Regimes of Polymer Behaviour in Drop-on-Demand Ink-Jetting

A recent model for the drop-on-demand (DoD) jetting of mono-disperse linear polymer solutions has identified three regimes of behaviour: with the polymer chains fully relaxed; with significant viscoelastic effects; and with the chains fully extended. These regimes have now been confirmed by experimental observations in several different systems. The model predicts the maximum jettable concentration of polymer as a function of molecular weight. The highly extensional flows in high-speed jetting with viscous solvents can fully stretch linear polymer chains outside the nozzle, permitting jetting of solutions with higher polymer contents than for purely elastic behaviour. These results are significant for DoD printing at high jet speeds and will be relevant to jet formation from linear polymer solutions for any DoD printhead. Introduction Drop-on-demand (DoD) ink-jet printing of dilute polymer solutions, with a fixed drive waveform to achieve a main drop tip speed of ~ 6 m/s at ~ 1 mm standoff had been studied by highresolution spark flash imaging [1]. Dilute solutions of polystyrene (PS) with near mono-disperse molecular weights (Mw) from 24,000 to 488,000 g/mol were prepared in a good but viscous solvent, diethyl phthalate (DEP). The dependence on polymer molecular length (L) of the maximum PS concentration for DoD jetting was determined. Zimm-Rouse theory for polymers in dilute solution provides a quantitative relation between L and Mw as described elsewhere [2]. These polymer solutions with low Mw and dilutions were expected to show jetting behaviour either controlled by linear viscosity tending towards the Newtonian solvent viscosity at the lowest polymer concentrations, or “elastic” behaviour, with the maximum jettable concentration falling rapidly (at least as an inverse square law) with Mw. However results from both our experiments [3, 4] and our numerical simulations [5] appeared to show the variation of this limiting concentration to be close to 1/Mw, which was unexplained by polymer theory. Earlier results, from de Gans et al [6], for slower (~ 2 m/s) jets of mono-disperse polystyrene solutions in another “good” solvent from a different print head type appeared to show “elastic” behaviour at high Mw. We have sought to understand these apparent discrepancies. Numerical simulations & models Numerical code developed by Morrison and Harlen has been validated separately for DoD and CIJ jetting conditions, using a large-scale drop generator system [7] as well as by various results on the scale of industrial inkjet printing [8]. This axi-symmetric code for fluid flow incorporates polymer Zimm models [5] and at high shear rate predicted a relationship between the required print head drive and Mw which was unexpected but which was consistent with the jetting data for PS in DEP obtained in earlier work. Recently [9], another possible regime for polymer behaviour was considered: if polymer chains become fully stretched, i.e. beyond the viscoelastic region, how does this affect the fluid jetting? Once the polymer chains are fully stretched they behave like rigid rods of length L in solution, with very high viscosity >100 times the viscosity at low extension rate. (The viscosity ratio between fully stretched chains and the Newtonian viscosity in this model [9] is given by 2⁄3L2, and for a linear polymer with Mw > 100,000 g/mol the value of L > ~15.) This higher viscosity suggests a power-law dependence on polymer concentration that still does not conform directly to all ink-jet measurements, since it neglects key aspects of high speed ink-jets. A previous approach to understanding polymer solution inkjets [10] was based on a simple model of an extending ligament combined with the polymer theory. This approach gave very good descriptions of viscoelastic behaviour and predicted the conditions under which fluid is initially ejected from the nozzle but never separates, and retracts back inside (“bungee jumpers”). However, this approach was applied only to relatively low extension rates (jet speeds) compared with those reported here. Figure 1 shows the model for the rapidly extending ink-jet. Figure 1: Basic model of extending ink-jet [9], following earlier approach [10]. Using the previous model approach [10] and the multi-mode Zimm model, it was found [9] that at high extension rates in fast jetting conditions a combination of fully stretched polymer chains and thinning of the ligament during fluid extension reduced the predicted overall power-law dependence on concentration close to that observed for ink-jets of PS in the (viscous) solvent diethyl phthalate (DEP), also providing some explanations for the scaling law that was found in the earlier simulation results [5]. Model results Three regimes of ink-jet jetting behaviour are predicted [9]: two are well-known but the third had not previously been clearly identified in the literature or experimental data. Transitions between these regimes depend on the initial Weissenberg number (Wi) given by the product of initial extension rate (ė) and polymer relaxation time (τ), where we approximate the extension rate from the initial jet speed (U0) and length scale (d) for the jet neck (see Figure 1):