Young–Laplace equation

In physics, the Young–Laplace equation describes the equilibrium pressure difference sustained across the interface between two static fluids, such as water and air, due to the phenomenon of surface tension. It relates the pressure difference to the shape of the surface and it is fundamentally important in quantifing the phenomena of capillary action and of soap bubbles.

The equation is named for Thomas Young, who developed the qualitative theory of surface tension in 1805, and Pierre-Simon Laplace who completed the mathematical description in the following year.

A surface can be characterised by its principal radii of curvature, R₁ and R₂ (say). For an interface with surface tension γ, the Young–Laplace equation gives the pressure difference sustained by the surface, Δp as:

<math>\Delta p = \gamma \left( \frac{1}{R_1} + \frac{1}{R_2}\right)</math>

— or, in terms of the average curvature, ρ:

<math>\Delta p = 2 \gamma \rho.</math>

Soap films

Template:Mainarticle If the pressure difference is zero, as in a soap film without gravity, the interface will assume the shape of a minimal surface.

Emulsions

The equation also explains the energy required to create an emulsion. To form the small, highly curved droplets of an emulsion, extra energy is required to overcome the large pressure that results from their small radius.

Capillary pressure with a spherical meniscus

File:Spherical meniscus.PNG Spherical meniscus with wetting angle less than 90°

In a sufficiently narrow tube of circular cross-secion, the interface between two fluids forms a meniscus in the form of a portion of the surface of a sphere with radius R and the Young–Laplace equation simplifies to:

<math>\Delta p = \frac{2 \gamma}{R}.</math>.:

In capillary action in a narrow tube of circular cross-section, radius a, the radius of the sphere of the meniscus will be a function only of the wetting angle, θ, which in turn depends on the exact properties of the fluids and the solids in which they are in contact.

<math>R = \frac{a}{\cos \theta}</math>

— and the equation then becomes:

<math>\Delta p = \frac{2 \gamma \cos \theta}{a}.</math>.

File:CapillaryAction.svg

Illustration of capillary rise. Red=contact angle less than 90°; blue=contact angle greater than 90°

In order to maintain hydrostatic equilibrium, the induced capillary pressure is balanced by a change in height, h, which can be positive or negative, depending on whether the wetting angle is less than or greater than 90°. For a fluid of density ρ:

<math>h = \frac{2 \gamma \cos \theta}{\rho g a}.</math>.

— where g is the gravitational acceleration. This is sometimes known as the Jurin rule or Jurin height after James Jurin who studied the effect in 1718.

For a water-filled glass tube in air at sea level:

γ = 0.0728 J/m² at 20 °C	θ = 20° (0.35 rad)
ρ = 1000 kg/m3	g = 9.8 m/s²

— and so the height of the water column is given by:

<math>h\approx {{1.4 \times 10^{-5}}\over a}</math> m.

Thus for a 2 mm wide (1 mm radius) tube, the water would rise 14 mm. However, for a capillary tube with radius 0.1 mm, the water would rise 14 cm (about 6 inches).

Capillary action in general

In the general case, for a free surface and where there is an applied "over-pressure", Δp, at the interface then, in equilibrium, there is a balance between the applied pressure, the hydrostatic pressure and the effects of surface tension. The Young–Laplace equation becomes:

<math>\Delta p = \rho g h - \gamma \left( \frac{1}{R_1} + \frac{1}{R_2}\right)</math>.

The equation can be non-dimensionalised in terms of its characteristic length-scale, the capillary length:

<math>L_{c} = \sqrt{\frac{\rho g}{\gamma}},</math>

— and characteristic pressure:

<math>p_{c} = \frac {\gamma} {L_{c}} = \frac {\gamma^{\frac{3}{2}}}{\sqrt{\rho g}}.</math>

For clean water at standard temperature and pressure, the capillary length is ~2 mm.

The non-dimensional equation then becomes:

<math>h^* - \Delta p^* = \left( \frac{1}{{R_1}^{*}} + \frac{1}{{R_2}^{*}}\right)</math>

Thus, the surface shape is determined by only one parameter, the over pressure of the fluid, Δp^* and the scale of the surface is given by the capillary length. The solution of the equation requires an initial condition for position, and the gradient of the surface at the start point.

File:YoungLaplace example drop.gif

A sessile drop is produced for an over pressure of Δp^*=3 and initial condition r₀=10^-4, z₀=0, dz/dr=0

File:YoungLaplace example bridge.gif A liquid bridge is produced for an over pressure of Δp*=3.5 and initial condition r0=0.25-4, z0=0, dz/dr=0

Axisymmetric equations

The (nondimensional) shape, r(z) of an axisymmetric surface can be found by substituting general expressions for curvature to give the hydrostatic Young-Laplace equations:

<math>\frac{r}{(1+r'^2)^{\frac{3}{2}}} - \frac{1}{r(z) \sqrt{1+r'^2} } = z - \Delta p^* </math>

<math>\frac{z}{(1+z'^2)^{\frac{3}{2}}} + \frac{z'}{r \sqrt{1+z'^2} } = z(r) - \Delta p^* </math>

Application in medicine

In medicine it is often referred to as the Law of Laplace, and it is used in the context of respiratory physiology, in particular alveoli in the lung, where a single alveolus is modeled as being a perfect sphere.

In this context, the pressure differential is a force pushing inwards on the surface of the alveolus. The Law of Laplace states that there is an inverse relationship between surface tension and alveolar radius. It follows from this that a small alveolus will experience a greater inward force than a large alveolus, if their surface tensions are equal. In that case, if both alveoli are connected to the same airway, the small alveolus will be more likely to collapse, expelling its contents into the large alveolus.

This explains why the presence of surfactant lining the alveoli is of vital importance. Surfactant reduces the surface tension on all alveoli, but its effect is greater on small alveoli than on large alveoli. Thus, surfactant compensates for the size differences between alveoli, and ensures that smaller alveoli do not collapse.

The Law of Laplace also explains various phenomena encountered in the pathology of vascular or gastrointestinal walls. The "surface tension" in this case represents the muscular tension on the wall of the vessel. For example, if an aneurysm forms in a blood vessel wall, the radius of the vessel has increased. This means that the inward force on the vessel decreases, and therefore the aneurysm will continue to expand until it ruptures. A similar logic applies to the formation of diverticuli in the gut.

History

Francis Hauksbee performed some of the earliest observations and experiments in 1709 and these were repeated in 1718 by James Jurin who observed that the height of fluid in a capillary column was a function only of the cross-sectional area at the surface, not of any other dimensions of the column.

Thomas Young laid the foundations of the equation in his 1804 paper An Essay on the Cohesion of Fluids where he set out in descriptive terms the principles governing contact between fluids (along with many other aspects of fluid behaviour). Pierre Simon Laplace followed this up in Mécanique Céleste with the formal mathematical description given above, which reproduced in symbolic terms the relationship described earlier by Young.

Laplace accepted the idea propounded by Hauksbee in the Philosophical Transactions for 1709, that the phenomenon was due to a force of attraction that was insensible at sensible distances. The part which deals with the action of a solid on a liquid and the mutual action of two liquids was not worked out thoroughly, but ultimately was completed by Gauss. Carl Neumann later filled in a few details.

Bibliography

• [Anon.] (1911) Capillary action, Encyclopedia Britannica
• Batchelor, G. K. (1967) An Introduction To Fluid Dynamics, Cambridge University Press
• Jurin, J. (1717/1719). "Template:Pdf". Philosophical Transactions of the Royal Society. 30: 739–747.
• Tadros T. F. (1995) Surfactants in Agrochemicals, Surfactant Science series, vol.54, Dekker