# Entropy Contribution to the Line Tension: Insights from Polymer Physics, Water String Theory, and the Three-Phase Tension

## Abstract

**:**

## 1. Introduction

^{−5}–10

^{−12}N were reported [6,7,8,9,10,11]. Very few methods allowing experimental measurement of line tension were developed [9,10,11,12,13,14]. Marmur estimated a line tension as $\Gamma \cong 4{d}_{m}\sqrt{{\gamma}_{SA}\gamma}\mathrm{cot}{\theta}_{Y}$, where d

_{m}is the molecular dimension, ${\gamma}_{SA},\gamma $ are surface energies of solid and liquid correspondingly, and θ

_{Y}is the Young angle. Marmur concluded that the magnitude of the line tension is less than 5 × 10

^{−9}N, and that it is positive for acute and negative for obtuse Young angles [15,16]. However, researchers reported negative values of line tension for hydrophilic surfaces [14]. As for the magnitude of line tension, the values in the range 10

^{−9}–10

^{−12}look realistic. Large values of $\Gamma $ reported in the literature are most likely due to contaminations of the solid surfaces [3].

## 2. “Water String Theory”, Insights from Polymer Physics and Entropy Contribution into Line Tension

_{m}. Regrettably, experimental data in the field are scarce, and do not enable clear conclusion on this item.

## 3. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Origin of the Entropic Elastic Forces

_{en}is given by Equation (A2):

**Figure A1.**The origin of the entropic force ${\overrightarrow{f}}_{en}$ is exemplified with the stretching of the polymer ribbon. Sketch (

**a**) depicts the non-stretched polymer ribbon. Molecules of polymer are non-stretched. Sketch (

**b**) depicts the same ribbon when stretched (the stretch is dl). Molecules of polymer are stretched; the entropy of the ribbon S is correspondingly decreased; $\left|{f}_{en}\right|=-T\frac{\Delta S}{dl}>0$.

## References

- Adamson, A.W.; Gast, A.P. Physical Chemistry of Surfaces, 6th ed.; Wiley Interscience Publishers: New York, NY, USA, 1990. [Google Scholar]
- Erbil, H.Y. Surface Chemistry of Solid and Liquid Interfaces. Wiley-Blackwell: Oxford, UK, 2006. [Google Scholar]
- De Gennes, P.G.; Brochard-Wyart, F.; Quéré, D. Capillarity and Wetting Phenomena: Drops Bubbles; Pearls, W., Ed.; Springer: New York, NY, USA, 2004. [Google Scholar]
- Bormashenko, Ed.Y. Wetting of Real Surfaces; Walter de Gruyter: Berlin, Germany, 2011. [Google Scholar]
- Gibbs, J.W. The Scientific Papers of J. W. Gibbs; Dover: New York, NY, USA, 1961; Volume 1, p. 288. [Google Scholar]
- Amirfazli, A.; Neumann, A.W. Status of the Three-Phase Line Tension. Adv. Colloid Interface Sci.
**2004**, 110, 121–141. [Google Scholar] [CrossRef] [PubMed] - Drelich, J. The Significance and Magnitude of the Line Tension in Three-Phase (Solid-Liquid-Fluid) Systems. Colloids Surf. A
**1996**, 116, 43–54. [Google Scholar] [CrossRef] - Das, S.K.; Egorov, S.A.; Virnau, P.; Winter, D.; Binde, R. Do the Contact Angle and Line Tension of Surface-Attached Droplets Depend on the Radius of Curvature? J. Phys. Condens. Matter
**2018**, 30, 255001. [Google Scholar] [CrossRef] [PubMed] - Zhelev, D.V.; Needham, D. Tension-stabilized Pores in Giant Vesicles: Determination of Pore Size and Pore Line Tension. Biochim. Biophys. Acta
**1993**, 1147, 89–104. [Google Scholar] [CrossRef] - De Feijter, J.A.; Vrij, I. Transition Regions, Line tensions and Contact Angles in Soap Films. J. Electroanal. Chem. Interfacial Electrochem.
**1972**, 37, 9–22. [Google Scholar] [CrossRef] - Law, B.M.; McBride, S.P.; Wang, J.Y.; Wi, H.S.; Paneru, G.; Betelu, S.; Ushijima, B.; Takata, Y.; Flanders, B.; Bresme, F.; et al. Line Tension and its Influence on Droplets and Particles at Surfaces. Prog. Surface Sci.
**2017**, 92, 1–39. [Google Scholar] [CrossRef] - Alexandrov, A.D.; Toshev, B.V.; Scheludko, A.D. Nucleation from Supersaturated Water Vapors on n-Hexadecane: Temperature Dependence of Critical Supersaturation and Line Tension. Langmuir
**1991**, 7, 3211–3215. [Google Scholar] [CrossRef] - Checco, A.; Guenoun, P. Nonlinear Dependence of the Contact Angle of Nanodroplets on Contact Line Curvature. Phys. Rev. Lett.
**2003**, 91, 186101. [Google Scholar] [CrossRef] [PubMed] - Pompe, T.; Fery, A.; Herminghaus, S. Measurement of Contact Line Tension by Analysis of the Three-Phase Boundary with Nanometer Resolution. In Apparent and Microscopic Contact Angles; Drelich, J., Laskowski, J.S., Mittal, K.L., Eds.; VSP: Utrecht, The Netherlands, 2000; pp. 3–12. [Google Scholar]
- Marmur, A. Line Tension and the Intrinsic Contact Angle in Solid-Liquid-Fluid Systems. J. Colloid Interface Sci.
**1997**, 186, 462–466. [Google Scholar] [CrossRef] [PubMed] - Marmur, A. Line Tension Effect on Contact Angles: Axisymmetric and Cylindrical Systems with Rough or Heterogeneous Solid surfaces. Colloids Surf. A
**1998**, 136, 81–88. [Google Scholar] [CrossRef] - Bormashenko, E.; Whyman, G. On the Role of the Line Tension in the Stability of Cassie Wetting. Langmuir
**2013**, 29, 5515–5519. [Google Scholar] [CrossRef] [PubMed] - Schimmele, L.; Napiórkowski, M.; Dietrich, S. Conceptual Aspects of Line Tensions. J. Chem. Phys.
**2007**, 127, 164715. [Google Scholar] [CrossRef] [PubMed] - Weijs, J.H.; Marchand, A.; Andreotti, B.; Lohse, D.; Snoeije, J.H. Origin of Line Tension for a Lennard-Jones Nanodroplet. Phys. Fluids
**2011**, 23, 022001. [Google Scholar] [CrossRef] - Khalkhali, M.; Kazemi, N.; Zhang, H.; Liu, Q. Wetting at the Nanoscale: A Molecular Dynamics Study. J. Chem. Phys.
**2017**, 146, 114704. [Google Scholar] [CrossRef] [PubMed] - Nosonovsky, M.; Ramachandran, R. Geometric Interpretation of Surface Tension Equilibrium in Superhydrophobic Systems. Entropy
**2015**, 17, 4684–4700. [Google Scholar] [CrossRef] [Green Version] - Wernet, P.; Nordlund, D.; Bergmann, U.; Cavalleri, M.; Odelius, M.; Ogasawara, H.; Näslund, L.Å.; Hirsch, T.K.; Ojamäe, L.; Glatzel, P.; et al. The Structure of the First Coordination Shell in Liquid Water. Science
**2004**, 304, 995–999. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Head-Gordon, T.; Johnson, M.E. Tetrahedral Structure or Chains for Liquid Water. PNAS
**2006**, 103, 7973–7977. [Google Scholar] [CrossRef] [PubMed] - Ball, P. Water: Water—An Enduring Mystery. Nature
**2008**, 452, 291–292. [Google Scholar] [CrossRef] [PubMed] - Scatena, L.F.; Brown, M.G.; Richmond, G.L. Water at Hydrophobic Surfaces: Weak Hydrogen Bonding and Strong Orientation Effects. Science
**2001**, 292, 908–912. [Google Scholar] [CrossRef] [PubMed] - Strazdaite, S.; Versluis, J.; Backus, E.H.G.; Bakker, H.J. Enhanced Ordering of Water at Hydrophobic Surfaces. J. Chem. Phys.
**2014**, 140, 054711. [Google Scholar] [CrossRef] [PubMed] - Strazdaite, S.; Versluis, J.; Bakker, H.J. Water Orientation at Hydrophobic Interfaces. J. Chem. Phys.
**2015**, 143, 084708. [Google Scholar] [CrossRef] [PubMed] - Rubinstein, M.; Colby, R.H. Polymer Physics; Oxford University Press: Oxford, UK, 2003. [Google Scholar]
- Rowlinson, J.S.; Widom, B. Molecular Theory of Capillarity; Clarendon: Oxford, UK, 1982. [Google Scholar]
- Pohl, R.W. Mechanic, Akystik und Wärmlehre; Springer: Berlin, Germany, 1964. [Google Scholar]
- Wang, J.Y.; Betelu, S.; Law, B.M. Line Tension Approaching a First-Order Wetting Transition: Experimental Results from Contact Angle Measurements. Phys. Rev. E
**2001**, 63, 031601. [Google Scholar] [CrossRef] [PubMed] - Tadmor, R. Line Energy and the Relation between Advancing, Receding, and Young Contact Angles. Langmuir
**2004**, 20, 7659–7664. [Google Scholar] [CrossRef] [PubMed] - Tadmor, R. Line Energy, Line Tension and Drop Size. Surf. Sci.
**2008**, 602, L108–L111. [Google Scholar] [CrossRef] - Schmitt, M.; Groß, K.; Grub, J.; Heib, F. Detailed Statistical Contact Angle Analyses; “Slow Moving” Drops on Inclining Silicon-Oxide Surfaces. J. Colloid Interface Sci.
**2015**, 447, 229–239. [Google Scholar] [CrossRef] [PubMed] - Schmitt, M.; Heib, F. High-Precision Drop Shape Analysis on Inclining Flat Surfaces: Introduction and Comparison of This Special Method with Commercial Contact Angle Analysis. J. Chem. Phys.
**2013**, 139, 134201. [Google Scholar] [CrossRef] [PubMed] - Bar-Ziv, R.; Frisch, T. Moses, Entropic Expulsion in Vesicles. Phys. Rev. Lett.
**1995**, 75, 3481–3484. [Google Scholar] [CrossRef] [PubMed] - Evans, E.; Rawicz, W. Entropy-Driven Tension and Bending Elasticity in Condensed-Fluid Membranes. Phys. Rev. Lett.
**1990**, 64, 2094–2097. [Google Scholar] [CrossRef] [PubMed] - Linde, M.; Sens, P.; Phillips, R. Entropic Tension in Crowded Membranes. PLoS Comput. Biol.
**2012**, 8, e100243. [Google Scholar]

**Figure 1.**Three-phase line of the sessile droplet is approximated by the polymer chain with a diameter of the monomer d

_{m}, where d

_{m}is the diameter of the liquid molecule.

© 2018 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Bormashenko, E.
Entropy Contribution to the Line Tension: Insights from Polymer Physics, Water String Theory, and the Three-Phase Tension. *Entropy* **2018**, *20*, 712.
https://doi.org/10.3390/e20090712

**AMA Style**

Bormashenko E.
Entropy Contribution to the Line Tension: Insights from Polymer Physics, Water String Theory, and the Three-Phase Tension. *Entropy*. 2018; 20(9):712.
https://doi.org/10.3390/e20090712

**Chicago/Turabian Style**

Bormashenko, Edward.
2018. "Entropy Contribution to the Line Tension: Insights from Polymer Physics, Water String Theory, and the Three-Phase Tension" *Entropy* 20, no. 9: 712.
https://doi.org/10.3390/e20090712