Dry Time Kinetics Overview

Inkjet Dry Time Kinetics
An Overview of the Mathematical Physics
Bob Cornell
Cornellr@Lexmark.com
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

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
EvaporationAbsorption
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.
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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).

AbsorptionVolumeperUnitSurfaceArea(mL/m2)
)(ms)( 0.5
mstime
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 50
5
10
15
20
25
time (ms
0.5
)
InkTransferred(mL/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

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20 C
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30 C
35 C
40 C
45 C
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55 C
60 C
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70 C75 C80 C
Comparison Between the Effect of Absorption and Evaporation
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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
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impactprior todiameterdroplet
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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.

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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.
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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 35
0
20
40
60
80
100
120
Droplet Volume (pL)
SpotDiameterontheMedia(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
Drymediais~100micronsthick

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)
36
10: mmLNote  

Diffusive Mass Transport  Ink Absorption Mechanism
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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
MoistureConcentration
LXK Model Validation
From: G. Desie & C. Van Roost, J. IS&T, 50, 294-303, (2006).

Already printed
Column1
Column2
Column3
Column4
30 in/s
18 KHz
1/600”
1/600” 1/600”
1/1800”
1/1800”
1/1800”
  pixeldpi600
pL
5.22
dot
pL
5.7
in
600
2
1800
3
dots6
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
/ms0.5
)
Penetrationdepth(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/m
2
/ms
0.5
)
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/m
2
/ms
0.5
)
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 30
10
-1
10
0
10
1
Ink Coverage (picoliters per 600 dpi)
FunctionalDryTime(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.
TypicalLXKMonoCoverage
MemjetMono(KK)Coverage@60ppm
1600dpix800dpimode
MemjetMono(KK)Coverage@30ppm
1600dpix1600dpimode
10%ReducedLXKMonoCoverage

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.themselvedropletsindividualnot thedry time,governsthatappliedvolumetotaltheisIt
droplets.smallerwithenhancednotisDry timeconstantiseunit volumperareadiffusion)(i.e.absorptionconstant~is
4
4
:I][Eq.intoIII][Eq.Substitute
III][Eq.
4
areafootprintpancake
4
lumedroplet vooftindependenisthicknessPancakeconstant.~is
4
2
:I][Eq.intoII][Eq.Substitute
impactatthicknesspancake
II][Eq.
24
:asdescribedbealsomaypancakeThe
lumedroplet vo
constant
I][Eq.*diameterpancake
*
2**
*
**
2*
*
2
**
*
*
*
*
*
*
2*



































d
d
d
dd
d
dd
A
kA
k
A
A
DA
D
H
k
HH
kD
D
H
H
D
H
D
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
FFK
t
C
t
C
K
n
n
nmn
mnn
n
n
nnnn
1
1
1
1
11
*
inversionmatrixbyforsolveThen
*modified
*][modified
][and][modifyingbyinterestofnodesat thevaluesconditionboundaryknownapply theNow
thatso;
:evaluatezone),impactinkat the(likevaluesnodalspecifiedforaccountTo
methoddifferenceBackward1
methodsGalerkin'
3
2
methoddifferenceCentral
2
1
methoddifferenceforwardEuler0
steptime
steplast timeatvariablefield
steptimenewatvariablefield
matrixforceglobal
matrixecapacitancglobal
matrixconductionglobal
:
11
























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





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










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
tzyx
D

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 90 100
10
20
30
40
50
60
70
80
90
100
Temperature (C)
SurfaceTension/Viscosity(m/s)
M1K-Mono Ink (/)
mediatheofsizeporesticcharacteri
interfacemedia-inkat theanglecontact
inktheofviscositydynamic
inktheoftensionsurface
cos
tcoefficienAbsorption
:thatknowWe











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 80
1
1.5
2
2.5
Media Temperature at the Print Zone (C)
AbsorptionCoefficientMultiplier
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-coordinate(nottoscale)
20 30 40 50 60 70 80 90 100 110
328
12.2
74

0 20 40 60 80 100 120 140 160 180
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Wrap Angle (degrees)
HeatTransferFactor()
Paper speed = 5 in/s
Ambient
Fuser
PZ
AmbientFuser
AmbientPZ
T
T
T
TT
TT






eTemperaturAmbient
eTemperaturControlFuser
Print Zonein theeTemperaturPaper
1][Eq.FactorTransferHeat 
Generic Temperature Response at the Print Zone for All Conditions

0 20 40 60 80 100 120 140 160 180
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0 20 40 60 80 100 120 140 160 180
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
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
Example #2:
For a fuser temperature of 130C, we can
achieve a 70C paper temperature in the
print zone with:

                 
 
 
 
 
 
       
           
 
   
 
     BA
BB
AA
BA
BABFP
FPA
or
t
F
C
K
where
FFK
t
C
t
C
K
n
n
nmn
mnn
n
n
nnnn
1
1
1
1
11
*
inversionmatrixbyforsolveThen
*modified
*][modified
][and][modifyingbyinterestofnodesat thevaluesconditionboundaryknownapply theNow
thatso;
:evaluateinlet),paperat the(likevaluesnodalspecifiedforaccountTo
methoddifferenceBackward1
methodsGalerkin'
3
2
methoddifferenceCentral
2
1
methoddifferenceforwardEuler0
steptime
steplast timeatvariablefield
steptimenewatvariablefield
matrixforceglobal
matrixecapacitancglobal
matrixconductionglobal
:
11











































See page 325 of “The Finite
See page 48-62 of “The Finite
Click to compute Paper Feed and Warming
t
T
CQQQ
y
T
K
yx
T
K
x
pConvectionRadiationFuseryx





















 


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.
mediatheofsizeporesticcharacteri
interfacemedia-inkat theanglecontact
inktheofviscositydynamic
inktheoftensionsurface
cos
tcoefficienAbsorption











c
ca
r
rK
q



q

0 20 40 60 80 100 120 140 160 180
0
0.5
1
1.5
2
2.5
3
Contact Angle (degrees)
PQScore
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 (SLSG), 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
mediamedia
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
interfaceliquid-solidat thetensionsurface
interfacegas-solidat thetensionsurface
interfacegas-liquidat thetensionsurface
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)
PotentialEnergy(Joules/mol)
Well depth (e)
Characteristic diameter of the molecule ()
Molecules repel each other
Molecules attract each other
ExcludedRegion
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)
LiquidDensity(kg/m3
)
20 40 60 80 100
0.1
0.2
0.3
0.4
0.5
Temperature (C)
VaporDensity(kg/m3
)
20 40 60 80 100
20
40
60
80
Temperature (C)
Misc Plots
Plot-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)
cosine(q)
We can now solve the sharp-kink equation

10 20 30 40 50 60 70 80 90 100
10
20
30
40
50
60
70
80
90
100
Temperature (C)
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)
cosine(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).

Asde-glycolgoesfrom
3060%absorptionincreases
Asde-glycolgoesfrom
20%10%absorption
decreases
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).
Thisbehaviorsuggestsaninflectionpoint

0
10
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).

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.
Mediathickness

X (not to scale)
Y(nottoscale)
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(nottoscale)
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
T
CQQQ
y
T
K
yx
T
K
x
pConvectionRadiationFuseryx





















 

Solve

Dry Time Kinetics Overview

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Dry Time Kinetics Overview