Colloidal Soft Sphere Crystallisation and Phase...

Colloidal Soft Sphere Crystallisation

and Phase Behaviour

Dissertation

Zum Erlangen des Grades

„Doktor der Naturwissenschaften“

am Fachbereich Physik

der Johannes Gutenberg-Universität Mainz

Jianing Liu

Mainz, May 2003

Germany

----------- Characterization in single component

and binary mixture

Dekan: Prof. K. Binder

Vorsitz: Prof. H. Backe

Gutachter: 1. Prof. Dr. T. Palberg

2. Prof. Dr. J. O. Rädler

Termin der mündlichen Prüfung: 18.09.2003

Contents

Contents

Chapter 1. Introduction to colloids, collodal phase behaviour

and crystallization kinetics

1.1. What are colloids? Why study colloidal model systems

and colloidal crystallization?

1.2. What is known about colloidal phase behaviour?

1.3. Solidification kinetics in single component systems

1.4. Crystallization behaviour in binary mixtures

Chapter 2. Colloidal interaction

2.1. Interaction potential, Debye parameter and effctive charge

2.2. Pair interaction in binary mixtures

2.3. State lines for characterising charged-stabilized spherical colloids

Chapter 3. Experimental techniques and corresponding theories

3.1. Standard preparation technique

3.2. Sample preparation under pump tubing circuit

3.3. Structure and concentration determined by static light scatting

3.4. Bragg microscopy

Chapter 4. Further developments for precise sample preparation

4.1. Experimental control of salt concentration by addition of CO2

4.2. Improved deionisation controlled via crystal growth

4.3. An improved empirical qmax –n relation for determining

n of fluid-like phase

Chapter 5. Fluid-crystal phase transition, crystal morphology

transition and phase diagram

5.1. Earlier experiment and theory for phase boundary

5.2. Fluid-crystal phase transition and crystal morphology transition

in single component system

5.3. A comparison to theoretical diagram

5.4. Fluid-crystal (FC) phase transition and twin domain morphology

transition in binary mixture

1

1

4

7

8

17

17

27

29

35

35

36

40

47

55

55

59

62

71

71

75

80

85

Contents

Chapter 6. Crystal growth kinetics

6.1. Experimental preview of crystal growth detected by Bragg microscopy

6.2. Wilson-Frenkel theory and experimental evaluation

6.3. Crystal growth and Wilson-Frenkel fits in PnBAPS68/PS100 binary mixture

6.4. Limiting crystal growth in PS120/PS156 binary mixture

6.5. Observation on the initial crystal thickness d0

6.6 Former shear influenced crystal structure and a discussion to the structure

of PnBAPS68/PS100 binary mixture

Summary

Appendix

Acknowledgement

Reference

98

98

103

111

117

121

130

140

141

145

146

Chapter 1. Introduction to colloids, colloidal phase behaviour and crystallization kinetics

1

Chapter 1

Introduction to colloids,

colloidal phase behaviour and

crystallization kinetics

1.1. What are colloids? Why study colloidal model systems and colloidal

crystallization?

The term ‘colloids’ was derived from the Greek words ó (glue). Its original meaning,

‘sticky stuff’ was coined in 1860’s by Thomas Graham1. The common characteristic is that

their particle size ranges from 1nm to 1m, which is larger than atoms or solvent molecules

but sufficiently small to undergo vivid Brownian motion. Most frequently applied

experimental tools for characterizing colloids are static and dynamic light scattering, electron-

microscopy, torsional resonance spectroscopy, Bragg microscopy, optical tweezers, atom

force microscopy, confocal microscopy etc. Colloids are abundant in daily life, like blood,

ink, smoke, oil etc., applied broadly in the chemical, pharmaceutical and food industries. A


2

large variety of colloidal dispersions with either one state of gas, liquid, or solid with some

examples have been listed in Tab.11.

Tab. 1. Some common examples as various types of colloidal dispersions. Table courtesy to1 .

Dispersion systems fall into two categories, namely lyophilic (solvent loving) and lyophobic

(solvent fearing)2 , 3

, where the latter requires a protective mechanism against their

agglomeration. Model colloids can be classified into ‘steric-stabilized’ hard spheres and

‘charge-stabilized’ soft spheres. The comparison of different potentials of hard spheres, nearly

hard spheres and soft spheres are shown in Fig.1.

For the hard sphere system, each particle is covered with a brush of flexible polymers where

the chains can be either adsorbed on the surface or chemically attached to it. These polymers

form an excluded volume resulting a repulsive interaction between the monomers.

For the soft sphere system, electric double layer arisen from surface charge ‘coating’ on each

sphere results a Coulomb repulsive interaction. This repulsive interaction is historically

described as Derjaguim-Laudau-Verwey-Overbeek (DLVO) pair-wise interaction2, 4

. My

thesis here will base on studying such soft sphere system including an additional comparison

to hard sphere colloids.

Like atom systems, disperse colloidal suspensions can exhibit several phases; unlike atomic

systems, due to the size reason, the structure relaxation of colloidal suspensions is much

slower ( 10-2

s) than that of atomic or molecular crystals ( 10-13

s). Colloidal crystals

typically have elastic constants 1010

weaker than that for atomic crystals. Therefore, non-

equilibrium states, like fluid-crystal phase transitions, metastable fluids and glass transitions

can be easily observed by experimental optical tools. The colloidal particle interaction can be

easily controlled by the choice of colloids, the preparation of samples, solvents in different

concentrations, etc. Colloidal crystal as a self-assembled long-range order shows a wide range

of highly ordered phases. It can be induced by thermal equilibrium5,6 ,7 ,8

, gravitational9,

Dispersion phase

Dispersion medium

Notation

Technical name

Example

Solid Liquid S/L Sol or dispersion Printing ink, paint

Liquid Liquid L/L Emusion Milk, mayonnaise

Gas Liquid G/L Foam Fire-extinguisher foam

Solid Solid S/S Solid dispersion Ruby glass, some alloys

Liquid Solid L/S Solid emulsion ice cream


3

convective10,11,12,13

, and electro hydrodynamic forces14

which can produce some periodically

patterned templates15

, and have some industry application like the creation of three-

dimensional photonic structures16,17,18,19,20,21,22

. Based on this, disperse colloids, their phase

behaviour, their crystallization kinetics are studied as model systems to characterize

condensed matter, especially soft matter. This is the basic motivation for this thesis.

Fig. 1. A sketch of different types of the pair-wise interaction potential U ( r ) versus r/a, where r is the center-

center distance of colloidal spheres, a is their radius: (up) hard spheres (e.g. billiard balls); (middle) nearly hard

spheres (e.g. PMMA particles); (down) soft spheres (e.g. Polystyrene spheres).

So far a lot of development, both in theories and experiments, has been achieved for single

component systems. However, less publication concern binary mixture systems, especially

charge-stabilized colloids. Thus, it motivates this thesis to explore soft sphere colloidal phase

behaviour and phase transition starting from single component to binary mixture systems.


4

1.2. What is known about colloidal phase behaviour?

In this thesis, I shall investigate the phase behaviour and the phase transition kinetics of

charged colloidal suspension and compare the results to those obtained for binary mixtures.

This study is motivated by a large number of previous experiments, which shown a strong

interest in these model systems, gives a qualitative description of the overall phenomenology

but also leaves open important details. To make this point explicit, it is instructive to recall the

state of the art in this field.

Colloidal suspension of different repulsive interactions are known for long time to show a

first order phase transition from the short-ranged order fluid to the long-ranged ordered

crystalline state7,23,24,25,26,27,28,29,30,31,32

. For hard spheres, the transition is driven by entropy

and located between f = 0.495, m = 0.545, where ‘f’ and ‘m’ denote freezing and melting,

respectively33,34,35,36

. In accordance with theoretical expectations, a similar phase behaviour is

observed for microgel particles37

or slightly charged hard spheres38,39,40

. For highly charged

particles in aqueous suspension a number of experimental and theoretical articles on the phase

behaviour and the transition kinetics are available29,41,42,43,44,45,46,47,48

. As an example, I show

Sirota et al’28

s phase diagram of PS91 in Fig. 2, and Würth ‘s phase diagram taken on a very

similar system (PS109) at much lower concentrations of particles and salt in Fig. 3. Note that,

the phase boundary of Fig. 2 approaches the hard sphere value at elevated salt concentration.

This demonstrates that for charged spheres the interaction may be varied between theoretical

limits of the single component plasma case and hard sphere case. Note further that Fig.1 was

taken using a batch deionisation procedure, while use of a pump circuit allowed for the

exploration of the lower region of the phase diagram.

As early as 1988, the phase diagram of charged particles was determined from computer

simulation. In their pioneering study, Robbins, Kremer and Guest45

used the Lindemann

criterion49

to distinguish different phases. The work was later improved quantitatively by

Meijer and Frenkel46

and also by Voegtli and Zukoski50

with perturbation theoretical

approaches, but without quanlitative changes in the overall behaviour.


5

Fig. 2. Volume fraction versus electrolyte concentration of HCl phase diagram for 91nm polyballs in 9:1

methanol/water suspension.(■) bcc crystal; (∆) fcc crystal; (□) bcc + fcc coexistence; (●) glass; (○) liquid. Solid

lines for phase boundary are “guided to eye”, Dashed line is the fcc-liquid theoretical phase boundary for a

similar point-charge Yukawa system. Figure courtesy to28

.

Fig. 3. Würth’s phase diagram for PS109, with 2a = 109nm, conductivity charge Z* = 450, c is salt

concentration. (■) denotes the fluid-crystal coexistence region. Figure courtesy to43

.

The trends for the phase boundaries are in qualitative agreement with experimental findings,

however a quantitative consistency has as yet not been found. As an example, the data of

Voegtli and Zukoski are shown in Fig. 4(a) and (b).


6

(a) (b)

Fig. 4. Comparison of perturbation model predictions with the experimental phase transition data, while (●)

represents ordered suspensions; (○)represents disordered phase; (◒)represents coexisting phases; The solid and

dashed lines represent model predictions of the melting and freezing curves, respectively. In (a) experimental

data of phase transition comes from Hachisu et al26

for 170nm diameter spheres. Comparisons are made for

dimensionless surface potentials (e0 / kBT) of 3.0 and 2.0. In (b), experimental data of phase transition is

reported by Monovoukas and Gast42

for particles with diameter 133.4 nm under dimensionless surface potential

of 2.49. Figure courtesy to50

.

While theory and simulations yield qualitatively similar results, the comparison to

experimental data based on a constant effective particle charge usually fails. In particular, the

theoretical investigations seem to systematically overestimate the stability of the colloidal

crystal.

Clearly this calls for systematic investigations using advanced preparation methods with on-

line access to effective charge in situ. These are performed in this thesis and will be described

in Chapter 4. A quantitative representation of the samples way in the two parameter phase

diagram of Robbins, Kremer and Guest45

turning systematic variation of n and c by the so-

called ‘state line’ is developed in Chapter 2.


7

1.3. Solidification kinetics in single component systems

In addition to the static phase behaviour, also the solidification kinetics of colloids have been

studied in great detail with most emphasis again on hard spheres51,52,53,54,55

, or hard sphere like

systems40, 56 , 57

. Comparably few studies exist for charged spheres and mostly for

growth30,43,58,59,60

. Up to now growth velocities were often found to correspond to reaction-

limited growth61

. For samples investigated above coexistence growth is linear in time both for

radial growth in homogeneous crystal and interfacial growth in heterogeneous crystal as Fig.

5. It further shows that the slope of homogeneous crystal (radial growth velocity vR) is larger

than the slope of heterogeneous crystal growth (interfacial growth in plane (110) v110).

Accordingly a description in terms of Wilson-Frenkel growth law62

was observed to apply

above coexistence shown in Fig.6 (a), (b)43,58

. This law states that at low super saturation or

“undercooling”, the growth velocity is proportional to the “undercooling”, while at infinite

“undercooling” it is given by the maximum attachment rate of particles to the crystal, hence

by the ratio of an appropriate diffusion coefficient to a typical length scale. Note that such a

description gives access to an estimate of the “undercooling” of the suspension via the

chemical potential difference between melt and solid.

Fig. 5. Comparison of the velocities of radial growth (○) to those measured for a planar (110) interface (□). Data

is based on PS109 (diameter 2a = 109nm, conductivity measured effective charge Z* = 450. They are measured

under volume fraction = 0.0022 and c = 0.5 M. The radial growth velocity vR = 9.6 ms-1

which is

considerably larger than v110 = 8.4 ms-1

for the planar interface. However, the growth velocities are found to be

independent of the sample history in both cases. Figure courtesy to32

.


8

Fig. 6. Crystal growth versus volume fraction , the difference of chemical potential and reduced energy

density *. (a) PS109* (diameter 2a = 109nm, conductivity measured effective charge Z* = 450); (b) PS109 (2a

= 109nm, Z* = 395) shows as (□) under = 0.003 with increased salt condition(0-2M); () for PS109* under

complete deionised condition and (○) for PS109* under = 0.0022 with increased salt (0 – 2 M). (∗)Shown in

(a) and (b) is the fluid-crystal coexistence. Figure courtesy to32

.

Up to now it remained unexplored, how growth proceeds across coexistence and how growth

velocities obtained there will be compared to the Wilson-Frenkel behaviour. This will be

studied in Chapter 6. In addition I shall test the different recipes to apply the Wilson-Frenkel

description and to obtain estimates of the chemical potential difference43,58,59, 63

.

1.4. Crystallization behaviour in binary mixtures

Most studies so far were conducted for monodisperse or slightly polydisperse single

component samples, but much less has been done on colloidal binary mixture

64,65,66,67,68,69,

70,71,72,73,74,75. Due to mixing ‘tracer particle’ into ‘host particle’, it results a so-called optical

polydispersity as scattered optical properties of ‘host particles’ is altered by the amount of

‘tracer particles’. When two monodispersed suspensions of different particle diameters are

mixed, the system which has a fixed size ratio and charge ratio, may evolve into any one of

the following phases: a liquid mixture, a disordered crystalline alloy, a compound of the type

of AB2, AB4, etc., a glass or a multi-phase system. The first ordered colloidal alloys were

found in naturally-occurring gem opals76

.


9

A binary mixture of hard spheres has been thought as the simplest model for a mixture of

simple molecular species, thus thermodynamic properties of binary hard sphere mixtures64

including the phase transitions of this model system by computer simulation has been studied

by Kranendonk and Frenkel65

, which two parameters (the ratio of diameters of two

spheres), composition p (here p is the mole ratio of the larger spheres, i.e. the mixing number

ratio of large spheres) are considered as the variations. By fixing the diameter ratio (=0.85)

but changing the composition p, it is found that the acceptance ratio Pacc for interchanging

small and larger particles increase with p increasing, but decrease with volume fraction

(shown in Fig. 7).

Fig. 7. Acceptance ratio Pacc for interchanging small and large particles as a function of the packing volume

fraction for the solid state. The diameter ratio = 0.85. Three compositions were given as: p = 0.2037 ();

0.5 (□); and 0.7963 () . The solid lines were a guide to the eye. Figure courtesy to 64

.

And a pressure dependence in the function of the composition p at constant packing fraction

is shown in Fig.8. The reduced pressure was found strongly different with p when is more

deviated from 1 (the large and small spheres become more dissimilar).


10

Fig. 8. Reduced pressure as a function of the composition p at a constant packing fraction = 0.5498 in the solid

state. Results from molecular dynamics simulation were shown for three diameter ratios: = 0.95 (□); 0.90();

0.85(). The estimated standard deviations were represented by the error bars. The results of the global fitting

was given by the solid lines. Figure courtesy to64

.

By considering the mechanical stability of creating a lattice with composition p = 0.2, and

diameter ratio = 0.2, they concluded that “we must either distort the lattice or introduce

some substitution order” .

The order formation in binary latexes was to be understood as a phase transition phenomenon

in the binary hard sphere system by Hachisu, Yoshimura75

, where they have taken several

alloy patterns by light microscopy. The patterns of these alloys included mostly a close-

packed stable structure conformed by large particles and an less stable structure (or even

simply a particle) in the centre conformed by small particles shown as Fig. 9, Fig. 10, Fig.11.


11

(a) (b)

Fig. 9. Alloy pattern appeared in 280nm / 800 nm binary mixture. The structure was initially determined to be

AlB2 type. (a) pattern microscopy; (b) Lattice of AlB2 structure. The pattern in (a) is the ABCD plane or ( 0110

)

plane of the structure in (b). Figure courtesy to75

.

(a) (b)

Fig. 10. Alloy pattern appeared in 250nm / 550 nm binary mixture. The structure was initially determined to be

NaZn13 type. (a) pattern microscopy; (b) Lattice structure of NaZn13 type . Figure courtesy to75

.


12

(a1) (a2)

Fig. 11. Alloy pattern appeared in 310 nm / 550 nm binary mixture. The structure was initially determined to be

CaCu5 type. The configuration of small particles differs between foot note ‘1’ and ‘2’ which ‘1’denotes the

lattice pattern and structure just close the cell boundary and ‘2’ denotes that towards inside. (a1) and (a2) are

pattern microscopes; (b1)and (b2) are lattice structure of CaCu5 type, where () represents as small particle, (○)as

large particle. Figure courtesy to75

.

The pattern in Fig. 9(a) was found in a binary mixture of 280/800nm latex. Larger particles

were packed closely in a square lattice with small particles in the center. The structure was

shown in Fig. 9(b), which a small particle situated in the center of a trigonal prism formed by

the large particles. So what was observed in Fig. 9a is the ( 0110 ) plane of Fig.9(b).

The pattern in Fig. 10(a) was found in 250/550nm latex mixture, where the staggering pattern

was noticed to be constituted by small particles. The lattice of the entire structure was shown

in Fig. 10(b), where shown that in each of the simple cubic cells of large particles resides an

icosahedrons of small particles, neighbour icosahedrons stagger by 90° and may be a bit

distorted to produce a better packing.

(b1)

(b2)


13

By moving the focus of microscope applied for observing 310/470nm latex mixture, a

different pattern was found shown in Fig. 11(a1) with pattern b1 just inside close the cell

boundary, and Fig. 11(a2) with pattern (b2) toward the inside of the structure. The entire

structure was an alternative stack of these net ABAB planes, each net plane and the whole

structure is shown in Fig. 11 (b1) and (b2).

Although pattern morphology formed by mixing small and large latex is different from the so-

called ‘ratio of effective diameters’ with some regularity, the ‘effective diameter’ what they

based on was just an arbitrary assumption, i.e. “the difference d between the effective

diameter and the actual core diameter is the same for small particles and for large particles in

the attending mixture”75

. The dominated particle interaction actually was not correlated to the

pattern formation in this case, which motivates for the further exploration.

10 12 14 16 18 20 22 24 26 28 30

514,5 nm

I(q

) sin

()

/ b

.E.

q / m-1

PS85:PS100

10:0

9:1

8:2

7:3

6:4

5:5

4:8

3:7

2:8

1:9

0:10

Fig. 12. Angle-corrected scattering intensity of the PS85/PS100 mixture as a function of composition. Figure

courtesy to77

.


14

At low-to-intermediate particle concentrations, randomly substituted alloys of body-centred

cubic structure for charge-stabilized binary mixture PS85/PS100 (their conductivity measured

effectively charge ratio almost 1:1) were found by Wette77

. This is demonstrated by static

light scattering patterns at different mixing number ratios in Fig. 12. Meanwhile, some other

experimental data, such as shear modulus G versus mixing number ratio, intermediate

scattering function f(q,) versus mixing number ratio, plateau height of f(q,) versus mixing

number ratio etc. supports this conclusion.

Fig. 13. The phase diagrams of inverse volume fraction -1

vs relative particle number density p (mixing

number ratio of small component) for three values of the size ratio . (a) spindle-type diagram ( = 0.87 0.03);

(b) isotropic-type diagram ( = 0.780.04); (c) eutectic-type diagram ( = 0.540.02). Solid lines between the

liquid (), glass (●), and crystal (○) were just a guide to the eye. Figure courtesy to78

.


15

Finally, for still different size ratio of charge-stabilized colloids, some completely different

phase diagrams were obtained using diffusing wave spectroscopy78

. They are shown in Fig.

13. it strongly supports the idea that the reduced pressure depends on the size ratio in effective

hard sphere systems64

. Data were shown as a phase diagram with the parameter of inverse of

volume fraction -1

versus mixing number ratio p and different phase boundaries were

obtained for different .

Indeed for systems of not too large size ratio, the above studies showed that the solid phases

have several structural possibilities: pure crystal of component A or B; substitutional crystals

in which the two species are distributed on a common lattice but without compositional order;

ordered alloys consisting of interpenetrated lattices of each species; binary glasses or order-

disorder coexistence regions. Still larger size ratios may in addition introduce entropic

attraction and further enrich the phase behaviour56

. Already the data presented above

demonstrate the importance and the interest in phase transition studies in particular for binary

mixtures. So far however data taken on highly charged systems are still rare. Neither

solidification kinetics nor morphological issues have been addressed. As a continuation of

such work, I shall investigate the phase behaviour, crystal growth kinetics, crystal

construction and resulting morphologies, systematically both for single component systems

and binary mixtures.

This survey identified a number of important open questions, which I then organize and sort

my thesis as following:

In Chapter 1, I gave a general introduction on colloidal model system and crystallization in

colloids. I have reviewed colloidal phase behaviour and crystal growth kinetics both for single

component and binary mixture. It motivates this thesis.

In Chapter 2, I introduce particle interaction potential, Debye parameter, and effective charge.

An averaged DLVO pair interaction is promoted also for binary mixtures, and the concept of

state lines are applied for characterizing the phase behaviour of charge-stabilized spherical

colloids.

In Chapter 3, I introduce our sample preparation using the pump tubing circuit, and other

experimental techniques, like static light scattering and Bragg microscopy both under aspects

of experiment and theory.

In Chapter 4, I report further technical developments for precise sample preparation. A novel

way of monitoring residual salt concentrations via crystal limiting growth is described. And a


16

novel empirical relation to determine the particle number density n in the fluid ordered state is

developed.

In Chapter 5, I show my experimental data of fluid-crystal phase transition, crystal

morphology transition both in single component and binary mixture, evaluate their phase

diagram, and correlate this with theoretical data. The observation of cloud-like and zig-zag

morphology for binary mixture is another point of interest here.

In Chapter 6, I describe crystal growth kinetics obtained with Bragg microscopy and compare

it to different evaluation prescriptions of Wilson-Frenkel growth law by fitting. Based on

PnBAPS68/PS100 and PS120/PS156 binary mixture, I give a correlation between the initial

crystal thickness d0 and ‘undercooling’. Further, I discuss their crystal structure influenced by

former shear.

Finally I will give the conclusions and outlook.


17

Chapter 2

Colloidal interaction

2.1. Interaction potential, Debye parameter and effective charge

By considering the strong interactions between ions/molecules in solution and the electrode

surface, Helmholtz79

first promoted the term 'electrical double layer' in the 1850's. Starting

from the idea of electrical double layer, one needs to build a theoretical model to describe the

distribution of macroions and microions caused by ions’ interaction. One of the main

theoretical tools which has been used to describe the physics of charged colloidal suspensions

is the Poisson-Boltzmann equation which reads

i

Biii

2

r0 Tψ/kezexpnzeψεε [2.01]

Here, 0 = 8.85410-12

C2/Nm

2 is the permittivity of the vacuum; r is dielectric constant of

the solvent, like water r 80, vacuum r = 1; elementary charge e = 1.60210-19

C; kB =

1.380662 × 10-23

J/K is called the Boltzmann constant, which is a ratio of the universal gas

constant to Avogadro's number; T is absolute temperature with kBT as thermal energy; zi is


18

the ion’s valence of type i; ni is the number density of ion i; is the potential and 2 as the

Laplace operator.

However, this equation can be solved analytically only in very special cases. e.g. within the

Gouy-Chapman model for plane surfaces80, 81

, the Debye-Hückel model82

where one gets an

often applied approximate solution83

. To be specific, the Gouy-Chapman diffusion double

layer model provides a solution to Eq. [2.01] based on the following assumptions: the phase

boundary is a non-limited plane with homogeneously distributed surface charges; the ions in

the diffusive region are considered as point ions; the dielectric properties of the electrolyte are

considered uniform in the diffusive region and the whole system is in electroneutrality. This

model has successfully explained the electric charge distribution in the diffusive region, the

distribution of the potential, and also quantitatively provides the relationship among the

valence of electrolyte, concentration of electrolyte, potential, and double layer thickness,

which is consistent with experimental results83, 84,

and computer simulations85, 86

. While this

solution is valid for arbitrarily large surface case as any other mean field theory, the Gouy

Chapman model finds its limits as ions do not behave as point charges; dielectric constants

are different between surface and bulk; there is no provision for surface complexes (specific

adsorption), etc. Through my thesis, as the investigated samples are charge-stabilized

spherical colloids, the Gouy-Chapman model is inapplicable and so I will use the Debye-

Hückel model as a solution for the Poisson Boltzmann equation. This model is valid for

potentials much smaller than the thermal potential kBT/e. By expanding the exponential (e-x

1-x) in Eq. [2.01] for the simple case of a symmetrical electrolyte, the Boltzmann Poisson

equation is expressed as

ψκψ 22 [2.02]

Considering the colloidal sphere size modification, the Debye-Hückel potential outside a

sphere of radius carrying charge Z reads87

r

κrexp

κa1

κaexp

εε4π

Zerψ

r0

[2.03]

In this approach, particles are considered to be monodisperse. At a distance -1

, the potential

has decayed to a factor of (1/e), where -1

is used as a measure of the extension of the double


19

layer and is often loosely called the thickness of the double layer, or Debye screening length,

calculated by via

)nZ (nTk εε

eκ s

Br0

22 [2.04]

Here n is the particle number density, ns = 2000 NA c is the number density of small ions with

NA being Avogadro´s number and c the molar concentration of salt. The mean average

distance between two particles is d = n-1/3

.

Our colloidal particles are negatively charged with hydrophobic tails gathered in the center

and hydrophilic negative charged heads of a chemical surface group (e.g. -SO4,-COOH ) on

the surface surrounded by the external water phase. Such surface groups may dissociate and

their counter-ions are either distributed in the diffusive part of the double layer or re-associate

to the inner Helmholtz-plane, which is assumed to be limited to the radius of the hydrated

ions. In addition, an outer Helmholtz-layer may form by adsorption of co-ions on top of the

inner one. Starting from the bare particle surface the potential may first drop and then rise

again depending on the small ion surface densities within the Helmholtz-layers. This

behaviour is similar to that of a simple plate capacitor. The potential at the outer Helmholtz-

layer is also called Stern-potential, and the corresponding charge Z is used as an in-put of Eq.

[2.03] and Eq. [2.04].

Combining electrical double layer theory and Debye-Hückel model, the potential versus

distance r to one charged colloidal sphere is shown in Fig. 14.


20

0

/e

I II

-1r

0

/e

I II

-1r

Fig. 14. Potential in the electrical double layer in the Debye-Hückel model for a charged colloidal sphere. is

the Debye-Hückel potential; r is the distance between the sphere centre; 0 is the sphere surface potential, is

the stern potential or outer Helmholtz layer potential, and -1

is the Debye screening length while ‘I’ shows the

region of compact part in the double layer, ‘II’ shows the region of the diffusive part of the double layer.

The bare charge Z, which takes into account the dissociated and undissociated end groups and

bound ions in the stern layer, determines what is known as the outer Helmholtz layer potential

88

calculated via

κa1εaε

e Zψ

r0

δ

[2.05]

For the case of Na+, H

+ as counter-ions, it however was observed that there is no specific

adsorption , thus the Helmholtz layer is empty and 0 = 89

.

The thickness of the double layer depends markedly on the ionic concentration as shown in

Fig. 15, that is, with increasing ionic strength (i.e. c1 < c2 < c3 corresponded to the potential

curve i, ii, and iii in Fig. 15, respectively), the thickness of the double layer decreases rapidly.


21

/0

r1-12

-13-1

1/e iiiiii

/0

r1-12

-13-1

1/e iiiiii

Fig. 15. Dependence of the ratio of /0 on the ion-strength c ( and 0 are the Debye-Hückel potentials at

distance r and at r = 0, respectively. Lines i, ii, iii correspond to the lower electrolyte concentration c1 (Debye

screening length 1-1

); the medium electrolyte concentration c2 (Debye screening length 2-1

) and the higher

electrolyte concentration c3 (Debye screening length 3-1

). The potential steeply decreases when c is increasing.

Due to the assumption of ‘point ions’ and the uniform charge distribution at the small

distances away from the surface, the Debye-Hückel approximation cannot be valid near the

surface of a highly charged sphere. In the Debye-Hückel approach, Eq.[2.03] is valid for a

potential << kBT/e, whereas for large Eq.[2.03] can be solved numerically in a Poisson

Boltzmann cell model 90, 91

92, 93

or a Poisson Boltzmann jellium model 93, 94

. Then Eq.[2.03]

and [2.04] are fitted to these solutions to yield an effective or renormalized charge Z* and an

effective screening constant * [Note 1]. Again one often speaks of counterion condensation, i.e.

counterions are assumed to be energetically confined to a narrow region outside the stern

layer.

Within this picture, it is customary to treat the bound and free counterions separately. The

effect of the bound counterions is to renormalize the charge of the colloids from its bare value

Z into a new value Z*, with Z* < Z. On the side of experiments, the phenomenological

approach is to consider Z* as a free parameter adjusted to experimental data, like scattering

profiles determined by light scattering95 , 96

. It is also possible to perform conductivity

Note

1: Later on, I simply write instead of * for simplicity and consistency with most publications.


22

measurements for Z* ( denotes the conductivity), elasticity measurements for Z*G (G

denotes the shear modulus) or electrophoresis measurements for Z* ( denotes the mobility)

97, 98. The general trend is that Z* increases with Z and saturates at a certain value for high

values of the bare charge as a result of counterion condensation. The saturation values,

however, are different for different experiments (shown in Fig. 16). For my experiments, Z*

is obtained from conductivity measurements [Note 2], and is given by

μμeZ nσσ *

0 [2.06]

Here 0 is the background conductivity stemming from the self dissociation of water and

residual impurities. At T = 297K, a typical conductivity value of the background (water) is

0 0.06µS/cm; µ+ = 36.510-8

m2V

-1s

-2 is the proton mobility and µ– = (2-12)10

-8 m

2V

-1s

-2

is the mobility of the particles measurable from electrophoresis99

.

0 100 200 300 400 500 6000

200

400

600

800

1000

1200

Z*

Z*G

Z* Water [Okubo]

Z* Water/Glycerol[Garbow]

Eff

ective

ch

arg

e Z

*

Diameter (nm)

Fig. 16. Effective charges from different experiments which were correlated to the particle diameter. They are

Z* from conductivity measurements, Z*G from shear modus experiments, Z* from electrophoresis

measurements with the data courtesy of T. Okubo. It was measured in the medium with water, and the data

was measured in the mixed medium of water/Glycerol courtesy of N. Garbow. Figure courtesy to100

.

Note

2: Later on, I simply write Z* instead of Z* as all my effective charges are calculated from conductivity

measurement.


23

As a function of radius, Z* seems to obey Alexander’s assumption101

Bλ

aAZ* [2.07]

where B =e2/(40rkBT) represents the Bjerrum length, which at room temperature (24°C),

B 0.714 nm. Throughout this thesis, the conductivity measured Z* will be used, so -1

can be calculated using Z*. It turns out that -1

decays with increasing n. As an example data,

-1

versus n of a deionised surfactant free polystyrene sample PS120 (with diameter 120nm,

effective charge Z* = 68520) is shown in Fig. 17. n are measured from static light scattering

experiments, whereas -1

are calculated from Eq. [2.04].

The Debye-Hückel equation has the basic feature of superposition inherent in linear

equations. Based on it, the two colloidal spheres’ interaction (pair-wise) follows from the

superimposed fields with a screened Coulomb repulsion, which is called the Yukawa

interaction energy3

r

e

εε4π

e*ZrU

rκ

r0

2

Yukawa

[2.08]

After considering the geometrical factor [exp(a)/(1+a)]2, Derjaguin, Landau

102 and Verwey,

Overbeek2 introduce their repulsive part of the DLVO pair-wise energy as

r

κr)exp(

κa1

a)exp(κ

εε4π

e)*(ZU(r)

2

r0

2

[2.09]


24

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5

200

300

400

500

600

-1(n

m)

n( m-3 )

Fig. 17. Debye screening length -1

dependence on n. Data is received by measurements of the polystyrene

PS120 sample (diameter 2a = 120nm, Z* = 685 20) under deionised condition. Data (sign ○) n and -1

are

obtained from the static light scattering experiments and calculation with Eq.[2.06], respectively. The connecting

line of these data points is a first order exponential decay received by fitting.

The omission of the geometrical factor does not make any significant difference between

Yukawa and DLVO interaction in very dilute suspensions if a << 129

. However if a is not

much less thand, the geometrical factor must be taken into account 103, 104, 105

. In this case, a

divergence is found when applying Yukawa interaction and DLVO pair-wise interaction to

predict some physical behaviour, like phase transition and osmotic pressure, etc. As Eq.[2.09]

is the repulsive part of DLVO pair-wise energy, the attraction part stemming from the van der

Waals force is much weaker than 0.01kBT. It can be masked by the long-range pure repulsive

part, and therefore can be neglected. The distribution of macroions, counterions, small ions

and corresponding parameters for an isolated pair of particles can be depicted in Fig. 18.


25

+

+-

-

+

+

-

+

+

+

+

+

+

-+

+

+

+

+

+

+

+

+

++

+

+

+

+

+

+

+

+

--

+

+

+

+

+

+

+

-

+-

+

+

+

+

+

+

+

+

--

+

+

+

-

+

+

+

+

+-

+

-

++

-

-

+

+

-+

+

+

-

-

++

+

+

+

+

+-

++

+

-

-

+

+

++++

++

+++

++

++

++

+

-

+-

+

++

-

+

++

+++

++

+++

++

++

++

+ +

-1d

+

+

+

+-

-

-

-

+

+

+

+

-

+

-

+

+

+

+

+

+

+

+

-+

+

+

+

+

+

+

+

+

-+

+

+

+

+

+

+

+

+

-+

-+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

++

+

+

+

+

+

+

+

+

+

+

++

+

+

+

+

+

+

+

+

+

+

+

+

+

+

--

+

+

+

+

+

+

+

+

--

+

+

+

+

+

+

+

+

-+

--

+

-

+

+

+

+

+

+

+

+

+

+

+

-

+-

+

+

+

+

+

+

+

-

+

-

+-

+

-

+

+

+

+

+

+

+

+

+

+

+

+

--

+

+

+

+

+

+

-+

--

+

-

+

+

+

+

+

-

+

+

+

+

+

+

+

+

+-

+

-

+

-

++

-+

-

-

+

+

-+

+

-

+

-

+

+

-+

-+

+

+

+

+

-

-

++

+

+

-+

-

-

+

-

++

+

+

+

+

+

+

+-

+

+

+

+

+

+

+

+

+-

+

-

++

+

+

+

-

--

-

+

+

+

+

++++

++

+++

++

++

++

+

-

+

-

+-

+

-

+

++

-+

-

+

+

+

++

+++

++

+++

++

++

++

+ +

-1d

Fig. 18. A simple pair-wise interaction model between two colloidal spheres, “–” in the middle of each large

sphere represents its negative charge Z, small “⊕”, are counter ions and “⊖” are coions. -1

is the Debye

screening length,d is the mean particle distance, and a is the radius of particle. The dark part between the two

large spheres shows the overlap of interaction fields, which results the repulsive interaction.

The DLVO theory has been shown to explain some experimental data32, 106, 107

, but seems to

fail to explain some other features108, 109

of charged colloids. The pair potential has to be

understood as an effective interaction obtained by integrating out the additional degrees of

freedom using a thermal canonical average110

. Since this average is highly non-linear, the

effective interactions involved many-body terms have been concluded through so-called

volume terms94, 111, 112, 113, 114, 115, 116

, which the contributions to the effective Hamiltonian of

the colloids are independent on coordinates but dependent on density. By simulating two

macroions separated in one direction in a periodic cubic box with length 1m at the

temperature T = 300K in water, Tehver, Ancilotto, etc.117

have compared the DLVO

prediction to the numerical local density approximation data shown in Fig. 19. They found

that the potentials were shifted to zero at a maximum macroion separation. It has been proved

that as the charge of a macroion increases, the potential is further away from a linear regime,

and the deviations from the DLVO prediction therefore becomes more pronounced.

Recently, the colloidal effective interaction has been measured with optical tweezers by

Crocker and Grier107, 108

in confined and unconfined geometries. They extract the two-particle

interaction from the dynamics of isolated pairs of particles moving away from artificially

created initial configurations. With this method, they have measured the effective pair-

interaction between monodisperse macroparticles plus pointlike small ions shown in Fig. 20.


26

Fig. 19. Total interaction potential energy of a macroion in a periodic system with two macroions and free

counterions in the primary simulation cell. The DLVO prediction (in solid line) is compared to the numerical

local density approximation data (open circles). The best fit of a DLVO-type potential is also plotted (dashed

line). The potentials are shifted to zero at a maximum macroion separation. Figure courtesy to 117

.

According to their detection, the pair-interaction is purely repulsive in an unconfined

geometry. They108

concluded that the attractive pairwise interactions in a confined geometry,

which is not found in the dilute-limit pair-interaction, may arise from many-body effects at a

finite volume fraction. This coincidence suggests that the strong coupling between the

counterion clouds of the spheres and the walls is necessary to produce the observed attraction.

The DLVO theory is not formulated for such conditions and its failure is not surprising. On

the other hand, their data provide strong evidence for the validity of the DLVO description on

the pair level. There is an ongoing discussion about the effects of many body forces for the

bulk level118, 119

. However, it seems to be justified to use this approach within this thesis.


27

Fig. 20. The confinement-induced attraction for two different sphere size populations measured in the same

electrolyte. 2a = 1.53 m for (a) unconfined, (b) d =3.0 0.5 m; 2a = 0.97 m for (c) unconfined, (d) d = 3.5

0.5 m. Curves are offset for clarity. Figure courtesy to108

.

2.2. Pair interaction in binary mixtures

For colloidal binary mixtures, I define the mixing number ratio of one component relative to

the whole system as pA, and pB, given as pA = nA/(nA+nB) = 1 - pB, where the indices ‘A’ and

‘B’ represent the two components. By extending Eq. [2.06] to the case of binary mixtures,

Wette et al98

have rewritten the equation as

0BBAA

0B

*

BBA

*

AA

σσpσp

σ)μμZpμμZne(pσ

[2.10]

which predicts a linear variation between and n for binary mixture systems. This was

experimentally verified and for instance shown in Fig. 21 for PS90/100 binary mixture at p90

= 0.50. Another linear variation between and pA (or pB), which was also experimentally

verified, for instance shown in Fig. 22 for binary mixture PS90/100 at n = 20m-3

. Here nA,

nB are particle number densities, A, B are conductivities, ZA*, ZB* are effective charges of

sample A and B, respectively. Then an averaged effective charge Z* is given by


28

Z* = pAZA* + pBZB* [2.11]

Whereas the averaged Debye screening length -1

can be calculated via

s

*

BB

*

AA

Br0

22 nZpZpn

Tkεε

eκ [2.12]

Further, the DLVO pair-interaction is rewritten in averaged terms as

2

B

B2*

B

2

B

BA

BA*

B

*

ABA

2

A

A2*

A

2

A

r0

2

BA,aκ1

aκ expZp

aκ1aκ1

aκaκ expZZp2p

aκ1

)a(κ expZp

r

rκ exp

ε ε 4π

eU(r)

[2.13]

Here aA, aB are the radii of component A and B, respectively.

5 10 15 20 25

2

4

6

8

10

PS100/PS90 1:1

(-

0)

(

S/c

m)

n (µm-3)

Fig. 21. Background corrected conductivities - 0 of a PS100/PS90 mixture at p90 = 0.50 as a function of

particle number density n. Figure courtesy to98

.


29

0.0 0.2 0.4 0.6 0.8 1.06.5

6.6

6.7

6.8

6.9

7.0

7.1

(-

0)

(

S/c

m)

p90

1.0 0.8 0.6 0.4 0.2 0.0

p100

Fig. 22. Background corrected conductivities - 0 of a PS100/PS90 mixture as a function of mixing number

ratio p90 and p100 at a constant n (n = 20m-3

). Figure courtesy to98

.

Eq. [2.13] is found to be consistent with previous experimental results, like Wette etc.’s98

shear modulus measurements of a mixture of PS90/PS100, and Lindsay and Chaikin’s66

shear

modulus measurements, where they concluded that the shear modulus is not very structure

dependent, but rather relates to the averaged particle interactions and the particle density.

2.3. State lines for characterising charged-stabilized spherical colloids

In an experiment using charged colloidal spheres, a number of different parameters determine

the interaction between the colloids and hence the suspension properties. The electrostatic

repulsion between charged spheres is in most cases well described using Eq. [2.09] or Eq.

[2.13]. In systematic measurements, both the particle number density n and c are conveniently

varied. Phase diagrams therefore are usually presented in the n - c plane. For some systems

also a variation of the particle effective charge Z* is possible via titration or

adsorption/desorption processes. Furthermore, a systematic variation of radii at constant n and


30

c may in principle be possible for microgel-particles made of e.g. Poly-N-

Isopropylacrylamide.

On the other hand, a compact 2D representation of the system state may be obtained by

plotting the reduced pair energy of interaction dT/Uk B at the average particle separation

versus the coupling parameter ( = d )45

. In this case the named experimental parameters

enter in a complex way which is not readily visualised. It is therefore quite instructive to

explore the pathways of suspensions in the λdT/UkB plane upon changes of the

experimental parameters106

. In what follows, I term such pathways as ‘state lines’.

I shall first discuss variations of the particle concentration under complete deionised

conditions. In Fig. 23, I show the results of calculations for increasing n at a fixed residual salt

concentration of c = 0.2 M, corresponding to the background concentration given by the

self-dissociation of water. The particle diameter is fixed to 100nm. The plot contains five state

lines corresponding to increasing effective charges Z* of 100, 200, 500, 1000 and 2000 (from

left to right).

2 4 6 8 10 12 140.0

0.1

0.2

0.3

0.4

kBT

/U(d

)

Fig. 23. Charge - dependence of state lines of particles with d = 100 nm in λdT/UkB diagram ( dκλ ) at

deionised condition (c = 0.2 M, n ~ 0.001---1000 m-3

). Curves are shown for effective charge of Z*: 100 (—

• —); 200 (— —); 500 (— —); 1000 (—▼—); 2000 (— ◊ —). The details can be found in the text.


31

The interesting qualitative features of a state line under variation of n are best demonstrated

by discussing the curve of the largest charge (Z*=2000). This rightmost curve may be

separated into three regions. Starting from high dilution, an increase of n results in an

increased overlap of double layers. Therefore, the repulsive pair energy increases and the

curve proceeds downwards. Also the coupling parameter decreases as the salt concentration

is dominated by the background concentration, so stays constant, whereas the mean particle

separation d decreases due to d = n-1/3

. Note that the Z*2 dependence of dU , the pair

energy at a given rapidly increases with increased Z* and the curves therefore appear to be

shifted to the right.

This behaviour changes as becomes dominated by the contribution of the counter-ions. In

this case the salt concentration increases linearly with n and increases with n1/2

, whereas

increases with n1/6

. Thus the curve bends rightward. At the same time the potential becomes

steeper due to the additional self-screening. At still large separations the pair energy is

reduced and the curve bends upward. A second change is observed at elevated particle

densities, where the pair energy again increases at somewhat smaller separations. This time

the increase of continues as the counter-ions keep dominating the screening.

In summary, the overall increase of the interaction energy with n is suspended at medium n,

where the effects of self-screening dominate. This effect becomes more pronounced as the

effective charge increases or the particle radius decreases [ref. Fig. 16]. Alexander et al.101

gave an estimate of the magnitude of the effective (renormalized) charge as Z* = Aa/B,

where A is a constant of order 10. In our calculations the maximum takes larger values for

increasing A, while the minimum decreases. Vice versa, systems with small A (of say below

2) only show a shoulder-like feature in their state lines. For the samples investigated

experimentally by me, the latter two situations are not met. As in our extreme version, A is

found between 7 and 10. The A value of my sample are calculated with Eq. [2.07] and listed

together with radius and effective charge of the sample in Tab. 2.


32

Tab. 2. Particle data. 2aNOM is the nominal diameter from TEM measurements as given by the manufacturer. ah is

the hydrodynamic radius from dynamic light scattering. For PS120 in addition, the geometric radius is given

from static light scattering (S) and for PnBAPS68 the radius from ultracentrifugation (UZ). Z* is the effectively

transported charge from conductivity measurement. A is the empirically constant in the relation Z* = A a/B,

calculated using Z* and a Bjerrum length of B = 0.72nm (c.f. Fig. 16).

I now shortly sketch the influence of the other experimental parameters. The salt

concentration dependence of state lines is shown in Fig. 24 for Z* = 500, a = 50nm and 10-3

µm-3

n 103µm

-3, c increases from left to right. I note that the salt concentrations used are

far below the critical coagulation concentration, and pair interactions are still well described

using repulsive terms only. With increasing salt concentration the upper part of the curves is

shifted right towards larger values of and the maximum gradually disappears. This is due to

an increased but almost constant . Once the counter-ions dominate all curves again

coincide. Larger open circles and squares represent state lines for constant n respectively of n

= 0.5µm-3

and n = 5µm-3

. Both curves ascend with increasing salt concentration showing the

pair energy decreases as the screening is increased, and this screening effect appears more

pronounced for dilute suspension (e.g. n = 0.5 m-3

).

# Batch No. 2aNOM

/nm

ah /nm Z* A

PnBAPS68 BASF 68 34 (UZ) 45016 9.7

PS90 Bangs Lab

3012

90 49.5 51038 8

PS100 Bangs Lab

3067

100 55.9 53050 7.6

PS120 IDC

10-202-66

120 64.1

60.6 (S)

68520 8.2

PS156 IDC 2-179-4 156 - 94570 8.7


33

2 3 4 5 6 7

0.05

0.10

0.15

0.20

0.25

0.30

kBT

/U(d

)

Fig. 24. Salt concentration dependence of state lines for sample PS120 (2a = 120nm, Z* 685). Salt

concentrations are: (— ■ —) 0.2 M; (— ● —) 0.37 M; (— —) 0.70 M; (— ◊ —) 1.03 M; (— —) 1.36

M. The two upward lines connect points of same particle number density under difference salt concentrations:

(—○—), n = 0.5 m-3

; (------ ), n = 2.0 m-3

. For a detailed discussion see text.

For completeness, also the particle size dependence of state lines is shown in Fig. 25 . Here I

fix Z* = 2000, c = 0.2 µM, and 10-3

µm-3

n 103µm

-3. It shows that all the state lines in the

left part of very dilute n are almost coincident at the same pair energy level for a given n, and

almost the same energy interval for a given interval of n. It proves that the geometrical factor

in the DLVO pair interaction (i.e. [exp(a)/(1+a)]2

) has less significance at very dilute

suspensions29

. Also these state lines show that with increasing particle radius the height of the

maximum decreases, i.e. self-screening becomes less important.

Since the applied effective charge Z* in this thesis are taken from conductivity

measurement83

, Z* may have some uncertainty mediated by the particle mobility -.

However, under my calculation, the possible maximum error of state lines due to - reason for

)dT/U(kB and is less than 10 % and 1 %, respectively (shown in Fig. 26). This proves that

state lines under the model of DLVO pair interaction can be safely used to describe here the

charged colloidal system.


34

4 6 8 10 12 14

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

KBT

/U(d

)

Fig. 25. Particle size dependence of state lines for fixed effective charge Z* = 2000 and c = 0.2 M, n~ 0.001---

1000m-3

. Particle diameters d: (— • —)50nm; (— ● —)100nm; (— —)200nm; (—▼—)500nm. For a

detailed discussion see text.

2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0

0.02

0.04

0.06

0.08

0.10

0.12

0.14

PS120

-=12*10

-8m²V

-1s

-1

-=6*10

-8m²V

-1s

-1

-=2*10

-8m²V

-1s

-1

kBT

/U(d

)

Fig. 26. Particle mobility dependent state lines in the sample of PS120. A varying mobility causes a variation of

effective charges for a certain range of n. State lines are calculated for these ranges of effective charges. The

possible variation of kBT/U( d ) and due to the uncertainty of the particle’s mobility are 10% and 1%,

respectively.


35

Chapter 3

Experimental techniques

and corresponding theories

3.1. Standard preparation technique

Standing preparation for colloidal crystals has been commonly applied for a long time. In the

standing preparation the colloidal suspension is kept in a sealed cell together with ion-

exchange resin (IEX). Impurities and small salt ions then are gradually deleted by the IEX.

One obtains colloidal crystals after completion of the deionisation process for a sufficiently

large value of particle number density n (see Fig. 27), which an opalline sample results in

beautiful colours. For such crystal powders, the differences in colour can be traced back to the

sample structure and the crystallite orientations. For the theoretical description (see Chapter

4.3) where I shortly recall the theoretical background which is similar to the description of x-

ray or neutron scattering120, 121

. For a known crystal structure, one can easily determine n from

a static light scattering measurement. However, there is no way to measure the conductivity

and therefore the salt concentration c within this type of sealed cell. Other shortcomings also


36

cannot be neglected, like a long time sample preparation for a fully deionisation (up to months

for dilute suspension); particle sedimentation, which diminishes n in the suspension, etc.

Therefore an alternative method for sample preparation has been developed by our group

using a pump-tubing circuit122

. My sample preparation is based on that set up with some

experimental modification (see Fig. 28), Further I will introduce some theoretical methods for

a developed sample preparation in Chapter 4.

Fig. 27. Polycrystalline colloidal solids obtained under standing preparation. Sample cells are all put upside

down to immediately catch CO2 leakage in the surroundings of IEX. Note the different colours of the

polycrystals originating from different lattice spacing and different crystal orientations.Image courtesy to AK.

Komet 336, Physik. Institue, Uni. Mainz website.

3.2. Sample preparation under pump tubing circuit

Different to standing preparation, this pump tubing circuit can delete the impurities very fast

simply by pumping the suspension through an IEX column, and repeating this procedure by a

tubing circuit. In addition, one can control the salt concentration by introducing a

conductivity meter in the circuit and pumping the sample through a by-pass.

The pump tubing circuit is a closed Teflon tubing system containing several components: a

pump for continuously pumping the sample through the circuit; a reservoir for adding

additional salt, suspension or water (for preventing CO2 leakage from the air into the

suspension, usually the reservoir is filled with Argon gas above the suspension); IEX

(Amberlite UP 604, Rohm & Haas, Chancy, F) is filled within a column in the circuit where

two nylon film (0.2- 0.5 m filters, Millipore, USA) are fixed above and below the column

for isolating the IEX from other components, especially the sample cell; a by-pass is for


37

deleting air bubbles or as a passage for nondeionising sample at a certain amount of salt

concentration; a conductivity meter (WTW 2001, electrode LTA 01, Weilheim, D) is

connected for controlling the salt concentration in suspension; different sample cells for

different experiments can be linked in one circuit. Notice that here in Fig. 28, I only show the

rectangular cell for Bragg microscopy measurement.

Fig. 28. Sample preparation in pump tubing system connected by several component: pump, reservoir, By-pass,

IEX (ion-exchange resin) cell, measuring sample cell, conductivity meter cell. (--) denotes the direction of

sample flux.

To interpret conductivity measurements, I use the Hessinger´s model 97

, which allows for an

exchange of ions between the inner and outer part of the electric double layer as long as the

overall radial charge distribution is retained. If salt is added, salt ions may exchange with

counterions of equal charge sign, while coions are assumed to stay outside the proposed inner

shell due to electrostatic repulsion. Within this model it is convenient to introduce the number

concentration M = c1000 NA/n of small ions per particle and the arithmetic mean small ion

mobilities

i

i

i

ii

M

Mμ

μ ;

i

-

i

i

ii

M

Mμ

μ [3.01]

Br

agg


38

Assuming the additivity of all conductivity contributions ( = inieziµi with zi =1 in the case

of monovalent salt and counter-ions), one may formulate

0-

* σ + μμMμμ Zneσ [3.02]

For comparison, the mobility of H+, OH

-, Na

+, Cl

-, are 36.5x10

-8 m²/Vs, 15.8 5.02x10

-8 m²/Vs,

7.1x10-8

m²/Vs, respectively123

.

In the case of complete deionisation, Eq.[3.02] reduces to Eq. [2.06], i.e.

μμeZ nσσ *

0

Thus, in many of the samples a linear function of - n is found. This is shown in Fig. 29.

0 1 2 3 4 5 6 7 8 9 10 11 12 130

1

2

3

4

5

6

coex. fccbcc

fluid

(

S/c

m)

n (µm-3)

Fig. 29. Linear relation between conductivity and particle number density n as measured under deionised

conditions. Note that the - n relation is a linear function independent on the sample phase (Here what is shown

is sample PS120). The solid line is a fit of Eq.[2.06] yielding a conductivity effective charge Z* = 68520.

Figure courtesy to124

.

According to Fig. 29, taking PS120 as an example, a linear function - n is found under

deionised condition, it is independent of the colloidal phase. One can determine n directly

from conductivity measurement if one knows the effective charge Z* of sample. This method


39

is found also suitable to be applied in binary mixtures (see theory in Chapter 1), where for ,

n and Z* the average values are applied.

The deionisation process with the pump tubing circuit is very fast, it can be measured via

conductivity shown in Fig. 30.

0.0 4.0 8.0 12.0 16.0

0

10

20

30

40

50

60

70

80

min

(

S/c

m)

t(hour)

Fig. 30. Conductivity versus deionising time. It first shows a sharp decrease for fast deionising then a stable

minimum conductivity of min, giving the conductivity of the background. Sample PS90 (2a = 90nm, Z* =

51038) at n =11.1m-3

. Full deionisation only needs a few hours.

This method shows some benefits in sample preparations as following: (1) fast deionising

thus sample preparation can be realized in a short time; (2) c and n can be continuously

controlled with conductivity meter; (3) By linking several sample cells in one circuit, one can

characterize the sample detected under different set-ups; (4) an experimental condition simply

by diluting or concentrating suspension and adjusting c is reproducible; (5) IEX is kept in the

IEX column by the Nylon-films, so it won’t disturb the experimental detection in the sample

cell. In addition, One may prepare metastable melt states as well as single crystals or

polycrystalline as the suspension is easily shear molten and readily re-crystallising once shear

(pumping) is aborted.

Usually, the sample is considered as deionised once = min, however, there is still a small

amount of salt left in the suspension, which leaves difficulties to precisely control c and n, and

in turn presents a difficulty to obtain sample crystallisation in dilute suspensions. Therefore,


40

except for a experimental modification to the pump tubing circuit (shown in Fig. 28), further

theoretical developments for a careful sample preparation are required.

3.3. Structure and concentration determined by static light scattering

Fig. 31. Sketch of light scattering through sample

A typical static light scattering geometry is shown in Fig. 31. A vertically polarized and

monochromatic beam of incident light is used with vacuum wave length and wave vector

ik (λ

γ2πk i ). The scattered light has a wave vector

fk (f

fλ

γ2πk , f is the wave

length in solution, and f ). In a static light scattering experiment the scattered intensity IS

is measured as a function of scattering angle connected to the modulus of the scattering

vector q via fi kkq

. = 1.333 is the refractive index of the suspending medium

(water) and is the laser wave length in vacuum. VS is the illuminated scattering volume

(shown in parallel dotted oblique lines), which contains NS particles in illumination. The

samples can be divided into many small subregions, which are all polarized by the alternating

incident electric field iE , and

iE can be mathematically described as a plane wave. If the

largest dimension of the particle is small compared to the wavelength of the light, all

subregions see an identical incident electric field. The amplitude Ef of the scattered field can

then be written as a simple sum of the contributions from the individual scatters.


41

We assume a suspension to be illuminated by a plane electromagnetic wave. The amplitude of

the light scattered by a particle i is bi q . The modulus q of the scattering vector q is related

to the scattering angle and wavelength as

2

θsin

λ

π4q [3.03]

The instantaneous amplitude E(q) of the field from light scattered by an assembly of N

particles is

N

1i

ii rqiexpqbqE

[3.04]

Since here only static properties are concerned, the time dependence of E is dropped. The

instantaneous intensity of the scattered light is proportional to the square of the scattered field.

2

t,qEt,qI [3.05]

Its average value is therefore given as

N

1i

N

1j

ijji rδqiexpqbqbqI

[3.06]

For spherical monodisperse particles, all bi(q) are the same, so bi(q) = b(q). Then Eq.[4.05]

can be written as:

)S(qP(q)0bNqI2

[3.07]

which b(0) = (4/3)a³ (refractive index variation = particle - medium), P(q) is the form

factor of a single particle, where

P

2

0b

qbq

[3.08]


42

For homogeneous spheres with radius a, b q

can be resolved in the Raley and Debye-Gans

approximation as

qaqacosqasinqa

3δγaqb

3

3 [3.09]

So the form factor can be deduced as

26

qaqacosqasinqa

9qP [3.10]

The structure factor S(q) is defined as

N

1i

N

1j

ijrδqiexpN

1)qS(

[3.11]

The case of light scattering from crystal lattice planes is illustrated in Fig. 32.

Fig. 32. Bragg reflection in a crystal lattice. is the scattering angle, dhkl is the lattice distance, ik is the incident

wavevector, fk is the reflected wavevector.


43

For the crystalline state, one observes Bragg-peaks at positions 222hkl lkh

g

2πq ,

where h, k, l are the Miller indices and g is the lattice constant of the crystal. For a body

centred cubic (bcc) crystal g = (2/n)1/3

and for a face centred cubic (fcc) crystal g = (4/n)1/3

.

From the Bragg peak position, one can determine crystal structures and orientations and

particle number densities according to their Miller indices (h, k, l), which is simply

conclusively drawn in Fig. 33125, 126, 127

. For a reflection from bcc structure, h+k+l should be

an even number, while for a reflection of fcc structure, the numbers h, k, l should either be all

even, or all odd. But for a refection from sc structure, it shows all the possibility of h, k, l.

Fig. 33. Normal lattice structures and their correlated faces: SC (simple cubic) has the face (100), (110), (111),

(200), (210), (211), (220), (221), (300), (301), (311), (222), (302), (321); bcc (body centred cubic) has the face

(110), (200), (211), (220), (301), (222), (321); fcc (face centred cubic) has the face (111), (200), (220), (311),

(222). They are prospected to be detected by static light scattering with increasing scattering angle from left to

right. Figure courtesy to125

.

A sketch of static light scattering set-up is shown in Fig. 34.


44

Fig. 34. Configuration of light scattering spectroscopy

The sample is illuminated by a diode laser (Vast technologies 3mW variable power, = 690

nm). A /2 – plate (LHP) is added immediately after the laser which is able to rotate the

polarization of the laser, and then a high quality polarizer (POL) which selects vertically

polarized light only. The beam is then focused by lens L1 into the middle of the sample cell.

Beyond the sample cell, the detection optics are mounted on a goniometer rotating around the

axis of the sample holder. Lens L2 converts the scattered light again into parallel light, and

then passes it through two Apertures AP1, AP2 defining the observation volume and deleting

parasitic stray light from the cell and the index matching bath. The scattered light intensity is

collected and amplified by a photomultiplier (PM). The data is further processed by a counter,

correlator and computer system. The correlator is chosen as a digital correlator (ALV,

ALV5000), which the details were described in128, 129

. The home-built goniometer has an

angular range of = 20° - 150° with a resolution of approximately 0.1°.

The suspension under study is contained in a glass cell of 10 mm in diameter (see Fig. 35),

held in a sample holder (see Fig. 36) and positioned at the centre of a refractive index =

1.458, height 70mm, inner-diameter 80mm, outer-diameter 85mm for cylindrical water bath.

This bath is filled with a mixture of decalin and tetralin, the proportions of which are chosen


45

to give the same refractive index as the suspension. The sample temperature is held within 24

± 1°C of the target temperature during the course of an experiment.

A typical example of a static structure measurement is given in Fig. 37. Five Bragg peaks of

the polycrystalline sample are clearly distinguishable. To identify the crystal structure and

evaluate the particle number density the square root of the sum of cubed Miller indices is

plotted versus the scattering vector q. In this case a bcc lattice constant of g = 658 nm and a

particle number density of n = 7.01 µm-3

results.

Fig. 35. Sample cell in light scattering. Image courtesy to130

.


46

Hohlschraube

Küvette

Schrägkugellager

innerer Zylinder

äußerer Zylinder

Mutter zum Verspannen der Kugellager

Teflonkonus

hollow screw

cuvette

interior cylinder

teflon cone

oblique ball beds

outside cylinder

nut wedge for ball beds

Fig. 36. Cut image of sample holder. Image courtesy to130

.

10 15 20 25 30 35

222

301

220

211

200

I (q

) /

P(q

) (a

.u.)

q (µm-1

)

300 600 900 12000

2

4

6

8

10

12

110

h2 +

k2 +

l2

q2 (µm

-2)

Fig. 37. Left: Scattered intensity divided by measured particle form factor for sample PTFE260 at n = 7.01m-3

.

Right: Evaluation for lattice constant gbcc = 658 nm. Figure courtesy to130

.


47

3.4. Bragg microscopy

According to Fig. 28, one may connect a Bragg microscopy cell in the same pump tubing

circuit, then under the Bragg microscopy measurements, the phase morphologies of the

sample and growth velocities of wall crystals can be detected by a video CCD camera. A

sketch of the experimental set-up of Bragg microscopy is shown in Fig. 38. Here I use an

inverted microscope with a low resolution objective (Laborlux 12, Leitz, Wetzlar, Germany).

Images are recorded by CCD-camera attached to our microscope’s video port.

This Bragg microscopy set-up is convenient for observing colloidal crystal nucleation in

three-dimensional view by turning the illumination light (white light) and the angle of the cell

(in angle and ). Here in this thesis, the 2x10x100 mm3 or 1x10x100mm

3 rectangular cells

are used in the measurements.

Fig. 38. Set up of Bragg microscopy with several components

Video images must be converted into digital format before they can be analysed. Digitising

video frames requires a dedicated frame grabber which typically takes the form of an add-on

board for a computer. The frame grabber used in this study is Type Oculus TCi-SE (Coreco

Inc., St-Laurent, Quebec, Canada) installed in a 586-class personal computer. Frame grabbers

convert the analogy video stream to digital images in real time, a process which requires more

than 12 million analogy to digital (A/D) conversions per second. Video tape decks with


48

computer interfaces such as the SONY EVO-9650 can be controlled by the same computer

which hosts the frame grabber card. A fairly straightforward program (PCI-Bus,

Programmable Communication Interface, Coreco Inc., St-Laurent, Quebec, Canada) then can

direct the tape deck to seek out and pause at a particular video frame, guide the frame grabber

digitise the paused image, and store the result to disk. Repeating this process permits

digitising any sequence of video frames131

. The line-engraved slab (Spindler & Hoyer,

Göttingen, Deutschland, Best.-Nr. 063510) takes the scale of the microscope picture which I

show it in Fig. 39.

Fig. 39. A line-engraved slab scaled with 200 strips in 5 mm under a 5x-objektiv microscope .

Due to different crystal colours correlated with different crystal lattice surface, suitable

technique for adjusting the direction of illuminating and observing to get the better contrast of

crystal colour is very important. In my experiment, I use two methods for observing crystal

morphology, one is called top-view, another is called side-view. The cell profile and crystal

growth direction can be depicted as Fig. 40. This is a 2x10x100mm3 rectangular quartz glass

cell. If 10x100mm2 plan faces the objective of Bragg microscopy, I call it top-view, I observe

the twin domain morphology normally in this way; If 2*100mm2 plan faces the objective of

Bragg microscopy, I call it side-view, which I speculate the wall crystal growth also the fluid-

crystal phase transition. The arrows show the direction of wall crystal growth in the cell. In

Fig. 41, the sketched plane image together with some crossed blue lines in the cell simply

shows the fully contacted wall crystal in the cell.


49

Fig. 40. Fully wall crystal growth in the 2mm*10mm*100mm cell. By observing the face 2mm*100mm (left

image), wall crystals stop the growth up and down (sign ⇧⇩) in the middle of the cell shown as one line (- - - -),

where wall crystals contact each other. However, if observing in the plan 10mm*100mm (right image), wall

crystals are observed stopping growth shown in two lines (- - - -) without contacting each other. This behaviour

is due to that wall crystals abort to growth in the face of 2mm*100mm.

Thus by side-view, crystals either homogeneous crystal (c.f. Fig. 42) or heterogeneous sheet-

like wall crystal in (c.f. Fig. 43) can be observed. In Fig. 42 from (a) to (b), it shows the large

bulk homogeneous crystal growth with time, and its shape shows its favourable orientation.

As a contrast, the dark background is wall crystal however in an unfavourable illumination

direction.

Fig. 41. Sketches (blue lines) show cross section of wall crystal growth in the cell.


50

(a) (b)

(c) (d)

Fig. 42. Homogeneous crystal growth with time increase. By illuminating a fully crystallized sample with white

light at a certain angle and , a large bulk homogeneous crystal shown pink colour is observed. (a) to (d) show

the shape variation of this crystal versus time due to the time-dependent favourable orientation. As a contrast, the

dark background is the wall crystal.

Fig. 43. Heterogeneous sheet–like wall crystal growth versus time (time increases from left to right) observed by

a side-view (the distance in between is 2mm). By illuminating the sample at a certain angle and , wall crystals

show pink colour. The colour of dark yellow in the bulk shows the fluid phase of the suspension. The black lines

above and below are the cell boundary The wall crystals finally contact each other as fully crystallization shown

in the most right part of the image.

In Fig. 43, Samples are shear molten and after stop of shear readily solidify via heterogeneous

nucleation at the cell wall with subsequent quasi-epitaxial growth. The formerly applied shear

orients the nuclei and oriented twinned crystals result. Their (110) plane is parallel to the cell

wall with the <111> direction parallel to the formerly applied flow direction. Growth

proceeds inward in the <110> direction. Here the sheet-like wall crystals grow up and down

versus time (time increases from left to right), finally they contact each other due to fully


51

crystallization, therefore, no fluid leaves in the bulk (the colour of dark yellow disappears in

the most right part of the image).

By side-view, I also observe fluid-crystal phase transition and morphology transition, it will

be shown in Fig. 61, Fig. 62, Fig. 67 together with detail discussion concerning the crystal

morphologies.

By top-view, I observe the twin domain crystal morphology shown in Fig. 44. This is the

most common morphology of bcc twin domain, further observation will be also shown in Fig.

68, Fig. 69, Fig. 70, Fig. 71.

Fig. 44. The twin domain observed by top-view. The cloudy-like twin domain is found for most of the single

component samples also binary mixture at some range of n. It hints two orientation of crystal lattice plane

existed in this twinned crystal.

Except the normal Bragg microscopy illuminated with white light, I also observe the wall

crystal growth illuminating the sample cell (2x100mm2 plane) with a laser source (laser diode

= 690 nm, Vast technologies, 3mW variable power). A couple of lens and pinholes are used

to get a collimated light, a good alignment and also a decreased incident power intensity. In

this case, sample should be tiled a little bit (angle ) in case to observe the colour contrast

between fluid (bright, due to higher scattered intensity) and wall crystal (dark, due to lower

scattered intensity) (shown in Fig. 46). The set-up profile is shown in Fig. 45. The Fig. 46

from left to right part, shows the crystal growth versus time.


52

Fig. 45. A sketch of set-up for observing crystal growth illuminated with laser source. Notice that 2x100mm2

plane of the sample cell is not perpendicular to the microscopy objective, it tilts a small angle ( angle), which

follows () 110. For obtaining a collimated, aligned, focused, and then parallel light source, a couple of lens

and pinholes are used.

Fig. 46. Crystal growth illuminated with laser source as Fig. 45. The white bright rod in the most left image

shows only fluid phase in the cell. From the left image to the right, wall crystal (dark rod) gradually grows,

whereas fluid phase (white bright rod becomes less amount in the bulk with time increases.. The most right

image leaves no fluid in the bulk, wall crystals contact each other up and down, which shown as the bright rod in

the bulk only. The black part shows the non-illuminated suspension.

In Fig. 46, a couple of bright and dark “rods” are observed. Different to normal Bragg

microscopy but combined the observation of sheet-like wall crystal growth as Fig. 43, I

conclude that these bright parts of “rods” mean fluid phase which has a higher scattered


53

intensity, while these dark parts of “rods” mean crystal phase. The theory is shown by static

light scattering measurement in Fig. 47.

3 0 6 0 9 0 1 2 0 1 5 00

2

4

6

8

1 0

1 2

1 4

1 6

1 8

9 5 .2 °

( )

s h e a r

m e lt

b c c

s o lid

I() (

a.u

.)

(° )

Fig. 47. The static light scattering measurements for the crystal and melts of PS100 sample at n = 16.4m-3

. The

experimental points for crystals are shown as (--), that of melts is shown as (- -). The blue line crossing the

scattering curves gives a contrast of the scattering intensities of crystals and melts at () 110.

In Fig. 47, the sample (PS100, n = 16.4 m-3

) is measured twice. One is the crystallisation

measurement, which simply measure the scattering intensity of sample after stop pump

shearing and sample is fully crystallized. It shows a sharp peak at bcc (110) face with

scattering angle 110 = 95.2°; Another is the shear molten test, which I first shear the sample

by pumping, then stop shear for a measurement to the scattering intensity of shear molten

sample. By repeating the same procedure, I measure the scattering intensity of shear molten

sample scanning from 30° to 140° by rotating the Goniometer, the structure factor (non-

normalized) of this sample is obtained in this way. If I tilt the rectangular sample cell suitably

( angle) shown in Fig. 45, then the bright and dark ‘rods’ shown in Fig. 46 are actually due

to the different light scattering intensities of the crystals and melts guided by the blue line in


54

Fig. 47. Due to the structure factor of fluid phase has a broad first peak than the (110) crystal

phase, at scattering angle () 110, a higher scattering intensity is observed for the melts

than that of the crystal phase, therefore, the bright part of “rod” is detected for the melts,

whereas the crystal phase shows a dark “rod” in Fig. 46.

By Bragg microscopy illuminated either with white light or laser source, crystal growth can

then be detected according to Fig. 43 and Fig. 46. At fully crystal phase, the height of wall

crystal shows a linear function with time shown in Fig. 48. Notice that the fit does not

extrapolate to zero, this allows to define an initial thickness d0 which will be investigated with

detail in Chapter 6.

0 20 40 60 80 100 120 1400

200

400

600

800

1000

d(

m)

t(s)

Fig. 48. Crystal growth height. d as a function of the nucleation time t after stop pump shearing. The

experimental data show as (--) , whereas the black line is the linear fit. The growth velocity v110 of the crystal

(110) face obtains from the slope of the linear fit.


55

Chapter 4

Further developments

for precise sample preparation

4.1. Experimental control of salt concentration by addition of CO2

Pump tubing circuit as an improved method of sample preparation has been introduced in the

last chapter. This technique has been known as a method for sample preparation with many

benefits. However, it is still found difficult to precisely control c by conductivity as the

relation between c and strongly depends on the type of electrolyte97

(see Fig. 49). Even at

c 10-5

M, d/dc was found nonlinearly by Hessinger et al.97

. In particular, when NaCl is

added, there always may be a background of airborne contamination. So I consider to alter c

by adding CO2. CO2 reacts to give small amount of dissociated H2CO3, which yields identical

counter-ions and was found to have no surface chemical influence to particles97

.

To implement the CO2- addition, I modify the original pump tubing circuit47

(c.f. Fig. 28) as

following: (1) Two switches are added before and after the soft pump tubing, they are closed


56

immediately after stop pump shearing; (2) A larger volume of reservoir (about 400ml) is used

to increase the total volume of suspension. Thus small changes of c and n due to leakage from

the connecting switches in the circuit, or due to the evaporation during pumping have less

influence to ; (3) After adding addition air (CO2 is inside) by a syringe into the suspension

of the reservoir, the suspension is pumped through the by-pass, within a few minutes, a stable

value of (correlated to c) is obtained.

I further check the mixing of additives with the original suspension by the following test. A

pump circuit is built and filled with water only. Then under by-pass conditions, a small

amount of particles are added to the reservoir and their dispersion under continuous pumping

are observed by microscopy. Fig. 50 shows that the evolution of homogeneous distribution of

particles in the sample cell can be completed within two minutes.

0 1 2 30.5

1.0

1.5

2.0

2.5

3.0

3.5

HCl

NaCl

NaOH

(

µS

cm

-1)

c ( µM)

Fig. 49. The dependence of the conductivity on the kind of electrolyte applied in the sample PS109 at n =54.36

m-3

. Notice that for additional neutral electrolytes, shows a pronounced nonlinear effect via c. Figure courtesy

to97

.


57

Fig. 50. The time dependence of a sample flux through the cell observed by Bragg microscopy. The

homogeneous blue colour distribution in the cell shows that a homogeneous density distribution needs less than

2 min after abortion of shear. The dark colour at t = 0 shows that only water in the cell.

By pumping through by-pass, a conductivity measurement after stepwise adding CO2 is

shown in Fig. 51. After the additional damped oscillation corresponded to the mixing process,

After the oscillation’s decay, a stable value of is obtained with duration time about 10min or

more (shown the position of A, B and C in Fig. 51). It is much longer than the duration time

for the particle density distribution (< 2min). Therefore, the number of A, B, and C in Fig.

51 can be taken for the calculation of salt concentration c. Once I get a stable (like A, B and

C), I stop shear, close the two switches before and after the soft tubing (c. f. Fig. 28), and then

start to record the sample’s phase morphologies and the crystal growth by a video CCD

connected to the Bragg microscopy.


58

15 20 25 30 35 40 45 50 55

1.15

1.20

1.25

1.30

1.35

1.40

1.45

1.50

1.55 C

B

A

add CO2 and pump through by-pass

pump through IEX for deionizing

(

S/c

m)

t(min)

Fig. 51. Salt control in the sample cell by a conductivity meter. At t =15 min, sample is pumped through the IEX

cell for a fast deionising, At 17.5 min the IEX is by-passed. Then CO2 is added into the reservoir and pumping

proceeds through the by-pass. After each addition, a stable conductivity at positions A, B and C is obtained

with duration time of about 10 min or more.

To calculate the actual salt concentration I proceed as following. First, I measure min after

complete deionisation of this suspension by pumping the sample through IEX column;

second, I add air into the reservoir by a syringe and pump the suspension through the by-pass,

continuously pump until getting a stable (c.f. conductivities at the position A, B and C in

Fig. 51). The salt concentration c can be calculated from corresponding by the following

equation within the precision in the order of 0.1 M123

c = ( - min) / H2CO3

, [4.01]

where H2CO3

= 39.41510-4

(S·l)/(cm·mol) (T=298.15K) denotes the limiting molar

conductivity at infinite dilution. Thirdly, I stop shear and perform Bragg microscopy or light

scattering measurements to the sample with a duration time less than the time scale of further

contamination, then these experiments are realized under a controlled ionic strength. Each

measurement can be repeated simply by the same procedure as above.


59

4.2. Improved deionisation controlled via crystal growth

Besides the experimental control of c for an additional CO2, an improved theoretical method

for sample deionising via crystal growth experiments is developed. Crystal growth is

observed with side-view by Bragg microscopy (c.f. Fig. 41, Fig. 42, Fig. 43). The control

procedure is demonstrated in the two commercially available species of charged Polystyrene

spheres in aqueous suspension (PS120 and PS156). Samples are investigated in a binary

mixture of size ratio 1.3:1 (PS156/120). Before preparing mixtures of these, the individual

components are carefully characterised by various experiments. Tab. 3. compiles the most

important results.

Tab. 3. The properties of the pure components. 2anom: nominal diameter; Z*: effectively transported charge from

conductivity; v: limiting growth velocity at infinite “undercooling” .

Experiments are conducted under deionised conditions and at a particle concentration where

all samples including the pure suspensions are completely solidified at equilibrium. Two

boundary conditions should meet in that choice. First n should be large enough to be very

close to the limiting growth velocity v for the pure systems. On the other hand, n should be

lower than the bcc-fcc transition for PS 120 in order to keep the experiment conceptionally

simple. For n = 0.47 µm-3

both pure systems are of bcc structure124

, v120 is only slightly lower

than v,120. Experiments then are conducted at this fixed n, i.e. PS120 particles are stepwise

replaced by PS 156 without altering the lattice constant. I stepwise replace the small by the

big particles up to the minority fractions of p156 0.18 (note: p156 = n156 /(n120 +n156)).

Experiments are further intended to be conducted at thoroughly deionised conditions. To

control the deionisation process, I monitor the conductivity. As demonstrated before the

conductivity shows a first sharp drop and a constant low value is reached after some 30 to 60

minutes (c.f. Fig. 30). In studies reported previously and in this thesis, deionisation usually is

Sample Source

Batch No

2anom

/nm

Titrated

charge Z

Z*

v

(m.s-1

)

PS120

IDC

10-202-66

120

3600

68520

4.1

PS156

IDC

2-179-4

156

5180

94570

3


60

continued for at least four times that period. Sometimes, and in particular at low n even a

shallow minimum is visible77,

132

. Achievement of minimum conductivity, however, is not

necessarily identical to reach the point of complete deionisation. To make this point explicit I

show measured growth velocities as a function of deionisation time in Fig. 52.

The initially observed growth velocity drops with increasing p156. For p156 = 0.18, no growth

is observed immediately after reaching minimum conductivity. In general however, v110

increases with continued deionisation time to saturate at a constant value after some hours.

For the measurements presented later on, v110 is taken as the growth velocity under thoroughly

deionised conditions. I note that once this state is reached, further contamination proceeds

mainly via CO2 leaking in through fittings etc. Interestingly that impurity can be removed on

a much faster time scale (c.f. Fig. 30) for deleting multiple impurity within more than half

hour. In Fig. 51, min is achieved within a few minutes by deleting only CO2 through IEX.

Even more important, the times to reach minimum conductivities and maximum growth

velocities in most different mixing number ratio of p156 are coincident.

0 20 40 60 80 100 120

2.2

2.4

2.6

2.8

3.0

3.2

3.4

p156

=0.05

p156

=0.08

p156

=0.11

p156

=0.14

p156

=0.18

V1

10(

m/s

)

time after reaching minimum conductivity(min)

Fig. 52. The developments of crystal growth velocities (v110) for different binary mixtures as a function of

deionisation time (the time after reaching their conductivity minimum).


61

The unexpected increase of v110 for an increased duration time at constant conductivity during

continued deionisation needs further clarification. Since the particle number density stays

constant, It is known that any changes in v110 are attributed to the changes in the salt

concentration43

. In particular, I have to suspect a continuous decrease of salt concentration at

constant or slightly increased conductivity. To further explore this first point I carry out

deionisation experiments on aqueous electrolytes to find the rate of anion exchange to

significantly exceed the rate of cation exchange, which is possibly understood from the

mobility of Cl- (7.1x10

-8 m²/Vs) and and Na

+ (5.02x10

-8 m²/Vs) from chemical hand book

123.

Thus during deionisation from a pH neutral electrolyte (NaCl) a transient excess

concentration of Na+ appears. Deionisation of a suspension therefore corresponds to the

backward performance of an acid-base titration.

Not in all cases, however, a pronounced shallow of conductivity minimum is observed77, 132

.

In particular, for cases of low charge ratio Z*/Z, the conductivity drops monotonous in time.

(Here Z* is the number of counter-ions visible in the conductivity experiment and Z is the

number of dissociated surface groups). To explain this finding I resort to Hessinger's model of

conductivity97

. It divides the electrical double layer into an outer part (where all ionic species

freely migrate with their bulk mobility) and an inner part (where the counter-ions move

together with the particle). The total conductivity is then given by the sum over all particles.

Further only the number Z- Z*/Z of bound counter-ions is conserved, but an exchange is

allowed between the two parts of the double layer. Consequently at a low charge ratio Z*/Z,

an effective mobility close to that of H+ will show up with added Na

+, while at large Z*/Z the

effective mobility is close to the lower value of Na+. Co-ions are assumed to stay outside the

inner region and always contribute with their bulk mobility.

The transient excess of Na+ lowers the average counter-ion mobility and at large charge ratios

a minimum is observed. It disappears once as a pure protonic counter-ion cloud is established.

Therefore in CO2-contaminated samples without further electrolyte, the growth velocities

obtain the final large value at the same time as the minimum conductivity. At low charge ratio

the effect is less pronounced and compensated by the continuous decrease in anion

concentration. Therefore in most of the present cases the conductivity is observed to stay

constant during further decrease of the salt concentration. The latter then translates into an

increase in v110.

Referring to Hessinger's conductivity model the interesting observation of a changed growth

velocity at constant conductivity thus is explained with a transient excess of cationic


62

impurities. Vice versa, it can be used as a parameter for control the transient excess of

cationic impurities.

4.3. An improved empirical qmax – n relation for determining n of fluid-like

phase

Light scattering has been used as an important means to determine the particle number density

n of the crystalline state. This can be achieved by calculating the lattice constant from the

Bragg peak position once the crystal structure is identified133

. The most common alternative

way of determining n, however, is a calculation from the volume of added water by diluting a

sample with known volume fraction (e.g. from a drying experiment) and relating the

resulting volume fraction to n via n = / (4/3)a3 where a is the particle radius. This

procedure faces the difficulty of both dilution and evaporation errors and errors in the exact

determination of the geometrical particle radius134

. Therefore different results are obtained for

a, if it is determined by Transmission Electron Microscopy or dynamic light scattering.

Another way is by weighing the mass difference of the suspended and dried particles if the

mass densities are known to sufficient accuracy (PS = 1.05g/cm3). Again the particle radius

has to be known135

. This technique is well suitable for concentrated samples of say 0.1

but faces large weighing errors for dilute samples. So detecting particle number density to a

high accuracy needs a well defined procedure. In a recent paper of our group the use of

conductivity was proposed97

which was calibrated by light scattering in the crystalline regime.

The only additional quantity required is the knowledge of the particle mobility available from

electrophoresis measurements with sufficient accuracy99,122

. Then the particle number density

can be obtained from measurements in the deionised state with an error of less than 2%. This

method is very suitable for homogeneous samples.

The focus of recent research, however, has shifted from homogeneous systems in equilibrium

to inhomogeneous systems as observed e.g. during crystallization, at coexistence, under

conditions of phase separation or under shear. For instance, it is well known from the hard

sphere system, that across the coexistence region a pronounced difference in particle

concentration may appear136

. This is also true for charged and other soft potential systems116

.

Further in some cases of dilute near salt free suspensions gas-liquid-like or gas-solid-like

phase separations were encountered109, 137,

138, 139, 140

. In such situations conductivity cannot be


63

applied, as it only yields the average particle number density but not the densities of the

individual phases. For characterising phase behaviour even under very dilute suspensions,

determining n becomes important. This motivates the finding of an empirical q formula for a

theoretical calculation of n. It is especially suitable to estimate n under disordered fluid-like

phase, fluid-crystal coexistence region without pronounced Bragg peaks or the cases without

reliable conductivity measurement, etc.

For the fluid state one may try to estimate n from the position of the main peak of the static

structure factor S(q). This position should relate to the wavelength L of the most common

spatial variation of the refractive index and hence of the particle positions. An often used

estimate of this length scale is the mean interparticle spacing L = d = n-1/3

. It has been shown

that for very dilute fluids this is a good approximation141

. However, this has not been

undertaken for more concentrated samples. As a solution, first I shortly introduce the

experimental procedures and illustrate the discrepancy between the simple estimates based on

d and my experimental findings, then I conduct systematic measurements to derive the

improved relation between the positions of the maximum of the static structure factor and n.

Finally I discuss the short range order in colloidal fluids and melts and the application of the

technique to further experiments.

As shown above, samples are prepared in a pump circuit. Here, I use commercially available

Polystyrene spheres of different diameter and charge, they are PS120, PS100 and PS90 with

sample data appeared in Tab.2. All experiments are performed at a temperature of 24 1°C.

The tubing system contains a cylindrical flow through cell of outer diameter 10mm (c.f.

Fig.35) used for the static light scattering experiment (c.f. Fig. 34). Simultaneously the

scattered angle is recorded and min is determined.

A typical example of a static structure measurement is given in Fig. 37. Five Bragg peaks of

the polycrystalline sample are clearly distinguished. To identify the crystal structure and

evaluate for the particle number density the square root of the sum of cubed Miller indices is

plotted versus the scattering vector q99

. In this case a bcc lattice constant of g = 658 nm and a

particle number density of n = 7.01 µm-3

results. As nucleation and growth of crystallites is a

complicated procedure, we can not always observe ideal Bragg peaks in our scattered

diagram. In particular, close to the phase boundary only few crystallites are contained in the

scattering volume reducing the statistical accuracy of Debye-Scherrer powder technique. At

the largest concentrations only the main peak is available. Both cause the calculated n to have


64

a changed error. A lower boundary for this error is on the order of 1% for cases shown in Fig.

53. A realistic estimate for the cases discussed below is 2%.

For liquid-like state the structure factor shows a main peak followed by some oscillations, but

no Bragg-peaks are visible. Since the structure factor S(q) is the Fourier transform of the pair

correlation function g(r) the position of the main maximum qmax is related to a wave length L

of the most pronounced oscillation in particle density. An approximate relation between this

length scale L and qmax is then given by

d

2

L

2πq axm

[4.02]

where L has been identified with the average inter-particle distance d = n-1/3

. The latter

relation does not necessarily hold for arbitrary pair potentials120

. For very dilute charged

colloids of the fluid-like order, the theoretical considerations show that it gives the correct

cubic dependence of qmax on n141

. I now may calculate n from the angular positions max of the

first peaks in S(q) measured for both fluid and crystalline suspensions (bcc: (110);

bcc: (110)):

2

θsin

λ2

γ22n max3

33

[4.03a]

fcc: (111)):

2

θsin

λ3

γ42n max3

33

[4.03b]

fluid:

2

θsin

λ

2γn max3

3

[4.03c]

I draw the three linear relations in Fig. 53.

From this plot, I can deduce that e.g. at n = 16.4µm-3

a difference of some 13° in advance as

compared to the bcc (110) peak position is expected for the fluid first peak position. I

prepared a sample at this n and compare the measurements of its crystalline and shear molten

fluid-like state (shown in Fig. 47). As the suspension rapidly re-solidified, care is taken to

keep the sample homogeneous. To this end data are collected in different runs. In each, the

sample is first shear molten by pumping the suspension through the tubing system, then the

shear is aborted and a few points for the scattered intensity I() are measured. The observed


65

scattering angle difference between first maximum of S(q) in the shear melt and the bcc(110)

Bragg-peak of the crystalline state is rather small (about 1°). This experimental result is

different with the conclusion of Fig. 53.

0.0

0.1

0.2

0.3

0.4

0 5 10 15 20

Fluid 1st peak

FCC(111) peak

BCC(110) peak

n (m-3 )

sin

³( m

ax/2

)

Fig. 53. Plot of the expected linear relations of sin3(max/2) versus n for fluid, fcc and bcc ordered suspensions

Also for the other polystyrene samples and for different n, the angle differences are less than

2°. Since the data were taken on homogeneous samples, either fully shear molten or fully

crystalline, both have the same n. Thus the use of Eq. [4.03c] would lead to an error in n of

some 20%. This finding motivates the following systematic investigation to derive an

empirical relation between max and n.

In a first step I measure different bcc (110) Bragg peak positions from a stepwise diluted

sample of PS100 and the sample conductivity . At each dilution I further measure the

position of the main peak of the shear molten fluid state, respectively. Care was taken to

ensure completely deionised conditions and constant temperature for all measurements.


66

I plot my data in Fig. 54 as versus n. For the crystalline phase n was calculated from Eq.

[4.03a]. The data observe a linear relation described well by μμeZ nσσ *

0 , where

in the case of PS100, I obtain Z* = 530 50.

0 5 10 15 20 25 300

2

4

6

8

10

(S

/cm

)

n ( m-3)

Fig. 54. Conductivity versus n for PS100: in crystalline state and fluid phase. n for crystalline phase are

calculated using Eq. [4.03a] signed () and n in fluid state are calculated from the dilution ratios signed (). The

solid line is a fit of Eq. [2.06] to the data using 0 = 0.06µS/cm, µ- 510-8

m2V

-1s

-1 and Z* 530.

Note the presence of a statistical scatter, and the absence of any systematic deviations for the

crystalline state data. This indicates that there are no systematic errors in our dilution

procedure. The dilution is continued into the equilibrium fluid phase. There the conductivity

is plotted versus n as calculated from the dilution ratios. All data are again well described by

the linear relationship of Eq. [2.06].

My data and further recent measurements have shown that Eq. [2.06] holds for both solid and

fluid state 97, 99. 142, 143

. In particular, the conductivity is independent on the suspension’s phase

state, but strongly depends on the material through Z*. I therefore, in a third step, use the fit

of Fig 54 as a calibration to obtain the particle number density of the shear molten and

equilibrium fluid states. Accordingly I plot sin3(max/2) versus the particle number density n in

Fig. 55. Data in Fig. 54 and Fig. 55 are for PS100. I repeated these measurements for the


67

other samples. Again the conductivity is well described by Eq. [2.06] using Z* = 51038 and

Z* = 68520 for PS90 and PS120, respectively. I note that the main contribution to the error

of Z* is given by the error of the particle mobility, which is smaller in the case of PS12099

.

0 4 8 12 16 20 24 28 32

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

bcc (110)

fluid qmax

sin

³(

max/2

)

n ( m-3)

Fig. 55. PS100: Position of the first maximum for crystalline () and fluid or shear molten state () as a function

of the particle number density n. The latter is inferred from the calibrated conductivity. Lines are linear fits to

the data.

Fig. 56 shows the results for PS120 taken in a much more diluted regime. While PS120 shows

a bcc-fcc phase transition at around 2.8µm-3

, I note that all my samples here are in a bcc

crystalline state as determined from combined measurements of static structure and static

shear modulus124

. Within each measurement the same linear sin3(θmax/2) - n relations are

obtained for both fluid and shear molten states. In other words, the position of the first peak is

inversely proportional to a typical length scale which scales as the cube root of the particle

density. The length scale, however is observed to be much shorter than d . Thus the original

estimate qmax = 2/ d has to be modified as qmax = 2/b d with b < 1 being an (empirical)

constant. In Tab. 4, I compile my findings.


68

0.00

0.01

0.02

0.03

0.04

0.05

0.0 0.5 1.0 1.5 2.0

bcc (110)

fluid qmax

n ( m-3)

sin

³(

max/2

)

Fig. 56. PS120: Position of the first maximum for crystalline () and shear molten state () as a function of the

particle number density. The latter is inferred from the calibrated conductivity. Lines are linear fits to the data.

PS90 PS100 PS120 Average Value Theory bcc Theory fcc

b 0.900.01 0.920.02 0.910.01 0.910.010.01 0.89 0.92

Tab. 4. Numerical constants b for the improved empirical relation for the position of the first maximum of the

static structure factor of fluid or shear molten charged and deionised colloidal suspensions as compared to the

crystalline cases of bcc and fcc, respectively. The errors of the average values denote the statistical and residual

systematic uncertainties, respectively.

Accordingly I modify Eq.[4.03c] for the evaluation of scattering angles for particle number

densities.

fluid:

2

θsin

λ

1.82γn max3

3

[4.03d]

The relations are compared again in Fig. 57.

I note again, that also in the improved empirical formula the original length scale dependence

is retained. The length scale, however has significantly changed. In fact, now it is much closer


69

to the values of the nearest neighbour spacing of the crystalline structures. This is in line with

an argument already given for atomic substances, stating that the first peak in S(q) appears at

the same position as the first diffraction peak in the corresponding solid phase (e.g. bcc[110]

phase)144

. Also Kesavamoorthy et al.145

have argued that way in their investigation of

crystallisation under the influence of µ-gravity. The authors further concluded that this

implies the local structure (at least the co-ordination in the first shell) in the liquid phase is not

drastically different from that for the crystalline phase. Cotter and Clark146

further reported

investigations of a sample prepared at equilibrium coexistence of bcc and fluid phase to show

the first scattering maximum at practically the same position and cross correlations appear in

the dynamic light scattering q = qmax and azimuthal positions corresponding to a bcc (110)

local environment.

0.0

0.1

0.2

0.3

0.4

0 5 10 15 20

FCC(111) peak

Fluid 1st peak

BCC(110) peak

n(m-3)

sin

³(

ma

x/2

)

Fig. 57. Comparison of the improved linear relation of n versus sin3(max/2) for fluid (Eq.4.03d) to those of the

fcc and bcc ordered suspensions(Eq.4.03a and b)

While in my experiment I cannot distinguish between the two crystal structures possibly

constituting the local structure of the shear melt and of the highly correlated fluid, this study


70

nevertheless provides additional evidence for the closeness of fluid and crystal short range

order. In particular, our systematic investigation is free of gravitational influences and

influences of the ion exchange resin. Further I use the well founded conductivity

determination for a calibration of the crystalline particle density and thus the density of the

homogeneous shear melt. It is the observed extension of the relation between scattering angles

and densities into the equilibrium fluid regime that provides a well founded support of the

arguments given earlier on a much more qualitative basis. What remains to be tested is the

position of peaks in the vicinity of the equilibrium bcc-fcc transition, and the influence of the

salt concentration.

The study again show a close structural analogy between atomic and colloidal melts. In

addition, I demonstrated an improved procedure for precise density determination. In contrast

to measurements of conductivity, it is applicable also for samples prepared at standing

preparation and to cases of phase coexistence. This will further be exploited now in careful

measurements of particle densities several interesting equilibrium and non-equilibrium

situations. Like in atomic systems and for hard sphere melts a significant density difference is

expected at equilibrium coexistence for the case that the particle’s ion clouds possess

translational degrees of freedom not coupled to the particle centres116

. Further during

solidification also transient compression of the crystal and dilution was observed for hard

spheres32

. This should also be the case for soft spheres, but hasn’t been shown explicitly.

Finally density determinations of systems with coexisting shear induced phases shall be

possible.

According to Chapter 3 and Chapter 4, my sample preparation with a careful control of c and

n can be realized under further developed experimental and theoretical methods. Therefore, an

extension to a careful control of interparticle interaction and colloidal phase behaviour based

on a pump tubing circuit can be available.

Chapter 5. Fluid-crystal phase transition, crystal morphology transition and phase diagram

71

Chapter 5

Fluid-crystal phase transition, crystal

morphology transition and phase diagram

5.1. Earlier experiment and theory for phase boundary

Due to the complexity of phase behaviour in soft sphere colloids, the whole systematic phase

behaviour can be described in a phase diagram. A complete phase diagram pioneer concerning

charged sulphate polystyrene as a model colloids was contributed by Sirota et al.28

in 1989

(shown in Fig. 2). It was derived from scattering experiment, describing a phase diagram

depending on concentration of HCl. A comparison to the theoretical phase boundary45, 147

was

also available in this phase diagram but with less discussion for the less good agreement

between each other at higher volume fraction for different cHCl.

With the help of molecular dynamics, Yukawa system provides a testing ground for general

ideas about phase transitions. As the shape of the potential varies continuously with the

screening length -1

, also the relative stability of bcc and fcc structures generally found

increase substantially as temperature T increases148

. Phase diagram for the dimensionless


72

inverse of energy kBT/U( d ) versus coupling parameter is found very valid to correlate the

pair-interaction energy with the phase behaviour. This is due to the coupling parameter =

d combines the most two important parameters: ion strength and particle concentration

which dominate the phase behaviour of suspension.

By computer simulation of Monte Carlo, Meijer and Frenkel46

have located the melting line

of Yukawa system by determining the free energy of both fluid and solid phases. Based on the

theory that if two phases in a pure system are in thermodynamic coexistence, they will have

equal pressure and Gibbs free energy per particle. So they calculated the fluid free energy by

thermodynamic integration with a polynomial fit to the density - pressure data, while the solid

free energy was calculated with the modification of the Frenkel - Ladd method149, 150

referring

the lattice-coupling expansion method. The free energy difference during expansion was

given by the integration of a modified pressure as a function of the density. They found that at

a high pair energy the fluid freezes into a bcc solid, and the Linderman ratio of the melting

line is equal to 0.19, whereas for a low pair energy it freezes into a fcc solid, and the

Linderman ratio of the melting line is smaller than 0.16. In addition, Young and Alder151

have

calculated their melting point with density-functional theory. A kBT/U( d ) - phase diagram

combining several theoretical methods45, 46, 151

above is shown in Fig. 58.


73

Fig. 58. Theoretical kBT/U(

d ) - Phase diagram: (○) indicate the calculated fluid-crystal coexistence with the

size indicating the statistical errors by Meijer and Frenkel46

. The solid line is the Meijer and Frenkel melting line.

() denotes the solid state with the Linderman ratio of 0.19, the dashed line indicate the Robbins, Kremer, Guest

melting line and the bcc-fcc phase boundary, where the melting line was based on the Linderman criterion with

ratio of 0.19. (∆) denotes melting results obtained with Young and Alder’s density-functional theory151

. Figure

courtesy to45

.

Within the kBT/U( d ) - phase diagram, Monovoukas and Gast42

have found that their

experimental phase boundary has large deviation with Robbins et al.’s theoretical results45

,

however, their phase boundary can be shifted to be consistent with theoretical data under the

method of theoretical charge renormalization by Alexsander et al. method101

. This results get

much progress in correlating experimental phase boundary with theory.

Up to now, experimental fluid/solid coexistence region both for single component and binary

mixture system are still rarely systematic. This leaves most phase diagram incompleteness.

Since some of the theoretical criteria, like the Lindemann’s melting criteria49

, Hansen-Verlet’s

freezing criteria152

and Löwen-Palberg-Simon-criteria’s kinetic freezing criteria153, 154

have

predicted the existence of freezing transition and melting transition. It’s pretty much

necessary to develop an systematical experimental method to find the fluid-crystal phase

transition and complete the phase diagram. It is then one of my main topic in the following.

The preliminary tests for fluid-crystal coexistence region have been done in static light

scattering. Samples (PS100) are all prepared in a developed way with detail see Chapter 3 and


74

Chapter 4. Here for testing crystal, I first pump the sample through IEX column for deionising

and then stop pump shearing, wait for some minutes for solidification and then test. For

testing melted fluid, I first pump shearing the sample through IEX column, stop shear,

immediately test, and repeat this procedure. In this way, I get the structure factor of melted

sample. Sample cell what I used is a kind of cylinder cell. The detail of experimental set-up

and the cell profile see Fig. 34 and Fig. 35.

First, I test the PS100 sample at particle number density n 4.5m-3

by static light scattering

shown in Fig.59. It shows that some scattered peak are not pronounced where their

background is consistent with the profile of fluid structure factor. This behaviour possibly

hints the coexistence of crystal/fluid two phase.

40 60 80 100 120 140

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

Crystalling test (twait

~30 min)

Shear Melted test

I(a

.u.)

(°)

Fig. 59. Static light scattering measurement for PS100 under n 4.5m-3

data are taken after 30min of stop (ٱ) .

shear; () got from the procedure: shear stop shear measure, it results the structure factor of shear-melted

sample. Both are the same sample but under different way of testing. (ٱ) data show that except for the first peak

at (110) plane, other peaks have very low intensity and base on the background of fluid structure factor (),

which hints that a possible coexisting of crystal /fluid phase.

Second, I test several deionised samples of PS100 under step-wise dilution by static light

scattering in several measurements. They are combined in Fig. 60, where a comparable large

difference of the scattering intensity between fluid and solid phase are seen, and the scattered

intensity of the prospected fluid-crystal coexistence is in between. Therefore, the fluid-crystal

coexistence can be roughly estimated between 3.8 and 4.5 m-3

. However, both fluid phase


75

and freezing point show fluid-like structure factor, which is difficult to distinguish between

each other. Thus, other experimental techniques are required for determining the fluid-crystal

phase transition with more exactness.

40 50 60 70 80 90 100 110 120 130 140

5

10

15

20

25

30

35

40

Crystal

fluid

fluid-crystal coexistence

I(a

.u.)

Fig. 60. A comparison of scattering intensities for the deionised sample PS100 under different n by static light

scattering. Fluid-like states only show the first peak of their static structure factors with a broad but lower

intensities,. The fully crystal phase shows a sharply higher intensity at the first maximum. The first peak

intensity of the prospected fluid-crystal coexistence is in between. The first peak positions move a larger value of

with n increasing. For consistency, all samples are tested after 30 minutes of stop shear, which all samples are

assured to be in equilibrium phases. Two black lines guide the different phases according to their scattering

intensities.

5.2. Fluid-crystal phase transition and crystal morphology transition in

single component system

Except for some theoretical45,46,147,149,153

and experimental approach32,43

, however, the

morphologies of soft sphere fluid-crystal phase transition and some crystal morphologies

transitions has not been clearly reported before this thesis. The difficulties can be concluded

as following: (1) The soft sphere fluid-crystal phase transitions happen at very low volume

fractions ( ~ 0.001 order) under deionised condition, one faces the difficulty to exactly

adjust n; (2) Crystallites in fluid-crystal phase transition region is easily melt under small


76

amount of salt, which one finds it may come from the small amount of CO2 leakage to the

suspension in the tubing system; (3) The variation of temperature can be another difficulty to

distinguish the fluid-crystal phase transition in a stable range of n; (4) The common top-view

by optical microscopy is found not suitable to observing the fluid-crystal phase transition.

As a solution, here the samples are prepared in developed techniques of sample preparation,

the fluid-crystal phase transitions are observed by side-view (i.e. illuminating in

10mm*100mm plane, and observing in 2mm*100mm plane). The top-view (illuminating in

2mm*100mm plane, and observing in 10mm*100mm plane) is applied for observing the twin

domain morphologies. Images are recorded by a video CCD camera connected with Bragg

microscopy. Therefore, I observed the homogeneous to heterogeneous crystal morphology

transition and fluid-crystal phase transition of PS120 illuminated with cold white light source

shown in Fig. 61. Fig. 61 (a) to (h) guide the process of salt c increasing by adding additional

CO2 under a certain n (n = 3.95m-3

for PS120).

(a) (b) (c) (d) (e) (f) (g) (h)

Fig. 61. A series of Bragg microscopy images (a) - (h) show the crystal morphology transition and the fluid-

crystal phase transition with increasing salt concentration c at fixed particle number density n = 3.95 m-3

for

PS120. In image (a) under c < 0.2 M, the small-size homogeneously nucleated crystals dominate. The thickness

of the heterogeneously nucleated wall crystals at each side of the cell (white parts) are observed very thin; (b)

under c = 1.8 M, the heterogeneous nucleation and growth dominates nucleation (dark part in b) and

occasionally a large-size lens-like homogeneously nucleated crystal appears (blue part); (c) at c = 2.2 M, only

the sheet-like wall crystals exist; (d) at c = 2.4 M, the tooth-like wall crystals appear; ( e) at c = 2.5 M, the

tooth-like wall crystal with bulk fluid (dark part) both exist; (f) at c = 2.6 M, tooth-like wall crystal with more

fluid left in the bulk (blue part); (g) at c = 2.7 M, less cap-like small wall crystals and more bulk fluid leave;

(h) at c = 2.9 M, only fluid state exhibit (grey colour). The different colours from fluid and crystal phase come

from the light Bragg scattering, they are obtained by varying the illuminated direction for clarity. The black lines

at top and bottom of each image show the boundary of cell with height of 2mm in between.

Fig. 61 from (a) to (h) compares the resulting morphologies for the samples of PS120 at

constant n and increased salt concentration c. Thus one observes two wall nucleated crystals

growing from top and bottom. Homogeneously nucleated crystals are found in the cell

interior. At thoroughly deionised conditions, the latter channel dominates the solidification

process. Accordingly, the growth of wall crystals is stopped after short times and the thin wall

crystals result shown in (a). In (b), under increased c, the rates of homogeneous nucleation


77

decrease and the thicker wall crystals result. If the rates of homogeneous nucleation are low

enough, the wall crystal growth is terminated by intersection of two wall crystals, and maybe

also some very few amount of bulk homogeneous crystals. Then the typical shape of bulk

homogeneous crystal is lenticular due to termination of growth by intersection with the wall

crystals. Here the lenticular homogeneous crystal shows bright blue, while wall crystals show

black, which I illuminate the sample for a better contrast to observe their competing growth.

In (c), the monolithic sheet-like appearance of the wall crystals is obvious. Further, my

investigations have shown that during growth a planar compact interface advances into the

melt43

. I therefore term this morphology sheet-like. In (d), exceeding a certain critical salt

concentration, the morphology changes rather abruptly from a compact sheet-like appearance

to a tooth-like appearance of individually growing columns. Across coexistence, individual

columns are observed to separate as (e). In (f), close to freezing, the cap-like appearance is

shown. The latter transition in (g), however, is gradual. In (h), for salt concentrations larger

than the freezing concentration cf, the system stays fluid. I note that the same sequence of

phase transitions and morphological transitions is also observed for other particles at fixed n

with increasing c.

For a constant c, diluting process (n decrease) exhibits the same morphology trends (see Fig.

62).

(a) (b) (c) (d) (e) (f) (g)

Fig. 62. Morphology transition and fluid-crystal phase transition under step diluted particle number density n.

Images from (a) to (g) show the process of n decreasing (here I take deionised sample PS100 as an example).

Details for the description of each image see the figure capture and the text of Fig. 61.

The freezing and melting transition under deionised monodispersed samples are then detected

by Bragg microscopy and compiled in Tab. 5.


78

Tab. 5. Data for monodispersed single component polystyrene sample. Data include particle number density at

freezing point (nf), particle number density at melting point (nm).

Since I get FC phase transition by altering c for several n (or inverse, by altering n for a

certain c) applied with the technique of Bragg microscopy, the n - c phase diagram then can

be used to conclude experimental crystal morphology transition and fluid-crystal phase

transition shown in Fig. 63 (a) and (b).

0 1 2 3 4 5 6 7

0

2

4

6

8

10

n (

m-3)

c (M)

(a)

Sample Source

Batch No

nf

(m-3

)

nm

(m-3

)

PnBAPS68 BASF ZK2168/7387 6.10.4 6.20.4

PS100 Bangs Lab 3067 3.80.3 4.40.2

PS120

IDC

10-202-66

0.340.05

0.440.05

PS156

IDC

2-179-4

0.280.05

0.400.05


79

0 1 2 3 4 5 6 7 8

0

2

4

6

8

10

n (

m-3)

c(M)

(b)

Fig. 63. The n-c phase diagram of sample PS120, including (a) (●)denote the homogeneous crystal to sheet-like

wall crystal morphology transition; (■) denote the sheet-like wall crystal to tooth-like wall crystal morphology

transition. (b) (△) denote the melting transition; and ( ) denote the freezing transition; (○) shown as the fluid-

crystal coexistence region within additional NaCl as electrolyte (data (○) courtesy of D. Hessinger).

Starting from a concentrated deionised suspension of PS120, CO2 was successively added by

syringe into the reservoir and samples are first pumped through by-pass until reaching a stable

conductivity number which records the salt concentration c in suspension, I stop pump

shearing, and take the image of sample morphology by video CCD camera. Experiments are

repeated in the same way with sample diluting step by step. All the measured fluid-crystal

phase transition and crystal morphology transition then can be concluded in the n - c diagram.

In n - c phase diagram of Fig. 63, I show the results of such systematic measurements to the

positions of the freezing transition (), melting transition (△), the sheet/tooth morphological

transition (■) and homogeneously crystals morphology to sheet-like wall crystal morphology

transition (●). I also include the FC coexistence region under additional NaCl as electrolyte

(○) courtesy of Hessinger32

.

Freezing lines measured with different electrolytes agree reasonably well. The sheet/tooth

transition appears at slightly larger n or lower c. At even larger n or smaller c, homogeneously


80

nucleated crystals appear. At these points the induction time of homogeneous nucleation has

become shorter than the time needed for wall crystals growing at velocities of 5µm/s to reach

a thickness of half the cell depth of 2mm: I < 200s. This effect thus strongly depends on the

geometry used. Note that the morphological transitions are approximately straight lines in this

diagram which slightly diverge for larger n and c.

5.3. A comparison to theoretical diagram

I then plot the experimental fluid-crystal phase boundary for several monodispersed samples

into a λdT/UkB diagram including state lines of each sample. State lines are calculated

for a residual salt concentration of c = 0.2 M and using conductivity measured effective

charge Z* . Note that the typical form as introduced in Fig. 64 is observed for all samples. The

interaction energy first increases, then decreases and finally increases again. At the same time

the coupling parameter first decreases then increases.

In Fig. 64, all coexistence regions are observed to be near the turning point of the state lines,

i.e. close to the switch point of small ions dominating and counter ions dominating case. This

somehow expects that the descending part of the state line represents the non-interacting case

and the region around the maximum represents the effects of self-screening at large overlap of

double layers. Further, if the predictions of Robbins, Kremer, Grest45

and Meijer, Frenkel46

apply universally, all coexistence regions for samples of similar are expected to be confined

to a narrow region in dT/Uk B . With the exception of PS156, this is in fact not the case. I

suspect that the deviation of PS156 is due to a larger polydispersity of that particular sample.

From an intuitive argument one would expect the free energy of a slightly disordered solid to

be larger than that of a perfect crystal. Such disorder is naturally introduced, when the system

is somewhat polydisperse. Therefore one would expect a larger concentration or compression

to be necessary to introduce crystallisation. In fact, for hard spheres quantitative studies have

shown that no crystallisation is possible for polydispersity larger than 12 %136,155

. In addition

one also has to discuss the deviation of all samples from the theoretical prediction. Two

possible explanations are at hand. First, all phase transition portions may appear shifted to

larger n as expected for monodisperse samples due to their polydispersity. With PS156, it

may be shifted further than the others. Second, independent of this, the charge number used as

an input to calculate the positions of phase transition may be wrong. e.g. for PS100 the


81

effective charge needed to make the predictions and the experimental data coincide is Z*

330. This is much lower than Z* 530 which already falls below Alexander´s estimate of

700. While there is convincing evidence from various experiments that the charge to be used

here should be much smaller than the titrated charge. At present, there is no unequivocal way

to discriminate between alternative experimental and theoretical methods of its determination.

Like other authors before44

, here I may state that the electrokinetic charge from conductivity

meets the right order of magnitude but does not give quantitative agreement.

2.0 2.5 3.0 3.5 4.0 4.50.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.10

0.11

0.12

kBT

/U(d

)

Fig. 64. State lines and position of fluid-crystal phase transition from the parameters of different experimental

samples: (— ○ —) PnBAPS68 (2a = 68nm, Z* 450); (—٭—) PS90 (2a= 90nm, Z* 510); (— —) PS100

(2a = 100nm, Z* 530); (— ◊ —) PS120 (2a = 120nm, Z* 685); ( — ٱ —) PS156 (2a = 156nm, Z* 945).

The large-marks are experimental freezing points and melting points, (- - -)lines is the Robbins, Kremer, Guest

melting line, (-.-.-.) line is the Meijer-Frenkel melting line. Samples are all under deionised condition, i.e. c =

0.2 M.

As a complement, I would like to introduce an alteration of phase diagram by charge

renormalization developed by Wette, Schöpe and Liu156, 157

(see Fig. 65).


82

1.5 2.0 2.5 3.0 3.5 4.0 4.50.04

0.06

0.08

0.10

0.12

0.14

kBT

/ U

(d)

(a)

1.5 2.0 2.5 3.0 3.5 4.0 4.5

0.06

0.08

0.10

0.12

0.14

kBT

/ U

(d)

(b)

Fig. 65. Shifting experimental fluid-crystal phase transition by charge renormalization (taken sample PnBAPS68

as a represent, it’s conductivity measured effective charge Z* = 450 16, shear modulus measured effective

charge Z*G = 326 7).(- - -) shows the Robbins, Kremer, Guest’s melting line; (-.-.-) shows the Meijer, Frenkel’s

melting line; (a) state line and fluid-crystal phase transition calculated with Z*; (b) state line and fluid-crystal

phase transition calculated with Z*G; Position of fluid-crystal phase transition is found shifted to be close the

theoretical melting lines. Sample is under complete deionised condition, i.e. c = 0.2 M. Figure courtesy to157

.

In Fig. 65, the state line and fluid-crystal phase transition of PnBAPS68 are respectively

calculated with effective charge Z* by conductivity measurement(shown in (a)) and Z*G by

shear modulus measurement (shown in (b)). Meanwhile, theoretical melting lines45,46

are

shown in the same phase diagram. Experimental melting transition has been found shifted to


83

be close the theoretical melting lines by applying Z*G shown as (b), which hints that charge

renormalization may be available by experimental determined shear modulus. However, this

conclusion has only been proved to be applicable in single components under deionised

condition, other conditions, like nondeionised case and polydispersed case have not been

proved yet. The detail concerning charge renormalization is still developing which is out of

the topic of this thesis. For consistency later on, all my calculation still only use effective

charge Z* from conductivity measurements.

A comment concerns the absolute position of the observed phase boundaries. The freezing

line from the simulations is found at considerably larger values of dT/UkB, i.e. at lower

pair energies (see also Fig. 66). As compared to the experiments, the simulation thus predicts

crystals to be stable even at much lower particle concentrations. It may be argued that close to

the phase boundary, the kinetics of the solidification processes are too slow to reach the

equilibrium phase. This is true, if homogeneous nucleation is the only possible solidification

scenario. Here, however crystal nuclei are instantaneously provided by the presence of the

quartz container wall54

. Thus I may safely rule out non-equilibrium conditions.

Further I translate the n - c phase diagram in Fig. 63 into kBT/U( d ) - phase diagram as Fig.

66.

Based on PS120, Fig. 66 compares experimental data to the Robbins Kremer and Grest45 ‘

s

melting line by computer simulation and Meijer, Frenkel46

’s melting line by thermodynamic

integration. Data are taken in successive runs at fixed n increasing the salt concentration c.

State lines calculated from the parameters of all samples cross the predicted phase boundary

and all samples crystallize somewhat below the predicted melting line (see Fig. 64). As

discussed above this implies that the experimental solids are less stable than those theoretical

conclusion. Note that the experimental transition points run approximately parallel to these

lines irrespective of the nature of the transition. This indicates a correlation of the phase

boundary with the morphological transitions via the pair interaction and the coupling

parameter.


84

2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.20

k

BT

/U(d

)

Fig. 66. Comparison of experimental data PS120 (2a =120, Z* 685) to the fluid-bcc phase boundary predicted

by Robbins, Kremer, Guest from computer simulation (……) and Meijer, Frenkel’s melting line by

thermodynamic integration (·-·-·). Above shown are the positions of ( ) freezing transition,; ( )melting

transition; (●) as appearance of homogeneous nucleation. I also include the state lines ( - - - ) follow the

course of the experiments, where for fixed n the salt concentration c is increased and experiments are repeated

after dilution.

As a short conclusion above, I have determined the positions of the fluid-crystal phase

transition and crystal morphological transition for monodisperse charge stabilised colloidal

spheres in aqueous suspension. Systematic measurements in dependence of electrolyte

concentration c and particle number density n are performed for a sample of fixed effective

charge Z* 685 and diameter 2a = 120nm. I further present the calculations of pair

interaction energies and the coupling parameters over a wide range of experimental

parameters bracketing the conditions accessed experimentally. This enables me to visualise

the complex pathway of our experimental samples under changed n and c in the

corresponding λdT/UkB phase diagram. Both observed transitions (phase transition and

morphology transition) run approximately parallel to the melting transition predicted from

computer simulation and thermodynamic integration. This last point deserves some further

discussion.


85

In addition, I would like to comment on an important difference between the two observed

morphological transitions. The appearance of homogeneous nucleation is connected to an

induction or waiting time. This time involves the evolution of the stationary distribution of

fluctuations in the metastable melt starting from the sheared state, where fluctuations are

suppressed. This mechanical equilibration is similar to the thermal equilibration in computer

simulations starting from a high temperature fluid configuration. On the other hand the

sheet/tooth transition is connected to the wetting properties and thus surface tension of the

wall crystal phase. In this case the observed correlation might also be explained in terms of a

correlation between a bulk freezing transition and a wetting transition. However, I expect

kinetic aspects to be involved as well, like the relaxation of shear induced structures close to

the wall and the competition between reorientation of the wall based nucleus and growth

kinetics. In this sense, I hope to have added a fascinating aspect also to the discussion of

colloidal systems under the influence of external fields.

5.4. Fluid - crystal phase transition and twin domain morphology transition

in binary mixture

Since Bragg microscopy has been successfully applied in single component system for

detecting fluid - crystal phase transition and crystal morphology transition. With the aim of

characterising phase behaviour and phase morphology in binary mixture system, this

technique may be also applicable. Here two commercially available species of charged

Polystyrene spheres with size ratio of 1:1.47 with controlled total particle number density n

and the mixing number ratio p68 (p68=n68/(n68+n100)) of the small amount (PnBAPS68) in

aqueous suspension (PnBAPS68 and PS100) are investigated. Within the same way of sample

preparation as single components, I also observe the fluid-crystal phase transition and the

crystal morphology transition by side view. I find that these binary mixture under different

mixing number ratio p68 shows the same phase behaviour and morphology behaviour

influenced by n and c as monodispersed sample. I turn the way inverse, i.e. increasing n under

fixed c, or decreasing c under fixed n, the morphology trends is just the inverse of Fig. 61 and

Fig. 62. Taken a binary mixture of PnBAPS68/PS100 under mixing number ratio p68 = 0.50, c

= 0.2 M as an example, with the procedure of increasing n, the cap-like, the tooth-like, then


86

the sheet-like wall crystals also appear as the morphologies of the fluid-crystal phase

transition (see Fig. 67).

(a) (b) (c) (d) (e) (f)

Fig. 67. Fluid-crystal phase transition morphology trends in binary mixture (PnBAPS68/PS100). Fig.(a) - (f)

show the morphology trends with n increasing under p68 = 0.50 in PnBAPS68/PS100 binary mixture under

deionised condition, i.e. c = 0.2M. The fluid-crystal phase transition is thus concluded in between nf 4.4m-3

,

nm 4.5m-3

.

Tab. 6. The freezing and melting points of some monodispersed samples and the binary mixtures of

PnBAPS68/100 at several mixing number ratios of PnBAPS68. i.e. p68 , respectively, at 0, 0.15, 0.20, 0.25, 0.50,

0.75, 1. The effective charge Z* for monodispersed samples are measured from conductivity measurements, Z*

for binary mixture are the averaged numbers of effective charges calculated by Z* = p68Z*68 + p100Z*100. nf, nm

are the particle number densities of freezing transition and melting transition observed from Bragg microscopy,

respectively.

So the developing trends of fluid-crystal phase transition morphologies are consistent between

single components and binary mixtures. It can be reproduced by altering n (either diluting or

sample Z* nf nm

P68 = 0

(PS100)

53050

3.80.3

4.40.2

p68 = 0.15

(PnBAPS68/PS100)

51835

3.20.5

3.40.5

p68 = 0.20

(PnBAPS68/PS100)

51434

2.70.5

3.10.5

p68 = 0.25

(PnBAPS68/PS100)

51033

3.80.5

4.00.5

p68 = 0.50

(PnBAPS68/PS100)

49027

4.40.5

4.50.5

p68 = 0.75

(PnBAPS68/PS100)

47022

5.30.5

5.50.5

P68=1

(PnBAPS68)

45016

6.10.4

6.20.4


87

concentrating) at fixed c, or by altering c (increasing c or decreasing c) at fixed n. I then get

the data of nf , nm for binary mixture and compiled them in Tab. 6. Data of conductivity

measured effective charge Z* for different p68 are also introduced. The average effective

charge Z* for PnBAPS68/100 binary mixture are got from the calculation with Eq. [2.11], i.e.

Z* = pAZA* + pBZB*. So the averaged particle number density n of binary mixture can be

calculated with Eq.[2.06], i.e. μμeZ nσσ *

0 by using the calculated averaged Z*

and measured for binary mixture system.

In Tab. 6, I find that nf, nm at some mixing number ratio of p68, e.g. p68 0.15, 0.20, are lower

than their single components (PnBAPS68 and PS100). This effects is quite different with hard

sphere case. Thus I further test crystal morphology of PnBAPS68/PS100 binary mixture.

By top-view, I observe the crystal twin domain morphology. I change the mixing number

ratio p68 at constant n (n 8.90.7m-3

), the morphology transitions of twin domain then are

shown in Fig. 68.

(a) (b) (c) (d) (e)

Fig. 68. Twin domain variation under different mixing number ratio p68 of binary mixture PnBAPS68/100 but

constant particle number density n 8.9 m-3

. Images from (a) to (e) are at p68 = 0, 0.20, 0.50, 0.75, and 1,

respectively. In the binary mixture of PnBAPS68/100, zig-zag domain pattern is observed in (b), (c) and (d),

which quite different with monodispersed samples of (a) PS100 and (e) PnBAPS68.

Still on top-view, I change n but fix p68, at p68 = 0.20, their twin domain morphologies are

shown in Fig. 69 from (a) to (e), respectively.


88

(a) (b) (c) (d) (e)

Fig. 69. Twin domain variations under the mixing number ratio p68 0.20 in binary mixture of PnBAPS68/100

at different n. From left to right, they are n = 16.6 2.14 m-3

; 7.56 0.97 m-3

; 6.74 0.87 m-3

; 4.82 0.62

m-3

; 4.02 0.53 m-3

, respectively. In (a), the twin domain morphology shows net-like or cloudy-like; (b) at n

= 7.56 0.97 m-3

, a zig-zag domain can be observed, but occasionally, a large homogeneous crystal(lens-like)

can also be observed. Further diluting, only the zig-zag domain can be observed (green colour), however, this

zig-zag domain gradually becomes obscure with further diluting (like (d)), and finally only left cloudy-like

morphology (like (e)).

As a description to above, the twin domain morphology transition of PnBAPS68/100 binary

mixture under different mixing number ratio p68 and different particle number density n are

shown in Fig. 68, Fig. 69, respectively. Twin domain morphologies show a cloudy-likezig-

zag patterncloudy-like transition either by altering p68 (but fixing n) or by altering n (but

fixing p68). The zig-zag pattern is observed in different mixing number ratio p68 and in some

range of n. For instance at the mixing number ratio p68 = 0.20 (see Fig. 69), (a) at higher n (n

= 16.62.14 m-3

), a cloudy-like twin domain is observed; (b) at n = 7.56 0.97 m-3

, a large

sized homogeneous crystal (shown red ellipse) is observed coexisting with the zig-zag

domain; (c) further diluting, at n = 6.74 0.87 m-3

, no homogeneous crystal appears, and the

zig-zag domain pattern shows more clearly. However in (d), this zig-zag pattern shows

blurred with further diluting. Finally in (e) at n = 4.02 0.53 m-3

, no zig-zag pattern can be

observed, instead, again only the cloudy-like twin domain appears. This domain morphology

transition from cloudy-like twin domain, zig-zag pattern, then again cloudy-like twin domain

under step diluting are found in other mixing number ratio of p68 in PnBAPS68/PS100 binary

mixture, for instance, p68 = 0.50 (see Fig. 70).


89

(a) (b) (c) (d)

Fig. 70. Twin - domain morphology transition under step diluting in p68 = 0.50 of PnBAPS68/PS100 binary

mixture. The images from (a) to (d), are at n: 8.83 1.19 m-3

; 6.29 0.85m

-3; 5.35 0.72 m

-3 ; 5.16 0.69

m-3

, respectively.

With much interest to the special zig-zag domain pattern in binary mixture of PnBAPS68/100,

I detect its kinetic growth process. Fig. 71 shows the zig-zag pattern growth with time

(sample is taken at p68 = 0.20 , n = 6.74 0.87m-3

)

At small t, little dark and bright areas are arranged under a mutual angle . The pattern

coarsens in terms of area extension and contrast. From Fig. 71 (a) to Fig. 71 (d), zig-zag

pattern shows a coarsening process with increasing time. The same Bragg colour for all

domains indicates that the crystals are orientated in the same direction, which the pattern’s

length scale (observed by top-view) is found extending with a speed of about 4 m/s, while in

<110> direction is about 8 m/s observed by sheet-like wall growth by side-view. Detail

analysis will be shown in Chapter. 6.


90

(a) (b)

(c) (d)

Fig. 71. Top view of the nucleation time-dependent twin domain patterns for the sample of the

PnBAPS68/PS100 binary mixture at p68 = 0.20 and n = 6.74 0.87m-3

. Images are taken at nucleation time t:

(a) 9s; (b) 20s; (c) 30s; (d) 69s. The black bar represents 100m. A regularly distributed zig-zag pattern is

observed to coarsen with time.

I have preformed several samples at mixing number fraction p68 = 0, 0.15, 0.2, 0.25, 0.5, 0.75,

and 1, and widely observed the cloudy-like twin domainzag-zag domain patterncloudy-

like twin domain morphology transitions. Experiments are all conducted under the deionised

conditions and at a particle concentration where samples are completely or partly (fluid-

crystal coexistence) solidified at equilibrium including a few experiment under nondeionized

condition. As a conclusion, a n - p68 phase diagram concerning twin domain morphology

transition and fluid-crystal phase transition then can be depicted in Fig. 72, Fig. 73. Notice

that, due to my sample preparation for different mixing number ratio p68 is taken from the

calculated mixing number ratio p68 = n68/(n68 + n100) according to the original n number in

single component mother suspension, the systematic errors for the sample preparation at

different p68 should be consistent. For clarity, the data of Fig. 73 are shown without error bar,

however we should notice that errors exist both for p68 and n.

From Fig. 73, again I show the concave loop of nf and nm in the range of p68 = 0.15 , 0.20

and 0.25. In addition, I show the morphology transition of zig-zag pattern to cloudy-like twin

domain at lower n. With very liming data, I see a roughly parallel trend between the twin

domain morphology transition and fluid-crystal phase transition. And in Fig. 72, zig-zag


91

domain pattern at p68 = 0.20 dominates the most range of n, which I also find a minimum of nf

and nm at this mixing number ratio. This may hint a special crystal structure with this domain

morphology. Based on this thinking, I would like to give my analysis concerning the particle

components in the crystal with zig-zag domain morphology.

0.2 0.3 0.4 0.5 0.6 0.7 0.8

4

5

6

7

8

9

10

11

12

13

zig-zag domain pattern

cloudy-like twin domain

n (

m-3)

p68

Fig. 72. n - p68 domain morphology diagram. (▲) guides the zig-zag domain morphology; () guides the

cloudy-like twin domain morphology; (- - - -) are the connecting lines of twin domain morphology transitions

guided by eye.


92

-0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

2.5

3.0

3.5

4.0

4.5

5.0

5.5

6.0

6.5

7.0

the last appeared

zig-zag domain pattern

Freezing transition

Melting transition

n (

m-3)

p68

Fig. 73. Fluid-crystal phase transition and zig-zag domain to cloudy - like twin domain morphology transition in

n – p68 phase diagram. (--▲--) show the zig-zag domain to cloudy-like morphology transition, and ()show the

freezing transition.(△) show the melting transition.

In Fig. 71, I have shown the kinetic growth of zig-zag domain morphology at p68 = 0.20, n =

6.74 0.87 m-3

. By side view, I have got the melting transition of one components

(PnBAPS68 and PS100), respectively as nm68 = 6.2 0.4 m-3

and nm100 = 4.4 0.2 m-3

(c.f.

Tab. 5, Fig. 73). At the binary mixture of p68 = 0.20, n = 6.74 0.87 m-3

, the one component

particle number density should be n68 = 1.35 m-3

and n100 = 5.39 m-3

, respectively. So for

such a low particle number density n68 (< < nm68), PnBAPS68 can not form a crystal lattice by

its own. But I do not observe fluid voids in these twin domains. In addition, an even lower

melting point nmp0.2 ( 3.1 0.5 m-3

) (see Fig. 73) is found for p68 = 0.20. A possible

explanation for these behaviours could be that compound crystals are formed. Furthermore,

the zig-zag pattern, which was only found in the mixture may correspond to a microscopic

morphology of this compound crystal.

For this zig-zag pattern, I would also like to give an analogy to metal nanocrystaline (nc). In

general, the most important crystal defect determining the mechanical properties of materials

is dislocation, which is explained by the traditional Hall–Petch relationship158, 159, 160

using the

concept of dislocation pileups in crystallites. However, in the nc materials, dislocations are


93

seldom seen in individual crystallites and even those that are seen have some special

immovable configurations. Therefore, the contribution to plasticity in nc materials from

mobile dislocations is not possible. Some authors once pointed out that the stress field from

grain boundaries could be the main source of drastic changes in dislocation density and the

appearance of the special dislocation configurations in crystallites161

. In most cases, the lattice

planes in the crystallites of nc materials near the grain boundaries are slightly distorted, which

indicates the existence of a strong stress field in the grain boundaries. Qin et al 162, 163

once

proposed that the stress field induced by the vacancies and the vacancy clusters in the grain

boundaries of nc materials can affect the lattice structure of crystallites and quantitatively

grain mated the deviation of the lattice parameters of the crystallites of nc materials from the

standard values, also indicates that the vacancies and the vacancy clusters in the grain

boundaries of nc materials are the dominant sources of stress field existing in the crystallites

of nc materials. Again they concluded that these kinds of stress fields also affect dislocation

stability and configuration in dislocations. I find the twin domain structure of my samples

altered with particle number density n and mixing number ratio p68, which may be correlated

with ‘stress field’ from grain boundary. Later on in Chapter 6, the described initial growth d0

influenced by n and p, also shearing speed may be another prove for the existence of stress

field near the grain boundary. Ordered particle arrangement induced by AC electric field and

magnetic field is recently found164, 165

, which in another way prove the existing of stress field

induced ordering. Further exploration can be expected to be done by some other experiments,

like ordering morphology observed by supermicroscopy experiment166,167

in nanoscale, lattice

structure detected by static or neutron light scattering and x-ray diffraction168

, and their

correlated relation in structure and morphology. And I may expect a very promising

application in industry for a controlled crystal plane - plane growth if it can be realized by

carefully controlled parameter of interaction.

Up to now, it is still a difficult task to correlate FC phase transition with theory in binary

mixture. One of the difficulty may come from the complexity to discriminate the pair

interaction from A-A, B-B, or A-B particle in the binary mixture within A and B components,

and mixing at different mixing number ratio provides more complexity. However, if one

assumes a random composition bcc lattice formed in the binary mixture, then Eq.[2.13], i.e.

2

B

B2*

B

2

B

BA

BA*

B

*

ABA

2

A

A2*

A

2

A

r0

2

BA,aκ1

aκ expZp

aκ1aκ1

aκaκ expZZp2p

aκ1

)a(κ expZp

r

rκ exp

ε ε 4π

eU(r)


94

and Debye parameter in Eq.[2.12], i.e.

s

*

BB

*

AA

Br0

22 nnZpZp

Tkεε

eκ

can be used to calculate the average value of pair-interaction dU and coupling parameter

( = dκ ) under different mixing number ratio of binary mixture. Therefore, I get the state

lines at different mixing number ratio p and show the positions of fluid-crystal phase

boundary of binary mixture in dUTk B --- diagram (shown in Fig. 74).

2.2 2.4 2.6 2.8 3.0 3.2 3.4

0.05

0.06

0.07

0.08

kBT

/U(d

)

Fig. 74. State lines and fluid-crystal phase boundaries under several mixing number ratio p68 of binary mixture

PnBAPS68/PS100. State lines for different mixing number ratios are signed as: (ٱ) for p68 = 0 (i.e. PS100); ()

for p68 = 0.15; () for p68 = 0. 20; () for p68 = 0.25; (⋄)for p68 = 0.50 () for p68 0.75; (▽) for p68 = 1 (i.e.

PnBAPS68), respectively. The large symbols on each state lines are their fluid-crystal phase transitions

respectively. (- - -) is Robbins, Kremer, Guest’s melting line; (-.-.-) is Meijer, Frenkel’s melting line. (─) line

shows the possible trends for a random bcc packing.

In Fig. 74, I take the samples of PnBAPS68/PS100 binary mixture at the mixing number

ratios of p68 = 0, 0.15, 0.20, 0.25, 0.50, 0.75, and 1. I draw their state lines also their fluid-


95

crystal phase transitions (larger marks on each state lines). Robbins, Kremer, Guest’s45

melting line and Meijer, Frenkel’s46

melting line are also shown as a reference. The same as

finding from single component system, the fluid-crystal phase transitions of binary mixture

are less stable than theoretical data45, 46

. Fluid-crystal phase transition can be prospected as a

linear tendency from p68 = 0 to p68 = 1 if assuming a random bcc structure formed at each

mixing number ratio. The random bcc structure is shown as Fig. 75. For clarity, I draw the

dimensionless inverse of pair interaction kBT/U( d ) at fluid-crystal phase boundaries versus

the mixing number ratio p68 in Fig. 76, and the coupling parameter versus p68 in Fig. 77.

Fig. 75. Assumed random bcc packing for PnBAPS68/PS100 binary mixture. (●) show the larger particle, while

(●) shows the small particle. The dashed square shows a bcc lattice constituted by different particles.


96

0.0 0.2 0.4 0.6 0.8 1.0

0.050

0.052

0.054

0.056

0.058

0.060

kBT

/U(d

) FC

ph

ase b

oun

da

ry

p68

Fig. 76. Inverse of pair-interaction energy at fluid-crystal phase boundary versus mixing number ratio p68 with

(▼) shown as the melting transition and (△) shown as the freezing transition.

0.0 0.2 0.4 0.6 0.8 1.0

2.65

2.70

2.75

2.80

2.85

p68

Fig. 77. Coupling parameter( ) versus mixing number ratio p68 with (▼) shown as the melting point, (∆) shown

as the freezing point at different mixing number ratio, respectively.


97

It can be found that the fluid-crystal phase boundary under averaged DLVO pair-interaction

by assuming random bcc packing tends almost linearly as a function of the mixing number

ratio p68. However, the coupling parameter shows a deep minimum at p68 0.20, which

means a non-random binary compound possibly formed around this region. A pronounced

zig-zag domain pattern (c.f. Fig. 71) gives the crystal morphology of non-random binary

compound at this mixing region. Further structure discussion can be found in Chapter 6,

which crystal growth kinetics is included. The quantitative determination can be prospected

done under a new designed set-up described in the part of “Conclusion and outlook”.

In this Chapter, I get fluid-crystal phase transition and crystal morphology transition by Bragg

microscopy. I have combined my experimental data into several phase diagram. A correlation

between pair-interaction, fluid-crystal phase transition and crystal morphology transition is

provided for single component system in a kBT/U( d ) - phase diagram. However, This

correlation is still lack for binary mixture system. By applying with the average pair

interaction assumption, the further correlation could be possible as it neglects the complexity

from A-A, B-B or A-B interaction. The further correlation between experiment and theory can

be expected in the same way as single component case. i.e. by altering c under a fixed p and n,

detecting the fluid crystal phase transition and phase behaviour, translating a n - c phase

diagram at certain p into kBT/U( d ) - phase diagram etc., but the experimental work is out

of my thesis here.


98

Chapter 6

Crystal growth kinetics

6.1. Experimental preview of crystal growth detected by Bragg microscopy

In Chapter 5, I have shown the images of fluid-crystal phase transitions and crystal

morphology transitions both in single components and binary mixtures. A suppressed

heterogeneous crystal growth dominated by homogeneous crystal growth is always observed

in infinite ‘undercooling’ (see Fig. 62 (a)) for single component. I also find this behaviour in

Fig. 78 for PnBAPS68/100 binary mixture at p68 = 0.50 with multiple coloured homogeneous

crystal in the bulk and blue coloured heterogeneous wall crystal near the cell boundary.

With c increasing or n decreasing, homogeneous crystal nucleation gradually loses its

dominant role both for single components and binary mixtures. A competing growth between

homogenous crystal and heterogeneous crystal has been found also by Aastuen169

. As a

technical improvement based on the observation by Aastuen169

, I adjust the illumination

direction to observe wall crystal, homogeneous crystal and metastable fluid-like phase all

under the Bragg reflection condition as shown in Fig. 79. It is taken by side view from a


99

PnBAPS68/PS100 binary mixture at mixing number ratio p68 = 0.50 and n=11.12 1.50m-3

.

It shows that a heterogeneously nucleated wall crystal grows both from top and bottom of the

cell (black and blue part), whereas a homogeneously nucleated crystal grows in the central

part (yellow part), a metastable fluid-like region also exist at the initial time (orange colour

part in the center of the cell). At t = 37s, I observe the central homogeneous crystal almost

perfect round shape only in contact with metastable fluid, and the wall crystal is still far away

from it. At t = 65s, wall crystals have contacted the centre crystal, thus both growth is

hindered. With time further increasing, the central crystal can only grow perpendicular to the

direction of wall crystal growth, therefore, a final lens-like crystal is formed enclosed by wall

crystals without fluid left. As the inside thickness of the cell is 2mm, I can then get the crystal

growth velocity v110 in <110> direction by simply calculating the slope from the height of

wall crystal growth versus time.

Fig. 78. A suppressed heterogeneous crystal growth dominated by homogeneous crystal in the bulk. Here the

image is taken by side view from a binary mixture of PnBAPS68/100 at p68 = 0.50 and n = 20.44 2.75 m-3

.

The homogeneous crystal shows multiple colour in the bulk, while the heterogeneous crystal shows blue close to

the cell boundary by a certain way of illumination with cold white light.


100

(a) (b) (c) (d)

Fig. 79. A competing growth between homogeneous crystal and heterogeneous crystal. Images are taken from

PnBAPS68/100 binary mixture at p68 0.50 and n = 11.12 1.50 m-3

. Images from (a) - (d) are observed at

nucleation time t = 37s, 65s, 77s, 98s, respectively. They are observed as homogeneous crystal (yellow and

green in the bulk), fluid-like (orange in the bulk) and heterogeneous crystal (black and blue at the cell boundary).

Start from stop pump shear, both homogeneous crystal and heterogeneous crystal grow with t increases,

meanwhile less and less fluid is left in the bulk. Afterwards, up and below crystals contact each other, their

growth is stopped limited by the thickness of the cell. Therefore, a lens-like shaped homogeneously nucleated

crystal sounds fixed in the bulk, whereas, no yellow coloured fluid-like phase leaves in the cell.

By altering particle number density n and mixing number ratio p, different crystal growth

behaviours may be observed (shown in Fig. 80, and Fig. 81). For the first time here, we are

able to observe a sub-linear crystal growth across the coexistence region. For increased n,

crystal growth in <110> direction is found to be linear in time t. For further increased n,

heterogeneous crystal growth aborts upon contacting with the bulk homogeneously nucleated

crystals (c.f. Fig. 78, Fig. 79). According to the phase diagram of PS100 and the

PnBAPS68/PS100 binary mixture at p68 = 0.20, both show a wide coexistence region.

Therefore the metastable melt has to separate into two phases of equal chemical potential,

which these are at different densities. In other words, density differences have to be

established from an originally homogeneous situation. This is in contrast to the situation

above melting, where the original density is retained. Consequently above melting the self

diffusion coefficient is sufficient to describe the dynamics, while across coexistence the

(smaller) collective diffusion coefficient is also needed to describe the phase separation

kinetics. This would result in a kinetic slowing of growth.


101

0.0 83.3 166.7 250.0 333.3 416.7 500.00

200

400

600

800

1000

d(

m)

t(min)

Fig. 80. Heterogeneous wall crystal height d in <110> direction versus time t. It is taken from single component

PS100. Across coexistence crystal growth is non-linear in time: (◊) n 3.8 m-3

, (□) n 4.0 m-3

; (○) n

4.2m-3

; Above melting the crystal grows linearly with time: () at n 4.5 m-3

; but when n increase further, an

abortion of wall crystal growth is found at () n 4.9 m-3

.

0.0 50.0 100.0 150.0 200.0

200

400

600

800

1000

d(

m)

t(min)

Fig. 81. Heterogeneous crystal growth in <110> direction versus time t. Data are taken from PnBAPS68/PS100

binary mixture at mixing number ratio p68 = 0.20 under different particle number density n. Similar as single

component shown in Fig. 80, across fluid-crystal coexistence, crystal growth tends non-linearly with time t: (○)

n 2.95 m-3

; ()n 3.0 m-3

; () n 3.2 m-3

. Notice the saturation of d in this case for long times. Above

melting, the crystals grow linearly with time t: (♦)n 3.5 m-3

; but further increasing n, an abortion of crystal

growth is also observed at (●)n 8.67 m-3

.


102

In addition, there also may be a thermodynamic reason. The final state is inhomogeneous.

This means that the density difference between the final fluid state and the state of the

remaining melt continuously decreases. This is depicted in Fig. 82 under the assumption of an

exponential decrease of that difference and exclusion of any transient crystal compression and

of compositional fluctuations. Upper and lower horizontal lines give the melting and freezing

densities, respectively. Now, as n(t) approaches nf (freezing density) the thermodynamic

driving force vanishes. This again is different to the case above melting, where the chemical

potential difference for particles in the melt state as compared to particles in the crystalline

state of the same density remains constant until the completion of solidification.

0 5 10 15 20 25 30

0.5

0.6

0.7

0.8

0.9

1.0

norm

aliz

ed m

elt

densi

ty n

/nm

time (a.u.)

Fig. 82. A prospected time dependent density variation from crystal phase to fluid phase. (…..) shows the

melting line, where above this line represents the crystal phase; (----) shows the freezing line. In between the two

lines shows the crystal/fluid coexistence region. The exponential decay curve is an assumed line and the numbers

of scale are arbitrary values..

A quantitative modelling of the growth behaviour across coexistence is rather difficult, as it

demands knowledge of the collective diffusion coefficient and an estimate of the changing

chemical potential difference. In what follows I thus restrict myself to the behaviour above

melting, where linear growth is observed in all cases. Fig. 83 shows a typical growth curve

present under such conditions with the schematic definition of the determination of the


103

growth velocity and the initial height d0. In what follows, I shall first address the measured

growth velocities and then discuss the results for d0 and their interpretation.

0,0 5,0 10,0 15,0 20,0 25,00

200

400

600

800

1000

d0

v = dd / dt

d(

m)

t(min)

Fig. 83. schematic growth curve. Notice that above coexistence, growth is linear allowing the assignment of a

constant growth velocity. Notice further that there is an initial thickness of the wall based nucleus d0 (drawn in

red).

6.2. Wilson-Frenkel theory and experimental evaluation

This Chapter is concerned with the classical theory for reaction controlled growth of a flat

crystalline surface against an adjacent melt. One assumes that particles have to overcome a

diffusion barrier in order to make the transition from the liquid to the solid phase. Then the

rate of incorporation of particles into a crystal lattice is given by Wilson62

as

Tk

Qexp lνv

B

crystal [6.01]

with l as some characteristic length, as the attempt frequency and Q the diffusion barrier. In

verse, this process is counter-acted by particles that move from the crystal to the liquid. Since

the Gibbs free energy per molecule (the chemical potential) = G/N is higher in the liquid


104

than in the crystal, the rate of melting will be smaller than the rate of crystallization by a

factor TΔμ/kexp B , so

Tk

Δμexp

Tk

Qexplνv

BB

fluid [6.02]

Taken to be equal to the frequency of crystal lattice vibrations, the net rate of growth is thus

as

Tk

Δμexp-1

Tk

Qexplνv-vv

BB

fluidcrystal [6.03]

According to Frenkel 62

’s derivation

self

B

2 D Tk

Qexp νl

[6.04]

which both Wilson and Frenkel’s formula is connected with Stokes-Einstein’s formula. Here

D0 is free-diffusion coefficient associated with the ‘tracer’ particle surrounded by a bulk of

like particles. Then finally Wilson-Frenkel formula come to transform as

Tk

Δμexp1

l

Dv

B

selfWF [6.05]

For atomic systems the self diffusion coefficient Dself is equal to the tracer diffusion

coefficient, and the typical length scale l equals the diffusion mean free path d. For colloids,

some modifications have been proposed adapting this law to the specific circumstances. Since

the diffusion is strongly time dependent for interacting colloidal particles either the short time

self diffusion coefficient DSS D0

169 or the long time self-diffusion coefficient DL

S 0.1D0

may be used153, 170

. Here the Stokes Einstein diffusion coefficient D0 of free particles in a

solvent given as


105

a η 6π

TkD B

0 [6.06]

Second the typical length scale corresponds to the mean particle separationd n-1/3

. Thirdly,

Eq.[6.05] assumes a mono-layer interface. One may in addition consider that not every

attempt is successful by using a factor f < 1, and further that growth may speed up

considerably, if (according to Würth 43

) the interfacial thickness is dinterf /

d > 1. This leads to

Tk

Δμexp1

d

Dfdv

B

2

selfinterf

WF [6.07]

Using the limiting growth velocity v for infinite chemical potential difference between

melt and solid:

2

selfinterf

d

Dfdv [6.08]

one finally arrives at:

v = v (1 - exp(-µ/kBT)) [6.09]

A test of the Wilson-Frenkel may be performed, if an estimate of is available. A first

suggestion was provided by Aastuen169

, who rewrite the Wilson-Frenkel law as:

m

m

B

2

1

31

0n

nn

Tk

Aexp1

A

n4Dv [6.10]

with the assumption that increases linearly with n-nm, i.e. = A2[(n-nm)/nm], where A1

and A2 are constant got from nonlinear least square fit to experimental data of v versus n.

This description catches the direct density dependence. however, there is also some density

dependence of the pair potential. To cover this, Würth et al.43

suggested to use a properly

rescaled energy density m* = (-m)/m. Here, = (1/2)nV(r) is the actual energy density


106

summed over to all pairs in a given volume and having the dimension of an osmotic pressure,

is an effective coordination number and V(r) is the DLVO pair-wise interaction energy. The

suffix ‘m’ denotes melting. Thus the difference of chemical potential can be expressed as

m

mm B*BΔμ

[6.11]

where B is in units of kBT. This choice yields good fits even for samples of broad coexistence

regions. However, as pointed out in32

, it is thermodynamically inconsistent. We therefore

chose

f

ff 'B*B'Δμ

[6.12]

with

f

f

ff

f

f

ff

'

nd

dκexp

nd

dκexpn

d

dκexp

BΔμ [6.13]

Here B’ has the same function as B but applied in Eq.[6.12] and ‘f’ denotes freezing.

This latter method of fitting has been applied to several samples and the results are shown in

Fig. 84. In this plot, freezing is at µ = 0 and melting is indicated by the arrows. Note that the

description obtained is quite good for samples of narrow coexistence region (PS120), but data

fall short of the theoretical expectation across coexistence, if the coexistence region is wider

(PS109). Notice that Aastuen et al.’s data show a much worse statistical accuracy. This

resulted in a 15% improvement of the error in B for the PS109 data.


107

Fig. 84. Crystal growth velocity versus the difference of chemical potential . () denotes the experimental

data from sample of PS91 (diameter 2a = 91nm, effective charge Z* = 800); (○)denotes the experimental data

from sample of PS109 (diameter 2a = 109nm, effective charge Z* = 395); (●) denotes the experimental data of

PS120 (diameter 2a =120nm, effective charge Z*=685). They are respectively fitted with Wilson-Frenkel

formula as correlated lines. Two arrows shows the possible melting transition of sample PS109 (left) and PS120

(right). Figure courtesy to32

.

The actual fitting procedure is as following. Experimentally, v is calculated as v = dd/dt,

which holds while crystal growth is linear in t. The Debye parameter is correlated with the

effective charge Z* and the particle number density n, and the mean particle distance

31

nd

. Thus once n and Z* of the particles are known, and v110 as the crystal growth

velocity at (110) plane is measured, one can get the v110 - experimental diagram and

directly compare to Wilson-Frenkel growth law by fitting. The free parameters are v and B’.

I shall now compare my own experimental data to the Wilson-Frenkel laws of Eq.[6.10] and

Eq.[6.13]. They are respectively shown as Fig. 85 for PnBAPS68, and Fig. 86 for PS100.

The results are compiled in Tab. 7.

Tab. 7. Parameters in Wilson - Frenkel law and their statistic error and systematic error for single component of

PnBAPS68 and PS100.

sample v v

(m/s)

B’ B’

(kBT)

A2 A2

(kBT)

PnBAPS68 15.900.400.01 2.00.10.66 1.800.100.01

PS100 6.900.400.01 1.60.30.38 3.400.500.03


108

6 8 10 12 14 16

0

4

8

12

16

v110 (

m/s

)

n ( m-3)

0.0 0.5 1.0 1.5 2.0 2.5

0

4

8

12

16

v110 (

m/s

)

(kBT)

Fig. 85. PnBAPS68: v110 – n experimental data () and Wilson - Frenkel fits v110 = 15.9(1-exp(-0.29(n-6.1)) (-.-.-

.) (left image); PnBAPS68: v110 – experimental data ()and Wilson-Frenkel fits v110 = 15.9(1-exp(-/kBT))

(-.-.-.) (right image) with assuming =B’(-f)/f . Both v110 – n and v110 - data fit Wilson-Frenkel

formula very well. Here sample PnBAPS68 has a very small fluid-crystal coexistence region (nf = 6.1 0.4m-3

,

nm = 6.2 0.4m-3

).

3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5

0

2

4

6

8

v110(

m/s

)

n( m-3)

0.0 0.5 1.0 1.5 2.0 2.5

0

2

4

6

v110 (

m/s

)

(kBT)

Fig. 86. PS100: experimental data v110 – n ()and (-.-.-.)Wilson-Frenkel fits v110 = 6.9(1-exp(-0.89(n-3.8))) (left

image): PS100: experimental data () v110 – and (-.-.-.)Wilson-Frenkel fits v110 = 6.9(1-exp(-/kBT))) (right

image) with assuming = B’(-f)/f . Both v110 – n and v110 - data fit Wilson-Frenkel formula very bad

especially close the fluid-crystal coexistence. Here sample PS100 has a very large fluid-crystal coexistence

region (nf =3.8 0.3m-3

, nm = 4.4 0.2m-3

).

Compared with Fig. 85 and Fig. 86, also with previous work in Fig. 84, I can easily get a

conclusion that Wilson-Frenkel fit either applied with the style of Eq. [6.09] or the style of

Eq. [6.10] both shows a good fit for small fluid-crystal phase transition (c.f. PS109 in Fig. 84,

and PnBAPS68 in Fig. 85) but a bad fit for large fluid-crystal phase transition (c.f. sample


109

PS120 in Fig. 84 and sample PS100 in Fig. 85). This possibly hints that Wilson-Frenkel law

do not fit to be applied in fluid-crystal phase transition region. This is possibly due to the

additional function of density dependent collective diffusion that the simple description using

a constant v will not suffice to describe the crystallization behaviour below melting where

significant density difference appears. This assumption has been given first by Würth et al43

,

and it is more pronouncedly convinced by the Wilson-Frenkel fits to my experimental data.

Furthermore, the nonlinearly crystal growth behaviour below melting transition (c.f. Fig. 80,

Fig. 81) also prove the different growth mechanics correlated to more complicated diffusion

behaviour compared with that above melting.

For comparison I shall also quote some results on a system with different type of repulsive

interaction. For a hard sphere system, the Wilson– Frenkel law reads

Tk

ΦΔμexp1

d

ΦDKv

B

eff [6.14]

where the density dependent effective diffusion coefficiency Deff() is available from

measurements53, 171

and the difference of chemical potential between fluid and solids ()

are taken from simulation and theory 172, 173

. K is a fitting parameter accounting for the

interfacial thickness and the effective factor. For the data of Stipp et al.174

K = 40 (shown in

Fig. 87), roughly speaking, each particle attaching to the crystal has to move over a distance

on the order of 1/40 of the interparticle distance, which is on the order of the distance between

the particle surfaces.


110

Fig. 87. Wilson-Frenkel fit to hard sphere sample PMMA890. The best fit is found at K = 40. Figure courtesy

to32

.

Also the hard sphere sample a larger deviation between Wilson-Frenkel fit and experimental

data below melting. Again two kinetic process responsible for the growth velocity is

prospected to this case concluded by Palberg32

, i.e., the self-diffusion of a particle towards its

target place and the proceeding transport towards the interface which may be much slower or

over a longer distances and in fact limit v; the latter may be expected from the results of

computer simulations of fcc/melt interfaces61

, where for the degree of undercooling via

freezing T* = (T-Tf)/T = 0, growth was facilitated through the release of latent heat and the

local density variations provided by the density difference between crystal and melt.

After discussing the qualitative differences between the expectations and the data taken across

coexistence I shall now turn to the quantitative estimation of the uncertainty of B’ and v in

general and for PnBAPS68 in particular. For soft sphere systems, the deviation by applying

Wilson-Frenkel growth law can also arise from the following parameter. One is the estimation

of particle number density n especially close to the fluid - crystal phase boundary; second is

the uncertainty of effective charge Z* estimated from it mobility; third is the fluctuation of

salt concentration c, which has a pronounced effect to the phase behaviour close to the fluid-

crystal phase boundary. These uncertainties will result in a deviation of diffusion behaviour


111

and then crystal growth. Applying with these experimental parameters to an experimental

measured v and calculated will result in a deviation of fitting parameter B’ (unit: kBT) and

limiting growth velocity v, while actually 1 kBT corresponds to 2.5 kJ mol-1

crystallization

free energy roughly32

. Here I table the most important parameters in different style of Wilson-

Frenkel law and their error bars by calculating the partial deviations of Eq. [6.09] shown in

Tab. 7.

6.3. Crystal growth and Wilson-Frenkel fits in PnBAPS68/PS100 binary

mixture

Up to now, the Wilson-Frenkel law has been tested for several single component charged

systems32, 169, 170

. Here I shall for the first time present such a comparison for bimodal

mixtures. This comparison will be facilitated by assuming an average DLVO pair-wise

interaction and Eq. [2.13] is rewritten as

With

s

*

BB

*

AA

Br0

22 nZpZpn

Tkεε

eκ

Both are used in the calculation of * with Eq. [6.09] and Eq.[6.12]. The fitting with Wilson-

Frenkel law both for v110 - n and v110 - data then can be performed for under several

mixing number ratios pA or pB of the binary mixture. Here I shows the data and Wilson-

Frenkel fits for PnBAPS68/PS100 binary mixture with mixing number ratio, respectively, as

p68 = 0.20, 0.50, 0.75 .

2

B

B2*

B

2

B

BA

BA*

B

*

ABA

2

A

A2*

A

2

A

r0

2

BA,aκ1

aκ expZp

aκ1aκ1

aκaκ expZZp2p

aκ1

)a(κ expZp

r

rκ exp

ε ε 4π

eU(r)


112

2 4 6 8 10 12 14-2

0

2

4

6

8

10

12

p68

= 0.2

W F - Fit

v1

10(

m/s

)

n ( m-3)

(a1)

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

0

2

4

6

8

10

p68

= 0.2

W F - Fit

v 110 (

m/s

)

(kBT)

(b1)


113

4 6 8 10 12 14 16-2

0

2

4

6

8

10

12

p68

= 0.5

W F - Fit

v110(

m/s

)

n (m-3)

(a2)

0 1 2 3 4

0

2

4

6

8

10

12

p68

= 0.5

W F - Fit

v110 (

m/s

)

(kBT)

(b2)


114

4 6 8 10 12 14 16-3

0

3

6

9

12

p68

= 0.75

W F - Fit

v110 (

m/s

)

n (m-3)

(a3)

0 1 2 3 4 5 6

0

2

4

6

8

10

12

p 68

=0.75

W F - Fit

v110 (

m/s

)

(kBT)

(b3)

Fig. 88. v110 – n and v110 - /kBT experimental data and Wilson-Frenkel fits for PnBAPS68/PS100 binary

mixture at mixing number ratio of p68 = 0.20, 0.50, and 0.75. At p68 = 0.20, get the fitting v110 =10.04(1-exp(-

0.45(n-2.74)) in (a1) and v110 =10.04(1-exp(-/kB T)) in (b1); At p68 = 0.50, get the fitting v110 = 10.25(1-exp(-

0.48(n-4.4) ) in (a2) and v110 =10.25(1-exp(- /kB T)) in (b2); At p68 = 0.75, get the fitting v110 = 11.2(1-exp(-

0.61(n-5.32)) in (a3) and v110 = 11.2(1-exp(- /kB T)) in (b3).


115

The best fits obtained were v110 = 10.04(1-exp(-0.45(n-2.74)) for p68 = 0.20; v110 = 10.25(1-

exp(-0.48(n-4.4)) for p68 = 0.50; v110 = 11.2(1-exp(-0.61(n-5.32)) for p68 = 0.75. Also I fit v110

as a function of *f, where I obtain the same v. The fits are of good quality, except for p68 =

0.20 where a relative large deviation is visible below melting. In conclusion the findings

indicate that the comment for single component below melting may also be applied in binary

mixture system; On the other hand, the assumption of averaged DLVO pair interaction shows

efficient to evaluate Wilson-Frenkel fit to binary mixture. In Tab. 8, I compare the values

obtained as following.

Tab. 8. Some parameters, their statistic errors and systematic errors for PnBAPS68/PS100 binary mixture

appeared in different style of Wilson-Frenkel law.

It is interesting to compare the observed behaviour to the phase behaviour as observed in

Chapter 6. In Fig. 88, I show again the fit curves for the pure components and the three

investigated mixing ratios. The n dependence of the growth velocity is shown as a function of

p68 in Fig. 89. Here the data of Fig. 88 are replotted

In accordance with the observed phase behaviour, at lower p68, it is found that the starting

point of crystal growth shifts to lower n. At low n, v110 first increases but then decreases again

as p68 increases. At larger n, the growth velocity increases monotonously with p68 and

approaches the limiting growth velocity v. This is shown in Fig. 90 where one also notices

that v varies approximately linearly with p68. As v corresponds to the average diffusion

coefficient divided by a typical length squared and multiplied by the interfacial thickness, I

may here conclude that in the case of PnBAPS68/PS100, I see the composition dependent

change of the diffusion coefficient rather than a change of typical length scale (n stays

constant) or of the interface thickness.

PnBAPS68/PS100

binary mixture

v v

(m/s)

B’ B’

(kBT)

A2 A2

(kBT)

P68 = 0.20 10.040.250.01 1.270.070.72 1.240.060.01

P68 = 0.50 10.250.200.01 2.230.120.81 2.090.130.02

P68 = 0.75 11.200.100.01 1.000.070.41 3.230.250.02


116

2 4 6 8 10 12 14 16-3

0

3

6

9

12

15

PnBAPS68/PS100 growth fitting

in exponetial first order decay

P68

=0 (PS100)

p68

=0.20

p68

=0.50

p68

=0.75

p68

=1.0 (PnBAPS68)

v110 (

m/s

)

n (m-3)

Fig. 89. Exponential first order decay fitting to growth velocity v110 versus n for PnBAPS68/PS100 single

components and binary system. The fitting curves show that: at p68 = 0.20, melting transition shifts to lower n,

however liming growth velocity v and v110 shifts higher than the same n of PS100 single component. Further

increasing p68, crystal growth starts at higher n (higher nm than PS100), and v grows with increasing p68 further,

however it is not so pronounced as the case of adding small amount of p68 in the binary mixture.

0.0 0.2 0.4 0.6 0.8 1.00

2

4

6

8

10

12

14

16

n=5 (m-3) n=6 (m

-3) n=7 (m

-3)

n=8 (m-3) n=9 (m

-3) n=10 (m

-3)

n=11 (m-3) n=12 (m

-3) n=13 (m

-3)

v110 (

m/s

)

p68

Fig. 90. n dependent growth velocity v110 versus mixing number ratio p68 for PnBAPS68/PS100 binary mixture.

A radial curve hints a switched n dependent and p68 dependent binary compound, and different dominating effect

for the mixing number ratio.


117

6.4. Limiting crystal growth in PS120/PS156 binary mixture

Additional measurements are carried out on a mixture of PS120 and PS156 up to p156 = 0.2.

As described before, samples prepared in a closed pump tubing system and observed with a

microscope equipped with a low resolution objective video CCD camera. Here I take the

rectangular cell (1x10x100mm3) simply for a parallel comparison with the previous work by

A. Stipp. By side-view, heterogeneous crystal is found dominating the solidification after

diluting the suspension to some n. Results are shown in Fig. 91. Here a completely different

behaviour is observed. At constant n the growth velocity decreases much stronger than

expected on the basis of an average diffusion coefficient. These data are confirmed and

extended to larger p156 by A. Stipp (c.f. Fig. 93).

-0.05 0.00 0.05 0.10 0.15 0.20 0.250

1

2

3

4

5

6

v ( m

/s)

p156

Fig. 91. Growth velocities in a mixture of PS120 with PS156 under deionised conditions. The growth velocity

decreases with increasing fraction of larger particles. The upper line gives the expectation for the growth of a

random composition bcc crystal with the kinetic prefactor change dominated by the decreasing average diffusion

coefficient.


118

The case of PS120/PS156 is different to that of PnBAPS68/PS100. There I observe

indications of a possible compound formation. Here this is not the case. In addition,

experiments by Wette indicate the formation of a random composition bcc material. In

particular, there is no significant deviation between experimentally determined shear modules

and the expectation based on the assumption of a random composition bcc crystal. This is

shown in Fig. 92. Other samples (like PS90/PS100) however showed significant deviations

from the expectation which were interpreted as indication of the formation of compositional

fluctuations.

0.0 0.2 0.4 0.6 0.8 1.0

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

G (

Pa)

p156

1.0 0.8 0.6 0.4 0.2 0.0

p120

Fig. 92. Shear modulus measurement at PS120/PS156 binary mixture. Within experimental error, the data are

compatible with the formation of random composition bcc crystals. The upper and lower solid lines give the

limits of that expectation for two different residual concentrations of background electrolyte. Data was provided

by Wette presented in the poster of ECIS 2000 .

Now in the case of PS120/PS156, there is no clear evidence of the formation of such

fluctuations from elasticity. Accordingly the findings of Fig. 91 may be explained by some

different mechanism. Without further knowledge the simplest explanation based on Wette´s

finding would assume formation of random composition bcc wall crystals. In that case the


119

addition of a small fraction of large particles will lead to a decrease of the average diffusion

coefficient. The theoretical expectation for this case was already shown in Fig. 91 and is again

shown in Fig. 93, which compares my data to those obtained by A. Stipp. The larger error bar

of v at larger p156 is due to homogeneous nucleation starting to dominate and further due to

the increase of turbidity upon exchanging more and more small by large particles. Clearly the

observed decrease is stronger than expected. In addition above p156 = 0.15 the decrease slows

and proceeds parallel to the expectation but with values much below. Note further that at p156

= 0.15 the growth velocity is smaller than of any of the pure components.

0.0 0.2 0.4 0.6 0.8 1.0

1

2

3

4

Part IIPart I

v(

m/s

)

p156

Fig. 93. Comparison of growth velocities as obtained by A. Stipp () and me. (

) links the predicted trend of

limiting growth velocity with p156 increase from one component ( PS120) to another component (PS156). (

)

short dash line in Part I indicates the trend of experimental data decrease of v before p156 = 0.14; (

)

long dash in Part II indicates the trend of experimental data decrease of v after p156 0.143, and it runs roughly

parallel with the theoretically predicted line (

). Part I and Par II is separated by (

).

I therefore conclude that the decrease is not solely due to the change of the diffusion

coefficient. Three alternative possibilities shall be discussed. Firstly the decrease may be

caused by thermodynamics. Fig. 90 have shown that the growth velocity decreases as the

distance to the freezing transition is decreased. This could be possible, if the phase boundary

as a function of p would have the appearance qualitatively sketched below. Then the reduced


120

thermodynamic drive would be the dominating effect. At present, however, the complete

phase diagram is not yet determined for this mixture.

0.0 0.2 0.4 0.6 0.8 1.03

4

5

6

7

8

9

10

freezing

solid

nf(a.u

)

p156

Fig. 94. Schematic phase diagram to explain the results of Fig. 93 on a thermodynamic basis.

Second there may be a change in the kinetic prefactor due to either a change of the relevant

length scale or of the interfacial thickness. Since v = fdinterfDself/2d our data are consistent

with either a larger length scale d , a reduced efficiency factor f or a reduced interfacial

thickness dinterf. The first case may be ruled out, as n stays constant. The second case could be

explained by a reduced energy gain upon crystallization for the crystals with “impurities”.

Therefore the successful occupation of a lattice site will need more attempts. The third

possibility would involve a layered growth of pure component systems and a reduced layering

if impurities are present to disturb the interfacial structure.

Finally there also may be a change of the diffusive mechanism itself, if one assumes that only

suitable particles are incorporated into the growing crystal. In that case, the growth velocity

would be limited by the transport of impurity particles away from the interface. As the

impurities in this case are the larger particles, this diffusion process will be slower than the

self diffusion. In addition it would build up a concentration gradient of impurities which may

be thermodynamically unfavourable. This would further slow the separational diffusion.


121

At present these alternatives may not be discriminated decisively. Before doing so, one would

need a determination of the position of the phase boundary, some information on the twin

domain morphology and quantitative growth measurements independence of n.

Concluding, I have measured the growth velocities in two different binary mixtures of

charged spheres. I have observed two different scenarios. In the case of PnBAPS68/PS100,

the mixing ratio dependent change of the growth velocities is found to be strongly correlated

to the distance of the preparation conditions to the phase boundary, i.e. correlated to the

thermodynamic driving force. Only the limiting growth velocity is observed to correlate with

the change in the diffusion properties. The observations of Chapter 5 in addition indicated the

possibility of compound formation, which is analysed to be consistent with the observed

growth behaviour. For this mixture, the addition of a second component favours increased

crystal stability.

For the other mixture, the present data indicate that the addition of a second component may

rather be discussed in terms of impurities or an enhanced polydispersity. Polydispersity

reduces thermodynamic stability but also may slow crystallization kinetics. Both the

thermodynamic analysis and the kinetic analysis performed above would allow for such

interpretation.

Thus I face with the observation, that mixtures of colloidal spheres may show quite different

solidification behaviour closely related to their phase behaviour. This may be regarded as

starting point for further systematic investigations, which may be successful to correlate the

observed trends to parameters like the size ratio, the charge ratio etc.

6.5. Observations on the initial crystal thickness d0

A. Stipp made an interesting observation in his investigations of the PS120/PS156 binary

mixture. He found at constant composition and particle concentration that the shear conditions

applied to melt the suspension before growth experiments have a significant influence on the

solidification scenario. An example is shown in Fig. 95. This mixture at p156 = 0.143 was

investigated after shear melting at different pumping speeds. In both cases growth was linear

in time and with the same growth velocity. Extrapolation of the crystal dimension to t = 0,

however yielded different values for the initial crystal thickness d0. For the faster pumping

speed the larger d0 was observed. A systematic survey on the composition dependence further


122

showed that d0 increased approximately linearly with composition p156 shown in Fig. 96, and

particle number density n for PS100 and PnBAPS68/PS100 binary mixture before

(approximate up-linear)and after the occurrence of homogeneous crystal (approximate down

linear) shown in Fig. 97, respectively. Notice in Fig. 96, the left set of data was taken at high

pumping speed, and the right set at low pumping speed.

0 20 40 60 80 100-50

0

50

100

150

200

250

300

350

400

450

d (m

)

t(s)

Fig. 95. Time dependent growth of the binary mixture PS120/PS156 at p156 = 0.143 after abortion different

pumping speed. (▲) fast: v = (2.6 0.01) µm/s, (□) slow: v = (2.55 0.02) µm/s. By the crossing of deduced

line (----) and (…….

) from heterogeneously nucleated crystal growth with t = 0 line (

); (Ο) marks the initial

nucleation height d0 after abortion fast pumping shear, (■) marks the initial nucleation height d0 after abortion

slow pumping shear. Data courtesy of A. Stipp.


123

0.0 0.2 0.4 0.6 0.8 1.0

50

100

150

200

250

d0(

m)

p156

Fig. 96. Initial nucleation height d0 (t = 0) shows a linear relation with increasing p156 at slow pumping speed

(guided by line (

) ) and at fast pumping speed (guided by line (─ ─ ─)). d0 shows more pronounced with fast

pumping speed. (prepared sample are at a constant particle number density n = 0.40 0.05 m-3

). Data courtesy

of A. Stipp.

Thus there seems to be an influence of both the shear rate and the interaction parameters on

the thickness of the initial wall based layer serving as heterogeneous nucleus for later crystal

growth. To check for this, I first determined d0 for a pure component (PS100) as a function of

n and at fixed pumping speed. As shown in Fig. 97(a), the initial thickness increases with

increasing particle concentration. Working with the PnBAPS68/PS100 mixture, I could

reproduce this effect: d0 first increases approximately linearly with n, however it goes through

a maximum and then decreases again.


124

4.8 5.0 5.2 5.4 5.6 5.8 6.0 6.2 6.4 6.6-40

-20

0

20

40

60

80

100

120

140

d0 versus n for PS100 single component.

The dash-dot-doted line shows the trends of d0

d0(

m)

n (m-3)

(a)

4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5

40

60

80

100

120

140

160

180

d0 versus n at PnBAPS68/PS100 binary mixture at p

68=0.2

The dotted line guides the trends of d0

d0(

m)

np0.2

(m-3)

(b)


125

5 6 7 8 9 10 110

20

40

60

80

100

120

140

160

d0 versus n for PnPS68/PS100 binary mixture at p

68=0.5.

The dash lines guide the devlopping of d0

d0(

m)

np0.5

(m-3)

(c)

5 6 7 8 9 10 11 12-20

0

20

40

60

80

100

120

140

160

180

d0 versus n for PnBAPS68/PS100 binary mixture

at p68

=0.75.The dash-dot lines guide the developping of d0

d0(

m)

np0.75

(m-3)

(d)

Fig. 97. d0 versus n for (a) in single component PS100, for (b) in PnBAPS68/PS100 binary mixture at p68 = 0.20;

for (c) at p68 = 0.50; and for (d) at p68 = 0.75, respectively. All plots, d0 shows a linear increase versus n at wall

crystal region, however d0 goes approximately linearly down versus n after homogeneous crystals appear. The

larger error bar and some deviation of data again shows that d0 value greatly depends on the altered shearing

speed.


126

Due to the interference of homogeneous nucleation, I am not able to determine the existence

of a maximum for the pure PS100 system. For the three mixtures investigated the maximum

height is about 140 - 160µm. Its position changes with composition. In Fig. 98, I replot the

data taken at constant n = 6.5 µm-3

versus composition p68. In this case, a pronounced

maximum is observed at p68 = 0.20. I notice that this corresponds to the minimum of the

freezing density nf at p68 comparing with other composition (c.f. Fig.73). This indicates the

possibility of a common thermodynamic drive behind the behaviour of all examined systems.

Accordingly I tried an evaluation in terms of the differences of chemical potential between

melt and solid. For this qualitative check, I use the simple estimate of µ n-nf. I replot the

data of Fig 98 in Fig. 99 to obtain a roughly linear relationship.

-0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.840

60

80

100

120

140

d0(

m)

p68

Fig. 98. Initial wall crystal height d0 versus composition p68 at a constant n (n = 6.5µm-3

). (○) are the

experimental data or the cross points of guided lines with n = 6.5 µm-3

in Fig.97, respectively for p68 at 0, 0.20,

0.50, and 0.75. Here the connecting line between each point guides the trends of d0 with p68.


127

0 1 2 3 4 50

50

100

150

Experimental Data

Linear fit

d0(

m)

n-nf (m

-3)

Fig. 99. Initial wall crystal height d0 of PnBAPS68/PS100 single and binary mixture system (p68 = 0, 0.2, 0.5,

0.75) versus ‘distance’ of ‘undercooling’ to the phase boundary (n-nf) .

The same is possible with the pure component. For the data of the PS120/PS156 mixture, I

again assume a linear variation of the freezing and melting density between the values

observed for the pure components. Here the distance to the phase boundary is estimated from

the actual constant n and the interpolated values plotted in Fig 100.

0.0 0.2 0.4 0.6 0.8 1.00.20

0.25

0.30

0.35

0.40

0.45

0.50

PS120 and PS156

nf

nm

n(

m-3)

p156

Fig. 100. The estimate of the phase boundary for the mixture of PS120/PS156 based on the measured melting

and freezing densities and the observation of random substitution crystal formation by Wette.


128

In this case, an increase of the composition parameter p156 corresponds to an increase of

‘undercooling’. Fig. 101 summarises the results. In each case the initial thickness increases

with increasing the ‘distance’ to the freezing density (n-nf). Therefore I conclude that the

larger the ‘undercooling’, the larger will be the wall based shear induced layer serving as

nucleus for wall crystal growth. In addition the data of A. Stipp clearly indicate the

pronounced influence of the increased shear rate.

0 1 2 3 4

30

60

90

120

150

180

210

PS120/PS156 slow shear

PS120/PS156 fast shear

PnBAPS68/PS100 slow shear

PS100 slow shear

d0(

m)

n-nf (m

-3)

Fig. 101. Initial wall crystal height d0 versus the ‘distance’ of ‘undercooling’ to the phase boundary (n-nf ).

Samples include PS120/PS156 single components and binary mixtures after abortion fast and slow shear,

PnBAPS68/PS100 single component and binary mixtures after abortion slow shear only. Dotted lines guide the

developing of d0 started from freezing.

To further investigate this point, I shall now turn to some microscopic observations made by

Bragg microscopy. Fig. 102 shows several micrographs as obtained after abortion different

pumping shearing rates. After abortion low shear rate, one recognizes a wall based Bragg

scattering crystal. Whereas, after abortion of fast shear, the region of Bragg scattering crystal

is confined to the cell interior. I recognize from the growth experiments that this non

scattering region close to the cell boundary more or less coincides with d0. This means that

the part of the wall crystal formed under shear already has a different structure and/or

0.06 0.07 0.08 0.09 0.10 0.11 0.12

60

90

120

150

180

210

240


129

orientation than the centre part which was epitaxially grown from the melt. Note that this

applies both to the pure component and the mixture. It is now tempting to correlate this region

of different morphology to d0. This will be justified below. In Addition it is further tempting

to correlate the non-scattering region to the regions of twin domain formation as observed by

top view. For this a more detailed model of the structure will be needed. However, the data so

far do not allow for a unequivocal correlation to the composition dependent formation of zig-

zag domain morphologies. Clearly we need more systematic experiments on this point.

Fig. 102. Heterogeneously nucleated crystal after abortion different pumping shear rates at the same n. (A1), (A2)

are of PS120 at n 0.67 0.05m-3

after abortion slow and fast shear, respectively. (B1), (B2) are PS120/PS156

mixture at p156=0.02 and n 0.60 0.05 m-3

after abortion slow and fast shear, respectively. The maximum

height of the wall crystals is about 500 m shown (↕). The black base is the cell boundary. The micrographs are

taken by top view.

I shall nevertheless now present some speculation about this issue. In order to do so I shall

first introduce two related topics, namely layering. From recent computer simulations83

, it is

known that there is some modulation of a fluid structure next to a rigid wall. This effect was

termed layering.


130

(a) (b) (c)

Fig. 103. (a) Liquid density profile at a vapour-liquid interface. is the bulk liquid density and 0 is the width

or molecular-scale ’roughness’ of the interface. (b) Liquid density profile at an isolated solid-liquid interface.

s() is the ‘contact’ density at the surface. (c) Liquid density profile between two hard walls with distance D

apart. The contact density s(D) is a function of D. is the diameter of the particle. Figure courtesy to83

.

At very dilute condition (similar as just above vapour - liquid interface shown in Fig.103(a)),

particles fill the cell loosely, which leaves almost no particle density difference between the

cell boundary and the bulk ( bulk); In the medium n, the density profile difference

between the surface of cell boundary and bulk shows very large (c.f. Fig.103(b)); In the

higher n, particles packed very close, similar as the two cell walls are very close. Thus the

density profile is pressed from left side to right side, induced again less density difference

between the surface of cell boundary and the bulk (c.f. Fig.103(c) ). This phenomenon is

known as fluid modulation and is an effect of confining geometry. It may be enhanced, if

further wall particle interaction is present. In addition, I give a comparison to Biehl’s

experiment in the next part of this Chapter. The conclusion is then left to that part.

6.6. Former shear influenced crystal structure and a discussion to the

structure of PnBAPS68/PS100 binary mixture

From recent experiments on monodisperse samples under shear in a plate-plate geometry one

further knows that there is also in-layer orientation occurring at the like charged cell wall175

.


131

Starting from a bcc solid, layers of hexagonal in-plane structure are formed parallel to the cell

wall, with the orientation of their densest packed direction <111> parallel to the formerly

applied shear. Upon abortion of shear, the layers were observed to register and form a solid.

Only on long time scales, this metastable solid would transform back to the bcc equilibrium

structure via nucleation and growth. Interestingly, the shear induced orientation was found to

retain.

Shear-induced particle alignment in a dispersion of hard discs has also been found by Brown

and Rennie176

who reported that at low shear rates (below 1s-1

), the discs aligned normal in

the flow direction, whereas at high shear rates (above 18 s-1

), the discs were aligned with their

normal in the gradient direction.

Interestingly layers may be induced also where the equilibrium structure is fluid. The

stationary layer thickness under shear first decreases with increasing shear rate but then

increases again. The phenomenon was interpreted as the shear induced formation of registered

sliding random hcp crystal layers in the first case and of the formation of free sliding

hexagonal planes in the second case. This is shown in Fig. 104 and Fig. 105.

(a) (b)

(c) (d)

(a) (b)

(c) (d)

Fig. 104. Time domain averaged position correlation diagrams (PCD) of suspensions of PS310 at n = 0.15 µm-3

,

and (a) equilibrium bcc crystal with (110) plane parallel to the wall. (b) wall based registered sliding r-hcp layers

sheared with 0.89 Hz. (c) wall based hexagonal layers sheared with 7.1Hz and sliding independently over each

other. (d) Registered planes after stop of shear. Images courtesy to Biehl ‘s PhD thesis177

.


132

A time domain averaged position correlation diagrams (PCD) of suspensions of PS310 is

shown above, each PCD was constructed from successive but statistically independent video

frames at n = 0.15 µm-3

. Here (a) shows the equilibrium bcc crystal with (110) plane parallel

to the wall. (b) shows the wall based registered sliding r-hcp layers sheared with 0.89 Hz. (c)

shows the wall based hexagonal layers sheared with 7.1Hz and sliding independently over

each other. (d) shows the registered planes after stop of shear. An r-hcp crystal may be

reached from both (b) and (c). (a) occurs when the relaxation back to the equilibrium state on

much longer time scales.

The occurrence of registered sliding planes is confined to low shear rates. Upon increasing the

salt concentration, the suspension shear melts. With increasing shear rate the region of

existence is confined to ever smaller salt concentrations. In contrast the free sliding plane

structure occurs at larger shear rates and its stability against shear melting upon increase of

the salt concentration is enhanced with increasing shear rate. This is shown in Fig. 105.

0.0 0.2 0.4 0.6 0.8 1.0

1

1E1

sh

ea

r ra

te (

Hz)

added salt (µM)

wall middle

fluid

hexagonal

Fig. 105. Non-equilibrium phase diagram for a suspension of charged spheres subjected to linear shear in a plate-

plate cell of gap height 30µm. The diagram was given in terms of concentration of added H2CO3 and shear rate.

Up (down) triangles denote hexagonal in plane order observed in the first three layers off the cell wall (in the cell

centre). The dashed vertical line denoted the equilibrium bulk fluid-bcc phase boundary. Hexagonal in-plane

order was obtained under shear even where the bulk equilibrium structure was fluid. With increasing shear rate

the boundary between fluid and hexagonal order first shifted to lower c then to higher c. In all cases, the

transition to fluid order was observed to occur at lower c for layers in the cell centre. Figure courtesy to177

.


133

With this said, I suggest that both for the pure components and the mixture of PS120/PS156

the situation is equivalent to that observed by Biehl. I thus correlate the microscopically

observed non-scattering region to the initial wall crystal extension d0 and in addition to a r-

hcp structure. The bcc crystal then grows on top of this r-hcp crystal. It herits the original

orientation, i.e. the former <111> direction of r-hcp now is the <111> direction in a (110)

plane. This has previously been confirmed for vanishing d0 by Maaroufi178

.

In addition, Fig. 105 shows that the stability against salt induced shear melting and the

extension of wall based free sliding layer phases increases with increasing shear rate. If I

identify the non-scattering region with an r-hcp crystal originating from free sliding planes

registered after abortion of shear, this also is in line with my analysis of d0 above.

The analysis of the morphology of the second mixture PnBAPS68/ PS100 is more difficult.

The angle between the <100> and the <111> direction in a bcc structure is 54.73°. This angle

is found in the top-view as a zig-zag pattern. There, it is formed between the principal

direction of stripes’ orientation and the former shear direction. This indicates that the zig-zag

pattern is a part of bcc structure. The situation is sketched in Fig. 106.

54.73°

<111>

54.73°

<111>

Fig. 106. A comparison between the orientation of a bcc based (110) frame lines and the strips of zig-zag pattern

with the <111> direction of former shear They are found almost having an equal angle of 54.73° .

In this case I have to locate this pattern in the upper scattering part of Fig. 102. Thus again

bcc would grow on top of r-hcp. The reason for the appearing contrast, however cannot be

explained sufficiently by twinning, as the stripe pattern is visible only within a single

conventional twin domain as was demonstrated particular in Fig. 70.

Therefore I suggest further that the contrast mechanism for the zig-zag pattern morphology is

not a twinning but possibly some distortion in regions of a single twin domain. To illustrate

this idea, I sketch distorted and undistorted (110) plane in Fig. 107, where I colour an

undistorted unit cell in light blue and a distorted one in light green.


134

Fig. 107. Domain distortion of binary alloy under shear. Here I assume this domain is constituted with several

(110) planes based on a bcc alloy structure.

While this clarifies the possibility of contrast within one twin domain, it still does not explain

the possibility of a bcc like structure in the wall crystal and in addition the persistence of the

zig-zag structure with changing composition. I shall therefore now look at possible structures.

Some examples are given in Fig. 108.


135

(a) (b)

(c) (d)

(a) (b)

(c) (d)

Fig. 108. Models for different structures. (a) pure component bcc structure with the (110) plane indicated by the

dashed lines. (b) random composition bcc structure, (c) AB3 structure. Note that in this case each (111) plane

contains both species. (d) AB4 structure at p68 = 0.20 with four PS100 particles forming the central particle of the

bcc structure. For p68 = 0.75, this structure may appear inverted; for p68 = 0.50, it may reduce to an AB structure.

A possible structure of the upper scattering part of the wall crystals may, of course, be a

random composition bcc (Fig. 108 (b)). With increasing density such a structure would suffer

more elastic tension due to the different geometrical properties of both species. Then a

distortion as sketched in Fig. 107 will become more favourable, a tendency in fact observed

from Fig. 70. Further there is no reason for distortion of the pure component twins, where no

zig-zag patterns are observed. On the other hand, I would expect no anomaly in the phase

boundary, but observe a strongly decreased nf at p68 = 0.20, which may indicate alloy


136

formation or at least some compositional fluctuation. I shall investigate such a possibility

now.

The formation of a regular compositional fluctuation has the formation of an alloy as the final

extreme. An AB3 alloy (Fig. 108(c)) can be excluded as the observed angles are not

compatible with an underlying hexagonal structure. A more interesting candidate would be an

AB4 structure with four PS100 particles forming a central unit and its derivates. This structure

is most favourable for p68 = 0.20, where the minimum in the freezing density was observed.

While from this stoichiometric aspect this structure may exist, one may argue, that it may not

exist for geometrical reasons, as PS100 has a larger diameter than PnBAPS68, and thus the

latter should form the central unit.

However this argument may be weakened using an effective hard sphere model. I consider the

effective particle size to be deff = d + 2-1

. If I choose to calculate this with the average -1

, I

obtain a strong mismatch for my packing. I therefore choose to introduce local -1

. In

principle, this will lead to non-spherical objects, as the local -1

on the line connecting sphere

centres will depend on the nature of the bond: AA, BB or AB. Such a calculation seems

extremely interesting, but is out of the scope of the present thesis. For a first semi-quantitative

estimate leaving the constituents spherical, I only calculate local -1

for the AA interaction

and the BB interaction. For each calculation, I use the small ion concentration which would

persist throughout each single phase of A and B, if the system would be phase separated. I.e. I

use the respective number concentrations of species A and B at fixed n and p. An example

calculation is given in the Table below. Here n is fixed to n = 6.740.87 µm-3

, thus the cubic

lattice constant constituted by the ‘effective large’ particles (here, it’s PnBAPS68) is L =

(1/n68)1/3

= 905.2642.67 nm, and p68 is fixed to p68 = 0.20. The effective radius of the

PnBAPS68 is nearly double than that of PS100.


137

-1

(nm) aeff = a+-1

(nm)

PS100, 2a=100nm

Z* = 53050

n100 =5.3920.696m-3

194.8021.85

244.8021.85

PnBAPS68, 2a = 68nm,

Z* = 450±16

n68 = 1.3480.174m-3

392.8227.81

426.8227.81

Tab. 9. local effective radii for PnBAPS68 and PS100 as calculated with Z* = 53050 and n100 = 5.3920.696

m-3

for PS100; with Z* = 450±16, n68 = 1.3480.174 m-3

. Here the whole system of the binary mixture is at n

= 6.740.87 µm-3

and p68 = 0.20.

I now try to fit these objects onto my bcc lattice. Three more sketches of crystal planes are

shown drawn to scale in Fig. 109(a). The first shows the situation, if a close packed tetraeder

of PS100 with effective diameters is formed and approximated by a single sphere

circumventing this object. Its radius was calculated as R = deff/2sin(54.73°) + deff/2. There is a

large overlap between the effective central sphere and the effective PnBAPS68 spheres. This

construction does not fit. I therefore have to further assume a specific orientation of the

tetraeder with PS100 now filling the interstitials between PnBAPS68 spheres. This is

indicated in Fig. 109(b). The view along a (100) plane for this situation is sketched in Fig.

109(c). This still does not fit perfectly, as the PS100 effective radii are still somewhat too

large, but is a significant improvement over Fig. 109(a). It may be well possible but is out of

reach of this thesis to go one step further and also include distortions of the lattice and/or local

AB interactions. Possibly this will yield an optimised packing.

At present my considerations have simply shown that a packing of effective hard spheres with

individual local screening lengths is not completely contradicting possible sphere packings for

the AB4 structure at p68 = 0.20. At larger p the ratio of effective radii, if calculated as above,

will inverse. Therefore at p68 = 0.50 nearly equal effective spheres may construct an AB

structure (c.f. Fig. 109(b), but now with nearly equal radii) and at p68 = 0.75 one may

anticipate an AB4 structure again, but now with PS100 having the much larger radius. The

intermediate non-stoichiometric cases would then have to be formed by substitution of

effective spheres and thus will be closer again to the random composition bcc. Unfortunately,

however, these compositions were not investigated. Since without the knowledge of local -1

,


138

a consequent persuasion of this local -1

effective hard sphere model is out of the scope of my

thesis, the central weak point of that compositional ordering model remains. I therefore favour

the random composition model of Fig. 108(b) for its simplicity.

21/2 L=1280 60nm

L=90543nm 2R=109098nm

21/2 L=1280 60nm

L=90543nm 2R=109098nm

21/2 L=1280 60nm

L=90543nm

21/2 L=1280 60nm

L=90543nm

L=90543nmL=90543nm

(a)

(b)

(c)

21/2 L=1280 60nm

L=90543nm 2R=109098nm

21/2 L=1280 60nm

L=90543nm 2R=109098nm

21/2 L=1280 60nm

L=90543nm

21/2 L=1280 60nm

L=90543nm

L=90543nmL=90543nm

(a)

(b)

(c)

Fig. 109. Three more sketches of crystal planes in effective hard core model by a single sphere circumventing

this object. (a) shows the situation, if a close packed tetraeder of PS100 with effective diameters is formed and

approximated. Its radius is calculated as R = deff/2sin(54.73°) + deff/2. There is a large overlap between the

effective central sphere and the effective PnBAPS68 spheres. (b) shows another specific orientation of the

tetraeder with PS100 filling the interstitials between PnBAPS68 spheres. For this situation in (100) pane is

sketched in (c).


139

In conclusion to Chap. 6.5 and Chap. 6.6, I have investigated the initial wall crystal thickness

for two binary mixtures and the corresponding pure components. I am able to confirm the

trends reported by A. Stipp and extend the measurements to a system with a more complex

phase behaviour. A common description is possible in terms of an increase of d0 with

increased undercooling.

It further is observed that the initial thickness derived from extrapolations of growth curves to

t = 0 yielded values well compatible with the extension of wall based non scattering regions

as observed by Bragg microscopy. From a careful discussion in the light of recent

microscopic shear experiments it seems well possible that these regions are composed of

(111) layers originally formed under shear and registered after abortion of shear. If that

applies, one would expect d0 to increase with increasing shear rate and increased

‘undercooling’ as it was indeed observed in Fig. 101.

Finally, I argued that the observed zig-zag morphology should be restricted to the scattering

upper part of the wall crystal and originates from crystal distortions. As most likely structure I

identified a random composition bcc lattice, but I also discussed possible realisations of

compositional fluctuations leading to an overall bcc-like structure in terms of a modified

effective hard sphere model.

Summary

140

Summary

Colloidal suspensions are valuable model systems for condensed matter and statistical physics

in general and soft matter in particular. In this thesis the phase behaviour, the morphology and

the crystallization kinetics of charged spheres in aqueous suspensions were investigated.

These were characterized by a long ranged screened Coulomb interaction which is

conveniently adjusted via the experimental parameters salt and particle concentration, c and n,

respectively. On the other hand their precise control affords a continuous advance of

preparation and characterisation techniques. In this thesis an improved empirical relation to

determine the particle density from light scattering was developed [1]. A new method to

monitor the deionisation process beyond conductivity measurements which is based on the

salt concentration dependence of crystal growth velocities was introduced [2]. For a compact

representation of the development of the interaction energy U(r)/kBT and the coupling

parameter upon changes in c or n the concept of state lines was developed and successfully

applied to facilitate detailed comparisons of observed and predicted phase behaviour [3]. With

these means suspensions of pure components and of binary mixtures were prepared and

investigated by microscopic methods and light scattering. For the pure components the c and

n dependent fluid-bcc phase boundary was observed to run parallel to the theoretical

predictions of Robbins, Kremer, Grest, and also Meijer, Frenkel. It was further observed to be

correlated to the morphological transition from columnar to sheet like wall crystal growth and

the onset of homogeneous nucleation leading to polycrystaline materials [3]. Growth

velocities were observed to follow a Wilson Frenkel law only above the coexistence region

[4, 5]. Across coexistence growth was sub-linear indicating a limitation by diffusive transport

to the interface. For sufficiently small coexistence regions the obtained differences in the

chemical potentials µ was used to interpret nucleation kinetics [5,6]. In the

PnBAPS68/PS100 mixtures I found indications of compound formation from analysis of the

Summary

141

mixture specific zig-zag domain patterns and the phase behaviour. Under the effective hard-

core model, a random bcc structure with a fluctuation of two components (possible an AB4

alloy at some n and p68) was suggested. Growth could nevertheless be understood in terms of

a Wilson-Frenkel law under the assumption of forming a random composition bcc crystal. A

detailed analysis of the increase of the initial wall crystal height d0 with increasing µ led to a

model of shear induced formation of oriented heterogeneous nuclei of stoichiometry. A

different growth scenario was observed for the PS120/PS156 mixture, which is presently not

fully understood yet. Experiments and theories to further test and develop the proposed

models and clarify the second scenarios are suggested.

[1] J. Liu, H. J. Schöpe, T. Palberg: Part. Part. Syst. Charact. 17, 206 – 212 (2000) and (E)

ibid. 18 50 (2000). An improved empirical relation to determine the particle number

density in charged colloidal fluids

[2] J. Liu, A. Stipp, T. Palberg: Prog. Colloid Polym. Sci. 118, 91 – 95 (2001).

Crystal growth kinetics in deionised two-component colloidal suspensions

[3] J. Liu, H. J. Schöpe, T. Palberg: J. Chem. Phys. 116, 5901 – 5907 (2001).

Correlations between morphology, phase behaviour and pair interaction in soft sphere

solids

[4] J. Liu, T. Palberg: Prog. Colloid Polym. Sci. 122, (accepted 2002).

Crystal growth and crystal morphology of charged colloidal binary mixtures

[5] P. Wette, H. J. Schöpe, J. Liu, T. Palberg: Prog. Colloid Polym. Sci. 122, (accepted

2002). Characterization of colloidal solids

[6] P. Wette, H. J. Schöpe, J. Liu, T. Palberg: Europhys. Lett. (revised 2003)

Solidification in model systems of spherical particles with density dependent

interactions

Appendix

142

Appendix

(Estimating the system errors of B’ and A2)

v =fDselfdinterf/d2

= C n1/3

(C is a constant)

v / v = (1/3)n/n , and I assume v / v (1/3)n/n, 110 = 0.1° = 1.74533*10-3

As

3

110

λ

2

θsin 2γ

n

, so

2

θtg

10*2.618

2

θtg

Δθ

2

3

2

θsin

2

θsinΔ

n

Δn

110

3-

110

110

1103

1103

According to v = v (1-exp(-/kBT) ,

And exp(-/kBT) 1+(-/kBT)/1!+ (-/kBT)2/2!+… 1-/kBT

so /kBT v/v

Tkv

v

Un

UnnUB'Δμ B

ff

ff

, ff

ffB

UnnU

Un

v

vTKB'

and

m

m2

n

nnAμ

The following are known

110

(°)

110f

(°)

nf

(m-3

)

110m

(°)

nm

(m-3

)

110

(°)

n

(m-3

) vm

(m/s)

v

(m/s)

Uf

(kBT) Uf

(kBT) U

(kBT)

B’ A2

PnBAPS68 0.1 64.1 6.1 64.5 6.2 91.5 15 0.67 15.9 17.16 2 15.74 2.0 1.8

PS100 0.1 53.9 3.8 56.9 4.4 68.2 7.2 0.64 6.9 19.76 2 19.26 1.6 3.4

p68=0.20 0.1 47.8 2.7 50.1 3.1 83 12.1 0.12 10.04 16.69 2 18.10 1.27 1.24

p68=0.50 0.1 56.8 4.4 57.3 4.5 87.2 13.4 0.08 10.25 18.43 2 17.11 2.23 2.09

p68=0.75 0.1 60.8 5.3 61.7 5.5 86.5 13.1 0.04 11.20 13.82 2 16.96 1.00 3.23

Therefore

Appendix

143

nf

(m-3

)

nf/nf nm

(m-3

)

nm/nm n

(m-3

)

n/n vm/vm v/v

PnBAPS68 2.5510*10-2 4.1820*10-3 2.5726*10-2 4.1493*10-3 3.8254*10-2 2.5503*10-3 1.3831*10-3 8.501*10-4

PS100 2.0031*10-2 5.2712*10-3 2.1260*10-2 4.8318*10-3 2.7841*10-2 3.8668*10-3 1.6106*10-3 1.2889*10-3

p68=0.20 1.5951*10-2 5.9079*10-3 1.7365*10-2 5.6016*10-3 3.5805*10-2 2.9591*10-3 1.8672*10-3 9.8637*10-4

p68=0.50 2.1304*10-2 4.8419*10-3 2.1563*10-2 4.7918*10-3 3.6839*10-2 2.7492*10-3 1.5973*10-3 9.1640*10-4

p68=0.75 2.3650*10-2 4.4623*10-3 2.4107*10-2 4.3830*10-3 3.6457*10-2 2.7830*10-3 1.4610*10-3 9.2767*10-4

For estimating the maximum value of B’ and A2, I do some approximation:

n n , U(r) Uf(r) 2kBT, U U , nU - nfUf nfUf,

It can be prospected from above that

nf/nf > nm/nm > n/n > n/n, vm/vm > v/v > v/v,

So I can do the approximation

(n-nm)/(n-nm) nf/nf, nm/nm nf/nf, v/v vm/vm

According to the system error formula and the above approximation,

taken PnBAPS68 as an example,

I. Calculating B’ :

109345.00958.001358.0

0958.001358.010*7489.110*8260.3

68.104

21.602551.016.1721503826.074.15

16.17

210*4.182010*3831.12

Un

ΔUnΔnUΔUnΔnU

U

ΔU

n

Δn

v

Δv2

UnnU

UnΔnUΔ

U

ΔU

n

Δn

v

Δv2

UnnU

UnnUΔ

U

ΔU

n

Δn

v

Δv

v

Δv

B'

ΔB'

56

2

22222222223-23

2

ff

2

f

2

f

2

f

2

f

2

f

2222

f

f

2

f

f

2

m

m

2

ff

2

ff

22

f

f

2

f

f

2

m

m

2

ff

ff

2

f

f

2

f

f

222

Therefore for B’=2.00 kBT, I get B’ 0.66 kBT

According to the above calculation, I can further apply the approximation

Appendix

144

2

f

2

f

2

f

f

2

f

f

2

f

2

ff

2

f

2

f

2

f

2

f

2

f

2222

f

f

2

ff

2

ff

22

f

f

2

ff

ff

2

f

f

2

f

f

222

U

U

n

Δn

n

Δn

U

ΔU

n

n2

Un

ΔUnΔnUΔUnΔnU

U

ΔU

UnnU

UnΔnUΔ

U

ΔU

UnnU

UnnUΔ

U

ΔU

n

Δn

v

Δv

v

Δv

B'

ΔB'

II. Calculating A2 :

5-

2323

2

f

f

2

m

m

2

f

f

2

m

m

2

m

m

2

m

m

2

m

m

222

2

2

10*3.8804

10*4.1820210*1.38312

n

Δn2

v

Δv2

n

Δn

n

Δn

v

Δv2

nn

nnΔ

n

Δn

v

Δv

v

Δv

A

ΔA

As A2=1.80kBT, so A2= 0.01kBT

Therefore

B’

(kBT)

A2

(kBT)

PnBAPS68 0.66 0.01

PS100 0.38 0.03

p68=0.20 0.72 0.01

p68=0.50 0.81 0.02

p68=0.75 0.41 0.02

Acknowledgement

145

Acknowledgement

Especially, I would like to thank my project leader and Professor Thomas Palberg. Every step

of my progress in the thesis was supported by his talent guide and deep thinking.

Also I would like to deeply thank the kind financial support from DFG (Deutsche

Forschungsgemeinschaft), Prof. Heinz Decker in Institut für Molecular Biophysik, Prof.

Wolfgang Knoll in Max-Planck-Institut für Polymerforschung during this thesis.

Very many thanks to Dr. H. J. Schöpe, Dr. R. Biehl, Dr. M. Evers, A. Stipp, R. Niehüser, T.

Preis for their kind help in my PhD study. Thanks to M. Medbach, H. Reiber, and P. Wette for

the discussion and collaboration.

Many thanks to Prof. H. Löwen, Prof. O. Glatter, Prof. K. A Dawson, Prof. J. O. Rädler, Dr.

E. Weeks, Dr. U. Felderhof, Dr. G. Nägele, Dr. J. Haucker, Dr. E. Bartsch, Dr. W. Erker et al.

scientists who are very nice to provide me their favourable idea in discussion during my PhD.

Very many thanks to Dr. T. Kreer for his careful English checking.

146

Reference

1 R. J. Hunter, Foundations of colloidal science, Vol.1, Clarendon Press, Oxford (1986)

2 E. J. W. Verwey and J. Th. G. Overbeek, Theory of the stability of Lyophobic Colloids (Elsevier, New York,

1948).

3 J. Th. G. Overbeek, in “Physics of Comlex and Supermolecular Fluids” (S. A: Safran and N. A. Clark, eds.). p.

3. Wiley, New York, 1987.

4 B. V. Derjaguim, L. D. Landau, Acta Physicochim.USSR 14, 633 (1941)

5 A. D. Dinsmore, J. C. Crocker, and A. G. Yodh, Curr. Opin. Colloid Interf. Sci. 3, 5 (1998).

6 A. P. Gast and W. B. Russel, Phys. Today 51, No. 12, 24 (1998).

7 P. N. Pusey and W. van Megan, Nature (London) 320, 340 (1986)

8 B. J. Ackerson and P. N. Pusey, Phys. Rev. Lett. 61, 1033 (1988).

9 A. van Blaaderen, R. Ruel, and P. Wiltzius, Nature, 385, 321 (1997).

10 R. Micheletto, H. Fukuda, and M. Ohtsu, Langmuir, 11, 3333 (1995)

11 N. D. Denkov , O. D. Velev, P. A. Kralchevsky, I. B. Ivanov, H. Yoshimura, K. Nagoyama, Nature, 361, 26

(1993)

12 I. Peterson, Science 148, 296 (1995)

13 Peter A. Kralchevsky and K. Nagayama, Langmuir 10, 23 (1994)

14 M. Holgado, F. García-Santamaría, A. Blanco, M. Ibisate, A. Cintas, H. Míguez, C. J. Serna, C.

Molpeceres, J. Requena, A. Mifsud, F. Meseguer, and C. López et al., Langmuir 15, 4701 (1999)

15 K. H. Lin, J. C. Crocker, V. Prasad, A. Schofield, D. A. Weitz, Phys.Rev. Lett. 85, 1770 (2000)

16 I. I. Tarhan and G. H. Watson, Phys. Rev. Lett. 76, 315 (1996)

17 A. Imhof and D. J. Pine, Nature 389, 948 (1997)

18 J. E. G. J. Wijnhoven and W. L. Vos, Science 281, 802 (1998)

19 O. D. Velev, P. M. Tessier, A. M. Lenhoff, E. W. Kaler, Nature (London) 401, 548 (1999)

20 A. Y. Vlasov, N. Yao, and D. J. Norris, Adv. Mater. 11, 165 (1999)

21 G. Subramania, K. Constant, R. Biswas, M. M. Sigalas, K.-M. Ho, Appl. Phys. Lett. 74, 3933 (1999)

22 E. Yablonovitch, Nature (London) 401, 539 (1999).

23 W. Luck, M. Klier, and H. Wesslau, Ber. Bunseges, Phys. Chem. 67, 75 (1963)

24 I. M. Krieger and P.A. Hiltner, in “Polymer Colloids” , R. M. Fitch, ed, p.63. Plenum, New York, 1971

25 P. A. Hiltner, and I. M. Krieger, J. Phys. Chem. 73, 2686 (1969)

26 S. Hachisu, Y. Kobayashi, and A. Kose, J. Coll. Interface Sci. 42, 342 (1973)

27 S. Hachisu, and Y. Kobayashi, J. Coll. Interface Sci. 46, 470 (1974)

147

28

E. B. Sirota, H. D. Ou-Yang, S. K. Sinha, P. M. Chaikin, J. D. Axe, Y. Fujii, Phys. Rev. Lett. 62, 1524 (1989).

29 A. K. Sood: Solid State Physics 45, 1 (1991).

30 D. J. W. Aastuen, N. A. Clark, J. C. Swindal, C. D. Muzny in B. J. Ackerson (ed.): Phase Transitions 21 (2-4),

pp139, (1990).

31 Atelier d´Hiver sur les Cristeaux colloideaux: J. Phys. (France) C-3, 46 (Les Houches 1985).

32 T. Palberg, J. Phys: Condens Matt 11, R323 (1990)

33 B. J. Alder, T. E. Wainwright, J. Chem. Phys. 27, 1208 (1957).

34 P. N. Pusey, and W. van Megen, Phys. Rev. Letters, 59, 2083 (1987b)

35 P. Bartlett, W. v. Megen in: A. Mehta, Granular Matter 195. (Springer New York, 1994),

36 W. van Megen: Transport Theory and Statistical Phys. 24, 1017 (1995).

37 E. Bartsch: Transport Theory and Statistical Phys. 24, 1125 (1995).

38 J. K. G. Dhont, C. Smits, H. N. W. Lekkerkerker: J. Coll. Interface Sci. 152, 386 (1992)

39 H. N. W. Lekkerkerker, J. K. G. Dhont, H. Verduin, C. Smits, J. S. van Duijneveldt: Physica A 213, 18(1995).

40 U. Gasser, E. Weeks, A. Schofield, P. N. Pusey, D. A. Weitz, Science 292, 258 (2001)

41 A. P. Gast, Y. Monovoukas, Nature 351, 552 (1991)

42 Y. Monovoukas, A. P. Gast: J. Coll. Interface Sci. 128, 533 (1989).

43 M. Würth, J. Schwarz, F. Culis, P. Leiderer, and T. Palberg, Phys. Rev. E 52, 6415 (1995)

44 L. P. Voegtli, C. F. Zukoski IV: J. Coll. Interface Sci. 141, 79 (1991). Pasendiagramme per Störungsrechnung

45 M. Robbins, K. Kremer and G. S. Grest, J. Chem. Phys., 88, 3286 (1988)

46 E. J. Meijer and D. Frenkel, J. Chem. Phys. 94, 2269 (1991)

47 T. Palberg, W. Mönch, F. Bitzer, L. Bellini, R. Piazza, Phys. Rev. Lett. 74, 4555 (1995)

48 F. El. Azhar, M. Baus, J.-P. Ryckaert, J. Chem. Phys. 112, 5121 (2000)

49 F. A. Lindemann, Physik. Zeitschr. 11, 609 (1910)

50 L. P. Voegtli and C. F. Zukoski, J. Coll. Interface Sci. 141, 92 (1991).

51 Y. He, B. J. Ackerson, W. van Megen, S. M. Underwood, K. Schätzel: Phys. Rev. E 54, 5286 (1996).

52 B. J. Ackerson, K. Schätzel: Phys. Rev. E 52, 6448 (1995).

53 J. L. Harland, W. van Megen: Phys. Rev. E 55, 3054 (1996).

54 A. Heymann, A. Stipp, Chr. Sinn, T. Palberg: J. Coll. Interface Sci. 207, 119 (1998).

55 W. v. Megen: J. Phys. Condens. Matter 14, 7699 (2002).

56 V. J. Anderson, H. N. W. Lekkerkerker, Nature 416, 811 (2002)

57 K.G. Jan Dhont, M.P. Lettinga, Z. Dogic, T. A.J.Lenstra, H. Wang, S. Rathgeber, P. Carletto, L. Willner, H.

Frielinghaus and P. Lindner, Faraday Discuss. 123, 1 (2002), 58

D. J. W. Aastuen, N. A. Clark, L. K. Cotter, and B. J. Ackerson, Phys. Rev. Lett. 57, 2772 (1986)

59 H. -J. Schöpe, T. Palberg: J. Non-Cryst. Mater 307, 613 (2002).

60 M. R. Maaroufi, A. Stipp, T. Palberg: Prog. Colloid Polym. Sci. 108; 83 (1998).

61 E. Burke, J. Q. Broughton, G. H. Gilmer: J. Chem. Phys. 89, 1030 (1988)

62 H. A. Wilson, Philos Mag 50: 238 (1900) and J. Z. Frenkel, Sowjetunion 1, 498(1932)

63 P. Wette, H. J. Schöpe, J. Liu, T. Palberg: Europhys. Lett. (submitted 2002)

64 W. G. T. Kranendonk, D. Frenkel, Molecular Phys.72, 715 (1991)

65 W. G. T. Kranendonk, D. Frenkel, Molecular Phys.72, 679 (1991)

148

66

H. M. Lindsay, P.M. Chaikin, J. Chem. Phys. 76, 3774 (1982).

67 W. Härtl and H. Versmold, J. Chem. Phys. 80, 1387 (1984)

68 R. Kesavamoorthy, A. K. Sood, B. V. R. Tata and A. K. Arora, J. Phys.C: Solid State Phys. 21, 4737 (1988)

69 S. Yoshimura, and S. Hachisu, J. Phys., Paris, 46, Colloque 3, 115 (1985)

70 T. Okubo, J. Chem. Phys. 87, 5528 (1987)

71 B. V. R. Tata, R. Kesavamoorthy and A. K. Sood, Mol. Phys. 61, 943 (1987)

72 W. H. Shih, D. Stroud, J. Chem. Phys., 80, 4429 (1984)

73 R. Kesavamoorthy and A.K. Arora, J. Phys. C, 19, 2833 (1986)

74 J. M. Méndez-Alcaraz, B. D’Aguanno and R. Klein, Physica A, 178, 421 (1991)

75 S. Hachisu, S. Yoshimura, “Physics of complex and supermolecular fluids” (Ed. S. A. Safran and N. A. Clark)

Published by John Wiley & Sons. Inc. (1987)

76 J. V. Sanders and M. J. Murray, Nature 275, 201 (1978)

77 P. Wette, H. J. Schöpe, T. Palberg: Prog. Colloid Polym. Sci. 118, 260 (2001).

78 A. Meller, J. Stavans, Phys. Rev. Lett. 68, 3646 (1992)

79 R. J. Hunter, S. K. Nicol, J. Coll. Interface Sci. 28, 250 (1968)

80 G. Gouy, J. Phys. Radium 9, 457 (1910)

81 D. L. Chapman, Phil. Mag. 25, 457 (1913)

82 P. W. Debye, E. Hückel, Phys. Z. 24, 185 (1923)

83 J. N. Israelachvili, Intermolecular and surface forces, Academic Press (1985)

84 D. A. Haydon, “The electrical double-layer and electrokinetic phenomena”, in Progress an Surface Science,

p94-158, Academic Press (1964).

85 G.M. Torrie, J. P. Valleau, Chem. Phys. Lett. 65, 343 (1979)

86 B. Jonsson, H. Wennerstrom, B. Halle, J. Phys. Chem., 84, 2179 (1980)

87 G. M. Bell, S. Levine, and L. N. McCartney, J. Coll. Interface Sci. 33, 335 (1970)

88 H. J. van den Hul and J. W. Vanderhoff, in “Polymer Colloids”, R. M. Fitch, Edited, p.1. Plenum, New York,

1971.

89 J. Lykelema, “Foundamentals of interface and colloidal science” Volume II (Solid-Liquid Interfaces), ISBN:

0-12-460524-9, Academic Press Ltd. Harcourt Brace, Publishers, London, United Kingdom (1995)

90 S. Alexander, P. M. Chaikin, P. Grant, G. J. Morales, P. Pincus, and D. Hone, J. Chem. Phys., 101, 9924

(1994)

91 L. Belloni, Colloids Surf. A, 140, 227 (1998)

92 L. Belloni, M. Drifford, and P. Turq, Chem. Phys. 83, 147 (1984)

93 W. Härtl, H. Versmold, J. Chem. Phys. 88, 7157 (1988)

94 B. Beresford-Smith, D. Y. C. Chan, D. J. Mitchell, J. Coll. Interface Sci., 105, 216 (1985)

95 S. H. Chen, Annu. Rev. Phys. Chem. 37, 351 (1986).

96 W. Härtl, H. Versmold, U. Wittig, Langmuir 8, 2885 (1992)

97 D. Hessinger, M. Evers, T. Palberg, Phys. Rev. E 61, 5493 (2000)

98 P. Wette, H.-J. Schöpe, R. Biehl, T. Palberg, J. Chem. Phys. 114, 7556 (2001)

99 M. Evers, N. Garbow, D. Hessinger and T. Palberg, Phy. Rev. E57, 6774 (1998)

149

100

T. Palberg, H. J. Schöpe, P. Wette, N. Garbow, M. Medbach, „Experimental determination of colloidal

effective charges“ Poster in Liquid Matter Conference, Konstanz, Sep. 2002

101 S. Alexander, P. M. Chaikin, P. Grant, G. J. Morales, and P. Pincus, J. Chem. Phys. 80, 5776 (1984)

102 B. V. Derjaguin, L. Landau, Acta Physicochim, (USSR) 14, 633 (1941)

103 W. H. Shih, D. Stroud, J. Chem. Phys., 79, 6254 (1982)

104 J. O. M. Blockris, A. K. Reddy, “Modern Electrochemistry”, Plenum, New York, vol. I, Chap. 3, (1970)

105 R.O. Rosenberg, D. Thirumalai, Phys. Rev. A36, 5690 (1987)

106 J. Liu, H. J. Schöpe, T. Palberg, J. Chem. Phys., 116, 5901 (2001)

107 J. C. Crocker and D. G. Grier, Phys. Rev. Lett., 73, 352 (1994)

108 J. C. Crocker, D. G. Grier, Phys. Rev. Lett. 77, 1897 (1996)

109 A. E. Larsen and D. G. Grier, Phys. Rev. Lett. 76, 3862 (1996)

110 J. S. Rowlinson, Mol. Phys. 52, 567 (1984)

111 D. Y. C. Chan, Phys. Rev. E 63, 061806 (2001)

112 R. van Roij, J. P. Hansen, Phys. Rev. Lett. 79, 3082 (1997)

113 R. van Roij, M. Dijkstra, J. P. Hansen, Phys. Rev. E 59, 1010 (1999)

114 P. B. Warren, J. Chem. Phys. 112, 4683 (2000)

115 M. J. Grimson, M. Silbert, Mol. Phys. 74, 397 (1991)

116 H. Graf and H. Löwen, Phys. Rev. E57, 5774 (1998)

117 R. Tehver, F. Ancilotto, F. Toigo, J. Koplik, J. R. Banavar, Phys. Rev. E59, R1335 (1999)

118 J. Dobnikar, R. Rzehak, H. H. von Grünberg, Europhys. Lett. (submitted) (2002)

119 J. J. Gray, B. Chiang, R. T. Bonnecaze, Nature, 402, 139 (1999)

120 J.-P. Hansen, I. R. McDonald, „ The theory of simple liquids “, London, Academic Press

121 J. K. G. Dhont, “An introduction to dynamics of colloids”, Elsevier Science, 1996

122 T. Palberg, W. Härtl, U. Wittig, H. Versmold, M. Würth, E. Simnacher,. J. Phys. Chem. 96, 8180 (1992)

123 D. R. Lide (Editor), “Handbook of Chemistry and Physics”, 73

rd ed., CRC_press, Boca Raton (1992).

124 H. -J. Schöpe, Th. Decker, T. Palberg: J. Chem. Phys. 109, 10068 (1998).

125 B. D. Cullity, Elements of X-Ray Diffraction, Addison Wesley, Reading, Massachusetts, Menlo Park,

California, London, Amsterdam, 1978

126 C. Hammond, The Basics of Crystallography and Diffraction, Oxford University Press, Oxford, New York,

1998

127 M. A. Krivoglaz, X-Ray and Neutron Diffraction in Nonideal Crytals, Springer, Berlin, 1996

128 K. Schätzel, “Single-photon correlation techniques in Dynamic Light Scattering”, Edi. by W. Brown,

Oxford, Clarendon Press, 1993

129 R. Peters, “Noise on photon correlation function and its effects on data reduction algorithms in Dynamic

Light Scattering”, edited by W. Brown, Oxford, Clarendon Press, 1993

130 H.- J. Schöpe, „Physikalische Eigenschaften kolloidaler Festkörper“, PhD Dissertation, Universität Mainz,

2001

131 Coreco, User’s Manual for Oculus TCi-SE

132 T. Palberg, W. Härtl. M. Deggelmann, E. Simnacher, R. Weber, Prog. Colloid. Polym. Sci. 84, 352 (1991)

133V. W. Luck, M. Klier, H. Wesslau, Naturwissenschaften 50, 485 (1963)

134 H. Lange, Part. Part. Syst. Charact. 12, 148 (1995).

150

135

M. K. Udo, M. F. de Souza, Solid State Comm.35 (12), 907 (1980)

136 P. N. Pusey in J. P. Hansen, D. Levesque, J. Zinn-Justin, Liquids, freezing and glass transition, 51st summer

school in theoretical physics, Les Houches (F) 1989, pp 763, (Elsevier, Amsterdam 1991)

137 A. K. Arora, B. V. R. Tata, “Ordering and phase transitions in charged colloide”, p18, VCH Publishers, New

York, 1996

138 B. V. R. Tata, E. Yamahara, P. V. Rajamani, N. Ise, Phys. Rev. Lett. 78, 2660 (1997).

139 A. E. Larsen, D. A. Grier, Nature 385, 230 (1997).

140 P. D. Kaplan, J. L. Rouke, A. G. Yodh, D. J. Pine, Phys. Rev. Lett. 72, 582 (1994).

141 G. Nägele, Phys. Rep. 272, 215(1996)

142 D. W. Schaefer, J. Chem. Phys. 66,3980 (1977).

143 M. Deggelmann, T. Palberg, M. Hagenbüchle, E. E. Maier, R. Krause, Ch. Graf, R. Weber:, J. Coll. Interface

Sci. 143, 318 (1991)

144 B. E. Warren: X-Ray Diffraction (Massachusetts: Addison-Wesley) (1969).

145 R. Kesavamoorthy, A. K. Arora, J. Phys. A: Math. Gen. 18, 3389 (1985).

146 L. K. Cotter, N. A. Clark: J. Chem. Phys. 86, 6616 (1987).

147 K. Kremer, G. S. Grest, M. O. Robbins, Phys. Rev. Lett. 57, 2694 (1986)

148 R. Hultgren, R. L. Orr, P. D. Anderson, and K. K. Kelley, Selected values of thermodynamic properties of

metals and alloys , Wiley-VCH, New York (1063)

149 D. Frenkel, and A. J. C. Ladd, J. Chem. Phys. 81, 3188 (1984)

150 D. Frenkel, Phys. Rev. Lett. 56, 858 (1984)

151 D. A. Young and B. J. Alder, J. Chem. Phys. 60, 1254 (1974)

152 J. P. Hansen and L. Verlet, Phys. Rev. 184, 151 (1969)

153 H. Löwen, T. Palberg, R. simon, Phys. Rev. Lett. 70, 1557 (1993).

154 F. Bitzer, T. Palberg, H. Löwen, R. Simon, P. Leiderer, Phys. Rev. E 50, 2821 (1994)

155 D. A. Kofke, P. G. Bolhuis, Phys. Rev. E 59, 618 (1999).

156 P. Wette, H. J. Schöpe, J. Liu, Phys.. Rev. Lett. (2002 submitted)

157 P. Wette, H. J. Schöpe, J. Liu, T. Palberg, Progr. Colloid. & Interf. Sci. (2002 accepted)

158 A. H. Chokshi, A. Rosen, J. Karch, H. Gleiter, Scripta Metall. Mater. 23, 1679 (1989)

159 G. E. Fougere, J.R. Weertman, R.W. Siegel, S. Kim, Scripta Metall. Mater. 26, 1879 (1992)

160 K. Lu, W.D. Wei, J.T. Wang, Scripta Metall. Mater. 24, 2319 (1990).

161 W. Wunderlich, Y. Ishida, R. Maurer, Scripta Metall. Mater. 24, 403. (1990)

162 W. Qin, Z.H. Chen, P.Y. Huang, Y.H. Zhuang, J. Alloys. Comp. 292, 230 (1999)

163 W. Qin , Y.W. Du , Z.H. Chen , W.L. Gao, J. Alloys. Comp. 337, 168 (2002)

164 T. Riste, and D. Sherrington (Edited), “Phase transitions in soft condensed matter”, Plenum Press, New Yor

and London, p23 (1989)

165 H.-J. Schöpe, Poster and report in DPG (2002)

166 S. Hachisu, S. Yoshimura, Nature 283, 188 (1980).

167 S. Yoshimura, S. Hachisu, Prog. Colloid. & Interf. Sci. 68, 59 (1983)

168 P. Bartlett, R. H. Ottewill, J. Chem. Phys. 96, 3306 (1992)

169 D. J. W. Aastuen, N. A. Clark, L. K. Cotter, and B. J. Ackerson, Phys. Rev. Lett. 57, 1733 (1986)

170 T. Palberg, M. Würth, R. Simon, and P. Leiderer, Prog. Colloid. Polys. Sci., 96,62 (1994)

151

171

W. van Megan, Transport Theory Stat. Phys. 24, 1017 (1995)

172 N. F. Carnahan, K. E. Starling, J. Chem. Phys.,51, 635 (1979)

173 K. R. Hall, J. Chem. Phys., 57, 2252 (1972)

174 A. Stipp, A. Heymann, Chr. Sinn, T. Palberg: Prog. Colloid Polym. Sci. 118, 266 (2001).

175 R. Biehl, T. Palberg: Proc. Roy. Chem. Soc. Faraday Disc. 123, 133 (2003).

176 A. B. D. Brown, A. R. Rennie, Chem. Engi. Sci. 56, 2999 (2001)

177 R. Biehl, „ Optische Mikroskopie an kolloidalen Suspensionen unter Nichtgleichgewichtsbedingungen“, PhD

Thesis, Uni. Mainz.

178 M. R. Maaroufi, A. Stipp, T. Preis, T. Palberg, Sci. Tech. Information CDR 5, 35 – 51 (2001).

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