# 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

(scale bars: <math>1 mm, 150 \mu m, 10 \mu m</math>)

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 briefly extended to a system using actual ink and paper.

(scale bars: <math>1 mm, 80 \mu m, 15 \mu m</math>)

## 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 above.

(scale bar: <math>1 mm</math>)

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 first is the nondimensionalized blot spreading as a function of time, which can be seen to be a good linear fit. The reason the line does not necessarily intersect the origin is presumably because the radius does not start out at zero when the tube contacts the surface.

This second plot is a fit to the equation of the nondimensionalized parabolic wavefront (which was not derived above, for brevity), but the plot is produced here to show that the analytic solution for the shape is a good one.

Finally, here is the fit to the line width as a function of various experimental parameters, which are specified in the supplemental materials of the paper. Again, for their ideal system, their analytic solutions describe the system well.

The authors do not do an in-depth analysis of the real pen-on-paper system, but merely state that using the image shown at the top of the page, it is possible to provide approximate values for the pore properties and compare them with predictions. The predictions mostly match, but the one place where the strongly non-ideal system of ink and paper seems to break down is that the theory overestimates blot radius (but not line width), suspected to be due to paper swelling.