Physics of Ink-jet Printing

Mitja BlazincicFaculty of Mathematics and Physics, University of Ljubljana

Kodak European Research

Mentor: Dr. Andrew Clarke, F. Inst. P.Dept. of Applied Math., University of Birmingham

Kodak European Research

Abstract

Ink-jet is a dot-matrix printing technology in which the drops of ink are jetted through a small orifice to a specificposition on a substrate. The mechanism by which a liquid stream breaks up into droplets was described by LordRayleigh in 1878[1]. There are two qualitatively different ink-jet printing methods - drop on demand (DOD) andcontinuous ink-jet printing (CIJ). The underlying mechanism for CIJ printing is breaking a continuous jet intodroplets and was ”discovered” first, with first patent of a working practical Rayleigh break-up ink-jet device filed in1951[5]. The DOD printer, in which we actually ”squirt” a jet of liquid that joins in a drop under surface tensiondriven forces, was not made until 1977[5].

In this seminar I will try to outline basic governing physics of ink-jet process, current challenges and possible futureapplications utilizing this technology.

1 June 2008

1. Rayleigh breakup and Plateau-Rayleigh instability

Plateau-Rayleigh instability is a physics phenomena where a thread of liquid breaks up into droplets inorder to minimize it’s surface energy. Consequences of this instability can be seen in everyday life: a threadof liquid dripping from a tap breaks into drops, condensation on a spider’s web in the morning forms dropsinstead of uniform cylindrical film, etc. Plateau-Rayleigh instability is the underlying phenomena that isexploited in continuous ink-jet printing process and is purely consequence of surface tension.

We know, that the increase in hydrostatic pressure (∆p) in a drop of liquid is

∆p =2σR,

where σ is surface tension and R is radius of the drop. For a general surface between two fluids with R1

and R2 principle radii of curvature, the increase in hydrostatic pressure is described by the Young-Laplaceequation

∆p = σ

(1R1

+1R2

). (1)

1.1. Intuitive picture

In a cylinder of liquid we have two distinctive radii of curvature, which create differences in hydrostaticpressure - radius of the jet itself R1 and radius of curvature of the perturbation R2. Radius of the jet itselfis always positive, thus creating positive increase in pressure. The smaller this radius is, the higher is theincrease in hydrodynamic pressure. The radius of curvature of disturbance can also be negative, thereforedecreasing the pressure exactly in the region where radius of the jet is smaller. We will show that if thedisturbance will grow or not depends entirely on the wavelength of the disturbance.

z

R2 is negative

R2 is positive

R1

Fig. 1. Intermediate stage of jet breaking up into droplets. R1 is here denoted as radius of the axisymmetric jet and R2 as the

the radius of the curvature of the perturbation.

1.2. Condition for Stability

Let’s say we have a cylindrical jet of liquid, taking the axis of z along the axis of the cylinder (see fig.1) with a spatially periodic perturbation. By Fourier’s theorem, any such perturbation can be resolved intosinusoidal components; therefore we can write radius of the cylinder as

R1 = R+ ζ cos kz ,

where ζ is small compared to R, variable with time and k = 2π/λ; λ is the wavelength of the perturbation.The potential energy due to surface energy is proportional with surface area of the cylinder. On average perunit length along the axis, the area is ∗

A = 2πR+π

2Rk2ζ2 . (2)

From condition that V the volume enclosed per unit length is given

V = πR2 +π

2ζ2 ,

∗ Because this is actually average of the surface of revolution, we have to calculate it as 1L

∫ L/2

−L/22πr

√1 +(

drdz

)2dz instead

of 1L

∫ L/2

−L/22πr dz

2

Fig. 2. Photograph of an unstable liquid jet. Perturbation amplitude is increasing until the drop finally pinches off from therest of the jet at the breakoff length. A thread of liquid between main drops that eventually comes together forming satellite

drops is a nonlinear phenomenon and its existence cannot be predicted by a linear theory of instability.

we can substitute in Eq. 2 approximate R =√V/π(1 − πζ2/(4V )). With sufficient approximation we can

then get

A = 2√πV +

πζ2

2R(k2R2 − 1

),

or if we denote A0 as the value of undisturbed A

A−A0 =πζ2

2R1

(k2R2 − 1

). (3)

From this we can see, that if kR > 1 the surface (and also potential energy) is greater after perturbationthan before; therefore for perturbations with wavelengths λ < 2πR the system is stable. Plateau, a Belgianphysicist was the first to study the instability of cylindrical films. He was the first that understood thatthe cylinder distorts itself spontaneously in order to lower surface energy as soon as the wavelength of thedistortion exceeds the perimeter of the cylinder[2] - when λ > 2πR.

1.3. Linear Stability of an Inviscid jet

With respect to instability, the principal problem is determination of the number of masses into whichthe cylinder with given length may be expected to distribute itself.Let us adopt a cylindrical coordinate system fixed in the axisymmetric jet, that is moving with velocity U .If we rewrite Navier-Stokes equation for incompressible Newtonian fluid

ρ

(∂u∂t

+ u · ∇u)

= −∇p+ µ∇2u + f ,

in that system, drop nonlinear terms from the left side and set viscosity to 0 we get, written by components

∂ur∂t

= −1ρ

∂p

∂r,

∂uz∂t

= −1ρ

∂p

∂z. (4)

Under the same conditions, the continuity equation takes the form

1r

∂

∂rrur +

∂uz∂z

= 0 . (5)

Let φ be a velocity potential function with properties

ur =∂φ

∂z, uz =

∂φ

∂r. (6)

Substituting this into dynamic equations (Eq. 4) gives a pair of equations for pressure that are satisfied by

p0 =σ

R− ρ∂φ

∂z.

We took σ/R as an integration constant so that the pressure in an undisturbed jet (φ = 0) is everywhereuniform and strictly due to surface tension.Substituting Eq. 6 into continuity equation we find that φ must be a solution to Laplace’s equation

1r

∂

∂r

(r∂φ

∂r

)+∂2φ

∂z2= 0 . (7)

From experimental observation we know that the disturbance is periodic along z and it grows monotonically

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Fig. 3. (a) Plot of modified Bessel functions. Because K0(kr) diverges at r = 0 it is not good solution. (b) Dimensionless

growth rate α(ρR3/σ)1/2 as a function of dimensionless wave number.

in time. We are therefore looking for a solution of the form

φ = Φ(r)eikzeαt .

Putting this into Eq. 7, we find that Φ has to satisfy following condition

1r

∂

∂r

(r∂Φ∂r

)− k2Φ = 0 ,

which is solved by modified Bessel functions I0(kr) and K0(kr). Because K0 diverges at r = 0 it is not goodsolution since it does not satisfy the condition that velocity is bounded at r = 0. We therefore write thesolution as

Φ = AI0(kr) .

Using this, we can now write for pressure

p0 = AραI0(kr)eikz+αt +σ

R. (8)

Let us now write perturbation to the jet radius as

R0 = R+ ζ0(z, t) , (9)

where initial perturbation ζ0 is connected with radial velocity at the surface by

∂ζ0

∂t= u0

r at r = R .

This permits us to write the surface disturbance as

ζ0 = Ak

αI1(kR)eikzeαt . ∗ (10)

The boundary condition for pressure is given by the Young-Laplace equation (Eq. 1)

p0 = γ

(1R1

+1R2

)at r = R . (11)

One radius of curvature is the radius of the jet (Eq. 9), hence

1R1

=1

R+ ζ0≈ 1− ζ0/R

R,

and the second one is the radius of the curvature of the perturbation

1R2

= −∂2ζ0

∂z2.

∗ We have used the fact that dI0/dr = I1

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Fig. 4. Dimensionless breakup length L′/D (D is the jet diameter) as a function of Weber number We = ρU2D/σ. The

measurements correlate really well (except one set) with the theory, which predicts dimensionless breakup length to be a linearfunction of (We)1/2. Outlying set of data is for viscosity 0.16Pa · s whereas the other well correlated data is for liquids with

viscosities ranging from 0.001Pa · s to 0.026Pa · s. Picture taken from [3].

Using Eq.10 for ζ0 we get an expression for p0 which, if equated with Eq. 8 yields following expression forgrowth rate α

α2 =σk

ρR2

(1− k2R2

) I1(kR)I0(kR)

.

From Eq. 5 we can see that perturbation will only grow if α is positive and real, that is if kR < 1, as wehave already found out. For given kR the perturbation will grow at rate α.The jet is subject to environmental noise either though environmental noise in nozzle or air. The noisehas a certain frequency spectrum and for a certain frequency in the spectrum the disturbance will growwith a corresponding growth rate α(ν). Some frequencies will therefore damp out and some will grow. Thedisturbance will grow most rapidly at frequency that corresps to the maximum growth rate αmax, which isat kR = 0.69. Observations show that we can reasonably assume that a jet breaks down from most rapidgrowing disturbance. We can therefore expect to observe uniformly spaced drops spaced 2πR/L = 0.69 or

L ≈ 9.1R ,

which Lord Rayleigh has demonstrated in his paper in 1878[1]. Since we assumed that the observed distur-bance corresponds to αmax, the radius of the disturbance is expected to grow as

ζ0 = ζieαmaxt , (12)

where ζi is the amplitude of the initial disturbance. This is too small to observe, but if we assume that a jettraveling with constant velocity U breaks up at time t - at a breakoff length L from nozzle - when ζ0 = R,then we can obtain from Eq. 12

t =L′

U=

1αmax

lnR

ζi=

C

αmax.

This model shows good alignment with observations for low-viscosity liquids (see Fig. 4) and even up to anorder of magnitude greater than that of water.

1.4. Breakup of a Viscous jet

The theory can be quite easily extended to viscous liquids using same assumptions and approximationsand yields good approximation for a growth rate as[3]

α2 + µ3(kR)2

ρR2α︸ ︷︷ ︸

additional ’viscous’ term

=σ

2ρR3

(1− (kR)2

)(kR)2 .

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The viscous term extends the breakoff length which now becomes

L′

D= C ·We1/2(1 + 3Oh) .

Notice additional term 3We1/2 ·Oh compared to the inviscid analysis.

1.5. Nonlinear analysis

Fig. 5. Dependence of drop radius on wave number of disturbance. Taken from [4]. In their notation k is the dimensionless wave

number k = 2πR0/λ and rd is Rdrop/R0.

An important feature of jet breakup is the existence of sattelite drops - the thread of liquid between twoprimary drops that comes together forming smaller drops (see Fig. 2). Calculations in the linear regime failto predict their existence. One way to do nonlinear analysis is to use a full numerical solution of the Navier-Stokes equation. The alternative approach using a one-dimensional approximation to the jet equations isalso possible and gives some good results (Bousfield 1986)[4]. This approach, for instance, correctly predictsradius of satellite and primary drops (Fig. 5).

2. Continuous Ink-Jet

As we have already pointed out, fundamentally, ink-jet printing is divided into drop-on-demand andcontinuous ink-jet methods. The latter uses just described mechanism of breaking up a jet of liquid intothe droplets through introduction of a perturbation. How exactly the jet will break up is down to nozzledesign, the type of perturbation we are actually using and physical properties of the ink. In our discussionwe assumed the disturbance is a perturbation in radius but in reality it is really hard to achieve perturbationsolely on radius. Mechanical perturbation using piezoceramic, or thermal perturbation using various heatersetups are most common. Both methods eventually lead to a perturbation in radius ∗ and from there onour analysis is valid. It is important to point out that the amplitude of initial perturbation ζi that we haveused in our analysis, now becomes just a proxy quantity that describes gross effect of nozzle setup andperturbation mechanism on the jet breakup.Whichever perturbation mechanism we eventually choose to use, the scale of the perturbation has to be

∗ For instance when using thermal perturbation, the heater around nozzle increases temperature of the liquid, which for

usual Newtonian liquids results in decreased surface tension thus decreased hydrodynamic pressure in that region and throughdifferences in pressures drives flows towards lower pressure zones, thus resulting in perturbation in radius.

Fig. 6. Example of an nozzle plate with 10µm nozzle size and split ring heaters. Courtesy of Kodak Eastman Company[7]

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Fig. 7. (a) Spontaneous jet breakup due to environment noise. (b) Short (≈ 1µs), low energy (≈ 6nJ) pulses provide

enought thermal perturbation for controlled jet breakup. Courtesy of Kodak Eastman Company[7]

greater than environment noise and sufficient to create controlled breakup of the jet.Using the described method, drops of controlled size can be generated † . Because we constantly create

Fig. 8. Scheme of CIJ with electrostatic deflection method, recirculating charged drops. Note that a charging electrode isshown in a cross section. Actual electrical field is established between charging electrode and a drop generator - due to voltage

difference. Voltage is applied to the charging electrode before drop pinches off from the main jet so that the drop retains the

charge. Taken from[5].

drops, we need some sort of control over which drops to print. The drops and are separated to printingand non-printing drops using some sort of deflection mechanism. Non-printing drops are caught in a gutterand recirculated, while printing drops are left to hit the paper. The deflection mechanism can be eitherelectrostatic, airflow deflection or even thermal steering.In electrostatic deflection we control the charge on the drops. Charged drops are deflected by use of anelectrostatic field and depending on the printer design, we print with either deflected or undeflected dropsand recirculate the others.The air deflection mechanism involves making drops of two different sizes and deflecting them by use oflaminar air flow. Smaller drops are generated by short pulses of high frequency; to generate bigger drops wejust increase the time between two pulses, thus ”pinching” off a longer thread of fluid column. The largerdrops are obviously deflected less than smaller ones. Again, the size we chose to print with is down to printerdesign.If we are perturbing the jet using heaters around the nozzle, that also enables thermal jet steering. In thismethod only a part of the jet is heated - decreasing surface tension only in that part. Subsequently the jetsteers away from that direction.With typical rate of 105 generated drops per second per nozzle, CIJ printing method is faster than DOD

(typically 104 drops per second). The drop generation rate intrinsically depends on jet velocity (typically20m/s). Increased jet velocity enables us to increase drop generation rate, but that also increases the velocitywith which the drop impacts the substrate. Drop impact dynamics is thus one of the limiting factors - wedon’t really want drops to splash as this reduces image quality.In standard design of CIJ printer the we use four print heads the size of the width of the substrate sequentiallyfor each base color and black. Instead of a moving printhead, the substrate moves and makes only onepass under the printhead. At 640 dpi image quality, the substrate typically moves as fast as several m/s.Additionally, the long stable stream of drops allows the printing surface to be far away (up to several cm)

† In first approximation drop volume is Vd = πR2jUj/ν, where Rj is jet radius, Uj jet velocity and ν frequency of perturbation.

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a b

Fig. 9. (a) In an air deflection system we use two different drop sizes. To produce larger drops we reduce perturbation

frequency and consequently ”pinch off” longer thread of liquid jet, which we can see on the picture as well, come together intoa drop due to surface tension. (b) Separation of drops by air deflection mechanism - large drops are deflected less. In this

picture air flows in direction from bottom to the top of the picture. Courtesy of Kodak Eastman Company[7].

from the nozzle thus making it useful to print on uneven surfaces. CIJ printing is for instance used to printproduction date on eggs, fast, personalised printing of advertisements, bills and bank statements.

3. Drop On Demand

a b

Fig. 10. (a) Thermal DOD method with roof-shooter configuration. Rapid increase in the temperature of the heater causesformation of a vapor bubble, the growth of which ejects a drop.[5] (b) Side-shooter configuration.

Drop on demand is a printing method where drops are produced impulsively, due to a very short durationpressure rise in the cavity. The pressure can be generated by a piezoelectric source or by rapidly increasingthe temperature of the surface in contact with reservoir, which causes formation of a vapor bubble, thegrowth of which ejects a drop.A thermal ink-jet can be by their cavity configuration either a roof-shooter (Fig. 10a) with an nozzle locatedon top of the heater, or a side-shooter (Fig. 10b) with an nozzle located on a side. With a current pulse ofless than a few microseconds through the heater, heat is transferred from the surface of the heater to theink. The ink becomes superheated to the critical temperature and starts vaporising. For water-based ink,this temperature is around 300◦C [5]. A water vapor bubble instantaneously expands and forces the ink outof the nozzle. Once all the heat stored in the ink is used, the bubble begins to collapse on the surface ofthe heater, which results in the droplet break off. The whole process of bubble formation and collapse takesplace in less than 10µs. The ink then refills the cavity and the process is ready to begin again. Dependingon the cavity geometry and ink’s physical properties, the ink refill time can be from 80 to 300µs.[5]. Dropgeneration rate is dominated by this time. The whole process is very ”rich” in physics phenomena andproblems that are not well understood. For example: what exactly occurs during bubble creation in respectof vaporisation and fluid flows; when the bubble finally collapses, how exactly is the chamber refiled; etc.In piezoelectric ink-jet depending on the piezoceramic deformation mode four main types exist: squeeze,

bend, push, and shear. Whichever design we use, deformation of the piezoceramic material causes changein the volume of the cavity . This generates a pressure wave that propagates toward the nozzle. Acousticpressure wave has to be sufficent to overcome the viscous pressure loss (nozzle is very small), and thesurface tension force due to the ink meniscus and still propel the droplet towards substrate. One of the mainproblems to solve when designing a cavity is the acoustic behavior and the interaction between neighboring

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a b

Fig. 11. (a) In a push-mode design, as the piezoceramic rods expand, they push against ink to eject the droplets. (b)

In a bend-mode design, the piezoceramic plates are bonded to the diaphragm bending of which is used to eject the ink droplets.

Fig. 12. Droplets generated by DOD with a trail of liquid following the main drop.

nozzle cavities. Piezoelectric ink-jet is noisier than thermal thus resulting in lower resolution, but offersgreater printing speeds and is beginning to compete for same market as CIJ.DOD - as the name implies - ejects ink droplets only when they are used in imaging on the media. Theprinthead prints in scanning a motion - moving left and right on substrate (as we are used to in desktopprinters) pausing whenever we want. This eliminates the complexity of both the drop deflection hardwareand the ink recirculation systems required for the CIJ and enables the printhead to be small and cheap.Many home desktop printers use thermal DOD printing method. For DOD there is not much we can calculatesemiquantitatively and full numerical solution is usually required. The underlying physics is however thesame - because of surface tension, the squirted jet of liquid comes together forming a drop. A very importantfeature is the trail of liquid behind the main drop (Fig. 12), which under Plateau-Rayleigh instability breaksup into trail of satellite droplets that reduce quality of the printed picture.

4. Challenges in the technology and Conclusions

Today ink-jet printing technology offers cheap, digitized printout of images. Because the process is digital,it enables production of any kind of images and the printing process itself does not distinguish betweenimages - as analog printing processes such as printing press does.Recently DOD technology has been pushed to its boundaries with the release of the fastest (but only 108mmwide) color printhead[6] with full-color print speeds of up to 3m/s. CIJ technology still has a lot of room fordevelopment in it’s performance and printout quality - print drop generation rate and placement precision.Printout quality also depends on the ink-substrate interaction. When a liquid ink droplet contacts the surfaceof a paper, it tends to spread along paper fiber lines as well as penetrating into paper pores. The spreadingof ink droplets on paper could be too irregular to maintain the resolution required. The penetration of inkinto the paper is usually too slow to absorb multiple ink drops on the same spot within very short timeintervals. Poor color image quality is due to ink intercolor mixing and surface flows of ink driven by pressuredifferences (due to differences in liquid film thickness) or by Marangoni flows (due to the presence of agradient in surface tension). To obtain a high-quality color ink-jet image, the surface of the media requires aspecial coating. The special ink-jet-coated media must balance many design parameters such as drop volume,evaporation rate, penetration rate, coating thickness, porosity, etc. Another approach to obtaining betterimage quality without relying on special media is the use of smart inks (or hot melt or phase-shift ink). Inoperation, the ink is jetted as molten liquid drops. On contact with the media, the ink material solidifies,

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Fig. 13. A SEM photograph of phase-change ink drops on the surface of a bond paper. Reader should note that usual ink wouldwet the paper and we cannot see dry drops of ink on porous surface as spherical caps. Taken from[5].

very little spreading or absorption occurs so that brilliant color and high resolution can be realized almostindependent of the substrate properties.

Ink-jet technology can utilize inks with wide selection of properties. In future we might use it for low costdigitized production of conductive circuits (using conductive inks) and printed electronics, or printing offood packaging using water-based, smart, phase-switching inks on plastic, hydrophobic and non-impregnablesurfaces. Rapid prototyping (layer by layer 3D printing) and printing organic tissue (using piezoelectric DODprinter) are also areas that show much promise for future utilization of ink-jet technology.

References

[1] Lord Rayleigh, ”On the Instability of Jets”, Proc. Lond. math. Soc. 10, 4 (1878)

[2] J. Plateau, ”Statique experimentale et theorique des liquides soumis aux seules forces moleculaires” (Experimental andTheoretical Equilibrium State of Liquids Subjected to Molecular Forces Only), Gauthiers-Villars (1873)

[3] Stanley Middleman, ”Modeling Axisymmetric Flows: Dynamics of Films, Jets, and Drops”, Academic Press, Inc. (1995)[4] D.W. Bousfield, R.Keunings, G. Marrucci, M.M. Denn, ”Nonlinear Analysis of the Surface Tension Driven Breakup of

Viscoelastic Filaments”, J. of Non-Newtonian Fluid mechanics 21,79− 97 (1986)

[5] Hue P. Le, ”Progress and Trends in Ink-jet Printing Technology”, Journal of Imaging Science and Technology, Vol. 42,Number 1 (1998)

[6] http://www.dcviews.com/press/Kyocera-KJ4.htm

[7] Gilbert A. Hawkins, Eastman Kodak Company, ”A New Ink Jet Technology for Commercial Printing”, IMI 13th AnnualInk Jet Printing Conference (2004)

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