Modelling piezo-driven inkjet printheads is a rewarding task. A lot of insight can already be obtained by considering one single pump out of a manifold of many pumps in a multi-nozzle printhead. By starting at first principles and reducing the system to one degree of freedom, namely a mass-spring-damper ordination most of the characteristic behavior of a single inkjet pump can be retrieved. First principles are the laws of Newton, Hagen-Poiseuille, Newton-Laplace and Young-Laplace all derived in the 18th and 19th centuries. A mass-spring system was probably analyzed for the first time by le Rond d’Alembert (1717-1783) in order to try to understand the compositions of Rameau (1683-1764). Consulting the works of Rayleigh (1842-1919), all the ingredients can be found to learn about acoustics e.g. of Helmholtz type of resonators, break-up of fluid jets and vibrations of droplets.
The book of Helmholtz (1821-1894) on the Sensations of Tone (Die Lehre von den Tonempfindungen, 1885) can be considered as the scientific foundation of the understanding of musical instruments, the physiology of hearing and harmony. An inkjet pump can be considered as a small sized musical instrument with its own timbre given by its key tone and overtones, the material it is built of and the properties of the ink that must be driven in a well-tuned way to sound well, to generate droplets in a stable manner. Standard engineering mathematics suffices to solve the governing mass-spring-damper equation and to discuss the response of such a system in the time and frequency domain and to investigate the effects of high frequency pulsing. Adding complexity in terms of more degrees of freedom, by taking into account e.g. a connecting duct to the ink supply, surface tension, the compliance of the structure of the pump, different nozzle shapes and relying on more advanced mathematics gives an increased view on the behavior of a print head in the time and frequency domain. As long as the wave speed effects can be left out, the method of adding more degrees of freedom can be used to investigate the complex interaction of many pumps integrated in a multi-nozzle printhead and communicating with each other through the ink supply duct.
Many piezo inkjet pumps are integrated in a multi-nozzle printhead and designed at the smallest pitch possible to arrive at a high native “dots per inch” number. To end up with a sufficient large volume displacement of the actuator the pumps have to be long and wave effects cannot be ruled out. Upon actuation waves start to travel back and forth through the long channel, their development in course of time depending on the reflection properties at both ends; one end being the nozzle and the other end being the connection to the ink supply. Damping is caused by viscous drag everywhere along the inside of the pump. To model this effect the wave equation must be solved, increasing the complexity of the mathematics needed. The wave equation type of description allows for analyzing complex designs as far as the geometrical layout of the structure of the pump is concerned. Wave speed and delay effects enter the picture and the analysis of the interaction between pumps integrated in a multi-nozzle print head can only be done by considering spatial symmetry in driving such a printhead.
Droplet formation is a highly dynamic and complex phenomenon. Before a droplet is released it stays connected to the fluid inside the nozzle for a short while by a stretching fluid filament. To describe the effect of the fluid filament on droplet speed and size the investigation of creating moving free surfaces, variable mass effects, elongational viscosity and details of the flow between tail and droplet are needed. After a short flight the droplet lands on the substrate, spreads, and dries either by evaporation or permeation into the substrate. In case the ink is viscoelastic the droplet formation is strongly influenced the increased elongational viscosity.
Actuation by pulse-wise charging the piezoelectric actuator is key for driving the printhead in a stable manner. Handling different pulse shapes and analyzing their effects on the flow dynamics inside the pump and on the droplet formation will be an essential part of the course.
All models that will be presented are basically analytical models; most of the solutions can easily be programmed in Excel to investigate the behavior of a specific printhead filled with a specific ink. For some of the non-linear models numerical integration is needed and/or matrix methods to solve large systems of linear algebraic equations.
by Prof Dr J. Frits Dijksman, University of Twente, Faculty of Science and Technology
You can learn more about how printheads work at the Fluid Dynamics & Acoustics course presented by Prof Dr J. Frits Dijksman, part of the IMI Europe Inkjet Summer School, 10-14 June 2019 in Cambridge, UK.