Crystallization kinetics and phase behaviour of molecular solutions

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

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

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

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

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

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-workers76The 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, γc and γe, respectively, are approximately equal .36 Table 3.4 Calculated values of surface tensions σ and thermodynamic parameters

B from solubility ce, solid density ncr, molecular size d, and equilibrium activity

coefficient γe 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 D2 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

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/H2O (50/50 w/w%) at 20 °C using 1H NMR Areas of peaks are normalized by the area associated with the –CH2– 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/H2O (50/50 w/w%) at 20 °C .116

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-ln3Sc) versus (ln2Sc) for different compounds According

to Eq (3.5), the slope of this plot is equal to ln(A/Jc) 41

Figure 3.6 Comparison of experimental data and model predictions for critical

supersaturation as a function of dimensionless surface tension, σd2/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/Jc) = 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

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 (D2 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 H2O at 5, 25, 40 and

75 °C (a) Self diffusivities are plotted against absolute concentration of glycine in H2O (b) The absolute concentration is converted to particle volume

fraction as described in the text Values of D2 are given by the slopes of the

linear fits .81

Figure 5.5 Phase diagram for a variety of solutes in D2 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

insert shows how a cloud point temperature is determined .94

Figure 6.2 Plots of D2 as a function of ε/kT for different λ .97 Figure 6.3 Scaled long-time self diffusivities of ibuprofen in EtOH/H2O (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 D2 are given by the slopes of the best linear fits.98 Figure 6.4 D2 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/H2O; (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/H2O (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 H2O (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 D2 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 literature81,90,157,183,185 and D2 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 literature47 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 literature126 and expressed into D2 The open crosses presents LLPS data

of Veesler's API47 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

1 GENERAL INTRODUCTION

Over the past century the field of crystallization has evolved from crystals of simple inorganic salts1 to supramolecular complexes2, from the classical nucleation theory3 originally developed for liquid droplet formation in the vapor phase to multi-environment simulation of mixing effects in antisolvent crystallization4, from primitive copper crystallizing pans5 to modern crystallizers equipped with sophisticated process analytical technology (PAT)6, from large scale industrial crystallizers to nanoscale crystallizing reservoirs in microfluidic chips7-11, from

Edisonian experimental protocols to a priori crystal structural predictions by

molecular modeling12,13 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 processes14-16 and predicting solution phase behavior17-23 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

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

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

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

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

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 Weber3 from condensing vapor studies, and has been further developed by Volmer,27 Gibbs,28

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

n

A

A A A

A A A

⇔+

⇔+

⇔+

− 1 1

3 1 2

2 1 1

where A1, A2 and A3 represent the monomer, dimer and trimer of the solute molecule,

respectively; and An 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 ΔGS, and a negative

term, the excess free energy for the formation of a new bulk solid phase ΔGV:

2 4 34

3

where σ is the surface tension (surface energy) and ΔGv 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 ΔGS 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 ΔGV and further increase of r will reduce ΔG This

critical nuclei size rc can be easily determined by taking the first derivative of the

total excess energy with respect to r and equating it to zero:

33

In the supersaturated homogeneous mother solution, the critical radius rc 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:

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:

2ln

where a and ae 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/ae is approximately equal to c/ce, where c and ce are the

actual concentration and the equilibrium solubility of the solute, respectively This

ratio is generally defined as the supersaturation ratio S:

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:

3 2 2

where J has the unit of number per unit volume per time, and the pre-exponential

factor A is usually written as14:

where h is Planck’s constant, ΔGa is energy barrier for diffusion from the bulk

solution to the cluster and N1 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

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

Sc, 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

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

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

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

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

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

complex liquid behavior in mathematically simplified form Among them, the

simplest is the hard sphere models:

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:

( )

, ,

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:

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:

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

coefficient B2 provides a dilute solution measurement of the intermolecular

interactions The second virial coefficient is linked to the particle pair potential by

( )

2 2

0

Thus the parameters of relevant model potential can be calculated from B2, 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-B2/B2HS)], where B2HS = 2πd3/3 is the hard sphere second virial

coefficient In this manner, B2 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 B2/B2HS.21,23,43,44 Hence simple fluid

models can be used to develop a universal phase diagram where B2/B2HS is plotted

against the particle volume fraction

The above studies21,23,43,44 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 B2/B2HS 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

observed to occur below the solid-liquid transition, suggesting that the phase separation is metastable Based on simple fluid theories, liquid-liquid phase separation is a consequence of short range of intermolecular interactions.18-20 Even though experimental methods for detecting and altering the metastable liquid-liquid phase boundary have been explored extensively,45-50 linking these methods to delicate control over parameters of intermolecular interactions has received much less attention to date

As mentioned above, the second virial coefficient B2 can be used as

dimensionless temperature in characterizing the thermodynamics of protein solutions This being an effective and efficient method is due largely to the straightforward

measurement of B2 of nanoparticle with diameters of 1 nm or larger (e.g protein

molecules) using light scattering

2.3 Present Work

2.3.1 Rate of Supersaturation

Research on the role of operating parameters and/or external stimuli in determining the crystal morphology and quality, has gained attention due to the significance of solution crystallization on the final product characteristics in pharmaceutical, biotechnological and fine chemical industries.46,50-56 A variety of research activities have studied the effects of temperature, concentration, pH, solvents, presence of precipitating agents and impurities on nucleation and growth

kinetics and polymorph selectivity, however, the effects of the rate of supersaturation

on the kinetics and thermodynamics of crystallization have barely been studied 57 The work described in this thesis intends to fill this gap using a wide range of model compounds such as amino acids, pharmaceuticals and biological molecules The crystallization experiments have been performed using an evaporation-based crystallization platform.58-60 The process parameters and the resulting crystals are

characterized by in situ or offline instruments The results are discussed in light of

the classical nucleation theory and equilibrium thermodynamics, and possible industrial applications are highlighted

Most of the work linking simple fluid models to phase behavior of molecules in solution has been carried out on nanoparticles, where light scattering provides a

facile method for measuring second virial coefficients B2 For relatively small

organic molecules used in our studies, such as amino acids and pharmaceutical

compounds, B2 is difficult to measure due to the small sizes of the solute molecules

In search of alternative probe of solute pair interactions we use the long-time self diffusivity determined by pulsed-field gradient spin-echo nuclear magnetic resonance (PGSE NMR) The pair contribution of the scaled long-time self

diffusivity D2 is used as a measure for the strength of solute interactions and aids in

the development of a generalized phase diagram that is applicable for a variety of molecules Determination of the self diffusivity with NMR provides an easier and

faster route to measure the interaction potentials compared to measurement of B2

with light scattering technique Using the generalized phase diagram in the diffusivity space, studies have been conducted on how the microscale molecular interactions influence equilibrium phase behavior such as solubility boundary and nonequilibrium phase transitions such as liquid-liquid phase separation and gel formation

3 EVAPORATION-DRIVEN CRYSTALLIZATION: EFFECTS OF

SUPERSATURATION ON CRYSTALLIZATION KINETICS

Equation Section 3

3.1 Introduction

Active pharmaceutical ingredients (APIs) are usually purified through crystallization processes in solution.61 These molecules are “difficult” to crystallize due to their complex molecular structures, often requiring high levels of supersaturation and long induction times for crystals to form To further complicate the manufacturing process, these APIs typically crystallize into a number of polymorphs that have different efficacies, and therefore require a high level of process control General industrial techniques used to crystallize these molecules include cooling crystallization, where the thermodynamic state of the solution is changed by reducing the temperature, often reducing the solubility of the API; evaporative crystallization, where the concentration of the solution is increased through regulated evaporation of solvent; and addition of an anti-solvent that is carefully chosen to suppress the solubility of the API The common purpose of the above techniques is to supersaturate the solution by either physical or chemical means, resulting in spontaneous formation of crystals Control of crystallization in any of these processes is often limited by an inadequate knowledge of crystal nucleation and growth kinetics, although sophisticated process analytical technology (PAT) could be used to monitor properties such as solution concentration, pH, temperature, crystal shape and morphology, and crystal size distribution (CSD)

The in situ monitoring capability of PAT provides ample information used for feedback control of the crystallization process; however, any a priori operating

strategies still rely on fundamental understanding of the underlying physics and chemistry – the mechanism and kinetics of crystal nucleation and growth For the past few decades, many experimental and theoretical research efforts have been geared towards this end.14,16,62-65

Nucleation, the first step of crystallization, primarily determines the rate of crystal formation and the polymorph of the crystals formed As the rate of nucleation is often difficult to measure directly, alternative techniques are employed

to obtain this kinetic information.66 For example, several indirect techniques have been implemented to measure the nucleation rate of the protein hen egg white lysozyme (HEWL):64,67-69 Galvin and Vekilov67,68 used a temperature-jump

technique to obtain nucleation kinetics of HEWL; Bhamidi et al.69 used a particle counting instrument to obtain the number of molecular clusters formed with respect

to time, estimating the nucleation rates of HEWL; Kulkarni and Zukoski64characterized the nucleation rate by measuring the induction time for crystallization using the dynamic light scattering technique

Measurement of the induction time, the time one must wait to observe crystals after the solution has been supersaturated, is one of the primary approaches used to determine nucleation kinetics The induction time is often interpreted as the sum of the times for formation of critical nuclei and growth of the nuclei to a detectable size.70 Using a modern instrument that can detect particles with dimensions on the

nanometer scale, the time for the growth of critical nuclei is negligible compared to the time of formation of such critical nuclei This time of nuclei formation is semi-empirically considered to be inversely proportional to the rate of nucleation, and thus is extremely sensitive to the magnitude of supersaturation.70 Analogously, measurement of the metastable zone width (MZW) is another method used to acquire useful information about crystallization processes The MZW is the region above the solubility boundary (supersaturation = 1) and below the metastable limit, and in this region crystals do not form within a protracted period of time Crystals form rapidly, however, at supersaturation levels above the metastable limit Only crystal growth occurs for solutions that lie within the MZW; however, for solutions that cross the metastable limit, spontaneous nucleation and crystal growth take place competitively Several experimental studies have estimated the width of the metastable zone by cooling crystallization, in which the cooling rates could be considered as instantaneous relative to the kinetics of the physical processes resulting in crystal formation.71-73 The width of the metastable zone determined in the above studies was found to depend on the method or path by which supersaturation was achieved, suggesting that the rates of both heat and mass transfer are always important in determining the rates of crystal nucleation and growth

In this chapter the effects of the rate of supersaturation on the induction time and the MZW are explored The rate of supersaturation is systematically varied by regulating the rate of solvent evaporation with an evaporation-based crystallization platform58 In accordance with earlier studies,71-73 the MZW increases with an

increasing rate of supersaturation Moreover, starting with larger initial concentrations of solute, the MZW becomes less dependent on the rate of supersaturation and approaches a critical value, corresponding to an induction time

of zero The initial supersaturation level corresponding to an induction time of zero

is referred to as the critical supersaturation A broad range of materials, including two amino acids, an API, an inorganic salt, and a protein, is discovered to display a similar dependence of the induction time and the MZW on the rate of supersaturation and initial solute concentration This dependence is explained in terms of molecular properties through the solute solubility and the surface tension of the solid/liquid interface

In section 3.2, the evaporation technique used in our studies to characterize the induction time as a function of initial solute concentration and rate of evaporation is described in detail In section 3.3, the experimental results establishing the presence of a critical supersaturation value and its dependence on material properties are reported Also, the classical nucleation theory is used to link the critical supersaturation to solubility and crystal surface tension Finally, the key results of these controlled evaporation experiments are summarized in section 3.4

3.2 Experimental Systems and Methods

Glycine (Fluka, >99.5%), L-histidine (Fluka, >99.5%) and paracetamol

(N-Acetyl-4-aminophenol, Sigma, 98%) were dissolved in deionized water (18 MΩ-cm, E-pure, Barnstead) Silicotungstic acid (Sigma, >99.9%) was dissolved in deionized water in the presence of lithium chloride (Fluka, Certified A.C.S.) Hen egg white lysozyme (Seikagaku, 6× recrystallized) was dissolved in sodium acetate buffer (0.1 M, pH = 4.50) in the presence of sodium chloride (Fluka, Certified A.C.S.) All chemicals were used without further purification

3.2.2 Evaporation-Based Crystallization Platform

3.2.2.1 Rationale of Design

To rapidly generate kinetic data, Kenis and co-workers58 have developed an evaporation-based crystallization platform that controls the rate of evaporation of solvent from a crystallizing droplet to the outside environment through a channel of predefined geometry and dimensions (Figure 3.1) Gradual evaporation of the solvent drives the solution of the droplet to a condition conducive to a phase transition A phase transition is ensured in every experiment since the final state of the droplet will always be a completely desiccated drop Arrays of 10 of these crystallization compartments are micro-machined in a polypropylene block so that multiple experiments can be performed simultaneously

The aforementioned crystallization platform offers several advantages Firstly, each set of experiments takes less than a week and often less than 3 days; hence, many experiments can be performed in a short period of time Secondly, the evaporation method precisely regulates the rate of solvent evaporation by altering the

dimensions of the evaporation channel, thus permitting the systematic study of the effect of the rate of solvent evaporation (and thus the rate of supersaturation) on nucleation Thirdly, this platform requires only minimal amounts of material because of the small sample volumes (2-5 μl) required for each experiment

3.2.2.2 Calculation of the Rate of Evaporation

In light of the work done by Fowlis et al.74, four sequential transport processes relevant to evaporative crystallization are identified: (1) diffusion of the solute from the interior of the droplet to the surface; (2) evaporation from the droplet surface to the air space of the crystallization compartment; (3) diffusion across the air space to the entry of the microchannel; and (4) diffusion through the microchannel to the

ambient atmosphere To determine the overall mass transfer rate of water, the kinetics of each mass transfer process must be known Since the mass transfer rates

of the four sequential processes are not of the same order of magnitude, the calculation can be simplified by identifying the rate determining step (RDS) to estimate the overall mass transfer rate The assumptions made in the procedure of identifying the RDS can then be justified by comparing the experimental rates of evaporation with the calculated rates

For process 1, water molecules from the interior of the droplet need to diffuse across the concentration boundary layer at the surface of the droplets, which is created by the gradual evaporation of water In a solution droplet under the earth’s gravity, this mass transfer is governed simultaneously by the diffusion of water molecules in the liquid phase and the bulk flow convection due to the density gradient within the droplet This convection can greatly shorten the relaxation time

of concentration gradient within the droplet, thus process 1 is assumed to be much faster than processes 3 and 4 In process 2, the liquid and vapor phases are assumed to reach thermodynamic equilibrium instantaneously when they are in contact with each other, giving rise to identical chemical potentials of the two phases Therefore, the evaporation of water molecules from the liquid phase to vapor phase

is assumed to be instantaneous Processes 3 and 4 are identical physical processes whose relative mass transfer rates depend solely on the geometrical constraints; given that the transport distances are comparable in both processes, the cross-sectional area become the exclusive determining factor in the relative mass

transfer rate Because the diameter of the crystallization compartment is much

greater than that of the microchannel (Figure 3.1), it is concluded that process 4 is

the RDS for the overall transport process

According to the model of molecular diffusion for water diffusing through

stagnant, non-diffusing air, the mass transfer rate of process 4 can be written as75

WA

1ln

where EWA is the molar flow rate of evaporating water out of the crystallization

compartment, DWA is the diffusivity of water vapor in air, R is the gas constant, T is the

absolute temperature, P is the total pressure while P1 is the saturated vapor pressure of

water, and P2 is the partial pressure of water vapor in the laboratory environment that

is calculated from multiplying relative humidity by P1 γ is the activity coefficient of

water in the droplet, and AC and L are the cross-sectional area and length of the

evaporation channel, respectively

Eq (3.1) was validated by measuring the drying times of several droplets of

known volume These experimental drying times of the droplets are in good

agreement (Table 3.1) with the calculated times from Eq (3.1) Thus the mass

transfer equation describes the evaporation process adequately

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

range from 0.5 to 2 h) using an optical microscope (Leica MZ12.5) The limit of observation for this microscope is about 5 µm Given known rapid crystal growth rates from literature,77-80 the time between the actual onset of crystal nucleation and the crystal reaching the minimum observable size can be neglected After setting

up the experiments, if no crystals were seen at, say, m hours, but crystals were apparent at n hours, then the nucleation time was taken as (m+n)/2 hours Typical

numbers of crystals observed in our experiments ranged from 1 to 4 The experiments were carried out under different combinations of temperature and relative humidity (RH), and the rates of evaporation under each set of experimental conditions were controlled by changing the cross-sectional area of the evaporation channel Three replicates of each experiment were performed to account for reproducibility and experimental uncertainty

3.3 The Effects of Rate of Supersaturation on Crystallization Kinetics

The crystallization behavior of several compounds in aqueous solutions was explored Firstly, the nucleation times were determined as a function of the initial concentration of the solute (glycine, L-histidine, paracetamol, STA or HEWL) and the precipitating agent (LiCl and NaCl for STA and HEWL, respectively) at different evaporation rates (Figure 3.2 and Figure 3.3) Note that the concentrations of solute and the precipitating agent (where present) increase with time as evaporation progresses For systems of glycine, L-histidine and paracetamol, the increase in

supersaturation was due solely to the increase in solute concentration; however, for systems of STA and HEWL, supersaturation increased as a joint result of the increase in solute concentration and the increase in precipitant concentration Since the concentrations of the solute and the precipitating agent increased proportionally, the ratio of concentrations of the two species remained constant

For the range of rates of evaporation used in our studies, a linear relationship between the nucleation time and the initial solute concentration is observed Each straight line in Figure 3.2 and Figure 3.3 represents a different constant evaporation rate of water The experiments were very reproducible (Table 3.2)

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

Nucleation Time (h) Channel

Diameter

Standard Deviation (h)

1.5 17.5 17.5 15.0 16.7 1.44 1.4 16.5 18.0 18.0 17.5 0.87 1.3 20.0 20.0 20.0 20.0 0.00 1.2 22.0 23.0 26.0 23.7 2.08 1.1 26.0 26.0 26.0 26.0 0.00 1.0 32.0 32.0 34.0 32.7 1.15 0.9 38.0 38.0 40.0 38.7 1.15 0.8 48.0 48.0 46.0 47.3 1.15

Figure 3.2 shows experimental results for glycine crystallization at three different sets of temperature and humidity For each rate of evaporation, a straight line can

be drawn through the experimental induction times and extrapolated to an induction time of zero, where the line intersects with the horizontal axis An induction time

of zero for a given glycine concentration implies that a solution will nucleate

spontaneously without any water evaporation if prepared at that initial glycine concentration When all the lines for various evaporation rates are extrapolated to the point of zero induction time for a temperature of 21°C, they all intersect the

horizontal axis at nearly the same glycine concentration cc ≈ 225.5 g/l (Figure 3.2b)

At this temperature the solubility of glycine ce is 202 g/l,81 and the critical

supersaturation Sc (= cc/ce) is therefore equal to 1.12 Similar results are obtained

from crystallization experiments with the amino acid L-histidine, the pharmaceutical paracetamol, the inorganic compound STA, and the protein HEWL (Figure 3.3) For each of these compounds, all the extrapolated lines for a given temperature and humidity intersect at a single concentration for an induction time of zero, leading to

unique values of Sc for each compound at a given temperature and humidity (Table

3.3)

0 20 40 60 80 100 120

0.276 0.246 0.226 0.188 0.162 0.134 0.114 0.093