Rheol Acta (2013) 52:313–325 DOI 10.1007/s00397-013-0688-4
ORIGINAL CONTRIBUTION
Generation of inkjet droplet of non-Newtonian fluid
Hansol Yoo · Chongyoup Kim
Received: 3 June 2012 / Revised: 28 January 2013 / Accepted: 31 January 2013 / Published online: 17 February 2013 © Springer-Verlag Berlin Heidelberg 2013
Abstract In this study, the generation of inkjet droplets Keywords Drop-on-demand inkjet · Elasticity · Shear of xanthan gum solutions in water–glycerin mixtures was thinning · Infinite shear viscosity · Jeffery–Hamel flow · investigated experimentally to understand the jetting and Strain hardening · DWS microrheology drop generation mechanisms of rheologically complex flu- ids using a drop-on-demand inkjet system based on a piezo- electric nozzle head. The ejected volume and velocity of Introduction droplet were measured while varying the wave form of bipo- lar shape to the piezoelectric inkjet head, and the effects of As the inkjet printing technology has widen its application the rheological properties were examined. The shear prop- to bio and electronic industries beyond household or office erties of xanthan gum solutions were characterized for wide inkjet printers (Basaran 2002; Schubert 2005;deGansetal. ranges of shear rate and frequency by using the diffusive 2004), many different kinds of inks have to be handled. In wave spectroscopy microrheological method as well as the most cases, inks are suspensions or polymeric liquids (de conventional rotational rheometry. The extensional prop- Gans et al. 2005), and hence most of the inks are rheolog- erties were measured with the capillary breakup method. ically complex fluids showing shear-dependent viscosities The result shows that drop generation process consists of and/or elastic characteristics. Some additives such as sur- two independent processes of ejection and detachment. The factants are usually added in suspensions for stabilization ejection process is found to be controlled primarily by high and better performances. This can make the rheology of or infinite shear viscosity. Elasticity can affect the flow suspension more complex. However, inks have not been through the converging section of inkjet nozzle even though characterized properly especially at the operating conditions the effect may not be strong. The detachment process is of inkjet printing, and the processing conditions have been controlled by extensional viscosity. Due to the strain hard- sought mostly through trial and error basis. ening of polymers, the extensional viscosity becomes orders To generate inkjet droplets, either the continuous jet- of magnitude larger than the Trouton viscosities based on ting or drop-on-demand (DOD) method can be used (Derby the zero and infinite shear viscosities. The large extensional 2010). In the DOD method, droplets are generated by apply- stress retards the extension of ligament, and hence the stress ing a pressure wave to a liquid-filled nozzle. Then, a portion lowers the flight speed of the ligament head. The viscoelas- of liquid is squeezed out of the nozzle overcoming the sur- tic properties at the high-frequency regime do not appear face tension force, and the liquid element is detached from to be directly related to the drop generation process even the nozzle tip by inertia and capillary force. In this stage, though it can affect the extensional properties. the fluid element becomes elongated before detachment, and the elongated liquid thread is either contracted so that a single drop is generated or divided into the leading drop H. Yoo · C. Kim () and some smaller satellite drops by instability mechanisms. Department of Chemical and Biological Engineering, In some cases, satellite drops can be merged into the lead- Korea University, Anam-dong, Sungbuk-ku, Seoul 136–713, South Korea ing drop. It is known that the formation of satellite drops e-mail: [email protected] should be avoided for better printing quality, and hence the 314 Rheol Acta (2013) 52:313–325 determination of the proper window on the operating param- noted that the drop generation is a highly nonlinear process eters for a single drop generation is one of the most impor- of large extension and high extensional rate, and therefore tant issues in inkjet droplet generation. It has been reported linear viscoelastic properties may not be correlated quan- that drop generation characteristics are governed by Ohne- titatively with the nonlinear process. Morrison and Harlen sorge number (Oh) of the drop, which is the ratio of viscous (2010) investigated the effects of viscoelasticity numerically time scale and surface tension time scale and defined as on drop formation in inkjet printing by using viscoelastic follows (Derby 2010): fluids represented by the single-mode FENE-CR constitu- η tive equation (Chilcott and Rallison 1988). They showed Oh = √ , (1) that the ligament became longer for elastic liquids and the ρRγ formation of satellite drops was suppressed by elasticity. where η is viscosity, ρ is density, R is radius, and γ is Also, they argued that the lowering of drop speed was due surface tension. For Newtonian fluids, Oh is unequivocally to elasticity. Recently, Hoath et al. (2012) presented a quan- defined since fluid properties are independent of flow con- titative model which predicted three different regimes of ditions. But in the case of shear thinning fluids, viscosity behavior depending upon the jet Weissenberg number (Wi) is a function of shear rate, and hence the operating window and extensibility of polymer molecule. They predicted New- cannot be predicted based on the theory for Newtonian flu- tonian regime (Wi < 1/2), viscoelastic regime with partial ids (Lai et al. 2010; Tai et al. 2008). It has been the usual extension (1/2 < Wi
Inkjet system and imaging
The inkjet droplet was generated by a piezo-type nozzle pur- chased from MicroFab Co. (Model # MJ-AT). The nozzle diameter at the exit was 50 μm. In Fig. 2, the inside geome- try of the nozzle is shown. In taking the picture, the nozzle filled with air was immersed in a decalin-filled square box. Since decalin has the same refractive index as the glass, the Fig. 3 Pulse wave form for generating inkjet droplets: a expansion of refraction at the curved nozzle surface can be avoided. nozzle chamber, b delay for pressure wave propagation, c compres- sion of fluid for ejection, d delay for pressure wave propagation, and e To generate droplets, a bipolar wave form was used as nozzle chamber expansion to the initial state shown in Fig. 3. During the rise period (a), the piezo- element expands for fluid intake from the reservoir and this state continues during the dwell period (b). During the fall period (c), the piezo-element shrinks and fluid is ejected out of pixels per frame has to be small (512 × 48 pixels). The of the nozzle. This state continues during the echo period pixel size was 3.76 μm. One may use the flash videography (d). Finally, the piezo-element expands to return to the initial method to get the better quality as demonstrated by van Dam state while completing a cycle (e). Depending on the time and Le Clerc (2004) and Dong et al. (2006). But, in the case intervals and the voltages imposed on the piezo-element, a of non-Newtonian drops, the reproducibility of drop gener- drop or drops of different sizes and velocities are generated. ation was not as good as in case of Newtonian fluids; hence, In the present research, the rise and fall times in the volt- we decided not to use the flash videography method used in age pulse to the nozzle were set at 2 μs. The dwell and echo the literature. Considering that the drop diameter and travel times were in the range of 4–32 μs, and the dwell and echo distance are in the order of 50 μm and 1 mm, respectively, voltages were in the range of 12–50 V. We performed drop the numbers of pixels to cover these sizes are about 15 and generation experiments by varying operating conditions and 150. Therefore, the image resolution was enough to measure measured the drop size and velocity. All the experimental drop diameter and velocity. runs were performed at the room condition. The high-speed camera (IDT, XS-4) was triggered by the jetting driver as a Materials droplet was ejected from the inkjet nozzle. The camera was equipped with a microscopic objective lens (Mitutoyo, M Three kinds of Newtonian fluids were prepared with dif- × Plan Apo) with the magnification of 5. As the illumina- ferent viscosities by mixing deionized water and glycerin tion source, a back lighting system (Stocker Yale, # 21 AC, (Sigma-Aldrich Co.). Shear-thinning fluids were prepared 180 W) was installed. by dissolving xanthan gum (Sigma-Aldrich Co.) in deion- The CCD camera can capture 50,000 frames per second ized water or in one of the water–glycerin mixtures. Table 1 and the pictures were taken with this mode. The exposure shows the composition of the fluids tested here. The time was 1 μs. When this fast mode was used, the number shear viscosity and linear viscoelastic properties of liquids were measured by a rotational rheometer with a Couette fixture (AR2000, TA Instrument). High-frequency linear
Table 1 Newtonian base fluids
Sample name 1 cP 4.5 cP 10 cP 16.5 cP
Deionized water, wt% 100 55 40 32 Glycerin, wt% 0 45 60 68 Viscosity, mPa s 1 4.5 10 16.5 Fig. 2 Inside geometry of the nozzle. The inlet diameter is 456 m μ Surface tension, mN/m 71.6 66.0 67.5 66.9 and the exit diameter is 50 μm Rheol Acta (2013) 52:313–325 317 viscoelastic properties were measured by a diffusive wave spectroscopy (DWS) microrheology rheometer (RheoLab, LS Instrument) using polystyrene spheres of 520 nm in diameter as tracer particles. The cuvette thickness was 2 mm and the properties were measured under the transmission mode. The extensional viscosity was estimated by using a capillary breakup apparatus (CaBER, ThermoHaake Co.). To handle low-viscosity fluids, two small plates of diameter 2 mm were machined from titanium. The initial gap dis- tance was the same as the radius of the plate and the initial deformation was imposed to 2.5 mm for 20 ms. The surface tension was measured by using the Du Nouy¨ ring method (K9 Tensiometer, KRUSS¨ GmbH).
Result and discussion Fig. 4 Viscosities of xanthan gum solutions for differing solvents and xanthan gum concentrations. The symbols are measured values. The solid lines are the Carreau model fit Figure 4 shows the shear viscosities of two sets of fluids tested in the present research. The mixtures of DI water and glycerin have shear-independent viscosities while all xan- than solutions have shear-thinning viscosities. The viscosity of the solvent. In the following discussion, we will com- of xanthan solution is fitted to the Carreau model (Bird et al. pare the drop generation characteristics of the fluids with 1987): the same base fluid systematically. Figure 5 shows linear − viscoelastic properties of some of the fluids listed in Table η η∞ = 1 − , (7) 2. Figure 5 shows that the shapes of G and G for the η − η∞ 1 n 0 1 + (βγ˙)2 2 xanthan gum solutions are not similar to those of poly- mer solutions of flexible polymers such as Boger fluids in and the model parameters are listed in Table 2. In the table that the typical slope of 2 for G at low-frequency regime (alsoinFig.4), we note that the infinite shear viscosities does not appear yet at the lowest frequency of 0.1 s−1. of xanthan gum solutions of the same base solvent are only The data obtained from the DWS microrheology appear to slightly changed from or almost the same as the viscosity be reasonably extended to the conventional rheometry data.
Table 2 Xanthan gum solutions studied here and their 1.0 cP 4.5 cP Carreau model parameters and surface tension 50 ppm 100 ppm 200 ppm 50 ppm 100 ppm 200 ppm
η0 (mPa s) 2.2 3.9 7.09 6.89 12.6 28.8 η∞ (mPa s) 1.2 1.4 1.6 4.7 4.0 4.3 β,s−1 0.25 0.22 0.24 0.28 0.47 0.68 n 0.63 0.38 0.52 0.61 0.71 0.41 σ (mN/m) 71.4 71.6 71.4 65.7 66.5 66.0
10 cP 16.5 cP
50 ppm 100 ppm 200 ppm 50 ppm 100 ppm 200 ppm
η0 (mPa s) 17.2 29.4 71.8 29.3 48.7 136.2 η∞ (mPa s) 9.3 8.7 10.6 16.0 16.5 14.0 β,s−1 0.77 1.30 1.85 1.66 0.95 10.3 n 0.73 0.72 0.61 0.75 0.64 0.69 σ (mN/m) 67.3 67.2 67.4 66.8 67.2 66.9 318 Rheol Acta (2013) 52:313–325
Fig. 5 Viscoelastic properties of some xanthan gum solutions: a 10-cP-based solutions, b 16.5-cP-based solutions. Symbols were obtained by rotational rheometry. Solid lines were obtained by DWS microrheology
One important characteristic is that the xanthan gum solu- Fig. 6 Typical drop generation patterns. The time interval between tions have smaller G than G for all the frequency regimes two adjacent frames is 20 μs. a Single-drop formation for the 100-ppm tested here including the DWS microrheology measurement solution in 4.5-cP solvent. The tail is shrunk to the main drop. Driving voltage is 30 V. b A satellite drop generated by the separation from the regime. This implies that the elastic effect should not be tail for the 50-ppm solution in 10-cP solvent. Driving voltage is 36 V. c large even at the inkjet drop generation condition (shear rate A satellite drop generated by reflected acoustic waves for the 50-ppm of 105 s−1). solution in water. In this case, the tail is shrunk to the main drop, but a Figure 6 shows some typical drop generation patterns for new drop is generated behind the main drop differing fluids at differing conditions. In Fig. 6a, a thin liquid ligament with a spherical head is formed, and then 1994; Stone et al. 1986). These drops may be called “satel- the ligament tail contracts to the head eventually. Therefore, lites from tail.” In Fig. 6c, a satellite drop is generated by only one drop is generated. In Fig. 6b, the situation is much a different mechanism. In this case, the satellite drop is not the same as Fig. 6a, but the tail is separated from the main linked to the original ligament and appears to be generated head and becomes a satellite drop. More than one satellite by the reflected acoustic wave inside the nozzle. These satel- drop can be generated and these drops can be coalesced lite drops may be called “satellites from reflected wave.” depending upon the relative velocity between the main and In the following discussion, we will consider the cases for satellite droplets. The separation of the satellite drop is which only one single drop is generated. To generate one caused by the capillary instability or end pinching (Stone single drop without satellites, a proper operating window Rheol Acta (2013) 52:313–325 319 for voltage and time of dwell and echo has to be chosen as performed experiments while varying dwell/echo voltage well as rise and fall time. Since there are too many com- and collected data when only a single drop was generated. binations for operating parameters, we confine ourselves to Figure 7 shows the drop velocity variations for the flu- the following cases. First, the rise and fall times were fixed ids listed in Table 1. First, we note that, for all cases, at 2 μs. Dwell and echo voltages were set to be equal, and as dwell/echo voltage (voltage hereafter) increases, drop then we changed dwell/echo time to find out the condition velocity increases. Also, the minimum voltage required to at which the drop velocity became the maximum value. In generate a single drop increases with xanthan gum con- most of the cases, the drop velocity became the maximum centration. At a fixed voltage value, drop velocity becomes or close to the maximum value when dwell/echo time was smaller with an increase in xanthan gum concentration. For 24 μs. Therefore, we fixed dwell/echo time at 24 μs. The example, when the viscosity of solvent is 4.5 cP, the drop optimum dwell/echo time (the condition at which the drop velocity is 4.5, 3.2, and 0.6 m s−1 when the driving voltage velocity attains the maximum value) is closely related to the is 24 V and the xanthan gum concentration is 0, 50, and 100 length of the nozzle and acoustic velocity of fluid (Bogy ppm, respectively. At 24 V, the 200-ppm solution could not and Talke 1984). Since the nozzle length is fixed and the be ejected due to the strong pull back of the ligament to the acoustic velocity is not much different (between 1,481 m nozzle. Similar patterns are observed for all cases shown in s−1 (water) and 1,980 m s−1 (100 % glycerin) at 20 ◦C), Fig. 7. Figure 8 shows the drop volume changes with driv- the optimum dwell/echo time appears to have similar val- ing voltage. Even though we have only a limited number ues. After choosing the dwell/echo and rise/fall times, we of data points for each fluid with the same base fluid, we
Fig. 7 Velocity of drop for differing driving voltage. The solvent is a water, b 4.5 cP, c 10 cP, and d 16.5 cP 320 Rheol Acta (2013) 52:313–325
Fig. 8 Total ejected volume for differing driving voltage. The solvent is a water, b 4.5 cP, c 10 cP, and d 16.5 cP can note that the drop volume is a function of driving volt- the inertia of the liquid and the pull-back action of the piezo- age only regardless of xanthan gum concentration since the element, and hence the drop velocity is determined by the data points are continued along a single curve (with some ejection velocity at the nozzle tip and the extensional char- overlaps) for solutions from the same base fluid except for acteristics of fluid. In the next discussion, we have described the 200-ppm solutions. In the case of the 200-ppm solu- these two steps sequentially. tions in 16.5-cP solvent, it was impossible to generate a To understand the fluid ejection step, we need to know drop for the full range of driving voltage. The concentra- the amount of fluid that is ejected by the acoustic pres- tion independency was also reported by Hoath et al. (2009) sure wave and its relation with fluid properties. Since it (the deviation of the 200-ppm solutions from the general is very difficult to measure the velocity profile at the exit trend will be considered later in this report). The indepen- accurately, we estimated it by numerical simulation using dency of concentration on drop volume indicates that drop Fluent™, a commercial software package based on the finite volume is a function of infinite shear viscosity only. This volume method. We used 6,320 elements and 6,657 nodes is quite in contrast to the fact that drop velocity is a strong in the simulation. The operating conditions were obtained function of xanthan gum concentration. The two contrasting by the following method. From the two consecutive images observations imply that drop generation process consists of on jetting of a Newtonian fluid, the flow rate through the two separate stages, and each stage is governed by differ- nozzle was obtained for a typical pressure wave form. From ent physical properties: At the first stage, a certain amount the numerical simulation results, we read the pressure drop of liquid is ejected from the nozzle and the drop volume between the manifold and the exit of the nozzle that gave is mainly determined by this step. At the second stage, the the same flow rate as the experimentally observed flow ejected liquid is pinched off from the fluid in the nozzle by rate value for the Newtonian fluid. We assumed that the Rheol Acta (2013) 52:313–325 321
observed the velocity blunting of water more clearly. As vis- cosity increases, the velocity profile becomes close to the parabolic profile. Next, we performed the simulation for the Carreau model fluids with the experimentally fitted numer- ical parameters. In Fig. 10, we have plotted the velocity profiles for two different sets of fluids with the same infi- nite shear viscosities. It is seen that the velocity profiles are almost the same regardless of zero shear viscosity of fluids tested here, in other words, xanthan gum concentration. This is because at the central region, the inertial term is dominant while shear rate is extremely large near the boundary as in the case of a Newtonian fluid. Therefore, the shear rates at which viscosity changes appreciably occur for a very narrow region near the center. This result means that the exit veloc- ity is almost the same regardless of zero shear viscosity as Fig. 9 Velocity profiles of Newtonian fluids at the nozzle exit at a long as infinite shear viscosities are the same for differing typical pressure difference of 20,000 Pa between the nozzle inlet and fluids. The dependence of drop size only on driving voltage exit acoustic velocity of a dilute polymer solution was the same as the solvent. Then, the pressure drop between the man- ifold and the exit should be the same regardless of fluid for the given wave form. Using the same pressure drop, we calculated the velocity profile while varying fluids of dif- ferent zero shear viscosities. Since xanthan gum solutions show elasticity, it has to be considered in velocity profile calculation for the flow through the converging geometry of nozzle due to Lagrangian unsteadiness. Until now, the converging flow of elastic fluids has been treated only for Oldroyd B and upper convected Maxwell fluids (Hull 1981; Evans and Hagen 2008). Since the fully nonlinear constitu- tive model for xanthan gum solutions is not available at the present time, we were not able to include the elastic effect in the numerical simulation properly. Rather, we neglected the elasticity of xanthan gum solutions since the viscoelas- tic measurements showed that G was substantially smaller than G even at high-frequency regimes. Therefore, even though the velocity profile obtained by the present method may not represent the physics of the problem exactly, we can obtain at least semiquantitative result. The effect of elasticity will be considered later in this study. Figure 9 shows the velocity profiles of Newtonian fluids of differ- ing viscosities for a typical pressure difference of 20,000 Pa between the exit and the entrance of nozzle. When vis- cosity is 1 cP, the exit velocity is severely blunted. The blunting of velocity profile is caused by the inertial effect in the Jeffery–Hamel flow (Batchelor 1967). In the blunted ∂vr region, the inertial term ρ vr ∂r dominates over the viscous term and is balanced with the pressure term, while near the solid boundary, the viscous term dominates over the iner- tial term. The blunted velocity profile at the nozzle exit can Fig. 10 Numerical simulation result on the effect of xanthan gum con- centration on the velocity profile at the nozzle exit for different solvent: be surmised from the experiment for a low-viscosity fluid a water and b 16.5 cP. In each case, the driving voltage is different to (please see the third frame of Fig. 6a). Dong et al. (2006) simulate real situations 322 Rheol Acta (2013) 52:313–325 has been already observed in the experiment as described above. Therefore, we can argue that drop volume is mainly determined by infinite shear viscosity. Also, from the match between experimental and simulation results on drop size, we may argue that elasticity is not important in the present case except for the 200-ppm solution. Experimentally, the ejected volume of the 200-ppm solutions is smaller than the pure solvent. It appears that the difference in experimental results is due to the elasticity of fluids which has not been taken into account in the Carreau model. This point will be considered more in this paper. Fig. 12 Filament shapes during thinning for the 100-ppm xanthan Next, we consider the detachment of a drop. During the gum solution in 16.5-cP solvent. From the left: after the loading, just detachment process, the ejected fluid is elongated and the after the pulling apart, establishment of the cylindrical shape, and just neck becomes thinned. From the images shown in Fig. 6,we before the breakup can estimate the order of the extension rate. Knowing that the frame rate is 50,000 s−1 and the ligament length changes extensional rate cannot be measured by a commercial exten- from 0 to 400 μm between four frames, the average exten- sional rheometer as of now. Therefore, we used the CaBER −1 sional rate (( L/ t)/Laverage) is 25,000 s . As far as the (ThermoHaake Co.) to estimate the extensional viscosity authors are aware of, the extensional viscosity at this high based on the liquid bridge stretching. In Fig. 11,wehave shown the changes in the diameter of the stretched liquid bridge as a function of time for some of the samples tested in the present study. In Fig. 11, we note that the time evolution of thread diameter is substantially delayed when xanthan gum concentration increases, meaning that the extensional viscosity increases with the increase in xanthan gum con- centration. Just before the breakup, the diameter decrease pattern changes to an exponential shape. It appears that, just before breakup, the dominant resistance to filament thinning is changed to inertia. In the figure, we note that the diameter changes become linear after a transient period and until they become exponential. We find that the filament has a cylin- drical shape during the thinning process as shown in Fig. 12. From the cylindrical filament shape, we can confirm that the thinning process follows the elasto-capillary thinning regime. Also, from the linear decrease in R(t) with time, we can confirm that the extensional viscosity is the same during the linear decrease. Hence, we can calculate the extensional viscosity and extensional rate by using Eqs. 6 and 3, respec- tively. In Table 3, we have listed extensional viscosities and extensional rates obtained by this method. In all cases, the xanthan gum solutions show much larger extensional vis- cosities compared with the corresponding solvents. In the
Table 3 Extensional viscosity of some selected samples
−1 Sample Range of ε˙,s ηE,Pas
10 cP + 100 ppm 108 ± 17–256 ± 112 4.2 ± 0.6 10 cP + 200 ppm 47 ± 14–142 ± 83 6.3 ± 1.1 16.5 cP + 100 ppm 87 ± 26–172 ± 96 4.0 ± 0.6 + ± ± ± Fig. 11 Typical results on the diameter change of liquid filament in 16.5 cP 200 ppm 29 9–58 34 9.0 2.0 the capillary breakup experiment. The piston movement stopped at t = 0; a 10-cP-based solutions and b 16.5-cP-based solutions For each sample, more than seven runs were averaged Rheol Acta (2013) 52:313–325 323
2 table, in the case of the 200-ppm xanthan gum solution in Vl/ πd /4 to a cylinder of length l(t). Then, the Hencky 10-cP solvent, the extensional viscosity is 6.3 Pa s for exten- strain is given as follows: −1 sional rates between approximately 47 and 142 s .The πd2l(t) extensional viscosity of 6.3 Pa s is much larger than the ε (t) = ln (9) l 4V Trouton viscosity (three times the shear viscosity) based on l the zero shear viscosity of 215 mPa s and the Trouton vis- In estimating the strain of the filament, the drop with the cosity based on the infinite shear viscosity of 30 mPa s. The lowest velocity we observed is used for a conservative esti- substantially larger extensional viscosity than either of the mate. For drops with higher velocities, the strain rate will be Trouton viscosities is due to the strain-hardening behavior higher. In the case of the 100-ppm solution in 16.5-cP sol- of the xanthan gum polymer. Strain hardening occurs when vent, the Hencky strain when the ligament length becomes polymers are stretched and aligned. The degree of alignment 240 μm just before the breakup is ln 20 while the strain of should be much larger for xanthan gum (stiff polymer) than the filament during the CaBER experiment is ln 16 when that for flexible polymers. The easiness of alignment can be the filament diameter begins to decrease linearly (0.02 s in seen from the strong shear-thinning viscosity. This means Fig. 11a). At another case of the 200-ppm solution in 10-cP that the extensional viscosity obtained in this study can be solvent, when the ligament length is 214 μm, these values quite close to the true value at the drop generation condi- are ln 33 and ln 16, respectively. Considering that the strain tion. The true values can be larger than the values measured rate is much higher for the ligament stretching during drop here, but the measured values can be useful enough to con- generation than the filament stretching during the CaBER firm that the extensional stress reduces the velocity of the experiment, one can confirm that the polymers in the liga- inkjet droplet. ment stretching are almost fully stretched, and therefore the Since the extensional rate at the pinch-off condition of extensional viscosity obtained by CaBER can be applied to the drop generation process (in the order of 25,000 s−1 as the ligament stretching. described above) is much larger than the value at the mea- As the extensional viscosity of the fluid thread is much suring condition of 47–140 s−1, the extensional viscosity larger than that of the Newtonian fluid with the same infi- at the pinch-off condition should not be smaller than 6.3 nite shear viscosity, during the extension of the thread, the Pa s considering the extensional hardening characteristics extensional stress will strongly retard the deformation or of polymers. The value 6.3 Pa s in the present case may be breakup. Also, before the pinch off, the extensional stress the value at the fully extended state and strain hardening is retards the flight of the drop head. Therefore, even if drop already saturated. If this is the case, the extensional viscos- sizes are the same regardless of concentration of xanthan ity at the pinch-off condition will be at least 6.3 Pa s. Here, gum as long as the infinite shear viscosity is the same, flight one may raise a question whether the polymers inside the velocity is strongly dependent on xanthan gum concentra- ligament are fully extended, and hence the extensional vis- tion. Due to the limited spatial resolution of the image and cosity obtained by CaBER can be applied to the ligament time interval between two subsequent frames, we have not stretching. To warrant the application, we have compared been able to estimate the force acting on the head quantita- two strains as follows: First, the Hencky strain of the fila- tively to estimate the velocity change. As a rough estimate, dv ≈ πD3 v ment from the unstretched state in the CaBER experiment the inertial force acting on the drop head is m dt ρ 6 t is (m and v are the mass and the velocity of the ligament head, respectively) while the force due to extensional stress l(t) D2 is πR2 × η ε˙. If we insert a set of numerical values at ε(t) = ln = ln 0 , (8) neck E 2 v = −2 = l0 D(t) 30-V driving voltage ( t 50,000 m s , D 50 μm, −1 Rneck = 5 μm, ηE = 6.3Pas,ε˙ = 25,000 s ),the where l and D are the length and diameter of the fila- inertial and extensional forces are 6 × 10−6 and 1 × 10−5 ment at time t and the subscript 0 denotes the value before N, respectively. In this case, the drop velocity is substan- stretching. Next, the total strain of the inkjet ligament can tially decreased from the ejection velocity at the nozzle tip, be estimated as follows: Before the breakup from the noz- and hence it should be impossible to detach the ligament zle, the ejected liquid element can be divided into two parts: from the nozzle. In this case, the surface tension force is ∼ −6 drop head and ligament. The volume of the ligament (Vl) πDσ = 1.0 × 10 N and is almost 1 order smaller than can be calculated from the difference between the drop vol- the inertia or extensional force. Hence, the surface tension ume (Vd) and the drop head volume (Vh): Vl = Vd − Vh. force does not strongly retard the deformation. As shown in The ligament volume Vl is assumed to be maintained. When Fig. 7c, for the 200-ppm solution, no drop is generated at liquid comes out of the nozzle, the diameter of the liquid this condition of 30 V and the minimum driving voltage is element is the same as the nozzle diameter (d), and hence 36 V. For the case of the 100-ppm solution, the extensional the liquid volume is extended from a cylinder of length viscosity is 4.2 Pa s and the extensional force is 6 × 10−6 324 Rheol Acta (2013) 52:313–325
N. In this case, the inertial force and the extensional force shear rate or frequency. The result shows that drop gen- are balanced and the drop is barely generated as shown in eration process consists of two independent processes of Fig. 7c. Of course, this calculation is just for an example, ejection and detachment. The ejection process is found to and depending on diameter and other factors, there should be controlled primarily by high or infinite shear viscosity. be wide variations in forces. Elasticity can affect the flow rate (drop size) through the Following the argument of Hoath et al. (2009), we have converging section of an inkjet nozzle. However, the elas- checked whether viscoelastic properties at high-frequency tic effect may not strongly affect the flow rate since the regime are correlated with the drop generation character- residence time of polymer molecules of the section is too istics. In the present case, the G |G∗| values at 5 kHz short for the elastic stress to grow to a significant level. The (31,400 s−1) are almost the same for two different flu- detachment process is controlled by the extensional viscos- ids in Fig. 5a, b, and hence the correlation could not be ity. Due to the strain hardening of polymers, the extensional found. This implies that the linear viscoelastic property can- viscosity becomes orders of magnitude larger than the Trou- not affect the drop generation process directly and the effect ton viscosities based on zero and infinite shear viscosities. of added polymers manifests itself as increased extensional The large extensional stress retards the extension of liga- viscosity which primarily affects the detachment process. ment and hence lowers the flight speed of the ligament head. The increased extensional viscosity may play a role also in The viscoelastic properties at high-frequency regime do not the converging flow inside the nozzle at high xanthan gum appear to be directly related to the drop generation pro- concentrations, too. As seen in Fig. 8, the drop size of the cess even though it is surmised that the elastic effect should 200-ppm solutions is smaller than those from less concen- strongly affect the extensional properties. trated solutions. This difference appears to be caused by We have not performed the numerical simulation on the the increase in extensional viscosity of the 200-ppm xan- whole drop generation process since, first of all, there exists than gum solutions inside the nozzle. This is because the no constitutive relation which is reasonably well matched flow through the conical region of the nozzle is strongly to the experimental data for xanthan solutions. Also, the extensional, especially at the central region. However, it is detailed numerical analysis is far beyond the scope of the expected that the contribution of extensional stress may not present paper. But it should be worth doing by considering be as large as in the case of ligament extension. This is the flow inside the nozzle and detachment process together. because the mean residence time of polymer chains within This is especially important in non-Newtonian liquids since the region is very short, and hence the polymer chains may polymers are elongated during the converging flow inside not be fully extended. It can be confirmed from the fact the nozzle and the elongated polymers cannot be relaxed that the reduction of ejection velocity is not noticeable for until they come out of the nozzle exit considering the pro- less concentrated solutions. We are sure that the analysis of cess time of 100 μs and the relaxation time of the same order the detailed process should be performed through an elab- for most polymers. orate constitutive modeling on the xanthan gum solutions Even though the present research has been performed and possibly numerical solutions of the governing equa- with polymeric liquids only, the same principle should be tions based on the model. Another issue is the location of applied to inks from other materials. Especially, the present detachment. Depending on extensional properties, the loca- result can also be used even for predicting the jetting behav- tion of detachment can vary, which results in the droplet size ior of the solution of a flexible polymer which can be change. regarded as a different class of fluids from the xanthan gum solution. Considering that suspensions do not show strong strain hardening, the detachment will be much easier. How- Summary and conclusion ever, since most inks contain polymers and/or surfactants for suspension stability, the strain hardening can affect the In this study, the generation of inkjet droplets of non- detachment to a certain degree. As many different kinds of Newtonian fluids has been investigated experimentally. polymers and large molecules of various shapes are used Noting that most of inks used in inkjet technology are rhe- in electronic industries such as organic light-emitting diode ologically complex fluids, xanthan gum solutions in water– displays and polymer light-emitting diodes, the results of glycerin mixtures have been chosen as model inks. The the present research will be valuable information for those rheological properties of xanthan gum solutions have been industries. characterized by the diffusive wave spectroscopy microrhe- ological method for high-frequency viscoelastic properties and the capillary breakup method for extensional viscosity Acknowledgement This work was partially supported by Mid- as well as conventional rotational rheometry for viscosity career Researcher Program through NRF grant funded by the Ministry and linear viscoelastic properties for moderate values of of Education, Science and Technology, Korea (no. 2010–0015186). Rheol Acta (2013) 52:313–325 325
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