Capillary length
It is a fundamental physical property that governs the behavior of menisci, and is found when body forces (gravity) and surface forces (Laplace pressure) are in equilibrium.
[2] In the case of a fluid–fluid interface, for example a drop of water immersed in another liquid, the capillary length denoted
The term capillary constant is somewhat misleading, because it is important to recognize that
is a composition of variable quantities, for example the value of surface tension will vary with temperature and the density difference will change depending on the fluids involved at an interface interaction.
However if these conditions are known, the capillary length can be considered a constant for any given liquid, and be used in numerous fluid mechanical problems to scale the derived equations such that they are valid for any fluid.
[3] For molecular fluids, the interfacial tensions and density differences are typically of the order of
For a soap bubble, the surface tension must be divided by the mean thickness, resulting in a capillary length of about
[6] One way to theoretically derive the capillary length, is to imagine a liquid droplet at the point where surface tension balances gravity.
, The above derivation can be used when dealing with the Eötvös number, a dimensionless quantity that represents the ratio between the gravitational forces and surface tension of the liquid.
The Bond number can be written such that it includes a characteristic length- normally the radius of curvature of a liquid, and the capillary length[7] with parameters defined above, and
The capillary length can also be found through the manipulation of many different physical phenomenon.
One method is to focus on capillary action, which is the attraction of a liquids surface to a surrounding solid.
[8] Jurin's law is a quantitative law that shows that the maximum height that can be achieved by a liquid in a capillary tube is inversely proportional to the diameter of the tube.
One can reorganize to show the capillary length as a function of surface tension and gravity.
This property is usually used by physicists to estimate the height a liquid will rise in a particular capillary tube, radius known, without the need for an experiment.
When the characteristic height of the liquid is sufficiently less than the capillary length, then the effect of hydrostatic pressure due to gravity can be neglected.
[10] Another way to find the capillary length is using different pressure points inside a sessile droplet, with each point having a radius of curvature, and equate them to the Laplace pressure equation.
This time the equation is solved for the height of the meniscus level which again can be used to give the capillary length.
The shape of a sessile droplet is directly proportional to whether the radius is greater than or less than the capillary length.
Microdrops are droplets with radius smaller than the capillary length, and their shape is governed solely by surface tension, forming a spherical cap shape.
If a droplet has a radius larger than the capillary length, they are known as macrodrops and the gravitational forces will dominate.
[11] The investigations in capillarity stem back as far as Leonardo da Vinci, however the idea of capillary length was not developed until much later.
Fundamentally the capillary length is a product of the work of Thomas Young and Pierre Laplace.
At the turn of the 19th century they independently derived pressure equations, but due to notation and presentation, Laplace often gets the credit.
The equation showed that the pressure within a curved surface between two static fluids is always greater than that outside of a curved surface, but the pressure will decrease to zero as the radius approached infinity.
[12] This was a mathematical explanation of the work published by James Jurin in 1719,[13] where he quantified a relationship between the maximum height taken by a liquid in a capillary tube and its diameter – Jurin's law.
[14] Like a droplet, bubbles are round because cohesive forces pull its molecules into the tightest possible grouping, a sphere.
[15] This pressure difference can be calculated from Laplace's pressure equation, For a soap bubble, there exists two boundary surfaces, internal and external, and therefore two contributions to the excess pressure and Laplace's formula doubles to The capillary length can then be worked out the same way except that the thickness of the film,
must be taken into account as the bubble has a hollow center, unlike the droplet which is a solid.
As above, the Laplace and hydrostatic pressure are equated resulting in Thus the capillary length contributes to a physiochemical limit that dictates the maximum size a soap bubble can take.
