Research ArticleDrop-on-Demand Inkjet Printhead PerformanceEnhancement by Dynamic Lumped Element Modelingfor Printable Electronics Fabrication

Maowei He12 Liling Sun12 Kunyuan Hu1 Yunlong Zhu1 Lianbo Ma1 and Hanning Chen3

1Department of Information Service amp Intelligent Control Shenyang Institute of Automation Chinese Academy of SciencesFaculty Office VII Nanta Street No 114 Shenhe District Shenyang 110016 China2University of Chinese Academy of Sciences Beijing 100049 China3School of Computer Science and Software Tianjin Polytechnic University Tianjin 300387 China

Correspondence should be addressed to Hanning Chen perfect chnhotmailcom

Received 10 October 2014 Accepted 26 November 2014 Published 24 December 2014

Academic Editor Victor Sreeram

Copyright copy 2014 Maowei He et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The major challenge in printable electronics fabrication is the print resolution and accuracy In this paper the dynamic lumpedelement model (DLEM) is proposed to directly simulate an inkjet-printed nanosilver droplet formation process and used forpredictively controlling jetting characteristics The static lumped element model (LEM) previously developed by the authors isextended to dynamic model with time-varying equivalent circuits to characterize nonlinear behaviors of piezoelectric printheadThe model is then used to investigate how performance of the piezoelectric ceramic actuator influences jetting characteristics ofnanosilver ink Finally the proposed DLEM is applied to predict the printing quality using nanosilver ink Experimental resultsshow that compared to other analytic models the proposed DLEM has a simpler structure with the sufficient simulation andprediction accuracy

1 Introduction

The ability of inkjet technology to deposit materials withdiverse physical and chemical properties on a substrate hasmade it an exciting technology for mechanical engineeringlife sciences and electronics industry [1] Low cost metalcoating and rapid prototyping are popular applications ofinkjet technology in mechanical engineering [2] In the lifescience field it has been used for printingDNAstructures andmaking artificial skin and other important organs by jettinglive cells [3 4] In the past decade electronics fabricationusing the drop-on-demand (DoD) piezoelectric (PZT) inkjetprinthead to print conductive ink on flexible substrates is anenabling technology that will provide desired high-volumelow-cost production of flexible electronic devices rangingfrom radio frequency identification (RFID) tags solar cellsand organic light emitting diodes (OLED) to packagingflexible electronic devices such as healthcare equipment [5ndash7]

A typical DoD piezoelectric inkjet printhead comprises anumber of ink channels that are arranged in parallel [8] Eachink channel is connected with a piezoelectric actuator Whenapplying a standard actuation voltage pulse the actuatorcan generate pressure oscillations inside the ink channelto push the ink droplet out of the nozzle Nowadays thedevelopments of DoD printing are moving towards higherproductivity and quality which require adjustable smalldroplet sizes fired at high jetting frequencies Generallythe print quality delivered by an inkjet printhead dependson jetting characteristics that is the jetting direction thedroplet velocity and the droplet volume However in orderto meet the challenging requirements of printable electronicsfabrication the jetting characteristics of the conductive inkhave to be tightly controlled for higher inkjet performance[9] The performance improvement which is associated withthe design and operation of DoD printhead is severelyhampered by several operational issues such as residualvibrations and cross talk [10]

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 270679 16 pageshttpdxdoiorg1011552014270679

2 Mathematical Problems in Engineering

Researchers have developed many approaches to solvethese issues The most typical approaches include usinganalytical or numerical techniques to model the DoD piezo-electric inkjet printhead [12 13] Based on solving the nonlin-ear Navier-Stokes equations numerical modeling of dropletformation process has been focused on the liquid filamentevolution with known pressure input boundary conditionsassumed to be either known or an input correction forcertain specific droplet generator Numerical models areoften programmed in CFD packages using complex meshesand solving techniques This approach provides visualizedinformation about the acoustic pressure wave travellinginside the channel [14] However numerical models are verycomplex and therefore computationally expensive In the lastdecade analytical modeling approach is gradually used todescribe the ink channel dynamics although the accuracyof analytical models is lower than numerical models [15]Based on several assumptions and simplifications analyticalmodels can provide simple and timesaving ways to model-based control droplet generator with a sufficient accuracy Inthis category lumped elementmodeling [16] and feedforwardtwo-port network modeling [17] technique are two typicalanalytic printheadmodels to analyze the acoustic-mechanicalbehaviors of droplet generator

The lumped element modeling (LEM) approach intro-duces an equivalent electric circuit including capacitorsresistors and inductors in series or parallel to describe thedynamics of the ink channel [19] The LEM results explainthe driving force difference between different operatingconditions Lumped element modeling technique has beena suitable method to simulate the operating mechanism ofdroplet generator due to its outstanding advantages suchas moderate model complexity and high simulation speedHowever the accuracy of these classical LEM models islow for the improved performance requirements of printableelectronics fabrication [20] The two-port model is proposedto describe the ink channel dynamics utilizing the conceptof bilaterally coupled systems [21] Each part of dropletgenerator is respectively modeled as a two-port subsystemand finally cascaded into the whole system Compared withLEM technique two-port network owns the advantage of sat-isfactory simulation accuracy and the disadvantages of morecomplex structures and inevitably low simulation speed

This work is aimed at building a dynamic LEM (DLEM)model to characterize nonlinear behaviors of piezoelectricinkjet printhead In the proposed DLEM the linear equiva-lent circuit of piezoceramic in original LEM is replaced intononlinear time-varying capacitance inductance and resistorin series Compared to other analytic models such as two-port network the DLEM has a simpler structure with thesufficient simulation accuracy Additionally the DLEM isused to search the optimal combinations of high-frequencydriving waveform to effectively minimize the residual vibra-tion for nanosilver ink With comparisons of simulation andexperimental results the significant merits of the proposedmethods lie in the following aspects

(1) By applying dynamic nonlinear structure the simula-tion accuracy of LEM is remarkably improvedThat is

the simulation curves of droplet volume and velocitycan well overlap the experimental measurements

(2) The obvious overshoot observed in the outflow curvescan sufficiently indicate the break-off phenomenonwhich divides the droplet formation process intothe meniscus protruding filament breaking-off anddroplet falling-down stages

(3) TheproposedDLEMcan effectively predicate optimalcombinations of high-frequency driving waveformwith high jetting characteristics

The rest of this paper is organized as follows In Section 2a review of original lumped element modeling droplet gener-ator is present and three equivalent circuits are given to char-acterize the nonlinear behaviors of piezoelectric printheadNext a schematic of experimental method is introduced inSection 3 After that the simulation results of droplet volumeand velocity decide the most suitable model in Section 4Theprediction under restrictive condition is built in Section 5Finally discussing and concluding remarks are collected inSection 6

2 Modeling

21 Original Lumped Element Model LEM provides a simplemethod to estimate the low-frequency response of printheadwhich assumes that the characteristic length scales of thegoverning physical phenomena are larger than geometricdimension For example the pressure wave length (in theorder of 100mm) of KM series IJ Head of Konica which isdetermined by the sonic speed of the fluid (about 1400ms)and the highest operation frequency of the droplet generator(128 kHz) is much larger than nozzle diameter (27120583m) Sothis assumption for LEM application is well satisfied

A classical LEM is given to simulate jetting character-istics of PZT printhead by Gallas as shown in Figure 1The equivalent circuit model is constructed by the energystorage elements and ideal dissipative terminal In thiselectroacoustic system pressure and voltage are indepen-dent variables while current and volumetric flow rate aredependent variables Model structure shows that the energyconverts from electrical energy to mechanical energy thento fluidicacoustic energy and finally to kinetic energy asdescribed in Figure 1(b) The droplet generator structure canbe characterized by equivalent acoustic mass (representingstored kinetic energy) and acoustic compliance (represent-ing stored potential energy) in which the correspondingequivalent circuit models are supported by various fluidmechanisms as represented in Figures 1(a) and 1(c) Further-more the piezoceramic model is constructed based on theelectrofluidicacoustic theory [22] The neck model is builton the velocity profile function [23] The nozzle model isconstructed according to the end correction for an open tubetheory [24]

As the equivalent circuit model shown in Figure 1 exci-tation voltage 119881ac is applied to piezoelectric ceramic to createmechanical deformation Coupling coefficient 120601

119886represents

a conversion from mechanical domain to acoustic domain

Mathematical Problems in Engineering 3

Channel Neck Nozzle

Piezoceramic

Piezoceramic

(a)

DropformationNozzleNeckChannel

Drivingwaveform Piezoceramic

Electricaldomain

Mechanicaldomain domain domain domain domain

Fluidicacoustic Fluidicacoustic Fluidicacoustic Kinetic

(b)

Electrical Fluidicacoustic Kineticdomaindomaindomain

sim

Vac

CaCCeb MaRad

QoutMaNRaNMaDRaDCaD1 Na

RaO

(c)

Figure 1 Schematic overview of lumped-element modeling for PZT printhead (a) droplet generator structure (b) model structure and (c)equivalent circuit model

119862eb is the blocked electrical capacitance of the piezoelectricmaterial In the acoustic domain 119862aD and 119862aC representthe acoustic compliance of piezoceramic and channel 119877aD119877aN and 119877aO are the acoustic resistance due to structuraldamping neck tapering and fluid flowing out of nozzlerespectively 119872aD 119872aN and 119872aRad represent the acousticmass of piezoceramic neck and nozzle in proper orderrespectively

The dimensions of droplet generator and physical proper-ties of nanosilver ink provided by the LEED-PV Corporationare listed in Table 1 and calculation formulas of LEM modelparameters are provided in Table 2

According to above lumped element model a transferfunction describing the response relationship between theflow rate119876out and input AC voltage119881ac is obtained as follows

119876out (119904)

119881ac (119904)=

120601119886119862aD119904

11988641199044 + 119886

31199043 + 119886

21199042 + 119886

11199041 + 1

(1)

where

1198861= 119862aD (119877aO + 119877aN + 119877aD) + 119862aC (119877aO + 119877aN)

1198862= 119862aD (119872aRad +119872aN +119872aD) + 119862aC (119872aRad +119872aN)

+ 119862aC119862aD119877aD (119877aO + 119877aN)

1198863= 119862aC119862aD [119872aD (119877aO + 119877aN) + (119872aRad +119872aN) 119877aD]

1198864= 119862aC119862aD119872aD (119872aO +119872aN)

(2)

The frequency response of droplet generator driven bythe sinusoidal signal as shown in Figure 2 reveals thatnatural frequency (around 486 kHz) is higher than operating

2 4 6 8 100

5

10

15

20

Frequency (Hz) times104

Vm

axV

norm

Figure 2 Frequency response of lumped element model

frequency (below 128 kHz) This means that the meniscuscan obtain sufficient energy to overcome viscosity and surfacetension and then to generate consistent droplets

As the above simulation models 119862aD 119877aD and 119872aD areregarded as static parameters based on the approximate linearbehaviors of piezoelectric ceramic with a fix path of P-Ehysteresis loop However in fact the piezoelectric ceramic isa nonlinear material with multiple electric fieldstrain curves(parallel and perpendicular to the field) as shown in Figure 3It has been proved that the above original static modelhas insufficient ability to simulate the piezoelectric ceramicdynamic response in time-domain below linear voltage [11]

Our research finds that the maximum error between theexperimental measurements and simulation results can reacheight percent Additionally when the break-off phenomenon

4 Mathematical Problems in Engineering

Table 1 Droplet generator parameters

Parameter Unit Value

Geometry

Channel volume 1198810

Pl 166000Channel length 119871

0Um 7200

Nozzle radius 1198860

Um 135Neck length 119871 Um 45

PZT Piezoceramic thickness ℎ Um 25Free electrical capacitance 119862ef F 28 nF

Fluid

Surface tension 120590 Nm 45 lowast 10minus3

Viscosity 120583 Kg(mlowasts) 15Density 120588

0Kgm3 105 lowast 103

Sonic speed 1198880

ms 1400

Table 2 Lumped elements modeling parameter estimation

LEM Description

Neck 119877aN 119877aN = (2 +1198860

119903)radic2120583120588

0120596

1205871199032 where 119906 is viscosity 119908 is wave frequency and 119886

0119903 is gradients ratio

119872aN 119872aN =1205880(119905 + Δ119905)

1205871199032 where Δ119905 = 085119903(1 minus 07(119903119886

0)) + 085119903

Cavity 119862aC 119862aC =1198810

120588011988820

where 1198810is volume of the cavity

Piezoelectric119862aD 119862aD =

1205871199036

(1 minus V2)16119864ℎ3

where 119864 is the elastic modulus V is Poissonrsquos ratio and ℎ is the thickness

119872aD 119872aD =41205880119871

312058711988620

119877aD 119877aD = 2120585radic119872aD +119872aRad

119862aD where 120585 is the experimentally determined damping factor [18]

Nozzle119877aO 119877aO =

12058801198880

1205871199032[1 minus

21198691(2119896119903)

2119896119903] where 119869

1is the Bessel function of the first kind and 119896 = 119908119888

0

119872aRad 119872aRad =81205880

312058721198860

119862eB 119862eB = 119862eF(1 minus 1205812

) where 1205812 is electroacoustic coupling factor

1 2

e

1

2

3

4

abc

d

x1

x3

minus1

minus1

minus2

times10minus3

Δll

E (kVmm)

Figure 3 Possible paths of P-E hysteresis loop for piezoceramicmaterial [11]

occurs part of filament would go back into the nozzle andthe droplet volume will have a certain decrease But thesimulation curves of droplet volume with original modelhave no obvious sign to show that the break-off phenomenonoccurs (please refer to the simulation results in Section 42)

C0 R1

L1

C1

Figure 4 Equivalent electrical circuitmodel of PZTnonlinear effectwith proportional coefficient

22 Improved LEM with Nonlinear Equivalent Circuits Inorder to resolve above problems in this section threeequivalent circuits with different characters are incorporatedinto the classical LEM to characterize the nonlinear effect ofthe piezoelectric ceramics

(1) Proportional Model In order to describe the nonlinearrelationship between electric field and strain a proportionalmodel (illustrated in Figure 4) as a resonance was defined inthe IEEE standards publication [25] This analytical model is

Mathematical Problems in Engineering 5

C0

C2

C1 R1 L1

R2 L21 N

Figure 5 Equivalent electrical circuitmodel of PZTnonlinear effectwith multiple parallel resonances

derived based on the constitutive relations of piezoelectricityand the following assumptions (a) Euler-Bernouli beamassumption [26] (b) negligible external excitation from airdamping (c) proportional damping (ie the strain ratedamping and viscous damping are assumed to be propor-tional to the bending stiffness and mass per length of thebeam) (d) uniform electric field through the piezoelectricthickness Then the circuit element values of the proposedlumped parameter model are formulated in (1) where T isthe adjustment coefficient

1198711=

119871119898

T2

1198771=

119877119898

T2

1198621= T2 lowast 119862

119898

(3)

(2) Multiple-Resonator Model Yang and Tang [27] givethe multiple-resonator model to characterize piezoelectricceramic in piezoelectric energy harvesters as shown inFigure 5 His research finds that the network of piezoelectricceramic driving circuit consists of infinite branches and eachis composed of simplified equivalent circuit with an inductora capacitor and a resistorWhen the voltage signals from sev-eral branches drive the piezoelectric ceramic simultaneouslythe higher order vibrationmodes are excited It is realized thatworking at higher order modes causes the nonlinear effectof piezoelectric ceramic To characterize the different higherorder modes a circuit of multiple resonators is constructedto match the external branches In this literature to modelthe droplet generator nearly a double circuit may meet thesimulation accuracy Then the circuit element values of theproposed lumped parameter model are formulated in (3)where 120572 and N are both the adjustment coefficients

1198622=

1198621

1205722

1198772= 120572 lowast 119877

1

1198712= 1198711

(4)

(3) Dynamic Model Based on the works in [28 29] an equiv-alent electrical circuit with nonlinear resonance oscillationsin series namely the dynamic model is given to analyzepower BAW (bulk acoustic wave) resonators as shown inFigure 6Three major nonlinear effects are considered in this

C0 C0

C1

C2

C3

R1

L1

L1

R2

R3

C(I)

R(I)

Figure 6 Equivalent electrical circuitmodel of PZTnonlinear effectwith dynamic coefficients

equivalent circuit the amplitude-frequency effect (A-F effect)[30] the intermodulation effect [31] and the bias-frequencyeffect (B-F effect) [32] The dynamic values of resistance andcapacitance in the motional branch are dependent on themotional current in the branchThen derivation formulas aregiven in (4) where A

1 A2 B1 and B

2are the adjustment

coefficients

1198622= 12057211198682

1198623= 12057221198684

1198772= 12057311198682

1198773= 12057321198684

997904rArr 119862 (119868) = 119862

1(1 + A

11198682

+ A21198684

)

119877 (119868) = 1198771(1 + B

11198682

+ B21198684

) (5)

Then the improved LEM structure can be illustrated as inFigure 7

3 Experimental Method

31 Experimental Setup In this experiment the investigatedprinthead is KM series IJ Head of Konica and the used ink isnanosilver ink provided by the LEED-PV Corporation andthe according parameters are listed in Table 1 in Section 2 Aschematic overview of the experimental setup to observe thenanosilver droplet formation process is shown in Figure 8The visualization system consists of a cold source of halogens(XAO LG150) and a CCD camera (SVS ECO424) whichcan synchronize to the driving signal The volume andvelocity of droplet and the profile of meniscus are measuredthrough the images captured by CCD with exposure time of1 us The high-frequency driving waveform is programmedby arbitrary wave generator (RIGO DG2014A) In orderto repeatedly generate uniform droplets ink supply unitconnects air pressure unit which offers suitable negative forceto prevent the ink leaking from the printhead

32 Edge Detection and Image Analysis as a QualitativeMeasure The leading edge detection is to determine thenozzle position and meniscus position A straight line isgenerated along the nozzle in the region of interest (ROI)with

6 Mathematical Problems in Engineering

sim

Vac

CaCCeb

MaRadMaNRaN

MaDRaDCaD1 Na

RaO

(a)

sim

Vac

CaCCeb

MaRadMaNRaN

1 NaRaDT

2MaDT

2

RaO

CaDT2

(b)

sim

Vac

CaC

Ceb

MaRadMaNRaN

MaD

MaD

RaDCaD

1 Na

1 N

CaDa2

aRaD

RaO

(c)

sim

Vac

CaCCeb

MaRadMaNRaN

MaD1 Na C(I) R(I)

RaO

(d)

Figure 7 (a) Original LEM (b) proportional coefficient model (c) multiple resonators model and (d) dynamic coefficients model

Temperature unit

Air pressure unit

Ink supply

Parallel

Halogens light source

CCDPC

Wave generator

Drivingboard

Image

(a)

CCD

Halogens light

sourceArbitrary waveform generator

Drivingboard

PrintheadInk supply

(b)

Figure 8 Experimental setup (a) schematic of experimental setup and (b) image of experimental setup

CCD camera image analysis By setting a suitable thresholdgradient of brightness themeniscus location can be detectedThe satellite droplets and filament can be detected throughfarther image processing such as cropping filtering andthreshold recontrasting (shown as in Figure 9)

In image analysis system the droplet volume is calculatedby

Vol = sum120587 sdot (119889

2)

2

sdot 120575119911 (6)

which hypothesizes that liquid jet is axis-symmetric alongnozzle centerline And the droplet velocity is calculated basedon the position change of the moving-forward leading edgein two consecutive images with specified sampling intervalNote that the information is lost if the meniscus is hiddeninside the nozzle so edge detection only measures theprotruding meniscus

33 High-Frequency Driving Waveform as a System Excita-tion The high-frequency driving waveform is composed of

positive and negative periods including three pulse edges andtwo dwell times as shown in Figure 10 The first rising edgeexpands the channel to get the inkmeniscus into the channelThen the falling edge works to eject the droplet from thenozzle

This period of waveform design is based on the halfperiod of channel resonance Thus the waveform designissue is focused on calculating the optimum value of dwelltime 119879

1= 1198710119862 where 119871

0is the length of channel and 119862

is the sonic speed of the used ink The falling edge of thesecond negative voltage generates larger pressure in channelto accelerate the droplet with higher speed than conventionalsingle trapezoidal waveform The second rising edge whichsets the voltage back to 0 works to restrain the meniscusvibration [33]

Theoretically the suitable slops of the rising edge andfalling edge can restrain the meniscus vibration [34] How-ever in practical applications these two values are too largeto control accurately

Asmentioned above theoretical optimum value for dwelltime 119879

1is set as 52 us The corresponding dwell time 119879

2

Mathematical Problems in Engineering 7

(a) (b) (c) (d)

Figure 9 Image processing steps (a) original image (b) cropped image (c) threshold image and (d) edge image

Time

Voltage

Dwell

Dwell

time T1

time T2

Figure 10 Applied driving waveform of printhead

is recommended as 104 us (twice of 1198791) and the voltage

magnitude is set as 12 V according to the instruction ofKonica printhead In this paper the driving waveform withthese parameters is called the ldquostandardrdquo waveform [35]

4 Simulation Result

41 Coefficient Identification The coefficients of originalLEM model can be determined experimentally Howeverthe three improved models in this work are somewhatcomplicated and their coefficients reveal some importantfeatures of these equivalent circuits For this reason thissection is dedicated to identify the adjustment coefficients inthese corresponding circuits

In this experiment a group of experimental measure-ments with printhead driven by the standard high-frequencywaveform are chosen as the basic reference The coeffi-cients of each proposed improved LEM model should beadjusted to minimize the deviations between the simulationresults and the selected experimental measurements shownin Figure 11 The nonlinear least squares method [36] usingthe Levenberg-Margquardt and Gauss-Newton Algorithms

Table 3 Coefficients for three lumped element models

Model Coefficient ValueProportional 119879 0837

Multiple resonators 120572 51119873 228

Dynamic coefficients

1198601

133 lowast 103

1198611

353 lowast 103

1198602

234 lowast 104

1198612

411 lowast 104

is applied to find the optimal coefficients set for each pro-posed equivalent circuit which makes the simulation curvesoverlap the experimental measurements as far as possibleThe resulting coefficients are listed in Table 3

42 Droplet Volume Simulation In many applications of ink-jet printing the volume of droplet is a critical restrictivecondition Droplets with controllable volume affect not onlythe printing resolution but also the depositional thicknessTherefore how to accurately control droplet volume is a keyproblem to be solved by the proposed method

According to experimental measurements in Figure 11the comparison between the original LEM model and theproposed three improved models is presented in Figure 12In this figure the solid line is the outflow simulation curvesbased on the LEM technique and the star line is the actualmeasured volume values during the droplet formation pro-cess It should be noted that in Figures 12(a)ndash12(d) the firstthree points of actual measurement curve are default to zerobecause no fluid flowing outside the nozzle can be capturedby our image system

As the actual experimental measurements indicated inFigure 11 the droplet formation process is divided into threestages the meniscus protruding stage (ie 119879

1st = 15sim30 us)the filament breaking-off stage (ie 1198792nd = 30sim35 us) andthe droplet falling-down stage (ie 1198793rd = 35sim60 us) That

8 Mathematical Problems in Engineering

3us0pL0ms

6us0pL0ms

9us0pL0ms

12us 15us 18us 21us 24us

30us 33us 36us 39us 42us 45us 48us27us

678pL876ms

994pL730ms

1221 pL684ms

1401pL624ms

1521pL618ms

1533pL600ms

1567pL600ms

1509pL606ms

1492pL618ms

1489pL624 ms

1480 pL624 ms

1479pL618ms

1480pL618ms

Figure 11 Pictures of nanosilver droplet formation process driven by standard waveform

is ideally the outflow curve simulated by analytic modelsshould include overshoot phase which can clearly demon-strate that the part of filament is inhaled back into nozzle orsplits into satellite droplets by viscosity and surface tensionin the filament break-off stage However according to thesimulation result of the original LEM in Figure 12(a) there

is no obvious decreasing after reaching the maximum in theoutflow curve and the outflow curve of the original modelcannot well overlap the actual measurement curve

In Figures 12(b) and 12(d) the overshoot phase canbe observed clearly This performance improvement can bestraightforwardly explained by the fact that poles and zeros

Mathematical Problems in Engineering 9

0 20 40 60

0

10

20

Time (us)

Volu

me (

pL)

ExperimentOriginal

minus10

(a)

0 20 40 60

0

10

20

Time (us)

Volu

me (

pL)

ExperimentProportional

minus10

(b)

0 20 40 60

0

10

20

Time (us)

Volu

me (

pL)

ExperimentMultiple

minus10

minus20

(c)

0 20 40 60

0

10

20

Time (us)

Volu

me (

pL)

ExperimentDynamic

minus10

(d)

Figure 12 Outflow change ofmeasurements and simulations of (a) original model (b) proportional coefficientmodel (c)multiple resonatorsmodel and (d) dynamic coefficient model

of the transfer functions are manipulated to speed up thedynamic response speed of the proportional and dynamicmodels From Figure 12 both the multiple-resonator anddynamic models can obtain precise simulation results Thatis more well-designed coefficients are used to characterizethe nonlinear behaviors of piezoelectric printhead and theadjustable ranges are extended to a satisfied degree of bothmodels Additionally it should be noted that the outflowcurve of dynamic model has not only the overshoot phasebut also high overlap ratio to the experimental measurementMoreover the simulation curves reveal that the meniscusmoves inward before the droplet appears

43 Flow and Droplet Velocity Simulation The velocity ofdroplet is the other important aspect to evaluate the printingeffect Droplets with a high speed can remarkably improve

the printing efficiency and the quality of impact with sub-strate Therefore this experiment gives the velocity simula-tion using the proposed LEM models on the same actualexperimental measurement in Section 41

The comparisons between the flow velocity curves sim-ulated by the original LEM and three improved models areillustrated in Figure 13 The excitation is the standard drivingwaveform For the actual measured velocity curves the firsttwo points are also default to zero because no fluid flowingoutside the nozzle can be captured by our image system

It should be noted that all the values of actual measuredvelocity are positive However from the flow velocity curvesin Figure 13 the values of flow velocity in nozzle can bepositive or negative in different phases This positive-to-negative velocity variation indicates the changes of the flowdirection that leads to the break-off phenomenon Hence in

10 Mathematical Problems in Engineering

0 20 40 60

0

5

10

Time (us)

Velo

city

(ms

)

ExperimentFlow velocityOriginal

minus10

minus5

(a)

0 20 40 60

0

5

10

Time (us)

Velo

city

(ms

)

ExperimentFlow velocityProportional

minus10

minus5

(b)

0 20 40 60

0

5

10

Time (us)

Velo

city

(ms

)

ExperimentFlow velocityMultiple

minus10

minus5

(c)

0 20 40 60

0

5

10

Time (us)

Velo

city

(ms

)

ExperimentFlow velocityDynamic

minus10

minus5

(d)

Figure 13 Comparison of measured droplets velocity and the flowaverage velocity simulated by (a) original model (b) ldquoproportionalcoefficientrdquo model (c) ldquomultiple resonatorsrdquo model and (d) ldquodynamic coefficientsrdquo model

order to correctly simulate the velocity in leading edge wepropose a new calculation formula which combines both ofoutflow curve and flow velocity curve

119881average (119905) = radicsum1199052

119905=1199051

(12) 119876out (119905) sdot 119881 (119905)2

sum1199052

119905=1199051

(12)119876out (119905) (7)

where119876out is the outflow value and119881 is the flow velocity 1199051is

the moment when the sum of 119876out changes from negative topositive in outflow curve and 119905

2after 119905

1is the moment when

flow velocity changes to negative in nozzleFrom Figure 13 it can be obviously observed that in

the beginning of droplet formation process the multiple-resonator and dynamic models exhibit better performance insimulating the droplet velocity rather than the originalmodelThat is the average velocity curves obtained by these two

models are almost coincident to the actual measured velocitycurves

Based on the above analysis and comparisons thedynamicmodel based LEMhas themost accuracy in simulat-ing the droplet volume and velocity and can carry sufficientinformation to distinguish the break-off phenomenon occur-ing in the nanosilver droplet formation processs Thereforeamong these three improved models the dynamic lumpedelement model (DLEM) is the most suitable improved equiv-alent structure for simulating jetting characteristics

5 Prediction

For industry applications in printable electronics fabricationthe printhead must work at a certain status to meet manyrestrictive conditions Among them the most important

Mathematical Problems in Engineering 11

0 20 40 60

0

5

10

15

20

Time (us)

Volu

me (

pL)

minus25

minus20

minus15

minus10

minus5

T1 = 3usT1 = 4usT1 = 5us

T1 = 6usT1 = 7usT1 = 8us

(a)

0 20 40 604

5

6

7

8

9

10

Velo

city

(ms

)

Time (us)

T1 = 3usT1 = 4usT1 = 5us

T1 = 6usT1 = 7usT1 = 8us

(b)

Figure 14 Predictive droplet volume and velocity with different and various dwell times 1198792= 21198791 (a) simulation curves of outflow change

and (b) simulation curves of droplet average velocity

4 6 813

14

15

16

17

SimulationExperiment

Volu

me (

pL)

Time (us)

(a)

SimulationExperiment

4 6 8

45

5

55

6

Time (us)

Velo

city

(ms

)

(b)

Figure 15 Predictive droplet volume and velocity compared with measured volume and velocity

problem is choosing the appropriate combinations of thedriving waveform parameters for the used conductive inksHowever an exhausting manual selective process inevitablywastes a lot of time Therefore the computer-aided methodsare urgently needed for searching these appropriate combi-nations efficiently and robustly In this section the proposed

DLEM is applied to predict jetting characteristics for nanosil-ver ink

51 General Prediction Procedure The general procedure ofdroplet volumevelocity prediction based on DLEM is asfollows

12 Mathematical Problems in Engineering

0us0pL0ms

3us0pL0ms

6us0pL0ms

9us0pL0ms

12us6353pL736ms

15us9318pL612ms

18us11441pL572ms

18us 24us

30us14364pL498ms

33us14046pL510ms

36us13980pL520ms

39us13952pL525ms

42us13858pL525ms

45us13869pL520ms

48us13891pL524ms

51us13924pL522ms

54us

524ms13877pL

57us13840pL530ms

13127 pL548ms

14252pL520ms

27us14786pL504ms

Figure 16 A sequence of pictures of droplet falling from nozzle

Step 1 According to droplet generator parameters and prop-erties of fluid calculate most of the parameters in DLEM

Step 2 According to experimental measurements of dropletvolume driven by a standard waveform determine the rest ofthe adjustment coefficients and damping coefficients

Step 3 Based on the constructed DLEM predicate the valuesof droplet volume and velocity

Step 4 According to the need of applications and restrictiveconditions find out several groups of the appropriate param-eters of driving waveform frompredictive data lists of volumeand velocity

Step 5 After making a subtle adjustment in practice decidethe most appropriate parameters of driving waveform

52 Prediction Case 1 Standard Waveform with DifferentDwell Time Asmentioned above theDLEMmodel is chosento simulate jetting characteristics for the nanosilver inkHere an illustrative example is given to display the effect of

predicting the typical combinations of waveform parameterswith different dwell times (119879

1= 3sim8 us 119879

2= 2119879

1) and

the same amplitude (12V) The properties of nanosilver inkprovided by the LEED-PV Corporation have been listed inTable 1

In Figure 14 the predicted outflow and average velocitycurves with various dwell times are depicted We can observefrom Figure 14(a) that before the droplets rush out of thenozzle the depth of the inhaled meniscus is proportionalto dwell time From Figure 14(b) the time of the dropletappearing is postponed with the increase of dwell timeThat is both two simulation phenomena consist with thetheoretical analysis results in [37]

The comparisons between actual measured and predictedjetting characteristics with various dwell times are given inFigures 15(a) and 15(b) which both illustrate a parabolicchange rule with the increase of dwell time It should benoted that at the dwell time 119879

1= 56 us near the theoretical

optimum value 11987910158401= 52 us the droplet volume and velocity

simultaneously reach the maximum From the performanceof prediction accuracy listed in Table 4 the deviation of thepredicted jetting characteristics is within acceptable range

Mathematical Problems in Engineering 13

45

67

68

1012

1416

10121416

Volu

me (

pL)

91011121314151617

T1 (us)

T2(us)

(a)

45

67

68

1012

1416

3

4

5

6

Velo

city

(ms

)

3

35

4

45

5

55

6

T1 (us) T2

(us)

(b)

Figure 17 Predictive droplet volume and velocity with various dwell times1198791and119879

2 (a) predictive droplet volume and (b) predictive droplet

velocity

53 Prediction Case 2 Driving Waveform under RestrictiveCondition In this experiment an illustrative example underrestrictive conditions is presented to demonstrate the feasibil-ity and effectiveness of the proposed method for predictingjetting characteristics The dimensions of droplet generatorand properties of the nanosilver ink are listed in Table 1 Inorder to reach a high printing resolution and good quality ofdroplet impact with the substrate the restrictive conditionsinvolve that the droplet volume is smaller than 14 pL and thedroplet velocity is larger than 5ms

Based on the prediction procedure the predictive valuesof drop volumevelocity with various dwell times 119879

1and

1198792are listed in Table 5 According to the restrictive con-

ditions three combinations of parameters are found out1198791

= 5 us1198792

= 13 us 1198791

= 55 us1198792

= 14 us and1198791= 5 us119879

2= 75sim8 us After a small range search around

predictive values in the actual test the combination of 1198791=

49 us1198792

= 77 us is finally chosen The dynamic effect ofjetting characteristics driven by this combination is shown inFigure 16 From this figure the jetting characteristics satisfythe specified restrictive conditions quite well Meanwhilealmost no satellite droplet emerges after jetting the maindroplet

The 3D plot of predictive jetting characteristics withvarious dwell times 119879

1and 119879

2is shown in Figure 17 With

the increase of dwell time 1198791 peak positions of volume and

velocity have a tiny shift in 95sim115 us With the same dwelltime 119879

2 there is also a parabolic profile in predictive volume

and velocity

6 Conclusion

The lumped element modeling based on linear model ofpiezoelectric ceramic has been successfully used to simu-late piezoelectric droplet generator However the nonlinearmodel of piezoelectric ceramic is necessary to diminishdeviation between the simulated and measured results Inorder to characterize the nonlinear behavior of piezoelectric

Table 4 Performance of prediction accuracy

Deviation Value

Volume

Max 07135 (499)Min 0036 (013)Mean 02636Std 03316

Velocity

Max 02465 (502)Min 00143 (025)Mean 01095Std 01350

ceramic three models are introduced to simulate dropletgeneration In this work the droplet volumevelocity curvessimulated by dynamic model can carry sufficient informa-tion to distinguish nanosilver droplet formation processclearly Although the complexity of system increases theoutflow curves can obviously reflect the filament breaking-off stage that occurs which is coincident with sampledimages Such nonlinear LEM model is also successful inpredicting jetting characteristics under restrictive conditionsComparing the droplet volumevelocity curves with differentdwell time of high-frequency driving waveform the linearrelationship between dwell time and meniscus appearingmoment is discovered Meanwhile a parabolic change ruleof volumevelocity with the increase of dwell time is in goodagreement with present theoretical analysis

It should be noted that the proposed nonlinear model isfocused on relationship between the droplet volumevelocityand the dwell time of driving waveform while the voltageamplitude is also a decisive factor to droplet formationprocess This should be discussed in our future work

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

14 Mathematical Problems in Engineering

Table5Predictiv

edropletvolume(pL

)and

velocity(m

s)

1198791

1198792

665

775

885

995

10105

11115

12125

13135

14145

15155

4119881

11119

127

134

139

143

146

149

151

15147

142

Default

Default

Default

Default

Default

Default

Default

Default

119878371

395

417

439

459

478

491

500

489

471

451

435

Default

Default

Default

Default

Default

Default

Default

Default

45 119881

Default

117

125

134

143

15155

159

162

16157

153

147

139

Default

Default

Default

Default

Default

Default

119878Default

421

451

476

498

516

532

548

560

543

523

501

482

466

Default

Default

Default

Default

Default

Default

5119881

Default

Default

Default

133

143

151

157

163

167

168

166

163

157

15142

133

Default

Default

Default

Default

119878Default

Default

Default

480

507

531

553

572

587

600

583

563

540

525

509

498

Default

Default

Default

Default

55 119881

Default

Default

Default

Default

Default

151

159

165

168

169

171

169

165

16154

146

138

129

Default

Default

119878Default

Default

Default

Default

Default

524

550

572

593

609

624

601

571

551

537

523

509

496

Default

Default

6119881

Default

Default

Default

Default

Default

Default

Default

158

163

164

165

165

165

158

1495

141

130

119

108

97119878

Default

Default

Default

Default

Default

Default

Default

495

515

533

549

564

544

519

494

474

458

445

432

421

65 119881

Default

Default

Default

Default

Default

Default

Default

Default

Default

155

159

16155

150

143

134

125

114

103

95119878

Default

Default

Default

Default

Default

Default

Default

Default

Default

473

485

495

503

490

469

446

420

400

390

383

7119881

Default

Default

Default

Default

Default

Default

Default

Default

Default

Default

125

135

139

141

135

127

119

110

102

94119878

Default

Default

Default

Default

Default

Default

Default

Default

Default

Default

445

465

479

463

444

422

401

385

370

363

Defaults

areb

ecause

thec

ombinatio

nsof

parametersindrivingwaveform

areo

utof

theo

peratio

nrange

Mathematical Problems in Engineering 15

Authorsrsquo Contribution

Maowei He and Liling Sun contributed equally to this work

Acknowledgment

This research is partially supported by National Natural Sci-ence Foundation of China under Grants 61105067 7100107261174164 and 71271140

References

[1] M SinghHMHaverinen PDhagat andG E Jabbour ldquoInkjetprinting-process and its applicationsrdquo Advanced Materials vol22 no 6 pp 673ndash685 2010

[2] E Macdonald R Salas D Espalin et al ldquo3D printing for therapid prototyping of structural electronicsrdquo IEEE Access vol 2pp 234ndash242 2014

[3] S Umezu T Kitajima H Murase H Ohmori K Katahiraand Y Ito ldquoFabrication of living cell structure utilizing electro-static inkjet phenomenardquo in Proceedings of the 22nd IEEEInternational Conference on Micro Electro Mechanical Systems(MEMS rsquo09) pp 419ndash422 IEEE Sorrento Italy January 2009

[4] P Cosseddu L Basirico A Loi et al ldquoInkjet printed OrganicThin Film Transistors based tactile transducers for artificialrobotic skinrdquo in Proceedings of the 4th IEEE RAS and EMBSInternational Conference on Biomedical Robotics and Biomecha-tronics (BioRob rsquo12) pp 1907ndash1912 June 2012

[5] J Hwang A Wan and A Kahn ldquoEnergetics of metal-organicinterfaces new experiments and assessment of the fieldrdquoMate-rials Science and Engineering R Reports vol 64 no 1-2 pp 1ndash312009

[6] B J Kang and J H Oh ldquoGeometrical characterization of inkjet-printed conductive lines of nanosilver suspensions on a polymersubstraterdquoThin Solid Films vol 518 no 10 pp 2890ndash2896 2010

[7] F Villani P Vacca G Nenna et al ldquoInkjet printed polymerlayer on flexible substrate for OLED applicationsrdquo The Journalof Physical Chemistry C vol 113 no 30 pp 13398ndash13402 2009

[8] J-C Liou and F-G Tseng ldquoMulti-dimensional data registrationCMOSMEMS integrated inkjet printheadrdquo MicroelectronicEngineering vol 88 no 6 pp 888ndash901 2011

[9] J Miettinen K Kaija M Mantysalo et al ldquoMolded substratesfor inkjet printed modulesrdquo IEEE Transactions on Componentsand Packaging Technologies vol 32 no 2 pp 293ndash301 2009

[10] K Silverbrook ldquoPrinthead with multiple actuators in eachchamberrdquo US7708387 B2 2010

[11] V Piefort Finite element modelling of piezoelectric active struc-tures [PhD thesis] Department ofMechanical Engineering andRobotics Universite Libre de Bruxelles Brussels Belgium 2001

[12] L Sang Y Hong and F Wang ldquoInvestigation of viscosity effecton droplet formation in T-shaped microchannels by numericaland analytical methodsrdquo Microfluidics and Nanofluidics vol 6no 5 pp 621ndash635 2009

[13] J Liu S-H Tan Y F Yap M Y Ng and N-T NguyenldquoNumerical and experimental investigations of the formationprocess of ferrofluid dropletsrdquo Microfluidics and Nanofluidicsvol 11 no 2 pp 177ndash187 2011

[14] F Sarrazin K Loubiere L Prat C Gourdon T Bonometti andJ Magnaudet ldquoExperimental and numerical study of dropletshydrodynamics in MicroChannelrdquo AIChE Journal vol 52 no12 pp 4061ndash4070 2006

[15] X Q Xing D L Butler S H Ng Z Wang S Danyluk and CYang ldquoSimulation of droplet formation and coalescence usinglattice Boltzmann-based single-phasemodelrdquo Journal of Colloidand Interface Science vol 311 no 2 pp 609ndash618 2007

[16] Q Gallas R Holman T Nishida B Carroll M Sheplak and LCattafesta ldquoLumped element modeling of piezoelectric-drivensynthetic jet actuatorsrdquoAIAA Journal vol 41 no 2 pp 240ndash2472003

[17] A A Khalate X Bombois R Babuska H Wijshoff and RWaarsing ldquoOptimization-based feedforward control for a Drop-on-Demand inkjet printheadrdquo in Proceedings of the AmericanControl Conference (ACC rsquo10) pp 2182ndash2187 Baltimore MdUSA July 2010

[18] L Meirovitch Fundamentals of Vibrations McGraw-Hill NewYork NY USA 2001

[19] H Seitz and J Heinz ldquoModelling of a microfluidic device withpiezoelectric actuatorsrdquo Journal of Micromechanics and Micro-engineering vol 14 no 8 pp 1140ndash1147 2004

[20] S Kim J Sung andMH Lee ldquoPressure wave and fluid velocityin a bend-mode inkjet nozzle with double PZT actuatorsrdquoJournal of Thermal Science vol 22 no 1 pp 29ndash35 2013

[21] G Wassink Inkjet printhead performance enhancement by feed-forward input design based on two-port modeling [PhD thesis]Delft University of Technology 2007

[22] S Prasad SHorowitz QGallas B Sankar L Cattafesta andMSheplak ldquoTwo-port electroacoustic model of an axisymmet-ric piezoelectric composite platerdquo in Proceedings of the 43rdAIAAASMEASCEAHS Structures Structural DynamicsandMaterials Conference AIAA Paper 2002-1365 AIAA DenverColo USA April 2002

[23] D T BlackstockFundamentals of Physical Acoustics JohnWileyamp Sons New York NY USA 2000

[24] F M White Fluid Mechanics McGraw-Hill New York NYUSA 1979

[25] H Schempt andD R Yoerger ldquoStudy of dominant performancecharacteristics in robot transmissionsrdquo Journal of MechanicalDesign Transactions Of the ASME vol 115 no 3 pp 472ndash4821993

[26] E A Witmer ldquoElementary Bernoulli-Euler beam theoryrdquoMITUnified Engineering Course Notes vol 5 pp 114ndash164 1991-1992

[27] Y Yang and L Tang ldquoEquivalent circuit modeling of piezoelec-tric energy harvestersrdquo Journal of Intelligent Material Systemsand Structures vol 20 no 18 pp 2223ndash2235 2009

[28] F Constantinescu A G Gheorghe and M Nitescu ldquoNewcircuit models of power BAW resonatorsrdquo in Proceedings of theEuropean Microwave Integrated Circuit Conference October2007

[29] A Albareda and R PerezNon-Linear Behaviour of PiezoelectricCeramics vol 140ofSpringer Series inMaterials Science Springer2011

[30] J Nosek ldquoDrive level dependence of the resonant frequency inBAW quartz resonators and his modelingrdquo IEEE TransactionsonUltrasonics Ferroelectrics and Frequency Control vol 46 no4 pp 823ndash829 1999

[31] R Aigner N-H Huynh M Handtmann and S MarksteinerldquoBehavior of BAW devices at high power levelsrdquo in Proceedingsof the IEEE MTT-S International Microwave Symposium pp429ndash432 Long Beach Calif USA June 2005

[32] R S Ketcham G R Kline and K M Lakin ldquoPerformanceof TFR filters under elevated power conditionsrdquo in Proceedingsof 42nd the Annual Frequency Control Symposium pp 106ndash111Baltimore Md USA June 1988

16 Mathematical Problems in Engineering

[33] K-S Kwon ldquoWaveform design methods for piezo inkjet dis-pensers based on measured meniscus motionrdquo Journal of Micro-electromechanical Systems vol 18 no 5 pp 1118ndash1125 2009

[34] H-C Wu and H-J Lin ldquoEffects of actuating pressure wave-forms on the droplet behavior in a piezoelectric inkjetrdquoMateri-als Transactions vol 51 no 12 pp 2269ndash2276 2010

[35] KMinolta InkjetHeadApplicationNote-KM1024 Series KonicaMinolta Ij Technologies

[36] G Golub andV Pereyra ldquoSeparable nonlinear least squares thevariable projection method and its applicationsrdquo Inverse Prob-lems vol 19 no 2 pp R1ndashR26 2003

[37] H Wijshoff ldquoThe dynamics of the piezo inkjet printheadoperationrdquo Physics Reports vol 491 no 4-5 pp 77ndash177 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

Researchers have developed many approaches to solvethese issues The most typical approaches include usinganalytical or numerical techniques to model the DoD piezo-electric inkjet printhead [12 13] Based on solving the nonlin-ear Navier-Stokes equations numerical modeling of dropletformation process has been focused on the liquid filamentevolution with known pressure input boundary conditionsassumed to be either known or an input correction forcertain specific droplet generator Numerical models areoften programmed in CFD packages using complex meshesand solving techniques This approach provides visualizedinformation about the acoustic pressure wave travellinginside the channel [14] However numerical models are verycomplex and therefore computationally expensive In the lastdecade analytical modeling approach is gradually used todescribe the ink channel dynamics although the accuracyof analytical models is lower than numerical models [15]Based on several assumptions and simplifications analyticalmodels can provide simple and timesaving ways to model-based control droplet generator with a sufficient accuracy Inthis category lumped elementmodeling [16] and feedforwardtwo-port network modeling [17] technique are two typicalanalytic printheadmodels to analyze the acoustic-mechanicalbehaviors of droplet generator

The lumped element modeling (LEM) approach intro-duces an equivalent electric circuit including capacitorsresistors and inductors in series or parallel to describe thedynamics of the ink channel [19] The LEM results explainthe driving force difference between different operatingconditions Lumped element modeling technique has beena suitable method to simulate the operating mechanism ofdroplet generator due to its outstanding advantages suchas moderate model complexity and high simulation speedHowever the accuracy of these classical LEM models islow for the improved performance requirements of printableelectronics fabrication [20] The two-port model is proposedto describe the ink channel dynamics utilizing the conceptof bilaterally coupled systems [21] Each part of dropletgenerator is respectively modeled as a two-port subsystemand finally cascaded into the whole system Compared withLEM technique two-port network owns the advantage of sat-isfactory simulation accuracy and the disadvantages of morecomplex structures and inevitably low simulation speed

This work is aimed at building a dynamic LEM (DLEM)model to characterize nonlinear behaviors of piezoelectricinkjet printhead In the proposed DLEM the linear equiva-lent circuit of piezoceramic in original LEM is replaced intononlinear time-varying capacitance inductance and resistorin series Compared to other analytic models such as two-port network the DLEM has a simpler structure with thesufficient simulation accuracy Additionally the DLEM isused to search the optimal combinations of high-frequencydriving waveform to effectively minimize the residual vibra-tion for nanosilver ink With comparisons of simulation andexperimental results the significant merits of the proposedmethods lie in the following aspects

(1) By applying dynamic nonlinear structure the simula-tion accuracy of LEM is remarkably improvedThat is

the simulation curves of droplet volume and velocitycan well overlap the experimental measurements

(2) The obvious overshoot observed in the outflow curvescan sufficiently indicate the break-off phenomenonwhich divides the droplet formation process intothe meniscus protruding filament breaking-off anddroplet falling-down stages

(3) TheproposedDLEMcan effectively predicate optimalcombinations of high-frequency driving waveformwith high jetting characteristics

The rest of this paper is organized as follows In Section 2a review of original lumped element modeling droplet gener-ator is present and three equivalent circuits are given to char-acterize the nonlinear behaviors of piezoelectric printheadNext a schematic of experimental method is introduced inSection 3 After that the simulation results of droplet volumeand velocity decide the most suitable model in Section 4Theprediction under restrictive condition is built in Section 5Finally discussing and concluding remarks are collected inSection 6

2 Modeling

21 Original Lumped Element Model LEM provides a simplemethod to estimate the low-frequency response of printheadwhich assumes that the characteristic length scales of thegoverning physical phenomena are larger than geometricdimension For example the pressure wave length (in theorder of 100mm) of KM series IJ Head of Konica which isdetermined by the sonic speed of the fluid (about 1400ms)and the highest operation frequency of the droplet generator(128 kHz) is much larger than nozzle diameter (27120583m) Sothis assumption for LEM application is well satisfied

A classical LEM is given to simulate jetting character-istics of PZT printhead by Gallas as shown in Figure 1The equivalent circuit model is constructed by the energystorage elements and ideal dissipative terminal In thiselectroacoustic system pressure and voltage are indepen-dent variables while current and volumetric flow rate aredependent variables Model structure shows that the energyconverts from electrical energy to mechanical energy thento fluidicacoustic energy and finally to kinetic energy asdescribed in Figure 1(b) The droplet generator structure canbe characterized by equivalent acoustic mass (representingstored kinetic energy) and acoustic compliance (represent-ing stored potential energy) in which the correspondingequivalent circuit models are supported by various fluidmechanisms as represented in Figures 1(a) and 1(c) Further-more the piezoceramic model is constructed based on theelectrofluidicacoustic theory [22] The neck model is builton the velocity profile function [23] The nozzle model isconstructed according to the end correction for an open tubetheory [24]

As the equivalent circuit model shown in Figure 1 exci-tation voltage 119881ac is applied to piezoelectric ceramic to createmechanical deformation Coupling coefficient 120601

119886represents

a conversion from mechanical domain to acoustic domain

Mathematical Problems in Engineering 3

Channel Neck Nozzle

Piezoceramic

Piezoceramic

(a)

DropformationNozzleNeckChannel

Drivingwaveform Piezoceramic

Electricaldomain

Mechanicaldomain domain domain domain domain

Fluidicacoustic Fluidicacoustic Fluidicacoustic Kinetic

(b)

Electrical Fluidicacoustic Kineticdomaindomaindomain

sim

Vac

CaCCeb MaRad

QoutMaNRaNMaDRaDCaD1 Na

RaO

(c)

Figure 1 Schematic overview of lumped-element modeling for PZT printhead (a) droplet generator structure (b) model structure and (c)equivalent circuit model

119862eb is the blocked electrical capacitance of the piezoelectricmaterial In the acoustic domain 119862aD and 119862aC representthe acoustic compliance of piezoceramic and channel 119877aD119877aN and 119877aO are the acoustic resistance due to structuraldamping neck tapering and fluid flowing out of nozzlerespectively 119872aD 119872aN and 119872aRad represent the acousticmass of piezoceramic neck and nozzle in proper orderrespectively

The dimensions of droplet generator and physical proper-ties of nanosilver ink provided by the LEED-PV Corporationare listed in Table 1 and calculation formulas of LEM modelparameters are provided in Table 2

According to above lumped element model a transferfunction describing the response relationship between theflow rate119876out and input AC voltage119881ac is obtained as follows

119876out (119904)

119881ac (119904)=

120601119886119862aD119904

11988641199044 + 119886

31199043 + 119886

21199042 + 119886

11199041 + 1

(1)

where

1198861= 119862aD (119877aO + 119877aN + 119877aD) + 119862aC (119877aO + 119877aN)

1198862= 119862aD (119872aRad +119872aN +119872aD) + 119862aC (119872aRad +119872aN)

+ 119862aC119862aD119877aD (119877aO + 119877aN)

1198863= 119862aC119862aD [119872aD (119877aO + 119877aN) + (119872aRad +119872aN) 119877aD]

1198864= 119862aC119862aD119872aD (119872aO +119872aN)

(2)

The frequency response of droplet generator driven bythe sinusoidal signal as shown in Figure 2 reveals thatnatural frequency (around 486 kHz) is higher than operating

2 4 6 8 100

5

10

15

20

Frequency (Hz) times104

Vm

axV

norm

Figure 2 Frequency response of lumped element model

frequency (below 128 kHz) This means that the meniscuscan obtain sufficient energy to overcome viscosity and surfacetension and then to generate consistent droplets

As the above simulation models 119862aD 119877aD and 119872aD areregarded as static parameters based on the approximate linearbehaviors of piezoelectric ceramic with a fix path of P-Ehysteresis loop However in fact the piezoelectric ceramic isa nonlinear material with multiple electric fieldstrain curves(parallel and perpendicular to the field) as shown in Figure 3It has been proved that the above original static modelhas insufficient ability to simulate the piezoelectric ceramicdynamic response in time-domain below linear voltage [11]

Our research finds that the maximum error between theexperimental measurements and simulation results can reacheight percent Additionally when the break-off phenomenon

4 Mathematical Problems in Engineering

Table 1 Droplet generator parameters

Parameter Unit Value

Geometry

Channel volume 1198810

Pl 166000Channel length 119871

0Um 7200

Nozzle radius 1198860

Um 135Neck length 119871 Um 45

PZT Piezoceramic thickness ℎ Um 25Free electrical capacitance 119862ef F 28 nF

Fluid

Surface tension 120590 Nm 45 lowast 10minus3

Viscosity 120583 Kg(mlowasts) 15Density 120588

0Kgm3 105 lowast 103

Sonic speed 1198880

ms 1400

Table 2 Lumped elements modeling parameter estimation

LEM Description

Neck 119877aN 119877aN = (2 +1198860

119903)radic2120583120588

0120596

1205871199032 where 119906 is viscosity 119908 is wave frequency and 119886

0119903 is gradients ratio

119872aN 119872aN =1205880(119905 + Δ119905)

1205871199032 where Δ119905 = 085119903(1 minus 07(119903119886

0)) + 085119903

Cavity 119862aC 119862aC =1198810

120588011988820

where 1198810is volume of the cavity

Piezoelectric119862aD 119862aD =

1205871199036

(1 minus V2)16119864ℎ3

where 119864 is the elastic modulus V is Poissonrsquos ratio and ℎ is the thickness

119872aD 119872aD =41205880119871

312058711988620

119877aD 119877aD = 2120585radic119872aD +119872aRad

119862aD where 120585 is the experimentally determined damping factor [18]

Nozzle119877aO 119877aO =

12058801198880

1205871199032[1 minus

21198691(2119896119903)

2119896119903] where 119869

1is the Bessel function of the first kind and 119896 = 119908119888

0

119872aRad 119872aRad =81205880

312058721198860

119862eB 119862eB = 119862eF(1 minus 1205812

) where 1205812 is electroacoustic coupling factor

1 2

e

1

2

3

4

abc

d

x1

x3

minus1

minus1

minus2

times10minus3

Δll

E (kVmm)

Figure 3 Possible paths of P-E hysteresis loop for piezoceramicmaterial [11]

occurs part of filament would go back into the nozzle andthe droplet volume will have a certain decrease But thesimulation curves of droplet volume with original modelhave no obvious sign to show that the break-off phenomenonoccurs (please refer to the simulation results in Section 42)

C0 R1

L1

C1

Figure 4 Equivalent electrical circuitmodel of PZTnonlinear effectwith proportional coefficient

22 Improved LEM with Nonlinear Equivalent Circuits Inorder to resolve above problems in this section threeequivalent circuits with different characters are incorporatedinto the classical LEM to characterize the nonlinear effect ofthe piezoelectric ceramics

(1) Proportional Model In order to describe the nonlinearrelationship between electric field and strain a proportionalmodel (illustrated in Figure 4) as a resonance was defined inthe IEEE standards publication [25] This analytical model is

Mathematical Problems in Engineering 5

C0

C2

C1 R1 L1

R2 L21 N

Figure 5 Equivalent electrical circuitmodel of PZTnonlinear effectwith multiple parallel resonances

derived based on the constitutive relations of piezoelectricityand the following assumptions (a) Euler-Bernouli beamassumption [26] (b) negligible external excitation from airdamping (c) proportional damping (ie the strain ratedamping and viscous damping are assumed to be propor-tional to the bending stiffness and mass per length of thebeam) (d) uniform electric field through the piezoelectricthickness Then the circuit element values of the proposedlumped parameter model are formulated in (1) where T isthe adjustment coefficient

1198711=

119871119898

T2

1198771=

119877119898

T2

1198621= T2 lowast 119862

119898

(3)

(2) Multiple-Resonator Model Yang and Tang [27] givethe multiple-resonator model to characterize piezoelectricceramic in piezoelectric energy harvesters as shown inFigure 5 His research finds that the network of piezoelectricceramic driving circuit consists of infinite branches and eachis composed of simplified equivalent circuit with an inductora capacitor and a resistorWhen the voltage signals from sev-eral branches drive the piezoelectric ceramic simultaneouslythe higher order vibrationmodes are excited It is realized thatworking at higher order modes causes the nonlinear effectof piezoelectric ceramic To characterize the different higherorder modes a circuit of multiple resonators is constructedto match the external branches In this literature to modelthe droplet generator nearly a double circuit may meet thesimulation accuracy Then the circuit element values of theproposed lumped parameter model are formulated in (3)where 120572 and N are both the adjustment coefficients

1198622=

1198621

1205722

1198772= 120572 lowast 119877

1

1198712= 1198711

(4)

(3) Dynamic Model Based on the works in [28 29] an equiv-alent electrical circuit with nonlinear resonance oscillationsin series namely the dynamic model is given to analyzepower BAW (bulk acoustic wave) resonators as shown inFigure 6Three major nonlinear effects are considered in this

C0 C0

C1

C2

C3

R1

L1

L1

R2

R3

C(I)

R(I)

Figure 6 Equivalent electrical circuitmodel of PZTnonlinear effectwith dynamic coefficients

equivalent circuit the amplitude-frequency effect (A-F effect)[30] the intermodulation effect [31] and the bias-frequencyeffect (B-F effect) [32] The dynamic values of resistance andcapacitance in the motional branch are dependent on themotional current in the branchThen derivation formulas aregiven in (4) where A

1 A2 B1 and B

2are the adjustment

coefficients

1198622= 12057211198682

1198623= 12057221198684

1198772= 12057311198682

1198773= 12057321198684

997904rArr 119862 (119868) = 119862

1(1 + A

11198682

+ A21198684

)

119877 (119868) = 1198771(1 + B

11198682

+ B21198684

) (5)

Then the improved LEM structure can be illustrated as inFigure 7

3 Experimental Method

31 Experimental Setup In this experiment the investigatedprinthead is KM series IJ Head of Konica and the used ink isnanosilver ink provided by the LEED-PV Corporation andthe according parameters are listed in Table 1 in Section 2 Aschematic overview of the experimental setup to observe thenanosilver droplet formation process is shown in Figure 8The visualization system consists of a cold source of halogens(XAO LG150) and a CCD camera (SVS ECO424) whichcan synchronize to the driving signal The volume andvelocity of droplet and the profile of meniscus are measuredthrough the images captured by CCD with exposure time of1 us The high-frequency driving waveform is programmedby arbitrary wave generator (RIGO DG2014A) In orderto repeatedly generate uniform droplets ink supply unitconnects air pressure unit which offers suitable negative forceto prevent the ink leaking from the printhead

32 Edge Detection and Image Analysis as a QualitativeMeasure The leading edge detection is to determine thenozzle position and meniscus position A straight line isgenerated along the nozzle in the region of interest (ROI)with

6 Mathematical Problems in Engineering

sim

Vac

CaCCeb

MaRadMaNRaN

MaDRaDCaD1 Na

RaO

(a)

sim

Vac

CaCCeb

MaRadMaNRaN

1 NaRaDT

2MaDT

2

RaO

CaDT2

(b)

sim

Vac

CaC

Ceb

MaRadMaNRaN

MaD

MaD

RaDCaD

1 Na

1 N

CaDa2

aRaD

RaO

(c)

sim

Vac

CaCCeb

MaRadMaNRaN

MaD1 Na C(I) R(I)

RaO

(d)

Figure 7 (a) Original LEM (b) proportional coefficient model (c) multiple resonators model and (d) dynamic coefficients model

Temperature unit

Air pressure unit

Ink supply

Parallel

Halogens light source

CCDPC

Wave generator

Drivingboard

Image

(a)

CCD

Halogens light

sourceArbitrary waveform generator

Drivingboard

PrintheadInk supply

(b)

Figure 8 Experimental setup (a) schematic of experimental setup and (b) image of experimental setup

CCD camera image analysis By setting a suitable thresholdgradient of brightness themeniscus location can be detectedThe satellite droplets and filament can be detected throughfarther image processing such as cropping filtering andthreshold recontrasting (shown as in Figure 9)

In image analysis system the droplet volume is calculatedby

Vol = sum120587 sdot (119889

2)

2

sdot 120575119911 (6)

which hypothesizes that liquid jet is axis-symmetric alongnozzle centerline And the droplet velocity is calculated basedon the position change of the moving-forward leading edgein two consecutive images with specified sampling intervalNote that the information is lost if the meniscus is hiddeninside the nozzle so edge detection only measures theprotruding meniscus

33 High-Frequency Driving Waveform as a System Excita-tion The high-frequency driving waveform is composed of

positive and negative periods including three pulse edges andtwo dwell times as shown in Figure 10 The first rising edgeexpands the channel to get the inkmeniscus into the channelThen the falling edge works to eject the droplet from thenozzle

This period of waveform design is based on the halfperiod of channel resonance Thus the waveform designissue is focused on calculating the optimum value of dwelltime 119879

1= 1198710119862 where 119871

0is the length of channel and 119862

is the sonic speed of the used ink The falling edge of thesecond negative voltage generates larger pressure in channelto accelerate the droplet with higher speed than conventionalsingle trapezoidal waveform The second rising edge whichsets the voltage back to 0 works to restrain the meniscusvibration [33]

Theoretically the suitable slops of the rising edge andfalling edge can restrain the meniscus vibration [34] How-ever in practical applications these two values are too largeto control accurately

Asmentioned above theoretical optimum value for dwelltime 119879

1is set as 52 us The corresponding dwell time 119879

2

Mathematical Problems in Engineering 7

(a) (b) (c) (d)

Figure 9 Image processing steps (a) original image (b) cropped image (c) threshold image and (d) edge image

Time

Voltage

Dwell

Dwell

time T1

time T2

Figure 10 Applied driving waveform of printhead

is recommended as 104 us (twice of 1198791) and the voltage

magnitude is set as 12 V according to the instruction ofKonica printhead In this paper the driving waveform withthese parameters is called the ldquostandardrdquo waveform [35]

4 Simulation Result

41 Coefficient Identification The coefficients of originalLEM model can be determined experimentally Howeverthe three improved models in this work are somewhatcomplicated and their coefficients reveal some importantfeatures of these equivalent circuits For this reason thissection is dedicated to identify the adjustment coefficients inthese corresponding circuits

In this experiment a group of experimental measure-ments with printhead driven by the standard high-frequencywaveform are chosen as the basic reference The coeffi-cients of each proposed improved LEM model should beadjusted to minimize the deviations between the simulationresults and the selected experimental measurements shownin Figure 11 The nonlinear least squares method [36] usingthe Levenberg-Margquardt and Gauss-Newton Algorithms

Table 3 Coefficients for three lumped element models

Model Coefficient ValueProportional 119879 0837

Multiple resonators 120572 51119873 228

Dynamic coefficients

1198601

133 lowast 103

1198611

353 lowast 103

1198602

234 lowast 104

1198612

411 lowast 104

is applied to find the optimal coefficients set for each pro-posed equivalent circuit which makes the simulation curvesoverlap the experimental measurements as far as possibleThe resulting coefficients are listed in Table 3

42 Droplet Volume Simulation In many applications of ink-jet printing the volume of droplet is a critical restrictivecondition Droplets with controllable volume affect not onlythe printing resolution but also the depositional thicknessTherefore how to accurately control droplet volume is a keyproblem to be solved by the proposed method

According to experimental measurements in Figure 11the comparison between the original LEM model and theproposed three improved models is presented in Figure 12In this figure the solid line is the outflow simulation curvesbased on the LEM technique and the star line is the actualmeasured volume values during the droplet formation pro-cess It should be noted that in Figures 12(a)ndash12(d) the firstthree points of actual measurement curve are default to zerobecause no fluid flowing outside the nozzle can be capturedby our image system

As the actual experimental measurements indicated inFigure 11 the droplet formation process is divided into threestages the meniscus protruding stage (ie 119879

1st = 15sim30 us)the filament breaking-off stage (ie 1198792nd = 30sim35 us) andthe droplet falling-down stage (ie 1198793rd = 35sim60 us) That

8 Mathematical Problems in Engineering

3us0pL0ms

6us0pL0ms

9us0pL0ms

12us 15us 18us 21us 24us

30us 33us 36us 39us 42us 45us 48us27us

678pL876ms

994pL730ms

1221 pL684ms

1401pL624ms

1521pL618ms

1533pL600ms

1567pL600ms

1509pL606ms

1492pL618ms

1489pL624 ms

1480 pL624 ms

1479pL618ms

1480pL618ms

Figure 11 Pictures of nanosilver droplet formation process driven by standard waveform

is ideally the outflow curve simulated by analytic modelsshould include overshoot phase which can clearly demon-strate that the part of filament is inhaled back into nozzle orsplits into satellite droplets by viscosity and surface tensionin the filament break-off stage However according to thesimulation result of the original LEM in Figure 12(a) there

is no obvious decreasing after reaching the maximum in theoutflow curve and the outflow curve of the original modelcannot well overlap the actual measurement curve

In Figures 12(b) and 12(d) the overshoot phase canbe observed clearly This performance improvement can bestraightforwardly explained by the fact that poles and zeros

Mathematical Problems in Engineering 9

0 20 40 60

0

10

20

Time (us)

Volu

me (

pL)

ExperimentOriginal

minus10

(a)

0 20 40 60

0

10

20

Time (us)

Volu

me (

pL)

ExperimentProportional

minus10

(b)

0 20 40 60

0

10

20

Time (us)

Volu

me (

pL)

ExperimentMultiple

minus10

minus20

(c)

0 20 40 60

0

10

20

Time (us)

Volu

me (

pL)

ExperimentDynamic

minus10

(d)

Figure 12 Outflow change ofmeasurements and simulations of (a) original model (b) proportional coefficientmodel (c)multiple resonatorsmodel and (d) dynamic coefficient model

of the transfer functions are manipulated to speed up thedynamic response speed of the proportional and dynamicmodels From Figure 12 both the multiple-resonator anddynamic models can obtain precise simulation results Thatis more well-designed coefficients are used to characterizethe nonlinear behaviors of piezoelectric printhead and theadjustable ranges are extended to a satisfied degree of bothmodels Additionally it should be noted that the outflowcurve of dynamic model has not only the overshoot phasebut also high overlap ratio to the experimental measurementMoreover the simulation curves reveal that the meniscusmoves inward before the droplet appears

43 Flow and Droplet Velocity Simulation The velocity ofdroplet is the other important aspect to evaluate the printingeffect Droplets with a high speed can remarkably improve

the printing efficiency and the quality of impact with sub-strate Therefore this experiment gives the velocity simula-tion using the proposed LEM models on the same actualexperimental measurement in Section 41

The comparisons between the flow velocity curves sim-ulated by the original LEM and three improved models areillustrated in Figure 13 The excitation is the standard drivingwaveform For the actual measured velocity curves the firsttwo points are also default to zero because no fluid flowingoutside the nozzle can be captured by our image system

It should be noted that all the values of actual measuredvelocity are positive However from the flow velocity curvesin Figure 13 the values of flow velocity in nozzle can bepositive or negative in different phases This positive-to-negative velocity variation indicates the changes of the flowdirection that leads to the break-off phenomenon Hence in

10 Mathematical Problems in Engineering

0 20 40 60

0

5

10

Time (us)

Velo

city

(ms

)

ExperimentFlow velocityOriginal

minus10

minus5

(a)

0 20 40 60

0

5

10

Time (us)

Velo

city

(ms

)

ExperimentFlow velocityProportional

minus10

minus5

(b)

0 20 40 60

0

5

10

Time (us)

Velo

city

(ms

)

ExperimentFlow velocityMultiple

minus10

minus5

(c)

0 20 40 60

0

5

10

Time (us)

Velo

city

(ms

)

ExperimentFlow velocityDynamic

minus10

minus5

(d)

Figure 13 Comparison of measured droplets velocity and the flowaverage velocity simulated by (a) original model (b) ldquoproportionalcoefficientrdquo model (c) ldquomultiple resonatorsrdquo model and (d) ldquodynamic coefficientsrdquo model

order to correctly simulate the velocity in leading edge wepropose a new calculation formula which combines both ofoutflow curve and flow velocity curve

119881average (119905) = radicsum1199052

119905=1199051

(12) 119876out (119905) sdot 119881 (119905)2

sum1199052

119905=1199051

(12)119876out (119905) (7)

where119876out is the outflow value and119881 is the flow velocity 1199051is

the moment when the sum of 119876out changes from negative topositive in outflow curve and 119905

2after 119905

1is the moment when

flow velocity changes to negative in nozzleFrom Figure 13 it can be obviously observed that in

the beginning of droplet formation process the multiple-resonator and dynamic models exhibit better performance insimulating the droplet velocity rather than the originalmodelThat is the average velocity curves obtained by these two

models are almost coincident to the actual measured velocitycurves

Based on the above analysis and comparisons thedynamicmodel based LEMhas themost accuracy in simulat-ing the droplet volume and velocity and can carry sufficientinformation to distinguish the break-off phenomenon occur-ing in the nanosilver droplet formation processs Thereforeamong these three improved models the dynamic lumpedelement model (DLEM) is the most suitable improved equiv-alent structure for simulating jetting characteristics

5 Prediction

For industry applications in printable electronics fabricationthe printhead must work at a certain status to meet manyrestrictive conditions Among them the most important

Mathematical Problems in Engineering 11

0 20 40 60

0

5

10

15

20

Time (us)

Volu

me (

pL)

minus25

minus20

minus15

minus10

minus5

T1 = 3usT1 = 4usT1 = 5us

T1 = 6usT1 = 7usT1 = 8us

(a)

0 20 40 604

5

6

7

8

9

10

Velo

city

(ms

)

Time (us)

T1 = 3usT1 = 4usT1 = 5us

T1 = 6usT1 = 7usT1 = 8us

(b)

Figure 14 Predictive droplet volume and velocity with different and various dwell times 1198792= 21198791 (a) simulation curves of outflow change

and (b) simulation curves of droplet average velocity

4 6 813

14

15

16

17

SimulationExperiment

Volu

me (

pL)

Time (us)

(a)

SimulationExperiment

4 6 8

45

5

55

6

Time (us)

Velo

city

(ms

)

(b)

Figure 15 Predictive droplet volume and velocity compared with measured volume and velocity

problem is choosing the appropriate combinations of thedriving waveform parameters for the used conductive inksHowever an exhausting manual selective process inevitablywastes a lot of time Therefore the computer-aided methodsare urgently needed for searching these appropriate combi-nations efficiently and robustly In this section the proposed

DLEM is applied to predict jetting characteristics for nanosil-ver ink

51 General Prediction Procedure The general procedure ofdroplet volumevelocity prediction based on DLEM is asfollows

12 Mathematical Problems in Engineering

0us0pL0ms

3us0pL0ms

6us0pL0ms

9us0pL0ms

12us6353pL736ms

15us9318pL612ms

18us11441pL572ms

18us 24us

30us14364pL498ms

33us14046pL510ms

36us13980pL520ms

39us13952pL525ms

42us13858pL525ms

45us13869pL520ms

48us13891pL524ms

51us13924pL522ms

54us

524ms13877pL

57us13840pL530ms

13127 pL548ms

14252pL520ms

27us14786pL504ms

Figure 16 A sequence of pictures of droplet falling from nozzle

Step 1 According to droplet generator parameters and prop-erties of fluid calculate most of the parameters in DLEM

Step 2 According to experimental measurements of dropletvolume driven by a standard waveform determine the rest ofthe adjustment coefficients and damping coefficients

Step 3 Based on the constructed DLEM predicate the valuesof droplet volume and velocity

Step 4 According to the need of applications and restrictiveconditions find out several groups of the appropriate param-eters of driving waveform frompredictive data lists of volumeand velocity

Step 5 After making a subtle adjustment in practice decidethe most appropriate parameters of driving waveform

52 Prediction Case 1 Standard Waveform with DifferentDwell Time Asmentioned above theDLEMmodel is chosento simulate jetting characteristics for the nanosilver inkHere an illustrative example is given to display the effect of

predicting the typical combinations of waveform parameterswith different dwell times (119879

1= 3sim8 us 119879

2= 2119879

1) and

the same amplitude (12V) The properties of nanosilver inkprovided by the LEED-PV Corporation have been listed inTable 1

In Figure 14 the predicted outflow and average velocitycurves with various dwell times are depicted We can observefrom Figure 14(a) that before the droplets rush out of thenozzle the depth of the inhaled meniscus is proportionalto dwell time From Figure 14(b) the time of the dropletappearing is postponed with the increase of dwell timeThat is both two simulation phenomena consist with thetheoretical analysis results in [37]

The comparisons between actual measured and predictedjetting characteristics with various dwell times are given inFigures 15(a) and 15(b) which both illustrate a parabolicchange rule with the increase of dwell time It should benoted that at the dwell time 119879

1= 56 us near the theoretical

optimum value 11987910158401= 52 us the droplet volume and velocity

simultaneously reach the maximum From the performanceof prediction accuracy listed in Table 4 the deviation of thepredicted jetting characteristics is within acceptable range

Mathematical Problems in Engineering 13

45

67

68

1012

1416

10121416

Volu

me (

pL)

91011121314151617

T1 (us)

T2(us)

(a)

45

67

68

1012

1416

3

4

5

6

Velo

city

(ms

)

3

35

4

45

5

55

6

T1 (us) T2

(us)

(b)

Figure 17 Predictive droplet volume and velocity with various dwell times1198791and119879

2 (a) predictive droplet volume and (b) predictive droplet

velocity

53 Prediction Case 2 Driving Waveform under RestrictiveCondition In this experiment an illustrative example underrestrictive conditions is presented to demonstrate the feasibil-ity and effectiveness of the proposed method for predictingjetting characteristics The dimensions of droplet generatorand properties of the nanosilver ink are listed in Table 1 Inorder to reach a high printing resolution and good quality ofdroplet impact with the substrate the restrictive conditionsinvolve that the droplet volume is smaller than 14 pL and thedroplet velocity is larger than 5ms

Based on the prediction procedure the predictive valuesof drop volumevelocity with various dwell times 119879

1and

1198792are listed in Table 5 According to the restrictive con-

ditions three combinations of parameters are found out1198791

= 5 us1198792

= 13 us 1198791

= 55 us1198792

= 14 us and1198791= 5 us119879

2= 75sim8 us After a small range search around

predictive values in the actual test the combination of 1198791=

49 us1198792

= 77 us is finally chosen The dynamic effect ofjetting characteristics driven by this combination is shown inFigure 16 From this figure the jetting characteristics satisfythe specified restrictive conditions quite well Meanwhilealmost no satellite droplet emerges after jetting the maindroplet

The 3D plot of predictive jetting characteristics withvarious dwell times 119879

1and 119879

2is shown in Figure 17 With

the increase of dwell time 1198791 peak positions of volume and

velocity have a tiny shift in 95sim115 us With the same dwelltime 119879

2 there is also a parabolic profile in predictive volume

and velocity

6 Conclusion

The lumped element modeling based on linear model ofpiezoelectric ceramic has been successfully used to simu-late piezoelectric droplet generator However the nonlinearmodel of piezoelectric ceramic is necessary to diminishdeviation between the simulated and measured results Inorder to characterize the nonlinear behavior of piezoelectric

Table 4 Performance of prediction accuracy

Deviation Value

Volume

Max 07135 (499)Min 0036 (013)Mean 02636Std 03316

Velocity

Max 02465 (502)Min 00143 (025)Mean 01095Std 01350

ceramic three models are introduced to simulate dropletgeneration In this work the droplet volumevelocity curvessimulated by dynamic model can carry sufficient informa-tion to distinguish nanosilver droplet formation processclearly Although the complexity of system increases theoutflow curves can obviously reflect the filament breaking-off stage that occurs which is coincident with sampledimages Such nonlinear LEM model is also successful inpredicting jetting characteristics under restrictive conditionsComparing the droplet volumevelocity curves with differentdwell time of high-frequency driving waveform the linearrelationship between dwell time and meniscus appearingmoment is discovered Meanwhile a parabolic change ruleof volumevelocity with the increase of dwell time is in goodagreement with present theoretical analysis

It should be noted that the proposed nonlinear model isfocused on relationship between the droplet volumevelocityand the dwell time of driving waveform while the voltageamplitude is also a decisive factor to droplet formationprocess This should be discussed in our future work

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

14 Mathematical Problems in Engineering

Table5Predictiv

edropletvolume(pL

)and

velocity(m

s)

1198791

1198792

665

775

885

995

10105

11115

12125

13135

14145

15155

4119881

11119

127

134

139

143

146

149

151

15147

142

Default

Default

Default

Default

Default

Default

Default

Default

119878371

395

417

439

459

478

491

500

489

471

451

435

Default

Default

Default

Default

Default

Default

Default

Default

45 119881

Default

117

125

134

143

15155

159

162

16157

153

147

139

Default

Default

Default

Default

Default

Default

119878Default

421

451

476

498

516

532

548

560

543

523

501

482

466

Default

Default

Default

Default

Default

Default

5119881

Default

Default

Default

133

143

151

157

163

167

168

166

163

157

15142

133

Default

Default

Default

Default

119878Default

Default

Default

480

507

531

553

572

587

600

583

563

540

525

509

498

Default

Default

Default

Default

55 119881

Default

Default

Default

Default

Default

151

159

165

168

169

171

169

165

16154

146

138

129

Default

Default

119878Default

Default

Default

Default

Default

524

550

572

593

609

624

601

571

551

537

523

509

496

Default

Default

6119881

Default

Default

Default

Default

Default

Default

Default

158

163

164

165

165

165

158

1495

141

130

119

108

97119878

Default

Default

Default

Default

Default

Default

Default

495

515

533

549

564

544

519

494

474

458

445

432

421

65 119881

Default

Default

Default

Default

Default

Default

Default

Default

Default

155

159

16155

150

143

134

125

114

103

95119878

Default

Default

Default

Default

Default

Default

Default

Default

Default

473

485

495

503

490

469

446

420

400

390

383

7119881

Default

Default

Default

Default

Default

Default

Default

Default

Default

Default

125

135

139

141

135

127

119

110

102

94119878

Default

Default

Default

Default

Default

Default

Default

Default

Default

Default

445

465

479

463

444

422

401

385

370

363

Defaults

areb

ecause

thec

ombinatio

nsof

parametersindrivingwaveform

areo

utof

theo

peratio

nrange

Mathematical Problems in Engineering 15

Authorsrsquo Contribution

Maowei He and Liling Sun contributed equally to this work

Acknowledgment

This research is partially supported by National Natural Sci-ence Foundation of China under Grants 61105067 7100107261174164 and 71271140

References

[1] M SinghHMHaverinen PDhagat andG E Jabbour ldquoInkjetprinting-process and its applicationsrdquo Advanced Materials vol22 no 6 pp 673ndash685 2010

[2] E Macdonald R Salas D Espalin et al ldquo3D printing for therapid prototyping of structural electronicsrdquo IEEE Access vol 2pp 234ndash242 2014

[3] S Umezu T Kitajima H Murase H Ohmori K Katahiraand Y Ito ldquoFabrication of living cell structure utilizing electro-static inkjet phenomenardquo in Proceedings of the 22nd IEEEInternational Conference on Micro Electro Mechanical Systems(MEMS rsquo09) pp 419ndash422 IEEE Sorrento Italy January 2009

[4] P Cosseddu L Basirico A Loi et al ldquoInkjet printed OrganicThin Film Transistors based tactile transducers for artificialrobotic skinrdquo in Proceedings of the 4th IEEE RAS and EMBSInternational Conference on Biomedical Robotics and Biomecha-tronics (BioRob rsquo12) pp 1907ndash1912 June 2012

[5] J Hwang A Wan and A Kahn ldquoEnergetics of metal-organicinterfaces new experiments and assessment of the fieldrdquoMate-rials Science and Engineering R Reports vol 64 no 1-2 pp 1ndash312009

[6] B J Kang and J H Oh ldquoGeometrical characterization of inkjet-printed conductive lines of nanosilver suspensions on a polymersubstraterdquoThin Solid Films vol 518 no 10 pp 2890ndash2896 2010

[7] F Villani P Vacca G Nenna et al ldquoInkjet printed polymerlayer on flexible substrate for OLED applicationsrdquo The Journalof Physical Chemistry C vol 113 no 30 pp 13398ndash13402 2009

[8] J-C Liou and F-G Tseng ldquoMulti-dimensional data registrationCMOSMEMS integrated inkjet printheadrdquo MicroelectronicEngineering vol 88 no 6 pp 888ndash901 2011

[9] J Miettinen K Kaija M Mantysalo et al ldquoMolded substratesfor inkjet printed modulesrdquo IEEE Transactions on Componentsand Packaging Technologies vol 32 no 2 pp 293ndash301 2009

[10] K Silverbrook ldquoPrinthead with multiple actuators in eachchamberrdquo US7708387 B2 2010

[11] V Piefort Finite element modelling of piezoelectric active struc-tures [PhD thesis] Department ofMechanical Engineering andRobotics Universite Libre de Bruxelles Brussels Belgium 2001

[12] L Sang Y Hong and F Wang ldquoInvestigation of viscosity effecton droplet formation in T-shaped microchannels by numericaland analytical methodsrdquo Microfluidics and Nanofluidics vol 6no 5 pp 621ndash635 2009

[13] J Liu S-H Tan Y F Yap M Y Ng and N-T NguyenldquoNumerical and experimental investigations of the formationprocess of ferrofluid dropletsrdquo Microfluidics and Nanofluidicsvol 11 no 2 pp 177ndash187 2011

[14] F Sarrazin K Loubiere L Prat C Gourdon T Bonometti andJ Magnaudet ldquoExperimental and numerical study of dropletshydrodynamics in MicroChannelrdquo AIChE Journal vol 52 no12 pp 4061ndash4070 2006

[15] X Q Xing D L Butler S H Ng Z Wang S Danyluk and CYang ldquoSimulation of droplet formation and coalescence usinglattice Boltzmann-based single-phasemodelrdquo Journal of Colloidand Interface Science vol 311 no 2 pp 609ndash618 2007

[16] Q Gallas R Holman T Nishida B Carroll M Sheplak and LCattafesta ldquoLumped element modeling of piezoelectric-drivensynthetic jet actuatorsrdquoAIAA Journal vol 41 no 2 pp 240ndash2472003

[17] A A Khalate X Bombois R Babuska H Wijshoff and RWaarsing ldquoOptimization-based feedforward control for a Drop-on-Demand inkjet printheadrdquo in Proceedings of the AmericanControl Conference (ACC rsquo10) pp 2182ndash2187 Baltimore MdUSA July 2010

[18] L Meirovitch Fundamentals of Vibrations McGraw-Hill NewYork NY USA 2001

[19] H Seitz and J Heinz ldquoModelling of a microfluidic device withpiezoelectric actuatorsrdquo Journal of Micromechanics and Micro-engineering vol 14 no 8 pp 1140ndash1147 2004

[20] S Kim J Sung andMH Lee ldquoPressure wave and fluid velocityin a bend-mode inkjet nozzle with double PZT actuatorsrdquoJournal of Thermal Science vol 22 no 1 pp 29ndash35 2013

[21] G Wassink Inkjet printhead performance enhancement by feed-forward input design based on two-port modeling [PhD thesis]Delft University of Technology 2007

[22] S Prasad SHorowitz QGallas B Sankar L Cattafesta andMSheplak ldquoTwo-port electroacoustic model of an axisymmet-ric piezoelectric composite platerdquo in Proceedings of the 43rdAIAAASMEASCEAHS Structures Structural DynamicsandMaterials Conference AIAA Paper 2002-1365 AIAA DenverColo USA April 2002

[23] D T BlackstockFundamentals of Physical Acoustics JohnWileyamp Sons New York NY USA 2000

[24] F M White Fluid Mechanics McGraw-Hill New York NYUSA 1979

[25] H Schempt andD R Yoerger ldquoStudy of dominant performancecharacteristics in robot transmissionsrdquo Journal of MechanicalDesign Transactions Of the ASME vol 115 no 3 pp 472ndash4821993

[26] E A Witmer ldquoElementary Bernoulli-Euler beam theoryrdquoMITUnified Engineering Course Notes vol 5 pp 114ndash164 1991-1992

[27] Y Yang and L Tang ldquoEquivalent circuit modeling of piezoelec-tric energy harvestersrdquo Journal of Intelligent Material Systemsand Structures vol 20 no 18 pp 2223ndash2235 2009

[28] F Constantinescu A G Gheorghe and M Nitescu ldquoNewcircuit models of power BAW resonatorsrdquo in Proceedings of theEuropean Microwave Integrated Circuit Conference October2007

[29] A Albareda and R PerezNon-Linear Behaviour of PiezoelectricCeramics vol 140ofSpringer Series inMaterials Science Springer2011

[30] J Nosek ldquoDrive level dependence of the resonant frequency inBAW quartz resonators and his modelingrdquo IEEE TransactionsonUltrasonics Ferroelectrics and Frequency Control vol 46 no4 pp 823ndash829 1999

[31] R Aigner N-H Huynh M Handtmann and S MarksteinerldquoBehavior of BAW devices at high power levelsrdquo in Proceedingsof the IEEE MTT-S International Microwave Symposium pp429ndash432 Long Beach Calif USA June 2005

[32] R S Ketcham G R Kline and K M Lakin ldquoPerformanceof TFR filters under elevated power conditionsrdquo in Proceedingsof 42nd the Annual Frequency Control Symposium pp 106ndash111Baltimore Md USA June 1988

16 Mathematical Problems in Engineering

[33] K-S Kwon ldquoWaveform design methods for piezo inkjet dis-pensers based on measured meniscus motionrdquo Journal of Micro-electromechanical Systems vol 18 no 5 pp 1118ndash1125 2009

[34] H-C Wu and H-J Lin ldquoEffects of actuating pressure wave-forms on the droplet behavior in a piezoelectric inkjetrdquoMateri-als Transactions vol 51 no 12 pp 2269ndash2276 2010

[35] KMinolta InkjetHeadApplicationNote-KM1024 Series KonicaMinolta Ij Technologies

[36] G Golub andV Pereyra ldquoSeparable nonlinear least squares thevariable projection method and its applicationsrdquo Inverse Prob-lems vol 19 no 2 pp R1ndashR26 2003

[37] H Wijshoff ldquoThe dynamics of the piezo inkjet printheadoperationrdquo Physics Reports vol 491 no 4-5 pp 77ndash177 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

Channel Neck Nozzle

Piezoceramic

Piezoceramic

(a)

DropformationNozzleNeckChannel

Drivingwaveform Piezoceramic

Electricaldomain

Mechanicaldomain domain domain domain domain

Fluidicacoustic Fluidicacoustic Fluidicacoustic Kinetic

(b)

Electrical Fluidicacoustic Kineticdomaindomaindomain

sim

Vac

CaCCeb MaRad

QoutMaNRaNMaDRaDCaD1 Na

RaO

(c)

Figure 1 Schematic overview of lumped-element modeling for PZT printhead (a) droplet generator structure (b) model structure and (c)equivalent circuit model

119862eb is the blocked electrical capacitance of the piezoelectricmaterial In the acoustic domain 119862aD and 119862aC representthe acoustic compliance of piezoceramic and channel 119877aD119877aN and 119877aO are the acoustic resistance due to structuraldamping neck tapering and fluid flowing out of nozzlerespectively 119872aD 119872aN and 119872aRad represent the acousticmass of piezoceramic neck and nozzle in proper orderrespectively

The dimensions of droplet generator and physical proper-ties of nanosilver ink provided by the LEED-PV Corporationare listed in Table 1 and calculation formulas of LEM modelparameters are provided in Table 2

According to above lumped element model a transferfunction describing the response relationship between theflow rate119876out and input AC voltage119881ac is obtained as follows

119876out (119904)

119881ac (119904)=

120601119886119862aD119904

11988641199044 + 119886

31199043 + 119886

21199042 + 119886

11199041 + 1

(1)

where

1198861= 119862aD (119877aO + 119877aN + 119877aD) + 119862aC (119877aO + 119877aN)

1198862= 119862aD (119872aRad +119872aN +119872aD) + 119862aC (119872aRad +119872aN)

+ 119862aC119862aD119877aD (119877aO + 119877aN)

1198863= 119862aC119862aD [119872aD (119877aO + 119877aN) + (119872aRad +119872aN) 119877aD]

1198864= 119862aC119862aD119872aD (119872aO +119872aN)

(2)

The frequency response of droplet generator driven bythe sinusoidal signal as shown in Figure 2 reveals thatnatural frequency (around 486 kHz) is higher than operating

2 4 6 8 100

5

10

15

20

Frequency (Hz) times104

Vm

axV

norm

Figure 2 Frequency response of lumped element model

frequency (below 128 kHz) This means that the meniscuscan obtain sufficient energy to overcome viscosity and surfacetension and then to generate consistent droplets

As the above simulation models 119862aD 119877aD and 119872aD areregarded as static parameters based on the approximate linearbehaviors of piezoelectric ceramic with a fix path of P-Ehysteresis loop However in fact the piezoelectric ceramic isa nonlinear material with multiple electric fieldstrain curves(parallel and perpendicular to the field) as shown in Figure 3It has been proved that the above original static modelhas insufficient ability to simulate the piezoelectric ceramicdynamic response in time-domain below linear voltage [11]

Our research finds that the maximum error between theexperimental measurements and simulation results can reacheight percent Additionally when the break-off phenomenon

4 Mathematical Problems in Engineering

Table 1 Droplet generator parameters

Parameter Unit Value

Geometry

Channel volume 1198810

Pl 166000Channel length 119871

0Um 7200

Nozzle radius 1198860

Um 135Neck length 119871 Um 45

PZT Piezoceramic thickness ℎ Um 25Free electrical capacitance 119862ef F 28 nF

Fluid

Surface tension 120590 Nm 45 lowast 10minus3

Viscosity 120583 Kg(mlowasts) 15Density 120588

0Kgm3 105 lowast 103

Sonic speed 1198880

ms 1400

Table 2 Lumped elements modeling parameter estimation

LEM Description

Neck 119877aN 119877aN = (2 +1198860

119903)radic2120583120588

0120596

1205871199032 where 119906 is viscosity 119908 is wave frequency and 119886

0119903 is gradients ratio

119872aN 119872aN =1205880(119905 + Δ119905)

1205871199032 where Δ119905 = 085119903(1 minus 07(119903119886

0)) + 085119903

Cavity 119862aC 119862aC =1198810

120588011988820

where 1198810is volume of the cavity

Piezoelectric119862aD 119862aD =

1205871199036

(1 minus V2)16119864ℎ3

where 119864 is the elastic modulus V is Poissonrsquos ratio and ℎ is the thickness

119872aD 119872aD =41205880119871

312058711988620

119877aD 119877aD = 2120585radic119872aD +119872aRad

119862aD where 120585 is the experimentally determined damping factor [18]

Nozzle119877aO 119877aO =

12058801198880

1205871199032[1 minus

21198691(2119896119903)

2119896119903] where 119869

1is the Bessel function of the first kind and 119896 = 119908119888

0

119872aRad 119872aRad =81205880

312058721198860

119862eB 119862eB = 119862eF(1 minus 1205812

) where 1205812 is electroacoustic coupling factor

1 2

e

1

2

3

4

abc

d

x1

x3

minus1

minus1

minus2

times10minus3

Δll

E (kVmm)

Figure 3 Possible paths of P-E hysteresis loop for piezoceramicmaterial [11]

occurs part of filament would go back into the nozzle andthe droplet volume will have a certain decrease But thesimulation curves of droplet volume with original modelhave no obvious sign to show that the break-off phenomenonoccurs (please refer to the simulation results in Section 42)

C0 R1

L1

C1

Figure 4 Equivalent electrical circuitmodel of PZTnonlinear effectwith proportional coefficient

22 Improved LEM with Nonlinear Equivalent Circuits Inorder to resolve above problems in this section threeequivalent circuits with different characters are incorporatedinto the classical LEM to characterize the nonlinear effect ofthe piezoelectric ceramics

(1) Proportional Model In order to describe the nonlinearrelationship between electric field and strain a proportionalmodel (illustrated in Figure 4) as a resonance was defined inthe IEEE standards publication [25] This analytical model is

Mathematical Problems in Engineering 5

C0

C2

C1 R1 L1

R2 L21 N

Figure 5 Equivalent electrical circuitmodel of PZTnonlinear effectwith multiple parallel resonances

derived based on the constitutive relations of piezoelectricityand the following assumptions (a) Euler-Bernouli beamassumption [26] (b) negligible external excitation from airdamping (c) proportional damping (ie the strain ratedamping and viscous damping are assumed to be propor-tional to the bending stiffness and mass per length of thebeam) (d) uniform electric field through the piezoelectricthickness Then the circuit element values of the proposedlumped parameter model are formulated in (1) where T isthe adjustment coefficient

1198711=

119871119898

T2

1198771=

119877119898

T2

1198621= T2 lowast 119862

119898

(3)

(2) Multiple-Resonator Model Yang and Tang [27] givethe multiple-resonator model to characterize piezoelectricceramic in piezoelectric energy harvesters as shown inFigure 5 His research finds that the network of piezoelectricceramic driving circuit consists of infinite branches and eachis composed of simplified equivalent circuit with an inductora capacitor and a resistorWhen the voltage signals from sev-eral branches drive the piezoelectric ceramic simultaneouslythe higher order vibrationmodes are excited It is realized thatworking at higher order modes causes the nonlinear effectof piezoelectric ceramic To characterize the different higherorder modes a circuit of multiple resonators is constructedto match the external branches In this literature to modelthe droplet generator nearly a double circuit may meet thesimulation accuracy Then the circuit element values of theproposed lumped parameter model are formulated in (3)where 120572 and N are both the adjustment coefficients

1198622=

1198621

1205722

1198772= 120572 lowast 119877

1

1198712= 1198711

(4)

(3) Dynamic Model Based on the works in [28 29] an equiv-alent electrical circuit with nonlinear resonance oscillationsin series namely the dynamic model is given to analyzepower BAW (bulk acoustic wave) resonators as shown inFigure 6Three major nonlinear effects are considered in this

C0 C0

C1

C2

C3

R1

L1

L1

R2

R3

C(I)

R(I)

Figure 6 Equivalent electrical circuitmodel of PZTnonlinear effectwith dynamic coefficients

equivalent circuit the amplitude-frequency effect (A-F effect)[30] the intermodulation effect [31] and the bias-frequencyeffect (B-F effect) [32] The dynamic values of resistance andcapacitance in the motional branch are dependent on themotional current in the branchThen derivation formulas aregiven in (4) where A

1 A2 B1 and B

2are the adjustment

coefficients

1198622= 12057211198682

1198623= 12057221198684

1198772= 12057311198682

1198773= 12057321198684

997904rArr 119862 (119868) = 119862

1(1 + A

11198682

+ A21198684

)

119877 (119868) = 1198771(1 + B

11198682

+ B21198684

) (5)

Then the improved LEM structure can be illustrated as inFigure 7

3 Experimental Method

31 Experimental Setup In this experiment the investigatedprinthead is KM series IJ Head of Konica and the used ink isnanosilver ink provided by the LEED-PV Corporation andthe according parameters are listed in Table 1 in Section 2 Aschematic overview of the experimental setup to observe thenanosilver droplet formation process is shown in Figure 8The visualization system consists of a cold source of halogens(XAO LG150) and a CCD camera (SVS ECO424) whichcan synchronize to the driving signal The volume andvelocity of droplet and the profile of meniscus are measuredthrough the images captured by CCD with exposure time of1 us The high-frequency driving waveform is programmedby arbitrary wave generator (RIGO DG2014A) In orderto repeatedly generate uniform droplets ink supply unitconnects air pressure unit which offers suitable negative forceto prevent the ink leaking from the printhead

32 Edge Detection and Image Analysis as a QualitativeMeasure The leading edge detection is to determine thenozzle position and meniscus position A straight line isgenerated along the nozzle in the region of interest (ROI)with

6 Mathematical Problems in Engineering

sim

Vac

CaCCeb

MaRadMaNRaN

MaDRaDCaD1 Na

RaO

(a)

sim

Vac

CaCCeb

MaRadMaNRaN

1 NaRaDT

2MaDT

2

RaO

CaDT2

(b)

sim

Vac

CaC

Ceb

MaRadMaNRaN

MaD

MaD

RaDCaD

1 Na

1 N

CaDa2

aRaD

RaO

(c)

sim

Vac

CaCCeb

MaRadMaNRaN

MaD1 Na C(I) R(I)

RaO

(d)

Figure 7 (a) Original LEM (b) proportional coefficient model (c) multiple resonators model and (d) dynamic coefficients model

Temperature unit

Air pressure unit

Ink supply

Parallel

Halogens light source

CCDPC

Wave generator

Drivingboard

Image

(a)

CCD

Halogens light

sourceArbitrary waveform generator

Drivingboard

PrintheadInk supply

(b)

Figure 8 Experimental setup (a) schematic of experimental setup and (b) image of experimental setup

CCD camera image analysis By setting a suitable thresholdgradient of brightness themeniscus location can be detectedThe satellite droplets and filament can be detected throughfarther image processing such as cropping filtering andthreshold recontrasting (shown as in Figure 9)

In image analysis system the droplet volume is calculatedby

Vol = sum120587 sdot (119889

2)

2

sdot 120575119911 (6)

which hypothesizes that liquid jet is axis-symmetric alongnozzle centerline And the droplet velocity is calculated basedon the position change of the moving-forward leading edgein two consecutive images with specified sampling intervalNote that the information is lost if the meniscus is hiddeninside the nozzle so edge detection only measures theprotruding meniscus

33 High-Frequency Driving Waveform as a System Excita-tion The high-frequency driving waveform is composed of

positive and negative periods including three pulse edges andtwo dwell times as shown in Figure 10 The first rising edgeexpands the channel to get the inkmeniscus into the channelThen the falling edge works to eject the droplet from thenozzle

This period of waveform design is based on the halfperiod of channel resonance Thus the waveform designissue is focused on calculating the optimum value of dwelltime 119879

1= 1198710119862 where 119871

0is the length of channel and 119862

is the sonic speed of the used ink The falling edge of thesecond negative voltage generates larger pressure in channelto accelerate the droplet with higher speed than conventionalsingle trapezoidal waveform The second rising edge whichsets the voltage back to 0 works to restrain the meniscusvibration [33]

Theoretically the suitable slops of the rising edge andfalling edge can restrain the meniscus vibration [34] How-ever in practical applications these two values are too largeto control accurately

Asmentioned above theoretical optimum value for dwelltime 119879

1is set as 52 us The corresponding dwell time 119879

2

Mathematical Problems in Engineering 7

(a) (b) (c) (d)

Figure 9 Image processing steps (a) original image (b) cropped image (c) threshold image and (d) edge image

Time

Voltage

Dwell

Dwell

time T1

time T2

Figure 10 Applied driving waveform of printhead

is recommended as 104 us (twice of 1198791) and the voltage

magnitude is set as 12 V according to the instruction ofKonica printhead In this paper the driving waveform withthese parameters is called the ldquostandardrdquo waveform [35]

4 Simulation Result

41 Coefficient Identification The coefficients of originalLEM model can be determined experimentally Howeverthe three improved models in this work are somewhatcomplicated and their coefficients reveal some importantfeatures of these equivalent circuits For this reason thissection is dedicated to identify the adjustment coefficients inthese corresponding circuits

In this experiment a group of experimental measure-ments with printhead driven by the standard high-frequencywaveform are chosen as the basic reference The coeffi-cients of each proposed improved LEM model should beadjusted to minimize the deviations between the simulationresults and the selected experimental measurements shownin Figure 11 The nonlinear least squares method [36] usingthe Levenberg-Margquardt and Gauss-Newton Algorithms

Table 3 Coefficients for three lumped element models

Model Coefficient ValueProportional 119879 0837

Multiple resonators 120572 51119873 228

Dynamic coefficients

1198601

133 lowast 103

1198611

353 lowast 103

1198602

234 lowast 104

1198612

411 lowast 104

is applied to find the optimal coefficients set for each pro-posed equivalent circuit which makes the simulation curvesoverlap the experimental measurements as far as possibleThe resulting coefficients are listed in Table 3

42 Droplet Volume Simulation In many applications of ink-jet printing the volume of droplet is a critical restrictivecondition Droplets with controllable volume affect not onlythe printing resolution but also the depositional thicknessTherefore how to accurately control droplet volume is a keyproblem to be solved by the proposed method

According to experimental measurements in Figure 11the comparison between the original LEM model and theproposed three improved models is presented in Figure 12In this figure the solid line is the outflow simulation curvesbased on the LEM technique and the star line is the actualmeasured volume values during the droplet formation pro-cess It should be noted that in Figures 12(a)ndash12(d) the firstthree points of actual measurement curve are default to zerobecause no fluid flowing outside the nozzle can be capturedby our image system

As the actual experimental measurements indicated inFigure 11 the droplet formation process is divided into threestages the meniscus protruding stage (ie 119879

1st = 15sim30 us)the filament breaking-off stage (ie 1198792nd = 30sim35 us) andthe droplet falling-down stage (ie 1198793rd = 35sim60 us) That

8 Mathematical Problems in Engineering

3us0pL0ms

6us0pL0ms

9us0pL0ms

12us 15us 18us 21us 24us

30us 33us 36us 39us 42us 45us 48us27us

678pL876ms

994pL730ms

1221 pL684ms

1401pL624ms

1521pL618ms

1533pL600ms

1567pL600ms

1509pL606ms

1492pL618ms

1489pL624 ms

1480 pL624 ms

1479pL618ms

1480pL618ms

Figure 11 Pictures of nanosilver droplet formation process driven by standard waveform

is ideally the outflow curve simulated by analytic modelsshould include overshoot phase which can clearly demon-strate that the part of filament is inhaled back into nozzle orsplits into satellite droplets by viscosity and surface tensionin the filament break-off stage However according to thesimulation result of the original LEM in Figure 12(a) there

is no obvious decreasing after reaching the maximum in theoutflow curve and the outflow curve of the original modelcannot well overlap the actual measurement curve

In Figures 12(b) and 12(d) the overshoot phase canbe observed clearly This performance improvement can bestraightforwardly explained by the fact that poles and zeros

Mathematical Problems in Engineering 9

0 20 40 60

0

10

20

Time (us)

Volu

me (

pL)

ExperimentOriginal

minus10

(a)

0 20 40 60

0

10

20

Time (us)

Volu

me (

pL)

ExperimentProportional

minus10

(b)

0 20 40 60

0

10

20

Time (us)

Volu

me (

pL)

ExperimentMultiple

minus10

minus20

(c)

0 20 40 60

0

10

20

Time (us)

Volu

me (

pL)

ExperimentDynamic

minus10

(d)

Figure 12 Outflow change ofmeasurements and simulations of (a) original model (b) proportional coefficientmodel (c)multiple resonatorsmodel and (d) dynamic coefficient model

of the transfer functions are manipulated to speed up thedynamic response speed of the proportional and dynamicmodels From Figure 12 both the multiple-resonator anddynamic models can obtain precise simulation results Thatis more well-designed coefficients are used to characterizethe nonlinear behaviors of piezoelectric printhead and theadjustable ranges are extended to a satisfied degree of bothmodels Additionally it should be noted that the outflowcurve of dynamic model has not only the overshoot phasebut also high overlap ratio to the experimental measurementMoreover the simulation curves reveal that the meniscusmoves inward before the droplet appears

43 Flow and Droplet Velocity Simulation The velocity ofdroplet is the other important aspect to evaluate the printingeffect Droplets with a high speed can remarkably improve

the printing efficiency and the quality of impact with sub-strate Therefore this experiment gives the velocity simula-tion using the proposed LEM models on the same actualexperimental measurement in Section 41

The comparisons between the flow velocity curves sim-ulated by the original LEM and three improved models areillustrated in Figure 13 The excitation is the standard drivingwaveform For the actual measured velocity curves the firsttwo points are also default to zero because no fluid flowingoutside the nozzle can be captured by our image system

It should be noted that all the values of actual measuredvelocity are positive However from the flow velocity curvesin Figure 13 the values of flow velocity in nozzle can bepositive or negative in different phases This positive-to-negative velocity variation indicates the changes of the flowdirection that leads to the break-off phenomenon Hence in

10 Mathematical Problems in Engineering

0 20 40 60

0

5

10

Time (us)

Velo

city

(ms

)

ExperimentFlow velocityOriginal

minus10

minus5

(a)

0 20 40 60

0

5

10

Time (us)

Velo

city

(ms

)

ExperimentFlow velocityProportional

minus10

minus5

(b)

0 20 40 60

0

5

10

Time (us)

Velo

city

(ms

)

ExperimentFlow velocityMultiple

minus10

minus5

(c)

0 20 40 60

0

5

10

Time (us)

Velo

city

(ms

)

ExperimentFlow velocityDynamic

minus10

minus5

(d)

Figure 13 Comparison of measured droplets velocity and the flowaverage velocity simulated by (a) original model (b) ldquoproportionalcoefficientrdquo model (c) ldquomultiple resonatorsrdquo model and (d) ldquodynamic coefficientsrdquo model

order to correctly simulate the velocity in leading edge wepropose a new calculation formula which combines both ofoutflow curve and flow velocity curve

119881average (119905) = radicsum1199052

119905=1199051

(12) 119876out (119905) sdot 119881 (119905)2

sum1199052

119905=1199051

(12)119876out (119905) (7)

where119876out is the outflow value and119881 is the flow velocity 1199051is

the moment when the sum of 119876out changes from negative topositive in outflow curve and 119905

2after 119905

1is the moment when

flow velocity changes to negative in nozzleFrom Figure 13 it can be obviously observed that in

the beginning of droplet formation process the multiple-resonator and dynamic models exhibit better performance insimulating the droplet velocity rather than the originalmodelThat is the average velocity curves obtained by these two

models are almost coincident to the actual measured velocitycurves

Based on the above analysis and comparisons thedynamicmodel based LEMhas themost accuracy in simulat-ing the droplet volume and velocity and can carry sufficientinformation to distinguish the break-off phenomenon occur-ing in the nanosilver droplet formation processs Thereforeamong these three improved models the dynamic lumpedelement model (DLEM) is the most suitable improved equiv-alent structure for simulating jetting characteristics

5 Prediction

For industry applications in printable electronics fabricationthe printhead must work at a certain status to meet manyrestrictive conditions Among them the most important

Mathematical Problems in Engineering 11

0 20 40 60

0

5

10

15

20

Time (us)

Volu

me (

pL)

minus25

minus20

minus15

minus10

minus5

T1 = 3usT1 = 4usT1 = 5us

T1 = 6usT1 = 7usT1 = 8us

(a)

0 20 40 604

5

6

7

8

9

10

Velo

city

(ms

)

Time (us)

T1 = 3usT1 = 4usT1 = 5us

T1 = 6usT1 = 7usT1 = 8us

(b)

Figure 14 Predictive droplet volume and velocity with different and various dwell times 1198792= 21198791 (a) simulation curves of outflow change

and (b) simulation curves of droplet average velocity

4 6 813

14

15

16

17

SimulationExperiment

Volu

me (

pL)

Time (us)

(a)

SimulationExperiment

4 6 8

45

5

55

6

Time (us)

Velo

city

(ms

)

(b)

Figure 15 Predictive droplet volume and velocity compared with measured volume and velocity

problem is choosing the appropriate combinations of thedriving waveform parameters for the used conductive inksHowever an exhausting manual selective process inevitablywastes a lot of time Therefore the computer-aided methodsare urgently needed for searching these appropriate combi-nations efficiently and robustly In this section the proposed

DLEM is applied to predict jetting characteristics for nanosil-ver ink

51 General Prediction Procedure The general procedure ofdroplet volumevelocity prediction based on DLEM is asfollows

12 Mathematical Problems in Engineering

0us0pL0ms

3us0pL0ms

6us0pL0ms

9us0pL0ms

12us6353pL736ms

15us9318pL612ms

18us11441pL572ms

18us 24us

30us14364pL498ms

33us14046pL510ms

36us13980pL520ms

39us13952pL525ms

42us13858pL525ms

45us13869pL520ms

48us13891pL524ms

51us13924pL522ms

54us

524ms13877pL

57us13840pL530ms

13127 pL548ms

14252pL520ms

27us14786pL504ms

Figure 16 A sequence of pictures of droplet falling from nozzle

Step 1 According to droplet generator parameters and prop-erties of fluid calculate most of the parameters in DLEM

Step 2 According to experimental measurements of dropletvolume driven by a standard waveform determine the rest ofthe adjustment coefficients and damping coefficients

Step 3 Based on the constructed DLEM predicate the valuesof droplet volume and velocity

Step 4 According to the need of applications and restrictiveconditions find out several groups of the appropriate param-eters of driving waveform frompredictive data lists of volumeand velocity

Step 5 After making a subtle adjustment in practice decidethe most appropriate parameters of driving waveform

52 Prediction Case 1 Standard Waveform with DifferentDwell Time Asmentioned above theDLEMmodel is chosento simulate jetting characteristics for the nanosilver inkHere an illustrative example is given to display the effect of

predicting the typical combinations of waveform parameterswith different dwell times (119879

1= 3sim8 us 119879

2= 2119879

1) and

the same amplitude (12V) The properties of nanosilver inkprovided by the LEED-PV Corporation have been listed inTable 1

In Figure 14 the predicted outflow and average velocitycurves with various dwell times are depicted We can observefrom Figure 14(a) that before the droplets rush out of thenozzle the depth of the inhaled meniscus is proportionalto dwell time From Figure 14(b) the time of the dropletappearing is postponed with the increase of dwell timeThat is both two simulation phenomena consist with thetheoretical analysis results in [37]

The comparisons between actual measured and predictedjetting characteristics with various dwell times are given inFigures 15(a) and 15(b) which both illustrate a parabolicchange rule with the increase of dwell time It should benoted that at the dwell time 119879

1= 56 us near the theoretical

optimum value 11987910158401= 52 us the droplet volume and velocity

simultaneously reach the maximum From the performanceof prediction accuracy listed in Table 4 the deviation of thepredicted jetting characteristics is within acceptable range

Mathematical Problems in Engineering 13

45

67

68

1012

1416

10121416

Volu

me (

pL)

91011121314151617

T1 (us)

T2(us)

(a)

45

67

68

1012

1416

3

4

5

6

Velo

city

(ms

)

3

35

4

45

5

55

6

T1 (us) T2

(us)

(b)

Figure 17 Predictive droplet volume and velocity with various dwell times1198791and119879

2 (a) predictive droplet volume and (b) predictive droplet

velocity

53 Prediction Case 2 Driving Waveform under RestrictiveCondition In this experiment an illustrative example underrestrictive conditions is presented to demonstrate the feasibil-ity and effectiveness of the proposed method for predictingjetting characteristics The dimensions of droplet generatorand properties of the nanosilver ink are listed in Table 1 Inorder to reach a high printing resolution and good quality ofdroplet impact with the substrate the restrictive conditionsinvolve that the droplet volume is smaller than 14 pL and thedroplet velocity is larger than 5ms

Based on the prediction procedure the predictive valuesof drop volumevelocity with various dwell times 119879

1and

1198792are listed in Table 5 According to the restrictive con-

ditions three combinations of parameters are found out1198791

= 5 us1198792

= 13 us 1198791

= 55 us1198792

= 14 us and1198791= 5 us119879

2= 75sim8 us After a small range search around

predictive values in the actual test the combination of 1198791=

49 us1198792

= 77 us is finally chosen The dynamic effect ofjetting characteristics driven by this combination is shown inFigure 16 From this figure the jetting characteristics satisfythe specified restrictive conditions quite well Meanwhilealmost no satellite droplet emerges after jetting the maindroplet

The 3D plot of predictive jetting characteristics withvarious dwell times 119879

1and 119879

2is shown in Figure 17 With

the increase of dwell time 1198791 peak positions of volume and

velocity have a tiny shift in 95sim115 us With the same dwelltime 119879

2 there is also a parabolic profile in predictive volume

and velocity

6 Conclusion

The lumped element modeling based on linear model ofpiezoelectric ceramic has been successfully used to simu-late piezoelectric droplet generator However the nonlinearmodel of piezoelectric ceramic is necessary to diminishdeviation between the simulated and measured results Inorder to characterize the nonlinear behavior of piezoelectric

Table 4 Performance of prediction accuracy

Deviation Value

Volume

Max 07135 (499)Min 0036 (013)Mean 02636Std 03316

Velocity

Max 02465 (502)Min 00143 (025)Mean 01095Std 01350

ceramic three models are introduced to simulate dropletgeneration In this work the droplet volumevelocity curvessimulated by dynamic model can carry sufficient informa-tion to distinguish nanosilver droplet formation processclearly Although the complexity of system increases theoutflow curves can obviously reflect the filament breaking-off stage that occurs which is coincident with sampledimages Such nonlinear LEM model is also successful inpredicting jetting characteristics under restrictive conditionsComparing the droplet volumevelocity curves with differentdwell time of high-frequency driving waveform the linearrelationship between dwell time and meniscus appearingmoment is discovered Meanwhile a parabolic change ruleof volumevelocity with the increase of dwell time is in goodagreement with present theoretical analysis

It should be noted that the proposed nonlinear model isfocused on relationship between the droplet volumevelocityand the dwell time of driving waveform while the voltageamplitude is also a decisive factor to droplet formationprocess This should be discussed in our future work

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

14 Mathematical Problems in Engineering

Table5Predictiv

edropletvolume(pL

)and

velocity(m

s)

1198791

1198792

665

775

885

995

10105

11115

12125

13135

14145

15155

4119881

11119

127

134

139

143

146

149

151

15147

142

Default

Default

Default

Default

Default

Default

Default

Default

119878371

395

417

439

459

478

491

500

489

471

451

435

Default

Default

Default

Default

Default

Default

Default

Default

45 119881

Default

117

125

134

143

15155

159

162

16157

153

147

139

Default

Default

Default

Default

Default

Default

119878Default

421

451

476

498

516

532

548

560

543

523

501

482

466

Default

Default

Default

Default

Default

Default

5119881

Default

Default

Default

133

143

151

157

163

167

168

166

163

157

15142

133

Default

Default

Default

Default

119878Default

Default

Default

480

507

531

553

572

587

600

583

563

540

525

509

498

Default

Default

Default

Default

55 119881

Default

Default

Default

Default

Default

151

159

165

168

169

171

169

165

16154

146

138

129

Default

Default

119878Default

Default

Default

Default

Default

524

550

572

593

609

624

601

571

551

537

523

509

496

Default

Default

6119881

Default

Default

Default

Default

Default

Default

Default

158

163

164

165

165

165

158

1495

141

130

119

108

97119878

Default

Default

Default

Default

Default

Default

Default

495

515

533

549

564

544

519

494

474

458

445

432

421

65 119881

Default

Default

Default

Default

Default

Default

Default

Default

Default

155

159

16155

150

143

134

125

114

103

95119878

Default

Default

Default

Default

Default

Default

Default

Default

Default

473

485

495

503

490

469

446

420

400

390

383

7119881

Default

Default

Default

Default

Default

Default

Default

Default

Default

Default

125

135

139

141

135

127

119

110

102

94119878

Default

Default

Default

Default

Default

Default

Default

Default

Default

Default

445

465

479

463

444

422

401

385

370

363

Defaults

areb

ecause

thec

ombinatio

nsof

parametersindrivingwaveform

areo

utof

theo

peratio

nrange

Mathematical Problems in Engineering 15

Authorsrsquo Contribution

Maowei He and Liling Sun contributed equally to this work

Acknowledgment

This research is partially supported by National Natural Sci-ence Foundation of China under Grants 61105067 7100107261174164 and 71271140

References

[1] M SinghHMHaverinen PDhagat andG E Jabbour ldquoInkjetprinting-process and its applicationsrdquo Advanced Materials vol22 no 6 pp 673ndash685 2010

[2] E Macdonald R Salas D Espalin et al ldquo3D printing for therapid prototyping of structural electronicsrdquo IEEE Access vol 2pp 234ndash242 2014

[3] S Umezu T Kitajima H Murase H Ohmori K Katahiraand Y Ito ldquoFabrication of living cell structure utilizing electro-static inkjet phenomenardquo in Proceedings of the 22nd IEEEInternational Conference on Micro Electro Mechanical Systems(MEMS rsquo09) pp 419ndash422 IEEE Sorrento Italy January 2009

[4] P Cosseddu L Basirico A Loi et al ldquoInkjet printed OrganicThin Film Transistors based tactile transducers for artificialrobotic skinrdquo in Proceedings of the 4th IEEE RAS and EMBSInternational Conference on Biomedical Robotics and Biomecha-tronics (BioRob rsquo12) pp 1907ndash1912 June 2012

[5] J Hwang A Wan and A Kahn ldquoEnergetics of metal-organicinterfaces new experiments and assessment of the fieldrdquoMate-rials Science and Engineering R Reports vol 64 no 1-2 pp 1ndash312009

[6] B J Kang and J H Oh ldquoGeometrical characterization of inkjet-printed conductive lines of nanosilver suspensions on a polymersubstraterdquoThin Solid Films vol 518 no 10 pp 2890ndash2896 2010

[7] F Villani P Vacca G Nenna et al ldquoInkjet printed polymerlayer on flexible substrate for OLED applicationsrdquo The Journalof Physical Chemistry C vol 113 no 30 pp 13398ndash13402 2009

[8] J-C Liou and F-G Tseng ldquoMulti-dimensional data registrationCMOSMEMS integrated inkjet printheadrdquo MicroelectronicEngineering vol 88 no 6 pp 888ndash901 2011

[9] J Miettinen K Kaija M Mantysalo et al ldquoMolded substratesfor inkjet printed modulesrdquo IEEE Transactions on Componentsand Packaging Technologies vol 32 no 2 pp 293ndash301 2009

[10] K Silverbrook ldquoPrinthead with multiple actuators in eachchamberrdquo US7708387 B2 2010

[11] V Piefort Finite element modelling of piezoelectric active struc-tures [PhD thesis] Department ofMechanical Engineering andRobotics Universite Libre de Bruxelles Brussels Belgium 2001

[12] L Sang Y Hong and F Wang ldquoInvestigation of viscosity effecton droplet formation in T-shaped microchannels by numericaland analytical methodsrdquo Microfluidics and Nanofluidics vol 6no 5 pp 621ndash635 2009

[13] J Liu S-H Tan Y F Yap M Y Ng and N-T NguyenldquoNumerical and experimental investigations of the formationprocess of ferrofluid dropletsrdquo Microfluidics and Nanofluidicsvol 11 no 2 pp 177ndash187 2011

[14] F Sarrazin K Loubiere L Prat C Gourdon T Bonometti andJ Magnaudet ldquoExperimental and numerical study of dropletshydrodynamics in MicroChannelrdquo AIChE Journal vol 52 no12 pp 4061ndash4070 2006

[15] X Q Xing D L Butler S H Ng Z Wang S Danyluk and CYang ldquoSimulation of droplet formation and coalescence usinglattice Boltzmann-based single-phasemodelrdquo Journal of Colloidand Interface Science vol 311 no 2 pp 609ndash618 2007

[16] Q Gallas R Holman T Nishida B Carroll M Sheplak and LCattafesta ldquoLumped element modeling of piezoelectric-drivensynthetic jet actuatorsrdquoAIAA Journal vol 41 no 2 pp 240ndash2472003

[17] A A Khalate X Bombois R Babuska H Wijshoff and RWaarsing ldquoOptimization-based feedforward control for a Drop-on-Demand inkjet printheadrdquo in Proceedings of the AmericanControl Conference (ACC rsquo10) pp 2182ndash2187 Baltimore MdUSA July 2010

[18] L Meirovitch Fundamentals of Vibrations McGraw-Hill NewYork NY USA 2001

[19] H Seitz and J Heinz ldquoModelling of a microfluidic device withpiezoelectric actuatorsrdquo Journal of Micromechanics and Micro-engineering vol 14 no 8 pp 1140ndash1147 2004

[20] S Kim J Sung andMH Lee ldquoPressure wave and fluid velocityin a bend-mode inkjet nozzle with double PZT actuatorsrdquoJournal of Thermal Science vol 22 no 1 pp 29ndash35 2013

[21] G Wassink Inkjet printhead performance enhancement by feed-forward input design based on two-port modeling [PhD thesis]Delft University of Technology 2007

[22] S Prasad SHorowitz QGallas B Sankar L Cattafesta andMSheplak ldquoTwo-port electroacoustic model of an axisymmet-ric piezoelectric composite platerdquo in Proceedings of the 43rdAIAAASMEASCEAHS Structures Structural DynamicsandMaterials Conference AIAA Paper 2002-1365 AIAA DenverColo USA April 2002

[23] D T BlackstockFundamentals of Physical Acoustics JohnWileyamp Sons New York NY USA 2000

[24] F M White Fluid Mechanics McGraw-Hill New York NYUSA 1979

[25] H Schempt andD R Yoerger ldquoStudy of dominant performancecharacteristics in robot transmissionsrdquo Journal of MechanicalDesign Transactions Of the ASME vol 115 no 3 pp 472ndash4821993

[26] E A Witmer ldquoElementary Bernoulli-Euler beam theoryrdquoMITUnified Engineering Course Notes vol 5 pp 114ndash164 1991-1992

[27] Y Yang and L Tang ldquoEquivalent circuit modeling of piezoelec-tric energy harvestersrdquo Journal of Intelligent Material Systemsand Structures vol 20 no 18 pp 2223ndash2235 2009

[28] F Constantinescu A G Gheorghe and M Nitescu ldquoNewcircuit models of power BAW resonatorsrdquo in Proceedings of theEuropean Microwave Integrated Circuit Conference October2007

[29] A Albareda and R PerezNon-Linear Behaviour of PiezoelectricCeramics vol 140ofSpringer Series inMaterials Science Springer2011

[30] J Nosek ldquoDrive level dependence of the resonant frequency inBAW quartz resonators and his modelingrdquo IEEE TransactionsonUltrasonics Ferroelectrics and Frequency Control vol 46 no4 pp 823ndash829 1999

[31] R Aigner N-H Huynh M Handtmann and S MarksteinerldquoBehavior of BAW devices at high power levelsrdquo in Proceedingsof the IEEE MTT-S International Microwave Symposium pp429ndash432 Long Beach Calif USA June 2005

[32] R S Ketcham G R Kline and K M Lakin ldquoPerformanceof TFR filters under elevated power conditionsrdquo in Proceedingsof 42nd the Annual Frequency Control Symposium pp 106ndash111Baltimore Md USA June 1988

16 Mathematical Problems in Engineering

[33] K-S Kwon ldquoWaveform design methods for piezo inkjet dis-pensers based on measured meniscus motionrdquo Journal of Micro-electromechanical Systems vol 18 no 5 pp 1118ndash1125 2009

[34] H-C Wu and H-J Lin ldquoEffects of actuating pressure wave-forms on the droplet behavior in a piezoelectric inkjetrdquoMateri-als Transactions vol 51 no 12 pp 2269ndash2276 2010

[35] KMinolta InkjetHeadApplicationNote-KM1024 Series KonicaMinolta Ij Technologies

[36] G Golub andV Pereyra ldquoSeparable nonlinear least squares thevariable projection method and its applicationsrdquo Inverse Prob-lems vol 19 no 2 pp R1ndashR26 2003

[37] H Wijshoff ldquoThe dynamics of the piezo inkjet printheadoperationrdquo Physics Reports vol 491 no 4-5 pp 77ndash177 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

Table 1 Droplet generator parameters

Parameter Unit Value

Geometry

Channel volume 1198810

Pl 166000Channel length 119871

0Um 7200

Nozzle radius 1198860

Um 135Neck length 119871 Um 45

PZT Piezoceramic thickness ℎ Um 25Free electrical capacitance 119862ef F 28 nF

Fluid

Surface tension 120590 Nm 45 lowast 10minus3

Viscosity 120583 Kg(mlowasts) 15Density 120588

0Kgm3 105 lowast 103

Sonic speed 1198880

ms 1400

Table 2 Lumped elements modeling parameter estimation

LEM Description

Neck 119877aN 119877aN = (2 +1198860

119903)radic2120583120588

0120596

1205871199032 where 119906 is viscosity 119908 is wave frequency and 119886

0119903 is gradients ratio

119872aN 119872aN =1205880(119905 + Δ119905)

1205871199032 where Δ119905 = 085119903(1 minus 07(119903119886

0)) + 085119903

Cavity 119862aC 119862aC =1198810

120588011988820

where 1198810is volume of the cavity

Piezoelectric119862aD 119862aD =

1205871199036

(1 minus V2)16119864ℎ3

where 119864 is the elastic modulus V is Poissonrsquos ratio and ℎ is the thickness

119872aD 119872aD =41205880119871

312058711988620

119877aD 119877aD = 2120585radic119872aD +119872aRad

119862aD where 120585 is the experimentally determined damping factor [18]

Nozzle119877aO 119877aO =

12058801198880

1205871199032[1 minus

21198691(2119896119903)

2119896119903] where 119869

1is the Bessel function of the first kind and 119896 = 119908119888

0

119872aRad 119872aRad =81205880

312058721198860

119862eB 119862eB = 119862eF(1 minus 1205812

) where 1205812 is electroacoustic coupling factor

1 2

e

1

2

3

4

abc

d

x1

x3

minus1

minus1

minus2

times10minus3

Δll

E (kVmm)

Figure 3 Possible paths of P-E hysteresis loop for piezoceramicmaterial [11]

occurs part of filament would go back into the nozzle andthe droplet volume will have a certain decrease But thesimulation curves of droplet volume with original modelhave no obvious sign to show that the break-off phenomenonoccurs (please refer to the simulation results in Section 42)

C0 R1

L1

C1

Figure 4 Equivalent electrical circuitmodel of PZTnonlinear effectwith proportional coefficient

22 Improved LEM with Nonlinear Equivalent Circuits Inorder to resolve above problems in this section threeequivalent circuits with different characters are incorporatedinto the classical LEM to characterize the nonlinear effect ofthe piezoelectric ceramics

(1) Proportional Model In order to describe the nonlinearrelationship between electric field and strain a proportionalmodel (illustrated in Figure 4) as a resonance was defined inthe IEEE standards publication [25] This analytical model is

Mathematical Problems in Engineering 5

C0

C2

C1 R1 L1

R2 L21 N

Figure 5 Equivalent electrical circuitmodel of PZTnonlinear effectwith multiple parallel resonances

derived based on the constitutive relations of piezoelectricityand the following assumptions (a) Euler-Bernouli beamassumption [26] (b) negligible external excitation from airdamping (c) proportional damping (ie the strain ratedamping and viscous damping are assumed to be propor-tional to the bending stiffness and mass per length of thebeam) (d) uniform electric field through the piezoelectricthickness Then the circuit element values of the proposedlumped parameter model are formulated in (1) where T isthe adjustment coefficient

1198711=

119871119898

T2

1198771=

119877119898

T2

1198621= T2 lowast 119862

119898

(3)

(2) Multiple-Resonator Model Yang and Tang [27] givethe multiple-resonator model to characterize piezoelectricceramic in piezoelectric energy harvesters as shown inFigure 5 His research finds that the network of piezoelectricceramic driving circuit consists of infinite branches and eachis composed of simplified equivalent circuit with an inductora capacitor and a resistorWhen the voltage signals from sev-eral branches drive the piezoelectric ceramic simultaneouslythe higher order vibrationmodes are excited It is realized thatworking at higher order modes causes the nonlinear effectof piezoelectric ceramic To characterize the different higherorder modes a circuit of multiple resonators is constructedto match the external branches In this literature to modelthe droplet generator nearly a double circuit may meet thesimulation accuracy Then the circuit element values of theproposed lumped parameter model are formulated in (3)where 120572 and N are both the adjustment coefficients

1198622=

1198621

1205722

1198772= 120572 lowast 119877

1

1198712= 1198711

(4)

(3) Dynamic Model Based on the works in [28 29] an equiv-alent electrical circuit with nonlinear resonance oscillationsin series namely the dynamic model is given to analyzepower BAW (bulk acoustic wave) resonators as shown inFigure 6Three major nonlinear effects are considered in this

C0 C0

C1

C2

C3

R1

L1

L1

R2

R3

C(I)

R(I)

Figure 6 Equivalent electrical circuitmodel of PZTnonlinear effectwith dynamic coefficients

equivalent circuit the amplitude-frequency effect (A-F effect)[30] the intermodulation effect [31] and the bias-frequencyeffect (B-F effect) [32] The dynamic values of resistance andcapacitance in the motional branch are dependent on themotional current in the branchThen derivation formulas aregiven in (4) where A

1 A2 B1 and B

2are the adjustment

coefficients

1198622= 12057211198682

1198623= 12057221198684

1198772= 12057311198682

1198773= 12057321198684

997904rArr 119862 (119868) = 119862

1(1 + A

11198682

+ A21198684

)

119877 (119868) = 1198771(1 + B

11198682

+ B21198684

) (5)

Then the improved LEM structure can be illustrated as inFigure 7

3 Experimental Method

31 Experimental Setup In this experiment the investigatedprinthead is KM series IJ Head of Konica and the used ink isnanosilver ink provided by the LEED-PV Corporation andthe according parameters are listed in Table 1 in Section 2 Aschematic overview of the experimental setup to observe thenanosilver droplet formation process is shown in Figure 8The visualization system consists of a cold source of halogens(XAO LG150) and a CCD camera (SVS ECO424) whichcan synchronize to the driving signal The volume andvelocity of droplet and the profile of meniscus are measuredthrough the images captured by CCD with exposure time of1 us The high-frequency driving waveform is programmedby arbitrary wave generator (RIGO DG2014A) In orderto repeatedly generate uniform droplets ink supply unitconnects air pressure unit which offers suitable negative forceto prevent the ink leaking from the printhead

32 Edge Detection and Image Analysis as a QualitativeMeasure The leading edge detection is to determine thenozzle position and meniscus position A straight line isgenerated along the nozzle in the region of interest (ROI)with

6 Mathematical Problems in Engineering

sim

Vac

CaCCeb

MaRadMaNRaN

MaDRaDCaD1 Na

RaO

(a)

sim

Vac

CaCCeb

MaRadMaNRaN

1 NaRaDT

2MaDT

2

RaO

CaDT2

(b)

sim

Vac

CaC

Ceb

MaRadMaNRaN

MaD

MaD

RaDCaD

1 Na

1 N

CaDa2

aRaD

RaO

(c)

sim

Vac

CaCCeb

MaRadMaNRaN

MaD1 Na C(I) R(I)

RaO

(d)

Figure 7 (a) Original LEM (b) proportional coefficient model (c) multiple resonators model and (d) dynamic coefficients model

Temperature unit

Air pressure unit

Ink supply

Parallel

Halogens light source

CCDPC

Wave generator

Drivingboard

Image

(a)

CCD

Halogens light

sourceArbitrary waveform generator

Drivingboard

PrintheadInk supply

(b)

Figure 8 Experimental setup (a) schematic of experimental setup and (b) image of experimental setup

CCD camera image analysis By setting a suitable thresholdgradient of brightness themeniscus location can be detectedThe satellite droplets and filament can be detected throughfarther image processing such as cropping filtering andthreshold recontrasting (shown as in Figure 9)

In image analysis system the droplet volume is calculatedby

Vol = sum120587 sdot (119889

2)

2

sdot 120575119911 (6)

which hypothesizes that liquid jet is axis-symmetric alongnozzle centerline And the droplet velocity is calculated basedon the position change of the moving-forward leading edgein two consecutive images with specified sampling intervalNote that the information is lost if the meniscus is hiddeninside the nozzle so edge detection only measures theprotruding meniscus

33 High-Frequency Driving Waveform as a System Excita-tion The high-frequency driving waveform is composed of

positive and negative periods including three pulse edges andtwo dwell times as shown in Figure 10 The first rising edgeexpands the channel to get the inkmeniscus into the channelThen the falling edge works to eject the droplet from thenozzle

This period of waveform design is based on the halfperiod of channel resonance Thus the waveform designissue is focused on calculating the optimum value of dwelltime 119879

1= 1198710119862 where 119871

0is the length of channel and 119862

is the sonic speed of the used ink The falling edge of thesecond negative voltage generates larger pressure in channelto accelerate the droplet with higher speed than conventionalsingle trapezoidal waveform The second rising edge whichsets the voltage back to 0 works to restrain the meniscusvibration [33]

Theoretically the suitable slops of the rising edge andfalling edge can restrain the meniscus vibration [34] How-ever in practical applications these two values are too largeto control accurately

Asmentioned above theoretical optimum value for dwelltime 119879

1is set as 52 us The corresponding dwell time 119879

2

Mathematical Problems in Engineering 7

(a) (b) (c) (d)

Figure 9 Image processing steps (a) original image (b) cropped image (c) threshold image and (d) edge image

Time

Voltage

Dwell

Dwell

time T1

time T2

Figure 10 Applied driving waveform of printhead

is recommended as 104 us (twice of 1198791) and the voltage

magnitude is set as 12 V according to the instruction ofKonica printhead In this paper the driving waveform withthese parameters is called the ldquostandardrdquo waveform [35]

4 Simulation Result

41 Coefficient Identification The coefficients of originalLEM model can be determined experimentally Howeverthe three improved models in this work are somewhatcomplicated and their coefficients reveal some importantfeatures of these equivalent circuits For this reason thissection is dedicated to identify the adjustment coefficients inthese corresponding circuits

In this experiment a group of experimental measure-ments with printhead driven by the standard high-frequencywaveform are chosen as the basic reference The coeffi-cients of each proposed improved LEM model should beadjusted to minimize the deviations between the simulationresults and the selected experimental measurements shownin Figure 11 The nonlinear least squares method [36] usingthe Levenberg-Margquardt and Gauss-Newton Algorithms

Table 3 Coefficients for three lumped element models

Model Coefficient ValueProportional 119879 0837

Multiple resonators 120572 51119873 228

Dynamic coefficients

1198601

133 lowast 103

1198611

353 lowast 103

1198602

234 lowast 104

1198612

411 lowast 104

is applied to find the optimal coefficients set for each pro-posed equivalent circuit which makes the simulation curvesoverlap the experimental measurements as far as possibleThe resulting coefficients are listed in Table 3

42 Droplet Volume Simulation In many applications of ink-jet printing the volume of droplet is a critical restrictivecondition Droplets with controllable volume affect not onlythe printing resolution but also the depositional thicknessTherefore how to accurately control droplet volume is a keyproblem to be solved by the proposed method

According to experimental measurements in Figure 11the comparison between the original LEM model and theproposed three improved models is presented in Figure 12In this figure the solid line is the outflow simulation curvesbased on the LEM technique and the star line is the actualmeasured volume values during the droplet formation pro-cess It should be noted that in Figures 12(a)ndash12(d) the firstthree points of actual measurement curve are default to zerobecause no fluid flowing outside the nozzle can be capturedby our image system

As the actual experimental measurements indicated inFigure 11 the droplet formation process is divided into threestages the meniscus protruding stage (ie 119879

1st = 15sim30 us)the filament breaking-off stage (ie 1198792nd = 30sim35 us) andthe droplet falling-down stage (ie 1198793rd = 35sim60 us) That

8 Mathematical Problems in Engineering

3us0pL0ms

6us0pL0ms

9us0pL0ms

12us 15us 18us 21us 24us

30us 33us 36us 39us 42us 45us 48us27us

678pL876ms

994pL730ms

1221 pL684ms

1401pL624ms

1521pL618ms

1533pL600ms

1567pL600ms

1509pL606ms

1492pL618ms

1489pL624 ms

1480 pL624 ms

1479pL618ms

1480pL618ms

Figure 11 Pictures of nanosilver droplet formation process driven by standard waveform

is ideally the outflow curve simulated by analytic modelsshould include overshoot phase which can clearly demon-strate that the part of filament is inhaled back into nozzle orsplits into satellite droplets by viscosity and surface tensionin the filament break-off stage However according to thesimulation result of the original LEM in Figure 12(a) there

is no obvious decreasing after reaching the maximum in theoutflow curve and the outflow curve of the original modelcannot well overlap the actual measurement curve

In Figures 12(b) and 12(d) the overshoot phase canbe observed clearly This performance improvement can bestraightforwardly explained by the fact that poles and zeros

Mathematical Problems in Engineering 9

0 20 40 60

0

10

20

Time (us)

Volu

me (

pL)

ExperimentOriginal

minus10

(a)

0 20 40 60

0

10

20

Time (us)

Volu

me (

pL)

ExperimentProportional

minus10

(b)

0 20 40 60

0

10

20

Time (us)

Volu

me (

pL)

ExperimentMultiple

minus10

minus20

(c)

0 20 40 60

0

10

20

Time (us)

Volu

me (

pL)

ExperimentDynamic

minus10

(d)

Figure 12 Outflow change ofmeasurements and simulations of (a) original model (b) proportional coefficientmodel (c)multiple resonatorsmodel and (d) dynamic coefficient model

of the transfer functions are manipulated to speed up thedynamic response speed of the proportional and dynamicmodels From Figure 12 both the multiple-resonator anddynamic models can obtain precise simulation results Thatis more well-designed coefficients are used to characterizethe nonlinear behaviors of piezoelectric printhead and theadjustable ranges are extended to a satisfied degree of bothmodels Additionally it should be noted that the outflowcurve of dynamic model has not only the overshoot phasebut also high overlap ratio to the experimental measurementMoreover the simulation curves reveal that the meniscusmoves inward before the droplet appears

43 Flow and Droplet Velocity Simulation The velocity ofdroplet is the other important aspect to evaluate the printingeffect Droplets with a high speed can remarkably improve

the printing efficiency and the quality of impact with sub-strate Therefore this experiment gives the velocity simula-tion using the proposed LEM models on the same actualexperimental measurement in Section 41

The comparisons between the flow velocity curves sim-ulated by the original LEM and three improved models areillustrated in Figure 13 The excitation is the standard drivingwaveform For the actual measured velocity curves the firsttwo points are also default to zero because no fluid flowingoutside the nozzle can be captured by our image system

It should be noted that all the values of actual measuredvelocity are positive However from the flow velocity curvesin Figure 13 the values of flow velocity in nozzle can bepositive or negative in different phases This positive-to-negative velocity variation indicates the changes of the flowdirection that leads to the break-off phenomenon Hence in

10 Mathematical Problems in Engineering

0 20 40 60

0

5

10

Time (us)

Velo

city

(ms

)

ExperimentFlow velocityOriginal

minus10

minus5

(a)

0 20 40 60

0

5

10

Time (us)

Velo

city

(ms

)

ExperimentFlow velocityProportional

minus10

minus5

(b)

0 20 40 60

0

5

10

Time (us)

Velo

city

(ms

)

ExperimentFlow velocityMultiple

minus10

minus5

(c)

0 20 40 60

0

5

10

Time (us)

Velo

city

(ms

)

ExperimentFlow velocityDynamic

minus10

minus5

(d)

Figure 13 Comparison of measured droplets velocity and the flowaverage velocity simulated by (a) original model (b) ldquoproportionalcoefficientrdquo model (c) ldquomultiple resonatorsrdquo model and (d) ldquodynamic coefficientsrdquo model

order to correctly simulate the velocity in leading edge wepropose a new calculation formula which combines both ofoutflow curve and flow velocity curve

119881average (119905) = radicsum1199052

119905=1199051

(12) 119876out (119905) sdot 119881 (119905)2

sum1199052

119905=1199051

(12)119876out (119905) (7)

where119876out is the outflow value and119881 is the flow velocity 1199051is

the moment when the sum of 119876out changes from negative topositive in outflow curve and 119905

2after 119905

1is the moment when

flow velocity changes to negative in nozzleFrom Figure 13 it can be obviously observed that in

the beginning of droplet formation process the multiple-resonator and dynamic models exhibit better performance insimulating the droplet velocity rather than the originalmodelThat is the average velocity curves obtained by these two

models are almost coincident to the actual measured velocitycurves

Based on the above analysis and comparisons thedynamicmodel based LEMhas themost accuracy in simulat-ing the droplet volume and velocity and can carry sufficientinformation to distinguish the break-off phenomenon occur-ing in the nanosilver droplet formation processs Thereforeamong these three improved models the dynamic lumpedelement model (DLEM) is the most suitable improved equiv-alent structure for simulating jetting characteristics

5 Prediction

For industry applications in printable electronics fabricationthe printhead must work at a certain status to meet manyrestrictive conditions Among them the most important

Mathematical Problems in Engineering 11

0 20 40 60

0

5

10

15

20

Time (us)

Volu

me (

pL)

minus25

minus20

minus15

minus10

minus5

T1 = 3usT1 = 4usT1 = 5us

T1 = 6usT1 = 7usT1 = 8us

(a)

0 20 40 604

5

6

7

8

9

10

Velo

city

(ms

)

Time (us)

T1 = 3usT1 = 4usT1 = 5us

T1 = 6usT1 = 7usT1 = 8us

(b)

Figure 14 Predictive droplet volume and velocity with different and various dwell times 1198792= 21198791 (a) simulation curves of outflow change

and (b) simulation curves of droplet average velocity

4 6 813

14

15

16

17

SimulationExperiment

Volu

me (

pL)

Time (us)

(a)

SimulationExperiment

4 6 8

45

5

55

6

Time (us)

Velo

city

(ms

)

(b)

Figure 15 Predictive droplet volume and velocity compared with measured volume and velocity

problem is choosing the appropriate combinations of thedriving waveform parameters for the used conductive inksHowever an exhausting manual selective process inevitablywastes a lot of time Therefore the computer-aided methodsare urgently needed for searching these appropriate combi-nations efficiently and robustly In this section the proposed

DLEM is applied to predict jetting characteristics for nanosil-ver ink

51 General Prediction Procedure The general procedure ofdroplet volumevelocity prediction based on DLEM is asfollows

12 Mathematical Problems in Engineering

0us0pL0ms

3us0pL0ms

6us0pL0ms

9us0pL0ms

12us6353pL736ms

15us9318pL612ms

18us11441pL572ms

18us 24us

30us14364pL498ms

33us14046pL510ms

36us13980pL520ms

39us13952pL525ms

42us13858pL525ms

45us13869pL520ms

48us13891pL524ms

51us13924pL522ms

54us

524ms13877pL

57us13840pL530ms

13127 pL548ms

14252pL520ms

27us14786pL504ms

Figure 16 A sequence of pictures of droplet falling from nozzle

Step 1 According to droplet generator parameters and prop-erties of fluid calculate most of the parameters in DLEM

Step 2 According to experimental measurements of dropletvolume driven by a standard waveform determine the rest ofthe adjustment coefficients and damping coefficients

Step 3 Based on the constructed DLEM predicate the valuesof droplet volume and velocity

Step 4 According to the need of applications and restrictiveconditions find out several groups of the appropriate param-eters of driving waveform frompredictive data lists of volumeand velocity

Step 5 After making a subtle adjustment in practice decidethe most appropriate parameters of driving waveform

52 Prediction Case 1 Standard Waveform with DifferentDwell Time Asmentioned above theDLEMmodel is chosento simulate jetting characteristics for the nanosilver inkHere an illustrative example is given to display the effect of

predicting the typical combinations of waveform parameterswith different dwell times (119879

1= 3sim8 us 119879

2= 2119879

1) and

the same amplitude (12V) The properties of nanosilver inkprovided by the LEED-PV Corporation have been listed inTable 1

In Figure 14 the predicted outflow and average velocitycurves with various dwell times are depicted We can observefrom Figure 14(a) that before the droplets rush out of thenozzle the depth of the inhaled meniscus is proportionalto dwell time From Figure 14(b) the time of the dropletappearing is postponed with the increase of dwell timeThat is both two simulation phenomena consist with thetheoretical analysis results in [37]

The comparisons between actual measured and predictedjetting characteristics with various dwell times are given inFigures 15(a) and 15(b) which both illustrate a parabolicchange rule with the increase of dwell time It should benoted that at the dwell time 119879

1= 56 us near the theoretical

optimum value 11987910158401= 52 us the droplet volume and velocity

simultaneously reach the maximum From the performanceof prediction accuracy listed in Table 4 the deviation of thepredicted jetting characteristics is within acceptable range

Mathematical Problems in Engineering 13

45

67

68

1012

1416

10121416

Volu

me (

pL)

91011121314151617

T1 (us)

T2(us)

(a)

45

67

68

1012

1416

3

4

5

6

Velo

city

(ms

)

3

35

4

45

5

55

6

T1 (us) T2

(us)

(b)

Figure 17 Predictive droplet volume and velocity with various dwell times1198791and119879

2 (a) predictive droplet volume and (b) predictive droplet

velocity

53 Prediction Case 2 Driving Waveform under RestrictiveCondition In this experiment an illustrative example underrestrictive conditions is presented to demonstrate the feasibil-ity and effectiveness of the proposed method for predictingjetting characteristics The dimensions of droplet generatorand properties of the nanosilver ink are listed in Table 1 Inorder to reach a high printing resolution and good quality ofdroplet impact with the substrate the restrictive conditionsinvolve that the droplet volume is smaller than 14 pL and thedroplet velocity is larger than 5ms

Based on the prediction procedure the predictive valuesof drop volumevelocity with various dwell times 119879

1and

1198792are listed in Table 5 According to the restrictive con-

ditions three combinations of parameters are found out1198791

= 5 us1198792

= 13 us 1198791

= 55 us1198792

= 14 us and1198791= 5 us119879

2= 75sim8 us After a small range search around

predictive values in the actual test the combination of 1198791=

49 us1198792

= 77 us is finally chosen The dynamic effect ofjetting characteristics driven by this combination is shown inFigure 16 From this figure the jetting characteristics satisfythe specified restrictive conditions quite well Meanwhilealmost no satellite droplet emerges after jetting the maindroplet

The 3D plot of predictive jetting characteristics withvarious dwell times 119879

1and 119879

2is shown in Figure 17 With

the increase of dwell time 1198791 peak positions of volume and

velocity have a tiny shift in 95sim115 us With the same dwelltime 119879

2 there is also a parabolic profile in predictive volume

and velocity

6 Conclusion

The lumped element modeling based on linear model ofpiezoelectric ceramic has been successfully used to simu-late piezoelectric droplet generator However the nonlinearmodel of piezoelectric ceramic is necessary to diminishdeviation between the simulated and measured results Inorder to characterize the nonlinear behavior of piezoelectric

Table 4 Performance of prediction accuracy

Deviation Value

Volume

Max 07135 (499)Min 0036 (013)Mean 02636Std 03316

Velocity

Max 02465 (502)Min 00143 (025)Mean 01095Std 01350

ceramic three models are introduced to simulate dropletgeneration In this work the droplet volumevelocity curvessimulated by dynamic model can carry sufficient informa-tion to distinguish nanosilver droplet formation processclearly Although the complexity of system increases theoutflow curves can obviously reflect the filament breaking-off stage that occurs which is coincident with sampledimages Such nonlinear LEM model is also successful inpredicting jetting characteristics under restrictive conditionsComparing the droplet volumevelocity curves with differentdwell time of high-frequency driving waveform the linearrelationship between dwell time and meniscus appearingmoment is discovered Meanwhile a parabolic change ruleof volumevelocity with the increase of dwell time is in goodagreement with present theoretical analysis

It should be noted that the proposed nonlinear model isfocused on relationship between the droplet volumevelocityand the dwell time of driving waveform while the voltageamplitude is also a decisive factor to droplet formationprocess This should be discussed in our future work

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

14 Mathematical Problems in Engineering

Table5Predictiv

edropletvolume(pL

)and

velocity(m

s)

1198791

1198792

665

775

885

995

10105

11115

12125

13135

14145

15155

4119881

11119

127

134

139

143

146

149

151

15147

142

Default

Default

Default

Default

Default

Default

Default

Default

119878371

395

417

439

459

478

491

500

489

471

451

435

Default

Default

Default

Default

Default

Default

Default

Default

45 119881

Default

117

125

134

143

15155

159

162

16157

153

147

139

Default

Default

Default

Default

Default

Default

119878Default

421

451

476

498

516

532

548

560

543

523

501

482

466

Default

Default

Default

Default

Default

Default

5119881

Default

Default

Default

133

143

151

157

163

167

168

166

163

157

15142

133

Default

Default

Default

Default

119878Default

Default

Default

480

507

531

553

572

587

600

583

563

540

525

509

498

Default

Default

Default

Default

55 119881

Default

Default

Default

Default

Default

151

159

165

168

169

171

169

165

16154

146

138

129

Default

Default

119878Default

Default

Default

Default

Default

524

550

572

593

609

624

601

571

551

537

523

509

496

Default

Default

6119881

Default

Default

Default

Default

Default

Default

Default

158

163

164

165

165

165

158

1495

141

130

119

108

97119878

Default

Default

Default

Default

Default

Default

Default

495

515

533

549

564

544

519

494

474

458

445

432

421

65 119881

Default

Default

Default

Default

Default

Default

Default

Default

Default

155

159

16155

150

143

134

125

114

103

95119878

Default

Default

Default

Default

Default

Default

Default

Default

Default

473

485

495

503

490

469

446

420

400

390

383

7119881

Default

Default

Default

Default

Default

Default

Default

Default

Default

Default

125

135

139

141

135

127

119

110

102

94119878

Default

Default

Default

Default

Default

Default

Default

Default

Default

Default

445

465

479

463

444

422

401

385

370

363

Defaults

areb

ecause

thec

ombinatio

nsof

parametersindrivingwaveform

areo

utof

theo

peratio

nrange

Mathematical Problems in Engineering 15

Authorsrsquo Contribution

Maowei He and Liling Sun contributed equally to this work

Acknowledgment

This research is partially supported by National Natural Sci-ence Foundation of China under Grants 61105067 7100107261174164 and 71271140

References

[1] M SinghHMHaverinen PDhagat andG E Jabbour ldquoInkjetprinting-process and its applicationsrdquo Advanced Materials vol22 no 6 pp 673ndash685 2010

[2] E Macdonald R Salas D Espalin et al ldquo3D printing for therapid prototyping of structural electronicsrdquo IEEE Access vol 2pp 234ndash242 2014

[3] S Umezu T Kitajima H Murase H Ohmori K Katahiraand Y Ito ldquoFabrication of living cell structure utilizing electro-static inkjet phenomenardquo in Proceedings of the 22nd IEEEInternational Conference on Micro Electro Mechanical Systems(MEMS rsquo09) pp 419ndash422 IEEE Sorrento Italy January 2009

[4] P Cosseddu L Basirico A Loi et al ldquoInkjet printed OrganicThin Film Transistors based tactile transducers for artificialrobotic skinrdquo in Proceedings of the 4th IEEE RAS and EMBSInternational Conference on Biomedical Robotics and Biomecha-tronics (BioRob rsquo12) pp 1907ndash1912 June 2012

[5] J Hwang A Wan and A Kahn ldquoEnergetics of metal-organicinterfaces new experiments and assessment of the fieldrdquoMate-rials Science and Engineering R Reports vol 64 no 1-2 pp 1ndash312009

[6] B J Kang and J H Oh ldquoGeometrical characterization of inkjet-printed conductive lines of nanosilver suspensions on a polymersubstraterdquoThin Solid Films vol 518 no 10 pp 2890ndash2896 2010

[7] F Villani P Vacca G Nenna et al ldquoInkjet printed polymerlayer on flexible substrate for OLED applicationsrdquo The Journalof Physical Chemistry C vol 113 no 30 pp 13398ndash13402 2009

[8] J-C Liou and F-G Tseng ldquoMulti-dimensional data registrationCMOSMEMS integrated inkjet printheadrdquo MicroelectronicEngineering vol 88 no 6 pp 888ndash901 2011

[9] J Miettinen K Kaija M Mantysalo et al ldquoMolded substratesfor inkjet printed modulesrdquo IEEE Transactions on Componentsand Packaging Technologies vol 32 no 2 pp 293ndash301 2009

[10] K Silverbrook ldquoPrinthead with multiple actuators in eachchamberrdquo US7708387 B2 2010

[11] V Piefort Finite element modelling of piezoelectric active struc-tures [PhD thesis] Department ofMechanical Engineering andRobotics Universite Libre de Bruxelles Brussels Belgium 2001

[12] L Sang Y Hong and F Wang ldquoInvestigation of viscosity effecton droplet formation in T-shaped microchannels by numericaland analytical methodsrdquo Microfluidics and Nanofluidics vol 6no 5 pp 621ndash635 2009

[13] J Liu S-H Tan Y F Yap M Y Ng and N-T NguyenldquoNumerical and experimental investigations of the formationprocess of ferrofluid dropletsrdquo Microfluidics and Nanofluidicsvol 11 no 2 pp 177ndash187 2011

[14] F Sarrazin K Loubiere L Prat C Gourdon T Bonometti andJ Magnaudet ldquoExperimental and numerical study of dropletshydrodynamics in MicroChannelrdquo AIChE Journal vol 52 no12 pp 4061ndash4070 2006

[15] X Q Xing D L Butler S H Ng Z Wang S Danyluk and CYang ldquoSimulation of droplet formation and coalescence usinglattice Boltzmann-based single-phasemodelrdquo Journal of Colloidand Interface Science vol 311 no 2 pp 609ndash618 2007

[16] Q Gallas R Holman T Nishida B Carroll M Sheplak and LCattafesta ldquoLumped element modeling of piezoelectric-drivensynthetic jet actuatorsrdquoAIAA Journal vol 41 no 2 pp 240ndash2472003

[17] A A Khalate X Bombois R Babuska H Wijshoff and RWaarsing ldquoOptimization-based feedforward control for a Drop-on-Demand inkjet printheadrdquo in Proceedings of the AmericanControl Conference (ACC rsquo10) pp 2182ndash2187 Baltimore MdUSA July 2010

[18] L Meirovitch Fundamentals of Vibrations McGraw-Hill NewYork NY USA 2001

[19] H Seitz and J Heinz ldquoModelling of a microfluidic device withpiezoelectric actuatorsrdquo Journal of Micromechanics and Micro-engineering vol 14 no 8 pp 1140ndash1147 2004

[20] S Kim J Sung andMH Lee ldquoPressure wave and fluid velocityin a bend-mode inkjet nozzle with double PZT actuatorsrdquoJournal of Thermal Science vol 22 no 1 pp 29ndash35 2013

[21] G Wassink Inkjet printhead performance enhancement by feed-forward input design based on two-port modeling [PhD thesis]Delft University of Technology 2007

[22] S Prasad SHorowitz QGallas B Sankar L Cattafesta andMSheplak ldquoTwo-port electroacoustic model of an axisymmet-ric piezoelectric composite platerdquo in Proceedings of the 43rdAIAAASMEASCEAHS Structures Structural DynamicsandMaterials Conference AIAA Paper 2002-1365 AIAA DenverColo USA April 2002

[23] D T BlackstockFundamentals of Physical Acoustics JohnWileyamp Sons New York NY USA 2000

[24] F M White Fluid Mechanics McGraw-Hill New York NYUSA 1979

[25] H Schempt andD R Yoerger ldquoStudy of dominant performancecharacteristics in robot transmissionsrdquo Journal of MechanicalDesign Transactions Of the ASME vol 115 no 3 pp 472ndash4821993

[26] E A Witmer ldquoElementary Bernoulli-Euler beam theoryrdquoMITUnified Engineering Course Notes vol 5 pp 114ndash164 1991-1992

[27] Y Yang and L Tang ldquoEquivalent circuit modeling of piezoelec-tric energy harvestersrdquo Journal of Intelligent Material Systemsand Structures vol 20 no 18 pp 2223ndash2235 2009

[28] F Constantinescu A G Gheorghe and M Nitescu ldquoNewcircuit models of power BAW resonatorsrdquo in Proceedings of theEuropean Microwave Integrated Circuit Conference October2007

[29] A Albareda and R PerezNon-Linear Behaviour of PiezoelectricCeramics vol 140ofSpringer Series inMaterials Science Springer2011

[30] J Nosek ldquoDrive level dependence of the resonant frequency inBAW quartz resonators and his modelingrdquo IEEE TransactionsonUltrasonics Ferroelectrics and Frequency Control vol 46 no4 pp 823ndash829 1999

[31] R Aigner N-H Huynh M Handtmann and S MarksteinerldquoBehavior of BAW devices at high power levelsrdquo in Proceedingsof the IEEE MTT-S International Microwave Symposium pp429ndash432 Long Beach Calif USA June 2005

[32] R S Ketcham G R Kline and K M Lakin ldquoPerformanceof TFR filters under elevated power conditionsrdquo in Proceedingsof 42nd the Annual Frequency Control Symposium pp 106ndash111Baltimore Md USA June 1988

16 Mathematical Problems in Engineering

[33] K-S Kwon ldquoWaveform design methods for piezo inkjet dis-pensers based on measured meniscus motionrdquo Journal of Micro-electromechanical Systems vol 18 no 5 pp 1118ndash1125 2009

[34] H-C Wu and H-J Lin ldquoEffects of actuating pressure wave-forms on the droplet behavior in a piezoelectric inkjetrdquoMateri-als Transactions vol 51 no 12 pp 2269ndash2276 2010

[35] KMinolta InkjetHeadApplicationNote-KM1024 Series KonicaMinolta Ij Technologies

[36] G Golub andV Pereyra ldquoSeparable nonlinear least squares thevariable projection method and its applicationsrdquo Inverse Prob-lems vol 19 no 2 pp R1ndashR26 2003

[37] H Wijshoff ldquoThe dynamics of the piezo inkjet printheadoperationrdquo Physics Reports vol 491 no 4-5 pp 77ndash177 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

C0

C2

C1 R1 L1

R2 L21 N

Figure 5 Equivalent electrical circuitmodel of PZTnonlinear effectwith multiple parallel resonances

derived based on the constitutive relations of piezoelectricityand the following assumptions (a) Euler-Bernouli beamassumption [26] (b) negligible external excitation from airdamping (c) proportional damping (ie the strain ratedamping and viscous damping are assumed to be propor-tional to the bending stiffness and mass per length of thebeam) (d) uniform electric field through the piezoelectricthickness Then the circuit element values of the proposedlumped parameter model are formulated in (1) where T isthe adjustment coefficient

1198711=

119871119898

T2

1198771=

119877119898

T2

1198621= T2 lowast 119862

119898

(3)

(2) Multiple-Resonator Model Yang and Tang [27] givethe multiple-resonator model to characterize piezoelectricceramic in piezoelectric energy harvesters as shown inFigure 5 His research finds that the network of piezoelectricceramic driving circuit consists of infinite branches and eachis composed of simplified equivalent circuit with an inductora capacitor and a resistorWhen the voltage signals from sev-eral branches drive the piezoelectric ceramic simultaneouslythe higher order vibrationmodes are excited It is realized thatworking at higher order modes causes the nonlinear effectof piezoelectric ceramic To characterize the different higherorder modes a circuit of multiple resonators is constructedto match the external branches In this literature to modelthe droplet generator nearly a double circuit may meet thesimulation accuracy Then the circuit element values of theproposed lumped parameter model are formulated in (3)where 120572 and N are both the adjustment coefficients

1198622=

1198621

1205722

1198772= 120572 lowast 119877

1

1198712= 1198711

(4)

(3) Dynamic Model Based on the works in [28 29] an equiv-alent electrical circuit with nonlinear resonance oscillationsin series namely the dynamic model is given to analyzepower BAW (bulk acoustic wave) resonators as shown inFigure 6Three major nonlinear effects are considered in this

C0 C0

C1

C2

C3

R1

L1

L1

R2

R3

C(I)

R(I)

Figure 6 Equivalent electrical circuitmodel of PZTnonlinear effectwith dynamic coefficients

equivalent circuit the amplitude-frequency effect (A-F effect)[30] the intermodulation effect [31] and the bias-frequencyeffect (B-F effect) [32] The dynamic values of resistance andcapacitance in the motional branch are dependent on themotional current in the branchThen derivation formulas aregiven in (4) where A

1 A2 B1 and B

2are the adjustment

coefficients

1198622= 12057211198682

1198623= 12057221198684

1198772= 12057311198682

1198773= 12057321198684

997904rArr 119862 (119868) = 119862

1(1 + A

11198682

+ A21198684

)

119877 (119868) = 1198771(1 + B

11198682

+ B21198684

) (5)

Then the improved LEM structure can be illustrated as inFigure 7

3 Experimental Method

31 Experimental Setup In this experiment the investigatedprinthead is KM series IJ Head of Konica and the used ink isnanosilver ink provided by the LEED-PV Corporation andthe according parameters are listed in Table 1 in Section 2 Aschematic overview of the experimental setup to observe thenanosilver droplet formation process is shown in Figure 8The visualization system consists of a cold source of halogens(XAO LG150) and a CCD camera (SVS ECO424) whichcan synchronize to the driving signal The volume andvelocity of droplet and the profile of meniscus are measuredthrough the images captured by CCD with exposure time of1 us The high-frequency driving waveform is programmedby arbitrary wave generator (RIGO DG2014A) In orderto repeatedly generate uniform droplets ink supply unitconnects air pressure unit which offers suitable negative forceto prevent the ink leaking from the printhead

32 Edge Detection and Image Analysis as a QualitativeMeasure The leading edge detection is to determine thenozzle position and meniscus position A straight line isgenerated along the nozzle in the region of interest (ROI)with

6 Mathematical Problems in Engineering

sim

Vac

CaCCeb

MaRadMaNRaN

MaDRaDCaD1 Na

RaO

(a)

sim

Vac

CaCCeb

MaRadMaNRaN

1 NaRaDT

2MaDT

2

RaO

CaDT2

(b)

sim

Vac

CaC

Ceb

MaRadMaNRaN

MaD

MaD

RaDCaD

1 Na

1 N

CaDa2

aRaD

RaO

(c)

sim

Vac

CaCCeb

MaRadMaNRaN

MaD1 Na C(I) R(I)

RaO

(d)

Figure 7 (a) Original LEM (b) proportional coefficient model (c) multiple resonators model and (d) dynamic coefficients model

Temperature unit

Air pressure unit

Ink supply

Parallel

Halogens light source

CCDPC

Wave generator

Drivingboard

Image

(a)

CCD

Halogens light

sourceArbitrary waveform generator

Drivingboard

PrintheadInk supply

(b)

Figure 8 Experimental setup (a) schematic of experimental setup and (b) image of experimental setup

CCD camera image analysis By setting a suitable thresholdgradient of brightness themeniscus location can be detectedThe satellite droplets and filament can be detected throughfarther image processing such as cropping filtering andthreshold recontrasting (shown as in Figure 9)

In image analysis system the droplet volume is calculatedby

Vol = sum120587 sdot (119889

2)

2

sdot 120575119911 (6)

which hypothesizes that liquid jet is axis-symmetric alongnozzle centerline And the droplet velocity is calculated basedon the position change of the moving-forward leading edgein two consecutive images with specified sampling intervalNote that the information is lost if the meniscus is hiddeninside the nozzle so edge detection only measures theprotruding meniscus

33 High-Frequency Driving Waveform as a System Excita-tion The high-frequency driving waveform is composed of

positive and negative periods including three pulse edges andtwo dwell times as shown in Figure 10 The first rising edgeexpands the channel to get the inkmeniscus into the channelThen the falling edge works to eject the droplet from thenozzle

This period of waveform design is based on the halfperiod of channel resonance Thus the waveform designissue is focused on calculating the optimum value of dwelltime 119879

1= 1198710119862 where 119871

0is the length of channel and 119862

is the sonic speed of the used ink The falling edge of thesecond negative voltage generates larger pressure in channelto accelerate the droplet with higher speed than conventionalsingle trapezoidal waveform The second rising edge whichsets the voltage back to 0 works to restrain the meniscusvibration [33]

Theoretically the suitable slops of the rising edge andfalling edge can restrain the meniscus vibration [34] How-ever in practical applications these two values are too largeto control accurately

Asmentioned above theoretical optimum value for dwelltime 119879

1is set as 52 us The corresponding dwell time 119879

2

Mathematical Problems in Engineering 7

(a) (b) (c) (d)

Figure 9 Image processing steps (a) original image (b) cropped image (c) threshold image and (d) edge image

Time

Voltage

Dwell

Dwell

time T1

time T2

Figure 10 Applied driving waveform of printhead

is recommended as 104 us (twice of 1198791) and the voltage

magnitude is set as 12 V according to the instruction ofKonica printhead In this paper the driving waveform withthese parameters is called the ldquostandardrdquo waveform [35]

4 Simulation Result

41 Coefficient Identification The coefficients of originalLEM model can be determined experimentally Howeverthe three improved models in this work are somewhatcomplicated and their coefficients reveal some importantfeatures of these equivalent circuits For this reason thissection is dedicated to identify the adjustment coefficients inthese corresponding circuits

In this experiment a group of experimental measure-ments with printhead driven by the standard high-frequencywaveform are chosen as the basic reference The coeffi-cients of each proposed improved LEM model should beadjusted to minimize the deviations between the simulationresults and the selected experimental measurements shownin Figure 11 The nonlinear least squares method [36] usingthe Levenberg-Margquardt and Gauss-Newton Algorithms

Table 3 Coefficients for three lumped element models

Model Coefficient ValueProportional 119879 0837

Multiple resonators 120572 51119873 228

Dynamic coefficients

1198601

133 lowast 103

1198611

353 lowast 103

1198602

234 lowast 104

1198612

411 lowast 104

is applied to find the optimal coefficients set for each pro-posed equivalent circuit which makes the simulation curvesoverlap the experimental measurements as far as possibleThe resulting coefficients are listed in Table 3

42 Droplet Volume Simulation In many applications of ink-jet printing the volume of droplet is a critical restrictivecondition Droplets with controllable volume affect not onlythe printing resolution but also the depositional thicknessTherefore how to accurately control droplet volume is a keyproblem to be solved by the proposed method

According to experimental measurements in Figure 11the comparison between the original LEM model and theproposed three improved models is presented in Figure 12In this figure the solid line is the outflow simulation curvesbased on the LEM technique and the star line is the actualmeasured volume values during the droplet formation pro-cess It should be noted that in Figures 12(a)ndash12(d) the firstthree points of actual measurement curve are default to zerobecause no fluid flowing outside the nozzle can be capturedby our image system

As the actual experimental measurements indicated inFigure 11 the droplet formation process is divided into threestages the meniscus protruding stage (ie 119879

1st = 15sim30 us)the filament breaking-off stage (ie 1198792nd = 30sim35 us) andthe droplet falling-down stage (ie 1198793rd = 35sim60 us) That

8 Mathematical Problems in Engineering

3us0pL0ms

6us0pL0ms

9us0pL0ms

12us 15us 18us 21us 24us

30us 33us 36us 39us 42us 45us 48us27us

678pL876ms

994pL730ms

1221 pL684ms

1401pL624ms

1521pL618ms

1533pL600ms

1567pL600ms

1509pL606ms

1492pL618ms

1489pL624 ms

1480 pL624 ms

1479pL618ms

1480pL618ms

Figure 11 Pictures of nanosilver droplet formation process driven by standard waveform

is ideally the outflow curve simulated by analytic modelsshould include overshoot phase which can clearly demon-strate that the part of filament is inhaled back into nozzle orsplits into satellite droplets by viscosity and surface tensionin the filament break-off stage However according to thesimulation result of the original LEM in Figure 12(a) there

is no obvious decreasing after reaching the maximum in theoutflow curve and the outflow curve of the original modelcannot well overlap the actual measurement curve

In Figures 12(b) and 12(d) the overshoot phase canbe observed clearly This performance improvement can bestraightforwardly explained by the fact that poles and zeros

Mathematical Problems in Engineering 9

0 20 40 60

0

10

20

Time (us)

Volu

me (

pL)

ExperimentOriginal

minus10

(a)

0 20 40 60

0

10

20

Time (us)

Volu

me (

pL)

ExperimentProportional

minus10

(b)

0 20 40 60

0

10

20

Time (us)

Volu

me (

pL)

ExperimentMultiple

minus10

minus20

(c)

0 20 40 60

0

10

20

Time (us)

Volu

me (

pL)

ExperimentDynamic

minus10

(d)

Figure 12 Outflow change ofmeasurements and simulations of (a) original model (b) proportional coefficientmodel (c)multiple resonatorsmodel and (d) dynamic coefficient model

of the transfer functions are manipulated to speed up thedynamic response speed of the proportional and dynamicmodels From Figure 12 both the multiple-resonator anddynamic models can obtain precise simulation results Thatis more well-designed coefficients are used to characterizethe nonlinear behaviors of piezoelectric printhead and theadjustable ranges are extended to a satisfied degree of bothmodels Additionally it should be noted that the outflowcurve of dynamic model has not only the overshoot phasebut also high overlap ratio to the experimental measurementMoreover the simulation curves reveal that the meniscusmoves inward before the droplet appears

43 Flow and Droplet Velocity Simulation The velocity ofdroplet is the other important aspect to evaluate the printingeffect Droplets with a high speed can remarkably improve

the printing efficiency and the quality of impact with sub-strate Therefore this experiment gives the velocity simula-tion using the proposed LEM models on the same actualexperimental measurement in Section 41

The comparisons between the flow velocity curves sim-ulated by the original LEM and three improved models areillustrated in Figure 13 The excitation is the standard drivingwaveform For the actual measured velocity curves the firsttwo points are also default to zero because no fluid flowingoutside the nozzle can be captured by our image system

It should be noted that all the values of actual measuredvelocity are positive However from the flow velocity curvesin Figure 13 the values of flow velocity in nozzle can bepositive or negative in different phases This positive-to-negative velocity variation indicates the changes of the flowdirection that leads to the break-off phenomenon Hence in

10 Mathematical Problems in Engineering

0 20 40 60

0

5

10

Time (us)

Velo

city

(ms

)

ExperimentFlow velocityOriginal

minus10

minus5

(a)

0 20 40 60

0

5

10

Time (us)

Velo

city

(ms

)

ExperimentFlow velocityProportional

minus10

minus5

(b)

0 20 40 60

0

5

10

Time (us)

Velo

city

(ms

)

ExperimentFlow velocityMultiple

minus10

minus5

(c)

0 20 40 60

0

5

10

Time (us)

Velo

city

(ms

)

ExperimentFlow velocityDynamic

minus10

minus5

(d)

Figure 13 Comparison of measured droplets velocity and the flowaverage velocity simulated by (a) original model (b) ldquoproportionalcoefficientrdquo model (c) ldquomultiple resonatorsrdquo model and (d) ldquodynamic coefficientsrdquo model

order to correctly simulate the velocity in leading edge wepropose a new calculation formula which combines both ofoutflow curve and flow velocity curve

119881average (119905) = radicsum1199052

119905=1199051

(12) 119876out (119905) sdot 119881 (119905)2

sum1199052

119905=1199051

(12)119876out (119905) (7)

where119876out is the outflow value and119881 is the flow velocity 1199051is

the moment when the sum of 119876out changes from negative topositive in outflow curve and 119905

2after 119905

1is the moment when

flow velocity changes to negative in nozzleFrom Figure 13 it can be obviously observed that in

the beginning of droplet formation process the multiple-resonator and dynamic models exhibit better performance insimulating the droplet velocity rather than the originalmodelThat is the average velocity curves obtained by these two

models are almost coincident to the actual measured velocitycurves

Based on the above analysis and comparisons thedynamicmodel based LEMhas themost accuracy in simulat-ing the droplet volume and velocity and can carry sufficientinformation to distinguish the break-off phenomenon occur-ing in the nanosilver droplet formation processs Thereforeamong these three improved models the dynamic lumpedelement model (DLEM) is the most suitable improved equiv-alent structure for simulating jetting characteristics

5 Prediction

For industry applications in printable electronics fabricationthe printhead must work at a certain status to meet manyrestrictive conditions Among them the most important

Mathematical Problems in Engineering 11

0 20 40 60

0

5

10

15

20

Time (us)

Volu

me (

pL)

minus25

minus20

minus15

minus10

minus5

T1 = 3usT1 = 4usT1 = 5us

T1 = 6usT1 = 7usT1 = 8us

(a)

0 20 40 604

5

6

7

8

9

10

Velo

city

(ms

)

Time (us)

T1 = 3usT1 = 4usT1 = 5us

T1 = 6usT1 = 7usT1 = 8us

(b)

Figure 14 Predictive droplet volume and velocity with different and various dwell times 1198792= 21198791 (a) simulation curves of outflow change

and (b) simulation curves of droplet average velocity

4 6 813

14

15

16

17

SimulationExperiment

Volu

me (

pL)

Time (us)

(a)

SimulationExperiment

4 6 8

45

5

55

6

Time (us)

Velo

city

(ms

)

(b)

Figure 15 Predictive droplet volume and velocity compared with measured volume and velocity

problem is choosing the appropriate combinations of thedriving waveform parameters for the used conductive inksHowever an exhausting manual selective process inevitablywastes a lot of time Therefore the computer-aided methodsare urgently needed for searching these appropriate combi-nations efficiently and robustly In this section the proposed

DLEM is applied to predict jetting characteristics for nanosil-ver ink

51 General Prediction Procedure The general procedure ofdroplet volumevelocity prediction based on DLEM is asfollows

12 Mathematical Problems in Engineering

0us0pL0ms

3us0pL0ms

6us0pL0ms

9us0pL0ms

12us6353pL736ms

15us9318pL612ms

18us11441pL572ms

18us 24us

30us14364pL498ms

33us14046pL510ms

36us13980pL520ms

39us13952pL525ms

42us13858pL525ms

45us13869pL520ms

48us13891pL524ms

51us13924pL522ms

54us

524ms13877pL

57us13840pL530ms

13127 pL548ms

14252pL520ms

27us14786pL504ms

Figure 16 A sequence of pictures of droplet falling from nozzle

Step 1 According to droplet generator parameters and prop-erties of fluid calculate most of the parameters in DLEM

Step 2 According to experimental measurements of dropletvolume driven by a standard waveform determine the rest ofthe adjustment coefficients and damping coefficients

Step 3 Based on the constructed DLEM predicate the valuesof droplet volume and velocity

Step 4 According to the need of applications and restrictiveconditions find out several groups of the appropriate param-eters of driving waveform frompredictive data lists of volumeand velocity

Step 5 After making a subtle adjustment in practice decidethe most appropriate parameters of driving waveform

52 Prediction Case 1 Standard Waveform with DifferentDwell Time Asmentioned above theDLEMmodel is chosento simulate jetting characteristics for the nanosilver inkHere an illustrative example is given to display the effect of

predicting the typical combinations of waveform parameterswith different dwell times (119879

1= 3sim8 us 119879

2= 2119879

1) and

the same amplitude (12V) The properties of nanosilver inkprovided by the LEED-PV Corporation have been listed inTable 1

In Figure 14 the predicted outflow and average velocitycurves with various dwell times are depicted We can observefrom Figure 14(a) that before the droplets rush out of thenozzle the depth of the inhaled meniscus is proportionalto dwell time From Figure 14(b) the time of the dropletappearing is postponed with the increase of dwell timeThat is both two simulation phenomena consist with thetheoretical analysis results in [37]

The comparisons between actual measured and predictedjetting characteristics with various dwell times are given inFigures 15(a) and 15(b) which both illustrate a parabolicchange rule with the increase of dwell time It should benoted that at the dwell time 119879

1= 56 us near the theoretical

optimum value 11987910158401= 52 us the droplet volume and velocity

simultaneously reach the maximum From the performanceof prediction accuracy listed in Table 4 the deviation of thepredicted jetting characteristics is within acceptable range

Mathematical Problems in Engineering 13

45

67

68

1012

1416

10121416

Volu

me (

pL)

91011121314151617

T1 (us)

T2(us)

(a)

45

67

68

1012

1416

3

4

5

6

Velo

city

(ms

)

3

35

4

45

5

55

6

T1 (us) T2

(us)

(b)

Figure 17 Predictive droplet volume and velocity with various dwell times1198791and119879

2 (a) predictive droplet volume and (b) predictive droplet

velocity

53 Prediction Case 2 Driving Waveform under RestrictiveCondition In this experiment an illustrative example underrestrictive conditions is presented to demonstrate the feasibil-ity and effectiveness of the proposed method for predictingjetting characteristics The dimensions of droplet generatorand properties of the nanosilver ink are listed in Table 1 Inorder to reach a high printing resolution and good quality ofdroplet impact with the substrate the restrictive conditionsinvolve that the droplet volume is smaller than 14 pL and thedroplet velocity is larger than 5ms

Based on the prediction procedure the predictive valuesof drop volumevelocity with various dwell times 119879

1and

1198792are listed in Table 5 According to the restrictive con-

ditions three combinations of parameters are found out1198791

= 5 us1198792

= 13 us 1198791

= 55 us1198792

= 14 us and1198791= 5 us119879

2= 75sim8 us After a small range search around

predictive values in the actual test the combination of 1198791=

49 us1198792

= 77 us is finally chosen The dynamic effect ofjetting characteristics driven by this combination is shown inFigure 16 From this figure the jetting characteristics satisfythe specified restrictive conditions quite well Meanwhilealmost no satellite droplet emerges after jetting the maindroplet

The 3D plot of predictive jetting characteristics withvarious dwell times 119879

1and 119879

2is shown in Figure 17 With

the increase of dwell time 1198791 peak positions of volume and

velocity have a tiny shift in 95sim115 us With the same dwelltime 119879

2 there is also a parabolic profile in predictive volume

and velocity

6 Conclusion

The lumped element modeling based on linear model ofpiezoelectric ceramic has been successfully used to simu-late piezoelectric droplet generator However the nonlinearmodel of piezoelectric ceramic is necessary to diminishdeviation between the simulated and measured results Inorder to characterize the nonlinear behavior of piezoelectric

Table 4 Performance of prediction accuracy

Deviation Value

Volume

Max 07135 (499)Min 0036 (013)Mean 02636Std 03316

Velocity

Max 02465 (502)Min 00143 (025)Mean 01095Std 01350

ceramic three models are introduced to simulate dropletgeneration In this work the droplet volumevelocity curvessimulated by dynamic model can carry sufficient informa-tion to distinguish nanosilver droplet formation processclearly Although the complexity of system increases theoutflow curves can obviously reflect the filament breaking-off stage that occurs which is coincident with sampledimages Such nonlinear LEM model is also successful inpredicting jetting characteristics under restrictive conditionsComparing the droplet volumevelocity curves with differentdwell time of high-frequency driving waveform the linearrelationship between dwell time and meniscus appearingmoment is discovered Meanwhile a parabolic change ruleof volumevelocity with the increase of dwell time is in goodagreement with present theoretical analysis

It should be noted that the proposed nonlinear model isfocused on relationship between the droplet volumevelocityand the dwell time of driving waveform while the voltageamplitude is also a decisive factor to droplet formationprocess This should be discussed in our future work

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

14 Mathematical Problems in Engineering

Table5Predictiv

edropletvolume(pL

)and

velocity(m

s)

1198791

1198792

665

775

885

995

10105

11115

12125

13135

14145

15155

4119881

11119

127

134

139

143

146

149

151

15147

142

Default

Default

Default

Default

Default

Default

Default

Default

119878371

395

417

439

459

478

491

500

489

471

451

435

Default

Default

Default

Default

Default

Default

Default

Default

45 119881

Default

117

125

134

143

15155

159

162

16157

153

147

139

Default

Default

Default

Default

Default

Default

119878Default

421

451

476

498

516

532

548

560

543

523

501

482

466

Default

Default

Default

Default

Default

Default

5119881

Default

Default

Default

133

143

151

157

163

167

168

166

163

157

15142

133

Default

Default

Default

Default

119878Default

Default

Default

480

507

531

553

572

587

600

583

563

540

525

509

498

Default

Default

Default

Default

55 119881

Default

Default

Default

Default

Default

151

159

165

168

169

171

169

165

16154

146

138

129

Default

Default

119878Default

Default

Default

Default

Default

524

550

572

593

609

624

601

571

551

537

523

509

496

Default

Default

6119881

Default

Default

Default

Default

Default

Default

Default

158

163

164

165

165

165

158

1495

141

130

119

108

97119878

Default

Default

Default

Default

Default

Default

Default

495

515

533

549

564

544

519

494

474

458

445

432

421

65 119881

Default

Default

Default

Default

Default

Default

Default

Default

Default

155

159

16155

150

143

134

125

114

103

95119878

Default

Default

Default

Default

Default

Default

Default

Default

Default

473

485

495

503

490

469

446

420

400

390

383

7119881

Default

Default

Default

Default

Default

Default

Default

Default

Default

Default

125

135

139

141

135

127

119

110

102

94119878

Default

Default

Default

Default

Default

Default

Default

Default

Default

Default

445

465

479

463

444

422

401

385

370

363

Defaults

areb

ecause

thec

ombinatio

nsof

parametersindrivingwaveform

areo

utof

theo

peratio

nrange

Mathematical Problems in Engineering 15

Authorsrsquo Contribution

Maowei He and Liling Sun contributed equally to this work

Acknowledgment

This research is partially supported by National Natural Sci-ence Foundation of China under Grants 61105067 7100107261174164 and 71271140

References

[1] M SinghHMHaverinen PDhagat andG E Jabbour ldquoInkjetprinting-process and its applicationsrdquo Advanced Materials vol22 no 6 pp 673ndash685 2010

[2] E Macdonald R Salas D Espalin et al ldquo3D printing for therapid prototyping of structural electronicsrdquo IEEE Access vol 2pp 234ndash242 2014

[3] S Umezu T Kitajima H Murase H Ohmori K Katahiraand Y Ito ldquoFabrication of living cell structure utilizing electro-static inkjet phenomenardquo in Proceedings of the 22nd IEEEInternational Conference on Micro Electro Mechanical Systems(MEMS rsquo09) pp 419ndash422 IEEE Sorrento Italy January 2009

[4] P Cosseddu L Basirico A Loi et al ldquoInkjet printed OrganicThin Film Transistors based tactile transducers for artificialrobotic skinrdquo in Proceedings of the 4th IEEE RAS and EMBSInternational Conference on Biomedical Robotics and Biomecha-tronics (BioRob rsquo12) pp 1907ndash1912 June 2012

[5] J Hwang A Wan and A Kahn ldquoEnergetics of metal-organicinterfaces new experiments and assessment of the fieldrdquoMate-rials Science and Engineering R Reports vol 64 no 1-2 pp 1ndash312009

[6] B J Kang and J H Oh ldquoGeometrical characterization of inkjet-printed conductive lines of nanosilver suspensions on a polymersubstraterdquoThin Solid Films vol 518 no 10 pp 2890ndash2896 2010

[7] F Villani P Vacca G Nenna et al ldquoInkjet printed polymerlayer on flexible substrate for OLED applicationsrdquo The Journalof Physical Chemistry C vol 113 no 30 pp 13398ndash13402 2009

[8] J-C Liou and F-G Tseng ldquoMulti-dimensional data registrationCMOSMEMS integrated inkjet printheadrdquo MicroelectronicEngineering vol 88 no 6 pp 888ndash901 2011

[9] J Miettinen K Kaija M Mantysalo et al ldquoMolded substratesfor inkjet printed modulesrdquo IEEE Transactions on Componentsand Packaging Technologies vol 32 no 2 pp 293ndash301 2009

[10] K Silverbrook ldquoPrinthead with multiple actuators in eachchamberrdquo US7708387 B2 2010

[11] V Piefort Finite element modelling of piezoelectric active struc-tures [PhD thesis] Department ofMechanical Engineering andRobotics Universite Libre de Bruxelles Brussels Belgium 2001

[12] L Sang Y Hong and F Wang ldquoInvestigation of viscosity effecton droplet formation in T-shaped microchannels by numericaland analytical methodsrdquo Microfluidics and Nanofluidics vol 6no 5 pp 621ndash635 2009

[13] J Liu S-H Tan Y F Yap M Y Ng and N-T NguyenldquoNumerical and experimental investigations of the formationprocess of ferrofluid dropletsrdquo Microfluidics and Nanofluidicsvol 11 no 2 pp 177ndash187 2011

[14] F Sarrazin K Loubiere L Prat C Gourdon T Bonometti andJ Magnaudet ldquoExperimental and numerical study of dropletshydrodynamics in MicroChannelrdquo AIChE Journal vol 52 no12 pp 4061ndash4070 2006

[15] X Q Xing D L Butler S H Ng Z Wang S Danyluk and CYang ldquoSimulation of droplet formation and coalescence usinglattice Boltzmann-based single-phasemodelrdquo Journal of Colloidand Interface Science vol 311 no 2 pp 609ndash618 2007

[16] Q Gallas R Holman T Nishida B Carroll M Sheplak and LCattafesta ldquoLumped element modeling of piezoelectric-drivensynthetic jet actuatorsrdquoAIAA Journal vol 41 no 2 pp 240ndash2472003

[17] A A Khalate X Bombois R Babuska H Wijshoff and RWaarsing ldquoOptimization-based feedforward control for a Drop-on-Demand inkjet printheadrdquo in Proceedings of the AmericanControl Conference (ACC rsquo10) pp 2182ndash2187 Baltimore MdUSA July 2010

[18] L Meirovitch Fundamentals of Vibrations McGraw-Hill NewYork NY USA 2001

[19] H Seitz and J Heinz ldquoModelling of a microfluidic device withpiezoelectric actuatorsrdquo Journal of Micromechanics and Micro-engineering vol 14 no 8 pp 1140ndash1147 2004

[20] S Kim J Sung andMH Lee ldquoPressure wave and fluid velocityin a bend-mode inkjet nozzle with double PZT actuatorsrdquoJournal of Thermal Science vol 22 no 1 pp 29ndash35 2013

[21] G Wassink Inkjet printhead performance enhancement by feed-forward input design based on two-port modeling [PhD thesis]Delft University of Technology 2007

[22] S Prasad SHorowitz QGallas B Sankar L Cattafesta andMSheplak ldquoTwo-port electroacoustic model of an axisymmet-ric piezoelectric composite platerdquo in Proceedings of the 43rdAIAAASMEASCEAHS Structures Structural DynamicsandMaterials Conference AIAA Paper 2002-1365 AIAA DenverColo USA April 2002

[23] D T BlackstockFundamentals of Physical Acoustics JohnWileyamp Sons New York NY USA 2000

[24] F M White Fluid Mechanics McGraw-Hill New York NYUSA 1979

[25] H Schempt andD R Yoerger ldquoStudy of dominant performancecharacteristics in robot transmissionsrdquo Journal of MechanicalDesign Transactions Of the ASME vol 115 no 3 pp 472ndash4821993

[26] E A Witmer ldquoElementary Bernoulli-Euler beam theoryrdquoMITUnified Engineering Course Notes vol 5 pp 114ndash164 1991-1992

[27] Y Yang and L Tang ldquoEquivalent circuit modeling of piezoelec-tric energy harvestersrdquo Journal of Intelligent Material Systemsand Structures vol 20 no 18 pp 2223ndash2235 2009

[28] F Constantinescu A G Gheorghe and M Nitescu ldquoNewcircuit models of power BAW resonatorsrdquo in Proceedings of theEuropean Microwave Integrated Circuit Conference October2007

[29] A Albareda and R PerezNon-Linear Behaviour of PiezoelectricCeramics vol 140ofSpringer Series inMaterials Science Springer2011

[30] J Nosek ldquoDrive level dependence of the resonant frequency inBAW quartz resonators and his modelingrdquo IEEE TransactionsonUltrasonics Ferroelectrics and Frequency Control vol 46 no4 pp 823ndash829 1999

[31] R Aigner N-H Huynh M Handtmann and S MarksteinerldquoBehavior of BAW devices at high power levelsrdquo in Proceedingsof the IEEE MTT-S International Microwave Symposium pp429ndash432 Long Beach Calif USA June 2005

[32] R S Ketcham G R Kline and K M Lakin ldquoPerformanceof TFR filters under elevated power conditionsrdquo in Proceedingsof 42nd the Annual Frequency Control Symposium pp 106ndash111Baltimore Md USA June 1988

16 Mathematical Problems in Engineering

[33] K-S Kwon ldquoWaveform design methods for piezo inkjet dis-pensers based on measured meniscus motionrdquo Journal of Micro-electromechanical Systems vol 18 no 5 pp 1118ndash1125 2009

[34] H-C Wu and H-J Lin ldquoEffects of actuating pressure wave-forms on the droplet behavior in a piezoelectric inkjetrdquoMateri-als Transactions vol 51 no 12 pp 2269ndash2276 2010

[35] KMinolta InkjetHeadApplicationNote-KM1024 Series KonicaMinolta Ij Technologies

[36] G Golub andV Pereyra ldquoSeparable nonlinear least squares thevariable projection method and its applicationsrdquo Inverse Prob-lems vol 19 no 2 pp R1ndashR26 2003

[37] H Wijshoff ldquoThe dynamics of the piezo inkjet printheadoperationrdquo Physics Reports vol 491 no 4-5 pp 77ndash177 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

sim

Vac

CaCCeb

MaRadMaNRaN

MaDRaDCaD1 Na

RaO

(a)

sim

Vac

CaCCeb

MaRadMaNRaN

1 NaRaDT

2MaDT

2

RaO

CaDT2

(b)

sim

Vac

CaC

Ceb

MaRadMaNRaN

MaD

MaD

RaDCaD

1 Na

1 N

CaDa2

aRaD

RaO

(c)

sim

Vac

CaCCeb

MaRadMaNRaN

MaD1 Na C(I) R(I)

RaO

(d)

Figure 7 (a) Original LEM (b) proportional coefficient model (c) multiple resonators model and (d) dynamic coefficients model

Temperature unit

Air pressure unit

Ink supply

Parallel

Halogens light source

CCDPC

Wave generator

Drivingboard

Image

(a)

CCD

Halogens light

sourceArbitrary waveform generator

Drivingboard

PrintheadInk supply

(b)

Figure 8 Experimental setup (a) schematic of experimental setup and (b) image of experimental setup

CCD camera image analysis By setting a suitable thresholdgradient of brightness themeniscus location can be detectedThe satellite droplets and filament can be detected throughfarther image processing such as cropping filtering andthreshold recontrasting (shown as in Figure 9)

In image analysis system the droplet volume is calculatedby

Vol = sum120587 sdot (119889

2)

2

sdot 120575119911 (6)

which hypothesizes that liquid jet is axis-symmetric alongnozzle centerline And the droplet velocity is calculated basedon the position change of the moving-forward leading edgein two consecutive images with specified sampling intervalNote that the information is lost if the meniscus is hiddeninside the nozzle so edge detection only measures theprotruding meniscus

33 High-Frequency Driving Waveform as a System Excita-tion The high-frequency driving waveform is composed of

positive and negative periods including three pulse edges andtwo dwell times as shown in Figure 10 The first rising edgeexpands the channel to get the inkmeniscus into the channelThen the falling edge works to eject the droplet from thenozzle

This period of waveform design is based on the halfperiod of channel resonance Thus the waveform designissue is focused on calculating the optimum value of dwelltime 119879

1= 1198710119862 where 119871

0is the length of channel and 119862

is the sonic speed of the used ink The falling edge of thesecond negative voltage generates larger pressure in channelto accelerate the droplet with higher speed than conventionalsingle trapezoidal waveform The second rising edge whichsets the voltage back to 0 works to restrain the meniscusvibration [33]

Theoretically the suitable slops of the rising edge andfalling edge can restrain the meniscus vibration [34] How-ever in practical applications these two values are too largeto control accurately

Asmentioned above theoretical optimum value for dwelltime 119879

1is set as 52 us The corresponding dwell time 119879

2

Mathematical Problems in Engineering 7

(a) (b) (c) (d)

Figure 9 Image processing steps (a) original image (b) cropped image (c) threshold image and (d) edge image

Time

Voltage

Dwell

Dwell

time T1

time T2

Figure 10 Applied driving waveform of printhead

is recommended as 104 us (twice of 1198791) and the voltage

magnitude is set as 12 V according to the instruction ofKonica printhead In this paper the driving waveform withthese parameters is called the ldquostandardrdquo waveform [35]

4 Simulation Result

41 Coefficient Identification The coefficients of originalLEM model can be determined experimentally Howeverthe three improved models in this work are somewhatcomplicated and their coefficients reveal some importantfeatures of these equivalent circuits For this reason thissection is dedicated to identify the adjustment coefficients inthese corresponding circuits

In this experiment a group of experimental measure-ments with printhead driven by the standard high-frequencywaveform are chosen as the basic reference The coeffi-cients of each proposed improved LEM model should beadjusted to minimize the deviations between the simulationresults and the selected experimental measurements shownin Figure 11 The nonlinear least squares method [36] usingthe Levenberg-Margquardt and Gauss-Newton Algorithms

Table 3 Coefficients for three lumped element models

Model Coefficient ValueProportional 119879 0837

Multiple resonators 120572 51119873 228

Dynamic coefficients

1198601

133 lowast 103

1198611

353 lowast 103

1198602

234 lowast 104

1198612

411 lowast 104

is applied to find the optimal coefficients set for each pro-posed equivalent circuit which makes the simulation curvesoverlap the experimental measurements as far as possibleThe resulting coefficients are listed in Table 3

42 Droplet Volume Simulation In many applications of ink-jet printing the volume of droplet is a critical restrictivecondition Droplets with controllable volume affect not onlythe printing resolution but also the depositional thicknessTherefore how to accurately control droplet volume is a keyproblem to be solved by the proposed method

According to experimental measurements in Figure 11the comparison between the original LEM model and theproposed three improved models is presented in Figure 12In this figure the solid line is the outflow simulation curvesbased on the LEM technique and the star line is the actualmeasured volume values during the droplet formation pro-cess It should be noted that in Figures 12(a)ndash12(d) the firstthree points of actual measurement curve are default to zerobecause no fluid flowing outside the nozzle can be capturedby our image system

As the actual experimental measurements indicated inFigure 11 the droplet formation process is divided into threestages the meniscus protruding stage (ie 119879

1st = 15sim30 us)the filament breaking-off stage (ie 1198792nd = 30sim35 us) andthe droplet falling-down stage (ie 1198793rd = 35sim60 us) That

8 Mathematical Problems in Engineering

3us0pL0ms

6us0pL0ms

9us0pL0ms

12us 15us 18us 21us 24us

30us 33us 36us 39us 42us 45us 48us27us

678pL876ms

994pL730ms

1221 pL684ms

1401pL624ms

1521pL618ms

1533pL600ms

1567pL600ms

1509pL606ms

1492pL618ms

1489pL624 ms

1480 pL624 ms

1479pL618ms

1480pL618ms

Figure 11 Pictures of nanosilver droplet formation process driven by standard waveform

is ideally the outflow curve simulated by analytic modelsshould include overshoot phase which can clearly demon-strate that the part of filament is inhaled back into nozzle orsplits into satellite droplets by viscosity and surface tensionin the filament break-off stage However according to thesimulation result of the original LEM in Figure 12(a) there

is no obvious decreasing after reaching the maximum in theoutflow curve and the outflow curve of the original modelcannot well overlap the actual measurement curve

In Figures 12(b) and 12(d) the overshoot phase canbe observed clearly This performance improvement can bestraightforwardly explained by the fact that poles and zeros

Mathematical Problems in Engineering 9

0 20 40 60

0

10

20

Time (us)

Volu

me (

pL)

ExperimentOriginal

minus10

(a)

0 20 40 60

0

10

20

Time (us)

Volu

me (

pL)

ExperimentProportional

minus10

(b)

0 20 40 60

0

10

20

Time (us)

Volu

me (

pL)

ExperimentMultiple

minus10

minus20

(c)

0 20 40 60

0

10

20

Time (us)

Volu

me (

pL)

ExperimentDynamic

minus10

(d)

Figure 12 Outflow change ofmeasurements and simulations of (a) original model (b) proportional coefficientmodel (c)multiple resonatorsmodel and (d) dynamic coefficient model

of the transfer functions are manipulated to speed up thedynamic response speed of the proportional and dynamicmodels From Figure 12 both the multiple-resonator anddynamic models can obtain precise simulation results Thatis more well-designed coefficients are used to characterizethe nonlinear behaviors of piezoelectric printhead and theadjustable ranges are extended to a satisfied degree of bothmodels Additionally it should be noted that the outflowcurve of dynamic model has not only the overshoot phasebut also high overlap ratio to the experimental measurementMoreover the simulation curves reveal that the meniscusmoves inward before the droplet appears

43 Flow and Droplet Velocity Simulation The velocity ofdroplet is the other important aspect to evaluate the printingeffect Droplets with a high speed can remarkably improve

the printing efficiency and the quality of impact with sub-strate Therefore this experiment gives the velocity simula-tion using the proposed LEM models on the same actualexperimental measurement in Section 41

The comparisons between the flow velocity curves sim-ulated by the original LEM and three improved models areillustrated in Figure 13 The excitation is the standard drivingwaveform For the actual measured velocity curves the firsttwo points are also default to zero because no fluid flowingoutside the nozzle can be captured by our image system

It should be noted that all the values of actual measuredvelocity are positive However from the flow velocity curvesin Figure 13 the values of flow velocity in nozzle can bepositive or negative in different phases This positive-to-negative velocity variation indicates the changes of the flowdirection that leads to the break-off phenomenon Hence in

10 Mathematical Problems in Engineering

0 20 40 60

0

5

10

Time (us)

Velo

city

(ms

)

ExperimentFlow velocityOriginal

minus10

minus5

(a)

0 20 40 60

0

5

10

Time (us)

Velo

city

(ms

)

ExperimentFlow velocityProportional

minus10

minus5

(b)

0 20 40 60

0

5

10

Time (us)

Velo

city

(ms

)

ExperimentFlow velocityMultiple

minus10

minus5

(c)

0 20 40 60

0

5

10

Time (us)

Velo

city

(ms

)

ExperimentFlow velocityDynamic

minus10

minus5

(d)

Figure 13 Comparison of measured droplets velocity and the flowaverage velocity simulated by (a) original model (b) ldquoproportionalcoefficientrdquo model (c) ldquomultiple resonatorsrdquo model and (d) ldquodynamic coefficientsrdquo model

order to correctly simulate the velocity in leading edge wepropose a new calculation formula which combines both ofoutflow curve and flow velocity curve

119881average (119905) = radicsum1199052

119905=1199051

(12) 119876out (119905) sdot 119881 (119905)2

sum1199052

119905=1199051

(12)119876out (119905) (7)

where119876out is the outflow value and119881 is the flow velocity 1199051is

the moment when the sum of 119876out changes from negative topositive in outflow curve and 119905

2after 119905

1is the moment when

flow velocity changes to negative in nozzleFrom Figure 13 it can be obviously observed that in

the beginning of droplet formation process the multiple-resonator and dynamic models exhibit better performance insimulating the droplet velocity rather than the originalmodelThat is the average velocity curves obtained by these two

models are almost coincident to the actual measured velocitycurves

Based on the above analysis and comparisons thedynamicmodel based LEMhas themost accuracy in simulat-ing the droplet volume and velocity and can carry sufficientinformation to distinguish the break-off phenomenon occur-ing in the nanosilver droplet formation processs Thereforeamong these three improved models the dynamic lumpedelement model (DLEM) is the most suitable improved equiv-alent structure for simulating jetting characteristics

5 Prediction

For industry applications in printable electronics fabricationthe printhead must work at a certain status to meet manyrestrictive conditions Among them the most important

Mathematical Problems in Engineering 11

0 20 40 60

0

5

10

15

20

Time (us)

Volu

me (

pL)

minus25

minus20

minus15

minus10

minus5

T1 = 3usT1 = 4usT1 = 5us

T1 = 6usT1 = 7usT1 = 8us

(a)

0 20 40 604

5

6

7

8

9

10

Velo

city

(ms

)

Time (us)

T1 = 3usT1 = 4usT1 = 5us

T1 = 6usT1 = 7usT1 = 8us

(b)

Figure 14 Predictive droplet volume and velocity with different and various dwell times 1198792= 21198791 (a) simulation curves of outflow change

and (b) simulation curves of droplet average velocity

4 6 813

14

15

16

17

SimulationExperiment

Volu

me (

pL)

Time (us)

(a)

SimulationExperiment

4 6 8

45

5

55

6

Time (us)

Velo

city

(ms

)

(b)

Figure 15 Predictive droplet volume and velocity compared with measured volume and velocity

problem is choosing the appropriate combinations of thedriving waveform parameters for the used conductive inksHowever an exhausting manual selective process inevitablywastes a lot of time Therefore the computer-aided methodsare urgently needed for searching these appropriate combi-nations efficiently and robustly In this section the proposed

DLEM is applied to predict jetting characteristics for nanosil-ver ink

51 General Prediction Procedure The general procedure ofdroplet volumevelocity prediction based on DLEM is asfollows

12 Mathematical Problems in Engineering

0us0pL0ms

3us0pL0ms

6us0pL0ms

9us0pL0ms

12us6353pL736ms

15us9318pL612ms

18us11441pL572ms

18us 24us

30us14364pL498ms

33us14046pL510ms

36us13980pL520ms

39us13952pL525ms

42us13858pL525ms

45us13869pL520ms

48us13891pL524ms

51us13924pL522ms

54us

524ms13877pL

57us13840pL530ms

13127 pL548ms

14252pL520ms

27us14786pL504ms

Figure 16 A sequence of pictures of droplet falling from nozzle

Step 1 According to droplet generator parameters and prop-erties of fluid calculate most of the parameters in DLEM

Step 2 According to experimental measurements of dropletvolume driven by a standard waveform determine the rest ofthe adjustment coefficients and damping coefficients

Step 3 Based on the constructed DLEM predicate the valuesof droplet volume and velocity

Step 4 According to the need of applications and restrictiveconditions find out several groups of the appropriate param-eters of driving waveform frompredictive data lists of volumeand velocity

Step 5 After making a subtle adjustment in practice decidethe most appropriate parameters of driving waveform

52 Prediction Case 1 Standard Waveform with DifferentDwell Time Asmentioned above theDLEMmodel is chosento simulate jetting characteristics for the nanosilver inkHere an illustrative example is given to display the effect of

predicting the typical combinations of waveform parameterswith different dwell times (119879

1= 3sim8 us 119879

2= 2119879

1) and

the same amplitude (12V) The properties of nanosilver inkprovided by the LEED-PV Corporation have been listed inTable 1

In Figure 14 the predicted outflow and average velocitycurves with various dwell times are depicted We can observefrom Figure 14(a) that before the droplets rush out of thenozzle the depth of the inhaled meniscus is proportionalto dwell time From Figure 14(b) the time of the dropletappearing is postponed with the increase of dwell timeThat is both two simulation phenomena consist with thetheoretical analysis results in [37]

The comparisons between actual measured and predictedjetting characteristics with various dwell times are given inFigures 15(a) and 15(b) which both illustrate a parabolicchange rule with the increase of dwell time It should benoted that at the dwell time 119879

1= 56 us near the theoretical

optimum value 11987910158401= 52 us the droplet volume and velocity

simultaneously reach the maximum From the performanceof prediction accuracy listed in Table 4 the deviation of thepredicted jetting characteristics is within acceptable range

Mathematical Problems in Engineering 13

45

67

68

1012

1416

10121416

Volu

me (

pL)

91011121314151617

T1 (us)

T2(us)

(a)

45

67

68

1012

1416

3

4

5

6

Velo

city

(ms

)

3

35

4

45

5

55

6

T1 (us) T2

(us)

(b)

Figure 17 Predictive droplet volume and velocity with various dwell times1198791and119879

2 (a) predictive droplet volume and (b) predictive droplet

velocity

53 Prediction Case 2 Driving Waveform under RestrictiveCondition In this experiment an illustrative example underrestrictive conditions is presented to demonstrate the feasibil-ity and effectiveness of the proposed method for predictingjetting characteristics The dimensions of droplet generatorand properties of the nanosilver ink are listed in Table 1 Inorder to reach a high printing resolution and good quality ofdroplet impact with the substrate the restrictive conditionsinvolve that the droplet volume is smaller than 14 pL and thedroplet velocity is larger than 5ms

Based on the prediction procedure the predictive valuesof drop volumevelocity with various dwell times 119879

1and

1198792are listed in Table 5 According to the restrictive con-

ditions three combinations of parameters are found out1198791

= 5 us1198792

= 13 us 1198791

= 55 us1198792

= 14 us and1198791= 5 us119879

2= 75sim8 us After a small range search around

predictive values in the actual test the combination of 1198791=

49 us1198792

= 77 us is finally chosen The dynamic effect ofjetting characteristics driven by this combination is shown inFigure 16 From this figure the jetting characteristics satisfythe specified restrictive conditions quite well Meanwhilealmost no satellite droplet emerges after jetting the maindroplet

The 3D plot of predictive jetting characteristics withvarious dwell times 119879

1and 119879

2is shown in Figure 17 With

the increase of dwell time 1198791 peak positions of volume and

velocity have a tiny shift in 95sim115 us With the same dwelltime 119879

2 there is also a parabolic profile in predictive volume

and velocity

6 Conclusion

The lumped element modeling based on linear model ofpiezoelectric ceramic has been successfully used to simu-late piezoelectric droplet generator However the nonlinearmodel of piezoelectric ceramic is necessary to diminishdeviation between the simulated and measured results Inorder to characterize the nonlinear behavior of piezoelectric

Table 4 Performance of prediction accuracy

Deviation Value

Volume

Max 07135 (499)Min 0036 (013)Mean 02636Std 03316

Velocity

Max 02465 (502)Min 00143 (025)Mean 01095Std 01350

ceramic three models are introduced to simulate dropletgeneration In this work the droplet volumevelocity curvessimulated by dynamic model can carry sufficient informa-tion to distinguish nanosilver droplet formation processclearly Although the complexity of system increases theoutflow curves can obviously reflect the filament breaking-off stage that occurs which is coincident with sampledimages Such nonlinear LEM model is also successful inpredicting jetting characteristics under restrictive conditionsComparing the droplet volumevelocity curves with differentdwell time of high-frequency driving waveform the linearrelationship between dwell time and meniscus appearingmoment is discovered Meanwhile a parabolic change ruleof volumevelocity with the increase of dwell time is in goodagreement with present theoretical analysis

It should be noted that the proposed nonlinear model isfocused on relationship between the droplet volumevelocityand the dwell time of driving waveform while the voltageamplitude is also a decisive factor to droplet formationprocess This should be discussed in our future work

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

14 Mathematical Problems in Engineering

Table5Predictiv

edropletvolume(pL

)and

velocity(m

s)

1198791

1198792

665

775

885

995

10105

11115

12125

13135

14145

15155

4119881

11119

127

134

139

143

146

149

151

15147

142

Default

Default

Default

Default

Default

Default

Default

Default

119878371

395

417

439

459

478

491

500

489

471

451

435

Default

Default

Default

Default

Default

Default

Default

Default

45 119881

Default

117

125

134

143

15155

159

162

16157

153

147

139

Default

Default

Default

Default

Default

Default

119878Default

421

451

476

498

516

532

548

560

543

523

501

482

466

Default

Default

Default

Default

Default

Default

5119881

Default

Default

Default

133

143

151

157

163

167

168

166

163

157

15142

133

Default

Default

Default

Default

119878Default

Default

Default

480

507

531

553

572

587

600

583

563

540

525

509

498

Default

Default

Default

Default

55 119881

Default

Default

Default

Default

Default

151

159

165

168

169

171

169

165

16154

146

138

129

Default

Default

119878Default

Default

Default

Default

Default

524

550

572

593

609

624

601

571

551

537

523

509

496

Default

Default

6119881

Default

Default

Default

Default

Default

Default

Default

158

163

164

165

165

165

158

1495

141

130

119

108

97119878

Default

Default

Default

Default

Default

Default

Default

495

515

533

549

564

544

519

494

474

458

445

432

421

65 119881

Default

Default

Default

Default

Default

Default

Default

Default

Default

155

159

16155

150

143

134

125

114

103

95119878

Default

Default

Default

Default

Default

Default

Default

Default

Default

473

485

495

503

490

469

446

420

400

390

383

7119881

Default

Default

Default

Default

Default

Default

Default

Default

Default

Default

125

135

139

141

135

127

119

110

102

94119878

Default

Default

Default

Default

Default

Default

Default

Default

Default

Default

445

465

479

463

444

422

401

385

370

363

Defaults

areb

ecause

thec

ombinatio

nsof

parametersindrivingwaveform

areo

utof

theo

peratio

nrange

Mathematical Problems in Engineering 15

Authorsrsquo Contribution

Maowei He and Liling Sun contributed equally to this work

Acknowledgment

This research is partially supported by National Natural Sci-ence Foundation of China under Grants 61105067 7100107261174164 and 71271140

References

[1] M SinghHMHaverinen PDhagat andG E Jabbour ldquoInkjetprinting-process and its applicationsrdquo Advanced Materials vol22 no 6 pp 673ndash685 2010

[2] E Macdonald R Salas D Espalin et al ldquo3D printing for therapid prototyping of structural electronicsrdquo IEEE Access vol 2pp 234ndash242 2014

[3] S Umezu T Kitajima H Murase H Ohmori K Katahiraand Y Ito ldquoFabrication of living cell structure utilizing electro-static inkjet phenomenardquo in Proceedings of the 22nd IEEEInternational Conference on Micro Electro Mechanical Systems(MEMS rsquo09) pp 419ndash422 IEEE Sorrento Italy January 2009

[4] P Cosseddu L Basirico A Loi et al ldquoInkjet printed OrganicThin Film Transistors based tactile transducers for artificialrobotic skinrdquo in Proceedings of the 4th IEEE RAS and EMBSInternational Conference on Biomedical Robotics and Biomecha-tronics (BioRob rsquo12) pp 1907ndash1912 June 2012

[5] J Hwang A Wan and A Kahn ldquoEnergetics of metal-organicinterfaces new experiments and assessment of the fieldrdquoMate-rials Science and Engineering R Reports vol 64 no 1-2 pp 1ndash312009

[6] B J Kang and J H Oh ldquoGeometrical characterization of inkjet-printed conductive lines of nanosilver suspensions on a polymersubstraterdquoThin Solid Films vol 518 no 10 pp 2890ndash2896 2010

[7] F Villani P Vacca G Nenna et al ldquoInkjet printed polymerlayer on flexible substrate for OLED applicationsrdquo The Journalof Physical Chemistry C vol 113 no 30 pp 13398ndash13402 2009

[8] J-C Liou and F-G Tseng ldquoMulti-dimensional data registrationCMOSMEMS integrated inkjet printheadrdquo MicroelectronicEngineering vol 88 no 6 pp 888ndash901 2011

[9] J Miettinen K Kaija M Mantysalo et al ldquoMolded substratesfor inkjet printed modulesrdquo IEEE Transactions on Componentsand Packaging Technologies vol 32 no 2 pp 293ndash301 2009

[10] K Silverbrook ldquoPrinthead with multiple actuators in eachchamberrdquo US7708387 B2 2010

[11] V Piefort Finite element modelling of piezoelectric active struc-tures [PhD thesis] Department ofMechanical Engineering andRobotics Universite Libre de Bruxelles Brussels Belgium 2001

[12] L Sang Y Hong and F Wang ldquoInvestigation of viscosity effecton droplet formation in T-shaped microchannels by numericaland analytical methodsrdquo Microfluidics and Nanofluidics vol 6no 5 pp 621ndash635 2009

[13] J Liu S-H Tan Y F Yap M Y Ng and N-T NguyenldquoNumerical and experimental investigations of the formationprocess of ferrofluid dropletsrdquo Microfluidics and Nanofluidicsvol 11 no 2 pp 177ndash187 2011

[14] F Sarrazin K Loubiere L Prat C Gourdon T Bonometti andJ Magnaudet ldquoExperimental and numerical study of dropletshydrodynamics in MicroChannelrdquo AIChE Journal vol 52 no12 pp 4061ndash4070 2006

[15] X Q Xing D L Butler S H Ng Z Wang S Danyluk and CYang ldquoSimulation of droplet formation and coalescence usinglattice Boltzmann-based single-phasemodelrdquo Journal of Colloidand Interface Science vol 311 no 2 pp 609ndash618 2007

[16] Q Gallas R Holman T Nishida B Carroll M Sheplak and LCattafesta ldquoLumped element modeling of piezoelectric-drivensynthetic jet actuatorsrdquoAIAA Journal vol 41 no 2 pp 240ndash2472003

[17] A A Khalate X Bombois R Babuska H Wijshoff and RWaarsing ldquoOptimization-based feedforward control for a Drop-on-Demand inkjet printheadrdquo in Proceedings of the AmericanControl Conference (ACC rsquo10) pp 2182ndash2187 Baltimore MdUSA July 2010

[18] L Meirovitch Fundamentals of Vibrations McGraw-Hill NewYork NY USA 2001

[19] H Seitz and J Heinz ldquoModelling of a microfluidic device withpiezoelectric actuatorsrdquo Journal of Micromechanics and Micro-engineering vol 14 no 8 pp 1140ndash1147 2004

[20] S Kim J Sung andMH Lee ldquoPressure wave and fluid velocityin a bend-mode inkjet nozzle with double PZT actuatorsrdquoJournal of Thermal Science vol 22 no 1 pp 29ndash35 2013

[21] G Wassink Inkjet printhead performance enhancement by feed-forward input design based on two-port modeling [PhD thesis]Delft University of Technology 2007

[22] S Prasad SHorowitz QGallas B Sankar L Cattafesta andMSheplak ldquoTwo-port electroacoustic model of an axisymmet-ric piezoelectric composite platerdquo in Proceedings of the 43rdAIAAASMEASCEAHS Structures Structural DynamicsandMaterials Conference AIAA Paper 2002-1365 AIAA DenverColo USA April 2002

[23] D T BlackstockFundamentals of Physical Acoustics JohnWileyamp Sons New York NY USA 2000

[24] F M White Fluid Mechanics McGraw-Hill New York NYUSA 1979

[25] H Schempt andD R Yoerger ldquoStudy of dominant performancecharacteristics in robot transmissionsrdquo Journal of MechanicalDesign Transactions Of the ASME vol 115 no 3 pp 472ndash4821993

[26] E A Witmer ldquoElementary Bernoulli-Euler beam theoryrdquoMITUnified Engineering Course Notes vol 5 pp 114ndash164 1991-1992

[27] Y Yang and L Tang ldquoEquivalent circuit modeling of piezoelec-tric energy harvestersrdquo Journal of Intelligent Material Systemsand Structures vol 20 no 18 pp 2223ndash2235 2009

[28] F Constantinescu A G Gheorghe and M Nitescu ldquoNewcircuit models of power BAW resonatorsrdquo in Proceedings of theEuropean Microwave Integrated Circuit Conference October2007

[29] A Albareda and R PerezNon-Linear Behaviour of PiezoelectricCeramics vol 140ofSpringer Series inMaterials Science Springer2011

[30] J Nosek ldquoDrive level dependence of the resonant frequency inBAW quartz resonators and his modelingrdquo IEEE TransactionsonUltrasonics Ferroelectrics and Frequency Control vol 46 no4 pp 823ndash829 1999

[31] R Aigner N-H Huynh M Handtmann and S MarksteinerldquoBehavior of BAW devices at high power levelsrdquo in Proceedingsof the IEEE MTT-S International Microwave Symposium pp429ndash432 Long Beach Calif USA June 2005

[32] R S Ketcham G R Kline and K M Lakin ldquoPerformanceof TFR filters under elevated power conditionsrdquo in Proceedingsof 42nd the Annual Frequency Control Symposium pp 106ndash111Baltimore Md USA June 1988

16 Mathematical Problems in Engineering

[33] K-S Kwon ldquoWaveform design methods for piezo inkjet dis-pensers based on measured meniscus motionrdquo Journal of Micro-electromechanical Systems vol 18 no 5 pp 1118ndash1125 2009

[34] H-C Wu and H-J Lin ldquoEffects of actuating pressure wave-forms on the droplet behavior in a piezoelectric inkjetrdquoMateri-als Transactions vol 51 no 12 pp 2269ndash2276 2010

[35] KMinolta InkjetHeadApplicationNote-KM1024 Series KonicaMinolta Ij Technologies

[36] G Golub andV Pereyra ldquoSeparable nonlinear least squares thevariable projection method and its applicationsrdquo Inverse Prob-lems vol 19 no 2 pp R1ndashR26 2003

[37] H Wijshoff ldquoThe dynamics of the piezo inkjet printheadoperationrdquo Physics Reports vol 491 no 4-5 pp 77ndash177 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

(a) (b) (c) (d)

Figure 9 Image processing steps (a) original image (b) cropped image (c) threshold image and (d) edge image

Time

Voltage

Dwell

Dwell

time T1

time T2

Figure 10 Applied driving waveform of printhead

is recommended as 104 us (twice of 1198791) and the voltage

magnitude is set as 12 V according to the instruction ofKonica printhead In this paper the driving waveform withthese parameters is called the ldquostandardrdquo waveform [35]

4 Simulation Result

41 Coefficient Identification The coefficients of originalLEM model can be determined experimentally Howeverthe three improved models in this work are somewhatcomplicated and their coefficients reveal some importantfeatures of these equivalent circuits For this reason thissection is dedicated to identify the adjustment coefficients inthese corresponding circuits

In this experiment a group of experimental measure-ments with printhead driven by the standard high-frequencywaveform are chosen as the basic reference The coeffi-cients of each proposed improved LEM model should beadjusted to minimize the deviations between the simulationresults and the selected experimental measurements shownin Figure 11 The nonlinear least squares method [36] usingthe Levenberg-Margquardt and Gauss-Newton Algorithms

Table 3 Coefficients for three lumped element models

Model Coefficient ValueProportional 119879 0837

Multiple resonators 120572 51119873 228

Dynamic coefficients

1198601

133 lowast 103

1198611

353 lowast 103

1198602

234 lowast 104

1198612

411 lowast 104

is applied to find the optimal coefficients set for each pro-posed equivalent circuit which makes the simulation curvesoverlap the experimental measurements as far as possibleThe resulting coefficients are listed in Table 3

42 Droplet Volume Simulation In many applications of ink-jet printing the volume of droplet is a critical restrictivecondition Droplets with controllable volume affect not onlythe printing resolution but also the depositional thicknessTherefore how to accurately control droplet volume is a keyproblem to be solved by the proposed method

According to experimental measurements in Figure 11the comparison between the original LEM model and theproposed three improved models is presented in Figure 12In this figure the solid line is the outflow simulation curvesbased on the LEM technique and the star line is the actualmeasured volume values during the droplet formation pro-cess It should be noted that in Figures 12(a)ndash12(d) the firstthree points of actual measurement curve are default to zerobecause no fluid flowing outside the nozzle can be capturedby our image system

As the actual experimental measurements indicated inFigure 11 the droplet formation process is divided into threestages the meniscus protruding stage (ie 119879

1st = 15sim30 us)the filament breaking-off stage (ie 1198792nd = 30sim35 us) andthe droplet falling-down stage (ie 1198793rd = 35sim60 us) That

8 Mathematical Problems in Engineering

3us0pL0ms

6us0pL0ms

9us0pL0ms

12us 15us 18us 21us 24us

30us 33us 36us 39us 42us 45us 48us27us

678pL876ms

994pL730ms

1221 pL684ms

1401pL624ms

1521pL618ms

1533pL600ms

1567pL600ms

1509pL606ms

1492pL618ms

1489pL624 ms

1480 pL624 ms

1479pL618ms

1480pL618ms

Figure 11 Pictures of nanosilver droplet formation process driven by standard waveform

is ideally the outflow curve simulated by analytic modelsshould include overshoot phase which can clearly demon-strate that the part of filament is inhaled back into nozzle orsplits into satellite droplets by viscosity and surface tensionin the filament break-off stage However according to thesimulation result of the original LEM in Figure 12(a) there

is no obvious decreasing after reaching the maximum in theoutflow curve and the outflow curve of the original modelcannot well overlap the actual measurement curve

In Figures 12(b) and 12(d) the overshoot phase canbe observed clearly This performance improvement can bestraightforwardly explained by the fact that poles and zeros

Mathematical Problems in Engineering 9

0 20 40 60

0

10

20

Time (us)

Volu

me (

pL)

ExperimentOriginal

minus10

(a)

0 20 40 60

0

10

20

Time (us)

Volu

me (

pL)

ExperimentProportional

minus10

(b)

0 20 40 60

0

10

20

Time (us)

Volu

me (

pL)

ExperimentMultiple

minus10

minus20

(c)

0 20 40 60

0

10

20

Time (us)

Volu

me (

pL)

ExperimentDynamic

minus10

(d)

Figure 12 Outflow change ofmeasurements and simulations of (a) original model (b) proportional coefficientmodel (c)multiple resonatorsmodel and (d) dynamic coefficient model

of the transfer functions are manipulated to speed up thedynamic response speed of the proportional and dynamicmodels From Figure 12 both the multiple-resonator anddynamic models can obtain precise simulation results Thatis more well-designed coefficients are used to characterizethe nonlinear behaviors of piezoelectric printhead and theadjustable ranges are extended to a satisfied degree of bothmodels Additionally it should be noted that the outflowcurve of dynamic model has not only the overshoot phasebut also high overlap ratio to the experimental measurementMoreover the simulation curves reveal that the meniscusmoves inward before the droplet appears

43 Flow and Droplet Velocity Simulation The velocity ofdroplet is the other important aspect to evaluate the printingeffect Droplets with a high speed can remarkably improve

the printing efficiency and the quality of impact with sub-strate Therefore this experiment gives the velocity simula-tion using the proposed LEM models on the same actualexperimental measurement in Section 41

The comparisons between the flow velocity curves sim-ulated by the original LEM and three improved models areillustrated in Figure 13 The excitation is the standard drivingwaveform For the actual measured velocity curves the firsttwo points are also default to zero because no fluid flowingoutside the nozzle can be captured by our image system

It should be noted that all the values of actual measuredvelocity are positive However from the flow velocity curvesin Figure 13 the values of flow velocity in nozzle can bepositive or negative in different phases This positive-to-negative velocity variation indicates the changes of the flowdirection that leads to the break-off phenomenon Hence in

10 Mathematical Problems in Engineering

0 20 40 60

0

5

10

Time (us)

Velo

city

(ms

)

ExperimentFlow velocityOriginal

minus10

minus5

(a)

0 20 40 60

0

5

10

Time (us)

Velo

city

(ms

)

ExperimentFlow velocityProportional

minus10

minus5

(b)

0 20 40 60

0

5

10

Time (us)

Velo

city

(ms

)

ExperimentFlow velocityMultiple

minus10

minus5

(c)

0 20 40 60

0

5

10

Time (us)

Velo

city

(ms

)

ExperimentFlow velocityDynamic

minus10

minus5

(d)

Figure 13 Comparison of measured droplets velocity and the flowaverage velocity simulated by (a) original model (b) ldquoproportionalcoefficientrdquo model (c) ldquomultiple resonatorsrdquo model and (d) ldquodynamic coefficientsrdquo model

order to correctly simulate the velocity in leading edge wepropose a new calculation formula which combines both ofoutflow curve and flow velocity curve

119881average (119905) = radicsum1199052

119905=1199051

(12) 119876out (119905) sdot 119881 (119905)2

sum1199052

119905=1199051

(12)119876out (119905) (7)

where119876out is the outflow value and119881 is the flow velocity 1199051is

the moment when the sum of 119876out changes from negative topositive in outflow curve and 119905

2after 119905

1is the moment when

flow velocity changes to negative in nozzleFrom Figure 13 it can be obviously observed that in

the beginning of droplet formation process the multiple-resonator and dynamic models exhibit better performance insimulating the droplet velocity rather than the originalmodelThat is the average velocity curves obtained by these two

models are almost coincident to the actual measured velocitycurves

Based on the above analysis and comparisons thedynamicmodel based LEMhas themost accuracy in simulat-ing the droplet volume and velocity and can carry sufficientinformation to distinguish the break-off phenomenon occur-ing in the nanosilver droplet formation processs Thereforeamong these three improved models the dynamic lumpedelement model (DLEM) is the most suitable improved equiv-alent structure for simulating jetting characteristics

5 Prediction

For industry applications in printable electronics fabricationthe printhead must work at a certain status to meet manyrestrictive conditions Among them the most important

Mathematical Problems in Engineering 11

0 20 40 60

0

5

10

15

20

Time (us)

Volu

me (

pL)

minus25

minus20

minus15

minus10

minus5

T1 = 3usT1 = 4usT1 = 5us

T1 = 6usT1 = 7usT1 = 8us

(a)

0 20 40 604

5

6

7

8

9

10

Velo

city

(ms

)

Time (us)

T1 = 3usT1 = 4usT1 = 5us

T1 = 6usT1 = 7usT1 = 8us

(b)

Figure 14 Predictive droplet volume and velocity with different and various dwell times 1198792= 21198791 (a) simulation curves of outflow change

and (b) simulation curves of droplet average velocity

4 6 813

14

15

16

17

SimulationExperiment

Volu

me (

pL)

Time (us)

(a)

SimulationExperiment

4 6 8

45

5

55

6

Time (us)

Velo

city

(ms

)

(b)

Figure 15 Predictive droplet volume and velocity compared with measured volume and velocity

problem is choosing the appropriate combinations of thedriving waveform parameters for the used conductive inksHowever an exhausting manual selective process inevitablywastes a lot of time Therefore the computer-aided methodsare urgently needed for searching these appropriate combi-nations efficiently and robustly In this section the proposed

DLEM is applied to predict jetting characteristics for nanosil-ver ink

51 General Prediction Procedure The general procedure ofdroplet volumevelocity prediction based on DLEM is asfollows

12 Mathematical Problems in Engineering

0us0pL0ms

3us0pL0ms

6us0pL0ms

9us0pL0ms

12us6353pL736ms

15us9318pL612ms

18us11441pL572ms

18us 24us

30us14364pL498ms

33us14046pL510ms

36us13980pL520ms

39us13952pL525ms

42us13858pL525ms

45us13869pL520ms

48us13891pL524ms

51us13924pL522ms

54us

524ms13877pL

57us13840pL530ms

13127 pL548ms

14252pL520ms

27us14786pL504ms

Figure 16 A sequence of pictures of droplet falling from nozzle

Step 1 According to droplet generator parameters and prop-erties of fluid calculate most of the parameters in DLEM

Step 2 According to experimental measurements of dropletvolume driven by a standard waveform determine the rest ofthe adjustment coefficients and damping coefficients

Step 3 Based on the constructed DLEM predicate the valuesof droplet volume and velocity

Step 4 According to the need of applications and restrictiveconditions find out several groups of the appropriate param-eters of driving waveform frompredictive data lists of volumeand velocity

Step 5 After making a subtle adjustment in practice decidethe most appropriate parameters of driving waveform

52 Prediction Case 1 Standard Waveform with DifferentDwell Time Asmentioned above theDLEMmodel is chosento simulate jetting characteristics for the nanosilver inkHere an illustrative example is given to display the effect of

predicting the typical combinations of waveform parameterswith different dwell times (119879

1= 3sim8 us 119879

2= 2119879

1) and

the same amplitude (12V) The properties of nanosilver inkprovided by the LEED-PV Corporation have been listed inTable 1

In Figure 14 the predicted outflow and average velocitycurves with various dwell times are depicted We can observefrom Figure 14(a) that before the droplets rush out of thenozzle the depth of the inhaled meniscus is proportionalto dwell time From Figure 14(b) the time of the dropletappearing is postponed with the increase of dwell timeThat is both two simulation phenomena consist with thetheoretical analysis results in [37]

The comparisons between actual measured and predictedjetting characteristics with various dwell times are given inFigures 15(a) and 15(b) which both illustrate a parabolicchange rule with the increase of dwell time It should benoted that at the dwell time 119879

1= 56 us near the theoretical

optimum value 11987910158401= 52 us the droplet volume and velocity

simultaneously reach the maximum From the performanceof prediction accuracy listed in Table 4 the deviation of thepredicted jetting characteristics is within acceptable range

Mathematical Problems in Engineering 13

45

67

68

1012

1416

10121416

Volu

me (

pL)

91011121314151617

T1 (us)

T2(us)

(a)

45

67

68

1012

1416

3

4

5

6

Velo

city

(ms

)

3

35

4

45

5

55

6

T1 (us) T2

(us)

(b)

Figure 17 Predictive droplet volume and velocity with various dwell times1198791and119879

2 (a) predictive droplet volume and (b) predictive droplet

velocity

53 Prediction Case 2 Driving Waveform under RestrictiveCondition In this experiment an illustrative example underrestrictive conditions is presented to demonstrate the feasibil-ity and effectiveness of the proposed method for predictingjetting characteristics The dimensions of droplet generatorand properties of the nanosilver ink are listed in Table 1 Inorder to reach a high printing resolution and good quality ofdroplet impact with the substrate the restrictive conditionsinvolve that the droplet volume is smaller than 14 pL and thedroplet velocity is larger than 5ms

Based on the prediction procedure the predictive valuesof drop volumevelocity with various dwell times 119879

1and

1198792are listed in Table 5 According to the restrictive con-

ditions three combinations of parameters are found out1198791

= 5 us1198792

= 13 us 1198791

= 55 us1198792

= 14 us and1198791= 5 us119879

2= 75sim8 us After a small range search around

predictive values in the actual test the combination of 1198791=

49 us1198792

= 77 us is finally chosen The dynamic effect ofjetting characteristics driven by this combination is shown inFigure 16 From this figure the jetting characteristics satisfythe specified restrictive conditions quite well Meanwhilealmost no satellite droplet emerges after jetting the maindroplet

The 3D plot of predictive jetting characteristics withvarious dwell times 119879

1and 119879

2is shown in Figure 17 With

the increase of dwell time 1198791 peak positions of volume and

velocity have a tiny shift in 95sim115 us With the same dwelltime 119879

2 there is also a parabolic profile in predictive volume

and velocity

6 Conclusion

The lumped element modeling based on linear model ofpiezoelectric ceramic has been successfully used to simu-late piezoelectric droplet generator However the nonlinearmodel of piezoelectric ceramic is necessary to diminishdeviation between the simulated and measured results Inorder to characterize the nonlinear behavior of piezoelectric

Table 4 Performance of prediction accuracy

Deviation Value

Volume

Max 07135 (499)Min 0036 (013)Mean 02636Std 03316

Velocity

Max 02465 (502)Min 00143 (025)Mean 01095Std 01350

ceramic three models are introduced to simulate dropletgeneration In this work the droplet volumevelocity curvessimulated by dynamic model can carry sufficient informa-tion to distinguish nanosilver droplet formation processclearly Although the complexity of system increases theoutflow curves can obviously reflect the filament breaking-off stage that occurs which is coincident with sampledimages Such nonlinear LEM model is also successful inpredicting jetting characteristics under restrictive conditionsComparing the droplet volumevelocity curves with differentdwell time of high-frequency driving waveform the linearrelationship between dwell time and meniscus appearingmoment is discovered Meanwhile a parabolic change ruleof volumevelocity with the increase of dwell time is in goodagreement with present theoretical analysis

It should be noted that the proposed nonlinear model isfocused on relationship between the droplet volumevelocityand the dwell time of driving waveform while the voltageamplitude is also a decisive factor to droplet formationprocess This should be discussed in our future work

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

14 Mathematical Problems in Engineering

Table5Predictiv

edropletvolume(pL

)and

velocity(m

s)

1198791

1198792

665

775

885

995

10105

11115

12125

13135

14145

15155

4119881

11119

127

134

139

143

146

149

151

15147

142

Default

Default

Default

Default

Default

Default

Default

Default

119878371

395

417

439

459

478

491

500

489

471

451

435

Default

Default

Default

Default

Default

Default

Default

Default

45 119881

Default

117

125

134

143

15155

159

162

16157

153

147

139

Default

Default

Default

Default

Default

Default

119878Default

421

451

476

498

516

532

548

560

543

523

501

482

466

Default

Default

Default

Default

Default

Default

5119881

Default

Default

Default

133

143

151

157

163

167

168

166

163

157

15142

133

Default

Default

Default

Default

119878Default

Default

Default

480

507

531

553

572

587

600

583

563

540

525

509

498

Default

Default

Default

Default

55 119881

Default

Default

Default

Default

Default

151

159

165

168

169

171

169

165

16154

146

138

129

Default

Default

119878Default

Default

Default

Default

Default

524

550

572

593

609

624

601

571

551

537

523

509

496

Default

Default

6119881

Default

Default

Default

Default

Default

Default

Default

158

163

164

165

165

165

158

1495

141

130

119

108

97119878

Default

Default

Default

Default

Default

Default

Default

495

515

533

549

564

544

519

494

474

458

445

432

421

65 119881

Default

Default

Default

Default

Default

Default

Default

Default

Default

155

159

16155

150

143

134

125

114

103

95119878

Default

Default

Default

Default

Default

Default

Default

Default

Default

473

485

495

503

490

469

446

420

400

390

383

7119881

Default

Default

Default

Default

Default

Default

Default

Default

Default

Default

125

135

139

141

135

127

119

110

102

94119878

Default

Default

Default

Default

Default

Default

Default

Default

Default

Default

445

465

479

463

444

422

401

385

370

363

Defaults

areb

ecause

thec

ombinatio

nsof

parametersindrivingwaveform

areo

utof

theo

peratio

nrange

Mathematical Problems in Engineering 15

Authorsrsquo Contribution

Maowei He and Liling Sun contributed equally to this work

Acknowledgment

This research is partially supported by National Natural Sci-ence Foundation of China under Grants 61105067 7100107261174164 and 71271140

References

[1] M SinghHMHaverinen PDhagat andG E Jabbour ldquoInkjetprinting-process and its applicationsrdquo Advanced Materials vol22 no 6 pp 673ndash685 2010

[2] E Macdonald R Salas D Espalin et al ldquo3D printing for therapid prototyping of structural electronicsrdquo IEEE Access vol 2pp 234ndash242 2014

[3] S Umezu T Kitajima H Murase H Ohmori K Katahiraand Y Ito ldquoFabrication of living cell structure utilizing electro-static inkjet phenomenardquo in Proceedings of the 22nd IEEEInternational Conference on Micro Electro Mechanical Systems(MEMS rsquo09) pp 419ndash422 IEEE Sorrento Italy January 2009

[4] P Cosseddu L Basirico A Loi et al ldquoInkjet printed OrganicThin Film Transistors based tactile transducers for artificialrobotic skinrdquo in Proceedings of the 4th IEEE RAS and EMBSInternational Conference on Biomedical Robotics and Biomecha-tronics (BioRob rsquo12) pp 1907ndash1912 June 2012

[5] J Hwang A Wan and A Kahn ldquoEnergetics of metal-organicinterfaces new experiments and assessment of the fieldrdquoMate-rials Science and Engineering R Reports vol 64 no 1-2 pp 1ndash312009

[6] B J Kang and J H Oh ldquoGeometrical characterization of inkjet-printed conductive lines of nanosilver suspensions on a polymersubstraterdquoThin Solid Films vol 518 no 10 pp 2890ndash2896 2010

[7] F Villani P Vacca G Nenna et al ldquoInkjet printed polymerlayer on flexible substrate for OLED applicationsrdquo The Journalof Physical Chemistry C vol 113 no 30 pp 13398ndash13402 2009

[8] J-C Liou and F-G Tseng ldquoMulti-dimensional data registrationCMOSMEMS integrated inkjet printheadrdquo MicroelectronicEngineering vol 88 no 6 pp 888ndash901 2011

[9] J Miettinen K Kaija M Mantysalo et al ldquoMolded substratesfor inkjet printed modulesrdquo IEEE Transactions on Componentsand Packaging Technologies vol 32 no 2 pp 293ndash301 2009

[10] K Silverbrook ldquoPrinthead with multiple actuators in eachchamberrdquo US7708387 B2 2010

[11] V Piefort Finite element modelling of piezoelectric active struc-tures [PhD thesis] Department ofMechanical Engineering andRobotics Universite Libre de Bruxelles Brussels Belgium 2001

[12] L Sang Y Hong and F Wang ldquoInvestigation of viscosity effecton droplet formation in T-shaped microchannels by numericaland analytical methodsrdquo Microfluidics and Nanofluidics vol 6no 5 pp 621ndash635 2009

[13] J Liu S-H Tan Y F Yap M Y Ng and N-T NguyenldquoNumerical and experimental investigations of the formationprocess of ferrofluid dropletsrdquo Microfluidics and Nanofluidicsvol 11 no 2 pp 177ndash187 2011

[14] F Sarrazin K Loubiere L Prat C Gourdon T Bonometti andJ Magnaudet ldquoExperimental and numerical study of dropletshydrodynamics in MicroChannelrdquo AIChE Journal vol 52 no12 pp 4061ndash4070 2006

[15] X Q Xing D L Butler S H Ng Z Wang S Danyluk and CYang ldquoSimulation of droplet formation and coalescence usinglattice Boltzmann-based single-phasemodelrdquo Journal of Colloidand Interface Science vol 311 no 2 pp 609ndash618 2007

[16] Q Gallas R Holman T Nishida B Carroll M Sheplak and LCattafesta ldquoLumped element modeling of piezoelectric-drivensynthetic jet actuatorsrdquoAIAA Journal vol 41 no 2 pp 240ndash2472003

[17] A A Khalate X Bombois R Babuska H Wijshoff and RWaarsing ldquoOptimization-based feedforward control for a Drop-on-Demand inkjet printheadrdquo in Proceedings of the AmericanControl Conference (ACC rsquo10) pp 2182ndash2187 Baltimore MdUSA July 2010

[18] L Meirovitch Fundamentals of Vibrations McGraw-Hill NewYork NY USA 2001

[19] H Seitz and J Heinz ldquoModelling of a microfluidic device withpiezoelectric actuatorsrdquo Journal of Micromechanics and Micro-engineering vol 14 no 8 pp 1140ndash1147 2004

[20] S Kim J Sung andMH Lee ldquoPressure wave and fluid velocityin a bend-mode inkjet nozzle with double PZT actuatorsrdquoJournal of Thermal Science vol 22 no 1 pp 29ndash35 2013

[21] G Wassink Inkjet printhead performance enhancement by feed-forward input design based on two-port modeling [PhD thesis]Delft University of Technology 2007

[22] S Prasad SHorowitz QGallas B Sankar L Cattafesta andMSheplak ldquoTwo-port electroacoustic model of an axisymmet-ric piezoelectric composite platerdquo in Proceedings of the 43rdAIAAASMEASCEAHS Structures Structural DynamicsandMaterials Conference AIAA Paper 2002-1365 AIAA DenverColo USA April 2002

[23] D T BlackstockFundamentals of Physical Acoustics JohnWileyamp Sons New York NY USA 2000

[24] F M White Fluid Mechanics McGraw-Hill New York NYUSA 1979

[25] H Schempt andD R Yoerger ldquoStudy of dominant performancecharacteristics in robot transmissionsrdquo Journal of MechanicalDesign Transactions Of the ASME vol 115 no 3 pp 472ndash4821993

[26] E A Witmer ldquoElementary Bernoulli-Euler beam theoryrdquoMITUnified Engineering Course Notes vol 5 pp 114ndash164 1991-1992

[27] Y Yang and L Tang ldquoEquivalent circuit modeling of piezoelec-tric energy harvestersrdquo Journal of Intelligent Material Systemsand Structures vol 20 no 18 pp 2223ndash2235 2009

[28] F Constantinescu A G Gheorghe and M Nitescu ldquoNewcircuit models of power BAW resonatorsrdquo in Proceedings of theEuropean Microwave Integrated Circuit Conference October2007

[29] A Albareda and R PerezNon-Linear Behaviour of PiezoelectricCeramics vol 140ofSpringer Series inMaterials Science Springer2011

[30] J Nosek ldquoDrive level dependence of the resonant frequency inBAW quartz resonators and his modelingrdquo IEEE TransactionsonUltrasonics Ferroelectrics and Frequency Control vol 46 no4 pp 823ndash829 1999

[31] R Aigner N-H Huynh M Handtmann and S MarksteinerldquoBehavior of BAW devices at high power levelsrdquo in Proceedingsof the IEEE MTT-S International Microwave Symposium pp429ndash432 Long Beach Calif USA June 2005

[32] R S Ketcham G R Kline and K M Lakin ldquoPerformanceof TFR filters under elevated power conditionsrdquo in Proceedingsof 42nd the Annual Frequency Control Symposium pp 106ndash111Baltimore Md USA June 1988

16 Mathematical Problems in Engineering

[33] K-S Kwon ldquoWaveform design methods for piezo inkjet dis-pensers based on measured meniscus motionrdquo Journal of Micro-electromechanical Systems vol 18 no 5 pp 1118ndash1125 2009

[34] H-C Wu and H-J Lin ldquoEffects of actuating pressure wave-forms on the droplet behavior in a piezoelectric inkjetrdquoMateri-als Transactions vol 51 no 12 pp 2269ndash2276 2010

[35] KMinolta InkjetHeadApplicationNote-KM1024 Series KonicaMinolta Ij Technologies

[36] G Golub andV Pereyra ldquoSeparable nonlinear least squares thevariable projection method and its applicationsrdquo Inverse Prob-lems vol 19 no 2 pp R1ndashR26 2003

[37] H Wijshoff ldquoThe dynamics of the piezo inkjet printheadoperationrdquo Physics Reports vol 491 no 4-5 pp 77ndash177 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

3us0pL0ms

6us0pL0ms

9us0pL0ms

12us 15us 18us 21us 24us

30us 33us 36us 39us 42us 45us 48us27us

678pL876ms

994pL730ms

1221 pL684ms

1401pL624ms

1521pL618ms

1533pL600ms

1567pL600ms

1509pL606ms

1492pL618ms

1489pL624 ms

1480 pL624 ms

1479pL618ms

1480pL618ms

Figure 11 Pictures of nanosilver droplet formation process driven by standard waveform

is ideally the outflow curve simulated by analytic modelsshould include overshoot phase which can clearly demon-strate that the part of filament is inhaled back into nozzle orsplits into satellite droplets by viscosity and surface tensionin the filament break-off stage However according to thesimulation result of the original LEM in Figure 12(a) there

is no obvious decreasing after reaching the maximum in theoutflow curve and the outflow curve of the original modelcannot well overlap the actual measurement curve

In Figures 12(b) and 12(d) the overshoot phase canbe observed clearly This performance improvement can bestraightforwardly explained by the fact that poles and zeros

Mathematical Problems in Engineering 9

0 20 40 60

0

10

20

Time (us)

Volu

me (

pL)

ExperimentOriginal

minus10

(a)

0 20 40 60

0

10

20

Time (us)

Volu

me (

pL)

ExperimentProportional

minus10

(b)

0 20 40 60

0

10

20

Time (us)

Volu

me (

pL)

ExperimentMultiple

minus10

minus20

(c)

0 20 40 60

0

10

20

Time (us)

Volu

me (

pL)

ExperimentDynamic

minus10

(d)

Figure 12 Outflow change ofmeasurements and simulations of (a) original model (b) proportional coefficientmodel (c)multiple resonatorsmodel and (d) dynamic coefficient model

of the transfer functions are manipulated to speed up thedynamic response speed of the proportional and dynamicmodels From Figure 12 both the multiple-resonator anddynamic models can obtain precise simulation results Thatis more well-designed coefficients are used to characterizethe nonlinear behaviors of piezoelectric printhead and theadjustable ranges are extended to a satisfied degree of bothmodels Additionally it should be noted that the outflowcurve of dynamic model has not only the overshoot phasebut also high overlap ratio to the experimental measurementMoreover the simulation curves reveal that the meniscusmoves inward before the droplet appears

43 Flow and Droplet Velocity Simulation The velocity ofdroplet is the other important aspect to evaluate the printingeffect Droplets with a high speed can remarkably improve

the printing efficiency and the quality of impact with sub-strate Therefore this experiment gives the velocity simula-tion using the proposed LEM models on the same actualexperimental measurement in Section 41

The comparisons between the flow velocity curves sim-ulated by the original LEM and three improved models areillustrated in Figure 13 The excitation is the standard drivingwaveform For the actual measured velocity curves the firsttwo points are also default to zero because no fluid flowingoutside the nozzle can be captured by our image system

It should be noted that all the values of actual measuredvelocity are positive However from the flow velocity curvesin Figure 13 the values of flow velocity in nozzle can bepositive or negative in different phases This positive-to-negative velocity variation indicates the changes of the flowdirection that leads to the break-off phenomenon Hence in

10 Mathematical Problems in Engineering

0 20 40 60

0

5

10

Time (us)

Velo

city

(ms

)

ExperimentFlow velocityOriginal

minus10

minus5

(a)

0 20 40 60

0

5

10

Time (us)

Velo

city

(ms

)

ExperimentFlow velocityProportional

minus10

minus5

(b)

0 20 40 60

0

5

10

Time (us)

Velo

city

(ms

)

ExperimentFlow velocityMultiple

minus10

minus5

(c)

0 20 40 60

0

5

10

Time (us)

Velo

city

(ms

)

ExperimentFlow velocityDynamic

minus10

minus5

(d)

Figure 13 Comparison of measured droplets velocity and the flowaverage velocity simulated by (a) original model (b) ldquoproportionalcoefficientrdquo model (c) ldquomultiple resonatorsrdquo model and (d) ldquodynamic coefficientsrdquo model

order to correctly simulate the velocity in leading edge wepropose a new calculation formula which combines both ofoutflow curve and flow velocity curve

119881average (119905) = radicsum1199052

119905=1199051

(12) 119876out (119905) sdot 119881 (119905)2

sum1199052

119905=1199051

(12)119876out (119905) (7)

where119876out is the outflow value and119881 is the flow velocity 1199051is

the moment when the sum of 119876out changes from negative topositive in outflow curve and 119905

2after 119905

1is the moment when

flow velocity changes to negative in nozzleFrom Figure 13 it can be obviously observed that in

the beginning of droplet formation process the multiple-resonator and dynamic models exhibit better performance insimulating the droplet velocity rather than the originalmodelThat is the average velocity curves obtained by these two

models are almost coincident to the actual measured velocitycurves

Based on the above analysis and comparisons thedynamicmodel based LEMhas themost accuracy in simulat-ing the droplet volume and velocity and can carry sufficientinformation to distinguish the break-off phenomenon occur-ing in the nanosilver droplet formation processs Thereforeamong these three improved models the dynamic lumpedelement model (DLEM) is the most suitable improved equiv-alent structure for simulating jetting characteristics

5 Prediction

For industry applications in printable electronics fabricationthe printhead must work at a certain status to meet manyrestrictive conditions Among them the most important

Mathematical Problems in Engineering 11

0 20 40 60

0

5

10

15

20

Time (us)

Volu

me (

pL)

minus25

minus20

minus15

minus10

minus5

T1 = 3usT1 = 4usT1 = 5us

T1 = 6usT1 = 7usT1 = 8us

(a)

0 20 40 604

5

6

7

8

9

10

Velo

city

(ms

)

Time (us)

T1 = 3usT1 = 4usT1 = 5us

T1 = 6usT1 = 7usT1 = 8us

(b)

Figure 14 Predictive droplet volume and velocity with different and various dwell times 1198792= 21198791 (a) simulation curves of outflow change

and (b) simulation curves of droplet average velocity

4 6 813

14

15

16

17

SimulationExperiment

Volu

me (

pL)

Time (us)

(a)

SimulationExperiment

4 6 8

45

5

55

6

Time (us)

Velo

city

(ms

)

(b)

Figure 15 Predictive droplet volume and velocity compared with measured volume and velocity

problem is choosing the appropriate combinations of thedriving waveform parameters for the used conductive inksHowever an exhausting manual selective process inevitablywastes a lot of time Therefore the computer-aided methodsare urgently needed for searching these appropriate combi-nations efficiently and robustly In this section the proposed

DLEM is applied to predict jetting characteristics for nanosil-ver ink

51 General Prediction Procedure The general procedure ofdroplet volumevelocity prediction based on DLEM is asfollows

12 Mathematical Problems in Engineering

0us0pL0ms

3us0pL0ms

6us0pL0ms

9us0pL0ms

12us6353pL736ms

15us9318pL612ms

18us11441pL572ms

18us 24us

30us14364pL498ms

33us14046pL510ms

36us13980pL520ms

39us13952pL525ms

42us13858pL525ms

45us13869pL520ms

48us13891pL524ms

51us13924pL522ms

54us

524ms13877pL

57us13840pL530ms

13127 pL548ms

14252pL520ms

27us14786pL504ms

Figure 16 A sequence of pictures of droplet falling from nozzle

Step 1 According to droplet generator parameters and prop-erties of fluid calculate most of the parameters in DLEM

Step 2 According to experimental measurements of dropletvolume driven by a standard waveform determine the rest ofthe adjustment coefficients and damping coefficients

Step 3 Based on the constructed DLEM predicate the valuesof droplet volume and velocity

Step 4 According to the need of applications and restrictiveconditions find out several groups of the appropriate param-eters of driving waveform frompredictive data lists of volumeand velocity

Step 5 After making a subtle adjustment in practice decidethe most appropriate parameters of driving waveform

52 Prediction Case 1 Standard Waveform with DifferentDwell Time Asmentioned above theDLEMmodel is chosento simulate jetting characteristics for the nanosilver inkHere an illustrative example is given to display the effect of

predicting the typical combinations of waveform parameterswith different dwell times (119879

1= 3sim8 us 119879

2= 2119879

1) and

the same amplitude (12V) The properties of nanosilver inkprovided by the LEED-PV Corporation have been listed inTable 1

In Figure 14 the predicted outflow and average velocitycurves with various dwell times are depicted We can observefrom Figure 14(a) that before the droplets rush out of thenozzle the depth of the inhaled meniscus is proportionalto dwell time From Figure 14(b) the time of the dropletappearing is postponed with the increase of dwell timeThat is both two simulation phenomena consist with thetheoretical analysis results in [37]

The comparisons between actual measured and predictedjetting characteristics with various dwell times are given inFigures 15(a) and 15(b) which both illustrate a parabolicchange rule with the increase of dwell time It should benoted that at the dwell time 119879

1= 56 us near the theoretical

optimum value 11987910158401= 52 us the droplet volume and velocity

simultaneously reach the maximum From the performanceof prediction accuracy listed in Table 4 the deviation of thepredicted jetting characteristics is within acceptable range

Mathematical Problems in Engineering 13

45

67

68

1012

1416

10121416

Volu

me (

pL)

91011121314151617

T1 (us)

T2(us)

(a)

45

67

68

1012

1416

3

4

5

6

Velo

city

(ms

)

3

35

4

45

5

55

6

T1 (us) T2

(us)

(b)

Figure 17 Predictive droplet volume and velocity with various dwell times1198791and119879

2 (a) predictive droplet volume and (b) predictive droplet

velocity

53 Prediction Case 2 Driving Waveform under RestrictiveCondition In this experiment an illustrative example underrestrictive conditions is presented to demonstrate the feasibil-ity and effectiveness of the proposed method for predictingjetting characteristics The dimensions of droplet generatorand properties of the nanosilver ink are listed in Table 1 Inorder to reach a high printing resolution and good quality ofdroplet impact with the substrate the restrictive conditionsinvolve that the droplet volume is smaller than 14 pL and thedroplet velocity is larger than 5ms

Based on the prediction procedure the predictive valuesof drop volumevelocity with various dwell times 119879

1and

1198792are listed in Table 5 According to the restrictive con-

ditions three combinations of parameters are found out1198791

= 5 us1198792

= 13 us 1198791

= 55 us1198792

= 14 us and1198791= 5 us119879

2= 75sim8 us After a small range search around

predictive values in the actual test the combination of 1198791=

49 us1198792

= 77 us is finally chosen The dynamic effect ofjetting characteristics driven by this combination is shown inFigure 16 From this figure the jetting characteristics satisfythe specified restrictive conditions quite well Meanwhilealmost no satellite droplet emerges after jetting the maindroplet

The 3D plot of predictive jetting characteristics withvarious dwell times 119879

1and 119879

2is shown in Figure 17 With

the increase of dwell time 1198791 peak positions of volume and

velocity have a tiny shift in 95sim115 us With the same dwelltime 119879

2 there is also a parabolic profile in predictive volume

and velocity

6 Conclusion

The lumped element modeling based on linear model ofpiezoelectric ceramic has been successfully used to simu-late piezoelectric droplet generator However the nonlinearmodel of piezoelectric ceramic is necessary to diminishdeviation between the simulated and measured results Inorder to characterize the nonlinear behavior of piezoelectric

Table 4 Performance of prediction accuracy

Deviation Value

Volume

Max 07135 (499)Min 0036 (013)Mean 02636Std 03316

Velocity

Max 02465 (502)Min 00143 (025)Mean 01095Std 01350

ceramic three models are introduced to simulate dropletgeneration In this work the droplet volumevelocity curvessimulated by dynamic model can carry sufficient informa-tion to distinguish nanosilver droplet formation processclearly Although the complexity of system increases theoutflow curves can obviously reflect the filament breaking-off stage that occurs which is coincident with sampledimages Such nonlinear LEM model is also successful inpredicting jetting characteristics under restrictive conditionsComparing the droplet volumevelocity curves with differentdwell time of high-frequency driving waveform the linearrelationship between dwell time and meniscus appearingmoment is discovered Meanwhile a parabolic change ruleof volumevelocity with the increase of dwell time is in goodagreement with present theoretical analysis

It should be noted that the proposed nonlinear model isfocused on relationship between the droplet volumevelocityand the dwell time of driving waveform while the voltageamplitude is also a decisive factor to droplet formationprocess This should be discussed in our future work

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

14 Mathematical Problems in Engineering

Table5Predictiv

edropletvolume(pL

)and

velocity(m

s)

1198791

1198792

665

775

885

995

10105

11115

12125

13135

14145

15155

4119881

11119

127

134

139

143

146

149

151

15147

142

Default

Default

Default

Default

Default

Default

Default

Default

119878371

395

417

439

459

478

491

500

489

471

451

435

Default

Default

Default

Default

Default

Default

Default

Default

45 119881

Default

117

125

134

143

15155

159

162

16157

153

147

139

Default

Default

Default

Default

Default

Default

119878Default

421

451

476

498

516

532

548

560

543

523

501

482

466

Default

Default

Default

Default

Default

Default

5119881

Default

Default

Default

133

143

151

157

163

167

168

166

163

157

15142

133

Default

Default

Default

Default

119878Default

Default

Default

480

507

531

553

572

587

600

583

563

540

525

509

498

Default

Default

Default

Default

55 119881

Default

Default

Default

Default

Default

151

159

165

168

169

171

169

165

16154

146

138

129

Default

Default

119878Default

Default

Default

Default

Default

524

550

572

593

609

624

601

571

551

537

523

509

496

Default

Default

6119881

Default

Default

Default

Default

Default

Default

Default

158

163

164

165

165

165

158

1495

141

130

119

108

97119878

Default

Default

Default

Default

Default

Default

Default

495

515

533

549

564

544

519

494

474

458

445

432

421

65 119881

Default

Default

Default

Default

Default

Default

Default

Default

Default

155

159

16155

150

143

134

125

114

103

95119878

Default

Default

Default

Default

Default

Default

Default

Default

Default

473

485

495

503

490

469

446

420

400

390

383

7119881

Default

Default

Default

Default

Default

Default

Default

Default

Default

Default

125

135

139

141

135

127

119

110

102

94119878

Default

Default

Default

Default

Default

Default

Default

Default

Default

Default

445

465

479

463

444

422

401

385

370

363

Defaults

areb

ecause

thec

ombinatio

nsof

parametersindrivingwaveform

areo

utof

theo

peratio

nrange

Mathematical Problems in Engineering 15

Authorsrsquo Contribution

Maowei He and Liling Sun contributed equally to this work

Acknowledgment

This research is partially supported by National Natural Sci-ence Foundation of China under Grants 61105067 7100107261174164 and 71271140

References

[1] M SinghHMHaverinen PDhagat andG E Jabbour ldquoInkjetprinting-process and its applicationsrdquo Advanced Materials vol22 no 6 pp 673ndash685 2010

[2] E Macdonald R Salas D Espalin et al ldquo3D printing for therapid prototyping of structural electronicsrdquo IEEE Access vol 2pp 234ndash242 2014

[3] S Umezu T Kitajima H Murase H Ohmori K Katahiraand Y Ito ldquoFabrication of living cell structure utilizing electro-static inkjet phenomenardquo in Proceedings of the 22nd IEEEInternational Conference on Micro Electro Mechanical Systems(MEMS rsquo09) pp 419ndash422 IEEE Sorrento Italy January 2009

[4] P Cosseddu L Basirico A Loi et al ldquoInkjet printed OrganicThin Film Transistors based tactile transducers for artificialrobotic skinrdquo in Proceedings of the 4th IEEE RAS and EMBSInternational Conference on Biomedical Robotics and Biomecha-tronics (BioRob rsquo12) pp 1907ndash1912 June 2012

[5] J Hwang A Wan and A Kahn ldquoEnergetics of metal-organicinterfaces new experiments and assessment of the fieldrdquoMate-rials Science and Engineering R Reports vol 64 no 1-2 pp 1ndash312009

[6] B J Kang and J H Oh ldquoGeometrical characterization of inkjet-printed conductive lines of nanosilver suspensions on a polymersubstraterdquoThin Solid Films vol 518 no 10 pp 2890ndash2896 2010

[7] F Villani P Vacca G Nenna et al ldquoInkjet printed polymerlayer on flexible substrate for OLED applicationsrdquo The Journalof Physical Chemistry C vol 113 no 30 pp 13398ndash13402 2009

[8] J-C Liou and F-G Tseng ldquoMulti-dimensional data registrationCMOSMEMS integrated inkjet printheadrdquo MicroelectronicEngineering vol 88 no 6 pp 888ndash901 2011

[9] J Miettinen K Kaija M Mantysalo et al ldquoMolded substratesfor inkjet printed modulesrdquo IEEE Transactions on Componentsand Packaging Technologies vol 32 no 2 pp 293ndash301 2009

[10] K Silverbrook ldquoPrinthead with multiple actuators in eachchamberrdquo US7708387 B2 2010

[11] V Piefort Finite element modelling of piezoelectric active struc-tures [PhD thesis] Department ofMechanical Engineering andRobotics Universite Libre de Bruxelles Brussels Belgium 2001

[12] L Sang Y Hong and F Wang ldquoInvestigation of viscosity effecton droplet formation in T-shaped microchannels by numericaland analytical methodsrdquo Microfluidics and Nanofluidics vol 6no 5 pp 621ndash635 2009

[13] J Liu S-H Tan Y F Yap M Y Ng and N-T NguyenldquoNumerical and experimental investigations of the formationprocess of ferrofluid dropletsrdquo Microfluidics and Nanofluidicsvol 11 no 2 pp 177ndash187 2011

[14] F Sarrazin K Loubiere L Prat C Gourdon T Bonometti andJ Magnaudet ldquoExperimental and numerical study of dropletshydrodynamics in MicroChannelrdquo AIChE Journal vol 52 no12 pp 4061ndash4070 2006

[15] X Q Xing D L Butler S H Ng Z Wang S Danyluk and CYang ldquoSimulation of droplet formation and coalescence usinglattice Boltzmann-based single-phasemodelrdquo Journal of Colloidand Interface Science vol 311 no 2 pp 609ndash618 2007

[16] Q Gallas R Holman T Nishida B Carroll M Sheplak and LCattafesta ldquoLumped element modeling of piezoelectric-drivensynthetic jet actuatorsrdquoAIAA Journal vol 41 no 2 pp 240ndash2472003

[17] A A Khalate X Bombois R Babuska H Wijshoff and RWaarsing ldquoOptimization-based feedforward control for a Drop-on-Demand inkjet printheadrdquo in Proceedings of the AmericanControl Conference (ACC rsquo10) pp 2182ndash2187 Baltimore MdUSA July 2010

[18] L Meirovitch Fundamentals of Vibrations McGraw-Hill NewYork NY USA 2001

[19] H Seitz and J Heinz ldquoModelling of a microfluidic device withpiezoelectric actuatorsrdquo Journal of Micromechanics and Micro-engineering vol 14 no 8 pp 1140ndash1147 2004

[20] S Kim J Sung andMH Lee ldquoPressure wave and fluid velocityin a bend-mode inkjet nozzle with double PZT actuatorsrdquoJournal of Thermal Science vol 22 no 1 pp 29ndash35 2013

[21] G Wassink Inkjet printhead performance enhancement by feed-forward input design based on two-port modeling [PhD thesis]Delft University of Technology 2007

[22] S Prasad SHorowitz QGallas B Sankar L Cattafesta andMSheplak ldquoTwo-port electroacoustic model of an axisymmet-ric piezoelectric composite platerdquo in Proceedings of the 43rdAIAAASMEASCEAHS Structures Structural DynamicsandMaterials Conference AIAA Paper 2002-1365 AIAA DenverColo USA April 2002

[23] D T BlackstockFundamentals of Physical Acoustics JohnWileyamp Sons New York NY USA 2000

[24] F M White Fluid Mechanics McGraw-Hill New York NYUSA 1979

[25] H Schempt andD R Yoerger ldquoStudy of dominant performancecharacteristics in robot transmissionsrdquo Journal of MechanicalDesign Transactions Of the ASME vol 115 no 3 pp 472ndash4821993

[26] E A Witmer ldquoElementary Bernoulli-Euler beam theoryrdquoMITUnified Engineering Course Notes vol 5 pp 114ndash164 1991-1992

[27] Y Yang and L Tang ldquoEquivalent circuit modeling of piezoelec-tric energy harvestersrdquo Journal of Intelligent Material Systemsand Structures vol 20 no 18 pp 2223ndash2235 2009

[28] F Constantinescu A G Gheorghe and M Nitescu ldquoNewcircuit models of power BAW resonatorsrdquo in Proceedings of theEuropean Microwave Integrated Circuit Conference October2007

[29] A Albareda and R PerezNon-Linear Behaviour of PiezoelectricCeramics vol 140ofSpringer Series inMaterials Science Springer2011

[30] J Nosek ldquoDrive level dependence of the resonant frequency inBAW quartz resonators and his modelingrdquo IEEE TransactionsonUltrasonics Ferroelectrics and Frequency Control vol 46 no4 pp 823ndash829 1999

[31] R Aigner N-H Huynh M Handtmann and S MarksteinerldquoBehavior of BAW devices at high power levelsrdquo in Proceedingsof the IEEE MTT-S International Microwave Symposium pp429ndash432 Long Beach Calif USA June 2005

[32] R S Ketcham G R Kline and K M Lakin ldquoPerformanceof TFR filters under elevated power conditionsrdquo in Proceedingsof 42nd the Annual Frequency Control Symposium pp 106ndash111Baltimore Md USA June 1988

16 Mathematical Problems in Engineering

[33] K-S Kwon ldquoWaveform design methods for piezo inkjet dis-pensers based on measured meniscus motionrdquo Journal of Micro-electromechanical Systems vol 18 no 5 pp 1118ndash1125 2009

[34] H-C Wu and H-J Lin ldquoEffects of actuating pressure wave-forms on the droplet behavior in a piezoelectric inkjetrdquoMateri-als Transactions vol 51 no 12 pp 2269ndash2276 2010

[35] KMinolta InkjetHeadApplicationNote-KM1024 Series KonicaMinolta Ij Technologies

[36] G Golub andV Pereyra ldquoSeparable nonlinear least squares thevariable projection method and its applicationsrdquo Inverse Prob-lems vol 19 no 2 pp R1ndashR26 2003

[37] H Wijshoff ldquoThe dynamics of the piezo inkjet printheadoperationrdquo Physics Reports vol 491 no 4-5 pp 77ndash177 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

0 20 40 60

0

10

20

Time (us)

Volu

me (

pL)

ExperimentOriginal

minus10

(a)

0 20 40 60

0

10

20

Time (us)

Volu

me (

pL)

ExperimentProportional

minus10

(b)

0 20 40 60

0

10

20

Time (us)

Volu

me (

pL)

ExperimentMultiple

minus10

minus20

(c)

0 20 40 60

0

10

20

Time (us)

Volu

me (

pL)

ExperimentDynamic

minus10

(d)

Figure 12 Outflow change ofmeasurements and simulations of (a) original model (b) proportional coefficientmodel (c)multiple resonatorsmodel and (d) dynamic coefficient model

of the transfer functions are manipulated to speed up thedynamic response speed of the proportional and dynamicmodels From Figure 12 both the multiple-resonator anddynamic models can obtain precise simulation results Thatis more well-designed coefficients are used to characterizethe nonlinear behaviors of piezoelectric printhead and theadjustable ranges are extended to a satisfied degree of bothmodels Additionally it should be noted that the outflowcurve of dynamic model has not only the overshoot phasebut also high overlap ratio to the experimental measurementMoreover the simulation curves reveal that the meniscusmoves inward before the droplet appears

43 Flow and Droplet Velocity Simulation The velocity ofdroplet is the other important aspect to evaluate the printingeffect Droplets with a high speed can remarkably improve

the printing efficiency and the quality of impact with sub-strate Therefore this experiment gives the velocity simula-tion using the proposed LEM models on the same actualexperimental measurement in Section 41

The comparisons between the flow velocity curves sim-ulated by the original LEM and three improved models areillustrated in Figure 13 The excitation is the standard drivingwaveform For the actual measured velocity curves the firsttwo points are also default to zero because no fluid flowingoutside the nozzle can be captured by our image system

It should be noted that all the values of actual measuredvelocity are positive However from the flow velocity curvesin Figure 13 the values of flow velocity in nozzle can bepositive or negative in different phases This positive-to-negative velocity variation indicates the changes of the flowdirection that leads to the break-off phenomenon Hence in

10 Mathematical Problems in Engineering

0 20 40 60

0

5

10

Time (us)

Velo

city

(ms

)

ExperimentFlow velocityOriginal

minus10

minus5

(a)

0 20 40 60

0

5

10

Time (us)

Velo

city

(ms

)

ExperimentFlow velocityProportional

minus10

minus5

(b)

0 20 40 60

0

5

10

Time (us)

Velo

city

(ms

)

ExperimentFlow velocityMultiple

minus10

minus5

(c)

0 20 40 60

0

5

10

Time (us)

Velo

city

(ms

)

ExperimentFlow velocityDynamic

minus10

minus5

(d)

Figure 13 Comparison of measured droplets velocity and the flowaverage velocity simulated by (a) original model (b) ldquoproportionalcoefficientrdquo model (c) ldquomultiple resonatorsrdquo model and (d) ldquodynamic coefficientsrdquo model

order to correctly simulate the velocity in leading edge wepropose a new calculation formula which combines both ofoutflow curve and flow velocity curve

119881average (119905) = radicsum1199052

119905=1199051

(12) 119876out (119905) sdot 119881 (119905)2

sum1199052

119905=1199051

(12)119876out (119905) (7)

where119876out is the outflow value and119881 is the flow velocity 1199051is

the moment when the sum of 119876out changes from negative topositive in outflow curve and 119905

2after 119905

1is the moment when

flow velocity changes to negative in nozzleFrom Figure 13 it can be obviously observed that in

the beginning of droplet formation process the multiple-resonator and dynamic models exhibit better performance insimulating the droplet velocity rather than the originalmodelThat is the average velocity curves obtained by these two

models are almost coincident to the actual measured velocitycurves

Based on the above analysis and comparisons thedynamicmodel based LEMhas themost accuracy in simulat-ing the droplet volume and velocity and can carry sufficientinformation to distinguish the break-off phenomenon occur-ing in the nanosilver droplet formation processs Thereforeamong these three improved models the dynamic lumpedelement model (DLEM) is the most suitable improved equiv-alent structure for simulating jetting characteristics

5 Prediction

For industry applications in printable electronics fabricationthe printhead must work at a certain status to meet manyrestrictive conditions Among them the most important

Mathematical Problems in Engineering 11

0 20 40 60

0

5

10

15

20

Time (us)

Volu

me (

pL)

minus25

minus20

minus15

minus10

minus5

T1 = 3usT1 = 4usT1 = 5us

T1 = 6usT1 = 7usT1 = 8us

(a)

0 20 40 604

5

6

7

8

9

10

Velo

city

(ms

)

Time (us)

T1 = 3usT1 = 4usT1 = 5us

T1 = 6usT1 = 7usT1 = 8us

(b)

Figure 14 Predictive droplet volume and velocity with different and various dwell times 1198792= 21198791 (a) simulation curves of outflow change

and (b) simulation curves of droplet average velocity

4 6 813

14

15

16

17

SimulationExperiment

Volu

me (

pL)

Time (us)

(a)

SimulationExperiment

4 6 8

45

5

55

6

Time (us)

Velo

city

(ms

)

(b)

Figure 15 Predictive droplet volume and velocity compared with measured volume and velocity

problem is choosing the appropriate combinations of thedriving waveform parameters for the used conductive inksHowever an exhausting manual selective process inevitablywastes a lot of time Therefore the computer-aided methodsare urgently needed for searching these appropriate combi-nations efficiently and robustly In this section the proposed

DLEM is applied to predict jetting characteristics for nanosil-ver ink

51 General Prediction Procedure The general procedure ofdroplet volumevelocity prediction based on DLEM is asfollows

12 Mathematical Problems in Engineering

0us0pL0ms

3us0pL0ms

6us0pL0ms

9us0pL0ms

12us6353pL736ms

15us9318pL612ms

18us11441pL572ms

18us 24us

30us14364pL498ms

33us14046pL510ms

36us13980pL520ms

39us13952pL525ms

42us13858pL525ms

45us13869pL520ms

48us13891pL524ms

51us13924pL522ms

54us

524ms13877pL

57us13840pL530ms

13127 pL548ms

14252pL520ms

27us14786pL504ms

Figure 16 A sequence of pictures of droplet falling from nozzle

Step 1 According to droplet generator parameters and prop-erties of fluid calculate most of the parameters in DLEM

Step 2 According to experimental measurements of dropletvolume driven by a standard waveform determine the rest ofthe adjustment coefficients and damping coefficients

Step 3 Based on the constructed DLEM predicate the valuesof droplet volume and velocity

Step 4 According to the need of applications and restrictiveconditions find out several groups of the appropriate param-eters of driving waveform frompredictive data lists of volumeand velocity

Step 5 After making a subtle adjustment in practice decidethe most appropriate parameters of driving waveform

52 Prediction Case 1 Standard Waveform with DifferentDwell Time Asmentioned above theDLEMmodel is chosento simulate jetting characteristics for the nanosilver inkHere an illustrative example is given to display the effect of

predicting the typical combinations of waveform parameterswith different dwell times (119879

1= 3sim8 us 119879

2= 2119879

1) and

the same amplitude (12V) The properties of nanosilver inkprovided by the LEED-PV Corporation have been listed inTable 1

In Figure 14 the predicted outflow and average velocitycurves with various dwell times are depicted We can observefrom Figure 14(a) that before the droplets rush out of thenozzle the depth of the inhaled meniscus is proportionalto dwell time From Figure 14(b) the time of the dropletappearing is postponed with the increase of dwell timeThat is both two simulation phenomena consist with thetheoretical analysis results in [37]

The comparisons between actual measured and predictedjetting characteristics with various dwell times are given inFigures 15(a) and 15(b) which both illustrate a parabolicchange rule with the increase of dwell time It should benoted that at the dwell time 119879

1= 56 us near the theoretical

optimum value 11987910158401= 52 us the droplet volume and velocity

simultaneously reach the maximum From the performanceof prediction accuracy listed in Table 4 the deviation of thepredicted jetting characteristics is within acceptable range

Mathematical Problems in Engineering 13

45

67

68

1012

1416

10121416

Volu

me (

pL)

91011121314151617

T1 (us)

T2(us)

(a)

45

67

68

1012

1416

3

4

5

6

Velo

city

(ms

)

3

35

4

45

5

55

6

T1 (us) T2

(us)

(b)

Figure 17 Predictive droplet volume and velocity with various dwell times1198791and119879

2 (a) predictive droplet volume and (b) predictive droplet

velocity

53 Prediction Case 2 Driving Waveform under RestrictiveCondition In this experiment an illustrative example underrestrictive conditions is presented to demonstrate the feasibil-ity and effectiveness of the proposed method for predictingjetting characteristics The dimensions of droplet generatorand properties of the nanosilver ink are listed in Table 1 Inorder to reach a high printing resolution and good quality ofdroplet impact with the substrate the restrictive conditionsinvolve that the droplet volume is smaller than 14 pL and thedroplet velocity is larger than 5ms

Based on the prediction procedure the predictive valuesof drop volumevelocity with various dwell times 119879

1and

1198792are listed in Table 5 According to the restrictive con-

ditions three combinations of parameters are found out1198791

= 5 us1198792

= 13 us 1198791

= 55 us1198792

= 14 us and1198791= 5 us119879

2= 75sim8 us After a small range search around

predictive values in the actual test the combination of 1198791=

49 us1198792

= 77 us is finally chosen The dynamic effect ofjetting characteristics driven by this combination is shown inFigure 16 From this figure the jetting characteristics satisfythe specified restrictive conditions quite well Meanwhilealmost no satellite droplet emerges after jetting the maindroplet

The 3D plot of predictive jetting characteristics withvarious dwell times 119879

1and 119879

2is shown in Figure 17 With

the increase of dwell time 1198791 peak positions of volume and

velocity have a tiny shift in 95sim115 us With the same dwelltime 119879

2 there is also a parabolic profile in predictive volume

and velocity

6 Conclusion

The lumped element modeling based on linear model ofpiezoelectric ceramic has been successfully used to simu-late piezoelectric droplet generator However the nonlinearmodel of piezoelectric ceramic is necessary to diminishdeviation between the simulated and measured results Inorder to characterize the nonlinear behavior of piezoelectric

Table 4 Performance of prediction accuracy

Deviation Value

Volume

Max 07135 (499)Min 0036 (013)Mean 02636Std 03316

Velocity

Max 02465 (502)Min 00143 (025)Mean 01095Std 01350

ceramic three models are introduced to simulate dropletgeneration In this work the droplet volumevelocity curvessimulated by dynamic model can carry sufficient informa-tion to distinguish nanosilver droplet formation processclearly Although the complexity of system increases theoutflow curves can obviously reflect the filament breaking-off stage that occurs which is coincident with sampledimages Such nonlinear LEM model is also successful inpredicting jetting characteristics under restrictive conditionsComparing the droplet volumevelocity curves with differentdwell time of high-frequency driving waveform the linearrelationship between dwell time and meniscus appearingmoment is discovered Meanwhile a parabolic change ruleof volumevelocity with the increase of dwell time is in goodagreement with present theoretical analysis

It should be noted that the proposed nonlinear model isfocused on relationship between the droplet volumevelocityand the dwell time of driving waveform while the voltageamplitude is also a decisive factor to droplet formationprocess This should be discussed in our future work

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

14 Mathematical Problems in Engineering

Table5Predictiv

edropletvolume(pL

)and

velocity(m

s)

1198791

1198792

665

775

885

995

10105

11115

12125

13135

14145

15155

4119881

11119

127

134

139

143

146

149

151

15147

142

Default

Default

Default

Default

Default

Default

Default

Default

119878371

395

417

439

459

478

491

500

489

471

451

435

Default

Default

Default

Default

Default

Default

Default

Default

45 119881

Default

117

125

134

143

15155

159

162

16157

153

147

139

Default

Default

Default

Default

Default

Default

119878Default

421

451

476

498

516

532

548

560

543

523

501

482

466

Default

Default

Default

Default

Default

Default

5119881

Default

Default

Default

133

143

151

157

163

167

168

166

163

157

15142

133

Default

Default

Default

Default

119878Default

Default

Default

480

507

531

553

572

587

600

583

563

540

525

509

498

Default

Default

Default

Default

55 119881

Default

Default

Default

Default

Default

151

159

165

168

169

171

169

165

16154

146

138

129

Default

Default

119878Default

Default

Default

Default

Default

524

550

572

593

609

624

601

571

551

537

523

509

496

Default

Default

6119881

Default

Default

Default

Default

Default

Default

Default

158

163

164

165

165

165

158

1495

141

130

119

108

97119878

Default

Default

Default

Default

Default

Default

Default

495

515

533

549

564

544

519

494

474

458

445

432

421

65 119881

Default

Default

Default

Default

Default

Default

Default

Default

Default

155

159

16155

150

143

134

125

114

103

95119878

Default

Default

Default

Default

Default

Default

Default

Default

Default

473

485

495

503

490

469

446

420

400

390

383

7119881

Default

Default

Default

Default

Default

Default

Default

Default

Default

Default

125

135

139

141

135

127

119

110

102

94119878

Default

Default

Default

Default

Default

Default

Default

Default

Default

Default

445

465

479

463

444

422

401

385

370

363

Defaults

areb

ecause

thec

ombinatio

nsof

parametersindrivingwaveform

areo

utof

theo

peratio

nrange

Mathematical Problems in Engineering 15

Authorsrsquo Contribution

Maowei He and Liling Sun contributed equally to this work

Acknowledgment

This research is partially supported by National Natural Sci-ence Foundation of China under Grants 61105067 7100107261174164 and 71271140

References

[1] M SinghHMHaverinen PDhagat andG E Jabbour ldquoInkjetprinting-process and its applicationsrdquo Advanced Materials vol22 no 6 pp 673ndash685 2010

[2] E Macdonald R Salas D Espalin et al ldquo3D printing for therapid prototyping of structural electronicsrdquo IEEE Access vol 2pp 234ndash242 2014

[3] S Umezu T Kitajima H Murase H Ohmori K Katahiraand Y Ito ldquoFabrication of living cell structure utilizing electro-static inkjet phenomenardquo in Proceedings of the 22nd IEEEInternational Conference on Micro Electro Mechanical Systems(MEMS rsquo09) pp 419ndash422 IEEE Sorrento Italy January 2009

[4] P Cosseddu L Basirico A Loi et al ldquoInkjet printed OrganicThin Film Transistors based tactile transducers for artificialrobotic skinrdquo in Proceedings of the 4th IEEE RAS and EMBSInternational Conference on Biomedical Robotics and Biomecha-tronics (BioRob rsquo12) pp 1907ndash1912 June 2012

[5] J Hwang A Wan and A Kahn ldquoEnergetics of metal-organicinterfaces new experiments and assessment of the fieldrdquoMate-rials Science and Engineering R Reports vol 64 no 1-2 pp 1ndash312009

[6] B J Kang and J H Oh ldquoGeometrical characterization of inkjet-printed conductive lines of nanosilver suspensions on a polymersubstraterdquoThin Solid Films vol 518 no 10 pp 2890ndash2896 2010

[7] F Villani P Vacca G Nenna et al ldquoInkjet printed polymerlayer on flexible substrate for OLED applicationsrdquo The Journalof Physical Chemistry C vol 113 no 30 pp 13398ndash13402 2009

[8] J-C Liou and F-G Tseng ldquoMulti-dimensional data registrationCMOSMEMS integrated inkjet printheadrdquo MicroelectronicEngineering vol 88 no 6 pp 888ndash901 2011

[9] J Miettinen K Kaija M Mantysalo et al ldquoMolded substratesfor inkjet printed modulesrdquo IEEE Transactions on Componentsand Packaging Technologies vol 32 no 2 pp 293ndash301 2009

[10] K Silverbrook ldquoPrinthead with multiple actuators in eachchamberrdquo US7708387 B2 2010

[11] V Piefort Finite element modelling of piezoelectric active struc-tures [PhD thesis] Department ofMechanical Engineering andRobotics Universite Libre de Bruxelles Brussels Belgium 2001

[12] L Sang Y Hong and F Wang ldquoInvestigation of viscosity effecton droplet formation in T-shaped microchannels by numericaland analytical methodsrdquo Microfluidics and Nanofluidics vol 6no 5 pp 621ndash635 2009

[13] J Liu S-H Tan Y F Yap M Y Ng and N-T NguyenldquoNumerical and experimental investigations of the formationprocess of ferrofluid dropletsrdquo Microfluidics and Nanofluidicsvol 11 no 2 pp 177ndash187 2011

[14] F Sarrazin K Loubiere L Prat C Gourdon T Bonometti andJ Magnaudet ldquoExperimental and numerical study of dropletshydrodynamics in MicroChannelrdquo AIChE Journal vol 52 no12 pp 4061ndash4070 2006

[15] X Q Xing D L Butler S H Ng Z Wang S Danyluk and CYang ldquoSimulation of droplet formation and coalescence usinglattice Boltzmann-based single-phasemodelrdquo Journal of Colloidand Interface Science vol 311 no 2 pp 609ndash618 2007

[16] Q Gallas R Holman T Nishida B Carroll M Sheplak and LCattafesta ldquoLumped element modeling of piezoelectric-drivensynthetic jet actuatorsrdquoAIAA Journal vol 41 no 2 pp 240ndash2472003

[17] A A Khalate X Bombois R Babuska H Wijshoff and RWaarsing ldquoOptimization-based feedforward control for a Drop-on-Demand inkjet printheadrdquo in Proceedings of the AmericanControl Conference (ACC rsquo10) pp 2182ndash2187 Baltimore MdUSA July 2010

[18] L Meirovitch Fundamentals of Vibrations McGraw-Hill NewYork NY USA 2001

[19] H Seitz and J Heinz ldquoModelling of a microfluidic device withpiezoelectric actuatorsrdquo Journal of Micromechanics and Micro-engineering vol 14 no 8 pp 1140ndash1147 2004

[20] S Kim J Sung andMH Lee ldquoPressure wave and fluid velocityin a bend-mode inkjet nozzle with double PZT actuatorsrdquoJournal of Thermal Science vol 22 no 1 pp 29ndash35 2013

[21] G Wassink Inkjet printhead performance enhancement by feed-forward input design based on two-port modeling [PhD thesis]Delft University of Technology 2007

[22] S Prasad SHorowitz QGallas B Sankar L Cattafesta andMSheplak ldquoTwo-port electroacoustic model of an axisymmet-ric piezoelectric composite platerdquo in Proceedings of the 43rdAIAAASMEASCEAHS Structures Structural DynamicsandMaterials Conference AIAA Paper 2002-1365 AIAA DenverColo USA April 2002

[23] D T BlackstockFundamentals of Physical Acoustics JohnWileyamp Sons New York NY USA 2000

[24] F M White Fluid Mechanics McGraw-Hill New York NYUSA 1979

[25] H Schempt andD R Yoerger ldquoStudy of dominant performancecharacteristics in robot transmissionsrdquo Journal of MechanicalDesign Transactions Of the ASME vol 115 no 3 pp 472ndash4821993

[26] E A Witmer ldquoElementary Bernoulli-Euler beam theoryrdquoMITUnified Engineering Course Notes vol 5 pp 114ndash164 1991-1992

[27] Y Yang and L Tang ldquoEquivalent circuit modeling of piezoelec-tric energy harvestersrdquo Journal of Intelligent Material Systemsand Structures vol 20 no 18 pp 2223ndash2235 2009

[28] F Constantinescu A G Gheorghe and M Nitescu ldquoNewcircuit models of power BAW resonatorsrdquo in Proceedings of theEuropean Microwave Integrated Circuit Conference October2007

[29] A Albareda and R PerezNon-Linear Behaviour of PiezoelectricCeramics vol 140ofSpringer Series inMaterials Science Springer2011

[30] J Nosek ldquoDrive level dependence of the resonant frequency inBAW quartz resonators and his modelingrdquo IEEE TransactionsonUltrasonics Ferroelectrics and Frequency Control vol 46 no4 pp 823ndash829 1999

[31] R Aigner N-H Huynh M Handtmann and S MarksteinerldquoBehavior of BAW devices at high power levelsrdquo in Proceedingsof the IEEE MTT-S International Microwave Symposium pp429ndash432 Long Beach Calif USA June 2005

[32] R S Ketcham G R Kline and K M Lakin ldquoPerformanceof TFR filters under elevated power conditionsrdquo in Proceedingsof 42nd the Annual Frequency Control Symposium pp 106ndash111Baltimore Md USA June 1988

16 Mathematical Problems in Engineering

[33] K-S Kwon ldquoWaveform design methods for piezo inkjet dis-pensers based on measured meniscus motionrdquo Journal of Micro-electromechanical Systems vol 18 no 5 pp 1118ndash1125 2009

[34] H-C Wu and H-J Lin ldquoEffects of actuating pressure wave-forms on the droplet behavior in a piezoelectric inkjetrdquoMateri-als Transactions vol 51 no 12 pp 2269ndash2276 2010

[35] KMinolta InkjetHeadApplicationNote-KM1024 Series KonicaMinolta Ij Technologies

[36] G Golub andV Pereyra ldquoSeparable nonlinear least squares thevariable projection method and its applicationsrdquo Inverse Prob-lems vol 19 no 2 pp R1ndashR26 2003

[37] H Wijshoff ldquoThe dynamics of the piezo inkjet printheadoperationrdquo Physics Reports vol 491 no 4-5 pp 77ndash177 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

0 20 40 60

0

5

10

Time (us)

Velo

city

(ms

)

ExperimentFlow velocityOriginal

minus10

minus5

(a)

0 20 40 60

0

5

10

Time (us)

Velo

city

(ms

)

ExperimentFlow velocityProportional

minus10

minus5

(b)

0 20 40 60

0

5

10

Time (us)

Velo

city

(ms

)

ExperimentFlow velocityMultiple

minus10

minus5

(c)

0 20 40 60

0

5

10

Time (us)

Velo

city

(ms

)

ExperimentFlow velocityDynamic

minus10

minus5

(d)

Figure 13 Comparison of measured droplets velocity and the flowaverage velocity simulated by (a) original model (b) ldquoproportionalcoefficientrdquo model (c) ldquomultiple resonatorsrdquo model and (d) ldquodynamic coefficientsrdquo model

order to correctly simulate the velocity in leading edge wepropose a new calculation formula which combines both ofoutflow curve and flow velocity curve

119881average (119905) = radicsum1199052

119905=1199051

(12) 119876out (119905) sdot 119881 (119905)2

sum1199052

119905=1199051

(12)119876out (119905) (7)

where119876out is the outflow value and119881 is the flow velocity 1199051is

the moment when the sum of 119876out changes from negative topositive in outflow curve and 119905

2after 119905

1is the moment when

flow velocity changes to negative in nozzleFrom Figure 13 it can be obviously observed that in

the beginning of droplet formation process the multiple-resonator and dynamic models exhibit better performance insimulating the droplet velocity rather than the originalmodelThat is the average velocity curves obtained by these two

models are almost coincident to the actual measured velocitycurves

Based on the above analysis and comparisons thedynamicmodel based LEMhas themost accuracy in simulat-ing the droplet volume and velocity and can carry sufficientinformation to distinguish the break-off phenomenon occur-ing in the nanosilver droplet formation processs Thereforeamong these three improved models the dynamic lumpedelement model (DLEM) is the most suitable improved equiv-alent structure for simulating jetting characteristics

5 Prediction

For industry applications in printable electronics fabricationthe printhead must work at a certain status to meet manyrestrictive conditions Among them the most important

Mathematical Problems in Engineering 11

0 20 40 60

0

5

10

15

20

Time (us)

Volu

me (

pL)

minus25

minus20

minus15

minus10

minus5

T1 = 3usT1 = 4usT1 = 5us

T1 = 6usT1 = 7usT1 = 8us

(a)

0 20 40 604

5

6

7

8

9

10

Velo

city

(ms

)

Time (us)

T1 = 3usT1 = 4usT1 = 5us

T1 = 6usT1 = 7usT1 = 8us

(b)

Figure 14 Predictive droplet volume and velocity with different and various dwell times 1198792= 21198791 (a) simulation curves of outflow change

and (b) simulation curves of droplet average velocity

4 6 813

14

15

16

17

SimulationExperiment

Volu

me (

pL)

Time (us)

(a)

SimulationExperiment

4 6 8

45

5

55

6

Time (us)

Velo

city

(ms

)

(b)

Figure 15 Predictive droplet volume and velocity compared with measured volume and velocity

problem is choosing the appropriate combinations of thedriving waveform parameters for the used conductive inksHowever an exhausting manual selective process inevitablywastes a lot of time Therefore the computer-aided methodsare urgently needed for searching these appropriate combi-nations efficiently and robustly In this section the proposed

DLEM is applied to predict jetting characteristics for nanosil-ver ink

51 General Prediction Procedure The general procedure ofdroplet volumevelocity prediction based on DLEM is asfollows

12 Mathematical Problems in Engineering

0us0pL0ms

3us0pL0ms

6us0pL0ms

9us0pL0ms

12us6353pL736ms

15us9318pL612ms

18us11441pL572ms

18us 24us

30us14364pL498ms

33us14046pL510ms

36us13980pL520ms

39us13952pL525ms

42us13858pL525ms

45us13869pL520ms

48us13891pL524ms

51us13924pL522ms

54us

524ms13877pL

57us13840pL530ms

13127 pL548ms

14252pL520ms

27us14786pL504ms

Figure 16 A sequence of pictures of droplet falling from nozzle

Step 1 According to droplet generator parameters and prop-erties of fluid calculate most of the parameters in DLEM

Step 2 According to experimental measurements of dropletvolume driven by a standard waveform determine the rest ofthe adjustment coefficients and damping coefficients

Step 3 Based on the constructed DLEM predicate the valuesof droplet volume and velocity

Step 4 According to the need of applications and restrictiveconditions find out several groups of the appropriate param-eters of driving waveform frompredictive data lists of volumeand velocity

Step 5 After making a subtle adjustment in practice decidethe most appropriate parameters of driving waveform

52 Prediction Case 1 Standard Waveform with DifferentDwell Time Asmentioned above theDLEMmodel is chosento simulate jetting characteristics for the nanosilver inkHere an illustrative example is given to display the effect of

predicting the typical combinations of waveform parameterswith different dwell times (119879

1= 3sim8 us 119879

2= 2119879

1) and

the same amplitude (12V) The properties of nanosilver inkprovided by the LEED-PV Corporation have been listed inTable 1

In Figure 14 the predicted outflow and average velocitycurves with various dwell times are depicted We can observefrom Figure 14(a) that before the droplets rush out of thenozzle the depth of the inhaled meniscus is proportionalto dwell time From Figure 14(b) the time of the dropletappearing is postponed with the increase of dwell timeThat is both two simulation phenomena consist with thetheoretical analysis results in [37]

The comparisons between actual measured and predictedjetting characteristics with various dwell times are given inFigures 15(a) and 15(b) which both illustrate a parabolicchange rule with the increase of dwell time It should benoted that at the dwell time 119879

1= 56 us near the theoretical

optimum value 11987910158401= 52 us the droplet volume and velocity

simultaneously reach the maximum From the performanceof prediction accuracy listed in Table 4 the deviation of thepredicted jetting characteristics is within acceptable range

Mathematical Problems in Engineering 13

45

67

68

1012

1416

10121416

Volu

me (

pL)

91011121314151617

T1 (us)

T2(us)

(a)

45

67

68

1012

1416

3

4

5

6

Velo

city

(ms

)

3

35

4

45

5

55

6

T1 (us) T2

(us)

(b)

Figure 17 Predictive droplet volume and velocity with various dwell times1198791and119879

2 (a) predictive droplet volume and (b) predictive droplet

velocity

53 Prediction Case 2 Driving Waveform under RestrictiveCondition In this experiment an illustrative example underrestrictive conditions is presented to demonstrate the feasibil-ity and effectiveness of the proposed method for predictingjetting characteristics The dimensions of droplet generatorand properties of the nanosilver ink are listed in Table 1 Inorder to reach a high printing resolution and good quality ofdroplet impact with the substrate the restrictive conditionsinvolve that the droplet volume is smaller than 14 pL and thedroplet velocity is larger than 5ms

Based on the prediction procedure the predictive valuesof drop volumevelocity with various dwell times 119879

1and

1198792are listed in Table 5 According to the restrictive con-

ditions three combinations of parameters are found out1198791

= 5 us1198792

= 13 us 1198791

= 55 us1198792

= 14 us and1198791= 5 us119879

2= 75sim8 us After a small range search around

predictive values in the actual test the combination of 1198791=

49 us1198792

= 77 us is finally chosen The dynamic effect ofjetting characteristics driven by this combination is shown inFigure 16 From this figure the jetting characteristics satisfythe specified restrictive conditions quite well Meanwhilealmost no satellite droplet emerges after jetting the maindroplet

The 3D plot of predictive jetting characteristics withvarious dwell times 119879

1and 119879

2is shown in Figure 17 With

the increase of dwell time 1198791 peak positions of volume and

velocity have a tiny shift in 95sim115 us With the same dwelltime 119879

2 there is also a parabolic profile in predictive volume

and velocity

6 Conclusion

The lumped element modeling based on linear model ofpiezoelectric ceramic has been successfully used to simu-late piezoelectric droplet generator However the nonlinearmodel of piezoelectric ceramic is necessary to diminishdeviation between the simulated and measured results Inorder to characterize the nonlinear behavior of piezoelectric

Table 4 Performance of prediction accuracy

Deviation Value

Volume

Max 07135 (499)Min 0036 (013)Mean 02636Std 03316

Velocity

Max 02465 (502)Min 00143 (025)Mean 01095Std 01350

ceramic three models are introduced to simulate dropletgeneration In this work the droplet volumevelocity curvessimulated by dynamic model can carry sufficient informa-tion to distinguish nanosilver droplet formation processclearly Although the complexity of system increases theoutflow curves can obviously reflect the filament breaking-off stage that occurs which is coincident with sampledimages Such nonlinear LEM model is also successful inpredicting jetting characteristics under restrictive conditionsComparing the droplet volumevelocity curves with differentdwell time of high-frequency driving waveform the linearrelationship between dwell time and meniscus appearingmoment is discovered Meanwhile a parabolic change ruleof volumevelocity with the increase of dwell time is in goodagreement with present theoretical analysis

It should be noted that the proposed nonlinear model isfocused on relationship between the droplet volumevelocityand the dwell time of driving waveform while the voltageamplitude is also a decisive factor to droplet formationprocess This should be discussed in our future work

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

14 Mathematical Problems in Engineering

Table5Predictiv

edropletvolume(pL

)and

velocity(m

s)

1198791

1198792

665

775

885

995

10105

11115

12125

13135

14145

15155

4119881

11119

127

134

139

143

146

149

151

15147

142

Default

Default

Default

Default

Default

Default

Default

Default

119878371

395

417

439

459

478

491

500

489

471

451

435

Default

Default

Default

Default

Default

Default

Default

Default

45 119881

Default

117

125

134

143

15155

159

162

16157

153

147

139

Default

Default

Default

Default

Default

Default

119878Default

421

451

476

498

516

532

548

560

543

523

501

482

466

Default

Default

Default

Default

Default

Default

5119881

Default

Default

Default

133

143

151

157

163

167

168

166

163

157

15142

133

Default

Default

Default

Default

119878Default

Default

Default

480

507

531

553

572

587

600

583

563

540

525

509

498

Default

Default

Default

Default

55 119881

Default

Default

Default

Default

Default

151

159

165

168

169

171

169

165

16154

146

138

129

Default

Default

119878Default

Default

Default

Default

Default

524

550

572

593

609

624

601

571

551

537

523

509

496

Default

Default

6119881

Default

Default

Default

Default

Default

Default

Default

158

163

164

165

165

165

158

1495

141

130

119

108

97119878

Default

Default

Default

Default

Default

Default

Default

495

515

533

549

564

544

519

494

474

458

445

432

421

65 119881

Default

Default

Default

Default

Default

Default

Default

Default

Default

155

159

16155

150

143

134

125

114

103

95119878

Default

Default

Default

Default

Default

Default

Default

Default

Default

473

485

495

503

490

469

446

420

400

390

383

7119881

Default

Default

Default

Default

Default

Default

Default

Default

Default

Default

125

135

139

141

135

127

119

110

102

94119878

Default

Default

Default

Default

Default

Default

Default

Default

Default

Default

445

465

479

463

444

422

401

385

370

363

Defaults

areb

ecause

thec

ombinatio

nsof

parametersindrivingwaveform

areo

utof

theo

peratio

nrange

Mathematical Problems in Engineering 15

Authorsrsquo Contribution

Maowei He and Liling Sun contributed equally to this work

Acknowledgment

This research is partially supported by National Natural Sci-ence Foundation of China under Grants 61105067 7100107261174164 and 71271140

References

[1] M SinghHMHaverinen PDhagat andG E Jabbour ldquoInkjetprinting-process and its applicationsrdquo Advanced Materials vol22 no 6 pp 673ndash685 2010

[2] E Macdonald R Salas D Espalin et al ldquo3D printing for therapid prototyping of structural electronicsrdquo IEEE Access vol 2pp 234ndash242 2014

[3] S Umezu T Kitajima H Murase H Ohmori K Katahiraand Y Ito ldquoFabrication of living cell structure utilizing electro-static inkjet phenomenardquo in Proceedings of the 22nd IEEEInternational Conference on Micro Electro Mechanical Systems(MEMS rsquo09) pp 419ndash422 IEEE Sorrento Italy January 2009

[4] P Cosseddu L Basirico A Loi et al ldquoInkjet printed OrganicThin Film Transistors based tactile transducers for artificialrobotic skinrdquo in Proceedings of the 4th IEEE RAS and EMBSInternational Conference on Biomedical Robotics and Biomecha-tronics (BioRob rsquo12) pp 1907ndash1912 June 2012

[5] J Hwang A Wan and A Kahn ldquoEnergetics of metal-organicinterfaces new experiments and assessment of the fieldrdquoMate-rials Science and Engineering R Reports vol 64 no 1-2 pp 1ndash312009

[6] B J Kang and J H Oh ldquoGeometrical characterization of inkjet-printed conductive lines of nanosilver suspensions on a polymersubstraterdquoThin Solid Films vol 518 no 10 pp 2890ndash2896 2010

[7] F Villani P Vacca G Nenna et al ldquoInkjet printed polymerlayer on flexible substrate for OLED applicationsrdquo The Journalof Physical Chemistry C vol 113 no 30 pp 13398ndash13402 2009

[8] J-C Liou and F-G Tseng ldquoMulti-dimensional data registrationCMOSMEMS integrated inkjet printheadrdquo MicroelectronicEngineering vol 88 no 6 pp 888ndash901 2011

[9] J Miettinen K Kaija M Mantysalo et al ldquoMolded substratesfor inkjet printed modulesrdquo IEEE Transactions on Componentsand Packaging Technologies vol 32 no 2 pp 293ndash301 2009

[10] K Silverbrook ldquoPrinthead with multiple actuators in eachchamberrdquo US7708387 B2 2010

[11] V Piefort Finite element modelling of piezoelectric active struc-tures [PhD thesis] Department ofMechanical Engineering andRobotics Universite Libre de Bruxelles Brussels Belgium 2001

[12] L Sang Y Hong and F Wang ldquoInvestigation of viscosity effecton droplet formation in T-shaped microchannels by numericaland analytical methodsrdquo Microfluidics and Nanofluidics vol 6no 5 pp 621ndash635 2009

[13] J Liu S-H Tan Y F Yap M Y Ng and N-T NguyenldquoNumerical and experimental investigations of the formationprocess of ferrofluid dropletsrdquo Microfluidics and Nanofluidicsvol 11 no 2 pp 177ndash187 2011

[14] F Sarrazin K Loubiere L Prat C Gourdon T Bonometti andJ Magnaudet ldquoExperimental and numerical study of dropletshydrodynamics in MicroChannelrdquo AIChE Journal vol 52 no12 pp 4061ndash4070 2006

[15] X Q Xing D L Butler S H Ng Z Wang S Danyluk and CYang ldquoSimulation of droplet formation and coalescence usinglattice Boltzmann-based single-phasemodelrdquo Journal of Colloidand Interface Science vol 311 no 2 pp 609ndash618 2007

[16] Q Gallas R Holman T Nishida B Carroll M Sheplak and LCattafesta ldquoLumped element modeling of piezoelectric-drivensynthetic jet actuatorsrdquoAIAA Journal vol 41 no 2 pp 240ndash2472003

[17] A A Khalate X Bombois R Babuska H Wijshoff and RWaarsing ldquoOptimization-based feedforward control for a Drop-on-Demand inkjet printheadrdquo in Proceedings of the AmericanControl Conference (ACC rsquo10) pp 2182ndash2187 Baltimore MdUSA July 2010

[18] L Meirovitch Fundamentals of Vibrations McGraw-Hill NewYork NY USA 2001

[19] H Seitz and J Heinz ldquoModelling of a microfluidic device withpiezoelectric actuatorsrdquo Journal of Micromechanics and Micro-engineering vol 14 no 8 pp 1140ndash1147 2004

[20] S Kim J Sung andMH Lee ldquoPressure wave and fluid velocityin a bend-mode inkjet nozzle with double PZT actuatorsrdquoJournal of Thermal Science vol 22 no 1 pp 29ndash35 2013

[21] G Wassink Inkjet printhead performance enhancement by feed-forward input design based on two-port modeling [PhD thesis]Delft University of Technology 2007

[22] S Prasad SHorowitz QGallas B Sankar L Cattafesta andMSheplak ldquoTwo-port electroacoustic model of an axisymmet-ric piezoelectric composite platerdquo in Proceedings of the 43rdAIAAASMEASCEAHS Structures Structural DynamicsandMaterials Conference AIAA Paper 2002-1365 AIAA DenverColo USA April 2002

[23] D T BlackstockFundamentals of Physical Acoustics JohnWileyamp Sons New York NY USA 2000

[24] F M White Fluid Mechanics McGraw-Hill New York NYUSA 1979

[25] H Schempt andD R Yoerger ldquoStudy of dominant performancecharacteristics in robot transmissionsrdquo Journal of MechanicalDesign Transactions Of the ASME vol 115 no 3 pp 472ndash4821993

[26] E A Witmer ldquoElementary Bernoulli-Euler beam theoryrdquoMITUnified Engineering Course Notes vol 5 pp 114ndash164 1991-1992

[27] Y Yang and L Tang ldquoEquivalent circuit modeling of piezoelec-tric energy harvestersrdquo Journal of Intelligent Material Systemsand Structures vol 20 no 18 pp 2223ndash2235 2009

[28] F Constantinescu A G Gheorghe and M Nitescu ldquoNewcircuit models of power BAW resonatorsrdquo in Proceedings of theEuropean Microwave Integrated Circuit Conference October2007

[29] A Albareda and R PerezNon-Linear Behaviour of PiezoelectricCeramics vol 140ofSpringer Series inMaterials Science Springer2011

[30] J Nosek ldquoDrive level dependence of the resonant frequency inBAW quartz resonators and his modelingrdquo IEEE TransactionsonUltrasonics Ferroelectrics and Frequency Control vol 46 no4 pp 823ndash829 1999

[31] R Aigner N-H Huynh M Handtmann and S MarksteinerldquoBehavior of BAW devices at high power levelsrdquo in Proceedingsof the IEEE MTT-S International Microwave Symposium pp429ndash432 Long Beach Calif USA June 2005

[32] R S Ketcham G R Kline and K M Lakin ldquoPerformanceof TFR filters under elevated power conditionsrdquo in Proceedingsof 42nd the Annual Frequency Control Symposium pp 106ndash111Baltimore Md USA June 1988

16 Mathematical Problems in Engineering

[33] K-S Kwon ldquoWaveform design methods for piezo inkjet dis-pensers based on measured meniscus motionrdquo Journal of Micro-electromechanical Systems vol 18 no 5 pp 1118ndash1125 2009

[34] H-C Wu and H-J Lin ldquoEffects of actuating pressure wave-forms on the droplet behavior in a piezoelectric inkjetrdquoMateri-als Transactions vol 51 no 12 pp 2269ndash2276 2010

[35] KMinolta InkjetHeadApplicationNote-KM1024 Series KonicaMinolta Ij Technologies

[36] G Golub andV Pereyra ldquoSeparable nonlinear least squares thevariable projection method and its applicationsrdquo Inverse Prob-lems vol 19 no 2 pp R1ndashR26 2003

[37] H Wijshoff ldquoThe dynamics of the piezo inkjet printheadoperationrdquo Physics Reports vol 491 no 4-5 pp 77ndash177 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 11

0 20 40 60

0

5

10

15

20

Time (us)

Volu

me (

pL)

minus25

minus20

minus15

minus10

minus5

T1 = 3usT1 = 4usT1 = 5us

T1 = 6usT1 = 7usT1 = 8us

(a)

0 20 40 604

5

6

7

8

9

10

Velo

city

(ms

)

Time (us)

T1 = 3usT1 = 4usT1 = 5us

T1 = 6usT1 = 7usT1 = 8us

(b)

Figure 14 Predictive droplet volume and velocity with different and various dwell times 1198792= 21198791 (a) simulation curves of outflow change

and (b) simulation curves of droplet average velocity

4 6 813

14

15

16

17

SimulationExperiment

Volu

me (

pL)

Time (us)

(a)

SimulationExperiment

4 6 8

45

5

55

6

Time (us)

Velo

city

(ms

)

(b)

Figure 15 Predictive droplet volume and velocity compared with measured volume and velocity

problem is choosing the appropriate combinations of thedriving waveform parameters for the used conductive inksHowever an exhausting manual selective process inevitablywastes a lot of time Therefore the computer-aided methodsare urgently needed for searching these appropriate combi-nations efficiently and robustly In this section the proposed

DLEM is applied to predict jetting characteristics for nanosil-ver ink

51 General Prediction Procedure The general procedure ofdroplet volumevelocity prediction based on DLEM is asfollows

12 Mathematical Problems in Engineering

0us0pL0ms

3us0pL0ms

6us0pL0ms

9us0pL0ms

12us6353pL736ms

15us9318pL612ms

18us11441pL572ms

18us 24us

30us14364pL498ms

33us14046pL510ms

36us13980pL520ms

39us13952pL525ms

42us13858pL525ms

45us13869pL520ms

48us13891pL524ms

51us13924pL522ms

54us

524ms13877pL

57us13840pL530ms

13127 pL548ms

14252pL520ms

27us14786pL504ms

Figure 16 A sequence of pictures of droplet falling from nozzle

Step 1 According to droplet generator parameters and prop-erties of fluid calculate most of the parameters in DLEM

Step 2 According to experimental measurements of dropletvolume driven by a standard waveform determine the rest ofthe adjustment coefficients and damping coefficients

Step 3 Based on the constructed DLEM predicate the valuesof droplet volume and velocity

Step 4 According to the need of applications and restrictiveconditions find out several groups of the appropriate param-eters of driving waveform frompredictive data lists of volumeand velocity

Step 5 After making a subtle adjustment in practice decidethe most appropriate parameters of driving waveform

52 Prediction Case 1 Standard Waveform with DifferentDwell Time Asmentioned above theDLEMmodel is chosento simulate jetting characteristics for the nanosilver inkHere an illustrative example is given to display the effect of

predicting the typical combinations of waveform parameterswith different dwell times (119879

1= 3sim8 us 119879

2= 2119879

1) and

the same amplitude (12V) The properties of nanosilver inkprovided by the LEED-PV Corporation have been listed inTable 1

In Figure 14 the predicted outflow and average velocitycurves with various dwell times are depicted We can observefrom Figure 14(a) that before the droplets rush out of thenozzle the depth of the inhaled meniscus is proportionalto dwell time From Figure 14(b) the time of the dropletappearing is postponed with the increase of dwell timeThat is both two simulation phenomena consist with thetheoretical analysis results in [37]

The comparisons between actual measured and predictedjetting characteristics with various dwell times are given inFigures 15(a) and 15(b) which both illustrate a parabolicchange rule with the increase of dwell time It should benoted that at the dwell time 119879

1= 56 us near the theoretical

optimum value 11987910158401= 52 us the droplet volume and velocity

simultaneously reach the maximum From the performanceof prediction accuracy listed in Table 4 the deviation of thepredicted jetting characteristics is within acceptable range

Mathematical Problems in Engineering 13

45

67

68

1012

1416

10121416

Volu

me (

pL)

91011121314151617

T1 (us)

T2(us)

(a)

45

67

68

1012

1416

3

4

5

6

Velo

city

(ms

)

3

35

4

45

5

55

6

T1 (us) T2

(us)

(b)

Figure 17 Predictive droplet volume and velocity with various dwell times1198791and119879

2 (a) predictive droplet volume and (b) predictive droplet

velocity

53 Prediction Case 2 Driving Waveform under RestrictiveCondition In this experiment an illustrative example underrestrictive conditions is presented to demonstrate the feasibil-ity and effectiveness of the proposed method for predictingjetting characteristics The dimensions of droplet generatorand properties of the nanosilver ink are listed in Table 1 Inorder to reach a high printing resolution and good quality ofdroplet impact with the substrate the restrictive conditionsinvolve that the droplet volume is smaller than 14 pL and thedroplet velocity is larger than 5ms

Based on the prediction procedure the predictive valuesof drop volumevelocity with various dwell times 119879

1and

1198792are listed in Table 5 According to the restrictive con-

ditions three combinations of parameters are found out1198791

= 5 us1198792

= 13 us 1198791

= 55 us1198792

= 14 us and1198791= 5 us119879

2= 75sim8 us After a small range search around

predictive values in the actual test the combination of 1198791=

49 us1198792

= 77 us is finally chosen The dynamic effect ofjetting characteristics driven by this combination is shown inFigure 16 From this figure the jetting characteristics satisfythe specified restrictive conditions quite well Meanwhilealmost no satellite droplet emerges after jetting the maindroplet

The 3D plot of predictive jetting characteristics withvarious dwell times 119879

1and 119879

2is shown in Figure 17 With

the increase of dwell time 1198791 peak positions of volume and

velocity have a tiny shift in 95sim115 us With the same dwelltime 119879

2 there is also a parabolic profile in predictive volume

and velocity

6 Conclusion

The lumped element modeling based on linear model ofpiezoelectric ceramic has been successfully used to simu-late piezoelectric droplet generator However the nonlinearmodel of piezoelectric ceramic is necessary to diminishdeviation between the simulated and measured results Inorder to characterize the nonlinear behavior of piezoelectric

Table 4 Performance of prediction accuracy

Deviation Value

Volume

Max 07135 (499)Min 0036 (013)Mean 02636Std 03316

Velocity

Max 02465 (502)Min 00143 (025)Mean 01095Std 01350

ceramic three models are introduced to simulate dropletgeneration In this work the droplet volumevelocity curvessimulated by dynamic model can carry sufficient informa-tion to distinguish nanosilver droplet formation processclearly Although the complexity of system increases theoutflow curves can obviously reflect the filament breaking-off stage that occurs which is coincident with sampledimages Such nonlinear LEM model is also successful inpredicting jetting characteristics under restrictive conditionsComparing the droplet volumevelocity curves with differentdwell time of high-frequency driving waveform the linearrelationship between dwell time and meniscus appearingmoment is discovered Meanwhile a parabolic change ruleof volumevelocity with the increase of dwell time is in goodagreement with present theoretical analysis

It should be noted that the proposed nonlinear model isfocused on relationship between the droplet volumevelocityand the dwell time of driving waveform while the voltageamplitude is also a decisive factor to droplet formationprocess This should be discussed in our future work

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

14 Mathematical Problems in Engineering

Table5Predictiv

edropletvolume(pL

)and

velocity(m

s)

1198791

1198792

665

775

885

995

10105

11115

12125

13135

14145

15155

4119881

11119

127

134

139

143

146

149

151

15147

142

Default

Default

Default

Default

Default

Default

Default

Default

119878371

395

417

439

459

478

491

500

489

471

451

435

Default

Default

Default

Default

Default

Default

Default

Default

45 119881

Default

117

125

134

143

15155

159

162

16157

153

147

139

Default

Default

Default

Default

Default

Default

119878Default

421

451

476

498

516

532

548

560

543

523

501

482

466

Default

Default

Default

Default

Default

Default

5119881

Default

Default

Default

133

143

151

157

163

167

168

166

163

157

15142

133

Default

Default

Default

Default

119878Default

Default

Default

480

507

531

553

572

587

600

583

563

540

525

509

498

Default

Default

Default

Default

55 119881

Default

Default

Default

Default

Default

151

159

165

168

169

171

169

165

16154

146

138

129

Default

Default

119878Default

Default

Default

Default

Default

524

550

572

593

609

624

601

571

551

537

523

509

496

Default

Default

6119881

Default

Default

Default

Default

Default

Default

Default

158

163

164

165

165

165

158

1495

141

130

119

108

97119878

Default

Default

Default

Default

Default

Default

Default

495

515

533

549

564

544

519

494

474

458

445

432

421

65 119881

Default

Default

Default

Default

Default

Default

Default

Default

Default

155

159

16155

150

143

134

125

114

103

95119878

Default

Default

Default

Default

Default

Default

Default

Default

Default

473

485

495

503

490

469

446

420

400

390

383

7119881

Default

Default

Default

Default

Default

Default

Default

Default

Default

Default

125

135

139

141

135

127

119

110

102

94119878

Default

Default

Default

Default

Default

Default

Default

Default

Default

Default

445

465

479

463

444

422

401

385

370

363

Defaults

areb

ecause

thec

ombinatio

nsof

parametersindrivingwaveform

areo

utof

theo

peratio

nrange

Mathematical Problems in Engineering 15

Authorsrsquo Contribution

Maowei He and Liling Sun contributed equally to this work

Acknowledgment

This research is partially supported by National Natural Sci-ence Foundation of China under Grants 61105067 7100107261174164 and 71271140

References

[1] M SinghHMHaverinen PDhagat andG E Jabbour ldquoInkjetprinting-process and its applicationsrdquo Advanced Materials vol22 no 6 pp 673ndash685 2010

[2] E Macdonald R Salas D Espalin et al ldquo3D printing for therapid prototyping of structural electronicsrdquo IEEE Access vol 2pp 234ndash242 2014

[3] S Umezu T Kitajima H Murase H Ohmori K Katahiraand Y Ito ldquoFabrication of living cell structure utilizing electro-static inkjet phenomenardquo in Proceedings of the 22nd IEEEInternational Conference on Micro Electro Mechanical Systems(MEMS rsquo09) pp 419ndash422 IEEE Sorrento Italy January 2009

[4] P Cosseddu L Basirico A Loi et al ldquoInkjet printed OrganicThin Film Transistors based tactile transducers for artificialrobotic skinrdquo in Proceedings of the 4th IEEE RAS and EMBSInternational Conference on Biomedical Robotics and Biomecha-tronics (BioRob rsquo12) pp 1907ndash1912 June 2012

[5] J Hwang A Wan and A Kahn ldquoEnergetics of metal-organicinterfaces new experiments and assessment of the fieldrdquoMate-rials Science and Engineering R Reports vol 64 no 1-2 pp 1ndash312009

[6] B J Kang and J H Oh ldquoGeometrical characterization of inkjet-printed conductive lines of nanosilver suspensions on a polymersubstraterdquoThin Solid Films vol 518 no 10 pp 2890ndash2896 2010

[7] F Villani P Vacca G Nenna et al ldquoInkjet printed polymerlayer on flexible substrate for OLED applicationsrdquo The Journalof Physical Chemistry C vol 113 no 30 pp 13398ndash13402 2009

[8] J-C Liou and F-G Tseng ldquoMulti-dimensional data registrationCMOSMEMS integrated inkjet printheadrdquo MicroelectronicEngineering vol 88 no 6 pp 888ndash901 2011

[9] J Miettinen K Kaija M Mantysalo et al ldquoMolded substratesfor inkjet printed modulesrdquo IEEE Transactions on Componentsand Packaging Technologies vol 32 no 2 pp 293ndash301 2009

[10] K Silverbrook ldquoPrinthead with multiple actuators in eachchamberrdquo US7708387 B2 2010

[11] V Piefort Finite element modelling of piezoelectric active struc-tures [PhD thesis] Department ofMechanical Engineering andRobotics Universite Libre de Bruxelles Brussels Belgium 2001

[12] L Sang Y Hong and F Wang ldquoInvestigation of viscosity effecton droplet formation in T-shaped microchannels by numericaland analytical methodsrdquo Microfluidics and Nanofluidics vol 6no 5 pp 621ndash635 2009

[13] J Liu S-H Tan Y F Yap M Y Ng and N-T NguyenldquoNumerical and experimental investigations of the formationprocess of ferrofluid dropletsrdquo Microfluidics and Nanofluidicsvol 11 no 2 pp 177ndash187 2011

[14] F Sarrazin K Loubiere L Prat C Gourdon T Bonometti andJ Magnaudet ldquoExperimental and numerical study of dropletshydrodynamics in MicroChannelrdquo AIChE Journal vol 52 no12 pp 4061ndash4070 2006

[15] X Q Xing D L Butler S H Ng Z Wang S Danyluk and CYang ldquoSimulation of droplet formation and coalescence usinglattice Boltzmann-based single-phasemodelrdquo Journal of Colloidand Interface Science vol 311 no 2 pp 609ndash618 2007

[16] Q Gallas R Holman T Nishida B Carroll M Sheplak and LCattafesta ldquoLumped element modeling of piezoelectric-drivensynthetic jet actuatorsrdquoAIAA Journal vol 41 no 2 pp 240ndash2472003

[17] A A Khalate X Bombois R Babuska H Wijshoff and RWaarsing ldquoOptimization-based feedforward control for a Drop-on-Demand inkjet printheadrdquo in Proceedings of the AmericanControl Conference (ACC rsquo10) pp 2182ndash2187 Baltimore MdUSA July 2010

[18] L Meirovitch Fundamentals of Vibrations McGraw-Hill NewYork NY USA 2001

[19] H Seitz and J Heinz ldquoModelling of a microfluidic device withpiezoelectric actuatorsrdquo Journal of Micromechanics and Micro-engineering vol 14 no 8 pp 1140ndash1147 2004

[20] S Kim J Sung andMH Lee ldquoPressure wave and fluid velocityin a bend-mode inkjet nozzle with double PZT actuatorsrdquoJournal of Thermal Science vol 22 no 1 pp 29ndash35 2013

[21] G Wassink Inkjet printhead performance enhancement by feed-forward input design based on two-port modeling [PhD thesis]Delft University of Technology 2007

[22] S Prasad SHorowitz QGallas B Sankar L Cattafesta andMSheplak ldquoTwo-port electroacoustic model of an axisymmet-ric piezoelectric composite platerdquo in Proceedings of the 43rdAIAAASMEASCEAHS Structures Structural DynamicsandMaterials Conference AIAA Paper 2002-1365 AIAA DenverColo USA April 2002

[23] D T BlackstockFundamentals of Physical Acoustics JohnWileyamp Sons New York NY USA 2000

[24] F M White Fluid Mechanics McGraw-Hill New York NYUSA 1979

[25] H Schempt andD R Yoerger ldquoStudy of dominant performancecharacteristics in robot transmissionsrdquo Journal of MechanicalDesign Transactions Of the ASME vol 115 no 3 pp 472ndash4821993

[26] E A Witmer ldquoElementary Bernoulli-Euler beam theoryrdquoMITUnified Engineering Course Notes vol 5 pp 114ndash164 1991-1992

[27] Y Yang and L Tang ldquoEquivalent circuit modeling of piezoelec-tric energy harvestersrdquo Journal of Intelligent Material Systemsand Structures vol 20 no 18 pp 2223ndash2235 2009

[28] F Constantinescu A G Gheorghe and M Nitescu ldquoNewcircuit models of power BAW resonatorsrdquo in Proceedings of theEuropean Microwave Integrated Circuit Conference October2007

[29] A Albareda and R PerezNon-Linear Behaviour of PiezoelectricCeramics vol 140ofSpringer Series inMaterials Science Springer2011

[30] J Nosek ldquoDrive level dependence of the resonant frequency inBAW quartz resonators and his modelingrdquo IEEE TransactionsonUltrasonics Ferroelectrics and Frequency Control vol 46 no4 pp 823ndash829 1999

[31] R Aigner N-H Huynh M Handtmann and S MarksteinerldquoBehavior of BAW devices at high power levelsrdquo in Proceedingsof the IEEE MTT-S International Microwave Symposium pp429ndash432 Long Beach Calif USA June 2005

[32] R S Ketcham G R Kline and K M Lakin ldquoPerformanceof TFR filters under elevated power conditionsrdquo in Proceedingsof 42nd the Annual Frequency Control Symposium pp 106ndash111Baltimore Md USA June 1988

16 Mathematical Problems in Engineering

[33] K-S Kwon ldquoWaveform design methods for piezo inkjet dis-pensers based on measured meniscus motionrdquo Journal of Micro-electromechanical Systems vol 18 no 5 pp 1118ndash1125 2009

[34] H-C Wu and H-J Lin ldquoEffects of actuating pressure wave-forms on the droplet behavior in a piezoelectric inkjetrdquoMateri-als Transactions vol 51 no 12 pp 2269ndash2276 2010

[35] KMinolta InkjetHeadApplicationNote-KM1024 Series KonicaMinolta Ij Technologies

[36] G Golub andV Pereyra ldquoSeparable nonlinear least squares thevariable projection method and its applicationsrdquo Inverse Prob-lems vol 19 no 2 pp R1ndashR26 2003

[37] H Wijshoff ldquoThe dynamics of the piezo inkjet printheadoperationrdquo Physics Reports vol 491 no 4-5 pp 77ndash177 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 Mathematical Problems in Engineering

0us0pL0ms

3us0pL0ms

6us0pL0ms

9us0pL0ms

12us6353pL736ms

15us9318pL612ms

18us11441pL572ms

18us 24us

30us14364pL498ms

33us14046pL510ms

36us13980pL520ms

39us13952pL525ms

42us13858pL525ms

45us13869pL520ms

48us13891pL524ms

51us13924pL522ms

54us

524ms13877pL

57us13840pL530ms

13127 pL548ms

14252pL520ms

27us14786pL504ms

Figure 16 A sequence of pictures of droplet falling from nozzle

Step 1 According to droplet generator parameters and prop-erties of fluid calculate most of the parameters in DLEM

Step 2 According to experimental measurements of dropletvolume driven by a standard waveform determine the rest ofthe adjustment coefficients and damping coefficients

Step 3 Based on the constructed DLEM predicate the valuesof droplet volume and velocity

Step 4 According to the need of applications and restrictiveconditions find out several groups of the appropriate param-eters of driving waveform frompredictive data lists of volumeand velocity

Step 5 After making a subtle adjustment in practice decidethe most appropriate parameters of driving waveform

52 Prediction Case 1 Standard Waveform with DifferentDwell Time Asmentioned above theDLEMmodel is chosento simulate jetting characteristics for the nanosilver inkHere an illustrative example is given to display the effect of

predicting the typical combinations of waveform parameterswith different dwell times (119879

1= 3sim8 us 119879

2= 2119879

1) and

the same amplitude (12V) The properties of nanosilver inkprovided by the LEED-PV Corporation have been listed inTable 1

In Figure 14 the predicted outflow and average velocitycurves with various dwell times are depicted We can observefrom Figure 14(a) that before the droplets rush out of thenozzle the depth of the inhaled meniscus is proportionalto dwell time From Figure 14(b) the time of the dropletappearing is postponed with the increase of dwell timeThat is both two simulation phenomena consist with thetheoretical analysis results in [37]

The comparisons between actual measured and predictedjetting characteristics with various dwell times are given inFigures 15(a) and 15(b) which both illustrate a parabolicchange rule with the increase of dwell time It should benoted that at the dwell time 119879

1= 56 us near the theoretical

optimum value 11987910158401= 52 us the droplet volume and velocity

simultaneously reach the maximum From the performanceof prediction accuracy listed in Table 4 the deviation of thepredicted jetting characteristics is within acceptable range

Mathematical Problems in Engineering 13

45

67

68

1012

1416

10121416

Volu

me (

pL)

91011121314151617

T1 (us)

T2(us)

(a)

45

67

68

1012

1416

3

4

5

6

Velo

city

(ms

)

3

35

4

45

5

55

6

T1 (us) T2

(us)

(b)

Figure 17 Predictive droplet volume and velocity with various dwell times1198791and119879

2 (a) predictive droplet volume and (b) predictive droplet

velocity

53 Prediction Case 2 Driving Waveform under RestrictiveCondition In this experiment an illustrative example underrestrictive conditions is presented to demonstrate the feasibil-ity and effectiveness of the proposed method for predictingjetting characteristics The dimensions of droplet generatorand properties of the nanosilver ink are listed in Table 1 Inorder to reach a high printing resolution and good quality ofdroplet impact with the substrate the restrictive conditionsinvolve that the droplet volume is smaller than 14 pL and thedroplet velocity is larger than 5ms

Based on the prediction procedure the predictive valuesof drop volumevelocity with various dwell times 119879

1and

1198792are listed in Table 5 According to the restrictive con-

ditions three combinations of parameters are found out1198791

= 5 us1198792

= 13 us 1198791

= 55 us1198792

= 14 us and1198791= 5 us119879

2= 75sim8 us After a small range search around

predictive values in the actual test the combination of 1198791=

49 us1198792

= 77 us is finally chosen The dynamic effect ofjetting characteristics driven by this combination is shown inFigure 16 From this figure the jetting characteristics satisfythe specified restrictive conditions quite well Meanwhilealmost no satellite droplet emerges after jetting the maindroplet

The 3D plot of predictive jetting characteristics withvarious dwell times 119879

1and 119879

2is shown in Figure 17 With

the increase of dwell time 1198791 peak positions of volume and

velocity have a tiny shift in 95sim115 us With the same dwelltime 119879

2 there is also a parabolic profile in predictive volume

and velocity

6 Conclusion

The lumped element modeling based on linear model ofpiezoelectric ceramic has been successfully used to simu-late piezoelectric droplet generator However the nonlinearmodel of piezoelectric ceramic is necessary to diminishdeviation between the simulated and measured results Inorder to characterize the nonlinear behavior of piezoelectric

Table 4 Performance of prediction accuracy

Deviation Value

Volume

Max 07135 (499)Min 0036 (013)Mean 02636Std 03316

Velocity

Max 02465 (502)Min 00143 (025)Mean 01095Std 01350

ceramic three models are introduced to simulate dropletgeneration In this work the droplet volumevelocity curvessimulated by dynamic model can carry sufficient informa-tion to distinguish nanosilver droplet formation processclearly Although the complexity of system increases theoutflow curves can obviously reflect the filament breaking-off stage that occurs which is coincident with sampledimages Such nonlinear LEM model is also successful inpredicting jetting characteristics under restrictive conditionsComparing the droplet volumevelocity curves with differentdwell time of high-frequency driving waveform the linearrelationship between dwell time and meniscus appearingmoment is discovered Meanwhile a parabolic change ruleof volumevelocity with the increase of dwell time is in goodagreement with present theoretical analysis

It should be noted that the proposed nonlinear model isfocused on relationship between the droplet volumevelocityand the dwell time of driving waveform while the voltageamplitude is also a decisive factor to droplet formationprocess This should be discussed in our future work

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

14 Mathematical Problems in Engineering

Table5Predictiv

edropletvolume(pL

)and

velocity(m

s)

1198791

1198792

665

775

885

995

10105

11115

12125

13135

14145

15155

4119881

11119

127

134

139

143

146

149

151

15147

142

Default

Default

Default

Default

Default

Default

Default

Default

119878371

395

417

439

459

478

491

500

489

471

451

435

Default

Default

Default

Default

Default

Default

Default

Default

45 119881

Default

117

125

134

143

15155

159

162

16157

153

147

139

Default

Default

Default

Default

Default

Default

119878Default

421

451

476

498

516

532

548

560

543

523

501

482

466

Default

Default

Default

Default

Default

Default

5119881

Default

Default

Default

133

143

151

157

163

167

168

166

163

157

15142

133

Default

Default

Default

Default

119878Default

Default

Default

480

507

531

553

572

587

600

583

563

540

525

509

498

Default

Default

Default

Default

55 119881

Default

Default

Default

Default

Default

151

159

165

168

169

171

169

165

16154

146

138

129

Default

Default

119878Default

Default

Default

Default

Default

524

550

572

593

609

624

601

571

551

537

523

509

496

Default

Default

6119881

Default

Default

Default

Default

Default

Default

Default

158

163

164

165

165

165

158

1495

141

130

119

108

97119878

Default

Default

Default

Default

Default

Default

Default

495

515

533

549

564

544

519

494

474

458

445

432

421

65 119881

Default

Default

Default

Default

Default

Default

Default

Default

Default

155

159

16155

150

143

134

125

114

103

95119878

Default

Default

Default

Default

Default

Default

Default

Default

Default

473

485

495

503

490

469

446

420

400

390

383

7119881

Default

Default

Default

Default

Default

Default

Default

Default

Default

Default

125

135

139

141

135

127

119

110

102

94119878

Default

Default

Default

Default

Default

Default

Default

Default

Default

Default

445

465

479

463

444

422

401

385

370

363

Defaults

areb

ecause

thec

ombinatio

nsof

parametersindrivingwaveform

areo

utof

theo

peratio

nrange

Mathematical Problems in Engineering 15

Authorsrsquo Contribution

Maowei He and Liling Sun contributed equally to this work

Acknowledgment

This research is partially supported by National Natural Sci-ence Foundation of China under Grants 61105067 7100107261174164 and 71271140

References

[1] M SinghHMHaverinen PDhagat andG E Jabbour ldquoInkjetprinting-process and its applicationsrdquo Advanced Materials vol22 no 6 pp 673ndash685 2010

[2] E Macdonald R Salas D Espalin et al ldquo3D printing for therapid prototyping of structural electronicsrdquo IEEE Access vol 2pp 234ndash242 2014

[3] S Umezu T Kitajima H Murase H Ohmori K Katahiraand Y Ito ldquoFabrication of living cell structure utilizing electro-static inkjet phenomenardquo in Proceedings of the 22nd IEEEInternational Conference on Micro Electro Mechanical Systems(MEMS rsquo09) pp 419ndash422 IEEE Sorrento Italy January 2009

[4] P Cosseddu L Basirico A Loi et al ldquoInkjet printed OrganicThin Film Transistors based tactile transducers for artificialrobotic skinrdquo in Proceedings of the 4th IEEE RAS and EMBSInternational Conference on Biomedical Robotics and Biomecha-tronics (BioRob rsquo12) pp 1907ndash1912 June 2012

[5] J Hwang A Wan and A Kahn ldquoEnergetics of metal-organicinterfaces new experiments and assessment of the fieldrdquoMate-rials Science and Engineering R Reports vol 64 no 1-2 pp 1ndash312009

[6] B J Kang and J H Oh ldquoGeometrical characterization of inkjet-printed conductive lines of nanosilver suspensions on a polymersubstraterdquoThin Solid Films vol 518 no 10 pp 2890ndash2896 2010

[7] F Villani P Vacca G Nenna et al ldquoInkjet printed polymerlayer on flexible substrate for OLED applicationsrdquo The Journalof Physical Chemistry C vol 113 no 30 pp 13398ndash13402 2009

[8] J-C Liou and F-G Tseng ldquoMulti-dimensional data registrationCMOSMEMS integrated inkjet printheadrdquo MicroelectronicEngineering vol 88 no 6 pp 888ndash901 2011

[9] J Miettinen K Kaija M Mantysalo et al ldquoMolded substratesfor inkjet printed modulesrdquo IEEE Transactions on Componentsand Packaging Technologies vol 32 no 2 pp 293ndash301 2009

[10] K Silverbrook ldquoPrinthead with multiple actuators in eachchamberrdquo US7708387 B2 2010

[11] V Piefort Finite element modelling of piezoelectric active struc-tures [PhD thesis] Department ofMechanical Engineering andRobotics Universite Libre de Bruxelles Brussels Belgium 2001

[12] L Sang Y Hong and F Wang ldquoInvestigation of viscosity effecton droplet formation in T-shaped microchannels by numericaland analytical methodsrdquo Microfluidics and Nanofluidics vol 6no 5 pp 621ndash635 2009

[13] J Liu S-H Tan Y F Yap M Y Ng and N-T NguyenldquoNumerical and experimental investigations of the formationprocess of ferrofluid dropletsrdquo Microfluidics and Nanofluidicsvol 11 no 2 pp 177ndash187 2011

[14] F Sarrazin K Loubiere L Prat C Gourdon T Bonometti andJ Magnaudet ldquoExperimental and numerical study of dropletshydrodynamics in MicroChannelrdquo AIChE Journal vol 52 no12 pp 4061ndash4070 2006

[15] X Q Xing D L Butler S H Ng Z Wang S Danyluk and CYang ldquoSimulation of droplet formation and coalescence usinglattice Boltzmann-based single-phasemodelrdquo Journal of Colloidand Interface Science vol 311 no 2 pp 609ndash618 2007

[16] Q Gallas R Holman T Nishida B Carroll M Sheplak and LCattafesta ldquoLumped element modeling of piezoelectric-drivensynthetic jet actuatorsrdquoAIAA Journal vol 41 no 2 pp 240ndash2472003

[17] A A Khalate X Bombois R Babuska H Wijshoff and RWaarsing ldquoOptimization-based feedforward control for a Drop-on-Demand inkjet printheadrdquo in Proceedings of the AmericanControl Conference (ACC rsquo10) pp 2182ndash2187 Baltimore MdUSA July 2010

[18] L Meirovitch Fundamentals of Vibrations McGraw-Hill NewYork NY USA 2001

[19] H Seitz and J Heinz ldquoModelling of a microfluidic device withpiezoelectric actuatorsrdquo Journal of Micromechanics and Micro-engineering vol 14 no 8 pp 1140ndash1147 2004

[20] S Kim J Sung andMH Lee ldquoPressure wave and fluid velocityin a bend-mode inkjet nozzle with double PZT actuatorsrdquoJournal of Thermal Science vol 22 no 1 pp 29ndash35 2013

[21] G Wassink Inkjet printhead performance enhancement by feed-forward input design based on two-port modeling [PhD thesis]Delft University of Technology 2007

[22] S Prasad SHorowitz QGallas B Sankar L Cattafesta andMSheplak ldquoTwo-port electroacoustic model of an axisymmet-ric piezoelectric composite platerdquo in Proceedings of the 43rdAIAAASMEASCEAHS Structures Structural DynamicsandMaterials Conference AIAA Paper 2002-1365 AIAA DenverColo USA April 2002

[23] D T BlackstockFundamentals of Physical Acoustics JohnWileyamp Sons New York NY USA 2000

[24] F M White Fluid Mechanics McGraw-Hill New York NYUSA 1979

[25] H Schempt andD R Yoerger ldquoStudy of dominant performancecharacteristics in robot transmissionsrdquo Journal of MechanicalDesign Transactions Of the ASME vol 115 no 3 pp 472ndash4821993

[26] E A Witmer ldquoElementary Bernoulli-Euler beam theoryrdquoMITUnified Engineering Course Notes vol 5 pp 114ndash164 1991-1992

[27] Y Yang and L Tang ldquoEquivalent circuit modeling of piezoelec-tric energy harvestersrdquo Journal of Intelligent Material Systemsand Structures vol 20 no 18 pp 2223ndash2235 2009

[28] F Constantinescu A G Gheorghe and M Nitescu ldquoNewcircuit models of power BAW resonatorsrdquo in Proceedings of theEuropean Microwave Integrated Circuit Conference October2007

[29] A Albareda and R PerezNon-Linear Behaviour of PiezoelectricCeramics vol 140ofSpringer Series inMaterials Science Springer2011

[30] J Nosek ldquoDrive level dependence of the resonant frequency inBAW quartz resonators and his modelingrdquo IEEE TransactionsonUltrasonics Ferroelectrics and Frequency Control vol 46 no4 pp 823ndash829 1999

[31] R Aigner N-H Huynh M Handtmann and S MarksteinerldquoBehavior of BAW devices at high power levelsrdquo in Proceedingsof the IEEE MTT-S International Microwave Symposium pp429ndash432 Long Beach Calif USA June 2005

[32] R S Ketcham G R Kline and K M Lakin ldquoPerformanceof TFR filters under elevated power conditionsrdquo in Proceedingsof 42nd the Annual Frequency Control Symposium pp 106ndash111Baltimore Md USA June 1988

16 Mathematical Problems in Engineering

[33] K-S Kwon ldquoWaveform design methods for piezo inkjet dis-pensers based on measured meniscus motionrdquo Journal of Micro-electromechanical Systems vol 18 no 5 pp 1118ndash1125 2009

[34] H-C Wu and H-J Lin ldquoEffects of actuating pressure wave-forms on the droplet behavior in a piezoelectric inkjetrdquoMateri-als Transactions vol 51 no 12 pp 2269ndash2276 2010

[35] KMinolta InkjetHeadApplicationNote-KM1024 Series KonicaMinolta Ij Technologies

[36] G Golub andV Pereyra ldquoSeparable nonlinear least squares thevariable projection method and its applicationsrdquo Inverse Prob-lems vol 19 no 2 pp R1ndashR26 2003

[37] H Wijshoff ldquoThe dynamics of the piezo inkjet printheadoperationrdquo Physics Reports vol 491 no 4-5 pp 77ndash177 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 13

45

67

68

1012

1416

10121416

Volu

me (

pL)

91011121314151617

T1 (us)

T2(us)

(a)

45

67

68

1012

1416

3

4

5

6

Velo

city

(ms

)

3

35

4

45

5

55

6

T1 (us) T2

(us)

(b)

Figure 17 Predictive droplet volume and velocity with various dwell times1198791and119879

2 (a) predictive droplet volume and (b) predictive droplet

velocity

53 Prediction Case 2 Driving Waveform under RestrictiveCondition In this experiment an illustrative example underrestrictive conditions is presented to demonstrate the feasibil-ity and effectiveness of the proposed method for predictingjetting characteristics The dimensions of droplet generatorand properties of the nanosilver ink are listed in Table 1 Inorder to reach a high printing resolution and good quality ofdroplet impact with the substrate the restrictive conditionsinvolve that the droplet volume is smaller than 14 pL and thedroplet velocity is larger than 5ms

Based on the prediction procedure the predictive valuesof drop volumevelocity with various dwell times 119879

1and

1198792are listed in Table 5 According to the restrictive con-

ditions three combinations of parameters are found out1198791

= 5 us1198792

= 13 us 1198791

= 55 us1198792

= 14 us and1198791= 5 us119879

2= 75sim8 us After a small range search around

predictive values in the actual test the combination of 1198791=

49 us1198792

= 77 us is finally chosen The dynamic effect ofjetting characteristics driven by this combination is shown inFigure 16 From this figure the jetting characteristics satisfythe specified restrictive conditions quite well Meanwhilealmost no satellite droplet emerges after jetting the maindroplet

The 3D plot of predictive jetting characteristics withvarious dwell times 119879

1and 119879

2is shown in Figure 17 With

the increase of dwell time 1198791 peak positions of volume and

velocity have a tiny shift in 95sim115 us With the same dwelltime 119879

2 there is also a parabolic profile in predictive volume

and velocity

6 Conclusion

The lumped element modeling based on linear model ofpiezoelectric ceramic has been successfully used to simu-late piezoelectric droplet generator However the nonlinearmodel of piezoelectric ceramic is necessary to diminishdeviation between the simulated and measured results Inorder to characterize the nonlinear behavior of piezoelectric

Table 4 Performance of prediction accuracy

Deviation Value

Volume

Max 07135 (499)Min 0036 (013)Mean 02636Std 03316

Velocity

Max 02465 (502)Min 00143 (025)Mean 01095Std 01350

ceramic three models are introduced to simulate dropletgeneration In this work the droplet volumevelocity curvessimulated by dynamic model can carry sufficient informa-tion to distinguish nanosilver droplet formation processclearly Although the complexity of system increases theoutflow curves can obviously reflect the filament breaking-off stage that occurs which is coincident with sampledimages Such nonlinear LEM model is also successful inpredicting jetting characteristics under restrictive conditionsComparing the droplet volumevelocity curves with differentdwell time of high-frequency driving waveform the linearrelationship between dwell time and meniscus appearingmoment is discovered Meanwhile a parabolic change ruleof volumevelocity with the increase of dwell time is in goodagreement with present theoretical analysis

It should be noted that the proposed nonlinear model isfocused on relationship between the droplet volumevelocityand the dwell time of driving waveform while the voltageamplitude is also a decisive factor to droplet formationprocess This should be discussed in our future work

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

14 Mathematical Problems in Engineering

Table5Predictiv

edropletvolume(pL

)and

velocity(m

s)

1198791

1198792

665

775

885

995

10105

11115

12125

13135

14145

15155

4119881

11119

127

134

139

143

146

149

151

15147

142

Default

Default

Default

Default

Default

Default

Default

Default

119878371

395

417

439

459

478

491

500

489

471

451

435

Default

Default

Default

Default

Default

Default

Default

Default

45 119881

Default

117

125

134

143

15155

159

162

16157

153

147

139

Default

Default

Default

Default

Default

Default

119878Default

421

451

476

498

516

532

548

560

543

523

501

482

466

Default

Default

Default

Default

Default

Default

5119881

Default

Default

Default

133

143

151

157

163

167

168

166

163

157

15142

133

Default

Default

Default

Default

119878Default

Default

Default

480

507

531

553

572

587

600

583

563

540

525

509

498

Default

Default

Default

Default

55 119881

Default

Default

Default

Default

Default

151

159

165

168

169

171

169

165

16154

146

138

129

Default

Default

119878Default

Default

Default

Default

Default

524

550

572

593

609

624

601

571

551

537

523

509

496

Default

Default

6119881

Default

Default

Default

Default

Default

Default

Default

158

163

164

165

165

165

158

1495

141

130

119

108

97119878

Default

Default

Default

Default

Default

Default

Default

495

515

533

549

564

544

519

494

474

458

445

432

421

65 119881

Default

Default

Default

Default

Default

Default

Default

Default

Default

155

159

16155

150

143

134

125

114

103

95119878

Default

Default

Default

Default

Default

Default

Default

Default

Default

473

485

495

503

490

469

446

420

400

390

383

7119881

Default

Default

Default

Default

Default

Default

Default

Default

Default

Default

125

135

139

141

135

127

119

110

102

94119878

Default

Default

Default

Default

Default

Default

Default

Default

Default

Default

445

465

479

463

444

422

401

385

370

363

Defaults

areb

ecause

thec

ombinatio

nsof

parametersindrivingwaveform

areo

utof

theo

peratio

nrange

Mathematical Problems in Engineering 15

Authorsrsquo Contribution

Maowei He and Liling Sun contributed equally to this work

Acknowledgment

This research is partially supported by National Natural Sci-ence Foundation of China under Grants 61105067 7100107261174164 and 71271140

References

[1] M SinghHMHaverinen PDhagat andG E Jabbour ldquoInkjetprinting-process and its applicationsrdquo Advanced Materials vol22 no 6 pp 673ndash685 2010

[2] E Macdonald R Salas D Espalin et al ldquo3D printing for therapid prototyping of structural electronicsrdquo IEEE Access vol 2pp 234ndash242 2014

[3] S Umezu T Kitajima H Murase H Ohmori K Katahiraand Y Ito ldquoFabrication of living cell structure utilizing electro-static inkjet phenomenardquo in Proceedings of the 22nd IEEEInternational Conference on Micro Electro Mechanical Systems(MEMS rsquo09) pp 419ndash422 IEEE Sorrento Italy January 2009

[4] P Cosseddu L Basirico A Loi et al ldquoInkjet printed OrganicThin Film Transistors based tactile transducers for artificialrobotic skinrdquo in Proceedings of the 4th IEEE RAS and EMBSInternational Conference on Biomedical Robotics and Biomecha-tronics (BioRob rsquo12) pp 1907ndash1912 June 2012

[5] J Hwang A Wan and A Kahn ldquoEnergetics of metal-organicinterfaces new experiments and assessment of the fieldrdquoMate-rials Science and Engineering R Reports vol 64 no 1-2 pp 1ndash312009

[6] B J Kang and J H Oh ldquoGeometrical characterization of inkjet-printed conductive lines of nanosilver suspensions on a polymersubstraterdquoThin Solid Films vol 518 no 10 pp 2890ndash2896 2010

[7] F Villani P Vacca G Nenna et al ldquoInkjet printed polymerlayer on flexible substrate for OLED applicationsrdquo The Journalof Physical Chemistry C vol 113 no 30 pp 13398ndash13402 2009

[8] J-C Liou and F-G Tseng ldquoMulti-dimensional data registrationCMOSMEMS integrated inkjet printheadrdquo MicroelectronicEngineering vol 88 no 6 pp 888ndash901 2011

[9] J Miettinen K Kaija M Mantysalo et al ldquoMolded substratesfor inkjet printed modulesrdquo IEEE Transactions on Componentsand Packaging Technologies vol 32 no 2 pp 293ndash301 2009

[10] K Silverbrook ldquoPrinthead with multiple actuators in eachchamberrdquo US7708387 B2 2010

[11] V Piefort Finite element modelling of piezoelectric active struc-tures [PhD thesis] Department ofMechanical Engineering andRobotics Universite Libre de Bruxelles Brussels Belgium 2001

[12] L Sang Y Hong and F Wang ldquoInvestigation of viscosity effecton droplet formation in T-shaped microchannels by numericaland analytical methodsrdquo Microfluidics and Nanofluidics vol 6no 5 pp 621ndash635 2009

[13] J Liu S-H Tan Y F Yap M Y Ng and N-T NguyenldquoNumerical and experimental investigations of the formationprocess of ferrofluid dropletsrdquo Microfluidics and Nanofluidicsvol 11 no 2 pp 177ndash187 2011

[14] F Sarrazin K Loubiere L Prat C Gourdon T Bonometti andJ Magnaudet ldquoExperimental and numerical study of dropletshydrodynamics in MicroChannelrdquo AIChE Journal vol 52 no12 pp 4061ndash4070 2006

[15] X Q Xing D L Butler S H Ng Z Wang S Danyluk and CYang ldquoSimulation of droplet formation and coalescence usinglattice Boltzmann-based single-phasemodelrdquo Journal of Colloidand Interface Science vol 311 no 2 pp 609ndash618 2007

[16] Q Gallas R Holman T Nishida B Carroll M Sheplak and LCattafesta ldquoLumped element modeling of piezoelectric-drivensynthetic jet actuatorsrdquoAIAA Journal vol 41 no 2 pp 240ndash2472003

[17] A A Khalate X Bombois R Babuska H Wijshoff and RWaarsing ldquoOptimization-based feedforward control for a Drop-on-Demand inkjet printheadrdquo in Proceedings of the AmericanControl Conference (ACC rsquo10) pp 2182ndash2187 Baltimore MdUSA July 2010

[18] L Meirovitch Fundamentals of Vibrations McGraw-Hill NewYork NY USA 2001

[19] H Seitz and J Heinz ldquoModelling of a microfluidic device withpiezoelectric actuatorsrdquo Journal of Micromechanics and Micro-engineering vol 14 no 8 pp 1140ndash1147 2004

[20] S Kim J Sung andMH Lee ldquoPressure wave and fluid velocityin a bend-mode inkjet nozzle with double PZT actuatorsrdquoJournal of Thermal Science vol 22 no 1 pp 29ndash35 2013

[21] G Wassink Inkjet printhead performance enhancement by feed-forward input design based on two-port modeling [PhD thesis]Delft University of Technology 2007

[22] S Prasad SHorowitz QGallas B Sankar L Cattafesta andMSheplak ldquoTwo-port electroacoustic model of an axisymmet-ric piezoelectric composite platerdquo in Proceedings of the 43rdAIAAASMEASCEAHS Structures Structural DynamicsandMaterials Conference AIAA Paper 2002-1365 AIAA DenverColo USA April 2002

[23] D T BlackstockFundamentals of Physical Acoustics JohnWileyamp Sons New York NY USA 2000

[24] F M White Fluid Mechanics McGraw-Hill New York NYUSA 1979

[25] H Schempt andD R Yoerger ldquoStudy of dominant performancecharacteristics in robot transmissionsrdquo Journal of MechanicalDesign Transactions Of the ASME vol 115 no 3 pp 472ndash4821993

[26] E A Witmer ldquoElementary Bernoulli-Euler beam theoryrdquoMITUnified Engineering Course Notes vol 5 pp 114ndash164 1991-1992

[27] Y Yang and L Tang ldquoEquivalent circuit modeling of piezoelec-tric energy harvestersrdquo Journal of Intelligent Material Systemsand Structures vol 20 no 18 pp 2223ndash2235 2009

[28] F Constantinescu A G Gheorghe and M Nitescu ldquoNewcircuit models of power BAW resonatorsrdquo in Proceedings of theEuropean Microwave Integrated Circuit Conference October2007

[29] A Albareda and R PerezNon-Linear Behaviour of PiezoelectricCeramics vol 140ofSpringer Series inMaterials Science Springer2011

[30] J Nosek ldquoDrive level dependence of the resonant frequency inBAW quartz resonators and his modelingrdquo IEEE TransactionsonUltrasonics Ferroelectrics and Frequency Control vol 46 no4 pp 823ndash829 1999

[31] R Aigner N-H Huynh M Handtmann and S MarksteinerldquoBehavior of BAW devices at high power levelsrdquo in Proceedingsof the IEEE MTT-S International Microwave Symposium pp429ndash432 Long Beach Calif USA June 2005

[32] R S Ketcham G R Kline and K M Lakin ldquoPerformanceof TFR filters under elevated power conditionsrdquo in Proceedingsof 42nd the Annual Frequency Control Symposium pp 106ndash111Baltimore Md USA June 1988

16 Mathematical Problems in Engineering

[33] K-S Kwon ldquoWaveform design methods for piezo inkjet dis-pensers based on measured meniscus motionrdquo Journal of Micro-electromechanical Systems vol 18 no 5 pp 1118ndash1125 2009

[34] H-C Wu and H-J Lin ldquoEffects of actuating pressure wave-forms on the droplet behavior in a piezoelectric inkjetrdquoMateri-als Transactions vol 51 no 12 pp 2269ndash2276 2010

[35] KMinolta InkjetHeadApplicationNote-KM1024 Series KonicaMinolta Ij Technologies

[36] G Golub andV Pereyra ldquoSeparable nonlinear least squares thevariable projection method and its applicationsrdquo Inverse Prob-lems vol 19 no 2 pp R1ndashR26 2003

[37] H Wijshoff ldquoThe dynamics of the piezo inkjet printheadoperationrdquo Physics Reports vol 491 no 4-5 pp 77ndash177 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

14 Mathematical Problems in Engineering

Table5Predictiv

edropletvolume(pL

)and

velocity(m

s)

1198791

1198792

665

775

885

995

10105

11115

12125

13135

14145

15155

4119881

11119

127

134

139

143

146

149

151

15147

142

Default

Default

Default

Default

Default

Default

Default

Default

119878371

395

417

439

459

478

491

500

489

471

451

435

Default

Default

Default

Default

Default

Default

Default

Default

45 119881

Default

117

125

134

143

15155

159

162

16157

153

147

139

Default

Default

Default

Default

Default

Default

119878Default

421

451

476

498

516

532

548

560

543

523

501

482

466

Default

Default

Default

Default

Default

Default

5119881

Default

Default

Default

133

143

151

157

163

167

168

166

163

157

15142

133

Default

Default

Default

Default

119878Default

Default

Default

480

507

531

553

572

587

600

583

563

540

525

509

498

Default

Default

Default

Default

55 119881

Default

Default

Default

Default

Default

151

159

165

168

169

171

169

165

16154

146

138

129

Default

Default

119878Default

Default

Default

Default

Default

524

550

572

593

609

624

601

571

551

537

523

509

496

Default

Default

6119881

Default

Default

Default

Default

Default

Default

Default

158

163

164

165

165

165

158

1495

141

130

119

108

97119878

Default

Default

Default

Default

Default

Default

Default

495

515

533

549

564

544

519

494

474

458

445

432

421

65 119881

Default

Default

Default

Default

Default

Default

Default

Default

Default

155

159

16155

150

143

134

125

114

103

95119878

Default

Default

Default

Default

Default

Default

Default

Default

Default

473

485

495

503

490

469

446

420

400

390

383

7119881

Default

Default

Default

Default

Default

Default

Default

Default

Default

Default

125

135

139

141

135

127

119

110

102

94119878

Default

Default

Default

Default

Default

Default

Default

Default

Default

Default

445

465

479

463

444

422

401

385

370

363

Defaults

areb

ecause

thec

ombinatio

nsof

parametersindrivingwaveform

areo

utof

theo

peratio

nrange

Mathematical Problems in Engineering 15

Authorsrsquo Contribution

Maowei He and Liling Sun contributed equally to this work

Acknowledgment

This research is partially supported by National Natural Sci-ence Foundation of China under Grants 61105067 7100107261174164 and 71271140

References

[1] M SinghHMHaverinen PDhagat andG E Jabbour ldquoInkjetprinting-process and its applicationsrdquo Advanced Materials vol22 no 6 pp 673ndash685 2010

[2] E Macdonald R Salas D Espalin et al ldquo3D printing for therapid prototyping of structural electronicsrdquo IEEE Access vol 2pp 234ndash242 2014

[3] S Umezu T Kitajima H Murase H Ohmori K Katahiraand Y Ito ldquoFabrication of living cell structure utilizing electro-static inkjet phenomenardquo in Proceedings of the 22nd IEEEInternational Conference on Micro Electro Mechanical Systems(MEMS rsquo09) pp 419ndash422 IEEE Sorrento Italy January 2009

[4] P Cosseddu L Basirico A Loi et al ldquoInkjet printed OrganicThin Film Transistors based tactile transducers for artificialrobotic skinrdquo in Proceedings of the 4th IEEE RAS and EMBSInternational Conference on Biomedical Robotics and Biomecha-tronics (BioRob rsquo12) pp 1907ndash1912 June 2012

[5] J Hwang A Wan and A Kahn ldquoEnergetics of metal-organicinterfaces new experiments and assessment of the fieldrdquoMate-rials Science and Engineering R Reports vol 64 no 1-2 pp 1ndash312009

[6] B J Kang and J H Oh ldquoGeometrical characterization of inkjet-printed conductive lines of nanosilver suspensions on a polymersubstraterdquoThin Solid Films vol 518 no 10 pp 2890ndash2896 2010

[7] F Villani P Vacca G Nenna et al ldquoInkjet printed polymerlayer on flexible substrate for OLED applicationsrdquo The Journalof Physical Chemistry C vol 113 no 30 pp 13398ndash13402 2009

[8] J-C Liou and F-G Tseng ldquoMulti-dimensional data registrationCMOSMEMS integrated inkjet printheadrdquo MicroelectronicEngineering vol 88 no 6 pp 888ndash901 2011

[9] J Miettinen K Kaija M Mantysalo et al ldquoMolded substratesfor inkjet printed modulesrdquo IEEE Transactions on Componentsand Packaging Technologies vol 32 no 2 pp 293ndash301 2009

[10] K Silverbrook ldquoPrinthead with multiple actuators in eachchamberrdquo US7708387 B2 2010

[11] V Piefort Finite element modelling of piezoelectric active struc-tures [PhD thesis] Department ofMechanical Engineering andRobotics Universite Libre de Bruxelles Brussels Belgium 2001

[12] L Sang Y Hong and F Wang ldquoInvestigation of viscosity effecton droplet formation in T-shaped microchannels by numericaland analytical methodsrdquo Microfluidics and Nanofluidics vol 6no 5 pp 621ndash635 2009

[13] J Liu S-H Tan Y F Yap M Y Ng and N-T NguyenldquoNumerical and experimental investigations of the formationprocess of ferrofluid dropletsrdquo Microfluidics and Nanofluidicsvol 11 no 2 pp 177ndash187 2011

[14] F Sarrazin K Loubiere L Prat C Gourdon T Bonometti andJ Magnaudet ldquoExperimental and numerical study of dropletshydrodynamics in MicroChannelrdquo AIChE Journal vol 52 no12 pp 4061ndash4070 2006

[15] X Q Xing D L Butler S H Ng Z Wang S Danyluk and CYang ldquoSimulation of droplet formation and coalescence usinglattice Boltzmann-based single-phasemodelrdquo Journal of Colloidand Interface Science vol 311 no 2 pp 609ndash618 2007

[16] Q Gallas R Holman T Nishida B Carroll M Sheplak and LCattafesta ldquoLumped element modeling of piezoelectric-drivensynthetic jet actuatorsrdquoAIAA Journal vol 41 no 2 pp 240ndash2472003

[17] A A Khalate X Bombois R Babuska H Wijshoff and RWaarsing ldquoOptimization-based feedforward control for a Drop-on-Demand inkjet printheadrdquo in Proceedings of the AmericanControl Conference (ACC rsquo10) pp 2182ndash2187 Baltimore MdUSA July 2010

[18] L Meirovitch Fundamentals of Vibrations McGraw-Hill NewYork NY USA 2001

[19] H Seitz and J Heinz ldquoModelling of a microfluidic device withpiezoelectric actuatorsrdquo Journal of Micromechanics and Micro-engineering vol 14 no 8 pp 1140ndash1147 2004

[20] S Kim J Sung andMH Lee ldquoPressure wave and fluid velocityin a bend-mode inkjet nozzle with double PZT actuatorsrdquoJournal of Thermal Science vol 22 no 1 pp 29ndash35 2013

[21] G Wassink Inkjet printhead performance enhancement by feed-forward input design based on two-port modeling [PhD thesis]Delft University of Technology 2007

[22] S Prasad SHorowitz QGallas B Sankar L Cattafesta andMSheplak ldquoTwo-port electroacoustic model of an axisymmet-ric piezoelectric composite platerdquo in Proceedings of the 43rdAIAAASMEASCEAHS Structures Structural DynamicsandMaterials Conference AIAA Paper 2002-1365 AIAA DenverColo USA April 2002

[23] D T BlackstockFundamentals of Physical Acoustics JohnWileyamp Sons New York NY USA 2000

[24] F M White Fluid Mechanics McGraw-Hill New York NYUSA 1979

[25] H Schempt andD R Yoerger ldquoStudy of dominant performancecharacteristics in robot transmissionsrdquo Journal of MechanicalDesign Transactions Of the ASME vol 115 no 3 pp 472ndash4821993

[26] E A Witmer ldquoElementary Bernoulli-Euler beam theoryrdquoMITUnified Engineering Course Notes vol 5 pp 114ndash164 1991-1992

[27] Y Yang and L Tang ldquoEquivalent circuit modeling of piezoelec-tric energy harvestersrdquo Journal of Intelligent Material Systemsand Structures vol 20 no 18 pp 2223ndash2235 2009

[28] F Constantinescu A G Gheorghe and M Nitescu ldquoNewcircuit models of power BAW resonatorsrdquo in Proceedings of theEuropean Microwave Integrated Circuit Conference October2007

[29] A Albareda and R PerezNon-Linear Behaviour of PiezoelectricCeramics vol 140ofSpringer Series inMaterials Science Springer2011

[30] J Nosek ldquoDrive level dependence of the resonant frequency inBAW quartz resonators and his modelingrdquo IEEE TransactionsonUltrasonics Ferroelectrics and Frequency Control vol 46 no4 pp 823ndash829 1999

[31] R Aigner N-H Huynh M Handtmann and S MarksteinerldquoBehavior of BAW devices at high power levelsrdquo in Proceedingsof the IEEE MTT-S International Microwave Symposium pp429ndash432 Long Beach Calif USA June 2005

[32] R S Ketcham G R Kline and K M Lakin ldquoPerformanceof TFR filters under elevated power conditionsrdquo in Proceedingsof 42nd the Annual Frequency Control Symposium pp 106ndash111Baltimore Md USA June 1988

16 Mathematical Problems in Engineering

[33] K-S Kwon ldquoWaveform design methods for piezo inkjet dis-pensers based on measured meniscus motionrdquo Journal of Micro-electromechanical Systems vol 18 no 5 pp 1118ndash1125 2009

[34] H-C Wu and H-J Lin ldquoEffects of actuating pressure wave-forms on the droplet behavior in a piezoelectric inkjetrdquoMateri-als Transactions vol 51 no 12 pp 2269ndash2276 2010

[35] KMinolta InkjetHeadApplicationNote-KM1024 Series KonicaMinolta Ij Technologies

[36] G Golub andV Pereyra ldquoSeparable nonlinear least squares thevariable projection method and its applicationsrdquo Inverse Prob-lems vol 19 no 2 pp R1ndashR26 2003

[37] H Wijshoff ldquoThe dynamics of the piezo inkjet printheadoperationrdquo Physics Reports vol 491 no 4-5 pp 77ndash177 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 15

Authorsrsquo Contribution

Maowei He and Liling Sun contributed equally to this work

Acknowledgment

This research is partially supported by National Natural Sci-ence Foundation of China under Grants 61105067 7100107261174164 and 71271140

References

[1] M SinghHMHaverinen PDhagat andG E Jabbour ldquoInkjetprinting-process and its applicationsrdquo Advanced Materials vol22 no 6 pp 673ndash685 2010

[2] E Macdonald R Salas D Espalin et al ldquo3D printing for therapid prototyping of structural electronicsrdquo IEEE Access vol 2pp 234ndash242 2014

[3] S Umezu T Kitajima H Murase H Ohmori K Katahiraand Y Ito ldquoFabrication of living cell structure utilizing electro-static inkjet phenomenardquo in Proceedings of the 22nd IEEEInternational Conference on Micro Electro Mechanical Systems(MEMS rsquo09) pp 419ndash422 IEEE Sorrento Italy January 2009

[4] P Cosseddu L Basirico A Loi et al ldquoInkjet printed OrganicThin Film Transistors based tactile transducers for artificialrobotic skinrdquo in Proceedings of the 4th IEEE RAS and EMBSInternational Conference on Biomedical Robotics and Biomecha-tronics (BioRob rsquo12) pp 1907ndash1912 June 2012

[5] J Hwang A Wan and A Kahn ldquoEnergetics of metal-organicinterfaces new experiments and assessment of the fieldrdquoMate-rials Science and Engineering R Reports vol 64 no 1-2 pp 1ndash312009

[6] B J Kang and J H Oh ldquoGeometrical characterization of inkjet-printed conductive lines of nanosilver suspensions on a polymersubstraterdquoThin Solid Films vol 518 no 10 pp 2890ndash2896 2010

[7] F Villani P Vacca G Nenna et al ldquoInkjet printed polymerlayer on flexible substrate for OLED applicationsrdquo The Journalof Physical Chemistry C vol 113 no 30 pp 13398ndash13402 2009

[8] J-C Liou and F-G Tseng ldquoMulti-dimensional data registrationCMOSMEMS integrated inkjet printheadrdquo MicroelectronicEngineering vol 88 no 6 pp 888ndash901 2011

[9] J Miettinen K Kaija M Mantysalo et al ldquoMolded substratesfor inkjet printed modulesrdquo IEEE Transactions on Componentsand Packaging Technologies vol 32 no 2 pp 293ndash301 2009

[10] K Silverbrook ldquoPrinthead with multiple actuators in eachchamberrdquo US7708387 B2 2010

[11] V Piefort Finite element modelling of piezoelectric active struc-tures [PhD thesis] Department ofMechanical Engineering andRobotics Universite Libre de Bruxelles Brussels Belgium 2001

[12] L Sang Y Hong and F Wang ldquoInvestigation of viscosity effecton droplet formation in T-shaped microchannels by numericaland analytical methodsrdquo Microfluidics and Nanofluidics vol 6no 5 pp 621ndash635 2009

[13] J Liu S-H Tan Y F Yap M Y Ng and N-T NguyenldquoNumerical and experimental investigations of the formationprocess of ferrofluid dropletsrdquo Microfluidics and Nanofluidicsvol 11 no 2 pp 177ndash187 2011

[14] F Sarrazin K Loubiere L Prat C Gourdon T Bonometti andJ Magnaudet ldquoExperimental and numerical study of dropletshydrodynamics in MicroChannelrdquo AIChE Journal vol 52 no12 pp 4061ndash4070 2006

[15] X Q Xing D L Butler S H Ng Z Wang S Danyluk and CYang ldquoSimulation of droplet formation and coalescence usinglattice Boltzmann-based single-phasemodelrdquo Journal of Colloidand Interface Science vol 311 no 2 pp 609ndash618 2007

[16] Q Gallas R Holman T Nishida B Carroll M Sheplak and LCattafesta ldquoLumped element modeling of piezoelectric-drivensynthetic jet actuatorsrdquoAIAA Journal vol 41 no 2 pp 240ndash2472003

[17] A A Khalate X Bombois R Babuska H Wijshoff and RWaarsing ldquoOptimization-based feedforward control for a Drop-on-Demand inkjet printheadrdquo in Proceedings of the AmericanControl Conference (ACC rsquo10) pp 2182ndash2187 Baltimore MdUSA July 2010

[18] L Meirovitch Fundamentals of Vibrations McGraw-Hill NewYork NY USA 2001

[19] H Seitz and J Heinz ldquoModelling of a microfluidic device withpiezoelectric actuatorsrdquo Journal of Micromechanics and Micro-engineering vol 14 no 8 pp 1140ndash1147 2004

[20] S Kim J Sung andMH Lee ldquoPressure wave and fluid velocityin a bend-mode inkjet nozzle with double PZT actuatorsrdquoJournal of Thermal Science vol 22 no 1 pp 29ndash35 2013

[21] G Wassink Inkjet printhead performance enhancement by feed-forward input design based on two-port modeling [PhD thesis]Delft University of Technology 2007

[22] S Prasad SHorowitz QGallas B Sankar L Cattafesta andMSheplak ldquoTwo-port electroacoustic model of an axisymmet-ric piezoelectric composite platerdquo in Proceedings of the 43rdAIAAASMEASCEAHS Structures Structural DynamicsandMaterials Conference AIAA Paper 2002-1365 AIAA DenverColo USA April 2002

[23] D T BlackstockFundamentals of Physical Acoustics JohnWileyamp Sons New York NY USA 2000

[24] F M White Fluid Mechanics McGraw-Hill New York NYUSA 1979

[25] H Schempt andD R Yoerger ldquoStudy of dominant performancecharacteristics in robot transmissionsrdquo Journal of MechanicalDesign Transactions Of the ASME vol 115 no 3 pp 472ndash4821993

[26] E A Witmer ldquoElementary Bernoulli-Euler beam theoryrdquoMITUnified Engineering Course Notes vol 5 pp 114ndash164 1991-1992

[27] Y Yang and L Tang ldquoEquivalent circuit modeling of piezoelec-tric energy harvestersrdquo Journal of Intelligent Material Systemsand Structures vol 20 no 18 pp 2223ndash2235 2009

[28] F Constantinescu A G Gheorghe and M Nitescu ldquoNewcircuit models of power BAW resonatorsrdquo in Proceedings of theEuropean Microwave Integrated Circuit Conference October2007

[29] A Albareda and R PerezNon-Linear Behaviour of PiezoelectricCeramics vol 140ofSpringer Series inMaterials Science Springer2011

[30] J Nosek ldquoDrive level dependence of the resonant frequency inBAW quartz resonators and his modelingrdquo IEEE TransactionsonUltrasonics Ferroelectrics and Frequency Control vol 46 no4 pp 823ndash829 1999

[31] R Aigner N-H Huynh M Handtmann and S MarksteinerldquoBehavior of BAW devices at high power levelsrdquo in Proceedingsof the IEEE MTT-S International Microwave Symposium pp429ndash432 Long Beach Calif USA June 2005

[32] R S Ketcham G R Kline and K M Lakin ldquoPerformanceof TFR filters under elevated power conditionsrdquo in Proceedingsof 42nd the Annual Frequency Control Symposium pp 106ndash111Baltimore Md USA June 1988

16 Mathematical Problems in Engineering

[33] K-S Kwon ldquoWaveform design methods for piezo inkjet dis-pensers based on measured meniscus motionrdquo Journal of Micro-electromechanical Systems vol 18 no 5 pp 1118ndash1125 2009

[34] H-C Wu and H-J Lin ldquoEffects of actuating pressure wave-forms on the droplet behavior in a piezoelectric inkjetrdquoMateri-als Transactions vol 51 no 12 pp 2269ndash2276 2010

[35] KMinolta InkjetHeadApplicationNote-KM1024 Series KonicaMinolta Ij Technologies

[36] G Golub andV Pereyra ldquoSeparable nonlinear least squares thevariable projection method and its applicationsrdquo Inverse Prob-lems vol 19 no 2 pp R1ndashR26 2003

[37] H Wijshoff ldquoThe dynamics of the piezo inkjet printheadoperationrdquo Physics Reports vol 491 no 4-5 pp 77ndash177 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

16 Mathematical Problems in Engineering

[33] K-S Kwon ldquoWaveform design methods for piezo inkjet dis-pensers based on measured meniscus motionrdquo Journal of Micro-electromechanical Systems vol 18 no 5 pp 1118ndash1125 2009

[34] H-C Wu and H-J Lin ldquoEffects of actuating pressure wave-forms on the droplet behavior in a piezoelectric inkjetrdquoMateri-als Transactions vol 51 no 12 pp 2269ndash2276 2010

[35] KMinolta InkjetHeadApplicationNote-KM1024 Series KonicaMinolta Ij Technologies

[36] G Golub andV Pereyra ldquoSeparable nonlinear least squares thevariable projection method and its applicationsrdquo Inverse Prob-lems vol 19 no 2 pp R1ndashR26 2003

[37] H Wijshoff ldquoThe dynamics of the piezo inkjet printheadoperationrdquo Physics Reports vol 491 no 4-5 pp 77ndash177 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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