COAGULATION OF CALCIUM OXALATE MONOHYDRATE SUSPENSIONS

By

JAMES HANSELL ADAIR

A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF

THE UNIVERSITY OF FLORIDA

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE

DEGREE OF DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

1981

To Susan Bailey Adair

ACKNOWLEDGEMENTS

Thanks are gratefully extended to the faculty and staff

of the Department of Materials Science and Engineering for

their support. A special thank you is given to Mr. E.J.

Jenkins for his help in the electron microscopy studies.

I would also like to thank the members of my

supervisory committee for their guidance throughout the

course of this work. In particular, the many hours given in

consultation by the chairman, Professor George Onoda, and

Professor Birdwell Finlayson are deeply appreciated.

Thanks are given to Professor Peter Curreri for lending

me his experience and knowledge of calcium oxalate

monohydrate without which this work could not have been

undertaken. Thanks also go to Professor Michael Sacks for

his encouragement and guidance during the preparation of

this manuscript.

Finally, I would like to thank my colleagues in

Professor Onoda's Ceramic Particulate Science Group for the

many hours spent in fruitful conversation; Mr. David

Hoelzer and Mr. William Carter for assistance; Mr. Arthur

Smith for technical assistance; and Mrs. Paulette Senior in

manuscript preparation.

This work was supported by National Institutes of

Health SCOR Urolithiasis Grant No. AM20586.

iii

TABLE OF CONTENTS

PAGE

ACKNOWLEDGEMENTS ....................................... i ii

ABSTRACT............................................... vii

CHAPTER

I INTRODUCTION .... ..................... ...... 1

II DEVELOPMENT AND PHYSICAL CHARACTERIZ-

ATION OF PARTICLES IN CALCIUM OXALATE

MONOHYDRATE SUSPENSIONS USED FOR

COAGULATION STUDIES......................... 9

Introduction ............................. 9

Materials and Methods..................... 10

Results and Discussion.................... 13

Calculation of the Lifshitz-van der

Waals Constant.......................... 23

Conclusions ..................... ......... 29

III DETERMINATION OF SURFACE AND SOLUTION

CHEMISTRY OF PARTICLES IN THE MODEL

SUSPENSION SYSTEM USED IN COAGULATION

EXPERIMENTS..... ....... .................. . 30

Introduction............................. 30

Materials and Methods..................... 31

Results and Discussion.................... 42

Conclusions .............................. ... 49

IV DEVELOPMENT OF AN ANALYTICAL METHOD TO

DETERMINE THEORETICAL COAGULATION

BEHAVIOR ................................... 50

Introduction ........................... 50

Theoretical Models and Thermodynamic

Relationships for Interacting

Particles .............................. 51

Calculation of Total Interaction Energies

and Stability Ratios.................... 60

Hypothetical Calculations using DLVO...... 64

Table of Contents continued:

CHAPTER PAGE

V PROCEDURES TO DETERMINE EXPERIMENTAL

COAGULATION BEHAVIOR OF COM

SUSPENSIONS.......... ...................... 70

Introduction. ... ............. .... ....... 70

Theoretical and Experimental

Considerations...... ............ ...... 70

Materials, Methods, and Procedures in

Experimental Determinations of

Coagulation Behavior ................... 74

Outline of Experimental Procedures to

Determine Coagulation Behavior.......... 85

VI STABILITY OF COM SUSPENSIONS IN SOLUTIONS

OF INDIFFERENT AND POTENTIAL DETERMINING

ELECTROLYTES............................... 88

Introduction............................... 88

Materials and Methods..................... 94

Results and Discussion.................... 98

Conclusions ....... ....................... 125

VII STABILITY OF COM SUSPENSIONS IN SOLUTIONS

OF SPECIFICALLY ADSORBED ELECTROLYTES

AND MACROMOLECULES.......................... 129

Introduction...............................129

Materials and Methods ........ .............132

Results and Discussion....................136

Conclusions .... ....... .................... 166

VIII CONCLUSIONS AND SUGGESTIONS FOR FUTURE

STUDIES..................................... 168

Conclusions .............................. 168

Suggestions for Future Studies............177

APPENDIX

A DLVO: COMPUTER PROGRAM USED TO GENERATE

TOTAL INTERACTION ENERGY AND THEORETICAL

STABILITY RATIOS FOR LARGE PARTICLE

SUSPENSIONS OF CALCIUM OXALATE

MONOHYDRATE .. ... ......................... 181

Table of Contents continued:

APPENDIX PAGE

B DEVELOPMENT OF REPULSIVE ENERGY EQUATIONS

USED IN DLVO FOR SUSPENSIONS IN SOLUTIONS

OF ARBITRARY COMPLEXITY .................... ..210

C VERIFICATION OF EQUATIONS USED IN DLVO........229

D EQUIL AND DOUBLE LAYER: COMPUTER PROGRAM

USED TO CALCULATE IONIC EQUILIBRIA AND

THEORETICAL STERN POTENTIALS ...............236

E COMPUTER PROGRAM USED TO CALCULATE ANGLES

BETWEEN CRYSTALLOGRAPHIC PLANES FOR

CALCIUM OXALATE MONOHYDRATE... ..............263

BIBLIOGRAPHY ........................... ................265

BIOGRAPHY.................................... ....... .271

Abstract of Dissertation Presented to the Graduate

Council of the University of Florida in Partial

Fulfillment of the Requirements for the

Degree of Doctor of Philosophy

COAGULATION OF CALCIUM OXALATE MONOHYDRATE SUSPENSIONS

By

JAMES HANSELL ADAIR

JUNE, 1981

Chairman: George Y. Onoda, Jr.

Major Department: Materials Science and Engineering

Calcium oxalate monohydrate suspensions were used as a

model system to test coagulation theory with experiment.

Experimental and theoretical methods were developed to

quantitatively analyze the coagulation behavior of calcium

oxalate monohydrate suspensions. This work was partly

motivated by the hypothesis that coagulation is an important

mechanism in human kidney stone disease. To test this

hypothesis, experimental methods were designed to utilize

calcium oxalate monohydrate particles similar to those found

in human urine of sizes greater than 1 um. Methods were

developed to theoretically analyze the coagulation behavior

of calcium oxalate monohydrate suspensions in solutions of

arbitrary complexity. This allowed us to analyze the

vii

coagulation behavior of suspensions in urine-like solutions

composed of many ionic species.

Model calcium oxalate particles which were essentially

flat plates were obtained from a series of crystallization

experiments in which supersaturation of calcium and oxalate

ions were varied. A standard procedure was developed to

prepare calcium oxalate monohydrate suspensions containing

the model particle system. The Lifshitz-van der Waals

constant for calcium oxalate monohydrate particles

interacting in water was calculated using optical dispersion

data. Measurements were performed to ensure that the

solution chemistry and the solid-solution interfaces of

standard calcium oxalate monohydrate suspensions were

consistent with other studies on calcium oxalate monohydrate

suspensions.

Theoretical and experimental techniques were developed

to quantitatively analyze primary minimum or secondary

minimum coagulation of calcium oxalate monohydrate

suspensions. Since theoretical techniques required analyses

in solutions of arbitrary complexity, general equations were

developed from the theoretical models presented by Verwey

and Overbeek. A computer program was written to numerically

analyze the theoretical expressions for the coagulation

behavior of suspensions in solutions of arbitrary ionic

complexity. Experimental techniques were developed using

orthokinetic rate theories for the coagulation of calcium

viii

oxalate monohydrate suspensions. From experimentally

determined coagulation rates, experimental stability ratios

were calculated for either primary minimum or secondary

minimum coagulation.

In primary minimum coagulation studies, critical

coagulation concentrations from experimental stability

ratios for primary minimum coagulation as a function of

electrolyte concentration were used to determine

experimental Lifshitz-van der Waals constants from

theoretical stability ratio curves. Experimental values

were compared to the calculated Lifshitz-van der Waals

constant to assess the validity of the models used in

theoretical calculations. The possibility that substantial

sodium ion adsorption occurs at sodium ion concentrations

greater than 1 x 10-2 M in calcium oxalate monohydrate

suspensions was investigated in primary minimum coagulation

studies.

In secondary minimum coagulation studies, experimental

stability ratios for secondary minimum coagulation were

determined for suspensions over a wide range of theoretical

Stern potentials and ionic strengths. Suspension

stabilities were also determined for specifically adsorbing

ionic species and macromolecules in urine-like solutions.

Stability ratios for primary minimum coagulation were

determined in these suspensions.

ix

Theoretically predicted secondary minimum coagulation

agreed with experimental coagulation behavior. Our

experimental results indicate calcium oxalate monohydrate

suspensions typically coagulated. Secondary minimum

coagulation was implicated as a mechanism, because

experimental coagulation was observed over a wide range of

Stern potentials and ionic strengths where stability to

primary minimum coagulation was predicted. Qualitatively,

there was agreement between the magnitudes of interaction

energies at secondary minima and experimental stability

ratios for secondary minimum coagulation in the large

particle suspensions (i.e., particles greater than 1 um

in size). However, little agreement between theory and

experiment for primary minimum coagulation was observed.

The studies implicate coagulation as a possible mechanism in

kidney stone disease.

CHAPTER I

INTRODUCTION

Calcium oxalate monohydrate (COM) is a material with

many of the features required to test classical coagulation

theory (1) with experiment. It is a sparingly soluble

salt. In a previous study, it was shown that for many

solutions with different ionic equilibria in COM

suspensions, the Nernst-Gouy-Stern (NGS) model analytically

described the COM-solution interface (2,3). While the NGS

model does not give an analytical description of the

adsorption of macromolecules on COM surfaces, it has been

shown that the absorption of some large molecules on COM

surfaces follows Langmuir adsorption isotherms (4). An

analytical method is available to establish ionic equilibria

of COM suspensions containing simple and complex ions in

solution (5). Therefore, the major objective of this work

was to test classical coagulation theory with experiment

using COM suspensions as a model system.

These studies were partly motivated by hypotheses that

coagulation is an important mechanism in human kidney stone

disease (6). Since COM is one of the major constituents of

stones in the United States (7), a study of COM coagulation

behavior may give insight into the relative importance of

coagulation in stone formation. The development of stones

is essentially a phase transformation (6). After

precipitation, particle sizes must increase by growth,

aggregation, or Ostwald ripening (8). The majority of

studies on size increase mechanisms have focused on

precipitation and growth. However, crystal growth rates are

far too low to produce free-particle stone disease by a

growth mechanism alone (6). Furthermore, one of the few

points of common agreement in stone formation is that stones

are polycrystalline aggregates (6). A few investigators

have attempted to study particle aggregation (9,10,

11,12). It was shown that urine from stone-formers promoted

size increases of COM particles more than urine from non-

stone-formers (9). However, the authors were uncertain

whether size increases were due to aggregation or Ostwald

ripening. Furthermore, the investigations did not attempt

to relate the experimental findings to coagulation theory.

To test theoretical coagulation theory by experiment,

both thermodynamic and kinetic features of particle

collisions in a suspension must be known, or at least

controlled (13). Thermodynamic theories predict when a

collision will result in coagule formation (1), whereas

kinetic theories predict the number of collisions per unit

time (14). Classical coagulation theory gives total

interaction energy as a function of separation distance

between two approaching particles (1). A typical total

interaction energy curve is shown schematically in

Figure I-1. Total interaction energy is the superposition

of repulsive and attractive energies between the

particles. Repulsive energies arise from the increasing

overlap of adsorbed ion layers as particles approach one

another. Attractive energies arise from mutual van der

Waals interactions between the particles. If repulsive

energies are small, total interaction energies occupy the

line in the figure indicated for attractive energies.

Coagulation in the deep minimum at dm in the figure is

primary minimum coagulation (pmc). If repulsive energies

are present, as indicated in Figure I-1, total energies are

given by the solid line with a repulsive energy barrier with

a maximum of Vm. However, coagulation may still occur if

the magnitude of the minimum, shown as Vs, located at ds in

the figure, is large enough (15). Coagulation of this

nature is secondary minimum coagulation (smc).

Using AgI or polystyrene model particles in

suspension, many investigators have demonstrated the basic

validity of classical coagulation theory (16,17,18,19,20).

However, only a few studies have examined the relative

contribution of smc to coagulation of these model systems

(15,21,22). More often than not, smc was a nuisance to be

avoided in pmc studies (23). However, it was shown that smc

may be significant if particle sizes exceed 1 um (21), which

is often the case for urinary crystalluria. Furthermore,

s - ~ - - - - ----

V - -

U ,

z

W V

e /

II-

III

d d

m s

SEPARATION DISTANCE

Figure I-1.

Potential energy as a function of separation

distance between two particles approaching

one another, Repulsive energies, attractive

energies and their superposition giving total

energies are shown.

5

separation distances of crystallites in stones are 50 A or

greater (6). These distances are too large for stones to be

formed exclusively by pmc (1). Therefore, it suggests that

smc may play a role in stone disease.

To achieve the objectives, the following studies were

performed:

Chapter II

To relate theoretical and experimental findings to the

role coagulation may play in stone disease, we needed COM

suspensions in which particle sizes were consistent with

those found in human urine (9,10). In non-stone-formers

urine, particle size distributions usually have means

from 3 um to 10 pm equivalent spherical diameters (9). To

obtain a model system of COM particles with mean sizes

similar to those found in human urine, we performed a series

of crystallization experiments. A standard preparation

procedure for the model particle system was developed to

give reproducible suspensions. The physical features of

model particles important to coagulation theory were

determined.

Chapter III

To test coagulation theory, a quantitative description

of the double layer is required. Therefore, the NGS model

of the double layer and the analytical method establishing

ionic equilibria were tested for COM suspensions containing

the model particle system.

Chapter IV

An analytical, numerical method was developed to assess

theoretical coagulation behaviors. Historically,

coagulation theory was developed assuming simplified ionic

equilibria conditions. The assumption that only two ionic

species of opposite charge were present gave analytical

solutions from the differential equations describing double

layer overlap. However, a more generalized repulsive energy

expression had to be used to accommodate the multi-ion

solutions present in COM suspensions. Even the simplest COM

suspension contains more than two ionic species.

Furthermore, Matijevic (24) has demonstrated that it is

necessary to understand the role of complex chemistry in

coagulation behavior. Therefore, expressions were

implemented following the development of Verwey and Overbeek

(1) for repulsive energies which gave a theoretical

treatment of repulsive energies for ionic solutions of

arbitrary complexity (Appendices B and C). To solve these

expressions, numerical analysis was performed by a computer

program designed for that purpose. We compared theoretical

coagulation behavior given by total interaction energy

curves with experimental coagulation behaviors of COM

suspensions under various ionic equilibria and adsorbed ion

conditions.

Chapter V

Early in this work, we decided to focus on the

thermodynamic aspects, while maintaining kinetic features as

a constant. Therefore, experimental coagulation procedures

were developed so that kinetic aspects were constant and

normalized, to focus on changes in thermodynamic features.

Using the model particles in standard suspensions,

experimental techniques were developed to independently

determine primary minimum or secondary minimum coagulation

behavior.

Chapter VI

Theoretical and experimental coagulation studies were

made with COM suspensions in relatively simple indifferent

and potential determining ion solutions. Special attention

was given to identifying the presence of primary or

secondary minimum coagulation over a wide range of Stern

potentials and ionic strengths.

Chapter VII

Theoretical and experimental coagulation studies were

made with COM suspensions in artificial urine with additions

of specifically absorbing small ions or macromolecules.

8

One of the important findings of these studies is that

theoretical calculations predicted the experimental

coagulation properties of most of the suspensions

investigated. Theoretical calculations indicate secondary

minimum coagulation must be considered to understand

experimental coagulation behaviors of suspensions containing

COM particles greater than 1 um in diameter.

CHAPTER II

DEVELOPMENT AND PHYSICAL CHARACTERIZATION OF PARTICLES

IN CALCIUM OXALATE MONOHYDRATE SUSPENSIONS USED FOR

COAGULATION STUDIES

Introduction

The objective of the work in this chapter was to

develop a calcium oxalate monohydrate (COM) particle system

to be used in suspensions for the coagulation studies

(Chapters VI and VII). Classical coagulation theory is

based on particles with either spherical or flat plate

morphologies (1). Therefore, to experimentally test

coagulation theory, COM particles with either spherical or

flat plate morphologies would be preferred. To relate

experimental coagulation behavior to its possible role in

stone disease, COM particle size distributions were desired

which were similar to size distributions found in human

urine (i.e., mean sizes about 5 pm in diameter) (9).

Furthermore, to perform quantitative coagulation

experiments, the suspensions of particles needed to be

dense, washable, and have reproducible properties (25).

Since previous investigators (9,10) have not fully

established the physical features of the surfaces of COM

precipitates, it was not known which preparation procedures

produced particles with the desired characteristics.

Therefore, to obtain COM particles with different surfaces

and different morphologies (26), precipitation studies were

performed by varying initial calcium and oxalate ion

concentrations in supersaturated solutions.

To develop a standard suspension, preparation procedure

required establishment of: (1) a procedure to collect model

system particles from reacted solutions; (2) a washing step

to remove excess ions from precipitates; and (3) a

concentration procedure to yield COM suspensions with enough

particles (e.g., at least 1 x 106 particles ml"1) to ensure

an effective number of collisions between particles in

coagulation experiments. This assured reproducibility in

particle number concentration, size, and shape.

In concert with the experimental work in this chapter,

the Lifshitz-van der Waals constant (AL) was calculated for

COM particles interacting across water. This constant is

needed for calculating the attractive energies between

particles in suspensions (Chapter IV).

Materials and Methods

Solution Preparation

All solutions were prepared with water which was

deionized1 and then distilled2 (specific conductivity less

1. Culligan Water Conditioner, Culligan International Co.,

Northbrook, Ill.

2. Corning Mega-Pure Six Liter Automatic Still, Corning

Glass Works, Corning, N.Y.

than 1.5 mmho cm-1). Reagent grade3 calcium chloride

dihydrate (CaC12*2H20) and potassium oxalate monohydrate

(K2C204'H20) were used to prepare stock 1.0 M CaC12 (aq) and

1.0 M K2C204 (aq) at 25C. After equilibration, stock

solutions were passed through a 0.22 um Millipore filter4

From stock solutions, dilutions of CaC12 (aq) and K2C204

(aq) were volumetrically prepared. After equilibration,

dilutions were filtered (0.22 um). The CaC12 solutions were

stored under dry N2 (g) to minimize CaCO3 formation through

CO2 from the surroundings. Solution concentrations were

verified using atomic absorption5

Precipitation Studies

Precipitation studies were performed by mixing equal

volumes of various concentrations of CaC12 and K2C204

solutions at 250C. For identification purposes,

precipitated COM particles were designated by the negative

logarithms of the initial C2042 and Ca2+ molar

concentrations, respectively (e.g., particles prepared from

1 x 10-3 M C2042- and 1 x 10-2 M Ca2+ were designated "32"

3. Mallinkrodt Chemical Co., via American Scientific

Products, Ocala, Fla.

4. Sterifil Aseptic System and CF filter, Millipore

Intertech, Inc., Bedord, Mass.

5. Perkin-Elmer, 306 Atomic Absorption Spectrometer,

Perkin-Elmer Instrument Division, Norwalk, Conn.

particles). In this manner, CaC12 and K2C204 concentrations

were mixed to yield "00", "11", "22", "32", and "33"

particles. After 24 hours equilibration, a general idea of

particle morphologies was obtained by examining precipitates

in an optical microscope6 (OM). Particles with promising

morphologies, as determined with OM, were observed at higher

magnifications using a scanning electron microscope7

(SEM). X-ray diffraction analyses8 (XRD) were performed on

precipitates to verify the presence of the crystalline form

of COM.

Characterization of COM Particles and Standard Suspensions

Particle sizes and number concentrations

A Coulter Counter9 was used to determine particle

volume distributions and number concentrations in selected

COM suspensions. Supporting electrolyte needed for the

Coulter Counter was a pre-filtered (i.e., 0.22 pm) 1 weight

per cent KC1 solution which was 90 per cent saturated with

CaC204"H20 to minimize dissolution of particles during

volume measurements.

6. Nikon Optiphot, Nippon Kogaku (U.S.A.) Inc., Garden

City, N.Y.

7. JSM-35C, JEOL Ltd., Tokyo, Japan.

8. Norelco Diffractometer, Phillips Electronics, Inc.,

Mahwah, N.J.

9. Coulter CounterR TAII with Population Accessory,

Coulter Electronics Inc., Hialeah, Fla.

Particle sizes were monitored using a 100 um aperture

such that volumes were measured from 8.378 m3 (equivalent

spherical diameter (esd) = 2.52 pm) to 6.863 x

104 m3 (esd = 50.8 um). Mean particle volumes and standard

deviations for particles in suspensions were calculated from

the volume-number distributions. Particle number

concentrations for suspensions were calculated from total

number of particles counted.

Determination of habit planes on COM model particles

Crystallographic planes associated with habit planes on

the model particles were determined by comparison between

measured angles from SEM photomicrographs of model particles

and predicted angles between crystal planes for COM (27).

Since analysis of the monoclinic crystal structure (28)

involved laborious calculations (29), a computer program

(listed in Appendix E) was written to generate predicted

angles.

Results and Discussion

Model Particle System

All mixtures of CaC12 and K2C204 gave precipitated

particles. XRD indicated precipitates were COM. The "11"

and "22" particles did not meet the guidelines given above

for the model particle system. For example, "11" particles

were obvious aggregates of smaller particles. The "22"

particles were dendritic-shaped. However, optical

microscopy observations revealed that "00", "32", and "33"

particles had many of the desired features for the model

system. SEM examinations of these particles are given in

Figure II-1. The SEM revealed that "00" particles

are 5 um aggregates composed of particles with esd from 0.1

to 1.0 um. The "32" and "33" particles have rectangular

plate-like morphologies with esd between 5 and 15 pm. The

ratio of edge lengths is approximately 10:6:3. There is a

groove present in the large face of "33" particles. Its

surfaces are relatively smooth. The large faces of "32"

particles are rough.

Schematic engineering views for a typical "33" particle

are given in Figure 11-2. Since "33" and "32" particles had

similar habit planes and morphologies, the simpler "33"

particle is given in the figure for illustration purposes.

From a front view, both types of particles are similar to

biconvex lenses. From an endview, both morphologies appear

rectangular with the long axis at the top and bottom edges

adjacent to the large faces.

The "32" particles were selected as the model system.

The "00" particles were not as fully dense as required.

Both "32" and'"33" particles had at least one major

deviation from the preferred physical features. The "32"

particles were not ideal flat plates because of surface

roughness. The "33" particles were not fully dense because

of the groove bisecting. them along their long axes. The

(a) (b)

Figure II-1.

SEM photomicrographs of: (a) "00"; (b) "32"; and

(c) "33" particles at 5,400 X, 4,000 X and 9,400 X,

respectively. Samples were coated with Au-Pd and

imaged using secondary electrons with a 30 degree

tilt from normal.

(O1iT)

TOPVIEW

FRONTVIEW SIDEVIEW

[010]

Figure 11-2.

Schematic representations of model particles with

crystallographic and optical orientations, (Top)

Engineering drawings of top, front and side views.

(Bottom) Perspective drawing with orientations.

For clarity, a "33" particle is shown.

"32" particles were selected because suspensions containing

"32" particles were more easily dispersed using ultrasonic

disruption.

Standard Suspension Preparation Procedure

By trial and error, a standard preparation procedure

was developed to yield reproducible COM suspensions.

Precipitates were prepared by mixing 10 liters each of 1 mM

K2C204 with 10 mM CaC12. After at least 12 hours

equilibration, precipitates were collected from reacting

vessels using 3.0 um Millipore filters. Filtered particles

were washed with at least 10 liters of deionized water.

Washed filtrates were transferred to a 200 ml

volumetric flask. Deionized, distilled water was added to

obtain final suspension volumes equal to 200 ml. If

required, appropriate amounts of electrolyte were added

prior to addition of the water.

After particle size distribution determination and

optical microscopy examination to assure model particle

morphology, suspensions were placed in a temperature

controlled glove box at 380C 10C. COM suspensions were

stored in sealed containers until used.

Characterization of Particles and Standard Suspensions

Morphologies of particles in standard suspensions

To simplify the discussion, the particles shown in

Figure II-l(b) and (c) are referred to as "32" and "33"

particles, respectively, regardless of the solution ion

conditions used to prepare the particles. Figure 11-3 gives

SEM photomicrographs of particles present in a standard

suspension. As indicated in the figure, both "32" and "33"

particles are present in suspensions prepared using the

standard procedure. Attempts to obtain suspensions with one

morphology were not effective, except by lowering the

concentrations of Ca2+, which resulted in "33" particles.

However, it was difficult to disperse suspensions only

containing "33" particles. Since "32" and "33" particles

were similar in terms of size and shape, the standard

preparation procedure outlined above was eventually left

unaltered.

Size distributions of particles in standard suspensions

The size distribution of particles in a typical

suspension is given in the histogram on Figure 11-4.

Probability analysis (30) of the cumulative number of

particles as a function of equivalent spherical radius (esr)

indicated a normal distribution with a slight skew toward

smaller particle sizes. Mean esr was 5.06 um 0.01 um.

The increase in percentage of total number at

1.0 um to 1.3 um compared to 1.3 pm to 2.5pm indicates the

particle size distribution for particles in the suspension

(b)

(c) (d)

Figure 11-3.

SEM photomicrographs of particles in a standard COM

suspension at different magnifications: (a) 480 X;

(b) 1,000 X; (c) 2,000 X; and (d) 10,000 X. Sample was

coated with Au-Pd, tilted 30 degrees from normal, and

imaged using secondary electrons.

S25.0

-j 20.0

I-

o

15.0

O

u-

0 10.0 -

w

I---

U

0 5.0

1Q-

0.0

1.0 1.3 1.6 2.0 2.5 3.2 4.0 5.0 6.4 8.0 10.1 12.7

EQUIVALENT SPHERICAL RADIUS (pm)

I I I I I I I I I I

4.19 8.38 16.8 33.5 67.0 134 268 536 1070 2150 4290 8580

VOLUME (m3 )

Figure II-4,

Percentage of total number of particles in a given size

range for a typical suspension of model particles. Mean

radius and 95% confidence interval are indicated.

I____

was bimodal. Since coagulation rates are proportional to

particle volumes (Chapter V), the disproportionate number of

smaller particles would not effect coagulation behavior

much. Furthermore, more than 75% of the particles were

larger than 2.5 um. Like most of the suspensions prepared,

the particles in this suspension were relatively

monodispersed in the size range of interest (i.e. esr

> 2.5 um). Therefore, to relate theoretical and

experimental coagulation behaviors, mean esr were used in

theoretical calculations.

Reproducibility of particle sizes in standard preparation

procedure

A histogram showing number of standard suspensions as a

function of average size ranges is given in Figure 11-5.

The mean esr for the distribution in the histogram is

5.31 um 0.21 um. The average esr ranged from 3.6 um to

6.4 pm. The reproducibility of suspensions prepared by the

standard procedure is indicated by the narrow distribution

of average esr.

Habit plane determination of particles in COM suspensions

Comparisons between measured and calculated angles are

based upon the premise that plane C, indicated in Figure II-

2, is a crystallographic twin plane (31). The angular

relationships give planes A and C as (010)

and (101), respectively. Assuming a 90 degree angle exists

between planes A and B results in (001) for plane B.

22

I I I I I I

Mean Radius = 5.32 pm 1.43 pm

o

8

C, 5

o 4

Z

LU

z

c5

U-

04

2 ---- ----- -

3.6 4.2 4,6 4,9 5.2 5.5 5.8 6.0 6.2 6.4

AVERAGE EQUIVALENT SPHERICAL RADIUS (Pm)

I I I I I I I I I I

200 300 400 500 600 700 800 900 1000 1100

AVERAGE VOLUME (m 3)

Figure 11-5.

Number of suspensions containing model particles

in a given average size range for 27 different

suspensions precipitated over a period of three

months. Mean radius and 95% confidence interval

are indicated.

However, it is difficult to determine this angle since the

ends of "32" and "33" particles are irregular. Fortunately,

other evidence lends strong support to the orientations

given for planes A and C. Filtered particles are primarily

oriented with plane C parallel to the filter surface as

indicated by Figures II-1 and 11-3. Enhanced x-ray

diffraction intensities for (101) of filtered particles

supports the contention that plane C is (101) (29).

Calculation of the Lifshitz-van der Waals Constant

Background

Let the difference of separation for two identical

macroscopic bodies, composed of material T, interacting

across a vacuum, be 1000 A or less, and assume a single

optical absorption frequency, then the Lifshitz-van der

Waals constant (AL(r r)) as given by Gregory (32) is

(8 1)2

AL(T T) = 0.230 hv (II- )

L (E + 1)3/2 ( + 2)1/2'

The h is Planck's constant and vv is the characteristic

optical absorption frequency for the static dielectric

permeability, e .

It is possible to combine AL(T T) to obtain

AL(T H20 x) (i.e., AL for dissimilar materials

interacting across water) to within at least one-half an

order of magnitude (33). Utilizing the relationships given

by Visser.(34)

AL(T H20 T) = c[A (T T) + AL(H20 H20) (11-2)

-2AL(T H20)],

and

AL(T H20 x) = c[AL(T x) + AL(H20 H20) (11-3)

AL(T H20) AL(x H20)],

with

AL(T H20) = [AL(T T) AL(H20 H20)]2

where c = 1.6 for aqueous systems.

To calculate AL(T H20 -r) or AL(r H20 x), static

dielectric permeabilities and corresponding dispersion

frequencies for COM and water must be known. These values

are readily obtainable for water (32). However, since COM

is crystallographically anisotropic (i.e., monoclinic

crystal structure), three static dielectric permeabilities

and dispersion frequencies have to be determined (35).

Since E = n 2 (36) (i.e., the square of the refractive

index of a material at the characteristic optical absorption

frequency), e0 and vv may be found by analyzing an

expression given by Gregory (32) and Visser (34) with

2 e N

n 1 p o r ( II-4)

2 r (11-4)

n + 2 M 3m e V- v )

where n is the refractive index at frequency v in

Hertz, Mw is the molecular weight of the material in grams

mole"1, and p is the density in grams cm"3. No is

Avogadro's number, me is the electron rest mass in grams,

and r is the effective number of electrons of characteristic

optical absorption frequency, v in each molecule of the

material.

Optical dispersion10 tables (37) give indices of

refraction for the optical directions in COM as a function

of frequency. A plot of (n2 + 2)/(n2 1) as a function

of v2 gives straight lines for each optical direction.

Values of e and vv can be estimated from these plots.

Once e and u are established for a particular optical

direction, use of Equations II-i, 11-2, and 11-3 gives

Lifshitz-van der Waals constants for the interactions of two

COM particles along all combinations of optical directions

(e.g., since there are three optical directions for each

particle, six AL are calculated).

10. "Dispersion" is used in physics (36) to describe

dependence of any variable upon frequency (e.g., for

the above, the index of refraction). It should not be

confused with "dispersion" as used in surface chemistry

(13) to describe reduction of coagules to primary

particles. To avoid confusion, "dispersion" in the

colloid sense will not be used. Instead, peptizationn"

(13) will be used to describe reduction of coagules.

Orientation of Lifshitz-van der Waals constants with habit

pl anes

The relationship between optical directions and habit

planes in COM model particles gives the orientation of

Lifshitz-van der Waals constants with habit planes. The

orientation of optical directions with habit planes for COM

(31) are indicated in Figure 11-2. The a optical direction

is parallel to [010] (i.e., perpendicular to plane A).

The y optical direction is aligned 30 degrees away from

[001] and 9 degrees away from [101]. Since optical

directions in biaxial crystals are mutually perpendicular

(38) a and y give the 8 optical direction 90 degrees away

from [010] and 60 degrees away from [001].

Lifshitz-van der Waals constants for COM

Static dielectric permeabilities and dispersion

frequencies for COM as a function of optical direction are

given in Table II-1. Using the values in Table II-1 and

1.75 and 3.35 x 1015 Hz respectively, for e and v for

water, Lifshitz-van der Waals constants were calculated for

different orientations of COM particles. The results of

these calculations are given in Table 11-2. Values ranged

from 7.13 x 10-14 ergs for a H20 a to 16.26 x 10-14 ergs

for y H20 y interactions. The mean AL is 11.35 x 10-14

ergs 3.41 x 10-14 ergs. These values are consistent with

Lifshitz-van der Waals constants found for similar materials

interacting in water (34).

TABLE II-1

STATIC DIELECTRIC PERMEABILITIES AND OPTICAL ABSORPTION

FREQUENCIES FOR CALCIUM OXALATE MONOHYDRATE

Static Optical Absorption

Optical Dielectric Frequency (x 1015

Direction Permeability Hertz)

a 2.183 3.33

8 2.361 3.19

y 2.642 2.88

TABLE 11-2

LIFSHITZ-VAN DER WAALS CONSTANTS FOR DIFFERENT

ORIENTATIONS OF INTERACTING COM PARTICLES IN WATER

Type of AL from Equation

Interaction (x 10-14 erg)

a H20 a 7.13

B H20 8 11.34

y H20 y 16.26

a H20 8 8.99

a H20 y 10.77

8 H20 y 13.58

10-14 + 3.41 x 10-14 erg

Mean AL 11.35 x

Conclusions

A reproducible batch method for producing COM particles

that are essentially flat plates was developed. Particle

size distributions in suspensions prepared from a standard

procedure were narrow, with mean equivalent spherical

diameters similar to COM crystals spontaneously occurring in

human urine. Particle number concentrations were

sufficiently high in standard suspensions for aggregation

measurements.

The Lifshitz-van der Waals constant was calculated for

all combinations of the a, B, and y optical directions of

COM particles interacting in water, using approximate

equations given by Gregory. The mean Lifshitz-van der Waals

constant was consistent with Lifshitz-van der Waals

constants found for similar materials interacting in

water. Crystallographic and optical direction orientations

were determined for the habit planes in model COM particles.

CHAPTER III

DETERMINATION OF SURFACE AND SOLUTION CHEMISTRY OF

PARTICLES IN THE MODEL SUSPENSION SYSTEM USED IN

COAGULATION EXPERIMENTS

Introduction

The surface characteristics and ionic equilibria of COM

suspensions had to be determined in order to compare

theoretical and experimental coagulation behavior (1). To

make theoretical calculations (Chapters VI and VII), the

Stern potential (*,), characterizing the solid-solution

interface, and the activities of ionic species present in

solution must be known (1). Experimentally, it is difficult

to determine Stern potentials at the high ionic strengths

required for the coagulation studies. However, previous

investigations (2,3) have shown that theoretical and

experimental Stern potentials for COM suspensions with low

ionic strengths are in agreement as long as the pertinent

surface and solution chemistry features are known.

Therefore, theoretical Stern potentials could be used to

characterize suspensions at high ionic strengths. We needed

to show that the surface and solution chemistry of standard

COM suspensions (Chapter II) are consistent with the

findings of previous studies.

Materials and Methods

Solutions and COM Suspensions

Electrolyte solutions were prepared using deionized,

distilled water and reagent grade chemicals. COM

suspensions were prepared by the procedure given in Chapter

II. Suspensions were prepared with additions of potential

determining ions (i.e., Ca2+ and C242-), indifferent

electrolytes (i.e., NaC1 and KC1), and specifically

adsorbing ions, pyrophosphate as Na4P207'10H20 and citrate

as Na3C6H507"2H2O (2). COM suspensions were equilibrated in

their suspending medium for at least 24 hours at 380C before

use.

Solution pH1,2 of the COM suspensions was measured at

380C 1C after experimental determination of

electrophoretic mobility of the COM particles. Total

concentrations of added electrolytes were measured by atomic

absorption (Chapter II).

Ionic Equilibria

Equilibrium activities of the ionic species dissolved

in COM suspensions were obtained using a computer program,

1. Radiometer PHM82 Standard pH Meter, The London Company,

Cleveland, Ohio.

or

2. Metrohm pH Meter E512, Brinkmann Instruments, Westbury,

N.Y.

EQUIL (5) (See Appendix D for a listing). The stability

constants used to determine the activities of ionic species

are given in Table III-1. Based on total concentrations of

ionic species and measured pH values, EQUIL first calculates

ion equilibrium with total solution of salts, and then

adjusts for the precipitation of COM until the solution is

saturated. It uses the Davies modification of Debye-Huckel

theory to obtain activity coefficients. At ionic strengths

greater than 0.1 M, some of the assumptions for activity

coefficients start to fail (e.g., in this work, the maximum

error in calculating a divalent activity coefficient at

higher ionic strengths is about 5%).

Theoretical Stern Potentials

The NGS model used for calculations of theoretical Stern

potentials

In a previous study (2), the composite NGS model

(39,40,41) was suitable for calculations of theoretical

Stern potentials. A schematic representation of the NGS

model is given in Figure III-i. This figure shows the

double layer developed when the surface potential (4 ) is

positive, and specifically adsorbing negative ions are

present. However, any combination of negative or positive

surface potential with any combination of negative or

positive chemically adsorbed ions may apply.

Charge at the surface in the NGS model arises from the

adsorption of potential determining ions. These are the

TABLE III-1

STABILITY CONSTANTS USED IN

ION EQUILIBRIUM CALCULATIONSa(5)

Reaction

Ca2+ + C204 CaC204

Na+ + C2 4 + NaC204

2-+-

Ca2+ + CaC204 + Ca2C204 2

H + CO2- +

H + C204 + HC204

Ca2+ + OH" + CaOH

K + C204 + KC204

+ 2- +

NH4 + C204 + NH4C204

Ca2++ HPO4 2 CaHPO4

Ca2+ + H2PO4 + CaH2PO4

2+ +

H+ + HPO42" 4 + H2PO4"

2- +

HPO4 + H2PO4

Na+ + HPO 2 + NaHPO4

4 4

+ 3- + 2-

H + PO + HPO

continued

continued

Stability

Constant

(M-1)

0.274 x 10

0.134 x 10

0.714 x 10

0.215 x 10

0.295 x 10

0.134 x 10

0.130 x 10

0.319 x 10

0.319 x 10

0.152 x 10

0.129 x 10

0.152 x 10

4

2

2

5

2

2

3

2

2

8

2

13

Table III-1 continued:

2+

Ca2

H +

Mg2+

2+

Mg2+

Ca2+

Na+

H+ +

Ca2+

H+ +

Ca2+

H +

H+ +

Reaction

3- +

+ P04 + CaPO4

H2PO4 + H3P04

2- +

+ C204 + MgC204

+ 2+

+ MgC204 + Mg2C2042

+ OH- + MgOH+

2- +

+ SO4 + CaSO4

2- +

+ SO4 + NaSO4

2- +

S042 HSO -

3- +

+ C6H507 + CaC6H'07

3- + 2-

C6H5 07 HC6H5 07

2- +

+ HC6H507 + CaHC6H507

2- +

HCG6H507 H2C6H507-

H2C6H507 + H3C6H507

Stability

Constant

(M-1)

0.346 x 107

0.164 x 10

0.402 x 104

0.475 x 101

0.380 x 103

0.200 x 103

0.525 x 101

0.100 x 103

0.600 x 105

0.272 x 107

0.505 x 103

0.561 x 106

0.127 x 10

continued

Table III-1 continued:

Stability

Constant

Reaction (M-1)

2+ + + 2

Ca + H2C6H507 + CaH2C6H507 0.125 x 10

H++ P207 + HP2 07 0.615 x 1010

+ 3- + 2- 7

H + HP2073 + H2P 2072 0.615 x 10

H + H2P207- H3P207 0.190 x 102

2+ 4- + 2- 5

Ca + P207 +4 CaP2072 0.562 x 105

Ca2 + P207 4 + CaHP207 0.550 x 10

Ca0H+ + P2074" + Ca0HP2073 0.269 x 108

a The complex CaC204 has a concentration in solution at

380C of (6.16 0.38) x 10-6 M.

I-

I-

z 0

U -

Stern Gouy-Chapman

Layer Layer

DISTANCE FROM SURFACE X

Figure III-1.

Schematic of NGS model of the double layer.

Specifically adsorbed ions are shown in the

Stern Layer reversing charge to a negative

value at 6 followed by an exponential

decrease in potential with increasing dis-

tance in the Gouy-Chapman Layer.

ions which make up the solid (e.g., Ca2+ and C2042- for

COM). The Nernst equation (1) gives the relationship

between o and activity of the potential determining ions in

solution such that

kT (Ca2+)

o =- In 2 (III-1)

ze (Ca )z

pzc

where (Ca2+)pzc is calcium ion activity when (o = 0, z is

the valence of the potential determining ions, and k, T, and

e are the Boltzmann constant, absolute temperature, and

electronic charge, respectively.

The Stern layer (i.e., the layer in which specific

adsorption occurs) is composed of up to a monolayer of

adsorbed ions. For the example given in Figure III-1, the

potential is negative when distance from the surface equals

6 (the thickness of the Stern layer) where the Stern plane

occurs. One of the Stern equations as given by Grahame (42)

is

I

DD

a = (o 9 ) (III-2)

47iS

where o is total surface charge density in C and cm-2, Do is

equal to 1.112 x 10-12CV'1cm-1, and D' is the dielectric

constant of the Stern layer. Another Stern expression gives

the charge density in the Stern plane (aS) due to

specifically adsorbed species (42) such that

N1zie

S = [ ]. (111-3)

S 1 N z e(6 -@i

1 + --- exp

niMS kT

N1.is the number of adsorption sites per square centimeter

of surface, N is Avogadro's number, MS is the molecular

weight of solvent, ni is the number of ions per cubic

centimeter of specifically adsorbing species i, and 4i is

the specific adsorption potential of species i.

The Gouy-Chapman layer extends from the Stern plane

into the bulk solution. Unlike ions in the Stern layer,

ions in the Gouy-Chapman layer are only held by

electrostatic forces. The charge in this region (oGC) is

given as (42)

DD kT zie*6 ,

G = T /2 Eni(exp(- kT 1)]2, (III-4)

S2w 1

where D is the dielectric constant of the bulk solution.

The sign of aGC is opposite that of .

Last, the requirement of charge neutrality in the

system of solids and solvent is expressed as (1)

a + aS + aGC = 0. (II1-5)

Approach used in calculating theoretical Stern potentials

Theoretical Stern potentials were calculated using the

et al. (3). A computer program (listed in Appendix D as a

subroutine (Double Layer) in EQUIL) similar to the program

written by Curreri (2) was used to numerically analyze the

simultaneous equations for the NGS model (i.e., equations

(III-1) through (111-5)). Activities of ionic species

required in Double Layer calculations were provided using

EQUIL.

Experimental Stern Potentials

Calculation of zeta potentials from electrophoresis

measurement

Stern potentials are difficult to directly determine

experimentally. Usually electrophoretic mobilities of

particles in suspensions are determined. From such values

zeta potentials (ZP) are calculated using the Smoluchowski

equation (43)

4 iTnU

ZP = (III-6)

D

ZP is in volts, n is absolute suspension viscosity in poise,

D is relative dielectric constant of the suspension, and U

is electrophoretic mobility in um sec -/V cm-1. The

Smoluchowski equation was derived for spherical particles.

In this work, it is assumed that ZP is equal to

*9 While the validity of this assumption is debatable,

Lyklema (44) has given strong support to it. More recently,

40

TABLE III-2

CONSTANTS USED IN THEORETICAL CALCULATIONS

OF STERN POTENTIALS (2)a

Adsorption Potentials

Ionic Species

Ca2+

Mg2+

C2042-

S042-

HP042-

P2074-, MP2073-, M2P2072"

Adsorption Potentials

i (mV)

130

112

100

93

110

113

C6H5073- 107

3- 2-

P043-, MC6H507 and all 0

monovalent species

Other Constants Used In Theoretical Calculations

of 6 :

N1 = 4.06 x 1013 Adsorption sites cm-2

(Ca2+)pzc = 7.85 x 10-6 M

D = 8.21 x 10-11 F cm-1

D' = 6.5

6 = 5 x 10-8 cm

a. M is a univalent cation given in the examples.

Curreri et al. (3) found that ZP for COM suspensions

compared favorably to theoretical 6 when adsorption

potentials for specifically adsorbed species were used as

adjustable parameters. Therefore, zeta potentials are

referred to as experimental Stern potentials in the

remainder of the discussion.

Retardation and relaxation effects should cause only

minor errors by using the Smoluchowski equation to calculate

zeta potentials when ZP < 50 mV, or when

-I

Ka > 100 (45,46,47). K1 is the Debye thickness of the

double layer and a is particle radius. Other effects, such

as surface conductance, may give rise to inaccuracies using

the Smoluchowski equation. However, these effects are

difficult to analyze.

Zeta potentials calculated using the Smoluchowski

equation are inaccurate if particle anisotropy is

appreciable. Modifications to the equation for rod-shaped

particles predict a decrease in electrophoretic mobility for

a given ZP (48,49,50). An average U based upon possible

orientations a rod-like particle has in an electric field is

given by (48)

1 D 1 2

__ - + ] ZP, (III-7)__

3 en 4 F(Ka')

where is the average electrophoretic mobility

and F(Ka') is a function dependent on K and the radius of

the rod (a'). F(Ka') varies between 4 and 8.

Equation III-7 is related to the Smoluchowski equation

1 2

by G(Ka') = 3/(4[4 + F(-a')]). Depending upon the degree of

anisotropy, G(Ka') varies from 1.0 to 1.5 (e.g., for

spherical particles and long narrow rods, respectively)

(50). In the absence of other deviations, reasonable values

may be found for G(Ka') to account for the effect of shape

anisotropy by comparison of theoretical and experimental

Stern potentials.

Determination of electrophoretic mobilities of COM

suspensions

A commercial instrument3 was used to determine

electrophoretic mobilities. All components of the apparatus

except the power source were contained in a controlled

temperature box maintained at 380C 10C. The mobilities of

at least twenty particles in a suspension were averaged to

yield each data point.

Results and Discussion

Potential Determining Ions

Stern potential- as a function of the logarithm of

calcium ion activity for standard suspensions is given in

Figure 111-2. The data points give experimental Stern

potentials calculated using the Smoluchowski equation.

3. Zeta-MeterR Zeta-Meter, Inc., New York, N.Y.

43

+40

S 0 Experimental

+30 Theoretical

S+20-

-10

-10 /i

-6 -5 -4 -3 -2

LOG (Ca 2+)

Figure III-2.

Stern potential as a function of the loga-

rithm of calcium ion activity in molar

units for COM suspensions. Data points

indicate experimental Stern potentials

determined using the Smoluchowski equation.

Solid line gives theoretical values.

Values of pH ranged from 6.3 to 7.2.

potentials calculated using the Smoluchowski equation.

Theoretical Stern potentials are indicated by the solid

line. The values of pH for the experimental data ranged

from 6.3 to 7.2. Experimental Stern potentials are not

given for a wider range of (Ca2+) because it was difficult

to measure U in suspensions other than those reported.

A lack of quantitative agreement between experimental

and theoretical *, is indicated by the figure. Solution

analysis by atomic absorption indicated suspensions were

properly prepared and Ca2+ concentrations were consistent

with those predicted by EQUIL. Assessment of solid loading

on equilibrium conditions indicated suspensions were at

equilibrium. Experimental Stern potentials using COM

particles similar to those used by Curreri indicated the

experimental procedure was correct.

Since no major differences were found in the solution

chemistries of standard suspensions and theoretical values,

the effect of particle anisotropy on experimental Stern

potentials was assessed. Using G(Ka') = 1.5, experimental

Stern potentials were calculated from electrophoretic

mobilities. These values and theoretical Stern potentials

as a function of log(Ca2+) are summarized in Figure III-3.

Much better agreement was found when corrections for

particle anisotropy were used.

Since Curreri used isotropic-shaped aggregates, he was

fully justified in using the Smoluchowski equation to

+40 -- a I

0 Experimental .

,Theoretical

+30

S+20

+

z +10

0

0 /

-10

-10 ----__-- ---- ---- ------------ ---

-6 -5 -4 -3 -2

LOG (Ca2+)

Figure 111-3.

Stern potential as a function of the loga-

rithm of calcium ion activity in molar units

for COM suspensions. Data points are experi-

mental values using G(Ka') = 1.5 for the

modified Smoluchowski equation. Solid line

gives theoretical values. The pH values

ranged from 6.3 to 7.2.

calculate zeta potentials. The results give good agreement

between zeta potentials for Curreri's work on isotropic

particles and our work on flat plate-shaped particles so

long as anisotropic effects are taken into account in zeta

potential calculations.

Indifferent Ions

Figure III-4 summarizes theoretical and experimental

Stern potentials for COM suspension as a function of added

NaCl and KC1 using the modified Smoluchowski equation

(i.e., G(Ka') = 1.5). Values of pH ranged from 6.8 to 5.7

and 6.8 to 5.8 for NaCl and KC1, respectively. With errors

in experimental Stern potentials up to 5 mV, reasonably good

agreement between theory and experiment is indicated, except

at higher concentrations of NaCl (i.e., 9 x 10-3 M). Since

Curreri (2) also observed this behavior at higher NaC1

concentrations, it may be due to specific adsorption at

higher Na+ concentrations. However, sodium ion adsorption

is not taken into account in calculations using Double Layer

(i.e.,
Na

Specifically Adsorbed Ions

Figure III-5 summarizes theoretical and experimental

Stern potentials (i.e., using the modified Smoluchowski

equation) as a function of added Na4P207 or Na3C6H507

Additions of HC1 or NaOH were used to control pH values from

7.00 to 7.85 and 6.95 to 7.40 for Na4P207 and Na3C6H507

E

W Experimental Theoretical

+NaC

-J

0

D NaCl ________

-10 0 KC1

-0 -5 -4 -3 -2

LOG [CONCENTRATION]

Figure 111-4.

Stern potential as a function of the logarithm of

electrolyte concentration in molar units for COM

suspension. Correction factor of G(Ka') = 1.5 for

the Smoluchowski equation was used to calculate

values indicated by data points. Solid line gives

theoretical Stern potentials. Values of pH ranged

from 6.8 to 5.7 and 6.8 to 5.8 for NaCl and KC1,

respectively.

48

+20 i

7.0 Na3C6H507

+10 Na4 0

- 0

_ j 27.

27.2

S-10 \

C 7.2

-20

7.3

7.4 7.1

-30 7.

-o -5 -4 -3

LOG [CONCENTRATION]

Figure 111-5.

Stern potential as a function of the loga-

rithm of concentration in molar units for

sodium citrate and sodium pyrophosphate in

COM suspensions. Data points indicate

experimental Stern potentials calculated

using a correction factor of G(Ka') = 1.5

in the Smoluchowski equation. Lines give

theoretical values. Values of pH controlled

by additions of HC1 or Na0H are indicated.

additions, respectively. Theoretical and experimental Stern

potentials are in agreement within the errors present for

experimental data (i.e., up to 5 mV).

Conclusions

There was good agreement between the zeta potentials of

Curreri's studies on isotropic-shaped COM aggregates and

model particle COM suspensions as long as particle shape

anisotropies of model particles were accommodated in the

Smoluchowski equation. Also, there was good agreement

between theoretical and experimental Stern potentials using

a Smoluchowski equation modified to correct for the particle

shape anisotropy. Measured ion concentrations of

equilibrated standard suspensions were consistent with

theoretical predictions and validate the ion equilibrium

calculations. The results indicate the NGS model gives an

adequate description of the double layer surrounding the

model particle system. The computational techniques were

adequate to calculate COM Stern potential if the supporting

electrolyte solution was specified.

CHAPTER IV

DEVELOPMENT OF AN ANALYTICAL METHOD TO DETERMINE

THEORETICAL COAGULATION BEHAVIOR

Introduction

The objective of the work presented in this chapter was

to develop an analytical method to calculate theoretical

coagulation behavior in solutions of arbitrary complexity.

Historically, implementation of classical coagulation theory

has been limited to simple ionic equilibria (1). The simple

thermodynamic expressions for coagulation assuming two ionic

species in solution were analytic, such that they could be

solved without resorting to numerical analysis using a

digital computer. However, to theoretically calculate

coagulation behavior in urine-like solutions which contain

many ionic species, we needed to use thermodynamic

expressions for solutions of arbitrary complexity.

Therefore, a procedure was developed which used generalized

equations (i.e. obtained using the assumptions and models

from classical coagulation theory (1) in Appendix B and

verified in Appendix C) to describe theoretical coagulation

behavior in these solutions.

In this chapter, the thermodynamic expressions utilized

to describe theoretical coagulation behavior are given. The

strategies, constants, and variables which are used in a

computer program to analyze the thermodynamic expressions

are discussed. Finally, a hypothetical calculation is

presented showing theoretical coagulation behavior as a

function of ionic strength. In the example, emphasis is

given to the theoretical parameters which relate theory to

experiment.

Theoretical Models and Thermodynamic Relationships

for Interacting Particles

Physical Models

To compare experimental and theoretical coagulation

behavior required physical modelling of particles present in

suspensions (1). Semi-infinite flat plates and spherical

morphologies lend themselves to quantitative, theoretical

descriptions. The model particles in COM suspensions are

essentially rectangular plates (Chapter II).

Experimentally, particles in COM suspensions are likely to

interact at edges and corners, as well as faces. It is

difficult to theoretically analyze particle interactions due

to interactions at particle corners or edges (51). However,

consideration of interactions at corners or edges may not be

needed to compare theory and experiment because it is

probable that coagulated particles will re-orient themselves

into a face to face configuration to obtain the lowest total

interaction energy (52). Particle surface roughness and its

effect on total interaction energies is difficult to analyze

(53). In theoretical calculations, the magnitude of total

interaction energy is dependent on the surface area of

interaction (1). Minimum and maximum interacting surface

areas are given by spherical and flat plate interactions,

respectively (1). Ignoring edge and corner interactions and

surface roughness effects, particles modelled as spheres or

flat plates will give the minimum and maximum total

interactions energies, respectively. Therefore, theoretical

calculations'to be compared to experimental values were made

using spheres and flat infinitely thick plates to model COM

particles in suspensions. Radius of the spheres and cubic

face areas were obtained from Coulter Counter volume data

for theoretical calculations.

Thermodynamic Relationships

Attractive energies between particles

Two particles separated by a given distance in a

suspension undergo mutual van der Waals interactions (54).

Only the London energy contributions to the van der Waals

interactions are important in an analysis of particle

coagulation (55). London-van der Waals interactions occur

when charge fluctuations in atoms or molecules of one

particle create induced dipoles in atoms or molecules of

another particle (33). These interactions produce a net

attractive energy between the particles (33). London

interactions are retarded by the finite propagation time of

the electromagnetic fluctuations if separation distances are

equal to or greater than the London wavelength (XL) (e.g.,

usually about 1000A) (56). An adsorbed layer (e.g., the

Stern plane of thickness, 6 (Chapter III)) also reduces

attractive energies (57). Including retardation and Stern

layer effects, attractive energies between two semi-infinite

flat plates (VA-pl(d + y)) as a function of separation

distance are given by (58,59)

AL 2r(d + y)

VA (d + y) = ---- 2 [1.01 0.28

A-pl 127(d + )2 XL

20(d + y) 2 0 2(d + y) 3

+ 0.1735 ( ) 0.028 ( )

XL xL

2ir(d + y) 4

+ 0.00193 ( ) )

xL

3

for 0 < d < L (IV-1(a))

2T

and

A 0.098X

V (d + y) = L L

Apl -12n(d + Y)2 (2n(d + y))

3

for X, < d < (IV-1(b))

2 L

where VA-pl is in ergs cm-2 at the surface to surface

separation distance, d + 6,, AL is the Lifshitz-van der

Waals constant equal to 1.135 x 10-13 erg for COM (Chapter

II), d is the separation distance between the Stern planes

of the two particles, and y = 26. Unless otherwise

specified' the cgs system of units is used for all

variables.I If the surface areas of interacting flat plates

are known, attractive energies for a pair of particles may

be calculated in ergs (i.e., for given faces of flat plates)

assuming the lines of force remain parallel at face edges

(60).

The attractive energies from interactions between two

large spherical particles (VA-sp(Ho + y)) (i.e.,

Ka > 2.5) are given as a function of the distance of closest

approach (Ho) between the spheres (60). Again including

retardation and Stern layer effects, and assuming Ho<
(particle radius), VA-sp(H + y) is given by (1,60)

ALa 1

VA sp(H0 + y) = [ -,

12(Ho + y) 1 + 1.77p

0 < p < 1 (IV-2(a))

and

ALa 2.45 2.17 0.59

VA-sp(H0 + Y) = + -

A-s (H0 + y) 60p 180p2 420p

1 < p < (IV-2(b))

with p = [2(Ho + y)]/XL'

where V -sp(H0 + y) is in ergs.

Repulsive energies between particles

In coagulation theory (1), repulsive energies arise

from the impingement of Gouy-Chapman layers. Overlap of

Stern layers cannot occur because of Born repulsions arising

from adsorbed ion interactions. Contemporary developments

of theoretical expressions for repulsive energies were made

assuming simple solutions (e.g., two electrolytes with equal

valences) (1). Therefore, one of our major goals was to

utilize more general equations and develop a computer

program to analyze the equations to yield repulsive energies

in multi-ion solutions such as urine.

A detailed discussion of the general equations utilized

for repulsive energies is given in Appendix B. The

repulsive energy for flat semi-infinite plates (VR-pl(d)) as

a function of separation distance is given by (1)

VR-pl(d) = 2(F(d) F(-)). (IV-3)

VR-pl(d) is in ergs cm-2, F(d) is the free energy of one of

the Gouy-Chapman layers at a given separation distance,

and F(-) is the free energy of the undisturbed double

layer. An absolute repulsive energy for plates may be

calculated if the dimensions of finite flat plates are known

assuming parallel lines of force at the edges (60).

When Gouy-Chapman layers of two large spherical

particles (1) (i.e., Ka > 2.5 (Appendix C)) impinge, the

repulsive energy is obtained from an integration of

repulsive energies over their surfaces (60). Assuming lines

of force between particles are parallel, VR-pl(d) gives the

repulsive energy for the interaction of localized areas on

the faces of two spheres separated by a distance d. A

summation of all repulsive energies for the curved surfaces

gives the total repulsive potential for the spheres. The

repulsive energy as a function of minimum separation

distance (Ho) for two large spherical particles (VR-sp(Ho))

is given by (60)

VR-sp(H ) = ia f VR pl(d) dd (IV-4)

Ho

where VR-sp(Ho) is in ergs, and Ho and radius (a) are in cm.

The free energy of a single, planar interacting Gouy-

Chapman layer as a function of separation distance (i.e., d

varied from zero to -) is given by

kTd zie d/2

F(d) = [--][zn.(exp(- /) 1)]

2 kT

DD kT 12, ziep(x)

[ ] 2 [En ((exp(- )

2xr ,d/2 n(exp kT

exp(- ed/2 )) d (IV-5)

kT

where id/2 is the electric potential at the midpoint between

the Stern planes. For an undisturbed double layer extending

a distance approaching infinity into the bulk solution,

*d/2 = 0, and equation (IV-5) reduces to

DD kT V.6 z.e (x) 1

F() = [] [n (exp(- 1) 2 d (IV-6)

2w1 o kT

The relationship between separation distance and *d/2'

needed to calculate F(d) in equation (IV-5), is given by

d i 2d*

d = 8rkT zi(x) z z /2 /

Dd/2 E-DD-1 [Eni(exp( exp( /2))]2

kT kT

(IV-7)

The endpoint where i(x) approaches *d/2 may be adequately

represented by an extrapolation function.

Total interaction energy between particles

Total interaction energy as a function of separation

distance is given by a superposition of attractive and

repulsive energies. For semi-infinite flat plates, this is

given by (1)

VT-pl(d + y) = VA-pl(d + y) + VRpl(d), (IV-8)

where VT-pl(d + -y) is total interaction energy as a function

of separation distance. VT-pl(d + y) and VA-pl(d + y) are

dependent on separation distances between solid surfaces,

while VR-pl(d) is dependent on Stern plane separation

distances.

For large spherical particles, total interaction energy

(VT-sp(Ho + y)) is given by (1)

VT-sp(H + Y) = VA-sp(Ho + Y) + VR-sp(Ho). (IV-9)

Theoretical Stability Ratios

Theoretical stability ratios for coagulation provide a

convenient way to compare theory and experiment (1).

Stability ratios predict whether or not a suspension

undergoes primary minimum coagulation. Stability.ratios are

related to VTsp(H + 6) and VA-sp(H + 6) in an expression

given by McGown and Parfitt (61) where

Sexp(VT-p(s)/kT)

Wt 2 s (IV-10)

t exp(VAsp(s)/kT)

f 2 ds

2 s

The s is a change of separation distance variable from

H + y

(H + 6) to a dimensionless value given by s = 2 + 0

a

Theoretically, stability ratios are related to

coagulation rates of suspension. When there is no energy

barrier present in total interaction energy curves, rapid

coagulation rates are theoretically possible equal to

(dN/dt)fast. However, when an energy barrier is present

(e.g., as in Figure I-1), decreased coagulation rates equal

(dN/dt)slow. The coagulation rate (dN/dt)fast occurs when

Wt = 1, such that no energy barrier is present and VT-sp(s)

= VA-sp(s) in Equation IV-10. The coagulation rate

(dN/dt)slow occurs when Wt > 1, such that an energy barrier

is present between two approaching particles and every

collision between particles does not create a coagule, as

opposed to Wt = 1 which theoretically does. Theoretical

stability ratios in terms of coagulation rates are given by

(16)

(dN/dt)fast

W = (IV-11)

(dN/dt)slow

where coagulation rates are given in particles ml1.

Equation IV-11 is an important relationship used in

Chapter V to theoretically relate theoretical and

experimental stability ratios analyzing primary minimum

coagulation. Equations IV-10 and IV-11 theoretically relate

thermodynamic factors to kinetic aspects, such that

theoretical stability ratios may be compared to experimental

stability ratios obtained from experimentally determined

coagulation rates. The details of the comparison between

theoretical and experimental stability ratios are given in

Chapter V.

Calculations of Total Interaction Energies

and Stability Ratios

Equations IV-4, IV-5, IV-6, IV-7, and IV-10 cannot be

analytically integrated (1). Therefore, a computer program,

DLVO (Appendix A), was written to numerically integrate the

expressions needed to give total interaction energies and

theoretical stability ratios. Although a different computer

program has been developed by previous investigators to

calculate total interaction energies (62), it was based on

less general equations (1) restricted to two ionic

species. DLVO calculations in this work are not restricted

by the solution chemistry since they are based on the

general repulsive energy equations given above.

Constants Used in DLVO

Table IV-1 summarizes the constants used in DLVO. The

Lifshitz-van der Waals constant for COM was calculated in

Chapter II. The absolute distance of closest approach (ya)

is the average interplanar distance for the habit planes of

the model particles (Chapter II). The London wavelength

(XL) was calculated from the average optical absorption

frequency for COM (Chapter II). The values of other

constants were taken from the appropriate references for a

temperature equal to 380C (i.e., normal human body

temperature).

TABLE IV-1

CONSTANTS USED IN DLVO FOR COM SUSPENSIONS

Constant Value Reference

Lifshitz-van der Waals 1.135 x 10-13 ergs (Chapter II)

Constant (AL)

Absolute Distance of 6.306 x 10-8 cm (Chapter IV)

Closest Approach (ya)

London Wavelength (x) 9.73 x 10-6 cm (Chapter IV)

Boltzmann Constant (k) 1.38054 x 10-23 J K-1

Temperature (T) 3.1116 x .102 OK (Chapter IV)

Relative Dielectric 7.3825 x 101

Constant of Water

(D) at 38C

Dielectric Permit- 1.112 x 10"12

tivity (Do)

Electron Charge (e) 1.6021 x 10-19 C

Data Necessary for Calculations Using DLVO

Calculations of theoretical coagulation behavior using

DLVO require: number, concentrations, and valences of ionic

species in solution; Stern potential; and volume of

particles. Number, concentrations, and valences of ion

species are obtained from EQUIL calculations (Chapter

III). Stern potentials are obtained experimentally or,

preferably, from Double Layer calculations (Chapter III).

Mean volumes of particle size distributions in COM

suspensions are given by Coulter Counter analysis

(Chapter II).

Computer Strategies Used in DLVO

Main program

DLVO first numerically integrates the second term in

Equations IV-5 and IV-7 using Simpson's Rule (63) for one

hundred dJ/2 values in equal increments from zero to *,.

This establishes the relationship between d and *d/2 so that

the first term in Equation IV-5 may be calculated. VR-pl(d)

may then be evaluated from Equation IV-3 using F(-) as a

constant given by Equation IV-6. The trapezoid rule (63) is

used to integrate VR-sp(d) from VR-pl(d) values (i.e., by

using Equation IV-4). Attractive energies are calculated as

functions of d and ya (i.e., using Equations IV-1 and IV-

2). Summations of repulsive and attractive energies

yield (VT-pl(d + y) and (VT-sp(H + y) as functions of

separation distances (i.e., Equations IV-8 and IV-9). The

stability ratio is calculated using the trapezoid rule (63)

to integrate Equation IV-10 with the change of variable

H + Y

(i.e., s = 2 + (- ) ) and VT-sp(s). A typical output

a

of values calculated using DLVO is given in Appendix A

preceded by a listing of DLVO.

Errors for integration using Simpson's Rule were

estimated from the difference between two successive

iterations (63). Errors using the trapezoid rule are

estimated by the difference between integration using 100

points and 50 points (63).

Optional plotting subroutine in DLVO

DLVO contains an optional subroutine, J1PLOT1, to graph

total interaction energies as a function of separation

distance. Ionic strength, Stern potential, and stability

ratio are included in a separate heading on the graphs.

Graphs generated by J1PLOT allow rapid analysis of the total

interaction energies as a function of separation distances.

1. Author: Robert Gould, Jr., Software Consultants, Inc.,

Gainesville, Fl.

Hypothetical Calculations Using DLVO

Total Interaction Energy Curves

Hypothetical interaction energy curves calculated using

DLVO as a function of ionic strength (I) are given in Figure

IV-1. The spherical radius was the mean esr for COM

suspensions (Chapter II). Since Stern potential varies as a

function of ionic strength (Chapter III), this example is

unrealistic in using = 20 mV for all the solutions

analyzed. However, it gives a general idea of the effect of

ionic strength on total interaction energy curves. Other

parameters pertinent to the calculations are indicated in

the figure. Most of the curves (i.e., I = 10-2 M to 10-1 M)

indicate resistance to primary minimum coagulation (i.e.,

the minimum located at d m 1 nm). Primary minimum

coagulation occurs when the positive energy barrier, located

between 1 nm and 4 nm, is not present as indicated by the

total interaction energy curve for I = 1.8 x 10-2 M. The

secondary minima which occur at d = 5 nm to about 20 nm for

I = 10-2 M to 10-1 M solutions and their relationship to

coagulation are discussed below.

Theoretical Stability Ratios

Using total interaction energy curves to directly

assess primary minimum coagulation is unwieldy if a large

number of suspensions need to be analyzed. A more

convenient method is to use theoretical stability ratios

which give a single value for each interaction energy

- +450 -2 (1-1) .Electrolyte

1 x 10 M a = 5.31 rm -

-13

A = 1.135 x 10 ergs

>_ T = 38C

S+30 1.8 x 10 M = 6.306

u +300

w

2 --3 x 10-2 M

+150

C--2

U

LT 4-5.6 x102 M

-1

0 -1

A- 1.8 x 10 M

-150 -- -- i *- I '- *

0 20 40 80

DISTANCE OF SEPARATION (nm)

Figure IV-1.

Hypothetical total interaction energy in units of kT as a

function of particle-particle separation distance in nano-

meters for various ionic strengths in molar units. Vari-

ables held constant are indicated. The spherical model

was used in DLVO calculations of total interaction energies.

curve. Figure IV-2 giving Wt as a function of ionic

strength more concisely summarizes primary minimum

coagulation behavior of the hypothetical examples. The

figure indicates that primary minimum coagulation occurs

for I > 1.8 x 10-1M (i.e., Wt = 1). Since large absolute

interaction energies result in exponential overflows or

underflows2 in stability ratio calculations using DLVO,

large VT sp(Ho) are made equal to 150 ergs. Therefore, no

Wt are reported greater than 1 x 1077. Since we are

interested in the ionic strength which causes Wt to equal 1

(for primary minimum coagulation) and Wt for stable

suspensions are generally greater than 1 x 1064, the errors

introduced in Wt calculations by this convention are

unimportant.

Secondary Minimum Energies

The greater the depth of the secondary minimum which

occurs at intermediate separation distances (e.g., Figure

IV-1), the higher the probability that this type of

coagulation will occur (1). The magnitudes of total

interaction energies at secondary minima (Vs) and surface to

surface separation distances (ds) as a function of ionic

strength are summarized in Figure IV-3 taken from

interaction energy curves given in Figure IV-1. Particles

2. An Amdahl 470 with a numerical range equal to 1 x 1077

was used to make calculations using DLVO.

64 1 1 1 1

6 = +20 mV

(1-1) Electrolyte

a = 5.31 p -m 13

A = 1.135 x 10 ergs

48 T\ = 380C o

y = 6.306 A

, 32

.J

16

-2.0 -1.5 -1.0 -0.5 0.0

LOG [IONIC STRENGTH]

Figure IV-2.

Theoretical stability ratio as a function

of ionic strength in molar units on a double

logarithm scale. Variables held constant

are indicated. The spherical particle model

was used in DLVO calculations.

6.0

5.0 -

4.0 -;

-J

S.0

2.0

1.0 I i

2.5---

26 = +20 mV

(1-1) Electrolyte

a = 5.31 um 13

2.0 A = 1.135 x 10 ergs

2.0, 2.O T = 38'0

I = ,= 6.306

S1.5

1.0 I

-2.0 -1.5 -1.0 -0.5 0.0

LOG [IONIC STRENGTH]

Figure IV-3.

Theoretical secondary minimum interaction

energy magnitude (Vs) in units of kT and

secondary minimum separation distance (ds)

in nanometers as a function of ionic strength

in molar units on a double logarithm scale.

Values are given for the spherical model

(-- ) and for the flat plate model (- ).

Variables held constant are indicated.

were modelled as spheres or cubes in DLVO calculations with

volume equal to 630 im3 (i.e., radius for spheres is equal

to 5.31 Pm and cubes interact with surface areas equal to

73 um2). For both morphologies, Vs increases with

increasing ionic strength. However, Vs from spherical

interactions are substantially smaller than from flat

surface interactions. All reported values of Vs are great

enough to lead to secondary minimum coagulation (1). Values

of ds for the hypothetical example decrease with increasing

ionic strength. The ds values for the flat plate model are

slightly greater (i.e., 5 A to 30 A) than ds values for

spherical models. The values in general range

from 180 A to 13 A.

CHAPTER V

PROCEDURES TO DETERMINE EXPERIMENTAL COAGULATION

BEHAVIOR OF COM SUSPENSIONS

Introduction

This discussion presents the procedures employed to

determine experimental coagulation behavior of COM

suspensions. To compare theory and experiment, techniques

are required to separately assess aggregation by primary

minimum coagulation (pmc) or secondary minimum coagulation

(smc) (14,16,64). Previous investigations of COM

coagulation behavior (9,10,11,12) were qualitative and could

not be used to quantitatively compare theory with

experiment. Therefore, we established an experimental

procedure with a theoretical motivation in order to

quantitatively compare theory and experiment. After general

theoretical and experimental considerations are presented,

solution and suspension preparations, equipment, and

detailed experimental procedures will be discussed.

Theoretical and Experimental Considerations

Orthokinetic Coagulation Theory

To compare theory with experiment, the relationship

between coagulation rates of particles in suspensions and

stability ratios (Chapter IV) must be established (16). For

the simplest situation (65), the basic assumptions are:

1. short range attractive energies are present which

have no influence over the motion of particles

until contact between particles occurs;

2. the particle diameters are large compared with the

mean free path of the fluid molecules; and

3. particles in the suspension are monodisperse

spheres.

Theoretical coagulation rate of large particles greater than

dN

1 um in a suspension (-t) undergoing laminar shear (i.e.,

orthokinetic coagulation) is given by (14)

dN 4G(

-= - N(t). (V-1)

dt T

TheN is in particles ml-sec-1, G is shear rate in sec-1

dt ml ec e1

is volume fraction of solids, and N(t) is number

concentration in particles ml-1 at time t. This is limited

to initial stages of coagulation because with increasing

coagulation, the suspension becomes polydisperse.

Integration of equation (V-l) yields (14)

N(t) 4G8

In[- ] t, (V-2)

N(O) 1T

where N(O) is initial particle number concentration.

,-Several investigators have experimentally demonstrated

the validity of the orthokinetic coagulation rate equations

(14,66,67). ISuspension systems were used which had

properties meeting the requirements of the basic assumptions

used to develop the orthokinetic rate equations such as

monodispersity and high salt concentrations to give only

attractive energies between approaching particles. For

monodisperse suspensions in high salt concentrations, dN/dt

as a function of N(t)2, where 0 = N(t) in Equations V-1

and V-2, was a linear function for coagulation times where

monodispersity applied. Therefore, within the constraints

of the assumptions, the orthokinetic rate equations are

useful in theoretically predicting experimental coagulation

rates.

If instead of assumption (1), an energy barrier is

present between the particles (i.e., Wt > 1), coagulation

rate is given by (68)

dN 4G4

= -- N(t) (V-3)

dt Wt

where Wt is theoretical stability ratio (Chapter IV). When

Wt = 1 for a rapidly coagulating suspension, Equation V-3

reduces to Equation V-1. Therefore, Equation V-1 gives the

dN

"fast" coagulation rate ((-)fast), while Equation V-3

gives the "slow" coagulation rate ((N )slow) which were

discussed in Chapter IV.

Determination of Experimental Stability Ratios

Experimental stability ratios (We) are obtained from

experimentally determined coagulation rates. Since

dN dN

Wt = ( )fast/( t)slow (Chapter IV), coagulation rates

determined, for example, at high ionic strengths (i.e., no

energy barrier) and low ionic strengths give (17)

AN

(A- )fast

N We, (V-4)

(AN)slow

where (-)fast and (-7)slow are experimental coagulation

rates. The rates are determined from changes in particle

number concentration (AN) as a function of time (At).

The assumption of monodispersity is maintained by

extrapolation to t = 0.

Deviations from theory including polydispersity (69),

morphological anisotropy (70), and hydrodynamic retardation

(71) are minimized when fast and slow coagulation rates are

measured for samples from the same suspension (17).

However, coagulation convolutes with particle coarsening,

and particle-container wall adhesion. These other factors

must be eliminated or taken into account (72).

Materials, Methods, and Procedures in Experimental

Determinations of Coagulation Behavior

The experimental determination of coagulation behavior

requires:

1. solution and suspension preparation;

2. knowledge of the time evolution of particle number

concentrations in sheared suspensions; and

3. equipment to reproducibly shear suspensions.

Procedures were developed to independently determine pmc and

smc. Simultaneous rate events were taken into account by

separate determinations or eliminated in the procedures

discussed below.

Preparation of Solutions and COM Suspensions

Electrolyte solutions and COM suspensions were prepared

by previously described procedures (Chapters II and III) for

secondary minimum coagulation studies. To study primary

minimum coagulation as a function of ionic strength, COM

suspensions were only equilibrated 4 to 6 hours after

preparation. Ionic strength was adjusted immediately before

shear was induced. These procedures minimized pmc prior to

shearing.

Coarsening of particles in COM suspensions

Experimental determinations of coarsening rates

indicated that coarsening rates in COM suspensions were low

enough to be ignored in coagulation rate experiments. Five

different COM suspensions containing 8.0 x 10-5M CaC12 were

aged for times from 25 hours up to 432 hours. Mean volumes

(Chapter II) were determined immediately after preparation,

at 24 hours, and at designated end times. The average

coarsening rate for all suspensions from zero time

3 -1

was 0.5 pm hr-. The 25 hour suspension had the highest

3 -1

coarsening rate equal to 1.3 pm hr Coagulation rate

experiments never exceeded 5 hours, which would give a total

increase in particle mean volumes of 6.5 pm using the

fastest coarsening rate given by the 25'hour suspension.

Since 95 per cent confidence intervals of mean volumes were

around 20 m 3, inclusion of the low coarsening rate as a

simultaneous rate event in coagulation rate experiments was

not necessary.

These findings were supported by optical microscope

examinations of suspensions before and after shearing (see

below for shear procedure). Figure V-1 gives the results of

one such examination for a COM suspension in 4.5 x 10'1M

KC1. Figure V-l(a) was obtained following sonication.

Figure V-l(b) is a photomicrograph of the same suspension

following shear at a shear rate equal to 3.43 sec-1 for 1200

seconds. Mean esr for the particles in Figures V-l(a) and

(b) were 6.3 pm and 10.1 pm, respectively. Visual

inspection of the photomicrographs indicates the difference

in particle size was due to aggregation, not coarsening.

(a)

Figure V-1.

Optical micrographs of a COM suspension

in 4.5 x 10-1 M KC1. Maanification is

100 X. (a) Immediately after sonication

and (b) after 1200 seconds at G = 3.43

sec- Mean equivalent spherical diameters

determined by Coulter Counter analyses were:

(a) 6.3 vm; and (b) 10.1 um.

Particle Number Concentrations in Sheared Suspensions

Possible sources of error were considered in using a

Coulter Counter to determine particle number concentrations

and particle size distribution. Previous coagulation

studies (14,73,74) using Coulter Counters demonstrated

supporting electrolytes and dilution of suspension into

supporting electrolytes did not interfere with accurate

counting or volume determinations of aggregated particles.

Also, flow at the aperture did not interfere with particle

counting. However, shear at pipet tips used for suspension

transfer broke down aggregates if tip radius was too small

(i.e., < 0.5 mm) (73). Therefore, pipet tips were cut to a

3 mm radius.

In pmc studies, particle number concentrations were

determined from particle number counts (Chapter II).

Sonication after shear and before Coulter Counter

measurements (procedure described below), eliminated errors

in particle counting due to smc and particle adhesion to

container walls (23).

For smc studies, particle number concentrations were

calculated using mean volume determinations and initial

particle number concentrations. Errors due to wall adhesion

were minimized by taking samples from the suspension body

(72).

Peptization and Reduction of Aggregation in Primary Minimum

Coagulation

Procedure

Sonication was used to peptize suspensions or eliminate

smc and wall adhesion. Two ultrasonicators1 were used in

the studies because the model W-375 was disabled after the

primary minimum coagulation studies given in Chapter VI.

Each was separately calibrated to yield the sonication

procedures discussed below. The ultrasonicators were

designed so that suspensions were maintained at 380C by a

temperature control bath and circulator2. To maintain

temperatures at 380C while peptizing, model W-220F sonicated

suspensions using maximum intensity for 30 seconds. Model

W-375 sonicated suspensions using the maximum intensity

setting for 45 seconds. As long as sonication intervals

were not less than 4 minutes, suspension temperatures

remained at 380C. Particle number concentrations indicated

maximum peptization of saturated COM suspensions which had

not undergone pmc. To fully peptize saturated COM

suspensions which had undergone pmc required multiple

sonications of 5 10 sonications, 4 minutes apart.

1. Models W-220F or W-375, Heat Systems-Ultrasonics, Inc.,

Plainview, N.Y.

2. Model 2095, Forma Scientific, Inc., Marietta, Ohio.

Estimation of sonication energy input

An estimation of the minimum energy required to peptize

suspensions was obtained from heat capacity calculations.

Even with the refrigerated bath, sonication at maximum

intensity for 90 seconds increased the temperature of a 25

ml suspension to 390C. Using 4.1784 J oC-1 g-1 for the heat

capacity of water at 380C, the temperature increase

corresponds to a 104.5 J energy increase. For a 25 ml

suspension containing 5 x 106 particles ml-1 with mean esr =

5.31 pm, the power per particle is 1 x 10-1 ergs particle-1

s-1 (i.e., 2 x 1014 kT particle-1 s-1). Even though we

ignored the energy lost to the water bath, this value

indicates sufficient energy to disrupt all aggregates except

those formed by pmc. Theoretically, energies of interacting

particles in primary minima approach -- (1). However, in

practice, primary minima energies may be much less because

of surface roughness and adsorbed layer effects, and

coagules in a suspension formed by this phenomenon may be

disrupted by sonication (75).

Shear Equipment and Procedure

Equipment

A commercial stirrer3 with variable speed control was

used to produce shear in suspensions. Steel paddles were

machined to 11/2 x 1 x 1/32 inch to conform dimensions to

suspension containers To minimize contamination from the

steel, paddles were dip-coated with 5% polymethacrylate.

After curing at 750 for 24 hours, paddles were placed in

water for 48 hours to toughen coatings. The stirrer with

paddles was placed in a temperature-controlled box

maintained at 380C. To minimize suspension evaporation

during shear, lids with punches holes through their centers

were placed on paddle stirring rods to cover the suspension.

Estimations of shear rates (G)

Shear rates were estimated from an equation given by

Ives (76)

S= [Cd Ap p(v v)3 5)

G d [ P ]/2 (V-5)

2VP

where G is shear rate in sec-1, Cd is the drag coefficient

of a paddle blade, Ap is the area of a paddle blade normal

to. motion in m2, and p is density of the liquid in Kg m-3

3. Six Paddle Floc-Stirrer, Phipps and Bird, Inc.,

Richmond, Va.

4. Falcon 4020 Containers, obtained via American

Scientific Products, Inc., Ocala, Fl.

The mean velocity of the paddle blade, in m sec-1, is vp, v

is the mean velocity of the liquid suspension in m sec1, V

is the volume of the suspension in m3, and p is the dynamic

viscosity in Kg m-1 sec-1.

The variables, Cd and v, are difficult to evaluate.

Bholes (77) found Cd varied with shear rate from 0.8 to 2.0

with an average value of 1.17. He found that the mean

velocity of the suspension varied from 25% to 53% of the

paddle blade velocity. Using the extreme values for Cd and

v, the geometry of the paddle blades, and a suspension

volume equal to 25 ml, average rates were calculated.

Angular velocities in rpm of the paddle blades and

corresponding average shear rates are given in Table V-1.

All values are well under the conditions necessary for

turbulent flow (78).

The shear rates of 3.43 sec-1 and 4.80 sec-1 have been

calculated to be the probable range of shear rates of urine

in kidney tubules (79). Because of experimental difficulty

in maintaining a constant stirring speed of 4.80 sec-1

experiments were performed with a standard shear rate equal

to 3.43 sec1.

Analysis of deviations from theory for the shear procedure

Equation V-2 predicts a straight line for In[N(t)/N(0)]

as a function of time in the initial stages of

coagulation. To test this, 25 ml samples from a COM

suspension in 4.5 x 10-1 M KC1 were each sheared at 300

TABLE V-1

ANGULAR VELOCITY IN RPM AND SHEAR RATES IN RECIPROCAL

SECONDS FOR PADDLE STIRRER USED IN ESTABLISHING

COAGULATION RATES OF COM SUSPENSIONS

Angular Velocity (rpm) Shear Rate (sec-1)

0 0

20 0.43

40 1.21

60 2.23

80 3.43

100 4.80

83

second increments up to 1200 seconds at various shear

rates. Figure V-2 gives In[N(t)/N(0)] as a function of time

for shear rates equal to 0 sec-1, 1.21 sec-1, and 3.43

sec-1. None of the curves are linear, although generally

In[N(t)/N(0)] decreases with shear rate. These deviations

were probably due to polydispersity (69), inability to make

measurements less than .300 seconds apart, coagule breakup

(80), and/or sedimentation of the particles. Coagule

breakup and sedimentation effects are difficult to analyze

(80,81). However, polydispersity reduces theoretical

orthokinetic coagulation rates by a factor from four to ten

(69). Using Equation V-l, N(t) = 2.50 x 106, N(O) = 3.91 x

6 a3

106, = 2.5 x 10 and G = 3.43, theoretical coagulation

time varies from 160 to 410 seconds. The experimental

coagulation time calculated from the data in Figure V-2 for

G = 3.43 sec, equals 300 seconds. This value is within the

range given by theory. The inability to make measurements

less than 300 seconds apart was due to turnaround times on

the Coulter Counter and therefore unavoidable. Fortunately,

these inherent deviations are minimized by calculating

experimental stability ratios (13).

84

-0.8

-0.6

/-1

Shear Rate

/1.21 sec-1

3.43 sec-I

0.0 I

0 300 600 900 1201

TIME OF SHEAR (sec)

Figure V-2.

Natural logarithm of N(t)/N(0) as a function

of shear time for different shear rates:

0.0 sec-1; 1.21 sec-1; and 3.43 sec-1. Six

paddle stirrer was used to induce shear in

COM suspensions containing 4.5 x 10-1 M KC1.

Outline of Experimental Procedures to Determine

Coagulation Behavior

Primary Minimum Coagulation Studies

The procedure to determine coagulation rates in COM

suspensions due to primary .minimum coagulation was:

1. following a six hour equilibration at 380C 10C,

200 ml COM suspensions with given amounts of CaC12

were removed from the temperature control box.

Well sonicated suspensions (i.e., multiple

sonications) were separated into 20 ml aliquots.

The aliquots were placed back into the temperature

control box for 30 minutes to re-establish their

temperature at 380C;

2. a suspension aliquot was removed and sonicated;

3. Coulter Counter analysis was performed on 0.25 ml

of the suspension aliquot;

4. immediately upon initiation of Coulter Counter

analysis, the aliquot was transferred to the

temperature-controlled box. Five milliliters of

the prescribed concentration of electrolyte were

pipetted into the aliquot;

5. after swirling to mix electrolyte and suspension

and resuspend solids, shear was started using 80

rpm (i.e., shear rate equal to 3.43 sec-1);

6. after 300 seconds of shear, the aliquot was removed

and resonicated; and

7. Coulter Counter analysis was again performed on the

suspension as in step 3.

Steps 2 through 7 were performed on each suspension aliquot

with different electrolyte concentrations. Experimental

stability ratios were calculated using particle number

concentrations.

In the early stages of the primary minimum coagulation

study, the second sonication in step 6 tended to break up

the particles slightly. As the ionic strength got less,

breakage got less. However, in the range of electrolyte

concentrations where the stability ratios became independent

of electrolyte concentration, in the region of rapid

coagulation, it created some deviations in the data.

Therefore, all calculated number concentrations from Coulter

Counter number data were divided by an experimentally

determined breakage factor of 10 6 per cent. This had

little effect on the stability ratio values in the region of

slow coagulation, but tended to normalize the stability

ratio values to one in the region of fast coagulation.

Secondary Minimum Coagulation Studies

The procedure to assess coagulation due to smc

was:

1. a 200 ml COM suspension was separated into equal

volume aliquots. On addition of the prescribed

concentrations of electrolytes, final volumes of 25

ml were obtained;

2. as a matter of convenience, aliquots were

equilibrated for 24 hours at 380C;

3. after equilibration, an aliquot was sonicated and

Coulter Counter analysis performed on a 0.25 ml

sample;

4. within 20 seconds after sonication, shear at 80 rpm

was started; and

5. after 300 seconds of shear, a 0.25 ml sample was

used for Coulter Counter analysis.

Steps 3 through 5 were repeated for each 25 ml aliquot.

Experimental stability ratios were determined from mean

volumes as a function of time. In a given set of samples

from a suspension, one was prepared with high electrolyte

concentration. This sample was used to obtain (AN/At)fast.

Sa s t

CHAPTER VI

STABILITY OF COM SUSPENSIONS IN SOLUTIONS OF

INDIFFERENT AND POTENTIAL DETERMINING ELECTROLYTES

Introduction

The objective of the studies presented in this chapter

was to establish baseline stability criteria for COM

suspensions. With previous investigations using COM

suspensions being strictly qualitative (9,10,11), it was

unknown under what surface and solution conditions COM

suspensions would coagulate. Therefore, coagulation studies

were designed over a wide range of surface and solution

compositions to experimentally investigate primary and

secondary minima coagulation. Since one of our major goals

was to test coagulation theory (Chapter IV) with experiment,

studies compared experimental coagulation behavior with

theoretical predictions calculated using DLVO. To obtain

maximum simplicity in surface chemistries for these initial

studies, only indifferent electrolytes and potential

determining ions were added to suspensions (Chapter III).

For primary minimum coagulation studies of COM

suspensions, experimental stability ratios were determined

as a function of ionic strength. Historically, experimental

stability ratios as a function of ionic strength have been

used to experimentally determine Lifshitz-van der Waals

constants (13,16,17,18,19). To determine "experimental"

88

Lifshitz-van der Waals constants, the size of particles in

suspensions, the Stern potentials, and the critical

coagulation concentration at which the stability ratio

becomes independent of ionic strength, such that We = 1,

were used as independent parameters in approximations to the

thermodynamic expressions given in Chapter IV (58). Using a

mean particle radius and assuming a constant Stern potential

over the range of ionic strengths, the approximate equations

were used to solve for the "experimental" Lifshitz-van der

Waals constant which gave the same critical coagulation

concentration as that given by the experimental stability

ratio curves. However, using the procedure described above,

"experimental" Lifshitz-van der Waals constants determined

for different batches of the same suspension system often

vary as much as two or three orders of magnitude (34).

Furthermore, the slopes of theoretical and experimental

stability ratios as a function of ionic strength usually do

not agree (13).

Several theoretical possibilities offered to explain

the discrepancies between theoretical and experimental

stability ratios as functions of ionic strength, have been

implemented with only limited success. Spielman (71) showed

that theoretical corrections to stability ratio calculations

for hydrodynamic retardation as two particles approach one

another cannot account for differences in theoretical and

experimental stability ratio curves. Reerink and Overbeek

(16) and Ottewill and Shaw (17) have shown that experimental

stability ratios as functions of ionic strength are

independent of the particle sizes in suspensions, a result

inconsistent with coagulation theory. Other factors

effecting theoretical stability ratios calculations which

may give deviations between experiment and theory such as

surface roughness, particle anisotropy, and adsorbed ion

effects, are more difficult to determine (34).

However, the methods used to obtain Stern potentials

for theoretical stability ratio calculations do not take

into account the effect of ionic strength on Stern

potentials. This could explain the discrepancies between

theoretical and experimental stability ratios as functions

of ionic strength. Zeta potentials usually cannot be

determined at ionic strengths greater than 10-2 M. Since

most coagulation studies are performed at ionic strengths of

10-2 M or greater, most investigators determine zeta

potentials at ionic strengths less than 10-2 M. These zeta

potentials, equated to Stern potentials, are used to

calculate theoretical stability ratios in coagulation

studies of suspensions without taking into account the

effect of increased ionic strength or Stern potentials.

Lyklema (20) has pointed out theoretically that Stern

potentials decrease as a function of'ionic strength.

Therefore, the assumption that Stern potentials are constant

with increasing ionic strength may lead to errors in