Simulating the different stages of the inkjet printing process

In order to utilize fully the potential of the inkjet printing process, more understanding and better prediction of droplet behaviour, during flight and upon impact, is needed. The behaviour of the substrate due to droplet depositioning (for example local temperature changes and deformation due to absorption) is of interest too, since this influences the precision of the jetting process. Modelling is a good way of gaining insight into the phenomena which drive the different stages of the printing process. The capability to predict the behaviour of the droplet and substrate is of high value.

Jakko Nieuwenkamp of Reden will make a presentation based on this work at the IMI Europe Inkjet Engineering Conference in Lausanne, 14 March 2017.

The different phases of printing

In the printing process four phases can be distinguished: droplet actuation, droplet flight, droplet impact and substrate behaviour after impact (see Figure 1).

Figure 1. The phases in the printing process

Figure 1. The phases in the printing process

In all four phases, different phenomena occur. In the next sections, the different phases will be discussed. An overview will be given of  the methods used to gain insight in the phenomena, resulting in design rules.

The actuation phase

In this phase, a closer look is taken at the droplet formation. In piezo inkjet (PIJ), droplet generation is driven by a pressure wave in the nozzle chamber. The liquid in this case is a water-based ink, with density ρ  = 1000 kg/m3 and dynamic viscosity μ = 10*10-3 Pa*s. The air is taken into account with density ρ  = 1.2 kg/m3 and dynamic viscosity μ = 18*10-6 Pa*s. The droplet actuation is modelled using the commercial finite element package Comsol Multiphysics [1], in which use is made of the implemented level set function and the Navier-Stokes equations.

In Wijshoff [12], the droplet actuation mechanism, using a pressure pulse is studied experimentally by photographing the droplet during the actuation phase. The same conditions as used by Wijshoff for the experimental droplet actuation are used here. The nozzle diameter is 32 µm, from which the length scales can be calculated. As can be seen in the figure, our simulation agreed well with the experimental work of Wijshoff [2].

Figure 2. Droplet actuation. Experimental work of Wijshoff [2] and our simulation results. In the lower left the pressure pulse for actuation is shown.

Figure 2. Droplet actuation. Experimental work of Wijshoff [2] and our simulation results. In the lower left the pressure pulse for actuation is shown.

Using this model, the droplet formation using different pressure pulses can be investigated. The formation of satellite droplets due to the actuation mechanism, as well as the influence of the actuation mechanism on the final droplet speed. Also, the formation of one droplet from several smaller droplets can be researched.

Droplet flight

During the flight of the droplet between nozzle and substrate, the droplet undergoes temperature loss and drag (deflection and/or deceleration). For this phase, analytical equations with regard to the impact velocity, time to impact and temperature loss have been derived. Note that the input for this phase, such as droplet speed, is the output of the previous phase. In this phase, the droplet is assumed to be spherical.

These equations have been implemented as design rules into the software tool MrReves [3]. Using this software tool, the equations can be solved (fully coupled) for any set of parameters which the user provides. Also, a domain can be specified, and the effect of parameters in this domain can be shown by MrReves.

Figure 3. Solutions found by MrReves for different droplet parameters. Left upper corner: impact temperature versus impact temperature, upper middle: volume versus impact temperature, upper right: density versus impact temperature, bottom left: impact temperature versus flight time, bottom middle: volume versus flight time, bottom right: volume versus density. The red solutions are the solutions with impact temperatures above 94 ° C.

Figure 3. Solutions found by MrReves for different droplet parameters. Left upper corner: impact temperature versus impact temperature, upper middle: volume versus impact temperature, upper right: density versus impact temperature, bottom left: impact temperature versus flight time, bottom middle: volume versus flight time, bottom right: volume versus density. The red solutions are the solutions with impact temperatures above 94 ° C.

As an example, in Figure 3 different parameters are plotted against each other for the following jetting parameters:  a printhead velocity (parallel to the substrate) of 200 mm/s, at a distance of 2 mm above the substrate, the printhead (initial drop) temperature is 100 ° C. The drops will leave the nozzle with a speed of 4 m/s, the surrounding air has a temperature of 20 ° C (specific heat capacity of the ink 4180 J/kg/K, thermal conductivity 0.6 W/(mK)). The volume and density are varied between 2 and 30 picolitre and 800 and 1600 kg/m3  respectively.

MrReves generated solutions until 2000 valid solutions were found for the given parameters. The results of MrReves can be visualised in a two dimensional plane, in which the user can choose the different variables. In Figure 3, these different parameters are plotted. In the solutions found by MrReves, subsolutions with an impact temperature (Tprint) of at least 94 degrees Celsius are selected. These solutions are highlighted in red. In the first plot (left top) the Tprint as a function of the droplet volume is plotted. One can see that not all volumes can acquire the necessary Tprint. There is a first regime which shows that none of the droplets will make it with a 94 °C Tprint. A second regime shows that some of the droplets will fulfil the impact temperature, some of them won’t. When the volume is big enough, all the droplets with such a volume, no matter which other parameters from the chosen domain, will reach the substrate with Tprint ≥ 94 °C. In the top right plot, the impact temperature as a function of the density is shown. The influence of the density is clearly less in the chosen domain than that of the volume. For all densities, there seems to be a solution such that a droplet will impact with Tprint ≥  94 °C onto the surface. At the left bottom side, the impact time as a function of the flight time is plotted. It is clear that the flight time should be as low as possible to obtain as high an impact temperature as possible. As all solutions which fulfil the Tprint ≥  94 °C are only at the side where the lowest flighttimes are, this shows that the flight time is a dominant factor in the given domain. In the last figure, bottom right, the density versus the volume of the droplet is plotted. As the flight time dominates the impact temperature, one can also couple the selected group as flight time. The flight time seems a clear function of the volume and density. However, not only the mass (rho*V) is describing this phase. Since, those with a lower density, must still have a higher mass to keep the same flight time. This is due to the increase in volume and the drag coefficient which changes due to the frontal area of the droplet .

The use of MrReves on the solutions generated by analytical equations (or design rules) makes it easy to find new relations, for different regimes, between printing parameters. Design rules and/or analytical equations are coupled and solved by MrReves.  

Droplet impact

Figure 4. Impact of a water droplet with a diameter of 46 micrometer on substrate with a contact angle of 50 ° and one with a contact angle of 90 °. Left the experimental data of Son et al. [5] and at the right the Comsol Multiphysics simulations.

Figure 4. Impact of a water droplet with a diameter of 46 micrometer on substrate with a contact angle of 50 ° and one with a contact angle of 90 °. Left the experimental data of Son et al. [5] and at the right the Comsol Multiphysics simulations.

The impact phase is the time between first impact and equilibrium (or, for bouncing droplets, between first impact and detachment from the surface). The impact phase depends on the Weber and Reynolds numbers which determine whether or not the droplets splash (see for example the work of Bussmann [4]). The contact angle between liquid and substrate can play an important role in the droplet dynamics and final state of the droplet.

The FEM model used to investigate jetting is also used to look at impact. This time, a water-based ink droplet with a size of 46 µm is investigated. The Reynolds and Weber are chosen such that no splashing should occur. The material properties are the same as used in the jetting phase and the droplet velocity is approximately 1.44 m/s. The exact properties can be found in Son et al. [5], who took images of the droplet impacting on a smooth substrate. Their work will be used as a validation of the FEM model.

The contact angle is varied, and the droplet spreading due to this variation is investigated. In Figure 4, the impact behavior of a water-based droplet on a smooth substrate with contact angles of 50° and 90° is shown.  The simulations show good agreement with the experimental work of Son et al. [5].

Influence toward the substrate

Figure 5. Lines printed, during paper transport along a heated area (see Figure 6), in deformed stage (blue) and the same line after deformation is gone (red). This shows the error in placement of droplets due to the in-plane deformation of the paper from a temperature change.

Figure 5. Lines printed, during paper transport along a heated area (see Figure 6), in deformed stage (blue) and the same line after deformation is gone (red). This shows the error in placement of droplets due to the in-plane deformation of the paper from a temperature change.

Not only the actuation, flight and impact determine the droplet placement and final geometry, but also the substrate position. Substrate deformation during the printing process will cause misplacement of the droplets on the substrate.

In this example, the deformation and wrinkling of paper due to absorption of water-based ink and due to a temperature gradient is examined. These simulations are build within the finite element package Abaqus [6]. First, the influence of in-plane deformation due to temperature gradients is examined. The paper substrate is moved underneath a printhead, which will heat a smaller area of the substrate. Due to the temperature increase the substrate will deform (see Figure 6). When printing lines, these lines are straight in the deformed situation, but will change when the paper is out of the heated area, which results in curved lines (Figure 5). The second is the out-of-plane wrinkling due to ink load on the paper. This causes local expansion of the paper, resulting in wrinkles (Figure 7). These deformations result in incorrect positioning of the droplets on the substrate.