# 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 becomes

$F_d = 2 \pi \gamma (f-1)r$

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