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

Tank B

otto

m

Tank T

op

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

How do we minimize pigment settling? 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

How do we minimize pigment settling? 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

How do we minimize pigment settling? 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]:

0at constant 0,

:condition Initial

,0at ,

:conditionsBoundary

density liquid -density particle

viscosityliquid

radius particle

etemperatur

constant sBoltzmann'

9

2 ;

6

surface bottom @ ;surface top@ 0 position;

time

density particle volumetric,

0

2

2

2

tnyn

LyBny

nA

a

T

K

gaB

a

KTA

Lyyy

t

tyn

y

nB

y

nA

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

161 1

2222

21

'

4'2

2

/1

/

0

22

mmt

tB

L

BL

Ah

L

yh

m

hmmhmeme

ee

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

surface bottom @ ;surface top@ 0 position;

time

density particle volumetric,

2

2

2

lpBga

Ba

TKA

Lyyy

t

tyn

y

nB

y

nA

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.

density liquid -density particle

gravity todueon accelerati

radius particle spherical

3

4 particle on the forcebuoyancy - force nalgravitatio

etemperatur


position at density particle

statesteady at on distributi colloid volumetric

3

11

13

1

lp

lpgb

B

TK

yyF

g

a

gaF

T

K

yyn

enynm

particlesB

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

location toposition reference from move torequiredenergy

position reference fromnt displaceme

potential micThermodynaeTemperaturconstant sBoltzmann'

position referenceat on distributi particle volumetricconstant

particles theofon distributi volumetric

:case specific In this

functionon distributi sBoltzmann' of form generic

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

6particle theof velocity terminal

63

4

0 force drag viscous

2

3

lpd

lp

dgb

dgb

ga

a

Fy

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

Ja

nFyn d

secflux particle downward

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

Fn

y

n

eTK

Fn

y

n

FFenyn

B

d

B

d

TKyyF

B

d

dgbTK

yyF

B

d

B

d

m6

for 4] [Eq. Substitute

that Recall

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

mm

mm

mm

mm

m

9

2

6

3

4

6 ;

6

66

unit timeper gain particle tricnet volume

/

|

66

:flux particle downwardnet a is therestate, transientIn the

066

0

:downward diffusing are thereas upward diffusing particlesmany as are therestate,steady At

6

:flux term for the 6] [Eq. Solving

22

2

2

2

2

2

0

lplp

dB

dB

y

dB

dB

B

ga

a

ga

a

FB

a

TKA

y

nB

y

nA

y

n

a

F

y

n

a

TK

t

n

t

n

m

s

mparticle

y

J

y

J

a

nF

y

n

a

TKJ

a

nF

y

n

a

TK

JJJJ

Jy

n

a

TKyn

J

dyy

JJ

The Mason & Weaver equation

is thus produced

dy

Mason & Weaver Equation Overview

m

rr

m 9

2 ;

6

2

2

2

lpBga

Ba

TKA

y

nB

y

nA

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)].

LyBny

nA

or

nga

y

n

a

TK

a

nF

y

n

a

TK

LyJJ

lpBdB

,0at timesallfor

09

2

60

66

,0at timesallfor 0

:ConditionsBoundary

2

m

rr

mmm

[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

lpBga

Ba

TKA

y

nB

y

nA

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

Tv

y

T

t

T

Then

CLet

y

Tv

t

T

y

T

C

Q

Qv

Cyt

T

constvelocityvy

Tv

t

TCQ

y

T

v

v

v

v

2

2

2

2

2

2

:

ydiffusivit Thermal:

:0for

sourceheat internal velocity;

heat; specific olumeconstant v position; time;

density; :etemperatur :tyconductivi thermal

.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

Ta

nk h

eig

ht

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

TKl

rr

3

)(

4

3lengthelement minimum

Finite Element Solution (cont.) The time marching finite element solution of a generic field problem takes the following form:

)(

2)(

)(

)(

)()(

*)(

)(

)(

)(

)()(

1213)1(321

*

*

9

2 termconvectionelement

6termdiffusion element

lengthelement area;element

1 ;21

12

6

11

11

211

11

...., nodesat ion concentrat m

particles........

---2)Appendix (see shown that becan it problem, settlingpigment For the---

function forcing

at time variablefield

at time variablefield

step time

termcapacitive

termdiffusion and convection

22

e

lpe

eBe

ee

ee

e

e

ee

NN

newold

new

old

oldnew

gaB

a

TKA

lArea

lArea

B

l

AArea

nnnnnnn

FFF

tt

t

t

Ftt

m

rr

m

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]


.1rank ofmatrix square a be willelements

of system afor matrices global theso unknowns), two(i.e. nodes twohaselement Each

.matrix ecapacitanc global theand matrix convection-diffusion global the

into assembled are These term.capactive for the matrix -subanother and

termsconvection-diffusion for the matrix -sub a have llelement wiEach

)(

)(

NN

GG

e

e

*****

***

***

*****

*****

shown. asmatrix global the toadded are 4-3 nodesat uesmatrix val-sub theso

4,-node and 3-node are (3)element with associated nodes The

;

:(3)element for exampleby below dillustrate as matrices, global theof indices

ingcorrespond at the placed are matrices-sub thefrom valuesnodal The

5.5 be lmatrix wil globaleach Then .4let example,For

(3)

ww

wwG

ww

ww

N

This procedure is continued until all

elements are accounted for


.conditionsboundary for theaccount must we12], [Eq. solving before However,

matrixidentity thesuch that inverse,matrix theis where

:is equations of system a osolution t that thealgebramatrix from Recall

11 ;11 ;11

:sizes following thehave matrices The

:form thehas 9] [Eq. that Note

11

1

IHHH

RH

NNRNNNH

RH

new

new

new

rrr

Boundary Conditions

[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

)(

)(

)(

)()(

1

)1(

)1()1(

)1(

)1()1(

2112

)(12

21

21

:node bottom at the valuenodal set thesimilarly

21

21

:node at top valuenodal set the so (1)number element for ;2

:as stated is ally thisMathematic

lost).or gained is mass no (i.e. boundaries at theexist can flux net nothat

is problem settling for thecondition boundary the theRecall

N

NN

N

NN

NN

e

AlB

AlB

nn

AlB

AlB

nnnn

Bl

nnA

or

Bny

nA

[Eq. 13b]

[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.

1213)1(321 ...., nodesat ion concentrat m

particles........

NN nnnnnnn

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.

particles solid theoffraction volume

solids he without tliquid theofviscosity

0.711 used; bemay equation Dougherty -Krieger theely,alternativ ;

2

51

:Einstein fromequation another by given is viscosityeffective themixture, solid-liquid aFor

(C) etemperatur

s)-(mPaor (cP) viscositymixture

55.22

:is response re)(temperatu viscosity theink, mono-PK1For

0

924.1P

00

711.0

P

P

f

ff

T

T

m

mmmm

m

m [Eq. 14]

[Eq. 15]

Varying Viscosity Field (cont.)

[Eq. 16]

cP 38.2itselfby liquid theofviscosity

38.20204.02

515.2

compute to15] [Eq. usemay we22C),(at cP 2.5 is viscositymixture that theknow weSince

0204.0mixture in the particles solid offraction volume

components liquid036.1

11

component particle solid8.1

11

mixture of grams 100per pigment of grams 3.5 i.e. 035.0

ink mono-PK1For Ex.

1mixture cm

particles solid cm

1:

continuityfor ;1

component liquid component particle solid

cm ; -component of volumespecific

essdimensionl )1(0 ; -component offraction mass

cm mixture theof volumespecific

0

00

0

3

3

3

3

3

3

m

mm

m

r

r

P

lL

pp

pm

pmLppm

ppm

m

ppmP

pmLppmm

pmLm

LLmppmm

i

imim

iimm

f

gcmv

gcmv

f

fvvf

vf

v

vff

fvvfvSo

ff

vfvfv

giv

fif

vfg

v


zero at time

cm

particlespigment 1093.2

10553

4

0204.0

3

4

. zero at time particlespigment ofion concentrat volumetric

shomogeneou initial, theestimatemay weThus mixture. in the 0.0204 offraction volume

a consume particles that theknow weand nm, 110 is mono-PK1 of size particlemedian that theknow We

3

13

3730

0

a

fn

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. of useby bottom-top

from map viscosity thedetermine order toin nodeevery toapply this nowcan weSo

3

4

mixture cm

particlespigment cm

:bygiven is particlespigment ofion concentrat c volumetri that theknow We

...., nodesat ion concentrat m

particles........

3

3

3

1213)1(321

anf

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/K

T =

(va

n d

er

Wa

als

att

raction

+ E

lectr

osta

tic r

ep

uls

ion

po

ten

tia

l)

with

re

sp

ect to

th

erm

od

yn

am

ic p

ote

ntia

l (K

T)

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:

1112

112

:Then ;1 :Let

1

44

1

284

2

22

241

2ln2

1

1

2

1

12

2

1

nt)displaceme-(x respect to with force attraction der Waalsvan

force a into expression above the transformuslet effect, derWaal van theof magnitude theillustrate To

Joules 100.4 ink) (aqueous #2 & #1between #3 material ofconstant Hamaker

Joules 101 particle)pigment (carbon #2 surface ofconstant Hamaker

Joules 101 particle)pigment (carbon #1 surface ofconstant Hamaker

Joules 1035.1constantHamaker

distance particle-particle

radius particle

2

1

2ln2

1

1

2

1

12energy attraction der Waalsvan

3

2

22

132

23

2

22

1322

2

2132

20-33

19-22

19-11

2033223311132

22132

a

AF

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

dExF

A

A

A

AAAAA

x

a

a

xb

b

bb

bbb

AE

A

A

AAA

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

Bm

RR

m

B

c

R

TK

ze

TK

ze

kx

c

BmR

eze

TKak

dx

dEF

x

k

a

T

K

e

zz

E

e

e

eze

TKaE

B

c

B

c

2

0

1-

9-

2

212-

0

23-

19-

1-1

2

2

2

0

32forcerepulsion ticelectrosta

(meter) particles ebetween th distance

(meter) ;EDL theoflength Debye

1

meter 1055 radius particle

78.5 waterof const. dielectric particles ebetween th medium theofconstant dielectric

meterNewton

Coulombs 108.9space free ofty permittivi

Kelvin 298 etemperatur

CoulombJoules101.381 constant sBoltzmann'

nformulatio PK1 in thepigment monofor mV 15-about e.g particle; theof potential zeta

Coulombs 101.6 chargeunit

1in water Cl and NaNaCl e.g. valence);of valueabsolute (i.e.number charge eelectrolyt

overlap. begin to fields ticelectrosta theandother each approach particles

theas ions-counter of cloud EDLan with particles chargedbetween energy repulsive

1

1

32

:by edapproximat isother each gapproachin particles charged Two

z

z

z

[Eq. 20]

[Eq. 21]

(the Debye length will be discussed a bit later)

[Eq. 22]

Debye Length

nanometers 31

inkspigment LXK for mol/Liter 01.0 estimates MingFrank

1number charge eelectrolyt

)(mol/Literion concentrat eelectrolyt

are units :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

Repuls

ion

Repuls

ion

Attra

ction

Attra

ction

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]

Repuls

ion

A

ttra

ction

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

Repuls

ion

Att

raction

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

PK

1 form

ula

tion

Pig

ment

concentr

ate

[PK1 zeta values from Frank Ming]

variablesabove theof definitionfor 22] and 21 18a, [Eq. see

Liter

mol units

10329.0

1

444

2

10

2

132

0

ncoagulatio

c

Bmncoagulatio

k

zccc

ze

TK

Ak

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

af

f

P

S

SPS

particlepigment theofdensity

layerion stabilizat steric polymeric theofdensity

1density composite

:shown that becan It

3

r

r

rrrr

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

lpBga

Ba

TKA

y

nB

y

nA

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).

kx

c

Bm

LT

Tkx

c

BmL

dRAgb

eze

TKk

a

AgaBy

yaeze

TKak

a

Aga

FFFF

Forces

onacceleratimassForces

2

03

2

222131

2

2

03

2

22

1313

161121

143

2

3

1for Solving

6321121

123

4

:Then

drag. viscousandrepulsion ticelectrosta aremotion particle thebrake to tendingforces The

der Waals van andbuoyancy -gravity are together particles theclump to tendingforces The

0 :Then ion.sedimentatfor equation aldifferenti partial

governing s Weaver'&Mason of derivation theintoheavily plays assumption thisIndeed

velocity.r terminalreach theiinstantly they assume topracticecommon isit small, so are colloids Because

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.

CU

MU

LA

TIV

E D

EN

SIT

Y F

UN

CT

ION

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

Tank B

otto

m

Tank T

op

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

What if the Tank is Upside-Down for 30 Days

Exit Port

Lid Side

Co

nce

ntr

atio

n a

t th

e lid

Concentr

atio

n a

t th

e e

xit p

ort

Now let us flip the tank without shaking it

Now Flip the Tank Exit Port Down

Exit Port

Lid Side

Concentr

atio

n a

t th

e e

xit p

ort

Concentr

atio

n a

t th

e lid

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

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

How do we minimize pigment settling? 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

How do we minimize pigment settling? 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

How do we minimize pigment settling? 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

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

onaccelerati nalgravitatio

density liquid density, particle,

radius particle

viscositydynamic

etemperatur


4

3

:lengthelement minimum for thedirectly solvemay weThus,

9

2 velocity terminalparticle ;

6tcoefficiendiffusion Einstein -Stokes

:, know wecase,ion sedimentat For the

dim :number essdimensionl a is that Note

tcoefficiendiffusion length;element velocity;convective

22

:problemion sedimentat For the

lengthelement ty;conductivi thermal velocity;convective heat; specific density;

ydiffusivit

lengthvelocity that Note ;2

2numberPeclet fer Heat trans

:sother wordIn 2. bemust element any for number Peclet the

thatruleelement finiteknown - wella isit problem,fer heat trans aIn

3

)(

2

2

)(

)(

g

a

T

K

ga

KTl

gau

a

KTD

uD

ensionless

sm

ms

m

Pe

Dlu

D

ulPe

luC

ulCulCPe

lp

lp

e

lp

e

p

pe

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

rrm

d

m

rrdm

d

m

particle theof potential nalgravitatio

particle theof potential micthermodyna1

.density massbuoyant and volumeparticle toconnection theseeinstantly can We

terms? theseof meaning physical theis What further. thisexamine uslet so ,accidental becannot This

result. above theand Ia] [Eq.solution stability ebetween th similarity theNotice

1

4

31

92

6

; ; velocityterminal9

2 ;tcoefficiendiffusion Einstein -Stokes

6

)(

)(3)(2)(

2

e

elp

ee

lp

lg

KT

lga

KT

lag

aKT

Bl

A

agB

a

KTA

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

)()(

)()(

)(

0 0

)()(

2

2

)()(

2

2

2

11

)(element for numbers node, ;1function shape

areaelement

function shape Where;element over theion concentrat particle

9

2 termconvective ;

6 termdiffusive ;

)( )(

)()(

)(

)(

)(

ee

ji

eeji

e

l lee

x

x

x

ii

x

ix

ee

lpxx

llx

N

x

N

x

N

ejil

x

l

xNNN

A

ndxx

NNMAdx

x

N

x

NAD

t

n

dnx

NNMdn

x

N

x

ND

dnx

NNMn

x

N

x

ND

dx

nMN

x

n

x

ND

dNx

nM

x

nD

t

n

Nnx

N

x

nnNn

gaM

a

KTD

x

nM

x

nD

t

n

e e

ee

e

e

e

m

rr

m[Eq. IIa]

[Eq. IIb]

[Eq. IIc]

Element Equations for Colloid Settling (cont.)

variablefield,,

termstorage material

termgeneration internal

domain theof properties material diffusive,,

:problem fielddependent timegeneric a of form at thelook uslet IIc] [Eq. of termdependent timeFor the

10] [Eq. of for equation element convective11

11

2

111

:IIc] [Eq. of termsecond The

10] [Eq. of for equation element diffusive11

11

11

1

1

:IIc] [Eq. of first term The

)(

0 0)()(

)(

)()()(

)(

)(

0)()(

)(

)()(

0

)(

)( )(

)()(

zyx

Q

KKK

tQ

zK

zyK

yxK

x

t

n

MA

dxll

l

x

l

x

MAdxx

NNMA

l

AD

dxll

l

lADdxx

N

x

NAD

zzyyxx

zzyyxx

e

l l

ee

e

eee

e

ex

l

eee

eex

le

x

e e

ee

Element Equations for Colloid Settling (cont.)

1 :problem settling particlefor

heat specificdensity :problemfer heat transfor

termstorage

11] 9][Eq. [Eq. of termcapactive21

12

6

1

1

Let ;

:derivative time theTaking

valuesnodal theand factor shape thefrom computed is variablefield theelement, D-1each For

elements. discrete intodomain theup breakssolution element finite The

)()()(

0)()(

)(

)()(

0

)()(

)(

)()(

)()(

)((e)

)()(

eee

l

ee

e

eel

Tee

ei

Tjie

e

j

iee

jie

lArea

dxl

x

l

x

l

xl

x

AreadxNNArea

dNNdt

NNdNt

tN

t

N

N

ee

Appendix-3

Newton-Raphson method

Newton-Raphson

N

NNN

xf

xfxx

'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.

xN xN+1

f(xN)

f(x)

Mathematical Physics of Pigment Settling

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