The manipulation of the trajectory of high-pressure micro water jets has the potential to greatly improve the accuracy of water jet related manufacturing processes. An experimental study was conducted to understand the basic static and dynamic responses of high-pressure micro water jet systems in the presence of nonuniform electric fields. A single electrode was employed to create a nonuniform electric field to deflect a high-pressure micro water jet toward the electrode by the dielectrophoretic force generated. The water jet's motions were precisely recorded by a high-speed camera with a 20× magnification and the videos postprocessed by a LabVIEW image processing program to acquire the deflections. The experiments revealed the fundamental relationships between three experimental parameters, i.e., voltage, pressure, and the distance between the water jet and the electrode and the deflection of the water jet in both nonuniform static and dynamic electric fields. In the latter case, electric signals at different frequencies were employed to experimentally investigate the jet's dynamic response, such as response time, frequency, and the stability of the water jet's motion. A first-order system model was proposed to approximate the jet's response to dynamic input signals. The work can serve as the basis for the development of closed-loop control systems for manipulating the trajectory of high-pressure micro water jets.

## Introduction

The unique properties of water jet technology, such as no significant heat effect, no chemical reaction, and environmental friendliness, are some of the advantages responsible for its widespread use in various manufacturing processes [1]. Among all the applications of high-pressure water jets, sheet metal cutting is the most popular one. The technology can also be utilized for peening and cleaning purposes. In combination with other manufacturing methods like grinding and turning, water jets can also be applied for cooling [2]. Recently, high-pressure water jets were adopted in sheet metal forming processes and specifically water jet incremental sheet metal forming [2]. Water jets with abrasive particles used in abrasive water jet (AWJ) machining [3] constitute the most commonly used method with the capability to cut, drill or mill a large variety of materials, including different kinds of metals and alloys, composites (carbon fiber composites, glass fiber composites, and other-reinforced plastics), brittle materials (glasses and ceramics), soft materials (plastics and rubber), etc. The major limitations of the AWJ technology include a short working length and the positioning accuracy of water jet impingement.

As the demand for micro products is steadily increasing in various areas, especially in IT, medical, automotive and biomedical industries, renewed focus is being placed on water jetting technologies in the field of micromanufacturing [4]. Although numerous innovations and improvements of this manufacturing technique already meet most of the associated stringent requirements at the macroscale, water jet technology is still quite limited when it comes to fulfilling existing demands in the domain of micromanufacturing. As a result, extensive studies have been done to improve the performance of AWJ technology at the microscale. Park et al. machined micro grooves on glass with micro abrasive water jets [5]. Shin et al. also used micro AWJ to create micro holes a few hundreds of microns in diameter on silicon carbide, which is a highly wear resistant material with excellent mechanical properties [6]. Pang et al. machined microchannels on glass by an abrasive slurry water jet process [7]. The minimum feature size for micro AWJ of about 100 μm is primarily limited by the size of the particles, the orifice, and the flow conditions of the particles through the orifice. The development of micro AWJ techniques, which are still in their infancy, is aimed at achieving even smaller feature sizes and higher accuracy as compared to current capabilities [1].

To overcome the disadvantages caused by the nature of AWJ, pure water jets with smaller orifices were used to cut soft materials in the micro domain. However, without abrasive particles, pure micro water jets, as the working tool, are not capable of machining a large variety of materials, which is a huge advantage of AWJ. High-pressure micro water jets can only be utilized for machining soft materials. In some application areas, traditional physical mechanisms like electric and magnetic fields, acoustics, and optics are frequently introduced into manufacturing processes to enhance their performance. Different manufacturing techniques are also combined to eliminate the limitations of each individual technique. Richerzhagen proposed a hybrid manufacturing process called waterjet-guided laser processing in which a laser beam instead of abrasive particles is coupled into a very fine water jet to take advantages of both laser and water jet technologies and overcome the material limitation of pure water jets [8,9]. Currently, this hybrid process has been implemented as a tool in the manufacture of semiconductors [10], solar cells [11], electronics, tooling, high-brightness light-emitting diode [12], watchmaking, and automotive industries [13]. As a process in the micromanufacturing area, high-speed thin Ga–As wafer dicing and cutting [14] and grooving of metals with high precision are the major applications. The typical size of the orifice varies from 25 to 100 μm in diameter [1]. The minimum size of the jet is constrained by the difficulties in manufacturing small diamond or sapphire orifices and the beam waist of the laser beam at the focal position.

Although the proposed waterjet-guided laser processing method does solve the material restriction of pure micro water jets, the accuracy of the position of water jet impingement is still not greatly improved. The accuracy of water jet technologies could be further increased if the water jet trajectory could be manipulated. However, available literature does not offer any effective way of controlling the trajectory of high-pressure micro water jets. Therefore, the aim of this work is to explore methods for jet manipulation and control. Among numerous possible methods, dieletrophoresis (DEP), which exerts a translational motion component by polarizing a neutral body in a nonuniform electric field, has the potential to achieve this goal [15]. This particular technique has successfully been adopted for manipulating, separating, or sorting bioparticles in a fluid by different electric field configurations [16]. In addition, Chiarot et al. and Doak et al. observed dielectrophoretic action on water droplets and continuous ink jets and revealed the basic relationships between some experimental parameters and the jet's deflection at low water pressures through fitted empirical equations [17,18]. Hokmabad et al. experimentally studied the rhythmic motion of a liquid jet with both dielectrophoresis and electrophoresis and the motion of the emanated droplets caused by jet instability in a traverse electric field [19]. The motion, deformation, and instability of charged liquid jets were examined by using a nozzle-ring electrode apparatus by Hokmabad et al. [20]. A preliminary study on manipulating the trajectory of a low-pressure water jet with corresponding modeling in comsol was performed by Mohanty et al. [21]. Furthermore, Jones discussed the possibility, capability, and applications of DEP liquid actuation at the microscale [22]. Nevertheless, the study of dielectrophoresis applied to high-pressure micro water jets for manufacturing purposes has not yet been addressed in the literature. High-pressure jets have the potential to markedly widen the pragmatic application domains of micro water jet-based technologies, especially in waterjet-guided laser processing. Hence, it is necessary to study and understand the static and dynamic motion responses of high-pressure micro water jets as a function of various parameters so that a precise control of the water jet's motion characterized by short response time and submicron accuracy of the impingement position can be realized.

The overarching objective of this paper is to investigate the influence of possible control parameters on the behavior of water jets in nonuniform electric fields to serve as guidance for establishing real-time closed-loop control mechanisms that precisely manipulate the trajectory of the water jet. To accomplish this goal, the motion of high-pressure micro water jets will be observed and recorded by a high-speed camera. To prevent the influence of complex two phase flow conditions and sophisticated electric field configurations on the water jet's motion, only a single electrode will be used to generate nonuniform electric fields. The study will start with simple direct current (DC) signals applied to the electrode to define an empirical equation for calculating the deflection in static nonuniform electric fields. However, with the ultimate goal of controlling the water jet's deflections, time variant nonuniform electric fields will be applied to ascertain the dynamic response characteristics of the jet. Alternating current signals of different waveforms and frequencies will be applied to the electrode to cause dynamic jet motions. Based on the observations of the jet's behavior in both static and dynamic nonuniform electric fields, a dynamic model of its behavior that can serve as a basis for formulating real time closed-loop control strategies will be suggested.

The rest of the paper is organized as follows: First of all, the fundamentals of dielectrophoresis are briefly introduced with a corresponding model of the water jet's motions in Sec. 2. The experimental setup used is explained with adequate details in Sec. 3. The static motion relationships between water jet deflection and relevant experimental parameters are discussed in Sec. 4, while Sec. 5 illustrates the experimental results on dynamic responses under different signals. Based on the experimental results, a model for the jet's response is proposed. Conclusions followed by future works are given in Sec. 6.

## Fundamental Theories

The deflection of water jets in the nonuniform electric field is the result of the DEP force, which is a force caused by polarization of dielectric materials in electric fields. The body does not necessarily need to be charged to have DEP forces acting on it in both uniform and nonuniform electric fields [15]. In a uniform electric field, a neutral body is merely polarized to have a possible torque, but no net translational force. As a result, the torque generated by anisotropic polarization realigns the neutral body in the electric field. Consequently, to have a net translational force on a neutral body, a nonuniform electric field is required. It is noteworthy that the direction of the net translational force will not change if the polarity of the electric field is switched. Polarized dielectric objects move toward the region of greatest field intensity [15]. In the case of a water jet, the purpose of the nonuniform electric field, to be used in this study, is to introduce a variable intensity DEP force acting on the jet leading to its bending. As a result, the control of the nonuniform electric field is essential to accurately manipulate the jet's trajectory.

The schematic representation of a water jet in a nonuniform electric field that is induced by a single electrode is shown in Fig. 1. Assuming that the DEP force is the only force exerted on the water jet in the traverse direction (direction of the water jet deflection), the governing equations for the motion in the traverse direction can be easily expressed by Newton's second law. No force will act on the water jet in the traverse direction when the water jet moves out from the region in which a strong electric field exists. In this section, the fundamental theory of dielectrophoresis will be introduced followed by the mathematical derivation of the water jet motion to reveal the relationships between processes parameters and water jet deflection.

### Dielectrophoresis.

For ideal dielectrics, under certain assumptions, the net DEP force on a small neutral body in a static field can be expressed as Refs. [15] and [23]
$FDEP=αvE⋅∇E=12αv∇|E|2$
(1)
where $v$ is the volume of the neutral body, $E$ is the external electric field, and $α$ is the dipole moment per unit volume in a unit field. The magnitude of the DEP force depends on the strength of the electric field, the gradient of the electric field, the volume of the neutral body, and its dipole moment. The electric field is the negative gradient of the potential filed $φ$, which can be obtained by solving the Laplace equation [24]
$∇2φ=0$
(2)
It is not always possible to solve the Laplace equation analytically for arbitrary configurations of the electric field. To easily understand the influence of the external voltage source applied to the electrode on the electric field, an expression for the electric field needs to be written as a function of voltage U. In the experimental setup to be used in this work, a single electrode is the only source of the electric field. Under these conditions, the electrical field expression can be simplified to Ref. [17]
$E=−∇φ=Uf(x,y,z)$
(3)

where $f(x,y,z)$ is a function, which can be solved from Eq. (2) to determine the strength of the electric field at any location.

### Modeling of Water Jet Deflection.

A high-pressure water jet in air poses a sophisticated two-phase flow problem, which is very hard to model. For the present purpose, the understanding and modeling of the deflection of the water jet in a single direction is sufficient. The objective of this section is to provide an analysis that can reveal which parameters are the key experimental parameters that significantly influence jet deflection and to formulate the corresponding relationships. Figure 1 shows the schematic of the problem to be considered in which the water jet before breaking up into water droplets is assumed to be a perfect cylindrical jet.

By discretizing the water jet into small segments, each segment is considered to follow the exactly same path at the velocity of water jet Vj, which can be calculated from Ref. [25] as:
$Vj=K2Pρ$
(4)
where K is the discharge coefficient accounting for the friction in the orifice, P is the water pressure from the pump, and $ρ$ is the density of water. The value of K is typically in the range of 0.95–0.99 for cone-down orifices [10]. The discretized water jet segment can be considered as a water droplet so that a physical model similar to the models used for continuous streams of water droplets proposed in Refs. [17] and [18] can be used. In the z-direction in Fig. 1, the space can be divided into two regions: (a) region of length l with a strong electric field and corresponding gradient around the electrode and (b) a region of length L in which the DEP force caused by the electric field is weak and can be neglected. For the single electrode configuration, the DEP force drops to less than 10% of its peak magnitude, viewed along the axis of the jet, at a distance of approximately three times the diameter of the electrode in either direction from its center. In the region in which the electric field exists, a net DEP force, FDEP, acts in the traverse direction (x-direction) while gravity, FG, acts in the vertical direction (z-direction). However, the water jet speed is so large under high pressures that the change of jet velocity due to gravity is negligible. As a result, only a DEP force is exerted on the water jet in the electric field region and no other forces act on the jet. Since the direction of the water jet trajectory is determined by the DEP force direction, and the effect of gravity can be neglected for high-speed water jets, this technique can be applied to different water jet nozzle orientations. Consequently, the water jet maintains its velocity in both the x- and z-directions after passing through the electric field region. Therefore, based on the force analysis in both regions in Fig. 1, the governing equations of the water jet's motion in the traverse direction can be written as:
$mx¨=FDEP (0≤z
(5)
where m is the mass of each small water jet segment, L is the length of the region without the electric field, and l is the length of the electric field region. Under the assumption that the water jet velocity remains the same and equal to Vj in the z-direction, the total travel time from the nozzle to a fixed location (e.g., camera position) can be easily obtained. Therefore, by integrating Eq. (5) twice, the total deflection, D, of the jet can be expressed as
$D=1mVj2(∬0→lFDEPdz2+L∫ll+LFDEPdz)$
(6)
Substituting Eqs. (1) and (4) into Eq. (6) and taking only the x-component of the dielectrophoretic force, the total deflection can be rewritten as a function of the voltage, pressure, and $f(x,y,z)$ from Eq. (3) as
$D=αU24K2P(∬0→l∇x|f|2dz2+L∫ll+L∇x|f|2dz)$
(7)

From Eq. (7), it is obvious that jet deflection is mainly affected by voltage, pressure, and the electric field configuration. Deflection is a quadratic function of the voltage applied to the electrode and is inversely proportional to water pressure. As for the electric field configuration, for the single electrode setup, the only experimental parameter that affects the electric field configuration is the distance between the jet and the electrode. In addition, the deflection of the jet will decrease as the overall distance, l + L, between the nozzle exit and the working surface, i.e., the working distance, is reduced as shown in Eq. (7).

## Experimental Apparatus

The objective of the experiments is to systematically understand the water jet behavior in both static and dynamic nonuniform electric fields. Figure 2 schematically shows the experimental setup designed for the study in this paper. The high-pressure water is generated by a MAXPRO (Fairview, PA) customized air driven pump, which transfers air pressure to water pressure with a magnification ratio of 796:1. The maximal design pressure output for this air driven pump is 450 MPa. The high-pressure water output is connected to a AccuStream cutting assembly, which contains a cone-down diamond orifice with an inner dimeter of 60 μm from SUGINO Machine Limited (Uozu City, Japan) to form the high-pressure micro water jet needed in the experiments, via a stainless-steel pipe with an outer diameter of 6.35 mm. An air switch is included in the cutting assembly to switch on and off the high-pressure output by air pressure (operating air pressure threshold 0.41 MPa). The water jet that emanates from the diamond orifice is only 48 μm in diameter, which is smaller than the size of the orifice because of the vena contracta effect [26]. A copper wire with a diameter of 1 mm is mounted on a XYZ three axes manual stage with a 10 μm resolution to control the position of the electrode right below the nozzle assembly. The electrode is connected to an amplifier (Trek PZD700A M/S, Lockport, NY) that generates voltages from 0 to 1400 V with a slew rate of 370 V/μs and a 125 kHz bandwidth for large signals. The input signal to the amplifier is generated by a National Instruments data acquisition card (USB6221, NI) with 16-bit resolution. The Trek amplifier has an output to input ratio (DC voltage gain) of 200:1. A 240 W halogen lamp is installed to illuminate the water jet profile. A 20× objective (Mitutoyo Plan Apo SL infinity corrected objective, Sakado, Japan) is mounted on a high-speed camera (Photron FASTCAM Mini UX100, Tokyo, Japan) whose operating frame rate can be regulated up to 200,000 fps with a 1.3-megapixel (1280 × 1024 pixels) sensor to provide a 0.5 μm/pixel actual resolution for monitoring the water jet motion. To properly focus the camera system, it is mounted on a XY manual stage with a 0.5 μm resolution to obtain a sharp image within the 3.5 μm depth of focus of the 20× objective.

To achieve ideal machining performance, the working distance must be shorter than the water jet breakup length. The breakup length is defined as the distance from the nozzle exit to the point where water jet becomes unstable and starts breaking up. The breakup length in the current system, within the range of pressures used, is slightly higher than 27 mm. In processes like water jet incremental forming and laser guided water jet manufacturing, the actual working distance should be maintained at any value within the breakup length [8,27]. Therefore, the motion of the water jet is filmed at a fixed position, i.e., H = 26 mm below the center of the electrode where the water jet is about to break up at which point the jet's deflection is maximal. The high-speed camera videos are processed by a LabVIEW program using the NIVision module to acquire the water jet deflection frame-by-frame. The experimental parameter d is defined as the distance between the center line of the water jet and the electrode surface as shown in Fig. 2. Figure 3(a) is a photo of the entire actual setup with most of the equipment mentioned above shown. Figure 3(b) is the photo of the enlarged area under the nozzle assembly.

## Water Jet Deflection in a Static Electric Field

Before studying the water jet motion in dynamic nonuniform electric fields, a parametric study is essential to investigate the relationships between the control parameters and water jet deflection in a static electric field. The objective of this particular section is to propose an equation based on the experimental results that express the water jet deflection as a function of control parameters. An analysis, based on the models formulated in Sec. 2, of the experimental configuration in Fig. 2, indicates that the voltage applied to the electrode, U (V), the output water pressure from the air driven pump, P (MPa), and the distance, d (μm), between the electrode and the water jet, as defined in Sec. 3, exert the most significant influence on the deflection of the jet. Therefore, a simple design of experiments (DOE) is implemented with the values used for each of the three parameters listed in Table 1. The reasons for the choice of these parameters in the regions shown in Table 1 are the limitations imposed by the amplifier, resolution of the high-speed camera system, and the variation of the water jet position. There are two major reasons for water jet variations/instability in the current experimental setup—fluctuations of the output water pressure caused by the air driven pump and orifice geometry imperfections. From the experiments performed without turning on the amplifier to create an electric field, the variation of the water jet position was measured to be up to 1.5 μm. The variation of the water jet itself restricts the minimum deflection, which can be accurately measured to around 1–1.5 μm. Since the maximal voltage that can be applied to the electrode is 1400 V, the upper limit of the distance d and water pressure P are set to 400 μm and 51.72 MPa to make sure that the water jet deflections for all the scenarios listed in Table 1 are larger than 1.5 μm.

A total of 64 sets of experiments were conducted to experimentally study the influence of each parameter. The flow rate of the high pressure system is directly measured to obtain the true velocity of the water jet Vj for fixed values of the water pressure P. The discharge coefficient in Eq. (4) is then determined to be 0.97 for the diamond orifice used in all experiments. The value of the discharge coefficient was obtained by substituting the measured actual flow rate at the orifice and the corresponding water jet pressure indicated on the gauge into Eq. (4). The minimum water jet velocity of 179.4 m/s for the lowest pressure used in this study (17.24 MPa) can then be obtained from Eq. (4). Considering that the discharge coefficient does not change for each specific orifice, the water jet velocities for other pressures can be calculated the same way by substituting the water jet pressure P into the Eq. (4).

Due to the variation of the water jet position in the traverse direction, several measurements were taken to quantify the uncertainty caused by experimental error, pressure variations caused by the air driven pump, and the inherent unstable characteristics of the fine water jet under high pressure. A series of photos, shown in Fig. 4, were taken at the same pressure of 17.24 MPa and electrode position of 300 μm. The dark lines are the boundaries of the water jet. The difference in the gray level of these two lines is the result of different light intensities reflected by the water jet at the two boundaries due to the relative position of the jet to the light source. An evident change of the water jet deflection can be noticed from left to right in the second row as the voltage increases.

To acquire the static deflection of the jet for different values of the parameters, a series of images were taken by the camera to monitor the jet's stable positions before and after the electric field was applied. After having each set of images for each parameter setting, the deflection measurements were processed by the previously-mentioned LabVIEW program, which basically tracks the location of the edge of the jet one image at a time. The final deflection is the difference between the stable jet positions with and without the electric field. The same experimental procedure was repeated ten times to calculate the mean and standard deviation of the deflection for every single parameter set. Figure 5 presents the mean values of the deflection with respect to all three experimental parameters. Since the design of experiments has three individual experimental parameters, three sets of plots are needed to clearly illustrate the results for each individual experimental parameter (as shown in Fig. 5). The error bars in Fig. 5 indicate the standard deviation of ten deflection measurements.

As expressed by Eq. (7), water jet deflection should be inversely proportional to water pressure and have a quadratic relationship with the voltage on the electrode. The experimental results shown in Figs. 5(b) and 5(c) support the conclusions related to the influences of the pressure and voltage, respectively. The relationship between water jet deflection and distance can be empirically formulated from the experimental results shown in Fig. 5(a). It is obvious that even for different values of the voltage, data points in the four plots in Fig. 5(a) exhibit the same trend for different pressures. An inverse relationship was found to be the most accurate fitting model for the experimental data shown in Fig. 5 between water jet deflection and the distance between the water jet and the electrode. The three subplots in Fig. 5 facilitate the formulation of an empirical equation between the jet's deflection and the three key parameters identified. The following empirical relation for jet deflection can be assumed:
$D=CU2dP$
(8)

where the constant C depends on the electric field, dipole moment, and the discharge coefficient and can be obtained by the least square method based on the experimental data.

To nondimensionalize both sides of Eq. (8), the jet deflection is normalized by the diameter of the electrode $Φe$ while the right side of Eq. (8) is normalized by the permittivity of the air $ε1$ and the radius of the water jet R. Consequently, the final empirical function for water jet deflection for the current particular single electrode configuration is
$DΦe=Cε1U2RdP$
(9)

where $Φe$, R, and $ε1$ are also constants introduced only for nondimensionalization purposes. Linear regression can now be applied to Eq. (9) to find the value of the constant C. The fitted line is shown in Fig. 6 with a constant C = 90 and a R2 value 0.9968. This analysis confirms that the relationships between water jet deflection and voltage and water pressure are quadratic and inverse, respectively. Equation (9) with the fitted coefficient C (dashed line in Fig. 6) is a suitable empirical equation to predict jet deflection as a function of the three given experimental parameters.

In conclusion, a useful and simple empirical equation for water jet deflection with a single electrode is obtained to further assist in the establishment of a control strategy for water jet trajectory control in the future. This equation includes all the relationships between the three basic process parameters and water jet deflection and is verified by experimental data. From another perspective, the study of the static response of the water jet motion within a nonuniform electric field is still not quite sufficient for real-time closed-loop control of the entire system. Research on the dynamic response of the water jet deflection is necessary. The objective of Sec. 5 is to explore the dynamic properties of the water jet system and the limitations of the water jet motion in the high-frequency domain.

## Dynamic Response of Water Jets

With the final goal to precisely control the position of water jet impingement, a real-time closed-loop control system is essential. After understanding the behavior of the water jet motion in a static electric field, the next step is to understand the dynamic properties of this unknown system. Based on the dynamic input and output relationship of the system, information like system type and corresponding parameters can be identified. The analysis of the dynamic response also helps in determining how fast the water jet motion be controlled in the future.

To understand the dynamics of the jet's responses, instead of just measuring the final stable water jet deflections, the high-speed camera videos were processed by the LabVIEW program to track the water jet position as a function of time. The maximal frame rate of the high-speed camera is 200,000 fps. However, limited by the internal storage space in the camera and the light intensity of the external light source, the frame rate in the current experiments was set to 20,000 fps for low frequency studies and 50,000 fps for high frequencies to record videos with a clear water jet profile, high enough resolution, and sufficient time duration. Like in typical dynamic response tests, several typical types of waveforms were utilized. Four types of waveforms, i.e., square, sinusoidal, triangular, and sawtooth signals, with different frequencies were applied to the electrode to observe the jet's responses. To successfully capture the entire water jet motion at the rising and falling side of the square signal, a duty cycle that guarantees that the water jet has enough time to respond to a sudden change of the voltage is desirable. As a result, a duty cycle of 50% was selected for the square signals to investigate the jet's behavior. A sampling frequency of 500 kHz was used by the National Instruments (Austin, TX) data acquisition card, which is much higher than the frame rate of the high-speed camera, to generate the input signal for the Trek amplifier. The Trek amplifier's slew rate of 380 V/μs ensures that the voltage output changes from zero to its maximal value within 5 μs. The transient response time of the DEP force for water droplets in air is governed by the charge relaxation time, which is around 1 μs [18]. Therefore, it is reasonable to believe that the electric field induced DEP force is instantaneously exerted on the water jet.

To characterize the dynamics of an unknown system, its frequency response is commonly obtained as a quantitative measurement of the output spectrum of the system's response to the input signals. Test signals with a wide frequency spectrum are usually used as input signals to the system to observe any phase or magnitude change and possible time delay in the responses. In the current experiments, a frequency range from 0 to 2000 Hz was selected.

Starting with a low frequency of 10 Hz and a peak-to-peak amplitude of Upp = 1000 V, Fig. 7 illustrates the water jet deflection by dashed lines for the four types of waveform inputs selected (solid lines). The only changing parameter is voltage. The frame rate was set to 20,000 fps to record low the frequency response of the water jet deflection. The distance between the electrode and the water jet was fixed at 300 μm and the pressure at 17.24 MPa. Since the voltage on the electrode is too high to be directly measured, the input signals in Fig. 7 are the measured signals from the National Instruments data acquisition card multiplied by the DC voltage gain of the Trek amplifier. There is no evident time delay observable with such low frequency input signals. Different waveforms do not have any influence on the maximal deflection of the water jet when the peak-to-peak amplitude of the input signals is set to a fixed value. The small variations shown in Fig. 7 are mainly because of the variations of the water jet which are, as mentioned in Sec. 4, approximately 1∼ 1.5 μm.

Considering that the water jet deflection has a quadratic relationship with the voltage applied to the electrode, a simple model can be formed to fit the curves in Fig. 7. When d and P are fixed and only the voltage, U, is changing, Eq. (9) can be simplified to:
$D=CΦeε1RdPU2=CvU2$
(10)

where $Cv$ is a constant. For the current experimental setup, given that parameters d and P in Eq. (9) are fixed to 300 μm and 17.24 MPa, respectively, in all three dynamic experiments described in this section, $Cv$ can be calculated to be 6 × 10−12 m/V2.

The black dotted line in Fig. 8 represents the predictions by Eq. (10). The root-mean-square error (RMSE) value is commonly used to quantify the difference between the model predictions and real measurement data. The calculated RMSE values were normalized by the total water jet deflection to ensure that the criteria are comparable between different conditions. Table 2 lists all the normalized RMSE values for each waveform. The small RMSE values in Table 2 and the fitted model (dotted lines) in Fig. 8 indicate that Eq. (10), as a simple mathematical model, is good enough for representing the water jet dynamic response at low frequencies.

Identical experiments were repeated at higher frequencies as well, i.e., 100 Hz and 500 Hz. Figure 9 shows the experimental results for 100 Hz using the same prediction model, i.e., Eq. (10). All four subplots contain four consecutive periods. The value of the constant $Cv$ is the same as the value for the experiments with the 10 Hz frequency. In Fig. 9, the vertical shift of the water jet deflection (0.5–1 μm) between the different periods is normal and reasonable due to the water jet variation. The values of the RMSE for different waveforms are calculated over these four periods in Table 2. Despite the variation of the water jet, the motion of the jet still follows the prediction model as expected. Compared to the results for 10 Hz, a very short time delay can be found for the square signal when the electric field is suddenly turned on. The static model is still applicable for jet motions at low frequencies such as 10 Hz and 100 Hz.

To investigate the time delay in the higher frequency range, the deflection of the water jet to 500 Hz input signals is illustrated in Fig. 10. For the frequency of 500 Hz, the time delay is even more obvious for the square signal. The RMSE value for the square signal also reflects that the measurement deviates from the model predictions. Therefore, for high frequencies, the static relationship between the voltage and water jet deflection needs to be improved with dynamic modeling. The delay of the water jet motion is especially apparent when a sudden change in the voltage occurs. For continuously changing waveforms like a sinusoid and triangle, the delay of the motion is not quite as severe. The RMSE is still quite small for the sinusoid and the triangular signals at 500 Hz.

From Fig. 10(b), it can be seen that the response of the water jet to square signals is very similar to the response of a typical first-order system. However, such kind of response only happens at the rising edge, not the falling edge. The reason for this difference in behavior on the rising and falling edges of the signals is that the water emanating from the nozzle always remains in its original position. When the electric field is suddenly turned off, the pressurized water coming directly from the nozzle replaces the deflected water jet. Considering that the jet velocity is > 180 m/s and that the effective length of the electric field is around 3–4 mm, the deflected water jet will be replaced by a new water jet in its original position in less than 0.02 ms, which is the time for only one frame. The behavior on the rising edge of the water jet deflection is a different situation as compared to the falling edge. When an electric signal is suddenly applied to the electrode, the effects of the complex two-phase flow condition at high-speed, the air drag involved in the system, and the changing fluid conditions around the electrode can cause a delay in the electric field to reach a stable condition. The delay of the electric field then further causes a delay behavior of the rising edge of the water jet deflection.

For a typical first-order system, only one parameter, the time constant $τ$, is needed to characterize the system. By definition, the total time for the system to reach 99.5% of its total response is 5 times the time constant [28]. To better understand the time delay phenomenon for the square signal, the frame rate of the high-speed camera was increased to 50,000 fps, which is the highest frame rate at which the water jet profile could be clearly recorded under the lighting conditions used in the experiments. The light intensity was not enough to let the water jet reflects enough light back into high speed camera with the 20× objective at frame rates higher than 50,000 fps. Several parameters, like the peak-to-peak voltage, frequency, and water pressure, were changed to observe any influence on the delay. Water pressure was found to have a significant influence on the delay. To have an in-depth understanding of the influence of water jet pressure on the delay, five different water pressure settings in a range from 17.24 to 137.9 MPa were chosen. Water jet deflection becomes extremely small when water pressure is set to 137.9 MPa. To ensure that the variation of the water jet position does not jeopardize the accuracy of the water jet deflection measurement, the electrode was moved to the closest fixed location from the water jet (d = 100 μm) to have the largest water jet deflection. Based on the total delay time at the rising edge for each pressure, the time constant was obtained and plotted with respect to water pressure in Fig. 11(a). Figures 11(b)11(f) shows all the responses of the water jet motion for the five different water pressures for the same input square signal at 500 Hz.

Several types of fitting models were tried to fit the data in Fig. 11(a). The following simple model was found to fit the data:
$τ=Ct1P$
(11)

where $Ct$ is an unknown constant and P is the water jet pressure. By using the method of least squares, $Ct$ was obtained to be 0.2126 with a R2 value to be 0.9889. The response of the assumed first-order system was simulated in the matlab Simulink module. The time constant was calculated from the fitted model in Fig. 11(a) for each water pressure. Table 3 includes all the normalized RMSE values for different pressures. The prediction capability of the new model is much better than that of the static model in Fig. 10(b). However, a small discrepancy is observable between the simulated first-order response and measured water jet deflection. This is principally caused by the imaging method used. In each frame, the water continues moving fast during the camera's exposure time that leads to blurry water jet boundaries in the images taken. The observed measurement error is induced by this blurry boundary of the water jet in the LabVIEW program because there is no longer possible to locate an exact water jet position just by tracking the boundary. The system may also not be a perfect first-order system and contain other components with higher-order or even nonlinear parts. However, the response is still first-order dominant. Therefore, a first-order system can be used as an approximate model for the high-pressure micro water jet system under high frequencies. It can be seen that the same first-order system is also valid for low frequencies input signals, which makes it suitable for modeling the dynamic response of the water jet deflection in changing electric fields. Overall, the proposed method is good enough for controlling the water jet trajectory for manufacturing purposes.

To further test the water jet dynamic response at even higher frequencies, experiments were repeated with 1 kHz and 2 kHz square signals at 17.24 MPa water jet pressure and a distance between the electrode and water jet equal to 300 μm. From Sec. 4, the maximal water jet deflection should be able to reach 6 μm when the water jet is finally stabilized in the electric filed. From the previous dynamic analysis of the water jet motion, it is also known that the water jet needs at least 0.27 ms to reach its final position at a 17.24 MPa water jet pressure (Fig. 11(b)). Therefore, the water jet should still be able to reach its final position with a 1 kHz square signal because the time of the half period of the input signal (5 ms) is long enough for the water jet to stabilize in the electric field. However, since the total rise time (0.27 ms) is larger than the time of the half period of the 2 kHz square signal (0.25 ms), the water jet should fail to reach its final position. Figure 12 plots the experimental results that verify this analysis. The signal at 2 kHz, however, still provides enough time for the water jet to reach more than 90% of its total reachable deflection as shown in Fig. 12. It can be expected that the total reachable deflection will be smaller when higher frequency square signals are used. In conclusion, the maximal frequency of the square signals, which can be applied to the electrode to let the water jet reach its ideal maximal deflection that is the same as the static response under the same circumstance, is mainly limited by the time delay at the rising edge.

In conclusion, the control of the dynamic deflection of the water jet by dielectrophoresis adds a secondary mechanism for controlling the motion to the jet and its point of impingement, superimposed onto the motion of the machine's motion stages. However, these motions may take place at one or two orders of magnitude higher frequencies. In future applications, this secondary high frequency manipulation of the jet's motion may support micro pattern generation on parts and surfaces at very high rates and considered as an alternative for laser texturing [29] and vibration-assisted cutting processes [30].

## Conclusion and Discussion

In this paper, a high-pressure micro water jet is manipulated by dielectrophoresis in one dimension with both static and dynamic nonuniform electric fields. By simplifying the physical model of the water jet's deflection in nonuniform electric fields, its deflection was expressed as a function of several important parameters. The main significant results from the static electric field experimental study can be summarized as follows:

• The water jet deflection in a single electrode configuration was found to be inversely proportional to the distance between the water jet and electrode.

• Both theoretical derivation and experimental results support the fact that deflection has a quadratic relationship with voltage on the electrode while water pressure has an inverse linear relationship.

• It is the first time that experiments on controlling water jet trajectory by dielectrophoresis at such high pressure (17.24–137.90 MPa) were conducted.

• An empirical equation for water jet deflection was presented to support the parametric study results.

The static experimental study of the water jet deflection in nonuniform electric fields offers the possibility of controlling the high-pressure micro water impingement for increasing the accuracy of micro water jet manufacturing processes. To form a real-time closed-loop control system in the future, the dynamic response of the water jet motion in changing electric fields was also studied. The important findings can be listed as follows:

• For low frequency signals, the water jet deflection basically follows the static model. The static model is sufficient to predict the water jet motion in response to different types of signals.

• The static model is not suitable for describing jet motion at high frequencies (> 100 Hz).

• According to the experimental results, a simple first-order system can be used as a model to predict the jet's motion.

• Water pressure has the largest impact on the time constant of the system. A possible relationship between time constant and water pressure is proposed.

It has been shown that a first-order system response model is sufficiently accurate to represent water jet motion in the 0–2000 Hz frequency range. This provides a starting point for the implementation of a real-time closed-loop control system for manipulating the jet's trajectory.

Future work needs to focus on implementing a jet control method in an actual manufacturing processes. Water jet deflection induced by nonuniform electric fields can be adopted as an additional motion control mechanism integrated into traditional water jet processes. The accuracy of the water jet deflection can be improved by building a closed-looped control mechanism along with stabilizing the water jet trajectory by the DEP forces. In addition to that, a more in-depth analytical analysis and modeling needs to be performed to explain the time delay phenomena associated with high frequency signal inputs.

## Funding Data

• Division of Civil, Mechanical and Manufacturing Innovation, the National Science Foundation (NSF) Grant CMMI No. 1234491.

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