Generation of inkjet droplet of non-newtonian fluid

Transcription

1 Rheol Acta (2013) 52: DOI /s 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 of xanthan gum solutions in water glycerin mixtures was investigated experimentally to understand the jetting and drop generation mechanisms of rheologically complex fluids using a drop-on-demand inkjet system based on a piezoelectric nozzle head. The ejected volume and velocity of droplet were measured while varying the wave form of bipolar shape to the piezoelectric inkjet head, and the effects of the rheological properties were examined. The shear properties of xanthan gum solutions were characterized for wide ranges of shear rate and frequency by using the diffusive wave spectroscopy microrheological method as well as the conventional rotational rheometry. The extensional properties were measured with the capillary breakup method. The result shows that drop generation process consists of two independent processes of ejection and detachment. The ejection process is found to be controlled primarily by high or infinite shear viscosity. Elasticity can affect the flow through the converging section of inkjet nozzle even though the effect may not be strong. The detachment process is controlled by extensional viscosity. Due to the strain hardening of polymers, the extensional viscosity becomes orders of magnitude larger than the Trouton viscosities based on the zero and infinite shear viscosities. The large extensional stress retards the extension of ligament, and hence the stress lowers the flight speed of the ligament head. The viscoelastic properties at the high-frequency regime do not appear to be directly related to the drop generation process even though it can affect the extensional properties. H. Yoo C. Kim ( ) Department of Chemical and Biological Engineering, Korea University, Anam-dong, Sungbuk-ku, Seoul , South Korea cykim@grtrkr.korea.ac.kr Keywords Drop-on-demand inkjet Elasticity Shear thinning Infinite shear viscosity Jeffery Hamel flow Strain hardening DWS microrheology Introduction As the inkjet printing technology has widen its application to bio and electronic industries beyond household or office inkjet printers (Basaran 2002; Schubert 2005;de Gans et al. 2004), many different kinds of inks have to be handled. In most cases, inks are suspensions or polymeric liquids (de Gans et al. 2005), and hence most of the inks are rheologically complex fluids showing shear-dependent viscosities and/or elastic characteristics. Some additives such as surfactants are usually added in suspensions for stabilization and better performances. This can make the rheology of suspension more complex. However, inks have not been characterized properly especially at the operating conditions of inkjet printing, and the processing conditions have been sought mostly through trial and error basis. To generate inkjet droplets, either the continuous jetting or drop-on-demand (DOD) method can be used (Derby 2010). In the DOD method, droplets are generated by applying a pressure wave to a liquid-filled nozzle. Then, a portion of liquid is squeezed out of the nozzle overcoming the surface tension force, and the liquid element is detached from the nozzle tip by inertia and capillary force. In this stage, 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 and some smaller satellite drops by instability mechanisms. In some cases, satellite drops can be merged into the leading drop. It is known that the formation of satellite drops should be avoided for better printing quality, and hence the

2 314 Rheol Acta (2013) 52: determination of the proper window on the operating parameters for a single drop generation is one of the most important issues in inkjet droplet generation. It has been reported that drop generation characteristics are governed by Ohnesorge number (Oh) of the drop, which is the ratio of viscous time scale and surface tension time scale and defined as follows (Derby 2010): Oh = η ρrγ, (1) where η is viscosity, ρ is density, R is radius, and γ is surface tension. For Newtonian fluids, Oh is unequivocally defined since fluid properties are independent of flow conditions. But in the case of shear thinning fluids, viscosity is a function of shear rate, and hence the operating window cannot be predicted based on the theory for Newtonian fluids (Lai et al. 2010; Tai et al. 2008). It has been the usual notion that the rheological behaviors at large strain rates control the generation of inkjet droplet (Hoath et al. 2009) since the drop generation process is an ultrahigh shear rate process with an average shear rate of 10 5 s 1 order. Actually, inks show finite viscosities as shear rate goes either to zero or a very large value. In this case, some questions should arise naturally. Is the zero shear viscosity not relevant to the generation of inkjet drop generation? If this is so, is the infinite shear viscosity the only variable that affects drop generation? If not, what other properties are relevant to drop generation? In the present paper, we have tried to answer these questions by using a class of fluids (as model inks) which show various rheologically complex behaviors such as elasticity, shear thinning, and strain hardening. Shore and Harrison (2005) reported that the presence of a small amount of polymer in a Newtonian solvent can have a significant change in the inkjet drop generation characteristics. Especially, satellite drop formation is suppressed and the drop velocity is significantly lowered by the addition of polymer. Using two different types of polymers (linear and star polymers), de Gans et al. (2005) reported that the distance traveled by the primary droplet was dependent only on input voltage to the piezo-element and independent of polymer concentration, molecular weight, and topological architecture. They also reported that the rupture of the ligament was dependent on the rheological properties of the solution. Hoath et al. (2009) noticed that, in the generation of inkjet droplet of elastic polymer solutions, the final main drop size was independent of polymer concentration even though the length of the ligament increased markedly with the elasticity of the fluid. In the meantime, Hoath et al. (2009) did not observe any correlation between low shear viscosity and jetting behavior for the fluids they investigated, but the jetting behavior was well correlated with high-frequency rheological properties measured at 5 khz using a piezoelectric axial vibrator rheometer. Here, it is noted that the drop generation is a highly nonlinear process of large extension and high extensional rate, and therefore linear viscoelastic properties may not be correlated quantitatively with the nonlinear process. Morrison and Harlen (2010) investigated the effects of viscoelasticity numerically on drop formation in inkjet printing by using viscoelastic fluids represented by the single-mode FENE-CR constitutive equation (Chilcott and Rallison 1988). They showed that the ligament became longer for elastic liquids and the formation of satellite drops was suppressed by elasticity. Also, they argued that the lowering of drop speed was due to elasticity. Recently, Hoath et al. (2012) presented a quantitative model which predicted three different regimes of behavior depending upon the jet Weissenberg number (Wi) and extensibility of polymer molecule. They predicted Newtonian regime (Wi < 1/2), viscoelastic regime with partial extension (1/2 < Wi <L,whereL is the extensibility of polymer chain), and fully extended regime (Wi >L). They also gave the scaling law for the maximum polymer concentration at which a jet of a certain speed could be formed as a function of molecular weight of polymer. Their analysis is based on the FENE-CR model which is valid for solutions of flexible polymers. Also, their analysis is limited to the detachment process, and hence the model cannot predict the drop size. Therefore, more studies on drop size and drop velocity should be still required to understand the mechanism of inkjet drop generation of viscoelastic fluids. All of these papers argued that the elasticity has a significant effect on the generation of inkjet droplet of elastic solution. However, they did not give the detailed reason why elasticity could affect the drop formation. In the present paper, we have examined the elongation characteristics and linear viscoelasticity of inks along with the flow of inks inside the nozzle and their effects on inkjet drop generation. The elongation of liquid thread has been an important issue in rheology. In continuous jetting, dripping, necking, and breakup of liquid bridge, the thinning of a liquid filament is driven by capillarity and resisted by inertia, viscosity, and elasticity. On the other hand, the stretching of the ligament in the early stage of drop formation is primarily driven by the inertia (in the main flight direction) of the ligament head and resisted by surface tension, inertia (in the perpendicular direction to the main flight direction), viscosity, and elasticity of fluid. Hence, the detailed flow should not be the same. At the final stage of filament breakup, it is known that the breakup process is determined by the natural variables of surface tension and fluid properties regardless of boundary and initial conditions (Eggers 1993; Renardy 2004; McKinley 2005). Therefore, we may gain useful information from the capillary breakup test. The visco-elasto-capillary thinning of complex fluids has been studied extensively since Eggers (1993) first found the similarity solution for the one-dimensional governing

3 Rheol Acta (2013) 52: equation and subsequent reports on the similarity solutions for various non-newtonian models. Detailed reviews on the self-similar solutions for non-newtonian models were given by Renardy (2004), and comprehensive reviews were given by McKinley (2005) on capillary thinning of liquid bridges and its applications to extensional rheometry. The studies on capillary thinning have shown that, in the case of Stokes flow of a Newtonian fluid, the filament breaks off at a finite time t c and the radius of the filament changes with time as follows (Papageorgiou 1995): R mid = σ (t c t) (2) R 0 η s R 0 Then, the extensional rate at the middle of the filament is given as follows: ε mid (t) = 2 dr mid = 2 (3) R mid dt t t c In the above equations, ε is extension rate, R is radius, η s is viscosity, and t is time. Also, subscripts mid and 0 denote the value at the midpoint of the filament and initial radius, respectively. In the case of elastic fluid (McKinley 2005), the capillary thinning flow becomes a homogeneous extensional flow with Wi = λ 1 ε = 2/3 (4) and the radius changes with time as follows: ( ) R 1/3 mid GR0 = exp ( t/(3λ 1 )) (5) R 0 2σ where G is modulus and λ 1 is the longest relaxation time. In this case, there is no finite breakup time and there is a long tail. This equation is valid for a dilute solution of infinitely extensible polymers. But in a real polymeric liquid, polymers cannot be extended infinitely. Renardy (2002) and Fontelos and Li (2004) have shown that, for viscoelastic fluids of Giesekus and FENE-P types, the jet diameter decreases linearly with time when close to the breakup: [ ] R(t) σ = (t c t) (6) R 0 2η E R 0 This result means that, as polymer molecules are fully stretched at a sufficiently large strain rate, the extensional viscosity approaches a constant value and the fluid behaves as a Newtonian fluid with a constant extensional viscosity of η E (McKinley 2005; Stelter et al. 2002, 1999). From this relationship, we may obtain the extensional viscosity of inkjet fluid from the capillary breakup experiment. If we can observe a linearly decreasing filament radius while exhibiting a cylindrical filament, η E can be obtained from the slope. The η E value obtained here can be close to the true extensional viscosity at the inkjet drop generation condition since polymer coils can be almost fully stretched at the high strain rate, and hence η E value approaches the limiting value. In the present research, we have attempted to correlate the relevant variables to each drop-generating step by performing inkjet drop generation experiments with xanthan gum solutions together with numerical simulations on the flow inside the nozzle. Since xanthan gum solutions are less elastic than most of the flexible polymer solutions, the result presented here can describe the practical inkjet problems more realistically. In the analysis of experimental data, we used the rheological properties at the real processing condition measured by the diffusive wave spectroscopy microrheology method for high-frequency linear viscoelastic properties and capillary breakup method for extensional viscosities at high strain rate as well as the conventional rotational rheometry. The result shows that drop generation process consists of two separate stages: At the first stage, a certain amount of liquid is ejected from the nozzle and the drop volume is determined by this step. Especially, it has been found that drop volume is determined mainly by the infinite shear viscosity of the xanthan gum solution. At the second stage, the ejected liquid is pinched off from the fluid inside the nozzle by the inertia of the liquid and the pulling-back action of piezo-element, and the drop velocity is determined by the ejection velocity and the extensional viscosity of fluid. As xanthan gum solutions show most of the important characteristics of non-newtonian fluids such as elasticity, shear thinning, and extensional thickening, the present study can give an insight into the processing of non- Newtonian fluids for various applications by using a DOD inkjet printing system. The present result can be also used even for predicting the jetting behavior of the solution of a flexible polymer which can be regarded as a different class of fluids from the xanthan gum solution. Experiment To investigate the generation of inkjet drops, we set up an inkjet system as shown in Fig. 1, which is the same as the set that one of the authors used for the previous studies on Fig. 1 Schematic diagram of the experimental setup

4 316 Rheol Acta (2013) 52: spreading of inkjet drop (Son et al. 2008). In the present case, there is no such part as the solid surface. The system consists of an inkjet nozzle, a jetting driver (pulse generating system), a high-speed camera, and an illumination source. Inkjet system and imaging The inkjet droplet was generated by a piezo-type nozzle purchased from MicroFab Co. (Model # MJ-AT). The nozzle diameter at the exit was 50 μm. In Fig. 2, the inside geometry 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 refraction at the curved nozzle surface can be avoided. To generate droplets, a bipolar wave form was used as shown in Fig. 3. During the rise period (a), the piezoelement 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 the nozzle. This state continues during the echo period (d). Finally, the piezo-element expands to return to the initial state while completing a cycle (e). Depending on the time intervals and the voltages imposed on the piezo-element, a drop or drops of different sizes and velocities are generated. In the present research, the rise and fall times in the voltage pulse to the nozzle were set at 2 μs. The dwell and echo times were in the range of 4 32 μs, and the dwell and echo voltages were in the range of V. We performed drop generation experiments by varying operating conditions and measured the drop size and velocity. All the experimental runs were performed at the room condition. The high-speed camera (IDT, XS-4) was triggered by the jetting driver as a droplet was ejected from the inkjet nozzle. The camera was equipped with a microscopic objective lens (Mitutoyo, M Plan Apo) with the magnification of 5. As the illumination source, a back lighting system (Stocker Yale, # 21 AC, 180 W) was installed. The CCD camera can capture 50,000 frames per second and the pictures were taken with this mode. The exposure time was 1 μs. When this fast mode was used, the number Fig. 3 Pulse wave form for generating inkjet droplets: a expansion of nozzle chamber, b delay for pressure wave propagation, c compression of fluid for ejection, d delay for pressure wave propagation, and e nozzle chamber expansion to the initial state of pixels per frame has to be small ( pixels). The pixel size was 3.76 μm. One may use the flash videography method to get the better quality as demonstrated by van Dam and Le Clerc (2004) and Dong et al. (2006). But, in the case of non-newtonian drops, the reproducibility of drop generation was not as good as in case of Newtonian fluids; hence, we decided not to use the flash videography method used in the literature. Considering that the drop diameter and travel distance are in the order of 50 μm and 1 mm, respectively, the numbers of pixels to cover these sizes are about 15 and 150. Therefore, the image resolution was enough to measure drop diameter and velocity. Materials Three kinds of Newtonian fluids were prepared with different viscosities by mixing deionized water and glycerin (Sigma-Aldrich Co.). Shear-thinning fluids were prepared by dissolving xanthan gum (Sigma-Aldrich Co.) in deionized water or in one of the water glycerin mixtures. Table 1 shows the composition of the fluids tested here. The 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 Fig. 2 Inside geometry of the nozzle. The inlet diameter is 456 μm and the exit diameter is 50 μm Deionized water, wt% Glycerin, wt% Viscosity, mpa s Surface tension, mn/m

5 Rheol Acta (2013) 52: 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 distance 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 Noüy ring method (K9 Tensiometer, KRÜSS GmbH). Result and discussion 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 xanthan solutions have shear-thinning viscosities. The viscosity of xanthan solution is fitted to the Carreau model (Bird et al. 1987): η η 1 = η 0 η [ 1 + (β γ ) 2 ], (7) 1 n 2 and the model parameters are listed in Table 2. In the table (alsoinfig.4), we note that the infinite shear viscosities of xanthan gum solutions of the same base solvent are only slightly changed from or almost the same as the viscosity 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 of the solvent. In the following discussion, we will compare the drop generation characteristics of the fluids with the same base fluid systematically. Figure 5 shows linear viscoelastic properties of some of the fluids listed in Table 2. Figure 5 shows that the shapes of G and G for the xanthan gum solutions are not similar to those of polymer solutions of flexible polymers such as Boger fluids in that the typical slope of 2 for G at low-frequency regime does not appear yet at the lowest frequency of 0.1 s 1. The data obtained from the DWS microrheology appear to be reasonably extended to the conventional rheometry data. Table 2 Xanthan gum solutions studied here and their Carreau model parameters and surface tension 1.0 cp 4.5 cp 50 ppm 100 ppm 200 ppm 50 ppm 100 ppm 200 ppm η 0 (mpa s) η (mpa s) β,s n σ (mn/m) cp 16.5 cp 50 ppm 100 ppm 200 ppm 50 ppm 100 ppm 200 ppm η 0 (mpa s) η (mpa s) β,s n σ (mn/m)

6 318 Rheol Acta (2013) 52: 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 solutions have smaller G than G for all the frequency regimes tested here including the DWS microrheology measurement regime. This implies that the elastic effect should not be large even at the inkjet drop generation condition (shear rate of 10 5 s 1 ). Figure 6 shows some typical drop generation patterns for differing fluids at differing conditions. In Fig. 6a, a thin liquid ligament with a spherical head is formed, and then the ligament tail contracts to the head eventually. Therefore, only one drop is generated. In Fig. 6b, the situation is much the same as Fig. 6a, but the tail is separated from the main head and becomes a satellite drop. More than one satellite drop can be generated and these drops can be coalesced depending upon the relative velocity between the main and satellite droplets. The separation of the satellite drop is caused by the capillary instability or end pinching (Stone Fig. 6 Typical drop generation patterns. The time interval between two adjacent frames is 20 μs. a Single-drop formation for the 100-ppm 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 tail for the 50-ppm solution in 10-cP solvent. Driving voltage is 36 V. c A satellite drop generated by reflected acoustic waves for the 50-ppm solution in water. In this case, the tail is shrunk to the main drop, but a new drop is generated behind the main drop 1994; Stone et al. 1986). These drops may be called satellites from tail. In Fig. 6c, a satellite drop is generated by a different mechanism. In this case, the satellite drop is not linked to the original ligament and appears to be generated by the reflected acoustic wave inside the nozzle. These satellite drops may be called satellites from reflected wave. In the following discussion, we will consider the cases for which only one single drop is generated. To generate one single drop without satellites, a proper operating window

7 Rheol Acta (2013) 52: for voltage and time of dwell and echo has to be chosen as well as rise and fall time. Since there are too many combinations for operating parameters, we confine ourselves to the following cases. First, the rise and fall times were fixed at 2 μs. Dwell and echo voltages were set to be equal, and then we changed dwell/echo time to find out the condition at which the drop velocity became the maximum value. In most of the cases, the drop velocity became the maximum or close to the maximum value when dwell/echo time was 24 μs. Therefore, we fixed dwell/echo time at 24 μs. The optimum dwell/echo time (the condition at which the drop velocity attains the maximum value) is closely related to the length of the nozzle and acoustic velocity of fluid (Bogy and Talke 1984). Since the nozzle length is fixed and the acoustic velocity is not much different (between 1,481 m s 1 (water) and 1,980 m s 1 (100 % glycerin) at 20 C), the optimum dwell/echo time appears to have similar values. After choosing the dwell/echo and rise/fall times, we performed experiments while varying dwell/echo voltage and collected data when only a single drop was generated. Figure 7 shows the drop velocity variations for the fluids listed in Table 1. First, we note that, for all cases, as dwell/echo voltage (voltage hereafter) increases, drop velocity increases. Also, the minimum voltage required to generate a single drop increases with xanthan gum concentration. At a fixed voltage value, drop velocity becomes smaller with an increase in xanthan gum concentration. For example, when the viscosity of solvent is 4.5 cp, the drop velocity is 4.5, 3.2, and 0.6 m s 1 when the driving voltage is 24 V and the xanthan gum concentration is 0, 50, and 100 ppm, respectively. At 24 V, the 200-ppm solution could not be ejected due to the strong pull back of the ligament to the nozzle. Similar patterns are observed for all cases shown in Fig. 7. Figure 8 shows the drop volume changes with driving voltage. Even though we have only a limited number 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

8 320 Rheol Acta (2013) 52: 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 voltage only regardless of xanthan gum concentration since the data points are continued along a single curve (with some overlaps) for solutions from the same base fluid except for the 200-ppm solutions. In the case of the 200-ppm solutions in 16.5-cP solvent, it was impossible to generate a drop for the full range of driving voltage. The concentration independency was also reported by Hoath et al. (2009) (the deviation of the 200-ppm solutions from the general trend will be considered later in this report). The independency of concentration on drop volume indicates that drop volume is a function of infinite shear viscosity only. This is quite in contrast to the fact that drop velocity is a strong function of xanthan gum concentration. The two contrasting observations imply that drop generation process consists of two separate stages, and each stage is governed by different physical properties: At the first stage, a certain amount of liquid is ejected from the nozzle and the drop volume is mainly determined by this step. At the second stage, the ejected liquid is pinched off from the fluid in the nozzle by the inertia of the liquid and the pull-back action of the piezoelement, and hence the drop velocity is determined by the ejection velocity at the nozzle tip and the extensional characteristics of fluid. In the next discussion, we have described these two steps sequentially. To understand the fluid ejection step, we need to know the amount of fluid that is ejected by the acoustic pressure wave and its relation with fluid properties. Since it is very difficult to measure the velocity profile at the exit accurately, we estimated it by numerical simulation using Fluent, a commercial software package based on the finite volume method. We used 6,320 elements and 6,657 nodes in the simulation. The operating conditions were obtained by the following method. From the two consecutive images on jetting of a Newtonian fluid, the flow rate through the nozzle was obtained for a typical pressure wave form. From the numerical simulation results, we read the pressure drop between the manifold and the exit of the nozzle that gave the same flow rate as the experimentally observed flow rate value for the Newtonian fluid. We assumed that the

9 Rheol Acta (2013) 52: Fig. 9 Velocity profiles of Newtonian fluids at the nozzle exit at a typical pressure difference of 20,000 Pa between the nozzle inlet and exit observed the velocity blunting of water more clearly. As viscosity increases, the velocity profile becomes close to the parabolic profile. Next, we performed the simulation for the Carreau model fluids with the experimentally fitted numerical parameters. In Fig. 10, we have plotted the velocity profiles for two different sets of fluids with the same infinite 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 velocity is almost the same regardless of zero shear viscosity as long as infinite shear viscosities are the same for differing fluids. The dependence of drop size only on driving voltage acoustic velocity of a dilute polymer solution was the same as the solvent. Then, the pressure drop between the manifold 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 different 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 constitutive 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 viscoelastic 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 differing viscosities for a typical pressure difference of 20,000 Pa between the exit and the entrance of nozzle. When viscosity 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 v region, the inertial term ρ v r r r dominates over the viscous term and is balanced with the pressure term, while near the solid boundary, the viscous term dominates over the inertial term. The blunted velocity profile at the nozzle exit can be surmised from the experiment for a low-viscosity fluid (please see the third frame of Fig. 6a). Dong et al. (2006) Fig. 10 Numerical simulation result on the effect of xanthan gum concentration on the velocity profile at the nozzle exit for different solvent: a water and b 16.5 cp. In each case, the driving voltage is different to simulate real situations

10 322 Rheol Acta (2013) 52: 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. Next, we consider the detachment of a drop. During the detachment process, the ejected fluid is elongated and the neck becomes thinned. From the images shown in Fig. 6,we can estimate the order of the extension rate. Knowing that the frame rate is 50,000 s 1 and the ligament length changes from 0 to 400 μm between four frames, the average extensional rate (( L/ t)/l average ) is 25,000 s 1. As far as the authors are aware of, the extensional viscosity at this high Fig. 12 Filament shapes during thinning for the 100-ppm xanthan gum solution in 16.5-cP solvent. From the left: after the loading, just after the pulling apart, establishment of the cylindrical shape, and just before the breakup extensional rate cannot be measured by a commercial extensional rheometer as of now. Therefore, we used the CaBER (ThermoHaake Co.) to estimate the extensional viscosity 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 concentration. 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 cylindrical 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, respectively. 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 viscosities compared with the corresponding solvents. In the Table 3 Extensional viscosity of some selected samples Sample Range of ε,s 1 η E,Pas Fig. 11 Typical results on the diameter change of liquid filament in the capillary breakup experiment. The piston movement stopped at t = 0; a 10-cP-based solutions and b 16.5-cP-based solutions 10 cp ppm 108 ± ± ± cp ppm 47 ± ± ± cp ppm 87 ± ± ± cp ppm 29 ± 9 58 ± ± 2.0 For each sample, more than seven runs were averaged

11 Rheol Acta (2013) 52: table, in the case of the 200-ppm xanthan gum solution in 10-cP solvent, the extensional viscosity is 6.3 Pa s for extensional rates between approximately 47 and 142 s 1.The extensional viscosity of 6.3 Pa s is much larger than the Trouton viscosity (three times the shear viscosity) based on the zero shear viscosity of 215 mpa s and the Trouton viscosity based on the infinite shear viscosity of 30 mpa s. The substantially larger extensional viscosity than either of the Trouton viscosities is due to the strain-hardening behavior of the xanthan gum polymer. Strain hardening occurs when polymers are stretched and aligned. The degree of alignment should be much larger for xanthan gum (stiff polymer) than that for flexible polymers. The easiness of alignment can be seen from the strong shear-thinning viscosity. This means that the extensional viscosity obtained in this study can be quite close to the true value at the drop generation condition. The true values can be larger than the values measured here, but the measured values can be useful enough to confirm that the extensional stress reduces the velocity of the inkjet droplet. Since the extensional rate at the pinch-off condition of the drop generation process (in the order of 25,000 s 1 as described above) is much larger than the value at the measuring condition of s 1, the extensional viscosity at the pinch-off condition should not be smaller than 6.3 Pa s considering the extensional hardening characteristics of polymers. The value 6.3 Pa s in the present case may be the value at the fully extended state and strain hardening is already saturated. If this is the case, the extensional viscosity at the pinch-off condition will be at least 6.3 Pa s. Here, one may raise a question whether the polymers inside the ligament are fully extended, and hence the extensional viscosity obtained by CaBER can be applied to the ligament stretching. To warrant the application, we have compared two strains as follows: First, the Hencky strain of the filament from the unstretched state in the CaBER experiment is ( ) ( ) l(t) D 2 ε(t) = ln = ln 0 D(t) 2, (8) l 0 where l and D are the length and diameter of the filament at time t and the subscript 0 denotes the value before stretching. Next, the total strain of the inkjet ligament can be estimated as follows: Before the breakup from the nozzle, the ejected liquid element can be divided into two parts: drop head and ligament. The volume of the ligament (V l ) can be calculated from the difference between the drop volume (V d ) and the drop head volume (V h ): V l = V d V h. The ligament volume V l is assumed to be maintained. When liquid comes out of the nozzle, the diameter of the liquid element is the same as the nozzle diameter (d), and hence the liquid volume is extended from a cylinder of length V l / ( πd 2 /4 ) to a cylinder of length l(t). Then, the Hencky strain is given as follows: ( πd 2 ) l(t) ε l (t) = ln (9) 4V l In estimating the strain of the filament, the drop with the lowest velocity we observed is used for a conservative estimate. For drops with higher velocities, the strain rate will be higher. In the case of the 100-ppm solution in 16.5-cP solvent, the Hencky strain when the ligament length becomes 240 μm just before the breakup is ln 20 while the strain of the filament during the CaBER experiment is ln 16 when the filament diameter begins to decrease linearly (0.02 s in Fig. 11a). At another case of the 200-ppm solution in 10-cP solvent, when the ligament length is 214 μm, these values are ln 33 and ln 16, respectively. Considering that the strain rate is much higher for the ligament stretching during drop generation than the filament stretching during the CaBER experiment, one can confirm that the polymers in the ligament stretching are almost fully stretched, and therefore the extensional viscosity obtained by CaBER can be applied to the ligament stretching. As the extensional viscosity of the fluid thread is much larger than that of the Newtonian fluid with the same infinite shear viscosity, during the extension of the thread, the extensional stress will strongly retard the deformation or breakup. Also, before the pinch off, the extensional stress retards the flight of the drop head. Therefore, even if drop sizes are the same regardless of concentration of xanthan gum as long as the infinite shear viscosity is the same, flight velocity is strongly dependent on xanthan gum concentration. Due to the limited spatial resolution of the image and time interval between two subsequent frames, we have not been able to estimate the force acting on the head quantitatively to estimate the velocity change. As a rough estimate, the inertial force acting on the drop head is m dv dt ρ πd3 v 6 t (m and v are the mass and the velocity of the ligament head, respectively) while the force due to extensional stress is πrneck 2 η E ε. If we insert a set of numerical values at 30-V driving voltage ( v t = 50,000 m s 2, D = 50 μm, R neck = 5 μm, η E = 6.3 Pas, ε = 25,000 s 1 ),the inertial and extensional forces are and N, respectively. In this case, the drop velocity is substantially decreased from the ejection velocity at the nozzle tip, and hence it should be impossible to detach the ligament from the nozzle. In this case, the surface tension force is πdσ = N and is almost 1 order smaller than the inertia or extensional force. Hence, the surface tension force does not strongly retard the deformation. As shown in Fig. 7c, for the 200-ppm solution, no drop is generated at this condition of 30 V and the minimum driving voltage is 36 V. For the case of the 100-ppm solution, the extensional viscosity is 4.2 Pa s and the extensional force is

12 324 Rheol Acta (2013) 52: N. In this case, the inertial force and the extensional force are balanced and the drop is barely generated as shown in Fig. 7c. Of course, this calculation is just for an example, and depending on diameter and other factors, there should be wide variations in forces. Following the argument of Hoath et al. (2009), we have checked whether viscoelastic properties at high-frequency regime are correlated with the drop generation characteristics. In the present case, the G / G values at 5 khz (31,400 s 1 ) are almost the same for two different fluids in Fig. 5a, b, and hence the correlation could not be found. This implies that the linear viscoelastic property cannot affect the drop generation process directly and the effect of added polymers manifests itself as increased extensional viscosity which primarily affects the detachment process. The increased extensional viscosity may play a role also in the converging flow inside the nozzle at high xanthan gum concentrations, too. As seen in Fig. 8, the drop size of the 200-ppm solutions is smaller than those from less concentrated solutions. This difference appears to be caused by the increase in extensional viscosity of the 200-ppm xanthan gum solutions inside the nozzle. This is because the flow through the conical region of the nozzle is strongly extensional, especially at the central region. However, it is expected that the contribution of extensional stress may not be as large as in the case of ligament extension. This is because the mean residence time of polymer chains within the region is very short, and hence the polymer chains may not be fully extended. It can be confirmed from the fact that the reduction of ejection velocity is not noticeable for less concentrated solutions. We are sure that the analysis of the detailed process should be performed through an elaborate constitutive modeling on the xanthan gum solutions and possibly numerical solutions of the governing equations based on the model. Another issue is the location of detachment. Depending on extensional properties, the location of detachment can vary, which results in the droplet size change. Summary and conclusion In this study, the generation of inkjet droplets of non- Newtonian fluids has been investigated experimentally. Noting that most of inks used in inkjet technology are rheologically complex fluids, xanthan gum solutions in water glycerin mixtures have been chosen as model inks. The rheological properties of xanthan gum solutions have been characterized by the diffusive wave spectroscopy microrheological method for high-frequency viscoelastic properties and the capillary breakup method for extensional viscosity as well as conventional rotational rheometry for viscosity and linear viscoelastic properties for moderate values of shear rate or frequency. The result shows that drop generation process consists of two independent processes of ejection and detachment. The ejection process is found to be controlled primarily by high or infinite shear viscosity. Elasticity can affect the flow rate (drop size) through the converging section of an inkjet nozzle. However, the elastic effect may not strongly affect the flow rate since the residence time of polymer molecules of the section is too short for the elastic stress to grow to a significant level. The detachment process is controlled by the extensional viscosity. Due to the strain hardening of polymers, the extensional viscosity becomes orders of magnitude larger than the Trouton viscosities based on zero and infinite shear viscosities. The large extensional stress retards the extension of ligament and hence lowers the flight speed of the ligament head. The viscoelastic properties at high-frequency regime do not appear to be directly related to the drop generation process even though it is surmised that the elastic effect should strongly affect the extensional properties. We have not performed the numerical simulation on the whole drop generation process since, first of all, there exists no constitutive relation which is reasonably well matched to the experimental data for xanthan solutions. Also, the detailed numerical analysis is far beyond the scope of the present paper. But it should be worth doing by considering the flow inside the nozzle and detachment process together. This is especially important in non-newtonian liquids since polymers are elongated during the converging flow inside the nozzle and the elongated polymers cannot be relaxed until they come out of the nozzle exit considering the process time of 100 μs and the relaxation time of the same order for most polymers. Even though the present research has been performed with polymeric liquids only, the same principle should be applied to inks from other materials. Especially, the present result can also be used even for predicting the jetting behavior of the solution of a flexible polymer which can be 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. However, since most inks contain polymers and/or surfactants for suspension stability, the strain hardening can affect the detachment to a certain degree. As many different kinds of polymers and large molecules of various shapes are used in electronic industries such as organic light-emitting diode displays and polymer light-emitting diodes, the results of the present research will be valuable information for those industries. Acknowledgement This work was partially supported by Midcareer Researcher Program through NRF grant funded by the Ministry of Education, Science and Technology, Korea (no ).

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