown via either mathematical or physical reasoning.

1.3.1 Measurement of the Contact AngleThe contact angle is not simply the tangential

angle at the point of contact. In the region of the triple-point line (the line of contact between the substrate, air, and liquid), the liquid exhibits three distinct regions: the molecular region, the transition region, and the capillary region.

Within the molecular region, only a few molecules cover the surface, and therefore interfacial tensions overcome the intraspecific molecular interactions, resulting in a concave shape. At the transition region, the number of molecules increases to the point where intraspecific interactions begin to dominate. The result is a relatively straight surface at the transition region. Finally, at the capillary region molecular interactions and liquid-air surface tension dominate over the now relatively weak substrate-air and small liquid-air interfacial tensions, forcing the surface into a convex shape.

2

It is at the transition region where we measure our angle. The angle that the transition region makes with the substrate is the true contact angle to be used in the Law of Young-Dupré4.

1.3.2 Contact Angle HysteresisSurfaces, while ideally smooth and uniform,

often have minute flaws and are marred with imperfections. The result is a seemingly different substrate depending on where the triple-phase line is taken. Local uniformity can no longer be assumed and the contact angle between the liquid and substrate is no longer uniquely defined, as the forces that act upon the contact line become vastly more complicated due to imperfections in the surface. Liquid-substrate interactions take on a range of values, corresponding to a range of angles. The triple-point line also deforms and is no longer a perfect circle, but rather a curve alternately pinned and released by the dips and rises in the blemished surface5.

However, though surfaces are not usually smooth, they are usually uniform. Because of this, the range of contact angle remains relatively constant as the droplet expands and recedes. The angle at which the advancing triple-point contact line advances is called the advancing angle, and the angle at which the line recedes is called the receding angle. On “good” surfaces, the hysteresis reaches about 5o, but can exceed 50o on some common surfaces. Hysteresis is the phenomenon in which radius and angle changes follow specific paths independent of volume rates.

1.4 MenisciLiquids contained within a vessel maintain a

horizontal surface, with the exception of near the walls, where adhesion forces (surface tension) produce a deformation. The liquid rises from the horizontal if it is wetting (i.e. S is not very negative) and falls otherwise. The overall shape is called the meniscus, and its curvature is determined by an equilibrium between capillary forces and gravity forces.

2. Methodology

Overview of materials and methods used.

2.1 Contact Angle Hysteresis

2.1.1 Liquid/Substrate Composition and ImagingDistilled water dyed with methylene-blue II

1% (1 drop/10 mL) is added to a single-side pre-frosted glass slide (PEARL CAT 7105) using a micropipette set to 20.2 μL. Dye makes the liquid more visible and thus easier to analyze. The slide is washed with Chromasolv® heptane > 99% HPLC and wiped with Kimwipes®. Heptane is used in order to reduce the attractive forces between the liquid and substrate, thus increasing the contact angle, reducing irregularities in the substrate, and making angle identification easier.

Pipette tips are fully filled to 20.2 μL each time but liquid is only dispensed until the drop falls or touches (and is absorbed by) the droplet on the slide. The micropipette is kept centered over the droplet in an attempt to distribute the force of the falling drop evenly. Liquid is added to the slide 6 times and removed 7 times.

Images are captured from both the top and the side. The top view utilizes the Olympus Q-Color3 3.2 megapixel digital camera with a 1”x1.8” Baver color sensor and high 2080x1542 px resolution at 30 fps connection to a desktop computer on which is captured images of the droplet after each addition and removal of liquid. The side view utilizes a Canon DSLR EOS Rebel XT camera at 1/40th second shutter speed and a 57 aperture held at a distance of approximately 1 foot. Images are captured before, throughout, and after every addition and removal of liquid from the droplet. The droplet is illuminated using four high-power tube lights oriented to eliminate shadows in order to make measurements easier. A white plastic ruler showing centimeters is included in the top view photos to help determine the size of the droplets. The cameras are oriented as perfectly as possible along the vertical and horizontal axes to the slide.

2.1.2 Droplet Image AnalysisAnalysis of the images utilizes the program

ImageJ, which includes a function that performs measurement of angles6. Only images that show the droplet after it stabilizes are used, for a total of 75 images. Images are zoomed in 400-800% and converted to black and white. Transition regions are identified via definition of a threshold for color between the dye and the background for the black and white images and identified by hand for color images. All 150 black/white and color images are analyzed by hand using ImageJ’s angle measurement tool. The results are averaged for each step for the left and right contact angles respectively (13 pairs of data total; 6 of advancing angles, 7 of receding angles). A plot is made of angle vs radius of droplet to exhibit the hysteresis effects. Figs 2 and 3 are example images.

Fig 2: Example side-view image.

3

Fig 3. Example top-view image.

2.2 Surface Tension Calculation

2.2.1 Pipette Tip ImagingA pipette tip with a constant diameter of 5

millimeters is used. The pipette tip is washed in the same Chromasolv® heptane > 99% HPLC solution used before and then dipped into a container of the methylene-blue II 1% dyed water and pulled back out. The resulting droplet is imaged using the Canon DSLR EOS Rebel XT camera with the white plastic ruler as a scale placed as close to the tip as possible. Fig. 4 is an example pipette image.

Fig. 4: Example pipette tip image.

2.2.2 Image Capture Angle CompensationNote that these images are not captured at a

completely horizontal angle. In order to account for this, the viewing angle must be determined and a conversion in coordinate systems must be performed.

Due to the capture angle, the contact line between the liquid and the tube will not form a straight line as if from the side, but rather an ellipse that becomes more and more like a circle as the capture angle increases. This ellipse’ major and minor axes are determined by the capture angle by the following formula:

(8)

sin (θ )=ab

(8)

where a is the minor axis and b the major. The capture angle is thus determined. Fig 5 illustrates an example.

Fig. 5: Illustration of viewing angle determination. The upper contact line forms an ellipse from whose axes the viewing angle may be determined by equation (8).

A coordinate system conversion must now be performed. In the Cartesian system, the horizontal axis x (or r in our case) is the rotational axis. Furthermore, a projection onto the viewing plane of the rotated point must be done. For an actual point α on the x-z cross-section at y=0 and the corresponding measured pointβ, the conversion is as follows:

α=[ 0y

f ( y )]8) β=[0 0 0

0 1 00 0 1] [ cos (θ) 0 sin (θ)

0 1 0−sin (θ) 0 cos (θ)]α=[ 0

ycos (θ ) f ( y)](9

)

Therefore, in order to convert measured coordinates to the actual, we need merely divide the z coordinate by the cosine of the viewing angle. As for measured angles, conversion to actual angles follows the following formula:

(10)

cos ( θactual )=cos (θm ) cos (θ) (10)

where θm is the measured angle and θ the viewing angle. All measured angles and coordinates were first converted to actual and then analyzed.

2.2.2 Meniscus and Liquid Volume ApproximationThe program PlotDigitizer is used to overlay a

Cartesian grid on the pipette tip images based on pixel locations. The top and bottom meniscus profiles are then obtained by recording pixel locations along the menisci. Conversion to meters from number of pixels is done by measuring the number of pixels that corresponds to 1 centimeter on the scale. The coordinates are then shifted to center the lowest top meniscus y-coordinate and the highest bottom meniscus y-coordinate along the y-axis. Parabolic approximation is used to fit the menisci profiles according to the following formula:

4

θθθθθθθθθθθθ

x=¿ x1 , x2 , …, xn>¿, y=¿ y1 , y2 ,… yn>¿

if y=a x2+bx+c , then:

(8) [ y1

y2

…yn

]=[ x12

x22

x1❑

x2❑

11

⋮xn

2⋮❑

xn❑

⋮1][abc ] (11

)

To calculate the volume taken up by each of the menisci, the parabola is then rotated for a radius R according to the following equation:

V m=∫y(0)

y(R )

x2 πdy

(11) ¿ ∫

y(0)

y (R) (√ y−c+b2

4 aa

−b

2 a)

2

πdy(12)

This result follows from a simple complete-the-square method. Now we compute the volume of the liquid without accounting for volume taken up by the menisci. This is quite simple; a cylinder approximates this quite well, and the height and radius are readily obtained:

(12)

V cyl=h r2 π

¿ ( y top (R )− ybot ( R ) ) R2 π(13)

Now subtracting the total volume of the menisci from the cylinder volume should give us the volume of the liquid. 2.2.3 Surface Tension Value Calculation

In the pipette, the weight of the liquid is countered by the forces of surface tension holding it up. The equation is as follows:

(11)

ρgV =2 πRγ (cos θt−cosθb)(14)

From here, it is easy to calculate the surface tension value.

2.3 Comparison of Predicted Profile by the Young-Laplace Equation and Data from PlotDigitizer

Using our calculated surface tension value, we integrate the Young-Laplace equation to determine a theoretical profile for the meniscus. The theoretical

profile is compared against the original to confirm its validity.

2.3.1 Mean Curvature Define the function f describing the meniscus,

and let it be in cylindrical coordinates, i.e. z=f (r ,θ ) . Because the Young-Laplace equation depends on the curvature of the surface, the surface tension and the pressure difference, we need to find each of these individually. First we find the mean curvature of the surface f.

Begin with the following equation describing 3D surfaces:

(12) 2 H=−∇ ∙ n (15

)

where n is the unit normal to the surface and ∇ is the vector differential operator Nabla. The equation can also be written as: Because f is defined on cylindrical coordinates, we rewrite ∇f as:

∇ f =¿ f r ,1r

f θ , f z>¿

√1+¿∇ f ¿2=√1+f r2+ 1

r2 f θ2+f z

2

¿ :α=r √1+f r2+ 1

r2 f θ2+f z

2

∇ f

√1+¿∇ f ¿2=¿

r f r

α,

f θ

α,

r f z

α>¿

∇ ∙∇ f

√1+¿∇ f ¿2=∇ ∙<

r f r

α,f θ

α,r f z

α>¿

¿∂( r2 f r

α )∂ r

+1r

∂( f θ

α )∂ θ

+∂( r f z

α )∂ z

¿2f r

α+r

αrr− f r α r

α2 +1r (α f θθ−f θ

α 2 )+rα f zz− f z α z

α2

Now, remember that z=f (r , θ). This means

that any partial derivative of f with respect to z is equal to 0. However, the partial derivatives of α with respect to each variable must first be derived:

α r=r (1+ f r

2 )+r2 f r f rr+ f θ f θr

α

α θ=r2 f r f θr+ f ❑θ f θθ

α5

(13) 2 H=∇ ∙( ∇ f

√1+¿∇ f ¿2 ) (16)

α z=0

∇⋅ ∇ f

√1+|∇ f|2¿−r2 f r−r2 f r

3−r 2 f r2 f rr−r f r f θ f θr

α 3

+α 2f θθ−f θ (r 2 f r f θr+ f θ f θθ )r (α 3 )

+2 f r+r f rr

α

¿ 1

α3 (−2 r2 f θ f r f θr+r f θθ (1+ f r2 )+ f rr (r3+r f θ

2)+f r (r2+r2 f r2+2 f θ

2 ) )

where α=r √1+f r2+

f θ2

r2.

If we assume f to be axisymmetric (which it is around the center of the meniscus), any partial derivatives of f with respect to θ will disappear, leaving us with:

(14)

Δ Pγ

¿∇⋅ ∇ f

√1+|∇ f|2

¿ f ' '

(1+( f ' )2 )32

+ 1r

f '

(1+( f ' )2 )12

(17)

2.3.2 Conversion to 1st order ODE systemWe need to integrate this formula to obtain the

profile for the meniscus. However, it is impossible to obtain an explicit formula for the meniscus. Additionally, integration of a second-order equation of the above form is quite hard and unwieldy. It is much easier to convert the equation to a system of first order ordinary differential equations and perform integration on those.

Let z1=f and ¿❑b a rrow rmula z2=f '. We then have, after some basic algebraic manipulation:

(15) [ z1 '

z2 ' ]=[ z2

Δ P (r )γ

(1+ z22 )

32−1

rz2(1+z2

2)] (18)

We can represent the left hand side as z ' and

the right as f (z ,r ). We can find an expression for

Δ P (r ) using hydrostatics equations:

(16)

P (r )+ρgf =P (0 )P (r )=P (0 )−ρgf

Δ P (r )=P (r )−Patm

¿ P (0 )− ρgf −Patm

Δ P (r )=Δ P (0 )−ρgf

(19)

where P(r ) is the pressure of the liquid at r and

Δ P (r ) the Laplace pressure at r . Substitution into the ODE gives us a system dependent only on the initial condition Δ P (0) and the value of surface tension γ . Surface tension has already been determined. The pressure difference at r=0 can easily be determined using equation (14) and the meniscus approximation parabola as f .

2.3.3 Fourth Order Runge-Kutta IntegrationWe use the fourth order Runge-Kutta method

to integrate our ODE system7. The method is as follows:

(19)

z (r+h )=z (r )+ 16(F1+2 F2+2 F3+F4)

F1=hf (r , z)

F2=hf (r+ 12

h , z+ 12

F1)F3=hf (r+ 1

2h , z+ 1

2F2)

F4=hf (r+h , z+F3)

(20)

where h is our step size of integration. The resulting

integration gives us a set of values for z1 and z2 against

a set of values for r . We can then plot the result z1

against r to get the meniscus profile, and with small step sizes for h, numerical error may be minimized. Comparison of this profile against the actual meniscus will determine the validity of the Young-Laplace equation.

3. Results

3.1 Contact Angle HysteresisContact angle measurement is done on a total

of 150 images, half black and white and half color. The left and right contact angles are measured for each image using ImageJ. The left and right contact angles are consistently within 2o of each other. Transition regions are identified by taking the best-fitting tangent line. Figures 6 and 7 show the results of the left and right contact angle measurements respectively. Average angles are plotted against the diameter of the droplets in inches.

Table 1 details the average angles with the number of images per stage. A clear contact hysteresis can be seen here with an approximate advancing angle at 45o and a receding angle at 12o. Note however that the range of advancing angles spans about 5o.

3.2 Surface Tension CalculationAnalysis of the ellipse gave a viewing angle of

19.73o. Parabolic approximation of data points taken

6

along the menisci, after centering, yields 82.96 r2 for

the top meniscus and −12.04 r 2 for the bottom meniscus.

Difference in constant terms is equal to 2.57e-03. Integration of horizontal cross-sections of the rotated parabolas yields volumes of 9.09e-09 m3 and 8.34e-10 m3 respectively. The cylinder yields a volume of 6.60e-08 m3, giving a final liquid volume value of 5.60e-08 m3. Contact angles for the menisci are measured multiple times to ensure

0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

0

5

10

15

20

25

30

35

40

45

50

Left Contact Angle

Droplet Diameter (In)

Angl

e (D

egre

es)

Fig. 6: Graph of the droplet’s left contact angle vs the droplet’s

diameter.

0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

0

5

10

15

20

25

30

35

40

45

50

Right Contact Angle

Droplet diameter (In)

Ang

le (d

egre

es)

Fig. 7: Graph of the droplet’s right contact angle vs the droplet’s diameter. Table 1: Left and right contact angles and droplet diameters for each advancing and receding stage.

accuracy and are analyzed for surface tension values separately.

Assuming a 1000 kg/m3 density for water and a local gravity of 9.81 m/s2, we use equation (14) for each set of contact angles to obtain a post-calculations

average surface tension value of approximately 0.0727 N/m, approximately 0.15% lower than the literature value8. Table 2 lists the angles obtained and their corresponding surface tensions.

Top Contact Angle Lower Contact Angle Surface Tension55.76 86.61 0.073855.59 85.38 0.076757.93 86.55 0.079054.16 88.45 0.066655.43 87.17 0.071856.65 87.14 0.074454.19 85.06 0.074554.85 87.00 0.071057.45 87.26 0.075856.59 89.08 0.069554.83 88.55 0.067555.44 88.14 0.069556.75 88.69 0.070858.11 87.63 0.0763Table 2: List of contact angles and their corresponding surface tension values.

3.3 Comparison of Predicted Profile by the Young-Laplace Equation and Data from PlotDigitizer

The first derivative of a y-axis centered parabola is simply 2 ax, and the second 2 a. Substitution into equation (17) at r=0 and simplification gives us:

(14)

Δ Pγ

=4 a (21)

Using the value of surface tension we previously derived and the meniscus approximations also previously made, we calculate a value of 23.83 N/m2

for the upper meniscus Laplace pressure and a 3.46 N/m2 for the bottom. Substitution into equation (18) and integration using the fourth order Runge-Kutta method with a step size of h=R/10000 and initial values

¿ z1 , z2≥¿0,0>¿ gives us the profile of each meniscus. Figures 8 and 9 show the integrated profile overlaid on the meniscus profiles for the top and the bottom menisci respectively.

The difference in the upper profiles is 5.46 μm, and the lowers 4.05 μm, yielding percent differences of +0.99% and -5.07% respectively. Percent differences (p) are calculated by the following formula:

7

Stage Avg. Left Angle Avg. Right Angle DistanceA1 43.7225o 41.9000o 0.5892 inA2 45.4843o 45.3057o 0.6459 inA3 45.6064o 44.9614o 0.7127 inA4 42.6287o 44.5733o 0.7899 inA5 46.2553o 45.6120o 0.9045 inA6 45.4900o 45.4450o 1.0052 inR1 40.3033o 40.2500o 1.0051 inR2 36.9600o 36.1950o 1.0076 inR3 36.2633o 35.0100o 0.9953 inR4 28.2350o 28.3100o 0.9997 inR5 19.1350o 18.6850o 0.9951 inR6 12.4333o 12.1533o 0.8984 inR7 12.6000o 12.6250o 0.6589 in

p=

max0 ≤ r ≤ R

(h theoretical (r )−hactual(r ))max

0≤ r ≤ R(hactual(r ))

4. Discussion

4.1 Contact Angle HysteresisA clear hysteresis effect can be seen in both

graphs. Deviations of about 5o can be seen in the advancing angle, as well as a small deviation of about 0.02 inches in the receding diameter measurements (diameters in advancing measurements and angles in receding measurements are impossible to analyze for deviation, as these are constantly changing). Note that side view measurements were slightly hindered by the reflections from the glass slide and the resolution of the images, making it difficult to identify for certain the triple-point contact line

Fig. 8: Overlay of the RK4 integrated Young-Laplace profile on the upper meniscus profile.

Fig. 9: Overlay of the RK4 integrated Young-Laplace profile on the lower meniscus profile.

and the transition regions, thus producing the above advancing angle errors. As for top-view photos, though heptane proved an effective hydrophobic substrate, as the substrate itself was not perfectly smooth, diameter measurements were difficult, though values still proved relatively constant until the droplet started receding, as expected of hysteresis. Were measurements ideal, contact angles would be relatively constant upon reaching the advancing angle, and receding diameters would be constant until the receding angle is reached. However, they could not be ideal, given the above

reasons. Varying droplet addition locations on the droplet surface and striations of heptane produced by wiping with Kimwipes likely contributed to distortion of the circular contact line, thus producing the variance in contact angles and droplet diameters and contact line shape. Overall, a variance in contact angles of 5o and in diameters of < 0.02 in was seen. The rectangles drawn on figures 6 and 7 above show the extrapolated ideal hysteresis effects.

Future hysteresis measurements should ideally use a mounted high-resolution camera at ranges of less than one inch, a thin unfrosted glass slide with an air-dried heptane wash, and deionized and distilled water with a highly diluted yet dark dye, coupled with just enough ambient illumination to eliminate significant shadows. A white background and a scale located just behind the droplet combined with a few measurements by hand will be used for confirmation of accuracy in digital analysis.

4.2 Surface Tension CalculationWhile the nonzero viewing angle was

accounted for, approximation of the viewing angle itself may have been flawed, as again low-resolution images and unclear boundaries for the ellipse produced by refractions due to the viewing angle hindered measurements. Leeway in measurements of the ellipse’s minor and major axes produces possible viewing angles between 18-21o. Our coordinate conversion also assumed that the tangential cross-section of the meniscus could be assumed as equivalent as the rotated horizontal cross-section, but this is not true. A small difference will theoretically exist in dimensions, though how large is a question of surface tension magnitude and radius of the pipette. Nonetheless, considering how large menisci are, such differences should be avoided. Given our small pipette and relatively small viewing angle, this did not become a large problem. This error will also be eliminated if the viewing angle were reduced to 0o.

While transparency was obviously requisite in the pipette to allow imaging of the droplet inside, it also caused refractions to blur the boundaries of the menisci and pipette walls. Manual measurement of the pipette radius (and thus the droplet’s radius) compensated for unclear images, but height and boundaries remained ambiguous. A “halfway” approach was taken in determining the exact boundaries and height by taking the halfway point between the upper and lower parts of the blurred boundaries.

Such blurring also interfered in exporting of the meniscus’ pixels to a Cartesian grid using PlotDigitizer, though the halfway approach proved fairly accurate, as shown by the > 0.99 R2 values in parabola approximations (0.9981 for upper, 0.9968 in lower). More concerning was the low resolution image itself, producing few data points to work with due to the

8

low amount of pixels. Furthermore, any tilting of the pipette (which could easily have been done by accident; 1o or 2o tilting was highly likely) would skew the menisci to either end, though shapes should still remain about the same.

Such tilting would additionally produce a small normal force from the pipette’s walls, skewing the calculations of surface tension from equation (14), as there would be a missing force.

Even with these limitations, a surface tension value of 0.0727 N/m was produced. Literature values for surface tension of distilled water at 20o Celsius are approximately 0.0728 N/m, giving a -0.15% error. It should be noted that surface tension values would be expected to be lower than 0.0728 N/m here, due to the presence of a dye surfactant, lowering the strength of molecular interactions of the water at the surface and increasing adhesion forces (γ sl and γ al would be very slightly increased). However, 1 drop of dye per 10 mL is miniscule, and should not significantly affect the surface tension of the water.

Future surface tension calculations using this method should utilize a mounted straight, thin glass pipette calibrated to be exactly vertical using a level and a perfectly horizontal mounted high-resolution camera at ranges of less than one inch. Water used should be deionized and distilled water dyed with a highly dilute yet dark dye. A container of water should be held up to the end of the pipette to allow capillary action to pull a small droplet inside, at which point the outside of the pipette should be dried. Illumination should be uniform on all sides and kept low enough to reduce refractive effects yet high enough to allow for close-range focusing and imaging. A scale should be imaged next to the pipette and manual measurements should be taken to confirm values obtained by digital analysis. Exporting of pixels to a Cartesian grid should be done using as many data points as possible.

4.3 Young-Laplace Theoretical vs Experimental Comparison

Profiles centered on the same axis, while useful in calculations, are impractical in relation to real life situations. Ideally, though, menisci profiles should be already centered on account of no tilting of the pipettes. Nonetheless, a slight difference in axes of the profiles was calculated at 2e-06 m, though easily small enough to be ignored.

Calculations of Laplace pressures at the origins of both parabola approximations assume the x2

coefficient and calculated surface tension values to be accurate. The previous meniscus approximations and surface tension calculations must therefore be correct, or the high proportionality constant ρg=9810 kg/m2s2

will magnify any error and alter the Laplace pressure calculations. The step size of h=R/10000 should easily be sufficient to integrate an accurate profile for the meniscus.

The theoretical profiles calculated using this method produced an error of only 0.99% in the upper meniscus and a larger error of -5.07% in the lower meniscus. However, given the slew of limitations in the method, materials and equipment we used, this error is expected.

5. ConclusionsContact angle hysteresis in capillary effect is

tested and confirmed using a DSLR camera and dyed, distilled water. A new method for calculation of surface tension is also tested here. Previous well-known methods included Wilhelmy’s method of thin plate-liquid forces, measurements of the rise of liquid in a capillary, characterization of drops by shape and comparison to theoretical models, and even laser detection of capillary waves. In this case, all that was required was a thin tube, the liquid being measured, and a camera. Additionally, measurements of the surface tension included data from both surfaces formed by the droplet, instead of focusing on one (such as in the pendant drop method). Finally, a novel approach for verification of the Young-Laplace equation is presented, by calculation of theoretical profiles based on a two parameters (Laplace pressure at the vertex of each meniscus and the surface tension of the liquid) and comparison against the actual meniscus. Verification was successful and the method produced profiles almost identical to the real menisci. Additionally, in the process, using fourth-order Runge-Kutta integration (RK4) and parameterization, a method for calculation of a theoretical meniscus profile for any liquid was created. Future development can improve this method to create meniscus profiles for not only vertical tubes but any container. Overall, RK4 integration of a first-order ODE curvature provides a very accurate method for meniscus profile calculation and confirmation of the Young-Laplace equation, the basis for the derivation of the ODE system.

AcknowledgementsThe authors thank the Clark Scholars Program

at Texas Tech University. We also gratefully acknowledge the help provided by Mary Catherine Hastert and Steven Platten in conducting the experiments described in this report.

6. References1. Gennes, Pierre-Gilles De., Francoise Brochard-Wyart, and

David Quéré. Capillarity and Wetting Phenomena: Drops. Bubbles, Pearls, Waves. New York: Springer, 2004.

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2. Stalder, Aurélien F., Tobias Melchior, Michael Müller, Daniel Sage, Thierry Blu, and Michael Unser. "Low-bond Axisymmetric Drop Shape Analysis for Surface Tension and Contact Angle Measurements of Sessile Drops." Colloids and Surfaces A: Physicochemical and Engineering Aspects 364.1-3 (2010): 72-81. Lamb, Horace. Hydrodynamics. New York: Dover Publications, 1945.

3. Gao, Lichao, and Thomas J. McCarthy. "Contact Angle Hysteresis Explained." Langmuir 22.14 (2006): 6234-237.

4. Merchant, Gregory J., and Joseph B. Keller. "Contact Angles." Physics of Fluids A: Fluid Dynamics 4.3 (1992): 477.

5. Lin, Shi-Yow, Hong-Chi Chang, Lung-Wei Lin, and Pao-Yao Huang. "Measurement of Dynamic/advancing/receding Contact Angle by Video-enhanced Sessile Drop Tensiometry." Review of Scientific Instruments 67.8 (1996): 2852.

6. Kincaid, David, and E. W. Cheney. Numerical Analysis: Mathematics of Scientific Computing. Pacific Grove, CA: Brooks/Cole, 1991.

7. Lin, Shi-Yow, Ting-Li Lu, and Woei-Bor Hwang. "Adsorption Kinetics of Decanol at the Air-Water Interface." Langmuir 11.2 1995): 555-62.

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