## Ohnesorge for Inkjet

### Quick Start

The formulation space for inkjet inks is very restricted, thanks to the combination of the Ohnesorge and Reynolds numbers.

If you play with your variables, viscosity, surface tension, density, drop velocity and "Length" (nozzle diameter) then you find that it is quite hard to get your formulation into the green zone where drops can form, they don't splatter and don't have terrible satellites.

### Ohnesorge Number

_{max}/r

The Ohnesorge number is a dimensionless constant that describes the tendency for a drop to either stay together or fly apart, by comparing viscous forces with inertial and surface tension forces.

For inkjet the key variables are viscosity η, scale length (e.g. nozzle diameter) l, density ρ and surface tension σ. The formula is:

`Oh=η/sqrt(lρσ) = "Viscosity"/sqrt("inertia"."surface_tension")`

The Ohnesorge number is related to the Reynolds number, Re=Vρl/η, and Weber number, We=V²ρl/σ by:

`Oh=sqrt(We)/"Re"`

It turns out to be convenient to use Z=1/Oh to characterise drops in inkjet printing as shown in the graph. As you change your variables, the values for your ink appear as a red dot. You can also see the ink plotted as Oh versus Re, showing similar trends with different descriptions for the good and bad areas.

By playing with the app you will see why inkjet inks are normally in the 2-15cP and 20-40 dyne/cm range, with nozzle velocities in the 3-10m/s range and nozzle diameters (i.e. the "Length" in Ohnesorge) of a few 10s of μm.

The effect of the drop properties on its radius on impact with the surface is shown as r_{max}/r and is explained below.

The graph is based on the work of Prof Brian Derby^{1} of U. Manchester. The area to the right of the "Satellites" line is characterised by Z>10 and is a zone where satellite drops are likely to dominate the printing process (though recent work shows that this limit might be closer to 14). The area to the left of the "Viscosity" line is characterised by Z<1 where the ink is too viscous to print.

Finally, where We<4 there is insufficient energy to create a drop and at high values of We there is too much drop splashing on contact with the surface. *Note that till 1 May 2019 the Splashing line was rather too high. An attentive reader kindly pointed out the error.*

The sweet zone between all these bad areas is shown in green and the aim is to alter your settings so the green dot is safely inside that zone. We all know that it is hard to create an inkjet ink in terms of particle sizes, dispersion/solubility, tendency to dry out and (partially) block the nozzle. This consideration of dimensionless numbers shows why the limits on viscosity, surface tension and driving energy are extra challenges to the formulator.

An equivalent way to look at it plots Oh versus Re. The diagram is often quoted but seems (see, coincidentally, the note below with respect to the Z-Re diagram) to be out by a factor of 10, fixed for this version. The zones are:

- Rayleigh break-up
- Sinusoidal wave break-up
- Wavel-like break-up with air friction
- Atomisation

The line between Sine and Wave is Ohnesorge's 1936 line.

### Drop size

Most of us think that the drop smashing into the surface will have a change shape dramatically and give a rather large radius. In fact, most drops arrive very gently and give rather little enlargement of the drop radius. For a drop of radius r then its maximum radius, r_{max} is given by:

`r_"max"/r = sqrt[("We"^2+12)/(3(1-cosθ)+4"We"^2/sqrt("Re"))]`

For simplicity, the contact angle θ is assumed to be 60° because its effect is rather small.

^{1}Brian Derby, *Inkjet Printing of Functional and Structural Materials: Fluid Property Requirements, Feature Stability, and Resolution*, Annu. Rev. Mater. Res. 2010. 40:395–414.

Note that Fig 4 in the paper contains an editorial factor of 10 error in the Re axis, and that the version shown in this app has been kindly confirmed as correct by Prof Derby.