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<p>The pendant drop method is a widely used technique for measuring boundary tension (the term involves both surface and interfacial tension). The method is based on the evaluation of the shape of a drop in mechanical equilibrium. The method was often declared suitable for viscous materials. Contrary to this, an opposite opinion can sometimes be encountered. The explanation is that the higher limit of the viscosity range where the method is applicable (units or tens of Pa·s) is higher than that for many traditional methods. However, the measurement of materials with a viscosity of hundreds of Pa·s or more still brings complications. Waiting for equilibrium can be inconvenient or impossible. The relaxation of a pendant drop has not been mathematically described. However, the relaxation of an ellipsoid was described; it forms the principle of the deformed drop retraction (DDR) measurement, which is more appropriate for highly viscous materials. This paper brings together the findings from various papers dealing with errors in boundary tension measurements by the pendant drop method caused by mechanical disequilibrium. It also explores how an understanding of the relaxation process of the ellipsoid drop can help to estimate boundary tension in the pendant drop method when reaching mechanical equilibrium is a problem. The shape evolution of the pendant drop has been approximated using equations derived from the ellipsoid drop relaxation model. Recently, it has been found that the equations based on the DDR model can describe a small subset of drop behaviour, and that the parameters found in the equations vary across individual drops of the same material. Consequently, the tested equations do not permit the evaluation of boundary tension from drop shape evolution using the principles of the DDR method. Nonetheless, they can be used to extrapolate the equilibrium shape parameters for the pendant drop method. To determine boundary tension using the DDR approach, a specific mathematical model describing the relaxation of the pendant drop has to be developed. In addition, a model for non-Newtonian fluids would be useful for polymers. Such a model would require capturing the complex behaviour of these materials to improve measurement accuracy.</p> Graphical Abstract <p></p>

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Boundary tension measurement by pendant drop method: relaxation of drop shape and errors caused by mechanical disequilibrium

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年1月15日

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  • Josef Jůza,
  • Martina Nevoralová,
  • Zdeněk Starý

摘要

The pendant drop method is a widely used technique for measuring boundary tension (the term involves both surface and interfacial tension). The method is based on the evaluation of the shape of a drop in mechanical equilibrium. The method was often declared suitable for viscous materials. Contrary to this, an opposite opinion can sometimes be encountered. The explanation is that the higher limit of the viscosity range where the method is applicable (units or tens of Pa·s) is higher than that for many traditional methods. However, the measurement of materials with a viscosity of hundreds of Pa·s or more still brings complications. Waiting for equilibrium can be inconvenient or impossible. The relaxation of a pendant drop has not been mathematically described. However, the relaxation of an ellipsoid was described; it forms the principle of the deformed drop retraction (DDR) measurement, which is more appropriate for highly viscous materials. This paper brings together the findings from various papers dealing with errors in boundary tension measurements by the pendant drop method caused by mechanical disequilibrium. It also explores how an understanding of the relaxation process of the ellipsoid drop can help to estimate boundary tension in the pendant drop method when reaching mechanical equilibrium is a problem. The shape evolution of the pendant drop has been approximated using equations derived from the ellipsoid drop relaxation model. Recently, it has been found that the equations based on the DDR model can describe a small subset of drop behaviour, and that the parameters found in the equations vary across individual drops of the same material. Consequently, the tested equations do not permit the evaluation of boundary tension from drop shape evolution using the principles of the DDR method. Nonetheless, they can be used to extrapolate the equilibrium shape parameters for the pendant drop method. To determine boundary tension using the DDR approach, a specific mathematical model describing the relaxation of the pendant drop has to be developed. In addition, a model for non-Newtonian fluids would be useful for polymers. Such a model would require capturing the complex behaviour of these materials to improve measurement accuracy.

Graphical Abstract