Evaporation effects on jetting performance

27
Print Physics Notes: Evaporation Effects on Inkjet Performance and Image Quality December 2008 1 Evaporation Effects on Inkjet Performance and Print Quality Robert W. Cornell; Print Systems Science; Lexmark International; Lexington, Kentucky Abstract As nozzles sit idle, water evaporation causes large, highly localized viscosity variations in an inkjet ejector. This has long been a topic of qualitative discussion and empirical studies. This analysis goes to the root of the problem, covering the underlying multi-physics at work, the mathematical solution techniques and model validations at each step along the way. It is shown that significant viscosity field variations occur in the ejector on a time scale less than one second. Idle time evaporation induced viscosity variations have a negative impact on jetting performance and print quality. This article examines this phenomenon quantitatively. It begins with a discussion on the properties of moist air and continues on to the topics; heat and mass convection/diffusion. The boundary layer equations for heat and mass transfer in the gap between a swathing print-head and a media surface are derived, as are the resultant convection coefficients. It is shown that the mass convection coefficient is inversely proportional to media gap a result that helps explain machine-machine idle time variation. Also, a method is presented to predict viscosity and mass diffusivity properties for multi- component ink- mixtures as a function of formulation, temperature and water loss. Next an overview of the finite element method is presented to illustrate the solution means for the field equations. Evaporative flux exiting an inkjet ejector is then discussed from experimental and theoretical viewpoints. It will be shown that a heretofore undiscovered mechanism is at work during the first few seconds of evaporation. This mechanism tends to cause the evaporation rate from an inkjet nozzle to be self-limiting. The experimental and simulation results agree quite well on this phenomenon. The evaporation and mass diffusion results are then merged with the jetting model. The temporal and spatial variations of viscosity are accounted for, enabling the jetting model to predict the idle time print quality defects drop placement error and reflective luminance variation. The predicted idle time defects are very much on target with experimental results. Lastly, the pigment- dye dilemma is quantitatively discussed as are the paths that some competitors have taken to enable idle time robustness with pigment inks. Introduction An uncapped, idle nozzle will suffer water loss by evaporation. After a few seconds, or less, the remnant mixture in the nozzle consists primarily of highly viscous co-solvents. While all inks are affected by evaporation, the negative impact on performance is magnified with pigment inks. HP has expressed similar thoughts on this. “Pigment inks cannot be de-capped for more than a few seconds without significant property changes, so Edgeline spits all nozzles after ~800ms of idle time.” [1] Within seconds, the physical properties of the ink change well beyond the range needed for consistent jetting. The resulting weak and misdirected droplets are easily visible and highly objectionable from a print quality viewpoint. Fig.I illustrates this effect. Figure I: Idle Time Effect This article examines and quantifies the interrelationships between the ejector flow features, local environmental conditions, media gap and ink formulation choices that impact the transient mass diffusion and viscosity fields, and ultimately how these affect jetting and imaging performance. The goal of all mathematical models is to predict performance of some variable set [ as a function of a wide variety of inputs. In some simple cases, the model results lead to optimum, bulletproof designs. However, like most things inkjet, there is no “optimum” design for idle time. Rather, there exists a series of tradeoffs where one parameter suffers at the expense of another. That is not to say modeling is an esoteric exercise. Indeed, exactly the opposite is the case. When a simple system is designed for optimum performance it is often easy to do so with minimal experimentation and computation. However, when the system involves a complex set of field variables covering many scientific disciplines, it may take decades of experimentation to build a disconnected set of empirical do/don’t rules. When the modeling approach is not easily accomplished because of complex mathematical physics, the variables also are usually so endless that full-factorial experiments are utterly impractical. In such cases, taking the time to develop a mathematical model is extremely important, and well-worth the effort, because it permits tradeoffs to be studied without decades of empiricism. That said, the value of the multi-physics model described herein is a means of quantifying idle time tradeoffs between: ink formulation, environmental effects, thermo-hydrodynamic ejector effects, jetting and image quality. Properties of Moist Air Evaporation may be rate-limited by any of the following [2]: Vapor diffusion across a stagnant, or non-receptive gas layer above the liquid

Transcript of Evaporation effects on jetting performance

Page 1: Evaporation effects on jetting performance

Print Physics Notes: Evaporation Effects on Inkjet Performance and Image Quality

December 2008 1

Evaporation Effects on Inkjet Performance and Print Quality

Robert W. Cornell; Print Systems Science; Lexmark International; Lexington, Kentucky

Abstract As nozzles sit idle, water evaporation causes large, highly

localized viscosity variations in an inkjet ejector. This has long

been a topic of qualitative discussion and empirical studies. This

analysis goes to the root of the problem, covering the underlying

multi-physics at work, the mathematical solution techniques and

model validations at each step along the way.

It is shown that significant viscosity field variations occur in

the ejector on a time scale less than one second. Idle time

evaporation induced viscosity variations have a negative impact on

jetting performance and print quality. This article examines this

phenomenon quantitatively. It begins with a discussion on the

properties of moist air and continues on to the topics; heat and

mass convection/diffusion. The boundary layer equations for heat

and mass transfer in the gap between a swathing print-head and a

media surface are derived, as are the resultant convection

coefficients. It is shown that the mass convection coefficient is

inversely proportional to media gap – a result that helps explain

machine-machine idle time variation. Also, a method is presented

to predict viscosity and mass diffusivity properties for multi-

component ink- mixtures as a function of formulation, temperature

and water loss. Next an overview of the finite element method is

presented to illustrate the solution means for the field equations.

Evaporative flux exiting an inkjet ejector is then discussed from

experimental and theoretical viewpoints. It will be shown that a

heretofore undiscovered mechanism is at work during the first few

seconds of evaporation. This mechanism tends to cause the

evaporation rate from an inkjet nozzle to be self-limiting. The

experimental and simulation results agree quite well on this

phenomenon. The evaporation and mass diffusion results are then

merged with the jetting model. The temporal and spatial variations

of viscosity are accounted for, enabling the jetting model to predict

the idle time print quality defects – drop placement error and

reflective luminance variation. The predicted idle time defects are

very much on target with experimental results. Lastly, the pigment-

dye dilemma is quantitatively discussed as are the paths that some

competitors have taken to enable idle time robustness with pigment

inks.

Introduction An uncapped, idle nozzle will suffer water loss by evaporation.

After a few seconds, or less, the remnant mixture in the nozzle

consists primarily of highly viscous co-solvents. While all inks are

affected by evaporation, the negative impact on performance is

magnified with pigment inks. HP has expressed similar thoughts

on this. “Pigment inks cannot be de-capped for more than a few

seconds without significant property changes, so Edgeline spits all

nozzles after ~800ms of idle time.” [1]

Within seconds, the physical properties of the ink change well

beyond the range needed for consistent jetting. The resulting weak

and misdirected droplets are easily visible and highly objectionable

from a print quality viewpoint. Fig.I illustrates this effect.

Figure I: Idle Time Effect

This article examines and quantifies the interrelationships

between the ejector flow features, local environmental conditions,

media gap and ink formulation choices that impact the transient

mass diffusion and viscosity fields, and ultimately – how these

affect jetting and imaging performance.

The goal of all mathematical models is to predict performance

of some variable set [ as a function of a wide variety of inputs.

In some simple cases, the model results lead to optimum,

bulletproof designs. However, like most things inkjet, there is no

“optimum” design for idle time. Rather, there exists a series of

tradeoffs where one parameter suffers at the expense of another.

That is not to say modeling is an esoteric exercise. Indeed, exactly

the opposite is the case. When a simple system is designed for

optimum performance it is often easy to do so with minimal

experimentation and computation. However, when the system

involves a complex set of field variables covering many scientific

disciplines, it may take decades of experimentation to build a

disconnected set of empirical do/don’t rules. When the modeling

approach is not easily accomplished because of complex

mathematical physics, the variables also are usually so endless that

full-factorial experiments are utterly impractical. In such cases,

taking the time to develop a mathematical model is extremely

important, and well-worth the effort, because it permits tradeoffs to

be studied without decades of empiricism. That said, the value of

the multi-physics model described herein is a means of quantifying

idle time tradeoffs between: ink formulation, environmental

effects, thermo-hydrodynamic ejector effects, jetting and image

quality.

Properties of Moist Air Evaporation may be rate-limited by any of the following [2]:

Vapor diffusion across a stagnant, or non-receptive gas

layer above the liquid

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Print Physics Notes: Evaporation Effects on Inkjet Performance and Image Quality

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A barrier that impedes molecular transport at the liquid-

vapor surface

Evaporation of a volatile species may generate

concentration gradients in the liquid such that mass

transport within the liquid is rate-limiting

Since vapor diffusion is dependent upon the gas properties existing

in the boundary layer at the liquid-air interface, a good starting

point for this analysis is psychrometrics [3].

Moist air is a binary mixture of dry air and water vapor.

Humidity ratio (W) is the mass of water vapor per unit mass of dry

air. Since it is known that dry air has a molecular weight of 29

grams/mol and water vapor has a molecular weight of 18

grams/mol, it is convenient to write the following expression for

humidity ratio.

W

W

W

W

x

x

x

xW

1622.0

129

18 (Eq.1)

xW = mol-fraction water vapor contained in the moist air

Relative humidity () is also conveniently related to water vapor

mol fraction:

SW

W

x

x

,

(Eq.2)

xW = mol-fraction water vapor contained within the air at Troom

Troom = room temperature

xW,S = mol-fraction water vapor at saturation at Troom

Using Dalton’s Rule, (Eqs.1-2) can be written in terms of pressure.

pressure catmospheri

airmoist in or water vapof pressure partial

622.0

P

P

PP

PW

W

W

W

(Eq.3)

Standard atmospheric pressure = 101300 (N/m2) = 101300 (Pa)

SW

W

P

P

,

(Eq.4)

PW,S = saturated vapor pressure(TROOM , 100% relative humidity)

Then the saturated humidity ratio (WS) can be written:

SW

SW

SPP

PW

,

,622.0

(Eq.5)

Pressure values may be obtained from look-up steam tables, or

they may be computed from the ideal gas law approximation of the

Clausius-Clapeyron equation, shown below.

TTbR

hfgMPTP

11exp)( (Eq.6)

P(T) = vapor pressure (Pascal, or N/m2) at temperature (T)

hfg = latent heat of evaporation = 2.35 x 106 (J/Kg) for water

M = molecular weight = 0.018 (Kg/mol) for water

R = universal gas constant = 8.315 (J/mol-K)

Tb = normal boiling temperature = 373 (K) for water

T = temperature (K)

Consequently, saturated temperature values may be computed by

rearranging (Eq.6):

P

PRTbMhfg

MhfgTbPT

ln

)( (Eq.7)

T(P) = saturated vapor temperature (K) at vapor pressure [P(Pa)]

Eq.6 compares favorably to steam table values, as shown in Fig.1.

Figure 1: Vapor pressure versus water temperature

Example If the room temperature is 25C, and the relative humidity is 35%;

find the saturated vapor pressure (PW,S), the partial pressure of

water vapor in the moist air (PW), the humidity ratio (W), the

saturated humidity ratio (WS) and the dew point temperature

(TDEW).

By (Eq.6) the saturated vapor pressure at 25C is:

PW,S(298K) = 3272 Pa = 24 mm-Hg

By (Eq.4) the partial pressure of water vapor at 25C and 35%

relative humidity is:

PW(298K, 0.35) = 1145 Pa = 8.6 mm-Hg

By (Eq.3) the humidity ratio of air at 25C and 35% relative

humidity is:

W = 0.007 Kg-water/Kg-dry air

By (Eq.5) the saturated humidity ratio at 25C is:

Ws = 0.0208 Kg-water/Kg-dry air

By (Eq.7) the dew point temperature at a dry bulb temperature of

25C and 35% relative humidity is:

TDEW = 281K = 8C

These values may also be extracted from a psychrometric chart

(Fig. 2-3), but for mathematical convenience it is more desirable to

use (Eq.1-7).

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Print Physics Notes: Evaporation Effects on Inkjet Performance and Image Quality

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Figure 2: Psychrometric Chart

Figure 3: Psychrometric Chart-Expanded

Heat and Mass Convection Now let us discuss the kinetics of water evaporation into moist air.

During typical evaporation, heat is transferred from the air into the

water, and mass (water vapor) is transferred from the liquid into

the air. The energy transfer process occurring at the liquid-air

interface is heat-convection and heat-conduction, while the

evaporative transfer process is mass-convection and mass-

diffusion.

Heat transfer at the vapor-liquid interface obeys the following

relationship:

Surfacey

SROOMCy

TkTThq

(Eq.8)

q = heat flux (W/m2)

hC = heat transfer convection coefficient (W/m2-C)

TROOM = room temperature (C)

TS = temperature at the liquid surface (C)

k = thermal conduction coefficient of air (W/m-C)

Similarly, mass transfer at the vapor-liquid interface is described

by:

flux mass areaunit per unit timeper transfer mass

:Where

0

A

m

y

WDWWh

A

m

y

AIRAIRSD

(Eq.9)

hD = mass convection coefficient (Kg/m2-s)

DAIR = water vapor-air diffusion coefficient (m2/s)

AIR = density of air (Kg/m3)

y = 0 is the vapor-liquid interface

Note: it is obvious from (Eq.3) and (Eq.5) that (Eq.9) can

also be written in terms of vapor pressure and pressure

gradient, as often seen in the literature.

It can be shown that the mass diffusion coefficient (hD) is related to

the heat transfer convection coefficient (hC) by [4]:

32

3/2

LeDCph

h

AIR

AIR

AIRD

C

(Eq.10)

CpAIR = specific heat of air (J/Kg-C)

AIR = thermal diffusivity of air (m2/s)

The ratio of thermal diffusivity () to mass diffusivity (D) is the

Lewis number (Le).

So if the heat transfer convection coefficient (hC) is known, the

mass convection coefficient (hD) may be computed from (Eq.10).

Unfortunately, (Eq.8-9) are deceptively simple. From a

psychrometric analysis we may know the saturated water vapor

pressure at the liquid-air interface and the partial pressure of water

vapor beyond the boundary layer in the free stream (i.e. the room).

However, without a priori knowledge of the boundary layer (y)

characteristics (Fig.4), the convection coefficients are intractable.

Often this dilemma leads to an expedition into the literature –

looking for pre-solved special cases that mimic, or approximate the

nature of the boundary layer at hand, e.g. flat infinite plates,

spheres, circular tubes, etc.

For the case of an inkjet print head swathing at 30 inches per

second, a few millimeters above a media surface, it is expected that

forced convection is at work. Table 1 indicates a vast range, from

10-500 W/m2-C, to be expected for forced air convection. Perhaps

a boundary layer analysis will yield a more narrow range for (hC)

than this.

Table 1: Approximate heat transfer convection coefficients [5]

Mode hC (W/m2-C)

Free convection, air 5-20

Forced convection, air 10-500

Forced convection, water 100-15000

Boiling water 2500-25000

Condensing water vapor 5000-100000

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Print Physics Notes: Evaporation Effects on Inkjet Performance and Image Quality

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Figure 4: Typical Evaporation Boundary Layer

Boundary Layer Analysis The boundary layer for the case at hand is formed between a

moving nozzle plate and a fixed media surface (Fig. 5).

Figure 5: Velocity Profile Between Nozzle-Media

The velocity boundary layer of Fig.5 is the well-known

Couette flow condition. It can be shown that the velocity profile

reaches steady state within 60-70 ms, when the nozzle-media gap

is 1.65mm. At a carrier velocity of 30 in/s, steady state is achieved

in the first two inches of the print zone. Thus it may be argued that

the majority of the print zone sees the steady state velocity profile

shown in Fig.5. So the velocity profile (U) is simply:

L

yUyU C)( (Eq.11)

Furthermore, since the carrier velocity is much less than the speed

of sound in air and the Reynolds number is on the order of 102, we

may consider this problem domain as laminar, incompressible

flow. Then the energy equation for the condition shown in Fig.5 is: 2

2

2

0

y

U

y

Tk (Eq.12)

Integrating (Eq.12) twice and introducing the boundary condition:

[T(y = L) = TNOZ] where TNOZ is the nozzle plate temperature, leads

to an expression for temperature in the gap.

22

12

)(L

y

k

UTyT C

NOZ

(Eq.13)

= dynamic viscosity of air ~ 18.5 x 10-6 Pa-s

k = thermal conductivity of air ~ 0.0265 W/m-C

From Fourier’s Law, heat flux (q) at the nozzle surface is:

Lyy

Tkq

(Eq.14)

Taking the derivative of (Eq.13) and solving it at (y = L) says that

heat flux at the nozzle plate is:

L

ULyq C

2

(Eq.15)

From Newton’s Law of Cooling:

LyTyThq C 0 (Eq.16)

Solving (Eq.13) for T(y = 0) and T(y = L) and inserting those

values into (Eq.16) and combining that with (Eq.15) leads to the

convective heat transfer coefficient for the nozzle-gap case shown

in Fig.5.

L

kh

k

ULyTyT

C

C

2

20

2

(Eq.17)

Combining (Eq.17) with (Eq.10) leads to a value to mass

convection coefficient for an inkjet print head swathing back and

forth over a media surface.

3/2

,

3/2

,

2

LeC

L

k

DC

hh

AIRPAIRAIRAIRP

C

D

(Eq.17a)

Eq.17 returns an unexpected result (hC = 2k/L). Surprisingly, it

shows that the convective heat transfer coefficient in the nozzle-

media gap is independent of carrier velocity (UC). Instead it shows

that (hC) is a linear function of the conductive heat transfer

coefficient of air (k) and inversely proportional to the nozzle-media

gap (L). Since (Eq.17) was unexpected, an order of magnitude

analysis is called for to check on its form.

The Prandtl number (Pr) characterizes the relationship

between momentum diffusivity (i.e. kinematic viscosity ) and

thermal diffusivity ().

Pr

Physically this implies that as Pr 1.0 the momentum and

thermal boundary layers become identical. For air, the Prandtl

number is 0.71. Thus it may be argued that an order of magnitude

estimation of the temperature and momentum boundary layers are

nearly identical (for liquids the Prandtl number is much greater

than one, so this simplification would not apply).

Since the nozzle-media gap is filled with air, the

simplification applies. So, it is reasonable to state that a first order

approximation of the temperature field in the gap mimics the linear

Couette flow field. That said - it follows:

tcoefficien convection ofion approximatorder first

L

kh

ThL

Tk

y

Tkq

C

C

Ly

Page 5: Evaporation effects on jetting performance

Print Physics Notes: Evaporation Effects on Inkjet Performance and Image Quality

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Note that the first order approximation and (Eq.17) have the same

form. So from two different perspectives it can be shown that the

convective heat transfer coefficient for the nozzle-media gap is

proportional to thermal conductivity and inversely proportional to

nozzle-media gap.

The thermal conductivity of air is basically constant over the

temperature range of interest, so for the special case of a swathing

inkjet print head, the convective heat transfer coefficient is:

)(W/m 1.2665.1

gap media nozzle 1.65mm aFor

(m)

C)-(W/m 043.02

2 Cmmh

LL

kh

C

C

(Eq.17a)

This falls into the low range for forced air convection, as shown in

Table 1.

Eq.10 showed an inverse relationship between (hC) and (hD). This

implies that any variability in nozzle-media gap will have a direct

relationship on evaporation rate. Historically, idle time is a metric

with a lot of variability.

So an unexpected result of this boundary layer analysis is that

some of the observed idle time variability can be attributed to

machine-machine gap variation.

Before moving on, it is important to note that if idle time

evaporation was simply like water vapor coming from a pool, or a

lake, we would have the mass flux solution in hand at this point.

Recall (Eq.9), shown repeated below:

WWhA

mSD

(Eq.9)

From the boundary layer analysis (hD) is known. So if the water

temperature is known and the psychrometric properties of air are

known (as they usually are), the mass flux of water vapor due to

evaporation is solved directly from this equation. However, the

inkjet problem is not so simple. Presumably the pool has just one

component – water. That said, there is no concentration gradient on

the liquid side of the interface, so knowing the mass convection

coefficient and the humidity ratio difference between the surface

and the free stream – computing mass flux is just an algebra

problem. It will be shown that the limiting factor for water

evaporation from an inkjet nozzle is the concentration gradient on

the liquid side of the ink-air interface. Computing that gradient is a

calculus problem.

Heat and Mass Diffusion For many field problems in mathematical physics (heat transfer,

mass diffusion, electric field, flow in porous media, torsion, etc.),

the partial differential equation has the following form [6]:

tQ

zK

zyK

yxK

xzyX

= field variable of interest

KX, KY, KZ are material properties

is a storage term

Q is an internal generation term

(x,y,z) = spatial coordinates

t = time

Because the ejectors are placed side by side there is negligible heat

flux in the z-direction. Also, because the ejectors are separated by

physical flow feature walls, the diffusive mass flux between

ejectors may be ignored. These conditions reduce the heat and

mass diffusion fields, as related to inkjet, to two spatial dimensions

(x,y).

For the heat transfer problem [with internal heat generation

(Q)] the field variable is temperature (T); and the material

properties of interest are thermal conductivity (k), density () and

constant pressure specific heat (CP). For the inkjet evaporation-

mass diffusion problem, the field variable is water concentration

(cW); the material property of interest is mass diffusivity (D), and

because there is no species generation, the Q term equals zero in

the evaporation-mass diffusion problem. Considering these

physical descriptors, the partial differential equations describing

the heat and mass diffusion fields are written:

t

TCQ

y

Tk

yx

Tk

xP

(Eq.18)

t

c

y

cD

yx

cD

x

WWW

(Eq.19)

Because these equations have the same form, the same finite-

element, solution technique works for both. The heat transfer

solutions, as related to nucleation, bubble growth and jetting will

be described in a later article. This article will focus on the mass

diffusion problem (Eq.19).

For the evaporative, mass flux problem, as related to nozzle idle

time, a typical finite element mesh with initial and boundary

conditions is shown in Fig.6. Each element has an associated width

that depends upon the spatial location of the element centroid.

Element width is considered during the solution phase. Also, note

the boundary condition enforced along the wall:

0

WALLZ

W

z

c

This technique permits diffusion in the z-direction within the mesh,

but stops it at the ejector walls, as in the actual device. This is an

important point. The governing equation (Eq.19) is in just two

dimensions, but since each element in the mesh has three

dimensions associated with it, the solution to the mass diffusion

field variable (cW) does account for flow feature variations in the z-

direction. Because the mesh is 3-D the field solution is essentially

3-D within the ejector even though it was generated from a 2-D

governing equation.

Figure 6: Typical mass diffusion field with evaporative boundary condition

Page 6: Evaporation effects on jetting performance

Print Physics Notes: Evaporation Effects on Inkjet Performance and Image Quality

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The solution technique for (Eq.19) will be discussed in a later

section. It was simply introduced at this point to show the

mathematical relationship between water concentration (cW) and

diffusivity (D). Before solving the mass diffusion field equation it

is appropriate to quantify viscosity of a multi-component ink

formulation as a function of water concentration and mass

diffusivity as a function of viscosity.

Ink Property: Viscosity Solving the evaporation problem for an inkjet ejector presents

several dilemmas. In order to quantify how evaporation affects

jetting performance, it is required to know how evaporation

impacts ink viscosity. This is dilemma number one – predicting the

viscosity of a multi-component liquid mixture.

Unlike gases, a kinetic theory of viscosity does not exist.

Yet there is a need to predict how mixture viscosity

varies with formulation, solid loading, temperature and

various evaporation conditions and nozzle idle times.

The viscosity model described herein was created to

address this dilemma. The genesis of this model was

during the Lexmark’s short excursion into printed

electronics. At that time, it was discovered that silver ink

could be loaded up to 27 wt.% Ag and still possess

reasonably low viscosity. The viscosity model, combined

with experimental results, provided the needed teachings

to obtain a US patent on this topic (US 7,316,475).

Viscosity Model The viscosity model is an extension of the Teja-Rice method [7]

developed at the Thermodynamic Research Center. Mixture

viscosity is computed from the thermodynamic properties of the

mixture components.

• Tb = normal boiling temperature

• Tc = critical temperature

• Pc = critical pressure

• Vc = critical volume

• Zc = critical compressibility factor

• Mw = molecular weight

• (Tref) = viscosity at reference temperature (25C)

• (Tamb) = liquid density at ambient temperature

• Vb = molar volume at the normal boiling point

• = acentric factor

Note: the thermodynamic properties for the chemicals of interest

come from the DIPPR database (licensed yearly from Brigham

Young University - $750/PC install).

The model is best described in a series of steps.

Step 1: Describe the formulation. Identify each component and its

mass fraction.

Step 2: Compute the mol-fraction of each liquid component in the

mixture.

Step 3: Compute the viscosity of each ink component at the

temperatures of interest [(T)]. This step utilizes the Lewis-Squires

equation [7].

758.3

,2661.0

,233

iREF

iREFi

TTT (Eq.20)

REF,i = component-i viscosity (mPa-s) at temperature TREF,i

T = temperatures of interest = 15, 20, 25,…70 C

Note: 1 milli-Pascal (mPa-s) = 1 centipoise (cP)

Step 4: Select two reference liquids (R1, R2) from the mixture.

These are the two components with the largest mol-fraction.

Step 5: If this is a pigment ink, account for the solid particles. The

Einstein equation is only valid up to 5% volume fraction solid, so

the model make use of the Krieger-Dougherty equation [8], as it is

reportedly better for cases involving high particle packing and non-

spherical particles.

917.1

71.01

fpF

v

m

fp

mmv

MIX

P

P

ii

i

P

P

MIX

(Eq.21)

vMIX = specific volume of the mixture

(mP, P) = particle mass fraction and density

(mi, i) = liquid component-i mass fraction, density

fP = volume fraction of the pigment particles

F = viscosity multiplication factor due to solid particles

Step 6: Compute the critical volume (VCM) of the mixture using

the quadratic mixing rule [7][9].

n

i

n

jijCjiCM VxxV , (Eq.22)

(xi, xj) = mol-fraction of components i, j

VC,ij = critical volume of components i, j

n = number of liquid components in the ink mixture

The inks used by Lexmark are multi-component, usually

containing five liquids (n = 5). The model’s algorithm for (Eq.22)

accounts for such mixtures. It is easy to get confused over the i’s

and j’s of (Eq.22), so let us illustrate how the algorithm works by

showing a binary mixture example (Table 2).

Table 2: Binary Mixture Example

i j xixj VC,ij

1 1 x1x1 x12 VC11 VC1

1 2 x1x2 x1x2 VC12 VC12

2 1 x2x1 VC21

2 2 x2x2 x22 VC22 VC2

8

33/1

2

3/1

1

12

CC

C

VVV

Applying the values listed in Table 2 to (Eq.22) the critical volume

of a binary mixture may be written as:

82

33/1

2

3/1

1

212

2

21

2

1

CC

CCCM

VVxxVxVxBinaryV

Step 7: Compute the critical temperature of the mixture using the

quadratic mixing rule [7][9].

CM

n

i

n

jijCijCji

CMV

VTxx

T

,,

(Eq.23)

(xi, xj) = mol-fraction of components i, j

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December 2008

7

TC,ij = critical temperature of components i, j

VC,ij = critical volume of components i, j

jCiCjCiCijCijC VVTTVT ,,,,,, (Eq.24)

= interaction parameter (on the order of unity, and is generally

determined from experimental results – over 50 comparisons

between inkjet lab data and simulations have led to an estimate of

between 1.65 and 1.85 for the ink viscosity model)

Again, the ink formulation used in Lexmark products typically

consists of five or six liquids, and the model’s algorithm handles

such multi-dimensional mixtures; however, for clarification let’s

illustrate how the algorithm for (Eq.23-24) works by using a binary

mixture example (Table 3).

Table 3: Binary Mixture Example

i j xixj TC,ijVC,ij

1 1 x1x1 x12 TC1TC1VC1VC1)

0.5 TC1VC1

1 2 x1x2 x1x2 TC1TC2VC1VC2)0.5 TC1TC2VC1VC2)

0.5

2 1 x2x1 TC2TC1VC2VC1)0.5

2 2 x2x2 x22 TC2TC2VC2VC2)

0.5 TC2VC2

Applying the values listed in Table 3 to (Eq.23-24) the critical

temperature of a binary mixture may be written as:

CM

CCCCCCCC

CMV

VVTTxxVTxVTxT

21212122

2

211

2

1 2

Step 8: Compute the molecular weight of the mixture (MM).

n

iiiM MxM (Eq.25)

xi = mol-fraction of component-i

Mi = molecular weight of component-i

Step 9: Compute the acentric factor of the mixture (M). Note that

the acentric factor is a thermodynamic property related to the ratio

of critical pressure and vapor pressure at a temperature equal to

0.7X the critical value.

n

iiiM x (Eq.26)

i = acentric factor of component-i

Step 10: Compute (M) for the mixture and (R1, R2) for the two

reference liquids of step 4.

5.0

22

2

3/2

25.0

11

1

3/2

1

3/2

;RRC

RC

R

RRC

RC

R

MCM

CM

M

MT

V

MT

V

MT

V

(Eq.27)

Step 11: Compute the mixture viscosity (M) for each temperature

of interest (T).

11

22

12

111 lnlnlnRR

RR

RR

RM

M

RR

T

TT

eFTM (Eq.28)

(R1, R2) were determined in steps 3-4

(F was determined in step 5 to account for solid particles

Step 12: Reduce the water content by 1%. Return to step 2.

Repeat this loop until water content is gone.

Viscosity Model Validation Figs.7-8 illustrate that the model mimics published results for

binary mixtures of water-glycerol and water-methanol.

Figure 7: Data-Model comparison water-glycerol mixture

Figure 8: Data-Model comparison water-methanol mixture

Fig.9 illustrates that the model also mimics lab data across a range

temperature values for ink DS1-C (experimental results courtesy of

Agnes Zimmer).

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Figure 9: Data-Model comparison DS1-C ink viscosity versus temperature

Finally, Fig.10 illustrates the model results are well correlated over

a range of 57 test cases (the viscosity measurements and

formulations were performed by Agnes Zimmer).

Figure 10: Data-Model correlation over 57 test cases

Ink Property: Mass Diffusivity The validity of the viscosity model has been established, and

demonstrated to be a relatively accurate means of predicting

viscosity as a function of formulation, temperature and

evaporation. Now can it be used to estimate idle time

performance? No, not just yet - to estimate idle time performance

we also need a predictive means for the evaporation rate. The

evaporation rate is dependent upon knowing the mass diffusivity of

the mixture. That is the topic of this section. It is worth noting that

this is dilemma number two. Predicting the mass diffusivity of a

multi-component liquid mixture is (and historically has been)

problematic.

Mass diffusivity in many solid materials is generally

predictable due to long range crystal order [19].

Mass diffusivity in gas is well understood due to kinetic

theory [19].

Many scientists have worked on developing a kinetic

theory for mass diffusion in liquid:

• Einstein, Stokes, Born, Deybe, Eyring

Unfortunately, much like the viscosity dilemma, kinetic

theory does not extend to mass diffusion in liquid [7]

Another complication is that commonly published mass

diffusivity values only hold for binary mixtures at

infinite dilution.

Mass diffusivity, over a wide solvent-solute range, is

published for only a few binary aqueous mixtures.

Our ink formulations are not binary, nor are they

infinitely dilute.

Molecular modeling holds the promise of solving multi-

component mixture diffusivity someday….but published

molecular modeling predictions (USA-ORNL, Spain,

Korea, Norway) show significant deviations from

experiment.

In the meantime, we must rely on semi-empirical models

to estimate mass diffusivity.

Mass Diffusivity Model The diffusivity model makes use of the Wilke-Chang method

[7][8]. This is said to be an empirical modification of the Stokes-

Einstein relation.

6.0

8104.7

BMIX

AMIX

ABV

MTD

(Eq.29)

DAB = mutual diffusion coefficient (cm2/s) between components A

and B

TMIX = temperature (K) of the liquid mixture

MIX = viscosity (cP) of the mixture at TMIX

MA = molecular weight of water (g/mol)

VB = molar volume of the non-aqueous ink components (cm3/mol)

= association factor of the solvent

[Wilke-Chang recommends ( = 2.6) when the solvent is water]

The Wilke-Chang method is intended for use in binary mixtures.

As mentioned earlier, ink mixtures are not binary. However, they

may be generally classified as mixtures of water (M = 18 g/mol)

and a cocktail of high molecular weight co-solvents (M ~ 100-200

g/mol). So for the purposes of this model, component-A is water

and component-B is treated as a quadratic mixture of the co-

solvents.

Note that the mass diffusion coefficient (DAB) is inversely

proportional to the mixture viscosity. This should be expected

since mass transport in liquids requires the molecules to squeeze

past each other as they move from point(x) to point(x+x). Thus it

is quite natural that there should be a connection between viscosity

and mass diffusivity. Mixture viscosity increases as evaporation

removes water (leaving a higher content of viscous glycols), and a

corresponding DAB reduction is expected.

Diffusivity Model Validation Table 4 shows an excellent correlation between published DAB

values [10] and (Eq.29) for several binary mixtures.

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Table 4: Computed DAB Compared to Reported Experimental values [10]

Solute in an infinite

dilution of water

T(C)

DAB (10-5 cm2/s)

Experimental

Value

Wilke-Chang

Equation

Acetone 25 1.28 1.28

Ethanol 25 1.24 1.41

Ethylene glycol 25 1.16 1.28

Glycerol 25 1.06 1.02

Methanol 15 1.28 1.41

1-propanol 15 0.87 0.9

Sucrose 25 0.52 0.58

Cyclohexane 20 0.84 0.83

3-methyl-1-butanol 10 0.69 0.59

Aniline 25 0.91 1.06

Eq.29 does a good job of predicting DAB for binary mixtures that

are infinitely dilute. However, it is known that DAB varies with

viscosity. So it begs the question: If it is known how viscosity

varies as a function of the mixture, can W-C be used to predict DAB

over a wide range of concentrations? This is important to know

because as water evaporates from the ink the water concentration

varies widely. Let us answer this by studying an example in the

literature.

Transport Canada funded a research project to evaluate the

effectiveness of new de-icing fluids [11]. Fundamental to solving

the problem was an understanding of diffusion characteristics of

the candidate mixtures. The University of Quebec measured

diffusion coefficients as part of the project. To validate their

diffusion measurement technique, they first calibrated it against

published values for water-EG mixtures. Their experimental results

along with the values computed with (Eq.29) are shown in Fig.11.

Figure 11: Data-Model correlation for water-EG mixtures

Fig.11 illustrates an important point - mass diffusivity is not a

constant. It varies greatly with water concentration. Any attempt to

simulate evaporation rate and idle time performance must account

for this.

Failing to account for mass diffusivity as a variable can easily

cause an order of magnitude error in the solution of (Eq.19).

Solution Procedure The groundwork has now been laid, so it is time to apply all of the

teachings to a multi-physics solution.

Ink Formulation

Flow Feature Geometry

Ink Temperature

Water Evaporation

Mass Diffusivity

Viscosity

Ink Simulation Example The model workspace contains thermodynamic information [Tb,

Tc, Pc, Vc, Zc, Mw,(Tref), (Tamb), Vb, ] for a wide variety of

chemicals. These thermodynamic properties are used, as described

previously, to simulate ink viscosity as a function of temperature

and water loss.

Table 5: Gen1 ink formulation

Component Wt.%

C M Y

DI water 67.79 69.54 69.21

Glycerol 12.5 6 6

Tripropylene glycol 7.5 --- ---

Triethylene glycol 5.5 7 7

1,3-Propanediol --- 8 8

Surfactant 0.75 0.9 0.8

Biocide* 0.13 0.13 0.13

Wax emulsion 0.5 0.5 0.5

Pigment** 4 6 6.5

Dispersant** 1.33 1.93 1.86

*The biocide is ignored in the model because it has a negligible

mol-fraction.

**The pigment and its encapsulating dispersant are considered

solids and are handled with the aforementioned Krieger-Dougherty

equation.

Figure 12: Data-Model correlation for Gen1-Magenta

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Figure 13: Mass diffusivity versus viscosity Gen1-Magenta

Fig.12-13 illustrate the simulation results for Gen-1 magenta. Note

the exponential rise in viscosity as ink evaporates from the

mixture. This mimics the actual response of lab data as measured

by Shirish Mulay (Fig. 14).

Figure 14: Lab results – exponential viscosity increase with evaporation

Also note how mass diffusivity plummets as viscosity increases

(Fig 13). If the ink was pure water its diffusivity would be 300

m2/s, and if the ink contained just 10% water, the remaining, high

viscosity components drive DAB to 8 m2/s.

To put these values into perspective consider that the mass

diffusivity of moist air at 25C is 25,000,000 m2/s. That said, the

limiting factor is the diffusivity in the liquid. Once a water

molecule makes it to the air-liquid interface there is no waiting line

to transport it into the free stream. In other words for the special

case of inkjet:

Evaporation of the volatile species generates concentration

gradients in the liquid such that mass transport within the

liquid is rate-limiting.

Finite Element Analysis Using the finite element method, Fick’s 2nd law can now be

solved. Recall (Eq.19) and the finite element mesh of the mass

diffusion domain.

t

c

y

cD

yx

cD

x

WWW

Fick’s 2nd Law (Eq.19)

Figure 6: Typical mass diffusion field with evaporative boundary condition

The finite element method is a powerful numerical procedure

that can be applied to a wide variety of application areas. It had its

beginnings in the aerospace industry about 50 years ago, and was

primarily used for structural and solid mechanics problems. It did

not gain wide acceptance until computing power became

ubiquitous (and cheap). Today the finite element method is

commonly used to solve problems in all areas of mathematical

physics. If one can write a set of governing partial differential

equations for the phenomena, it can be solved via the finite

element method. Therefore, it is well suited to the multi-physics

problems so common in inkjet.

There are several excellent books on this topic, and they

should be studied by anyone interested in print physics modeling

[6][12][13]. This section will provide a brief overview of the

numerical procedure that is the finite element method. In this

overview the field variable will be referred to as (). This makes

the discussion generic, applying equally to heat transfer, mass

diffusion, electric field, solid mechanics, etc.

Fig.15 shows the basic 3-node triangular element. The mesh

consists of (N) such elements interconnected at the nodes. The

meshing routine is an exercise in analytical geometry. It will be

covered in a future article.

100% Water

10% Water remaining In the mixture

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Figure 15: Triangular finite element

Diffusive Element Equations Over the element, variable () is given by:

yx

NNN

NNN

mji

mji

m

j

i

mji

e

, variableof valuesnodal

functionsion interpolat

)(

(Eq.30a)

It can be shown that the shape functions are:

yx

yx

yx

AN

N

N

mmm

jjj

iii

e

m

j

i

)(2

1 (Eq.30b)

Where:

areaelement

1

1

1

det2

)(

)(

e

mm

jj

ii

e

A

yx

yx

yx

A (Eq.30c)

ijmmijjmi

jimimjmii

iiiimmimijmjmii

xxxxxx

yyyyyy

xyyxyxxyxyyx

(Eq.30d)

Since field problems like those described by (Eq.18-19) are often

gradient dependent, it is convenient to write the gradient term as:

y

xg (Eq.30e)

Taking derivatives of (Eq.30a):

m

j

i

mji

mji

m

j

i

mji

mji

A

y

N

y

N

y

Nx

N

x

N

x

N

g

2

1

In a more compact form the gradient term may be restated as:

Bg (Eq.30g)

Field problems having the form of (Eq.18-19) have a diffusive

material property associated with them:

B-A speciesbetween y diffusivit mass mutual

0

0

:ydiffusivit isotropicy with diffusivit massFor

directions y)(x,in ty conductivi thermal,

0

0

:problemfer heat trans For the

AB

AB

AB

MATL

yx

y

x

MATL

D

D

DD

KK

K

KD

(Eq.30h)

Field problems having the form of (Eq.18-19) have diffusive-like

and convective-like properties. The diffusive-like term is handled

as:

element theof thickness

equationelement diffusive

:Where

)(

)(

)()()(

e

e

D

MATL

Teee

D

thk

k

BDBAthkk

(Eq.30i)

Convective Element Equations The convective terms occur at domain boundaries, as shown in

Fig.16. Since convection is a function of exposed area, the element

equations must take that into account.

miEXP

mjEXP

jiEXP

e

EXPconve

conv

LL

LL

LL

thkLhk

and;

201

000

102

:m)-(i is sideelement exposed theif

and;

210

120

000

:m)-(j is sideelement exposed theif

and;

000

021

012

:j)-(i is sideelement exposed theif

6

)()(

(Eq.30j)

hconv = convective coefficient;

- if [ = temperature (T)]; hconv = hc of (Eq.17)

- if [ = concentration (cW)]; hconv = hD/air of (Eq.17a)

LEXP = exposed element length = Li-m in (Fig.16)

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Figure 16: Arbitrary mesh showing a convection boundary condition

Global Matrices The mesh consists of (NNODE) nodes and (NELE) elements. For

each element (Eq.30i) is written, and for elements that have a

convective boundary (Eq.30j) is added to it.

)()()( e

conv

e

D

e kkk (Eq.30k)

The individual element equations are assembled into a global

matrix {K}. Since the finite element method had its roots in

structural mechanics, the {K} matrix is often referred to as the

global stiffness matrix.

Breaking up the domain into an interconnected mesh of finite

elements allows the numerical solution to take the form of

algebraic matrix equations, like (Eq.30m), instead of the often

intractable partial differential equations when applied to complex

geometries and boundary conditions.

0

FK

tC (Eq.30m)

(Eq.18) aldifferenti partial For the

(Eq.19) aldifferenti partial For the 1

211

121

112

12

matrix ecapacitanc Global

matrix stiffness Global

:Where

)()()(

1

)(

P

eee

NNODEe

C

Athkc

cC

K

(Eq.30n)

For elements with a convective boundary condition:

0

1

1

2

:(Eq.19) aldifferenti partialFor

0

1

1

2

:(Eq.18) aldifferenti partialFor

matrix force Global

)(

)(

)(

)(

1

)(

AIR

e

EXPDe

e

EXPCe

NNODEe

thkLWhf

thkLThf

fF

(Eq.30m)

Note (Eq.30m), as shown, is for elements with a convective

boundary along element side i-j. If the convective boundary is on

side j-m the (3 x 1) matrix is [0 1 1]T. If convection is along side i-

m the (3 x 1) matrix is [1 1 0]T.

Matrix Equation Solution Eq.30m may be solved by many numerical techniques. The central

difference method is shown below.

FPS

KCt

P

Ct

KA

OLD

2

2

(Eq.30n)

[OLD] = nodal values of the field variable at the last time step

t = time step

To account for convective and fixed boundary conditions, [A] and

[S] must be modified. There are many techniques to do this (e.g.

see Appendix 3 of reference [6]). A convenient mathematical trick

is to identify the node numbers that are affected by fixed and/or

convective conditions. Then multiply the diagonal terms of those

nodes in [A] by a very large number (e.g. 1015). Next set the

affected nodes of [S] equal to the boundary condition value and

multiply those terms by the same very large number. The set of

equations may now be solved using various matrix reduction

techniques.

MODMODNEW SA

1 (Eq.30p)

[NEW] = nodal values of field variable at time (t+t)

[AMOD] [SMOD] are the matrices of (Eq.30n) that have been

modified for fixed and/or convective boundary conditions.

i

j

m

Li-m

j

i m

Free Stream

Convection Boundary

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Because the mesh usually contains thousands of nodes, matrix

inversion is an impractical means of solving (Eq.30p). Matrix

inversion is storage and CPU intensive. The LXK model uses node

renumbering to minimize the bandwidth of the [AMOD] matrix, and

since the matrix is symmetrical, only the upper, non-zero terms,

are stored and manipulated. The details of the bandwidth reduction

method and the rectangular matrix solver algorithm will be

covered in a future article.

After each time step (t), the field variables are laminated

onto a results matrix {}.

nNNODENNODENNODENNODE

n

n

n

tntttttt

,3,2,1,

,32,31,30,3

,23,21,20,2

,13,11,10,1

0 2

For each element, at each time step, there is an associated gradient

as shown in (Eq.30g). Gradients of the field variable are the prime

movers of some flux variable. Flux is the flow rate of some

physical property per unit area. The generic flux over each element

(e) is given by:

)()()()( eee

MATL

e BDFlux

Where:

format.equation matrix element, finitein

:and form,equation aldifferenti partialin

)()()( eee Bg

Grad

In particular, the energy flux in a heat transfer problem (Eq.18) is

given by:

22

)()()()(

ms

Joules

m

W attsTBDq eee

MATL

e (Eq.30q)

Similarly for the mass diffusion problem (Eq.19), the mass flux is

given by:

2

)()()()(

)(

ms

gramscBDm

e

W

ee

MATL

e

e

(Eq.30r)

[DMATL] is given by (Eq.30h)

(e) is the density of the material in the element

To determine the evaporative mass flux leaving the ejector

(Eq.30r) is summed over the elements at the nozzle exit.

Evaporative Flux Exiting an Inkjet Nozzle The groundwork has now been laid for computing the evaporation

of water vapor from an inkjet ejector and the resultant, transient

viscosity field in the flow features. Reviewing the steps to get to

this point, it was necessary to quantify:

- The thermodynamic properties of moist air

- Boundary layer analysis for (hC, hD) convection coefficients

- Heat and mass diffusion

- Ink viscosity simulation method

- Mass diffusivity (DAB) for multi-component ink mixture

- Finite element analysis overview

Demonstration Example The example chosen here is the evaporation of water from a

Romulan ejector filled with Puddy2-CMY dye inks. This ink-

ejector combination was chosen because evaporation lab data

exists to check the model veracity. The CMY-flow field is shown

in Fig.17. The K1-K2 flow fields are not shown because those

nozzles were not part of this experiment (they were intentionally

covered with tape so that PM1-mono ink would not confound the

experiment with its clumping, nozzle-blocking tendency). The

Puddy2 ink formulation is shown in Table 6. The simulated

viscosity versus temperature results are shown in Fig.18 along with

the measured values [(T) measurements courtesy of Agnes

Zimmer]. Except at very low temperatures, the model-lab data

comparisons are good for all three inks. Fig.19 shows how

viscosity is expected to vary with temperature and evaporative

mass loss.

Figure 17: Romulan-PINP color ejector

Table 6: Puddy2 ink formulation

Component Wt.%

C M Y

DI water 72.4 72.4 72.3

1,2-propanediol 7.0 --- ---

1,3-propanediol 7.0 6.0 7.0

Triethylene glycol 6.0 --- 6.0

1,2-hexanediol 3.0 3.0 3.0

trimethylolpropane --- 6.0 ---

1,5-pentanediol --- 8.0 ---

Tripropylene glycol --- --- 7.0

Silwet 0.5 0.5 0.5

Biocide 0.1 0.1 0.1

triethanolamine --- --- 0.1

Dye 4.0 4.0 4.0

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Figure 18:Puddy2 Ink – Viscosity versus temperature

Figure 19:Puddy2 Ink – Viscosity versus temperature and evaporation

To test the veracity of the evaporation model, the Romulan

print head was placed, nozzles-up, on a microbalance and weighed

numerous times over a 24 hour period. The top cover of the

balance housing was left open slightly to avoid a build up of water

vapor. The lab environment was constant (20C), as was the relative

humidity (25%) over this one day time frame. The measured

evaporative mass loss versus time is shown in Fig.20. The head

had 1920 exposed nozzles, and each nozzle had an exit area of 143

m2. As shown in Fig.20, the measured evaporative mass flux was

0.0038 ng/s/m2.

The Romulan-CMY ejectors were designed to shoot 4pL

droplets. Given the measured evaporative mass flux, for every 7.4

seconds left idle, each nozzle lost the equivalent mass of one jetted

droplet. In other words, if the 1920 CMY nozzles on a ½”

swathing head are left uncapped for one day, the print-head will

have enough evaporative mass loss exiting the nozzles to equate to

22-million jetted droplets.

How does the measured evaporative mass flux (0.0038

ng/s/m2) compare to the numerical model described thus far? To

answer this, the evaporation model was run for the inks described

in Table 6 and the flow features of Fig.17. All of the techniques

described earlier were used, except for the boundary layer analysis.

Since this experiment did not have a shuttling print head above a

fixed media gap (Eq.17a) would not apply. So the convective

boundary condition was computed for the well-known case -

natural mass convection from a stationary flat surface. The results

are shown in Fig.21.

Figure 20: Measured evaporative mass flux

Self-Limiting Evaporation Effect The measured value and the computed evaporation rate shown in

Fig.21 compare quite favorably. Note that the model computes an

initial evaporative flux of 0.037 ng/m2-s, and within about 10

seconds it drops exponentially to 1/10th that value. The reason why

evaporative flux has a highly transient response is due the rapidly

changing value of DAB. As evaporation drives viscosity higher it

becomes more difficult for the molecules to slide past each other,

and a natural consequence of this is that mass diffusivity (and

evaporation rate) decreases.

This is a sweet and sour phenomenon. It is sweet because high

viscosity at the nozzle exit reduces mass diffusivity and

evaporation rate slows. If it were not for this self-limiting effect,

the evaporation rate would be 10X higher. Unfortunately, it is a

sour phenomenon because the remnant, high viscosity mixture at

the nozzle exit becomes very difficult to jet.

This self-limiting evaporation effect is a new discovery for

inkjet. It has been discussed qualitatively, but it appears nowhere

quantitatively in the literature until now. Perhaps that is because

most attempts to model evaporation assume constant properties.

Furthermore, transient evaporation rates, on the order of a few

seconds, defy measurement capability. The oft-used laboratory

microbalance cannot capture transients (like the exponential decay

of Fig.21) because it takes a few seconds just to stabilize the

sensing means from the induced vibrations of setting the print head

on the scale pan. While we cannot capture the transient portion of

the curve in Fig.21, it is obvious that the computed evaporative

mass flux response curve is asymptotically approaching the steady-

state, measured value, indicated by the red dot.

Fig.21 is exceptionally good and gratifying news for the

author because it validates all of the steps of the evaporative

modeling process described so far, and it provides insight into the

self-limiting evaporation discovery cited above.

Evaporation

Temperature

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At this point, we can infer the model treats the following correctly:

- The thermodynamic properties of moist air

- Boundary layer analysis for (hC, hD) convection coefficients

- Heat and mass diffusion

- Ink viscosity simulation method

- Mass diffusivity [DAB()] for multi-component ink mixture

- Finite element analysis of the mass diffusion domain

Figure 21: Evaporative mass flux-measured and simulated

Mass Diffusion Field Results Recall the governing partial differential equation (Eq.19).

Everything is now in place to solve it for any given ejector design,

ink formulation, idle time and environmental condition.

Let us shift focus to Newman-mono. It will be beneficial to

use these teachings to quantitatively show why this ejector-ink

system is more prone to have idle time problems than Newman-

CMY (and Canon-mono).

Fig.22 shows the Newman-mono water concentration field

after just one second of idle time evaporation from a chip held at

42C in an atmosphere of 25C and 25% R.H. Also note the insert in

the upper right corner of Fig.22 showing viscosity versus nozzle

location. According to these results, liquid at the top 2 microns of

the nozzle has a viscosity of 70mPa-s after just one second of idle

time evaporation. That is an increase of 35X over the initial

mixture at 42C.

Recall the earlier comment that HP spits their Edgeline head

every 800 ms because they said substantial changes to

pigment ink properties occur within seconds. Fig.22-23

confirms - substantial changes indeed.

Figure 22: Water concentration after one second of evaporation

Figure 23: Water concentration after ten seconds of evaporation

85,000 s

Evaporative Mass Flux Experimental Result

0.0038 ng/s-m2

Simulated evaporative flux

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Evaporation Effects on Jetting:

Merging the Models

Fig.I shows the dramatic effect that idle time has on print quality.

After just 3 seconds, the droplets become weak, misdirected and

tiny as evidenced by the poorly formed, wiggly gray line in Fig.I.

The information presented thus far quantifies exactly what happens

to the liquid during those idle seconds – the ink in the nozzle

becomes highly viscous (Fig.24).

The previous discussion focused on the evaporative mass flux

and the resultant impact on ink viscosity. It remains to be shown

how the transient viscosity field affects jetting performance. That

is the topic to be covered in this section – merging the

evaporation/viscosity model with the jetting model.

Figure I: Idle Time Effect

Figure 24: Newman-mono viscosity heater-nozzle after 3 idle seconds

Jetting Model - FEAJET Overview Over the last two decades, the jetting model has evolved from a

simple heat transfer model into a complex multi-physics ejector

design tool with the following capabilities:

- Current density distribution

- Transient heat transfer

- Bubble nucleation

- Thermal boundary layer in the ink

- Phase change and heat transfer at a moving bubble wall

- Vapor pressure pulse and explosive bubble growth

- Inertance and hydraulic resistance of the flow field

- Outflow from the nozzle and blowback at the choke

- Non-traditional heater shapes

- Thermal stress in the thin films

Now we will add a module to accommodate idle time evaporation

and the resultant effect on jetting.

A typical simulation summary is shown in Fig.25. This particular

set of input variables defines Newman-mono: flow features, ink

formulation, pulse train, electrical properties and heater thin films.

The output variables predicted by the model are on target with the

experimental results. For instance, ejected mass (5.21 ng) and

velocity (454 in/s) are in line with experimental results. Also, the

predicted onset of nucleation (1345-1432 ns) is in agreement with

lab data. Furthermore, the simulation predicts a refill time (first

meniscus crossing) at 50 s – indicating that the maximum fire

frequency is 20 KHz. This too is very much in line with the actual

hardware (within microseconds) because it is known that mono

refills faster than the 18 KHz required.

Having established that the jetting model is a good predictor

of Newman-mono performance, let us use this ejector-ink system

as a test case to illustrate the merged models.

Figure 25: Newman-mono ejector

Hydraulic Resistance The phase change pressure pulse must overcome inertance and

viscous resistance to accelerate the ink in the ejector. Because ink

density is relatively insensitive to water evaporation, idle time does

not affect inertance. However, idle time evaporation greatly affects

viscous resistance. Therefore, there is no reason to focus on

anything other than the shift of hydraulic resistance as the water

evaporates. Hydraulic resistance may be defined as:

Page 17: Evaporation effects on jetting performance

Print Physics Notes: Evaporation Effects on Inkjet Performance and Image Quality

December 2008

17

dzP

tz

P

HYD

(Eq.31)

Where:

s/m rate flow volumetric

)(mfactor resistance hydraulic geometric

s)-(Paexit nozzle-heater from viscositydynamic

nozzle)-(heaterdirection -z in the s/m)-(Pagradient pressure

(Pa) resistance hydraulic todue drop pressure

3

4-

t

z

z

P

PHYD

FEAJET solves for volumetric flow rate by iteration. As the bubble

grows, liquid is converted to water vapor. This phase change

requires energy. The source of the phase-change energy is the

thermal boundary layer in the ink at the onset of nucleation. So

phase-change energy pulled from the thermal boundary layer tends

to cool liquid-vapor interface. Also, at nucleation the temperature

gradient in the thermal boundary layer is on the order of 300

million Kelvin per meter. This gigantic gradient effects rapid

diffusion of thermal energy into the cool region of the ink.

Combining the diffusion effect with the phase-change effect causes

a very rapid cooling of the ink. With the rapid cooling comes a

rapid decrease in the phase-change pressure pulse. The phase-

change pressure pulse and the resultant bubble growth act as a

virtual piston – displacing liquid in the ejector and shooting ink

onto the media. The nature of the explosive phase-change pressure

pulse is seen by comparing Fig.26a and Fig.26b. In both of these

cases the end-point is when the bubble reaches its maximum

displacement, i.e. the onset of bubble collapse. When the pressure

pulse must overcome a highly viscous flow field, the pressure

pulse is spent more quickly. For example, the 3 second idle time

case (Fig.26a) expends its pressure pulse in about 1.6 s, while the

0 second idle time pressure pulse (Fig.26b) lasts about 3.6 s.

These pressure-temperature response curves are a function of how

hard the bubble has to work to move liquid. There is a fixed

amount of available energy in the thermal boundary layer for

phase-change. So when the viscous resistance is high, the pressure

pulse is quickly dissipated.

Figure 26a: Newman-mono phase change pressure pulse after 3 seconds of

idle time evaporation

Figure 26b: Newman-mono phase change pressure pulse after 0 seconds of

idle time evaporation

Fig.26 illustrates the effect of PHYD. As for the other terms in

(Eq.31), the evaluation of dz can be problematic because, like

many MEMS flow structures, inkjet ejectors are non-circular. This

presents a problem for micro-flow fields, but a solution

methodology was derived and published in 2007 [14]. Instead of

repeating those teachings here, interested parties may consult [14].

The result for the Newman-mono flow features dz is given by:

0.00579 (m-3) shelf region

0.00677 (m-3) choke region

0.00133 (m-3) chamber region

0.00884 (m-3) nozzle region

Page 18: Evaporation effects on jetting performance

Print Physics Notes: Evaporation Effects on Inkjet Performance and Image Quality

December 2008

18

While the flow field inertance is independent of evaporation, as are

the dz values, the dz values are very dependent upon

evaporation and jetting. The evaporation model computes

(x,y,z,t), as previously described. As the bubble grows, it

displaces ink at the nozzle exit. So, each time step uses a new

viscosity field. The highly viscous portion at the nozzle exit is

pushed out of the ejector during the first few hundred nanoseconds

of bubble growth.

Examination of (Eq.31) indicates that (t) should decrease as

the bubble pushes the more viscous liquid out of the nozzle during

the early stages of bubble growth. However, this PHYD lowering

effect is offset because as the liquid in the chamber accelerates due

to the explosive pressure pulse, the volumetric flow rate increases

– tending to increase PHYD. Then, as the pressure pulse dissipates,

the volumetric flow rate decreases. Thus, it is expected that PHYD

should be non-linear in time. The non-linear nature of the

hydraulic resistance pressure drop is shown in Fig.27a-b. While

both curves have the same shape, they differ greatly in time and

pressure magnitude. Obviously, when idle time evaporation causes

an increase in hydraulic resistance, the phase-change pressure

pulse has less ability to push ink out of the nozzle. As expected,

Fig.28 illustrates a dramatic decrease in droplet momentum after 3

seconds of idle time evaporation. Droplet volume goes from 5.2pL

to 1.4pL, and velocity goes from 454 in/s to 322 in/s.

Figure 27a: Newman-mono hydraulic resistance pressure drop after 3

seconds of idle time evaporation

Figure 27b: Newman-mono hydraulic resistance pressure drop after 0 seconds of idle time evaporation

.

Figure 28: Newman-mono ejector after 3 seconds idle time evaporation

Imaging Effects of Idle Time Evaporation Quantifying the droplet momentum effect of idle time evaporation,

while useful, is not the end game. Idle time evaporation manifests

itself as print quality defects. Quantifying these print quality

defects, as they relate to ejector design, ink formulation and idle

time evaporation is the most poignant output of this study. The

defects are due to droplet misplacement and visible shifts in color,

or black-white contrast. It is possible to use basic imaging laws to

quantify the L* effect of drop size variation. An overview follows:

As a minimum set of requirements, it is appropriate to consider

that the mono droplet should:

(1) Provide the largest number of just-noticeable gray

levels between white-black. Oversized dots ensure

black is black, but do poorly with this criterion.

(2) Provide the largest dynamic range by saturating the

media with black colorant. Undersized dots do poorly

with this criterion.

Page 19: Evaporation effects on jetting performance

Print Physics Notes: Evaporation Effects on Inkjet Performance and Image Quality

December 2008

19

Criteria (1) and (2) compete with each other. This suggests that the

mono droplet characteristics need to be considered carefully. So it

begs the question, can we mathematically describe the transition

from white to black? To answer that, let us begin with the

relationship between area coverage and digital grayscale levels (0-

255).

The relationship between area coverage and digital grayscale

begins with an exercise in analytical geometry. Localized regions

of the image can be made white, black, or any discrete gray level

in between, but individual pixels can only be white or black. This

fact suggests that grayscale imaging involves the art of distributing

errors. The art of how individual droplets are laid down to

minimize visible patterns is constantly being improved. For the

purpose of this analysis the Floyd-Steinberg error diffusion pattern

will be applied to determine the grayscale dot patterns. For F-S

error diffusion, Fig.29 shows the pixel overlap types. For each of

the 10 types, the region of interest includes a 3 x 3 box of pixels.

The center pixel (crosshatched) is the one being considered. It is

surrounded by pixels that have already been considered (the solid

colored circles and rectangles). When an inkjet device is

considering whether to print the center pixel in this box it may, or

may not print the dot. This binary decision creates an error because

these choices amount to a pixel grayscale of 0 or 255. If the region

under consideration is supposed to be a gray level in between, an

error is created. The Floyd-Steinberg error diffusion algorithm

distributes this error to the pixels to the right and below, as they

have yet to be considered. The incremental area generated by

printing the center, crosshatched dot is a function of the

surrounding neighbors that have already been considered. For the

Floyd-Steinberg error diffusion algorithm, there are 10 unique

overlap types to consider for computing the incremental area of the

center pixel. The analytical geometry functions are made

applicable to any pixel resolution by normalizing the dot size to the

pixel size (Eq.32).

)0.20( ; size pixel

diameterSpot FF (Eq.32)

There are many error diffusion techniques. They all have the same

goal – grayscale reproduction while minimizing visible pattern

artifacts. We are simply using this particular error diffusion

method to estimate the incremental printed area as the digital

grayscale level goes from 0-255. We are not using it to study

pattern visibility and Fourier analysis. Surprisingly, the Floyd-

Steinberg error diffusion algorithm produces nearly the same

incremental area coverage as the antiquated Bayer 16 x 16 halftone

algorithm, and it will be shown later that it well describes the

incremental area function of the halftone algorithm used in

Rushmore. Fig.30 shows the F-S results: incremental area coverage

as a function of digital halftone level for (F) values from 0.5 to 2.0.

Type A

Type B

Type C

Type D

Type E

Type F

Type G

Type H

Blank neighbor from

previous scan

Filled neighbor from

previous scan

Current pixel

being printedError diffusion

neighbor

x

yScan directions

Pel

Size

Spot

Diameter size Pel

diameterSpot F

Type J

Type K

Analytical geometry functions are derived to

determine the dot fill area for each overlap type

(F < 2.0)

Computing The Area Fill Fraction By Analyzing

Pixel Overlap Types In An Error Diffusion Pattern

This

is m

ath

em

atically

tra

cta

ble

with a

naly

tical g

eom

etr

y

Figure 29: Pixel overlap types

Pel

Size

Spot

Diameter size Pel

diameterSpot F

F = 1

.4

F =

2.0

Figure 30: Area coverage as a function of F and digital halftone level

The blank page has a reflective luminance Ymax. The solid fill

region of the page has a reflective luminance Ymin. Between Ymax

and Ymin the gray levels exist. An 8-bit white-black transition has

256 digital levels. A representative (0-255) gray ladder is shown in

Fig.31. This curve shows the measured values of reflective

luminance (Y) for a spot size value of (F = 1.74).

Figure 31a: Measured reflectance for pigment-mono ink with normalized

spot size F = 1.74

Ymax

Blank page

Ymin

Solid Fill

For F = 1.74

Page 20: Evaporation effects on jetting performance

Print Physics Notes: Evaporation Effects on Inkjet Performance and Image Quality

December 2008

20

The cira-1936 Murray-Davies equation relates area coverage to

reflective luminance. Unfortunately, as shown in Fig.31b, it does a

poor job of matching the lab data over much of the experimental

space.

Figure 31b: Measured reflectance for pigment-mono ink with normalized

spot size F = 1.74 along with values computed by Murray-Davies equation

So the obvious solution relating luminance and grayscale, based

simply upon incremental dot area coverage, is the wrong one.

However, it is interesting to note that the Murray-Davies equation

matches the experimental data in the solidly filled region, but it

does a poor job in the regime where dot coverage is low. This

observation provides a clue to the source of the discrepancy. The

discrepancy is explained by examining what happens to the light

when it enters the non-dot regions (Fig.32). Because the media

scatters light in the non-dot regions, some of the diffused light

exits in the dot regions. Thus sparsely placed dots absorb more

light than their area alone would suggest. This discovery is

attributed to Yule and Nielsen (1957) [15].

Figure 32: Internal light reflections in the non-dot regions of the media

tcoefficienNielsen -Yule

fraction coverage area

fill solid ofdensity dot theofdensity

paperblank ofdensity

density

1011log

:isequation Nielsen -Yule theof formdensity The

10

n

c

DD

D

D

cnDD

a

Pd

P

ndD

aP

(Eq.33)

16100

116*

:by *familiar more the toconverted then is form ereflectanc The

0.10;10

:by ereflectanc toconverted is formdensity The

3/1

YL

L

RR D

DP and Dd are known from densitometry measurements (solid fill

and blank paper). Coverage area (ca) is computed by analytical

geometry. What value is to be placed upon (n)?

Physical arguments behind the derivation of (Eq.33) suggest

that n is contained within the range of 1 to 2. Furthermore, it can

be shown that as the dot coverage (ca) goes to unity, n goes to

unity. In other words, when there is little to no non-dot area there

is no need to account for the scattering effect shown in Fig.32. The

literature abounds with various means of estimating (n). Since

there is no general agreement within the image science community,

the sigmoid function proposed here (Eq.34) is both easy and

reasonable.

0.05factor steepness slope

n transitioba theofpoint -mid

)0.10(fraction coverage area

1.0 regions fill-solid in thefactor Nielsen -Yule

2.0 regionsdot -non in thefactor Nielsen -Yule

exp1

0

0

w

x

xcx

b

a

w

xx

baan

a

(Eq.34)

Figure 33: Sigmoid function for Yule-Nielsen factor (n)

RY

RcRR daP

100

11

R = image reflectivity

Rp = paper reflectance

Rd= dot reflectance

ca = area coverage fraction

Ymax

Blank page

Ymin

Solid Fill

F = 1.74

Dot Dot Dot

Media

Light Rays

The media scatters the light entering the non-dot regions. Some of the scattered light is absorbed by nearby dots.

The sigmoid function is used in this analysis to determine n as a function of area coverage*

n 2

n 1

Page 21: Evaporation effects on jetting performance

Print Physics Notes: Evaporation Effects on Inkjet Performance and Image Quality

December 2008

21

Figure 33: Measured reflectance for pigment-mono ink with normalized

spot size F=1.74 along with values computed by Yule-Nielsen equation

Fig.33 shows that when the Yule-Nielsen effect is accounted for

with the sigmoid function, the simulation results more closely

match the measured values. However, as a further test to the

veracity of (Eq.34) it is useful to apply it to a few other cases.

Fig.34 shows that (Eq.33-34) closely follows the data from

Rushmore for both pigment-black and process-black dye inks.

Figure 34: Measured L* for Rushmore pigment and process black inks along with values computed by Yule-Nielsen equation

Now that we have a means to compute the reflectance response

curve, we want to introduce the droplet size variable into the

analysis. This will permit us to quantify the idle time evaporation

effect on the darkness (L*) of the printed dots.

There are various equations that describe the spot size –

droplet size dependence. The following equation is the regression

resulting from an experiment performed by the author using a

pigment black ink on Hammermill Laser Print – a plain paper.

s)(picoliter lumedroplet vo

(microns)diameter spot

8.1841.0

pL

D

pLD

SPOT

SPOT

(Eq.35)

With all the emphasis on photo, what is the relevance of plain

paper? Let’s look to HP for the answer:

“mono-pigment is optimized for high quality black on

plain-paper”(Ref: Science of Inkjet Printing, by Ross Allen

HP-IPG)

Combining (Eq.35) with the analytical geometry method described

earlier, the coverage area for any spot diameter/pixel size (F) may

be computed as a function of digital level 0-255. Image reflectance

for any coverage area may be computed by use of the

aforementioned Yule-Nielsen equation. So knowing the

relationship between droplet size and spot size, along with the

above steps allow us to compute the L* response curve for any

combination of:

- digital level (0-255)

- droplet volume (1, 2, 3, 4,….,etc.)pL

- pixel resolution (1/300”, 1/600”, 1/1200”, 1/1800”, etc.)

Fig.35 illustrates the dependence of L* on droplet size. Note that a

nominal 5.5pL Newman-mono droplet (22pL/600dpi) is

appropriately sized because ordinary droplet size variations will

have little effect on the perceived darkness. According to Fig.35,

the edge of the response curve is around 18pL/600 dpi pixel (i.e.

4.5pL for each 1200 dpi Newman droplet). Because the nominal

Newman-mono droplet is bigger than 4.5pL this provides an

insurance policy for the tolerance range of mass produced print-

heads and their expected Gaussian variations. However, idle time

evaporation can cause the first mono droplet shot out after a 3

second idle time to be as small as 1.4pL (5.6pL/600dpi).

According to Fig.35 this will cause that first droplet to be gray, not

black. The printed single-pel lines, shown in Fig.35 leave little

room for debate: the slice printed after just 3 idle seconds is middle

gray, not black.

1/600”

1/1200”(4X) 5.5pL

Or

(1X) 22 pL

Newman-mono

No idle time evaporation

(4X) 5.5pL per

600 dpi pixel

Newman-mono after 3 seconds

Idle time evaporation

(4X) 1.4pL per 600 dpi pixel

Figure 35: L* variation with droplet size

Fig.35 shows that when idle time evaporation is not at work,

the ejectors have the ability to fully fill a 600 dpi pixel with (4X)

5.5 pL droplets. Fully filled pixels have an L* value of 21. This is

in good agreement with actual, plain-paper printing results. Fig.35

also shows that when 3 seconds of idle time have elapsed, the

resultant small droplets form light gray images, and according to

the response curve, have L* values of 55. Incidentally, the just-

noticeable-difference (JND) of L* values is on the order of 1-2%,

so an L* shift from 21 to 55, after just 3 seconds of idle time

evaporation, is remarkable, extremely visible and real.

F = 1.74 size Pel

diameterSpot F

Process Black Rushmore Bigs Only

Pigment Black Rushmore Mono Only

Lab Results

Simulation Results

Process Black Rushmore Bigs Only (10pl)

Pigment Black Rushmore Mono Only (24pl)

(Lab data from Richard Reel’s department)

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Print Physics Notes: Evaporation Effects on Inkjet Performance and Image Quality

December 2008

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Now the model has the ability to go beyond the multi-physics of

evaporative mass diffusion, ink formulation, heat transfer, bubble

nucleation, phase-change bubble growth and jetting. By using a

few basic imaging laws, combined with analytical geometry, the

L* response to spot size has been demonstrated. Thus the effect of

idle time evaporation has been carried from the micron,

microsecond scale, hardware domain of the ejector to the

macroscopic, psychophysical domain of visible print quality

defects.

Idle Time Evaporation and Dot Placement Error As mentioned earlier, print quality defects may also come from dot

placement error. That aspect of idle time evaporation will now be

studied. This will also be a good time to bring Newman-color inks

and ejectors into the analysis.

The Puddy2 ink formulation was described in Table 6, and the

evaporation-viscosity response was shown in Fig.18-21. The

FEAJET results for the Newman color ejector with Puddy2 ink is

shown in Fig.36. The simulated droplet size (4.2pL), droplet

velocity (504 in/s), refill time (44s) are all in good agreement

with lab data. So the base model is a good starting point for

comparative studies on idle time jetting.

Figure 36: Newman-color ejector

Figure 37 shows the results of the evaporation-diffusion model at 3

seconds of idle time for Puddy2 ink filled Newman-color ejectors.

Note that this response curve is similar to that of Newman-mono

ink (Fig.24). The top 3 microns of the nozzle is greatly more

viscous than the rest of the ejector. However, the viscosity scale is

greatly reduced compared to the pigment-mono response curve. At

3 seconds of idle time evaporation, mono ink at the nozzle exit is

70 mPa-s, greatly exceeding the color ink’s 25 mPa-s value.

Figure 37: Newman-color viscosity heater-nozzle after 3 idle seconds

Importing the Newman-color viscosity response curve into

FEAJET leads to the idle time jetting response shown in Fig.38. As

expected, the droplet momentum is reduced due to idle time

evaporation. However, the reduction is not as pronounced as it was

with the mono ejector. This is clearly illustrated when comparing

drop placement error between mono and color. Drop placement

error is computed by:

IDLEJETJET

IDLEJETJET

CARRIERVV

VVLV

0

0 (Eq.36)

= drop placement error

VCARRIER = carrier velocity

L = nozzle-media gap

VJET-0 = jet velocity at zero idle time

VJET-IDLE = jet velocity after idle time evaporation

The drop placement error curves of Fig.39 are very close to the

actual idle time response data on drop placement error. Note that

the simulation shows mono should have a one pixel error after

three seconds. For this same one pixel error, the color ejector can

remain idle for 9 seconds. Fig.39 looks simple and anticlimactic,

but it has a powerful message – all of the preceding numerical and

physics exercises come into play to produce this ordinary looking

plot.

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Figure 38: Newman-color ejector after 3 seconds idle time evaporation

Figure 39: Drop placement error Newman mono and color

Discussion The results shown in Fig.39 beg the question: Is the difference in

dot placement error an artifact of the ejector design, or is it due to

ink formulation?

At 20C, the dye-based color ink has a viscosity of 2.6cP. The

pigment-mono ink viscosity is 2.8cP. These starting values are

very close to each other and in no way could be responsible for the

idle time response difference between Newman color and mono.

Rather, the idle time response is a function of the ink viscosity in

its evaporated state, not its original state. Fig.24 and 37 show this

effect more clearly. The pigment-mono ink viscosity after 3

seconds of idle time is greatly different than the dye-based ink.

Qualitatively and historically this has always been the case.

Pigment-mono sets the maintenance frequency not the dye-based

colors. One may still argue that the viscosity response curves of

Fig.24 and 37 are confounded because they are dependent not just

upon ink formulation, but flow features as well. True enough, so

let us simply swap the inks and re-run the evaporation and jetting

models to see if idle time sensitivity follows the ink, or whether it

follows the flow features.

Fig.40 clearly shows that idle time sensitivity follows the

inks. In fact, when the pigment-mono ink is coupled with the

Newman-color flow features, the model indicates that this ejector-

ink combination would have near-zero idle time before the drop

placement error exceeded one pixel. It also shows that loading

Newman-mono flow features with dye-based color ink would push

idle time to ten seconds, or more, before exceeding one pixel of

drop placement error. This not only confirms that idle time

sensitivity follows the ink, but it also shows that pigment ink in a

small-droplet, color ejector will have horrific maintenance

problems (as we have already experienced in prototype 2.5pL

ejectors). This is easily explained. Eq.31 teaches that hydraulic

resistance is a function of viscosity and geometry effects: (x,y,z,t)

and . The ink swap performed here not only accounted for the

higher evaporated viscosity of pigment ink, it also accounted for

the higher geometric flow resistance factor () of the color ejector.

Because the flow features of the Newman color ejector are smaller

than those of the mono ejector -color is 2.3X greater than -

mono. Thus pigment inks, with their extremely high viscosity in

the evaporated state are especially problematic in ejectors with

small flow features, designed for small droplets.

Figure 40: Drop placement error Newman after swapping inks

The Pigment Ink Dilemma Pigment inks are notoriously more prone to maintenance problems

than dye-based inks. This analysis quantified the “why” with

respect to the Newman inks. When water evaporates, leaving

behind the high boiling point co-solvents, the mixture viscosity

increases rapidly. The evaporated viscosity increase of the

Newman-pigment formulation is 200% greater than the Newman-

dye formulation. Part of the problem with pigment inks is the

viscosity multiplying effect of solid particles (Eq.21). Thus

lowering pigment and dispersant loading is beneficial to idle time;

however, there is an image contrast price to pay for low pigment

load. Low pigment loading in mono ink produces low OD-solid fill

regions. This not only affects the appearance of text, but it reduces

the dynamic range of the media-ink system. Reducing the dynamic

range of the media-ink system makes it difficult to produce

continuous grayscale ramps that are so vital in producing realistic

looking shadow regions of images. Low pigment loading in color

inks produce images that lack the gamut needed to produce photo

Page 24: Evaporation effects on jetting performance

Print Physics Notes: Evaporation Effects on Inkjet Performance and Image Quality

December 2008

24

quality. For example, to satisfy photo-centric imaging scientists,

the Gen1-pigment color formulations had between 6-6.5 wt.%

colorant to achieve the gamut obtained with 4 wt.% dye

formulations. The Gen1-pigment colors also used a polymeric

dispersant that consumed another 1.9 wt.% of the mix (typically in

the industry, pigment-dispersant ratios are about 15:4). Because the

dispersants attach themselves to the pigment particle, the charged

particle shell is physically larger than the dry pigment particle

itself. This effect is illustrated as (D) in Fig.41. That said, when

polymeric dispersants are used in conjunction with pigment

particles, they both go into the Krieger-Dougherty equation, and

the solid volume fraction increases – increasing the viscosity

multiplier of Eq.21.

Pigment Particle Kinetics The solids account for some of the pigment-dye viscosity

difference. The other part of the issue is that the remnant co-

solvent blend used in pigment formulations are very often more

viscous than those used in the dye formulation. This slows the

pigment retreat as water evaporates – a good thing. Charged

pigment particles seek water to balance the electrostatic field

formed by the double charge layer surrounding them. When

evaporation leaves a humectant-rich, water-poor mixture in the

nozzle, the pigment particles retreat to the water-rich region of the

ink via.

Pigment retreat is a form of electro-kinetics. If a charged

particle is accelerated with respect to the surrounding fluid when

an electric field is applied, the particle kinetics are defined as

electrophoresis. If the fluid moves under the influence of an

electric field, the flow is defined as electro-osmotic. Observations

of pigment particle retreat show that the particles move thru a

stationary liquid, thus the kinetics of pigment retreat is a form of

electrophoresis. When an electric field is placed across a pigment

loaded ink, electrophoretic flow occurs. This is easily seen when

an anode-cathode probe pair is placed in the ink because pigment

particles travel towards one of the probes. The source of the

electric field in capillary electrophoresis may be anode-cathode

pairs as often used in bioMEMS applications [18] (DNA micro-

arrays, Lab-on-chip, hemoglobin electrophoresis, protein analysis,

etc.). Or, the electric field source may be due to ionic and pH

gradients that come about from evaporation driven concentration

gradients of water in the ejector.

Figure 41: Charge field around an electrophoretic particle

Fig.41 shows the double charge layer surrounding a particle. Under

the influence of an electric field, the electrophoretic velocity is

[17]:

Ev L

EP

0 (Eq.37)

vEP = electrophoretic velocity

0 = zeta potential = f(chemistry, pH, temperature)

L = dielectric constant of the liquid

= viscosity

E = electric field strength (Volt/m)

If (vEP) is large, the pigment retreat is fast, and the ejected droplets

lack colorant. Pigment retreat is a serious problem – what good is a

jettable ink formulation if the first few droplets have no colorant

due to pigment retreat. Pigment particles move more slowly thru a

high viscosity mixture. Note in (Eq.37), electrophoretic velocity is

inversely proportional to viscosity.

So there is a particle-kinetics benefit to using more viscous

co-solvents in pigment inks. If co-solvents are chosen such that

the mixture has low viscosity in the evaporated state, the idle time

jetting may be unimpeded, but the ejected droplet will have low

colorant loading because the low viscosity mixture allows rapid,

electrophoretic transport of pigment particles away from the water-

poor nozzle towards the water-rich ink via. Yet, if the co-solvents

are viscous enough to slow the pigment kinetics, they form a

highly viscous plug, a few microns long, that impedes jetting.

There may be a middle ground between dye-based and

pigment-based formulations that offer the best set of tradeoffs

between image permanence, gamut, pigment retreat

(electrophoresis) and idle time jetting. If so, the viscosity response

curves will likely lie between those of Fig.19 (dye) and Fig.42

(pigment).

Figure 19:Puddy2-dye-ink – Viscosity versus temperature and evaporation

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Figure 42:894 Pigment-mono ink – Viscosity versus temperature and evaporation

Epson Pigment-Color Ink In Lexmark’s market segments, the nearly universal rule is that

mono ink is pigment (with droplets >> 4pL) and color is dye-based

(with droplets < 4pL). An exception to this rule is Epson. With

their piezo-electric technology they have chosen to incorporate

pigment-color inks. Let’s examine their formulation to see if they

have discovered the magic where image permanence, gamut

pigment particle kinetics and idle time jetting live happily in

unison. Tab.7 (courtesy of Agnes Zimmer) lists our best analytical

chemistry estimate of the Epson C88 Pigment-Magenta ink

formulation. Fig.43 shows the viscosity response curves of this ink.

The simulation shows that this ink has a room temperature (20C)

viscosity of 4.36cP. Epson, on their MSDS [16], reports this ink

having a viscosity less than 5cP, so the simulation results agree

with the published information. The Epson C88 pigment color

viscosity response curves look nearly identical to the Gen1-

pigment magenta results shown earlier in Fig.12. The Gen1

pigment-CMY formulations were simply horrible from an idle time

viewpoint. That said, it is clear that Epson may have a pigment ink

formulation for image permanence and probably not prone to

pigment retreat; however, the enormous viscosity in the evaporated

state means that the C88 formulation has no magical idle time

properties.

Table 7: Epson C88 pigment-magenta ink formulation

Component Wt.%

2-pyrollidone 3.2

Glycerol 12.6

Triethylene glycol mono butyl ether 2.5

Trimethyol propane 4.3

1,2-Hexanediol 2

Water 70

Colorant – dispersant load Bal (5.4)

Figure 43:Epson C88 Pigment-color ink – Viscosity versus temperature

and evaporation

HP Pigment-Mono Ink The mono-pigment formulation from HP is shown in Table 8

(courtesy Agnes Zimmer), and its viscosity response curves are

shown in Fig.44. According to the simulation the HP-mono ink

should have a viscosity of 3.5cP at 20C. This agrees very well with

the experimental value of 3.3cP at 22C. It is interesting to note that

the HP pigment mono formulation has much lower viscosity in the

evaporated state than all other pigment inks examined in this

document. In the evaporated state the HP pigment mono ink looks

more like Puddy2-dye inks than 894-mono pigment, Gen1-color

pigment and Epson-color pigment. This does not imply that HP has

found the magic spot where pigment inks are optimized, but it

clearly shows that they understand the issues and the importance of

not allowing the evaporated state viscosity to leap up to values

greater than dye-based inks.

Table 8: HP88 pigment-mono ink formulation

Component Wt.%

2-pyrollidone 18.8

2-methyl-1,3-propanediol 2

Tetraethylene glycol 2

Water 71.3

Colorant – dispersant load Bal (6)

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Figure 44:HP88 Pigment-mono ink – Viscosity versus temperature and

evaporation

Canon Pigment-mono Ejector One last set of simulations is worth a short discussion here to

quantify how the ejector design knobs may be used to enable

robust idle time with pigment ink. Let us examine the pros and

cons of Canon’s mono ejector. This is an interesting design point

because it is a throwback to where Lexmark ejectors were a decade

ago – 600 dpi, large droplet. To isolate the effect of this ejector

design, allowing direct comparison to Newman-mono, let us fill it

with 894-pigment mono ink. Fig.45 shows the simulation summary

for this design point. Canon does not publish the size of their mono

droplets; however, it is quite likely that the predicted value is

correct because at 23pL per 600dpi pixel, this ejector shoots a very

good droplet size according to Lexmark criteria (and Fig.35).

Figure 45: Canon mono-pigment 600 dpi ejector

Fig.46 shows that the Canon ejector with pigment mono ink is far

less prone than Newman to produce pixel-sized drop placement

errors even after 10 seconds of idle. It is more likely that Canon-

mono idle time is set by loss of colorant due to droplet size

reduction, as shown in Fig.47. This plot clearly illustrates the value

of a 600dpi ejector when pigment-mono inks are used. After 5

seconds of idle, the Canon droplet size falls to about 14pL. At

14pL the first droplet shot is in the vicinity of one JND- L* unit (a

just noticeable difference of reflective luminance).

The primary use of pigment mono is for text. At 600 dpi, text

is about as good as it needs to be. There is really no need for

pigment mono in images as long as the process black is dark

enough to ensure high contrast and dynamic range. By coupling a

600dpi mono ejector that shoots 22+pL, with 1200 dpi, multi-level

color ejectors (1.5 – 4.5pL), Canon has chosen to design their

mono-pigment ejector for text and the color-dye ejectors for

images. In so doing, they avoid the idle time issues that are so

problematic when one tries to use pigment inks in conjunction with

high resolution (1200dpi) and small droplets (<5pL).

As mentioned at the outset of this report, idle time physics is

not about using models to find bulletproof, optimized designs –

rather it is about using models to identify the proper tradeoffs. In

Canon’s case, the tradeoffs were system architecture decisions, not

just tweaks to hardware designs.

Figure 46: Drop placement error Newman and Canon mono

Figure 47: L* variation due to idle time Newman and Canon mono

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Print Physics Notes: Evaporation Effects on Inkjet Performance and Image Quality

December 2008

27

Conclusions Idle time brings to mind idyllic Summer scenes of sipping lemonade

on the porch swing. However, like most things inkjet, idle time is

anything but an easy, mindless exercise. This document has shown

that to quantify inkjet idle time one needs to make use of multiple

areas of science; ink formulations, heat and mass diffusion on both

sides of the liquid-air interface, thermodynamic interactions

between moist air and water, the thermo-hydrodynamics of jetting

and some aspects of image science. Even with all these fields

covered there are gaps in the unknown – for instance the

quantitative electro-kinetics of pigment retreat. There are always

gaps in knowledge, but that does not mean what is known is

irrelevant. This article has shown consistent veracity between

experimental results and simulations. Also, the analysis has shown

the self-limiting characteristic of idle time evaporation from a

quantitative viewpoint. Furthermore, it has shown direct cause and

effect between idle time evaporation and the resultant print quality

defects due to L* variations and drop placement error. As mixed

viscosity is important – indeed the ejector flow features are designed

and tuned to this value. However, this analysis clearly shows that

viscosity in the evaporated state is the dominant idle time lever.

Idle time has always been a topic of great concern and debate,

but it never got beyond the qualitative and empirical realms. For the

first time in nearly two decades of inkjet development at Lexmark,

we are now poised to start speaking of idle time issues and tradeoffs

quantitatively.

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