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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 its 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 spiders 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 =2R,

where is surface tension and R is radius of the drop. For a general surface between two fluids with R1and R2 principle radii of curvature, the increase in hydrostatic pressure is described by the Young-Laplaceequation

p = (

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

Lets 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 Fouriers theorem, any such perturbation can be resolved intosinusoidal components; therefore we can write radius of the cylinder as

R1 = R+ cos kz ,

where is small compared to R, variable with time and k = 2/; 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 = 2R+

2Rk22 . (2)

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

V = R2 +

22 ,

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

L/2L/2 2r

1 +(

drdz

)2dz instead

of 1L

L/2L/2 2r 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/(1 2/(4V )). With sufficient approximation we can

then get

A = 2V +

2

2R(k2R2 1

),

or if we denote A0 as the value of undisturbed A

AA0 =2

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 < 2R 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 > 2R.

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

(ut

+ u u)

= p+ 2u + f ,

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

urt

= 1

p

r,

uzt

= 1

p

z. (4)

Under the same conditions, the continuity equation takes the form

1r

rrur +

uzz

= 0 . (5)

Let be a velocity potential function with properties

ur =

z, uz =

r. (6)

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

p0 =

R

z.

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

1r

r

(r

r

)+2

z2= 0 . (7)

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

3

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

growth rate (R3/)1/2 as a function of dimensionless wave number.

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

= (r)eikzet .

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

1r

r

(rr

) k2 = 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

= AI0(kr) .

Using this, we can now write for pressure

p0 = AI0(kr)eikz+t +

R. (8)

Let us now write perturbation to the jet radius as

R0 = R+ 0(z, t) , (9)

where initial perturbation 0 is connected with radial velocity at the surface by

0

t= u0r at r = R .

This permits us to write the surface disturbance as

0 = Ak

I1(kR)eikzet . (10)

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

p0 = (

1R1

+1R2

)at r = R . (11)

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

1R1

=1

R+ 0 1

0/R

R,

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

1R2

= 20

z2.

We have used the fact that dI0/dr = I1

4

Fig. 4. Dimensionless breakup length L/D (D is the jet diameter) as a function of Weber number We = U2D/. Themeasurements 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 withviscosities ranging from 0.001Pa s to 0.026Pa s. Picture taken from [3].

Using Eq.10 for 0 we get an expression for p0 which, if equated with Eq. 8 yields following expression forgrowth rate

2 =k

R2(1 k2R2

) I1(kR)I0(kR)

.

From Eq. 5 we can see that perturbation will only grow if is positive and real, that is if kR < 1, as wehave already found out. For given kR the perturbation will grow at rate .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 (). Some frequencies will therefore damp out and some will grow. Thedisturbance will grow most rapidly at frequency that corresps to the maximum growth rate max, 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 2R/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 max, the radius of the disturbance is expected to grow as

0 = iemaxt , (12)

where i 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 0 = R,then we can obtain from Eq. 12

t =L