Optimization of Experimental Parameters To Determine the Jetting

Oct 9, 2013 - In electrohydrodynamic (EHD) printing where droplets or jets are generated by the electrostatic force, the physical as well as the elect...
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Optimization of Experimental Parameters To Determine the Jetting Regimes in Electrohydrodynamic Printing Ayoung Lee,† Howon Jin,† Hyun-Woo Dang,‡ Kyung-Hyun Choi,§ and Kyung Hyun Ahn*,† †

School of Chemical and Biological Engineering, Institute of Chemical Process, Seoul National University, Seoul 151-744, Korea School of Electronics Engineering and §School of Mechatronics Engineering, Jeju National University, Jeju 690-756, Korea

‡

ABSTRACT: The harmony of ink and printing method is of importance in producing ondemand droplets and jets of ink. Many factors including the material properties, the processing conditions, and the nozzle geometry affect the printing quality. In electrohydrodynamic (EHD) printing where droplets or jets are generated by the electrostatic force, the physical as well as the electrical properties of the fluid should be taken into account to achieve the desired performance. In this study, a systematic approach was suggested for finding the processing windows of the EHD printing. Six dimensionless parameters were organized and applied to the printing system of ethanol/terpineol mixtures. On the basis of the correlation of the dimensionless voltage and the charge relaxation length, the jet diameter of cone-jet mode was characterized, and the semicone angle was compared with the theoretical Taylor angle. In addition, the ratio of electric normal force and electric tangential force on the charged surface of the Taylor cone was recommended as a parameter that determines the degree of cone-jet stability. The cone-jet became more stable as this ratio got smaller. This approach was a systematic and effective way of obtaining the Taylor cone of the cone-jet mode and evaluating the jetting stability. The control of the inks with optimized experimental parameters by this method will improve the jetting performance in EHD inkjet printing.

1. INTRODUCTION Electrohydrodynamic (EHD) printing is a complex process in which the charge of the fluid induced by the electric field influences the momentum flux of the fluid. Applying high electric potential between the capillary tip and the counter electrode creates an electrostatic force on the fluid, and the electric tangential force induces the formation of a Taylor cone. Since a typical jet diameter formed by this method is smaller than the size of the nozzle by more than 1 order of magnitude, the EHD printing allows submicrometer resolution, mitigating the drawback of piezo-type inkjet printing, where the droplet size is almost twice the nozzle size.1−3 One practical advantage in EHD printing is that the probability of the nozzle clogging is minimized and the charged particles become self-dispersed. It allows the solid content of the ink to be higher than the conventional ink and suppresses flow-induced aggregation, which is often caused in piezo-type inkjet printing.4,5 Thus, it is applicable in various fields such as biological systems,6,7 food,8,9 electronics,10 solar cells,11−13 and many more.14,15 The harmony of ink material and printing method is of great importance in producing on-demand droplet and jet. In contrast to the piezo-type inkjet printing, where jetting property is determined by the rheological properties of the fluid, in EHD printing, where the droplet or the jet is generated by the electrostatic force, the physical and electrical properties of the fluids should also be taken into account in addition to the rheological properties. The electrical and mechanical processing conditions are also crucial variables influencing the quality of the jetting and the printing. The printhead can make a © 2013 American Chemical Society

significant difference to jetting performance, and thus, the reliability of printhead design should be attained.16 Ever since the pioneering research on the droplet and jet formation induced by the electric field by Zeleny17,18 and by Taylor,19 there have been a lot of analytical and numerical studies on this subject.20−24 Experimental studies have also been done for various fluid properties and processing conditions. The size of the droplet after jetting with respect to the fluid viscosity,25,26 the effect of permittivity on the droplet size and emitted current,27 the deposition of the polymer precursor with respect to the flow rates,28 and the deposition of different sizes of particles or functional particles29,30 have been studied experimentally. However, previous studies have mostly been focused on the printing performance or simple comparison of the output, rather than on the optimization of the printing system in terms of the fluid properties or processing conditions. Few have attempted to systematically study the Taylor cone of the conejet mode,31,32 which is regarded as the ideal jetting mode for its stable jetting and patterning in EHD printing. In addition, even though the electric tangential force is a decisive factor in forming the Taylor cone and it was simulated by Hayati et al.,22,23,33 direct comparison with the experimental results for the cone-jet stability has not been performed. Received: August 23, 2013 Revised: October 8, 2013 Published: October 9, 2013 13630

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During this process, the surface charge of the fluid experiences two types of charge transportcharge conduction and convectionand constant surface charge is maintained by their balance.35 Here, the charges move toward the surface and causes the electric normal stress (τE,n). The supplied flow rate (Qs) and the electrical charging time, which is related to the resistance (∼K−1) and the capacitance (∼ε0ε′) of the system, are the variables influencing the charge conduction (where ε0 is the permittivity of free space, ε′ is the fluid permittivity, and K is the fluid conductivity).10 The surface charges induced by charge conduction lead to electric tangential stress (τE,t) along the surface meniscus with acceleration toward the cone apex.33 The charge acceleration speeds up the surrounding fluid sequentially, K referred to as the charge convection. At this moment, the fluid experiences viscous stress, and as a result, a jet is formed by these charge and fluid motions. Then, the variables influencing the charge convection are the electric field (E = Va/L, where Va is the applied voltage and L is the distance between nozzle and counter electrode), the permittivity ε′ representing the charge amount, and viscosity η representing the flow resistance. Here, 10 variables listed below determine the electrodynamics and fluid dynamics in the EHD printing

In this study, jetting maps were drawn as a processing window, in terms of the rheological and the electrical properties of the fluid, the processing conditions, and the geometrical configuration. In addition, with the help of the numerical simulation, the electrical normal stress (τE,n) and tangential stress (τE,t) acting on the charged surface of a Taylor cone were calculated to evaluate the degree of cone-jet stability, and the results were compared with the experimental data. This approach is a systematic and effective way of getting the Taylor cone of cone-jet mode as a target jetting mode. The control of the inks with optimized experimental parameters will improve the jetting performance in EHD printing. The present paper is organized as follows. Six dimensionless numbers that determine the printing system are first defined. Then, the model fluid, characterization, and printing apparatus are introduced in the experimental part. Lastly, the results are presented and discussed in terms of three concepts: (1) jetting system, (2) jetting map, and (3) jet stability. When discussing jet stability, the cone-jet stability is analyzed in terms of the dimensionless parameters by comparing the experimental results with numerical simulation.

2. THEORY 2.1. Variables. In the electrohydrodynamic capillary tip (or nozzle exit), when the fluid meniscus is under the electric field, the meniscus deforms, and the droplet is formed at the capillary tip. This droplet is distorted as the electric potential increases by the surface charge, and the conical meniscus is maintained over a certain critical limit of electric potential. At this limit, the fluid meniscus is balanced by three forces: the hydrodynamic force (Fh), which supplies more fluid to the emerging droplet; the capillary force (Fγ), which hangs the droplets on the capillary tip; and the electrostatic force (FE), which is caused by the electric field (see Figure 1).1,34

ρ , γ , ε0 , ε′, K , η , d , L , Q s , Va

(1)

where ρ is the density, γ is the surface tension, ε0 is the permittivity of free space, ε′ is the permittivity of the fluid, K is the conductivity, η is the viscosity, d is the nozzle diameter, L is the distance between nozzle and counter electrode, Qs is the supplied flow rate, and Va is the applied voltage. 2.2. Dimensional Analysis. We designed the dimensionless numbers to systemize the variables affecting the printing system. The aforementioned 10 variables (m = 10) contain four fundamental dimensions (n = 4): [M], [L], [T], and [V]. Using the Buckingham π theorem, the number of dimensionless groups can be determined as the number of variables minus the number of fundamental dimensions; m − n = 6. The six dimensionless numbers (D1−D6) were derived as follows ⎛ ⎜ ε0ε′Q D1−D6 = ⎜ 2 s , ⎜ d LK ⎝

1/3 2 ε0ε ′ K

(γ ρ ) η

, ε′ ,

d ρKQ s , , L γε0ε′

⎞ ε0 Va ⎟ ⎟ γd ⎟ ⎠

(2)

and the designed dimensionless numbers are summarized in Table 1 with additional information. Now let us briefly examine the physical meaning of D1−D6 = (Tq/Th, χ, ε′, d/L, α, β).

Figure 1. Forces acting on the fluid surface. S, L, and G signify the solid phase, liquid phase, and the gas phase, respectively.

Table 1. Design of Dimensionless Numbers (D1−D6) Governing the System dimensionless group expression

D1 ε0ε′Qs/d2LK

D2

D3 ε′

⎛ 2 ε0ε ′ ⎞1/3 ⎜γ ρ ⎟ ⎝ K ⎠

D4 d/L

D5 ρKQs/γε0ε′

D6 γd

η

in present notation representation

Tq/Th jetting system

ε0 Va

χ

ε′ material property 13631

d/L geometry

α

β processing condition

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Table 2. Material Properties and the Characteristic Numbers of Ethanol/Terpineol Mixtures E10T0, E8T2, E4T6, E2T8, E0.5T9.5, and E0T10a η (mPa·s) γ (mN·m−1) ρ (kg·m−3) ε′ (−) K (S·m−1) Tq (s) χ (−) Qc (m3·s−1) Vc (kV) a

E10T0

E8T2

E4T6

E2T8

E0.5T9.5

E0T10

1.47 22.8 789 29.1 1.50 × 10−4 1.72 × 10−6 6.05 4.96 × 10−11 0.878

2.08 26.3 819 22.8 1.10 × 10−4 1.84 × 10−6 4.87 5.89 × 10−11 0.943

6.11 29.5 879 14.0 2.30 × 10−5 5.39 × 10−6 2.62 1.80 × 10−10 0.999

13.5 31.5 909 9.20 1.50 × 10−6 5.43 × 10−5 2.71 1.88 × 10−9 1.03

28.1 33.0 932 4.80 2.30 × 10−7 1.86 × 10−4 2.04 6.56 × 10−9 1.05

38.9 33.3 940 3.90 5.00 × 10−11 6.94 × 10−1 23.0 6.94 × 10−1 1.06

For example, E4T6 means a mixture with ethanol 40% and terpineol 60% in volume.

On the basis of the six dimensionless numbers D1−D6 = (Tq/ Th, χ, ε′, d/L, α, β), we will explore the processing maps in EHD printing. It is again to be noted that the aim of this study is to develop a systematic method for obtaining the Taylor cone of the cone-jet mode and new tools to evaluate the degree of cone-jet stability, based on which the printing process can be optimized.

D1 = Tq/Th is the ratio of two characteristic times that determine the jetting system. Tq is the characteristic time of charge transport determined by the electrical properties of the fluid, and Th is the characteristic time of the fluid supply:36 Tq =

ε0ε′ K

Th =

Ld 2 Qs

(3)

3. EXPERIMENTAL SECTION 3.1. Materials. Ethanol (bp 78 °C, dipole moment = 1.6 D) and terpineol (bp 219 °C, dipole moment = 1.6−2.0 D) were used as model fluids (purchased from Sigma Aldrich). By mixing a different ratio of ethanol (E) and terpineol (T), we could design Newtonian fluids with a wide range of properties. In this paper, we prepared six fluidsE10T0, E8T2, E4T6, E2T8, E0.5T9.5, and E0T10as a mixture of the two. Here, for example, E4T6 means a mixture with ethanol 40% and terpineol 60% in volume. We did not choose E6T4 as the model system, since it was obvious that the properties of E4T6 existed in between those of E8T2 and E4T6.38 3.2. Characterization. The viscosity was measured with a straincontrolled rheometer (ARES, TA Instruments, U.S.A.), and the surface tension was measured by surface tensionmat (Fisher Scientific, U.K.). The permittivity and conductivity were measured with a dielectric analyzer (SI 1260 impedance/gain-phase analyzer and dielectric interface, Solartron, U.K.). All measurements were carried out at 25 °C and the equipment was calibrated with standard samples before use. The measured values of viscosity (η), surface tension (γ), density (ρ), permittivity (ε′), conductivity (K), and calculated charge relaxation time (Tq), dimensionless velocity (χ), critical flow rate (Qc), and critical voltage (Vc) of each fluid are presented in Table 2. The error bounds of measurements for the material properties were in the range of ±2%. 3.3. Printing Apparatus. The fluid was injected into a stainless steel capillary tube or nozzle (300 μm o.d., 180 μm i.d.) at a constant volumetric flow rate that was adjusted by a digitally controlled syringe pump (Longer Pump, Model LSP02-1B). A high-voltage power source was connected to the stainless steel nozzle and copper plate ground electrode. To observe the jetting behavior, a high-speed camera (Photron FASTCAM ultima 512) with 512 × 512 resolution and microzoom lens (6.5×) were used. The light source (MORITEX, 250 W metal halide lamp) was located opposite to the camera. A schematic drawing of the printing setup is shown in Figure 2.

(4)

For the EHD printing system, a charge layer should be developed in the surface; thus, the condition Tq/Th < 1 should be satisfied.31 This will be explained in detail later in the Results and Discussion. D2 and D3 are dimensionless numbers linked with the material properties. D2 = χ is the ratio of two characteristic velocities; the characteristic velocity of the fluid (Qc/d2) and the propagation velocity of a perturbation by viscous diffusion (η/ρd),

χ=

Q c/d2 η / ρd

1/3 2 ε0ε ′ K

(γ ρ ) = η

=

(γ 2ρTq)1/3 η

(5)

where Qc is the critical flow rate required for cone-jet formation, and it is defined as follows:

Qc =

γTq γε0ε′ = ρK ρ

(6)

As suggested in eq 6, Qc is the value determined by the properties of the fluid. This is not simply the flow rate due to the upstream pressure but rather the specific flow rate that causes electrical stress to strip off (or shear) the surface charge layer of the fluid.37 D3 = ε′ is the permittivity of fluid determined by the amount of the alignment of dipoles. D4 = d/ L is the ratio of nozzle diameter (d) and nozzle−counter electrode distance (L). D5 and D6 are the dimensionless numbers linked with processing conditions. D5 = Qs/Qc = α is the dimensionless flow rate defined as the ratio of supplied flow rate (Qs) and critical flow rate (Qc). D6 = Va/Vc = β is the dimensionless voltage, which is the ratio of applied voltage (Va) and critical voltage (Vc), where Vc is the critical voltage that supports the meniscus on the capillary tube of diameter d as follows: Vc =

γd ε0

4. RESULTS AND DISCUSSION 4.1. Jetting System. The jetting system is determined by the magnitude of Tq/Th, which is the ratio of the two characterization times, the charge relaxation time (Tq = ε0ε′/K) and the hydrodynamic time (Th = Ld2/Qs). If the fluid has a sufficient concentration of charge carriers with large enough electrical mobility compared to other velocities of the system, enough free charge, and the induced charge buildup below the

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system in this paper, the condition of Tq/Th < 1 should be satisfied. 4.2. Jetting Map. The fluid is supplied inside the nozzle by a syringe pump and is simultaneously taken out of the nozzle by an applied electric field between the nozzle and the counter electrode. This supply and loss balance of the fluid affects the shape and volume of the cone-jet. Processing parameters are D5 (=Qs/Qc = α, where Qc = γε0ε′/ρK) and D6 [=Vs/Vc = β, where Vc = (γd/ε0)1/2]. Qc is the key parameter for cone-jet formation.11,45 When α = 1, the stable jet could be generated from the balance of flux strength and electric field strength. Thus, by setting α = 1, we found the range of electric voltage β in which the stable jet was generated. Then β was chosen for clear and stable cone-jet shape at α = 1 for E10T0, E8T2, E4T6, and E2T8 (and at α = 0.1 for E0.5T9.5). The jetting behaviors in terms of α and β are shown in Figure 4. Since E10T0, E8T2, and E4T6 belong to the EHD jetting range (Tq/Th < 1) as shown in Figure 3c, the Taylor cone of the stable cone-jet mode was generated under the condition of specific electric voltage β at α = 1. On the other hand, under the same condition, the jet was not formed or was not straight but tilted for α < 1. The Taylor cone was generated, but the thickening of jet diameter was observed for α > 1 (see Figure 4). In the case of E2T8, both systems of EHD jetting (Tq/Th < 1) and electrically forced jet (Tq/Th > 1) are allowed, as shown in Figure 3c. The region of α > 1 corresponded to a nonclassical EHD system, and it showed the microsized thick thread (see Figure 4). For E0.5T9.5, the cone-jet mode appeared (with spray in lower part) at α = 0.1 and the ball-cone was observed for α larger than 0.5 (see Figure 4). That is, two jetting systems appeared dependent on α, and they are wellmatched with the jetting window of Figure 3c. When α is larger than 0.1, the fluid climbed up the nozzle wall (see the dotted line in the figure). For E0T10, the system corresponds to an electrically forced jet (Tq/Th > 1); thus, the ball-cone should appear at all α. However, in this experiment, we could confirm the ball-cone formation only at small α due to the limit of Qs. In Figure 4, in terms of cone-jet shape, when it satisfies the condition of Tq/Th > 1, E10T0 and E8T2, which have relatively high conductivity (K ∼ 10−4 S·m‑1), showed clear jet formation at the meniscus apex. On the other hand, for E4T6 and E2T8, which have relatively lower conductivity (10−5−10−6 S·m‑1), the distinction was not clear between cone and jet.32 When it satisfies the condition Tq/Th < 1, either a jet whose diameter is the same as the nozzle diameter or a ball-cone was formed. The indicated β of Figure 4 is the controlled value for a stable Taylor cone of the cone-jet mode at α = 1. In practice, the electric potential as well as the flow rate affects the jetting behavior significantly. In addition, the jetting behavior depends on the rheological and electrical properties of the fluid.

Figure 2. Schematic diagram of the EHD printing device. Spray was generated beyond a certain distance after jet formation from the cone apex, which is indicated by the light gray color.

liquid−gas interface, a thin layer of charges form in the direction normal to the interface. If the fluid is slowly flowing as in a quasiequilibrium state, the charge relaxation time (Tq) is shorter than the hydrodynamic time (Th); thus, EHD-driven jet can be generated (Tq/Th < 1). This is categorized as the classical EHD jetting system.11,39 Here, a stable cone-jet mode can be generated when physical conditions are optimized (see Figure 3a). When the condition for EHD jetting is satisfied as Tq/Th < 1, electrospinning can also take place for polymer solutions or melts.40−42 On the other hand, if the conductivity of the fluid is very low, as in the case of a dielectric fluid (K ∼ 10−10 S·m‑1), the supply of the fluid prevails over the charge transport to the fluid surface; thus, a layer at the liquid−gas interface does not exist. In this case, the applied electric field and the supplied flow rate provoke an acceleration of the fluid, leading to a ball-cone jet with the microsized thick thread (Tq/Th > 1). Thus, it is categorized as a nonclassical EHD system, which is a newly defined electrically forced jet (see Figure 3b).43,44 In Tq/Th, which determines the jetting system, when the geometrical configuration is fixed (in this study d = 300 μm and L = 25 mm), the supplied flow rate Qs (=αQc) is the crucial factor that determines the jetting system. Since the critical flow rate Qc (=γε0ε′/ρK) is determined by the fluid properties, the dimensionless flow rate eventually determines the jetting system, as presented in eq 8. D1 =

Tq Th

=

ε0ε′/K Ld 2/Q s (=αQ c)

=

2 γ ⎛ ε0ε′ ⎞ ⎜ ⎟ α Ld 2ρ ⎝ K ⎠

(8)

The magnitude of Tq/Th is calculated with respect to α by eq 8, and as a result, the jetting system can be categorized as in Figure 3c. Here, the white region is for the classical EHD jetting (Tq/Th < 1) and gray region is for the forced jet system (Tq/Th > 1). As our attention is directed to the classical EHD jetting

Figure 3. Representative shape of (a) cone-jet (Tq/Th < 1) and (b) ball-cone (Tq/Th > 1). (c) Jetting window with respect to α for each fluid; the white region is for a classical EHD system and the gray region is for a forced jet system. 13633

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Figure 4. Jetting behavior of the fluids in terms of the processing parameters dimensionless flow rate α and dimensionless voltage β. To distinguish the position of the nozzle exit, dotted white lines were marked.

Figure 5. EHD jetting modes with respect to electric potential.

the imbalance of the surface tension and the electric charge of the ions. Cone-jet mode is an ideal jetting mode in EHD printing when the Taylor cone is formed consistently from the cone apex. In addition, the increase in electric field results in an imbalance between the forces, causing the jet to lean toward one side, which is called the tilted-jet mode. Next, the twin-jet mode where two jets are formed occurs. Further increase of the electric potential causes excessive charges on the cone, and as a result, the accumulated charges reach the limit and the multi-jet mode is generated to minimize the energies by rearranging them to the large areas.30,46

In the classical EHD jetting system (Tq/Th < 1), a variety of jetting modes can be generated with respect to the electric potential, as shown in Figure 5. The increase of electric potential generally leads to a sequence of jetting modes: dripping, pulsating, cone-jet, tilted-jet, twin-jet, and multi-jet. In the dripping mode, the pendant drops pinch off from the nozzle end due to the gravitational force. Additional electrostatic force from the electric field counters the surface tension force and helps the gravitational force to pinch off the droplets more rapidly. In the pulsating mode (or spindle mode), the hemispherical shape and Taylor cone shape are repeated due to 13634

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Figure 6. Jetting maps in terms of the dimensionless numbers D1−D6= (Tq/Th, χ, ε′, d/L, α, β) for (a) E10T0 (χ = 6.05, ε′ = 29.1), (b) E8T2 (χ = 4.87, ε′ = 22.8), (c) E4T6 (χ = 2.62, ε′ = 14.0), and (d) E2T8 (χ = 2.71, ε′ = 9.2). Classified jetting modes a−f of Figure 5 are indicated in the map.

when (β1=) 4.4 ≤ βcone‑jet ≤ 5.9 (=β2) for E4T6 (see Figure 6c) and when (β1=) 3.8 ≤ βcone‑jet ≤ 4.8 (=β2) for E2T8 (see Figure 6d). In other words, at α = 1, the dimensionless voltage βcone‑jet for cone-jet formation shifted to higher values as ε′ increased when χ was similar. Depending on α, low limit β1 and high limit β2 were varied. As a result, the cone-jet modes were generated when 0.5 ≤ α ≤ 3 for E4T6 (Figure 6c) and for all α in the case of E2T8 (if Tq/Th < 1) (Figure 6d). Gray-shaded regions in Figure 6 correspond to the cone-jet mode. Accordingly, E2T8 showed the most optimized performance for cone-jet mode formation in this study. On the basis of these results, we will examine the jetting physics in order to understand the nature of the cone-jet mode formation and then suggest the evaluating tools for cone-jet stability. 4.3. Jet Stability. 4.3.1. Charge Relaxation Length. The cone-jet mode is the most ideal jetting mode for stable EHD printing, and the surface charge layer should be built up by the charge conduction. The thickness of this layer refers to the charge relaxation length (r*)27 and it is defined as

Let us now draw jetting/operating maps based on various jetting modes of classical EHD in terms of six dimensionless parameters D1−D6 = (Tq/Th, χ, ε′, d/L, α, β). The points where the transition of the jetting mode occurs were found experimentally by performing the experiments at α values of 0.1, 0.5, 1 3, 5, 7, and 10 and various β values. The processing map was drawn by connecting these points. Jetting maps for the Newtonian fluids E10T0, E8T2, E4T6, and E2T8 including classified jetting modes a−f (see Figure 5) are shown in Figure 6. It should be noted that not all the jetting modes appear in all the fluid systems due to the difference in processing parameters and the fluid properties. For α < 1 in E10T0 and E8T2, the jet never appeared at low β and the cone-jet mode was not generated. The increase of β ultimately led to the multi-jet mode for all the fluids, but for α > 1 in E2T8, the tilted-jet mode predominated without the multi-jet mode within the applied β. In addition, E2T8 showed a wide range of cone-jet mode among the fluids we have covered. In experiments, it was not easy to control the parameters independently. Among the data set, we found E4T6 (χ = 2.62, ε′ = 14.0) and E2T8 (χ = 2.71, ε′ = 9.2) had similar χ with different ε′, and we tried to discuss the effect of ε′. For the cone-jet formation at α = 1, the cone-jet mode was generated

⎡⎛ γε ε′ ⎞⎛ ε ε′ ⎞⎤ r * = (Q sTq)1/3 = ⎢⎜ 0 ⎟⎜ 0 ⎟⎥α1/3 ⎣⎝ ρ K ⎠⎝ K ⎠⎦ 13635

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where Qs is the supplied flow rate and Tq is the charge relaxation time. This supports that the charge conduction is determined by the supplied flow rate Qs and charge relaxation time ∼ε0ε′/K. The charge relaxation length (r*), which was calculated from eq 9, is shown in Figure 7.

Figure 8. (a) Definition of jet diameter (djet) and semicone angle (θ) in cone-jet mode. Cone-jet images for (b) E4T6 (χ = 2.62, ε′ = 14.0) and (c) E2T8 (χ = 2.71, ε′ = 9.2) with respect to α and β.

Figure 7. Charge relaxation length (r*) as a function of dimensionless flow rate α for fluids: E10T0 (χ = 6.05, ε′ = 29.1), E8T2 (χ = 4.87, ε′ = 22.8), E4T6 (χ = 2.62, ε′ = 14.0), and E2T8 (χ = 2.71, ε′ = 9.2).

There is a significant difference in r* between the fluids E4T6 (χ = 2.62, ε′ = 14.0) and E2T8 (χ = 2.71, ε′ = 9.2) due to different ε′ values in spite of similar χ ∼ 2.7. E4T6 had a thinner (smaller r*) and denser (higher ε′ charge layer than E2T8. Thus, in order to strip off the generated r* of E4T6 by accelerating the charges (composed of ions) and the surrounding fluids toward the cone apex, relatively large electric potential needs to be applied. This result supports that E4T6 had a larger β1 (low limit β for cone-jet formation) than E2T8 in Figure 6. In other words, r*, which is the thickness of the surface charge layer generated by charge conduction, and the applied voltage βcone‑jet that strips off the charge layer are decisive parameters for cone-jet stability. 4.3.2. Characterization of Cone-Jet Mode. We investigated the jet diameter and the cone angle in the cone-jet mode with respect to the processing conditions and the fluid properties. The measurements were conducted with the software Image J. In this study, djet was defined as the jet diameter at a distance of 500 μm from the nozzle exit and θ was designated as the semicone angle of the cone apex (see Figure 8a). The cone-jet images for E4T6 (χ = 2.62, ε′ = 14.0) and E2T8 (χ = 2.71, ε′ = 9.2) with respect to the processing parameters α and β are given in Figure 8b,c. The jet diameter djet was analyzed on the basis of the images in Figure 8. For E4T6 and E2T8, the djet was increased as α was increased and was decreased with the increase in β (see Figure 9a, top). In addition, the normalized djet with charge relaxation length r* was close to the unity (djet/r* ∼ 1) (see Figure 9a, bottom). So djet can be predicted and the printed pattern with fine resolution can be obtained by tuning the material properties once r* is known. The Taylor angle (θT) is known to be 49.3°.1,19 This was theoretically calculated from the balance of capillary and electric pressures under no fluid motion (zero hydrostatic pressure). However, the practical system is put under flow and electric fields, so that it has a discrepancy compared to the theoretical one. The semicone angle θ increased with the increase in β (see Figure 9b, top), and the

Figure 9. (a) Jet diameter (djet) (top) and normalized jet diameter with charge relaxation length (r*) as a function of dimensionless voltage β (bottom), (b) semicone angle (θ) (top) and normalized semicone angle with the Taylor angle (θT) (bottom) for E4T6 (χ = 2.62, ε′ = 14.0) and E2T8 (χ = 2.71, ε′ = 9.2).

normalized θ with θT approached 1 (θ/θT ∼1) with a slight variation (see Figure 9b, bottom). 4.3.3. Ratio of Electrostatic Forces (Numerical Simulation). We conducted a numerical simulation in order to estimate the electric normal and tangential forces (FE,n, FE,t) and compared the results with the experiments. The charge transport originates from both the conduction by the electric field and the convection by the flow field. The cone angle, the electric field, and the surface stress change simultaneously depending on the electric potential. To take all these factors into account, we have to solve the conservation equations of the momentum and the electrodynamic equations at the same time. In this simulation, only the effect of the electric field was investigated by using the information on geometry that was obtained experimentally from the given potential difference. To calculate the electric field in Figure 10a, we solved the following equations by using commercial software COMSOL Multiphysics (COMSOL 4.2, Comsol Inc., U.S.A.) E = −∇ϕ 13636

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FE,n(t ) =

(10b)

∇·D = ρe

(10c)

where E is the electric field, ϕ is the electric potential, D is the dielectric displacement, ε is the permittivity, and ρe is the charge density. We refined the mesh in the cone part, and the total number of elements was about 12 000. The simulation domain in the r-direction was set to 10 times the size of the nozzle radius, and only a part of it is shown in Figure 10b. The permittivity was 9.2 for E2T8 and 14.0 for E4T6 in fluid phase (green) and 1.0 in gas phase (blue), respectively. The simulation was performed in a cylindrical coordinate; Er and EZ were obtained at the surface of the cone. They were transformed to the normal (En) and tangential (Et) component with the information on cone angle, and then the electric stresses due to the electric field at the surface were obtained as follows:23 τE,n =

1 1 ε0En 2 + (ε − ε0)Et 2 2 2

τE,t = ε0EtEn

τE,n(t) dA

(12)

where A is the area of the cone. Here, FE,n is the normal component of the electrostatic force, which tends to destabilize the cone-jet by acting on the meniscus outward, and FE,t is the tangential component of the electrostatic force, which tends to stabilize the cone-jet by acting along the meniscus toward the cone apex. Therefore, the ratio of two forces FE,n/FE,t can be used as an index of the degree of jet stability. As FE,n/FE,t becomes larger, the cone-jet becomes less stable, whereas the cone-jet becomes more stable when FE,n/FE,t gets smaller. In addition, even though the fluids have cone-jet mode outwardly, we can predict which cone-jet mode is more stable and tends to promote unstable jets (e.g., tilted, twin, and multi-jets) more rapidly by comparing the slope of FE,n/FE,t with respect to β. As the slope of FE,n/FE,t with respect to the dimensionless voltage β becomes larger, the cone-jet may proceed to an unstable jet more rapidly. Compared with the experimental results in Figure 5, the cone-jet mode will be transformed to be tilted, twin, and multijet mode as β increases. The behavior of FE,n/FE,t and the slope of E2T8 (χ = 2.71, ε′ = 9.2) and E4T6 (χ = 2.62, ε′ = 14.0) with respect to α and β are given in Figure 10c. Figure 10c shows that the ratio between electric normal force (FE,n) and electric tangential force (FE,t) was larger than 3 (FE,n/ FE,t ≥ 3) for both E4T6 and E2T8. When this ratio was calculated for the data from the paper of Hayati et al.,23 it turned out that FE,n was almost 3.7 times larger than FE,t (FE,n/ FE,t > 3.7), independent of flow rate. The slope of FE,n/FE,t was 5.45 for E4T6, 0.34 for E2T8 at α = 0.5, and 0.82 for E4T6 and 0.14 for E2T8 at α = 1. The slope of FE,n/FE,t at α = 0.5 for E4T6 is larger than for the other fluids. Thus, it tends to proceed to the unstable jet more rapidly than the others. On the other hand, for E2T8 at α = 3, there was no change in FE,n/ FE,t, which means that β contributes to the jet acceleration rather than to the jet stability at large flow rate. This may suggest that the range of β for a stable cone-jet area will become narrower as α decrease, which is actually shown in Figure 6. In other words, whether to form stable jet or to proceed to unstable jet can be determined with the magnitude and the slope of FE,n/FE,t according to β at the same α. The ratio of electrostatic forces can be a new evaluating parameter for conejet stability.

Figure 10. (a) Simulation domain. The nozzle size and the length of nozzle head are 300 μm and 13 mm, repectively. The distance between the nozzle and counter electrode was set to be 25 mm. (b) Mesh: green and blue colors correspond to fluid and gas phase, respectively. (c) FE,n/FE,t, the ratio of electric normal force FE,n and electric tangential force FE,t for E4T6 (χ = 2.62, ε′ = 14.0) and E2T8 (χ = 2.71, ε′ = 9.2), with respect to α and β. The slopes of FE,n/FE,t are displayed in the graph.

D = εE

∫

5. CONCLUSIONS In this study, with the Buckingham π theorem, six dimensionless numbers D1−D6 = (Tq/Th, χ, ε′, d/L, α, β) governing EHD printing system were systemized, and jetting maps were drawn for ethanol/terpineol mixtures on the basis of these parameters. For a classical EHD printing system of Tq/Th < 1, at α = 1, the cone-jet mode was generated at 4.4 ≤ βcone‑jet ≤ 5.9 for E4T6 and 3.8 ≤ βcone‑jet ≤ 4.8 for E2T8, respectively. The E4T6 and E2T8 have similar χ ∼ 2.7 (the ratio of characteristic velocity of the bulk fluid and propagation velocity of the perturbation by viscous diffusion) but different ε′ (fluid permittivity). Since E4T6 has a thinner (smaller r*) and denser (higher ε′) charge layer than E2T8, to strip off the generated r* of E4T6 by accelerating charges and surrounding fluids toward the cone apex, relatively large electric potential may have to be applied. In the cone-jet mode, the normalized djet with r* approached unity. This suggests that it is highly probable that djet can be predicted from the information on r* and that it is

(11a) (11b)

The forces acting on the surface were calculated by integrating the electric stress over the free surface 13637

dx.doi.org/10.1021/la403111m | Langmuir 2013, 29, 13630−13639

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possible to optimize the value by adjusting the fluid properties. In addition, the normalized θ with θT approached 1 with a little variation. By numerical simulation, we could suggest the ratio of two electrostatic forces, FE,n/FE,t, as an evaluation parameter for the cone-jet stability. The smaller the ratio, the more stable the cone-jet. In addition, as the slope of the ratio with respect to β becomes larger, the cone-jet may proceed to an unstable jet more rapidly. These were matched well with the experimental results for a stable cone-jet area. By setting up the dimensionless numbers and the evaluating tools of the degree of cone-jet stability, the target of a stable Taylor cone in conejet mode can be obtained in a systematic and effective way. The control of the inks with the optimized experimental parameters will contribute to the design of improved jetting performance in the EHD inkjet printing.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by the National Research Foundation of Korea (NRF) grant (No. 20100026139) funded by the Korea government (MEST).

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NOMENCLATURE ρ = density (kg·m−3) ρe = charge density (C·m−3) γ = surface tension (N·m−1) ε0 = permittivity of free space (F·m−1) ε′ = permittivity of fluid K = conductivity (S·m−1) η = viscosity (Pa·s) d = nozzle diameter (m) L = distance of nozzle and counter-electrode (m) χ = dimensionless characteristic velocity Qs = supplied flow rate (m3·s−1) Qc = critical flow rate (m3 ·s−1) α = dimensionless flow rate Va = applied voltage (V) Vc = critical voltage (V) β = dimensionless voltage Th = hydrodynamic time (s) Tq = charge relaxation time (s) E = electric field (V·m−1) ϕ = electric potential (V) D = dielectric displacement (C·m−2) τE,n = electric normal stress (Pa) τE,t = electric tangential stress (Pa) FE = electrostatic force (N) FE,n = electric normal force (N) FE,t = electric tangential force (N) r* = charge relaxation length (m) djet = jet diameter (m) θT = Taylor angle (rad or deg) θ = semicone angle (rad or deg) [M] = mass [L] = length [T] = time [V] = voltage 13638

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