Abstract

Electrohydrodynamic (EHD) printing is an alternative method to fabricate high-resolution micro- and nanostructures with high efficiency, low cost, and low pollution. Numerical simulation is an effective approach to systematically investigate the formation process of EHD jet. However, there are a few articles performing this work. In this study, a finite element model was established. The jet formation process and jetting modes were analyzed. The influence of applied voltage and printing distance on the maximum electric field near the nozzle tip was investigated. The effect of flow rate on the jet diameters was studied. Comparison between numerical and experimental results demonstrated that the proposed simulation model had a high potential for EHD jet analysis. According to the optimized printing conditions (printing distance of 200–300 μm, applied voltage of ∼1100 V, and flow rate of 0.1–0.3 ml/h), stable EHD jet can generate and polyvinyl pyrrolidone (PVP) lines with minimum line-width of 0.9 μm can be printed onto the glass slide.

1 Introduction

Direct-writing techniques have exhibited advantages over traditional micro-electro-mechanical systems fabrication approaches. They are low cost, compatible to traditional fabrication methods, and printable for all kinds of substrates. Ink jet printing, a conventional and well-developed direct-writing method, is usually based on thermal or mechanical deformation resulting in the ejection of liquid droplets from the nozzles. Its resolution depends on the inner diameter of the nozzle. However, it is reported that the highest resolution could be 20 μm [1]. Although with assistant methods, such as wetting-and-dewetting to confine printed droplets [2], it remains a challenge to further improve the printing resolution up to submicroscale.

Another direct-writing approach, called electrohydrodynamic (EHD) jet printing, was proposed in recent years. The electrohydrodynamic phenomenon and its mechanism were first introduced by John's team in the year of 1917 [3]. The well development of EHD jet printing was made by Rogers' group [46]. Until now, complex two-dimensional and three-dimensional structures could be printed with the highest resolution up to nanoscale [79]. During EHD jet printing, a high voltage is applied on the conductive nozzle. Under such high electric field, the charges in the ink move to the nozzle tip and then aggregate quickly, forming a meniscus at the nozzle tip. When the electric field force is larger than the surface tension of the meniscus, an EHD jet is dragged out from the meniscus.

Lots of works have been carried out to analyze the EHD printing process. Choi investigated the influence of inner diameter of the nozzle on the size of ejected droplets and proposed an estimation model for printed line-width [10]. Lee studied the effect of flow rate and applied voltage on the jetting modes, and a chart was presented to describe the condition for each jetting mode [11]. Kira developed a feedback control system for achieving better printing stability. The relationship between the quantity of electricity and the size of the charged droplet was analyzed, and a prediction model was proposed to estimate the printed line-width [12]. Zou developed a tip-assisted EHD jet printing method and analyzed its electric field enhancement effect [7]. The influence of working parameters (applied voltage, nozzle inner diameter, and printing distance) on electric field intensity was analyzed, and it demonstrated that the printing resolution can increase by nearly five times [7].

With the development of numerical methods, researchers also used commercial software to emulate and analyze the formation of EHD jets. Lastow used a commercial computational fluid dynamics code to simulate the EHD atomization process [13]. The jet formation process as a function of applied voltage was analyzed, and the velocity field inside the liquid cone was also studied. It was demonstrated that their simulation results were consistent with experimental results in terms of droplet size prediction. Kim investigated the influence of a nonconductive tip inserted into a nozzle on jet generation and printed line-width during EHD jet printing [14]. Simulation results indicated that an inserted nonconductive tip conduced to the decrease of threshold voltage and printing resolution because of the backflow elimination near the apex of Taylor cone. Pan proposes a numerical simulation model for whole process of droplet generation, containing jet generation, jet break, and jet retraction stages [15]. The charge density, velocity, and volume fraction distribution of the jet in three above stages was investigated by numerical simulation. Lee investigated the interference and distortion of the electric field affected by the multinozzles [16]. To control each nozzle independently, the printing conditions were optimized. Their results showed a strong correlation between the simulated and experimental results. Zou developed a tip-assisted EHD printing method, and numerical simulation was carried out to investigate the electric field enhancement effect [7]. By using the optimized condition optimized by both simulation and experiments, microdroplet arrays and microwires down to sizes of 2.3 μm could be printed. Pannier established a simulation model coupling a spherical cap volume conservation law to a molecular kinetic relationship [17]. This model has been validated by experiments, and it can be used to predict the drop shape dynamics and dynamic base radius. Singh presented a simulation model to predict the transient ejection of micro-/nanoscopic jets from the nozzles [18]. This model was based on the Taylor–Melcher leaky dielectric model and phase field method, and it can observe the deformation of a flat liquid meniscus into a Taylor cone. Rahmat presented a numerical model based on traditional volume of fluid method [19]. By introducing three dimensionless numbers (Reynolds, electro-Weber, and Weber numbers), the influence of competing forces on the printing process was analyzed. The results indicated that an increase in Reynolds and electro-Weber numbers could result in the generation of a smaller jet flow. The above simulation works help us to understand the forming of EHD jets and printing condition dependence effect. However, to the best of our knowledge, there is few work on systematic numerical simulation of EHD jet printing. In addition, a quantitative study has not been performed to investigate the relationship between printing conditions and diameter of the jets. This work is crucial to estimate the printed line-width and optimize the printing parameters. Thus, a systematic study on EHD jet printing should be carried out by numerical simulation.

In this work, the electric field distribution near the nozzle was analyzed and the leaning jet phenomenon was thus explained. The influence of printing conditions on the diameter of formed jets was investigated and the jetting modes were also analyzed.

2 Experiments

In this work, a commercial stainless-steel nozzle was used for EHD jet printing. The inner diameter of the nozzle is 160 μm. Figure 1 shows a schematic diagram of the EHD jet printing system, including syringe pump, voltage supply, camera, computer, and XY stage. Syringe pump is used for continuous ink supply to the nozzle. Voltage supply could generate high electric field between the nozzle and the substrate. The camera and the computer are used for jet observation, recording, and measurement. XY stage, a motion platform, is used for trajectory generation according to the commands sent by the computer.

For numerical simulation and experiments, polyvinyl pyrrolidone (PVP) ink with a concentration of 2% (mass ratio) was used. The ink was home-made by dissolving PVP powder into the de-ionized water. The ink has the following properties: density = 1098.25 kg/m3, viscosity = 0.33 Pa·s, surface tension = 0.036 N/m, dielectric constant = 25.7, and electrical conductivity = 2.1 × 10−5 S/m. Glass slides with thickness of 1 mm was used as the substrate. The printing conditions were changed during EHD jet printing to analyze the influence of printing conditions on the diameter of formed jets. The changed printing conditions contains printing distance and ink flow rate.

3 Results and Discussion

3.1 Governing Equations for Numerical Simulation.

The formation of EHD jet printing is the balance between hydrodynamic and electrostatic forces. The diameter of the jets mainly depends on electric field intensity, ink flow rate, and ink properties (such as density, viscosity, conductivity, and permittivity). Considering the force difference caused by surface tension (Fs), electrical force (Fe), and gravitational force, Navier–Stokes equation can be expressed as [20]
ut=Pρ+μ2uρ+g+Fsρ+Feρ
(1)
·u=0
(2)

where u is the velocity of the ink, ρ is the density of the ink, P is the pressure provided by the syringe pump, and μ is the viscosity of the ink.

The electrical force could be expressed as follows [20]:
Fe=12E·Eε+σE+Ps
(3)

where E is the electric field between the nozzle and the substrate, ε is the permittivity of the ink, σ is the charge density on the meniscus, and Ps is the electrostrictive force.

Fs could be calculated as follows according to the Laplace formula [21]:
Fs=γ(1R1+1R2)
(4)

where γ is the surface tension, and R1 and R2 are the curvature radius of the ink meniscus.

The lowest electric field to generate an EHD jet is called critical electric field (Ec), while the corresponding voltage is called critical voltage (Vc). It is demonstrated that EHD jet can be formed when the electrostatic force is equal to the surface tension of the ink; therefore, Ec can be estimated as follows [22]:
Ec=2Vcdln(4hd)
(5)
where d is the inner diameter of the nozzle and h is the printing distance (the distance between nozzle tip to the substrate). The critical voltage can be calculated by the following equation:
Vc=dγcosθ2ε0ln(4hd)
(6)

where θ is the half angle of the meniscus (Taylor cone) and ε0 is the vacuum permittivity.

By combining Eqs. (5) and (6), the following equation can be obtained:
Ec=2γcosθdε0
(7)

This equation can be used to estimate the required lowest electric field (critical electric field). Under this condition, an EHD jet can be generated from the nozzle.

3.2 The Formation Process of Electrohydrodynamic Jet.

In this work, a two-dimensional model was used to analyze the EHD printing process. The governing equations for numerical simulation have been explained in Sec. 3.1. Figure 2(a) shows the model used for simulations. The stainless steel nozzle with inner diameter of 160 μm is connected to the high voltage. The substrate is connected to the ground voltage. The red region in Fig. 2(a) presents PVP ink. The blue dot line on the nozzle is set as the fluid inlet with a given flow rate, while the dot line on the substrate is set as the fluid outlet with pressure of 0 MPa. Since the nozzle inner diameter is in microscale, nonslip boundary condition is applied to the inner wall of the nozzle.

The images in Figs. 2(b)2(d) show the jet formation process. PVP ink is made by PVP powder and purified water. In the mixed solution, there are lots of microscale or nanoscale particles. When high voltage is applied, the particles can be charged. The positive charged ones move to the nozzle tip and gathers together. The ink at nozzle tip is pulled to the substrate under electric field force, forming a circular meniscus (Fig. 2(b)). With the increase of charged particles, the ink is further stretched and the meniscus is sharpened (Fig. 2(c)). When the electric charge improves to a threshold value, the electric field force becomes high enough. At this point, electric field force overcomes the surface tension of the ink. The EHD jet generates from the nozzle tip, as shown in Fig. 2(d).

3.3 The Electric Field Distribution Near the nozzle tip.

The electric field intensity between the nozzle and the substrate is an important factor for EHD jet formation. The changes of both printing distance and applied voltage can affect the electric field distribution. For EHD jet printing, a voltage of 1100 V is applied on the conductive nozzle first, and then a pressure is applied on the ink by the syringe pump. Under pump pressure and electroosmotic force, the ink flows into the nozzle. Then the printing distance is decreased from 1 cm. When the electric field intensity is high enough, the EHD jet could generate. The formation of the jet depends on the electric field intensity, so we investigated the electric field distribution near the nozzle tip by numerical simulation. Figures 3(a) and 3(b) show the electric field distribution near the nozzle tip before and after ink filling. It is known that the EHD jet can more easily generate under higher electric field. For the above two conditions, a high electric field is formed on the edge of the nozzle (the blue area in Figs. 3(a) and 3(b). This indicates that the jet may generate at the edge of the nozzle, rather than the center of the nozzle. The EHD jet printing experiments have demonstrated this phenomenon, as shown in Fig. 3(c).

The critical electric field can be calculated after measuring the required parameters in Eq. (7), containing the surface tension (γ), the angle of the meniscus (2θ), and the inner diameter of the nozzle (d). In our study, the values for γ, ε0, and d are 0.036 N/m, 8.85 × 10−12 C2/J, and 160 μm, respectively. Since during EHD jet printing we used Taylor cone mode, the angle for Taylor cone is 98.6 deg [13]. According to Eq. (7), the critical electric field should be 5.76 × 106 V/m.

The applied voltage and printing distance are two critical factors which can affect the electric field distribution during EHD printing. Thus, the influence of applied voltage and printing distance on the maximum electric field near the nozzle tip was also investigated. Figure 4 shows the numerical simulation results. Figure 4(a) shows the relationship between maximum electric field and applied voltage that follows a simple linear regression based on applied voltage. A larger maximum electric field is associated with a higher applied voltage. However, a larger maximum electric field correlates with a lower printing distance, as shown in Fig. 4(b). In addition, the maximum electric field has a nonlinear relationship with the printing distance. Equation (5) can be used to explain the above linear and nonlinear relationships. It is obvious that the electric field and the applied voltage have a linear relation when the inner diameter and printing distance are constant. Since the applied voltage and inner diameter are fixed, the relationship between electric field and printing distance is nonlinear (exponential).

3.4 The Influence of Electric Field on the Jetting Modes.

It is reported that there are mainly three jetting mode for EHD jet printing. They are fine-jet mode, cone-jet mode, and multijet mode [11,23,24]. The jetting modes depend on the electric field between nozzle tip and the substrate. In this work, we changed the electric field by changing the printing distance. For our numerical simulation, the applied voltage and ink flow rate are fixed at 1100 V and 0.1 ml/h.

Figure 5 shows the influence of printing distance on the jetting modes at different printing distances. Fine-jet mode occurs at low electric field condition, as shown in Figs. 5(a) and 6(a). Due to the low electric field, the electric force applied on the ink meniscus is close to the surface tension. Pulled by the gravity and pump pressure, a jet can generate with a large diameter. At this mode, the jet is not stable and its diameter may change. The start and stop of the jet cannot be well controlled at this mode. However, when the electric field continuously increases, the electric force is sufficient enough to overcome the surface tension. Then the cone-jet mode appears (Figs. 5(b) and 6(b)). Under cone-jet mode, the formed jet is stable and controllable. The meniscus at the nozzle tip, called Taylor cone, has a constant taper angle of 98.6 deg. The enlarged view in Figs. 6(a) and 6(b) shows the shape of the meniscus at the nozzle tip. For fine-jet mode, there is a half-circular meniscus, while for cone-jet mode there is a conical meniscus at the nozzle tip. As there is continuous increase of the electric field, the velocity of the jet becomes faster. Since the ink flow rate is constant, the volume of ejected ink will be larger than that provided by the pump. The jet will shrink into the nozzle, as shown in Fig. 5(c). Then the electric field changes and the jet are apt to be unstable. The jet will be finally terminated, when all the ink in the nozzle is ejected out. Figures 5(d) and 6(c) show the multijet mode with more than one ejection at the nozzle tip under extremely high electric field or low surface tension. The position and number of jetting occurs randomly, leading to the irregular pattern printing. At such high electric field, discharge and breakdown will occur between the nozzle and the substrate. Since the printing distance is always at microscale, the air breakdown or the discharge will easily damage the voltage-supply module, current detection module, and even the XY stage. Thus, multijet mode should be avoided during practical EHD jet printing process.

The simulation and experimental results were compared to verify the established simulation model. It is hard to observe the jet flow and accurately measure the diameter of the EHD jet using the camera. Therefore, we assume the diameter of the EHD jet is equal to the line-width of printed PVP lines. Figures 5(a)5(d) are the simulation results, while Figs. 5(e)5(h) are the experimental ones. It can be seen that the diameter of EHD jet drops with the decrease of printing distance. According to Fig. 5(h), one can see that multiple PVP lines are printed on the substrate, since multijet mode occurs at printing distance of 200 μm (Fig. 5(d)). Figure 7 shows a comparison of EHD jet diameters obtained from simulation and experiments. The comparison between simulation and experiments also proves the simulation can agree with the experiments.

3.5 The Influence of Ink Flow Rate on the Jet Diameters.

We have discussed the effect of electric field on the EHD jet printing. Another vital factor, influencing EHD jet printing process, is ink flow rate. It can significantly affect the diameter of formed EHD jet. In this work, the influence of ink flow rate on the jet diameters was analyzed by numerical simulation. Figure 8 shows the numerical simulation results. In our numerical simulation, printing distance, applied voltage, and nozzle inner diameter were fixed at 200 μm, 1100 V, and 160 μm. The flow rates for Figs. 8(a)8(e) are 0.02, 0.1, 0.2, 0.3, and 0.6 ml/h, respectively. At small flow rate, the meniscus cannot form, and then EHD jet cannot generate either. Even though by improving the electric field the ink can eject from the nozzle, the unstable EHD jet can easily terminate because of insufficient ink supply. As the flow rate increases, the flow rate exceeds a threshold value. Then stable EHD jets can generate without termination. However, the diameter of EHD jet varies depending on the flow rate. This can be demonstrated by numerical simulations. Our simulation results (Figs. 8(b)8(e)) indicate that a larger diameter is associated with higher flow rate.

The simulation and experimental results were also compared. Figures 8(a)8(e) are the simulation results, while Figs. 8(f)8(j) are the experimental results. At flow rate of 0.02 ml/h, EHD jet cannot generate (Fig. 8(a)), and thus there is no printed line on the substrate (Fig. 8(f)). From 0.1 to 0.6 ml/h, it obvious that a larger diameter is associated with higher flow rate. Figure 9 shows EHD jet diameters obtained from simulation and experiments. According to the comparison, we can see that the numerical simulation can agree with the experiments.

Ganon-Calvo scaling law is a theory to calculate the diameter of formed EHD jet [25]. It is assumed that the printing distance, applied voltage, and the inner diameter of the nozzle are constant. Ganon-Calvo scaling law only describes the relationship between the printing condition (the flow rate and ink properties) and the jet diameter. For different solution, the calculation equation is different. In this study, the ink was made by PVP and water. Both water and PVP are polar material. Thus, the jet diameter (D) can be estimated by the following equation [25]:
D=(ρε0Q3γG)16
(8)

where ρ is the density of the ink, Q is ink flow rate, γ is the surface tension, and G is the conductivity of the ink. The above equation demonstrates that a larger jet diameter is associated with a higher ink flow rate.

According to the above discussion, suitable printing conditions can be obtained as follows: printing distance of 200–300 μm, applied voltage of ∼1100 V, and flow rate of 0.1–0.3 ml/h. Under those conditions, the cone-jet mode can occur and the formed EHD jet is continuous and stable. By changing the printing conditions, different line-width PVP patterns can be printed on glass slides. Figure 10 shows the printed PVP patterns. The line-widths of PVP lines are 0.9, 2.0, and 5.0 μm, respectively. It indicates that the developed numerical simulation method can be used to optimize printing conditions and EHD jet printing has a high potential to fabricate micro- and even nanoscale patterns.

4 Conclusion

In this work, a finite element model was established according to the presented EHD theory and verified by the experiments. The influence of printing distance and flow rate on the line-width of printed PVP lines was investigated by both simulation and experiments. The results showed good agreement between simulation and experiments. The electric distribution near the nozzle tip was analyzed. Fine-jet, cone-jet, and multijet modes can occur as different voltage applied. Simulation results also indicated the incline jet could easily generate due to the nonuniform electric field near the nozzle edge. The above simulation and experimental results demonstrated the practicability of proposed simulation method for optimizing and understanding EHD jet printing process.

Acknowledgment

This project is supported by the National Natural Science Foundation of China (No. 51705198), Open Fund of State Key Laboratory of Geohazard Prevention and Geoenvironment Protection (No. SKLGP2020K020), and Science and Technology Project of Jilin Provincial Education Department (No. JJKH20200962KJ).

Funding Data

  • Education Department of Jilin Province (JJKH20200962KJ; Funder ID: 10.13039/501100010211).

  • National Natural Science Foundation of China (51705198; Funder ID: 10.13039/501100001809).

  • State Key Laboratory of Geohazard Prevention and Geoenvironment Protection (SKLGP2020K020; Funder ID: 10.13039/501100011171).

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