Hydrodynamics of Writing with Ink

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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)


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 <math>f</math> be the roughness (total surface area over projected area, <math>f \ge 1</math>), the effective driving force (as a function of r) due to the surface energies is <math>F_d = 2 \pi \gamma (f-1) r</math>

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

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

with <math>\phi \def \frac{(f-1)}{f}</math>.

Experimental Results