Mathematical Physics of Pigment Settling

An Introduction
to the
Mathematical
Physics of
Pigment Settling
Bob Cornell
Print Systems Science
October 26, 2009

Analysis
Overview
PK1-mono Pigment Sedimentation

When a tank is motionless for
extended periods of time, the
solid particles tend to accumulate
towards the tank bottom. This
pigment migration will effect
decreased L*, darker dots,
(for awhile) because the solid
volume at the tank bottom is
abnormally high.
PK1-mono Sedimentation Response
TankBottom
TankTop

PK1-mono Viscosity Response to Pigment Sedimentation
towards the tank bottom. As the
solid volume fraction increases at
the tank’s exit port, the ink delivered
to the ejectors becomes increasingly
viscous – causing jetting problems.
Until the highly viscous region is
either jetted out, or pumped out,
the ejectors are functionally
constipated.
Ink Exit
Port

How do we minimize pigment settling?
Let us look at zeta potential (z).
Control
PK1-mono
Test Case 1
4X Higher Zeta Potential
The zeta potential is important for electrostatic repulsion - ensuring that the
particles do not clump and grow; however, (z) has virtually no affect on sedimentation
|z| = 15 mV |z| = 60 mV

Let us look at ink viscosity.
Control
PK1-mono
Test Case 2
10X Higher Viscosity
Greatly increasing the viscosity puts the brakes on pigment sedimentation. The solid particles still
settle over time, but the rate is much reduced. Unfortunately, increasing viscosity by 10X creates
other, more severe, problems than settling.
m = 2.5 cP m = 25 cP

Let us look at pigment particle size.
Control
PK1-mono
Test Case 3
Particle size: 50-65 nm
Small particles are less likely to settle. If the ink consisted primarily of 50-65 nm solids instead of
110 nm solids, pigment settling would not be an issue. Unfortunately, the LXK ink formulators
state that a mixture consisting of all small particles produces low OD.
Particles distributed
from ~50-260 nm
Particles = 50 nm
Particles = 65 nm

Let us look at particle density.
Control
PK1-mono
Test Case 4
Lower Particle Density
Another way to negate the effect of pigment sedimentation is to use solid particles that have
nearly the same density as the liquid. [In a 10/23/09 meeting with the MEMJET ink formulator, he
cited using a mono pigment particle with a density of 1.0 g/cm3 - he stated that it has no settling].
r = 1.8 g/cm3 r = 1.1 g/cm3

Recommendations
• Pigment sedimentation must be addressed when using off-carrier tank systems to supply ink
to the print head.
– The print quality defects associated with settling include varying L* and varying viscosity.
– The pigment rich layer that occurs after sedimentation delivers ink into the ejectors that is
incompatible with consistent jetting. The ejectors are finely tuned MEMS devices that must
refill at 18-24 KHz without meniscus puddles forming, so they cannot tolerate wide viscosity
swings. High viscosity  slow jets, very slow refill, poor start-up,…,poor PQ.
• Small particles ~50-65 nm in size will drastically slow down the sedimentation effect;
however, the LXK ink chemists state that particles this small effect low OD.
• Particle density is another ink parameter that can dramatically slow down pigment
sedimentation.
– If pigment particles can be produced in the range of 1.0 - 1.1 g/cc, the settling effect is
minimized.
– The MEMJET colloidal dispersion expert cites that he uses solids with a density of 1.0 g/cc
and his pigment ink does not settle.
• If lower density solids, or smaller pigment particles cannot be implemented, the printer will
need some mechanical means to prevent the print quality defects that accompany pigment
sedimentation.
– The simulations indicate that the particles move very slowly – on the order of 1 nanometer
per second. That said, the mechanical agitation means does not need to be very vigorous.

Part-1
Gravity
Buoyancy
Viscous Drag
Brownian Motion
Buoyancy + Viscous Drag
Gravity
Random molecular
bombardment

Pigment Particles = Colloids
• Colloids are generally described as particles whose size ranges from 1
nanometer to 1 micrometer. This size range makes colloids incredibly interesting
from a physics viewpoint.
• They are big enough to be described by the classical Newtonian physics of
macroscopic matter and small enough to be significantly affected by atomic forces
too. When they are electrically charged, things really get interesting.
• The study of colloidal mechanics touches many areas of science. Charged colloid
particles immersed in a pool of liquid feel the following effects:
– (1) Gravitational pull versus buoyancy effects
– (2) Viscous drag
– (3) Brownian motion due to molecular bombardment
– (4) van der Waals attraction
– (5) Electrostatic and/or steric repulsion
• Effects (1 – 3) will be covered in Part-1.
• Effects (4 – 5) will be covered in Part-2.
• In Part-3 we will apply all the above teachings to the specific case of PK1-mono,
and then we will explore the parameters best used to minimize sedimentation

Mason & Weaver
Max Mason and Warren Weaver published a paper in 1923 that addressed the problem of
settling of small (uncharged) particles in a fluid. Their solution is shown below in [Eq. 1]:
 
   
 
   
 
  0atconstant0,
:conditionInitial
,0at,
:conditionsBoundary
densityliquid-densityparticle
viscosityliquid
radiusparticle
etemperatur
constantsBoltzmann'
9
2
;
6
surfacebottom@;surfacetop@0position;
time
densityparticlevolumetric,
0
2
2
2






















tnyn
LyBn
y
n
A
a
T
K
ga
B
a
KT
A
Lyyy
t
tyn
y
n
B
y
n
A
t
n
lp
lp
rr
m
m
rr
m
Historical footnote:
Mason and Weaver, both mathematicians,
were refugees from the quantum mechanics
world that was becoming the leading edge
of research in many prestigious universities
during the 1920’s.
Because both men were ardent supporters
of the classical, deterministic view of nature
(and they lacked the fame of Einstein),
they were considered outsiders by the
mainstream world of mathematical physics.
[Eq. 1]
M. Mason & W. Weaver, The Settling of Small Particles in a Fluid, Phys. Rev. 23, (1923).

Mason & Weaver
When Mason & Weaver postulated [Eq. 1] as the governing partial differential
equation for colloid settling, they did not have high speed digital computers available
to seek a numerical solution. By some impressive mathematical gymnastics, they
arrived at the following analytical solution for [Eq. 1].
   
 
    
 
     ..5,3,1for,..6,4,2for;';;;10;
41
cos2sin1
16
1 1
2222
2
1
'
4
'2
2
/1
/
0
22






























 





mm
t
t
B
L
BL
A
h
L
y
h
m
hmmhmeme
e
e
e
n
n
m
tm
thh












[Eq. 2]
To illustrate the utility of [Eq. 2], Mason and Weaver checked its veracity against
experimental data gathered by the French physicist Jean Perrin.
That solution set is shown on the next page.

Model Comparison
Figure 6
From Mason & Weaver’s 1923 article
LXK Solution of [Eq. 2]
For Perrin’s experiment having:
a = 212 nm; rP = 1.194 g/cc; rL = 1.0 g/cc
L = 100 mm; m = 1 cP; A = 9.03 x 10-9
B = 1.902 x 10-6;  = 5260;  = 0.475

Moving Forward
• While it is gratifying to be able to reproduce Mason & Weaver’s results, it is a
solution of limited utility towards our present problem of pigment sedimentation.
• To move forward, we must be able to develop a model that answers questions
that deal with non-uniform starting conditions and colloid stability. For example:
– If the tank is allowed to sit upside down for several weeks – how long does it
take for ink at the exit port to return to its normal pigment concentration?
– What if the tank is stored upside-down at 40C, 50C, 60C – how long does
restoration take?
– As pigment settles, the viscosity of the liquid is no longer constant – how do
we account for a non-uniform viscosity field in the tank?
– How do we account for the electrostatic repulsion and van der Waals
attraction as the pigment particles settle and move closer together?
• None of these questions can be addressed with Mason & Weaver’s equation.
Their solution is based upon a uniform concentration field at time zero, and they
have no means of accounting for the forces of attraction and repulsion.
• Because these questions, and others, are important in our R&D odyssey, we
need to go beyond Mason & Weaver’s analysis.

Going Beyond Mason & Weaver
If we are to move beyond the Mason & Weaver analysis, we need to have a solid
understanding of the derivation of [Eq. 1]. Studying the derivation of a partial
differential equation not only brings the underlying physics to life - it also illuminates
the path we need to take to add the physics to the p.d.e. needed to answer
the questions posed on the last page.
That said, let us tear into the derivation of [Eq. 1].
 
   
 
m
rr
m 9
2
;
6
surfacebottom@;surfacetop@0position;
time
densityparticlevolumetric,
2
2
2
lpB
ga
B
a
TK
A
Lyyy
t
tyn
y
n
B
y
n
A
t
n













[Eq. 1]

Steady State Particle Distribution
Left standing long enough, a steady state concentration profile is achieved.
The steady state volumetric concentration function (n = particles/cm3) will
take on an exponential shape. While this may seem like an obvious result,
Jean Perrin was awarded the Physics Nobel Prize in 1926 for this discovery.
The discovery of this exponential function in itself was not so monumental;
however, from his work on colloid sedimentation, Perrin was able to
experimentally verify the atomic kinetics theory of Boltzmann and to confirm
Avogadro’s number during a time when many physicists did not believe in the
still unseen atom.  
 
 
  densityliquid-densityparticle
gravitytodueonaccelerati
radiusparticlespherical
3
4
particleon theforcebuoyancy-forcenalgravitatio
etemperatur
constantsBoltzmann'
positionatdensityparticle
statesteadyatondistributicolloidvolumetric
3
11
13
1














lp
lpgb
B
TK
yyF
g
a
gaF
T
K
yyn
enyn
m
particles B
gb
rr
rr
[Eq. 2]
Note that [Eq. 2] is a form of the
Boltzmann distribution

A Boltzmann Side Trip
The Boltzmann distribution is ubiquitous in the scientific literature, so it is no
surprise that it makes an appearance here. It will make another appearance
in Part-2, during the discussion about the particle’s charge field. The equation takes
on various forms, but it always relates some form of energy to the thermodynamic
potential (KBT).  
 
   
 
    yyyyF
yy
TK
yyn
ynf
ef
gb
B
TKB
locationtopositionreferencefrommovetorequiredenergy
positionreferencefromntdisplaceme
potentialmicThermodynaeTemperaturconstantsBoltzmann'
positionreferenceatondistributiparticlevolumetricconstant
particlestheofondistributivolumetric
:casespecificIn this
functionondistributisBoltzmann'offormgeneric
11
1
110
0











Boltzmann, an Austrian, was the father of statistical thermodynamics and the kinetic theory of gases.
This put him at odds with the reining German scientists of the 1870’s because Boltzmann’s
theories required the existence of atoms. Many others believed in nothing they could not see and
measure. Boltzmann was brilliant, but terribly insecure. He wanted to be accepted into the “club”,
but his research was not accepted. The depressed Boltzmann hung himself.

Back to the Derivation
 
 
 
m
rr
m
mrr
9
2
6
particletheofvelocityterminal
6
3
4
0forcedragviscous
2
3
lpd
lp
dgb
dgb
ga
a
F
y
yaga
FF
FF








Colloid particles falling in a gravity field, being opposed by viscous drag
have a Reynolds number << 1; therefore Stokes Drag Law applies.
The particle reaches its terminal velocity very quickly. When the terminal
velocity is reached, particle acceleration goes to zero; thus the sum of the
forces acting upon the particle go to zero. So it follows that:
[Eq. 4]
When we multiply both sides of [Eq. 4] by (n = particles/cm3) we get a term
that is equivalent to the downward particle flux passing a unit area.










 

ond
cm
particles
J
a
nF
yn d
sec
fluxparticledownward
6
2
m
[Eq. 5]

Brownian Motion
Gravity and buoyancy forces tend to drive the particle in a straight line from the top
of the tank to the bottom. This is the convective contribution to pigment settling.
A colloid particle is small enough to be affected by molecular bombardment.
This bombardment is random, forcing the particle to take a random,
drunken sailor-like, walk on its sedimentation path. This is Brownian motion.
Brownian motion is the diffusive contribution to pigment settling. Like other diffusive
effects, a concentration gradient is needed to account for Brownian motion.
Taking the derivative of [Eq. 2] yields the needed gradient term.
 
 
 
 
TK
yan
y
n
F
TK
F
n
y
n
e
TK
F
n
y
n
FFenyn
B
d
B
d
TK
yyF
B
d
dgb
TK
yyF
B
d
B
d





























m6
for4][Eq.Substitute
thatRecall
1
1
1
1
[Eq. 6]

A Brownian Motion Side Trip
Brownian motion is easily observed in the lab under
a microscope. An inkjet nozzle that is filled with
pigment ink will show the particles doing their random
walk. This simple observation has a powerful impact:
It confirms the kinetic theory of matter. The pigment
particles are being pushed around by random
molecular bombardment.
In 1826, Robert Brown, a Scottish botanist, observed that pollen grains were moving
around when he viewed them under his microscope. He had no explanation for this
magical motion. The physics behind Brownian motion was revealed by Einstein in a
paper published in 1905. Einstein proved that Brownian motion was due to the
kinetic theory of matter. Boltzmann would have been pleased had he lived.

Back to the Derivation
   
m
rr
m
rr
mm
mm
mm
mm
m
9
2
6
3
4
6
;
6
66
unit timepergainparticletricnet volume
/
|
66
:fluxparticledownwardnetaistherestate,transientIn the
0
66
0
:downwarddiffusingarethereasupwarddiffusingparticlesmanyasaretherestate,steadyAt
6
:flux termfor the6][Eq.Solving
22
2
2
2
2
2
0
lplp
dB
dB
y
dB
dB
B
ga
a
ga
a
F
B
a
TK
A
y
n
B
y
n
A
y
n
a
F
y
n
a
TK
t
n
t
n
m
s
mparticle
y
J
y
J
a
nF
y
n
a
TK
J
a
nF
y
n
a
TK
JJJJ
J
y
n
a
TK
yn













































































































J
dy
y
J
J



The Mason & Weaver equation
is thus produced
dy

Mason & Weaver Equation Overview
 
m
rr
m 9
2
;
6
2
2
2
lpB
ga
B
a
TK
A
y
n
B
y
n
A
t
n
























There is an unstated beauty in Mason & Weaver’s equation that the mathematicians
either did not recognize, or appreciate. An engineer would expect that the governing
partial differential equation for uncharged colloid settling would have three terms:
- a transient term
- a diffusion term
- a convection term
The left hand side of [Eq. 1] supplies the transient term. The variable (A) is easily
recognized as the diffusion coefficient of the Stokes-Einstein equation, and
variable (B) is the terminal velocity of the particle. Thus, the form of the Mason-Weaver
equation is intuitively satisfying to an engineer.
[Eq. 1]

Boundary Conditions
Perhaps the advantage of taking an engineering view of the partial
differential equation is that the boundary conditions become obvious.
Since there can be no net particle flux at the boundaries, we know that
the diffusive flux term (J-) must equal the convective flux term (J+)
at the boundaries [y = (0,L)].
 
 
 LyBn
y
n
A
or
nga
y
n
a
TK
a
nF
y
n
a
TK
LyJJ
lpBdB
,0attimesallfor
0
9
2
6
0
66
,0attimesallfor0
:ConditionsBoundary
2



























 
m
rr
mmm
[Eq. 7]

Finite Element Solution to Mason-Weaver
Now that we’ve discussed the derivation and beauty of [Eq. 1], the foundation is
laid for moving forward to a model that can handle terms not included in Mason-Weaver
but are fundamental to inkjet applications.
We know that additional features must be added to the Mason-Weaver equation to
make it suitable for answering our questions about charged pigment settling, that may,
or may not, have uniform colloidal concentrations at time zero. Now that we
understand the derivation of the governing partial differential equation, these terms
are easily added to the pde; however, seeking a new analytical solution, such as [Eq. 2],
is exceedingly difficult. Since high speed computers are the tools of the trade today,
perhaps Mason & Weaver would not mind if we use numerical methods to solve their
pde. That said, it will now be shown how to use the finite element technique to
solve [Eq. 1] numerically. Once that is done, we will proceed to adding the additional
physics (variable viscosity, electrostatic repulsion and van der Waals attraction) in Part-2.

Finite Element Solution
 
m
rr
m 9
2
;
6
2
2
2
lpB
ga
B
a
TK
A
y
n
B
y
n
A
t
n
























An extensive literature search did not yield any articles teaching the application
of finite element analysis to the pigment sedimentation problem. Thus we will be
forced to derive our own finite element solution
to [Eq. 1]. (see Appendix-2)
If one stares at [Eq. 1] long enough, it becomes obvious that it has the same form as
the one dimensional pde for heat transfer with convection and diffusion.
That revelation jump starts the solution.
 

















































y
T
v
y
T
t
T
Then
C
Let
y
T
v
t
T
y
T
C
Q
Qv
Cyt
T
constvelocityv
y
T
v
t
T
CQ
y
T
v
v
v
v
2
2
2
2
2
2
:
ydiffusivitThermal:
:0for
sourceheatinternalvelocity;
heat;specificolumeconstant vposition;time;
density;:etemperatur:tyconductivithermal
.for;

r


r

r
r
[Eq. 1]
[Eq. 8]
The similarity between [Eq. 1] and [Eq. 8] is
obvious. The fact that the units of  and A are
diffusion related (m2/s) - and v and B are both
velocity terms makes this a powerful connection
because the well-known finite element methods
of heat transfer can be used to solve the
pigment settling problem.

Finite Element Solution (cont.)
y = 0
y = L
Element (1)
Element (2)
Element (3)
Element (N)
l(e)
Area(e)
Node nN+1
Node n4
Node n3
Node n2
Node n1
The first step in the finite element solution is to
discretize the domain. The pigment settling problem
is well described by a 1D model, as shown on the
right. We know the free surface height and we know
how the tank width and length varies over the
tank height.
Tank length
Tankheight
It is true that minimizing nodes and elements
minimizes the computation time; however, the
discretization cannot be arbitrary. If the element
length is too great, numerical instability results.
Appendix-1 derives the minimum element length for the
settling problem. The end result is shown below.
 lp
Be
ga
TK
l
rr 
 3
)(
4
3
lengthelementminimum

The time marching finite element solution of a generic field problem takes the following form:
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21
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---2)Appendix(seeshown thatbecanitproblem,settlingpigmentFor the---
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y = 0
y = L
Element (1)
Element (2)
Element (3)
Element (N)
l(e)
Area(e)
Node nN+1
Node n4
Node n3
Node n2
Node n1
* In the heat transfer case, the storage term:  = density x specific heat
[Eq. 9]
[Eq. 10]
[Eq. 11]

 
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[Eq. 12]
y = 0
y = L
Element (1)
Element (2)
Element (3)
Element (N)
l(e)
Node nN+1
Node n4
Node n3
Node n2
Node n1
 
   
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[Eq. 13a]

The boundary conditions are used to modify the system of equations [Eq. 9] by
fixing [1]new = n1 and [N+1]new = n(N+1) from [Eq. 13]. The first guess at n2 and nN is
[n2, nN]old. This is a dilemma because the boundary condition is unknown, yet it is
required to know the boundary conditions to solve the system of equations.
This dilemma dictates an iterative solution for each time step. Using the Newton-
Raphson method, the iteration continues until [n2, nN]new – [n2, nN]guess  0. The system
converges quickly because of the predictor-corrector nature of Newton-Raphson (see
Appendix-3 for more information).
For each time step, the sub-matrices are computed and assembled into the global
matrices. Then the system of equations represented by [Eq. 9] is solved for the
unknown variable [] at each node.
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m
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The groundwork is now
laid, so let us compare
the results for
uncharged particle
settling between
Mason & Weaver’s
analytical approach
and the finite element
solution.

Comparison of Results --- Test Case A
100 nm Particle
Solution to the Mason-Weaver Equation
by the Analytical Form [Eq. 2]
100 nm Particle
by the Finite Element Method
Concentration profile at 1 week Concentration profile at 1 week
Concentration profile
at 20 weeks
Concentration profile
at 20 weeks
Input conditions: Temperature = 22C; viscosity = 2.5 cP; uncharged particles; particle diameter = 100nm;
solid particle density = 1.8 g/cc; liquid density = 1.036 g/cc; tank height = 2.23 cm
element length for numerical stability = 0.101 cm  22 elements
Other results: Brownian diffusion coef. = 1.725 x 10– 8 cm2/s; Terminal particle velocity = 1.66 nm/s
t = 0 t = 0

Comparison of Results --- Test Case B
40 nm Particle
by the Analytical Form [Eq. 2]
Input conditions: Temperature = 22C; viscosity = 2.5 cP; uncharged particles; particle diameter = 40nm;
solid particle density = 1.8 g/cc; liquid density = 1.036 g/cc; tank height = 2.23 cm
Other results: Brownian diffusion coef. = 4.31 x 10– 8 cm2/s; Terminal particle velocity = 0.27 nm/s
40 nm Particle
by the Finite Element Method
Concentration profile at 1 week
Concentration profile at 20 weeks Concentration profile at 20 weeks
Concentration profile at 1 week
Observation: 40 nm particles settle much slower than 100 nm particles

The analytical results and the finite element solution are
in excellent agreement with each other for these test cases.
Now let us add some additional physics to the finite element
model so that we may more accurately simulate the
settling dynamics of charged pigment particles.
Conclusion of Part-1

Part-2
Varying viscosity field
Electrostatic repulsion
van der Waals attraction
Poly-steric repulsion
ions
counter-ions
Poly-steric repulsion site

Varying Viscosity Field
The analytical solution to the Mason-Weaver equation was given in [Eq. 2]. In this
form, there is no way to account for viscosity varying throughout the mixture. Yet
it is well-known that the effective ink viscosity is directly related to the solid content
of the mixture. So a fundamental limitation of the analytical solution is that it cannot
account for this braking effect.
 
particlessolidtheoffractionvolume
solidshewithout tliquidtheofviscosity
0.71
1used;bemayequationDougherty-Kriegertheely,alternativ;
2
5
1
:Einsteinfromequationanotherbygivenisviscosityeffectivethemixture,solid-liquidaFor
(C)etemperatur
s)-(mPaor(cP)viscositymixture
55.22
:isresponsere)(temperatuviscositytheink,mono-PK1For
0
924.1
P
00
711.0




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









P
P
f
f
f
T
T
m
mmmm
m
m [Eq. 14]
[Eq. 15]

Varying Viscosity Field (cont.)
[Eq. 16]
 
 
cP38.2itselfbyliquidtheofviscosity
38.20204.0
2
5
15.2
computeto15][Eq.usemaywe22C),(atcP2.5isviscositymixturethat theknowweSince
0204.0mixturein theparticlessolidoffractionvolume
componentsliquid
036.1
11
componentparticlesolid
8.1
11
mixtureofgrams100perpigmentofgrams3.5i.e.035.0
inkmono-PK1ForEx.
1mixturecm
particlessolidcm
1:
continuityfor;1
componentliquidcomponentparticlesolid
cm
;-componentofvolumespecific
essdimensionl)1(0;-componentoffractionmass
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mixturetheofvolumespecific
0
00
0
3
3
3
3
3
3
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m
mm
m
r
r
P
l
L
p
p
pm
pmLppm
ppm
m
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pmLppmm
pmLm
LLmppmm
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imim
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f
g
cmv
g
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f
fvvf
vf
v
vf
f
fvvfvSo
ff
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g
iv
fif
vf
g
v

 
 
zeroat time
cm
particlespigment
1093.2
1055
3
4
0204.0
3
4
.zeroat timeparticlespigmentofionconcentratvolumetric
shomogeneouinitial,theestimatemayweThusmixture.in the0.0204offractionvolume
aconsumeparticlesthat theknowweandnm,110ismono-PK1ofsizeparticlemedianthat theknowWe
3
13
373
0
0





a
f
n
n
P
Now let us assume that good mixing ensures that the particles are equidistant from each
other at time zero. Then each d3 unit volume contains 2 particles (1/4 particle on each corner).
d
d
d
Therefore at time zero, dimension (d) is:
d = [2/2.93x1013]1/3 (cm)
d = 409 nm and
 = 299 nm


     
15][Eq.ofusebybottom-top
frommapviscositythedetermineorder toinnodeeverytoapply thisnowcanweSo
3
4
mixturecm
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:bygivenisparticlespigmentofionconcentratcvolumetrithat theknowWe
....,nodesationconcentrat
m
particles
........
3
3
3
1213)1(321












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



a
nf
nnnnnnn
iP
NN

For each time step, the finite element solution of [Eq. 9] provides a map of pigment particle
concentration at each node, as shown below:
Initial viscosity
PK1-Mono Ink Settling
After 1 Week
This illustrates why pigment ink has a start-up
problem when the tanks are left in idle state, and
it indicates why it is a transient effect – clearing up
after a bit of ink is pumped out, or jetted out.
[Eq. 17]

• It is not possible to obtain the m(y,t) response by an
analytical solution of the governing differential equation.
– Mason and Weaver’s solution required a constant value for
viscosity at all times and positions.
• The finite element model derived here is thus an
improvement to the mathematical physics governing pigment
sedimentation.
• Now let us move on to other important effects that cannot be
addressed with the Mason-Weaver equation – van der Waals
attraction and electrostatic repulsion between the particles.
Varying Viscosity Field -Summary

Derjagin-Landau-Verway-Overbeek Theory
Let us create a response curve
like this for PK1-mono ink
VT/KT=(vanderWaalsattraction+Electrostaticrepulsionpotential)
withrespecttothermodynamicpotential(KT)

Van der Waals Force
• In a homogeneous mixture of PK1 mono, the average distance between
pigment particle centers is ~409 nm. Since the particle diameter is ~110 nm
the particles collide when the distance between their surfaces is ~299 nm.
• For a stationary ink tank with no mechanical stirring means, the particles
are constantly moving due to Brownian motion and gravity.
• With just 299 nm to move before crashing into a neighbor, physical
encounters between the colloid particles occur frequently. The stability of the
mixture is determined by the nature of these interactions.
• Van der Waals force (VDWF) is a weakly attractive force at large distances;
however, over a short distances (such as during collisions between particles
suspended in a liquid), the VDWF is very strong.
• Thus in the absence of any counteracting force, every collision between
suspended particles could effect irreversible coagulation. Eventually, the
entire solid content of the mixture will be clumped together at the bottom of
the container: where particles not only settle – they grow due to VDWF.
• To quantitatively identify the enemy of pigment inks, VDWF will be
discussed over the next few pages. Then we will move on to describe the
electrokinetic countermeasures (EDL) used to prevent irreversible pigment
coagulation (often called flocculation).

Black Gunk = A Form of Colloid Instability
When an electrolyte concentration swamps the mixture, e.g. ionic dye components
pollute the mixture, the Debye length of the electric double layer (EDL) goes to zero.
When this happens the EDL disappears, so van der Waals forces are not held in check
by electrostatic forces, and the pigment irreversibly coagulates.
Coagulated pigment particles
of Newman black gunk infamy

van der Waals Attraction
For equal size spheres, the van der Waals energy between particles is given by:
   
 
 
  
 
           
 
 

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1112
112
:Then;1:Let
1
44
1
284
2
22
241
2
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1
1
2
1
12
2
1
nt)displaceme-(xrespect towithforceattractionder Waalsvan
forceaintoexpressionabovethetransformusleteffect,derWaalvantheofmagnitudetheillustrateTo
Joules100.4ink)(aqueous#2&#1between#3materialofconstantHamaker
Joules101particle)pigment(carbon#2surfaceofconstantHamaker
Joules101particle)pigment(carbon#1surfaceofconstantHamaker
Joules1035.1constantHamaker
distanceparticle-particle
radiusparticle
2
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1
12
energyattractionder Waalsvan
3
2
22
132
23
2
22
132
2
2
2
132
20-
33
19-
22
19-
11
20
33223311132
22
132
a
A
F
b
b
b
b
bb
bb
b
a
A
b
bb
db
d
bdb
d
bbdb
dA
db
dE
adx
db
dx
db
db
dE
dx
dE
xF
A
A
A
AAAAA
x
a
a
x
b
b
bb
bbb
A
E
A
A
AA
A
A [Eq. 18]
[Eq. 19]
[Eq. 18a]

Electrostatic Repulsion
1/k
For a self-dispersed pigment (like PK1-mono), the surface
of the pigment particle itself is covered with an adsorbed
layer of negative charge. When the particle is immersed in
a liquid solution, anions (+) will surround the particle due to
electrostatic attraction. This is called the electric double layer
(EDL) and it ensures that the negatively charged particle plus
this cloud of anions is electrically neutral when viewed from
a distance. However, when two such charged particles and
their surrounding EDL are brought into close proximity, the
electric fields will interact and repulse each other with a
Coulomb-like force. The ingredients of the EDL are shown in
the figure to the left. An exact, quantitative treatment of the
EDL is still a fruitful area of research; however, some
well-accepted approximations (e.g. the Debye-Huckel and
Smoluchowski approximations) do a very good job of
describing the behavior of the EDL and the
resultant force field it produces.

Electrostatic Repulsion
   
 kx
c
B
m
R
R
m
B
c
R
TK
ze
TK
ze
kx
c
B
mR
e
ze
TK
ak
dx
dE
F
x
k
a
T
K
e
zz
E
e
e
e
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TK
aE
B
c
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c

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

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2
0
1-
9-
2
2
12-
0
23-
19-
1-1
2
2
2
0
32forcerepulsionticelectrosta
(meter)particlesebetween thdistance
(meter);
EDLtheoflengthDebye
1
meter1055radiusparticle
78.5waterofconst.dielectricparticlesebetween thmediumtheofconstantdielectric
meterNewton
Coulombs
108.9spacefreeoftypermittivi
Kelvin298etemperatur
Coulomb
Joules101.381constantsBoltzmann'
nformulatioPK1in thepigmentmonoformV15-aboute.gparticle;theofpotentialzeta
Coulombs101.6chargeunit
1in waterClandNaNaCle.g.valence);ofvalueabsolute(i.e.numberchargeeelectrolyt
overlap.begin tofieldsticelectrostatheandothereachapproachparticles
theasions-counterofcloudEDLanwithparticleschargedbetweenenergyrepulsive
1
1
32
:byedapproximatisothereachgapproachinparticleschargedTwo




z



z
z
[Eq. 20]
[Eq. 21]
(the Debye length will be discussed a bit later)
[Eq. 22]

Debye Length
nanometers3
1
inkspigmentLXKformol/Liter01.0estimatesMingFrank
1numberchargeeelectrolyt
)(mol/Literionconcentrateelectrolyt
areunits:10329.0 110




 
k
c
z
c
mczk
Debye length (1/k) is where the potential
decreases by an exponential factor. The Debye
length is generally considered the thickness of
the diffuse portion of the electric double layer.
It is a function of temperature, electrolyte
concentration and the dielectric constant of the
medium. For a 1-1 electrolyte (z = 1) in an aqueous
solution at 25C, the value of (k) is:
Another way to estimate (c) is by the use of
conductivity measurements. For PK1 ink having
s ~ 0.7 mS/cm  c ~ 0.008 mol/Liter
1/k ~ 3.6 nanometers
[Eq. 22]
Since (1/k) is between 3.0 and 3.6 nm for PK1, we
may state that the electrostatic repulsion force field
has a reach of ~3.5 nanometers.

Debye Length (cont.)
Note that [Eq. 22], of the previous page, teaches that as the electrolyte concentration increases,
the Debye length decreases. What this means is - if the pigment ink mixture gets polluted with
ions, the reach of the electrostatic repulsion force field gets smaller and smaller. When the
electrostatic force field shrinks, the particles can get closer and closer before the repulsion force
has any effect. Remember what was taught earlier – the van der Waals attraction force field
becomes exponentially stronger when the particles come within a few nanometers of each other.
So when a charged pigment particle is overdosed with liquid containing as little as 0.05 mol/liter of
a 1:1 electrolyte, the Debye length drops to about 1.4 nm, i.e. the particles need to get
within 1.4 nm of each other before the repulsion force is felt; however, by this time the
van der Waals force will dominate – causing irreversible coagulation when the pigment particles
randomly bump into each other. Soon it is no longer a stable ink mixture – it is black gunk.
MMMMMM NNNNNN LLLLLLLLLLL NNNNNN MMMMMM
Therefore, in any inkjet architecture that plumbs dye and pigment inks in close proximity to each
other, i.e. on the same chip - black gunk is going to be a serious problem unless the electrolyte
content of the dye inks is filtered down to less than 0.01 mol/liter.

Interaction Energy Response
Combining [Eq. 18] with [Eq. 20] over a particle-particle distance (0  x) is an
illustrative means of looking at pigment stability as a function of electrolyte
concentration (c) and zeta potential (z).
z  75 mV
z  60 mV
z  45 mV
z  30 mV
z  15 mV
z  0 mV
c = 0.1 mol/L
c = 0.025 mol/L
c = 0.01 mol/L
c = 0.0001 mol/L
c = 0.005 mol/L
c = 0.0025 mol/L
c = 0.001 mol/L
c = 0.01 mol/Liter
(like PK1-mono)
z = 15 mV
(like PK1-mono)
Particle size = 110 nm; Temperature = 298 K; A11 = 1 x 10-19 J; A33 = 4 x 10-20 J;
Dielectric constant of the liquid = 78,5
Repulsion
Repulsion
Attraction
Attraction
In any colloid system, |z| is rarely in excess of 75 mV.
In the concentrate, |z| ~ 32 mV.
In the PK1 formulation |z| ~ 15 mV*
The electrolyte concentration in the PK1 formulation
c ~ 0.01 mol/Liter *Ref. Frank Ming

Interaction Energy Response - Observations
The plots on the previous page illustrate the effect of the zeta potential and
electrolyte concentration. They show that if one wants to maximize the stability
of the colloid system, such that particle growth is improbable, then one needs
to maximize the magnitude of the zeta potential and minimize the electrolyte
concentration of the ink mixture (i.e. no salts, or metallic ions).

PK1-mono Interaction Energy Response
The stability of the particles, i.e. their ability to fight off coagulation, is a function of the potential
energy during particle interaction (ET). Values of ET greater than zero indicate that the system is
stable. Think of the ET -maxima as the energy barrier to coagulation. At a zeta potential of ~|15 mV|
and an electrolyte concentration of ~0.01 mol/L, the PK1 formulation is stable*.
*Note: in this context, “stable”, means that the particles will not coagulate. They’ll still be subject to gravitational settling, but the
particles won’t stick together. Think of them as passengers on a crowded bus, they may become tightly packed, but they will
not stick together.
ER = Electrostatic repulsion (|z| = 15 mV)
EA = Van der Waals attraction
ET = repulsion + attraction = DLVO response curve
Energy barrier
[Derjagin-Landau-Verway-Overbeek]
RepulsionAttraction

PK1-mono Interaction Force Response
z = 15 mV  pigment in the PK1 formulation
z = 25 mV
z = 30 mV
z = 20 mV
z = 32 mV  mono pigment in the concentrate
This illustrates the fact that the repulsive nature of charged colloids acts
over a distance on order of the Debye length scale , i.e. the thickness
of the EDL. Charged colloids will settle over time due to gravitational
force, but the nature of the interaction force will prevent them from
getting within ~3-10 nm of each other – thus avoiding the clumping,
coagulation effect of the van der Waals force.
[Eq. 22] minus [Eq. 19] = interaction force response
RepulsionAttraction

Critical Coagulation Concentration (c.c.c)
The previous few pages showed the importance of electrolyte concentration and the zeta potential
for producing a stable colloidal mixture. Of great interest is the critical coagulation concentration
(c.c.c) because it indicates the electrolyte concentration at which the colloids crash (stick together
during collisions and grow into clumps). According to Deryagin-Landau-Verway-Overbeek theory
setting ER = EA = 0; and dET/dx = 0; produces an equation for c.c.c.
110 nm carbon particles in
an aqueous solution at 298K z = 1
z = 2
z = 3
PK1formulation
Pigmentconcentrate
[PK1 zeta values from Frank Ming]
variablesabovetheofdefinitionfor22]and2118a,[Eq.see
Liter
mol
units
10329.0
1
444
2
10
2
132
0



















ncoagulatio
c
Bm
ncoagulatio
k
z
ccc
ze
TK
A
k

Electrolyte charge number
This is another way to quantify why black gunk
occurred when PK1 mono was swept into
CMY-dye nozzles driving c > c.c.c.
[Eq. 23]

Steric Stabilization
The pigment used in PK1 is self-dispersed. As such, it effects colloid stability by incorporating the
electrostatic repulsion effect that was just covered. However, the PK1-mono formulation also
contains a polymeric dispersant. The polymeric dispersant has a molecular
weight > 10000 D. These long chain molecules have lengths greater than the attraction range
of the van der Waals force. The dispersant molecules attach to the surface of the pigment, forming
a fuzzy, spring-like coating that prevents the particles from getting close enough for the van der
Waals force to become problematic. This stabilization mechanism is called “steric stabilization.”
When a mixture uses both electrostatic and steric means to stabilize the colloids (as does PK1), it
is called electro-steric stabilization. The picture below is illustrative of electro-steric stabilization.
Note: electro-steric stabilization provides some well-known
advantages over steric, or electrostatic means alone;
however, an often overlooked advantage of bonding a
fuzzy cloud of polymer molecule chains to the surface is
a reduction of the particle density. Since polymers are less
dense than carbon, the steric layer reduces the effect of
the gravity-buoyancy force.

Gravity-Buoyancy Effect of Steric Stabilization
a a
Pigment particle of radius (a)
with a polymeric dispersant shell of thickness a
a
 
a
a
f
f
P
S
SP
S







particlepigmenttheofdensity
layerionstabilizatstericpolymerictheofdensity
1
densitycomposite
:shown thatbecanIt
3
r
r
rr
rr
So for a carbon black pigment particle with a radius of 55 nm, having a density of 1.8 g/cm3,
and a 4 nm thick polymeric, steric stabilization layer, having a density of 1.1 g/cm3 – the
composite density is ~1.65 g/cm3. Since this is closer to the liquid density (~1.0 g/cm3), than
carbon black, the composite particle is more buoyant than the uncoated particle.
[Eq. 24]

Gravity-Buoyancy Effect of Steric Stabilization
The effect of composite density is illustrated in the plots below. In both
cases the electrostatic repulsion and van der Waals attraction are not considered.
The simulation takes into account: gravity-buoyancy effect, viscous drag and
Brownian motion.
Composite Density = 1.65 g/cm3
Density = 1.8 g/cm3
Gravity-buoyancy effect
3.5 wt.% solids, viscosity = 2.4 cP, carbon-black particle size = 110 nm, wetted height = 2.23 cm

Accounting for the Repulsive and Attractive
Forces During Pigment Sedimentation
Recall that the derivation of the partial differential equation [Eq. 1]
governing sedimentation of small particles began with a study
of the forces acting upon the colloid, as shown to the right.
 
m
rr
m 9
2
;
6
2
2
2
lpB
ga
B
a
TK
A
y
n
B
y
n
A
t
n
























[Eq. 1]
Now we wish to account for the forces that
have just been introduced:
- electrostatic repulsion [Eq. 22]
- van der Waals attraction [Eq. 19]
Mason & Weaver did not account for these
forces in their model, so we must now
figure out how to modify [Eq. 1] to incorporate
[Eq. 19 & 22]. This will enable us to more accurately account for the
multi-physics involved in pigment settling.

Derive the Terminal Velocity Term
During the derivation of [Eq. 1] it was noted that the variable (A) was the diffusion coefficient
of the Stokes-Einstein equation. Its purpose is to account for Brownian motion. The variable
(B) was identified as the terminal velocity of the particle when the forces acting upon it
were gravity-buoyancy and viscous drag. Since we want to account for electrostatic repulsion
and van der Waals attraction (i.e. forces acting upon the colloid), it is obvious that we need to
account for them in variable (B), not the diffusion term (A).
 
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kx
c
B
m
L
T
T
kx
c
B
mL
dRAgb
e
ze
TK
k
a
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By
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ze
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ak
a
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ga
FFFF
Forces
onacceleratimassForces
2
03
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222
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03
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22
1313
16
1121
143
2
3
1
forSolving
632
1121
123
4
:Then
drag.viscousandrepulsionticelectrostaaremotionparticlethebraketotendingforcesThe
der Waalsvanandbuoyancy-gravityaretogetherparticlestheclumptotendingforcesThe
0:Thenion.sedimentatforequationaldifferentipartial
governingsWeaver'&MasonofderivationtheintoheavilyplaysassumptionthisIndeed
velocity.r terminalreach theiinstantlytheyassumetopracticecommonisitsmall,soarecolloidsBecause







rr
m
m






rr
[Eq. 25]
Inserting [Eq. 25] into [Eq. 1] accounts for repulsion and attractive forces as well as
viscous drag and gravity-buoyancy

Part-3
The mathematical physics of
charged particle sedimentation
applied to the PK1-mono
formulation.

PK1-mono Particle Size Distribution
The particle counts follow a lognormal distribution
having a mean at 110 nm and a shape factor (s)
equal to 0.333.
CUMULATIVEDENSITYFUNCTION

PK1-mono Particle Size Distribution
Particle Distribution in PK1-mono Formulation With 3.5 wt.% Solids
Bin # Particle size
(nm)
Probability of this
particle size
Particles of this
size per pL in the
mixture
Wt.% of this particle
size in the mixture
1 46 0.0241 448 0.0040
2 68 0.1301 2416 0.0691
3 89 0.2299 4269 0.2738
4 111 0.2378 4415 0.5493
5 132 0.1656 3074 0.6432
6 154 0.0999 1855 0.6165
7 175 0.0551 1023 0.4986
8 197 0.0287 533 0.3705
9 219 0.0144 268 0.2561
10 240 0.0071 132 0.1662
11 262 0.0035 65 0.1056

towards the tank bottom. This
pigment migration will effect
increased OD (for awhile)
because the solid volume at
the tank bottom is abnormally high.
PK1-mono Overall Sedimentation Response
TankBottom
TankTop

What if the Tank is Upside-Down for 30 Days
Exit Port
Lid Side
Concentrationatthelid
Concentrationattheexitport
Now let us flip the tank without shaking it

Now Flip the Tank  Exit Port Down
Exit Port
Lid Side
Concentrationattheexitport
Concentrationatthelid
10 days after flipping to port-down, the pigment concentration is
nearly uniform, reversing the effect of 30 days upside-down

Settling Response to Sit  Flip  Sit
A
A
A
A
Concentration at line A-A

References
• M. Mason & W. Weaver, “The Settling of Small Particles in a Fluid”, Phys.
Rev., 23, p412-426, (1924).
• D.J. Shaw, “Colloid and Surface Chemistry”, Elsevier Sci. Ltd., 4th ed., (2003).
• K. Huebner, et.al., “The Finite Element Method for Engineers”, Wiley & Sons,
(1975).
• Bird, Stewart & Lightfoot, “Transport Phenomena”, Wiley & Sons, 2nd ed.
(2002).
• C.F. Gerald, “Applied Numerical Analysis”, Addison-Wesley, (1973).
• J. Shi, “Steric Stabilization – Literature Review”, Ohio State Univ. Center for
Industrial Sensors & Measurements, (2002).
• J.L. Li, et.al., “Use of Dielectric Functions in the Theory of Dispersion
Forces”, Phys. Rev. B, 71, (2005).
• G. Ahmadi, “London-van der Waals Force”, Clarkson Univ.

Appendix-1
Derivation of the minimum
element length for numerical
stability

Minimum Element Length
 
 
 
 
 
onacceleratinalgravitatio
densityliquiddensity,particle,
radiusparticle
viscositydynamic
etemperatur
constantsBoltzmann'
4
3
:lengthelementminimumfor thedirectlysolvemayweThus,
9
2
velocityterminalparticle;
6
tcoefficiendiffusionEinstein-Stokes
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tcoefficiendiffusionlength;elementvelocity;convective
2
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:problemionsedimentatFor the
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ydiffusivit
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thatNote;2
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numberPecletferHeat trans
:sother wordIn2.bemustelementanyfornumberPecletthe
thatruleelementfiniteknown-wellaisitproblem,ferheat transaIn
3
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2
2
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
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g
a
T
K
ga
KT
l
ga
u
a
KT
D
uD
ensionless
s
m
m
s
m
Pe
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ul
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lp
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e
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e
p
p
e
p
rr
m
rr
m
rr
m
r

r

r
[Eq. Ia]

Minimum Element Length (cont.)
The Mason & Weaver article contains an interesting comment. After presenting their
analytical solution {[Eq. 2] in this document}, they state that it, “converges very slowly
for small values of t, especially if  be so small that it is unsuitable for calculation.”
In their paper  = A/Bl
 
 
   

d
d
rr
m
d
m
rrd
m
d
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e
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lg
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lga
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lag
a
KT
Bl
A
ag
B
a
KT
A
Perhaps one way to view this result is that the solution becomes unstable if
during any given time step, the gravitational displacement of the particle greatly exceeds
the Brownian motion of the particle.

Appendix-2
Derivation of the element
equations for pigment settling

Derivation of Element Equations for the
Settling of Uncharged Colloid Particles
It was recognized in this analysis that the partial differential equation describing
uncharged colloid particle settling has the same form as the pde for convective-
conductive heat transfer. Finite element textbooks show the element equations for
convective-conductive, but not colloid settling.
Since they do not appear in textbooks, the derivation of the element equations
for colloid settling are described here.

Element Equations for Colloid Settling
 
      
     
     
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9
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[Eq. IIb]
[Eq. IIc]

Element Equations for Colloid Settling (cont.)
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 
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Element Equations for Colloid Settling (cont.)
     
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       
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Appendix-3
Newton-Raphson method

Newton-Raphson
 
 N
N
NN
xf
xf
xx
'
1 
Given a function f(x) and its derivative f ’(x) - if we have these values at point (xN), we can
estimate the position of the root, i.e. f(x) should go to zero in the vicinity of xN+1. The equation
shown above is used to predict the value of xN+1 to use on the next iteration to find the root.
For the sedimentation problem, this method is used to iterate on values of the boundary condition
until the guessed values at nodes [2] and [NNODE-1] equal the true values obtained by solving
[Eq. 9] with the finite element method.
xNxN+1
f(xN)
f(x)

Mathematical Physics of Pigment Settling

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