A Model for Jet Shortening in Drop-On-Demand Ink-Jet Printing

A new model has been developed for the surface energydriven shortening of a free, cone-shaped fluid ligament of finite length, as a function of ligament diameter, length, mass and head speed. It differs significantly from classical models based on infinitely long cylindrical (Taylor) or conical (Keller) shapes, but leads to overall shortening speeds which are very similar to those provided by Taylor’s model for typical drop-on-demand fluids. However, if a realistic initial velocity distribution along the length of the ligament is included, the model predicts more rapid shortening, by as much as 2 m/s for a jet speed of 6 m/s. Such effects should be taken into account when analyzing the behavior of real jets. The model’s predictions of shortening speeds for free dropon-demand jets fail to account for all experimental observations, which for some polymer solutions can be as much as 2-3 times as high. This effect is attributed to elastic retraction, and may be a general feature linked to the polymer relaxation time. Previous models Higher drop-on-demand (DoD) ink-jet speeds generally involve longer fluid ligaments and later detachment of the jet from the nozzle. Reliable estimates of the time required to form a final drop would be valuable in the design of high speed printing or deposition processes, in setting-up or specifying parameters such as drop velocity, substrate position and printing rate, for almost all types of ink-jet fluid. We focus here on the speed with which the fluid ligament length shortens. Theoretical treatments of free ligament shortening after break-off based on static cylinders or cones have continued to be used, despite the availability of precise measurements of the shapes and speeds of extending ink-jets [1-3]. Reasons for this have included their simplicity and the apparent success of the classical results for the shortening speed, derived from Taylor’s [4] model for the case of the bounding rim formed on a thin fluid sheet formed by impacting jets. Another reason is the attention paid to fluids with low viscosity, where the ligament tends to pinch off at both ends, leaving a thin fluid body that can reasonably be represented by a cylindrical shape during the early recoil time. However, ligament shapes measured for more viscous fluids are approximately conical and comprise fluid with an internal, axial velocity distribution prior to break-off. Keller [5] surmised that, for the breaking of threads (and films) of various geometries, some useful relations could be deduced on the basis of a power law in the radial (relevant) dimension. Power laws assume the same behavior right down to vanishing radius, so that after break-off the broken tip has zero mass and infinite recoil speed, for a conically shaped ligament. The subsequent tail shortening speed always decreases with increasing time in this model and therefore never reaches a constant value. Such results have been recently used without comment [1], although the dynamics of pinching and breaking of viscous threads involve some very complex phenomena [6]. In this work we explore how finite jet length and more realistic assumptions of jet shape and internal velocity distribution can be incorporated into a model for ligament shortening, and compare its predictions with those from the Keller and Taylor models. Shortening speeds are then compared with those measured in jetting experiments with Newtonian and viscoelastic fluids. Jetting fluid is often characterized by its density ρ, surface tension σ and viscosity η, but influences of viscosity or elasticity are not included in any of the models. The shortening of the fluid ligament is driven by surface tension and mass is conserved. Figure 1 shows the geometries assumed in (a) the classical model for a cylindrical ligament (Taylor), and (b) the model for a conical ligament (Keller) [5]. In the Taylor model the diameter D is constant but for cones it increases linearly with x. Figure 1(a): The basis of the classical (Taylor) model for shortening of a cylindrical ligament. The ligament is assumed to be infinitely long and stationary (U=0), with no internal velocity distribution. The end mass m grows linearly with time, and the ligament exhibits a constant shortening speed given by vT = 2(σ/ρD). Figure 1(b): The basis of the (Keller) model for shortening of a conical ligament (where D′ = 2bx). The shortening speed is given by v(t) = (8σ/5ρbt) and thus varies with time. The mean speed over time T is = (3/2) v(T). Proposed model Figure 2 shows the model proposed for a truncated conical ligament with initial length L, including an initial axial speed distribution which varies linearly along the ligament. The total ligament mass M and the tail end mass m experience equal and opposite surface forces at the ligament diameter D′, and the remaining part has mass (M-m), here shown lumped into the head. Both ends are assumed to remain attached to the ligament throughout shortening; the viscosity and elasticity are also ignored. NIP25 and Digital Fabrication 2009 Technical Program and Proceedings 75 Figure 2: The basis of the present model for shortening of a truncated conical ligament with initial length L. The total (conserved) mass is M and there is an initial axial velocity profile due to ligament extension between the break-off position (close to the nozzle) and the tip. Further details are described in [7]. The acceleration for the tail (and head) can be expressed in terms of the physical and the geometrical parameters of the problem, and a full derivation is given elsewhere [7]. The classical Taylor model for an infinite cylindrical ligament (Fig. 1a) predicts a constant jet shortening speed vT given by: