2.4 Capillaries

Generally, people think of capillaries to be tiny round pipes. However, they can be of any cross-section as long as their dimensions are narrow enough so that the characteristics of the liquid combined with the material properties of the wall cause the liquid inside the pipe to rise or fall. This observation is often called capillary action instigated by capillary forces.

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The entire topic of capillaries is vast. In the following, I will talk about conditions needing to prevail for the function of the feed. Hence, I will concentrate on situations where capillary action causes the liquid to rise.  I will endeavour to provide sufficient information for you to understand the function of the feed.

Capillary Forces

The material properties of a capillary and its interacting liquid result in a contact angle θ causing a capillary pull, which forces the liquid to rise.  I include the highly truncated approximation of the formula for the calculation of the rise of water at a given radius (half the diameter) of the capillary.  The unit of measure for the rise h is the same as for the radius r, generally [mm]:

h ≈ approx  1.48 x 10^-5 ÷ r

Indubitably, this formula calculates the theoretical maximal rises, achievable under favourable laboratory conditions, however, the highest rise I have ever achieved in my laboratory in a 0.5 mm glass capillary was about 30mm.  During investigations of feeds made from various plastics and other material, the rise in a 0.4 mm wide canal was about 25mm.  That’s it.  At 0.3mm width canals achieved 3mm more, but canals of such width are impracticable; firstly, they don’t let go of the ink at the cross-over to the nib slit, secondly, they dry out quickly because of the stagnant ink.

Click on Capillary Action to read more on the Wikipedia site.

Diagram 1 — Calculated liquid height due to capillary force in glass tubes of various diameters for different contact angles. (Geoff Martin)

The graph in diagram 1 shows results correlating with my experience.  I have seen feeds with overflow slits at heights exceeding this limit, but, in my experience, they are useless because they don’t fill under “normal” circumstances.  When optimising my feed, I have observed more than once, the ink drips off the nib as soon as the 25mm mark is reached, and before the slits above could fill.  It’s hard to obey the laws of nature, sometimes.

Only when the fountain pen is in a nib-up position, and it experiences air pressure changes, then those upper slits could fill if the air pressure increase would not escape through the air inlet.  Even if the upper slits would be filled, they would release their content as soon as the pen is turned into its writing position. Splosh!

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Sketch 2 shows the influence of the width of the diameter of the capillary with constant, equal parameters.  All capillary surface reaction is the same, causing the same contact angle θ and therefore the same capillary pull.  Also note: The rise h₁ does not change when the capillary is tilted.

Sketch 2 — Capillary rise in tilted columns and various diameters

Allow me to discuss sketch 2 which helps to solidify your understanding:

One could deduce that the rise in the thin tube (1) is higher because of the smaller cross-sectional area A₁ and therefore, the air-pressure causes less force pushing against the capillary force of the liquid.  Consequently, the rise in the tube (2) is less because of the larger cross-sectional area A₂. 

However, as plausible as it may sound, the above explanation is incorrect. The air-pressure is constant all over the arrangement. It does not only act inside the capillaries but also in the container.

Sketch 3 — A section of a capillary tube

The difference in column height exists purely because of capillarity.  The strength of the capillary force is proportional to the size of the area of contact (mauve coloured surface in sketch 3) between the fluid and the solid.  The formula for a tubular section is:

A (the mauve part only)  =  2r π h    … where h is the rise of the liquid

The force acting against the capillary force is caused by the weight of the liquid, which is proportional to the volume.

V = r² π h

Comparing two capillaries with one being twice as thick, the surface area doubles and so does the surface action to support the liquid column; however, the volume and therefore the weight quadruples.

Capillary Force ∝ A ∝ r     but      weight ∝ V  ∝ r²

Therefore, as the diameter increases, the rise reduces as shown in diagram 1, because the weight grows by the square of the radius.  Got it?  Interesting, very interesting, that’s where the magic begins, IMHO.

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Sketch 4 — Radius of a spherical meniscus

Let’s continue on.  When talking about surface tension and capillaries, the term meniscus is quite common.   It is the shape the top liquid layer takes on when in contact with enclosing surfaces. In a narrow tubular capillary (less than 2mm, also depends on liquid), the shape approximates a section of a sphere with a radius R, see sketch 4.  It is calculated:

R = a ÷ tan θ

Where a is half the diameter, the radius of the capillary.

The rise of the liquid inside the capillary is often referred to as the height of the meniscus.

Round, Square or Rectangle

As I said before, and allow me to repeat it, when we talk about or think of a capillary, in general, it is a thin round tube.  In a fountain pen feed (and many other items containing capillaries), they are square and most often rectangular.

Mainly, it is the consequence of the manufacturing process: it is much simpler to cut an accurate groove (0.3 to 0.5mm wide to a tolerance of ± 0.02 to 0.05mm) than to drill a small hole of the same length (approx. 25mm) and accuracy, I consider rather impossible.  A drill bit of such dimensions would wander off; it is almost impossible to cool the bit and remove the debris.  If the feed is entirely produced through injection moulding (which is now the standard process) a groove is the only way.

During most of my research on ink, I used glass tubes. They come in all sizes, with high accuracies and constant surface properties. The available feeds were not reliable enough, even when produced under laboratory conditions.

Here only briefly about my work on ink: Once I had noticed the considerable variation of the ink’s parameters, and before I could start working on the feed, I had to standardise an ink for my tests and learn to produce it within a narrow range of variation of its characteristics.  Otherwise, when investigating any changes during the feed’s development, I would not have known whether they were caused by the ink’s inaccuracies or the variations I wanted to test on the feed.

Besides the width of the canal and the resulting capillarity, the other dictating criterion was the volume of ink that needed to be supplied to the nib when writing.  Therefore, in my feed design, I chose a deep rectangular cross-section and two grooves in parallel.

Sketch 5 — Cross-section through a rectangular capillary

An added benefit of injection moulding is the sharpness of the corners at the bottom of the capillary groove.  They cause a much stronger capillary pull than the capillary itself. This effect helps the ink to advance in an empty or even dried out ink canal.  More about this later.

Sketch 5 shows the cross-section of a rectangular capillary.  The liquid towards the corner experiences a higher capillary pull.

One way to explain this behaviour is by imagining that the liquid closer to the edge experiences the pull like in tubular capillaries with reducing diameters D to A and therefore, it rises higher. The more contact between the liquid and its solid surrounds increases the capillarity.

Sketch 6 – Rectangular capillary (my own creation)

Helping you with understanding what I try to depict, you may recognise in the next daring picture, sketch 6, a perspective image of the above, sketch 5.

Allow me to explain the various parts and expressions:

A is the gutter or arch meniscus which rises the most,

B is the meniscus along the wall of the wider side a at a rise h2

C is the smaller meniscus along the wall of the narrower side b at a rise h1

D shows the bottom of the trough with its rise h3 which is created by the interaction between the three (A, B, C).

Knowing of the sneaky behaviour of fluid in edgy capillaries helps to coerce the fluid to move where the designer wants it to go.  By now, you know more than many pen designers; congratulations.

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In the next chapter, with our current understanding, we can now look more closely at the design of the feed… Want to have a look?

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Amadeus W.
Ingeneer

20 March 2016

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