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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
Chapter 1. Introduction to colloids, colloidal phase behaviour and crystallization kinetics
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
Chapter 1. Introduction to colloids, colloidal phase behaviour and crystallization kinetics
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.
Chapter 1. Introduction to colloids, colloidal phase behaviour and crystallization kinetics
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.
Chapter 1. Introduction to colloids, colloidal phase behaviour and crystallization kinetics
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).
Chapter 1. Introduction to colloids, colloidal phase behaviour and crystallization kinetics
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.
Chapter 1. Introduction to colloids, colloidal phase behaviour and crystallization kinetics
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
.
Chapter 1. Introduction to colloids, colloidal phase behaviour and crystallization kinetics
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
.
Chapter 1. Introduction to colloids, colloidal phase behaviour and crystallization kinetics
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).
Chapter 1. Introduction to colloids, colloidal phase behaviour and crystallization kinetics
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.
Chapter 1. Introduction to colloids, colloidal phase behaviour and crystallization kinetics
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
.
Chapter 1. Introduction to colloids, colloidal phase behaviour and crystallization kinetics
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)
Chapter 1. Introduction to colloids, colloidal phase behaviour and crystallization kinetics
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
.
Chapter 1. Introduction to colloids, colloidal phase behaviour and crystallization kinetics
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
.
Chapter 1. Introduction to colloids, colloidal phase behaviour and crystallization kinetics
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
Chapter 1. Introduction to colloids, colloidal phase behaviour and crystallization kinetics
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.
Chapter 2. Colloidal interaction
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
Chapter 2. Colloidal interaction
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
Chapter 2. Colloidal interaction
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.
Chapter 2. Colloidal interaction
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.
Chapter 2. Colloidal interaction
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.
Chapter 2. Colloidal interaction
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.
Chapter 2. Colloidal interaction
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]
Chapter 2. Colloidal interaction
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.
Chapter 2. Colloidal interaction
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.
Chapter 2. Colloidal interaction
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.
Chapter 2. Colloidal interaction
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
Chapter 2. Colloidal interaction
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
.
Chapter 2. Colloidal interaction
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
Chapter 2. Colloidal interaction
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.
Chapter 2. Colloidal interaction
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.
Chapter 2. Colloidal interaction
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
Chapter 2. Colloidal interaction
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.
Chapter 2. Colloidal interaction
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.
Chapter 3. Experimental techniques and corresponding theories
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
Chapter 3. Experimental techniques and corresponding theories
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
Chapter 3. Experimental techniques and corresponding theories
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
Chapter 3. Experimental techniques and corresponding theories
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
Chapter 3. Experimental techniques and corresponding theories
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,
Chapter 3. Experimental techniques and corresponding theories
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.
Chapter 3. Experimental techniques and corresponding theories
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]
Chapter 3. Experimental techniques and corresponding theories
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.
Chapter 3. Experimental techniques and corresponding theories
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.
Chapter 3. Experimental techniques and corresponding theories
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
Chapter 3. Experimental techniques and corresponding theories
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
.
Chapter 3. Experimental techniques and corresponding theories
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
.
Chapter 3. Experimental techniques and corresponding theories
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
Chapter 3. Experimental techniques and corresponding theories
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.
Chapter 3. Experimental techniques and corresponding theories
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.
Chapter 3. Experimental techniques and corresponding theories
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
Chapter 3. Experimental techniques and corresponding theories
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.
Chapter 3. Experimental techniques and corresponding theories
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
Chapter 3. Experimental techniques and corresponding theories
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
Chapter 3. Experimental techniques and corresponding theories
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.
Chapter 4. Further developments for precise sample preparation
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
Chapter 4. Further developments for precise sample preparation
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
.
Chapter 4. Further developments for precise sample preparation
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.
Chapter 4. Further developments for precise sample preparation
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.
Chapter 4. Further developments for precise sample preparation
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
Chapter 4. Further developments for precise sample preparation
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).
Chapter 4. Further developments for precise sample preparation
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
Chapter 4. Further developments for precise sample preparation
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
Chapter 4. Further developments for precise sample preparation
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
Chapter 4. Further developments for precise sample preparation
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
Chapter 4. Further developments for precise sample preparation
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.
Chapter 4. Further developments for precise sample preparation
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
Chapter 4. Further developments for precise sample preparation
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.
Chapter 4. Further developments for precise sample preparation
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
Chapter 4. Further developments for precise sample preparation
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
Chapter 4. Further developments for precise sample preparation
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
Chapter 5. Fluid-crystal phase transition, crystal morphology transition and phase diagram
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.
Chapter 5. Fluid-crystal phase transition, crystal morphology transition and phase diagram
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
Chapter 5. Fluid-crystal phase transition, crystal morphology transition and phase diagram
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
Chapter 5. Fluid-crystal phase transition, crystal morphology transition and phase diagram
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
Chapter 5. Fluid-crystal phase transition, crystal morphology transition and phase diagram
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
Chapter 5. Fluid-crystal phase transition, crystal morphology transition and phase diagram
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.
Chapter 5. Fluid-crystal phase transition, crystal morphology transition and phase diagram
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
Chapter 5. Fluid-crystal phase transition, crystal morphology transition and phase diagram
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
Chapter 5. Fluid-crystal phase transition, crystal morphology transition and phase diagram
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
Chapter 5. Fluid-crystal phase transition, crystal morphology transition and phase diagram
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).
Chapter 5. Fluid-crystal phase transition, crystal morphology transition and phase diagram
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
Chapter 5. Fluid-crystal phase transition, crystal morphology transition and phase diagram
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.
Chapter 5. Fluid-crystal phase transition, crystal morphology transition and phase diagram
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.
Chapter 5. Fluid-crystal phase transition, crystal morphology transition and phase diagram
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
Chapter 5. Fluid-crystal phase transition, crystal morphology transition and phase diagram
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
Chapter 5. Fluid-crystal phase transition, crystal morphology transition and phase diagram
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.
Chapter 5. Fluid-crystal phase transition, crystal morphology transition and phase diagram
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).
Chapter 5. Fluid-crystal phase transition, crystal morphology transition and phase diagram
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.
Chapter 5. Fluid-crystal phase transition, crystal morphology transition and phase diagram
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
Chapter 5. Fluid-crystal phase transition, crystal morphology transition and phase diagram
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.
Chapter 5. Fluid-crystal phase transition, crystal morphology transition and phase diagram
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
Chapter 5. Fluid-crystal phase transition, crystal morphology transition and phase diagram
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)
Chapter 5. Fluid-crystal phase transition, crystal morphology transition and phase diagram
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-
Chapter 5. Fluid-crystal phase transition, crystal morphology transition and phase diagram
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.
Chapter 5. Fluid-crystal phase transition, crystal morphology transition and phase diagram
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.
Chapter 5. Fluid-crystal phase transition, crystal morphology transition and phase diagram
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.
Chapter 6. Crystal growth kinetics
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
Chapter 6. Crystal growth kinetics
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.
Chapter 6. Crystal growth kinetics
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.
Chapter 6. Crystal growth kinetics
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
.
Chapter 6. Crystal growth kinetics
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
Chapter 6. Crystal growth kinetics
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
Chapter 6. Crystal growth kinetics
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
Chapter 6. Crystal growth kinetics
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
Chapter 6. Crystal growth kinetics
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.
Chapter 6. Crystal growth kinetics
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
Chapter 6. Crystal growth kinetics
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
Chapter 6. Crystal growth kinetics
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.
Chapter 6. Crystal growth kinetics
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
Chapter 6. Crystal growth kinetics
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)
Chapter 6. Crystal growth kinetics
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)
Chapter 6. Crystal growth kinetics
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)
Chapter 6. Crystal growth kinetics
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).
Chapter 6. Crystal growth kinetics
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
Chapter 6. Crystal growth kinetics
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.
Chapter 6. Crystal growth kinetics
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.
Chapter 6. Crystal growth kinetics
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
Chapter 6. Crystal growth kinetics
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
Chapter 6. Crystal growth kinetics
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.
Chapter 6. Crystal growth kinetics
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
Chapter 6. Crystal growth kinetics
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.
Chapter 6. Crystal growth kinetics
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.
Chapter 6. Crystal growth kinetics
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)
Chapter 6. Crystal growth kinetics
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.
Chapter 6. Crystal growth kinetics
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.
Chapter 6. Crystal growth kinetics
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.
Chapter 6. Crystal growth kinetics
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
Chapter 6. Crystal growth kinetics
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.
Chapter 6. Crystal growth kinetics
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
.
Chapter 6. Crystal growth kinetics
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
.
Chapter 6. Crystal growth kinetics
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
.
Chapter 6. Crystal growth kinetics
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.
Chapter 6. Crystal growth kinetics
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.
Chapter 6. Crystal growth kinetics
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
Chapter 6. Crystal growth kinetics
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.
Chapter 6. Crystal growth kinetics
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
,
Chapter 6. Crystal growth kinetics
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).
Chapter 6. Crystal growth kinetics
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
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