Rheol Acta (2013) 52:313–325DOI 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 dropletsof xanthan gum solutions in water–glycerin mixtures wasinvestigated experimentally to understand the jetting anddrop 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 ofdroplet were measured while varying the wave form of bipo-lar shape to the piezoelectric inkjet head, and the effects ofthe rheological properties were examined. The shear prop-erties of xanthan gum solutions were characterized for wideranges of shear rate and frequency by using the diffusivewave spectroscopy microrheological method as well as theconventional rotational rheometry. The extensional prop-erties were measured with the capillary breakup method.The result shows that drop generation process consists oftwo independent processes of ejection and detachment. Theejection process is found to be controlled primarily by highor infinite shear viscosity. Elasticity can affect the flowthrough the converging section of inkjet nozzle even thoughthe effect may not be strong. The detachment process iscontrolled by extensional viscosity. Due to the strain hard-ening of polymers, the extensional viscosity becomes ordersof magnitude larger than the Trouton viscosities based onthe zero and infinite shear viscosities. The large extensionalstress retards the extension of ligament, and hence the stresslowers the flight speed of the ligament head. The viscoelas-tic properties at the high-frequency regime do not appearto be directly related to the drop generation process eventhough it can affect the extensional properties.

H. Yoo · C. Kim (�)Department of Chemical and Biological Engineering,Korea University, Anam-dong, Sungbuk-ku, Seoul 136–713,South Koreae-mail: cykim@grtrkr.korea.ac.kr

Keywords Drop-on-demand inkjet · Elasticity · Shearthinning · Infinite shear viscosity · Jeffery–Hamel flow ·Strain hardening · DWS microrheology

Introduction

As the inkjet printing technology has widen its applicationto bio and electronic industries beyond household or officeinkjet printers (Basaran 2002; Schubert 2005; de Gans et al.2004), many different kinds of inks have to be handled. Inmost cases, inks are suspensions or polymeric liquids (deGans et al. 2005), and hence most of the inks are rheolog-ically complex fluids showing shear-dependent viscositiesand/or elastic characteristics. Some additives such as sur-factants are usually added in suspensions for stabilizationand better performances. This can make the rheology ofsuspension more complex. However, inks have not beencharacterized properly especially at the operating conditionsof inkjet printing, and the processing conditions have beensought mostly through trial and error basis.

To generate inkjet droplets, either the continuous jet-ting or drop-on-demand (DOD) method can be used (Derby2010). In the DOD method, droplets are generated by apply-ing a pressure wave to a liquid-filled nozzle. Then, a portionof liquid is squeezed out of the nozzle overcoming the sur-face tension force, and the liquid element is detached fromthe 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 thata single drop is generated or divided into the leading dropand some smaller satellite drops by instability mechanisms.In some cases, satellite drops can be merged into the lead-ing drop. It is known that the formation of satellite dropsshould 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-eters for a single drop generation is one of the most impor-tant issues in inkjet droplet generation. It has been reportedthat drop generation characteristics are governed by Ohne-sorge number (Oh) of the drop, which is the ratio of viscoustime scale and surface tension time scale and defined asfollows (Derby 2010):

Oh = η√ρRγ

, (1)

where η is viscosity, ρ is density, R is radius, and γ issurface tension. For Newtonian fluids, Oh is unequivocallydefined since fluid properties are independent of flow con-ditions. But in the case of shear thinning fluids, viscosityis a function of shear rate, and hence the operating windowcannot be predicted based on the theory for Newtonian flu-ids (Lai et al. 2010; Tai et al. 2008). It has been the usualnotion that the rheological behaviors at large strain ratescontrol the generation of inkjet droplet (Hoath et al. 2009)since the drop generation process is an ultrahigh shear rateprocess with an average shear rate of 105 s−1 order. Actu-ally, inks show finite viscosities as shear rate goes eitherto zero or a very large value. In this case, some questionsshould arise naturally. Is the zero shear viscosity not rele-vant to the generation of inkjet drop generation? If this is so,is the infinite shear viscosity the only variable that affectsdrop generation? If not, what other properties are relevantto drop generation? In the present paper, we have tried toanswer these questions by using a class of fluids (as modelinks) which show various rheologically complex behaviorssuch as elasticity, shear thinning, and strain hardening.

Shore and Harrison (2005) reported that the presence ofa small amount of polymer in a Newtonian solvent can havea significant change in the inkjet drop generation character-istics. Especially, satellite drop formation is suppressed andthe drop velocity is significantly lowered by the addition ofpolymer. Using two different types of polymers (linear andstar polymers), de Gans et al. (2005) reported that the dis-tance traveled by the primary droplet was dependent onlyon input voltage to the piezo-element and independent ofpolymer concentration, molecular weight, and topologicalarchitecture. They also reported that the rupture of the lig-ament was dependent on the rheological properties of thesolution. Hoath et al. (2009) noticed that, in the genera-tion of inkjet droplet of elastic polymer solutions, the finalmain drop size was independent of polymer concentrationeven though the length of the ligament increased markedlywith the elasticity of the fluid. In the meantime, Hoathet al. (2009) did not observe any correlation between lowshear viscosity and jetting behavior for the fluids they inves-tigated, but the jetting behavior was well correlated withhigh-frequency rheological properties measured at 5 kHzusing a piezoelectric axial vibrator rheometer. Here, it is

noted that the drop generation is a highly nonlinear processof large extension and high extensional rate, and thereforelinear viscoelastic properties may not be correlated quan-titatively with the nonlinear process. Morrison and Harlen(2010) investigated the effects of viscoelasticity numericallyon drop formation in inkjet printing by using viscoelasticfluids represented by the single-mode FENE-CR constitu-tive equation (Chilcott and Rallison 1988). They showedthat the ligament became longer for elastic liquids and theformation of satellite drops was suppressed by elasticity.Also, they argued that the lowering of drop speed was dueto elasticity. Recently, Hoath et al. (2012) presented a quan-titative model which predicted three different regimes ofbehavior depending upon the jet Weissenberg number (Wi)and extensibility of polymer molecule. They predicted New-tonian regime (Wi < 1/2), viscoelastic regime with partialextension (1/2 < Wi < L, where L is the extensibility ofpolymer chain), and fully extended regime (Wi > L). Theyalso gave the scaling law for the maximum polymer concen-tration at which a jet of a certain speed could be formed asa function of molecular weight of polymer. Their analysisis based on the FENE-CR model which is valid for solu-tions of flexible polymers. Also, their analysis is limited tothe detachment process, and hence the model cannot predictthe drop size. Therefore, more studies on drop size and dropvelocity should be still required to understand the mech-anism of inkjet drop generation of viscoelastic fluids. Allof these papers argued that the elasticity has a significanteffect on the generation of inkjet droplet of elastic solution.However, they did not give the detailed reason why elastic-ity could affect the drop formation. In the present paper, wehave examined the elongation characteristics and linear vis-coelasticity of inks along with the flow of inks inside thenozzle and their effects on inkjet drop generation.

The elongation of liquid thread has been an importantissue in rheology. In continuous jetting, dripping, neck-ing, and breakup of liquid bridge, the thinning of a liquidfilament is driven by capillarity and resisted by inertia, vis-cosity, and elasticity. On the other hand, the stretching ofthe ligament in the early stage of drop formation is primar-ily driven by the inertia (in the main flight direction) ofthe ligament head and resisted by surface tension, inertia(in the perpendicular direction to the main flight direc-tion), viscosity, and elasticity of fluid. Hence, the detailedflow should not be the same. At the final stage of filamentbreakup, it is known that the breakup process is determinedby the natural variables of surface tension and fluid prop-erties regardless of boundary and initial conditions (Eggers1993; Renardy 2004; McKinley 2005). Therefore, we maygain useful information from the capillary breakup test.The visco-elasto-capillary thinning of complex fluids hasbeen studied extensively since Eggers (1993) first foundthe similarity solution for the one-dimensional governing

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equation and subsequent reports on the similarity solutionsfor various non-Newtonian models. Detailed reviews on theself-similar solutions for non-Newtonian models were givenby Renardy (2004), and comprehensive reviews were givenby McKinley (2005) on capillary thinning of liquid bridgesand its applications to extensional rheometry. The studieson capillary thinning have shown that, in the case of Stokesflow of a Newtonian fluid, the filament breaks off at a finitetime tc and the radius of the filament changes with time asfollows (Papageorgiou 1995):

Rmid

R0= 0.0709

σ

ηsR0(tc − t) (2)

Then, the extensional rate at the middle of the filament isgiven as follows:

εmid(t) = − 2

Rmid

dRmid

dt= 2

t − tc(3)

In the above equations, ε is extension rate, R is radius,ηs is viscosity, and t is time. Also, subscripts mid and 0denote the value at the midpoint of the filament and initialradius, respectively. In the case of elastic fluid (McKinley2005), the capillary thinning flow becomes a homogeneousextensional flow with

Wi = λ1ε = 2/3 (4)

and the radius changes with time as follows:

Rmid

R0=

(GR0

2σ

)1/3

exp (−t/ (3λ1)) (5)

where G is modulus and λ1 is the longest relaxation time.In this case, there is no finite breakup time and there isa long tail. This equation is valid for a dilute solution ofinfinitely extensible polymers. But in a real polymeric liq-uid, polymers cannot be extended infinitely. Renardy (2002)and Fontelos and Li (2004) have shown that, for viscoelas-tic fluids of Giesekus and FENE-P types, the jet diameterdecreases linearly with time when close to the breakup:

R(t)

R0=

[σ

2ηER0

](tc − t) (6)

This result means that, as polymer molecules are fullystretched at a sufficiently large strain rate, the extensionalviscosity approaches a constant value and the fluid behavesas a Newtonian fluid with a constant extensional viscos-ity of ηE (McKinley 2005; Stelter et al. 2002, 1999). Fromthis relationship, we may obtain the extensional viscosity ofinkjet fluid from the capillary breakup experiment. If we canobserve a linearly decreasing filament radius while exhibit-ing a cylindrical filament, ηE can be obtained from the slope.The ηE value obtained here can be close to the true exten-sional viscosity at the inkjet drop generation condition sincepolymer coils can be almost fully stretched at the high strainrate, and hence ηE value approaches the limiting value.

In the present research, we have attempted to correlatethe relevant variables to each drop-generating step by per-forming inkjet drop generation experiments with xanthangum solutions together with numerical simulations on theflow inside the nozzle. Since xanthan gum solutions areless elastic than most of the flexible polymer solutions,the result presented here can describe the practical inkjetproblems more realistically. In the analysis of experimentaldata, we used the rheological properties at the real process-ing condition measured by the diffusive wave spectroscopymicrorheology method for high-frequency linear viscoelas-tic properties and capillary breakup method for extensionalviscosities at high strain rate as well as the conventionalrotational rheometry. The result shows that drop generationprocess consists of two separate stages: At the first stage,a certain amount of liquid is ejected from the nozzle andthe drop volume is determined by this step. Especially, ithas been found that drop volume is determined mainly bythe infinite shear viscosity of the xanthan gum solution. Atthe second stage, the ejected liquid is pinched off from thefluid inside the nozzle by the inertia of the liquid and thepulling-back action of piezo-element, and the drop velocityis determined by the ejection velocity and the extensionalviscosity of fluid. As xanthan gum solutions show most ofthe important characteristics of non-Newtonian fluids suchas elasticity, shear thinning, and extensional thickening, thepresent study can give an insight into the processing of non-Newtonian fluids for various applications by using a DODinkjet printing system. The present result can be also usedeven for predicting the jetting behavior of the solution of aflexible polymer which can be regarded as a different classof fluids from the xanthan gum solution.

Experiment

To investigate the generation of inkjet drops, we set up aninkjet system as shown in Fig. 1, which is the same as theset that one of the authors used for the previous studies on

Fig. 1 Schematic diagram of the experimental setup

316 Rheol Acta (2013) 52:313–325

spreading of inkjet drop (Son et al. 2008). In the presentcase, there is no such part as the solid surface. The systemconsists of an inkjet nozzle, a jetting driver (pulse generatingsystem), a high-speed camera, and an illumination source.

Inkjet system and imaging

The inkjet droplet was generated by a piezo-type nozzle pur-chased from MicroFab Co. (Model # MJ-AT). The nozzlediameter at the exit was 50 μm. In Fig. 2, the inside geome-try of the nozzle is shown. In taking the picture, the nozzlefilled with air was immersed in a decalin-filled square box.Since decalin has the same refractive index as the glass, therefraction at the curved nozzle surface can be avoided.

To generate droplets, a bipolar wave form was used asshown in Fig. 3. During the rise period (a), the piezo-element expands for fluid intake from the reservoir and thisstate continues during the dwell period (b). During the fallperiod (c), the piezo-element shrinks and fluid is ejected outof the nozzle. This state continues during the echo period(d). Finally, the piezo-element expands to return to the initialstate while completing a cycle (e). Depending on the timeintervals and the voltages imposed on the piezo-element, adrop or drops of different sizes and velocities are generated.In the present research, the rise and fall times in the volt-age pulse to the nozzle were set at 2 μs. The dwell and echotimes were in the range of 4–32 μs, and the dwell and echovoltages were in the range of 12–50 V. We performed dropgeneration experiments by varying operating conditions andmeasured the drop size and velocity. All the experimentalruns were performed at the room condition. The high-speedcamera (IDT, XS-4) was triggered by the jetting driver as adroplet was ejected from the inkjet nozzle. The camera wasequipped with a microscopic objective lens (Mitutoyo, MPlan Apo) with the magnification of ×5. As the illumina-tion source, a back lighting system (Stocker Yale, # 21 AC,180 W) was installed.

The CCD camera can capture 50,000 frames per secondand the pictures were taken with this mode. The exposuretime was 1 μs. When this fast mode was used, the number

Fig. 2 Inside geometry of the nozzle. The inlet diameter is 456 μmand the exit diameter is 50 μm

Fig. 3 Pulse wave form for generating inkjet droplets: a expansion ofnozzle chamber, b delay for pressure wave propagation, c compres-sion of fluid for ejection, d delay for pressure wave propagation, and enozzle chamber expansion to the initial state

of pixels per frame has to be small (512 × 48 pixels). Thepixel size was 3.76 μm. One may use the flash videographymethod to get the better quality as demonstrated by van Damand Le Clerc (2004) and Dong et al. (2006). But, in the caseof non-Newtonian drops, the reproducibility of drop gener-ation was not as good as in case of Newtonian fluids; hence,we decided not to use the flash videography method used inthe literature. Considering that the drop diameter and traveldistance are in the order of 50 μm and 1 mm, respectively,the numbers of pixels to cover these sizes are about 15 and150. Therefore, the image resolution was enough to measuredrop diameter and velocity.

Materials

Three kinds of Newtonian fluids were prepared with dif-ferent viscosities by mixing deionized water and glycerin(Sigma-Aldrich Co.). Shear-thinning fluids were preparedby dissolving xanthan gum (Sigma-Aldrich Co.) in deion-ized water or in one of the water–glycerin mixtures. Table 1shows the composition of the fluids tested here. Theshear viscosity and linear viscoelastic properties of liquidswere measured by a rotational rheometer with a Couettefixture (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

Surface tension, mN/m 71.6 66.0 67.5 66.9

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viscoelastic properties were measured by a diffusive wavespectroscopy (DWS) microrheology rheometer (RheoLab,LS Instrument) using polystyrene spheres of 520 nm indiameter as tracer particles. The cuvette thickness was 2 mmand the properties were measured under the transmissionmode. The extensional viscosity was estimated by using acapillary breakup apparatus (CaBER, ThermoHaake Co.).To handle low-viscosity fluids, two small plates of diameter2 mm were machined from titanium. The initial gap dis-tance was the same as the radius of the plate and the initialdeformation was imposed to 2.5 mm for 20 ms. The surfacetension was measured by using the Du Nouy ring method(K9 Tensiometer, KRUSS GmbH).

Result and discussion

Figure 4 shows the shear viscosities of two sets of fluidstested in the present research. The mixtures of DI water andglycerin have shear-independent viscosities while all xan-than solutions have shear-thinning viscosities. The viscosityof xanthan solution is fitted to the Carreau model (Bird et al.1987):

η − η∞η0 − η∞

= 1[1 + (βγ )2] 1−n

2

, (7)

and the model parameters are listed in Table 2. In the table(also in Fig. 4), we note that the infinite shear viscositiesof xanthan gum solutions of the same base solvent are onlyslightly changed from or almost the same as the viscosity

Fig. 4 Viscosities of xanthan gum solutions for differing solvents andxanthan gum concentrations. The symbols are measured values. Thesolid lines are the Carreau model fit

of the solvent. In the following discussion, we will com-pare the drop generation characteristics of the fluids withthe same base fluid systematically. Figure 5 shows linearviscoelastic properties of some of the fluids listed in Table2. Figure 5 shows that the shapes of G′ and G′′ for thexanthan gum solutions are not similar to those of poly-mer solutions of flexible polymers such as Boger fluids inthat the typical slope of 2 for G′ at low-frequency regimedoes not appear yet at the lowest frequency of 0.1 s−1.The data obtained from the DWS microrheology appear tobe reasonably extended to the conventional rheometry data.

Table 2 Xanthan gumsolutions studied here and theirCarreau model parameters andsurface tension

1.0 cP 4.5 cP

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: a10-cP-based solutions, b 16.5-cP-based solutions. Symbols wereobtained by rotational rheometry. Solid lines were obtained by DWSmicrorheology

One important characteristic is that the xanthan gum solu-tions have smaller G′ than G′′ for all the frequency regimestested here including the DWS microrheology measurementregime. This implies that the elastic effect should not belarge even at the inkjet drop generation condition (shear rateof 105 s−1).

Figure 6 shows some typical drop generation patterns fordiffering fluids at differing conditions. In Fig. 6a, a thinliquid ligament with a spherical head is formed, and thenthe ligament tail contracts to the head eventually. Therefore,only one drop is generated. In Fig. 6b, the situation is muchthe same as Fig. 6a, but the tail is separated from the mainhead and becomes a satellite drop. More than one satellitedrop can be generated and these drops can be coalesceddepending upon the relative velocity between the main andsatellite droplets. The separation of the satellite drop iscaused by the capillary instability or end pinching (Stone

Fig. 6 Typical drop generation patterns. The time interval betweentwo adjacent frames is 20 μs. a Single-drop formation for the 100-ppmsolution in 4.5-cP solvent. The tail is shrunk to the main drop. Drivingvoltage is 30 V. b A satellite drop generated by the separation from thetail for the 50-ppm solution in 10-cP solvent. Driving voltage is 36 V. cA satellite drop generated by reflected acoustic waves for the 50-ppmsolution in water. In this case, the tail is shrunk to the main drop, but anew drop is generated behind the main drop

1994; Stone et al. 1986). These drops may be called “satel-lites from tail.” In Fig. 6c, a satellite drop is generated bya different mechanism. In this case, the satellite drop is notlinked to the original ligament and appears to be generatedby the reflected acoustic wave inside the nozzle. These satel-lite drops may be called “satellites from reflected wave.”In the following discussion, we will consider the cases forwhich only one single drop is generated. To generate onesingle 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 aswell as rise and fall time. Since there are too many com-binations for operating parameters, we confine ourselves tothe following cases. First, the rise and fall times were fixedat 2 μs. Dwell and echo voltages were set to be equal, andthen we changed dwell/echo time to find out the conditionat which the drop velocity became the maximum value. Inmost of the cases, the drop velocity became the maximumor close to the maximum value when dwell/echo time was24 μs. Therefore, we fixed dwell/echo time at 24 μs. Theoptimum dwell/echo time (the condition at which the dropvelocity attains the maximum value) is closely related to thelength of the nozzle and acoustic velocity of fluid (Bogyand Talke 1984). Since the nozzle length is fixed and theacoustic velocity is not much different (between 1,481 ms−1 (water) and 1,980 m s−1 (100 % glycerin) at 20 ◦C),the optimum dwell/echo time appears to have similar val-ues. After choosing the dwell/echo and rise/fall times, we

performed experiments while varying dwell/echo voltageand collected data when only a single drop was generated.

Figure 7 shows the drop velocity variations for the flu-ids listed in Table 1. First, we note that, for all cases,as dwell/echo voltage (voltage hereafter) increases, dropvelocity increases. Also, the minimum voltage required togenerate a single drop increases with xanthan gum con-centration. At a fixed voltage value, drop velocity becomessmaller with an increase in xanthan gum concentration. Forexample, when the viscosity of solvent is 4.5 cP, the dropvelocity is 4.5, 3.2, and 0.6 m s−1 when the driving voltageis 24 V and the xanthan gum concentration is 0, 50, and 100ppm, respectively. At 24 V, the 200-ppm solution could notbe ejected due to the strong pull back of the ligament to thenozzle. Similar patterns are observed for all cases shown inFig. 7. Figure 8 shows the drop volume changes with driv-ing voltage. Even though we have only a limited numberof 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-age only regardless of xanthan gum concentration since thedata points are continued along a single curve (with someoverlaps) for solutions from the same base fluid except forthe 200-ppm solutions. In the case of the 200-ppm solu-tions in 16.5-cP solvent, it was impossible to generate adrop for the full range of driving voltage. The concentra-tion independency was also reported by Hoath et al. (2009)(the deviation of the 200-ppm solutions from the generaltrend will be considered later in this report). The indepen-dency of concentration on drop volume indicates that dropvolume is a function of infinite shear viscosity only. Thisis quite in contrast to the fact that drop velocity is a strongfunction of xanthan gum concentration. The two contrastingobservations imply that drop generation process consists oftwo separate stages, and each stage is governed by differ-ent physical properties: At the first stage, a certain amountof liquid is ejected from the nozzle and the drop volumeis mainly determined by this step. At the second stage, theejected liquid is pinched off from the fluid in the nozzle by

the inertia of the liquid and the pull-back action of the piezo-element, and hence the drop velocity is determined by theejection velocity at the nozzle tip and the extensional char-acteristics of fluid. In the next discussion, we have describedthese two steps sequentially.

To understand the fluid ejection step, we need to knowthe amount of fluid that is ejected by the acoustic pres-sure wave and its relation with fluid properties. Since itis very difficult to measure the velocity profile at the exitaccurately, we estimated it by numerical simulation usingFluent™, a commercial software package based on the finitevolume method. We used 6,320 elements and 6,657 nodesin the simulation. The operating conditions were obtainedby the following method. From the two consecutive imageson jetting of a Newtonian fluid, the flow rate through thenozzle was obtained for a typical pressure wave form. Fromthe numerical simulation results, we read the pressure dropbetween the manifold and the exit of the nozzle that gavethe same flow rate as the experimentally observed flowrate value for the Newtonian fluid. We assumed that the

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Fig. 9 Velocity profiles of Newtonian fluids at the nozzle exit at atypical pressure difference of 20,000 Pa between the nozzle inlet andexit

acoustic velocity of a dilute polymer solution was the sameas the solvent. Then, the pressure drop between the man-ifold and the exit should be the same regardless of fluidfor the given wave form. Using the same pressure drop, wecalculated the velocity profile while varying fluids of dif-ferent zero shear viscosities. Since xanthan gum solutionsshow elasticity, it has to be considered in velocity profilecalculation for the flow through the converging geometryof nozzle due to Lagrangian unsteadiness. Until now, theconverging flow of elastic fluids has been treated only forOldroyd 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 thepresent time, we were not able to include the elastic effectin the numerical simulation properly. Rather, we neglectedthe elasticity of xanthan gum solutions since the viscoelas-tic measurements showed that G′ was substantially smallerthan G′′ even at high-frequency regimes. Therefore, eventhough the velocity profile obtained by the present methodmay not represent the physics of the problem exactly, wecan obtain at least semiquantitative result. The effect ofelasticity will be considered later in this study. Figure 9shows the velocity profiles of Newtonian fluids of differ-ing viscosities for a typical pressure difference of 20,000Pa between the exit and the entrance of nozzle. When vis-cosity is 1 cP, the exit velocity is severely blunted. Theblunting of velocity profile is caused by the inertial effectin the Jeffery–Hamel flow (Batchelor 1967). In the bluntedregion, the inertial term ρ vr

∂vr

∂rdominates over the viscous

term and is balanced with the pressure term, while near thesolid boundary, the viscous term dominates over the iner-tial term. The blunted velocity profile at the nozzle exit canbe surmised from the experiment for a low-viscosity fluid(please see the third frame of Fig. 6a). Dong et al. (2006)

observed the velocity blunting of water more clearly. As vis-cosity increases, the velocity profile becomes close to theparabolic profile. Next, we performed the simulation for theCarreau model fluids with the experimentally fitted numer-ical parameters. In Fig. 10, we have plotted the velocityprofiles for two different sets of fluids with the same infi-nite shear viscosities. It is seen that the velocity profiles arealmost the same regardless of zero shear viscosity of fluidstested here, in other words, xanthan gum concentration. Thisis because at the central region, the inertial term is dominantwhile shear rate is extremely large near the boundary as inthe case of a Newtonian fluid. Therefore, the shear rates atwhich viscosity changes appreciably occur for a very narrowregion near the center. This result means that the exit veloc-ity is almost the same regardless of zero shear viscosity aslong as infinite shear viscosities are the same for differingfluids. The dependence of drop size only on driving voltage

Fig. 10 Numerical simulation result on the effect of xanthan gum con-centration 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 tosimulate real situations

322 Rheol Acta (2013) 52:313–325

has been already observed in the experiment as describedabove. Therefore, we can argue that drop volume is mainlydetermined by infinite shear viscosity. Also, from the matchbetween experimental and simulation results on drop size,we may argue that elasticity is not important in the presentcase except for the 200-ppm solution. Experimentally, theejected volume of the 200-ppm solutions is smaller than thepure solvent. It appears that the difference in experimentalresults is due to the elasticity of fluids which has not beentaken into account in the Carreau model. This point will beconsidered more in this paper.

Next, we consider the detachment of a drop. During thedetachment process, the ejected fluid is elongated and theneck becomes thinned. From the images shown in Fig. 6, wecan estimate the order of the extension rate. Knowing thatthe frame rate is 50,000 s−1 and the ligament length changesfrom 0 to 400 μm between four frames, the average exten-sional rate ((L/t)/Laverage) is 25,000 s−1. As far as theauthors are aware of, the extensional viscosity at this high

Fig. 11 Typical results on the diameter change of liquid filament inthe capillary breakup experiment. The piston movement stopped at t =0; a 10-cP-based solutions and b 16.5-cP-based solutions

Fig. 12 Filament shapes during thinning for the 100-ppm xanthangum solution in 16.5-cP solvent. From the left: after the loading, justafter the pulling apart, establishment of the cylindrical shape, and justbefore the breakup

extensional rate cannot be measured by a commercial exten-sional rheometer as of now. Therefore, we used the CaBER(ThermoHaake Co.) to estimate the extensional viscositybased on the liquid bridge stretching. In Fig. 11, we haveshown the changes in the diameter of the stretched liquidbridge as a function of time for some of the samples tested inthe present study. In Fig. 11, we note that the time evolutionof thread diameter is substantially delayed when xanthangum concentration increases, meaning that the extensionalviscosity increases with the increase in xanthan gum con-centration. Just before the breakup, the diameter decreasepattern changes to an exponential shape. It appears that, justbefore breakup, the dominant resistance to filament thinningis changed to inertia. In the figure, we note that the diameterchanges become linear after a transient period and until theybecome 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 thatthe thinning process follows the elasto-capillary thinningregime. Also, from the linear decrease in R(t) with time, wecan confirm that the extensional viscosity is the same duringthe linear decrease. Hence, we can calculate the extensionalviscosity and extensional rate by using Eqs. 6 and 3, respec-tively. In Table 3, we have listed extensional viscosities andextensional rates obtained by this method. In all cases, thexanthan gum solutions show much larger extensional vis-cosities compared with the corresponding solvents. In the

Table 3 Extensional viscosity of some selected samples

Sample Range of ε, s−1 ηE, Pa s

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

16.5 cP + 200 ppm 29 ± 9–58 ± 34 9.0 ± 2.0

For each sample, more than seven runs were averaged

Rheol Acta (2013) 52:313–325 323

table, in the case of the 200-ppm xanthan gum solution in10-cP solvent, the extensional viscosity is 6.3 Pa s for exten-sional rates between approximately 47 and 142 s−1. Theextensional viscosity of 6.3 Pa s is much larger than theTrouton viscosity (three times the shear viscosity) based onthe zero shear viscosity of 215 mPa s and the Trouton vis-cosity based on the infinite shear viscosity of 30 mPa s. Thesubstantially larger extensional viscosity than either of theTrouton viscosities is due to the strain-hardening behaviorof the xanthan gum polymer. Strain hardening occurs whenpolymers are stretched and aligned. The degree of alignmentshould be much larger for xanthan gum (stiff polymer) thanthat for flexible polymers. The easiness of alignment can beseen from the strong shear-thinning viscosity. This meansthat the extensional viscosity obtained in this study can bequite close to the true value at the drop generation condi-tion. The true values can be larger than the values measuredhere, but the measured values can be useful enough to con-firm that the extensional stress reduces the velocity of theinkjet droplet.

Since the extensional rate at the pinch-off condition ofthe drop generation process (in the order of 25,000 s−1 asdescribed above) is much larger than the value at the mea-suring condition of 47–140 s−1, the extensional viscosityat the pinch-off condition should not be smaller than 6.3Pa s considering the extensional hardening characteristicsof polymers. The value 6.3 Pa s in the present case may bethe value at the fully extended state and strain hardening isalready saturated. If this is the case, the extensional viscos-ity at the pinch-off condition will be at least 6.3 Pa s. Here,one may raise a question whether the polymers inside theligament are fully extended, and hence the extensional vis-cosity obtained by CaBER can be applied to the ligamentstretching. To warrant the application, we have comparedtwo strains as follows: First, the Hencky strain of the fila-ment from the unstretched state in the CaBER experimentis

ε(t) = ln

(l(t)

l0

)= ln

(D2

0

D(t)2

), (8)

where l and D are the length and diameter of the fila-ment at time t and the subscript 0 denotes the value beforestretching. Next, the total strain of the inkjet ligament canbe estimated as follows: Before the breakup from the noz-zle, the ejected liquid element can be divided into two parts:drop head and ligament. The volume of the ligament (Vl)

can be calculated from the difference between the drop vol-ume (Vd) and the drop head volume (Vh): Vl = Vd − Vh.The ligament volume Vl is assumed to be maintained. Whenliquid comes out of the nozzle, the diameter of the liquidelement is the same as the nozzle diameter (d), and hencethe liquid volume is extended from a cylinder of length

Vl/(πd2/4

)to a cylinder of length l(t). Then, the Hencky

strain is given as follows:

εl(t) = ln

(πd2l(t)

4Vl

)(9)

In estimating the strain of the filament, the drop with thelowest velocity we observed is used for a conservative esti-mate. For drops with higher velocities, the strain rate will behigher. In the case of the 100-ppm solution in 16.5-cP sol-vent, the Hencky strain when the ligament length becomes240 μm just before the breakup is ln 20 while the strain ofthe filament during the CaBER experiment is ln 16 whenthe filament diameter begins to decrease linearly (0.02 s inFig. 11a). At another case of the 200-ppm solution in 10-cPsolvent, when the ligament length is 214 μm, these valuesare ln 33 and ln 16, respectively. Considering that the strainrate is much higher for the ligament stretching during dropgeneration than the filament stretching during the CaBERexperiment, one can confirm that the polymers in the liga-ment stretching are almost fully stretched, and therefore theextensional viscosity obtained by CaBER can be applied tothe ligament stretching.

As the extensional viscosity of the fluid thread is muchlarger than that of the Newtonian fluid with the same infi-nite shear viscosity, during the extension of the thread, theextensional stress will strongly retard the deformation orbreakup. Also, before the pinch off, the extensional stressretards the flight of the drop head. Therefore, even if dropsizes are the same regardless of concentration of xanthangum as long as the infinite shear viscosity is the same, flightvelocity is strongly dependent on xanthan gum concentra-tion. Due to the limited spatial resolution of the image andtime interval between two subsequent frames, we have notbeen able to estimate the force acting on the head quantita-tively to estimate the velocity change. As a rough estimate,

the inertial force acting on the drop head is mdvdt

≈ ρ πD3

6vt

(m and v are the mass and the velocity of the ligamenthead, respectively) while the force due to extensional stressis πR2

neck × ηEε. If we insert a set of numerical values at30-V driving voltage (v

t= 50,000 m s−2, D = 50 μm,

Rneck = 5 μm, ηE = 6.3 Pa s, ε = 25,000 s−1), theinertial and extensional forces are 6 × 10−6 and 1 × 10−5

N, respectively. In this case, the drop velocity is substan-tially decreased from the ejection velocity at the nozzle tip,and hence it should be impossible to detach the ligamentfrom the nozzle. In this case, the surface tension force isπDσ ∼= 1.0 × 10−6 N and is almost 1 order smaller thanthe inertia or extensional force. Hence, the surface tensionforce does not strongly retard the deformation. As shown inFig. 7c, for the 200-ppm solution, no drop is generated atthis condition of 30 V and the minimum driving voltage is36 V. For the case of the 100-ppm solution, the extensionalviscosity 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 forceare balanced and the drop is barely generated as shown inFig. 7c. Of course, this calculation is just for an example,and depending on diameter and other factors, there shouldbe wide variations in forces.

Following the argument of Hoath et al. (2009), we havechecked whether viscoelastic properties at high-frequencyregime are correlated with the drop generation character-istics. In the present case, the G′ /|G∗| values at 5 kHz(31,400 s−1) are almost the same for two different flu-ids in Fig. 5a, b, and hence the correlation could not befound. This implies that the linear viscoelastic property can-not affect the drop generation process directly and the effectof added polymers manifests itself as increased extensionalviscosity which primarily affects the detachment process.The increased extensional viscosity may play a role also inthe converging flow inside the nozzle at high xanthan gumconcentrations, too. As seen in Fig. 8, the drop size of the200-ppm solutions is smaller than those from less concen-trated solutions. This difference appears to be caused bythe increase in extensional viscosity of the 200-ppm xan-than gum solutions inside the nozzle. This is because theflow through the conical region of the nozzle is stronglyextensional, especially at the central region. However, it isexpected that the contribution of extensional stress may notbe as large as in the case of ligament extension. This isbecause the mean residence time of polymer chains withinthe region is very short, and hence the polymer chains maynot be fully extended. It can be confirmed from the factthat the reduction of ejection velocity is not noticeable forless concentrated solutions. We are sure that the analysis ofthe detailed process should be performed through an elab-orate constitutive modeling on the xanthan gum solutionsand possibly numerical solutions of the governing equa-tions based on the model. Another issue is the location ofdetachment. Depending on extensional properties, the loca-tion of detachment can vary, which results in the droplet sizechange.

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 rhe-ologically complex fluids, xanthan gum solutions in water–glycerin mixtures have been chosen as model inks. Therheological properties of xanthan gum solutions have beencharacterized by the diffusive wave spectroscopy microrhe-ological method for high-frequency viscoelastic propertiesand the capillary breakup method for extensional viscosityas well as conventional rotational rheometry for viscosityand linear viscoelastic properties for moderate values of

shear rate or frequency. The result shows that drop gen-eration process consists of two independent processes ofejection and detachment. The ejection process is found tobe controlled primarily by high or infinite shear viscosity.Elasticity can affect the flow rate (drop size) through theconverging section of an inkjet nozzle. However, the elas-tic effect may not strongly affect the flow rate since theresidence time of polymer molecules of the section is tooshort for the elastic stress to grow to a significant level. Thedetachment process is controlled by the extensional viscos-ity. Due to the strain hardening of polymers, the extensionalviscosity becomes orders of magnitude larger than the Trou-ton viscosities based on zero and infinite shear viscosities.The large extensional stress retards the extension of liga-ment and hence lowers the flight speed of the ligament head.The viscoelastic properties at high-frequency regime do notappear to be directly related to the drop generation pro-cess even though it is surmised that the elastic effect shouldstrongly affect the extensional properties.

We have not performed the numerical simulation on thewhole drop generation process since, first of all, there existsno constitutive relation which is reasonably well matchedto the experimental data for xanthan solutions. Also, thedetailed numerical analysis is far beyond the scope of thepresent paper. But it should be worth doing by consideringthe flow inside the nozzle and detachment process together.This is especially important in non-Newtonian liquids sincepolymers are elongated during the converging flow insidethe nozzle and the elongated polymers cannot be relaxeduntil they come out of the nozzle exit considering the pro-cess time of 100 μs and the relaxation time of the same orderfor most polymers.

Even though the present research has been performedwith polymeric liquids only, the same principle should beapplied to inks from other materials. Especially, the presentresult can also be used even for predicting the jetting behav-ior of the solution of a flexible polymer which can beregarded as a different class of fluids from the xanthan gumsolution. Considering that suspensions do not show strongstrain hardening, the detachment will be much easier. How-ever, since most inks contain polymers and/or surfactantsfor suspension stability, the strain hardening can affect thedetachment to a certain degree. As many different kinds ofpolymers and large molecules of various shapes are usedin electronic industries such as organic light-emitting diodedisplays and polymer light-emitting diodes, the results ofthe present research will be valuable information for thoseindustries.

Acknowledgement This work was partially supported by Mid-career Researcher Program through NRF grant funded by the Ministryof Education, Science and Technology, Korea (no. 2010–0015186).

Rheol Acta (2013) 52:313–325 325

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