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The determination of the surface free energy $\gamma_{\text{SV}}$ of solid materials provides information to explain and understand a wide variety of phenomena in surface science. One of the most widely used methods to determine the value of $\gamma_{\text{SV}}$ is the Neumann equation of state (EQS), whose application requires the values of the contact angle $(\theta)$ and the surface tension of a probe liquid $\gamma_{\text{LV}}$. Both parameters are determined experimentally, and their values may be subject to considerable uncertainty. In the present work, the uncertainties of the EQS are analyzed by means of the Taylor series method, using a polynomial adjustment to the EQS previously reported. This analysis allowed us to determine the effects of the ${\theta}$ and $\gamma_{\text{LV}}$ uncertainties on $\gamma_{\text{SV}}$ values. The $\gamma_{\text{SV}}$ of polyoxymethylene (POM) and polytetrafluoroethylene (PTFE) was calculated using the EQS from four probe liquids and taking into account the propagation of the ${\theta}$ and $\gamma_{\text{LV}}$ uncertainties. Generally, the comparison of the $\gamma_{\text{SV}}$ values calculated from the different probe fluids revealed significant differences. As a solution to this inconsistency in the EQS method, we proposed taking into account a systematic standard uncertainty associated with the method equal to $1.3 \, \text{mJ/m}^2$ . This allowed the differences between the $\gamma_{\text{SV}}$ values for the different probe liquids to be non-significant and therefore the method to be consistent., Comment: 46 pages, 6 tables, 10 figures

The Neumann Equation of State (EQS) allows obtaining the value of the surface free energy of a solid ${\gamma}_{SV}$ from the contact angle $({\theta})$ of a probe liquid with known surface tension ${\gamma}_{LV}$. The value of ${\gamma}_{SV}$ is obtained by numerical methods solving the corresponding EQS. In this work, we analyzed the discrepancies between the values of ${\gamma}_{SV}$ obtained using the three versions of the EQS reported in the literature. The condition number of the different EQS was used to analyze their sensitivity to the uncertainty in the ${\theta}$ values. Polynomials fit to one of these versions of EQS are proposed to obtain values of ${\gamma}_{SV}$ directly from contact angles $({\gamma}_{SV} ({\theta}))$ of particular probe liquids. Finally, a general adjusted polynomial is presented to obtain the values of ${\gamma}_{SV}$ not restricted to a particular probe liquid $({\gamma}_{SV}({\theta},{\gamma}_{LV}))$. Results showed that the three versions of EQS present non-negligible discrepancies, especially at high values of ${\theta}$. The sensitivity of the EQS to the uncertainty in the values of ${\theta}$ is very similar in the three versions and depends on the probe liquid used (greater sensitivity at higher ${\gamma}_{LV})$ and on the value of ${\gamma}_{SV}$ of the solid (greater sensitivity at lower ${\gamma}_{SV})$. The discrepancy of the values obtained by numerical resolution of both the fifth-order fit polynomials and the general fit polynomial was low, no larger than ${\pm}0.40\,mJ/m^{2}$. The polynomials obtained allow the analysis and propagation of the uncertainty of the input variables in the determination of ${\gamma}_{SV}$ in a simple and fast way., Comment: 37 pages, 13 figures

Here, we present the construction and calibration of a low-cost goniometer to measure contact angles by the sessile drop method. Besides, we propose a simple and fast method to calculate the uncertainty in the determination of the surface free energy (SFE) and its polar and dispersive components through the Owens-Wendt model and tested it by using two testing liquids. The goniometer performance and the SFE uncertainty were determined on two polymers: polytetrafluorethylene (PTFE) and polyoxymethylene (POM), by using water and methylene iodide. The values of contact angle measured were used to calculate the SFE and its components with their errors. The SFE values obtained for PTFE were 17.57-17.91 mJ/m^2, with a relative error lower than 5.5 %, whereas those for POM were 42.80-43.23 mJ/m^2, with a relative error lower than 4.3%. Both the SFE values and the errors were in the range of those previously reported. Based on the mathematical analysis of the uncertainty propagation in the determination of SFE, we concluded that the uncertainty is minimized when the testing liquids are an apolar liquid and water.