Capillary length

The capillary length or capillary constant is a length scaling factor that relates gravity and surface tension. 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.

The pressure of a static fluid does not depend on the shape, total mass or surface area of the fluid. It is directly proportional to the fluid's specific weight – the force exerted by gravity over a specific volume, and its vertical height. However, a fluid also experiences pressure that is induced by surface tension, commonly referred to as the Young–Laplace pressure.^[1] Surface tension originates from cohesive forces between molecules, and in the bulk of the fluid, molecules experience attractive forces from all directions. The surface of a fluid is curved because exposed molecules on the surface have fewer neighboring interactions, resulting in a net force that contracts the surface. There exists a pressure difference either side of this curvature, and when this balances out the pressure due to gravity, one can rearrange to find the capillary length.^[2]

In the case of a fluid–fluid interface, for example a drop of water immersed in another liquid, the capillary length denoted ${\displaystyle \lambda _{\rm {c}}}$ or ${\displaystyle l_{\rm {c}}}$ is most commonly given by the formula,

${\displaystyle \lambda _{\rm {c}}={\sqrt {\frac {\gamma }{\Delta \rho g}}}}$,

where ${\displaystyle \gamma }$ is the surface tension of the fluid interface, ${\displaystyle g}$ is the gravitational acceleration and ${\displaystyle \Delta \rho }$ is the mass density difference of the fluids. The capillary length is sometimes denoted ${\displaystyle \kappa ^{-1}}$ in relation to the mathematical notation for curvature. The term capillary constant is somewhat misleading, because it is important to recognize that ${\displaystyle \lambda _{\rm {c}}}$ 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 ${\displaystyle 10-100}$ mN m⁻¹ and ${\displaystyle 0.1-1}$g mL⁻¹ respectively resulting in a capillary length of ${\displaystyle \sim 3}$mm for water and air at room temperature on earth.^[4] On the other hand, the capillary length would be ${\displaystyle {\lambda \scriptscriptstyle c}=6.68}$mm for water-air on the moon. For a soap bubble, the surface tension must be divided by the mean thickness, resulting in a capillary length of about ${\displaystyle 3}$ meters in air!^[5] The equation for ${\displaystyle \lambda _{\rm {c}}}$ can also be found with an extra ${\displaystyle {\sqrt {2}}}$ term, most often used when normalising the capillary height.^[6]