This surface phenomenon gets a good play in school physics, and is a valuable and interesting study

It is wonderful the way the free surface of a liquid seems to be covered by a thin film in tension, that makes drops spherical, jets cylindrical, and climbs up the wall of a container. Soap films are a related wonder, forming iridescent spheres so light they float in the air, and thick foams when agitated. The way it is presented in school physics often reinforces this view, though the film has properties unlike those of a material substance.

We are dealing with a surface phenomenon, one of the rare ones that is mentioned in school physics, and one with a great number of interesting applications. The molecules on the surface have water on one side, and something else on the other. Let us say we are dealing with familiar water, though it could be any fluid. These molecules are less well nestled among their kind, and so there is a deficit in attractive energy. If we make new surface, we must put some energy in when we bring molecules from the comfort of the interior to the challenges of the borders. This is not just individual molecules, but in the case of water a change of structure, since the order of bulk water cannot apply on the frontiers. Liquid water has a curious structure, in which dodecahedrons probably play a role.

Free energy F = U - TS, where U is the internal energy, T the absolute temperature, and S the entropy. Then dF = dU - TdS - SdT = pdV + TdS - TdS - SdT = pdV - SdT. At constant temperature, dT = 0, dF = pdV, where pdV represents the work done on the system (it is only one kind of work, but stands here for them all, everything but heat). In the present case, the work in making new surface of area dA at constant temperature is γdA, so dF = γdA. γ is the free energy per unit area of new surface. Dimensionally, this is erg/cm^{2} or, what is the same thing, dyne/cm. Indeed, if dA = Ldx, then dF = (γL)dx, as if γL were a tension in dynes. This is the usual definition of surface tension, though even school physics makes it clear that it is really a free energy per unit area. When a surface is extended, it cools, and heat must flow in to maintain isothermal conditions.

However, the free energy F is something quite different from the usual energy U. In the present case, γ = U + T(dγ/dT). Surface tension decreases with temperature, so F is less than U. In fact, U and γ are quite different, though both become zero at the critical temperature. The Figure shows how U and F vary with temperature. If you are considering what happens to molecules on the surface, U is probably closer to what you need. U is not too far from constant, until the surface itself disappears. Under normal conditions, we do not get near the critical temperature, of course, since the water boils first.

Water molecules are not only attracted to other water molecules. If they are more strongly attracted to the molecules of another substance, that substance is said to be wetted, or hydrophilic, and surface molecules tend to cuddle up to it. If they are less attracted, the substance is hydrophobic, and surface molecules tend to shy away from it. The water-air surface meets another at an angle of contact θ. This angle is small for a hydrophilic surface, large for a hydrophobic surface. It is 90° if the attraction is the same as that of water itself. All of this is strongly affected by the purity of the surfaces involved.

Surfaces postively attract hydrophobic impurities. If a molecule or body is not wetted, it is best consigned to the surface so that it cannot get in the way of molecules who want to be close to one another. Soap is an interesting case. The soap molecule has one end that is polar, like water, and to which water molecules are attracted. The other end is fatty and saturated, strongly hydrophobic (but equally strongly fatophilic!). Soap molecules in water tend to make a carpet on the surface, fatty ends outward, polar ends inward. This lessens the expense of making a surface, so the surface tension falls. In addition, the surface is now rendered somewhat permanent, and the evaporation of water through it is discouraged. The surface will now attract fats and oils, which is why the soap is used.

Consider a spherical ball of water. This occurs when the water is free from gravity, as when falling freely (even at terminal velocity in air, small drops are quite spherical). Cut the sphere neatly in half in your imagination. The tension at the surface is 2πrγ, and this must be balanced by a higher pressure p over the diametral plane. Thus 2πrγ = πr^{2}p, from which p = 2γ/r. This is the difference in pressure inside and outside the sphere, due to the surface energy. The surface energy is made a minimum by the spherical shape, which has the least surface area for a given volume. In terms of the principal radii of curvature R and R', p = γ(1/R + 1/R').

If we have a soap bubble instead, the film is an inner and an outer surface carpeted with soap molecules, with water in between. Hence, the pressure is 4γ/r higher inside the soap bubble than outside. The γ is certainly not the value for pure water, 73 dyne/cm (290 K), but something less. One reference gives 25 dyne/cm. Cgs units have a very pleasant size for this quantity. Mercury has the high surface tension of 470 dyne/cm. The surface tension of water falls with increasing temperature, to 59 dyne/cm at the boiling point. Rayleigh obtained 74 dyne/cm for clean water, 53 for greasy water, 41 with an olive oil surface, and 25 for a surface of sodium oleate (soap).

The capillary rise in a narrow tube can be estimated by using the angle of contact. We have 2πrγcos θ = πr^{2}hρg, so the capillary elevation is h = 2γ cos θ/ρgr. With a zero contact angle, the rise in a 1 mm diameter capillary is 3 cm. This is, of course, an upper limit.

Surface tension encourages the evaporation of small droplets (the added pressure inside a convex surface), and the collapse of small bubbles, for the same reason. This can lead to supersaturation on condensation in the one case, and to superheating on boiling in the second.

Surface tension can be measured by capillary rise (θ must also be measured), or by the pressure necessary to produce a small bubble (Jaeger's method). A method not as simple as it looks measures the weight of a droplet detached from a thick-walled capillary of radius r. If you write mg = 2πrγ, you will be in error about 100%, as a better balance of forces will show. The correct relation turns out to be about mg = 3.8rγ instead, as determined empirically by Lord Rayleigh.

If the surface of a body of water is deformed in a wavelike manner, the higher parts will create a positive pressure from surface tension and their curvature, and the lower parts a negative pressure for the same reason. This will add to the gravitational force acting against the inertia of the water, and wave propagation will be the result. Therefore, we are led to an investigation of waves on water. We describe the motion by its velocity potential φ, whose negative derivative with respect to distance (gradient) is the particle velocity. The motion is irrotational, and φ satisfies Laplace's equation. These matters are explained in detail in Wave Dynamics. Consider plane waves travelling in the direction x, with z the depth from the surface of the water, z = 0. A suitable solution of Laplace's equation is φ = e^{i(ωt - kx)}(Ae^{kz} + Be^{-kz}). Because dφ/dz = 0 at the bottom, z = L, A and B are chosen so that φ = Ce^{i(ωt - kx)} cosh k(z - L).

The relation between ω and k depends on the dynamics. If p is the overpressure, p/ρ = gz - dφ/dt. Therefore, at z = -h, the surface, (T/ρ)d^{2}h/dx^{2} = gh + dφ/dt. Now, dh/dt = -dφ/dz (the vertical velocity), so (T/ρ)d^{3}φ/dx^{2}dz = g dφ/dz - d^{2}φ/dt^{2}. If we substitute our form for φ into this equation, we find the dispersion relation

c^{2} = (ω/k)^{2} = (g/k + Tk/ρ)tanh kL.

The g/k applies to pure gravity waves, and the Tk/ρ to pure capillary waves. In deep water, tanh kL = 1, and the dispersion relation is Kelvin's formula, published in 1871. The phase velocity of gravity waves is proportional to the square root of the wavelength, and the phase velocity of capillary waves is inversely proportional to the wavelength. There is a minimum phase velocity (4Tg/ρ)^{1/4}. For water, with T = 76 dyne/cm, this velocity is 23.1 cm/s, the wavelength 1.71 cm, and the frequency 13.6 Hz.

Capillary waves were studied by Faraday in 1831, who termed them crispations. Scott Russell observed capillary waves made by an obstacle in flowing water in 1844. These are called stationary waves, but are different from the usual stationary waves formed by wave trains moving in opposite directions. Their velocities are fixed by the velocity of the stream and the angle of their wavefronts. The stream velocity must be greater than 23 cm/s (1/2 mph) for such waves to form, since this is their minimum phase velocity. Capillary waves form on the upstream side, gravity waves on the downstream.

P. M. Whelan and M. J. Hodgson, *Essential Principles of Physics* (London: John Murray, 1978).

Lord Rayleigh, *Theory of Sound*, 2nd ed. (London: Macmillan, 1926), Vol. II, Chapter XX.

Return to Physics Index

Composed by J. B. Calvert

Created 25 June 2000

Last revised 30 July 2000