mani sivaramakrishnan and peter boltryk - comsol · pdf filemultiphysics modelling of piezo...

Multiphysics Modelling of Piezo Inkjet Printheads

Mani Sivaramakrishnan andPeter Boltryk

Multiphysics Modelling of Piezo Inkjet Printheads – COMSOL Conference 2014 (Cambridge) – September 2014

Outline

• Xaar - company and products

• Multiphysics

• FEA

• 3D coupled models

• jetting model

• Piezoelectric physics

• Narrow channel equations

• SPICE modelling

2


Xaar – the Company

• Xaar is a world-class manufacturer of industrial inkjet printheads

• Develops world-leading piezoelectric drop-on-demand inkjet

technologies

• Our technologies enable the most effective way to lay down inks and

other fluids with

• Precise drop volumes

• Highly accurate drop placement

• High frequency jetting

• Variable drop size capability

• Unrivalled reliability

3


Current application markets

4


Role of modelling in Xaar’s R&D

• Physics underlying actuation and jetting

• Guide low level actuator design • Optimize design parameters (material, geometry, etc.) with respect to

some target performance

Drop volume, drop velocity, frequency response, cross-talk, reliability/failure mode analysis, etc.

• Design guide at a higher level• Interface with electronics, fluidic manifold, etc.

• Drive waveform optimization

• Guide testing and understand testing results• Test results in turn are used to fine tune/calibrate models

5


Multiphysics in the actuator

Piezo-physics / structural mechanics

Fluid-structure interaction (FSI)

Fluid Physics (CFD)Fluid supply

Through-flow, thermalAcoustics

Travelling waves, Helmholtz frequencyRheologyDroplet ejection

6


Modelling at Xaar – modelling strategies

Models range from:

A. Phenomenological model (seconds)

• Detailed structural mechanics analytical calculations, transmission line acoustic model,

droplet ejection calculations

B. Fully coupled multiphysics FEA models of actuation chamber (Highly optimised, ~8 hours)

• FSI, piezoelectric physics, free surface modelling

C. FEA and FV jetting models (1 - 2 hours)

• Standalone nozzle models taking externally-generated acoustic pressure as inlet

boundary condition, for investigating droplet formation process (satellites, misting,

ligature break-off)

D. Reduced order models – narrow channel equations, SPICE (seconds)

• Analytical expressions, and

• some parameters taken from FEA models

7


Actuator – heart of the printhead

8


Actuator – printhead acoustic-mode operation

1. Waveform’s first edge moves two walls apart, rapidly rarefying fluid along channel length

2. Reduced pressure draws meniscus inwards

3. Positive pressure waves propagate from supply manifolds towards nozzle

4. Superposition of two waves from each end, plus waveform timed to move walls together creates large positive pressure behind nozzle

5. Positive nozzle velocity, driven by acoustic pressure creates droplet

6. Residual acoustic fluctuations couple to meniscus motion until damping mechanisms return system to quiescent state

9


Actuator – cross talk

Nozzle flow rates, generated using COMSOL Multiphysics®

3D FSI model

10

FEA (and FV) modelling


FEA modelling of chamber in 3D

12

3D compressible FSI

piezoelectric physics

free surface modelling (sometimes jetting models)

2 axis of symmetry

• Initial COMSOL Multiphysics® models ~7 days

• Optimised modelling ~8 hours

• With BLADE servers, can run up to 8 simultaneous simulations


Standalone FEA (or FV) jetting modelling

• Various jetting codes available to

Xaar for investigating jetting and

drop formation

• 2D axisymmetric or 3D

• Simple method is to couple a

nozzle inlet pressure boundary

condition, calculated from a

cheaper method such as the

NCE or SPICE

• No two-way coupling

14


Narrow channel equations (NCE)

• Narrow channel theory used by Wijshoff †

• Viscothermal wave propagation theory developed by Beltman‡ (Twente)

𝐵𝜕2𝑝

𝜕𝑥2+

𝜔2

𝑐0+ 𝜌0𝜔

2𝛽 𝑝 = −𝜌0𝜔2𝛼𝑈

• 𝐵 is frequency dependent velocity profile of wave – dependent on geometry, fluid viscosity

• 𝛼 and 𝛽 specify electrical and pressure compliance

• in multichannel models capture direct and indirect structural cross-talk

• Calculate parameters using detailed COMSOL Multiphysics® models

• Frequency domain solution – use FFT methods to transform to time domain

† H. Wijshoff (2010) ‘The dynamics of the piezo inkjet printhead operation’, Physics Reports , 491(4):77-177.‡ W.M. Beltman (1998) 'Viscothermal wave propagation including acoustoelastic interaction‘, PhD thesis, Uni. of Twente

15



• Segment chamber into regions of constant properties (cavity with constant geometry and compliance)

• Calculate velocity profiles for each segment

• V(ω)=fft(V(t))

• Calculate complex wave amplitudes in each segment (frequency domain) using system of equations

• IFFT to time domain

• Nozzle pressure time history coupled to FEA jetting model or a simple energy balance used to estimate drop speed and volume

P1P3P2 P4 P5

16

𝑃1 = −𝐴1𝑒−𝜆𝑘𝑥1 + 𝐴2𝑒

𝜆𝑘𝑥1 −𝜌0ω

2𝛼1𝑉(ω)

𝐵𝑐ℎ1

𝑃2 = −𝐵1𝑒−𝜆𝑘𝑥2 + 𝐵2𝑒


2𝛼2𝑉(ω)

𝐵𝑐ℎ2

⋮ ⋮ ⋮

𝑃5 = −𝐸1𝑒−𝜆𝑘𝑥5 + 𝐸2𝑒


2𝛼5𝑉(ω)

𝐵𝑐ℎ5

−𝑥1 𝑥1

−𝑥2 𝑥2

−𝑥3 𝑥3

−𝑥4 𝑥4

−𝑥5 𝑥5



• Profiles generated by two methods

show good agreement

• FEA: 8 hrs

• NCE: 4 sec

• COMSOL Multiphysics®

• remains useful for full 3D

validation, particularly new

geometries

• critical for providing numerical

constants for structural

compliance

• NCE model is *linear*

• Meniscus position is currently

constant

17


Piezoelectric physics

18

Piezoelectric and elastic material properties are tightly coupled.

Stress-Charge form

𝑇𝑚 = 𝐶𝑚𝑛𝑋𝑛 + 𝑒𝑚𝑗𝐸𝑗; 𝐷𝑖 = 𝑒𝑖𝑛𝑋𝑛+∈𝑖𝑗 𝐸𝑗

Strain-Charge form

𝑋𝑚 = 𝑆𝑚𝑛𝑇𝑛 + 𝑑𝑚𝑗𝐸𝑗; 𝐷𝑖 = 𝑑𝑖𝑛𝑇𝑛+∈𝑖𝑗 𝐸𝑗

m,n:=1,2,…6; i,j =1,2,3

T: Stress, X: Strain, E: Electric field, D: Electric Polarization C: Stiffness coefficients, d: Compliance

coefficients e: Piezoelectric coupling coefficients, e: di-electric constants

COMSOL Multiphysics® provides very convenient implementations of both types of

piezoelectric physics, as needed.


Substrate effects on piezoelectric response

-100

-50

0

50

100

150

200

250

0.0E+00 5.0E-04 1.0E-03 1.5E-03 2.0E-03 2.5E-03

Z-d

ispla

cem

ent (p

m)

Electrode size (mm)

delta_Si

delta_PZT

delta_tot (d33,f)

1 1.5 2 2.50.50

S. Sivaramakrishnan et al. Applied Physics Letters, 103,132904 (2013)

COMSOL Multiphysics® modelling was used to isolate the substrate clamping effects.

19

Reduced Order Modelling (SPICE)


General features of SPICE modelling

Lumped parameter elements created using finite element or analytical

models

The components are then assembled into a ‘system’ using a circuit

System effects (e.g. fluid-structure interaction) are captured at the

system level using SPICE

‘Circuit’ simulations are extremely fast (~ seconds)

Easy to interface with electronics

• Waveform optimization, frequency response, etc.

Easy to investigate cross-talk effects (mechanical and fluidic)

21


Generic SPICE model

Rr_i n

'R in '

Lr_i n

'L in '

0V1

PW L (0 'P act' 10e-6 'Pact' 10. 3e-6 0 11.3e-6 0 11. 6e-6 'Pact' 1e-4 'P act')

V2

Lch

'L c_4'

Rch

'Rc_4'

R_n

'Rn'

L_n

'L n'

Cm1

'Cm'

'Cc_2'

C_ch 2

1. 859e-21

C_p zt2

D1

di od e1

Rch 1

'Rc_4'

Lch 1

'L c_4'

Lch 2

'L c_4'

Rch 2

'Rc_4'

Lch 3

'L c_4'

Rch 3

'Rc_4'

Li n1

'L in '

Ri n1

'R in '

'Cc_2'

C_ch 1

1. 859e-21

C_p zt1

0V3

;op

. tran 0 3e-5 1e-7

. mo del d io de1 D(IS =1e-11 N=65000 BV= 5e6)

. in clu de param_73. txt

Piezo actuation

Chamber/2 Chamber/2

Piezo parameters: Extracted from FEM simulations in COMSOL Multiphysics®

Fluidics parameters: Derived using analytical expressions from Morris paper at the Helmholtz frequency.

22


Draw-release operation: typical time response

0 µ s 3 µ s 6 µ s 9 µ s 1 2 µ s 1 5 µ s 1 8 µ s 2 1 µ s 2 4 µ s 2 7 µ s

-5 M V

-4 M V

-3 M V

-2 M V

-1 M V

0 M V

1 M V

2 M V

3 M V

4 M V

5 M V

6 M V

7 M V

8 M V

9 M V

1 0 M V

-1 n A

0 n A

1 n A

2 n A

3 n A

4 n A

5 n A

6 n A

7 n A

8 n A

9 n A

1 0 n A

1 1 n A

1 2 n A

1 3 n AV (n 0 0 1 ) V (n 0 0 8 ) I (D 1 )

Drive waveform

Pressure at mid-chamber (behind nozzle)

Ejected drop

23


Printing frequency 33 kHz

0µs 30µs 60µs 90µs 120µs 150µs 180µs 210µs 240µs 270µs 300µs

-4.8MV

-4.0MV

-3.2MV

-2.4MV

-1.6MV

-0.8MV

0.0MV

0.8MV

1.6MV

2.4MV

3.2MV

4.0MV

4.8MV

-2nA

0nA

2nA

4nA

6nA

8nA

10nA

12nA

14nA

16nA

18nA

20nAV(n008) I(D1)

1st drop: 5.6 pL

9th drop 5.6 pL

24


Printing frequency 200 kHz

0µs 6µs 12µs 18µs 24µs 30µs 36µs 42µs 48µs 54µs

-5.4MV

-4.5MV

-3.6MV

-2.7MV

-1.8MV

-0.9MV

0.0MV

0.9MV

1.8MV

2.7MV

3.6MV

4.5MV

-2nA

0nA

2nA

4nA

6nA

8nA

10nA

12nA

14nA

16nA

18nA

20nAV(n008) I(D1)

1st drop: 5.6 pL

9th drop 2.8 pL

25


Conclusions

• Modelling plays pivotal role in Xaar for

• understanding complex and coupled Multiphysics problems

• optimising design and control of printheads

• FEA and FV models are used extensively for in-depth

investigations of new geometries, or significant changes in

parameters

• Reduced order models (NCE and SPICE), with input from FEA,

show exciting promise for rapid parametric sweeps with no

topological changes

26

mani sivaramakrishnan and peter boltryk - comsol · pdf filemultiphysics modelling of piezo...

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