DesignofPiezoInkjetPrintHeads Design of Piezo Inkjet Print Heads FromAcousticstoApplications J.FritsDijksman Author AllbookspublishedbyWiley-VCH arecarefullyproduced.Nevertheless, J.FritsDijksman authors,editors,andpublisherdonot UniversityofTwente warranttheinformationcontainedin PhysicsofFluids thesebooks,includingthisbook,to P.O.Box217 befreeoferrors.Readersareadvised 7500AEEnschede tokeepinmindthatstatements,data, Netherlands illustrations,proceduraldetailsorother itemsmayinadvertentlybeinaccurate. LibraryofCongressCardNo.:appliedfor BritishLibraryCataloguing-in-Publication Data Acataloguerecordforthisbookis availablefromtheBritishLibrary. 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PrintISBN:978-3-527-34266-2 ePDFISBN:978-3-527-80688-1 ePubISBN:978-3-527-80696-6 oBookISBN:978-3-527-80687-4 Typesetting SPiGlobal,Chennai,India PrintingandBinding Printedonacid-freepaper 10 9 8 7 6 5 4 3 2 1 v Contents Preface xi ListofSymbols xv 1 Introduction 1 References 10 2 SingleDegreeofFreedomSystem 13 2.1 Introduction 13 2.2 GoverningEquationsandSolutionforSquarePulseDriving 15 2.2.1 EntranceandExitEffects(EntrancePressureDrop,ExitLoss) 22 2.2.2 CorrectedSpeedofSound 34 2.2.3 EffectofSurfaceTensiononResonanceFrequency 36 2.2.4 Rayleigh’sMethodforCalculatingtheResonanceFrequency 37 2.2.5 LogarithmicDecrementMethodtoEstimateDamping 38 2.2.6 BulkViscosity 40 2.2.7 FirstEstimateontheFrequencyDependenceofDamping 41 2.3 SolutionforRampedPulseDriving 42 2.4 SolutionforExponentialPulseDriving 47 2.5 SolutionforHarmonicDrivingandFourierAnalysis 50 2.5.1 Frequency-dependentDamping(FullSolution) 56 2.6 Non-linearEffectsAssociatedwithNon-completeFillingofthe Nozzle 61 References 71 3 TwoDegreesofFreedomSystem 75 3.1 Introduction 75 3.1.1 Rayleigh’sMethodtoDetermineApproximatelytheResonance FrequenciesofaTwoDegreesofFreedomSystemfortheCase withSurfaceTension 79 3.1.2 CalculationoftheDampingofTwoDegreesofFreedomSystem withLowViscosityUsingtheLogarithmicDecrementMethod 84 3.1.3 FlowThroughaConicalNozzle 87 3.1.4 FlowThroughaBell-mouth-shapedNozzle 91 3.2 GoverningEquationsandSolutionsforSquarePulseDriving 98 vi Contents 3.2.1 SpecialCases 101 3.2.2 SolutionsfortheLowViscosityInkstoSquarePulseDriving 105 3.2.3 SolutionsforInkswithaModerateViscositytoSquarePulse Driving 111 3.2.4 SolutionsforaHighViscosityInktoSquarePulseDriving 115 3.3 SolutionsforRampedPulseDriving 119 3.3.1 SolutionsforLowViscosityInkstoRampActuation 121 3.3.2 SolutionsforModerateViscosityInkstoRampActuation 122 3.3.3 SolutionforLargeViscosityInkstoRampActuation 122 3.3.4 SolutiontoRampedPulseDriving 123 3.4 SolutionsforExponentialPulseDriving 128 3.4.1 SolutionforLowViscosityInkstoExponentialRampDriving 130 3.4.2 SolutionforModerateViscosityInkstoExponentialRamp Driving 131 3.4.3 SolutionforLargeViscosityInkstoExponentialRampActuation 131 3.4.4 SolutionstoExponentialPulseDriving(PulseConsistingofTwo ExponentialRamps) 132 3.5 SolutionforHarmonicDrivingandFourierAnalysis 134 3.5.1 FrequencyDependentDamping(FullSolution) 144 3.6 Non-linearAnalysis 148 3.6.1 CapillaryPressureandForceinConicalNozzle 157 3.6.2 CapillaryPressureandForceinBell-mouth-shapedNozzle 161 References 163 4 Multi-cavityHelmholtzResonatorTheory 167 4.1 Introduction 167 4.2 GoverningEquations 169 4.2.1 SpeedofSoundinMainSupplyChannel 172 4.3 SolutionsforRampedPulseDrivingforLowViscosityInks 174 4.4 SolutionforHarmonicDrivingandFourierAnalysis 183 References 192 5 WaveguideTheoryofSingle-nozzlePrintHead 193 5.1 Introduction 193 5.2 LongWaveguideTheory 197 5.2.1 CharacteristicsofaClosedEnd/ClosedPumpoftheWaveguideType WithoutConnectingDucts 202 5.2.2 CharacteristicsofanOpenEnd/ClosedEndPumpoftheWaveguide TypeWithoutConnectingDucts 204 5.2.3 ViscousDraginNon-circularChannels 206 5.3 SolutionsforRampedPulseDrivingoftheWaveguide-typeInkjet Pump 207 5.3.1 TheClosedEnd/ClosedEndCase 207 5.3.2 DampingoftheClosedEnd/ClosedEndPrintHead 216 5.3.3 OpenEnd/ClosedEndCase 219 5.4 SolutionsforHarmonicDrivingandFourierAnalysisIncludingthe EffectofDamping 221 Contents vii 5.4.1 SolutionofWaveEquationwithPoiseuilleDampinginNozzle andThrottle 224 5.4.2 SampleCalculationandResultsforClosedEnd/ClosedEndPrintHead ChannelArrangement 227 5.4.3 SampleCalculationandResultsforOpenEnd/ClosedEndPrintHead ChannelArrangement 230 5.4.4 FullSolutionofWaveEquationIncludingFrequency-dependent Damping 233 5.4.5 ClosedEnd/ClosedEndCase 238 5.4.6 OpenEnd/ClosedEndCase 240 5.5 Non-linearAnalysisoftheWaveguideTypeofPrintHeadIncluding Inertia,Viscous,andSurfaceTensionEffectsintheNozzle 243 5.5.1 ResultsfortheClosedEnd/ClosedEndArrangement 245 5.5.2 ResultsfortheOpenEnd/ClosedEndTypeofWaveguidePump 246 5.5.3 HighFrequencyPulsing,Start-up,andNozzleFrontFlooding 249 5.5.4 EffectofanAirBubbleontheInternalAcousticsofaPrintHead 252 5.5.5 HigherOrderMeniscusOscillations 254 5.6 MeansandMethodstoEnhanceFluidVelocityinNozzle 258 References 259 6 Multi-cavityWaveguideTheory 263 6.1 IntroductiontoMulti-cavityAcoustics 263 6.2 AnalysisofCross-talkinanOpenEnd/ClosedEndLinearArrayPrint HeadwithAlternatelyActivatedandNon-activatedPumps 266 6.3 AnalysisofCross-talkinanOpenEnd/ClosedEndLinearArrayPrint HeadwithAlternatelyOnePumpActivatedandTwoPumps Idling 277 6.4 AnalysisofCross-talkinanOpenEnd/ClosedEndLinearArrayPrint HeadwithAlternatelyOnePumpActivatedandThreePumps Idling 285 6.5 AnalysisofCross-talkinanOpenEnd/ClosedEndLinear Array-sharedWallShear-modePrintHeadwithAlternatelyOnePump ActivatedandTwoPumpsNon-activated 297 6.6 AnalysisofCross-talkinaClosedEnd/ClosedEndLinearArrayPrint HeadwithAlternatelyActivatedandNon-activatedPumps 302 References 307 7 DropletFormation 309 7.1 Introduction 309 7.2 AnalysisofDropletFormation(PositivePulse) 312 7.2.1 Force(Impulse)Consideration 313 7.2.2 EnergyConsideration 316 7.2.3 DropletFormationCriterionfromaRetractedMeniscus 319 7.3 AnalysisofDropletFormation(NegativePulse) 320 7.3.1 ForceConsideration 321 7.3.2 EnergyConsideration 324 viii Contents 7.4 DecelerationDuetoElongationalandSurfaceTensionEffectsPrior toPinchingOff 326 7.5 Non-linearTwoDegreesofFreedomAnalysisIncludingtheEffects ofDropletFormation 332 7.6 Non-linearWaveguideTheoryIncludingtheEffectsofDroplet Formation 335 7.6.1 ResultsfortheClosedEnd/ClosedEndArrangement 336 7.6.2 ResultsfortheOpenEnd/ClosedEndTypeofWaveguidePump 340 References 344 8 DropletFlight,Evaporation,Impact,Spreading,Permeation, andDrying 347 8.1 Introduction 347 8.2 EvaporationofaFree-flyingDropletExposedtoStillAir 348 8.3 CoolingofaFree-flyingDropletDuringFlightThroughStillAir 353 8.4 DecelerationofaFree-flyingDropletduetoAirFriction 355 8.5 Spreading 357 8.5.1 StaticSpreading 359 8.5.2 Surface-tension-drivenSpreading 362 8.5.3 Inertia-controlledSpreading 366 8.6 PermeationintoPorousSubstrates 389 8.7 EvaporationofDome-shapedBlobsofFluid 391 References 393 AppendixA:SolvingAlgebraicEquations 399 A.1 Second-orderAlgebraicEquation 399 A.2 Third-orderAlgebraicEquation 399 A.3 Fourth-orderAlgebraicEquation 402 References 404 AppendixB:FourierDecompositionofaPulse 407 B.1 PulsewithTwoRamps 407 B.2 ExponentialPulse 409 B.3 PulsewithThreeRampsandTwoStationaryLevels 413 References 416 AppendixC:ToroidalCo-ordinateSystem 417 C.1 Introduction 417 C.2 DefinitionwithRespecttoRectangularCo-ordinateSystem 417 C.3 ScaleFactors 417 C.4 ElementaryLineElement 418 C.5 UnitVectors 418 C.6 NablaOperator∇ 419 C.7 GradientofScalar 419 C.8 DivergenceofaVectorField 419 C.9 DyadicProduct∇v 420