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`Lupin Ex. 1030 (Page 1 of 19)
`Lupin Ex. 1030 (Page 1 of 19)
`
`

`

`ISBN: 0-8247-0237-9
`
`This book is printed on acid-free paper.
`
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`Copyright © 1999 by Marcel Dekker, Inc. All Rights Reserved.
`
`m
`
`Neithe~ this book nor any part may be reproduced or transmitted in any form or by
`any means, electronic or mechanical, including photocopying, microfilming, and re-
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`
`Current printing (last digit):
`10 9 8 7 6 5 4 3 2 I
`
`PRINTED IN THE UNITED STATES OF AMERICA
`
`Preface
`
`Since the middle of the last century, it has been noted that organic
`molecules can be obtained in more than one distinct crystal form, a
`property that became known as polymorphism. Once experimental
`methods based on the diffraction of x-rays were developed to determine
`the structures of crystalline substances, it was quickly learned that an
`extremely large number of molecuies were capable of exhibiting the
`phenomenon. In addition, numerous compotmds were shown to form
`other nonequivalent crystzlline structures through the inclusion of sol-
`vent molecuIes in the lattice.
`It was also established that the structure adopted by a given com-
`pound upon crystallization would exert a profound effect on the solid-
`state properties of that system. For a given material, the heat capacity,
`conductivity, volume, density, viscosity, surface tension, diffttsivity;
`crystal hardness, crystal shape and color, refractive index, electrolytic
`conductivity, melting or sublimation properties, latent heat of fusion,
`heat of solution, solubility, dissolution rate, enthalpy of transitions,
`phase diagrams, stability, hygroscopicity, and rates of reactions are all
`determined primarily by the nature of the crystal structure.
`
`iii
`
`

`

`1
`
`Theory and Origin. of Polymorphism
`
`David J. W. Grant
`
`University of Minnesota
`Minneapolis, Minnesota
`
`I.
`
`INTRODUCTION
`
`II. THERMODYNAMICS OF POLYMORPHS
`
`l-fL ENANTIOTROPY AND MONOTROPY,
`
`IV, KINETICS OF CRYSTALLIZATION
`
`V. NUCLEATION O1~ POLYMORPHS
`
`VI. NEW OR DISAPPEARING POLYMORPHS
`
`REFERENCES
`
`t
`
`10
`
`18
`
`19
`
`25
`
`31
`
`I.
`
`INTRODUCTION
`
`Many pharmaceutical solids exhibit polymorphism, which is frequently
`defined as the ability of a substance to exist as two or more crystalline
`phases that have different arrangements and/or conformations of the tool-
`
`m
`
`

`

`Grant
`
`(a)
`
`2
`
`(b)
`
`(b)
`
`Fig. 1 Molecular structure of (a) acetarninophen and (b) spiperone.
`
`ecutes in the crystal lattice [ 1-3]. Thus, in the strictest sense, polymorphs
`are different crystalline forms of the same pure substance in which the
`molecules have different arrangements and/or different conformations
`of the molecules. As a result, the polymorphic solids have different unit
`cells and hence display different physicalproperties, including those due
`to packing, and various thermodynamic, speclroscopic, interracial, and
`mechanical properties, as discussed below [1-3].
`For example, acetaminophen (paracetamol, 4-acetamidophenol,
`4-hydroxyacetanilide, shown in Fig. la) can exist as a monoclinic form,
`of space group P21/n [4], which is thermodynamically stable under
`ambient conditions. The compound can also be obtained as a less stable
`o~daorhombic form, of space group Pbca, and which has a higher den-
`sity indicative of closer packing [5-7]. The unit celts of these two forms
`are compared in Fig. 2 and Table 1. The molecule of acetaminophen
`is rigid on account of resonance due to conjugation involving the by-
`
`Fig. 2 View of the unit cell contents for two polymorphs of acetaminophen:
`(a) orthorhombic form (b) monoclinic form [4,5,7]. (Reproduced w~th permis-
`sion of the copyright owner, the American Crystallographic Association,
`Washington, DC.)
`
`m
`x
`
`

`

`4
`
`Grant
`
`Theory and Origin of Polymorphism
`
`5
`
`Table 1 Crystal Data for Two Polymorphs of Acetaminophen
`
`Form I
`
`Crystal data and
`structure refinement Orthorhombic phase Monoclhaic phase
`
`Empirical formula
`Formula weight
`Crystal system
`Space group
`Unit cell dimensions
`
`Volume
`Z
`Density (calculated)
`Crystal s~ze
`Refinement method
`
`Hydrogen bond
`length§ and angles
`
`C~HgNO~
`151.16
`Orthorhombic
`Pbca
`a = 17.1657(12) ~
`b = 11.7773(11) ,~
`c = 7,212(2) 2~
`ot = 90.000°
`,6 = 90,000°
`7/ = 90.000°
`1458.1(4) ,~3
`8
`1.377 g/cm3
`0.28 × 0.25 × 0.15 mm
`Pull-matrix least-squares
`on F2
`
`CsH~NO~
`151.16
`Monoclinic
`P21/n
`a = 7.0941(12) ~
`b = 9.2322(1i) ~
`c = 11.6196(10)
`~ ~- 90,000°
`,6--- 97.821(10)°
`7’ = 90.000°
`753.9(2) 2~~
`4
`1.332 g/cm3
`0.30 × 0.30 × 0.15 mm
`Full-matrix least-squares
`on F~
`
`H(5)O(2)
`H(6)O(1)
`O(1)--H(5)O(2)
`N(1)--H(6)O(1)
`
`1.852(26) ~ 1.772(20) ~
`2.072(28) ~
`2.007(18) ~
`170.80(2.35)°
`166.15(1.75)°
`163.93(1.51)°
`163.52(2.19)°
`
`Source: Refs. 4, 5, and 7. Reproduced with permission of the copyright owner, the
`American Crystallographic Association, Washington, DC.
`
`droxyl group, the benzene ring, and the amido group. Therefore the
`conformation of the molecule is virtually identical in the two poly-
`morphs of acetaminophen. On the other hand, the spiperone molecule
`(8-[3-(p-fluorobenzoyl)-propyl]-l-phenyl-l,3,8-triazaspiro[4,5]decan-
`4-one, shown in Fig. lb) contains a flexibIe -CH2-CH2-CH2- chain and
`is therefore capable of existing in different molecular conformations
`[8]. Two such conformations, shown in Fig. 3, give rise to two different
`conformafional polymorphs (denoted Forms I and EO, which have dif-
`ferent unit cells (one of which is shown in Fig. 4) and densities, even
`
`m
`x
`
`Form II
`
`Fig, 3 The molecular conformations of the spiperone molecule in potymor-
`phic forms I and II [8], (Reproduced with permission of the copyright owner,
`the American Pharmaceutical Association, Washington, DC.)
`
`though their space groups are the same, both being P21/n, monoclirdc,
`as shown in Table 2 [8].
`As mentioned above, the various poIymorphs of a substance can
`exhibit a variety of different physical properties. Table 3 lists some of
`the many properties that differ among different polymorphs [1-3,9].
`Because of differences in the dimensions, shape, symmetry, capacity
`
`

`

`6
`
`Grant
`
`Theory and Origin of Polymorphism
`
`7
`
`Table 3 List of Physical Properties that Differ Among VazSous
`Polymorphs
`
`1. Packing properties
`a. Molar volume and density
`b. Refractive index
`c. Conductivity, electrical and thermal
`d. Hygroscopicity
`2. Thermodynamic properties
`a. Melting and sublimation temperatures
`Internal energy (i.e., Structural energy)
`b.
`c. Enthalpy (i.e., Heat content)
`d. Heat capacity
`e. Entropy
`f. Free energy and chemical potential
`g. Thermodynamic activity
`lx Vapor presstlre
`i.
`Solubility
`3. Spectroscopic properties
`Electronic transitions (i.e., ultraviolet-visible absorption spectra)
`Vibrational transitions (i.e., infrared absorption spectra and Raman
`spectra)
`Rotational transitions (i.e., far infrared or microwave absorption
`spectra)
`Nuclear spin transitions (i.e., nuclear magnetic resonance spectra)
`d.
`4. Kinetic properties
`a. Dissolution rate
`b. Rates of solid state reactions
`c. Stability
`5. Surface p~operties
`a. Surface free energy
`Interracial tensions
`b.
`c. Habit (i.e., shape)
`6. Mechanical properties
`a. Hardness
`b. Tensile strength
`c. Compactibility, tableting
`d. Handling, flow, and blending
`
`b.
`
`Fig. 4 View of the unit celJ contents for the form I polymorph of spiperone
`[8]. (Reproduced with permission of the copyright owner, the American Phar-
`maceutical Association, Washington, DC.)
`
`Table 2 Crystal Data for Two Polymorphs of Spiperone
`
`Empirical formula
`Molecular weight
`.Crystal system
`Space group
`Unit cell dimensions
`
`Unit cell volume
`Z
`
`Form i
`
`C:3H:6FN~O2
`395.46
`Monoclinic
`P21/a
`a = 12.722 ~
`b = 7.510 ~
`c = 21.910 ~
`a = 90.00°
`fl = 95.08°
`7’= 90.00°
`2085.1 ~
`4
`
`Form
`
`C23H26FN302
`395.46
`Monoclinie
`P21/c
`a = 18.571 ,~
`b = 6.072 ~
`c = 20.681 ,~
`~ = 90.00°
`/3 = 118.69°
`~/= 90.00°
`2045.7 ~3
`4
`
`Source: Ref. 8. Reprodnced with permission of the copyright owner, the American
`Pharmaceutical Association, Washington, DC.
`
`m
`x
`
`o
`
`

`

`8
`
`Grant
`
`Theory and Origin of Polymorphism
`
`9
`
`(number of molecules), and void volumes of their unit cells, the differ-
`ent polymorphs of a given substance have different physical properties
`arising from differences in molecular packing. Such properties include
`molecular volume, molar volume (which equals the molecular volume
`multiplied by Avogadro’s number), density (which equals the molar
`mass divided by the molar volume), refractive index in a given direc-
`tion (as a result of the interactions of light quanta with the vibrations
`of the electrons in that direction), thermal conductivity (as a result of
`the interaction of infrared quanta with the intramolecular and intermo-
`lecular vibrations and rotations of the molecules), electrical conductiv-
`ity (as a result of movement of the electrons in an electric field), and
`hygroscopicity (as a result of access of water molecules into the crystal
`and their interactions with the molecules of the substance). Differences
`in melting point of the various polymorphs arise from differences of
`the cooperative interactions of the molecules in the solid state as com-
`pared with the liquid state. Differences in the other thermodynamic
`properties among the various polymorphs of a given substance are dis-
`cussed below. Also involved are differences in spectroscopic proper-
`ties, kinetic properties, and some surface properties. Differences in
`packing properties and in the energetics of the intermolecular interac-
`tions (thermodynamic properties) among polymorphs give rise to dif-
`ferences in mechanical properties.
`Many pharmaceutical solids can exist in an amorphous form,
`which, because of its distinctive properties, is sometimes regarded as
`a polymorph. However, unlike true polymorphs, amorphous forms are
`not crystalline [ 1,2,10]. In fact, amorphous solids consist of disordered
`arrangements of molecules and therefore possess no distinguishable
`crystal lattice nor unit cell and consequently have zero crystallinity. In
`amorphous forms, the molecules display no long-range order, although
`the-short-range intermolecular forces give rise to the short-range 6~Ier
`typical of that between nearest neighbors (see Fig. 5). Thermodynami-
`cally, the absence of stabilizing lattice energy causes the molar internal
`energy or molar enthalpy of the amorphous form to exceed that of the
`crystalline state. The absence of long-range order causes the molar en-
`tropy of the amorphous form to exceed that of the crystalline state.
`Furthermore, the lower stability and greater reactivity of the amorphous
`form indicates that its molar Gibbs free energy exceeds that of the crys-
`
`m
`x
`
`(a)
`
`Fig. 5 Schematic diagram showing the difference in long-range order of
`silicon dioxide in (a) the crystnlline state (crystobalite) and (b) the amorphous
`state (silica glass) [2]. The two forms have the same short-range order. (Repro-
`duced with permission of the copyright owner, the American Pharmaceutical
`Association, Washington, DC.)
`
`

`

`10
`
`Grant
`
`Theory and Origin of Polymorphism
`
`11
`
`talline state. This observation implies that the increased molar enthalpy
`of the amorphous form outweighs the TAS term that arises from its
`increased molar entropy.
`
`I!. THERMODYNAMICS OF POLYMORPHS
`
`The energy of interaction between a pair of molecules in a solid, liquid,
`or real gas depends on the mean intermolecular distance of separation
`according to the Morse potential energy curve shown in Fig. 6 [11,12].
`For a given pair of molecules, each polymorph, liquid or real gas has
`its own characteristic interaction energies and Morse curve. These in-
`termolecular Morse curves are similar in shape but have smaller ener-
`gies and greater distances than the Morse potential energy curve for the
`interaction between two atoms linked by a covalent bond in a diatomic
`molecule or within a functional group of a polyatomic molecule. The
`Morse potential energy curve in Fig. 6 is itself the algebraic sum of a
`curve for intermolecular attraction due to van der Waals forces or hy-
`drogen bonding and a curve for intermotecular electron-electron and
`nucleus-nucleus repulsion at closer approach. The convention em-
`ployed is that attraction causes a decrease in potential energy, whereas
`repulsion causes an increase in potential energy. At the absolute zero
`of temperature, the pair of molecules would occupy the lowest or zero
`point energy level. The Heisenberg uncertainty principle requires that
`the molecules have an indeterminate position at a defined momentum
`or energy. This indeterminate position corresponds to the familiar vi-
`bration of the molecules about the mean positions that define the mean
`intermolecular distance. At a temperature T above the absolute zero,
`a proportion of the molecules will occupy higher energy levets ac-
`cording to the Boltzmann equation:
`
`where N~ is the number of molecules occupying energy level 1 (for
`which the potential energy exceeds the zero point level by the energy
`difference A~I), No is the number of molecules occupying the zero point
`
`(1)
`
`rn
`x
`
`0.0
`0
`
`Fig, 6 Morse potential energy curve of a given condensed phase, solid or
`liquid [11]. The potential energy of interaction V is plotted against the mean
`intermolecular distance d. (Reproduced with permission of the copyright
`owner, Oxford University Press, Oxford, UK.)
`
`

`

`12
`
`Grant
`
`Theory and Origin of Polymorphism
`
`13
`
`level, and k is the Boltzmann constant (1.381 × 10-a3 J/K, or 3.300
`× 10-2~ calIK, i.e. the gas constant per molecule).
`With increasing temperature, increasing numbers of molecules
`occupy the higher energy levels so that the distribution of the molecules
`among the various energy levels (lcnown as the Boltzmanu distribution)
`becomes broader, as shown in Fig. 7. At any given temperature, the
`number of distinguishable arrangements of the molecules of the system
`among the various energy levels (and positions in space) available to
`them is termed the .th_e.rm0.dynamic pr.obability_~.l-2~. With increasing tem-
`perature, ffZ increases astron0mically. According to the Boltzmann
`equation,
`
`S = k. In f2
`
`(2)
`
`where the entropy S is a logarithmic function of ~2, so increasing tem-
`perature causes a steady rise, though not an astronomical rise, in the
`entropy. In a macroscopic system, such as a given polymorph, the prod-
`uct T ¯ S represents the energy of the system that is associated with
`
`the disorder of the molecules. This _energy_is ~e bound energy_.gf...th_e_ .....
`sy~s_t~em that is unavailable for doing work.
`The sum of ~£Jindividual ~ii~rgi~~m~eracfion between nearest
`neighbors, next nearest neighbors, and so on, throughout the entire
`crystal lattice, Liquid, or real gas can be used to define the internal
`energy E (i.e., the intermolecular structural energy) of tli~ phase. Nor-
`mally the interactions beyond next nearest neighbors are weak enough
`to be approximated or even ignored. For quantitative convenience one
`mole of substance is considered, corresponding to molar thermody-
`namic quantities. At constant pressure P (usually equal to atmospheric
`pressure), the total energy of a phase is represented by the enthalpy H:
`H = E + P. V
`(3)
`where V is the volume of the phase (the other quantifies have already
`been defined). With increasing temperature, E, V, and H tend to in-
`crease.
`Figure 8 shows that the enthalpy H and the entropy S of a phase
`
`rn
`x
`
`L
`|
`
`¯
`¯
`
`¯
`
`I
`
`¯
`
`i
`~
`
`Go;Ho
`
`-
`
`H
`
`TS
`
`j G-H-~-$
`
`0 AbsoluLe Temperature, T
`
`Fig. 7 Populations of molecular states at various temperatures [11]. The
`temperature is increasing from left to right. (Reproduced with permission of
`the copyright owner, Oxford University Press, Oxford, UK.)
`
`Fig. 8 Plots of various thermodynamic quantities against the absolute tem-
`perature T of a given solid phase (polymorph) or liquid phase at constant
`pressure. H = enthalpy, S = entropy, and G = Gibbs free energy.
`
`

`

`14
`
`Grant
`
`Theory and Origin of Polymorphism
`
`15
`
`tend to increase with increasing absolute temperature T. According to
`the third law of thermodynamics, the entropy of a perfect, pure crys-
`talline solid is zero at the absolute zero of temperature. The product
`T. S increases more rapidly with increasing temperature than does H.
`Hence the Gibbs free energy G, which is defined by~
`
`G =H- T’S
`
`:
`
`(4)
`
`tends to decrease with increasing temperature (Fig. 8)’. This decrease
`also corresponds to the fact that the slope (6G/ST) ofthe plot of G
`against T is negative according to the equation
`
`P
`
`@
`I1
`t
`!
`
`I
`
`E
`n
`e
`r
`g
`y
`
`Liquid
`
`A
`
`B
`Potymorphs ,
`
`C
`
`(5)
`
`blean Intermolecular Distance
`
`As already stated, the entropy of a perfect, pure crystalline solid is zero
`at the absolute zero of temperature. Hence the value of G at T = 0
`(termed Go) is equal to the value of H at T = 0, termed H0 (Fig. 8).
`Each polymorph yieIds an energy diagram similar to that of Fig. 6,
`although the values of G, H, and the slopes of the curves at a given
`temperature are expected to differ between different polymorphs.
`~’Becanse each polymorph has its own distinctive crystal lattice, it
`has its own distinctive Morse potential energy curve for the dependence
`of the intermolecular interaction energies with intermolecular distance.
`The liquid state has a Morse curve with greater intermolecular energies
`and distances, because the liquid state has a higher energy and molar
`volume (lower density) than does the solid state. Figure 9 presents a
`series of Morse curves, one for each polymorph (A, B, and C) and for
`the liquid state of a typical substance of pharmaceutical interest. The
`composite curve in Fig. 9 is the algebraic sum of the Morse curves for
`each phase (polymorph or liquid). The dashed line corresponds to the
`potential energy of the separated, non_interacting molecules in the gas-
`eous state. The increase in potential energy from the zero point value
`of a given polymorph to the dashed line corresponds to the lattice en-
`ergy of that polymorph or energy of sublimation (if at constant pres-
`sure, the enthalpy of vaporization). For the liquid state the increase in
`potential energy from the average value in the liquid state to the dashed
`line for the gaseous molecules corresponds to the energy of vaporiza-
`
`m
`x
`
`Fig. 9 Composite Morse potential energy curve of a series of polymorphs,
`A, B, and C, and of the corresponding liquid phase.
`
`tion (if at constant pressure, the enthalpy of vaporization). The increase
`in potential energy from the zero point value of a given polymorph to
`the average value for the liquid state corresponds to the energy of fusion
`(if at constant pressure, the enthalpy of fusion).
`When comparing the thermodynamic properties of polymorph 1
`and polymorph 2 (or of one polymorph 1 and the liquid state 2) the
`difference notation is used:
`
`AG= G2- G1
`
`AS= $2-S~
`
`AV = Vz - Vl
`
`(6)
`
`(7)
`
`(9)
`
`¯ In discussions of the relative stability of polymorphs and the driving
`force for polymorphic transformation at constant temperature and pres-
`sure (usually ambient conditions), the difference in Gibbs free energy
`is the decisive factor and is given by
`
`AG = AH - T AS
`
`(10)
`
`

`

`16
`
`Grant
`
`Theory and Origin of Polymorphism
`
`17
`
`of temperature increase must be slow enough to allow polymorph 1 to
`change completely to polymorph 2 over a few degrees. Because in Fig.
`10, Ha > H~, M!is positive and the transition is endothermic in nature.
`Figure 10 shows that, below T~, polymorph 1 (or the solid) has
`the Iower Gibbs free energy and is therefore more stable (i.e., G2 >
`G1). On the other hand, above Tt, polymorph 2 (or the Hquid) has the
`lower Gibbs free energy and is therefore more stable (i.e., G2 < G1).
`Under defined conditions of temperature and pressure, only one poly-
`morph can be stable, and the other polymorph(s) are unstable. If a phase
`is unstable but transforms at an imperceptibly low rate, then it is some-
`times said to be metastable.
`The Gibbs free energy difference AG between two phases reflects
`the ratio of "e.scaping tendencies" of the two phases. The escaping
`tendency is termed the fugacity f-fffid is approximated by the saturated
`vapor pressure, p. Therefore
`
`where the subscripts 1 and 2 refer to the respective phases, R is the
`universnl gas constant, and T is the absolute temperature. The fugacity
`is proportionai to the thermodynamic activity a (where the constant of
`proportionality is defined by the standard state), whiIe thermodynamic
`activity is approximately proportional to the solubility s (in any given
`solvent) provided the laws of dilute solution apply. Therefore
`
`~G= RTln(~)
`
`(14)
`
`i
`
`!
`
`61
`
`~bsoluLe Temper’aLu~’e. T
`
`Fig, 10 Plots of the Gibbs free energy G and the enthalpy H at constant
`pressure against the absolute temperature T for a system consisting of two
`polymorphs, i and 2 (or a solid, 1, and a liquid, 2). Tz is the transition tempera-
`ture (or melting temperature) and S is the entropy.
`
`Figure 10 shows the temperature dependence of G and H for two
`different polymorphs 1 and 2 (or for a solid 1, corresponding to any
`polymorph, and a liquid 2) [13]. In Fig. 10 the free energy curves cross.
`At the point of intersection, known as the transition temperature Tt (or
`the melting point for a solid and a liquid), the Gibbs free energies of
`the two phases are equnl, meaning that the phases 1 and 2 are in equilib-
`rium (i.e., AG = 0). However, at Tt Fig. 10 shows that polymorph 2
`(or the liquid) has an enthalpy H2 that is higher than that of polymorph
`1 (or the solid), so that H2 > H1. Equations 10 and 6 show that, if AG
`= 0, polymorph 2 (or the liquid) atso has a higher entropy S: than does
`polymorph 1 (or the solid), so that S~ > $1. Therefore according to
`Equation 10, at T~,
`
`~,/-/t = Tt ASt
`
`(11)
`
`i’n
`x
`
`where i~/t = H~ - Ht and ASt = Sz - $l at Tt. By means of differential
`scanning calorimetry, the enthalpy transition AHt (or the enthalpy of
`fusion M-!f) may be determined. For a polymorphic transition, the rate
`
`in which the symbols have been defined above. Hence, because the
`most stabIe polymorph under defined conditions of temperature and
`pressure has the lowest Gibbs free energy, it also has the Iowest values.
`
`

`

`18
`
`Grant
`
`Theory and Origin of Polymorphism
`
`19
`
`of fagacity, vapor pressure, thermodynamic activity, and solubility in
`
`trolled under sink conditions and under constant conditions of hydrody-
`namic flow, the dissolution rate per unit surface area J is proportional to
`the solubility according to the Noyes-Whitney [14] equation; therefore
`
`AG= RTln(~) (16)
`
`According to the law of mass action, the rate r of a chemical reaction
`(including the decomposition rate) is proportional to the thermody-
`namic activity of the reacting substance. Therefore
`
`AG= RTln(r~) (17)
`
`To summarize, the most stable polymorph has the lowest Gibbs free
`energy, fugacity, vapor pressure, thermodynamic activity, solubility,
`and dissolution rate per unit surface area in any solvent, and rate of
`reaction, including decomposition rate.
`
`I!1. ENANTIOTROPY AND MONOTROPY
`
`If as shown in Fig. 10 one polymorph is stable (i.e., has the lower
`free energy content and solubility over a certain temperature range and
`pressure), while another polymorph is stable (has a lower free energy
`and solubility over a different temperature range and pressure), the two
`polymorphs are said to be enantiotrope_~, and the system of the two
`solid phases is said to be ena~.tiotropico For an enantiotropic system a
`reversible transition can be observed at a definite transition tempera-
`ture, at which the free energy curves cross before the melting point
`is reached. Examples showing such behavior include acetazolamide,
`carbamazepine, metochlopramide, and tolbutamide [9,14,15].
`Sometimes only one polymo~rph is stable at all temperatures below
`the melting point, with alt other polymorphs being, therefore-marble.
`These polymorphs are said to be monotropes, and the system of the
`two solid phases is said to be monotropic. For a monotropic system
`
`the free energy curves do not cross, so no reversible transition can be
`observed below the meIting point. The polymorph with the higher free
`energy curve and solubility at any given temperature is, of course, al-
`ways the unstable polymorph. Examples of this type of system include
`chloramphenicol palrnitate and metolazone [9,14,15].
`To help decide whether two polymorphs are enhntiotropes or
`monotropes, Burger and Ramberger developed four thermodynamic
`rules [14]. The application of these rules was extended by Yn [15].
`The most useful and applicable of the thermodynamic rules of Burger
`and Ramberger are the heat of transition rule and the heat of fusion
`rule. Figure 11, which includes the liquid phase as well as the two
`polymorphs, illustrates the use of these rules. The heat of fusion rule
`states that, if an endothermic polymorphic transition is observed, the
`two forms are enantiotropes. Conversely, if an exothermic polymorphic
`transition is observed, the two forms are monotropes.
`The heat of fusion rule states that, if the higher melting polymorph
`has the lower heat of fusion, the two forms are enantiotropes. Con-
`versely, if the higher melting polymorph has the higher heat of fusion,
`the two forms are monotropes. Figure 11, which inciudes the liquid
`phase as we11 as the two polymorphs, is necessary to illustrate the heat
`of fusion rule.
`The above conditions, that are implicit in the thermodynamic
`rules, are summarized in Table 4. The last two rules in Table 4, the
`infrared rule and the density rule, were found by Burger and Ramberger
`[14] to be significantly Iess reliable than the heat of transition rule and
`the heat of fusion rule and are therefore not discussed here.
`
`IV. KINETICS OF CRYSTALLIZATION
`
`Among the various methods for preparihg different polymorphs are
`sublimation, crystallization from the melt, crystallization from super-
`critical fluids, and crystallization from liqNd solutions. In the pharma-
`ceuticai sciences, different polymorphs are usually prepared by crys-
`tallization from soIution employing various solvents and various
`temperature regimes, such as initial supersaturation, rate of de-super-
`saturation, or final supersaturation. The supersaturation of the solution
`
`rn
`x
`
`

`

`2O
`
`(a)
`
`(b)
`
`m
`x
`
`Grant
`
`Theory and Or|g!n of Polymorphism
`
`21
`
`Table 4 Thermodynamic Rules for PoIymorphic Transitions According to
`Burger and Ramberger [14],AVhere Form I is the Higher-Melting Form
`
`En~nfiotropy-
`
`Transition, < melting I
`I Stable > transition
`II Stable .< transition
`Transition reversible
`Solubility i higher < transition
`,Solubility 1 lower > transition
`Transition II -~ I is endothermic
`
`Monotropy
`
`Transition > melting I
`I always stable ..
`
`Transition irreversible
`Solubikity I always lower than II
`
`Transition II --~ I is exothermic
`
`IR peak I before II
`Density I < density II
`
`IR peak I after II
`Density I > density Ii
`
`Source: Reproduced from Refer. 9 with permission.of the copyright owner, Elsevier,
`Amsterdam~ The Netherlands.
`
`that is necessary for crystalhzation may be achieved by evaporation of
`the solvent (although any impurities will be concentrated), coohng the
`solution from a known initial supersaturation (or heating the solution if
`the-heat of solution is exothermic), addition of a Poor solvent (sometimes
`termed a precipitant), chemical reaction between two or more solubte
`species, or variation of pH to produce a less solubIe acid or base from a
`salt or vice versa (while minimizing other changes in composition).
`During the 19~th century, Gay Lussac observed that, during crys-
`tallization, an unstabIe form is frequently obtained first that subse-
`quently Iransforms into a stable form [13]. This observation was later
`explained thermodynamically by Ostwald [13,16-19], who formulated
`the law of successive reactions, also known as Ostwald’s step rule. This
`
`Fig. 11 Plots of the Gibbs free energy G and the enthalpy H at constant
`pressure against the absolute temperature T for a system consisting of two
`polymorphs, A and B, and a liquid phase, i [14]. T~ is the transition tempera-
`ture, Tf is the melting temperature, and S is the entropy for (a) an enanfiotropic
`system and (b) a monotropic system. (Reproduced with permission of the
`copyright owner, Springer Verlag, Vienna~ Austria.)
`
`

`

`22
`
`Grant
`
`Theory and Origin of Polymorphism
`
`23
`
`rule may be stated as, ’ ’In all processes, it is not the most stable state with
`the lowest amount of free energy that is initially formed, but the least
`stable state lying nearest in free energy to the original state [ 13]."
`Ostwald’s step rule [13,16-19] is illustrated by Fig. 12. Let an
`enantiotropic system (Fig. 12a) be initially in a state represented by
`point X, corresponding to an unstable vapor or liquid or to a supersatu-
`rated solution. If this system is cooled, the Gibbs free energy will de-
`
`(a)
`
`TF~PERATORE
`
`rn
`x
`
`TEMPERATURE
`
`Fig. 12 Relationship between the Gibbs f~ee energy G and the temperature
`T for two polymorphs for (a) an enantiotropic system and (b) a monolropic
`system in which the system is cooled from point X [9]. The arrows indicate
`the direction of change. (Reproduced with permission of the copyright owner,
`Elsevier, Amsterdam, The NetherIands.)
`
`crease as the temperature decreases. When the state of the system
`reaches point Y, form 13 will tend to be formed instead of form A,
`because according to Ostwald’s step ruIe Y (not Z) is the least stable
`state lying nearest in free energy to the original state. Similarly, let a
`monotropic system (Fig. 12b) be initially in a state represented by point
`X, corresponding to an unstable vapor or liquid or to a~supersaturated
`solution. If this system is cooled, the Gibbs free energy will decrease
`as the temperature decreases. When the state of the system reaches
`point Z, form A will tend to be formed instead of form B, because
`according to Ostwald’s step rule Z (not Y) is now the 1east stable state
`Iying nearest in free energy to the original state. This rule is not an
`invariable thermodynamic law but a useful practical rule that is based
`on kinetics, and it is not always obeyed.
`An understanding of the kinetics of the crystallization process
`involves consideration of the various steps involved. In the first step
`(termed nucleation) tiny crystallites of the smallest size capable of inde-
`pendent existence (termed nuclei) are formed in the supersaturated
`phase. Molecules of the crystallizing phase then progressively attach
`themseIves to the nuclei, which then grow to form macroscopic crystals
`in the process known as crystaI growth, until the crystallization medium
`is no longer supersaturated because saturation equilibrium has now
`been achieved. If the crystals are now allowed to remain in the saturated
`medium, the smaller crystals, which have a slightly greater solubility
`according to the Thomson (Kelvin) equation [11,20], tend to dissolve.
`At the same time, the larger crystals, which consequently have a lower
`solubility, tend to grow. This process of the growth of larger crystN_~..at
`the expense of smaller crystals is sometimes termed Ostwald ripening.
`The nucleation step is the most critical for the production of dif-
`ferent polymorphs and is therefore discussed in some detail below. Nu-
`cleation may be primary (which does not require preexisting crystals
`of the substance that crystallizes) or secondary (in which nucleation is
`induced by preexisting crystals of the substance). Primary nucleation
`may be homogeneous, whereby the nuclei of the crystallizing substance
`arise spontaneously in the medium in which crystal