Capillary length

One way to theoretically derive the capillary length, is to imagine a liquid droplet at the point where surface tension balances gravity.

Let us consider a spherical droplet with radius ${\displaystyle \lambda _{c}}$,

The characteristic Laplace pressure ${\displaystyle P_{\gamma }}$, due to surface tension, is equal to

${\displaystyle P_{\gamma }=2{\frac {\gamma }{\lambda _{\rm {c}}}}}$,

where ${\displaystyle \gamma }$ is the surface tension. The pressure due to gravity (hydrostatic pressure) ${\displaystyle P_{\rm {h}}}$ of a column of liquid is given by

${\displaystyle P_{\rm {h}}=\rho gh=2\rho g\lambda _{\rm {c}}}$,

where ${\displaystyle \rho }$ is the droplet density, ${\displaystyle g}$ the gravitational acceleration, and ${\displaystyle h=2\lambda _{\rm {c}}}$ is the height of the droplet.

At the point where the Laplace pressure balances out the pressure due to gravity ${\displaystyle P_{\rm {h}}=P_{\gamma }}$, we obtain that

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

We can use the above derivation when dealing with the Eötvös number, a dimensionless quantity that represents the ratio between the buoyancy forces and surface tension of the liquid. Despite being introduced by Loránd Eötvös in 1886, he has since become fairly dissociated with it, being replaced with Wilfrid Noel Bond such that it is now referred to as the Bond number in recent literature.

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]

${\displaystyle \mathrm {Bo} ={\frac {\Delta \rho \,g\,L^{2}}{\gamma }}}$,

with parameters defined above, and ${\displaystyle L}$ the radius of curvature.

Therefore we can write the bond number as

${\displaystyle \mathrm {Bo} =\left({\frac {L}{\lambda _{\rm {c}}}}\right)^{2}}$,

with ${\displaystyle \lambda _{\rm {c}}}$ the capillary length.

If the bond number is set to 1, then the characteristic length is the capillary length

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. The law can be illustrated mathematically during capillary uplift, which is a traditional experiment measuring the height of a liquid in a capillary tube. When a capillary tube is inserted into a liquid, the liquid will rise or fall in the tube, due to an imbalance in pressure. The characteristic height is the distance from the bottom of the meniscus to the base, and exists when the Laplace pressure and the pressure due to gravity are balanced. One can reorganize to show the capillary length as a function of surface tension and gravity.

${\displaystyle \lambda _{\rm {c}}^{2}={\frac {hr}{2\cos \theta }}}$,

with ${\displaystyle h}$ the height of the liquid, ${\displaystyle r}$ the radius of the capillary tube, and ${\displaystyle \theta }$ the contact angle.

The contact angle is defined as the angle formed by the intersection of the liquid-solid interface and the liquid–vapour interface.[2] The size of the angle quantifies the wetability of liquid, i.e the interaction between the liquid and solid surface. Here we will consider a contact angle of ${\displaystyle \theta =0}$, perfect wetting.

${\displaystyle \lambda _{\rm {c}}^{2}={\frac {hr}{2}}}$.

Thus the ${\displaystyle \lambda _{\rm {c}}^{2}}$ forms a cyclical 3 factor equation with ${\displaystyle r,h}$.

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.[9]

Using the same premises of capillary rise, one can find the capillary length as a function of the volume increase, and wetting perimeter of the capillary walls.[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. Macrodrops will be 'flattened' by gravity and the height of the droplet will be reduced.[11]

The capillary length against radii of a droplet

The size of soap bubbles are limited by the capillary length

Like a droplet, bubbles are round because cohesive forces pull its molecules into the tightest possible grouping, a sphere. Due to the trapped air inside the bubble, it is impossible for the surface area to shrink to zero, hence the pressure inside the bubble is greater than outside, because if the pressures were equal, then the bubble would simply collapse.[15] This pressure difference can be calculated from Laplace's pressure equation,

${\displaystyle \Delta P={\frac {2\gamma }{R}}}$.

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

${\displaystyle \Delta P={\frac {4\gamma }{R}}}$.[16]

The capillary length can then be worked out the same way except that the thickness of the film, ${\displaystyle e_{0}}$ must be taken into account as the bubble has a hollow center, unlike the droplet which is a solid. Instead of thinking of a droplet where each side is ${\displaystyle \lambda _{c}}$ as in the above derivation, for a bubble ${\displaystyle m}$ is now

${\displaystyle m=\Delta \rho R^{2}e_{0}}$,

with ${\displaystyle R}$ and ${\displaystyle e_{0}}$ the radius and thickness of the bubble respectively.

As above, the Laplace and hydrostatic pressure are equated resulting in

${\displaystyle R={\frac {\gamma }{\Delta \rho ge_{0}}}={\frac {\lambda _{\rm {c}}^{2}}{e_{0}}}}$.

Thus the capillary length contributes to a physiochemical limit that dictates the maximum size a soap bubble can take.[5]