EQUILIBRIUM VOLUME OF A SMALL SESSILE DROP

  • Aslan A. Sokurov Junior Researcher, Institute of Applied Mathematics and Automation of Kabardin-Balkar Scientific Centre, RAS, Nalchik, Russia; tel.: +7(965) 4958002, e-mail: asokuroff@gmail.com
Keywords: sessile drop, surface tension, capillary pressure, Laplace equation, contact angle, capillary constant, equilibrium capillary surface, size dependence, Tolman length, mean curvature, radius of curvature, nano-droplet.

Abstract

In the current paper we consider a small liquid drop resting on a horizontal smooth surface with the effect of gravity when it is in thermodynamic equilibrium with its own vapor. An equation that expresses the main condition for the mechanical equilibrium of the droplet surface is obtained taking into account the size dependence of the surface tension. This equation is an analog of the Bashforth – Adams equation that is well known from the mathematical theory of equilibrium capillary surfaces. Based on the analog of the Bashforth – Adams equation systems of nonlinear first-order differential equations describing the drop profile are obtained. The Cauchy problem for the resulting systems of differential equations has an analytical solution only in the trivial case when the parameter associated with the gravitational field strength equals zero. In other cases it is not possible to find an exact solution which makes it necessary to use numerical methods and software complexes. It is established that the Runge – Kutta fourth order method may be used as an effective numerical method for finding the approximate solution of the formulated problems. An integral relation between the coordinates of an arbitrary point on the droplet surface and the volume of the enclosed liquid is found. In the general case this relation is not expressed in terms of elementary functions. For this reason an approximation formula, an asymptotic equality and estimates (indications of the accuracy are given) are obtained for it. The computational experiment on the effect of the volume of a liquid on the droplet shape is presented. All equations and formulas go over to the earlier known if the parameter responsible for the size effect equals zero. The results obtained in the work may find application in the development of methods for the determination of surface tension and in studying the second kind capillary phenomena.

 

 

Downloads

Download data is not yet available.

References

1. Matyukhin S. I., Frolenkov K. Yu. Condensed Matter and Interphases, 2013, vol. 15, no. 3, pp. 292–304. Available at: http://www.kcmf.vsu.ru/resources/t_15_3_2013_012.pdf (in Russ.)
2. Del Rı́o O. I., Neumann A. W. Journal of Colloid and Interface Science, 1997, vol. 196, no. 2, pp. 136–147. DOI: 10.1006/jcis.1997.5214
3. Rotenberg Y., Boruvka L., Neumann A. W. Journal of Colloid and Interface Science, 1983, vol. 93, no. 1, pp. 169–183. DOI: 10.1016/0021-9797(83)90396-X
4. Aurélien F. S., et al. Colloids and Surfaces A: Physicochemical and Engineering Aspects, 2010, vol. 364, no. 1–3, pp. 72–81. DOI: 10.1016/j.colsurfa.2010.04.040
5. Faour G., et al. Journal of Colloid and Interface Science, 1996, vol. 181, no. 2, pp. 385–392. DOI: 10.1006/jcis.1996.0395
6. Rusanov А. I., Prokhorov V. А. Mezhfaznaya tenziometriya [Interfacial Tensiometry]. Saint Petersburg, Khimiya Publ., 1994, 400 p. (in Russ.)
7. Tolman R. C. The Journal of Chemical Physics, 1949, vol. 17, no. 3, pp. 333–337. DOI: 10.1063/1.1747247
8. Rekhviashvili S. Sh., Kishtikova E. V. Zhurnal tekhnicheskoj fiziki [Technical Physics. The Russian Journal of Applied Physics], 2011, vol. 81, no. 1, pp. 148–152. Available at: http://journals.ioffe.ru/articles/viewPDF/10213 (in Russ.)
9. Rekhviashvili S. Sh., Kishtikova E. V. Fizikokhimiya poverkhnosti i zashhita materialov [Protection of Metals and Physical Chemistry of Surfaces], 2014, vol. 50, no. 1, pp. 3–7. DOI: 10.7868/S0044185614010112 (in Russ.)
10. Kalová J., Mareš R. International Journal of Thermophysics, 2015, vol. 36, no. 10–11, pp. 2862–2868. DOI: 10.1007/s1076
11. Burian S. Physical review E, 2017, vol. 95, no. 6, p. 062801. DOI: 10.1103/PhysRevE.95.062801
12. Wente H. C. Pacifc J. Math., 1980, vol. 88, no. 2, pp. 387–397.
13. Sokurov А. А., Rekhviashvili S. Sh. Condensed Matter and Interphases, 2013, vol. 15, no. 2, pp. 173–178. Available at: http://www.kcmf.vsu.ru/resources/t_15_2_2013_014.pdf (in Russ.)
14. Bashforth F., Adams J. C. An Attempt to Test the Theories of Capillary Action by Comparing the Theoretical and Measured Forms of Drops of Fluid, University Press, Cambridge, 1883, 158 p.
15. Kanchukoev V. Z. Pis'ma v Zhurnal tekhnicheskoj fiziki [Technical Physics Letters], 2004, vol. 3, no. 2, pp. 12–16. Available at: http://journals.ioffe.ru/articles/viewPDF/11286 (in Russ.)
16. Markov I. I., et al. Vestnik Severo-Kavkazskogo gosudarstvennogo tekhnicheskogo universiteta [Journal Newsletter of North-Caucasus State Technical University], 2009, vol. 19, no. 2, pp. 51–58. Available at: https://elibrary.ru/item.asp?id=12833518 (in Russ.)
17. Finn R. Equilibrium Capillary Surfaces. New York, Springer, 1986, 284 p. DOI: 10.1007/978-1-4613-8584-4
Published
2018-09-12
How to Cite
Sokurov, A. A. (2018). EQUILIBRIUM VOLUME OF A SMALL SESSILE DROP. Kondensirovannye Sredy I Mezhfaznye Granitsy = Condensed Matter and Interphases, 20(3), 460-467. https://doi.org/10.17308/kcmf.2018.20/583
Section
Статьи