Drop Spread Modeller

Quick Start

If you look at most drops spreading on a surface they seem to spread very slowly. But if you'd used a high-speed camera you would find a rapid initial velocity that slows down rapidly.

Spreading theory (Tanner) is well-known but hard to model. To build up understanding, start with a simple spherical cap drop of your chosen radius, and choose viscosity, surface tension and select a starting contact angle θStart of 90°, a perfect fresh hemisphere.

You may have to tweak Tmax to get the right time-scale and Rmax to get the right Radius scale. But you will quickly see the fast initial spread and the rapid slow-down.

How do you stop a drop from spreading? With a non-zero equilibrium contact angle, θEquilibrium!

Drop Spread

Viscosity Pa.s
ST Dyne/cm
Radius or Wedge µm
TMax s
RMax µm
Height µm
Width µm
End Rad. µm
Start Height µm
End Height µm
Start µm
Start Angle°
End Angle°
End µm

When a spherical drop, a cylindrical drop or a line (modelled here as a wedge, explained below) are printed, they want to spread from their initial size. This makes life difficult for the printer as the printed feature needs to be finer than the desired value. The modeller calculates the growth of each feature and provides some important before and after values.

The take-home message is that drop spread can be astonishingly fast at first, so by the time it is measured most of the spreading is already over. The physics of the spreading are remarkably easy to describe, though the modelling itself is rather complex.

The key ideas are based on Tanner spreading theory which, in its basic form, can only be applied to hemispherical caps, i.e. simple drops. The velocity, v, of spreading depends on contact angle θ, surface tension σ, and viscosity η:


The ability to calculate the effects for drops, cylinders and wedges (e.g. a line) comes from key papers by Prof Glen McHale who was then at Nottingham Trent University. His help in putting these models together is gratefully acknowledged.

Why do drops spread?

People often think that spreading is due to gravity. For normal printed drops gravitational effects are irrelevant by many orders of magnitude. Spreading is driven purely by surface tension. There are 3 key facts behind the theory.

  1. The rate at which a drop spreads is proportional to the cube of the contact angle. This means that a "sharp" printed dot, with a high initial angle from the printing step will spread very rapidly at first.
  2. As the drop spreads, the contact angle decreases (the height decreases because the volume is constant) so the rate of spreading decreases rapidly. The width grows as the 1/10th power of time.
  3. The spreading stops either when the ink dries/cures or the contact angle reaches the equilibrium value

Using the Drop Spread Modeller

You need to enter 3 key properties of your system:

  1. Viscosity (in Pa.s)
  2. Surface Tension (in dyne/cm or mN/m)
  3. Equilibrium contact angle of the liquid with your substrate.

Then you need to specify the geometry, choosing from:

  • Drop (i.e. a Spherical Cap) and its initial radius
  • Cylinder (i.e. a one-dimensional Spherical Cap) and its initial radius
  • Wedge, an object, such as a line, defined by its height, width (along the top) and, using the Radius input, a wedge shape either side.

The Height and Width inputs are only enabled in Wedge mode. Wedge is a simple approximation that makes the calculation possible. The initial line of a starting Height cannot be perfectly vertical. A cross section will be the rectangle of your given Width plus a triangular "Wedge" either side acting as a sort of buttress, specified via the Radius or Wedge input. The starting contact angle of the wedge shape is ArcTan(Height/Wedge). The total starting width, shown as the Start value, is Width + 2*Wedge. Small errors in Wedge aren't all that important because narrower Wedges have a higher contact angle so spread much faster.

Finally, you need to specify TMax - the time, in seconds, to run the simulation (this needs to be similar to the timescale of drying on your press) and RMax, the maximum "radius" to scale the plot. Although the plot could be autoscaled it turns out that having an absolute size makes it much easier to work out the effects of changing input parameters.

From those inputs the modeller plots how the radius (or wedge) changes over time (i.e. it plots Δr, the change from the given radius) and also calculates the final height, width and angle, including the total width (2*radius or width+2*wedge) at the start and finish.

Tooltips appear over the input/output boxes to remind you what they are and moving the mouse over the graph gives you the data at any given time.

How can I stop drop spreading?

Because of the cubic dependence on angle, changes in viscosity or surface tension make remarkably little difference. Therefore there are only 3 things you can do to avoid drop spreading:

  1. Print features with small heights and, therefore, small contact angles
  2. Print onto a substrate with a high equilibrium contact angle
  3. Dry/cure amazingly quickly after printing

Printing with a higher-solids ink to give a smaller dot has the double benefit (generally) of a higher viscosity - though for many printing techniques (e.g. ink jet) it is that higher viscosity which makes it impossible to print higher solids. Dry/curing faster is usually not practical - especially as a lot of spreading takes place within 100msec. Therefore everyone reaches the same conclusion - that they have to find an ink/substrate combination with a higher equilibrium contact angle. There is an irrational fear of this because people wrongly associate higher contact angles with poor adhesion, on the grounds that adhesion depends on surface energy. For a thorough debunking of this myth and, therefore, a strong encouragement to seek higher equilibrium contact angles to reduce drop spread, see the first few pages of Practical Adhesion