Theory of Surface Tension

An interface is the region at the boundary between two immiscible phases. Ideally, the interfacial boundary is considered to be the plane dividing the phases. In real systems, the interface is not a plane but a region which has a composition different from that of the bulk phase (Fig. 4.1). The boundary may be solid-solid, solid-liquid, solid-vapor, liquid-liquid, or liquid-vapor immiscible phases. If one of the two phases in contact with each other is a gas or vapor, the term surface is used to denote the boundary. 

Energetics predict that the free energy of interface formation has to be positive. If the free energy is negative, the interface is unstable or metastable. If the free energy is zero, the two phases are miscible and a boundary cannot exist. 

Because the free energy of interface formation is positive, work has to be done in order to expand the interface. The work, required to expand the surface of a liquid reversibly and isothermally by a unit area is called the surface tension, y, 

As a consequence of surface tension, a liquid tends to contract its surface area. In the absence of other forces, a drop assumes a spherical shape. 

The cause of surface tension is the difference in attractive forces acting on molecules at the surface. From the energetics point of view, molecules at or near 

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Tolman, R.C., Consideration of the Gibbs theory of surface tension, J. Chem. Phys. 1948,16, 758-774... 

The theory of surface tension, in other words, the problem how certain known facts and certain assumptions about the liquid state can be made to account for the existence of a surface tension, has been treated exhaustively by Laplace, by Gauss, and more recently by Van der Waals. The mathematical apparatus employed is very considerable, and we must confine ourselves to a statement of the... 

In the preceding pages we have availed ourselves of only one of the theories of surface tension, that of Laplace. It has led us directly to recognise an important property of liquids—their cohesion or intrinsic pressure—and has enabled us to establish... 

Molecular Theory of Surface Tension (Harasima) Molecules, Barriers to Internal Rotation in (Wilson) Molecules, Convex, in Gaseous and Crystalline States (Kihara). ... 

MSE.I8. I. Prigogine, The molecular theory of surface tension, in Cavitation in Real liquids, R. Davies, ed., Elsevier, Amsterdam, 1964, pp. I47-I63. 

The molecular theory of surface tension was dealt with by Laplace (1749-1827). But, as a result of the clarification of the nature, of intermolecular forces by quantum mechanics and of the more recent developments in the study of molecular distribution in liquids, the nature and value of surface tension have been better understood from a molecular viewpoint. Surface tension is closely associated with a sudden, but continuous change in the density from the value for bulk liquid to the value for die gaseous state in traversing the surface. See Fig. 2. As a result of this inhomogeneity, the stress across a strip parallel to the boundary—pu per unit area—is different from that across a strip perpendicular to die boundary—pr per unit area. This is in contrast with die case of homogeneous fluid in which the stress across any elementary plane has the same value regardless of the direction of die plane,... 

Ono S., and Kondo S. Molecular theory of surface tension in liquids, in "Encyclopedia of Physics" (S. Fliigge, Ed.), Springer, Berlin (1960). 

Tn recent years, developing interests in surface energetics and adhesion of liquid-like polymers, or polymer liquids, have prompted both theoretical and experimental work on surface tension. Unlike low molecular weight liquids, polymer liquids have not been extensively studied. Bondi and Simkin (1) mentioned surface tension in their study on high molecular weight liquids. Roe (28) applied both the cell theory of polymer liquids and the hole theory of surface tension of simple liquids to develop an approximate theory of surface tensions of polymer liquids. His approach has met some degree of success. Notably, both Bondi s and Roes work are somewhat related to the cell theory introduced by Prigogine and... 

J.F. Padday, Surface Tension I. The Theory of Surface Tension, in Surface and Colloid Science, E. Matijevic, Ed., Vol. 1, Wiley (1969) 39. (Considers some aspects of our sections 2.9-2.11.)... 

Theberge SM, Luther GW, Rozan TF, Rickard DT (2000) Evidence for aqueous clusters as intermediates during copper sulfide formation. Abstr Am Chem Soc 220 353 Thompson DN, Sayer RL, Noah KS (2000) Sawdust-supported passive bioremediation of western United States acid rock drainage in engineered wetland systems. Minerals Metall Process 17 96-104 Tolman RC (1966) Consideration of Gibbs theory of surface tension. J Chem Phys 16 758-774 Tolman RC (1949) The effect of droplet size on surface tension. J Chem Phys 17 333-337 Tomino H, Kusaka I, Nishioka K, Takai T (1991) Interfacial tension for small nuclei in binary nucleatioa J Crystal Growth 113 633-636... 

As a result of the clarification of the nature of intermolecular forces by quantum mechanics and of the investigation, both from the experimental and the theoretical points of view, of the distribution of atoms or molecules in liquids, the situation has become more favorable for the study of the theory of surface tension. 

In this paper recent developments in the theory of surface tension will be reported with special reference to statistical mechanics, and problems which are to be solved in the future will be presented. 

In any attempt to establish a theory of surface tension, it is very natural to assume, as a first approximation, that the transition from the liquid phase to the gas phase occurs discontinuously at a mathematical plane. As stated in Section V, Fowler introduced this assumption and obtained Eq. V.9 for the surface tension. The nature of this assumption will now be studied more closely by calculating the normal pressure and the tangential pressure />x.14-21 The assumption is stated as... 

Another approach to the theory of surface tension by an approximate procedure was proposed by Tolman30 and improved by Hill.12 The latter assumed that the contribution to surface tension which arises from entropy considerations is determined only by the local density and that the contributions from energy considerations can be obtained by integrating the potential energies from all the... 

Before entering into the problem of quantum effects in the theory of surface tension or surface energy, some considerations based on the principle of corresponding states for these quantities will be presented. In order to consider the surface tension from the standpoint of corresponding states, it is not necessary to resort to expressions such as Eq. V.ll. The principle of corresponding states for surface tension can be derived directly from Eq. V.10 for the free energy. If the area of the interface between the liquid and the vapor is A, the surface tension is given by... 

This makes the early papers of Irvin Langmuir interesting. Langmuir (1918) wrote "The theory of surface tension which I developed in 1916 and which was later elaborated by Harkins, also furnishes striking proof that the surface layers in pure liquids are normally of the thickness of a... 

It may be of interest to note that Mandelstamm s analysis forms the basis of the capillary wave theory of surface tension. ... 

During the 1870 s, Carlo Marangoni, who was apparently aware of Carra-dori s work but not of Thompson s, formulated a rather complete theory of surface tension driven flow (M2, M3). He noted that flow could result from surface tension variations as they are caused by differences in temperature and superficial concentration, and that, conversely, variations in temperature and concentration could be induced by an imposed surface flow. Marangoni ascribed several new rheological properties to the surface (notably surface viscosity, surface elasticity, and even surface plasticity), while remarking that perhaps some of these properties could be associated only with surface contamination. Most present-day authors ascribe the first explanation of surface tension driven flow to Marangoni, and term such flow a Maragoni effect. ... 

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