Comment on “Discussion on a mechanical equilibrium condition of a sessile drop on a smooth solid surface” [J. Chem. Phys. 130, 144106 (2009)] H. Ghasemi and C. A. Ward Citation: J. Chem. Phys. 134, 247101 (2011); doi: 10.1063/1.3605662 View online: http://dx.doi.org/10.1063/1.3605662 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v134/i24 Published by the American Institute of Physics.

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THE JOURNAL OF CHEMICAL PHYSICS 134, 247101 (2011)

Comment on “Discussion on a mechanical equilibrium condition of a sessile drop on a smooth solid surface” [J. Chem. Phys. 130, 144106 (2009)] H. Ghasemi and C. A. Warda) Thermodynamics and Kinetics Laboratory, Department of Mechanical and Industrial Engineering, University of Toronto, 5 King’s College Road, Toronto M5S 3G8, Canada

(Received 11 January 2011; accepted 9 June 2011; published online 29 June 2011) [doi:10.1063/1.3605662] NECESSARY CONDITIONS FOR EQUILIBRIUM

The conditions for equilibrium of a sessile droplet have recently been considered in two papers. One appeared in 2009 (Ref. 1) and the other in 2010.2 Both papers had the objective of deriving the Young equation, but ironically the second paper did not reference the first, and both are based on the same mathematical errors. The authors find two conditions for equilibrium at threephase line. If the pressure in the vapor phase is denoted Pg , the liquid phase pressure as Pl , the mean liquid-vapor interface curvature as Hlg , the liquid-vapor surface tension as σlg , and the contact angle as θ , their first condition may be written (Eq. (20) of Ref. 1) 0 = [(Pg − Pl ) + 2Hlg σlg ] sin θ.

(1)

Note that without approximation, this first condition gives the Laplace equation (Pl − Pg ) = 2Hlg σlg .

(2)

Their second condition can be written (Eq. (25) of Ref. 1 and Eq. (16) of Ref. 2) 0 = [(Pg − Pl ) + 2Hlg σlg ] sin θlw + σlg cos θ − σsg + σsl . (3) Thus, if they had taken into account their first condition, Eq. (2), their second condition would have reduced to the Young equation, σsg − σsl = σlg cos θ.

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(4)

EFFECT OF CURVATURE ON THE LAPLACE PRESSURE DIFFERENCE

The authors make a further unjustified assumption. They say: “In the present study, it is assumed that the droplet on the solid surface is very small (Pl ≈ Pg ).” But for very small droplets, the mean curvature in the Laplace equation becomes very large, not zero. Thus, in this limit Eq. (2) indicates Pl  Pg . Thus, their modification of the Young equation is not valid: they keep the curvature term, 2Hlg σlg , in Eq. (2) while neglecting (Pl − Pg ) and obtain as the “modified” Young equation: σsg = 2Hlg σlg sin θ + σlg cos θ + σsl .

(5)

The authors then introduce fitting parameters and compare Eq. (5) with selected data, but Eq. (5) has no basis. We note that three other investigations of the necessary conditions for equilibrium of an isolated liquid-vapor-solid system have been reported.3–5 All three obtained the Laplace and the Young equations as necessary conditions for equilibrium. They also included both the effects of gravity and allowed for molecular exchange at the interfaces. a) Electronic

mail: [email protected]. Yonemoto and T. Kunugi, J. Chem. Phys. 130, 144106 (2009). 2 Y. Yonemoto and T. Kunugi, Phys. Rev. E 81, 056310 (2010). 3 J. W. Gibbs, in The Scientific Papers of J. Willard Gibbs, edited by H. A. Bumstead and R. G. V. Name (Dover, New York, 1961), Vol. 1, pp. 55–349. 4 C. A. Ward and M. R. Sasges, J. Chem. Phys. 109, 3651 (1998). 5 O. Voitcu and J. A. W. Elliott, J. Phys. Chem. B 112, 11981 (2008). 1 Y.

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