Development of limited-view and three-dimensional reconstruction method for analysis of electrohydrodynamic jetting behavior

Yeonghyeon Gim; Dong Ho Shin; Do Yoon Moh; Han Seo Ko

doi:10.1364/OE.25.009244

1. Introduction

In general, the electrohydrodynamic (EHD) spray represents a dynamic combination of electrically charged fluids. The ejection principle was established based on this phenomenon, which is expressed as a formation of micro and nanometer sized droplets from an electrostatic nozzle. This is an interesting and useful issue in various fields such as ink-jet printing and drug delivery [1–3].

EHD jetting behavior has been studied by many researchers. Hayati et al., investigated phenomena of various modes for various types of liquids according to applied voltages and liquid flow rates [4,5]. Jaworek and Krupa also observed phenomena and defined them systematically according to the geometric forms of the ejecting pattern [6–8]. However, the phenomena were typically observed by using only one camera. Therefore, this method was limited for validating the theory of the phenomena using the visualization technique.

Especially, unstable phenomena such as the multi-spindle mode [7] and precession mode [8] need to be visualized in three-dimensions (3D) by using more than two angles of views in order to observe the exact locations of the ejected droplets or jets according to the position of the nozzle, which is not able to be viewed with only one camera. In order to solve this limitation, Chen et al, represented the unstable ejection phenomena of the droplets using 3D tomographic technique [9]. In addition, Nguyen et al. described the unstable jetting behaviors in the 3D field using shadowgraphic tomography technique [10–12]. However, the general tomographic technique using the binary images with several iterations is inappropriate to reconstruct the 3D jetting behavior because the locations of non-zero voxels from the reconstructed results are almost same as those by the multiplicative line-of-sight (MLOS) estimation with only one iteration. Also, it is impossible to reduce the reconstruction error caused by an invisible area. Thus, a method for reconstructing the 3D phenomena is developed in this paper in order to solve the invisible area problem. The algorithm used in the method includes a MLOS estimation and an ellipse estimation. The MLOS estimation which was proposed by Atkinson et al. reconstructed an object from the multi-directional projection data of the object [13]. The valuable voxels were estimated by the MLOS estimation to reconstruct the volume. The method was then generally applied in a tomographic particle image velocimetry (Tomo-PIV) with a simultaneous multiplicative algebraic reconstruction technique (SMART). However, the reconstruction error still cannot be reduced by the invisible area using the multi-directional binary projection images. In order to solve this problem, the MLOS estimation was only conducted by using the multi-directional binary projection images for removing poor pixels caused by the noise of the detector and by estimating the center of whole layer that corresponded to the whole layer of the real field.

Following this procedure, the ellipse estimation was performed to reconstruct the 3D field because a section of the droplet behavior could be assumed as an ellipse shape. The ellipse estimation was based on the formula of the ellipse in the rectangular coordinate. In this procedure, any ellipse was estimated by the multi-directional orthogonal projections which were obtained from the whole layer of the multi-directional binary projection images modified by the MLOS estimation in the same layer.

In this study, the developed method, which includes both the MLOS estimation and the ellipse estimation, was verified by the numerical simulation used for application to the reconstruction of the EHD jetting behavior. The method was then revaluated by comparing the reference flow rate caused by a micro syringe pump with the flow rate which was calculated using the reconstructed volume of the ejection behavior. Finally, a multi-spindle mode and precession mode which were reconstructed in 3D were analyzed. All the experimental data were obtained from the multi-directional images of the ejection behavior by three high-speed cameras arranged at offset angles of 45° to each other.

2. Reconstruction technique

Multiplicative line-of-sight (MLOS) estimation was proposed by Atkinson et al. [13], and the method was commonly used for tomographic particle image velocimetry (Tomo-PIV) with the simultaneous multiplicative algebraic reconstruction technique (SMART). In this work, binary images gained from several angles of view were used. During the MLOS estimation process, valuable voxels were estimated by multiplying the value of these corresponding pixels which were included in the binary images. The estimated valuable voxel’s coordinates were saved after using the MLOS estimation process. At the same time, any mismatched pixels caused by noise or dust on the detector were removed.

In the case of reconstructing the jetting behavior from the nozzle using multi-directional binary projection data, the reconstructed results using the MLOS estimation showed a similar quality to those reconstructed by the tomography technique because the method cannot consider the differences of light intensity. However, the reconstructing time using the MLOS method was shorter than that using the tomography technique because the iteration was not required for the MLOS method. Also, the improvement of the reconstruction quality is still important to analyze the jetting behavior. In order to solve the problem, an interpolating method using the ellipse estimation was developed because a horizontal section of the droplet behavior generally represents the ellipse shape. In the case of reconstructing the object using the projection data obtained by the binary images, it is almost impossible to eliminate the reconstruction error caused by the invisible area, even when using the tomographic technique. Thus, it is necessary to carry out the interpolation procedure using the ellipse estimation to increase the quality of the reconstruction results with the multi-directional binary images. The interpolation procedure was based on the ellipse formula as follows:

\frac{{((x - x_{0}) \cos t - (y - y_{0}) \sin t + x_{0})}^{2}}{a^{2}} + \frac{{((x - x_{0}) \sin t + (y - y_{0}) \cos t + y_{0})}^{2}}{b^{2}} = 1

where x and y are variables in rectangular coordinates, x₀ and y₀ are center points of the ellipse shape, and t is the rotating angle of the ellipse shape. The ellipse in the rectangular coordinate has two perpendicular axes: a major axis a and a minor axis b. In this procedure, a section of the object was assumed to be the ellipse shape because the surface of the jetting behavior was affected by the surface tension. Thus, an orthogonal projection can be substituted for a layer of the binary image. For estimating the mathematical form of any ellipse that is rotated at an unknown angle w, Eq. (2) must be satisfied at all of the multi-directional angles.

L_{n} = \frac{2 \sqrt{a^{2} m {(α_{n}, w)}^{2} + b^{2}}}{\sqrt{1 + m {(α_{n}, w)}^{2}}} \to m (α_{n}, w) = - \frac{1}{\tan (α_{n} + w)}

where L_n is the orthogonal projection from the multi-directional angle α_n. Equation (2) can be represented as a way of determining b², then it is possible to link all of the orthogonal projections as shown in Eq. (3).

4 b^{2} = (L_{i}^{2} - 4 a^{2}) m {(α_{i}, w)}^{2} + L_{i}^{2} = (L_{j}^{2} - 4 a^{2}) m {(α_{j}, w)}^{2} + L_{j}^{2}

Thus, the major axis a can be estimated using the matrix equation form of Eq. (3), which is represented in Eq. (4) for three multi-directional angles. The calculated results; $a {(w)}_{1}^{2}$ , $a {(w)}_{2}^{2}$ , and $a {(w)}_{3}^{2}$ are changed according to the unknown angle w and they have the same value in this equation. Thus, Eq. (5) was used to find the angle w and evaluate the similarity among the results.

\begin{array}{l} [\begin{matrix} \frac{m {(α_{1}, w)}^{2} + 1}{m {(α_{1}, w)}^{2} - m {(α_{2}, w)}^{2}} & - \frac{m {(α_{2}, w)}^{2} + 1}{m {(α_{1}, w)}^{2} - m {(α_{2}, w)}^{2}} & 0 \\ 0 & \frac{m {(α_{2}, w)}^{2} + 1}{m {(α_{2}, w)}^{2} - m {(α_{3}, w)}^{2}} & - \frac{m {(α_{3}, w)}^{2} + 1}{m {(α_{2}, w)}^{2} - m {(α_{3}, w)}^{2}} \\ - \frac{m {(α_{1}, w)}^{2} + 1}{m {(α_{3}, w)}^{2} - m {(α_{1}, w)}^{2}} & 0 & \frac{m {(α_{3}, w)}^{2} + 1}{m {(α_{3}, w)}^{2} - m {(α_{1}, w)}^{2}} \end{matrix}] [\begin{matrix} L_{1}^{2} \\ L_{2}^{2} \\ L_{3}^{2} \end{matrix}] \\ = 4 [\begin{matrix} a {(w)}_{1}^{2} \\ a {(w)}_{2}^{2} \\ a {(w)}_{3}^{2} \end{matrix}] \end{array}

Φ_{s i m i l a r i t y} = \frac{\sum_{i = 1}^{N} | \frac{\sum_{j = 1}^{N} a {(w)}_{j}^{2}}{N} - a {(w)}_{i}^{2} |}{N}

where N is the total number of layers in the binary projection image.

If the similarity has the minimum value at any angle w, the values of a_k and b_k are selected using Eq. (3). In addition, the area of the ellipse in the layer plane A_k can be calculated using Eq. (6).

A_{k} = π a_{k} b_{k}

Finally, the total volume of object V can be obtained by Eq. (7).

V = \sum_{k = 1}^{N} A_{k}

3. Numerical simulation

In order to evaluate the developed method, numerical simulation was conducted. As shown in Fig. 1(a), a 3D twisted phantom was generated in the (x,y,z) coordinate using Eq. (8). The phantom is represented in the volume size of 120 × 120 × 181 (pixels) and the field impedance is 0 for the outside portion of the model and 1 for its inside portion. Also, the 3D binary projection images were obtained from the phantom at 0°, 45°, and −45° as shown in Figs. 1(b), 1(c) and 1(d), respectively. The size of the images is 120 × 181 (pixels).

\frac{x^{2}}{30 + z} + \frac{y^{2}}{{(40 - \sqrt{30 + z})}^{2}} - 1 \leq 0, {\begin{matrix} x = 20 \sin \frac{π}{180} z - 60.5 \\ y = 20 \cos \frac{π}{180} z - 60.5 \\ 0 \leq z \leq π \end{matrix}

where x, y, and z are cartesian coordinates to express the phantom.

Fig. 1 Twisted phantom in 3D field; (a) isosurface of twisted phantom, (b) binary projection image of phantom at 0°, (c) 45°, and (d) −45°.

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During this procedure, the calibration error between the detectors and the phantom was assumed to be zero and the obtained images had no noise. From these images, the reconstructed results using the MLOS estimation without and with the ellipse estimation are shown in Figs. 2(a) and 2(b), respectively. Also, the reconstruction errors were calculated to confirm the discrepancy between the reconstructed results and the real phantom. The average error between the reconstructed function $\hat{f}$ and the reference phantom function f can be obtained as follows [4].

Φ_{a v g} = \frac{\sum_{j = 1}^{N_{v}} | f (x_{j}, y_{j}, z_{j}) - \hat{f} (x_{j}, y_{i}, z_{i}) |}{N_{v}}

where

N_{v}

is the total number of voxels. The phantom reconstructed by the MLOS estimation with the ellipse estimation is more similar to the original phantom than to that without the ellipse estimation. The average errors shown in Figs. 2(a) and 2(b) are 0.0464 and 0.0195, respectively. From this simulation, the method which included the ellipse estimation was validated and applied to reconstruct the EHD jetting behavior.

Fig. 2 Reconstructed results in 3D field using MLOS estimation; (a) without ellipse estimation and (b) with ellipse estimation.

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4. Experimental setup

The multi-directional binary images of the jetting behavior could be obtained by setting three high-speed cameras which were positioned in front of the nozzle and the light sources as shown in Fig. 3. These cameras were positioned at an offset angle of 45° to each other and recorded the images at 5000fps. In Fig. 3, a micro-sized nozzle with a diameter of 250 μm was connected to a micro syringe pump and ethanol was supplied to the nozzle to observe the jetting behavior. The relative permittivity of the liquid was 24.3, the density was 0.789 (g/cm³), the viscosity was 0.0012 (Pa s), and the surface tension was 22.39 (mN/m). To confirm the developed method, the ejection behavior without the electric force on the nozzle was reconstructed, and the flow rate was calculated to compare with the real flow rate caused by the micro syringe pump. The size of each projection view was 100 × 430 (pixels), and the images were converted from the gray images to the binary images using an image toolbox in the Matlab program. The noise caused by dust on the camera lens was then removed using the MLOS estimation, and the ellipse estimation was conducted using the modified images.

Fig. 3 Schematic diagram of experimental setup.

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5. Results and discussion

The volume of the meniscus V and the flow rate Q were calculated by Eqs. (7) and (10), respectively.

Q \approx \frac{V (t + Δ t) - V (t)}{Δ t}

The calculated flow rate ranged from 9.63 to 9.83 μl/s considering the standard error. Thus, the developed method was revaluated because the reference flow rate caused by the micro syringe pump was 10 μl/s. The procedure of obtaining the projection data from three projection angles of view using three cameras could make the discrepancy of the flow rate between the measured and the reconstructed values because the real data could not be gained perfectly by the cameras which converted the data to the pixel unit.

To observe the EHD jetting behavior, an electric force was applied to the nozzle tip on the substrate using the high voltage controller. The distance from the nozzle tip to the substrate was 30 mm, and the applied voltage was increased to observe the various EHD spraying modes. The reconstructed results of the multi-spindle and precession modes are shown in Figs. 4 and 5, respectively. In the side views of the multi-spindle mode (Figs. 4(a) and 4(b)) with the applied voltage of 3.5 kV, two droplets ejected from the nozzle were observed to be moving in a similar direction. However, in the 3D view, the droplets were moving at a different direction to that of the red dotted line as shown in Fig. 4(c). As can be seen in the side views of the precession mode at the applied voltage of 4 kV (Figs. 5(a) and 5(b)), the trajectory of the precession mode resembles a spiral shape. The curvature angle was observed to be varied by time in the 3D field compared with the red line for the previous trajectory, as shown in Fig. 5(c). Using the developed method, the volume of the meniscus, the flow rate of the unstable jetting behavior, and the locations of the ejected droplets or jets according to the nozzle position could be exactly obtained in the 3D field.

Fig. 4 Multi-spindle mode of ethanol ejection; (a) multi-directional projection images, (b) projection images by time, and (c) reconstructed result using ellipse estimation.

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Fig. 5 Precession mode of ethanol; (a) multi-directional projection images, (b) projection images by time, and (c) reconstructed result using ellipse estimation.

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6. Conclusion

Analysis of the EHD jet behavior has progressed in various fields. To increase the accuracy of the reconstruction of the EHD jetting behavior, variables such as the direction of the meniscus, the mass flux, and the stability of the EHD jet need to be analyzed. Also, the EHD jetting behavior was reconstructed in the 3D field because the considered variables cannot be observed in the two-dimensional projection images. However, the reconstruction error of the multi-directional binary images caused by the invisible area could not be removed even using the 3D tomographic technique. Thus, a 3D reconstruction method which includes both the MLOS estimation and the ellipse estimation using the limited multi-directional binary projection images was developed in this study. The developed method was verified by numerical simulation using a twisted phantom. The method was then successfully applied to observe the flow rate, the locations of the ejected droplet, and the trajectory of the unstable EHD jetting behavior from the nozzle according to the voltage in the 3D field.

Funding

National Research Foundation (NRF) of Korea, funded by the Korean government (2016R1A2B4011087).

References and links

1. J. U. Park, M. Hardy, S. J. Kang, K. Barton, K. Adair, D. K. Mukhopadhyay, C. Y. Lee, M. S. Strano, A. G. Alleyne, J. G. Georgiadis, P. M. Ferreira, and J. A. Rogers, “High-resolution electrohydrodynamic jet printing,” Nat. Mater. 6(10), 782–789 (2007). [CrossRef] [PubMed]

2. W. D. Luedtke, U. Landman, Y. H. Chiu, D. J. Levandier, R. A. Dressler, S. Sok, and M. S. Gordon, “Nanojets, electrospray, and ion field evaporation: molecular dynamics simulations and laboratory experiments,” J. Phys. Chem. A 112(40), 9628–9649 (2008). [CrossRef] [PubMed]

3. A. Jaworek and A. T. Sobczyk, “Electrospraying route to nanotechnology: an overview,” J. Electrost. 66(3-4), 197–219 (2008). [CrossRef]

4. I. Hayati, A. I. Bailey, and T. H. F. Tadros, “Investigations into the mechanisms of electrohydrodynamic spraying of liquids: I. Effect of electric field and the environment on pendant drops and factors affecting the formation of stable jets and atomization,” J. Col. Inter. Sci. 117(1), 205–221 (1987). [CrossRef]

5. I. Hayati, A. I. Bailey, and T. H. F. Tadros, “Investigations into the mechanisms of electrohydrodynamic spraying of liquids: II. Mechanism of stable jet formation and electrical forces acting on a liquid cone,” J. Col. Inter. Sci. 117(1), 222–230 (1987). [CrossRef]

6. A. Jaworek and A. Krupa, “Main modes of electrohydrodynamic spraying of liquids,” in Third International Conference on Multiphase Flow (1998).

7. A. Jaworek and A. Krupa, “Jet and drops formation in electrohydrodynamic spraying of liquids. A systematic approach,” Exp. Fluids 27(1), 43–52 (1999). [CrossRef]

8. A. Jaworek and A. Krupa, “Generation and characteristics of the precession mode of EHD spraying,” J. Aerosol Sci. 27(1), 75–82 (1996). [CrossRef]

9. C. Chen, Y. J. Kim, and H. S. Ko, “Three-dimensional tomographic reconstruction of unstable ejection phenomena of droplets for electrohydrodynamic jet,” Exp. Therm. Fluid Sci. 35(3), 433–441 (2011). [CrossRef]

10. X. H. Nguyen, S.-H. Lee, and H. S. Ko, “Comparative study on basis functions for projection matrix of three-dimensional tomographic reconstruction for analysis of dropletbehavior from electrohydrodynamic jet,” Appl. Opt. 51(24), 5834–5844 (2012). [CrossRef] [PubMed]

11. X. H. Nguyen, S.-H. Lee, and H. S. Ko, “Analysis of electrohydrodynamic jetting behaviors using three-dimensional shadowgraphic tomography,” Appl. Opt. 52(19), 4494–4504 (2013). [CrossRef] [PubMed]

12. X. H. Nguyen, Y. Gim, and H. S. Ko, “Multifunctional, three-dimensional tomography for analysis of eletrectrohydrodynamic jetting,” Opt. Lasers Eng. 68, 235–243 (2015). [CrossRef]

13. C. Atkinson and J. Soria, “An efficient simultaneous reconstruction technique for tomographic particle image velocimetry,” Exp. Fluids 47(4-5), 553–568 (2009). [CrossRef]

Development of limited-view and three-dimensional reconstruction method for analysis of electrohydrodynamic jetting behavior

Abstract

1. Introduction

2. Reconstruction technique

3. Numerical simulation

4. Experimental setup

5. Results and discussion

6. Conclusion

Funding

References and links

Cited By

Figures (5)

Equations (10)

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