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# The contrast of the optical contact angle measuring instruments’ testing principles and methods

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Presently, the main contact angle measurement methods are as follows:

(1)    The simple geometric methods: protractor method and WH method or θ/ 2 method.

The protractor method analyzes the contact angle by using a slant straight line which is tangent to the droplet profile, while the WH or the θ/2 method assumes that the droplet profile follows a circular curve, that is, the droplet is assumed to be a part of the spherical crown. Thus, the WH or the θ/2 method is sometimes referred to as the little spherical tank method. At this point, through the width and height of the arc the contact angle could be calculated by the inverse trigonometric function.

The advantages of this method: (1) It has been invented and used for a long time; (2) The measurement does not need complex instruments Such as the US Kino earlier vison SL100; could be observed by eyes directly. (3) The calculation is very fast when using computer assisted software.

The disadvantages of this method include: (1) measurement error is large the protractor method is usually 2 degrees or even higher; poor repeatability and low accuracy. (2) human factors affect a lot, especially the protractor method, different people have different standards, thus there is no unitary quality criterion. The WH method is under the influence of the number of the pixels. The accuracy would not be very high. (3) the noises effect, especially in the use of WH method, the background noises will significantly affect the measurement of the pixels; (4) the droplet volume effect. Because of the obvious gravitational effect, there is a deviation measurement results. In another case, when the contact angles greater than 80 degrees or measuring the super hydrophobic materials (above 150 degrees), the droplet shape (2uL or 1uL) actually changes due to gravity.

(2)   The complex Advanced Mathematics: equation fitting method such as circle fitting, ellipse fitting, tangent method (quadratic curve or compound curve), and no equation fitting like Spline curve fitting, real droplet method.

The testing process of this series advanced mathematic method is: taking the droplet contour image, using the image recognition technology to fit the edge of the image (such as Canny Operator), extracting the coordinates of the edge curve, and then obtaining the final equation curve by the least squares fitting of the coordinate curve and curve equation. After that, the contact angle is obtained by deriving at both ends of the contact.

The characteristic of these series of method is using the curve algorithm to calculate the tangent angle of a curve. While as lack the support of the interface chemistry correlation knowledge, the results are only the apparent angle value of the droplet profile, in other words, it is only the geometry value of the contour, but not the real solid-liquid-gas or solid- Liquid three-phase system value. In fact, due to gravity, buoyancy, chemical diversity and the existence of heterogeneous, the contact angle system is very complex.

The advantages of this series of algorithms are: (1) easy to understand and accept. This series of methods involves limited chemical expertise, so it is easy to be understood by the user; (2) high test accuracy, generally, in the control of the amount of liquid droplets, when the fitting curve and contour curve coincides with each other , the accuracy would be up to 0.5 ° (circle fitting or ellipse fitting); (3) Both the circle fitting method and the ellipse fitting method are less susceptible to the background noise points, and the success rate of the automatic fitting is high. In particular, the circle fitting method is the preferred method when the contact angle within 5 degrees.

The shortcomings of these method are: (1) the impact of droplet size and gravity is big, especially the circle fitting method; (2) for the ellipse fitting method when the contact angle is within 10 degrees or the contact angle is 150 degrees above (such as the super-hydrophobic material, etc.), at this time the impact of gravity has affected the contour, especially the center line below the contour is no longer symmetrical with the above part; (3) tangent method is easily affected by the noises at the contact point, especially in the advancing contact angle and receding contact angle test, the success rate is low; (4) unable to represent the interface of chemical phenomena truly, such as chemical diversity, the embodiment of gravity, the effect of interfacial chemistry sandwiches, etc.; (5) unable to fit non-axisymmetric images, except the ellipse fitting and tangent method can partly fit the non-axisymmetric images.

(3)   Young-Laplace equation fitting method.

The Young-Laplace equation is introduced into the measurement of contact angle and interfacial tension (surface tension), which takes into account the influence of various factors such as gravity, buoyancy and interfacial tension, and also more truly characterizes the solid-liquid-gas or solid-liquid-liquid three- phase system of the interface chemical phenomenon. Compared with the above two categories of methods, the measurement accuracy and the repeatability are relatively high. The process of the method is as follows: taking the droplet contour image, using the image recognition technology to extract the edge of the image and get the coordinate points, and then the coordinate points are fitted to the Young-Laplace equation to get the surface tension value, volume value, surface value, contact angle and other parameters. Its core technology is the algorithm of Bond Number and the Young-Laplace equation. According to the Bond Number different Young-Laplace equation fitting technology is divided into two categories: ADSA and Select Plane.

The most well-known algorithms of the Young-Laplace methods is the ADSA algorithm which invented by Professor A.W. Neumann. The ADSA includes ADSA-P, ADSA-D, ADSA-CD, ADSA-NA and ADSA-Real Drop. The ADSA algorithm is an axisymmetric image analysis method developed by Professor Neumann in 1983Rotenberg, Y., Boruvka, L. and Neumann, A.W. Determination of Surface Tension and Contact Angle from the Shapes of Axisymmetric Fluid Interfaces J. Colloid Interface Sci. 93 p.169-183, and published in 1987 Spelt, J.K., Rotenberg, Y., Absolom, D.R. and Neumann, A.W. Sessile Drop Contact Angle Measurements Using Axisymmetric Drop Shape Analysis (ADSA) Colloids Surfaces 24 p.127-137, 1987, Neumanns team in 1997 summarize the ADSA algorithm and form the theoretical system (OI del R? O and AW Neumann, Axisymmetric Drop Shape Analysis: Computational Methods for the Measurement of Interfacial Properties from the Shape and Dimensions of Pendant and Sessile Drops, JOURNAL OF COLLOID AND INTERFACE SCIENCE 196, P136-147, 1997). ADSA-Real Drop algorithm is based on ADSA-NA, and its core technology is the bond number’s fitting using the joint vertex radius of curvature and surface tension values to establish the relationship, and then quadratic fit the Young -Laplace equation. Compared to ADSA-P this algorithm is closer to the real shape of the droplet profile, which is known as Real Drop technology, can also be called non-axis of the image analysis method.

The other three well-known algorithms are as follows:

These algorithms are commercialized by other many instrument manufacturers, and thus have some certain popularity. Its core of the Bond Number is using Select Plane algorithm.

(1) Young-Laplace equation fitting method of the Song Bihai teamBIHAI SONG AND JU¨ RGEN SPRINGER, Determination of Interfacial Tension from the Profile of a Pendant Drop Using Computer-Aided Image Processing, JOURNAL OF COLLOID AND INTERFACE SCIENCE 184, P6476 ,1996）。

(2) Young-Laplace equation fitting method of the HansenF. K. HANSEN AND G. RODSRUD Surface Tension by Pendant Drop I. A Fast Standard Instrument Using Computer Image AnalysisJournal of Colloid and Interface Science, Vol. 141, No. I, p1-9, January 1991

(3) Young-Laplace equation fitting method of the J. W. Jennings teamJ. W. Jennings N. R. Pallasan Efficient Method for the Determination of Interfacial Tensions from Drop ProfilesLangmuir. Vol. 4, No. 4, 1988P959-967

The disadvantages of this method are as follows: (1) except the ADSA NA and ADSA REAL DROP, all the other algorithms have the assumption of axisymmetric droplet, that is the contour of the droplet’s left, right, front and back are symmetric. Therefore, in practical tests, only fit the contours of the center of the droplet and copy the fitting curve to the other side. But in fact, there is little solid surface of the droplets can form axisymmetric. (2) for small contact angle values, such as less than 3 degrees, due to the use of the fitting edge is limited, the accuracy would not be very high.

The advantages of this method are as follows: (1) the influence of gravity and buoyancy on the contact angle measurement can be corrected. Droplet volume would not affect the test result. The precision is high and the repeatability is good. (2) the contact angle can be used in super hydrophobic material, especially the contact angle is larger than 80 degree, the fitting quality is very good; (3) can truly reflect the contact angle of solid-liquid-gas or solid-liquid-liquid three-phase system.

Among them, ADSA-Real Drop algorithm, because of its non-axis symmetry combined with Wensel-Cassie model, its advantages are more obvious: (1) can be used to analyze the 3D contact angle, especially chemical diversity, heterogeneity, contact angle hysteresis (2) is completely independent of the droplet size, from 0.1uL to 400uL, and the contact angle is kept within 2 degrees; (3) it is very quick to judge the chemical diversity of samples, cleaning degrees, etc., without the need to measure the multiple droplets.

 NO. Contact angle measurement methods Fundamentals Advantages Disadvantages 1 Protractor method Artificial measurement 1. invented and used for a long time 2. able to measure dynamic contact angle 1. Big measurement error 2. Poor repeatability 3. Easily affected by the noises 2 WH method or θ/2 method Geometric conversion, calculation of inverse trigonometric functions 1. invented and used for a long time 2. fast and easy to understand 1. affected by gravity and buoyancy 2. hard to control the droplet size 3. measure static contact angle only 4. test angle should less 80 degree 3 Circle fitting method Advanced mathematics, fitting circular curve equation after seeking derivative 1. able to measure small contact angle 2. easy to understand 1. affected by gravity and buoyancy 2. hard to control the droplet size 3. test angle should less 100 degree； 4. axisymmetric image only 5. unable to represent the interface of chemical phenomena truly 4 Ellipse fitting method Advanced mathematics, fitting circular curve equation after seeking derivative 1. A symmetric droplet image with a central axis can be tested with the effect of gravity； 2. easy to understand 1. affected a little by gravity and buoyancy； 2.unable to measure the contact angle less than 10 degree； 3. unable to measure the contact angle bigger than 140 degree especially the hydrophobic materials； 4. unable to represent the interface of chemical phenomena truly 5 Tangent method (compound curve method - Germany Kruss), curve method (KINO) Higher mathematics, fitting quadratic curve equation or polynomial curve equation, then derivative 1. can measure dynamic contact angle especially the advancing and receding contact angle 2. easy to use 1. big measurement error 2. poor repeatability 3. affected by the noises 4. unable to represent the interface of chemical phenomena truly 6 Spline curve real drop method Interpolation fit; limit equation principle 1. can measure dynamic contact angle especially the advancing and receding contact angle 2. easy to use 1. affected by the noises 2. unable to represent the interface of chemical phenomena truly 7 Young-Laplace equation fitting Select Plane Image analysis method Fitting the Young-Laplace equation 1. Can test the contact angle of most of the axisymmetric droplets; 2. Especially for more than 30 degrees above the contact angle measurement, can modify the impact of gravity coefficient； 3. widely used in super-hydrophobic materials 1.hard to measure hydrophilic materials, contact angle less than 3 degree 2. Cannot test non-axisymmetric droplets, especially 3D contact angle； 3. cannot measure dynamic contact angle 4. The test error of the contact angle of the hydrophobic material, especially some non - axisymmetric materials, is too large. 5. For a nearly circular image greater than 160 degrees, the test error is too large 8 Young-Laplace equation fitting ADSA－P Image analysis method Fitting the Young-Laplace equation 1. Can test the contact angle of most of the axisymmetric droplets; 2. Especially for more than 30 degrees above the contact angle measurement, can modify the impact of gravity coefficient； 3. widely used in super-hydrophobic materials 1.hard to measure hydrophilic materials, contact angle less than 3 degree 2. Cannot test non-axisymmetric droplets, especially 3D contact angle； 3. cannot measure dynamic contact angle 9 Young-Laplace equation fitting ADSA－RealDrop Non-axis symmetry Image analysis method Fitting the Young-Laplace equation； 1. can used to measure 3D contact angle 2. Can test the contact angle of most of the axisymmetric droplets 3. widely used in super-hydrophobic materials 4. an measure dynamic contact angle especially the advancing and receding contact angle 1. hard to measure hydrophilic materials, contact angle less than 3 degree