# Capillary length

Capillary Length basics

The Capillary Length or Capillary Constant, is a one dimensional scaling factor that relates gravity and surface tension at a liquid interface. It is most easily derived when there exists an equilibrium between the Laplace and Hydro-static pressure, i.e when the pressure difference between the inside and outside of a curved boundary, is equal to the pressure exerted by the fluid.

Being a scaling factor, the capillary length is a constant for any given liquid, and is used to in numerous fluid mechanical problems to scale the derived equations such that they are valid for any fluid, such as giving a normalised value of the capillary height[1] . Appearing in nearly all analytical formulas where surface tension is of interest, the capillary length describes the curvature of the meniscus level of a liquid at a boundary, for example a drop of water on a countertop.

Denoted by ${\displaystyle \lambda \scriptscriptstyle c}$ or ${\displaystyle k^{-1}}$, the capillary length is

${\textstyle \lambda _{c}={\sqrt {\frac {\gamma }{\rho g}}}}$
${\textstyle \lambda _{c}={\sqrt {\frac {2\gamma }{\rho g}}}}$

where ${\displaystyle g}$ is the gravitational acceleration and ${\displaystyle \rho }$is the mass density of the fluid, and ${\displaystyle \gamma }$ is the surface tension of the fluid interface. The factor of two arises because when dealing with capillary systems in a gravitational field, the density differences between phases and interfacial tensions are the fundamental properties in enabling dimensionless quantities, rather that the actual quantities used[2].

## History

Fundamentally, the Capillary length is a product of the work Thomas Young and Pierre Laplace and their pressure equation that shows 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.

## Derivation

• The capillary length is found through the manipulation of many different physical phenomenon. One method is using the idea that when a circular capillary tube is inserted into a liquid, the liquid will rise due to the imbalance in pressure. By equating the Laplace and hydro static pressure and solving for the height of the capillary tube, one can reorganise to show the capillary length as a function of surface tension and gravity. See Jurin's Law

• Another way to find the capillary length is using different pressure points inside a sessile droplet, with each point having a radii 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. See Sessile droplet

• The capillary length can also be derived theoretically by using the Bond number, a dimensionless number that represents the ratio between the buoyancy forces and surface tension. By setting the Bond number to one, it can rearrange to solve for the capillary length. See Eötvös number

• The capillary length can also be found through capillary uplift. When rearranging the equation, one can find the capillary length as a function of the volume increase ${\displaystyle V}$ and wetting perimeter of the capillary walls, ${\displaystyle l}$ .

${\displaystyle \gamma l=V\rho g}$  therefore ${\displaystyle {\frac {\gamma }{\rho g}}={\frac {V}{l}}=\lambda _{c}^{2}}$ [3]

## Effects and Consequences

The capillary length is fundamental in the shaping of liquid droplets/menisci, and above or below the Capillary length, there is an imbalance of forces;

• Traditionally raindrops are pictured shaped as a pendent. Imagine a raindrop falling steadily through the air with it's weight being balanced by drag. With the drag forces balanced by gravity, it must be that the hydro static pressure inside the drop is negligible compared to the pressure exerted due to surface tension. In other words the radii of the drop is << the capillary length and gravity is negligible. The drop will fall will near perfect spherical shape.[4]

Illustration of sessile drops with radii above and below the capillary length.
• Microdrops are sessile droplets with radii smaller than the capillary length. Here the shape of the droplet is governed solely by surface tension and they form a Spherical cap shape. If a droplet has a radii larger than the Capillary length, they are known as Macrodrops and the gravitational forces will dominate. Macrodrops will be 'flattened' by gravity and the height of the droplet will be reduced. This effect is mirrored by the effect that the Bond number will have on a sessile drop[5].

• The capillary length can also be used as an extension of the Young Laplace formula in physical phenomenon. Imagining a droplet substantially smaller than the capillary length of a non wetting fluid on a hydrophobic surface, such as mercury on plastic, it is theoretically agreed that when the surface is tilted, the droplet will 'roll' will near perfect spherical shape in cylindrical motion until the droplet merges with the solid at a finite contact angle.[6]

Wetting

## References

1. ^ Bastian E. Rapp, Microfluidics: Modelling, Mechanics and Mathematics, Elsevier, 2017, Pages 445-451
2. ^ E A Boucher 1980 Rep. Prog. Phys. 43 497, https://iopscience.iop.org/article/10.1088/0034-4885/43/4/003/pdf
3. ^ V. V. Kashin, K. M. Shakirov, and A. I. Poshevneva, The Capillary Constant in Calculating the Surface Tension of Liquids, Steel in Translation, 2011, Vol. 41, No. 10, pp. 795–798. © Allerton Press, Inc., 2011.
4. ^ B. Lautrup, Physics of Continuous Matter: Exotic and Everyday Phenomena in the Macroscopic World, CRC Press Published March 22, 2011, p72
5. ^ Berthier, J, & Silberzan, P 2009, Microfluidics for Biotechnology, Artech House, Norwood. Available from: ProQuest Ebook Central. [19 February 2019].
6. ^ Yves Pomeau and Emmanuel Villermaux, 200 years of Capillary research, Physics Today, March 2006, P40