The maximum force on a vertical rod supporting a stable meniscus, at equilibrium, formed at the free surface of a liquid, is shown to be a characteristic property of the system. This maximum force depends only on the rod radius, the density of the liquid, the gravitational acceleration and the surface tension.The maximum force represents the product of the maximum volume of liquid held above the general level, the gravitational acceleration and the density. The maximum volume of the meniscus has been derived theoretically as a function of rod radius in the form of a parametric equation with a table of coefficients. Thus from the measured rod radius and maximum force, the surface tension is derived.The surface tensions of water and of other liquids have been measured using this method and are found to agree very well with other methods given in the literature. However, this method is believed to be accurate to kO.1 mN m-I, and it is an absolute method in that it requires no arbitrary end corrections involving a prior knowledge of the surface tension.It is important that the rod should be level and have a radius not so large that the angle of contact became a limitation on the maximum force measurable.The method does not involve detachment of the rod from the surface. It is an equilibrium method the position of which may be approached from both directions.Gay-Lussac,' attempted to determine the surface tension of a liquid by the measurement of the maximum force exerted on a horizontal plate as it was pulled away from the free flat surface of a liquid. He realised it was necessary to use a large plate (1 18.4 mm diameter) and he assumed that the meniscus formed at the edge of the plate approached that of cylindrical symmetry : in fact he assumed that the angle of the meniscus with the underside of the plate was zero.These experiments captured the attention of Young,' L a p l a ~e , ~ P o i ~s o n , ~ Kir-~h o f f , ~ Cantor,6 Lohn~tein,~ Gallenkarnpp8 Quincke, Ferguson,lO Bouasse '' and Bakker,12 all of whom attempted to obtain a relation between the force on the rod or plate, the radius of the plate and the surface tension of the liquid. Of these studies only that of Lohnstein reveals the understanding necessary to obtain this relation.The measurement of surface tension with solids of other shapes, such as the Du Nouy ring l 3 has been studied by Freud and Freud,14 and Harkins and Jordan,ls and the sphere has been studied by Huh and Mason The sphere problem also arises in flotation studies,l87 l 9 and equations used in these studies are identical with those used in surface tension measurement.However, despite this extensive area of previous investigation it appears that the solution to the rod or plate problem has not been established previously.In this study we attempt to set out the physical conditions that govern the maximum pull on a rod as it is drawn from the free surface of a liquid to form a stable meniscus. We then show that the force reaches a stable maximum value well before breakaway occurs, and that ...

The conditions that govern the equilibrium and stability of a meniscus have been obtained from the first and second derivatives of the energy of the meniscus when it undergoes axisymmetric deformation. The energy of forming a meniscus is defined in thermodynamic terms and methods are given for calculating the free energy of a mensicus in the perturbed and unperturbed state. The stable, critically stable and unstable equilibrium states of a meniscus are all defined in terms of an energy profile, that is, the variation of energy with degree of perturbation. The variational problem of defining parameters for a critically stable meniscus is solved graphically by using a three-dimensional cluster of energy profiles, and it is shown that certain properties of the meniscus, notably volume or pressure, reach limiting values at critical conditions. Four types of stability are considered for each of three forms of axisymmetric menisci. The stability types are those limited by volume or pressure, in conjunction with limitation by the size of the supporting solid surface or the angle of contact. The three forms of menisci are pendant drops, sessile drops and rod-in-free-surface menisci. Detailed stability criteria are given for each of the twelve different combinations of stability type and meniscus form. The stability criteria of this study are all derived by numerical interpolation methods applied to the tables of equilibrium meniscus shapes - they are thus theoretical. Where possible they have been compared with experiment and with other studies, and are found to predict critically stable states with an accuracy greater than that likely to be found in the normal course of experiments.

The shape and forces of axisymmetric menisci have been calculated for sessile drops, pendant drops and liquid bridge profiles. The tables of Bashforth & Adams have been extended into the liquid bridge region by generating profiles beyond the 180° angle of their study, and into regions covered by a much wider range of shape factor. Numerical integration of the Laplace equation was performed by using a first-order method originally proposed by Lord Kelvin but adapted and modified for use with high-speed computers. Tables have also been generated, by the same techniques, of profiles of a wide range of asixymmetric menisci that do not cross the axis of symmetry. Such tables include the shape of a meniscus formed by a rod at a free liquid surface. This second group of tables greatly extended the region over which liquid bridge shapes could be obtained. Closed menisci of the type of Bashforth & Adams’s tables are defined by one shape factor B, but open menisci require two parameters to identify their shape. The general properties of these shape factors are discussed. The volume of any part of the meniscus bounded by two horizontal planes has also been derived and its relationship to the forces acting between the planes is given. The tables (400 pages) are not reproduced here but the main features of these profile shapes are summarized and discussed with the aid of graphs.

Pendant liquid bridges are defined as pendant drops supporting a solid axisymmetric endplate at their lower end. The stability and shape properties of such bridges are defined in terms of the capillary properties of the system and of the mass and radius of the lower free-floating endplate. The forces acting in the pendant liquid bridge are defined exactly and expressed in dimensionless form. Numerical analysis has been used to derive the properties of a given bridge and it is shown that as the bridge grows by adding more liquid to the system a maximum volume is reached. At this maximum volume, the pendant bridge becomes unstable with the length of the bridge increasing spontaneously and irreversibly at constant volume. Finally the bridge breaks with the formation of a satellite drop or an extended thread. The bifurcation and breakage processes have been recorded using a high-speed video camera with a digital recording rate of up to 6000 frames per second. The details of the shape of the bridge bifurcation and breakage for many pendant bridge systems have been recorded and it is shown that satellite drop formation after rupture is not always viscosity dependent. Bifurcation and breakage in simulated low gravity demonstrated that breakage was very nearly symmetrical about a plane through the middle of the pendant bridge.

The process of forming and rupturing a thin liquid film at a solid surface is described thermodynamically for both high and low energy solid surfaces. In part 1 the build-up of thin films on high-energy surfaces from the first monolayer is considered and reviewed. Components of the surface free energy of formation of the thin film (disjoining pressure) are defined. For curved surfaces the disjoining forces should be combined with the Laplace capillary pressure to give a correct form of the Kelvin equation. It is suggested from the early work of Bangham and Deryaguin that thin liquid layers have anomalous physical properties. These studies are discussed in relation to the thickness of the liquid films.