Dynamic organic templating as a means of controlling sizes and shapes of colloidal semiconductor nanocrystals

Chapter 1
Introduction
1.1 Introduction and Motivation
One of the main interests of current research in science is the understanding and the
engineering of condensed matter at low dimensions.
1-7
 Low dimensional materials lie between
bulk matter and molecular and atomic sizes, in a size scale of nanometers. Materials in this
regime exhibit novel physical and chemical properties. Research devoted to this field is
interdisciplinary, covering physics, chemistry, biology, and many areas of technological interest.
Among these materials, semiconductors are nowadays the most investigated, mainly because
they are greatly affected by the consequences of low dimensionality.
4,8
The aim of this thesis is to give a contribution to the existing preparation techniques for
colloidal nanocrystals, in particular of II-VI semiconductors. Research in this area is intense, but
much has still to be done, as we will be able to understand when the discussion will proceed
through the next chapters.
1. Introduction
2
1.2 The Description of Solids at Low Dimensions
In a metal, electrons and holes are generally free to move along the whole structure or, in
other words, they are delocalized. The same happens in a semiconductor, provided electrons are
given sufficient energy to jump into the conduction band. From the point of view of a physicist,
electrons and holes in a solid are described as a superposition of plane waves, extended
throughout the solid. By means of theoretical models, it is possible to calculate the band
structure of the solid and to predict some of its physical properties. Two basic assumptions, the
conservation of translational symmetry and the neglect of contribution from the surface by
assuming an infinite solid (periodic boundary conditions), tremendously reduce the complexity
of the theoretical problem.
9
When one or more dimensions of the solid become smaller than the wavelengths
associated to these carriers, an additional contribution of energy is required to confine the
component of the motion of the carriers along this dimension. This extra energy is called
confinement energy and can be evaluated from the quantum mechanical model of the particle in
a box. Confinement of carriers in one or more directions introduces dramatic changes in
electronic structures of the solid.
10
 It is also responsible for a number of new physical effects,
such as luminescence from porous silicon,
11
 fractional quantum Hall effect,
12
 metal-insulator
transition,
13
 to cite some examples. In a low dimensional structure and especially in
nanocrystals, the assumptions of translational symmetry and infinity of the crystal are not valid
any more. New theoretical approaches have been developed in order to understand the behavior
of matter at low dimensions.
14-19
 We will not go into the details of these methods. Instead, we
will try to give a simple explanation of ‘what happens to a solid’ when its dimensions shrink.
From the point of view of a chemist, the electronic structure of solids can be described in
terms of combinations of atomic orbitals, as is done with molecules.
20-22
 When several atomic
orbitals are combined, the resulting orbitals are merged into bands. If, in the ground state,
electrons are partially occupying one band, the solid has a metal character. In this case, the
Fermi level lies inside the band. If two bands are separated by an energy gap and electrons are
fully occupying the lower band, the resulting solid can be either an insulator or a semiconductor,
depending on how big is the gap. When this happens, the Fermi level lies in the middle of the
gap. If one or more of the dimensions of the solid is extremely thin, of the order of few
nanometers, the density of electronic states changes. The spacing between the allowed energy
1. Introduction
3
levels in the bands increases, because fewer levels are now present: this is more dramatic at the
edges of the bands. The change influences semiconductors more than metals, as in
semiconductors electronic properties are strongly related to the edges of the bands. The
corresponding densities of states in a semiconductor show different features, depending on how
many dimensions collapse (figure 1.1, b-d). In the limiting case, when all dimensions shrink, the
bands converge again to atom-like energy states, with the oscillator strength compressed into
few transitions (fig 1.1d).
Figure 1.1. Density of electronic states in a bulk solid (a) and in
quantum confined structures (b-d)
1. Introduction
4
1.3 Scaling Laws
In low dimensional systems, surface atoms correspond to a non-negligible percentage of
the whole number of atoms. These atoms have a chemical environment completely different
from the ones in the interior of the structure: for instance they have unsaturated valences. The
combination of low dimensionality and the contribution from the surface are responsible for the
existence of some scaling laws,
23
 summarized as follows.
i) Size dependence of the optical transitions. One of the most striking effects in low
dimensional materials (and remarkably of 0D semiconductors), is the widening of the gap
between the highest occupied electronic states (the top of the original valence band) and the
lowest unoccupied states (the bottom of the original conduction band). In addition, the oscillator
strength for transitions between these states increases. The widening of the gap can be easily
modeled in terms of quantum confinement of the electron and the hole in a box. From this
simple picture a size dependence of the energy gap over 1/r
2
 is found (r is the size of the
dimension that shrinks), in approximate accordance with experiments.
19
ii) Size dependence of the melting temperature. In every phase, surface atoms have a
higher energy than bulk atoms. This is readily seen in liquid were surface tension can be easily
measured. In a dynamic environment, like in liquids, surface atoms tend to decrease their energy
by continuously diffusing into the bulk. In solids, surface atoms have a very high energy and
cannot modify their status by diffusion. When surface atoms correspond to a significant amount
of the whole object, their contribution to the overall behavior of the solid is remarkable. In
nanocrystalline solids, the melting temperature lowers.
4,24,25
 This happens because surface atoms
prefer to reach earlier the liquid state, where can minimize their energy. The decrease in melting
temperature is approximately dependent on the inverse of nanocrystals radius.
iii) Size dependence of the structural phase transitions. When a pressure is progressively
applied to a crystal a structural phase transition can occur: the crystal becomes more stable in a
new, more compact phase. Real crystals have a certain number of defects (point, linear and
planar). It is documented that structural phase transitions nucleate on one or more of these
defects and then propagate along the solid. Defects act as catalyst for the phase transition. This
mechanism contributes to significantly lower the pressure that must be applied in order for the
phase transition to occur. Nanocrystals are usually so small that the probability of occurrence of
1. Introduction
5
defects is very low. Phase transitions occur via different mechanisms and require higher
pressures.
4,23,26-28
iv) Charging energy. In an extended solid, the addition of one extra charge, i.e. an
electron, does not lead to any significant change in the electronic structure. This in turn does not
influence the further addition of another electron. In highly confined structures, as molecules,
nanocrystals,
29-31
 nanowires,
32
 and nanotubes,
33
 the increasing coulomb repulsion caused by the
successive addition of charges tends to increase the energy of charging. This coulombic energy
scales as 1/r, r being the radius of the confined structure.
v) Superparamagnetism in nanomagnets. When the size of ferromagnetic materials
decreases below a certain threshold, there is an increasing chance that it will be composed of a
single ferromagnetic domain. Nanomagnets tend to be they perfectly aligned along a magnetic
field. Their magnetization curve has no hysteresis, that is, there is no memory of the field when
it is turned off. An ensemble of nanomagnets behaves as a single, giant, paramagnetic atom and
the direction of magnetization is easily influenced by thermal fluctuations of the local
environment (superparamagnetism).
34-36
From the above examples, it becomes clear that the size is an important parameter. The
possibility to better understand the laws of physical chemistry and their machinery opens
fascinating scenarios in basic and applied research.
1.5 Impact of Low Dimensional Materials in Present and Future Technology
The progress in microelectronic industry has led to a continuous miniaturization of the
components that can be assembled on a single chip. The development has been so tremendous in
the last years that the conventional lithographic techniques can draw features as small as few
tens of nanometers on a chip (down to 0.13 microns in the next generation of Pentium 4 
TM
chips). Increasing technological problems will soon limit the advance of conventional
lithographic techniques in this up-bottom approach. Two nanofabrication technologies, electron-
beam lithography and atomic-beam holography, will probably play a relevant role in the near
future.
37
 Moreover, when dimensions will shrink down to few nanometers, new laws will govern
the behavior of the devices, as quantum effects will become predominant. The difficulty of
making and assembling devices that are smaller and smaller (and making them work!) and the
1. Introduction
6
increasing demand of computational speed are urging the development of new ideas. New
devices, working with single molecules, quantum dots, nanowires and nanotubes will eventually
be the components of the electronic industry of the future.
38-42
Nanostructures will also influence some other areas of technology: new devices for solar
energy conversions based on semiconductor nanocrystals are becoming competitive with
traditional, silicon based technology.
43
 Nanoparticles can be useful as catalysts due to their high
surface to volume ratio and their novel electronic properties.
1
 Data storage devices could use in
the future ordered arrays of nanomagnets, provided a suitable probe for their reading/writing will
be available: also problems connected with loss of memory due to thermalization of spins have
to be solved.
44
 High quantum fluorescence yields from some class of semiconducting nanocrystals
make them potentially useful as chromophores in a number of applications, such as light
emitting diodes,
45
 and quantum dot lasers.
46
 Recently, the possible use of quantum dots as
biological labels is under intense investigation.
47,48
One dimensional structures, as nanowires, nanorods, organic and inorganic nanotubes,
are also very promising candidates in a number of applications, like in electronic devices, as new
probes for scanning probe microscopy,
49
 in nanomechanics,
50
 and in photovoltaic devices.
51
1.4 A Brief Overview on the Preparation of Low Dimensional Structures
The ability to systematically manipulate the sizes and the shapes of nanoscale inorganic
materials is an important goal of modern materials science. There is of course an extensive
research devoted to the preparation of low dimensional structures. The main aim of all the
methods is to obtain objects with a high crystallinity, controlled size, size distribution and shape.
The approaches generally followed by physicists have as a common background the
controlled deposition of thin films (one atomic layer at a time) on a substrate following
decomposition of gaseous precursors. This allows the production of 2D confined structures, and,
by using a combination of different deposition techniques and lithography with X-ray, ion or
electron beams, or the specific interactions of the growing film with the substrate, it is also
possible to produce 0- and 1- dimensional structures.
1. Introduction
7
Chemists, on the other hand, have a tendency to produce size and shape controlled
nanostructures with solutions-based methods, without the requirements of high vacuum and high
temperatures needed in deposition techniques. Chemists see nanostructures as big molecules,
52
and try to synthesize them in a way such that they behave as single objects, which can be further
processed and manipulated. Their preparative approaches can be grouped in two main classes:
a) Methods that make use of static templates
53
 such as molecular sieves,
54
 zeolites
hosts,
55
 porous membranes,
56
 and carbon nanotubes.
57
 The nanopores of the host material behave
as both reaction chambers and static templates, which dictate the size and the shape of the
particle. Sometimes the material of the host can be one of the reactants.
b) Methods that make use of organic materials to control the growth. This approach can
be further divided in two main sub-classes: i) the first sub-class includes techniques that make
use of special molecules that behave as terminating agents.
53
 These molecules direct the growth
of nanostructures, by coordinating their surface. In addition, crystal morphology can be
controlled by the deliberate introduction of additional molecules in the growing medium. These
‘impurities’ selectively enhance the growing rate of one face over another by modifying the
chemical interactions at the crystal-solution interface:
58,59
 ii) the second sub-class includes
techniques that make use of organic templates, obtained by the self-assembly of organic
molecules. Special classes of molecules, called amphiphiles, under particular conditions, tend to
assemble into soluble aggregates.
60,61
 Depending on the physical-chemical parameters, these
assembles can have spherical shapes or evolve into rod-like or cylindrical aggregates, flexible
bilayers, and planar bilayers. By carrying out chemical reactions in the hydrophilic region of
these assemblies, it is possible to synthesized nanoscale materials with good control in both size
and shape.
The above classification into physical and chemical methods is of course rather simple,
and does not take into account various other areas of research. Production of nanoparticles with
aerosol techniques such as combustion flame, plasma, laser ablation, spray pyrolysis,
electrospray, plasma spray, is also relevant, and has marked the beginning of research in the field
of nanocrystalline materials.
62
 In all these approaches the nanoparticles are generated following
reactions in the gas phase. Other methods include sonochemical processing, high-energy ball
milling, and so on.
62
 The main drawback of most of these techniques is that they tend to produce
nanocrystalline powders with a broad distribution of sizes, rather than confined structures with
homogeneous properties, and sometimes their further processability can be problematic.
1. Introduction
8
We will now give a brief overview of the most popular techniques in use for each type
of low dimensional structure (0, 1, 2 D), along with some specific examples.
2D structures or quantum wells can be prepared via conventional technological growth
techniques like molecular beam epitaxy (MBE) and metal-organic chemical vapor deposition
(MOCVD),
63,64
 when there is favorable epitaxy on a substrate. The high quality of films obtained
has allowed basic science to make much progress in the understanding of 2D structures (much
more than for 1D and 0D).
Synthesis of layered (2D) II-VI semiconductor and metal (iron, tin, titanium) oxide
materials is possible by controlled nucleation and growth in the hydrophilic interlayer of LB
films of fatty acids, or in interlamellar phases composed by alternating layers of water molecules
and surfactants.
53,65
 The use of self-assembled monolayers of amphiphilic molecules on a solid
substrate to direct mineral growth is also of great relevance.
66
 The functional groups at the tails
of the molecules (such as sulfate, phosphate, or carboxylate) induce the nucleation of the
inorganic material. The high control in oriented nucleation derives from the molecular
recognition between the two-dimensional organic template and the inorganic ions, leading to
crystals with well-defined structure and morphology.
Also macromolecules can manipulate the growth of 2D structures, as in the case of
nucleation and growth of apatite minerals in solutions containing organic macromolecules, such
as peptides.
67
 The ability of these molecules to selectively interact to some planes of the growing
crystals leads to the formation of 2D single crystals of few micrometers in cross section and few
nanometers in thickness.
0D structures (or quantum dots) are now prepared in a variety of ways. High quality
clusters of various materials were first obtained via laser vaporization in the gas phase.
68
 This
technique was soon abandoned because the amount of clusters afforded was small. Recently,
nanosized particles of various materials can be produced by a combination of laser vaporization
and controlled diffusion in a cloud chamber.
69,70
0D crystals are also synthesized in glass matrices:
71,72
 glasses doped with suitable atoms
are transformed by heat treatment (i.e. at around 600 °C) into a composite material consisting of
a vitreous matrix in which semiconductor nanocrystals are embedded. This particular preparative
approach has been developed extensively in companies that manufacture accessories for optical
1. Introduction
9
instrumentation, as nanocrystals-doped glasses are used as cut-off filters in optical experiments.
At the temperatures used for the preparation, the glass can be considered in a molten state.
Dopant impurities diffuse and cause nucleation and subsequent growth of the nanocrystals.
Intensive studies on the reaction conditions allowed the preparation of good quality nanocrystals,
in terms of size and size distribution, and provided both experimentalists and theoreticians with
‘toys’ on which they could develop new spectroscopic techniques and refine theories. In
nanocrystals embedded in a vitreous matrix, the confinement of carriers originates from two
causes, namely the barrier to the carriers at the interface between the matrix and the nanocrystal,
and the dielectric confinement. Nowadays, much of what is known on semiconductor
nanocrystals comes from the availability of these samples.
Sol-gel techniques provide nanocrystals embedded in a rigid, transparent, porous glass
matrix (for instance SiO
2
 or Al
2
O3-SiO
2
), but with a lower temperature route.
54
 As a general
approach, the precursors are dissolved in a solvent like water or an alcohol, in the presence of
organic molecules containing Si-O bonds and/or molecules containing Al-O bonds, usually
metal alkoxides. Gelation of this mixture leads to the formation of an extended three-
dimensional network of metal-oxygen bonds, in which the precursors are trapped. A dehydration
procedure removes the solvent from the pores and the residual Me-OH groups. Finally,
annealing of the glass leads to the formation of nanocrystals inside the pores. The simplicity of
this technique is balanced by the scarce control of the growth conditions and by the wide
distribution of nanocrystal sizes.
0D structures are also obtained by epitaxially growing a thin layer of a semiconductor
material over another semiconductor, using MBE techniques.
63,73,74
 The two materials must have
suitable band gaps and electron affinities. Moreover, the respective crystal faces in contact must
have a high lattice mismatch, as in the case of InAs on GaAs 
75
 and Ge on Si.
76,77
 A thin, strained
film, called ‘wetting layer, initially grows. The strain accumulated during further growth of the
material on the top is partly released through formation of an array of pyramidal islands
(Stranski – Krastanov regime). The final step consists in the growth, on the top of the pyramids,
of several layers of the substrate material, so that the dots are completely buried. The relative
alignment of the band gaps creates a confining potential for the carriers that form inside the
pyramids, which behave as quantum dots. Localized strain fields in the proximity of the dots,
due to the lattice mismatch between the two materials, create additional potentials that modify
the band gap of the material of the island. Nanocrystals prepared with this technique are of good
1. Introduction
10
quality in terms of size distribution (around 10%) and optical activity: most of the surface trap
states are in fact passivated. Photoluminescence from strain-induced quantum dots is
characterized by several, narrow emission lines, related to different exciton states in the dots,
and is reminiscent of the emission from atoms. For this reasons quantum dots gained the
nickname of artificial atoms: following investigations showed that many parallelisms could be
drawn between atoms and quantum dots.
78,79
 Nevertheless, the distribution in island shapes and
sizes, and the occurrence of larger, dome-shaped structures in addition the pyramids caused
these peaks to broaden, and much work is currently in progress to produce better samples.
The main drawback of dots embedded in a matrix or grown on a substrate is that they
have no possibility of being further processed (i.e. isolated and/or embedded in another material
or deposited onto another substrate).
The advent of chemical methods has opened new possibilities in the preparation and
processing of nanocrystals. All chemical strategies previously described have been used to
produce 0D materials. The peculiarity of most chemical methods resides in the fact that they
produce nanocrystals that are easily processable, and that can be further manipulated.
Nanocrystals can be dispersed in a variety of solvents, embedded in a polymeric matrix,
immobilized on a metal, integrated into electrical circuits, or they can have attached on their
surface some organic molecules or another inorganic material. Nanocrystals directly prepared in
solution are called colloidal nanocrystals. 
*
 They will we treated extensively in this dissertation.
1D structures can be prepared in a variety of ways.
80
 A combination of MBE and
nanolitography was the first attempt to grow quantum wires on a substrate. The intrinsic
resolution limit of lithography and the presence of high density of surface states, due to etching
and subsequent growth, always led to samples of low quality.
The discovery of carbon nanotubes boosted the research in the direction of nanotubes
and nano-wires. Nowadays, a variety of materials, organic and inorganic, can be obtained in the
form of nanotubes or nano-wires, via condensation from hot plasmas or the vapor phase,
81,82
 and
via laser ablation.
83
                                                
*
 A colloidal solution is a system well known to chemists; it consists of a solution in which very
small solid particles are dispersed. The sizes of the particles are much smaller than the wavelength of the
visible light and consequently a colloidal solution appears optically clear to a human eye.
1. Introduction
11
Perhaps the most beautiful example of a nanotemplate is a carbon nanotube. Carbon
nanotubes are extremely stable structures, resistant to high temperature treatments.
Consequently, a wide range of materials, such as gallium nitride,
57
 boron nitride,
84
 silicon
carbide,
85
 germanium oxide,
86
 nickel,
87
 and cobalt,
88
 can be synthesized in the form of nanotubes
and nanowires stating from carbon nanotubes. The nanotubes either can behave as nanoreactors,
by confining a reaction in their cavities, yielding a carbon nanotube filled with a specific
material, or can behave as reactants, when carbon is involved in the specific reaction. In this
case, the product is often a nanorod or a nanotube.
Nanorods and nanowires of some metals and II-VI semiconductors (Au, Ag, CdSe, CdS)
are produced via electrochemical deposition in porous membranes. Porous anodic aluminum
oxide is often the membrane of choice, due to the high regularity in the size of the pores.
Rod-like nanoparticles and nanowires of various shapes, sizes and of different materials,
such as copper,
89
 gold,
90,91
 calcium sulfate,
92
 barium sulfate,
93
 barium carbonate,
94
 have been
prepared by confined reaction in cylinder-shaped micelles.
The vapor-liquid-solid growth mechanism in which a solid rod grows out of a
supersaturated droplet on a substrate has been very successful in creating one-dimensional
materials 
95
, and has been translated into growth of (insoluble) nanorods of InP, InAs, GaAs, and
Si in a liquid medium.
96,97
Coordinating solvents can behave as dynamic molecular templates, as in the case of the
solvothermal production of nanorods of different materials, such as ME (M = Zn, Cd, E = S, Se,
Te),
98,99
 tin phosphide,
100
 CuInE
2
 (E = S, Se),
101
 NiE
2
 (E = Se, Te),
102
 and CoTe
2
.
103
 In the case of
ZnSe and CdSe nanorods,
99
 the authors proposed that ethylenediamine, the solvent in which rods
were grown, is bound to Zn
2+
 (or Cd
2+
) and Se in a line, as a consequence of its structure. This
proto-structure then dictates the anisotropic growth.
Many of these techniques have some drawbacks. The recovery process from the
substrate or the template can be problematic. The preparation methods also tend to yield these
structures in the form of aggregates, and they are usually insoluble. In many cases, their
diameters are often too large, out of the quantum confinement regime.
1. Introduction
12
1.6 Contribution of This Thesis and Outline of the Dissertation
The contribution brought by this doctoral thesis can be separated into two components:
a) From the point of view of the size control of colloidal II-VI semiconductor
nanocrystals, we introduce a new synthetic medium, consisting of a quaternary CTAB/n-
pentanol/n-hexane/water micellar system. We test this system by preparing CdS nanocrystals,
we compare the synthesis with similar approaches reported in the literature. We deduce that this
microemulsion offers a good control over nanocrystals size.
b) From the point of view of the shape control of colloidal II-VI semiconductor
nanocrystals, we presents a new method for the production of rod-like quantum confined CdSe
via modification of a previously reported synthesis of CdSe nanocrystals.
104
 These rods can be as
long as 100 nm or longer, possess defined optical absorption and emission spectra, and maintain
good solubility. The method can be extended to produce nanocrystals with more exotic shapes.
The outline of the dissertation is as follows:
In chapter 2 we describe a simple model of crystal growth of colloidal particles. Theory
often gives us some general hints on how to prepare monodispersed, quantum confined, colloidal
particles. We review the state of the art in two preparative approaches that are pertinent to this
dissertation: the synthesis in reverse micelles and the synthesis by thermal decomposition of
precursors in hot coordinating solvents. For both techniques, we will give some examples.
In chapter 3 we report the synthesis and the characterization of CdS nanoparticles
prepared in the CTAB/n-pentanol/n-hexane/water microemulsion. We analyze its possible
advantages and its drawbacks.
In Chapter 4 we report the process that allows shape control of CdSe nanocrystals. The
chapter covers a detailed analysis of the anisotropic growth of CdSe nanocrystals in a binary
mixture of hot coordinating solvents. In the discussion section, we propose a mechanism of
growth that explains the experimental results.
In Chapter 5 we report the synthesis and the optical characterization of core/shell
structures starting from CdSe quantum rods.
A conclusion paragraph ends the dissertation.
2. Preparation  of Colloidal Nanocrystals: State of the Art
13
Chapter 2
Preparation of Colloidal Nanocrystals: State of the Art
2.1 Introduction
In this chapter, we will describe a general model of growth of colloidal particles.
Following theoretical considerations, we will get some suggestions on how to obtain colloidal
particles with uniform size. The aim of obtaining particles with uniform size is mainly to
correlate the collective physical properties of an ensemble of particles to the features of a single
particle, the same as is done with atoms and molecules. Colloidal particles have a great interest
in industrial processes such as catalysis, manufacturing of ceramic materials and pigments, drug
release systems, and so on. Control over size allows the production of higher quality products,
with well-tailored properties. In the recent years, size control in the nanometer regime is opening
new scenarios. In this chapter, we will give a detailed description of two synthetic approaches to
nanosized colloidal particles, namely the synthesis in reverse micelle and in hot coordinating
solvents, as both of them are the main topic of this dissertation.
2. Preparation  of Colloidal Nanocrystals: State of the Art
14
2.2 A Model for the Growth of Colloidal Particles
In the theory of crystal growth, the formation of a crystal is described in terms of a
sequence of idealized stages. These concepts apply also to the case of colloidal crystals.
Nucleation. Let us imagine that in a given reaction environment (for instance in a
solution) the concentration of reactants is progressively increased by some means. Passed a
concentration threshold, called critical supersaturation, the system becomes unstable and the
reactants tend to form nuclei. The kinetics of nucleation is difficult to study and we will not go
into the details.
Growth. In a supersaturated solution, nuclei grow at the expense of the monomer
species. For the moment, we will neglect the complications connected with the relative reactivity
of different crystal faces: we will assume that the particles are spherical and that an averaged
surface tension γ can be defined. The growth of crystals depends on a number of parameters,
such as surface energy, concentration of reactants in solution, crystal size, and can be modeled
according to the relative speed of the various processes. A general expression for the free energy
of addition of monomers to a crystal is the following:
( )
iiBsolcrystgrowth
dAcTkG γµµ +−−=∆ ln
cryst
µ  is the standard state chemical potential of a monomer in the crystal, 
sol
µ  is the standard
state chemical potential of the monomer in solution, 
i
c  is the monomer concentration in the
solution adjacent to the crystal, γ  is the surface tension of the crystal and 
i
dA  is the increase of
the surface area of the crystal upon addition of the monomer. If the free energy is negative, the
crystal will grow, if it is positive, it will shrink. The first term of the expression is a constant: the
growth is governed by the competition between the concentration term (
iB
cTk ln− ), and the
surface term (
i
dAγ ). Usually, during the first stages of crystal growth, the concentration term
dominates, as the concentration of monomer is high, and the crystal grows. Over time, the
monomer concentration drops and the surface term prevails. Under these conditions, smaller
crystals will start shrinking while larger ones will still be growing. The dissolution of small
crystals will then refurnish larger crystals with new monomers, allowing them to grow further
(Ostwald ripening).
We now define the solubility of a crystal as the concentration of monomer in solution at
which a given crystal size is stable, that is, it has no tendency to grow nor to dissolve.