Research: Science and Education
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The Gibbs Treatment of Interfaces Allan J. B. Spaull Brunel University, Uxbridge, Middlesex, UB8 3 PH, United Kingdom;
[email protected] In his approach to interfaces, Gibbs employs rigorous methods, which makes his treatment difficult to understand. These difficulties are compounded by his classical style. His treatment is divided into two sections: plane and curved interfaces. There is a considerable literature aimed at simplifying and modernizing his treatment of plane interfaces; however, there is a paucity of similar literature concerning the second part his treatment, that is, curved interfaces. Compared with a plane interface, a curved interface is complicated by having to include with the term Aγ curvature terms C1兾r1 and C2兾r2, where γ is the surface tension, A the area of the surface, C ’s the curvature constants, expressing the variation of tension in the hypothetical Gibbs (dividing) surface with variation in its location, and r’s the principal radii of curvature. The problem that arises is where to locate the curved Gibbs surface so that the C’s vanish. The pivotal equation Gibbs uses is,
As far as we are aware there is no literature illustrating how Gibbs dealt with the problem of vanishing C ’s, compared with the large literature illustrating his treatment of plane interfaces. The importance of Gibbs’ method is that he uses a purely continuum method to show the location of the Gibbs surface where the C ’s vanish. This location is close to or coincident with the surface of tension. Many authors, in discussing curved surfaces, start at the point where Gibbs finishes, that is, by locating their Gibbs surface at the surface of tension without giving any reason why their C ’s vanish. We give an overview of Gibbs’ treatment showing his transition from plane to curved interfaces, in which we show how Gibbs probably derived the above pivotal equation, and emphasize the difference between the properties of the Gibbs surface and the surface of tension. Our main aim is to show how with a little help Gibbs may be read in the original so that we can appreciate the elegance of his method. This is suitable for undergraduate and research students.
C1′ + C 2′ = a γ λ + C1′ ′ + C 2′ ′ W
where the superscripts ′ and ″ denote different locations of the Gibbs surface, a an area, and λ a constant; little is given by Gibbs as to how this equation is derived.
www.JCE.DivCHED.org
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Supplemental Material
Detailed discussion of Gibbs’ treatment of interfaces is available in this issue of JCE Online.
Vol. 81 No. 3 March 2004
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Journal of Chemical Education
423
