Dry Time Kinetics Overview

Inkjet Dry Time Kinetics An Overview of the Mathematical Physics

Bob Cornell [email protected] Print Systems Science

February 17, 2012

(after April 30, 2012)

To move into the space traditionally dominated by laser printers, the speed and

print quality of inkjet printers needs to be on par with EP. The speed and print quality

attributes have been addressed; however, along the way a new artifact has emerged.

As we move forward into even faster devices - and even more demanding customer

environments - dry time kinetics needs to be engineered into the products as

robustly as all the other features and functions.

Motivation

Contents • Evaporation versus absorption

• Functional dry time

• Absorption = diffusive mass transport

• Interceptor case studies

• The small droplet zombie

• Paper warming to supercharge dry time

• Appendix #1: The Cosine(q) effect

• Appendix #2: The wetting delay phenomena

Admittedly, this is a lengthy document to call an Overview. However, this document is intended to be a stand-alone epistle that does not require verbal, stand-up narration. Hopefully, it captures all of the analysis and research on this topic that has evolved over the course of this lengthy investigation, and hopefully, it presents it in a manner that is interesting and useful to future LXK investigations on this topic. Lastly, it is hoped that this document gives timely, quantitative guidance to the system and product engineers responsible for defining the architecture of our next generation product lines. The models and literature described herein are stored on the Public Drive: inkjet-data(\\lexfile15) Q: PUBLIC/Dry Time Kinetics

Preface

//lexfile15

//lexfile15

//lexfile15

Let’s begin the investigation by examining just what goes on at the ink-paper and ink-air interfaces.

Paper Surface Roughness

Ink Surface at Time Zero

Evapora

tion

Absorp

tion

Time Zero - Just After Droplet Impact

The Onset of Evaporation and Absorption

Ink Layer

Water Management • Many people assume that water management implies water evaporation. Thus sizing

a drier is simply:

• In 1996, HP’s Ross Allen presented a paper at IS&T-NIP12, where he discussed his calculations for the drier requirements of a page-wide machine.

• This article stated that the drier needed to be an astonishing 4300 Watts. – Sidebar: HP’s Edgeline woes may have had their genesis with the back-of-the-envelope calculations

contained within the Ross Allen article. It is well-known that they nearly required 220V to supply power to this beast. They barely made it under the max power capability of 115V wall plugs.

• In a 1990 presentation at the SPIE Conference, Arthur Gooray, of Xerox, presented his evaporation calculations that cited a need for 3000 Watts to dry 50 ppm.

• Both of these calculations are supposedly based upon the laws of thermodynamics. • However, they are both absurd and misleading. We will show that attacking the

problem from the absorption front reduces power by an order of magnitude. • Lastly, ink may be mostly water from a mol-fraction viewpoint, but from a wt.%

viewpoint it is only ~70% water. Typical inkjet inks contain 15-20 wt.% low vapor pressure liquids from the glycol family. These liquids have a boiling point ~300C, so they are not affected by attempts to evaporate them with heated driers.

kg

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HTCmP

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re temperatuRoom -point Boiling Rise eTemperatur

ink theofheat specific

dryingfor available time

evaporate tomassink

(Watts) RequiredPower n Evaporatio

Water Management is Simply Evaporation, Right?

Ambient pressure = Pamb (Pa)

Ambient temperature = Tamb (C)

Relative humidity = RH (%)

Normal boiling point = Tboil (C)

Latent heat of evaporation = hfg (J/g)

Liquid molecular weight = Mliquid (g/mol)

Liquid density = rho_liquid (g/cm^3)

Air molecular weight = Mair (g/mol)

Air-water diffusivity = Dab (cm^2/s)

Absorption coefficient = Ka (mL/m^2/ms^0.5)

Evaporation-Absorption Variables

Click to compute Absorption/Evaporation

Model Developed to Quantify the Evaporation – Absorption Effects

Air-Ink Interface

Variables

Ink-Media Interface

Variable

The Air-Ink-Interface is described by the laws of thermodynamics, but how do we quantify the Ink-Media-Interface?

The Ink-Media Interface Response is Typically Measured With a Bristow Tester

Historical footnote: J.A. Bristow developed this tester while working for the Swedish Packaging

Research Institute in Stockholm (circa 1967).

Absorp

tion V

olu

me p

er

Unit S

urf

ace A

rea (

mL/m

2)

)(ms )( 0.5mstime

Vo

Vp

wetting

delay *

Vo = amount of liquid that fills the surface voids

before capillary forces drive absorption

Typical Bristow Test: Absorption Response for Paper-Ink Pair

* The wetting delay is discussed in Appendix #2

Typical Bristow Test Results*

* Selim, Yesavage, Al-Ubaidi, Sung, Drying of Water-Based Inks on Plain Paper,

Colorado School of Mines Report for IBM, (1989).

Typical Results From LXK-Built Absorption Tester*

* Bill Conners, John Writt, Paul Sacoto

0 5 10 15 20 25 30 35 40 45 505

10

15

20

25

time (ms0.5

)

Ink

Tra

ns

ferr

ed

(m

L/m

2)

Tidal Paper With M1K Ink

Ref: S. Bares, K. Rennels, Paper Compatibility With Next Generation Ink-Jet Printers, TAPPI Journal, (1990).

Common Theme in the Literature:

The Absorption Coefficient (Ka) Must be Less Than ~0.25 to Achieve Good PQ

humidity relative

K 373point boiling

298Kat kg

J10 2.35n evaporatio ofheat latent

equationClapyron -Clausius ;11

exp

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

100

200

300

400

500

600 15 C

20 C

25 C

30 C

35 C

40 C

45 C

50 C 55 C 60 C 65 C 70 C 75 C 80 C

Comparison Between the Effect of Absorption and Evaporation

Relative Humidity

=

Ab

so

rpti

on

Ra

te/E

va

po

rati

on

Ra

te

Ka = 0.25 (mL/m2/ms

0.5)

Absorption/Evaporation = 96: at Tamb = 25 (C), RH = 40 %

Dry-Time-Kinetics

14-Feb-2012

Evaporation-Vs-Absorption-GUI

For typical environments and typical ink-media pairs, the absorption effect

is about two orders of magnitude more dominant than the evaporation effect.

Thus if we want to supercharge the dry time kinetics, it is best to work with

the parameter having the biggest lever - absorption.

Model Results

With this preamble finished, let us now focus upon how we may enhance absorption. It is important to note that HP8600 chose to address this problem via ink formulation. Not surprisingly, their faster penetrating ink option also resulted in very noticeable PQ degradation with an L* value reminiscent of process black (on non-color lock media). Trading PQ for speed would be a step backwards for LXK after our hard fought (and won) multi-year battle to produce PQ at speed.

Ambient pressure = Pamb (Pa)

Ambient temperature = Tamb (C)

Relative humidity = RH (%)

Normal boiling point = Tboil (C)

Latent heat of evaporation = hfg (J/g)

Liquid molecular weight = Mliquid (g/mol)

Liquid density = rho_liquid (g/cm^3)

Air molecular weight = Mair (g/mol)

Air-water diffusivity = Dab (cm^2/s)

Absorption coefficient = Ka (mL/m^2/ms^0.5)

Evaporation-Absorption Variables

Single droplet (pL)

Solid ink coverage (pL/600 dpi) Enter a value


Select the Ink Coverage

Now Let’s Use the Model to Compute Functional Dry-Time

Air-Ink Interface

Ink-Media Interface

Ink Coverage

Paper Surface

Ink Has Dropped Below The Surface Roughness Level

Ink Penetration Front

Functional Dry-Time*

* Note: The ink is still subject to smearing if rubbed by a finger at this point

in the drying process; however, it is dry enough to prevent offset-smear, or snow-plowing

onto the next print in the queue.

According to the literature, functional dry-time is a measure of when offset smear does not occur.

The point at which this occurs is when the ink layer drops below the microscopic hill tops that

exist on all papers.

When the droplets hit the media they make a transition from sphere to pancake. Since ink impact is the genesis of the dry-time event, let us begin our quest by closely examining the physics of this event.

Pancake Formation at Droplet Impact

*Ref: A. Asai, et.al., Impact of Ink Drop on Paper, IS&T-NIP7, (1991).

This study* evaluated the droplet spreading response of various ink formulations across

a range of media types. It concluded that spreading was entirely due to dimensionless

properties of the droplet: Weber number (We), Reynolds number (Re). Media type

(copy, bond, transparent film) had no effect, nor did surface coating or roughness.

While these parameters affect the capillary response, Asai found that the spread factor

(i.e. the initial conditions for the ensuing capillary event) of any inkjet droplet are

determined solely by We and Re.

Asai’s Inertial Spreading Response

impact prior todiameter droplet

impactafter diameter pancake final*

1] [Eq. Re48.1exp48.01*

:equation regression sAsai'

21.022.0

d

D

W eW ed

D

Historical Footnote: Akria Asai is one of the world’s original pioneers in the science of thermal inkjet.

Dr. Asai authored many scientific articles during his career at Canon. I had the privilege of

meeting him at an IS&T function in 1990.

impact @locity droplet ve

impact @ tension surfacedroplet

impact @scosity droplet vi

densitydroplet

lumedroplet vo

6

impactdroplet after diameter pancake final*

2] [Eq. 5.1exp2

*

22.0

21

31

f

d

fd

fd

U

a

D

UaUaD

Asai’s equation may be re-arranged as shown below. In this form, it shows us that the

droplet’s spreading response at impact should behave somewhat like a function of

droplet volume to the 1/3 to 1/2 power. For impact velocity equal to zero, the 1/3 power

dominates, and as impact velocity increases, the ½ power term dominates.

This observation will be important when we later discuss the small droplet zombie.

D*

As droplets travel between the ejector and the media surface - aerodynamic drag effects a

non-trivial deceleration. Aerodynamic deceleration is a nonlinear function of droplet size.

Mist investigations conducted during the Newman timeframe led to an aerodynamic model* to

evaluate the forces acting upon the droplet in-flight. Using that model, we can estimate (Uf)

at impact. It can be shown that the deceleration effect may be estimated by [Eq. 3].

This equation represents a regression on the simulation results for droplets ranging from

1.5 to 150 picoliters (pL) traversing a media gap of 1.65 mm.

Sidebar: For typical droplets during typical inkjet flight time, it can be shown that droplet

evaporation is negligible – on the order of 1%. Therefore, it is reasonable to ignore the

in-flight evaporation effect.

(pL) lumedroplet vo

locitydroplet ve initial

ocityimpact veldroplet

3] [Eq. log85.0

0

2.010

0

d

f

df

U

U

U

U

*Ref. R. Cornell, Misting Analysis.ppt (6/2009).

Accounting for Aerodynamic Drag in the Spread Factor Droplet Velocity Term (Uf)

It has been this author’s experience that ink-media spread does tend to behave in a

fashion somewhat like a (Vol)1/2 power. Others have reported similar results.

On the occasions that droplet size-spot size experiments have been run, it has been

the collective LXK experience that ink-media spread does tend to follow a response curve

that is described by a regression equation to the ~ 1/2 power.

A particular example is illustrated below, along with the response predicted by [Eq. 2-3]

0 5 10 15 20 25 30 350

20

40

60

80

100

120

Droplet Volume (pL)

Sp

ot

Dia

me

ter

on

th

e M

ed

ia (

m

)

Typical Droplet Impact Response

[Eq. 2-3]

LXK-JP/RG Experiment

Dry-Time-Kinetics

20-Jan-2012

Plot-Impact-Spread

[Eq. 2-3]

Ink-media spread factor regression result:

D* = 18.8(pL)0.5

Ref: Jim Powers & Rich Goin 1/29/2001

The Ink Pancake Provides for us a Starting Condition for Drying/Absorption

Media with an initial moisture content (~5%)

Ink pancake @ impact

Centerline 100% moisture content

at nodes along this line

until the ink volume

drains into the media. Initial volume

The ink pancake may be an

axisymmetric puddle due to

droplet impact, or a stripe of

ink due to printing a solid patch

Initial height

Dry

med

ia i

s ~

10

0 m

icro

ns t

hic

k

When one studies the literature, a common variable keeps popping up that

others use to describe the ink-media penetration mechanics.

It is Ka --- the Absorption coefficient.

Indeed the determination of Ka is the primary objective of Bristow Testers.

The variable Ka is always reported in units of mL/m2/ms0.5

With a small amount of manipulation it becomes apparent that Ka2 has units

of m2/s. This is important because all diffusion equations have a m2/s term.

Thermal diffusivity has units of m2/s. Mass diffusivity has units of m2/s.

Viscous diffusivity has units of m2/s. Even charge carrier flow per volt in a

semiconductor has units of m2/s.

This m2/s observation is important because it says that if we can measure Ka

experimentally – which we can – we may then solve the diffusion equation to

quantitatively determine the ink-media penetration behavior. This will allow us

to estimate functional dry time a-priori – just from the Ka experiment . Having a

means of predicting dry-time from an offline piece of test equipment, will avert

the late-DVT smear surprises like we had on Interceptor.

The Absorption Coefficient (Ka)

3610 : mmLNote

Diffusive Mass Transport Ink Absorption Mechanism

time

ionconcentrat species

t coefficiendiffusion

:Law 2nd sFick'

22

2

2

2

2

2

2

2

t

Ks

mD

Dtzyx

D

aa

aa

We can solve this equation by any convenient numerical means.

The most powerful and flexible method of solving any given p.d.e.

is arguably – the finite element method.

Historical Footnote:

Adolf Fick (Germany 1829-1901) was studying to become a physician, but he was also very

interested in mathematics and physics. Fick’s laws of diffusion were similar to Fourier’s laws of

heat transfer. Fourier (1768-1830) showed that heat transfer was a function temperature gradient,

while Fick showed that mass transport was a function of concentration gradient. Fick’s laws were

experimentally validated 25 years after he posed it theoretically.

LXK Ink Penetration Model

matches Agfa’s

experimental results.

Agfa Experimental Result

3.2 m

Penetration time = 984 ms

Mo

istu

re C

on

ce

ntr

ation

LXK Model Validation

From: G. Desie & C. Van Roost, J. IS&T, 50, 294-303, (2006).

Dry Time Examples

Already printed

Colu

mn 1

Colu

mn 2

Colu

mn 3

Colu

mn 4

30 in/s

18 KHz 1/600”

1/600” 1/600”

1/1800”

1/1800”

1/1800”

pixel dpi 600

pL5.22

dot

pL5.7

in 600

21800

3

dots 6 Coverage

2

Mariner Mono Ink Coverage

22.5 pL / 600 dpi

During full swath printing

the chip warms and actually

jets ~8 pL/fire, so the solid

fill coverage may actually

be closer to 24 pL/600 dpi.

Typical L* Versus Coverage Response Yellowstone Mono Dot Size vs. L* Study; Hammermill Laser Print

20

22

24

26

28

30

32

34

36

38

5.00 15.00 25.00 35.00 45.00 55.00 65.00 75.00 85.00

Ink Area Density [ ng / (1/600 in.)² ]

L*

7-9 ng 10-11 ng 12-13 ng 18-20 ng 21-23.5 ng 24-26 ng

Interceptor 7.5 pL droplet(2012)

Babbage 28 pL droplet (1997)

Monet 112 pL droplet (1995)

Picasso 135 pL droplet (1992)

Over a 20 year time frame, mono-droplet size has changed by ~20X; however, the solid fill

ink-volume per unit area has changed very little…because customers want black, not gray.

Solid Fill Ink Coverage (pL per 600 dpi pixel)

Representative L* versus coverage data from Colin Maher

Solid Fill for Mono Inkjet

Canon and HP’s inkjet history maps into this envelope as well

Example 1: Hot Interceptor Head 24 pL/600 dpi Pixel Solid Patch of Ink

24

0 20 40 60 80 100 120 140 160-100

-90

-80

-70

-60

-50

-40

-30

-20

-10

0

Horizontal Distance From Centerpoint Position (m)

24 (pL/600 dpi) Solid Patch; Absorption Coef = Ka = 0.2 (mL/m2/ms

0.5)

Pe

ne

tra

tio

n d

ep

th (

m

)

10

20

30

40

50

60

70

80

90

100

Moisture Concentration Field at Functional Dry-time

Functional dry time = Absorption time = 4.4078 (s)

Moisture penetration depth at functional dry time = 42 (m)

Initial pancake thickness = 13.392 (m)

10% Surface concentration at X = 122 (m)

Initial ink patch width X = 100 (m)

For this example, the functional dry time is 4.41 seconds.

Thus the ink-media pair is capable of ~ 14 pages per minute, i.e.

A Newman-like machine would probably be smear-free, but Interceptor would not.

Example 2: Nominal Interceptor Head 22.5 pL/600 dpi Pixel

Solid Patch of Ink

22.5

For this example, the functional dry time is 3.93 seconds.

Thus the ink-media pair is capable of ~ 15 pages per minute, i.e.

A Newman-like machine would probably be smear-free, but Interceptor would not.

0 20 40 60 80 100 120 140 160-100

-90

-80

-70

-60

-50

-40

-30

-20

-10

0


22.5 (pL/600 dpi) Solid Patch; Absorption Coef = Ka = 0.2 (mL/m2/ms

0.5)

Pe

ne

tra

tio

n d

ep

th (

m

)

10

20

30

40

50

60

70

80

90

100







Example 2: Reduce Coverage by 10% 20 pL/600 dpi Pixel Solid Patch of Ink

0 20 40 60 80 100 120 140 160-100

-90

-80

-70

-60

-50

-40

-30

-20

-10

0


20 (pL/600 dpi) Solid Patch; Absorption Coef = Ka = 0.2 (mL/m2/ms

0.5)

Pe

ne

tra

tio

n d

ep

th (

m

)

10

20

30

40

50

60

70

80

90

100







Reducing the solid ink fill by 10% reduces the functional dry time to 3.17 seconds.

So this ink-media pair is capable of ~19 pages per minute. While, not quite

smear-free at 20 ppm, reducing solid fill by 10% is a big improvement.

Obviously, Reducing Coverage is not a Sustainable Strategy to Address Dry Time Kinetics on Even Faster Machines

Yellowstone Mono Dot Size vs. L* Study; Hammermill Laser Print

20

22

24

26

28

30

32

34

36

38

5.00 15.00 25.00 35.00 45.00 55.00 65.00 75.00 85.00

Ink Area Density [ ng / (1/600 in.)² ]

L*

7-9 ng 10-11 ng 12-13 ng 18-20 ng 21-23.5 ng 24-26 ng

100%

Interceptor

90%

Interceptor

Visibly lighter print

Representative L* versus coverage data from Colin Maher

So reducing coverage by 10% effects 1-2 L* units of variation. Is this important?

Thus it may be asked: “How much L* variation is needed to be detectable by humans?”

The answer is that our vision system can detect ~ 0.4 L* units of change in solid-fill black regions.

So, the answer is – yes – reducing solid fill coverage is detectable by the human vision system.

What about the effect of droplet size? -------------------------------------------------- Doesn’t Memjet state that smaller droplets are their key to dry prints at 60ppm? -------------------------------------------------- Let us use the model to put their statement to the test.

5 10 15 20 25 3010

-1

100

101

Ink Coverage (picoliters per 600 dpi)

Fu

nc

tio

na

l D

ry T

ime

(s

)

Functional Dry Time Versus Ink Coverage and Absorption Coefficient

60 ppm max

30 ppm max

20 ppm max

15 ppm max

12 ppm max

Ka = 0.10 mL/m2/ms0.5

Ka = 0.15 mL/m2/ms0.5

Ka = 0.20 mL/m2/ms0.5

Ka = 0.25 mL/m2/ms0.5

Ka = 0.30 mL/m2/ms0.5

Ka = 0.35 mL/m2/ms0.5

Matlab-Drying Mechanics

12-Feb-2012

Plot-Ka-time

Model Summary Results

For a solid area fill of 22.5 pL/600 dpi pixel, a 60 ppm machine needs

an absorption coefficient greater than 0.35 mL/m2/ms0.5. Memjet simply lays down less

ink to produce “dry” images on plain paper (Ka ~ 0.2) at high speed.

Typic

al LX

K M

ono C

overa

ge

Mem

jet

Mono (

KK

) C

overa

ge @

60 p

pm

16

00

dp

i x 8

00

dp

i m

od

e

Mem

jet

Mono (

KK

) C

overa

ge @

30 p

pm

16

00

dp

i x 1

60

0 d

pi m

od

e

10%

Reduced L

XK

Mono C

overa

ge

The plot on the previous page illustrates that Memjet’s dry time has nothing to do with their droplet size choice. Rather it is simply the expected result of putting down far less ink than we do (which manifests itself in gamut an OD deficiencies). ------------------------------------------------------------------------------- Still, there is a persistent belief that smaller droplets should be faster drying than larger droplets. This leads to the widely held postulate that 60 ppm dry time can only be achieved with small, Memjet-like droplets. -------------------------------------------------------------------------------- Let us examine this postulate from yet another quantitative viewpoint.

The existing evidence shows us that a jetted droplet pancakes into a flat shape

upon impact with the paper. Furthermore, the existing evidence points to a relation

showing that the pancake diameter is a function of (droplet volume)1/2.

From this it follows that:

s. themselvedroplets individual not the dry time, governs that applied volume total theisIt

droplets.smaller with enhancednot is Dry timeconstant is eunit volumper area diffusion) (i.e. absorptionconstant~ is

4

4 :I] [Eq. into III] [Eq. Substitute

III] [Eq. 4

areafootprint pancake4

lumedroplet vo oft independen is thicknessPancake constant. ~ is

4

2 :I] [Eq. into II] [Eq. Substitute

impactat thicknesspancake

II] [Eq. 24

:as described be alsomay pancake The

lumedroplet vo

constant

I] [Eq. *diameter pancake

*

2**

***

2*

*

2

***

*

*

**

*

2*

d

d

d

dd

d

dd

A

kAk

A

ADA

D

H

kHH

kDD

H

HD

HD

k

kD

Undoubtedly, some will still insist that smaller droplets are the key to dry time improvements. Unfortunately, LXK mythology is harder to kill than zombies. To this, I offer no further analysis, and simply advise the purchase of this survival kit.

BA

BB

AA

BA

BABFP

FPA

or

t

F

C

K

where

FFKt

C

t

CK

n

n

nmn

mnn

n

n

nnnn

1

1

1

1

11

*

inversionmatrix by for solveThen

* modified

*][ modified

][ and ][ modifyingby interest of nodes at the valuescondition boundary known apply the Now

that so ;

:evaluate zone),impact ink at the (like valuesnodal specifiedfor account To

method difference Backward1

method sGalerkin'3

2

method difference Central2

1

method difference forwardEuler 0

step time

step last timeat variablefield

step timenewat variablefield

matrix force global

matrix ecapacitanc global

matrix conduction global

:

11

See page 325 of “The Finite

Element Method for Engineers”

See page 48-62 of “The Finite



2

2

2

2

2

2

2

aa D

tzyxD

Using Paper Warming to Effect Increased Absorption

Motivation: Absorption is several orders of magnitude more effective

than brute force evaporation.

10 20 30 40 50 60 70 80 90 10010

20

30

40

50

60

70

80

90

100

Temperature (C)

Su

rface T

en

sio

n/V

isco

sit

y (

m/s

)

M1K-Mono Ink (/)

media theof size pore sticcharacteri

interface media-ink at the anglecontact

ink theof viscositydynamic

ink theof tension surface

cost coefficien Absorption

: thatknow We

c

ca

r

rK

q

q

Postulate: We may increase absorption (Ka) by taking advantage of the response shown below.

It can be shown that that the response of (/) to temperature is 400X stronger than the response of cos(q) to temperature. The cos(q) effect is discussed more fully in Appendix #1.

20 30 40 50 60 70 801

1.5

2

2.5

Media Temperature at the Print Zone (C)

Ab

so

rpti

on

Co

eff

icie

nt

Mu

ltip

lie

r

Absorption Coefficient Multiplier as a Function of Media Warming

Thus we should expect ink to absorb ~2X faster if printed onto a 70 C media surface

Lexmark Confidential 9

Smear Score of Tidal Paper

Off the chart

Experiments have validated this approach.

Ref. Chad Young & Jancy Bonewits

Ambient temperature (C)

Fuser control temperature (C)

Fuser OD (mm)

Fuser wall thickness (mm)

Fuser roll length (mm)

Maximum fuser power (W)

Fuser wrap angle (10 - 170 degrees)

Paper speed (in/s)

Fuser Roll Input Parameters

Click to compute Paper Feed and Warming

Fuser warm-up time (s)

Paper temperature at the print zone (C)

Average power (W)

Heat flux at the fuser inside wall (W/m2)

Contact length (mm)

Contact time (ms)

System Outputs

Select the MATLAB Workspace Dry-Time-Kinetics

Then enter on the command line:

Fuser_Paper_Warming_GUI

The GUI shown here will appear. It will be loaded with

default values, but you can change them to suit

your simulation needs. Once you have entered the

values of interest click the green button to execute

the calculations.

Paper thickness (mm)

20

110

30.3

1.0

236

550

80

24483

12

0.1

21

69

-50 0 50

-40

-30

-20

-10

0

10

20

30

40

50

60

x-coordinate (not to scale)

Temperature Field

y-c

oo

rdin

ate

(n

ot

to s

ca

le)

20 30 40 50 60 70 80 90 100 110

328

12.2

74

0 20 40 60 80 100 120 140 160 1800

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Wrap Angle (degrees)

He

at

Tra

ns

fer

Fa

cto

r (

)

Paper speed = 5 in/s





Ambient

Fuser

PZ

AmbientFuser

AmbientPZ

T

T

T

TT

TT

eTemperaturAmbient

eTemperatur ControlFuser

Print Zone in the eTemperaturPaper

1] [Eq. FactorTransfer Heat

Generic Temperature Response at the Print Zone for All Conditions

0 20 40 60 80 100 120 140 160 1800

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8


He

at

Tra

ns

fer

Fa

cto

r (

)






0 20 40 60 80 100 120 140 160 1800

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8


He

at

Tra

ns

fer

Fa

cto

r (

)






Exclusion Zone for 110C Fuser No wrap angle solutions exist in the

red zone that effect a print zone temperature of 70C

Exclusion Zone for 130C Fuser No wrap angle solutions exist in the

red zone that effect a print zone temperature of 70C

Example #1: For a fuser temperature of 110C, we can achieve a 70C paper temperature in the print zone with: 5 in/s paper speed and wrap greater than 25o

10 in/s paper speed and wrap greater than 55o


Example #2: For a fuser temperature of 130C, we can achieve a 70C paper temperature in the print zone with: 5 in/s paper speed and wrap greater than 15o



BA

BB

AA

BA

BABFP

FPA

or

t

F

C

K

where

FFKt

C

t

CK

n

n

nmn

mnn

n

n

nnnn

1

1

1

1

11

*

inversionmatrix by for solveThen

* modified

*][ modified

][ and ][ modifyingby interest of nodes at the valuescondition boundary known apply the Now

that so ;

:evaluate inlet),paper at the (like valuesnodal specifiedfor account To

method difference Backward1

method sGalerkin'3

2

method difference Central2

1

method difference forwardEuler 0

step time

step last timeat variablefield

step timenewat variablefield

matrix force global

matrix ecapacitanc global

matrix conduction global

:

11

See page 325 of “The Finite


See page 48-62 of “The Finite


Click to compute Paper Feed and Warming

t

TCQQQ

y

TK

yx

TK

xpConvectionRadiationFuseryx

Appendix #1

The cosine(q) effect

Ref: S. Bares, K. Rennels, Paper Compatibility With Next Generation Ink-Jet Printers, TAPPI Journal, (1990).

The literature has many examples that indicate PQ degrades as Ka increases.

We had initially worried that warming the media to effect increased Ka would cause

us to degrade PQ too. Yet that did not happen. Indeed it was found that our PQ

improved as we warmed the media. That said, let us revisit the components of Ka

to see how this phenomenon can be explained.

media theof size pore sticcharacteri

interface media-ink at the anglecontact

ink theof viscositydynamic

ink theof tension surface

cost coefficien Absorption

c

ca

r

rK

q

q

0 20 40 60 80 100 120 140 160 1800

0.5

1

1.5

2

2.5

3

Contact Angle (degrees)

PQ

Sco

re

Data of Shimormura (1990)

Canon found that there was not a consistent trend between Ka and PQ, but there was a definite, general trend between PQ and contact angle. This may seem contrary to the other studies, but we must keep in mind that the other studies related to Ka and PQ were not monitoring contact angle. They only reported Ka effects which could have been caused by /, but since they reported degraded PQ its more likely their variable was cos(q)

excellent

good

acceptable

unacceptable

q

LG

LG

q

For the same value of (SLSG), an interface having a large contact angle (q)

has a surface tension vector (LG) of higher magnitude than the low contact angle case.

Thus ink at the interface having a higher contact angle will tend to remain more

droplet-like during the absorption process. While the low contact angle interface

will be more prone to lateral spreading along the media surface.

The ink-media interface does not have a homogeneous surface energy at

the perimeter of the liquid (paper structure is random, not crystalline). Thus, the more

lateral spreading involved at the ink-media interface, the more non-uniform

(i.e. ragged) the image will appear.

An Heuristic Explanation of Simormura’s Results

media media

ink

ink

The characteristic pore size of the media is in the hands of the paper manufacturers. That leaves us two components to work with in our attempt to effect increased absorption by media warming. - / - cos(q There is evidence in the literature that PQ is more directly correlated to cos(q than it is to Ka. As contact angle decreases so does PQ. We have demonstrated that dry time is enhanced when we take advantage of the d(//dT effect. We have also seen that PQ does not degrade when we increase Ka by warming the media. This begs the question: What is the effect of temperature on contact angle? If we can show that d(cosq)/dT is flat over our temperature range then we will be able to show that our anomaly is not an anomaly at all – physics will tell us that it is an expected outcome.

LG

SG

SL

q

q

cos

interface liquid-solid at the tension surface

interface gas-solid at the tension surface

interface gas-liquid at the tension surface

anglecontact

LGSLSG

SL

SG

LG

Unfortunately, SL and SG are generally unknown. Thus it is common practice to measure the contact angle rather than attempt to compute it. However, we need to know the behavior of cos(q) as a function of temperature if we going to explain the unexpected results of our smear experiments, i.e. why PQ did not degrade with paper temperature increases. ------------------------------------------------------------------------------------------------------------------------------ We will take advantage of a recent development in wetting theory called the sharp-kink approximation to help us in our quest. The beauty of the sharp-kink approximation is that it estimates contact angle from mathematically tractable values: - the liquid-gas surface tension (LG) - the difference in density between the liquid and its vapor phase (L- V) - the Lennard-Jones potential ()

Gas

Liquid

Solid

[Young’s equation (circa 1805)]

2 4 6 8 10 12 14 16-1000

-500

0

500

1000

1500

Intermolecular Spacing (Angstroms)

Po

ten

tial E

nerg

y (

Jo

ule

s/m

ol)

Well depth (e)

Characteristic diameter of the molecule ()

Molecules repel each other

Molecules attract each other

Exclu

ded R

egio

n

zmin

When envisioning the physics at the ink-media interface we need to consider the fact

that atoms cannot approach each other closer than (zmin).

The sharp-kink approximation takes advantage of this fact.

Z

For the wetting problem at hand, describes

the preference of the adsorbing molecule to

adhere to the surface instead of forming a droplet.

Typical Lennard-Jones Potential

Gas

Gas

Liquid

Solid

Z

zmin

The sharp-kink approximation assumes that the solid has an adsorbed layer of gas

at the surface. Since the liquid cannot enter the excluded region, the closest that the

liquid can be to the solid is zmin, at the well-depth, just beyond the van der Waals

radius. By setting the derivative of the Lennard-Jones function to zero, we may

compute the closest approach distance zmin.

The region contained in the red block above can be considered having three regimes:

(1) solid-gas surface tension, (2) liquid-gas surface tension, (3) a term that accounts

for the van der Waals attraction of the liquid to the solid substrate.

Given the heuristic explanation on the previous page, the sharp-kink approximation

provides for the following mathematical description of the liquid-gas-solid interface.

Rearranging:

When combined with Young’s equation yields*

We know how liquid and vapor density vary with temperature, and we also

know how liquid-gas surface tension varies with temperature. This just leaves

us with the integral of the potential energy function to determine.

* Ref: Gatica, Zhao, Johnson, Cole, J. Phys. Chem. B., 2004, 108.

In the vicinity of the contact line, it is assumed that the fluid regime

consists of saturated liquid and vapor.

(in the vicinity of zmin, density varies from that of gas to liquid – thus the term)

sharp-kink equation*

Lennard-Jones Potential Energy Function:

= characteristic diameter of the molecule

e = maximum energy of attraction = well depth

Wetting is a function of molecular

kinetics. Ink is mostly water at

the molecular level (mfH20 > 0.9).

So let:

= 2.7 Angstroms

e = 650 Joules/mol

zmin = 3.5 Angstroms

20 40 60 80 100

1080

1100

1120

1140

1160

Temperature (C)

Liq

uid

De

ns

ity

(k

g/m

3)

20 40 60 80 100

0.1

0.2

0.3

0.4

0.5

Temperature (C)

Va

po

r D

en

sit

y (

kg

/m3)

20 40 60 80 100

20

40

60

80

Temperature (C)

Su

rfa

ce

Te

ns

ion

/Vis

co

sit

y (

m/s

)

Misc PlotsPlot-M1K-Properties

M1K-Mono Ink Properties

These were computed in FEAJET for the mixture:

Water 74.68 wt.%

Triethylene glycol 5.0 wt.%

1,3-Propylene glycol 5.0 wt.%

Glycerol 10.0 wt.%

Surfactant 0.2 wt.%

Pigment+dispersant = 5.0 wt.%

L V

/

10 20 30 40 50 60 70 80 90 100

0.35

0.4

0.45

0.5

0.55

0.6

Temperature (C)

co

sin

e( q

)

We can now solve the sharp-kink equation

10 20 30 40 50 60 70 80 90 10010

20

30

40

50

60

70

80

90

100

Temperature (C)

Su

rface T

en

sio

n/V

isco

sit

y (

m/s

)

M1K-Mono Ink (/)

10 20 30 40 50 60 70 80 90 100

0.35

0.4

0.45

0.5

0.55

0.6

Temperature (C)

co

sin

e( q

)

We are now at a point where we can separate the contributions of (/)

and cosine(q) to the absorption coefficient (Ka) versus temperature.

/

Cosine (q) Summary • The absorption coefficient is a function of (/) and cosine(q).

• There is evidence in the literature that PQ is a stronger function of cosine(q) than Ka. As contact angle decreases – so does PQ.

• We achieve increased absorption by warming the paper to increase the (/) effect. Yet we see no degradation in PQ with higher media temperatures.

• Using the sharp-kink approximation, we have shown that cosine(q) varies negligibly over our temperature range of interest.

• Since cosine(q) is relatively invariant over our temperature range of interest we should not expect PQ to degrade with warmed media.

• Therefore when we quantitatively examine the fundamental physics of wetting at the atomic level, we do not expect PQ to degrade when we warm the media to effect an increased absorption coefficient (Ka).

Appendix #2 What about the wetting delay?

We have already discussed the absorption coefficient Ka and how it can be enhanced via warming the paper. We know that this

is well founded in theory as well as in practice. However, what about wetting delay?

If the wetting delay becomes a dominant fraction of the available dry-time, do we also need to figure out how to control it too?

What drives this phenomenon? The answer follows.

Discussion on the wetting delay

After the droplet impacts the paper, it flattens into a pancake-like puddle. This is an inertial event - not dictated by the contact angle.

According to Asai, et.al., this stage of the event is dictated by the Weber and Reynolds numbers. However, after the pancake

oscillations dampen out (microseconds), the wetting-capillary-spreading motion (milliseconds) of the pancake along the paper

surface becomes strongly dependent upon contact angle. While static contact angle is interesting, it is the dynamic contact angle

that actually governs this stage of the ink-media event. So how does the wetting delay, shown in the typical Bristow tester plots,

relate to dynamic contact angle?

I was looking thru one of my books (Capillary Flows With Forming Interfaces, by Y.D. Shikhumurzaev). On page 123 he shows the

following plot. The significance of this plot is that it shows a nice relationship between advancing-dynamic contact angle (qd) and the

capillary number (Ca). The data clusters together nicely across a wide range of water-glycerol mixtures. Since the capillary number

for a system is equal to [viscosity x velocity/surface tension] = (U/) this captured my attention - because as we've seen - Ka is a

function of (/). This implies that wetting delay is probably related to /; therefore, wetting delay should be inversely related to

Ka since Ka follows /, in other words, we should expect wetting delay to decrease when we warm the paper.

Wetting delay (cont.)

Anyway, that was the postulate that appeared in my head when I saw this plot relating dynamic (advancing) contact angle to Ca.

Postulates and hypothesis are easily created and destroyed, so how can we determine whether this postulate is on the right track?

Luckily we can find the validation within the literature.

The study by Selim, et.al. showed the plot (next page). Their data shows Bristow test results for various mixtures of water and

ethylene glycol. They show a similar plot for mixtures of water-diethylene glycol. Notice that some of their ink mixtures reduced

the wetting time and some increased it. There is also an obvious relationship between the mixtures and the absorption

coefficient (Ka = the slope of the line past the wetting delay). While Selim did mention that the Ka effect was probably due

to the / relationship, he had no explanation for the wetting delay effect.

This is where the relationship identified in Shikhmurzaev's book helps us.

Plot from: Y.D. Shikhmurzaev, Capillary Flows With Forming Interfaces, Chapman-Hall/CRC, Boca Raton, (2008).


As d

e-g

lyco

l go

es fro

m

30

60%

ab

so

rptio

n in

cre

ases

As d

e-g

lyco

l go

es fro

m

20

%

10

% a

bso

rptio

n

decre

ases

Wetting delay is clearly

affected by formulation

The effect of diethylene glycol

on absorption coefficient

* Selim, Yesavage, Al-Ubaidi, Sung, Drying of Water-Based Inks on Plain Paper,

Colorado School of Mines Report for IBM, (1989).

This

behavio

r suggests

an in

flectio

n p

oin

t

010

20 30 40 50 60DEG Concentration (%)


Using FEAJET to simulate the characteristics of water-EG and water-DEG mixtures, we see that indeed we should see

an inflection point for both absorption and wetting delay. This is exactly what is shown in Selim’s data.

Interestingly, the simulations shown above illustrate the expected inverse relationship between absorption and wetting delay.

Thus we may feel confident that as we use temperature to increase Ka, the same temperature rise will tend to decrease

the wetting delay; therefore, warming the paper gives us two levers by which dry time is enhanced.

Given this, we may use the FEAJET model to simulate various ink formulations to examine this effect.

The plot above illustrates those results.

This suggests that wetting delay can be positively (or negatively) affected by ink formulation. Since we want wetting delay to go

down and Ka to go up, we need to ensure that our ink formulations place us on the right hand side of these response curves.

Wetting delay falls off rapidly once we get past the peak. For enhancing Ka (increasing absorption) and minimizing wetting delay,

these plots suggest that we desire an ink formulation having a ratio of viscosity/surface tension greater than 0.03, and ideally

greater than 0.04 (the units of this metric are in seconds per meter.....i.e. the inverse of velocity).


The plot below shows some data from an old Xerox study (ca. 1992). If one looks at it in light of this discussion, it provides

validation of the postulate relating wetting delay to /. Xerox ran a Bristow-like test on a heated platen. They found that

indeed Ka increased with temperature - and - if you look at their plot below it is evident that wetting delay decreases with

temperature. So warming increases absorption - and - it reduces wetting delay for all the reasons cited in this document.


Ref: Carriera, Agbezuge & Gooray, Rates of Aqueous Ink Penetration into Papers and Their Effects on

Printability, IS&T-NIP8, (1992).

Appendix #3

Miscellaneous

Boundary points for the initial ink contact zone

are determined by the function: Identify_Bnodes

Mesh the Domain and Identify the Boundary Nodes

The function: Mesh2D breaks the domain into discrete nodes and elements.

It uses the principles described in the Segerlind book for mesh generation.

Y

R

(radial) Centerline

The function: Stiffness_Diffusion turns these

2D elements into 3D by applying an

axisymmetric term to the calculation.

Media

thic

kness

X (not to scale)

Y (

no

t to

scale

)

Heat Flux Input Nodes

Are on the Inside Surface

of the Fuser Roll

Radiation and

Convection Surfaces

Functions:

Mesh_Fuser2

Mesh2D

Identify_F_Nodes2

Identify_Paper_Nodes2

Translation_Nodes2

X (not to scale)

Y (

no

t to

scale

)

Paper Outlet

These nodes

are set to

Tambient

Each time step involves computing the transient temperature field with [Eq. 1] and

rotating/translating the paper and fuser nodes.

Functions:

Solve_Fuser_Tfield2

Stiffness_Tfield_Fuser2

Force_Fuser2

1] [Eq. t

TCQQQ

y

TK

yx

TK

xpConvectionRadiationFuseryx

Solve

Dry Time Kinetics Overview

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Transcript of Dry Time Kinetics Overview