Meniscus Oscillations Driven by Flow Focusing Lead to Bubble Pinch-Off and Entrainment in a Piezoacoustic Inkjet Nozzle

The stability of high-end piezoacoustic drop-on-demand (DOD) inkjet printing is sometimes compromised by the entrainment of an air bubble inside the ink channel. Here, bubble pinch-off from an oscillating meniscus is studied in an optically transparent DOD printhead as a function of the driving waveform. We show that bubble pinch-off follows from low-amplitude high-frequency meniscus oscillations on top of the global high-amplitude low-frequency meniscus motion that drives droplet formation. In a certain window of control parameters, phase inversion between the low- and high-frequency components leads to the enclosure of an air cavity and bubble pinch-off. Although phenomenologically similar, bubble pinch-off is not a result of capillary-wave interaction such as observed in drop impact on a liquid pool. Instead, we reveal geometrical-flow focusing as the mechanism through which, at first, an outward jet is formed on the retracted concave meniscus. The subsequent high-frequency velocity oscillation acts on the now toroidal-shaped meniscus and it accelerates the toroidal ring outward, resulting in the formation of an air cavity that can pinch off. Through incompressible boundary-integral simulations, we reveal that bubble pinch-off requires an unbalance between the capillary and inertial time scales and that it does not require acoustics. The critical control parameters for pinch-off are the pulse timing and amplitude. To cure the bubble entrainment problem, the threshold for bubble pinch-off can be increased by suppressing the high-frequency driving through appropriate waveform design. The present work therefore aids the improvement of the stability of inkjet printers through a physical understanding of meniscus instabilities.

Figure 1

(a) Bubble pinch-off and entrainment in a 70-$\mu \mathrm{m}$-diameter nozzle of a piezo drop-on-demand inkjet printhead. The piezo actuation pulse is a rectangular push-pull pulse with a 150-V amplitude and a 30-$\mu \mathrm{s}$ width. The images are recorded using 8-ns single-flash stroboscopic imaging with illumination by laser-induced fluorescence (iLIF) [31]. (b) Details of the bubble pinch-off process: the center of the meniscus moves inward while the outer region of the meniscus moves outward, leading to the formation of an air cavity that eventually pinches off as an entrained air bubble. The scale bars represent $50\mu \mathrm{m}$.

Figure 2

(a) A rectangular piezo actuation pulse, with its amplitude $A$ and length $\mathrm{\Delta }t$ as the control parameters. (b),(c) The window of bubble pinch-off for (b) a pull-push pulse with a pulse width of $30\mu \mathrm{s}$ and a varying amplitude and (c) a pull-push pulse with an amplitude of 94 V and a varying pulse width. The scale bars represent $50\mu \mathrm{m}$.

Figure 3

(a) The experimental setup employed to image meniscus motion and bubble pinch-off in the drop-on-demand piezo-acoustic inkjet nozzle, using illumination by iLIF [31]. (b) A schematic layout of the functional acoustic part of the inkjet printhead. It consists of a glass capillary tube (blue) that is tapered toward the 70-$\mu \mathrm{m}$-nozzle exit. A cylindrical piezo (gray) can be actuated to drive the ink-channel acoustics and the resulting droplet formation.

Figure 4

An example of an image-analysis result, showing the meniscus positions $y_o$ of the outer region $x_o$ and $y_i$ of the inner region $x_i$, the bubble position $y_b$, and the bubble diameter $d$. The scale bar represents $50\mu \mathrm{m}$.

Figure 5

(a) The meniscus outer, $y_o$, and inner, $y_i$, positions and the bubble position, $y_b$, as functions of time after the start of piezo actuation. The black dashed line and the black dash-dotted line show the low-frequency (low-pass-filtered red curve) motion of the meniscus outer region and that of the bubble (low-pass-filtered green curve), respectively. (b) The high-frequency [high-pass-filtered red curve in (a)] movement of the meniscus outer-region position, $\mathrm{\Delta }{y}_o$, and that of the bubble position, $\mathrm{\Delta }{y}_b$. The dark green curve shows the bubble diameter $d$ as a function of time.

Figure 6

(a) A piezo actuation pulse with amplitude $A$, FWHM pulse width $\mathrm{\Delta }t$, and rise-and-fall time $\mathrm{\Delta }e$. (b),(c) Piezo (b) ring-down measurements with (c) the corresponding Fourier spectra, for a pull-push pulse with $A=10$ V, $\mathrm{\Delta }t=72\mu \mathrm{s}$, and values of $\mathrm{\Delta }e$ as indicated in the legend. (d) The meniscus motion for three pulses with $\mathrm{\Delta }t=72\mu \mathrm{s}$ and with $\mathrm{\Delta }e$ and $A$ as given in the legend. Bubble entrainment takes place in the gray-shaded time window. (e)–(g) Images of the nozzle taken during the time window indicated by the gray-shaded time window in (d), showing whether or not bubble pinch-off takes place. The numbers under the images correspond to the time in $\mu \mathrm{s}$ in (d). The scale bar represents $50\mu \mathrm{m}$.

Figure 7

Bubble pinch-off for (a) a rectangular push-pull pulse with an amplitude of 160 V and a width of $30\mu \mathrm{s}$ and (b) a rectangular pull-push pulse with an amplitude of 150 V and a width of $30\mu \mathrm{s}$. (c),(d) Details of the meniscus shape deformation process prior to, during, and after bubble pinch-off for the push-pull and the pull-push pulse, respectively. (e),(f) The meniscus outer-region position $y_o$, inner-region position $y_i$, and bubble position $y_b$ as a function of time for the push-pull and pull-push pulse, respectively. The thin blue dashed line is added to guide the eye in the parts of the jet-formation and jet-recoil process where the position of the inner region of the meniscus cannot be tracked. The scale bars represent $50\mu \mathrm{m}$.

Figure 8

(a) The numerical setup for the BI simulation, where the left boundary is subject to a stream-velocity boundary condition. The initial meniscus shape is a parabola with a depth of 0.75 times the nozzle diameter. (b) The stream-velocity boundary condition $v$ for the numerical setup as a function of time with a 100-kHz velocity oscillation followed by an outward-directed flow. (c) The meniscus shape deformation process prior to bubble pinch-off, simulated using the BI method. (d) A schematic summary of the main steps in (c).

Figure 9

(a) Toroidal bubble pinch-off as experimentally recorded for a rectangular pull-push pulse with an amplitude of 92 V and a width of $53\mu \mathrm{s}$. The scale bar represents $50\mu \mathrm{m}$. (b) A simulation using the BI method of the meniscus shape deformation process eventually leading to toroidal bubble pinch-off.