MULTICOMPONENT DIFFUSION OF MACROMOLECULE-ADDITIVE AND DRUG-SURFACTANT AQUEOUS TERNARY SYSTEMS

Transcription

1 MULTICOMPONENT IFFUSION OF MACROMOLECULE-AITIVE AN RUG-SURFACTANT AQUEOUS TERNARY SYSTEMS By HUIXIANG ZHANG Master of Science, 988 Central South University Changsha, Hunan, China Master of Science, 996 University of Minnesota Minneapolis, Minnesota Submitted to the Graduate Faculty of the College of Science and Engineering Texas Christian University In partial fulfillment of the requirements for the degree of octor of Philosophy May 9

2 ACKNOWLEGEMENTS I gratefully acknowledge the help of many people during my graduate study. First and the foremost, I sincerely thank my advisor, r. Onofrio Annunziata, for his guidance and insightful advice along the way. I would like to thank r. John G. Albright, who is always supportive on the instrumentation of Gosting diffusiometer and provides help on diffusion experiments. I thank my committee members, rs. Jeff Coffer, Sergei zyuba, and Youngha Ryu for their valuable comments. I thank my fellow graduate students, past and present, who have made my journey enjoyable. A special thank goes to Ying Wang and Cong Tan for their help. support. I also would like to thank my friend Haresh Bhagat for his encouragement and I would like to thank my great family, my husband and children for their support and love to make this happen. The funding for this project was provided by TCU RCAF and ACS PRF funds. ii

3 TABLE OF CONTENTS Acknowledgements..... ii List of Figures viii List of Tables..... xi Chapter Introduction Chapter Theoretical Background Introduction Isothermal diffusion in liquids iffusion in binary systems iffusion in ternary systems Frictional formalism Macromolecule-additive interactions Thermodynamics of macromolecule-additive-solvent ternary systems Relation of preferential-interaction coefficient to ternary diffusion The role of macromolecule solvation on onsager transport coefficients.. iii

4 .4 Micelles and solubilization Surfactants and micelles Partitioning model for solubilizates Chapter 3 Materials and Methods Materials Solution Preparation ensity Measurements Rayleigh Interferometry and the Gosting iffusiometer ynamic Light Scattering rug Solubility Measurements Chapter 4 Effect of Macromolecular Polydispersity on iffusion Coefficients Measured by Rayleigh Interferometry Introduction Theory Effect of macromolecule polydispersity on diffusion coefficients of macromolecule-solvent system iffusion equations iffusion moments and polydispersity indices Molecular-weight polydispersity iv

5 4...4 Procedure for the characterization of diffusion polydispersity Effect of macromolecular polydispersity on ternary diffusion coefficients of macromolecule-additive-solvent systems Monodisperse macromolecular solute Polydisperse macromolecular solute Corrective procedure Results and discussion Effect of macromolecule polydispersity on diffusion coefficients of macromolecule-solvent systems etermination of A, ω, and ξ for polymer-solvent system Comparison between Rayleigh Interferometry and ynamic Light Scattering Simulation on a model polydisperse Conclusions Chapter 5 Effect of Solute Solvation and Preferential Interaction on Coupled iffusion of Macromolecule-Osmolyte-Water Ternary System Introduction Results v

6 5.. iffusion coefficients.. 84 ( m) ( m) 5.. etermination of µ / µ and ( L ) /( L ) iscussion.. 9 ( m) ( m) 5.3. Examination of µ / µ Examination of ( L ) /( L ) Examination of cross-diffusion coefficients Conclusions Chapter 6 Characterization of Macromolecule-Salt Interactions by Ternary iffusion Introduction Results and iscussion Experimental diffusion coefficients Preferential-interaction coefficients Conclusions and future work. 7 Chapter 7 Multicomponent iffusion in Nonionic rug-surfactant Aqueous Mixtures Introduction Theory Relationship between drug partitioning constant and drug solubility vi

7 7.. iffusion model Results iffusion in the tyloxapol-water system rug solubility in tyloxapol-water solutions iffusion in the hydrocortisone-tyloxapol-water system iscussion Conclusions Chapter 8 iffusion of an Ionic rug in Micellar Aqueous Solutions Introduction Results Naproxen solubility as a function of tyloxapol concentration Ternary diffusion coefficients iffusion model Thermodynamic factors Onsager transport coefficients iffusion coefficients iscussion Conclusions... 7 Chapter 9 Summary.. 73 vii

8 Appendix Appendix A Appendix B... 9 Appendix C... Appendix Appendix E. References. 7 Vita Abstract viii

9 LIST OF FIGURES The two-domain model Chemical structure of Triton X Micelle formation Graphic representation of the two-phase model Scheme of the optical apparatus in Rayleigh configuration (A) Scheme of the Rayleigh interferometric pattern (B) Picture of the Rayleigh interferometric pattern taken from the Gosting diffusiometer Schematic drawing of the diffusiometer Tiselius cell Scheme of the dynamic light scattering apparatus etermination of the parameters, a, a and a by linear extrapolation to y = of the functions: s (A), ( s a )/ y (B) and ( s a a y )/ y 4 (C) Mean diffusion coefficients, LS and A for the PEGk-water system as a function of polymer concentration, C... 7 Mean diffusion coefficients, LS, (f) LS, (s) LS and A for the PVA-water system Volume-fixed diffusion coefficients as a function of EG concentration, C, for the PEG-EG-H O system at C =.5 mm and 5 ºC ix

10 ( m) ( m) 4 Thermodynamic ratio, µ / µ, and transport-coefficient ratio, ( L ) /( L ), as a function of EG concentration, C, for the PEG-EG-H O system ex 5 Water excess, N, as functions of EG concentration, C, for the PEG-EG-H O system iffusion-coefficient ratios, [( ) ( ) ]/ C and [( ) ( ) ] / C as a function of EG concentration, C, for the PEG-EG-H O. 7 Volume-fixed diffusion coefficients as functions of NaCl concentration, C, for the PEG-NaCl-H O system at C =.5 mm and 5 ºC... 8 Volume-fixed diffusion coefficients as functions of Na SO 4 concentration, C, for the PEG- Na SO 4 -H O system at C =.5 mm and 5 ºC.. 9 Viscosity-corrected ( ) V ( η η) as a function of salt ionic strength for the PEG-Na SO 4 -H O, PEG-NaCl-H O, and PEG-KCl-H O ternary systems 3 ( m) ( m) Thermodynanic ratio, µ / µ, as a function of salt concentration, C, for (a) PEG-NaCl-H O and PEG-KCl-H O ternary systems, (b) PEG-Na SO 4 -H O ternary system Chemical structures of hydrocortisone (A) and tyloxapol (B)... Mutual diffusion coefficients for the binary tyloxapol-water system as a function of tyloxapol volume fraction measured by Rayleigh inteferometry and dynamic light scattering at 5 C Chemical structures of dexamethasone (A) and rimexolone (B)... 3 x

11 4 Solubility of hydrocortisone, dexamethasone and rimexolone in tyloxapol-water mixtures as a function of tyloxapol volume fraction at 5 C Hydrocortisone and tyloxapol main-diffusion coefficients, ( ) V and ( ) V as a function of tyloxapol volume fraction measured by Rayleigh inteferometry at 5 C Ratio of hydrocortisone cross-diffusion coefficient, ( ) V, to hydrocortisone concentration, C, as a function of tyloxapol volume fraction measured by Rayleigh inteferometry at 5 C Schematic diagram of one-dimensional steady-state diffusion occurring between two compartments (L and R) Ratio ψ / ψ for case A as a function of ω at several values of γ Chemical structure of naproxen anion Solubility of naproxen in tyloxapol-water mixtures as a function of tyloxapol volume fraction at 5 C and ph Ternary inter-diffusion coefficients for the naproxen()-tyloxapol()-water system as a function of tyloxapol volume fraction at 5 C and ph Ternary diffusion ratios for the naproxen()-tyloxapol()-water system Ternary diffusion ratios for the naproxen()-tyloxapol()-water system... 7 xi

12 LIST OF TABLES Operating conditions for hydrocortisone and dexamethasone HPLC measurement. 46 Operating conditions for rimexalone HPLC measurement Operating conditions for naproxen HPLC measurement Mean diffusion coefficients, A, and polydispersity indices, ω and ξ for polymer-water systems Experimental (Exp) and calculated (Calc) values of A, ω and ξ for PEG-water polydisperse systems Relative percent differences for pp, ps, sp and ss with / = / = /. 77 ps ss sp ss 7 Relative percent differences on sp for pp / ss = / and / = /.. 78 ps ss 8 Relative percent differences on sp for pp / ss = / and / = / = /. 79 ps ss sp ss 9 Relative percent differences on sp for sp / ss = /.. 79 Ternary diffusion coefficients for the PEG-EG-H O system at 5 C. 85 Ternary diffusion coefficients for the PEG-NaCl-H O system at 5 C 9 Ternary diffusion coefficients for the PEG-Na SO 4 -H O system at 5 C... 3 Values of fitting parameters (a and b) and water excess ( N ex ) for PEG-additive-water ternary systems.. 7 xii

13 4 Solubility of hydrocortisone as a function of tyloxapol concentration at 5. C 33 5 Solubility of rimexolone as a function of tyloxapol concentration at 5 C Solubility of dexamethasone as a function of tyloxapol concentration at 5 C Solubility parameters iffusion coefficients for the hydrocortisone-tyloxapol-water ternary system Mathematical expressions for drug fluxes and concentration profiles.. 4 Naproxen solubility as a function of tyloxapol concentration at ph 7 at 5 C.. 48 iffusion coefficients for the naproxen-tyloxapol-water ternary system... 5 xiii

14 Chapter Introduction

15 Mutual-diffusion is the net transport of a chemical component from a region of a higher concentration to a lower concentration by random molecular motion. Understanding diffusion in liquid mixtures is important for many industrial and biochemical processes in the presence of concentration gradients. -3 Mutual-diffusion coefficients are fundamental for modeling controlled release of drugs, 4-7 drying behavior of polymers in mixed solvents, 8 transport near interfaces, 9 in vivo transport processes involving biomacromolecules, - protein crystallization and mixing inside microfluidic devices. 3-6 Majority of these applications involve aqueous solutions containing more than one solute component. Therefore these are known as multicomponent systems. iffusion in multicomponent systems with N solutes is described by the generalized Fick s first law 3 as following: N J = C (with i, j =,,..., N) i ij j i= where J i and is the molar flux of solute i, C j is the molar concentration of solute j, and the ij are the multicomponent mutual-diffusion coefficients. Note that the diffusion process in a binary system (one solute) is characterized by one diffusion coefficient. However, diffusion in a ternary system (two solutes) becomes fundamentally more complicated since it requires four diffusion coefficients. N In general, the complete description of a system with N solutes requires an N matrix of diffusion coefficients, ij, relating the flux of each solute component to the gradients of all solute components. 3 The N diagonal main coefficients, ii,

16 characterize the flux of a solute due to its own concentration gradient while the remaining N( N ) cross coefficients, ij with i j, characterize the flux of a solute due to the concentration gradient of the other solute. These cross terms describe the phenomenon of coupled diffusion. The knowledge of all the diffusion coefficients is required to accurately model and predict diffusion-based transport of solutes. However, assumptions and simplification are often made in analyzing transport processes, due to the difficulties of obtaining all the diffusion coefficients. For example, many diffusion problems utilize the assumption that the N( N ) cross-diffusion coefficients can be ignored., This assumption may not be valid for multicomponent systems with strongly interacting components or for processes occurring in the presence of large concentration gradients. Since cross-diffusion coefficients describe the net interaction between two different solutes, experimental and theoretical investigations on ternary systems have played a major role in comprehension of coupled diffusion. It has been demonstrated that cross-diffusion coefficients can be quite significant, even exceeding normal diagonal diffusion coefficients in magnitude in systems. 7-9 Large and positive cross-diffusion coefficients have been observed in systems where the solutes tend to salt-out. Large and negative cross-diffusion coefficients can occur in systems with large attractive interactions between solutes. This happens in the systems that involve ions, chemical association and excluded-volume interactions. 8-9 Even though the effect of coupled diffusion in relation to the behaviors of salting-in and salting-out are qualitatively known, a quantitative understanding of coupled diffusion and its relation to molecular interactions is not well understood, especially in macromolecular and colloidal systems. The general purpose of this dissertation is to examine multicomponent diffusion in 3

17 macromolecular or colloidal ternary systems using Rayleigh interferometry. - Rayleigh interferometry is an optical method that can be used to measure diffusion coefficients with a high precision (relative errors of.%). This technique probes diffusion-based mixing that is occurring between two initial solutions separated by a flat horizontal interface. The first type of ternary system that we have investigated is the macromoleculeadditive-water system in which macromolecule-additive salting-out interactions occur. Aqueous solutions containing macromolecules typically contain other additives such as salts, osmolytes, organic solvents and denaturants. -7 These components are necessary for modulating the thermodynamic state (chemical potential) of macromolecules in water so that processes such as crystallization, 8,9 aggregation, 3 conformational changes 3-33 and enzymatic activity 34 would be either promoted or inhibited. Additives not only change the diffusion coefficients of macromolecules in solution but also introduce complexity in the transport processes because additive diffusion is coupled to that of the macromolecules according to the generalized Fick s first law shown above. -7 To better understand the effect of additives, we have experimentally characterized multicomponent diffusion for poly(ethylene glycol) (PEG) with a molecular weight of kg mol - in the presence of di(ethylene glycol) (EG). PEG is a hydrophilic nonionic polymer used in many biochemical and industrial applications. ue to its nontoxic character, this chemical can be found in cosmetics, food, and pharmaceutical products. The mild action of PEG on the biological activity of cell components explains the success of this polymer in biotechnological applications. PEG is commonly used for liquid liquid partitioning 4 and precipitation of biomacromolecules. 4 In protein crystallography, PEG is considered 4

18 the most successful precipitating agent for the production of protein crystals, 4 the crucial step for the determination of the molecular structure of a protein. All these applications make PEG by far the most widely used polymer in aqueous solutions of biological molecules. The osmolyte EG has been chosen as an additive. Since the chemical structure of the macromolecule and the additive are similar, we believe that their ternary aqueous mixtures represent a reference system for a basic understanding of multicomponent diffusion in aqueous solutions. The main objective of this work is to examine the relation between multicomponent diffusion and macromolecule-additive net interactions in solution. In fact, we have shown that the four ternary diffusion coefficients can be used to extract the dependence of the macromolecule chemical potentials on additive concentration. In addition, we have examined multicomponent diffusion of PEG in aqueous solutions in the presence of salts (KCl, NaCl and Na SO 4 ). These studies have allowed us to relate multicomponent diffusion to the ranking of salts as salting-out agents (Hofmeister series) Our results will also help to understand the behavior of multicomponent diffusion coefficients previously measured for the more complex ternary systems containing lysozyme, a model protein, in the presence of aqueous chloride. 46 Another important feature related to macromolecules and colloidal particles is polydispersity. For example, all synthetic polymers are characterized by a molecularweight distribution In this dissertation, we introduce a theoretical framework for examining macromolecular polydispersity using Rayleigh interferometry. We experimentally characterize the diffusion of two synthetic polymers, poly(ethylene glycol) and poly(vinyl alcohol), in aqueous solution. We then examine the experimental 5

19 results with the proposed polydispersity theory. An important outcome of this investigation has been the implementation of a new algorithm that removes polydispersity effects from experimental diffusion results on macromolecule-additivewater ternary systems. We also compare our diffusion coefficients obtained using Rayleigh interferometry with those obtained using dynamic light scattering (LS), 49 another optical technique available in our laboratory. LS, also known as quasielastic light scattering or photon correlation spectroscopy, is a versatile method used for measuring mutual diffusion coefficients of particles with characteristic size ranging from nm to µm in solution. This technique probes relaxation times of microscopic concentration fluctuations in solution. Compared to the Rayleigh interferometric method, 5-5 it has an advantage of requiring a small sample size and a short run time. Hence, many LS instruments are commercially available for routine measurements of diffusion coefficients. Moreover, the demand for LS instruments has grown significantly over the past few years because its application in pharmaceutics and biotechnology Most of the particles of interest in these fields are proteins, polymers, viruses, micelles, liposomes, and inorganic nanoparticles. Rayleigh interferometry, compared to LS, is more directly connected to Fick s law, thereby offering superior accuracy in the determination of mutual diffusion coefficients. Rayleigh interferometry can also be used -, to determine the whole diffusion coefficient matrix of a multicomponent system, which is another advantage with respect to LS. A direct comparison between LS and Rayleigh interferometry provides a means to reveal the accuracy of diffusion coefficients measured by LS and validate or refine LS theories. 6

20 The second type of ternary system we have investigated is the drug-micelle-water system in which drug-micelle salting-in interactions occur. Since drug formulations typically include other additives, drug transport is expected to depend on the additive concentration gradients. Micelles and other nanocarriers are often used to increase drug solubility and enhance drug bioavailability They are also used for drug protection and targeted drug delivery As a model nano-carrier system, a tyloxapol micelle is considered. Tyloxapol is a nonionic oligomeric surfactant that forms very stable micelles in solution and possesses poly(ethylene glycol) hydrophilic groups. 64 The main objective of this work is to examine the effect of concentration and concentration gradient of micelles on diffusion-based transport of a nonionic drug (hydrocortisone) as well as an ionic drug (naproxen). The dissertation is organized as follows. The basic theoretical background needed to understand the investigated multicomponent systems is discussed in Chapter. Materials, experimental methods and related data analysis are described in Chapter 3. The role of polydispersity on macromolecular solutions is discussed in Chapter 4. iffusion studies on the PEG-EG-water ternary system are presented and discussed in Chapter 5. Preliminary analysis of diffusion studies on PEG-salt-water ternary systems is presented in Chapter 6. iffusion studies on hydrocortisone-tyloxapol-water and naproxentyloxapol-water ternary systems are presented and discussed in Chapters 7 and 8 respectively. Finally, Chapter 9 summarizes the results. 7

21 Chapter Theoretical Background 8

22 . Introduction In this Chapter, theoretical relevant to binary and ternary diffusion, macromolecule-additive interactions, as well as micelles and solubilization are described.. Isothermal iffusion in Liquids.. iffusion in Binary Systems iffusion of a solute in a binary solute()-solvent() system is given by Fick s first law: 3 J = C () where C is molar concentration of the solute, J is the corresponding flux and is the mutual-diffusion coefficient. An important aspect of diffusion is that the flux can be described relative to different reference frames. 65 This is related to the fact that the flux of the molecules can be defined only if a given reference frame is introduced. iffusion coefficients are normally reported with respect to the so-called volume- and solvent-fixed frames. In the volume-fixed frame, the fluxes of the components of a binary system satisfy ( J = ) V V + ( J) V V ; in the solvent-fixed frame, it follows ( J ) =. Here, J and V are the molar flux and partial molar volume of the solute, respectively. The subscript V denotes the volume-fixed frame. The subscript denotes the solvent component when appended directly to a flux, and denotes the solvent-fixed frame when appended outside the parentheses to an already-subscripted flux or diffusion coefficient. iffusion 9

23 measurements normally correspond to the volume-fixed frame, ( ) V. This is because the volume-fixed frame corresponds to the laboratory reference frame to an excellent approximation. 65 The solvent-frame diffusion coefficient ( ) can be calculated from ( ) V using: 66,67 ( ) = ( ) /( CV ) () V At infinite dilution, the tracer-diffusion coefficient is obtained as: * = ( ) = ( ) V. In general, ( ) > ( ) V. This is because solvent molecules move in the direction opposite to the solute molecules. Thus, the mobility of the solute molecules appears to be higher relative to an observer at rest with respect to the solvent counter-flux. According to non-equilibrium thermodynamics, the actual driving force of diffusion is the gradient of the solute chemical potential, µ. The solvent-frame diffusion coefficient, ( ), is more simply related to the solute chemical potential compared to ( ) V In this reference frame, we can write: ( J ) = ( L ) µ (3) where ( L ) is called the Onsager diffusion coefficient in the solvent-reference frame. Combining Eq. with Eq. 3 yields:

24 ( ) = ( L ) µ (4) ( c) where µ ( µ C ), 66-7 where T is the absolute temperature and p the pressure. ( c) T, p The superscript (c) indicates that the derivative is taken with respect to the molar concentration. This can be distinguished from those based on molality, which is introduced later in this chapter. It is important to remark that the diffusion coefficient depends not only on the intrinsic mobility of the solute molecules in solution but also on the system thermodynamic behavior through the thermodynamic factor µ shown in Eq. ( c) 4. This is one of the reasons why diffusion measurements can be used for the determination of thermodynamic parameters. At infinite dilution, we have: µ = RT / C, where R is the ideal-gas constant ( c) and T is the absolute temperature. Therefore, the infinite-dilution diffusion coefficient, *, is given by ( L ) = RT (5) C * Since * does not vanish as C, ( L ) is directly proportional to C. Brownian-motion theory 7 provides the molecular basis for understanding the physical meaning of *. The most important result of this theory is the Stokes-Einstein equation: 7

25 kt * B = 6π Rh η (6) where k B is the Boltzmann constant, R h is the equivalent hydrodynamic radius of the solute particle, and η is the viscosity of the solvent assumed to be a continuum fluid. Stokes-Einstein equation shows that the diffusion coefficient of macromolecule is small compared to that of the small ions or additive molecules. It also shows that * decreases as the viscosity of the surrounding continuum fluid increases. This occurs because viscosity provides a resistance to the motion of the macromolecules in solution... iffusion in Ternary Systems For a ternary solute()- solute ()-solvent() system, Fick s first law becomes: J = C+ C (7a) J = C+ C (7b) Here, C and C are the molar concentrations of the two solutes, J and J are the corresponding fluxes and the s (with i, j =,) are the four multicomponent diffusion ij coefficients. Main-diffusion coefficients, and, describe the flux of a solute due to its own concentration gradient, while cross-diffusion coefficients, and, describe the flux of a solute due to the concentration gradient of the other solute. We note that the cross-term ij (with i j) in Eqs. 7a,b vanishes as C, otherwise, the i

26 paradoxical result: J = C is obtained even in the absence of i. The ij matrix i ij j is normally referred as the multicomponent-diffusion matrix. Similar to binary systems, the diffusion coefficients in Eqs. 7a,b can be described relative to different reference frames. 65 In the volume-fixed frame, the fluxes of the components of a ternary system satisfy ( J) VV + ( J) VV + ( J) VV = ; in the solventfixed frame, we have ( J ) =. Here, J i and V i are the molar flux and partial molar volume of component i, respectively. The solvent-frame diffusion coefficients, ( ij ), are more simply related to chemical-potential derivatives compared to the ( ) as in the case of the binary system. Thus, thermodynamic analysis on diffusion data is usually performed on ( ij ). 54 These diffusion parameters are calculated from the measured ( ) using 66,67 ij V ij V ( ) = ( ) + [ C /( CV CV )] [ V ( ) + V ( ) ] (8a) V V V ( ) = ( ) + [ C /( CV CV )] [ V ( ) + V ( ) ] (8b) V V V ( ) = ( ) + [ C /( CV C V )] [ V ( ) + V ( ) ] (8c) V V V ( ) = ( ) + [ C /( CV C V )] [ V ( ) + V ( ) ] (8d) V V V Isothermal diffusion is driven by the gradient of chemical potential, µ i, of the system components. For a ternary system, we can write: 3

27 ( J ) = ( L ) µ + ( L ) µ (9a) ( J ) = ( L ) µ + ( L ) µ (9b) where the ( L ij ) are the solvent-frame Onsager coefficients in the solvent-frame reference system. These coefficients obey the Onsager Reciprocal Relations: ( L ) = ( L ) () Comparing Eqs. 9a,b and Eqs. 7a,b, we can express the solvent-frame diffusion coefficients in terms of Onsager transport coefficients and thermodynamic factors: ( ) = ( L ) µ + ( L ) µ (a) ( ) ( c) ( ) = ( L ) µ + ( L ) µ (b) ( ) ( c) ( ) = ( L ) µ + ( L ) µ (c) ( ) ( c) ( ) = ( L ) µ + ( L ) µ (d) ( ) ( c) where ( c) µ ij ( µ i C j ) T, p, Ck, k j, are the thermodynamic factors characterizing the interactions between the solution components...3 Frictional Formalism The Onsager transport coefficients can be examined in term of frictional coefficients. Indeed a diffusion process can be thought to occur in a quasi-stationary regime in which the thermodynamic driving forces equal the opposing frictional force. 4

28 Based on frictional forces, the following linear laws for diffusion have been proposed by Bearman: 75 N µ = C ζ ( u u ) (with i, j =,..., N) () i j ij i j j= where ζ ij are frictional coefficients that describe the frictional force between species i and species j, and u i is the species velocity of component i relative to the center of mass of the system. The frictional coefficients satisfy the condition: ζij = ζ with i j. 76 ji Eq. is equivalent to resistance-based equations previously reported by Onsager 54 and to the Stefan-Maxwell equations. 78 Contrary to ( L ik ), ζ ij is frame independent and have a direct physical interpretation in terms of friction between the system components. 77 Hence, several theoretical examinations of diffusion coefficients used frictional coefficients. Generally, all ζ ij values (with i j) are assumed to be positive, which is consistent with the physical concept of friction. However, this assumption is not a necessary condition for the second law of thermodynamics to be respected. The relation of the solvent-frame Onsager coefficients, ( L ij ) for a ternary system. In this case, Eq. reduces to, to ζ ij is examined µ = C ζ ( u u ) + C ζ ( u u ) (3a) µ = C ζ ( u u ) + Cζ ( u u ) (3b) 5

29 where ζ = ζ. Here, the expression for µ has been omitted because it can be recovered from Eqs. a,b using the Gibbs-uhem equation: 3 C µ + C µ + C µ =. If u i u j is replaced with ( Ji) / Ci ( J j) / Cj in Eqs. 3a,b, it yields: C µ = ( C ζ + C ζ )( J ) Cζ ( J ) (4a) C µ = ( C ζ + Cζ )( J ) C ζ ( J ) (4b) By comparing Eqs. 4a,b with Eqs. 9a,b, the following expressions for ( L ij ) are obtained: 3 ( ) = C ζ + Cζ L C C ζζ + C ζ C ζ + C ζ ( ) (5) ( ) ( ) L = L = CC C ζζ + C ζ C ζ + C ζ ζ ( ) (6) ( ) C ζ + C ζ L = C C ζζ + C ζ C ζ + C ζ ( ) (7) Physical insight on the behavior of both ( L ij ) can be obtained by examining Eqs. 5-7 for dilute solutions. In this limiting case, we obtain: 6

30 ( L ) = C ( V / ζ ) = C (8) * * ( L ) = C ( V / ζ ) = C (9) * * ( L ) = CC ( V / ζ )( V / ζ ) ζ = CC ζ () * * * * where * V is the molar volume of pure solvent, and = V / ζ is the tracer-diffusion * * i i coefficient of solute i. Eqs. 8- describe the first-order dependence of the Onsager coefficients on the solute concentrations..3 Macromolecule-Additive Interactions.3. Thermodynamics of Macromolecule-Additive-Solvent Ternary Systems The thermodynamics of macromolecule-additive-water mixtures is commonly described using molality-based chemical potential derivatives: ( m) µ ij ( µ i m j ) T, p, mk, k j, where m i is the chemical potential of the i-th component,,3,7,8 and µ = µ. 8 The ( m) ( m) µ s are related to the ( m ) ij µ s by the following linear relations: 7 ( c ) ij ( C / m)( CV CV) µ = ( CV) µ + CV µ (a) ( c) ( m) ( m) ( C / m )( CV CV ) µ = ( CV) µ + CV µ (b) ( c) ( m) ( m) ( C / m )( CV CV) µ = ( CV) µ + CV µ (c) ( c) ( m) ( m) ( C / m )( CV CV ) µ = ( CV) µ + CV µ (d) ( c) ( m) ( m) 7

31 Macromolecule-additive preferential-interaction coefficients, which are thermodynamically linked to the dependence of the macromolecule chemical potential on additive concentration, are the primary thermodynamic parameters used to characterize macromolecule-additive interactions. The preferential interaction coefficient, Γ, is defined as: 8,83 m µ Γ = m µ ( m) lim lim m ( ) m m µ, T, p () These coefficients have been interpreted in terms of models based on the existence of local-bulk domains. 3,8 As shown in Fig., the local domain inside the dashed circle is represented by the water-additive layer surrounding individual macromolecule. This dashed circle represents an imaginary membrane permeable to the additive shown as small dots, and solvent molecules. The bulk domain, which has exactly the same properties of the binary additive-solvent system, is represented by the area outside the imaginary membrane. This local domain is in chemical equilibrium with a bulk domain, representing the water-additive remaining solution. Since macromolecules interact with the additive and water molecules in their vicinity, the concentration of additive in the local domain is different from that of the unperturbed bulk domain. If the additive concentration in the local domain is lower than that of the bulk domain, preferential hydration of the macromolecules occurs. In this case, the preferential-interaction coefficient is negative. On the other hand, a positive value of this coefficient is obtained if the macromolecules preferentially interact with the additive. 8

32 Figure. The two-domain model. The coil represents the macromolecule while the black circles represent the additive molecules. The solvent molecules are omitted for clarity. The local domain is inside the dashed circle, while the bulk domain outside the dashed circle. The molecules of the additive partition between the local domain and the bulk domain. This chemical equilibrium can be described by osmolyte partitioning constant: ( N / N ) α = (3) ( b) ( b) ( n / n ) where N and N are respectively the number of additive and water molecules inside the local domain, and ( b n ) and ( b n ) are respectively the number of additive and water molecules in the bulk domain. When α <, the additive is preferentially excluded from the local domain. On the other hand, when α >, the additive preferentially interacts with the macromolecule. 9

33 The relation between osmolyte partitioning constant α and preferentialinteraction coefficients Γ is given by: m Γ = N ( α ) (4) m where m 55.5mol kg =. Eq. 4 shows that Γ < if the additive is preferentially excluded from the local domain. On the other hand, when Γ >, the additive preferentially interacts with the macromolecule. Since the solvent is the component present in large excess, N does not significantly change with additive concentration. It is important to note that measurements of Γ as a function of m allows us to determine ex the solvent excess: N ( ) N α..3. Relation of Preferential-Interaction Coefficient to Ternary iffusion In order to relate preferential-interaction coefficient, Γ, to the solvent-frame diffusion coefficients, we first solve Eqs. a,d with respect to ( L ) and ( L ). We then use Eq. to obtain: ( ) µ ( ) µ = ( ) µ ( ) µ (5) ( ) ( ) ( ) ( ) ( m) ( m) By inserting Eqs. a-d into Eq. 5, and solving with respect to µ / µ, we obtain:

34 µ µ ( ) ( CV ) ( ) CV ( µ / µ )[( ) ( CV ) ( ) CV ] ( m) ( m) ( m) = ( m) ( ) ( CV ) ( ) ( CV ) + ( ) CV ( ) CV (6) Eq. 6 shows that ternary diffusion coefficients can be used to determine µ / µ, provided that ( m) ( m) µ and ( m) µ are known. Notably, the accuracy of ( m) µ does ( m) not need to be high, i.e. % error in µ results in only % error in ( m) µ / µ. ( m) ( m) This implies that ( m µ ) = RT / m is an excellent approximation for relatively low macromolecule concentrations ( mg/ml). Indeed, several terms contribute only marginally to the value of µ / µ and Eq. 6 can be approximately written as: ( m) ( m) µ µ ( ) CV (7) ( m) ( m) ( ).3.3 The Role of Macromolecule Solvation on Onsager Transport Coefficients Although it is well established that solute molecules move together with their solvation shells in solution, this aspect has not been previously examined in relation to frictional coefficients. ue to solvent binding, the chemical potentials of the solvated solutes ˆi µ are: 84 ˆµ = µ + ν µ (8a) ˆµ = µ + ν µ (8b)

35 where ν i is the number of solvent molecules bound to solute i. This has motivated us to apply the frictional-coefficient formalism not to the solutes but to the solvated solutes. Based on the frictional-coefficient formalism, Eqs. 3a,b are rewritten as: ˆ µ = Cˆ ˆ ζ ( uˆ u ) + C ˆ ζ ( u u ) (9a) ˆ µ = Cˆ ˆ ζ ( uˆ u ) + C ˆ ζ ( u u ) (9b) ˆ i where ˆi µ = µ i + ν i µ, ζ describes the frictional force between solvated solute i and free solvent, and ˆ ζ that between the two solvated solutes. In Eqs. 9a,b, Ĉ = C νc νc is the concentration of free solvent and û is the velocity the freesolvent molecules relative to the center of mass of the system. The relation between ζ ij and ˆij ζ will now be derived. Since the free-solvent flux is given by Ĵ = J ν J ν J, the following relation between û and u can be obtained: uˆ Cu ν Cu ν Cu = C ν C ν C (3) Eq. 3 represents the difference between the free-solvent and solvent reference frames. If we insert Eq. 3 into Eqs. 9a,b and use C µ + C µ + C µ =, we obtain:

36 [( C ν C ) µ ν C µ ] / C = C ˆ ζ ( u u ) + C ( ˆ ζ ν ˆ ζ )( u u ) (3a) [( C ν C ) µ ν C µ ] / C = C ˆ ζ ( u u ) + C ( ˆ ζ ν ˆ ζ )( u u ) (3b) Eqs. 3a,b can be used to obtain explicit expressions for µ and µ. The comparison to Eqs. 3a,b yields: ζ ζ ζ ( C ν C ) ˆ ζ + ν C ˆ ζ = C ν C ν C ( C ν C ) ˆ ζ + ν C ˆ ζ = C ν C ν C ( ) ˆ ( ) ˆ ˆ ν C ν C ζ + ν C ν C ζ = ζ C ν C ν C (3) (33) (34) For dilute solutions, we obtain the following relations between the Onsager transport coefficients and ˆij ζ : ( L ) = C ( V / ˆ ζ ) = C (35) * * ( L ) = C ( V / ˆ ζ ) = C (36) * * ( L ) = CC [ ˆ ζ V ( ν + ν )] (37) * * * * * Eq. 37 shows that negative values of ζ can be obtained in the presence of significant solute solvation. For solvated macromolecules with high molecular weight, ν 3

37 may become large. This implies that ˆ ζ may be neglected in Eq. 34 and ζ ν ζˆ. We obtain: ( L ) C ( L ) V ν (38) *.4 Micelles and Solubilization.4. Surfactants and Micelles Surfactants are organic compounds that are amphiphilic, meaning they contain both hydrophobic groups and hydrophilic groups. 58,59 Figure shows the chemical structure of a common surfactant, Triton X-. 8 Hydrophilic Hydrophobic Triton X- Figure. Chemical structure of Triton X-. Surfactants in aqueous solution form micelles as a result of spontaneous association of their monomeric molecules (see Fig. 3). A micelle is a spherical particle 4

38 characterized by an inner core consisting of surfactant hydrophobic groups and an outer surface consisting of hydrophilic groups interacting with aqueous solution. The formation of micelles occurs above a critical concentration of surfactant called critical micelle concentration (CMC). 59 At a concentration below CMC, surfactant molecules exist predominately as monomeric units, while micelles form when surfactant concentration is above its CMC. The number of surfactant monomers that self-associate to form a micelle is called aggregation number. In this dissertation, micelles are assumed to be monodisperse with aggregation number, m. According to Stokes-Einstein equation (Eq. 6), micelles diffuse relatively slow due to their large size. Hydrophobic tail Hydrophilic head Surfactant monomers Micelle Figure 3. Micelle formation..4. Partitioning Model for Solubilizates Micelles can be used to solubilize organic molecules with low affinity to water. In this dissertation, we focus on solubilization of drugs with micelles. Solubilization of drug molecules into micelles can be described using the following chemical-equilibrium scheme: 5

39 M+ M M + M M + M i i where denotes the free drug molecule in aqueous solution, M is the micelle and M i is the micelle-drug complex containing i drug molecules in one micelle. For a drugsurfactant-water ternary system at a constant temperature, the composition is characterized by drug molar concentration, C, and surfactant molar concentration, C, assuming all components are neutral. When the critical micellar concentration is low and the surfactant in solution is essentially present as micelles only. Concentrations of individual species are related to the component concentrations through the following mass balances: C = C + ic (38a) Mi i= C m = C + C (38b) / M Mi i= where C, C M and C M i are the molar concentrations of drug, micelle and drug-micelle complexes, respectively. The thermodynamic characterization of the drug-micelle interaction requires the measurements of a set of equilibrium constants. Since their experimental determination is unfeasible, drug solubilization can be described by employing a well-established two-phase partitioning model Within this model, drug molecules are assumed to partition between the micelle-free aqueous pseudo-phase (free 6

40 drug) and the micellar pseudo-phase (bound drug), as shown in Fig. 4. This partitioning equilibrium is described by the following condition: K C C C φ = = (39) C φ (M) (W) C where K is the partitioning constant, (M) C and (W) C are the drug molar concentrations in the micellar and water pseudo-phases, respectively. C is the free drug molar concentration in the total volume, and φ = CV is the volume fraction of micelle in solution. Eq. 39 assumes ideal-dilute behavior with respect to the drug component. Figure 4. Graphic representation of the two-phase model. 7

41 Chapter 3 Materials and Methods 8

42 3. Materials Poly(ethylene glycol) (PEG) with average molecular weights of, 8 and kg mol - (PEGk, PEG8k, and PEGk respectively) having 99% purity were purchased from Fluka. They were used without further purification. PEG stock solutions in purified water were used for sample preparation. Poly(vinyl alcohol) (PVA, 99% hydrolyzed) was purchased from Celanese Chemicals. The viscosity-average molecular weight of polyvinyl alcohol, M V, was determined by viscosity measurements using the Mark-Houwink-Sakurada equation: a KM V [ η ] =, where K = 4 g L -, a =.5, and [ η ] is the intrinsic viscosity. 34 For our sample, [ η ] =.75 L g - and M V = 57 kg mol -. i(ethylene glycol) (EG) ReagentPlus with purity of 99%, was purchased from Sigma-Aldrich and used without further purification. The molecular weight of EG is 6. g mol -, and its density as.4 g cm -3 for buoyancy corrections. A stock solution was made by weight. Mallinckrodt AR NaCl with 99.9% purity, was heated at 4 C for 4 hours and used without further purification. The molecular weight of NaCl was taken to be g mol -, and its density to be.6 g cm -3 for buoyancy correction. Solution were prepared by mass from samples of NaCl (cr) that had previously been dried in air at 4 C. Stock solution of Na SO 4 was prepared from Baker Analyzed ACS reagent (J. T. Baker) Na SO 4 anhydrous with purity of 99.7% in purified water, and followed by filtration through a. µm Nalgene nylon filter membrane. The density of the stock solution was measured in order to get the stock concentration using a density equation available in 9

43 literature. 36 The molecular weight of Na SO 4 was taken as 4.37 g mol -, and its density as.68 g cm -3 for buoyancy corrections. Hydrocortisone (purity >98%) was purchased from Sigma and used without further purification. examethasone (purity %) and rimexolone (purity 98%) were kindly donated by Alcon Research Ltd. Naproxen was purchased from TCI America (Portland, OR). All the drug materials were used without further purification. The molecular weights for hydrocortisone, dexamethasone, rimexolone and naproxen were taken to be 36.47, 39.46, 37.5 and 3.6 g mol -, respectively. Tyloxapol (SigmaUltra grade) was purchased from Sigma Chemical Co. (St. Louis, MO) and used without further purification. The molecular weight for tyloxapol was taken to be 4.5 kg mol -. Tyloxapol stock solutions in purified water were used for sample preparation. Potassium hydroxide was purchased from Sigma Chemical Co. (St. Louis, MO). Glacial acetic acid and acetonitrile were purchased from EMScience and EM Chemicals Inc. (Gibbstown, NJ), respectively. Materials were used as received from the manufacturers. HPLC-grade phosphoric acid and acetonitrile were purchased from EM Chemicals Inc. (Gibbstown, NJ). eionized water was passed through a four-stage Millipore filter system to provide high purity water for all the experiments. The molecular weight of water was taken as 8.5 g mol -. 3

44 3. Solution Preparation All solutions were prepared by mass with appropriate buoyancy corrections. All weighings were performed with a Mettler Toledo AT4 electrobalance. Mass and stock solutions were weighed to. mg. Stock concentrated solutions were made by weight. ensity of the solutions were accurately measured and used to obtain the molarity of the stock solution. Stock solutions of tyloxapol-water, hydrocortisone-water, PEG-water, EGwater, and Na SO 4 -water were made by weight to. mg. The pairs of solutions for each diffusion experiment were prepared by weight. For binary systems other than the NaCl-water system, precise mass of stock solution was diluted with pure water to reach the final target concentrations. For the NaCl-water binary system, precise masses of pure salt were added to flask and diluted with pure water to reach the final target NaCl concentration. Similarly, ternary solutions, except those containing NaCl, were made by weighing precise masses of stock solutions to flasks and diluted with pure water to reach the final target concentrations of the solutions. PEG-NaCl-water ternary solutions were prepared by weighing precise masses of the PEG stock solution and pure salt to a flask and diluting with pure water to reach the final target concentrations of the solutions. Stock solutions of naproxen-water were made by weight to. mg. To prepare potassium naproxenate, the ph of the naproxen stock solution was increased to ~ 7 using KOH. Precise masses of stock solutions were added to flasks and diluted with pure water to reach the final target concentrations of the solutions used for the diffusion experiments. ph was measured for all final solutions used for diffusion experiments. The solution ph was measured using an Orion Expandable IonAnalyzer EA94 ph meter with a Ross ph probe 3

45 (Orion Ross Semi Micro Comb ph by Thermo Fisher). The ph meter was calibrated with standard ph 7. and ph 4. buffers. The density of each solution was measured and the final molar concentrations calculated. 3.3 ensity Measurements Molar concentrations of the solutions were obtained from density. All density measurements were made with a Mettler-Paar MA4 density meter, thermostated with water from a large, well-regulated water bath, with an RS-3 output to an Apple II+. By time averaging the output, a precision of ±. g cm -3 or better could be achieved. Accuracy is somewhat lower because it is related to the composition uncertainty of the solutions. The temperature of the vibrating tube in the density meter was controlled with water from a large well-regulated water bath at temperature of 5. ±. C. The solution density d is related to the vibration period T of cell tube by the following relation: d A + BT = () where A, B are two instrumental constants. Two reference periods are needed for the determination of the instrumental parameters. ensities of the water ( d = g cm -3 ) and the air ( d air =.5 g cm -3 ) were chosen as references. An accurate value of the air density was estimated by a state equation that shows explicit dependence on the pressure, temperature (5. C) and humidity. water 3

46 ensity measurements are used for calculating both molar concentrations and partial molar volumes, Vi ( V / ni) T, p, nj, i j with i, j =,,. For the diffusion experiments, several solutions with different concentrations are prepared and the final transport characterization is referred to the average composition. Since the compositions of those solutions exhibit relatively small differences respect to the mean, linear concentration dependence for the density, is usually assumed: dc (, C) = dc (, C) + H ( C C) + H ( C C) () where the H i s represent the constant density derivatives with respect to concentration. The solute partial molar volumes corresponding to the mean composition, can be calculated by Mi Hi Vi = dc (, C) HC H C with i =, (3) where M i is the molar weight of the component i. condition: The solvent partial molar volume can be calculated by applying the following CV + CV + CV = (4) 33

47 3.4 Rayleigh Interferometry and the Gosting iffusiometer Among all techniques used to measure mutual-diffusion coefficients, the most precise and accurate methods are the Rayleigh and Gouy interferometric techniques (the precision on the diffusion coefficient values was estimated to be equal to about.%). The versatile and high-precision Rayleigh interferometry was employed for the characterization of the diffusion properties of multicomponent systems reported in this work. The Rayleigh interferometric method together with the fundamental equations employed for the determination of the multicomponent diffusion coefficients is reviewed next. The Rayleigh method yields the one-dimensional profile of the refractive index of a liquid system contained in a cell with rectangular geometry. This refractive index profile, under given initial and boundary conditions, can be used for the determination of the diffusion coefficients. Light generated from a laser source and emanated from a point (spatial filter) is rendered convergent by a main spherical lens that focuses the slit image to a "camera" plane (see Fig. 5). Two coherent beams are generated by two narrow vertical slits positioned between the convergent lens and the cell. One beam goes through the diffusion channel and the other through the reference channel. The interference pattern (Young's interference) is then collected on the "camera" plane. The distance between the fringes is determined by the separation of the two slits while the absolute shift of the fringes is proportional to the difference in the optical path between the two beams. If the refractive index is uniform in the diffusion channel along the Z direction (see Fig. 6), the value of the position Y of the maxima at the camera plane is independent of Z. The straight interference vertical lines are produced. 34

48 Reference Channel Light Source Main Lens iffusion Channel Cylinder Lenses Figure 5. Scheme of the optical apparatus in Rayleigh configuration. (A) Y (B) Z Figure 6. (A) Scheme of the Rayleigh interferometric pattern; the solid lines correspond to maxima positions. (B) Picture of the Rayleigh interferometric pattern taken from the Gosting diffusiometer. 35

49 If the refractive index inside the diffusion channel changes along the Z direction, the position of the maxima will shift along the same direction of a quantity proportional to the shift in the refractive index. Note that, since the rays are deflected by the gradient of refractive index, a system of cylinder lenses is needed to effectively reverse the deflection and focus the cell to the detector plane. A picture of a Rayleigh pattern taken from the Gosting diffusiometer operating in the Rayleigh interferometric mode is shown in Fig. 6. The Gosting diffusiometer, located at TCU, is considered to be the best optical diffusiometer in the world. A side view of the diffusiometer (optical-bench length of 8.84 m) is shown in Fig. 7. The light source is a.5mw, λ =543.4nm (in air) Uniphase He-Ne laser. The main lens (focal length of 45.6±.3 cm) is installed in a lens mount on the source slit side of the water bath so that the diffusion cell is in the converging light between the lens and the camera. The cylinder lens consists of two plane-convex borosilicate lenses, each 7.5 cm square and.3 cm thick at its thickest part. A cell holder is used to locate the cell in the bath and to support a mask located between the cell and the light source. The mask consists of a double window allowing the beam to split in two parts one going through the diffusion channel and the other passing directly in the water bath (reference channel). The cell is a glass Tiselius type. Fig. 8 shows the main features of the cell. The cell is composed of three pieces in contact by very smooth plane surfaces. Grease is used to seal the plates and lubricate sliding relative to each other. The shift of the plates allows the solutions, filled in all three parts, to be either in contact or isolated from each other. The right side of the middle part contains the solution that is then going to be optically investigated (diffusion channel). 36