Crystallization kinetics and phase behaviour of molecular solutions

Acknowledgements

i
ACKNOWLEDGEMENTS
First and foremost, I would like to express my heartful gratitude to my advisors,
A/Prof. Reginald B. H. Tan from the National University of Singapore (NUS),
A/Prof. Paul J. A. Kenis and Prof. Charles F. Zukoski from the University of Illinois
at Urbana-Champaign (UIUC) for their guidance, patience and inspiration, which
have gone far beyond my graduate study.
I thank Prof. Richard D. Braatz, UIUC, for his stimulating discussion and
integral role in the success of this study. Also thanks to A/Prof. Sing Bor Chen,
A/Prof Zhi Li, NUS, for generously spending his precious time to offer help and be
part of the thesis committee. Further thanks to the past and present members of
various research groups, Dr. Ann P. S. Chow, Chin Lee Tan, Dr. Zaiqun Yu and Dr.
Xing Yi Woo, Institute of Chemical & Engineering Science (ICES); Sendhil K.
Poornachary and Nicholas C. S. Kee, NUS; Dr. Venkateswarlu Bhamidi, Dr. Sameer
Talreja, Dr. Amir Y. Mirarefi, Dr. Ranga S. Jayashree, Ashish Kapoor, Pedro Lopez,
Joshua Tice, Sarah L. Perry, Dr. Michael Mitchell, Eric B. Mock, Dr. Vijay
Gopalakrishnan, Dr. Paul Molitor, Dr. Vera V. Mainz, Dr. Scott R. Wilson and Dr. Yi
Gui Gao, UIUC; Dr. Subramanian Ramakrishnan, Florida State University; for their
friendship and assistance during my stay in Singapore and the United States.
Financial support for this work was provided by the Agency of Science,
Technology and Research (A*STAR). I wish to thank both universities for offering
this challenging Joint Ph.D. program, which gears me up with invaluable exposure
and experience.
Heartfelt gratitude to my parents, without whose inculcation I will not be the
real me today. Last but not least, I am grateful to my wife Che Chang for her
unconditional love and moral support. Her continuing encouragement drives me to
move forward in my study as well as my life.
Table of Contents

ii
TABLE OF CONTENTS
Acknowledgements i
Table of Contents ii
Summary iv
List of Tables v
List of Figures vii
1 General Introduction 1
2 Literature Review 4
2.1 Why Pharmaceutical Crystallization? 4
2.2 Solution Crystallization and Phase Behavior 5
2.2.1 Nucleation and Growth 5
2.2.2 Induction Time, Metastable Zone Width and Critical
Supersaturation
9
2.2.3 Polymorphism 11
2.2.4 Phase Behavior and Phase Diagram 12
2.3 Present Work 17
2.3.1 Rate of Supersaturation 17
2.3.2 Generalized Phase Diagram 18
3 Evaporation-Driven Crystallization: Effects of Supersaturation on
Crystallization Kinetics 20
3.1 Introduction 20
3.2 Experimental Systems and Methods 23
3.2.1 Experimental Systems 23
3.2.2 Evaporation-Based Crystallization Platform 24
3.2.3 Experimental Methods 28
3.3 The Effects of Rate of Supersaturation on Crystallization Kinetics 29
3.3.1 Experimental Results 29
3.3.2 Origins of the Critical Supersaturation 37

3.4 Concluding Remarks 43
4 Evaporation-Driven Crystallization: Effects of Supersaturation on Polymorph
Selectivity
44
4.1 Introduction 44
4.2 Experimental Systems and Methods 47
4.2.1 Experimental Systems 47
4.2.2 Methods of Analysis 50
4.3 The Effects of Rate of Supersaturation on Polymorph Selectivity 51
4.3.1 Experimental Results and Discussion 51
4.3.2 Polymorphic Transformation 58
4.3.3 Polymorphic Selectivity 60
4.4 Concluding Remarks 61
5 Generalized Phase Diagram of Molecular Solutions 63
5.1 Introduction 63
5.2 Experimental Systems and Methods 66
Table of Contents

iii
5.2.1 Experimental Systems 66
5.2.2 Pulsed-Field Gradient Spin-Echo (PGSE) NMR 67
5.3 Linking Experiments to Theory 71
5.3.1 Equilibrium Thermodynamic Model 72
5.3.2 Self Diffusivity 74
5.3.3 Results and Discussion 78
5.4 Concluding Remarks 84
6 Metastable States of Molecular Solutions 86
6.1 Introduction 86
6.2 Experimental Systems and Methods 92
6.2.1 Experimental Systems 92

6.2.2 Turbidity Meter 93
6.2.3 Nuclear Magnetic Resonance (NMR) 95
6.3 Results and Discussion 95
6.3.1 Solution Phase Behavior 95
6.3.2 Molecular Self Diffusion 96
6.3.3 Generalized State Diagrams 103
6.3.4 Discussion on the Presence and Absence of Metastable States 108
6.3.5 Rate of Nucleation 114
6.3.6 Partitioning in Ibuprofen/Ethanol/Water Solutions 116
6.4 Concluding Remarks 117
7 Conclusion and Recommendation 118
7.1 Conclusions 118
7.2 Future Directions 119
Bibliography 121

Summary

iv
SUMMARY
The goals of this research are: (i) to study the effects of the rate of
supersaturation on crystallization kinetics and polymorph selectivity; and (ii) to
develop a generalized phase diagram from first principles and verify its applicability
to a wide range of molecular solutions. This thesis begins with highlights to the
importance of pharmaceutical crystallization (Chapter 1), then summarizes current
state-of-the-art of solution crystallization research (Chapter 2), followed by
describing the progressive aspects of this research in detail, from how the
macroscopic phase transitions (Chapter 3) and final crystal properties (Chapter 4) are
affected by the rate of supersaturation to how the microscopic particle interactions
influence both the equilibrium solution phase behavior such as solubility boundary
(Chapter 5) and the nonequilibrium phase transitions such as liquid-liquid phase

separation and gel formation (Chapter 6). Last but not least, conclusions are drawn
and future directions are prospected in Chapter 7.
List of Tables

v
LIST OF TABLES
Table 3.1 Comparison of experimental and calculated drying times and rates of
evaporation for aqueous solutions of glycine. Initial volume of solution droplet =
5 μl, initial concentration of glycine = 191 g/l, temperature = 18 °C, pressure =
101325 Pa, saturated vapor pressure of water = 2063 Pa, relative humidity = 52%,
and the length of the microchannel = 7 mm. The activity coefficients are
calculated from an empirical correlation developed by Myerson and co-workers
76

The size of the cross-sectional area varies for different experiments 28
Table 3.2 Standard deviation of nucleation times of aqueous glycine solutions.
Experimental conditions and crystallization platform specifications are the same
as those in Table 3.1 30
Table 3.3 The average extrapolated critical concentration and critical
supersaturation for different compounds crystallizing under various conditions:
glycine (in water), STA (2 M LiCl, water), L-histidine (water), paracetamol
(water), and HEWL (4.06 %(w/v) NaCl, 0.1 M acetate buffer, pH = 4.50). The
activity coefficients for water in solutions with critical and saturated
concentrations of solute, γ
c
and γ
e
, respectively, are approximately equal. 36
Table 3.4 Calculated values of surface tensions σ and thermodynamic parameters
B from solubility c

e
, solid density n
cr
, molecular size d, and equilibrium activity
coefficient γ
e
using Christoffersen’s correlation.
85
41
Table 4.1 Experimental Conditions and Results for Crystallization of Aqueous
Glycine Solutions by Slow Evaporation 49
Table 4.2 Calculated Supersaturation Values at Onset of Nucleation for both α
and γ Glycine Polymorphs for Typical Crystallization Conditions.
52
Table 4.3 Calculated Rates of Polymorphic Transformation from α Glycine to γ
Glycine at Different Experimental Conditions.
59
Table 5.1 Solvent compositions and temperatures used for the self diffusivity
measurement of different solutes used in this study.
67
Table 5.2 The particle sizes of molecules studied that are derived from spheres
whose volumes are estimated as described in the text 81
Table 6.1 Solvent compositions and temperatures used for the self diffusivity
measurement of the five different solutes used in this study. Values of D
2
are
obtained from PGSE NMR experiments. The sizes of the molecules are
estimated as described in the text 93
Table 6.2 Values of
ε

/k extracted from Eq. (6.3) for ibuprofen in different
List of Tables

vi
solvents 103
Table 6.3 Characterization of compositions of ibuprofen, ethanol and water of the
upper and lower liquid layers formed by liquid-liquid phase separation of
solution of ibuprofen (274.3 g/kg solvent) in EtOH/H
2
O (50/50 w/w%) at 20 °C
using
1
H NMR. Areas of peaks are normalized by the area associated with the
–CH
2
– group in EtOH 116
Table 6.4 Compositions of ibuprofen, ethanol and water of the two liquid phases
formed by liquid-liquid phase separation of solutions of ibuprofen in EtOH/H
2
O
(50/50 w/w%) at 20 °C. 116

List of Figures

vii
LIST OF FIGURES
Figure 3.1 (a) A typical array of evaporation-based crystallization compartments
in a polypropylene platform made by micro-machining; (b) Schematic diagram
of an individual crystallization compartment. Typical dimensions for channel
diameter d range from 0.6 to 1.5 mm 25

Figure 3.2 Nucleation time as a function of initial solute concentration of
aqueous glycine solution for different evaporation rates at three different
combinations of temperature and relative humidity (RH): (a) 18 ºC and 52 %RH;
(b) 21 ºC and 50 %RH; and (c) 36 ºC and 19 %RH. In some cases, the error
bars are smaller than the size of the data points. Normalized rate of evaporation
1.0 = 446 μg/h 32
Figure 3.3 Nucleation time vs. initial solute concentration of four compounds for
different rates of evaporation under different combinations of temperature and
RH: (a) L-histidine, 18 ºC and 50 %RH; (b) Paracetamol, 21 ºC and 16% RH; (c)
Silicotungstic acid (STA), 21 ºC and 18 %RH. The LiCl concentrations range
from 0.6 to 1.5 M; and (d) Hen egg white lysozyme (HEWL), 21 ºC and 24 %RH.
The NaCl concentrations range from 1.09 to 3.28 %(w/v). The ratios of the
solute and salt concentrations, which stay inherently constant throughout each
experiment, are specified in panels c and d.
35
Figure 3.4 Supersaturation at nucleation time S(t
n
) as a function of evaporation
rate for different initial glycine concentrations (T = 21ºC, RH = 50 %).
Normalized rate of evaporation 1.0 = 446 μg/h 37
Figure 3.5 Plot of (B-ln
3
S
c
) versus (ln
2
S
c
) for different compounds. According
to Eq.

(3.5), the slope of this plot is equal to ln(A/J
c
) 41
Figure 3.6 Comparison of experimental data and model predictions for critical
supersaturation as a function of dimensionless surface tension, σd
2
/kT.
Experimental conditions are: glycine (18, 21, and 36ºC, in water), STA (21ºC,
LiCl, water), L-histidine (18ºC, water), paracetamol (21ºC, water), and HEWL
(21ºC, NaCl, 0.1M acetate buffer, pH = 4.50). The calculated curve is obtained
by setting ln(A/J
c
) = 5.15. In most cases, the error bars are smaller than the size
of the data points 42
Figure 4.1 pH values of the aqueous glycine solution as a function of glycine
concentration at 21 °C. The solid lines connecting the data points are drawn to
guide the eye. 48
Figure 4.2 The effects of rate of evaporation (rate of supersaturation) on
polymorph formation of solution crystallization of glycine. 54
List of Figures

viii
Figure 4.3 Optical micrographs of γ glycine crystals formed in aqueous solution
droplets at different experimental conditions: (a) temperature = 18 ºC, relative
humidity = 52%, and rate of evaporation = 0.090 mg/h; (b) 21 ºC, 22%, 0.159
mg/h; (c) 21 ºC, 22%, 0.189 mg/h; (d) 21 ºC, 22%, 0.221 mg/h; and (e) 21 ºC,
22%, 0.256 mg/h 55
Figure 4.4 Optical micrographs of α glycine crystals formed in aqueous solution
droplets crystallized on silanized glass slides, open to the laboratory environment
(21 ºC, 32% RH, evaporation rate ~5.0 mg/h) 55

Figure 4.5 Powder X-ray diffraction data for: (a) raw glycine powder (Fluka);
and (b) glycine crystals grown in aqueous solutions at 21 ºC, 22% RH by slow
evaporation of water at a rate of 0.189 mg/h. In both (a) and (b), the top
diffraction pattern is the actual experimental data and the bottom pattern is
simulated as described in the text.
56
Figure 4.6 Rate coefficients of polymorphic transformation from α to γ glycine
for different starting amount of γ glycine at different temperatures at 70 %
relative humidity. 59
Figure 4.7 Change of supersaturation at different rates of evaporation as a
function of time. Supersaturation of glycine in aqueous solution is calculated
with respect to the solubility of different polymorphs: (a) α glycine; and (b) γ
glycine. The curves terminate at the onset of nucleation events 61
Figure 5.1 The sample set-up in PGSE NMR self diffusion experiments 71
Figure 5.2. Phase diagram (ε/kT as a function of
φ
) showing solubility
boundaries for different ranges of interaction λ
74
Figure 5.3 Phase diagram (D
2
as a function of
φ
) showing solubility boundaries
for different ranges of interaction λ 78
Figure 5.4 Scaled long-time self diffusivities of glycine in H
2
O at 5, 25, 40 and
75 °C. (a) Self diffusivities are plotted against absolute concentration of
glycine in H

2
O. (b) The absolute concentration is converted to particle volume
fraction as described in the text. Values of D
2
are given by the slopes of the
linear fits.
81
Figure 5.5 Phase diagram for a variety of solutes in D
2
space. The different
symbols correspond to experimental data obtained for molecular systems as
specified in Table 5.1. Solubility data for various systems are obtained from the
literature,
81,88,90,156,157
then converted to volume fraction as described in the text.
The solid line is the model solid-liquid phase boundary for range of interaction λ
of 1.1.
83
Figure 6.1 Temperature and turbidity are plotted as a function of time. The
List of Figures

ix
insert shows how a cloud point temperature is determined 94
Figure 6.2 Plots of D
2
as a function of
ε
/kT for different
λ
. 97

Figure 6.3 Scaled long-time self diffusivities of ibuprofen in EtOH/H
2
O (60/40
w/w%) as a function of solute volume fraction at 10, 15, 20 and 25 °C. The
absolute concentration of ibuprofen is converted to particle volume fraction as
described in the text. Values of D
2
are given by the slopes of the best linear fits.98
Figure 6.4 D
2
of various solute molecules in different solutions at different
temperatures: (a) glycine in water; (b) citric acid in water; (c) ibuprofen in
different solvent compositions of EtOH/H
2
O; (d) hen egg white lysozyme in 0.1
M NaAc buffer (pH = 4.5) in the presence of different concentrations of NaCl; (e)
trehalose in water; and (f) an API in EtOH/H
2
O (54.2/45.8 wt%). Note that
(a)-(e) present experimental data obtained in this work and (f) presents data
extracted from literature as described in the text.
47
102
Figure 6.5 Solution of ibuprofen (200 g/kg solvent) in mixture of EtOH and H
2
O
(50/50 w/w%) at 20 °C. (a) Opaque solution, when the stirrer is on; (b) two
distinct homogenous liquid layers, when the stirrer is off. The arrow indicates
the liquid-liquid interface 105
Figure 6.6 Cloud point temperatures of solutions of ibuprofen in different

mixtures of ethanol and water 105
Figure 6.7 Generalized phase diagram for a variety of molecules in D
2
space.
Various symbols are experimental (a) solubility data, and (b) data corresponding
to metastable states. The closed upper triangles, circles, diamonds, lower
triangles, and squares are pairs of solubility data in terms of volume fraction
from literature
81,90,157,183,185
and D
2
data measured in this study of ibuprofen,
glycine, trehalose, citric acid and lysozyme The closed right-angle triangles are
fitted solubility data of Veesler's API extracted from literature
47
as described in
the text. The open circles stand for conditions where glycine crystals form in
aqueous solution. The open upper, lower and right-angle triangles correspond
to LLPS data of ibuprofen in ethanol/water mixtures with ethanol content of 40,
50 and 60 w%, respectively. The open diamonds represent the glass transition
point for aqueous trehalose solutions. The upper and lower half-filled squares
correspond to LLPS data of lysozyme solutions in the presence of 3 and 5 w/v%
NaCl, respectively.
45
The filled squares are gelation data of lysozyme taken
from literature
126
and expressed into D
2
. The open crosses presents LLPS data

of Veesler's API
47
as described in the text. The solid, short-dashed and
long-dashed lines are the model solid-liquid, liquid-liquid and MCT gel
boundaries for ranges of interaction λ = 1.1. Experimental conditions are
specified in Table 6.1. 106

General Introduction

1
1 GENERAL INTRODUCTION
Over the past century the field of crystallization has evolved from crystals of
simple inorganic salts
1
to supramolecular complexes
2
, from the classical nucleation
theory
3
originally developed for liquid droplet formation in the vapor phase to
multi-environment simulation of mixing effects in antisolvent crystallization
4
, from
primitive copper crystallizing pans
5
to modern crystallizers equipped with
sophisticated process analytical technology (PAT)
6
, from large scale industrial
crystallizers to nanoscale crystallizing reservoirs in microfluidic chips

7-11
, from
Edisonian experimental protocols to a priori crystal structural predictions by
molecular modeling
12,13
. This interdisciplinary area of research has already greatly
impacted society with its applications in the pharmaceutical, biotechnological and
fine chemical industries. Yet detailed physical insight is lacking for many of these
processes.
The field of crystallization research can roughly be divided in two categories,
fundamental and industrial research, although with ambiguous boundary and much
overlap between them. Fundamental research sheds light into both theoretical
development and experimental advances. Theoretical approaches towards
developing sound philosophies of fundamental crystallization processes
14-16
and
predicting solution phase behavior
17-23
have never lost their research edge ever since
the importance of the crystallization technique was recognized. Experimental
methods, often developed to facilitate the testing of hypotheses, are constrained
within microbatch to bench scale environment for easier implementation, better
General Introduction

2
control and greater reproducibility. Industrial research typically focuses on large
scale process development from pilot plant to full scale production including on-line
analysis techniques, sensor development, specialty process development, and an
escalating need for molecular modeling and polymorph prediction.
24

Despite all
the progress that has been made, tremendous gaps still exist in our understanding of
how to control nucleation rates, the presence of impurity, habit and morphology, how
to predict the size distribution of crystals, and how to monitor the degree of
crystallinity. These gaps provokes a strong need for deeper collaboration between
those focusing on the fundamentals of phase transitions to those interested in control
of phase changes at large scale. Thus fundamental and industrial research
initiatives can and do provide mutual benefits and supplemental research strengths
and focus.
This thesis focuses on (i) the study of the effects of the rate of supersaturation
on crystallization kinetics and polymorph selectivity; and (ii) the development of
generalized phase diagrams from first principles and verification of their
applicability to a wide range of molecular solutions.
In chapter 2, relevant literature will be discussed in detail. The fundamentals
of crystallization, including nucleation and growth kinetics, induction time and
metastable zone width, polymorphism, will be summarized in terms of past and
present advances, and industrial applications. Theoretical progress in generating
phase diagrams based on simple fluid models and experimental measurements
General Introduction

3
interaction potentials between molecules to facilitate verification and justification
of proposed models will also be reviewed.
Chapters 3 and 4 cover detailed experimental studies on solution
crystallization at different rates of supersaturation, which are created by different
rates of solvent evaporation using an in-house evaporation-based crystallization
platform. The effects of the rate of supersaturation on nucleation kinetics and
polymorph selectivity will be discussed in chapters 3 and 4, respectively.
Chapters 5 and 6 outline the importance of phase diagrams and their ability of
making a priori predictions of solution phase behavior such as conditions conducive

to crystal nucleation and liquid-liquid phase separation. In chapter 5, a novel phase
diagram will be developed based on self diffusion coefficients, whose universality
and applicability in predicting the solubility boundary is experimentally verified and
justified from sub-nanometer amino acid molecules to biomacromolecules.
Chapter 6 will describe the metastable states of small molecule solutions and their
similarities with those of nanoparticle suspensions.
In chapter 7, Conclusion and Recommendation, major results in this thesis will
be reviewed, and potential future research directions of crystallization at the
microscale and further development and application of the generalized phase
diagrams will be discussed.
Literature Review

4
2 LITERATURE REVIEW
Equation Section 2
2.1 Why Pharmaceutical Crystallization?
Pharmaceutical solids (active pharmaceutical ingredients or APIs) can be
classified as either crystalline or amorphous.
25
Different crystallization processes,
spray drying and lyophilization, and post crystallization treatment lead to various
degree of crystallinity of the final product.
26
Crystalline solids, i.e. crystals,
possess remarkably symmetrical arrangement of molecules and therefore a regular
internal structure.
1
Typically, crystalline is a preferred state of most APIs and
marketed drugs because of ease of processing, a high level of purity and stability,
particularly when the crystal is the most thermodynamic stable polymorphic form.

Polymorphism is the ability of crystalline materials to exist in different molecular
packing yet the same chemical compositions.
5
The polymorphic nature of many
small organic molecules and salts has significant impact on various industries,
because different polymorphs possess different physical properties such as solubility,
dissolution rate, compressibility, and bioavailability that impact the final product
performance.
Pharmaceuticals require exceptionally pure, stable, well-defined, and
well-characterized crystalline materials. Crystallization stands out as the most
widely used separation and purification step for API manufacturing, with several
extraordinary advantages that are appealing in the pharmaceutical industry.
Crystallization processes usually operate at low temperature which eradicates
Literature Review

5
thermal degradation of heat-sensitive products; they also commonly operate at high
concentration so that unit costs are minimized and separation factors are maximized.
Solution crystallization produces solid particles of defined shape and size from
mother liquor. Moreover, the crystal size distribution (CSD) is feasibly controllable
through fine tuning of the operating conditions, which greatly facilitates subsequent
downstream processing such as filtration and drying. Last but not least, crystals are
efficient in storing and packing.
The details of the crystallization process substantially influence critical physical
properties such as crystal habit, morphology, size distribution, etc, either predictably
or unpredictably. Thus, crystallization of pharmaceutical compounds has been a
very active area of research for many decades, in view of its influence on the final
product performance.

2.2 Solution Crystallization and Phase Behavior

2.2.1 Nucleation and Growth
Crystallization from solution consists of two consecutive processes: nucleation,
which is the formation of a new solid phase from a supersaturated homogeneous
mother phase, followed by crystal growth, which is the further addition and
integration of growth units to the pre-existing nuclei/crystals.
Classical nucleation theory was first introduced by Volmer and Weber
3
from
condensing vapor studies, and has been further developed by Volmer,
27
Gibbs,
28

Literature Review

6
Becker and Döring.
29
The theory considers both kinetic and thermodynamic
aspects of the formation of nuclei, and it is applicable to any first order phase
transitions. In the kinetic treatment of theory, cluster growth and decay are due to
the net effects of addition and detachment of monomers,
1,14
which will eventually
lead to the formation of critical clusters if the concentration of solute is sufficiently
high. This process is an analog to a sequential chemical reaction and can be
represented by a series of elementary reactions

nn
AAA

AAA
AAA
⇔+
⇔+
⇔
+
− 11
312
211
M
(2.1)
where A
1
, A
2
and A
3
represent the monomer, dimer and trimer of the solute molecule,
respectively; and A
n
represents the critical-sized cluster. Phase transition takes
place by monomeric addition to various clusters with different sizes; clusters
exceeding the critical size are more likely to grow and subcritical clusters tend to
redissolve to the solution. Therefore, the cluster size distribution is evolving with
time. The overall excess free energy of formation of the critical cluster is usually
treated as the energy barrier of the nucleation process, which is the central concept of
the thermodynamic treatment of the classical nucleation theory. This overall excess
free energy for the formation of a solid particle of solute (assumed to be spherical of
radius r) from the homogeneous solution ΔG, equals to the summation of a positive
term, the excess free energy for the creation of a new surface ΔG

S
, and a negative
term, the excess free energy for the formation of a new bulk solid phase ΔG
V
:
Literature Review

7

23
4
4
3
SV v
GG G r rG
πσ π
Δ=Δ +Δ = + Δ
(2.2)
where σ is the surface tension (surface energy) and ΔG
v
is the excess free energy of
bulk phase formation per unit volume. If the size of the new solid particle r is
small, the positive energy term ΔG
S
will dominate so that the total excess energy will
increase as r increases. As r increases to a critical value, the dominance will shift
to the negative energy term ΔG
V
and further increase of r will reduce ΔG. This
critical nuclei size r

c
can be easily determined by taking the first derivative of the
total excess energy with respect to r and equating it to zero:

2
84 0
ccv
dG
rrG
dr
πσ π
Δ
=
+Δ=
(2.3)
Rearranging gives:

2
c
v
r
G
σ
=−
Δ
(2.4)
Substituting Eq.
(2.4) into Eq. (2.2) gives the critical total excess free energy ΔG
c
:

()
3
2
2
16 4
3
3
cc
v
Gr
G
πσ
π
σ
Δ= =
Δ
(2.5)
In the supersaturated homogeneous mother solution, the critical radius r
c
represents
the size above which the newly formed solid particles can grow spontaneously to
form the final crystalline phase. Below this size they will dissolve back into the
solution. The growth of the clusters is governed by the Gibbs-Thomson equation,
which was originally derived from the study of condensing vapor. This equation
shows that the vapor pressures p of a liquid in a droplet of radius r is greater than the
saturated vapor pressure p* of liquid in a planar surface:
Literature Review

8

2
ln
*
p
p
kTr
σ
ν
⎛⎞
=
⎜⎟
⎝⎠
(2.6)
where ν is the molecular volume, k is the Boltzmann constant and T is the absolute
temperature. Ostwald modified the Gibbs-Thomson relation to determine the
equilibrium solubility of the crystalline clusters in the supersaturated solution:

2
ln
e
a
akTr
σ
ν
⎛⎞
=
⎜⎟
⎝⎠
(2.7)

where a and a
e
are the activities for the solute in the supersaturated and saturated
solutions, respectively. For a dilute solution that can be practically considered as an
ideal solution, the ratio a/a
e
is approximately equal to c/c
e
, where c and c
e
are the
actual concentration and the equilibrium solubility of the solute, respectively. This
ratio is generally defined as the supersaturation ratio S:

ee
ac
S
ac
=
≈ (2.8)
Combining Eqs.
(2.4), (2.5), (2.7) and (2.8) gives:

()
32
2
16
3ln
c
G

kT S
π
σν
Δ= (2.9)
Finally, if the growth and decay of the molecular clusters in the solution are assumed
to be at steady state and the cluster size distribution is assumed to follow the
Boltzmann distribution, an Arrhenius-type equation can be used to express the rate
of nucleus formation, i.e. the rate of nucleation J:

()
32
2
33
16
exp exp
3ln
c
G
JA A
kT
kT S
πσ ν
⎛⎞
Δ
⎛⎞
⎜⎟
=−=−
⎜⎟
⎜⎟
⎝⎠

⎝⎠
(2.10)
Literature Review

9
where J has the unit of number per unit volume per time, and the pre-exponential
factor A is usually written as
14
:

1
2
exp
a
G
kT
AN
hkT
νσ
−Δ
⎛⎞
=
⎜⎟
⎝⎠
(2.11)
where h is Planck’s constant, ΔG
a
is energy barrier for diffusion from the bulk
solution to the cluster and N
1

is number of monomeric species.
Crystal growth is the subsequent growth of the nuclei that exceed the critical size.
Similar to nucleation, the overall growth rate depends on the fluxes of the growth
units joining and leaving the crystal surface.
30
A positive net flux (joining >
leaving) results in growth of the crystal when the concentration of the growth units
in the solution is greater than the equilibrium solubility. After the growth units
diffuse to the crystal surface they move along the surface and finally integrate into
the crystal lattice. The crystal growth rate is limited by the slowest step of the
following three processes: bulk diffusion, surface diffusion, and integration. The
nature of the actual rate equation is determined by the mechanism of crystal growth,
which depends on the roughness of the crystal surface and the degree of
supersaturation.
1

2.2.2 Induction Time, Metastable Zone Width and Critical Supersaturation
Induction time is defined as the time that elapses between the creation of
supersaturated state of the mother phase and the onset of nucleation. This time
interval results from the molecules needing time to build up clusters until at least one
reaches critical size even though a sufficient driving force (supersaturation) for
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10
nucleation is present. Induction time is a kinetic parameter that is substantially
influenced by factors including the degree of supersaturation and temperature.
Experimentally, induction time can be determined by visually observing the phase
transition, measuring the concentration of the solution, or measuring some other
concentration-related properties such as turbidity.
31

For a chosen induction time, the metastable zone boundary of the parent phase is
defined as the supersaturation, below which the parent phase can stay homogeneous
without any phase transitions for a protracted period of time. The region between
the metastable zone boundary and the equilibrium solubility is generally defined as
the metastable zone. The metastable zone width (MZW) is affected by numerous
factors including the rate of supersaturation (equivalent to the cooling rate in cooling
crystallization, or the rate of solvent evaporation in vapor diffusion crystallization),
the thermal history of the solution, mechanical effects, and the amount of impurities
present.
As the initial supersaturation of the solute in the solution increases, the induction
time of the system drops. This supersaturation will eventually reach a critical value
S
c
, above which the nucleation can occur in a very short period of time. However,
critical supersaturation is not a fundamental characteristic of the system because it
depends heavily on the equipment’s sensitivity to the detection of nuclei. Hence,
critical supersaturation is a kinetic definition of metastability.
16

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11
2.2.3 Polymorphism
Polymorphism is a general phenomenon existing in solid-state crystalline
material such as small organic compounds and salts due to different arrangement
and/or conformation of constituent molecules in the crystal lattice. Polymorphs are
crystals of the same material with distinct structures giving rise to different
mechanical, thermodynamic and physical properties such as compressibility, density,
solubility and bioavailability.

5

Typically, only one polymorph is thermodynamically stable at a given
temperature and pressure except at the transition temperature between two
enantiotropic polymorphs where the relative stability of two polymorphs switches.
Unstable polymorphs may crystallize at the same physical conditions that can
transform over time to the stable polymorph because they are not the
thermodynamically preferred forms. The difference in free energies of the
metastable and stable forms can be considered as a generic thermodynamic driving
force for the polymorphic transformation.
5
Although thermodynamics compel
crystals to form the most stable polymorphs as the final products, crystallization of
metastable polymorphs is not uncommon, indicating that kinetic effects cannot be
neglected. In fact, experimental observations of crystallization of several organic
and inorganic systems often obey Ostwald's Rule of Stages.
32-35
The Rule states
that an unstable phase (say, a liquid phase) will transform to its stable form (say, a
solid phase) in steps, each step involving a formation of a metastable form and a
minimum change of free energy.
36

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12
Crystal morphology, habit, and size have tremendous practical and commercial
impact on research and development as well as the eventual mass production of
pharmaceuticals.
5

Knowledge of all possible different crystalline structures or
polymorphs of an API is important to companies for the intellectual property
protection of their products and also for the approval of the potential drug materials
by the Food and Drug Administration (FDA). Since different polymorphs exhibit
different drug delivery release profiles, it is very important to obtain only the desired
polymorph in the actual manufacturing process. Unfortunately, the number and
identity of polymorphs of an organic compound cannot be predicted through ab
initio algorithms or rules. Trial-and-error crystallization of an organic material by
screening a multidimensional parameter space of experimental conditions is the sole
option to identify possible polymorphs. Control over polymorph formation
requires physical insight into the thermodynamic stability and phase diagram of the
crystallizing system, which makes polymorphism a challenging issue in the
pharmaceutical industry.
2.2.4 Phase Behavior and Phase Diagram
All types of phase transitions such as nucleation, precipitation, gel/glass
formation, and liquid/liquid phase separation can be attributed to the consequences
of the interactions between molecules in the solution. These interactions play a
major role in determining the surface homogeneity and preferred orientation of the
molecules, and therefore control the solution phase behavior.
Over the last few years, numerous theoretical and experimental efforts have
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13
focused on interaction forces between protein molecules and how these interactions
alter solution phase behavior.
17
Many, if not most, of the theoretical studies that
predict phase behavior mainly use statistical mechanical models of simple fluids.
18-23

In this highly simplified, protein molecules are treated as spheres interacting with
isotropic potentials.
The interaction between particles is traditionally separated into a short-range
repulsion and a long-range attraction. The repulsive contribution, which comes
from the overlapping of the outer electronic shell when the particles get close to each
other, characterizes the short-range order of the particles, and thus the structure of
the thermodynamic phase. The attractive potential, which acts over a relatively
longer range and varies in a smoother way, mainly determines the physical properties
of the system. This particular fashion of separation originates from the ideas of van
der Waals, and invokes many well-established empirical expressions such as the
Lennard-Jones potential
37
:

()
12 6
4
dd
ur
rr
ε
⎡
⎤
⎛⎞ ⎛⎞
=−
⎢
⎥
⎜⎟ ⎜⎟
⎝⎠ ⎝⎠
⎢

⎥
⎣
⎦
(2.12)
where r is the center-to-center distance of separation of the particles, u(r) is the
interaction potential, ε is the depth of the potential well, and d is the collision
diameter (hard-sphere diameter). Though the Lennard-Jones potential sufficiently
characterizes the behavior of simple liquids, its mathematical complexity greatly
hinders its application in theoretical studies. Therefore, several other simplified
models have been developed for the purpose of being able to capture the essence of
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complex liquid behavior in mathematically simplified form. Among them, the
simplest is the hard sphere models:

()
,
0,
rd
ur
rd
∞
<
⎧
=
⎨
≥
⎩
(2.13)

Theoretical studies including molecular dynamics and Monte Carlo simulations have
already shown that this simple model does not differ significantly from other more
sophisticated interaction potentials in terms of structure determination of the liquid.
37

Another commonly used interaction potential of simple liquids is the square well
potential. The potential u(r) is given by:

()
,
,
0,
rd
ur d r d
rd
ε
λ
λ
∞<
⎧
⎪
=− ≤<
⎨
⎪
≥
⎩
(2.14)
where ε and λ are the strength and range of interaction, respectively, and are usually
referred to as the depth and width of the potential well. The square well potential
offers several advantages because it has been extensively studied, and the

correlations between thermodynamic and transport properties of the square well
liquids have been well described in the literature.
38-40

The limiting case of the square well potential is typically defined when the well
becomes infinitely deep while the well width is infinitely narrow such that the
potential can be written as:
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()
()
1
,
12 1
lim ln ,
0,
rd
ur d r d
d
rd
λ
τλ
λ
λ
λ
→
∞<
⎧

⎪
⎡⎤
⎛⎞
−
⎪
=≤<
⎢⎥
⎨⎜ ⎟
⎢⎥
⎝⎠
⎪
⎣⎦
⎪
≥
⎩
(2.15)
This model is called the adhesive hard sphere (AHS) and was initially introduced by
Baxter.
41
The parameter τ is a measure of the strength of attraction in the system,
and is sometimes known as the stickiness parameter. Lower values of τ indicate
greater attraction in the system while higher values designate weaker attraction, and
vice versa. The major advantage of the AHS potential is that Baxter has calculated
an exact analytical form of the equation of state that is directly beneficial to the
theoretical study of the liquid. This has been impossible for the square well
potential mentioned above as well as the Yukawa potential in which the interaction
potential is treated as a repulsive hard core with an attractive tail.
42
The Yukawa
potential is defined as:

()
()
,
exp 1
,
rd
ur
drd
rd
rd
κ
ε
∞
<
⎧
⎪
=
⎡⎤
−
⎨
⎣⎦
−
≥
⎪
⎩
(2.16)
Similar to the square well potential, the Yukawa potential a three-parameter model.
The parameter ε characterizes the depth of the attractive well at r = d, while κd,
called the decay length, specifies the range of attraction.

The parameters associated with the above-mentioned model potentials, the well
depth ε, the well width λ, and the stickiness factor τ, are not known a priori and must
be determined experimentally. For this purpose, the osmotic second virial
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16
coefficient B
2
provides a dilute solution measurement of the intermolecular
interactions. The second virial coefficient is linked to the particle pair potential by

()
()
2
2
0
21
ur kT
B
re dr
π
∞
−
=−
∫
(2.17)
Thus the parameters of relevant model potential can be calculated from B
2
, which is
an integral measure of the pair interactions between molecules. For example, in the

AHS model, the stickiness factor can be calculated from the second virial coefficient
as τ = 1/[4(1-B
2
/B
2
HS
)], where B
2
HS
= 2πd
3
/3 is the hard sphere second virial
coefficient. In this manner, B
2
can be used to characterize the pair potential u(r),
yet not to separately identify the strength and range of interactions because it is only
an integral measure. Several studies have shown that when particles interact with a
small range of attraction compared to the particle size, solubilities fall into a narrow
range of values when written as a function of B
2
/B
2
HS
.
21,23,43,44
Hence simple fluid
models can be used to develop a universal phase diagram where B
2
/B
2

HS
is plotted
against the particle volume fraction.
The above studies
21,23,43,44
show that the solubility behavior predicted from the
square well and Yukawa fluid theories captures the solid-liquid phase boundary quite
accurately. Also in the B
2
/B
2
HS
space, the solubility depends weakly on the particle
type, indicating that the solubility behavior is insensitive to the finer details of the
particle interaction potentials. On the contrary, the location of the liquid-liquid
transition is found to be strongly correlated to the range of intermolecular
interactions. In protein solutions, liquid-liquid phase separation has often been