# 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!

Capillary bridges exert forces between two substrates. Both surface tensions and laplace pressures contribute to this force. In the case of parallel plates, as the distance between the two places approaches 0, the laplace pressure terms dominates over the surface tension terms and <math> F = \gamma_{lv} V (\cos(\Theta_1)+\cos(\Theta_2))/D^2 + O(D^{-1/2}),</math>

where the <math>\Theta</math> are the contact angles, D is the distance between the plates, and <math>O(D^{-1/2})</math> is the contribution from surface tension terms..

(E J De Souza, M Brinkmann, C Mohrdieck, A Crosby, E. Arzt. Capillary Forces between Chemically Different Substrates. Langmuir 2008, 24, 10161-10168.)

## 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.

Therefore the densities of the liquid, object and air are important. These determine the curvature of the interaction at the surface of the object as well as other properties.

A water strider "floats" because it bends the surface of the water to support its weight. Floating by de Gennes means bouyancy.

In the figure from de Gennes the object must have been submerged and have a lower density than water so it would float.

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!)

**This image looks similar to the one above. It shows how colloids self-assemble through evaporation and capillarity:**

Source: Nagayama et al., "Two-Dimensional Crystallization", *Nature*, **361** (1993)

## 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

An interesting paper on calculating meniscus profiles in nanoparticle-surface interactions.

Pakarinen et al. "Towards an accurate description of the capillary force in nanoparticle-surface interactions," Modelling and Simulation in Materials Science and Engineering 13, 1175-1186 (2005).

http://lib.tkk.fi/Diss/2007/isbn9789512285693/article4.pdf

The authors model the situation of a nanoparticle interacting with a surface in a humid atmosphere, since the capillary force formed by the meniscus can be one of the most important interactions in the system. These interactions are often studied with AFM which has forced soft matter physicists to move beyond simple models as we can know obtain nanoscale measurements. The authors consider particles beyond just the simple spherical model to model particles of different shapes and sizes, different humidity levels, and different particle-surface separation levels. Many of the concepts we discussed in class are covered here.

## Cool Applications of Capillary Action

Elecrophoresis is a very commonly used technique in biology. Charged particles move through a liquid under the influence of an applied electric field. This allows species to be separated, because some move down the liquid medium faster than others. There is another type of electrophoresis technique called capillary electrophoresis (CE). The basic idea of CE is that species can be separated depending on their size to charge ratio when put inside a small capillary filled with electrolyte solution. This gives unprecedented separation. For example, proteins differing only by one amino acid can be resolved using CE! Capillaries provide better modes of separation then normal gel electrophoresis.

Charged species is introduced into a capillary via capillary action, and the movement of the particles up the capillary is started by the application of an electric field. This migration is called electroosmotic flow. The species become separated because they have different eletrophoretic mobilities, and this can be detected over time. The capillaries have to be stable enough to be detected with methods such as fluorescence, UV or UV-Vis absorbance. This can be tricky, because for some methods, such as UV absorbance, the capillary must be optically transparent, meaning they have a tendency of breaking upon detection. Fluorescence is used to detect samples that fluoresce naturally or which have fluorescent tags attached to them. Once detection has been made, the identity of each sample is made, generally with mass spectrometry or surface enhanced Raman spectroscopy.

There are other types of capillary electrophoresis. One example capillary gel electrophoresis (GE), which is an adaptation of the traditional gel electrophoresis. Polymers in solution are put inside the capillary which creates a molecular sieve which mimics the gel used in GE. The beauty of this technique is that analytes with similar charge to mass rations can also be resolved inside of the capillary by molecular size.

Another example of CE is called capillary isoelectric focusing. This method requires putting amphoteric molecules inside of a capillary and then generating a pH gradient inside. This causes the solute to migrate until it has is no net charge: this is the isoelectric point. These species can then be resolved by mobilizing them past a detector by using pressure or other chemical means. Yet another example of CE is called capillary electrochromatography.

## Below the Capillary Length

For some applications, it is possible to go below the capillary length, but generally, the droplets are then in an unstable state and would tend to coalesce to form larger droplets.

Droplets of water, in rain for example, can typically be expected to be on the order of the capillary length, i.e. on the order of a millimeter; and that is the case when rain is falling from the sky at normal, above-freezing temperature. However, in fog and sometimes when it drizzles, the drops of water are below the capillary length; but that is only a transient state. When water condenses on the lid of a pot of boiling water, the same holds: the droplets are initially very small, but then coalesce to form capillary-size droplets. One can easily access this transient state of small water droplets at room temperature by using an ordinary spray-bottle.

For typical cooking oil at room temperature, surface tension is a couple of times smaller than that of water and density is between 80% and 90% that of water. So effectively, the capillary length is the same as that of water, again, on the order of a millimeter. When cooking on a pan, one often desires to have a thin layer of oil covering the pan; if you just pour cold oil in the pan until it covers the whole surface, the thickness of the layer will be on the order of the capillary length and your meal may just end up being very unhealthy that way. Several easy solutions are available:

- pour a few drops of oil, heat up the pan, and then spread the liquid by rocking the pan back and forth in various directions
- pour a few drops of oil and spread it via mechanical action, e.g. rubbing it around with a paper towel.
- use cooking spray!

Here again, the spray will create tiny droplets of oil that spread all over fairly uniformly. If you let it sit for a while, you can expect pure oil droplets to coalesce; however, it is probable - though I haven't checked - that cooking sprays contain some surfactant as well to prevent this from happening too fast. The rate at which it does happen will depend on temperature, the amount of spray you've applied, i.e. the density or proximity of the droplets, as well as the concentration of surfactant. You can find interesting uses for cooking spray online, for example at [2].

There are plenty of examples for other liquids. The point of the above was just to remind ourselves that the capillary length is only a lower limit on the dimensions of a static droplet or liquid film, if it is not forced mechanically to adopt some other temporary shape.