Crystallization kinetics and phase behaviour of molecular solutions

Acknowledgements 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. i Table of Contents TABLE OF CONTENTS Acknowledgements .i Table of Contents .ii Summary .iv List of Tables v List of Figures vii General Introduction .1 Literature Review 2.1 Why Pharmaceutical Crystallization? 2.2 Solution Crystallization and Phase Behavior .5 2.2.1 Nucleation and Growth 2.2.2 Induction Time, Metastable Zone Width and Critical Supersaturation 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 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 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 Generalized Phase Diagram of Molecular Solutions 63 5.1 Introduction 63 5.2 Experimental Systems and Methods 66 ii Table of Contents 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 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 Conclusion and Recommendation 118 7.1 Conclusions 118 7.2 Future Directions 119 Bibliography 121 iii Summary 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. iv List of Tables 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 = 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 = mm. The activity coefficients are calculated from an empirical correlation developed by Myerson and co-workers76 The size of the cross-sectional area varies for different experiments. .28 Table 3.2 Standard deviation of nucleation times of aqueous glycine solutions. Experimental conditions and crystallization platform specifications are the same as those in Table 3.1 .30 Table 3.3 The average extrapolated critical concentration and critical supersaturation for different compounds crystallizing under various conditions: glycine (in water), STA (2 M LiCl, water), L-histidine (water), paracetamol (water), and HEWL (4.06 %(w/v) NaCl, 0.1 M acetate buffer, pH = 4.50). The activity coefficients for water in solutions with critical and saturated concentrations of solute, 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 v List of Tables 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 vi List of Figures 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(tn) 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 vii List of Figures 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 viii List of Figures 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 and 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 ix General Introduction 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 Metastable States of Molecular Solutions The partitioning test shows that the upper layer of the liquid contain more ibuprofen due to relatively lower water content in the solvent. It can be seen from Table 6.4 that a large amount of water is "expelled" to the lower layer. Interestingly, note also that the ethanol content is relatively constant in both upper and lower liquid layers, independent of the starting ibuprofen concentration. 6.4 Concluding Remarks In this work the existence of metastable phase transitions such as liquid-liquid phase separation and gel formation in small molecule solutions is explored. The results are aided by the ability to compare states of solutions at similar values of the strength of interparticle attraction as characterized by D2. As a result one can develop generalized phase diagrams, which allow comparison of both small molecule and nanoparticle solutions. These studies show that while the equilibrium behavior of the solutes is similar, i.e., they have similar solubilities at similar D2 values, the metastable states seen on quenching below the solubility boundary are variable. Several explanations drawn largely from studies motivated by colloidal and nanoparticle systems are seen to potentially capture the observed trends. These include anisotropic and limited valency interactions. However, much more work is required to understand how these models can be linked to the observed states. These studies also point to the limited understanding one has of the nucleation of equilibrium states in nanoparticle and small molecule solutions. 117 Conclusion and Recommendation CONCLUSION AND RECOMMENDATION 7.1 Conclusions Based on the research work and results presented in this thesis, the following conclusions can be drawn: ♦ Strong experimental evidence has been found for the existence of a state of critical supersaturation, above which spontaneous nucleation occurs instantaneously in solutions. The critical supersaturation has been measured for a wide range of molecules including the amino acids glycine and L-histidine, the pharmaceutical paracetamol, the inorganic molecule silicotungstic acid (STA), and the protein hen egg white lysozyme (HEWL), and it is independent of process kinetics, i.e. it is a thermodynamic phenomenon. This critical supersaturation is correlated to the solubility of the compound and the surface tension between the crystal and the surrounding media through well-established concepts of the classical nucleation theory and equilibrium thermodynamics. ♦ The rate of supersaturation generation has a profound impact on polymorph selectivity. By introducing a slow rate of solvent evaporation, the thermodynamically most stable γ form of the glycine crystals were grown from an aqueous solution without introducing chemical additives or applying external electromagnetic stimuli. It is hypothesized that this slow rate of evaporation allows the solution to sample a broad range of energy states and 118 Conclusion and Recommendation thus the system is not trapped in a local-minimum energy state that results in a metastable solid form, the α polymorph. The slow rate of supersaturation leads to redissolution of the metastable nuclei of α polymorph in favor of the lower free energy molecular configuration that leads to the formation of the nuclei of the γ polymorph. ♦ It has been found that the pair contribution of the scaled long-time self diffusivity D2 can be used as a measure for the strength of solute interactions and aids in the development of a generalized phase diagram. Using this phase diagram, a corresponding-states solubility behavior has been identified for both small molecules and nanoparticles including glycine, L-histidine, L-phenylalanine, paracetamol, ibuprofen, citric acid, trehalose, and hen egg white lysozyme, when compared on the same basis of particle interactions. This suggests that advances made in understanding the equilibrium phase behavior and kinetic phase transitions with nanoparticle suspensions as model systems can be applied to a wide variety of molecular solutions. 7.2 Future Directions Over the past century the knowledgebase development of crystallization processes has received much attention and has already impacted society with its applications in the chemical, agrochemical, pharmaceutical and biotechnological industries. Among the sub-areas of crystallization studies, this thesis focuses on the understanding of crystallization kinetics and phase behavior of molecular solutions. 119 Conclusion and Recommendation While the data in this thesis show promise, much more work is still required to gain the physical insights of phase transitions. ♦ While small molecule and nanoparticle solutions display a corresponding-states solubility behavior, the presence and absence of metastable states of such solutions are variable. It is found that both temperature and solvent composition play an important sole in tuning particle interactions, which drives solution phase behavior more unpredictable. Apart from innovative experimental advances, only detailed theoretical and modeling studies involving atomic bonding, molecular interactions, and cluster formation and distribution will assist in solving these puzzles. 120 Bibliography BIBLIOGRAPHY 1. Mullin, J. W., Crystallization. 4th ed. 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Soc., Perkin Trans. 2001, 824-826. 132 [...]... 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... 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 17 Literature Review kinetics and polymorph selectivity, however, the effects of the rate of supersaturation on the kinetics and thermodynamics of crystallization have... Evaporation-Driven Crystallization: Effects of Supersaturation on Crystallization Kinetics 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... 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 11 Literature Review Crystal morphology, habit, and size have tremendous practical and commercial impact on research and. .. 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... and nonequilibrium phase transitions such as liquid-liquid phase separation and gel formation 19 Evaporation-Driven Crystallization: Effects of Supersaturation on Crystallization Kinetics 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... condensing vapor studies, and has been further developed by Volmer,27 Gibbs,28 5 Literature Review 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... 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... 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... 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 . Compositions of ibuprofen, ethanol and water of the two liquid phases formed by liquid-liquid phase separation of solutions of ibuprofen in EtOH/H 2 O (50/50 w/w%) at 20 °C. 116 List of Figures. Values of ε /k extracted from Eq. (6.3) for ibuprofen in different List of Tables vi solvents 103 Table 6.3 Characterization of compositions of ibuprofen, ethanol and water of the upper and. 7. List of Tables v LIST OF TABLES Table 3.1 Comparison of experimental and calculated drying times and rates of evaporation for aqueous solutions of glycine. Initial volume of solution