# 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 <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 \equiv \frac{(f-1)}{f}</math>.

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 <math>\Delta p_t = p_0 - p_t = 2\gamma/R - \rho g H</math>, where <math>H</math> 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:

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

where the latter numbers are for their particular conditions. The important thing to note is that for a smooth substrate, the liquid does not spread beyond a small radius, due to the additional Laplace pressure that gets set up with the droplet spreading from the capillary tube, as in the figure.

For a moving pen, the authors solve the same balance equations in a moving frame and a non-radial profile, and show that the resulting profile is a parabola. Of course, the liquid doesn't spread across the surface forever; far behind the parabolic leading edge, the distance saturates due to contact line pinning. Since the pinning is a constraint external to the surface tension and viscosity, it is expressed as two free parameters in the final equation for the width:

<math>\frac{w_f}{R} = \alpha \frac{\eta^2 h}{R} + \beta</math>

## Experimental Results

The equations derived above are tested rigorously for the micropillar array, in particular, the predictions for the blot size as a function of time, the parabolic shape of the leading edge for a moving pen, and the final line width for different experimental parameters.

The authors do not do an in-depth analysis of the real pen-on-paper system, but merely state that the image shown
The authors express concern that the paper structure has strong anisotropy out-of-plane, but that the data seems to demonstrate that the effect is not large enough to have a significant impact. The one place where the strongly non-ideal system of ink and paper seems to break down is that it overestimates blot radius (but not line width), suspected to be due to paper swelling.