## Stefan's Law

How a drop of liquid deforms under squeeze pressure affects both printing (dot gain) and UV embossing (residual layer).

The same basic formula with its dependence on starting thickness h_{1}, viscosity η and drop radious R works for both, with the R^{4} dependence dominating things, followed by the h² term. The force F is important for both, though for printing we use the nip pressure.

An alternative view of squeeze flow of a drop of adhesive can be found at Drop Squeeze within Practical Adhesion

### Stefan's Law for Printing

_{1}nm

_{1}μm

_{2}

### Stefan's Law for UV embossing

_{1}nm

_{2}nm

In both printing and UV embossing, a feature of radius R and thickness h_{1} is squashed with a force F and will go to thickness h_{2} after a time t that depends also on viscosity η. The time, t, taken is given by Stefan's law (this is a different Stefan from the black body radiation formula):

t=3πR^{4}η/(16F)(1/h_{2}²-1/h_{1}²)

For the printing situation, you apply force F for time t on a dot with radius R_{1}, with an initial thickness h_{1}. From this you can calculate h_{2} and, by conservation of volume you can calculate R_{2} and the % in area gain.

1/h_{2}²=1/h_{1}²+16Ft/(3πR_{1}^{4}η)

R_{2}=√h_{1}R_{1}²/h_{2}

Rather than input F and t directly, we take a (hopefully) measured nip pressure in MPa (pressures in the 0.5-1MPa range are normal) and a known printing speed V. F is MPa times Dot_Area (πR²) and t=2R/V.

We can take 3 lessons from this in terms of dot gain. Confusingly there are three possible measures: the gain in radius, the gain in printed area and, not shown here, the value quoted in dot gain curves^{1}.

- Halving pressure, Doubling viscosity, Doubling speed each cause a 1.4 decrease in radius dot gain or 2x reduction in the area (measured) gain. Offset's super-high viscosities are clearly an advantage compared to flexo, though flexo pressures will be lower.
- In the formula, larger dots have a larger F (R²) and t (R) giving an R
^{3}increase, outweighed by an R^{4}decrease, giving a 1/R dependence. The net result is that halving the radius increases gain by 1.4 (or 2 for area). - Halving the ink thickness halves the radius gain or reduces area gain by a factor of 4. So going to a more pigmented ink requiring a smaller thickness gives significant dot gain benefits.

### UV embossing

In UV embossing a feature of radius R is pushed into a UV coating of thickness h_{1} and viscosity η with a force F. The idea is to get down to a thickness of h_{2} as quickly as possible, with h_{2} ideally being close to zero.There are four key points for UV embossing:

- Going to 0 thickness requires infinite time - so UV embossing always leaves a residual thickness and you never get infinite dot gain.
- Halving the desired residual quadruples the time required for UV embossing
- Doubling the radius of the feature gives a 16x increase in time.
- Viscosity and Force only have a linear effect so are not so critical.

So UV embossing is easy with lots of small features and an acceptance of a significant residual thickness. It is hard if broad features with little residual are required.

^{1}If your plate has a 20% dot and it prints as a 30% dot, then the convention is to say that this is a 10% dot gain, even though it is clearly a 50% increase relative to its original area.