Regimes of steady jetting in electrohydrodynamic jet printing

We present an experimental and theoretical investigation of steady jetting regimes in the electrohydrodynamic (EHD) jet printing process, wherein a liquid jet issuing from a needle is stretched by applying a high electric field between the needle and a collector substrate. While EHD jetting is well understood in the context of the electrospray process, we show that the jetting characteristics can differ significantly in the printing configuration in which the needle is kept close to the collector to prevent spraying. Through experimental visualization of EHD jets and current measurements, we show that steady EHD jetting can occur in three regimes, namely, the cone-jet, moderately stretched jet, and thick-jet regimes, depending on the flow rate, applied potential difference, and needle-to-collector gap. To elucidate the underlying physics of these experimentally observed regimes, we perform order-of-magnitude analysis for the current transfer region of the jet, where the conduction and surface convection currents are of comparable magnitude. Based on the relative size of the current transfer region and the jet diameter, we derive the criteria for observing the three steady EHD jetting regimes. For each of these regimes, we also describe the dependence of the jet diameter and the current carried by jet on flow rate, potential difference, and aspect ratio in the form of scaling relations. The theoretical criteria for various steady jetting regimes and the corresponding scaling relations for the jet diameter and current show good agreement with the experimental observations. These theoretical relations can be useful for the design, performance monitoring, and control of EHD jet printing.

Figure 1

Schematic illustration of an experimental setup in the EHD jet printing. A liquid jet issuing from the needle is stretched by applying a large potential difference between the needle and the collector. The current carried by the EHD jet is measured through a current measuring probe connected between the collector and the ground. Simultaneously, the jet is visualized through a horizontally mounted microscope.

Figure 2

Jetting behavior for 1-octanol at various dimensionless flow rates $\tilde{Q}$, at a fixed dimensionless potential difference $\tilde{\varphi }=22.2$ and aspect ratio $\chi =6$. Here $\tilde{Q}=Q/Q_0$ and characteristic flow rate $Q_0=\gamma {\epsilon }_0/\rho k$ = 14.4 $\mu \mathrm{l}{\min }^{-1}$.

Figure 3

Variation of current and jet diameter of an EHD jet of 1-octanol for varying values of flow rate and aspect ratio. (a) Log-log plot of the dimensionless current $\tilde{I}$ for varying dimensionless flow rate $\tilde{Q}$. At $\tilde{Q}\lesssim 40$, the current scales as $\tilde{I}\sim {\tilde{Q}}^{1/2}$, whereas at $\tilde{Q}\gtrsim 500,$ the current weakly depends on flow rate. Here (i) to (vii) correspond to the conditions for which snapshots of the jet are shown in Fig. 2. Inset of (a) shows $\tilde{I}$ vs $\tilde{Q}$ for low values of $\tilde{\varphi }$, wherein current is nearly independent of flow rate for $\tilde{Q}\gtrsim 500$. Here hollow and filled symbols correspond to $\chi =6$ and $\chi =3$, respectively. (b) Log-log plot of the dimensionless jet diameter $\tilde{d}$ for varying $\tilde{Q}$ at different values of $\tilde{\varphi }$ and $\chi$. The jet diameter increases monotonically with flow rate, and at high $\tilde{Q}$ it tends to a constant value. Moreover, at flow rates, $\tilde{Q}\lesssim 40$, the jet diameter increases with the square root of the $\tilde{Q}$, whereas at $\tilde{Q}\gtrsim 40$, the jet diameter linearly increases with $\tilde{Q}$.

Figure 4

EHD jetting behavior of 1-octanol for varying values of potential difference $\tilde{\varphi }$ at three flow rates: (a) $\tilde{Q}=14$, (b) $\tilde{Q}=208$, and (c) $\tilde{Q}=694$. The cone-jet regime is observed at low flow rate $\tilde{Q}=14$. The jet is moderately stretched at intermediate flow rate ($\tilde{Q}=208$) and at high flow rate with high $\tilde{\varphi }$. At high flow rate and low potential difference, the jet diameter is almost equal to the diameter of the needle.

Figure 5

Current characteristics of an EHD jet of 1-octanol for varying $\tilde{\varphi }$, at the different values $\tilde{Q}$ for a fixed aspect ratio $\chi =4$. In the cone-jet regime ($\tilde{Q}\lesssim 40$), the measured current $\tilde{I}$ increases linearly with $\tilde{\varphi }$. In the thick-jet regime at high $\tilde{Q}$ and low $\tilde{\varphi }$, the current increases linearly with $\tilde{\varphi }$. At other conditions where the jet is moderately stretched, upon increasing the $\tilde{\varphi }$, the current decreases, and thereafter it begins to increase with $\tilde{\varphi }$.

Figure 6

Various regimes of an electrified jet of 1-octanol in the EHD jet printing configuration. An electrified jet shows three distinct regimes: (i) cone-jet regime, where the current transfer region is very small in comparison to the diameter of the needle, $L_{ct}/d_n\lesssim 1$, (ii) thick-jet regime, in which the diameter of the jet is approximately equal to the diameter of the needle, $d\approx d_n$, and (iii) moderately stretched jet regime, where the length of the current transfer region is the order of the needle diameter and needle to collector distance, $L_{ct}/d_n\gtrsim 1$, while the jet diameter is smaller than the needle diameter.

Figure 7

The dependence of the measured current on the aspect ratio $\chi$ in the thick jet regime for 1-octanol at $\tilde{Q}=833$ at low $\tilde{\varphi }$. (a) The dimensionless current $\tilde{I}$ scales linearly with $\tilde{\varphi }/ln4\chi$ in this regime at various $\chi$. In the inset, (b) $\tilde{I}$ is plotted against ${\tilde{I}}_{bm}$ collapses in a single curve, which validates Eq. (14). Here ${\tilde{I}}_{bm}$ is the total current in the thick jet regime predicted by Eq. (14).

Figure 8

The dependence of measured current on the aspect ratio $\chi$ in the moderately stretched jet regime for 1-octanol at $\tilde{Q}=208$. (a) The dimensionless current $\tilde{I}$ shows a nonmonotonic dependence on $\tilde{\varphi }/ln4\chi$, which is in qualitative agreement with Eq. (16). (b) Replotting the data on a $\tilde{I}\tilde{\varphi }/ln4\chi$ vs ${(\tilde{\varphi }/ln4\chi )}^4$ plot shows that the intercept, corresponding to the conduction current, decreases upon increasing $\chi$, as predicted by Eq. (17).

Figure 9

Current characteristics of an EHD jet of 1-butanol for varying operating parameters ($\tilde{Q}$ and $\tilde{\varphi }$). (a) Dimensionless current $\tilde{I}$ vs dimensionless applied potential difference $\tilde{\varphi }$ for various dimensionless flow rates $\tilde{Q}$ at $\chi =4$. The inset of (a) shows $\tilde{I}$ vs $\tilde{Q}$ for fixed $\tilde{\varphi }=32$ and $\chi =4$. (b) Various regimes of EHD jetting of 1-butanol in the printing configuration. EHD jetting of 1-butanol shows similar behavior as that for 1-octanol.