# Capillarity

## Contents

## Capillarity

**Capillarity**, or **capillary motion** is the ability of a substance to draw another substance into it. It occurs when the adhesive intermolecular forces between the liquid and a substance are stronger than the cohesive intermolecular forces inside the liquid. The effect causes a concave meniscus to form where the substance is touching a vertical surface. The same effect is what causes porous materials such as sponges to soak up liquids.

A common apparatus used to demonstrate capillary action is the *capillary tube*. When the lower end of a vertical glass tube is placed in a liquid such as water, a concave meniscus forms. Surface tension pulls the liquid column up until there is a sufficient mass of liquid for gravitational forces to overcome the intermolecular forces. The contact length (around the edge) between the liquid and the tube is proportional to the diameter of the tube, while the weight of the liquid column is proportional to the square of the tube's diameter, so a narrow tube will draw a liquid column higher than a wide tube. For example, a glass capillary tube 0.5mm in diameter will lift approximately a 2.8 mm column of water.

With some pairs of materials, such as mercury and glass, the interatomic forces within the liquid exceed those between the solid and the liquid, so a convex meniscus forms and capillary action works in reverse. The term capillary flow is also used to describe the flow of carrier gas in a silica capillary column of a gas-liquid chromatography system. This flow can be calculated by Poiseuille's equation for compressible fluids.

## Capillary length

(From de Gennes, 2004, 0.33f)

For scales smaller than the capillary length, gravity hardly affects the movement of a liquid. As a result, liquids exhibit many extraordinary behaviors including moving up inclined planes and creeping up the sides of a small capillary tube. Gravity begins to affect a liquid when the LaPlacian pressure and hydrostatic pressure are equal. The Laplace pressure can be written as: <math>\Delta p=\sigma \left( \frac{1}{R_{1}}+\frac{1}{R_{2}} \right)=\kappa \sigma </math> where <math>\kappa ^{-1}</math> is a curvature.

Hydrostatic pressure can be written similarly: <math>\Delta p=\rho g\kappa ^{-1}</math> where <math>\kappa ^{-1}</math> is a height. Equating these two pressures yields the capillary length scale <math>\kappa</math>: <math>\begin{align}

& \frac{\sigma }{\kappa ^{-1}}=\rho g\kappa ^{-1} \\ & \text{or }\kappa ^{-1}=\sqrt{{\sigma }/{\rho g}\;} \\

\end{align}</math>

Typical values for these constants are: <math>\sigma \approx 30\times 10^{-3}{J}/{m^{3}}\;</math>, <math>\rho \approx 1\text{ }gm/cm^{3}\approx 10^{3}\text{kg/}m^{3}</math>, <math>g=9.8\text{ }m/s^{2}</math>

For most real world systems <math>\kappa ^{-1}\sim 1\text{ }</math> is on the millimeter scale.

Sources: de Gennes, Ch.2

## Capillary bridges

Liquid bath rising to form a capillary bridge (From "Nucleation radius and growth of a liquid meniscus" by G. Debregeas and F. Brochard-Wyart in *JCIS*, *190*, 134, **1997**.)

As the bridge grows, the curvature decreases and the Laplace pressure decreases – a form of capillary rise without a capillary!

## Using the capillary length

In the image on the left wouldn't that object be submerged? Maybe I am just thinking about water but for something to be floating wouldn't the surfaces have to be pointing in an upward direction to counteract gravity? [1]

I think the difference between the picture you showed and the one originally on the wiki is: The newer diagram shows a subject staying on top of the liquid because of the SURFACE TENSION of the liquid. However, the older diagram shows a subject floating because of its BUOYANCY.

The capillary length can be though of as a "screening" length - a surface perturbation decays in that distance.

The curvature in one dimension is <math>-\frac{\partial ^{2}z}{\partial x^{2}}</math>.

The Laplace pressure at any height is: <math>\Delta p=p_{atm}-p_{x}=\sigma \frac{\partial ^{2}z}{\partial x^{2}}</math>

The hydrostatic pressure is: <math>p_{x}=p_{atm}-\rho gz</math>

A little algebra gives: <math>\frac{\partial ^{2}z}{\partial x^{2}}=\frac{\rho g}{\sigma }z=\kappa ^{2}z</math>

Which has the solution: <math>z=z_{0}\exp \left( -\kappa x \right)</math>

The perturbation decreases exponentially with a decay constant of the capillary length. (Of course!)

## Capillary rise (Thermodynmaics)

The height of liquid in a capillary can be derived by a thermodynamic argument (The credit is given by de Gennes to Jurin, but I haven't seen others do so.)

The area covered, ignoring the area covered by the meniscus,is : <math>A=2\pi Rh</math>

The driving force per unit area is: <math>I=\sigma _{sv}-\sigma _{sl}=\sigma _{lv}\cos \theta _{E}</math>

The energy of liquid in the column is: <math>\frac{1}{2}\pi R^{2}h^{2}\rho g</math>

The energy of the system at height h is: <math>E=-2\pi Rh\cdot I+\frac{1}{2}\pi R^{2}h^{2}\rho g</math>

Substituting and finding the H that minimizes the energy gives: <math>H=\frac{2\sigma _{lv}\cos \theta _{E}}{\rho gR}</math>

## Capillary rise (Mechanics)

The liquid meniscus has a curvature:
<math>C=\frac{1}{R_{1}}+\frac{1}{R_{2}}=\frac{2}{R}=\frac{2\cos \theta }{\text{R}}</math>

The pressure inside the liquid at A is: <math>p_{A}=p_{0}-\frac{2\sigma _{lv}\cos \theta _{E}}{\text{R}}</math>

Mechanical equilibrium is: <math>p_{0}-\frac{2\sigma _{lv}\cos \theta _{E}}{\text{R}}=p_{0}-\rho gH</math>

which gives the same result as the “thermodynamic” result except it is less dependent on tube geometry: <math>H=\frac{2\sigma _{lv}\cos \theta _{E}}{\rho g\text{R}}</math>

## Fun Example of Capillary Action

If you have played with a straw wrapper in a restaurant, you may have already observed this example of capillary action. Scrunch the wrapper to one end of the straw and remove straw. You are now left with a straw wrapper that is roughly folded like an accordian. If you place a few drops of water on one end the straw, it will slowly unfold and straighten out as the water moves through the straw via capillary action. By the time the wrapper has stopped moving, most of it will be wet even though you've only placed drops of water on a small portion of the wrapper.

Source: APS outreach
http://www.physicscentral.com/experiment/physicsathome/mealtime.cfm