# Difference between revisions of "Hydrodynamics of Writing with Ink"

Original entry: Tamas Szalay (APPHY225 2012)

"Hydrodynamics of Writing with Ink"

Jungchul Kim, Myoung-Woon Moon, Kwang-Ryeol Lee, L. Mahadevan, and Ho-Young Kim

Phys. Rev. Lett., 107, 264501 (2011)

## Summary

In this paper, the authors examine the hydrodynamics of writing with ink on paper with a capillary source, eg. a fountain pen. The experiment is performed with a model system of superhydrophilic silicon micropillars to create the rough surface, and a simple glass capillary tube with various solutions inside to simulate the pen. In this model system, they calculate the expected rate of spreading for a stationary pen, and use this to derive the line width for a moving pen, which they then compare to experimental results. The analysis is then extended to a system using actual ink and paper.

## Soft matter keywords

Wetting, Laplace pressure, surface tension, spreading, capillary

## Physics of Writing

The spreading force caused by the energetic gain of wetting the surface has to be balanced with the viscous shear force holding it back. Letting $f$ be the roughness (total surface area over projected area, $f \ge 1$), the effective driving force (as a function of r) due to the surface energies is $F_d = 2 \pi \gamma (f-1) r$

This can be balanced with the resisting force due to viscous shear, which scales as $F_r \sim \mu U (r^2 - R^2)f/h$, where $R$ is the radius of the droplet-surface contact (roughly the radius of the tube itself), and $U = \frac{dr}{dt}$. Solving and integrating for $U$ (at late times and small tube radius) gives:

$r \sim (\phi \frac{\gamma}{\mu}h)^{1/2}t^{1/2}$

with $\phi \equiv \frac{(f-1)}{f}$.

One of the key considerations for spreading, however, is the depletion of the liquid from the pen/tube as time passes. The Laplace pressure inside the capillary sets up an equilibrium rise height, and as the liquid flows out of the tube, this difference sets up a pressure drop $\Delta p_t = p_0 - p_t = 2\gamma/R - \rho g H$, where $H$ is the difference in height from equilibrium. There is a minimum roughness for the liquid to spread on its own, from balancing the driving force to get past the tube radius and the resulting pressure drop:

$f_{min} \approx 1 + 2h/R - Hh/l_c^2 \in (1.04-1.07)$

On a smooth substrate, the liquid does not spread beyond a small radius, due to the additional Laplace pressure that gets set up