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Fundamentals of Polymorphism: The Phase Rule and Thermodynamic Relations Lian Yu University of Wisconsin – Madison, School of Pharmacy (608) 263 2263 lyu@pharmacy.wisc.edu

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Gibbs Findlay Westrum and McCullough McCrone Burger …

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[This Erice course] will provide a.the theoretical basis for the existence of these diverse structural forms, b.the methodology to control the form, from the nucleation to macroscopic growth, c.the techniques used the characterize the variety of products obtained, d.the advantages resulting by this way of surveying structure/property relations for the design and preparation of new materials.

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a.the theoretical basis for the existence of these diverse structural forms, The stability of a polymorph is determined by G = H - TS, not just H or S. Energy-entropy compensation is important

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b. the methodology to control the form, from the nucleation to macroscopic growth Thermodynamics tells us the direction and driving force of transformations that yield the desired form (but not the rate)

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c. the techniques used the characterize the variety of products obtained Calorimetry and thermal analysis are key techniques of polymorph characterization

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d. the advantages resulting by this way of surveying structure/property relations for the design and preparation of new materials Property = stability, solubility Structure/stability relations: The Close Packing Principle The Density Rule The greater stability of racemic compounds over conglomerates

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Polymorphs are different solid phases of the same component(s)

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An Example of Polymorphism in One- Component System R P-1 mp 106.2 o C = 21.7° ROY ORP Pbca = 39.4° OP P2 1 /c mp 112.7 o C = 46.1° ON P2 1 /c mp 114.8 o C = 52.6° YN P-1 = 104.1° Y P2 1 /c mp 109.8 o C = 104.7° J. Am. Chem. Soc. 2000, 122, 585

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An Example of Polymorphism in Two- Component System Henck, J.-O. et al. J. Am. Chem. Soc. 2001, 123, 1834

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x R-tazofeloneS-tazofelone Two-Component Polymorphs of Racemic Compounds Racemic CompoundSpace Groupmp, ºC Form I P21/c156.6 Form II Pbca154.7 Reutzel, S.; Russell, V.; Yu, L. J. Chem. Soc. Perkin Trans 2 2000, 913

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Two-Component Polymorphs: Racemic Compounds and Conglomerates R S R R R R R R S R S R R R S S S S R R S RSRSRSRSRSRSRSRS SRSRSRSRSRSRSRSR RSRSRSRSRSRSRSRS SRSRSRSRSRSRSRSR RRRRRRR SSSSSSS + racemic compound (single phase) racemic liquid conglomerate (two phases) R S R S S R R S S R polymorphs ?

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The Phase Rule F = C – P + 2 P = the number of phases C = the number of components F = the degree of freedom

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The Gibbs Free Energy G = H – TS H = enthalpy energy S = entropy G determines the stability of a phase at constant pressure The relative stability of two polymorphs depends on their enthalpy difference and entropy difference

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For a one-component system at constant pressure, the transition temperature T t between two polymorphs is unique C = 1 (one component) P = 2 (two polymorphs) F = C – P + 2 = 1 The condition of constant p removes one more degree of freedom, making the system invariant (F = 0).

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Can two polymorphs have more than one transition temperature? Buerger, M. J. Chapter 6. Crystallographic Aspects of Phase Transitions. In Phase Transitions in Solids; Smoluchowski, R. ; Mayer, J. E.; Weyl, W. A., Eds.; John Wiley & Sons Inc.: New York, 1951.

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Enantiotropy G A A B B L T mA T mB A stableB stable transition point T t L stable T G A B L T mA T mB B stable virtual transition point T tv L stable T Monotropy Stability Relation between Two Polymorphs (Constant Pressure)

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LT-to-HT transition is endothermic HT-to-LT transition is exothermic LT: low-temp. stable phase HT: high-temp. stable phase This result leads to HTR (Heat of Transition Rule) and HFT (Heat of Fusion Rule): see Henck and Griesser

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Quantitative Determination of H, S, and G at Constant Pressure Low-temperature calorimetry Solubility Heat of solution and heat of transition Melting and eutectic melting data

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H and G of 1-Heptene Polymorphs Data from McCullough, J. P. et al. J. Phys. Chem. 1957, 61, 289 Form T m, K I 154.3 II 153.9 H or G, cal/mole HIHI H II GIGI G II TtTt TmTm T, K H = CpdCpd S = C p dln G = H - TS

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Solubility G i – G j = RTln(x i /x j ) x i and x j = solubility of i and j in mole fraction T = temperature in K

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Heat of Solution Heat of Transition These measurements yield the enthalpy difference between polymorphs (H i – H j ), which gives the temperature slope of their free-energy difference: d[(G i – G j )/T]d(1/T) = (H i – H j ) If (G i – G j ) and (H i – H j ) are known at one temperature, (G i – G j ) at nearby temperatures can be estimated

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Melting Data Widely available for organic polymorphs because of their sluggish solid-solid transitions Easily measured by DSC T Heat flow T m,A T m,B H m,A H m,B

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A B AB A B enantiotropy A B monotropy TtTt The Heat of Fusion Rule DSC data G - T curves Burger, A.; Ramberger, R. Mikrochimica Acta [Wien] 1979 II, 259-271 and 273-316.

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G 0 = H m,B (T m,A /T m,B - 1)+ C p term T GG T m,A T m,B d G 0 /dT = - S 0 = - H m,A /T m,A + H m,B /T m,B C p term slope value TtTt Quantitative Analysis of Melting Data Yu, L. J. Pharm. Sci., 1995, 84, 966 B A extrapolation

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= 369 K (H I - H III ) = d[(G I - G III )/T]/d(1/T) = 7.1 kJ/mol Solubility vs. Melting Data: Sulfathiazole

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Solubility, Heat of Solution and Melting Data Melting data Solubility data (37 o C) Heat of solution data (25 o C) provide the slope G (kJ/mole) Form A Form B T, K Reinterpretation of data of Lindenbaum, S. et al. Int. J. Pharmaceutics 1985, 26, 123-132. Auranofin 1

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Eutectic Melting Data McCrone, W. C. Fusion Methods in Chemical Microscopy; Interscience Publishers, Inc.: New York, 1957. Measured below pure melting points: T e < T m T e changes with additive Standard technique of chemical microscopy

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Teetsov, A. S.; McCrone, W. C. Microscope & Crystal Front 1965, 5, 13 Haleblian, J.; McCrone, W. C. J. Pharm. Sci. 1969, 58, 911 HMX Polymorphs Studied through Eutectic Melting “Free energy-temperature diagram for HMX. The intersection temperatures are measured points, but the actual slopes are unknown.”`

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DSC Signal +thymol +azobenzene +benzil +acetanilide pure forms Y Y ON Y Y Y G ON -G Y, kJ/mol melting eutectic melting TtTt T, o C T m Y T m ON L L-sc Y Y ON Eutectic Melting Measured by DSC Yu, L. et al. J. Am. Chem. Soc. 2000, 122, 585. ROY

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G 0 = H m,B (T m,A /T m,B - 1)+ C p term T GG T m,A T m,B d G 0 /dT = - S 0 = - H m,A /T m,A + H m,B /T m,B C p term slope value x e2 (G 1 -G 2 )(T e1 )= H me2 (T e2 -T e1 )/T e2 + RT e1 {x e2 ln(x e1 /x e2 ) + (1-x e2 )ln[(1-x e1 )/(1- x e2 )]} C p term x e1 (G 1 -G 2 )(T e2 )= H me1 (T e1 -T e2 )/T e1 -RT e2 {x e1 ln(x e2 /x e1 ) + (1-x e1 )ln[(1-x e2 )/(1- x e1 )]} C p term T e1 T e2 x x

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G-G Y,kJ/mol R Y OP ON OP L L-sc YN Y T, o C Relative Thermodynamic Stability of ROY Polymorphs

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Melting/Eutectic Melting Method Applied to Pairs of Racemic Compounds and Conglomerates RI, RII: racemic compounds A = enantiomorph (+ or -) C = conglomerate T, K G-G RII, kJ/mole RII C, A T mRII LRLR LALA T mA TgTg T mC A RI T mRI TtTt tazofelone

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R = Racemic Compound C = Conglomerate Jacques, J.; Collet, A.; Wilen, S. H. Enantiomers, Racemates, and Resolutions; Krieger Publishing Company: Malabar, Florida, 1991.

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Summary Thermodynamic studies provides the relative stability of polymorphs driving forces of crystallization and polymorph conversion the basis for structure-stability studies Thermodynamics does not address kinetic and structural aspects of polymorphism. Many behaviors of polymorphic systems require non- thermodynamic explanations Combining thermodynamic, kinetic, and structural studies is necessary for understanding and controlling polymorphism

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The fascination of a growing science lies in the work of the pioneers at the very borderland of the unknown, but to reach this frontier one must pass over well traveled roads; of these one of the safest and surest is the broad highway of thermodynamics. G. N. Lewis and M. Randall, 1923

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Figure 4.1 Solubility as a function of temperature for paracetamol in three solvents (data from Granberg and Rasmuson (1999)).

Figure 4.1 Solubility as a function of temperature for paracetamol in three solvents (data from Granberg and Rasmuson (1999)).

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