3. General introduction
This study provides a review on the effects of impurities during the process of
crystallization. However, as the subject is complex and the documentation of
it is quite voluminous, it appeared not so easy to make a selection of the most
important effects and to discuss them in a limited number of pages. Therefore,
this study has become more lengthy than at first sight was expected.
An impurity can be regarded as any substance other than the material being
crystallized. Even the solvent from which the crystals are grown can therefore be
considered to be an impurity. When impurities are added specifically with the
purpose of achieving a specific effect, they are referred to as additives. Such addi-
tives are for instance ions, small molecules or macromolecules (F¨uredi-Milhofer et
al. (1988, 1990) [1, 2]). The presence of impurities or additives in a crystallization
system can have a dramatic effect on nucleation, crystal growth, macrostep for-
mation, agglomeration and on the uptake of foreign ions in the crystal structure.
Selected impurities are for example, widely used in industry to change the shape
of crystals and to improve the quality of crystalline products, powders or granular
materials.
The possible effects of additives on crystallization processes are summarized in
Figure 1. It is seen that depending on the nature and concentration of the additive,
nucleation and crystal growth may be retarded and/or accelerated, with significant
consequences for the properties of the nascent solid phase(s). Some additives may
even exert a highly selective effect, acting only on certain crystallographic faces.
Aggregation, Ostwald ripening and/or phase transformation may commence at
a very early stage of the crystallization process and may also be significantly
influenced by additives.
The understanding of the mechanisms involved and the possible consequences
is of utmost importance as they underlie the intimate mineral/matrix relationships
in biological and pathological mineralization. Biological macromolecules often
exert a very specific control of mineralization processes in living organisms by
influencing the type and/or morphology of the crystallizing polymorph, as well
as the crystal size and orientation of the mineral phase within the organic matrix
1
4. 2 Contents
Figure 1: Schematic presentation illustrating the possible role of
additives in the precipitation of slightly soluble salts, initiated by
heterogeneous nucleation. Figure is taken from Reference [1].
(Addadi and Weiner (1989) [3]; Lowenstam and Weiner (1989) [4]). Interactions
of ionic crystals with natural polymorphs are also thought to be at the roots of
the formation of pathologic concrements such as kidney stones, dental calculus
and deposits formed in gout and atherosclerosis. In the industry, water-soluble
synthetic polyelectrolytes are widely used as scale inhibitors and/or dispersants
for the prevention of scale deposits in cooling water systems (Fivizzani et al.
(1991) [5]). In addition, there is an enormous potential for materials scientists to
apply knowledge on polymer/crystal interactions to the design of low-temperature
technologies for the preparation of novel composite materials.
It is the objective of this study to examine the most important features of im-
purities in general and their effects on nucleation and crystal growth in particular.
In order to explain the actions of impurities on these phenomena clearly, Chap-
ter 1 first considers the fundamentals of nucleation and describes the different
mechanisms of crystal growth. Then, Chapter 2 discusses the effects of impurities
on nucleation and, finally, Chapter 3 discusses their effects on crystal growth.
Although a wide variety of papers and books have been studied, it has to be
stated that this review will not cover the whole range available. For instance,
the morphology change of crystals, due to the addition of selected impurities, is
not discussed in this study. Neither is the phenomenon of macrostep formation
5. Contents 3
discussed. Undoubtedly, there are a large number of other events, caused by the
presence of impurities, which are not discussed here. However, it has been tried
to cover a spectrum as broad as possible and hopefully, this study provides a good
starting point for anybody who is interested in this extensive field.
7. CHAPTER 1
Nucleation and crystal growth
1.1 Supersaturation
A homogeneous solution is formed by the addition of a solid solute to the solvent.
At a given temperature, there is a maximum amount of solute that can dissolve
in a given amount of solvent and when this maximum is reached, the solvent is
said to be saturated. The amount of solute required to make a saturated solution
at a given condition is called the solubility limit of that solute.
The solubility of materials depends on properties such as temperature and pH,
but it is under specified conditions always a defined amount. Consequently, the
solubility limit provides the concentration at which the solid solute and the liquid
solution are at equilibrium. At this stage, there is no net gain in free enthalpy
upon transition from the solid to the solution state and reverse. Crystals therefore,
cannot grow from or dissolve in a solution that is saturated.
The crystallization of any solid from solution requires a net gain in free en-
thalpy upon transformation of dissolved solute molecules into the solid state (i.e.
a driving force for crystallization). To achieve this, the solubility limit of the
solute must be exceeded. A solution in which the solute concentration exceeds
the equilibrium (saturation) solute concentration at a given temperature is known
as a supersaturated solution and represents a thermodynamically unstable state.
This situation is best described by using a phase (or solubility) diagram as is
illustrated in Figure 1.1.
Formally, the supersaturation σ can be defined in thermodynamic terms as
the dimensionless difference in chemical potential between a molecule in an equi-
librium state and a molecule in its supersaturated state:
(µfluid − µsolid)
kT
=
∆µ
kT
= ln
a
aeq
≃ ln
f C
feqCeq
≈ ln
C
Ceq
, (1.1)
where, ∆µ
kT
is the driving force for crystal growth, a is the activity of the solute, f is
the activity coefficient, C is the concentration of the solute and the subscript ”eq”
indicates the property at saturation. In most situations, the activity coefficients
5
8. 6 CHAPTER 1. Nucleation and crystal growth
Figure 1.1: A typical solubility diagram for crystallization from solution, plot-
ting the solute concentration as function of system temperature. The solid line
labeled B represents the saturation limit of the material in the solvent. Any-
where below this limit (region A), the system is undersaturated and no crystal
will form or grow from these conditions. Above the saturation limit (C & D),
the system is supersaturated. In region D, the solution is relatively high su-
persaturated. Spontaneous nucleation and growth, both may occur. However,
in region C the system is closer to the saturation limit meaning that nucle-
ation cannot occur, but growth of existing nuclei can occur. This region is more
commonly known as the metastable zone. Figure is taken from Reference [6].
are not known and the dimensionless chemical potential difference is approximated
by a dimensionless concentration difference
σ =
C − Ceq
Ceq
. (1.2)
This approximation is only accurate when f ≈ feq and when C - Ceq ≪ Ceq
and it has been shown that it is generally a poor approximation at σ > 0.1 [7].
However, it is still normally used because the needed thermodynamic data are
usually unavailable.
1.2 Metastability and nucleation
Supersaturated solutions are metastable. This means that supersaturating a so-
lution some amount will not necessarily result in crystallization. This can be
9. 1.2. Metastability and nucleation 7
explained by referring to the solubility diagram that is illustrated in Figure 1.2. If
a solution with concentration and temperature at point A is cooled down to point
B just below saturation, the solution would be supersaturated. If this solution is
left undisturbed, it might take days before crystals are formed. If another sample
is taken, cooled to point C and is left undisturbed, it might crystallize in a matter
of hours. Eventually one gets to a point where the solution crystallizes rapidly
and no longer appears to be stable. As can be seen from this experiment, the
metastability of a solution decreases as the supersaturation increases.
Figure 1.2: Metastable zone width for KCl-
water system. Data is taken from Reference [6].
If crystals of the solute are placed in any supersaturated solution, they will
start to grow and the solution will eventually reach equilibrium. The obvious
question that comes to mind is why supersaturated solutions are metastable. It
seems reasonable to think that if the solubility is exceeded in a solution, crystals
should form. To understand why they do not, an event called nucleation needs to
be discussed.
Nucleation is the start of the crystallization process and involves the birth of
a new crystal. Nucleation theory tells us that when the solubility of a solution
is exceeded and it is supersaturated, the molecules start to associate and form
aggregates (clusters), or concentration fluctuations. If it is assumed that these
aggregates are spherical, the following equation can be written for the Gibbs free
energy change required to form a cluster
∆G = 4πr2
γ −
4πr3
3Ω
∆µ, (1.3)
10. 8 CHAPTER 1. Nucleation and crystal growth
where r is the cluster radius, γ is the solid-liquid surface free energy and Ω is the
specific volume of a solute molecule [8]. The first term accounts for the increase
in energy due to the surface of the formed nucleus. The second term accounts for
the negative bulk contribution that lowers the energy, due to the transformation
of N fluid particles to solid particles. For small numbers of molecules the total
Gibbs free energy change is positive, which means that the clusters are unstable
and will dissolve. A plot of ∆G as a function of cluster size (see Figure 1.3) shows
that as the cluster size increases, a point is reached where the Gibbs free energy
change is negative and the cluster would grow spontaneously. When this happens,
nucleation will occur.
The reason that supersat-
Figure 1.3: Free energy change (∆G) versus clus-
ter radius (r). Data is taken from Reference [9].
urated solutions are metastable
is because ∆G needs to pass
through a maximum, ∆Gcrit,
which corresponds to the crit-
ically sized cluster, rc. From
Eq. (1.3), an expression for
the critical size can be derived
by setting the derivative
d∆G/dr = 0 (the maximum
in Figure 1.3), which yields
rc =
2Ωγ
∆µ
, (1.4)
and from Eqs. (1.3) and (1.4)
it follows that ∆G has a max-
imum value of
∆Ghomo
crit =
16πΩ2
γ3
3∆µ2
=
4πγ r2
c
3
. (1.5)
Although it can be seen from the energy diagram why a particle of size greater
then rc is stable, it does not explain how the amount of energy, ∆Ghomo
crit , necessary
to form a stable nucleus is produced. This may be explained as follows: The
energy of a fluid system at constant temperature and pressure is constant, but
this does not mean that the energy level is the same in all parts of the fluid.
There will be fluctuations in the energy about the constant mean value, i.e. there
will be a statistical distribution of energy, or molecular velocity, in the molecules
constituting the system. In those supersaturated regions where the energy level
rises temporarily to a high value, nucleation will be favoured.
11. 1.2. Metastability and nucleation 9
From Eq. (1.4) it can be seen that as the supersaturation increases, the critical
size, rc decreases. It is for this reason that solutions become less and less stable as
the supersaturation is increased. Unfortunately, Eqs. (1.3)-(1.5) are not useful for
practical calculations because one of the parameters, γ, is not available or difficult
to be measured but it has a very significant effect on the calculation.
Every solution has a maximum amount that it can be supersaturated before
it becomes unstable. The zone between the saturation curve and this instability
boundary is called the metastable zone. Every supersaturated solution exhibits
a metastable zone in which nucleation is usually not spontaneous. When the
supersaturation is increased, eventually a supersaturation will be reached at which
nucleation occurs spontaneously. This is called the metastable limit. A precise
thermodynamic definition of the metastable limit is the locus of points where
∂2G
∂x2
P,T
= 0. Here, x represents the molfraction of the solute. This set of points
is known as the spinodal curve and marks the boundary between the metastable
region (where nucleation and crystal growth are the phase separation mechanisms)
and the unstable region (where phase separation is governed by a phenomena
known as homogeneous nucleation). Knowledge of the width of the metastable
zone is important in crystallization because it aids in understanding of nucleation
behavior of each system. Unfortunately, the width of the metastable zone is
influenced by the solution history (how it was made and stored), the cooling
rate employed, the impurities present (including dust and dirt) and the use of
agitation. The measurement of effective metastable limits and a tabulation of
results for several inorganic species is found in N´yvlt et al. (1985) [10].
Nucleation can be classified into various mechanisms as is illustrated in Fig-
ure 1.4. Primary nucleation occurs in the absence of crystalline surfaces, whereas
secondary nucleation involves the active participation of these surfaces. The dif-
ferent mechanisms of nucleation are shortly discussed below.
1.2.1 Primary nucleation
1.2.1.1 Homogeneous nucleation
Exactly how a stable crystal nucleus is formed within a homogeneous fluid is not
known with any degree of certainty, but the formation of it is a very difficult
process to envisage. Not only have the constituent molecules to coagulate, re-
sisting the tendency to redissolve, but they also have to become oriented into a
fixed lattice. The number of molecules in a critical crystal nucleus can vary from
about ten to several thousand molecules. However, a critical nucleus could hardly
result from the simultaneous collision of the required number of molecules since
this would constitute an extremely rare event. More likely, it could arise from a
sequence of bimolecular additions according to the scheme:
12. 10 CHAPTER 1. Nucleation and crystal growth
Figure 1.4: Mechanisms of nucleation. Figure is taken from Reference [8].
A
A2
A3
(A2
An−1
+ A
+ A
+ A
+ A2
. . . . . .
. . . . . .
+ A
⇀↽
⇀↽
⇀↽
⇀↽
⇀↽
A2
A3
A4
A4)
An (critical cluster)
Further molecular additions to the critical cluster would result in a stable nucleus
and its subsequent growth. Similarly, ions or molecules in solution can interact to
form short-lived clusters. Initially, short chains or flat monolayers may be formed
and eventually a crystalline lattice structure is built up. The construction pro-
cess can only continue in local regions of very high supersaturation, and many
of the embryos fail to achieve maturity; they simply redissolve because they are
extremely unstable. If the nucleus grows beyond a certain critical size, as was
explained in Section 1.2, it becomes stable under the average conditions of super-
saturation in the bulk of the fluid. Unfortunately, the structure of the assembly
of molecules or ions is not known and usually they are too small to be observed
directly. The morphology of very small atomic clusters has, for instance, been
discussed by Hoare and McInnes [11].
13. 1.2. Metastability and nucleation 11
Homogeneous nucleation appears to be a rarely occurring event and is difficult
to be observed in practice, due to the presence of dissolved impurities and physical
features such as crystallizer walls, stirrers and baffles. However, it forms the basis
of several nucleation theories [12, 13] and has been studied for several inorganic
systems (KNO3 [14], NH4Br [15], NH4Cl [16]) using the dispersed phase method
proposed by Vonnegut [17]. Critical reviews of nucleation mechanisms have been
made by Nancollas and Purdie (1964) [18], Nielsen (1964) [13], Walton (1967) [19],
Strickland-Constable (1968) [20], Zettlemoyer (1969) [21], N´yvlt et al. (1985) [10]
and S¨onel and Garside (1992) [22]. The recent publication by Kashchiev (2000)
[23] is noteworthy for its in-depth analysis of the thermodynamics and kinetics of
homogeneous -and heterogeneous nucleation.
1.2.1.2 Heterogeneous nucleation
Heterogeneous nucleation is usually induced by the presence of solid particles
or gas bubbles in the liquid. A foreign substance present in a supersaturated
solution is generally known to reduce the energy required for nucleation. As
nucleation in a heterogeneous system generally occurs at a lower supersaturation
level than in a homogeneous system, the Gibbs free energy change associated
with the formation of a critical nucleus under heterogeneous conditions, ∆Ghet
crit
must be less than the corresponding free energy change, ∆Ghomo
crit , associated with
homogeneous nucleation, i.e.
∆Ghet
crit = φ∆Ghomo
crit , (1.6)
where φ is less than unity.
It has been indicated by for instance Eqs. (1.3)-(1.5), that the surface free
energy, γ has a very significant effect on the nucleation process. Figure 1.5 shows
an interfacial energy diagram for three phases in contact; in this case however,
the three phases are not the more familiar solid, liquid and gas, but two solids
and a liquid. The three surface free energies are denoted by γcl (between the solid
crystalline phase, c and the liquid), γsl (between another foreign solid surface, s
and the liquid) and γcs (between the solid crystalline and the foreign solid surface).
Resolving these forces in a horizontal direction results in
cos θ =
γsl − γcs
γcl
, (1.7)
where the angle of contact between the crystalline deposit and the foreign solid
surface, θ corresponds to the angle of wetting in liquid-solid systems.
14. 12 CHAPTER 1. Nucleation and crystal growth
Figure 1.5: Surface free energies at the boudaries between three
phases (two solids, one liquid). Figure is taken from Reference [9].
Volmer (1939) [12] found that the factor φ in Eq. (1.6) depends on the wetting
angle, θ of the solid phase, according to
φ =
1
4
(2 + cos θ)(1 − cos θ)2
. (1.8)
Consequently, when θ = 180
◦
and cos θ = -1, Eq. (1.6) becomes
∆Ghet
crit = ∆Ghomo
crit . (1.9)
When θ lies between 0 and 180
◦
, φ < 1; therefore
∆Ghet
crit < ∆Ghomo
crit . (1.10)
When θ = 0, φ = 0, and
∆Ghet
crit = 0. (1.11)
The three situations represented by Eqs. (1.9) -(1.11) can be interpreted as fol-
lows: For the case of complete non-affinity between the crystalline solid and the
foreign solid surface (corresponding to that of complete non-wetting in liquid-solid
systems), θ = 180
◦
and Eq. (1.9) applies, i.e. the overall free energy of nucleation
is the same as that required for homogeneous nucleation.
For the case of partial affinity (cf. the partial wetting of a solid with a liquid),
0 < θ <180
◦
, Eq. (1.10) applies, which indicates that nucleation is easier to
achieve because the overall excess free energy is less than that for homogeneous
nucleation. Partial affinity is possible in a case where the foreign substance and the
crystal have a similar atomic arrangement. It was shown by Preckshot and Brown
[24] that the energy for nucleus formation is reduced, only if the difference in
isomorphism between the crystal and the foreign particle is <15%. For differences
>15%, the energy requirements are similar to that for a homogeneous system.
For the case of complete affinity (cf. complete wetting), θ=0 and the free
energy of nucleation becomes zero too. This situation corresponds to the seeding
15. 1.2. Metastability and nucleation 13
of a supersaturated solution with crystals of the required crystalline product, i.e.
no nuclei have to be formed. The relationship between φ and θ is illustrated in
Figure 1.6. A review of experimental work on heterogenous nucleation is given by
Turnbull and Vonnegut [25].
Figure 1.6: Ratio of critical free energies between ho-
mogeneous and heterogeneous nucleation as a function
of contact angle θ. Figure is taken from Reference [9].
1.2.2 Secondary nucleation
Secondary nucleation involves the presence of crystals and its interactions with
the environment (crystallizer walls, impellers, etc.). The parent crystals have a
catalyzing effect on the nucleation phenomena and secondary nucleation therefore
requires a lower supersaturation level than spontaneous nucleation. Although
a number of investigations is applied on secondary nucleation, the mechanisms
and kinetics are still poorly understood. Comprehensive reviews on this subject
have been made by Strickland-Constable (1968) [20], Botsaris (1976) [26], de Jong
(1979) [27], Garside and Davey (1980) [28], Garside (1985) [29] and N´yvlt et al.
(1985) [10].
Several theories have been proposed to explain secondary nucleation. These
theories fall into two categories. One traces the origin of the secondary nuclei to
the parent crystal that include: (1) initial or dust breeding; (2) needle breeding;
and (3) collision breeding. Secondary nuclei can also originate from the solute in
16. 14 CHAPTER 1. Nucleation and crystal growth
the liquid phase and the theories that take this into account include: (1) impurity
concentration gradient nucleation and (2) nucleation due to fluid shear.
Strickland-Constable (1968) [30], for instance, described several possible mech-
anisms of secondary nucleation, such as ’initial’ breeding (crystalline dust swept off
a newly introduced seed crystal), ’needle’ breeding (the detachment of weak out-
growths), ’polycrystalline’ breeding (the fragmentation of a weak polycrystalline
mass) and ’collision’ breeding (a complex process resulting from the interaction
of crystals with one another or with parts of the crystallization vessel). A more
thorough description of the above mentioned nucleation mechanisms is presented
in References [8, 9].
1.3 Crystal growth
Before we can explain the effects of impurities on crystal growth clearly, we first
need to consider the mechanisms by which crystals are growing. Once the nu-
cleation step has been overcome crystals of macroscopic dimensions may develop.
For crystal growth to proceed, the surface of the crystal must first be able to cap-
ture growth units arriving from solution and subsequently integrate them into the
crystal lattice. Whilst this process is dependent on a number of factors, the most
important is the availability of so-called kinked sites on the developing crystal
surface.
This situation is best described by the so-called Terrace-Ledge-Kink model
of crystal growth [31] which is schematized in Figure 1.7. As illustrated in this
figure, the model divides the crystal interface into regions having unique structural
attributes:
1. Flat surfaces, or terraces, which are atomically smooth.
2. Steps, which separate terraces.
3. Kinks sites formed from incomplete regions on steps.
The growth unit is first transported from the bulk solution to the crystal surface.
Once arrived, the solute molecule will be in one of three situations: (i) it will be
able to integrate into the crystal lattice directly because it has landed at a site
with an available kink position (growth unit A in Figure 1.7), (ii) it will first
adsorb on the crystal surface terrace, migrate to a step and finally adhere at a
kink position (growth unit B in Figure 1.7), or (iii) it will desorb from the surface
and return to the fluid phase (growth unit C in Figure 1.7).
Kink positions are crucial to the crystal growth process, as they offer ener-
getically preferred binding conditions to a growth unit. Indeed, a kink site offers
interactions in three dimensions to the arriving growth unit, namely from the
17. 1.3. Crystal growth 15
Figure 1.7: A schematic illustration of the Terrace-Ledge-Kink model. In this
model, growth units adsorb to the crystal terrace and either attach to a kink position
directly after adsorption to the crystal surface (growth unit A), adsorb onto a terrace
and migrate across to a step, where they locate a kink site and subsequently become
incorporated into the crystal lattice (growth unit B) or desorb from the surface
(growth unit C). The repeated addition of such growth units to kink sites results in
the progression of the step across the terrace. Figure is taken from Reference [31].
surface beneath it, the step behind it and its neighbouring solute molecule that
had previously been integrated into the lattice. Energetically, this situation is
preferred as compared to the two interactions offered by the step position, and
one by the ledge position.
1.3.1 Crystal growth mechanisms
Most crystal growth theories are based on atomistic models of the crystal struc-
ture. From that point of view, phenomena such as thermal and kinetic roughening,
steps, 2D nucleation and the growth of spirals can be understood. Without giving
a complete overview of crystal growth theory, some important aspects relevant for
this study will be treated in this and the next sections.
As discussed previously, the presence of steps and kink sites on growing crystal
surfaces are vitally important in the crystal growth process. However, it is equally
important to understand how these steps actually come into existence.
Three possibilities are known to dominate almost all forms of crystal growth:
Wilson-Frenkel growth (or normal growth as it is sometimes termed), 2D nu-
cleation growth and spiral growth (e.g. Davey and Garside, 2000) [32]. Less
18. 16 CHAPTER 1. Nucleation and crystal growth
common step sources on a crystal surface may be stacking faults creating perma-
nent partial steps, foreign particles acting as heterogeneous nuclei and 3D nuclei
originating from the mother phase landing on the crystal surface. All these growth
mechanisms will be discussed below, but only the three main mechanisms (i.e.,
Wilson-Frenkel growth, 2D nucleation and spiral growth), will be discussed in
more detail. For a more detailed view on crystal growth theories, growth kinetics
and surface morphology, one is referred to References [33–35].
1.3.1.1 Wilson-Frenkel or normal growth
Most of the crystal surfaces that
Figure 1.8: Kinetic and thermal roughen-
ing as a function of bond strength and su-
persaturation. The roughening transition
at Tr is defined only for zero supersatura-
tion. Figure is taken from Reference [36].
are discussed in this study may be
regarded as molecularly flat. This
means that the crystals are neither
thermally nor kinetically roughed.
However, as roughened growth is
one of the main growth mechanisms,
it is briefly treated anyway. Both
types of roughening are schemat-
ically illustrated in Figure 1.8. If
the crystal bond energy, φ, between
growth units is high, it costs much
energy to create steps and kinks on
a crystal surface and therefore it will be flat. Thermal roughening occurs if the
thermal energy, kT of the growth units roughly equals the bond energy, φ. Then,
the step free energy vanishes and new steps are created without energy barrier.
In this case the surface roughens even at equilibrium [36, 37]. The roughening
transition occurs at a dimensionless temperature, kTr/φ, which can be calculated
using different methods. The most important ones are Monte Carlo simulations
for three-dimensional (3D) crystals and the application of statistical mechanics to
two-dimensional (2D) Ising models. Often the Kossel crystal is used as a model
system in which it is assumed that no overhangs can occur on a crystal surface
(solid-on-solid condition).
If the surface is thermally rough, it does no longer grow layer-by-layer but con-
tinuously. This also occurs for flat surfaces if the driving force for crystallization is
increased to a high extent: kinetic roughening. On a molecular level such surfaces
will be extremely rough, providing all arriving growth units a lattice integration
point [32]. The maximum growth rate is then given by the Wilson-Frenkel law
[39, 40] for continuous rough growth
19. 1.3. Crystal growth 17
Figure 1.9: Impression of a Wilson-Frenkel or normal
growth mechanism. When the surface is kinetically or ther-
mally roughed, it does not longer grow layer-by-layer as can
be seen from the image. Figure is taken from Reference [38].
R ∝ exp
∆µ
kT
− exp
∆µ∗
kT
, (1.12)
with ∆µ∗
the critical driving force at which the surface becomes rough. For
thermally roughened surfaces, ∆µ∗
= 0. In deriving this expression it is assumed
that the surface configuration, and therefore also the evaporation and dissolution
processes, is independent of the driving force. For solution and melt growth
experiments, the driving force for crystallization is often low. In these cases the
growth rate is approximately linear with the driving force, which explains why
Wilson-Frenkel growth is often referred to as ”normal growth”:
R ∝ J0 exp
∆µ
kT
− 1
= J0
C − Ceq
Ceq
≈ J0∆µ. (1.13)
Below the roughening temperature and at low supersaturation, crystal surfaces
grow, in the absence of linear defects, by the formation of two dimensional nuclei.
This is the subject of the next section.
1.3.1.2 2D nucleation and birth-and-spread growth
The general way in which a perfect crystal grows from solution is by two-dimensional
nucleation [41]. This process is often termed the Birth and Spread model and is
illustrated schematically in Figure 1.10.
The mechanism consists of two parts: Namely, the birth or nucleation of new
islands on the surface and the spreading of these islands. It occurs when, unlike
for Wilson-Frenkel growth, most growth units arriving at the surface do not imme-
diately find a growth site. Such units either return to the fluid phase or join other
adsorbed molecules on the crystal surface to form the characteristic 2D islands
as seen in Figure 1.10. Newly adsorbed growth units add to the kinks formed at
the growth steps encircling the 2D nuclei. These islands expand until a complete
monomolecular layer has covered the entire crystal surface.
20. 18 CHAPTER 1. Nucleation and crystal growth
Figure 1.10: The formation and expansion of two-dimensional (2D) nucleation is-
lands on a crystal surface. In (a), an island is formed and expands in all directions,
and is joined by a second island in (b). Growth proceeds by solute addition to both
of these islands, and eventually a completely new growth layer will be formed. Dur-
ing that period new islands are nucleated on top of the previous islands initiating
a new growth layer, as illustrated in (c). Figure is taken from Reference [31]
The following expression for the change in Gibbs free energy for the formation
of a circular 2D nucleus of radius r can be formulated:
∆G = −
πr2
hst∆µ
Ω
+ 2πrγst, (1.14)
where Ω is the volume of one growth unit, hst is the step height and γst is the
edge free energy. The first term accounts for the negative bulk contribution which
lowers the energy due to the conversion of N fluid particles to solid particles. The
second term accounts for the increase in energy due to the edge of the nucleus
formed.
Figure 1.11 shows this expression as a function of the nucleus radius, r. At a
certain critical radius
rc =
Ωγst
hst∆µ
(1.15)
21. 1.3. Crystal growth 19
the Gibbs free energy has a maximum of
∆Gc =
πΩγ2
st
hst∆µ
. (1.16)
Once this radius is reached, adding
Figure 1.11: The Gibbs free energy of a 2D
nucleus as a function of its size r. ∆Gedge
represents the increase in energy due to the
formation of the edge of a nucleus which is
proportional to r. ∆Gbulk represents the bulk
contribution which lowers the energy due to
the formation of N fluid particles to solid par-
ticles. This term is proportional to r2. At r
= rc the Gibbs free energy has a maximum.
Figure is taken from Reference [38].
new growth units is rewarding. The
rate limiting step for nucleation is
to overcome the barrier ∆Gc. At
very low driving force, ∆Gc is large
and no nucleation occurs. At slightly
higher driving force, nuclei are formed
one at a time, which subsequently
spread across the surface (mononu-
cleation). At higher driving force
but below the critical value for ki-
netic roughening, many nuclei are
formed and often additional nucle-
ation occurs on these islands too
before the layer is completely filled
(Birth-and-spread).
Many attempts have been made
to find an analytical expression for
this birth and spread model and
many models have been considered.
A brief overview of the different models is given by S¨ohnel and Garside [22].
As growth proceeds, in a closed system the driving force for crystallization
slowly diminishes due to the loss of solute molecules to the expanding crystalline
surface. Theoretically, growth on a crystal surface will come to a halt when the
level of ∆µ
kT
becomes too low to support the formation of new 2D nuclei. However,
because the vast majority of crystals are not perfect, another growth mechanism,
namely spiral growth [41, 42], becomes prominent.
1.3.1.3 Spiral growth
The growth of perfect single crystals typically proceeds through 2D nucleation
[41], which is extremely slow at low temperatures and low driving forces. In
reality, most crystals have a considerable amount of defects like dislocations [43–
45], vacancies and stacking faults [46–48], twinning [48, 49] and incorporation of
micro-crystals [50–52].
Spiral growth occurs at relatively low supersaturations if screw dislocations
terminate at the facet. The concept of spiral growth was introduced by Frank
22. 20 CHAPTER 1. Nucleation and crystal growth
[53]. Scientists often observed that crystals, growing at low supersaturations, grew
much faster than as would be expected from 2D nucleation theory. Frank was the
first one to realize that screw dislocations provide an inexhaustible source of steps
and that the formation of nuclei was therefore not necessary. These steps allow a
crystal facet to grow at low driving forces far below the roughening temperature.
Figure 1.12: A schematic representation of spiral crystal growth. (a) A
perfect crystal surface, (b) the creation of a screw dislocation by lifting one
region of the crystal one (or more) unit cells relative to another region. As
indicated in the image: The resulting step height, hst corresponds to the
length of the Burgers vector component perpendicular to the surface. (c)
Spiral growth at the dislocation outcrop. Figure is taken from Reference [31].
A screw dislocation is a line defect that is formed when one region of the
crystal is ”pushed up” one (or more) unit cells relative to another region. The
displacement of these two regions corresponds to the Burgers vector b, as can be
seen from Figure 1.12. This vector has a component, b⊥, perpendicular to the
crystal plane, which is equal to one or several times the interplanar distance, dhkl.
When the screw dislocation ends at the crystal surface, it produces a step running
from the dislocation (see Figure 1.12 (b)). Since one end of the step is pinned by
the screw dislocation, the step winds up around the dislocation during growth,
forming a continuous source of steps. At these steps solute molecules attach,
23. 1.3. Crystal growth 21
permitting a continuation of growth, even at reduced levels of supersaturation.
After each rotation of the spiral the crystal grows with a step height of hst = |b⊥|.
This process is schematically illustrated in Figure 1.12.
Over the years several papers have treated the behaviour of growth spirals
emerging from screw dislocations using a continuum description. Burton, Cabr-
era and Frank [41] described the growth from screw dislocations analytically, ap-
proaching the spirals as Archimedean spirals. Cabrera and Levine [54] included
elastic strain energy. Kaishev [54] discussed the low-temperature shape of the
spiral and M¨uller-Krumbhaar et al. [55] took the anisotropy of the spirals into
account. These studies showed that a spiral is isotropic at high temperatures and
that if surface diffusion is not rate limiting, spirals are anisotropic at low temper-
atures, reflecting the symmetry of the crystal surface. This general behaviour is
confirmed in experimental studies of spiral growth [42].
Alternatively, spiral growth is also studied by using an atomistic description.
For instance, Swendsen et al. [56], Gilmer [57] and Xiao et al. [58] performed
Monte Carlo simulations on simple cubic surfaces with screw dislocations. Swend-
sen et al. neglected 2D nucleation in their study and found for low ∆µ
kT
that the
pitch of spiral hillocks is inversely proportional to the driving force and that the
angular frequency is proportional to (∆µ
kT
)2
. Both observations were predicted from
continuum theory. Gilmer studied the contribution of screw dislocations to the
growth rate of low index faces. Xiao et al. looked at the growth morphology of
surfaces with screw dislocations and investigated the influence of surface and bulk
diffusion. His results show smooth and polygonized spirals for low temperatures
and driving forces. When the temperature or driving force is increased, the spi-
rals get more rounded. Cuppen et al. [59] investigated the interaction between
dislocation growth, 2D nucleation and misorientation of the step flow for a wide
range of driving forces by means of Monte Carlo simulations. She found that the
interactions between different growth mechanisms are in agreement with a general
model for velocity source behaviour, which allows for a simple analytical expres-
sion of the growth rate. Consequently, this expression can be used in a continuum
description of crystal growth.
1.3.1.4 Other step sources
There are a small number of less common mechanisms by which step sources are
formed. One of them is a type of surface nucleation, termed three-dimensional
(3D) nucleation. This mechanism is believed to be unique to macromolecular crys-
tal growth [60]. At higher levels of supersaturation in such systems it appears that
large quantities of solute molecules are able to aggregate in solution and subse-
quently sedimentate onto the crystal surface, at which point they form misaligned
microcrystals or multilayered stacks. The steps originating form these aggregates
24. 22 CHAPTER 1. Nucleation and crystal growth
typically expand over the surface as step bunches of several monolayers high. This
mechanism is discussed in further detail in References [45, 46, 50, 51, 61–63].
25. CHAPTER 2
The influence of impurities on
nucleation
2.1 Introduction
Nucleation is one of the most fundamental aspects of phase transition in general
and crystal growth in particular. Depending on the role of foreign bodies (i.e. dust
particles, impurities or additives), nucleation can occur either via homogeneous
nucleation or heterogeneous nucleation. In homogeneous nucleation the potential
barrier, which a system must overcome to create a (crystalline) nucleus in the ho-
mogeneous mother phase and which determines the rate of nucleation, is defined
by the interfacial energy between the crystallizing and ambient phase and the
thermodynamic driving force. On the other hand, the occurrence of foreign bod-
ies may exert an additional influence on the nucleation barrier and rate. There
are for instance, hundreds of reports that describe the effects of foreign bodies
on crystallization and work on this area is summarized in recent reviews [8, 9].
Nucleation promoted by foreign bodies is regarded as heterogeneous nucleation.
It has been widely accepted [64–68] that nucleation plays a key role in con-
trolling polymorphism, ripening, spherulitic crystallization, size distribution of
crystals, crystal network formation, and in the growth of large and high quality
single crystals, including protein crystals. Many modern and robust technologies,
such as epitaxial growth, templated crystallization, biomineralization, molecular
and nanoparticle formation and self-assembly, and macromolecular crystallization,
are virtually based on the effective control of nucleation and it follows that nu-
cleation needs to be promoted and controlled in many cases. For instance, the
structure of protein molecules, which is very essential for the human genome pro-
gram, can be determined via X-ray diffraction techniques [69]. However, for this
purpose protein crystals of sufficient size and quality are required and promotion
and effective control of nucleation rate is therefore essential.
On the other hand, the inhibition of nucleation is also crucial in many situa-
tions. For the pharmaceutical industry for instance, different polymorphs (struc-
23
26. 24 CHAPTER 2. The influence of impurities on nucleation
tures) of a drug may lead to significant differences in the stability and dissolution
rate of the drug [70]. From this point of view, the control of polymorphism and
the size of the crystals will directly affect the bio-efficacy and bio-availability of
the drug. This can be achieved if a certain crystal structure is inhibited. For
the petroleum industry, the prevention of wax crystallization in pipelines is very
important for the transport of crude oil, particularly in a cold winter [71]. Again,
one of the key issues is to inhibit wax nucleation. Apart from this, the antifreezing
function of certain plants and animals, which allows them to survive severe winter
conditions, is directly associated with preventing the nucleation of ice [72].
Nowadays, fabrication of the micro -or nanostructure of complex materials and
devices is one of the most important factors for modern sciences and technolo-
gies. The techniques of epitaxial growth of semiconducting materials on certain
substrates play a key role in the electronic and computer industries. In the field
of life science, hard tissue engineering via biomineralization is controlled by bio-
matrixes, so-called collagens. In this case, how bio-matrixes serve as precursors
during biomineralization and how other proteins interact with nucleating biomin-
erals will determine the microstructure and, consequently, the mechanical and
elastic properties of hard tissues.
To engineer materials with complex structures, one should be able to predict
and control nucleation via some novel manners [64, 73–78]. This requires a decent
understanding on the interfacial process of nucleation. Nevertheless, such an
understanding has not been well-established yet.
Although many models have been published to describe the kinetics of nu-
cleation [64–67] in the past thirty years, much confusion remains. The major
issues associated with this subject are the effect of foreign bodies on the general
kinetics for heterogeneous nucleation. In most cases of crystallization, it is almost
impossible to remove foreign bodies, ranging from solid or liquid particles, gas
bubbles, macromolecules and the wall of crystallization vessels, completely from
nucleating systems. Then some simple but crucial questions come to mind: (i)
How do foreign bodies affect the nucleation kinetics, and (ii) when can we obtain
genuine homogeneous nucleation? Associated with these questions, it is noted
that one of most effective ways to obtain the surface free energy of crystals is via
nucleation experiments, based on the assumption of homogeneous nucleation [64].
If the effect of foreign particles cannot be avoided, the surface free energy will not
be measured accurately. Actually, this has been a standing issue for a long period
of time.
Since nucleation phenomena were identified several decades ago [9, 13], con-
siderable attention has been paid to the formulation of phenomenological theories
including density functional theories, which try to predict nucleation rates quan-
titative, starting from macroscopic, measurable properties of crystals and fluids
27. 2.2. Nucleation kinetics 25
(for a review, see Reference [79]). In addition to these, some efforts were devoted
to more fundamental theoretical approaches in describing the properties of nu-
cleating clusters on a molecular level [80]. However, as pointed out by McGraw
[81], at present, it is not clear that any of these efforts is overall more successful
than the classical nucleation theory [9, 13]. The problem is that in the concern-
ing models developed so far, most attention was focussed solely on the effect of
foreign bodies, on the nucleation barrier and on the formation of clusters. How
foreign particles with different sizes and interactions with the nucleating phase
affect the general kinetic process, including the surface process and the transport
of structural units, has not been systematically considered yet. This insufficient
understanding has significantly restricted our capability in controlling nucleation
in general and in a proper interpretation of experimental results.
It is the purpose of this chapter to describe the effects of additives or im-
purities on nucleation in general and on the kinetics of nucleation in particular.
First, Section 2.2 describes the kinetics of homogeneous nucleation and examines
the measurements of true homogeneous nucleation in previous studies. Finally,
three models to describe the kinetics of heterogeneous nucleation are presented.
Section 2.3 describes the most important features of metastable zone widths and
considers the effects of impurities on it. Finally, Section 2.4 describes the patterns
of the action of impurities on nucleation that are emerging and considers the dif-
ferent molecular interactions between impurities and nuclei of the host molecule
that may appear.
2.2 Nucleation kinetics
In Chapter 1 we have seen that a progressive buildup of supersaturation in a
solution results in the formation of a new solid phase, i.e., nucleation. It was
found that this phenomenon may proceed by two mechanisms: primary nucleation,
occurring in the absence of the crystallizing material, or secondary nucleation,
which is brought about by the presence of the concerning material in solid state.
Primary nucleation is termed homogeneous or spontaneous if the new solid phase
formation is not caused by the presence of any solid phase. If the new solid phase
formation is induced by the presence of a foreign solid phase, the process is said to
be heterogeneous primary nucleation. In this section, I shall consider the kinetics
of homogeneous and heterogeneous nucleation more closely.
2.2.1 Homogeneous nucleation
The kinetics of nucleation is virtually characterized by the nucleation rate, J,
which is defined as the number of nuclei formed per unit time per unit volume.
28. 26 CHAPTER 2. The influence of impurities on nucleation
The nucleation rate can be expressed in the form of the Arrhenius reaction velocity
equation, which is commonly used for describing the rate of a thermally activated
process:
J = B exp −
∆Ghomo
crit
kT
, (2.1)
where B is a kinetic constant that depends of the system under investigation,
∆Ghomo
crit represents the amount of energy required to form a stable nucleus (see
Eq. (1.5)) under homogeneous conditions and k is the Boltzmann constant. For
instance, Nielsen (1964) [13] developed the following expression for the homoge-
neous nucleation rate:
J =
D
d5
mNcrit
64πΩ2
γ3
3π(kT)3(ln S)2
1/2
exp −
16πΩ2
γ3
(kT)3(ln S)2
≡ B exp −
16πΩ2
γ3
(kT)3(ln S)2
,
(2.2)
where D is the diffusion constant of the solute, dm is the molecular diameter,
Ncrit is the number of molecules in the critically sized nucleus, Ω is the volume
of one growth unit, γ is the surface free energy and ln S = ln(1 + σ) is the
supersaturation ratio. This equation indicates that three main variables govern
the rate of nucleation: temperature, T; degree of supersaturation, σ; and surface
free energy, γ. A plot of Eq. (2.2), as shown by the solid curve in Figure 2.1,
indicates the extremely rapid increase in nucleation rate once some critical level
of supersaturation is exceeded.
Figure 2.1: Effect of supersaturation on the nu-
cleation rate. Figure is taken from Reference [9].
The dominant effect of the degree of supersaturation on the time required for
the spontaneous appearance of nuclei in supercooled water vapor was calculated
29. 2.2. Nucleation kinetics 27
by Volmer (1925) as
Supersaturation ratio, S Time
1.0 ∞
2.0 1062
years
3.0 103
years
4.0 0.1 s
5.0 10−13
s
In this case, a ”critical” supersaturation could be said to exist in the region of S
∼ 4.0, but it is clear that nucleation would have occurred at any value of S > 1
if sufficient time is allowed to elapse. By rearranging the last part of Eq. (2.2)
and choose, arbitrarily, the critical supersaturation ratio, S to correspond with
a nucleation rate, J of one nucleus per second per unit volume, one finds the
following expression:
ln Scrit =
16πΩ2
γ3
3(kT)3 ln B
1/2
. (2.3)
2.2.2 Measurement techniques
It is only in recent years that suitable techniques have been devised for study-
ing the kinetics of homogeneous nucleation. The main difficulties have been the
preparation of systems free from impurities, which might act as nucleation cata-
lysts, and the elimination of the effects of retaining vessel walls, which frequently
catalyse nucleation.
An early attempt to study homogeneous nucleation was made by Vonnegut
(1948) [17] who dispersed a liquid system into a large number of discrete droplets,
exceeding the number of heteronuclei present. A significant number of droplets
were therefore entirely mote-free and could be used for the study of true homo-
geneous nucleation. The dispersed droplet method, however, has many attendant
experimental difficulties: concentrations and temperatures must be measured with
precision for critical supersaturations to be determined; the tiny droplets (< 1 mm)
must be dispersed into an inert medium, e.g. an oil, which does not act as a nu-
cleation catalyst; and any nuclei that form in the droplets have to be observed
microscopically.
Variations of the droplet method have since been developed to overcome the
above difficulties (White and Frost (1959) [14]; Melia and Moffitt (1964) [15];
Komarov, Garside and Mullin (1976) [82]), but the reliability of homogeneous
nucleation studies is still difficult to judge. For example, experimental values of
the ”collision factor” (the pre-exponential factor B in Eq. (2.1) and (2.2)) has
30. 28 CHAPTER 2. The influence of impurities on nucleation
frequently been reported to be in the range of 103
to 105
cm−3
s−1
, but as these
are well outside the range predicted from the Gibbs-Volmer theory (∼ 1025
) it
is probable that true homogeneous nucleation was not observed in these cases.
Another point to note is that the surface free energy, γ, which appears in Eq. (2.2)
to the third power, cannot be assumed to be independent of temperature.
An interesting technique was reported by Garten and Head (1963, 1966) [83,
84] who showed that crystalloluminescence occurs during the formation of a three
dimensional nucleus in solution, and that each pulse of light emitted lasting less
than 10−7
s corresponds to a single nucleation event. Measured nucleation rates
were close to those predicted from classical theory, with collision factors in the
range of 1025
to 1030
cm−3
s−1
. In their work on the precipitation of sodium
chloride in the presence of lead impurities, true homogeneous nucleation only
occurred at very high supersaturation (S > 14). The nucleation process was
envisaged as the development of a molecular cluster in the solution as a quasi-
liquid, which after attaining critical size suddenly ’clicks’ into crystalline form.
As a result of this high-speed rearrangement, the surface of the newly formed
crystalline particle may be expected to contain large numbers of imperfections
that would encourage further rapid crystalline growth. As a nucleus appears to
be generated in < 10−7
s, its steady build-up as a crystalline body by diffusion
is ruled out (a diffusion coefficient for NaCl of 10−5
cm2
s−1
gives a formation
time more than ten times greater than the pulse period). These obsevations
may, therefore, be taken as strong evidence for the existence and development of
molecular clusters in supersaturated solutions.
From their work on sodium chloride Garten and Head suggested that a critical
nucleus can be as small as about 10 molecules. A different order of magnitude
was proposed by Otpushchennikov (1962) [85], who estimated the sizes of critical
nuclei by observing the behaviour of ultrasonic waves in melts just kept above
their freezing point. For phenol, naphthalene and azobenzene for example, he
suggested that fewer than 1000 molecules constitute a stable nucleus. In contrast
to this, the work of Adamski (1963) [86] with relatively insoluble barium salts
led to the conclusion that a critical nucleus was about 10−15
g, and as small as
this mass may appear it still represents several million molecules. It is therefore
obvious that there are still some widely diverging views on the question of the size
of a critical nucleus, but this is not surprising as the critical size is supersaturation-
dependent (Eq. (1.4)) and no consideration is given to this important variable by
any of the above authors.
There has long been an interest in the potential effects on the nucleation
process of externally applied electrostatic or magnetic fields. There is evidence
that both homogeneous nucleation and the duration of the nucleation induction
31. 2.2. Nucleation kinetics 29
period can be influenced∗
. However, the relevance of experimental data, obtained
from small-scale investigations under controlled laboratory conditions, to bulk
solutions in flow or agitated conditions normally encountered in industrial practice
is still the subject of considerable controversy (S¨ohnel and Mullin (1988) [87]). A
detailed account of recent theoretical studies on the effect of electric fields on
nucleation has been given by Kashchiev (2000) [23].
2.2.3 Heterogeneous nucleation
The nucleation rate of a solution or melt can be affected considerably by the
presence of mere traces of impurities in the system. However, an impurity that acts
as a nucleation inhibitor in one case may not necessarily be effective in another;
indeed it may even act as an accelerator. Actually, the influence of impurities on
nucleation is poorly understood and each case should be considered separately.
Nevertheless, one must realize that when the effect of impurities on nucleation is
studied in practice, the effect on heterogeneous nucleation is actually seen.
Many reported cases of spontaneous (homogeneous) nucleation are found on
careful examination to have been induced in some way. A supercooled system for
example, can be seeded unknowingly by the presence of atmospheric dust, which
may contain ”active” particles (heteronuclei). Aqueous solutions as normally pre-
pared in the laboratory may contain > 106
solid particles per cm3
of sizes < 1 µm.
Consequently, it is virtually impossible to achieve a solution completely free from
foreign bodies, although careful filtration can reduce the numbers to < 103
cm−3
and may render the solution more or less immune to spontaneous nucleation.
In Section 1.2.1.2 we have seen that the presence of solid foreign particles
influences the Gibbs free energy change, ∆Ghet
crit, due to the change in total surface
free energy, which is caused by the angle of wetting, θ in liquid-solid systems (see
Eqs. (1.6)-(1.8)). Consequently, it was noticed that for solutions containing crystal
nuclei wetting the surface of foreign particles, ∆Ghet
crit < ∆Ghomo
crit . Turnbull and
Vonnegut (1952) [25] proposed to evaluate the rate of heterogeneous nucleation
by an equation whose form corresponds to that of Eq. (2.1) for homogeneous
nucleation
Jhet = Bhet exp
−∆Ghet
crit
kT
, (2.4)
∗
The period of time between the achievement of supersaturation and the appearance of
crystals is generally referred as an ’induction period’ (τ ∼= 1/J) and is considerably influenced
by the level of supersaturation, state of agitation, presence of impurities, viscosity, etc. The
existence of an induction period in a supersaturated system is contrary to the expectations from
the classical theory of homogeneous nucleation (Section 2.2.1), which assumes ideal steady-
state conditions and predicts immediate nucleation once supersaturation is achieved. More
information about ’induction and latent periods’ is given in Reference [9]
32. 30 CHAPTER 2. The influence of impurities on nucleation
where Bhet < B and ∆Ghet
crit is defined by Eq. (1.6). They observed that Eq. (2.4)
was in excellent agreement with experimental results of the nucleation frequency
of mercury crystals in supercooled mercury droplets coated with mercury acetate.
Further they found that the nucleation frequency per droplet is proportional to
its area.
Figure 2.2: Schematic representation of nucleation on active centers (a) in the
absence and (b) in the presence of additive molecules behaving as second-type
active centers: Open circles are original active centers, closed circles are nuclei,
shaded circles are additive molecules. Figure is taken from Reference [88].
Another model was proposed by van der Leeden et al. (1993) [88]. They
assumed that in the absence of additives, nucleation is heterogeneous and occurs
on so-called active centers (e.g. microscopic foreign particles) in the solution, as
schematically illustrated in Figure 2.2 (a). These centers are regarded as being
of equal activity and producing only one nucleus each. A general formula for the
nucleation rate in the absence of additives is given by [21]
J(S) = Zf0N3DS exp −
βΩ2
γ3
δ2(kT)3(ln S)2
, (2.5)
in which Z=0.1 to 1 is the Zeldovich factor, f0 is a practically S-independent
frequency factor, N3D is the number density of the active centers, S is the super-
saturation ratio, β is a numerical shape factor (e.g. 16π/3 for spheres), Ω is the
molecular volume, γ < γhomo is an effective specific surface free energy accounting
for the center activity (γhomo is the specific surface free energy for homogeneous
nucleation) and δ is the number of ions in the formula unit.
They also assumed that when an additive is present in the solution, the additive
molecules behave as second-type active centers (see Figure 2.2 (b)), which can
be less, equally or more active than the original heterogeneous nucleation centers.
As it is supposed that there is no additive adsorption on the nuclei, because
their lifetime is too short to be reached by additive molecules due to diffusion
33. 2.2. Nucleation kinetics 31
[89] and/or their surface area is too small, the effective surface free energy of the
nuclei contacting both types of active centers does not depend on the additive
concentration, C. The nucleation rate, JC on the additive molecules is then given
by
JC(S) = CZf0S exp −
βΩ2
γ3
a
δ2(kT)3(ln S)2
, (2.6)
where γa is the effective surface free energy of the nuclei formed on the additive
molecules, which can be different from the effective surface free energy, γ of the
nuclei on the original active centers. Consequently, the nucleation rate, Ja in the
presence of additives is supposed to be the sum of the nucleation rates on the two
types of active centers; Ja = J + JC. Substituting Eqs. (2.5) and (2.6) into this
expression, one obtains
Ja(S, C) = Zf0N3DS exp −
βΩ2
γ3
δ2(kT)3(ln S)2
+
C
N3D
exp −
βΩ2
γ3
a
δ2(kT)3(ln S)2
.
(2.7)
Eq. (2.7) was used to investigate the precipitation of unseeded BaSO4 with and
without a co-polymer of maleic acid and vinyl sulphonic acid (PMA-PVS) being
present. It was found that the surface free energy, γ and the edge free energy, γl of
the nuclei increased in the presence of the additive, possibly because the additive
molecules provide extra active centers for 2D and 3D nucleation. Furthermore,
it appeared that Eq. (2.7) described the currently available experimental data of
the unseeded precipitation of BaSO4 in the presence of PMA-PVS well.
Recently, Liu presented an even more extensive kinetic model, capable of de-
scribing a wide spectrum of nucleation phenomena [90–95]. This model not only
takes into account the effects of foreign bodies on the nucleation barrier, but
also makes it possible to consider the transport and molecular surface integration
process during nucleation.
Liu correctly noticed that expressions like Eq. (2.4) have the implication that
heterogeneous nucleation is always kinetically more favourable than homogeneous
nucleation. However, this does not explain the fact that in many cases, nucle-
ation in different supersaturation regimes is controlled by different processes in the
same system [96]. It is generally believed [64, 67, 97] that at low supersaturations,
heterogeneous nucleation will be dominant and at high supersaturations, homoge-
neous nucleation will occur. This is explained by the fact that the pre-exponential
factor in Eq. (2.4) is several orders of magnitude larger for homogeneous nucle-
ation. Furthermore, Eq. (2.4) does not explain the following observations (see
Reference [91], Section III):
1. Nucleation can be controlled by more than two independent processes if the
supersaturation changes in a much wider range;
34. 32 CHAPTER 2. The influence of impurities on nucleation
2. The relationship between lnJ and 1/[ln(1 + σ)]2
is nonlinear for some sys-
tems at low supersaturations (see Reference [91], Figure 7);
Associated with these questions, another question to be addressed is whether
or not we have genuine homogeneous nucleation at high supersaturations under
normal conditions. Correct answers to these questions are not only important for
the control of nucleation in general, but also for the measurement of the surface
free energy.
Liu started to look at the nucleation barrier for a system having foreign par-
ticles of a spherical shape with an average radius and density of Rs
and N◦
(see
Figure 2.3 (a)). The assumption of the spherical shape is necessary for mathe-
matical simplicity, but is also physically reasonable, since in the size range where
shape effects enter, particles occurring in many processes tend to be still spherical.
Following a similar approach as considered by Fletcher [97], the formation energy
of a critical embryo on the nucleating particle is given as
∆Ghet
crit = ∆Ghomo
crit f(m, x), (2.8)
where ∆Ghomo
crit is given by Eq. (1.5) and the interfacial correlation function, f(m,x)
can be expressed as
f(m, x) =
1
2
+
1
2
1 − mx
w
2
+
1
2
x3
2 − 3
x − m
w
+
x − m
w
3
+
3
2
mx2 x − m
w
− 1 ,(2.9)
with
w = (1 + x2
− 2xm)1/2
, (2.10)
x = Rs
/rc ≡ Rs
∆µ/Ωγcl ≡ Rs
kT ln(1 + σ)/Ωγcl, (2.11)
rc is given by Eq. (1.4) and all other variables are defined before. The variable
”m” depends on the interaction and the structural match between the crystalline
phase and the foreign particles, and is related to the surface free energy between
different phases according to
m ≡ cos θ =
γsl − γcs
γcl
, (2.12)
which is equivalent to Eq. (1.7). Here, the variables γsl, γcs and γcl correspond to
the surface free energy between substrate and liquid, crystal and substrate, and
crystal and liquid, respectively.
35. 2.2. Nucleation kinetics 33
Figure 2.3: (a) Schematic illustration of heterogeneous 3D nucleation on a foreign
particle with a radius of Rs and a ”contact angle” of θ with the nucleating phase.
Embryo: ”c”, foreign particle: ”s”; mother phase: ”l”. (b) Dependence of factor
f(m,x) on the relative particle size x=Rs/rc and contact angle, θ. x→0 implies that
foreign particles as the nucleating substrate vanish completely. θ=π corresponds to the
case where foreign particles and embryos are repulsive to each other. This corresponds
to the situation where foreign particles do not catalyse the nucleation any more. With
increasing x (x≥0.01) and decreasing θ (0≤ θ ≤2π), f(m,x) decreases correspondingly.
Referring to Eq. (2.8), this implies that the nucleation barrier ∆Ghet
crit is ranging from
∆Ghomo
crit to 0 (0≤ ∆Ghet
crit ≤ ∆Ghomo
crit ). In this regime, foreign particles play a crucial role
in lowering the nucleation barrier. Figure (a) & (b) are taken from Reference [93].
The factor f(m,x), varying between 1 and 0, describes quantitatively the re-
duction of the nucleation barrier from ∆Ghomo
crit to ∆Ghet
crit, due to the occurrence
of foreign bodies. Notice that the function f(m,x) is dependent of two variables
(”m ≡ cos Θ” and ”x”) instead of one (θ), as was the case with function φ(θ)
in Chapter 1. Namely, it also takes the relative size of the foreign particle into
account. Figure 2.3 (b) illustrates f(m,x) as a function of x for various values
of the contact angle, θ. One can see from the graph that when x→0 or m→ -1,
f(m,x)=1. This is the case where homogeneous nucleation takes place. In the
case of m→1, foreign particles reduce to embryos of the crystallizing phase. It
follows from Figure 2.3 (b) that when x ≥ 1 and with smaller contact angles,
f(m,x) or ∆Ghet
crit decreases to a large extent. For x ≪ 1 or θ = 0 the situation
corresponds to crystal growth. In the case of x →∞, a round nucleating particle
reduces to a planar substrate. Then Eq. (2.8) describes the nucleation on a planar
surface.
36. 34 CHAPTER 2. The influence of impurities on nucleation
To derive the effect of foreign particles on the nucleation rate, one should obtain
the distribution function that describes the concentrations of embryos of various
sizes on the substrate surface, for which it is assumed that a quasi-”dynamic
equilibrium” state exist between the monomers and the g-mers (the so-called
steady-state kinetics [90, 96]). One takes a basis of a unit surface at the substrate
and defines the number of monomers, dimers, . . . , and g-mers on this volume to
be n1, n2, . . . , ng. These equilibrium surface concentrations are time-invariant:
n1 monomers ⇀↽ . . . ⇀↽ g-mer. Regarding the ensemble of monomers and g-mers
on the surface of the substrate and approximating the equilibrium with the above
quasi-chemical reaction equilibrium [65, 67, 90], one obtains the nucleation rate
on a foreign particle
J′
= 4aβkinkf′′
(m, x)Ω(n1)2 γcl
kT
f(m, x)
1/2
exp −
∆Ghomo
crit
kT
f(m, x) (2.13)
with
f′′
(m, x) =
1 + (1 − xm)/w
2
, (2.14)
where a is the dimension of structural units in the direction parallel to the crystal
surface, βkink denotes the kink kinetic coefficient [64] and n1 is the density of the
growth units in the system.
The average nucleation rate in the solution depends on the density and size
of foreign particles occurring in the system, and is given, according to J =
4πaN◦
(Rs
)2
J′
, by
J = 4πaf′′
(m, x)N◦
(Rs
)2
[f(m, x)]1/2
B exp −
16πΩ2
γ3
cl
3kT[kT ln(1 + σ)]2
f(m, x) (2.15)
with
B = 4aβkinkN◦
Ω(n1)2 γcl
kT
1/2
. (2.16)
Introducing the term 4πaN◦
(Rs
)2
is based on the fact that heterogeneous nu-
cleation only takes place in the liquid layers adjacent to the foreign particles.
Evidently, only for this part of the solutions, the nucleation rate will be effec-
tively influenced by the foreign particles. The relative effective volume fraction
for heterogeneous nucleation is therefore equal to the volume of the liquid adjacent
to the foreign particles, which is proportional to the density and surface area of
nucleating particles occurring in the system, namely 4πaN◦
(Rs
)2
.
The factor f′′
(m,x) is a function of the ”contact angle”, θ and the relative size of
the foreign particles, x, and gives a similar plot as the function f(m,x) in Figure 2.3
37. 2.2. Nucleation kinetics 35
(b). In the case of homogeneous nucleation, one has f′′
(m,x)=f(m,x)=1, and
4πaN◦
(Rs
)2
→ 1, according to Liu [90]. Then Eq. (2.15) is reduced to Eq. (2.1)
As indicated in Eqs. (2.13) and (2.15), f(m,x) and f′′
(m,x) characterize the
major difference between homogeneous and heterogeneous nucleation kinetics. It
is shown above that the occurrence of foreign particles will lower the nucleation
barrier, resulting in an increase in nucleation rate. This effect, characterized by
f(m,x), will be escalated by lowering the ”contact angle”, θ between the crystal
phase and the substrate and/or by increasing the relative size of foreign particles,
x=Rs
/rc. On the other hand, foreign particles exert also a negative impact on
the nucleation kinetics. As shown in Figure 2.3 (a), nucleation on a foreign
particle will cause a reduction in the ”effective surface of embryos”, where the
growth units are incorporated into the embryos. This tends to slow down the
nucleation kinetics, which gives rise to a counter effect against the lowering of the
nucleation barrier caused by f(m,x) and f′′
(m,x) in the pre-exponential term of
J, given by Eq. (2.15). Therefore, in terms of the effective collisions, this implies
that lowering of the contact angle, θ and/or an increase of x will slow down the
nucleation kinetics (see Eq. (2.15)).
Figure 2.4: (a) The effect of contact angle, θ between foreign particles and the nucleat-
ing phase on the nucleation rate. As shown, foreign particles with a low θ will control the
kinetics at low supersaturations, while those with a high θ will control the kinetics at high
supersaturations. Rs = 1000a, γclΩ2/3/kT = 1.5 (a is the dimension of growth units).
(b) The effect of the size of foreign particles on the nucleation rate. Foreign particles
with a large radius of curvature will control the kinetics at low supersaturations, while
those with a small radius of curvature will control the kinetics at high supersaturations.
θ = π/5, γcl/kT = 1.5. Figure (a) & (b) are taken from Reference [90].
Since f(m,x) and f′′
(m,x) are functions of both θ and Rs
, the effect of f(m,x)
and f′′
(m,x) on nucleation can be expressed in terms of θ and Rs
, respectively (see
38. 36 CHAPTER 2. The influence of impurities on nucleation
Figure 2.3 (b)). In Figure 2.4 (a) the relative nucleation rate, J/(4πaN◦
(Rs
)2
B)
is plotted as a function of supersaturation, σ for different ”contact angles”, θ
and (b) for different radii of foreign particles, Rs
. As expected, nucleation on
foreign particles with a strong interaction and a good structural match between the
crystalline phase and the substrate (θ → 0 and/or a large Rs
, f(m,x) and f′′
(m,x)
→ 0) will be dominant at low supersaturations, whereas nucleation on those with a
weak interaction and/or a poor structural match (θ → π and/or a small Rs
, f(m,x)
and f′′
(m,x) → 1) become kinetically favourable at high supersaturations. This
implies that for a nucleating system, different foreign particles having different
sizes and distinct surface properties or contact angles with the crystalline phase,
will control nucleation at different supersaturation regimes. In the case of a strong
interaction and a good structural match, it follows that crystallization only occurs
at the surface of foreign bodies and that the growth of crystals will be compact and
well oriented. In the case of a weak interaction and/or a poor structural match,
foreign bodies exert almost no influence on nucleation and the crystals occurring
in the system will be random and uncorrelated to the substrate. This situation
frequently leads to the false identification of homogeneous nucleation. Genuine
homogeneous nucleation can be regarded as an up-limited case of heterogeneous
nucleation, where one has f(m,x)=f′′
(m,x)=1. As illustrated in Figure 2.4 (a) &
(b), this occurs only at very high supersaturations under normal conditions.
The change of nucleation process in different supersaturation regimes can be
understood as follows: At low supersaturations, the nucleation barrier is very high
(cf. Eq. (1.5)). To lower the nucleation barrier is the top priority to accelerate
the kinetics. Then heterogeneous nucleation will occur favourably on the particles
with a small f(m,x). Conversely, at higher supersaturations, the exponential term
associated with the nucleation barrier becomes unimportant. Due to the pre-
exponential factors f(m,x) and f′′
(m,x), nucleation on the substrates having a
larger f(m,x) and f′′
(m,x) becomes kinetically favourable.
The above mentioned model was used to examine the generic heterogeneous
effect of foreign particles on 3D nucleation. It was shown that nucleation ob-
served under normal conditions includes a sequence of progressive heterogeneous
processes, which can be characterized by different interfacial correlation functions,
f(m,x). Furthermore it was noticed that the size, Rs
and contact angle, θ exert a
very strong effect on the free energy barrier of nucleation, as well as on the trans-
port of growth units to the surface of embryos. It appears that most nucleation
observed under gravity is likely to be heterogeneous and that genuine homoge-
neous nucleation may not be easy to achieve under gravity. These results are
verified by nucleation experiments of N-lauroyl-L-glutamic acid di-n-butylamide
from isostearyl alcohol solutions and microgravity experiments for CaCO3 (cal-
cite) nucleation [92, 95]. The experimental results appeared to be in excellent
39. 2.3. Metastable zone widths 37
agreement with the theoretical predictions.
The current model can also be used to explain the effects of impurities or
additives on nucleation and to describe the ordering of fluid molecules induced
by the substrate. However, these effects are not further discussed in this study.
Additional information about these subjects is given in References [92, 95].
2.3 Metastable zone widths
Unlike the successes of the proposed rate expressions in the previous section, there
is still a lack of success in explaining the behaviour of real systems. This has led a
number of authors to suggest that most primary nucleation in industrial crystal-
lizers is heterogeneous rather than homogeneous and that empirical relationships
such as
J = kn∆Cn
, (2.17)
are the only ones that can be justified†
. Here is J the nucleation rate, kn is
the nucleation rate constant and ∆C ≡ C − Ceq is the supersaturation. The
exponent n, which is frequently referred to as the apparent order of nucleation,
has no fundamental significance. It does not give an indication of the number of
elementary species involved in the nucleation process.
However, Eq. (2.17) is not entirely empirical as it can be derived from the
classical nucleation relationship (see Eq. (2.2)) (Nielsen (1964) [13]; N´yvlt (1968)
[98]). The nucleation rate may be expressed in terms of the rate at which super-
saturation is created by cooling, e.g.
J = ε ˙T
dCeq
dT
, (2.18)
where ˙T = -dT/dt is the cooling rate, the derivative term represents the slope of
the solubility curve and ε=R/[1 − C(R − 1)]2
[9]. R is the ratio of the molecular
weights of anhydrous salt and hydrate, and C is the solute concentration, expressed
as mass of anhydrous solute per unit mass of solvent at a given temperature.
The maximum allowed supersaturation, ∆Cmax ≡ Cmax − Ceq, may be ex-
pressed in terms of the maximum allowed undercooling, ∆Tmax ≡ Teq − Tmax,
according to
†
Nucleation is, in a strict sense, not a chemical reaction of a definite order, since the velocity is
not proportional to a power of the concentration. However, as Figure 2.1 in Reference [13] shows,
the curves log J against log S are almost straight lines over several decades of J. Therefore, it
is a fairly good approximation to write J = kn∆Cn
, which is the so-called ”Power Law”, in
intervals of restricted lengths. More information about this subject can be found in Reference
[13].
40. 38 CHAPTER 2. The influence of impurities on nucleation
∆Cmax =
dCeq
dT
∆Tmax. (2.19)
Consequently, Eq. (2.17) (using ∆C = ∆Cmax) and (2.18) can be rewritten as
ε ˙T
dCeq
dT
= kn
dCeq
dT
∆Tmax
n
, (2.20)
or by taking logarithms
log ˙T = (n − 1) log
dCeq
dT
− log ε + log kn + n log ∆Tmax, (2.21)
which indicates that the dependence of log ˙T on log ∆Tmax is linear with a line of
slope n. Hence, the value of the nucleation order, n can be obtained from a log-log
plot of the maximum allowed undercooling, ∆Tmax versus different cooling rates,
˙T.
Although Eq. (2.21) can be useful for characterizing the metastability of crys-
tallizing systems, it is no longer regarded as a reliable indicator of the nucleation
kinetics alone. The over-simplification in the above analysis is that it assumes that
at the moment when nuclei are first detected, the rate of supersaturation change
is equal to the rate of nucleation, but the true situation is rather more complex.
The created supersaturation is dissipated in two ways, partly by growth on ex-
isting crystalline particles and partly by the formation of new nuclei. Further, in
the experimental determination of the metastable limit, nuclei are not detected
at the moment of their creation but at some later time when they have grown
to visible size (about 10 µm). In other words, the results of such measurements
are dependent not only on nucleation but also on the subsequent crystal growth
process.
As a result, N´yvlt (1983) [99] proposed a refinement of the theoretical analysis.
He concluded that for unseeded solutions the slope of the
log ∆Tmax vs. log ˙T
line is not equal to n, but to (3g + 4 + n)/4, where g is the apparent ’order’
of the growth process [9].
However, Janse and de Jong (1978) [99] warned that attempts to evaluate
crystallization kinetics from metastable zone widths, evidence should be treated
with caution, while Kubota, Kawakami and Tadaki (1986) [100] suggested that
the cooling rate dependence of ∆Tmax can be reasonably explained by a random
nucleation model. Other detailed analysis of metastable zone width measurements
and their relationship to nucleation and growth kinetics are given by Mullin and
Janˇci´c (1979) [101] and by S¨ohnel and Mullin (1988) [102].
41. 2.3. Metastable zone widths 39
Figure 2.5: Apparatus for measuring metastable
limits: A, cooling water-bath; B, pump; C, flow me-
ter; D, magnetic stirrer; E, Perspex water jacket; F,
thermometer. Figure is taken from Reference [9].
The apparatus shown in Figure 2.5 (Mullin, Chakraborty and Mehta (1970)
[103]), based on an earlier one devised by N´yvlt, (1968), [98], can be used to deter-
mine equilibrium solubilities as well as metastable zone widths. At present, there
are much more advanced systems to determine solubilities and metastable zone
widths [104, 105], but compared with measurements from the past, the proceedings
are still very much the same. A nearly saturated solution of known concentration
is placed in the flask and rapidly cooled until nucleation commences. The content
of the flask is then slowly heated. During all experiments the solution is stirred.
The cooling and heating sequences may be effected by means of the water jacket,
or by an externally operated cold/hot air blower. On approaching the saturation
temperature the heating rate is reduced to about 0.2
◦
C/min. The temperature at
which the last crystalline particle disappears is taken as the saturation tempera-
ture, Teq.
The nucleation temperature is measured in a similar way. The flask containing
the solution of known concentration is warmed about 4 or 5
◦
higher than the sat-
uration temperature. A steady rate of cooling is maintained and the temperature
at which nuclei first appear is recorded. The difference between the saturation
and nucleation temperatures is the maximum allowed undercooling, ∆Tmax, cor-
responding to a particular cooling rate, ˙T.
Nucleation temperatures in the presence of crystalline materials can be deter-
mined by a procedure similar to that for the measurement of unseeded data by
introducing two small crystals (∼2 mm in size) into the flask when the solution
42. 40 CHAPTER 2. The influence of impurities on nucleation
has cooled to its predetermined saturation temperature.
The variation of the maximum allowed undercooling, ∆Tmax with the cooling
rate, ˙T for aqueous solutions of ammonium sulphate (Mullin, Chakraborty and
Mehta (1970) [103]) is shown in Figure 2.6. The lines of seeded and unseeded
solutions are not parallel; the seeded points lie approximately 1.5 - 2
◦
C below the
unseeded. The slopes of the lines for seeded and unseeded solutions are approxi-
mately 2.6 and 6.4, respectively, which indicates that the mechanisms of primary
and secondary nucleation are different. The best straight lines through the data
yield the relationships
˙T = (1.38 ± 0.9)∆T2.64±0.92
seeded (secondary)
and
˙T = (1.28 ± 0.91)×102
∆T6.43±1.62
unseeded (primary),
which gives a measure of the scatter of the data. The maximum allowed un-
dercoolings for seeded and unseeded solutions are more or less independent of the
saturation temperature over the range 20 - 40
◦
C, but do depend on the rate of
cooling. At low rates of cooling (∼5
◦
C/h) the values are about 1.8 and 3.8
◦
C
for seeded and unseeded solutions of ammonium sulphate, respectively, compared
with 3.5 and 5
◦
C for a cooling rate of 30
◦
C.
Figure 2.6: Nucleation characteristics of aqueous ammonium sulphate solution:
(a) pure solutions, seeded and unseeded; (b) effect of impurities in seeded solu-
tions. The broken line represents data from (a). Figure is taken from Mullin,
Chakraborty and Mehta, (1970), [103].
43. 2.3. Metastable zone widths 41
Unfortunately, undercooling data obtained from unseeded solutions have lit-
tle or no industrial relevance. In fact, it is often impossible to obtain consistent
’unseeded’ values for many aqueous solutions, e.g. sodium acetate, sodium thio-
sulphate and citric acid. Therefore, for crystallizer design purposes, the lowest
’seeded’ value should be taken as the maximum allowed undercooling, and the
working value of the undercooling should be kept well below this.
Some typical maximum allowed undercoolings in seeded solutions are given in
Table 2.1. Although the values of ∆Tmax for any two substances may be similar,
the values of the supersaturation, ∆Cmax and S, may be very different. The
relationship between the two quantities is given by Eq. (2.19). For example, ∆Tmax
= 1
◦
C for both sodium chloride and sodium thiosulphate, but the corresponding
values of ∆Cmax are 0.25 and 18 g of crystallizing substance per kg of solution
and S ∼ 1.01 and 1.4, respectively.
Table 2.1: Maximum allowed undercooling‡, ∆Tmax, for some common aqueous salt
solutions at 25◦C (Measurements made in the presence of crystals under conditions of
slow cooling ∼5◦C/h) and moderate agitation. Table is taken from Reference [9].
Substance
◦
C Substance
◦
C Substance
◦
C Substance
◦
C
NH4 alum 3.0 MgSO4 · 7H2O 1.0 NaI 1.0 KBr 1.1
NH4Cl 0.7 NiSO4 · 7H2O 4.0 NaHPO4 · 12H2O 0.4 KCl 1.1
NH4NO2 0.6 NaBr·2H2O 0.9 NaNO3 0.9 KI 0.6
(NH4)2SO4 1.8 Na2CO3 · 10H2O 0.6 NaNO2 0.9 KH2PO4 9.0
NH4H2PO4 2.5 Na2CrO4 · 10H2O 1.6 Na2SO4 · 10H2O 0.3 KNO3 0.4
CuSO4 · 5H2O 1.4 NaCl 1.0 Na2S2O3 · 5H2O 1.0 KNO2 0.8
FeSO4 · 7H2O 0.5 Na2B4O7 · 10H2O 4.0 K alum 4.0 K2SO4 6.0
Metastable zone widths can be greatly affected by the thermal history of the
solution. A solution that has been kept for an hour at a temperature sufficiently
higher than the saturation temperature will be found to have a wider metastable
zone than if it had been kept only slightly above the saturation temperature. The
higher the preheating and the longer the solution is maintained at that tempera-
ture, the higher the supersaturation at which nucleation commences. Preheating
also increases the induction period and decreases the number of crystals formed
(S¨ohnel and Garside (1992) [22]). The influence of the thermal history has often
been attributed to the deactivation of heteronuclei in the solution, but an alter-
native view is that preheating changes the solution structure and influences the
subcritical cluster sizes (N´yvlt et al. (1985) [10]).
‡
The working value for normal crystallizer operation may be 50% of these values, or lower.
The relation between ∆Tmax and ∆Cmax is given by Eq. (2.19).
44. 42 CHAPTER 2. The influence of impurities on nucleation
Metastable zone widths can be widened and narrowed by the addition of dif-
ferent impurities [104, 106–111]. For instance, it was observed that organic com-
pounds, like EDTA (ethylene diamine tetra acetic acid), urea and thiourea, lead
to an increase of the metastable zone width and to an improvement of crystal
quality [106–109]. In most of the cases, an increase of the metastable zone width
with impurity concentration has been reported [105, 112, 113]. Myerson and
Jang (1995) [113] noticed a certain trend between the binding energy for adipic
acid with various alkanoic acids and the metastable zone width. They found the
binding energy to increase with increasing carbon number to a maximum of C14
(myristic acid), to decrease from C14 to C16 (palmitic acid) and then to increase
again. A similar trend was observed for the metastable zone width, indicating
a correlation between the two. Sangwal et al. [114–117] observed that the sol-
ubility of ammonium oxolate monohydrate [(NH4)2C2O4 · H2O; AO] in all of the
investigated cases increases linearly with an increase of impurity concentration
and that the extent of increase depends on the type of impurity. More recently,
Sangwal et al. (2004) [118] applied a study on the effect of different bi- and triva-
lent cation impurities on the metastable zone width for the growth of AO crystals
from aqueous solutions. He found that (1) Mn2+
, Co2+
and Ni2+
impurities lead
to an increase of the supersaturation ratio, Smax, corresponding to the metastable
zone width, the effect of Co2+
and Ni2+
being more pronounced than that of
Mn2+
, and (2) that Cu2+
, Fe3+
and Cr3+
impurities have practically no effect on
Smax. Using the theory for heterogeneous three-dimensional nucleation and the
concept of appearance of supersaturation barriers, analysis of the observations
revealed that: (1) the dependence of the metastable zone width on the concentra-
tion, cimp of an impurity is in agreement with the predictions of the theory, where
adsorbed impurity particles block the active growth sites on the surfaces of nuclei
developing in the bulk solution, (2) the value of Smax is associated with the dis-
tance between active adsorption sites, which is related with the average distance
between adsorbed impurity molecules on the growing surface of the crystal and
(3) an impurity leads to an increase of the metastable zone width of ammonium
oxalate solutions when the complex adsorbing on the growing surface is not very
stable. Other parameters influencing the metastable zone width are impurity con-
centration, concentration of the solute, nature of the solution, pH, cooling rate
and mechanical effects [104, 110].
The measurement of industrially meaningful metastable zone widths can be
very time consuming and be subject to experimental errors. For this reason Mers-
mann and Bartosch (1998) [119] have proposed a theoretical model and claimed
to be able to predict working values for the design of seeded batch crystallizers.
However, they made a number of basic assumptions. First, that secondary nucle-
ation is not caused by attrition between seed crystals, but by surface nucleation
45. 2.4. Effect of impurities 43
on the seeds, which develop into outgrowths and then detach§
. This mode of
behaviour was first analyzed by Nielsen (1964) [13] and given the name ’needle
breeding’ by Strickland-Constable (1979) [20]. It is further assumed that the de-
velopment of outgrowths is controlled by the integration step [9] and that the
shower of detectable nuclei that marks the onset of secondary nucleation occurs
when the volumetric hold-up of crystals in the vessel is between 10−4
and 10−3
(m3
crystals/m3
suspension), corresponding to a detectable size of ∼10 µm.
2.4 Effect of impurities
It is well known that the presence of small amounts of impurities in a system can
affect the nucleation behaviour very considerably. This effect is best manifested
experimentally by an appreciable broadening of the metastable zone, as discussed
in the previous section. For instance, it has long been known that the presence
of small amounts of colloidal substances such as gelatin can suppress nucleation
in aqueous solution, and that certain surface-active agents also exert a strong in-
hibiting effect. Traces of foreign ions, especially Fe3+
and Cr3+
, can have a similar
action on inorganic salts, as can be seen from the data recorded in Figure 2.6 (b).
The mechanism of this inhibition is by far less understood than the mechanism
of impurity effects on crystal growth and it is therefore unwise to attempt a
general explanation for nucleation suppression by added impurities with so little
quantitative evidence available yet. However, certain patterns are beginning to
emerge. For example, the higher the charge on the cation, the more powerful
the inhibiting effect, e.g. Cr3+
> Fe3+
> Al3+
> Ni2+
> Na+
. Furthermore,
there often appears to be a ’threshold’ concentration of impurity above which the
inhibiting effect may actually diminish (Mullin, Chakraborty and Mehta (1970)
[103]). The modes of action of high molecular weight substances and cations
are probably quite different. The former may have their main action on the
heteronuclei, rendering them inactive by adsorbing on their surfaces, whereas the
latter may act as structure-breakers in the solution phase.
The causes of inhibition due to adsorption, lie in a combination of the ease
by which an impurity or guest molecule can be incorporated into the nucleus of
the host, and the subsequent ease by which a further host molecule can be incor-
porated onto or next to the guest molecule. The ease of incorporation depends
§
Attrition refers to the process by which asperities and fines are removed from the surface
of parent crystals and it is recognized as a major cause of secondary nucleation in industrial
crystallizers. In dilute suspensions, it is mainly a consequence of collisions of particles with
parts of the equipment (impeller or walls) [28, 120, 121]. The production rate of the attrition
fragments depends on the fluid dynamics of the suspension, on the mechanical properties of the
solid material and on the physico-chemical properties of the liquid.
46. 44 CHAPTER 2. The influence of impurities on nucleation
upon the similarity of the size, shape and intermolecular interactions of the guest
molecule with the host molecule (Hendriksen and Grant (1995) [78]). For instance,
impurities recognized as forming strong bonds on the surface will effectively lessen
the nucleation rate and hence, increase the metastable zone width (Rauls et al.
(2000) [113]). Structurally related additives may influence the nucleation and
crystal growth of a host molecule in a number of ways. They may block adsorp-
tion of solute molecules and thereby give rise to morphological changes; they may
dock onto the surface and may then become incorporated into the crystal lattice;
they may disrupt the emerging nucleus and so inhibit the nucleation step. Each
of these interactions will potentially depend upon the molecular similarity, steric
and energetic, to the host molecule. Such reasoning has formerly been applied
to the modification of habit by structurally related substances (Weissbuch et al.
(1986) [122]; He et al. (1994) [123]). Although these possible interactions are
interrelated, it is convenient to consider each separately (Hendriksen et al. (1998)
[124]).
(a) ”Blocking” is the ability of an already adsorbed molecule to hinder the
subsequent adsorption of further layers of host molecules. A high propensity to
block will manifest itself in morphological changes as the relative growth rates of
the various crystal faces change in comparison to the control. This effect has been
directly measured for the entire crystal though not for the individual faces.
(b) ”Docking” is the ability of a molecule in solution to be adsorbed onto the
growing crystal surface in an orientation appropriate for ultimate inclusion into
the crystal lattice, by the familiar lock-and-key mechanism prevalent in the uptake
of tailor-made additives. Studies of adsorption from solution would be required
to directly measure this uptake, but have not yet been attempted. A molecule
which successfully docks onto a crystal surface may or may not block the arrival
of further host molecules; if it does block effectively (see above) then the overall
level of incorporation into the crystal lattice will be less than if it were a poor
blocker. The level of incorporation, provided the blocking ability is considered, is
a reasonable measure of the docking ability.
(c) ”Disruption” is the ability of an incorporated molecule to reduce the sta-
bility of the emerging nucleus such that the critical nucleus size (for growth rather
than dissolution) rises, meaning, in effect, that nucleation will be inhibited. The
incorporated molecule does this by a combination of steric hindrance and even
disruption of for instance, the hydrogen bond network. The propensity to inhibit
nucleation has been reported previously (Hendriksen and Grant (1995) [78]). Since
the formation of the critically sized nucleus is analogous to the growth of a macro-
scopic crystal, this incorporation of foreign molecules could explain why there are
much higher supersaturations required as compared to impurity free solutions.
Other suggestions for the action of impurities have been made as well. For
47. 2.4. Effect of impurities 45
example, Botsaris, Denk and Chua (1972) [125] suggested that if the impurity
suppresses primary nucleation, secondary nucleation can occur if the uptake of
impurity by the growing crystals is significant; the seed crystal creates an impurity
concentration gradient by itself; the concentration of the impurity near the crystal
surface becomes lower than that in the bulk solution; and if it is reduced enough,
nucleation can occur. Another possibility is that certain impurities could enhance
secondary nucleation by adsorbing at defects on existing crystal surfaces and, by
initiating crack propagation, render the crystals prone to disintegration (Sarig
and Mullin (1980) [126]). Kubota, Ito and Shimizu (1986) [127] on the other
hand, interpreted the effect of ionic impurities on contact nucleation by a random
nucleation model.
The influence of soluble impurities can also affect the induction period, tind, but
it is virtually impossible to predict the effect. Ionic impurities, especially Fe3+
and
Cr3+
, may increase the induction period in aqueous solutions of inorganic salts.
Some substances, such as sodium carboxymethylcellulose or polyacrylamide, can
also increase tind, whereas others may have no effect at all. The effects of soluble
impurities may be caused by changing the equilibrium solubility or the solution
structure, by adsorption or chemisorption on nuclei or heteronuclei, by chemical
reaction or complex formation in the solution and so on. The effects of insoluble
particles are also unpredictable.
The effects of soluble and insoluble impurities on crystal growth processes are
discussed in detail in Chapter 3
48. 46 CHAPTER 2. The influence of impurities on nucleation
