RS 201
`.S57 P64
`1999
`Copy 2
`
`Liquid
`
`Argentum EX1010
`
`Page 1
`
`

`

`Polymorphism
`in
`Pharmaceutical
`Solids
`
`Page 2
`
`

`

`Contents
`
`Preface
`Contributors
`
`1. Theory and Origin of Polymorphism
`David J. W. Grant
`
`2. Application of the Phase Rule to the Characterization
`of Polymorphic Systems
`Harry G. Brittain
`
`3. Structural Aspects of Polymorphism
`Harry G. Brittain and Stephen R. Byrn
`
`4. Structural Aspects of Hydrates and Solvates
`Kenn eth R. Morris
`
`5. Generation of Polymorphs, Hydrates, Solvates, and
`Amorphous Solids
`J. Keith Guillory
`
`iii
`lX
`
`35
`
`73
`
`125
`
`183
`
`vii
`
`Page 3
`
`

`

`viii
`6. Methods for the Characterization of Polymorphs and
`Solvates
`Harry G. Brittain
`7. Effects of Polymorphism and Solid-State Salvation
`on Solubility and Dissolution Rate
`Harry G. Brittain and David J. W. Grant
`8. Effects of Pharmaceutical Processing on Drug
`Polymorphs and Solvates
`Harry G. Brittain and Eugene F. Fiese
`9. Structural Aspects of Molecular Dissymmetry
`Harry G. Brittain
`Impact of Polymorphism on the Quality of
`Lyophilized Products
`Michael J. Pikal
`
`10.
`
`Index
`
`Contents
`
`227
`
`279
`
`331
`
`363
`
`395
`
`42/
`
`Page 4
`
`

`

`s
`
`7
`
`'9
`
`n
`
`53
`
`95
`
`21
`
`Contributors
`
`Harry G. Brittain Discovery Laboratories, Inc., Milford, New
`Jersey
`
`Stephen R. Bryn Department of Medicinal Chemistry and Phama(cid:173)
`cognosy, Purdue University, West Lafayette, Indiana
`
`Eugene F. Fiese Pharmaceutical Research and Development, Pfizer
`Central Research, Groton, Connecticut
`
`David J. W. Grant Department of Pharmaceutics, College of Phar(cid:173)
`macy, University of Minnesota, Minneapolis, Minnesota
`
`J. Keith Guillory College of Pharmacy, The University of Iowa,
`Iowa City, Iowa
`
`Kenneth R. Morris Department of Industrial and Physical Phar(cid:173)
`macy, Purdue University, West Lafayette, Indiana
`
`Michael J. Pikal School of Pharmacy, University of Connecticut,
`Storrs, Connecticut
`
`ix
`
`Page 5
`
`

`

`1
`Theory and Origin of Polymorphism
`
`David J. W. Grant
`University of Minnesota
`Minneapolis, Minnesota
`
`INTRODUCTION
`I.
`II. THERMODYNAMICS OF POL YMORPHS
`III. ENANTIOTROPY AND MONOTROPY
`IV. KINETICS OF CRYSTALLIZATION
`V. NUCLEATION OF POL YMORPHS
`VI. NEW OR DISAPPEARING POL YMORPHS
`
`REFERENCES
`
`1
`
`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 mol-
`
`1
`
`Page 6
`
`

`

`Grant
`
`I ,
`
`2
`
`(a)
`
`(b)
`
`HO~
`
`O
`
`~NJlCH
`
`H
`
`3
`
`Fig. 1 Molecular structure of (a) acetaminophen and (b) spiperone.
`ecules 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 physical properties, including those due
`to packing, and various thermodynamic, spectroscopic, interfacial, 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,
`[4], which is thermodynamically stable under
`of space group P2i/n
`ambient conditions. The compound can also be obtained as a less stable
`orthorhombic form, of space group Pb a, and which has a higher den(cid:173)
`sity indicative of cl ser packing [5-7). Th unit cells of these Lwo forms
`are compared in Fig. 2 and Table l. The molecule of acetaminophen
`is rigid on account of resonance due to conjugation involving the hy-
`
`Fig. 2 View of the unit cell con.rents for two polymorphs of acetaminophen:
`(a) orthorhombic fonn (b mono clinic form [4,5,7). (Reproduced with penuis(cid:173)
`sion of the copyright owner. the American Crystallographic Asso iation,
`Washington, DC.)
`
`Page 7
`
`

`

`int
`
`(a)
`
`(b)
`
`rphs
`1 the
`ions
`unit
`~due
`, and
`
`enol,
`'orm,
`mder
`;table
`· den-
`:arm s
`>phen
`te hy-
`
`:>phen:
`ermis(cid:173)
`:iation
`
`Page 8
`
`

`

`Grant
`
`4
`Table 1 Crystal Data for Two Polymorphs of Acetaminophen
`Crystal data and
`Monoclinic phase
`structure refinement
`CsH9NO2
`Empirical formula
`151.16
`Formula weight
`Monoclinic
`Crystal system
`P2i/n
`a = 1.0941(12) A
`Space group
`Unit cell dimensions
`b = 9.2322(11) A
`c = 11.6196(10) A.
`a= 90.000°
`/3 = 97.821(10) 0
`r= 90.000°
`753.9(2) A.3
`4
`1.332 g/cm 3
`0.30 X 0.30 X 0.15 mm
`Full-matrix least-squares
`on F 2
`
`Orthorhombic phase
`CsH9NO2
`151.16
`Orthorhombic
`Pbca
`a = 17.165702) A
`b = 11.111301) A
`c = 7.212(2) A.
`a= 90.000°
`/3 = 90.000°
`r= 90.00~0
`1458.1(4) A3
`8
`1.377 g/cm 3
`0.28 X 0.25 X 0.15 mm
`Full-matrix least-squares
`on F 2
`
`Volume
`z
`Density (calculated)
`Crystal size
`Refinement method
`
`Hydrogen bond
`lengths and angles
`1.112(20) A
`1.852(26) A
`H(5)O(2)
`2.001(18) A
`2.072(28) A
`H(6)O(1)
`166.15(1.75)0
`170.80(2.35)0
`O(l)-H(5)O(2)
`163.93(1.51)0
`163 .52(2.19)0
`N(l)-H(6)O(1)
`Source: Refs. 4, 5, and 7. Reproduced with penn.is.sion of the copyright owner, the
`American Crystallographic Association, Washlngton, DC.
`droxyl group, the benzene ring, and the amido group. Therefore the
`conformation of the molecule is virtually identical in the two poly(cid:173)
`morphs of acetaminophen. On the other hand, the spiperone molecule
`(8-[3-(p-fluorobenzoyl )-propyl]- l -phenyl-1,3,8-tri azaspiro[ 4,5]decan-
`4-one, shown in Fig. lb) ontains a flexible -CH2-CHr CHr chain and
`is therefore capable of existing i n different molecular conformations
`[8]. Two such confonnation , shown in Fig. 3, give rise to two different
`conformational pol ymorphs ( denoted Forms I and II), which have dif(cid:173)
`ferent unit cells (one of which is shown in Fig. 4) and densities, even
`
`Page 9
`
`

`

`Theory and Origin of Polymorphism
`
`5
`
`Form I
`
`Form II
`
`Fig. 3 The molecular conformations of the spiperone molecule in polymor(cid:173)
`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 P2i/n, monoclinic,
`as shown in Table 2 [8].
`As mentioned above, the various polymorphs 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
`
`imm
`uares
`
`1er, the
`
`re the
`poly-
`1lecule
`focan(cid:173)
`Llll and
`1ations
`fferenl
`ve dif(cid:173)
`s, even
`
`Page 10
`
`

`

`6
`
`Grant
`
`Fig. 4 View of the unit cell onten ts for the form I polymorph of spiperone
`[8]. (Reprodu oed witb perrn.ission of the copyri ght owner, the American Phar(cid:173)
`mac eutical Associa tion, Washington, DC. )
`
`Table 2 Crystal Data for Two Polymorphs of Spiperone
`Form I
`
`Empirical formula
`Molecular weight
`Crystal system
`Space group
`Unit cell dimensions
`
`Form II
`C23H26FN102
`C23H26FN.102
`395.46
`395.46
`Monoclinic
`Monoclinic
`P2i/c
`P2 1/a
`a= 18.571 A
`a= 12.722 A
`b = 6.072 A
`b = 1.510 A
`c = 20.681 A
`c = 21.910 A
`a= 90.00°
`a= 90.00°
`/J = 118.69°
`/3 = 95.08°
`r= 90.00°
`r= 90.00°
`2045.7 A.3
`2085 . 1 A.3
`Unit cell volume
`4
`4
`z
`Source: Ref . 8. Reproduced with permission of the copyright owner, the American
`Pharmaceutical Association , Washington, DC.
`
`Page 11
`
`

`

`int
`
`Theory and Origin of Polymorphism
`
`7
`
`Table 3 List of Physical Properties that Differ Among Various
`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
`h. Vapor pressure
`i. Solubility
`3. Spectroscopic properties
`a. Electronic transitions (i.e., ultraviolet-visible absorption spectra)
`b. Vibrational transitions (i.e., infrared absorption spectra and Raman
`spectra)
`C. Rotational transitions (i.e., far infrared or microwave absorption
`spectra)
`d. Nuclear spin transitions (i.e., nuclear magnetic resonance spectra)
`4. Kinetic properties
`a. Dissolution rate
`b. Rates of solid state reactions
`C. Stability
`5. Surface properties
`a. Surface free energy
`Interfacial tensions
`b.
`C. Habit (i.e., shape)
`6. Mechanical properties
`a. Hardness
`b. Tensile strength
`c. Compactibility, tableting
`d. Handling, flow, and blending
`
`rone
`'har-
`
`II
`
`~302
`
`1ic
`
`;11 A
`12A
`,s1 A
`00°
`:.69°
`)00
`\ J
`
`nerican
`
`Page 12
`
`

`

`Grant
`
`8
`(number of molecules), and void volumes of their tmit cells, the differ(cid:173)
`ent polymorphs of a given substa nce have diff erent physi cal properties
`arising from differ ences in molecular packing. Such prop e1ties include
`molecular volume, molar volume (which equals Lhe molecular volume
`multipl ied by Avogadro's numb er), densily (which equals Lhe molar
`mass divid ed by the molar volume), refractive index in a gjven direc(cid:173)
`tion (as a result of the interactions of light quanla with the vibrations
`of the electrons in that direction), lhermal conductivity (as a result of
`th e interaction of infrared quanta with the intramo lecular and intermo(cid:173)
`lecular vibrations and rotation s of Lhe molecul es) . electri cal conductiv(cid:173)
`ity (as a result of movement of the electro ns in an elect ric field), and
`bygroscop icity (as a resull of access of water molecules into the crystal
`and their interact ions with the molecules of the substance). Diff erences
`in melting point of the various polymorphs arise from diff erences of
`the cooperative interactions of the molecules in the solid sta te as com(cid:173)
`pared with the liquid state. Differences in the other thermodynamic
`properties among the various polymorphl:i of a give n substance are dis(cid:173)
`cussed below . Also involved arc differences in spectroscopic proper(cid:173)
`in
`ties, kinetic prop erties, and some surface properties. Differences
`packing properties and in the energetics of the intermolecular interac(cid:173)
`tions (thennodynamic properties) among polymorpbs give rise to dif(cid:173)
`ferences in mechanical properties.
`form,
`Many pharmaceutica l solid s can exist in an amorphous
`whjch, because of its distinctive properties, is sometim es regarde d as
`a po lymorph. However , unlik e true polymorphs, amorphous forms are
`not crystalline [ 1,2, 1 OJ. In fact, amorphous solids consist of disordered
`an-angements of molecules and Lherefore possess no di stinguis hable
`crystal lattice nor unit ce1J and conseq uently have zero crysta llinily. ln
`amorpho us forms, the molecules disp lay no long-ra nge order , although
`the short-ra nge intcnnolecular forces give rise to the short- range order
`typical of that between nearest neighbor s (see Fig. 5). Thennodynami(cid:173)
`cally, the absence of stabili zing lattice energy causes the molar internal
`energy or molar enthalpy of the amorphou s form to exceed that of the
`crysta lline state . The absence of long-range order causes the molar en(cid:173)
`tropy of the amorphous form lo exceed that of the crysta lline state.
`Furthermore, the lower stability and grea ter reactivity of the amorphous
`form indicates that its molar Gibbs free energy exceeds that of the crys-
`
`Page 13
`
`

`

`Theory and Origin of Polymorphi sm
`
`9
`
`(ll)
`
`(b)
`
`Fig. 5 Schematic diagram show ing the difference in long-range order of
`silico n dioxide in (a) 1he crystalli ne state (crystoba lite) and (b) 1he amorphous
`state (silico glass) [2J. The two form~ have the same short-range order. (Repro(cid:173)
`duced with permission o f the copyright owner, the American Pharmaceutical
`Association. Washington, DC.)
`
`nt
`
`.es
`de
`ne
`lar
`!C-
`ms
`of
`10-
`iv(cid:173)
`md
`stal
`ces
`; of
`)tn(cid:173)
`rnic
`dis(cid:173)
`?er(cid:173)
`s in
`rac(cid:173)
`dif-
`
`)rm,
`d as
`, are
`ered
`table
`y. ln
`::mgh
`)rder
`,am:i(cid:173)
`ernal
`>f the
`u- en(cid:173)
`stare.
`)bous
`crys-
`
`Page 14
`
`

`

`10
`talline state. This observation implies that the increased molar enthalpy
`of the amorphous form outweighs the T liS term that arises from its
`increased molar entropy.
`
`Grant
`
`II. THERMODYNAMICS OF POL YMORPHS
`The energy of interaction between a pair of molecules in a solid, Liquid
`or real gas depend. on the mean intermolecular distance of separation
`according to the Morse potential energy curve hown in Fig. 6 Ll 1,12).
`For a given pair of molecules each polymorph liquid or rea l ga has
`its own characteristic interaction energies and Morse curve. These in(cid:173)
`termolecular Morse curves are similar in shape but have smaller ener(cid:173)
`gies and greater distances than the Morse potential energy curve for the
`interaction between two atoms !·inked by a covalent bond in a diatomic
`molecule or within a functional group of a polyatomic molecule. T.he
`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(cid:173)
`drogen bonding and a curve for intermolecular electron-electron and
`repulsion at closer approach. The convention em(cid:173)
`nucleus-nucleus
`ployed i 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
`that
`po.int energy level. The Hei senberg uncertainty principle require
`the molecules have an jndeterminate position at a defined momentum
`or energy. This indeterminate positi, n corresponds to the familiar vi(cid:173)
`bration of the molecule 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 levels ac(cid:173)
`cording to the Boltzmann equation:
`(-lie)
`N: = exp ,;j
`where N, is the number of molecules occupying energy level 1 (for
`the zero point level by the energy
`which the potential energy exceed
`difference ~E, ), N0 istbe number of molecules occupying the zero point
`
`(1)
`
`N
`
`Page 15
`
`

`

`Theory and Origin of Polymorphism
`
`11
`
`2.0
`
`::::i.. 1.0
`
`O.OL---'---;ai,r;,--""--------------..._-
`d
`
`0
`
`Fig. 6 Morse potential energy curve of a given condensed phase, solid or
`liquid [11]. The potential energy of interaction Vis plotted against the mean
`intermolecular distance d. (Reproduced with permission of the copyright
`owner, Oxford University Press, Oxford, UK.)
`
`ant
`
`llpy
`L its
`
`1uid
`1tion
`,12].
`; ha
`e in(cid:173)
`!ner(cid:173)
`If th
`omic
`. Th e
`t of a
`,r hy-
`1 and
`em-
`1ereas
`: zero
`r zero
`.s that
`mtum
`.ar vi(cid:173)
`mean
`. zero,
`:ls ac-
`
`(1)
`
`1 (for
`energy
`o point
`
`Page 16
`
`

`

`Grant
`
`12
`level, and k is the Boltzmann constant (1.381 X 10- 23 J/K, or 3.300
`X 10- 26 cal/K, i.e. the gas constanL per molecule) .
`With increasing temperature, increasil1g numbers of molecules
`occupy the hjgher energy levels so Urnt the distiibution of the molecules
`among the various energy levels (known as the Boltzmann distribution)
`the
`becomes broader as shown in Fig. 7 . At any given temperature
`number of di tinguisbable arrangements of the molecules of the system
`among the variou energy levels (and positions in space) available to
`them i termed the thermodynamic probability n. With increasing tem(cid:173)
`to the Boltzmann
`perature n increase
`astronom ically. According
`equation,
`(2)
`s = k. ln n
`where the entropy S is a logarithmic function of n, so increasing tem(cid:173)
`though not an aslronomi a l ri e in the
`perature cause. a steady rise
`entropy . ln a macroscopic system such a a given polymorph, the prod(cid:173)
`the energy of the sy tern that i ru sociated with
`ucL T · S represent
`
`I
`
`I
`
`I
`
`-
`
`I
`
`•
`
`-
`
`I • I •
`--
`•
`•
`
`[1 L]. The
`Fig. 7 Populatioas of molecular sta les at various temperatures
`temperature is increasing from left to right. (Reproduced with permission of
`the copyright owner, Oxford Unjve rsii-y Press , Oxford , -UK.)
`
`Page 17
`
`

`

`ant
`
`300
`
`1les
`Jles
`ion)
`the
`.tern
`e to
`:em(cid:173)
`rnnn
`
`(2)
`
`tem-
`1 the
`irod(cid:173)
`with
`
`Theory and Origin of Polymorphism
`
`13
`
`the disorder of the molecules. This energy is the bound energy of the
`system that is unavailable for doing work.
`The sum of the individual energies of interaction 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 the phase. Nor(cid:173)
`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(cid:173)
`namic quantities. At constant pressure P (usually equal to atmospheric
`pressure), the total energy of a phase is represented by the enthalpy H:
`(3)
`H=E+P·V
`where Vis the volume of the phase (the other quantities have already
`been defined). With increasing temperature, E, V, and H tend to in(cid:173)
`crease.
`Figure 8 shows that the enthalpy H and the entropy S of a phase
`
`>(cid:173)C)
`
`L.
`Q)
`C:
`LU
`
`lJ. The
`sion of
`
`Fig. 8 Plots of various thermodynamic quantities against the absolute tem(cid:173)
`perature T of a given solid phase (polymorph) or liquid phase at constant
`pressure. H = enthalpy, S = entropy, and G = Gibbs free energy.
`
`Page 18
`
`

`

`I.
`
`Grant
`
`(5)
`
`14
`tend to increase with increasing absolute temperature T. According to
`the third law of thermodynamics, the entropy of a perfect, pure crys(cid:173)
`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
`(4)
`G=H-T·S
`tends to decrease with increasing temperatur e (Fig. 8). This decrease
`also corresponds to the fact that the slope (fJG/OT) of the plot of G
`against T is negative according to the equation
`(oG) _ s
`-
`8T ,, -
`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 G 0 ) is equal to the value of Hat T = 0, termed H 0 (Fig. 8).
`Each polymorph yields an energy diagram similar to that of Fig. 6,
`although the values of G, H, and the slopes of the curves at a given
`temp erature are expected to differ between different polymorphs.
`B ecause each polymorph bas its own distinctive crystal lattice, it
`has its own distinctive Morse potenti al ene1·gy curve for the dep endence
`of the intermolecular interaction energies with i ntermol ecular distance.
`The liquid state has a Mor se curve with greater intermol ecul ar energies
`and distances , because the liquid state ha · 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 typica l substanc e of pharma ceutical interest. Th e
`compo sit curve in Fig. 9 is the algebraic sum of the Morse curves for
`eacb phase (polymor ph or liquid ). The dashed line orresponds to the
`pot ential energy of the . eparaled, nonint eracting molecules in the gas(cid:173)
`tate. The increase in potential energy from the zero point value
`eous
`of a given polymorph to the dashed line corresponds to the lattice en(cid:173)
`ergy of lbal polymorph or energy of sublimation (if at constant pres(cid:173)
`tate d1e increase in
`f v aporization . For the liquid
`sure, the enth alpy
`late to the dashed
`potential energy from the average value in the liquid
`line for the gaseous molecules con esponds to 'the e nergy of vaporiza-
`
`Page 19
`
`

`

`rant
`
`tg to
`;rys(cid:173)
`,duct
`:sH .
`
`(4)
`
`rease
`of G
`
`(5)
`
`s zero
`~ = 0
`tg. 8).
`•ig. 6,
`given
`lS.
`tice, it
`tdence
`;tance.
`tergies
`mol ar
`,ents a
`md for
`st. Th e
`ves for
`; to th e
`he gas(cid:173)
`,t va lue
`cice cn-
`1t pres(cid:173)
`:ea. e in
`dashed
`tpor iza-
`
`Theory and Origin of Polymorphism
`
`15
`
`p
`0
`t
`e
`n
`t
`I
`a
`I
`E
`n
`e
`r
`g
`y
`
`Liquid
`
`D
`C Pol mor hs
`
`Mean Intermolecular Distance
`
`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:
`
`(6)
`
`(7)
`
`(8)
`
`(9)
`
`In discussions of the relative stability of polymorphs and the driving
`force for polymorphic transformation at constant temperature and pres(cid:173)
`sure (usually ambient conditions), the difference in Gibbs free energy
`is the decisive factor and is given by
`!).G = !).H-
`
`TM
`
`(10)
`
`Page 20
`
`

`

`I •
`
`I
`
`1·
`
`16
`
`Grant
`
`_ ..
`_.,. .. .-- ...
`-...
`........
`......
`
`>- 2
`Cl
`'-
`Q)
`C w
`
`1
`
`Absolute Temperature, T
`Fig. 10 PlOls of the Gibbs free energy G and the enthalpy Hat constant
`prcsstu·e again t the abso lute temperature T for a system consisting of two
`polymorphs, I and 2 or a solid , I, and a liquid, 2). T, is the transition tempera(cid:173)
`the entropy .
`ture (or melting tempe rature) and Sis
`
`Figure 10 show s the tem pera ture dependence of G and H for two
`to any
`l and 2 (or for a solid 1, corresponding
`differ ent polymorphs
`po lymorph and a liquid 2) [13). ln Fig . 10 the free energy curves cross.
`At the point of intersection k11own as the transition temperature T1 (or
`the melting point for a olid and a liquid ), the Gibb s free energies of
`the two phase s are equal , meaning that the phase 1 and 2 are in equilib(cid:173)
`rium (i.e., 6.G = 0). However al Tl Fig. IO show s that polymorph 2
`(or the liquid has an enthalpy H 1 tbat is higher tban that of polymorph
`J (or the solid ) so that H 1 > H 1• Equations 10 and 6 show that if 6.G
`= 0, polymorph 2 (or the Liquid) al. o ha s a higher entr opy S2 than do e
`polymorph 1 (or the so lid), so that S 2 > S 1• Therefore according to
`Equation 10, at T1,
`
`(11)
`where f:J/ 1 = H 2 - H 1 and t!.S1 = S 2 - S 1 al T1• By means of differe11tial
`tbe enthalpy transition 11/11 (or Lhe enthalpy of
`canning calorimetry,
`fusio n Mfr ) may b det er mined. For a polymo rphi c transjtion, the rat
`
`Page 21
`
`

`

`·ant
`
`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, H 2 > H 1, mis positive and the transition is endothermic in nature.
`Figure 10 shows that, below T,, polymorph 1 (or the solid) has
`the lower Gibbs free energy and is therefore more stable (i.e., G2 >
`G 1). On the other hand, above T;, polymorph 2 (or the liquid) has the
`lower Gibbs free energy and is therefore more stable (i.e., G2 < G 1).
`Under defined conditions of temperature and pressure, only one poly(cid:173)
`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(cid:173)
`times said to be metastable.
`The Gibbs free energy difference ~G between two phases reflects
`tendencies" of the two phases. The escaping
`the ratio of "escaping
`tendency is termed the fugacity f and is approximated by the saturated
`vapor pressure, p. Therefore
`
`J.G - RT In~)
`-RT!n(;:)
`
`(12)
`
`(13)
`
`where the subscripts 1 and 2 refer to the respective phases, R is the
`universal gas constant, and Tis the absolute temperature. The fugacity
`is proportional to the thermodynamic activity a (where the constant of
`proportionality is defined by the standard state), while thermodynamic
`activity is approximately proportional to the solubility s (in any given
`solvent) provided the laws of dilute solution apply. Therefore
`- RT!n(::)
`<I.G
`
`(14)
`
`in which the symbols have been defined above. Hence, because the
`most stable polymorph under defined conditions of temperature and
`pressure has the lowest Gibbs free energy, it also has the lowest values
`
`(15)
`
`1stant
`ftwo
`1pera-
`
`,r two
`o any
`cross.
`T1 (or
`;ies of
`1uilib(cid:173)
`)rpb 2
`morph
`if ~G
`n doe
`ting to
`
`(11)
`
`:rential
`tlpy of
`be rat
`
`Page 22
`
`

`

`Grant
`
`18
`of fugacity, vapor pressure thermodynamic activity, and solubility in
`any given solvent. Durin g the dissolution process, if transport-con(cid:173)
`lTOI led tmder sink condition s and under constant conditions of hydrody(cid:173)
`namic flow, the dissolution rate per unit surface area J is proportional to
`the solubility according to the Noyes-Whitney [14) equation; therefore
`In e:)
`AG~ RT
`(16)
`According to the law of mass action the rate ,. of a chemical reaction
`(including the decomposition rate) is proporLionaJ to the thermody(cid:173)
`namic activity of the reacting substance. Therefore
`AG~ RT{:)
`the low est Gibbs free
`To summarize, the most stable polymorph ha
`energy, fugacity, vapor pressur e, thermodynamic activity, solubility,
`and dissolution rate per unit surface area in any solvent, and rate of
`reaction, including decomposition rate .
`
`(17)
`
`Ill. 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
`polymorph are said to be enantiotropes, and the system of the two
`soli d phases is said to be enantiotropic. For an enantiotropic system a
`reversible transition can be observed at a definite transition tempera(cid:173)
`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 polymorph is stable at all temperatures below
`the melting point, with all other polymorphs being therefore unstable.
`These polymorphs are said to be monotropes, and the system of the
`two solid phases is said to be monotropic. For a monotropic system
`
`Page 23
`
`

`

`rant
`
`:yin
`con(cid:173)
`ody(cid:173)
`.al to
`~fore
`
`(16)
`
`ction
`10dy-
`
`(17)
`
`free
`bility
`ate of
`
`lower
`ge and
`~nergy
`he two
`1e two
`stem a
`npera(cid:173)
`~ point
`amide,
`
`; below
`1stable.
`. of the
`system
`
`Theory and Origin of Polymorphism
`
`19
`
`the free energy curves do not cross, so no reversible transition can be
`observed below the melting point. The polymorph with the higher free
`energy curve and solubility at any given temperature is, of course, al(cid:173)
`ways the unstable polymorph. Examples of this type of system include
`chloramphenicol palmitate and metolazone [9,14,15].
`To help decide whether two polymorphs are enantiotropes or
`monotropes, Burger and Ramberger developed four thermodynamic
`rules [14]. The application of these rules was extended by Yu [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(cid:173)
`versely, if the higher melting polymorph has the higher heat of fusion,
`the two forms are monotropes. Figure 11, which includes the liquid
`phase as well 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 less 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 preparing different polymorphs are
`sublimation, crystallization from the melt, crystallization from super(cid:173)
`critical fluids, and crystallization from liquid solutions. In the pharma(cid:173)
`ceutical sciences, different polymorphs are usually prepared by crys(cid:173)
`tallization from solution employing various solvents and various
`temperature regimes, such as initial supersaturation, rate of de-super(cid:173)
`saturation, or final supersaturation. The supersaturation of the solution
`
`Page 24
`
`

`

`Grant
`
`20
`
`(a)
`
`(b)
`
`B
`
`A 6Ho
`
`,,
`
`...
`
`.,.,
`
`HA
`
`,, -
`
`---
`-.-r:::: __ - --
`8
`' A ClHo
`
`::..
`~
`Q:;
`~
`lu
`
`TcMP£RATURE (KJ
`
`,,
`
`',,s ',,A
`
`Page 25
`
`

`

`·ant
`
`Theory and Origin of Polymorphism
`
`21
`
`Table 4 Thennodynamic Rules for Polymorphic Transitions According to
`Burger and Ramberger [14], Where Form I is the Higher-Melting Form
`Monotropy
`Transition > melting I
`I always stable
`
`Enantiotropy
`Transition < melting I
`I Stable > transition
`II Stable < transition
`Transition reversible
`Solubility I higher < transition
`Solubility I lower > transition
`Transition II --, I is endothermic
`MI} < Af/}1
`IR peak I before II
`Density I < density II
`Source: Reproduced from Refer. 9 with permission of the copyright owner, Elsevier,
`Amsterdam, The Netherlands .
`
`Transition irreversible
`Solubility I always lower than II
`
`Transition II --, I is exothermic
`Ml)> Aff}1
`IR peak I after II
`Density I > density II
`
`-
`
`Tt,A
`
`that is necessary for crystallization may be achieved by evaporation of
`the solvent (although any impurities will be concentrated), cooling 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 soluble
`species, or variation of pH to produce a less soluble acid or base from a
`salt or vice versa (while minimizing other changes in composition).
`During the 19th century, Gay Lussac observed that, during crys(cid:173)
`tallization, an unstable form is frequently obtained first that subse(cid:173)
`quently transforms 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, 1 [14]. T, is the transition tempera(cid:173)
`ture, Tr is the melting temperature, and Sis the entropy for (a) an enantiotropic
`system and (b) a monotropic system. (Reproduced with permission of the
`copyright owner, Springer Verlag, Vienna, Austria.)
`
`Page 26
`
`

`

`I.
`
`Grant
`
`22
`rule may be stated as, ''In all processes, it is not the most stable state with
`the lowe t amount of free energy that is initially formed, but the least
`stable state lying near es t in free en ergy to U1e original stale [13].
`ig. 12. L eL an
`Ostwald 's step rule [13,16-19) is illu stra ted by
`enantiotr pie sys tem (Fig. 12a) be initially in a state repr esented by
`to an un stable vapor or liquid or to a supersatu(cid:173)
`point X corresponding
`free energy will de-
`rnted solution. If thj · system js cooled, the Gibb
`
`(a)
`
`(b)
`
`TEMPERATURE
`
`A 'y
`~elt
`
`G
`
`~.
`
`•
`
`... . '.
`
`t
`
`,
`
`\
`
`'
`T f
`B
`
`T f
`A
`TEMPERATURE
`
`free e nergy G and the temp eratme
`Fig. 12 Relation sh ip between the Gibb
`T for two poJymorphs for (a) an cna ntiotropi c sys tem and (b) a monotropic
`sys tem in which the syste m is coo led from point X [9]. The arrows indicate
`the direction of change. (Reprodu ced with permi sion of the copyrig ht owner,
`Elsevier, Amsterdam, The Netherlands.)
`
`Page 27
`
`

`

`-
`
`rant
`
`with
`least
`
`:Lan
`ll by
`satu-
`1 de-
`
`)eratme
`1otropic
`indicate
`t own er,
`
`Theory and Origin of Polymorphism
`
`23
`
`crease as the temperature decreases. When the state of the