The theoretical model and verification of electric-field-driven jet 3D printing for large-height and conformal micro/nano-scale parts

ABSTRACT Electrohydrodynamic (EHD) jet printing, as one of the most popular micro/nano-scale additive manufacturing methods, is still facing challenges in large-height printing and conformal printing due to poor electric field stability. The newly proposed electric field-driven (EFD) jet 3D printing has claimed better electric field stability. To reveal changing behaviour and generation mechanism of the electric field in 3D printing, an electric field model for EFD jet 3D printing was built and further validated by simulation and experiments (line width and critical voltage vs. printing height). Then, the advantage of the EFD method over EHD was confirmed by a case application of conformal printing with a height difference of larger than 9 mm and a multi-layer structure with a height of 5 mm and a line width of 20 μm. Therefore, the EFD jet 3D printing offers the possibility of achieving 3D printing in a larger height range with better electric field stability.


Introduction
Electrohydrodynamic (EHD) jet printing, as one of the most popular micro/nano-scale additive manufacture methods, has been widely used in the fields of microelectronics, biomedicine, tissue engineering, and microoptical device (Fang et al. 2018;Reiser et al. 2019;Zhang et al. 2019;Zhao et al. 2016), because of its capability to print user-specific micro/nano-scale patterns with high efficiency, high resolution, and low cost.The high-resolution printing capability of the EHD jet printing is mainly ascribed to the action of the electric field generated between an electrode pair (a conductive nozzle and a conductive substrate) leading to the elongation of the liquid materials on the tip of the nozzle and the subsequent breakout of the liquid materials into micro/ nano-sized droplets or jet (Park et al. 2007).
Over the past decades, EHD jet printing has been successfully applied to fabricate 2D and 2.5D micro/nanoscale patterns.And many researchers have proposed theoretical models to describe the physical process of EHD jetting.A leakage dielectric model (LDM) was proposed to describe the tiny deformation of droplets under the action of a uniform weak electric field (Saville 1997).Then, the LDM was extended to describe the behaviour of the large droplet in DC electric fields, which can predict the data of droplet distortions quantitatively (Bentennitis and Krause 2005).Furthermore, the electric fields in and outside the cone, and the surface charge density at the liquid surface have been calculated (Hartman et al. 1999).Recently, for guiding experimental optimisation, a theoretical model for predicting the feature size of the printing dots or lines has been established (Qian, Lan, and Zhang 2018).However, due to the inherent mechanism of conventional EHD-based printing with two counter electrodes (nozzle electrode and a grounded electrode under the substrate), the electric field between two electrodes will be weakened with the stacking up of 3D structure and the increase of printing height, thus, EHD jet printing is still facing great challenges to achieve the large-height printing (larger than 5 mm) and the conformal printing (uneven substrate) (Zhang et al. 2016;Mkhize and Bhaskaran 2021;Kwon et al. 2021;Gao and Zhou 2019).
To overcome these weaknesses, two kinds of methods have been proposed.One is increasing the voltage to compensate for the electric field (He et al. 2022;Wunner et al. 2018).However, it is very complicated to control the electric field intensity stable in the real complex 3D printing process.The other method is to restructure the electrode such as the needle-ring electrode system (Ohyama and Ohyama 2011) and single potential EHD printing (Yudistira et al. 2010).For example, the electric field-driven (EFD) jet micro/nano-scale 3D printing, which only uses a nozzle electrode and cancels the grounded electrode (Zhang et al. 2020) was proposed to fabricate the multi-layer scaffolds (Li et al. 2020), transparent glass heaters (Zhu et al. 2019), the highaspect-ratio transparent electrodes (Zhu et al. 2021) and cylindrical microlens array (Zhang et al. 2021).It has proven the potential for 3D printing and conformal printing.However, due to the lack of a theoretical model for the new method with a modified electrode structure, the generation mechanism and changing behaviour of the electric field of EFD jet 3D printing in the 3D printing process has not been revealed systematically.
In this paper, a theoretical model was built to describe the electric field intensity for EFD jet 3D printing.The feasibility of the proposed theoretical model for EFD jet 3D printing was verified by a series of experimental results and simulation results, which were also compared with the EHD method with two counter electrodes to demonstrate their differences in maintaining electric field stability.Finally, to verify the capability and stability of EFD jet 3D printing for large-height printing and conformal printing non-planar surfaces, the scaffold structure with a height of 5 mm and the micro-scale lines on the non-planar surfaces were successfully realised without adjusting the voltage.

Theoretical and experimental methods
For the theoretical investigation, the theoretical model of electric field intensity was built based on the electrostatic induction and electrical polarisation of a spatial model of three layers of different dielectrics.For the numerical simulation, the electric field was modelled and simulated in a 2D space using the COMSOL Multiphysics software (Zhang et al. 2020).The standoff height is set as 1 mm, the printing height is in the range of 1 mm to 10 mm, the applied voltage on the nozzle electrode is set in the range of 900-1350 V.The medium between the nozzle and the substrate was set as air with relative permittivity of 1, and the relative permittivity of the printing materials (polylactic acid) and glass were set as 4 and 7, respectively.
For the experimental investigation, firstly, the effect of voltage and printing height on the line width was investigated by measuring the line width of printed polylactic acid (PLA) using a nozzle of 200 µm internal diameter, air pressure of 6 kPa, moving speed of 30 mm/s, heating temperature of 120°C, and standoff height of 200 μm.The voltage (ranging from 900 to 1400 V) was used to build the relationship between voltage and line width.And line width of printed PLA on different printing heights ranging from 1 to 10 mm was investigated.Due to the different mechanisms of the electric field formed by the EHD and EFD, different voltages need to be applied to ensure the same initial line width, which is set to 1200 V (EHD) and 1400 V (EFD), respectively.And the line width of the printed PLA was measured using a digital microscope (DSX510, OLYMPUS, Japan).
Secondly, the jet behaviour at different printing heights was conducted with the parameters of constant standoff height (Hs) of 750 μm, printing materials of alcohol, air pressure of 6 kPa.The shape of the Taylor cone was recorded using a CCD camera (iSpeed-221).The statistical results of the cone angle and cone length of the Taylor cone were measured.Then, the critical voltage of keeping the similar Taylor cone with printing height ranging from 1 to 10 mm was counted.

Theoretical model for electric field intensity
To calculate the electric field intensity, the related parameters have been defined in Figure 1(c).The nozzle electrode (define radius as r c ) can be considered as a semi-infinite linear structure with a large number of free charges distributed on its surface.The thickness of the printed structure is defined as h d , and the distance between the top surface of the printed structure and the nozzle tip is defined as h s (this distance is generally kept constant during printing).It is worth mentioning that the sum of the two heights (h d and h s ) is the printing height.To meet the actual situation in the printing process, the calculation area is considered as a space composed of three layers of dielectric materials, which are set as region I with relative permittivity of ε 1 , region II with ε 2 , and region III with ε 3 , respectively.Due to the distribution of free charges in region I (a large number of positive charges are distributed on the surface of the nozzle electrode), the electric field distribution satisfies the Poisson equation, while there is no free charge distribution in region II and region III, and the electric field distribution satisfies the Laplace equation.Considering the simplest case that the charge element q is located at z = h s in region I, then the electric field distribution in the three regions satisfies the following equation.
In the cylindrical coordinate system, the Bessel function expansion can be written as.
The electric field distribution in region I can be written as follows.
At the interface of the three regions, the boundary conditions are expressed as Synthesise the resulting boundary conditions and coefficient equations concerning the original equation.
where M 21 , M 23 are ratio of relative permittivity, Combining the above conditions, the electric field distribution in region I can be expressed as.
Replacing the charge element in region I with a semiinfinite long-charged linear structure, and solving the integral of the obtained electric potential along the Zaxis direction, then, the electric potential of the composite electric field can be obtained.
In the area near the tip of the nozzle, when z = h s , ρ = r c , the potential is equal to that of the nozzle surface.And taking into account the geometric conditions in the printing process (h s ≫r c ), when the voltage applied at the nozzle electrode is set to w 0 , it can be substituted into and calculated the linear charge density of the linearly charged structure.
According to E = −∇w, the electric field intensity can be expressed as Then, the electric field intensity model of EFD jet 3D printing has been presented.And for EHD with two counter electrodes, the electric field intensity near the nozzle has been demonstrated in a theoretical model (Jones and Thong 1971) of: an infinite earthed plate at a distance z 0 from a semi-infinite line of charge.The classical equation of electric field intensity can be written as.

Simulation investigation
In order to understand the changing behaviour of electric field intensity in EFD jet 3D printing, electric field intensity was calculated using the proposed theoretical model with the voltage (from 950 to 1350 V) and printing height (from 1 to 10 mm) change, as shown in Figure 2(a).Then, to verify the correctness of the proposed theoretical model, the numerical simulation by finite element model was applied to calculate the electric field intensity under the same conditions (shown in Figure 2(b)).It can be seen from Figure 2(a,b) that the theoretical results present a very high consistency with simulation results.To further investigate the matching degree of the changing behaviour of electric field intensity obtained by the theoretical model and the numerical simulation, the electric field intensity was calculated when only changing the printing height with a certain voltage of 1000 V or only changing the voltage with the certain printing height of 1 mm, respectively.As shown in Figure 2(c,d), the results obtained by the theoretical model and the finite element model are always in the same order of magnitude and the specific values differ by about 8%.More importantly, the change rate of the electric field intensity for these two methods with the increase of the printing height and the voltage are almost the same.Therefore, the calculated results of electric field intensity match with the simulation results obtained from the finite element model.

Experimental investigation
To further verify the reliability of the proposed model, the change of electric field intensity with the increasing height of the printed structure in the 3D printing process has been investigated.On the one hand, due to the difficulty in measurement of the electric field intensity, the feature size (line width) was used to represent the changing trend of electric field intensity because of the determination of electric field intensity on the volume of ejected materials (Qian, Lan, and Zhang 2018;Chen, Saville, and Aksay 2006).On the other hand, to observe an obvious change trend of the line width in a large-height range, the insulating glass slides with a thickness of 1 mm can be inserted layer by layer between the nozzle and the substrate for imitating the quick increase of the printed structure height (Figure 3(a)), instead of the slowly depositing materials with several millimeters by micro/nano-scale printing.Then, the changing trend of line width in   these two methods is demonstrated in Figure 3(b).The line width decreases gradually with the increasing thickness of the glass slide in both EHD and EFD jet 3D printing.Specifically, in EFD jet 3D printing, when the thickness of the glass slide increases from 1 mm to 3 and 1 mm to 6 mm, the line width is reduced by 16.9% (decreases from 21.2 to 17.6 μm) and 41.9% (decreases from 21.2 to 12.3 μm), respectively.While the line width reduced by 41.6% in EHD jet printing (decreases from 23.3 to 13.6 μm) with the thickness of the glass slide increasing from 1 to 3 mm, and the line becomes unstable and even disappears when the printing height exceeds 3 mm.Therefore, the result of the slower decreasing trend of line width proves a more stable electric field with the increasing height of the printed structure in EFD jet 3D printing.
As well known that the increase in voltage often leads to higher electric field intensity, increasing the size of the deposited features (Zhang et al. 2016).As shown in Figure 3 Considering that the electric field intensity is the main parameter that affects the shape and size of the Taylor cone (Vaseashta 2007), thus, the shape of the Taylor cone can help us understand the change in electric field intensity.Figure 4(a,b) demonstrates the morphology of the Taylor cone at different heights.Figure 4(c) shows the statistical results of the cone angle and cone length  of the Taylor cone, indicating that the shape of the obtained Taylor cone is similar.As shown in Figure 4(d), when the printing height increases from 1 to 10 mm, the growth rate of voltage value for the EFD method is 44.2% (increases from 1592 to 2296 V), while that for EHD method is 100.8%(increases from 1210 to 2430 V).
Compared with the EHD method, the EFD method has a smaller voltage variation range, which means that the electric field intensity decays more slowly.
Then, the equation of electric field distribution for the EFD jet 3D printing was compared with the classical equation of the EHD jet printing (Jones and Thong 1971), which was based on the model of a semi-infinite wire perpendicular and an infinite grounded metal plane to plot the electric field distribution at the nozzle tip.The experimental parameters used in the calculation are shown in Table 1. Figure 5(a) shows the theoretically predicted change trend of electric field intensity and the line width from the experimental result with the voltage increasing from 900 to 1400 V.And Figure 5(b) shows the calculated electric field intensity and the line width with the printing height ranging from 1 mm to 5 mm.It can be seen that the calculated electric field intensity changes in the same way as the experimental line width for both the variables of voltage and printing height, which confirms that the equation is believable.It should be noted that the result of the experimental line width does not match the electric field intensity calculated by the theoretical model of the EHD method very well.It can be explained that the existing theoretical model of the EHD method did not include the effect of electric field contributed by printed structure.In addition, by substituting the critical voltage value for maintaining a similar Taylor cone shape at different printing heights into the equation, the electric field intensity can be obtained as shown in Figure 5(c).It can be seen that the calculated electric field intensity of EFD jet 3D printing is almost unchanged, which is consistent with the experimental results.

Case application
To further verify the better electric field stability of EFD jet 3D printing and its applicability in practical processes, we show the printed linear structures on highly variable structures without changing the voltage by these two methods.During the printing process, the parameters were maintained at the initial value (voltage of 1200 V for EHD and voltage of 1400 V for EFD, air pressure of 3 kPa).As shown in Figure 6(a), when the heights differ by 2 mm, the lines formed by the two printing methods remain uniform with the same line width.When the heights differ by 5 mm (Figure 6(b)), EFD jet 3D printing can still form a continuous and uniform line on the surface, while the line width on the right surface begins to change sharply and the formed line is no longer uniform in the EHD printing method due to the weakening of the electric field.Figure 6(c) shows the results when the height difference is 10 mm, the width of the line formed by the EFD jet printing also begins to decrease.Compared to the left surface, the line width of the right surface is reduced by 41% (44 to 26 mm), but the uniformity of the structure can still be maintained.However, the printing results even show obvious fractures in the EHD printing method, because the electric field cannot keep the material ejection in a cone-jet mode on the right surface, but a dripping mode appears.
Furthermore, to prove the improved electric field stability in printing a large-height structure, a scaffold with heights of 5 cm and a line width of 20 μm has been successfully printed without changing the voltage in the printing process (Figure 6(d)).Figure 6 (e)-(g) shows the surface appearances of the lower part to the upper part on the different heights of one side of the scaffold, the structure of the scaffold remained intact and uniform, indicating that the jet did not fluctuate greatly during the printing process.In addition, Figure 6(h) shows the printed simple circuits on non-planar surfaces with a maximum height difference of 9 mm without changing voltage using silver paste.Therefore, it demonstrates that EFD jet 3D printing method can maintain a more stable electric field in a wider range of height variation.

Conclusions and prospects
In summary, a theoretical model of electric field intensity in the EFD jet 3D printing was built based on a spatial model of three layers of different dielectrics, and the reliability of the proposed equation was confirmed by the simulation and experimental results (line width and critical voltage for Taylor cone).The calculated results of electric field intensity match with the simulation results obtained from the finite element model.And the experimental results show that the line width in EFD jet 3D printing decreased 16.9% (1 to 3 mm) and 41.9% (1 to 6 mm), respectively, while that in EHD method decreased 41.6% (1 to 3 mm) and even disappeared when the printing height exceeds 3 mm.And the critical voltage for keeping the same Taylor cone shape with increasing printing height from 1mm to 10mm increased by 44.2% in the EFD method, and 100.8% in the EHD method.The results of printing on structures with variable thickness show that the EFD jet 3D printing method can be used in a wider range of heights than the EHD method.As a result, the different electric field generation methods in the EFD method demonstrated better electric field stability compared to the EHD method, which will facilitate the progress of micro/nano-scale 3D printing with largeheight and conformal printing.

Figure 1 .
Figure 1.(a) Schematic diagram of EFD jet 3D printing.(b) Charges distribution in the electric field between the nozzle and printed structure of EFD jet 3D printing.(c) A spatial model of three layers of different dielectrics.

Figure 2 .
Figure 2. Electric field intensity of: (a) the calculation results by proposed theoretical model, (b) the simulation results, (c) comparison of calculation results and simulation results when the printing height changes, (d) comparison of calculation results and simulation results when the voltage changes.

Figure 3 .
Figure 3. (a) The multi-layer 3D printing process, (b) The effect of printing height on the line width, (c) The effect of voltage on the line width, (d) The fitting results of voltage and printing height.

Figure 4 .
Figure 4.The stable Taylor cones with same shape at different heights in the two methods of (a) EFD method and (b) EHD method, (c) Geometric parameters of Taylor cone at different heights in the EFD and EHD method, (d) The critical voltages for getting stable Taylor cones with same shape at different heights in the EFD and EHD method.
(c), as the voltage increases, the line width in EHD changes drastically, while the change in EFD is relatively smooth.And the relationship between voltage and height can be obtained by fitting the longitudinal (height change in Figure 3(b)) and lateral (voltage change in Figure 3(c)) data.It can be seen from Figure 3(d), the two fitted lines are nearly parallel, which means that the change in height in the two printing methods is equivalent to the change in voltage.However, the changes in line width that characterise the electric field intensity show different trends.It can be concluded that the mechanisms of generating an electric field between EFD jet 3D printing and EHD jet printing are different.

Figure 6 .
Figure 6.Printing on the surface of the thickness-variable structure: (a) from 5 mm to 7 mm, (b) from 5 mm to 10 mm, (c) from 5 mm to 15 mm, (d) print a large-height and well-controlled scaffold without increasing voltage, (e) lower part on the side of the scaffold, (f) middle part on the side of the scaffold, (g) upper part on the side of the scaffold, (h) conformal printing on the non-planar surface.

Table 1 .
Parameters in the actual printing process.
Team in Universities of Shandong Province, China [grant number 2021KJ044], and Natural Science Foundation of Shandong Province, China [grant number ZR2020ZD04].
Hui Huang is Postgraduate researcher in the School of Mechanical and Automotive Engineering, Qingdao University of Technology, China.His research activities are mainly focused on 3D printing and micro-nano manufacturing.Guangming Zhang is an associate professor in the School of Mechanical and Automotive Engineering, Qingdao University of Technology, China.His research activities are mainly focused on the research and development of Micro/Nanoscale 3D printing process and equipment, Bio 3D printing, and Powder metallurgy and ceramic 3D printing.Wenhai Li is Postgraduate researcher in the School of Mechanical and Automotive Engineering, Qingdao University of Technology, China.His research activities are mainly focused on 3D printing and micro-nano manufacturing.Zun Yu is Postgraduate researcher in the School of Mechanical and Automotive Engineering, Qingdao University of Technology, China.His research activities are mainly focused on 3D printing and micro-nano manufacturing.Zilong Peng is a full professor in the School of Mechanical and Automotive Engineering, Qingdao University of Technology, China.Fei Wang is an associate professor in the School of Mechanical and Automotive Engineering, Qingdao University of Technology, China.Xiaoyang Zhu is an associate professor in the School of Mechanical and Automotive Engineering, Qingdao University of Technology, China.Hongbo Lan is a full professor in the School of Mechanical and Automotive Engineering, Qingdao University of Technology, China.He is now a director in Shandong Engineering Research Center for Additive Manufacturing; Qindao Engineering Research Center for 3D Printing.His current research interests include micro-and nano-scale 3D printing, additive manufacturing, large-area nanoimprint lithography, and micro/nano-manufacturing.He is a committee member of Additive Manufacturing Standards-ISO/TC 261, committee member of Additive Manufacturing Standards -SAC/TC562.He was awarded the Expert with Special Government Allowances from the State Council, Young and Middle-aged Experts with Outstanding Contributions in Shandong Province, New Century Excellent Talents in University of Ministry of Education, etc.