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thesis entitled

THE DYNAMICS AND SCIENTIFIC VISUALIZATION
FOR THE ELECTROPHORETIC DEPOSITION PROCESSING
OF SUSPENDED COLLOIDAL PARTICLES
ONTO A REINFORCEMENT FIBER
presented by
Peter Timothy Robinson

has been accepted towards fulfillment
of the requirements for

Masters degree in Engineering

 

 

 
    
 

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MSU Is An Affirmative Action/Equal Opportunity institution
cMMth

THE DYNAMICS AND SCIENTIFIC VISUALIZATION
FOR THE ELECTROPHORETIC DEPOSITION PROCESSING
OF SUSPENDED COLLOIDAL PARTICLES
ONTO A REINFORCEMENT FIBER

By

Peter Timothy Robinson

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

MASTER OF SCIENCE

Department of Material Science and Mechanics

1993

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ABSTRACT

THE DYNAMICS AND SCIENTIFIC VISUALIZATION
FOR THE ELECTROPHORETIC DEPOSITION PROCESSING
OF SUSPENDED COLLOIDAL PARTICLES
ONTO A REINFORCEMENT FIBER

By

Peter Timothy Robinson

To meet the demands for new, innovative and more efficient manufacturing tech-
niques of matrix composite materials, a method based on the ideas of colloid science has
been introduced. The method relys on maximizing the electrophoretic deposition of sus-
pended colloidal matrix particles onto a reinforcement fiber. A numerical algorithm has
been developed to simulate the many body problem for the colloidal system. The algorithm
uses numerical integration to solve the dynamical equations of motion. Motion of the par-
ticles are due to London - van der Waal forces, Coulombic forces, gravitational forces and
Stoke ’5 drag.

Visualization of the algorithm in two dimensions has been attempted on a personal
computer. A menu user interface allows flexibility and efficiency for modifying the initial
condition parameters such that the optimal initial condition parameters that maximize the
matrix - collector deposition may be determined.

The colloidal suspension simulator algorithm was intended to be tested with param-
for Fe-40Al matrix particles using an A1203 reinforcement fiber. This thesis presents

eters
the foundation work necessary for the construction of a functioning colloidal suspension

simulator.

 

LIST OF TABLE

LIST OF HOW

LIST OF SIM“-

I

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NROD‘
LUIRAT
FORML’I
IMPLEM
RESULT.
CONCLL

RECOM)

GEXERAL REF

APPENDIX B

TABLE OF CONTENTS

LIST OF TABLES

LIST OF FIGURES
LIST OF SYMBOLS
I INTRODUCTION
II LITERATURE REVIEW
III FORMULATION OF EQUATIONS
IV IMPLEMENTATION
V RESULTS / DISSCUSSION
VI CONCLUSIONS
VII RECOMMENDATIONS
LIST OF REFERENCES
GENERAL REFERENCES

APPENDIX A CSS PROGRAM LISTINGS

APPENDIX B CSSDISP PROGRAM LISTINGS

APPENDIX C CSSRUN PROGRAM LISTINGS

iii

iv

viii

25

48

59

60

63

65

68

130

142

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On.

Table

2.1

2.2

4.1

5.1

LIST OF TABLES

Colloidal Systems.

Everett. D. H.. W Royal 300i-
ety of Chemistry, London, 1988.

Forces in a colloidal system.

The table shows the possible forces that may be present in a col-
loidal suspension system and lists the variables on which the
forces depend.

Russel, William B.. W The
University of Msconsin Press, 1987.
Default parameters used to initiate the CSS software.

Number of changeable variables using the CSS software.

iv

Page

58

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Figure
1.1

2.1

2.2

2.3

2.4

LIST OF FIGURES

Fiber electrophoretic deposition processing of Fe-40Al/A1203.
The schematic diagram shows the FeAl fiber being pulled
through the suspension basin. The colloidal particles adhere to
the fiber by maximizing the attractive forces between particle
and fiber and by minimizing the homocoagulation between Fe-
4OA1 - Fe-40Al and A1203 -A1203. This schematic shows only
the idea of the production method.

Free body diagram for a colloidal particle in suspension.
This figure shows possible force vectors acting on the colloid
particle, where,

FDblis the double layer force,

FSteer the steric repulsive force,

FBris the force due to Brownian motion,

Vel.is the velocity vector of the particle,

FStruciS the structural force,

FLoniS the attractive London - van der Waals force,

Fvais the force due to gravity,

Hyd

Gouy - Chapman Double Layer.
This figure shows an expanded view of the surface of the colloi-
dal particle. The ions and coions migrate to the surface as shown.

Ratio of the particle radius to the double layer thickness.
The magnitudes of an vary in this figure. Most ceramic colloids
have magnitudes of arc in the proximity of 50 to 100.

Myscls. Karol 1.. W Robert E-
Krieger Publishing Company, Huntington, New York, 1978.

The Stern electrical double layer.

This figure shows the small scale surface of a colloid particle,
represented as a vertical line. The ions and coions are depicted
as + and - symbols. The lower graph shows the shape of decay
of the electric potential moving away from the surface of the
particle.

Myscls. Karol 1.111me Robert E.
Krieger Publishing Company, Huntington, New York, 1978.

Page

11

16

18

(J!
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5,4

5.6

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n'fifir
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var. d

2.5

3.1

3.2

4.1

5.1

5.2

5.3

5.4

5.5

5.6

vi

Molecule near a sphere.

This figure depicts a molecule at point P at a distant OP from a
sphere centered at point 0.

Hamaker, H. C., Physica 4, 1058 - 1072, 1937, “The London -
van der Waals Attraction Between Spherical Particles”.

Flow chart for solving the many body problem.

This figure begins in the upper left corner and progresses
through the necessary steps required to solve multiple colloidal
bodies interacting with each other.

Graphic display used to visualize the suspended colloids.

This figure shows the display screen for the simulation and the x
and y dimensions.The grid shown is not actually displayed. Each
particle is centered in one of the boxes to prevent initial overlap
of particles.

Communication links between the three sub - programs.
This figure shows the direction of flow between the CSS, CSS-
RUN and CSSDISP sub - programs.

Zeta potential data for Fe-40Al.
The measured data was obtained using 5.0 wt% Fe-40Al (0.935
volume%) powder dispersed into 0.001N KNO3.

Zeta potential data for A1203 - FP.
The measured data was obtained using 1.075 wt% (0.272 vol-
ume%) A1203 -FP dispersed into 0.001 N KNO3.

Zeta potential data for A1203 - PRD-l66.
The measured data was obtained using 1.075 wt% (0.272 vol-
ume%) A1203 - PRD-166 in 0.001N KNO3.

Zeta potential data for Fe-4OAl.

The measured data was obtained using 0.5 volume% A1203
dispersed into 0.001 N KNO3. This data was measured by Bret
Mlson [19].

Zeta potential data for A1203 .

The measured data was obtained using 0.2 volume% Fe-40Al
powder dispersed into 0.001N KN 03. This data was measured
by Bret Wilson [19].

Data file created with the CSSRUN software.
This figure displays an example data file that is generated by the
CSSRUN sub - program and is read by the CSSDISP sub - pro-

22

29

33

41

49

50

51

52

53

55

7.1

vii

gram. The first number is the number of particles, followed by
the radius and color number of each of the three particles. The
rest of the data format is repeated showing the frame number fol-
lowed by the x and y positions of each particle at that time frame.

Depiction of accumulated mass. 62
This figure shows an example of data that could be calculated
using the colloidal suspension simulator.

 

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LIST OF SYMBOLS

Hamakcr constant

Electric point charge

Electronic charge = 1.60217733E-19 Coulomb
Dielectric constant

Coulombic, double layer force

Force due to gravity

Hydrodynamic force

Total force acting on a colloid

London - van der Waals force

Electric potential

Boltzmann constant = 1.380658E-23 J/K
Temperature

Repulsive potential energy

Charge density

Bulk concentration of positive and negative ions
Electric potential at the surface of a colloid
Electric potential at the Stem layer

Electric potential at the surface of shear, the zeta potential
Debye-Hiickel reciprocal length parameter
Concentration of ions

Bulk concentration of an ionic species

Viscosity

Electrophoretic mobility or mean

London — van der Waals constant

x position

y position

x component of velocity

y component of velocity

Valence number of an ion

viii

 

 

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INTRODUCTION

The manufacturing of an intermetallic matrix composite using a process based on
the ideas of colloid science has been proposed [1]. The proposed process involves electro-
phoretic deposition of Fe-40Al matrix particles onto a bundled A1203 fiber. The bundled
fiber is pulled through a concentrated suspension of the particles, Figure 1.1, and the parti-
cles adhere to the fiber due to adhesive, London - van der Waal and Coulombic forces. A
major advantage of this technique over other production methods is the uniformity of
matrix particles covering the fiber. The uniformity produced by this process results in a
composite material that theoretically has improved mechanical properties and has greater
resistance to fatigue.

To understand the electrophoretic deposition process, investigation into the field of
colloid chemistry is required. Physically, the colloidal domain is the size range of particles
that lie between one nanometer and one micron. The domain of colloid chemistry lies
between the microscopic size range, in which the strong and weak nuclear forces dominate
and the macroscopic size range, in which gravitational forces dominate.

A complete description of the dynamics of the colloidal electrophoretic deposition
process involves two steps. The first step is the transport step in which the colloid particles
are transported through the suspension medium and come into contact with each other or
with the collector. The second step is the surface interaction step in which the particles are
close enough to each other or to the collector such that surface interactions occur. Inquiry
into the nature of the colloidal interactions between the surfaces of the colloids leads to the

classical theoretical description developed independently by Derjaguin and Landau, and by

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suggests that the stability of a colloidal suspension is determined by the total surface inter-
action energy acting between the colloids. The total surface interaction energy possessed
by a colloid is the sum of the electrodynamic attractive energy plus the electrodynamic
repulsive energy. The electrodynamic attractive energy is the direct consequence of the
London-van der Waal attractive force. The electrodynamic repulsive energy arises from the
Coulombic electrostatic force. Other possible forces that may also be present in a colloidal
system will be described in greater detail later in this thesis.

Several variables dictate the quality and efficiency of the electrophoretic deposition
process. Changes in the processing pH level, the initial electrolyte concentration, the parti-
cle size or the dielectric and Hamakcr constants for different types of materials can lead to
a composite that is more or less uniformly distributed than a composite that has been pro-
cessed with different starting conditions. This situation and the question of how accurately
the DLVO theory models the physical world has lead to the construction of a computer
model.

The computer model is the focus of this thesis. The model is an algorithm imple-
mented as a computer program and designed to provide scientific visualization of the dis-
persion behavior of the composite components while they are in suspension. The program
may be initialized with information for both matrix particles and collector and allowed to
run. For the initial conditions provided, a measurement of developed mass onto the collec-
tor over a time period may be recorded. The recorded developed mass on the collector for
a specified time period indicates a measure of the success of the electrophoretic deposition
process for a set of initial conditions. The program is flexible to allow the user to change
any initial conditions that are required. A goal for the program is that it will be able to pre-
dict optimal initial conditions that maximize the electrophoretic deposition for an arbitrary
set of composite components.

The program uses a numerical algorithm to simulate the many body problem for

 

 

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the suspended particles as time evolves. The numerical algorithm is based on a “predictor
- corrector” method for numerically integrating the equations of motion. For the “predictor”
part of the algorithm, the Euler modified method is used. For the “corrector” part of the
algorithm, the fourth order Adams — Bashforth method is used. The dynamical equations of
motion are derived by incorporating the total forces that act on the system of particles.

In order to obtain the total surface energy described by the DLVO theory, the elec-
tric surface potential of the matrix particles and the electric surface potential of the collector
must be characterized. Electrokinetic sonic amplitude (ESA) measurements were made for
iron aluminide powder and for alumina fiber. The ESA measurements provide the experi—
mental approximation to the electric surface potential in the form of the measured zeta
potential. Using the zeta potential data as an input into the computer model, analysis of the
dynamics of the electrophoretic deposition process may be conducted. Questions as to how
well the algorithm describes the physical world will be addressed in order to gain better
insight into the workings of the electrophoretic deposition phenomenon.

This thesis provides the algorithm necessary to generate a computer tool for analyz-
ing a colloidal suspension. A semi functioning computer program is included both in binary

form on a 3 1/2 inch floppy disk and in printed form in the appendices.

 

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LITERATURE REVIEW

A colloid is a particle that has at least one of its three dimensions in the size range
from 1.0 x 10'9 meters to 1.0 x 10'6 meters. The branch of science that studies these mac-
roscopic objects is called colloid chemistry. D.H. Everett [4] discusses several familiar
examples of colloidal systems including the following: fogs, mists, tobacco smoke, milk,
butter, jellies, stained glass, photographic “emulsions”, blood, paints, muds and slurries. In
general, a colloidal system is composed of a disperse phase; a gas, solid or liquid, and a
dispersion medium; a gas, solid or liquid. A colloidal system that has a liquid disperse
phase in a liquid dispersion medium, for example, is termed an emulsion. A colloidal sys-
tem that has a liquid disperse phase in a gas dispersion medium is called a liquid aerosol.
Table 2.1 outlines several types of colloidal systems.

The class of colloidal system in which a solid is dispersed in a liquid is referred to
as a colloidal suspension or a sol. For the electrophoretic deposition process under investi-
gation, the disperse phase will be iron aluminide particles and the dispersion medium will
be deionized, distilled water. Therefore, an investigation into the dynamics of a colloidal
suspension is essential in order to formulate a physical model of the electrophoretic depo-
sition process.

To attempt a complete description of the overall dynamics of a colloidal suspension,
many contributing factors need to be considered. The particle size as well as how the par-
ticle size is distributed are two such factors. Vifrese and Healy [5] found experimental evi-
dence correlating the particle size with colloid stability. The particle shape is another
important factor. Intuitively, the motion of a spherical particle will behave unlike some
Other geometrically shaped object, such as a cube or needle, while in suspension. Other fac-
tors to be considered are the particles surface properties, such as the surface composition

and the amount of electrical charge on the particles surface Finally, the primary factor con-

 

 

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Table 2.1

Everett, D. H., W, Royal Society of Chemistry, London,

1988.

Colloidal Systems.

 

 

  

 

 

Dispersion
Medium

Class

 

Examples

    

 

 

 

 

 

 

 

 

 

 

Liquid Gas Liquid aerosol Fog, mist, tobacco
smoke, “aerosol”
sprays

Solid Gas Solid aerosol Industrial smokes

Liquid Liquid Emulsions Milk, butter, may-
onnaise, asphalt

Solid Liquid Colloidal Suspension Silver iodide, paints

Solid Liquid Paste Clay slurries, tooth-
paste, muds

Solid Solid Solid Suspension Opal, pearl, stained
glass

Gas Liquid Foam Froths, foams

 

 

 

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tributing to the dynamics of a colloidal suspension is the total force acting on an individual
colloid particle. The total force can be derived from knowledge of the particles size, shape
and surface properties as well as from the dispersion medium properties.

Several forces collectively define the total force acting on an individual colloid par-
ticle. Electrodynamic forces include the attractive London-van der Waals force and the
repulsive electrostatic double layer force. Hydrodynamic forces, that obey Stoke ’5 law,
arise from viscosity and are proportional to the velocity of the particle moving in the dis-
persion medium. Steric forces are repulsive forces that may be present from the overlap of
adsorbed polymer layers. Brownian motion is erratic particle motion driven from fluctua-
tions in the density of the liquid. Structural forces are strong repulsive forces that act over
a very short range and result from changes in the dispersion medium structure in the vicin-
ity of the surface or interface [6,7]. The gravitational force is an attractive force obeying Sir
Isaac Newton’s law of gravitation. Lastly, the presence of an external electric or magnetic
field can affect the motion of a charged colloid particle. Table 2.2 outlines the possible
forces that may be present in a colloidal system. Figure 2.1 shows a free body diagram for
a suspended colloidal particle.

Electrodynamic forces, comprised of the repulsive double layer overlap force and
the attractive London-van der Waal force, are present only if the colloid particles have elec-
trically charged surfaces. Although some special types of colloidal suspensions exist in
which the colloid particles possess no surface charge, the majority of colloidal suspensions
do, in fact, contain electrically charged particles.

A colloid in suspension can obtain a surface charge through several mechanisms.
Many of these mechanisms are described by Ross and Morrison and by Hinze] and Raja-
gopalan [2,6] and include preferential adsorption of ions, accumulation of electrons at the
interface and adsorption of polyelectrolytes.

Realizing that suspended colloids possess a surface charge led to the idea of the
electrostatic double layer. The double layer concept, originated by Helmholtz (1879) and

 

 

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Table 2.2 Forces in a colloidal system.

The table shows the possible forces that may be present in a colloidal suspension system
and lists the variables on which the forces depend.

Russel, William B., WW3, The University of Wisconsin
Press, 1987.

 

I FORCE l FUNCTION OF

 

 

 

 

 

 

Electrodynamic
- London - van der Waals Displacement, Material, Surface Charge
- Electrostatic (Double Layer) Displacement, Material, Surface Charge
Hydrodynamic
- Stokes Drag Viscosity, Velocity
Steric Displacement (Position of adsorbed polymer
layers)
Brownian Thermal Motion (Density Fluctuations)
Structural Displacement
Gravitation Mass, Displacement
External Field
- Magnetic Surface Charge, Displacement, Field Strength
- Electric Surface Charge, Displacement, Field Strength

 

 

 

 

 

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This figure shows possible force vectors acting on the colloid particle, where,

FDbl is the double layer force,

FSter is the steric repulsive force,

F3, is the force due to Brownian motion,

Vel. is the velocity vector of the particle,

FStruc is the structural force,

FLon is the attractive London - van der Waals force,
FGrav is the force due to gravity,

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investigated in greater detail by Louis George Gouy (1910) and David Leonard Chapmann
(1913), suggests that a cloud of ions gather around a suspended colloid in an organized
fashion. The fixed charge on the colloids surface attracts free ions of opposite sign, referred
to as counterions. The attracted counterions, in turn, attract ions of the same sign as the ions
on the surface of the colloid, called coions. This cycle repeats outward from the surface of
the particle creating integrated layer upon layer of attracted coions and counterions. Figure
2.2 shows an exploded view of the Gouy-Chapman layer surrounding a particle.

The cloud of diffuse ions that surround the colloid exactly neutralizes the fixed
charge on the surface of the colloid. Collectively, the diffuse ion region and the colloids
fixed surface charge region make up what is called the electrical double layer. In the
absence of thermal agitation, counterions would migrate to the surface of a charged particle
and would completely cover it, exactly neutralizing its charge [8]. The double layer in this
situation would be extremely compact. In reality, thermal motion has a tendency to uni-
formly distribute the free ions in the dispersion medium. As counterions are attracted to the
colloids surface, the counterions produce a screening effect that blocks further attraction of
other counterions. The combined effects of thermal agitation and screening of ions causes
the charged ions in the diffuse region of the double layer to have a distinct distribution.

Before a description of how the charged ions in the diffuse region are distributed,
other important characteristic parameters of the double layer need to be defined, namely,
the electrical potential energy and the electric potential. Potential energy, in general, is
defined as the energy that a system possesses as a result of its configuration. In the case of
the electrical double layer, ions and counterions that surround the colloid may be assumed
to be electric point charges. These elecu'ic charges are separated by varying distances and
exert Coulombic forces upon each other of the form:

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Figure 2.2 Gouy - Chapman Double Layer.

This figure shows an expanded view of the surface of the colloidal particle. The
ions and coions migrate to the surface as shown.

 

1;»: a: ma: PC
“”594 to file“ 5

.w-‘e‘u .

1.2135236 eieCf—‘T

m z is the Va
union (2,. desc
time potenual.

As outline
ifise pm of the
tier: particle rel

river -
trial potentia.‘

12
where, q and qj are two electric point charges, a is the dielectric constant of the medium,
and r is the distance between the charges. As a consequence of these forces, an electric
potential scalar field exists between the charges. It is important to note that an electric
potential, w, is associated with each point in space, whether or not there is any electric
charge at that point. A change in the electric potential is equal to the amount of work
required to move an electric charge from one point to another point. This latter statement

defines the electrical potential energy and is defined as:

VR = zqul, (2)

where, z is the valence of the ion, q is the electronic charge and W is the electric potential.
Equation (2) describes the general relation between the electrical potential energy and the
electric potential.

As outlined by Shaw [9], the Gouy-Chapman description suggests that ions in the
diffuse part of the double layer obey Boltzmann’s distribution law. Boltzmann’s law for a
colloid particle relates the probability of ions being at a given point at which they have an
electrical potential energy relative to the surface of the colloid, i.e.:

-Z.-q\v
n,- = niocxp (W) (3)

Where ni are the concentrations of positive and negative ions at points where the electric
potential energy of these ions are zqw and -zq\u, respectively, “i0 is the bulk concentration
of each ionic species, 2 is the valence number of the ions and q is the elecu'onic charge.
Now that the concentration of ions can be calculated at points where the electric
POtefltial is w, another definition, namely, the charge density, follows. The charge density,
P- at a point where the electric potential is w, is defined as the sum of the charged ions, at

that point, per unit volume. Mathematically, the charge density is:

A.
_ ... ‘ r
‘ ' .«ouhf't mflub‘
\.\..LU““: A
i ‘v

T)

Thenexts
:i; potent. ‘ in o

zaauxchz-e

"’1

“mefimemi
ill-3mm} and m6

@3330“ ('8). the

13
p = zqni (4)

Substituting equation (3) into equation (4), then,

. "2W — fl 5)
zq[n,o (exp[ kT ] exp|:kT:|):l

=_2 ,. .1. fl- WW (6)
p z...[2(..p[n] ...[HD]

. 24W
p = -22eqnioSlnh(-fi) . (7)

'0
II
A

 

 

The next step is to find another equation that relates the charge density with the elec-
tric potential in order to obtain an equation explicitly containing w. The Poisson equation

relates the charge density to the Laplacian of the electric potential as:
Vzv = 3. (8)

where e is the dielectric constant of the medium, p is the charge density, ‘l’ is the electric
potential and the inverted triangle is the gradient operator. Substituting equation (7) into
equation (8), the well known Poisson-Boltzmann equation is obtained:

Zan- 2
2 - —‘o . q_w
V w — e srnh( kT

) . (9)

The Poisson-Boltzmann equation is a second order, non-linear differential equation
for the electric potential. A solution to this equation will provide a quantitative scalar field
description for which the magnitude of the electric potential at any location in the diffuse
part of the electrical double layer can be calculated. Unfortunately, no exact analytical solu-
tion to this equation is known to exist and numerical methods must be used.

If, however, the assumption is made that the value of

 

I130 The ass;

+5 .3 fl‘n. ‘3‘!
.I;NCAC~L‘ \ r ‘5‘

:azor. simplit

i‘hm.

14
24“,
(fi— « 1 (10)

in equation (9), then the following approximation holds:

24W

24W) = (777) - (11)

Sinh ( '77-..—

Equation (11) follows from the Taylor expansion of sinh and setting the higher order terms
to zero. The assumption in equation (10) implies that at room temperature, i.e. T = 25° C,

the electric potential has the value of,
2v « 5:: = 25.69mV. . (12)

Substituting the right hand side of equation (11) into equation (9), the Poisson-Boltzmann

equation simplifies to,

V21) = sz, (13)

where, 222 2n-
x2 = ———q ‘ON” , (14)
ekT

\V = the electric potential,

K = the Debye - Hiickel length,

2 = valence number of ions,

q = the electronic charge,

q = 1.60217733 E—19 Coulomb,
“10 = the bulk concentration of ions,
nio = ni0(1000)(mole/meter3),

N A = Avagadros number,

N A = 6.0221367 E+23 ( 1/mole),

e = dielectric constant of the medium,
8 = £02,,

£0 = permittivity of free space,

so = 8.8541878 E-12 (Farad/meter),
e, = the relative permittivity,

k = Boltzmann constant,

Tie ass-terrace. us
Tribe-Hui:
:0: 121V < :5.
32m: reciproc 212
it a}: The :52.

wrist: the e.-ec

To traders

5 972.3; out

F0? ‘3ng

"Tier _ .
S'V01$m<

3:11:

CleCm'c DOLC

A fume
mentioned abox

f .
Amn

U

15

k = 1.380658 E-23 (Joules/Kelvin),
T = Temperature in Kelvins.

The assumption used to derive equation (13) is known as the Debye-Hiickel approximation.
The Debye-Hiickel approximation is valid only for small electric potentials, i.e. from equa-
tion (12) \y < 25.69 mV. The constant term K in equation (13) is defined as the Debye-
Hiickel reciprocal length parameter and is an indicator of the thickness of the electrical dou-
ble layer. The thickness of the double layer, UK, is the distance in the diffuse double layer
in which the electric potential decays by a factor of 1/q for low potentials.

To understand the geometry of the double layer, consider a spherical colloid particle
of radius, a, and the ratio of this particle radius to the double layer thickness, arc, see Figure
2.3 [8]. When are is large, the double layer is nearly flat When are is small, the double layer
is spread out.

For variations in the potential in the x-direction, equation (13) takes the form,

a it K2

— = w . (15)
8x2

Equation (15) is a second order linear differential equation. Letting the boundary conditions

be t]! = ‘l’o at x = O and u! = 0 at x = infinity, and assuming low potentials at room tempera-

ture, the solution to equation (15) is,

where, Va is the electric potential at the surface of the particle. Equation (16) shows that
the electric potential decays exponentially from the surface of the colloid.

A further attribute of the electrical double layer was introduced by Otto Stern. As
mentioned above, thermal agitation prevents counterions in the suspension medium from

forming a very compact layer surrounding the charged colloid. If, however, the electrostatic

16

 

 

arc=1 ax=0.5

arc=10 arc=2

 

 

 

Figure 2.3 Ratio of the particle radius to the double layer thickness.

The magnitudes of art vary in this figure. Most ceramic colloids have magnitudes of
at: in the proximity of 50 to 100.

Myscls. Karol 1.. Wmnmmemim Robert E. Kricgcr Publishing

Company, Huntington, New York, 1978.

 

.5.b

Tue ClCCC
.,..' 1
3:: 30151.53; El
vvéa S 1'2 ' '
L hl5 m ib}c
Era-«M A. ans
w*...-..€b 11511.5
had by ele,
IG‘Q 1 A I q ‘
.-._.g ere-.tm i“-
t‘rrces a force t
itl‘lly. A laser
Verifies.
l
The CICC'
dentude of rh

17
forces are too strong near the colloids surface, then thermal agitation will not be able to
overcome them. The result is a semi—compact layer of counterions surrounding the colloid

called the Stern layer.

The new picture of the electrical double layer in terms of the elecuical potential is
shown in Figure 2.4 [8].

The electric potential at the plane of shear near the Stern potential is defined as the
zeta potential. Exactly how close the zeta potential lies in relation to the electric potential
at the Stern layer is a topic of current research. The zeta potential can be experimentally
determined using various techniques such as micro-electrophoresis or acoustophoresis as
measured by electrokinetic sonic amplitude (ESA). Micro-electrophoresis applies an oscil-
lating electric field to a colloid suspension. The presence of the external electric field
induces a force that acts on each colloid causing the colloid particles to move with a certain
velocity. A laser beam and a detector are used to optically measure the colloid particles
velocities.

The electrophoretic mobility is determined by dividing the observed velocity by the
magnitude of the elecuic field. The electrophoretic mobility is directly related to the zeta

potential by the following equations [2]:

 

 

6

g: “2“ whenaK<0.1 (17)
4

g = in“ when arc>100. (18)

Where, C is the zeta potential, 1] is the viscosity, p. is the electrophoretic mobility and e is
the dielectric constant Equation (17) is known as the Hiickel equation and equation (18) is
knOWn as the Helmholtz - Smoluchowski equation.

The acoustophoresis technique using an electrokinetic sonic amplitude (ESA) mea-
sur<38 the zeta potential by applying a one megahertz oscillating electric field to a colloid

suspension. Again, the particles move with a certain velocity due to the effects of the elec-

 

1“,-
3+
'+
1++
+ -

+

( ‘nlluid Surface
+

+
‘T
1+

 

 

+
+-
+
§
1"“

 

 

lik‘t‘n'u.‘ l 'nu-nliul

 

 

 

We 2.4 n“
TrQSfi Sho
1:1 . V)
5:: .nlC [OnS an
“PC of 0C3

18

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

+
+ +
+ -
+ _ -
i3 + ' + Positive Ion
g + ' - Negative Ion
E + - +
"o‘ - : . .
U + + g + W0 = Electric potential at the
+ _ - + - surface
+ +
+ - - . .
3 Vs = Electric potential at the
—> <— Stern layer Stern layer
“’0 Plane of shear t; = Electric potential at the
surface of shear,
the zeta potential
'g “’3 _1
8
a.
'é’
.13. C
:1:
Vs
° § 1h: h
>
(1

Distance from the Surface

 

 

 

l:“igure 2.4 The Stern electrical double layer.

This figure shows the small scale surface of a colloid particle, represented as a vertical
11118. The ions and coions are depicted as + and - symbols. The lower graph shows the
shape of decay of the electric potential moving away from the surface of the particle.

Mysers, Karol 1., W Robert E. Krieger Publishing
Company, Huntington, New York, 1978.

 

r5]; ficid. AS 1
aressure vl aw
one deteetoi

may be deter?

zer: porenual
“he:

res;lt is an el
energy heme
the potential 1
of concentric.
(ELF) [ll]. S
Local potentia

tion energy be

l9
tric field. As the colloid particles vibrate back and forth in the suspension medium, a sonic
pressure wave is produced. The frequency of the sound wave is measured by a sensitive
sonic detector. Once the frequency of the sound wave is known, the velocity of the particle
may be determined as well as the electrophoretic mobility. Using equations (17) or (18), the
zeta potential is then obtained.

When the diffuse regions of two double layers surrounding two colloids overlap, the
result is an electrical potential energy of interaction. Derjaguin [10] derived the potential
energy between two parallel plates of unequal charge. Using the calculations of Derjaguin,
the potential energy between two unequal spheres was determined using a summation idea
of concentric, parallel plates. This method was employed by Hogg, Healy and Fuerstenau
(HHF) [12]. Starting with the electric potential and relating the electric potential to the elec-
trical potential energy, Hogg, Healy and Fuerstenau [12] obtained the electrostatic interac-

tion energy between two dissimilar spherical particles,

 

 

_ 30102(\l’(2)1+‘l’(2)2) [[ 2W01‘V02 J (1+ exp (—KH0)

- +1 1— -2 H
R 4(dl-l-02) “[31+ng l-CXP(_KH0)) n( exp( K (9)]

(19)
where, e is the dielectric constant of the medium, a1, a2 are the respective radii of particle
one and particle two, v01, rpm are the elecuic potentials at the surface of each particle, K is
the Debye-Hilckel reciprocal length parameter and H0 is distance between the two parti-
cle’s surfaces. When the initial conditions for two spherical colloids in suspension are spec-
ified, equation (19) has the displacement between the two particles surfaces as its only
independent variable. HHF [11] also show that equation (19) is a valid approximation for
surface potentials less than approximately 50 to60 mV. Other methods for obtaining the
potential energy relation between two dissimilar spherical particles are given by Bell,
Levine and McCartney [12] and by Bell and Peterson [13].

The next topic to consider is the London - van der Waals attraction force between

 

ccloit‘l pamcies. .
' n 4 r
h Marion a... .
The conce
3:: Elm mole;
“.1: dtVClOpcj
4","‘1‘ “.3993 t
nun-5L1 . ‘~U
iot- energy of a g
W0 mOICCUECS' '

iticlgpcd bV I..(

35.3: A is the
cries. The cal cu
titles has bCCn
my ”Mel of
Shaded 0n Pal:
COHOid pmfle
lemmas and r1

Hawker mOde

20

colloid particles. A brief summary of the evolution of the nature of this force is described
by Mahanty and Ninham [14] and is outlined below.

The concept of a force field existing between any pair of molecules whose range is
larger than molecular dimensions was first investigated by van der Waals in 1873. van der
Waals deve10ped an equation of state for a gas in which a constant term appeared that was
directly related to the strength of the intermolecular forces. By averaging the interaction
between two dipoles over all orientations, van der Waal and others found that the interac-
tion energy of a dipolar molecule was proportional to l/r6, where r is the distance between
two molecules. The explanation of the force between a pair of non-polar molecules was
developed by London in 1930. The attractive interaction energy between two molecules
due to the London - van der Waal force was determined to be:

E (r) = -— (20)
where, A is the London - van der Waals constant and r is the distance between the mole-
cules. The calculation of the electrodynanric attraction force between two macroscopic par-
ticles has been approached by two different methods. The Lifshitz model is based on a
molar model of condensed media and uses quantum electrodynamics. The Hamaker model
is based on pairwise summation of the attractive energies between the molecules of each
colloid particle, ignoring multibody perturbations. Due to the complexity of the Lifshitz
formulas and the necessity for numerical methods for determining material functions, the
Hamaker model will be considered.

Integration of equation (20) over the total volumes of two colloid particles provides
the potential energy of interaction between two particles containing q atoms per cm3 and is
given by Hamaker [15] as:

I:
_<n_'..

dvj 9-?de , (21)
V2

 

tiff!- d‘l‘ dy} \

' 0F
'. l-
T'S'ciulcil' r 15 '“

The sphere

attic worn: P w:

r

“left. 60 is gix-cn

“533.21% mark

The yOlUmc ClCmc

The POtentiaI Cner

21

where, dvl, dvz, V1 and V2 are volume elements and total volumes of the two particles
respectively, r is the distance between dvl and dvz and A is the London - van der Waals
constant. Hamakcr derived an equation for the attractive interaction energy between two
spherical particles. His derivation begins by investigating a molecule near a sphere.
Consider a sphere of radius a1 with center at point 0 and a point P at a distance OP = R>a1.
(Figure 2.5).

The sphere around 0 cuts out a surface, S(ABC) out of a second sphere centered

around point P with radius r. The surface S(ABC) is:

00
5 (ABC) = 2x I rzsinOdO . (22)
o
where, 60 is given by the law of cosines:
a1 = R2 + r2 + 2chosOo . (23)

Integrating equation (22), the surface ABC is:
S(ABC) = 112% (of- (R-r)2) . (24)
The volume element dvl is given by:
dvl = S (ABC) dr . (25)

The potential energy of a molecule at P may then be written as:

(R+afi
A
EP=— I %(%)(a§-(R—r)2)dr. (26)

(R-q)r

 

 

 

 

 

Figure 2.5 3

This fig“ do;
mint 0.

”maker. H. c
IlOI‘i BCIWCCn S

22

 

 

Figure 2.5 Molecule near a sphere.

 

This figure depicts a molecule at point P at a distant OP from a sphere centered at
point 0.

Hamakcr, H. C., Physica 4, 1058 - 1072, 1937, “The London - van der Waals Attrac-
tion Between Spherical Particles”.

 

 

 

 

Cflf.’

Tie pom: al

£23111“. centers be

nods
l4 : -A I
o 6 . I

r‘r -
Shires.
The P016:

L f

23
The potential energy of interaction between two spheres, the second sphere having radius
a2, with centers being a distance C apart is obtained from the following:
(C + 02) R
VA = 1 mo? (ag— (C-R)2)dR. (27)
(C - 02)
The result of this integration for the potential energy of interaction between two spheres

yields,

200 200 Cz—(a +a)2
vA=-é(2 ’2 +2 ‘2 +ln(2 1 2 D (28)
6 C -(al-t-c12)2 C -(al—az)2 C -(al—az)2

 

 

 

where A is the Hamakcr constant and C is the distance between the centers of the two
spheres.

The potential energy of interaction for a sphere and a plane can be calculated by let-
ting one of the sphere’s radius go to infinity and has the form [14]:
x

1
+§-+—x+IN(f+—x)) 9 (29)

(

CM):-

1

where,
_ (C‘al)
_ —0_1—— ,

x (30)
A is the Hamakcr constant, C is the distance between the center of the sphere and the sur—
face of the plane, and a1 is the radius of the sphere.

The development of the potential energy equations due to both the double layer and
the London - van der Waals force will be used to determine the dynamics of the surface
interaction step of the electrophoretic deposition process. The transport step dynamics
involve the forces listed in table 2.2 that may be present in the system, excluding electro-

dl’narnic forces. The dynamical equations of motion and the numerical algrithm that may

24

be implemented on a computer will be derived in the following chapter.

 

To cumin
eecijed in order
son will Clarify \s
at: Ht 1.1565 10 (it:

Because rl
tric forces that
be included. Alrhl

he“
a»

. .ngie. for an

'7')

combined ms 0
motive force u
of no electrical 1

For the F
the repulsive sre
Finally. external
obtained from th

Other ini-

hiding

Vie-4 (T—I

91—1 ’13—: m

FORMULATION OF EQUATIONS

To examine the dynamics of a colloidal suspension, initial conditions need to be
specified in order to isolate what forces will be present in the system. The following discus-
sion will clarify what forces will be used for the derivation of the equations of motion that
will be used to describe the dynamics of the colloidal system.

Because the suspended particles will be moving in a viscous medium, hydrody-
namic forces that obey Stoke’s law will be included in the system. Gravitational forces will
be included. Although the magnitude of the gravitational force on an individual colloid is
negligible, for an unstable system in which the colloids flocculate to form agglomerates, the
combined mass of several particles can lead to sedimentation. The London - van der Waal
attractive force will be included. The electrostatic repulsive force arising from the overlap
of two electrical double layers will be included as well.

For the FED process, no polymer chains will be added to the system and therefore
the repulsive steric force will not be included. Brownian motion will not be included.
Finally, external magnetic or electric fields will not be present and so the resulting forces
obtained from these fields will not be included.

Other initial conditions that will be imposed on the colloidal system include the fol-
lowing:

1. The colloidal particles will be assumed to be spherical in shape and insolu-
ble.

2. The surface of each particle will be assumed to have a constant charge den-
srty.

3. Each spherical particle will be assumed to be infinitely hard and smooth.

4. The zeta potential will be used as the numerical equivalent of the surface
potential in calculations.

5. The frame of reference used to specify the particles coordinates will be
assumed to be an inertial frame of reference.

25

 

. ti fir
Tn: alt-3“

1"”1‘3'1 15 {1‘51

rec. in is "re
“ : one
:' ‘ Lhe Le .
332:} L
‘ '- l
driganc Fg is t.

‘ 4,. tr
Tie :orce one .t

there. r is the ;

densities and g

“‘bm~ 1'l is the

Velma;

The CICCtrod .Vn

What F iS Lhc |

dient operamn ‘

26

The equation of motion for an individual spherical, colloidal particle suspended in a
medium is given by Newton’s second law of motion,
di) .5 A J A A
"13-,- : FDBL+FVan+FHYD+Fg = F771,, (31)
where, m is the mass of the colloid particle, FDBL is the Coulombic, double layer force,
FVan is the London - van der Waals force, FHYD is the hydrodynamic force due to Stoke’s
drag and F8 is the force due to gravity.

The force due to gravity for a sphere in a medium is given as:
fi-4 3 * 32
g ‘ -31". (pz-p1)g 9 ( )

where, r is the particle radius, p2, p1 are the respective particle and suspension medium

densities and g is the acceleration due to gravity. The Stoke’s drag force for a sphere is:
73mm = 61mm? . (33)

where, 11 is the suspension medium viscosity, r is the particle radius and v is the particle’s
velocity.

The electrodynamic forces can be obtained from the following relation that is valid for a
conserved system:

F=—VU, an

Where F is the total force, U is the potential energy equation and the dc] symbol is the gra-
dient operator. Applying equation (34) to the potential equations, (19) and (28) the follow-

ing electrodynamic force equations are:

;.
Th: $831115“: H.

‘ r ‘. -
Fhs' LPP. '
b o

35:76.

"the attractive

Where.

27

The repulsive force between two non-identical, spherical particles is:

 

)(1-exp(-KH))

2 rtexp(-KH) _(l+exp(—rtH))kexp(-KH)
9‘ ( “am-K”) (hem-«1m2

 

 

 

‘ P =—s s 35
PM [P l ‘ (ef+e§)(1+exp(-KH)) + 2 ( )
where,
_ £r1r2(g§+g§) 36
1 " 4rl+4r2 ( )
2xex -2xH
2 = p( ) . (37)
1 - exp (-2KH)
The attractive force between two non-identical, spherical particles is:
_, A r2T2 r272 271 T2 T3T2
Fm, PP =—-——-— — —-— 38
V I l 12( '17:; ”73+ Ts (T1 T} D ( )

where,
H2 ”'2 H "2

T =—+—+—+— (39)
1 4r? 2r? 2”1 r1
H ’2 l
T =_+_+_ (40)
2 2r? 2r? 2"1
2 Hr
T3: H +—3+” (41)

4r? r? 2'1

and H is the distance between the two particles surfaces.
The system of n colloidal particles produces N equations of motion, one for each particle,
Where N is an integer.
The many body problem involves determining the position and velocity of each par-
ticle as time progresses, provided that the initial position and velocity of each particle are

k“OWn. A general solution to the many body problem for N greater than three is unknown

flllfflelCélr r

.
a»

n
\I

CCSS‘

iii: tilt DC

'““= SVSICTI‘L.

.31“-

‘ l
Deane L

r
P

A

((folarllrlal.l|l

11243456700

((I‘l'

28

and numerical methods must be employed. The flow chart presented in Figure 3.1 shows

the basic process used to solve the many body problem. The following algorithm will pro-

vide the necessary steps that outline a solution to the many body problem applied to a col-

loidal system.

Many body problem Algorithm:

1. Define the initial values of the colloidal system.

A-Eanifilzinfmmamm

Common Name

Chemical Name

Shape

Density

Diameter mean

Diameter Standard Deviation
Number of Particles
Hamaker Constant

”\IO‘MhWNr—n

E

Common Name
Chemical Name
Density

Viscosity

Relative permittivity
Temperature
Hamakcr Constant

\lOsUrthv-e

CW

1 Common Name
2 Chemical Name
3 Concentration

DQQIleatsrLInfotmatiaa

Common Name
Chemical Name

Shape

Density

Diameter (Aspect Ratio)
Hamaker Constant

OsthNv-t

gm/cm3
um
um

Joules
Imus

grit/cm3
gm/(cm sec)

Degrees Celsius
Joules

Joules

 

l Define

1 Possible

j Coormnates
l ofPanicles

 

Calculate
Debye -
Huckel
length

3:

Define
initial
POSonn

 

 

 

Obtain
Initial
Values

1

Define
Arrays

 

 

 

 

Define
Possible
Coordinates
of Particles

i

Calculate
Debye -
Huckel
Length

1

 

 

 

 

 

 

 

 

Define
Initial
Position

and Velocity

 

 

 

I

Assign
Radii to
Particles

l

 

 

 

 

 

Calculate
Mass and
Volume
of each
Particle

 

 

 

 

Figure 3.1

29

 

Output the
Initial Position,
and Radius for
the zero State

 

 

 

 
  

Loop for the
Number of States

 

Calculate
Distances
Between
Particles

 

 

 

  
  
 
 

If
Particles
are in Contact
With each
other

  

 

Calculate
van der Waals
Force

__'__.

 

 

 

   
  
   
   
 

 

 

Calculate
Hydrodynamic
Force

 

 

 

 

i

 

Calculate
new position
and velocity

due to the ‘—

forces

 

 

V

 

Output

new position
and radius for
each particle

 

 

 

V

Increment
the time
step AT

 

 

 

 

 

 

 

Check for
Collisions

 

l...—

 

Calculate
Double Layer ‘
Force

 

 

Calculate
Gravitational
Force

 

 

 

Flow chart for solving the many body problem.

This figure begins in the upper left comer and progresses through the necessary steps
recIllired to solve multiple colloidal bodies interacting with each other.

 

.0
9
l

l

'JJ.J.-"‘

:11
u—- r1;

All
number
ticle de

1.H
Suspeng
to filua
aSpher
thWec

panic}:

VEnwc)
knda]]
"search
indjvid
“tree m

30

E. W LLLni s
1 pH Level -
2 Zeta potential of the particle mV
3 Zeta potential of the Collector mV
F. W Lani
1 AT seconds
0 mull/1.93121.
1 DLVO
2 Acid / Base
3 Random
H. Simulau’oaim

1 Particle A - Collector
2 Particle A - Particle B

All of the initial values will be referenced throughout this algorithm by the step
number proceeded by a letter proceeded by an index number. For example, the par-
ticle density will be referenced by l.A.4.

l.H describes a simulation type. Simulation l.H.l involves colloidal particles in
suspension interacting with a plane-shaped collector. Equation (34) may be applied
to equation (29) to obtain a force equation for the van der Waals attraction between
a sphere and a plate. A similar formula may be obtained for the double layer force
between a sphere and a plane. Simulation l.H.2 involves the interaction of colloidal
particles of type A collecting onto particles of type B.

l.G are the possible dynamic models available. 1.6.] is the Derjaguin, Landau,
Verwey and Overbeek model. 1.G.2 is an acid/base model that claims that when col-
loidal particles come into close proximity to one another the particles coagulate
regardless of what forces may be present. 1.G.3 is a random model that allows the
individual colloids to sample from a time dependent force distribution. Each of the

three models affth the magnitude of the electrodynamic forces.

—J

‘ xi
)1

do‘dt

dygiit

de/dr
dry/dz

DlSlanCe

Define the

“65 will tx-

 

Define the necessary arrays to hold the pertinent information for each colloidal par-

ticle.

Let N be the number of particles obtained from (l.A.7).
Amalia:

Density
Mass
Radius

Volume

dy/dt

dvx/dt
dvy/dt

Distance

imnin

le
le
le
le
le
le
le
le
4xN

4xN

4xN

4xN

NxN

NxN

le
le

Comment
Particle Density
Particle Mass
Particle Radius
Particle Volume
x Position

y Position

x Velocity

y Velocity

The four previous values
Of in

The four previous values
Of Vyi
The four previous values

of acceleration.

The four previous values
of acceleration.

The distance between par-
ticle i and particle j

The total force acting on
particle i due to particle j

The x component of Fm
The x component of FT'I'L

Define the display dimensions and the possible coordinates where the initial parti-

cles will be placed.

32

The display size may be sized as desired but for this thesis the following values will

be used:

Display Dimensions: 150 units x 100 units
Maximum Number of Particles: 120

Maximum Particle Diameter: 2 units

Define an 8 row by 15 column grid centered in the middle of the display. Let each
row and each column be separated by 3 units. The 8 row by 15 column grid defines
120 boxes, each box having dimension 3 units by 3 units.

Figure 3.2 shows the graphic display that will be used to visualize the suspended
colloidal particles.

To prevent the particles from overlapping one another, each particle is initially posi-
tioned at the center of an unoccupied box in the grid.

Calculate the Debye-Huckel length using equation (14):

22
22 n
K2=_4_°_’Xe,

ekT (42)

Define the initial position and velocity of each particle at time t = to.

Xxiao) = xxio’

xyi(t0) = xyioa
in(to) = ijo.
Vyi(to)= Vyio.

Assign a radius to each particle assuming that the particle diameters obey a Gauss-

ian distribution.

To calculate a gaussian random number from two uniform random numbers do the
following:

Obtain the Diameter mean, u, from l.A.S and obtain the Diameter standard devia-

 

 

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34
tion, 0, from l.A.6. Let 111 and uz be two uniform random numbers whose values

range from 0 to 1. Define the parameter G as:

G = -2 x logul x cos (21tu2) . (43)
Assign the radius to be:
ri = u + 00. (44)
Calculate the Volume and Mass of each particle.
V, = gm? , (45)
"'1' = ini r (46)

where, Vi is the volume of the i-th particle using the radius from equation (44), mi
is the mass of the i-th particle, pi is the density of the i-th particle obtained from
l.A.4 and i is an integer, i.e. i= 1, 2, ..., N.
Assign the initial value conditions for each particle.
(dXIdt)0,i = ini (dvx/dt)” = O,
(dy/dt)o,i = vyi, (dvy/dt)oj = 0
Output the initial x, y positions and the corresponding radii.
Calculate the state of the colloidal system at each time increment The state of the
system is determined by calculating the position and velocity of each particle at a
specified time. The time between states is incremented by At obtained from 1.F.
Loop until the number of states have been generated. (Each loop through the fol-
lowing sub-algorithm will generate a state of the system.) Generally, 30 states
are needed for one second of animation.
Let j = 1
Loop from i = 1 to N
If j 0 i then calculate the total distance between particles i and j.
(D), = Refresh (icy-.15..)2 (47)
If the distance between particle i and particle j is less than 6 units then

 

Calculate the relative Hamaker constant

”-
»-....-..

 

 

 

 

35
A123 = (Alf-435) (Ali—AS?) (48)
A123 is the relative Hamakcr constant of substance 1 and 3 that are
separated by substance 2. All is the Hamakcr constant for particle
A, A33 is the Hamakcr constant for particle B and A22 is the
Hamaker constant for the medium.
Calculate the London van der Waals force between particle i and
particle j using equations (38), (39), (40) and (41).
If the distance between particle i and the collector is less than 8
units then
Calculate the London van der Waals force between particle i and
the collector.
Calculate the dielectric constant of the medium using 1.B.5, the
medium permittivity and the following equation:
Dielectric Constant = (1.13.5) x (8.854187799 x 10'”.
Use the zeta potential data from (1.E.2) and (1.E.3).
Calculate the double layer repulsive force between particle i and par-
ticle j using equations (35), (36) and (37).
If the distance between particle i and the collector is less than 8
units then
Calculate the double layer repulsive force between particle i and
the collector.
Calculate the total force acting on particle i due to particle j.
75711. = 1“} Van + 1"; DBL (49)

Calculate the x and y force components on particle i:

(50)

. ”M- -2

36

where D is the distance between the particles.

M (51)

F - = F x
r T71,- (D)ij

y
Else if the distance between particle i and particle j is equal to It + rj then
Use the following elastic collision algorithm:
For a perfectly elastic collision, the coefficient of restitution, e = 1.
The magnitudes of the velocity are known from in and vyi. The
. direction of the velocity may be obtained from xxi, xx“, xyi and xyi,
1. Define a vector, n, that is normal to the centers of the two particles.
Define a tangent vector, t, that is perpendicular to n. Resolve the
velocity into components along the t and n vectors.The impulse
forces are directed along the vector n. The t components of velocity
are unchanged after the collision. Use the following two equations

to determine the new direction and magnitudes of the velocity after

the collision [l6]:
m;(V,')n+mj (Vj)n = m;(V,P)n+mj(VJP)n (52)
(v5),- (vi), = e((v,-),,- (van) (53)

Add the force due to gravity using equation (32). (Note that the force of

gravity acts only on the y component.)

..‘I A

4 .8
F,.- = F,1-1— 1-3nr3(p2—p,)g) (54)

where p2 is the particle density from (l.A.4) and p1 is the medium den-
sity from (1.8.3).

37

Add the Hydrodynamic force obeying Stoke’s law using equation (33).

A

in = fixi—l ‘ (61t11r,‘71,-_1) (55)

'11
II
in

yi ' (6“n’rl’yi- 1) (56)

Use the Euler modified method to find the first three values of position

and velocity.

 

Ath
x“. - xx, 1+At(vm_.1+——2m"") (57)
Athy.
xyi - xy,_ l+At(vy‘_ l+ 2m; y') (58)
F
v“ = v 1+At(;x ii) (59)
v - +At(fl:) (60)
yr vyi- l m
dx _
(E)k,i - vxi (61)
dy _
(2,7)“ — v,. (62)
dvx in
a ..- ' (‘7) ‘63)
dv F .
_J' = 1
dt )k,t' (mi) (64)

where k =1, 2, 3.

 

38
Use the fourth order Adams - Bashforth method to find the new values

of position and velocity for the remaining particles.

{ti—1+; 4(.—55( (1:3) i-5.9(a—t dxz) i+3.7(2- (“)1 i.CT-09( dx)0 .') (65)

xi-

1: - =1

yr yi— 1+2A4

_ dy dy dy
(5.5(— f)“ 5‘9(H?)2,+37(d_t)1,’ 0'9(E)o,1) (66)

V-=V

5.5 59 dv" 37 dv‘ 09 dv" 67
x; xi- 1+2—4( (d—v :3)’i— ' (a )2,i+ ° (E)l’i— ' (E )O’J') ( )

dv (IV
'.::(55( :y) -5.9(—’) +3.7(—’) -0.9(—

‘9": ye 1+A_2.4 3,,- dr 2,,- dt 1,,- dt v50 ,) (68)
Preserve the past three values of va and vyJ.
Loop from k = 0 to 2
(dXIdt)k J = (dx/dt)k+1J (69)
(dY/dt)k,i = (dY/dt)k+l,i (70)
(dvx/dt)k J = (dvx/dt)k+1 J (71)
(dvy/dt)k J = (dvy/dt)k+1 J (72)
End the loop
Set the new values
(ax/down. (73)
(dy/dt)3 J = vyi (74)
(dvx/dt)3 J = (in/mi) (75)
(dvy/dt)3J = (FyJ/mJ) (76)

EndtheLoopfromi=1toN

Output the new position and radius for the new state of the colloidal system.

39

Count the number of particles that have accumulated onto the collector.

End the Loop for the number of states.
Display the calculated states and display vital data to the display.

The above algorithm provides the necessary instructions to visualize the colloidal
system on a computer. The following chapter describes the implementation of the algorithm

on a personal computer.

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IMPLEMENTATION

The algorithm outlined in chapter three was implemented on an PC - clone type
computer utilizing the Intel X86 series microprocessor chip. The program was written in
Microsoft Quick Basic version 4.5 [17] and was linked with an additional graphics library
[18]. Requirements for the program are only a color video graphics adapter (VGA) monitor
and approximately 200 kilobytes of memory.

The computer program is actually composed of three sub-programs. The first sub—

pro gram is the human-machine interface consisting of several pop up menus. The pop up
menus allow a user flexibility to change the initial values defined in step one of the algo-
rithm outlined in chapter three. The second sub-program consists of the algorithm defined
in step ten in chapter three. This program does all of the calculations to generate the new
states of the colloidal system. The third sub-program provides the graphic display allowing
scientific visualization to take place.

The names of the three sub-programs are respectively, CSS, CSSRUN and CSS-
DISP. The acronym CSS stands for colloidal suspension simulator. Figure 4.1 shows the
communication links between the three sub-programs.

The following discussion will provide instruction for using the colloidal suspension
Simulator software. The floppy disk included with this thesis contains the following infor-
mation:

1. BIN: This directory contains the executable files that run the soft-

ware. The names of the files residing in this directory are:
A. CSS.EXE

B. CSSDISPEXE

C. CSSRUNEXE

2. INCLUDE: This directory contains the include files that hold common

40

41

 

CSS
Colloidal Suspension
Simulator
Menu Interface

 

 

 

 

 

 

 

 

 

 

 

CSSRUN CSSDISP
R al 1 ti Display and
un c on a ons Animate
A
DATA
FILE

 

 

 

 

Figure 4.1 Communication links between the three sub - programs.

This figure shows the direction of flow between the CSS, CSSRUN and CSSDISP

sub - programs.

 

42
information and initialization parameters that are needed to
run the software. The names of the files residing in this direc-
tory are:
A. CSSCOM.INC
B. CSSINITJNC
C. MENUPARJN C

3. SOURCE: This directory contains all of the binary computer source
code for use with a Quick Basic compiler. The names of the
files residing in this directory are:

A. CSS.BAS
B. CSSDISP.BAS
C. CSSRUN.BAS

4. TXT: This directory contains all of the human-readable, ASCII
computer source code listings of the software. The names of
the files residing in this directory are:

A. CSS.TXT
B. CSSDISP.TXT
C. CSSRUN.TXT

The files necessary to run the software reside in the BIN directory. To install the
software on a computer, the three executable files need to be copied from the BIN directory
to a directory on the computer’s hard drive.

To run the software, enter the command CSS. A menu will appear entitled, “Colloi-
dal Suspension Simulator” with a second menu entitled, “Main Menu”. To select any of the

options listed in the main menu, the high-lighted letter or number need only be pressed. By

43
pressing the letter “H”, for example, the help menu for the main menu will appear.

The main menu allows for the choice of a simulation type, to save or load a config-
uration and to display a previously calculated simulation. The first simulation involves the
interactions between two types of particles, i. e., between particles of type A and particles
of type B. The second simulation type involves the interaction between one type of particle
and a fiber.

A configuration is the set of current values that describe information about the par-
ticles, the suspension medium, the fiber, the electrolyte, the surface parameters and the
choice of a dynamic model. When the program is run for the first time, a default configura-
tion is loaded. The default configuration is shown in table 4.1.

Once a simulation type is specified by pressing “l” or “2” at the main menu, a menu
entitled either, “Particle A - Particle B” or “Particle A - Fiber” will be displayed. This menu
will be referred to as the simulation menu here after. In either case, the selections for the
simulation menu allow for the modification of material parameters, the modification of sur-
face parameters, the choice of a dynamic model and a gateway for running the CSSRUN
sub-program.

By pressing “2” at the simulation menu, the “Surface Parameter” menu will appear.
This menu allows the pH of the medium to be changed and allows for the input of the zeta
potential data for each particle or fiber.

By pressing “3” at the simulation menu, the “Dynamic Model” menu will appear.
The three possible model types to run are named, DLVO Theory Model, The Acid/Base
Model and the Random Model. Only the DLVO model is implemented at this time. The
DLVO Theory Model uses the theory stating that the stability of a colloidal suspension is
based on the sum of the electrostatic repulsions due to the overlap of electrical double lay-
ers plus the attractive potential due to the London - van der Waals forces.

By pressing “4” at the “Dynamic Model” menu, the time increment can be modi-

fied. The time increment specifies the time interval between the calculations of each state

...__.._

Table 4.1

E . l I E .
Common Name

Chemical Name

Shape

Density

Average Initial Velocity
Mean Diameter

Diameter Standard Deviation
Number of Particles
Hamakcr Constant

E] I E .
Common Name
Chemical Name
Shape

Density

Diameter
Hamakcr Constant

Wanna
Conunon Name
Chemical Name
Density

Viscosity

Hamakcr Constant
Permeability
Temperature

1 Inf ' n
Common Name
Chemical Name
Concentration

WW
pH Level

Zeta Potential, particle A

Zeta Potential, particle B

Zeta Potential, Fiber

Time Increment
Dynamic Model

44

MM

Iron Aluminum
FeAl

Spherical

5.56 gm/cm3
30.0 urn/sec

1.0 pm

0.001

5

3.08E-19 Joules

Elm:

Alumina

A1203
Cylindrical
3.97 gm/cm3
25.0 um
1.54E-19 Joules

Mgfiigm

Deionized Water

H20
1.0 gm/cm3

0.01 gm/(cm sec)

4.35E-20 Joules
78.54

25.0 Degrees Celsius

mm

Potassium Nitrate

KNO3

0.0001 Normality

W

8.0
5.0
18.00
18.00

0.0000001 sec
DLVO

Default parameters used to initiate the CSS software.

Baniszlifi
Alumina

A1203
Spherical

3.97 gm/cm3
30.0 tun/sec
0.9 pm

0.001

5

1.54E-19 Joules

 

 

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45

of the system.

Returning to the simulation menu and pressing “1” results in the “Material Param-
eters” menu to be displayed. The “Material Parameters” menu allows for the modification

of information for particle A, for particle B, for the fiber, for the suspension medium and

for the electrolyte.

By pressing “1” at the “Material Parameters” menu, the “Particle Information”
menu appears. The following particle information may be modified:

Common Name

Chemical Name

Shape

Maximum Diameter

Mean Diameter

Diameter Standard Deviation
Density

Number of Particles
Hamaker Constant

pwflgwéwww

If the “Particle A - Fiber” simulation was chosen, then pressing “2” at the “Material

Parameters” menu causes the “Fiber Information” menu to appear. The following fiber data

may be modified:

Common Name
Chemical Name
Shape

Diameter

Density

Hamakcr Constant

QMPPNP

By pressing “3” at the “Material Parameters” menu, the “Medium Information”
menu appears. The following medium data may be modified:

Common Name
Chemical Name
Density

Viscosity
Hamakcr Constant
Permeability
Temperature

oawewwr

By pressing “4” at the “Material Parameters” menu, the “Electrolyte Information”

menu appears. The following electrolyte data may be modified:

_ _.-——-.~._-._-. -

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46

1. Common Name

2. Chemical Name

3. Concentration

Returning to the simulation menu and pressing “R”, the “Run Model” menu will
appear. An information box will appear at the bottom of the screen and will state: “Enter
the Output File Name (with no extension): The requested file name will be used to write
out the x and y positions of the colloidal particles at each time frame. The user may press
enter without typing a file name and will be prompted with “Do You “fish to Quit [N]?”.
Upon entering a valid output file name at the initial prompt, a second prompt will appear
stating: “Enter the Number of Frames to be Calculatedz”. The number of frames to be cal-
culated correspond to the number of states of the system that will be animated with the
CSSDISP sub program.

Once a valid number of frames to be calculated has been entered, the CSSRUN sub
program will perform the calculations to generate each frame. Status messages will be dis-
played to the screen to allow the user to keep track of how the calculations are progressing.
Upon successful completion of the sub program, the user will be asked to press “C” to con-
tinue. The file containing the newly calculated data MUST be written down and remem-
bered at this point. The name of the data file will be used later in the display sub program

The main menu will reappear. By pressing “D” at the main menu, the “Display
Model” menu will appear. A prompt at the bottom of the screen will appear stating, “Enter
the Input File Name (with no extension):”. The user may quit from this menu by pressing
enter without entering a file name. The input file name that the software is looking for is the
name of a file that was created by the CS SRUN sub program. If a valid file name is entered,
the “Colloidal Suspension Simulator” display menu will appear. The right column of the
screen displays the choice of dynamic model, the pH level of the medium, the zeta poten-
tials of each material, the Debye length, the concentration, the time increment between each
frame and the current frame number.

By pressing any key, the simulation will begin. The simulation can be paused by

 

 

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pressing the “enter” key or stopped entirely by pressing any other key. The up and down
“arrow” keys adjust the replay speed of the simulation, level 20 being the fastest speed and
level 0 being the slowest speed. The simulation may be replayed again and again by press-

ing the space bar.

Simulations that have been run previously may be viewed again by choosing the

desired file at the “Display Model” menu.

 

 

 

RESULTS / DISCUSSION

The algorithm derived in chapter three requires some empirical data, namely, the
zeta potential data. The colloidal suspension software requires the user to input the pH of
the medium and the corresponding zeta potentials of each particle or fiber. Zeta potential ‘
data was obtained for the interrnetallic, Fe-40Al, particles and for two types of Alumina,
A1203 - PP and A1203 PRD-166. The data was obtained at E. I. DuPont in \Vrlmington,

Delaware under the supervision of Dr. Rulon Johnson and Mr. Jerry Hughes. The zeta

 

potentials were measured using the acoustophoresis technique on an electrokinetic sonic
amplitude (BSA) device engineered by Matec Inc. The data obtained from the Matec 8000
is displayed in Figure 5.1, in Figure 5.2 and in Figure 5.3. The zeta data for the A1203 fiber
obtained at Dupont, however, does not agree with the known zeta potential data for Alu-
mina. The inconsistency with the A1203 fiber was most likely attributed to the large, dis- .
continuous fiber sizes that were not entirely eliminated by the grinding process that was }
used. The zeta potential data shown in Figure 5.4 and in Figure 5.5 were performed with

greater accuracy using a Matec 8000 by Brett \Vrlson [19].
The primary equations that determine the stability of the colloidal suspension

 

according to the DLVO theory were found in the literature and were presented in chapter
two by equations (19) and (28). The corresponding equations for the repulsive force, equa-

tions (35) through (37), and the attractive force, equations (38) through (41), were derived

from the equations given in the literature.
The algorithm described in chapter three was based on the general method for solv-

ing a many body system. The algorithm was tailored and customized for the explicit intent

of solving the many body problem for a colloidal suspension system.
The implementation of the algorithm in the form of the colloidal suspension simu—

lator (CS S) software was the intended result of this thesis. The software in its delivered

form provides a flexible tool for understanding colloidal suspensions. The CSS software

48

 

 

 

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54

presents all of the necessary parameters and a priori data that are required to define the sys-
tem. Any of the parameters that describe the system may be modified easily and efficiently
to maximize the ability to control the system in any desired way. Upon finding acceptable
parameters that suit a users needs, the entire configuration of the system as well as the
actual simulation data may be recorded. The flexibility and control of the colloidal system
is a major advantage for using the CSS software versus conducting tedious and perhaps
costly experimental procedures for understanding the suspension mechanics.

Five tests were run with the CSS software to determine the influence of certain
parameters on the outcome of a simulation. Each of the tests resulted in several data files
that are included in this thesis on two floppy disks labeled, “Test Data”. The test data may
be viewed by copying the contents of each “test” directory on the floppy disk to the location
on the computer’s hard drive where the CSS software resides. An example of the format of
a data file that was generated by the CSSRUN sub - program is displayed in Figure 5.6. The
data file shown in Figure 5.6 contains the state information for three particles for four time
frames. The following discussion will outline the conducted tests.

The first test was an attempt to determine how the time increment between calcula-
tions affects the dynamics of the simulation. The default parameters defined in table 4.1
were used. The names of the output files are, respectively, A, B and C and each simulation
ran for 500 frames. The data in A used a time increment of 0.001 seconds. The data in B

used a time increment of 0.01 seconds. The data in C used a time increment of 0.1 seconds.
Upon viewing each simulation with the CSSDISP software, it was evident that the time
increment is quite crucial for accurate position calculations to be determined. The error for
determining the new position at each time frame starts out very small, since the initial posi-
tions are given, but the error appears to propagate as time progresses. The smaller time
increment minimizes the accumulation of errors during the calculation of each time frame.
The second test was an attempt to determine the affect of a variable concentration

level while maintaining the other parameters. The default parameters defined in table 4.1

55

CSS DATA FILE
3
0.500706. 15
0.500760, 15
0.450549. 10
0
93.000000, 57.500000
78.000000. 45.500000
78.000000, 39.500000
1
92.966286, 57.473743
77.973236, 45.557690
77.989143, 39.527519
2
92.932709, 57.447628
77.946426, 45.615337
77.977898, 39.554565
3
92.899361, 57.421753
77.919540, 45.672920
77.966003, 39.580826

Figure 5.6 Data file created with the CSSRUN software.

This figure displays an example data file that is generated by the CSSRUN sub - program
and is read by the CSSDISP sub - program. The first number is the number of particles,
followed by the radius and color number of each of the three particles. The rest of the data
format is repeated showing the frame number followed by the x and y positions of each
particle at that time frame.

‘—-——— -—-————_~ —4~—4- 44

-_- 1.- fiam - ‘—

 

56

were used. The data files for this test are called, respectively, D, E, F and G.The time incre-
ment was held at 0.001 seconds and each simulation ran for 500 time frames. The data in
D used a concentration of 0.001 N. The data in E used a concentration of 0.01 N. The data
in F used a concentration of 0.1 N. The data in G used a concentration of 1.0 Normality. No
noticeable difl‘erences were apparent between each of the simulations. This is an unex-
pected result, and provides the first indication that the dynamics of the software is not
behaving as in nature. Colloidal theory predicts that as the electrolyte concentration
increases, the double layer length, UK, will decrease. The decrease in the double layer
length reduces the magnitude of the repulsive force between the colloids. In this case, the
colloids are able to flocculate and form agglomerates. This phenomenon was not observed
in simulations D, E, F or G.

Test number three was an experiment to determine how a variable particle size
would affect the colloidal system. The default parameters defined in table 4.1 were used
except that it was discovered experimentally that the a time increment of 0.00001 seconds
was required. The data files for test three are called, respectively, H, I, J, K and L. The data
in H used a mean diameter of 0.1 microns and ran for 20 time frames. The data in I used a
mean diameter of 0.5 microns and ran for 50 frames. The data in J used a mean diameter of
0.3 microns and ran for 50 frames. The data in K used a mean diameter of 0.15 microns and
ran for 200 frames. The data in L used a mean diameter of 0.50 microns and ran for 10
frames. In experiments H and L, the particles move very rapidly ofl’ the screen. In experi-
ments I and J, the particles are relatively motionless. In experiment K, the particles are
slightly mobile. These experiments suggest that as the particle size becomes smaller, a
smaller time increment is required to control the dynamics of the suspension.

Test number four was an experiment to determine how a change in the viscosity of
the medium would affect the colloidal system. The default parameters defined in table 4.1
were used except that the particles average initial velocities were given a value of 50.0

micTOMS/second. The data file for this test is called M. The viscosity of the medium, the

 

 

57
deionized water, was intentionally set to 999 gm/(cm sec). The particles moved very slowly
as the hydrodynamic forces would suggest.

Test number five was used to observe the affects of the suspension due to a variable
pH level and consequently, variable zeta potential data. The default parameters defined in
table 4.1 were used. The data files for this test are called, respectively, N, O and P. Each
simulation ran for 300 time frames. The data in N used a pH level of 5.0 with zeta potentials
of 5.0 mV and 18.0 mV for Fe-40Al and A1203, respectively. The data in 0 used a pH level
of 7.8 with zeta potentials of 5.0 mV and 12.0 mV for Fe-40Al and A1203, respectively.
The data in P used a pH level of 10.0 with zeta potentials of 5.0 mV and -l9.0 mV for Fe-
40Al and A1203, respectively. None of the data files exhibited expected results. This pro-
vides the second indication that the dynamics of the software is not behaving as nature. For
the suspension under investigation, colloidal theory predicts that as the pH level varies

causing the zeta potential data to fluctuate, the stability of the suspension should change.

The test data files show some of the experimental capabilities that are possible using
the CSS software for probing many unanswered questions concerning colloidal suspen-
sions. It would appear that many combinations of changing one, two or more initial param-
eters to conduct an experiment are possible with the CSS software. The actual number of
possible experiments is calculated by summing the combinations of all of the variables
taken one at a time, two at a time,..., n at a time, where n is the number of variables. The
itnportant changeable variables with their degrees of freedom are listed below in table 5.1.
The total number of experiments possible using 19 variables is 524,287. This large number

of combinations shows how the colloidal suspension simulator can be used as an important

experimental tool.

58

Table 5.1 Number of changeable variables using the CSS software.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Variable 0333;“
mi
Initial Velocity 2
Number of Particles 2
Hamakcr Constant 4
Density 3
Viscosity 1
Temperature 1
Concentration 1
pH Level 2
Trme Increment 1
(Dynamic Model) (3)
Total: 19 Variables

 

 

 

 

CONCLUSIONS

The zeta potential data obtained at Dupont as described in chapter five provided the
first clues of the surface characterization of the Iron Aluminum particles and the A1203
fiber. The zeta potential data shown in Figures 5.4 and 5.5 are the most accurate zeta poten-
tial data obtained at the present time.

The repulsive and attractive force equations, equations (35) through (37) and equa-
tions (38) through (41), provide an alternative method for viewing a colloidal system.

The fundamental forces arising in a colloidal suspension were defined. A detailed
. description for the solution to a many body dynamical system was tailored to describe the
dynamics of a colloidal suspension system. The algorithm for describing the colloidal sus-
pension was implemented on a personal computer.

The colloidal suspension simulator (CSS) software was tested by modifying certain
initial parameters and observing the animated suspended particles. The choice of a small
time increment was found to be an important factor for ensuring accuracy during calcula-
tions. Experimentation with the CSS showed that as particle diameters become smaller than
0.1 microns, a time increment on the order of 10'4 or smaller is required. The CSS software
is a working prototype, but with some discrepancies. No noticeable differences were
observed between the experimental data for different electrolyte concentrations. Changes
in the surface parameters produced no noticeable change to the system for different exper-
iments. These two points suggest that the equations describing the electrodynamic forces
need some attention. The movement of the colloidal particles were severely damped when
the viscosity of the medium was increased which agrees with the hydrodynamic expecta-
tions.

The CSS software is flexible and provides thousands of possible ways to analyze

and to understand the dynamics of colloidal suspensions.

59

 

 

 

RECOMMENDATIONS

The computer program does not function as well as desired. Problems encountered

with the implementation and recommendations include the following:

- Particle collisions were not included in the computer program. By not properly
accounting for collisions, the particles are able to penetrate each other and the
integrity of the net forces is corrupted.

- Units consistency needs to be further verified to ensure that the order of mag-
nitude difference between forces, velocities and positions are within reason.

- The time increment between calculations could be chosen with greater sophis-
tication. A variable time step might eliminate the problem that arises with par-
ticle impacts of three or more. Also, a consistently small enough time step will
eliminate a particle “jumping” over another particle that possibly prevents some
instances of collisions from occurring at all.

- The coagulation of particles needs to be thought out and handled appropriately.
One method could be such that if two particles of radius r1 and r2 coagulate,
then the two particles coalesce into a new larger particle of radius r1+ r2.
Another possible method is to use rigid body mechanics for an agglomerate.

Once proper implementation of the colloidal suspension simulator has been

achieved, useful experiments may be conducted. For specified parameters, the stability
ratio of the suspension may be used to determine the critical coagulation concentration. The
stability ratio is the total number of collisions divided by the total number of collisions
r‘3-‘3‘11ting in adhesion. The pH level regions at which the particles flocculate, becoming
unstable may be determined. An accumulative mass function could be added to the program
‘0 keep track of the mass per unit area per unit time that collects onto the fiber. An interest-
ing three dimensional graph would be to plot time versus pH versus the amount of accumu-

lated mass on the collector. The accumulated mass functionality would provide an

60

61

indication of the rate at which the particles collect onto the fiber for a specified pH level.

Figure 7.1 depicts such a possible image.

 

62

>

Accumulated Mass onto the Fiber

 

 

 

Time

Figure 7.1 Depiction of accumulated mass.

This figure shows an example of data that could be calculated using the colloidal
SUSpension simulator.

LIST OF REFERENCES

. Crimp, M. A. and Crimp, M.J., REF Proposal, July 1990, “Fiber Electrophoretic Dep-
osition Processing of FeAl/A1203 Interrnetallic Matrix Composite”.

. Heimenz, WW Marcel Dekkcr. New York,
2“dedition,1986.

. Ross, Sydney and Morrison, Ian Douglas, W, John
Wiley and Sons, New York, 1988.

. Everett, D. H., W, Royal Society of Chemistry, Lon-
don,1988.

. Wiese, G. R. and Healy, T. W., Transactions of the Faraday Society, 1970, 66, 490 -
499, “Efiect of Particle Size on Colloid Stability”.

. Hinzcl. C. S. and Rajagopalan. Raj. CollmdaljhenmmAdxancedlonms Noyes
Publications, New Jersey, 1985.

. Mahanty, J. and Ninham, B. W., W, Academic Press, Inc., London,
1976.

. Mysels, Karol 1., W Robert E. Krieger Publishing
Company, Huntington, New York, 1978.

. Shaw. Duncan 1.. WW Buttcrworths. Lon-
don, 1966.

10. Derjaguin, B. V., Discussions of the Faraday Society,Vol. 18, 85 - 98, 1954, “A Theory

of the Hetercoagulation, Interaction and Adhesion of Dissirrrilar Particles in Solutions
of Electrolytes.”

11. Hogg, R., Healy, T. w. and Fuerstenau, D. W., Transactions of the Faraday Society, 62,

1638, 1966, “Mutual Coagulation of Colloidal Dispersions”.

12. Bell, G. M., Levine, S. and McCartney, L N., Journal of Colloid and Interface Science,
Vol. 33, No. 3, 335 - 359, July 1970, “Approximate Methods of Determining the Double
- Layer Free Energy of Interaction Between Two Charged Colloidal Spheres”.

13. Bell, G.M. and Peterson, G. C., Journal of Colloid and Interface Science, Vol. 41, No.

3. 342 - 366, December 1972, “Calculation of the Electric Double - Layer Between
Unlike Spheres”.

63

64

14. Mahanty, J. and Ninham, B. W., Disfigign Fgmes, Academic Press, Inc., London,
1976.

15. Hamakcr, H. C., Physica 4, 1058 - 1072, 1937, “The London - van der Waals Attraction
Between Spherical Particles”.

16. Beer, Ferdinand P. and Johnston, E. Russel, Vggtgr Mmhgigs for; Engineers; Dynam-
ics, McGraw-Hill Book Company, Ney York, 1988.

17. Hammerly, Wayne M. and Hanlin III, Thomas G., ADVAS Proramnring Library, Share-
ware, 1988.

18. Microsoft Corporation, QuickBASIC 4.50, 1985-1988.
19. Wilson, Brett, Masters Thesis, “Prediction of Colloidal Suspension Stability for SiC/

Si3N4 and FeAl/A1203 Fiber Systems Using Material and System Parameters”,Michi-
gan State University, 1992.

GENERAL REFERENCES

. Heimcnz. WW Marcel Dckkcr. New York.
2“dedition,1986.

. Crimp, M. A. and Crimp, M.J., REF Proposal, July 1990, “Fiber Electrophoretic Dep-
osition Processing of FeAl/Ale3 Interrnetallic Matrix Composite”.

. Hamakcr, H. C., Physica 4, 1058 - 1072, 1937, “The London - van der Waals Attraction
Between Spherical Particles”.

. Hogg, R., Healy, T. w. and Fuerstenau, D. W., Transactions of the Faraday Society, 62,
1638, 1966, “Mutual Coagulation of Colloidal Dispersions”.

. Bierman, Arthur, Journal of Colloid and Interface Science, 10, 231 - 245, 1955, “Elec-
trostatic Forces Between Nonidentical Colloidal Particles”.

. Barouch, Eytan, Matijevic, Egon, Ring, Terry A. and Finlan, J. Michael, Journal of Col-
loid and Interface Science, Vol. II, No. 1, October 1978, “Hetercoagulation II. Interac-
tion Energy of Two Unequal Spheres”.

. Bell, G. M., Levine, S. and McCartney, L N., Journal of Colloid and Interface Science,
Vol. 33, No. 3, 335 - 359, July 1970, “Approximate Methods of Determining the Double
- Layer Free Energy of Interaction Between Two Charged Colloidal Spheres”.

. Bell, G.M. and Peterson, G. C., Journal of Colloid and Interface Science, Vol. 41, No.
3, 342 - 366, December 1972, “Calculation of the Electric Double - Layer Between
Unlike Spheres”.

. Schenkel, J. K. and Kitchener, Transactions of the Faraday Society, September 1959,
161 - 173, “A Test of the Derjaguin — Landau - Verwey - Overbeek theory with a C01-
loidal Suspension”.

10. Hall, Simon Bemers, Duffield, John Ralph and Williams, David Raymond, Journal of

Colloid and Interface Science, Vol. 143, No. 2, 416 - 422, May 1991, “A Stochastic
Computer Simulation of Emulsion Coalescence”.

ll. Derjaguin, B. V., Discussions of the Faraday Society,Vol. 18, 85 - 98, 1954, “A Theory

of the Hetercoagulation, Interaction and Adhesion of Dissimilar Particles in Solutions
of Electrolytes.”

12. Wiese, G. R. and Healy, T. w., Transactions of the Faraday Society, 1970, 66, 490 -

499. “Effect of Particle Size on Colloid Stability”.

65

66

13. Matijevic, Egon, Pure and Appl. Chem, Vol. 53, 2167 - 2179, “Interactions in Mixed
Colloidal Systems (Hetercoagulation, Adhesion, Microflotation)”.

14. Tamura, Hiroki, Matijevic, Egon and Meites, Louis, Journal of Colloid and Interface
Science, Vol. 92, No. 2, 303- 314, April 1983, “Adsorption of Co 2* Ions on Spherical
Magnetite Particles”

15. Rajagopalan and Chu, Richard Q., Journal of Colloid and Interface Science, Vol. 86,
No. 2, 299 - 317, April 1982, “Dynamics of Adsorption of Colloidal Particles in Packed
Beds”.

16. Derjaguin, V. M., Muller and Toporov, Yu. B, Journal of Colloid and Interface Science,
Vol. 53, No. 2, 314 - 326, November 1975, “Effect of Contact Deformations on the
Adhesion of Particles”.

17. Kar, G, Chander, S. and Mika, T. 8., Journal of Colloid and Interface Science, Vol. 44,
No. 2, August 197 3, 347 - 355, “The Potential Energy of Interaction Between Dissim-
ilar Electrical Double Layers”.

18. Visser, J ., Advan. Colloid Interface Sci., 3, 331 - 363, 1972, “On Hamaker Constants:
A comparison Between Hamakcr Constants and Lifshitz - van der Waals Constants”.

19. Verwey. E. J. W., Overbeek. J. T. G.. W.
Elsevier, Amsterdam, 1948.

20. Derjaguin, B. V., Landau, L. D., Acta Physichim, URSS, 14, 633, 1941.

21. Ross, Sydney and Morrison, Ian Douglas, W5, John
Wiley and Sons, New York, 1988.

22. Russel, VVrlliam B., W, The University of Wisconsin
Press, 1987.

23. Vold, Robert and Vold, Marjorie J., W Addison - Wesley
Publishing Company, Inc., Reading, Massachusetts, 1983.

24. Mysels. Karol 1.. mmmmmmunamim Robert E. Kricgcr Publishing
Company, Huntington, New York, 197 8.

25. ShaW. Duncan J., Wmmmmmmmm ButtcrworthS. Lon-
don. 1966.

26. Edited by Kerker, Milton, Zettlemoyer, Albert C. and Rowell, Robert L., W

A,cademic Press Inc, London,
1988.

67

27. Everett, D. H., W, Royal Society of Chemistry, Lon-
don,1988.

28. Lynch, C. T. and Kershaw, J. P., W, CRC Press, The Chenrical
Rubber Co., Cleveland, Ohio, 1988.

29. Mahanty, J. and Ninham, B. W., W, Academic Press, Inc., London,
1976.

30. Marciniak. Andrzej. W. D. Reidel Publish-
ing Company, Dordrecht, Holland, 1985.

31. Becher. Paul. W Marcel Dekker. Inc., New
York, 1990.

32. Hirtzel. C. S. and RajagOPalan. Raj. ColloidaLEhenomenaaAdxancedenics. Noyes
Publications, New Jersey, 1985.

33. Crimp, Melissa J ., Masters Thesis, Case Western Reserve University, 1985, “Colloidal
Characterization of Silicon Carbide and Silican Nitride”.

34. me CRC Press. Inc., Flor-
ida, 1989.

35. Mattuck. Richard. D.. AW
McGraw - Hill Book Company, London, 1967.

APPENDIX A

CSS PROGRAM LISTINGS

APPENDIX A

CSS PROGRAM LISTINGS

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APPENDIX B

CSSDISP PROGRAM LISTINGS

APPENDIX B

CSSDISP PROGRAM LISTINGS

 

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