Fountain Pen Design

Function, Development, Construction and Fabrication

Appendix 1: Jurin’s Law

Surface Tension Revisited

Fig. 1. Water molecule

Pure water is comprised of only two elements: oxygen and hydrogen. These atoms are held together because they share electrons in their outer shells, forming a very tight bond, and resulting in a group of two Hydrogens atoms locked to one Oxygen atom, hence the “formula”, H2O.

Fig. 2. Hydrogen bonds between adjacent molecules shown as dotted arrows.

When two or more water molecules are located near each other, there is a weak electrostatic attraction (“electrostatic” attraction is a force caused by differences in electrical charge between two things; where one object has too many electrons (and therefore has a negative charge) and another has too few (and therefore has a positive charge). This is the same force that causes your hair to be attracted to a ballon.) between the oxygen atom in one molecule and a hydrogen atom in an adjacent molecule. This attraction, known as a hydrogen bond, causes the molecules themselves to be slightly attracted to each other, resulting in a larger effect called cohesion. This results in the tendency for water to “clump”. Watch a water drop running down a window pane: when it comes near another drop, they will suddenly merge into one bigger drop. This is a visible result of that cohesion.

That cohesion is also the reason a water drop, falling in air, will try to stay together; and, since all of the molecules are equally attracted to each other, they tend to form a sphere, so that all those forces are equally distributed throughout the entire drop. (If you’ve already read this page, it should not come as a surprise to find out that raindrops, while falling, are not “raindrop shaped” but spherical. This is why spherical shot for guns was originally made by pouring lead from high towers like the one this page.)

The figure above shows a small group of water molecules, held together by the hydrogen bonds between them, shown as dotted arrows. In this case, I’ve drawn a small portion of a larger amount of water, so the hydrogen bonds are all pointing in different directions, and they are all roughly equal in strength, since each molecule has the same number of adjacent molecules to be attracted to.

However, if you are on Earth, and if you don’t have an infinite amount of water, (both of these conditions are very probable) then it will have a surface somewhere. Let’s look at what happens at the molecular level where the water meets the air.

Fig. 3. The hydrogen bonds between the molecules at the surface (the grey area) are stronger, forming a “skin”.

At the surface of the water, the last “layer” of molecules have fewer adjacent molecules to be attracted to, because there are none above them. In this layer, the hydrogen bonds between the molecules are stronger, since they don’t have to “share” the forces with as many molecules as the ones deeper in the water. This is shown in Figure 3, above as solid arrows in the grey area near the top. That extra-strong attraction is called surface tension which forms a kind of “skin” on the surface of the water. If you then have something small and light enough (like an insect or even a very small rock), it can rest on the surface of the water without breaking that skin. It’s a little important to understand that this is not the same as an object floating in the water (where the water is displaced by the object, like wood, a duck, or a witch for example). The object is resting on the water’s surface.

Contact angle

Fig. 4. The contact angles of the same liquid, shown on two different horizontal surfaces.

The electrostatic attraction between adjacent water molecules produces the cohesion that, on a human scale, causes waterdrops to hold together. However, the same water molecules can also be attracted to other substances. For example, there is an attraction between water and glass, seen when a rain drop is “stuck” on a window pane. The same forces that hold the water drop together hold the drop itself to the glass. However, when we’re talking about a liquid being attracted to a solid, it’s called adhesion instead of cohesion. (In other words, cohesion occurs between molecules of the same type, whereas adhesion occurs between molecules of different types.)

The strength of the adhesion between a liquid and a solid (for this example, water and glass) can easily be observed in different ways, but one of the simplest ones is to put a drop of water on a flat, horizontal piece of glass and looking at the shape of the drop where it meets the surface. The angle of the water at that intersection is an indication of whether the water is attracted to the glass or if it’s repelled by it. If the angle is less than 90º, then this is an indication that the solid is attracting the water, so it’s said to be hydrophilic or “water loving”. If the angle is greater than 90º, then it means that the solid is repelling the water, so it’s called hydrophobic or “water fearing”.

As was discussed in Surface Tension, that angle is called the contact angle, abbreviated θ, and is shown in Figure 4, above.

This angle is dependent on the relationship between all three substances in question: the liquid, the solid, and the gas. However, since we normally assume that we’re working in air, we will only concern ourselves with the liquid (in our case, ink) and the solid (the feeder, and/or the nib).

Fig. 5. Measuring the sessile contact angle from a photo.

One of the earliest ways to determine the contact angle for a liquid on a solid was to use a combination of a magnifying glass to view the drop and a protractor to measure the angle. Nowadays, this is easier to do by simply taking a close-up photo of the drop with the camera level with the surface, and then determine the contact angle in the photo. An example of this is shown in the figure above. This way of doing the measurement is called the sessile drop method (“sessile” is just a fancy word meaning “fixed to one spot”), and the magnifying glass / protractor combination is called a contact angle goniometer.

When the liquid is stationary, the contact angle does not change, however, it is not the same as when there is some kind of movement in the liquid. However, this can also be measured by changing the volume of the drop on the surface. If the volume is increased by adding more liquid, then the contact angle will increase as the diameter of the drop expands. This is called the advancing contact angle.

Fig. 6. Measuring the advancing contact angle.

Of course, if there’s an advancing contact angle, then there is also a \textit{receding contact angle}, which is typically a lower value, and is measured by reducing the volume of the drop. This is the contact angle of the liquid at the position where it pulls away from the solid’s surface as the drop’s diameter decreases. It might be interesting to note that a drop of liquid on a vertical surface (for example, a raindrop on a window pane) displays both the advancing contact angle (at the bottom) and the receding contact angle (at the top).

Fig. 7. Measuring the receding contact angle.

There are many more methods for measuring the contact angle of a liquid on the surface of a solid – but we’ll leave those out of this discussion.

Capillary action

As was already discussed in the page on Capillary Forces, if a glass tube is partially submerged into water, then the water will climb up the inside of the tube. But why does this happen?

Fig. 8. The water can be seen creeping up the surface of the glass.

As we’ve already seen, there is an electrostatic force of adhesion between the water and the glass: in other words, the water is attracted to the glass. The water molecules on the surface “see” the surface of the glass nearby, and are attracted to it, not only sideways, but also upwards (in the same way that the water molecules below are attracted upwards to the water molecules above them as shown in Figure 3, above).

As a result, if we put a piece of glass in a bed of water, we can see the water surface rising upwards at the contact point of the glass, as shown in Figure 8.

Fig. 9. The force on the water pulling in the direction of the black arrow.

The adhesive force that causes the water to climb up the surface of the glass is pulling in the direction that can be determined from the contact angle. Looking at Figure 9, the direction of the force is shown as the black arrow. If the contact angle, θ, were less than 90º, then the water would be pulled vertically upwards. If θ = 90º, then the liquid would not be pulled upwards or downwards at all.

The lower the contact angle of the liquid, the more the liquid will rise at the surface of the solid because the force is pulling more vertically. This relationship between the vertical force and the contact angle can easily be calculated using cos(θ), which is 1 when θ = 0º, 0 when θ = 90º.

Pulling downwards against this force is the weight of the water itself. The more water, the heavier it is; but this is also dependent on gravity. If we know the mass of the water and the force that gravity is pulling downwards on it, then we can calculate the weight of the water. In other words, if we were to take the same photo shown in Figure 8 on the moon, the water would be much higher on the glass because there’s less weight pulling it downwards.

If we submerge a tube (for example, a glass pipe) in a liquid (for example, water) then the force shown in Figure 9 exists for the entire circular line where the surface of the water touches the inside of the tube. The total length of that line can be calculated using the equation for the circumference of a circle (the inside of the tube).

l = 2 π r

The total force pulling on the water below can then be calculated by multiplying that length by the coefficient of the water’s surface tension in air, which is specified in Newtons per metre. In the case of water and glass, this value, labelled γla, is 0.0728 N/m. (In other words, for every metre where the water touches the glass, it is pulling with a force of 0.0728 N.) However, we’ve already seen that the force is not directly vertical, it’s scaled by the cosine of the contact angle. This means that the total vertical force pulling upwards (FU) is

FU = 2 π r * γla * cos(θ)

We now know how to calculate the Force that’s pulling the liquid upwards in the tube. The question then is “how high will it go?” This height is the point where the FU equals the force pulling the liquid downwards (FD) caused by the mass of water in the tube that’s being pulled upwards. So, how do we calculate this?

Let’s start by saying that the liquid has already risen up the tube and reached some height that we can measure. If we know this, then we can take the mass density of water (abbreviated ρ), which is 1000 kg per cubic metre, and multiply it by the total volume of the liquid in the tube. This is the volume of a cylinder, so it’s the cross sectional area of the tube multiplied by the height of the liquid, therefore π r2 * h. If we multiply this by the acceleration due to Earth’s gravity (abbreviated g) of 9.81 m/s2, we get the total downwards force of the liquid in Newtons.

FD = ρ * π r2 *h * g

The liquid will rise in the tube until the upwards and the downwards forces are equal, but opposite, as is represented by the illustration in Figure 10, below.

Fig. 10. The capillary forces pulling upwards balance with the mass of the water being pulled down by gravity.


2 π r * γla * cos(θ) = ρ * π r2 *h * g
2 π r * γla * cos(θ) = ρ * π r *r *h * g
2 γla cos(θ) = ρ r h g
(2 γla cos(θ) ) / h = ρ r g
(2 γla cos(θ) ) / (ρ r g) = h

This means that we can calculate the height of the liquid in the tube where the two forces balance each other; in other words, the height that will be reached by the liquid as a result of the upwards force acting on it.

This equation is known as Jurin’s Law, named after the English scientist and physician James Jurin who not only discovered it (around 1718), but he also did some of the earliest statistical work proving the effectiveness of smallpox variolation, a kind of precursor to vaccination. This law is more commonly stated as follows:

h = (2 γla cos(θ) ) / (ρ g r0)


  • h is the height of the liquid in the tube, in the same unit as r0 (typically m)
  • γla is the surface tension of the liquid in air in N/m (or J/m2) at a specified temperature (typically 20 ºC)
  • θ is the contact angle of the liquid measured relative to the tube wall
  • ρ is the mass density of the liquid in kg/m3
  • g is the acceleration due to gravity in m/s2
  • r0 is the radius of the tube, in the same unit as h (typically m)

Note that this equation is true only if the radius r0 is less than γ / (ρ g), known as the “capillary length” of the liquid. (As an extreme example, there will be no capillary action for water in a tube 1 m in diameter.)

If we want to find the height of a water column in a glass tube on Earth with an ambient temperature of 20 ºC, we can plug the following values into the equation:

  • θ = 0º for water in a glass tube
  • γla = 0.0728 N/m at 20 ºC for water and air
  • ρ = 1000 kg/m3 for water
  • g = 9.81 m/s2 on Earth

which simplifies it to:

h = (1.4842 * 10-5) / r0

So, for example, if you have a glass tube with a diameter of 1 mm, then r0 = 0.5 mm = 0.0005 m, and h = 0.0297 m or 29.7 mm.

Fig. 11. Scale representations of the liquid heights achieved in capillary tubes. On the left are tubes with identical diameters, but made of four different materials that result in different contact angles. On the right are three tubes made of identical materials (and therefore with matching contact angles) but with different diameters.

Figure 11 plots the relationship between the capillary tube diameter and the height of the liquid column for three different contact angles. It also shows a to-scale example of this information for capillary tubes with different diameters.

On the left are four tubes with identical diameters of 1 mm, but made of different materials that result in different contact angles. You may notice that the tube with the 30º contact angle has half the height of the one with the 90º contact angle. This is because cos(90º) = 1 and cos(30º) = 0.5.

On the right are three tubes made of identical materials resulting in a contact angle of 0º, but with different diameters.