A Method for Calculating the Critical Velocity of Microdroplets Produced by Circular Nozzles

Abstract

Piezoelectric print-heads (PPHs) are used with a variety of fluid materials with specific functions; thus, matching the drive waveform of the PPH and the physical characteristics of the fluid can greatly improve deposition accuracy and quality. In this study, a distribution equation for the critical point for producing droplets defined in a dimensionless plane composed of Weber and Reynolds numbers was proposed. Computational fluid dynamics (CFD) was used to calculate the critical point distribution of the droplets generated by various materials in the dimensionless plane. Simulation results showed that the proposed distribution equation could describe the varying laws of the critical points of different materials. Process parameters in the distribution equation were identified using a fitting method based on CFD simulation results. The formula for calculating the critical characteristic velocity and dimensionless coordinates of the generated droplets was proposed based on the identified process parameters. The critical characteristic velocity and dimensionless coordinates of water, ethanol, aniline, and glycol were calculated using the proposed method. The average relative error between the calculated results and those from CFD is less than 5%. The drive waveforms of PPH are designed using the critical characteristic velocity of four different materials. Their droplet generation processes were recorded through a droplet watch system. The experimental results indicate that the flight velocity of the droplets generated at the critical characteristic velocity is close to zero, which indicates that the method proposed in this study is accurate.

Introduction

A piezoelectric print-head (PPH) is a manufacturing device that deposits microdroplets, of less than 17 pL when required,¹ onto a surface. With the advantages of high deposition precision and good control of the droplets' size and speed, the PPH has become an effective device for accurate material deposition.² Therefore, PPHs have been widely used in 3D printers as a high-precision material deposition component; for example, Meng et al.³ and Zhang et al.⁴ used PPH to print nanometer-sized silver suspensions to manufacture conformal curved antennas. Kyobula et al.⁵ used PPH to manufacture a drug with a unique 3D structure. Nanodimension has developed a 3D printer (DragonFly LDM™ System) to manufacture printed circuit boards.⁶ The plastic 3D printers of 3D Systems (MJP 2500) also used the droplet deposition technology.⁷

In addition, PPH has been used to manufacture displays,^8,9 electronic devices,¹⁰ and sensors.^11,12 As the fields of application increase, more and more materials with special functions have been developed, and the applicability of PPH to multiple materials has become an urgent problem to solve. PPH can produce higher quality droplets that can further improve the accuracy of 3D printing.

For PPHs, droplets are formed by propagating a pressure pulse in a fluid held in a chamber behind a nozzle.¹³ Free surfaces at nozzles respond to this pressure pulse and produce a velocity which, when averaged over time, is called characteristic velocity. This characteristic velocity allows the fluid at the nozzle to overcome surface tension and viscous forces and form droplets. However, satellite droplets are produced when the characteristic velocity is extremely high; conversely, they cannot be produced when the characteristic velocity is extremely low.

Normally, when this characteristic velocity is sufficiently high to produce droplets, the fluid splits and forms droplets at the nozzle with zero flight velocity, the characteristic velocity is called critical characteristic velocity; therefore, no satellite droplets are produced because the tail does not elongate and produce new splits, even in water with high surface tension and low viscosity. Thus, the critical characteristic velocity required for producing droplets is an important reference for high-quality droplet production.

Many existing studies focused on estimating a threshold based on a dimensionless number to analyze the forming quality of microdroplets; we primarily referred to a review article.¹⁴ The earliest significant work attempting to understand the mechanisms of drop generation was conducted by Fromm.¹⁵ He used the Ohnesorge number, Oh, to characterize drop formation. The parameter Z = 1/Oh for Z > 2 is used for stable droplet production. This study was further refined by Reis and Derby¹⁶; they used computational fluid dynamics (CFD) to simulate the droplet generation process and obtained a new condition, 10 > Z > 1, for stable droplet formation. However, these threshold conditions are merely empirical summaries of the results. An accurate calculation of the critical point is still a challenge.

Duineveld et al.¹⁷ provided a formula for calculating the minimum speed required to form droplets without considering the viscous force. This condition can be rewritten as We = 4, where We is the Weber number. For materials with a negligible viscosity, this formula provides accurate results; however, in most new applications, the viscosity of the fluid material cannot be ignored, and the calculation error in Duineveld's formula is unacceptable. According to the characteristics of droplet formation, Poozesh et al.¹⁸ deduced a dimensionless N–S (Navier–Stokes) equation describing droplet formation and indicated that the quality of forming droplets was determined by the Weber and Reynolds numbers. They developed a coordinate plane where the Weber number denoted the horizontal axis, and the Reynolds number denoted the vertical axis. This plane was divided into three parts, as follows: single, double, and multiple droplets. The dimensionless point distribution of any material for different characteristic velocities can be plotted in this plane; furthermore, the characteristic velocity corresponding to the dimensionless points distributed in the single droplet part can be obtained. However, this process is extremely tedious, and the characteristic velocity is obtained manually from the graph; furthermore, the final results contain a few manual measurement errors.

To determine the critical conditions more succinctly and rapidly according to the physical characteristics of fluid materials and nozzle size, we deduce the dimensionless N–S equation; the results show that the quality of forming the droplets can be determined using any two of the Weber, Reynolds, and Capillary numbers. The Reynolds and Weber numbers were selected to characterize the droplets formed. According to the dimensionless N–S equation, the flow of droplets is similar to flow processes with the same dimensionless number.

As the dimensionless quantity is the ratio of a real physical quantity to a characteristic physical quantity, the dimensionless process is performed to replace the real physical quantity with the product of characteristic physical and dimensionless quantities. Characteristic physical quantities are lumped together to obtain the corresponding dimensionless number, and dimensionless quantities are lumped together to obtain process parameters. Therefore, the parameters in the dimensionless N–S equation can be divided into dimensionless numbers and process parameters. The dimensionless numbers set the flow conditions, such as the characteristic size and physical properties of materials and velocity; the process parameters set the flow pattern. For PPHs, any fluid material forming droplets through extrusion, pull, and fracture involves three steps; under critical conditions, the three steps for different fluid materials have very high similarities. Therefore, the process parameters representing these three steps can be considered constants under the critical conditions. The dimensionless N–S equation now becomes an algebraic equation with the Weber and Reynolds numbers as variables. This new equation can describe the dimensionless coordinate distribution rules of similar flow processes. To assign actual values to process parameters, we obtained a set of critical dimensionless points that produced droplets based on CFD. These critical dimensionless points are used to estimate the process parameters in the new equation. Finally, we derive the formula for calculating the point and dimensionless coordinates of droplet generation for any material and nozzle size.

To verify the calculation accuracy of the proposed formula, we used CFD to obtain the critical dimensionless coordinates and characteristic velocity of water, ethanol, aniline, and glycol for varying nozzle diameters. Meanwhile, we also calculated the critical dimensionless coordinates and characteristic velocities of the four materials using the proposed method; compared with CFD results, the average relative error of the former is less than 5%. The drive waveforms of the PPH with 80 μm nozzle diameters were designed according to the critical characteristic velocity of the four materials, and their droplet production processes were recorded by a droplet watch system; a detailed description of the droplet watch system can be found in Refs.^19,20 Experimental results indicate that the proposed method exhibits good computational accuracy.

This article proposes a method for calculating the critical characteristic velocity of the droplet generation of arbitrary materials. In the Mathematical Model section, the dimensionless N–S equation is derived, and the formulas for calculating the critical characteristic velocity and dimensionless coordinates of droplet generation are provided. In the Results and Discussion section, the critical dimensionless coordinates and characteristic velocities of four materials are calculated using CFD and the proposed method; the results indicate that the latter exhibits a good calculation accuracy. To further verify the accuracy, the drive waveforms of PPH were set according to the calculated characteristic velocity, and the droplet generation processes of four materials at critical characteristic velocity were recorded using the droplet watch system. The experimental results indicate that the method proposed in this study is accurate. The study is summarized in the Conclusions section.

Mathematical Model

Critical characteristic velocity

Droplet generation is a classic gas–liquid two-phase flow process that can be described using N–S and continuity equations. As the droplet generation process is conducted on a micron scale and only relates to the free surface at the nozzle, the process can be assumed to be incompressible and gravity negligible. Therefore, the continuity and N–S equations can be written as follows²¹:

$$\left\{\begin{array}{c}\frac{\partial u_i}{\partial x_i}=0\\ \frac{\partial u_i}{\partial t}+u_j\frac{\partial u_i}{\partial x_j}=-\frac{1}{\rho }\frac{\partial p}{\partial x_i}+\frac{\mu }{\rho }\frac{{\partial }^2{u}_i}{\partial x_j\partial x_j}-\sigma \frac{1}{\rho }\frac{\partial }{\partial x_i}\left(\frac{1}{R1}+\frac{1}{R2}\right)\\ \end{array}\right.$$ (1)

where i = 1, 2, 3 and j = 1, 2, 3 are tensors subindices, u_i is velocity tensor, x_i or x_j is the coordinate tensor, p is pressure, ρ is density, μ is viscosity, $\sigma$ is surface tension coefficient, and R1 and R2 are a set of orthogonal curvature radii of any point on the free surface. The surface tension term exists only at the free surface. To obtain the dimensionless N–S equation, the dimensionless quantity is expressed as the ratio of a real physical quantity to characteristic physical quantity.¹⁸

$$\begin{array}{cc}& x^0=\frac{x}{L_0},{\mu }^0=\frac{\mu }{{\mu }_0},u^0=\frac{u}{V_0},t^0=\frac{t{V}_0}{L_0},{\rho }^0=\frac{\rho }{{\rho }_0},p^0=\frac{p}{{\rho }_0{V}_0^2},\\ & {\sigma }^0=\frac{\sigma }{{\sigma }_0},R{1}^0=\frac{R1}{L_0},R{2}^0=\frac{R2}{L_0}\end{array}$$

where the superscripts and subscripts represent the dimensionless and characteristic quantities, respectively. L₀ is the nozzle diameter, and V₀ is the characteristic velocity; for droplet formation, the characteristic velocity is the average velocity at the nozzle during the extrusion process. By replacing the real physical quantity in the N–S equation with the product of the dimensionless and characteristic physical quantities, Equation (1) can be rewritten as follows:

$$\begin{array}{cc}& \frac{V_0^2}{L_0}\left(\frac{\partial u_i^0}{\partial t^0}+u_j^0\frac{\partial u_i^0}{\partial x_j^0}+\frac{1}{{\rho }^0}\frac{\partial p_i^0}{\partial x_i^0}\right)\\ & =\frac{{\sigma }_0}{L_0^2{\rho }_0}\left[{\sigma }^0\frac{1}{{\rho }^0}\frac{\partial }{\partial x_i^0}\left(\frac{1}{R{1}^0}+\frac{1}{R{2}^0}\right)\right]+\frac{V_0{\mu }_0}{L_0^2{\rho }_0}\left(\frac{{\mu }^0}{{\rho }^0}\frac{{\partial }^2{u}_i^0}{\partial x_j^0\partial x_j^0}\right)\end{array}$$ (2)

In Equation (2), the left-hand term represents the inertial force per unit volume, the first term on the right represents the surface tension per unit volume, and the second term represents the viscous force per unit volume. Therefore, the formation process of the droplets is determined by inertial force, surface tension, and viscous force. To obtain the dimensionless number associated with the droplet forming process, we multiply the coefficient $L_0∕V_0^2$ by Equation (2) to obtain the Weber number, $We=L_0{V}_0^2{\rho }_0∕{\sigma }_0$, and the Reynolds number, $Re=L_0{V}_0{\rho }_0∕{\mu }_0$. Similarly, we multiply the coefficient $L_0^2{\rho }_0∕{\sigma }_0$ by Equation (2) to obtain the Weber and capillary numbers, $Ca={\mu }_0{V}_0∕{\sigma }_0$, and we multiply the coefficient $L_0^2{\rho }_0∕V_0{\mu }_0$ by Equation (2) to obtain the Reynolds and capillary numbers. This result indicates that any two of the Weber, Reynolds, and capillary numbers can describe the process to generate microdroplets. In this study, the Reynolds and Weber numbers are selected to describe the process of generating microdroplets. Therefore, Equation (2) can be rewritten as follows:

$$A=\frac{1}{We}B+\frac{1}{Re}C$$ (3)

$$\begin{array}{cc}& A=\left(\frac{\partial u_i^0}{\partial t^0}+u_j^0\frac{\partial u_i^0}{\partial x_j^0}+\frac{1}{{\rho }^0}\frac{\partial p_i^0}{\partial x_i^0}\right),B={\sigma }^0\frac{1}{{\rho }^0}\frac{\partial }{\partial x_i^0}\left(\frac{1}{R{1}^0}+\frac{1}{R{2}^0}\right),\\ & C=\frac{{\mu }^0}{{\rho }^0}\frac{{\partial }^2{u}_i^0}{\partial x_j^0\partial x_j^0}\end{array}$$

where A, B, and C are called process parameters and describe the fluid flow patterns at the nozzle during droplet formation. As the droplets are formed by inertial forces overcoming the surface tension and viscous forces, the fluid at the nozzle experiences extrusion, stretching, breaking, and thus forms a droplet. Through CFD and experimental analysis, we found that the process is very similar for any material under critical conditions. Therefore, the process parameters A, B, and C can be considered constants. These constants only mean that the flow pattern is constant, and the values of the constants have no real meaning. For further analysis, Equation (3) can be rewritten as follows:

$$Re=\frac{1}{\frac{A}{C}-\frac{B}{CWe}}$$ (4)

When the values of A/C and B/C are determined, Equation (4) will also be determined. Therefore, process parameters A, B, and C may have multiple sets of constants, as long as the ratio between them remains constant. The Reynolds number is a positive real number, Re ≥ 0; therefore, the denominator in Equation (4) must be greater than zero. Thus, the Weber number has to satisfy the inequality We ≥ B/A. As the Weber number approaches B/A, the Reynolds number approaches infinity. The Reynolds number is the ratio of inertial force to viscous force; therefore, when the viscous force is considerably smaller than the inertial force, the Reynolds number approaches infinity.

The formation process of droplets is determined by the surface tension and inertial force since the viscous force is much smaller than the surface tension and inertial forces. Meanwhile, as the Weber number approaches infinity, the Reynolds number approaches C/A. The Weber number is the ratio of inertial force to surface tension; therefore, when the surface tension is considerably smaller than the inertial force, the Weber number approaches infinity. The formation process of droplets is determined by the viscous and inertial forces and the surface tension is much smaller than the viscous and inertial forces. Therefore, the dimensionless threshold of droplet generation in the two extreme cases can be written as follows:

$$We\ge \frac{B}{A}$$ (5)

$$Re\ge \frac{C}{A}$$ (6)

The critical ejection dimensionless boundary distribution law is described by Equation (4) for fluids with similar flow patterns. By determining the process parameters A, B, and C, a critical ejection dimensionless boundary distribution curve with a similar droplet generation process in the dimensionless plane can be obtained. The asymptotes of this curve are provided by Equations (5) and (6). To understand the influence mechanism of material parameters and nozzle size on the droplet generation process, the characteristic physical quantity is included in Equation (4), and is rewritten as follows:

$$A=\frac{1}{\psi V_0^2}\left({\sigma }_{0}B+V_0{\mu }_{0}C\right)$$ (7)

where ψ = ρ₀L₀. For the droplet forming process with critical conditions, process parameters A, B, and C are constants. According to Equation (7), the characteristic velocity increases with the surface tension coefficient to maintain process parameter A constant; furthermore, the effect of viscous force also increases with characteristic velocity. Similarly, increasing the viscosity of the material will increase the corresponding characteristic velocity. The nozzle diameter and material density have the same influence mechanism; when the density or nozzle diameter increases, the corresponding characteristic velocity will decrease, and when this happens, the corresponding characteristic velocity will increase.

Equation (7) can be viewed as a quadratic function with characteristic velocity V₀ as the independent variable. Therefore, the formula for calculating the critical characteristic velocity that can produce droplets can be written as follows:

$$V_c=\frac{{\mu }_{0}C+\sqrt{{\left({\mu }_{0}C\right)}^2+4A\psi {\sigma }_{0}B}}{2A\psi }$$ (8)

where V_c is the critical characteristic velocity. Meanwhile, the relationship between the Weber number and the Reynolds number can be written as follows:

$$We=\frac{{\mu }_0^2}{{\sigma }_0{\rho }_0{L}_0}{Re}^2$$ (9)

Combined with Equations (9) and (4), the formula for calculating the critical Weber number that produces droplets can be written as follows:

$$W{e}_c=\frac{B}{A}+\frac{{\left({\mu }_{0}C\right)}^2+{\mu }_{0}C\sqrt{4L_0{\rho }_0{\sigma }_{0}AB+{\left({\mu }_{0}C\right)}^2}}{2L_0{\rho }_0{\sigma }_0{A}^2}$$ (10)

where We_c is the critical Weber number. The critical Reynolds number, Re_c, can be obtained by replacing We in Equation (9) with We_c.

Identification of process parameters

To assign a specific numerical value to the process parameters, we used a CFD tool (COMSOL) to calculate the critical dimensionless coordinates of a group of theoretical fluid materials. Figure 1 shows the CFD simulation model using a level set formulation of the free surface and incompressible N–S equations. As our study focused on the critical condition of droplet formation, we only simulated the droplet forming process close to the nozzle. The critical conditions for droplet formation indicate that inertial forces overcome the surface tension and viscous forces to form zero kinetic energy droplets, meaning that the spatial position of the droplet does not change after its formation (ignoring gravity). Therefore, we can determine whether the droplet is in the critical condition based on its flight velocity. As shown in Figure 1, when t = 95 μs, a droplet is generated close to the nozzle, and the position of the droplet does not change significantly later. At this time, we believe that the corresponding parameters of this process are parameters under critical conditions.

FIG. 1.

CFD simulation model. CFD, computational fluid dynamics.

The volume flow rate curve shown in Figure 2a was selected at the nozzle to generate microdroplets. The volume flow rate curve can be divided into three parts. The first forms an internal concave surface on the nozzle, the second extrudes the fluid and produces the liquid column, and the third breaks the liquid column and forms the droplets. As the droplet formation in the second part is the main driving force, the characteristic velocity of the corresponding material is considered the velocity averaged over time of the second part. Therefore, the amplitude of the volume flow rate curve can be determined according to the critical characteristic velocity of the materials as follows.

FIG. 2.

(a) Volume flow rate at the nozzle. (b) Fitting result of Equation (4).

The material parameters for a virtual material are listed in Table 1. These material parameters are designed such that the critical dimensionless points are more evenly distributed in the plane. All material parameters have the same density and surface tension coefficient; only the viscosity varies. The characteristic velocity can be obtained by finding the volume flow curve under the critical condition through multiple numerical calculations. For PPHs, the material with 1–5 cP viscosity listed in Table 1 has an extremely low viscosity, meaning that the viscous force is very low and that the conditions described in Equation (5) apply to these material parameters, which represent a class of materials in which the inertial force and surface tension play a predominant role in the droplet forming process. For material with a viscosity of 6.5–20 cP, as listed in Table 1, the viscous force and surface tension have similar effects and represent a class of materials in which inertial and viscous force and surface tension play a predominant role in the droplet forming process. The viscous and inertial forces of most fluid materials have similar effects in practical applications. The material with a viscosity of 30–60 cP in Table 1 has an extremely high viscosity, meaning that the effect of surface tension is very minimal and the conditions described in Equation (6) can be applied to these material parameters; therefore, these material parameters represent a class of materials in which the inertial and viscous forces play a predominant role in the droplet forming process. Therefore, the materials listed in Table 1 represent almost all fluid materials applied to a PPH.

Table 1.

Critical Dimensionless Coordinates for Different Fluid Parameters

Nozzle parameters	Fluid material parameters			Critical dimensionless coordinates calculated using CFD
Diameter, μm	Density, kg/m3	Surface tension, N/m	Viscosity, cP	Weber number	Reynolds number
80	1000	0.0725	1	4.8893	168.4
			1.25	4.865	134.38
			1.5	4.8928	112.31
			2	5.1261	86.214
			2.5	5.2505	69.803
			3	5.1721	57.733
			4	5.428	44.358
			5	5.3888	35.358
			6.5	6.1165	28.977
			10	6.9971	20.145
			13	8.144	16.718
			20	11.638	12.99
			30	16.86	10.424
			40	21.87	8.9023
			50	26.437	7.837
			60	32.086	7.1898
			80	43.066	6.2473

CFD, computational fluid dynamics.

Critical dimensionless coordinate points obtained using CFD are used as data points, and Equation (4) is used to fit these. Process parameters A, B, and C, are undetermined coefficients. In this study, a PPH with a nozzle diameter of 80 μm was used, and the process parameters after fitting were A = 1935, B = 8967, and C = 11,910 (of the several groups of process parameters, the group with the highest fitting accuracy is selected). The fitting results are shown in Figure 2b. The hollow circles are the critical dimensionless point obtained by CFD in Table 1. The blue line is the critical ejection dimensionless boundary distribution curve fitted by Equation (4). The fitting results show that Equation (4) can describe the critical droplet generation process, indicating that process parameters can be considered constants under critical conditions.

Results and Discussion

CFD results

Substituting the identified process parameters into Equations (5) and (6), and the critical condition for droplet generation when the viscous force is negligible, we obtain We ≥ 4.64. This conclusion is similar to that obtained by Duineveld et al.¹⁷ (We > 4). Similarly, when the surface tension is very low, the critical condition for droplet generation is Re ≥6.16. This condition is suitable for the injection of high-viscosity fluids; for example, a droplet generator squeezes high-viscosity fluids through a push rod.²² However, for most fluid materials, the viscosity and surface tension of the droplet generation process cannot be neglected. The generation of droplets is determined by inertial and viscous forces and surface tension. Material parameters and changes in nozzle size will lead to considerable variations in Re and We; therefore, it is difficult to analyze the microdroplet generation process based on a single dimensionless number.

In this case, Equations (9) and (10) provide the method for calculating critical dimensionless coordinates, and Equation (8) provides the calculation method for critical characteristic velocity. For any material and nozzle diameter, when the jet velocity is equal to the critical characteristic velocity, the material can form droplets at the nozzle with zero flight velocity. When the injection velocity is greater than the critical characteristic velocity, the material can form droplets at the nozzle, and the flight velocity of the droplet is determined by the difference between the jet and critical characteristic velocity. To verify the accuracy of the proposed method, we substituted the identified process parameters (A = 1935, B = 8967, C = 11,910) into Equations (8–10). Equation (8) calculates the critical characteristic velocity, while Equations (9) and (10) calculate the critical dimensionless coordinate points. The physical parameters of water, ethanol, aniline, and glycol for different nozzle sizes (80, 70, and 60 μm), thus calculated, are listed in Table 2.

Table 2.

Material Parameters

Material name	Density, kg/m3	Viscosity, cP	Surface tension, N/m
Water	1000.0	1.0	0.0725
Alcohol	789.0	1.074	0.02255
Aniline	1021.7	3.71	0.04483
Glycol	1115.5	19.153	0.04649

The critical dimensionless coordinates and characteristic velocities of the four fluids with different nozzle diameters were calculated using Equations (8–10) and CFD, respectively, and the results are summarized in Table 3. Comparing the computational results of CFD and Equations (9) and (10) shows that the proposed method exhibits good computational accuracy.

Table 3.

Theoretical and Computational Fluid Dynamics Results of the Four Materials for Different Nozzle Diameters

Nozzle diameter, μm	Material name	Equation (10), Wec	CFD, We	Equation (9), Rec	CFD, Re	Equation (8), Vc	CFD, Vc
80	Water	4.8066	4.5292	166.9679	162.08	2.087	2.02598
	Alcohol	5.0394	5.6828	78.8575	83.74	1.3418	1.42485
	Aniline	5.5752	5.6225	38.5253	38.688	1.7487	1.75607
	Glycol	11.2189	15.871	11.2641	13.398	2.4347	2.87545
70	Water	4.8201	4.3607	156.4027	150.96	2.2343	2.15657
	Alcohol	5.0701	4.9391	73.9882	75.538	1.4388	1.46891
	Aniline	5.6481	5.5929	36.2721	36.092	1.8816	1.87238
	Glycol	11.879	14.845	10.8425	12.12	2.6826	2.97292
60	Water	4.8368	4.2365	145.052	137.89	2.4175	2.29812
	Alcohol	5.1083	5.0443	68.7577	68.326	1.5599	1.5501
	Aniline	5.7398	5.3857	33.8530	32.792	2.048	1.98457
	Glycol	12.7413	13.188	10.3958	10.576	3.0024	2.97292

Experiment result

The practical implications of the proposed method are verified by calculating the results of the droplets from four materials, namely water, ethanol, aniline, and glycol using a PPH (MJ-AL-80) with a nozzle diameter of 80 μm; the schematic representation of the experimental system is shown in Figure 3. This system consists of a USB Camera (Microvision MV-VD040SM), an LED strobe, a PPH (MicroFab MJ-AL-80), a droplet watch system controller, pressure controller, and personal computer. The pressure controller maintains negative pressure inside the PPH to balance the gravity of the fluid and consists of a peristaltic pump, differential pressure sensor, and an embedded system (STMicroelectronics STM32F103). The embedded system controls the peristaltic pump to adjust the negative pressure inside the PPH.

FIG. 3.

Schematic of the experimental system.

The droplet watch system controller is used to generate the camera trigger signal, LED switch signal, and drive the waveform of the print head synchronously. It consists of an embedded system (STMicroelectronics STM32F103), a field-programmable gate array (FPGA) (Altera EP1C3T144C8N), high-speed digital-to-analog (DA) converters (Analog Devices AD9767), a high-power amplifier (APEX PA84), and an LED driver. The embedded system is applied to create a jetting trigger signal to FPGA. When the FPGA detects the jetting trigger signal, it drives the DA convertor (output range: ±5 V) to generate low-voltage waveforms for the PPH; meanwhile, the trigger signals of the USB camera and LED are generated by FPGA. The low-voltage drive waveform is magnified (output range: ±100 V) by the high-power amplifier to drive the PPH. The personal computer is used to store the images captured by the USB camera. Detailed theory on the experiment can be found in Refs.^19,20

To ensure that the desired volume flow curve is generated at the nozzle of the print head, the drive waveforms of four materials were designed based on a previous study (J.J. Wang, J. Huang, J. Peng, et al., unpublished). The drive waveforms (one cycle represents the generation of one droplet) corresponding to jetting the four materials with MJ-AL-80 PPH are shown in Figure 4.

FIG. 4.

Drive waveforms of four materials.

In Figure 4, the red, blue, black, and magenta curves represent, respectively, the drive waveforms of water, ethanol, aniline, and glycol that were used to jet the corresponding fluid materials. Figure 5a–d represents the droplet generation process of water, ethanol, aniline, and glycol, respectively, at the critical characteristic velocity. It shows that the distance between the droplet and nozzle does not change after the droplet formation and that the flight velocity of the droplets is close to zero. Therefore, the proposed method can accurately calculate the critical velocity of the droplets generated.

FIG. 5.

Droplet formation process of different materials: (a) water, (b) alcohol, (c) aniline, and (d) glycol.

Conclusions

This study proposed a method for calculating the critical characteristic velocity and dimensionless coordinates of generated droplets. When the viscous force of the fluid is considerably inferior to the surface tension, We ≥ 4.6 is used to determine whether droplets will be produced. When the surface tension of the fluid is considerably inferior to the viscous force, Re ≥6.16 is used to determine whether a droplet will be produced. However, for most fluid materials, the forces of inertia, viscosity, and surface tension have similar effects. In such cases, the method proposed in this study can be used to calculate the critical dimensionless number or characteristic velocity that determines whether or not the droplet will be produced.

Because we only analyzed the limited material parameters, we recommend that the range of fluid material properties suitable for this method is “ viscosity: 1–100 cp, density: 700–1400 kg/m³, tension coefficient: 0.01–0.1 N/m” and the range of circular nozzle diameter is “10–200 μm.” Beyond this parameter range, this method should be used with caution. Even so, this parameter range covers virtually all fluid materials commonly used for injection. Even though this method describes only the formation of droplets, it is also applicable for calculating the critical characteristics of other microdroplet deposition devices (e.g., piston, pneumatic, and thermal bubble).

Author Disclosure Statement

No competing financial interests exist.

Funding Information

This work was supported by the National Natural Science Foundation of China (Grant Nos. 51575419 and 51705387) and the National 111 Project of China (Grant No. B14042); the authors kindly acknowledge these supports.