Investigation of Amorphous Solid Dispersions of Poorly Water-soluble Drugs in Poly(2-Hydroxyethyl Methacrylate) Hydrogels for Enhanced Solubility and Controlled Release
by
Dajun Sun
A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy
Pharmaceutical Sciences University of Toronto
© Copyright by Dajun Sun (2014)
ii
Amorphous Solid Dispersions of Poorly Water-soluble Drugs in Poly(2-Hydroxyethyl Methacrylate) Hydrogels for Enhanced Solubility and
Controlled Release
Dajun Sun
Doctor of Philosophy
Pharmaceutical Sciences
University of Toronto
2014
Abstract
The purpose of this study was to investigate the potential of applying amorphous solid dispersions
(ASD) in crosslinked PHEMA hydrogels to enhance the dissolution behavior of poorly water-
soluble drugs. The first part of the study identifies physicochemical properties affecting the solid
state and physical stability of ASD of the model drug indomethacin (IND) in PHEMA hydrogels.
The results of the second part show that ASD based on water-insoluble crosslinked PHEMA can
maintain a high level of supersaturation over a prolonged duration via a diffusion-controlled
feedback mechanism, thus avoiding the initial surge of supersaturation followed by a sharp decline
in drug concentration, which is typically encountered with ASD based on water-soluble polymers
(e.g., PVP, HPMCAS) under nonsink dissolution conditions. A subsequent study examines the
effect of supersaturation generation rate on the resulting kinetic solubility profiles of amorphous
pharmaceuticals and delineates the interplay between dissolution and precipitation processes from
iii
a mechanistic viewpoint. In the absence of any dissolved polymer to inhibit drug precipitation
from the supersaturated state, both our experimental and predicted results confirm that the faster
rise of the kinetic solubility profile of an amorphous drug will inevitably lead to an earlier but
higher maximum kinetic solubility and a sharper decline in the de-supersaturation phase, and vice
versa. The relationship between the achievable maximum supersaturation and the rate of
supersaturation generation in the observed kinetic solubility profiles has been described for the
first time by our comprehensive mechanistic model taking into account the role of supersaturation
in both the nucleation and crystallization processes as well as the associated competitive particle
growth and ripening effects. Finally, this theoretical framework was further employed to semi-
quantitatively predict the evolution of supersaturation of amorphous pharmaceuticals generated
from nonlinear dissolution profiles. The effects of initial degree of supersaturation, dissolution of
amorphous drug and that from the IND-PHEMA ASD under nonsink dissolution conditions were
subsequently examined in detail. The comparison of dissolution behaviors between amorphous
IND and IND-PHEMA ASD demonstrates the advantage of the diffusion-controlled feedback
mechanism that makes crosslinked PHEMA a unique and desirable carrier for amorphous drug
delivery systems.
iv
Acknowledgements
For those who studied in the disciplines of science and engineering, the term “problem set” should
sound familiar. The journey of my Ph.D. study feels like a very long problem set in which an
answer to a question indefinitely leads to and expands to more, larger and harder questions. This
daunting task would not have been achievable without a great amount of guidance, support and
assistance from many wonderful individuals. First and foremost, I would like to express my
greatest gratitude to my Ph.D. supervisor Professor Ping Lee for his endless advice, support and
supervision throughout my research project. His scientific insights and comprehensive expertise
in pharmaceutical science has not only provided immense leadership to this project but also shaped
me as an independent researcher. I have learnt many great personal characteristics from him,
especially diligence and professionalism. I will always remain thankful to his generous education.
Also, I am greatly thankful to my advisory committee members, Professors Edgar Acosta,
Christine Allen and Shirley Wu, for providing me with valuable guidance and direction to my
research as well as precious career advice. My thanks to Professor Carolyn Cummins for
completing my experience as a graduate student by giving me a teaching opportunity. I am grateful
to Dr. Rob Ju from Abbvie for a delightful experience of academia-industry collaboration. Special
thanks are due to the helpful staff and students in the Department of Geology, the Department of
Chemistry, University Health Network pre-formulation lab and Professor Allen’s lab for kindly
assisting me to use their laboratory equipment.
This research work was supported by research funding from Abbvie and Natural Sciences and
Engineering Research Council of Canada (NSERC), and I was also supported by a University of
Toronto Fellowship Award.
v
I would like to thank the past and present colleagues in the PIL research group, Dr. Beibei Qu, Dr.
Hui Zhao, Dr. Yan Li, Dr. Hongliang Jiang, Dr. Yanhong Luo, Sammi Liu, Arthur Li, Giovanna
Medeiros and many others, for their continuous assistance and advice. Working together with them
is always a nice and memorable experience. Lastly, I would like to thanks my family and friends,
especially my parents, my brother Peter and Che-chien Wang for their unreserved faith in me. I
could not have reached my goals without their unconditional love and support.
vi
Publications
- Dajun D. Sun and Ping I. Lee “Crosslinked hydrogel – a promising class of insoluble solid
molecular dispersion carriers for enhancing the delivery of poorly soluble drugs” Acta
Pharmaceutica Sinica B, Volume 4, Issue 1, pp 26-36 (2014). [Invited Review and Cover Story]
- Dajun D. Sun and Ping Lee, “Evolution of supersaturation of amorphous pharmaceuticals: the
effect of rate of supersaturation generation” Molecular Pharmaceutics, Volume 10, Issue 11, pp.
4330-4346 (2013).
- Dajun D. Sun, Tzu-chi Rob Ju, Ping I. Lee, “Enhanced kinetic solubility profiles of indomethacin
amorphous solid dispersions in poly(2-hydroxyethyl methacrylate) hydrogels” European Journal
of Pharmaceutics and Biopharmaceutics, Volume 81, Issue 1, pp. 149-158 (2012).
vii
Table of Contents
Abstract ii
Acknowledgments iv
Publications vi
Table of contents vii
List of tables xii
List of figures xiv
List of appendices xxi
List of symbols xxiii
List of abbreviations xxv
Chapter 1: Introduction
1.1 Solubility enhancement of poorly water-soluble drugs for oral drug delivery 1
1.2 Pharmaceutical significance of amorphous solid dispersions in polymeric carriers 3
1.2.1 Water-soluble carriers 7
1.2.2 Water-insoluble carriers 14
1.3 Crosslinked PHEMA hydrogels for amorphous solid dispersions carriers 16
1.4 Crystallization of amorphous pharmaceuticals in the solid state 19
1.4.1 Solubility advantage of amorphous solids 19
1.4.2 Classical nucleation theory (solid state) 21
1.4.3 Crystal growth (solid state) 23
1.4.4 Kolmogorov-Johnson-Mehl-Avrami (KJMA) theory 24
1.5 Crystallization of supersaturated drug solutions 25
1.5.1 Classical nucleation theory (solution state) 25
1.5.2 Crystal growth (solution state) 26
viii
1.6 Overview of the Ph.D. research 27
1.6.1 Hypothesis 27
1.6.2 Research objectives 27
Chapter 2: Indomethacin amorphous solid dispersions in PHEMA
2.1 Introduction 29
2.2 Materials and methods 31
2.2.1 Materials 31
2.2.2 Synthesis of PHEMA hydrogel beads 31
2.2.3 Preparation of amorphous solid dispersion systems 34
2.2.4 Scanning electron microscopy (SEM) 36
2.2.5 X-ray diffraction (XRD) 36
2.2.6 Differential scanning calorimetry (DSC) 37
2.2.7 Fourier-transformed infrared (FTIR) spectroscopy 37
2.2.8 Solubility parameter estimation 37
2.3 Results and discussion 40
2.3.1 PHEMA hydrogel beads synthesis 40
2.3.2 Physical properties of ASD IND in PHEMA, PVP and HPMCAS 44
2.3.3 IND-polymer interactions 55
2.3.4 Solubility parameters 57
2.4 Conclusion 59
Chapter 3: Physical stability of amorphous indomethacin in PHEMA
3.1 Introduction 60
3.2 Materials and methods 62
3.2.1 Materials 62
3.2.2 Preparation of amorphous solid dispersion systems 63
3.2.3 Stability study 63
3.2.4 Preparation of physical mixtures of amorphous, crystalline - and -indomethacin
in polymeric carriers 63
3.2.5 Raman spectroscopy 64
3.2.6 Multivariate data analysis 65
ix
3.2.7 Isothermal crystallization kinetics 65
3.2.8 Water sorption isotherm 66
3.3 Results and discussion 66
3.3.1 Storage stability study 66
3.3.2 Quantification of ternary mixtures of different solid-state forms of indomethacin in
polymeric carriers 68
3.3.3 Crystallization kinetics of amorphous indomethacin in polymeric carriers 74
3.3.4 Estimation of drug solubility in polymers 78
3.3.5 Intermolecular forces between amorphous indomethacin and polymers 80
3.3.6 Analysis of isothermal water vapor absorption 82
3.4 Conclusion 86
Chapter 4: Enhanced kinetic solubility profiles of amorphous indomethacin in PHEMA
4.1 Introduction 87
4.2 Materials and methods 88
4.2.1 Materials 88
4.2.2 Dissolution testing of amorphous solid dispersion systems under nonsink
dissolution conditions 88
4.3 Results and discussion 90
4.3.1 Comparing solubility advantages of amorphous IND in PHEMA with that in water-
soluble polymers 90
4.3.2 Diffusion-controlled release of amorphous IND from PHEMA hydrogels 96
4.4 Conclusion 101
Chapter 5: Effect of rate of supersaturation generation on the kinetic solubility profiles
5.1 Introduction 103
5.2 Theory 106
5.3 Materials and methods 111
5.3.1 Materials 111
5.3.2 Measurement of kinetic solubility profiles 111
5.3.3 X-ray diffraction (XRD) 113
5.3.4 Differential scanning calorimetry (DSC) 114
x
5.3.5 Scanning electron microscopy (SEM) 114
5.3.6 Particle size distribution 114
5.3.7 Simulation of modeling equations 115
5.4 Results and discussion 115
5.4.1 Effect of rate of supersaturation generation on the kinetic solubility profiles 115
5.4.2 Amorphous solid dispersions in polymeric carriers 127
5.4.3 Kinetic solubility advantage of amorphous solids 129
5.4.4 Crystallization kinetics and particle size distribution 132
5.4.5 Evolution of concentration-time profiles due to dissolution and recrystallization
processes 140
5.5 Conclusion 142
Chapter 6: Semi-quantitative prediction of the kinetic solubility profiles of amorphous
indomethacin
6.1 Introduction 144
6.2 Theory 146
6.2.1 Effect of linear rate of supersaturation generation 146
6.2.2 Effect of initial degree of supersaturation 146
6.2.3 Effect of first-order supersaturation generation 147
6.2.4 Supersaturation generation with diffusion-controlled feedback mechanism –
Dissolution of amorphous solid dispersions from PHEMA hydrogel beads 148
6.3 Materials and methods 152
6.3.1 Materials 152
6.3.2 Preparation of amorphous indomethacin 152
6.3.3 Dissolution testing of amorphous IND and ASD IND-PHEMA under nonsink
dissolution conditions 152
6.3.4 Dissolution testing of solid-state amorphous IND under sink dissolution
conditions 153
6.4 Results and discussion 154
6.4.1 Effect of initial degree of supersaturation 154
6.4.2 Dissolution of amorphous IND from the solid state 158
6.4.3 Dissolution of amorphous IND from PHEMA hydrogels 163
xi
6.5 Conclusion 170
Chapter 7: Summary and future research directions
7.1 Summary 172
7.2 Future research directions 175
Appendices 177
References 202
xii
List of Tables
Chapter 1
Table 1.1: Selected studies of water-soluble carriers for amorphous drugs... 11
Table 1.2: Selected examples of insoluble carriers for amorphous drugs …. 13
Chapter 2
Table 2.1: Chemical structures of IND, PVP and HPMCAS and their
potential hydrogen bonding sites ………………………………. 36
Table 2.2: Solubility parameters of component group contribution from
van Krevenlen/Hoftyzer and Hoy’s methods ………………….. 40
Table 2.3: Measured and calculated solubility parameters of IND, PVP,
HPMCAS, PHEMA and PHEMA copolymers ………………… 58
Chapter 3
Table 3.1: Comparison of existing technology of quantification of drug
crystallinity in ASD ……………………………………………. 62
Table 3.2 Physical mixtures of crystalline -, - and amorphous IND …… 64
Table 3.3 Stability study of IND solid dispersions in PHEMA beads …..... 67
xiii
Table 3.4 Raman molecular assignment for the C=O stretching group …... 70
Table 3.5 KJMA isothermal crystallization parameters for amorphous
IND ……………………………………………………………... 77
Table 3.6 IND solubility in PHEMA, PVP and HPMCAS ……………….. 80
Table 3.7 Characteristics of the model drug and polymers relevant for
hydrogen bonding ……………………………………………… 81
Chapter 5
Table 5.1: Physicochemical properties of model poorly water-soluble IND,
NAP and PIR …………………………………………………… 112
Table 5.2 Summary of relevant physical constants for the numerical
simulation of IND crystallization kinetics …………………… 120
Chapter 6
Table 6.1: Dissolution rate constants of amorphous IND of various particle
size ranges ……………………………………………………… 161
xiv
List of Figures
Chapter 1
Figure 1.1: Biopharmaceutics Classification System (BCS) of drugs ………………. 2
Figure 1.2: Thermodynamic description of different solid states …………………… 4
Figure 1.3: Gibb’s free energy levels of the amorphous state (metastable), crystalline
(stable) and unstable state …..………………..………………………….. 5
Figure 1.4: Schematics of substitutional, interstitial and polymeric solid solution ….. 6
Figure 1.5: Classification of solid dispersion/solution of drug molecules in
polymeric carrier matrix …………………………………………………. 7
Figure 1.6: Dissolution performance of ASD containing a model poorly water-
soluble compound ………………………………………………………. 9
Figure 1.7 “Spring” and “parachute” dissolution behaviors ………………………… 10
Figure 1.8: Free energy diagram for nucleation process ……………………………. 22
Chapter 2
Figure 2.1: Experimental apparatus for PHEMA hydrogel beads synthesis ……….... 33
Figure 2.2: Microscopic images of PHEMA hydrogel beads ……………………….. 34
xv
Figure 2.3: Quantification of IND by UV-spectrometer …………………………….. 35
Figure 2.4: Particle size distributions of PHEMA hydrogel beads .………………….. 42
Figure 2.5: Images of failed batches of crosslinked PHEMA hydrogels…………….. 43
Figure 2.6: Equilibrium solvent content of PHEMA and IND solubility in
DMSO/ethanol mixtures for IND loading process ……………………… 45
Figure 2.7: IND loading levels in PHEMA hydrogel beads as a function of loading
solution concentration ………………………….. ……………………… 46
Figure 2.8: Residual solvent content in crosslinked PHMEA hydrogel after
equilibrium sorption …………………………………………………….. 47
Figure 2.9: Microscopic images of IND-loaded PHEMA hydrogel beads ………….. 48
Figure 2.10: Microscopic images of IND-loaded PHEMA-co-MMA, -co-EMA and -
co-BMA hydrogel beads ………………………………………………… 49
Figure 2.11: Microscopic images of IND-loaded PVP and HPMCAS cast films ……. 50
Figure 2.12: SEM images of surface morphology of IND-loaded PHEMA hydrogel
beads …………………............................................................................... 51
Figure 2.13: SEM images of cross-section of IND-loaded PHEMA hydrogel beads … 51
Figure 2.14: XRD spectra of IND-loaded PHEMA, PVP and HPMCAS …………….. 52
Figure 2.15: DSC thermograms of IND-loaded PHEMA, PVP and HPMCAS ………. 54
Figure 2.16: Carbonyl stretch region of FTIR spectra of ASD and physical mixture
IND-loaded PHEMA, PVP and HPMCAS ……………………………… 56
Figure 2.17: Determination of solubility parameters of PHEMA hydrogels by Gee’s
method of equilibrium swelling …………………………………………. 58
xvi
Chapter 3
Figure 3.1: Evolution of physical appearance of IND-PHEMA discs ………………. 67
Figure 3.2: XRD spectra of IND-PHEMA ASD after 8-month stability study ……… 68
Figure 3.3: XRD spectra of IND of different solid state forms and polymer carriers .. 69
Figure 3.4: Raman spectra of IND of different solid state forms and polymer carriers 70
Figure 3.5: Raman spectra and their statistical peak fittings of physical mixtures of
crystalline -. - and amorphous IND in PHEMA, PVP and HPMCAS in
the carbonyl stretching region ………………………………………….. 71
Figure 3.6: Raman spectra and their statistical peak fittings of physical mixtures of
different amount of -IND and amorphous IND in PHEMA …………… 72
Figure 3.7: Calibration standard curves of intensity ratio between IND polymorphs
and polymers …………………………………………………………….. 73
Figure 3.8: Effect of relative humidity on crystallization kinetics of IND in PHEMA
hydrogel (0-54% RH) ……………………………………………………. 75
Figure 3.9: Effect of relative humidity on crystallization kinetics of IND in PHEMA
hydrogel discs (76 and 95% RH) ……………………………………….. 75
Figure 3.10: Effect of temperature on crystallization kinetics of IND in PHEMA …… 76
Figure 3.11: Effect of the polymer type on crystallization kinetics of IND ………….. 78
Figure 3.12: Water sorption isotherms of -form crystalline and amorphous IND,
PHEMA, PVP and HPMCAS …………………………………………… 83
Figure 3.13: Water sorption isotherm of ASD of IND (25% drug loading) in polymers 84
Figure 3.14: DSC endotherms of polymers in various RH storage conditions ……….. 85
xvii
Chapter 4
Figure 4.1: Effect of drug loading on kinetic solubility profiles of ASD IND in
PHEMA, PVP and HPMCAS …………………………………………… 91
Figure 4.2: Effect of polymer type on kinetic solubility profiles of ASD IND in
PHEMA, PVP and HPMCAS …………………………………………… 94
Figure 4.3: Effect of particle size on kinetic solubility profiles of ASD IND-
PHEMA …………………………………………………………………. 95
Figure 4.4: Distribution of IND in the hydrogel and dissolution medium after 24 h of
dissolution from IND-PHEMA ASD …………………………………… 97
Figure 4.5: Effect of crosslinked PHEMA hydrogels in the dissolution medium on
the equilibrium solubility of crystalline IND.…………………………… 98
Figure 4.6: Amorphous drug release mechanism: crosslinked hydrogels vs. water-
soluble polymers ………………………………………………………… 100
Chapter 5
Figure 5.1: Schematic depiction of the nucleation and crystallization events due to
supersaturation generation by the infusion experiment …………………. 106
Figure 5.2: Representative UV absorbance spectra and second derivative spectra for
IND, NAP and PIR ………………………………………………………. 116
Figure 5.3 Calibration curve of drug concentration and the second-derivative UV
spectra for model drugs ………………………………………………….. 117
Figure 5.4: Elimination of spectral interference by second-derivate UV technique ….. 118
Figure 5.5: Experimental kinetic solubility profiles of IND, NAP and PIR as a
function of supersaturation rate generated with infusion rates ………….. 119
xviii
Figure 5.6: Comparison of experimental data and simulation results of kinetic
solubility profiles of IND as a function of supersaturation rate …………. 123
Figure 5.7: Predicted kinetic solubility profiles of IND as a function of
supersaturation rate ……………………………………………………… 124
Figure 5.8: Effect of rate of supersaturation generation on Cmax and AUC of kinetic
solubility profiles …………………………….………………………….. 125
Figure 5.9: Effect of rate schedule of supersaturation on kinetic solubility profiles.... 127
Figure 5.10: Effect of rate schedule of solid addition on kinetic solubility profiles of
IND-PVP ASD …………………………………………………………... 129
Figure 5.11: Cmax measured during the infusion experiment as a function of the
inverse of infusion rate raised to an exponent α …………………………. 130
Figure 5.12: Crystallization kinetics of IND, NAP and PIR as a function of
supersaturation rate generated with various drug solution infusion rates .. 133
Figure 5.13: XRD spectra of precipitated IND from the infusion experiment ..………. 135
Figure 5.14: DSC thermograms of precipitated IND from the infusion experiment ..… 136
Figure 5.15: SEM images of precipitated IND from the infusion experiment ..………. 137
Figure 5.16: Particle size distributions of precipitated IND from the infusion
experiment ……………………………………………………………….. 139
Figure 5.17: Simulated time-dependent growth of IND particle size under various
rates of supersaturation generation ………………………………………. 139
Figure 5.18: Conceptual concentration-time profile during dissolution and
precipitation of amorphous sparingly soluble drugs …………………….. 141
xix
Chapter 6
Figure 6.1: Schematics of diffusion-controlled release of amorphous drug from a
PHEMA hydrogel bead into a finite volume of dissolution medium ……. 149
Figure 6.2: Kinetic solubility profiles of IND at various initial degrees of
concentration …………………………………………………………….. 154
Figure 6.3: COMSOL simulation of kinetic solubility profiles of various initial
degrees of supersaturation ……………………………………………….. 156
Figure 6.4: Cmax and AUC of in vitro dissolution data and simulation results of the
kinetic solubility profiles of various initial degrees of supersaturation ….. 158
Figure 6.5: Kinetic solubility profiles of amorphous IND of different particle size
ranges under nonsink dissolution conditions…………………………….. 159
Figure 6.6: Dissolution of amorphous IND with various particle size ranges under a
sink dissolution condition ……………………………………………….. 160
Figure 6.7: COMSOL simulation of kinetic solubility profiles of amorphous IND
with various particle size ranges under nonsink dissolution conditions ... 162
Figure 6.8: Comparison of IND dissolution profiles from purely amorphous IND
versus that of IND-PHEMA ASD (10 wt % IND loading) ……………… 164
Figure 6.9: Dissolution profiles over an extended time period from IND-PHEMA
ASD (10 wt% drug loading) and crystalline IND ………..……………… 165
Figure 6.10: Determination of the partition coefficient p between PHEMA hydrogel
and external dissolution medium ………………………………………... 166
Figure 6.11: Illustration of one-dimensional lengths for one hydrogel bead and the
external dissolution medium in COMSOL simulation …………….….. 166
Figure 6.12: Diffusion coefficient of IND in PHEMA hydrogels …….………………. 168
xx
Figure 6.13: Comparison of COMSOL simulation results of kinetic solubility profiles
between amorphous IND solids and IND-PHEMA ASD system ……….. 170
xxi
List of Appendices
Chapter 2
Figure A2.1 XRD spectra of physical mixtures of IND and PHEMA with
various IND weight percentage ………………………………. 177
Figure A2.2 DSC thermograms of physical mixtures of IND and PHEMA
with various IND weight percentage …………………………. 178
Chapter 5
Table A5.1 Parameter input in COMSOL 3.5a (Figures 5.6 and 5.7) ….…. 180
Chapter 6
Table A6.1 Parameter input in COMSOL 3.5a (Figure 6.3) ……………… 184
Table A6.2 Parameter input in COMSOL 3.5a (Figure 6.7) ……………… 188
Table A6.3 Parameter input in COMSOL 3.5a (Figure 6.13) …………….. 192
Figure A6.1 COMSOL simulation results of time-dependent concentration
profiles of IND inside PHEMA hydrogel beads under various
sink conditions ……………………………………………….. 196
xxii
Figure A6.2 IND release profiles in the external dissolution medium from
Figure A6.2 …………………………………………………… 199
xxiii
List of Symbols
C concentration
Cb bulk concentration
Ceq concentration in equilibrium with a precipitate particle of radius r
Cmax maximum concentration
CS equilibrium solubility
Ct concentration assuming all drugs have dissolved
D diffusion coefficient
d molecular size (twice the molecular diameter)
Dose dose required
f molar concentration of precipitate
Fi molar attraction constant
G crystallization kinetics coefficient
J nucleation rate
Jc nucleation rate constant
k rate constant
kB Boltzmann’s constant
mo repeating monomer’s MW
MW molecular weight
N average particle number density
No Avogadro’s number
p partition coefficient
Q equilibrium swelling ratio
r particle size
xxiv
R rate of drug input
rn critical particle size
s supersaturation
s’ measured supersaturation at time t”
T temperature
t time
t’ time in which addition of dissolved drugs in solvent stops
t” time in which dissolution stops
u precipitation rate
V (molar) volume
wd dry weight of hydrogels
ws swollen weight of hydrogels
δ solubility parameter
ε thickness of Gibbs surface
ρ density
σ interfacial tension
Σ average surface of the precipitate particles per unit volume
υ molar volume of precipitate
Φ average radius of the particles per unit volume
ϕ volumetric fraction
Ψ radius dependence of the surface tension
ω capillary length
xxv
List of Abbreviations
AIBN Azobisisobutyronitrile [2,2′-Azobis(2-methylpropionitrile), 2-(azo(1-cyano-1-
methylethyl))-2-methylpropane nitrile]
API Active pharmaceutical ingredient
ASD Amorphous solid dispersion
AUC Area under the curve
BCS Biopharmaceutics classification system
BMA Butyl methacrylate
CAP Cellulose acetate phthalate
CNT Classical nucleation theory
DSC Differential scanning calorimetry
DMSO Dimethyl sulfoxide
ESC Equilibrium solvent content
EMA Ethyl methacrylate
EDGMA Ethylene glycol dimethylacrylate [2-(2-Methyl-acryloyloxy)ethyl 2-methyl-
acrylate]
FDA Food and Drug Administration
FTIR Fourier-transform infrared
GFA Glass forming ability
GI Gastrointestinal
GS Glass stability
ICH International conference on harmonization
Tg Glass transition temperature
HEMA Hydroxyethyl methacrylate [2-Hydroxyethyl 2-methylprop-2-enoate]
xxvi
HPMC Hydroxypropyl methylcellulose
HPMCAS Hydroxypropyl methylcellulose acetate succinate
HPMCP Hydroxypropylmethyl cellulose phthalate
IND Indomethacin [2-{1-[(4-chlorophenyl)carbonyl]-5-methoxy-2-methyl-1H-indol-3-
yl}acetic acid]
IR Infrared
MMA Methyl methacrylate
MW Molecular weight
NAP Naproxen [(+)-(S)-2-(6-methoxynaphthalen-2-yl) propanoic acid]
NMR Nuclear magnetic resonance
ODE Ordinary differential equation
PDE Partial differential equation
PHEMA Poly(2-hydroxyethyl methacrylate)
PIR Piroxicam [(8E)-8-[hydroxy-(pyridin-2-ylamino)methylidene]-9-methyl-10,10-
dioxo-10λ6-thia-9-azabicyclo[4.4.0]deca-1,3,5-trien-7-one]
PLS Partial least square
PVP Polyvinylpyrrolidone
RH Relative humidity
SEM Scanning electron microscopy
SI Sink index
SNV Standard normal variate
ss Solid state
UV Ultraviolet
XRD X-ray diffraction
1
Chapter 1
Introduction
1.1 Solubility enhancement of poorly water-soluble drugs for oral drug
delivery
Recent advances in combinatorial chemistry, automated synthesis and high-throughput screening
have significantly improved effectiveness of the drug discovery process (White, 2000). Although
many of these new chemical entities exhibit promising therapeutic potential, one major dilemma
in developing these candidate compounds into oral dosage forms, the most popular route of
administration, is their poor aqueous solubility and/or permeability across the intestinal villi in the
gastrointestinal (GI) tract. According to the Biopharmaceutics Classification System (BCS), drug
substances are classified into four categories according to their solubility and permeability
properties, as shown in Figure 1.1. Absorption of orally administered medications involves
solubilization of drug molecules in the GI fluid and transport across membranes of the epithelial
cells in the GI tract. Therefore, poorly water-soluble and/or poorly permeable drugs with
suboptimal bioavailability require enabling formulation approaches to develop oral dosage forms
with an enhanced solubility and/or permeability to reach systemic circulation. Improving drug
permeability often requires chemical modifications of the drug, which may fundamentally alter its
toxicity profile and therapeutic potential. On the other hand, various techniques are available for
the enhancement of the aqueous solubility of poorly water-soluble drugs without chemically
altering the molecular structure of the drug. It is estimated that approximately 60-70% of newly
discovered therapeutic compounds are classified in BCS II (high permeability, low solubility)
during the drug discovery process (Douroumis and Fahr, 2012; Lipinski et al., 1997). Hence, a
2
significant amount of current research focuses on enhancing the aqueous solubility of these BCS
II, small molecular weight (MW less than 1000 Da) active pharmaceutical ingredients (API).
Figure 1.1: The Biopharmaceutics Classification System (BCS) of drugs is based on the solubility
threshold above which the highest required dose is soluble in less than 250-mL aqueous media
over the pH 1.0 to 7.5 and the permeability threshold above which the extent of absorption in
humans is determined to be more than 90% of the dose administered by i.v.
(http://www.fda.gov/AboutFDA/CentersOffices/OfficeofMedicalProductsandTobacco/CDER/uc
m128219.htm).
In previous attempts to address this issue, methods commonly used to increase drug solubility have
practical limitations and may not always accomplish the desired enhancement in drug solubility
and bioavailability (Fahr and Liu, 2007; Pouton, 2006). For example, particle size reduction (or
increase in surface area) often has a threshold of achievable size reduction; creating stable salt
forms or pro-drug of therapeutic agents is not always feasible; introducing surfactants or co-
solvents may lead to liquid formulations that are usually poor in patient acceptability and
undesirable for commercialization (Serajuddin, 1999). Among various known approaches,
incorporating a poorly water-soluble compound in a suitable polymeric carrier to form an
amorphous solid dispersion (ASD) has become an increasingly important strategy in the solubility
and bioavailability enhancement for oral delivery of poorly water-soluble compounds. Various
approaches for the preparation, characterization and stabilization of ASDs for oral drug delivery
http://www.fda.gov/AboutFDA/CentersOffices/OfficeofMedicalProductsandTobacco/CDER/ucm128219.htmhttp://www.fda.gov/AboutFDA/CentersOffices/OfficeofMedicalProductsandTobacco/CDER/ucm128219.htm
3
have been reviewed comprehensively (Chiou and Riegelman, 1971; Craig, 2002; Kawakami, 2009;
Leuner and Dressman, 2000; Serajuddin, 1999; Yu, 2000). Poorly water-soluble drugs in their
stabilized amorphous form can generate a transient but highly supersaturated solution
concentration (i.e., kinetic solubility) significantly greater than the equilibrium saturation
concentration of their crystalline counterparts. Since drug supersaturation increases the driving
force for oral absorption, maintaining an elevated and sustained level of drug supersaturation is
critical to improving the bioavailability of poorly water-soluble drugs. The causality between
increased kinetic solubility from ASDs and improved oral bioavailability has been demonstrated
in many in vivo studies (Kohri et al., 1999; Law et al., 2004; Newa et al., 2007; Six et al., 2005;
Verreck et al., 2004; Yüksel et al., 2003).
1.2 Pharmaceutical significance of amorphous solid dispersions in polymeric
carriers
Solids in the amorphous state (also referred to as “unstable form”, “high-energy state”) are
structurally defined as the lack of a long-range order of molecular packing or the lack of a
crystalline state. The amorphous state has a higher internal energy and specific volume compared
to the crystalline state (Figure 1.2). Due to “loose” molecular packing, the amorphous state has a
high level of free energy, which leads to various physicochemical characteristics such as an
elevated aqueous solubility, higher vapor pressure, greater molecular mobility and higher chemical
reactivity than their crystalline counterpart (Figure 1.3). In contrast to the equilibrium solubility,
which is an intrinsic thermodynamic property of the crystalline drug, the dissolution of amorphous
pharmaceuticals achieves a transient supersaturation (i.e., a kinetic drug solubility that is higher
than the equilibrium solubility), as a result of the lack of crystalline lattice energy of the solids.
High-energy formulations based on amorphous pharmaceuticals can therefore improve oral
bioavailability of poorly water-soluble drugs by generating supersaturated drug solutions in the GI
tract. However, the metastable amorphous structure will be eventually converted to the equilibrium
crystalline state, provided that there is a thermodynamic driving force sufficient to overcome the
Gibb’s free energy barrier. The threshold drug loading level in ASD systems in different polymeric
matrices above which an amorphous-to-crystalline transition tends to occur has typically been
identified empirically. Stressed conditions including high relative humidity, elevated temperature
and aging are known to accelerate the kinetics of transformation from metastable amorphous drugs
4
into the crystalline state (Konno and Taylor, 2008). Unfortunately, questions on their solid-state
structure, mechanisms of dissolution enhancement, and criteria of solid dispersion stability upon
storage have remained mostly unanswered (Craig, 2002). Recurrence of crystallization in many of
these systems still represents the primary factor affecting product stability. The poor predictability
of ASD stability is due to the lack of a more fundamental understanding of their physical properties
and parameters which govern the stability of amorphous structures in retarding the initiation of
nucleation and the propagation of drug crystallization. The formation of stabilized ASD in an inert
carrier has been shown to be very effective in delaying nucleation and crystallization of amorphous
drugs to achieve a reasonable shelf life of pharmaceutical products.
Figure 1.2: Thermodynamic description of different solid states (crystalline, amorphous and
supercooled liquid).
5
Figure 1.3: Gibb’s free energy levels of the amorphous state (metastable), crystalline state (stable)
and unstable state.
From a list of pharmaceutically acceptable excipients that are FDA-approved in human oral
delivery, ideal carrier matrix systems have to demonstrate the ability to maintain amorphous drug
in solid dispersions, enhancing drug dissolutions and subsequent bioavailability, and a potential of
programmable release rates through the GI tract. Earlier investigations of incorporating poorly
water-soluble drugs in solubilizing agents were mostly concerned with identifying rapidly
dissolving small MW carriers (e.g., sulfathiazole-urea (Sekiguchi and Obi, 1961)) and polymeric
carriers (e.g., sulfathiazole-polyvinylpyrrolidone PVP (Simonelli et al., 1969)) to improve the drug
dissolution rate, the degree of supersaturation and the oral bioavailability. The solute molecules
(amorphous drug) can either be substituted for solvent molecules or be fitted into the interstices
between the solvent molecules in a eutectic mixture (Figure 1.4 A&B) (Leuner and Dressman,
2000). Recent research interests have gradually been shifting to relevant physicochemical
properties of drug-carrier composites that can enhance the stabilization of metastable amorphous
drugs, particularly in the form of solid solutions in polymeric carriers because of their ability to
inhibit nucleation and crystal growth in the solid state (Figure 1.4 C) (Bhugra and Pikal, 2008;
Khougaz and Clas, 2000b; Konno and Taylor, 2007). This generally involves the preparation of
solid solutions in polymers that have a glass transition temperature (Tg) higher than room
temperature as the low molecular mobility of which contributes to a slow crystallization rate at
ambient conditions. A variety of these carrier matrices can be categorized as hydrophilic,
6
hydrophobic, enteric and an association polymer system. In some cases, a ternary (polymer A-
polymer B-drug) system might be necessary to achieve both stabilization of an amorphous solid
state and controlled release. More recently, refinement of the solid dispersion approach has been
pursued to include surfactants, plasticizer, alkalizer, copolymer and disintegratants to form the
desired carrier matrices (Broman et al, 2001; Ghebremeskel et al, 2006; Tran et al, 2009).
Figure 1.4: Schematics of (A) substitutional (B) interstitial and (C) polymeric solid solution. Figure
adapted from Leuner and Dressman (Leuner and Dressman, 2000) (reproduced with permission
from the European Journal of Pharmaceutics and Biopharmaceutics, Copyright Elsevier 2000).
Entrapped drug molecules in polymeric carrier matrices can be classified as crystalline solid
dispersion (2-phase system), amorphous solid dispersion (2-phase system) and solid solution (1-
phase system) as illustrated in Figure 1.5. Compared to the 1-phase solid solution in which drug
molecules are molecularly dispersed in the carrier, 2-phase solid dispersion systems contain a
separate phase of either crystalline (long-range molecular order) or amorphous drugs (short-range
molecular order). Pharmaceutical solid dispersion systems are rarely in a completely amorphous
state. In most cases, pharmaceutical solids are in a state between crystalline and amorphous solids
(i.e., a mixture or hybrid of both the 2-phase and 1-phase systems). The magnitude of crystallinity
can be measured by x-ray diffraction (XRD) analysis, spectroscopic analysis (e.g., IR and Raman)
and thermal analysis (e.g., differential scanning calorimetry DSC and thermally stimulated current
TSC).
(A) (B) (C)
7
Figure 1.5: Classification of solid dispersion/solution of drug molecules in polymeric carrier
matrix. Figure adapted from Sun and Lee (Sun and Lee, 2014) (reproduced with permission from
Acta Pharmaceutica Sinica B, Copyright Elsevier 2014).
1.2.1 Water-soluble carriers
It is a common practice to employ water-soluble polymers as carriers in conventional ASDs to
enhance solubility and dissolution rate. The conventional design of oral dosage forms based on
amorphous solid dispersions for poorly water-soluble drugs typically focuses on increasing the
dissolution rates, elevating the degree of supersaturation and extending its duration following the
dissolution of various ASD systems. Table 1.1 summarizes recent studies of water-soluble ASD
carriers such as hydrophilic polymers and enteric polymers utilized to convert poorly water-soluble
model drugs into amorphous formulations. In this case, hydrophilic or hydrocolloid matrices form
solid dispersion systems from which water-soluble polymers and entrapped amorphous drugs can
dissolve in an aqueous medium. Available references commonly show enhanced dissolution rates
of poorly soluble drugs from hydrophilic polymers including PVP and derivatives of cellulose
such as hydroxypropyl methyl cellulose (HPMC or hypromellose), methylcellulose (MC),
carboxymethyl cellulose sodium, and hydroxyethyl cellulose (HEC). In addition, enteric polymers
are preferentially soluble in a pH environment above pH 5.5 to pH 6 of the intestine relative to the
acidic gastric fluid. For instance, rapid disintegration of enteric polymers like hydroxypropyl
8
methylcellulose phthalate (HPMCP) and hydroxypropyl methylcellulose acetate succinate
(HPMCAS) can be achieved in an elevated pH environment. Moreover, it is worth noting that the
frequently employed solid dispersion carrier polyethylene glycol (PEG) has a low glass transition
temperature (Tg) and is therefore more “rubbery” at ambient temperature. This unfortunately does
not offer much retardation in the rates of drug diffusion and crystallization. In fact, most PEG-
based solid dispersions are truly dispersions of micro-crystalline drug particles. A typical example
of this micro-crystalline drug dispersion is a product called Gris-PEG®, a solid crystalline
dispersion of griseofulvin in PEG. Ideal candidates of polymeric carriers should prevent
amorphous pharmaceutical solids from nucleating and becoming crystalline in order to achieve the
desired stability during shelf life.
Over the past decade, numerous in vitro and in vivo studies have displayed a variety of
combinations of carrier polymers and solid dispersion systems of poorly water-soluble drugs to
demonstrate the effectiveness of amorphous pharmaceuticals in solubility and bioavailability
enhancement. Regrettably, almost none of the research results drew any proper mechanistic
conclusions. Many of the literature reports investigated methods of preparation and evaluated
concomitant pharmacokinetic improvement of the drug-polymer amorphous solid dispersions by
means of in vivo studies. Well-characterized water-insoluble drugs such as indomethacin,
nifedipine, felodipine, and itraconazole were frequently investigated for their enhanced dissolution
rates in various polymer carriers. The common methods can generally be classified as solvent-
based (e.g., dissolution in co-solvent followed by solvent evaporation, spray-drying,
electrospinning), temperature-based (e.g., hot-stage melting, hot-melt extrusion), and physically
mixing (e.g., granulation, milling) in order to achieve a homogenous binary, ternary (Janssens et
al., 2008) or multi-component solid dispersion system (Yoo et al., 2009).
9
Figure 1.6: Dissolution performance of ASD containing a model poorly water-soluble compound
in various polymers at 10% drug loading. Dissolution was carried out using the microcentrifuge
tube under nonsink dissolution conditions in PBS at 37oC with 200 g/mL total drug concentration
(dissolved plus undissolved drug). Figure adapted from Curatolo et al. (Curatolo et al., 2009)
(reproduced with permission from Pharmaceutical Research, Copyright Springer 2009).
Typically, the dissolution of ASDs based on water-soluble polymers under nonsink dissolution
conditions is very rapid, resulting in an initial surge of drug concentration in the dissolution
medium followed by a decline in drug concentration due to the nucleation and crystallization
events triggered by the rapid buildup of drug supersaturation. Depending on the ability of the
dissolved polymer to inhibit drug precipitation from the supersaturated state, such a decline in drug
concentration can be retarded to different degrees. In general, the more gradual the decline in drug
concentration, the greater its effectiveness in inhibiting drug precipitation and in maintaining drug
supersaturation (Alonzo et al., 2011; Alonzo et al., 2010). In this regard, amphiphilic HPMCAS
has been identified as the most effective in achieving and maintaining drug supersaturation among
several available water-soluble polymers commonly employed in ASD-based oral drug products
(Figure 1.6) (Curatolo et al., 2009; Friesen et al., 2008). Typical dissolution profiles of ASDs
showing a rapid initial buildup of drug supersaturation and subsequent retardation of precipitation
have been qualitatively characterized as the “spring and parachute” approach (Figure 1.7)
(Brouwers et al., 2009; Guzman et al, 2007; Warren et al., 2010). This combination of a rapidly
dissolving and supersaturating “spring” with a precipitation-retarding “parachute” has been
pursued as an effective formulation strategy to enhance the rate and extent of oral absorption.
10
Although such “spring and parachute” dissolution data have been fitted to empirical rate equations
to estimate the time constants for the “spring” and “parachute” portions of the dissolution profiles
(Kawakami, 2012), the interplay between these two rate processes in achieving and maintaining
supersaturation remains inadequately understood.
Figure 1.7: Illustration of the time evolution of kinetic solubility profiles of a crystalline drug,
amorphous drug and ASD in water-soluble carriers under nonsink dissolution conditions. Profile
2 represents the dissolution of a higher energy “spring” form of the drug in absence of any
crystallization inhibitor; Profile 3 displays the combination of the rapidly supersaturating “spring”
form and precipitation-inhibiting “parachute” form. Cs indicates the equilibrium solubility. Figure
adapted from Brouwers et al. (Brouwers et al., 2009) (reproduced with permission from the
Journal of Pharmaceutical Sciences, Copyright John Wiley and Sons, 2009).
11
Model Drug Carrier* Additives (copolymer,
surfactant, plasticizers) Preparation method**
Ref
Felodipine PVP None C/SE (Marsac et al., 2010)
Tacrolimus HPMC
PEG6000, PEG4000, Poloxamer 188, Poloxamer 407, SDS SD (Park et al., 2009)
Indomethacin PVP None C/SE (Telang et al., 2009)
AMG 517 (VR1 antagonist) HPMCAS, HPMC None SD
(Kennedy et al., 2008)
Compound C35H35N5O3 PVP poloxamer 188 C/SE, HE
(Lakshman et al., 2008)
Itraconazole mannitol/lecithin None FD (URF), PM (Yang et al., 2008)
Itraconazole PEG / HPMC None SD (Janssens et al., 2008)
UC 781 (anti-HIV) PVP/VA, HPMC
TPGS (d-alpha-tocopheryl polyethylene glycol 1000 succinate) C/SE
(Goddeeris et al., 2008)
Felodipine PVP, HPMCAS, HPMC None C/SE
(Konno et al., 2008)
Nifedipine PVP, HPMC, PHPA None SD (Aso et al., 2007)
Ibuprofen
PVP (Kollidon 25, Kollidon 30, Kollidon VA64, Kollidon CL) None SD (Xu et al., 2007)
Itraconazole HPMC, HPMCP
Polysorbate 80, anhydrous silicic acid, croscarmellose sodium, magnesium stearate C/SE, HE
(Oshima et al., 2007)
Itraconazole PEG, HPMC None SD, HE (Janssens et al., 2007)
Itraconazole HPMCP, Eudragit (L100) None FD (URF)
(Overhoff et al., 2007)
Piroxicam
Polyoxyethylene 40 Stearate, Eudragit (E100) Mannitol, dextrin C/SE
(Valizadeh et al., 2007)
Compound (MW400)
PVP, Plasdone (S630), HPMC
Tween-80 and Docusate sodium HE
(Ghebremeskel et al., 2007)
Meloxicam PEG None C/SE, PM (Kumar and Mishra, 2006)
KRN633 (VEGF tyrosine kinase inhibitor) PVP None C/SE
(Matsunaga et al., 2006)
Nifedipine / Felodipine PVP None C/SE
(Marsac et al., 2006)
Acetaminophen HPMCP, chitosan None SD (Chen et al., 2006)
Bicalutamide PVP None C/SE, PM (Ren et al., 2006)
Felodipine PVP, HPMC Poloxamer 127 C/SE, PM (Kim et al., 2006)
Felodipine PVP, PEG sodium docusate C/SE (Karavas et al., 2005)
Nitrendipine HPMCP None HE (Wang et al., 2005)
12
Model Drug Carrier* Additives (copolymer,
surfactant, plasticizers) Preparation method** Ref
Felodipine HPMC
Poloxamer 188, Poloxamer 407, HCO-60 C/SE (Won et al., 2005)
Nifedipine HPMCAS, HPMCP, MAEA, PVP None C/SE (Tanno et al., 2004)
Nifedipine / Phenobarbital PVP None C/SE (Aso et al., 2004)
Ketoprofen PEO CrosPVP, SDS PM, ME (Schachter et al., 2004)
Ritonavir PEG None C/SE (Law et al., 2004)
Indomethacin PVP None PM (Watanabe et al., 2003)
Itraconazole HPMC None PM, HE (Verreck et al. 2003a,b,c)
Indomethacin Na-indomethacin / PVP None PM, C/SE
(Tong and Zografi, 2001)
Probucol PVP, PAA, PEO Tween 80 C/SE, ME (Broman et al., 2001b)
Furosemide PVP calcium alginate Floating multi-unit system
(Iannuccelli et al., 2000)
Itraconazole lactose Disintegrants: Primogel, Kollidon CL, and Ac-Di-Sol PM
(Chowdary and Rao, 2000)
*Carrier: HPMC hydroxypropyl methyl cellulose; HPMCAS hydroxypropyl methyl cellulose acetate succinate; HPMCP hydroxypropyl methyl
cellulose phthalate; MAEA methacrylic acid ethyl acrylate; PAA poly(acrylic acid); PEG polyethylene glycol; PEO polyethylene oxide; PHPA
poly(N-5-hydroxypentyl) aspartamide; PVP polyvinyl pyrrolidone; PVP/VA polyvinyl pyrrolidone/vinyl acetate; SDS sodium dodecyl sulfate.
**Preparation method: C/SE cosolvent/solvent evaporation; SD spray-drying; FD freeze-drying; URF ultra-rapid freezing; ME melting; HE hot-
melt extrusion; PM physical mixing
Table 1.1: Selected studies of water-soluble carriers for amorphous drugs.
13
Category Examples Carrier* Model Drug Preparation
method** Release mode***
Ref
a. Crosslinked hydrogel
cr-PHEMA hydrogel
diclofenac sodium, naproxen, piroxicam, indomethacin
S/SE IR/CR (Sun et al., 2012; Zahedi and Lee, 2007)
cr-PEO hydrogel
progesterone S/SE CR (Carelli et al., 1993)
Carbopol® phenacetin S/SE CR (Ozeki et al, 2000)
I. Non-porous
b. Water-insoluble polymer
Ethylcellulose indomethacin C/SE CR (Ohara et al, 2005)
Eudragit® RS, RL
indomethacin, dipyridamole
C/SE CR (Beten et al, 1994; Oth et al, 1989)
c. Lipid
Labrasol and Gelucire 44/14
piroxicam HM IR (Yüksel et al, 2003)
Gelucire 44/14 and Gelucire 50/13
gilbenclamide C/SE IR (Chauhan et al, 2005)
“popcorn” cr-PVP
griseofulvin, indomethacin I/SE, HG IR
(Fujii et al, 2005; Shibata et al, 2007; Carli et al, 1986)
II. Porous
silica
fenofibrate, carbamazepine, cinnarizine, danazol, ibuprofen diazepam, griseofulvin, indomethacin, ketoconazole, nifedipine, phenylbutazone
I/SE, SD IR
(Van Speybroeck et al, 2009; Van Speybroeck et al, 2010; Shen et al, 2010)
starch foam lovastatin I/SE IR (Wu et al, 2011)
carbon celecoxib I/SE IR (Zhao et al, 2012)
*Carrier: cr-PHEMA poly(2-hydroxyethyl methacrylate) crosslinked with ethylene glycol dimethacrylate; Carbopol® (910, 971P, 934P, 974P,
940) polyacrylic acid (lightly crosslinked with allyl sucrose or allyl pentacrythritol); cr-PEO poly(ethylene oxide) crosslinked with hexamethylene
diisocyanate (HMDIC) or tolylene-2,4-diisocyanate (TDIC); cr-PVP crosslinked polyvinylpyrrolidone; Gelucire 44/14, 50/13 polyethylene glycol
glycerides; Labrasol caprylocaproyl macrogolglycerides.
**Preparation method: S/SE swelling/solvent evaporation; C/SE cosolvent/solvent evaporation; I/SE immersion/solvent evaporation, HM hot melt;
HG heated granulation (
14
1.2.2 Water-insoluble carriers
In contrast to studies of water-soluble ASD carriers, previous efforts in applying water-insoluble
polymers as ASD carriers have primarily aimed to achieved controlled release (Tran et al. 2011;
Zhu et al. 2006). Table 1.2 summarizes available water-insoluble carriers utilized to convert poorly
water-soluble model drugs into amorphous formulations categorized by their physicochemical
characteristics such as porosity, location of amorphous drug and carrier chemical composition. In
the case of nonporous ASD carriers where amorphous drug molecules are completely dissolved
(i.e., one-phase solid solution), carriers with extremely low drug diffusivity such as “glassy
polymers” (i.e., Tg much higher than ambient temperature) will exhibit better ASD stability due
to hindered drug diffusion and inhibition of drug precipitation in the glassy matrix. The presence
of specific drug-carrier intermolecular interactions due to hydrogen bonding, dipole-dipole
attraction and van der Waals forces further stabilizes the entrapped amorphous drug, preventing it
from nucleating and becoming crystallite. By contrast, in porous carriers where the incorporated
amorphous drug is localized in interstitial pore space (i.e., not molecularly dispersed), the
nucleation and crystallization of this aggregated amorphous drug can give rise to stability issues.
However, the drug nucleation and crystallization rates can be reduced if the size of the pore is
sufficiently small compared to that of the critical nuclei, rendering it energetically unfavorable for
nuclei to grow (Rengarajan et al., 2008; Van Speybroeck et al., 2009; Van Speybroeck et al., 2010).
For inorganic mesoporous materials such as silica and silicon (Xu et al., 2012), their large surface
area and pore volume can accommodate a large amount of drug payload. Drug nucleation and
crystal growth in the porous channels will be energetically unfavorable if the size of the critical
nucleus is larger than that of the pore (Jackson and McKenna, 1996; Wang et al., 2006). In other
words, the spatial constraint of a capillary imposed on the amorphous drug below the critical
nucleus size has a stabilizing effect. In addition, a strong interaction with the pore walls (e.g.,
through hydrogen bonding) further stabilizes the confined drug molecules in the amorphous state
and this typically occurs in nano-sized pores with a pore diameter smaller than approximately 10
nm (Rengarajan et al., 2008). The drug loading process in such mesoporous materials commonly
involves immersing the carriers in a concentrated drug solution to fill the pores followed by the
evaporative removal of solvents. One study claims to have successfully accomplished solid
dispersion by means of a melt-mixing method, during which a physical mixture of nitrendipine
and mesoporous silica particles is heated above the melting point of the drug (Wang et al., 2006).
15
Nonetheless, in this case the high viscosity of the melted drug can interfere with the capillary
action within the pores, potentially causing an incomplete drug loading. Similar constraints also
apply to macroporous polymeric carriers such as crosslinked polyvinylpyrrolidone (crospovidone)
which exhibits a “popcorn” structure, containing many macroscopic cavities, as clearly shown
under the SEM (Carli et al., 1986; Fujii et al., 2005; Shibata et al., 2007). In this case, in addition
to the conventional loading method of sorption from a concentrated drug solution, an alternative
method of drug loading based on blending or milling of the drug and porous crospovidone particles
without solvent under high mechanical shear for an extended period of time has been proposed
(Fujii et al., 2005; Shibata et al., 2007). However, the “amorphous” material produced in this could
be a result of the well-known shear-induced phase transformation at the shear-fractured surfaces
on drug crystals (Greco and Bogner, 2010; Koike et al., 1990). As such, the long-term stability of
such mechanically generated amorphous systems is questionable since without any mechanism for
crystallization inhibition this exposed surface amorphous content can easily be converted to the
crystalline phase under accelerated temperature and/or in higher humidity.
Alternatively, in a nonporous insoluble carrier system, amorphous drugs can exist as surface
adsorbed or molecularly dissolved/dispersed in the matrix depending on the drug loading process.
For example, solvent or melt granulation of a crystalline drug with a nonporous insoluble carrier
typically results in an amorphous drug either adsorbed on the carrier surface or unevenly
distributed throughout the carrier-drug granule (i.e., a two-phase amorphous solid dispersion).
Such pure amorphous drug, either surface adsorbed or unevenly distributed lacks the benefit of
crystallization inhibition conferred by dissolving or molecularly dispersing the drug in a protective
carrier matrix and, therefore, is prone to undesirable nucleation and crystallization in the solid
state. Nonetheless, the creation of uniformly dissolved or molecularly dispersed drug in ASD
polymeric carriers (i.e., one-phase solid solution) through either drug sorption (e.g., from a good
swelling solvent) or co-precipitating from a common organic solvent (e.g., via spray drying or
freeze drying) is understandably more advantageous than producing dispersed two-phase ASD
systems in terms of its effectiveness in crystallization inhibition and its effect in enhancing the
physical stability. Therefore, applicable nonporous water-insoluble carriers include crosslinked
polymers (e.g., PHEMA), cellulose derivatives (e.g., ethylcellulose), and lipids (e.g., PEG-
glycerides). The common methods of preparation of ASD based on water-insoluble carriers can
be generally classified as supercooling of the melt (e.g., hot-stage melting, hot-melt extrusion), co-
16
precipitating from a common organic solvent (e.g., spray-drying, freeze-drying), and equilibrium
sorption (e.g., from a concentrated drug solution prepared in a good swelling solvent) to produce
a homogenous binary one-phase system. In this case, the carrier plays an essential role in
preserving the entrapped drug molecules in the amorphous state despite the fact that the exact
mechanisms of crystallization inhibition in solid molecular dispersions are still not completely
understood. Various factors such as molecular mobility, thermodynamic properties, and drug-
polymer interactions have been identified as responsible for inhibiting drug crystallization from
the amorphous state (Bhugra and Pikal, 2008; Hancock and Zografi, 1997; Khougaz and Clas,
2000). Stressed conditions involving high relative humidity and high temperature in combination
with ageing may accelerate the transformation of metastable amorphous drugs in an ASD into a
more thermodynamically stable crystalline state (Hancock and Zografi, 1997; Konno and Taylor,
2008).
1.3 Crosslinked PHEMA hydrogels for amorphous solid dispersions carriers
Poly (2-hydroxyethyl methacrylate) (PHEMA) has been widely utilized in controlled drug delivery
systems particularly for water-soluble drugs and in biomedical application such as contact lenses,
wound dressing and tissue engineering. Crosslinked PHEMA is probably the most extensively
studied gel-forming water-insoluble polymer among several other synthetic hydrogels based on
nonionic hydrophilic monomers (e.g., hydroxyalkyl acrylate, N-substituted methacrylamides and
N-vinyl-2-pyrrolidone) commonly used in swelling-controlled oral drug delivery (Gehrke, 2000).
The drug release kinetics and release rate modulation from such a spherical monolithic diffusion-
controlled system have previously been characterized in detail (Lee and Kim, 1991; Lee and
Peppas, 1987). Chemically, PHEMA has hydrophilic side groups which can promote the sorption
of a significant amount of polar solvent, thereby providing desired conditions to create solid
dispersions with high drug loading. PHEMA can also be easily copolymerized with other
hydrophobic monomers, such as methyl methacrylate (MMA) or its homologs, to improve the
compatibility of the hydrogel matrix with hydrophobic drugs. Furthermore, the crosslinker chain
length and crosslinking density can be adjusted to affect the network mesh size which provides
further control of the diffusional drug release (Hoare and Kohane, 2008). Thus, PHEMA is a good
candidate drug carrier because of the ease in regulating drug release by controlling particle size,
17
crosslinking density, chemical composition (e.g., copolymers) and water sorption rate to achieve
different drug loading levels and release profiles suitable for oral drug delivery.
Orally administered excipients must be stable, non-toxic, biocompatible in the GI tract and
compatible to API and other excipients in the formulation in order to satisfy regulatory
requirements. Full in vivo studies such as biodistribution, immune response, clearance, and chronic
toxicology studies for different durations are generally required to demonstrate the excipient’s
safety for use in human drug products. The approval process to introduce a new pharmaceutical
excipient onto the market requires expensive toxicological investigations and typically takes a
similar period of time as that of a new API (Katdare and Chaubal, 2006). Due to the complexity
and high cost of long-term toxicological and clinical testing, PHEMA has not yet been used as an
oral excipient in the FDA approved drug products (note: PHEMA is a FDA-approved inactive
ingredient for only topical and dressing application as on September 16, 2013). Nevertheless,
PHEMA hydrogel has a long history of human use as biomedical implants and for controlled drug
release from medical devices. A number of in vivo tests using PHEMA hydrogel delivery systems
have produced favorable results in blood and tissue compatibility (Imai and Masuhara, 1982).
Moreover, no toxic abnormalities were reported when crosslinked PHEMA hydrogels loaded with
diclofenac sodium and theophylline were administered orally and rectally, respectively, to human
subjects in clinical pharmacokinetic studies (De Leede et al., 1986; Thakker et al., 1992). Since
crosslinked hydrogels for the purpose of oral delivery are generally of micron sizes and insoluble
in the GI fluid, cellular internalization or absorption through the mucosal membrane of the GI tract
is very unlikely. Therefore, the above mentioned lines of evidence suggest that crosslinked
PHEMA hydrogel beads would be an attractive candidate excipient for oral drug delivery. To
promote the practical application of PHEMA hydrogels as an excipient for oral dosage forms,
long-term oral toxicity studies should be further investigated and established.
In selecting an appropriate polymer matrix for an ASD system, the carrier polymer must provide
sufficient stabilization effect in the solid state to prevent the entrapped amorphous drug from
crystallizing, thereby achieving a reasonable shelf life. From a mechanistic point of view,
nucleation and subsequent crystallization of an amorphous drug in a polymer matrix deplete the
local drug concentration around the crystallite and thus triggering drug diffusion from the
surrounding amorphous drug region to the crystallite surface to sustain crystal growth. The reduced
segmental mobility of polymer chains in glassy PHEMA hydrogels at ambient temperature well
18
below its Tg of 115 oC (Roorda et al., 1988) should lengthen the polymer molecular relaxation
time and retard the diffusional mobility as well as reduce the nucleation and crystallization rates
of the entrapped molecularly dispersed drug, thus resulting in enhanced stability of such hydrogel-
based solid dispersion systems. An additional advantage is that PHEMA hydrogels exist as
optically clear glassy solids at ambient temperature, which is suitable for direct microscopic
examination for any phase transformation and for the determination of the drug loading threshold
in the hydrogel matrix above in which amorphous-to-crystalline transition may occur.
In addition to the physical stability of ASDs, the extent of solubility and dissolution enhancement
is another key criterion to consider in selecting an appropriate ASD carrier. For a long while, the
kinetics of drug release from crosslinked water-insoluble hydrogels was characterized only in the
context of achieving controlled drug release under perfect sink dissolution conditions (Lee, 1985).
The swelling kinetics and diffusional drug release kinetics are important physical characteristics
of hydrogel-based drug delivery. The swelling kinetics of crosslinked hydrogels is important not
only to reach an adequate drug loading but also to regulate the rate of drug release. During drug
release, the time evolution of the solvent-penetrating front and normalized dimensional changes
of drug-loaded PHEMA hydrogel beads is a function of crosslinking density, drug solubility and
initial drug loading. For example, two moving boundaries (swelling front and drug dissolution
front) were generally observed for poorly water-soluble diclofenac sodium in PHEMA hydrogels
at a high drug loading range of 25-30% w/w as the aqueous medium penetrates into the matrix,
whereas a single fast-moving swelling front was observed below this range of drug loading (Lee
and Kim, 1991). In terms of dimensional transformation, the swelling PHEMA beads generally
first expand to a maximum in radius followed by a gradual decrease to an equilibrium dimension
during the drug release (Lee, 1983; Lee and Kim, 1991). Moreover, the dissolution of incorporated
drug from crosslinked PHEMA hydrogels is regulated by diffusion where the hydrogel matrix is
insoluble and the dissolved or dispersed drug slowly diffuses out of the hydrogel network.
Although the application of PHEMA hydrogel as a carrier for ASDs of model poorly water-soluble
drugs such as diclofenac sodium, naproxen and piroxicam have been proposed recently (Zahedi
and Lee, 2007), the effect of the hydrogel matrix on the maintenance of supersaturation was
virtually unknown. Crosslinked hydrogels with a unique diffusion-controlled mechanism of drug
release will hopefully provide an adequate alternative to the rapidly dissolving ASD carriers in
enhancing the solubility and dissolution rate of poorly water-soluble drugs.
19
1.4 Crystallization of amorphous pharmaceuticals in the solid state
1.4.1 Solubility advantage of amorphous solids
Several theoretical models present amorphous solids as having “excess” (Yu, 2000)
thermodynamic properties including higher volume, enthalpy, entropy and free energy, as
compared to its crystalline state. The change in free energy levels between amorphous and
crystalline states accounts for their significant differences in physiochemical properties such as the
apparent solubility (or the kinetic solubility) in water. Parks et al (Parks et al, 1934) for the first
time theoretically estimated the difference in kinetic solubility between amorphous and crystalline
glucose based on thermodynamic analysis, and experimentally confirmed their theoretical
predictions. The free energy difference between the amorphous and crystalline phases is expressed
by:
∆𝐺𝑎,𝑐 = −𝑅𝑇𝑙𝑛 (𝑎𝑎
𝑎𝑐) = −𝑅𝑇𝑙𝑛 (
𝛾𝑎𝐶𝑆𝑎
𝛾𝑐𝐶𝑆𝑐 ) Equation 1.1
where a is activity of the solute in the saturated solution (superscript: a for amorphous and c for
crystalline), R the gas constant, T the temperature, 𝛾 the activity coefficient (close to unity when
the system is in a diluted form) and Cs the concentration at saturation (i.e., solubility). One
important assumption of this equation is that an equilibrium solubility exits in both crystalline and
amorphous states. For crystalline materials, the equilibrium solubility refers to the physical state
when the molecules in the solid state are in chemical equilibrium with those in the solution state
in a close system. In other words, the rates of dissolution and precipitation of individual molecules
between the solid and solution phases are equal to each other. However, amorphous solids are in
a non-equilibrium state (i.e., metastable) at which the disordered molecular structure does not
require the breaking of crystal lattice upon dissolution. The apparent solubility of amorphous solids
generated by time-dependent rates of dissolution and precipitation is in fact a kinetic property.
Nevertheless, Parks et al (Parks et al, 1934) treated amorphous drug as either an equilibrium
supercooled liquid or a pseudo-equilibrium glass and estimated the ratio of solubility enhancement
between the amorphous and crystalline states by calculating the difference in Gibbs free energy
expressed in enthalpy (H) and entropy (S):
20
∆𝐺𝑎,𝑐 = ∆𝐻𝑎,𝑐 − 𝑇∆𝑆𝑎,𝑐 Equation 1.2
According to Kirchhoff’s law, ∆𝐻𝑎,𝑐and ∆𝑆𝑎,𝑐 can be calculated by:
∆𝐻𝑎,𝑐 = ∆𝐻𝑓𝑐 + ∫ ∆𝐶𝑝
𝑇
𝑇𝑚𝑑𝑇 = ∆𝐻𝑓
𝑐 − (𝐶𝑝𝑎 − 𝐶𝑝
𝑐) × (𝑇𝑚 − 𝑇) Equation 1.3
∆𝑆𝑎,𝑐 = ∆𝑆𝑓𝑐 + ∫
∆𝐶𝑝
𝑇
𝑇
𝑇𝑚𝑑𝑇 = ∆𝑆𝑓
𝑐 − (𝐶𝑝𝑎 − 𝐶𝑝
𝑐) × 𝑙𝑛 (𝑇𝑚
𝑇) Equation 1.4
where ∆𝐻𝑓𝑐 is the entropy of fusion, Cp the isobaric heat capacity, Tm melting temperature and
∆𝑆𝑓𝑐 entropy of fusion (calculated by ∆𝐻𝑓
𝑐/𝑇𝑚). The difference in isobaric heat capacity (∆𝐶𝑝)
has been estimated to be (1) zero (i.e., ∆𝐶𝑝 ≈ 0), (2) the heat capacity gained upon going through
the glass transition (i.e., ∆𝐶𝑝 ≈ ∆𝐶𝑝,𝑇𝑔), (3) the entropy of fusion (i.e., ∆𝐶𝑝 ≈ ∆𝑆𝑓), and (4) a non-
zero constant in which the difference in enthalpy between the two states vanishes at a temperature
T∞, slightly below Tg (i.e., ∆𝐶𝑝 ≈∆𝐻𝑓
𝑇𝑚−𝑇∞). The last approach can be further utilized to derive a
simple mathematical express of the Gibb’s free energy difference, also known as the Hoffman
equation:
∆𝐺𝑎,𝑐 =∆𝐻𝑓(𝑇𝑚−𝑇) 𝑇
𝑇𝑚2 Equation 1.5
This theoretical consideration has been the basis for many attempts on the thermodynamic
prediction of solubility enhancement ratios between the amorphous and crystalline forms of poorly
water-soluble drugs (Hancock and Parks, 2000; Alonzo et al, 2010; Alonzo et al, 2011). Additional
rigorous modifications including the consideration of configurational heat capacity, water sorption
during dissolution and different degrees of drug ionization in amorphous and crystalline forms
have provided further improvements in producing a closer agreement with the measured data
(Aceves‐Hernandez et al., 2009; Hoffman, 1958; Matteucci et al., 2008; Murdande et al., 2010a,
b, 2011a, b; Ogino et al., 1990). Nevertheless, both the theoretical prediction of the solubility
enhancement using thermodynamic analysis and the accurate experimental measurements of the
solubility of the amorphous state has proven to be very challenging.
21
1.4.2 Classical nucleation theory (solid state)
Due to the non-equilibrium nature of the amorphous solids, the amorphous state is less
thermodynamically stable than any crystalline form, leading to an inevitable tendency for the
amorphous materials to transform to a crystalline phase. The higher free energy level of the
amorphous form relative to the crystalline state provides the thermodynamic driving force for
nucleation and crystallization, causing physical instability (i.e., recrystallization) in the dosage
forms. Nonetheless, acceptable drug stability can still be achieved in amorphous formulations for
oral dosage forms during pharmaceutical development if the kinetics of amorphous-to-crystalline
phase transformation can be delayed to an adequate extent. The physical process of
recrystallization from a high-energy amorphous form in the solid state is a complex phenomenon
from a mechanistic viewpoint based on both nucleation and crystal growth. The Classical
Nucleation Theory (CNT) first describes the kinetics of the homogenous nucleation process by
considering the difference in the free energies of the crystalline and liquid phases (note: amorphous
materials which lack crystalline lattice energy have a molecular structure similar to that of a liquid).
The overall difference of Gibb’s free energy between the crystalline and amorphous phases is equal
to the sum of two surface excess energies, Gs (the difference between the surface and the bulk of
the crystalline phase) and GV (the difference between a very large crystalline particle (r = ∞) and
the amorphous phase). Considering the growth of a spherical crystalline particle, the overall excess
free energy is given:
∆𝐺 = ∆𝐺𝑆 + ∆𝐺𝑉 = 4𝜋𝑟2𝛾 +
4
3𝑟3∆𝐺𝜈 Equation 1.6
where r is the radius of the crystalline particle, γ the interfacial tension and ∆𝐺𝜈 the free energy
change of the transformation per unit volume. The overall ∆𝐺 as a sum of positive Gs and
negative GV has a local maximum corresponding to the critical nucleus, rc, as illustrated in Figure
1.8. This local maximum can be obtained by taking the first-derivative of Equation 1.6:
𝑑∆𝐺
𝑑𝑟= 8𝜋𝑟𝛾 + 4𝜋𝑟2∆𝐺𝜈 = 0 Equation 1.7
Therefore, the critical radius of a nucleus, rc, and the free energy difference for such a nucleus,
∆𝐺𝑐𝑟𝑖𝑡𝑖𝑐𝑎𝑙, are given by:
22
𝑟𝑐 =−2𝛾
∆𝐺𝜈 Equation 1.8
∆𝐺𝑐𝑟𝑖𝑡𝑖𝑐𝑎𝑙 =16𝜋𝛾3
3(∆𝐺𝜈)2=
4𝜋𝛾𝑟𝑐2
3 Equation 1.9
Figure 1.8: Free energy diagram for nucleation process. Figure adapted from Crystallization 3rd
edition (Mullin, 2001) (reproduced with permission from the Crystallization 3rd edition, Copyright
Butterworth-Heinemann, 1993).
The nucleation rate J (i.e., the number of nuclei formed per unit volume per unit time) is expressed
in the form of an Arrhenius equation, commonly used for the rate process of a thermally activated
reaction:
𝐽 = 𝐴𝑒(−∆𝐺
𝑘𝑇) Equation 1.10
where A is the pre-exponential frequency factor, k the Boltzmann constant and R the gas constant.
For homogenous nucleation from the melt (i.e., liquid phase), the free energy change is estimated
by:
∆𝐺𝜈 = Δ𝐻𝑓(𝑇𝑚−𝑇)
𝑇𝑚 Equation 1.11
23
where Δ𝐻𝑓 is the latent heat of fusion and Tm the melting temperature (solid-liquid equilibrium
temperature). Therefore, the critical radius of a nucleus and the critical free energy difference are
given as:
𝑟𝑐 =−2𝛾𝑇𝑚
Δ𝐻𝑓(𝑇𝑚−𝑇) Equation 1.12
∆𝐺𝑐𝑟𝑖𝑡𝑖𝑐𝑎𝑙 =16𝜋𝛾3
3(∆𝐺𝜈)2=
16𝜋𝛾3𝑇𝑚2
3Δ𝐻𝑓2(𝑇𝑚−𝑇)2
Equation 1.13
and the nucleation rate for the amorphous-to-crystalline transformation in the solid state may be
expressed by substituting Equation 1.13 into Equation 1.10:
𝐽 = 𝐴𝑒(
−16𝜋𝛾3
3𝑘𝑇Δ𝐻𝑓2(𝑇/𝑇𝑚)(1−𝑇/𝑇𝑚)
2 )
Equation 1.14
The analytical solution describing the nucleation kinetics in the amorphous materials is a function
of heat of fusion, melting temperature and interfacial tension. It is worth noting that the
determination of specific interfacial tension between the developing crystalline surface and the
amorphous bulk is not easily measured experimentally.
1.4.3 Crystal growth (solid state)
After the barrier to nucleation has been overcome, a number of newly formed stable nuclei will
continue to grow into crystals of microscopic size. The process of crystal growth is generally
described by models of (1) normal or continuous growth, (2) two-dimensional growth and (3)
growth mediated by screw dislocation (Gutzow and Schmelzer, 2013). The crystal growth rate can
be described by the following equation in a general form:
𝑈 =𝐶𝑇𝜔
𝜂(1 − 𝑒(
−Δ𝐺𝑉𝑘𝑇
)) Equation 1.15
where C is a constant, ω the constant which depends on the mechanism of growth, η the viscosity
of the system and Δ𝐺𝑉 the free energy difference between the amorphous and crystalline phases.
The combination of equations calculating the nucleation rate (Equation 1.14) and the crystal
24
growth rate (Equation 1.15) can assess the overall crystallization kinetics of amorphous-to-
crystalline phase transformation. However, it is worth highlighting that many of the parameters
listed in the above mentioned equations are not easily accessible experimentally.
1.4.4 Kolmogorov-Johnson-Mehl-Avrami (KJMA) Theory
The kinetics of overall crystallization taking into account nucleation, crystallization and the lag
time in both of these events can be described by the Kolmogorov-Johnson-Mehl-Avrami (KJMA)
theory which reduces the complicated mechanism into a simple mathematical correlation
describing the process of overall crystallization as a function of time (Avrami, 1939; Avrami 1940),
which is convenient to use for the physical stability study of amorphous pharmaceuticals (Andronis
and Zografi, 2000; Miyazaki 2004, Bhugar and Pikal, 2000; Gutzow Schmelzer 2013). The
empirical equation is presented by an exponential term:
𝑥(𝑡) = 1 − exp [−𝑘𝑡𝑛] Equation 1.16
where x is the percentage of the crystalline phase, t the time, and k and n Avrami crystallization
constants. One limitation of the KJMA theory is that time-dependent crystallization kinetics is
only an approximation in which the constants of k or n may not have any physical significance.
Despite the fact that the KJMA theory provides limited mechanistic information on crystallization
kinetics, comparing the crystallization rate to the Avrami equation is very convenient. Therefore,
the use of the KJMA theory offers an easy tool of direct comparison of crystallization rates between
two systems.
25
1.5 Crystallization of supersaturated drug solutions
1.5.1 Classical nucleation theory (solution state)
Amorphous pharmaceuticals can generate a supersaturated concentration in aqueous solution
significantly higher than the equilibrium saturation concentration of their crystalline counterparts.
In a solute-solvent binary system (e.g., drug-water), solute concentration above the equilibrium
solubility (i.e., supersaturation) provides the thermodynamic driving force for nucleation and
crystallization. In a supersaturated drug solution, the free energy is higher in the solution phase
than any newly formed crystalline phase. Classical nucleation theory based on the free energy
difference can also be applied on the precipitation process of solute from a supersaturated solution.
Equations 1.6 to 1.10 describe the nucleation rate for homogenous nucleation of the solute from a
supersaturated solution. The free energy change ∆𝐺𝜈 in Equation 1.10 becomes
∆𝐺𝜈 =−2𝛾
𝑟𝑐=
−kTln(S)
𝜐 Equation 1.17
where υ is the molecular volume and s the supersaturation. This equation is derived from the basic
Gibbs-Thomson relationship for a non-electrolyte:
ln(𝑠) =2𝛾𝜐
𝑘𝑇𝑟 Equation 1.18
The substitution of Equation 1.17 into Equations 1.9 and 1.10 gives:
∆𝐺𝑐𝑟𝑖𝑡𝑖𝑐𝑎𝑙 =16𝜋𝛾3
3(∆𝐺𝜈)2=
16𝜋𝛾3𝜈2
3(𝑘𝑇𝑙𝑛𝑆)2 Equation 1.19
𝐽 = 𝐴𝑒(
−16𝜋𝛾3𝜈2
3𝑘3𝑇3(𝑙𝑛𝑆)2 )
Equation 1.20
where described here is the interfacial tension between the evolving crystalline surface and
aqueous solution (note: different from described in Equations 1.12 to 1.14). Equations 1.19 and
1.20 are for nucleation from a supersaturated solution which significantly depends on the degree
of supersaturation are analogous to Equations 1.13 and 1.14 for nucleation from the melt.
26
1.5.2 Crystal growth (solution state)
Once stable nuclei have formed (i.e., r > rc) in a supersaturated solution, these particles start to
grow into crystals of visible size. Crystallization in a supersaturated solution is the reverse process
of dissolution, and both phenomena are governed by diffusion in which the concentration gradient
between the solid surface and the bulk of solution provide the driving force. During the
precipitation process from a supersaturated solution, crystal growth requires both long-range
transport of solute to the growing particles (i.e., diffusion) and local atomic rearrangement as the
solutes near the particle surface (i.e., interface) are integrated into the crystal lattice. Depending
on the rate-limiting step, crystal growth kinetics can be either diffusion-controlled or interface-
controlled. In a diffusion-controlled crystallization process, the growth rate of a crystal can be
predicted by the molecular flux which is related to the concentration gradient between the bulk
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