Pen Nib - Deeper Dive | Journal

(Diagram 1 – A droplet on glass showing γ acting tangentially at the surface)

1. Surface Tension — the Liquid’s Invisible Skin

Watch a bead of water balance on a leaf and you’ll glimpse the physics behind every fountain pen.
The droplet holds together because of surface tension (γ). This is a boundary force that acts like a stretched film across the liquid’s surface.
Inside the liquid, every molecule feels equal pull from its neighbours.
At the surface, the upward neighbours are missing, so the remaining molecules tug inward.
That imbalance produces tension, measured in newtons per metre:
γ=F/L

Surface tension is the energy stored in that “skin.”
It is the foundation of every capillary effect that follows.

2. Cohesion and Adhesion — What holds water together and draws it to surfaces?

If we zoom in on a single water molecule, its shape explains everything that follows.
A water molecule is not symmetrical: oxygen draws shared electrons slightly toward itself, leaving the hydrogen side faintly positive.
The result is a molecular dipole — one end negative, one end positive.
These tiny charges attract neighbouring molecules, forming hydrogen bonds that hold the liquid together.
That attraction is cohesion.

(Diagram 2A – A water molecule showing partial charges and dipole moment.)

When water meets a solid surface, the outcome depends on the surface’s chemistry:

Surfaces with charged or polar groups attract the dipoles in the water — this is adhesion, and such surfaces are hydrophilic (“water-loving”).
Surfaces without polarity or charge offer nothing to grip. Cohesion wins, the liquid tightens into droplets, and the surface is hydrophobic (“water-fearing”).

(Diagram 2B – Two droplets: Left – hydrophilic surface, θ < 90°; Right – hydrophobic surface, θ > 90°.)

The contact angle (θ) expresses that balance precisely:

Small θ → strong adhesion → wetting.
Large θ → strong cohesion → non-wetting.

In a tube, this manifests as either a concave (small θ) or convex (large θ) meniscus.

(Diagram 2C – Cross-section of a tube showing concave vs convex menisci.)

Hydrophilicity in Pen Feeds and Converters

In a fountain pen, these same forces decide whether ink flows or fails.
Both the feed and the converter or cartridge interior must be — or be treated to behave as — hydrophilic.
Only then will a continuous film of ink form along the walls, allowing capillary pressure to act.
If the surfaces are too hydrophobic, ink retracts into droplets: the pen starves.
If they are too hydrophilic, ink spreads instantly and completely, the feed floods.
That is why materials such as ebonite, ABS, and POM are engineered to achieve a delicate middle ground — just hydrophilic enough for stable flow.

(Diagram 2D – Feed cross-section: One showing a concave meniscus (hydrophilic) and one a convex meniscus (hydrophobic)

3. Adhesion’s Vertical Component — Where Capillary Force Begins

At the contact line, the vertical component of surface tension provides lift:

F_”γ” is the liquid surface tension force per unit length
F_(adh, vertical) is the the vertical component of the liquid surface tension force per unit length dl is the unit length of the liquid-capillary contact line
Around a circular rim:;

This is the capillary force — the sum of those vertical pulls acting around the wall.

4. Capillary vs Hydrostatic Force — The Balance of Heights

Gravity resists through the hydrostatic weight of the liquid column:

At equilibrium:

In our illustration, two tubes of different radii extend down from the bottom of a sealed reservoir.

r is the radius of the relevant pipe
ρ is the density of the waterwater
g is gravity
h is the distance the water travels down each pipe

The smaller pipe, with its tighter radius, sustains a higher capillary pressure and therefore pulls the column of water farther down.

(Diagram 3 – Sealed container with two downward pipes, r₁ < r₂, showing h₁ > h₂.)

5. Capillary and Hydrostatic Pressure — The Same Balance in a Different Form

The key driver in this process is actually the pressure. Dividing by area gives the pressure relationship:

This shows that capillary pressure increases as radius decreases - and the central reason why fine channels can pull ink so effectively.

6. Meniscus Formation — Concave and Convex

Inside the sealed reservoir, air pressure pushes upward while surface tension resists.
At each pipe mouth, the liquid surface curves to hold back the air:

That curvature is defined by the Laplace relation:

Smaller radius → tighter curvature → larger resisting pressure.

Small pipe: the meniscus bows outward — convex — able to withstand higher pressure.
Large pipe: the meniscus curves inward — concave — and yields sooner.

(Diagram 4 – Convex meniscus at small pipe, concave at large.)

7. Bubble Formation — The Threshold Rule

When the holding pressure inside the reservoir drops below what the water surface tension can hold at a pipe mouth, air enters and forms a bubble.
The simple condition:

Because the large pipe has a greater r, the trigger to release a bubble happens at a lower pressure difference and so happens first.
Air pushes through, a bubble rises, and the internal pressure resets — the gentle pump that keeps the equilibrium in the converter.

(Diagram 5 – Pressure-threshold bars; larger pipe triggers first, bubble shown entering.)

8. From Pipes to Feeds — Designing Flow in Layers

A fountain-pen feed is a miniature version of that system.
Each fin channel has its own width (which can be modeled as a radius r_i) and thus its own capillary pressure:

By graduating those widths, we determine the order in which the fins fill:

The central capillary channel is narrowest and fills the first.
Narrow fins (highest P_c) then fill next, anchoring ink firmly.
Wider fins store excess and buffer flow.

The feed thus self-regulates — a hierarchy of channels orchestrating ink and air so quietly that we rarely notice it.

(Diagram 6 – Feed cross-section showing fin width gradient, ink-flow arrows, air-return path.)

Takeaway

Surface tension creates the pull.
Adhesion gives it direction.
Geometry establishes its strength.
By adjusting the chemistry and width of every channel, the feed can control the movement of ink and air — one bubble, one line, one steady flow at a time.