Acta Polytechnica https://doi.org/10.14311/AP.2025.65.0276 Acta Polytechnica 65(3):276–281, 2025 © 2025 The Author(s). Licensed under a CC-BY 4.0 licence Published by the Czech Technical University in Prague COMPARISON BETWEEN THE WILHELMY SURFACE TENSION MEASUREMENT METHOD AND THE PENDANT DROP SHAPE ANALYSIS METHOD Estela Cristina Cavalcante de Farias, Ricardo Kropf Santos Fermam∗, Amsterdam de J. Souza Marques de Mendonça National Institute of Metrology, Quality and Technology (INMETRO), Av. Nossa Senhora das Graças 50, Xerém, Duque de Caxias, 25250-020 Rio de Janeiro, Brazil ∗ corresponding author: rkfermam@inmetro.gov.br Abstract. Surface tension plays an essential role in various laboratory and industrial processes. The Fluid Metrology Laboratory (Laflu) of the National Institute of Metrology, Quality and Technology (Inmetro) uses the Wilhelmy and DuNoüy methods and has a tensiometer for determining surface tension by the drop shape analysis method in use. One way to ensure the reliability of surface tension measurement results is to compare the methods used. A comparison was made between the Wilhelmy method and the drop shape analysis method. The comparison involved measurements of the surface tension of these liquids: bidistilled water, n-dodecane, and Perfluorocarbon (FC-40), and used the calculation of the Normalized Error (EN), presenting results according to acompatible criterion. Analysing the uncertainties involved, the contribution of the uncertainty of the regression used in the correction of the tensiometer indication was the most relevant. Keywords: Surface tension, Wilhelmy method, drop shape analysis method. 1. Introduction Surface tension, as an inherent characteristic of the liquid-air interface, plays a significant role in various segments of industry, science, and technology through wettability, capillarity, atomisation, jetting, and the dynamics of liquid surfaces [1]. These phenomena are intensely present in production and industrial pro- cesses [2], such as the manufacturing of chemicals, semiconductor manufacturing, steel production, gal- vanisation processes, and the exploration and refining of oil, in addition to a variety of applications rele- vant to health, the economy, and the environment, such as the manufacturing of automotive components, electronic devices, food, beverages, and pharmaceuti- cals [3]. For the measurement of the surface tension of liq- uids, the most commonly used methods are the Wil- helmy method, also called the plate method, which is based on the force that prevents the removal of the plate from the surface of the liquid [4], the DuNoüy method, also called the ring method, which is based on the force required to remove a metal ring from the surface of a liquid [5], and the pendant drop shape analysis method, in which the surface tension is calcu- lated from known parameters, theoretical images are generated and compared with experimental images [6]. The last one has only become more accurate and faster with the advance of computer image analysis [7]. The DuNoüy and Wilhelmy methods are commonly used for the determination of surface tension in vari- ous sectors of industry, calibration laboratories, and research institutes. However, an increase in the use of the pendant drop shape analysis method has been observed. The Fluid Metrology Laboratory (Laflu) at the National Institute of Metrology, Quality, and Technol- ogy (INMETRO), in addition to conducting research in the field of fluid property measurement [8–11], is responsible for maintaining metrological traceability and disseminating the magnitude of surface tension in Brazil [9], using the Wilhelmy and DuNoüy meth- ods and a tensiometer for determining the surface tension by the pendant drop shape analysis method in use. Laflu provides calibration services for surface tension instruments using the Wilhelmy and DuNoüy methods, having expertise in the use of these methods. 2. Materials and methods One way to ensure the reliability of surface tension measurements using the pendant drop shape analysis method is through method comparison [12]. A com- parison was made between the Wilhelmy measurement method and the pendant drop shape analysis method with measurement uncertainty analysis. The ring method was not used because the Wilhelmy method is the most used method in the Laflu, but it may be used in future studies to complement the analyses.This comparison involved surface tension measurements of these liquids: bidistilled water, n-dodecane, and per- fluorocarbon (FC-40), these liquids were chosen to cover a wide range of surface tension with stability [8], allowing for a more comprehensive evaluation of the 276 https://doi.org/10.14311/AP.2025.65.0276 https://creativecommons.org/licenses/by/4.0/ https://www.cvut.cz/en vol. 65 no. 3/2025 Comparison between the Wilhelmy surface tension measurement . . . Influence quantities Unit Estimate Distribution Divisor Sensitivity coefficient Contributions Indicated temperature °C 0.012 Normal 2.000 1.000 10.4 % Correction of the apparent tension mN m−1 0.037 Normal 2.000 1.000 31.8 % Regression deviation mN m−1 0.033 Rectangular 3.464 1.000 91.2 % Resolution mN m−1 5 × 10−4 Rectangular 3.464 1.000 1.3 % Mass g 12 × 10−6 Normal 2.000 243.2 7.1 % Acceleration of gravity m s−2 5 × 10−5 Normal 2.000 2.898×10−4 0.0 % Wetted perimeter m 2 × 10−2 Normal 2.000 −7.046×105 0.4 % Regression deviation mN m−1 0.163 Triangular 1.732 1.000 57.7 % Table 1. Contributions considered in the estimation of uncertainty of the measurement of the correction regression determined in the surface calibration of the tensiometer by the Wilhelmy method. methods used.The temperatures of 20 °C and 25 °C were chosen because they are the most requested tem- peratures for calibration at Laflu by clients, thus en- suring the practical relevance of the results obtained. The Kruss K100 model tensiometer was used for the Wilhelmy method, and the Kruss DSA100 model for the pendant drop shape analysis method. To perform the comparison of the results, the Nor- malized Error (EN ) comparison parameter was used, where the results are considered satisfactory if EN is less than or equal to 1 [13]. 3. Measurement of surface tension by the Wilhelmy method The calibration certificate provides a regression for obtaining surface tension by the Wilhelmy method in mN m−1 (Equation (1)), for measurements of liq- uids in the surface tension range of 15 mN m−1 to 75 mN m−1, in the temperature range of 15 °C to 40 °C, with an expanded uncertainty of 0.077 mN m−1, and a coverage factor of 2.000: γc = C0 + C1 · γi + C2 · (TR − TL), (1) where γc is the corrected surface tension, in mN m−1; γi is the indicated surface tension, in mN m−1; C0, C1, and C2 are the parameters of the regression equation; TR is the reference temperature, in °C; and TL is the measurement temperature, in °C. The apparent sur- face tension was calculated according to Equation (2), being the surface tension that the tensiometer should indicate as a function of the applied force and the perimeter of the plate: γa = m · g Lw · cos θ , (2) where γa is the apparent surface tension, in mN m−1; m is the mass of the calibrated standard weights, ap- plied on the tensiometer, in grams; g is the local grav- ity acceleration, in m s−2; Lw is the wetted perimeter Figure 1. Plate used in the Wilhelmy method mea- surements [14]. of the plate, in m, and θ is the contact angle between the liquid and the plate. The regression uncertainty provided by the calibra- tion certificate of the tensiometer for the Wilhelmy method considers the uncertainties of temperature, the regression deviation, and the correction of the ap- parent surface tension (which is also given by a regres- sion). The apparent surface tension is the verification of the tensiometer’s response with the application of a known force, and considers the uncertainties of the tensiometer’s resolution, the standard masses used, the acceleration of gravity, the wet perimeter of the plate, and the regression deviation. Table 1 presents the contributions of the correction regression uncer- tainty provided in the calibration certificate of the tensiometer considered in the calculation; the shaded part in italics are the contributions to the uncertainty of the apparent surface tension. The plate used in the measurements (Figure 1) is of the Pt-Ir (platinum-iridium) type, with a support rod, 19.912 mm long and 0.212 mm thick. 277 E. C. C. de Farias, R. K. S. Fermam, A. de J. S. M. de Mendonça Acta Polytechnica The flask with the liquid was placed in the thermo- static bath adjusted to the measurement temperature, and a temperature sensor was inserted into the liquid inside the flask to monitor the liquid’s temperature. The plate was attached to the tensiometer, and the tensiometer software parameters were entered, such as plate or ring dimensions, reference temperature, air density, liquid to be measured, number of mea- surements, etc. Once the temperature of the liquid has stabilised, which takes an hour and a half on av- erage to reach 20 °C, and two hours to reach 25 °C, an aliquot of the fluid was transferred to a glass cru- cible (the volume should be slightly above half of the crucible, usually around 50 mL). The tank supporting the crucible is connected to a thermostatic bath via hoses to ensure the solution inside the crucible remains at the measurement temperature. Subsequently, the tank with the crucible was brought close to the plate without touching the liquid’s surface, and a sequence of twenty measurements was initiated, with the first five discarded. Thus, the liquid measurements were performed using the Wilhelmy method at tempera- tures of 20 °C and 25 °C, ensuring rigorous cleaning of the plate between measurements and using a clean crucible for each liquid. 4. Surface tension measurement by the pendant drop shape analysis method The calibration data include the regression for obtain- ing the surface tension in mN m−1 in the temperature range from 18 °C to 30 °C and the surface tension mea- surement range was from 16 mN m−1 to 73 mN m−1 with an expanded uncertainty of 0.28 mN m−1 and a coverage factor k = 2.00, according to: γc = C0 + C1 · TL + C2 · γi, (3) where γc is the corrected surface tension, in mN m−1; γi is the indicated surface tension, in mN m−1; C0, C1 and C2 are the parameters of the regression equation, and TL is the measurement temperature, in °C. The correction regression (Equation (3)), used to correct the tensiometer reading by the pendant drop shape analysis method, considers the contributions from the standard tensiometer uncertainties (23 %), tensiometer resolution (2 %); liquid temperature (7 %) and regression deviation (69 %) in the calculation of measurement uncertainty. The uncertainty in this method is influenced by the repeatability in the drop formation process, which becomes more sensitive to changes in surface tension as the measured liquid has a lower surface tension [15]. Additionally, the resolu- tion in the pendant drop shape method (0.01 mN m−1) is higher than the resolution of the Wilhelmy method (0.001 mN m−1). A thermostatic bath was used to stabilise the tem- perature of the measurement liquid. This bath was Figure 2. Syringe connected to the tensiometer posi- tioner [14]. Figure 3. Captured images of the formation of the pendant drop. connected to the measurement chamber via hoses (Fig- ure 2). The chamber, which has a double glass wall, allows the circulation of water from the thermostatic bath, keeping the measurement environment, where the drop forms, more stable. An aliquot (5 mL) of the liquid was taken with a syringe after the stabili- sation to the bath’s temperature (approximately one and a half hours on average to reach 20 °C and two hours to reach 25 °C), and the syringe was fixed in the holder of the tensiometer. The adjustment was made to visualise the drop on the computer screen and to start the program for forming and visualising the pendant drop (Figure 3). 278 vol. 65 no. 3/2025 Comparison between the Wilhelmy surface tension measurement . . . Liquid Wilhelmy Drop shape analysis method γ Uγ γ Uγ FC-40 16.820 0.077 16.36 0.51 n-dodecane 25.743 0.077 25.77 0.25 Water 72.970 0.078 72.76 0.24 Table 2. Measurement results for the temperature of 20 °C. Liquid Wilhelmy Drop shape analysis method γ Uγ γ Uγ FC-40 16.502 0.078 16.16 0.68 n-dodecane 25.435 0.077 25.32 0.28 Water 71.382 0.077 72.27 0.26 Table 3. Measurement results for the temperature of 25 °C. The tensiometer software uses the Young-Laplace equation [7] (Equation (4)) to determine the curve that provides the best fit to the contour of the formed drop (Figure 3): ∆P = γ ( 1 R1 + 1 R2 ) , (4) where γ is the surface tension, in mN m−1; R1 and R2 are the two principal radii of the curvature; ∆P is the pressure difference at the interface, given by: ∆P = ∆P0 + (∆ρ)gz, (5) where ∆P0 is the pressure difference at a reference plane; ∆ρ is the density difference between the two phases; g is the acceleration due to gravity; and z is the vertical height of the given point on the drop surface, measured from the reference level. With the known parameters, the software calculates the theoretical images and compares them with the experimental images. 5. Results and discussions The results of the corrected surface tension of the liquids (γ), according to the equation provided in the calibration certificate and the measurement un- certainty calculated according to GUM [16] (Uγ), are provided in Tables 2 and 3. In the measurement uncertainty of the results by the Wilhelmy method, approximately 88 % of the con- tribution to the uncertainty comes from the correction regression, while the remaining 12 % are attributed to repeatability. In the measurements performed by the pendant drop shape analysis method, the con- tributions to the measurement uncertainty were ap- proximately 77.3 % from the standard used in the calibration, 0.1 % from the temperature, 0.1 % from the temperature variation, and 22.5 % from repeata- bility. In the Wilhelmy method, the correction of the ap- parent surface tension was one of the main contri- butions to the measurement uncertainty in the ten- siometer calibration. For the tensiometer used in this work, it is the second largest contribution. The largest contribution was from the regression deviation that corrects the indication and adjusts it to the reference temperature. In this regard, the pendant drop shape analysis method presents an advantage, as there is no need for an apparent tension correction in the cali- bration. The pendant drop shape analysis method is more suitable for small sample volumes. In this work, 5 mL was used for each liquid measurement, while for the Wilhelmy method, approximately 50 mL of each liquid was used. In a temperature-controlled environment, as in the measurements conducted in this study, the impact of temperature on the uncertainty calculations is min- imal. The liquids were stabilised at measurement temperatures in a bath, and the measurement tank of the tensiometer has a jacket that circulates the bath water, ensuring better temperature stabilisation during measurements. Some tensiometer models do not have temperature stabilisation features, requiring an understanding of how the surface tension of the liquid changes with temperature. It is essential to con- duct measurements close to the desired temperature to apply tension corrections accurately. This is par- ticularly important when the tensiometer’s reference temperature differs from the measured temperature. Table 4 presents the results of the calculation of the normalised error (EN ) between the measurement results obtained by the Wilhelmy method and the pendant drop shape analysis method according to: |EN | = Vp − VR√ Up 2 + UR 2 , (6) where Vp is the measured value, VR is the reference value, Up is the measurement uncertainty of the mea- 279 E. C. C. de Farias, R. K. S. Fermam, A. de J. S. M. de Mendonça Acta Polytechnica Liquid EN (20 °C) EN (25 °C) FC-40 0.90 0.57 n-dodecane 0.11 0.38 Water 0.82 0.41 Table 4. Normalised error. sured value, and UR is the measurement uncertainty of the reference value. Comparing the EN results of liquid measurements between methods using the normalised error criterion, the results are compatible. The result for water at 20 °C was the closest to the failure limit. The mea- surement of water is more susceptible to variations due to the characteristics of its molecular interactions. It is important to note that the measurement uncer- tainty for the FC-40 results was higher compared to the other liquids measured, as there was a greater variation in the measurement. With lower surface tension, the drop shape becomes more sensitive to changes in surface tension, as was the case here, as FC-40 has the lowest surface tension of the liquids measured. 6. Conclusion The results of the comparison between the meth- ods were compatible, which ensures the reliability of measurements with the pendant drop shape analy- sis method. The tensiometers used have a regression in their calibration certificate to correct the indicated surface tension values. An advantage is the ease of correcting the instrument indication across the cal- ibrated range. However, the disadvantage is the in- crease in the measurement uncertainty due to the regression deviation, which, as shown in the results, is the largest contribution. In the case of the pendant drop shape analysis tensiometer, which has a reso- lution of 0.01 mN m−1, the uncertainty was about 0.25 mN m−1 in the best case, which was the measure- ment of dodecane. Regression can be an alternative for cases where the measured values are intermediate to the calibration points. Recalibrating the tensiome- ters, providing the measured results as an alternative to the adjustment regression, can be a viable option to improve the measurement uncertainty. Each method has peculiarities, with inherent advantages and dis- advantages. 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